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WILEY SERIES IN PURE AND APPLIED OPTICS Founded by Stanley S. Ballard, University of Florida EDITOR: Bahaa E. A. Saleh, Boston University BARRETT AND MYERS · Foundations of Image Science BEISER · Holographic Scanning BERGER-SCHUNN · Practical Color Measurement BOYD. Radiometry and The Detection of Optical Radiation BUCK. Fundamentals of Optical Fibers, Second Edition CATHEY · Optical Information Processing and Holography CHUANG · Physics of Optoelectronic Devices DEL ONE AND KRAINOV · Fundamentals of Nonlinear Optics of Atomic Gases DERENIAK AND BOREMAN · Infrared Detectors and Systems DERENIAK AND CROWE · Optical Radiation Detectors DE VANY. Master Optical Techniques ERSOY · Diffraction, Fourier Optics and Imaging GASKILL · Linear Systems, Fourier Transform, and Optics GOODMAN · Statistical Optics HOBBS · Building Electro-Optical Systems: Making It All Work HUDSON · Infrared System Engineering IIZUKA · Elements of Photonics, Volume I: In Free Space and Special Media IIZUKA · Elements of Photonics, Volume II: For Fiber and Integrated Optics JUDD AND WYSZECKI · Color in Business, Science, and Industry, Third Edition KAFRI AND GLATT · The Physics of Moire Metrology KAROW · Fabrication Methods for Precision Optics KLEIN AND FURTAK · Optics, Second Edition MALACARA · Optical Shop Testing, Second Edition MILONNI AND EBERLY · Lasers NASSAU. The Physics and Chemistry of Color: The Fifteen Causes of Color, Second Edition NIETO- VESPERINAS · Scattering and Diffraction in Physical Optics OSCHE · Optical Detection Theory for Laser Applications O'SHEA. Elements of Modern Optical Design OZAKTAS · The Fractional Fourier Transform SALEH AND TEICH · Fundamentals of Photonics, Second Edition SCHUBERT AND WILHELMI · Nonlinear Optics and Quantum Electronics SHEN. The Principles of Nonlinear Optics UDD. Fiber Optic Sensors: An Introductionfor Engineers and Scientists UDD · Fiber Optic Smart Structures VANDERLUGT · Optical Signal Processing VEST · Holographic Interferometry VINCENT · Fundamentals of Infrared Detector Operation and Testing WILLIAMS AND BECKLUND · Introduction to the Optical Transfer Function WYSZECKI AND STILES · Color Science: Concepts and Methods, Quantitative Data and Formulae, Second Edition XU AND STROUD · Acousto-Optic Devices YAMAMOTO · Coherence, Amplification, and Quantum Effects in Semiconductor Lasers YARIV AND YEH · Optical Waves in Crystals YEH · Optical Waves in Layered Media YEH. Introduction to Photorefractive Nonlinear Optics YEH AND GU · Optics of Liquid Crystal Displays
FUNDAMENTALS OF PHOTONICS
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FUNDAMENTALS OF PHOTONICS SECOND EDITION BAHAA E. A. SALEH Boston University MALVIN CARL TEICH Boston University Columbia University 1807 @0WILEY 2007 ; WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication (ffJ)
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PREFACE TO THE SECOND EDITION Since the publication of the First Edition in 1991, Fundamentals of Photonics has been reprinted some 20 times, translated into Czech and Japanese, and used worldwide as a textbook and reference. During this period, major developments in photonics have continued apace, and have enabled technologies such as telecommunications and applications in industry and medicine. The Second Edition reports some of these developments, while maintaining the size of this single-volume tome within practical limi ts. In its new organization, Fundamentals of Photonics continues to serve as a self- contained and up-to-date introductory-level textbook, featuring a logical blend of the- ory and applications. Many readers of the First Edition have been pleased with its abundant and well-illustrated figures. This feature has been enhanced in the Second Edition by the introduction of full color throughout the book, offering improved clarity and readability. While each of the 22 chapters of the First Edition has been thoroughly updated, the principal feature of the Second Edition is the addition of two new chapters: one on photonic-crystal optics and another on ultrafast optics. These deal with developments that have had a substantial and growing impact on photonics over the past decade. The new chapter on photonic-crystal optics provides a foundation for understand- ing the optics of layered media, including Bragg gratings, with the help of a matrix approach. Propagation of light in one-dimensional periodic media is examined using Bloch modes with matrix and Fourier methods. The concept of the photonic bandgap is introduced. Light propagation in two- and three-dimensional photonic crystals, and the associated dispersion relations and bandgap structures, are developed. Sections on photonic-crystal waveguides, holey fibers, and photonic-crystal resonators have also been added at appropriate locations in other chapters. The new chapter on ultrafast optics contains sections on picosecond and femtosec- ond optical pulses and their characterization, shaping, and compression, as well as their propagation in optical fibers, in the domain of linear optics. Sections on ultrafast non- linear optics include pulsed parametric interactions and optical solitons. Methods for the detection of ultrafast optical pulses using available detectors, which are relatively slow, are reviewed. In addition to these two new chapters, the chapter on optical interconnects and switches has been completely rewritten and supplemented with topics such as wave- length and time routing and switching, FBGs, WGRs, SOAs, TOADs, and packet switches. The chapter on optical fiber communications has also been significantly updated and supplemented with material on WDM networks; it now offers concise descriptions of topics such as dispersion compensation and management, optical am- plifiers, and soliton optical communications. Continuing advances in device-fabrication technology have stimulated the emer- gence of nanophotonics, which deals with optical processes that take place over subwavelength (nanometer) spatial scales. Nanophotonic devices and systems include quantum-confined structures, such as quantum dots, nanoparticles, and nanoscale periodic structures used to synthesize metamaterials with exotic optical properties such as negative reflactive index. They also include configurations in which light (or its interaction with matter) is confined to nanometer-size (rather than micrometer- size) regions near boundaries, as in surface plasmon optics. Evanescent fields, such as those produced at a surface where total internal reflection occurs, also exhibit v
VI PREFACE such confinement. Evanescent fields are present in the immediate vicinity of sub- wavelength-size apertures, such as the open tip of a tapered optical fiber. Their use allows imaging with resolution beyond the diffraction limit and forms the basis of near-field optics. Many of these emerging areas are described at suitable locations in the Second Edition. New sections have been added in the process of updating the various chapters. New topics introduced in the early chapters include: Laguerre-Gaussian beams; near-field imaging; the Sellmeier equation; fast and slow light; optics of conductive media and plasmonics; doubly negative metamaterials; the Poincare sphere and Stokes parame- ters; polarization mode dispersion; whispering-gallery modes; microresonators; optical coherence tomography; and photon orbital angular momentum. In the chapters on laser optics, new topics include: rare-earth and Raman fiber amplifiers and lasers; EUV, X-ray, and free-electron lasers; and chemical and random lasers. In the area of optoelectronics, new topics include: gallium nitride-based struc- tures and devices; superluminescent diodes; organic and white-light LEDs; quantum- confined lasers; quantum-cascade lasers; microcavity lasers; photonic-crystal lasers; array detectors; low-noise APDs; SPADs; and QWIPs. The chapter on nonlinear optics has been supplemented with material on parametric- interaction tuning curves; quasi-phase-matching devices; two-wave mixing and cross- phase modulation; THz generation; and other nonlinear optical phenomena associated with narrow optical pulses, including chirp pulse amplification and supercontinuum light generation. The chapter on electro-optics now includes a discussion of electroab- sorption modulators. Appendix C on modes of linear systems has been expanded and now offers an overview of the concept of modes as they appear in numerous locations within the book. Finally, additional exercises and problems have been provided, and these are now numbered disjointly to avoid confusion. In this full-color edition, we have used the color code illustrated in the following chart for most of the illustrations. Light beams and field distributions are colored red (except when light beams of multiple colors are involved, as in nonlinear optics). Glass and glass fibers are depicted in light blue. Semiconductors are cast in green, with various shades representing different doping levels, and metal is indicated by the color of copper. Energy diagrams are marked in blue and forbidden photonic bandgaps in pink, as indicated. Glass pJ I O . I · ptica ray Semiconductor Optical beam Energy levels Dielectric waveguide < Optical wave .0.. Metal Photonic bandgap Fiber Color chart Organization In its new incarnation, Fundamentals of Photonics comprises 24 chapters compart- mentalized into six parts, as depicted in the diagram below. The form of the book is modular so that it can be used by readers with different needs; it also provides
PREFACE VII instructors an opportunity to select topics for different courses. Essential material from one chapter is often briefly summarized in another to make each chapter as self- contained as possible. For example, at the beginning of Chapter 24 (Optical Fiber Communications), relevant material from earlier chapters that describe fibers, light sources, detectors, and amplifiers is briefly reviewed. This places the important features of the various components at the disposal of the reader before the chapter proceeds with a discussion of the design and performance of the overall communication system that makes use of these components. Fundamentals Wave Propagation Laser Optics Lightwave Devices 1. Ray Optics 7. Photonic-Crystal Optics 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Semiconductor Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Semiconductor Detectors 24. Optical Communications Optoelectronics Lightwave Systems Recognizing the different degrees of mathematical sophistication of the intended readership, we have endeavored to present difficult concepts in two steps: at an intro- ductory level providing physical insight and motivation, followed by a more advanced analysis. This approach is exemplified by the treatment in Chapter 20 (Electro-Optics), in which the subject is first presented using scalar notation and then treated again using tensor notation. Commonly accepted notation and symbols have been used wherever possible. Be- cause of the broad spectrum of topics covered, however, there are a good number of symbols that have multiple meanings; a list of symbols and units is provided at the end of the book to clarify symbol usage. Throughout the book, important equations are highlighted by boxes to facilitate future retrieval. Sections dealing with material of a more advanced nature are indicated by asterisks and may be omitted if desired. Summaries are provided throughout at points where a recapitulation is deemed useful because of the involved nature of the material. Each chapter also contains exercises, problem sets, and updated selected reading lists. Examples of real systems are included to emphasize the concepts governing appli- cations of current interest, and appendixes summarize the properties of one- and two- dimensional Fourier transforms, linear-systems theory, and modes of linear systems. Representative Courses The chapters of this book may be combined in various ways for use in semester or quarter courses. Representative examples of such courses are provided below. Some of these courses may be offered as part of a sequence. Other selections may be made to suit the particular objectives of instructors and students. Optics/Photonics 1. Ray Optics 2. Wave Optics 3. Beam Optics 4. Fourier Optics 5. Electromagnetic Optics 6. Polarization Optics --- - - 7. Photonic Crystals 8. Guided-Wave Optics 9. Fiber Optics 10. Resonator Optics 11. Statistical Optics 12. Photon Optics 13. Photons and Atoms 14. Laser Amplifiers 15. Lasers 16. Semiconductor Optics 17. Sources 18. Detectors 19. Acousto-Optics 20. Electro-Optics 21. Nonlinear Optics 22. Ultrafast Optics 23. Interconnects/Switches :- 24. Optical Communications -
VIII PREFACE The first six chapters of the book are suitable for an introductory course on Optics or Photonics. These may be supplemented by Chapter 11, Statistical Optics, to introduce incoherent and partially coherent light, or by the introductory sections of Chapters 8 and 9, Guided- Wave Optics and Fiber Optics, which offer applications. Optical Information Processing 1. Ray Optics 7. Photonic Crystals 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Detectors 24. Optical Communications A course on Optical Information Processing may begin with a background of wave and beam optics, and cover Fourier Optics (including coherent image formation and processing), along with incoherent and partially coherent imaging in Statistical Optics. This may be followed by material on devices used for analog data processing, such as Acousto-Optics, and end with switches and gates (Chapter 23), which are used for digital data processing. Guided-Wave Optics 1. Ray Optics 7. Photonic Crystals 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Detectors 24. Optical Communications A course on Guided-Wave Optics may begin with an introduction to wave propagation in layered and periodic media (Chapter 7, Photonic-Crystal Optics) and follow with the chapters on Guided- Wave Optics, Fiber Optics, and Resonator Optics. Additional topics may include Electro-Optics and Optical Interconnects and Stvitches. Lasers 1. Ray Optics 7. Photonic Crystals 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Detectors 24. Optical Communications A course on Lasers could begin with Beam Optics and Resonator Optics, and follow with the theory of interaction of light with matter (Chapter 13) and laser amplification and oscillation (Chapters 14 and 15), and include semiconductor LEDs and lasers (Chapters 16 and 17). An introduction to femtosecond lasers can be provided by including appropriate sections from Ultrafast Optics. Optoelectronics l 1. Ray Optics 7. Photonic Crystals 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Detectors 24. Optical Communications : The three chapters covering semiconductor optics, sources/amplifiers, and detectors form a suitable basis for a course on Optoelectronics. This material may be supplemented with optics background from earlier chapters, and extended to include topics such as liquid-crystal devices (Secs. 6.5 and
PREFACE IX 20.3), semiconductor electroabsorption modulators (Sec. 20.5), and an introduction to the use of photonic devices for switching and/or communications (Chapters 23 and 24, respectively). Photonic Devices 1. Ray Optics 7. Photonic Crystals 13. Photons and Atoms 19. Acousto-Optics 2. Wave Optics 8. Guided-Wave Optics 14. Laser Amplifiers 20. Electro-Optics 3. Beam Optics 9. Fiber Optics 15. Lasers 21. Nonlinear Optics 4. Fourier Optics 10. Resonator Optics 16. Semiconductor Optics 22. Ultrafast Optics 5. Electromagnetic Optics 11. Statistical Optics 17. Sources 23. Interconnects/Switches 6. Polarization Optics 12. Photon Optics 18. Detectors 24. Optical Communications Photonic Devices is another possible topic for a course that combines photonic-crystal and guided- wave devices with electro-optic, acousto-optic, and nonlinear optical devices, and includes ultrafast optics and optical interconnects/switches. Fiber-Optic Communications 1. Ray Optics 2. Wave Optics 3. Beam Optics 4. Fourier Optics 5. Electromagnetic Optics 6. Polarization Optics 7. Photonic Crystals 8. Guided-Wave Optics 9. Fiber Optics 10. Resonator Optics 11. Statistical Optics 12. Photon Optics 13. Photons and Atoms 14. Laser Amplifiers 15. Lasers 16. Semiconductor Optics 17. Sources 18. Detectors 19. Acousto-Optics 20. Electro-Optics 21. Nonlinear Optics 22. Ultrafast Optics 23. Interconnects/Switches 24. Optical Communications : A course on Fiber-Optic Communications could include optical waveguides and fibers, semiconduc- tor sources and amplifiers (possibly also Sees. 14.3C and 14.3D on optical-fiber and Raman-fiber amplifiers), as background material for the chapter on Optical Fiber Communications (Chapter 24). If fiber-optic networks are to be emphasized, Sec. 23.3 on photonic switches may also be included. Acknowledgments We are grateful to many colleagues for providing us with valuable comments about draft chapters for the Second Edition and for drawing our attention to errors in the First Edition: Mete Atatiire, Michael Bar, Silvia Carrasco, Thomas Daly, Gianni Di Giuseppe, Adel EI-Nadi, John Fourkas, Majeed Hayat, Tony Heinz, Erich Ippen, Mar- tin Jaspan, Gerd Keiser, Jonathan Kane, Paul Kelley, Ted Moustakas, Magued Nasr, Roy Olivier, Roberto Paiella, Alexander Sergienko, Peter W. E. Smith, Stephen P. Smith, Kenneth Suslick, and Tommaso Toffoli. We extend our special thanks to those colleagues who graciously provided us with in-depth critiques of various chapters: Ayman Abouraddy, Luca Dal Negro, and Paul Prucnal. We are indebted to the legions of students and postdoctoral associates who have posed so many excellent questions that helped us hone our presentation. In particular, many improvements were initiated by suggestions from Mark Booth, Jasper Cabalu, Michael Cunha, Darryl Goode, Chris LaFratta, Rui Li, Eric Lynch, Nan Ma, Nishant Mohan, Julie Praino, Yunjie Tong, and Ranjith Zachariah. We are especially grateful to Mohammed Saleh, who diligently read much of the manuscript and provided us with excellent suggestions for improvement throughout. Wai Yan (Eliza) Wong provided logistical support and a great deal of assistance in crafting diagrams and figures. Many at Wiley, including George Telecki, our Editor, and Rachel Witmer have been most helpful, patient, and encouraging. We appreciate the attentiveness and thoroughness that Melissa Yanuzzi brought to the production pro- cess. Don DeLand of the Integre Technical Publishing Company provided invaluable assistance in setting up the Latex style files. We are most appreciative of the financial support provided by the National Sci- ence Foundation (NSF), in particular the Center for Subsurface Sensing and Imaging
X PREFACE Systems (CenSSIS), an NSF-supported Engineering Research Center; the Defense Advanced Research Projects Agency (DARPA); the National Reconnaisance Office (NRO); the U.S. Army Research Office (ARO); the David & Lucile Packard Founda- tion; the Boston University College of Engineering; and the Boston University Photon- ics Center. Photo Credits. Most of the portraits were carried forward from the First Edi- tion with the benefit of permissions provided for all editions. Additional credits are: Godfrey Kneller 1689 portrait (Newton); Siegfried Bendixen 1828 lithograph (Gauss); Engraving in the Small Portraits Collection, History of Science Collections, University of Oklahoma Libraries (Fraunhofer); Stanford University, Courtesy AlP Emilio Segre Visual Archives (Bloch); Eli Yablonovitch (Yablonovitch); Sajeev John (John); Charles Kao (Kao); Philip St John Russell (Russell); Ecole Poly technique (Fabry); Observa- toire des Sciences de l'Univers (Perot); AlP Emilio Segre Visual Archives (Born); Lagrelius & Westphal 1920 portrait (Bohr); AlP Emilio Segre Visual Archives, Weber Collection (W. L. Bragg); Linn F. Mollenauer (Mollenauer); Roger H. Stolen (Stolen); and James P. Gordon (Gordon). In Chapter 23, the Bell Symbol was reproduced with the permission of BellSouth Intellectual Property Marketing Corporation, the AT&T logo is displayed with the permission of AT&T, and Lucent Technologies permitted us use of their logo. Stephen G. Eick kindly provided the image used at the beginning of Chapter 24. The photographs of Saleh and Teich were provided courtesy of Boston University. BAHAA E. A. SALEH MALVIN CARL TEICH Boston, Massachusetts December 19,2006
PREFACE TO THE FIRST EDITION Optics is an old and venerable subject involving the generation, propagation, and de- tection of light. Three major developments, which have been achieved in the last thirty years, are responsible for the rejuvenation of optics and for its increasing importance in modern technology: the invention of the laser, the fabrication of low-loss optical fibers, and the introduction of semiconductor optical devices. As a result of these de- velopments, new disciplines have emerged and new terms describing these disciplines have come into use: electro-optics, optoelectronics, quantum electronics, quantum optics, and lightwave technology. Although there is a lack of complete agreement about the precise usages of these terms, there is a general consensus regarding their meanIngs. Photonics Electro-optics is generally reserved for optical devices in which electrical effects playa role (lasers, and electro-optic modulators and switches, for example). Optoelectronics, on the other hand, typically refers to devices and systems that are essentially electronic in nature but involve light (examples are light-emitting diodes, liquid-crystal display devices, and array photodetectors). The term quantum electronics is used in connection with devices and systems that rely principally on the interaction of light with matter (lasers and nonlinear optical devices used for optical amplification and wave mixing serve as examples). Studies of the quantum and coherence properties of light lie within the realm of quantum optics. The term lightwave technology has been used to describe devices and systems that are used in optical communications and optical signal pro- cessIng. In recent years, the term photonics has come into use. This term, which was coined in analogy with electronics, reflects the growing tie between optics and electronics forged by the increasing role that semiconductor materials and devices play in optical systems. Electronics involves the control of electric-charge flow (in vacuum or in matter); photonics involves the control of photons (in free space or in matter). The two disciplines clearly overlap since electrons often control the flow of photons and, conversely, photons control the flow of electrons. The term photonics also reflects the importance of the photon nature of light in describing the operation of many optical devices. Scope This book provides an introduction to the fundamentals of photonics. The term pho- tonics is used broadly to encompass all of the aforementioned areas, including the following: . The generation of coherent light by lasers, and incoherent light by luminescence sources such as light-emitting diodes. . The transmission of light in free space, through conventional optical components such as lenses, apertures, and imaging systems, and through waveguides such as optical fibers. . The modulation, switching, and scanning of light by the use of electrically, acous- tically, or optically controlled devices. . The amplification and frequency conversion of light by the use of wave interac- tions in nonlinear materials. . The detection of light. XI
xii PREFACE These areas have found ever-increasing applications in optical communications, signal processing, computing, sensing, display, printing, and energy transport. Approach and Presentation The underpinnings of photonics are provided in a number of chapters that offer concise introductions to: . The four theories of light (each successively more advanced than the preceding): ray optics, wave optics, electromagnetic optics, and photon optics. . The theory of interaction of light with matter. . The theory of semiconductor materials and their optical properties. These chapters serve as basic building blocks that are used in other chapters to describe the generation of light (by lasers and light-emitting diodes); the transmission of light (by optical beams, diffraction, imaging, optical waveguides, and optical fibers); the modulation and switching of light (by the use of electro-optic, acousto-optic, and nonlinear-optic devices); and the detection of light (by means of photo detectors). Many applications and examples of real systems are provided so that the book is a blend theory and practice. The final chapter is devoted to the study of fiber-optic communica- tions, which provides an especially rich example in which the generation, transmission, modulation, and detection of light are all part of a single photonic system used for the transmission of information. The theories of light are presented at progressively increasing levels of difficulty. Thus light is described first as rays, then scalar waves, then electromagnetic waves, and finally, photons. Each of these descriptions has its domain of applicability. Our approach is to draw from the simplest theory that adequately describes the phenomenon or intended application. Ray optics is therefore used to describe imaging systems and the confinement of light in waveguides and optical resonators. Scalar wave theory provides a description of optical beams, which are essential for the understanding of lasers, and of Fourier optics, which is useful for describing coherent optical systems and holography. Electromagnetic theory provides the basis for the polarization and dispersion of light, and the optics of guided waves, fibers, and resonators. Photon optics serves to describe the interactions of light with matter, explaining such processes as light generation and detection, and light mixing in nonlinear media. Intended Audience Fundamentals of Photonics is meant to serve as: . An introductory textbook for students in electrical engineering or applied physics at the senior or first-year graduate level. . A self-contained work for self-study. . A text for programs of continuing professional development offered by industry, universities, and professional societies. The reader is assumed to have a background in engineering or applied physics, including courses in modern physics, electricity and magnetism, and wave motion. Some knowledge of linear systems and elementary quantum mechanics is helpful but not essential. Our intent has been to provide an introduction to photonics that emphasizes the concepts governing applications of current interest. The book should, therefore, not be considered as a compendium that encompasses all photonic devices and systems. Indeed, some areas of photonics are not included at all, and many of the individual chapters could easily have been expanded into separate monographs.
PREFACE xiii Problems, Reading Lists, and Appendices A set of problems is provided at the end of each chapter. Problems are numbered in accordance with the chapter sections to which they apply. Quite often, problems deal with ideas or applications not mentioned in the text, analytical derivations, and numerical computations designed to illustrate the magnitudes of important quantities. Problems marked with asterisks are of a more advanced nature. A number of exer- cises also appear within the text of each chapter to help the reader develop a better understanding of (or to introduce an extension of) the material. Appendices summarize the properties of one- and two-dimensional Fourier trans- forms, linear-systems theory, and modes of linear systems (which are important in polarization devices, optical waveguides, and resonators); these are called upon at appropriate points throughout the book. Each chapter ends with a reading list that includes a selection of important books, review articles, and a few classic papers of special significance. Acknowledgments We are grateful to many colleagues for reading portions of the text and providing helpful comments: Govind P. Agrawal, David H. Auston, Rasheed Azzam, Nikolai G. Basov, Franco Cerrina, Emmanuel Desurvire, Paul Diament, Eric Fossum, Robert J. Keyes, Robert H. Kingston, Rodney Loudon, Leonard Mandel, Leon McCaughan, Richard M. Osgood, Jan Perina, Robert H. Rediker, Arthur L. Schawlow, S. R. Se- shadri, Henry Stark, Ferrel G. Stremler, John A. Tataronis, Charles H. Townes, Patrick R. Trischitta, Wen I. Wang, and Edward S. Yang. We are especially indebted to John Whinnery and Emil Wolf for providing us with many suggestions that greatly improved the presentation. Several colleagues used portions of the notes in their classes and provided us with invaluable feedback. These include Etan Bourkoff at Johns Hopkins University (now at the University of South Carolina), Mark O. Freeman at the University of Colorado, George C. Papen at the University of Illinois, and Paul R. Prucnal at Princeton Univer- sity. Many of our students and former students contributed to this material in various ways over the years and we owe them a great debt of thanks: Gaetano L. Aiello, Mohamad Asi, Richard Campos, Buddy Christyono, Andrew H. Cordes, Andrew David, Ernesto Fontenla, Evan Goldstein, Matthew E. Hansen, Dean U. Hekel, Conor Heneghan, Adam Heyman, Bradley M. Jost, David A. Landgraf, Kanghua Lu, Ben Nathanson, Winslow L. Sargeant, Michael T. Schmidt, Raul E. Sequeira, David Small, Kraisin Songwatana, Nikola S. Subotic, Jeffrey A. Tobin, and Emily M. True. Our thanks also go to the legions of unnamed students who, through a combination of vigilance and the desire to understand the material, found countless errors. We particularly appreciate the many contributions and help of those students who were intimately involved with the preparation of this book at its various stages of completion: Niraj Agrawal, Suzanne Keilson, Todd Larchuk, Guifang Li, and Philip Tham. We are grateful for the assistance given to us by a number of colleagues in the course of collecting the photographs used at the beginnings of the chapters: E. Scott Barr, Nicolaas Bloembergen, Martin Carey, Marjorie Graham, Margaret Harrison, Ann Kot- tner, G. Thomas Holmes, John Howard, Theodore H. Maiman, Edward Palik, Martin Parker, Aleksandr M. Prokhorov, Jarus Quinn, Lesley M. Richmond, Claudia Schuler, Patrick R. Trischitta, J. Michael Vaughan, and Emil Wolf. Specific photo credits are as follows: AlP Meggers Gallery of Nobel Laureates (Gabor, Townes, Basov, Prokhorov, W. L. Bragg); AlP Niels Bohr Library (Rayleigh, Fraunhofer, Maxwell, Planck, Bohr, Einstein in Chapter 12, W. H. Bragg); Archives de l' Academie des Sciences de Paris (Fabry); The Astrophysical Journal (Perot); AT&T Bell Laboratories (Shockley, Brat-
xiv PREFACE tain, Bardeen); Bettmann Archives (Young, Gauss, Tyndall); Bibliotheque Nationale de Paris (Fermat, Fourier, Poisson); Burndy Library (Newton, Huygens); Deutsches Mu- seum (Hertz); ETH Bibliothek (Einstein in Chapter 11); Bruce Fritz (Saleh); Harvard University (Bloembergen); Heidelberg University (Pockels); Kelvin Museum of the University of Glasgow (Kerr); Theodore H. Maiman (Maiman); Princeton University (von Neumann); Smithsonian Institution (Fresnel); Stanford University (Schawlow); Emil Wolf (Born, Wolt). Corning Incorporated kindly provided the photograph used at the beginning of Chapter 8. We are grateful to GE for the use of their logotype, which is a registered trademark of the General Electric Company, at the beginning of Chapter 16. The IBM logo at the beginning of Chapter 16 is being used with special permission from ffiM. The right-most logotype at the beginning of Chapter 16 was supplied courtesy of Lincoln Laboratory, Massachusetts Institute of Technology. AT&T Bell Laboratories kindly permitted us use of the diagram at the beginning of Chapter 22. We greatly appreciate the continued support provided to us by the National Sci- ence Foundation, the Center for Telecommunications Research, and the Joint Services Electronics Program through the Columbia Radiation Laboratory. Finally, we extend our sincere thanks to our editors, George Telecki and Bea Shube, for their guidance and suggestions throughout the course of preparation of this book. BAHAA E. A. SALEH Madison, Wisconsin MALVIN CARL TEICH New York, New York April 3, 1991
CONTENTS PREFACE TO THE SECOND EDITION v PREFACE TO THE FIRST EDITION xi 1 RAY OPTICS 1 1.1 Postulates of Ray Optics 3 1.2 Simple Optical Components 6 1.3 Graded-Index Optics 17 1.4 Matrix Optics 24 Reading List 34 Problems 35 2 WAVE OPTICS 38 2.1 Postulates of Wave Optics 40 2.2 Monochromatic Waves 41 *2.3 Relation Between Wave Optics and Ray Optics 49 2.4 Simple Optical Components 50 2.5 Interference 58 2.6 Polychromatic and Pulsed Light 66 Reading List 72 Problems 73 3 BEAM OPTICS 74 3.1 The Gaussian Beam 75 3.2 Transmission Through Optical Components 86 3.3 Hermite-Gaussian Beams 94 *3.4 Laguerre-Gaussian and Bessel Beams 97 Reading List 100 Problems 100 4 FOURIER OPTICS 102 4.1 Propagation of Light in Free Space 105 4.2 Optical Fourier Transform 116 4.3 Diffraction of Light 121 4.4 Image Formation 127 4.5 Holography 138 Reading List 145 Problems 147 xv
XVI CONTENTS 5 ELECTROMAGNETIC OPTICS 150 5.1 Electromagnetic Theory of Light 152 5.2 Electromagnetic Waves in Dielectric Media 156 5.3 Monochromatic Electromagnetic Waves 162 5.4 Elementary Electromagnetic Waves 164 5.5 Absorption and Dispersion 170 5.6 Pulse Propagation in Dispersive Media 184 *5.7 Optics of Magnetic Materials and Metamaterials 190 Reading List 193 Problems 195 6 POLARIZATION OPTICS 197 6.1 Polarization of Light 199 6.2 Reflection and Refraction 209 6.3 Optics of Anisotropic Media 215 6.4 Optical Activity and Magneto-Optics 228 6.5 Optics of Liquid Crystals 232 6.6 Polarization Devices 235 Reading List 239 Problems 240 7 PHOTONIC-CRYSTAL OPTICS 243 7.1 Optics of Dielectric Layered Media 246 7.2 One-Dimensional Photonic Crystals 265 7.3 Two- and Three-Dimensional Photonic Crystals 279 Reading List 286 Problems 288 8 GUIDED-WAVE OPTICS 289 8.1 Planar-Mirror Waveguides 291 8.2 Planar Dielectric Waveguides 299 8.3 Two-Dimensional Waveguides 308 8.4 Photonic-Crystal Waveguides 311 8.5 Optical Coupling in Waveguides 313 8.6 Sub-Wavelength Metal Waveguides (Plasmonics) 321 Reading List 322 Problems 323 9 FIBER OPTICS 325 9.1 Guided Rays 327 9.2 Guided Waves 331 9.3 Attenuation and Dispersion 348 9.4 Holey and Photonic-Crystal Fibers 359 Reading List 362 Problems 363 10 RESONATOR OPTICS 365 10.1 Planar-Mirror Resonators 367 10.2 Spherical-Mirror Resonators 378 10.3 Two- and Three-Dimensional Resonators 390 10.4 Microresonators 394 Reading List 400 Problems 400
CONTENTS XVII 11 STATISTICAL OPTICS 403 11.1 Statistical Properties of Random Light 405 11.2 Interference of Partially Coherent Light 419 * 11.3 Transmission of Partially Coherent Light Through Optical Systems 427 11.4 Partial Polarization 436 Reading List 440 Problems 442 12 PHOTON OPTICS 12.1 The Photon 12.2 Photon Streams * 12.3 Quantum States of Light Reading List Problems 13 PHOTONS AND ATOMS 13.1 Energy Levels 13.2 Occupation of Energy Levels 13.3 Interactions of Photons with Atoms 13.4 Thermal Light 13.5 Luminescence and Light Scattering Reading List Problems 14 LASER AMPLIFIERS 14.1 Theory of Laser Amplification 14.2 Amplifier Pumping 14.3 Common Laser Amplifiers 14.4 Amplifier Nonlinearity * 14.5 Amplifier Noise Reading List Problems 15 LASERS 15.1 Theory of Laser Oscillation 15.2 Characteristics of the Laser Output 15.3 Common Lasers 15.4 Pulsed Lasers Reading List Problems 16 SEMICONDUCTOR OPTICS 444 446 458 471 476 478 482 483 499 501 517 522 528 530 532 535 539 547 556 562 564 565 567 569 575 590 605 621 624 16.1 Semiconductors 16.2 Interactions of Photons with Charge Carriers Reading List Problems 627 629 660 675 677
xviii CONTENTS 17 SEMICONDUCTOR PHOTON SOURCES 680 17.1 Light-Emitting Diodes 682 17.2 Semiconductor Optical Amplifiers 702 17.3 Laser Diodes 716 17.4 Quantum-Confined and Microcavity Lasers 728 Reading List 741 Problems 745 18 SEMICONDUCTOR PHOTON DETECTORS 748 18.1 Photodetectors 749 18.2 Photoconductors 758 18.3 Photodiodes 762 18.4 Avalanche Photodiodes 767 18.5 Array Detectors 775 18.6 Noise in Photodetectors 777 Reading List 798 Problems 800 19 ACOUSTO-OPTICS 804 19.1 Interaction of Light and Sound 806 19.2 Acousto-Optic Devices 819 * 19.3 Acousto-Optics of Anisotropic Media 828 Reading List 832 Problems 832 20 ELECTRO-OPTICS 834 20.1 Principles of Electro-Optics 836 *20.2 Electro-Optics of Anisotropic Media 849 20.3 Electro-Optics of Liquid Crystals 856 *20.4 Photorefractivity 863 20.5 Electroabsorption 868 Reading List 869 Problems 871 21 NONLINEAR OPTICS 873 21.1 Nonlinear Optical Media 875 21.2 Second-Order Nonlinear Optics 879 21.3 Third-Order Nonlinear Optics 894 *21.4 Second-Order Nonlinear Optics: Coupled-Wave Theory 905 *21.5 Third-Order Nonlinear Optics: Coupled-Wave Theory 917 *21.6 Anisotropic Nonlinear Media 924 *21.7 Dispersive Nonlinear Media 927 Reading List 932 Problems 934 22 ULTRAFAST OPTICS 936 22.1 Pulse Characteristics 937 22.2 Pulse Shaping and Compression 946 22.3 Pulse Propagation in Optical Fibers 960
CONTENTS XIX 22.4 Ultrafast Linear Optics 973 22.5 Ultrafast Nonlinear Optics 984 22.6 Pulse Detection 999 Reading List 1011 Problems 1013 23 OPTICAL INTERCONNECTS AND SWITCHES 23.1 Optical Interconnects 23.2 Passive Optical Routers 23.3 Photonic Switches 23.4 Optical Gates Reading List Problems 1016 1018 1030 1038 1058 1069 1071 24 OPTICAL FIBER COMMUNICATIONS 24.1 Fiber-Optic Components 24.2 Optical Fiber Communication Systems 24.3 Modulation and Multiplexing 24.4 Fiber-Optic Networks 24.5 Coherent Optical Communications Reading List Problems 1072 1074 1084 1101 1106 1112 1118 1120 A FOURIER TRANSFORM A.1 One-Dimensional Fourier Transform A.2 Time Duration and Spectral Width A.3 Two-Dimensional Fourier Transform Reading List 1122 1122 1124 1128 1131 B LINEAR SYSTEMS 8.1 One-Dimensional Linear Systems 8.2 Two-Dimensional Linear Systems Reading List 1132 1132 1135 1136 C MODES OF LINEAR SYSTEMS 1137 SYMBOLS AND UNITS 1142 AUTHORS 1159 INDEX 1161
FUNDAMENTALS OF PHOTONICS
CHAPTER . .. 1.1 POSTULATES OF RAY OPTICS 1.2 SIMPLE OPTICAL COMPONENTS A. Mirrors B. Planar Boundaries C. Spherical Boundaries and Lenses D. Light Guides 1.3 GRADED-INDEX OPTICS A. The Ray Equation B. Graded-Index Optical Components *C. The Eikonal Equation 1.4 MATRIX OPTICS A. The Ray-Transfer Matrix B. Matrices of Simple Optical Components C. Matrices of Cascaded Optical Components D. Periodic Optical Systems 3 6 17 24 .,., , t - , .. Sir Isaac Newton (1642-1727) set forth a theory of optics in which light emissions con- sist of collections of corpuscles that propagate rectilinear! y. Pierre de Fermat (160]-1665) enunciated the principle that light travels along the path of least time.
Light can be described as an electromagnetic wave phenomenon governed by the same theoretical principles that govern all other forms of electromagnetic radiation, such as radio waves and X-rays. This conception of light is called electromagnetic optics. Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric-field wave and a magnetic-field wave. Nevertheless, it is possible to describe many optical phenomena using a simplified scalar wave theory in which light is described by a single scalar wavefunction. This approximate way of treating light is called scalar wave optics, or simply wave optics. When light waves propagate through and around objects whose dimensions are much greater than the wavelength of the light, the wave nature is not readily discerned and the behavior of light can be adequately described by rays obeying a set of geomet- rical rules. This model of light is called ray optics. From a mathematical perspective, ray optics is the limit of wave optics when the wavelength is infinitesimally small. Thus, electromagnetic optics encompasses wave optics, which, in turn, encompasses ray optics, as illustrated in Fig. 1.0-1. Ray optics and wave optics are approximate theo- ries that derive their validity from their successes in producing results that approximate those based on the more rigorous electromagnetic theory. Quantum Optics Ray Optics Figure 1.0-1 The theory of quantum optics provides an explanation of virtually all optical phe- nomena. The electromagnetic theory of light (elec- tromagnetic optics) provides the most complete treatment of light within the confines of classical optics. Wave optics is a scalar approximation of electromagnetic optics. Ray optics is the limit of wave optics when the wavelength is very short. Electromagnetic Optics Wave Optics Although electromagnetic optics provides the most complete treatment of light within the confines of classical optics, certain optical phenomena are characteristically quantum mechanical in nature and cannot be explained classically. These phenomena are described by a quantum version of electromagnetic theory known as quantum electrodynamics. For optical phenomena, this theory is also referred to as quantum optics. Historically, the theories of optics developed roughly in the following sequence: (1) ray optics > (2) wave optics > (3) electromagnetic optics > (4) quantum optics. These models are progressively more complex and sophisticated, and were devel- oped successively to provide explanations for the outcomes of increasingly subtle and precise optical experiments. The optimal choice of a model is the simplest one that satisfactorily describes a particular phenomenon, but it is sometimes difficult to know a priori which model will achieve this. Fortunately, however, experience often provides a good guide. For pedagogical reasons, the initia] chapters in this book follow the historical order indicated above. Each model of light begins with a set of postulates (provided without proof), from which a large body of results are generated. The postulates of each model are shown to arise in special cases of the next-higher-Ievel model. In this chapter we begin with ray optics. 2
1.1 POSTULATES OF RAY OPTICS 3 This Chapter Ray optics is the simplest theory of light. Light is described by rays that travel in different optical media in accordance with a set of geometrical rules. Ray optics is therefore also called geometrical optics. Ray optics is an approximate theory. Al- though it adequately describes most of our daily experiences with light, there are many phenomena that ray optics does not adequately describe (as amply attested to by the remaining chapters of this book). Ray optics is concerned with the location and direction of light rays. It is therefore useful in studying image formation the collection of rays from each point of an object and their redirection by an optical component onto a corresponding point of an image. Ray optics permits us to determine conditions under which light is guided within a given medium, such as a glass fiber. In isotropic media, optical rays point in the direction of the flow of optical energy. Ray bundles can be constructed in which the density of rays is proportional to the density of light energy. When light is generated isotropically from a point source, for example, the energy associated with the rays in a given cone is proportional to the solid angle of the cone. Rays may be traced through an optical system to determine the optical energy crossing a given area. This chapter begins with a set of postulates from which the simple rules that govern the propagation of light rays through optical media are derived. In Sec. 1.2 these rules are applied to simple optical components such as mirrors and planar or spher- ical boundaries between different optical media. Ray propagation in inhomogeneous (graded-index) optical media is examined in Sec. 1.3. Graded-index optics is the basis of a technology that has become an important part of modem optics. Optical components are often centered about an optical axis, about which the rays travel at small inclinations. Such rays are called paraxial rays. This assumption is the basis of paraxial optics. The change in the position and inclination of a paraxial ray as it travels through an optical system can be efficiently described by the use of a 2 x 2-matrix algebra. Section 1.4 is devoted to this algebraic tool, called matrix optics. . 1.1 POSTULATES OF RAY OPTICS Postulates of Ray Optics . Light travels in the form of rays. The rays are emitted by light sources and can be observed when they reach an optical detector. . An optical medium is characterized by a quantity n > 1, called the refractive index. The refractive index n Co c where Co is the speed of light in free space and c is the speed of light in the medium. Therefore, the time taken by light to travel a distance d is d c nd CO. It is proportional to the product nd, which is known as the optical pathlength. . In an inhomogeneous medium the refractive index n r is a function of the position r x, y, z . The optical path length along a given path between two points A and B is therefore Optical pathlength B n r ds, A (1.1-1) where ds is the differential element of length along the path. The time taken by light to travel from A to B is proportional to the optical pathlength.
4 CHAPTER 1 RAY OPTICS . . Fermat's Principle. Optical rays traveling between two points, A and B, fol- Iowa path such that the time of travel (or the optical pathlength) between the two points is an extremum relative to neighboring paths. This is expressed mathematically as B 8 n r ds 0, A (1.1-2) where the symbol 8, which is read "the variation of," signifies that the optical pathlength is either minimized or maximized, or is a point of inflection. It is, however, usually a minimum, in which case: Light rays travel along the path of least time. Sometimes the minimum time is shared by more than one path, which are then all followed simultaneously by the rays. An example in which the pathlength is maximized is provided in Probe 1.1-2. In this chapter we use the postulates of ray optics to determine the rules governing the propagation of light rays, their reflection and refraction at the boundaries between different media, and their transmission through various optical components. A wealth of results applicable to numerous optical systems are obtained without the need for any other assumptions or rules regarding the nature of light. Propagation in a Homogeneous Medium In a homogeneous medium the refractive index is the same everywhere, and so is the speed of light. The path of minimum time, required by Fermat's principle, is therefore also the path of minimum distance. The principle of the path of minimum distance is known as Hero's principle. The path of minimum distance between two points is a straight line so that in a homogeneous medium, light rays travel in straight lines (Fig. 1. 1- 1 ). , -- Figure 1.1-1 Light rays travel in straight lines. Shadows are perfect projections of stops.
1.1 POSTULATES OF RAY OPTICS 5 Reflection from a Mirror Mirrors are made of certain highly polished metallic surfaces, or metallic or dielectric films deposited on a substrate such as glass. Light reflects from mirrors in accordance with the law of reflection: The reflected ray lies in the plane of incidence; the angle of reflection equals the angle of incidence. The plane of incidence is the plane formed by the incident ray and the normal to the mirror at the point of incidence. The angles of incidence and reflection, 8 and 8', are defined in Fig. 1.1-2(a). To prove the law of reflection we simply use Hero's principle. Examine a ray that travels from point A to point C after reflection from the planar mirror in Fig. 1.1-2(b) . A cc ordi ng to Hero's principle, for a mirror of infinitesimal thic knes s, th e distanc e A B + BC must be minimum. If C' is a mirror image of C, then BC BC', so that AB + BC' must be a minimum. This occurs when ABC' is a straight line, i.e., when B coincides with B' so that 8 8'. Plane of incidence Mirror Mirror c c' Reflected ray , , , , , , , , , , , , , , , ", " " " , " " " , , B ' ., , " , ., " " , , ., , , , , , , , '" , ()' " ,,' () ........_'.1 B ...- ...... ...- ...... - Normal to mirror ()' () A Incident ray (a) (b) Figure 1.1-2 (a) Reflection from the surface of a curved mirror. (b) Geometrical construction to prove the law of reflection. Reflection and Refraction at the Boundary Between Two Media At the boundary between two media of refractive indexes n1 and n2 an incident ray is split into two a reflected ray and a refracted (or transmitted) ray (Fig. 1.1-3). The reflected ray obeys the law of reflection. The refracted ray obeys the law of refraction: The refracted ray lies in the plane of incidence; the angle of refraction 8 2 is related to the angle of incidence 8 1 by Snell's law, nl sin 8 1 n2 sin 8 2 . (1.1-3) Snell's law The proportion in which the light is reflected and refracted is not described by ray . optIcs.
6 CHAPTER 1 RAY OPTICS Reflected ray ! -- Nonnal to boundary OJ OJ Refracted I '2 ray Incident ray , Plane of incidence n2 nl Figure 1.1-3 Reflection and refraction at the boundary between two media. EXERCISE 1.1-1 Proof of Snell's Law. The proof of Snell's law is an exercise in the ap plic ation of Fermat's principle. Referring to Fig. 1.1-4, we seek to minimize the optical pathlength nlAB +n2BC between points A and C. We therefore have the following optimization problem: Minimize nl d 1 see ()l + n2 d 2 see f)2 with respect to the angles f)l and f)2, subject to the condition d 1 tan f)l + d 2 tan f)2 d. Show that the solution of this constrained minimization problem yields Snell's law. n} n2 d 2 C ----------------. O} °2 B d. d} }\ ----------------- Figure 1.1-4 Construction to prove Snell's law. The three simple rules propagation in straight lines and the laws of reflection and refraction are applied in Sec. 1.2 to several geometrical configurations of mirrors and transparent optical components, without further recourse to Fermat's principle. 1.2 SIMPLE OPTICAL COMPONENTS A. Mirrors Planar Mirrors A planar mirror reflects the rays originating from a point PI such that the reflected rays appear to originate from a point P2 behind the mirror, called the image (Fig. 1.2-1). Paraboloidal Mirrors The surface of a paraboloidal mirror is a paraboloid of revolution. It has the useful property of focusi ng a ll incident rays parallel to its axis to a single point called the fo- cus. The distance P F f defined in Fig. 1.2-2 is called the focal length. Paraboloidal .
1.2 SIMPLE OPTICAL COMPONENTS 7 mirrors are often used as light-collecting elements in telescopes. They are also used for making parallel beams of light from point sources such as in flashlights. Mirror PI Pz " " " " " " " " " " " " " " " " " " " " , " " " " " ---------------- -------------- Figure 1.2-1 Reflection of light from a planar Figure 1.2-2 Focusing of light by a mIrror. paraboloidal mirror. Elliptical Mirrors An elliptical mirror reflects all the rays emitted from one of its two foci, e.g., PI, and images them onto the other focus, P 2 (Fig. 1.2-3). In accordance with Hero's principle, the distances traveled by the light from PI to P 2 along any of the paths are equal. Figure 1.2-3 Reflection from an elliptical mIrror. Spherical Mirrors A spherical mirror is easier to fabricate than a paraboloidal mirror or an elliptical mirror. However, it has neither the focusing property of the paraboloidal mirror nor the imaging property of the elliptical mirror. As illustrated in Fig. 1.2-4, parallel rays meet the axis at different points; their envelope (the dashed curve) is called the caustic curve. Nevertheless, parallel rays close to the axis are approximately focused onto a single point F at distance (- R) /2 from the mirror center C. By convention, R is negative for concave mirrors and positive for convex mirrors. Paraxial Rays Reflected from Spherical Mirrors Rays that make small angles (such that sin () () with the mirror's axis are called paraxial rays. In the paraxial approximation, where only paraxial rays are consid- ered, a spherical mirror has a focusing property like that of the paraboloidal mirror and an imaging property like that of the elliptical mirror. The body of rules that results from this approximation forms paraxial optics, also called first-order optics or Gaussian optics.
8 CHAPTER 1 RAY OPTICS " " z " " , , , , I I I , , I I , , \ \ \ , , , , ... ... c z ------ ------ Spherical mIrror , , I I , \ \ \ \ \ , , , ... ... ------ " "----;-RH Figure 1.2-4 Reflection of parallel rays from a concave spherical mirror. (-2 R ) I (R) Figure 1.2-5 A spherical mirror approxi- mates a paraboloidal mirror for paraxial rays. A spherical mirror of radius R therefore acts like a paraboloidal mirror of focal length f == R/2. This is in fact plausible since at points near the axis, a parabola can be approximated by a circle with radius equal to the parabola's radius of curvature (Fig. 1.2-5). All paraxial rays originating from each point on the axis of a spherical mirror are reflected and focused onto a single corresponding point on the axis. This can be seen (Fig. 1.2-6) by examining a ray emitted at an angle ()l from a point PI at a distance ZI away from a concave mirror of radius R, and reflecting at angle ( -()2) to meet the axis at a point P 2 that is a distance Z2 away from the mirror. The angle ()2 is negative since the ray is traveling downward. Since the three angles of a triangle add to 180 0 , we have ()I == ()o - () and (-()2) == ()o + (), so that ( -()2) + ()I == 2()o. If ()o is sufficiently small, the approximation tan ()o ()o may be used, so that ()o Y / ( - R), from which 2y ( -( 2 ) + 0 1 (- R) , (1.2-1) where y is the height of the point at which the reflection occurs. Recall that R is negative since the mirror is concave. Similarly, if ()I and ()2 are small, ()l Y / Zl and (-()2) == y/Z2' so that (1.2-1) yields y/Zl + y/Z2 2y/( -R), whereupon 1 1 -+- ZI Z2 2 (- R) . (1.2-2) z T y 1 -..;( z Zl (-R) Z2 (-R)/2 o Figure 1.2-6 Reflection of paraxial rays from a concave spherical mirror of radius R < O.
1.2 SIMPLE OPTICAL COMPONENTS 9 This relation holds regardless of y (i.e., regardless of 0 1 ) as long as the approxima- tion is valid. This means that all paraxial rays originating from point PI arrive at P2. The distances ZI and Z2 are measured in a coordinate system in which the z axis points to the left. Points of negative z therefore lie to the right of the mirror. According to (1.2-2), rays that are emitted from a point very far out on the z axis ZI ()() are focused to a point F at a distance Z2 R 2. This means that within the paraxial approximation, all rays coming from infinity (parallel to the axis of the mirror) are focused to a point at a distance f from the mirror, which is known as its focal length: f , R' , ) 2 , (1.2-3) Focal Length Spherical Mirror Equation (1.2-2) is usually written in the form 1 1 + Zl Z2 1 f' ( 1.2 -4 ) Imaging Equation (Paraxial Rays) which is known as the imaging equation. Both the incident and the reflected rays must be paraxial for this equation to hold. EXERCISE 1.2-1 Image Formation by a Spherical Mirror. Show that within the paraxial approximation, rays originating from a point PI (y!, Zl) are reflected to a point P 2 (Y2, Z2), where Zl and Z2 satisfy (1.2-4) and Y2 YIZ2/ Zl (Fig. 1.2-7). This means that rays from each point in the plane Z Zl meet at a single corresponding point in the plane Z Z2, so that the mirror acts as an image-formation system with magnification Z2/ Z1 . Negative magnification means that the image is inverted. y P I =(y I' Z I) c z Pz=(yz, zz) Figure 1.2-7 Image formation by a spherical mirror. Four particular rays are illustrated. B. Planar Boundaries The relation between the angles of refraction and incidence, O 2 and 0 1 , at a planar boundary between two media of refractive indexes nl and n2 is governed by Snel]'s law (1.1-3). This relation is plotted in Fig. 1.2-8 for two cases: . External Refraction nl < n2 . When the ray is incident from the medium of smaller refractive index, O 2 < 0 1 and the refracted ray bends away from the boundary.
1 0 CHAPTER 1 RAY OPTICS . Internal Refraction ni > n2 . If the incident ray is in a medium of higher refractive index, ()2 > ()I and the refracted ray bends toward the boundary. n I n 2 n} n z °e 90° n 2 /n}= 2/3 O 2 00 I Oe OJ 3/2 External refraction Internal refraction 0 1 90° Figure 1.2-8 Relation between the angles of refraction and incidence. The refracted rays bend in such a way as to minimize the optical pathlength, i.e., to increase the pathlength in the lower-index medium at the expense of pathlength in the higher-index medium. In both cases, when the angles are small (i.e., the rays are paraxial), the relation between ()2 and ()I is approximately linear, nl()1 n2()2, or ()2 nl n2 ()I. Total Internal Reflection For internal refraction ni > n2 , the angle of refraction is greater than the angle of incidence, ()2 > ()I, so that as ()I increases, ()2 reaches 90 0 first (see Fig. 1.2-8). This occurs when ()I ()e (the critical angle), with ni sin ()e n2 sin 7r 2 n2, so that ()e · -1 n2 SIn . nl ( 1.2-5) Critical Angle When ()I > ()e, Snells' law (1.1-3) cannot be satisfied and refraction does not occur. The incident ray is totally reflected as if the surface were a perfect mirror [Fig. 1.2- 9(a)]. The phenomenon of total internal reflection is the basis of many optical de- vices and systems, such as reflecting prisms [see Fig. 1.2-9(b)] and optical fibers (see Sec. 1.2D). It can be shown using electromagnetic optics (Fresnel's equations in Chap- ter 6) that all of the energy is carried by the reflected light so that the process of total internal reflection is highly efficient. n 1 n 2 o o n 1 ocr n 2 = 1 (a) (b) (c) Figure 1.2-9 (a) Total internal reflection at a planar boundary. (b) The reflecting prism. If n] > 2 and n2 1 (air), then Oe < 45°; since 0 1 45°, the ray is totally reflected. (c) Rays are guided by total internal reflection from the internal surface of an optical fiber.
1.2 SIMPLE OPTICAL COMPONENTS 11 Prisms A prism of apex angle a and refractive index n (Fig. 1.2-1 0) deflects a ray incident at an angle e by an angle e d e Q + sin- 1 n 2 sin 2 e sin Q sin e cos Q . (1.2-6) This may be shown by using Snell's law twice at the two refracting surfaces of the prism. When Q is very small (thin prism) and e is also very small (paraxial approxima- tion), (1.2-6) is approximated by L ed n 1 Q. (1.2-7) 60° 40° Q' = 45° Q' = 30° , , , , , , (}d (}d () Q = 10° 20° n n=l 0° 0° () 90° Figure 1.2-1 0 Ray deflection by a prism. The angle of deflection ()d is a function of the angle of incidence () for different apex angles a when n 1.5. When both a and () are small ()d (n l)a, which is approximately independent of (). When a 45° or () 0°, total internal reflection occurs, as illustrated in Fig. ] .2-9(b). Beamsplitters The beamsplitter is an optical component that splits the incident beam into a reflected beam and a transmitted beam, as illustrated in Fig. 1.2-11. Beamsplitters are also frequently used to combine two light beams into one [Fig. t.2-11(e)]. Beamsplitters are often constructed by depositing a thin semitransparent metallic or dielectric film on a glass substrate. A thin glass plate or a prism can also serve as a beamsplitter. (a) Partially reflective mirror ( b) Thin glass plate (c) Beam combiner Figure 1.2-11 Beamsplitters and combiners.
12 CHAPTER 1 RAY OPTICS c. Spherical Boundaries and Lenses We now examine the refraction of rays from a spherical boundary of radius R between two media of refractive indexes nl and n2. By convention, R is positive for a convex boundary and negative for a concave boundary. The results are obtained by applying Snell's law, which relates the angles of incidence and refraction relative to the normal to the surface, defined by the radius vector from the center e. These angles are to be distinguished from the angles ()I and ()2, which are defined relative to the z axis. Considering only paraxial rays making small angles with the axis of the system so that sin () () and tan () (), the following properties may be shown to hold: . A ray making an angle ()I with the z axis and meeting the boundary at a point of height y where it makes an angle ()o with the radius vector [see Fig. 1.2-12(a)] changes direction at the boundary so that the refracted ray makes an angle ()2 with the z axis and an angle ()3 with the radius vector. The angle of incidence is therefore 0 1 + ()2 while the angle of refraction is ()3, so that nl ()2 ()I n2 n2 n 1 y n2 R' ( 1.2-8) , R , , (a) OJ ( -(2) y - - - - - - - - - - - - - - - - - -- PI C P 2 n j n 2 z \ \ , , y PI =(YJ,zl) - - - - - --. - - - - - - - .. C (b) o - - P 2 = (Y2' Z2) Z Z Figure 1.2-12 Refraction at a convex spherical boundary (R > 0). . All paraxial rays originating from a point PI Y1 , Zl in the z Zl plane meet at a point P 2 Y2, Z2 in the z Z2 plane, where Zl Z2 R and nl Z2 Yl. n2 Zl (1.2-1 0) Y2
1.2 SIMPLE OPTICAL COMPONENTS 13 The Z ZI and Z Z2 planes are said to be conjugate planes. Every point in the first plane has a corresponding point (image) in the second with magnifi- cation nl n2 Z2 ZI . Again, negative magnification means that the image is inverted. By convention PI is measured in a coordinate system pointing to the left and P2 in a coordinate system pointing to the right (e.g., if P 2 lies to the left of the boundary, then Z2 would be negative). The similarities between these properties and those of the spherical mirror are evi- dent. It is important to remember that the image formation properties described above are approximate. They hold only for paraxial rays. Rays of large angles do not obey these paraxial laws; the deviation results in image distortion called aberration. EXERCISE 1.2-2 Image Formation. Derive (1.2-8). Prove that paraxial rays originating from PI pass through P 2 when (1.2-9) and (1.2-1 0) are satisfied. EXERCISE 1.2-3 Aberration-Free Imaging Surface. Determine the equation of a convex aspherical (nonspheri- cal) surface between media of refractive indexes nl and n2 such that all rays (not necessarily paraxial) from an axial point PI at a distance Zl to the left of the surface are imaged onto an axial point P2 at a distance Z2 to the right of the surface [Fig. 1.2-12(a)]. Hint: In accordance with Fermat's principle the optical pathlengths between the two points must be equal for all paths. Lenses A spherical lens is bounded by two spherical surfaces. It is, therefore, defined com- pletely by the radii R 1 and R 2 of its two surfaces, its thickness , and the refractive index n of the material (Fig. 1.2-13). A glass lens in air can be regarded as a combina- tion of two spherical boundaries, air-to-glass and glass-to-air. , ., , " , " \ , " \ ( ) " \ I -R2/" \ " , , , , , , , , , , , , , , , , , , , , , / / , , , , , , , , R I ,,/ , , , , , , , , , , , , , , , " ',' , " " , \ , \ I , I I / / , " Figure 1.2-13 A biconvex spherical lens. A ray crossing the first surface at height y and angle 8 1 with the Z axis [Fig. 1.2- 14(a)] is traced by applying (1.2-8) at the first surface to obtain the inclination angle 8 of the refracted ray, which we extend until it meets the second surface. We then use (1.2-8) once more with 8 replacing 8 1 to obtain the inclination angle ()2 of the ray after refraction from the second surface. The results are in general complicated. When the lens is thin, however, it can be assumed that the incident ray emerges from the lens at about the same height y at which it enters. Under this assumption, the following relations follow:
14 CHAPTER 1 RAY OPTICS . The angles of the refracted and incident rays are related by ()2 ()I Y f' (1.2-11) where f, called the focal length, is given by 1 f 1 RI 1 R 2 n 1 . (1.2-12) Focal Length Thin Spherical Lens OJ ( -(}2) P 1 =(y 1 ,Z 1 ) y - - - - - - - - - - - - - . - - F PJ P2 P 2=(Y2'Z2) Zl o Z2 ZJ o f Z2 (a) (b) Figure 1.2-14 (a) Ray bending by a thin lens. (b) Image formation by a thin lens. . All rays originating from a point PI [Fig. 1.2-14(b)], where YI, ZI meet at a point P 2 Y2, Z2 1 1 + Zl Z2 1 f (1.2-13) Imaging Equation and Y2 Z2 Yl- Zl (1.2-14) Magnification These results are identical to those for the spherical mirror [see (1.2-4) and Exer- cise 1.2-1]. These equations indicate that each point in the Z Zl plane is imaged onto a corresponding point in the Z Z2 plane with the magnification factor Z2 ZI. The magnification is unity when Zl Z2 2f. The focal length f of a lens therefore completely determines its effect on paraxial rays. As indicated earlier, PI and P 2 are measured in coordinate systems pointing to the left and right, respectively, and the radii of curvatures RI and R 2 are positive for convex surfaces and negative for concave surfaces. For the biconvex lens shown in Fig. 1.2-13, RI is positive and R 2 is negative, so that the two terms of (.1.2-12) add and provide a positive f.
1.2 SIMPLE OPTICAL COMPONENTS 15 EXERCISE 1.2-4 Proof of the Thin Lens Formulas. Using (1.2-8), along with the definition of the focal length given in (1.2-12), prove (1.2-11) and (1.2-13). It is emphasized once again that the foregoing relations hold only for paraxial rays. The presence of nonparaxial rays results in aberrations, as illustrated in Fig. 1.2-15. f Figure 1.2-15 Nonparaxial rays do not meet at the paraxial focus. The dashed envelope of the refracted rays is called the caustic curve. D. Light Guides Light may be guided from one location to another by use of a set of lenses or mirrors, as illustrated schematically in Fig. 1.2-16. Since refractive elements (such as lenses) are usually partially reflective and since mirrors are partially absorptive, the cumula- tive loss of optical power will be significant when the number of guiding elements is large. Components in which these effects are minimized can be fabricated (e.g., antireflection-coated lenses), but the system is generally cumbersome and costly. (a) =--------------------- - (b) - - ----------- ---------- ----------- ------ ) --- - -- --- Figure 1.2-16 Guiding light: (a) lenses; (b) mirrors; (c) total internal reflection. An ideal mechanism for guiding light is that of total internal reflection at the bound- ary between two media of different refractive indexes. Rays are reflected repeatedly without undergoing refraction. Glass fibers of high chemical purity are used to guide light for tens of kilometers with relatively low loss of optical power. An optical fiber is a light conduit made of two concentric glass (or plastic) cylinders (Fig. 1.2-17). The inner, called the core, has a refractive index nl, and the outer, called
16 CHAPTER 1 RAY OPTICS the cladding, has a slightly smaller refractive index, n2 < nl. Light rays traveling in the core are totally re fl ected from the cladding if their angle of incidence is greate r than the critical angle, () > (}c sin- 1 n2 nl . The rays making an an gl e () 90 ° () with the optical axis are therefore confined in the fiber core if () < (}c, where (}c 90° (}c cos- 1 n2 nl . Optical fibers are used in optical communication systems (see Chapters 9 and 24). Some important properties of optical fibers are derived in Exercise 1.2-5. Cladding Core n2 nl - - - -- - Figure 1.2-17 The optical fiber. Light rays are guided by multiple total internal reflections. Here - () represents the angle measured from the axis of the optical fiber so that its complement () 90° () is the angle of incidence at the dielectric interface. EXERCISE 1.2-5 Numerical Aperture and Angle of Acceptance of an Optical Fiber. An optical fiber is illuminated by light from a source (e.g., a light-emitting diode, LED). The refractive indexes of the core and cladding of the fiber are nl and n2, respectively, and the refractive index of air is 1 (Fig. 1.2- 18). Show that the half-angle ()a of the cone of rays accepted by the fiber (transmitted through the fiber without undergoing refraction at the cladding) is given by NA sin ()a n 2 1 2 n 2 · (1.2-15) Numerical Aperture Optical Fiber The angle () a is called the acceptance angle and the parameter N A sin () a is known as the numerical aperture of the fiber. Calculate the numerical aperture and acceptance angle for a silica-glass fiber with nl 1.475 and n2 1.460. Air Cladding n2 Core n I - - - - - - - - - - - ()a n2 Figure 1.2-18 Acceptance angle of an optical fiber. Trapping of Light in Media of High Refractive Index It is often difficult for light originating inside a medium of large refractive index to be extracted into air, especially if the surfaces of the medium are parallel. This occurs since certain rays undergo multiple total internal reflections without ever refracting into air. The principle is illustrated in Exercise 1.2-6.
1.3 GRADED-INDEX OPTICS 17 EXERCISE 1.2-6 Light Trapped in a Light-Emitting Diode. (a) Assume that light is generated in all directions inside a material of refractive index n cut in the shape of a parallelepiped (Fig. 1.2-19). The material is surrounded by air with unity refractive index. This process occurs in light-emitting diodes (see Chapter 17). What is the angle of the cone of light rays (inside the material) that will emerge from each face? What happens to the other rays? What is the numerical value of this angle for GaAs (n 3.6)? I Figure 1.2-19 Trapping of light in a paral- lelepiped of high refractive index. (b) Assume that when light is generated isotropically the amount of optical power associated with the rays in a given cone is proportional to the solid angle of the cone. Show that the ratio of the optical power that is extracted from the material to the total generated optical power is 3 1 1 1/n2, provided that n > 2. What is the numerical value of this ratio for GaAs? 1.3 GRADED-INDEX OPTICS A graded-index (GRIN) material has a refractive index that varies with position in accordance with a continuous function n r . These materials are often fabricated by adding impurities (dopants) of controlled concentrations. In a GRIN medium the opti- cal rays follow curved trajectories, instead of straight lines. By appropriate choice of n r , a GRIN plate can have the same effect on light rays as a conventional optical component, such as a prism or lens. A. The Ray Equation To determine the trajectories of light rays in an inhomogeneous medium with refractive index n r , we use Fermat's principle, B 8 n r ds 0, A (1.3-1) where ds is a differential length along the ray trajectory between A and B. If the trajectory is described by the function x s , y s , and z s , where s is the length of the trajectory (Fig. 1.3-1), then using the calculus of variations it can be shown t that x s , t This derivation is beyond the scope of this book; see, e.g., R. Weinstock, Calculus of Variations, Dover, 1974.
18 CHAPTER 1 RAY OPTICS Y ( s ), and z ( s) must satisfy three partial differential equations, d ( dX ) an - n- ==- ds ds ax ' d ( dY ) an ds n ds == oy ' d ( dZ ) an ds n ds == oz . (1.3-2) By defining the vector r(s), whose components are x(s), y(s), and z(s), (1.3-2) may be written in the compact vector form d ( dr ) ds n ds == V n, (1.3-3) Ray Equation where Vn, the gradient of n, is a vector with Cartesian components an/ax, on/oy, and an/oz. Equation (1.3-3) is known as the ray equation. yt B A Figure 1.3-1 The ray trajectory is described parametrically by three functions x( s), y( s), and z(s), or by two functions x(z) and y(z). One approach to solving the ray equa tion is to describe the traje ctory by two func- tions x(z) and y(z), write ds == dz -J l + (dx/dz)2 + (dy/dz)2, and substitute in (1.3-3) to obtain two partial differential equations for x(z) and y(z). The algebra is generally not trivial, but it simplifies considerably when the paraxial approximation is used. The Paraxial Ray Equation In the paraxial approximation, the trajectory is almost parallel to the z axis, so that ds dz (Fig. 1.3-2). The ray equations (1.3-2) then simplify to d ( dX ) an dz n dz ax ' d ( dY ) an dz n dz oy . ( 1.3-4) Paraxial Ray Equations Given n == n(x, y, z), these two partial differential equations may be solved for the trajectory x(z) and y(z). In the limiting case of a homogeneous medium for which n is independent of x, y, z, (1.3-4) gives d 2 x/dz 2 == 0 and d 2 y/dz 2 == 0, from which it follows that x and yare linear functions of z, so that the trajectories are straight lines. More interesting cases will be examined subsequently.
1.3 GRADED-INDEX OPTICS 19 y z Figure 1.3-2 Trajectory of a paraxial ray in a graded-index medium. B. Graded-Index Optical Components Graded-Index Slab Consider a slab of material whose refractive index n n y is uniform in the x and z directions but varies continuously in the y direction (Fig. 1.3-3). The trajectories of paraxial rays in the y z plane are described by the paraxial ray equation d dy dz n dz dn dy' (1.3-5) from which d 2 y dz 2 1 dn y n y dy . (1.3-6) Given n y and initial conditions (y and dy dz at z 0), (1.3-6) can be solved for the function y z , which describes the ray trajectories. y y+L\y y O(y + L\y) O(y) -- - - dn n(y)+ d L\y n(y) Y z Refractive index Figure 1.3-3 Refraction in a graded-index slab. D Derivation of the Paraxial Ray Equation in a Graded-Index Slab Using Snell's Law. Equation (1.3-6) may also be derived by the direct use of Snell's law (Fig. 1.3-3). Let O(y) dyJdz be the angle that the ray makes with the z axis at the position (y, z). After traveling through a layer of thickness y the ray changes its angle to O(y + y). The two angles are related by Snell's law where 0, as defined in Fig. 1.3-3, is the complement of the angle of incidence (refraction): n(y) cos O(y) n(y + y) cos O(y + y) dn dO. cosO(y) (1.3-7) where we have applied the expansion f (y + y) f (y) + (df J dy) y to the functions f (y) n(y) and f(y) cosO(y). In the limit y ) 0, after eliminating the term in (y)2, we obtain the differential equation dn dO dy (1.3-8) For paraxial rays 0 is very small so that tan 0 O. Substituting 0 ( 1.3-6). dy J dz in (1.3-8), we obtain .
20 CHAPTER 1 RAY OPTICS EXAMPLE 1.3-1. Slab with Parabolic Index Profile. An important particular distribution for the graded refractive index is n 2 (y) n6 1 O?y2. ( 1.3-9) This is a symmetric function of y that has its maximum value at y 0 (Fig. 1.3-4). A glass slab with this profile is known by the trade name SELFOC. Usually, a is chosen to be sufficiently small so that a 2 y2 « 1 for all y of interest. Under this condition, n(y) no 1 a 2 y2 no(l a2y2); i.e., n(y) is a parabolic distribution. Also, because n(y) no «::: no, the fractional change of the refractive index is very small. Taking the derivative of (1.3-9), the right-hand side of (1.3-6) is (l/n)dn/dy (no/n)2a 2 y Q2y, so that (1.3-6) becomes d 2 y Q2y. dz 2 (1.3-10) The solutions of this equation are harmonic functions with period 2n / Q. Assuming an initial position y(O) Yo and an initial slope dy/dz Bo at z 0 inside the GRIN medium, (}o · Yo CDS Q'Z + SIn Q'Z, Q' y(z) (1.3-11) from which the slope of the trajectory is B(z) dy dz yoa sin QZ + Bo cos az. (1.3-12) The ray oscillates about the center of the slab with a period ( distance) 2n / Q known as the pitch, as illustrated in Fig. 1.3-4. y r 27r a y Yo 0 0 z no n(y) Figure 1.3-4 Trajectory of a ray in a GRIN slab of parabolic index profile (SELFOC). The maximum excursion of the ray is Ymax Y5 + (Bo/a)2 and the maximum angle is Bmax aYmax. The validity of this approximate analysis is ensured if (}max « 1. If 2Ymax is smaller than the thickness of the slab, the ray remains confined and the slab serves as a light guide. Figure 1.3- 5 shows the trajectories of a number of rays transmitted through a SELFOC slab. Note that all rays have the same pitch. This GRIN slab may be used as a lens, as demonstrated in Exercise 1.3- I . z 7r Q' d -i Figure 1.3-5 Trajectories of rays from an external point source in a SELFOC slab.
1.3 GRADED-INDEX OPTICS 21 EXERCISE 1.3-1 The GRIN Slab as a Lens. Show that a SELFOC slab of length d < 7r /2a and refractive index given by (1.3-9) acts as a cylindrical lens (a lens with focusing power in the y-z plane) of focal length nod a sin a (1.3-13) Show that the principal point (defined in Fig. 1.3-6) lies at a distance from the slab edge AH (l/noa) tan(ad/2). Sketch the ray trajectories in the special cases d 7r /a and 7r /2a. Y f Yo ------------ I "'" "" "'" "" I ..... "" I I I A F z H r- d Figure 1.3-6 The SELFOC slab used as a lens; F is the focal point and H is the principal point. Graded-Index Fibers A graded-index fiber is a glass cylinder with a refractive index n that varies as a function of the radial distance from its axis. In the paraxial approximation, the ray trajectories are governed by the paraxial ray equations (1.3-4). Consider, for example, the distribution n 2 n 2 1 o a? x 2 + y2 . (1.3-14) Substituting (1.3-14) into (1.3-4) and assuming that Q2 x 2 + y2 «1 for all x and y of interest, we obtain d 2 x r"..J r"..J dz 2 Q 2 X , tPy r"..J r"..J dz 2 Q2y. (1.3-15) Both x and yare therefore harmonic functions of z with period 27r Q. The initial positions Xo, Yo and angles (B xo dx dz and Byo dy dz at z 0 determine the amplitudes and phases of these harmonic functions. Because of the circular symmetry, there is no loss of generality in choosing Xy O. The solution of (1.3-15) is then Bxo . x z Sinaz a Byo . SIn az + Yo CDS az. a (1.3-16) y z If Bxo 0, Le., the incident ray lies in a meridional plane (a plane passing through the axis of the cylinder, in this case the y z plane), the ray continues to lie in that plane following a sinusoidal trajectory similar to that in the GRIN slab [Fig. 1.3-7(a)]. On the other hand, if Byo 0, and Bxo ayo, then x z . Yo SIn az ( 1.3-17) y z Yo CDS az,
22 CHAPTER 1 RAY OPTICS (a) 27r Q ()o - - .... ..... - - - - z \ Yo - - , , -- , , - - / , /- , .,v (\ / - , - --".- , z ., ., J (b) I \ \ Figure 1.3-7 (a) Meridional and (b) helical rays in a graded-index fiber with parabolic index profile. so that the ray follows a helical trajectory lying on the surface of a cylinder of radius Yo [Fig. 1.3- 7 (b)]. In both cases the ray remains confined within the fiber, so that the fiber serves as a light guide. Other helical patterns are generated with different incident rays. Graded-index fibers and their use in optical communications are discussed in Chap- ters 9 and 24. EXERCISE 1.3-2 Numerical Aperture of the Graded-Index Fiber. Consider a graded-index fiber with the index profile provided in (1.3-] 4) and radius a. A ray is incident from air into the fiber at its center, which then makes an angle (}o with the fiber axis in the medium (see Fig. 1.3-8). Show, in the paraxial approximation, that the numerical aperture is NA sin (}a noaa, ( 1.3-18) Numerical Aperture Graded-Index Fiber where () a is the maximum acceptance angle for which the ray trajectory is confined within the fiber. Compare this to the numerical aperture of a step-index fiber such as the one discussed in Exercise 1.2- 5. To make the comparison fair, t ake the refr active indexes of the core and cladding of the step- index fiber to be nl no and n2 no V I a 2 a 2 no (1 !a 2 a 2 ), respectively. y T (}a eo a -1- - - - - - - I e a eo z Figure 1.3-8 Acceptance angle of a graded-index optical fiber.
1.3 GRADED-INDEX OPTICS 23 *c. The Eikonal Equation The ray trajectories are often characterized by the surfaces to which they are normal. Let 8 r be a scalar function such that its equilevel surfaces, 8 r constant, are everywhere normal to the rays (Fig. 1.3-9). If 8 r is known, the ray trajectories can readily be constructed since the normal to the equilevel surfaces at a position r is in the direction of the gradient vector \7 8 r . The function 8 r , called the eikonal, is akin to the potential function V r in electrostatics; the role of the optical rays is played by the lines of electric field E \7V. Rays ,. S(r) = constant Figure 1.3-9 Ray trajectories are normal to the surfaces of constant S(r). To satisfy Fermat's principle (which is the main postulate of ray optics) the eikonal 8 r must satisfy a partial differential equation known as the eikonal equation, 08 2 ax + 08 2 ay 08 2 az n 2 , (1.3-19) + which is usually written in the vector form \78 2 n 2 , (1.3-20) Eikonal Equation where \7 8 2 \7 8 · \7 8. The proof of the eikonal equation from Fermat's principle is a mathematical exercise that lies beyond the scope of this book. t Fermat's principle (and the ray equation) can also be shown to follow from the eikonal equation. There- fore, either the eikonal equation or Fermat's principle may be regarded as the principal postulate of ray optics. Integrating the eikonal equation (1.3-20) along a ray trajectory between points A and B gives B 8 rB 8 rA \7 8 ds B n ds optical pathlength between A and B. A A (1.3-21) This means that the difference 8 r B 8 r A represents the optical pathlength be- tween A and B. In the electrostatics analogy, the optical pathlength plays the role of the potential difference. To determine the ray trajectories in an inhomogeneous medium of refractive index n r , we can either solve the ray equation (1.3-3), as we have done earlier, or solve the eikonal equation for 8 r , from which we calculate the gradient \7 8. t See, e.g., M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002.
24 CHAPTER 1 RAY OPTICS If the medium is homogeneous, i.e., n r is constant, the magnitude of \7 S is constant, so that the wavefront normals (rays) must be straight lines. The surfaces S r constant may be parallel planes or concentric spheres, as illustrated in Fig. 1.3- 10. S(r) = constant Rays / - Rays - Figure 1.3-10 Rays and surfaces of constant S(r) in a homogeneous medium. The eikonal equation is revisited from the point-of-view of the relation between ray optics and wave optics in Sec. 2.3. 1.4 MATRIX OPTICS Matrix optics is a technique for tracing paraxial rays. The rays are assumed to travel only within a single plane, so that the formalism is applicable to systems with planar geometry and to meridional rays in circularly symmetric systems. A ray is described by its position and its angle with respect to the optical axis. These variables are altered as the ray travels through the system. In the paraxial approxima- tion, the position and angle at the input and output planes of an optical system are related by two linear algebraic equations. As a result, the optical system is described by a 2 x 2 matrix called the ray transfer matrix. The convenience of using matrix methods lies in the fact that the ray-transfer matrix of a cascade of optical components (or systems) is a product of the ray-transfer matrices of the individual components (or systems). Matrix optics therefore provides a formal mechanism for describing complex optical systems in the paraxial approximation. A. The Ray-Transfer Matrix Consider a circularly symmetric optical system formed by a succession of refracting and reflecting surfaces all centered about the same axis (optical axis). The z axis lies along the optical axis and points in the general direction in which the rays travel. Consider rays in a plane containing the optical axes, say the y z plane. We proceed to trace a ray as it travels through the system, i.e., as it crosses the transverse planes at different axial distances. A ray crossing the transverse plane at z is completely characterized by the coordinate of y of its crossing point and the angle () (Fig. 1.4- 1 ). An optical system is a set of optical components placed between two transverse planes at Zl and Z2, referred to as the input and output planes, respectively. The system is characterized completely by its effect on an incoming ray of arbitrary position and
1.4 MATRIX OPTICS 25 y Ray e z Figure 1.4-1 A ray is charac- terized by its coordinate y and its angle (). Optical . aXIs Input (YI, ( 1 ) Optical system Output (Y2, (2) Y Input plane ° 1 Output plane °2 Figure 1.4-2 A ray enters an YI Yz optical system at location Zl with Optical position Yl and angle ()l and leaves Zl Zz . z aXIs at position Y2 and angle ()2. direction Yl, 0 1 . It steers the ray so that it has new position and direction Y2, O 2 at the output plane (Fig. 1.4-2). In the paraxial approximation, when all angles are sufficiently small so that sin 0 0, the relation between Y2, O 2 and YI, Ol is linear and can generally be written in the form Y2 AYl + BO I O 2 CYl + 001, ( 1.4- 1 ) (1.4-2) where A, B, C, and 0 are real numbers. Equations (1.4-1) and (1.4-2) may be conve- niently written in matrix form as Y2 O 2 A B C 0 Yl Ol · (1.4-3) The matrix M, whose elements are A, B, C, and 0, characterizes the optical system completely since it permits Y2, O 2 to be determined for any Yl, Ol . It is known as the ray-transfer matrix. As will be seen in the examples provided in Sec. 1.4B, angles that turn out to be negative point downward from the z axis in their direction of travel. Radii that turn out to be negative indicate concave surfaces whereas those that are positive indicate convex surfaces. EXERCISE 1.4-1 Special Forms of the Ray- Transfer Matrix. Consider the following situations in which one of the four elements of the ray-transfer matrix vanishes: (a) Show that if A 0, all rays that enter the system at the same angle leave at the same position, so that parallel rays in the input are focused to a single point at the output. (b) What are the special features of each of the systems for which B 0, C 0, or 0 O?
26 CHAPTER 1 RAY OPTICS B. Matrices of Simple Optical Components Free-Space Propagation Since rays travel along straight lines in a medium of uniform refractive index such as free space, a ray traversing a distance d is altered in accordance with Y2 YI + 0 1 d and O 2 0 1 . The ray-transfer matrix is therefore M 1 d o 1 · (1.4-4) d Refraction at a Planar Boundary At a planar boundary between two media of refractive indexes nl and n2, the ray angle changes in accordance with Snell's law nl sin 0 1 n2 sin O 2 . In the paraxial approximation, nlOl n202. The position of the ray is not altered, Y2 Yl. The ray-transfer matrix is nl n2 I M 1 0 o nl · n2 (1.4-5) Refraction at a Spherical Boundary The relation between 0 1 and O 2 for paraxial rays refracted at a spherical boundary between two media is provided in (1.2-8). The ray height is not altered, Y2 Yl. The ray-transfer matrix is R n} n 2 1 0 (n2- n l) nl · n2 R n2 ( 1.4-6) M Convex: R > 0; concave: R< 0 Transmission Through a Thin Lens The relation between 0 1 and O 2 for paraxial rays transmitted through a thin lens of focal length f is given in (1.2-11). Since the height remains unchanged Y2 Yl, we have 1 1 - f o 1 · (1.4-7) f M Convex: f> 0; concave: f < 0 Reflection from a Planar Mirror Upon reflection from a planar mirror, the ray position is not altered, Y2 Yl. Adopting the convention that the z axis points in the general direction of travel of the rays, i.e., toward the mirror for the incident rays and away from it for the reflected rays, we
1.4 MATRIX OPTICS 27 conclude that ()2 ()l. The ray-transfer matrix is therefore the identity matrix z z M 1 0 o 1 · ( 1.4-8) Reflection from a Spherical Mirror Using (1.2-1), and the convention that the z axis follows the general direction of the rays as they reflect from mirrors, we similarly obtain (-R) M 1 2 R o 1 · ( 1.4-9) Concave: R < 0; convex: R > 0 Note the similarity between the ray-transfer matrices of a spherical mirror (1.4-9) and a thin lens (1.4-7). A mirror with radius of curvature R bends rays in a manner that is identical to that of a thin lens with focal length f R 2. C. Matrices of Cascaded Optical Components A cascade of N optical components or systems whose ray-transfer matrices are M 1 , M 2 , . . . , MN is equivalent to a single optical system of ray-transfer matrix ). M 1 ). M 2 >- . . . ). MN >- M MN · · · M 2 M 1 . ( 1.4-1 0) Note the order of matrix multiplication: The matrix of the system that is crossed by the rays is first placed to the right, so that it operates on the column matrix of the incident ray first. A sequence of matrix multiplications is not, in general, commutative, although . . .. It IS aSSocIative. EXERCISE 1.4-2 A Set of Parallel Transparent Plates. Consider a set of N parallel planar transparent plates of refractive indexes nI, n2,.. · nN and thicknesses d 1 , d 2 ,. . . d N , placed in air (n 1) normal to the z axis. Using induction, show that the ray-transfer matrix is I n 1 n2 . . . nN 1 h z M o 1 ( 1.4-11 ) d} d 2 d N Note that the order in which the plates are placed does not affect the overall ray-transfer matrix. What is the ray-transfer matrix of an inhomogeneous transparent plate of thickness do and refractive index n(z)?
28 CHAPTER 1 RAY OPTICS EXERCISE 1.4-3 A Gap Followed by a Thin Lens. Show that the ray-transfer matrix of a distance d of free space followed by a lens of focal length 1 is f 1 1 - f 1 d d · - f ( 1.4-12) M d I EXERCISE 1.4-4 Imaging with a Thin Lens. Derive an expression for the ray-transfer matrix of a system com- prised of free space/thin lens/free space, as shown in Fig. 1.4-3. Show that if the imaging condition (1/ d 1 + 1/ d 2 1/ f) is satisfied, all rays originating from a single point in the input plane reach the output plane at the single point Y2, regardless of their angles. Also show that if d 2 f, all parallel incident rays are focused by the lens onto a single point in the output plane. f d 1 I--c d 2 Figure 1.4-3 Single-lens imaging system. EXERCISE 1.4-5 Imaging with a Thick Lens. Consider a glass lens of refractive index n, thickness d, and two spherical surfaces of equal radii R (Fig. 1.4-4). Determine the ray-transfer matrix of the system between the two planes at distances d l and d 2 from the vertices of the lens. The lens is placed in air (refractive index 1). Show that the system is an imaging system (i.e., the input and output planes are conjugate) if 1 1 + 1 f or 1 2 , 81 8 2 (1.4-13) Zl Z2 where ZI d l + h, Z2 d 2 + h, 81 ZI f 82 Z2 f (1.4-14) (1.4-15) and h (n l)fd nR (1.4-16) n Id n R . ( 1.4-17) 1 f R The points F I and F 2 are known as the front and back focal points, respectively. The points PI and P 2 are known as the first and second principal points, respectively. Show the importance of these points by tracing the trajectories of rays that are incident parallel to the optical axis.
1.4 MATRIX OPTICS 29 ......c: d Sl f n . FI . PI . P2 f ......c . F2 S2 d 1 h ...... h d 2 Z2 ZI Figure 1.4-4 Imaging with a thick lens. PI and P 2 are the principal points and FI and F 2 are the focal points. . D. Periodic Optical Systems A periodic optica] system is a cascade of identical unit systems. An example is a sequence of equally spaced identical relay lenses used to guide light, as shown in Fig. 1.2-16( a). Another example is the reflection of light between two mirrors that form an optical resonator (see Sec. 10.2A); in that case, the ray repeatedly traverses the same unit system (a round trip of reflections). Even a homogeneous medium, such as a glass fiber, may be considered as a periodic system if it is divided into contiguous identical segments of equal length. We proceed to formulate a general theory of ray propagation in periodic optical systems using matrix methods. Difference Equation for the Ray Position A periodic system is composed of a cascade of identical unit systems (stages), each with a ray-transfer matrix A, B, G, D , as shown in Fig. 1.4-5. A ray enters the system with initial position Yo and slope ()o. To determine the position and slope Ym, ()m of the ray at the exit of the mth stage, we apply the ABGD matrix m times, Yrn ()m A B m Yo G D ()o' ( 1.4-18) We can also iteratively apply the relations Ym+l AYm + B()m ()m+l GYm + D()m (1.4-19) (1.4-20) to determine (Yl, ()l from Yo, ()o , then Y2, ()2 from Yl, ()l , and so on, using a software routine. Yo ° 0 A B Yl C D OJ 1 A B C D . .. A B C D A B Y m A B C D Om C D Ym+l Om + I 2 m-l m m+ 1 Figure 1.4-5 A cascade of identical optical systems. It is of interest to derive equations that govern the dynamics of the position Ym, m 0,1, . . . , irrespective of the angle ()m. This is achieved by eliminating ()m from
30 CHAPTER 1 RAY OPTICS (1.4-19) and (1.4-20). From (1.4-19) Om Ym+ 1 AYm B . ( 1.4- 21 ) Replacing m with m + 1 in (1.4-21) yields AYm+ 1 B Substituting (1.4-21) and (1.4-22) into (1.4-20) gives Om+l Yrn+2 . (1.4-22) Ym+2 2bYm+l F2Ym, (1.4-23) Recurrence Relation for Ray Position where b A+D 2 AD BC det M , (1.4-24) (1.4-25) F 2 and det M is the determinant of M. Equation (1.4-23) is a linear difference equation governing the ray position Ym. It can be solved iteratively by computing Y2 from Yo and Yl, then Y3 from Yl and Y2, and so on. The quantity Yl may be computed from Yo and 0 0 by use of (1.4-19) with m O. It is useful, however, to derive an explicit expression for Ym by solving the difference equation (1.4-23). As with linear differential equations, a solution satisfying a linear difference equation and the initial conditions is a unique solution. It is therefore appro- priate to make a judicious guess for the solution of (1.4-23). We use a trial solution of the geometric form Ym Yohm, (1.4-26) where h is a constant. Substituting (1.4-26) into (1.4-23) immediately shows that the trial solution is suitable provided that h satisfies the quadratic algebraic equation h 2 2bh + F 2 0, (1.4-27) from which h b -X j p2 b 2 . ( 1.4-28) The results can be presented in a more compact form by defining the variable <p cos- 1 b F , (1.4-29) so that b F cos <p, F2 b 2 F sin <p, and therefore h F cos <p -X j sin <p F exp -xj<p , whereupon (1.4-26) becomes Ym yoFm exp -xjm<p . A general solution may be constructed from the two solutions with positive and negative signs by forming their linear combination. The sum of the two exponential functions can always be written as a harmonic (circular) function, so that Ym Ymax Fm sin m<p + <Po , (1.4-30)
1.4 MATRIX OPTICS 31 where Ymax and 'Po are constants to be determined from the initial conditions Yo and Yl. In particular, setting m 0 we obtain Ymax Yo sin 'Po. The parameter F is related to the determinant of the ray-transfer matrix of the unit system by F det M . It can be shown that regardless of the unit system, det M ni n2, where ni and n2 are the refractive indexes of the initial and final sections of the unIt system. This general result is easily verified for the ray-transfer matrices of all the optical components considered in this section. Since the determinant of a product of two matrices is the product of their determinants, it follows that the relation det M ni n2 is applicable to any cascade of these optical components. For exam- ple, if det M 1 ni n2 and det M 2 n2 n3, then det M 2 M I n2 n3 nl n2 ni n3. In most applications the first and last stages are air (n 1) ni n2, so that det M 1 and F 1, in which case the solution for the ray position is . , , Ym Ymax SIn, m'P + 'Po) · ( 1.4- 31 ) Ray Position Periodic System We shall assume henceforth that F 1. The corresponding solution for the ray angle is obtained by use of the relation ()m Ym+1 A Ym B, which is derived from (1.4-19). Condition for a Harmonic Trajectory For Ym to be a harmonic (instead of hyperbolic) function, 'P cos- I b must be real. This requires that b < 1 (1.4-32) Stability Condition or If, instead, b > 1, 'P is then imaginary and the solution is a hyperbolic function (cosh or sinh), which increases without bound, as illustrated in Fig. 1.4-6(a). A harmonic solution ensures that Ym is bounded for all m, with a maximum value of Ymax. The bound b < 1 therefore provides a condition of stability (boundedness) of the ray trajectory. Since Ym and Ym+1 are both harmonic functions, so too is the ray angle corre- sponding to (1.4-31), by virtue of (1.4-21) and trigonometric identities. Thus, ()m ()max sin m<p + 'PI , where the constants ()max and 'PI are determined by the initial conditions. The maximum angle ()max must be sufficiently small so that the paraxial approximation, which underlies this analysis, is applicable. Condition for a Periodic Trajectory The harmonic function (1.4- 31) is periodic in m if it is possible to find an integer s such that Ym+s Ym for all m. The smallest integer is the period. The ray then retraces its path after s stages. This condition is satisfied if S'P 21fq, where q is an integer. Thus, the necessary and su ffi cient condition for a periodic traj ec t or y is that 'P 21f is a is periodic with period s 11 stages. This case is illustrated in FIg. 1.4-6(b). Periodic optical systems will be revisited in Chapter 7.
32 CHAPTER 1 RAY OPTICS Ym (a) o 10 20 m Ym (b) m Ym (c) 10 m Figure 1.4-6 Examples of trajectories in periodic optical systems: (a) unstable trajectory (b > 1); (b) stable and periodic trajectory (<.p 67r /11; period 11 stages); (c) stable but nonperiodic trajectory (<.p 1.5). < Summary < : A paraxial ray (Omax « 1) traveling through a cascade of identical unit optical -. : systems, each with a ray-transfer matrix with elements A, B, C, D such that . AD BC 1, follows a harmonic (and therefore bounded) trajectory if the -. j. -. ; at th e m th stage is then Ym Ymax sin mcp + CPo , m 0, 1, 2, . . ., where I f : positions Yo and Yl .. Ayo + BOo, where (}o is the initial ray inclination. The : ray angles are related to the positions by Om Ym+1 AYm B and follow a ; harmonic function Om Oroax sin m<p + CPI . The ray trajectory is periodic with I ; period s if cp 21r is a rational number q s. -- EXAMPLE 1.4-1. A Sequence of Equally Spaced Identical Lenses. A set of identical lenses of focal length 1 separated by distance d, as shown in Fig. 1.4-7, may be used to relay light between two locations. The unit system, a distance of d of free space followed by a lens, has a ray-transfer matrix given by (1.4-12); A 1, B d, C 1/ I, D 1 d/ I. The parameter b ! (A + D) 1 d /2/ and the determinant is unity. The condition for a stable ray trajectory, Ibl < 1 or 1 < b < 1, is therefore o < d < 41, (1.4-33) so that the spacing between the lenses must be smaller than four times the focal length. Under this condition the positions of paraxial rays obey the harmonic function Ym Ymax sin( m<.p + <.po), <.p cos- 1 1 d . 21 ( 1.4-34) 1- - - - - - - - - - - r - .. - ... - - - ... - - ..... - - - - - - - - - ... .. : jf : jf : jf : I I I I I I I I I I I I I 1- _ _ _ _ _ _ _ _ _ _ I -------- ..----------- J.. d + d + d Figure 1.4-7 A periodic sequence of lenses.
1.4 MATRIX OPTICS 33 When d 2f, 'P n /2, and 'P /21r , so that the trajectory of an arbitrary ray is periodic with period equal to four stages. When d f, 'P 7r /3, and 'P /2n t, so that the ray trajectory is periodic and retraces itself each six stages. These cases are illustrated in Fig. 1.4-8. (a) d I (b) 1 --< d I Figure 1.4-8 Examples of stable ray trajectories in a periodic lens system: (a) d 2f; (b) d f. EXERCISE 1.4-6 A Periodic Set of Pairs of Different Lenses. Examine the trajectories of paraxial rays through a periodic system comprising a sequence of lens pairs with alternating focal lengths fl and f2, as shown in Fig. 1.4-9. Show that the ray trajectory is bounded (stable) if o < 1 d 2fl 1 d 2f2 < 1. ( 1.4-35) r lr 1r 1 - - II 1 2 II 1 2 II 1 2 II I I I II II I I I II I I I I II II I I I II II I I I II l I - I -f d .c d d 4l1li( d d d .. Figure 1.4-9 A periodic sequence of lens pairs. EXERCISE 1.4-7 An Optical Resonator. Paraxial rays are reflected repeatedly between two spherical mirrors of radii R 1 and R 2 separated by a distance d (Fig. 1.4-10). Regarding this as a periodic system whose unit system is a single round trip between the mirrors, determine the condition of stability for the ray trajectory. Optical resonators will be studied in detail in Chapter 10. ----- .------- - ,,; --... ... ... ..... , - , , .... .... .... .... .... " " " " "", R2 ..... '" ..... ., , , , , , , , " RI " , , , , , , , , , , , , " .... " .... " " , " " \ \ \ \ , , , , , I I I I Z , , , , , , I I I I , , , , , , , , , , Figure 1.4-1 0 The optical resonator as a periodic optical system. '" , , , , , I I I I I I I , , : Yo I I I , , , \ Y2 \ \ \ \ " " , " " " " " " " " , -01 - , , , , " " " " " " " d .... " '" '" .... '" '" , .......... -... ...- ... ..._-......."....._----.--... " '" .... ..... ....
34 CHAPTER 1 RAY OPTICS READING LIST General F. L. Pedrotti, L. M. Pedrotti, and L. S. Pedrotti, Introduction to Optics, Prentice Hall, 3rd ed. 2006. K. K. Sharma, Optics: Principles and Applications, Academic Press, 2006. A. Walther, The Ray and Wave Theory of Lenses, Cambridge University Press, 1995, paperback 00. 2006. K. D. Moeller, Optics: Learning by Computing with Examples Using Maple, MathCad, Mathematica, and MATLAB, Springer-Verlag, 2nd ed. 2006. T.-C. Poon and T. Kim, Engineering Optics with MATLAB, World Scientific, 2006. A. Siciliano, Optics: Problems and Solutions, World Scientific, 2006. G. Chartier, Introduction to Optics, Springer- Verlag, 2005. J. Strong, Concepts of Classical Optics, Freeman, 1958; Dover, paperback ed. 2004. G. Brooker, Modem Classical Optics, Oxford University Press, 2003. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002. E. Hecht, Optics , Addison-Wesley, 4th ed. 2002. M. Mansuripur, Classical Optics and Its Applications, Cambridge University Press, 2002. M. P. Keating Geometric, Physical, and Visual Optics, Butterworth-Heinemann, 2nd ed. 2002. M. Young, Optics and Lasers Including Fibers and Optical Waveguides, Springer-Verlag, 1977, 5th ed. 2000. J. R. Meyer-Arendt, Introduction to Classical and Modem Optics, Prentice Hall, 1972, 4th ed. 1995. J. W. Blaker and W. M. Rosenblum, Optics An Introduction for Students of Engineering, Macmillan, 1993. D. T. Moore, ed., Selected Papers on Gradient-Index Optics, SPIE Optical Engineering Press (Mile- stone Series Volume 67), 1993. F. A. Jenkins and H. E. White, Fundamentals of Optics, McGraw-Hill, 1937, 4th revised ed. 1991. P. P. Banerjee and T.-C. Poon, Principles of Applied Optics, Aksen Associates, 1991. R. D. Guenther, Modern Optics, Wiley, 1990. E. Hecht and A. Zajac, Optics, Addison-Wesley, 1974, 2nd ed. 1990. W. T. Welford, Optics, Oxford University Press, 1976, 3rd ed. 1988. R. W. Wood, Physical Optics, Macmillan, 3rd ed. 1934; Optical Society of America, 1988. M. V. Klein and T. E. Furtak, Optics, Wiley, 1982, 2nd ed. 1986. E. W. Marchand, Gradient-Index Optics, Academic Press, 1978. F. P. Carlson, Introduction to Applied Optics for Engineers, Academic Press, 1977. R. W. Ditchburn, Light, Academic Press, 3rd ed. 1976. E. Hecht, Schaum's Outline of Optics, McGraw-Hili, paperback ed. 1974. B. B. Rossi, Optics, Addison-Wesley, 1957, reprinted 1965. J. M. Stone, Radiation and Optics, McGraw-Hili, 1963. A. Sommerfeld, Lectures on Theoretical Physics: Optics, Academic Press, paperback ed. 1954. Geometrical Optics Yu. A. Kravtsov, Geometrical Optics in Engineering Physics, Alpha Science, 2005. J. E. Greivenkamp, Field Guide to Geometrical Optics, SPIE Optical Engineering Press, 2004. K. B. Wolf, Geometric Optics on Phase Space, Springer-Verlag, 2004. M. Katz, Introduction to Geometrical Optics, World Scientific, 2002. R. Ditteon, Modern Geometrical Optics, Wiley, 1998. F. Colombini and N. Lerner, eds., Geometrical Optics and Related Topics, Birkhauser, 1997. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design, Oxford University Press, 1997 . D. S. Loshin, The Geometrical Optics Workbook, Butterworth-Heinemann, 1991. G. A. Fry, Geometrical Optics, Chilton, 1969, reprinted 1981.
PROBLEMS 35 W. T. Welford and R. Winston, The Optics of Non imaging Concentrators, Academic Press, 1978 O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, Academic Press, 1972. H.-G. Zimmer, Geometrical Optics, Springer-Verlag, 1970. A. Nussbaum, Geometric Optics: An Introduction, Addison-Wesley, 1968. R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics, University of California Press, 1964, reprinted 1966. Optical System Design H. Gross, ed., Handbook of Optical Systems, Wiley, 2005. D. Malacara and Z. Malacara, Handbook of Optical Design, Marcel Dekker, 1994, 2nd ed. 2004. R. E. Fischer and B. Tadic-Galeb, Optical System Design, McGraw-Hill, 2000. W. J. Smith, Modern Optical Engineering: The Design of Optical Systems, McGraw-Hill, 1966, 3rd ed.2000. A. Nussbaum, Optical System Design, Prentice Hall, 1998. D. C. O'Shea, Elements of Modern Optical Design, Wiley, 1985. R. Kingslake, Optical System Design, Academic Press, 1983. L. Levi, Applied Optics: A Guide to Optical System Design, Wiley, Volume 1, 1968; Volume 2, 1980. Matrix Optics A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics, Wiley, 1975; Dover, paperback ed. 1994. J. W. Blaker, Geometric Optics: The Matrix Theory, Marcel Dekker, 1971. W. Brouwer, Matrix Methods in Optical Instrument Design, Benjamin, 1964. Popular and Historical R. J. Weiss, A Brief History of Light and Those that Lit the Way, World Scientific, 1996. A. R. Hall, All was Light: An Introduction to Newton's Opticks, Clarendon Press/Oxford University Press, 1993. R. Kingslake, A History of the Photographic Lens, Academic Press, 1989. M. I. Sobel, Light, University of Chicago Press, 1987. A. I. Sabra, Theories of Light from Descartes to Newton, Cambridge University Press, 1981. I. Newton, Opticks or a Treatise of the Reflections, Refractions, Inflections & Colours of Light, 4th ed. 1704; Dover, reissued [979. A. C. S. van Heel and C. H. F. Velze, What is Light?, McGraw-Hill, 1968, reprinted 1978. V. Ronchi, The Nature of Light: An Historical Survey, Harvard University Press, 1970. S. Tolansky, Revolution in Optics, Penguin, 1968. S. Tolansky, Curiosities of Light Rays and Light Waves, Elsevier, 1965. "- W. H. Bragg, Universe of Light, Dover, paperback ed. 1959. E. Riichardt, Light, Visible and Invisible, University of Michigan Press, 1958. PROBLEMS 1.1-2 Fermat's Principle with Maximum Time. Consider the elliptical mirror shown in Fig. PI. 1- 2(a), whos e foci are denoted A and B. Geomet rical p roper ties of the ellipse dictate that the pathlength AP B is identical to the pathlengths AP' B and AP" B for adjacent points on the ellipse. (a) Now consider another mirror with a radius of curvature smaller than that of the ell iptical mirror, but tangent to it at P, as displayed in Fig. P 1.1- 2( b). Show that the path AP B followed by the light ray in traveling b etwee n poi nts A an d B is a path of maximum time, i.e., is greater than the adjacent paths AQ' Band AQ" B. (b) Finally, consider a mirror that crosses the ellipse, b ut is ta n gent t o it at P, as i llustrated in Fig. P1.1-2(c). Show that the possible ray paths AQ' B, APB, and AQ" B exhibit a point of inflection.
36 CHAPTER 1 RAY OPTICS ----------------. ...- ...... -'" .'" ............- ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ! A . ,.. " .... , ........ " .... " ........ , ........ " .... " ........ " .... .... " .... , " ...." , ...., .... , , ,,-.... , ,.... , , " , , ......-.----. ----.---.-.. - .__ _.._ . ..a. __. -"". w P -. .- -... -- .. ...... -- ..- ... ... .. -,. .. .. -. ..... -.. ..... -.. ... ...... ... .. .. "" ..... """"... .- -.. .. ..... ...... ,. .. .. .. .. ...- '1IIiI "411 "'.. ... ...... ... .... ..... ... .. 1IiII.... .. _ .. ".. ... ... .., . .. .. - ... # "I- ... . .. , .. .. .. . .. .. . .. . 1IIi . -. . -. ... #I 'II . .. .. . , . .. .. . .. . . .. . .. . . B \ fA B \ lAB . . . . . .----------------------------.-- ..---...... ---- ------------------------.---... -...-........ ---- -------------..-.--------..---- ---,..... .. . I. . I . " ,.... I .. . .. '\ .... .... .. . . . . . .. . . " ,.... .. .. . . . "........ .. .. " . . , .... .... . . -. . .. . .. I .. " ".... . .. .. .. " '\ .... .... .. . ... . . . .. ... . .. '\ .... . .. .. .. '\ .... .. .. .." ... .... .. ... ..... '\ .... .. .. -.. . ... .' -.."" , .- ...... ,-...." .. , -- .., .411 .... " _ ..... """, .... ,,".. .. I .......... -.",,'" ...... Q ._ eo..,"._ ..... ..... ..,... "-.. J."'''' -.L" J.-'" -. .... ---, .-., P QH I P Q" P' , ,," , , , ,," , , , ,,' , , , " , , , , , , , , , , ......-. I . . . I . " .. " " .. ... p pit 1.2-7 (a) (b) (c) Figure P1.1-2 (a) Reflection from an elliptical mirror. (b) Reflection from an inscribed tangential mirror with greater curvature. (c) Reflection from a tangential mirror with curvature changing from concave to convex. Transmission through Planar Plates. (a) Use Snell's law to show that a ray entering a planar plate of thickness d and refractive index nl (placed in air; n 1) emerges parallel to its initial direction. The ray need not be paraxial. Derive an expression for the lateral displacement of the ray as a function of the angle of incidence O. Explain your results in terms of Fermat's principle. (b ) If the plate instead comprises a stack of N parallel layers stacked against each other wi th thicknesses d 1 , d 2, . . . , d N and refractive indexes n 1 , n2, . . . , n N, show that the transmitted ray is parallel to the incident ray. If Om is the angle of the ray in the mth layer, show that n m sin (}m sin (), m 1, 2, . . . . Lens in Water. Determine the focal length f of a biconvex lens with radii 20 em and 30 em and refractive index n 1.5. What is the focal length when the lens is immersed in water (n )? Numerical Aperture of a Cladless Fiber. Determine the numerical aperture and the accep- tance angle of an optical fiber if the refractive index of the core is nl 1.46 and the cladding is stripped out (replaced with air n2 1). Fiber Coupling Spheres. Tiny glass balls are often used as lenses to couple light into and out of optical fibers. The fiber end is located at a distance f from the sphere. For a sphere of radius a 1 mm and refractive index n 1.8, determine f such that a ray parallel to the optical axis at a distance y O. 7 mm is focused onto the fiber, as illustrated in Fig. P 1.2-1 O. f 1.2-8 1.2-9 1.2-1 0 2a .. y \. \. /" /' - - - - - - - Lens Fiber Figure P1.2-1 0 Focusing light into an optical fiber with a spherical glass ball. 1.2-11 Extraction of Light from a High-Refractive-Index Medium. Assume that light is gener- ated isotropically in all directions inside a material of refractive index n 3. 7 cut in the shape of a parallelepiped and placed in air (n 1) (see Exercise 1.2-6). (a) If a reflective material acting as a perfect mirror is coated on all sides except the front side, determine the percentage of light that may be extracted from the front side. (b) If another transparent material of refractive index n 1.4 is placed on the front side, would that help extract some of the trapped light? 1.3-3 Axially Graded Plate. A plate of thickness d is oriented normal to the z axis. The refractive index n(z) is graded in the z direction. Show that a ray entering the plate from air at an incidence angle (}o in the y-z plane makes an angle (}(z) at position z in the medium given by n(z) sin O(z) sin (}o. Show that the ray emerges into air parallel to the original incident ray. Hint: You may use the results of Prob. 1.2-7. Show that the ray position y(z) inside the plate obeys the differential equation (dyjdz)2 (n 2 j sin 2 0 1)-1. 1.3-4 Ray Trajectories in GRIN Fibers. Consider a graded-index optical fiber with cylindrical symmetry about the z axis and refractive index n(p), p x 2 + y2. Let (p, 4J, z) be the position vector in a cylindrical coordinate system. Rewrite the paraxial ray equations, (1.3- 4), in a cylindrical system and derive differential equations for p and 4J as functions of z.
PROBLEMS 37 1.4-8 Ray-Transfer Matrix of a Lens System. Determine the ray-transfer matrix for an optical system made of a thin convex lens of focal length f and a thin concave lens of focal length f separated by a distance f. Discuss the imaging properties of this composite lens. 1.4-9 Ray-Transfer Matrix of a GRIN Plate. Determine the ray-transfer matrix of a SELFOC plate [i.e., a graded-index material with parabolic refractive index n(y) no (1 ex2y2)] of thickness d. 1.4-10 The GRIN Plate as a Periodic System. Consider the trajectories of paraxial rays inside a SELFOC plate normal to the z axis. This system may be regarded as a periodic system made of a sequence of identical contiguous plates, each of thickness d. Using the result of Probe 1.4-9, determine the stability condition of the ray trajectory. Is this condition dependent on the choice of d? 1.4-1 ] Recurrence Relation for a Planar-Mirror Resonator. Consider a planar-mirror optical resonator, with mirror separation d, as a periodic optical system. Determine the unit ray- transfer matrix for this system, demonstrating that b 1 and F 1. Show that there is then only a single root to the quadratic equation (1.4-27) so that the ray position must then take the form ex + m{3, where ex and {3 are constants. 1.4-12 4 x 4 Ray-Transfer Matrix for Skewed Rays. Matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (non- circularly symmetric) systems. A ray crossing the plane z 0 is generally characterized by four variables the coordinates (x, y) of its position in the plane, and the angles (Ox, ()y ) that its projections in the x-z and y-z planes make with the z axis. The emerging ray is also characterized by four variables linearly related to the initial four variables. The optical system may then be characterized completely, within the paraxial approximation, by a 4 x 4 matrix. x (a) Determine the 4 x 4 ray-transfer matrix of a distance d in free space. (b) Determine the 4 x 4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction. The cylindrical lens has focal length f for rays in the y-z plane, and no focusing power for rays in the x-z plane. z \ y
CHAPTER 2.1 2.2 POSTULATES OF WAVE OPTICS MONOCHROMATIC WAVES A. Complex Representation and the Helmholtz Equation B. Elementary Waves C. Paraxial Waves RELATION BETWEEN WAVE OPTICS AND RAY OPTICS SIMPLE OPTICAL COMPONENTS A. Reflection and Refraction B. Transmission Through Optical Components C. Graded-Index Optical Components INTERFERENCE A. Interference of Two Waves B. Multiple-Wave Interference POLYCHROMATIC AND PULSED LIGHT A. Temporal and Spectral Description B. Light Beating 40 41 *2.3 2.4 49 50 2.5 58 2.6 66 t " t .- . " , . ... " . . . . . '" .. , " . " Christiaan Huygens (1629-1695) advanced several new concepts concerning the propaga- tion of light waves. Thomas Young (1773-1829) championed the wave theory of light and discovered the princi- ple of optical interference. 38
Light propagates in the form of waves. In free space, light waves travel with a constant speed, Co 3.0 x 10 8 m s (30 cm/ns or 0.3 mm/ps or 0.3 /-lmlfs). As illustrated in Fig. 2.0-1, the range of optical wavelengths contains three regions: infrared (0.76 to 300 /-lm), visible (390 to 760 nm), and ultraviolet (10 to 390 nm). The corres onding range of optical frequencies stretches from 1 THz in the far-infrared to 3 x 10 6 Hz in the extreme ultraviolet. -58 \D 00 8 8 o-5s 0 t" C\ 0 COEf") . ('f"') ('f"') C""I N N 0 ....-.4 ....... CJ) C s:: - - --...... ... . - ::k:JI:-- -. - - - . s::'-"" (1)"-"" - . ... ... - OJ . , . , OJ OJ ;> > > > > ;> e= ...... ::> ::J ::> . z :E z Frequency I N N f X I X f....c c.. UJ ...... CL>,...... > . , ........ ex: . U'.:I ::> · ...-4 .> . I Wavelength (in vacuum) e e c - , ...... . , Q.) s:: CJ) ..c:: "tj OJ OJ OJ 0 OJ ::s . , b.O - , 0 - , . . s:: 0 . ...-4 (1) 0 > . , OJ",......., 0 ('.1 f'. f'. ('.1 V') 0 \D ('.1 C\ f'. 0\ V') C\ t' \0 V') V') ('f"') Figure 2.0-1 Optical frequencies and wavelengths. The infrared (IR) region of the spectrum comprises the near infrared (NIR), mid infrared (MIR), and far infrared (FIR) bands while the ultraviolet (UV) region comprises the near ultraviolet (NUV), mid ultraviolet (MUV), far ultraviolet (FUV), and extreme ultraviolet (EUV or XUV) bands. Radiation in the EUV band is also known as soft X-rays (SXR). The vacuum ultraviolet (VUV) consists of the and EUV bands. The infrared, visible, and ultraviolet regions are all termed "optical" since they make use of similar types of components (e.g., lenses and mirrors). The wave theory of light encompasses the ray theory (Fig. 2.0-2). Strictly speaking, ray optics is the limit of wave optics when the wavelength is infinitesimally short. However, the wavelength need not actually be zero for the ray-optics theory to be useful. As long as the light waves propagate through and around objects whose dimen- sions are much greater than the wavelength, the ray theory suffices for describing most optical phenomena. Because the wavelength of visible light is much shorter than the dimensions of the usual objects encountered in our daily lives, the manifestations of the wave nature of light are not apparent without careful observation. This Chapter In this chapter, light is described by a scalar function, called the wave function, that obeys a second-order differential equation known as the wave equation. A discussion of the physical significance of the wave function is deferred to Chapter 5, where we consider electromagnetic optics; we will see that the wave function represents any of the components of the electric or magnetic fields. The wave equation, and a relation between the optical power density and the wavefunction, constitute the postulates of the 39
40 CHAPTER 2 WAVE OPTICS Wave Optics Ray Optics Figure 2.0-2 Wave optics encompasses ray optics. Ray optics is the limit of wave optics when the wavelength is very short. scalar-wave model of light known as wave optics. The consequences of these simple postulates are manifold and far reaching. Wave optics constitutes a basis for describing a whole host of optical phenomena that fall outside the confines of ray optics, including interference and diffraction, as demonstrated in this and the following two chapters. Wave optics does have its limitations, however. It is not capable of providing a complete picture of the reflection and refraction of light at the boundaries between dielectric media, nor can it account for optical phenomena that require a vector formu- lation, such as polarization effects. Those problems will be addressed in Chapter 5, as will the conditions under which scalar wave optics provides a good approximation to electromagnetic optics. This chapter begins with the postulates of wave optics (Sec. 2.1). In Secs. 2.2 2.5 we consider monochromatic waves; polychromatic light is discussed in Sec. 2.6. Elementary waves, such as the plane wave and the spherical wave, are introduced in Sec. 2.2. Section 2.3 establishes that ray optics can be derived from wave optics. The interaction of optical waves with simple optical components such as mirrors, prisms, lenses, and gratings is examined in Sec. 2.4. Interference, an important manifestation of the wave nature of light, is the subject of Sees. 2.5 and 2.6. 2.1 POSTULATES OF WAVE OPTICS The Wave Equation Light propagates in the form of waves. In free space, light waves travel with speed Co. A homogeneous transparent medium such as glass is characterized by a single constant, its refractive index n ( > 1). In a medium of refractive index n, light waves travel with a reduced speed Co . (2.1-1 ) Speed of Light in a Medium C n An optical wave is described mathematically by a real function of position r x, y, z and time t, denoted u r, t and known as the wavefunction. It satisfies a partial differ- ential equation called the wave equation, yr 2 u 1 8 2 u c 2 8t 2 0, (2.1-2) Wave Equation where V'2 is the Laplacian operator, which is V'2 8 2 8x 2 + 8 2 8y2 + 8 2 8z2 in Cartesian coordinates. Any function that satisfies (2.1-2) represents a possible optical wave.
2.2 MONOCHROMATIC WAVES 41 Because the wave equation is linear, the principle of superposition applies: if U1 r, t and U2 r, t represent possible optical waves, then u r, t U1 r, t +U2 r, t also represents a possible optical wave. At the boundary between two different media, the wavefunction changes in a way that depends on their refractive indexes. However, the. laws that govern this change depend on the physical significance assigned to the wavefunction which, as will be seen in Chapter 5, is an electromagnetic-field component. The underlying physical origin of the refractive index derives from electromagnetic optics (Sec. 5.5B). The wave equation is also approximately applicable for media with refractive in- dexes that are position dependent, provided that the variation is slow within distances of the order of a wavelength. The medium is then said to be locally homogeneous. For such media, n in (2.1-1) and c in (2.1-2) are simply replaced by the appropriate position-dependent functions n rand c r , respectively. Intensity, Power, and Energy The optical intensity I r, t , defined as the optical power per unit area (units of ,vatts cm 2 ), is proportional to the average of the squared wavefunction: I r,t 2u 2 r,t . (2.1-3) Optical Intensity The operation · denotes averaging over a time interval much longer than the time of an optical cycle, but much shorter than any other time of interest (such as the duration of a pulse of light). The duration of an optical cycle is extremely short: 2 x 10- 15 S 2 fs for light of wavelength 600 nm, as an example. This concept is further elucidated in Sec. 2.6. Although the physical significance of the wavefunction u r, t has not been explic- itly specified, (2.1-3) represents its connection with a physically measurable quantity the optical intensity. There is some arbitrariness in the definition of the wavefunction and its relation to the intensity. For example, (2.1-3) could have been written without the factor 2 and the wavefunction scaled by a factor 2, in which case the intensity would remain the same. The choice of the factor 2 in (2.1- 3) will later prove convenient, however. The optical power P t (units of watts) flowing into an area A normal to the direc- tion of propagation of light is the integrated intensity Pt I r,t dA. A (2.1-4 ) The optical energy (units of joules) collected in a given time interval is the integral of the optical power over the time interval. 2.2 MONOCHROMATIC WAVES A monochromatic wave is represented by a wavefunction with harmonic time depen- dence, u r,t a r cas 21fvt + f{J r , (2.2-1)
42 CHAPTER 2 WAVE OPTICS as illustrated in Fig. 2.2-1 (a), where a(r) == amplitude cp(r) == phase v == frequency (cycles/s or Hz) w == 27rv == angular frequency (radians/s or S-l) T == l/v == 27r / W == period (s). Both the amplitude and phase are generally position dependent, but the wavefunction is a harmonic function of time with frequency v at all positions. Optical waves have frequencies that lie in the range 3 x 1011 to 3 X 10 16 Hz, as depicted in Fig. 2.0-1. u(t) Im{ U} Im{U} a Re{ U} Re{U} t ................. ................. w w Figure 2.2-1 Representations of a monochromatic wave at a fixed position r: (a) the wave-function u(t) is a harmonic function of time; (b) the complex amplitude U = aexp(jcp) is a fixed phasor; (c) the complex wavefunction U(t) = U exp(j27rvt) is a phasor rotating with angular velocity w = 27rV radians/so A. Complex Representation and the Helmholtz Equation Complex Wavefunction It is convenient to represent the real wavefunction u(r, t) in (2.2-1) in terms of a complex function U(r, t) == a(r) exp[jcp(r)] exp(j27rvt) , (2.2-2) so that u(r, t) == Re{U(r, t)} == [U(r, t) + U*(r, t)] , (2.2-3) where the symbol * signifies complex conjugation. The function U(r, t), known as the complex wavefunction, describes the wave completely; the wavefunction u(r, t) is simply its real part. Like the wavefunction u(r, t), the complex wavefunction U(r, t) must also satisfy the wave equation '\J2U _ fj2U = o. c 2 8t 2 (2.2-4) Wave Equation The two functions satisfy the same boundary conditions.
2.2 MONOCHROMATIC WAVES 43 Complex Amplitude Equation (2.2-2) may be written in the form U r,t U r exp j27rvt , (2.2-5) where the time-independent factor U r a r exp jep r is referred to as the com- plex amplitude of the wave. The wavefunction u r, t is therefore related to the com- plex amplitude by u r, t Re U r exp j27rvt j27rvt . (2.2-6) At a given position r, the complex amplitude Uris a complex variable [depicted in Fig. 2.2-I(b)] whose magnitude U r a r is the amplitude of the wave and whose argument arg U r ep r is the phase. The complex wavefunction U r, t , shown in Fig. 2.2-1 (c), is represented graphically by a phasor that rotates with angular velocity w 27rv radians/so Its initial value at t 0 is the complex amplitude U r . The Helmholtz Equation Substituting U r, t U r exp j27rvt from (2.2-5) into the wave equation (2.2-4) leads to a differential equation for the complex amplitude U r : \J2U + k 2 U 0 , (2.2-7) Helmholtz Equation which is known as the Helmholtz equation, where k 27rV w c c (2.2-8) Wavenumber is referred to as the wavenumber. Different solutions are obtained from different boundary conditions. Optical Intensity The optical intensity is determined by inserting (2.2-1) into (2.1- 3): 2u 2 r, t 2a 2 r cos 2 27rvt + ep r U r 2 1 + cos 2 27rvt + ep r . (2.2-9) Averaging (2.2-9) over a time longer than an optical period, 1 v, causes the second term of (2.2-9) to vanish, whereupon Ir U r 2. (2.2-1 0) Optical Intensity The optical intensity of a monochromatic wave is the absolute square of its complex amplitude. The intensity of a monochromatic wave does not vary with time.
44 CHAPTER 2 WAVE OPTICS Wavefronts The wavefronts are the surfaces of equal phase, <p r constant. The constants are often taken to be multiples of 21r so that <p r 21rq, where q is an integer. The wavefront normal at position r is parallel to the gradient vector \7 <p r (a vector that has components 8<p 8x, 8<p 8y, and 8<p 8z in a Cartesian coordinate system). It represents the direction at which the rate of change of the phase is maximum. :: -. .. -: :: Summary . A monochromatic wave of frequency v is described by a complex wavefunction U r, t U r exp j27rvt , which satisfies the wave equation. , . The complex amplitude U r satisfies the Helmholtz equation; its magnitude U r and argument arg U r are the amplitude and phase of the wave, respectively. The optical intensity is I r . U r 2.. The wavefronts are the surfaces of constant phase, <p r .. arg . U r 27rq (q integer). . . The wavefunction u r, t is the real part of the complex wavefunction, u r, t Re U r, t . The wavefunction also satisfies the wave equation. . ; -, :i , , , , , B. Elementary Waves The simplest solutions of the Helmholtz equation in a homogeneous medium are the plane wave and the spherical wave. The Plane Wave The plane wave has complex amplitude Ur Aexp jk.r Aexp j kxx + ky y + k z Z , (2.2-11) where A is a complex constant called the complex envelope and k k x , ky, k z is called the wavevector. Subs tit uting (2.2-11) into the Helmholtz equation (2.2-7) wavenumber k. Since the phase of the wave is arg U r arg A k · r, the surfaces of constant phase (wavefronts) obey k. r kxx + kyY + kzz 21rq + arg A with q integer. This is the equation describing parallel planes perpendicular to the wavevector k (hence the name "plane wave"). Consecutive planes are separated by a distance A 21r k, so that A c , (2.2-12) Wavelength v where A is called the wavelength. The plane wave has a constant intensity I r A 2 everywhere in space so that it carries infinite power. This wave is clearly an idealization since it exists everywhere and at all times. If the z axis is taken along the direction of the wavevector k, then U r A exp j kz and the corresponding wavefunction obtained from (2.2-6) is . u r, t A cos 21rvt kz + arg A A cos 21rV t z c + arg A . (2.2-13)
2.2 MONOCHROMATIC WAVES 45 The wavefunction is therefore periodic in time with period 1 v, and periodic in space with period 21r k, which is equal to the wavelength A (see Fig. 2.2-2). Since the phase of the complex wavefunction, arg U r, t 21rV t z C + arg A , varies with time and position as a function of the variable t z C (see Fig. 2.2-2), C is called the phase velocity of the wave. x A I ..... u(x, z, tl) u(x, Z, t) 1/ v l .....c z x u(x, Z, t2) t z Figure 2.2-2 A plane wave traveling in the z direction is a periodic function of z with spatial period A and a periodic function of t with temporal period 1/v. In a medium of refractive index n, the wave has phase velocity C Co nand wavelength A C v Co nv, so that A AD n where AD CO v is the wavelength in free space. Thus, for a given frequency v, the wavelength in the medium is reduced relative to that in free space by the factor n. As a consequence, the wavenumber k 21r A is increased relative to that in free space ko 21r AD by the factor n. As a monochromatic wave propagates through media of different refractive in- dexes itsfrequency remains the same, but its velocity, wavelength, and wavenum- ber are altered: , A AD Co , k nko . (2.2-14) c n n The wavelengths displayed in Fig. 2.0-1 are in free space n 1. The Spherical Wave Another simple solution of the Helmholtz equation (in spherical coordinates) is the spherical wave Ur Ao exp jkr, (2.2-15) r where r is the distance from the origin, k 21rV C W c is the wavenumber, and Ao is a constant. The intensity I r Ao 2 r 2 is inversely proportional to the square of the distance. Taking arg Ao 0 for simplicity, the wavefronts are the surfaces kr 21rq or r qA, where q is an integer. These are concentric spheres separated by a radial distance A 21r k that advance radially at the phase velocity c (Fig. 2.2-3). A spherical wave originating at the position ro has a complex amplitude U r Ao r ro exp jk r ro . Its wavefronts are spheres centered about rOe A wave with complex amplitude U r Ao r exp + j kr is a spherical wave traveling inwardly (toward the origin) instead of outwardly (away from the origin).
46 CHAPTER 2 WAVE OPTICS x z Figure 2.2-3 Cross section of the wave- fronts of a spherical wave. Fresnel Approximation of the Spherical Wave: The Paraboloidal Wave Let us examine a spherical wave (originating at r 0) at points r x, y, z that are sufficiently close to the z axis but far from the origin, so that x 2 + y2 « z. The paraxial approximation of ray optics (Sec. 1.2) would be applicable were these points the endpoints of rays beginning at the origin. Denoting ()2 x 2 + y2 z2« 1, we use an approximation based on the Taylor-series expansion: ()4 + . . . 8 r x 2 + y2 + z2 Z 1 + ()2 ()2 Z 1 + 2 x2 + y2 z+ 2z z ()2 1+ 2 . (2.2-16) This expression, r z + x 2 + y2 2z, is now substituted into the phase of U r in (2.2-15). A less accurate expression, r z, can be substituted for the magnitude since it is less sensitive to errors than is the phase. The result is known as the Fresnel approximation of a spherical wave: z jkz exp . x2 + y2 Ur Ao exp . (2.2-17) Fresnel Approximation of a Spherical Wave This approximation plays an important role in simplifying the theory of optical-wave transmission through apertures (diffraction), as discussed in Chapter 4. The complex amplitude in (2.2-17) may be viewed as representing a plane wave Ao exp jkz modulated by the factor 1 z exp jk x 2 + y2 2z, which involves the phase k x 2 + y2 2z. This phase factor serves to bend the planar wavefronts of the plane wave into paraboloidal surfaces (Fig. 2.2-4), since the equation of a paraboloid of revolution is x 2 + y2 Z constant. In this region the spherical wave is well approximated by a paraboloidal wave. When z becomes very large, the paraboloidal phase factor in (2.2-17) approaches 0 so that the overall phase of the wave becomes kz. Since the magnitude Ao z varies slowly with z, the spherical wave eventually approaches the plane wave exp j kz , as illustrated in Fig. 2.2-4. The condition of validity for the Fresnel approximation is not simply that ()2 « 1, however. Although the third term of the series expansion, ()4 8, may be very small in comparison with the second and first terms, when multiplied by kz it can become com arable to 1r. The a Qroximation used in the foregoing is therefore valid when kz() 8« 1r, or x 2 + y 2« 4z3 A. For points x, y lying within a circle of radius a
2.2 MONOCHROMATIC WAVES 47 x :::::: Parapoloidal .A... r z Figure 2.2-4 A spherical wave may be approximated at points near the z axis and sufficiently far from the origin by a paraboloidal wave. For points very far from the origin, the spherical wave approaches a plane wave. \.... y ::::: Planar Spherical centered about the z axis, the validity condition is thus a 4 « 4z3 A or N 0 2 F m « 1 4 ' (2.2-18) where Om a z is the maximum angle and N F a 2 AZ (2.2-19) Fresnel Number is known as the Fresnel number. EXERCISE 2.2-1 Validity of the Fresnel Approximation. Determine the radius of a circle within which a spher- ical wave of wavelength A G33 nm, originating at a distance 1 m away, may be approximated by a paraboloidal wave. Determine the maximum angle (}m and the Fresnel number N F . c. Paraxial Waves A wave is said to be paraxial if its wavefront normals are paraxial rays. One way of constructing a paraxial wave is to start with a plane wave A exp j kz , regard it as a "carrier" wave, and modify or ''"modulate'' its complex envelope A, making it a slowly varying function of position, A r , so that the complex amplitude of the modulated wave becomes Ur A r exp j kz . (2.2-20) The variation of the envelope A r and its derivative with position z must be slow within the distance of a wavelength A 27r k so that the wave approximately main- tains its underlying plane-wave nature. The wavefunction of a paraxial wave, u r, tAr cos 27rvt kz + arg A r , is sketched in Fig. 2.2-5(a) as a function of z at t 0 and x y O. It is a sinusoidal function of z with amplitude A 0,0, z and phase arg A 0,0, z , both of which vary slowly with z. Since the phase arg A x, y, z changes little within the distance of a wavelength, the planar wavefronts kz 27rq of the carrier plane wave bend only slightly, so that their normals form paraxial rays [Fig. 2.2-5(b)].
48 CHAPTER 2 WAVE OPTICS x Wavefronts / c Rays u(O,O,z) AI " IAI ...- - - - .... ".... ........ -- z z ...- ...- .,," '" ....'" .... - - -- ..",. (a) (b) Figure 2.2-5 (a) Wavefunction of a paraxial wave at point on the z axis as a function of the axial distance z. (b) The wave fronts and wavefront normals of a paraxial wave in the x-z plane. The Paraxial Helmholtz Equation For the paraxial wave (2.2-20) to satisfy the Helmholtz equation (2.2-7), the complex envelope A r must satisfy another partial differential equation that is obtained by substituting (2.2-20) into (2.2-7). The assumption that A r varies slowly with respect to z signifies that within a distance /}.z A, the change /}.A is much smaller than A itself, i.e., A « A. This inequality of complex variables applies to the magnitudes of the real and imaginary parts separately. Since A 8A 8z z 8A 8z A, it follows that 8A 8z A A Ak 27r, so that 8A (2.2-21) The derivative 8A 8z itself must also vary slowly within the distance A, so that 8 2 A 8z2 « k 8A 8z, which provides 8 2 A (2.2- 22) Substituting (2.2-20) into (2.2-7), and neglecting 8 2 A 8z2 in comparison with k 8A 8z or k 2 A, leads to a partial differential equation for the complex envelope A r : V T 2 A J 8z 0, (2.2- 23) Paraxial Helmholtz Equation where V'} 8 2 8x 2 + 8 2 8y2 is the transverse Laplacian operator. Equation (2.2-23) is the slowly varying envelope approximation of the Helmholtz equation. We shall simply call it the paraxial Helmholtz equation. It bears some similarity to the Schrodinger equation of quantum physics [see (13.1-1)]. The simplest solution of the paraxial Helmholtz equation is the paraboloidal wave (Exercise 2.2-2), which is the paraxial approximation of a spherical wave. One of the most interesting and useful solutions, however, is the Gaussian beam, to which Chapter 3 is devoted. EXERCISE 2.2-2 The Paraboloidal Wave and the Gaussian Beam. Verify that a paraboloidal wave with the complex envelope A(r) (Ao/z)exp[ jk(x 2 + y2)/2z] [see (2.2-17)] satisfies the paraxial Helmholtz equation (2.2-23). Show that the wave with complex envelope A(r) [Al/q(Z)] exp[ jk(x 2 +
2.3 RELATION BETWEEN WAVE OPTICS AND RAY OPTICS 49 y2) /2q( Z ) ], where q( z ) z + j Zo and Zo is a constant, also satisfies the paraxial Helmholtz equation. This wave, caned the Gaussian beam, is the subject of Chapter 3. Sketch the intensity of the Gaussian beam in the plane z O. . *2.3 RELATION BETWEEN WAVE OPTICS AND RAY OPTICS We proceed to show that ray optics emerges as the limit of wave optics when the wavelength Ao O. Consider a monochromatic wave of free-space wavelength Ao in a medium with refractive index n r that varies sufficiently slowly with position so that the medium may be regarded as locally homogeneous. We write the complex amplitude in (2.2-5) in the form Ur a r exp j ko 8 r , (2.3-1 ) where a r is its magnitude, k o 8 r is its phase, and ko 27r Ao is the free-space wavenumber. We assume that a r varies sufficiently slowly with r that it may be regarded as constant within the distance of a wavelength Ao. The wavefronts are the surfaces 8 r constant and the wavefront normals point in the direction of the gradient vector \7 8. In the neighborhood of a given position ro, the wave can be locally regarded as a plane wave with amplitude a ro and wavevector k with magnitude k n ro ko and direction parallel to the gradient vector \7 8 at rOe A different neighborhood exhibits a local plane wave of different amplitude and different wavevectof. In ray optics it was shown that the optical rays are normal to the equilevel sur- faces of a function 8 r called the eikonal (see Sec. 1.3C). We therefore associate the local wavevectors (wavefront normals) in wave optics with the ray of ray optics and recognize that the function 8 r , which is proportional to the phase of the wave, is nothing but the eikonal of ray optics (Fig. 2.3-1). This association has a formal mathematical basis, as will be demonstrated shortly. With this analogy, ray optics can serve to determine the approximate effects of optical components on the wavefront normals, as illustrated in Fig. 2.3-]. (a) ..... ({C:- .. (b) Figure 2.3-1 (a) The rays of ray optics are orthogonal to the wavefronts of wave optics (see also Fig. 1.3-1 0). (b) The effect of a lens on rays and wavefronts. . The Eikonal Equation Substituting (2.3-1) into the Helmholtz equation (2.2-7) provides k n 2 \78 2 a + V 2 a jko 2 \78. Va + a V 2 8 0, (2.3-2) where a a rand 8 8 r . The real and imaginary parts of the left-hand side of (2.3-2) must both vanish. Equating the real part to zero and using ko 27r Ao, we
50 CHAPTER 2 WAVE OPTICS obtain vs 2 n 2 + Ao 2 V 2 Q 27r Q . (2.3-3) The assumption that Q varies slowly over the distance Ao means that AV2Q Q « 1, so that the second term of the right-hand side may be neglected in the limit Ao 0, whereupon vs 2 n 2 . (2.3-4) Eikonal Equation This is the eikonal equation (1.3-20), which may be regarded as the main postulate of ray optics (Fermat's principle can be derived from the eikonal equation and vice versa). Thus, the scalar function S r , which is proportional to the phase in wave optics, is the eikonal of ray optics. This is also consistent with the observation that in ray optics S rB S rA equals the optical path length between the points rA and rB. The eikonal equation is the limit of the Helmholtz equation when Ao ) O. Given n r we may use the eikonal equation to determine Sr. By equating the imaginary part of (2.3-2) to zero, we obtain a relation between Q and S, thereby permitting us to determine the wavefunction. 2.4 SIMPLE OPTICAL COMPONENTS In this section we examine the effects of optical components, such as mirrors, trans- parent plates, prisms, and lenses, on optical waves. A. Reflection and Refraction Reflection from a Planar Mirror A plane wave of wavevector k 1 is incident onto a planar mirror located in free space in the z 0 plane. A reflected plane wave of wavevector k 2 is created. The angles of incidence and reflection are 0 1 and O 2 , as illustrated in Fig. 2.4- L. The sum of the two waves satisfies the Helmholtz equation if the wavenumber is the same, i.e., if k 1 k 2 ko. Certain boundary conditions must be satisfied at the surface of the mirror. Since these conditions are the same at all points x, y , it is necessary that the wavefronts of the two waves match, i.e., k 1 · r k 2 · r for all r x,y,O . (2.4-1 ) Substituting r x,y,O , k 1 kosinOl,O,kocoSOl , and k 2 kosin02,O, ko cos O 2 into (2.4-1), we obtain kox sin 0 1 kox sin (}2, from which (}l (}2, so that the angles of incidence and reflection must be equal. Thus, the law of reflection of optical rays is applicable to the wavevectors of plane waves. Reflection and Refraction at a Planar Dielectric Boundary We now consider a plane wave of wavevector k 1 incident on a planar boundary between two homogeneous media of refractive indexes nl and n2. The boundary lies in the
2.4 SIMPLE OPTICAL COMPONENTS 51 x k2 (}2 ..................... Z .................... , Figure 2.4-1 Reflection of a plane wave from a planar mirror. Phase matching at the surface of the mirror requires that the angles of incidence and reflection be equal. 6 1 k 1 z 0 plane (Fig. 2.4-2). Refracted and reflected plane waves of wavevectors k 2 and k3 emerge. The combination of the three waves satisfies the Helmholtz equation everywhere if each of the waves has the appropriate wavenumber in the medium in which it propagates (k 1 k3 nlko and k 2 n2 k O). x nl n2 k2 () .................. ... Figure 2.4-2 Refraction of a plane wave at a dielectric boundary. Match ing th e wavefronts at the boundary: the distance PI P 2 for the incident wave, Al/ sinO! Ao/n! sinOl, equals that for the refracted wave, A2/ sin O 2 Ao/n2 sin O 2 , from which Snell's law follows. z ................... (}l k J Since the boundary conditions are invariant to x and y, it is necessary that the wavefronts of the three waves match, i.e., k 1 · r k 2 · r k3 · r for all r x,y,O . (2.4-2) Since k 1 nlko sin (}l, 0, nlko cos (}l , k3 nlko sin {}3, 0, nlko cos (}3 , and k 2 n2ko sin {}2, 0, n2 k o cos 8 2 , where 8 1 , 8 2 , and 8 3 are the angles of incidence, refraction, and reflection, respectively, it follows from (2.4-2) that (}l (}3 and nl sin (}l n2 sin 8 2 . These are the laws of reflection and refraction (Snell's law) of ray optics, now applicable to the wavevectors. It is not possible to determine the amplitudes of the reflected and refracted waves using scalar wave optics since the boundary conditions are not completely specified in this theory. This will be achieved in Sec. 6.2 using electromagnetic optics (Chapters 5 and 6). B. Transmission Through Optical Components We now proceed to examine the transmission of optical waves through transparent optical components such as plates, prisms, and lenses. The effect of reflection at the surfaces of these components will be ignored, since it cannot be properly accounted for using the scalar wave theory of light. Nor can the effect of absorption in the material, which is relegated to Sec. 5.5. The principal emphasis here is on the phase shift introduced by these components and on the associated wavefront bending.
52 CHAPTER 2 WAVE OPTICS Transmission Through a Transparent Plate Consider first the transmission of a plane wave through a transparent plate of refractive index n and thickness d surrounded by free space. The surfaces of the plate are the planes Z 0 and z d and the incident wave travels in the z direction (Fig. 2.4-3). Let U x, y, z be the complex amplitude of the wave. Since external and internal reflec- tions are ignored, U x, y, z is assumed to be continuous at the boundaries. The ratio t x, Y U x, y, dUx, y, 0 therefore represents the complex amplitude trans- mittance of the plate; it permits us to determine U x, y, d for arbitrary U x, y, 0 at the input. The effect of reflection is considered in Sec. 6.2 and the effect of multiple internal reflections within the plate is examined in Sec. 10.]. Ao A Figure 2.4-3 Transmission of a plane wave through a transparent plate. Once inside the plate, the wave continues to propagate as a plane wave with wavenumber nko, so that U x, y, z is proportional to exp jnkoz. Thus, the ratio U x, y, dUx, y, 0 exp jnkod, so that t : x, y: exp : j nko d: . (2.4-3) Transmittance Transparent Plate The plate is seen to introduce a phase shift nkod 27r d A . If the incident plane wave makes an angle () with respect to the z axis and has wavevector k (Fig. 2.4-4), the refracted and transmitted waves are also plane waves with wavevectors k 1 and k and angles (}1 and (), respectively, where (}1 and () are related by Snell's law: sin () n sin 0 1 . The complex amplitude U x, y, z inside the plate is now proportional to exp jk 1 . r exp jnk o z COS 0 1 + x sin 0 1 , so that the complex amplitude transmittance of the plate U x, y, dUx, y, 0 is t x,y exp jnkod cos 0 1 . (2.4-4) If the angle of incidence () is small (i.e., if the incident w ave is paraxial), then exp jnkod exp jko(}2d 2n . If the plate is sufficiently thin, and the angle () is suffi- ciently small such that ko(}2d 2n« 27r [or d Ao (}2 2n« 1], then the transmittance of the plate may be approximated by (2.4-3). Under these conditions the transmittance of the plate is approximately independent of the angle (). Thin Transparent Plate of Varying Thickness We now determine the amplitude transmittance of a thin transparent plate whose thick- ness d x, y varies smoothly as a function of x and y, assuming that the incident wave is an arbitrary paraxial wave. The plate lies between the planes z 0 and z do, which are regarded as the boundaries encasing the optical component (Fig. 2.4-5).
2.4 SIMPLE OPTICAL COMPONENTS 53 n ..................... ................... () k Figure 2.4-4 Transmission of an oblique plane wave through a thin transparent plate. x ,- - /1 I / I / I L I d(x,y) I I / I / / I L I I I I I I I I I I I I.,l__- I / I Z .I I / I I / I / I I / I / I / I ' :...y/ Y Figure 2.4-5 A transparent plate of vary- ing thickness. In the vicinity of the position x, y, 0 the incident paraxial wave may be regarded locally as a plane wave traveling along a direction that makes a small angle with the z axis. It crosses a thin plate of material of thickness d x, y surrounded by thin layers of air of tota] thickness do d x, y . In accordance with the approximate relation (2.4-3), the local transmittance is the product of the transmittances of a thin layer of air of thickness do d x, y and a thin layer of material of thickness d x, y , so that t x, y exp jnkod x, y exp jko do d x, y , from which t x, Y ho exp . J n 1 ko d x, y , (2.4-5) Transmittance Variable-Thickness Plate where ho exp jkod o is a constant phase factor. This relation is valid in the paraxial approximation (where all angles () are small) and when the thickness do is sufficiently small so that do Ao ()2 2n« 1. EXERCISE 2.4-1 Transmission Through a Prism. Use (2.4-5) to show that the complex amplitude transmittance of a thin inverted prism with small apex angle a « 1 and thickness do (Fig. 2.4-6) is t( x, y ) ko exp[ j(n l)k o ax], where ho exp( jkod o ). What is the effect of the prism on an incident plane wave traveling in the z direction? Compare your results with those obtained via the ray-optics model [see (1.2-7)].
54 CHAPTER 2 WAVE OPTICS x do Q o z Figure 2.4-6 Transmission of a plane wave through a thin prism. Thin Lens The general expression (2.4-5) for the complex amplitude transmittance of a thin trans- parent plate of variable thickness is now applied to the planoconvex thin lens shown in Fig. 2.4- 7. Since the lens is t he cap of a sph ere of radius R, the thickness at the point x, y is d x, y do PQ do R QC, or d x,y do R R2 x2 + y2 . (2.4-6) This expression may be simplified by considering onl points for which x and yare sufficiently small in comparison with R so that x 2 + y «R 2 . In that case R2 x2 + y2 R 1 x2 + y2 1 x2 + y2 2R2 , (2.4-7) where we have used the same Taylor-series expansion that led to the Fresnel approx- imation of a spherical wave in (2.2-17). Using this approximation in (2.4-6) then provides d x, y do x2 + y2 2R . (2.4-8) Finally, substitution into (2.4-5) yields I , h t\x, YJ 0 exp ... x2 + y2 - , (2.4-9) Transmittance Thin Lens - - where f R n 1 (2.4-1 0) is the focal length of the lens (see Sec. 1.2C) and ho exp jnkod o is another constant phase factor that is usually of no significance. Since the lens imparts a phase proportional to x 2 + y2 to the incident plane wave, it transforms the planar wavefronts into wavefronts of a paraboloidal wave centered at a distance f from the lens, as demonstrated in Exercise 2.4-3.
2.4 SIMPLE OPTICAL COMPONENTS 55 x .. .. .- .. ,. . , d(x,y) P Q c z .. .. .. .. do .. Figure 2.4-7 A planoconvex lens. EXERCISE 2.4-2 Double-Convex Lens. Show that the complex amplitude transmittance of the double-convex lens (also called a spherical lens) shown in Fig. 2.4-8 is given by (2.4-9) with (n 1) 1 Rl 1 R 2 . (2.4-11) 1 f You may prove this either by using the general formula (2.4-5) or by regarding the double-convex lens as a cascade of two planoconvex lenses. Recall that, by convention, the radius of a convex/concave surface is positive/negative, so that R] is positive and R 2 is negative for the lens displayed in Fig. 2.4- 8. The parameter f is recognized as the focal length of the lens [see (1.2-12)]. R 2 _-'''--- .-- ",- -- -- --- --- --- --- --- --.... --- --- --.... _....- -- ----------.... R ] _....- Figure 2.4-8 A double-convex lens. EXERCISE 2.4-3 Focusing of a Plane Wave by a Thin Lens. Show that when a plane wave is transmitted through a thin lens of focal length f in a direction parallel to the axis of the lens, it is converted into a paraboloidal wave (the Fresnel approximation of a spherical wave) centered about a point at a distance f from the lens, as illustrated in Fig. 2.4-9. What is the effect of the lens on a plane wave incident at a small angle B? ,. z . . ...... , f Figure 2.4-9 A thin lens transforms a plane wave into a paraboloidal wave. EXERCISE 2.4-4 Imaging Property of a Lens. Show that a paraboloidal wave centered at the point PI (Fig. 2.4- 1 0) is converted by a lens of focal length f into a paraboloidal wave centered about P 2 , where 1/ ZI + 1/ Z2 1/ f (this is known as the imaging equation). · z ..... Zl . . ...... . Z2 Figure 2.4-10 A lens transforms a paraboloidal wave into another paraboloidal wave. The two waves are centered at distances that satisfy the imaging equation.
56 CHAPTER 2 WAVE OPTICS Diffraction Gratings A diffraction grating is an optical component that serves to periodically modulate the phase or amplitude of an incident wave. It can be made of a transparent plate with periodically varying thickness or periodically graded refractive index (see Sec. 2.4C). Repetitive arrays of diffracting elements such as apertures, obstacles, or absorbing elements (see Sec. 4.3) can also be used for this purpose. A reflection diffraction grating is often fabricated from a periodically ruled thin film of aluminum that has been evaporated onto a glass substrate. Consider a diffraction grating made of a thin transparent plate placed in the z 0 plane whose thickness varies periodically in the x direction with period A (Fig. 2.4- 11). As will be demonstrated in Exercise 2.4-5, this plate converts an incident plane wave of wavelength A A, traveling at a small angle (}i with respect to the z axis, into several plane waves at small angles with respect to the z axis: Oq A (2.4-12) Grating Equation where q 0,1:1, ::f:2, . . . , is called the diffraction order. Successive diffracted waves are separated by an angle () A A, as shown schematically in Fig. 2.4-11. 1...i\"LLL"""'" . . . . t . . t 1 1 1 1 1 I I ....t4t.. ,.t. '........, · . t , I I I I I Figure 2.4-11 A thin transparent plate with periodically varying thickness serves as a diffraction grating. It splits an incident plane wave into multiple plane waves travel- ing in different directions. EXERCISE 2.4-5 Transmission Through a Diffraction Grating. (a) The thickness of a thin transparent plate varies sinusoidally in the x direction, d(x, y) do [1 + cos(27rx/ A)], as illustrated in Fig. 2.4-11. Show that the complex amplitude transmittance is t(x, y) ho exp[ j (n l)kod o cos(27rx/ A)] where ho exp[ j (n + l)kod o ]. (b) Show that an incident plane wave traveling at a small angle Oi with respect to the z direction is transmitted in the form of a sum of plane waves traveling at angles Oq given by (2.4-12). Hint: Expand the periodic function t( x, y) in a Fourier series. Equation (2.4-12) is valid only in the paraxial approximation (when all angles are small). This approximation is applicable when the period A is much greater than the wavelength A. A more general analysis of thin diffraction gratings, without the use of the paraxial approximation, shows that the incident plane wave is converted into several plane waves at angles Oq satisfying t sin () q . A (2.4-13) t See, e.g., E. Hecht and A. Zajac, Optics, Addison-Wesley, 2nd ed. 1990.
2.4 SIMPLE OPTICAL COMPONENTS 57 Diffraction gratings are used as filters and spectrum analyzers. Since the angles Oq depend on the wavelength A (and therefore on the frequency v), an incident poly- chromatic wave is separated by the grating into its spectral components (Fig. 2.4-12). Diffraction gratings have found numerous applications in spectroscopy. R R G B R+G+B R+G+B . B G R Figure 2.4-12 A diffraction grating directs two waves of different wavelengths, '\1 and '\2, into two different directions, (}1 and (}2. It therefore serves as a spectrum analyzer or a spectrometer. .. C. Graded-Index Optical Components The effect of a prism, lens, or diffraction grating on an incident optical wave lies in the phase shift it imparts, which serves to bend the wavefront in some prescribed manner. This phase shift is controlled by the variation in the thickness of the material with the transverse distance from the optical axis (linearly, quadratically, or periodically, in the cases of a prism, lens, and diffraction grating, respectively). The same phase shift may instead be introduced by a transparent planar plate of fixed thickness but with varying refractive index. This is a result of the fact that the thickness and refractive index appear as a product in (2.4-3). The complex amplitude transmittance of a thin transparent planar plate of thickness do and graded refractive index n x, y is, from (2.4-3), t : x, y: exp: j n : x, y: ko do: . (2.4-14) Transmittance Graded-Index Thin Plate By selecting the appropriate variation of n x, y with x and y, the action of any constant-index thin optical component can be reproduced, as demonstrated in Exer- cise 2.4-6. EXERCISE 2.4-6 Graded-Index Lens. Show that a thin plate of unifonn thickness do (Fig. 2.4-13) and quadrati- cally graded refractive index n(x, y) no [1 a2(x2 + y2)], with ado « 1, acts as a lens of focal length f 1/n o d o a. 2 (see Exercise 1.3-1). . Figure 2.4-13 A graded-index plate acts as a lens.
58 CHAPTER 2 WAVE OPTICS 2.5 INTERFERENCE When two or more optical waves are simultaneously present in the same region of space and time, the total wavefunction is the sum of the individual wavefunctions. This basic principle of superposition follows from the linearity of the wave equation. For monochromatic waves of the same frequency, the superposition principle carries over to the complex amplitudes, which follows from the linearity of the Helmholtz equation. The superposition principle does not apply to the optical intensity since the intensity of the sum of two or more waves is not necessarily the sum of their intensities. The disparity is associated with interference. The phenomenon of interference cannot be explained on the basis of ray optics since it is dependent on the phase relationship between the superposed waves. In this section we examine the interference between two or more monochromatic waves of the same frequency. The interference of waves of different frequencies is discussed in Sec. 2.6. A. Interference of Two Waves When two monochromatic waves with complex amplitudes U 1 rand U 2 r are super- posed, the result is a monochromatic wave of the same frequency that has a complex amplitude Ur U 1 r + U 2 r . (2.5-1 ) In accordance with (2.2-10), the intensities of the constituent waves are II 1 2 U 2 2, while the intensity of the total wave is U 1 2 and U 1 + U 2 2 U 1 2 + U 2 2 + U;U 2 + U 1 U;. (2.5-2) I U 2 The explicit dependence on r has been omitted for convenience. Substituting U 1 II exp j'Pl and U 2 1 2 exp j'P2 (2.5- 3) into (2.5-2), where 'PI and 'P2 are the phases of the two waves, we obtain 1 II + 1 2 + 2 II 1 2 COS 'P , (2.5-4) Interference Equation with 'P 'P2 'PI · (2.5-5) This relation, called the interference equation, can also be understood in terms of the geometry of the phasor diagram displayed in Fig. 2.5-1(a), which demonstrates that the magnitude of the phasor U is sensitive not only to the magnitudes of the constituent phasors but also to the phase difference 'P. It is clear, therefore, that the intensity of the sum of the two waves is not the sum of their intensities [Fig. 2.5-1(b)]; an additional term, attributed to interference between the two waves, is present in (2.5-4). This term may be positive or negative, corresponding to constructive or destructive interference, respectively. If II 1 2 1 0 , for example, then (2.5-4) yields I 210 1 + COS'P 41 0 cos 2 'P 2 , so that for 'P 0, I 41 0 (i.e., the total intensity is four times the intensity of each of the
2.5 INTERFERENCE 59 1 (a) U2 C{J I] + 12 ------ - -- - - - - -- - -- ---. (b) C{JI UI C{J2 I I I I -47r -27r 0 27r 47r C{J Figure 2.5-1 (a) Phasor diagram for the superposition of two waves of intensities II and 1 2 and phase difference <P <P2 <Pl. (b) Dependence of the total intensity I on the phase difference <po superposed waves). For cp 7r, on the other hand, the superposed waves cancel one another and the total intensity 1 O. Complete cancellation of the intensity in a region of space is generally not possible unless the intensities of the constituent superposed waves are equal. When cp 7r 2 or 37r 2, the interference term vanishes and 1 21 0 ; for these special phase relationships the total intensity is the sum of the constituent intensities. The strong dependence of the intensity I on the phase difference cp permits us to measure phase differences by detecting light intensity. This principle is used in numerous optical systems. Interference is accompanied by a spatial redistribution of the optical intensity with- out a violation of power conservation. For example, the two waves may have uniform intensities II and 1 2 in a particular plane, but as a result of a position-dependent phase difference cp, the total intensity can be smaller than II + /2 at some positions and larger at others, with the total power (integral of the intensity) conserved. Interference is not observed under ordinary lighting conditions since the random fluctuations of the phases CPl and CP2 cause the phase difference cP to assume random values that are uniformly distributed between 0 and 27r, so that cos cp averages to 0 and the interference term washes out. Light with such randomness is said to be partially coherent and Chapter 11 is devoted to its study. We limit ourselves here to the study of coherent light. Interferometers Consider the superposition of two plane waves, each of intensity /0, propagating in the z direction, and assume that one wave is delayed by a distance d with respect to the other so that U 1 /0 exp jkz and U 2 10 exp jk z d. The intensity 1 of the sum of these two waves can be determined by substituting II 1 2 10 and cp kd 27rd A into the interference equation (2.5-4), d 1 210 1 + cos . (2.5-6) The dependence of / on the delay d is sketched in Fig. 2.5-2. When the delay is an integer multiple of A, complete constructive interference occurs and the total intensity 1 410. On the other hand, when d is an odd integer multiple of A 2, complete destructive interference occurs and 1 O. The average intensity is the sum of the two intensities, i.e., 210. An interferometer is an optical instrument that splits a wave into two waves us- ing a beamsplitter, delays them by unequal distances, redirects them using mirrors, recombines them using another (or the same) beamsplitter, and detects the intensity of their superposition. Three important examples are illustrated in Fig. 2.5-3: the Mach Zehnder interferometer, the Michelson interferometer, and the Sagnac interfer- ometer.
60 CHAPTER 2 WAVE OPTICS I 41 0 21 0 ,\ 2,\ 3,\ d Figure 2.5-2 Dependence of the intensity I of the superposition of two waves, each of intensity 1 0 , on the delay distance d. When the delay distance is a multiple of '\, the interference is constructive; when it is an odd multiple of '\/2, the interference is destructive. Uo -------.. -------.. -i\. 1 1 "'- U2 ' I I VI ' (a) Mach-Zehnder t t VI -------.. Uo U -------.. ....-- -------.. -------.. U2 ....-- U; ....-- (b) Michelson Ull 1 U2 Uo -------.. U; 1--- 1--- Figure 2.5-3 Interferometers: A wave U o is split into two waves U 1 and U 2 (they are shown as shaded light and dark for ease of vi- sualization but are actually congruent). After traveling through different paths, the waves are recombined into a superposition wave U = U 1 + U 2 whose intensity is recorded. The waves are split and recombined using beamsplitters. In the Sagnac interferometer the two waves travel through the same path, but in opposite directions. f 1 = 1--- U -------.. (c) Sagnac Ull 1 U2 Since the intensity I is sensitive to the phase c.p == 27rd / A == 27rnd / AO == 27rnvd / Co, where d is the difference between the distances traveled by the two waves, the interferometer can be used to measure small changes in the distance d, the refractive index n, or the wavelength Ao (or frequency v). For example, if d / Ao == 10 4 , a change of the refractive index of only !::::.n == 10- 4 corresponds to an easily observable phase change !::::.c.p == 27r. The phase c.p also changes by a full 27r if d changes by a wavelength A. An incremental change of the frequency v == c/ d has the same effect. Interferometers have numerous applications. These include the determination of distance in metrological applications such as strain measurement and surface profil- ing; refractive-index measurements; and spectrometry for the analysis polychromatic light (see Sec. 11.2B). In the Sagnac interferometer the optical paths are identical but opposite in direction, so that rotation of the interferometer results in a phase shift c.p proportional to the angular velocity of rotation. This system can therefore be used as a gyroscope. Because of its precision, optical interferometry is also being co-opted to detect the passage of gravitational waves. Finally, we demonstrate that energy conservation in an interferometer requires that the phases of the waves reflected and transmitted at a beamsplitter differ by 7r /2. Each of the interferometers considered in Fig. 2.5-3 has an output wave U == U 1 + U 2 that exits from one side of the beamsplitter and also another output wave U' == Uf + U that exits from the opposite side. Energy conservation dictates that the sum of the intensities of these two waves must equal the intensity of the incident wave, so that if one output wave has high intensity by virtue of constructive interference, the other
2.5 INTERFERENCE 61 must have low intensity by virtue of destructive interference. This complementarity can only be achieved if the phase differences <p and <p', associated with the components of output waves U and U', respectively, differ by 7r. Since the components of U and the components of U' experience the same pathlength differences, and the same numbers of reflections from mirrors, the 7f phase difference must be attributable to different phases introduced by the beamsplitter upon reflection and transmission. Examination of the three interferometers in Fig. 2.5-3 reveals that for one output wave, each of the components is transmitted through the beamsplitter once and reflected from it once, so that no phase difference is introduced. However, for the other output wave, one component is transmitted twice and the other is reflected twice, thereby introducing the phase difference of 7f. It follows that the phases of the reflected and transmitted waves at a beamsplitter differ by 7r 2. This important property of the beamsplitter is described in more detail in Sec. 7.1 (see Example 7.1-2). Interference of Two Oblique Plane Waves Consider now the interference of two plane waves of equal intensities: one propagating in the z direction, U 1 10 exp j kz ; the other propagating at an angle () with respect to the z axis, in the x z plane, U 2 10 exp j k cos () z + k sin () x , as illustrated in Fig. 2.5-4. At the z 0 plane the two waves have a phase difference <p k Sill () x, for which the interference equation (2.5-4) yields a total intensity I 210 1 + cos k sin () x . (2.5-7) This pattern varies sinusoidally with x, with period 27r k sin () ,X sin (), as shown in Fig. 2.5-4. If () 30° , for example, the period is 2'x. This suggests a method of printing a sinusoidal pattern of high resolution for use as a diffraction grating. It also suggests a method of monitoring the angle of arrival () of a wave by mixing it with a reference wave and recording the resultant intensity distribution. As discussed in Sec. 4.5, this is the principle that lies behind holography. \\\\\\\,\\\,\ \\\\\,\\\,\\ \,\\\,\\\\\\,\\\ "\"\,\\\\ Figure 2.5-4 The interference of two plane waves traveling at an angle () with respect to each other results in a sinusoidal intensity pattern in the x direction with period AI sin (). EXERCISE 2.5-1 Interference of a Plane Wave and a Spherical Wave. A plane wave traveling along the z direction with complex alnplitude Al exp( jkz), and a spherical wave centered at z 0 and approximated by the paraboloidal wave of complex amplitude (A 2 Iz) exp( jkz) exp[ jk(x 2 + y2) /2z] [see (2.2-17)], interfere in the z d plane. Derive an expression for the total intensity lex, y, d). Assuming that the two waves have the same intensities at the z d plane, verify that the locus of points of zero intensity is a set of concentric rings, as illustrated in Fig. 2.5-5.
62 CHAPTER 2 WAVE OPTICS x x z y Figure 2.5-5 The interference of a plane wave and a spherical wave creates a pattern of concentric rings (iHustrated at the plane z == d). EXERCISE 2.5-2 Interference of Two Spherical Waves. Two pherical waves of equal intensity 1 0 , originating at the points (-a, 0, 0) and (a, 0, 0), interfere in the plane z == d as iHustratcd in Fig. 2.5-6. This double-pinhole syten1 is similar to that used by Thomas Young in his celebrated double-slit exper- iment in which he demonstrated interference. Use the paraboloidal approxiInation for the spherical waves to how that the intensity at the plane z == d is ( 2 7rXe ) I(x,y,d)21o l+cos ' (2.5-8) where the angle subtended by the centers of the two waves at the observation plane is e 2a/ d. The intensity pattern is periodic with period 'A/e. A --., (J I z --.,' (JA=1T .....:=- '" ... . .' 2a ...., ..... I .. ./ /1 d x . . 1 · t1\j\ · T 2G ; u::}JiJ)ijjj]jj!!!!\;IIII!!I) I .. jI, !11I1/j'1111111I/J .. / ., . d . e x Figure 2.5-6 [nterference of two spherical waves of equal intensities originating at the points PI and P2. The two waves can be obtained by permitting a plane wave to impinge on two pinholes in a screen. The light intensity at an observation plane a large distance d froIn the pinholes take the form of a sinusoidal interference pattern, with period 'A/(J, along the direction of the line connecting the pinholes. B. Multiple-Wave Interference The superposition of I monochromatic waves of the same frequency, with cOlnplex amplitudes VI, V 2 , . . . , VAl gives rise to a wave whose frequency renlains the same and whose complex amplitude is given by U == U 1 + U 2 + . . . + U AI. Knowledge of the intensities of the individual waves, 1 1 ,1 2 , . . . , I A !, is not sufficient to determine the total intensity I == I UI 2 since the relative phases must also be known. The role played by the phase is dralnatical1y illustrated in the foHowing exanlple.
2.5 INTERFERENCE 63 Interference of M Waves with Equal Amplitudes and Equal Phase Differences We first examine the interference of !!vI waves with complex amplitudes Urn 10 exp j m 1 <p , m 1, 2, . . . , it! . (2.5-9) The waves have equal intensities 1 0 , and phase difference <p between successive waves, as illustrated in Fig. 2.5-7(a). To derive an expression for the intensity of the super- position, it is convenient to introduce the quantity h exp j<p whereupon Urn 10 h m - 1 . The complex amplitude of the superposed wave is then u 10 1 + h + h 2 + · · · + hM-l h M h o 1 exp jMcp . , exp J cp (2.5-1 0) o 1 which has the corresponding intensity I U 2 exp 10 exp jM<p 2 j<p 2 exp j<p 2 , (2.5-11) whence I 02. sin <p 2 (2.5-12) Interference of M Waves UM I - l .......... 27r -Ji I J ,<{J Wo . . . . . - . . I -- - ----- - ---- - ------ - - . . . . . 27r M t..c V3 VI V2 o 27r M 27r 47r 67r <(J Figure 2.5-7 (a) The sum of !vI phasors of equal magnitudes and equal phase differences. (b) The intensity I as a function of <p. The peak intensity occurs when all the phasors are aligned; it is then - M times greater than the mean intensity I MIo. In this example M 5. The intensity I is evidently strongly dependent on the phase difference <p, as illus- trated in Fig. 2.5-7(b) for M 5. When <p 21fq, where q is an integer, all the phasors are aligned so that the amplitude of the total wave is M times that of an individual component, and the intensity reaches its peak v a lue of M2Io. The m ean intensity is the same as the result obtained in the absence of interference. The peak intensity is therefore M times greater than the mean intensity. The sensitivity of the intensity to the
64 CHAPTER 2 WAVE OPTICS phase is therefore dramatic for large M. At its peak value, the intensity is magnified by a factor M over the mean but it decreases sharply as the phase difference <p deviates slightly from 27fq. In particular, when <p 27f M the intensity becomes zero. It is instructive to compare Fig. 2.5-7(b) for M 5 with Fig. 2.5-2 for Al 2. EXERCISE 2.5-3 Bragg Reflection. Consider light reflected at an angle () from M parallel reflecting planes separated by a distance A, as shown in Fig. 2.5-8. Assume that only a small fraction of the light is reflected from each plane, so that the amplitudes of the M reflected waves are approximately equal. Show that the reflected waves have a phase difference c.p k(2A sin ()) and that the angle () at which the intensity of the total reflected light is maximum satisfies sin () A 2A · (2.5-13) Bragg Angle This equation defines the Bragg angle (). Such reflections are encountered when light is reflected from a multilayer structure (see Sec. 7.1) or when X-ray waves are reflected from atomic planes in crystalline structures. It also occurs when light is reflected from a periodic structure created by an acoustic wave (see Chapter 19). An exact treatment of Bragg reflection is provided in Sec. 7.1C. VI V2 . 12..M . .U M , , '\ ." A Figure 2.5-8 Reflection of a plane wave from AJ parallel planes separated from each other by a distance A. The reflected waves interfere constructively and yield maximum intensity when the angle () is the Bragg angle. Note that () is defined with respect to the parallel planes. () Interference of an Infinite Number of Waves of Progressively Smaller Amplitudes and Equal Phase Differences We now examine the superposition of an infinite number of waves with equal phase differences and with amplitudes that decrease at a geometric rate: U 1 10 , U 2 hUt, U 3 hU2 h 2 U t , . . . , (2.5-14) . where h h eJ<P, h < 1, and 10 is the intensity of the initial wave. The amplitude of the mth wave is smaller than that of the mIst wave by the factor h and the phase differs by <po The phasor diagram is shown in Fig. 2.5-9(a). The superposition wave has a complex amplitude U U t + U 2 + U 3 + · · · 10 1 + h + h 2 + · · ·
2.5 INTERFERENCE 65 I Imax . <pI I I VI U2 <P U3 <P " " P=2 27r 'F I 'F = 10 o 27r 47r <P Figure 2.5-9 (a) The sum of an infinite number of phasors whose magnitudes are successively reduced at a geometric rate and whose phase differences cp are equal. (b) Dependence of the intensity I on the phase difference cp for two values of :F. Peak values occur at cp 27rq. The full width at half maximum of each peak is approximately 27r /:F when :F » 1. The sharpness of the peaks increases with increasing :F. 10 1 h 10 h e jep · (2.5-15) 1 The total intensity is then 10 h e jep 2 1 1 10 2 h COscp + h 2 · 2 ' SIn cp (2.5-16) 1 u 2 from which 1 10 h 2 + 4 h sin 2 cp 2 · (2.5-17) I It is convenient to write this equation in the form 1 I max. 1 + 2:7 7r 2 sin 2 cp 2 ' I max. 10 1 h 2' (2.5-18) I ntensity of an I nfin ite Number of Waves where the quantity 1f h 1 h (2.5-19) Finesse is a parameter known as the finesse. The intensity 1 is a periodic function of cp with period 21f, as illustrated in Fig. 2.5- 9(b). It reaches its maximum value Imax. when cp 27rq, where q is an integer. This occurs when the phasors align to form a straight line. (This result is not unlike that displayed in Fig. 2.5-7(b) for the interference of M waves of equal amplitudes and equal phase differences.) en the finesse :7 is large (Le., the factor h is close to 1), I becomes a sharply peaked function of cp. Consider values of cp near the cp 0 peak,
66 CHAPTER 2 WAVE OPTICS as a representative example. For cp «1, sin cp 2 cp 2 whereupon (2.5-18) can be wri tten as 1 + 1"' 1r 2cp2 (2.5- 20) The intensity I then decreases to half its peak value when cp width at half maximum ( HM) of the peak becomes 1f 1"', so that the full 21r cp . 1"' (2.5-21) Width of Interference Pattern In the regime 1"' » 1, we then have cp « 21f and the assumption that cp 1 is applicable. The finesse 1"' is the ratio of the period 21f to the HM of the peaks in the interference pattern. It is therefore a measure of the sharpness of the interference function, i.e., the sensitivity of the intensity to deviations of <p from the values 27rq corresponding to the peaks. A useful device based on this principle is the Fabry Perot interferometer. It consists of two parallel mirrors within which light undergoes multiple reflections. In the course of each round trip, the light suffers a fixed amplitude reduction h r , arising from losses at the mirrors, and a phase shift <p k2d 41rvd C 21rV C 2d associated with the propagation, where d is the mirror separation. The total light intensity depends on the phase shift cp in accordance with (2.5-18), attaining maxima when cp 2 is an integer multiple of 1f. The proportionality of the phase shift cp to the optical frequency v shows that the intensity transmission of the Fabry Perot device will exhibit peaks separated in frequency by c 2d. The width of these peaks will be c 2d 1"', where the finesse :f is governed by the loss via (2.5-19). The Fabry Perot interferometer, which also serves as a spectrum analyzer, is considered further in Sec. 7.1 B. It is commonly used as a resonator for lasers, as discussed in Secs. 10.1 and 15.1A. 2.6 POLYCHROMATIC AND PULSED LIGHT Since the wavefunction of monochromatic light is a harmonic function of time extend- ing over all time (from 00 to (0), it is an idealization that cannot be met in reality. This section is devoted to waves of arbitrary time dependence, including optical pulses of finite time duration. Such waves are polychromatic rather than monochromatic. A more detailed introduction to the optics of pulsed light is provided in Chapter 22. A. Temporal and Spectral Description Although a polychromatic wave is described by a wavefunction u r, t with nonhar- monic time dependence, it may be expanded as a superposition of harmonic func- tions, each of which represents a monochromatic wave. Since we already know how monochromatic waves propagate in free space and through various optical components, we can determine the effect of optical systems on polychromatic light by using the principle of superposition. Fourier methods permit the expansion of an arbitrary function of time u t , repre- senting the wavefunction u r, t at a fixed position r, as a superposition integral of
2.6 POLYCHROMATIC AND PULSED LIGHT 67 harmonic functions of different frequencies, amplitudes, and phases: .. ex) u t v v exp j27rvt dv, (2.6-1) -00 where v v is determined by carrying out the Fourier transform 00 vv u t exp j27rvt dt. (2.6-2) -00 A review of the Fourier transform and its properties is presented in Sec. A.I of Appendix A. The expansion in (2.6-1) extends over positive and negative frequencies. However, since u t is real, v v v* v (see Sec. A.l). Thus, the negative- frequency components are not independent; they are simply conjugated versions of the corresponding positive-frequency components. Complex Representation It is convenient to represent the rea] function u t in (2.6-1) by a complex function ex) Ut 2 v v exp j27rvt dv (2.6-3) o that includes only the positive-frequency components (multiplied by a factor of 2), and suppresses aU the negative frequencies. The Fourier transform of U t is therefore a function V v 2v v for v > 0, and 0 for v < o. The real function u t can be determined from its complex representation U t by simply taking the real part, u t Re U t (2.6-4) The complex function U t is known as the complex analytic signal. The validity of (2.6-4) can be verified b y breaking the integral in (2.6-1) into two parts, with limits whereas the second is given by o 00 v v exp j27rvt dv v v exp j27rvt dv -ex) o 00 v* v exp j27rvt dv o The first step above reflects a simple change of variable from v to v, while the second step uses the symmetry relation v v v* v . The net result is that u t can be 4). As a simple example, the complex representation of the real harmonic function u t cos wt is the complex harmonic function U t exp jwt . This is the complex representation introduced in Sec. 2.2A for monochromatic waves. In fact, the complex representation of a polychromatic wave, as described in this section, is simply a superposition of the complex representations of each of its monochromatic Fourier components.
68 CHAPTER 2 WAVE OPTICS The complex analytic signa] corresponding to the wavefunction u r, t is called the complex wavefunction U r, t . Since each of its Fourier components satisfies the wave equation, so too does the complex wavefunction U r, t , v 2 u 1 a 2 u c 2 8t 2 o. (2.6-6) Wave Equation Figure 2.6-1 shows the magnitudes of the Fourier transforms of the wavefunction u r, t and the complex wavefunction U r, t . In this illustration the optical wave is quasi-monochromatic, i.e., it has Fourier components with frequencies confined within a narrow band of width f:,.v surrounding a central frequency vo, such that f:,.v « Vo. Iv(r, v)1 IV(r, v)1 ---------.------- -- (a) (b) __ _e._ae...___..____ ___________________ -vo 0 Vo v 0 Vo v Figure 2.6-1 (a) The magnitude Iv(r, v)1 of the Fourier transform of the wavefunction u(r, t). (b) The magnitude I V (r, v) I of the Fourier transform of the corresponding complex wavefunction U(r, t). Intensity of a Polychromatic Wave The optical intensity is related to the wavefunction by (2.1-3): Ir,t 2u 2 r,t 2 2 + U* r, t U 2 r, t + U*2 r, t + U r, t U* r, t . (2.6-7) For a quasi-monochromatic wave with central frequency va and spectral width D..v « va, the average · is taken over a time interval much longer than the time of an optical cycle 1 va but much shorter than 1 D..v (see Sec. 2.1). Since U r, t is given by (2.6-4), the term U 2 in (2.6-7) has components oscillating at frequencies 2vo. Similarly, the components of U*2 oscillate at frequencies 2vo. These terms are therefore washed out by the averaging operation. The third term, however, contains only frequency differences, which are of the order of f:,.v «:: va. It therefore varies slowly and is unaffected by the time-averaging operation. Thus, the third term in (2.6- 7) survives and the light intensity becomes I r,t 2 U r, t . (2.6-8) Optical Intensity The optical intensity of a quasi-monochromatic wave is the absolute square of its complex wavefunction.
2.6 POLYCHROMATIC AND PULSED LIGHT 69 The simplicity of this result is, in fact, the rationale for introducing the concept of the complex wavefunction. Pulsed Plane Wave The simplest example of pulsed light is a pulsed plane wave. The complex wavefunc- tion has the form U r,t A t z C exp j27fvo t z C , (2.6-9) where the complex envelope A t is a time-varying function and Vo is the central optical frequency. The monochromatic plane wave is a special case of (2.6-9) for which A t is constant, i.e., U r, t A exp j27fvo t z c A exp jkoz exp jwot , where ko Wo c and Wo 27rvo. Since U r, t in (2.6-9) is a function of t z c it satisfies the wave equation (2.6-6) regardless of the form of the function A · (provided that d 2 A dt 2 exists). This can be verified by direct substitution. If A t is of finite duration 7, then at any fixed position z the wave lasts for a time period 7, and at any fixed time t it extends over a distance C7. It is therefore a wavepacket of fixed extent traveling in the z direction (Fig. 2.6-2). As an example, a pulse of duration 7 1 ps extends over a distance C7 0.3 mm in free space. The Fourier transform of the complex wavefunction in (2.6-9) is V r,v . A v Vo exp j27fvZ C , (2.6-10) where A v is the Fourier transform of At. This may be shown by use of the fre- quency translation property of the Fourier transform (see Sec. A.I of Appendix A). The complex envelope A t is often slowly varying in comparison with an optical cycle, so that its Fourier transform A v has a spectral width llv much smaller than the central frequency Yo. The spectral width llv is inversely proportional to the temporal width 7. In particular, if A t is Gaussian, then its Fourier transform A v is also Gaussian. If the temporal and spectral widths are defined as the power-rms widths, then their product equals 1 47r (see Sec. A.2 of Appendix A). For example, if 7 1 ps, then llv 80 GHz. If the central frequency Vo is 5 X 10 14 Hz (corresponding to Ao 0.6 MID), then llv Vo 1.6 X 10- 4 , so that the light is quasi-monochromatic. Fig. 2.6-2 illustrates the temporal, spatial, and spectra] characteristics of the pulsed plane wave in terms of the wavefunction. IA(t) I at t , , , " CT IA(v) I I V(v) I .... " " " T " , , , ...c , o " z V V " , , t at t + T ..... " , , " o cT ' , z " , .... , ' 0 1/ 0 1/0 V .... (a) (b) (c) (d) Figure 2.6-2 Temporal, spatial, and spectral characteristics of a pulsed plane wave. (a) The wavefunction at a fixed position has duration T. (b) The wavefunction as a function of position at times t and t + T. The pulse travels with speed C and occupies a distance CT. (c) The magnitude IA(v)1 of the Fourier transform of the complex envelope. (d) The magnitude IV(v)1 of the Fourier transform of the complex wavefunction is centered at Vo.
70 CHAPTER 2 WAVE OPTICS The propagation of a pulsed plane wave through a medium with frequency- dependent refractive index (i.e., with a frequency-dependent speed of light C Co n) is discussed in Sec. 5.5B while Chapter 22 covers other aspects of pulsed optics. B. Light Beating The dependence of the intensity of a polychromatic wave on time may be attributed to interference among the monochromatic components that constitute the wave. This concept is now demonstrated by means of two examples: interference between two monochromatic waves and interference among a finite number of monochromatic waves. Interference of Two Monochromatic Waves with Different Frequencies An optical wave composed of two monochromatic waves of frequencies VI and V2 and intensities II and 1 2 has a complex wavefunction at some point in space Ut II exp j27rVIt + 1 2 exp j 27rV 2 t , (2.6-11 ) where the phases are taken to be zero and the r dependence has been suppressed for convenience. The intensity of the total wave is determined by use of the interference equation (2.5-4), I t II + 1 2 + 2 1 1 1 2 cas 27r V2 vI t . (2.6-12) The intensity therefore varies sinusoidally at the difference frequency v2 vI, which is known as the beat frequency. This phenomenon goes by a number of names: light beating, optical mixing, photomixing, and optical heterodyning. Equation (2.6-12) is analogous to (2.5-7), which describes the spatial interference of two waves of the same frequency traveling in different directions. This can be understood in terms of the phasor diagram in Fig. 2.5-1. The two phasors U 1 and U 2 rotate at angular frequencies WI 27rVI and W2 27rV2, so that the difference angle <P <P2 <PI 27r V2 VI t, in accord with (2.6-] 2). Waves of different frequencies traveling in different directions exhibit spatiotemporal interference. In electronics, beating or mixing is said to occur when the sum of two sinusoidal signals is detected by a nonlinear (e.g., quadratic) device called a mixer.. producing signals at the difference and sum frequencies. This device is used in heterodyne radio receivers. In optics, photodetectors are responsive to the optical intensity (see Chap- ter 18), which, in accordance with (2.6-8), is proportional to the absolute square of the complex wavefunction. Optical detectors are therefore sensitive only to the difference frequency. Much as (2.5- 7) provides the basis for determining the direction of a wave via the spatial interference pattern at a screen, (2.6-12) provides a way of determining the frequency of an optical wave by measuring the temporal interference pattern at the output of a photodetector. The use of optical beating in optical heterodyne receivers is discussed in Sec. 24.5. Other forms of optical mixing make use of nonlinear media to generate optical-frequency differences and sums, as described in Chapter 21. EXERCISE 2.6-1 Optical Doppler Radar. As a result of the Doppler effect, a monochromatic optical wave of frequency v, reflected from an object moving with a velocity component v along the line of sight from an observer, undergoes a frequency shift l::1v i:(2v Ie) v, depending on whether the object is
2.6 POLYCHROMATIC AND PULSED LIGHT 71 moving toward (+) or away ( ) from the observer. Assuming that the original and reflected waves are superimposed, derive an expression for the intensity of the resultant wave. Suggest a method for measuring the velocity of a target using such an arrangement. If one of the mirrors of a Michelson interferometer [(Fig. 2.5-3(b)] moves with velocity i:v, use (2.5-6) to show that the beat frequency is i:(2v / c) v. Interference of M Monochromatic Waves with Equal Intensities and Equally Spaced Frequencies The interference of a large number of monochromatic waves with equal intensities, equal phases, and equally spaced frequencies can result in the generation of brief pulses of light. Consider an odd number of waves, AI 2L + 1, each with intensity fo and zero phase, and with frequencies L, . . . , 0, . . . L , (2.6-13) V q Vo + qVF, q centered about frequency Vo and spaced by frequency VF « yo. At a given position, the total wave has a complex wavefunction L Ut fo exp j27r Vo + qVF t . q -L (2.6-14) This represents the sum of itI phasors of equal magnitudes and successive phases that differ by cp 27rVFt. Results for the intensity are immediately available from the analysis carried out in Sec. 2.5B, which is mathematically identical to the case at hand. Referring to (2.5-12) and Fig. 2.5-7, and using the substitution cp 27ft T F with T F 1 VF, the total intensity is I t U t 2 . (2.6-15) !(t) - MI T F -I I V(lI) I VF ] 2...M - I -- - ----- - ----- -------- - -- -I I .... T F M Figure 2.6-3 Time dependence of the optical intensity I{t) of a polychromatic wave comprising A! monochromatic waves of equal intensities, equal phases, and successive frequencies that differ by VF. The intensity I(t) is a periodic train of pulses of period T F l/VF with a peak that is Al - times greater than the mean I. The duration of each pulse is Al times shorter than the period. In this example 1\1 5. These graphs should be compared with those in Fig. 2.5-7. The magnitude of the Fourier transform IV(v)1 is shown in the lower graph. t Vo v "As illustrated in Fig. 2.6-3, the intensity I t is a periodic sequence of optical pulses with period T F , peak intensity 111 2 1 0 , and mean intensity I AIIo. The peak intensity is therefore !vI times greater than the mean intensity. The duration of each pulse is approximately T F AI so that the pulses become very short when AI is large. If VF 1 GHz, for example, then T F 1 TIS; for A-l 1000, pulses of I-ps duration are generated.
72 CHAPTER 2 WAVE OPTICS This example provides a dramatic demonstration of how it! monochromatic waves can conspire to produce a train of very short optical pulses. We shall see in Sec. 15.4D that the modes of a laser can be phase locked in the fashion described above to produce sequences of ultrashort laser pulses. READING LIST Books on Wave Optics and Interferometry See also the general reading list in Chapter 1. J. R. Pierce, Almost All About Waves, MIT Press, 1974; Dover, reissued 2006. H. J. Pain, The Physics of Vibrations and Waves, Wiley, 6th ed. 2005. R. H. Webb, Elementary Wave Optics, Academic Press, 1969; Dover, 2005 P. Hariharan, Optical Interferometry, Academic Press, 2nd ed. 2003. M. Mansuripur, Classical Optics and Its Applications, Cambridge University Press, 2002. S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics, Cambridge University Press 3rd ed. 1998. S. A. Akhmanov and S. Yu. Nikitin, Physical Optics, Oxford University Press, 1997. A. R. Mickelson, Physical Optics, Van Nostrand Reinhold, 1992. J. M. Vaughan, The Fabry-Perot Interferometer, Adam Hilger 1989. H. D. Young, Fundan1entals of Waves, Optics, and Moden1 Physics, McGraw-Hill, paperback 2nd ed. 1976. S. Tolansky, An Introduction to Interferometry, Wiley, 2nd ed. 1973. M. Franon, N. Krauzman, J. P. Matieu and M. May, Experilnents in Physical Optics, Gordon and Breach, 1970. M. Franon, Optical Interferometry, Academic Press, 1966. Books on Spectroscopy J. M. Hollas, Modern Spectroscopy, Wiley, 4th ed. 2004. J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy, Wiley-VCH, 2001. A. A. Christy, Y. Ozaki, and V. G. Gregoriou, Modern Fourier Transforl1l Illfrared Spectroscopy, Elsevier, 2001. D. L. Pavia, G. M. Lampman, and G. S. Kriz, Introduction to Spectr()scopy Brooks/Cole, paperback 3rd ed. 2000. B. C. Smith, Fundalnentals of Fourier Transform Infrared Spectroscopy, CRC Press, 1996. Books on Diffraction Gratings C. Palmer, Diffraction Grating Handbook, Richardson Grating Laboratory (Newport Corporation/Spectra- Physics, Irvine, CA) 4th ed. 2000. E. G. Loewen and E. Popov, Diffraction Gratings and Application,, Marcel Dekker, 1997. Popular and Historical J. Z. Buchwald, The Rise of the Wave Theory of Light: Optical Theory and Experi1nent in the Early Nineteenth Century, University of Chicago Press, paperback ed. 1989. W. E. Kock, Sound Waves and Light Waves, Doubleday/Anchor, 1965. C. Huygens, Treatise on Light, 1690, University of Chicago Press, 1945. Articles T. E. Bell, Waiting for Gravity, IEEE Spectrum, vol. 43, no. 7, pp. 40-46 2006. G. W. Kamerman, ed., Selected Papers on Laser Radar, SPIE Optical Engineering Press (Milestone Series Volume 133), 1997. D. Maystre, ed., Selected Papers 011 Diffraction Gratings, SPIE Optical Engineering Press (Milestone Series Volume 83), 1993. P. Hariharan, ed., Selected Papers on Interferolnetry, SPIE Optical Engineering Press (Milestone Series Volume 28), 1991.
PROBLEMS 73 PROBLEMS , 2.2-3 Spherical Waves. Use a spherical coordinate system to verify that the complex amplitude of the spherical wave (2.2-15) satisfies the Helmholtz equation (2.2-7). 2.2-4 Intensity of a Spherical Wave. Derive an expression for the intensity I of a spherical wave at a distance r from its center in terms of the optical power P. What is the intensity r 1 ill for P 100 W? 2.2-5 Cylindrical Waves. Derive expressions for the complex amplitude and intensity of a monochromatic wave whose wavefronts are cylinders centered about the y axis. 2.2-6 Paraxial Helmholtz Equation. Derive the paraxial Helmholtz equation (2.2-23) using the approximations in (2.2-21) and (2.2-22). 2.2-7 Conjugate Waves. Compare a monochromatic wave with complex amplitude U(r) to a monochromatic wave of the same frequency but with complex amplitude U*(r), with respect to intensity, wavefronts, and wavefront normals. Use the plane wave U(r) A exp[ jk(x + y)1 2] and the spherical wave U(r) (Air) exp( jkr) as examples. 2.3-1 Wave in a GRIN Slab. Sketch the wavefronts of a wave traveling in the graded-index SELFOC slab described in Example 1.3-1. 2.4- 7 Reflection of a Spherical Wave from a Planar Mirror. A spherical wave is reflected from a planar mirror sufficiently far from the wave origin so that the Fresnel approximation is sat- isfied. By regarding the spherical wave locally as a plane wave with slowly varying direction, use the law of reflection of plane waves to determine the nature of the reflected wave. 2.4-8 Optical Path Length. A plane wave travels in a direction normal to a thin plate made of N thin parallel layers of thicknesses d q and refractive indexes nq, q 1, 2, . . . N. If all reflections are ignored, determine the complex amplitude transmittance of the plate. If the plate is replaced with a distance d of free space, what should d be so that the same complex amplitude transmittance is obtained? Show that this distance is the optical path length defined in Sec. 1.1. 2.4-9 Diffraction Grating. Repeat Exercise 2.4-5 for a thin transparent plate whose thickness d(x, y) is a square (instead of sinusoidal) periodic function of x of period A » A. Show that the angle () between the diffracted waves is still given by () AlA. If a plane wave is incident in a direction normal to the grating, determine the amplitudes of the different diffracted plane waves. 2.4-1 0 Reflectance of a Spherical Mirror. Show that the complex amplitude reflectance r( x, y) (the ratio of the complex amplitudes of the reflected and incident waves) of a thin spherical mirror of radius R is given by r(x, y) ho exp[ jk o (x 2 + y2)1 R], where ho is a constant. Compare this to the complex amplitude transmittance of a lens of focal length f R12. 2.5-4 Standing Waves. Derive an expression for the intensity I of the superposition of two plane waves of wavelength A traveling in opposite directions along the z axis. Sketch I versus z. 2.5-5 Fringe Visibility. The visibility of an interference pattern such as that described by (2.5-4) and plotted in Fig. 2.5-1 is defined as the ratio V (Imax Imin)/(Imax + I min ), where Imax and I min are the maximum and minimum values of I. Derive an expression for V as a function of the ratio II 11 2 of the two interfering waves and determine the ratio III 1 2 for which the visibility is maximum. 2.5-6 Michelson Interferometer. If one of the mirrors of the Michelson interferometer [Fig. 2.5- 3(b)] is misaligned by a small angle (), describe the shape of the interference pattern in the detector plane. What happens to this pattern as the other mirror moves? 2.6-2 Pulsed Spherical Wave. (a) Show that a pulsed spherical wave has a complex wavefunction of the form U(r, t) (l/r)a(t rlc), where a(t) is an arbitrary function. (b) An ultrashort optical pulse has a complex wavefunction with central frequency corre- sponding to a wavelength Ao 585 nm and a Gaussian envelope of RMS width of at 6 fs (1 fs 10- 15 s). How many optical cycles are contained within the pulse width? If the pulse propagates in free space as a spherical wave initiated at the origin at t 0, describe the spatial distribution of the intensity as a function of the radial distance at time tIps.
CHAPTER - . 3.1 THE GAUSSIAN BEAM A. Complex Amplitude B. Properties C. Beam Quality TRANSMISSION THROUGH OPTICAL COMPONENTS A. Transmission Through a Thin Lens B. Beam Shaping C. Reflection from a Spherical Mirror *0. Transmission Through an Arbitrary Optical System HERMITE GAUSSIAN BEAMS LAGUERRE GAUSSIAN AND BESSEL BEAMS 75 3.2 86 3.3 *3.4 94 97 . .. \ " .... .. If. , """ , . " .. ..... "'\.' ....r.... . \!, . . " \\-, . ... .. . . . . " - - . , .... .. . , .. . \ J " .. .. 'J .. . ..,-, .. .., . , . . . ". . -- ..A ,.. "'\. ' . . . , .. ..... -.. .I y r: , - ... lit ' '- +... . . . ,-' -"" . 1 , .. , - ":.,. \'- A ..... '"<to..."., The Gaussian be,lm t,lkes the n£lme of the cele- br£lted Germ£ln m£lthem£ltician Carl Friedrich Gauss (1777-1855). Lord Rayleigh (John William Strutt) (1842-1919) contributed to many areas of optics. The depth of focus of the Gaussian beam is named in his honor. 74
Can light be spatially confined and transported in free space without angular spread? Although the wave nature of light precludes the possibility of such idealized transport, light can, in fact, be confined in the form of beams that come as close as possible to spatially localized and nondiverging waves. The two extremes of angular and spatial confinement are the plane wave and the spherical wave. The wavefront normals (rays) of a plane wave coincide with the di- rection of travel of the wave so that there is no angular spread, but the energy extends spatially over all of space. The spherical wave, in contrast, originates from a single spatial point, but has wavefront normals (rays) that diverge in all angular directions. Waves whose wavefront normals make small angles with the z axis are called parax- ial waves. They must satisfy the paraxial Helmholtz equation, which was derived in Sec. 2.2C. The Gaussian beam is an important solution of this equation that exhibits the characteristics of an optical beam, as attested to by the following features. The beam power is principally concentrated within a small cylinder that surrounds the beam axis. The intensity distribution in any transverse plane is a circularly symmetric Gaussian function centered about the beam axis. The width of this function is mini- mum at the beam waist and gradually becomes larger as the distance from the waist increases in both directions. The wavefronts are approximately planar near the beam waist, gradually curve as the distance from the waist increases, and ultimately become approximately spherical far from the waist. The angular divergence of the wavefront normals assumes the minimum value permitted by the wave equation for a given beam width. The wavefront normals are therefore much like a thin pencil of rays. Under ideal conditions, the light from many types of lasers takes the form of a Gaussian beam. This Chapter An expression for the complex amplitude of the Gaussian beam is set forth in Sec. 3.1 and a detailed discussion of its physical properties (intensity, power, beam width, beam divergence, depth of focus, and phase) is provided therein. The shaping of Gaussian beams (focusing, relaying, collimating, and expanding) by the use of various optical components is the subject of Sec. 3.2. In Sec. 3.3 we introduce a more general family of optical beams called Hermite Gaussian beams, of which the simple Gaussian beam is a member. Finally, in Sec. 3.4, Laguerre Gaussian and Bessel beams are discussed. 3.1 THE GAUSSIAN BEAM A. Complex Amplitude The concept of paraxial waves was introduced in Sec. 2.2C. A paraxial wave is a plane wave traveling along the z direction e- jkz (with wavenumber k 21r A and wave- length A), modulated by a complex envelope A r that is a slowly varying function of position (see Fig. 2.2-5), so that its complex amplitude is Ur A r exp jkz. (3.1-1) The envelope is taken to be approximately constant within a neighborhood of size A, so that the wave locally maintains its plane-wave nature but exhibits wavefront normals that are paraxial rays. In order that the complex amplitude U r satisfy the Helmholtz equation, \72U + k 2 U 0, the complex envelope A r must satisfy the paraxial Helmholtz equa- 75
76 CHAPTER 3 BEAM OPTICS tion (2.2-23) \7 T 2 A J 8z 0, (3.1-2) where \7} {)2 ox 2 + {)2 {)y2 is the transverse Laplacian operator. A simple solution to the paraxial Helmholtz equation yields the paraboloidal wave (see Exercise 2.2-2), for which exp 2 "k P J 2z ' p2 x2 + y2 , (3.1-3) Ar Al z where Al is a constant. The paraboloidal wave is the paraxial approximation of the spherical wave UrAl r exp j kr when x and yare much smaller than z (see Sec. 2.2B). Another solution of the paraxial Helmholtz equation leads to the Gaussian beam. It is obtained from the paraboloidal wave by use of a simple transformation. Since the complex envelope of the paraboloidal wave (3.1- 3) is a solution of the paraxial Helmholtz equation (3.1-2), so too is a shifted version of it, with z replacing z where ( is a constant: Ar Al q Z exp 2 2q z , q z z . (3.1-4) This represents a paraboloidal wave centered about the point z ( instead of about z O. Equation (3.1-4) remains a solution of (3.1-2) even when is complex, but the solution acquires dramatically different properties. In particular, when is purely imaginary, say (, jzo where Zo is real, (3.1-4) yields the complex envelope of the Gaussian beam Ar Al q Z exp 2 2q z , q z z + j Zo. (3.1-5) Complex Envelope The quantity q z is called the q-parameter of the beam and the parameter Zo is known as the Rayleigh range. To separate the amplitude and phase of this complex envelope, we write the complex function 1 q z 1 z + j Zo in terms of its real and imaginary parts by defining two new real functions, R z and W z , such that 1 q z 1 Rz " A J 1rT2 Z · (3.1-6) It will be shown subsequently that W z and R z are measures of the beam width and wavefront radius of curvature, respectively. Expressions for T z and R z as functions of z and Zo are provided in (3.1-8) and (3.1-9). Substituting (3.1-6) into (3.1- 5) and using (3.1-1) leads directly to an expression for the complex amplitude U r of
3.1 THE GAUSSIAN BEAM 77 the Gaussian beam: exp 2 P TV2 Z exp jkz 2 Ok P J 2R z +j( z (3.1-7) Complex Amplitude Ur W o v z W o 1+ z 2 (3.1-8) Rz Zo Zo 2 Z 1 + z (3.1-9) ( z z tan- I Zo (3.1-10) W o '\zo . (3.1-11) Beam Parameters 7r A new constant Ao Al jzo has been defined for convenience. The expression for the complex amplitude of the Gaussian beam provided above is centra] to this chapter. It is described by two independent parameters, Ao and zo, which are determined from the boundary conditions. All other parameters are related to the Zo and the wavelength ,\ by (3.1-8) to (3.1-11). The significance of these parameters will become clear in the sequel. B. Properties Equations (3.1-7) to (3.1-11) will now be used to determine the properties of the Gaussian beam. Intensity The optical intensity 1 r U r 2 is a function of the axial and radial positions, z and P x 2 + y2 , respectively 1 P,Z tV o 10 Wz 2 exp 2p 2 TV2 Z , (3.1-12) where 10 Ao 2. At any value of z the intensity is a Gaussian function of the radial distance p hence the appellation "Gaussian beam." The Gaussian function has its peak on the z axis, at p 0, and decreases monotonically as p increases. The beam width W z of the Gaussian distribution increases with the axial distance z as illustrated in Fig. 3.1-1. On the beam axis p 0 the intensity in (3.1-12) reduces to 2 10,z o TT Z 10 1 + z Zo 2 ' (3.1-13)
78 CHAPTER 3 BEAM OPTICS y y y ... X . X X 1/10 1/10 1/10 .- o Wo X 0 Wo X 0 Wo X Figure 3.1-1 The normalized beam intensity 1/10 as a function of the radial distance p at different axial distances: (a) z 0; (b) z Zo; (c) z 2zo. which has its maximum value 10 at Z ° and decays gradually with increasing z, reaching half its peak value at z ::f:zo (Fig. 3.1-2). When z » zo, I 0, z Ioz5 z2, so that the intensity decreases with distance in accordance with an inverse- square law, as for spherical and paraboloidal waves. Overall, the beam center z 0, p ° is the location of the greatest intensity: I 0, 0 10. 1/10 1 .------- 0.5 . . . I . I ---------- I I I . . I I - 0 z Figure 3.1-2 The normalized beam intensity 1/10 at points on the beam axis (p 0) as a function of distance along the beam axis., z. Power The total optical power carried by the beam is the integral of the optical intensity over any transverse plane (say at position z), ex) p Ip,z 27rpdp, (3.1-14) o which yields p (3.1-15) The beam power is thus half the peak intensity multiplied by the beam area. The result is independent of z, as expected. Since optical beams are often described by their power
3.1 THE GAUSSIAN BEAM 79 P, it is useful to express 10 in terms of P via (3.1-15), whereupon (3.1-12) can be rewritten in the form 2P [ 2 p2 ] I(p, z) = 7rW 2 (z) exp - W2(z) . (3.1-16) Beam Intensity The ratio of the power carried within a circle of radius Po in the transverse plane to the total power, at position z, is 1 (PO [ 2 P6 ] p Jo I(p, z) 27rpdp = 1 - exp - W2(z) . (3.1-17) The power contained within a circle of radius Po == W (z) is therefore approximately 86% of the total power. About 99% of the power is contained within a circle of radius 1.5 W(z). Beam Width At any transverse plane, the beam intensity assumes its peak value on the beam axis, and decreases by the factor 1/ e 2 0.135 at the radial distance p == W (z). Since 86% of the power is carried within a circle of radius W (z ), we regard W (z) as the beam radius (or beam width). The RMS width of the intensity distribution, on the other hand, is a == ! W (z) (see Appendix A, Sec. A.2, for the different definitions of width). The dependence of the beam width on z is governed by (3.1-8), W(z) = W o 1 + ( y. (3.1-18) Beam Width (Beam Radius) It assumes its minimum value, W o , at the plane z == O. This is the beam waist and W o is thus known as the waist radius. The waist diameter 2W o is also called the spot size. The beam width increases monotonically with z, and assumes the value .J2w o at z == ::f:zo (Fig. 3.1-3). - ........................ -- -- ........................ -- --- ....................... --- -- ............ ;";" ........................ -'..,.., e W(z) -- -- -- -- -- --- -- -- --- -3Z0 -2Z0 -Zo 0 Zo 2Z0 3Z0 Z Figure 3.1-3 The beam width W(z) assumes its minimum value W o at the beam waist (z == 0), reaches V2W o at z == :!:zo, and increases linearly with z for large z.
80 CHAPTER 3 BEAM OPTICS Beam Divergence For z » Zo the first term of (3.1-18) may be neglected, which results in the linear relation W o W(z) -z == eoz. Zo (3.1-19) As illustrated in Fig. 3.1-3, the beam then diverges as a cone of half-angle W o A eo == - == Zo 7rW o ' (3.1-20) where we have made use of (3.1-11). Approximately 86% of the beam power is con- fined within this cone, as indicated following (3.1-17). Rewriting (3.1-20) in terms of the spot size, the angular divergence of the beam becomes 4 A 2()o == - -. 7r 2W o (3.1-21) Divergence Angle The divergence angle is directly proportional to the wavelength A and inversely propor- tional to the spot size 2W o . Squeezing the spot size (beam-waist diameter) therefore leads to increased beam divergence. It is clear that a highly directional beam is con- structed by making use of a short wavelength and a thick beam waist. Depth of Focus Since the beam has its minimum width at z == 0, as shown in Fig. 3.1-3, it achieves its best focus at the plane z == O. In either direction, the beam gradually grows "out of focus." The axial distance within which the beam width is no greater than a factor J2 times its minimum value, so that its area is within a factor of 2 of the minimum, is known as the depth of focus or confocal parameter (Fig. 3.1-4). It is evident from (3.1-18) and (3.1-11) that the depth of focus is twice the Rayleigh range: 27rWc? 2zo == A . (3.1-22) Depth of Focus --- - ] : =-_c:= ::: ::: --------- - -- J - wo I __ I _ ---- I __ I -2Zo _=------------- 0 --------_______z 2Zo _-- _------------------- I ( 2zo >1 -------------------______ --- --- z Figure 3.1-4 Depth of focus of a Gaussian beam. The depth of focus is therefore directly proportional to the area of the beam at its waist, 7r WJ, and inversely proportional to the wavelength, A. A beam focused to a
3.1 THE GAUSSIAN BEAM 81 small spot size thus has a short depth of focus; locating the plane of focus thus requires increased accuracy. Small spot size and long depth of focus can be simultaneously attained only for short wavelengths. As an example, at Ao == 633 nm (a common He- Ne laser-line wavelength), a spot size 2W o == 2 cm corresponds to a depth of focus 2zo 1 km. A much smaller spot size of 20 J-lm corresponds to a much shorter depth of focus of 1 mm. Phase The phase of the Gaussian beam is, from (3.1- 7), k p 2 cp(p, z) = kz - ((z) + 2R(z) ' (3.1-23) On the beam axis (p == 0) the phase comprises two components: <p(0, z) == kz - ((z). (3.1-24) The first, kz, is the phase of a plane wave. The second represents a phase retardation ((z) given by (3.1-10), which ranges from -7r/2 at z == -00 to +7r/2 at z == 00, as illustrated in Fig. 3.1-5. This phase retardation corresponds to an excess delay of the wavefront in relation to a plane wave or a spherical wave (see also Fig. 3.1-8). The total accumulated excess retardation as the wave travels from z == -00 to z == 00 is 7r. This phenomenon is known as the Gouy effect. t - 2Z 0 (z) n/2 n/4 , . -3Z 0 2zo 3Z 0 z Figure 3.1-5 The function (( z) represents the phase retardation of the Gaussian beam relative to a uniform plane wave at points on the beam axis. Wavefronts The third component in (3.1-23) is responsible for wavefront bending. It represents the deviation of the phase at off-axis points in a given transverse plane from that at the axial point. The surfaces of constant phase satisfy k[z + p2/2R(z)] - ((z) == 27rq. Since ((z) and R(z) are relatively slowly varying functions, they are effectively constant at points within the beam width on each wavefront. We may therefore write z + p2 /2R qA + (A/27r, where R == R(z) and ( == ((z). This is the equation of a paraboloidal surface with radius of curvature R. Thus, R(z), plotted in Fig. 3.1-6, is the radius of curvature of the wavefront at position z along the beam axis. As illustrated in Fig. 3.1-6, the radius of curvature R( z) is infinite at z == 0, so that the wavefronts are planar, i.e., they have no curvature. The radius decreases to a minimum value of 2zo at z == Zo, where the wavefront has the greatest curvature (Fig. 3.1-7). The radius of curvature subsequently increases as z increases further until R(z) z for z » zoo The wavefronts are then approximately the same as those of a t See, for example, A. E. Siegman, Lasers, University Science, 1986.
82 CHAPTER 3 BEAM OPTICS R(z) 2Z0 Zo 2zo 3Zo z -3Zo -2Zo Figure 3.1-6 The radius of curvature R( z) of the wavefronts of a Gaussian beam as a function of position along the beam axis. The dashed line is the radius of curvature of a spherical wave. -2Zo xi o 2Zo z Figure 3.1-7 Wavefronts of a Gaussian beam. spherical wave. The pattern of the wavefronts is identical for negative z, except for a change in sign (Fig. 3.1-8). We have adopted the convention that a diverging wavefront has a positive radius of curvature whereas a converging wavefront has a negative radius of curvature. (a) ... z (b) z z Figure 3.1-8 Wavefronts of (a) a uniform plane wave; (b) a spherical wave; (c) a Gaus- sian beam. At points near the beam center, the Gaussian beam resembles a plane wave. ' At large z the beam behaves like a spherical wave except that its phase is retarded by 7r /2 (a quarter of the distance between two adjacent wavefronts). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . (c) Parameters Required to Characterize a Gaussian Beam Assuming that the wavelength A is known, how many parameters are required to describe a plane wave, a spherical wave, and a Gaussian beam? The plane wave is
3.1 THE GAUSSIAN BEAM 83 completely specified by its complex amplitude and direction. The spherical wave is specified by its complex amplitude and the location of its origin. The Gaussian beam, in contrast, requires more parameters for its characterization - its peak amplitude [determined by Ao in (3.1-7)], its direction (the beam axis), the location of its waist, and one additional parameter, such as the waist radius W o or the Rayleigh range zoo Thus, if the beam peak amplitude and the axis are known, two additional parameters are required for full specification. If the complex q-parameter, q( z) == z + j zo, is known, the distance to the beam . waist z and the Rayleigh range Zo are readily identified as the real and imaginary parts thereof. As an example, if q(z) is 3 + j4 cm at some point on the beam axis, we infer that the beam waist lies at a distance z == 3 cm to the left of that point and that the depth of focus is 2zo == 8 cm. The waist radius W o may then be determined via (3.1-11). The quantity q(z) is therefore sufficient for characterizing a Gaussian beam of known peak amplitude and beam axis. Given q( z) at a single point, the linear dependence of q on z permits it to be determined at all points: if q( z) == ql and q( z + d) == q2, then q2 == ql + d. Using the example provided immediately above, at z == 13 cm it is evident that q == 13 + j4. If the beam width W (z) and the radius of curvature R( z) are known at an arbitrary point on the beam axis, the beam can be fully identified by solving (3.1-8), (3.1-9), and (3.1-11) for z, zo, and WOe Alternatively, the beam can be identified by determining q(z) from W(z) and R(z) using (3.1-6). Summary: Properties of the Gaussian Beam at Special Locations . At the location z == Zo. At an axial distance Zo from the beam waist, the wave has the following properties: - The intensity on the beam axis is ! the peak intensity. - The beam width is a factor of vI2 greater than the width at the beam waist, and the beam area is larger by a factor of 2. - The phase on the beam axis is retarded by an angle 1r / 4 relative to the phase of a plane wave. - The radius of curvature of the wavefront achieves its minimum value, R == 2zo, so that the wavefront has the greatest curvature. . Near the beam center. At locations for which Iz( « Zo and p W o , the quantity exp[-p2 /W 2 (z)] exp( _p2 /W6) 1, so that the beam intensity, which is proportional to the square of this quantity, is approximately constant. Also, R(z) z5/ z and «z) 0, so that the phase k[z + p2 /2R(z)] kz(l + p2 / 2z 5) kz, by virtue of (3.1-11) when Zo A. The Gaussian beam may therefore be approximated near its center by a plane wave. . Far from the beam waist. At transverse locations within the waist radus (p < W o ), but far from the beam waist (z :» zo), the wave behaves approximately like a spherical wave. In this domain W (z) Woz / Zo » W o and p < W o , so that exp[ - p2 /W 2 (z)] 1 and the beam intensity is approximately uniform. Since R(z) z in this regime, the wavefronts are approximately spherical. Thus, except for the Gouy phase retardation «z) 1r/2, the complex ampli- tude of the Gaussian beam approaches that of the paraboloidal wave, which in turn approaches that of the spherical wave in the paraxial approximation.
84 CHAPTER 3 BEAM OPTICS EXERCISE 3.1-1 Parameters of a Gaussian Laser Beam. A l-mW He-Ne laser produces a Gaussian beam at a wavelength of A = 633 nm with a spot size 2fV o = 0.1 mm. (a) Determine the angular divergence of the beam, its depth of focus, and its diameter at z = 3.5 X 10 5 km (approximately the distance to the moon). (b) What is the radius of curvature of the wavefront at z = 0, z = Zo, and z = 2zo? (c) What is the optical intensity (in W/cm 2 ) at the beam center (z = 0, p = 0) and at the axial point z = zo? Compare this with the intensity at z = Zo of a 100- W spherical wave produced by a small isotropically emitting light source located at z = o. EXERCISE 3.1-2 Validity of the Paraxial Approximation for a Gaussian Beam. The complex envelope A(r) of a Gaussian beam is an exact solution of the paraxial Helmholtz equation (3.1-2), but its corresponding complex amplitude U(r) = A(r) exp( -jkz) is only an approximate solution of the Helmholtz equation (2.2-7). This is because the paraxial Helmholtz equation is itself approximate. The approximation is satisfactory if the condition (2.2-21) is satisfied. Show that if the divergence angle eo of a Gaussian beam is small (eo « 1), the necessary condition (2.2-21) for the validity of the paraxial Helmholtz equation is indeed satisfied. EXERCISE 3.1-3 [Determination of a Beam with Given Width and Curvature. Consider a Gaussian beam whose width TiT and radius of curvature R are known at a particular point on the beam axis (Fig. 3.1- 9). Show that the beam waist is located to the left at a distance R z= 1 + (AR/7rV2)2 (3.1-25) and that the waist radius is W o = V I + (7r TT2 / AR)2 TT (3.1-26) ;; ;; ;; ;; ;; w [% R ---........ -- -- Ie :- -------.1 Figure 3.1-9 Given TV and R, determine z and TV o . EXERCISE 3.1-4 Determination of the Width and Curvature at One Point Given the Width and Curva- ture at Another Point. Assume that the width and radius of curvature of a Gaussian beam of wavelength A = 1 J-Lm at some point on the beam axis are fTl = 1 mm and Rl = 1 m, respectively (Fig. 3.1-10). Determine the beam width Vl T 2 and radius of curvature R 2 at a distance d = 10 em to the right. ...... ...... ......... --- --- WI W2 -_....-- Rl R2 --....... --- --- -- -; Figure 3.1-10 Given fV 1 , Rl and d, determine V2 and R 2 .
3.1 THE GAUSSIAN BEAM 85 EXERCISE 3.1-5 Identification of a Beam with Known Curvatures at Two Points. A Gaussian beam has radii of curvature Rl and R 2 at two points on the beam axis separated by a distance d, as illustrated in Fig. 3.1-11. Verify that the location of the beam center and its depth of focus may be determined from the relations -d(R 2 - d) Zl == R 2 - Rl - 2d 2 -d (R 1 + d)(R 2 - d)(R 2 - Rl - d) Zo == (R 2 - Rl - 2d)2 (3.1-27) (3.1-28) W o = J >';0 . _2 -- _..---- .1 --- -- -- ---- --- R2 r:-;I Figure 3.1-11 Given R 1 , R 2 , and d, determine Zl, Z2, Zo, and Woo C. Beam Quality The Gaussian beam is an idealization that is only approximately met, even in well- designed laser systems. A measure of the quality of an optical beam is the deviation of its profile from Gaussian form. For a beam of waist diameter 2ff l m and angular divergence 20m, a useful numerical measure of the beam quality is provided by the M 2 _ factor, which is defined as the ratio of the waist-diameter-divergence product, 2V m . 20m (usually measured in units of mm.mrad), to that expected for a Gaussian beam, which is 21V o .20 0 == 4)../7r. Thus, M 2 == 2V m . 20m 4)../7r . (3.1-29) If the two beams have the same waist diameter, the M 2 -factor is simply the ratio of their angular divergences, M 2 == Om/O O , (3.1-30) where 0 0 == )../7r1V o == )../7rW m [see (3.1-21)]. Since the Gaussian beam enjoys the smallest possible divergence angle of all beams with the same waist diameter, M 2 > 1. The specification of the M 2 -factor of an optical beam thus signifies a divergence angle that is M 2 times greater than that of a Gaussian beam of the same waist diameter. Optical beams produced by commonly available Helium-Neon lasers usually ex- hibit M 2 < 1.1. For ion lasers, M 2 is typically in the range 1.1-1.3. Collimated TEMoo diode-laser beams usually exhibit M 2 1.1-1.7, whereas high-energy mul- timode lasers display M 2 factors as high as 3 or 4. For an optical beam that is approximately Gaussian, the M 2 -factor may be de- termined by making use of a charge-coupled device (CCD) camera to measure the
86 CHAPTER 3 BEAM OPTICS intensity profile of the beam at various locations along the axis of the beam. The beam is focused, by a high-quality lens with a long focal length and large P#, to a size that is roughly the same as that of the CCD array. First, the beam center is located by finding the plane at which the spot size is minimized; the waist diameter 2TV m is then measured. The axial distance from the beam center to the plane at which the beam diameter increases by a factor of V2 provides the Rayleigh range Zm. An estimate of the a ngular divergence 2e m is obtained by using the Gaussian-beam relation em == V >"'/1rZm, which is obtained from (3.1-11) and (3.1-20). Finally, the M 2 -factor is computed by means of (3.1-29). 3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS We proceed now to a discussion of the effects of various optical components on a Gaussian beam. We demonstrate that if a Gaussian beam is transmitted through a set of circularly symmetric optical components aligned with the beam axis, the Gaussian beam remains a Gaussian beam, provided that the overall system maintains the parax- ial nature of the wave. The beam is reshaped, however - its waist and curvature are altered. The results of this section are of importance in the design of optical instruments that rely on Gaussian beams. A. Transmission Through a Thin Lens The complex amplitude transmittance of a thin lens of focal length f is proportional to exp(jkp2/2f) [see (2.4-9)]. When a Gaussian beam traverses such a component, its complex amplitude, given in (3.1-7), is multiplied by this phase factor. As a result, although the beam width is not altered (TTT' == TT)'I the wavefront is. To be specific consider a Gaussian beam centered at Z == 0, with waist radius lV o , transmitted through a thin lens located at position z, as illustrated in Fig. 3.2-1. The phase of the incident wave at the plane of the lens is kz + k p 2 /2R - (, as prescribed by (3.1-23), where R == R(z) and ( == ((z) are given in (3.1-9) and (3.1-10), respectively. The phase of the emerging wave therefore becomes p2 p2 p2 kz + k 2R - ( - k 21 = kz + k 2R' - (, (3.2-1) where 1 R' 1 1 --- R f (3.2-2) We conclude that the transmitted wave is itself a Gaussian beam with width Ml' == IT! and radius of curvature R', where R' satisfies the imaging equation 1/ R -1 / R' == 1/ f. The sign of R is positive since the wavefront of the incident beam is diverging whereas the opposite is true of R'. The parameters of the emerging beam are determined by referring to the outcome of Exercise 3.1-3, in which the parameters of a Gaussian beam are determined from its width and curvature at a given point. Equation (3.1-26) provides that the waist radius IS TT TV == V I + (7rW 2 /AR,)2 (3.2-3)
3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS 87 z .1. z z Figure 3.2-1 Transmission of a Gaussian beam through a thin lens. whereas (3.1-25) provides that the beam center is located at a distance from the lens given by R' - z' == 1 + (,XR' /7rW2)2 . (3.2-4) The minus sign in (3.2- 4) indicates that the beam waist lies to the right of the lens. Substituting W == Wo V I + (z/zO)2 and R == z[I + (zO/z)2] from (3.1-8) and (3.1-9) into (3.2-2) to (3.2-4) yields a set of formulas that relate the unprimed parameters of the Gaussian beam incident on the lens to the primed parameters of the Gaussian beam that emerges from the lens, as represented in Fig. 3.2-1: Waist radius W == MW o (3.2-5) Waist location (z' - f) == M 2 (z - f) (3.2-6) Depth of focus 2zb == M2( 2z 0) (3.2-7) Divergence angle 20 = 20 0 (3.2-8) M Magnification M== M r (3.2-9) v I + r 2 Zo M r == f (3.2-9a) r== z - f' z-f Parameter Transformation by a Lens The magnification factor M evidently plays an important role. The waist radius is magnified by M, the depth of focus is magnified by M 2 , and the divergence angle is minified by M. Limit of Ray Optics Consider the limiting case in which (z - f) » Zo, so that the lens is well outside the depth of focus of the incident beam (Fig. 3.2-2). The beam may then be approximated by a spherical wave, and, in accordance with (3.2-9) and (3.2-9a), r « 1 so that M Mr. In this case (3.2-5)-(3.2-9a) reduce to W MW o 111 -+-- z, z f (3.2-10) (3.2-11)
88 CHAPTER 3 BEAM OPTICS f M M r == z-f (3.2-12) Equations (3.2-10)-(3.2-12) are precisely the relations provided by ray optics for the location and size of a patch of light of diameter 2W o located at a distance z to the left of a thin lens (see Sec. 1.2C). Indeed, the magnification factor M r is identically that based on ray optics. Since (3.2-9) provides that M < M r , the maximum Gaussian-beam magnification attainable is the ray-optics limit Mr. As r 2 increases, the magnification is reduced and the deviation from ray optics widens. Equations (3.2-10)-(3.2-12) also correspond to the results obtained from wave optics for the focusing of a spherical wave in the paraxial approximation (see Sec. 2.4B). z -T- z'1 Figure 3.2-2 Beam imaging in the ray-optics limit. B. Beam Shaping A lens, or sequence of lenses, may be used to reshape a Gaussian beam without compromising its Gaussian nature. Of course, graded-index components can serve this purpose as well. Beam Focusing For a lens placed at the waist of a Gaussian beam, as illustrated in Fig. 3.2-3, the appropriate parameter-transformation formulas are obtained by simply substituting z == 0 in (3.2-5) to (3.2-9a). The transmitted beam is then focused to a waist radius W at a distance z' given by H/:' _ W o o - V I + (zo/ f)2 , f z == - 1 + (f / zo) 2 . (3.2-13) (3.2-14) In the special case when the depth of focus of the incident beam 2zo is much longer than the focal length f of the lens, as illustrated in Fig. 3.2-4, (3.2-13) reduces to W6 (f / zo)W o . Using Zo == 7rWc? / A from (3.1-11), along with (3.1-20), then leads to the simple result , A W o f == Oof 7rW o z' f. (3.2-15) (3.2-16)
3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS 89 Zo z' ---.J 2w o l I 1-1-1 Figure 3.2-3 Focusing a Gaussian beam with a lens at the beam waist. The transmitted beam is then focused in the focal plane of the lens as would be expected for a collimated beam of parallel rays impinging on the lens. This result emerges because, at its waist, the incident Gaussian beam is well approximated by a plane wave. Wave optics provides that the focused waist radius W6 is directly proportional to the wavelength and the focal length, and inversely proportional to the radius of the incident beam. The spot size expected from ray optics is, of course, zero, a result that is indeed obtained from the wave-optics formulas as A o. Zo » f Figure 3.2-4 Focusing a colli- mated beam. In many applications, such as laser scanning, laser printing, compact-disc (CD) burning, and laser fusion, it is desired to generate the smallest possible spot size. It is clear from (3.2-15) that this is achieved by making use of the shortest possible wavelength, the thickest incident beam, and the shortest focal-length lens. Since the lens must intercept the incident beam, its diameter D should be at least 2W o . Taking D == 2W o , and making use of (3.2-15), the diameter of the focused spot is given by , 4 2W o -AF# 7r f F# == D ' (3.2-17) Focused Spot Size where F # is the F-number of the lens. A microscope objective with small F-number is often used for this purpose. A caveat is in order: since (3.2-15) and (3.2-16) are approximate their validity must always be confirmed before use. EXERCISE 3.2-1 Beam Relaying. A Gaussian beam of radius W o and wavelength A is repeatedly focused by a sequence of identical lenses, each of focal length 1 and separated by a distance d (Fig. 3.2-5). The focused waist radius is equal to the incident waist radius, i.e., W = Woo Using (3.2-6), (3.2-9), and (3.2-9a) show that this condition can arise only if the inequality d < 41 is satisfied. Note that this is the same as the ray-confinement condition for a sequence of lenses derived in Example 1.4-1 using ray optics.
90 CHAPTER 3 BEAM OPTICS .... 1\ fl\ d -+-1 J\ J\ /\ z Figure 3.2-5 Beam relaying. EXERCISE 3.2-2 Beam Collimation. A Gaussian beam is transmitted through a thin lens of focal length I. (a) Show that the locations of the waists of the incident and transmitted beams, z and z', respectively, are related by z' z / I - 1 - -1 = I (z/ I - 1)2 + (zo/I)2 . (3.2-18) This relation is plotted in Fig. 3.2-6. -2 , -] 1 Zo , -=0 ,I \ 0.25 2 -1 1 Figure 3.2-6 Relation between the waist locations of the incident and transmitted beams. (b) The beam is collimated by making the location of the new waist z' as distant as possible from the lens. This is achieved by using the smallest possible ratio zo/ I (short depth of focus and long focal length). For a given ratio zo/ I, show that the optimal value of z for collimation is z = I + zoo (c) Given A = 1 /-Lm, Zo = 1 cm, and I = 50 cm, determine the optimal value of z for collimation, and the corresponding magnification M, distance z', and width W of the collimated beam. EXERCISE 3.2-3 Beam Expansion. A Gaussian beam may be expanded and collimated by using two lenses of focal lengths 11 and 12, as illustrated in Fig. 3.2-7. Parameters of the initial beam (W o , zo) are modified by the first lens to (W', zg) and subsequently altered by the second lens to (W, zb). The first lens, which has a short focal length, serves to reduce the depth of focus 2zg of the beam. This prepares it for collimation by the second lens, which has a long focal length. The system functions as an inverse Keplerian telescope.
3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS 91 I z d--.J c Zl I+- Z2 ---.t Z' I 2W O 2' o Figure 3.2-7 Beam expansion using a two-lens system. (a) Assuming that 11 « z and z - 11 » Zo, use the results of Exercise 3.2-2 to determine the optical distance d between the lenses such that the distance z' to the waist of the final beam is as large as possible. (b) Determine an expression for the overall magnification M == W /W o of the system. C. Reflection from a Spherical Mirror We now examine the reflection of a Gaussian beam from a spherical mirror. The complex amplitude reflectance of the mirror is proportional to exp( - j k p 2 / R) (see Prob. 2.4-10), where by convention R > 0 for convex mirrors and R < 0 for concave mirrors. The action of the mirror on a Gaussian beam of width WI and radius of curvature R I is therefore to reflect the beam and to modify its phase by the factor -k p 2 / R, while leaving the beam width unaltered. The reflected beam therefore re- mains Gaussian, with parameters W 2 and R 2 given by W 2 == WI 1 1 2 R 2 == R I + R . Equation (3.2-20) is identical to (3.2-2) provided f - R/2. Thus, the Gaussian beam is modified in precisely the same way as it is by a lens, except for a reversal of the direction of propagation. Three special cases, illustrated in Fig. 3.2-8, are of interest: (3.2-19) (3.2-20) I --- ..--- ... ...... ...... '---- ...... ......... .......... -------; -- -- --- ... -............... ,,--- .... ........ (a) (b) (c) Figure 3.2-8 Reflection of a Gaussian beam with radius of curvature R 1 from a mirror with radius of curvature R: (a) R == 00; (b) R 1 == 00; (c) R 1 == - R. The dashed curves show the effects of replacing the mirror by a lens of focal length I == - R/2.
92 CHAPTER 3 BEAM OPTICS . If the mirror is planar, i.e., R == 00, then R 2 == R 1 , so that the mirror reverses the direction of the beam without altering its curvature, as illustrated in Fig. 3 .2-8( a). . If Rl == 00, i.e., if the beam waist lies on the mirror, then R 2 == R/2. If the mirror is concave (R < 0), R 2 < 0 so that the reflected beam acquires a negative curvature and the wavefronts converge. The mirror then focuses the beam to a smaller spot size, as illustrated in Fig. 3.2-8(b). . If Rl == - R, Le., if the incident beam has the same curvature as the mirror, then R 2 == R. The wavefronts of both the incident and reflected waves then coincide with the mirror and the wave retraces its path as shown in Fig. 3 .2-8( c). This is expected since the wavefront normals are also normal to the mirror so that the mirror reflects the wave back onto itself. In the illustration in Fig. 3.2-8(c) the mirror is concave (R < 0); the incident wave is diverging (R 1 > 0) and the reflected wave is converging (R 2 < 0). EXERCISE 3.2-4 Variable-Reflectance Mirrors. A spherical mirror of radius R has a variable intensity re- flectance characterized by (p) == Ir(p)12 == exp( -2 p 2 /W), which is a Gaussian function of the radial distance p. The reflectance is unity on axis and falls by a factor 1/ e 2 when p == W m" Determine the effect of the mirror on a Gaussian beam with radius of curvature R I and beam width WI at the mIrror. *D. Transmission Through an Arbitrary Optical System In the paraxial ray-optics approximation, an optical system is completely characterized by the 2 x 2 ray-transfer matrix relating the position and inclination of the transmitted ray to those of the incident ray (see Sec. 1.4). We now consider how an arbitrary paraxial optical system, characterized by a matrix M of elements (A, B, C, D), modifies a Gaussian beam (Fig. 3.2-9). [ :] Figure 3.2-9 Modification of a Gaussian beam by an arbitrary paraxial system described by an ABCD matrix. The ABCD Law The q-parameters, ql and q2, of the incident and transmitted Gaussian beams at the input and output planes of a paraxial optical system described by the (A, B, C, D) matrix are related by Aql + B q2 == Cql + D. (3.2-21) The ABCD Law
3.2 TRANSMISSION THROUGH OPTICAL COMPONENTS 93 Because the complex q parameter identifies the width T and radius of curvature R of the Gaussian beam (see Exercise 3.1-3), this simple expression, called the ABCD law, governs the effect of an arbitrary paraxial system on a Gaussian beam. The ABCD law will be established by verification in special cases; its generality will ultimately be proved by induction. Transmission Through Free Space When the optical system is a distance d of free space (or of any homogeneous medium), the elements of the ray-transfer matrix M are A == 1, B == d, C == 0, D == 1 [see (1.4-4)]. Since it has been established earlier that q == z + j Zo in free space, the q-parameter is modified by the optical system in accordance with q2 == ql + d. This is, in fact, is equal to (1 . ql + d) / (0 . ql + 1) so that the ABCD law is seen to apply. Transmission Through a Thin Optical Component An arbitrary thin optical component does not affect the ray position so that Y2 == YI, (3.2-22) but does alter the inclination angle in accordance with ()2 == CYI + D()l, (3.2-23) as illustrated in Fig. 3.2-10. Thus, A == 1 and B == 0, but C and D are arbitrary. However, in all of the thin optical components described in Sec. 1.4B, D == nl/n2. By virtue of the vanishing thickness of the component, the beam width does not change, I.e., W 2 == WI. (3.2-24) Moreover, if the beams at the input and output planes of the component are ap- proximated by spherical waves of radii RI and R 2 , respectively, then in the paraxial approximation, when ()l and ()2 are small, ()l YI/ RI and e 2 Y2/ R 2 . Substituting these expressions into (3.2-23), with the help of (3.2-22) we obtain 1 D -==C+-. R 2 RI (3.2-25) Using (3.1-6), which is the expression for q as a function of Rand W, and noting that D == nl/n2 == A2/ AI, (3.2-24) and (3.2-25) can be combined into a single equation, 1 D -==C+-, q2 ql (3.2- 26) from which q2 == (1 . ql + 0) / (Cql + D), so that the ABC D law again applies. Invariance of the ABCD Law to Cascading If the ABCD law is applicable to each of two optical systems with matrices M i (Ai, B i , C i , D i ), i == 1,2, it must also apply to a system comprising their cascade (a system with matrix M == M 2 M 1 ). This may be shown by straightforward substitution.
94 CHAPTER 3 BEAM OPTICS Rl /- -- --:--=-\\ _-:::--- Yl ---- -::::-- ....-:- () .- I R2 (---- H- -_ _ Y ---- 2 --_ -_ ---- --.:::::--- - - :-.. () 2 ::::: ::::: -- ..... Optical component z Figure 3.2-10 Modification of a Gaussian beam by a thin optical component. GenemfflyofeABCDLaw Since the ABCD law applies to thin optical components as well as to propagation in a homogeneous medium, it also applies to any combination thereof. All of the paraxial optical systems of interest are combinations of propagation in homogeneous media and thin optical components such as thin lenses and mirrors. It is therefore apparent that the ABCD law is applicable to all of these systems. Furthermore, since an inhomogeneous continuously varying medium may be regarded as a cascade of incremental thin elements followed by incremental distances, we conclude that the ABCD law applies to these systems as well, provided that all rays (wavefront normals) remain paraxial. EXERCISE 3.2-5 Transmission of a Gaussian Beam Through a Transparent Plate. Use the ABCD law to examine the transmission of a Gaussian beam from air, through a transparent plate of refractive index n and thickness d, and again into air. Assume that the beam axis is normal to the plate. 3.3 HERMITE-GAUSSIAN BEAMS The Gaussian beam is not the only beam-like solution of the paraxial Helmholtz equa- tion (3.1-2). Of particular interest are solutions that exhibit non-Gaussian intensity distributions but share the paraboloidal wave fronts of the Gaussian beam. Such beams have the salutary feature of being able to match the curvatures of spherical mirrors of large radius, such as those that form an optical resonator, and reflect between them without being altered. Such self-reproducing waves are called the modes of the res- onator. The optics of resonators is discussed in Chapter 9. Consider a Gaussian beam of complex envelope [see (3.1-5)] Al [ . x 2 + y2 ] Ac(x, y, z) = q(z) exp -Jk 2q(z) , (3.3-1) where q( z) == z + j zoo Expressions for the beam width W (z) and the wavefront radius of curvature R(z) are provided in (3.1-8) and (3.1-9), respectively. Now consider a second wave whose complex envelope is a modulated version of the Gaussian beam, A(x,y,z) = X[J2 wZ) ] 1J[J2 z) ] exp[jZ(z)] Ac(x,y,z), (3.3-2)
3.3 HERMITE-GAUSSIAN BEAMS 95 where X(.), }j (.), and Z(.) are real functions. This wave, should it be shown to exist, has the following two properties: 1. The phase is the same as that of the underlying Gaussian wave, except for an excess phase Z(z) that is independent of x and y. If Z(z) is a slowly varying function of z, both waves have paraboloidal wavefronts with the same radius of curvature R( z). These two waves are therefore focused by thin lenses and mirrors in precisely the same manner. 2. The magnitude [ X ] [ Y ] [ W o ] [ X2 + y2 ] AoX V2 V(z) 11 V2 TV (z) W(z) exp - W2(Z) · (3.3-3) where Ao == At/jzo, is a function of X/fT(Z) and y/TTT(Z) whose widths in the x and y directions vary with z in accordance with the same scaling factor fT T (z ). As z increases, the intensity distribution in the transverse plane remains fixed, except for a magnification factor TV (z). This distribution is a Gaussian function modulated in the x and y directions by the functions X 2 ( .) and }j 2 ( . ), respectively. The modulated wave therefore represents a beam of non-Gaussian intensity distri- bution, but it shares the same wavefronts and angular divergence as the underlying Gaussian wave. The existence of this wave is assured if three real functions X(.), }j (.), and Z(z) can be found such that (3.3-2) satisfies the paraxial Helmholtz equation (3.1-2). Substitut- ing (3.3-2) into (3.1-2).. using the fact that Ac itself satisfies (3.1-2), and defining two new variables u == V2 x/TV (z) and v == V2 y /W (z), we obtain 1 ( 82X 8X ) 1 ( 8 2 }j 8}j ) T2 8Z_ X au 2 - 2'11 au + 11 av2 - 2v av + kH (z) az - O. (3.3-4) Since the left-hand side of this equation is the sum of three terms, each of which is a function of a single independent variable, u, v, and z, respectively, each of these terms must be constant. Equating the first term to the constant - 2!-Ll and the second to - 2!-L2, the third must be equal to 2(!-Ll + !-L2). This technique of "separation of variables" per- mits us to reduce the partial differential equation (3.3-4) into three ordinary differential equations, for X ( u), }j ( v), and Z( z), respectively: 1 d 2 X dX - 2 du 2 + u du == !-LIX 1 d 2 }j d}j - 2 dv 2 + v dv == !-L2}j Zo [1+ ( )2] =Ml+M2. (3.3-5a) (3.3-5b) (3.3-5c) where we have used the expression fT7(Z) given in (3.1-8) and (3.1-11). Equation (3.3-5a) represents an eigenvalue problem whose eigenvalues are !-Ll == l, where l == 0,1,2,... and whose eigenfunctions are the Hermite polynomials X(u) == IHIl( u), l == 0, 1,2, . . .. These polynomials are defined by the recurrence relation IHIl+l (u) == 2u IHIl (u) - 2l IHI l - 1 (u) (3.3-6)
96 CHAPTER 3 BEAM OPTICS with IHIo ( u) == 1, IHI1(U) == 2u. (3.3-7) Thus, IHI2(U) == 4u 2 - 2, IHI3 (u) == 8u 3 - 12u, (3.3-8) Similarly, the solutions of (3.3-5b) are JL2 == m and (v) == IHIm(v), where m == 0,1,2, . ... There is therefore a family of solutions labeled by the indexes (l, m). Substituting JL1 == land JL2 == m in (3.3-5c), and integrating, we obtain Z(z) == (l + m) «(z), (3.3-9) where «(z) == tan- 1 (z/ zo). The excess phase Z(z) thus varies slowly between -(l + m) 7r /2 and +(l + m) 7r /2, as z varies between -00 and 00 (see (3.1-10) and Fig. 3.1- 5). Complex Amplitude Finally, substitution into (3.3-2) yields an expression for the complex envelope of the beam labeled by the indexes (l, m). Rearranging terms and multiplying by exp( -jkz) provides the complex amplitude [ W o ] [ V2 x ] [ V2 y ] Ul,m(X, y, z)= Al,m W(z) Gl W(z) G m W(z) x exp [-jkZ - jk X;R2 + j(l + m + 1) ((z)] (3.3-10) Hermite- Gaussian Beam where Gl(U) = JHIl(U) ex p ( ; 2), l == 0,1,2,... (3.3-11) is known as the Hermite-Gaussian function of order l, and Al m is a constant. , Since IHIo( u) == 1, the Hermite-Gaussian function of order 0 is simply the Gaussian function. Continuing to higher order, <G 1 (u) == 2u exp( -u 2 /2) is an odd function, G 2 (u) == (4u 2 - 2) exp( -u 2 /2) is even, <G 3 (u) == (8u 3 - 12u) exp(-u 2 /2) is odd, and so on. These functions are displayed schematically in Fig. 3.3-1. Go( u) t Gl(U) G2(U) G3(U) U U u Figure 3.3-1 Low-order Hermite-Gaussian functions: (a) G o ( u); (b) G 1 (u); (c) G 2 (u); (d) G 3 (u).
3.4 LAGUERRE-GAUSSIAN AND BESSEL BEAMS 97 An optical wavc \vith conlplex anlplitude given by (3.3-10) is known as the Hernlite-Gaussian bean1 of order (1. Ill). The Hennite-Gaussian beanl of order (0.0) is the silllple Gaussian bealll. Intensity Distribution The optical intensity of the (1. In) Hernlite-Gaussian beam 1 1 . n1 == IUI.11I1:2 is given by [ fTT ] :2 [ 0 ] [ 0 ] .) () .) V.!...r ,) V L. Y hilI (.r. !J. ) = 1 A , . III 1- Ie (-:; ) <G, II' ( .:; ) <G II n- ( .:; ) . (3.3-12) Figure 3.3-2 illustrates the dependence of the intensity on the nonnalized transverse distances II == J2.r/1\ (:) and I' == J2 Ij/1TT(z) for several values of 1 and Ill. Beams of higher order have larger widths than those of lower order as is evident from Fig. 3.3- 1. Regardless of the order however the width of the beanl is proportional to 1 T T (z) so that as -: increases the spatial extent of the intensity pattern is magni fled by the factor 1T(:)/1\o hut otherwise nlaintains its profile. The only circularly symmetric 1l1elllber alllong the falllily of I-Iernlite-Gaussian beanls is the eleJllentary Gaussian beanl itself. (0.0) (0.1) (0.2) (1,1) (1,2) (2,2) Figure 3.3-2 Intensity distrihutions of several lo\v-order Hernlite-Gaussian bealns in the transverse plane. The order (1. Ill) is indicated in each case. EXERCISE 3.3-1 The Donut Beam. Consider a wave that is a superposition of two Hennite-Gaussian bealns of orders (1.0) and (0. I) \vith equal intensities. The two beanls have independent and randOlll phases so that their intensities add with no interference. Show that the total intensity is described by a donut- shaped circularly synlnletric function. Assunling that H() = 1 mIn. detennine the radius of the circle of peak intensity and the radii of the two circles of 1/(''2 times the peak intensity at the beanl waist. *3.4 LAGUERRE-GAUSSIAN AND BESSEL BEAMS Laguerre-Gaussian Beams The Hernlite-Gaussian beallls fornl a cOlllplete set of solutions to the paraxial l-Ielnlholtz equation. Any other solution can be written as a superposition of these beanls. An alternate conlplete set of solutions, known as Laguerre-Gaussian beams, is obtained by writing the paraxial Helmholtz equation in cylindrical coordindte (p. 6. :) and then using the separation-of-variables technique in p and 9, rather than in .1' and .lJ.
98 CHAPTER 3 BEAM OPTICS The complex amplitude of the Laguerre-Gaussian beam is UI,m(P, 4Y, z) = Al,m [ :;:) ] ( z) Y L ( :;(2Z) ) ex p ( - :zJ x eXp[-jkZ-jk 2Z) -j Z 4Y+j(Z+2m+l)((Z)], (3.4-1) where IL ( .) is the generalized Laguerre polynomial function, t and fV (z ), R( z ), (( z ), and Wo are given by (3.1-8)-(3.1-11). The lowest-order Laguerre-Gaussian beam (l == m == 0) is again the Gaussian beam. The intensity of the Laguerre-Gaussian beam is a function of p and z, so that it is circularly symmetric. For l f= 0, the beam has zero intensity at the center (p == 0) and an annular intensity pattern. The phase has the same dependence on p and z as the Gaussian beam, but has an additional term proportional to the azimuthal angle cp, and also a Gouy phase that is greater by the factor (l + 2m + 1). Because of the linear dependence of the phase on cp (for l f= 0) the wavefront tilts helically as the wave travels in the z direction, as illustrated in Fig. 3.4-1. Beams with such spiral phase are of interest since they carry orbital angular momentum (see Secs. 5.1 and 12.1D) that can impart torque to the illuminated system. y y x x --Y---t z (a) Intensity (b) Wavefront Figure 3.4-1 Intensity distribution and wavefront of a Laguerre-Gaussian beam with I == 1. Bessel Beams and Bessel-Gaussian Beams In the search for beam-like waves, it is natural to attempt to construct waves whose wavefronts are planar but whose intensity distributions are nonuniform in the trans- verse plane. Consider, for example, a wave with complex amplitude U (r) == A(x, y) e- j {3z. (3.4-2) In order that this wave satisfy the Helmholtz equation (2.2-7), V' 2 U + k 2 U == 0, the quantity A( x, y) must satisfy V'A + kA == 0, (3.4-3) t The generalized Laguerre polynomials are defined by Rodrigues' formula lL (x) (x-lex 1m!) (d m Idxm)(xl+me- X ). For example, lLb(x) = 1; lL?(x) = 1 - x; lL(x) = 1 - 2x + x 2 /2.
3.4 LAGUERRE-GAUSSIAN AND BESSEL BEAMS 99 where k + {32 == k 2 and \7 == 8 2 /8x 2 + 8 2 /8 y 2 is the transverse Laplacian operator. Equation (3.4-3), known as the two-dimensional Helmholtz equation, may be solved by employing the method of separation of variables. Using polar coordinates (x == P cos cp, Y == P sin cp), the result turns out to be A(x, y) == Am Jrn(kTP) e jm 4>, m == 0, :1::1, :1::2,..., (3.4-4) where J rn ( .) is the Bessel function of the first kind and mth order, and Am is a constant. Solutions of (3.4-4) that are singular at P == 0 are not included. For m == 0, the wave has a complex amplitude U(r) == Ao JO(kTP) e- j {3z (3.4-5) and therefore has planar wavefronts. The wvefront normals (rays) are all parallel to the z axis. The intensity distribution I(p, cp, z) == IAoI2J5(kTP) is circularly symmetric, varies with P as illustrated in Fig. 3.4-2, and is independent of z, so that there is no spread of the optical power. This wave is called the Bessel beam. p z Figure 3.4-2 The intensity distribu- tion of the Bessel beam in the transverse plane is independent of z; the beam does not diverge. It is useful to compare the Bessel beam with the Gaussian beam. Whereas the complex amplitude of the Bessel beam is an exact solution of the Helmholtz equation, the complex amplitude of the Gaussian beam is only an approximate solution thereof (its complex envelope is an exact solution of the paraxial Helmholtz equation). The intensity distributions of these two beams are compared graphically in Fig. 3.4-3. It is apparent that the asymptotic behavior of these distributions in the limit of large radial distances is significantly different. The intensity of the Gaussian beam decreases expo- nentially with P as exp[ - 2p 2 /W 2 (z)]. The intensity of the Bessel beam, on the other hand, decreases as J5(k T P) (2/1rk T P) cos 2 (kTP - 1r/4), which is an oscillatory function superimposed on a slow inverse-power-Iaw decay with p. As a consequence, the transverse RMS width of the Gaussian beam, (J" == W(z), is finite, while the transverse RMS width of the Bessel beam is infinite for all z (see Appendix A, Sec. A.2 for the definition ofRMS width), and the beam carries infinite power. Evidently there is a tradeoff between minimum beam size and divergence; although the divergence of the Bessel beam is zero, its RMS width is infinite. Whereas the generation of Bessel beams requires special schemes, t Gaussian beams are the modes of spherical resonators and are therefore created naturally by lasers that make use of such resonators. Yet another class of beams are Bessel-Gaussian beams,:!: which are Bessel beams modulated by a Gaussian function of the radial coordinate p. The Gaussian serves as a windowing function that accelerates the slow radial decay of the Bessel beam. t See P. w. Milonni and J. H. Eberly, Lasers, Wiley, 1988, Sec. 14.14. :f: See F. Gori, G. Guattari, and C. Padovani, Bessel-Gauss Beams, Optics Communications, vol. 64, pp. 491- 495, 1987.
100 CHAPTER 3 BEAM OPTICS I Figure 3.4-3 Comparison of the ra- dial intensity distributions of a Gaussian beam and a Bessel beam. Parameters are selected such that the peak intensities and 1/ e 2 widths are identical in both P cases. READING LIST Books See also the books on lasers in Chapter 15. F. M. Dickey, S. C. Holswade, and D. L. Shealy, Laser Beam Shaping Applications, CRC Press, 2006. F. M. Dickey and S. C. Holswade, eds., Laser Bealn Shaping: Theory and Techniques, Marcel Dekker, 2000. P. F. Goldsmith, Quasioptical Systems: Gaussian Beam Quasioptical Propagation and Applications, Wiley, 1998. A. N. Oraevskiy, Gaussian Beams and Optical Resonators, Nova Science, 1996. J. A. Arnaud, Beam and Fiber Optics, Academic Press, 1976. Articles Special issue on propagation and scattering of beam fields, Journal of the Optical Society of America A, vol. 3, no. 4, 1986. H. Kogelnik and T. Li, Laser Beams and Resonators, Proceedings of the IEEE, vol. 54, pp. 1312- 1329, 1966. G. D. Boyd and J. P. Gordon, Confocal Multimode Resonator for Millimeter Through Optical Wave- length Masers, Bell System Technical Journal, vol. 40, pp. 489-508, 1961. A. G. Fox and T. Li, Resonant Modes in a Maser Interferometer, Bell Systenl Technical Journal. vol. 40, pp. 453-488, 1961. PROBLEMS 3.1-6 Beam Parameters. The light emitted from aNd: YAG laser at a wavelength of 1.06 J-Lm is a Gaussian beam of ] - W optical power and beam divergence 2()u == 1 mrad. Determine the beam waist radius, the depth of focus, the maximum intensity, and the intensity on the beam axis at a distance z == 100 em from the beam waist. 3.1-7 Beam Identification by Two Widths. A Gaussian beam of wavelength Ao == 10.6 /-Lm (emitted by a CO 2 laser) has widths UTI == 1.699 mm and IT2 == 3.380 mm at two points separated by a distance d == 10 cm. Determine the location of the waist and the waist radius. 3.1-8 The Elliptic Gaussian Beam. The paraxial Helmholtz equation admits a Gaussian beam with intensity I(x, y. 0) == IAo/2 exp[-2(x 2 /Wx + y2 /Wy)] in the z == 0 plane, with the beam waist radii rox and TTOy in the x and y directions, respectively. The contours of constant intensity are therefore ellipses instead of circles. Write expressions for the beam depth of focus, angular divergence, and radii of curvature in the x and y directions, as functions of Vox, VOy, and the wavelength A. If TV ox == 2W oy , sketch the shape of the beam spot in the z == 0 plane and in the far field (z much greater than the depths of focus in both transverse directions). 3.2-6 Beam Focusing. An argon-ion laser produces a Gaussian beam of wavelength A == 488 nm with waist radius W o == 0.5 mm. Design a single-lens optical system for focusing the light to
PROBLEMS 101 a spot of diameter 100 /-Lm. What is the shortest focal-length lens that may be used? 3.2-7 Spot Size. A Gaussian beam of Rayleigh range Zo == 50 cm and wavelength A == 488 nm is converted into a Gaussian beam of waist radius W using a lens of focal length f == 5 cm at a distance z from its waist, as illustrated in Fig. 3.2-2. Write a computer program to plot W as a function of z. Verify that in the limit z - f» Zo, (3.2-10) and (3.2-12) hold; and that in the limit z « Zo, (3.2-13) holds. 3.2-8 Beam Refraction. A Gaussian beam is incident from air (n == 1) into a medium with a planar boundary and refractive index n == 1.5. The beam axis is normal to the boundary and the beam waist lies at the boundary. Sketch the transmitted beam. If the angular divergence of the beam in air is 1 mrad, what is the angular divergence in the medium? *3.2-9 Transmission of a Gaussian Beam Through a Graded-Index Slab. The ABCD matrix of a SELFOC graded-index slab with quadratic refractive index (see Sec. 1.3B) n(y) no(1 - a2y2) and length disA == cosad, B == (lja) sin ad, C == -asinad, D == casad for paraxial rays along the z direction. A Gaussian beam of wavelength Ao, waist radius W o in free space, and axis in the z direction enters the slab at its waist. Use the ABCD law to determine an expression for the beam width in the y direction as a function of d. Sketch the shape of the beam as it travels through the medium. 3.3-2 Power Confinement in Hermite-Gaussian Beams. Determine the ratio of the power con- tained within a circle of radius W (z) in the transverse plane to the total power in the Hermite- Gaussian beams of orders (0,0), (1,0), (0, 1), and (1, 1). What is the ratio of the power contained within a circle of radius 1 W(z) to the total power for the (0,0) and (1, 1) Hermite-Gaussian beams? 3.3-3 Superposition of1\vo Beams. Sketch the intensity of a superposition of the (1,0) and (0,1) Hermite-Gaussian beams assuming that the complex coefficients A 1 ,o and A o ,1 in (3.3-10) are equal. 3.3-4 Axial Phase. Consider the Hermite-Gaussian beams of all orders (l, m) with Rayleigh range Zo == 30 cm in a medium of refractive index n == 1. Determine the frequencies within the band v == 10 14 :i: 2 X 10 9 Hz for which the phase retardation between the planes z == - Zo and z == Zo is an integer multiple of 7r on the beam axis. These frequencies are the modes of a resonator comprising two spherical mirrors placed at the z == :i:zo planes as described in Sec. 10.2D.
CHAPTER 4.1 PROPAGATION OF LIGHT IN FREE SPACE A. Spatial Harmonic Functions and Plane Waves B. Transfer Function of Free Space C. Impulse Response Function of Free Space D. Huygens Fresnel Principle 4.2 OPTICAL FOURIER TRANSFORM A. Fourier Transform in the Far Field B. Fourier Transform Using a Lens 4.3 DIFFRACTION OF LIGHT A. Fraunhofer Diffraction * B. Fresnel Diffraction 4.4 IMAGE FORMATION A. Ray-Optics of a Single-Lens Imaging System B. Wave-Optics of a 4-f Imaging System C. Wave Optics of a Single-Lens Imaging System D. Near-Field Imaging 4.5 HOLOGRAPHY 105 116 121 127 138 .... .. , II. .. Josef yon Fraunhofer (1787- 1826) developed the diffrac- tion grating and contributed to our understanding of diffrac- tion. His epitaph reads Approx- imavit sidera (he brought the stars closer). Jean-Baptiste Joseph Fourier (1768-1830) demonstrated that periodic functions could be constructed from sums of sinu- soids. Harmonic analysis is the basis of Fourier optics; it has many applications. Dennis Gabor (1900-1979) invented holography and con- tributed to its development. He made the first hologram in 1947 and received the Nobel Prize in 1971 for carrying out this body of work. 102
Fourier optics provides a description of the propagation of light waves based on har- monic analysis (the Fourier transform) and linear systems. The methods of harmonic analysis have proven to be useful in describing signals and systems in many disciplines. Harmonic analysis is based on the expansion of an arbitrary function of time f t as a superposition (a sum or an integral) of harmonic functions of time of different frequencies (see Appendix A, Sec. A.I). The harmonic function F v exp j27rvt , which has frequency v and complex amplitude F v , is the building block of the theory. Several of these functions, each with its own value of F v , are added to construct the function f t , as illustrated in Fig. 4.0-1. The complex amplitude F v , as a function of frequency, is called the Fourier transform of ft. This approach is useful for the description of linear systems (see Appendix B, Sec. B.l). If the response of the system to each harmonic function is known, the response to an arbitrary input function is readily determined by the use of harmonic analysis at the input and superposition at the output. j(t) t + + + . . . t t t Figure 4.0-1 An arbitrary function f(t) may be analyzed as a sum of harmonic functions of different frequencies and complex amplitudes. An arbitrary complex function f x, y of the two variables x and y, representing the spatial coordinates in a plane, may similarly be written as a superposition of har- monic functions of x and y, each of the form F vx' v y exp j27r VxX + vyy "where F v x , v y is the complex amplitude and V x and v y are the spatial frequencies (cycles per unit length; typically cycles/mm) in the x and y directions, respectively.t The harmonic function F Vx v y exp j27r VxX + vyy is the two-dimensiona] building block of the theory. It can be used to generate an arbitrary function of two variables f x, y , as illustrated in Fig. 4.0-2 (see Appendix A, Sec. A.3). v .. I I , , - " .. + + + ... x . fix, y) Figure 4.0-2 An arbitrary function f(x, y) may be analyzed as a sum of harmonic functions of different spatial frequencies and complex amplitudes, drawn here schematically as graded blue lines. The plane wave U x y, z A exp j kxx + kyY + kzz plays an important role in wave optics. The coefficients k x , ky, k z are components of the wavevector k and .i 1 is a complex constant. At points in an arbitrary plane, U x, y, z is a spatial harmonic function. In the z 0 plane, for example, U x, y, 0 is the harmonic function f .E Y A exp j27r VxX + vyy , where V x kx 27r and v y ky 27r are the t The spatial harmonic function is defined with a minus sign in the exponent, in contrast to the plus sign used in the definition of the tenlporal harmonic function (see Appendix A, Sec. A.3). These signs match those of a forward-traveling plane wave. 103
1 04 CHAPTER 4 FOURIER OPTICS spatial frequencies (cycles/mm) and kx and ky are the spatial angular frequencies (ra- dians/mm). There is a one-to-one correspondence between the plane wave U x, y, z and the spatial harmonic function f x, y U x , y, 0 sinc e kx and ky are sufficient will be subsequently explained, kx and ky may not be greater than w c; i.e. the spatial frequencies V x and v y may not exceed the inverse wavelength 1 A. Since an arbitrary function f x, y can be analyzed as a superposition of harmonic functions, an arbitrary traveling wave U x, y, z may be analyzed as a sum of plane waves (Fig. 4.0-3). The plane wave is the building block used to construct a wave of arbitrary complexity. Furthermore, if it is known how a linear optical system modifies plane waves, the principle of superposition can be used to determine the effect of the system on an arbitrary wave. = z z Figure 4.0-3 The principle of Fourier optics: an arbitrary wave in free space can be analyzed as a superposition of plane waves. Because of the important role Fourier analysis plays in describing linear systems, it is useful to describe the propagation of light through linear optical components, including free space, using a linear-systems approach. The complex amplitudes in two planes normal to the optic z axis are regarded as the input and output of the system (Fig. 4.0-4). A linear system may be characterized by either its impulse response function (the response of the system to an impulse, or a point, at the input) or by its transfer function (the response to spatial harmonic functions), as described in Appendix B. y Input plane z = 0 x U(X,y,z) x Figure 4.0-4 The transmission of an optical wave U (x, y, z) through an opti- cal system between an input plane z o and an output plane z d. This is regarded as a linear system whose input and output are the functions of f(x,y) U(x,y,O) and g(x,y) U(x, y, d), respectively. f(x,y) Optical system g( x,y ) Y Output plane z = d This Chapter The chapter begins with a Fourier description of the propagation of light in free space (Sec. 4.1). The transfer function and impulse response function of the free-space prop- agation system are determined. In Sec. 4.2 we show that a lens may perform the operation of the spatial Fourier transform. The transmission of light through apertures is discussed in Sec. 4.3; this is a Fourier-optics approach to the diffraction of light, a subject usually presented in introductory textbooks from the perspective of the Huy- gens principle. Section 4.4 is devoted to image formation and spatial filtering. Finally, an introduction to holography the recording and reconstruction of optical waves, is presented in Sec. 4.5. Knowledge of the basic properties of the Fourier transform and linear systems in one and two dimensions (reviewed in Appendixes A and B) is necessary for understanding this chapter.
4.1 PROPAGATION OF LIGHT IN FREE SPACE 1 05 4.1 PROPAGATION OF LIGHT IN FREE SPACE A. Spatial Harmonic Functions and Plane Waves Consider a plane wave of complex amplitude U x, y, z A exp j kxx+kyy+kzz with wavevector k k x , ky, k z , wavelength ,x, wavenumber k k + k + k; 27r A, and complex envelope A. The vector k makes angles ()x sin- 1 kx k and ()y SiIl- 1 ky k with the y z and x z planes, respectively, as illustrated in Fig. 4.1- I. Thus, if ()x 0, there is no component of k in the x direction. The complex amplitude in the z 0 plane, U x, y, 0 , is a spatia] harmonic function f x, Y A exp j21f VxX + vyy with spatial frequencies V x kx 21f and v y ky 21f (the spatial frequency v k 21f is specified in cycles/mm, whereas the optical frequency v kc 27r is specified in cycles/see or Hz, as discussed in Sec. 2.2). The angles of the wavevector are therefore related to the spatial frequencies of the harmonic function by ()x · -1 \ SIn /\V x , ()y · -1 \ SIn /\V y . (4.1-1 ) Spatial Frequencies and Angles Recognizing Ax 1 V x and Ay 1 v y as the periods of the harmonic functions in the x and y directions (mm/cycle), we see that the angles ()x sin- 1 ,X Ax and ()y sin -1 A Ay are governed by the ratios of the wavelength of light to the period of the harmonic function in each direction. These geometrical relations follow from matching the wavefronts of the wave to the periodic pattern of the harmonic function in the z 0 plane, as illustrated in Fig. 4.1-1. " "..' k " . - x xt kx , " , " Plane wave , " ,,' "", kv .., Ax = l/vx Ox = in-I AVx Harmonic function f(x,y) Figure 4.1-1 A harmonic function of spatial frequencies I/x and I/y at the plane z 0 is consistent with a plane wave traveling at angles ()x sin- 1 AI/x and ()y sin- 1 AI/ y . k z ... ...... ... ... ...... ...... "', z - 4., If kx « k and ky « k, so that the wavevector k is paraxial, the angles ()x and ()y are small (sin ()x ()x and sin ()y ()y) and ()x 'xv:r, ()y AV y . (4.1-2) Spatial Frequencies and Angles (Paraxial Approximation) The angles of inclination of the wavevector are then directly proportional to the spatial frequencies of the corresponding harmonic function. Apparently, there is a one-to-one correspondence between the plane wave U x, y, z and the harmonic function f x, y .
1 06 CHAPTER 4 FOURIER OPTICS Given one, the other can be readily determined, provided the wavelength A is known: the harmonic function f x, y is obtained by sampling at the Z 0 plane, f x, y U x, y, 0 . Given the harmonic function f x, y , on the other hand, the wave U x, y, Z is constructed by using the relation U x, y, z f x, y exp jkzz with k z :f: k 2 k k, k 27r A. (4.1-3) is real. This condition implies that AV x < 1 and AV y < 1, so that the angles ex and By defined by (4.1-1) exist. The + and signs in (4.1- 3) represent waves traveling in the forward and backward directions, respectively. We shall be concerned with forward waves only. SpanalSpecualAnalys When a plane wave of unity amplitude traveling in the z direction is transmitted through a thin optical element with complex amplitude transmittance f x, y exp j27r VxX + lI. Y the wave is modulated by the harmonic function, so that U x, y, 0 f x, y . The incident wave is then converted into a plane wave with a wavevector at angles ex sin- 1 AV x and e y sin- 1 AV y (see Fig. 4.1-2). The element thus acts much as a prism, bending the wave upward in this illustration. If the complex amplitude transmittance is f x, y exp +j27f VxX + vyy , the wave is converted into a plane wave whose wavevector makes angles ex and e y with the z axis, so the wave is bent downward instead. x A A k () x = sin- 1 Avx z Ax = Il v x f(x,y) Figure 4.1-2 A thin element whose complex amplitude transmittance is a harmonic function of spatial frequency V x (period Ax Ilv x ) bends a plane wave of wavelength A by an angle Ox sin- 1 AV x sin- 1 (A/Ax). The blue color is used to indicate that the element is a phase grating (changing only the phase of the wave). The wave-deflection property of an optical element with harmonic-function trans- mittance may be understood as an interference phenomenon. In a direction making an angle ex, two points on the element separated by a the period A 1 v x , have a relative pathlength difference of A sin ex 1 V x AV x A, i.e., equal to a wavelength. Hence, all segments separated by a period interfere constructively in this direction. If the transmittance of the optical element f x, y is the sum of several harmonic functions of different spatial frequencies, the transmitted optical wave is also the sum of an equal number of plane waves dispersed into different directions; each spatial frequency is mapped into a corresponding direction, in accordance with (4.1-1). The amplitude of each wave is proportional to the amplitude of the corresponding harmonic component of f x, y ·
4.1 PROPAGATION OF LIGHT IN FREE SPACE 1 07 Examples. . A com pl ex amplitude transmittance of the form f x, Y cas 27rll x x ponents traveling at angles:!: sin- 1 AlIx , i.e., in both the upward and downward directions. . An element with a transmittance that varies as 1 + cas 27rll y Y behaves as a diffraction grating (see Exercise 2.4-5); the incident wave is bent into right and left components, and a portion of it travels straight through. . An element with transmittance 1 x, y 1L cas 27rll x x , where 1L x is the unit step function [ 1L x 1 if x > 0, and 1L x 0 if x < 0], represents a periodic set of slits, wherein 1 x, Y 1, in an opaque screen [I x, yO]. This periodic function may be analyzed in a Fourier series as a sum of harmonic functions of frequencies 0, :1:lI x , :1:2v x ,..., corresponding to waves at angles 0,:1: sin- 1 AlIx, :!: sin- 1 2AlI x , . . ., with amplitudes proportional to the coefficients of the Fourier series. At these angles, the waves transmitted through the slits interfere constructively. More generally, if f x, Y is a superposition integral of harmonic functions, 00 f X,Y F V x , v y exp j27r lIxX + lIyY dll x dll y , ( 4.1-4 ) -00 with frequencies lI x , lIy and amplitudes F lI x , v y , the transmitted wave U x, Y, z is the superposition of plane waves, 00 U x,y,Z F lI x , lIy exp j 27rll x x + 27rv y y exp jkzz dv x dv y , -00 ( 4.1-5) with complex envelopes F lI x , v y where k z k 2 k k 27r A- 2 lI; lI. Note that F lI x , v y is the Fourier transform of f x, Y (see Appendix A, Sec. A.3. Since an arbitrary function may be Fourier analyzed as a superposition integral of the form (4.1-4), the light transmitted through a thin optical element of arbitrary trans- mittance may be written as a superposition of plane waves (see Fig. 4.1-3), provided x z Figure 4.1-3 A thin optical element of amplitude transmittance I(x, y) de- composes an incident plane wave into many plane waves. The plane wave traveling at the angles Ox sin -1 AV x and Oy sin -1 AV y has a complex en- velope F(v x , v y ), the Fourier transform of I(x, y). y f(x,y) This process of "spatial spectral analysis" is akin to the angular dispersion of differ- ent temporal-frequency components (wavelengths) provided by a prism. Free-space
1 08 CHAPTER 4 FOURIER OPTICS propagation serves as a natural "spatial prism," sensitive to the spatial rather than temporal frequencies of the optical wave. Amplitude Modulation Consider a transparency with complex amplitude transmittance 10 x, y . If the Fourier transfonn Fo v x , v y extends over widths :i::vx and :i:Vy in the x and y directions, the transparency wilJ defle c t an incident plane wave by angles ()x and By in the range :i:: sin -1 Avx and:i:: sin 1 Avy , respectively. Consider a second transparency of complex amplitude transmittance I x, Y 10 x, y exp j27r vxox + vyoy , where 10 x, y is slowly varying compared to exp j27r VxoX + vyoy so that vx Vxo and Vy vyo. We may regard 1 x, y as an amplitude-modulated function with a carrier frequency Vxo and vyo and modulation function 10 x, y . The Fourier transfonn of I x, y is Fo V x Vxo, v y vyo , in accordance with the frequency-shifting property of the Fourier transform (see Appendix A). The transparency will deflect a plane wave to directions centered about the angles B xO sin -1 Avxo and Byo sin -1 Avyo (Fig. 4.1-4). This can also be readily seen by regarding 1 x, y as a transparency of transmittance 10 x, y in contact with a grating or prism of transmittance exp j27r VxOX + vyoy that provides the angular deflection ()xo and Byo. x x j ,," " " ,. \ sin- 1 AlIxQ .... z z , y ; y !o(x,y) exp(-j27rllxQx) Figure 4.1-4 Deflection of light by the transparencies fo(x, y) and fo(x, y) exp( j27rll x ox). The "carrier" harmonic function exp( j21fll x ox) acts as a prism that deflects the wave by an angle ()xo · -1 \ SIn /\lI x o. This idea may be used to record two images 11 x, y and 12 x, y on the same transparency using the spatial-frequency multiplexing scheme I x, y 11 X, Y exp j27r vxlX + VylY + 12 x, y exp j27r vx2X + V y 2Y . The two images may be easily separated by illuminating the transparency with a plane wave, whereupon the two images are deflected at different angles and are thus separated. This principle will prove useful in holography (Sec. 4.5), where it is often desired to separate two images recorded on the same transparency. Frequency Modulation We now examine the transmission of a plane wave through a transparency comprising a "collage" of several regions, the transmittance of each of which is a harmonic function of some spatial frequency, as illustrated in Fig. 4.1-5. If the dimensions of each region are much greater than the period, each region acts as a grating or prism that deflects the wave in some direction, so that different portions of the incident wavefront are deflected into different directions. This principle may be used to create maps of optical interconnections. A transparency may also have a harmonic transmittance with a spatial frequency that varies continuously and slowly with position (in comparison with A), much as the
4.1 PROPAGATION OF LIGHT IN FREE SPACE 1 09 -.-...-.-.-.-..-.-.-.---.-. .. . Figure 4.1-5 Deflection of light by a trans- parency made of several hannonic functions (phase gratings) of different spatial frequen- . Cles. . , . . -------..------------...-.---------. - frequency of a frequency-modulated (PM) signal varies slowly with time. Consider, for example, the phase function f x, y exp j27r<jJ x, y , where <jJ x, y is a continuous slowly varying function of x and y. In the neighborhood of a point Xo, Yo , we may use the Taylor-series expansion <jJ x, y <jJ Xo, Yo + x Xo V x + Y Yo v y , where the derivatives V x 8<jJ 8x and v y 8<jJ 8y are evaluated at the position xo, Yo · The local variation of f x, y with x and y is therefore proportional to the quantityexp j27r VxX + vyy , which is a harmonic function with spatial frequencies V x a<jJ ax and v y 8<jJ 8y. Since the derivatives 8<jJ 8x vary with x and y, so do the spatial frequencies. The transparency f x, y exp j27r<jJ x, y therefore deflects the portion of the wave at the position x, y by the position-dependent angles ()x sin- 1 Aa<jJ 8x and ()y sin- 1 A8<jJ 8y . EXAMPLE 4.1-1. Scanning. A thin transparency with complex amplitude transmittance f(x, y) exp(j7rx 2 /Af) introduces a phase shift 27r4;(x, y) where 4;(x, y) x 2 /2Af, so that the wave is deflected at the position (x, y) by the angles Ox sin- 1 (Ao4;/ox) sin- 1 ( xl I) and Oy O. If Ix / II 1, Ox x I I and the deflection angle ex is directly proportional to the transverse distance x. This transparency may be used to deflect a narrow beam of light. If the transparency is moved at a uniform speed, the beam is deflected by a linearly increasing angle as illustrated in Fig. 4.1-6. x t " \ I - ---. II ! -ox ; - . ----------- . . . . . . . . . / - /' -. ./ .. -. - ----------------- - --- I / / z ./ .// /-// // // f Figure 4.1-6 Using a frequency- modulated transparency to scan an optical beam. Figure 4.1-7 A transparency with transmit- tance f ( x, y ) exp (j 7rX 2 I AI) bends the wave at position x by an angle ex x / f so that it acts as a cylindrical lens with focal length f. EXAMPLE 4.1-2. Imaging. If the transparency in Example 4.1-1 is illuminated by a plane wave, each part of the wave is deflected by a different angle and as a result the wavefront is altered. The local wavevector at position x bends by an angle x I I so that all wavevectors meet at a single point on the optical axis a distance I from the transparency, as illustrated in Fig. 4.1-7. The transparency acts as a cylindrical lens with a focal length f. Similarly, a transparency with the transmittance I(x, y) exp[f7r(x 2 + y2)/ AI] acts as a spherical lens with focal length I. Indeed, this is the expression for the amplitude transmittance of a thin lens [see (2.4-9)].
11 0 CHAPTER 4 FOURIER OPTICS EXERCISE 4.1-1 Binary-Plate Cylindrical Lens. Use harmonic analysis near the position x to show that a trans- parency with complex amplitude transmittance equal to the binary function f(x,y) x 2 , ( 4.1-6) where ti( x) is the unit step function [ti( x) 1 if x > 0, and ti( x) 0 if x < 0], acts as a cylindrical lens with multiple focal lengths equal to 00, -3=f, -3=f /2, .... x - - f z Figure 4.1-8 Binary plate as a cylindrical lens with multiple foci. Fresnel Zone Plate A two-dimensional generalization of the binary plate in Exercise 4.1-1 is a circularly symmetric transparency of complex amplitude transmittance x2 + y2 f x,y ( 4.1- 7) known as the Fresnel zone plate. It is a set of ring apertures of increasing radii, decreasing widths, and equal areas (see Fig. 4.1-9). The structure serves as a spherical lens with multiple focal lengths. A ray incident at each point is split into multiple rays, and the transmitted rays meet at multiple foci with focal lengths :l=f, :l=f 2, . . . , together with a component transmitted without deflection. The operation of the Fresnel zone plate may also be described as an interference effect (see Sec. 2.5B). The center of the mth ring has a radius Pm at the mth peak of the cosine function, i.e., 1rp Af m21r. At a focal point z f, the distance Rm to the mth ring is given by R f2 + P, so that Rm f2 + 2mAf. If f is sufficiently large so that the angles subtended by the rings are small, then Rm f +mA. Thus, the waves transmitted through consecutive rings have pathlengths differing by a wavelength, so that they interfere constructively at the focal point. A similar argument applies for the other foci. x I I , . Rm I I . ! I L r f z Figure 4.1-9 The Fresnel zone plate.
4.1 PROPAGATION OF LIGHT IN FREE SPACE 111 B. Transfer Function of Free Space We now examine the propagation of a monochromatic optical wave of wavelength A and complex amplitude U (x, y, z) in the free space between the planes z == 0 and z == d, called the input and output planes, respectively (see Fig. 4.1-10). Given the complex amplitude of the wave at the input plane, f(x, y) == U(x, y, 0), we shall determine the complex amplitude at the output plane, g(x, y) == U(x, y, d). U(x,y,z) g(x,y) o d Z -L I h H / y Figure 4.1-10 Propagation of light between two planes is regarded as a linear system whose input and output are the complex amplitudes of the wave in the two planes. We regard f (x, y) and g( x, y) as the input and output of a linear system. The system is linear since the Helmholtz equation, which U (x, y, z) must satisfy, is linear. The system is shift-invariant because of the invariance of free space to displacement of the coordinate system. A linear shift-invariant system is characterized by its impulse response function h(x, y) or by its transfer function H(v x , v y ), as explained in Ap- pendix B, Sec. B.2. We now proceed to determine expressions for these functions. The transfer function H(v x , v y ) is the factor by which an input spatial harmonic function of frequencies V x and v y is multiplied to yield the output harmonic function. We therefore consider a harmonic input function f(x, y) == A exp[-j27f(v x x + vyY)]. As explained earlier, this corresponds to a plane wave U(x, y, z) == A exp[-j(kxx + kyY + kzz)] where kx == 27fv x , ky == 27fv y , and k = V k2 - k 2 - k 2 = 21f V >..-2 - v 2 - v 2 z x Y x y. (4.1-8) The output g(x, y) == Aexp[-j(kxx + kyy + kzd)], so that we can write H(v x , v y ) == g(x, y)/ f(x, y) == exp( -jkzd), from which H(vx,v y ) = ex p ( -j21fd V >..-2 - v; - v) . (4.1-9) Transfer Function of Free Space The transfer function H(v x , v y ) is therefore a circularly symmetric complex function of the spatial frequencies V x and v y . Its magnitude and phase are sketched in Fig. 4.1-11. For spatial frequencies for which v; + v; < A -2 (i.e., frequencies lying within a circle of radius 1/A.) the magnitude IH(v x , vy)1 == 1 and the phase arg{H(v x , v y )} is a function of V x and v y . A harmonic function with such frequencies therefore undergoes a spatial phase shift as it propagates, but its manitude is not altered. At higher spatial frequencies, v; + v; > A. - , the quantity under the square root in (4.1-9) is negative so that the exponent is real and the transfer function exp[ - 27fd(v; +
112 CHAPTER 4 FOURIER OPTICS IH IA 1 Harmonic function I Plane wave 1 Vx I I I I A -- 2d 2 x 1 l/x : ,t I A-I Vy Shifted harmonic function -arg {H} I I I I I 2nd/ AI I I I I I I I Vx d A-I Vy Vx Figure 4.1-11 Magnitude and phase of the transfer function H(v x , v y ) for free-space propagation between two planes separated by a distance d. sharply when the spatial fre uency slightly exceeds A -1 , as illustrated in Fig. 4.1-11. We may therefore regard A- as the cutoff spatial frequency (the spatial bandwidth) of the system. Thus, the spatial bandwidth of light propagation in free space is approximately A -1 cycles/mm. Features contained in spatial frequencies greater than A -1 (corresponding to details of size finer than A) cannot be transmitted by an optical wave of wavelength A over distances much greater than A. Fresnel ApproximaUon The expression for the transfer function in (4.1-9) may be simplified if the input func- tion f x, y contains only spatial frequencies that are much smaller than the cutoff light then make s mall angles ()x AI/x and ()y AI/ y corresponding to paraxial rays. the phase factor in (4.1-9) is 1/2 x 1/2 Y d 1 ()2 21rd A- 2 t The sign in (4.1-3) was used since the + sign would have resulted in an exponentially growing function, which is physically unacceptable since the system is passive.
4.1 PROPAGATION OF LIGHT IN FREE SPACE 113 d ()2 ()4 . (4.1-10) . . . Neglecting the third and higher terms of this expansion, (4.1-9) may be approximated by H ' ' H - · d (2 2' - ,l/x, l/y) 0 exp _J7r A ....l/x + l/y) _ ' (4.1-11) Transfer Function of Free Space (Fresnel Approximation) where Ho exp j kd . In this approximation, the phase is a quadratic function of l/x and l/y, as illustrated in Fig. 4.1-12. This approximation is known as the Fresnel approximation. H -rarg {H} A 1 L , A-I 1/y I A-I Vy Figure 4.1-12 The transfer function of free-space propagation for low spatial frequencies (much less than 1/ A cycles/nun) has a constant magnitude and a quadratic phase. The condition of validity of the Fresnel approximation is that the third term in (4.1- 1 0) is much smaller than 7r for all (). This is equivalent to ()4d (4.1-12) If a is the largest radial distance in the output plane, the largest angle ()m a d, and (4.1-12) may be written in the form [see (2.2-18)] ()2 (4.1-13) Fresnel Approximation Condition of Validity ..... - -... - --- a -- --------r----- Om z d NF a 2 Ad' (4.1-14) Fresnel Number 1 cm, d 100 cm, and A 4 5 x 10 3. In this case the where N F is the Fresnel number. For example, if a 0.5 /-Lm, then Om 10 2 radian, N F 200, and N F ()2 Fresnel approximation is applicable.
114 CHAPTER 4 FOURIER OPTICS Input Output Relation Given the input function f x, y , the output function 9 x, Y may be determined as follows: (1) we determine the Fourier transform (X) F lI x , lIy f x, y exp j27r lIxX + lIyY dx dy, ( 4.1-(5) -(X) which represents the complex envelopes of the plane-wave components in the input plane; (2) the product H lIx,l/y F l/x, lIy gives the complex envelopes of the plane- wave components in the output plane; and (3) the complex amplitude in the output plane is the sum of the contributions of these plane waves, (X) 9 x,y H lI x , lIy F l/x, lIy exp j27r lIxX + lIyy dllxdll y . ( 4.1-16) -(X) Using the Fresnel approximation for H lI x , lIy , which is given by (4.1-11), we have 00 9 x,y Ho F l/x, lIy exp j27r lIxX + l/yY dll x dll y -00 (4. ] -17) Equations (4.1-1 7) and (4.1-] 5) serve to relate the output function 9 x, Y to the input function f x, y · c. Impulse Response Function of Free Space The impulse response function h x, y of the system of free-space propagation is the response 9 x, Y when the input f x, Y is a point at the origin 0,0. It is the inverse Fourier transform of the transfer function H l/x, lIy . Using the results of Sec. A.3 and Table A.2-1 of Appendix A, together with k 27r A, the inverse Fourier transform of (4.1-11) turns out to be h : x, y: ho exp - . x2 + y2 - , ( 4.1-18) Impulse Response Function Free Space (Fresnel Approximation) - - where ho j Ad exp j kd . This function is proportional to the complex ampli- tude at the z d plane of a paraboloidal wave centered about the origin 0,0 [see (2.2-17)]. Thus, each point in the input plane generates a paraboloidal wave; all such waves are superimposed at the output plane. Free-Space Propagation as a Convolution An alternative procedure for relating complex amplitudes f x, y and 9 x, Y is to regard f x, y as a superposition of different points (delta functions), each producing a paraboloidal wave. The wave originating at the point x', y' has an amplitude f x', y'
4.1 PROPAGATION OF LIGHT IN FREE SPACE 115 and is centered about x', y' so that it generates a wave with amplitude f x', y' h X x' , y y' at the point x, y in the output plane. The sum of these contributions is the two-dimensional convolution 00 9 x,y f x', y' h x I x,y y' dx' dy', (4.1-19) -00 which, in the Fresnel approximation, becomes (X) 9 x,y ho f x', y' exp . J1r x x' 2 + y Ad y' 2 dx' dy', (4.1-20) -(X) where ho j Ad exp jkd. In summary: within the Fresnel approximation, there are two approaches to deter- mining the complex amplitude 9 x, y in the output plane, given the complex amplitude f x, y in the input plane: (1) Equation (4.1-20) is based on a space-domain approach in which the input wave is expanded in terms of paraboloidal elementary waves; and (2) Equation (4.1-17) is a frequency-domain approach in which the input wave is expanded as a sum of plane waves. EXERCISE 4.1-2 Gaussian Beams Revisited. If the function f(x, y) Aexp[ (x 2 + y2)/W] represents the complex amplitude of an optical wave U(x, y, z) in the plane z 0, show that U(x, y, z) is the Gaussian beam discussed in Chapter 3, (3.1-7). Use both the space- and frequency-domain methods. D. Huygens Fresnel Principle The Huygens Fresnel principle states that each point on a wavefront generates a spherical wave (Fig. 4.1-13). The envelope of these secondary waves constitutes a new wavefront. Their superposition constitutes the wave in another plane. The system's impulse response function for propagation between the planes z 0 and z d is 1 h x, y ex: exp j kr , r r x 2 + y2 + d 2 . (4.1-21) x+ Spherical wave ( . IIII ..." I .... z Figure 4.1-13 The Huygens- Fresnel principle. Each point on a wavefront generates a spherical wave. In the paraxial approximation, the spherical wave given by (4.1-21) is approximated by the paraboloidal wave in (4.1-18) (see Sec. 2.2B). Our derivation of the impulse response function is therefore consistent with the Huygens Fresnel principle o y Wavefront Wavefront
116 CHAPTER 4 FOURIER OPTICS 4.2 OPTICAL FOURIER TRANSFORM As has been shown in Sec. 4.1, the propagation of light in free space is described conveniently by Fourier analysis. If the complex amplitude of a monochromatic wave of wavelength .A in the z 0 plane is a function f x, y composed of harmonic components of different spatial frequencies, each harmonic component corresponds to a plane wave: the plane wave traveling at angles ()x sin- 1 .AV x , ()y sin- 1 >..v y corresponds to the components with spatial frequencies V x and v y and has an ampli- tude F v x , v y , the Fourier transform of f x, y . This suggests that light can be used to compute the Fourier transform of a two-dimensional function f x, y , simply by making a transparency with amplitude transmittance f x, y through which a uniform plane wave of unity magnitude is transmitted. Because each of the plane waves has an infinite extent and therefore overlaps with the other plane waves, however, it is necessary to find a method of separating these waves. It will be shown that at a sufficiently long distance, only a single plane wave contributes to the total amplitude at each point in the output plane, so that the Fourier components are eventually separated naturally. A more practical approach is to use a lens to focus each of the plane waves into a single point, as described subsequently. A. Fourier Transform in the Far Field We now proceed to show that if the propagation distance d is sufficiently long, the only plane wave that contributes to the complex amplitude at a point x, y in the output plane is the wave with direction making angles ()x x d and ()y y d with the optical axis (see Fig. 4.2-1). This is the wave with wavevector components kx x d k and ky y d k and amplitude F v x , v y with V x x >"d and v y x .Ad. The complex amplitudes 9 x, y and f x, y of the wave at the z d and z 0 planes are related by (4.2-1) Free-Space Propagation as Fourier Transform (Fraunhofer Approximation) where F v x , v y is the Fourier transform of f x, y and ho j .Ad exp jkd. Contributions of all other waves cancel out as a result of destructive interference. This approximation is known as the Fraunhofer approximation. x y - -. b :t o ..... ---__ Ox - -- - f( x,y) ........ ...... ...... ...... ...... ...... ...... ...... ...... Oy ---___- ...... ........ ...... ...... ...... --- -- -- -- y -- -- --- -- a l x,y ) z g(x,y) Figure 4.2-1 When the distance d is sufficiently long, the complex amplitude at point (x, y) in the z d plane is proportional to the complex amplitude of the plane-wave component with angles Ox xjd AV x and Oy yjd AV y , i.e., to the Fourier transform F(v x , v y ) of f(x, y), with V x xj Ad and v y yj Ad.
4.2 OPTICAL FOURIER TRANSFORM 117 As noted in the following proofs, the conditions of validity of Fraunhofer approxi- . matlon are: (4.2-2) Fraunhofer Approximation Condition of Validity N F a 2 / Ad, N b 2 / Ad The Fraunhofer approximation is therefore valid whenever the Fresnel numbers N F and N are small. The Fraunhofer approximation is more difficult to satisfy than the Fresnel approximation" which requires that NF()?'n 4« 1 [see (4.1-13)]. Since ()m « 1 in the paraxial approximation, it is possible to satisfy the Fresnel condition NF()'?n 4« 1 for Fresnel numbers N F not necessarily « 1. N F 1 and N« 1, D Proofs of the Fourier Transform Property of Free-Space Propagation in the Fraunhofer Approximation. We begin with the relation between g(x, y) and f(x, y) in (4.1-20). The phase in the argument of the exponent is (7r / Ad) [(x X')2 + (y y')2] (7r / Ad) [(x 2 +y2) + (X,2 +y'2) 2(xx' + yy')]. If f (x , y) is confined to a small area of radius b, and if the distance d is sufficiently large so that the Fresnel number N b 2 / Ad is small. then the phase factor (7r / Ad) (X,2 + y'2) < 7r(b 2 / Ad) is negligible and (4.1-20) may be approximated by 00 g(;r,y) hoexp . x2 + y2 J7r Ad . xx' + yy' dx' dy'. (4.2-3) -00 The factors x / Ad and y / Ad may be regarded as the frequencies V x x / Ad and v y y / Ad, so that g(x,y) hoexp . x2 + y2 J7r Ad x Y F Ad ' Ad ' (4.2-4) where F(v x , v y ) is the Fourier transform of f(x, y). The phase factor given by exp[ j7r(x 2 + y2)/Ad] in (4.2-4) may also be neglected and (4.2-1) obtained if we also limit our interest to points in the output plane within a circle of radius a centered about the z-axis so that 7r(X2 + y2)/Ad < 7ra 2 / Ad « 7r. This is applicable when the Fresnel number N F a 2 / Ad « 1. Another proof is based on (4.1-17). which expresses the complex amplitude g(x, y) as an integral of plane waves of different frequencies. If d is sufficiently large so that the phase in the integrand is much greater than 27r, it can be shown using the method of stationary phase t that only one value of V x contributes to the integral. This is the value for which the derivative of the phase 7r Adv; 27rv x x with respect to V x vanishes; i.e., V x x / Ad. Similarly, the only value of v y that contributes to the integral is v y y / Ad. This proves the assertion that only one plane wave contributes to the far field at a given point. . EXERCISE 4.2-1 Conditions of Validity of the Fresnel and Fraunhofer Approximations: A Comparison. Demonstrate that the Fraunhofer approximation is more restrictive than the Fresnel approximation by taking A 0.5 /-Lm, and assuming that the object points lie within a circular aperture of radius b 1 em and the observation points lie within a circular aperture of radius a 2 cm. Determine the range of distances d between the object plane and the observation plane for which each of these approximations is applicable. t See, e.g., M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, Appendix III.
118 CHAPTER 4 FOURIER OPTICS Summary In the Fraunhofer approximation, the complex amplitude 9 x, y of a wave of wavelength A in the z d plane is proportional to the Fourier transform F V x , v y of the complex amplitude ! x, y in the z 0 plane, evaluated at the spatial frequencies V x x Ad and v y . y Ad. The approximation is valid if I x, y at the input plane is confined to a circle of radius b satisfying b 2 Ad« 1, . and at points in the output plane within a circle of radius a satisfying a 2 Ad« 1. ' 00'- ..,. .. - -... -. . "". .. - -.. . , . B. Fourier Transform Using a Lens The plane-wave components that constitute a wave may also be separated by use of a lens. A thin spherical lens transforms a plane wave into a paraboloidal wave focused to a point in the lens focal plane (see Sec. 2.4 and Exercise 2.4-3). If the plane wave arrives at small angles Ox and 0Y' the paraboloidal wave is centered about the point Oxf,Oyf , where! is the focal length (see Fig. 4.2-2). The lens therefore maps each direction Ox,Oy into a single point Ox I, Oyl in the focal plane and thus separates the contributions of the different plane waves. Ox .. x = Ox! z Figure 4.2-2 Focusing of a plane wave into a point. A direction ( Ox, 0 y ) is mapped into a point (x,y) (Oxf,Oyf). (see Exer- cise 2.4-3.) ", *"""'" l-c ! Focal plane In reference to the optical system shown in Fig. 4.2-3, let I x, y be the complex amplitude of the optical wave in the z 0 plane. Light is decomposed into plane waves, with the wave traveling at small angles Ox AV x and Oy >..v y having a complex amplitude proportional to the Fourier transform F v x , v y . This wave is focused by the lens into a point x, y in the focal plane where x Oxl >"Iv x and y Oy! >..fv y . The complex amplitude at point x, y in the output plane is therefore proportional to the Fourier transform of I x, y evaluated at V x x >"i and v y y >"1, so that x y >"1 ' >"i 9 x, y DC F . (4.2-5) To determine the proportionality factor in (4.2-5), we analyze the input function I x, y into its Fourier components and trace the plane wave corresponding to each component through the optical system. We then superpose the contributions of these waves at the output plane to obtain 9 x, Y . Assuming that these waves are paraxial and using the Fresnel approximation, we obtain: 9 x,y h z exp . x 2 + y2 d f x y , (4.2-6)
4.2 OPTICAL FOURIER TRANSFORM 119 x !(x,y) , (Ox. Oy ) -, .1" g(x,y) y tHHH j f Poi III ..... (x,y) = (Ox/' Oy/) - 11111 1 .. "'" .... z l:::. f Focal plane d z=o Figure 4.2-3 Focusing of the plane waves associated with the harmonic Fourier components of the input function f(x, y) into points in the foca] plane. The amplitude of the plane wave with direction (Ox,Oy) (AV x , AV y ) is proportional to the Fourier transform F(v x , v y ) and is focused at the point (x, y) (Oxf,Oyf) (Afv x , Afv y ). where hl Hoho j Af exp j k d + f . Thus, the coefficient of proportionality in (4.2-5) contains a phase factor that is a quadratic function of x and y. Since h l 1 Af it follows from (4.2-6) that the optical intensity at the output plane is 1 x Y Af 2 2 I x,y . (4.2-7) The intensity of light at the output plane (the back focal plane of the lens) is therefore proportional to the squared absolute value of the Fourier transform of the complex amplitude of the wave at the input plane, regardless of the distance d. The phase factor in (4.2-6) vanishes if d f, so that 9 x,y x y , (4.2-8) Fourier- Transform Property of a Lens where hl j >..f exp j2kf. In this geometry, known as the 2-1 system (see Fig. 4.2-4), the complex amplitudes at the front and back focal planes of the lens are related by a Fourier transform, both magnitude and phase. x f (x, y) z Figure 4.2-4 The 2- f system. The Fourier component of f ( x, y ) with spatial frequencies V x and v y gener- ates a plane wave at angles 0 x AV x and Oy AV y and is focused by the lens to the point ( x, y ) (fOx, fOy) (Afv x , Afv y ) so that g( x, y) is proportional to the Fourier transform F(x/ Af, y/ Af). ",,,,'" - Ox I...c Focal plane f tc f
120 CHAPTER 4 FOURIER OPTICS Summary The complex amplitude of light at a point x, y in the back focal plane of a lens of focal length f is proportional to the Fourier transform of the complex amplitude in the front focal plane evaluated at the frequencies V x x AI, v y Af. This relation is valid in the Fresnel approximation. Without the lens, the Fourier transformation is obtained only in the Fraunhofer approximation, which is more restrictive. D *Proof of the Fourier Transform Property of the Lens in the Fresnel Approximation. The proof takes the following four steps. 1. The plane wave with angles Ox AV x and Oy AV y has a complex amplitude U(x, y, 0) F(v x , v y ) exp[ j27r(v x x + vyy)] in the z 0 plane and U(x, y, d) H(v x , vy)F(v x , v y ) exp[ j27r(v x x + vyY)] in the z d plane, immediately before crossing the lens, where H(v x , v y ) Ho exp[j7rAd(v; + v;)] is the transfer function of a distance d of free space and Ho exp( jkd). 2. Upon crossing the lens, the complex amplitude is multiplied by the lens phase factor exp[j7r(x 2 + y2)/Af] [the phase factor exp( jk), where is the width of the lens, has been ignored]. Thus, . x2 + y2 x exp j7r Ad v; + v; F(v x , v y ) exp [ j27r(v x x + vyY)] . This expression is simplified by writing 2v x X+X 2 /Af (X2 2v x Afx)/Af [(x XO)2 x5]/ Af, with Xo Avxf; a similar relation for y is written with Yo Avyf, so that U(X,y, d +) (4.2-9) XO)2 + (y Af YO)2 U(X, y, d + ) , (4.2-10) where A(v x , v y ) Ho exp j7r A( d f) v; + v; F(v x , v y ). (4.2-11) Equation (4.2-10) is recognized as the complex amplitude of a paraboloidal wave converging toward the point (xo, Yo) in the lens focal plane, z d + D. + f. 3. We now examine the propagation in the free space between the lens and the output plane to determine U(x, y, d + + f). We apply (4.1-20) to (4.2-10), use the relation exp[j27r(x xo)x' / Af] dx' Af8(x xo), and obtain U(x, y, d + + f) h O (Af)2 A(v x , v y )8(x xo)8(y Yo) (4.2-12) where ho (j / Af) exp( jkf). Indeed, the plane wave is focused into a single point at Xo Avxf and Yo Avyf. 4. The last step is to integrate over all the plane waves (all V x and v y ). By virtue of the sifting property of the delta function, 8(x xo) 8(x Afv x ) (1/ Af)8(v x x / Af), this integral gives g(x, y) hoA(x/ Af, y/ Af). Substituting from (4.2-11) we finally obtain (4.2-6). . EXERCISE 4.2-2 The Inverse Fourier Transform. In the single-lens optical system depicted in Fig. 4.2-4, the field distribution in the front focal plane (z 2f) is a scaled version of the Fourier transform of the field distribution in the back focal plane (z 0). Verify that if the coordinate system in the front focal plane is inverted, i.e., (x, y) ) ( x, y), then the resultant field distribution yields the inverse Fourier transform.
4.3 DIFFRACTION OF LIGHT 121 4.3 DIFFRACTION OF LIGHT When an optical wave is transmitted through an aperture in an opaque screen and travels some distance in free space, its intensity distribution is called the diffraction pattern. If light were treated as rays, the diffraction pattern would be a shadow of the aperture. Because of the wave nature of light, however, the diffraction pattern may deviate slightly or substantially from the aperture shadow, depending on the distance between the aperture and observation plane, the wavelength, and the dimensions of the aperture. An example is illustrated in Fig. 4.3-1. It is difficult to determine exactly the manner in which the screen modifies the incident wave, but the propagation in free space beyond the aperture is always governed by the laws described earlier in this chapter. Figure 4.3-1 Diffraction patterns of the teeth of a saw. (From M. Cagnet, M. Franon, and J. C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, 1962.) The simplest theory of diffraction is based on the assumption that the incident wave is transmitted without change at points within the aperture, but is reduced to zero at points on the back side of the opaque part of the screen. If U x, y and f x, yare the complex amplitudes of the wave immediately to the left and right of the screen (Fig. 4.3-2), respectively, then in accordance with this assumption, f x,y U x, y p x, y , (4.3-1) where p x,y 1 inside the aperture 0, outside the aperture (4.3-2) is called the aperture function. x U(x,y) d z Figure 4.3-2 A wave U(x, y) is transmitted through an aperture of amplitude transmittance p(x, y), gen- erating a wave of complex am- plitude f(x, y) U(x, y)p(x, y). After propagation a distance d in free space, the complex amplitude is g(x, y) and the diffraction pattern is the intensity I(x, y) Ig(x, y)12. Observation plane f(x,y) y Aperture plane g( x,y ) Given f x, y , the complex amplitude 9 x, y at an observation plane a distance d from the screen may be determined using the methods described in Sees. 4.1 and 4.2. The diffraction pattern I x, y 9 x, Y 2 is known as Fraunhofer diffraction or
122 CHAPTER 4 FOURIER OPTICS Fresnel diffraction, depending on whether free-space propagation is described using the Fraunhofer approximation or the Fresnel approximation, respectively. Although this approach gives reasonably accurate results in most cases, it is not exact. The validity and self-consistency of the assumption that the complex amplitude f x, y vanishes at points outside the aperture on the back of the screen are question- able since the transmitted wave propagates in all directions and therefore reaches those points as well. A theory of diffraction based on the exact solution of the Helmholtz equation under the boundary conditions imposed by the aperture is mathematically difficult. Only a few geometrical structures have yielded exact solutions. However, different theories of diffraction have been developed using a variety of assumptions, leading to results with varying accuracies. Rigorous diffraction theory is beyond the scope of this book. A. Fraunhofer Diffraction Fraunhofer diffraction is the theory of transmission of light through apertures, as- suming that the incident wave is multiplied by the aperture function and that the Fraunhofer approximation determines the propagation of light in the free space beyond the aperture. The Fraunhofer approximation is valid if the propagation distance d between the aperture and observation planes is sufficiently large so that the Fresnel number N b 2 >"d« 1, where b is the largest radial distance within the aperture. Assuming that the incident wave is a plane wave of intensity Ii traveling in the z direction so that U x, Y Ii , then f x, Y Ii P x, Y . In the Fraunhofer approximation [see (4.2-1)], 9 x, Y x y (4.3-3) where CX) p 1/x, 1/y P x, Y exp j27r 1/ x X + 1/yy dx dy (4.3-4) -CX) is the Fourier transform of p x, y and ko pattern is therefore j >"d exp j kd . The diffraction Ii X Y P >"d 2 Ad ' Ad 2 I x,y . (4.3-5) In summary: the Fraunhofer diffraction pattern at the point x, y is proportional to the squared magnitude of the Fourier transform of the aperture function p x, y evaluated at the spatial frequencies 1/x x >"d and 1/y y >"d. EXERCISE 4.3-1 Fraunhofer Diffraction from a Rectangular Aperture. Verify that the Fraunhofer diffraction pattern from a rectangular aperture, of height and width Dx and Dy respectively, observed at a distance d is I(x,y) · 2 DxX · 2 DyY (4.3-6) where Io(DxDyjAd)2 Ii is the peak intensity and sinc(x) sin(7rx)j(7rx). Verify that the first
4.3 DIFFRACTION OF LIGHT 123 zeros of this pattern occur at x :tAd / Dx and y :tAd / Dy, so that the angular divergence of the diffracted light is given by Ox A Dx' Oy A Dy . ( 4.3-7) If Dy < Dx, the diffraction pattern is wider in the y direction than in the x direction, as illustrated in Fig. 4.3-3. Diffraction pattern z x Dx m I ..", , , ..", ..", , ..............., ..... Oy 1(0, y) Aperture ... y d Dy o y Figure 4.3-3 Fraunhofer diffraction from a rectangular aperture. The central lobe of the pattern has half-angular widths Ox A/ Dx and Oy A/ Dy. EXERCISE 4.3-2 Fraunhofer Diffraction from a Circular Aperture. Verify that the Fraunhofer diffraction pattern from a circular aperture of diameter D (Fig. 4.3-4) is 2 o 7r Dp/ Ad ' where 10 (7r D 2 / 4Ad)2 Ii is the peak intensity and J 1 (.) is the Bessel function of order 1. The Fourier transform of circularly symmetric functions is discussed in Appendix A, Sec. A.3. The circularly symmetric pattern (4.3-8), known as the Airy pattern, consists of a central disk surrounded by rings. Verify that the radius of the central disk, known as the Airy disk, is Ps 1.22Ad / D and subtends an angle x 2 + y2, ( 4.3-8) l(x,y) p o A 1.22 . D (4.3-9) Airy Disk Half Angle Diffraction pattern z ......., x I(p) Aperture ..", ..", ..", ..", ..", ..", ..", ..", ..", , , ..... ..... ..... ..... " 0 - y d D---1 I---- ops p Figure 4.3-4 The Fraunhofer diffraction pattern from a circular aperture produces the Airy pattern with the radius of the central disk subtending an angle 0 1.22A/ D.
124 CHAPTER 4 FOURIER OPTICS The Fraunhofer approximation is valid for distances d that are usually extremely large. They are satisfied in applications of long-distance free-space optical communi- cation such as laser radar (lidar) and satellite communication. However, as shown in Sec. 4.2B, if a lens of focal length / is used to focus the diffracted light, the intensity pattern in the focal plane is proportional to the squared magnitude of the Fourier transform of P x, Y evaluated at lJ x x)../ and lJ y Y Af. The observed pattern is therefore identical to that obtained from (4.3-5), with the distance d replaced by the focal length /. EXERCISE 4.3-3 Spot Size of a Focused Optical Beam. A beam of light is focused using a lens of focal length f with a circular aperture of diameter D (Fig. 4.3-5). If the beam is approximated by a plane wave at points within the aperture, verify that the pattern of the focused spot is 1(x,y) x2 + y2, (4.3-10) P o 7rDp/Af where 10 is the peak intensity. Compare the radius of the focused spot Ps f 1.22A D ' (4.3-1]) to the spot size obtained when a Gaussian beam of waist radius Vo is focused by an ideal lens of infinite aperture [see (3.2-15)]. x Aperture f Diffraction pattern Lens - y -----.. , ... ... ..... y D Figure 4.3-5 Focusing of a plane wave transmitted through a circular aperture of diameter D. * B. Fresnel Diffraction The theory of Fresnel diffraction is based on the assumption that the incident wave is multiplied by the aperture function P x, y and propagates in free space in accordance with the Fresnel approximation. If the incident wave is a plane wave traveling in the z-direction with intensity Ii, the complex amplitude immediately after the aperture is / x, Y liP x, Y . Using (4.1-20), the diffraction pattern I x, y 9 x, Y 2 at a distance d is 00 2 1- P x', y' x x' 2 + y y' 2 dx' dy' I x,y 1, . Ad 2 exp J1r )"d . -00 (4.3-12)
4.3 DIFFRACTION OF LIGHT 125 It is convenient to normalize all distances using Ad as a unit of distance, so that X x Ad and X' x' Ad are the normalized distances (and similarly for y and y'). Equation (4.3-12) then gives (X) 2 I X,Y Ii p X', y' exp j7r X X' 2 + y y' 2 dX'dY' . -ex::> (4.3-13) The integral in (4.3-13) is the convolution of p X, Y and exp j7r X 2 + y 2 . The real and imaginary parts of exp j1f X 2 , COS 7r X 2 and sin 7r X 2 , are plotted in Fig. 4.3-6. They oscillate at an increasing frequency and their first lobes lie in the intervals X < 1 2 and X < 1, respectively. The total area under the function exp j7r X 2 is 1, with the main contribution to the area coming from the first few lobes, since subsequent lobes cancel out. If a is the radius of the aperture, the radius of the normalized function p X, Y is a Ad . The result of the convolution, which depends on the relative size of the two functions, is therefore governed by the Fresnel number N F a 2 Ad. COS 7rX2 ) sin 7rX2 1 -1 2 3 X -3 -2 -1 3 X -3 -2 ) -1 -)- Figure 4.3-6 The real and imaginary parts of exp( j7r X 2 ). If the Fresnel number is large, the normalized width of the aperture a Ad is much greater than the width of the main lobe, and the convolution yields approximately the wider function p X, Y . Under this condition the Fresnel diffraction pattern is a shadow of the aperture, as would be expected from ray optics. Note that ray optics is applicable in the limit A 0, which corresponds to the limit NF 00. In the opposite limit, when NF is small, the Fraunhofer approximation becomes applicable and the Fraunhofer diffraction pattern is obtained. EXAMPLE 4.3-1. Fresnel Diffraction from a Slit. Assume that the aperture is a slit of width D 2 a, so that p(x, y) 1 when Ixl < a, and 0 elsewhere. The normalized coordinate X xl J >"d and P(X, Y) 1, N F (4.3-14) 0, elsewhere, where N F a 2 />"d is the Fresnel number. Substituting into (4.3-13), we obtain I(X, Y) Iilg(X)12, where g(X) VNF exp j7r(X X')2 dX' -VNF X+VNF exp( j7r X'2) dX'. (4.3-15) X-VNF
126 CHAPTER 4 FOURIER OPTICS This integral is usually written in terms of the Fresnel integrals x 7ra 2 cas do 2 x 7ra 2 sin d( 2 (4.3-16) C(x) Sex) o o which are available in the standard computer mathematical libraries. The complex function g( X) may also be evaluated us ing Fourier-transform techniques. Since g( x) is the convolution of a rectangular function of width V NF and exp( j7r X 2 ), its Fourier transform G(v x ) ex: sinc( N F v x ) exp(j7rV;) (see Table A.2-1 in Appendix A). Thus_ g(X) may be computed by determining the inverse Fourier transform of G(v x ). If N F » 1, the width of sinc( N F v x ) is muc h n arrower than the width of the first lobe of exp(j7rv;) (see Fig. 4.3-6) so that G(v x ) sinc( V NF v x ) and g(X) is the rectangular function representing the aperture shadow. The diffraction pattern from a slit is plotted in Fig. 4.3-7 for different Fresnel numbers corre- sponding to different distances d from the aperture. At very small distances (very large N F )- the diffraction pattern is a perfect shadow of the slit. As the distance increases (N F decreases), the wave nature of light is exhibited in the form of small oscillations around the edges of the aperture (see also the diffraction pattern in Fig. 4.3-1). For very small N F , the Fraunhofer pattern described by (4.3-6) is obtained. This is a sine function with the first zero subtending an angle AI D A/2a. xi. A 2a ----- --- ------ -- ---- -- --- --- -- --- ------ (a) 2a .... - - ... -- ... .-c.... - .... ... -- ..... t t NF= 10 I ---- -- ---- --- -- --- -- z 0.5 ---- N 01 ------- F = · --- ---- - . x xi. x ..t NF= 10 NF= I NF=0.5 NF=O.I ..-------- (b) 2a -------- --...-- ----...... ---. - - . . . 211 - - ...... ...--.. ..,,....-- Figure 4.3-7 Fresnel diffraction from a slit of width D 20-. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam. (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers N F 10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at Ixl (AI D)d represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number N F a 2 I Ad 0.5. EXAMPLE 4.3-2. Fresnel Diffraction from a Gaussian Aperture. If the aperture func- tion p(x, y) is the Gaussian function p(x, y) exp[ (x 2 + y2)IV{/5], the Fresnel diffraction equa- tion (4.3-12) may be evaluated exactly by finding the convolution of exp[ (x 2 + y2)/1V"c?] with ho exp[ j7r(.r 2 + y2) I Ad] using, for example, Fourier transform techniques (see Appendix A). The resultant diffraction pattern is I(x,y) {V o Ii fT( d) 2 exp x2 + y2 , (4.3-17) where Tf T2 ( d) {fT 5 + 06 d 2 and 0 0 AI7r UTo. The diffraction pattern is a Gaussian function of 1/c 2 half-width TI T (d). For small d, IT(d) Vo; but as d increases, 1TT( d) increases and approaches 1fT( d) od when d is sufficiently large for
4.4 IMAGE FORMATION 127 the Fraunhofer approximation to be applicable, so that the angle subtended by the Fraunhofer diffrac- tion pattern is Bo. These results are illustrated i n Fig. 4.3-8, which is analogous to the illustration in Fig. 4.3-7 for diffraction from a slit. The wave diffracted from a Gaussian aperture is the Gaussian beam described in detail in Chapter 3. xi e - A 0- 7rWo _--- --- - ---- --- - --. --- ---- --- -- ---- ---- (a) 2W o - NF= 10 1 0.5 --- Z -- --- -- --- -- ---- ---- N 01 --------- P = . ---- ---. xi xi x x Np=O.l Np=IO NF=1 NF = 0.5 (b) 2W o . . . 2a - - - Figure 4.3-8 Fresnel diffraction pattern for a Gaussian aperture of radius W o at distances d such that the parameter (7r /2)lfTJ / Ad, which is analogous to the Fresnel number N F in Fig. 4.3-7, is 10, 1, 0.5, and 0.1. These values correspond to W( d)/W o 1.001, 1.118, 1.414, and 5.099, respectively. The diffraction pattern is Gaussian at all distances. Summary In the order of increasing distance from the aperture, the diffraction pattern is: 1. A shadow of the aperture. 2. A Fresnel diffraction pattern, which is the convolution of the normalized aperture function with exp j7r X 2 + y2 . 3. A Fraunhofer diffraction pattern, which is the squared-absolute value of the Fourier transform of the aperture function. The far field has an angular divergence proportional to AD, where D is the diameter of the aperture. 4.4 IMAGE FORMATION An ideal image formation system is an optical system that replicates the distribution of light in one plane, the object plane, into another, the image plane. Since the optical transmission process is never perfect, the image is never an exact replica of the object. Aside from image magnification, there is also blur resulting from imperfect focusing and from the diffraction of optical waves. This section is devoted to the description of image formation systems and their fidelity. Methods of linear systems, such as the impulse response function and the transfer function (Appendix B), are used to characterize image formation. A simple ray-optics approach is presented first, then a treatment based on wave optics is subsequently developed.
128 CHAPTER 4 FOURIER OPTICS - A. Ray-Optics of a Single-Lens Imaging System Consider an imaging system using a lens of focal length f at distances d 1 and d 2 from the object and image planes, respectively, as shown in Fig. 4.4-1. When 1 d 1 + 1 d 2 1 f, the system is focused so that paraxial rays emitted from each point in the object plane reach a single corresponding point in the image plane. Within the ray theory of light, the imaging is "ideal," with each point of the object producing a single point of the image. The impulse response function of the system is an impulse function. Lens Object Image ... ..... f ... d 2 Figure 4.4-1 Rays in a focused imaging system. d 1 f Suppose now that the system is not in focus, as illustrated in Fig. 4.4-2, and assume that the focusing error is E 1 1 + d 1 d 2 1 f. (4.4-1) Focusing Error A point in the object plane generates a patch of light in the image plane that is a shadow of the lens aperture. The distribution of this patch is the system's impulse response function. For simplicity, we shall consider an object point lying on the optical axis and determine the distribution of light h x, y it generates in the image plane. x I I I I Ps I I . '- ...... I ........ h(x, y) P .......... .. I I I hex, y) I I - ....,,----. . Ps y d20 d2 .... x d) ... (a) (b) Figure 4.4-2 (a) Rays in a defocused imaging system. (b) The impulse response function of an imaging system with a circular aperture of diameter D is a circle of radius Ps Ed 2 D /2.. where E is the focusing error. Assume that the plane of the focused image lies at a distance d 20 satisfying the imaging equation 1 d 20 + 1 d 1 1 f. The shadow of a point on the edge of the aperture at a radial distance p is a point in the image plane with radial distance Ps where the ratio Ps P d 20 d 2 d 20 1 d 2 d 20 1 d 2 1 f 1 d l 1 d 2 1 d 2 f fd 2 . If P x, y is the aperture function, also called the pupil .
4.4 IMAGE FORMATION 129 function [p x, y 1 for points inside the aperture, and 0 elsewhere], then h x, y is a scaled version of p x, y magnified by a factor Ps P Ed 2 , so that x Y Ed 2 ' Ed 2 h x, Y DC P . (4.4-2) Impulse Response Function (Ray-Optics) As an example, a circular aperture of diameter D corresponds to an impulse re- sponse function confined to a circle of radius Ps 1 Ed 2 D, 2 (4.4-3) Blur Spot Radius as illustrated in Fig. 4.4-2. The radius Ps of this "blur spot" is an inverse measure of re- solving power and image quality. A small value of Ps means that the system is capable of resolving fine details. Since Ps is proportional to the aperture diameter D, the image quality may be improved by use of a small aperture. A small aperture corresponds to a reduced sensitivity of the system to focusing errors, so that it corresponds to an increased "depth of focus." B. Wave-Optics of a 4-f Imaging System Consider now the two-lens imaging system illustrated in Fig. 4.4-3. This system, called the 4-1 system, serves as a focused imaging system with unity magnification, as can be easily verified by ray tracing. I ..... x f + f ..... f + f p(X,y) x Object plane Fourier plane Image plane Figure 4.4-3 The 4- f imaging system. If an inverted coordinate system is used in the image plane, the magnification is unity. The analysis of wave propagation through this system becomes simple if we rec- ognize it as a cascade of two Fourier-transforming subsystems. The first subsystem (between the object plane and the Fourier plane) performs a Fourier transform, and the second (between the Fourier plane and the image plane) performs an inverse Fourier transform since the coordinate system in the image plane is inverted (see Exercise 4.2- 2). As a result, in the absence of an aperture the image is a perfect replica of the object. Let f x, y be the complex amplitude transmittance of a transparency placed in the object plane and illuminated by a plane wave exp j kz traveling in the z direction, as illustrated in Fig. 4.4-4, and let 9 x, Y be the complex amplitude in the image plane. The first lens system analyzes f x, y into its spatial Fourier transform and separates
130 CHAPTER 4 FOURIER OPTICS its Fourier components so that each point in the Fourier plane corresponds to a single spatial frequency. These components are then recombined by the second lens system and the object distribution is perfectly reconstructed. Lens -.- y -- - -....- Z Fourier plane x .::'.':. - g(x, y) "'" ..........-. Lens .. : . ........... . -- . --- - - --=... f{x, y) - ." . . y f f Image x plane , . . .i_I-. -- .\ .-- t , . .... ., .._0- - -- -- -;.. ,.- Plane wave Object f plane Figure 4.4-4 The 4- 1 imaging system performs a Fourier transform followed by an inverse Fourier transform, so that the image is a perfect replica of the object. The 4- f imaging system can be used as a spatial filter in which the image 9 x, y is a filtered version of the object f x, y . Since the Fourier components of f x, yare avail- able in the Fourier plane, a mask may be used to adjust them selectively, blocking some components and transmitting others, as illustrated in Fig. 4.4-5. The Fourier component of f x, y at the spatial frequency l/x, l/y is located in the Fourier plane at the point x Afl/x, y Afl/y. To implement a filter of transfer function H l/x, l/y , the complex amplitude transmittance p x, y of the mask must be proportional to H x Af, y Af . Thus, the transfer function of the filter realized by a mask of transmittance p x, y is H V x , l/y P Afl/x, Afl/y , (4.4-4) Transfer Function 4-1 System where we have ignored the phase factor j exp j2kf associated with each Fourier transform operation [the argument of h z in (4.2-8)]. The Fourier transforms G l/x, v y and F l/x, l/y of 9 x, y and f x, yare related by G l/x, l/y H l/x, v y F l/x, l/y . This is a rather simple result. The transfer function has the same shape as the pupil function. The corresponding impulse response function h x, y is the inverse Fourier transform of H l/x, l/y , h x,y 1 x Y Af ' Af , (4.4-5) Impulse Response Function 4-1 System where P l/x, l/y is the Fourier transform of p x, y .
4.4 IMAGE FORMATION 131 g(x, y) Lens y -- ...- -....- Z Mask - - _l p(x,y) ...- x Lens - -- , .... .... '-.... f(x, y) - y f f f Image x plane - - Fourier I plane Plane wave - Object plane - f -- Figure 4.4-5 Spatial filtering. The transparencies in the object and Fourier planes have complex amplitude transmittances f(x, y) and p(x, y). A plane wave traveling in the z direction is modulated by the object transparency, Fourier transformed by the first lens, multiplied by the transmittance of the mask in the Fourier plane, and inverse Fourier transformed by the second lens. As a result, the complex ampJitude in the image plane g(x, y) is a filtered version of f(x, y). The system has a transfer function H(v x , v y ) p(Afv x , >..fv y ). Examples of Spatial Filters . The ideal circularly symmetric l ow -pass filter has a transfer function H v x , v y are smaller than the cutoff frequency V s and blocks higher frequencies. This filter is implemented by a mask in the form of a circular aperture of diameter D, with D 2 vsAf. For example, if D 2 em, A 1 /L m , and f 100 cm, the cutoff frequency (spatial bandwidth) V s D 2AI 10 lines/mm. This filter eliminates spatial frequencies that are greater than 1 0 lines/mm, so that the smallest size of discernible detail in the filtered image is approximately 0.1 mm. . The high-pass filter is the complement of the low-pass filter. It blocks low fre- quencies and transmits high frequencies. The mask is a clear transparency with an opaque central circle. The filter output is high at regions of large rate of change and small at regions of smooth or slow variation of the object. The filter is therefore useful for edge enhancement in image-processing applications. . The vertical-pass filter blocks horizontal frequencies and transmits vertical fre- quencies. Only variations in the x direction are transmitted. If the mask is a vertical slit of width D, the highest transmitted frequency is v y D 2 Af. Examples of these filters and their effects on images are illustrated in Fig. 4.4-6.
132 CHAPTER 4 FOURIER OPTICS Object Mask Image - - -1 (a) ; (b) \ - - (c) ......... ............. ............... ................. ................... ..................... ..................... ....................... ....................... ....................... ....................... ....................... ....................... ....................... ....................... ....................... ..................... ..................... ................... ................. ............... ............. ......... Figure 4.4-6 Examples of object, mask, and filtered image for three spatia] filters: (a) low-pass filter; (b) high-pass filter; (c) vertical-pass filter. Black means the transmittance is zero and white means the transmittance is unity. c. Wave Optics of a Single-Lens Imaging System We now consider image formation in the single-lens imaging system shown in Fig. 4.4- 7, using a wave-optics approach. We first determine the impulse response function, and then derive the transfer function. These functions are determined by the defocusing error E, given by (4.4-1), and by the pupil function p x, y (the transmittance of the aperture in the lens plane). The pupil function in this single-lens imaging system plays the same role of the mask function in the 4- f imaging system described in the previous section. x d] v -- J VI o y p(x,y) Aperture plane Lens Object plane y h( x,y ) x Image plane Figure 4.4-7 Single-lens imaging system.
4.4 IMAGE FORMATION 133 Impulse Response Function To determine the impulse response function we consider an object composed of a single point (an impulse) on the optical axis at the point 0,0 , and follow the emitted optical wave as it travels to the image plane. The resultant complex amplitude is the impulse response function h x, y . An impulse in the object plane produces in the aperture plane a spherical wave approximated by [see (4.1-18)] U x, y hI exp . x2 + y2 (4.4-6) where hI j >"d i exp jkd l . Upon crossing the aperture and the lens, U x, y is multiplied by the pupil function p x, y and the lens quadratic phase factor exp j k x 2 + y2 2 f , becoming U I X, Y U x,y exp . x2 + y2 p x, y . (4.4-7) The resultant field U I x, y then propagates in free space a distance d 2 . In accordance with (4.1-20) it produces the impulse response function (X) h x,y h 2 U I I 1 X , Y exp . J1r x x' 2 + Y >"d 2 I 2 (4.4-8) -(X) where h 2 j >"d 2 exp j kd 2 . Substituting from (4.4-6) and (4.4-7) into (4.4-8) and casting the integrals as a Fourier transform, we obtain h x,y h I h2 ex P . x 2 + y2 x Y , (4.4-9) where P v x , v y is the Fourier transform of the function PI x, Y p x,y exp . x2 + y2 J1rE >.. ' (4.4-10) Generalized Pupil Function known as the generalized pupil function. The factor f. is the focusing error given by (4.4-1). For a high-quality imaging system, the impulse response function is a narrow func- tion, extending only over a small range of values of x and y. If the phase factor 7r x 2 + y2 >"d 2 in (4.4-9) is much smaller than 1 for all x and y within this range, it can be neglected, so that x h x,y >"d 2 ' >"d 2 , ( 4.4-11 ) Impulse Response Function
134 CHAPTER 4 FOURIER OPTICS where ho hI h 2 is a constant of magnitude 1 >"d l 1 >"d 2 . It follows that the system's impulse response function is proportional to the Fourier transform of the generalized pupil function PI x, Y evaluated at V x x >"d 2 and v y y >..d 2 . If the system is focused EO, then PI x, Y P x, Y , and x >"d 2 ' >"d 2 , (4.4-12) where P v x , v y is the Fourier transform of P x, Y . This result is similar to the corre- sponding result in (4.4-5) for the 4- f system. EXAMPLE 4.4-1. Impulse Response Function of a Focused Imaging System with a Circular Aperture. If the aperture is a circle of diameter D so that p( x, y ) 1 if P x 2 + y2 < D 1 2 .. and zero otherwise, then the impulse response function is h(x,y) x2 + y2 , ( 4.4-13) w Dp/ )"d 2 P and Ih(O,O)1 (wD 2 /4)..2d 1 d 2 ). This is a circularly symmetric function whose cross section is shown in Fig. 4.4-8. It drops to zero at a radius d Ps (4.4-14) and oscillates slightly before it vanishes. The radius Ps is therefore a measure of the size of the blur circle. If the system is focused at 00, d 1 00, d 2 f, and Ps 1.22)"F#, (4.4-15) Spot Radius where F # f 1 D is the lens F-number. Thus, systems of smaller F # (larger apertures) have better image quality. This assumes, of course, that the larger lens does not introduce geometrical aberrations. x d} d2 h(x,y) U U I o y D h{x,y) o .. ,- p Ps = ] .22).. d2 D x Figure 4.4-8 Impulse response function of an imaging system with a circular aperture
4.4 IMAGE FORMATION 135 Transfer Function The transfer function of a linear system can only be defined when the system is shift invariant (see Appendix B). Evidently, the single-lens imaging system is not shift invariant since a shift of a point in the object plane is accompanied by a different shift AI in the image plane, where AI == -d 2 / d 1 is the magnification. The image is different from the object in two ways. First, the image is a magnified replica of the object, i.e., the point (x, y) of the object is located at a new point (M x, My) in the image. Second, every point is smeared into a patch as a result of defocusing or diffraction. We can therefore think of image formation as a cascade of two systems - a system of ideal magnification followed by a system of blur, as depicted in Fig. 4.4-9. By its nature, the magnification system is shift variant. For points near the optical axis, the blur system is approximately shift invariant and therefore can be described by a transfer function. (a) x (b) I Magnification I Blur . Figure 4.4-9 The imaging system in (a) is regarded in (b) as a combination of an ideal imaging system with only magnification, followed by shift-invariant blur in which each point is blurred into a patch with a distribution equal to the impulse response function. The transfer function H(v x , v y ) of the blur system is determined by obtaining the Fourier transform of the impulse response function h( x, y) in (4.4-11). The result is H(v x , v y ) PI (Ad 2 v x , Ad 2 v y ), ( 4.4-16) Transfer Function where PI (x, y) is the generalized pupil function and we have ignored a constant phase factor exp( - j kd 1 ) exp( - j kd 2 ). If the system is focused, then H(v x , v y ) p(Ad 2 v x , Ad 2 v y ), ( 4.4-17) where p( x, y) is the pupil function. This result is identical to that obtained for the 4- f imaging system [see (4.4-4)]. If the aperture is a circle of diameter D, for example, then the transfer function is constant within a circle of radius vs, where D V s == 2Ad 2 ' (4.4-18) and vanishes elsewhere, as illustrated in Fig. 4.4-10.
136 CHAPTER 4 FOURIER OPTICS x dl v - d2 H(v x , v y ) y , VI o . D , .' h (x, y) lJ x lJ lJ s Y x Figure 4.4-1 0 Transfer function of a focused imaging system with a circular aperture of diameter D. The system has a spatia] bandwidth V s D /2Ad 2 . · If the lens is focused at infinity, i.e., d 2 f, V s 1 2A.F# ' (4.4-19) Spatial Bandwidth where F# f D is the lens F-number. For example, for an F-2lens (F# f D 2) and for A. 0.5 /-Lm, V s 500 lines/mm. The frequency V s is the spatial bandwidth, i.e., the highest spatial frequency that the imaging system can transmit. D. Near-Field Imaging It was shown in Sec. 4.1 B that the spatial bandwidth of light propagating in free space, at a wavelength ,x, is ,x -1 cycles mm. Fourier components of the object distribution with spatial frequencies greater than ,x -1 lead to evanescent waves that decay rapidly and diminish at distances from the object plane of the order of a wavelength, so that object features smaller than a wavelength cannot be transmitted. Moreover, it was shown in Sec. 4.4C that an imaging system using a lens with a specified F# has an impulse response function whose radius is 1.22A.F#, so that points separated by a distance smaller than 1.22'xF# cannot be discriminated [see Fig. 4.4-11(a)]. Another imaging modality that makes use of a laser beam focused by a lens to scan an object, as illustrated in Fig. 4.4-11(b), behaves similarly. The resolution of this system is dictated by the size of the focused spot, which has a radius of 1.22'xF #, as was shown in Exam- ple 4.4-1. In both of these cases, therefore.. object details whose dimensions are much smaller than a wavelength are washed out in the scanned image. This fundamental limit on the resolution of image-formation systems is often referred to as the diffraction limit. The diffraction limit may be circumvented, however. Light can be localized to a spot with dimensions much smaller than a wavelength within a single plane. The difficulty is that the evanescent waves fully decay at a short distance away from that plane, whereupon the spot diverges and acquires a size exceeding the wavelength. At yet greater distances, the wave ultimately becomes a spherical wave. Hence, the diffraction limit can be circumvented if the object is brought to the very vicinity of the sub-wavelength spot, where it is illuminated before the beam waist grows. This may be implemented in a scanning configuration by passing the illumination beam through an aperture of diameter much smaller than a wavelength, as depicted in Fig. 4.4-11(c).
4.4 IMAGE FORMATION 137 r ft f.\ , -- Aperture AT Impulse _ response "'I AT Focused spot Lens L . I.,. Illumination Object Image Illumination Object (a) (b) (c) Figure 4.4-11 In a single-lens imaging system, the sub-wavelength spatial details of an object are washed out in an image formed (a) by a single lens, or (b) by a focused laser scanning system. (c) A scanning imaging system that uses illumination transmitted through a sub-wavelength aperture preserves the subwavelength details of the object provided that the object plane is placed at a sub- wavelength distance from the aperture plane. The object is placed at a sub-wavelength distance from the aperture (usually less than half the diameter of the aperture) so that the beam illuminates a sub-wavelength-size area of the object. Upon transmission through the object, the traveling components of the wave form a spherical wave whose amplitude is proportional to the object transmittance at the location of the spot illumination. The resolution of this imaging system is therefore of the order of the aperture size, which is much smaller than the wavelength. An image is constructed by raster-scanning the illuminated aperture across the object and recording the optical response via a conventional far-field imaging system. The technique is known as near-field optical imaging or scanning near-field optical microscopy (SNOM). It falls within the domain of nanophotonics since the imaging takes place over a subwavelength (nanometer) spatial scale. SNOM is typically implemented by sending the illumination light through an optical fiber with an aluminum-coated tapered tip, as illustrated in Fig. 4.4-12. The light is guided through the fiber by total internal reflection. As the diameter of the fiber decreases, the light is guided by reflection from the metallic surface, which acts like a conical mirror. As the fiber diameter grows yet smaller in the region of the tip, the wave can no longer be guided (see Sec. 8.]) and becomes evanescent. The distribution of the illumination wave at, and beyond, the end of the tip is complex and must be determined numerically. Aperture diameters and spatial resolutions of the order of tens of nanometers are achieved in SNOM with visible light. Since the tip of the fiber scans the object at a distance of only a few nanometers, an elaborate feedback system must be employed to maintain the distance for a specimen of arbitrary topography. Applications of SNOM include non-destructive characterization of inorganic, organic, composite, and biological materials and nanostructures. . Metal I coating Incident light Object T A t 1 Glass fiber Tapered fiber tip Figure 4.4-12 An optical fiber with a ta- pered metal-coated tip for near-field imaging.
138 CHAPTER 4 FOURIER OPTICS 4.5 HOLOGRAPHY Holography involves the recording and reconstruction of optical waves. A hologram is a transparency that contains a coded record of the optical wave, including its am- plitude and phase properties. Consider a monochromatic optical wave whose complex amplitude in some plane, say the z 0 plane, is U o x, y . If, somehow, a thin optical element (call it a transparency) with complex amplitude transmittance t x, y equal to U o x, y were able to be made, it would provide a complete record of the wave. The wave could then be reconstructed simply by illuminating the transparency with a uniform plane wave of unit amplitude traveling in the z direction. The transmitted wave would have a complex amplitude in the z 0 plane U x, y 1 · t x, Y U o x, Y . The original wave would then be reproduced at all points in the z 0 plane, and therefore reconstructed everywhere in the space z > O. As an example, we know that a uniform plane wave traveling at an angle () with respect to the z axis in the x z plane has a complex amplitude U o x, y exp j kx sin (). A record of this wave would be a transparency with complex amplitude transmittance t x, y exp j kx sin () . Such a transparency acts as a prism that bends an incident plane wave exp jkz by an angle () (see Sec. 2.4B), thus reproducing the original wave. The question is how to make a transparency t x, y from the original wave U 0 x, y . One key impediment is that optical detectors, including the photographic emulsions used to make transparencies, are responsive to the optical intensity, U o x, y 2, and are therefore insensitive to the phase arg U o x, y . Phase information is obviously important and cannot be disregarded, however. For example, if the phase of the oblique wave U o x, y exp j kx sin () were not recorded, neither would the direction of travel of the wave. To record the phase of U o x, y , a code must be found that trans- forms phase into intensity. The recorded information could then be optically decoded in order to reconstruct the wave. . The Holographic Code The holographic code is based on mixing the original wave (hereafter called the object wave) U o with a known reference wave U r and recording their interference pattern in the z 0 plane. The intensity of the sum of the two waves is photographically recorded and a transparency of complex amplitude transmittance t, proportional to the intensity, is made [Fig. 4.5-I(a)]. The transmittance is therefore given by t ex: U o + U r 2 U r 2 + U o 2 + U;U o + UrU;, Ir + 10 + U;U o + UrU;, Ir + 10 + 2 IrIo cas arg U r arg U o , (4.5-1) where Ir and 10 are, respectively, the intensities of the reference wave and the object wave in the z 0 plane. The transparency, called a hologram, clearly carries coded information pertinent to the magnitude and phase of the wave U o . In fact, as an interference pattern the transmittance t is highly sensitive to the difference between the phases of the two waves, as was shown in Sec. 2.5 (the temporal analog to holography is heterodyning, discussed in Sec. 2.6). As indicated above, ordinary photography is responsive only to the intensity of the incident wave and therefore records no phase information. To decode the information in the hologram and reconstruct the object wave, the reference wave U r is again used to illuminate the hologram [Fig. 4.5-1(b)]. The result is a wave with complex amplitude U tU r ex: UrIr + UrIo + IrUo + U;U; (4.5-2)
4.5 HOLOGRAPHY 139 z x Object x Object (a) Recording Reference Reference Z Hologram Hologram (b) Reconstruction Figure 4.5-1 (a) A hologram is a transparency on which the interference pattern between the original wave (object wave) and a reference wave is recorded. (b) The original wave is reconstructed by illuminating the hologram with the reference wave. in the hologram plane z O. The third term on the right-hand side is the original wave multiplied by the intensity Ir of the reference wave. If Ir is uniform (independent of x and y), this term constitutes the desired reconstructed wave. But it must be separated from the other three terms. The fourth term is a conjugated version of the original wave modulated by U;. The first two terms represent the reference wave, modulated by the sum of the intensities of the two waves. If the reference wave is selected to be a uniform plane wave propagating along the z axis Ir exp j kz , then in the z 0 plane U f x, y Ir is a constant independent of x and y. Dividing (4.5-2) by U r Ir gives U x, y ex Ir + 10 x, y + Ir U o x, y + Ir U; x, y · (4.5-3) Reconstructed Wave in Plane of Hologram The significance of the various terms in (4.5-3), and the methods of extracting the original wave (the third term), are clarified by means of a number of examples. . EXAMPLE 4.5-1. Hologram of an Oblique Plane Wave. If the object wave is an oblique plane wave a t ang le 0 [Fig. 4.5-2(a)], Uo (x, y ) 10 exp( jkx sin 0), then (4.5-3) gives U(x, y) ex: Ir + 10 + V IrIo exp( jkxsinO) + V IrIo exp( jkx sin 0). Since the first two terms are constant, they correspond to a wave propagating in the z direction (the continuance of the reference wave). The third term corresponds to the original object wave, whereas the fourth term represents the conjugate wave, a plane wave traveling at an angle O. The object wave is therefore separable from the other waves. In fact, this hologram is nothing but a recording of the interference pattern formed from two oblique plane waves at an angle 0 (Sec. 2.5A). It serves as a sinusoidal diffraction grating that splits an incident reference wave into three waves at angles 0, 0, and 0 [see Fig. 4.5-2(b) and Sec. 2.4B]. EXAMPLE 4.5-2. Hologram of a Point Source. Here the object wave is a spherical wave originating at the point ro (0, 0, d), as illustrated in Fig. 4.5-3, so that Uo(x, y) ex: exp( jklr rol)/Ir rol, where r (x, y, 0). The first term of (4.5-3) corresponds to a plane wave traveling in the z direction, whereas the third is proportional to the amplitude of the original spherical wave originating at (0, 0, d). The fourth term is proportional to the amplitude of the conjugate wave U (x, y) ex: exp(jklr roD/lr rol, which is a converging spherical wave centered at the point (0, 0, d). The second term is proportional to l/lr rol2 and its corresponding wave therefore travels in the z direction with very small angular spread since its intensity varies slowly in the transverse plane.
140 CHAPTER 4 FOURIER OPTICS x x Object z Reference Reference \ \ \ \ \ \ \ \ \\\\\\' z -f44 1 t. .tt....Cf.. " ) ft.f t ,....... ,'......4fl " '44ft .t,,1 Hologram Hologram Object (a) Recording Conjugate (b) Reconstruction Figure 4.5-2 The hologram of an oblique plane wave is a sinusoidal diffraction grating. . x x Object .'1) Object Reference Reference z III.... ...111 Z Hologram d (a) Recording (b) Reconstruction Figure 4.5-3 Hologram of a spherical wave originating from a point source. The conjugate wave forms a real image of the point. Off-Axis Holography . One means of separating the four components of the reconstructed wave is to ensure that they vary at well-separated spatial frequencies, so that they have well-separated directions. This form of spatial frequency multiplexing (see Sec. 4.1 A) is assured if the object and reference waves are offset so that they arrive from well-separated directions. . Let us consider the case when the object wave has a complex amplitude U o x, y f x, y exp j kx sin e . This is a wave of complex envelope f x, y modulated by a phase factor equal to that introduced by a prism with deflection angle e. It is assumed that f x, y varies slowly so that its maximum spatial frequency V s corresponds to an angle Os sin- l AV s much smaller than e. The object wave therefore has directions centered about the angle e, as illustrated in Fig. 4.5-4. Equation (4.5-3) gives u x, y ex: Ir + f x, y 2 + Ir f x, y exp jkx sin () + Ir f* x, y exp +jkx sin () · (4.5-4) The third term is evidently a replica of the object wave, which arrives from a direction at angle e. The presence of the phase factor exp + j kx sin 0 in the fourth term indicates that it is deflected in the e direction. The first term corresponds to a plane wave traveling in the z direction. The second term, usually known as the ambiguity term, corresponds to a nonuniform plane wave in directions within a cone of small angle 2() s around the z direction. The offset of the directions of the object
4.5 HOLOGRAPHY 141 z Object x x Reference Ambiguity Reference . .. . .. .... . . . .....- ......-.- ff..".- "f, - Z Hologram Hologram () Conjugate Object (a) Recording (b) Reconstruction Figure 4.5-4 Hologram of an off-axis object wave. The object wave is separated from both the reference and conjugate waves. and reference waves results in a natural angular separation of the object and conjugate waves from each other and from the other two waves if () > 3()s, thus allowing the original wave to be recovered unambiguously. An alternative method of reducing the effect of the ambiguity wave is to make the intensity of the reference wave much greater than that of the object wave. The ambiguity wave [second term of (4.5-3)] is then much smaller than the other terms since it involves only object waves; it is therefore relatively negligible. Fourier- Transform Holography The Fourier transform F v x , v y of a function / x, y may be computed optically by use of a lens (see Sec. 4.2). If / x, y is the complex amplitude in one focal plane of the lens, then F x >.../, y >.../ is the complex amplitude in the other focal plane, where / is the focal length of the lens and>'" is the wavelength. Since the Fourier transform is usually a complex-valued function, it cannot be recorded directly. The Fourier transform F x >.../, y >.../ may be recorded holographically by regard- ing it as an object wave, U o x, y / x >.../, y >.../ , mixing it with a reference wave Uf x, y , and recording the superposition as a hologram [Fig. 4.5-5(a)]. Reconstruc- . tion is achieved by illumination of the hologram with the reference wave as usual. The reconstructed wave may be inverse Fourier transformed using a lens so that the original function / x, y is recovered [Fig. 4.5-5(b)]. Holographic Spatial Filters A spatial filter of transfer function H v x , v y may be implemented by use of a 4- / opti- cal system with a mask of complex amplitude transmittance p x, y H x >.../, y >.../ placed in the Fourier plane (see Sec. 4.4B). Since the transfer function H v x , v y is usually complex-valued, the mask transmittance p x, y has a phase component and is difficult to fabricate using conventional printing techniques. If the filter im- pulse response function h x, y is real-valued, however, a Fourier-transform holo- gram of h x, y may be created by holographically recording the Fourier transform U o x, y H x >.../, y >...f . Using the Fourier transform of the input / x, y as a reference, U r x, y F x >.../, y >.../ , the hologram constructs the wave U r x,y U o x,y F x >...f, y >...f H x >...f, y >.../ · (4.5-5)
142 CHAPTER 4 FOURIER OPTICS F , , , F* " ',,- Hologram f (a) Recording (b) Reconstruction Figure 4.5-5 (a) Hologram of a wave whose complex amplitude represents the Fourier transform of a function I(x, y). (b) Reconstruction of I(x, y) by use of a Fourier-transform lens. U r U r F ------ Hologram f The inverse Fourier transform of the reconstructed object wave, obtained with a lens of focal length I as illustrated in Fig. 4.5-6(b), therefore yields a complex amplitude 9 x, y with a Fourier transform G v x , v y H v x , v y F v x , v y . Thus, 9 x, y is the convolution of I x, y with h x, y . The overall system, known as the Vander I..Iugt filter, performs the operation of convolution, which is the basis of spatial filtering. ------ " " " " , ',,- Hologram g(x,y) f(x,y) (a) Recording (b) Reconstruction Figure 4.5-6 The Vander Lugt holographic filter. (a) A hologram of the Fourier transform of hex, y) is recorded. (b) The Fourier transform of I(x, y) is transmitted through the hologram and inverse Fourier transformed by a lens. The result is a function g(x, y) proportional to the convolution of I ( x, y) and h ( x, y ). The overall process provides a spatial filter with impulse response function hex, y). U r Hologram h(x,y) If the conjugate wave U r x, Y U; x, y F x AI, y AI H* x AI, y AI is, in- stead, inverse Fourier transformed, the correlation, instead of the convolution, of the functions I x, y and h x, y is obtained. The operation of correlation is useful in image-processing applications, including pattern recognition. The Holographic Apparatus An essential condition for the successful fabrication of a hologram is the availability of a monochromatic light source with minimal phase fluctuations. The presence of phase fluctuations results in the random shifting of the interference pattern and the washing out of the hologram. For this reason, a coherent light source (usually a laser) is a necessary part of the apparatus. The coherence requirements for the interference of light waves are discussed in Chapter 1 O.
4.5 HOLOGRAPHY 143 Figure 4.5-7 illustrates a typical experimental configuration used to record a holo- gram and reconstruct the optical wave scattered from the surface of a physical object. Using a beamsplitter, laser light is split into two portions; one is used as the reference wave, whereas the other is scattered from the object to form the object wave. The optical path difference between the two waves should be as small as possible to ensure that the two beams maintain a nonrandom phase difference [the term arg U r arg U o in (4.5-1)]. Laser ". Laser Reference Reference '" '" Object .... ......- ", """. """. ", - - .. Hologram t ''III -._ 'II .. I.. ........ - -- - -- - Object Hologram (a) Recording (b) Reconstruction Figure 4.5-7 Holographic recording and reconstruction. Since the interference pattern forming the hologram is composed of fine lines sep- arated by distances of the order of A sin (), where () is the angular offset between the reference and object waves, the photographic film must be of high resolution and the system must not vibrate during the exposure. The larger (), the smaller the distances between the hologram lines, and the more stringent these requirements are. The object wave is reconstructed when the recorded hologram is illuminated with the reference wave, so that a viewer see the object as if it were actually there, with its three-dimensional character preserved. Volume Holography It has been assumed so far that the hologram is a thin planar transparency on which the interference pattern of the object and reference waves is recorded. We now consider recording the hologram in a relatively thick medium and show that this offers an advantage. Consider the simple case when the object and reference waves are plane waves with wavevectors k r and ko. The recording medium extends between the planes z 0 and z , as illustrated in Fig. 4.5-8. The interference pattern is now a function of x, y, and z: I x,y,z 2 Ir exp jk r . r + 10 exp jko. r Ir + 10 + 2 IrIo cos ko · r k r · r Ir + 10 + 2 IrIo cos kg · r , (4.5-6) where kg ko k r . This is a sinusoidal pattern of period A 27r kg and with the surfaces of constant intensity normal to the vector kg. For example, if the reference wave points in the z direction and the object wave makes an angle () with the z axis, kg 2k sin () 2 and the period is A A 2 sin () 2 (4.5-7)
144 CHAPTER 4 FOURIER OPTICS () , \ , , " " " " " , " "" , , . , , , , , \ , , ,. \ \ \ \ \ \ ,\" \", \\\, ,", \\\\\ ",I' ,:,", \\\\\\,::",J \\\ \\\\\ ,',\\' kg k r ()/2 -072 .,.,.-- .,.,."'" .,.,.... x r -- . -- 1 A z ko A Figure 4.5-8 Interference pattern when the reference and object waves are plane waves. Since I k r I I ko I 27r / A and I kg I 27r / A, from the geometry of the vector diagram 27r / A 2(27r / A) sine () /2), so that A A/2 sine () /2). I as illustrated in Fig. 4.5-8. If recorded in emulsion, this pattern serves as a thick diffraction grating, a volume hologram. The vector kg is called the grating vector. When illuminated with the reference wave as illustrated in Fig. 4.5-9, the parallel planes of the grating reflect the wave only when the Bragg condition sin <p A 2A is satisfied, where <p is the angle between the planes of the grating and the incident reference wave (see Exercise (2.5- 3)). In our case <p () 2, so that sin () 2 A 2A. In view of (4.5-7), the Bragg condition is indeed satisfied, so that the reference wave is indeed reflected. As evident from the geometry, the reflected wave is an extension of the object wave, so that the reconstruction process is successful. x A z (J/2 A Figure 4.5-9 The reference wave is Bragg reflected from the thick hologram and the object wave is reconstructed. Suppose now that the hologram is illuminated with a reference wave of different wavelength A'. Evidently, the Bragg condition, sin () 2 A' 2A, will not be satisfied and the wave will not be reflected. It follows that the object wave is reconstructed only if the wavelength of the reconstruction source matches that of the recording source. If light with a broad spectrum (white light) is used as a reconstruction source, only the "correct" wavelength would be reflected and the reconstruction process would be successful. Although the recording process must be done with monochromatic light, the re- construction can be achieved with white light. This provides a clear advantage in many applications of holography. Other geometries for recording a reconstruction of a volume hologram are illustrated in Fig. 4.5-1 O. Another type of hologram that may be viewed with white light is the rainbow hologram. This hologram is recorded through a narrow slit so that the reconstructed
READING LIST 145 · J:.J O"{) Reference Reference ."" ."" -- - ,.". ", \, · G O'O ", ", ", Reference ", Reference ", .",. ", ", ", -- - - - - - - .» GC (a) Transmission hologram (b) Reflection hologram Figure 4.5-10 Two geometries for recording and reconstruction of a volume hologram. (a) This hologram is recorded with the reference and object waves arriving from the same side, and is reconstructed by use of a reversed reference wave; the reconstructed wave is a conjugate wave traveling in a direction opposite to the original object wave. (b) A reflection hologram is recorded with the reference and object waves arriving from opposite sides; the object wave is reconstructed by reflection from the grating. ", -- - - - - -- - -- - · J:\ O'O image, of course, also appears as if seen through a slit. However, if the wavelength of reconstruction differs from the recording wavelength, the reconstructed wave will ap- pear to be coming from a displaced slit since a magnification effect will be introduced. If white light is used for reconstruction, the reconstructed wave appears as the object seen through many displaced slits, each with a different wavelength (color). The result is a rainbow of images seen through parallel slits. Each slit displays the object with parallax effect in the direction of the slit, but not in the orthogonal direction. Rainbow holograms have many commercial uses as displays. READING LIST Fourier Optics and Optical Signal Processing J. W. Goodman, Introduction to Fourier Optics, Roberts, 3rd ed. 2005. E. G. Steward, Fourier Optics: An Introduction, Halsted Press, 2nd ed. 1987; Dover, reissued 2004. W. Lauterbom and T. Kurz, Coherent Optics: Fundamentals and Applications Springer- Verlag, 2nd ed. 2003. E. L. O'Neill, Introduction to Statistical Optics, Addison-Wesley, 1963; Dover, reissued 2003. M. A. Abushagur and H. Caulfield, eds., Selected Papers on Fourier Optics, SPIE Optical Engineering Press (Milestone Series Volume 105), 1995. P. W. Hooijmans, Coherent Optical System Design, Wiley, 1994. F. T. Yu and S. Yin, eds., Selected Papers on Coherent OpticaL Processing, SPIE Optical Engineering Press (Milestone Series Volume 52), 1992. G. Reynolds, 1. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics, SPIE Optical Engineering Press, 1989. J. L. Homer, ed., Optical Signal Processing, Academic Press, 1987. A. Papoulis, Systems and Transforms with Applications in Optics, McGraw-Hill, 1968; Krieger, reissued ] 986. F. T. S. Yu, White-Light Optical Signal Processing, Wiley, 1985.
146 CHAPTER 4 FOURIER OPTICS P. M. Duffieux, Fourier Transform and Its Applications to Optics, Wiley, 2nd ed. 1983. H. Stark, ed., Applications of Optical Fourier Transforms, Academic Press, 1982. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics, Wiley, 1978. F. P. Carlson, Introduction to Applied Optics for Engineers, Academic Press, 1977. G. Harbum, C. A. Taylor, and T. R. Welberry, Atlas of Optical Transforms, Cornell University Press, 1975. W. T. Cathey, Optical Information Processing and Holography, Wiley, 1974. H. S. Lipson, ed., Optical Transforms, Academic Press, 1972. M. Cagnet, M. Franon, and S. Mallick, Atlas of Optical Phenomena, Springer-Verlag, reprinted with supplement 1971. L. Mertz, Transformations in Optics, Wiley, 1965. M. Cagnet, M. Franon, and J. C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, 1962. Diffraction o. K. Ersoy.. Diffraction, Fourier Optics, and Imaging, Wiley, 2007. M. Nieto- Vesperinas, Scattering and Diffraction in Physical Optics, World Scientific, 2nd ed. 2006. A. Sommerfeld, Mathematical Theory of Diffraction, Mathematische Annalen, 1896; Birkhauser, 2004. D. C. 0' Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrica- tion, and Test, SPIE Optical Engineering Press, 2003. J. M. Cowley, Diffraction Physics, Elsevier, 3rd revised ed. 1995. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering, Cambridge University Press, 1992. K. E. Oughstun, ed., Selected Papers on Scalar Wave Diffraction, SPIE Optical Engineering Press (Milestone Series Volume 51), 1992. S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation, Academic Press, 1986. M. Franon, Diffraction: Coherence in Optics, Pergamon, ] 966. . Imaging L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge University Press, 2006. H. Barrett and K. Myers, Foundations of Image Science, Wiley, 2003. D. Courjon, Near-Field Microscopy and Near-Field Optics, Imperial College Press, 2003. C. S. Williams and O. A. Becklund, Introduction to the Optical Transfer Function, Wiley, 1989; SPIE Optical Engineering Press, 2002. S. Kawata, ed., Near-Field Optics and Surface Plasmon Polaritons, Springer, 2001. F. de Fomel, Evanescent Waves: From Newtonian Optics to Atomic Optics, Springer- Verlag, 2001. M. Gu, Advanced Optical Imaging Theory, Springer-Verlag, 1999. H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications, Taylor & Francis, 1997. M. Kufner and S. Kufner, Micro-Optics and Lithography, VUB Press, 1997. J. Fillard, Near Field Optics and Nanoscopy, World Scientific, 1996. M. Franon, Optical Image Formation and Processing, Academic Press, 1979. J. C. Dainty and R. Shaw, Image Science: Principles, Analysis and Evaluation Of Photographic-Type Imaging Processes, Academic Press, 1974. K. R. Barnes, The Optical Transfer Function, Elsevier, 1971. Holography G. Saxby, Practical Holography, Institute of Physics, 3rd ed. 2004. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Recon- struction, and Related Techniques, Springer-Verlag, 2004. L. Yaroslavsky, Digital Holography and Digital Image Processing: Principles, Methods, Algorithms, Kluwer, 2004. P. Hariharan, Basics of Holography, Cambridge University Press, 2002.
PROBLEMS 147 H. I. Bjelkhagen and HooJ. Caulfield, eds., Selected Papers on Fundamental Techniques in Hologra- phy, SPIE Optical Engineering Press (Milestone Series Volume 171), 2001. J. E. Kasper and S. A. FeUer, Complete Book of Holograms: How They Work and How to Make Them, Wiley, 1987; Dover, reissued 200 1. R. S. Sirohi and K. D. Hinsch, eds., Selected Papers on Holographic Interferometry Principles and Techniques, SPIE Optical Engineering Press (Milestone Series Volume 144), 1998. P. Hariharan, Optical Holography: Principles, Techniques and Applications, Cambridge University Press, 2nd ed. 1996. V. A. Soifer and M. V. Golub, Laser Beam Mode Selection by Computer Generated Holograms, CRC Press, 1994. H. M. Smith, Principles of Holography, Wiley, 2nd ed. 1975, reprinted 1988. W. Schumann, J.-P. Zurcher, and D. Cuche, Holography and Deformation Analysis, Springer-Verlag, 1985. N. Abramson, The Making and Evaluation of Holograms, Academic Press, 1981. Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography, Springer- Verlag, 1980. L. M. Soroko, Holography and Coherent Optics, Plenum, 1980. H. J. Caulfield, ed., Handbook of Optical Holography, Academic Press, 1979. W. Schumann and M. Dubas, Holographic Interferometry, Springer-Verlag, 1979. C. M. Vest, Holographic Interferometry, Springer-Verlag, 1979. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography, Academic Press, paperback ed. 1977 . M. Franon, Holography, Academic Press, 1974. H. J. Caulfield and L. Sun, The Applications of Holography, Wiley, 1970. PROBLEMS . Correspondence Between Harmonic Functions and Plane Waves. The complex ampli- tudes of a monochromatic wave of wavelength A in the z 0 and z d planes are f ( x, y ) and g(x, y), respectively. Assuming that d 10 4 A, use harmonic analysis to detennine g(x, y) in the following cases: (a) f(x, y) 1; (b) f(x, y) exp[( j7r I .A) (x + y)]; (c) f(x, y) cos(7rxI2A); (d) f(x,y) cos 2 (7ry/2A); (e) f(x, y) Lm rect[(xl10.A) 2m], m 0, :f:1, :f:2, . . . , where rect(x) 1 if Ixl < and 0, otherwise. Describe the physical nature of the wave in each case. In Probe 4. ] - 3, if f (x, y) is a circularly symmetric function with a maximum spatial frequency of 200 lines/mm, determine the angle of the cone within which the wave directions are confined. Assume that A 633 nm. Logarithmic Interconnection Map. A transparency of amplitude transmittance t(x, y) exp[ j27r4;(x)] is illuminated with a uniform plane wave of wavelength A 1 /-Lm. The transmitted light is focused by an adjacent lens of focal length f 100 cm. What must 4;(x) be so that the ray that hits the transparency at position x is deflected and focused to a position x' In(x) for all x > O? (Note that x and x' are measured in millimeters.) If the lens is removed, how should 4;( x) be modified so that the system perfonns the same function? This system may be used to perform a logarithmic coordinate transformation, as discussed in Chapter 21. Proof of the Lens Fourier-Transform Property. (a) Show that the convolution of f(x) and exp( j7rX 2 lAd) may be obtained in three steps: multiply f(x) by exp( j7rX 2 I Ad); evaluate the Fourier transform of the product at the 4.1-3 4.1-4 4.1-5 4.2-3
148 CHAPTER 4 FOURIER OPTICS frequency l/x X I Ad; and multiply the result by exp( j7fX 2 I Ad). (b) The Fourier transform system in Fig. 4.2-4 is a cascade of three systems propagation a distance f in free space, transmission through a lens of focal length f, and propagation a distance f in free space. Noting that propagation a distance d in free space is equivalent to convolution with exp( j7fX 2 lAd) [see (4.1-20)], and using the result in (a), derive the lens' Fourier transform equation (4.2-8). For simplicity ignore the y dependence. 4.2-4 Fourier Transform of the Line Functions. A transparency of amplitude transmittance t(x, y) is illuminated with a plane wave of wavelength A 1 {lm and focused with a lens of focal length f 100 cm. Sketch the intensity distribution in the plane of the transparency and in the lens focal plane in the following cases (all distances are measured in mm): (a) t(x, y) 8(x y); (b) t(x, y) 8(x + a) + 8(x a), a 1 mm; (c) t(x, y) 8(x + a) + j8(x a), a 1 mm, where 8(.) is the delta function (see Appendix A, Sec. A.I). 4.2-5 Design of an Optical Fourier-Transform System. A lens is used to display the Fourier transform of a two-dimensional function with spatial frequencies between 20 and 200 lines/mm. If the wavelength of light is A 488 nm, what should be the focal length of the lens so that the highest and lowest spatial frequencies are separated by a distance of 9 cm in the Fourier plane? 4.3-4 Fraunhofer Diffraction from a Diffraction Grating. Derive an expression for the Fraun- hofer diffraction pattern for an aperture made of M 2£ + 1 parallel slits of infinitesimal widths separated by equal distances a lOA, L p(x,y) 8(x ma). (4.5-8) m=-L Sketch the pattern as a function of the observation angle e x I d, where d is the observation distance. 4.3-5 Fraunhofer Diffraction with an Oblique Incident Wave. The diffraction pattern from an aperture with aperture function p(x, y) is proportional to IP(xIAd, yIAd)12, where pel/x, l/y) is the Fourier transform of p(x, y) and d is the distance between the aperture and observation planes. What is the diffraction pattern when the direction of the incident wave makes a small angle ex « 1, with the z-axis in the x-z plane? *4.3-6 Fresnel Diffraction from Two Pinholes. Show that the Fresnel diffraction pattern from two pinholes separated by a distance 2a, Le.,p(x, y) [8(x a)+8(x+a)]8(y), at an observation distance d is the periodic pattern, lex, y) (2/Ad)2 cos 2 (27faxIAd). *4.3-7 Relation Between Fresnel and Fraunhofer Diffraction. Show that the Fresnel diffraction pattern of the aperture function p(x, y) is equal to the Fraunhofer diffraction pattern of the aperture function p(x, y) exp[ j7f(X2 + y2)IAd]. 4.4-1 Blurring a Sinusoidal Grating. An object f (x, y ) cos 2 (27fx I a) is imaged by a defocused single-lens imaging system whose impulse response function hex, y) 1 within a square of width D, and 0 elsewhere. Derive an expression for the distribution of the image g(x, 0) in the x direction. Derive an expression for the contrast of the image in terms of the ratio Dla. The contrast (max min)/(max+min), where max and min are the maximum and minimum values of g(x, 0). 4.4-2 Image of a Phase Object. An imaging system has an impulse response function hex, y) rect( x )8 (y ). If the input wave is f(x,y) . 7r exp J 2 7r . exp J 2 for x > 0 (4.5-9) for x < 0, determine and sketch the intensity Ig(x, y)12 of the output wave g(x, y). Verify that even though the intensity of the input wave If (x, y ) 1 2 1, the intensity of the output wave is not uniform.
PROBLEMS 149 4.4-3 Optical Spatial Filtering. Consider the spatial filtering system shown in Fig. 4.4-5 with f 1000 mm. The system is illuminated with a uniform plane wave of unit amplitude and wavelength A 10- 3 mm. The input transparency has amplitude transmittance f(x, y) and the mask has amplitude transmittance p(x, y). Write an expression relating the complex amplitude g(x, y) of light in the image plane to f(x, y) and p(x, y). Assuming that all distances are measured in mm, sketch g(x, 0) in the following cases: (a) f(x, y) 8(x 5) and p(x, y) rect(x); (b) f(x, y) rect(x) and p(x, y) sinc(x). Determine p(x, y) such that g(x, y) V}I(x, y), where V} {)2 j8x2 + 8 2 j8y2 is the transverse Laplacian operator. 4.4-4 Optical Cross-Correlation. Show how a spatial filter may be used to perform the operation of cross-correlation (defined in Appendix A) between two images described by the real- valued functions II (x, y) and I2(x, y). Under what conditions would the complex amplitude transmittances of the masks and transparencies used be real-valued? *4.4-5 Impulse Response Function of a Severely Defocused System. Using wave optics, show that the impulse response function of a severely defocused imaging system (one for which the defocusing error f is very large) may be approximated by h(x, y) p(x/fd 2 , Y/fd 2 ), where p(x, y) is the pupil function. Hint: Use the method of stationary phase described on page 117 (second proof) to evaluate the integral that results from the use of (4.4-11) and (4.4-10). Note that this is the same result predicted by the ray theory of light [see (4.4-2)]. 4.4-6 1Wo-Point Resolution. (a) Consider the single-lens imaging system discussed in Sec. 4.4C. Assuming a square aperture of width D, unit magnification, and perfect focus, write an expression for the impulse response function h(x, y). (b) Determine the response of the system to an object consisting of two points separated by a distance b, Le., I(x, y) 8(x)8(y) + 8(x b)8(y). (4.5-10) 4.4- 7 (c) If Ad 2 j D 0.1 mm, sketch the magnitude of the image g(x, 0) as a function of x when the points are separated by a distance b 0.5, 1, and 2 mm. What is the minimum separation between the two points such that the image remains discernible as two spots instead of a single spot, i.e., has two peaks? Ring Aperture. (a) A focused single-lens imaging system, with magnification M 1 and focal length I 100 cm has an aperture in the form of a ring 1, a < x 2 + y2 < b, 0, otherwise, p(x,y) ( 4.5-11 ) 4.5-1 where a 5 mm and b 6 mm. Determine the transfer function H ( V x , v y ) of the system and sketch its cross section H(v x , 0). The wavelength A 1 J-Lm. (b) If the image plane is now moved closer to the lens so that its distance from the lens becomes d 2 25 cm, with the distance between the object plane and the lens d 1 as in (a), use the ray-optics approximation to determine the impulse response function of the imaging system h(x, y) and sketch h(x, 0). Holography with a Spherical Reference Wave. The choice of a uniform plane wave as a reference wave is not essential to holography; other waves can be used. Assuming that the reference wave is a spherical wave centered about the point (0,0, d), determine the hologram pattern and examine the reconstructed wave when: (a) the object wave is a plane wave traveling at an angle ()x; (b) the object wave is a spherical wave centered at ( xo,O, d 1 ). Approximate spherical waves by paraboloidal waves. Optical Correlation. A transparency with an amplitude transmittance given by I (x, y ) 11 (x a, y) + f2(X + a, y) is Fourier transformed by a lens and the intensity is recorded on a transparency (hologram). The hologram is subsequently illuminated with a reference wave and the reconstructed wave is Fourier transformed with a lens to generate the function 9 ( x, y ). Derive an expression relating g(x, y) to 11 (x, y) and I2(x, y). Show how the correlation of the two functions 11 (x, y) and I2(x, y) may be determined with this system. 4.5-2
CHAPTER 5.1 ELECTROMAGNETIC THEORY OF LIGHT 5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA A. Linear, Nondispersive, Homogeneous, and Isotropic Media B. Nonlinear, Dispersive, Inhomogeneous, or Anisotropic Media 5.3 MONOCHROMATIC ELECTROMAGNETIC WAVES 5.4 ELEMENTARY ELECTROMAGNETIC WAVES A. Plane, Spherical, and Gaussian Electromagnetic Waves B. Relation Between Electromagnetic Optics and Scalar Wave Optics C. Vector Beams 5.5 ABSORPTION AND DISPERSION A. Absorption B. Dispersion C. The Resonant Medium D. Optics of Conductive Media 5.6 PULSE PROPAGATION IN DISPERSIVE MEDIA *5.7 OPTICS OF MAGNETIC MATERIALS AND METAMATERIALS 152 156 , 162 164 170 184 190 ... ( * .. t , t .. . ..... ...... ". , . ... .J __ '\. a, . ... \. I " . ,. <:1 _. .- -- .--.: "'"'" ..- -" ,--.. I James Clerk Maxwell (1831-1879) advanced the theory that light is an electromagnetic wave phenomenon. - " -" - 150
It is apparent from the results presented in Chapters 2-4 that wave optics has a far greater reach than ray optics. Remarkably, both approaches provide similar results for many simple optical phenomena involving paraxial waves, such as the focusing of light by a lens and the behavior of light in graded-index media and periodic systems. But it is clear that wave optics offers something that ray optics cannot: the ability to explain phenomena such as interference and diffraction, which involve phase, and therefore lie hopelessly beyond the reach of a simple construct like ray optics. In spite of its many successes, however, wave optics, like ray optics, is unable to quantitatively account for some simple observations in an optics experiment, such as the division of light at a beamsplitter. The fraction of light reflected (and transmitted) turns out to depend on the polarization of the incident light, which means that the light must be treated in the context of a vector, rather than a scalar, theory. That's where electromagnetic optics enters the picture. In common with radio waves and X-rays, as shown in Fig. 5.0-1, light is an electromagnetic phenomenon that is described by a vector wave theory. Elec- tromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric-field wave and a magnetic-field wave. From this perspective, the wave-optics approach described in Chapter 2, and developed in Chapters 3 and 4, is merely a scalar approximation to the more complete electromagnetic theory. N N ::r:: :E N ::r:: c N ::r:: N ::r:: N ::r:: N ::r:: N Frequency .. .. .. . . , . m ,....,. 1 N I.Ij 00 ....... ("f') \C 0\ ....... ....... N 0 0 0 0 0 0 0 .-..4 ..... ..... .-.. .-.4 ....... ...... rJ:J C1) S ....... > ...J ::r:: ::r:: ::r:: Cd .D X rays 'Y rays ...J :E ::r:: s . --'I 1-c t t 00 > > en . ..... E-c > N ("f') ("f') b 9' b b 0 0 ......-4 ......-4 ....... ......... ..... I. -- I I I Hz .. I I I I s I I I I S I S I I .. I S ::l I I I I S s:: S Wavelength (in vacuum) .. , .. - Radiowave Microwave Optical Figure 5.0-1 The electromagnetic spectrum from low frequencies (long wavelengths) to high frequencies (short wavelengths). The optical region, shown as shaded, is displayed in greater detail in Fig. 2.0-1. Electromagnetic optics thus encompasses wave optics, which in turn reduces to ray optics in the limit of short wavelengths as shown in Chapter 2. This hierarchy is displayed in Fig. 5.0-2. Electromagnetic Optics Wave Optics Figure 5.0-2 Electromagnetic optics is a vector theory comprising an electric field and a magnetic field that vary in time and space. Wave optics is an approximation to electromagnetic optics that relies on the wavefunction, a scalar function of time and space. Ray optics is the limit of wave optics when the wavelength is very short. "' ... Ray Optics 151
152 CHAPTER 5 ELECTROMAGNETIC OPTICS Optical frequencies occupy a band of the electromagnetic spectrum that extends from the infrared through the visible to the ultraviolet, as shown in Fig. 5.0-1. The range of wavelengths generally considered to lie in the optical domain extends from 1 0 nm to 300 Mm (as is shown in greater detail in Fig. 2.0-1). Because these wavelengths are substantially shorter than those of radiowaves, or even microwaves, the techniques involved in their generation, transmission, and detection have traditionally been rather distinct. In recent years, however, the march toward miniaturization has served to blur these differences: wavelength-size lasers and optical waveguides, as well as tiny photodetectors, are now commonplace. This Chapter This chapter offers a brief review of those aspects of electromagnetic theory that are of paramount importance in optics. The fundamental theoretical construct Maxwell's equations is set forth in Sec. 5.1. The behavior of optical electromagnetic waves in dielectric media is examined in Sec. 5.2. Together, these sections layout the (un- damentals of electromagnetic optics and provide the set of rules that govern the re- maining sections of the chapter. These rules simplify considerably for the special case of monochromatic light, as discussed in Sec. 5.3. Elementary electromagnetic waves (plane waves, spherical waves, and Gaussian beams), introduced in Sec. 5.4, provide important examples that are often encountered in practice. Finally, Sec. 5.5 is devoted to a study of the propagation of light in dispersive media, which exhibit wavelength- dependent absorption and refraction as real media do. This topic will be revisited in Chapter 22. Chapter 6, which is based upon the theory of electromagnetic optics presented in this chapter, deals explicitly with the polarization of light and the interaction of polarized light with dielectric and anisotropic media, particularly liquid crystals. The material set forth here also forms the basis for the expositions provided in Chapters 8 11, which consider the optics of layered and periodic media, guided-wave optics, fiber optics, resonator optics, and statistical optics, respectively. Chapter 21 is devoted to the electromagnetic optics of nonlinear media. 5.1 ELECTROMAGNETIC THEORY OF LIGHT An electromagnetic field is described by two related vector fields that are functions of position and time: the electric field £ r, t and the magnetic field r, t . In general, therefore, six scalar functions of position and time are required to describe light in free space. Fortunately, these six functions are interrelated since they must satisfy the celebrated set of coupled partial differential equations known as Maxwell's equations. Maxwell's Equations in Free Space The electric- and magnetic-field vectors in free space satisfy Maxwell's equations: \7.£ \7 . a£ Eo at 8 Mo 8t o (5.1-1) \7x1i \7x£ 0, (5.1-2) (5.1-3) (5.1-4) Maxwell's Equations (Free Space)
5.1 ELECTROMAGNETIC THEORY OF LIGHT 153 where the constants Eo 1 367r X 10- 9 F/m and J-Lo 47r x 10- 7 HIm (MKS units) are, respectively, the Electric permittivity and the magnetic permeability of free space. The vector operators V. and V x represent the divergence and curl, respec- tively. t The Wave Equation A necessary condition for £ and to satisfy Maxwell's equations is that each of their components satisfy the wave equation V 2 u 1 8 2 u c 2 8t 2 o o. (5.1-5) Wave Equation (Free Space) Here 1 3 X 10 8 m/s (5.1-6) Speed of Light (Free Space) Co Eo J-Lo is the speed of light in vacuum, and the scalar function u r, t represents any of the three components G x , G y , G z of £ or the three components 9-C x , 9-C y , 9-C z of. The wave equation may be derived from Maxwell's equations by applying the curl operation V x to (5.1-2), making use of the vector identity V x V x £ V V. £ V 2 £, and then using (5.1-1) and (5.1-3) to show that each component of £ satisfies the wave equation. A similar procedure is followed for. Since Maxwell's equations and the wave equation are linear, the principle of superposition applies: if two sets of electric and magnetic fields are solutions to these equations separately, their sum is also a solution. The connection between electromagnetic optics and wave optics is now evident. The wave equation (2.1-2), which is the basis of wave optics, is embedded in the structure of electromagnetic theory; the speed of light is related to the electromagnetic constants Eo and J-Lo by (5.1-6); and the scalar wavefunction u r, t in Chapter 2 represents any of I the six components of the electric- and magnetic-field vectors. Electromagnetic optics reduces to wave optics in problems for which the vector nature of the electromagnetic fields is not of essence. As we shall see in this and the following chapters, the vector character of light underlies polarization phenomena and governs the amount of light reflected or transmitted through boundaries between different media, and therefore detennines the characteristics of light propagation in waveguides, layered media, and optical resonators. Maxwell's Equations in a Medium In a medium in which there are no free electric charges or currents, two additional vector fields are required the electric flux density (also called the electric displace- ment) 1) r, t and the magnetic flux density 23 r, t . The four fields, £, , 1), and 1J, are related by Maxwell's equations in a source-free medium: t In a Cartesian coordinate system V · [: 8Ex/8x + 8Ey/8y + 8Ez/8z whereas \7 x [: is a vector with Cartesian components (8Ez/8y 8E y /8z), (8Ex/8z 8Ez/8x), and (8Ey/8x 8Ex/8y).
154 CHAPTER 5 ELECTROMAGNETIC OPTICS Vx£ aTI at ap, at (5.1-7) Vx1{ V.TI v. o o. (5.1-8) (5.1-9) (5.1-10) Maxwell's Equations (Source-Free Medium) The relationship between the electric flux density TI and the electric field £ depends on the electric properties of the medium, which are characterized by the polarization density P. In a dielectric medium, the polarization density is the macroscopic sum of the electric dipole moments induced by the electric field. Similarly, the relation be- tween the magnetic flux density P, and the magnetic field 1{ depends on the magnetic properties of the medium, embodied in the magnetization density M, which is defined analogously to the polarization density. The equations relating the flux densities and the fields are 1) to£ + P 13 J.101{ + /loM. (5.1-11) (5.1-12) The vector fields P and M are in turn related to the externally applied electric and magnetic fields £ and 1{ by relationships that depend on the electric and magnetic character of the medium, respectively, as will be described in Sec. 5.2. Equations relating P and £, as well as M and 1{, are established once the medium is specified. When these latter equations are substituted into Maxwell's equations in a source-free medium, the flux densities disappear. In free space, P M 0, so that TI toG and 13 J.101{ whereupon (5.1-7) (5.1-10) reduce to the free-space Maxwell's equations, (5.1-1) (5.1-4). Boundary Conditions In a homogeneous medium, all components of the fields £, 1{, TI, and 13 are con- tinuous functions of position. At the boundary between two dielectric media, in the absence of free electric charges and currents, the tangential components of the electric and magnetic fields £ and 1{, and the normal components of the electric and magnetic flux densities TI and p" must be continuous (Fig. 5.1-1). Dielectric £ - - :J{ v Dielectric 'B fO Dielectric Perfect conductor Figure 5.1-1 Boundary conditions at: (a) the interface between two dielectric media; (b) the interface between a perfect conductor and a dielectric material. At the boundary between a dielectric medium and a perfectly conductive medium, the tangential components of the electric field vector must vanish. Since a perfect mirror is made of a perfectly conductive material (a metal), the component of the
5.1 ELECTROMAGNETIC THEORY OF LIGHT 155 electric field parallel to the surface of the mirror must be zero. This requires that at normal incidence the electric fields of the reflected and incident waves must have equal magnitudes and a phase shift of 7f so that their sum adds up to zero. These boundary conditions are an integral part of Maxwell's equations. They are used to determine the reflectance and transmittance of waves at various boundaries (see Sec. 6.2), and the propagation of waves in periodic structures (see Sec. 7.1) and waveguides (see Sec. 8.2). Intensity, Power, and Energy The flow of electromagnetic power is governed by the vector S £ x 1-C, (5.1-13) which is known as the Poynting vector. The direction of power flow is along the di- rection of the Poynting vector, i.e., orthogonal to both £ and j{. The optical intensity I r, t (power flow across a unit area normal to the vector S) t is the magnitude of the time-averaged Poynting vector S . The average is taken over times that are long in comparison with an optical cycle, but short compared to other times of interest. The wave-optics equivalent is given in (2.1-3). Using the vector identity \7. £ x j{ \7 x £ · j{ \7 x j{ · £, together with Maxwell's equations (5.1-7) (5.1-8) and (5.1-11) (5.1-12), we obtain a 1 2 1 2 ap aM at \7. S (5.1-14) . The first and second terms in parentheses in (5.1-14) represent the energy densities (per unit volume) stored in the electric and magnetic fields, respectively. The third and fourth terms represent the power densities delivered to the material's electric and mag- netic dipoles. Equation (5.1-14), known as the Poynting theorem, therefore represents conservation of energy: the power flow escaping from the surface of an incremental volume equals the time rate of change of the energy stored inside the volume. Momentum An electromagnetic wave carries linear momentum, which results in radiation pressure on objects from which the wave reflects or scatters. In free space, the linear momentum densjty (per unit volume) is a vector . Eo£ X 13 c 2 (5.1-15) Linear Momentum Density proportional to the Poynting vector S. The average momentum in a cylinder of length c and unit area is (S c 2 · C S c. This momentum crosses the unit area in a unit time, so that the average rate (per unit time) of momentum flow across a unit area oriented perpendicular to the direction of S is S c. An electromagnetic wave may also carry angular momentum and may therefore exert torque on an object. The average rate of angular momentum transported by an electromagnetic field is r x S c. For example, the Laguerre Gaussian beams intro- duced in Sec. 3.4 have helical wavefronts; the Poynting vector then has an azimuthal component, which leads to an orbital angular momentum. t For a discussion of this interpretation, see M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, pp. 7-10.
156 CHAPTER 5 ELECTROMAGNETIC OPTICS 5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA The character of the medium is embodied in the relation between the polarization and magnetization densities, P and M, on the one hand, and the electric and magnetic fields, £ and j{, on the other; these are known as the constitutive relation. In most media, the constitutive relation separates into a pair of constitutive relations, one be- tween P and E, and another between M and j{. The former describes the dielectric properties of the medium, whereas the latter describes its magnetic properties. With the notable exceptions of Sec. 5.7 and Sec. 6.4, the principal emphasis in this book is on the dielectric properties. We therefore direct our attention to the P-E relations for various dielectric media; the M-j{ relations for magnetic media obey similar relations under similar conditions. It is useful to regard the P-E constitutive relation as arising from a system in which E is the input and P is the output or response (Fig. 5.2-1). Note that E E r, t and P P r, t are functions of both position and time. £(r,t) P(r,t) Medium Figure 5.2-1 In response to an applied electric field £, the dielectric medium creates a polariza- tion density P. Definitions . A dielectric medium is said to be linear if the vector field P r, t is linearly related to the vector field E r, t . The principle of superposition then applies. . The medium is said to be nondispersive if its response is instantaneous, i.e., if P at time t is detennined by £ at the same time t and not by prior values of E Nondispersiveness is clearly an idealization since all physical systems, no matter how rapidly they may respond, do have a response time that is finite. . The medium is said to be homogeneous if the relation between P and £ is independent of the position r. . The medium is said to be isotropic if the relation between the vectors P and E is independent of the direction of the vector E, so that the medium exhibits the same behavior from all directions. The vectors P and E. must then be parallel. . The medium is said to be spatially nondispersive if the relation between P and E is local, i.e., if P at each position r is influenced only by E at the same position r. The medium is assumed to be spatially nondispersive throughout this chapter (optically active media, considered in Sec. 6.4A, are spatially dis- persive). A. Linear, Nondispersive, Homogeneous, and Isotropic Media Let us first consider the simplest case of linear, nondispersive, homogeneous, and isotropic dielectric media. The vectors P and E at every position and time are then parallel and proportional, so that P Eo X £, (5.2-1) where the scalar constant X is called the electric susceptibility (Fig. 5.2-2).
5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA 157 £, p x Figure 5.2-2 A linear, nondispersive, homoge- neous, and isotropic medium is fully characterized by a single constant, the electric susceptibility x. Substituting (5.2-1) in (5.1-11) shows that 1) and £ are also parallel and propor- tional, 1) f£, (5.2-2) where the scalar quantity E Eo 1 + X (5.2-3) is defined as the electric permittivity of the medium. The relative permittivity E Eo 1 + X is also called the dielectric constant of the medium. Under similar conditions, the magnetic relation can be written in the form 23 f-l'J{ , (5.2-4) where Jj is the magnetic permeability of the medium. With the relations (5.2-2) and (5.2-4), Maxwell's equations in (5.1-7) (5.1-10) relate oniy the two vector fields £ r, t and 'J{ r, t , simplifying to Vx'J{ o£ E ot Vx£ 0'J{ f-l ot V.£ 0 V.'J{ o. (5.2-5) (5.2-6) (5.2-7) (5.2-8) Maxwell's Equations (Linear, Nondispersive, Homogeneous, Isotropic, Source-Free Medium) It is apparent that (5.2-5) (5.2-8) are identical in form to the free-space Maxwell's equations in (5.1-1) (5.1-4) except that f replaces Eo and f-l replaces f-lo. Each compo- nent of £ and 'J{ therefore satisfies the wave equation V 2 u 1 02U c 2 ot 2 0, (5.2-9) Wave Equation (In a Medium) where the speed of light in the medium is denoted c: 1 . (5.2-10) Speed of Light (In a Medium) c . Ef-l
158 CHAPTER 5 ELECTROMAGNETIC OPTICS The ratio of the speed of light in free space to that in the medium, Co c, is defined as the refractive index n: Co € /-L c , (5.2-11) Refractive Index n Eo /-Lo where (5.1-6) provides 1 . (5.2-12) Co Eo/-Lo For a nonmagnetic material, /-L /-Lo and , E 1 + x, (5.2-13) Refractive Index (Nonmagnetic Media) n Eo so that the refractive index is the square root of the dielectric constant. These relations provide another point of connection with scalar wave optics (Sec. 2.1), as discussed further in Sec. 5.4B. Finally, the Poynting theorem (5.1-14) based on Maxwell's equations (5.2-5) and (5.2-6) takes the form of a continuity equation aw at (5.2-14) v.s where w IE £2 + 111.2 2 2 (5.2-15) is the energy density stored in the medium. B. Nonlinear, Dispersive, Inhomogeneous, or Anisotropic Media We now consider nonmagnetic dielectric media for which one or more of the properties of linearity, nondispersiveness, homogeneity, and isotropy are not satisfied. Inhomogeneous Media We first consider an inhomogeneous dielectric (such as a graded-index medium) that is linear, nondispersive, and isotropic. The simple proportionalities, P EoX£ and 1) E£, remain intact, but the coefficients X and E become functions of position: X X rand E E r (Fig. 5.2-3). The refractive index therefore also becomes position dependent so that n n r . £(r) x(r) P(r) Figure 5.2-3 An inhomogeneous (but linear, nondispersive, and isotropic) medium is character- ized by a position dependent susceptibility x(r).
5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA 159 Beginning with Maxwell's equations, (5.1-7) (5.1-10), and noting that E E r is position dependent, we apply the curl operation V x to both sides of (5.1-8). We then use (5.1-7) to write Eo V' X 'V x £' E \ J 1 f)2£ c 2 f)t 2 · o (5.2-16) Wave Equation (Inhomogeneous Medium) The magnetic field satisfies a different equation: V X Eo V X j{ E 1 f)2j{ c 2 f)t 2 · o (5.2-17) Wave Equation (Inhomogeneous Medium) Equation (5.2-16) may also be written in the form f)2£ ?toE f)t 2 o. (5.2-18) 1 V 2 £ + V VE. £ E The validity of (5.2-18) can be demonstrated by employing the following procedure. Use the identity V x V' x £ V V · £ V 2 £, valid for a rectilinear coordinate system. Invoke (5.1-9), which yields V · E£ 0, together with the identity V · E£ E"V · £ + VE. £, which provides V' · £ 1 E VE. £. Finally, substitute in (5.2-16) to obtain (5.2-18). For media with gradually varying dielectric properties, i.e., when E r varies suf- ficiently slowly so that it can be assumed constant within distances of the order of a wavelength, the second term on the left-hand side of (5.2-18) is negligible in compari- son with the first, so that V 2 £ 1 c 2 r f)2£ 2 0, f)t (5.2-19) where c r 1 ?toE Co n r is spatially varying and n r E r Eo is the refractive index at position r. This relation was invoked without proof in Chapter 2, but it is clearly an approximate consequence of Maxwell's equations. For a homogeneous dielectric medium of refractive index n perturbed by a slowly varying spatially dependent change n, it is often useful to write (5.2-19) in the form 1 f)2£ rv c 2 f)t 2 s, s V 2 c: f)2 p ?to f)t 2 ' p 2Eon n£, (5.2-20) where c Co n is the speed of light in the homogenous medium. Thus, £ satisfies the wave equation with a radiation source S created by a perturbation of the polarization density P, which in turn is proportional to n and £ itself. These equations may be verified by expanding the term 1 c 2 r in (5.2-19) as n+n 2 c n2+2nn c and bringing the perturbation term to the right-hand side of the equation. The term p is the perturbation in P, as can be shown by noting that P EoXC: Eo E Eo 1 c: Eo n 2 1 C:, so that p Eo n 2 1 £ 2Eonn£.
160 CHAPTER 5 ELECTROMAGNETIC OPTICS Anisotropic Media The relation between the vectors P and £ in an anisotropic dielectric medium depends on the direction of the vector £; the requirement that the two vectors remain parallel is not maintained. If the medium is linear, nondispersive, and homogeneous, each component of P is a linear combination of the three components of £: Pi foXijGj, (5.2-21) . J where the indexes i, j 1, 2, 3 denote the x, y, and z components, respectively. The dielectric properties of the medium are then described by a 3 x 3 array of constants Xij , which are elements of what is called the electric susceptibility tensor X (Fig. 5.2-4). A similar relation between 1) and £ applies: 1)i fij £ j , (.2-22) . J where fij are the elements of the electric permittivity tensor €. £1 XII X I2 X I 3 X 21 X 22 X 23 X 31 X 32 X 33 PI £2 Figure 5.2-4 An anisotropic (but linear, ho- mogeneous, and nondispersive) medium is char- acterized by nine constants, the elements of the susceptibility tensor Xij. Each component of P is a weighted superposition of the three components of £. ).- P2 £3 P3 The optical properties of anisotropic media are examined in Chapter 6. The relation between t and j{ t for anisotropic magnetic media takes a form similar to that of (5.2-22), under similar assumptions. Dispersive Media The relation between the vectors P and £ in a dispersive dielectric medium is dynamic rather than instantaneous. The vector £ t may be thought of as an input that induces the bound electrons in the atoms of the medium to oscillate, which then collectively give rise to the polarization-density vector P t as the output. The presence of a time delay between the output and the input indicates that the system possesses memory. Only when this time is short in comparison with other times of interest can the response be regarded as instantaneous, in which case the medium is approximately nondisper- . Slve. For dispersive media that are linear, homogeneous, and isotropic, the dynamic rela- tion between P t and £ t may be described by a linear differential equation such as that associated with a driven harmonic oscillator: al d 2 P dt 2 + a2 dP dt + a3 P £, where aI, a2, and a3 are constants. A simple analysis along these lines (see Sec. 5.5C) provides a physical rationale for the presence of dispersion (and absorption). More generally, the linear-systems approach provided in Appendix B may be used to investigate an arbitrary linear system, which is characterized by its response to an impulse (impulse response function). An electric-field impulse of magnitude 8 t ap- plied at time t 0 induces a time-dispersed polarization density of magnitude foX t , where X t is a scalar function of time with finite duration that begins at t O. Since the medium is linear, an arbitrary electric field £ t then induces a polarization density
5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA 161 Figure 5.2-5 In a dispersive (but linear, homoge- neous, and isotropic) medium the relation between P(t) and £(t) is governed by a dynamic linear system described by an impulse response function Eo x( t) that corresponds to a frequency dependent susceptibility X(v). x(t) P{t) E(t) that is a superposition of the effects of £ t' for all t' < t, so that the polarization density can be expressed as a convolution, as defined in Appendix A: 00 Pt Eo x t t' £ t' dt'. (5.2-23) -00 This dielectric medium is completely described by its impulse response function Eo X t . Alternatively, a dynamic linear system may be described by its transfer function, which governs the response to harmonic inputs. The transfer function is the Fourier transform of the impulse response function (see Appendix B). In the example at hand, the transfer function at frequency lJ is EoX lJ , where X lJ is the Fourier transform of x t so that it is a frequency-dependent susceptibility (Fig. 5.2-5). This concept is discussed further in Sees. 5.3 and 5.5. For magnetic media under similar assumptions, the relation between M t and 1{ t is analogous to (5.2-23). Nonlinear Media A nonlinear dielectric medium is defined as one in which the relation between P and £ is nonlinear, in which case the wave equation as written in (5.2-9) is not applicable. Rather, Maxwell's equations can be used to derive a nonlinear wave equation that electromagnetic waves obey in a such a medium. We first derive a general wave equation valid for homogeneous and isotropic non- magnetic media. Operating on Maxwell's equation (5.1-8) with the curl operator V x , and using the relation Mo1{ from (5.2-4) together with (5.1-7), we obtain V x V x £ M o ( 2 1) ot 2 . Making use of the vector identity V x V x £ V V. £ V 2 £ and the relation 1) Eo£ + P from (5.1-11) then yields 0 2 £ EoMo ot 2 02p Mo ot 2 · (5.2-24) . V V.£ V 2 £ For homogeneous and isotropic media 1) E£; thus \7 · 1) 0 from (5.1-9) is equivalent to V · £ O. Substituting this, along with EoMo 1 c from (5.1-6), into (5.2-24) therefore provides V 2 £ 1 0 2 £ c 2 ot 2 o 02p Mo ot 2 · (5.2-25) Wave Equation (Homogeneous and Isotropic Medium) Equation (5.2-25) is applicable for all homogeneous and isotropic dielectric media: nonlinear or linear, nondispersive or dispersive. Now, if the medium is nonlinear, nondispersive, and nonmagnetic, the polarization density P can be written as a memory less nonlinear function of £, say P W £ , valid at every position and time. (The simplest example of such a function is P
162 CHAPTER 5 ELECTROMAGNETIC OPTICS al £ + a2 £2, where al and a2 are constants.) Under these conditions (5.2-25) becomes a nonlinear partial differential equation for the electric-field vector £ r, t : V 2 £ 1 {)2£ c 2 {)t 2 o {)2\lJ £ J-to {)t 2 . (5.2-26) The principle of superposition is no longer applicable by virtue of the nonlinear nature of this wave equation. Nonlinear magnetic materials may be similarly described. Most dielectric media are approximately linear unless the optical intensity is sub- stantial, as in the case of focused laser beams. Nonlinear optics is discussed in Chap- ter 21. 5.3 MONOCHROMATIC ELECTROMAGNETIC WAVES , For the special case of monochromatic electromagnetic waves in an optical medium, all components of the electric and magnetic fields are harmonic functions of time with the same frequency v and corresponding angular frequency w 21fv. Adopting the complex representation used in Sec. 2.2A, these six real field components may be expressed as £ r, t j{ r, t Re E r exp jwt Re H r exp jwt , (5.3-1) where E rand H r represent electric- and magnetic-field complex-amplitude vec- tors, respectively. Analogous complex-amplitude vectors P, D, M, and B are similarly associated with the real vectors P, 1:>, M, and 13, respectively. Maxwell's Equations in a Medium Inserting (5.3-1) into Maxwell's equations (5.1-7) (5.1-10), and using the relation () {)t e jwt jw e jwt for monochromatic waves of angular frequency w, yields a et of equations obeyed by the field complex-amplitude vectors: VxH VxE V.D V.B jwD jwB o o. (5.3-2) (5.3-3) (5.3-4) (5.3-5) Maxwell's Equations (Source-Free Medium; Monochromatic Fields) Similarly, (5.1-11) and (5.1-12) give rise to D foE + P B J-toH + J-toM. (5.3-6) (5.3-7) Intensity and Power As indicated in Sec. 5.1, the flow of electromagnetic power is governed by the time average of the Poynting vector S £ x j{. Casting this expression in terms of complex
5.3 MONOCHROMATIC ELECTROMAGNETIC WAVES 163 amplitudes yields s Re Eei wt x Re He jwt ! Ee jwt + E*e- jwt x! He jwt + H*e- jwt 2 2 (5.3-8) The terms containing the factors e j2wt and e- j2wt oscillate at optical frequencies and are therefore washed out by the averaging process, which is slow in comparison with an optical cycle. Thus, s ! E x H* + E* x H 4 1 S + S* 2 Re S , (5.3-9) where the vector S ! E x H* 2 (5.3-10) Complex Poynting Vector may be regarded as a complex Poynting vector. The optical intensity is the magnitude of the vector Re S . Linear, Nondispersive, Homogeneous, and Isotropic Media For monochromatic waves, the relations provided in (5.2-2) and (5.2-4) become the material equations D fE and B p,H, (5.3-11) so that Maxwell's equations, (5.3-2) (5.3-5), depend solely on the interrelated complex- amplitude vectors E and H: \lxH \lxE \l.E \l.H jWfE jwp,H o o. (5.3-] 2) (5.3-13) (5.3-14) (5.3-15) Maxwell's Equations (Linear, Nondispersive, Homogeneous, Isotropic, Source-Free Medium; Monochromatic Light) Substituting the electric fJnd magnetic fields E and j{ given in (5.3-1) into the wave equation (5.2-9) yields the Helmholtz equation \l2U + k 2 U 0, k nko W Ep' (5.3-16) Helmholtz Equation where the scalar function U U r represents the complex amplitude of any of the three components (Ex, Ey, Ez) of E or three components (Hx, Hy, Hz) of H; and where n E Eo P, P,o , ko W Co, and c Co n. In the context of wave optics, the Helmholtz equation in (2.2- 7) was written in terms of the complex amplitude U r of the real wavefunction u r, t .
164 CHAPTER 5 ELECTROMAGNETIC OPTICS Inhomogeneous Media In an inhomogeneous nonmagnetic medium, Maxwell's equations (5.3-12) (5.3-15) remain applicable, but the electric permittivity of the medium becomes position de- pendent, E E r . For locally homogeneous media in which E r varies slowly with respect to the wavelength, the Helmholtz equation (5.3-16) remains approximately valid, subject to the substitutions k n r ko and n r Dispersive Media In a dispersive dielectric medium, P t and £ t are connected by the dynamic relation provided in (5.2-23). To determine the corresponding relation between the complex- amplitude vectors P and E, we substitute (5.3-1) into (5.2-23), which gives rise to .. P ' ' E EOX,V i (5.3-17) # where 00 xv x t exp j27fvt dt (5.3-18) -00 is the Fourier transform of x t . Equation (5.3-17) can also be directly inferred from (5.2-23) by invoking the convolution theorem: convolution in the time domain cor- responds to multiplication in the frequency domain (see Sees. A.l and B.l of Ap- pendixes A and B, respectively), and recognizing E and P as the components of £ and P of frequency v. The function EoX V may therefore be regarded as the transfer function of the linear system that relates P t to £ t . The relation between 1) and £ is similar, D EvE (5.3-19) where E 1/ Eo 1 + X v · (5.3-20) Therefore, in dispersive media the susceptibility X and the permittivity E are frequency-dependent and, in general, complex-valued quantities. The Helmholtz equation (5.3-16) is thus readily adapted for use in dispersive nonmagnetic media by taking k W E V Mo. (5.3-21 ) When X v and E v are approximately constant within the frequency band of interest, the medium may be treated as approximately nondispersive. The implications of the complex-valued nature of X and k in dispersive media are discussed further in Sec. 5.5. 5.4 ELEMENTARY ELECTROMAGNETIC WAVES A. Plane, Spherical, and Gaussian Electromagnetic Waves We now examine three elementary solutions to Maxwell's equations that are of sub- stantial importance in optics: plane waves and spherical waves, which were discussed
5.4 ELEMENTARY ELECTROMAGNETIC WAVES 165 in Sec. 2.2B in the context of wave optics, and the Gaussian beam, which was stud- ied in Chapter 3 using the wave-optics formalism. The medium is assumed to be linear, homogeneous, nondispersive, and isotropic, and the waves are assumed to be monochromatic. The Transverse Electromagnetic (TEM) Plane Wave Consider a monochromatic electromagnetic wave whose magnetic- and electric-field complex-amplitude vectors are plane waves with wavevector k (see Sec. 2.2B) so that Hr Er Hoexp jk. r Eo exp jk · r , (5.4-1) (5.4-2) where the complex envelopes Ho and Eo are constant vectors. All components of H r and E r satisfy the Helmholtz equation provided that the magnitude of k is k nko, where n is the refractive index of the medium. We now examine the conditions that must be obeyed by Ho and Eo in order that Maxwell's equations be satisfied. Substituting (5.4-1) and (5.4-2) into Maxwell's equa- tions (5.3-12) and (5.3-13), respectively, leads to , k x Ho W E Eo k x Eo W JL Ho. (5.4-3) (5.4-4) The other two Maxwell's equations, (5.3-14) and (5.3-15), are satisfied identically since the divergence of a uniform plane wave is zero. It follows from (5.4-3) that E must be perpendicular to both k and H and from (5.4-4) that H must be perpendicular to both k and E. Thus, E, H, and k are mutually orthogonal, as illustrated in Fig. 5.4-1. Since E and H lie in a plane normal to the direction of propagation k, the wave is called a transverse electromagnetic (TEM) wave. 'E k "--Wave fronts Figure 5.4-1 The TEM plane wave. The vectors E, H, and k are mutually orthogonal. The wave- fronts (surfaces of constant phase) are normal to k. H In accordance with (5.4-3), the magnitudes Ho and Eo are related by Ho WE k Eo. Similarly, (5.4-4) yields Ho k WJL Eo. For these two equations to be consistent, we must have WE k k WJL, or k W f JL W C nw Co nko. This is, in fact, the same condition required in order that the wave satisfy the Helmholtz . equatIon. The ratio between the amplitudes of the electric and magnetic fields is Eo Ho WJL k CJL JL f. This quantity is known as the impedance of the medium, 1} Eo Ho JL . (5.4-5) Impedance E
166 CHAPTER 5 ELECTROMAGNETIC OPTICS For nonmagnetic media JL JLo, whereupon 'f} the impedance of free space Tjo via JLo E may be defined in terms of Tj Tjo , (5.4-6) Impedance (Nonmagnetic Media) n where JLo 1207r 377 O. Eo (5.4-7) 'f}o The complex Poynting vector S 1 E x H* [see (5.3-1 0)] is parallel to the wavevec- tor k, so th at the power flows along a irection normal to the wavefronts. Its magnitude I Eo 2 2Tj . (5.4-8) Intensity The intensity of a TEM wave is thus seen to be proportional to the absolute-square of the complex envelope of the electric field. As an example, an intensity of 10 W/cm 2 in free space corresponds to an electric field of 87 V/cm. Note the similarity between (5.4-8) and the relation I U 2, which was defined for scalar waves in Sec. 2.2A. Equation (5.2-15) provides that the time-averaged energy density W W of the plane wave is W I E 2 2 E 0 , (5.4-9) The intensity in (5.4-8) and the time-averaged energy density in (5.4-9) are therefore related by I cW, (5.4-10) indicating that the time-averaged power density flow I results from the transport of the time-averaged energy density at the velocity of light c. This is readily visualized by considering a cylinder of area A and length c whose axis lies parallel to the direction of propagation. The energy stored in the cylinder, cAW, is transported across the area in one second, confirming that the intensity (power per unit area) is I cW. The linear momentum density (per unit volume) transported by a plane wave is """" 1 c 2 S 1 c 2 I k W c k. The Spherical Wave An oscillating electric dipole radiates a wave with features that resemble the scalar spherical wave discussed in Sec. 2.2B. This electromagnetic wave is readily con- structed from an auxiliary vector field A r , known as the vector potential, which is often used to facilitate the solution of Maxwell's equations in electromagnetics. For the case at hand we set Ar AoU r X, (5.4-11)
5.4 ELEMENTARY ELECTROMAGNETIC WAVES 167 where Ao is a constant and x is a unit vector in the x direction. The quantity U r represents a scalar spherical wave with the origin at r 0: 1 exp jkr. r (5.4-12) Ur Because U r satisfies the Helmholtz equation, as was established in Sec. 2.2B, A r will also satisfy the Helmholtz equation \72 A + k 2 A o. We now define the magnetic field in terms of the curl of this vector 1 \7 x A, Jl and determine the corresponding electric field from Maxwell's equation (5.3-12) H (5.4-13) E 1 \7 x H. . JW€ (5.4-14) The form of (5.4-13) and (5.4-14) ensures that \7 · E 0 and \7 · H 0, as required by (5.3-14) and (5.3-15), since the divergence of the curl of any vector field vanishes. Because A r satisfies the Helmholtz equation, it can readily be shown that the remaining Maxwell's equation, \7 x E jWJlH, is also satisfied. Thus, (5.4-11) to (5.4-14) define a valid electromagnetic wave that satisfies Maxwell's equations. To obtain explicit expressions for E and H, the curl operations in (5.4-13) and (5.4-14) must be carried out. This is conveniently accomplished by making use of the spherical coordinate system r, f), 4> defined in Fig. 5.4-2(a). For points at distances from the origin much greater than a wavelength (r » A or kr » 21r), the complex- amplitude vectors may be approximated by E r Eo sin f) U r e (5.4-15) (5.4-16) H r Ho sin f) U r cJ>, where Ho jk Jl Ao, Eo TJHo, f) cos 1 x r , and e and cJ> are unit vectors in spherical coordinates. The wavefronts are therefore spherical and the electric and magnetic fields are orthogonal to one another and to the radial direction r, as illustrated in Fig. 5.4-2(b). Unlike the scalar spherical wave, however, the magnitude of this vector wave varies as sin f). At points near the z axis and far from the origin, f) 7f 2 and 4> 1r 2, so that the wavefront normals are nearly parallel to the z axis (corresponding to paraxial rays) and sin f) 1. In a Cartesian coordinate system, e sin f) x + cos () cos 4> y + cas () sin 4> Z x + x z y z y + x z Z x + x z z, so that E r Eo """'" x""....., x+ z Ur, z (5.4-17) where Uris the paraxial approximation of the spherical wave, namely the paraboloidal wave discussed in Sec. 2.2B. For sufficiently large values of z, the term x z in (5.4- 17) may also be neglected, whereupon E r EoU r x H r Ho U r y. (5.4-18) (5.4-19) Under this approximation U r approaches 1 z e- jkz , so that a TEM plane wave ultimately emerges.
168 CHAPTER 5 ELECTROMAGNETIC OPTICS x x Wavefront " , \ \ \ \ -- -- H E H E \ \ H \ \ \ " Z " \ " " \ " E " H H E E ...... / " " I " I A " r /' I / " I /. A , I ct» \ \ /- -- -- -- \ , \ , r \ , \ , , J J / Z I <P "'" I '" " I ", .",,- -- -- ----- (a) (b) Figure 5.4-2 (a) Spherical coordinate system. (b) Electric- and magnetic-field vectors and wavefronts of the electromagnetic field, at distances r » , radiated by an oscillating electric dipole. The Gaussian Beam It was demonstrated in Sec. 3.1 that a scalar Gaussian beam is readily obtained from a paraboloidal wave (the paraxial approximation to a spherical wave) by replacing the coordinate z by z + j zo, where Zo is a real constant. The same transformation applied to the corresponding electromagnetic wave leads to the electromagnetic Gaussian beam. Replacing z in (5.4-17) by z + j Zo yields "'" x "'" x+ . z Ur, Z + JZo Er Eo (5.4-20) where U r now represents the scalar complex amplitude of a Gaussian beam provided in (3.1-7). The wavefronts of the Gaussian beam are illustrated in Fig. 5.4-3(a) (these are also shown in Fig. 3.1-7) whereas the E-field lines determined from (5.4-20) are displayed in Fig. 5.4-3(b). In this case, the direction of the E field is not spatially uniform. x (a) - 2z 0 -ZO o Zo 2Z0 z (b) z Figure 5.4-3 (a) Wavefronts of the scalar Gaussian beam U(r) in the x-z plane. (b) Electric-field lines of the electromagnetic Gaussian beam in the x-z plane. (Adapted from H. A. Haus, Waves and Fields in Optoelectronics, Prentice Hall, 1984, Fig. 5.3a.)
5.4 ELEMENTARY ELECTROMAGNETIC WAVES 169 B. Relation Between Electromagnetic Optics and Scalar Wave Optics The paraxial scalar wave, defined in Sec. 2.2C, has wavefront normals that form small angles with respect to the axial coordinate z. The wavefronts behave locally as plane waves while the complex envelope and direction of propagation vary slowly with z. This notion is also applicable to electromagnetic waves in linear isotropic media. A paraxial electromagnetic wave is locally approximated by a TEM plane wave. At each point, the vectors E and H lie in a plane that is tangential to the wavefront surfaces and normal to the wavevector k (Fig. 5.4-4). The optical power flows along the direction E x H, which is parallel to k and approximately parallel to the coordinate z. E k ( , z H Figure 5.4-4 Paraxial electromagnetic wave. A paraxial scalar wave of intensity I U 2 [see (2.2-10)] may be associated with a paraxial electromagnetic wave of the same intensity I E 2 2'T} [see (5.4-8)] by setting the complex amplitude to U E 2'T} and matching the wavefronts. As attested to by the extensive development provided in Chapters 2--4, the scalar-wave description of light provides a very good approximation for solving a great many problems involving the interference, diffraction, propagation, and imaging of paraxial waves. The Gaussian beam with small divergence angle, considered in Chapter 3, provides a case in point. Most features of such beams, such as their intensity, focusing by a lens, reflection from a mirror, and interference, are addressed satisfactorily within the context of scalar wave optics. Of course, when polarization comes into play, wave optics is mute and we must appeal to electromagnetic optics. It is of interest to note that U (as defined above) and E do not satisfy the same boundary conditions. For an electric field tangential to the boundary between two dielectric media, for example, E is continuous (Fig. 5.1-1), but U E 2'T} is dis- continuous since 'f} changes value at the boundary. Thus, problems involving reflection and refraction at boundaries cannot be addressed completely within the scalar wave theory, although the matching of phase that leads to the law of reflection and Snell's law is adequately carried out within its confines (Sec. 2.4). Indeed, calculations of reflectance and transmittance at a boundary depend on the polarization state of the light and therefore require electromagnetic optics (see Sec. 6.2). Similarly, problems involving the transmission of light through dielectric waveguides require an analysis . based on electromagnetic theory, as discussed in Chapters 8 and 9. c. Vector Beams Maxwell's equations in the paraxial approximation admit other cylindrically symmetric . beam solutions for which the direction of the electric field vector is spatially nonuni- form. One example is a beam for which the electric field is aligned in an azimuthal orientati9n with respect to the beam axis [see Fig. 5.4-5(a)], i.e., Er U p, z exp jkz $. (5.4-21)
170 CHAPTER 5 ELECTROMAGNETIC OPTICS The scalar function U p, z turns out to be the Bessel Gauss solution to the Helmholtz equation, as discussed in Sec. 3.4. This beam vanishes on-axis (p 0) and has a donut-like spatial distribution. The beam diverges in the axial direction and the spot size increases, much like the Gaussian beam. t Yet another cylindrically-symmetric beam has an azimuthally oriented magnetic field vector, so that the electric field vector is radial, as illustrated schematically in Fig. 5.4-5(b). It also has a spatial distribution with an on-axis null. The distribution of the vector field of this beam is similar to the electromagnetic field radiated by a dipole oriented along the beam axis. It has been shown that a vector beam with radial electric field vector may be focused by a lens of large numerical aperture to a spot of significantly smaller size than is pos- sible with a conventional scalar Gaussian beam.:f: Clearly, there are useful applications for such beams in high-resolution microscopy. y y - -- r \ " "-... -.. , E .k' "- , 1 " , E . , ......c.---- [ . \x " -x ., . J. -, . 1 . ,,", , (a) (b) Figure 5.4-5 Vector beams with cylindrical symmetry. (a) Electric-field vectors oriented in the azimuthal direction. (b) Electric-field vectors oriented in the radial direction. The shading indicates the spatial distribution of the optical intensity in the transverse plane. 5.5 ABSORPTION AND DISPERSION In this section, we consider absorption and dispersion in nonmagnetic media. A. Absorption The dielectric media considered thus far have been assumed to be fully transparent, i.e., not to absorb light. Glass is such a material in the visible region of the optical spectrum but it is, in fact, absorptive in the ultraviolet and infrared regions. Transmissive optical components in those bands are fabricated from other materials: examples are quartz and magnesium fluoride in the ultraviolet; and germanium and barium fluoride in the infrared. Fig. 5.5-1 illustrates the spectral windows within which some commonly encountered optical materials are transparent (see Sec. 13.1 C for further discussion). In this section, we adopt a phenomenological approach to the absorption of light in linear media. Consider a complex electric susceptibility X X' + jx", (5.5-1) t D. G. Hall, Vector-Beam Solutions of Maxwell's Wave Equation, Optics Letters, vol. 21, pp. 9-J I, 1996. t R. Dorn, S. Quabis, and G. Leuchs, Sharper Focus for a Radially Polarized Light Beam, Physical Review Letters, vol. 91, 233901, 2003.
5.5 ABSORPTION AND DISPERSION 171 00 Q =: o ..J Magnesium fluoride MgF2 Calcium fluoride CaF2 Barium fluoride BaF2 Quartz S i 02 UV fused silica Si02 IR fused silica Si02 Glass (BK-7) 00 00 00 < ..J 00 o u ;J Q Z o U t-4 00 Silicon Si Germanium Ge ... II!. Gallium arsenide GaAs Zinc sulfide ZnS Zinc selenide ZnSe Cadmium telluride CdTe 0.1 0.2 0.3 0.4 0.5 0.7 1 2 3 4 5 7 10 Wavelength (Jim) Figure 5.5-1 Spectral bands within which selected optical materials transmit light. 20 corresponding to a complex electric permittivity E Eo 1 + X and a complex dielectric constant E Eo 1 + X . For monochromatic light, the Helmholtz equation (5.3-16) for the complex amplitude U r remains valid, \72U + k 2 U 0, but the wavenumber k itself becomes complex-valued: k W EJ-Lo ko 1 + X ko 1 + X' + j X" , (5.5-2) where ko W Co is the wavenumber in free space. Writing k in terms of rea] and imaginary parts, k (3 related to the susceptibility components X' and X": (3 ko 1 + X' + j X" · (5.5-3) As a result of the imaginary part of k, a plane wave with complex amplitude U A exp j kz traveling through such a medium in the z-direction undergoes a change which corresponds to absorption in the medium, the envelope A of the original plane recognized as the absorption coefficient (also called the attenuation coefficient or extinction coefficient) of the medium. This simple exponential decay formula for the seen in Sec. 14.1 A that certain media, such as those used in lasers, can exhibit a < 0, in which case 'Y a is called the gain coefficient and the medium amplifies rather than attenuates light. Since the parameter (3 is the rate at which the phase changes with z, it represents the propagation constant of the wave. The medium therefore has an effective refractive
172 CHAPTER 5 ELECTROMAGNETIC OPTICS index n defined by j3 nko, (5.5-4) and the wave travels with a phase velocity c Co n. Substituting (5.5-4) into (5.5-3) therefore relates the refractive index n and the absorption coefficient ex to the real and imaginary parts of the susceptibility X' and X": 1 ex . n E Eo 1 + X' + j X" · (5.5-5) Absorption Coefficient and Refractive Index Note that the square root in (5.5-5) provides two complex numbers with opposite signs (phase difference of 7r). The sign is selected such that if X" is negative, i.e., the medium is absorbing, then ex is positive, i.e., the wave is attenuated. If 1 + X' is positive, then the complex number 1 + X' + jx" is in the fourth quadrant, and its square root can be in either the second or the fourth quadrant. By selecting the value in the fourth quadrant, we ensure that ex is positive and n is then also positive. Similarly, if 1 + X' is negative, then 1 + X' + j X" is in the third quadrant and its square root is selected to be in the fourth quadrant so that both ex and n are positive. The impedance associated with the complex susceptibility X, which is also complex, is given by 1} JLo E 1}o l+X . (5.5-6) Impedance In general, therefore, X, k, E, and 1} are complex quantities while ex, (3, and n are real. Weakly Absorbing Media In a weakly absorbing medium, X" « 1 + X' so th at that 1 + X' + j X" 1 + X' . It follows from (5.5-5) n 1 + X' ko II x. n (5.5-7) ex (5.5-8) Weakly Absorbing Medium Under these circumstances, the refractive index is determined by the real part of the susceptibility and the absorption coefficient is proportional to the imaginary part thereof. In an absorptive medium X" is negative so that ex is positive whereas in an amplifying medium X" is positive and ex is negative. EXERCISE 5.5-1 Dilute Absorbing Medium. A nonabsorptive medium of refractive index no serves as host to a dilute suspension of impurities characterized by susceptibility X X' + j X", where X' « 1 and
5.5 ABSORPTION AND DISPERSION 173 x" « 1. Determine the overall susceptibility of the medium and demonstrate that the refractive index and absorption coefficient are given approximately by x' nno+- 2no ko X" Q--. no (5.5-9) (5.5-10) Strongly Absorbing Media In a strongly abs orbing medium, I X" I » 1 1 + x'I , so that (5.5-5) yields n - ja/2ko vjx" == J jJ (-X") == :f: (I- j) J (-X")' whereupon n J (- X")/2 a 2ko J ( -X")/2. (5.5-11) (5.5-12) Strongly Absorbing Medium Since X" is negative for an absorbing medium, the plus sign of the square root was selected to ensure that a is positive, and this yields a positive value for n as well. B. Dispersion Dispersive media are characterized by a frequency-dependent (and therefore wavelength- dependent) susceptibility X(v), electric permittivity E(V), refractive index n(v), and speed co/n(v). Since the angle of refraction in Snell's law depends on refractive index, which is wavelength dependent, optical components fabricated from dispersive materials, such as prisms and lenses, bend light of different wavelengths by different angles. This accounts for the wavelength-resolving capabilities of refracting surfaces and for the wavelength-dependent focusing power of lenses (and the attendant chro- matic aberration in imaging systems). Polychromatic light is therefore refracted into a range of directions. These effects are illustrated schematically in Fig. 5.5-2. White / R G B White /\ >< (1) ] : I. c.= :: : (1) '. . p:: :: : White B G R R G B B G R Wavelength Figure 5.5-2 Optical components fabricated from dispersive materials refract waves of different wavelengths by different angles (B == blue, G == green, R == red). Moreover, by virtue of the frequency-dependent speed of light in a dispersive medium, each of the frequency components comprising a short pulse of light ex- periences a different time delay. If the propagation distance through a medium is substantial, as is often the case in an optical fiber, for example, a brief light pulse at the
174 CHAPTER 5 ELECTROMAGNETIC OPTICS Original pulse Delayed & broadened pulse R B o Dispersive Medium t o t Figure 5.5-3 A dispersive medium serves to broaden a pulse of light because the different frequency components that constitute the pulse travel at different velocities. In this illustration, the low-frequency component (long wavelength, denoted R) travels faster than the high-frequency component (short wavelength, denoted B) and therefore arrives earlier. input will be substantially dispersed in time so that its width at the output is increased, as illustrated in Fig. 5.5-3. The wavelength dependence of the refractive index of some common optical mate- rials is displayed in Fig. 5.5-4. 2.5 '-- --- AS2S3 glass SrTi03 - <U AgCI "0 2 s::: .- <U MgO > CsBr .- Quartz u ro BK7 0 Calcite 1.5 F. Silica e C aF 2 4 Ge 3.5 Si e GaAs 3 s::: .- <U > .- u 2.5 <U o ZnSe 2 o e e LiNb03 BBO - G KTP c=: C KDP 1.5 o e 1 0.1 1 10 Wavelength <Jlm) Figure 5.5-4 Wavelength dependence of the refractive index of selected optical materials, including glasses, crystals, and semiconductors.
5.5 ABSORPTION AND DISPERSION 175 Measures of Dispersion Material dispersion can be quantified in a number of different ways. For glass optical components and broad-spectrum light that covers the visible band (white light), a commonly used measure is the Abbe number V nd 1 np nc, where np, nd, and nc are the refractive indexes of the glass at three standard wavelengths: blue at 486.1 nm, yellow at 587.6 nm, and red at 656.3 nm, respectively. For flint glass V 38 whereas for fused silica V 68. On the other hand, if dispersion in the vicinity of a particular wavelength Ao is of interest, an often used measure is the magnitude of the derivative dn dAo at that wavelength. This measure is appropriate for prisms, for example, in which the ray deflection angle (}d is a function of n [see (1.2-6)]. The angular dispersion d(}d dAo d(}d dn dn dAo is then a product of the material dispersion factor, dn dAo, and another factor, d(}d dn, that depends on the geometry of the prism and the refractive index of the material of which it is made. The effect of material dispersion on the propagation of brief pulses of light is governed not only by the refractive index n and its first derivative dn dAo, but also by the second derivative d 2 n dA, as will be elucidated in Sec. 5..6 and Sec. 22.3. Absorption and Dispersion: The Kramers Kronig Relations Absorption and dispersion are intimately related. Indeed, a dispersive material, i.e., a material whose refractive index is wavelength dependent, must be absorptive and must exhibit an absorption coefficient that is also wavelength dependent. The relation between the absorption coefficient and the refractive index is a result of the Kramers Kronig relations, which relate the real and imaginary parts of the susceptibility of a medium, X' v and X" v : x'v 2 00 sX" S s2 2 ds (5.5-13) 7r 0 v X" v 2 00 vx ' s (5.5-14) v 2 2 ds. 7r 0 s Kramers Kronig Relations Given the real or the imaginary component of X v for all v, these powerful formulas allow the complementary component to be determined for all v. The Kramers Kronig relations connecting X" v and X' v translate into relations between the absorption coefficient Q v and the refractive index n v by virtue of (5.5-5), which relates Q and n to X" and X'. The Kramers Kronig relations are a special Hilbert-transform pair, as can be under- stood from linear systems theory [see Sec. B.I of Appendix B]. They are applicable for all linear, shift-invariant, causal systems with real impulse response functions. The linear system at hand is the polarization-density response of a medium P t to an applied electric field £ t set forth in (5.2-23). Since £ t and P t are real, so too is the impulse response function EoX t . As a consequence, its Fourier transform, the transfer function EoX V , exhibits Hermitian symmetry: X v X* v [see Sec. A.l of Appendix A]. This system therefore obeys all of the conditions required for the Kramers Kronig relations to apply. The real and imaginary parts of the transfer func- tion EoX V are therefore related by (B.1-6) and (B.1-7) and, in particular, by (5.5-13) and (5.5-14).
176 CHAPTER 5 ELECTROMAGNETIC OPTICS c. The Resonant Medium We now set forth a simple classical microscopic theory that leads to a complex suscep- tibility and provides an underlying rationale for the presence of frequency-dependent absorption and dispersion in an optical medium. The approach is known as the Lorentz oscillator model. A more thorough discussion of the interaction of light and matter is provided in Chapter 13. Consider a dielectric medium such as a collection of resonant atoms, in which the dynamic relation between the polarization density P t and the electric field £ t , considered for a single polarization, is described by a linear second-order ordinary differential equation of the form W5€oXo £, (5.5-15) Resonant Dielectric Medium where 0", Wo, and XO are constants. An equation of this form emerges when the motion of a bound charge associated with a resonant atom is modeled phenomenologically as a classical harmonic oscillator, in which the displacement of the charge x t and the applied force :.r t are related by d 2 X dx 2 :.r . (5.5-16) m Here m is the mass of the bound charge, Wo K m is its resonance angular fre- quency, K is the elastic constant of the restoring force, and 0" is the damping coefficient. If the dipole moment associated with each individual atom is p ex, the po- larization density of the medium as a whole is related to the displacement by P Np Nex, where e is the electronic charge and N is the number of atoms per unit volume of the medium. The electric field and force are related by [, :.r e . The quantities P and £ are therefore proportional to x and :.r, respectively, and comparison of (5.5-15) and (5.5-16) provides Ne 2 Xo 2 . €omwo (5.5-17) The applied electric field can thus be thought of as inducing a time-dependent electric dipole moment in each atom, as portrayed in Fig. 5.5-5, and hence a time-dependent polarization density in the medium as a whole. £[ t . .+ . - . + . Figure 5.5-5 A time-varying electric field f, applied to a Lorentz-oscillator atom induces a time-varying dipole moment p that contributes to the overall polarization density P. - The medium is completely characterized by its impulse response function foX t , an exponentially decaying harmonic function, or equivalently by its transfer function foX V , which is obtained by solving (5.5-15) one frequency at a time, as follows.
5.5 ABSORPTION AND DISPERSION 177 Substituting £ t yields Re Eexp jwt and P t Re P exp jwt into (5.5-15) w 2 + jCJw + W5 P w5 E oXoE, (5.5-18) from which P Eo XoW6 w6 EoX v E, and substituting w susceptibility, w 2 + jcrw E. Writing this relation in the form P 21I"v, yields an expression for the frequency-dependent ( " X\v) Xo 2 V o 2 o v 2 + jv v ' (5.5-19) Susceptibility (Resonant Medium) where Vo Wo 21I" is the resonance frequency and v The real and imaginary parts of X v , denoted X' v therefore given by () 21I". and X" v respectively, are I X V Xo 2 Vo v5 v5 v 2 v 2 2 + v v 2 v5 v v 2 · v 2 + v v 2 (5.5-20) x" v Xo 2 V o (5.5-21) These equations are plotted in Fig. 5.5-6. x'(v) -x"(v) 1/ XoQ lI lIo Xo II lIo II Figure 5.5-6 Real and imaginary parts of the susceptibility of a resonant dielectric medium. The real part X' (1/) is positive below resonance, zero at resonance, and negative above resonance. The imaginary part X" (1/) is negative so that X"(I/) is positive everywhere and has a peak value XoQ at 1/ I/o, where Q 1/0/ /).1/. The illustration portrays results for Q 10. At frequencies well below resonance v« Vo , X' v XO and X" v 0, so that the low-frequency susceptibility is simply Xo. At frequencies well above resonance v » Vo , X' v X" v 0 so that the medium behaves like free space. Precisely at resonance v Vo , X' Vo 0 and X" Vo reaches its peak value of XoQ, where Q Vo v. The resonance frequency Vo is usually much greater than v so that Q » 1. Thus, the magnitude of the peak value of X" v , which is xoQ, is much larger than the magnitude of the low-frequency value of X' v , which is Xo. The maximum and minimum values of X' v are ::f:Xo Q 2 =F 1 Q and occur at frequencies Vo 1 =F 1 Q, respectively. For large Q, X' swings between positive and negative values with a magnitude approximately equal to xoQ 2, i.e., one half of the
178 CHAPTER 5 ELECTROMAGNETIC OPTICS peak value of X". The signs of X' and X" determine the phase of \':, which simply determines the angle between the phasors P and E. The behavior of x(v) in the vicinity of resonance (v rv vo) is often of particular interest. In this region, we may use the approximation (v6 - v 2 ) == (vo + v)(vo - v) 2vo (vo - v) in the real part of the denominator of (5.5-19), and replace v with Vo in the imaginary part thereof, to obtain vo/2 X(v rv va) Xa (va _ v) + jtlv/2 ' (5.5-22) from which (5.5-24) Susceptibility (Near Resonance) The function X" (v) in (5.5-23), known as the Lorentzian function, drops to half its peak value when I v - Vo I == v /2. The parameter v therefore represents the full- width half-maximum (FWHM) value of X"(V). The behavior of X (v) far from resonance is also of interest. In the limit I (v - vo) I » 6v, the susceptibility given in (5.5-19) is approximately real, II vov 1 X (v) -Xa 4 (va - v)2 + (tlv/2)2 v - Vo X' (v) 2 tlv X" (v). (5.5-23) 2 X(v) Xo 2 0 2 ' V -v o (5.5-25) Susceptibility (Far from Resonance) so that the medium exhibits negligible absorption. The absorption coefficient and the refractive index of a resonant medium may be <:feterrnined by substituting the expressions for X' (v) and X" (v), e.g., (5.5-23) and (5.5-24) into (5.5-5). Each of these parameters generally depends on both y/(V) and x" (v). However, in the special case for which the resonant atoms are embedded in a nondispersive host medium of refractive index no, and are sufficiently dilute so that x" (v) and X'(V) are both« 1, this dependence is much simpler, namely, the refractive index and the absorption coefficient are dependent on X' and X", respectively. Using the results of Exercise 5.5-1, it can be shown that these parameters are related by: ( 27rv ) II a(v) - - X (v) noco X' ( v ) n(v)no+ . 2no (5.5-26) (5.5-27) The dependence of these quantities on v is illustrated in Fig. 5.5-7. Media with Multiple Resonances A typical dielectric medium contains multiple resonances corresponding to different lattice and electronic vibrations. The overall susceptibility arises from a superposition
5.5 ABSORPTION AND DISPERSION 179 Q(I/) n(l/) /:).1/ Xo/ 2n o -f--------- --------- -no I/o 1/ » 1/ Figure 5.5-7 Absorption coefficient a(v) and refractive index n(v) of a dielectric medium of refractive index no containing a dilute concentration of atoms of resonance frequency Vo. of contributions from these resonances. Whereas the imaginary part of the suscep- tibility is confined to frequencies near the resonance, the real part contributes at all frequencies near and below resonance, as shown in Fig. 5.5-6. This is exhibited in the frequency dependence of the absorption coefficient and the refractive index, as illustrated in Fig. 5.5-8. Absorption and dispersion are strongest near the resonance frequencies. Away from the resonance frequencies, the refractive index is constant and the medium is approximately nondispersive and nonabsorptive. Each resonance does, however, contribute a constant value to the refractive index at all frequencies below its resonance frequency. c:::-- .9 e. E o. (/) U .D «ti-< CI) o u v VI V2 V 3 Figure 5.5-8 Frequency dependence of absorption coefficient a(v) and re- fractive index n( v) for a medium with three resonances. V Other complex processes can also contribute to the absorption coefficient and the re- fractive index of a material, so that different patterns of frequency dependence emerge. Figure 5.5-9 shows an example of the wavelength dependence of the absorption co- efficient and refractive index for a dielectric material that is essentially transparent at visible wavelengths. The illustration shows a decreasing refractive index with increas- ing wavelength in the visible region by virtue of a nearby ultraviolet resonance. The material is therefore more dispersive at shorter visible wavelengths where the rate of decrease of the index is greatest. This behavior is not unlike that exhibited in Fig. 5.5-1 and Fig. 5.5-4 for various real dielectric materials. The Sellmeier Equation In a medium with multiple resonances, labeled i == 1,2,..., the susceptibility is approximately given by a sum of terms, each of the form of (5.5-25), for frequencies far from any of the resonances. Using the relation between the refractive index and the real susceptibility provided in (5.2-13), n 2 == 1 + x, the dependence of n on frequency
180 CHAPTER 5 ELECTROMAGNETIC OPTICS Ultraviolet: Infrared 1:01 '.;;;1 I> Absorption coefficient Q )..0 Figure 5.5-9 Typical wavelength de- pendence of the absorption coefficient and refractive index for a dielectric medium exhibiting resonant absorption in the ultraviolet and infrared bands, concomitant with low absorption in the visible band. Note that the abscissa is wavelength rather than frequency. 1 0.01 I 0.1 I 1 I 10 I 100 I )..0 (/lm) and wavelength assumes a form known as the Sellmeier equation: 2 '"'" v '"'" .x 2 n 1 + XOi 2 2 == 1 + XOi .x2 _ .x . v. - V i i (5.5-28) Sellmeier Equation Table 5.5-1 provides the Sellmeier equations for a few selected materials, extracted from measured data using a least-squares fitting algorithm. The Sellmeier equation pro- vides a good description of the refractive index for most optically transparent materials. Table 5.5-1 Sellmeier equations for the wavelength dependence of the refractive indexes for selected materials at room temperature. The quantities no and ne indicate the ordinary and extraordinary indexes of refraction, respectively, for anisotropic materials (see Sec. 6.3). The range of wavelengths. where the results are valid is indicated in the rightmost column. Material Sellmeier Equation Wavelength (Wavelength ..\ in /-Lm) Range (/-Lm) Fused silica 2 0.6962..\ 2 0.4079'\ 2 0.8975..\ 2 0.21-3.71 n == 1 + ..\2 _ (0.06840)2 + ..\2 _ (0.1162)2 + ..\2 - (9.8962)2 Si 2 10.6684..\2 0.0030,\2 1.5413..\2 1.36-11 n == 1 + ,\2 _ (0.3015)2 + ..\2 _ (1.1347)2 + ,\2 - (1104.0)2 GaAs 2 7.4969..\2 1.9347..\2 1.4-11 n == 3.5 + ..\2 _ (0.4082)2 + ..\2 _ (37.17)2 BBO n 2 == 2.7359 0.01878 - 0.01354,\2 0.22-1.06 o + ..\2 - 0.01822 2 0.01224 2 ne == 2.3753 +..\2 0 0 - 0.01516..\ - . 1667 KDP 2 1.2566..\2 33.8991..\2 0.4-1.06 no == 1 + ..\2 _ (0.09191)2 + ..\2 - (33.3752)2 2 1.1311..\2 5.7568..\2 n -1+ + e - ..\2 _ (0.09026)2 ..\2 - (28.4913)2 LiNb0 3 2 2.5112..\2 7.1333,\2 0.4-3.1 no == 2.3920 + ..\2 _ (0.217)2 + ..\2 _ (16.502)2 2 2.2565..\2 14.503,\2 n == 2.3247 + + e ..\2 _ (0.210)2 ,\2 - (25.915)2
5.5 ABSORPTION AND DISPERSION 181 At wavelengths for which A « Ai the ith term becomes approximately proportional to A 2 , and for A » Ai it becomes approximately constant. As an example, the dispersion in fused silica, illustrated in Example 5.6-1, is well described by three resonances. For some materials the Sellmeier equation is conveniently approximated by a power series. D. Optics of Conductive Media Conductive materials such as metals, semiconductors, doped dielectrics, and ionized gases have free electric charges and an associated electric current density (J. In such media, the first of the source-free Maxwell's equations, (5.1-7), must be modified by including the current density (J along with the displacement current density (1) /8t, so that (1) V' x 1( == 8t + (J. (5.5-29) The other three Maxwell's equations remain the same. For a monochromatic wave of angular frequency w, this equation takes the form V' x H == jwD + J, (5.5-30) which is a modified version of (5.3-2). For a medium with linear dielectric properties, D == EE == Eo(l + X)E. Similarly, for a medium with linear conductive properties and conductivity a, the electric current density is proportional to the electric field, J == aE, (5.5-31) which is a form of Ohm's law [see (18.1-13) and (18.1-14)]. The right-hand side of (5.5-30) then becomes (jwE + a)E == jw( E + a / jw )E, so that V' x H == jWEeE, (5.5-32) where the effective electric permittivity Ee is a Ee == E + -:- . JW (5.5-33) Effective Permittivity The effective permittivity Ee is a complex frequency-dependent parameter that repre- sents a combination of the dielectric and conductive properties of the medium. Since the second term in (5.5-33) varies inversely with frequency, the contribution of the conductive component diminishes as the frequency increases. Moreover, since (5.5-32) takes the same form as the analogous equation for a dielec- tric medium, the laws of wave propagation derived in Sees. 5.3-5.5 are applicable even in the presence of conductivity. Thus, the wavenumber in (5.5-2) and (5.5-3) b ecome s k == {3 - j 10, == w vEeJ-Lo , and the impedance in (5.5-6) becomes 'fJ == V J-Lo/ Ee , while the reG-active index n and the attenuat ion co efficient 0, in (5.5-5) are determined from the complex equation n- ja/2ko == V Ee/Eo. When a /w » E, conducti ve effect s dominate and Ee a / jw. We then have n-ja/ 2k o v a / jWE o and 'fJ vj wJ-Lo/a, from which we obtain
182 CHAPTER 5 ELECTROMAGNETIC OPTICS n Ja/2WE o Q J 2wJ-L o a 'f} (1 + j) J wJ-Lo/2a, (5.5-34) (5.5-35) (5.5-36) Conductive Medium where we have made use of the relation ko == W / Co == w . The optical intensity is attenuated by a factor e- 1 at a depth d p == 1/ Q == 1/ y 2wJ-L o a , which is known as the penetration depth or skin depth. t Both d p and n vary as 1/ VW. For metals, a is very large and therefore so is Q, indicating that optical waves are significantly attenuated as they cross the surface of the material. However, the impedance'f} is very small, so these materials are highly reflective (see Exercise 6.2-2). EXAMPLE 5.5-1. Penetration Depth and Refractive Index of Copper. Copper has a conductivity of a == 0.58 x 10 8 (O-m)-l, so that the penetration depth is a scant d p == 1.9 nm at a wavelength Ao == 1 /-lID. In accordance with (5.5-34) and (5.5-35), the refractive index is related to the penetration depth via n == a1J o d p , which, for the case at hand, turns out to be n == 41.6. The Drude Model Since the relation between (J and £ is dynamic, the conductivity a must be frequency dependent with a finite bandwidth. Treating the conduction electrons as independent particles in an ideal gas that move freely between scattering events, the Drude model prescribes a frequency-dependent conductivity ao a== 1 + jWT ' where ao is the low-frequency conductivity and T is a relaxation time. It then follows from (5.5-33) that (5.5-37) ao Ee == E + . jw(l + jWT) (5.5-38) For W » I/T, (5.5-38) provides Ee E - ao/w 2 T. It is apparent that the conductivity then reduces the real part of the permittivity of the medium, acting like a negative con- tribution to the dielectric constant, with a functional form that is inversely proportional to the square of the frequency. In particular, if the medium has free-space-like dielectric properties with E == EO, the effective permittivity can be written as Ee = EO(l- : ). (5.5-39) where w p == J ao/ EoT is known as the plasma frequency. A simple classical microscopic theory provides an underlying rationale for the re- sults of the Drude model. The construct is similar to that of the Lorentz model; since the t The penetration depth is sometimes defined as the distance over which the field, rather than the intensity, is attenuated by a factor e -1 .
5.5 ABSORPTION AND DISPERSION 183 electrons of interest in a conductive medium are free rather than bound, however, the restoring force is excluded (K == 0) as is the damping ({J == 0). Under these conditions, the Lorentz equation of motion (5.5-16) becomes md 2 x/dt 2 == -e£, so that the corresponding polarization density P == - Nex obeys the simple equation d 2p / dt 2 == (Ne 2 /m)£, where N is electron density of the medium. For a field oscillating at an angular frequency w, this gives rise to -w 2 P == (Ne 2 /m)E, which is equivalent to a conductivity-related reduction of the dielectric constant of magnitude P / EoE == - (Ne 2 / Eom) / w 2 . This is consistent with (5.5-39), with a plasma frequency given by - [f£ e 2 w p - - . Eom (5.5-40) Combining (5.5-40) with the Drude result w p == J ao/Eor yields ao Ne 2 r/m, which accords with (18.1-13). It is apparent from (5.5-39) that wave propagation in a medium described by the Drude model assumes distinctly different behavior below, at, and above the plasma frequency, as illustrated in Fig. 5.5-10. W I I I I I I I I I I -........--..-.1 1 I I W W W -1 0 1 Relative permittivity Ee / Eo / 2 2 2 w =VW p + c o !3 // '" "/ / // W = co!3 / // Plasmonic band wp -----7--------------------- / // Forbidden band / :/ 00 Propagation constant !3 I I I I I I I I I wp - --I I I I I 00 1 Refracti ve index n o o Attenuation coefficient a Figure 5.5-10 Frequency dependence of the relative permittivity Ee/Eo, propagation constant {3, refractive index n, and attenuation coefficient Q of a medium described by the Drude model. . At frequencies below the plasma frequency (w < w p ), the effective permittivity is negative, so that k == W vEeMo is imaginary, corresponding to attenuation without propagation. This spectral band may therefore be regarded as a forbidden band. The attenuation coefficient Q == 2ko(w/w2 _1)1/2 decreases monotonically with increasing frequency and vanishes at the plasma frequency. A negative permit- tivity also corresponds to an imaginary impedance. Therefore, at the boundary between an ordinary medium with real impedance and a conductive medium with imaginary impedance, the light is fully reflected (see Sec. 6.2) so that the interface serves as a perfect mirror. . At frequencies above the plasma frequency (w > w p ), the effective permittivity is positive and real so that the conductive medium behaves like a lossless dielectric, albeit with unique dispersion characteristics. The propagation constant becomes (3 == (w 2 - w)1/2 /c o while the refractive index n == (1 - w/w2)1/2 lies below unity and is very small near the plasma frequency. This spectral band is referred to as the plasmonic band. · At the plasma frequency, w == w p , the propagation constant (3 == 0 so that the wave does not travel in the conductive medium. However, the electric current density oscillates and the free electrons undergo longitudinal oscillations; the
184 CHAPTER 5 ELECTROMAGNETIC OPTICS quantum quasi-particle associated with these oscillations is called a plasmon (much as a photon is associated with an optical field, as discussed in Chapter 12). In most metals, the plasma frequency lies in the ultraviolet so that they are reflective and shiny in the visible band. Some metals, such as copper, have a plasma frequency in the visible band so that they reflect only a portion of the visible spectrum and therefore have a distinct color. In doped semiconductors, the plasma frequency is usually in the infrared. 5.6 PULSE PROPAGATION IN DISPERSIVE MEDIA The propagation of pulses of light in dispersive media is important in many applications including optical fiber communication systems, as will be discussed in detail in Chap- ters 9 and 24. As indicated above, a dispersive medium is characterized by a frequency- dependent refractive index and absorption coefficient, so that monochromatic waves of different frequencies travel through the medium with different velocities and undergo different attenuations. Since a pulse of light comprises a sum of many monochro- matic waves, each of which is modified differently, the pulse is delayed and broadened (dispersed in time); in general its shape is also altered. In this section we provide a simplified analysis of these effects; a detailed description is deferred to Chapter 22. Group Velocity Consider a pulsed plane wave traveling in the z direction through a lossless disper- sive medium with refractive index n(w). Following the example set forth in Sec. 2.6, assume that the initial complex wavefunction at z == 0 is U(O, t) == A(t) exp(jwot), where Wo is the central angular frequency and A(t) is the complex envelope of the wave. It will be shown below that if the dispersion is weak, i.e., if n varies slowly within the spectral bandwidth of the wave, then the complex wavefunction at a distance z is approximately U(z, t) == A(t - z/v) exp[jwo(t - z / c)], where c == co/n(wo) is the speed of light in the medium at the central frequency, and v is the velocity at which the envelope travels (see Fig. 5.6-1). The parameter v, called the group velocity, is given by ! = (3' = d(3 v dw' (5.6-1) Group Velocity where (3 == wn (w ) / Co is the frequency-dependent propagation constant and the deriva- tive in (5.6-1), which is often denoted (3', is evaluated at the central frequency WOo The group velocity is a characteristic of the dispersive medium, and generally varies with the central frequency. The corresponding time delay Td == z/v is called the group delay. Since the phase factor exp[jwo(t - z/c)] is a function of t - z/c, the speed of light c, given by 1/ c == (3 (wo) / Wo, is often called the phase velocity. In an ideal (nondispersive) medium, (3(w) == w/c whereupon v == c and the group and phase velocities are identical. D Derivation of the Formula for the Group Velocity. The proof of (5.6-1) relies on a Fourier de- composition of the envelope A (t) into its constituent harmonic functions. A component of frequency 0, assumed to have a Fourier amplitude A(O), corresponds to a monochromatic wave of frequency
5.6 PULSE PROPAGATION IN DISPERSIVE MEDIA 185 w == Wo + 0 traveling with propagation constant {3(wo + 0). This component of the pulsed plane wave therefore travels as A(O) exp{ -j[{3(wo + O)]z} exp[j(wo + O)t]. If {3(w) varies slowly near the central frequency wo, it may be approximately linearized via a two-term Taylor series expansion: {3(wo +0) {3(wo) +0 d{3/dw == wo/c+O/v. The 0 component of the complex wavefunction may therefore be approximated by A(O) exp[jO(t-z/v)] exp[jwo(t-z/c)]. It follows that, upon traveling a distance z, the envelope of the Fourier component A(O) exp(jOt) becomes A(O) exp[jO(t - z/v)] for every value of 0; thus the pulse envelope A(t) becomes A(t - z/v). The pulse therefore travels at the group velocity v == 1/(d{3/dw), in accordance with (5.6-1). . Pulse at z = 0 I o \ Weakly dispersive medium I z 1- A(t) expUwot) \/ Pulse at z \ A(t-z/v) expUwo(t-z/c)] z/v t Figure 5.6-1 An optical pulse traveling in a dispersive medium that is weak enough so that its group velocity is frequency independent. The envelope travels with group velocity v while the underlying wave travels with phase velocity c. Since the index of refraction of most materials is typically measured and tabulated as a function of optical wavelength rather than frequency, it is convenient to express the group velocity v in terms of n(A). Using the relations (3 == wn(Ao)/ Co == 27rn(Ao) / Ao and Ao == 27rc o /w in (5.6-1), along with the chain rule d(3/dw == (d(3/dA) (dA/dw), provides Co V ==- N dn N = n - Ao dAo . (5.6-2) Group Velocity and Group Index The parameter N is often called the group index. Group Velocity Dispersion (GVD) Since the group velocity v == 1/ ( d(3 / dw) is itself often frequency dependent, different frequency components of the pulse undergo different delays Td == z/v. As a result, the pulse spreads in time. This phenomenon is known as group velocity dispersion (GVD). To estimate the spread associated with GVD it suffices to note that, upon traveling a distance z, two identic a] pulses of central frequencies v and v + 8v suffer a differential group delay dTd d ( Z ) 8T == - 8v == - - 8v == Dv z8v, dv dv v (5.6-3) where the quantity Dv = ( ) == 27r(3" dv v (5.6-4) Dispersion Coefficient
186 CHAPTER 5 ELECTROMAGNETIC OPTICS is called the dispersion coefficient and {3" d 2 {3 / dw 2 l w o. This effect is actually associated with the higher-order terms in the Taylor series expansion of (3( w) that were neglected in the derivation of the group velocity carried out above; a more complete treatment will be provided in Chapter 22. If the pulse has an initial spectral width a v (Hz), in accordance with (5.6-3) a good estimate of its temporal spread is then provided by (5.6-5) Pulse Spread The dispersion coefficient Dv is a measure of the pulse-time broadening per unit distance per unit spectral width (s/m-Hz). This temporal broadening process is illus- trated schematically in Fig. 5.6-2. If the refractive index is specified in terms of the wavelength, n(Ao), then (5.6-2) and (5.6-4) give aT == IDvlav z . A 3 d 2 n Dv == --f d '2 . Co Ao (5.6-6) Dispersion Coefficient (s/m-Hz) t z t Z = Z2 . t Dispersive medium I . o Zl Z2 Z Figure 5.6-2 An optical pulse traveling in a dispersive medium is broadened at a rate proportional to the product of the dispersion coefficient Dv, the spectral width a v, and the distance traveled z. It is also common to define a dispersion coefficient D,\ in terms of the wave- length t instead of the frequency. Using D,\ dA == Dv dv yields D,\ == Dv dv / dAo == Dv (-c o / A), which leads directly to (5.6-7) Dispersion Coefficient (s/m-nm) In analogy with (5.6-5), for a source of spectral width a,\ the temporal broadening of a pulse of light is D).. = _ >'0 d2 . Co dAo (5.6-8) Pulse Spread As discussed in Sec. 9.3, Sec. 24.1, and Sec. 22.3, in fiber-optics applications D,\ is usually specified in units of pslkm-nm: the pulse broadening is measured in pi- coseconds, the length of the medium in kilometers, and the source spectral width in nanometers. aT == ID,\ la,\z. t An alternative definition of the dispersion coefficient, M = - D).., is also widely used in the literature.
5.6 PULSE PROPAGATION IN DISPERSIVE MEDIA 187 Normal and Anomalous Dispersion Although the sign of the dispersion coefficient Dv (or D,\) does not affect the pulse- broadening rate, it does affect the phase of the complex envelope of the optical pulse. As such, the sign can play an important role in pulse propagation through media consisting of cascades of materials with different dispersion properties, as examined in Chapter 22. If Dv > 0 (D,\ < 0), the medium is said to exhibit normal dispersion. In this case, the travel time for higher-frequency components is greater than the travel time for lower-frequency components so that shorter-wavelength components of the pulse arrive later than longer-wavelength components, as illustrated schematically in Fig. 5.6-3. If Dv < 0 (D,\ > 0), the medium is said to exhibit anomalous disper- sion, in which case the shorter-wavelength components travel faster and arrive earlier. Most glasses exhibit normal dispersion in the visible region of the spectrum; at longer wavelengths, however, the dispersion often becomes anomalous. I \ {3''J> 0, DvI> 0, D.xp::: 0 Nonnal Dispersion ,. Anomalous Dispersion , t {3'T< 0, Dv r < 0, D AI> 0 Figure 5.6-3 Propagation of an optical pulse through media with normal and anomalous dispersion. In a medium with norma] dispersion the shorter-wavelength components of the pulse (B) arrive later that those with longer wavelength (R). A medium with anomalous dispersion exhibits the opposite behavior. The pulses are said to be chirped since the instantaneous frequency is time varyIng. Single-Resonance Medium The group velocity and dispersion coefficient for an optical pulse propagating through a single-resonance medium is determined by substituting (5.5-20) and (5.5-21) into (5.5-5) and making use of (5.6-2) and (5.6-7). To illustrate the behavior of the pulse in this medium, the wavelength dependence of the refractive index n, the group index N, and the dispersion coefficient D,\, are plotted in Fig. 5.6-4 as a function of normalized wavelength A/ Ao, for a medium with parameters XO == 0.05 and v /vo == 0.1. In the vicinity of the resonance (shaded area in figure), n varies sufficiently rapidly with wavelength that the parameters Nand D,\, which are defined on the basis of a Taylor series approximation comprising a few terms, cease to be meaningful. Away from the resonance, on both sides thereof, the refractive index decreases monotonically with increasing wavelength and exhibits points of inflection (indicated by dots). The first derivative of the refractive index achieves local maxima at these locations so that the group index N attains its maximum values there. Moreover, the second derivative vanishes at these points so that the dispersion coefficient changes sign. As the wave- length approaches the resonance wavelength from below, the dispersion changes from anomalous to normal; the reverse is true as the wavelength approaches resonance from above, as is evident in Fig. 5.6-4.
188 CHAPTER 5 ELECTROMAGNETIC OPTICS 1.2 1.1 ..;= >< cu 1.0 tl::"'O 0.9 0.8 1.8 o.:" 1.4 :::J>< OCU '-''''0 1.0 Os:: - 0.6 c<lO s::Q 5 o ..... .V; .u 0 .- t+-o 5 Q 8 - U_ 1O 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Normalized wavelength AI AO Figure 5.6-4 Wavelength dependence of the optical parameters associated with a single-resonance medium plotted as a func- tion of the wavelength normalized to the wavelength at the resonance frequency, A / AO: the refractive index n == Co / c (dots indicate points of inflection), the group index N == co/v (dots indicate maxima), and the dispersion coefficient D.x (dots indicate zeros). The parameters Nand D.x 1.5 are not meaningful near resonance (shaded area) . Fast and Slow Light in Resonant Media As is evident in Fig. 5.6-4, in a resonant medium the refractive index n and the group index N undergo rapid changes near the resonance frequency, and may be substantially greater or smaller than unity. Consequently, the phase velocity C == coin and the group velocity v == coiN may be significantly less than, or greater than, the velocity of light in free space, Co. The group index, and hence the group velocity, may even be negative. This raises the question of a potential conflict with causality and the special theory of relativity, which provides that information cannot be transmitted at a velocity greater than Co. It turns out that there is no such conflict since neither the group velocity nor the phase velocity corresponds to the information velocity, which is the speed at which information is transmitted between two points. The information velocity may be determined by tracing the propagation of a nonanalytic point on the pulse, for example, the onset of a rectangular pulse. It cannot exceed Co. The concepts of phase and group velocity were considered earlier in the context of an optical pulse traveling in a weakly dispersive medium, i.e., a medium with propagation constant {3( w) that is approximately linear in the vicinity of the central frequency of the pulse, woo After traveling a distance z, the pulse is delayed by a time z I v and is modulated by a phase factor exp( - jwoz I c). This phase, which travels at the phase velocity c, carries no information. It is the group velocity v that governs the time of "arrival" of the pulse. Since, in this approximation, the pulse envelope maintains its shape as it travels (Fig. 5.6-1), the group velocity is a good approximation of the information velocity. In the resonant medium, this occurs at wavelengths far from resonance, where the group index is greater than unity and the group velocity is less than co. At frequencies closer to resonance, higher-order dispersion terms become appre- ciable. In the presence of second-order dispersion (GVD), but negligible higher-order dispersion, a Gaussian pulse, for example, remains Gaussian, albeit with an increased width; its peak travels at the group velocity v. However, since the Gaussian pulse has a continuous profile and infinite support (i.e., extends over all time), the velocity at which the peak travels is not necessarily the information velocity; indeed, it can be greater than the free-space speed of light. In the immediate vicinity of resonance, where the group velocity can be significantly greater than Co and can even be negative (Fig. 5.6-4), higher-order dispersion terms must be considered. The pulse shape can then be significantly altered and the group velocity can no longer be considered as a possible information velocity. For sufficiently short distances, however, the pulse may travel without a significant alteration in shape,
5.6 PULSE PROPAGATION IN DISPERSIVE MEDIA 189 and this may occur at a group velocity significantly higher than CO. The pulse can also travel at a negative group velocity, signifying that a point on the pulse, identified by a peak for example, arrives at the end of the medium before the corresponding point on the input pulse even enters the medium! In the opposite limit of slow light, certain special resonance media permit the group velocity of light to be made exceedingly small so that a light pulse may be substantially slowed or even halted. It should be emphasized, however, that in none of these situations does the information velocity exceed CO. Since fast- and slow-light phenomena can only be observed near resonance, where the absorption coefficient is large (and frequency dependent), a mechanism for optical amplification is necessary, and nonlinear effects are often exploited to enhance this phenomenon. EXAMPLE 5.6-1. Dispersion in a Multi-Resonance Medium: Fused Silica. In the region between 0.21 and 3.71 /-lm, the wavelength dependence of the refractive index n for fused silica at room temperature is well characterized by the Sellmeier equation (5.5-28). Expressing all wavelengths in /-lm, this is achieved using three resonance wavelengths at Al = 0.06840 /-lm, A2 = 0.1162 /-lm, and A3 = 9.8962 /-lm, with weights XOI = 0.6962, X02 = 0.4079, and X03 = 0.8975, respectively. Expressions for the group index N and the dispersion coefficient D A are readily derived from this equation by means of (5.6-2) and (5.6-7). The results of this calculation in the 600-1600-nm wavelength range are presented in Fig. 5.6-5. 1.48 1.47 a) "'0 .5 1.46 a) . ..... u cl:: 1.45 a) 1.44 40 0 ..... s:: a) -40 .u t;::-- c+-o E a) s:: -80 o I .9 -120 0.. 8. -- -160 v.J a -200 0.6 0.7 D).. Anomalous Normal dispersion dispersion 0.8 0.9 1 1.1 1.2 .3 1.4 1.5 1.6 Wavelength Ao (/lm) Figure 5.6-5 Wavelength depen- dence of optical parameters for fused silica calculated on the basis of the Sellmeier equation (5.5-28): the re- fractive index n = colc (dot indi- cates point of inflection), the group index N = Co I v (dot indicates min- imum), and the dispersion coefficient D A (dot indicates zero). The refractive index n is seen to decrease monotonically with increasing wavelength, and to exhibit a point of inflection at Ao = 1.276 /-lm. At this wavelength, the group index N is minimum so that the group velocity v = Co I N is maximum. Since the dispersion coefficient D).. is proportional to the second derivative of n with respect to Ao, it vanishes at this wavelength. Zero dispersion coefficient signifies minimal pulse broadening. At wavelengths shorter than 1.276 /-lm, D).. < 0 and the medium exhibits normal dispersion whereas at longer wavelengths, D).. > 0 and the dispersion is anomalous. The presence of a zero-dispersion wavelength offers significant advantages in the design of optical fiber communications systems in which optical pulses carry information, as will become evident in Secs. 9.3, 24.1, and 22.3. The silica-glass fibers used in such systems are doped and exhibit zero dispersion close to 1.312 /-lm.
190 CHAPTER 5 ELECTROMAGNETIC OPTICS *5.7 OPTICS OF MAGNETIC MATERIALS AND METAMATERIALS In this section we consider wave propagation in media that exhibit absorption and dispersion in both their dielectric and magnetic properties; these are media for which E and J-L are complex and frequency dependent. A monochromatic plane wave with wavevector k has electric and magnetic fields given by E(r) == Eo exp( -jk . r) and H(r) == Ho exp( -jk . r), respectively. These fields obey (5.4-3) and (5.4-4), reproduced here for convenience: k x Ho == -WE Eo k x Eo == W J-LHo. (5.7-1) (5.7-2) The associated wavenumber and impedance are, respectively, k == w , (5.7-3) and Eo WJ-L 1] == Ho == k . (5.7-4) If both E and J-L are real, then k == nko, where the refractive index {ff J-L n == -- == co. Eo J-Lo (5.7-5) When E and J-L are complex, following (5.5-5) we write k in the form nko - j !a, so that I nko - ja = wR-, (5.7-6) where the real numbers '!1 and a are the refractive index and the attenuation coefficient, respectively. The imaginary parts of E and J-L therefore contribute to the attenuation coefficient a. The dispersion properties of the medium are described by the frequency dependences of nand a. These quantities are in turn controlled by the frequency dependences of the permittivity E == E(V) and permeability J-L == J-L(v), which are governed by the dynamics of the electric and magnetic responses of the medium at the atomic and molecular levels. For example, a simple dielectric material obeys the resonant-medium model described in Sec. 5.5C. A new class of artificially structured composite materials, called metamaterials, has recently emerged (see Chapter 7). These materials are fabricated by using ele- ments patterned on a macroscopic scale, in place of the atoms or molecules that form patterns in natural materials. The electromagnetic properties of metamaterials can be engineered so that they exhibit tailor-made electric and magnetic parameters E(V) and J-L (v), and therefore prescribed dispersion properties. Metamaterials typically consist of matrices of conductive wires or lattices and ring-like metal loops. Such structures exhibit resonant-like behavior similar to that discussed in Sec. 5.5C. A doubly periodic array of pairs of parallel gold nanorods, for example, exhibits a negative refractive
5.7 OPTICS OF MAGNETIC MATERIALS AND METAMATERIALS 191 index in the near infrared, with n -0.3 at A == 1.5 Mm. In this case, the negative- index behavior results from the plasmon resonances in the pairs of nanorods for both the electric and the magnetic components of the field. t Doubly Negative Metamaterials Dielectric and magnetic materials with complex E and M support diverse forms of wave propagation, depending on the values of the real and imaginary components of these complex parameters. The signs of the real parts of these coefficients also playa crucial role (for absorbing media the imaginary parts are always positive, as dictated by causality). An unusual situation arises when both E and M are real and negative since, in the this case, a self-consistent and physically realizable solution of Maxwell's equations gives rise to a negative refractive index. Such materials are said to be doubly negative. For doubly negative materials with real E and M, i.e., E == -lEI and M == -IMI, (5.7-1) and (5.7-2) become k x Ho == w lEI Eo k x Eo == -w IMI Ho, (5.7-7) (5.7-8) respectively. The sign reversal in (5.7-7) and (5.7-8) [compare with (5.7-1) and (5.7-2) for an ordinary material with positive real E and M] is tantamount to exchanging the roJes of the electric and magnetic fields. As a result, Eo, Ho, and k form a left-handed set of vectors in a doubly negative material, whereas in ordinary materials they form a right-handed set. This has profound implications since the complex Poynting vector S == !Eo x Ho is then anti-parallel to the wavevector k. This means that the wavefront travels in a direction opposite to the flow of electromagnetic energy. Figures 5.7-1 (a) and (b) illustrate the directions of flow of the power and wavefronts in normal and doubly negative materials, respectively. E E k k s s H H (a) Ordinary material (b) Doubly negative material Figure 5.7-1 (a) Plane wave in an ordinary (doubly positive) material. The vectors E, H, and k form a right-handed set and the wavefronts travel in the same direction as the power flow. (b) Plane wave in a doubly negative material. The vectors E, H, and k form a left-handed set and the wavefronts travel in a direction opposite to that of the power flow. An example is provided by a plane electromagnetic wave traveling along the z axis in a doubly negative material, with E and H pointing in the x and y directions, respectively: E == Eo exp( -jkz) x H == Ho exp( -jkz) y. (5.7-9) (5.7-10) t See v. M. Shalaev, w. Cai, u. K. Chettiar, H.-K. Yuan, A. K. Sarychev, v. P. Drachev, and A. V. Kildishev, Negative Index of Refraction in Optical Metamaterials, Optics Letters, vol. 30, pp. 3356-3358, 2005.
192 CHAPTER 5 ELECTROMAGNETIC OPTICS Th e Poy nting vector S == EoHoz == (IE o I/2'l])z. Since the wave impedance 'l] == V I/-lI/IEI in (5.7-4) is positive, as can be verified by use of (5.7-7), the Poynting vector points in the +z direction so that the wavevector k must be in the -z direction. We conclude that the wavenumber k in (5.7-9) and (5.7-10) is negative. This peculiar situation corresponds to selecting the negative sign for the square root in (5.7-6), which results in a negative refractive index: n == -co V IEII/-lI. (5.7-11) The prospects for E and /-l both being real and negative at some frequency are remote. For example, if both parameters are described by a resonant model, such as that considered in Sec. 5.5C, then throughout the frequency range over which the real part is negative, the imaginary part cannot be zero. Fortunately, the condition of reality and negativity of E and /-l is sufficient but not necessary for left-handedness, and thus for negative index. Left-handedness can in fact be exhibited in conjunction with absorption, i.e., in materials with complex E and /-l. It can be shown that if the real parts of both E and /-l are negative, the materia] is indeed left-handed even in the presence of absorption (nonvanishing imaginary part).t For example, if E and /-l are both described by resonant models, then above both resonance frequencies a band of frequencies exists where this condition is met (see Fig. 5.5-6). For such materials, the wave is attenuated and its amplitude decays in the direction of the energy flow (direction of the Poynting vector S). Furthermore, requiring the real parts of both E and /-l to be negative again turns out to be sufficient but not necessary for achieving a negative refractive index. The class of left-handed media transcends doubly negative materials. The definitive necessary and sufficient condition for left-handedness is:!: (I E I - Re{ E } ) (I /-ll - Re{ /-l }) > 1m { E } 1m {/-l } . (5.7-12) Materials for which both parameters are real, but only one is negative, do not satisfy (5.7-12) and therefore cannot be left-handed, but they do support attenuated waves as can be seen by using (5.7-6). Nor do media for which one of the material parameters, E or /-l, is real and positive, whatever the real and imaginary values of the other. It is clear, then, that nonmagnetic materials cannot be left-handed. Optics of Negative-Index Materials Many ordinary optical phenomena behave quite differently in negative-index media. A simple example is provided by the refraction of light, which follows Snell's law at the boundary between two dielectric media, nl sin (}l == n2 sin {}2. If one of the media, say medium 2, has a negative refractive index, this law takes the form nl sin (}l == -l n 21 sin {}2 (5.7-13) Since nl and (}l are positive, the angle of refraction (}2 must be negative, and the refracted and incident rays both lie on same side of the normal to the boundary. The two forms of Snell's law are illustrated in Figs. 5.7-2(a) and (b), respectively. As a result, the optics of planar boundaries and lenses is altered significantly. For example a convex lens of negative-index material behaves like a concave lens, and vice-versa. More peculiarly, a planar boundary between positive- and negative-index t See, for example, M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, The Negative Index of Refraction Demystified, European Journal of Physics, vol. 23, pp. 353-359, 2002. t Ibid.
READING LIST 193 nl>O n2>O nl>O n2<O ,. :fF. ,. ,& , .,. . " -_'" if J. ".r -. (a) (b) Figure 5.7-2 (a) Refraction at the boundary between positive-index media. The directions of 8 2 and k 2 are the same. (b) Refraction at the boundary between positive- and negative-index media. The directions of 8 2 and k 2 are opposite. materials acts like a lens, as illustrated in Fig. 5.7-3 for the special case of a negative- index material n == -1 in free space (n == 1). Moreover, each of the two boundaries of a negative-index slab has focusing power, so that a second image is created beyond the slab. Moreover, this system has been shown to offer the remarkable property of unity transmittance for plane waves at any inclination, regardless of polarization. * From a Fourier-optics viewpoint, as discussed in Sec. 4.4, this means that all spatial frequencies of an image are transmitted through the slab, including evanescent waves. The slab, in principle, acts as an "ideal lens" that transmits information beyond the diffraction limit. . " '#'. n2 =-1 }Ji -.,. n3 = 1 nl = 1 ... ,., .... ," '. oj" 4' 'f ;\ .... Figure 5.7-3 Focusing of rays by a negative-index slab in free space. Each boundary acts as a lens so that images are formed both inside and outside the slab. READING LIST General See also the general reading list in Chapter 1. W. H. Hayt, Jr. and J. A. Buck, Engineering Electromagnetics, McGraw-Hill, 1958, 7th ed. 2006. M. N. O. Sadiku, Elements of Electromagnetics, Oxford University Press, 4th ed. 2006. J. A. Kong, Electromagnetic Wave Theory, Wiley, 2nd ed. 1990; EMW Publishing, 2005. * See J. B. Pendry, Negative Refraction Makes a Perfect Lens, Physical Review Letters, vol. 85, pp. 3966-3969, 2000.
194 CHAPTER 5 ELECTROMAGNETIC OPTICS V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research, Springer-Verlag, 2005. N. Narayana Rao, Elements of Engineering Electromagnetics, Prentice Hall, 1977. 6th ed. 2004. N. Ida, Engineering Electromagnetics, Springer-Verlag, 2nd ed. 2004. P. Lorrain, D. R. Corson, and F. Lorrain, Fundamentals of Electromagnetic Phenomena, Freeman, 2000. J. D. Jackson, Classical Electrodynamics, Wiley, 3rd ed. 1998. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, 4th revised ed. 1997. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, Wiley, 3rd ed. 1994. D. H. Staelin, A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves. Prentice Hall, 1994. V. V. Sarwate, Electromagnetic Fields and Waves, Wiley, 1993. F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Wiley, 1985; Krieger, reissued 1992. E. E. Kriezis, D. P. Chrissoulidis. and A. G. Papagiannakis, Electromagnetics and Optics. World Scientific, 1992. D. K. Cheng, Field and Wave Electromagnetics, Addison-Wesley, 1983, 2nd ed. 1989. H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy, Prentice Hall, 1989. H. A. Haus, Waves and Fields in Optoelectronics, Prentice Hall, 1984. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii. Electrodynamics of Continuous Media. Pergamon. 2nd revised ed. 1984. Optical Constants M. Bass, E. W. Van Stryland, D. R. Williams, and W. L. Wolfe, eds., Handbook of Optics, McGraw- Hill, 2nd ed. 1995. W. L. Wolfe and G. J. Zissis, eds., The Infrared Handbook, Environmental Research Institute of Michigan. 1993. E. D. Palik, ed., Handbook of Optical Constants of Solids II, Academic Press, 1991. W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, Wiley, 2nd ed. 1976. Plasmons M. L. Brongersma and P. G. Kik, eds., Suiface Plasmon Nanophotonics, Springer-Verlag, 2006. J. Tominaga and D. P. Tsai, eds., Optical Nanotechnologies: The Manipulation of Suiface and Local Plasmons, Springer-Verlag, 2003. S. Jutamulia, ed., Selected Papers on Near-Field Optics, SPIE Optical Engineering Press (Milestone Series Volume 172), 2002. S. Kawata, M. Ohtsu, and M. Irie, eds., Near-Field Optics and Suiface Plasmon Polaritons, Springer- Verlag, 2001. D. Pines, Elementary Excitations in Solids: Lectures on Phonons, Electrons, and Plasmons, West- view, paperback ed. 1999. A. Liebsch, Electronic Excitations at Metal Suifaces, Springer-Verlag, 1997. H. Raether, Suiface Plasmons on Smooth and Rough Suifaces and on Gratings, Springer-Verlag, 1988. Fast and Slow Light P. W. Milonni, Fast Light, Slow Light and Left-Handed Light. Institute of Physics, 2005. M. Stenner, D. J. Gauthier, and M. A. Neifeld, Fast Causal Information Transmission in a Medium with a Slow Group Velocity, Physical Review Letters, vol. 94, 053902, 2005. N. Brunner, V. Scarani, M. Wegmiiller, M. Legre, and N. Gisin, Direct Measurement of Superluminal Group Velocity and Signal Velocity in an Optical Fiber. Physical Review Letters. vol. 93, 203902. 2004. R. W. Boyd and D. Gauthier, Slow" and 'Fast" Light, in Progress in Optics, vol. 43, pp. 497-530, E. Wolf, ed., Elsevier, 2002.
PROBLEMS 195 R. Y. Chiao and A. M. Steinberg, Tunneling Times and Superluminality, in Progress in Optics, vol. 37, pp. 347-406, E. Wolf, ed., Elsevier, 1997. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics, Springer-Verlag, 1994, corrected ed. 1997. L. Brillouin, Wave Propagation and Group Velocity, Academic Press, 1960. Doubly Negative Materials G. V. Eleftheriades and K. G. Balmain, eds., Negative-Refraction Metamaterials: Fundamental Prin- ciples and Applications, Wiley, 2005. J. B. Pendry and D. R. Smith, Reversing Light with Negative Refraction, Physics Today, vol. 57, no. 6,pp. 37-43,2004. M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, The Negative [ndex of Refraction Demystified, European Journal of Physics, vol. 23, pp. 353-359, 2002. J. B. Pendry, Negative Refraction Makes a Perfect Lens, Physical Review Letters, vol. 85, pp. 3966- 3969, 2000. V. G. Veselago, The Electrodynamics of Substances with Simultaneously Negative Values of f and J-L, Soviet Physics-Uspekhi, vol. 10, pp. 509-514, 1968. PROBLEMS 5. 1-1 An Electromagnetic Wave. An electromagnetic wave in free space has an electric-field vector E = J(t - z/co)x, where x is a unit vector in the x direction, and f(t) = exp( -t 2 /7 2 ) exp(j27rV o t), where 7 is a constant. Describe the physical nature of this wave and determine an expression for the magnetic-field vector. 5.2-1 Dielectric Media. Identify the media described by the following equations, regarding lin- earity, dispersiveness, spatial dispersiveness, and homogeneity. Assume that all media are isotropic. (a) P = EoX£ - a'\l x £, (b) P + ap2 = foE, (c) al 82P/8t 2 + a2 8P/8t + P = foX£, (d) P = Eo {a] + a2 exp[-(x 2 + y2)]}£, where X, a, aI, and a2 are constants. 5.3-1 Traveling Standing Wave. The electric-field complex-amplitude vector for a monochro- matic wave of wavelength Ao traveling in free space is E(r) = Eo sin{3y exp( -j{3z) x. (a) Determine a relation between (3 and Ao. (b) Derive an expression for the magnetic-field complex-amplitude vector H(r). (c) Determine the direction of the flow of optical power. (d) This wave may be regarded as the sum of two TEM plane waves. Determine their direc- tions of propagation. 5.4-1 Electric Field of Focused Light. (a) 1 W of optical power is focused uniformly on a flat target of size 0.1 x 0.1 mm 2 placed in free space. Determine the peak value of the electric field Eo (V 1m). Assume that the optical wave is approximated as a TEM plane wave within the area of the target. (b) Determine the electric field at the center of a Gaussian beam (a point on the beam axis at the beam waist) if the beam power is 1 Wand the beam waist radius W o = 0.1 mm. Refer to Sec. 3.1. 5.5-2 Amplitude-Modulated Wave in a Dispersive Medium. An amplitude-modulated wave whose complex wavefunction takes the form A(t) = [1 + m cos(27rf st)] exp(j27rv o t) at z = 0, where fs « vo, travels a distance z through a dispersive medium of propagation constant (3(v) and negligible attenuation. If (3(vo) = {30, (3(vo - fs) = {3t, and (3(vo + fs) = {32, derive an expression for the complex envelope of the transmitted wave as a function of {30, {3}, {32, and z. Show that at certain distances z the wave is amplitude modulated with no phase modulation.
196 CHAPTER 5 ELECTROMAGNETIC OPTICS 5.6-1 Group Velocity Dispersion in a Medium Described by the Sellmeier Equation. (a) Derive expressions for the group index N and the group velocity dispersion coefficient D A for a medium whose refractive index is described by the Sellmeier equation (5.5-28). (b) Use a computer to plot the wavelength dependence of n, N, and D A for fused silica in the region between 0.25 and 3.5 /-lm. Make use of the parameters provided in Table 5.5-1 (and Example 5.6-1). Verify the curves provided in Fig. 5.6-5. (c) Construct a similar collection of plots for GaAs in the region between 1.5 and 10.5 J-Lm. As indicated in Table 5.5-1, GaAs is characterized by a 3-term Sellmeier equation with resonance wavelengths 0 J-Lm, 0.4082 J-Lm, and 37.17 J-Lm, whose weights are 3.5, 7.4969, and 1.9347, respectively, in the wavelength region between 1.4 and 11 J-Lm at room temperature. Compare and contrast the behavior of the dispersion properties of fused silica and GaAs. 5.6-2 Refractive Index of Air. The refractive index of air can be precisely measured with the help of a Michelson interferometer and a tunable light source. At atmospheric pressure and a temperature of 20° C, the refractive index of air differs from unity by n - 1 == 2.672 X 10- 4 at a wavelength of 0.76 J-Lm, by n - 1 == 2.669 X 10- 4 at a wavelength of 0.8] J-Lm, and by n - 1 == 2.665 X 10- 4 at a wavelength of 0.86 J-Lm. (a) Using a quadratic fit to these data, determine the wavelength dependence of the group velocity. (b) Obtain an expression for the dispersion coefficient D A in ps/km-nm and compare your result with that for a silica optical fiber. 5.6-3 Group Velocity in a Metal. Show that for a medium described by the Orude model, (5.5-39), the product of the phase velocity and the group velocity is equal to c.
CHAPTER 6 POLARIZATION OPTICS 6.1 POLARIZATION OF LIGHT A. Polarization B. Matrix Representation 6.2 REFLECTION AND REFRACTION 6.3 OPTICS OF ANISOTROPIC MEDIA A. Refractive Indexes B. Propagation Along a Principal Axis C. Propagation in an Arbitrary Direction D. Dispersion Relation, Rays, Wavefronts, and Energy Transport E. Double Refraction 6.4 OPTICAL ACTIVITY AND MAGNETO-OPTICS A. Optical Activity B. Magneto-Optics: The Faraday Effect 6.5 OPTICS OF LIQUID CRYSTALS 6.6 POLARIZATION DEVICES A. Polarizers B. Wave Retarders C. Polarization Rotators D. Nonreciprocal Polarization Devices 199 209 215 228 232 235 f "' . "... -- " I. "- Augustin Jean Fresnel (1788-1827) advanced a theory of light in which waves exhibit transverse vibrations. The equations describing the partial reflection and refraction of light are named in his honor. Fresnel also made important contributions to the theory of light diffraction. 197
The polarization of light at a fixed position is determined by the time course of the electric-field vector £(r, t). In a simple medium, this vector lies in a plane tangential to the wavefront at that position. For monochromatic light, any two orthogonal compo- nents of the complex-amplitude vector E(r) in that plane vary sinusoidally with time, with amplitudes and phases that are generally different, so that the endpoint of the vector E( r) traces an ellipse. Since the wavefront generally has different directions at different positions, the plane, the orientation, and the shape of the ellipse also vary with position, as illustrated in Fig. 6.0-1 (a). For a plane wave, however, the wavefronts are parallel transverse planes and the polarization ellipses are the same everywhere, as illustrated in Fig. 6.0-1 (b), although the field vectors are not necessarily parallel at any given time. The plane wave is therefore described by a single ellipse, and is said to be elliptically polarized. The orientation and ellipticity of the polarization ellipse determine the state of polarization of the plane wave, whereas the size of the ellipse is determined by the optical intensity. When the ellipse degenerates into a straight line or becomes a circle, the wave is said to be linearly polarized or circularly polarized, respectively. y y (a) Wavefronts z (b) ---- Wavefronts Figure 6.0-1 Time course of the electric field vector of monochromatic light at several positions: (a) arbitrary wave; (b) plane wave or paraxial wave traveling in the z direction. In paraxial optics, light propagates along directions that lie within a narrow cone centered about the optical axis (the z axis). Waves are approximately transverse electro- magnetic (TEM) and the electric-field vectors therefore lie approximately in transverse planes, and have negligible axial components. From the perspective of polarization, paraxial waves may be approximated by plane waves and described by a single polar- ization ellipse (or circle or line). Polarization plays an important role in the interaction of light with matter as attested to by the following examples: . The amount of light reflected at the boundary between two materials depends on the polarization of the incident wave. . The amount of light absorbed by certain materials is polarization dependent. . Light scattering from matter is generally polarization sensitive. . The refractive index of anisotropic materials depends on the polarization. Waves with different polarizations travel at different velocities and undergo different phase shifts, so that the polarization ellipse is modified as the wave advances (e.g., linearly polarized light can be transformed into circularly polarized light). This property is used in the design of many optical devices. 198
6.1 POLARIZATION OF LIGHT 199 . The polarization plane of linearly polarized light is rotated by passage through certain media, including those that are opticalJy active, liquid crystals, and certain substances in the presence of an external magnetic field. This Chapter This chapter is devoted to a description of elementary polarization phenomena and a number of their applications. Elliptically polarized light is introduced in Sec. 6.1 using a matrix formalism that is convenient for describing polarization devices. Sec. 6.2 describes the effect of polarization on the reflection and refraction of light at the boundaries between dielectric media. The propagation of light through anisotropic media (crystals), optically active media, and liquid crystals are the subjects of Secs. 6.3, 6.4, and 6.5, respectively. Finally, basic polarization devices (polarizers, retarders, and rotators) are discussed in Sec. 6.6. 6.1 POLARIZATION OF LIGHT A. Polarization Consider a monochromatic plane wave of frequency v and angular frequency w == 27rV traveling in the z direction with velocity c. The electric field lies in the x-y plane and is generally descri bed by £ (z, t) = Re { A exp [j W (t - : )]} , (6.1-1) where the complex envelope A == Axx + AyY, (6.1-2) is a vector with complex components Ax and Ay. To describe the polarization of this wave, we trace the endpoint of the vector £ (z, t) at each position z as a function of time. Polarization Ellipse Expressing Ax and A in terms of their magnitudes and phases, Ax == ax exp(jc.px) and Ay == a y exp(jc.py), and substituting into (6.1-2) and (6.1-1) we obtain £(z, t) == £xx + £yY, (6.1-3) where C x = ax cas [w (t - : ) + 'Px ] c y = a y cas [w (t - : ) + 'Py ] (6.1-4a) (6 .1-4b ) are the x and y components of the electric-field vector £(z, t). The components £x and £y are periodic functions of t - z / c that oscillate at frequency v. Equations (6.1-4) are the parametric equations of the ellipse £2 £2 £ £ + - 2 cos c.p x y == sin 2 c.p, ax a y axa y (6.1-5)
200 CHAPTER 6 POLARIZATION OPTICS where c.p == c.py - c.px is the phase difference. At a fixed value of z, the tip of the electric-field vector rotates periodically in the x-y plane, tracing out this ellipse. At a fixed time t, the locus of the tip of the electric- field vector follows a helical trajectory in space that lies on the surface of an elliptical cylinder (see Fig. 6.1-1). The electric field rotates as the wave advances, repeating its motion periodically for each distance corresponding to a wavelength A == c/v. y y Ie A -I I I T -7 \ I , I , " \ b) Z x (a) Figure 6.1-1 (a) Rotation of the endpoint of the electric-field vector in the x-y plane at a fixed position z. (b) Snapshot of the trajectory of the endpoint of the electric-field vector at a fixed time t. The state of polarization of the wave is determined by the orientation and shape of the polarization ellipse, which is characterized by the two angles defined in Fig. 6.1-2: the angle W determines the direction of the major axis, whereas the angle X determines the ellipticity, namely the ratio of the minor to major axes of the ellipse b / a. These angles depend on the ratio of the magnitudes r == Qy / U x , and on the phase difference c.p == c.py - c.px , in accordance with the following relations: x 2r tan 2W == 2 cas c.p , l-r . 2r. SIn 2x == 1 2 SIn c.p , +r U y r== - Qx (6.1-6) c.p == c.py - c.px. (6.1-7) ax--j Figure 6.1-2 Polarization ellipse. Equations (6.1-6) and (6.1-7) may be derived by finding the angle W that achieves a transformation of the coordinate system of Ex and Ey in (6.1-5) such that the rotated ellipse has no cross term. The size of the ellipse is determined by the intensity of the wave, which is proportional to IAxl2 + IAyl2 == u; + u. Linearly Polarized Light If one of the components vanishes (u x == 0, for example), the light is linearly po- larized (LP) in the direction of the other component (the y direction). The wave is also linearly polarized if the phase difference c.p == 0 or 7r, since (6.1-4) gives Ey == ::i::( Qy/ ux)Ex, which is the equation of a straight line of slope ::i::u y / U x (the + and - signs correspond to c.p == 0 or 7r, respectively). In these cases the elliptical cylinder in Fig. 6.1-1(b) collapses into a plane as illustrated in Fig. 6.1-3. The wave is therefore also said to have planar polarization. If U x == Qy, for example, the plane of polarization makes an angle 45° with the x axis. If U x == 0, the plane of polarization is the y-z plane.
(a) 6.1 POLARIZATION OF LIGHT 201 (b) x z Plane of polarization Figure 6.1-3 Linearly polarized light (also called plane polarized light). (a) Time course at a fixed position z. (b) A snapshot (fixed time t). Circularly Polarized Light If c.p == 7r/2 and ax == Qy == ao, (6.1-4) gives Ex == ao cos[w(t - z/c) + c.px] and Ey == =Faosin[w(t - z/c) + c.px], from which E; + E == aB, which is the equation of a circle. The elliptical cylinder in Fig. 6.1-1 (b) becomes a circular cylinder and the wave is said to be circularly polarized. In the case c.p == +7r /2, the electric field at a fixed position z rotates in a clockwise direction when viewed from the direction toward which the wave is approaching. The light is then said to be right circularly polarized (RCP). The case c.p == -7r /2 corresponds to counterclockwise rotation and left circularly polarized (LCP) light. t In the right circular case, a snapshot of the lines traced by the endpoints of the electric-field vectors at different positions is a right-handed helix (like a right-handed screw pointing in the direction of the wave), as illustrated in Fig. 6.1-4. For left circular polarization, a left-handed helix is followed. y (a) y z y z (b) Figure 6.1-4 Trajectories of the endpoint of the electric-field vector of a circularly polarized plane wave. (a) Time course at a fixed position z. (b) A snapshot at a fixed time t. The sense of rotation in (a) is opposite that in (b) because the traveling wave depends on t - z/c. Poincare Sphere and Stokes Parameters As indicated above, the state of polarization of a light wave can be described by two real parameters: the magnitude ratio r == a y / Qx and the phase difference c.p == c.py - c.px. These are sometimes lumped into a single complex number r exp(jc.p), called the com- plex polarization ratio. Alternatively, we may characterize the state of polarization t This convention is used in most optics textbooks. The opposite designation is often used in the engineering literature: in the case of right (left) circularly polarized light, the electric-field vector at a fixed position rotates counterclockwise (clockwise) when viewed from the direction toward which the wave is approaching.
202 CHAPTER 6 POLARIZATION OPTICS by the two angles wand X, which represent the orientation and ellipticity of the polarization ellipse, respectively, as defined in Fig. 6.1-2. The Poincare sphere (see Fig. 6.1-5) is a geometrical construct in which the state of polarization is represented by a point on the surface of a sphere of unit radius, with coordinates r 1, () 90° 2X, <p 2W in a spherical coordinate system. Each point on the sphere represents a polarization state. For example, points on the equator (X 0°) represent states of linear polarization, with the two points 21}J 0° and 2W 180° representing linear polarization along the x and y axes, respectively. The north and south poles (2X 900) represent right-handed and left-handed circular polarization, respectiv.ely. Other points on the sphere represent states of elliptical po- larization. y U3 U3 RCP , , , , , , , , , , , , , , , , x LP: 90° .. . - LP: 135° .. LP: 45° Polarization ellipse 2 U2 U2 LP: 0° Poincare sphere LCP ... (a) (b) Figure 6.1-5 (a) The orientation and ellipticity of the polarization ellipse are represented geometrically as a point on the Poincare sphere. (b) Points on the Poincare sphere representing linearly polarized (LP) light at various angles with the x direction, as well as right-circularly polarized (RCP) and left-circularly polarized (LCP) light. The two real quantities r, cp , or equivalently the angles X, W , describe the state of polarization but contain no information about the intensity of the wave. Another representation that does contain such information is the Stokes vector. This is a set of four re a l numbers 80, 81, 82, 83 , called the Stokes parameters. The first three, 81, 82, 83 , are the Cartesian coordinates of the point on the Poincare sphere, U1 , U2, U3 cos 2X cas 2W, cos 2X sin 2W, sin 2X , multiplied by 80, so that 81 80 cas 2X cas 211J 82 80 cas 2X sin 2W 83 80 sin 2X. (6.1-8a) (6.1-8b) (6.1-8c) Using (6.1-6) and (6.1-7), together with a few trigonometric identities, the Stokes parameters in (6.1-8) may be expressed in terms of the field parameters ax, a y , cp , and in terms of the components of the complex envelope Ax, Ay , as: 80 81 82 83 a 2 + a 2 x y a 2 a 2 x y 2a x a y cas cp 2a x Oy sin cp Ax 2 + Ay 2 2 Re," A * A ' x Y A * A 21m,.. x y... Ax 2 (6.1-9a) ( 6.1-9b ) (6.1-9c) (6.1-9d) Stokes Parameters
6.1 POLARIZATION OF LIGHT 203 Since si + s + s S6, only three of the four components of the Stokes vector are independent; they completely define the intensity and the state of polarization of the light. A generalization of the Stokes parameters suitable for describing partially coherent light is presented in Sec. 11.4. We conclude that there are three equivalent representations for describing the state of polarization of an optical field: (1) the polarization ellipse, (2) the Poincare sphere, and (3) the Stokes vector. Yet another equivalent representation, the Jones vector, is introduced in the following section. B. Matrix Representation The Jones Vector As indicated above, a monochromatic plane wave of frequency v traveling in the z direction is completely characterized by the complex envelopes Ax ax exp jcpx and Ay ayexp jcpy of the x and y components of the electric-field vector. These complex quantities may be written in the form of a column matrix known as the Jones vector: Ax A · y (6.1-10) J Given J, we can determine the total intensity of the wave, I Ax 2 + Ay 2 21], and use the ratio r a y ax Ay Ax and the phase difference cp Cpy CPx arg Ay arg Ax to determine the orientation and shape of the polarization ellipse, as well as the Poincare sphere and the Stokes parameters. The Jones vectors for some special polarization states are provided in Table 6.1-1. The intensity in each case has been normalized so that Ax 2 + Ay 2 1 and the phase of the x component is taken to be CPx O. Table 6.1-1 Jones vectors of linearly polarized (LP) and right- and left-circularly polarized (RCP, LCP) light. y y LP in x direction 1 o LP at angle e cose sine x x y y x 1 2 1 . J RCP 1 1 . 2 J LCP x OrlhogonalPomrizaUons Two polarization states represented by the Jones vectors J 1 and J 2 are said to be orthogonal if the inner product between J 1 and J 2 is zero. The inner product is defined by J 1, J 2 A 1x A;x + AlyA;y, (6.1-11 ) where A 1x and A 1y are the elements of J 1 and A 2x and A 2y are the elements of J 2 . An example of orthogonal Jones vectors are the linearly polarized waves in the x and
204 CHAPTER 6 POLARIZATION OPTICS y directions, or any other pair of orthogonal directions. Another example is provided by right and left circularly polarized waves. Expansion of Arbitrary Polarization as a Superposition of Two Orthogonal Polarizations An arbitrary Jones vector J can always be analyzed as a weighted superposition of two orthogonal Jones vectors, say J I and J 2, that form the expansion basis; thus J QI J 1 + Q2 J 2. If J 1 and J 2 are normalized such that J I, J 1 J 2 , J 2 1, the expansion coefficients are the inner products QI J , J I and Q2 J , J 2 . EXAMPLE 6.1-1. Expansions in Linearly Polarized and Circularly Polarized Bases. Using the x and y linearly polarized vectors and as an expansion basis, the expansion coefficients for a Jones vector of components Ax and Ay with lAx 1 2 + lAy 1 2 1 are, by definition, (Xl Ax and (X2 Ay. The same polarization state may be expanded in other bases. . In a basis of linearly polarized vectors at angles 45° and 135°, Le., J 1 and J 2 1 (Ax + Ay), A 135 2 A45 Ax). (6.1-12) 2 y are used as an expansion basis, the coefficients (Xl and Q2 are: 2 x 1 . (Ax + JAy). 2 (6.1-13) A R jAy), A L For example, a linearly polarized wave with a plane of polarization that makes an angle () with the x axis (i.e., Ax cos () and Ay sin ()) is equivalent to a superposition of right and left circularly polarized waves with coefficients e- j () and e j (), respectively. A linearly polarized wave therefore equals a weighted sum of right and left circularly polarized waves. EXERCISE 6. 1-1 Measurement of the Stokes Parameters. Show that the Stokes parameters defined in (6.1-9) for light with Jones vector components Ax and Ay are given by So lAx 1 2 + lAy 1 2 ( 6.1-14a) Sl I Ax 1 2 IAyl2 (6.1-14b) 82 IA4512 IA13512 (6.l-14c) S3 IARI2 IAL 1 2 , ( 6.1-14d) where A45 and A 135 are the coefficients of expansion in a basis of linearly polarized vectors at angles 45° and 135° as in (6.1-12), and A R and A L are the coefficients of expansion in a basis of the right and left circularly polarized waves set forth in (6.1-13). Suggest a method of measuring the Stokes parameters of light with arbitrary polarization.
6.1 POLARIZATION OF LIGHT 205 Matrix Representation of Polarization Devices Consider the transmission of a plane wave of arbitrary polarization through an optical system that maintains the plane-wave nature of the wave, but alters its polarization, as illustrated schematically in Fig. 6.1-6. The system is assumed to be linear, so that the principle of superposition of optical fields is obeyed. Two examples of such systems are the reflection of light from a planar boundary between two media, and the transmission of light through a plate with anisotropic optical properties. Optical system Figure 6.1-6 An optical system that alters the polarization of a plane wave. The complex envelopes of the two electric-field components of the input (incident) wave, A 1x and A 1y , and those of the output (transmitted or reflected) wave, A 2x and A 2y , are in general related by the weighted superpositions A 2x TIIAlx + T l2 A 1y A 2y T 2l A 1x + T 22 A 1y , (6.1-15) where T 11 , T 12 , T 21 , and T 22 are constants describing the device. Equations (6.1-15) are general relations that all linear optical polarization devices must satisfy. The linear relations in (6.1-15) may conveniently be written in matrix notation by defining a 2 x 2 matrix T with elements T 11 , T12, T 21 , and T 22 so that A 2x A 2y TII T l2 T 21 T 22 A 1x A 1y · ( 6.1-16) If the input and output waves are described by the Jones vectors J I and J 2 , respectively, then (6.1-16) may be written in the compact matrix form J 2 TJ 1 . (6.1-17) The matrix T, called the Jones matrix, describes the optical system, whereas the vectors J I and J 2 describe the input and output waves. The structure of the Jones matrix T of a given optical system determines its effect on the polarization state and intensity of the wave. The following is a compilation of the Jones matrices of some systems with simple characteristics. Physical devices that have such characteristics will be discussed subsequently in this chapter. Linear polarizers. The system represented by the Jones matrix T ... - 1 0 o 0 (6.1-18) Linear Polarizer Along x Direction .. - transforms a wave of components A lx, A ly into a wave of components A lx, 0 by eliminating the y component, thereby yielding a wave polarized along the x direction,
206 CHAPTER 6 POLARIZATION OPTICS as illustrated in Fig. 6.1-7. The system is a linear polarizer with its transmission axis pointing in the x direction. y - .. .- x , , ." .- Linearly polarized light Figure 6.1-7 The linear polarizer. The lines in the polarizer represent the field direction that is permitted to pass. .- Polarizer Wave retarders. The system represented by the matrix T - - 1 0 o e- jr (6.1-19) Wave-Retarder (Fast Axis Along x Direction) - - transforms a wave with field components A 1x , A 1y into another with components A 1x , e- Jr A 1y , thereby delaying the y component by a phase r while leaving the x · component unchanged. It is therefore called a wave retarder. The x and y axes are called the fast and slow axes of the retarder, respectively. The simple application of matrix algebra permits the results illustrated in Fig. 6.1-8 to be understood: F F x I I I x I I I ' 1r/2 y - / / 7T / S/ y / S/ F F x I I I x I I I y / S/ Tf/2 , / 'Tr , / , y s' (a) Quarter-wave retarder (b) Half-wave retarder Figure 6.1-8 Operations of quarter-wave (Tf /2) and half-wave (Tf) retarders on several particular states of polarization are shown in (a) and (b), respectively. F and S represent the fast and slow axes of the retarder, respectively.
6.1 POLARIZATION OF LIGHT 207 . When r 7r 2, the retarder (called a quarter-wave retarder) converts the 1 · , an d -J converts the right circularly polarized wave into the linearly polarized wave J 1 1 · . When r 7r, the retarder (called a half-wave retarder) converts the linearly plane of polarization by 90°. The half-wave retarder converts the right circularly polarized wave into the left circularly polarized wave ].. J J Polarization rotators. While a wave retarder can transform a wave with one form of polarization into another" a polarization rotator always maintains the linear polar- ization of a wave but rotates the plane of polarization by a particular angle. The Jones . matrIx T - cos () t;in () - Sill fJ cos () - - (6.1-20) Polarization Rotator SIn 1 of a linearly polarized wave by an angle (). Cascaded Polarization Devices The action of cascaded optical systems on polarized light may be conveniently deter- mined by using conventional matrix multiplication formulas. A system characterized by the Jones matrix T] followed by another characterized by T 2 are equivalent to a single system characterized by the product matrix T T2Tl. The matrix of the system through which light is first transmitted must stand to the right in the matrix product since it is the first to affect the input Jones vector. EXERCISE 6. 1-2 Cascaded Wave Retarders. Show that two cascaded quarter-wave retarders with parallel fast axes are equivalent to a half-wave retarder. What is the result if the fast axes are orthogonal? Coordinate Transformation The elements of the Jones vectors and Jones matrices are dependent on the choice of the coordinate system. However, if these elements are known in one coordinate system, they can be determined in another coordinate system by using matrix methods. If J is the Jones vector in the x y coordinate system, then in a new coordinate system x' y', with the x' direction making an angle () with the x direction, the Jones vector J' is given by J' R () J, (6.1-21)
208 CHAPTER 6 POLARIZATION OPTICS where R () is the matrix ,. y y\ \ \ \ \ \ , x , -' () ' ",,-, - - CDS () sin () sin () co'S () · ...". "", " " R'()' , ) \ X \ \ \ \ \ \ \ \ - - (6.1-22) Coordinate Transformation This can be verified by relating the components of the electric field in the two coordi- nate systems. The Jones matrix T, which represents an optical system, is similarly transformed into T', in accordance with the matrix relations T' R () TR e T R () T' R () (6.] -23) (6.] -24) where R () is given by (6.1-22) with () replacing e. The matrix R () is the in- verse of R () so that R () R () is a unit matrix. Equation (6.1-23) can be obtained by using the relation J 2 T J 1 and the transformation J R () J 2 R () T J 1. Since J 1 R () J, we have J R () T R () J; since J T' J, (6.1-23) follows. EXERCISE 6. 1-3 Jones Matrix of a Polarizer. Show that the Jones matrix of a linear polarizer with a transmission axis making an angle () with the x axis is T - cos 2 () Sill () cas () - sin e cos e · 2 () SIll . (6.1-25) Linear Polarizer at Angle () - - Derive (6.1-25) using (6. ] -18), (6.1-22), and (6.] -24). Normal Modes The normal modes of a polarization system are the states of polarization that are not changed when the wave is transmitted through the system (see Appendix C). These states have Jones vectors satisfying T J J..lJ , (6.1-26) where J--l is constant. The normal modes are therefore the eigenvectors of the Jones matrix T, and the values of J-l. are the corresponding eigenvalues. Since the matrix T is of size 2 x 2 there are only two independent normal modes, T J 1 J--l1J 1 and T J 2 J..l2 J 2. If the matrix T is a Hermitian.. Le., if T 12 T 21 , the normal modes are orthogonal: J 1, J 2 O. The normal modes are usually used as an expansion basis so that an arbitrary input wave J may be expanded as a superposition of normal modes: J a 1 J 1 + a2J2. The response of the system may then be easily evaluated since TJ T a1J 1 + a2 J 2 a1TJ 1 + a2TJ2 alJ..l1J 1 + a2J--l2J2 (see Appendix C).
6.2 REFLECTION AND REFRACTION 209 EXERCISE 6. 1-4 Normal Modes of Simple Polarization Systems. (a) Show that the norma] modes of the linear polarizer are linearly polarized waves. (b) Show that the norma] modes of the wave retarder are linearly polarized waves. (c) Show that the normal modes of the polarization rotator are right and left circularly polarized waves. What are the eigenvalues of the systems described above? 6.2 REFLECTION AND REFRACTION In this section we examine the reflection and refraction of a monochromatic plane wave of arbitrary polarization incident at a planar boundary between two dielectric media. The media are assumed to be linear, homogeneous, and isotropic with impedances 1]1 and 1]2, and refractive indexes n1 and n2. The incident, refracted, and reflected waves are labeled with the subscripts 1, 2, and 3, respectively, as illustrated in Fig. 6.2-1. As shown in Sec. 2.4A, the wavefronts of these waves are matched at the boundary if the angles of reflection and incidence are equal, ()3 ()1, and if the angles of refraction and incidence satisfy Snel]'s law, nIsin ()1 n2 sin ()2 · (6.2-1 ) To relate the amplitudes and polarizations of the three waves, we associate with each wave an x y coordinate system in a plane normal to the direction of propagation (Fig. 6.2-1). The electric-field envelopes of these waves are described by the Jones vectors J 1 A 1x A 1y , J 2 A 2x A 2y , J 3 A 3x A 3y · (6.2-2) We proceed to determine the relations between J 2 and J 1 and between J 3 and J 1. These relations are written in the form of matrices J 2 tJ 1 and J 3 r J 1, where t and rare 2 x 2 Jones matrices describing the transmission and reflection of the wave, respectively. The elements of the transmission and reflection matrices may be determined by imposing the boundary conditions required by electromagnetic theory, namely the continuity at the boundary of the tangential components of E and H and the normal components of D and B. The electric field associated with each wave is orthogonal to the magnetic field; the ratio of their envelopes is the characteristic impedance, which is 1]1 for the incident and reflected waves and 1]2 for the transmitted wave. The result is a set of equations that are solved to obtain relations between the components of the electric fields of the three waves. The algebra involved is reduced substantially if we observe that the two normal modes for this system are linearly polarized waves with polarizations along the x and y directions. This may be proved if we show that an incident, a reflected, and a refracted wave with their electric field vectors pointing in the x direction are self-consistent with the boundary conditions, and similarly for three waves linearly polarized in the y direction. This is indeed the case. The x and y polarized waves are therefore uncoupled.
21 0 CHAPTER 6 POLARIZATION OPTICS Reflected x wave x y k3 C\e e\{ :'o1e ()3 ()2 Plane of incidence y · Oe\ \e x ()t kl nl n2 Figure 6.2-1 Reflection and refraction at the boundary between two die1ectric media. The x-polarized mode is called the transverse electric (TE) polarization or the or- thogonal polarization, since the electric fields are orthogonal to the plane of incidence. The y-polarized mode is called the transverse magnetic (TM) polarization since the magnetic field is orthogonal to the plane of incidence, or the parallel polarization since the electric fields are parallel to the plane of incidence. The orthogonal and parallel polarizations are also called the s (for the German senkrecht, meaning "perpendicular") and p (for "parallel") polarizations, respectively. The y axes in Fig. 6.2-1 are arbitrarily defined such that their components parallel to the boundary between the dielectrics all point in the same direction. The independence of the x and y polarizations implies that the Jones matrices t and r are diagonal, t t x o o t ' y r Tx 0 o Ty (6.2-3) so that E 2x txElx, E 3x TxElx, E 2y tyEly E 3y T yEly · (6.2-4) (6.2-5) The coefficients t x and t y are the complex amplitude transmittances for the TE and TM polarizations, respectively; T x and T yare the analogous complex amplitude re- flectances. Applying the boundary conditions (i.e., equating the tangential components of the electric fields and the tangential components of the magnetic fields at both sides of the boundary) in each of the TE and TM cases, we obtain the following expressions for the reflection and transmission coefficients: Tx rJ2 see B 2 rJl sec Bl rJ2 sec B 2 + rJl see Bl ' rJ2 cas B 2 rJl cas Bl 'l}2 cas B 2 + 'l}l cas (}l ' t x 1 + Tx, (6.2-6) TE Polarization Ty t y + r y . \ ) cas B 2 (6.2-7) TM Polarization Reflection & Transmission The characteristic impedance 'l} J1 E is complex if E and or J-l are complex, as is the case for lossy or conductive media. For nonlossy, nonmagnetic, dielectric media, rJ rJo n is real, where rJo J-lo Eo and n is the refractive index. In this case,
6.2 REFLECTION AND REFRACTION 211 the reflection and transmission coefficients in (6.2-6) and (6.2-7) yield the following equations, known as the Fresnel equations: ry nl eos (}l n2 eos (}2 nl eos (}l + n2 eos (}2 ' nl see (}l n2 see (}2 nl see (}1 + n2 see (}2 ' t x 1 + r x , (6.2-8) TE Polarization r x t y + r y . \ ) eos (}2 (6.2-9) TM Polarization Fresnel Equations Given nl, n2, and (}l, the reflection coefficients can be determined the Fresnel equations by first determining (}2 using Snell's law, (6.2-1), from which eos (}2 1 sin 2 (}2 1 2 · 2 () n 1 n2 SIn 1. (6.2-10) Since the quantities under the square-root signs in (6.2-10) can be negative, the re- flection and transmission coefficients are in general complex. The magnitudes r x and r y , and the phase shifts C{Jx arg r x and C{Jy arg r y , are plotted in Figs. 6.2-2 to 6.2-5 for the two polarizations, as functions of the angle of incidence (}l. Plots are provided for external reflection nl < n2 as well as for internal reflection nl > n2 . TE Polarization The dependence of the reflection coefficient r x on (}l for the TE-polarized wave is given by (6.2-8): External reflection nl < n2. The reflection coefficient r x is always real and nega- tive, corresponding to a phase shift C{Jx 7r. The magnitude r x n2 n 1 n 1 + n2 at (}l 0 (normal incidence) and increases to unity at (}1 90° (grazing incidence), as shown in Fig. 6.2-2. I 0) --- OJ Irxl o n) n2 7r I I I . I I Figure 6.2-2 Magnitude and phase of the re- flection coefficient as a function of the angle of incidence for external reflection of the TE- polarized wave (n2/nl 1.5). <Px 00 ()t 90° Internal reflection nl > n2. For small (}l the reflection coefficient is real and positive. Its magnitude is nl n2 nl + n2 when (}l 0°, and increases grad- ually to a value of unity, which is attained when (}l equals the critical angle (}e sin- 1 n2 nl . For (}l > () e, the magnitude of r x remains at unity, which corresponds to total internal reflection. This may be shown by using (6.2-10) to write t eos (}2 t The choice of the minus sign for the square root is consistent with the derivation that leads to the Fresnel equation.
212 CHAPTER 6 POLARIZATION OPTICS 1 sin 2 ()1 sin 2 0 e j sin201 sin 2 0 e 1, and substituting into (6.2-8). To- tal internal reflection is accompanied by a phase shift C{Jx arg r x given by t C{Jx an 2 cos 2 () e cos 2 {)1 1 (6.2-11) TE-Reflection Phase Shift The phase shift C{Jx increases from 0 at 0 1 () e to 7r at ()1 90°, as illustrated in Fig. 6.2-3. This phase plays an important role in dielectric waveguides (see Sec. 8.2). I (}J --- --- (}l (}2 I rxl f- I I I I - t I I I I t -- o 7r I nl n2 I I Figure 6.2-3 Magnitude and phase of the re- flection coefficient as a function of the angle of incidence for internal reflection of the TE- polarized wave (nl/n2 1.5). c.px I I I 00 ()c () 90° J TM Polarization Similarly, the dependence of the reflection coefficient r y on 0 1 for the TM-polarized wave is provided by (6.2-9): External reflection nl < n2. The reflection coefficient r y is always real. It as- sumes a negative value of nl n2 n1 + n2 at 0 1 0 (normal incidence). Its magnitude then decreases until it vanishes when n1 see ()1 n2 see ()2, at an angle {)1 {)B, known as the Brewster angle: {)B tan -1 n2 n1 (6.2-12) Brewster Angle (see Probe 6.2-5 for other properties of the Brewster angle). For {)1 > {)B, ry reverses sign (C{Jy goes from 7r to 0) and its magnitude gradually increases until it approaches unity at {)1 90°. The absence of reflection of the TM wave at the Brewster angle is useful for making polarizers (see Sec. 6.6). 1 nl n2 /r) - (}l ...... (}l - -- - (}2 o 7r c.py Figure 6.2-4 Magnitude and phase of the re- flection coefficient as a function of the angle of incidence for external reflection of the TM- polarized wave (n2/nl 1.5). 00 ()B 90° OJ
6.2 REFLECTION AND REFRACTION 213 Internal reflection nl > n2. At ()1 0°, r y is positive and has magnitude nl n2 nl + n2 . As ()l increases, the magnitude decreases until it vanishes at the Brew- ster angle ()B tan -1 n2 nl . As ()1 increases beyond ()B, r y becomes negative and its magnitude increases until it reaches unity at the critical angle () c. For ()1 > () c the wave undergoes total internal reflection accompanied by a phase shift CPy arg r y given by I tan CPy 2 1 sin 2 ()c cos 2 ()c cos 2 () 1 1 . (6.2-13) TM-Reflection Phase Shift At normal incidence, evidently, the reflection coefficient is r whether the reflection is TE or TM, or external or internal. 1 nl n2 nl +n2 , o t I . I I I , I , I : I I 01 --- 01 -- I r I y nl n2 7r I I <;?y Figure 6.2-5 Magnitude and phase of the re- flection coefficient as a function of the angle of incidence for internal reflection of the TM- polarized wave (nl/n2 1.5). [ 00 (}B () c () 90° 1 EXERCISE 6.2-1 Brewster Windows. At what angle is a TM-polarized beam of light transmitted through a glass plate of refractive index n 1.5 placed in air (n 1) without suffering reflection losses at either surface? Such plates, known as Brewster windows (Fig. 6.2-6), are used in lasers, as described in Sec. 15.2D. (}o ---- -- Figure 6.2-6 The Brewster window transmits TM-polarized light with no reflection loss. Power Reflectance and Transmittance The reflection and transmission coefficients rand t represent ratios of complex ampli- tudes. The power reflectance J( and power transmittance 'J are defined as the ratios of power flow (along a direction normal to the boundary) of the reflected and transmitted waves to that of the incident wave. Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, it follows
214 CHAPTER 6 POLARIZATION OPTICS that T 2. (6.2-14) For both TE and TM polarizations, and for both external and internal reflection, the power reflectance at norma] incidence is therefore nl n2 nl + n2 2 . (6.2-15) Power Reflectance at Normal I ncidence At the boundary between glass n 1.5 and air n 1, for example, 0.04, so that 4% of the light is reflected at normal incidence. At the boundary between GaAs n 3.6 and air n 1, 0.32, so that 32% of the light is reflected at normal incidence. However, at oblique angles the reflectance can be much greater or much smaller than 32%, as illustrated in Fig. 6.2-7. n 1 n 3.6 1 () Q) t.> s::: u 0.5 Q) t.-.4 Q) o TE ------- TM - 00 20° 40° 60° 80° () Figure 6.2-7 Power reflectance ofTE- and TM-polarization plane waves at the boundary between air (n 1) and GaAs (n 3.6), as a function of the angle of incidence (). The power transmittance 'J' is determined by invoking the conservation of power, so that in the absence of absorption loss the transmittance is simply 'J' 1 . (6.2-16) It is important to note, however, that 'J' is generally not equal to t 2 since the power travels at different angles and with different impedances in the two media. For a wave traveling at an angle 0 in a medium of refractive index n, the power flow in the direction normal to the boundary is (, 2 2'l} cos () (, 2 21]0 n cos (). It follows that n2 cos O 2 2 t . nl cos Ol (6.2-17) 'J' Reflectance from a plate. The power reflectance at normal incidence from a plate with two surfaces is described by 1 + 'J"2 since the power reflected from the far surface involves a double transmission through the near surface. For a glass plate in air, the overall reflectance is 1 + 'J2 0.04 1 + 0.96 2 0.077, so that about 7.7% of the incident light power is reflected. However, this calculation does not include interference effects, which are washed out when the light is incoherent (see Sec.II.2), nor does it account for multiple reflections inside the plate. Optical transmission and reflectance from multiple boundaries in layered media are described in detail in Sec. 7 .1.
6.3 OPTICS OF ANISOTROPIC MEDIA 215 EXERCISE 6.2-2 Reflectance of a Conductive Medium. The equations for the reflection coefficients set forth in (6.2-6) and (6.2-7) can be used to determine the intensity reflectance at the boundary between a dielectric medium and a conductive medium. (a) Show that 1 if the conductivity of the conductive medium a is infinite. (b ) Show that at normal incidence, and for a » foW, the relation 1 2 2f o W / a, known as the Hagen-Rubens relation, emerges. Use this relation to determine the reflectance of copper at the wavelengths Ao 1.06 J.Lm and 10.6 J.Lm. Assume that the conductivity of copper is a 0.58 x 10 8 (O-m)-l. (c) Show that if the conductive medium is described by the Drude model, (5.5-39), then 1 at frequencies below the plasma frequency. 6.3 OPTICS OF ANISOTROPIC MEDIA A dielectric medium is said to be anisotropic if its macroscopic optical properties depend on direction. The macroscopic properties of a material are, of course, ultimately governed by its microscopic properties: the shape and orientation of the individual molecules and the organization of their centers in space. Optical materials have dif- ferent kinds of positional and orientational types of order, which may be described as follows (see Fig. 6.3-1): Isotropic Anisotropic ,'-,'/ - , - \, - \ " I - , - " " " - \ I , ' I \ '" -- " / " "' 1 \ /,' ."", I ' """ /' , I / " Gas, liquid, amorphous solid - . - - - - - . - - - - - - - . - - - - - - Polycrystalline Crystalline Liquid crystal Figure 6.3-1 Positional and orientational order in different types of materials. . If the molecules are located at totally random positions in space, and are them- selves isotropic or oriented along random directions, the medium is isotropic. Gases, liquids, and amorphous solids follow this prescription. . If the structure takes the form of disjointed crystalline grains that are randomly oriented with respect to each other, the material is said to be polycrystalline. The individual grains are, in general, anisotropic, but their averaged macroscopic behavior is isotropic. . If the molecules are organized in space according to a regular periodic pattern and they are oriented in the same direction, as in crystals, the medium is, in general, anisotropic.
216 CHAPTER 6 POLARIZATION OPTICS . If the molecules are anisotropic and their orientations are not totally random, the medium is anisotropic, even if their positions are totally random. This is the case for liquid crystals, which have orientational order but lack complete positional order. A. Refractive Indexes Permittivity Tensor In a linear anisotropic dielectric medium (a crystal, for example), each component of the electric flux density D is a linear combination of the three components of the electric field, D i EijEj · (6.3-1 ) . J The indexes i, j 1, 2, 3 refer to the x, y, and z components, respectively, as described in Sec. 5.2B. The dielectric properties of the medium are therefore characterized by a 3 x 3 array of nine coefficients, Eij , that form the electric permittivity tensor €, which is a tensor of second rank. The material equation (6.3-1) is usually written in the symbolic form D €E. (6.3-2) For most dielectric media, the electric permittivity tensor is symmetric, i.e., Eij Eji. This means that the relation between the vectors D and E is reciprocal, Le., their ratio remains the same if their directions are exchanged. This symmetry is obeyed for dielectric nonmagnetic materials that do not exhibit optical activity, and in the absence of an external magnetic field (see Sec. 6.4). With this symmetry, the medium is characterized by only six independent numbers in an arbitrary coordinate system. For crystals of certain symmetries, even fewer coefficients suffice since some vanish and some are related. Geometrical Representation of Vectors and Tensors A vector, such as the electric field E, for example, describes a physical variable with magnitude and direction. It is represented geometrically by an arrow pointing in that particular direction, whose length is proportional to the magnitude of the vector [Fig. 6.3-2(a)]. A vector, which is a tensor of first rank, is represented numerically by three numbers: its projections on the three axes of a particular coordinate system. Though these components depend on the choice of the coordinate system, the magni- tude and direction of the vector in physical space are independent of the choice of the coordinate system. A scalar, which is described by a single number, is a tensor of zero rank. " " , , (a) (b) Figure 6.3-2 Geometrical representation of (a) a vector and ( b) a symmetric second-rank tensor.
6.3 OPTICS OF ANISOTROPIC MEDIA 217 A second-rank tensor is a rule that relates two vectors. In a given coordinate system, it is represented nUl11erically by nine numbers. Changing the coordinate system yields a different set of nine numbers, but the physical nature of the rule is unchanged. A useful geometrical representation [Fig. 6.3-2(b)] of a symmetric second-rank tensor (the dielectric tensor €, for example), is a quadratic surface (an ellipsoid) defined by EijXiXj 1, (6.3-3) . . 'lJ which is known as the quadric representation. This surface is invariant to the choice of the coordinate system; if the coordinate system is rotated, both Xi and Eij are altered but the ellipsoid remains intact in physical space. The ellipsoid has six degrees of freedom and carries all information about the symmetric second-rank tensor. In the principal coordinate system, Eij is diagonal and the ellipsoid assumes a particularly simple form: 2 2 2 1 EIX I + E2 X 2 + E3 X 3 · (6.3-4 ) Its principal axes are those of the tensor, and its axes have half-lengths 1 EI , 1 E2 , and 1 E3. Principal Axes and Principal Refractive Indexes The elements of the permittivity tensor depend on how the coordinate system is chosen relative to the crystal structure. However, a coordinate system can always be found for which the off-diagonal elements of Eij vanish, so that DI EIEl, D 2 E2 E 2, D3 E3 E 3, (6.3-5) where EI EJI, E2 E22, and E3 E33. According to (6.3-1), E and D are parallel along these particular directions so that if, for example, E points in the x direction, then so too must D. This coordinate system defines the principal axes and principal planes of the crystal. Throughout the remainder of this chapter, the coordinate system X, y, z, which is equivalently denoted Xl, X2, X3, is assumed to lie along the principal axes of the crystal. This choice simplifies all analyses without loss of generality. The permittivities E I, E2'1 and E3 correspond to refractive indexes nl E 1 Eo , n2 E2 Eo, n3 E3 Eo , (6.3-6) respectively, where Eo is the permittivity of free space; these are known as the principal refractive indexes. Biaxial, Uniaxial, and Isotropic Crystals Crystals in which the three principal refractive indexes are different are termed biaxial. For crystals with certain symmetries, namely a single axis of threefold, fourfold, or sixfold symmetry, two of the refractive indexes are equal nl n2 and the crystal is called uniaxial. In this case, the indexes are usually denoted nl n2 no and n3 ne, which are known as the ordinary and extraordinary indexes, respectively, for reasons that will become clear shortly. The crystal is said to be positive uniaxial if ne > no, and negative uniaxial if ne < no. The z axis of a uniaxial crystal is called the optic axis. In certain crystals with even greater symmetry (those with cubic unit cells, for example), all three indexes are equal and the medium is optically isotropic.
218 CHAPTER 6 POLARIZATION OPTICS Impermeability Tensor The relation D €E can be inverted and written in the form E € -1 D, where €-1 is the inverse of the tensor €. It is also useful to define the electric impermeability tensor 11 Eo€-1 (not to be confused with the impedance of the medium 7]), so that EoE 11 D. Since € is symmetric, so too is 11. Both tensors, € and 11, share the same principal axes. In the princi al coordinate system, 11 is diagonal with principal values Eo El 1 ni, Eo E2 1 n2' and Eo E3 1 n. Either tensor, € or 11, fully describes the optical properties of the crystal. Index Ellipsoid The index ellipsoid (also called the optical indicatrix) is the quadric representation of the electric impermeability tensor 11 Eo€-I: llij XiXj 1, i,j 1,2,3. (6.3-7) . . 'lJ If the principal axes were to be used as the coordinate system, we would obtain x 2 X2 x 2 1 + 2 + 3 n 2 n 2 n 2 123 1 , ( 6.3-8) I ndex Ellipsoid with principal values 1 n!, 1 n, and 1 n, and axes of half-lengths nl, n2, and n3. The optical properties of the crystal (the directions of the principal axes and the values of the principal refractive indexes) are therefore completely described by the index ellipsoid (Fig. 6.3-3). For a uniaxial crystal, the index ellipsoid reduces to an ellipsoid of revolution; for an isotropic medium it becomes a sphere. X3 n3 I "". . 11* / ..... " , , I n2 X2 nl Figure 6.3-3 The index ellipsoid. The coor- dinates (Xl, X2, X3) are the principal axes while ( nl , n2, n3) are the principal refractive indexes of the crystal. Xl B. Propagation Along a Principal Axis The rules that govern the propagation of light in crystals under general conditions are rather complex. However, they become relatively simple if the light is a plane wave traveling along one of the principal axes of the crystal. We begin with this case. Normal Modes Let x y z be a coordinate system that coincides with the principal axes of a crystal. A plane wave traveling in the z direction and linearly polarized along the x direction [Fig. 6.3-4(a)] travels with phase velocity Co nl (wavenumber k nlko) without changing its polarization. The reason for this is that the electric field has only one ,
6.3 OPTICS OF ANISOTROPIC MEDIA 219 component, EI pointed along the x direction, so that D is also in the x direction with DI tIEl; the wave equation derived from Maxwell's equations therefore provides a velocity of light given by 1 Mot I Co nl. Similarly, a plane wave traveling in the z direction and linearly polarized along the y direction [Fig. 6.3-4(b)] travels with phase velocity Co n2, thereby experiencing a refractive index n2. Thus, the normal modes for propagation in the z direction are linearly polarized waves in the x and y directions. These waves are said to be normal modes because their velocities and polarizations are maintained as they propagate (see Appendix C). Other cases in which the wave propagates along one of the principal axes and is linearly polarized along another are treated similarly [Fig. 6.3-4(c)]. z z z H k H k E k H y y y (a) (b) (c) Figure 6.3-4 A wave traveling along a principal axis and polarized along another principal axis has phase velocity coin!, c o ln2, or c o ln3, when the electric field vector points in the x, y, or z directions, respectively. (a) k n!ko; (b) k n2ko; (c) k n3ko. Polarization Along an Arbitrary Direction We now consider a wave traveling along one principal axis (the z axis, for example) that is linearly polarized along an arbitrary direction in the x y plane. This case is addressed by analyzing the wave as a sum of the normal modes, namely the linearly polarized waves in the x and y directions. These two components travel with different phase velocities, Co nl and Co n2, respectively. They therefore undergo different phase shifts, <.px nlkod and <.py n2k o d, respectively, after propagating a distance d. Their phase retardation is thus cp Cpy CPx n2 nl kod. Recombination of the two components yields an elliptically polarized wave, as explained in Sec. 6.1 and illustrated in Fig. 6.3-5. Such a crystal can therefore serve as a wave retarder, a device in which two orthogonal polarizations travel at different phase velocities so that one is retarded with respect to the other (see Fig. 6.1-8). y y x y x / z + z / / z (a) (b) (c) Figure 6.3-5 A linearly polarized wave at 45° in the z 0 plane (a) is analyzed as a superposition of two linearly polarized components in the x and y directions (normal modes), which travel at velocities coin! and c o ln2 [(b) and (c), respectively]. As a result of phase retardation, the wave is converted from plane polarization to elliptical polarization (a). It is therefore clear that the initial linearly polarized wave is not a normal mode of the system.
220 CHAPTER 6 POLARIZATION OPTICS c. Propagation in an Arbitrary Direction We now consider the general case of a plane wave traveling in an anisotropic crystal in an arbitrary direction defined by the unit vector u. We demonstrate that the two normal modes are linearly polarized waves. The refractive indexes na and nb, and the directions of polarization of these modes, may be determined by use of a procedure based on the index ellipsoid: Index-Ellipsoid Construction for Determining Normal Modes Figure 6.3-6 illustrates a geometrical construction for determining the polariza- tions and refractive indexes na and nb of the normal modes of a wave traveling in the direction of the unit vector u in an anisotropic material characterized by the index ellipsoid: x 2 x 2 x 2 1 + 2 + 3 n 2 n 2 n 2 123 1. X3 Index n3 ellipse n a ..\. , , ., , , , . r " " Da , . , "\ "- u " -. n2 X2 ":;. "):---: Xl Index ellipsoid Figure 6.3-6 Determination of the normal modes from the index ellipsoid. . Draw a plane passing through the origin of the index ellipsoid, normal to n. The intersection of the plane with the ellipsoid is an ellipse called the index ellipse. . The half-lengths of the major and minor axes of the index ellipse are the refractive indexes na and nb of the two normal modes. . The directions of the major and minor axes of the index ellipse are the directions of the vectors Da and Db for the normal modes. These directions are orthogonal. . The vectors Ea and Eb may be determined from Da and Db with the help of (6.3-5). , D Proof of the Index-Ellipsoid Construction for Determining the Normal Modes. To determine the normal modes (see Sec. 6.1B) for a plane wave traveling in the direction ii, we cast Maxwell's equations (5.3-2)-(5.3-5), and the material equation D €E given in (6.3-2), as an eigenvalue problem. Since all fields are assumed to vary with the position r as exp( jk. r), where k kii, Maxwell's equations (5.4-3) and (5.4-4) reduce to k x H wD k x E wJ.LoH (6.3-9) (6.3-10)
6.3 OPTICS OF ANISOTROPIC MEDIA 221 Substituting (6.3-10) into (6.3-9) leads to w 2 JLoD . (6.3-11 ) k x (k x E) Using E €-1 D, we obtain w 2 JLoD . (6.3-12) k x (k x € -1 D ) This is an eigenvalue equation that D must satisfy. Working with D is convenient since we know that it lies in a plane normal to the wave direction fi. We now simplify (6.3-12) by using 11 Eo€-l, k kfi, n k/ko, and k w 2 JLoEo to obtain 1 2 D. n (6.3-13) fi x (fi x 11 D) The operation ii x (fi x l1D) may be interpreted as a projection of the vector l1D onto a plane normal to fi. We may therefore rewrite (6.3-13) in the form 1 2 D, n (6.3-14) Pul1 D where Puis an operator representing projection. Equation (6.3-14) is an eigenvalue equation for the operator P ul1 , with eigenvalue 1/n2 and eigenvector D. The two eigenvalues, l/n and l/n, and two corresponding eigenvectors, Da and Db, represent the two normal modes. The eigenvalue problem (6.3-]4) has a simple geometrical interpretation. The tensor 11 is represented geometrically by its quadric representation, the index ellipsoid. The operator P u 11 represents projection onto a plane normal to u. Solving the eigenvalue problem in (6.3-14) is thus equivalent to finding the principal axes of the ellipse formed by the intersection of the plane normal to fi with the index ellipsoid. This is precisely the construction set forth in Fig. 6.3-6 for determining the normal modes. . Special Case: Uniaxial Crystals In uniaxial crystals (n} n2 no and n3 ne) the index ellipsoid of Fig. 6.3-6 is an ellipsoid of revolution. For a wave whose direction of travel u forms an angle e with the optic axis, the index ellipse has half-lengths no and n () , where cos 2 e sin 2 e + n 2 n 2 o e 1 2 ' e ' n ... ) , (6.3-15) Refractive Index of Extraordinary Wave so that the normal modes have refractive indexes nb no and na n () . The first mode, called the ordinary wave, has a refractive index no regardless of (). In accor- dance with the ellipse shown in Fig. 6.3-7, the second mode, called the extraordinary wave, has a refractive index n e that varies from no when () 0°, to ne when () 90° . The vector D of the ordinary wave is normal to the plane defined by the optic axis (z axis) and the direction of wave propagation k, and the vectors E and D are parallel. The extraordinary wave, on the other hand, has a vector D that is normal to k and lies in the k z plane, and E is not parallel to D, as shown in Fig. 6.3-7. D. Dispersion Relation, Rays, Wavefronts, and Energy Transport We now examine other properties of waves in anisotropic media including the disper- sion relation (the relation between wand k).
222 CHAPTER 6 POLARIZATION OPTICS . I !o I I ne I / / / / "" .,,'" .----- ---- -- . E,Dj - !H I I . I . I . I k . -- .... -- --- ." ." / ,/ / / I I I I \ \ , , , " ......... ........ .............. ..... --- D E I . I . I . !e I . -0 I - I . I e wave U en I e . ...... . ....... I >< o I k no n( B) t ( I ,. , f o wave Figure 6.3-7 Variation of the refractive index n( B) of the extraordinary wave with () (the angle between the direction of propagation and the optic axis) in a uniaxial crystal, and directions of the electromagnetic fields of the ordinary (0) and extraordinary (e) waves. The circle with a dot at the center located at the origin signifies that the direction of the vector is out of the plane of the paper, toward the reader. The optical wave is characterized by the wave vector k, the field vectors E, D, H, and B, and the complex Poynting vector S ! E x H* (direction of power flow). These vectors are related by (6.3-9) and (6.3-1 0). It follows from (6.3-9) that D is normal to both k and H. Equation (6.3-10) similarly indicates that H is normal to both k and E. These geometrical conditions are illustrated in Fig. 6.3-8, which also shows the complex Poynting vector S, which is orthogonal to both E and H. Thus, D, E, k, and S lie in one plane to which Hand B are normal. In this plane D 1.. k and S 1.. E; but D is not necessarily parallel to E, and S is not necessarily parallel to k. D 1 I o. EI 0" I .J .. I 0" I I I I I I I I H,B I . I" / , k ' I I I I s: Figure 6.3-8 The vectors D, E, k, and S all lie in one plane to which Hand Bare normal. D 1.. k and E 1.. S. / Using the relation D €E in (6.3-11), we obtain k x k x E +w 2 J-Lo€E o. (6.3-16) This vector equation, which E must satisfy, translates to three linear homogeneous equations for the components E 1 , E 2 , and E3 along the principal axes, written in the matrix form n 2 k 2 k 2 1 0 2 k 2 k l k3 k l k5 k 1 k 2 n k ki k3 k 2 klk3 k5 k2 k 3 n k ki EI E 2 k E3 o o , o (6.3-17) where k 1 , k 2 , k3 are the components of k, ko w Co, and nl, n2, n3 are the principal refractive indexes given by (6.3-6). The condition for these equations to have a nontrivial solution is obtained by setting the determinant of the matrix to zero. The result is an equation that relates w to k 1 , k 2 , and k3 and that takes the form w w k 1 , k 2 , k3 , where w k 1 , k 2 , k3 is a nonlinear function. This relation, known as the dispersion relation, is the equation of a surface in the k l , k 2 , k3 space, known
6.3 OPTICS OF ANISOTROPIC MEDIA 223 as the normal surface or the k surface. The intersection of the direction u with the k surface determines the vector k whose magnitude k nw Co provides the refractive index n. There are two intersections corresponding to the two normal modes associated with each direction. The k surface is a centrosymmetric surface comprising two sheets, each correspond- ing to a solution (a normal mode). It can be shown that the k surface intersects each of the principal planes in an ellipse and a circle, as illustrated in Fig. 6.3-9. For biaxial crystals nl < n2 < n3 , the two sheets meet at four points, defining two optic axes. In the uniaxial case (nl n2 no, n3 n e ), the two sheets become a sphere and an ellipsoid of revolution that meet at only two points, thereby defining a single optic axis (the z axis). In the isotropic case nl n2 n3 n, the two sheets degenerate into a single sphere. k]/k o Q n2 l. n1 :Ie """ rl-t · ... ./8 " 1 "" k]/k o k]/k o UrJ:J . .-01 _ ...... .......... o no . ......... . ::-- n . .... " " . - " L nl ." .. n?t .. . .' . :: no ne : . k 2 / ko . y y .. k 2 / ko n k 2 / ko . , -- -- - n -. n] k/ko .. .- no :;'_._"" n. - . -.'-.' . - k/ko ne k 1/ ko (a) Biaxia] (b) Uniaxial (c) Isotropic Figure 6.3-9 One octant of the k surface for (a) a biaxial crystal (nl < n2 < n3); (b) a uniaxial crystal (nl n2 no, n3 n e ); and (c) an isotropic crystal (nl n2 n3 n). The intersection of the direction u a wavenumber k that satisfies Ul, U2, U3 with the k surface corresponds to U k 2 J k 2 n k 2 J 0 1. (6.3-18) j 1,2,3 This is a fourth-order equation in k (or second order in k 2 ). It has four solutions, -:tk a and -i:k b , of which only the two positive values are meaningful, since the negative . values represent a reversed direction of propagation. The problem is therefore solved: the wavenumbers of the normal modes are ka and kb and the refractive indexes are na ka ko and nb kb ko. To determine the directions of polarization of the two normal modes, we determine the components k l , k 2 , k3 ku 1, kU2, kU3 and the elements of the matrix in (6.3- 17) for each of the two wavenumbers k ka and k kb. We then solve two of the three equations in (6.3-17) to determine the ratios El E3 and E 2 E 3 , from which we determine the direction of the corresponding electric field E. The nature of waves in anisotropic media is best explained by examining the k surface w w k 1 , k 2 , k3 obtained by equating the determinant of the matrix in (6.3- 17) to zero, as illustrated in Fig. 6.3-9. The variation of the phase velocity c w k with the direction u can be determined from the k surface: the distance from the origin to the k surface in the direction of u is inversely proportional to the phase velocity. The group velocity may also be determined from the k surface. In analogy with the group velocity v dw dk that governs the propagation of light pulses (wavepack- ets), as discussed in Sec. 5.6, the group velocity for rays (localized beams or spatial
224 CHAPTER 6 POLARIZATION OPTICS wavepackets) is the vector v \7 kW k , the gradient of w with respect to k. Since the k surface is the surface w k 1 , k 2 , k3 constant, v must be normal to the k surface. Thus, rays travel along directions normal to the k surface. The wavefronts are perpendicular to the wavevector k since the phase of the wave is k · r. The wavefront normals are therefore parallel to the wavevector k. The complex Poynting vector S ! E x H* is also normal to the k surface. This can be demonstrated by choosing a value or wand considering two vectors k and k + k that lie on the k surface. By taking the differential of (6.3-9) and (6.3-10), and using certain vector identities, it can be shown that k · S 0, so that S is normal to the k surface. Consequently, S is also parallel to the group velocity vector v. If the k surface is a sphere, as it is for isotropic media, the vectors v, S., and k are all parallel, indicating that rays are parallel to the wavevector k and energy flows in the same direction, as illustrated in Fig. 6.3-10(a). On the other hand, if the k surface is not normal to the wavevector k, as illustrated in Fig. 6.3-1 O(b), the rays and the direction of energy transport are not orthogonal to the wavefronts. Rays then have the "extraordinary" property of traveling at an oblique angle to their wavefronts [Fig. 6.3- 1 O(b )]. k , s Ray k surface s Ray Wavefronts Wavefronts k surface k o o (a) Ordinary (b) Extraordinary Figure 6.3-10 Rays and wavefronts for (a) a spherical k surface, and (b) a nonspherical k surface. Special Case: Uniaxial Crystals In uniaxial crystals (n} n2 no and n3 w w k 1 , k 2 , k3 simplifies to n e ), the equation of the k surface k 2 + k 2 k 2 1 2 + 3 n 2 n 2 e 0 k 2 n 2 k 2 o 0 k 2 o O. (6.3-19) This equation has two solutions: a sphere, corresponding to the leftmost factor being zero: k no ko, (6.3-20) and an ellipsoid of revolution, corresponding to the rightmost factor being zero: k 2 + k 2 k 2 1 2 2 + k . n n e 0 (6.3-21 ) Because of symmetry about the z axis (optic axis), there is no loss of generality in assuming that the vector k lies in the y z plane. Its direction is then characterized by
6.3 OPTICS OF ANISOTROPIC MEDIA 225 kJI ko nn n( 0) k/k o no ne k Ik 2 0 Figure 6.3-11 Intersection of the k surfaces with the y-z plane for a positive uniaxial crystal (n e > no). the angle () it makes with the optic axis. It is thus convenient to draw the k-surfaces only in the y z plane, as a circle and an ellipse, as shown in Fig. 6.3-11. Given the direction ii of the vector k, the wavenumber k is determined by finding the intersection with the k surfaces. The two solutions define the two normal modes, the ordinary and extraordinary waves. The ordinary wave has wavenumber k noko regardless of the direction of u, whereas the extraordinary wave has wavenumber n () ko, where n e is given by (6.3-15), thereby confirming earlier results obtained from the index-ellipsoid geometrical construction. The directions of the rays, wave- fronts, energy flow, and field vectors E and D for the ordinary and extraordinary waves in a uniaxial crystal are illustrated in Fig. 6.3-12. E,D k kJ/ ko D E kJ/ ko no no s o E, D k/k o E o H no k 2 /k o ne k 2 / ko (a) Ordinary (b) Extraordinary Figure 6.3-12 The normal modes for a plane wave traveling in a direction k that makes an angle o with the optic axis z of a uniaxial crystal are: (a) An ordinary wave of refractive index no polarized in a direction normal to the k-z plane. (b) An extraordinary wave of refractive index n(O) [given by (6.3-15)] polarized in the k-z plane along a direction tangential to the ellipse (the k surface) at the point of its intersection with k. This wave is "extraordinary" in the following ways: D is not parallel to E but both lie in the k-z plane and S is not parallel to k so that power does not flow along the direction of k; the rays are therefore not normal to the wavefronts so that the wave travels "'sideways." E. Double Refraction Refraction of Plane Waves We now examine the refraction of a plane wave at the boundary between an isotropic medium (say air, n 1) and an anisotropic medium (a crystal). The key principle
226 CHAPTER 6 POLARIZATION OPTICS that governs the refraction of waves for this configuration is that the wavefronts of the incident and refracted waves must be matched at the boundary. Because the anisotropic medium supports two modes with distinctly different phase velocities, and therefore different indexes of refraction, an incident wave gives rise to two refracted waves with different directions and different polarizations. The effect is known as double refraction or birefringence. The phase-matching condition requires that Snell's law be obeyed, i.e., ko sin ()1 k sin (), (6.3-22) where ()1 and () are the angles of incidence and refraction, respectively. In an anisotropic medium, however, the wavenumber k n () ko is itself a function of (), so that sin ()1 n ()a + () sin (), (6.3-23) where ()a is the angle between the optic axis and the normal to the surface, so that ()a +() is the angle the refracted ray makes with the optic axis. Equation (6.3-23) is a modified version of Snell's law. To solve (6.3-22), we draw the intersection of the k surface with the plane of incidence and search for an angle () for which (6.3-22) is satisfied. Two solutions, corresponding to the two normal modes, are expected. The polarization state of the incident light governs the distribution of energy among the two refracted waves. Take, for example, a uniaxial crystal and a plane of incidence parallel to the optic axis. The k surfaces intersect the plane of incidence in a circle and an ellipse (Fig. 6.3- 13). The two refracted waves that satisfy the phase-matching condition are determined by satisfying (6.3-23): . An ordinary wave of orthogonal polarization (TE) at an angle () ()o for which sin ()1 no sin () 0 ; (6.3-24) . An extraordinary wave of parallel polarization (TM) at an angle () ()e, for which sin ()1 n ()a + ()e sin ()e , (6.3-25) where n () is given by (6.3-15). k surface ( crystal) , , I , I k surface ( air) Extraordinary wave Q . /c v..-r. ., /$ Q . :l'c v ., /$' Ordinary wave I kl I I I , Be , , I , I \ .. '" Crystal Air (h ' I I I ..... J I k · () k . I I 0 sin I 0 sin () I I Figure 6.3-13 Determination of the angles of refraction by matching projections of the k vectors in air and in a uniaxial crystal. If the incident wave carries the two polarizations, the two refracted waves will emerge, as shown in Fig. 6.3-13.
6.3 OPTICS OF ANISOTROPIC MEDIA 227 Refraction of Rays The analysis immediately above dealt with the refraction of plane waves. The refraction of rays is different in an anisotropic medium, since rays do not necessarily travel in directions normal to the wavefronts. In air, before entering the crystal, the wavefronts are normal to the rays. The refracted wave must have a wavevector that satisfies the phase-matching condition, so that Snell's law (6.3-23) is applicable, with the angle of refraction () determining the direction of k. However, since the direction of k is not the direction of the ray, Snell's law is not applicable to rays in anisotropic media. --9 Pf l c , is " I S I Os I k Extraordina ry () s ray · Ordinary ray k k surface "" ... Crystal " Air Figure 6.3-14 Double refraction at normal incidence. An example that dramatizes the deviation from Snell's law is that of normal in- cidence into a uniaxial crystal whose optic axis is neither parallel nor perpendicular to the crystal boundary. The incident wave has a k vector normal to the boundary. To ensure phase matching, the refracted waves must also have wavevectors in the same direction. Intersections with the k surface yield two points corresponding to two waves. The ordinary ray is parallel to k. But the extraordinary ray points in the direction of the normal to the k surface, at an angle () s with the normal to the crystal boundary, as illustrated in Fig. 6.3-14. Thus, normal incidence creates oblique refraction. The principle of phase matching is maintained, however: wavefronts of both refracted rays are parallel to the crystal boundary and to the wavefront of the incident ray. When light rays are transmitted through a plate of anisotropic material as described above, the two rays refracted at the first surface refract again at the second surface, creating two laterally separated rays with orthogonal polarizations, as illustrated in Fig. 6.3-15. Extraordinary ray - - - -:"'c ?J.\s - - - 09\\ Ordinary ray Figure 6.3-15 Double refraction through an anisotropic plate. The plate serves as a polarizing beamspli tter.
228 CHAPTER 6 POLARIZATION OPTICS 6.4 OPTICAL ACTIVITY AND MAGNETO-OPTICS A. Optical Activity Certain materials act as natural polarization rotators, a property known as optical ac- tivity. Their normal modes are waves that are circularly, rather than linearly polarized; waves with right- and left-circular polarizations travel at different phase velocities. We demonstrate below that an optically active medium with right- and left-circular- polarization phase velocities Co n+ and Co n- acts as a polarization rotator with an angle of rotation 1f n- n+ d Ao that is proportional to the thickness of the medium d. The rotatory power (rotation angle per unit length) of the optically active medium is therefore p 1f ( , n+) . (6.4-1) Rotatory Power The direction in which the polarization plane rotates is the same as that of the circularly polarized component with the greater phase velocity (smaller refractive index). If n+ < n_, p is positive and the rotation is in the same direction as the electric field vector of the right circularly polarized wave [clockwise when viewed from the direction toward which the wave is approaching, as illustrated in Fig. 6.4-1(a)]. Such materials are said to be dextrorotatory, whereas those for which n+ > n- are termed levorotatory. D Derivation of the Rotatory Power. Equation (6.4-1) may be derived by decomposing the incident linearly polarized wave into a sum of right and left circularly polarized components of equal amplitudes (see Exercise 6.1 B), cas () sin () J 1 . , J (6.4-2) where () is the initial angle of the plane of polarization. After propagating a distance d through the medium, the phase shifts encountered by the right and left circularly polarized waves are <P+ 21rn+ d / AD and <P- 21rn_ d / AD, respectively, resulting in a Jones vector 1 -j(J -jlp+ 1 + lej(J e- jlp - 2 e e j 2 . -J'Po e cos(() sin (() <P /2) , <P /2) (6.4-3) 1 . J where <Po !(<p+ + <p-) and <p <p- <p+ 21r(n_ n+)d /Ao. This Jones vector represents a linearly polarized wave with the plane of polarization rotated by an angle <p /2 7r( n- n+) d / AD, as provided in (6.4-1). . Optical activity occurs in materials with an intrinsically helical structure. Examples include selenium, tellurium, tellurium oxide (Te02), quartz (a-Si0 2 ), and cinnabar (HgS). Optically active liquids consist of so-called chiral molecules, which come in distinct left- and right-handed mirror-image forms. Many organic compounds, such as amino acids and sugars, exhibit optical activity. Almost all amino acids are levorota- tory, whereas common sugars come in both forms: dextrose (d-glucose) and levulose (fructose) are dextrorotatory and levorotatory, respectively, as their names imply. The rotatory power and sense of rotation for solutions of such substances are therefore sensitive to both the concentration and structure of the solute. A saccharimeter is used to determine the optical activity of sugar solutions, from which the sugar concentration is calculated.
6.4 OPTICAL ACTIVITY AND MAGNETO-OPTICS 229 k k .....c " , ',R L ......., '\ ',R L - -1- --- - -1- --- n#- n:t- I I I I I I I / / ,/ / , (a) Forward wave (b) Backward wave Figure 6.4-1 (a) The rotation of the plane of polarization by an optically active medium results from the difference in the velocities for the two circular polarizations. In this illustration, the right circularly polarized wave (R) is faster than the left circularly polarized wave (L), i.e., n+ < n_, so that p is positive and the material is dextrorotatory. (b) If the wave in (a) is reflected after traversing the medium, the plane of polarization rotates in the opposite direction so that the wave retraces itself. Material Equations A time-varying magnetic flux density B applied to an optically active structure induces a circulating current, by virtue of its helical character, that sets up an electric dipole moment (and hence a polarization) proportional to jwB \7 x E. The optically active medium is therefore spatially dispersive; i.e., the relation between D rand E r is not local. Drat position r is determined not only by E r , but also by E r' at points r' in the immediate vicinity of r, since it is dependent on the spatial derivatives contained in \7 x E r . For a plane wave, we have ErE exp jk · r and \7 x E jk x E, so that the dielectric permittivity tensor is dependent on the wavevector k. Spatial dispersiveness is analogous to temporal dispersiveness, which has its origin in the noninstantaneous response of the medium (see Sec. 5.2). While the permittivity of a medium exhibiting temporal dispersion depends on the frequency w, that of a medium exhibiting spatial dispersion depends on the wavevector k. An optically active medium is described by the k-dependent material equation D EE + j Eo k x E, (6.4-4) where is a quantity (called a pseudoscalar) that changes sign depending on the handedness of the coordinate system. This relation is a first-order approximation of the k dependence of the permittivity tensor, under appropriate symmetry conditions. t The first term represents the response of an isotropic dielectric medium whereas the second term accounts for the optical activity, as will be shown subsequently. This D- E relation is often written in the form D EE + j Eo G x E, (6.4-5) where G k is known as the gyration vector. In such media the vector D is clearly not parallel to E since the vector G x E in (6.4-5) is perpendicular to E. Normal Modes of the Optically Active Medium We proceed to show that the two normal modes of the medium described by (6.4-5) are circularly polarized waves, and we determine the velocities Co n+ and Co n- in terms of the constant G k. t See, for example, L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon, 2nd revised ed. 1984, Chapter 12.
230 CHAPTER 6 POLARIZATION OPTICS We assume that the wave propagates in the z direction, so that k 0,0, k and thus G 0,0, G . Equation (6.4-5) may then be written in matrix form as D] D 2 D3 n 2 Eo jG o jG 0 'f72 0 o n 2 EI E 2 E3 ( 6.4-6) where n 2 E Eo. The diagonal elements in (6.4-6) correspond to propagation in an isotropic medium with refractive index n, whereas the off-diagonal elements, propor- tional to G, represent the optical activity. To prove that the normal modes are circularly polarized, consider the two circularly polarized waves with electric-field vectors E Eo, jEo, 0 . The + and signs correspond to right and left circularly polarized waves, respectively. Substitution in (6.4-6) yields D Do, :f:jDo, 0 , where Do Eo n 2 :f: G Eo. It follows that D EonE, where n 71 2 :f: G . (6.4- 7) . Hence, for either of the two circularly polarized waves the vector D is parallel to the vector E. Equation (6.3-11) is satisfied if the wavenumber k nko. Thus., the right and left circularly polarized waves propagate without changing their state of polarization, with refractive indexes n+ and n_, respectively. They are therefore the normal modes for this medium. EXERCISE 6.4-1 Rotatory Power of an Optically Active Medium. Show that if G « n. the rotatory power of an opticaIIy active medium (rotation of the polarization plane per unit length) is approximately given by p';::j 7rG Aon · (6.4-8) The rotatory power is strongly dependent on the wavelength. Since G is proportional to k, as indicated by (6.4-5)., it is inversely proportional to the wavelength Ao. Thu, the rotatory power in (6.4-8) is inversely proportional to A. Moreover, the refractive index n is itself wavelength dependent. By way of example, the rotatory power p of quartz is 31 deg/mm at Ao 500 nm and 22 deg/mm at Ao GOO nm: for silver thiogallate AgGaS 2 ' p is 700 deg/mm at 490 nm and 5UO deg/mm at 500 nm. B. Magneto-Optics: The Faraday Effect Many materials act as polarization rotators in the presence of a static magnetic field, a property known as the Faraday effect. The angle of rotation is then proportional to the thickness of the material, and the rotatory power p (rotation angle per unit length) is proportional to the component of the magnetic flux density B in the direction of the . wave propagation, p 9JB, (6.4-9)
6.4 OPTICAL ACTIVITY AND MAGNETO-OPTICS 231 where Q1 is called the Verdet constant. The sense of rotation is governed by the direction of the magnetic field: for Q1 > 0, the rotation is in the direction of a right-handed screw pointing in the direction of the magnetic field [Fig. 6.4-2(a)]. In contrast to optical activity, however, the sense of rotation does not reverse with the reversal of the direction of propagation of the wave. Thus, when a wave travels through a Faraday rotator and then reflects back onto itself, traveling once more through the rotator in the opposite direction, it undergoes twice the rotation [Fig. 6.4-2(b)]. Materials that exhibit the Faraday effect include glasses, yt- B B ..... , , , , \ \ \ , , , I / , I / ./ I I I I I I I J I I I / --- --. -1- .., ... -- (a) Forward wave (b) Backward wave Figure 6.4-2 (a) Polarization rotation in a medium exhibiting the Faraday effect. (b) The sense of rotation is invariant to the direction of travel of the wave. trium iron garnet (YIG), terbium gallium garnet (TGG), and terbium aluminum garnet (TbAIG). The Verdet constant of TbAIG is Q1 1.16 min Oe-cm at Ao 500 nm. Thin films of these ferrimagnetic materials are used to make compact devices. Material Equations In magneto-optic materials, the electric permittivity tensor € is altered by the appli- cation of a static magnetic field H, so that € € H . This effect originates from the interaction of the static magnetic field with the motion of the electrons in the material in response to an optical electric field E. For the Faraday effect, in particular, the material equation is D EE + jEoG x E (6.4-10) with G 'YB. (6.4-11) Here, B JlH is the static magnetic flux density, and 'Y is a constant of the medium known as the magnetogyration coefficient. Equation (6.4-1 0) is identical to (6.4-5) so that the vector G 'Y B in Faraday rotators plays the role of the gyration vector G k in optically active media. For the Faraday effect, however, G does not depend on k, so that reversing the direction of propagation does not reverse the sense of rotation of the plane of polarization. This property is useful for constructing optical isolators, as explained in Sec. 6.6C. With this analogy, and using (6.4-8), we conclude that the rotatory power of the Faraday medium is p 'irG Aon 'ir'Y B Aon, from which the Verdet constant (rotatory power per unit magnetic flux density) is seen to be Q1 'ir'Y Aon · (6.4-12) The Verdet constant is clearly a function of the wavelength Ao.
232 CHAPTER 6 POLARIZATION OPTICS 6.5 OPTICS OF LIQUID CRYSTALS Liquid Crystals A liquid crystal comprises a collection of elongated organic molecules that are typ- ically cigar-shaped. The molecules lack positional order (like liquids) but possess orientational order (like crystals). There are three types (phases) of liquid crystals, as illustrated in Fig. 6.5-1: h / / ( I I I I I . I I '/ /I I I I I I / / v/ (a) Nematic (b) Smectic (c) Cholesteric Figure 6.5-1 Molecular organizations of different types of liquid crystals. . In nematic liquid crystals the orientations of the molecules tend to be the same but their positions are totally random. . In smectic liquid crystals the orientations of the molecules are the same, but their centers are stacked in parallel layers within which they have random positions; they therefore have positional order only in one dimension. . The cholesteric liquid crystal is a distorted form of its nematic cousin in which the orientations undergo helical rotation about an axis. Liquid crystallinity is a fluid state of matter. The molecules are able to change orien- tation when subjected to a force. When a thin layer of liquid crystal is placed between two parallel glass plates that are rubbed together, for example, the molecules orient themselves along the direction of rubbing. isted nematic liquid crystals are nematic liquid crystals on which a twist (sim- ilar to the twist that exists naturally in the cholesteric phase) is externally imposed. This can be achieved, for example, by placing a thin layer of nematic liquid crystal between two glass plates that are polished in perpendicular directions, as schematized in Fig. 6.5-2. This section is devoted to a discussion of the optical properties of twisted nematic liquid crystals, which are widely used in photonics, e.g., for liquid-crystal displays. The electro-optic properties of twisted nematic liquid crystals, and their use as optical modulators and switches, are described in Chapter 20. -- s:: o . ...... u (1) . "'C OJ) c: . ...... ..s::: VJ . ...... . . o -- -- -- -- .O'{\. se c \\! · 0 u.\ \ c;\\\1):t::) \>0\\" Figure 6.5-2 Molecular orientations of the twisted nematic liquid crystal.
6.5 OPTICS OF LIQUID CRYSTALS 233 Optical Properties of Twisted Nematic Liquid Crystals The twisted nematic liquid crystal is an optically inhomogeneous and anisotropic medium that acts locally as a uniaxial crystal, with the optic axis parallel to the elongated direction. The optical properties are conveniently analyzed by considering the material to be divided into thin layers perpendicular to the axis of twist, each of which acts as a uniaxial crystal; the optic axis is taken to rotate gradually, in a helical fashion, along the axis of twist (Fig. 6.5-3). The cumulative effects of these layers on the transmitted wave is then calculated. We show that, under certain conditions, the twisted nematic liquid crystal acts as a polarization rotator in which the plane of polarization rotates in alignment with the molecular twist. x e P!ic / aXIS I / , ----- ------ ---------- -..------------ Z I o ------ ----- --------------------- y Figure 6.5-3 Propagation of light in a twisted nematic liquid crystal. In this diagram the angle of twi st is 90° . Consider the propagation of light along the axis of twist (the z axis) of a twisted nematic liquid crystal and assume that the twist angle () varies linearly with z, e az, (6.5-1 ) where a is the twist coefficient (degrees per unit length). The optic axis is therefore parallel to the x y plane and makes an angle e with the x direction. The ordinary and extraordinary refractive indexes are no and ne, respectively (typically, ne > no), and the phase-retardation coefficient (retardation per unit length) is {3 ne no ko. (6.5-2) The liquid crystal cell is completely characterized by the twist coefficient a and the retardation coefficient {3. In practice, {3 a so that many cycles of phase retardation are introduced before the optic axis rotates appreciably. We show below that if this condition is satisfied, and the incident wave at z 0 is linearly polarized in the x direction, then the wave maintains its linearly polarized state but the plane of polarization rotates in alignment with the molecular twist, so that the angle of rotation is () az and the total rotation in a crystal of length d is the angle of twist ad. The liquid crystal cell then serves as a polarization rotator with rotatory power a. The polarization-rotation property of the twisted nematic liquid crystal is useful for making display devices, as explained in Sec. 20.3.
234 CHAPTER 6 POLARIZATION OPTICS D Proof that the 1\visted Nematic Liquid Crystal Acts as a Polarization Rotator. We proceed to show that the twisted nematic liquid crystal acts as a polarization rotator if (3 » a. We divide the overall width of the cell d into N incremental layers of equal widths z d / N. The mth layer.. located at the distance Z Zm mz, m 1, 2, . . . , N, is a wave retarder whose slow axis (the optic axis) makes an angle Om mO with the x axis, where O Qz. It therefore has a Jones matrix [see (6.1-24)] Tm R( Om) Tr R(Om)' (6.5-3) where .. Tr exp( jnekoz) o o exp( jnokoz) (6.5-4) is the Jones matrix of a wave retarder whose axis is along the x direction and R( 0) is the coordinate rotation matrix in (6.1-22). It is convenient to rewrite T r in terms of the phase-retardation coefficient (3 (ne no)ko, o exp (j(3z/2) , (6.5-5) Tr exp Jc.p z 0 where c.p (no + ne)ko/2. Since multiplying the Jones vector by a constant phase factor does not affect the state of polarization, we simply ignore the prefactor exp( jc.pz) in (6.5-5). The overall Jones matrix of the device is the product 1 1 T Tm R( Om) Tr R(Om). ( 6.5-6) m=N m=N Using (6.5-3) and noting that R(Om) R( Om-I) R(Om Om-I) R(O), we obtain T R( ()N) [TrR(())]N-l Tr R(Ol). (6.5-7) Substituting from (6.5-5) and (6.1-22), we obtain Tr R(O) exp ( j(3z/2) o o exp (j(3z/2) cas Q;Z sin QZ sin QZ CDS QZ · ( 6.5-8) Using (6.5-7) and (6.5-8), the Jones matrix T of the device can, in principle, be determined in terms of the parameters a, (3, and d N z. For a « (3, we may assume that the incremental rotation matrix R(O) is approximately the identity matrix, whereupon T R( ON) [Tr]N R(OI) o exp (j(3z/2) N j (3N z /2) o o exp (j (3N z /2) , ( 6.5-9) R( so that o exp (j (3d /2) · (6.5-10) T This Jones matrix represents a wave retarder of retardation (3d with the slow axis along the x direction, followed by a polarization rotator with rotation angle ad. If the original wave is linearly polarized along the x direction, the wave retarder imparts only a phase shift; the device then simply rotates the polarization by an angle ad equal to the twist angle. A wave linearly polarized along the y direction is .rotated by the same angle. .
6.6 POLARIZATION DEVICES 235 6.6 POLARIZATION DEVICES This section offers a brief description of a number of devices that are used to modify the state of polarization of light. The basic principles underlying the operation of these devices have been set forth earlier in this chapter. A. Polarizers A linear polarizer is a device that transmits the component of the electric field that lies along the direction of its transmission axis while blocking the orthogonal component. The blocking action may be achieved by selective absorption, selective reflection from isotropic media, or selective reflection/refraction in anisotropic media. Polarization by Selective Absorption (Dichroism) The absorption of light by certain anisotropic media, called dichroic materials, de- pends on the direction of the incident electric field (Fig. 6.6-1). These materials gener- ally have anisotropic molecular structures whose response is sensitive to the direction of the electric field. The most common dichroic materia] is Polaroid H-sheet, invented in 1938 and still in common use. It is fabricated from a sheet of iodine-impregnated polyvinyl alcohol that is heated and stretched in a particular direction. The analogous device in the infrared is the wire-grid polarizer, which comprises a planar configura- tion of closely spaced fine wires stretched in a single direction. The component of the incident electric field in the direction of the wires is absorbed whereas the component perpendicular to the wires passes through. 1.0 Maximum 0.8 Q) u s:: 0.6 ro ..... ..... f- oE (f'J 0.4 s:: ro 0.2 Polarizer 0 0 0 400 600 800 1000 1200 Wavelength (run) Figure 6.6-1 Power transmittances of a typical dichroic polarizer with the plane of polarization of the light aligned for maximum and minimum transmittance, as indicated. Polarization by Selective Reflection The reflectance of light at the boundary between two isotropic dielectric materials is dependent on its polarization, as discussed in Sec. 6.2. At the Brewster angle of inci- dence, in particular, the reflectance of TM-polarized light vanishes so that it is totally refracted (Fig. 6.2-4). At this angle, therefore, only TE-polarized light is reflected, so that the reflector serves as a polarizer. Polarization by Selective Refraction (Polarizing Beamsplitters) When light enters an anisotropic crystal, the ordinary and extraordinary waves refract at different angles and gradually separate from each other (see Sec. 6.3E and Fig. 6.3- 15). This provides an effective means for obtaining polarized light from unpolarized
236 CHAPTER 6 POLARIZATION OPTICS TE TE TM Figure 6.6-2 Brewster-angle polarizer. light, and it is commonly used. These devices usually consist of two cemented prisms comprising anisotropic (uniaxial) materials, often with different orientations, as il- lustrated by the examples in Fig. 6.6-3. These prisms therefore serve as polarizing beamsplitters. Opt}c @ aXIs Optic axis @ Optic t::\ axis \!I 0" Optic . . aXIs ........... ......... -. ..... ... ...... ...... .... . 0 0> .. -;..- "'. .. . -. .. . .,.... ......... ......... ........ .::::::-.0 .. > .... .JI'''. .... ......... ......... .. ........ .. ........ ... ...... .. .. e o ... '" u: u 0 .. .. -:- . . o . . e o .... . ...... ... ...... .. ...... . .... .. ..... .... .. ..: .. o 0 - - - -- Opt}c @ aXIs . .. o ..o ono_rno ........ It''''''': I no.._.... . Optic . aXIs (a) Wollaston prism (b) Rochon prism (c) Glan-Thompson prism Figure 6.6-3 Polarizing beam splitters. The directions and polarizations of the waves that exit differ for the three prisms. In this illustration, the crystals are negative uniaxial (e.g., calcite). The Glan- Thompson device has the merit of providing a large angular separation between the emerging waves. B. Wave Retarders A wave retarder serves to convert a wave with one form of polarization into another form. It is characterized by its retardation r and its fast and slow axes (see Sec. 6.1B). The normal modes are linearly polarized waves polarized along the directions of the axes. The velocities of the two waves differ so that transmission through the retarder imparts a relative phase shift r to these modes. Wave retarders are often constructed from anisotropic crystals in the form of plates. As explained in Sec. 6.3B, when light travels along a principal axis of a crystal (say the z axis), the normal modes are linearly polarized waves pointing along the two other principal axes (the x and y axes). These modes experience the principal refractive indexes nl and n2, and thus travel at velocities Co nl and Co n2, respectively. If nl < n2, the x axis is the fast axis. If the plate has thickness d, the phase retardation is r n2 nl kod 27f n2 nl d Ao. The retardation is thus directly proportional to the thickness d of the plate and inversely proportional to the wavelength Ao (note, however, that n2 nl is itself wavelength dependent). The refractive indexes of a thin sheet of mica, for example, are 1.599 and 1.594 at Ao 633 nm, so that r d 15.87f rad/mm. A sheet of thickness 63.3 J.Lm yields r 7f and thus serves as a half-wave retarder.
238 CHAPTER 6 POLARIZATION OPTICS D. Nonreciprocal Polarization Devices A device whose effect on the polarization state is invariant to reversal of the direction of propagation is said to be reciprocal. If a wave is transmitted through such a device in one direction and the emerging wave is retransmitted in the opposite direction, then it retraces the changes in the polarization state and arrives at the input in the very same initial polarization state. Devices that do not have this directional invariance are called nonreciprocal. All of the polarization systems described in this chapter are reciprocal, with the exception of the Faraday rotator (see Sec. 6.4B). A number of useful nonreciprocal polarization devices are obtained by combining the Faraday rotator with other reciprocal polarization components. Optical Isolator An optical isolator is a device that transmits light in only one direction, thereby acting as a "one-way valve." Optical isolators are useful for preventing reflected light from returning back to the source. Such feedback can have deleterious effects on the operation of certain devices, such as semiconductor lasers. An opticaJ isolator is constructed by placing a Faraday rotator between two po- larizers whose axes make a 45° angle with respect to each other. The magnetic flux density applied to the rotator is adjusted so that it rotates the polarization by 45° in the direction of a right-handed screw pointing in the z direction [Fig. 6.6-5(a)]. Light traveling through the system in the forward direction (from left to right) thus crosses polarizer A, rotates 45°, and is thence transmitted through polarizer B. Linearly polarized light with the polarization plane at 45° but traveling through the system in the backward direction [from right to left in Fig. 6.6-5(b)] successfully crosses polarizer B. However, on passing through the Faraday rotator, the plane of polarization rotates an additional 45° and is therefore blocked by polarizer A. Since the backward light might be generated by reflection of the forward wave from subsequent surfaces, the isolator serves to protect its source from reflected light. Note that the Faraday rotator is a necessary component of the optical isolator. An optically active, or liquid-crystal, polarization rotator cannot be used in its place. In those reciprocal components, the sense of rotation is such that the polarization of the reflected wave retraces that of the incident wave so that the light would be transmitted back through the polarizers to the source. 45° 45° i Transmitted 1 450 45° wave x jJ 45° Polarizer B 45° x Incident wave y Faraday Polarizer B rotator 5° Polarizer A y Faraday rotator (a) Polarizer A (b) 45° Figure 6.6-5 An optical isolator that makes use of a Faraday rotator transmits light in one direction. (a) A wave traveling in the forward direction is transmitted. (b) A wave traveling in the backward ( or reverse) direction is blocked. Faraday-rotator isolators constructed from yttrium iron garnet (YIG) or terbium gallium garnet (TGG) offer attenuations of the backward wave of up to 90 dB, over a relatively wide wavelength range. Thin films of these materials placed in permanent magnetic fields are used to make very compact optical isolators.
READING LIST 239 Nonreciprocal Polarization Rotation A combination of a 45° Faraday rotator followed by a half-wave retarder is another useful nonreciprocal device. As illustrated in Fig. 6.6-6(a), the state of polarization of a forward linearly polarized wave, with the plane of polarization oriented at 22.5° with the fast axis of the retarder, maintains its state of polarization upon transmission through the device (since it undergoes 45° rotation by the Faraday rotator, followed by 45° rotation by the retarder). However, for a wave traveling in the reverse direction, the plane of polarization is rotated by 45° + 45° 90°, as can be readily seen in Fig. 6.6-6(b). The device may therefore be used in combination with a polarizing beamsplitter to direct the backward wave away from the source of the forward wave and to access it independently. The system can be useful in implementing nonreciprocal interconnects, such as optical circulators, as described in Sec. 23.1. 22.5° .. .. .. .... .. .- .- 22.5° 45° - .- .- . .- . B 45° Retarder 7r Faraday rotator 45° Retarder 'iT (a) (b) Faraday rotator Figure 6.6-6 A nonreciprocal device that maintains the polarization state of a linearly polarized forward wave (a), but rotates the plane of polarization of the backward wave (b) by 90°. READING LIST General See also the general reading lists in Chapters 1 and 5. J. N. Damask, Polarization Optics in Telecommunications, Springer-Verlag, 2004. D. H. Goldstein, Polarized Light, Marcel Dekker, 2nd ed. 2003. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, reprinted 2003. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1957, reprinted with corrections and new material, 2001. S. Sugano and N. Kojima, eds., Magneto-Optics, Springer-Verlag, 2000. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, Wiley, 1998. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, Pergamon, 1971, reprinted 1996. S. Huard, Polarization of Light, Wiley, 1996. E. Collett, Polarized Light: Fundamentals and Applications, Marcel Dekker, 1993. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy, Academic Press, 1990. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland, 1977, reprinted 1989. P. Gay, An Introduction to Crystal Optics, Longmans, 1967, paperback ed. 1982.
240 CHAPTER 6 POLARIZATION OPTICS B. A. Robson, The Theory of Polarization Phenomena, Clarendon, 1974. W. A. Shurcliff, Polarized Light: Production and Use, Harvard University Press, 1962, reprinted 1966. L. Velluz, M. Le Grand, and M. Grosjean, Optical Circular Dichroism: Principles, Measurements, and Applications., Academic Press, 1965. W. A. Shurcliff and S. S. Ballard, Polarized Light, Van Nostrand, 1964. Books on Liquid Crystals P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Prop- erties Illustrated by Experiments, CRC Pressrraylor & Francis, 2005. L. Vicari, Optical Applications of Liquid Crystals, Institute of Physics, 2003. P. J. Collings, Liquid Crystals: Nature's Delicate Phase of Matter, Princeton University Press, 2nd ed. 2002. P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Wiley, 1999. V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications, Artech House, 1999. P. G. de Gennes, The Physics of Liquid Crystals., Clarendon Press, 1974; Oxford University Press, 2nd ed. 1995. S. Chandrasekhar, Liquid Crystals, Cambridge University Press, 2nd ed. 1992. J. L. Ericksen and D. Kinderlehrer, eds., Theory and Applications of Liquid Crystals, Springer-Verlag, 1987 . L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals, Wiley, 1983. W. H. de Jeu, Physical Properties of Liquid Crystalline Materials, Gordon and Breach, 1980. G. Meier, E. Sackmann, and J. G. Grabmaier, Applications of Liquid Crystals, Springer-Verlag, 1975. Articles K. Ando, W. Challener, R. Gambino, and M. Levy, eds., Magneto-Optical Materials for Photon- ics and Recording, Materials Research Society Symposium Proceedings Volume 834, Materials Research Society, 2005. M. Mansuripur, The Faraday Effect, Optics & Photonics News, vol. 10, no. ] 1, pp. 32-36, 1999. B. H. Billings, ed., Selected Papers on Applications of Polarized Light, SPIE Optical Engineering Press (Milestone Series Volume 57), 1992. S. D. Jacobs, ed., Selected Papers on Liquid Crystals for Optics., SPIE Optical Engineering Press (Milestone Series Volume 46), 1992. B. H. Billings, ed., Selected Papers on Polarization, SPIE Optical Engineering Press (Milestone Series Volume 23), 1990. A. Lakhtakia, ed., Selected Papers on Natural Optical Activity, SPIE Optical Engineering Press (Milestone Series Volume 15)., 1990. V. L. Ginzburg, On Crystal Optics with Spatial Dispersion, in Physics Reports, vol. 194, pp. 245-251, 1990. J. M. Bennett and H. E. Bennett, Polarization, in Handbook of Optics, W. G. Driscoll, ed., McGraw- Hill, 1978. W. Swindell, ed., Benchmark Papers in Optics: Polarized Light, Dowden, Hutchinson & Ross, 1975. v. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, in Progress in Optics, vol. 9, E. Wolf, ed., North-Holland, 197]. PROBLEMS 6.1-5 Orthogonal Polarizations. Show that if two elliptically polarized states are orthogonal, the major axes of their ellipses are perpendicular and the senses of rotation are opposite. 6.1-6 Rotating a Polarization Rotator. Show that the Jones matrix of a polarization rotator is invariant to rotation of the coordinate system.
PROBLEMS 241 6.1-7 Half-Wave Retarder. Consider linearly polarized light passed through a half-wave retarder. If the polarization plane makes an angle () with the fast axis of the retarder, show that the transmitted light is linearly polarized at an angle (), i.e., it is rotated by an angle 2(). Why is the half-wave retarder not equivalent to a polarization rotator? 6.1-8 Wave Retarders in Tandem. Write the Jones matrices for: (a) A 7r /2 wave retarder with the fast axis along the x direction. (b) A 7r wave retarder with the fast axis at 45° to the x direction. (c) A 7r /2 wave retarder with the fast axis along the y direction. If these three retarders are placed in tandem, with (c) following (b) following (a). show that the resulting device introduces a 90° rotation. What happens if the order of the three retarders is reversed? 6.1-9 Reflection of Circularly Polarized Light. Show that circularly polarized light changes handedness (right becomes left, and vice versa) upon reflection from a mirror. 6.1-10 Anti-Glare Screen. A self-luminous object is viewed through a glass window. An anti-glare screen is used to eliminate glare caused by reflection of background light from the window surfaces. Show that such a screen may be made of a combination of a linear polarizer and a quarter-wave retarder whose axes are at 45° with respect to the transmission axis of the polarizer. Can the screen be regarded as an optical isolator? 6.2-3 Derivation of Fresnel Equations. Derive the reflection equation (6.2-6), which is used to derive the Fresnel equation (6.2-8) for TE polarization. How would you go about obtaining the reflection coefficient if the incident light took the form of a beam rather than a plane wave? 6.2-4 Reflectance of Glass. A plane wave is incident from air (n 1) onto a glass plate (n 1.5) at an angle of incidence of 45 0 . Determine the power reflectances of the TE and TM waves. What is the average reflectance for unpolarized light (light carrying TE and TM waves of equal intensities)? 6.2-5 Refraction at the Brewster Angle. Use the condition nl see ()l n2 see O 2 and Snell's law, nl sin 0 1 n2 in O 2 , to derive (6.2-12) for the Brewster angle. Also show that at the Brewster angle, 0 1 + O 2 90° 'I so that the directions of the reflected and refracted waves are orthogonal, and hence the electric field of the refracted TM wave is parallel to the direction of the reflected wave. The reflection of light may be regarded as a scattering process in which the refracted wave acts as a source of radiation generating the reflected wave. At the Brewster angle, this source oscillates in a direction parallel to the direction of propagation of the reflected wave, so that radiation cannot occur and no TM light is reflected. 6.2-6 Retardation Associated with Total Internal Reflection. Determine the phase retardation between the TE and TM waves that is introduced by total internal reflection at the boundary between glass (n 1.5) and air (n 1) at an angle of incidence 0 1.2 0e, where Oe is the critical angle. 6.2- 7 Goos-Hanchen Shift. Consider two TE plane waves undergoing total internal reflection at angles 0 and 0 + dO, where dO is an incremental angle. If the phase retardation introduced between the reflected waves is written in the form d<p dO, find an expression for the coefficient . Sketch the interference patterns of the two incident waves and the two reflected waves and verify that they are shifted by a lateral distance proportional to . When the incident wave is a beam (composed of many plane-wave components), the reflected beam is displaced laterally by a distance proportional to . This is known as the Goos-Ranchen effect. 6.2-8 Reflection from an Absorptive Medium. Use Maxwell's equations and appropriate bound- ary conditions to show that the complex amplitude reflectance at the boundary between free space and a medium with refractive index n and absorption coefficient Q, at normal incidence, is r [(n jac/2w) l]/[(n jac/2w) + 1]. 6.3-1 Maximum Retardation in Quartz. Quartz is a positive uniaxial crystal with ne 1.553 and no ] .544. (a) Determine the retardation per mm at Ao 633 nDl when the crystal is oriented such that retardation is maximized. (b) At what thickness( es) does the crystal act as a quarter-wave retarder? 6.3-2 Maximum Extraordinary Effect. Determine the direction of propagation in quartz (ne 1.553 and no 1.544) at which the angle between the wavevector k and the Poynting vector S (which is also the direction of ray propagation) is maximum.
242 CHAPTER 6 POLARIZATION OPTICS 6.3-3 Double Refraction. An unpolarized plane wave is incident from free space onto a quartz crystal (n e 1.553 and no 1.544) at an angle of incidence 30°. The optic axis lies in the plane of incidence and is perpendicular to the direction of the incident wave before it enters the crystal. Determine the directions of the wavevectors and the rays of the two refracted components. 6.3-4 Lateral Shift in Double Refraction. What is the optimum geometry for maximizing the lateral shift between the refracted ordinary and extraordinary beams in a positive uniaxial crystal? Indicate all pertinent angles and directions. 6.3-5 Transmission Through a LiNb0 3 Plate. Examine the transmission of an unpolarized He- Ne laser beam (Ao 633 nm) normally incident on a LiNb0 3 plate (ne 2.29, no 2.20) of thickness 1 cm, cut such that its optic axis makes an angle 45° with the normal to the plate. Determine the lateral shift at the output of the plate and the retardation between the ordinary and extraordinary beams. *6.3-6 Conical Refraction. When the wavevector k points along an optic axis of a biaxial crystal an unusual situation occurs. The two sheets of the k surface meet and the surface can be approximated by a conical surface. Consider a ray normally incident on the surface of a biaxial crystal for which one of its optic axes is also normal to the surface. Show that multiple refraction occurs with the refracted rays forming a cone. This effect is known as conical refraction. What happens when the conical rays refract from the parallel surface of the crystal into air? 6.6-1 Circular Dichroism. Certain materials have different absorption coefficients for right and left circularly polarized light, a property known as circular dichroism. Determine the Jones matrix for a device that converts light with any state of polarization into right circularly polarized light. 6.6-2 Polarization Rotation by a Sequence of Linear Polarizers. A wave that is linearly po- larized in the x direction is transmitted through a sequence of N linear polarizers whose transmission axes are inclined by angles mB (m 1,2, . . . , N; B 7r /2N) with respect to the x axis. Show that the transmitted light is linearly polarized in the y direction but its amplitude is reduced by the factor cos N B. What happens in the limit N > oo? Hint: Use Jones matrices and note that R[(m + l)B] R( mB) R(B), where R( B) is the coordinate transformation matrix.
CHAPTER .. 7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 246 A. Matrix Theory of Multilayer Optics B. Fabry Perot Etalon C. Bragg Grating 7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 265 A. Bloch Modes B. Matrix Optics of Periodic Media C. Fourier Optics of Periodic Media D. Boundaries Between Periodic and Homogeneous Media 7.3 TWO- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS 279 A. Two-Dimensional Photonic Crystals B. Three-Dimensional Photonic Crystals ( .... .. , , . .. \.. , . ' ... 11 . Felix Bloch (J905-1983) de- veloped a theory that describes electron waves in the periodic structure of solids. Eli Yablonovitch (born 1946) coinvented the concept of the photonic bandgap; he made the first photonic bandgap crystal. Sajeev John (born 1957) in- voked the notion of photon lo- calization and was a coinventor of the photonic bandgap idea. 243
The propagation of light in homogeneous media and its reflection and refraction at the boundaries between different media are a principal concern of optics, as described in the earlier chapters of this book. Photonic devices often comprise multiple lay- ers of different materials arranged, for example, to suppress or enhance reflectance or to alter the spectral or the polarization characteristics of light. Multilayered and stratified media are also found in natural physical and biological systems and are responsible for the distinct colors of some insects and butterfly wings. Multilayered media can also be periodic, Le., comprise identical dielectric structures replicated in a one-, two-, or three-dimensional periodic arrangement, as illustrated in Fig. 7 .0-1. One-dimensional periodic structures include stacks of identical parallel planar multi- layer segments. These are often used as gratings that reflect optical waves incident at certain angles, or as filters that selectively reflect waves of certain frequencies. Two- dimensional periodic structures include sets of parallel rods as well as sets of parallel cylindrical holes, such as those used to modify the characteristics of optical fibers known as holey fibers (see Chapter 9). Three-dimensional periodic structures comprise arrays of cubes, spheres, or holes of various shapes, organized in lattice structures much like those found in natural crystals. .... ..... .... .. -.... ..... '. ID 2D 3D Figure 7.0-1 Periodic photonic structures in one-dimensional (1 D), two-dimensional (2D), and three-dimensional (3D) configurations. Optical waves, which are inherently periodic, interact with periodic media in a unique way, particularly when the scale of the periodicity is of the same order as that of the wavelength. For example, spectral bands emerge in which light waves cannot propagate through the medium without severe attenuation. Waves with frequencies lying within these forbidden bands, called photonic bandgaps, behave in a manner akin to total internal reflection, but applicable for all directions. The dissolution of the transmitted wave is a result of destructive interference among the waves scattered by elements of the periodic structure in the forward direction. Remarkably, this effect extends over finite spectral bands, rather than for just single frequencies. This phenomenon is analogous to the electronic properties of crystalline solids such as semiconductors. The periodic wave associated with an electron travels in a periodic crystal lattice, and energy bandgaps are commonly found. Because of this analogy, the photonic periodic structures have come to be called photonic crystals. Photonic crystals have found many applications, including use as special filters, waveguides, and resonators, and many more applications are in the offing. An electromagnetic-optics analysis is usually required to describe the optical prop- erties of inhomogeneous media such as multilayered and periodic media. For inhomo- geneous dielectric media, as we know from Sec. 5.2B, the permittivity E r is spatially varying and the wave equation takes the general forms of (5.2-16) and (5.2-17). For a harmonic wave of angular frequency w, this leads to generalized Helmholtz equations 244
CHAPTER 7 PHOTONIC-CRYSTAL OPTICS 245 for the electric and the magnetic fields expressed as 11r \7x \7xE w 2 2 E, Co W 2 2 H Co (7.0-1) \7x 11r \7xH (7.0-2) Generalized Helmholtz Equations where 11 r Eo E r is the electric impermeability (see Sec. 6.3A). One of these equations ITIay be solved for either the electric or the magnetic field.. and the other field may be directly determined by use of Maxwell's equations. Note that (7.0-1) and (7.0-2) are cast in the form of an eigenvalue problem: a differential operator applied on the field function equals a constant multiplied by the field function. The eigenvalues are w 2 c and the eigenfunctions provide the spatial distributions of the lTIodes of the propagating field (see Appendix C). For reasons to be explained in Sec. sec7n-2c and Sec. sec7n-3, we work with the magnetic field equation (7.0-2) instead of the electric field equation (7.0- J ). For nlultilayered media, E r is piecewise constant, i.e., it is uniform within any given layer but changes from one layer to another. Wave propagation can then be studied by using the known properties of optical waves in homogeneous media.. to- gether with the appropriate boundary conditions that dictate the laws of reflection and . . transmISSIon. Periodic dielectric media are characterized by periodic values of E rand 11 r . This periodicity ilnposes certain conditions on the optical wave. For example" the propagation constant deviates from simply proportionality to the angular frequency w, as is the case for a homogeneous medium. While the modes of propagation in a homogeneous medium are plane waves of the form exp jk. r , the modes of the periodic Inediunl, known as Bloch modes, are traveling waves modulated by standing waves. This Chapter Previous chapters have focused on the optics of thin optical components that are well separated, such as thin lenses, planar gratings, and image-bearing films across which the light travels. This chapter addresses the optics of bulk media comprising multiple dielectric layers and periodic 1 D, 2D, and 3D photonic structures. Section 7.1, in which ID layered media are considered, serves as a prelude to periodic media and photonic crystals. A matrix approach offers a systematic treatment of the multiple reflections that occur at the multiple boundaries of the medium. Section 7.2 introduces photonic crystals in their silnplest fornl 1 D periodic structures. Matrix methods are adopted to determine the dispersion relation and the band structure. An alternative approach, based on a Fourier-series representation of the periodic functions associated with the medium and the wave, is also presented. These results are generalized in Sec. 7.3 to two- and three-dimensional photonic crystals. Throughout this chapter, the various media are assumed to be isotropic, and there- fore described by a scalar permittivity E, although reflection and refraction at bound- aries have inherent polarization-sensitive characteristics. Photonic Crystals in Other Chapters By virtue of their omnidirectional reflection property, photonic crystals can be used as "perfect" dielectric mirrors. A slab of homogeneous medium embedded in a pho- tonic crystal may be used to guide light by multiple reflections from the boundaries.
246 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS Applications to optical waveguides are described in Sec. 8.4. Similarly, light may be guided through an optical fiber with a homogeneous core embedded in a cladding of the same material, but with cylindrical holes parallel to the fiber axis. Such "holey" fibers, described in Sec. 9.4, have a number of salutatory features not present in ordinary optical fibers. A cavity burrowed in a photonic crystal may function as an optical resonator since it has perfectly reflecting walls at frequencies within the photonic bandgap. Photonic-crystal microresonators will be described briefly in Secs. 10.4D and 17.4C. 7.1 OPTICS OF DIELECTRIC LAYERED MEDIA A. Matrix Theory of Multilayer Optics A plane wave normally incident on a layered medium undergoes reflections and trans- missions at the layer boundaries, which in turn undergo their own reflections and transmissions in an unending process, as illustrated in Fig. 7.1-1(a). The complex am- plitudes of the transmitted and reflected waves may be determined by use of the Fresnel equations at each boundary (see Sec. 6.2); the overall transmittance and reflectance of the medium can, in principle, be calculated by superposition of these individual waves. This technique was used in Sec. 2.5B to determine the transmittance of the Fabry-Perot interferometer. (a) (-) (-) VI V z (b) « « « ) .. ) vt) (+) V z Figure 7.1-1 (a) Reflections of a single wave from the boundaries of a multilayered medium. (b) In each layer, the forward waves are lumped into a .. forward collected wave U( +), and the backward waves are lumped into a back- ward collected wave U( - ) . When the number of layers is large, tracking the infinite number of "micro" re- flections and transmissions can be tedious. An alternative "macro" approach is based on the recognition that within each layer there are two types of waves: forward waves traveling to the right, and backward waves traveling to the left. The sums of these waves add up to a single forward collected wave U( +) and a single backward collected wave U( -) at any point, as illustrated in Fig. 7 .1-1 (b). Determining the wave propagation in a layered medium is then equivalent to determining the amplitudes of this pair of waves everywhere. The complex amplitudes of the four waves on the two sides of a boundary may be related by imposing the appropriate boundary conditions, or by simply using the Fresnel equations of reflection and transmission. Wave- Transfer Matrix Tracking the complex amplitudes of the forward and backward waves through the boundaries of a multilayered medium is facilitated by use of matrix methods. Consider two arbitrary planes within a given optical system, denoted plane 1 and plane 2. The amplitudes of the forward and backward collected waves at plane 1, uf +) and uf - ) ,
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 247 respectively, are represented by a column matrix of dimension 2, and similarly for plane 2. These two column matrices are related by the matrix equation V+) vi+) M u:(+) A B U(+) 2 1 (7 . 1-1) u:( ) . C D (-) (-) VI V 2 2 The matrix M, whose elements are A, B, C, and D, is called the wave-transfer matrix (or transmission matrix). It depends on the optical properties of the layered medium between the two planes. A multilayered medium is conveniently divided into a concaten48ation of basic elements described by known wave-transfer matrices, say M 1 , M 2 ,..., MN. The am- plitudes of the forward and backward collected waves at the two ends of the overall medium are then related by a single matrix that is the matrix product, M) M2 MN +- M MN · · · M 2 M 1 , (7.1-2) where the elements 1, 2, . . . , N are numbered from left to right as shown in the figure. The wave-transfer matrix cascade formula provided in (7 .1- 2) is identical to the ray- transfer matrix cascade formula given in (1.4-1 0), and it proves equally useful: Scattering Matrix An alternative to the wave-transfer matrix that relates the four complex amplitudes , often used to describe transmission lines, microwave circuits, and scattering systems. In this case, the outgoing waves are expressed in terms of the incoming waves, vj+) vi+} s u:( + ) 2 (-) U(-) VI 1 t12 T21 T12 t21 u(+) 1 u:( - ) , 2 (7 .1- 3 ) (-) V2 where the elements of the S matrix are denoted t 12, T21, T12, and t21. Unlike the wave- transfer matrix, these elements have direct physical significance. The quantities t12 and T12 are the forward amplitude transmittance and reflectance (i.e., the transmittance and reflectance of a wave incident from the left), respectively, while t21 and T21 are the amplitude transmittance and reflectance in the backward direction (i.e., a wave coming from the right), respectively. The subscript 12, for example, signifies that the light is incide n t from medium I into medium 2. This can b e easily verified b y noting th a t if b · u: (+) u: (-) d U (-) TT(-) we 0 taln 2 T21 2 an 1 t21 U 2. A distinct advantage of the S-matrix formalism is that its elements are directly related to the physical parameters of the system. On the other hand, a disadvantage is that the S matrix of a cascade of elements is not the product of the S matrices of the constituent elements. A useful systematic procedure for analyzing a cascaded system
248 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS therefore draws on both the wave-transfer and scattering matrix approaches: we use the handy multiplication formula of the M matrices and then convert to the S matrix to determine the overall transmittance and reflectance of the cascaded system. EXAMPLE 7.1-1. Propagation Through a Homogeneous Medium. For a homogeneous layer of width d and refractive index n, the complex amplitudes of the collected waves at the planes indicated by the arrows are related by UJ+) e-j<pUi+) and ui-) e-j<PUJ-), where <p nkod, so that in this case the wave-transfer matrix and the scattering matrix are: t . I I I , I I I I exp( j<p) 0 exp( j<p) 0 I n nkod. M ,5 I I ,<p I I 0 exp(j<p ) 0 exp( j<p) I I I I (7 .1-4 ) I I r d .. I Relation between Scattering Matrix and Wave- Transfer Matrix The elements of the M and S matrices are related by manipulating the defining equa- tions (7.1-1) and (7.1-3), whereupon the following conversion equations emerge: M A B C D 1 t21 t 12 t21 T12 T21 T12 r21 1 , (7 .1- 5) s t12 T12 T21 t21 1 AD D BC B C 1. (7.1-6) Conversion Relations These equations are not valid in the limiting cases when t21 0 or D O. , Summary - Matrix wave optics offers a systematic procedure for determining the amplitude transmittance and reflectance of a stack of dielectric layers with prescribed thick- nesses and refractive indexes: ". :? ", The stack is divided into a cascade of elements encompassing boundaries with homogeneous layers between them. 1-.. ,. . The M matrix is determined for each element. This may be achieved by using the Fresnel formulas for transmission and reflection to determine its S matrix, , and then using the conversion relation (7.1-5) to calculate the corresponding M matrix. . The M matrix for the full stack of elements is obtained by simply multiplying the M matrices for the individual elements, in accordance with the wave-transfer matrix formula provided in (7.1-2). i . Finally, the S matrix for the full stack is determined by conversion from the overall M matrix via (7.1-6). The elements of the S matrix then directly yield the amplitude transmittance and reflectance for the full stack of dielectric layers. " : :: . :" " .. :: )
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 249 Two Cascaded Systems: Airy's Formulas. Matrix methods may be used to derive explicit expressions for elements of the scatter- ing matrix of a composite system in terms of elements of the scattering matrices of the constituent systems. Consider a wave transmitted through a system described by an S matrix with elements t12, r21, r12, and t21, followed by another system with S matrix elements t23, r32, r23, and t32. By multiplying the two associated M matrices, and then converting the result to an S matrix, the following formulas for the overall forward transmittance and reflectance can be derived: t13 t 12 t23 1 r 21 r 23 ' r13 t 1 2 t 21 r23 r12 + 1 r21 r23 . (7 .1- 7) If the two cascaded systems are mediated by propagation through a homogeneous medium, as illustrated in Fig. 7 .1- 2, then by use of the wave-transmission matrix in (7.1-4), with the phase <p nkod, where d is the propagation distance and n is the refractive index of the medium, the following formulas for the overall transmittance and reflectance, known as the Airy's formulas, may be derived: t13 ( . , t12 t 23 exp, J<P) 1 r21 r23 exp: j2<P:' r13 t 12 t21 r23 exp: j2<p : r12 + , . , . 1 r21 r23 exp, J2<pJ (7.1-8) Airy's Formulas t l 2 f21 fl2 t 21 t23 f32 f23 t 32 ..... d . . . . ui+) .. -- . . \ ) U r U t Ul+ Figure 7 .1-2 Transmission of a plane wave through a cascade of two separated systems. U- I Ud+) The Airy's formulas may also be derived by tracking the multiple transmissions and reflections undergone by an incident wave between the two systems and adding up their amplitudes, as portrayed in Fig. 7.1-2. A plane wave of complex amplitu de t12Ui, which reflects back and forth between the two subsystems producing additional of the over all transmitted wa v e U t is re l at ed to th e t otal intern al a mplitude U ( +) is the round-trip multiplication factor, the overall amplitude transmittance t13 yields the Airy's formula in (7.1-8). Conservation Relations for Lossless Media If the medium between planes 1 and 2 is nonlossy, then the incoming and outgoing optical powers must be equal. Furthermore, if the media at the input and output planes have the same impedance and refractive index, then these powers are represented by
250 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS , (0,1), and (1, 1), the conservation formula above yields three equations that relate the elements of the S matrix. These equations can be used to prove the following formulas: t12 t21 t , r12 t 12 t;l r21 r , t 2 + r 2 1, (7.] -9) (7.1-10) * r12 r 21 . Equations (7 .1-9) relate the magni tudes of the elements of the 5 matrix for lossless media whose input and output planes see the same refractive index, whereas (7.1-1 0) relates their arguments. The formulas in (7.1-9) and (7.1-10) translate to the following relations among the elements of the M matrix: D A, C B, A 2 8 2 det M C a * A D * t t d t M 12 21 , e 1. (7 .1-11 ) (7.1-12) 1, These results can be derived by substituting the conservation relations for lossless media, (7 .1-9) and (7 .1-1 0), into the conversion relations between the wave-transfer and scattering matrices, (7. 1- 5) and (7 .1-6). Lossless Reciprocal Systems For lossless systems with reciprocal symmetry, namely systems whose transmis- sion/reflection in the forward and backward directions are identical, we have t21 t12 t and r21 r12 r. In this case, (7.1-9) and (7.1-] 0) yield t 2 +r 2 1 t r t r * arg t arg r :1:1f 2 (7 . 1 - 13) indicating that the phases associated with transmission and reflection differ by 1f 2. Under these conditions, the elements of the M matrix satisfy the following relations: A D* B C* , , A 2 a 2 1 , det M 1. (7 . 1- 14 ) The Sand M matrices then take the simple form S t r r t ' M 1 t* r t r* t* 1 t ' (7.1-15) Lossless Reciprocal System and the system is described by two complex numbers t and r related by (7.1-13). EXAMPLE 7.1-2. Partially Reflective Mirror (Beamsplitter). A 10ssJess partially reflec- tive mirror placed in a homogeneous medium is a reciprocal system with an S matrix given by (7.1-15). Assuming that the phase arg{t} O then (7.1-13) dictates that a.rg{r} 7r/2, so that r jlrl. Using the + sign, a model for the scattering matrix of the beamsplitter is: S It I jlrl jlrl It I ' Itl 2 + Irl 2 1. (7 .1-16)
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 251 The corresponding M matrix is: n n M 1 It I 1 jlrl jlrl 1 . (7 .1-17) An ideal mirror has an S matrix given by (7.1-16) with ITI 1 and It I O. In this limiting case, (7.1-17) is not applicable and the M matrix does not provide an appropriate representation since the two sides of the perfect mirror are isolated and independent. EXAMPLE 7.1-3. Single Dielectric Boundary. In this example, the system comprises a single boundary. In accordance with the Fresnel equations (see Sec. 6.2), the transmittance and reflectance at a boundary between two media of refractive indexes nl and n2 are defined by the S matrix S t 1 2 T12 r21 t 21 1 2nl n1 + n2 n1 n2 n2 n1 2n2 . (7.1-18) Substitution into (7. I -5) yields the M matrix n} n2 M 1 n2 + n1 n2 n1 2n2 n2 nl n2 + nl . (7.1-19) EXAMPLE 7.1-4. Propagation Followed by a Boundary. The M matrix of a homoge- neous layer of width d followed by a boundary is given by the M matrix for the boundary, (7 .1-19), multiplied by the M matrix for the homogeneous layer, (7.1-4): I I n) n2 I I (n2 + nl) e- jcp nl) e jcP : M 1 (n2 nlkod · (7.1-20) I <P I 2n2 (n2 nl) e- jcp (n2 + nl) e jcp , I I I d I EXAMPLE 7.1-5. Propagation Followed by Transmission Through a Slab. This sys- tem comprises a cascade of two subsystems, both of the type considered in Example 7 .1-4. In the first system the light travels from a medium of index nl to a medium of index n2, whereas in the second system the light travels from a medium of index n2 to a medium of index nl. By virtue of (7.1-2), the overall M matrix is a product of the constituent M matrices, with the matrix multiplication taking place in reverse order: I I I n) I I I I I I M 1 (nl + n2) e- jCP2 4nln2 (nl n2) e- jCP2 (n2 + nl) e- jCP1 x (n2 nl)e- jCP1 (nl n2) e jCP2 (nl + n2) e jCP2 (n2 nl) e jCP1 (n2 + nl) e jCP1 · (7.1-21) n2 n) d) ... d24 Here <PI nlkod l and <P2 n2k o d 2 , where d l and d 2 are the widths of the two regions, respectively. Elements of the matrix M, which are given by A D* 1 1 2 . (nl + n2) e- JCP2 4nln2 1 2 2 · n l ) e- JCP2 (n2 nl)2 e jCP2 . -J<{)l e , (7.1-22) t* B C* r . . e J <{)2 eJ «) 1 , (7 .1-23) - t satisfy the properties of a reciprocal and lossless system, as described by (7.1-14).
252 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS From (7.1-22) and (7.1-23) we can determine expressions for t and r. Thus, . 4nl n2 exp( j<P2) exp J<Pl · t (7.1-24) This expression can also be directly derived by regarding the system as a combination of two bound- aries mediated by propagation through a distance in a medium, and using the Airy's formula (7.1-8) with t 12 t32 2nl/(nl + n2), t21 t 23 2n2/(nl + n2), r12 T32 T21 T23 (nl n2)/(nl + n2). EXERCISE 7.1-1 Quarter-Wave Film as an Anti-Reflection Coating. Specially designed thin films are often used to reduce or eliminate reflection at the boundary between two media of different refractive indexes. Consider a thin film of refractive index n2 and thickness d sandwiched between media of refractive indexes nl and n3. Derive an expression for the B element of the M matrix for this multilay er m edium. Show that light incident from medium 1 has zero reflectance if d A/4 and n2 yl n ln3, where A Ao/n2- . n] n2 n3 n2 = v n l n 3 d = '\/4 Figure 7.1-3 Anti-reflection coating. Off-Axis Waves in Layered Media When an oblique wave is incident on a layered medium, the transmitted and reflected waves, along with their reflections and transmissions in turn, bounce back and forth between the layers, as illustrated by its real part as shown in Fig. 7.1-4(a). The laws of reflection and refraction ensure that, within the same layer, all of the forward waves are parallel, and all of the backward waves are parallel. Moreover, within any given layer the forward and backward waves travel at the same angle, when measured from the + z and z directions, respectively. --. -- ---- . (a) - .... (b) (-) VI (-) .... V 2 - - Figure 7.1-4 (a) Reflections of a single incident oblique wave from the boundaries of a multilayered medium. (b) In each layer, the forward waves are lumped into a collected forward wave, and the backward waves are lumped into a collected backward wave. - )II-- (+) VI (+) V2 The "macro" approach that was used earlier for normally incident waves is sitni- larly applicable for oblique waves. The distinction is that the Fresnel transmittances
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 253 and reflectances at a boundary, t12, T21, T12, and t21, are angle-dependent as well as polarization-dependent (see Sec. 6.2). The simplest example is propagation a distance d through a homogeneous medium of refractive index n, at an angle () measured from the z axis. The wave-transfer matrix M is then given by (7 .1-4), where the phase is now <p nko d cos (). Two other examples are presented below. EXAMPLE 7.1-6. Single Boundary: Oblique TE Wave. A wave transmitted through a planar boundary between media of refractive indexes nl and n2 at angles (}l and (}2, satisfying Snell's law (nl sin (}l n2 sin (}2), is described by an S matrix determined from the Fresnel equations (6.2-8) and (6.2-9), and its correspondingM matrix: nl n2 2a12rit - "'" t 12 T21 1 n2 n1 S (7 .1- 25) -.; """'-' ,.,..." 2a2l n2 , T12 t 21 n1 + n2 n1 n2 (}l ()2 (}l (}2 A B ,."""", ,.."." ,.,...... ,....."" 1 nl + n2 n2 nl M (7.1-26) 2a2l n2 - "..,."., ,.."." - . C D n2 n1 n1 + n2 These expressions are applicable for both TE and TM polarized waves with the following defini- tions: TE: TM: nl cos (}l , nl sec (}1, "'-' "..,.", n1 "'-' n1 n2 "'-' n2 n2 cos (}2, n2 sec (}2 , a12 a12 a21 1, cos (}ll cos (}2 1 I a2l · EXAMPLE 7.1-7. Propagation Followed by Transmission Through a Slab: Off-Axis Wave. This example deals with an oblique wave traveling through the system described in Exam- ple 7.1-5: a slab of thickness d 2 and refractive index n2 in a medium of refractive index nl. The wave travels a distance d l in the surrounding medium before it crosses into the slab. The wave-transfer matrix for an oblique wave is a generalization of the on-axis result: d nl n2 nl (nl + n2) e- j <P2 (nl n2) e jCP2 M 1 . 4nln2 (nl n2) e- j <P2 (nl + n2) e jCP2 ------_\_-- OJ ---- --....... (n2 + nl) e-j<PI (n2 nl) ej<Pl () (7.1-27) (}2 x nl) e-j<Pl (n2 + nl) ej<Pl , (n2 where <PI nlkod l COS(}1 and <P2 n2kod2cOS(}2, and, as in Example 7.1-6, ni ni COS(}l and n2 n2 cos ()2 for the TE polarization, and nl nl sec ()l and n2 n2 sec ()2 for the TM polarization. The expression for the matrix M in (7.1-27) is identical to that provided in (7.1-21), which describes the on-axis system, except that the parameters n}, n2, <pI, and <P2 are replaced by the angle- and polarization-dependent parameters ni and n2, and by the angle-dependent parameters <PI and <P2. Note that the factors aI2 and a21, which appear in (7.1-26) at each boundary, cancel out since a12a2l 1: With these substitutions, the expression (7.1-24) for the on-axis transmittance developed in Example 7.1-5 is generalized to the off-axis, polarization-dependent case, exp( Jc.pl (ni + n2)2 (ni n2)2 exp( j 2 <P2). (7.1-28) t
254 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS B. Fabry Perot Etalon The Fabry Perot etalon was introduced in Sec. 2.5B as an interferometer made of two parallel highly reflective mirrors that transmit light only at a set of specific uniformly spaced frequencies, which depend on the optical pathlength between the mirrors. It is used both as a filter and as a spectrum analyzer, and is controlled by varying the pathlength, e.g., by moving one of the mirrors. It is also used as an optical resonator, as discussed in Sec. 10.1. In this section, we examine this multilayer device using the matrix methods developed in this chapter. Mirror Fabry Perot Eta/on Consider two lossless partially reflective mirrors with amplitude transmittances tl and t2, and amplitude reflectances rl and r2, separated by a distance d filled with a medium of refractive index n. The overall system is described by the matrix product M 1 t* 1 r* t * 1 1 rl tl 1 tl . exp J C{J o o . exp J C{J 1 t* 2 r* t * 2 2 r2 t2 1 t2 ' (7.1-29) where <p nkod. Since the system is lossless and reciprocal, M takes the simplified form provided in (7 .1-15) and the amplitude transmittance t is therefore the inverse of the D element of M, so that tl t2 exp rl r2 exp 1 . JC{J j2C{J . (7.1-30) t This relation may also be derived by direct use of the Airy's formula (7.1-8). As a result, the intensity transmittance of the etalon is t 1 t2 2 T t 2 1 rlr2 exp j2<p 2. (7 . 1- 3 I ) This expression is similar to (2.5-16) for the intensity of an infinite number of waves with equal phase differences, and with amplitudes that decrease at a geometric rate, as described in Sec.2.5B. Assuming that arg rl r2 0, this expression can be written in the form t T Tmax 1 + 2 7r 2 sin 2 <p , (7.1-32) where t 1 t2 2 1 rl r2 2 1 rl 2 1 r2 2 1 rl r2 2 (7.1-33) Tmax and 7r rlr2 1 rl r2 . (7 .1- 34 ) Finesse t This expression reproduces (2.5-18) if cp is replaced by the round-trip phase 2cp.
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 255 The parameter :7, called the finesse, is a monotonic increasing function of the re- flectance product Tl T2, and is a measure of the quality of the etalon. For example, if fl T2 0.99, then 3=' 313. As described in Sec. 2.5B, the transmittance 'J is a periodic function of <p with period 7r. It reaches its maximum value of 'J mroo which equals unity if Tl T2 , when <p is an integer multiple of 7r. When the finesse is large (i.e., when Tl T2 1), 'J becomes a sharply peaked function of <p of approximate width 7f 3='. Thus, the higher the finesse :7, the sharper the peaks of the transmittance as a function of the phase <p. The phase <p nkod w c d is proportional to the frequency, so that the condition <p 7f corresponds to w Wp, or v Vp, where Vp c 2d ' 7fC d (7.1-35) Free Spectral Range Wp is called the free spectral range. It follows that the transmittance as a function of frequency, 'J v , is a periodic function of period Vp, 'Iv 'J max 1 + 2:7 7f 2 sin 2 7fV vp , (7 .1- 36) Transmittance (Fabry Perot Etalon) as illustrated in Fig. 7 .1- 5. It reaches its peak value of 'I max at the resonance frequencies v q qvp, where q is an integer. When the finesse :7 » 1, 'J v drops sharply as the frequency deviates slightly from v q , so that 'J v takes the form of a comb-like function. The spectral width of each of these high-transmittance lines is v vp ' (7.1-37) i.e., is a factor of 3=' smaller than the spacing between the resonance frequencies. C 1/F= 2d t) rl t 2 f2 ---.. 'III ,. .. .... T ....----.. .,.- ...... , , , , I I I . I · 1/F' uV = :F .---- -- , . . . . .-----.-. -- ...... , , , , , I , I . . I . . . . . . . I . I I . . .. . . . . . . . . . . . . I' . . . . . . . . I . . . I' d .... 1/ Figure 7 .1-5 Intensity transmittance and reflectance, T and :R 1 'J, of the Fabry-Perot etalon as a function of the angular frequency w. The Fabry Perot etalon may be used as a sharply tuned optical filter or a spectrum analyzer. Because of the periodic nature of the spectral response, however, the spectral width of the measured light must be narrower than the free spectral range vp c 2d in order to avoid ambiguity. The filter is tuned (i.e., the resonance frequencies are shifted) by adjusting the distance d between the mirrors. A slight change in mir- ror spacing d shifts the resonance frequency v q qc 2d by a relatively large amount Vq qc 2d 2 d v q d d. Although the frequency spacing Vp
256 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS also changes, it is by the far smaller amount vFd d. As an example, a mirror separation d 1.5 em leads to a free spectral range VF 10 GHz when n 1. For a typical optical frequency of v 10 14 Hz, corresponding to q 10 4 , a change of d by a factor of 10- 4 (d 1.5 Mm) translates the peak frequency by Vq 10 GHz, whereas the free spectral range is altered by only 1 MHz, becoming 9.999 GHz. Applications of the Fabry Perot etalon as a resonator are described in Sec. 10.1. The Dielectric-Slab as a Fabry Perot Etalon Based on (7 .1-24), the transmittance of a dielectric slab of width d and refractive index n2 in a medium of refractive index n 1 is 4nln2exp jep nl + n2 2 nl n2 2 exp j2<p' (7.138) t where <p n2k o d. This expression reproduces (7.1-30), which applies to the Fabry Perot etalon, if we substitute tl t2 4nl n2 nl + n2 and Tl f2 nl n2 2 nl + n2 2. It follows that the expressions for the intensity transmittance of the mirror etalon, (7.1-32) and (7.1-36), are applicable to the dielectric slab. Using (7.1-34), the finesse of the slab is given by 7r n ni 4 n 1 n2 . (7.1-39) Large values of are not normally obtained in slab etalons. For example, for nl 1.5 (the refractive index of Si0 2 ) and n2 3.5 (the refractive index of Si), 3=' 1.68. As illustrated in Fig. 7 .1-6, the frequency dependence of 'I in this case does not exhibit the sharp peaks seen in etalons with highly reflective mirrors, as displayed in Fig. 7.1-5. To obtain higher values of the surfaces of the slab must be coated to enhance internal reflection. 1 o I\ I\ I\ I \ ' \ I \ I \ I \ I \ I \ \ I \ I \ I \ I \ I \ I \ I \ I \ I \ I \ " " " , , , , '- n2 n} 0.5 'R nl d --.I l/F l/ Figure 7.1-6 Frequency dependence of the intensity transmittance and reflectance, 'J and 1(, respectively, of a slab with refractive index n2 3.5 (the refractive index of Si) in a medium with refractive index nl 1.5 (the refractive index of Si0 2 ). Off-Axis Transmittance of the Fabry Perot Etalon For an oblique wave traveling at an angle () with the axis of a mirror e ta lon, the ampli- tude transmittance is given by (7 .1- 30) with the phase ep replaced by (p' nko d cos (). It follows that the intensity transmittance in (7.1-36) is generalized to Tv Tmax 1 + 2 7r 2 sin 2 7rcos() v vF ' (7.1-40)
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 257 in the off-axis case. Maximum transmittance occurs at frequencies for which v q V F see () , q 1, 2, . . . , V F C 2 d. (7 .1-41 ) Resonance Condition If the finesse of the etalon is large, transmission occurs at these frequencies and is almost completely blocked at all other frequencies. The plot of this relation provided in Fig. 7 .1- 7 (c) shows that at each angle () only a set of discrete frequencies are transmitted. Likewise, a wave at frequency v is transmitted at only a set of angles, so that a cone of incident broad-spectrum (white) light creates a set of concentric rings spread like a rainbow, as illustrated in Fig. 7 .1-7 (b). For incident light with a spectral width smaller than the free spectral range VF, each frequency component corresponds to one and only one angle, so that the etalon can be used as a spectrum analyzer. \ erGO 80° 70° 60° 50° 40° 30° 20° 10° 0° o d - .... .... .. , . - --JI"'" 1 \ '- "\ \ - . 1 , . \ 1 . -- -..-------.. ,..---- ....--- _......... .-...- -.... . :a -------- - - () "" ..-- - .' . .. d r- 1 234 (c) 5vj V :F (a) (b) Figure 7.1-7 (a) An off-axis wave transmitted through a mirror Fabry-Perot etalon. (b) White light from a point source transmitted through the etalon creates a set of concentric rings of different frequencies (colors). (c) Frequencies and angles that satisfy the condition of peak transmittance, as set forth in (7. 1-41 ). c. Bragg Grating The Bragg grating was introduced in Exercise 2.5- 3 as a set of uniformly spaced parallel partially reflective planar mirrors. Such a structure has angular and frequency selectivity that is useful in many applications. In this section, we generalize the defini- tion of the Bragg grating to include a set of N uniformly spaced identical multilayer segments, and develop a theory for light reflection based on matrix wave optics. Simplified Theory The reflectance of the Bragg grating was determined in Exercise 2.5-3 under two assumptions: (1) the mirrors are weakly reflective so that the incident wave is not depleted as it propagates, and (2) secondary reflections (Le., reflections of the reflected waves) are negligible. In this approximation, the reflectance 1{N of an N -mirror grating is related to the reflectance R of a single mirror by the relation t 1{N sin 2 <p (7 .1-42) t Note that in Exercise 2.5-3 <p denotes the phase between successive phasors, while here that phase is denoted 2<p since it represents a round-trip phase.
258 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS As described in Sec. 2.5B, the factor sin 2 N <pI sin 2 <P represents the intensity of the sum of N phasors of unit amplitude and phase difference 2<p. This function has a peak value of N 2 when the Bragg condition is satisfied, i.e., when 2<p equals q27r, where q == 0, 1, 2, . . .. It drops away from these values sharply, with a width that is inversely proportional to N. In this simplified model, the intensity of the total reflected wave is, at most, a factor of N 2 greater than the intensity of the wave reflected from a single segment. For a Bragg grating comprising partially reflective mirrors separated from each other by a distance A and a round-trip phase 2<p == 2kA cas (), where () is the angle of incidence. Therefore, maximum reflection occurs when 2kA cos () == 2q7r or A W13 V13 cosf) == q- == q- == q-, 2A W v (7.1-43) Bragg Condition where C V13 == 2A ' 7rC W13 == A ' (7.1-44 ) Bragg Frequency is the Bragg frequency. 90° or 80° 70° -J I-- A 60° o z 50° ___________ 40° 30° 20° 10° 0° o 2 3 4 5 vjv'B Figure 7.1-8 Locus of frequencies v and angles () at which the Bragg condition is satisfied. For example, if v == 1.5 v, then () == 48.2°. This corresponds to a Bragg angle () == 41.8° (measured from the plane of the grating.) At normal incidence «() == 0°), peak reflectance occurs at frequencies that are integer multiples of the Bragg frequency, i.e., v == qV13. At angular frequencies v < V13, the Bragg condition cannot be satisfied at any angle. At frequencies V13 < v < 2V13, the Bragg condition is satisfied at one angle () == cos- 1 ('x/2A) == cos- 1 (V13/ V ). The complement of this angle, f)13 == 7r /2 - f), is the Bragg angle [see (2.5-13) and Fig. 2.5- 8], ()13 == sin- 1 ('x/2A). (7.1-45) Bragg Angle At angular frequencies v > 2V13, the Bragg condition is satisfied at more than one angle. Figure 7.1-8 illustrates the spectral and angular dependence of reflections from a Bragg grating, based on the simplified theory.
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 259 Matrix Theory We now use the matrix approach introduced in the previous section to develop an exact theory of Bragg reflection that includes multiple transmissions and reflections, as well as depletion of the incident wave. It turns out that the collaborative effects of the reflections, and the reflections of reflections, can lead to enhancement of the total reflected wave, and a phenomenon whereby total reflection occurs not only at single frequencies that are multiples of Vp> cos f), but over extended spectral bands surrounding these frequencies! Consider a grating comprising a stack of N identical generic segments (Fig. 7 .1-9), each described by a unimodular wave-transfer matrix Mo satisfying the conservation relations for a lossless, reciprocal system, so that Mo 1 t* T t T* t* 1 t ' (7.1-46) where t and r are complex amplitude transmittance and reflectance satisfying the conditions set forth in (7.] -13), and 'J' t 2 and T 2 are the corresponding intensity transmittance and reflectance. ----- Mo Mo Mo ----- Mo 1 2 N-l N Figure 7.1-9 Bragg grating made of N segments, each of which is described by a matrix Mo In accordan ce with (7.1-2), the wave-transfer matrix M for the N segments is simply M W NMo W N-II, (7 .1-4 7) where WN sin N <I> sin <I> ' (7 .1-48) cos <I> Re 1 t , (7 .1-49) and I is the identity matrix. Equation (7 .1-47) may be proved by induction (i.e., show that this relation is valid for N segments if it is valid for N 1 segments; this may be done by direct substitution and use of trigonometric identities). Since the N -segment system is also lossless and reciprocal, its matrix may be written in the form M N o 1 t* N r* t* N N TN tN 1 tN , (7.1-50)
260 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS where tN and rN are the N-segment amplitude transmittance and reflectance, re- spectively. Substituting from (7.1-46) and (7.1-50) into (7.1-47), and comparing the diagonal and off-diagonal elements of the matrices on both sides of the equation, leads to 1 tN rN tN Nt r WN-. t WN-l (7 .1- 51 ) (7.1-52) These two equations define tN and rN in terms of t and r. The intensity transmittance TN tN 2 is obtained by taking the squared absolute value of (7.1-52) and using the relation 1 T, TN T T + w1 1 . T (7 .1- 53) It follows that the intensity reflectance N 1 TN is given by N WJv 1 + W · (7.1-54) Bragg-Grating Reflectance - . . - - .=:':'. -" -.- - - - . . - - - -- --- - ..- - - -. - - - - - - - -- - - -. - . - "".... -. ...- - - -. - -"':"'_----.- - ---- -""- - -- ------- --- - - _:...- - .--_.:...:;.:. -- - .- -- . - . ......;...... - - - =-- .- - - - - - - -- ... . - - -- Summary The reflectance N of a medium made of N identical segments is related to the single-segment reflectance by a nonlinear relation, (7.1-54), which contains a factor q, N resulting from the interference effects associated with collective · j reflections from the N segments of the grating. Defined by (7 .1-48), q, N depends ' : on the number of segments N and another parameter , which is related to the : , single-segment complex amplitude transmittance t by (7.1-49). - . -t. The dependence of N on , described by (7.1-54), takes simpler forms in certain limits. If the single-segment reflectance is very small, Le., « 1, and if w1 is not too large so that w1 « 1, then (7 .1- 54) may be approximated by: N W7v sin 2 <I> (7 .1- 55) This relation is now similar in form to the approximate relation (7.1-42), with <I> playing the role of the phase <po In the opposite limit for which w1 » 1, the reflectance N w1 1 + w1 . This nonlinear relation between N and exhibits saturation and is typical of systems with feedback, which in this case results from multiple internal reflections at the seg- ment boundaries. Ultimately, if WJv 1, then N approaches its maximum value of unity, so that the N -segment device acts as perfect mirror even though the single segment is only partially reflective. A large interference factor W N accelerates the rise of N to unity as increases. The interference factor W N, which depends on <I> cos- 1 Re 1 t via (7 .1-48), follows two distinct regimes: i) a normal regime for which <I> is real and the grating ex- hibits partial reflection/transmission (including zero reflection, or total transmission),
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 261 and ii) an anomalous regime for which cI> is complex and \If N can be extremely large, corresponding to total reflection. Partial- and Zero-Reflection Regime This regime is defined by the condition Re 1 t :( 1, which ensures that <I> cos- 1 Re 1 t is real. In this case, N depends on 9{ and \If N in accordance with (7.1-48) and (7.1-54). Maximum reflectance occurs when \If N has its maximum value of N. In this case, N N2 1 + N2 . Therefore, N cannot exactly equal unity unless 1, exactly. For example, for N 10, if 0.5, then the maximum value of N 0.99. Zero reflectance, or total transmittance, is possible.. even if the reflectance R of the individual segment is substantial. This occurs when \If N 0, i.e., when sin N <I> 0, or <I> q7r N for q 0,1, . . . , N 1. The N frequencies at which this complete trans- parency occurs are resonance frequencies of the grating. The phenomenon represents some form of tunneling through the individually reflective segments. Total-Reflection Regime In this regime, Re 1 t cos <I> > 1 so that <I> is a complex variable <I> <I>R + j<I>I. Using the identity cos cI>R + j<I>I cos <I>R cosh <I>I j sin <I>R sinh <I>I, and equating the real and imaginary parts of both sides of (7 .1-49), we obtain sin <I>R 0 so that <:PR m7r and cos cI>R + 1, or 1, when m is an even or odd integer, respectively, which results in cosh <I>I Re 1 t . (7.1-56) Total-Reflection Regime The factor \If N sin N <I> sin cI> then becomes \IlN sinh N <I> I :f:: , sinh <I>I (7.1-57) Total-Reflection Regime where the :f:: sign is the sign of the factor cos N m7r cos m7r . Since sinh · increases exponentially with N for large N, \II N can be much greater than N. In this case, in accordance with (7 .1- 54), the reflectance N 1 and the grating acts as a total reflector. The forward waves become evanescent and do not penetrate the multisegment medium, much as occurs with total internal reflection. Because <I> depends on t, which depends on the frequency v, the two regimes correspond to distinct spectral bands, as illustrated in the following examples. The spectral bands associated with the total-reflection regime are called stop bands since they represent bands within which light transmission is almost completely blocked. The other regime corresponds to passbands. Total transmission (zero reflection) occurs at specific resonance frequencies within the passbands. EXAMPLE 7.1-8. Stack of Partially Reflective Mirrors. Consider a grating made of a stack of N identical partially reflective mirrors (beamsplitters) that are mutually separated by a distance A and embedded in a homogeneous medium of refractive index n, as illustrated in Fig. 7.1-10(a). A single segment comprises a distance A in a homogeneous medium, followed by a partially reflective mirror of amplitude transmittance t and amplitude reflectance T.
262 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS The wave-transfer matrix Mo for this segment is determined by multiplying the matrix in (7.1-17) by the matrix in (7.1-4): . -Jc.p e jlrle-j<p jlrlej<p Mo 1 It I . eJc.p , cP nkoA 7rV /vp" (7.1-58) where Vp, c/2A is the Bragg frequency. This provides t Itlej<p, and therefore 4> via 1 1 cosh 4>1 for I cos cpl > Itl. (7.1-60) The relationships between 4> and cp, and between 4>1 and cp, are nonlinear and unusual, as illus- trated in Fig. 7.1-10(b). The corresponding dependence of the intensity reflectance N on cp is shown in Fig. 7.1-10(c). In the normal regime (indicated by the shaded regions), 4> is real and the reflectance exhibits multiple peaks with zeros between. None of the peaks approaches unity, despite the fact that \l1 N reaches a maximum value of N 10. The situation is quite different in the total-reflection regime (unshaded regions), where 4> is complex. The factor W N reaches a value 3000 at the center of the band (cp 7r) when Itl 2 0.5. These regions represent ranges of cp where total reflection occurs (N 1). Since cp is proportional to the frequency v, Fig. 7 .1-1 O( c) is actually a display of the spectral reflectance, and the unshaded regions correspond to the stop bands. 7r cos 4> for I cos cpl < It I, (7.1-59) " , \ ,... , , I , 4> 3 I A I " --.... 4>. 4> I I I I I I I I I \.. .A J. J \.. i 2 3 ...r N (a) ....., " , \ cI> (b) 4>. 00 1 ,... ..... , , I , " " , \ 7r 2n <pV Rrvj v 13 (1) .....-4 N M U "'0 (c) s:: s:: s:: s:: (:) 0.5 .D .D .D Cl.t (1) 0 0 0 ..... (1) CI) CJ'.) CJ'.) o o V13 2V13 v Figure 7.1-10 (a) Bragg grating made of N 10 identical mirrors, each with an intensity reflectance Irl 2 0.5. (b) Dependence of <I> on the inter-mirror phase delay cp nkoA. Within the shaded regions, 4> is complex and its imaginary part 4>1 is represented by the dashed curves. (c) Reflectance as a function of frequency (in units of the Bragg frequency Vp, c/2A) . Within the stop bands, the reflectance is approximately unity. EXAMPLE 7.1-9. Dielectric Bragg Grating. A grating is made of N identical dielectric layers of refractive index n2, each of width d 2 , buried in a medium of refractive index nl and separated by a distance d I, as illustrated in Fig. 7 .1-11. This multi segment system is a stack of N identical double layers, each of the type described in Example 7.1- 5. The A 1 jt* element of the wave-transfer matrix Mo is given by (7.1-22), from which 1 Re - t 2 nl + n2 nln2 cos CPI CP2 ) , 4nI n 2 (7 .1-61 ) where CPI nikod l and CP2 n2k o d 2 are the phases introduced by the two layers of a segment. This result can be used in conjunction with (7.1-48), (7.1-49), (7.1-54), (7.1-56), and (7.1-57) to determine the reflectance of the grating. The spectral dependence of the reflectance can be computed as a function of v by noting that CPI + CP2 ko(nid l + n2 d 2) 7rv/vp" where Vp, (c o /n)/2A, and n (nIdI + n2d2)/A is
7.1 OPTICS OF DIELECTRIC LAYERED MEDIA 263 the average refractive index. The Bragg frequency Vp, is the frequency at which the single-segment round-trip phase 2ko(nl d 1 + n2d2) 27r. The phase difference <PI <P2 (,7rV /vp" with (, (nIdI n2 d 2)/(n 1 d l + n2 d 2), is also proportional to the frequency. Figure 7.1-11(b) provides an example of the spectral reflectance as a function of v. n l n) n 2 d. d 2 A n 1 I --.J / / I / . 1 C'1 '\j cu '\j u s:: s= c ft t\S A ft .,.D t\S A C\j .l:J (:)0.5 0.. 0 cu 0 '= en ....., en cu " 1. ° "- 4- "- ... y .... 0 2v'B 3v'B 1 2 N v'B v Figure 7.1-11 Intensity reflectance as a function of frequency for a dielectric Bragg grating made of N 10 segments, each of which has two layers of thickness d l d 2 and refractive indexes nl 1.5 and n2 3.5. The grating is placed in a medium with matching refractive index nl. The reflectance is approximately unity within the shaded stop bands centered about multiples of Vp, cj2A, where c coin and n is the mean refractive index. EXAMPLE 7.1-10. Dielectric Bragg Grating: Oblique Incidence. The results in Exam- ple 7 .1-9 may be generalized to oblique waves with angle of incidence ()l in medium 1, corresponding to angle ()2 in layer 2, where nl sin ()l n2 sin ()2. In this case, (7.1-61) becomes ( ,...., ,...., ) 2 nl + n2 "'-' nln2 1 Re - t ( ,...,., ,...,., ) 2 n2 nl 4ih n2 cos( <PI <P2 ) , (7.1-62) where <PI nlkod l cos ()l and <P2 n2k o d 2 cos ()2; nl nl cos ()l and n2 n2 cas ()2 for TE polarization; and n1 nl sec ()l and n2 n2 sec ()2 for TM polarization. This re- lation may be used to compute the spectral reflectance at any angle of incidence. Figure 7.1- 12 illustrates the dependence of the intensity reflectance RN on frequency and the angle of incidence for both TE and TM polarization for a high- contrast grating. The range of angles over which unity reflectance obtains increases with increasing refractive-index contrast ratio n2/nl' I 0 1 Q) TM CJ s::: 0.5 Cd CJ Q) 0 t+: 30° Q) I TE 0.5 0.5 o 1 0.5 o 1 0.5 o I 0.5 o 1 0.5 o o 0° TM 60° TE TM 70° TE V'B 2v'B v 3v'B Figure 7.1-12 Spectral dependence of the reflectance for the 10-segment dielectric Bragg grating shown in Fig. 7 .1-11 at several angles of incidence ()l and for TE and TM polarization.
264 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS Bragg Grating in an Unmatched Medium In the previous analysis, the Bragg grating was assumed to be made of N identical segments. If each segment is made of multiple dielectric layers, this requires that grating be placed in a matched medium, i.e., a medium with a refractive index equal to that of the front layer, so that the incident light undergoes no additional reflection at the front boundary, and reflects at the back boundary as if it were entering another layer of the grating. The device described in Example 7 .1-9 meets this condition. In most applications, the grating is placed in an unmatched medium, such as air, and boundary effects must be accounted for. This may be accomplished by writing the wave-transfer matrix M of the composite system, including all boundaries, and finding the corresponding scattering matrix S by use of the conversion relation. The reflectance of the composite system may be readily determined from S. the front layer, then the overall wave transfer function takes the form M MeM-lMi' (7.1-63) where M i is the wave-transfer matrix of the entrance boundary, and Me is the wave- transfer matrix of the Nth segment with a boundary into the unmatched medium. EXAMPLE 7.1-11. Reflectance of a Dielectric Bragg Grating in an Unmatched Medium. An N -segment Bragg grating is made of alternating layers of refractive indexes n] and n2, and widths d 1 and d 2 , placed in a medium of refractive index no. We wish to determine the reflectance for a wave incident at an angle 0 0 in the external medium corresponding to angles Ol and O 2 in the first and second layer of each segment, as determined by Snell's law (n! sin 0 1 n2 sin ( 2 ). In this case, (7.1-63) may be used with the following wave-transfer matrices: (1) M i represents a boundary between media of refractive indexes no and nl, as described in Example 7.1-6; (2) Mo represents a single segment of the grating, as described in Example 7.1-7; (3) Me represents propagation a distance d 1 in a medium with refractive index n] followed by a slab of width d 2 and refractive index n2, with boundary into a medium of refractive index no. Once the M matrix is determined, we use the conversion relation (7 .1-6) to determine the corresponding scattering matrix S. The overall reflectance is the element Tl2 in (7.1-4). I do) u c Crj u O . 5 cu r= cu 0 1 v/v'B = 0.9 TE -------------- -------------------.--------- v/v'B = 0.9 TE TM TM cu u c 50.5 u Q) r= cu 0 1 V/lI'B = I TE ---------------------- TE TM V/v'B = 0.97-1.18 TM I I I I I I I I v/v'B = 1.2 TE .-- ., , , , TE: , I , I , , I , , v/v'B = 1.2 --- , , , cu ' u : C : TM 0.5 I cu : r= , cu ' A " 0° 1 0° 20° 30° 40° 50° 60° 70° 80° 90° 0° I 0° 20° 30° 40° 50° 60° 70° 80° 90° o 8 (a) Grating in matched medium (b) Grating in air Figure 7.1-13 Intensity reflectance as a function of the angle of incidence 0 at fixed frequencies for the grating described in Fig. 7.1-11. (a) Grating is placed in a matched medium (n nl). (b) Grating is placed in air (n 1). In air, the grating has unity reflectance at all angles, for both TE and TM polarizations, at frequencies in the band O.97v13 to 1.18 V13.
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 265 7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS One-dimensional (lD) photonic crystals are dielectric structures whose optical prop- erties vary periodically in one direction, called the axis of periodicity, and are constant in the orthogonal directions. These structures exhibit unique optical properties, par- ticularly when the period is of the same order of magnitude as the wavelength. If the axis of periodicity is taken to be the z direction, then optical parameters such as the permittivity E z and the impermeability 11 z Eo E Z are periodic functions of z, satisfying 11z+A 11 z , (7.2-1) for all z, where A is the period. Wave propagation in such periodic media may be studied by solving the generalized Helmholtz equations (7.0-2), for periodic 11 z . For an on-axis wave traveling along the z axis and polarized in the x direction, the electric and the magnetic field components Ex and Hy are functions of z, independent of x and y, so that (7.0- 2) becomes w 2 2 Hy. Co (7.2-2) d d For an off-axis wave, i.e., a wave traveling in an arbitrary direction in the x z plane, the generalized Helmholtz equation has a more complex form. For example, for a TM- polarized off-axis wave, the magnetic field points in the y direction and (7.0-2) gives: 8 8 8 2 w 2 2 Hy. Co (7.2-3) Note that (7.2-2) and (7.2-3) are cast in the form of an eigenvalue problem from which the modes Hy x, z can be determined. · Before embarking on finding solutions to these eigenvalue problems, we first exam- ine the conditions imposed on the propagating modes by the translational symmetry associated with the periodicity. A. Bloch Modes Consider first a homogeneous medium, which is invariant to an arbitrary translation of the coordinate system. For this medium, an optical mode is a wave that is unaltered by such a translation; it changes only by a multiplicative constant of unity magnitude (a phase factor). The plane wave exp jkz is such a mode since, upon translation by a distance d, it becomes exp jk z + d exp jkd exp jkz. The phase factor exp j kd is the eigenvalue of the translation operation, as discussed in Appendix C. On-Axis Bloch Modes Consider now a J D periodic medium, which is invariant to translation by the distance A along the axis of periodicity. Its optical modes are waves that maintain their form upon such translation, changing only by a phase factor. These modes must have the form x . -..,.. ..... .n Uz PK z exp j K z , (7.2-4) Bloch Mode .. ....c . - ..... 1 z
266 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS where U represents any of the field components Ex, Ey, Hx, or Hy; K is a constant, and PK (z) is a periodic function of period A. This form satisfies the condition that a translation A alters the wave by only a phase factor exp( - j K A) since the periodic function is unaltered by such translation. This optical wave is known as a Bloch mode, and the parameter K, which specifies the mode and its associated periodic function PK(Z), is called the Bloch wavenumber. The Bloch mode is thus a plane wave exp( - j K z) with propagation constant K, modulated by a periodic function PK(Z), which has the character of a standing wave, as illustrated by its real part displayed in Fig. 7.2-1 (a). Since a periodic function of period A can be expanded in a Fourier series as a superposition of harmonic functions of the form exp( -jmg z), m == 0, :i::l, :i::2,. . ., with 9 == 27r I A, (7.2-5) it follows that the Bloch wave is a superposition of plane waves of multiple spatial frequencies K +mg . The fundamental spatial frequency 9 of the periodic structure and its harmonics mg, added to the Bloch wavenumber K, constitute the spatial spectrum of the Bloch wave, as shown in Fig. 7.2-1(b). The spatial frequency shift introduced by the periodic medium is analogous to the temporal frequency (Doppler) shift introduced by reflection from a moving object. Standing A ---+j,._ / wave _ ." L/ ." " " " " " (a) K-g K K+g (b) Spatial frequency Figure 7.2-1 (a) The Bloch mode. (b) Its spatial spectrum. z Two modes with Bloch wavenumbers K and K' == K + 9 are equivalent since they correspond to the same phase factor, exp( - j K' A) == exp( - j K A) exp( - j27r) == exp( - j KA). This is also evident since the factor exp( - j 9 z) is itself periodic and can be lumped with the periodic function PK (z). Therefore, for a complete specification of all modes, we need only consider values of K in a spatial-frequency interval of width 9 == 27r/A. The interval [-g 12, 9 12] == [-7r/A,7r/A], known as the first Brillouin zone, is a commonly used construct. Off-Axis Bloch Modes Off-axis optical modes traveling at some angles in the x-z plane assume the Bloch form xt """--- ......... U(x,y,z) ==PK(z)exp(-jKz)exp(-jkxx). (7.2-6) z Off-Axis Bloch Mode The uniformity of the medium in the x direction constrains the x dependence of the optical mode to the harmonic form exp( - j kxx), posing no other restriction on the transverse component kx of the wavevector. At a location where the refractive index
\ 7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 267 is n, kx nko sin (), where () is the inclination angle of the wave with respect to the z axis. As the wave travels through the various layers of the inhomogeneous medium, this angle changes, but in view of Snell's law, n sin () and kx are unaltered. Normal-to-Axis Bloch Modes When the angle of incidence in the densest medium is greater than the critical angle, the modes do not travel along the axis of periodicity (the z direction). Rather, they are normal-to-axis modes traveling along the lateral x direction that take the Bloch form (7.2-6) with K 0, x where Po z periodicity. (7.2-7) Normal-to-Axis Bloch Mode is a periodic function representing a standing wave along the axis of u x,y,z Po z exp j kxx , ---+- Z Eigenvalue Problem, Dispersion Relation, and Photonic Bandgaps Now that we have established the mathematical form of the modes, as imposed by the translational symmetry of the periodic medium, the next step is to solve the eigenvalue problem described by the generalized Helmholtz equation. For a mode with a Bloch wavenumber K, the eigenvalues w 2 c provide a discrete set of frequencies w. These values are used to construct the w K dispersion relation. The eigenfunctions help us determine the Bloch periodic functions PK z for each of the values of w associated with each K. The w K relation is a periodic multivalued function of K with period g, the funda- mental spatial frequency of the periodic structure; it is often plotted over the Brillouin zone 9 2 < k < 9 2 , as illustrated schematically in Fig. 7.2-2(a). When visualized as a monotonically increasing function of k, it appears as a continuous function with discrete jumps at values of K equal to integer multiples of 9 2. These discontinuities correspond to the photonic bandgaps, which are spectral bands not crossed by the dispersion lines, so that no propagating modes exist. The origin of the discontinuities in the dispersion relation lies in the special sym- metry that emerges when k 9 2, i.e., when the period of the traveling wave equals exactly half a period of the periodic medium. Consider the two modes with k g 2 and Bloch periodic functions PK z P-i:.g/2 Z . Since these modes travel with the same wavenumber, but in opposite directions, i.e. see inverted versions of the medium, P-g/2 z Pg/2 z. But these two modes are in fact one and the same, because their Bloch wavenumbers differ by g. It therefore follows that at the edge of a Brillouin zone, there are two Bloch periodic functions that are inverted versions of one another. Since the medium is inhomogeneous or piecewise homogeneous within a unit cell, these two distinct functions interact with the medium differently, and therefore have two distinct eigenvalues, i.e., distinct values of w. This explains the discontinuity that emerges as the continuous w K line crosses the boundary of the Brillouin zone. A similar argument explains the discontinuities that occur when K equals other integer multiples of 9 2. The variational principle (see Appendix C) is helpful in pointing out certain features of these eigenfunctions. Based on this principle, the eigenfunctions of a Hermitian op- erator are orthogonal distributions that minimize the variational energy. The variational energy associated with the linear operator £ in the eigenvalue equation (7.0-2) is Ev H, \7 x 11 r V' x H. D r 2 f r dr, so that minimization of Ev is achieved
268 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS by distributions for which higher displacement fields D r are located at positions of lower 1 f r , i.e. higher refractive index. For example, if the periodic medium is made of two alternating dielectric layers, as illustrated in Fig. 7 .2- 2(b), then at a discontinuity the eigenfunction of the lower frequency concentrates its displacement field in the layer with the greater refractive index, whereas the eigenfunction of the higher frequency has an inverted distribution for which the displacement field is concentrated in the layer with the lower refractive index. I I I I I I --....- I " I " I " I " I '... J '-_ I I I I __ . I ,", I " " : "" I ,,' -- I Brillouin zone : I I I --- I " I ...... " I " I " I "" : " I "" I -' I ' t ,- ...... w A n 1 n 2 I I I I ,--r--..... I ......, I " I , I " I " I , I '" _!g 0 z - I ..., , " I " I " I I B " iI / . . II · . u__.. i . . . . . . . . I . . . . : "..-...... : /' : / . / : / . - " . . . . . . . :, / . / :, / ., / : ,,, . ....... z I I : Bandgap 1 , / : , / . . ' - .- " . . . . . . . . . . . . . , , , /. - . . . . . . . . . . . A : --"'''''' : " I " I - I '" ,," I ',I,' I " I , I " I ' , , A ".-....." !g 9 z K Figure 7.2-2 (a) The dispersion relation is a multi valued periodic function with period 9 27r / A and discontinuities at k equals integer multiples of 9/2. (b) Bloch functions at the points A and B at the edge of the Brillouin zone for an alternating dielectric-layer periodic medium with n2 > nl. The challenging problem now is the solution of the eigenvalue problem associated with the Helmholtz equation. There are two approaches: . The first approach is based on expanding the periodic function 11 z of the medium and the periodic function PK z of the Bloch mode in Fourier series and con- verting the Helmholtz differential equation into a set of algebraic equation cast in the form of a matrix eigenvalue problem, which are solved numerically. This approach is called the Fourier Optics approach. . The second approach is applicable to layered (piecewise homogeneous) media with planar boundaries. Instead of solving the Helmholtz equation, we make direct use of the laws of propagation and reflection/refraction at boundaries, which are known consequences of Maxwell's equations. We then use the matrix methods developed for multilayer media in Sec. 7 .IA and applied to Bragg gratings in Sec. 7 .1 C. This Matrix Optics approach leads to a 2 x 2 matrix eigenvalue problem from which the dispersion relation and the Bloch modes are determined. The matrix-optics approach is discussed next, and the Fourier-optics approach is examined in Sec. 7.2C. B. Matrix Optics of Periodic Media . A one-dimensional periodic medium comprises identical segments, called unit cells, that are repeated periodically along one direction (the z axis) and separated by the period A (Fig. 7.2-3). Each unit cell contains a succession of lossless dielectric layers or partially reflective mirrors in some order, forming a reciprocal system represented by a generic unimodular wave- transfer matrix Mo 1 t* r t r* t* 1 t ' (7.2-8) where t and r are complex amplitude transmittance and reflectance satisfying the con- ditions set forth in (7 .1-15), and 'J' t 2 and 9( r 2 are the corresponding intensity
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 269 transmittance and reflectance. The medium is a Bragg grating, like that described in Sec. 7.1 C, with an infinite number of segments. A wave traveling through the medium undergoes multiple transmissions and reflections that add up to one forward and one backward wave at every plane. We now use the matrix method developed in Sec. 7.1A to determine the Bloch modes. (+) u( + ) U nl m+ I . . . Mo Mo Mo Mo Mo .. · L J (-) U m I mA (-) U m + l , (m+ l)A 1 I Z Figure 7.2-3 Wave-transfer ma- trix representation of a periodic medium. initial position z mA of unit cell m. Knowing these amplitudes, the amplitudes elsewhere within the cel] can be determined by straightforward application of the appropriate wave-transfer matrices, as described in Sec. 7 .1. We therefore direct our next. These dynamics are described by the recurrence relations u(+) m+l (-) U m + 1 Mo U+) U- ) , (7.2-9) which can be used to determine the amplitudes at a particular cell if the amplitudes at the previous cell are known. Eigenvalue Problem and Bloch Modes By definition, the modes of the periodic medium are self-reproducing waves, for which u(+) m+ 1 u(-) m+l u(+) _jq, m m 1,2,...; (7.2-10) after transmission through a distance A (in this case a unit cell), the magnitudes of the forward and backward waves remain unchanged and the phases are altered by a common shift <I>, called the Bloch phase. The corresponding Bloch wavenumber is K <I> A, so that <I> KA. (7.2-11 ) Bloch Phase the self-reproduction condition (7 .2-10) can be cast as an eigenvalue problem. This is accomplished by using (7 .2-9) with m 0 to write (7.2-10) in the form Mo u:(+) o u:( -) o u:( +) -jq, 0 o (7.2-12)
270 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS This is an eigenvalue problem for the 2 x 2 unit-cell matrix Mo. The factor e- j i1> is the The eigenvalues are determined by equating the determinant of the matrix Mo e- j i1>I to zer o. Noting that t 2 + r 2 1, the solution to the ensuing quadratic equation cas <I> 1 Re t . (7.2-13) Equation (7 .2-13) is identical to (7 .1-49) for the Bragg grating. This is gratifying inasmuch as the periodic medium at hand is an extended Bragg grating with an infinite number of segments. Since Mo is a 2 x 2 matrix, it has two eigenvalues. Hence, only two of the mul- tiple solutions of (7 .2-13) are independent. Since the cas-I. function is even, the two solutions within the interval 7f, 7f have equal magnitudes and opposite signs. They correspond to Bloch modes traveling in the forward and backward directions. Other solutions obtained by adding multiples of 27f are not independent since they are irrelevant to the phase factor e- j i1> . The associated eigenvectors of Mo are therefore u.( +) o ex u.( - ) o r t e-j<l> 1 t* , (7.2-14) as can be ascertained by operating on the right-hand side of (7 .2-14) with the Mo matrix; the result is again the right-hand side of (7.2-14) to within a constant. The periodic function PK z associated with the Bloch wave can be determined by initial layer in the unit cell is a homogeneous medium of refractive index nl and width d 1 , then the wave at distance z into this layer is PK z e- jKz (7.2-15) Using (7 .2-14) and (7 .2-11), (7 .2-15) then provides PK z ex re-jnlkoz + e- jKA 1 ejnlkoz e jKz , 0 < z < d I - (7 .2-16) The waves in (7 .2-16) may be propagated further into the subsequent layers within the cell by using the appropriate M matrices. Dispersion Relation and Photonic Band Structure The dispersion relation is an equation relating the Bloch wavenumber K and the angular frequency w. Equation (7.2-13), which provides the eigenvalues exp jif> of the unit-cell matrix, is the progenitor of the dispersion relation for the ID periodic medium. The phase if> K A is proportional to K, and t t w is related to w through the phase delay associated with propagation through the unit cell, so that (7 .2- 13), written in the form cas 9 Re 1 tw II) (7 .2-17) Dispersion Relation
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 271 is the w K dispersion relation. Here, 9 21r A is the fundamental spatial frequency of the periodic medium. The function cos 21r K 9 is a periodic function of K of period 9 21r A, so that for a given w, (7.2-17) has multiple solutions. However, solutions separated by the pe- riod 9 are not independent since they lead to identical Bloch waves. It is therefore com- mon to limit the domain of the dispersion relation to a period with values of K in the interval 9 2, 9 2 or 1r A,1r A, which is the Brillouin zone. This corresponds precisely to limiting the phase cI> to the interval 1r, 1r . Also, since cos 21r K 9 is an even function of K, at each value w there are two independent values of K of equal magnitudes and opposite signs within the Brillouin zone. They correspond to independent Bloch modes traveling in the forward and backward directions. The dispersion relation exhibits multiple spectral bands classified into two regimes: . Propagation regime. Spectral bands within which K is real correspond to prop- agating modes. Defined by the condition Re 1 t w < 1, these bands are numbered, 1, 2, . .., starting with the lowest-frequency band. . Photonic bandgap regime. Spectral bands within which K is complex corre- spond to evanescent waves that are rapidly attenuated. Defined by the condition Re 1 t w > 1, these bands correspond to the stop bands of the diffraction grating discussed in Sec. 7.1C. They are also called photonic bandgaps (PBG) or forbidden gaps since propagating modes do not exist. The dispersion relation is often plotted with K measured in units of 9 21r A, the fundamental spatial frequency of the periodic structure, whereas w is measured in units of the Bragg frequency w 1rC A, where C Co nand n is the average refractive index of the periodic medium. The ratio w 9 2 c, which is the slope of the dispersion relation w cK for propagation in a homogeneous medium with the average refractive index. EXAMPLE 7.2-1. Periodic Stack of Partially Reflective Mirrors. The dispersion relation for a wave traveling along the axis of a periodic stack of identical partially reflective lossless mirrors with intensity reflectance Irl 2 and intensity transmittance Itl 2 1 Ir1 2 , separated by a distance A, is determined directly from Example 7.1-8. Using the results obtained there, namely t Itle jcp with c.p nkoA (wlc)A, in conjunction with (7.2-13), provides the dispersion relation K cas 27r 9 1 W 7r Wp, , (7.2-18) where 9 27r 1 A, and w C7r / A is the Bragg frequency. This result is plotted in Fig. 7.2-4. W -- -. . . .. . -. -- - -- .- -- - A - -- - -- W'B ::. Photonic badgap ... .. .... .. _. -.. .- Photonic bandgap":: <... .. . . .. - . .- .. .- . . . ..- . z 3 2w'B .. .- .. .. . -. .. .. . .. . -. . 9. . -. . .. . .- . .- .- .. 2 .. .. .. .. . Figure 7 .2-4 Dispersion diagram of a pe- riodic set of mirrors, each with intensity transmittance Itl 2 0.5, separated by a distance A. Here, w 7rcl A and 9 27r 1 A. The dotted straight lines represent propagation in a homogeneous medium for which w 1 K w (g 12) c. o Photonic bandgap........ ........ -g/2 0 w.c K "-y....... -. .. . -. . . . . 1 . .... K g/2
272 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS The photonic bandgaps, which correspond to frequency regions where (7.2-18) does not admit a real solution, are centered at W'B, 2w'B, .... These frequency regions do not permit propagating modes; rather, they correspond to the stop bands that exhibit unity reflectance in Fig. 7 .1-1 O. In this system, the onset of the lowest photonic bandgap is at w o. EXAMPLE 7.2-2. Alternating Dielectric Layers. A periodic medium comprises alternating dielectric layers of refractive indexes nl and n2, with corresponding widths d 1 and d 2, and period A d 1 + d 2. This system is the dielectric Bragg grating described in Example 7.1-9 with N 00. For a wave traveling along the axis of periodicity, Re{l/t} Re{A} is given by (7.1-61). Using the relations 'PI + 'P2 k o (nl d l + n2 d 2) 1fWIW'B and 'PI 'P2 (,1fWI W 'B, where W'B ( Co In) ( 1f I A) is the Bragg frequency, n ( n 1 d 1 + n2 d 2) I A is the average refractive index and (, (nIdI n2 d 2)/(n 1 d 1 + n2 d 2), (7.2-13) provides the dispersion relation K COS 27r 9 1 W COS 7r t 12 t 21 W'B IT1212 cos 7f(, W Wp, , (7.2-19) where t12t21 4nln2/(nl + n2)2 and IT1212 (n2 nl)2/(nl + n2)2. An example of this dispersion relation is plotted in Fig. 7.2-5 for dielectric materials with nl 1.5 and n2 3.5, and d 1 d 2 . As with the periodic stack of partially reflective mirrors considered in Example 7.2-1, the photonic bandgaps are centered at the frequencies W'B and its multiples, and occur at either the center of the Brillouin zone (K 0) or at its edge (K 9 12). In this case, however, the frequency region surrounding W 0 admits propagating modes instead of a forbidden gap. Dielectric materials with lower contrast have bandgaps of smaller width, but the bandgaps exist no matter how small the contrast. . .. .. .. .. -. -. .. . n} n 2 / Aj.-- W . . . . .- .. .. . .. .. .- . . -.. .. -. ... .::: :::. Photonic bandgap ... ... .. -. .... "- --. .-.. . -. .. . - .. e. .. -. . - .. .. .. -. .. III d l d 2 z 2w'B -. " -. . - . ., ..... WIrC K ..: ..- " ... ... .. .. .. ..... . .. . o -g/2 o K g/2 Figure 7.2-5 Dispersion diagram of an alternating-layer periodic dielectric medium with nl 1.5 and n2 3.5, and d 1 d 2 . Here, W'B 1fC o l An and 9 21f I A. The dotted straight lines represent propagation in a homogeneous medium of mean refractive index n, so that W I K W'B I (g 12) coin c. ,. -. - - - - - . - ,. . .- .- W'B .:. Photonic bandgap .. ... .. .. . -. . -. .. -. .. .. . -. . .. Phase and Group Velocities The propagation constant K corresponds to a phase velocity w K and an effective refractive index neff coK w. The group velocity v dw dK, which governs pulse propagation in the medium, is associated with an effective group index Neff codK dw (see Sec.5.6). These indexes can be determined at any point on the w K dispersion curve by finding the slope dw dK, and the ratio w K, i.e., the slope of a line joining the point with the origin. Figure 7 .2-6 is a schematic illustration of a dispersion relation of an alternating-layer dielectric periodic medium, together with the effective index and group index, at frequencies extending over two photonic bands with a photonic bandgap in-between. At low frequencies within the first photonic band, neff is approximately equal to the average refractive index n. This is expected since at long wavelengths the material
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 273 . . . . . . . . . . . . . . . w - . . . . . . - . . . ,. . . . . . . . WtcrCK ,... . . . 41 - - .. Photonic bandgap . . . . . . . - . . . . . . . At Iii"""" . . . . " . . . I o o g/2 g K - n neff - n Neff Figure 7 .2-6 Frequency dependence of the effective refractive index neff, which determines the phase velocity, and the effective group index Neff, which determines the group velocity. behaves as a homogeneous medium with the average refractive index. With increase of frequency, neff increases above ii, reaching its highest value at the band edge. At the bottom of the second band, neff is smaller than fi but increases at higher frequencies, approaching ii in the middle of the band. This drop of neff from a value above average just below the bandgap to a value below average just above the bandgap is attributed to the significantly different spatial distributions of the corresponding Bloch modes, which are orthogonal. The mode at the top of the lower band, has greater energy in the dielectric layers with the higher refractive index, so that its effective index is greater than the average. For the mode at the bottom of the upper band, greater energy is localized in the layers with the lower refractive index, and the effective index is therefore lower than the average. The frequency dependence of the effective group index follows a different pattern, as shown in Fig. 7 .2-6. As the edges of the bandgap are approached, from below or above, this index increases substantially, so that the group velocity is much smaller, i.e., optical pulses are very slows near the edges of the bandgap. Off-Axis Dispersion Relation and Band Structure The dispersion relation for off-axis waves may be determined by using the same equa- tion, cos K A Re 1 t w , where Re 1 t w now depends on the angles of incidence within the layers of each segment and on the state of polarization (TE or TM). For example, for a periodic medium made of alternating dielectric layers, Re 1 t w takes the more general form in (7.1-62). Since the same transverse component kx of the wavevector determines the angles of incidence at the two layers (k x n1ko sin 0 1 n2ko sin ( 2 ), it is more convenient to express the dispersion relation as a function of k x , in the form of a three-dimensional surface w w K, kx . Every value of kx yields a dispersion diagram with bands and bandgaps similar to those of Fig. 7 .2-5. A simpler representation of the w K, kx 3D surface is the projected dispersion diagram, which displays in a two-dimensional plot of the edges of the bands and bandgaps at each value of k x , for both TE and TM polarizations, as illustrated in Fig. 7 .2- 7. This figure is constructed by determining the ranges of angular frequencies over which photonic bands and bandgaps exist in the dispersion diagram for a particular value of k x , and then projecting these onto corresponding vertical lines at that value of kx in the projected dispersion diagram. The loci of all such vertical lines for the bands at different values of kx correspond to the shaded (green) areas displayed in Fig. 7.2-7; the unshaded (white) areas represent the bandgaps. In this diagram, each angle of incidence is represented by a straight line passing through the origin. For example, the incidence angle 0 1 in layer 1 corresponds to the line kx w Cl sin 0 1 , i.e., w Cl sin 0 1 k x , where Cl Co nl. The line w C1kx, called the light line corresponds to 0 1 90°. Similar lines may be drawn for
274 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS W'B .. .. . . . .. . . .. .. .. . . . .. .. TM polarization TE polarization 90 . 0 \ ,I I I I : \ ' I I , .. 90 ° . \ I I ° * .. .. () B 40 0 \ I 60 : I 40 °.: . \ I. / _ -. \ \ 20 0 I I 0 0 20° I /: \\ \ , I I I / i .. , I I. -. , \ \ I I , :* e.. \ \ ' I I . - \ I I. -.\ \ ' I I I , /.: \ \ ' I " w=c 1 k X \ \ I , \ \ , I , '/. I \ I : o \' I I 7 : o. \ \ I I ,': / : \ I I /: .., \, I I /: -.. \ \, I I /:- .-\ I I /: ...' \ I " /: . . \, \, I I I': ... -.) \, I , '.: .. \ , I ' ..... \ I /:. .. \' I I,: .- .... \, 11/:* ... .. \ "1/: .* .... \, "1: .-* ... \\ III: .* .... \ It .... . . . x n} n 2 / / IA W ___ Z 2w'B . . .. .. . .. . . . . I 0) t-+-- °2 . * . . . . . . - . . * . . . . . Figure 7.2-7 Projected dispersion diagram for an alternating-layer periodic dielectric medium with nl 1.5, n2 3.5, and d l d 2 A/2. Here, W'13 'lrC o / An and 9 2'1r / A. Photonic bands are shaded (green). The dashed lines represent fixed angles of incidence 0 1 in layer I, including the Brewster angle OB 66.8°. Points within the region bounded by the light lines W CI kx and g W C2kx represent normal-to-axis waves. . . . * ...." .. w=c2 k x 0g kx kx the incidence angles in medium 2; Fig. 7.2-7 shows only the light line W C2kx, assuming that n2 > nI, i.e., C2 < CI. Points in the region bounded by the two light lines represent normal-to- axis modes, which travel in the lateral direction by undergoing total reflection in the denser medium (medium 2). The question arises as to whether there exists a frequency range over which prop- agation is forbidden at all angles of incidence (}I and (}2 and for both polarizations? This could occur if the forbidden gaps at all values of kx between the lines kx 0 and kx W C2, and for both polarizations, were to align in such a way as to create a common or photonic bandgap. This is clearly not the case in the example in Fig. 7 .2- 7. It turns out that this is not possible; complete photonic band gaps cannot exist within ID periodic structures. However, they can occur in 2D and 3D periodic structures, as we shall see in Sec. 7.3. Indeed, there is one special case in which a photonic bandgap cannot occur at all, and that is an oblique TM wave propagating at the Brewster angle () B tan -1 n2 nl in layer 1. As shown in Fig. 7 .2- 7, the line at the Brewster angle does not pass through a gap. This is not surprising since at this angle, the reflectance of a unit cell is zero, and the forward and backward waves are uncoupled so that the collective effect that leads to total reflection is absent. c. Fourier Optics of Periodic Media The matrix analysis of periodic media presented in the previous section is applicable to layered (i.e., piecewise homogeneous) media. A more general approach, applicable for arbitrary periodic media, including continuous media, is based on a Fourier-series representation of periodic functions and conversion of the Helmholtz equation into a set of algebraic equations whose solution provides the dispersion relation and the Bloch modes. This approach can also be generalized to 2D and 3D periodic media, as will be shown in Sec. 7.3. A wave traveling along the axis of a ID periodic medium (the z axis) and polarized in the x direction is described by the generalized Helmholtz equation (7.2-2). Since 11 z is periodic with period A, it can be expanded in a Fourier series, CX) 11 z 11£ exp j£g z , (7.2-20) f -CX)
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 275 where 9 27f A is the spatial frequency (rad/mm) of the periodic structure and 11£ is the Fourier coefficient representing the £th harmonic. The impermeability 11 z is real., so that 11-/ 11£ · The periodic portion of the Bloch wave PK z in (7 .2-4) may also be expanded in a Fourier series, CX) PK Z C m exp . Jmg z (7 .2- 21 ) Tn -CX) whereupon the Bloch wave representation of the magnetic field may be written as CX) Hy Z C m exp j K + mg z . (7.2-22) Tn - CX) For brevity, the dependence of the Fourier coefficients C m on the Bloch wavenumber K is suppressed. Substituting these expansions into the Helmholtz equation (7.2-2) and equating harmonic terms of the same spatial frequency, we obtain CX) w 2 2 Cm, FmP Co K + mg K + £g 11m-P (7.2-23) F mf Cp f -CX) where m 0, ::1:1, ::1:2, . . . . The differential equation (7.2-2) has now been converted into a set of linear equa- tions (7.2-23) for the unknown Fourier coefficients C m . These equations may be cast in the form of a matrix eigenvalue problem. For each K, the eigenvalues w 2 c correspond to multiple values of w, from which the w K dispersion relation may be constructed. The eigenvectors are sets of C m coefficients, which determine the periodic function PK z of the Bloch mode for each K. Posed as an eigenvalue problem for a matrix F with elements F mP, this set of coupled equations may be solved using standard numerical techniques. Since 11m-f l1£-m' the matrix F is Hermitian, i.e., FmP FPm. Note that if we were to use the electric-field Helmholtz equation instead of the magnetic-field Helmholtz equation (7 .2- 2), we would obtain another matrix representation of the eigenvalue problem, but the matrix would be non-Hermitian, and therefore more difficult to solve. This is why we elected to work with the Helmholtz equation for the magnetic field. t Approximate Solution of the Eigenvalue Problem In (7.2-23), the harmonics of the optical wave are coupled via the harmonics of the periodic medium. An optical-wave harmonic of spatial frequency K + £g mixes with a medium harmonic of spatial frequency m £ 9 and contributes to the optical-wave hannonic of spatial frequency K + £g + m £ 9 K + mg . The conditions under which strong coupling emerges can be determined by separat- ing out the mth term in (7.2-23), which leads to K+fg nw Co 2 C m l1'rn-f Pim 110 m 0, ::1:1, ::i:2,..., (7.2-24) t It can be shown that the differential operator in the generalized Helmholtz equation, (7.0-2), is a Hermitian operator, but that for the electric field is non-Hermitian
276 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS where n 1 110 is an average refractive index of the medium. Strong coupling between the mth harmonic of the wave and other harmonics exists if the denominator in (7.2-24) is small, i.e., wn Co K + my . (7.2-25) This equation represents a resonance condition for interaction between the harmonics. It can also be regarded as a phase matching condition. Figure 7.2-8 is a plot of (7.2-25) as an equality. For each value of m, the w K relation is a V-shaped curve. The intersection points of these curves represent common values of wand K at which (7 .2- 25) is simultaneously satisfied for two harmonics. The intersections between the m 0 curve (dashed) and the curves for m 1, m 2, . . ., are marked by filled circles; they correspond to the lowest-order bandgaps 1, 2, an integer multiple of the Bragg frequency WT> Co n 9 2 or WT> 27r 1/p, Co n 2A. This corresponds to the Bragg wavelength AT> 2A in the medium, and therefore to total reflection. Unmarked intersections in Fig. 7.2-8 are not independent since each of these has the same W as a marked intersection, and a value of K differing by a reciprocal lattice constant 9 . w m= 1 . . . . . . . . . . . ' . .. - -- ' . . . . . Figure 7.2-8 Plot of (7.2-25), as an equality, for various values of m. The m 0 curve is indicated as dashed. Strong coupling between the harmonics of the optical wave and those of the medium occurs at the intersection points 1, 2,. . . , which correspond to the lowest- order bandgaps. m=-3 - W1J m=O " , . . . . m-2 o 19 2 g 19 2 2g K The lowest-order bandgap occurs at the intersection of the m 0 and m 1 curves (point 1 in Fig. 7.2-8). In this case, only the coefficients Co and C- 1 are strongly coupled, so that (7.2-24) yields two coupled equations Co 111 'K 110 w 2 n 2 gK Co (7 .2- 26) C- 1 11i K K 9 110 w 2 n 2 C K 2 Co, 9 (7.2-27) where 11-1 TJi · These equations are self-consistent if 2 11 1 K 2 K 116 9 2 -2 2 n w 2 Co K 2 -2 2 n w 2 Co K 9 2 . (7.2-28) Dispersion Relation A plot of this equation (Fig. 7 .2-9) yields the w K dispersion relation near the edge of the bandgap, where the equation is valid. For K W:f: Wp, 1:f:: 111 110, (7.2-29)
7.2 ONE-DIMENSIONAL PHOTONIC CRYSTALS 277 representing the edges of the first photonic bandgap. The center of the bandgap is at the Bragg frequency Wp, Co fi g 2 'If A Co fi . The ratio of the gap width to the midgap frequency, which is called the gap-midgap ratio, grows with increasing impermeability contrast ratio 111 110. W ---.... t ".... , " , , I " ',I , ,,, " '4IIIt ' ...... . .. I ...., , I " I , I " , I Bandgap 2 W+ ........ " I , I " I " I ....., I ... W'B nnnnnn.non..- Bandgap 1 W . .... - I ....,.... I I', I I " I " I ',I,' I 'I" I " I ,," I , I ' , , !g g K Figure 7.2-9 Dispersion relation in the vicinity of photonic bandgaps. A similar procedure can be followed to determine the spectral width of higher-order bandgaps. The width of the mth bandgap is determined by a formula identical to (7 .2- 29), but the ratio 11m 110 replaces Tll Tlo, so that higher bandgaps are governed by higher spatial harmonics of the periodic function 11 z . Off-Axis Waves The dispersion relation for off-axis waves may be determined by use of the same Fourier expansion technique. For a TM-polarized off- axis wave traveling in an ar- bitrary direction in the x z plane, the Helmholtz equation is given by (7.2-3). The Bloch wave is a generalization of (7.2-22) obtained via (7.2-6), 00 Hy Z C m exp j K + mg z exp j kxx . (7.2-30) m -00 C ing out calculations similar to the on-axis case, leads to the following set of algebraic equations for the C m coefficients: 00 w 2 2 Cm, Fmt Co K +£g K + mg + k; 11m-t. (7.2-31) F mi Ct i -00 Equation (7.2-31) is a generalization of (7.2-23) for the off-axis wave. The dispersion relation may be determined by solving this matrix eigenvalue problem for the set of frequencies w associated with each pair of values of K and kx. .. D. Boundaries Between Periodic and Homogeneous Media The study of light wave propagation in periodic media has so far been limited to deter- mining the dispersion relation and its associated band structure, as well as estimating the phase and group velocity of such waves. By definition, the periodic medium extend indefinitely in all directions. The next step is to examine reflection and transmission at boundaries between periodic and homogeneous media. We first examine reflection from a single boundary and subsequently consider a slab of periodic medium embed- ded in a homogeneous medium. Other configurations made of homogeneous structures such slabs or holes embedded in extended periodic media are described in Sec. 9.4 and Sec. 10.4D.
278 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS Omnidirectional Reflection at a Single Boundary We examine the reflection and transmission of an optical wave at the boundary between a semi-infinite homogeneous medium and a semi-infinite one-dimensional periodic medium, as portrayed in Fig. 7 .2-1 O. We demonstrate that, under certain conditions and within a specified range of angular frequencies, the periodic medium acts as a perfect mirror, totally reflecting waves incident from any direction and with any polarization! Wave transmission and reflection at the boundary between two media is governed by the phase-matching condition. At the boundary between two homogeneous media, for example, the transverse components of the wavevector kx must be the same on both sides of the boundary. Since kx k sin () W Co n sin (), this condition means that the product n sin () is invariant. This is the origin of Snel]'s law of refraction, as explained in Sec. 2.4A. Similarly, for a wave incident from a homogeneous medium into a one-dimensional periodic medium, kx must remain the same. Thus, if the incident wave has angular frequency wand angle of incidence (), we have kx w Co n sin (), where n is the refractive index of the homogeneous medium. Knowing kx and w. we can use the dispersion relation w w K, kx for the periodic medium at the appropriate polariza- tion to determine the Bloch wavenumber K. If the angular frequency w lies within a forbidden gap at this value of k x , the incident wave will not propagate into the periodic medium but will, instead, be totally reflected. This process is repeated for all frequencies, angles of incidence, and polarizations of the incident wave. We now consider the possibility that the boundary acts as an omnidirectional reflec- tor (a perfect mirror). For this purpose, we use the projected dispersion diagram, which displays the bandgaps for each value of k x , as illustrated in the example provided in Fig. 7 .2-10. On the same diagram, we delineate by a red dashed line the w kx region that can be accessed by waves entering from the homogeneous medium. This region is defined by the equation kx w Co n sin (), which dictates that kx < w Co n or w > Co n k x ; it is thus bounded by the line w Co n k x , or w W13 fi n kx 9 2 , known as the light line. This line corresponds to an angle () 90° in the surrounding medium. Figure 7 .2-1 0 reproduces Fig. 7 .2- 7 with the light lines added, and the permissible w kx regions within the light lines highlighted. Waves incident from the homogeneous medium at all angles, and both polarizations, are represented by points within this region; points outside this region are not accessible by waves incident from the homo- geneous medium regardless of their angle of incidence or polarization. The spectral band bounded by the angular frequencies WI and W2, as defined in Fig. 7 .2-10, is of particular interest inasmuch as all w kx points lying in this band are within a photonic bandgap. In this spectral band, therefore, no incident wave, regardless of its angle or polarization, can be matched with a propagating wave in the periodic medium the boundary then acts as a perfect omnidirectional reflector. Also illustrated in Fig. 7 .2-1 0 is a second spectral band, at higher angular frequencies, that behaves in the same way. Slab of Periodic Material in a Homogeneous Medium A slab of I D periodic material embedded in a homogeneous medium is nothing but a 1 D Bragg grating with a finite number of segments. Reflection and transmission from the Bragg grating has already been examined in Sec. 7.1 C. One would expect that a Bragg grating with a large, but finite, number of seg- ments N captures the basic properties of a periodic medium made of the same unit cell. This is in fact the case since the passbands and stop bands of the grating are mathematically identical to the photonic bands and bandgaps of the extended periodic medium. However, the spectral transmittance and reflectance of the Bragg grating, which exhibit oscillatory properties sensitive to the size of the grating and the presence of its boundaries, do not have their counterparts in the extended periodic structure.
7.3 TWQ- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS 279 o g TM polarization " . . . . . ------- . . . . TE polarization / --------!WI=ICok x . . . . . x In] 7 2 I A j.- W . . . . . . . . . , . . . . n 2w'B () z . . . . . . . , . . -Id 1 -+- J.- d 2 . . . . . . . . . . . W'B , " -..- - - ...,...... -.: W 2 . * . . · I · -- ---: Wt . . . . . . . . ., . . . . . . . . . . . . kx o kx g Figure 7.2-10 Projected dispersion di- agram for an alternating-layer dielectric medium with ni 1.5, n2 3.5, and d 1 d 2 A/2. The dotted lines (red) are light lines for a homogeneous medium with refractive index n 1. In the spectral band between WI and W2, the medium acts as a perfect omnidirectional reflector for all polarizations. A similar band is shown at higher angular frequencies. . . . . . . . . . . . . . . . . . . . ". . . . . . Likewise, the phase and group velocities and the associated effective refractive in- dexes determined from the dispersion relation in the extended periodic medium do not have direct counterparts in the finite-size Bragg grating. Nevertheless, such parameters can be defined for a grating by determining the complex amplitude transmittance t w and matching it with an effective homogeneous medium of the same total thickness d such that arg tN w Co neff d . An effective group index Neff neff+wdneff dw is then determined [see (5.6-2)]. The dependence of these effective indexes on frequency is different from that shown in Fig. 7 .2-6 for the extended periodic medium in that it exhibits oscillations within the passbands. However, for sufficiently large N, say greater than 1 00, these oscillations are washed out and the effective indexes become nearly the same as those of the extended periodic medium. Another configuration of interest is a slab of homogeneous medium embedded in a periodic medium. In this configuration, the light may be trapped in the slab by omnidirectional reflection from the surrounding periodic medium, so that the slab becomes an optical waveguide. This configuration is discussed in Sec. 8.4. 7.3 TWO- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS The concepts introduced in Sec. 7.2 for the study of optical-wave propagation in ID periodic media can be readily generalized to 2D and 3D structures. These include Bloch waves as the modes of the periodic medium and w K dispersion relations with photonic bands and bandgaps. In contrast to ID structures, 2D photonic crystals have 2D-complete photonic bandgaps, i.e., common bandgaps for waves of both polarization traveling in any direction in the plane of periodicity. However, 3D-complete photonic bandgaps, i.e., common bandgaps for all directions and polarizations, can be achieved only in 3D photonic crystals. The mathematical treatment of 2D and 3D periodic media is more elaborate and the visualization of the dispersion diagrams is more difficult because of the additional degrees of freedom involved, but the concepts are essentially the same as those encountered for 1 D periodic media. This section begins with a simple treatment of 2D structures followed by a more detailed 3D treatment.
280 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS A. Two-Dimensional Photonic Crystals 2D Periodic Structures Consider a 2D periodic structure such as a set of identical parallel rods, tubes, or veins embedded in a homogeneous host medium [Fig. 7.3-1(a)] and organized at the points of a rectangular lattice, as illustrated in Fig. 7 .3-1 (b). The impermeability 11 x, y Eo E x, y is periodic in the transverse directions, x and y, and uniform in the axial direction z. If a1 and a2 are the periods in the x and y directions, then 11 x, Y satisfies the translational symmetry relation 11 x + m1 a 1, Y + m2 a 2 11 x, Y , (7.3-1) for all integers mI and m2. This periodic function is represented as a 2D Fourier series, CX) CX) 11 x,y 11£1,£2 exp j£1g IX exp j£2g 2 X , (7.3-2) i l -CX) £2 -CX) where 9 I 27f aI and 9 2 27r a2 are fundamental spatial frequencies (radians/mm) in the x and y directions, and £lg 1 and £2g 2 are their harmonics. The coefficients 11£1'£2 depend on the actual profile of the periodic function, e.g., the size of the rods. The 2D Fourier transform of the periodic function is composed of points (delta functions) on a rectilinear lattice, as shown in Fig. 7.3-1 (c). This Fourier-domain lattice is known to solid-state physicists as the reciprocal lattice. at are the optical modes of a medium with such symmetry? The answer is a simple generalization of the 1 D case given in (7 .2-4). For waves traveling in a direction parallel to the x y plane, the modes are 2D Bloch waves, u x,y PKx,Ky x, Y exp jKxx exp jKyy, (7.3-3) where PKx,Ky x, Y is a periodic function with the same periods as the medium. The wave is specified by a pair of Bloch wavenumbers K x, Ky. Another wave with Bloch wavenumbers Kx + g1, Ky + g2 is not a new mode. As shown in Fig. 7.3-1(c) a complete set of modes in the Fourier plane has Bloch wavenumbers located at points in a rectangle defined by 9 I 2 < K x 9 1 2 and 9 2 2 < K y 9 2 2, which is the first Brillouin zone. Other symmetries may be used to reduce the set of independent Bloch wavevectors within the Brillouin zone. When all symmetries are included, the result is an area called the irreducible Brillouin zone. For example, the rotational symmetry inherent in the square lattice results in an irreducible Brillouin zone in the form of a triangle, as shown in Fig 7.3-1(d). 2D Skew-Periodic Structures An example of another class of 2D periodic structures is a set of parallel cylindrical holes placed at the points of a triangular lattice, as illustrated in Fig. 7.3-2(a). Since the lattice points are skewed (not aligned with x and y axis), we use two primitive vectors aI and a2 [Fig. 7.3-2(b)] to generate the lattice via the lattice vector R mIaI + m2 a 2, where mI and m2 are integers. We also define a position vector r T x, Y so that the periodic function E r T E x, y satisfies the translational symmetry relation E r T + R E r T (the subscript "T" indicates transverse). The 2D Fourier series of such a function is a set of points on a reciprocal lattice defined by the vectors gI and g2, which are orthogonal to a1 and a2, respectively, and have magnitudes 9 1 27f a1 sin 0 and 9 2 27r a2 sin 0, where 0 is the angle between a1 and a2. The 2D reciprocal lattice is also a triangular lattice generated by
7.3 TWO- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS 281 y ky .. ... .. .. , .. .. .. I' .. .. .. , .. , .. ... .. ... , ... I' .. .. .. v - ,.-.... M C"-I .. ... ... .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. ... .. , .. , , , .. .. .. .. .. , , , kx x , .. , , I' , .. .. I' .. I' , , ... .. .. .. ... .. .. .. .. .. .. -:(.- x r - ". .:- . - I' , , I' .. , , , , , , .. , x ( a) 20 periodic structure at - (b) Lattice g2 ( c) Reciprocal lattice y (d) Irreducible Brillouin zone Figure 7.3-1 (a) A 20 periodic structure comprising parallel rods. (b) The rectangular lattice at which the rods are placed. (c) The 20 Fourier transform of the lattice points is another set of points forming a reciprocal lattice with periods 9 1 27r / al and 9 2 27r / a2. The shaded (yellow) area is the Brillouin zone. (d) For a square lattice (al a2 a), the irreducible Brillouin zone is the triangle rMX. { I \ , . \ \ I I . I x ky y , M K \ \ \ \ \ \ \ \ I I , , I , , , \ \ \ r \ , ( I I . y (d) Irreducible Brillouin zone Figure 7.3-2 (a) A 20 periodic structure comprising parallel cylindrical holes. (b) The triangular lattice at which the holes are placed. In this diagram the magnitudes al a2 a and () 120 0 . (c) Reciprocal lattice; the shaded (yellow) area is the Brillouin zone, a hexagon. (d) The irreducible Brillouin zone is the triangle rMK. x ( a) 20 periodic structure (b) Lattice ( c) Reciprocal lattice the vector G fIg l +£2g2, where f 1 and £2 are integers, as illustrated in Fig. 7.3-2(c). For waves traveling in a direction parallel to the x y plane, the Bloch modes are U r T PK r T exp j K T · r T , (7 .3-4 ) where K T Kx, Ky is the Bloch wavevector and PKT r T is a 2D periodic function on the same lattice. Two Bloch modes with Bloch wavevectors K T and K T + G are equivalent. To cover a complete set of Bloch wavevectors, we therefore need only consider vectors within the Brillouin zone shown in Fig. 7.3-2(c). The dispersion relation can be determined by ensuring that the Bloch wave in (7 .3- 3) or (7.3-4) satisfies the generalized Helmholtz equation. The calculations are facilitated by use of a Fourier series approach, as was done in the ID case and as will be described (in a more general form) in the 3D case. EXAMPLE 7.3-1. Cylindrical Holes on a Triangular Lattice. A 20 photonic crystal com- prises a homogeneous medium (n 3.6) with air-filled cylindrical holes of radius 0.48a organized at the points of a triangular lattice with lattice constant a. The calculated dispersion relation, shown in Fig. 7.3-3, forTE and TM waves traveling in the plane of periodicity (k z 0) exhibits a 2D-complete
282 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS photonic bandgap at frequencies near the angular frequency Wo == 7rC o / a. t As in the 1 D case, the gap can be made wider by use of materials with greater refractive-index contrast. Indeed, most geometries exhibit photonic bandgaps if the materials used have sufficiently high contrast. o: TE I' t.- ' <! /'1 M K (f) o cW TM \ :;g:s;;.' I I /<. --- w KT K KT r Figure 7.3-3 Calculated band structure of a 2D photonic crystal made of a homogeneous medium (n == 3.6) with air-filled cylindrical holes of radius O.48a organized at the points of a triangular lattice with lattice constant a. The abscissa spans Bloch wavevectors defined by points on the periphery of the irreducible Brillouin zone, the rMK triangle. The ordinate is plotted in units of Wo == 7rC o / a. The wave travels in the plane of periodicity and has TE polarization (left) and TM polarization (right). This photonic-crystal structure finds use as a "holey" optical fiber, which has a number of salutary properties (see Sec. 9.4). For an oblique wave traveling at an angle with respect to the x-y plane, the Bloch wave in (7.3-4) becomes U(r T ) ==PK(rT)exp(-jK T .rT)exp(-jkzz), (7.3-5) where k z is a constant. The band structure then takes the form of a set of surfaces of w == w(K T , k z ). A 3D-complete photonic bandgap is a range of frequencies w crossed by none of these surfaces, i.e., values of w that are not obtained by any combination of real K T and k z . While a 2D-complete photonic bandgap exists for k z == 0, as illustrated by the example in Fig. 7.3-3, a photonic bandgap for all off-axis waves is not attainable in 2D periodic structures. B. Three-Dimensional Photonic Crystals Crystal Structure A 3D photonic crystal is generated by placement of duplicates of a basic dielectric structure, such as a sphere or a cube, at points of a 3D lattice generated by the lattice vectors R == mial +m2a2+m3a3, where ml, m2, and m3 are integers, and aI, a2, and t See S. G. Johnson and J. D. Joannopoulos, Block-Iterative Frequency-Domain Methods for Maxwell's Equations in a Planewave Basis, Optics Express, vol. 8, pp. 173-190,2001.
7.3 TWO- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS 283 a3 are primitive vectors defining the lattice unit cell. The overall structure is periodic and its physical properties, such as the permittivity E r and the impermeability 11 r Eo E r , are invariant to translation by R, e.g., 11 r , (7.3-6) 11r+R for all positions r. This periodic functions may therefore be expanded in a 3D Fourier . senes, 11 G exp j G · r , (7.3-7) 11 r G where G £lgl + £2g2 + £3g3 is a vector defined by the primitive vectors gl, g2, and g3 of another lattice, the reciprocal lattice, and £1, £2, and £3, are integers. The g vectors are related to the a vectors via , g3 27r a1 x a2 al · a2 x a3 , (7.3-8) gl 27r a2 x a3 a 1 · a2 x a3 27r a3 x a 1 al · a2 x a3 , g2 so that gl · a1 27r , gl · a2 0 , and gl · a3 0, Le., gl is orthogonal to a2 and a3 aQd its length is inversely proportional to a1. Similar properties apply to g2 and g3. It can also be shown that G · R 21r · If aI, a2, and a3 are mutually orthogonal, then gl, g2, and g3 are also mutually orthogonal and the magnitudes 9 1 27r aI, 9 2 21r a2, and 9 3 27r a3 are the spatial frequencies associated with the periodicities in the three directions, respectively. An example of a 3D crystal lattice and its corresponding reciprocal lattice is shown in Fig. 7 .3-4. " ,. -- - --- z k z L z , . -- -- ... , I ... --......... _----- -.. --------.1\ ... "'y . ------_. ------- I : .7 ---fII\---'" : I I I I I I I I I I I I I I I I I I I I I . I . I I I I I I I I I I I I I #' -_ I " - "---- I ---", k kx .''" y WK &,: , -- -- , 0- , - - '0 T - : - ". "' \ - .,- I: 0 I I I I . I - . -- I .\ I Y X - " . I . h I X _L - -- - - --- " ---- I ... (a) 3D periodic structure (b) Lattice ( c) Reciprocal lattice (d) Irreducible Brillouin zone Figure 7.3-4 (a) A 3D periodic structure comprising dielectric spheres. (b) The spheres are placed at the points of a diamond (face-centered cubic) lattice for which al (aj 2)(x + y), a2 (aj 2)(y+z), and a3 (aj 2)(x+z), where a is the lattice constant. (c) The corresponding reciprocal lattice is a body-centered cubic lattice with a Brillouin zone indicated by the shaded volume, known as a Wigner-Seitz cell. (d) The irreducible Brillouin zone is the polyhedron whose comer points are marked by the crystallographic symbols rXULKW. Bloch Modes The modes of a 3D periodic medium are waves that maintain their shape upon trans- lation by a lattice vector R, changing only by a multiplicative constant of unity mag- nitude. These modes have the Bloch form PK r exp jK. r where PK r is a 3D periodic function, with the periodicity described by the same lattice vector R; K is the Bloch wavevector; and e is a unit vector in the direction of polarization. The Bloch mode is a traveling plane wave exp jK. r modulated by a periodic function PK r ·
284 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS Translation by R results in multiplication by a phase factor exp jK. R , which depends on K. Two modes with Bloch wavevectors K and K' K + G are equivalent since exp j K' · R exp j K. R , i.e., translation by R is equivalent to multiplication by the same phase factor. This is because exp jG. R exp j27r 1. Therefore, for the complete specification of all modes, we need only consider values of K within a finite volume of the reciprocal lattice, the Brillouin zone. The Brillouin zone is the volume of points that are closer to one specific reciprocal lattice point (the origin of the zone, denoted r) than to any other lattice point. Other symmetries of the lattice permit further reduction of that volume to the irreducible Brillouin zone, as illustrated by the example in Fig. 7.3-4. Photonic Band Structure To determine the w K dispersion relation for a 3D periodic medium, we begin with the eigenvalue problem described by the generalized Helmholtz equation (7.0-2). One approach for solving this problem is to generalize the Fourier method that was intro- duced in Sec. 7.2C for ID periodic structures. By expanding the periodic functions 11 rand PK r in Fourier series, the differential equation (7.0-2) is converted into a set of algebraic equations leading to a matrix eigenvalue problem that can be solved numerically using matrix methods. As discussed at the end of Sec. 7.2C, we work with the magnetic field to ensure Hermiticity of the matrix representation. Expanding the periodic function PK r in the Bloch wave into a 3D Fourier series PK r GG exp j G · r , (7.3-9) G we write the magnetic field vector in the Bloch form Hr PK r exp j K · r e GG exp j K + G · r e. (7.3-10) G . For notational simplicity, the dependence of the Fourier coefficients GG on the Bloch wavevector K is not explicitly indicated. Substituting (7 .3- 7) and (7 .3-1 0) into (7 .0- 2), using the relation V x exp jK. r e j K x e exp jK. r , and equating harmonic terms of the same spatial frequency yields w 2 2 GG e · Co (7.3-11) K + G x K + G' x e 11G-G' GG' G' Forming a dot product with e on both sides, and using the vector identity A. B x C B x A · C leads to w 2 C G FGG ' 2 ' Co K+G x e · K+G' x e 11G-G'. (7.3-12) F GG' GG' G' The Helmholtz differential equation has now been converted into a set of linear equations for the Fourier coefficients GG · Since 11 z is real, 11G-G' 11' -G' and the matrix F GG' is Hermitian. Hence, (7.3-12) represents an eigenvalue problem for a Hermitian matrix. For each Bloch wavevector K, the eigenvalues w 2 c provide multiple values of w, which are used to construct the w K diagram and the photonic band structure. The eigenvectors GG determine the periodic function PK r of the Bloch wave.
7.3 TWO- AND THREE-DIMENSIONAL PHOTONIC CRYSTALS 285 Examples Spherical holes on a diamond lattice. An example of a 3D photonic crystal that has been shown to exhibit a complete 3D photonic bandgap comprises air spheres embedded in a high-index material at the points of a diamond lattice (see Fig. 7.3-4). The radius of the air spheres is sufficiently large so that the spheres overlap, thereby creating intersecting veins. The calculated band structure shown in Fig. 7 .3-5 has a rel- atively wide complete 3D photonic bandgap between the two lowest bands. t Photonic crystals using spherical holes in silicon have been fabricated by growing silicon inside the voids of an opal template of close-packed silica spheres that are connected by small "necks" formed during sintering, followed by removal of the silica template. t w Complete photonic bandgap wo o X U L r K X w Figure 7.3-5 Calculated band structure of a 3D photonic crystal with a diamond lattice of lattice constant a. The structure comprises air spheres of radius 0.325 a embedded in a homogeneous material of refractive index n 3.6. The gap extends from approxi- K mately Wo 1fC o / a to 1.32 W0 0 Yablanovite. The first experimental observation of a 3D complete photonic bandgap was made by Eli Yablonovitch in 1991 using a variant of the diamond lattice structure, now known as the Yablonovite. This slanted-pore structure is fabricated by drilling a periodic array of cylindrical holes at specified angles in a dielectric slab. Three holes are drilled at each point of a 20 triangular lattice at the surface of the slab; the directions of the holes are parallel to three of the axes of the diamond lattice, as shown in Fig. 7.3-6(a). This structure has a complete gap with a gap-midgap ratio of 0.19 when the refractive index is n 3.6. Woodpile. Another 3D photonic-crystal structure, which is simpler to fabricate, is made of a 10 periodic stack of alternating layers, each of which is itself a 20 photonic crystal. For example, the woodpile structure illustrated in Fig. 7 .3-6(b) uses layers of parallel logs with a stacking sequence that repeats itself every four layers. The orientation of the logs in adjacent layers is rotated 90°, and the logs are shifted by half the pitch every two layers. The resulting structure has a face-centered-tetragonal lattice symmetry. Fabricated using silicon technology, at a minimum feature size of 180 nm this structure manifested a complete 30 photonic bandgap in the wavelength range.A 1.35 1.95 /-L m . * t See s. G. Johnson and J. D. Joannopoulos, Block-Iterative Frequency-Domain Methods for Maxwell's Equations in a Planewave Basis, Optics Express, vol. 8, pp. 173-190, 2001. t See A. Blanco et aI., Large-Scale Synthesis of a Silicon Photonic Crystal with a Complete Three-dimensional Bandgap Near 1.5 Micrometres, Nature, vol. 405, pp. 4370, 2000. * See J. G. Fleming and S.-Y. Lin, Three-Dimensional Photonic Crystal with a Stop Band from 1.35 to 1.95 J..Lm, Optics Letters, vol. 24, pp. 49-51, 1999.
286 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS (a) Yablanovite ---.... b '- (b) Woodpile (c) Holes and poles Figure 7.3-6 (a) The Yablonovite photonic crystal is fabricated by drilling cylindrical holes through a dielectric slab. At each point of a 2D triangular lattice at the surface, three holes are drilled along directions that make an angle of 35° with the normal and are separated azimuthally by 120°. (b) The woodpile photonic crystal comprises alternating layers of parallel rods, with adjacent layers oriented at 90°. (c) The holes-and-poles structure is made of alternating layers of 2D periodic structures: a layer of parallel cylindrical holes on a hexagonal lattice, followed by a layer of parallel rods lined up to fit between the holes. Holes and poles. Yet another example is the holes-and-poles structure illustrated in Fig. 7.3-6(c). Here, two complementary types of 2D-periodic photonic-crystal slabs are used: dielectric rods in air and air holes in a dielectric. Fabricated in silicon, this structure exhibited a stop-band for all tilt angles in the wavelength range A == 1.15-1.6 JLm t Both the holes-and-poles structure and the woodpile structure offer the opportunity of introducing arbitrary point defects, such as a missing hole or rod, which provide means for fabricating devices such as photonic crystal waveguides (see Sec. 8.5), photonic-crystal nano-resonators (see Sec. 10.4D), and specially controlled light emit- ters:/: (see Chapter 17). Indeed, the ability to insert a defect at will is the most valuable feature of 2D and 3D photonic structures since ID periodic media serve admirably as omnidirectional reflectors. READING LIST Books on Layered and Periodic Media s. Visnovsky, Optics in Magnetic Multilayers and Nanostructures, CRC Press, 2006. O. Stenzel, The Physics of Thin Film Optical Spectra: An Introduction, Springer-Verlag, 2005. P. Yeh, Optical Waves in Layered Media, Wiley, 2005. L. Brillouin, Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Dover, 2nd ed. 1953, reprinted 2003. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design, Marcel Dekker, 2003. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, Sec. 1.6. R. Kashyap, Fiber Bragg Gratings, Academic Press, 1999. W. C. Chew, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, 1990. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, 1985. t See M. Qi, et el., A Three-Dimensional Optical Photonic Crystal with Designed Point Defects, Nature, vol. 429, pp. 538-542, 2004. t See S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, Control of Light Emission by 3D Photonic Crystals, Science, vol. 305, pp. 227-229, 2004.
READING LIST 287 Books on Photonic Crystals K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals, CRC Press, 2006. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crys- tals: Towards Nanoscale Photonic Devices, Springer-Verlag, 2005. K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, 2nd ed. 2005. K. Busch, S. Lolkes, R. B. Wehrspohn, and H. FoIl, eds., Photonic Crystals: Advances in Design, Fabrication, and Characterization, Wiley, 2004. K. Inoue and K. Ohtaka, eds., Photonic Crystals: Physics, Fabrication and Applications, Springer- Verlag, 2004. S. Noda and T. Baba, eds., Roadmap on Photonic Crystals, Kluwer, 2003. V. Kochergin, Omnidirectional Optical Filters, Kluwer, 2003. R. E. Slusher and B. J. Eggleton, eds., Nonlinear Photonic Crystals, Springer-Verlag, 2003. S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice, Springer- Verlag, 2002. C. M. Soukoulis, ed., Photonic Band Gap Materials, Kluwer, 1996. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 1995. M. Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995. Articles Issue .on nanophotonics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 6, 2006. A. Adibi, S.- Y. Lin, and A. Scherer, eds., Photonic crystal materials and devices III, SPIE Proceed- ings, vol. 5733, 2005. A. Adibi, A. Scherer, and S.- Y. Lin, eds., Photonic crystal materials and devices II, SPIE Proceedings, vol. 5360, 2004. S. P. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, Control of Light Emission by 3D Photonic Crystals, Science, vol. 305, pp. 227-229, 2004. M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, A Three-Dimensional Optical Photonic Crystal with Designed Point Defects, Nature, vol. 429, pp. 538-542, 2004. A. Adibi, A. Scherer, and S.- Y. Lin, eds., Photonic crystal materials and devices I, SPIE Proceedings, vol. 5000, 2003. E. Yablonovitch, Photonic Crystals: Semiconductors of Light, Scientific American, vol. 285, no. 6, pp. 47-55, 2001. M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, Dielectric Omnidirectional Visible Reflector, Optics Letters, vol. 26, pp. 1197-1199, 2001. Focus issue on photonic bandgap calculations, Optics Express, vol. 8, no. 3, 2001. S. G. Johnson and J. D. Joannopoulos, Block-Iterative Frequency-Domain Methods for Maxwell's Equations in a Planewave Basis, Optics Express, vol. 8, pp. 173-190, 2001. M. Muller, R. Zentel, T. Maka, S. G. Romanov, and C. M. Sotomayor Torres, Photonic Crystal Films with High Refractive Index Contrast, Advanced Materials, vol. 12, pp. 1499-1503, 2000. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, Large-Scale Synthesis of a Silicon Photonic Crystal with a Complete Three-dimensional Bandgap Near 1.5 Micrometres, Nature, vol. 405, pp. 437-440, 2000. Y Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, A Dielectric Omnidirectional Reflector, Science, vol. 282, pp. 1679-1682, 1998. J. M. Bendickson, J. P. Dowling, and M. Scalora, Analytic Expressions for the Electromagnetic Mode Density in Finite, One-Dimensional, Photonic Band-Gap Structures, Physical Review E, vol. 53, pp.4107-4121,1996. P. R. Villeneuve and M. Piche, Photonic Bandgaps in Periodic Dielectric Structures, Progress in Quantum Electronics, vol. 18, pp. 153-200, 1994. S. John, Localization of Light, Physics Today, vol. 44, no. 5, pp. 32-40, 1991.
288 CHAPTER 7 PHOTONIC-CRYSTAL OPTICS E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Photonic Band Structures: The Face-Centered Cubic Case Employing Non-Spherical Atoms, Physical Review Letters, vol. 67, pp. 2295-2298, 1991. E. Yablonovitch and T. J. Gmitter, Photonic Band Structure: The Face-Centered-Cubic Case, Journal of the Optical Society of America A, vol. 7, pp. 1792-1800, 1990. S. John, Strong Localization of Photons in Certain Disordered Dielectric Superlattices, Physical Review Letters, vol. 58, pp. 2486-2489, 1987. E. Yablonovitch, Inhibited Spontaneous Emission in Solid-State Physics and Electronics, Physical Review Letters, vol. 58, pp. 2059-2062, 1987. PROBLEMS 7 .1-2 Beamsplitter Slab. A dielectric lossless slab of refractive index n and width d, oriented at 45° with respect to an incident beam, is used as a beamsplitter. Derive expressions for the transmittance and reflectance and sketch their spectral dependence for TE and TM polariza- tion. 7.1-3 Air Gap in Glass. Determine the transmittance through a thin planar air gap of width d A/2 in glass of refractive index n. Assume (a) normal incidence, and (b) a TE wave incident at an angle greater than the critical angle. Can the wave penetrate (tunnel) through the gap? 7 .1-4 Multilayer Device in an Unmatched Medium. The complex amplitude reflectance of a multilayer device is r m when it is placed in a medium with refractive index nl matching its front layer. If the device is instead placed in a medium with refractive index n, show that the amplitude reflectance is r (rb + r m )/(l + rbrm), where rb (n nl)/(n + nl) is the reflectance of the new boundary. Determine r in each of the following limiting cases: rb 0, rb 1, r m 0, and r m 1. 7.1-5 Quarter-Wave Film: Angular Dependence of Reflectance. Consider the quarter-wave an- tireflection coating described in Exercise 7 .1-1. Derive an expression for the reflectance as a function of the angle of incidence. 7 .1-6 Quarter-Wave and Half-Wave Stacks. Derive expressions for the reflectance of a stack of N double layers of dielectric materials of equal optical thickness, nl d l n2d2, equal to Ao/4 and Ao/2. 7 .1- 7 GaAsl AlAs Bragg Grating Reflector. A Bragg grating reflector comprises N units of al- ternating layers of GaAs (nl 3.57) and AlAs (n2 2.94) of widths d l and d 2 equal to a quarter wavelength in each medium. The grating is placed in an extended GaAs medium. Calculate and plot the transmittance and reflectance of the grating as functions of N, for N 1, 2, . . . , 10, at a frequency equal to the Bragg frequency. 7.1-8 Bragg Grating: Angular and Spectral Dependence of Reflectance. Write a computer program based on matrix algebra to determine the wave-transfer matrix and the reflectance of an N-Iayer alternating-layer dielectric Bragg grating. Use your program to verify the graphs presented in Fig. 7 .1-12 and Fig. 7 .1-13 for the spectral and angular dependence of the reflectance, respectively. 7.2-1 Gap-Midgap Ratio. Using a Fourier optics approach, determine the Bragg frequency and the gap-midgap ratio for the lowest bandgap of aID periodic structure comprising a stack of dielectric layers of equal optical thickness, with nl 1.5 and n2 3.5, and period A 2 J-Lm. Assume that the wave travels along the axis of periodicity. Repeat the process for nl 3.4 and n2 3.6. Compare your results. 7.2-2 OfT-Axis Wave in ID Periodic Medium. Derive equations analogous to those provided in (7.2-24)-(7.2-28) for an off-axis wave traveling through a ID periodic medium with a transverse wavevector kx. 7.2-3 Normal-to-Axis Wave in a ID Periodic Medium. Use the results of Prob. 7.2-2 to show that there are no bandgaps for a wave traveling along the lateral direction of aID periodic medium, Le., for K O. 7 .2-4 Omnidirectional Reflector. A periodic stack of double layers of dielectric materials with nl d l n2 d 2, n2 2nl and A d l + d 2 is to be used as an omnidirectional reflector in air. Plot the projected dispersion relation showing the light line for air (a diagram similar to Fig. 7.2-10). Determine the frequency range (in units of W13) for omnidirectional reflection.
C HAP T E R e . GUIDED-WAVE OPTICS 8.1 PLANAR-MIRROR WAVEGUIDES 291 8.2 PLANAR DIELECTRIC WAVEGUIDES 299 A. Waveguide Modes B. Field Distributions C. Dispersion Relation and Group Velocities 8.3 TWO-DIMENSIONAL WAVEGUIDES 308 8.4 PHOTONIC-CRYSTAL WAVEGUIDES 311 8.5 OPTICAL COUPLING IN WAVEGUIDES 313 A. Input Couplers B. Coupled Waveguides C. Periodic Waveguides 8.6 SUB-WAVELENGTH METAL WAVEGUIDES (PLASMONICS) 321 " 1-. 1:' '.. . '-' "\. ." . '100.' '- John Tyndall (1820-1893) was the first to demonstrate total internal reflection, the basis of guided-wave optics. 289
In traditional optical instruments and systems, light is transmitted between different locations in the form of beams that are collimated, relayed, focused, and scanned by mirrors, lenses, and prisms. The beams diffract and broaden as they propagate though they can be refocused by the use of lenses and mirrors. However, the bulk optical components that comprise such systems are often large and unwieldy, and objects in the paths of the beams can obstruct or scatter them. In many circumstances it is advantageous to transmit optical beams through di- electric conduits rather than through free space. The technology for achieving this is known as guided-wave optics. It was initially developed to provide long-distance light transmission without the necessity of using relay lenses. This technology now has many important applications. A few examples are: carrying light over long distances for lightwave communications, biomedical imaging where light must reach awkward locations, and connecting components within miniaturized optical and optoelectronic devices and systems. The underlying principle of optical confinement is simple. A medium of refractive index nl, embedded in a medium of lower refractive index n2 < nl, acts as a light "trap" within which optical rays remain confined by multiple total internal reflections at the boundaries. Because this effect facilitates the confinement of light generated inside a medium of high refractive index [see Exercise (1.2-6)], it can be exploited in making light conduits - guides that transport light from one location to another. An optical waveguide is a light conduit consisting of a slab, strip, or cylinder of dielectric material embedded in another dielectric material of lower refractive index (Fig. 8.0-1). The light is transported through the inner medium without radiating into the surrounding medium. The most widely used of these waveguides is the optical fiber, comprising two concentric cylinders of low-loss dielectric material such as glass (see Chapter 9). I L Figure 8.0-1 Optical waveguides. ( Integrated optics is the technology of combining, on a single substrate ("chip"), various optical devices and components useful for generating, focusing, splitting, com- bining, isolating, polarizing, coupling, switching, modulating, and detecting light. Op- tical waveguides provide the links among these components. Such chips (Fig. 8.0-2) are optical versions of electronic integrated circuits. Integrated optics has, as its goal, the miniaturization of optics in much the same way that integrated circuits have served to miniaturize electronics. This Chapter The basic theory of optical waveguides is presented in this and the following chapter. In this chapter, we consider rectangular waveguides, which are used extensively in integrated optics. Chapter 9 deals with cylindrical waveguides, i.e., optical fibers. If reflectors are placed at the two ends of a short waveguide, the result is a structure that traps and stores light - an optical resonator. These devices, which are essential to lasers, are described in Chapter 10. Other integrated-optic components and devices (such as semiconductor lasers, detectors, modulators, and switches) are considered in 290
8.1 PLANAR-MIRROR WAVEGUIDES 291 Modulator Fiber Figure 8.0-2 Example of an integrated-optic device used as an optical receiver/transmitter. Re- ceived light is coupled into a waveguide and directed to a pho- todiode where it is detected. Light from a laser is guided, modulated, and coupled into a fiber for trans- mISSIon. Laser Substrate Photodiode the chapters that deal specifically with those components and devices. Optical fiber communication systems are discussed in detail in Chapter 24. 8.1 PLANAR-MIRROR WAVEGUIDES We begin by examining wave propagation in a waveguide comprising two parallel infinite planar mirrors separated by a distance d (Fig. 8.1-1). The mirrors are assumed to reflect light without loss. A ray of light, say in the y-z plane, making an angle () with the mirrors reflects and bounces between them without loss of energy. The ray is thus guided along the z direction. This waveguide appears to provide a perfect conduit for light rays. It is not used in practical applications, however, principally because of the difficulty and cost of fabricating low-loss mirrors. Nevertheless, we study this simple example in detail because it provides a valuable pedagogical introduction to the dielectric waveguide, which we examine in Sec. 8.2, and to the optical resonator, which is the subject of Chapter 10. y x Mirrors 1-\- _\ 1 d z Figure 8.1-1 Planar-mirror waveguide. T Waveguide Modes The ray-optics picture of light being guided by multiple reflections cannot explain a number of important effects that require the use of electromagnetic theory. A simple approach for carrying out an electromagnetic analysis is to associate with each optical ray a transverse electromagnetic (TEM) plane wave. The total electromagnetic field is then the sum of these plane waves. Consider a monochromatic TEM plane wave of wavelength A == Aol n, wavenumber k == nk o , and phase velocity c == coin, where n is the refractive index of the medium between the mirrors. The wave is polarized in the x direction and its wavevector lies in the y-z plane at an angle () with the z axis (Fig. 8.1-1). Like the optical ray, the wave reflects from the upper mirror, travels at an angle -(), reflects from the lower mirror, and travels once more at an angle (), and so on. Since the electric-field vector is parallel
292 CHAPTER 8 GUIDED-WAVE OPTICS to the mirror, each reflection is accompanied by a phase shift 1r for a perfect mirror, but the amplitude and polarization are not changed. The 1r phase shift ensures that the sum of each wave and its own reflection vanishes so that the total field is zero at the mirrors. At each point within the waveguide we have TEM waves traveling upward at an angle 0 and others traveling downward at an angle 0; all waves are polarized in the x direction. We now impose a self-consistency condition by requiring that as the wave eflects twice, it reproduces itself [see Fig. 8.1-2(a)], so that we have only_t wo d istinct plane waves. Fields that satisfy this condition are called the modes (or eigenfunctions) of the waveguide (see Appendix C). Modes are fields that maintain the same transverse distribution and polarization at all locations along the waveguide axis. We shall see that self-consistency guarantees this shape invariance. In connection with Fig. 8.1-2, the phase shift <p encountered by the original wave in traveling from A to B must be equal to, or differ by an integer multiple of 21r, from that encountered when the wave reflects, travels from A to C, and reflect s on ce more. Acco unt ing for a phase shift of 1r at each reflection, w e h av e <p 21r AC A 21r 21r AB A 21rq, where q 0, 1, 2, . . ., so that 21r AC AB A 21r q + 1 . The geometry po rtr ayed in Fig. 8.1- 2(a), together with the identity cos 2x 1 2 sin 2 x, provides AC AB 2d sin 0, where d is the distance between the mirrors. Thus, 21r 2d sin 0 A 21r q + 1 so that 27r 21r m, m 1,2,.... (8.1-1) where m q + 1. The self-consistency condition is therefore satisfied only for certain bounce angles 0 Om satisfying sin Om A m 2d' m 1,2,.... (8.1-2) Bounce Angles Each integer m corresponds to a bounce angle Om, and the corresponding field is called the mth mode. The m 1 mode has the smallest angle, 0 1 sin- 1 A 2d ; modes with larger m are composed of more oblique plane-wave components. y ". 'tB ,( \ ,('\\ \ \ ( '\ \ \ \ ,\' \ \ \ \ \ ,( \ \ \ \ \\ \ \ A '0\ \ \ \ \ \ \ \ \ ... '\ . \ \. \ ... ,\ OriginaJ wave . A ... d c Z Twice- reflected wave , , , , , , i , j j , . , , ,. &. 1 """""""""""fTfTlfl"""l""""""'" ,tit t t,t,t,t,t,t,t,tN,t,t t t,t,t,t,t,H t t tat t t,t,t,t,t,t,t,t,t,t,t,t t t,t,tN,t,t,t t , , , , 'f1J T'"""" (a) (b) Figure 8.1-2 (a) Condition of self-consistency: as a wave reflects twice it duplicates itself. (b) At angles for which self-consistency is satisfied, the two waves interfere and create a pattern that does not change with z. When the self-consistency condition is satisfied, the phases of the upward and down- ward plane waves at points on the z axis differ by half the round-trip phase shift q1r,
8.1 PLANAR-MIRROR WAVEGUIDES 293 q 0,1, . . . , or m 1 7f, m 1,2, . . . , so that they add for odd m and subtract for even m. Since the y component of the propagation constant is given by ky nko sin 0, it is quantized to the values k ym nko sin Om 27f A sin Om. Using (8.1-2), we obtain k ym 7r m d , m 1,2,3..., (8.1-3) Wavevector Transverse Component so that the k ym are spaced by 7f d. Equation (8.1-3) states that the phase shift en- countered when a wave travels a distance 2d (one round trip) in the y direction, with propagation constant k ym , must be a multiple of 27f. Propagation Constants A guided wave is composed of two distinct plane waves traveling in the y Z plane at angles :f:O with the z axis. Their wavevectors have components 0, ky, k z and 0, ky, k z . Their sum or difference therefore varies with z as exp jkzz, so that the propagation constant of the guided wave is {3 k z k cos O. Thus, {3 is quantized to the values!3m kcosO m , from which!3?:n, k 2 1 sin 2 Om . Using (8.1-2), we obtain {3 k 2 m 2 7f2 d 2 · (8.1-4 ) Propagation Constants Higher-order (more oblique) modes travel with smaller propagation constants. The values of Om, k ym , and (3m for the different modes are illustrated in Fig. 8.1-3. sinO ] ky = nko sin 0 nko ---_.........--.._------.............................. M 0 0 ..........-.-._----- I I I I I I I I I I I I I I I OM I ----- I-- I I ---- -,-- T- I I I --- -I -- - m nko f)m 13m k ym ..-.-...........------,...............-- I )"/2d I I I .-.-. --- I I I I I I I I I I I I - I I I I I I I I I I I I I I I I I 0 Om el e2 e3 1rld 1r /2 0 /3 = nko cos f) Figure 8.1-3 The bounce angles Om and the wavevector components of the modes of a planar- mirror waveguide (indicated by dots). The transverse components k ym k sin Om are spaced uniformly at multiples of 1f / d, but the bounce angles Om and the propagation constants (3m are not equally spaced Mode m 1 has the smallest bounce angle and the largest propagation constant.
294 CHAPTER 8 GUIDED-WAVE OPTICS Field Distributions The complex amplitude of the total field in the waveguide is the superposition of the two bounci n g TEM plane waves . If Am exp j kymY j {3mz is the upw ard 0, the two waves differ by a phase shift m 1 1f]. There are therefore symmet- ric modes, for which the two plane-wave components are added, and antisymmetric modes, for which they are subtracted. The total field turns out to be Ex Y, z 2Am cos kymY exp j {3mz for odd modes and 2j Am sin kymY exp j (3mz for even modes. Using (8.1-3) we write the complex amplitude of the electric field in the form Ex Y, z UmU m Y exp j{3m z , (8.1-5) where 2 y Urn Y (8.1-6) 2. Y d SIn m 1f d ' m 2, 4, 6, . . . , with am 2d Am and j 2d Am' for odd m and even m, respectively. The functions U m Y have been normalized to satisfy d/2 u y dy 1. -dj2 (8.1-7) Thus, am is the amplitude of mode m. It can be shown that the functions U m Y also satisfy d/2 U m Y UI Y dy 0, -dj2 i.e., they are orthogonal in the d 2, d 2 interval. The transverse distributions U m yare plotted in Fig. 8.1-4. Each mode can be viewed as a standing wave in the y direction, traveling in the z direction. Modes of large m vary in the transverse plane at a greater rate ky and travel with a smaller propagation constant {3. The field vanishes at y ::I:d 2 for all modes, so that the boundary conditions at the surface of the mirrors are always satisfied. l i- m, (8.1-8) o . . -- y Mirrors Figure 8.1-4 Field distributions of the modes of a planar-mirror waveguide. d - 2 - - ./" z d -- 2 Since we assumed at the outset that the bouncing TEM plane wave is polarized in the x direction, the total electric field is also in the x direction and the guided wave is a transverse-electric (TE) wave. Transverse magnetic (TM) waves are analyzed in a similar fashion as will be seen subsequently.
8.1 PLANAR-MIRROR WAVEGUIDES 295 EXERCISE 8.1-1 Optical Power. Show that the optical power flow in the z direction associated with the TE mode Ex(Y, z) amum(y) exp( jf3m z ) is (la m I 2 /2'rJ) cos Om, where 'rJ 'rJo/n and'rJo J-to/Eo is the impedance of free space. Number of Modes Since sin ()m mA 2d, m 1,2, . . ., and taking sin ()m < 1, the maximum allowed value of m is the greatest integer smaller than 1 A 2d , 2d (8.1-9) Number of Modes The symbol . denotes that 2d A is reduced to the nearest integer. As examples, when 2d A 0.9, 1, and 1.1, we have M 0, 0, and 1, respectively. Thus, M is the number of modes of the waveguide. Light can be transmitted through the waveguide in one, two, or many modes. The actual number of modes that carry optical power depends on the source of excitation, but the maximum number is M. The number of modes increases with increasing ratio of the mirror separation to the wavelength. Under conditions such that 2d A < 1, corresponding to d < A 2, M is seen to be 0, which indicates that the self-consistency condition cannot be met and the waveguide cannot support any modes. The wavelength Ae 2d is called the cutoff wavelength of the waveguide. It is the longest wavelength that can be guided by the structure. It corresponds to the cutoff frequency 1/c C 2d' (8.1-10) Cutoff Frequency or the cutoff angular frequency We 27fV e 7fC d, the lowest frequency of light that can be guided by the waveguide. If 1 < 2d A < 2 (Le., d < A < 2d or 1/e < 1/ < 2v e ), only one mode is allowed. The structure is then said to be a single-mode waveguide. If d 5 {lm, for example, the waveguide has a cutoff wavelength Ae 10 J-Lm; it supports a single mode for 5 J-Lm < A < 10 J-Lm, and more modes for A < 5 J-Lm. Equation (8.1-9) can also be written in terms of the frequency v, M . v V e W We, so that the number of modes increases by unity when the angular frequency W is incremented by We, as illustrated in Fig. 8.1-5(a). Dispersion Relation The relation between the propagation constant (3 and the angular frequency W is an important characteristic of the waveguide, known as the dispersion relation. For a homogeneous medium, the dispersion relation is simply W c(3. For mode m of a planar-mirror waveguide, (3m and ware related by (8.1-4) so that (3 W C 2 m 2 'Jr2 d 2 . (8.1-11 )
296 CHAPTER 8 GUIDED-WAVE OPTICS This relation may be written in terms of the cutoff angular frequency We 27fll e 7f C d as , (3m W A 1 2 2 We m 2. W (8.1-12) Dispersion Relation c As shown in Fig. 8.1-5(b) for m 1,2, . . ., the propagation constant {3 for mode m is zero at angular frequency W mw e , increases monotonically with angular frequency, and ultimately approaches the linear relation {3 w c for sufficiently large values of {3. w w w ,.--- m=5 4 3 2 I I I I I I I I I I I I I I I 1 I C -- _.-__.- ..,._.... ---------- m=5 4 3 2 1 ----- ---------------- -_.. 3w c 2w c we , ., .. -... ...... - .....- ........ ----- _.-_.........,.-...- Light line w = c/3 -'---- °0 1 2 3 4 5 6 0 Number of modes M Propagation constant {3 (a) (b) Figure 8.1-5 (a) Number of modes M as a function of angular frequency w. Modes are not permitted for angular frequencies below the cutoff, W c wc/ d. M increments by unity as w increases by Wc. (b) Dispersion relation. A forbidden band exists for angular frequencies below Wc. (c) Group velocities of the modes as a function of angular frequency. --.-----._----------------------------- -,-,'" Forbidden band -----------------.----------- o Group velocity v (e) Group Velocities In a medium with a given w-{3 dispersion relation, a pulse of light (wavepacket) that has an angular frequency centered at w travels with a velocity v dw d{3, known as the group velocity (see Sec. 5.6). Taking the derivative of (8.1-12) and assuming that c is independent of w (i.e., ignoring dispersion in the waveguide material), we obtain 2{3m d{3m dw 2w c 2 , so that dw d{3m c 2 (3m W c 2 k cos Om W C COS Om, from which the group velocity of mode m is v rn c COS Om c 1 2 2 We m 2. W (8.1-13) Group Velocity It follows that more oblique modes travel with smaller group velocities since they are delayed by the longer paths of the zigzagging process. The dependence of the group velocity on angular frequency is illustrated in Fig. 8.1-5(c), which shows that for each mode, the group velocity increases monotonically from 0 to c as the angular frequency increases above the mode cutoff frequency. Equation (8.1-13) may also be obtained geometrically by examining the plane wave as it bounces between the mirrors and determining the distance advanced in the z
8.1 PLANAR-MIRROR WAVEGUIDES 297 direction and the time taken by the zigzagging process. For the trip from the bottom mirror to the top mirror (Fig. 8.1-6) we have v = dis.tance = d cot () = c cos () . tIme d csc () / c (8.1-14) / Figure 8.1-6 A plane wave bouncing at an angle () advances in the z direction by a distance d cot () in a time d csc () / c. The velocity is c cos (). T d 1 () I.. d cot () · I TM Modes Only TE modes (electric field in the x direction) have been considered to this point. TM modes (magnetic field in the x direction) can also be supported by the mirror waveguide. They can be studied by means of a TEM plane wave with the magnetic field in the x direction, traveling at an angle () and reflecting from the two mirrors (Fig. 8.1-7). The electric-field complex amplitude then has components in the y and z directions. Since the z component is parallel to the mirror, it behaves precisely like the x component of the TE mode (i.e., it undergoes a phase shift 7r at each reflection and vanishes at the mirrors). When the self-consistency condition is applied to this component the result is mathematically identical to that of the TE case. The angles (), the transverse wavevector components ky, and the propagation constants {3 of the TM modes associated with this component are identical to those of the TE modes. There are M . 2d/ A TM modes (and a total of 2M modes) supported by the waveguide. y x/ y TE TM Figure 8.1-7 TE and TM polarized guided waves. The z component of the electric-field complex amplitude of mode m, as previously, is the sum of an upward plane wave Am exp( -jkymY) exp( -j{3mz) and a downward plane wave e j (m-l)7r Am exp(j kymY) exp( - j {3mz), with equal amplitudes and phase shift (m - 1)7r, so that am cos (m1f ) exp( -jj3mz), m = 1,3,5,... Clm sin (m1f ) exp( -jj3mz), m = 2,4,6,..., Ez(Y,z) == (8.1-15) where am == V2d Am and j V2d Am for odd and even m, respectively. Since the electric-field vector of a TEM plane wave is normal to its direction of propagation, it
298 CHAPTER 8 GUIDED-WAVE OPTICS makes an angle 'IT /2 + Om with the z axis for the upward wave, and 7r /2 - Om for the downward wave. The Y components of the electric field of these waves are Am cot Om exp( - j kymY) exp( - j (3mz) and e jm1f Am cot Om exp(j kymY) exp( - j (3mz), (8.1-16) so that Ey(Y, z) Clm cot 8m cos (m1f ) exp( -j,Bm Z ) , m = 1,3,5,... Clm cot 8m sin (m1f ) exp( -j,Bmz), m = 2,4,6,. ... (8.1-17) Satisfaction of the boundary conditions is assured because Ez (y, z) vanishes at the mirrors. The magnetic field component Hx(Y, z) may be similarly determined by not- ing that the ratio of the electric to the magnetic fields of a TEM wave is the impedance of the medium 1]. The resultant fields Ey(Y, z), Ez(Y, z), and Hx(Y, z) do, of course, satisfy Maxwell's equations. Multimode Fields For light to be guided by the mirrors, it is not necessary that it have the distribution of one of the modes. In fact, a field satisfying the boundary conditions (vanishing at the mirrors) but otherwise having an arbitrary distribution in the transverse plane can be guided by the waveguide. The optical power, however, is then divided among the modes. Since different modes travel with different propagation constants and different group velocities, the transverse distribution of the field will alter as it travels through the waveguide. Fig. 8.1-8 illustrates how the transverse distribution of a single mode is invariant to propagation, whereas the multi mode distribution varies with z (the illustration is for the intensity distribution). (a) (b) (c) y z y z y z Figure 8.1-8 Variation of the intensity distribution in the transverse direction y at different axial distances z. (a) The electric-field complex amplitude in mode 1 is E(y, z) == Ul (y) exp( -j{31Z), where Ul (y) == /2/ d cos( 7rY / d). The intensity does not vary with z. (b) The complex amplitude in mode 2 is E(y, z) == U2(y) exp( -j{32Z), where U2(y) == /2/d sin(27ry/d). The intensity does not vary with z. (c) The complex amplitude in a mixture of modes 1 and 2, E(y, z) == Ul (y) exp( -j{31Z) + U2(y) exp( -j{32Z). Since {31 =1= {32, the intensity distribution changes with z.
8.2 PLANAR DIELECTRIC WAVEGUIDES 299 An arbitrary field polarized in the x direction and satisfying the boundary conditions can be written as a weighted superposition of the TE modes, M Ex(Y, z) == L amum(y) exp( -j(3m z ), m=O (8.1-18) where am, the superposition weights, are the amplitudes of the different modes. EXERCISE 8.1-2 Optical Power in a Multimode Field. Show that the optical power flow in the z direction associated with the multimode field in (8.1-18) is the sum of the powers (I am 1 2 /2'T]) cos em carried by each of the modes. 8.2 PLANAR DIELECTRIC WAVEGUIDES A planar dielectric waveguide is a slab of dielectric material surrounded by media of lower refractive indexes. The light is guided inside the slab by total internal reflection. In thin-film devices the slab is called the "film" and the upper and lower media are called the "cover" and the "substrate," respectively. The inner medium and outer media may also be called the "core" and the "cladding" of the waveguide, respectively. In this section we study the propagation of light in a symmetric planar dielectric waveguide made of a slab of width d and refractive index nl surrounded by a cladding of smaller refractive index n2, as illustrated in Fig. 8.2-1. All materials are assumed to be lossless. y x d 2: o nl d -2: n2 n2 z Figure 8.2-1 Planar dielectric ( slab) waveguide. Rays making an angle e < Be == cos-1(n2/nl) are guided by total internal reflection. Guided ray Unguided ray Light rays making angles () with the z axis, in the y-z plane, undergo multiple total internal reflections at the slab boundaries, provided that () is smaller than the complement of the critical angle ()c == 1T /2 - sin- 1 (n2/nl) == cos- 1 (n2/nl) [see (1.2- 5) and Figs. 6.2-3 and 6.2-5]. They travel in the z direction by bouncing between the slab surfaces without loss of power. Rays making larger angles refract, losing a portion of their power at each reflection, and eventually vanish. To determine the waveguide modes, a formal approach may be pursued by develop- ing solutions to Maxwell's equations in the inner and outer media with the appropriate boundary conditions imposed (see Prob. 8.2-6). We shall instead write the solution in terms of TEM plane waves bouncing between the surfaces of the slab. By imposing the
300 CHAPTER 8 GUIDED-WAVE OPTICS self-consistency condition, we determine the bounce angles of the waveguide modes from which the propagation constants, field distributions, and group velocities are determined. The analysis is analogous to that used in the previous section for the planar-mirror waveguide. A. Waveguide Modes Assume that the field in the slab is in the form of a monochromatic TEM plane wave of wavelength A == Aoln1 bouncing back and forth at an angle () smaller than the complementary critical angle () c. The wave travels with a phase velocity C1 == coin 1, has a wavenumber n1 ko, and has wavevector components kx == 0, ky == n1 ko sin (), and k z == n1ko cas (). To determine the modes we impose the self-consistency condition that a wave reproduces itself after each round trip. In one round trip, the twice-reflected wave lags behind the original wave by a - - distance AC - AB == 2d sin (), as in Fig. 8.1-2. There is also a phase CPr introduced by each internal reflection at the dielectric boundary (see Sec. 6.2). For self-consistency, the phase shift between the two waves must be zero or a multiple of 27r, 27r . T 2d SIn () - 2cpr == 27rm, m == 0,1,2, . . . (8.2-1 ) or 2k y d - 2cpr == 27rm. (8.2-2) The only difference between this condition and the corresponding condition in the mirror waveguide, (8.1-1) and (8.1-3), is that the phase shift 7r introduced by the mirror is replaced here by the phase shift CPr introduced at the dielectric boundary. 10 RHS LHS ,) m=O \ 1 \ \ \ \ \ , 2 3 4 5 6 7 8 o o I r- 2d sin fie sinO Figure 8.2-2 Graphical solution of (8.2-4) to determine the bounce angles Om of the modes of a planar dielectric waveguide. The RHS and LHS of (8.2-4) are plotted versus sin (). The intersection points, marked by filled circles, determine sin ()m. Each branch of the tan or cot function in the LHS corresponds to a mode. In this plot sin e c = 8(A/2d) and the number of modes is !vI = 9. The open circles mark sin ()m = mA/2d, which provide the bounce angles of the modes of a planar-mirror waveguide of the same dimensions. The reflection phase shift CPr is a function of the angle (). It also depends on the polarization of the incident wave, TE or TM. In the TE case (the electric field is in the
8.2 PLANAR DIELECTRIC WAVEGUIDES 301 x direction), substituting ()l == 7r /2 - () and ()c == 7r /2 - ()c in (6.2-11) gives CPr tan - == 2 sin 2 Bc -1 sin 2 () (8.2-3) so that CPr varies from 7r to 0 as () varies from 0 to () c. Rewriting (8.2-1) in the form tan(7rdsinB/'\ - m7r/2) == tan(cpr/2) and using (8.2-3), we obtain ( d. 7r ) tan 7r ,\ SIn () - m"2 sin 2 ()c - 1. sin 2 () (8.2-4) Self-Consistency Condition (TE Modes) This is a transcendental equation in one variable, sin B. Its solutions yield the bounce angles ()m of the modes. A graphic solution is instructive. The right- and left-hand sides of (8.2-4) are plotted in Fig. 8.2-2 as functions of sin (). Solutions are given by the intersection points. The right-hand side (RHS), tan( CPr/2), is a monotonic decreasing function of sin () that reaches 0 when sin () == sin Bc. The left-hand side (LHS) generates two families of curves, tan( (7rd / ,\) sin ()] and cot (( 7rd / ,\) sin ()], when m is even and odd, respectively. The intersection points determine the angles ()m of the modes. The bounce angles of the modes of a mirror waveguide of mirror separation d may be obtained from this diagram by using CPr == 7r or, equivalently, tan( CPr/2) == 00. For comparison, these angles are marked by open circles. The angles ()m lie between 0 and ()c . They correspond to wavevectors with compo- nents (0, nik o sin ()m, nlko cos ()m). The z components are the propagation constants !3m == nlko cos ()m. (8.2-5) Propagation Constants Since cos()m lies between 1 and cos ()c == n2/nl,!3m lies between n2ko and nlko, as illustrated in Fig. 8.2-3. ky n]ko n]ko k z =!3 Figure 8.2-3 The bounce angles ()m and the corresponding components k z and ky of the wavevector of the waveg- uide modes are indicated by_dots. The angles ()m lie between 0 and ()c, and the propagation constants !3m lie between n2ko and nl ko. These results should be compared with those shown in Fig. 8.1-3 for the planar-mirror waveguide. n2 k o n]kosinOc
302 CHAPTER 8 GUIDED-WAVE OPTICS The bounce angles {}m and the propagation constants (3m of TM modes can be found by using the same equation (8.2-1), but with the phase shift <{Jr given by (6.2-13). Similar results are obtained. Number of Modes To determine the number of TE modes supported by the dielectric waveguide we examine the diagram in Fig. 8.2-2. The abscissa is divided into equal intervals of width A 2d, each of which contain s a mode marked by a filled circle. This extends over angles for which sin {} < sin {} e. The number of TE modes is therefore the smallest integer greater than sin {} e A 2 d , so that sin {} e (8.2-6) The symbol · d enotes that sin {}e -A 2d is increased to the nearest integer. For example, if sin {}e A 2d 0.9, 1, or 1.1, then M 1, 2, and 2, respectively. Substituting cos Be n2 nl into (8.2-6), we obtain M ' Ao ' (8.2-7) Number of TE Modes where NA ,- <1 n 2 n 2 2 V 1 (8.2-8) Numerical Aperture is the numerical aperture of the waveguide (the NA is the sine of the angle of accep- tance of rays from air into the slab; see Exercise 1.2-5). A similar expression can be obtai n ed for the TM modes. If d Ao 10, nl 1.47, and n2 1.46, for example, then {}e 6.7°, NA 0.171, and M 4 TE modes. en A 2 d > sin {} c or 2 d Ao NA < 1, only one mode is allowed. The waveguide is then a single-mode waveguide. This occurs when the slab is sufficiently thin or the wavelength is sufficiently long. Unlike the mirror waveguide, the dielectric waveguide has no absolute cutoff wavelength (or cutoff frequency). In a dielectric waveguide there is at least one TE mode, since the fundamental mode m 0 is always allowed. Each of the modes m 1, 2, . . . has its own cutoff wavelength, however. Stated in terms of frequency, the condition for single-mode operation is that 1/ > 1/e, or W > We, where the mode cutoff frequency is 1/ e Wel 21r 1 Co NA 2d · (8.2-9) Mode Cutoff Frequency The number of modes is then M · 1/ 1/e W We, which is the relation illustrated in Fig. 8.2-4. M is incremented by unity as W increases by We. Identical expressions for the number of TM modes are obtained via a similar derivation.
8.2 PLANAR DIELECTRIC WAVEGUIDES 303 W 3wc 2w c 00 1 2 3 4 5 6 Number of modes M Figure 8.2-4 Number of TE modes as a function of frequency. Compare with Fig. 8.1-5(a) for the planar-mirror waveguide. There is no forbidden band in the case at hand. We EXAMPLE 8.2-1. Modes in an AIGaAs Waveguide. A waveguide is made by sandwiching a layer of AlxGal-xAs between two layers of AlyGal-yAs. By changing the concentrations of x, y of AI in these compounds their refractive indexes are controlled. If x and y are chosen such that at an operating wavelength AD 0.9 /-lm, nl 3.5, and nl n2 0.05, then for a thickness d 10 /-lm there are Al 14 TE modes. For d < O. 76 /-lm, only a single mode is allowed. B. Field Distributions We now determine the field distributions of the TE modes. Internal Field The field inside the slab is composed of two TEM plane waves traveling at angles Om and em with the z axis with wavevectorcomponents 0, :f::nlko sin em, nlko cos Om . They have the same amplitude and phase shift m7r (half that of a round trip) at the center of the slab. The electric-field complex amplitude is therefore Ex y, z am U m Y exp j {3m z , where {3m nl ko cos Om is the propagation constant, am is a constant, 27r sin em 0, 2, 4, . . . cos A Y , m d d Urn Y ex <y< 2' (8.2-1 0) sin em 2 . 27r 1,3,5,..., SIn A y , m and A Ao nl- Note that although the field is harmonic, it does not vanish at the slab boundary. As m increases, sin em increases, so that higher-order modes vary more rapidly with y. External Field The external field must match the internal field at all boundary points y :f::d 2. It must therefore vary with z as exp j {3m z . Substituting Ex y, z am U m Y exp j {3mz into the Helmholtz equation \72 + nk; Ex y, z 0, we obtain d 2 u m dy2 2 'Ym Um 0, (8.2-11)
304 CHAPTER 8 GUIDED-WAVE OPTICS where 1' {3 n k · (8.2-12) Since!3m > n2 k o for guided modes (See Fig. 8.2-3), 1' > 0, so that (8.2-11) is satisfied by the exponential functions exp l'mY and exp l'mY . Since the field must decay away from the slab, we choose exp l'mY in the upper medium and exp l'mY in the lower medium Urn Y ex exp l'mY, Y > d 2 exp l'mY, Y < d 2. (8.2-13) The decay rate l'm is known as the extinction coefficient. T he wave is said to be an evanescent wave. Substituting {3m nlko cos ()m and cos Be n2 nl into (8.2-12), we obtain , l'rn n2 k o A COs 2 ()m cos 2 () c 1 . (8.2-14) Extinction Coefficient As the mode number m increases, ()m increases, and l'm decreases. Higher-order modes therefore penetrate deeper into the cover and substrate. To determine the proportionality constants in (8.2-10) and (8.2-13), we match the internal and external fields at Y d 2 and use the normalization (X) U Y dy 1. (8.2-15) (X) This gives an expression for U m Y valid for all y. These functions are illustrated in Fig. 8.2-5. As in the mirror waveguide, all of the U m yare orthogonal, i.e., (X) Urn Y Ul Y dy 0, l =J m. (8.2-16) -(X) y m=O 1 2 3 8 d 2 o z d - 2 Figure 8.2-5 Field distributions for TE guided modes in a dielectric waveguide. These results should be compared with those shown in Fig. 8.1-4 for the planar-mirror waveguide.
8.2 PLANAR DIELECTRIC WAVEGUIDES 305 An arbitrary TE field in the dielectric waveguide can be written as a superposition of these modes: Ex y, z amU m y exp j{3mz, (8.2-17) Tn where am is the amplitude of mode m. EXERCISE 8.2-1 Confinement Factor. The power confinement factor is the ratio of power in the slab to the total power 0 00 u(y) dy · Derive an expression for r m as a function of the angle 8m and the ratio d / A. Demonstrate that the lowest-order mode (smallest 8m) has the highest power confinement factor. r m (8.2-18) The field distributions of the TM modes may be similarly determined (Fig. 8.2- 6). Since it is parallel to the slab boundary, the z component of the electric field behaves similarly to the x component of the TE electric field. The analysis may start by determining Ez y, z . Using the properties of the constituent TEM waves, the other components Ey y, z and Hx y, z may readily be determined, as was done for mirror waveguides. Alternatively, Maxwell's equations may be used to determine these fields. y TM . V IV'" , -- - . . . _ T z y TE J' /' ., E z H Figure 8.2-6 TE and TM modes in a dielectric planar waveguide. The field distribution of the lowest-order TE mode (m 0) is similar in shape to that of the Gaussian beam (see Chapter 3). However, unlike the Gaussian beam, guided light does not spread in the transverse direction as it propagates in the axial direction (see Fig. 8.2-7). In a waveguide, the tendency of light to diffract is compensated by the guiding action of the medium. --- --- --- .... --- .- _..-.---- --- -. -. z - - ---- --. -. --- --- "'_-1 A WI 1f!I Un'" .. . .- 'Y. 111M . I!M..... , Z --- -- --- - (a) (b) Figure 8.2-7 (a) Gaussian beam in a homogeneous medium. (b) Guided mode in a dielectric waveguide.
306 CHAPTER 8 GUIDED-WAVE OPTICS c. Dispersion Relation and Group Velocities The dispersion relation (w versus (3) is ob ta ined by w ri ting the self-consistency equa- w 2 c 2 1 {32 2cpr + 27rm. (8.2-19) 2d Since cos B {3 W Cl and cos Be n2 nl Cl C2, (8.2-3) becomes t 2 CPr an 2 {32 W 2 2 2 {3 2. W C l (8.2-20) Substituting (8.2-20) into (8.2-19) we obtain {32 W 2 2 2 r:l2. W C l fJ (8.2-21) Dispersion Relation 2 W 2 C 2 1 {32 7r m 2 This relation may be plotted by rewriting it in parametric form, n 2 1 n 2 1 n 2 2 n 2 2 m + tan- l 7r We n 2 n 2 1 n n 2 W , (3 nw Co, (8.2-22) in terms of the effective refractive index n defined in (8.2-22), where We 27r Co 2dNA is the mode-cutoff angular frequency. As shown in the schematic plot in Fig. 8.2-8(a), the dispersion relations for the different modes lie between the lines W c2{3 and W cl{3, the light lines representing propagation in homogeneous media with the refractive indexes of the surrounding medium and the slab, respectively. As the frequency increases above the mode cutoff frequency, the dispersion relation moves from the light line of the surrounding medium toward the light line of the slab, i.e., the effective refractive index n increases from n2 to nl. This effect is indicative of a stronger confinement of waves of shorter wavelength in the medium of higher refractive index. The group velocity is obtained from the dispersion relation by determining the slope v dw d{3 for each of the guided modes. The dependence of the group velocity on the angular frequency is illustrated schematically in Fig. 8.2-8(b). As the angular frequency increases above the mode cutoff frequency for each mode, the group velocity decreases from its maximum value C2, reaches a minimum value slightly below Cl, and then asymptotically returns back toward Cl. The group velocities of the allowed modes thus range from C2 to a value slightly below Cl. In propagating through a multimode waveguide, optical pulses spread in time since the modes have different velocities, an effect called modal dispersion. [n a single- mode waveguide, an optical pulse spreads as a result of the dependence of the group ve- locity on frequency. This effect is called group velocity dispersion (GVD). As shown in Sec. 5.6, GVD occurs in homogeneous materials by virtue of the frequency depen- dence of the refractive index of the material. Moreover, GVD occurs in waveguides even in the absence of material dispersion. It is then a consequence of the frequency dependence of the propagation coefficients, which are determined by the dependence of wave confinement on wavelength. As illustrated in Fig. 8.2-8(b), each mode has
8.2 PLANAR DIELECTRIC WAVEGUIDES 307 o o Propagation constant f3 (a) W W 4wc m=3 I . . . . . . . . . 4we Light line / W = C2{1 3wc , . , , . m=3 3wc . . . . I . I . 2wc . , . , . # . , # , , , , , , . , , , , , , 2we . . . . . I I I . . , . . , , # , , , 1# , , # # , Light line W = CI{3 2 1 I . . . . . . . We We o . . . . . . . . , # o CI C2 Group velocity V (b) Figure 8.2-8 Schematic representations of (a) the dispersion relation for the different TE modes, m 0, 1,2, . . .; and (b) the frequency dependence of the group velocity, which is the derivative of the dispersion relation, v dw / d{3. a particular angular frequency at which the group velocity changes slowly with fre- quency (the point at which v reaches its minimum value so that its derivative with respect to w is zero). At this frequency, the GVD coefficient is zero and pulse spreading is negligible. An approximate expression for the group velocity may be obtained by taking the total derivative of (8.2-19) with respect to (3, 2d 2w dw 2ky c d(3 2(3 (8.2-23) Substituting dw d(3 v, ky W Cl new parameters sin (), and ky (3 tan (), and introducing the l:1z O<pr 0(3 , l:17 Ocpr ow ' (8.2-24) we obtain v d cot () + l:1z d csc () Cl + l:1 7 · (8.2-25) As we recall from (8.1-14) and Fig. 8.1-6 for the planar-mirror waveguide, d cot () is the distance traveled in the z direction as a ray travels once between the two boundaries. This takes a time d csc () Cl. The ratio d cot () d csc () Cl Cl COS () yields the group velocity for the mirror waveguide. The expression (8.2-25) for the group velocity in a dielectric waveguide indicates that the ray travels an additional distance l:1z o<pr 0(3, a trip that lasts a time l:17 o<pr ow. We can think of this as an effective penetration of the ray into the cladding, or as an effective lateral shift of the ray, as shown in Fig. 8.2-9. The penetration of a ray undergoing total internal reflection is known as the Goos Hanchen effect (see Probe 6.2-7). Using (8.2-24) it can be shown that l:1z l:1 7 W (3 Cl COS () .
308 CHAPTER 8 GUIDED-WAVE OPTICS ... L1 z C; 6CJ I I I , I I I I I I - Figure 8.2-9 A ray model that replaces the reflection phase shift with an additional distance z traversed at velocity Cl / cos e. Idcot() EXERCISE 8.2-2 The Asymmetric Planar Waveguide. Examine the TE field in an asymmetric planar waveg- uide consisting of a dielectric slab of width d and refractive index nl placed on a substrate of lower refractive index n2 and covered with a medium of refractive index n3 < n2 < nl, as illustrated in Fig. 8.2-10. (a) Determine an expression for the maximum inclination angle e of plane waves undergoing total internal reflection, and the corresponding numerical aperture NA of the waveguide. (b) Write an expression for the self-consistency condition, similar to (8.2-4). (c) Determine an approximate expression for the number of modes M (valid when M is very large). n3 d n} n2 ............. - " yo Figure 8.2-10 Asymmetric planar waveguide. 8.3 TWO-DIMENSIONAL WAVEGUIDES The planar-mirror waveguide and the planar dielectric waveguide studied in the pre- ceding two sections confine light in one transverse direction (the y direction) while guiding it along the z direction. Two-dimensional waveguides confine light in the two transverse directions (the x and y directions). The principle of operation and the underlying modal structure of two-dimensional waveguides is basically the same as planar waveguides; only the mathematical description is lengthier. This section is a brief description of the nature of modes in two-dimensional waveguides. Details can be found in specialized books. Chapter 9 is devoted to an important example of two- dimensional waveguides, the cylindrical dielectric waveguide used in optical fibers. Rectangular Mirror Waveguide The simplest generalization of the planar waveguide is the rectangular waveguide (Fig. 8.3-1). If the walls of the waveguide are mirrors, then, as in the planar case, light is guided by multiple reflections at all angles. For simplicity, we assume that the cross section of the waveguide is a square of width d. If a plane wave of wavevector k x , ky, k z and its multiple reflections are to exist self-consistently inside the waveg-
8.3 TWO-DIMENSIONAL WAVEGUIDES 309 uide, it must satisfy the conditions: 2kx d 21rm x , 2k y d 21rmy, m x 1, 2, . . . my 1, 2, . . . , (8.3-1 ) which are obvious generalizations of (8.1-3). ky y Ilk o -----r-----1------r-----1------r-----1------: . . I I I I I I I I I I ---------- ----------t-----1------! I I I I I I I I . I I . I I . I I I I I I I 'I I I I ---------------- ----t-----------t d I I I . I I I , I I I . I I , I I . I I I I I . I I I -------------------- ----1------i '?r Id I I I I . I I II I I . I I . I I I I I I I I ----------------------- 1------: I I . I I . I I I I I I I I I I I I I I I I I I I ------------------------" ----I I I I f I I I I I I f I I I , I I I I I I I I I I I I I ------------------------- ---i I I I I I I I . . I . I f I I I I I I I . o I I I I I I I o 1r/d nko I I , ! I I I I I Mirror d x kx Figure 8.3-1 Modes of a rect- angular mirror waveguide are char- acterized by a finite number of discrete values of kx and ky repre- sented by dots. The propagation c on stant {3 k z can be determined by kx and ky by using the discrete values, yielding a finite number of modes. Each mode is identified by two indexes m x and my (instead of on e index m). All positive integer values of m x and my The number of modes M can be easily determined by counting the number of dots within a quarter circle of radius nko in the kx versusk y diagram (Fig. 8.3-1). If this number is large, it may be approximated by the ratio of the area 1r nko 2 4 to the area of a unit cell 1r d 2, M 1r 4 2d 2 A . (8.3-2) Since there are two polarizations per mode, the total number of modes is actually 2M. Comparing this to the number of modes in a one-dimensional mirror waveguide, M 2d A, we see that increase of the dimensionality yields approximately the square of the number of modes. The number of modes is a measure of the degrees of freedom. en we add a second dimension we simply multiply the number of degrees of freedom. The field distributions associated with these modes are generalizations of those in the planar case. Patterns such as those in Fig. 8.1-4 are obtained in each of the x and y directions depending on the mode indexes m x and my. Rectangular Dielectric Waveguide A dielectric cylinder of refractive index nl with square cross section of width d is embedded in a medium of slightly lower refractive index n2. The waveguide nodes can be determined using a similar t h eory. Comp o nents of th e w avev ec tor k x , ky, k z must and ky lie in the area shown in Fig. 8.3-2. The values of kx and ky for the different modes can be obtained from a self-consistency condition in which the phase shifts at the dielectric boundary are included, as was done in the planar case.
310 CHAPTER 8 GUIDED-WAVE OPTICS y ky nlko - nl kosin.Bc I ---.----.r:- I I I I I I I I .: . I . ----T----i--------- /d I I I I 'Ti I I I I II .:.:.: . : . --r------r . I I I I I I I I . - .: . I . :. : . . I I ----T------------,-------- e: .: .:.:. :. I I : I I I I. I I ___+__,L----- .. .. .:. I. I. _ I. I. I I I . I I . I : I I : ---.t---.ii--ifr-ii-i-it--it--1i- I I I I I I I I I I : O I I I I I . I. I I o 7r/d n2 nl d x d Figure 8.3-2 Geometry of a rectangular dielectric waveguide. The values of kx and ky for the waveguide modes are marked by dots. n I ko kx Unlike the mirror waveguide, kx and ky of the modes are not uniformly spaced. However, two consecutive values of kx (or ky) are separated by an average value of 7r d (the same as for the mirror waveguide). The number of modes can there- fore be approximated by counting the number of dots in the inner circle in the kx versusk y diagram of F ig. 8.3-2, assuming an average spacing of 7r d. The result is M 7r 4 nlko sinO c 2 7r d 2, from which !vI 7r 4 2d 2 Ao NA 2 , (8.3-3) Number of TE Modes where NA n n is the numerical aperture. The approximation is satisfactory when M is large. There is also an identical number M of TM modes. The number of modes is roughly the square of that for the planar dielectric waveguide (8.2-7). Geometries of Channel Waveguides Useful geometries for waveguides include the strip, the embedded-strip, the rib or ridge, and the strip-loaded waveguides illustrated in Fig. 8.3-3. The exact analysis for some of these geometries is not easy, and approximations are usually used. The reader is referred to specialized books for further information about this topic. Embedded strip Strip Rib or ridge Strip loaded Figure 8.3-3 Various waveguide geometries. The darker the shading, the higher the refractive index. The waveguide may be fabricated in different configurations, as illustrated in Fig. 8.3-4 for the embedded-strip geometry. S bends are used to offset the propagation axis. The Y branch plays the role of a beam splitter or combiner. Two Y branches may be used to make a Mach Zehnder interferometer. Two waveguides in close proximity, or intersecting, can exchange power and may be used as directional couplers, as we shall see in the next section.
8.4 PHOTONIC-CRYSTAL WAVEGUIDES 311 Straight S Bend Y branch Mach-Zehnder Directional coupler Intersection Figure 8.3-4 Different configurations for waveguides. Materials The most advanced technology for fabricating waveguides is Ti:LiNb0 3 . An embedded- strip waveguide is fabricated by diffusing titanium into a lithium niobate substrate to increase its refractive index in the region of the strip. GaAs strip waveguides are made by using layers of GaAs and AIGaAs of lower refractive index. Another semiconductor material that has recently gained importance in waveguides is InP. Glass waveguides are made by ion exchange. Polymer waveguides are also emerging as a viable technology. Waveguides can also be fabricated using silicon-on-insulator (Si-Si0 2 or SOl), and silicon and oxide etching tools, which are standards in the industry. This technology is also called silica-on-silicon. Since the refractive index of silica is rv 3.5 and that of silica is less than 1.5, this combination of materials exhibits a large index-of- refraction difference n. A typical SOl may take the form of a silicon rib waveguide (see Fig. 8.3-3) on top of a layer of silica, serving as a cladding, with a silicon sub- strate underneath. Silicon processing and fabrication has been well developed by the microelectronics industry, and compatibility with CMOS fabrication technology is an important advantage. Ti Si Silica Si substrate Figure 8.3-5 LiNb0 3 and silica-on- silicon waveguides. LiNb03 The ability to modulate the refractive index is an important requirement for materi- als used in integrated-optic devices, such as light modulators and switches, as we shall see in Chapters 20 and 23. 8.4 PHOTONIC-CRYSTAL WAVEGUIDES Bragg-Grating Waveguide We have seen so far that light may be guided by bouncing between two parallel reflec- tors e.g., planar mirrors as described in Sec. 8.1; or planar dielectric boundaries, at which the light undergoes total internal reflection, as described in Sec. 8.2. Alter- natively, Bragg grating reflectors (BGR) may be used (see Sec. 7.1 C), as illustrated in Fig. 8.4-1. The BGR is a stack of alternating dielectric layers that has special angle- and
312 CHAPTER 8 GUIDED-WAVE OPTICS frequency-dependent reflectance. For a given angle, the reflectance is close to unity at frequencies within a stop band. Similarly, at a given frequency, the reflectance is close to unity within a range of angles, but omnidirectional reflection is also possible. Thus, a wave with a given frequency can be guided through the waveguide by repeated reflec- tions within a range of bounce angles. Within this angular range, the self-consistency condition is satisfied at a discrete set of angles, each corresponding to a propagating mode. The field distribution of a propagating mode is confined principally to the slab; decaying (evanescent) tails reach into the adjacent grating layers, as illustrated in Fig. 8.4-1. . . J..,.".-oq . --... """ ,"",.,.. ""'f _ J I Ii..... "L. - .. - ,..,... 'L 1t:L."..k"".. - ,,- . y Waveguide BGR BGR . . Figure 8.4-1 Planar waveguide made of a dielectric slab sandwiched between two Bragg-grating reflectors (BGR). Bragg-Grating Waveguide as a Photonic Crystal with a Defect Layer If the upper and lower gratings of a Bragg-grating waveguide are identical, and the slab thickness is comparable to the thickness of the periodic layers constituting the gratings, then the entire medium may be regarded as a ID periodic structure, i.e., a ID photonic crystal, but with a defect. For example, the device shown in Fig. 8.4-1 is periodic everywhere except for the slab, which is a layer of different thickness and or different refractive index; the slab may therefore be viewed as a "defective" layer. As described in Sec. 7 .2, a perfect photonic crystal has a dispersion relation, or energy band diagram, containing bandgaps within which no propagating modes exist. In the presence of the "defective" layer, however, a mode whose frequency lies within the bandgap may exist, but it is confined primarily within the layer. Such a mode corresponds to a frequency in the dispersion diagram that lies within the photonic bandgap, as illustrated in Fig. 8.4- 2. Such a frequency is the analog of a defect energy level that lies within the bandgap of a semiconductor crystal. w Defect level :=J:.. Photonic bandgap o K Figure 8.4-2 Dispersion diagram of a photonic crystal with a defect layer. 2D Photonic-Crystal Waveguides Waveguides may also be created by introducing a path of defects in a 2D photonic crystal. In the example illustrated in Fig. 8.4-3(a), a 2D photonic crystal comprising a set of parallel cylindrical holes, placed in a dielectric material at the points of a periodic
8.5 OPTICAL COUPLING IN WAVEGUIDES 313 triangular lattice, exhibits a complete photonic bandgap for waves traveling along di- rections parallel to the plane of periodicity (normal to the cylindrical holes). The defect waveguide takes the form of a line of absent holes. A wave entering the waveguide at frequencies within the photonic bandgap does not leak into the surrounding periodic media so that the light is guided through the waveguide. A typical profile of the field distribution is illustrated in Fig. 8.4-3(a). - - x o 00 ,. -.... / 1//) -- / 17 / I ./ ...- ) /? \ /; /" "\ Z I - ) eee oo. Cv -.. .. -- .--- - . - _.,.:,- . . . ..&I-..- (a) Figure 8.4-3 (a) Propagating mode in a photonic-crystaJ waveguide. (b) L-shaped photonic-crystal waveguide. Moreover, because of the omnidirectional nature of the photonic bandgap, light may be guided through photonic-crystal waveguides with sharp bends and corners without losing energy into the surrounding medium, as illustrated by the L -shaped waveguide configuration shown in Fig. 8.4-3(b). Such behavior is not possible with conventional dielectric waveguides based on total internal reflection. 8.5 OPTICAL COUPLING IN WAVEGUIDES A. Input Couplers Mode Excitation As indicated in previous sections, light propagates in a waveguide in the form of modes. The complex amplitude of the optical field is generally a superposition of these modes, E y,z amu m y exp j{3mz, (8.5-1 ) m where am is the amplitude, U m Y is the transverse distribution (assumed to be real), and {3m is the propagation constant of mode m. The amplitudes of the different modes depend on the nature of the light source used to excite the waveguide. If the source has a distribution that is a perfect match to a spe- cific mode, only that mode will be excited. In general, a source of arbitrary distribution s y excites different modes at different levels. The fraction of power transferred from the source to mode m depends on the degree of similarity between s y and U m Y . To establish this, we write s y as an expansion (a weighted superposition) of the orthogonal functions U m Y , s y am Urn y , (8.5-2) m
314 CHAPTER 8 GUIDED-WAVE OPTICS where the coefficient al, which represents the amplitude of the excited mode l, is 00 al S Y Ul Y dy. (8.5- 3) -00 This expression can be derived by multiplying both sides of (8.5-2) by Ul Y , inte gr at- for l =I m along with the normalization condition. The coefficient al represents the degree of similarity (or correlation) between the source distribution s y and the mode distribution Ul y . Input Couplers Light may be coupled into a waveguide by directly focusing it at one end (Fig. 8.5-1). To excite a given mode, the transverse distribution of the incident light s y should match that of the mode. The polarization of the incident light must also match that of the desired mode. Because of the small dimensions of the waveguide slab, focusing and alignment are usually difficult and coupling using this method is inefficient. Lens )' - --- - ---- - - J.-..- - ........ n'J .... s(y) 1 um(y) .... z n) Figure 8.5-1 Coupling an opti- cal beam into an optical waveguide. In a multimode waveguide, the amount of coupling can be assessed by using a ray-optics ap p roach (Fig. 8.5-2). The guided rays within the waveguide are confined to an angle f)e cos- 1 n2 nl . Because of refraction at the input to the w a veg- uide, this corresponds to an external angle (Ja satisfying NA sin (Ja nl sin (Je nl 1 n2 nl 2 nr n ' where NA is the numerical aperture of the waveg- uide (see Exercise 1.2-5). For maximum coupling efficiency the incident light should be focused within the angle f)a. - Oc n2 --- ._-_._----------- -----_.---------- n} Figure 8.5-2 Focusing rays into a multimode waveguide. Light may also be coupled from a semiconductor source (a light-emitting diode or a laser diode) into a waveguide by simply aligning the ends of the source and the waveguide, leaving a small space that is selected for maximum coupling (Fig. 8.5-3). In light-emitting diodes, light originates from a semiconductor junction region and is emitted in all directions. In a laser diode, the emitted light is confined in a waveg- uide of its own (light-emitting diodes and laser diodes are described in Chapter 17). Other methods of coupling light into waveguides include the use of prisms, diffraction gratings, and other waveguides, as discussed below.
8.5 OPTICAL COUPLING IN WAVEGUIDES 315 -.....- Waveguide . -.... LED or Figure 8.5-3 End butt coupling from a light-emitting diode or laser diode into a waveguide. -- Light emitting . regIon laser diode Prism and Grating Side Couplers Can optical power be coupled into a guided mode of a waveguide by use of a source wave entering from the side at some angle ()i in the cladding, as shown in Fig. 8.5-4(a)? The condition for such coupling is that the axial component of the wavevector of the incident wave, n2ko cas ()i, equals the propagation constant (3m of the guided mode. Since (3m > n2 k o (see Fig. 8.5-4), it is not possible to achieve the required phase matching condition (3m n2 k o cas ()i. The axial component of the wavevector of the incident wave is simply too small. However, the problem may be alleviated by use of a prism or a grating. As illustrated in Fig. 8.5-4(b), a prism of refractive index np > n2 is placed at a short distance d p from the waveguide slab. The incident wave is refracted into the prism where it undergoes total internal reflection at an angle ()p. The incident and reflected waves form a wave traveling in the z direction with propagation constant (3p np ko cas ()p. The transverse field distribution extends into the space separating the prism and the slab, as an evanescent wave that decays exponentially. If the distance d p is sufficiently small, the wave is coupled into a mode of the slab waveguide with a matching propagation constant (3m (3p np ko cas ()p. Since np > n2, phase matching is possible, and if an appropriate interaction distance is selected, significant power can be coupled into the waveguide. The operation may also be reversed to make an output coupler, extracting light from the slab waveguide into free space. The grating [Fig. 8.5-4(c)] addresses the phase-matching problem by modifying the wavevector of the incoming wave. A grating with period A modulates the incom- ing wave by phase factors 27rq Az, where q :1:1, :1:2, . ... These are equivalent to changes of the axial component of the wavevector by factors 27rq A. The phase matching condition can now be satisfied if n2 ko cas ()i + 27rq A (3m, with q 1, for example. The grating may even be designed to enhance the q 1 component. (a) n2 nl n2 Op {3m (b) Prism coupler Figure 8.5-4 Prism and grating side couplers. Prism Incident wave Incident wave np Incident wave d p Grating (3m (c) Grating coupler B. Coupled Waveguides If two waveguides are sufficiently close such that their fields overlap, light can be coupled from one into the other. Optical power can then be transferred between the
316 CHAPTER 8 GUIDED-WAVE OPTICS waveguides, an effect that can be used to make optical couplers and switches. The basic principle of waveguide coupling is presented here; couplers and switches are discussed in Chapters 23 and 24. Consider two parallel planar waveguides made of two slabs of widths d, separation . 2a, and refractive indexes nl and n2, embedded in a medium of refraction index n that is slightly smaller than nl and n2, as illustrated in Fig. 8.5-5. Each of the waveguides is assumed to be single-mode. The separation between the waveguides is such that the optical field outside the slab of one waveguide (in the absence of the other) overlaps slightly with the slab of the other waveguide. y n T II) 2a n 1 d d T z Figure 8.5-5 Coupling between two parallel planar waveguides. At Zl light is mostly in waveguide 1, at Z2 it is divided equally between the two waveguides, and at Z3 it is mostly in waveguide 2. n - - f-c Z2 1..0 Z3 .., ZI The formal approach to studying the propagation of light in this structure is to write Maxwell's equations for the different regions and use the boundary conditions to determine the modes of the overall system. These modes are different from those of each of the waveguides in isolation. An exact analysis is not easy and is beyond the scope of this book. For weak coupling, however, a simplified approximate theory, known as coupled-mode theory, is often satisfactory. Coupled- mode theory assumes that the mode of each waveguide is determined as if the other waveguide were absent. In the presence of both waveguides, the modes are taken to remain approximately unchanged, say Ul y exp j{3lz and U2 y exp j{32Z. Coupling is assumed to modify only the amplitudes of these modes without affecting either their transverse spatial distributions or their propagation constants. The amplitudes of the modes of waveguides 1 and 2 are therefore functions of z, al Z , and a2 Z . The theory is directed toward determining al Z and a2 Z under appropriate boundary conditions. Coupling can be regarded as a scattering effect. The field of waveguide 1 is scattered from waveguide 2, creating a source of light that changes the amplitude of the field in waveguide 2. The field of waveguide 2 has a similar effect on waveguide 1. An analysis of this mutual interaction leads to two coupled differential equations that govern the variation of the amplitudes al Z and a2 Z . It can be shown (see the derivation at the end of this section) that the amplitudes al z and a2 z are governed by two coupled first-order differential equations dal dz da2 dz je 21 exp j {3 z a2 z (8.5-4a) je 12 exp j (3 z al Z , (8.5-4b) Coupled-Mode Equations
8.5 OPTICAL COUPLING IN WAVEGUIDES 317 where D.{3 {3I {32 (8.5-5) is the phase mismatch per unit length and k 2 a+d 1 n 2 n 2 e 2I 0 Ul Y U2 Y dy, - 2 2 {3I a (8.5-6) k 2 a 1 n 2 n 2 e I2 0 U2 Y UI Y dy - 2 1 {32 -a-d are coupling coefficients. We see from (8.5-4) that the rate of variation of al is pro- portional to a2, and vice versa. The coefficient of proportionality is the product of the coupling coefficient and the phase mismatch factor exp j D.{3 z . The coupled-mode equations may be solved by multiplying both sides of (8.5-4a) by exp jD.{3z, taking the derivative with respect to z, substituting from (8.5-4b), and solving the resultant second-order differential equation in al z . The result is: al Z A z al 0 + B z a2 0 C z al 0 + D z a2 0 , a2 z (8.5-7 a) (8.5-7b) where Az D* z j D.{3 z .D.{3 . (8.5-8a) exp 2 cas, z SIll, Z Bz e 2I .D.{3z . (8.5-8b) exp J 2 SIll, Z . J, Cz e I2 .D.{3z . (8.5-8c) exp J 2 SIll, Z . J, are elements of a transmission matrix T that relates the output and input fields and ,2 D.{3 2 2 + e 2 , e e I2 e 2I · (8.5-9) If we assume that no light enters waveguide 2 so that a2 0 powers PI z ex al z 2 and P 2 z ex a2 z 2 are 0, then the optical D.{3 2, 2 sin 2 , z (8.5-10a) PI Z PIO COS 2 ,z + P 2 Z PIO e 2I 2 . 2 2 SIll ,z. , (8.5-10b) Thus, power is exchanged periodically between the two waveguides, as illustrated in Fig. 8.5-6(a). The period is 7r ,.
318 CHAPTER 8 GUIDED-WAVE OPTICS {32, and {3 0, the two e, e I2 e 21 e, and When the waveguides are identical, i.e., ni n2, (3I guided waves are said to be phase matched. In this case, 'Y the transmission matrix takes the simpler form Az Cz Bz Dz cas ez j sin ez j sin ez cas ez . (8.5-11) T Equations (8.5-10) then simplify to PI Z P 2 Z PI 0 cas 2 ez PI 0 sin 2 ez. (8.5-12a) (8.5-12b) The exchange of power between the waveguides can then be complete, as illustrated in Fig. 8.5-6(b). PI (0) Waveguide 1 Waveguide 1 PI (0) Waveguide 2 Waveguide 2 , PI(O) , \ , / "-...1 /, I \ I \ I \ p] (z) , , ,/ \ , / "-...1 " Pl(Z) " ,/ \ ,/ \ , / "'-.J1 - Pl(O) \ \ \ , Lo I \ I \ I \ - P2(Z) \ I \ I \ P2(Z) \ I \ I \ u l I \.. k l \ 00 0 I , z 0 Lo z (a) (b) Figure 8.5-6 Periodic exchange of power between waveguides I and 2: (a) Phase mismatched case; (b) Phase matched case. We thus have a device capable of coupling any desired fraction of optical power from one waveguide into another. At a distance z Lo 7r 2e, called the coupling length or the transfer distance, the power is transferred completely from waveguide I into waveguide 2 [Fig. 8.5-7(a)]. At a distance Lo 2, half the power is transferred, so that the device acts as a 3-dB coupler, i.e., a 50/50 beamsplitter [Fig. 8.5-7(b)]. P P P Y Lo " P/2 (a) (b) P/2 Figure 8.5-7 Optical couplers: (a) switching power from one waveguide to another; (b) a 3-dB coupler.
8.5 OPTICAL COUPLING IN WAVEGUIDES 319 Switching by Control of Phase Mismatch A waveguide coupler of fixed length, Lo 1f 2e for example, changes its power- transfer ratio if a small phase mismatch /:).{3 is introduced. Using (8.5-1 Ob) and (8.5-9), the power-transfer ratio 'J P 2 Lo PI 0 may be written as a function of /:).{3, 'J 1f2 1 sinc 2 4 2 /:).{3 Lo 2 1+ , (8.5-13) Power- Transfer Ratio 7r where sinc x sin 1fX 1fX. Figure 8.5-8 illustrates the dependence of the power- transfer ratio 'J on the mismatch parameter /:).{3 Lo. The ratio achieves a maximum value of unity at /:).{3 Lo 0, decreases with increasing /:).{3 Lo, and then vanishes when /:).{3 Lo 3 7r. l o · fIIIIIt ro I--t I--t rJ'.J s::: ro I--t I--t Q) o 0 0 Figure 8.5-8 Dependence of the power transfer ratio T P 2 ( Lo ) / P 1 (0) on the phase mismatch parameter D..(3 Lo. The waveguide length is chosen such that for D..(3 0 (the phase-matched case), maximum power is transferred to waveguide 2, i.e., T 1. -Y31f Phase mismatch D..{3Lo The dependence of the transferred power on the phase mismatch can be utilized in making electrically activated directional couplers. If the mismatch /:).{3 Lo is switched between 0 and 3 1f, the light is switched from waveguide 2 to waveguide 1. Electrical control of /:).{3 can be achieved if the material of the waveguides is electro-optic (i.e., if its refractive index can be altered by applying an electric field). Such devices will be examined in Chapters 20 and 23 in connection with electro-optic switches. D *Derivation of the Coupled Wave Equations. We proceed to derive the differential equations (8.5-4) that govern the amplitudes al(z) and a2(z) of the coupled modes. When the two waveguides are not interacting they carry optical fields whose complex amplitudes are of the form El (y, z) E 2 (y, z) al Ul (y) exp( j (3lZ) a2 u 2(Y) exp( j(32Z). (8.5-14a) (8.5-14b) The amplitudes a land a2 are then constant. In the presence of coupling, we assume that the am- plitudes al and a2 become functions of z but the transverse functions Ul(Y) and U2(Y), and the propagation constants {3l and {32, are not altered. The amplitudes al and a2 are assumed to be slowly varying functions of z in comparison with the distance {3-1 (the inverse of the propagation constant, (3l or (32), which is of the order of magnitude of the wavelength of light. The presence of waveguide 2 is regarded as a perturbation of the medium outside waveguide 1 in the form of a slab of refractive index n2 n and width d at a distance 2a. The excess refractive index (n2 n) and the field E 2 correspond to an excess polarization density P (f2 f)E 2 fo(n n 2 )E2' which creates a source of optical radiation into waveguide I [see (5.2-25)] Sl ILo82p / 8t 2
320 CHAPTER 8 GUIDED-WAVE OPTICS with complex amplitude 8 1 JL o w 2 P JL o W 2 £o n n 2 E 2 k k 2 E 2 . n n 2 kE2 (8.5-15) Here £2 and £ are the electric permittivities associated with the refractive indexes n2 and n, respec- tively, and k 2 n2ko. This source is present only in the slab of waveguide 2. To determine the effect of such a source on the field in waveguide I, we write the Helmholtz equation in the presence of a source as k k 2 E 2 . (8.5-16a) \72 El + k El 8 1 We similarly write the Helmholtz equation for the wave in waveguide 2 with a source generated as a result of the field in waveguide 1, k k 2 El' (8.5-16b) \72 E 2 + kE2 8 2 where k 1 nl ko. Equations (8.5-16) are two coupled partial different equations that we solve to determine El and E 2 . This type of perturbation analysis is valid only for weakly coupled waveguides. We now write El (y, z) al (z) el (y, z) and E 2 (y, z) a2(z) e2(Y, z), where el (y, z) ul(y)exp( j{31Z) and e2(Y,z) u2(y)exp( j{32 Z ) and note that el and e2 must satisfy the Helmholtz equations, \7 2e l + kiel 0 \72 e2 + ke2 0, (8.5-17a) (8.5-17b) where k l k l k 2 nlko and k 2 n2ko for points inside the slabs of waveguides 1 and 2, respectively, and nko elsewhere. Substituting El aiel into (8.5-16a), we obtain d2al dz 2 el dal del 2 dz dz k 2 2 k 2 U2 e2 · (8.5-18) Noting that al varies slowly, whereas el varies rapidly with z, we neglect the first term of (8.5- 18) in comparison with the second. The ratio between these terms is [(dw /dz)el]/[2wdel/dz] [(d\II /dz)el]/[2w( j{3lel)] j(dw /W)/2{3l dz where \II dal/dz. The approximation is valid if d\II /\II « (3l dz, i.e., if the variation in a 1 (z) is slow in comparison with the length (31 1 . We proceed by substituting el Ul exp( j{31Z) and e2 U2 exp( j/32Z) into (8.5-18). Ne- glecting the first term leads to dal . k k 2 a2 U2(Y) e- j {32 Z . (8.5-19) Multiplying both sides of (8.5-19) by Ul (y), integrating with respect to y, and using the fact that ui (y) is normalized so that its integral is unity, we finally obtain dal -j{31 Z dz e where e 2l is given by (8.5-6). A similar equation is obtained by repeating the procedure for waveg- uide 2. These equations yield the coupled differential equations (8.5-4). . j e 21 a 2 ( z) e - j {32 Z , (8.5-20) c. Periodic Waveguides The analysis of light propagation in two coupled parallel planar waveguides may, in principle, be generalized to multiple waveguides, although the resultant coupled equations are difficult to solve. In the limit of a large number of parallel identical slabs separated by equal distances, the theory of light propagation in periodic media, which is
8.6 SUB-WAVELENGTH METAL WAVEGUIDES (PLASMONICS) 321 presented in Sec. 7.2, may be readily applied. It is instructive to compare the dispersion diagram for light propagation in a slab dielectric waveguide, as shown in Fig. 8.2-8( a), to that for light propagation in a periodic dielectric medium comprising a collection of parallel dielectric slabs, as shown in Fig. 7.2-7. These diagrams are reproduced in Fig. 8.5-9 for comparison. t d= t nz W .. t 3we 2we We o o Propagation constant {3 = k z (a) nz n} A t W t 2w1J ....... o. o o . .. W1J ...... W=C1 3 o 9 Propagation constant {3 = k z (b) Figure 8.5-9 Dispersion diagram of (a) slab waveguide with cutoff angular frequency We (7r/d)(c o /NA); (b) periodic waveguide with Bragg angular frequency Wp, = (7r/A)(c o /n). In the single-slab waveguide, light travels in modes, each with a dispersion line lying in the region between the light lines w == cl{3 and w == c2{3. At any frequency, there is at least one mode. In the periodic waveguide, the dispersion lines broaden into bands separated by photonic bandgaps. Here, we assume that the modes travel in a direction parallel to the layers (the z direction in Fig. 8.5-9, which corresponds to the x direction in Fig. 7.2-7), so that the bands also lie in the region between the light lines. 8.6 SUB-WAVELENGTH METAL WAVEGUIDES (PLASMONICS) As shown in earlier sections of this chapter, it is difficult to confine an optical wave to dimensions much smaller than a wavelength (see also Sec. 4.4D). In the mirror waveguide described in Sec. 8.1, for example, a wave of wavelength A cannot be guided if the mirror separation d is smaller than A/2 (since the wave frequency would then be smaller than the cutoff frequency c/2d). In the slab dielectric waveguide described in Sec. 8.2, if the width d is reduced below A/2, only a single mode can be supported, and if d is reduced further, there is substantial leakage of the guided wave into the cladding. Light can, however, be confined and guided at the sub-wavelength scale by the use of sub-wavelength metallic structures, such as thin films and metallic particles buried in dielectric media. This approach has become feasible in recent years as a result of advances in nanotechnology (nanostructures and nanoparticles), and the field is known as plasmonics.
322 CHAPTER 8 GUIDED-WAVE OPTICS The propagation of light in a bulk metal was described in Sec. 5.5D. It was shown that at frequencies below the plasma frequency, the optical wave decays with an at- tenuation coefficient that decreases as the frequency increases, and vanishes at the plasma frequency; the free electrons then undergo longitudinal oscillations associated with plasmons. Clearly, bulk metals cannot confine and guide optical waves. At a metal-dielectric interface, however, Maxwell's equations admit solutions in the form of charge-density waves coupled with optical waves, generally referred to as surface plasmon polaritons (SPPs). The conduction electrons oscillate in the longitudinal direction and the electromagnetic field is confined to sub-wavelength dimensions near the surface of the metal. These coupled waves can be excited at frequencies below the plasma frequency and become most localized at the plasma frequency. SPPs allow light to be controlled and manipulated at the nanometer spatial scale, while retaining the high temporal frequency associated with optical waves. Waveguides based on SPPs can, for example, be implemented by using a dielectric slab surrounded by metallic claddings. The width of the slab must be sufficiently small for the confined SPP waves at the claddings to overlap, thereby permitting the coupled SPP waves to be guided. The dispersion relation for such a structure may be obtained by matching the boundary conditions at the dielectric-metal interfaces using, e.g., the Drude model for the metal (see Sec. 5.5D). For sufficiently small slab thicknesses, large propagation constants can be achieved even for frequencies far below the bulk-metal plasma frequency. These plasmonic waveguides are made of metal/insulator/metal (MIM) heterostructures of submicrometer dimensions. Modes at near-infrared wave- lengths can be localized at the nanometer scale, but the propagation length is limited. Another class of plasmonic waveguides with subwavelength mode size makes use of arrays of metallic nanoparticles that are sufficiently close so that their localized plasmonic fields overlap. Such metamaterials (see Sec. 5.7) admit guided modes of submicrometer size at frequencies of the individual particle plasmons or at the inter- particle gap resonance. Plasmonics seeks to couple the domains of highly integrated electronics (with di- mensions < 100 nm) and optical-frequency photonics (with bandwidths> 100 THz). It is envisioned to have a number of valuable applications in nano-optics, including intrachip interconnects; the transmission of light through objects that are ordinarily opaque (as a result of plasmon excitations at nanosize holes in the material); the cre- ation of distributed point sources of light generated at the surfaces of metallic-coated nanosize objects; and devices such as nanoantennas, nanoresonators, and nanowaveg- uides that are analogous to electrical circuit elements but operate in the visible region of the spectrum. Biosensing applications, based on the sensitivity of plasmon excita- tions to the properties of a dielectric medium surrounding a metallic nanostructure, include measurements of the thickness of colloidal films as weB as the screening and quantifying of protein binding events. READING LIST Books C.-L. Chen, Foundations for Guided Wave Optics, Wiley, 2006. K. Iga and Y. Kokobun, eds., Encyclopedic Handbook of Integrated Optics, CRC Press, 2006. A. Sharma, ed., Guided Wave Optics, Anshan, 2006. K. Okamoto, Fundamentals of Optical Waveguides, Elsevier, 2nd ed. 2005. B. P. Pal, ed., Guided Wave Optical Components and Devices: Basics, Technology, and Applications, Academic Press, 2005. G. T. Reed and A. P. Knights, Silicon Photonics, Wiley, 2004.
PROBLEMS 323 G. Lifante, Integrated Photonics: Fundamentals, Wiley, 2003. C. Pollock and M. Lipson, Integrated Photonics, Kluwer, 2003. A. A. Barybin and V. A. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory: Application to Guided- Wave Optics, Rinton Press, 2002. R. G. Hunsperger, Integrated Optics: Theory and Technology, Springer-Verlag, 1982, 5th ed. 2002. K. Iizuka, Elements of Photonics, Volume 2: For Fiber and Integrated Optics, Wiley, 2002. R. W. Waynant and J. K. Lowell, Electronic and Photonic Circuits and Devices, IEEE Press, 1998. W. B. Leigh, Devices for Optoelectronics, Marcel Dekker, 1996. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, Wiley, 1995. H. P. Zappe, Introduction to Semiconductor Integrated Optics, Artech, 1995. R. G. Hunsperger ed. Photonic Devices and Systems Marcel Dekker, 1994. Y. Suematsu and A. R. Adams, eds., Handbook of Semiconductor Lasers and Photonic Integrated Circuits, Chapman & Hall, English ed. 1994. O. Wada, ed., Optoelectronic Integration: Physics, Technology, and Applications, Kluwer, 1994. K. 1. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors, Springer- Verlag, 1993. A. R. Mickelson, Guided Wave Optics, Springer-Verlag, 1993. L. A. Hornak, ed., Polymers for Lightwave and Integrated Optics: Technology and Applications, Marcel Dekker, 1992. J. E. Midwinter and Y. L. Guo, Optoelectronics and Lightwave Technology, Wiley, 1992. R. Syms and J. Cozens, Optical Guided Waves and Devices, McGraw-Hill, 1992. M. Young, Optics and Lasers: Including Fibers and Optical Waveguides, Springer-Verlag, 4th revised ed. 1992. D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press, 1974, 2nd ed. 1991. T. Tamir, ed., Guided- Wave Optoelectronics, Springer-Verlag, 2nd ed. 1990. D. Marcuse, Light Transmission Optics Van Nostrand Reinhold 1972 2nd ed. 1982; Krieger reis- sued 1989. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits, McGraw-Hill, 1989. S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation, Academic Press, 1986. Articles Issue on nanophotonics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 6, 2006. M. Paniccia and S. Koehl, The Silicon Solution, IEEE Spectrum, vol. 42, no. 10, pp. 38-43, 2005. Issue on integrated optics and optoelectronics, Part II, IEEE Journal of Selected Topics in Quantum Electronics, vol. 11, no. 2, 2005. Issue on integrated optics and optoelectronics, Part I, IEEE Journal of Selected Topics in Quantum Electronics, vol. 11, no. 1, 2005. Special issue on integrated optics, Applied Physics B: Lasers and Optics, vol. 73, no. 5-6, 2001. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. Issue on integrated optics and optoelectronics, IEEE Journal of Selected Topics in Quantum Elec- tronics, vol. 6, no. 1, 2000. D. G. Hall, ed., Selected Papers on Coupled Mode Theory in Guided- Wave Optics, SPIE Optical Engineering Press (Milestone Series Volume 84), 1993. PROBLEMS 8.1-3 Field Distribution. (a) Demonstrate that a single TEM plane wave Ex(Y, z)=A exp( -jkyY) exp( -j{3z) cannot satisfy the boundary conditions, Ex (-T-.d /2, z) == 0 at all z, in the mirror waveguide il1ustrated in Fig. 8.1-1.
324 CHAPTER 8 GUIDED-WAVE OPTICS (b) Show that the sum of two TEM plane waves written as Ex(Y, z) == Al exp( -jkyIY) exp( -j{3IZ) + A 2 exp( -jk y2 Y) exp( -j{32Z) does not satisfy the boundary conditions if Al == :l::A 2 , {3I == {32, and k yl == -k y2 == m7r / d where m == 1,2, . . .. 8.1-4 Modal Dispersion. Light of wavelength Ao == 0.633 J-Lm is transmitted through a mirror waveguide of mirror separation d == 10 J-Lm and n == 1. Determine the number ofTE and TM modes. Determine the group velocities of the fastest and the slowest mode. If a narrow pulse of light is carried by all modes for a distance 1 m in the waveguide, how much does the pulse spread as a result of the differences of the group velocities? 8.2-3 Parameters of a Dielectric Waveguide. Light of free-space wavelength Ao == 0.87 J-LID is guided by a thin planar film of width d == 2 J-Lm and refractive index ni == 1.6 surrounded by a medium of refractive index n2 == 1.4. (a) Determine the critical angle ()e and its complement ()e, the numerical aperture NA, and the maximum acceptance angle for light originating in air (n == 1). (b) Determine the number of TE modes. (c) Determine the bounce angle () and the group velocity v of the m == 0 TE mode. 8.2-4 Effect of Cladding. Repeat Prob. 8.2-3 if the thin film is suspended in air (n2 == 1). Compare the results. 8.2-5 Field Distribution. The transverse distribution U m (y) of the electric-field complex amplitude of a TE mode in a slab waveguide is given by (8.2-10) and (8.2-13). Derive an expression for the ratio of the proportionality constants. Plot the distribution of the m == a TE mode for a slab waveguide with parameters ni == 1.48, n2 == 1.46, d == 0.5 J-Lm, and Ao == 0.85 J-Lm, and determine its confinement factor (percentage of power in the slab). 8.2-6 Derivation of the Field Distributions Using Maxwell's Equations. Assuming that the electric field in a symmetric dielectric waveguide is harmonic within the slab and exponential outside the slab and has a propagation constant {3 in both media, we may write Ex (y, z) == u(y) e- j {3z, where ... { ACOS(k y y + <p), u(y) == Bexp(-,y), B exp(,y), -d/2 < Y < d/2, y > d /2, y < d /2. For the Helmholtz equation to be satisfied, k + {32 == ni k and _,2 + (32 == nk. Use Maxwell's equations to derive expressions for Hy(Y, z) and Hz(y, z). Show that the boundary conditions are satisfied if {3, " and ky take the values (3m, ,m, and k ym derived in the text and verify the self-consistency condition (8.2-4). 8.2-7 Single-Mode Waveguide. What is the largest thickness d of a planar symmetric dielectric waveguide with refractive indexes ni == 1.50 and n2 == 1.46 for which there is only one TE mode at Ao == 1.3 J-Lm? What is the number of modes if a waveguide with this thickness is used at Ao == 0.85 J-Lm instead? 8.2-8 Mode Cutoff. Show that the cutoff condition for TE mode m > a in a symmetric slab waveguide with ni n2 is approximately A 8nln d 2 /m2, where n == ni - n2. 8.2-9 TM Modes. Derive an expression for the bounce angles of the TM modes similar to (8.2-4). Use a computer to generate a plot similar to Fig. 8.2-2 for TM modes in a waveguide with sinO e == 0.3 and A/2d == 0.1. What is the number ofTM modes? 8.3-1 Modes of a Rectangular Dielectric Waveguide. A rectangular dielectric waveguide has a square cross section of area 10- 2 mm 2 and numerical aperture NA == 0.1. Use (8.3-3) to plot the number of TE modes as a function of frequency v. Compare your results with Fig. 8.2-4. 8.4-1 Coupling Coefficient Between Two Slabs. (a) Use (8.5-6) to determine the coupling coefficient between two identical slab waveguides of width d == 0.5 J-Lm, spacing 2a == 1.0 J-Lm, refractive indexes ni == n2 == 1.48, in a medium of refractive index n == 1.46, at Ao == 0.85 J-Lm. Assume that both waveguides are operating in the m == 0 TE mode and use the results of Prob. 8.2-5 to determine the transverse distributions. (b) Determine the length of the waveguides so that the device acts as a 3-dB coupler.
C HAP T E R . FIBER OPTICS 9.1 GUIDED RAYS 327 A. Step-Index Fibers B. Graded-Index Fibers 9.2 GUIDED WAVES 331 A. Step-Index Fibers B. Single-Mode Fibers *C. Quasi-Plane Waves in Step- and Graded-Index Fibers 9.3 ATTENUATION AND DISPERSION 348 A. Attenuation B. Dispersion 9.4 HOLEY AND PHOTONIC-CRYSTAL FIBERS 359 v .... , '" Charles Kao (born 1933) promulgated the concept of using low-loss optical fibers in practical telecommunications systems. Philip St John Russell (born 1953) invented the photonic-crystal fiber in 1991; it has found use in many applications. 325
An optical fiber is a cylindrical dielectric waveguide made of a low-loss material, such as silica glass. It has a central core in which the light is guided, embedded in an outer cladding of slightly lower refractive index (Fig. 9.0-1). Light rays incident on the core-cladding boundary at angles greater than the critical angle undergo total internal reflection and are guided through the core without refraction into the cladding. Rays at greater inclination to the fiber axis lose part of their power into the cladding at each reflection and are not guided. 1 a --- f - - . . t b __l__ C(( ))) Cladding Core n2 < n I fll Figure 9.0-1 An optical fiber is a cylindrical dielectric waveguide with an inner core and an outer cladding. Remarkable technological advances in the fabrication of optical fibers over the past two decades allow light to be guided through 1 km of glass fiber with a loss as low as 0.15 dB ( 3.4%) at the wavelength of maximum transparency. Because of this low loss, optical fibers long ago replaced copper coaxial cables as the preferred transmission medium for terrestrial and sub-oceanic voice and data communications. In this chapter we introduce the principles of light transmission in optical fibers. These principles are essentially the same as those applicable to planar dielectric waveg- uides (Chapter 8); the most notable distinction is that optical fibers have cylindrical geometry. In both types of waveguide, light propagates in the form of modes. Each mode travels along the axis of the waveguide with a distinct propagation constant and group velocity, maintaining its transverse spatial distribution and polarization. In a planar dielectric waveguide, each mode is described as the sum of the multiple reflections of a TEM wave bouncing within the slab, in the direction of an optical ray at a certain bounce angle. This approach is approximately applicable to cylindrical waveguides as well. When the core diameter is small, only a single mode is supported and the fiber is said to be a single-mode fiber. Fibers with large core diameters are multimode fibers. One of the difficulties as- sociated with the propagation of light in a multimode fiber arises from the differences among the group velocities of the modes. This results in a spread of travel times and results in the broadening of a light pulse as it travels through the fiber. This effect, called modal dispersion, limits how often adjacent pulses can be launched without resulting in pulse overlap at the far end of the fiber. Modal dispersion therefore limits the speed at which multi mode optical fiber communications systems can operate. Modal dispersion can be reduced by grading the refractive index of the fiber core from a maximum value at its center to a minimum value at the core-cladding boundary. The fiber is then called a graded-index fiber, or GRIN fiber, whereas conventional fibers with constant refractive indexes in the core and the cladding are known as step- index fibers. In a graded-index fiber the travel velocity increases with radial distance from the core axis (since the refractive index decreases). Although rays of greater inclination to the fiber axis must travel farther, they thus travel faster. This permits the travel times of the different modes to be equalized. In summary, optical fibers are classified as step-index or graded-index, and multi- mode or single-mode, as illustrated in Fig. 9.0-2. 326
9.1 GUIDED RAYS 327 Step- Index MMF n2 SMF n2 - ----- ----- n] GRIN MMF n2 n] Figure 9.0-2 Geometry, refractive-index profile, and typical rays in a step-index multimode fiber (MMF), a single-mode fiber (SMF), and a graded-index multimode fiber (GRIN MMF). This Chapter This chapter begins with ray-optics descriptions of step-index and graded-index fibers (Sec. 9.1). An electromagnetic-optics approach, emphasizing the nature of optical modes and single-mode propagation, follows in Sec. 9.2. The opticaJ properties of the fiber material (usually fused silica), including attenuation and material dispersion, as well as modal, waveguide, and polarization-mode dispersion, are discussed in Sec. 9.3. Since fibers are usually used to transmit information in the form of optical pulses, a brief introduction to pulse propagation in fibers is also provided in Sec. 9.3. Holey and photonic-crystal fibers, which have more complex refractive-index profiles, and unusual dispersion characteristics, are introduced in Sec. 9.4. We return to this topic in Chapters 22 and 24, which are devoted to ultrafast optics and optical fiber communications systems, respectively. 9.1 GUIDED RAYS A. Step-Index Fibers A step-index fiber is a cylindrical dielectric waveguide specified by the refractive indexes of its core and cladding, nl and n2, respectively, and their radii a and b (see Fig. 9.0-1). Examples of standard core-to-cladding diameter ratios (in units of /-lm/ /-lm) are 2a/2b == 8/125,50/125,62.5/125,85/125, and 100/140. The refractive indexes of the core and cladding differ only slightly, so that the fractional refractive- index change is small: 2 2 A == n 1 - n 2 nl - n2 u « 1. - 2ni nl (9.1-1) Most fibers used in currently implemented optical communication systems are made of fused silica glass (Si0 2 ) of high chemical purity. Slight changes in the refractive index are effected by adding low concentrations of doping materials (e.g., titanium, germanium, boron). The refractive index nl ranges from 1.44 to 1.46, depending on the wavelength, and typically lies between 0.001 and 0.02.
328 CHAPTER 9 FIBER OPTICS An optical ray in a step-index fiber is guided by total internal reflections within the fiber core if its angle of incidence at the core-cladding boundary is greater than the critical angle ()c == sin-1(n2/nl), and remains so as the ray bounces. Meridional Rays Meridional rays, which are rays confined to planes that pass through the fiber axis, have a particularly simple guiding condition, as shown in Fig. 9.1-1. These rays intersect the fiber axis and reflect in the same plane without changing their angle of incidence, behaving as if they were in a planar waveguide. Meridional rays are guided if the angle () they make with the fiber axis is smaller than the complement of the critical angle, i.e., if() < ()c == 1T/2 - ()c == cos-1(n2/nl). Since nl n2, ()c is usually small and the guided rays are approximately paraxial. Figure 9.1-1 The trajectory of a meridional ray lies in a plane that passes through the fiber axis. The ray is guided if () < Be == cos- 1 (n2/nl). Skewed Rays An arbitrary ray is identified by its plane of incidence, which is a plane parallel to the fiber axis through which the ray passes, and by the angle with that axis, as illustrated in Fig. 9.1-2. The plane of incidence intersects the core-cladding cylindrical boundary at an angle cp with respect to the normal to the boundary and lies at a distance R from the fiber axis. The ray is identified by its angle () with the fiber axis and by the angle <p of its plane. When <p f= 0 (R f= 0) the ray is said to be skewed. For meridional rays <p == 0 and R == O. A skewed ray reflects repeatedly into planes that make the same angle <p with the core-cladding boundary; it follows a helical trajectory confined within a cylindrical shell of inner and outer radii R and a, respectively, as illustrated in Fig. 9.1-2. The projection of the trajectory onto the transverse (x-y) plane is a regular polygon that is not necessarily closed. The condition for a skewed ray to always undergo total internal reflection is that its angle with the z axis be smaller than the complementary critical angle, i.e., () < () c. yt ( Figure 9.1-2 A skewed ray lies in a plane offset from the fiber axis by a distance R. The ray is identified by the angles () and cjJ. It follows a helical trajectory confined within a cylindrical shell with inner and outer radii R and a, respectively. The projection of the rayon the transverse plane is a regular polygon that is not necessarily closed.
9.1 GUIDED RAYS 329 Numerical Aperture A ray incident from air into the fiber becomes a guided ray if, upon refraction into the core, it makes an angle () with the fiber axis that is smaller than () c. As shown in Fig. 9.1-3(a), if Snell's law is applied at the air-core boundary, the angle ()a in - - air corresponding to the angle ()c in the co re is obt ained from 1 . sin Ba == n1 si n Bc, which leads to sin()a == n1 V I - cos 2 ()c == n1 V I - (n2/n1)2 == v ni - n (see Exercise 1.2-5). The acceptance angle of the fiber is therefore ()a == sin -1 NA, (9.1-2) where the numerical aperture (NA) of the fiber is given by NA = J n - n n 1 V2Ll (9.1-3) Numerical Aperture since n1 - n2 == nl and n1 + n2 2n1. The acceptance angle ()a of the fiber determines the cone of external rays that are guided by the fiber. Rays incident at angles greater than B a are refracted into the fiber but are guided only for a short distance since they do not undergo total internal reflection. The numerical aperture therefore describes the light-gathering capacity of the fiber, as illustrated in Fig. 9.1-3(b). When the guided rays arrive at the terminus of the fiber, they are refracted back into a cone of angle ()a. The acceptance angle is thus a crucial design parameter for coupling light into and out of a fiber. (a) (b) =>t==: 5 f:=><><><><><2K Figure 9.1-3 (a) The acceptance angle ()a of a fiber. Rays within the acceptance cone are guided by total internal reflection. The numerical aperture NA == sin ()a. The angles ()a and ()e are typically quite small; they are exaggerated here for clarity. (b) The light-gathering capacity of a large NA fiber is greater than that of a small NA fiber. EXAMPLE 9.1-1. Cladded and Uncladded Fibers. In a silica-glass fiber with nl == 1.46 and == (nl-n2)/nl == 0.01, the complementary critical angle Be == cos-1(n2/nl) == 8.1°, and the acceptance angle ()a == 11.9°, corresponding to a numerical aperture NA == 0.206. By comparison,
330 CHAPTER 9 FIBER OPTICS a fiber with silica-glass core (n! = 1.46) and a cladding with a much smaller refractive index n2 = 1.064 has Be = 43.2°, ()a = 90°, and NA = 1. Rays incident from all directions are guided since they eflect within a cone of angle Be = 43.2° inside the core. Likewise, for an uncladded fiber (n2 = 1), () e = 46.8°, and rays incident from air at any angle are also refracted into guided rays. Although its light-gathering capacity is high, the uncladded fiber is generally not suitable for use as an optical waveguide because of the large number of modes it supports, as will be explained subsequently. B. Graded-Index Fibers Index grading is an ingenious method for reducing the pulse spreading caused by differences in the group velocities of the modes in a multimode fiber. The core of a graded-index (GRIN) fiber has a refractive index that varies; it is highest in the center of the fiber and decreases gradually to its lowest value where the core meets the cladding. The phase velocity of light is therefore minimum at the center and increases gradually with radial distance. Rays of the most axial mode thus travel the shortest distance, but they do so at the smallest phase velocity. Rays of the most oblique mode zigzag at a greater angle and travel a longer distance, but mostly in a medium where the phase velocity is high. The disparities in distances are thus compensated by opposite disparities in the phase velocities. As a consequence, the differences in the travel times associated with a light pulse are reduced. In this section we examine the propagation of light in GRIN fibers. The core refractive index of a GRIN fiber is a function n( r) of the radial position r. As illustrated in Fig. 9.1-4, the largest value of n(r) is at the core center, n(O) == nl, while the smallest value occurs at the core radius, n( a) == n2. The cladding refractive index is maintained constant at n2. Cladding --------------------------- --Q ------- Core n 2 n 1 n Figure 9.1-4 Geometry and refractive- index profile of a graded-index optical fiber. A versatile refractive-index profile that exhibits this generic behavior is described as the power-law function n 2 (r) = ni [1 - 2 c r tl] , r < a, (9.1-4 ) where 2 2 n - n nl - n2 A == 1 2 '"'-' u 2 '"'-' 2n 1 nl (9.1-5) The grade profile parameter p determines the steepness of the profile. As illustrated in Fig. 9.1-5, n 2 (r) is a linear function of r for p == 1 and a quadratic function for p == 2 The quantity n 2 (r) becomes increasingly steep as p becomes larger, and ultimately approaches a step function for p --t 00. The step-index fiber is thus a special case of the GRIN fiber.
9.2 GUIDED WAVES 331 Core n 2 n 2 1 Figure 9.1-5 Power-law refractive-index profile n2 ( r) for various values of p. Cladding --------------------------- --Q ------- The transmission of light rays through a GRIN medium with parabolic-index profile was discussed in Sec. 1.3. Rays in meridional planes follow oscillatory planar trajec- tories, whereas skewed rays follow helical trajectories with the turning points forming cylindrical caustic surfaces, as illustrated in Fig. 9.1-6. Guided rays are confined within the core and do not reach the cladding. (a) I I I I I I ---- . Or[ Rf a r ---- (b) Figure 9.1-6 Guided rays in the core of a GRIN fiber. (a) A meridional ray confined to a meridional plane inside a cylinder of radius Ro. (b) A skewed ray follows a helical trajectory confined within two cylindrical shells of radii rl and Rlo The numerical aperture of a GRIN optical fiber may be determined by identifying the largest angle of the incident ray that is guided within the GRIN core without reaching the cladding. For meridional rays in a GRIN fiber with parabolic profile, the numerical aperture is given by (9.1-3) (see Exercise 1.3-2). 9.2 GUIDED WAVES We now proceed to develop an electromagnetic-optics theory of light propagation in fibers. We seek to determine the electric and magnetic fields of guided waves by using Maxwell's equations and the boundary conditions imposed by the cylindrical dielectric core and cladding. As in all waveguides, there are certain special solutions, called modes (see Appendix C), each of which has a distinct propagation constant, a char- acteristic field distribution in the transverse plane, and two independent polarization states. Since an exact solution is rather difficult, a number of approximations will be used.
332 CHAPTER 9 FIBER OPTICS Helmholtz Equation The optical fiber is a dielectric medium with refractive index n( r). In a step-index fiber, n(r) == nl in the core (r < a) and n(r) == n2 in the cladding (r > a). In a GRIN fiber, n(r) is a continuous function in the core and has a constant value n(r) == n2 in the cladding. In either case, we assume that the outer radius b of the cladding is sufficiently large so that it can be taken to be infinite when considering guided light in the core and near the core-cladding boundary. Each of the components of the monochromatic electric and magnetic fields obeys the Helmholtz equation, \12U + n2(r)kU == 0, where ko == 21r/Ao. This equation is obeyed exactly in each of the two regions of the step-index fiber, and is obeyed approximately within the core of the GRIN fiber if n( r) varies slowly within a wave- length (see Sec. 5.3). In a cylindrical coordinate system (see Fig. 9.2-1) the Helmholtz equation is written as [J2U 1 [JU 1 [J2U [J2U 8r 2 + r 8r + r 2 84J2 + 8z2 + n 2 k;U = 0, (9.2-1) where U == U(r, 1, z). The guided modes are waves traveling in the z direction with propagation constant (3, so that the z dependence of U is of the form e- j {3z. They are periodic in the angle 1 with period 21r, so that they take the harmonic form e-jl<p, where [ is an integer. Substituting U(r, 1, z) == u(r)e- jl <P e -j{3z, [ == 0, ::!:1, ::!:2, . . . (9.2-2) into (9.2-1) leads to an ordinary differential equation for the radial profile u( r ): d 2 u 1 du ( 2 2 2 [2 ) - + -- + n (r)k - (3 - - u == o. dr 2 r dr 0 r 2 (9.2-3) x Er '" --...." Z \ "',--...., / / I ' E " I Ez \ <P I r I I I , , I I I , I \'..... ",,,,//' I I I / ",/// z Figure 9.2-1 Cylindrical coordinate system. A. Step-Index Fibers As we found in Sec. 8.2B, the wave is guided (or bound) if the propagation constant is smaller than the wavenumber in the core ((3 < nl ko) and greater than the wavenumber in the cladding ((3 > n2ko). It is therefore convenient to define kf == nik - (32 (9.2-4a) and ry2 == (3 2 _ n 2 k 2 I 2 0' (9 .2-4b )
9.2 GUIDED WAVES 333 so that for guided waves k} and ')'2 are positive and k T and')' are real. Equation (9.2-3) may then be written in the core and cladding separately: d 2 u 1 du ( Z2 ) dr 2 + r dr + k - r 2 u = 0, d 2 u 1 du ( Z2 ) - + -- - ')'2 + _ u == 0, dr 2 r dr r 2 r < a (core), (9.2-5a) r > a (cladding). (9.2-5b) Equations (9.2-5) are well-known differential equations whose solutions are the family of Bessel functions. Excluding functions that approach 00 at r == 0 in the core, or r ---t 00 in the cladding, we obtain the bounded solutions: u ( r) ex: { Jl ( kTr ) , Kl ( ')'r ) , r < a (core) r > a (cladding), (9.2-6) where Jl (x) is the Bessel function of the first kind and order Z, and Kl (x) is the modified Bessel function of the second kind and order Z. The function Jl (x) oscillates like the sine or cosine function but with a decaying amplitude. The function Kl (x) decays exponentially at large x. Two examples of the radial distribution u(r) are displayed in Fig. 9.2-2. u(r) 1/=01 u(r) 1/= 3\ 0 0 a r 00 Figure 9.2-2 Examples of the radial distribution u(r) provided in (9.2-6) for I == 0 and I == 3. The shaded and un shaded areas represent the fiber core and cladding, respectively. The parameters k T and " and the two proportionality constants in (9.2-6), have been selected such that u(r) is continuous and has a continuous derivative at r == a. Larger values of k T and ,lead to a greater number of oscillations in u( r ). The parameters k T and')' determine the rate of change of u( r) in the core and in the cladding, respectively. A large value of k T means more oscillation of the radial distri- bution in the core. A large value of ')' means more rapid decay and smaller penetration of the wave into the cladding. As can be seen from (9.2-4), the sum of the squares of k T and')' is a constant, k 2 + '"'1/2 == ( n2 _ n 2 ) k2 == ( NA ) 2 . k 2 T I 1 2 0 0' (9.2-7) so that as k T increases, ')' decreases and the field penetrates deeper into the cladding. For those values of k T that exceed NA . ko, the quantity')' becomes imaginary and the wave ceases to be bound to the core.
334 CHAPTER 9 FIBER OPTICS Fiber V Parameter It is convenient to normalize k T and 'Y by defining the quantities x == kTa, Y == 'Ya. (9.2-8) In view of (9.2-7), X 2 + y 2 == V 2 , (9.2-9) where V == NA . koa, from which a V = 2n Ao NA. (9.2-10) V Parameter It is important to keep in mind that for the wave to be guided, X must be smaller than V. As we shall see shortly, V is an important parameter that governs the number of modes of the fiber and their propagation constants. It is called the fiber parameter or the V parameter. It is directly proportional to the radius-to-wavelength ratio aj .Ao, and to the numerical aperture NA. Equation (9.2-10) is not unlike (8.2-7) for the number of TE modes in a planar dielectric waveguide. Modes We now consider the boundary conditions. We begin by writing the axial components of the electric- and magnetic-field complex amplitudes, Ez and Hz, in the form of (9.2-2). The condition that these components must be continuous at the core-cladding boundary r == a establishes a relation between the coefficients of proportionality in (9.2-6), so that we have only one unknown for Ez and one unknown for Hz. With the help of Maxwell's equations, jWE o n 2 E == V' x Hand - jWJ.1oH == \7 x E [see (5.3-12) and (5.3-13)], the remaining four components, E4J, H4J, Er, and Hr, are determined in terms of Ez and Hz. Continuity of 4J and H4J at r == a yields two additional equations. One equation relates the two unknown coefficients of proportionality in Ez and Hz; the other provides a condition that the propagation constant (3 must satisfy. This condition, called the characteristic equation or dispersion relation, is an equation for (3 with the ratio aj.Ao and the fiber indexes nl, n2 as known parameters. For each azimuthal index l, the characteristic equation has multiple solutions yield- ing discrete propagation constants (3zm, m == 1,2,..., each solution representing a mode. The corresponding values of k T and 'Y, which govern the spatial distributions in the core and in the cladding, respectively, are determined by using (9.2-4) and are denoted k Tlm and 'YZm. A mode is therefore described by the indexes land m, characterizing its azimuthal and radial distributions, respectively. The function u( r) depends on both land m; l == 0 corresponds to meridional rays. Moreover, there are two independent configurations of the E and H vectors for each mode, corresponding to the two states of polarization. The classification and labeling of these configurations are generally quite involved (details are provided in specialized books in the reading Ii s t) . Characteristic Equation (Weakly Guiding Fiber) Most fibers are weakly guiding (i.e., nl n2 or « 1) so that the guided rays are paraxial, i.e., approximately parallel to the fiber axis. The longitudinal components of the electric and magnetic fields are then far weaker than the transverse components and
9.2 GUIDED WAVES 335 the guided waves are approximately transverse electromagnetic (TEM) in nature. The linear polarization in the x and y directions then form orthogonal states of polarization. The linearly polarized (l, m) mode is usually denoted as the LPz m mode. The two polarizations of mode (l, m) travel with the same propagation constant and have the same spatial distribution. For weakly guiding fibers the characteristic equation obtained using the procedure outlined earlier turns out to be approximately equivalent to the conditions that the scalar function u(r) in (9.2-6) is continuous and has a continuous derivative at r == a. These two conditions are satisfied if (kTa)J{(kTa) Jz(kTa) ('Ya)K{ ("fa) Kz ("fa) (9.2-11) The derivatives J{ and K{ of the Bessel functions satisfy the identities , (x) Jz (x) == ::I:: JZ=Fl (x) l x , Kz(x) Kz (x) == - KZ=Fl (x) l . x (9.2-12) (9.2-13) Substituting these identities into (9.2-11) and using the normalized parameters X == kTa and Y == "fa leads to the characteristic equation X Jl::l::1 (X) = ::I:: Y Kl::l::1 ( Y) Y = .J V 2 _ X 2 . Jz ( X) Kz (Y) , (9.2-14) Characteristic Equation Given V and l, the characteristic equation contains a single unknown variable X. Note that J-z(x) == (-l)zJ z (x) and K-z(x) == Kz(x), so that the equation remains unchanged if l is replaced by -l. The characteristic equation may be solved graphically by plotting its right- and left- hand sides (RHS and LHS, respectively) versus X and finding the intersections. As illustrated in Fig. 9.2-3 for l == 0, the LHS has multiple branches whereas the RHS decreases monotonically with increasing X until it vanishes at X == V (Y == 0). There are therefore multiple intersections in the interval 0 < X < V. Each intersection point corresponds to a fiber mode with a distinct value of X. These values are denoted X Zm , m == 1,2,..., Alz in order of increasing X. Once the X Zm are found, (9.2-8), (9.2-4), and (9.2-6) allow us to determine the corresponding transverse propagation constants k TZm , the decay parameters 'YIm, the propagation constants (JZm, and the radial distribution functions uzm(r). The graph in Fig. 9.2-3 is similar in character to that in Fig. 8.2-2, which governs the modes of a planar dielectric waveguide. Each mode has a distinct radial distribution. As examples, the two radial distribu- tions u(r) illustrated in Fig. 9.2-2 correspond to the LP ol mode (l == 0, m == 1) in a fiber with V == 5, and the LP34 mode (l == 3, m == 4) in a fiber with V == 25, respectively. Since the (l, m) and (-l, m) modes have the same propagation constant, it is of interest to examine the spatial distribution of their equal-weight superposition. The complex amplitude of the sum is proportional to uzm(r) cos lcp exp( - j (JZmz). The intensity, which is proportional to ufm (r) cos 2 lcp, is illustrated in Fig. 9.2-4 for the LP ol and LP 34 modes (the same modes for which u(r) is displayed in Fig. 9.2-2).
336 CHAPTER 9 FIBER OPTICS -------- 1": , , I I I I I , , , I ....---------- LHS = XJ.(X) .J>- Jo(X) Ii , ....--1....__ , -- I -_ , -- I I I I I I I I I I I RHS = YKd Y ) , Ko(Y) ----:-J y= J V 2 _X 2 , " : " I ' 8 V X o Figure 9.2-3 Graphical construction for solving the characteristic equation (9.2-14). The left- and right-hand sides are plotted as functions of X. The intersection points are the solutions. The LHS has multiple branches intersecting the abscissa at the roots of Jl1 (X). The RHS intersects each branch once and meets the abscissa at X = V. The number of modes therefore equals the number of roots of Jl1 (X) that are smaller than V. In this plot 1 = 0, V = 10, and either the - or + signs in (9.2-14) may be used. (a) (b) Figure 9.2-4 Intensity distributions of (a) the LP o1 and (b) the LP 34 modes in the transverse plane, assuming an azimuthal dependence of the form cos lcjJ. The distribution of the fundamental LP ol mode is similar to that of the Gaussian beam discussed in Chapter 3. Mode Cutoff It is evident from the graphical construction in Fig. 9.2-3 that as V increases, the number of intersections (modes) increases since the LHS of the characteristic equa- tion (9.2-14) is independent of V, whereas the RHS moves to the right as V increases. Considering the minus signs in the characteristic equation, branches of the LHS in- tersect the abscissa when J Z - I (X) == O. These roots are denoted xZ m , m == 1, 2, . . .. The number of modes J\;[z is therefore equal to the number of roots of JZ- I (X) that are smaller than V. The (l, m) mode is allowed if V > XZm. The mode reaches its cutoff point when V == XZm. As V decreases, the (l, m - 1) mode also reaches its cutoff point when a new root is reached, and so on. The smallest root of JZ- I (X) is XOI == 0 for l == 0 and the next smallest is XII == 2.405 for l == 1. The numerical values of some of these roots are provided in Table 9.2-1. Table 9.2-1 Cutoff V parameter for low-order LP lm modes. a l\m o 1 1 2 3.832 5.520 3 7.0]6 8.654 o 2.405 aThe cutoffs of the I = 0 modes occur at the roots of J -1 (X) = - J 1 (X). The I = 1 modes are cut off at the roots of Jo (X), and so on. When V < 2.405, all modes with the exception of the fundamental LP OI mode are cut off. The fiber then operates as a single-mode waveguide. The condition for single-
9.2 GUIDED WAVES 337 mode operation is therefore v < 2.405. (9.2-15) Single-Mode Condition Since V is proportional to the optical frequency [see (9.2- L 0)], the cutoff condition for the fundamental mode provided in (9.2-15) yields a corresponding cutoff frequency: / 1 Co V e == We 2n == - NA 2.61a (9.2-16) Cutoff Frequency By comparison, in accordance with (8.2-9), the cutoff frequency of the lowest-order mode in a dielectric slab waveguide of width d is V e == (l/NA) (c o /2d). Number of Modes A plot of the number of modes M z as a function of V therefore takes the form of a staircase function that increases by unity at each of the roots XZ m of the Bessel function JZ- 1 (X). A composite count of the total number of modes M (for all values of l), as a function of V, is provided in Fig. 9.2-5. Each root must be counted twice since, for each mode of azimuthal index l > 0, there is a corresponding mode -l that is identical except for opposite polarity of the angle cP (corresponding to rays with helical trajectories of opposite senses), as can be seen by using the plus signs in the characteristic equation. Moreover, each mode has two states of polarization and must therefore be counted twice. 10 8 I Figure 9.2-5 Total number of modes M versus the fiber parameter V == 27r(aj Ao)NA. Included in the count are the two helical polarities for each mode with I > 0 and the two polarizations per mode. For V < 2.405, there is only a single mode, the fundamental LP 01 mode with two polarizations. The dashed curve is the relation 1\;1 == 4 V 2 j 7r 2 + 2, which provides an approximate formula for the number of modes when V » 1. 6 V 4 2 o o 10 20 30 40 50 Number of modes M Number of Modes (Fibers with Large V Parameter) For fibers with large V parameters, there are a large number of roots of Jz(X) in the interval 0 < X < V. Since Jz(X) (2/nX)1/2cos[X - (l + )] when X» 1, its roots xZ m are approximately given by xZ m == (l + ) + (2m - 1). Thus, xZ m == (l + 2m - ) , so that when m is large the cutoff points of modes (l, m), which are the roots of JZ:f:l (X), are XZ m (l + 2m - :i: 1) (l + 2m), l == 0, 1, . . . ; m» 1. (9.2-17)
338 CHAPTER 9 FIBER OPTICS For fixed l, these roots are spaced uniformly at a distance 7r, so that the number of roots !VIz satisfies (l + 2Mz) == V, from which M z V/7r - l/2. Thus, M z decreases linearly with increasing l, beginning with Mz V /7r for l == 0 and ending at M z == 0 when l == lmax, where lmax == 2 V /7r, as illustrated in Fig. 9.2-6. Thus, the total number of modes is M E :o x Mz == E mo x ( V / 7r - l /2). Since the number of terms in this sum is assumed to be large, it may be readily evaluated by approximating it as the area of the unshaded triangle in Fig. 9.2-6: AI (2 V / 7r ) ( V / 7r) == V 2 / 7r 2 . Accommodating the two degrees of freedom for positive and negative l, and the two polarizations for each index (l, m), finally leads to 4 2 M 2"V . 7r (9.2-18) Number of Modes (V» 1) Note that (9.2-18) is valid only for large V. This approximate number of modes is compared with the exact number, obtained from the characteristic equation, in Fig. 9.2- 5. 1 2V/1r . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1 = 2(V/1r- m) . . . . . . it . . Ii . . . . Figure 9.2-6 The indexes of guided modes extend from m == 1 to m V /7r - 1/2 and from 1 == 0 to 1 2 V / 7r, as displayed by the unshaded area. o o V/1r m The expression for the number of modes M for the circular waveguide given in (9.2-18), M (4a/ Ao)2(NA)2, is analogous to the expression provided in (8.3-3) for the waveguide of rectangular cross section, M (7r / 4) (2d / Ao)2(NA)2. EXAMPLE 9.2-1. Approximate Numbe r of Mo des. A silica fiber with nl == 1.452 and l:1 == 0.01 has a numerical aperture NA == J ni - n nl yl2 0.205. If Ao == 0.85 /-Lm and the core radius a == 25 /-LID, then V == 27r ( a / Ao ) N A 37.9. There are therefore approximately M 4 V 2 / 7r 2 585 modes. If the cladding is stripped away so that the core is in direct contact with air, n2 == 1 and NA == 1, whereupon V == 184.8 and more than 13,800 modes are allowed. Propagation Constants (Fibers with Large V Parameter) As indicated earlier, the propagation constants can be determined by solving the char- acteristic equation (9.2-14) for the X Zm and using (9.2-4a) and (9.2-8) to obtain {JZm == (ni k - X;m/ a 2 )1/2. A number of approximate formulas for X Zm applicable in certain limits are available in the literature, but there are no explicit exact formulas. If V » 1, the crudest approximation is to assume that the X Zm are equal to the cutoff values XZm. This is equivalent to assuming that the branches in Fig. 9.2-3 are
9.2 GUIDED WAVES 339 approximately vertical lines, so that X Zm XZm. Since V » 1, the majority of the roots would then be large so that the approximation in (9.2-17) may be used to obtain {JZm 2 2 2 ( ) 2 7r n 1 ko - l + 2m . 4a (9.2-19) Since ]I.[ V 2 = (NA)2. a 2 k 2 (2n2 ) k2a2 7r2 7r 2 0 7r 2 1 0' (9.2-20) (9.2-19) and (9.2-20) yield J (1+2m)2 {JZm n1 k o 1 - 2 . AI (9.2-21) Because is small, we may use the approximation V I + 8 1 + 812 for /81 « 1 to obtain [ (l+2m)2 ] {JZm n1 k o 1 - AI . (9.2-22) Propagation Constants (V» 1) I == 0, 1, . . . , viM m == 1,2, . . . , (JM - I) Since l + 2m varies between 2 and 2 V 1 7r == VM (see Fig. 9.2-6), (JZm varies approximately between n1ko and n1ko(1 - ) n2ko, as illustrated in Fig. 9.2-7. 131m k n l 0 n2 k o 1M I . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . Figure 9.2-7 Approximate propagation con- stants {3lm of the modes of a fiber with large V parameter, as functions of the mode indexes I and m. m Group Velocities (Fibers with Large V Parameter) To determine the group velocity, VZm == dw 1 d{JZm, of the (l, m) mode we express {JZm as anexplicitfunctionofwbysubstitutingn1ko == WlC1 andAf == (4/7r2)(2nI)ka2 == (8/7r2)a2w2/cI into (9.2-22) and assume that C1 and are independent of w. The derivative dw 1 d{Jzm provides [ (l+2m)2 ] -1 VZm C1 1 + M (9.2-23)
340 CHAPTER 9 FIBER OPTICS Since « 1, the approximate expansion (1 + 8)-1 1 - 8 for 181 « 1, then leads to [ (l+2m)2 ] VIm Cl 1 - M . (9.2-24) Group Velocities (V» 1) Because the minimum and maximum values of (l + 2m) are 2 and m, respectively, and since AI » 1, the group velocity varies approximately between C1 and C1 (1- ) == Cl (n2/ nl). Thus, the group velocities of the low-order modes are approximately equal to the phase velocity of the core material, whereas those of the high-order modes are smaller. The fractional group-velocity change between the fastest and the slowest mode is roughly equal to , the fractional refractive index change of the fiber. Fibers with large , although endowed with a large NA and therefore large light-gathering capacity, also have a large number of modes, large modal dispersion, and consequently high pulse-spreading rates. These effects are particularly severe if the cladding is removed altogether. B. Single-Mode Fibers As discussed earlier, a fiber with core radius a and numerical aperture NA operates as a single-mode fiber in the fundamental LP ol mode if V == 27r(a/Ao)NA < 2.405. Single-mode operation is therefore achieved via a small core diameter and small nu- merical aperture (in which case n2 is close to nl), or by operating at a sufficiently low optical frequency [below the cutoff frequency V c == (1/NA)(c o /2.61a)]. The fundamental mode has a bell-shaped spatial distribution similar to the Gaussian [see Fig. 9.2-2 for l == 0 and 9 .2-4( a)]. It provides the highest confinement of light power within the core. EXAMPLE 9.2-2. Single-Mode Operation. A silica-glass fiber with nl = 1.447 and = 0.01 (NA = 0.205) operates at Ao = 1.3 /-LID as a single-mode fiber if V = 27r(a/Ao)NA < 2.405, i.e., if the core diameter 2a < 4.86 /-LID. If is reduced to 0.0025, single-mode operation is maintained for a diameter 2a < 9.72 /-LID. The dependence of the effective refractive index n == (3/ ko on the V parameter for the fundamental mode is shown in Fig. 9.2-8(a), and the corresponding dispersion relation (w versus (3) is illustrated in Fig. 9.2-8(b). As the V parameter increases, i.e., the frequency increases or the fiber diameter increases, the effective refractive index n increases from n2 to nl. This is expected since the mode is more confined in the core at shorter wavelengths. There are numerous advantages of using single-mode fibers in lightwave communi- cation systems. As explained earlier, the modes of a multimode fiber travel at different group velocities so that a short-duration pulse of multi mode light suffers a range of delays and therefore spreads in time. Quantitative measures of modal dispersion are examined in Sec. 9.3B. In a single-mode fiber, on the other hand, there is only one mode with one group velocity, so that a short pulse of light arrives without delay distortion. As explained in Sec. 9.3B, pulse spreading in single-mode fibers does result from other dispersive mechanisms, but these are significantly smaller than modal dispersion. Moreover, as shown in Sec. 9.3A, the rate of power attenuation is lower in a single- mode fiber than in a multimode fiber. This, together with the smaller rate of pulse
9.2 GUIDED WAVES 341 n-n2 n l -n2 n = n l 1 ............................................... ... ... .... w . ..... .... Light line ........ W = c2 a .... fJ .. ... --...... . . ... ..- ... .... \ . .. .. .- . .. ..- .... ........ ........... Light line :....... W = cl!3 .. ... o o (a) (b) o ............. ........!!.. ... ....... ............ o v 10 {JOl Figure 9.2-8 Schematic illustrations of the propagation characteristics of the fundamental LP ol mode. (a) Effective refractive index n == !3/ko as a function of the V parameter. (b) Dispersion relation (w versus !30l). spreading, permits substantially higher data rates to be transmitted over single-mode fibers than over multi mode fibers. This topic is considered further in Chapters 22 and 24. Another difficulty with multimode fibers stems from the random interference of the modes. As a result of uncontrollable imperfections, strains, and temperature fluc- tuations, each mode undergoes a random phase shift so that the sum of the complex amplitudes of the modes exhibits an intensity that is random in time and space. This randomness is known as modal noise or speckle. This effect is similar to the fading of radio signals resulting from multiple-path transmission. In a single-mode fiber there is only one path and therefore no modal noise. Polarization-Maintaining Fibers In a fiber with circular cross section, each mode has two independent states of polariza- tion with the same propagation constant. Thus, the fundamental LP ol mode in a single- mode weakly guiding fiber may be polarized in the x or y direction; the two orthogonal polarizations have the same propagation constant and the same group velocity. In principle, there should be no exchange of power between the two polarization components. If the power of the light source is delivered exclusively into one polariza- tion, the power should remain in that polarization. In practice, however, slight random imperfections and uncontrollable strains in the fiber result in random power transfer between the two polarizations. Such coupling is facilitated because the two polar- izations have the same propagation constant and their phases are therefore matched. Thus, linearly polarized light at the fiber input is generally transformed into elliptically polarized light at the fiber output. In spite of the fact that the total optical power remains fixed (see Fig. 9.2-9), the ellipticity of the received light fluctuates randomly with time as a result of fluctuations in the material strain and temperature, and of the source wavelength. The randomization of the power division between the two polarization components poses no difficulty if the object is solely to transmit light power, provided that the total power is collected. However, in many areas where fiber optics is used, e.g., integrated-optic devices, optical sensors based on interferometric techniques, and coherent optical communi- cations, the fiber must transmit the complex amplitude (magnitude and phase) of a specific polarization. Polarization-maintaining fibers are required for such applications. To construct a polarization-maintaining fiber, the circular symmetry of the conventional fiber must be abandoned, such as by using fibers with elliptical cross section or stress- induced anisotropy of the refractive index. This eliminates the polarization degeneracy, thereby making the propagation constants of the two polarizations different. The intro- duction of such phase mismatch serves to reduce the coupling efficiency.
342 CHAPTER 9 FIBER OPTICS Polarization C()) L (a) t ) Polarization-maintaing fiber $ ) t t L (C)) ) (b) t t ) 47 Conventional fiber ) t t Figure 9.2-9 (a) Ideal polarization-maintaining fiber. (b) Random transfer of power between two polarizations. *c. Quasi-Plane Waves in Step- and Graded-Index Fibers The modes of the graded-index fiber are determined by writing the Helmholtz equa- tion (9.2-1) with n == n( r), solving for the spatial distributions of the field components, and using Maxwell's equations and the boundary conditions to obtain the characteristic equation, as was done for the step-index case. This procedure is difficult, in general. In this section we use instead an approximate approach based on picturing the field distribution as a quasi-plane wave traveling within the core, approximately along the trajectory of the optical ray. A quasi-plane wave is a wave that is locally identical to a plane wave, but changes its direction and amplitude slowly as it travels. This approach permits us to maintain the simplicity of rays optics but at the same time retain the phase associated with the wave, so that the self-consistency condition to determine the propagation constants of the guided modes can be used (as was done for the planar dielectric waveguide in Sec. 8.2). This approximate technique, called the WKB (Wentzel-Kramers-Brillouin) method, is applicable only to fibers with a large number of modes (large V parameter). Quasi-Plane Waves Consider a solution of the Helmholtz equation (9.2-1) that takes the form of a quasi- plane wave (see Sec. 2.3) U(r) == a(r) exp [-jkoS(r)] , (9.2-25) where a( r) and S( r) are real functions of position that are slowly varying in compari- son with the wavelength Ao == 27r / ko. It is known from (2.3-4) that S( r) approximately satisfies the eikonal equation IV SI 2 n 2 , and that the rays travel in the direction of the gradient VS. If we take koS(r) == kos(r) + Zcp + (3z, where s(r) is a slowly varying function of r, the eikonal equation gives ( dS ) 2 2 Z2 2 2 ko dr + (3 + r 2 == n (r) ko. (9.2-26) The local spatial frequency of the wave in the radial direction is the partial derivative of the phase koS(r) with respect to r, ds k r == ko dr ' (9.2- 27)
9.2 GUIDED WAVES 343 so that (9.2-25) becomes U(r) = a(r) ex p ( -j Lrkrdr) e- jltP e- j {3z (9.2-28) Quasi-Plane Wave and (9.2-26) gives k; = n 2 (r) k _ {32 _ Z: . r (9.2- 29) Defining keJ> == Zir so that exp( -jZ1) == exp( -jkeJ>r1), and k z == (3, (9.2-29) yields k; + k + k; == n 2 (r) k. The quasi-plane wave therefore has a local wavevector k with magnitude n(r)k o and cylindrical-coordinate components (k r , keJ>, k z ). Since n(r) and keJ> are functions of r, k r is also generally position dependent. The direction of k changes slowly with r (see Fig. 9.2- L 0), following a helical trajectory similar to that of the skewed ray shown earlier in Fig. 9 .1-6(b). x y (a) Figure 9.2-10 (a) The wavevector k == (k r , kc/J, k z ) in a cylindrical coordinate system. (b) Quasi- plane wave following the direction of a ray. To establish the region of the core within which the wave is bound, we determine the values of r for which k r is real, or k; > O. For a given Z and (3 we plot k; == [n 2 (r) k - Z2/r 2 - (32] as a function of r. The term n 2 (r) k is first plotted as a function of r [thick solid curve in Fig. 9.2-11(a)]. The term Z2/r 2 is then subtracted, yielding the dashed curve. The value of (32 is marked by the thin solid vertical line. It follows that k; is represented by the difference between the dashed curve and the thin solid line, i.e., by the shaded area. Regions where k; is positive and negative are indicated by + and - signs, respectively." For the step-index fiber, we have n(r) == nl for r < a, and n(r) == n2 for r > a. In this case the quasi-plane wave is guided in the core by reflecting from the core- cladding boundary at r == a. As illustrated in Fig. 9.2-11 (a), the region of confinement is then rl < r < a, where 2 2 z2 2 n l ko - 2" - (3 == O. r l (9.2-30) The wave bounces back and forth helically like the skewed ray illustrated in Fig. 9.1-2. In the cladding (r > a), and near the center of the core (r < rl), k; is negative so that
344 CHAPTER 9 FIBER OPTICS k r is imaginary; the wave therefore decays exponentially in these regions Note that rz depends on (3. For large (3 (or large l), rz is large so that the wave is confined to a thin cy lindrical shell near the boundary of the core. For the graded-index fiber illustrated in Fig. 9.2-11(b), k r is real in the region rz < r < Rz, where rz and Rz are the roots of the equation n 2 (r) k - l: - (32 = o. r (9.2-31 ) It follows that the wave is essentially confined within a cylindrical shell of radii rz and Rz, just as for the helical ray trajectory shown in Fig. 9.1-6(b). r r I I I I a ----- rl 2 k 2 n (r) 0 I , 1 I a - ---( , , , , RZ------- , \ \ \ + , I k- k 2 r n2(r)k - P / ? n 2 (r)k;- P/r 2 k 2 r rz , ,," -- - o o 11 {32 ni n (32 ni (a) Step-index (b) Graded-index Figure 9.2-11 Dependence of n 2 (r) k, n 2 (r) k - [2/r2, and k; == n 2 (r) k - [2/r2 - {32 on the position r. At any r, k; is the width of the shaded area with the + and - signs denoting positive and negative values of k;, respectively. (a) Step-index fiber: k; is positive in the region rl < r < a. (b) Graded-index fiber: k; is positive in the region rl < r < Rl. Modes The modes of the fiber are determined by imposing the self-consistency condition that the wave reproduce itself after one helical period of travel between rz and Rz and back. The azimuthal path length corresponding to an angle 21T must correspond to a multiple of 21T phase shift, i.e., k4J21Tr == 21Tl; l == 0, ::i:l, ::i:2, . . .. This condition is evidently satisfied since k4J == l / r. Furthermore, the radial round-trip path length must correspond to a phase shift equal to an integer multiple of 21T: l Rl 2 k r dr == 21Tm, rl m == I, 2, . . . , M z , (9.2-32) where Rz == a for the step-index fiber. This condition, which is analogous to the self-consistency condition (8.2-2) for planar waveguides, provides the characteristic equation from which the propagation constants (3zm of the modes are determined. These values are indicated schematically in Fig. 9.2-12; the mode m == I has the largest value of (3 (approximately nlko) whereas m == Alz has the smallest value (approximately n2 k o ).
9.2 GUIDED WAVES 345 r n 2 (r) _[2/ r2 a n2(r)k 2 k 2 n2 0 Figure 9.2-12 The propagation constants and confinement regions of the fiber modes. Each curve corresponds to an index I, which stretches from 0 to 6 in this plot. Each mode (corresponding to a certain value of m) is marked schematically by two dots connected by a dashed vertical line. The ordinates of the dots mark the radii rl and Rl of the cylindrical shell within which the mode is confined. Values on the abscissa are the squared propagation constants of the modes, {32. 00 2 2 nl ko {32 Number of Modes The total number of modes can be determined by adding the number of modes Mz for [ == 0, 1, . . . , lmax. We consider this computation using a different procedure, however. We first determine the number q{3 of modes with propagation constants greater than a given value (3. For each l, the number of modes J\;I z ({3) with propagation constant greater than (3 is the number of multiples of 27r the integral in (9.2-32) yields, i.e., 1 1 Rl l 1 Rl J l2 Afz({3) == - k r dr == - n2(r)k - 2" - {32 dr, 7r 7r r (9.2-33) where rz and Rz are the radii of confinement corresponding to the propagation constant {3 as given by (9.2-31). Clearly, rz and Rz depend on {3, and Rz == a for the step-index fiber. The total number of modes with propagation constant greater than {3 is therefore zmax ({3) q{3 == 4 l: lv/z ((3) , z=o (9.2-34) where lmax({3) is the maximum value of l that yields a bound mode with propagation constants greater than (3, i.e., for which the peak value of the function n 2 (r) k _[2 / r 2 is greater than {32. The grand total mode count AI is q{3 for {3 == n2ko. The factor of 4 in (9.2-34) accommodates the two possible polarizations and the two possible polarities of the angle 1, corresponding to positive and negative helical trajectories for each ([, m). If the number of modes is sufficiently large, we can replace the summation in (9.2-34) by an integral, whereupon {Zmax ({3) q{3 ';::;j 4 } 0 AIL (fJ) dl. (9.2-35) For fibers with power-law refractive-index profiles, we insert (9.1-4) into (9.2-33),
346 CHAPTER 9 FIBER OPTICS and thence into (9.2-35). Evaluation of the integral then yields p+2 M [ 1 - ((3/n 1 k o )2 ] p q{3 2 (9.2-36) with AI P n 2 k 2 a 2 t:J.. = P V 2 p+2 1 0 p+2 2 ' where == (nl - n2)/nl and V == 27r(a/Ao)NA is the fiber V parameter. Since q{3 fYf at (3 == n2 k o, M is indeed the total number of modes. For step-index fibers (p (0), (9.2-36) and (9.2-37) become (9.2-37) M [ 1 - ((3/n 1 k o )2 ] q{3 2 (9.2-38) and (9.2-39) Number of Modes (Step-Index) respectively. This expression for M is nearly the same as that set forth in (9.2-18), M 4 V 2 /7r 2 0.41 V 2 , which was obtained in Sec. 9.2 using a different approximation. 1 M _V 2 2 ' Propagation Constants The propagation constant (3q for mode q is obtained by inverting (9.2-36), / ( q ) P/(P+2) /3q nlkoy 1 - 2 M t:J.. , where the index q{3 has been replaced by q, and (3 replaced by (3q. Since « 1, the approximation v I + 8 1 + 8 (applicable for 181 « 1) can be applied to (9.2-40), yielding q == 1, 2, . . . , AI, (9.2-40) [ ( q ) P / (p+ 2) ] (3q nlko 1 - AI . (9.2-41) Propagation Constants The propagation constant (3q therefore decreases from nlko (for q == 1) to n2ko (for q == fYI), as illustrated in Fig. 9.2-13. For the step-index fiber (p (0), (9.2-40) reduces to (9.2-42) Propagation Constants (Step-Index Fiber) This expression is identical to (9.2-22) if the index q == 1, 2, . . . , M is replaced by (l + 2m) 2 , with l == 0, 1, . . . , VM; m == 1, 2, . . . , (VM - l). /3q n 1 ko ( 1 - :1 t:J..) .
9.2 GUIDED WAVES 347 .... nlko .... VJ t:: o t) t:: .g n2 k o C\S blJ C\S o ct 0 M .... n lko .... VJ t:: o t) t:: .g n2 k o C\S blJ C\S e Graded-index fiber (p = 2) Step- index fiber Mode index q o Mode index q M Figure 9.2-13 Dependence of the propagation constants {3q on the mode index q == 1,2,..., M for a step-index fiber (p ---+ 00) and for an optimal graded-index fiber (p == 2). Group Velocities To determine the group velocity v q == dw / d{3q, we write {3q as a function of w by sub- stituting (9.2-37) into (9.2-41), substituting n1ko == w / C1 into the result, and evaluating v q == (d{3q / dw ) -1. With the help of the approximation (1 + 8) -1 1 - 8 (valid for 181 « 1), and assuming that C1 and are independent of w (i.e., ignoring material dispersion), we obtain [ p - 2 ( q ) P/(P+2) ] v C1 1 - - . q p+2 M (9.2-43) Group Velocities For the step-index fiber (p ---+ (0), (9.2-43) yields V q Cl ( 1 - :1 Ll ) , (9.2-44 ) which reproduces (9.2-24). The group velocity thus varies from approximately C1 to C1 (1 - ), as illustrated in Fig. 9.2-14(a). Optima/Index Profile Equation (9.2-43) indicates that the grade profile parameter p == 2 yields a group velocity v q C1 for all q, so that all modes travel at approximately the same velocity C1. This highlights the advantage of the graded-index fiber for multimode transmission. To determine the group velocity with better accuracy, we return to the derivation of v q from (9.2-4 0) for p == 2. Carrying the Taylor-series expansion to three terms instead of two, i.e., V I + 8 1 + 8 - 82, gives rise to v C1 ( 1-.!L Ll2 ) . q AI 2 (9.2-45) G roup Velocities (Graded-Index, p == 2) Thus, the group velocities vary from approximately C1 at q == 1 to approximately C1 (1 - 2 /2) at q == AI. Comparison with the results for the step-index fiber is provided in Fig. 9.2-14. The group-velocity difference for the parabolically graded fiber is f:,. 2 /2, which is substantially smaller than the group-velocity difference f:,. for the step-index fiber. Under ideal conditions, the graded-index fiber therefore reduces the group-velocity difference by a factor /2, thus realizing its intended purpose of .
348 CHAPTER 9 FIBER OPTICS equalizing the modal velocities. However, since the analysis leading to (9.2-45) is based on a number of approximations, this improvement factor is only a rough estimate - indeed it is not fully attained in practice. \::r< >. ..... .u ci o Q) :> ::3 o I-< o . ci (l -) \::r< >. .E cI o Q) :> ::3 o I-< o Graded-index fiber (p = 2) Step-index fiber : CI (1 - 2/2) o Mode index q M 0 Mode index q M Figure 9.2-14 Group velocities v q of the modes of a step-index fiber (p ---+ (0) and an optimal graded-index fiber (p == 2). The number of modes M in a graded-index fiber with grade profile parameter p is specified by (9.2-37). For p == 2, this becomes }.f V 2 . 4 (9.2-46) Number of Modes (Graded-Index, p == 2) Comparing this with the result for the step-index fiber provided in (9.2-39), AI V 2 /2, reveals that the number of modes in an optimal graded-index fiber is roughly half that in a step-index fiber with the same parameters nl, n2, and a. 9.3 ATTENUATION AND DISPERSION Attenuation and dispersion limit the performance of the optical-fiber medium as a data- transmission channel. Attenuation, associated with losses of various kinds, limits the magnitude of the optical power transmitted. Dispersion, which is responsible for the temporal spread of optical pulses, limits the rate at which such data-carrying pulses may be transmitted. A. Attenuation Attenuation Coefficient The power of a light beam traveling through an optical fiber decreases exponentially with distance as a result of absorption and scattering. The associated attenuation co- efficient is conventionally defined in units of decibels per kilometer (dB /km) and is denoted by the symbol cx, 1 1 cx == L 101og 10 'J ' (9.3-1) where'J == P( L) / P(O) is the power transmission ratio (ratio of transmitted to incident power) for a fiber of length L km. The conversion of a ratio to dB units is illustrated in
9.3 ATTENUATION AND DISPERSION 349 10 3 '" '" "'- ------ --- -- s I " -f- I , dB o 0.1 Ratio 0.5 1 Figure 9.3-1 The dB value of a ratio. For example, 3 dB is equivalent to a ratio of 0.5; 10 dB corresponds to 'J == 0.1; and 20 dB corresponds to 'J == 0.01. Fig. 9.3-1. An attenuation of 3 dB/lan, for example, corresponds to a power transmis- sion of'J == 0.5 through a fiber of length L == 1 km. For light traveling through a cascade of lossy systems, the overall transmission ratio is the product of the constituent transmission ratios. By virtue of the logarithm in (9.3- 1), the overall loss in dB therefore becomes the sum of the constituent dB losses. For a propagation distance of z km, the loss is exz dB. The associated power transmission ratio, which is obtained by inverting (9.3-1), is then P(z) == 10-£xz/10 e- 0 . 23 £xz P(O) , ex in dB /km. (9.3-2) Equation (9.3-2) applies when the quantity ex is specified in units of dB/km. However, that if the attenuation coefficient is specified in units of km -1, rather than in units of dB /km, then P(z)/P(O) == e- az , Q in km- 1 , (9.3-3) where Q 0.23 ex. The attenuation coefficient Q is usually specified in units of cm- 1 for components other than optical fibers, in which case the power attenuation is de- scribed by (9.3-3) with z in cm. Absorption The attenuation coefficient ex of fused silica (Si0 2 ) is strongly dependent on wave- length, as illustrated in Fig. 9.3-2. This material has two strong absorption bands: a mid-infrared absorption band resulting from vibrational transitions and an ultraviolet absorption band arising from electronic and molecular transitions. The tails of these bands form a window in the near infrared region of the spectrum in which there is little intrinsic absorption. Scattering Rayleigh scattering is another intrinsic effect that contributes to the attenuation of light in glass. The random localized variations of the molecular positions in the glass itself create random inhomogeneities in the refractive index that act as tiny scattering centers. The amplitude of the scattered field is proportional to w 2 , where w is the angular frequency of the light. t The scattered intensity is therefore proportional to w 4 , or to 1/ A, so that short wavelengths are scattered more than long wavelengths. Blue light is therefore scattered more than red (a similar effect, the scattering of sunlight from atmospheric molecules, is the reason the sky appears blue). t The scattering medium creates a polarization density T, which corresponds to a source of radiation proportional to d 2 T / dt 2 = -w 2 T; see (5.2-25).
350 CHAPTER 9 FIBER OPTICS S 3 --- 1:: 1 (1) .- t) E (1) 8 0.5 t:: .g 0.3 C\S ::j 0.2 < 0.1 0.6 , , , . " Rayleigh " I scattering " , , , " UV absorption ', band tail , , , I', 0.8 1.0 1.2 1.4 Wavelength >"0 (/Lm) I [nfrared / absorption I I OH absorption 1.6 1.8 Figure 9.3-2 Attenuation coefficient ex of silica glass versus wavelength Ao. There is a local minimum at 1.3 /-Lm (ex 0.3 dB/km) and an absolute minimum at 1.55 /-Lm (ex 0.15 dB/km). The functional form of Rayleigh scattering, which decreases with wavelength as 1/ .A, is known as Rayleigh's inverse fourth-power law. In the visible region of the spectrum, Rayleigh scattering is a more significant source of loss than is the tail of the ultraviolet absorption band, as shown in Fig. 9.3-2. However, Rayleigh loss becomes negligible in comparison with infrared absorption for wavelengths greater than 1.6 /-LID. We conclude that the transparent window in silica glass is bounded by Rayleigh scat- tering on the short-wavelength side and by infrared absorption on the long-wavelength side (indicated by the dashed curves in Fig. 9.3-2). Lightwave communication systems are deliberately designed to operate in this window. Extrinsic Effects Aside from these intrinsic effects there are extrinsic absorption bands that result from the presence of impurities, principally metallic ions and OH radicals associated with water vapor dissolved in the glass. Most metal impurities can be readily removed but OH impurities are somewhat more difficult to eliminate. Only recently have specialty fibers with significantly reduced OH absorption become available. In general, wave- lengths at which glass fibers are used for lightwave communication are selected to avoid the OH absorption bands. Light-scattering losses may also be accentuated when dopants are added, as they often are for purposes of index grading. The attenuation coefficient for guided light in glass fibers depends on the absorption and scattering in the core and cladding materials. Each mode has a different penetration depth into the cladding, causing the rays to travel different effective distances and rendering the attenuation coefficient mode dependent. It is generally higher for higher-order modes. Single-mode fibers therefore typically have smaller attenuation coefficients than multimode fibers (Fig. 9.3-3). Losses are also introduced by small random variations in the geometry of the fiber and by bends. Alternates to Silica Glass A number of materials other than silica glass are being examined for potential use in mid-infrared optical fiber systems. These include heavy-metal fluoride glasses, heavy- metal germanate glasses, and chalcogenide glasses. The infrared absorption band is located further into the infrared for these materials than it is for silica glass so that longer-wavelength operation, with its attendant reduced Rayleigh scattering (which .
9.3 ATTENUATION AND DISPERSION 351 (j I I I I I I I E 3 "'d '-' 1.0 1.2 1.4 Wavelength >"0 (J-Lm) Figure 9.3-3 Ranges of attenuation coefficients for silica-glass single-mode fibers (SMF) and multimode fibers (MMF). t: (I) .u b: (I) 8 0.5 t: o . 0.3 :3 t: 0.2 < 0.1 0.6 "" "" , , , "" , , Supressed /'........, OH absorption ................ I ...........j 0.8 1.6 1.8 decreases as 1 / .A), is possible. In particular, the optical attenuation for heavy-metal fluoride glass fibers is predicted to be about 10 times smaller than for silica fibers, reaching a minimum of 0.01 dB/km at .Ao 2.5 /-Lm. However, extrinsic loss mechanisms currently dominate fiber loss and these materials are generally far less durable than silica glass. Although quantum-cascade lasers offer room-temperature operation in the mid infrared, high-efficiency photodetectors are generally not available in this spectral region. B. Dispersion When a short pulse of light travels through an optical fiber, its power is "dispersed" in time so that the pulse spreads into a wider time interval. There are five principal sources of dispersion in optical fibers: . Modal dispersion . Material dispersion . Waveguide dispersion . Polarization-mode dispersion . Nonlinear dispersion The combined contributions of these effects to the spread of pulses in time are not necessarily additive, as will be subsequently shown. Modal Dispersion Modal dispersion occurs in multimode fibers as a result of the differences in the group velocities of the various modes. A single impulse of light entering an M -mode fiber at z == 0 spreads into !vI pulses whose differential delay increases as a function of z. For a fiber of length L, the time delays engendered by the different velocities are Tq == L/v q , q == 1, . . . , J\;I, where v q is the group velocity of mode q. If Vrnin and V rnax are the smallest and largest group velocities, respectively, the received pulse spreads over a time interval L/Vrnin - L/v rnax . Since the modes are usually not excited equally, the overalJ shape of the received pulse generally has a smooth envelope, as illustrated in Fig. 9.3-4. An estimate of the overall pulse duration (assuming a triangular envelope and using the FWHM definition of the width) is aT == (L/Vrnin - L/v rnax ), which represents the modal-dispersion response time of the fiber.
352 CHAPTER 9 FIBER OPTICS L o t o t a4n\ I A, . o t . z Figure 9.3-4 Pulse spreading caused by modal dispersion. In a step-index fiber with a large number of modes, Vrnin Cl (1 - ) and V rnax Cl [see Sec. 9.2C and Fig. 9.2-14(a)]. Since (1 - )-l 1 + for « 1, the response time turns out to be a fraction / 2 of the delay time L / Cl : (9.3-4) Response Time (Multimode Step-Index) Modal dispersion is far smaller in graded-index (GRIN) fibers than in step-index fibers since the group velocities are equalized and the differences between the delay times of the modes, Tq == L/v q , are reduced. It was shown in Sec. 9.2C and in Fig. 9.2- 14(b) that a graded-index fiber with an optimal index profile and a large number of modes has V rnax Cl and Vrnin cl(l - 2/2). The response time in this case is therefore a factor of /2 smaller than that in a step-index fiber: L aT - . - . Cl 2 L 2 aT - . - . Cl 4 (9.3-5) Response Time (Graded-Index) EXAMPLE 9.3-1. Multimode Pulse Broadening Rate. In a step-index fiber with l::J. == 0.01 and n == 1.46, pulses spread at a rate of approximately a T / L == l::J./2Cl == n 1 l::J./2c o 24 ns/km. In a IOO-Ian fiber, therefore, an impulse spreads to a width of 2.4 MS. If the same fiber is optimally index graded, the pulse broadening rate is approximately n 1 l::J. 2 /4c o 122 ps/km, a substantial reduction. The pulse broadening arising from modal dispersion is proportional to the fiber length L in both step-index and GRIN fibers. Because of mode coupling, however, this dependence does not necessarily apply for fibers longer than a certain critical length. Coupling occurs between modes that have approximately the same propagation constants as a result of small imperfections in the fiber, such as random irregularities at its surface or inhomogeneities in its refractive index. This permits optical power to be exchanged between the modes. Under certain conditions, the response time aT of mode-coupled fibers is proportional to L for small fiber lengths and to VI when a critical length is exceeded, whereupon the pulses are broadened at a reduced rate. t Material Dispersion Glass is a dispersive medium, i.e., its refractive index is a function of wavelength. As discussed in Sec. 5.6, an optical pulse travels in a dispersive medium of refractive t See, e.g., J. E. Midwinter, Optical Fibers for Transmission, Wiley, 1979; Krieger, reissued 1992.
9.3 ATTENUATION AND DISPERSION 353 index n with a group velocity v == c o / N, where N == n - >"'0 dn/ d>"'o. Since the pulse is a wavepacket, comprising a collection of components of different wavelengths, each traveling at a different group velocity, its width spreads. The temporal duration of an optical impulse of spectral width a,\ (nm), after traveling a distance L through a dispersive material, is aT == I ( d / d>"'o) (L / v ) I a,\ == I ( d / d>"'o) (LN / co) I a,\. This leads to a response time [see (5.6-2), (5.6-7), and (5.6-8)] aT == ID'\la,\L, (9.3-6) Response Time (Material Dispersion) where the material dispersion coefficient D,\ is >"'0 d 2 n D,\ == -- d '2 . Co Ao (9.3-7) The response time increases linearly with the distance L. Usually, L is measured in lan, aT in ps, and a,\ in nm, so that D,\ has units of ps/km-nm. This type of dispersion is called material dispersion. The wavelength dependence of the dispersion coefficient D,\ for a silica-glass fiber is displayed in Fig. 9.3-5. At wavelengths shorter than 1.3 /-LID the dispersion coefficient is negative, so that wavepackets of long wavelength travel faster than those of short wavelength. At a wavelength >"'0 == 0.87 /-lm, for example, the dispersion coefficient D,\ is approximately -80 ps/km-nm. At >"'0 == 1.55 /-lm, on the other hand, D,\ + 17 ps/km-nm. At >"'0 1.312 /-Lm the dispersion coefficient vanishes, so that aT in (9.3-6) vanishes. A more precise expression for aT that incorporates the spread of the spectral width a,\ about >"'0 == 1.312 /-LID yields a very small, but nonzero, width. 40 -- e t: CI:J -40 o Noal dipersio:n Anomalous . ... '.....!' .... ...,;.."::.......I.:::...::::::::::::[::::::::I::::::::::]:::::::::::Ir:":::::::: ..! .. :......... ::::: ::: ]:: :.:: :::::: r :::::::::: ....::: ::: ::: I: :::: ::: ::1: ::: ::: ::: :1::::::::: ::1 :::: ::: ::: _ooj -----umu-1 m - __m -- oot 00 "_m_ -oo-t-- mm m--1 m mmm_(moomoo_ mmmu--t-- m - mm -1--m-- m__ ..-.:::: Q t= -80 . u -120 (1) o u 6 -160 . CI:J I-t (1) -200 o 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Wavelength '\0 (JLm) Figure 9.3-5 Dispersion coefficient D).. for a silica-glass fiber as a function of wavelength Ao. The result is similar to, but distinct from, that of fused silica (see Fig. 5.6-5). EXAMPLE 9.3-2. Pulse Broadening Associated with Material Dispersion. The dis- persion coefficient D).. for a silica-glass fiber is approximately -80 pslkm-nm at Ao = 0.87 /-Lm. For a source of spectrallinewidth a).. = 50 nm (generated by an LED, for example) the pulse- spread rate in a single-mode fiber with no other sources of dispersion is ID)..la).. = 4 ns/km. An .\
354 CHAPTER 9 FIBER OPTICS impulse of light traveling a distance L == 100 krn in the fiber is therefore broadened to a width aT == IDAlaAL == 0.4 MS. The response time of the fiber is thus aT == 0.4 MS. As another example, an impulse with narrower spectrallinewidth a A == 2 nrn (generated by a laser diode, for example), operating near 1.3 Mrn where the dispersion coefficient is I ps/km-nm, spreads at a rate of only 2 ps/km. In this case, therefore, a IOO-km fiber has a substantially shorter response time, aT == 0.2 ns. Combined Material and Modal Dispersion The effect of material dispersion on pulse broadening in multimode fibers may be determined by returning to the original equations for the propagation constants {3q of the modes and determining the group velocities v q == (d{3q/ dw )-1 with n1 and n2 given as functions of w. Consider, for example, the propagation constants of a graded-index fiber with a large number of modes, which are given by (9.2-41) and (9.2-37). Although n1 and n2 are dependent on w, it is reasonable to assume that the ratio == (n1 - n2)/n1 is approximately independent of w. Using this approximation and evaluating v q == (d{3q/ dw )-1, we obtain r-...I Co [ P - 2 ( q ) p/(p+2) ] Vr-...I-l- - q N 1 P + 2 M ' (9.3-8) where N 1 == (d/ dw) (wn1) == n1 - Ao( dn1/ dAo) is the group index of the core material. Under this approximation, the earlier expression (9.2-43) for v q remains intact, except that the refractive index n1 is replaced with the group index N 1 . For a step-index fiber (p (0), the group velocities of the modes vary from c o /N 1 to (c o /N 1 )(1 - ), so that the resp<;>nse time is aT L (co/N l ) 2 (9.3-9) Response Time (Multimode Step-Index, Material Dispersion) This expression should be compared with (9.3-4), which is applicable in the absence of material dispersion. EXERCISE 9.3-1 Optimal Grade Profile Parameter. Use (9.2-41) and (9.2-37) to derive the following expression for the group velocity v q when both nl and are wavelength dependent: Co [ P - 2 - P s ( q ) p / (p+ 2) ] v-l- - q N 1 P + 2 M ' with Ps == 2(nl/Nl)(W/) d/dw. What is the optimal value of the grade profile parameter p for minimizing modal dispersion? q == 1, 2, . . . , AI (9.3-10) Waveguide Dispersion The group velocities of the modes in a waveguide depend on the wavelength even if material dispersion is negligible. This dependence, known as waveguide dispersion,
9.3 ATTENUATION AND DISPERSION 355 results from the dependence of the field distribution in the fiber on the ratio of the core radius to the wavelength (a / .Ao). The relative portions of optical power in the core and cladding thus depend on .Ao. Since the phase velocities in the core and cladding differ, the group velocity of the mode is altered. Waveguide dispersion is particularly impor- tant in single-mode fibers where modal dispersion is not present, and at wavelengths for which material dispersion is small (near .Ao == 1.3 /-lID in silica glass), since it then dominates. As discussed in Sec. 9.2A, the group velocity v == (d(3 / dw ) -1 and the propagation constant (3 are determined from the characteristic equation, which is governed by the fiber V parameter, V == 27r(a/ .Ao)NA == (a. NA/co)w. In the absence of material dispersion (i.e., when NA is independent of w), V is directly proportional to w, so that 1 d(3 dw d(3 dV -- dV dw a . NA d(3 Co dV. (9.3-11) v The pulse broadening associated with a source of spectral width a.x is related to the time delay L/v by aT == I (d/d.Ao)(L/v) la.x. Thus, aT == IDwla.x L , (9.3-12) where the waveguide dispersion coefficient Dw is given by Dw = do ( ) = - : ( ) . (9.3-13) Substituting (9.3-11) into (9.3-13) leads to ( 1 ) V2 J2(3 Dw = - 27rc o dV 2 ' (9.3-14) Thus, the group velocity is inversely proportional to d(3/ d V and the waveguide dispersion coefficient is proportional to V 2 d 2 (3/ d V 2 . The dependence of (3 on V is displayed in Fig. 9.2-8(a) for the fundamental LP 01 mode. Since (3 varies nonlinearly with V, the waveguide dispersion coefficient Dw is itself a function of V and is there- fore also a function of the wavelength. t The dependence of Dw on .Ao may be controlled by altering the radius of the core or, for graded-index fibers, the index grading profile. Combined Material and Waveguide Dispersion The combined effects of material dispersion and waveguide dispersion (which we refer to as chromatic dispersion) may be determined by including the wavelength depen- dence of the refractive indexes, nl and n2 and therefore NA, when determining d(3/ dw from the characteristic equation. Although generally smaller than material dispersion, waveguide dispersion does shift the wavelength at which the total chromatic dispersion . .. IS mInImum. Since chromatic dispersion limits the performance of single-mode fibers, more ad- vanced fiber designs aim at reducing this effect by using graded-index cores with refractive-index profiles selected such that the wavelength at which waveguide dis- persion compensates material dispersion is shifted to the wavelength at which the fiber is to be used. Dispersion-shifted fibers have been successfully fabricated by using a linearly tapered core refractive index and a reduced core radius, as illustrated in Fig. 9.3-6(a). This technique can be used to shift the zero-chromatic-dispersion t For further details on this ,topic, see the reading list, particularly the articles by Gloge.
356 CHAPTER 9 FIBER OPTICS wavelength from 1.3 /-LID to 1.55 /-LID, where the fiber has its lowest attenuation. Other grading profiles have been developed for which the chromatic dispersion vanishes at two wavelengths and is reduced for intermediate wavelengths. These fibers, called dispersion-flattened, have been implemented by using a quadruple-clad layered grad- ing, as illustrated in Fig. 9 .3-6(b). Note, however, that the process of index grading itself introduces losses since dopants are used. Fibers with other refractive index profiles may be engineered such that the combined material and waveguide dispersion coefficient is proportional to that of a conventional step-index fiber but has the opposite sign. This can be achieved over an extended wavelength band, as illustrated in Fig. 9.3-6(c). The pulse spread introduced by a conventional fiber can then be reversed by concatenating the two types of fiber. A fiber with a reversed dispersion coefficient is known as a dispersion compensating fiber (DCF). A short segment of the DCF may be used to compensate the dispersion introduced by a long segment of conventional fiber. (a) DSF d====== - [;--- . n s:: ..... o s:: ._ (I) Cl) .- u &b:J 0 . 8 o u , , , , , , ( -:--- , , , , , , o o M ...... o ...... o o V) ...... 8 >"0 (om) \Ci ...... ( } - ----- - ------ , - ---- s:: ..... o s:: ._ (I) fJ:J .- u &b:J 0 fJ:J (I) .- 0 o u , , , , , , , , , , , , (b) DFF . n >"0 (om) ...... ...... ...... -------------- o s:: ..... o s:: ._ (I) fJ:J .- u &b:J fJ:J (I) .- 0 o u ----- (c) DCF '/ . n o o V) ...... 8 >"0 (om) \Ci ...... Figure 9.3-6 Refractive-index profiles with schematic wavelength dependences of the material dispersion coefficient (dashed curves) and the combined material and waveguide dispersion coefficients (solid curves) for (a) dispersion-shifted fiber (DSF), (b) dispersion-flattened fiber (DFF), and (c) dispersion-compensating fiber (DCF). Polarization Mode Dispersion (PMD) As indicated earlier, the fundamental spatial mode (LP ol ) of an optical fiber has two polarization modes, say linearly polarized in the x and y directions. If the fiber has perfect circular symmetry about its axis, and its material is perfectly isotropic, then the two polarization modes are degenerate, i.e., they travel with the same velocity. How- ever, fibers exposed to real environmental conditions exhibit a small birefringence that
9.3 ATTENUATION AND DISPERSION 357 varies randomly along their length. This is caused by slight variations in the refractive indexes and fiber cross-section ellipticity. Although the effects of such inhomogeneities and anisotropies on the polarization modes, and on the dispersion of optical pulses, are generally difficult to assess, we consider these effects in terms of simple models. Consider first a fiber modeled as a homogeneous anisotropic medium with principal axes in the x and y directions and principal refractive indexes n x and ny. The third principal axis lies, of course, along the fiber axis (the z direction). The fiber material is assumed to be dispersive so that n x and ny are frequency dependent, but the principal axes are taken to be frequency independent within the spectral band of interest. If the input pulse is linearly polarized in the x direction, over a length of fiber L, it will undergo a group delay 7x == NxL / co; if it is linearly polarized in the y direction, the group delay will be 7y == NyL / CO. Here, N x and Ny are the group indexes associated with n x and ny (see Sec. 5.6). A pulse in a polarization state that includes both linear polarizations will undergo a differential group delay (DGD) 87 == 17y - 7x I given by 87 == NL/co, (9.3-15) Differential Group Delay where N == IN y - Nxl. Upon propagation, therefore, the pulse will split into two orthogonally polarized components whose centers will separate in time as the pulses travel (see Fig. 9.3-7). The DGD corresponds to polarization mode dispersion (PMD) that increases linearly with the fiber length at the rate N / Co, which is usually ex- pressed in units of ps/km. o t J o Tx Ty t Figure 9.3-7 Differential group delay (DGD) associated with polarization mode dispersion (PMD). Since a long fiber is typically exposed to environmental and structural factors that vary along its axis, the simple model considered above is often inadequate. Under these conditions, a more realistic model comprises a sequence of short homogeneous fiber segments, each with its own principal axes and principal indexes. The principal axes are taken to be slightly misaligned (rotated) from one segment to the next. Such a cascaded system is generally described by a 2 x 2 Jones matrix T, which is a product of the Jones matrices of the individual segments (see Sec. 6.1B). The polarization modes of the combined system are the eigenvectors of T and are not necessarily linearly polarized modes. If the fiber is taken to be lossless, the matrix T is unitary. Its eigenvalues are then phase factors exp(j<pl) and exp(j<p2), which may be written in the form exp(jnlkoL) and exp(jn2koL), where nl and n2 are the effective refractive indexes of the two polarization modes and L is the fiber length. The propagation of light through such a length of fiber may then be determined by analyzing the input wave into components along the two polarization modes; these components travel with effective refractive indexes n land n2. Since the fiber is dispersive, T is frequency dependent and so too are the indexes nl and n2 of the modes, as well as their corresponding group indexes N l and N 2 . An input pulse with a polarization state that is the same as that of the fiber's first polarization mode travels with an effective group index N l . Similarly, if the pulse is in the second polarization mode, it travels with an effective group index N 2 . However, an input pulse
358 CHAPTER 9 FIBER OPTICS with a component in each of the fiber's polarization modes suffers DGD, as provided in (9.3-15), with N == IN l - N 2 1. A statistical model describing the random variations in the magnitude and orienta- tion of birefringence along the length of the fiber leads to an expression of the RMS value of the pulse broadening associated with DGD. This turns out to be proportional to VI instead of L, apMD == DpMDVI, (9.3-16) Polarization-Mode Dispersion where D pMD is a dispersion parameter typically ranging from 0.1 to ] ps/ vlkm . Aside from DGD, higher-order dispersion effects are also present. Each of the polarization modes is spread by group-velocity dispersion (GVD) with dispersion co- efficients proportional to the second derivative of its refractive index (see Sec. 5.6). Another higher-order effect relates to the coupled nature of the spectral and polar- ization properties of the system. Since the matrix T is frequency dependent, not only are the eigenvalues (i.e., the principal indexes nl and n2) frequency dependent, but so too are the eigenvectors (i.e., the polarization modes). If the pulse spectral width is sufficiently narrow (i.e., the pulse is not too short), we may approximately use the polarization modes at the central frequency. For ultrashort pulses, however, a more detailed nalysis that includes a combined polarization and spectral description of the system is required. Polarization states may be found such that the group delays are frequency insensitive so that their associated GVD is minimal. However, these are not eigenvectors of the Jones matrix so that the input and output polarization states are not the same. t EXERCISE 9.3-2 Differential Group Delay in a Two-Segment Fiber. Consider the propagation of an optical pulse through a fiber of length 1 Ian comprising two segments of equal length. Each segment is a single-mode anisotropic fiber with principal group indexes N x == 1.462 and Ny == 1.463. The corresponding group velocity dispersion coefficients are Dx == Dy == 20 pslkm-nm. The principal axes of one segment are at an angle of 45° with respect to the other, as illustrated in Fig. 9.3-8. (a) If the input pulse has a width of 100 ps and is linearly polarized at 45° with respect to the fiber x and y directions, sketch the temporal profile of the pulse at the output end of the fiber. Assume that the pulse source has a spectrallinewidth of 50 nm. (b) Determine the polarization modes of the full fiber and determine the temporal profile of the output pulse if the input pulse is in one of the polarization modes. YI y' , I , ,: x' ..: ?o '...... x Figure 9.3-8 Two-segment birefringent fiber. t For more details on this topic, see C. D. Poole and R. E. Wagner, Phenomenological Approach to Polarization Dispersion in Long Single-Mode Fibers, Electronics Letters, vol. 22, pp. 1029-1030, 1986.
9.4 HOLEY AND PHOTONIC-CRYSTAL FIBERS 359 Nonlinear Dispersion Yet another dispersion effect occurs when the intensity of light in the core of the fiber is sufficiently high, since the refractive index then becomes intensity dependent and the material exhibits nonlinear behavior. Since the phase is proportional to the refractive index, the high-intensity portions of an optical pulse undergo phase shifts different from the low-intensity portions, resulting in instantaneous frequencies shifted by different amounts. This nonlinear effect, called self-phase modulation (SPM), contributes to pulse dispersion. Under certain conditions, SPM can compensate the group velocity dispersion (GVD) associated with material dispersion, and the pulse can travel without altering its temporal profile. Such a guided wave is known as a soliton. Nonlinear optics is introduced in Chapter 21 and optical solitons are discussed in Chapter 22. Summary The propagation of pulses in optical fibers is governed by attenuation and several types of dispersion. Figure 9.3-9 provides a schematic illustration in which the profiles of pulses traveling through different types of fibers are compared. . In a multimode fiber (MMF), modal dispersion dominates and the width of the pulse received at the terminus of the fiber. It is governed by the disparity in the group delays of the individual modes. . In a single-mode fiber (SMF), there is no modal dispersion and the transmission of optical pulses is limited by combined material and waveguide dispersion (called chromatic dispersion) The width of the output pulse is governed by group velocity dispersion (GVD). . Material dispersion is usually much stronger than waveguide dispersion. How- ever, at wavelengths where material dispersion is small, waveguide dispersion becomes important. Fibers with special index profiles may then be used to alter the chromatic dispersion characteristics, creating dispersion-flattened, dispersion-shifted, and dispersion-compensating fibers. . Pulse propagation in long single-mode fibers for which chromatic dispersion is negligible is dominated by polarization mode dispersion (PMD). Small anisotropic changes in the fiber, caused, for example, by environmental con- ditions, alter the polarization modes so that the input pulse travels in two . polarization modes with different group indexes. This differential group delay (DGD) results in a small pulse spread. . Under certain conditions an intense pulse, called an optical soliton, can render a fiber nonlinear and travel through it without broadening. This results from a balance between material dispersion and self-phase modulation (the depen- dence of the refractive index on the light intensity), as discussed in Chapter 22. 9.4 HOLEY AND PHOTONIC-CRYSTAL FIBERS A holey fiber is a pure silica-glass fiber that contains multiple cylindrical air holes parallel to, and along the length of, its axis. The holes are organized in a regular periodic pattern. As illustrated in Fig. 9.4-1, the core is defined by a defect, or fault,
360 CHAPTER 9 FIBER OPTICS (C)) I o o t MMF illt ' 7 aT , n . _ ". 'min 'max t (a) (()) o t SMF I o - aT . t (b) (()) I o -L t (c) o t SMF with PMD Soliton - '0 C()) '0 (d) I o t Nonlinear SMF 0 t Figure 9.3-9 Broadening of a short optical pulse after transmission through different types of fibers. (a) Modal dispersion in a multimode fiber (MMF). (b) Material and waveguide dispersion in a single-mode fiber (SMF). (c) Polarization mode dispersion (PMD) in a SMF. (d) Soliton transmission in a nonlinear SMF. in the periodic structure, such as a missing hole, a hole of a different size, or an extra hole. The holes are characterized by the spacing between their centers, A, and their diameters, d. The quantity A, which is also called the pitch, is typically in the range 1-10 /Jm. It is not necessary to include dopants in the glass. A d ----I t+- (a) CfSllRcW5NdP (J(I(IQODDOO 0000 (b) ,Wo ooD<?QQ °oo08 <booo@8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. -,..... .....,) ..... . . . . ......... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c) Figure 9.4-1 Various forms of holey fibers. (a) Solid core (dotted circle) surrounded by a cladding of the same material but suffused with a periodic array of cylindrical air holes whose diameters are much smaller than a wavelength. The average refractive index of the cladding is lower than that of the core. (b) Photonic-crystal holey fiber with cladding that contains a periodic array of large air holes and a solid core (dotted circle). (c) Photonic-crystal holey fiber with cladding that contains a periodic array of large air holes and a core that is an air hole of a different size (dotted circle). Holey fibers guide optical waves via one of two mechanisms: effective-index guid- ance and photonic-bandgap guidance, which we consider in turn below.
9.4 HOLEY AND PHOTONIC-CRYSTAL FIBERS 361 Effective-Index Guidance If the hole diameter is much smaller than the wavelength of light (d « A), then the periodic cladding behaves approximately as a homogeneous medium whose effective refractive index n2 is equal to the average refractive index of the holey material [see Fig. 9.4-1(a)]. Waveguiding is then achieved by making use of a solid core with index nl > n2, so that the light is guided by total internal reflection as with conventional fibers. In this configuration, the holes serve merely as distributed "negative dopants" that reduce the refractive index of the cladding below that of the solid core. The holes can therefore be randomly, rather than periodically, arrayed and they need not be axially continuous. If the size of the holes is not much smaller than the wavelength, then the ho- ley cladding must be treated as a two-dimensional periodic medium. The effective refractive index n2 is then equal to the average refractive index, weighted by the optical intensity distribution in the medium, and is therefore strongly dependent on the wavelength as well as on the size and the geometry of the holes. Since waves of shorter wavelength are more confined in the medium of higher refractive index, the effective refractive index of the cladding n2 (A) is a decreasing function of the wavelength. A similar effect occurs in a I D photonic crystal, for which the effective refractive index is an increasing function of frequency at frequencies in the lowest photonic band (see Fig. 7.2-6). The holey fiber is therefore endowed with strong waveguide dispersion, which can be an extremely useful feature. One consequence of the waveguide dispersion is that the holey fiber may operate as a single-mode structure over a broad range of wavelengths, possibly stretching from the infrared to the ultraviolet. t This property, called en dless singl e-mode guidance, results when the fiber V parameter, V == (27raj A) J ni - n(A), is approximately independent of A. This c ondition ari ses when the effective index n2 (A) decreases with A in such a way that J ni - n (A) ex A. For a conventional fiber, in contrast, V is inversely proportional to A so that single-mode behavior at a particular wavelength (V < 2.405) morphs into multi mode behavior for a sufficiently lower wavelength such that V increases above 2.405. Another interesting feature is the feasibility of large mode-area (LMA) single- mode operation. Optical fibers with large mode areas are useful for applications requir- ing high power delivery. In a conventional fiber the condition of single-mode operation (V == 27r( aj Ao)NA < 2.405) can be met for a large core diameter 2a by use of a small numerical aperture. Similarly, the guided mode size can be increased in holey fibers by having a larger hole-to-hole spacing A (thus resulting in a larger core diameter) and using holes with smaller diameter d (creating a lower numerical aperture and allowing the field to penetrate farther into the cladding). Dramatic increases in mode area for relatively small changes in the hole size are obtained and mode areas that are several order of magnitudes greater than in conventional fibers have been reported. Photonic-Bandgap Guidance The cladding of a holey fiber may be regarded as a two-dimensional photonic crys- tal. The triangular-hole microstructure shown in Fig. 9.4-1(b), for example, has a dispersion diagram with photonic bandgaps, as shown in Fig. 7.3-3 and discussed in Sec. 7.3A. If the optical frequency lies within the photonic bandgap, propagation through the cladding is prohibited and the fiber serves as a photonic-crystal waveguide (see Sec. 8.5). A photonic-crystal fiber (PCF) may have a solid or hollow core, as illustrated in Fig. 9.4-1(b) and (c), respectively. Fibers with a hollow core are novel since they cannot t See T. A. Birks, J. C. Knight, and P. St. J. Russell, Endlessly Single-Mode Photonic Crystal Fibre, Optics Letters, vol. 22, pp. 961-963, 1997.
362 CHAPTER 9 FIBER OPTICS operate by effective-index guidance; i.e., guidance cannot be based on total internal reflection. A guided wave traveling in an air-core PCF suffers lower losses and reduced nonlinear effects, and can carry increased amounts of optical power. As a result, PCFs offer many unique design possibilities. Dispersion flattening over broad wavelength ranges can be achieved as can dispersion shifting to wavelengths lower than the zero- material-dispersion wavelength. Powerful fiber lasers that operate over a broad range of wavelengths can be constructed. A whole raft of other applications are also possible, such as analyzing gases by introducing them into the fiber core. READING LIST Books See also the books on optical waveguides in Chapter 8. C. DeCusatis and C. J. Sher DeCusatis, Fiber Optic Essentials, Elsevier, 2005. J. Hecht, Understanding Fiber Optics, Prentice Hall, 5th ed. 2005. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres, Imperial College Press (London), 2005. A. Galtarossa and C. R. Menyuk, eds., Polarization Mode Dispersion, Springer-Verlag, 2005. R. P. Khare, Fiber Optics and Optoelectronics, Oxford University Press, 2004. J. A. Buck, Fundamentals of Optical Fibers, Wiley, 1995, 2nd ed. 2004. J. Hecht, City of Light: The Story of Fiber Optics, Oxford University Press, 2004. A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibers, Springer-Verlag, 2003. J. K. Petersen, Fiber Optics Illustrated Dictionary, CRC Press, 2003. I. Kaminow and T. Li, eds. Optical Fiber Telecommunications IVA: Components, Academic Press, 2002. C. Manolatou and H. A. Haus, Passive Components for Dense Optical Integration, Kluwer, 2002. D. R. Goff and K. S. Hansen. eds., Fiber Optic Reference Guide: A Practical Guide to the Technology, Focal, 3rd ed. 2002. R. Tricker, Optoelectronics and Fiber Optic Technology, Newnes, 2002. R. J. Bates, Basic Fiberoptics Technologies, McGraw-Hill, 2001. J. Crisp, Introduction to Fiber Optics, Newnes, 2nd ed. 2001. A. Othonos and K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunica- tions and Sensing, Artech, 1999. . A. Ghatak and K. Thyagarajan, An Introduction to Fiber Optics, Cambridge University Press, 1998. J. S. Sanghera and I. D. Aggarwal, eds., Infrared Fiber Optics, CRC Press, 1998. J. P. Powers, An Introduction to Fiber Optic Systems, Irwin, 2nd ed. 1997. M. H. Weik, Fiber Optics Standard Dictionary, Chapman & Hall, 3rd ed. 1997. A. Kumar, Antenna Design with Fiber Optics, Artech House, 1996. J. L. Miller and E. Friedman, Photonics Rules of Thumb: Optics, Electro-Optics, Fiber Optics, and Lasers, McGraw-Hill, 1996. C.-L. Chen, Elements of Optoelectronics and Fiber Optics, Irwin, 1995. R. B. Dyott, Elliptical Fiber Waveguides, Artech House, 1995. N. Kashima, Passive Optical Components for Optical Fiber Transmission, Artech, 1995. S. G. Krivoshlykov, Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides, Akademie- Verlag, 1994. G. Cancellieri, Single-Mode Optical Fiber Measurement: Characterization and Sensing, Artech House, 1993. J. E. Midwinter, Optical Fibers for Transmission, Wiley, 1979; Krieger, reissued 1992. K. Chang, ed., Fiber and Electro-Optical Components, Wiley, 1991. P. K. Cheo, Fiber Optics and Optoelectronics, Prentice Hall, 1985, 2nd ed. 1990.
PROBLEMS 363 P. Diament, Wave Transmission and Fiber Optics, Macmillan, 1989. D. Marcuse, Light Transmission Optics, Van Nostrand Reinhold, 1972, 2nd ed. 1982; Krieger, reis- sued 1989. C. K. Kao, Optical Fiber Systems: Technology, Design, and Applications, McGraw-Hill, 1982, reprinted 1986. D. Marcuse, PrincipLes of Optical Fiber Measurements, Academic Press, 1981. Articles M. Bayindir, A. F. Abouraddy, o. Shapira, J. Viens, D. Saygin-Hinczewski, F. Sorin, J. Arnold, 1. D. Joannopoulos, and Y. Fink, Kilometer-Long Ordered Nanophotonic Devices by Preform-to- Fiber Fabrication, IEEE Journal of SeLected Topics in Quantum Electronics, vol. 12, pp. 1202- 1213, 2006. P. Russell, Photonic Crystal Fibers, Science, vol. 299, pp. 358-362, 2003. Issue on novel and specialty fibers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 3, 2001. Millennium issue, IEEE Journal of Selected Topics in Quantum ELectronics, vol. 6, no. 6, 2000. J. P. Gordon and H. Kogelnik, PMD Fundamentals: Polarization Mode Dispersion in Optical Fibers, Proceedings of the National Academy of Sciences (USA), vol. 97, pp. 4541-4550, 2000. Issue on fiber-optic passive components, IEEE JournaL of Selected Topics in Quantum Electronics, vol. 5, no. 5, 1999. T. A. Birks, J. C. Knight, and P. St. J. Russell, Endlessly Single-Mode Photonic Crystal Fibre, Optics Letters, vol. 22, pp. 961-963, 1997. P. J. B. Clarricoats, Optical Fibre Waveguides-A Review, in Progress in Optics, vol. 14, E. Wolf, ed., North-Holland, 1977. D. Gloge, Weakly Guiding Fibers, Applied Optics, vol. 10, pp. 2252-2258, 1971. D. Gloge, Dispersion in Weakly Guiding Fibers, Applied Optics, vol. 10, pp. 2442-2445, 1971. PROBLEMS 9.1-1 Coupling Efficiency. (a) A source emits light with optical power Po and a distribution 1(0) == (1/ 7r ) Po cos 0, where 1(0) is the power per unit solid angle in the direction making an angle 0 with the axis of a fiber. Show that the power collected by the fiber is P == (NA)2 Po, so that the coupling efficiency is (NA)2, where NA is the numerical aperture of the fiber. (b) If the source is a planar light-emitting diode of refractive index ns bonded to the fiber, and the fiber cross-sectional area is larger than the LED emitting area, calculate the numerical aperture of the fiber and the coupling efficiency when nl == 1.46, n2 == 1.455, and ns == 3.5. 9.1-2 Numerical Aperture of a Graded-Index Fiber. Compare the numerical apertures of a step- index fiber with nl == 1.45 and == 0.01 and a graded-index fiber with nl == 1.45, == 0.01, and a parabolic refractive-index profile (p == 2). (See Exercise 1.3-2.) 9.2-1 Modes. A step-index fiber has radius a == 5 J-Lm, core refractive index nl == 1.45, and fractional refractive-index change == 0.002. Determine the shortest wavelength Ac for which the fiber is a single-mode waveguide. If the wavelength is changed to Ac/2, identify the indexes (1, m) of all the guided modes. 9.2-2 Modal Dispersion. A step-index fiber of numerical aperture NA == 0.16, core radius a == 45 J-Lm, and core refractive index nl == 1.45 is used at Ao == 1.3 J-Lm, where material dispersion is negligible. If a light pulse of very short duration enters the fiber at t == 0 and travels a distance of 1 lan, sketch the shape of the received pulse: (a) Using ray optics and assuming that only meridional rays are allowed. (b) Using wave optics and assuming that only meridiona] (1 == 0) modes are allowed. 9.2-3 Propagation Constants and Group Velocities. A step-index fiber with refractive indexes nl == 1.444 and n2 == 1.443 operates at Ao == 1.55 J-Lm. Determine the core radius at which
364 CHAPTER 9 FIBER OPTICS the fiber V parameter is 10. Use Fig. 9.2-3 to estimate the propagation constants of all the guided modes with I == O. If the core radius is now changed so that V == 4, use Fig. 9.2-8(a) to determine the phase velocity, the propagation constant, and the group velocity of the LP 01 mode. Ignore the effect of material dispersion. 9.2-4 Propagation Constants and Wavevector (Step-Index Fiber). A step-index fiber of radius a == 20 /-Lm and refractive indexes nl == 1.47 and n2 == 1.46 operates at Ao == 1.55 /-Lm. Using the quasi-plane wave theory and considering only guided modes with azimuthal index I == 1: (a) Determine the smallest and largest propagation constants. (b) For the mode with the smallest propagation constant, determine the radii of the cylindri- cal shell within which the wave is confined, and the components of the wavevector k at r == 5 /-Lm. 9.2-5 Propagation Constants and Wavevector (Graded-Index Fiber). Carry out the same calcu- lations as in Prob. 9.2-4, but for a graded-index fiber with parabolic refractive-index profile (p == 2). 9.3-3 Scattering Loss. At Ao == 820 nm the absorption loss of a fiber is 0.25 dB/km and the scattering loss is 2.25 dB/kIn. If the fiber is used instead at Ao == 600 nm, and calorimetric measurements of the heat generated by light absorption give a loss of 2 dB /km, estimate the total attenuation at Ao == 600 nm. 9.3-4 Modal Dispersion in Step-Index Fibers. Determine the core radius of a multi mode step- index fiber with a numerical aperture N A == 0.1 if the number of modes lvI == 5000 when the wavelength is 0.87 /-Lm. If the core refractive index nl == 1.445, the group index N 1 == 1.456, and is approximately independent of wavelength, determine the modal-dispersion response time aT for a 2-km-Iong fiber. 9.3-5 Modal Dispersion in Graded-Index Fibers. Consider a graded-index fiber with a/ Ao == 10, nl == 1.45, == 0.01, and power-law profile with index p. Determine the number of modes AI, and the modal-dispersion pulse-broadening rate a T / L, for p == 1.9, 2, 2.1, and 00. 9.3-6 Pulse Propagation. A pulse of initial temporal width 70 is transmitted through a graded- index fiber of length L km and power-law refractive-index profile with index p. The peak refractive index nl is wavelength-dependent with D). == -(Ao/c o ) d2nl/dA, l::J. is approx- imately independent of wavelength, a). is the spectral width of the source, and Ao is the operating wavelength. Discuss the effect of increasing each of the following parameters on the temporal width of the received pulse: L, 70, p, ID).I, a)., and Ao.
RESONATOR OPTICS 10.1 PLANAR-MIRROR RESONATORS A. Resonator Modes B. Off-Axis Resonator Modes 10.2 SPHERICAL-MIRROR RESONATORS A. Ray Confinement B. Gaussian Modes C. Resonance Frequencies D. Hermite-Gaussian Modes *E. Finite Apertures and Diffraction Loss 10.3 TWO- AND THREE-DIMENSIONAL RESONATORS A. Two-Dimensional Rectangular Resonators B. Circular Resonators and Whispering-Gallery Modes C. Three-Dimensional Rectangular Cavity Resonators 10.4 MICRORESONATORS A. Rectangular Microresonators B. Micropillar, Microdisk, and Microtoroid Microresonators C. Microsphere Microcavities D. Photonic-Crystal Microcavities CHAPTER 10 367 378 390 394 Charles Fabry Alfred Perot (1867-1945) (1863-1925) Fabry and Perot constructed an optical resonator for use as an interferometer. Now known as the Fabry-Perot etalon, it is used extensively in lasers. 365
An optical resonator is the optical counterpart of an electronic resonant circuit. It confines and stores light at resonance frequencies determined by its configuration. It may be viewed as an optical transmission system that incorporates feedback: light circulates or is repeatedly reflected within the resonator. Various optical resonator configurations are depicted in Fig. 10.0-1. The simplest of these, the Fabry-Perot resonator, comprises two parallel planar mirrors; light is repeatedly reflected between them while experiencing little loss. Other mirror configurations include spherical mir- rors, ring arrangements, and rectangular two- and three-dimensional cavities. Fiber- ring resonators and integrated-optic-ring resonators are also widely used. Dielectric resonators make use of total internal reflection at the boundary between two low-loss dielectric materials in lieu of mirrors. The confined rays skim around the inside rim of the resonator with an angle of incidence that is always greater than the critical angle, preventing them from refracting out of the resonator. In microdisks, mi- crotoroids, and microspheres, light circulates by reflecting at near-grazing incidence, in what are known as whispering-gallery modes. Periodic dielectric structures such as distributed Bragg reflectors (DBRs) play the role of mirrors in conventional Fabry- Perot resonators, providing feedback in structures such as the micropillar resonator. Two-dimensional photonic crystals with a defect are also used to make microcavities. )' II( )' Planar-mirror Spherical-mirror Ring-mirror Rectangular cavity ,/ Fiber-ring P::: o Integrated-optic-ring Jc:- ',I tt --------- Microdisk P::: o ,AY Microtoroid Microsphere Micropillar Photonic-crystal Figure 10.0-1 Storage of light in optical resonators via: multiple reflections from mirrors; propagation though closed-loop optical fibers and integrated-optic waveguides; whispering-gallery mode reflections near the surface of disks, toroids, and spheres; reflections from periodic structures such as Bragg gratings; and defects in photonic crystals. The optical resonator is characterized by two key parameters: . Modal volume V, which is the volume occupied by the confined optical mode. . Quality factor Q, which is proportional to the storage time in units of optical period. These parameters represent the degrees of spatial and temporal light confinement in the resonator. Improvement of spatial confinement has been achieved by the development 366
10.1 PLANAR-MIRROR RESONATORS 367 of microresonators of various geometries, while enhancement of temporal confinement has been realized by making use of low-loss materials and low-leakage configurations. Because of their frequency selectivity, optical resonators serve as optical filters or spectrum analyzers, as discussed in Chapter 7. Their most important use, however, is as a "container" within which laser light can be generated and built up. The laser comprises a medium that amplifies light inside an optical resonator; the resonator determines, in part, the frequency and spatial distribution of the laser beam produced. Because resonators have the capability of storing energy, they can also be used to generate pulses of laser energy. Lasers are discussed in Chapters 15 and 17; the material contained in this chapter is essential to their understanding. This Chapter Several theoretical approaches considered in previous chapters are useful for describ- ing the operation of optical resonators: . The simplest approach is based on Ray Optics (Chapter 1). Optical rays are traced as they repeatedly reflect within the resonator and geometrical conditions are established that assure that the rays are confined. . Wave Optics (Chapter 2) is used to determine the modes of the resonator, i.e., the resonance frequencies and wavefunctions of the optical waves that are permitted to exist self-consistently within the resonator. . The study of Beam Optics (Chapter 3) is useful for understanding the behavior of spherical-mirror resonators; the modes of a resonator with spherical mirrors are Gaussian and Hermite-Gaussian optical beams. . Fourier Optics and the theory of light propagation and diffraction (Chapter 4) determine how the finite sizes of the resonator mirrors affect resonator loss and the spatial characteristics of the modes. . Photonic-Crystal Optics and the optics of multilayer media (Chapter 7) is impor- tant for optical resonators, since they often make use of multiple dielectric layers and periodic media (e.g., distributed Bragg reflectors and photonic crystals) in lieu of mirrors. . The analysis of resonator modes is similar to that used in Guided- Wave Optics (Chapter 8) to determine the modes of planar-mirror and dielectric waveguides since the optical resonator may be regarded as an optical waveguide with reflec- tors at both ends - the propagating light is thus repeatedly reflected and confined with little leakage. The optical resonator evidently provides an excellent venue for applying the dif- ferent theories of light presented in earlier chapters. We begin with a study of planar- mirror resonators in Sec. 10.1 and spherical-mirror resonators in Sec. 10.2. We then introduce two- and three-dimensional resonators in Sec. 10.3 and consider microres- onators in Sec. 10.4. 10.1 PLANAR-MIRROR RESONATORS A. Resonator Modes In this section we examine the modes of an optical resonator constructed from two parallel, highly reflective, flat mirrors separated by a distance d (Fig. 10.1-1). This simple one-dimensional resonator is known as a Fabry-Perot etalon. We first consider an idealized version in which the mirrors are lossless; the effect of losses is included subsequently.
368 CHAPTER 10 RESONATOR OPTICS (a) (b) / « d )1 / « d -/ Figure 10.1-1 Two-mirror planar resonator (Fabry-Perot etalon). (a) Light rays perpendicular to the mirrors reflect back and forth without escaping. (b) Rays that are only slightly inclined eventually escape. Rays also escape if the mirrors are not perfectly parallel. Resonator Modes as Standing Waves As discussed in Sees. 2.2, 5.3, and 5.4, a monochromatic wave of frequency v has a wave function u ( r, t) == Re {U ( r) exp (j 27r vt )} , (10.1-1) representing a transverse component of the electric field. The complex amplitude U (r) satisfies the Helmholtz equation, \72U(r) + k 2 U(r) == 0, where k == 27rv/c is the wavenumber and c is the speed of light in the medium. The resonator modes are the solutions to the Helmholtz equation under the appropriate boundary conditions. For the lossless planar-mirror resonator, the transverse components of the electric field vanish at the mirror surfaces (see Sec. 5.1), so that U(r) == 0 at the planes Z == 0 and Z == d in Fig. 10.1-2. The standing wave U (r) == A sin kz, where A is a constant, satisfies the Helmholtz equation and vanishes at Z == 0 and Z == d if k satisfies the condition kd == q7r, where q is an integer. This restricts k to the values 7r kq == q d ' so that the modes have complex amplitudes U(r) == Aqsinkqz, q== 1,2,..., (10.1-2) (10.1-3) where the Aq are constants. Negative values of q do not constitute independent modes since sin k_qz == - sin kqz. Furthermore, the value q == 0 is associated with a mode that carries no energy since ko == 0 and sin koz == O. The modes of the resonator are therefore the standing waves Aq sin kqz, where the positive integer q == 1,2,... is called the mode number. An arbitrary wave inside the resonator can be written in terms of a superposition of the resonator modes: U(r) == L Aq sin kqz. (10.1-4) It follows from (10.1-2) that the assocPated frequencies v == ck/27r are restricted to the discrete values c v q == q 2d ' which are the resonance frequencies of the resonator. As illustrated in Fig. 10.1-3 adjacent resonance frequencies are separated by a constant frequency difference q== 1,2,..., (10.1-5) c Vp == 2d . (10.1-6) Frequency Spacing of Resonator Modes
10.1 PLANAR-MIRROR RESONATORS 369 -.1 A z z=o A/2 -.1 z=d Figure 10.1-2 (a) Complex amplitude of an ideal planar-mirror resonator mode. Since 14 half- wavelengths match the length of the resonator in this illustration, the mode number q = d/ (>-"/2) = 14. (b) Intensity distribution. The associated resonance wavelengths are Aq == clv q == 2d I q. The round-trip distance traversed at resonance must therefore precisely equal an integer number of wave- lengths: 2d == qAq, q== 1,2,.... (10.1-7) It is important to keep in mind that c coin is the speed of light in the medium between the two mirrors, and that the Aq represent wavelengths within that medium. d---1 -+J I d I 0 Resonator 4 VF= 2 (a) v q v q+ I v r- VF= 2 -1 I I . v3 v (b) vI V2 Resonant frequencies Figure 10.1-3 The adjacent resonance frequencies of a planar-mirror resonator are separated by VF = c/2d = c o /2nd, as illustrated by two examples: (a) A 30-cm long resonator (d = 30 em) with air between the mirrors (n = 1) has a frequency spacing between modes given by v F = 500 MHz. (a) A much shorter resonator with d = 3 /-Lm has v F = 50 THz, so that the first mode has a frequency corresponding to a wavelength of 6 /-LID and there are only two modes within the 700-900-nm optical band, which occupies a frequency range of 95 THz. Resonator Modes as Traveling Waves Alternatively, the resonator modes can be determined by following a wave as it travels back and forth between the two mirrors [Fig. 10.1-4(a)]. A mode is a wave that repro- duces itself after a single round trip (see Appendix C). The phase shift imparted by a single round trip of propagation (a distance 2d), c.p == k2d == 41TVd I c, must therefore be a multiple of 21T: c.p == k2d == q21T, q == 1, 2, . . . . (10.1-8) This result is not altered by an additional phase shift of 21T, which can be imparted by reflections at the two mirrors (see Sec. 6.2). As expected, we therefore obtain kd == q1T, as in (10.1-2), and the same resonance frequencies as set forth in (10.1- 5). Equation (10.1-8) may be viewed as a condition of positive feedback in the system
370 CHAPTER 10 RESONATOR OPTICS Ve -J<P Mirror 1 Mirror 2 VI e -j<p cr5tI127r p Va Va VI V2 . . ) ... (tg,27r (a) (b) (c) Figure 10.1-4 (a) A wave reflects back and forth between the resonator mirrors, suffering a phase shift cp on each round trip. (b) Block diagram of an optical feedback system with a phase delay cpo (c) Phasor diagram representing the sum U == U o + U 1 + . . . for cp =I q27r and for cp == q27r. displayed in Fig. 1 0.1-4(b); this requires that the output of the system be fed back in phase with the input. We now demonstrate that only self-reproducing waves, or combinations thereof, can exist within the resonator under steady-state conditions. Consider a monochromatic plane wave of complex amplitude U o at point P traveling to the right along the axis of the resonator [see Fig. 10.1-4(a)]. The wave is reflected from mirror 2 and propagates back to mirror 1 where it is again reflected. Its amplitude at P then becomes U 1 . Yet another round trip results in a wave of complex amplitude U 2 , and so on ad infinitum. Because the original wave U o is monochromatic, it is "eternal." Indeed, all of the partial waves, U o , U 1 , U 2 , . .. are monochromatic and perpetually coexist. Moreover, their magnitudes are identical because it has been assumed that there is no loss associated with reflection and propagation. The total wave U is therefore represented by the sum of an infinite number of phasors of equal magnitude, U == U o + U 1 + U 2 + . . . , (10.1-9) as shown in Fig. 10.1-4(c). The phase difference of two consecutive phasors imparted by a single round trip of propagation is c.p == k2 d. If the magnitude of the initial phasor is infinitesimal, the magnitude of each of these phasors must also be infinitesimal. The magnitude of the sum of this infinite number of infinitesimal phasors is itself infinitesimal unless they are aligned, i.e., unless c.p == q27r, as illustrated at the bottom of Fig. 10.1-4(c). Thus, an infinitesimal initial wave can result in the buildup of finite power in the resonator, but only if <p == q27r. Traveling-Wave Resonators In a traveling-wave resonator, an optical mode travels in one direction along a closed path representing a round trip and retraces itself without reversing direction. Examples are the ring resonator and the bow-tie resonator illustrated in Fig. 10.1-5. The reso- nance frequencies of the modes may be obtained by equating the round-trip phase shift to 27r. Each of the set of modes traveling in the clockwise direction has a corresponding mode of the same resonance frequency traveling in the counterclockwise direction, and the matching modes are said to be degenerate. EXERCISE 10.1-1 Resonance Frequencies of a Traveling-Wave Resonator. Derive expressions for the res- onance frequencies v q and their frequency spacing Vp for the three-mirror ring resonator and the
10.1 PLANAR-MIRROR RESONATORS 371 four-mirror bow-tie resonator shown in Fig. 10.1-5. Assume that each mirror reflection introduces a phase shift of 7r. T d 1 (a) Ring resonator (b) Bow-tie resonator Figure 10.1-5 Traveling-wave resonators. (a) Three-mirror ring resonator. (b) Four-mirror bow-tie resonator. Density of Modes The number of modes per unit frequency is the inverse of the frequency spacing be- tween modes, i.e., 1/ Vp == 2d / c in each of the two orthogonal polarizations. The density of modes M(v), which is the number of modes per unit frequency per unit length of the resonator, is therefore 4 M(v)==-. c (10.1-10) Density of Modes (1 D Resonator) The number of modes in a resonator of length d , in the frequency interval v, is thus ( 4/ c) d D:.v. This represents the number of degrees of freedom for the optical waves existing in the resonator, i.e., the number of independent ways in which these waves may be arranged. Losses and Resonance Spectral Width The strict condition on the frequencies of optical waves that are permitted to exist inside a resonator is relaxed when the resonator has losses. Consider again Fig. 10.1- 4(a) and follow the initial wave inside the resonator, Uo, in its excursions between the two mirrors. As discussed above, the result is the infinite sum of phasors shown in Fig. 10. I -4(c) and the phase difference imparted by propagation through a single round trip is c.p == 2kd == 47rvd / c . (10.1-11) Reflection at the two mirrors can impart an additional phase shift, usually 27r. However, in the presence of loss the phasors are not all of equal magnitude. Two successive phasors are related by a complex round-trip amplitude attenuation factor h == Irle-j<p resulting from losses associated with the two mirror reflections and the absorption in the medium (the corresponding intensity attenuation factor for a round trip is Irl 2 with Irl < 1). Thus, U 1 == hUo and, in fact, U 2 is related to U 1 by this same complex factor h, as are all consecutive phasor pairs. The net result is the superposition of an infinite number of waves, each distinguished from the previous one by a constant phase shift and an amplitude that is geometrically reduced. It is readily seen that U == U o + U 1 + U 2 +. . . == U o + hUo + h 2 U O +. . . == Uo(l + h + h 2 +. . .) == Uo/(l- h).
372 CHAPTER 10 RESONATOR OPTICS The net result, U == U o / (1 - h), is easily understood in terms of the simple feedback configuration pictured in Fig. 10.1-4(b). The intensity of the light in the resonator is therefore given by I = IUl 2 = IU o l 2 11 -lrle- jcp I 2 10 1 + Irl 2 - 21rl cos cp' (10.1-12) which can be written as I == Imax 1 + (2'Y/7r)2 sin 2 (cp/2) , 1 == 10 max (1 _ Irl)2 . (10.1-13) Here 10 == I U o 1 2 is the intensity of the initial wave and 9"= 1fJfTT l-Irl (10.1-14) finesse is the finesse of the resonator. The treatment offered above is nearly identical to that provided earlier in Sec. 2.5B, where the complex round-trip amplitude attenuation factor was chosen to be h == Ihle+ jcp . In the current context we instead select this factor to be h == Irle-jk == Irle-jcp by dint of the fact that successive phasors arise from the delay of the wave as it bounces between the mirrors. This distinction is superficial, however, and has no bearing on the results. Indeed, (10.1-13) is identical to (2.5-18), which is plotted in Fig. 2.5-9(b). The intensity I ( c.p) is a periodic function of c.p with period 27r. For large , I ( c.p) has sharp peaks centered about the values c.p == q27r, which correspond to the alignment of all phasors. The peaks have a full width at half maximum (FWHM) described by D,.c.p 27r /, in accordance with (2.5-20). The internal resonator intensity I ( c.p) in (10.1-13) can alternately be expressed as a function of the optical frequency of an internal monochromatic wave, I(v), by virtue of (10.1-11), which shows that c.p == 47rvd / c. This function then takes the form I == 1m ax 1 + (2/7r)2 sin 2 (7rV/Vp) , I _ 10 max - (1 _ Irl)2 , (10.1-15) with Vp == c/2d. This result is displayed in Fig. 10.1-6 and indeed it mirrors that depicted in Fig. 2.5-9. The maximum internal intensity I == Imax is attained when the second term in the denominator is zero, i.e., at the resonance frequencies v == V q == qvp, q== 1,2,.... (10.1-16) The minimum intensity, attained at the midpoints between the resonances, is Imax I min = 1 + (29" /1f)2 ' (10.1-17)
10.1 PLANAR-MIRROR RESONATORS 373 I C vF=- (a) 2d A v I (b) v Figure 10.1-6 (a) In the steady state, a lossless resonator (3=' == 00) sustains light waves only at the precise resonance frequencies v q . (b) A lossy resonator best sustains waves in the immediate vicinity of the resonance frequencies, but it can sustain waves at other frequen- cies as well. VF T v When the finesse is large ( » 1), it is clear that the spectral response of the resonator is sharply peaked about the resonance frequencies and Iminl Imax is small. In that case, the FWHM of the resonance peaks is 8v vp/ since 8v == (c/47rd)<p and <p 27r I in accordance with (2.5-20). This simple result provides the rationale for the definition of the finesse given in (10.1-14). In short, the spectral response of the Fabry-Perot optical resonator is characterized by two parameters: . The frequency spacing vp between adjacent resonator modes: c vp == 2d . (10.1-18) Frequency Spacing . The spectral width 8v of the individua] resonator modes: vp 8v . (10.1-19) Spectral Width Equation (10.1-19) is valid in the usual case when » 1. The spectral width 8v is inversely proportional to the finesse . As the loss increases, decreases and 8 v therefore increases. Sources of Resonator Loss The two principal sources of loss in optical resonators are: . Losses arising from imperfect reflection at the mirrors. There are two underlying sources of reduced reflection: (1) a partially transmitting mirror is often deliber- ately used in a resonator to permit laser light generated in the resonator to escape through it; and (2) the finite size of the mirrors causes a fraction of the light to leak around them and thereby to be lost. This latter effect also modifies the spatial distribution of the reflected wave by truncating it to the size of the mirror. The reflected light produces a diffraction pattern at the opposite mirror that is again truncated. Such diffraction loss may be regarded as an effective reduction of the mirror reflectance. Further details regarding diffraction loss are provided in Sec. 10.2E. . Losses attributable to absorption and scattering that occurs in the medium be- tween the mirrors. The round-trip power attenuation factor associated with these
374 CHAPTER 10 RESONATOR OPTICS effects is exp ( - 2a s d ), where as is the loss coefficient of the medium associated with absorption and scattering. For mirrors of reflectances 9(1 == Ir112 and 9(2 == Ir212, the wave intensity decreases by the factor 9(19<2 as a result of the two reflections associated with a single round trip. These are referred to as "lumped losses" since they occur only at the discrete locations where the mirrors are located. Accounting also for the "distributed losses" that take place within the intervening medium yields a round-trip intensity attenuation factor Irl 2 == 9(19(2 exp( -2a s d), (10.1-20) which is usually written in the form Irl 2 == exp( -2a r d), (10.1-21) where a r is an effective overall distributed-loss coefficient. Equating (10.1-20) and (10.1-21), and taking the natural logarithm of both sides, allows a r to be written in terms of the distributed and lumped loss parameters, as and 9<19(2, respectively: 1 1 a r = as + 2d In l2 . (10.1-22) Loss Coefficient This can also be written as a r == as + amI + a m 2, (10.1-23) where the quantities 1 1 aml = 2d In l ' 1 1 a m 2 = 2d In 2 (10.1-24) represent the effective distributed-loss coefficients associated with mirrors 1 and 2, respectively. These loss coefficients can be cast in a simpler form for mirrors of high reflectance. If 9<1 1, then In(l/9<I) == -In(9<I) == -In(l - (1 - 9(1)] 1 - 9(1, where we have used the Taylor-series approximation In(l - ) -, which is valid for II « 1. This allows us to write aml 1 - 9<1 2d (10.1-25) Similarly, if 9<2 1, we have a m 2 (1-9<2) /2d . If, furthermore, 9<1 == 9(2 == 9( 1, then a r as + 1- 9( d (10.1-26) The finesse :r can be expressed as a function of the effective loss coefficient a r by substituting (10.1-21) in (10.1-14). The result is 9=' == 1T exp( -ard /2) 1 - exp( -ard) , (10.1-27)
10.1 PLANAR-MIRROR RESONATORS 375 which is plotted in Fig. 10.1-7. It is clear that the finesse decreases as the loss increases. If the loss factor ard « 1, then exp( -ard) 1 - ard , whereupon 7r f1::" r-..; _ Jr-..;. ard (10.1-28) Finesse and Loss Factor This demonstrates that the finesse is inversely proportional to the loss factor a r d in this limit. 200 (1.) C/'.) C/'.) (1.) s::: 100 o 0.01 1 Figure 10.1-7 Finesse of an opti- cal resonator versus the loss factor Or d, where Or is the effective overall distributed-loss coefficient. The round- trip intensity attenuation factor ITI2 exp( -2o r d). EXERCISE 10.1-2 Resonator Modes and Spectral Width. Determine the frequency spacing, and spectral width, of the modes of a Fabry-Perot resonator whose mirrors have reflectances 0.98 and 0.99 and are separated by a distance d = 100 em. Assume that the medium has refractive index n = 1 and negligible losses. Is the approximation used to derive (10.1-28) appropriate in this case? Photon Lifetime The relationship between the resonance linewidth and resonator loss may be viewed as a manifestation of the time-frequency uncertainty relation, as we now demonstrate. Substituting (10.1-18) and (10.1-28) in (10.1-19), we obtain 8v c/2d == car . 7r/a r d 27r (10.1-29) Because a r is the loss per unit length, car represents the loss per unit time. Defining the characteristic decay time 1 Tp == - car (10.1-30)
376 CHAPTER 10 RESONATOR OPTICS as the resonator lifetime or photon lifetime, we obtain 1 8v == . 27rT p (10.1-31) The time-frequency uncertainty product is therefore 8v . Tp 1/27r. Resonance- line broadening may therefore be considered to be a consequence of optical-energy decay arising from resonator losses. An electric field that decays as exp( -t/2Tp), corresponding to an energy that decays as exp( -t/Tp), has a Fourier transform that is proportional to 1/(1 + j47rVT p ), which has a (FWHM) spectral width 8v == 1/27rTp. Quality Factor Q The quality factor Q is often used to characterize electrical resonance circuits and microwave resonators. This parameter is defined as Q = 21f stored energy . energy loss per cycle (10.1-32) Large values of Q are associated with low-loss resonators. A series RLC circuit has resonance frequency Va 1/27r vLC and quality factor Q == 27rvaL/ R, where R, L, and C are the resistance, inductance, and capacitance of the resonance circuit, respectively. The quality factor of an optical resonator is determined by observing that stored energy is lost at the rate car (per unit time), which is equivalent to the rate car / Va (per cycle), so that Q = 21f car / Va 27rVa (10.1-33) car Since 8v == ca r /27r, Q == Va 8v . ( 10.1- 34) By virtue of (10.1-33), the quality factor is related to the resonator lifetime (photon lifetime) Tp == 1/ car via Q == 27rVaTp. (10.1-35) Finally, combining (10.1-19) and (10.1-34) leads to a relationship between Q and the finesse of the resonator: Q == Va . Vp (10.1-36) Since optical resonator frequencies Va are typically much greater than the mode spacing Vp, we have Q » . Moreover, the quality factor of an optical resonator is typically far greater than that of a resonator at microwave frequencies.
10.1 PLANAR-MIRROR RESONATORS 377 Summary . Two parameters are convenient for characterizing the losses in an optical resonator: the loss coefficient a r (em -1) and the photon lifetime Tp == 1/ CQr (s). . Two dimensionless parameters characterize the quality of an optical res- onator of length d operated at frequency Vo: the finesse 3=' == 7r / a"d and the quality factor Q == 27rVQT p . . Two frequencies describe the spectral characteristics of an optical resonator: the frequency spacing between the modes Vp == c/2d, known as the free spectral range, and the spectral width 8v == Vp /3='. B. Off-Axis Resonator Modes An optical resonator with perfectly parallel planar mirrors of infinite dimensions can also support oblique, or off-axis, modes. A plane wave traveling at an angle () with respect to the axis of the resonator (the z direction) bounces back and forth between the mirrors [see Fig. 10.1-8(a)] as a guided wave traveling in the transverse direction (the x direction). Such guided waves were described in Sec. 8.1. The boundary conditions at the mirrors dictate that the axial component of the propagation constant, k z == k cas (), is an integer multiple of 7r / d. However, no such condition is imposed on the transverse component kx since the resonator is open in the x direction. The condition k cas () == q7r / d, where q is an integer, can be written in the form v == q v F sec () , q== 1,2,..., (10.1-37) where Vp == c/2d. This relation, which is plotted in Fig 10.1-8(b), is equivalent to the self-consistency condition for guided modes in planar-mirror waveguides (see Sec. 8.1). It is also identical to the condition (7.1-41) for the peak transmittance of an oblique wave through a Fabry-Perot etalon. As illustrated in Fig I 0.I-8( c), at a given frequency v, there are modes at a discrete set of angles ()q that satisfy the condition cas()q == qvp/v. These are the bounce angles of the guided modes of a waveguide. Also, at any fixed angle (), the modal frequencies are v q == qvp / cas (), as illustrated in Fig 1 0.1-8( d). The larger the inclination angle, the greater the spacing between the modal frequencies. xi 90° Ooo 70° 60° 50° 40° 30° 20° 10° d (a) 0° o 234 (b) 51//1/F '<'.:::=: 1/ F'jSec () 1--1 I- ) 1/ (c) (d) Figure 10.1-8 (a) Off-axis mode in a planar-mirror resonator. (b) Relation between mode angles and resonance frequencies. (c) Off-axis modes at a fixed frequency 1/ > 1/F. (d) Resonance frequencies of an off-axis mode of prescribed angle ().
378 CHAPTER 10 RESONATOR OPTICS 10.2 SPHERICAL-MIRROR RESONATORS The planar-mirror resonator configuration discussed in the preceding section is highly sensitive to misalignment. If the mirrors are not perfectly parallel, or the rays are not perfectly normal to the mirror surfaces, they undergo a sequence of lateral displace- ments that eventually causes them to wander out of the resonator (see Fig. 10.] -1). Spherical-mirror resonators, in contrast, provide a more stable configuration for the confinement of light that renders them less sensitive to misalignment under appropriate geometrical conditions. A spherical-mirror resonator is constructed from two spherical mirrors of radii Rl and R 2 , separated by a distance d (Fig. 10.2-1). A line connecting the centers of the mirrors defines the optical axis (z axis), about which the system exhibits circular symmetry. Each of the mirrors can be concave (R < 0) or convex (R > 0). The planar- mirror resonator is a special case for which Rl == R 2 == 00. Making use of the results set forth in Sec. 1.4D, we first examine the conditions required for ray confinement. Then, using the results derived in Chapter 3, we determine the resonator modes and resonance frequencies. Finally, we briefly discuss the implications of finite mirror size. z Figure 10.2-1 Geometry of a spherical-mirror resonator. In this illustration both mirrors are concave (their radii of curvature are negative). A. Ray Confinement We begin with ray optics to determine the conditions of confinement for light rays in a spherical-mirror resonator. We consider only meridional rays (rays lying in a plane that passes through the optical axis) and limit our consideration to paraxial rays (rays that make small angles with the optic axis). The matrix -optics methods introduced in Sec. 1.4, which are valid only for meridional and paraxial rays in a circularly symmetric system, are thus suitable for studying the trajectories of these rays as they travel inside the resonator. A resonator is a periodic optical system, since a ray travels through the same system after a round trip of two reflections. We may therefore make use of the analysis of periodic optical systems presented in Sec. 1.4D. Let Ym and em be the position and inclination of an optical ray after m round trips, as illustrated in Fig. 10.2-2. Given Ym and em, we determine Ym+l and e m + 1 by tracing the ray through the system. For paraxial rays, where all angles are small, the relation between (Ym+ 1, e m + 1) and (Ym, em) is linear and can be written in matrix form as [ Ym+l ] == [ A B ] [ Ym ] . e m + 1 C 0 em (10.2-1) Beginning at the bottom-left of Fig. 10.2-2 with Yo and eo, the round-trip ray-transfer matrix for the ray pattern depicted in Fig. 10.2-2 is [ g] = [ 1 ] [ ][ 1 ] [ ]. (10.2-2)
10.2 SPHERICAL-MIRROR RESONATORS 379 Rl R2 --.. Z Figure 10.2-2 The position and inclina- tion of a ray after m round trips are rep- resented by Ym and ()m, respectively, where m == 0, 1, 2, . . .. In this diagram, ()1 < 0 since the ray is directed downward. Angles are exaggerated for the purposes of illustration; all rays are paraxial so that sin () tan () () and the propagation distance of all rays between the mirrors is d . This cascade of ray-transfer matrices represents, from right to left [see (1.4-4) and (1.4-9)]: . Propagation a distance d through free space . Reflection from a mirror of radius R 2 . Propagation a distance d through free space . Reflection from a mirror of radius R 1 As shown in Sec. 1.4D, the solution of the difference euation (10.2-1) is Ym == YmaxFm sin(mcp + CPo), where F 2 == AD - BC, cp == cos- (bl F), b == (A + 0)/2, and Ymax and CPo are constants to be determined from the initial position and inclination of the ray. For the case at hand F == 1, so that Ym == Ymax sin (mcp + CPo), (10.2-3) cp == cos- 1 b, b=2(1+ :J (l+ :2 )-1. (10.2-4) The solution (10.2-3) is harmonic, and therefore bounded, provided cp == cos- 1 b is real. This is ensured if Ibl < 1, i.e., if -1 < b < 1, so that o < (1 + :J (1 + :2 ) < 1. (10.2-5) It is convenient to write this condition in terms of the quantities 91 == 1 + d / R 1 and 92 == 1 + d / R 2 , which are known as the 9 parameters: o < 91 92 < 1. (10.2-6) Confinement Condition The resonator is said to be stable when this condition is satisfied. This result also emerges from wave optics, as will be demonstrated subsequently [see (10.2-17)]. When the confinement condition (10.2-6) is not satisfied, cp is imaginary so that Ym in (10.2-3) becomes a hyperbolic sine function of m that increases without bound. The resonator is then said to be unstable. At the boundary of the confinement condition (when the inequalities are equalities), the resonator is said to be conditionally stable. A useful graphical representation of the confinement condition (Fig. 10.2-3) identi- fies each combination (91,92) of the two 9 parameters of a resonator as a point in a 92 versus 91 diagram. The left inequality in (10.2-6) is equivalent to {91 > 0 and 92 > 0; or 91 < 0 and 92 < O} so that all stable points (91, 92) must lie in the first or third
380 CHAPTER 10 RESONATOR OPTICS quadrants. The right inequality in (10.2-6) signifies that stable points (91, 92) must lie in a region bounded by the hyperbola 91 92 == 1. The unshaded area in Fig. 10.2- 3 represents the region for which both inequalities are satisfied, indicating that the resonator is stable. g2 1 (a) Planar (Rl = R2 = 00) (b) Symmetric confocal d (Rl = R2 = -cf) , , , I I , , I , , I , I (><) , , I (c) Symmetric , I , I , concentric , , 0 1 2 gl (Rl = R2 = -d/2) (d) Confocal/planar (Rl = d. R2 = 00) b -1 , , , , , , , , , , , , , , c --------- , , , , , , , , , 'V " #,, /.s :v ' C:,/'o , " " '" , , (e) Concave/convex t--- (Rl <O,R2>O) :::=---\ Figure 10.2-3 Resonator stability diagram. A spherical-mirror resonator is stable if the parameters 91 == 1 + d / R 1 and 92 == 1 + d / R 2 lie in the unshaded regions, which are bounded by the lines 91 == 0 and 92 == 0, and the hyperbola 92 == 1/91, R is negative for a concave mirror and positive for a convex mirror. Commonly used resonator configurations are indicated by letters and sketched at the right. All symmetric resonators lie along the line 92 == 91. Symmetric resonators, by definition, have identical mirrors (R 1 == R 2 == R) so that 91 == 92 == 9. Resonators in this class are thus represented in Fig. 10.2-3 by points lying along the line 92 == 91. The condition of stability then becomes 9 2 < 1, or -1 < 1 + d/ R < 1, which implies d o < < 2. (-R) (10.2-7) Confinement Condition (Symmetric Resonator) To satisfy (10.2-7) a stable symmetric resonator must use concave mirrors (R < 0) whose radii are greater than half the resonator length. Three examples within this class are of special interest: d / ( - R) == 0, 1, and 2, corresponding to planar, confocal, and concentric resonators, respectively. In the symmetric confocal resonator, (- R) == d so that the center of curvature of each mirror lies on the other. Thus, b == -1 and cP == 7f so that the ray position in (10.2-3) is prescribed to be Ym == Ymax sin(m7f + CPo), Le., Ym == (-1)myo. Rays initiated at position Yo, at any inclination, are thus imaged to position Y1 == -Yo, and then reimaged again to position Y2 == Yo, and so on, repeatedly. Each ray thus retraces itself after two round trips (Fig. 10.2-4). All paraxial rays are therefore confined,
10.2 SPHERICAL-MIRROR RESONATORS 381 whatever their original position and inclination. This is a substantial improvement in comparison with the planar-mirror resonator, for which only rays of zero inclination retrace themselves as schematized in Fig. 10.1-1. YI , '\. , '\. , '\. I \ I \ I \ Y" Y" Y" Yo 1 , 3 n , 5 4' , , , , , , : '. : '. , , , , ! \. , \ , , ! , , , , , , , , , , , , , , , ,I '. , , / , , \, \ \ ,I 'h ,I , Y , " Yo \,' Y2 \,' 4 \,' Figure 10.2-4 All paraxial rays in a symmetric confocal resonator retrace themselves after two round trips, whatever their original position and inclination. Angles are exaggerated in this drawing for purposes of illustration. Summary The confinement condition for paraxial rays in a spheric aI-mirror resonator, com- prising mirrors of radii R 1 and R 2 separated by a ditance d, is 0 < 9192 < 1, where 91 == 1 + d / R 1 and 92== 1 + d / R 2 . The confinement condition for symmetric resonators is 0 < d / (- R) < 2; this condition governs planar, symmetric confocal, and symmetric concentric mirror configurations. EXERCISE 10.2-1 Maximum Resonator Length for Confined Rays. A resonator is constructed using concave mirrors of radii 50 cm and 100 cm. Determine the maximum resonator length for which rays satisfy the confinement condition. B. Gaussian Modes Although the ray-optics approach considered in the preceding section is useful for de- termining the geometrical conditions under which rays are confined, it cannot provide information about the resonance frequencies and spatial intensity distributions of the resonator modes. For those quantities we must appeal to wave optics. We now proceed to show that Gaussian beams are solutions of the paraxial Helmholtz equation for the boundary conditions imposed by a pair of spherical mirrors in a resonator configura- tion. More generally, we demonstrate that Hermite-Gaussian beams are modes of the spherical-mirror resonator. In the course of our analysis, we obtain expressions for the resonance frequencies and spatial intensity distributions of the resonator modes. Gaussian Beams As discussed in Chapter 3, the Gaussian beam is a circularly symmetric wave whose energy is confined about its axis (the z axis) and whose wavefront normals are paraxial rays (Fig. 10.2-5). In accordance with (3.1-12), at an axial distance z from the beam
382 CHAPTER 10 RESONATOR OPTICS waist, the beam intensitl I varies in the transverse x-y plane as the Gaussian distribu- tion I == Io[Wo/W(z)] exp[-2(x 2 + y2)/W 2 (z)]. Its width is given by (3.1-8): W (z) = Wo 1 + ( y , (10.2-8) where Zo is the distance, known as the Rayleigh range, at which the beam wavefronts are most curved. The beam width (radius) W(z) increases in both directions from its minimum value W o at the beam waist (z == 0). The radius of curvature of the wavefronts, given by (3.1-9), R(z) = z [1 + ( r] (10.2-9) decreases from 00 at z == 0, to a minimum value at z Zo, and thereafter grows linearly with z for large z. For z > 0, the wave diverges and R(z) > 0; for z < 0, the wave converges and R(z) < O. The Rayleigh range Zo is related to the beam waist radius W o by (3.1-11): 7r T{l;2 o Zo == A . (10.2-10) The depth of focus is 2zo, i.e., twice the Rayleigh range. Beam radius z Figure 10.2-5 Gaussian beam wavefronts (solid curves) and beam width (dashed curve). The Gaussian Beam /s a Mode of the Spherical-Mirror Resonator A Gaussian beam reflected from a spherical mirror will retrace the incident beam if the radius of curvature of its wavefront is the same as that of the mirror radius (see Sec. 3.2C). Hence, if the radii of curvature of the wavefronts of a Gaussian beam, at planes separated by a distance d , match the radii of two mirrors separated by the same distance d, a beam incident on the first mirror will reflect and retrace itself to the second mirror, where it once again will reflect and retrace itself back to the first mir- ror, and so on. The beam can then exist self-consistently within that spherical-mirror resonator, satisfying the Helmholtz equation and the boundary conditions imposed by the mirrors. Provided that the phase also retraces itself, as discussed in Sec. 10.2C, the Gaussian beam is then said to be a mode of the spherical-mirror resonator. We now proceed to determine the Gaussian beam that matches a spherical-mirror resonator, whose mirrors have radii of curvature Rl and R 2 and are separated by the distance d . The task is illustrated in Fig. 10.2-6 for the special case when both mirrors are concave (R 1 < 0 and R 2 < 0).
10.2 SPHERICAL-MIRROR RESONATORS 383 ...... ...... I ( d Zl o Z2 z Figure 10.2-6 Fitting a Gaussian beam to two mirrors separated by a distance d . Their radii of curvature are Rl and R 2 . Both mirrors are taken to be concave so that Rl and R 2 are negative, as is Zl. The Z axis is defined by the centers of the mirrors. The center of the beam, which is yet to be determined, is assumed to be located at the origin Z == 0; mirrors Rl and R 2 are located at positions Zl and Z2 == Zl + d, (10.2-11) respectively. A negative value for Zl indicates that the center of the beam lies to the right of mirror I; a positive value indicates that it lies to the left. The values of Zl and Z2 are determined by matching the radius of curvature of the beam, R( z) == Z + z6 / z, to the radii Rl at Zl and R 2 at Z2. Careful attention must be paid to the signs. If both mirrors are concave, they have negative radii. But the beam radius of curvature was defined to be positive for Z > 0 (at mirror 2) and negative for Z < 0 (at mirror 1). We therefore equate Rl == R(Zl), but -R 2 == R(Z2), to obtain Rl == Zl + Z5/ Z 1 -R 2 == Z2 + z5/ Z2. (10.2-12) (10.2-13) Solving (10.2-11), (10.2-12), and (10.2-13) for Zl, Z2, and Zo leads to - d (R 2 + d) Z l - Z 2 == Zl + d , - R 2 + Rl + 2d ' 2 - d (R 1 + d) (R 2 + d) (R 2 + R 1 + d ) Zo == (R 2 + Rl + 2d)2 ' which accord with (3.1-27) and (3.1-28) (if R 2 is replaced with - R 2 ). Having determined the location of the beam center and the depth of focus 2 zo, ev - erything about the beam is known (see Sec. 3.1B). The waist radius is W o == V >"'zo/7r, and the beam radii at the mirrors are (10.2-14) (10.2-15) Wi=W o l+( ; Y, i == 1,2. (10.2-16) In order that the solution (10.2-14)-(10.2-15) indeed represents a Gaussian beam, Zo must be real. An imaginary value of Zo would signify that the Gaussian beam is a paraboloidal wave, which is an unconfined solution of the paraxial Helmholtz equation (see Sec. 3.IA). Using (10.2-15), it is not difficult to show that the condition z5 > 0 is
384 CHAPTER 10 RESONATOR OPTICS equivalent to 0 < (1+ :J (l+ :2 ) < 1. (10.2-17) This is precisely the confinement condition derived from ray optics as set forth in (10.2-5). EXERCISE 10.2-2 A Piano-Concave Resonator. When mirror 1 is planar (R 1 = 00), determine the confinement condition and the depth of focus, as well as the beam width at the waist and at each of the mirrors, as a function of d/IR 2 1. Gaussian Mode of a Symmetric Spherical-Mirror Resonator The results provided in (10.2-11)-(10.2-15) simplify considerably for symmetric res- onators with concave mirrors. Substituting RI == R 2 == -IRI into (10.2-14) provides ZI == -d /2 and Z2 == d /2. The beam center thus lies at the center of the resonator, and Z o = d V21RI 1 2 d ' H/; o 2 = Ad V 21RI 1 27r d ' W 2 == vv: 2 == >... d / 7r I 2 J (d/IRI)[2-(d/IRI)]. (10.2-18) (10.2-19) (10.2-20) The confinement condition (10.2-17) becomes d o < _ I I < 2. - R - (10.2-21) Given a resonator of fixed mirror separation d , we now examine the effect of in- creasing mirror curvature on the beam radius at the waist Wo, and at the mirrors WI == W 2 . (Increasing curvature corresponds to increasing d /IRI since the radius of curva- ture diminishes as the curvature increases.) The results are illustrated in Fig. 10.2-7. For a planar-mirror resonator, d /IRI == 0, so that W o and WI are infinite, correspond- ing to a plane wave rather than a Gaussian beam. As d/IRI increases, W o decreases until it vanishes for the concentric resonator (d /IRI == 2); at this point WI == W 2 == 00 and W o == O. In this limit, the resonator supports a spherical wave instead of a Gaussian beam. The width of the beam at the mirrors attains its minimum value, WI == W 2 J >"'d/7r, when d/IRI == 1, i.e., for the symmetric confocal resonator. In this case Zo == d /2, W o == J >"'d/27r, VI == W 2 == J2w o . (10.2-22) (10.2-23) (10.2-24)
Beam radius 2 /¥ /¥ WI =W 2 o 1 d/IRI 10.2 SPHERICAL-MIRROR RESONATORS 385 2 Figure 10.2-7 The beam width at the waist, W o , and at the mirrors, WI == W 2 , for a symmetric spherical- mirror resonator with concave mirrors, as a function of the ratio d fiRI. The planar-mirror resonator corresponds to d flRI == O. Symmetric confocal and concentric resonators correspond to d flRI == 1 and d flRI == 2, respectively. The depth of focus 2zo is then equal to the length of the resonator d, as shown in Fig. 10.2-8. This explains why the parameter 2zo is sometimes called the confocal parameter. A long resonator has a long depth of focus. The waist radius is proportional to the square root of the mirror spacing. A Gaussian beam at Ao == 633 nm (a Re- Ne-laser wavelen gth) in a resonator with d == 100 em, for example, has a waist radius W o == J Ad/21T == 0.32 mm, whereas a 25-cm-Iong resonator supports a Gaussian beam with a waist radius that is half as big at the same wavelength: 0.16 mm. The width of the beam at each of the mirrors is greater than it is at the waist by a factor of )2. rror 1 ------- i · Mirror T .J2w o z d=2Zo . Figure 10.2-8 Gaussian beam in a symmetric confocal resonator with concave mirrors. The depth of focus 2zo equals the length of the resonator d . The beam width at the mirrors is a factor of J2 greater than that at the waist. c. Resonance Frequencies As indicated in Sec. 10.2B, a Gaussian beam is a mode of the spherical-mirror res- onator provided that the wavefront normals reflect back onto themselves, always re- tracing the same path, and that the phase retraces itself as well. The phase of a Gaussian beam, in accordance with (3.1-23), is k p 2 <p(p, z) = kz - ((z) + 2R(z) , (10.2-25) where ((z) == tan- 1 (z/zo) and p2 == x 2 + y2. At points on the optical axis (p == 0), cp(O, z) == kz - ((z), so that the phase retardation relative to a plane wave is ((z). At the locations of the mirrors, Zl and Z2, we therefore have cp(O, Zl) == kZ 1 - ((Zl), cp(O, Z2) == kZ 2 - ((Z2). (10.2-26) (10.2-27)
386 CHAPTER 10 RESONATOR OPTICS Because the mirror surface coincides with the wavefronts, all points on each mirror share the same phase. As the beam propagates from mirror 1 to mirror 2, its phase changes by cp(O, Z2) - cp(O, Zl) == k(Z2 - Zl) - [((Z2) - ((Zl)] ==kd-(, (10.2-28) where ( == ((Z2) - ((Zl). (10.2-29) As the traveling wave completes a round trip between the two mirrors, therefore, its phase changes by 2kd - 2(. In order that the beam truly retrace itself, the round-trip phase change must be zero or a multiple of ::i::27r, i.e., 2kd - 2( == 27rq, q == 0, ::i::l, ::i::2, . ... Using the substitutions k == 27rV / c and Vp == c/2d, the frequencies v q that satisfy this condition are ( v q == qvp + - Vp. 7r (10.2-30) Resonance Frequencies Gaussian Modes The frequency spacing of adjacent modes is therefore Vp == c/2d, which is identical to the result obtained in Sec. 10.IA for the planar-mirror resonator. For spherical-mirror resonators, this frequency spacing is evidently independent of the curvatures of the mirrors. The second term in (10.2-30), which does depend on the mirror curvatures, simply represents a displacement of all resonance frequencies. EXERCISE 10.2-3 Resonance Frequencies of a Confocal Resonator. A symmetric confocal resonator has a length d = 30 em, and the medium has refractive index n = 1. Determine the frequency spacing Vp and the displacement frequency (( /7r) lip. Determine all resonance frequencies that lie within the band 5 x 10 14 ::f: 2 X 10 9 Hz. D. Hermite-Gaussian Modes In Sec. 3.3 it was shown that the Gaussian beam is not the only beam-like solution of the paraxial Helmholtz equation. The family of Hermite-Gaussian beams also pro- vides solutions. Although a Hermite-Gaussian beam of order (l, m) has an amplitude distribution that differs from that of the Gaussian beam, their wavefronts are identical. As a result, the design of a resonator that "matches" a given beam (or the design of a beam that "fits" a given resonator) is the same as for the Gaussian beam, whatever the values of (l, m). It follows that all members of the family of Hermite-Gaussian beams represent modes of the spherical-mirror resonator. The resonance frequencies of the (l, m) mode do, however, depend on the indexes (l, m). This is because of the dependence of the Gouy phase shift on land m. As is evident from (3.3-10), the phase of the (l, m) mode on the beam axis is cp(O, z) == kz - (l + m + l)((z). (10.2-31)
10.2 SPHERICAL-MIRROR RESONATORS 387 Again, the phase shift encountered by a traveling wave undergoing a single round trip through a resonator of length d must be set equal to zero or an integer multiple of ::l:27r in order that the beam retrace itself. Thus, 2kd - 2(l + m + 1)( == 27rq, q == 0, ::l:1, ::l:2,..., (10.2-32) where, as previously, ( == ((Z2) - ((Zl) and Zl, Z2 represent the positions of the two mirrors. With k == 27rv / c and Vp == c/2d, this yields the resonance frequencies ( Vl m q == qvp + (l + m + 1) - Vp. , , 7r (10.2-33) Resonance Frequencies Hermite-Gaussian Modes Modes of different q, but the same (l, m), have identical intensity distributions [see (3.3-12)]. They are known as longitudinal or axial modes. The indexes (l,m) label different spatial dependences on the transverse coordinates x, y; these therefore repre- sent different transverse modes, as illustrated in Fig. 3.3-2. Equation (10.2-33) dictates that the resonance frequencies of the Hermite-Gaussian modes satisfy the following properties: . Longitudinal modes corresponding to a given transverse mode have resonance frequencies spaced by Vp == c/2d since Vl,m,q+l - Vl,m,q == Vp. This result is the same as that obtained for the (0,0) Gaussian mode and for the planar-mirror resonator. . All transverse modes, for which the sum of the indexes l + m is the same, have the same resonance frequencies. . Two transverse modes (l, m), (l', m') corresponding to the same longitudinal mode q have resonance frequencies spaced by Vl,m,q - Vl',m',q = [(i + m) - (i' + m')] ( VF. (10.2-34) This expression determines the frequency shift between the sets of longitudinal modes of indexes (l, m) and (l', m'). EXERCISE 10.2-4 Resonance Frequencies of the Symmetric Confocal Resonator. Show that for a symmet- ric confocal resonator, the longitudinal modes associated with different transverse modes are either the same, or are displaced by VF /2, as illustrated in Fig. 10.2-9. j+vF+I (l m) v (I', m) v Figure 10.2-9 In a symmetric confocal res- onator, the longitudinal modes associated with two transverse modes of indexes (l, m) and (l', m') are either aligned or displaced by half a longitudinal mode spacing. v
388 CHAPTER 10 RESONATOR OPTICS *E. Finite Apertures and Diffraction Loss Since Gaussian and Hermite-Gaussian beams have infinite transverse extent whereas the resonator mirrors are of finite extent, a portion of the optical power leaks around the mirrors and escapes from the resonator on each pass. An estimate of the power loss may be obtained by calculating the fractional power of the beam that is not intercepted by the mirror. If the beam is Gaussian with width Wand the mirror is circular with radius a == 2W, for example, a small fraction, exp( -2a 2 jW 2 ) 3.35 X 10- 4 , of the beam power escapes on each pass [see (3.1-17)], the remainder being reflected (or transmitted through the mirror). Higher-order transverse modes suffer greater losses since they have greater spatial extent in the transverse plane. When the mirror radius a is smaller than 2W, the losses are greater. The Gaussian and Hermite-Gaussian beams then no longer provide good approximations for the res- onator modes. The problem of determining the modes of a spherical-mirror resonator with finite-size mirrors is difficult. A wave is a mode if it retraces its amplitude (to within a multiplicative constant) and reproduces its phase (to within an integer multiple of 27r) after completing a round trip through the resonator. One oft-used method for determining the modes involves following a wave repeatedly as it bounces through the resonator, thereby determining its amplitude and phase, much as we determined the position and inclination of a ray bouncing within a resonator. After many round trips this process converges to one of the modes. If U 1 (x, y) is the complex amplitude of a wave immediately to the right of mirror 1 in Fig. 10.2-10, and if U 2 (x, y) is the complex amplitude after one round trip of travel through the resonator, then U 1 (x, y) is a mode provided that U 2 (x, y) == J-lU l (x, y) and provided that arg{J-l} is an integer multiple of 27r (i.e., J-L is real and positive). After a single round trip, the mode intensity is attenuated by the factor J-L2, and the phase is reproduced. The methods of Fourier optics (Chapter 4) may be used to determine U 2 (x, y) from U1(x, y). These quantities may be regarded as the output and input, respectively, of a linear system (see Appendix B) characterized by an impulse response function h(x, y; x', y'), so that 00 U 2 (x,y) = 11 h(x,Yi X ',y')U 1 (x',y')dx'dy'. -00 (10.2-35) If the impulse response function h is known, the modes can be determined by solving the eigenvalue problem described by the integral equation (see Appendix C) 00 11 hex, Yi x', y') U(x', y') dx' dy' = J-L U(x, y). -00 (10.2-36) The solutions determine the eigenfunctions Ul,m (x, y), and the eigenvalues J-Ll,m, la- beled by the indexes (l, m). The eigenfunctions are the modes and the eigenvalues are the round-trip multiplicative factors. The squared magnitude lJ-Ll,mI 2 is the round-trip intensity reduction factor for the (l, m) mode. Clearly, when the mirrors are infinite in size and the paraxial approximation is satisfied, the modes reduce to the family of Hermite-Gaussian beams discussed earlier. It remains to determine h(x, y; x', y') and to solve the integral equation (10.2-36). A single pass inside the resonator involves traveling a distance d, truncation by the mirror aperture, and reflection by the mirror. The remaining pass, needed to comprise
Mirror ] Mirror 2 V2 T 2a VI 1 I c d -I 10.2 SPHERICAL-MIRROR RESONATORS 389 Figure 10.2-10 Propagation of a wave through a spherical-mirror resonator. The complex ampli- tude U 1 (x, y) corresponds to a mode if it repro- duces itself after a round trip, i.e., if U 2 (x, y) == JLU! (x, y) and arg{JL} == q27r. a single round trip, is similar. The impulse response function h( x, y; x' , y') can then be determined by applying the theory of Fresnel diffraction (Sec. 4.3B). In general, however, the modes and their associated losses can be determined only by numerically solving the integral equation (10.2-36). An iterative numerical solution begins with an initial guess U 1 , from which U 2 is computed and passed through the system one more round trip, and so on until the process converges. This technique has been used to determine the losses associated with the various modes of a spherical-mirror resonator with circular mirror apertures of radius a. The results are illustrated in Fig. 10.2-11 for a symmetric confocal resonator. The loss is governed by a single parameter, the Fresnel number N F == a 2 j .Ad . This is because the Fresnel number governs Fresnel diffraction between the two mirrors, as discussed in Sec. 4.3B. For the symmetric confocal reson ator d escribed by (10.2-23) and (10.2-24), the beam width at the mirrors is W == V .Ad j 7r, so that .Ad == 7r W 2 , from which the Fresnel number is readily determined to be N F == a 2 j 7r W 2 . N F is therefore proportional to the ratio a 2 jW 2 ; a higher Fresnel number corresponds to a smaller loss. From Fig. 10.2-11 we find that the loss per pass of the lowest-order symmetric- confocal-resonator mode (l, m) == (0,0) is about 0.1 % when N F 0.94. This Fresnel number corresponds to ajW == 1.72. If the beam were Gaussian with width W, the percentage of power contained outside a circle of radius a == 1.72W would be exp( -2a 2 jW 2 ) 0.27%. This is larger than the 0.1 % loss per pass for the actual resonator mode. Higher-order modes suffer from greater losses because of their greater spatial extent. d 100 T 2a 1 .. .... " ........... " " ", "- " " " " " \. " \. \. '\. " \ (1,0) " 2, 0) ,(0,0) \.. '\. I\. \. \. 1\ \ \ '-'10 en en 0.. Q) 0.. en en o 0.1 0.5 1.0 1.4 Fresnel number NF = a 2 / >"d Figure 10.2-11 Percent diffraction loss per pass (half a round trip) as a function of the Fresnel number N F == a 2 / >"d for the (0,0), (1,0), and (2,0) modes in a symmetric confocal resonator. (Adapted from A. E. Siegman, Lasers, University Science, 1986, Fig. 19.19 left.)
390 CHAPTER 10 RESONATOR OPTICS 10.3 TWO- AND THREE-DIMENSIONAL RESONATORS A. Two-Dimensional Rectangular Resonators A two-dimensional (2D) planar-mirror resonator is constructed from two orthogonal pairs of parallel mirrors, e.g., a pair normal to the z axis and another pair normal to the y axis. Light is confined in the z-y plane by a sequence of ray reflections, as illustrated in Fig. 10.3-1(a). T (a) d 1 (b) ! 1 I Figure 10.3-1 A two-dimensional planar-mirror resonator: (a) ray pattern; (b) standing-wave pattern with mode numbers qy = 3 and qz = 2. The boundary conditions establish the resonator modes, much as for the one- dimensional Fabry-Perot resonator. If the mirror spacing is d , then for standing waves the components of the wavevector k == (ky, k z ) are restricted to the values 7r ky == qy(j' 7r k z == qz d ' qy == 1, 2, . . . , qz == 1, 2, . . . , (10.3-1) where qy and qz are mode numbers for the y and z directions, respectively. These conditions are a generalization of (10.1-2). Each pair of integers (qy, qz) represents a resonator mode U(r) ex: sin(q1{7ryjd) sin(q z 7rzjd), as illustrated in Fig. 10.3-I(b). The lowest-order mode is (1,1) since the modes (qy,O) and (0, qz) have zero ampli- tude, i.e., U(r) == O. Modes are conveniently represented by dots that indicate their values of ky and k z on a periodic lattice of spacing 7r j d (Fig. 10.3-2). ky J . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 7r · · · · · · · · · · . T d . . .. ........ · · · · .. ... k = 27rV . . C . . . . . . . . . . . . . . . . . 00 k z Figure 10.3-2 Dots denote the endpoints of the wavevectors k = (ky, k z ) for modes in a two- dimensional resonator. The wavenumber k of a mode is the distance of the dot from the origin. The associ- ated frequency of the mode is v == ck j27r. The frequencies of the resonator modes are thus determined from k 2 = k; + k; = ( 2:V Y , (10.3-2)
10.3 TWO- AND THREE-DIMENSIONAL RESONATORS 391 so that V q = VF J q + q; , qy, qz = 1,2,. .., c Vp == 2d ' (10.3-3) Resonance Frequencies where q == (qy, qz). The number of modes in a given frequency band, VI < V < V2, is established by drawing two circles, of radii k i == 27rVI / c and k 2 == 27rV2/ c in the k diagram of Fig. 10.3-2, and counting the number of dots that lie within the annulus. This procedure converts the allowed values of the vector k into allowed values of the frequency v. EXERCISE 10.3-1 Density of Modes in a Two-Dimensional Resonator. (a) Determine an approximate expression for the number of modes in a two-dimensional resonator with frequencies lying between 0 and v, assuming that 21rv / C » 1r / d, Le., d » >-"/2, and allowing for two orthogonal polarizations per mode. (b) Show that the number of modes per unit area lying within the frequency interval between v and v + dv is M(v)dv, where the density of modes M(v) (modes per unit area per unit frequency) at frequency v is given by M(v) = 4v . C (10.3-4) Density of Modes (20 Resonator) The resonator modes described thus far in this section are in-plane modes, traveling in the plane of the 2D resonator (the y-z plane). Off-plane modes have a propagation constant with a component in the orthogonal direction (the x direction). These are guided modes traveling along the axis of a 2D waveguide such as that described in Sec. 8.3. Whereas the ky and k z components of the wavevector take discrete values dictated by the boundary conditions, the kx component takes continuous values since the 2D resonator is open in the x direction. B. Circular Resonators and Whispering-Gallery Modes Light may be confined in a two-dimensional circular resonator by repeated reflections from the circular boundary. As illustrated in Fig. 10.3-3, a ray that self-reproduces after N reflections traces a path with round-trip pathlength Nd, where d == 2a sin( 7r / N) and a is the radius. For a traveling-wave mode, the resonance frequencies are determined by equating the round-trip pathlength to an integer number of wavelengths, as in (10.1-7). Ignoring the phase shift associated with each reflection, this leads to Nd == q>... == qc/v, i.e., to resonant frequencies v q == qc/ Nd, where q == 1, 2, . . .. The spacing between these frequencies is therefore Vp == c/ Nd. For N == 2, we have Vp == c/2d == c/4a, which is identical to (10.1-5). Similarly, N == 3 yields Vp == c/3d == c/3v13 a, which coincides with the result for the three- mirror resonator (Exercise 10.1-1). In the limit N 00, the pathlength Nd approaches the cylindrical circumference 27ra and the corresponding spacing of the resonance
392 CHAPTER 10 RESONATOR OPTICS frequencies becomes c Vp == -. 27ra (10.3-5) Spacing of Resonance Frequencies The rays then hug the interior boundary of the resonator, reflecting at near-grazing incidence, as illustrated in Fig. 10.3-3. Such optical modes are known as whispering- gallery modes (WGM). The optical modes then behave similarly to acoustic modes in the familiar acoustical whispering gallery, so-named because of the ease with which an acoustic whisper can bounce along the convex surface of a church dome or gallery. , , , , , " ,,,.' " ,( , ' f "''' \: ,,,...-''- J,",,- a 7r/N Mirror resonator Dielectric resonator Figure 10.3-3 Reflections in a circular resonator. Two-dimensional resonators with other cross sections are also used. For example, the circular cross section can be squeezed into a stadium-shaped structure. This oblong configuration supports bow-tie modes [see Fig. 10.1-5(b)] in which the ray executes a round-trip path comprising localized reflections from the four locations on the perime- ter of the resonator that match the curvature of a conventional spherical-mirror confocal resonator (see Sec. 10.2A). C. Three-Dimensional Rectangular Cavity Resonators A three-dimensional (3D) planar-mirror resonator is constructed from three pairs of parallel mirrors forming the walls of a closed rectangular box of dimensions d x , d y , and d z . The structure is a three-dimensional resonator, as depicted in Fig. 10.3- 4(a). Standing-wave solutions within the resonator require that the components of the wavevector k == (k x , ky, k z ) are discretized to obey 7r kx = qx d x ' 7r ky == qYd' y 7r k z = qz d z ' qx, qy, qz == 1, 2, . . . , (10.3-6) where qx, qy, and qz are positive integers representing the respective mode numbers. Each mode q, which is characterized by the three integers (qx, qy, qz), is represented by a dot in (k x , ky, kz)-space. The spacing between these dots in a given direction is inversely proportional to the width of the resonator along that direction. Figure 10.3- 4(b) illustrates the concept of the k-space for a cubic resonator with d x == d y == d z == d. The values of the wavenumbers k, and the corresponding resonance frequencies v, satisfy k 2 = k; + k; + k; = ( 2:V Y . (10.3-7)
I d 1 II d 10.3 TWO- AND THREE-DIMENSIONAL RESONATORS 393 k z --- k= 27rv 7r c I ---+J d j.- ....-/ / ./ ../ ./ ./ ./ VI V .../' ../ .:;:. f..-:::? ,.-:::: 1 ...."J .... Y ../ V /' V V v...... ../ ...- ? l:::;:? 7r I--' ...-/ d /' v- v- V" ./ ..::::::> r:::;:;::::::> T ...,.."J /' .... /' ....-/ \ V v v...... v...... ...,.. .../' /.... { ........ y v....v....v ....-/ kx (a) (b) Figure 10.3-4 (a) Waves in a three-dimensional cubic resonator (d x == d y == d z == d). (b) The endpoints of the wavevectors (kx, ky, k z ) of the modes in a three-dimensional resonator are marked by dots. The wavenumber k of a mode is the distance from the origin to the dot. Each point in k-space occupies a volume (7r / d )3. All modes of frequency smaller than v lie inside the positive octant of a sphere of radius k == 27r V / c. The surface of constant frequency v is a sphere of radius k == 27fv / c. The resonance frequencies are determined from (10.3-6) and (10.3-7): _ J 22 + 22 + 22 v q - qx v px qy v Fy qz v pz , qx, qy, qz == 1, 2, . . . , (10.3-8) Resonance Frequencies where c VPx == 2d x ' c Vpy == 2d ' y c Vpz == - 2d z (10.3-9) are frequency spacings that are inversely proportional to the resonator widths in the x, y, and z direction, respectively. For resonators whose dimensions are much greater than a wavelength, the frequency spacing is much smaller than the optical frequency. For example, for d == 1 cm and n == 1, Vp == 15 GHz. This is not so for microresonators, however, as will be discussed in Sec. 10.4. Density of Modes When all dimensions of the resonator are much greater than a wavelength, the fre- quency spacing Vp == c/2d is small, and it is analytically difficult to enumerate the modes. In this case, it is useful to resort to a continuous approximation and introduce the concept of density of modes, the validity of which depends on the relative values of the bandwidth of interest and the frequency interval between successive modes. The number of modes lying in the frequency interval between 0 and v corresponds to the number of points lying in the volume of the positive octant of a sphere of radius k in the k diagram [Fig. 10.3-4(b)]. The number of modes in the positive octant of a sphere of radius k is 2(l)(7fk3)/(7f/d)3 == (k 3 /37f2)d 3 . The initial factor of 2 accounts for the two possible polarizations of each mode, whereas the denominator (7f / d)3 represents the volume in k space per point. Since k == 27fV / c, the number of
394 CHAPTER 10 RESONATOR OPTICS modes lying between 0 and v is [(27rv/C)3/37r 2 ]d 3 == (87rv 3 /3c 3 )d 3 . The number of modes in the incremental frequency interval lying between v and v + f1v is therefore given by (d/ dv) [(87rv 3 /3c 3 )d 3]f1v == (87rv 2 / c 3 )d 3 f1v. The density of modes M(v), i.e., the number of modes per unit volume of the resonator, per unit bandwidth surrounding the frequency v, is therefore (10.3-10) Density of Modes (3D Resonator) This formula was first derived by Rayleigh and Jeans in connection with the spectrum of blackbody radiation (see Sec. 13.4B). The quantity M(v) is a quadratically increas- ing function of frequency so that the number of modes within a fixed bandwidth f1v increases with the frequency v in the manner indicated in Fig. 10.3-5. At v == 3 X 10 14 (Ao == 1 /-Lm), M(v) == 0.08modes/cm 3 -Hz. Within a frequency band of width 1 GHz, there are therefore 8 x 10 7 modes/cm 3 . The number of modes per unit volume within an arbitrary frequency interval VI < V < V2 is simply the integral J2 M(v) dv. M(v) = 87f2 . C IIIIII!IIIIIIIII v M(v) Figure 10.3-5 (a) The frequency spacing between adjacent modes de- creases as the frequency increases. (b) The density of modes M(v) for a three-dimensional optical resonator is a quadratically increasing function of frequency. v The density of modes in two and three dimensions were derived on the basis of square and cubic geometry, respectively. Nevertheless, the results are applicable for arbitrary geometries, provided that the resonator dimensions are large in comparison with the wavelength. It is, perhaps, worthy of mention at this juncture that the enumeration of the elec- tromagnetic modes considered here is mathematically identical to the calculation of the allowed quantum states of electrons confined within perfectly reflecting walls. This latter model is of importance in determining the density of allowed electron states as a function of energy in semiconductor materials (see Sec. 16.1C). 10.4 MICRORESONATORS Microresonators are resonators in which one or more of the spatial dimensions as- sumes the size of a few wavelengths of light or smaller. The term microcavity res- onator, or microcavity for short, is usually reserved for a microresonator that has small dimensions in all spatial directions, so that the modes exhibit large spacings in all directions of k-space and the resonance frequencies are sparse. However, these terms are often used interchangeably. The absence of resonance modes in extended spectral bands can inhibit the emission of light from sources placed within a microcavity. At the same time, the emission of
10.4 MICRORESONATORS 395 light into particular modes of a high-Q, small-volume microcavity can be enhanced relative to emission into ordinary optical modes, as described in Sec. 13.3E. These effects can be important in the operation of microcavity lasers (see Sec. 17.4B). Microresonators can be fabricated using dielectric materials configured in various geometries, such as (1) micropillars with Bragg-grating reflectors; (2) microdisks and microspheres in which light reflects near the surface in whispering-gallery modes; (3) microtoroids, which resemble small fiber rings; and (4) 2D photonic crystals containing light-trapping defects that function as microcavities. These technologies have had two principal design objectives: . The reduction of the modal volume V, which is defined as the spatial integral of the optical energy density E £2 of the mode, normalized to its maximum value. . The enhancement of the quality factor Q. Typical modal volumes and quality factors for these structures are summarized in Table 10.4-1. Table 10.4-1 Normalized modal volume VI)..3 and quality factor Q for various microresonators. VI )..3 Q Micropillar Microdisk 5 5 10 3 10 4 Microtoroid 10 3 10 8 Microsphere 10 3 Photonic-Crystal 10 10 1 10 4 An exact analysis of the resonator modes of dielectric microresonators requires the full electromagnetic theory. The Helmholtz equation is solved in a coordinate system suitable for the geometry of the structure, and appropriate boundary conditions are applied to the electric and magnetic fields at the planar, cylindrical, or spherical boundaries. The solution yields the resonance frequencies of the modes and their spa- tial distributions, which may be used to determine the modal volume for each mode. Since the analysis is complex for all practical geometries, numerical solutions are often necessary. In the next section, we describe some of the properties of a simple rectangular (box) microresonator whose walls are made of perfect mirrors. A simple analysis of the modes of such a structure provides the resonance frequencies and the spatial distri- butions of the modes. High-Q microresonators do not make use of mirrors because of their relatively high losses, and the box structure is also not among the geometries typically used in practical microresonators. Nevertheless, the analysis is useful for elucidating the relation between the resonance frequencies and the dimensions of the resonator, and for illustrating the frequency dependence of the density of modes for boxes with different aspect ratios. A. Rectangular Microresonators The simplest microresonator structure is a rectangular (box) resonator made of planar parallel mirrors. The modes are then sinusoidal standing waves in all three directions and the resonance frequencies are given by (10.3-8). When the dimensions of the box are small, only the lowest order modes lie within the optical band. For a cubic resonator, the resonance frequencies are provided in Table 10.4-2 in units of Vp == c/2d. As an example, if d == 1 /-Lm and the medium has refractive index n == 1.5, we obtain VF == 100 THz. The frequencies of the lowest-order modes then correspond to the free-space wavelengths Ao == 2.13,1.73,1.34,1.22,1.06,1.00, and 0.87jLm, which are widely spaced.
396 CHAPTER 10 RESONATOR OPTICS Table 10.4-2 Resonance frequencies for the lowest-order modes of a cubic microcavity resonator. Mode (qx qy qz)(a) (011)(3) Frequency (units of vp) 1.41 (111)(1) 1.73 (012)(6) 2.24 (112)(3) 2.45 (022)(3) 2.83 (122)(3) 3 (222) (1) 3.46 a Superscripts in parentheses indicate the modal degeneracy, i.e., the number of modes of the same resonance frequency. As an example, three modes have the same resonance frequency 1.41vp: (011), (101), and (110). If the resonator has a mixture of dimensions both small and large, as with a box of large aspect ratio, the modes are placed at the points of an anisotropic grid in k- space [see Fig. lO.3-4{b)]. The grid is finely divided along the directions of the large dimensions and coarsely divided along the directions of the small dimensions. Mode counting may then be implemented by use of a continuous approximation only in those directions for which the grid is fine. The resultant modal density is displayed in Fig. 10.4-1 for various cases. zt d sl y Y d M(v) o vF 2VF (a) 3VF v zt .. y ds Yd s zt ds«l M(v _ M(v) t") c . - NO ON 8 s;:: N - N _ N N oN C: S:! 0 S:! S:! "-" --- NN ....: ON N N N :::, :::, o v - o s 2VF (b) v 0 2VF (c) vF vF Figure 10.4-1 Modal density M(v) for rectangular microresonators with (a) one, (b) two, and (c) three sides of small dimension d s « d. The frequency spacing associated with the small dimension is VF == c/2d s . When all dimensions are small, as in (c), the resonance frequencies are discrete and their values are those provided in Table 10.4-2 for the cubic microcavity resonator. The result shown in (b) represents a combination of discrete modes associated with a 2D microresonator and continuous modes associated with a ID large resonator, which has a uniform modal density [see (10.1-10)]. The result provided in (a) illustrates a combination of discrete modes associated with a I D microresonator and a continuum of modes associated with a 2D large resonator, which has a modal density that is linearly proportional to frequency [see (10.3-4)]. B. Micropillar, Microdisk, and Microtoroid Microresonators Dielectric microresonators have been fabricated in a number of configurations, includ- ing micropillars, microdisks, and microtoroids, as illustrated in Fig. 10.4-2. Light is confined in these structures by total internal reflection (see Fig. 10.3-3). The micropillar, or micropost, resonator is a cylinder of high-refractive-index ma- terial sandwiched between dielectric layers comprising distributed Bragg-grating re- flectors (DBRs), as illustrated in Fig. 10.4-2{a). Light is confined in the axial direction by reflection from the DBRs, as in a Fabry-Perot resonator; light is confined in the
10.4 MICRORESONATORS 397 0:: o t Dis k ----------------- ---- - - Silica ".- toroid 0:: o -. Silicon chip -..." Silicon - _____ pillar (a) Micropillar (b) Microdisk (c) Microtoroid Figure 10.4-2 Micropillar, microdisk and microtoroid resonators. lateral direction by total internal reflection from the walls of the cylinder. Micropillars are typically fabricated from compound semiconductors via conventional lithographic and etching processes; DBR layers are often made of AIAsjGaAs or AIGaAsjGaAs. The pillar itself can contain an active region such as a multi quantum-well structure that provides optical gain when pumped (see Sec. 17.4). The microdisk cavity displayed in Fig. 1 0.4-2(b) is a circular resonator in which light travels at near-grazing incidence in whispering-gallery modes and is confined by total internal reflection from the circular boundary (see Sec. 10.3B). Micropillar and microdisk sizes usually range from a few J-lm to tens of J-lm and their quality factors Q are substantially larger than those of mirror resonators since their losses are significantly lower (see Table 10.4-1). Still, their performance is limited by the surface quality of the material since the light travels near the boundary. The toroidal dielectric microresonator illustrated in Fig. 10.4-2(c) is much like a fiber-ring resonator, in which the resonator modes are circulating guided waves. These microresonators are usually fabricated from silica and are supported on a silicon chip by a silicon pillar. The toroid is formed by surface tension while the material is in a molten state; the outer boundary thus assumes a near atomic-scale surface finish and has significantly lower scattering losses than the microdisk resonator. Silica toroidal microresonators-on-a-chip exhibit exceptionally high values of the quality factor, Q > 10 8 (see Table 10.4-1). c. Microsphere Microcavities Dielectric spheres are used as three-dimensional optical microcavities. Certain modes are guided along trajectories (orbits) that are tightly confined near a great circle of the sphere, resulting in whispering-gallery modes. The modes of a dielectric sphere may be determined by solving the Helmholtz equa- tion (5.3-16) for the electric and magnetic field vectors, together with the appropriate boundary conditions. These modes are similar to the wavefunctions of an electron in a hydrogen atom (see Sec. 13.1 A) because of the spherical symmetry of both problems, but there are also differences in view of the vector nature of the electromagnetic field. The electric and magnetic vector fields are directly related to a scalar potentia] func- tion U that satisfies the Helmholtz equation. t For a sphere of radius a and refractive index n in air, the separation of variables method in a spherical coordinate system (r, B, 1) results in a solution of the form U(r, B, 1) ex: Vi Jf+l/2(nkor) P(cosB) exp(::l::jm1), r < a, (10.4-1) t For a detailed mathematical description, see, for example, A. N. Oraevsky, Whispering-Gallery Waves, Kvantovaya Elektronika (Quantum Electronics), vol. 32, pp. 377-400, 2002.
398 CHAPTER 10 RESONATOR OPTICS <X vr .fj1/2(nkor) p;. (cos 0) exp(::I::jmct», r > a, (10.4-2) where Jl(.) is the Bessel function of the first kind of order £, S)1) (.) is the Hankel function of the first kind of order £, P[n (.) is the adjoint Legendre function, and m and £ are nonnegative integers. The boundary conditions at r = a yield a characteristic equation that provides a discrete set of values for ko, corresponding to the resonance frequencies. These are indexed by a third integer n. In addition, there are two polariza- tion modes - an E mode for which Hr = 0 and an H mode for which Er = O. The modes are generally oscillatory functions of r, (), and c/J characterized by the radial, polar, and azimuthal mode numbers n, £, and m, respectively. There are n maxima in the radial direction within the sphere. The number of field maxima in the azimuthal direction is 2£, while the number of field maxima in the polar direction (between the two poles) is £ - m + 1. The fundamental mode (n = 1, m = £) has a single peak in the radial direction within the sphere, and a single peak in the polar direction at () = 7r /2. For large m == £, the modes are highly confined near the equator. This is because pi ( cos ()) sin i () vanishes rapidly at angles slightly different from () = 7r /2, and Jl( nkor) is small everywhere within the sphere except for a sharp peak near r = a. The mode therefore represents an optical beam traveling along the equator, as shown in Fig. 10.4-3(a), much like the whispering-gallery modes of the disk resonator displayed in Fig. 10.3-3. For sufficiently large £ = m, the resonance frequencies of these modes are approxi- mately equal to Vi £ c/27ra. This is to be expected since the angular mode number £ is close to the number of wavelengths that comprise the optical length of the equator. The whispering-gallery mode may be viewed from a ray-optics perspective in terms of quasi-plane waves with wavevectors parallel to the local rays (see Sec. 2.3 and Fig. 9.2-10) that zigz ag near th e equator, as shown in Fig. 10.4- 3(b). The wavevector k has magnitude k = J £ ( £ + 1) / a and azimuthal component k4J = m/ a. The inclination angle of the zigzagging rays is smallest ( 1/ Vi) for the fundamental mode m = £, while the m = 0 mode has a 90° inclination. t \ t (a) A B . a r (b) Figure 10.4-3 (a) Whispering-gallery mode in a microsphere resonator. (q) Ray mode] of the whispering-gallery mode. Microspheres fabricated from low-loss fused silica have been used as optical res- onators with ultrahigh values of Q. Like the toroidal resonator depicted in Fig. 10.4- 2(c), the shape and surface finish of the sphere are determined by the surface tension in the molten state during fabrication; the result is near atomic perfection in the surface finish. The reduced surface scattering losses lead to remarkably high quality factors, Q > 10 10 (see Table 10.4-1). Optical power may be coupled into the sphere via an optical fiber that is locally stripped of its cladding, as illustrated in Fig. 10.4-4.
10.4 MICRORESONATORS 399 Fiber \ Microsphere ./ Figure 10.4-4 Coupling optical power from an optical fiber into a microsphere resonator. D. Photonic-Crystal Microcavities As described in Chapter 7, photonic crystals are periodic dielectric structures exhibit- ing photonic bandgaps, i.e., spectral bands within which light cannot propagate. The Bragg grating reflector (BGR) is an example of a ID photonic crystal that serves as a reflector for frequencies within a photonic bandgap. For example, the micropillar resonator shown in Fig. 10.4-2(a) uses BGRs in lieu of mirrors. If the height of the microresonator equals one or just a few periods of the BGR, as illustrated in Fig. 10.4- 5(a), the structure may also be regarded as an extended photonic crystal with the cavity acting as a defect in the crystal structure. The resonator is then called a photonic- crystal resonator. This concept is also applicable to 2D photonic crystals. As schematized in Fig. 10.4- 5(b), a defect in the 2D periodic crystal structure is a local alteration such as a missing hole in a periodic array of air holes drilled in a slab. For wavelengths that fall within the photonic-crystal bandgap, the periodic structure surrounding the defect does not support light propagation, so that light is trapped within the defect, much like electrons or holes are trapped by a defect in a semiconductor crystal. The defect then serves as a microcavity resonator. Stated differently, the defect produces new resonance fre- quencies that lie within the bandgap and correspond to optical modes that have spatial distributions centered within the microcavity and that decay rapidly in the surrounding photonic crystal. Two-dimensional photonic crystals are fabricated by using e-beam lithography and reactive ion etching in semiconductor materials. Microcavities of dimensions close to a period of the photonic crystal, which can be of the order of a wavelength of light, can support modal volumes as small as A 3 . In comparison with other technologies, photonic-crystal microcavities have the smallest modal volume (see Table 10.4-1). The quality factors Q can also be as high as 10 4 . ..... , '-- , '- , ..... "" Photonic crystal Microcavity Microcavity Photonic crystal 000000 0000000 o 000000 00 000000 000 0000 0000 0000 0000 0000 000000000 00000000 0000000 000000 ... .. , .. '" ... " ... ... (a) (b) Figure 10.4-5 Photonic-crystal microresonators. (a) The micropillar resonator as a ID photonic crystal in which the microresonator acts as a defect. (b) A 2D photonic-crystal resonator may be fabricated by drilling holes in a dielectric slab at the points of a planar hexagonal lattice; a missing hole serves as the microcavity.
400 CHAPTER 10 RESONATOR OPTICS READING LIST Books See also the books on lasers in Chapter 15. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications, Springer-Verlag, 2nd ed. 2005. K. J. Vahala, ed., Optical Microcavities, World Scientific, 2004. K. Stalifinas and V. J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators, Springer- Verlag, 2003. R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities, World Scientific, 1996. A. N. Oraevskiy, Gaussian Beams and Optical Resonators, Nova, 1996. Yu. Anan'ev, Laser Resonators and the Beam Divergence Problem, Taylor & Francis, 1992. J. M. Vaughan, The Fabry-Perot Interferometer, Adam Hilger, 1989. G. Hernandez, Fabry-Perot Interferometers, Cambridge University Press, paperback ed. 1988. A. E. Siegman, Lasers, University Science, 1986. L. A. Weinstein, Open Resonators and Open Waveguides, Golem Press, 1969. Articles Issue on microresonators, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. I, 2006. K. J. Vahala, Optical Microcavities, Nature, vol. 424, pp. 839-846, 2003. Millennium issue, IEEE JournaL of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. J. U. Nackel and A. D. Stone, Ray and Wave Chaos in Asymmetric Resonant Optical Cavities, Nature, vol. 385, pp. 45-47, 1997. Y. Yamamoto and R. E. Slusher, Optical Processes in Microcavities, Physics Today, vol. 46, no. 6, pp.66-73,1993. H. Yokoyama, Physics and Device Applications of Optical Microcavities, Science, vol. 256, pp. 66- 70, 1992. A. E. Siegman, Unstable Optical Resonators, Applied Optics, vol. 13, pp. 353-367, 1974. H. Kogelnik and T. Li, Laser Beams and Resonators, Applied Optics, vol. 5, pp. 1550-1567, 1966 (published simultaneously in Proceedings of the IEEE, vol. 54, pp. 1312-1329, 1966). A. G. Fox and T. Li, Resonant Modes in a Maser Interferometer, Bell System Technical Journal, vol. 40, pp. 453-488, 1961. G. D. Boyd and J. P. Gordon, Confocal Multimode Resonator for Millimeter Through Optical Wave- length Masers, Bell System Technical Journal, vol. 40, pp. 489-508, 1961. PROBLEMS 10.1-3 Resonance Frequencies of a Resonator with an Etalon. (a) Determine the spacing between adjacent resonance frequencies in a resonator con- structed of two parallel planar mirrors separated by a distance d = 15 em in air (n = 1). (b) A transparent plate of thickness d 1 = 2.5 em and refractive index n = 1.5 is placed inside the resonator and is tilted slightly to prevent light reflected from the plate from reaching the mirrors. Determine the spacing between the resonance frequencies of the resonator. 10.1-4 Mirrorless Resonators. Semiconductor lasers are often fabricated from crystals whose surfaces are cleaved along crystal planes. These surfaces act as reflectors and therefore serve as the resonator mirrors. An expression for the intensity reflectance is provided in (6.2-15). Consider a crystal placed in air (n = 1) whose refractive index n = 3.6 and loss coefficient Os = 1 em -1. The light reflects between two parallel surfaces separated by a distance d = 0.2 mm. Determine the spacing between resonance frequencies VF, the overall distributed loss coefficient Or, the finesse , the spectral width 8v, and the quality factor
PROBLEMS 401 Q. Assuming that the free-space wavelength of the generated light is 1.55 /-Lm, estimate the longitudinal mode number q. 10.1-5 Fabry-Perot Etalon with Bragg Grating Reflectors. A Fabry-Perot etalon is made by sandwiching a layer of GaAs between two of the GaAs/ AlAs Bragg grating reflectors described in Prob. 7.1-7. Determine the finesse of the resonator and quality factor Q. Determine the transmittance of a Bragg grating reflector comprised of N == 10 alternating layers of GaAs (nl == 3.6) and AlAs (n2 == 3.2) of widths d l and d 2 equal to a quarter wavelength in each medium. Assume that the light is incident from an extended GaAs medium. 10.1-6 Resonator Spectral Response. The transmittance of a symmetric Fabry-Perot resonator was measured by using light from a tunable monochromatic light source. The transmittance versus frequency exhibits periodic peaks of period 150 MHz, each of width (FWHM) 5 MHz. Assuming that the medium within the resonator mirrors is a gas with n == 1, determine the length and finesse of the resonator. Assuming further that the only source of loss is associated with the mirrors, find their reflectances. 10.1- 7 Optical Energy Decay Time. How much time does it take for the optical energy stored in a resonator of finesse == 100, length d == 50 em, and refractive index n == 1, to decay to one-half of its initial value? 10.2-5 Stability of Spherical-Mirror Resonators. (a) Can a resonator with two convex mirrors ever be stable? (b) Can a resonator with one convex and one concave mirror ever be stable? 10.2-6 A Planar-Mirror Resonator Containing a Lens. A lens of focal length f is placed inside a planar-mirror resonator constructed of two flat mirrors separated by a distance d . The lens is located at a distance d /2 from each of the mirrors. (a) Determine the ray-transfer matrix for a ray that begins at one of the mirrors and travels a round trip inside the resonator. (b) Determine the condition of stability of the resonator. (c) Under stable conditions sketch the Gaussian beam that fits this resonator. 10.2-7 Self-Reproducing Rays in a Symmetric Resonator. Consider a symmetric resonator using two concave mirrors of radii R separated by a distance d == 3IRI/2. After how many round trips through the resonator will a ray retrace its path? 10.2-8 Ray Position in Unstable Resonators. Show that for an unstable resonator the ray position after m roun d trips is given by Y m == Q I hI + Q2h'2, where QI and Q2 are constants. Here hI == b + yl b 2 -1, h 2 == b - yl b 2 - 1, and b == 2(1 + d/R I )(1 + d/R 2 ) - 1. Hint: Use the results in Sec. 1.4D. 10.2-9 Ray Position in Unstable Symmetric Resonators. Verify that a symmetric resonator using two concave mirrors of radii R == -30 cm separated by a distance d == 65 em is unstable. Find the position YI of a ray that begins at one of the mirrors, at position Yo == 0 with an angle (}o == 0.1 0 , and undergoes one round trip. If the mirrors have 5-cm-diameter apertures, after how many round trips does the ray leave the resonator? Write a computer program to plot Ym, m == 2, 3, . . ., for d == 50 em and d == 65 em. You may use the results of Prob. 10.2-8. 10.2-10 Gaussian- Beam Standing Waves. Consider a wave formed by the sum of two identical Gaussian beams propagating in the + z and - z directions. Show that the result is a standing wave. Using the boundary conditions at two ideal mirrors placed such that they coincide with the wavefronts, derive the resonance frequencies (10.2-30). 10.2-11 Gaussian Beam in a Symmetric Confocal Resonator. A symmetric confocal resonator with mirror spacing d == 16 em, mirror reflectances 0.995, and n == 1 is used in a laser operating at Ao == 1 /-Lm. (a) Find the radii of curvature of the mirrors. (b) Find the waist of the (0,0) (Gaussian) mode. (c) Sketch the intensity distribution of the (1,0) modes at one of the mirrors and determine the distance between its two peaks. (d) Determine the resonance frequencies of the (0,0) and (1,0) modes. (e) Assuming that losses arise only from imperfect mirror reflectances, determine the dis- tributed resonator loss coefficient Qr.
402 CHAPTER 10 RESONATOR OPTICS . * 10.2-12 Diffraction Loss in a Symmetric Confocal Resonator. The percent diffraction loss per pass for the different low-order modes of a symmetric confocal resonator is given in Fig. 10.2-11, as a function of the Fresnel number N F a 2 / Ad (where d is the mirror spacing and a is the radius of the mirror aperture). Using the parameters provided in Probe 10.2-11, determine the mirror radius for which the loss per pass of the (1,0) mode is 1%. 10.3-2 Number of Modes in Resonators of Different Dimensions. Consider light of wavelength Ao 1.06 pm and spectral width v 120 GHz. How many modes have frequencies within this linewidth in the following resonators (n 1 ): (a) A one-dimensional resonator of length d 10 em? (b) A 10 em x 10 em two..,dimensional resonator? ( c ) A 10 em x 10 em x 10 em three-dimensional resonator?
CHAPTER 11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 405 A. Optical Intensity B. Temporal Coherence and Spectrum C. Spatial Coherence D. Longitudinal Coherence 11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT 419 A. Interference of Two Partially Coherent Waves B. Interference and Temporal Coherence C. Interference and Spatial Coherence k11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 427 A. Propagation of Partially Coherent Light B. Image Formation with Incoherent Light C. Gain of Spatial Coherence by Propagation 11.4 PARTIAL POLARIZATION 436 II -L ... . ' . I (.., . . . . . . J. . .... ,', I . - II . " -.:.. :." ......:..:O'.;-:O' II . . ", .." .. '4 '."'..oo '. .. , " . -. . ." " " , . . . '. . . , . . ':.' .:. .. . . , [T I , . . , ." . . ]" I l _ r .I '. .' .- I I .' . - ., , . , ," '. . '. , "" .- " " - , _ u . . . . . '. - ..... . . " Max Born (1882-1970) Emil Wolf (born 1922) The book Principles of Optics. first published in 1959 by Max Born and Emil Wolf, drew attention to the importance of coherence in optics. Emil Wolf is responsible for many advances in the theory of optical coherence. 403
Statistical optics is the study of the properties of random light. Randoillness in light arises because of unpredictable fluctuations of the light source or of the medium through which light propagates. Natural light, e.g., light radiated by a hot object, is random because it is a superposition of emissions from a very large number of atoms radiating independently and at different frequencies and phases. Randomness in light may also be a result of scattering from rough surfaces, diffused glass, or turbulent fluids, which impart random variations to the optical wavefront. The study of the random fluctuations of light is also known as the theory of optical coherence. In the preceding chapters it was assumed that light is deterministic or I.l. co herent." An example of coherent light is the monochromatic wave u r, t Re U r exp jwt , for which the complex amplitude Uris a deterministic complex function, e.g., U r A exp j kr r in the case of a spherical wave l Fig. 11.0-1 (a)]. The dependence of the wavefunction on time and position is perfectly periodic and predictable. On the other hand, for random light, the dependence of the wavefunction on time and position [Fig. 11.0-1 (b)] is not totally predictable and cannot generally be described without resorting to statistical methods. Time dependence t t \ ..\ Wavefronts -" "7 ,. " (a) (b) Figure 11.0...1 TilDe dependence and wavefronts of (a) a monochronlatic spherical wave, which is an example of coherent light; (b) random light. How can we extract from the fluctuations of a random optical wave SOll1e meaningful measures that characterize it and distinguish it from other random waves? Examine, for instance, the three random optical waves whose wavefunctions at SOll1e position vary with time as in Fig. 11.0-2. It is apparent that wave (b) is more I.l.intense" than wave (a) and that the envelope of wave ( c) fluctuates '"'faster" than the envelopes of the other two waves. I . I I I . I .1 III. II I 11111111111 1I11' ltllllll'ljllll 1'111 1 """11"" "'1'" II'! t . I I I I I I 111111 Id 11 111111 III Illf'II'IIIII'IIIIIIII"'III II! II t illdl.ILI Illd .lllldLLlill I , I I , I I I I ' II' I r " 'I '" I I ' II I I ' , , 'I t (a) (b) (c) Figure 11.0...2 TilDe dependence of the wavefunctions of three rand0l11 waves. 404
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 405 To translate these casual qualitative observations into quantitative measures, we use the concept of statistical averaging to define a number of nonrandom measures. Because the random function u r, t satisfies certain laws (the wave equation and boundary conditions) its statistical averages must also satisfy certain laws. The theory of optical coherence deals with the definitions of these statistical averages, with the laws that govern them, and with measures by which light is classified as coherent, incoherent, or, in general, partially coherent. This Chapter This chapter is an introduction to the theory of partial coherence. Familiarity with the theory of random fields (random functions of many variables space and time) is necessary for a full understanding of the theory of optical coherence. However, the ideas presented in this chapter are limited in scope, so that knowledge of the concept of statistical averaging is sufficient. In Sec. 11.1 we define two statistical averages used to describe random light: the op- tical intensity and the mutual coherence function. Temporal and spatial coherence are delineated, and the connection between temporal coherence and monochromaticity is established. The examples of partially coherent light provided in Sec. 11.1 demonstrate that spatially coherent light need not be temporally coherent, and that monochromatic light need not be spatially coherent. One of the basic manifestations of the coherence of light is its ability to produce visible interference fringes. Sec. 11.2 is devoted to the laws of interference of random light. The transmission of partially coherent light in free space and through different optical systems, including image-formation systems, is the subject of Sec. 11.3. A brief introduction to the theory of polarization of random light (partial polarization) is provided in Sec. 11.4. . 11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT An arbitrary optical wave is described by a wavefunction u r, t Re U, r, t , where U r, t is the complex wavefunction. For example, U r, t may take the form U r exp jwt for monochromatic light, or it may be a sum of many similar functions of different v for polychromatic light (see Sec. 2.6A for a discussion of the complex wavefunction). For random light, both functions, u r, t and U r, t , are random and are characterized by a number of statistical averages introduced in this section. A. Optical Intensity The intensity I r, t of coherent (deterministic) light is the absolute square of the complex wavefunction U r, t , I r,t 2 U r, t . (11.1-1) (see Sec. 2.2A and Sec. 2.6A). For monochromatic deterministic light the intensity is independent of time, but for pulsed light it is time varying. For random light, U r, t is a random function of time and position. The intensity U r, t 2 is therefore also random. The average intensity is then defined as I r,t U r t 2 , (11.1-2) Average Intensity
406 CHAPTER 11 STATISTICAL OPTICS where the symbol · now denotes an ensemble average over many realizations of the random function. This means that the wave is produced repeatedly under the same con- ditions, with each trial yielding a different wavefunction, and the average intensity at each time and position is determined. When there is no ambiguity we shall simply call I r, t the intensity of light (with the word average implied). The quantity U r, t 2 is called the random or instantaneous intensity. For deterministic light, the averaging operation is unnecessary since all trials produce the same wavefunction, so that (11. ] - 2) is equivalent to (11.1-1). The average intensity may be time independent or may be a function of time, as illustrated in Figs. 11.1-1 (a) and (b), respectively. The former case applies when the optical wave is statistically stationary; that is, its statistical averages are invariant to time. The instantaneous intensity U r, t 2 fluctuates randomly with time, but its average is constant. We will denote it, in this case, by I r . Stationarity does not necessarily mean constancy. It means constancy of the average properties. An example of stationary random light is that from an ordinary incandescent lamp heated by a constant electric current. The average intensity I r is a function of distance from the lamp, but it does not vary with time. However, the random intensity U r, t 2 fluctuates with both position and time, as illustrated in Fig. 11.1-1 (a). I U(r, t)1 2 I U(r, t)1 2 t t I(r, t) I(r, t) t t (a) Stationary (b) N on station ary Figure 11.1-1 (a) A statistically stationary wave has an average intensity that does not vary with time. (b) A statistically nonstationary wave has a time-varying intensity. These plots represent, e.g., the intensity of light from an incandescent lamp driven by a constant electric current in (a) and a pulse of electric current in (b). When the light is stationary, the statistical averaging operation in (11.1- 2) can usu- ally be determined by time averaging over a long time duration (instead of averaging over many realizations of the wave), whereupon Ir 1 lim T > 00 2 T - T T U r t 2 dt. , (11.1-3) B. Temporal Coherence and Spectrum Consider the fluctuations of stationary light at a fixed position r as a function of time. The stationary random function U r, t has a constant intensity I r U r, t 2 . For brevity, we drop the r dependence (since r is fixed), so that U r, t U t and I r I. The random fluctuations of U t are characterized by a time scale representing the "memory" of the random function. Fluctuations at points separated by a time interval longer than the memory time are independent, so that the process "forgets" itself. The function appears to be smooth within its memory time, but "rough" and "erratic"
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 407 when examined over longer time scales (see Fig. 11.0-2). A quantitative measure of this temporal behavior is established by defining a statistical average known as the autocorrelation function. This function describes the extent to which the wavefunction fluctuates in unison at two instants of time separated by a given time delay, so that it establishes the time scale of the process that underlies the generation of the wavefunc- . tlon. Temporal Coherence Function The autocorrelation function o( a stationary complex random function U t is the average of the product of U* t and U t + T as a function of the time delay T GT U* t U t + T ( 11.1-4 ) Temporal Coherence Function or GT . 1 11m T > 00 2 T T U* t U t + T dt -T (] 1.1- 5) (see Sec. A.I in Appendix A). To understand the significance of the definition in (11.1-4), consider the case in which the average value of the complex wavefunction U t O. This is applicable when the phase of the phasor U t is equally likely to have any value between 0 and 27r, as illustrated in Fig. 11.1- 2. The phase of the product U* t U t + T is the angle between phasors U t and U t + T . If U t and U t + Tare uncorrelated, the angle between their phasors varies randomly between 0 and 27r. The phasor U* t U t + T then has a totally uncertain angle, so that it is equally likely to take any direction, making its average, the autocorrelation function G T , vanish. On the other hand if, for a given T, U t and U t + T are correlated, their phasors will maintain some relationship. Their fluctuations are then linked together so that the product phasor U* t U t + T has a preferred direction and its average G T will not vanish. Im{U(t)} Re{U(t)} Figure 11.1-2 Variation of the phasor U(t) with time when its argument is uniformly dis- tributed between 0 and 2w. The average values of its real and imaginary parts are zero, so that (U(t)) O. In the language of optical coherence theory, the autocorrelation function G T is known as the temporal coherence function. It is easy to show that G T is a function with Hermitian symmetry, G T G* T , and that the intensity I, defined by (1 1.1- 2), is equal to G T when T 0, I GO. ( 11.1-6)
408 CHAPTER 11 STATISTICAL OPTICS Degree of Temporal Coherence The temporal coherence function G T carries information about both the intensity I G 0 and the degree of correlation (coherence) of stationary light. A measure of coherence that is insensitive to the intensity is provided by the normalized autocorre- lation function, 9 7 GT GO U* t U t + T U* t U t , (11.1 - 7) Complex Degree of Temporal Coherence which is called the complex degree of temporal coherence. Its absolute value cannot exceed unity, o < 9 T < 1. (11.1-8) The value of 9 T is a measure of the degree of correlation between U t and U t+ T . When the light is deterministic and monochromatic, i.e., UtA exp jwot , where A is constant, (11.1- 7) gives 9 T . exp JW07 , (11.1-9) so that 9 T 1 for all T. The variables U t and U t + T are then completely correlated for all time delays T . Usually, 9 T drops from its largest value 9 0 1 as T increases and the fluctuations become uncorrelated for sufficiently large T. Coherence Time If 9 T decreases monotonically with time delay, the value Tc at which it drops to a the fluctuations known as the coherence time (see Fig. 11.1-3). u(t) Tc-ll- I I I I I II I Ig( T) I I I I Ig( T) I 1 u(t) I I 1 4 Ie 1 -4t( I I. LILII.,Ld Ill.d.ltlLILLIIiI 11111"" 1'11'1" I" r 'I" 1'1" 1'1'11 t Ie III Jllldlllllll""11I11111.11111 """1'11" 11"'"1 1 11111 ' '' III t TC o T o T (a) (b) Figure 11.1-3 Illustrative examples of the wavefunction, the magnitude of the complex degree of temporal coherence Ig( T) I, and the coherence time Te for an optical field with (a) short coherence time and (b) long coherence time. The amplitude and phase of the wavefunction vary randomly with time constants approximately equal to the coherence time. In both cases the coherence time Ie is greater than the duration of an optical cycle. Within the coherence time, the wave is rather predictable and can be approximated as a sinusoid. However, given the amplitude and phase of the wave at a particular time, one cannot predict the amplitude and phase at times beyond the coherence time. For T < Tc the fluctuations are "strongly" correlated whereas for T > Tc they are "weakly" correlated. In general, Tc is the width of the function 9 T . Although the definition of the width of a function is rather arbitrary (see Sec. A.2 of Appendix A),
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 409 the power-equivalent width 00 Tc 9 T 2 dT (11.1-1 0) Coherence Time -00 is commonly used as the definition of coherence time [see (A.2-8) and note that 9 0 1]. The coherence time of monochromatic light is infinite since 9 T 1 everywhere. EXERCISE 11.1-1 Coherence Time. Verify that the fonowing expressions for the complex degree of temporal coherence are consistent with the definition of 7c given in (11.1-10): exp 17/ ( exponential) Tc g(T) (11.1-11) 7rT 2 2T 2 c exp (Gaussian) By what factor does Ig( T) I drop as 7 increases from 0 to 7c in each case? Light for which the coherence time Tc is much longer than the differences of the time delays encountered in the optical system of interest is effectively completely coherent. Thus, light is effectively coherent if the distance CT c is much greater than all optical path-length differences encountered. The distance l c CT c (] 1.1-12) Coherence Length is known as the coherence length. Power Spectral Density To determine the average spectrum of random light, we carry out a Fourier decompo- sition of the random function U t . The amplitude of the component with frequency v is the Fourier transform (see Appendix A) 00 Vv U t exp j27rvt dt. (11.1-13) -00 The average energy per unit area of those components with frequencies in the interval between v and v + dv is V V 2 dv, so that V v 2 represents the energy spectral density of the light (energy per unit area per unit frequency). Note that the complex wave function U t has been defined so that V v 0 for negative v (see Sec. 2.6A). Since a truly stationary function U t is eternal and carries infinite energy, we consider instead the povver spectral density. We first determine the energy spectral density of the function U t observed over a window of time width T by finding the truncated Fourier transform V T V T/2 U t exp j27rvt dt -T/2 (11.1-14)
41 0 CHAPTER 11 STATISTICAL OPTICS and then determine the energy spectral density V T v 2 . The power spectral density is the energy per unit time 1 T V T V 2 . We can now extend the time window to infinity by taking the limit T > 00. The result Sv 1 2 T > ex:> , (11.1-15) is called the power spectral density. It is nonzero only for positive frequencies. Be- cause U t was defined such that U t 2 represents power per unit area, or intensity (W/cm 2 , S v dv represents the average power per unit area carried by frequencies between v and v + dv, so that S v actually represents the intensity spectral density (W/cm 2 -Hz). It is often referred to simply as the spectral density or the spectrum. The total average intensity is the integral ex:> I S v dv. ( 11.1-16) o The autocorrelation function G T , defined by (11.1-4), and the spectral density S v defined by (11.1-15) can be shown to form a Fourier transform pair (see Probe 11.1-5), ex:> Sv G T exp j21TvT dT. (1 ] .1-17) Power Spectral Density -(X) This relation is known as the Wiener Khinchin theorem. An optical wave representing a color image, such as that illustrated in Fig. 11.1-4, has a spectrum that varies with position r; each spectral profile shown corresponds to a perceived color. >. . ..... .. .' ..::. .. ',I :. .! . . '. .. - . " . . :....... . ] . I .. . . . . , . .. . .. . ':. . .. , . .' . - , - " - ,:} -.- ... = a. -- "1 . I . 'oO - . . - r:.I'J t:: (],) "1; . . L r- . l . .r::;: l. I r 0'" .._:- , :'.' oO...: 1"1 . '" . .. L _ .' . 1- , , .\. ,.' .' I " -, r -:.. .t;. . - . - . .'c - . . , . . . . . . I , , , '. , .... . ',' , - , - I I I 400 500 600 700 Wavelength (nm) L >. ......... . r:.I'J C (1) . 1 ro u (],) Uj . , Red , . . - . . ... " -- . . " . " - . . , .... . ::. .. . . . -. . . .j-: .... . ", ,,,,.:. .,.: .. . 1 . ..;..:.. . .. .....-, . . - .. '. -" '. .'. .;'" .' ..:....-:::.. (,: :''-:.' . : ,> -..:.. '.: . . :.. . .:..;'.. . .'. . - -: .. . " .. .. . & .oO ",. .' . . ',' " .... - . . - . I -r . oOoO. r :.',:' ...f'! I . .: . , ..... . . . , . .- "11: I . =s. . , J- L [ . t ro u (1) CI1 400 500 600 700 Wavelength (nm) -- . . - . ..'. . 'oO "'L .,.-- - . . '. I . .' ... .- - . . .. . . _:. '. . . :..r"........ '" '. . -- - - ". '..'. 'f .n- f, .. ...t [ ,-;! o , .. . -. '- ..=- - -. . . i I >-- . ...... . . .. '3 .. - - . . ,.,,; .:. .... I. . '. r . UJ c:: (1) . Green. " -< - -- --. - - I .- .. ._... 'Lr-... }f, ., :i-- .' . - . r . '.' . . . -0 <: ..". - -- - Co . . :.... . . :..:;.:. :. .. ," . . . .. .. . . ., . . . . . . . .-..:. ........ . .... -{ .'. . . . .. - '. .' . ...... .. . . ..' -, ;.r. . '. ..Ii. . " " . 1 ro $..... U (1) CI1 400 500 600 700 Wavelength (nlTI) " ." .... - . - -[ :"r ':." \ Blue . .::, r :-. .' . ., . .. . r .. . .... . - .. . - . .. . '.. - - - . . . .;.... .... - .. . , .-. . ."I. -- - . ........... .. . ..."11 . "'. n...; .?,. ,.' .: '. . :. . .. .: '.' .... - . - . -. : ':.. -r. .". I '. . . - - . . ...... & :.':: . . , . .. .. - .' . . - . .' .' .... .... . . . . ';. ,:-; ;,'.. -. ., .. . . . . , .' . . .... . . . 1 '.. ...: ." .. .'. - - .. .' .' . . . . ., . . - . ... . ... .'. .. .............. .. .. '. Figure 11.1-4 Variation of the spectral density as a function of wavelength at three positions in a color image (Dahlias, Henri Matisse).
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 411 Spectral Width The spectrulll of light is often confined to a narrow band centered about a central frequency Va. The spectral width, or linewidth, of light is the width v of the spectral density S v . Because of the Fourier-transform relation between S v and G T , their widths are inversely related. A light source of broad spectrum has a short coher- ence time, whereas a light source with narrow linewidth has a long coherence time, as illustrated in Fig. 11.1-5. In the limiting case of monochromatic light, G T I exp jwoT , so that the corresponding intensity spectral density S v I 6 v Va contains only a single frequency component, Va. Thus, Tc 00 and v O. The coherence time of a light source can be increased by using an optical filter to reduce its spectral width. The resultant gain of coherence comes at the expense of losing light energy. I I g( T) I 1 S(v) ll( t) I TC --i l -c I I I I (a) I I I I I I I I I I I j I I I I I I i I I I i I I I I I "'1 11 '111 I 11"'1' "1'11'1'1'1 t ... Tc ....... 6v T o o Va V II I I g( T) I 1 S(v) ll(t) .1 Tc I .... I I. (b) j I I I I i I I I j I I j I . I I I I I I I I I J j I 1'111 1 '11" 1"'."1111111" 'I t I TC ..... 8 v T o o Va V Figure 11.1-5 Two random waves, the magnitudes of their complex degree of temporal coherence, and their spectral densities. There are severa] definitions for the spectral width. The most common is the full width of the function S v at half its maximum value (FWHM). The relation between the coherence time and the spectral width depends on the spectral profile, as indicated in Table 1 ] .1-] (see also Appendix A, Sec. A.2). . Table 11.1-1 Relation between spectral width and coherence time. Spectral Density Rectangular Lorentzian Gaussian r"'V r"'V 2ln 2/1r 0.66 r"'V r"....I Spectral Width l/F\\THM 1 1 0.32 . Tc 1TTr Tc Tc Tc Another convenient definition of the spectral width is CXJ 2 Sv dv Vc 0 (11.1-18) 00 . S2 V dv 0
412 CHAPTER 11 STATISTICAL OPTICS By this definition it can be shown that tJ.V c 1 (] 1.1-19) Spectral Width , Tc regardless of the spectral profile (see Exercise 11.1- 2). If S v is a rectangular function extending over a frequency interval from Va B 2 to Va + B 2, for example, then ( 11.1- 18) yields tJ.v c B. The two definitions of bandwidth, tJ.v c and tJ.VFWHM tJ.v, differ by a factor that ranges from 1 7r 0.32 to 1 for the profiles listed in Table] 1.1- 1. EXERCISE 11. 1-2 Relation Between Spectral Width and Coherence Time. Show that the coherence time Te defined by (11.1-1 0) is related to the spectral width v e defined in (11.1-18) by the simple inverse relation Te 1/ ve. Hint: Use the definition of ve and Te, the Fourier transform relation between S(v) and G(T), and Parseval's theorem [see (A. 1-7) in Appendix A]. Representative spectral bandwidths for different light sources, and their associated coherence times and coherence lengths lc CT c , are provided in Table 11.1- 2. Table 11.1-2 Spectral widths of a number of light sources together with their coherence times and coherence lengths in free space. Source ve (Hz) 3.74 x 10 14 1.5 X 10 13 5 X 1011 1.5 X 10 9 1 x 10 6 Te 1/ ve Ie CT e Filtered sunlight (A o 0.4-0.8/lID) Light-emitting diode (Ao 1 /lm, Ao 50 nm) Low-pressure sodium lamp Multimode He-Ne laser (A o 633 nm) Single-mode He-Ne laser (Ao 633 nm) 2.67 fs 67 fs 2 ps 0.67 os 1 JLS 800 nm 20 /lID 600 /ln1 20 em 300m EXAMPLE 11.1-1. A Wave Comprising a Random Sequence of Wavepackets. Light emitted from an incoherent source may be modeled as a sequence of wavepackets emitted at random times (Fig. 11.1-6). Each wavepacket has a random phase since it is emitted by a different atom. u(t) I g( T) I I Tc A A A A A TC A A A i , , . . , , , ' I i'..i, IA,. i'i... , r' , ., t , ,. I"" , III ,i, I, . J" I. 1" A, . 1'" '1 11 ' "'" "'" t o T Figure 11.1-6 Light comprised of wavepackets emitted at random times has a coherence time equal to the duration of a wavepacket.
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 413 The wavepackets may be sinusoidal with an exponentially decaying envelope, for example, so that a wavepacket emitted at t 0 has a complex wavefunction (at a given position) t Up(t) Apexp 0, exp(jwot), t > 0 Tc (1 1.1-20) t < O. The emission times are totally random, and the random independent phases of the different emissions are included in Ap. The statistical properties of the total field may be determined by performing the necessary averaging operations using the rules of mathematical statistics. The result yields a complex degree of coherence given by g(T) exp( ITI/Tc) exp(jwoT) whose magnitude is a double-sided exponential function. The corresponding power spectral density is Lorentzian, S(v) (v/21r)/[(v vo)2 + (V/2)2], where v l/1rTc (see Table A.2-1 in Appendix A). The coherence time Tc in this case is exactly the width of a wavepacket. The statement that this light is correlated within the coherence time therefore means that it is correlated within the duration of an individual wavepacket. C. Spatial Coherence Mutual Coherence Function An important descriptor of the spatial and temporal fluctuations of the random function U r, t is the cross-correlation function of U rl, t and U r2, t at pairs of positions rl and r2. Grl,r2,T U* rl, t U r2, t + T . (11.1-21) Mutual Coherence Function This function of the time delay T is known as the mutual coherence function. Its normalized form, 9 rl,r2,T G r}, r2, T , (11.1-22) Complex Degree of Coherence is called the complex degree of coherence. When the two points coincide so that rl r2 r, (11.1- 21) and (11.1- 22) reproduce the temporal coherence function and the complex degree of temporal coherence defined in (11.1-4) and (11.1-7) at the position r. Ultimately, when T 0, the intensity I r G r, r, 0 at the position r. The complex degree of coherence g rl, r2, T is the cross-correlation coefficient of the random variables U* rl, t and U r2, t + T . Its absolute value is bounded between zero and unity, o < 9 rl,r2,T < 1. (11.1-23) It is therefore considered a measure of the degree of correlation between the fluctua- tions at rl and those at r2 at a time T later. When the two phasors U rl, t and U r2, t fluctuate independently and their phases are totally random (each having equally probable phase between 0 and 27r), 9 rl, r2, T 0 since the average of the product U* rl, t U r2, t + T van- ishes. The light fluctuations at the two points are then uncorrelated. The other limit,
414 CHAPTER 11 STATISTICAL OPTICS 9 rl, r2, T 1, applies when the light fluctuations at rl, and at r2 a time T later, are fully correlated. Note that 9 rl, r2, 0 is not necessarily unity; however, by definition gr,r,O 1. The dependence of 9 rl, r2, T on time delay and on the positions characterizes the temporal and spatial coherence of light. Two examples of the dependence of 9 rl, r2, T on the distance rl r2 and the time delay T are illustrated in Fig. 11.1- 7. Ig(rl, r2, 7)1 Ig(rl, r2, 7)1 . - --. . .. .. - . . I I I I I I I . . - .-:: -0 -0 . Jot ____ __ .. I -. .:. 0 .-.- -« ---. -- . . ... '''-'''';' I ... ",,' . . . ..". -- .. --- Irl - r21 . .. ..... .. -"-''- -''"Iii; . 1: .. .. o 0 I ... o I 0 . .-- - 0 . . - - .. - ........ .. ....... ... .. ....... .- > . _......00- o ..... . . . .- o -.", Irl - r21 "'."..- I .... ',:.' "" I ::... ..- - ....0... . ,- I -:. " , .: " ., "" .0 T T " , - . .-.--..-... .. ...- - .. , , (a) (b) Figure 11.1-7 Two examples of Ig(rt, r2, 7)1 as a function of the separation Irt r21 and the time delay 7. In (a) the maximum correlation for a given Irt r21 occurs at 7 O. In (b) the maximum correlation occurs at Irt r21 CT. The temporal and spatial fluctuations of light are intimately related since light prop- agates in waves and the complex wavefunction U r, t must satisfy the wave equation. This imposes certain conditions on the mutual coherence function (see Exercise 11.1- 3). To illustrate this point, consider, for example, a plane wave of random light traveling in the z direction in a homogeneous and nondispersive medium with velocity c. Fluctu- ations at the points rIO, 0, Zl and r2 0,0, Z2 are completely correlated when the time delay is T TO Z2 Zl C, so that 9 rl, r2, TO 1. As a function of T, 9 rl, r2, T has a peak at T TO, as illustrated in Fig. 11.1-7 (b). This example will be discussed again in Sec. II.ID. EXERCISE 11. 1-3 Differential Equations Governing the Mutual Coherence Function. In free space, U (r, t) must satisfy the wave equation, \12U (1/c 2 )8 2 U /8t 2 O. Use the definition (11.1-21) to show that the mutual coherence function G (rt, r2, 7) satisfies the two partial differential equations \1G 18 2 G c 2 8T 2 18 2 G c 2 8T 2 o (l1.1-24a) \1G 0, (11.1-24b) where \1 and \1 are the Laplacian operators with respect to rt and r2, respectively. Mutual Intensity The spatial correlation of light may be assessed by examining the dependence of the mutual coherence function on position for a fixed time delay T. In many situations the
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 415 point T 0 is the most appropriate, as in the example in Fig. 11.1-7 (a). However, this need not always be the case, as in the example in Fig. ] 1.1-7(b). The mutual coherence function at T 0, G rl, r2, 0 U* rl, t U r2, t , ( 11.1- 25) is known as the mutual intensity and is denoted by G rl, r2 for simplicity. The diagonal values of the mutual intensity rl r2 r provide the intensity I r G r, r . When the optical path differences encountered in an optical system are much shorter than the coherence length Ie CT e , the light may be considered to effectively possess complete temporal coherence, so that the mutual coherence function is a harmonic function of time: G rl, r2, T G rl, r2 exp jwoT , (11.1- 26) where Vo is the central frequency. In this case the light is said to be quasi-monochromatic and the mutual intensity G rl, r2 describes the spatial coherence completely. The complex degree of coherence 9 rl, r2, 0 is similarly denoted by 9 rl, r2 . Thus, 9 rl,r2 G rl, r2 (11.1-27) Normalized Mutual Intensity is the normalized mutual intensity. The magnitude 9 rl, r2 is bounded between zero and unity and is regarded as a measure of the degree of spatial coherence (when the time delay T is zero). If the complex wavefunction U r, t is deterministic, 9 rl, r2 1 for all rl and r2, so that the light is completely correlated everywhere. Coherence Area The spatial coherence of quasi-monochromatic light in a given plane in the vicinity of a given position r2 is described by 9 rl, r2 as a function of the distance rl r2 . This function is unity when rl r2 and drops as rl r2 increases (but it need not be monotonic). The area scanned by the point rl within which the function 9 rl, r2 is It represents the spatial extent of 9 rl, r2 as a function of rl for fixed r2, as illustrated in Fig. 11.1-8. In the ideal limit of coherent light the coherence area is infinite. The coherence area is an important parameter that characterizes random light. This parameter must be considered in relation to other pertinent dimensions of the optical system. For example, if the area of coherence is greater than the size of the aperture through which light is transmitted, so that 9 rl, r2 1 at all points of interest, the light may be regarded as coherent, as if the coherence area were infinite. Similarly, if the coherence area is smaller than the resolution of the optical system, it can be regarded as infinitesimal, i.e., 9 rl, r2 0 for practically all rl -# r2. In this limit the light is said to be incoherent. Light radiated from an extended radiating hot surface has an area of coherence on the order of A 2 , where A is the central wavelength, so that for most practical cases it may be regarded as incoherent. Thus, complete coherence and incoherence are only idealizations representing the two limits of partial coherence.
416 CHAPTER 11 STATISTICAL OPTICS " . . .. -... .:;. 1 Ac Ig(rl, rz)1 Ig(rl, rz)1 . ...;.;... / .._\.....0.', ,a_ . . ";.::: .' ':<::..;.:.;:' ".. ...._.. -";. ..... ...... ..; ...;,.:,:.;:- '. "" - - . ... ,- " I' .- . :',':"'..."'-;f.;f . .:::::', ,- " ..... . .. ." ;t :-.:.... .. .. ....... ...... . .. .. .. ...... . . .... . .......:.. . ,. . ;.' . 1 r2 :r2 3.::.. -:..-"..... . :.i::. . . ......... Ac o o (a) (b) Figure 11.1-8 Two illustrative examples of the magnitude of the normalized mutual intensity as a function of rl in the vicinity of a fixed point r2. The coherence area in (a) is smaller than that in (b). Cross-Spectral Density The mutual coherence function G rl, r2, T describes the spatial correlation at each time delay T. The time delay T 0 is selected to define the mutual intensity G rl, r2 G rl, r2, 0 , which is suitable for describing the spatial coherence of quasi-monochromatic light. A useful alternative is to describe coherence in the frequency domain by examining the spatial correlation at a fixed frequency. The cross- spectral density (or the cross-power spectrum) is defined as the Fourier transform of G rl, r2, T with respect to T: c:x:) S rl,r2,V G rl, r2, T exp j27rVT dT. ( 11.1- 28) Cross-Spectral Density - c:x:) When rl r2 r, the cross-spectral density becomes the power-spectral density S v at position r, as defined in (11.1-17). The normalized cross-spectral density is defined by S rl,r2,V , S rl,rl,V S r2,r2,V S rl,r2,V (11.1-29) and its magnitude can be shown to be bounded between zero and unity, so that it serves as a measure of the degree of spatial coherence at the frequency v. It represents the degree of correlation of the fluctuation components of frequency v at positions rl and r2. In certain cases, the cross-spectral density factors into a product of one function of position and another of frequency, S rl, r2, v G rl, r2 S v , so that the spatial and spectral properties are separable. The light is then said to be cross-spectrally pure. The mutual coherence function must then also factor into a product of a function of position and another of time, G rl, r2, T G rl, r2 9 T , where 9 T is the inverse Fourier transform of s v . If the factorization parts are selected such that s v dv 1, then G rl, r2 G rl, r2, 0 , so that G rl, r2 is nothing but the mutual intensity. Cross- spectrally pure light has two important properties: 1. At a single position r, S r, r, v G r, r S v IrS v . The spectrum has the same profiles at all positions. If the light represents a visible image, it would appear to have the same color everywhere but with varying brightness.
11.1 STATISTICAL PROPERTIES OF RANDOM LIGHT 417 2. The normalized cross-spectral density S rl,r2,V G rl, r2 G rl, rl G r2, r2 9 rl,r2 (11.1-30) is independent of frequency. In this case the normalized mutual intensity 9 rl, r2 describes spatial coherence at all frequencies. D. Longitudinal Coherence In this section the concept of longitudinal coherence is introduced by taking examples of random waves with fixed wavefronts, such as planar and spherical waves. Partially Coherent Plane Wave Consider a plane wave U r,t z exp jwo t Z (11.1-31) a t c c traveling in the z direction in a homogeneous medium with velocity c. As shown in Sec. 2.6A, U r, t satisfies the wave equation for an arbitrary function at. If a t is a random function, U r, t represents partially coherent light. The mutual coherence function defined in (11.1- 21) is Z2 Zl Z2 Zl G rl, r2, T G a T . exp JWo T , (11.1-32) c c where Zl and Z2 are the z components of rl and r2 and GaT a * tat + T is the autocorrelation function of at, assumed to be independent of t. The intensity I r G r, r , 0 G a 0 is constant everywhere in space. Tem- poral coherence is characterized by the time function G r, r, T G a T exp jwoT , which is independent of position. The complex degree of coherence is 9 r, r, T ga T exp jwoT , where 9a T GaT G a 0 . The width of ga T 9 r, r, T , defined by an expression similar to (11.1-1 0), is the coherence time Tc. It is the same at all positions. The power spectral density is the Fourier transform of G r, r, T with respect to T. From (11.1-32), S v is seen to be equal to the Fourier transform of GaT shifted by a frequency Vo (in accordance with the frequency shift property of the Fourier transform defined in Appendix A, Sec. A.I. The wave therefore has the same power spectral density everywhere in space. The spatial coherence properties are described by G rl, r2, 0 G a Zl Z2 . Zl exp J Wo Z2 (11.1-33) c c and its nonnalized version 9 rl,r2,O Zl Z2 . Zl Z2 9a exp JWo (11.1-34) c c If the two points rl and r2 lie in the same transverse plane, i.e., Zl Z2, then 9 rl, r2, 0 9a 0 1. This means that fluctuations at points on a wavefront (a plane normal to the z axis) are completely correlated; the coherence area in any transverse plane is infinite (Fig. 11.1-9). On the other hand, fluctuations at two points
418 CHAPTER 11 STATISTICAL OPTICS fc = CTC I Ig( T) I S(v) ...... T C 8v U ncorrelated Figure 11.1-9 The fluctuations of a partially coherent plane wave at points on any wavefront (transverse plane) are completely correlated, whereas those at points on wavefronts separated by an axial distance greater than the coherence length Ie CT e are approximately uncorrelated. . . z . . T o o lIa II Correlated separated by an axial distance Z2 Zl such that Z2 Zl C > Te, or Z2 Zl > le, where lc eT c is the coherence length, are approximately uncorrelated. In summary: the partially coherent plane wave is spatially coherent within each transverse plane, but partially coherent in the axial direction. The axial (longitudinal) spatial coherence of the wave has a one-to-one correspondence with the temporal coherence. The ratio of the coherence length lc eT c to the maximum optical path difference lmax in the system governs the role played by coherence. If le » lmax, the wave is effectively completely coherent. The coherence lengths of a number of light sources are listed in Table 11.1-2. Partially Coherent Spherical Wave A partially coherent spherical wave is described by the complex wavefunction (see Sec. 2.2B and Sec. 2.6A) 1 a t r r exp jwo t r e , (11.1-35) U r, t C where a t is a random function. The corresponding mutual coherence function is r2 rl e T . exp JWo T r2 rl C G rl, r2, T , (11.1-36) rlr2 withG a T a* t a t+T . The intensity I r G a 0 r 2 varies in accordance with an inverse-square law. The coherence time Te is the width of the function 9a T G a T G a 0 . It is the same everywhere in space. So is the power spectral density. For T 0, fluctuations at all points on a wavefront (a sphere) are completely correlated, whereas fluctuations at points on two wavefronts separated by the radial distance r2 rl» lc eT c are uncorrelated (see Fig. 11.1-10). An arbitrary partially coherent wave transmitted through a pinhole generates a par- tially coherent spherical wave. This process therefore imparts spatial coherence to the incoming wave (points on any sphere centered about the pinhole become completely correlated). However, the wave remains temporally partially coherent. Points at differ- ent distances from the pinhole are only partially correlated. The pinhole imparts spatial coherence but not temporal coherence to the wave. Suppose now that an optical filter of very narrow spectral width is placed at the pinhole, so that the transmitted wave becomes approximately monochromatic. The wave will then have complete temporal, as well as spatial, coherence. Temporal co- herence is introduced by the narrowband filter, whereas spatial coherence is imparted
11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT 419 ..) cf(C \(c Uncorrelated wavefronts Figure 11.1-1 0 A partially coherent spherical wave has complete spatia] coherence at all points on a wavefront but not at points wi th different radial distances. by the pinhole, which acts as a spatial filter. The price for obtaining this ideal wave is, of course, the loss of optical energy introduced by the temporal and spatial filtering processes. 11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT The interference of coherent light was discussed in Sec. 2.5. This section is devoted to the interference of partially coherent light. A. Interference of Two Partially Coherent Waves The statistical properties of two partially coherent waves U 1 and U 2 are described not only by their own mutual coherence functions but also by a measure of the degree to which their fluctuations are correlated. At a given position r and time t, the intensities of the two waves are I] U] 2 and 1 2 U 2 2 , whereas their cross-correlation is described by the statistical average G 12 U;U 2 , and its normalized version UtU 2 1 1 1 2 912 . (11.2-1 ) When the two waves are superposed, the average intensity of their sum is U 1 + U 2 2 U 1 2 II + 1 2 + G 12 + G2 + U 2 2 + U{U 2 + U 1 U; II + 1 2 + 2 Re G 12 I II + 1 2 + 2 1 1 1 2 Re 912 , (11.2-2) from which I I] + 1 2 + 2 1 1 1 2 912 cas cp, (11.2-3) Interference Equation where cp arg 912 is the phase of 912. The third term on the right-hand side of (11.2- 3) represents optical interference. There are two illlportant limits: 1. For two cOlnJ}letely correlated waves with 912 exp jcp and 912 1, we re- cover the interference formula (2.5-4 ) for two coherent waves of phase difference cp.
420 CHAPTER 11 STATISTICAL OPTICS 2. For two uncorrelated waves with 912 0, we have I II + 1 2 so that there is no interference. In the general case, the normalized intensity I versus the phase cp assumes the form of a sinusoidal pattern, as shown in Fig. 11.2-1. The strength of the interference is measured by the visibility V (also called the modulation depth or the contrast of the interference pattern): V I max I min Imax + I min ' (11.2-4) where Imax and I min are, respectively, the maximum and minimum values that 1 takes as cp is varied. Since cos cp stretches between 1 and 1, inserting (11.2-3) into (11.2-4) yields V 2 1112 II + 1 2 912 · (11.2-5) The visibility is therefore proportional to the absolute value of the normalized cross- correlation 912 . In the special case when II 1 2 , we have V 912 · (11.2-6) Visibility 2 I 21 0 1 I g 12 'P Figure 11.2-1 Normalized intensity I / 21 0 of the sum of two partially coherent waves of equal intensities (II 1 2 1 0 ), as a function of the phase <p of their normalized cross-correlation 912. This sinusoidal pattern has visibility V 19121. o I I I I I I I I I - -- - - -- , - r- I - - I I I , I I I I I , I I f I I I I I - - -- -27r 0 27r The interference equation (11.2- 3) will now be considered in a number of specific contexts to illustrate the effects that temporal and spatial coherence have on the inter- ference of partially coherent light. B. Interference and Temporal Coherence Consider a partially coherent wave U t with intensity 10 and complex degree of temporal coherence 9 T U* t U t + T 10. If U t is added to a replica of itself delayed by the time T, U t + T , what is the intensity I of the superposition? Usingtheinterferenceformula(II.2-2)withU I U t ,U 2 U t+T ,II 1 2 10, and 912 U;U 2 fo U* t U t + T 10 9 T , we obtain 1 210 1 + Re 9 T 210 1 + 9 T COS cp T , ( 11.2- 7) where cp T arg 9 T . It is thus apparent that the ability of a \'vave to interfere with a tbne delayed replica of itself is governed by its cOlnplex degree of telnporal coherence at that tilne de lay.
11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT 421 Implementing the addition of a wave with a time-delayed replica of itself may be achieved by using a beamsplitter to generate two identical waves, one of which is made to traverse a longer optical path than the other, and then recombining them at another (or the same) beamsplitter. This can be effected, for example, with the help of a Mach Zehnder or a Michelson interferometer (see Fig. 2.5-3). Consider, as an example, the partially coherent plane wave introduced in Sec. 11.1D [see (11.1-31)], whose complex degree of temporal coherence is 9 T ga T exp jwoT . The spectral width of the wave is l:1v c 1 Tc, where Tc (the width of ga T ) is the coherence time. Substituting this into (11.2-7), we obtain 1 21 0 1 + 9a T COS WOT + CPa T , (11.2-8) where CPa T arg ga T · The relation between 1 and T, which is known as an interferogram, is illustrated in Fig. ] 1.2-2. Assuming that l:1v c 1 Tc« Yo, the functions ga T and CPa T vary slowly in comparison with the period 1 yo. The visibility of this interferogram in the vicinity of a particular time delay T is V 9 T ga T . It has a peak value of unity near T 0 and vanishes for T » Tc, i.e., when the optical path difference is much greater than the coherence length lc CTc. For the Michelson interferometer illustrated in Fig. 11.2-2, T 2 d 2 d 1 c. Interference occurs only when the optical path difference is smaller than the coherence length. - dl 2 )I.. I 21 0 " '" / '21 9 121 v d2 / , l 1 , ...... . , , I o , / VI +U2 o T 2(d2 - dl)/C Figure 11.2-2 The normalized intensity 1 /210, as a function of the time delay T, when a partially coherent plane wave is introduced into a Michelson interferometer. The visibility determines the magnitude of the complex degree of temporal coherence. The magnitude of the complex degree of temporal coherence of a wave, 9 T , may therefore be measured by monitoring the visibility of the interference pattern as a function of time delay. The phase of 9 T may be measured by observing the locations of the peaks of the pattern. Fourier- Transform Spectroscopy It is revealing to write (11.2- 7) in terms of the power spectral density of the wave S v . Using the Fourier-transform relation between G T and S v , GT log T 00 S v exp j27rVT dv, (11.2-9) o substituting into (11.2-7), and noting that S v obtain dv 1 0 , we 00 I 2 S v 1 + cos 27fVT dv. (11.2-10) o
422 CHAPTER 11 STATISTICAL OPTICS This equation can be interpreted as a weighted superposition of interferograms pro- duced by each of the monochromatic components of the wave. Each component v produces an interferogram with period 1 v and unity visibility, but the composite interferogram exhibits reduced visibility by virtue of the different periods. Equation (11.2-1 0) suggests that the spectral density S v of a light source can be determined by measuring the interferogram I versus T and then inverting the result by means of Fourier-transform methods. This technique is known as Fourier-transform spectroscopy. Optical Coherence Tomography Optical coherence tomography (OCT) is an interferometric technique for profiling a multilayered medium, i.e., for measuring the reflectance and depth of each of its boundaries. It makes use of a partially coherent light source of short coherence length and a Michelson interferometer. As illustrated in Fig. 11.2-3, a replica of the original wave, delayed by a movable mirror, is superposed with a collection of waves reflected from the multiple sample boundaries. Information about the sample profile is carried by the interferogram, which is the intensity measured at the detector as the movable mirror is translated. By virtue of the short coherence length of the source, the interferogram comprises sets of fringes centered at path delays of the movable mirror that match those of the reflecting boundaries. Movable . mIrror d __a 0 --- ..... 1 2 3 . , . - . - ..... .. .... , oJ ..... - ;;:: -- Sample ., 2 . . I 21 0 1 r} r2 r3 Detector o o 'L 72 T) I Figure 11.2-3 Optical coherence tomography. Let U t T be the wave reflected from the movable mirror, with its associated time delay T d Co, and let riU t Ti, i 1,2, . . ., be the waves reflected from the boundaries of the sample, where T i represents the amplitude reflectance at the ith boundary; the associated time delays are designated Ti. For a symmetric beamsplitter, the average intensity is then I T U t T + i ri U t Ti 2 , which may be written in normalized form as I 210 1 + TiRe 9 T Ti + TiT; Re 9 Tj Ti , (] 1.2-11) . 'l . . 'lJ since the complex degree of temporal coherence of the source is characterized by 9 T U* t U t + T U* t U t . The second term on the right-hand side of (11.2-11) is of paramount importance since it represents interference between the reference wave from the movable mirror and each of the waves reflected from the sample boundaries. The third term represents interference terms associated with multiple reflections from the sample; since these terms are independent of the path delay of the movable mirror, T d c, they may be regarded as background contributions and ignored.
11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT 423 For a light source of central frequency Vo, we have 9 T ga T exp jWOT , where the width of ga T is the coherence time Tc. Equation (11.2-11) then becomes I 210 1 + rig a T Ti COS Wo T Ti + cP a T Ti , (11.2-12) . where cpa T arg 9a T . If the source is of short coherence length, the function ga T is narrow. As illustrated in Fig. 11.2-3, the reflection from each sample boundary then generates a distinct set of interference fringes of brief duration Tc, centered about its corresponding time delay. Measurement of the OCT interferogram therefore permits the reflectance at each boundary, as well as the width of each of the sample layers, to be determined. " Optical coherence tomography has proven to be an effective imaging technique in clinical medicine as well as in engineering. c. Interference and Spatial Coherence The effect of spatial coherence on interference is demonstrated by considering the Young's double-pinhole interference experiment, discussed in Exercise 2.5-2 for co- herent light. A partially coherent optical wave U r, t illuminates an opaque screen with two pinholes located at positions rl and r2. The wave has mutual coherence function G rl, r2, T U* rl, t U r2, t + T and complex degree of coherence 9 rl, r2, T . The intensities at the pinholes are assumed to be equal. Light is diffracted in the form of two spherical waves centered at the pinholes. The two waves interfere, and the intensity I of their sum is observed at a point r in the observation plane a distance d from the screen sufficiently large so that the paraboloidal approximation is applicable. In Cartesian coordinates (Fig. 11.2-4) rl a, 0, 0 , r2 a, 0,0 , and r x, 0, d . The intensity is observed as a function of x. An important geometrical parameter is the angle e 2a d subtended by the two pinholes. A () 2a , Z l " ; \ . \ \ . I I \ I , 'r , . . . . . . , \; , , I 21 0 2 Ig(rl' r2)1 I -' \ _ , '1:' - x .E 1 - - - - - - " d . J , " o o x Figure 11.2-4 Young's double-pinhole interferometer. The incident wave is quasi-monochromatic and the nonnalized mutual intensity at the pinholes is g(rl' r2). The normalized intensity 1/210 in the observation plane at a large distance is a sinusoidal function of x with period Ale and visibility V Ig(rl, r2)1. In the paraboloidal (Fresnel) approximation [see (2.2-17)], the two diffracted spher-
424 CHAPTER 11 STATISTICAL OPTICS ical waves are approximately related to U r, t by r r1 e d + x + a 2 2d U 1 r, t ex: U r1, t U r1, t (II.2-13a) e U 2 r, t ex: U r2, t r r2 U r2, t d+ x a 2 2d , (11.2-13b) e e and have approximately equal intensities, 1 1 correlation between the two waves at r is 1 2 10. The normalized cross- 912 U{ r, t U 2 r, t 10 9 r1, r2, Tx , (11.2-14) where Tx r r1 r r2 e x+a 2 x a 2 2dc 2ax de () x e (11.2-15) is the difference in the time delays encountered by the two waves. Substituting (11.2-14) into the interference formula (11 .2- 3) gives rise to an ob- served intensity I I x : I f' \.x) 21 0 1 + 9 :r1, r2, Tx: cos <Px: , (11.2-16) where <Px arg 9 r1, r2, Tx . This equation describes the pattern of observed inten- sity as a function of position x in the observation plane, in terms of the magnitude and phase of the complex degree of coherence at the pinholes at time delay Tx ()x e. Quasi-Monochromatic Light If the light is quasi-monochromatic with central frequency Vo 9 r1, r2, T 9 r1, r2 exp jwoT , then (11.2-16) gives Wo 27r, i.e., if 21T() Ix 210 1 + Vcos , (11.2-17) where A C va, V 9 r1, r2 , Tx ex C, and <p arg 9 r1, r2 . The inter- ference fringe pattern is therefore sinusoidal with spatial period A e and visibility V. In analogy with the temporal case, the visibility of the interference pattern equals the magnitude of the complex degree of spatial coherence at the two pinholes (Fig. 11.2-4). The locations of the peaks depend on the phase <po Interference with Light from an Extended Source If the incident wave in Young's interferometer is a coherent plane wave traveling in the z direction, U r, t exp jkz exp jwot , then 9 r1, r2 1, so that 9 rl, r2 1, and arg 9 r1 r2 O. The interference pattern therefore has unity visibility and a peak at x O. But if the illumination is, instead, a tilted plane wave arriving from a direction in the x z plane making a small angle ()x with respect to the z axis,
11.2 INTERFERENCE OF PARTIALLY COHERENT LIGHT 425 i.e., U r, t exp j kz + kOxx exp jwot , then 9 rl, r2 exp jkO x 2a. The visibility remains V 1, but the tilt results in a phase shift cp kO x 2a 27rO x 2a A, so that the interference pattern is shifted laterally by a fraction 2aO x A of a period. When cp 27r, the pattern is shifted one period. Suppose now that the incident light is a collection of independent plane waves arriving from a source that subtends an angle Os at the pinhole plane (Fig. 11.2-5). The phase shift cp then takes values in the range ::l:27r Os 2 2a A ::l:27rO sa A and the fringe pattern is a superposition of displaced sinusoids. If Os A 2a, then cp takes on values in the range ::l:7r, which is sufficient to wash out the interference pattern and reduce its visibility to zero. z I Os . . --e o A e () 21 0 x 10 - Ac' .. -- . . . x Figure 11.2-5 Young's interference fringes are washed out if the illumination emanates from a source of angular diameter Os > A/2a. If the distance 2a is smaller than A/Os, the fringes become visible. We conclude that the degree of spatial coherence at the two pinholes is very small when the angle subtended by the source is Os A 2a (or greater). Consequently, the distance A Pc Os ( 11.2-18) Coherence Distance is a measure of the coherence distance in the plane of the screen and Ac A 2 Os (11.2-19) is a measure of the coherence area of light emitted from a source subtending an angle Os. The angle subtended by the sun, for example, is 0.5 0 , so that the coherence distance for filtered sunlight of wavelength A is Pc A Os 115A. At A 0.5 /-Lill, Pc 57.5/-L ill . A more rigorous analysis (see Sec. ] 1.3C) shows that the transverse coherence distance Pc for a circular incoherent light source of uniform intensity is Pc A 1.22 Os . (11.2-20)
426 CHAPTER 11 STATISTICAL OPTICS Effect of Spectral Width on Interference Finally, we examine the effect of the spectral width on interference in the Young's double-pinhole interferometer. The power spectral density of the incident wave is as- sumed to be a narrow function of width vc centered about Va, and vc « Va. The complex degree of coherence then has the form 9 rl,r2,T . ga rl, r2, T exp JWOT , (11.2-21) where ga rl, r2, I is a slowly varying function of I (in comparison with the period 1 va). Substituting (11.2-21) into (11.2-16), we obtain Ix 21a 1 + V x cos 21f() x + <{Jx A , ( 11.2-22) where V x ga rl, r2, IX , <{Jx arg ga rl, r2, IX , Tx ()x c, and A C Va. Thus, the interference pattern is sinusoidal with period A () but with a varying visibility V x and varying phase 'Px equal to the magnitude and phase of the complex degree of coherence at the two pinholes, respectively, evaluated at the time delay IX ()x c. If ga rl, r2, T 1 at I 0, decreases with increasing I, and vanishes for I » Ie, the visibility V x 1 and x 0, decreases with increasing x, and vanishes for x » Xc Clc (). The interference pattern is then visible over a distance lc ()' Xc (] 1.2-23) where lc Clc is the coherence length and () is the angle subtended by the two pinholes (Fig. 11.2-6). Observation plane I - A e , fc , I .. \ , Xc = I, 21 0 Ox;= a , l --c d , I - 2a I A £ -...- 2 e e 1 e d Incident wave Screen o o X Figure 11.2-6 The visibility of Young's interference fringes at position x is the magnitude of the complex degree of coherence at the pinholes at a time delay Tx ex/c. For spatiaIly coherent light the number of observable fringes is the ratio of the coherence length to the central wavelength, or the ratio of the central frequency to the spectrallinewidth. The number of ob s ervable fringes is thus Xc A () lc A CT c A Va vc. It equals the ratio lc A of the coherence length to the central wavelength, or the ratio Va v c of the central frequency to the linewidth. Clearly, if 9 rl, r2, 0 < 1, i.e., if the source is not spatially coherent, the visibility will be further reduced and even fewer fringes wilJ be observable.
11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 427 *11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS The transmission of coherent light through thin optical components, through apertures, and through free space was discussed in Chapters 2 and 4. In this section we pursue the same goal for quasi-monochromatic partially coherent light. We assume that the spectral width is sufficiently small so that the coherence length le CT e C ve is much greater than the differences of optical path lengths in the system. The mutual coherence function may then be approximated by G rl, r2, T G rl, r2 exp jWoT , where G rl, r2 is the mutual intensity and Vo is the central frequency. It is noted at the outset that the transmission laws that apply to the deterministic function U r , which represents coherent light, apply also to the random function U r , which represents partially coherent light. However, for partially coherent light our interest is in the laws that govern statistical averages: the intensity I r and the mutual intensity G rl, r2 . A. Propagation of Partially Coherent Light Transmission Through Thin Optical Components en a partially coherent wave is transmitted through a thin optical component char- acterized by an amplitude transmittance t x, y the incident and transmitted waves are related by U 2 r t r U 1 r , where r x, y is the position in the plane of the component (see Fig. 11.3-1). Using the definition of the mutual intensity, G rl, r2 U* rl U r2 , we obtain G 2 rl, r2 t* rl t r2 G 1 rl,r2 , (11.3-1) where G 1 rl, r2 and G 2 rl, r2 are the mutual intensities of the incident and trans- mitted light, respectively. f2 r] . . U 1 (r) Uz(r) Figure 11.3-1 The absolute value of the degree of spatial coherence is not altered by transmission through a thin optical component. .., Since the intensity at position r equals the mutual intensity at rl r2 r, 1 2 r t r 2 II r . (11.3-2) The normalized mutual intensities defined by (11.1-27) therefore satisfy 92 r1,r2 91 r1, r2 · (11.3-3) Although transmission through a thin optical component may change the intensity of partially coherent light, it does not alter the magnitude of its degree of spatial coherence. Naturally, if the complex amplitude transmittance of the component itself were random, the coherence of the transmitted light would be altered.
428 CHAPTER 11 STATISTICAL OPTICS Transmission Through an Arbitrary Optical System We next consider an arbitrary optical system one that includes propagation in free space or transmission through thick optical components. It was shown in Chapter 4 that the complex amplitude U 2 r at a point r x, y in the output plane of such a system is generally a weighted superposition integral comprising contributions from the complex amplitudes U 1 r at points r' x' y' in the input plane (see Fig. 11.3- 2), u 2 r h r; r' U 1 r' dr', (11.3-4) where h r; r' is the impulse response function of the system. The integral in (11.3-4) is a double integral with respect to r' x' , y' extending over the entire input plane. x U(X,y,z) , , her, r' U2(r) VI(r) Input plane Optical system y Output plane y Figure 11.3-2 An optical system is characterized by its impulse response function h(r; r'). To translate this relation between the random functions U 2 rand U 1 r into a relation between their mutual intensities, we substitute (11.3-4) into the definition G 2 rl, r2 U; rl U 2 r2 and use the definition GIrl, r2 U; rl U 1 r2 to obtain G 2 rl, r2 h * I h I G I I d I d ' rl;r 1 r2;r 2 1 r 1 ,r 2 r 1 r 2 . (11.3-5) Mutual Intensity If the mutual intensity G 1 rl, r2 of the input light and the impulse response function h r; r' of the system are known, the mutual intensity of the output light G 2 rl, r2 can be determined by carrying out the integrals in (11.3-5). The intensity of the output light is obtained by using the definition 1 2 r G 2 r, r , which reduces (11.3-5) to 1 2 r h * I h ' G I I d I d ' r;r 1 r;r 2 1 r 1 ,r 2 r 1 r2- (J 1.3-6) I mage Intensity To determine the intensity of the output light, we must know the mutual intensity of the input light. Kno)ivledge of the input intensity II r by itself is generally not sufficient to deternline the output intensity 1 2 r .
11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 429 B. Image Formation with Incoherent Light We now consider the special case when the input light is incoherent. The mutual intensity G 1 rl, r2 vanishes when r2 is only slightly separated from rl so that the coherence distance is much smaller than other pertinent dimensions in the system (for example, the resolution distance of an imaging system). The mutual intensity may then be written in the form G 1 rl, r2 II rl II r2 9 rl r2, where 9 rl r2 is a very narrow function. When G 1 rl, r2 appears under the integral in (11.3-5) or (11.3- 6) it is convenient to replace 9 rl r2 with a delta function, 9 rl r2 ab rl r2, where a 9 r dr is the area under 9 r , so that G 1 rl, r2 a II rl II r2 b rl r2. ( 11.3-7) Since the mutual intensity must remain finite and b 0 > 00, this equation is clearly not generally accurate. It is valid only for the purpose of evaluating integrals such as in (11.3-6). Substituting (11.3-7) into (11.3-6), the delta function reduces the double integral into a single integral and we obtain 1 2 r I 1 r' hi r; r' dr', (11.3-8) Imaging Equation (Incoherent Illumination) where hi r; r' a h r; r' 2 . (11.3-9) Impulse Response Function (Incoherent Illumination) Under these conditions, the relation between the intensities at the input and output planes describes a linear system of impulse response function hi r; r' , also called the point-spread function. When the input light is completely incoherent, therefore, the intensity of the light at each point r in the output plane is a weighted superposition of contributions from intensities at many points r' of the input plane; interference does not occur and the intensities simply add (Fig. 11.3-3). This is to be contrasted with the completely coherent system, for which the complex amplitudes rather than intensities are related by a superposition integral, as in (11.3-4). UI(r) U2(r) (a) her, r'x) II (r) /2(r) Figure 11.3-3 (a) The complex ampli- tudes of light at the input and output planes of an optical system illuminated by coherent light are related by a linear system with impulse response function h(r; r'). (b) The intensities of light at the input and output planes of an optical system illuminated by incoherent light are related by a linear system with impulse response function hi(r; r') alh(r; r') 1 2 . (b) hI (r, r'x)
430 CHAPTER 11 STATISTICAL OPTICS In certain optical systems the impulse response function h r; r' is a function of r r', say h r r'. The system is then said to be shift variant or isoplanatic (see Appendix B). In this case hi r; r' hi r r'. The integrals in (11.3-4) and (11.3-8) are then two-dimensional convolutions and the systems can be described by trans- fer functions H V x , v y and Hi v x , v y , which are the Fourier transforms of h r h x, y and hi r hi x, y , respectively. As an example, we apply the relations above to an imaging system. It was shown in Sec. 4.4C that with coherent illumination, the impulse response function of the single- lens focused imaging system illustrated in Fig. 11.3-4 in the Fresnel approximation . IS h r ex P x Y Ad 2 ' Ad 2 , ( 11.3-1 0) where P v x , v y is the Fourier transform of the pupil function p x, y and d 2 is the distance from the lens to the image plane. The pupil function is unity within the aperture and zero elsewhere. x dl p(X,y) - c:f j y Lens Object plane Lens y .... Aperture Image plane X Figure 11.3-4 A single-lens imaging system. When the illumination is quasi-monochromatic and spatially incoherent, the intensi- ties of light at the object and image plane are linearly related by a system with impulse response function 2 hi r uhr 2 exP x Y Ad 2 ' Ad 2 , (11.3-11 ) where A is the wavelength corresponding to the central frequency Va. EXAMPLE 11.3-1. Imaging System with a Circular Aperture. If the aperture is a circle of radius a, the pupil function p( x, y) 1 for x, y inside the circle, and 0 elsewhere. Its Fourier transform is P(v x , v y ) aJ 1 (21rv p a) , V p 2 + 2 V x v y , ( 11.3-12) V p
11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 431 where J 1 (.) is the Bessel function (see Appendix A, Sec. A.3). The impulse response function of the coherent system is obtained by substituting into (11.1-36), J 1 (27rv s p) h(x,y) ex: x2 + y2 , (11.3-13) , p 7rV s P where V s B 2A' B 2a d 2 · (11.3-14) For incoherent illumination, the impulse response function is therefore i x,y ex: · 7rV s P (11.3-]5) The response functions h(x, y) and hi(x, y) are illustrated in Fig. 11.3-5. Both functions reach their first zero when 27rv s p 3.832, or p Ps 3.832j27rv s 3.832>...j7rB, from which A Ps 1.22 B · (11.3-16) Two-Point Resolution Thus, the image of a point (impulse) in the input plane is a patch of intensity hi (x, y) and radius Ps. When the input distribution is composed of two points (impulses) separated by a distance Ps, the image of one point vanishes at the center of the image of the other point. The distance Psis therefore a measure of the resolution of the imaging system. <U h(p) hi(P) 00 0 00 1 1 c dS 0 .....-( 1.22>"'F # I <U U 00 d eB 8 00 00 t I Ps P P H(vp) Hi(V p) 1 1 00 t: o 1 r:J'1. u >"'F# cd c cE 00 Vs up 00 2vs up (a) Coherent (b) Incoherent Figure 11.3-5 Impulse response functions and transfer functions of a single-lens focused diffraction-limited imaging system with a circular aperture and F-number F# under (a) coherent and (b) incoherent illumination. The transfer functions of linear systems (see Appendix B) with the impulse response functions h(x, y) and hi(x, y) are the Fourier transforms (see Appendix A), 1, v p < V s H(vx,v y ) (11.3-17) 0, otherwise,
432 CHAPTER 11 STATISTICAL OPTICS and 2 v COS -1 P 1f 2v s 2 v p 2v s 1 v p 2v s , V p < 2v s Hi(V x , V y ) (11.3-18) 0, otherwise, where v p v; + V;. Both functions have been normalized such that their values at v p 0 are 1. These functions are illustrated in Fig. 11.3-5. For coherent illumination, the transfer function is flat and has a cutoff frequency V s () /2A lines/mm. For incoherent illumination, the transfer function drops approximately linearly with the spatial frequency and has a cutoff frequency 2v s () j A lines/mm. If the object is placed at infinity, i.e., d 1 00, then d 2 angle () 2aj f is then the inverse of the lens F-number, F# 2v s are related to the lens F-number by f, the focal length of the lens. The f /2a. The cutoff frequencies V s and Cutoff frequency (lines/mm) , 1 2AF# (coherent illumination) < (11.3-19) 1 '- AF # (incoherent ill umination). One should not draw the false conclusion that incoherent illumination is superior to coherent illumination since it has twice the spatial bandwidth. The transfer functions of the two systems should not be compared directly since one describes imaging of the complex amplitude, whereas the other describes imaging of the intensity. C. Gain of Spatial Coherence by Propagation Equation (11.3- 5) describes the change of the mutual intensity when the light prop- agates through an optical system of impulse response function h r; r' . When the input light is incoherent, the mutual intensity GIrl, r2 may be replaced by all rl II r2 b rl r2 and substituted in the double integral in (11.3-5) to obtain the single integral, G 2 rI, r2 a h* rI;r h r2;r II r dr. (11.3-20) Mutual Intensity It is evident that the received light is no longer incoherent. In general, light gains spatial coherence by the mere act of propagation. This is not surprising. Although light fluctuations at different points of the input plane are uncorrelated, the radiation from each point spreads and overlaps with that from the neighboring points. The light reaching two points in the output plane comes from many points of the input plane, some of which are common (see Fig. (11.3-6)). These common contributions create partial correlation between fluctuations at the output points. This is not unlike the transmission of an uncorrelated time signal (white noise) through a low-pass filter. The filter smooths the function and reduces its spectra] band- width, so that its coherence time increases and it is no longer uncorrelated. The prop- agation of light through an optical system is a form of spatial filtering that cuts the spatial bandwidth and therefore increases the coherence area.
11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 433 I / / / / 2 I Figure 11.3-6 Gain of coherence by propaga- tion is a result of the spreading of light. Although the light is completely uncorrelated at the source, the light fluctuations at points 1 and 2 share a common origin, the shaded area, and are therefore partially correlated. Incoherent source Van Cittert Zernike Theorem There is a mathematical similarity between the gain of coherence of initially inco- herent light propagating through an optical system, and the change of the amplitude of coherent light traveling through the same system. In reference to (11.3-20), if the observation point rI is fixed, for example at the origin 0, and the mutual intensity G 2 0, r2 is examined as a function of r2, then G 2 0, r2 a h* 0; r h r2; r II r dr. (11.3-21) Defining U 2 r2 G 2 0, r2 and U I r in the familiar form uh* 0, r II r , (11.3-21) may be written U 2 r2 h r2; r U I r dr, (1].3-22) which is exactly the integral (11.3-4) that governs the propagation of coherent light. Thus, the observed mutual intensity G 0, r2 at the output of an optical system whose input is incoherent is mathematically identical to the observed complex amplitude if a coherent wave of complex amplitude U I r ah* 0; r II r were the input to the same system. As an example, suppose that the incoherent input wave has uniform intensity and extends over an aperture p r p r 1 within the aperture, and zero elsewhere], i.e., II r p r ; and assume that the optical system is free space; i.e., h r'; r exp j k r' r r' r . The mutual intensity G 2 0, r2 is then identical to the amplitude U 2 r2 obtained when a coherent wave with input amplitude U I r ah* 0; r p r up r exp j kr r is transmitted through the same system. This is a spherical wave converging to the point 0 in the output plane and transmitted through the aperture. This similarity between the diffraction of coherent light and the gain of spatial coherence of incoherent light traveling through the same system is known as the Van Cittert Zernike theorem. Gain of Coherence in Free Space Consider the optical system of free-space propagation between two parallel planes sep- arated by a distance d (Fig. 11.3-7). Light in the input plane is quasi-monochromatic, spatially incoherent, and has intensity I x, y extending over a finite area. The distance d is sufficiently large so that for points of interest in the output plane the Fraunhofer approximation is valid. Under these conditions the impulse response function of the optical system is described by the Fraunhofer diffraction formula [see (4.2-3)] h r. r' , hoexp . x2 + y2 J1r Ad . xx' + yy' exp , (11.3-23)
434 CHAPTER 11 STATISTICAL OPTICS where r x, y, d and r' x' , y', 0 are the coordinates of points in the output and input planes, respectively, and ho j Ad exp j27rd A is a constant. x . (X2' Y2) x ./ (Xl, YI) . , .. ; . ' , (x, Y) .. . z . . . . . - .'" Source Y Y Observation plane d Figure 11.3-7 Radiation from an incoherent source in free space. To determine the mutual coherence function G Xl, YI, X2, Y2 at two points Xl, YI and X2, Y2 in the output plane, we substitute (11.3-23) into (11.3-20) and obtain 00 G Xl, YI, X2, Y2 27r . al YI Y I x, y dx dy , -00 where al determine I x, Y , (11.3-24) a ho 2 a A 2 d 2 is another constant. Given I x, Y , one can easily G Xl, YI, X2, Y2 in terms of the two-dimensional Fourier transform of CX) J v x , v y exp j27r VxX + vyy I x, y dx dy ( 11.3-25) -CX) evaluated at V x X2 Xl Ad and v y Y2 YI Ad. The magnitude of the corresponding normalized mutual intensity is X2 Xl Y2 YI Ad ' Ad J 0, 0 . (I 1.3-26) 9 Xl, YI, X2, Y2 J This Fourier transform relation between the intensity profile of an incoherent source and the degree of spatial coherence of its far field is similar to the Fourier transform relation between the amplitude of coherent light at the input and output planes (see Sec. 4.2A). The similarity is expected in view of the Van Cittert Zernike theorem. The implications of (11.3-26) are profound. If the area of the source, i.e., the spatial extent of I x, Y , is small, its Fourier transform J v x , v y is wide, so that the mutual intensity in the output plane extends over a wide area and the area of coherence in the output plane is large. In the extreme limit in which light in the input plane origi- nates from a point, the area of coherence is infinite and the radiated field is spatially completely coherent. This confirms our earlier discussions in Sec. 1 I .1 D regarding the coherence of spherical waves. On the other hand, if the input coherent light originates from a large extended source, the propagated light has a small area of coherence.
11.3 TRANSMISSION OF PARTIALLY COHERENT LIGHT THROUGH OPTICAL SYSTEMS 435 EXAMPLE 11.3-2. Radiation from an Incoherent Circular Source. For input light with unifonn intensity lex, y) /0 confined to a circular aperture of radius a, (11.3-26) yields Ig(xl, Yl, X2, Y2)1 2 J 1 ( 7r pO s / A) 7rpOs/ A , (11.3-27) where p (X2 Xl)2 + (Y2 Yl)2 is the distance between the two points, Os 2a/d is the angle subtended by the source, and J 1 ( .) is the Bessel function. This relation is plotted in Fig. 11.3-8. The Bessel function reaches its first zero when its argument is 3.832. We can therefore define the area of coherence as a circle of radius Pc 3.832(A/7rO s ), so that A pc (11.3-28) Coherence Distance A similar result, (11.2-18), was obtained using a less rigorous analysis. The area of coherence is inversely proportional to 0;. An incoherent light source of wavelength A 0.6 pm and radius 1 cm observed at a distance dIDO m, for example, has a coherence distance pc 3.7 mm. y P -. 1 I 9 I 1 A Pc = 1.22 Os 2 , , , , , , " , " , " , .,,,' , , , Incoherent ,,,,,, ,,' , , " , " , , , source "," ,,' , , ." , , , , , Os x o o -....1 2a d Pc P Figure 11.3-8 The magnitude of the degree of spatial coherence of light radiated from an incoherent circular light source subtending an angle Os, as a function of the separation p. Measurement of the Angular Diameter of Stars: The Michelson Stellar Interferometer Equation (] 1.3-28) is the basis of a method for measuring the angular diameters of stars. If the star is regarded as an incoherent disk of diameter 2a with uniform bril- liance, then at an observation plane a distance d away from the star, the coherence function drops to 0 when the separation between the two observation points reaches Pc 1.22A Os. Measuring Pc for a given A permits us to determine the angular diameter Os 2a d. As an example, taking the angular diameter of the sun to be 0.5°, Os 8.7 x 10- 3 radians, and assuming that the intensity is uniform, we obtain Pc 140A. For A 0.5 /-lID, Pc 70 /-lID. To observe interference fringes in a Young's double-slit apparatus, the holes would have to be separated by a distance smaller than 70 /-lID. Stars of smaller angular diameter have correspondingly larger areas of coherence. For example, the first star whose angular diameter was measured using this technique (a- Orion) has an angular diameter Os 22.6 x 10- 8 , so that for A 0.57 /-lID, Pc 3.1 ffi. A Young's interferometer can be modified to accommodate such large slit separations by using movable mirrors, as shown in Fig. 11.3-9.
436 CHAPTER 11 STATISTICAL OPTICS ... x I I I I I Ml Screen I Figure 11.3-9 Michelson stellar in- terferometer. The angular diameter of a star is estimated by measuring the mu- tual intensity at two points with variable separation P using Young's double-slit interferometer. The distance p between mirrors M 1 and ltI 2 is varied and the visibility of the interference fringes is measured. When p Pc 1.22)"'/()s, the visibility O. I I p I I I I I I M2 I . 11.4 PARTIAL POLARIZATION As we have seen in Chapter 6, the scalar theory of light is often inadequate and a vector theory that includes the polarization of light is necessary. This section provides a brief discussion of the statistical theory of random light, including the effects of polarization. The theory of partial polarization is based on characterizing the components of the optical field vector by correlations and cross-correlations similar to those defined earlier in this chapter. To simplify the presentation, we shall not be concerned with spatial effects. We therefore limit ourselves to light described by a transverse electromagnetic (TEM) plane wave traveling in the z direction. The electric-field vector has two components in the x and y directions with complex wavefunctions U x t and U y t that are generally random. Each function is characterized by its autocorrelation function (the temporal coherence function), G xx T G yy T U; t U x t + T U; t U y t + T . (11.4-1 ) (11.4-2) An additional descriptor of the wave is the cross-correlation function of U x t and U y t , G xy T u; t U y t + T . (11.4-3) The normalized function gxy T (11.4-4) is the cross-correlation coefficient of U x t and U y t + T . It satisfies the inequality o < gxy T < 1. When the two components are uncorrelated at all times, gxy T 0; when they are completely correlated at all times, gxy T 1. The spectral properties are, in general, tied to the polarization properties and the autocorrelation and cross-correlation functions can have different dependences on T. However, for quasimonochromatic light, all dependences on T in (11.4-1) to (11.4- 4) are approximately of the form exp jWoT , so that the polarization properties are described by the values at T O. The three numbers G xx 0 , G yy 0 , and G xy 0 , hereafter denoted G xx , G yy , and G xy , are then used to describe the polarization of the wave. Note that G xx Ix and G yy Iy are real numbers that represent the intensities of the x and y components, but G xy is complex and G yx G;y, as can easily be verified from the definition.
11.4 PARTIAL POLARIZATION 437 Coherency Matrix It is convenient to write the four variables G xx , G xy , G yx , and G yy in the form of a 2 x 2 Hermitian matrix G G xx G xy G yx G yy (11.4-5) called the coherency matrix. The diagonal elements are the intensities Ix and Iy, and the off-d ia onaI elements are the cross-correlations. The trace of the matrix, Tr G Ix + Iy I, is the total intensity. U x U' y defined in terms of the complex wavefunctions and complex amplitudes (instead of in terms of the complex envelopes as in Sec. 6.1B), The coherency matrix may also be written in terms of the Jones vector, J J*Jt U* x U x U y U;U x U;U x U;U y U;U y G, (11.4-6) U* y where t denotes the transpose of a matrix, and U x and U y denote U x t and U y t , respectively. The Jones vector is transformed by polarization devices, such as polarizers and retarders, in accordance with the rule J' T J [see (6.1-17)], where T is the Jones matrix representing the device [see (6.1-18) to (6.1-25)]. The coherency matrix is therefore transformed in accordance with G' T* J* TJ t T* J* JtTt T* J* Jt Tt, so that G' T*GTt. ( 11.4-7) We thus have a formalism for determining the effect of polarization devices on the coherency matrix of partially polarized light. Stokes Parameters and Poincare Sphere Representation The Stokes parameters were defined in Sec. 6.1 A for coherent light as a set of four real parameters related to the products of the x and y components of the complex envelope [see (6.1-9)]. This definition is readily generalized to partially coherent light as an average of these products: 80 U x 2 + u: 2 G xx + G yy (11.4-8a) y 81 U x 2 U 2 G xx G yy (11.4-8b) y 82 2Re U;U y 2Re G xy (11.4-8c) 83 21m U;U y 21m G xy · (11.4-8d) Stokes Parameters Thus, the Stokes parameters are directly related to elements of the coherency matrix G. The first parameter, 80, is simply the sum of the diagonal elements, which is the - total intensity I. The second, 81, is the difference of the diagonal elements, i.e., the
438 CHAPTER 11 STATISTICAL OPTICS difference between the intensities of the two polarization components. The third and fourth, 82 and 83, are proportional to the real and imaginary parts of the off-diagonal element, i.e., the cross-correlation function. Using these relations, it can be readily shown that the inequality G xy 2 < GxxG yy leads to the condition 8I + 8 + 8 < 86. For coherent light, these inequalities become equalities. The state of polarization of partially polarized light may be represented geometri- cally on the Poincare sphere as a point with Cartesian coordinates 81 80,82 80,83 80 . Since 8I + 8 + 8 < 86, such a point lies inside, or on, the surface of the sphere. To understand the significance of the coherency matrix and the Stokes parameters, we next examine two limiting cases. Unpolarized Light - Light of intensity I is said to be u npol a rized if its tw o components have the same then G - - 1 - 1 0 1 · ... - (11.4-9) Unpolarized Light By use of (11.4-7) and (6. I -22), it can be shown that (11.4-9) is invariant to rotation of the coordinate system, so that two components always have equal intensities and are uncorrelated. Unpolarized light therefore has an electric field vector that is statistically isotropic; it is equally likely to have any direction in the x y plane, as illustrated in Fig.ll.4-1(a). RCP ---.. .,.. ..... , " , " , , I , I I , , , , , , , . . I I . . . , , , , , , I , I , I , , " , " , .... "" ...... .... ------ Polarized Partially po larizea Un o1arized - (a) Unpolarized (b) Partially polarized (c) Polarized (RCP) (d) Poincare sphere Figure 11.4-1 Fluctuations of the electric field vector for (a) unpolarized light; (b) partially polarized light; (c) polarized light with circular polarization; (d) Poincare-sphere representation. When passed through a polarizer, unpolarized light becomes linearly polarized, but larized light since it only introduces a phase shift between two components that have a totally random phase to begin with. Similarly, unpolarized light transmitted through a polarization rotator remains unpolarized. These effects may be shown formally by use of (11.4-7) and (11.4-9) together with (6.1-18), (6.2-14), and (6.1- 20). _ The Stokes parameters describing unpolarized light are 80,81,82,83 1,0,0,0 as can be readily shown by use of (11.4-8) and (11.4-9). The corresponding representa- tion on the Poincare sphere is a point with cartesian coordinates 81 80,82 80, S3 So 0, 0, 0 , i.e., is located at the very origin of the sphere.
11.4 PARTIAL POLARIZATION 439 Polarized Light If the cross-correlation coefficient 9xy G xy IxIy has unit magnitude, 9xy 1, the two components of the optical field are perfectly correlated and the light is said to be completely polarized (or simply polarized). The coherency matrix then takes the form G Ix . IxIyeJ<P Iy , (11.4-10) . I 1 e -J<P x y where <p is the argument of 9xy. Defining U x Ix and U y I e J <P y , G U;U x U;U y U;U x U;U y J*Jt , (11.4-11 ) where J is a Jones matrix with components U x and U y . Thus, G has the same form as the coherency matrix of a coherent wave. Using the Jones vectors provided in Ta- ble 6.1- ], we can determine the coherency matrices for different states of polarization. Two examples are: r- - - 1 0 o - - 1 - -I 2 1 . J . J 1 in the x directi on polarized - - - - The Stokes parameters corresponding to (11.4-11) satisfy the relation 8I+8+8 86, so that polarized light is represented by a point on the surface, rather than inside, the Poincare sphere. It is instructive to examine the distinction between unpolarized light and circularly polarized light. In both cases the intensities of the x and y components are equal Ix Iy . For circularly polarized light the two components are completely corre- lated, but for unpolarized light they are uncorrelated. Circularly polarized light may be transformed into linearly polarized light by the use of a wave retarder, but unpolarized light remains unpolarized upon passage through such a device. Circularly polarized light is represented by a point at the north or south poles of the Poincare sphere, while unpolarized light is represented by a point at the origin. Degree of Polarization Partial polarization is a general state of random polarization that lies between the two ideal limits of unpolarized and polarized light. One measure of the degree of polarization is defined in terms of the determinant and the trace of the coherency matrix: JP> 1 4 det G TrG 2 (11.4-12) 1 4 IxIy Ix + Iy 2 1 2 9xy · ( 11.4-13) This measure is meaningful because of the following considerations:
440 CHAPTER 11 STATISTICAL OPTICS . It satisfies the inequality 0 < IP' < 1. . For polarized light, IP' has its highest value of 1, as can easily be seen by substitut- ing 9xy 1 into (11.4-13). For unpolarized light it has its lowest value JP> 0, since Ix Iy and gxy O. . It is invariant to rotation of the coordinate system (since the determinant and the trace of a matrix are invariant to unitary transformations). . The degree of polarization in (11.4-13) may also be expressed in terms of the Stokes parameters as: JP> S2 + S2 + S2 123 So , ( 11.4- 14) so that in the Poincare sphere representation, it is equal to the distance from the origin of the sphere. . It can be shown (Exercise 11.4-1) that a partially polarized wave can always be- regarded as a mixture of two uncorrelated waves: a completely polarized wave and an unpolarized wave, with the ratio of the intensity of the polarized component to the total intensity equal to the degree of polarization IP'. EXERCISE 11.4-1 Partially Polarized Light. Show that the superposition of unpolarized light of intensity (Ix + Iy)(l IP'), and linearly polarized light with intensity (Ix + Iy)IP', where IP' is given by {I 1.4-13), yields light whose x and y components have intensities Ix and Iy and normalized cross-correlation I 9 xy I. READING LIST General A. A. Kokhanovsky, Polarization Optics of Random Media, Springer-Verlag, 2003. E. L. O'Neill, Introduction to Statistical Optics, Addison-Wesley, 1963; Dover, reissued 2003. W. Lauterborn and T. Kurz, Coherent Optics: Fundamentals and Applications, Springer, 2nd ed. 2003. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, Chapter 10. B. R. Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach, Springer-Verlag, 1983, 3rd ed. 2001. J. W. Goodman, Statistical Optics, Wiley, 1985, paperback ed. 2000. H. E. Rowe, Electromagnetic Propagation in Multi-Mode Random Media, Wiley, 1999. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, Wiley, 1998. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995. H. Lefevre, The Fiber-Optic Gyroscope, Artech, 1993. G. Reynolds, J. B. DeVelis, G. B. Parrent, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics, SPIE Optical Engineering Press, 1989. J. Perina, Coherence of Light, Reidel, 1971, 2nd ed. 1985. J. C. Dainty, ed., Laser Speckle and Related Phenomena, Springer-Verlag, 1975, 2nd ed. 1984.
READING LIST 441 A. S. Marathay, Elements of Optical Coherence Theory, Wiley, 1982. B. E. A. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communica- tion, Springer-Verlag, 1978. B. Crosignani, P. Di Porto, and M. Bertolotti, Statistical Properties of Scattered Light, Academic Press, 1975. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence, Prentice Hall, 1964; SPIE Optical Engineering Press, reissued 1974. R. Hanbury-Brown, The Intensity Interferometer: Its Application to Astronomy, Taylor & Francis, 1974. G.1. Troup, Optical Coherence Theory, Methuen, 1967. Books on Random Functions A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1965, 4th ed. 2002. E. Parzen, Stochastic Processes, Holden-Day, 1962; Society for Industrial and Applied Mathematics (SIAM), reissued 1999. E. Parzen, Modern Probability Theory and Its Applications, Wiley, 1960, paperback ed. 1992. C. W. Helstrom, Probability and Stochastic Processes for Engineers and Scientists, Macmillan, 2nd. ed. [991. W. B. Davenport, Jr., and W. L. Root, An Introduction to the Theory of Random Signals and Noise, McGraw-HilI, 1958; IEEE Press, reissued 1987. E. Vanmarcke, Random Fields, MIT Press, 1983. 1. B. Thomas, An Introduction to Applied Probability and Random Processes, Wiley, 1971. Books on Optical Coherence Tomography M. E. Brezinski, Optical Coherence Tomography: Principles and Applications, Academic Press, 2006. W. Drexler, ed., Optical Coherence Tomography and Coherence Techniques, Volume 2, Progress in Biomedical Optics and Imaging, SPIE Optical Engineering Press, 2005. W. Drexler, ed., Optical Coherence Tomography and Coherence Techniques, Volume 1, Progress in Biomedical Optics and Imaging, SPIE Optical Engineering Press, 2003. B. E. Bouma and G. J. Teamey, eds., Handbook of Optical Coherence Tomography, Marcel Dekker, 2002. Articles P. H. Tomlins and R. K. Wang, Theory, Developments and Applications of Optical Coherence To- mography, Journal of Physics D: Applied Physics, vol. 38, pp. 2519-2535, 2005. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, Optical Coherence Tomography Principles and Applications, Reports on Progress in Physics, vol. 66, pp. 239-303, 2003. L. Mandel and E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (1850-1966), SPIE Optical Engineering Press (Milestone Series Volume 19), 1990. R. B. Smith, ed., Selected Papers on Fiber Optic Gyroscopes, SPIE Optical Engineering Press (Mile- stone Series Volume 8), 1989. Feature issues on applications of coherence and statistical optics, Journal of the Optical Society of America, no. 7, 1986 and no. 8, 1986. F. T. S. Yu, Principles of Optical Processing with Partially Coherent Light, in Progress in Optics, vol. 23, E. Wolf, ed., North-Holland, 1986. W. J. Tango and R. Q. Twiss, Michelson Stellar Interferometry, in Progress in Optics, vol. 17, E. Wolf, ed., North-Holland, 1980. G. O. Reynolds and J. B. DeVelis, Review of Optical Coherence Effects in Instrument Design, SPIE Proceedings, vol. 194, pp. 2-33, 1979. H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coherence, in Inverse Source Problems in Optics, H. P. Baltes, ed., Springer-Verlag, 1978. E. Wolf, Coherence and Radiometry, Journal of the Optical Society of America, vol. 68, pp. 6-17, 1978.
442 CHAPTER 11 STATISTICAL OPTICS L. Mandel and E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light, Volumes I and 2, Dover, 1970. B. J. Thompson, Image Fonnation with Partially Coherent Light, in Progress in Optics, vol. 7, E. Wolf, ed., North-Holland, 1969. L. Mandel and E. Wolf, Coherence Properties of Optical Fields, Reviews of Modem Physics, vol. 37, pp. 231-287, 1965. PROBLEMS 11.1-4 Lorentzian Spectrum. A light-emitting diode (LED) emits light of Lorentzian spectrum with a linewidth v (FWHM) 10 13 Hz centered about a frequency corresponding to a wavelength Ao O. 7 pm. Determine the linewidth Ao (in units of nm), the coherence time Tc, and the coherence length lco What is the maximum time delay within which the magnitude of the complex degree of temporal coherence Ig( 7) I is greater than 0.5? 11.1- 5 Proof of the Wiener-Khinchin Theorem. Use the definitions in (11.1-4), (] 1.1-14), and (11.1-] 5) to prove that the spectral density S(v) is the Fourier transfonn of the autocorre- lation function G ( T). Prove that the intensity 1 is the integral of the power spectral density S(v). 11. ] -6 Mutual Intensity. The mutual intensity of an optical wave at points on the x axis is given by G(X1, X2) 10 exp (x 2 + x 2 ) 1 2 ex p IV: 2 o (Xl X2)2 P , where 1 0 , W o , and Pc are constants. Sketch the intensity distribution as a function of x. De- rive an expression for the normalized mutual intensity g( Xl, X2) and sketch it as a function of Xl X2. What is the physical meaning of the parameters 1 0 , W o , and Pc? 11.1- 7 Mutual Coherence Function. An optical wave has a mutual coherence function at points on the X axis, G(X1, X2, T) exp exp[j27rU(X1, X2, )7] exp (Xl X2)2 p , 7rT 2 27 2 c where u(xl, X2) 5 x 10 14 S- 1 for Xl + X2 > 0, and 6 x 10 14 S- 1 for Xl + X2 < 0, Pc 1 mm, and Tc 1 ps. Detennine the intensity, the power spectral density, the coherence length, and the coherence distance in the transverse plane. Which of these quantities is position dependent? If this wave were recorded on color film, what would the recorded image look like? 11.1-8 Coherence Length. Show that light of narrow spectral width has a coherence length lc A 2 / A, where A is the linewidth in wavelength units. Show that for light of broad uniform spectrum extending between the wavelengths Amin and Amax 2A m in, the coherence length lc Amax. 11.1-9 Effect of Spectral Width on Spatial Coherence. A point source at the origin (0, 0, 0) of a Cartesian coordinate system emits light with a Lorentzian spectrum and coherence time Tc 10 ps. Determine an expression for the normalized mutual intensity of the light at the points (0,0, d) and (x, 0, d), where d 10 em. Sketch the magnitude of the normalized mutual intensity as a function of x. 11.1-10 Gaussian Mutual Intensity. An optical wave in free space has a mutual coherence function G(r1, r2, T) J(rl r2) exp(jwor). (a) Show that the function J(r) must satisfy the Helmholtz equation \J2J + kJ 0, where ko wo/ c. (b) An approximate solution of the Helmholtz equation is the Gaussian-beam solution J(r) 1 q(z) exp jk o (X2 + y2) 2q(z) exp( jkoz),
PROBLEMS 443 where q(z) z + jzo and Zo is a constant. This solution has been studied extensively in Chapter 3 in connection with Gaussian beams. Determine an expression for the coherence area near the z axis and show that it increases with I z I, so that the wave gains coherence with propagation away from the origin. 11.2-1 Effect of Spectral Width on Fringe Visibility. Light from a sodium lamp of Lorentzian spectrallinewidth v 5 x 1011 Hz is used in a Michelson interferometer. Determine the maximum path-length difference for which the visibility of the interferogram V > . 11.2-2 Number of Observable Fringes in Young's Interferometer. Determine the number of observable fringes in Young's interferometer if each of the sources in Table 11.1-2 is used. Assume full spatial coherence in all cases. 11.2-3 Spectrum of a Superposition of Two Waves. An optical wave is a superposition of two waves Ul(t) and U 2 (t) with identical spectra Sl(V) S2(V), which are Gaussian with spectral width v and central frequency Vo. The waves are not necessarily uncorrelated. Determine an expression for the power spectral density S(v) of the superposition U(t) Ul(t) + U 2 (t). Explore the possibility that S(v) is also Gaussian, with a shifted central frequency VI -I Vo. If this were possible, our faith in using the Doppler shift as a method to determine the velocity of stars would be shaken, since frequency shifts could originate from something other than the Doppler effect. * 11.3-1 Partially Coherent Gaussian Beam. A quasi-monochromatic light wave of wavelength A travels in free space in the z direction. Its intensity in the z 0 plane is a Gaussian function l(x) 10 exp( 2X2/TVJ) and its normalized mutual intensity is also a Gaus- sian function g(Xb X2) exp[ (Xl X2)2 / p]. Show that the intensity at a distance z satisfying conditions of the Fraunhofer approximation is also a Gaussian function lz(x) ex: exp[ 2X2 /W 2 (z)] and derive an expression for the beam width W (z) as a function of z and the parameters W o , Pc, and A. Discuss the effect of spatial coherence on beam divergence. * 11.3-2 Fourier-Transform Lens. Quasi-monochromatic spatially incoherent light of uniform in- tensity illuminates a transparency of intensity transmittance f (x, y) and the emerging light is transmitted between the front and back focal planes of a lens. Determine an expression for the intensity of the observed light. Compare your results with the case of coherent light in which the lens performs the Fourier transform (see Sec. 4.2). *11.3-3 Light from Two-Point Incoherent Source. A spatially incoherent quasi-monochromatic source of light emits only at two points separated by a distance 2a. Determine an expression for the normalized mutual intensity at a distance d from the source (use the Fraunhofer approximation) . * 11.3-4 Coherence of Light Transmitted Through a Fourier-Transform Optical System. Light from a quasi-monochromatic spatially incoherent source with uniform intensity is transmit- ted through a thin slit of width 2a and travels between the front and back focal planes of a lens. Determine an expression for the normalized mutual intensity in the back focal plane. 11.4-2 Partially Polarized Light. The intensities of the two components of a partially polarized wave are Ix ly !, and the argument of the cross-correlation coefficient gxy is 7r /2. (a) Plot the degree of polarization IP versus the magnitude of the cross-correlation coeffi- cient Igxy I. (b) Determine the coherency matrix if IP 0, 0.5, and 1, and describe the nature of the light in each case. (c) If the light is transmitted through a polarizer with its axis in the X direction, what is the intensity of the light transmitted?
CHAPTER 12.1 THE PHOTON A. Photon Energy B. Photon Polarization C. Photon Position D. Photon Momentum E. Photon Interference F. Photon Time 12.2 PHOTON STREAMS A. Mean Photon Flux B. Randomness of Photon Flow C. Photon-Number Statistics D. Random Partitioning of Photon Streams *12.3 QUANTUM STATES OF LIGHT A. Coherent-State Light B. Squeezed-State Light 446 458 471 .. -, ," " " - " -'.' . 4.' .' . ..J-'- -.. . .. . " . " , r' = -3.- I' ,':JI ,.... - ." .. - , .. , " . , I ,., " ", , .. . ,- ." . 4 . . . I . ' , ... . , ,., , . '00 " , .. <I' I .. . Max Planck (1858-1947) suggested that the emission and absorption of light by matter takes the form of quanta of energy. Albert Einstein (1879-1955) advanced the hypothesis that light itself comprises quanta of energy. 444
Electromagnetic optics (Chapter 5) provides the most complete treatment of light within the confines of classical optics. It encompasses wave optics, which in turn encompasses ray optics (Fig. 12.0-1). Although classical electromagnetic theory is capable of providing explanations for a great many effects in optics, as attested to by the earlier chapters in this book, it nevertheless fails to account for certain optical phenomena. This failure, which became evident at the beginning of the last century, ultimately led to the formulation of a quantum electromagnetic theory known as quantum electrodynamics. For optical phenomena, this theory is also referred to as quantum optics. Quantum electrodynamics (QED) is today accepted as a theory that is useful for explaining almost all known optical phenomena. Quantum Optics Electromagnetic Optics Wave Optics Ray Optics Figure 12.0-1 The theory of quantum optics provides an explanation for virtually all optical phenomena. It is more general than electromag- netic optics, which was shown earlier to encom- pass wave optics and ray optics. In the framework of QED, the electric and magnetic fields E and H are mathemat- ically treated as operators in a Hilbert space, rather than simply as vectors. They are assumed to satisfy certain operator equations and commutation relations that govern their time dynamics and their interdependence. The equations of QED describe the interactions of electromagnetic fields with matter in the same way that Maxwell's equations are used in classical electrodynamics. QED leads to results that are char- acteristically quantum in nature and cannot be explained classically. However, in spite of its vast successes, QED is not the final arbiter of all optical effects. That distinction currently belongs to electroweak theory, which combines quantum electrodynamics with the theory of weak interactions. The electroweak theory successfully explains unexpected (parity nonconserving) small rotations of the plane of polarization of light upon passage through certain materials. t There is hope that a combination of the elec- troweak theory with the theories of strong and gravitational interactions will ultimately lead to a general unified theory that accommodates all of the forces known in nature. This Chapter The formal treatment of QED is beyond the scope of this book. Nevertheless, it is possible to describe many of the quantum properties of light and its interaction with matter by supplementing electromagnetic optics with a few simple relationships drawn from QED that embody the corpuscularity, localization, and fluctuations of quantum fields and energy. This set of rules, which we call photon optics, permits us to deal with optical phenomena that lie beyond the reach of classical theory, yet retain classical optics as a limiting case. However, photon optics is not intended to be a theory capable of providing an explanation for all of the effects that can be explained by quantum optics. In Sec. 12.1 we introduce the concept of the photon and examine its properties. We use electromagnetic optics as a point of departure, imposing a number of rules that t See, for example, P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux, and E. N. Fortson, Precise Test of Electroweak Theory from a New Measurement of Parity Nonconservation in Atomic Thallium, Physical Review Letters, vol. 74, pp. 2658-2661, 1995. 445
446 CHAPTER 12 PHOTON OPTICS govern the behavior of photon energy, momentum, polarization, position, time, and interference. These rules take the form of deceptively simple relationships that have far- reaching consequences. This is followed, in Sec. 12.2, by a discussion of the properties of collections of photons and photon streams. The number of photons emitted by a light source in a given time interval is almost always random, exhibiting statistical properties that depend on the nature of the source. The photon-number statistics for several important optical sources, including lasers and thermal radiators, are set forth. The effect of simple optical components (such as beamsplitters and filters) on the randomness of a photon stream is also studied. In Sec. 12.3 we use quantum optics to study the random fluctuations of the magnitude and phase of the electromagnetic field. We provide a brief introduction to coherent, squeezed, and twin-beam light. The interactions of photons with atoms and semiconductors are described in Chapters 13 and 16, respectively. 12.1 THE PHOTON From a quantum perspective, light consists of particles called photons. A photon car- ries electromagnetic energy and momentum, as well as an intrinsic angular momentum (or spin) associated with its polarization properties. It can also carry orbital angular momentum. The photon has zero rest mass and travels at the speed of light in vacuum co; its speed in matter is reduced to c < CO. A photon also has a wavelike character that determines its localization properties in space and time, and the rules by which it interferes and diffracts. The notion of the photon initially grew out of an attempt by Max Planck in 1900 to resolve a long-standing riddle concerning the spectrum of blackbody radiation (this topic is discussed in Chapter 13). He finally achieved this goal by quantizing the allowed energy values of each of the electromagnetic modes in a cavity from which radiation was emanating. In 1905, Albert Einstein extended the notion of quantiza- tion by considered the light itself to be a collection of photons. This enabled him to successfully explain the photoelectric effect (this topic is discussed in Chapter 18). The concept of the photon and the rules of photon optics are introduced by consid- ering light inside an optical resonator (cavity). This is a convenient choice because it restricts the space under consideration to a simple geometry. However, the presence of the resonator turns out not to be an important restriction in the argument; the results can be shown to be independent of its presence. Electromagnetic-Optics Theory of Light in a Resonator In accordance with electromagnetic optics, light inside a lossless resonator of volume V is completely characterized by an electromagnetic field that takes the form of a su- perposition of discrete orthogonal modes of different spatial distributions, different fre- quencies, and different polarizations. The electric field vector is G r, t Re E r, t , where E r, t Aq U q r exp j27rv q t e q . (12.1-1 ) q The qth mode has complex envelope Aq, frequency v q , polarization along the direction of the unit vector e q , and a spatia] distribution characterized by the complex function U q r , which is normalized such that v U q r 2 dr 1. The expansion functions U q r , exp j27rv q t , and e q are not unique; other choices are possible including the use of polychromatic modes.
12.1 THE PHOTON 447 In a cubic resonator of dimension d, a convenient choice of the spatial expansion functions is the set of standing waves 2 3/2 . d SIn 7r 7r . qx d x SIn qy d Y 7r . SIn qz d Z , (12.1-2) U q r where qx, qy, and qz are integers denoted collectively by the index q qx, qy, qz [see Sec. 10.3 and Fig. 12.1-1(a)]. In accordance with (5.4-9), the energy density of mode Eq 1 f A 2 U. r 2 dr 2 q q v 1 A 2 2"f q , (12.1-3) where V is the modal volume. In classical electromagnetic theory, the energy Eq can assume an arbitrary nonnegative value, no matter how small. The total energy is the sum of the energies in all modes. 3 5 6 I 4 I 5 I I hVI 2 I 'Mode 1 4 3 I I hV2 Mode 2 t 3 1 2 f Q) 1 J 0 f 2 h V 3 I )--------- -- / 1 / / 1 /. / / Mode 3 0 /-- 0 0 / ./ (a) (b) Mode 1 Mode 2 Mode 3 Figure 12.1-1 (a) Three modes of different frequencies and directions in a cubic resonator. (b) Allowed energies of three modes of frequencies VI, V2, and V3. The solid circles represent the number of photons in each mode: modes 1, 2, and 3 contain 2, 0, and 3 photons, respectively. Photon-Optics Theory of Light in a Resonator The electromagnetic-optics theory described above is maintained in photon optics, but a restriction is placed on the energy that each mode is permitted to carry. Rather than assuming a continuous range, the modal energy is restricted to discrete values separated from each other by a fixed energy. The energy of a mode is quantized with only integral units of this fixed energy permitted. Each unit of energy is carried by one photon and the mode may carry an arbitrary number of photons. - iII!':- '-f.YIi----"'.'- -- -*..- ,," . $S'i. r.-'!;i:.'&-:otr.;;.:::--.--- ---:-ow ::-":::_:::::..:-:_£1ti -3;;."3:!: W! -::-:-:- : .wt.:-- *" .. ;s '" . Light in a resonator comprises a ser';or.modes each cQntalning an iIitegratnumber I of identical photons. Charactristics of the mQde&\lcb a:S \ts, Jreq!Jency spati] i - . - - di5tribution, direction of propagation, and polarization., are assigned'.to the photon. ; . . : _ n _n ....u n__-:""__-:"'.u___n.'....,_ _ .n nn._.",,......, -.,.. -7' -'- --'-=- -- _n- - _nn- - _nn_ _ _""_ n _ _ . _nn - -""-- 1- .. * ._ .v;:. ..5_ ......... ................VA........'"v..... :.. .. "Ie .......... ... A *"....:*::*"'ffi.:..: . ... .:;....:.:.....:.'S:: :\..-s:;'% .A...-.:..:.:..:: ::'"»xC-:v_" =::: -"......:x-..-...:....i:!..t:: ... .. ....:''"''''- .. ...:=:.-.y...Vt/'"......"I(.....6"JII....-.'v,-"''''=,....,.-)I\.v .. ....-..- ,..... ('.. .... . ..... ,,-"" ...... .....".... ...................?N .....v...................... ....... ...."r.-.. ....,p . ... .... ;"x.(.? A" ...........-.r..,::. ,........_....y.....+-..-:-.+..-y-............................ .........._..... 'V" ..v..."""'"_...............,.. ..............v...... )1"....... r............."""G"')% :.;..--w... ..V""- .. q ............
448 CHAPTER 12 PHOTON OPTICS A. Photon Energy Photon optics provides that the energy of an electromagnetic mode is quantized to discrete levels separated by the energy of a photon (Fig. 12.1-1). The energy of a photon in a mode of frequency v is E hv fiuJ, (12.1-4) Photon Energy where h 6.63 x 10- 34 J-s is Planck's constant and Ii h 27r. Energy may be added to, or taken from this mode only in units of hv. called the zero-point energy. When it carries n photons, therefore, the mode has total energy En n 0, 1, 2, . . . . (12.1-5) In most experiments the zero-point energy is not directly observable because only energy differences [for example, E 2 E 1 in (12.1-5)] are measured. However, the presence of the zero-point energy is manifested in subtle ways when matter is exposed to static fields. It also plays a crucial role in the process of spontaneous emission from an atom, as discussed in Chapter 13. The order of magnitude of photon energy is readily estimated. An infrared photon of wavelength Ao 1 /--lm in free space has a frequency 3 x 10 14 Hz by virtue of the relation Aov Co. Its energy is thus hv 1.99 x 10- 19 J 1.99 X 10- 19 1.6 X 10- 19 eV 1.24 eV (electron volts); this is the same as the kinetic energy imparted to an electron when it accelerates through a potential difference of 1.24 V. The conversion formula between wavelength pm and photon energy (e V) is therefore simply E eV 1.24 . Ao /--lm (12.1-6) Another example is provided by a microwave photon with a wavelength of 1 cm; the photon energy is therefore 10 4 times smaller, namely hv 1.24 x 10- 4 eVe The reciprocal wavelength is frequently used as a unit of energy, often in chemistry. It is specified in em -1 and is determined by expressing the wavelength in cm and simply taking the inverse. Thus, 1 cm- I corresponds to 1.24 10000 eV and 1 eV cor- responds to 8068.1 em-I. Conversions among photon frequency, wavelength, energy, and reciprocal wavelength are illustrated in Fig. 12.1- 2. Because photons of higher frequency carry larger energy, the particle nature of light becomes increasingly important as the frequency of the radiation increases. Wavelike effects such as diffraction and interference become more difficult to discern as the wavelength becomes shorter. X-rays and gamma-rays almost always behave like col- lections of particles, in contrast to radio waves, which almost always behave like waves. The frequency of light in the optical region is such that both particle-like and wavelike behavior occur, thus spurring the need for photon optics. B. Photon Polarization As indicated earlier, light is characterized by a set of modes of different frequen- cies, directions, and polarizations, each occupied by a number of photons. For each monochromatic plane wave traveling in some direction, there are two polarization modes. The polarization of a photon is that of the mode it occupies For example, the photon may be linearly polarized in the x direction, or right circularly polarized.
12.1 THE PHOTON 449 L ,:) . ". - - . ," ...-......-. ::.....- \ 10 nm Wavelength AO 100 nm 10 rn 100 rn Imrn lern Ii] .l i :( "rl( )NI( ts 10 em ---....... PII( ),I'() It s . ' . ." - . . ..... Frequency v ..- ;- :I'" " 1 1111111 . w.: 1 1 I I I I .. I = .:. ." :: I 1 . . . . . . -. -.. I J:PP 1 I 1 . :j: I 100 meV >- ;:;x t1) S '" N 10 - 20 II - - 1 O'meV' .. '.. ..... I - . - l ' THz . . . . . .: :oX '. ... ....... m . ,.,,"' ... I 100 GHz ..1Q:..QHz .' ...:.... ",'. :.:,. * St :l- S!! v 1 GHz 1 PHz j J.l)Q ta;:. 10 THz j eV 100 eV Energy I" J E = hv " 10-1-7 ..- cm- I 10 6 . . : .:. .-, '.,-,.-.'. . :-10 pe V . 10 eV I eV I t J i 1 0- 18 j 10-:l- Q -" 1 J l:;,':rneV n 1& rl'.... 10(J-:.peV 'iI . i . , i 10- 22 $I; - .,. 4) " :,"j 10 " .... '. i { . .......- .. tB 10- 24 10 5 10 4 10 3 10 2 10 1 10- 1 Figure 12.1-2 Relationships among photon wavelength .Ao, frequency v, and energy E (specified in units of eV, J, and reciprocal wavelength 1/.Ao in em-I). A photon of free-space wavelength .Ao 1 pm has frequency v 300 THz and energy E 1.24 eV = 1.99 x 10- 19 J = 10 4 em-I. The domains of photonics and electronics are indicated. Since the polarization modes of free space are degenerate, they are not unique. One may use modes with linear polarization in the x and y directions, linear polarization in two other orthogonal directions, say x' and y', or right- and left-circular polarizations. The choice of a particular set is a matter of convenience. A problem arises when a pho- ton occupying a given mode (say linear polarization in the x direction) is to be observed in a different set of modes (say linear polarization in the x' and y' directions). Since the photon energy cannot be split between the two modes, a probabilistic interpretation is necessary. In classical electromagnetic optics, the state of polarization of a plane wave is described by a Jones vector, whose components Ax, Ay are the components of the complex envelope in the x and y directions (see Sec. 6.1A). The very same wave may also be represented in a different coordinate system x', y' , e.g., one that makes a 45° angle with the initial coordinate system, by a Jones vector with components (12.1-7) Ay , Ay' Ax' as described in Sec. 6.1B. Therefore, a wave that is linearly polarized in the x direction is described by a Jones vector with components Ao,O in the x y coordinate system, where Ao is the c om plex e nvelope. In the x', y' coordinate system, the Jones vector .'-. " ... In photon optics, the state ofpolarization ofa;sing'e photon is described by a Jones .. vector with complex components. ... Ax, Ay., nonnaJized'such that. Ax 2 + Ay2 ",. . :.. 1. l1e coefficients Ax apd A y are interpreted a& complex; probability amplitudes, and their squared magnitudes .Ax. 2 and A y .... 2 represent the probabilities that the. photon is observed in the x and. y linear polarization modes, respectively. :!":x 't"'-.-... - The components Ax, Ay are transformed from one coordinate system to another like ordinary Jones vectors, and the new components represent complex probability amplitudes in the new modes. Thus, a single photon may exist, probabilistically, in more than one mode. This concept is illustrated by the following examples. Linearly Polarized Photons A photon is linearly polarized in the x direction. In terms of the x y linearly polarized modes, the photon is described by a Jones vector with components 1,0. In a set
450 CHAPTER 12 PHOTON OPTICS o f l in ea rly polarized modes in the x' and y' directions at 45°, these components are x X x' \XX x - - - - y' - - ....... .." ....... -' -' -' -' " " / /45 / \ \ "" / 45° , I I , I I I I I I I / I + I I I I I Z I Z , , Z I I I , I , I , I \ I \ I \ I / " / / " , / "- ,/ ./ ,/ " Y , )' ... '" ..... - - - - - - One x-polarized photon One x'-polarized photon (probability ) One y' -polarized photon (probability ) Figure 12.1-3 A photon in the x linear polarization mode is the same as a photon in a superposition of the x' linear polarization mode and the y' linear polarization mode with probability each. EXAMPLE 12.1-1. Transmission of Linearly Polarized Photon Through Polarizer. Consider the transmission of a photon that is linearly polarized in the x direction through a linear polarizer with a transmission axis along the x' direction at an angle e, as illustrated in Fig. 12.1- 4). The polarizer transmits light that is linearly polarized in the x' direction but blocks light in the orthogonal direction y'. To determine the probability that the photon is transmitted through the polarizer, we write the Jones vector of the photon polarization state in the x'-y' coordinate system as (cos e, sin e) [see (6.1-21) and (6.1-22)]. The probability of observing the photon in the mode with x' linear polarization is therefore cos 2 e so that this represents the probability of passage of the photon through the polarizer: p (e) cos 2 e. The probability that the photon is blocked is therefore 1 p (e) sin 2 e. It is known from classical polarization optics that the intensity transmittance of a polarizer in this same configuration is cos 2 e (see Sec. 6.IB). This tells us that the probability of transmission of a single photon is identical to the classical transmittance, namely p (e) 'J (e). x p(e) 1 00 7r 2 7r e Figure 12.1-4 Probability of a linearly polarized photon passing through a polarizer. The axis of the polarizer is at an angle e with respect to the photon polarization. Polarizer Circularly Polarized Photons where the + and signs corresponding to right- and left-handed polarizations, respectively. This description is based on an T y coordinate system, Le., linearly polarized modes. Therefore" the probability of the photon passing through a linear case, regardless of the direction of the linear polarizer. The circularly polarized photon may be regarded as equivalent to the probabilistic superposition of a photon with linear
12.1 THE PHOTON 451 Right- and left-circular polarizations may also be used as modes (as a coordinate system). In this description, a linearly polarized photon may be regarded as a proba- bilistic superposition of right- and left-circularly polarized photons, each with proba- bility !, as illustrated in Fig. 12.1-5. x x x OR z z z One LP photon One RCP photon One LCP photon (probability ) (probability ) Figure 12.1-5 A linearly polarized photon is equivalent to the superposition of a right- and a left-circularly polarized photon, each with probability . c. Photon Position Associated with each photon of frequency v is a wave described by the complex wavefunction U r exp j27rvt of the mode. However, when a photon impinges on a detector of small area dA located normal to the direction of propagation at the position r, its indivisibility causes it to be either wholly detected or not detected at all. The location at which the photon is registered is not precisely determined. It is governed by the optical intensity I r ex U r 2, in accordance with the following probabilistic law: . .:'. :-.: 'The probability- p , ,,: dA Qf opse:rving a'pbeton :at:' a'fPQi!it r :wit h il1 an: incretpental I :: area dA, 'atany,time, is proportional to' the: localqpticul intensity, /" 'P';,' ,OG; : :U:" r:,>"2"" i '.' " .' . '. . , " . "., ".. . -. : : .-: : " " p · 't' :dA rx': I "r,'u': dA. .:-.:. : 'Photon Position .. ...n....,..._. ..,..- ..... ... ...."' _ 'U """""'.'_ ' '- ... ,.---..,---. . :'O««i:.. '--- -'- ......_..n '._ -.. ,.-",., ... .'' ".""--.-' "" '.-"""'" m lli*"'"'" '.' .....,.,... ;:« .-....... . . . . ....,'....'.,.,' "-" .."" 'v-OOOOO; ""'.. .......-:«. ""''t'!:''_=:_- , -.-. :-... . _- _. -_ _ _.. _ __-:-_.. (oo.:!)o i"« -j.. ...:......................':....' ...-.:y.:........._..._.... ... ............... .... .... .. . -x:"-"i...".......... ... .... .;.;.:-:.:: , ""1 it -----_-..:-:.-;--- - --"-.-.ILI'W'..----- -h............................--h.--.--..-_-.........-.....__............._.........-........h................'II'>I".r....................._............ !(QtIC ....._-:..........................""........................_......_....-:"Iol'.AI""..'IoI'Io.":.I'V..,.._. ... .. ............",.... ....r .. -"-'"A.A...J'i.."\IU'!!____""....-......___ * .........-.--.-..: ..._......-,.-y-.--....r..'.. .. .---y--.-....-.-_-... ,--....;.A-...-.....--___'""'--.--.._-.,.-.-."".-.....--...-.......T-"o.'......".'WIIro.'to......-.- .-.- .. - - l".....-...,. ..........V'II....-...v"."'VI/VW"IoI"........,........:'V'V'II....--..":.... .. 4. _ .. rI'If"_. .... .._ ":...-__--..-."'Ir_.... ..;111,.,,-........--"'-"""--.&.....-..... .. The photon is therefore more likely to be found at those locations where the intensity is high. A photon in a mode described by a standing wave with the intensity distribution I x, y, z ex sin 2 7rZ d , where 0 < z < d , for example, is most likely to be detected at z d 2, but will never be detected at z 0 or Z d. In contrast to waves, which are extended in space, and particles, which are localized, optical photons behave as extended and localized entities. This behavior is called wave particle duality. The localized nature of photons becomes evident when they are detected. EXERCISE 12. 1-1 Photons in a Gaussian Beam. (a) Consider a single photon described by a Gaussian beam (the TEMo,o mode of a spherical-mirror resonator; see Sees. 3.1B, 5.4A, and 10.2B). What is the probability of detecting the photon at a point within a circle whose radius is the waist radius of the beam, W o ? Recall that at the waist (z 0), I(p, z 0) ex exp( 2 p 2/W5), where p is the radial coordinate. (b ) If the beam carries a large number n of independent photons, estimate the average number of photons that lie within this circle.
452 CHAPTER 12 PHOTON OPTICS Transmission of a Single Photon Through a Beamsplitter An ideal beamsplitter is an optical device that losslessly splits a beam of light into two beams that emerge at right angles. It is characterized by an intensity transmittance 'I and an intensity reflectance 1 'I. The intensity of the transmitted wave It and the intensity of the reflected wave Ir can be calculated from the intensity of the incident wave I using the electromagnetic relations It 'II and Ir 1 'J I. Because a photon is indivisible, it must choose between the two possible directions permitted by the beamsplitter. A single photon incident on the device follows one of the two possible paths in accordance with the probabilistic photon-position rule (12.1- 8). The probability that the photon is transmitted is proportional to It and is therefore equal to the transmittance 'J It I. The probability that it is reflected is 1 'I Ir I. From the point of view of probability, the problem is identical to that of flipping a biased coin. Figure 12.1-6 illustrates the process. Beamsplitter - - - ....... - - - - - One photon One photon (probability R = 1 -)(1) Figure 12.1-6 Probabilistic reflec- tion or transmission of a photon at a beamsplitter. D. Photon Momentum In classical electromagnetic optics, as discussed in Sec. 5.4A, an electromagnetic plane wave carries a linear momentum density (per unit volume) W c k, where W is the en- ergy density (per unit volume) and k is a unit vector in the direction of the wavevector k. In photon optics, the linear momentum of a photon is p fiw tick is the photon energy. Therefore: E c k where E 'ti ' .. i> ,,;,x.:.:. .'\v ..:....:..; '. If"r TheJinear momentum associated with a,ph9toh in a plane-wave mode of wavevec- k . e)..:. tor':: IS:- . ; Its magnitude is p c: 1ik .., ' w.c ' ... h21r, so that , > . =.='!-! p ' ... lik. ( 12.1-9) _ p o:. - . ",' E -.. p . -. - h '---' . . ....v - - A. (12.1-10) ::. ''''''''''''''''''.A.' ..'OW"..2,.......... G99G"'"..'ii: . v.-....*-_.- * Momentum of a Localized Wave A wave more general than a plane wave, with a complex wavefunction of the form U r exp j27rvt , can be expanded as a sum of plane waves of different wavevec- tors by using the techniques of Fourier optics (see Chapter 4). The component with wavevector k may be written in the form A k exp jk · r exp j27rvt , where A k is its amplitude.
12.1 THE PHOTON 453 .: _ .WQ(-._._.A.----: .x....-.c--:«<-..""'V:..,::.ww,x iK<: -X" ,"<j..:y v,:..w.-y:."...-.:-:-:.'l*-: " &. "! j.. f .: . i The momentum of a photon described 'by an arbitrary complex waefiInction: t > ; , .U.':r':.}{pj21f1lt,O is uncertain. It base. the 'value. I i , "" :.: ; : i-' .", :. " .:': ; :..: P . .. ti ....:.:;.: k; .'.. : ',' ,".: . :'::: , . . .-. _0'- (1'2.1.11) . .. ,-. .... -,"- plane-wave Fourier component of U.,.;;:.' .with wavevectork. ".' .. . :.- ,- , . '"'".'.'" ( :!C 1&1 ' - - -- --- .nn. . :. . r.,... .-. "'". .-".- "._ .'. . ,."..uu.".. ...nn .'" '.., . : .... .... .. . '. .. - ...... ... .. .. :'(?-,;........... .. .. . ______ .. .. .= ..'\0: .. ...... ..... ........ ;,;; -..y .... . ... .... ..:-;; _________ ... .. ... ..... ...... .. .... .........-... ..... .. .. .......... .. ... ............ ...... .. :s: .»Jo . , .J\N' 'VY'I....... YoI'."'V'Io.-v__vv- .."y.,.. ,\:..vv.....ru"""".,\:""... ;,0 ....... "".I . .._ .._ ... :.y",'V.. _ .. .... ...._...r ",...Vtt'ir.. ..V .-':r ...... .....y._......... ...._._._ ""''V''oo"':.r"v ."";'.Y..Y....V v... '\:_Y'..V..Y#'N'_Y'If'I.:.... .._...:I...:.".Y. V_.Y. ..,..y.......u :A....9,A.. If f x, y U x, y, 0 is the complex amplitude at the z 0 plane, the plane-wave Fourier component with wavevector k k x , ky, k z has an amplitude A k F kx 27r, ky 21T , where F v x , v y is the two-dimensional Fourier transform of f x, y (see Chapter 4). Because the functions f x, y and F v x , v y form a Fourier transform pair, their widths are inversely related and satisfy the duration bandwidth relation (see Appendix A, (A.2-6)). The uncertainty relation between the position of the photon and the direction of its momentum is established because the position of the photon at the z 0 plane is probabilistic ally determined by U r 2 f x, Y 2 , and the direction of its momentum is probabilistic ally determined by A k 2 F kx 27r, ky 27r 2. Thus if, at the plane z 0, a x is the positional uncertainty in the x direction, and a() sin- 1 akx k A 27r akx is the angular unc e rtainty about the to axa() > A 47r. A plane-wave photon has a known momentum (fixed direction and magnitude), so that (I() 0, but its position is totally uncertain ax 00; it is equally likely to be detected anywhere in the z 0 plane. When a plane-wave photon passes through an aperture, its position becomes localized at the expense of a spread in the direction of its momentum. The position momentum uncertainty therefore parallels the theory of diffraction described in Chapter 4. At the other extreme from the plane wave is the spherical-wave photon. It is well localized in position (at the center of the wave), but its momentum has a direction that is totally uncertain. Radiation Pressure Because a photon carries momentum, and momentum is conserved, the atom emitting the photon experiences a recoil of magnitude hv c. Moreover, the momentum associ- ated with a photon can be transferred to objects of finite mass, giving rise to a force and causing mechanical motion. As an example, light beams can be used to deflect atomic beams traveling perpendicularly to the photons. The term radiation pressure is often used to describe this phenomenon (pressure force/area). . EXERCISE 12.1-2 Photon-Momentum Recoil. Calculate the recoil velocity imparted to a 198Hg atom that has emitted a photon of energy 4.88 eV. Compare this with the root-mean-square thermal velocity v of the atom at a temperature of T 300 0 K (obtained by setting the average kinetic energy equal to the average thermal energy, mv2 kT, where k 1.38 x 10- 23 J /K is Boltzmann's constant). .
454 CHAPTER 12 PHOTON OPTICS Photon Spin Angular Momentum Photons possess intrinsic spin angular momentum associated with the circularly polar- ized states. The magnitude of the photon spin is quantized to two values, n, (12.1-12) Photon Spin where the plus (minus) signs are associated with right-handed (left-handed) circular polarization, respectively; the spin vector is parallel (antiparallel) to the linear momen- tum vector or the wavevector. Linearly polarized photons have an equal probability of exhibiting parallel and antiparalle] spin. In the same way that photons can transfer linear momentum to an object, circularly polarized photons can exert a torque on an object. For example, a circularly polarized photon will exert a torque on a half-wave plate. Photon Orbital Angular Momentum Aside from the spin angular momentum associated with circular polarization, an elec- tromagnetic wave may carry angular momentum by virtue of its spatial distribution. For example, the Laguerre Gaussian beam described by the wavefunction Ul,m P, cjJ, z in (3.4-1), which has an azimuthal phase dependence exp jlcjJ and a helical wavefront, has an angular momentum (for l i- 0) that is independent of its state of polarization. To distinguish it from the spin angular momentum, this is referred to as orbital angular momentum. A photon in such a spatial mode possesses an orbital angular momentum L In. Another example is provided by a photon in a whispering-gallery mode (WGM) of a cylindrical resonator (Sec. 10.3B). In the context of ray optics, the mode is described by a ray tracing the circular boundary of the resonator. In the context of wave optics, the wavelength satisfies the resonance condition 27ra qA, where a is the radius of the circle and q 1,2, . . .. The photon linear momentum is p nk fi27r A qfi a, and its angular momentum is therefore ap qn. Similarly, a photon in a WGM mode of a microsphere resonator (Sec. 10.4C) of radius a has an angular momentum L £fi, where the integer £ is associated with the resonance wavelength for an optical path tracing a great circle. This number may be regarded as an angular-momentum quantum number similar to that used to describe a hydrogen atom (see Sec. 13.1A). E. Photon Interference Young's double-pinhole interference experiment is generally invoked to demonstrate the wave nature of light (see Exercise 2.5-2). In fact, Young'ts experiment can be carried out even when there is only a single photon in the apparatus at a given time. The outcome of this experiment can be understood in the context of photon optics by using the photon-position rule. The intensity at the observation plane is calculated using electromagnetic (wave) optics and the result is converted to a probability density function that specifies the random position of the detected photon. The interference arises from phase differences associated with the two possible paths. Consider a plane wave illuminating a screen with two pinholes, as shown in Fig. 12.1- 7. On the other side of the screen, this generates two spherical waves that interfere at the observation plane. In the paraboloidal-wave approximation, these give rise to a sinusoidal intensity given by (see Exercise 2.5-2) I x 210 27rx() , (12.1-13)
12.1 THE PHOTON 455 where fo is the intensity of each of the waves at the observation plane, A is the wave- length, and e is the angle subtended by the two pinholes at the observation plane (Fig. 12.1-7). The line that joins the holes defines the x axis. The result in (12.1-13) describes the intensity pattern that is experimentally observed when the incident light is strong. r- A e x , , \ \ { \ z Single photon ..- . l , \ . . , 1 , , \ X , , .- --- , " . . \ , .,' , , , , ., ., ., , I"IIIIIIIIIf- , ., , , , , , I ", , I , I , , ------ - - 2a , "." " .(' " d I , Screen Observation Probability plane Figure 12.1-7 Young's double-pinhole experiment with a single photon. The interference pattern I (x) determines the probability density of detecting the photon at position x. Now if only a single photon is present in the apparatus, the probability of detecting it at position x is proportional to I x , in accordance with (12.1-8). It is most likely to be detected at those values of x for which I x is a maximum. It will never be detected at values for which I x O. If a histogram of the locations of the detected photon is constructed by repeating the experiment many times, as Taylor did in 1909, the classical interference pattern obtained by carrying out the experiment once with a strong beam of light emerges. The interference pattern thus represents the probability distribution of the position at which the photon is observed. The occurrence of the interference results from the extended nature of the photon, which permits it to pass through both holes of the apparatus. This gives it knowledge of the entire geometry of the experiment when it reaches the observation plane, where it is detected as a single entity. If one of the holes were to be covered, the interference pattern would disappear because the photon was forced to pass through the other hole, depriving it of knowledge of the whole apparatus. EXERCISE 12. 1-3 Single Photon in a Mach Zehnder Interferometer. Consider a plane wave of light of wavelength A that is split into two parts at a beamsplitter (see Sec. 12.1C) and recombined in a Mach- Zehnder interferometer, as shown in Fig. 12.1-8 [see also Fig. 2.5-3(a)]. If the wave contains only a single photon, plot the probability of finding it at the detector as a function of d / A (for 0 < d / A < 1), where d is the difference between the two optical paths of the light. Assume that the mirrors and beamsplitters are perfectly flat and lossless, and that the beamsplitters have 'J 1( . Where might the photon be located when the probability of finding it at the detector is not unity?
456 CHAPTER 12 PHOTON OPTICS Photon Detector Figure 12.1-8 Single photon in a Mach-Zehnder interferometer. F. Photon Time The modal expansion provided in (12.1-1) represents monochromatic (single-frequency) modes that are "eternal" harmonic functions of time. A photon in a monochromatic mode is equally likely to be detected at any time. However, as indicated previously, a modal expansion of the radiation inside (or outside) a resonator is not unique. A more general expansion may be made in terms of polychromatic modes (time-localized wavepackets, for example). The probability of detecting the photon described by the complex wavefunction U r, t (see Sec. 2.6A) at any position, in the incremental time interval between t and t + dt, is proportional to I r, t dt ex U r, t 2 dt. 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Y'" _,"; _ -.. _ _._ ...4.1".._........_".._- ___ _.... .. ,.._ r _ '\I("_. _-_-_ .... ... ..,... __ ___.. ______ ______ ___ _ ....--..... --_,..... ___ _ ___ __ _ _ __.._ __ ___ ______ __ _____ ____"'______________ ___"'I0:"Io.__ _..,.__ _..... 0£10. _....................",.. "V"to...--...........u' '_"\A._ ____ "'" ..-v .. _ -..-.. .............-- '...... ._ t" '9'... -........ ;r.. ."'I- ...I_.........:_..-"'( Time Energy Uncertainty The time during which a photon in a monochromatic mode of frequency v may be detected is totally uncertain, whereas the value of its frequency v (and its energy hv) is absolutely certain. On the other hand, a photon in a wavepacket mode with an intensity function I t of duration at must be localized within this time. Bounding the photon time in this way engenders an uncertainty in its frequency (and energy) as a result of the properties of the Fourier transform. The result is a polychromatic photon. Suppressing the r dependence for simplicity, the frequency uncertainty is readily determined by Fourier expanding U t in terms of its harmonic components, CX) Ut v v exp j27rvt dv, (12.1-15) -CX) where V v is the Fourier transform of U t (see Sec. A.I, Appendix A). The width a v of V v 2 represents the spectral width. If a t is the S width of the function U t 2 (namely the power-rms w idth), then at and a v must s at isfy the duration bandwidth for the definitions of at and a v that lead to this uncertainty relation).
12.1 THE PHOTON 457 The energy of the photon fiw cannot then be specified to an accuracy better than (IE naw. It follows that the energy uncertainty of a photon, and the time during which it may be detected, must satisfy n (12.1-16) Time Energy Uncertainty which is known as the time energy uncertainty relation. This relation is analogous to that between position and wavenumber (momentum), which sets a limit on the precision with which the posi tio n and momentum of a photon c an be si multaneously specified. The average energy E of this polychromatic photon is E hv !fj;j. To summarize: a monochromatic photon a v 0 has an eternal duration within which it can be observed at > (X) . In contrast, a photon associated with an optical wavepacket is localized in time and is therefore polychromatic with a corresponding energy uncertainty. Thus, a wavepacket photon can be viewed as a confined traveling packet of energy. EXERCISE 12. 1-4 Single Photon in a Gaussian Wavepacket. Consider a plane-wave wavepacket (see Sec. 2.6A) containing a single photon traveling in the z direction, with complex wavefunction U(r, t) z (12. 1-1 7) Q t - c where aCt) exp t 2 47 2 exp(j27rV o t). ( 12.1-18) (a) Show that the uncertainties in its time and z position are at T and a z cat, respectively. (b) Show that the uncertainties in its energy and momentum satisfy the minimum uncertainty rela- . tlons , aE at n/2 a z a p n/2. (12.1-19) (12.1-20) Equation (12.1-20) is the minimum-uncertainty limit of the Heisenberg position-momentum uncertainty relation [see (A.2-7) in Appendix A]. Electromagnetic radiation may 'be described as a sum of modes, e.g., monochro- . , matic uniform plane waves of tQe fo.rm :-. : -, ;. -> .-. -:. :.: . .- . , : -.- :-: .: , .-. - _-: o': .- ,: .:. ==: :, .- -: -: : Summary . ::: : .:- mfiI -- [ .. tWj'. :; jf E r, t uuuu Aq €XP . jkq ..r- c exp .j21tl/qt., €q. (1-2.1 21) q -.",- -: .;0 :-.' .. :.'P:.."$:' _ o' . ..:...:,;..-: ,,'\, . «...:-r..-:Jo:-:-. ....:-0......,.. -:---......",... ,on ::.:..:" .::::::
458 CHAPTER 12 PHOTON OPTICS - Each plane wave ,has: fwo, orthgon.al polarization states (e..g.., vertical/horizontaJ : linearly PQlacieQ, {ightllft'irulaty p()larized) ,xepresented by the vectors e q .. ,. When the .energy of a mode ;is measured, the result is an integer (in general, , random) number of energy quanta: (photons). Each of the photons associated with ,. the mode q has the fQIIDwjrtg pFQpeFtj$: <'. Energy E . ' " hV q . MomentUrhp ..; .. ..hkq . Spin S .'. .. " ::I:h, if it is cireul(irly P91lJ:i.eQ . The photon is equy likly to berfo,:!ndanywhere in space, and at any time, since the wavefunction of the mode isa monochromatic plane wave ,"The choice of modes;' is hot Wlique..- A modal expansion in terms of non- ; m:onochromatic (quasi...m()qQbrQma:tic), UQQ-pla{le waves, is also possible: -.: :. .:. .:. -': E '. ,- r . ._ - - -- - -- . -... . - . .r ,: t «. <0 ." ..' "A q U q " r te q -. - - -- , . . , -q (12.1-22) -, - .- : The photons associated, with the,.c.mode q then have the following properties: ': . Photon position and time" are: t.pvemed by the complex wavefunction U q . r, t . . .... The PfpbaQ\Iity oK q(dug, 'pb9to in the in«remental time between t and t + . dt, in an incremental area.dA atposition r., is proportional to U q r, t . 2 dA dt.. - . . If U q r, t ,has a finite- time <uration 'at, i.e., if the photon is localized in time, : then the photon energy hJ/q.has'an uncertainty hall > h 41rO"t. , . If Uq'r,. t has {l.:fi.piw $pa;tiJxtt;nt !A tbe transverse z . . 0 plane, i.e., ... if the photon is localized in ihe x direction, for example, then the direction . of photon momentum is uncertain.. The spread in photon momentum can be ..... determined by' analyzing [fq' p? t.t ltSa StUn of plane waves, the wave with , wavevectorkcortesPQnding to ,photon momentum fik. Spatial localization of , the photon in the transverse plane results in an increase in uncertainty of the .' photon-momentum directiort ,---, 12.2 PHOTON STREAMS In Sec. 12.1 we concentrated on the properties and behavior of single photons. We now consider the properties of collections of photons. As a result of the processes by which photons are created (e.g., emissions from atoms; see Chapter 13), the number of photons occupying any mode is generally random. The probability distribution obeyed by the photon number is governed by the quantum state of the light (see Sec. 12.3). Photon streams often contain numerous propagating modes, each carrying a random number of photons. If an experiment is carried out in which a weak stream of photons falls on a light- sensitive surface, the photons are registered (detected) at random localized instants of time and at random points in space, in accordance with (I 2.1-14). This space time process can be discerned by viewing a barely illuminated object with the naked dark- adapted eye. The temporal pattern of such photon registrations can be highlighted by examining the temporal and spatial behavior separately. Consider the use of a detector with good
12.2 PHOTON STREAMS 459 temporal resolution that integrates light over a finite area A, as illustrated in Fig. 12.2-1. Equation (12.1-14) tells us that the probability of detecting a photon in the incremental time interval between t and t + dt is then proportional to the optical power at time t: P t A I r, t dA. The photons are registered at random times. _.. ..'1......-.......".".,,,....1'.--... -. ...........,..,..".,.... . .. _7 -........,._..I'.-...._._--....-#'.AJ'!.._..-_..A1'._....-\ .- Figure 12.2-1 Photon registrations at random localized instants of time for a detector that integrates light over an area A. Light t Detector m. n -- ... _ _' 'CI(lJ1)JJII:Il!:Il't'I.I:I.IHI\)J_' _ : . i!-l!_: :::;J)":. . ;:.. "" Oscilloscope On the other hand, the spatial pattern of photon registrations is readily manifested by making use of a detector with good spatial resolution that integrates over a fixed exposure time T (e.g., photographic film). In accordance with (12.1-14), the prob- ability of observing a photon in an incremental area dA surrounding the point r is grainy photographic image of Max Planck provided in Fig. 12.2-2. This image was obtained by rephotographing, under very low light conditions, the picture of Max Planck presented on page 444. Each white dot in the photograph represents a random photon registration; the density of these registrations follows the local intensity. .. 1 - - ....,... - . . ... ' . . ..... . . .. ,..; J 0( > ,\-... ,. l . , . ..... ).<t _ . ,10 - .. -01 . .. fl . . Figure 12.2-2 The random photon registrations have a spatial density that follows the local optical intensity. This image of Max Planck under illumi- nation with a sparse stream of photons should be compared with the photograph on page 444 taken with high-intensity light. A. Mean Photon Flux We begin by introducing a number of definitions that relate the mean flow of photons to classical electromagnetic intensity, power, and energy. These definitions are inspired by (12.1-14), which governs the position and time at which a single photon is observed. We then discuss randomness in the photon flux and the photon-number statistics for different sources of light. Finally, we consider the random partitioning of a photon stream by a beamsplitter or detector.
460 CHAPTER 12 PHOTON OPTICS Mean Photon-Flux Density Monochromatic light of frequency v and classical intensity I r (watts cm 2 ) carries a mean photon-flux density q;r I r hv . (12.2-1) Mean Photon-Flux Density Since each photon carries energy hv, this equation provides a straightforward con- version from a classical measure (units of energy s-crn 2 ) into a quantum me a sure (units of photons s-crn 2 ). For quasi-monochromatic light of central frequency v, all photons have approximately the same energy hv, so that the mean photon-flux density is approximately q; r Ir . hv (12.2-2) Typical values of q; r for some common sources of light are provided in Table 12.2- 1. It is clear from these numbers that trillions of photons rain down on each square centimeter of us each second. Table 12.2-1 Mean photon-flux density for various sources of light. Source Mean Photon-Flux Density (photons/s-cm 2 ) 10 6 10 8 10 10 10 12 10 14 10 22 Starlight Moonlight Twilight Indoor light Sunlight Laser light a a A lO-mW He-Ne laser beam at Ao 633 nm focused to a 20-j.Lm-diameter spot. Mean Photon Flux The mean photon flux 'l> (units of photons s) is obtained by integrating the mean photon-flux density over a specified area, 'l> cP r dA A P hv ' (12.2-3) Mean Photon Flux where hv is again the average energy of a photon, and the optical power (watts) is p I r dA. A (12.2-4 ) As an example, 1 nW of optical power, at a wavelength Ao 0.2 /-lrn, delivers to an object an average photon flux 'l> 10 9 photons per second. Roughly speaking, one
12.2 PHOTON STREAMS 461 photon therefore strikes the object every nanosecond, i.e., 1 nW at Ao 0.2/Lill 1 photon ns. (12.2-5) A photon of wavelength Ao 1 /-lill carries one- fifth of the energy, in which case 1 n W corresponds to an average of 5 photons ns. Mean Number of Photons The mean number of photons n detected in the area A and in the time interval T is obtained by multiplying the mean photon flux 'l> in (12.2-3) by the time duration, whereupon n 'l>T E h v ' (12.2-6) Mean Photon Number where E PT is the optical energy Uoules). To summarize: The relations between the classical and quantum measures are: Classical Optical intensity I r Optical power P Optical energy E Quantum Photon-flux density cP r Photon flux 'l> Photon number n I r hv P hv E hv Spectral Densities of Photon Flux For polychromatic light of nonnegligible bandwidth, it is useful to define spectral densities of the classical intensity, power, and energy, and their quantum counterparts: the spectral photon-flux density, spectral photon flux, and spectral photon number: Classical Quantum W cm 2 -Hz W Hz J Hz cPv q>v nv Iv hv Pv hv Ev hv (photons s-cm 2 -Hz) (photons s- Hz) (photons Hz) Iv Pv Ev For example, P v dv represents the optical power in the frequency range v to v + dv whereas 'l>v dv indicates the flux of photons whose frequencies lie between v and v + dv. Time- Varying Light If the light intensity is time varying, the photon- flux density in (12.2-1) is a function of time, I r,t hv cP r, t . (12.2-7) Mean Photon-Flux Density
462 CHAPTER 12 PHOTON OPTICS The photon flux and optical power are then also functions of time, cI> t cp r, t dA A Pt h v ' ( 12.2-8) Mean Photon Flux where Pt I r, t dA. A ( 12.2-9) Consequently, the mean number of photons registered in a time interval between t 0 and t T, which is obtained by integrating the photon flux, also varies with time: n T cI> t dt E h v ' (12.2-10) Mean Photon Number o where E T P t dt T I r, t dA dt o A (12.2-11 ) o is the optical energy (intensity integrated over time and area). B. Randomness of Photon Flow When the classical intensity I r, t is constant, the time of arrival and position of regis- tration of a single photon is governed by (12.1-14), which provides that the probability density of detecting that photon at the space time point r, t is proportional to I r, t . The classical electromagnetic intensity I r, t governs the behavior of photon streams as well as single photons, but the interpretation ascribed to I r, t differs: For photon streams, the classical intensity I r, t deterlnines the lnean photon- flux density cp r, t . The properties of the light source deternline the fluctuations in cp r, t . Consider a detector that integrates over space, such as that illustrated in Fig. 12.2-1. If the intensity I is cons ta nt in time, then so too is the power P. T he mean photon-flux density is then cp I hv and the mean photon flux is cI> P hv. However, the times at which the photons are detected are random, their statistical behavior determined by the source, as illustrated in Fig. 12.2-3(a). For example, at AD 1 /-lID, an optical power PIn W carries an average of <I> 5 photons/ns, or 0.005 photons every picosecond. Of course, only integral numbers of photons may be detected. An average of 0.005 photons/ps means that if 10 5 time intervals are examined, each of duration TIps, most will be empty (no photons will be registered), about 500 intervals will contain one photon, and very few intervals will contain two or more photons. If the optical power P t does vary with time, the mean density of photon detections follows the func ti on P t , as schematically illustrated in Fig. 12.2-3(b). The mean flux <I> t P t hv, which accommodates the fact that there are more photon arrivals when the power is large than when it is small. This variation is in addition to the fluctuations in photon occurrence times associated with the source. The image of Max Planck in Fig. 12.2-2 illustrates the same behavior in the spatial domain. The locations of the detected photons generally follow the classical intensity
12.2 PHOTON STREAMS 463 P(t) P(t) . I . 1 $.-4 t1) (]) .;j .;: o o o o t t Photon Photon arrivals arrivals t t (a) (b) Figure 12.2-3 (a) Constant optical power and the corresponding random photon arrival times. (b) Time-varying optical power and the corresponding random photon arrival times. distribution, with a high density of photons where the intensity is large and a low pho- ton density where the intensity is small. But there is considerable graininess (noise) in the image, corresponding to the fluctuations in photon occurrence positions associated with the source of illumination. These fluctuations are most discernible when the mean photon-flux density is small, as in the case of Fig. 12.2-2. When the mean photon-flux density becomes large everywhere in the image, as in the picture of Max Planck on page 444, the graininess disappears and the classical intensity distribution is recovered. c. Photon-Number Statistics An understanding of photon-number statistics is important for applications such as reducing noise in weak images and optimizing optical information transmission. In an optical fiber communications system, for example, information is carried in the form of pulses of light (see Chapter 24). Only the mean number of photons per pulse is controlled at the source. The actual number of photons emitted is unpredictable and varies from pulse to pulse, resulting in errors in the transmission of information. The statistical distribution of the number of photons depends on the nature of the light source and must generally be treated by use of the quantum theory of light, as described briefly in Sec. 12.3. However, under certain conditions, the arrival of photons may be regarded as the independent occurrences of a sequence of random events at a rate equal to the photon flux, which is proportional to the optical power. The optical power may be deterministic (as in coherent light) or a time-varying random process (as in partially coherent light). For partially coherent light (see Chapter 11), the power fluctuations are correlated, so that the arrival of photons no longer forms a sequence of independent events; the photon statistics are then significantly altered. Coherent Light Coherent l ig ht has a constant optical power P. The corresponding mean photon flux tI> P hv (photons/s) is also constant, but the actual times of registration of the photons are random, as shown in Fig. 12.2-3(a) and Fig. 12.2-4. Given a time interval of duration T, let th e number of detec te d photons be n. We already know that the mean value of n is n tI> T PT hv. We seek an expression for the probability distribution p n , i.e., the probability p 0 of detecting no photons, the probability p 1 of detecting one photon, and so on. An expression for the probability distribution p n can be derived under the as- sumption that the registrations of photons are statistically independent. The result is the Poisson distribution n n I n ' exp I. ) n! In'; p \ ) , n 0,1,2,.... (12.2-12) Poisson Distribution
464 CHAPTER 12 PHOTON OPTICS n=9 n=8 n=7 n = ] 1 T T T T t Figure 12.2-4 Random arrival of photons in a light beam of constant power P within during intervals of duration T. Although the optical power is constant, the number n of photons arriving within each interval is random. This result, known as the Poisson distribution, is displayed on a semilogarithmic plot in Fig. 12.2-5 for several values of the mean n . The curves become progressively broader as n increases. 1 p(n) 10-] 10- 2 10- 3 o 5 10 15 20 n Figure 12.2-5 Poisson distribution p (n) of the photon number n. D Derivation of the Poisson Distribution. Divide the time interval T into a large number N of subintervals of sufficiently small duration T j N each, such that each interval carries one photon with probability p == n j N and no photons with probability 1 - p. The probability of finding n independent photons in the N intervals, like the flips of a biased coin, then follows the binomial distribution. N! n N-n p(n)== n!(N-n)! P (l-p) , TIN -1 r- N! n!(N-n)! ( y (1- r-n o I. T N ) I t In the limit as N -7 00, N!j(N - n)! N n -7 1, and [1- ( n jn)]N-n --t exp(- n ), which yields (12.2-12). . Mean and Variance Two important parameters characterize any random number n: its mean value, ex) n == Lnp(n), n=O (12.2-13) and its variance ex) a; == L(n - n )2p (n), n=O (12.2-14)
12.2 PHOTON STREAMS 465 which is the average of the squared deviation from the mean. The standard deviation an (the square root o f the variance) is a measure of the width of the distribution. The quantities p n , n, and an are collectively called the photon-number statistics. Although the function p n contains more information than just its mean and variance, these are useful measures. It is not difficult to show [by use of (1 2. 2-12) in (12.2-13) and (12.2-14)] that the mean of the Poisson distribution is indeed n and its variance is equal to its mean: "..2 n un · (12.2-15) Variance Poisson Distribution For example, when n 100, an 10; thus, the presence of 100 photons is accompa- nied by an inaccuracy of about ::l:10 photons. The Poisson photon-number distribution is applicable for an ideal laser emitting a beam of monochromatic coherent light in a single mode (see Chapter 15). This distribution corresponds to a quantum state of light known as the coherent state (see Sec. 12.3A). This distribution also provides an excellent approximation for the photon statistics of many other light sources, including multimode thermal light. Signal-to..Noise Ratio The randomness of the number of photons constitutes a fundamental source of noise that we have to cont en d with when using light to transmit a signal. Representing the mean of the signal as n, and its noise by the root mean square value an, a useful measure of the performance of light as an information-carrying medium is the signal-to-noise ratio (SNR). The SNR of the random number n is defined as n 2 2 · an (12.2-16) SNR (mean)2 . varIance For the Poisson distribution SNR n, (12.2-17) Signal-to-Noise Ratio Poisson Distribution so that the signal-to-noise ratio increases linearly with the mean number of photon counts. Although the SNR is a useful measure of the randomness of a signal, in some applications it is necessary to know the probability distrib u tion itself. For example, if one communicates by sending a mean number of photons n 20, according to (12.2- 12) the probability that no photons are received is p 0 2 x 10- 9 . This represents a probability of error in the transmission of information. This topic is addressed in Chapter 24. Thermal Light When the photon arrival times are not independent, as is the case for thermal light, the photon number statistics can obey distributions other than the Poisson. Consider an optical resonator whose walls are maintained at temperature T (OK), so that photons are emitted into the modes of the resonator. In accordance with the laws of statistical mechanics, under conditions of thermal equilibrium the probability of occupancy of energy level En in a mode satisfies the Boltzmann probability distribution
466 CHAPTER 12 PHOTON OPTICS P En ex: exp En kT , (12.2-18) Boltzmann Distribution where k is Boltzmann's constant k 1.38 x 10- 23 J OK). The origin of this distribution is discussed in more detail in Sec. 13.2. In thermal equilibrium, the energy associated with each mode is random. Higher energies are relatively less probable than lower energies, as provided by this sim- ple exponential law with parameter kT. The Boltzmann distribution is sketched in Fig. 12.2-6 with temperature as a parameter. The smaller the value of kT, the less likely it is that higher energies will be observed. At room temperature T 300° K , we have kT 0.026 e V, which is equivalent to 208 em -1. If we consider a collection . . y- T 2 > T] n En . . . . 2 E 2 TI 1 E 1 / Figure 12.2-6 Boltzmann prob- ability distribution P( En) versus energy En for two values of the temperature T. o Eo PeEn) . of photons in a resonator mode of frequency v as a gas in thermal equilibrium at temperature T, it follows from the Boltz m ann distribution (12.2-18) and the photon- n photons is p n ex: exp exp hv kT n nhv kT , n 0,1,2,.... (12.2-19) Using the condition that the probability distribution must sum to unity, i.e., c: 0 p n 1, the norm al ization constant is determined to be 1 exp hv kT . The zero-point accordance with the discussion in Sec. 12.1A. The probability distribution is most simply written in terms of its mean n as p n 1 n+1 n n+1 n , (12.2-20) Bose Einstein Distribution where n 1 exp hv kT 1 ' (12.2-21 ) as determined from (12.2-13). In the parlance of probability theory, this distribution is called the geometric distribution since p n is a geometrically decreasing function
12.2 PHOTON STREAMS 467 of n. In physics it is referred to as the Bose Einstein probability distribution. Equa- tion (12.2- 21) accords with the mean calculated for a collection of photons interacting with atoms in thermal equilibrium, as provided in (13.4-7). The Bose Einstein distribution is displayed in Fig. 12.2-7 for several values of n [or, equivalently, for several values of the temperature T via (12.2-21)]. Its exponential character is evidenced by the straight-line behavior in this semilogarithmic plot. Com- paring Figs. ] 2.2-7 with 12.2-5 demonstrates that the photon-number distribution for thennallight decreases monotonically and is far broader than that for coherent light. 1 10-] - n = 10 pen) 10- 2 5 1 10- 3 o 5 10 15 n 20 Figure 12.2-7 Bose-Einstein distribution p (n) of the photon number n. Using (12.2-14), the photon-number variance turns out to be a; n+ n 2 . (12.2-22) Variance Bose Einstein Distribution C omparing this expression to the variance for the Poisson distribution, which is simply n according to (12.2-17), we see that thermal light has a larger variance. This corre- sponds to more uncertainty and a greater range of fluctuations of the photon number. The signal-to-noise ratio of the Bose Ein st e in dis tribution is therefore n+l . (12.2-23) It is always smaller than unity no matter how large the optical power. The amplitude and phase of thermal light behave like random quantities, as described in Chapter 11. This randomness results in a broadening of the photon-number distribution. Indeed, this form of light is too noisy to be used in high-data-rate infonnation transmission. I EXERCISE 12.2-1 A verage Energy in a Resonator Mode. Show that the average energy of a resonator mode of frequency v, under conditions of thermal equilibrium at temperature T, is given by E exp(hv / kT) . (12.2-24) 1 Sketch the dependence of E on v for several values of kT jh. Us e a Taylor-series expansion of the denominator to obtain a simplified approximate expression for E in the limit hv j kT « 1. Explain the result on a physical basis.
468 CHAPTER 12 PHOTON OPTICS * Other Sources of Light As indicated earlier, for certain light sources the photon arrivals can be regarded as a sequence of independent events, arriving at a rate proportional to the optical power. For coherent light, the optical power P is deterministic, and the photon number obeys the Poisson distribution p n wne- w n!, where 1 I r, t dAdt. hv 0 A T hv 0 (12.2-25) w The quantity w, which is the integrated photon flux (nor m alized integrated optical power), is a constant representing the mean photon number n. For light sources in which the intensity I r, t itself fluctuates randomly in time and or space, the optical power P t also undergoes random fluctuations [see Fig. 12.2- 3(b)], and its integral w is thus also random. As a result, not only is the photon number random but so is its mean w. Because of this added source of randomness, the photon- number statistics for partially coherent light differ from the Poisson distribution. If the fluctuations in the mean photon number ware described by a probability density function p w , the unconditional probability distribution for partially coherent light is obtained by averaging the conditional Poisson distribution p n w wne- w n! over all permitted values of w, weighted by its probability density p w . The resultant photon-number distribution is then given by p n 00 wne- w p w dw, n! (12.2-26) Mandel's Formula o which is known as Mandel's formula. Equation (12.2-26) is also referred to as the dou- bly stochastic Poisson counting distribution because of the two sources of randomness that contribute to it: the photons themselves (which locally behave in Poisson fashion) and the intensity fluctuations arising from the noncoherent nature of the light (which must be specified). Note that this theory of photon statistics is applicable only to a certain class of light called classical light; a more general theory based on a quantum description of the state of light is described briefly in Sec. 12.3. The photon-number mean and variance for partially coherent light, which can be derived using (12.2-13) and (12.2-14) in conjunction with (12.2-26), are (12.2-27) n w and "..2 n + 2 Un o-w , (12.2-28) respectively. Here 0-;" signifies the variance of w. Note that the variance of the photon number is the sum of two contributions: the first term is the basic contribution of the Poisson distribution whereas the second is an additional contribution arising from the classical fluctuations of the optical power. In one important example, the fluctuations of the normalized integrated optical power w obey the exponential probability density function 1 exp w , w > 0 pw (12.2-29) w 0, w w < o.
12.2 PHOTON STREAMS 469 This distribution is appropriate for quasi-monochromatic spatially coherent light, when the real and imaginary components of the complex amplitude of the field are indepen- dent and have Gaussian probability distributions. It is applicable when the spectral width is sufficiently small so that the coherence time Tc is much greater than the counting time T, and the coherence area Ac is much larger than the area of the detector A (see Chapter 11). The photon-number distribution p n that corresponds to (12.2-29) can be obtained by substituting it into (12.2-26) and evaluating the integral. The result turns out to be the Bose Einstein distribution given in (12.2-20). The Gaussian-distributed optical field therefore has photon statistics identical to those of single-mode thermal light. When the area A and the time T are not small, the statistics are modified; they describe multimode thermal light (see Probs. 12.2-6 12.2-8). D. Random Partitioning of Photon Streams A photon stream is said to be partitioned when it is subjected to the removal of some of its photons. The photons removed may be either diverted or destroyed. The process is called random partitioning when they are diverted and random selection when they are destroyed. There are numerous ways in which this can occur. Perhaps the simplest example of random partitioning is provided by an ideallossless beamsplitter. Photons are randomly selected to join either of the two emerging streams (see Fig. 12.2-8). An example of random selection is provided by the action of an optical absorption filter on a light beam. Photons are randomly selected either to pass through the filter or to be destroyed (and converted into heat). Lossless beamsplitter . I . I I . I I I . I I I I I I . I I . I I I I I I . I I I . . I I I I I I I I I I I I I I I . I I I I I I . I Figure 12.2-8 Random . . I I . I I . . partltlon- I . I I I . I I I . I I I I I I I I . I I ing of photons by a beamsplitter. . . We restrict our treatment to situations in which the possibility of each photon being removed behaves in accordance with an independent random (Bernoulli) trial. In terms of the beamsplitter, this is satisfied if a photon stream impinges on only one of the input ports (Fig. 12.2-8). This eliminates the possibility of interlerence, which in general invalidates the independent-trial assumption. Although the results derived below are couched in terms of random partitioning, they apply equally well to random selection. Consider a lossless beamsplitter with transmittance 'I and reflectance J( 1 T. In electromagnetic optics, the intensity of the transmitted wave It is related to the intensity of the incident wave I by It T I. The result of a single photon impinging on a beamsplitter was examined in Sec. 12.1 C; it was shown that the probability of transmission is equal to the transmittance 'I. We now proceed to calculate the outcome w hen a photon stream of mean flux <P is incident, so that a mean number of photons n <P T strikes the beamsplitter in the time interval T. In accordance with (12.2-6), the mean number of photons in a beam is proportional to the optical energy. The mean number o f transmitted and reflected photons in this time must therefore be Tn and 1 Tn, respectively. We now consider a more general question: What happens to the photon-number statistics p n of the photon stream on partitioning by a beamsplitter? A single photon falling on the beamsplitter is transmitted with probability T and reflected with probability 1 'I (see Fig. 12.1-6). If the incident beam contains precisely
470 CHAPTER 12 PHOTON OPTICS n photons, the probability p m that m photons are transmitted is the same as that of flipping a coin n times, where the probability of achieving a head (being transmitted) is 'I. From elementary probability theory we know that the outcome is the binomial distribution pm n 'Jm 1 'I n-m m ' m 0, 1, . . . , n , (12.2-30) where; n! m! n m!. The mean number of transmitted photons is easily shown to be m 'In . (12.2-31) The variance for the binomial distribution is given by a 2 'I 1 'I n m 1 'I m. (12.2-32) Because of the symmetry of the problem, the results for the refl ec ted beam are obtained immediately. As the average nu mber o f t ran smi tt ed photons m increases, the signal- intensities, the photons will be partitioned between the two streams in good accord with 'I and 1 'I, indicating that the laws of classical optics are recovered. The expressions provided above are useful because they permit us to calculate the effect of a beamsplitter on photons obeying arbitrary photon-number statistics. The solution is obtained by recognizing that in these cases the number of photons n at the input to the beamsplitter is random rather than fixed. Let the probability that there are exactly n photons present be Po n . If we treat the photons as independent events, the photon-number probability distribution in the transmitted stream will be a weighted sum of binomial distributions, with n taking on the random value n. The weighting is in accordance with the probability that n photons were present. The probability of finding m photons transmitted through the beamsplitter, when the input photon- number distribution is P 0 n , is therefore given by p m n p m n Po n , where P m n ; 'Jm 1 'I n-m is the binomial distribution. Explicitly, then, 00 n m pm 'Jm 1 'J n-mpo n . (12.2-33) Photon-Number Statistics Under Random Partitioning n m When Po n is the Poisson distribution (coherent light) or the Bose Einstein dis- tribution (single-mode thermal light), the results turn out to be quite simple: p m has exactly the same form for the photon-number distribution as Po n . Both of these distributions retain their form under random partitioning. Thus, single-mode laser light transmitted through a beamsplitter remains Poisson and thermal light remains Bose Einstein, but of course the photon-number mean is reduced by the factor J. Light with a deterministic number of photons (see Sec. 12.3B), on the other hand, does not retain its form under random partitioning, and this unfortunate property is responsible for its lack of robustness. The signal-to-noise ratio of m is easily calculated for photon streams that have undergone partitioning or selection. For coherent light and single-mode thermal light, the results are, respectively,
12.3 QUANTUM STATES OF LIGHT 471 SNR T n Tn Tn+l coherent light (12.2-34) thermal light. (12.2-35) Since T < 1 it is clear that random partitioning decreases the signal-to-noise ratio. Another way of stating this is that random partitioning introduces noise. The effect is most severe for deterministic photon-number light. The same results are also applicable to the detection of photons. If every photon has an independent chance of being detected, then out of n incident photons, m photons would be detected where P m is related to Po n by (12.2-33). We will find this result useful in the theory of photon detection (Chapter 18). *12.3 QUANTUM STATES OF LIGHT The number of photons in an electromagnetic mode is generally a random quantity. In this section it will be shown that in the context of quantum optics the electric field itself is also generally random. Consider a monochromatic plane-wave electromagnetic mode in a volume V, described by the electric field £ r, t Re E r, t , where E r, t A exp jk · r exp j27rvt e. (12.3-1) According to classical electromagnetic optics, as provided in (12.1-3), the energy of hv a 2, thereby allowing a 2 to be interpreted as the energy of the mode in units of photon number. The electric field may then be written as E r, t 2hv jk · r exp j27rvt e, (12.3-2) . where the complex variable a determines the complex amplitude of the field. In classical electromagnetic optics, a exp j27rvt is a rotating phasor whose pro- jection on the real axis determines the sinusoidal electric field (see Fig. 12.3-1). The real and imaginary parts of a x + j p, which are x Re a and p 1m a respectively, are termed the quadrature components of the phasor a because they are a quarter cycle 90° out of phase with each other. They determine the amplitude and phase of the sine wave that represents the temporal variation of the electric field. The rotating phasor a exp j27rvt also describes the motion of a harmonic oscillator; the real component x is proportional to position and the imaginary component p to momentum. From a mathematical point of view, then, a classical monochromatic mode of the electromagnetic field and a classical harmonic oscillator behave identically. A parallel argument can be constructed to show that a quantum monochromatic electromagnetic mode and a one-dimensional quantum-mechanical harmonic oscillator have identical behavior. To facilitate the comparison, we review the quantum theory of a simple harmonic oscillator before proceeding. Quantum Theory of the Harmonic Oscillator where is the elastic constant, represents a harmonic oscillator of total energy
4 72 CHAPTER 12 PHOTON OPTICS T(t) w ---....----. .- --. . ." ........ .. ,., \ .. .... \ .. ..' \ .-' \ :- \ : \ . : \ : \ . I I I I I I I . I . I . . I . . . , .. . . . . .. I .. I . I .. .. ... ........ ...... - . --. ....- -....-.--..- -----_-.-._----- ...... .. .. .. ... .. .. .. .. I . '1... t == 0 lal . . , , . . I . I I . I I I I I I I I . I I I I . . . . " . . " " . " .. .. .. .. .. .. . .. .. .......... o t Figure 12.3-1 The real and imaginary parts of the variable aexp(j27rvt), which govern the complex amplitude of a classical electromagnetic field of frequency v. The time dynamics are identical to those of a classical harmonic oscillator with angular frequency w 27rv. In accordance with quantum mechanics, its behavior in a stationary state is de- scribed by a complex wavefunction 'l/J x satiscying the time-independent Schr6dinger equation fi2 d 2 'l/J 2m x E'l/J x , (12.3-3) where E is the energy of th e particle. The solutions of the Schr6dinger equation for the En n + ! hv 2 ' n 0,1,2, . .. . (12.3-4) Adjacent energy levels are separated by a quantum of energy hv fiw. The corre- sponding wavefunctions 'l/Jn x are normalized Hermite Gaussian functions, 2 n n! 7fn w n x exp wx 2 W 1/4 lH[n 'ljJ n X 1 2n , (12.3-5) where lH[n X is the Hermite polynomial of order n [see (3.3-6) (3.3-8) and (3.3-11)]. An arbitrary wavefunction 'l/J x may be expanded in terms of the orthonormal eigenfunctions 'ljJn x as the superposition 'l/J x n en 'l/Jn x . Given the wave- function 'l/J x , which governs the state of the system, the behavior of the particle is determined as follows: . The probability p n that the harmonic oscillator carries n quanta of energy is given by the coefficient c n 2. . The probability density of finding the particle at the position x is given by 'ljJ x 2 . . The probability density that the momentum of the particle is p is given by cp p 2 , where cp p is proportional to the inverse Fourier transform of 'l/J x evaluated at the (spatial) frequency p h, 1 h -00 00 x exp dx. (12.3-6) cpp As shown in Sec. A.2 of Appendix A, the Fourier transform relation between 'l/J x and cjJ p indicates that there is an uncertainty relation between the power- rms widths of x and p h given by
12.3 QUANTUM STATES OF LIGHT 4 73 ax a p 1 > h 47r n or (12.3-7) . This relation is the well-known Heisenberg position momentum uncertainty relation. Analogy Between an Optical Mode and a Harmonic Oscillator The energy of an electromagnetic mode is hv a 2 hv x 2 + p2 . The analogy with a x 1 2hw w x 1 2hw p. (12.3-8) and p as the energy of a harmonic oscillator. 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' x ':', ;: 1 . e ,. .-.. x , .. .. .:. . 0 :: W ".'}' .;: ,..'" x , "'.'-:. -' ::;.:::..:;:::Sx.::. .-.,;. :.:.::.:.: ;.4j t t:".n. ,».;,:;: $x-,:_:"::".,, ,:,:.:::::: -,: .. .:. n . .-' . -'.. . ..r. . . --'.' -. - -- n " .- . n . -' . - -...;,...:::-::.::"..:y-:".. . ::=z:::.......:" ....... ...i:r::-:-:", "'!."..vID* .-.:..;..-.":.:.:-;.V«..... ..'-Y: :t:ili. =::::::=: :z. :'!o .: ':- -., :. :', ::'. , . it -:..". .. . ". . ..: - .': . . n .. ,.' .:: " .' ., '. ,., '. .::: .. :.::..' -::::'%:':-:-y..:?':;::: B:;;::: ,. :.:;:: :'::::«fu *' ....-?- :,;:f) -. :. ..:.; - _ -. _ .- : . .. '.-. ..- . ." .r . :..: . mxx9:.:-:-- j:>: m...:X._.. : =*w. --::-..: :«..» -.: '. . -. _ __ -: ..",- :.-'."'.", .""..." .... ...".."._.,,,"".,r-;';;' ...".....;:- '-"" J!:.", .. .: . - ..... _ r=i-..:" ::..-=::=-;.'.: :_y :=-:....,, =::s.x... x.».. .:.::,, m x:..u:....:-;,:.= .: , P Y Y g D ;= "" -- .. -: . -'.' 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"= . ,' . : . -:' . :: . .' . =: . ,>:-0:( . _ . , . _ . :.. "._.,,-..... :."v;: :::::: "': ........_ ...=::;;;;,:rz...._. _.. "_"._K--::-5-j.,,. .. ._.. :_ ..: .;::"v".;: :-':-""_..;: .".". ..:..:.i: v...-:+:""" ':"'V.-V':---::-:"ij." :,-:.- :s.;: -- "»: ...::::::.5;;..- ::_....-............. :-- .......- - -:-:....- :---.- !.f,: :-i=i.:.X< :-.....ri: ».....-...{ »£;:...y......:--:-- :0000 .,. . .. . . ...... . . 00 - ." -:--.. .. ".." .. . .........."..,.::.............-.".......r.v::."...:.c,..:..:'.".:::.:...v.... ,..:'.;"':v-....::.:.":..:r.....: . xJ'J"-:.".........."._"x-:.... ,or... :.*:-:",-:-x-:..:.............: ,:. ':n..: 1 ;. :. . xx..«f;.; .;::-.': :'::::%- <:::::::-=::..;: :f.:.:::;:<; "* . .-. *.--." .....- ...."v.._y._.-x-.."'tt..._" ....*.._._ . . » . . .. - _ ' . '-.- .-:;. ..v..':":-;';:;' .:-Y"+: -:+::"',",,:-:.....--:X.:_J"h: "X"'".-i:;.."....v""....... ", - -::::-&:: - - - - - ,,-.5: 1t 4j ' ,'- ::;: p ' . .......:. 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474 CHAPTER 12 PHOTON OPTICS A. Coherent-State Light is Gaussian (see Sec. A.2 of Appendix A). In that case 'ljJ x ex exp x 2 Qx , (12.3-11) whereupon its Fourier transform is also Gaussian, so that cjJ p ex exp p 2 a p . (12.3-12) Here, ax and Qp are arbitrary values that represent the means of x and p, respectively. The quadrature uncertainties, determined from 'ljJ X 2 and cjJ p 2, are then given by ax a p 1 - 2 · (I 2.3-13) Under these conditions the electromagnetic field is said to be in a coherent state. The one-standard-deviation range of uncertainty in the quadrature components x and p , as well as in the complex amplitude a and in the electric field £ t , are illustrated in Fig. 12.3-2 for coherent-state light. The squared-magnitude c n 2 of the c oefficient of th e e xpansion of 'ljJ x in the Hermite Gaussian basis equals n n exp n n!, where tus in electromagnetic optics, in the context of quantum optics coherent light is not deterministic. 2a p 1 I 'E(t) 1 x 20- - 1 x :\ =:.: -=-:.:-=-=::::::.::::.- - - w - -- ---------------- p , t Figure 12.3-2 Uncertainties for the coherent state. Representative values of £(t) ex: aexp(j21rvt) are drawn by choosing several arbitrary points within the uncertainty circle. The coefficient of proportionality is chosen to be unity. The uncertainty of the coherent state is most pronounced when ax and Qp are small. The time behavior of the electric field is illustrated in Fig. 12.3-3 in the limit when ax Qp O. This corresponds to the case when the mode contains zero photons and B. Squeezed-State Light . Quadrature-Squeezed Light Although the uncertainty product a x a p cannot be reduced below its minimum value
12.3 QUANTUM STATES OF LIGHT 475 XA 'E( t) ._....,.._. ...., ...,............,...""""""'"."'I'I"'1II .............._............. ............... . ........ .........__ ........ ........_ ..... ..... ............. ............ ------------------------- ". - .... _0 _ . - -" ---------------- .-: . I II >< p t - - --- ---------------------- 0" ..._.....-..-.....-.....-...........- --------------------------- 2u p 1 Figure 12.3-3 Representative uncertainties for the vacuum state. light, which is distinctly nonclassical, is said to be quadrature squeezed. For example, a state for which 'ljJ X is a Gaussian function with a (stretched) width ax s 2 (s > 1) corresponds to a Gaussian cp p with a (squeezed) width a p 1 2s. The product a x a p into an ellipse, as shown in Fig. 12.3-4. The asymmetry in the uncertainties of the two quadrature components is manifested in the time course of the electric field by periodic occurrences of increased uncertainty followed, each quarter cycle later, by occurrences of decreased uncertainty. If the field were to be measured only at those times when its uncertainty is minimal, its noise would be reduced below that of the coherent state. The selection of those times may be achieved by heterodyning the squeezed field with a coherent optical field of appropriate phase (see Sec. 24.5). Because of its reduced noisiness, squeezed light has found a niche in precision measurements. It is not robust in the face of losses, however. X4 '£(1) l/s 2a x == 1/ - - -- --- ......==...a......===.=.__ w _______________________ .1 o. --------------------- j , t {/2a p =S p\\ Figure 12.3-4 Representative uncertainties for a quadrature-squeezed state. Photon-Number-Squeezed Light Quadrature-squeezed light exhibits an uncertainty in one of its quadrature components that is reduced relative to that of the coherent state. Another form of nonclassical light is photon-number-squeezed or sub-Poisson light. It has a photon-numb er variance that is "squeezed" below the coherent-state (Poisson) value, so that a < n. Photon- number fluctuations obeying this relation are nonclassical since (12.2-28) cannot be satisfied. Like quadrature-squeezed light, it enjoys some applications in precision mea- surements and is adversely affected by the presence of losses. Photon-number squeezed light can be generated by placing a quantum dot in a specially designed microcavity (see Example 17.4-3) or by making use of twin-beam light, as described below. An electromagnetic mode described by the harmonic oscillator eigenstate 'ljJ x 'ljJno X provides the quintessential description of photon-number-squeezed light. This is called a number state because p n C n 2 1 for n n 0, while all other
476 CHAPTER 12 PHOTON OPTICS coefficients vanish (c n == 0 for n -I- no). The number of photons carried by the mode is deterministic; it is precisely no. The mean photon number is obviously n == no and the variance is zero (since there are no photon-number fluctuations). The case no == 1 corresponds to the presence of precisely one photon. Many other states also exhibit photon-number squeezing. The uncertainties associated with number-state light are illustrated in Fig. 12.3-5. Although the quadrature components, as well as the phasor magnitude and phase, are all uncertain, the photon number is absolutely certain. T(t) t Figure 12.3-5 Representative uncertainties for the number state. This state is photon-number squeezed but not quadrature squeezed. Twin-Beam Light The question naturally arises as to whether photon-number-squeezed light can be gen- erated by manipulating coherent-state light in some manner. A first thought might be to monitor the photons from a coherent source in successive time intervals, and then to use the photons only in those time intervals where the desired photon number is observed. Unfortunately, this approach is generally doomed to failure because the very act of observing the photons annihilates them, rendering them unavailable for the purposes at hand. With the help of nonlinear optics and twin-beam light, however, coherent light can indeed be selectively manipulated to generate photon-number-squeezed light. Photons can be generated in correlated pairs by means of spontaneous parametric downconver- sion, a nonlinear-optical process in which some fraction of the photons incident on a crysta] are split into pairs of photons, while conserving energy and momentum (see Secs. 21.2C and 21.4C). Since the same number of photons is generated in each of the twin beams, the joint photon-number distribution has a width that is squeezed below its classical value and the light generated in this way can be viewed as two-mode photon- number-squeezed. Given such twin-beam light, information can be garnered from one of the beams by making measurements on it. Although the photons from this beam are annihilated in the measurement process, the information can nevertheless be used to control the photon number (as well as other features) of the surviving twin beam. READING LIST Books on Quantum Optics and Quantum Mechanics M. Fox, Quantum Optics: An Introduction, Oxford University Press, 2006. W. Vogel and D.-G. Welsch, Quantum Optics, Akademie-Verlag, 1994; Wiley-VCH, 3rd ed. 2006. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics, Benjamin, 1968; Dover, reissued 2006.
READING LIST 477 D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, 1995, paperback 2nd ed. 2006. V. Vedral, Introduction to Quantum Information Science, Oxford University Press, 2006. R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, 1985, reissued 2006. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume 3, Quantum Mechanics, 1965 and Volume 1, Mainly Mechanics, Radiation, and Heat, 1963, Addison-Wesley, 2nd ed. 2006. A. V. Sergienko, ed., Quantum Communications and Cryptography, Taylor & Francis, 2006. M. Planck, Planck's Columbia Lectures: Abridged and Unabridged Versions, with commentary by W. Vlasak, Adaptive Enterprises, 2005. Z. Ficek and S. Swain, Quantum Inteiference and Coherence: Theory and Experiments, Springer- Verlag, 2005. C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005. H. Paul, Introduction to Quantum Optics: From Light Quanta to Quantum Teleportation, Cambridge University Press, 2004. H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics, Wiley- VCH, paperback 2nd ed. 2004. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Springer-Verlag, 2003. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2002. R. Baierlein, Newton to Einstein: The Trail of Light, Cambridge University Press, paperback ed. 2001. J. Kim, S. Somani, and Y. Yamamoto, Nonclassical Light from Semiconductor Lasers and LEDs, Springer-Verlag, 2001. W. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001. R. Loudon, The Quantum Theory of Light, Oxford University Press, 3rd ed. 2000. Y. Yamamoto, F. Tassone, and H. Cao, Semiconductor Cavity Quantum Electrodynamics, Springer- Verlag, 2000. C. A. Mead, Collective Electrodynamics: Quantum Foundations of Electromagnetism, MIT Press, 2000. P. Meystre and M. Sargent HI, Elements of Quantum Optics, Springer-Verlag, 3rd ed. 1999. Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics, Wiley, 1999. R. P. Feynman, Quantum Electrodynamics, Benjamin, 1962; Addison-Wesley, 1998. V. Perinova, A. Luks, and J. Perina, Phase in Optics, World Scientific, 1998. S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics, Clarendon, 1997. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995. J. Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, Reidel, 2nd ed. 1991. W. H. Louisell, Quantum Statistical Properties of Radiation, Wiley, 1973, reprinted 1990. E. R. Pike and H. Walther, eds., Photons and Quantum Fluctuations, Adam Hilger, 1988. J. Penna, Coherence of Light, Reidel, 1971, 2nd ed. 1985. W. Heitler, The Quantum Theory of Radiation, Clarendon, 3rd ed. 1954; Dover, reissued 1984. E. Goldin, Waves and Photons: An Introduction to Quantum Optics, Wiley, 1982. H. Haken, Light: Waves, Photons, Atoms, Volume 1, North-Holland, 1981. B. Saleh, Photoelectron Statistics, Springer-Verlag, 1978. W. H. Louisell, Radiation and Noise in Quantum Electronics, McGraw-Hill, 1964; Krieger, reissued 1977. D. ter Haar, The Old Quantum Theory, Pergamon, 1967; contains English translations of key early papers by Planck, Einstein, Rutherford, and Bohr. C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds., Quantum Optics and Electronics, Gordon and Breach, 1965.
478 CHAPTER 12 PHOTON OPTICS Books on Statistical and Thermal Physics S. Blundell and K. Blundell, Concepts in Thermal Physics, Oxford University Press, paperback ed. 2006. M. Plischke and B. Bergersen, Equilibrium Statistical Physics, World Scientific, 3rd ed. 2006. C. Kittel, Elementary Statistical Physics, Wiley, 1958; Dover, reissued 2004. F. Reif, Customized Complete Statistical Physics: Berkeley Physics Course, Volume 5, McGraw-Hill, 1998. Articles R. J. Glauber, One Hundred Years of Light Quanta, in K. Grandin, ed., The Nobel Prizes 2005, Nobel Foundation, 2006, pp. 75-98. V. Jacques, E. Wu, T. Toury, F. Treussart, A. Aspect, P. Grangier, and J.-F. Roch, Single-Photon Wavefront-Splitting Interference: An Illustration of the Light Quantum in Action, The European Physical Journal D, vol. 35, pp. 561-565, 2005. Special issue on trends in quantum optics, Journal of Optics B, vol. 6, no. 3, 2004. C. Roychoudhuri and R. Roy, eds., The Nature of Light: What is a Photon?, OPN Trends (Optics & Photonics News), vol. 3, no. 1, 2003. D. V. Regelman, U. Mizrahi, D. Gershoni, E. Ehrenfreund, W. V. Schoenfeld, and P. M. Petroff, Semiconductor Quantum Dot: A Quantum Light Source of Multicolor Photons with Tunable Statistics, Physical Review Letters, vol. 87, 257401, 2001. G. S. Agarwal, ed., Selected Papers on Fundamentals of Quantum Optics, SPIE Optical Engineering Press (Milestone Series Volume 103), 1995. L. Mandel and E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light (1850-1966), SPIE Optical Engineering Press (Milestone Series Volume 19), 1990. M. C. Teich and B. E. A. Saleh, Squeezed and Antibunched Light, Physics Today, vol. 43, no. 6, pp. 26-34, 1990. M. C. Teich and B. E. A. Saleh, Squeezed States of Light, Quantum Optics, vol. 1, pp. 153-191, 1989. M. C. Teich and B. E. A. Saleh, Photon Bunching and Antibunching, in Progress in Optics, vol. 26, pp. 1-104, E. Wolf, ed., North-Holland, 1988. Special issue on squeezed states of the electromagnetic field, Journal of the Optical Society of America B, vol. 4, no. 10, 1987. Special issue on squeezed light, Journal of Modern Optics, vol. 34, no. 6/7, 1987. Special issue on quantum-limited imaging and image processing, Journal of the Optical Society of America A, vol. 3, no. 12, L986. M. C. Teich and B. E. A. Saleh, Observation of Sub-Poisson Franck-Hertz Light at 253.7 nm, Journal of the Optical Society of America B, vol. 2, pp. 275-282, 1985. E. Wolf, Einstein's Researches on the Nature of Light, Optics News, vol. 5, no. 1, pp. 24-39, 1979. S. Weinberg, Light as a Fundamental Particle, Physics Today, vol. 28, no. 6, pp. 32-37, 1975. L. Mandel and E. Wolf, eds., Selected Papers on Coherence and Fluctuations of Light, Volumes 1 and 2, Dover, 1970. L. Mandel and E. Wolf, Coherence Properties of Optical Fields, Reviews of Modern Physics, vol. 37, pp. 231-287, 1965. L. Mandel, Fluctuations of Light Beams, in Progress in Optics, vol. 2, E. Wolf, ed., North-Holland, 1963. PROBLEMS 12.1-5 Photon Energy. (a) What voltage should be applied to accelerate an electron from zero velocity in order that it acquire the same energy as a photon of wavelength Ao == 0.87 /Jm?
PROBLEMS 479 (b) A photon of wavelength 1.06 J-Lm is combined with a photon of wavelength 10.6 J-Lm to create a photon whose energy is the sum of the energies of the two photons. What is the wavelength of the resultant photon? Photon interactions of this type are discussed in Chapter 21. 12.1-6 Position of a Single Photon at a Screen. Consider a monochromatic light beam of wave- length Ao falling on an infini te scree n in the plane z == 0, with an intensity I (p ) == 10 exp( -pi Po), where p == y x2 + y2. Assume that the intensity of the source is reduced to a level at which only a single photon strikes the screen. (a) Find the probability that the photon strikes the screen within a radius Po of the origin. (b) If the beam contains exactly 10 6 photons, how many photons strike within a circle of radius Po on average? 12.1- 7 Momentum of a Free Photon. Compare the total momentum of the photons in a 10-J laser pulse with that of a I-g mass moving at a velocity of 1 cm/s and with an electron moving at a velocity c o /10. *12.1-8 Momentum of a Photon in a Gaussian Beam. (a) What is the probability that the momentum vector of a photon associated with a Gaus- sian beam of waist radius W o lies within the beam divergence angle eo? Refer to Sec. 3.1 for definitions. (b) Does the relation p == E I Co hold in this case? 12.1-9 Levitation by Light Pressure. Consider an isolated hydrogen atom of mass 1.66 x 10- 27 kg. (a) Find the gravitational force on this hydrogen atom near the surface of the earth (assume that at sea level the gravitational acceleration constant 9 == 9.8 m/s 2 ). (b) Let an upwardly directed laser beam emitting l-eV photons be focused in such a way that the ful1 momentum of each of its photons is transferred to the atom. Find the average upward force on the atom provided by one photon striking each second. (c) Find the number of photons that must strike the atom per second, and the corresponding optical power, for it not to fall under the effect of gravity, given idealized conditions in vacuum. (d) How many photons per second would be required to keep the atom from falling if it were perfectly reflecting? *12.1-10 Single Photon in a Fabry-Perot Resonator. Consider a Fabry-Perot resonator of length d == 1 em containing nonabsorbing material of refractive index n == 1.5 and perfectly reflecting mirrors. Assume that there is exactly one photon in the mode described by the standing wave sin(10 5 7rxld). (a) Determine the photon wavelength and energy (in eV). (b) Estimate the uncertainty in the photon's position and momentum (magnitude and direc- tion). Compare with the value obtained from the relation a x a p == n/2. 12.1-11 Single-Photon Beating (Time Interference). Consider a detector illuminated by a poly- chromatic plane wave consisting of two monochromatic waves that are superposed and traveling in the same direction. The constituent waves have complex wavefunctions given by U 1 (t) == J[; exp(j 27rV l t ) and U 2 (t) == .Ji; exp(j 27rV 2 t ), with frequencies VI and V2 and intensities II and 1 2 , respectively. According to wave optics (see Sec. 2.6B), the intensity of this wave is given by I(t) == II + 1 2 + 2 VII 1 2 eos[27r(V2- Vl)t]. Assume that the two constituent plane waves have equal intensities (II == 1 2 ) and that the wave is sufficiently weak so that only a single polychromatic photon reaches the detector during the time interval T == 1 I I V2 - vII. (a) Plot the probability density p (t) for the detection time of the photon for 0 < t < 1 I I V2 - vII. At what time instant during T is the probability density zero that the photon wi II be detected? (b) An attempt to discover from which of the two constituent waves the photon comes entails an energy measurement to a precision better than aE < hlv2 - vII.
480 CHAPTER 12 PHOTON OPTICS Use the time-energy uncertainty relation to show that the time required for such a measurement is of the order of the beat-frequency period. The very process of mea- surement thus washes out the interference and thereby precludes the interference from being observed. 12.1-12 Photon Momentum Exchange at a Beamsplitter. Consider a single photon, in a mode described by a plane wave, impinging on a lossless beamsplitter. What is the momentum vector of the photon before it impinges on the mirror? What are the possible values of the photon's momentum vector, and the probabilities of observing these values, after passage through the beamsplitter? 12.2- 2 Photon Flux. Show that the power of a monochromatic optical beam that carries an average of one photon per optical cycle is inversely proportional to the squared wavelength. 12.2-3 The Poisson Distribution. Verify that the Poisson probability distribution given by (12.2- 12) is normalized to unity and has mean n and variance a == n . 12.2-4 Photon Statistics of a Coherent Gaussian Beam. Assume that a 100-pW He-Ne single- mode laser emits light at 633 nm in a TEMo,o Gaussian beam (see Chapter 3). (a) What is the mean number of photons crossing a circle of radius equal to the waist radius of the beam W o in a time T == 100 ns? (b) What is the root-mean-square value of the number of photon counts in (a)? (c) What is the probability that no photons are counted in (a)? 12.2-5 The Bose-Einstein Distribution. (a) Verify that the Bose-Einstein probability distribution given by (12.2-20) is normalized and has a mean n and variance a == n + n 2. (b) If a beam of photons obeying Bose-Einstein statistics contains an average of == 1 photon per nanosecond, what is the probability that zero photons will be detected in a 20- ns time interval? *12.2-6 The Negative-Binomial Distribution. It is well known in the literature of probability theory that the sum of M identically distributed random variables, each with a geometric (Bose-Einstein) distribution, obeys the negative binomial distribution ( n + M - 1 ) ( n jM)n p (n) == n (1 + njM)n+M . Verify that the negative-binomial distribution reduces to the Bose-Einstein distribution for M == 1 and to the Poisson distribution as M 00. * 12.2- 7 Photon Statistics for MuItimode Thermal Light in a Cavity. Consider M modes of thermal radiation sufficiently close to each other in frequency that each can be considered to be occupied in accordance with a Bose-Einstein distribution of the same mean photon number Ij[exp(hv j kT) - 1]. Show that the variance of the total number of photons n is related to its mean by -2 2 - n an==n+ M ' indicating that multi mode thermal light has less variance than does single-mode thermal light. The presence of the multiple modes provides averaging, thereby reducing the noisiness of the light. * 12.2-8 Photon Statistics for a Beam of MuItimode Thermal Light. A multimode thermal light source that carries M identical modes, each with exponentially distributed (random) inte- grated rate, has an overall probability density p (w) describable by the gamma distribution ( ) M ( ) 1 M M-l Mw p (w) == (M _ I)! (w) w exp - (w) , w > O. Use Mandel's formula (12.2-26) to show that the resulting photon-number distribution assumes the form of the negative-binomial distribution defined in Probe 12.2-6. * 12.2-9 Mean and Variance of the Doubly Stochastic Poisson Distribution. Prove (12.2-27) and ( 12.2-28).
PROBLEMS 481 * 12.2-10 Random Partitioning of Coherent Light. (a) Use (12.2-33) to show that the photon-number distribution of randomly partitioned coherent light retains its Poisson form. (b) Show explicitly that the mean photon number for light reflected from a lossless beam- splitter is (1 - 'T) n . (c) Prove (L2.2-34) for coherent light. 12.2- L I Random Partitioning of Single-Mode Thermal Light. (a) Use (12.2-33) to show that the photon-number distribution of randomly partitioned single-mode thermal light retains its Bose-Einstein form. (b) Show explicitly that the mean photon number for light reflected from a lossless beam- splitter is (1 - 'T) n . (c) Prove (12.2-35) for single-mode thermal light. *12.2-12 Exponential Decay of Mean Photon Number in an Absorber. (a) Consider an absorptive material of thickness d and absorption coefficient a (cm- 1 ). If the average number of photons that enters the material is n o, write a differential equation to find the average number of photons n (x) at position x, where x is the depth into the material (0 < x < d). (b) Solve the differential equation. State the reason that your result is the exponential decay law obtained from electromagnetic optics (Sec. 5.5A). (c) Write an expression for the photon-number distribution, p (n), at an arbitrary position x in the absorber, when coherent light is incident on it. (d) What is the probability that a single photon incident on the absorber survives passage through it? * 12.3-1 Statistics of the Binomial Photon-Number Distribution. The binomial probability distri- bution is written as ( ) M! n ( ) M-n p n = (M-n)!n! p 1-p . It describes the counting statistics for certain sources of photon-number-squeezed light. (a) Indicate a possible mechanism for converting number-state light into light described by binomial photon statistics. (b) Prove that the binomial probability distribution is normalized to unity. (c) Find the count mean n and the count variance a of the binomial probability distribution in terms of its two parameters, p and M. (d) Find an expression for the SNR in terms of n and p. Evaluate it for the limiting cases p 0 and p 1. To what kinds of light do these two limits correspond? *12.3-2 Noisiness of a Hypothetical Photon Source. Consider a hypothetical light source that produces a photon stream with a photon-number distribution that is discrete-uniform, given by p(n)= { 2n1 ' 0, 0 < n < 2 n otherwise. (a) Verify that the distribution is normalized to unity and has mean n . Calculate the photon- number variance a and the signal-to-noise ratio (SNR) and compare them with those for the Bose-Einstein and Poisson distributions of the same mean. (b) In terms of SNR, would this source be quieter or noisier than an ideal single-mode laser when n < 2? When n = 2? When n > 2? ( c) By what factor is the SNR for this light larger than that for single-mode thermal light? Useful formulas: . ( . + 1 ) 1+2+3+...+ ) .=)) 2 ' 12+22+3 2 +...+j2= j(j+1)(2j+1) . 6
CHAPTER 13 PHOTONS AND ATOMS 13.1 ENERGY LEVELS 483 A. Atoms B. Molecules C. Solids 13.2 OCCUPATION OF ENERGY LEVELS 499 A. Boltzmann Distribution B. Fermi-Dirac Distribution 13.3 INTERACTIONS OF PHOTONS WITH ATOMS 501 A. Interaction of Single-Mode Light with an Atom B. Spontaneous Emission C. Stimulated Emission and Absorption D. Line Broadening *E. Enhanced Spontaneous Emission *F. Laser Cooling and Trapping of Atoms 13.4 THERMAL LIGHT 517 A. Thermal Equilibrium Between Photons and Atoms B. Blackbody Radiation Spectrum 13.5 LUMINESCENCE AND LIGHT SCATTERING 522 A. Forms of Luminescence B. Photoluminescence C. Light Scattering "_,,"r_ , c ,,' ", . ...- , \" .l' ';;.;' ;:,"',..':,' .. ..c-c 't. >- iI1' )'i ) If '\ iI "-1"1 · r .. . .", .< . .> '\: . ":", Niels Bohr (1885-1962) Albert Einstein (1879-1955) Bohr and Einstein laid the theoretical foundations for describing the interaction of light with matter. 482
Light interacts with matter because matter contains electric charges. The time-varying electric field of light exerts forces on the electric charges and dipoles in atoms, molecules, and solids, causing them to vibrate so that they undergo acceleration. Vibrating electric charges absorb and emit light. Atoms, molecules, and solids have specific allowed energy levels and bands that are determined by the rules of quantum mechanics. A photon may interact with an atom if its energy matches the difference between two atomic energy levels. If the atom is initially in the lower energy level, the photon may impart its energy to the atom and thereby raise it to the higher level; the photon is then said to be absorbed (or annihilated). Alternatively, if the atom is in the higher energy level, the photon may stimulate the atom to undergo a transition to the lower level, resulting in the emission (or creation) of a second photon whose energy is equal to the difference between the atomic energy levels. Under appropriate circumstances, stimulated emission can lead to the generation of laser light. Thermal excitations cause the atoms of matter to constantly undergo upward and downward transitions among their allowed energy levels via the absorption and emis- sion of photons. For blackbodies in thermal equilibrium, under steady-state conditions, the resulting collection of photons and atoms produces thermal light. All blackbodies whose temperatures lie above absolute zero radiate thermal light, which has a distribu- tion of frequencies known as the blackbody radiation spectrum. As the temperature of the object increases, the higher atomic energy levels become increasingly populated, causing the peak of the blackbody radiation spectrum to shift toward higher frequencies (shorter wavelengths). Photon emissions may also be instigated by external sources of energy other than thermal excitations. Exposure to ultraviolet radiation, sound waves, electric current, and chemical reactions can cause atoms to emit light called luminescence. Yet other processes can also result in the emission of light; examples include charged particles traveling faster than the velocity of light in a medium (Cherenkov radiation) and the deceleration of charged particles as they penetrate matter (Bremsstrahlung). A photon incident on a material can also have its direction and energy altered via light scattering, a process that can serve to elucidate the internal energy levels of the material, such as those associated with molecular vibrations. The application of the laws set forth in this chapter to the operation of laser ampli- fiers and oscillators is considered in Chapters 14 and 15, respectively. This Chapter The purpose of this chapter is to introduce the laws that govern the interaction of light with matter. These laws are responsible for the generation of laser, thermal, and luminescence light. The chapter begins with a brief review of the generic energy levels associated with different types of matter (Sec. 13.1), and the occupation of these energy levels (Sec. 13.2). In Sec. 13.3 we discuss the absorption and emission of photons by an atom. The interaction of many photons with many atoms, under conditions of steady state and thermal equilibrium, is considered in Sec. 13.4. Finally, an elementary description of luminescence light and light scattering is provided in Sec. 13.5. 13.1 ENERGY LEVELS The atoms of matter may exist in relative isolation, as in the case of a dilute atomic gas, or they may interact with neighboring atoms to form molecules, liquids, and solids. The 483
484 CHAPTER 13 PHOTONS AND ATOMS constituents of matter obey the laws of quantum mechanics. The behavior of a single nonrelativistic particle of mass m (an electron, for exam- ple), subject to a potential V(r, t), is governed by a complex wavefunction w(r, t) that satisfies the Schrodinger equation h 2 2 . 8w(r, t) -- V \]fer, t) + VCr, t)\]f(r, t) = -In 8 . 2m t (13.1-1) The potential characterizes the environment of the particle, including contributions from externally applied optical fields. The partial differential equation displayed in (13.1-1) thus has a great variety of solutions, depending on the form of V (r, t). Systems that comprise multiple particles, such as atoms, molecules, liquids, and solids, obey a more complex version of this equation in which the potential contains terms that accommodate interactions among the particles. Equation (13.1-1) is mathematically similar to the paraxial Helmholtz equation of wave optics (2.2-23) and to the slowly varying envelope equation of ultrafast optics (22.1-24). The Born postulate of quantum mechanics specifies that the probability of finding the particle within an incremental volume dV surrounding the position r, within the time interval between t and t + dt, is p(r, t) dV dt == Iw(r, t) 1 2 dV dt. (13.1-2) Equation (13.1-2) resembles (12.1-14) for the probability of finding a photon within an incremental area and time. In the absence of a time-varying potential, the allowed energy levels E of the particle are determined by using the technique of separation of variables. This leads to a solution of (13.1-1) of the form w(r, t) == 'ljJ(r) exp[j(E/h)t], where 'ljJ(r) satisfies the time-independent Schrodinger equation h 2 --V 2 'ljJ(r) + V(r)'ljJ(r) == E'ljJ(r). 2m (13.1-3) Equation (13.1-3), which is similar to the Helmholtz equation (2.2-7), may be regarded as an eigenvalue problem for which the allowed values of the energy E are the eigen- values, while the solutions 'ljJ(r) are the eigenfunctions. Systems of multiple particles obey a generalized form of (13.1-3). The solutions provide the allowed values of the energy of the system, E. These values can be discrete (as for an atom), or continuous (as for a free particle), or comprise sets of densely packed discrete levels called bands (as for a semiconductor). The presence of thermal excitation or an external field, such as light illuminating the material, can induce the system to move from one of its energy levels to another. It is by such means that the system exchanges energy with the outside world. In the following sections we schematically illustrate typical energy-level structures for selected atoms, molecules, and solids. A. Atoms Atomic energy levels are established by the potential energies of the electrons in the presence of the atomic nucleus and the other electrons, as well as by forces involving the orbital and spin angular momenta, which are usually much weaker than the inter- actions involving charges. Many atoms and ions are used as active laser media (see Sec. 15.3).
13.1 ENERGY LEVELS 485 Hydrogen The energy levels of a hydrogen-like atom comprising a nucleus of charge +Ze and a single electron of charge -e and mass m, where Z is the atomic number, are deter- mined by inserting the Coulomb potential, V(r) -Ze 2 /r, in the time-independent Schrodinger equation (13.1-3). Since V( r) is a function of the radial coordinate alone, the Laplacian may be written in spherical coordinates whereupon the partial differential equation splits into three ordinary differential equations via separation of variables. This enables us to solve the eigenvalue problem. The eigenvalues comprise an infinite number of discrete energy levels with values M r Z 2 e 4 1 E -- - n - ( 41fE o)2 2fi 2 n 2 ' n 1,2,3, . . . , ( 13.1-4) where the reduced mass of the atom M r replaces the electron mass m to accommodate the finite mass of the nucleus. The energy levels in (13.1-4), which are characterized by a single quantum number n called the principal quantum number, are displayed in Fig. 13.1-1 for Hand C 5 +. These levels can also be obtained by equating the eV eV H C 5 + 14 00 504 4 12 3 I 432 18.2-nm laser ,,-.... 10 2 t 360 CO ...... II II N N '-' 8 288 ';: ;;>-. OJ) Figure 13.1-1 Energy levels of hydrogen I-< Il) Il) = = (Z = 1; left ordinate) and C 5 + (a hydrogen- 6 216 like atom with Z = 6; right ordinate). The 4 144 n - 3 to n - 2 transition, indicated by an arrow, corresponds to the C 5 + extrerne- ultraviolet laser transition at 18.2 nrn, as 2 72 discussed in Sec. 15.3C. The arbitrary zero n = 1 of energy is taken to coincide with the n = 1 0 0 level. Coulomb force of attraction to the centrifugal force required to keep the electron in a circular orbit, while assuming that the electron orbital angular momentum is quantized to inte¥er multiples of fi. The radii of these Bohr orbits turn out to be r n (41fEo) n 2 fi /mZe 2 , n 1,2,3,...; the radius of the n 1 orbit is denoted rl. This is the basis of Bohr theory, which is part of the "old quantum theory." The eigenfunctions of the Schrodinger equation take the f9rm of products of three functions, 'l/JnRm (r, e, cjJ) IR nR ( r) eRm (e) <Pm (cjJ), where n 1, 2, 3, . . . is the prin- cipal quantum number; £ 0,1, 2, . . . , n - 1 is called the azimuthal quantum number; and m 0, ::t1, ::t2, . . . ,::t£ is called the magnetic quantum number. Here, IRnR(r) represent associated Laguerre functions (these are closely related to the generalized Laguerre polynomials discussed in the footnote on page 98), eRm (e) are associated Legendre functions, and the <Pm (cjJ) are phase functions. These solutions are similar to those for the spherical microcavity, as discussed in Sec. 10.4C. Incorporating the intrinsic spin of the electron requires an additional quantum num- ber known as the spin quantum number: s ::t . The interaction of the electron spin with its orbital angular momentum, which is referred to as the spin-orbit interaction, serves to split the energy levels into closely spaced, but distinct, components called fine
486 CHAPTER 13 PHOTONS AND ATOMS structure. The spin also interacts with the magnetic moment of the nucleus to produce yet finer splittings, called hyperfine structure. These effects cause the energy levels for hydrogen to differ from those specified in (13.1-4), but only slightly. Relativistic corrections to the energy levels, which are small but measurable, can also be taken into account via a formulation known as "Dirac theory." Indeed, the Dirac formulation au- tomatically leads to the notion of electron spin by virtue of its relativistically invariant form. Multielectron Atoms Multielectron atoms consist of a nucleus of charge +Ze surrounded by Z electrons, each of charge -e. The energy levels of multielectron atoms can be determined by using the Schrodinger theory, as long as relativistic effects can be ignored (such ef- fects are significant only for the lightest atoms). Because of the myriad Coulomb interactions involved among a collection of electrons, the Schrodinger equation is solved via an approximate self-consistent approach, known as the "Hartree method." Each electron is considered to move independently in a spherically symmetric net potential V (r), which is taken to be the sum of the spherically symmetric attractive Coulomb potential arising from the nucleus and a spherically symmetric repulsive potential representing the average effect of the Coulomb forces from all other electrons. Under these assumptions, the Z -electron Schrodinger equation splits into Z single- electron Schrodinger equations with an overall eigenfunction that is a product of the individual-electron eigenfunctions, and a total energy that is the sum of the energies of the individual electrons. Ultimately, perturbation theory is called upon to account for the deviations from spherical symmetry of the repulsive potential and for interactions involving electron spin. The resultant single-electron eigenfunctions are closely related to those for the hydrogen atom and are written in the same form. As the atomic number Z increases, the occupation of successive single-electron states proceeds by minimizing the total energy while satisfying the Pauli exclusion principle, which provides that no two electrons may have the same set of four quantum numbers. The states fill in the form of shells (designated by the principal quantum numbers n), each of which has the capacity to hold a specific number of electrons. Within each shell, subshells are designated by the pair of quantum numbers nR, where R is usually specified in spectroscopic notation (the letters s, p, d, f, g, h, i correspond to R == 0, 1, 2, 3, 4, 5, 6, respectively). The electron configuration nRu represents the arrangement of electrons in the subshells; the superscript u indicates the number of electrons present in each. For example, the configuration for the ground state of He (Z == 2) is ls 2 , its two electrons just filling the n == 1 shell (for which R == 0). Low-lying excited-state configurations of He include ls2s and ls2p, for which one of the electrons has been excited to the n == 2 shell. For Ne (Z == 10), the ground-state configuration is ls 2 2s 2 2 p 6; its 10 electrons just fill the n == 1 and n == 2 shells. Each of these electron configurations comprises a collection of closely spaced fine-structure energy-level splittings, called a manifold, as shown schematically in Fig. 13.1-2 for Ne. These are introduced principally by the spin-orbit interaction, which is the interaction between the overall spin and the overall orbital angular momentum of the atom. This coupling scheme, known as Russell-Saunders (or LS) coupling, is operative for all but the heaviest atoms. For a given electron configuration, the various atomic angular momenta are summarized by the term symbol 2S+1£8, where 8 is the total spin angular-momentum quantum number; 28 + 1 is the spin multiplicity (e.g., singlet, triplet); £ is the total orbital angular-momentum quantum number in spectroscopic notation (uppercase letters S, P, D, F,... represent £ == 0, 1, 2, 3,..., respectively); and a is the total overall angular-momentum quantum number. The term symbol for an atom or ion is often provided immediately after the electron configuration. For example, the lowest-lying excited singlet and triplet states
13.1 ENERGY LEVELS 487 He Ne 21 Is2s ISO 21 2p 5 5s 3.3 9-!lID laser 20 Is2s 3 S) 20 --- --- 2p 5 4s > > (l) (l) '--" '-' :>-. 19 :>-. 19 OJ) OJ) 2p 5 3p I-< I-< (l) (l) !::: !::: 18 18 17 17 2p 5 3s 16 16 Odd parity Even parity Figure 13.1-2 Selected energy levels of He and Ne atoms. Electron configurations and term symbols are indicated (the electron-configuration prefix for Ne, 18 2 28 2 , corresponding to filled subshells, is suppressed for brevity). The energy spacings between the fine-structure splittings, which are illustrated schematically, are greatly exaggerated. The Ne transitions marked by arrows correspond to wavelengths 3.39 /-LID and 632.8 nm, as indicated. These transitions, which lie in the mid-infrared and visible regions of the spectrum, respectively, are commonly used in He-Ne lasers (see Secs. 14.3E and 15.30). The close energy matches between the excited He and Ne levels facilitates excitation of the Ne atoms via collisions in a gas-discharge tube; hence the moniker "He- Ne laser." of He are denoted 1828 1 8 0 and 1828 3 8 1 , respectively, as shown in Fig. 13.1-2. When all occupied subshells are filled (as is the case for the ground states of all of the noble gases together with a number of other atoms such as Ca, Cd, Yb, and Hg), the term symbol is 18 0 . The magnitudes of the spin-orbit energy-level shifts are typically about 1 part in 10 4 for H and grow larger as the atomic number Z increases. Relativistic effects also contribute energy-level shifts of about 1 part in 10 4 , but they are independent of Z and can therefore safely be ignored in all but the lightest of atoms. H yperfine shifts are yet a factor of 10 3 smaller. Other interactions (e.g., spin-spin coupling) are also present, but are negligibly small. The larger the value of n, the less tightly bound is the electron to the atom because the Coulomb screening by the inner electrons moderates the nuclear potential. As a result, shells typically fill in the order n == 1, 2, 3, 4, . . .. Similarly, the larger the value of R, the less tightly bound is the electron because the electron probability density progressively shifts toward the atomic periphery. Hence, subshells typically fill in the order 8, p, d, f, . . .. As a consequence of these successive filling processes, many properties of the elements are periodic functions of Z, as exemplified by the periodic table displayed in Fig. 13.1-3. Successive rows of the table correspond to consecutive values of the principal quantum number n. Each column of the table contains elements whose physical and chemical properties bear a certain similarity to each other because they contain the same number of electrons in their outermost shells (valence electrons). Column VIII, for example, comprises the noble gases, including He and Ne, which are monoatomic and chemically inert because they have filled outer shells and a large energy difference between their filled p subshells and the next higher 8 subshells. Columns I and VII, in contrast, comprise elements that are highly active chemically, and easily form molecules. Each alkali-metal atom in column I, for example, contains a lone outer electron that it will readily share with any nearby halogen atom in column VII, which needs just such a lone electron to complete its outer shell. In general, multielectron atoms and ions exhibit an enormous variety of allowed
488 CHAPTER 13 PHOTONS AND ATOMS VIllA IlIA IVA VA VIA VIlA [ej D Gas D Liquid [] Solid t] t.11 H II j[j M 3 M8I IIIB IVB VB VIB VIIB - VIIIB - IB lIB W W LQJ .. 19'! . 2 . : . ' ... 2 . 4; .. : . ' . . \ . 28; . "' . 29 . . 4 LKJ I[!j W [Qd IMdI ll!U [ill l1kJ lK!J 5 mjlr7jl1 155 56 5772i ! 73 '117' [ 75 1 r 6 1 r 7 i 1 78 11 79 1 f80l t 81 ] rszl l 83 j 1 84 11 85 1 M 6 J=B 71 'Hf':.Ta tW:Re'Ps; il r ; ".Pt:Au l!!gJTllf!!) Bi' Po' At 18l . . . 88; 89 i;I04 . . . . [JO11 . 1 . O:H I07 . . . L08 . . ... . . HJ 09 .' ... . 111 IOH II I 11:HIII r t ffil rml fTI8l 7Pr; Rai 103, JDb$ '.'J3I1. s,MtDs R. ." uti JJut lJu. IA [!!J 'I' I' H IIA 2 [i &:J c : lmJIf Figure 13.1-3 Periodic table of the elements, with element abbreviations and atomic numbers Z indicated. Successive rows of the table comprise elements whose valence electrons have principal quantum numbers n indicated by the arabic numerals at left. Each column of the table, identified by a traditional roman-numeral designation, comprises elements with similar physical and chemical properties. Elements that take the form of gases, liquids, and solids at room temperature are indicated in blue, yellow, and silver, respectively. energy levels [however, energy formulas similar to (13.1-4) exist on]y for optically active electrons such as the valence electrons in alkali atoms]. Even though optical transitions typically involve only valence electrons, this abundance of energy levels in turn gives rise to a cornucopia of energy differences, many of which serve as viable laser wavelengths (see Secs. 14.3E and 15.3D). The energy differences between the excited atomic levels of Ne displayed in Fig. 13.1-2, for example, lie principally in the infrared and optical regions of the spectrum, typically extending up to energies of several e V (see Fig. 12.1-2 for relations among different energy units). B. Molecules Molecules can be formed by the combination of two or more atoms. Molecular energy levels are determined in part by the potential energies associated with the interatomic forces that bind the atoms. A stable molecule emerges when the sharing of valence electrons by the constituent atoms results in a reduction of the overall energy. The two principal types of molecular binding are ionic binding and covalent binding. For ionic ally bound molecules (such as HF), regions of positive and negative charges remain spatially separated so that the molecule exhibits a permanent electric dipole moment. For covalently bound molecules (such as H 2 ), the constituent atoms fully share the electrons so that the resulting molecule has no permanent dipole moment. For nonidentical nuclei, the bonding may be partly ionic and partly covalent. The form of the bonding plays a role in determining the energy-level structure of the molecule. The energy levels of a molecule arise from three distinct interactions, and have transitions that fall in different wavelength regions: rotational transitions lie in the microwave and far-infrared, vibrational transitions lie in the infrared, and electronic transitions lie in the visible and ultraviolet. The time scales of these features therefore differ considerably; hence, to first approximation, they may be analyzed separately. Molecules ranging from simple gases to dyes in a solvent serve as active laser media
13.1 ENERGY LEVELS 489 (see Secs. 15.3B and 15.3C). Rotating Diatomic Molecule The rotation of a diatomic molecule with moment of inertia J about its center of mass can be considered as the rotation of a rigid rotor about an axis perpendicular to its internuclear axis. The classical rotational energy for such a system is E-c == L 2 /2J, where L is the angular momentum of the system about the axis of rotation. According to quantum mechanics, therefore, the square-magnitude of the angular momentum of such a system is quantized in accordance with L 2 == t( t + 1) ti 2 , where t is the rotational quantum number. The allowed energy levels of the rotating diatomic molecule are thus 1 2 E-c == 2J t(t + 1)ti , t == 0, 1, 2, . . . . (13.1-5) The energy separations tiw of rotational energy levels typically lie in the range 10- 4 - 10- 2 eV, corresponding to photons in the microwave and far-infrared regions of the spectrum. The energy spacing between successive rotational energy levels increases with increasing quantum number t, in contrast to the spacing between successive electronic energy levels of the hydrogen atom, which decrease with increasing quantum number in accordance with (13.1-4). Diatomic molecules with identical nuclei (such as N 2 ) have no permanent electric dipole moment; they therefore do not exhibit pure rotational spectra. Vibrating Diatomic Molecule The vibrations of a diatomic molecule (such as N 2 , CO, or HCI) are governed by an intermolecular attraction subject to a restoring force that is approximately proportional to the change in the internuclear distance x. The system may therefore be modeled as two masses M 1 and M 2 , joined by a spring, with reduced mass M r == MIM2/(Ml + M 2 ). A molecular spring constant /'1; can be defined such that the potential energy is V ( x) == ! /'1;X 2 . These molecular vibrations therefore take on the energy levels of a quantum- mechanical harmonic oscillator. As discussed in Sec. 12.3, these levels are quantized in accordance with ED == (0 + !)tiw, o == 0,1,2, . . . , (13.1-6) where w == vi /'1;/ M r is the (angular) oscillation frequency and !tiw is the zero-point energy. Equation (13.1-6) is identical to the expression for the allowed energies of a mode of the electromagnetic field, as provided in (12.1-5). Typical values of tiw for molecular vibrations lie in the range 0.05-0.5 e V, corresponding to the infrared region of the spectrum (see the N 2 energy levels displayed in Fig. 13.1-4). Unlike the energy levels of the hydrogen atom and the rotating diatomic molecule, the vibrational energy levels of the diatomic molecule are equally spaced. In practice, however, the potential-energy curves for most molecules become anharmonic as the energy increases (see Sec. 21.7), resulting in a diminution of energy-level separations as 0 increases. In the course of undergoing a vibrational transition, the molecule may simultaneously alter its rotational state, so that both 0 and t change; this is character- ized by a vibrational-rotational spectrum. Vibrating Triatomic Molecule A triatomic molecule of great importance in photonics is carbon dioxide, in no small part because it serves as a highly useful active laser medium. Since it comprises
490 CHAPTER 13 PHOTONS AND ATOMS eV CO 2 eV N 2 (050) 0.4 04 (200) (040) 0.3 tJ=1 (001) 0.3 >-> (030) bI} <l) 10.6-Jlm laser 0.2 (100) 0.2 0.1 (010) 0.1 tJ=O (000) (000) (000) 0 Symmetric Asymmetric Symmetric Bending 0 stretch stretch stretch -- -- Figure 13.1-4 Lowest vibrational energy levels of the N 2 and CO 2 molecules (the zero of energy is arbitrarily chosen at t1 == 0). The transitions indicated by arrows represent energy exchanges between different normal modes, and correspond to Ao == 10.6 J-LID and Ao == 9. 6 J-LID as indicated. These transitions are used in CO 2 lasers (see Sees. 14.3E and I5.3D). Each CO 2 vibrational level has a manifold of finely spaced associated rotational energy levels (not shown). three atoms and is linear, the CO 2 molecule may undergo independent vibrations of three kinds, as illustrated in Fig. 13.1-4: asymmetric stretch (AS), symmetric stretch (SS), and bending (B). Each of these normal modes has the features of a quantum- mechanical harmonic oscillator, with its own spring constant and hence its own value of hw. The allowed energy levels of the molecule are thus characterized by a sum of three terms, each of the form (13.1-6), corresponding to the three modal quantum numbers (tJl, tJ2, tJ3) that characterize the vibrations (see Fig. 13.1-4). As with diatomic molecules, each vibrational level is split into many closely spaced rotational levels (not shown), whose energies are given approximately by (13.1-5). Dye Molecule Organic dyes are large and complex molecules. As a result, they may undergo elec- tronic, vibrational, and rotational transitions, and typically have a vast array of energy levels. Levels exist in both singlet (S) and triplet (T) states (see Sec. 13.1 A). Singlet states have an excited electron whose spin lies antiparallel to that of the remainder of the dye molecule; triplet states have parallel spins. The differences between energy levels correspond to wavelengths that cover broad regions of the optical and ultraviolet. Figure 13.1-5 provides a schematic illustration of a portion of the energy-level structure for Rhodamine-6G, which becomes an ion when dissolved in a solvent such as water or alcohol. This particular dye is sometimes used as a lasing medium in the yellow region of the spectrum. The organic dye laser is briefly discussed in Sec. 15.3C. c. Solids The molecules (or atoms) of solids lie in close proximity to each other and typically coalesce into a periodic arrangement comprising a crystal lattice. The strength of the forces holding the atoms together is roughly of the same magnitude as the forces that bind atoms into molecules. Consequently, the energy levels of solids are determined not only by the potentials associated with individual atoms, but also by the potentials
13.1 ENERGY LEVELS 491 5 ..--..., >- Q) '-" Sl 3 ec eH eN eO Dye \ ./ T 2 \. " ,# 4 >-. bJ) $.-< Q) t:: \ ./ T . ,,# 1 . # '' 2 1 Figure 13.1-5 Structure of the Rhodamine- 60 lon, which has the chemical formula C2sH31N20j. At left is a schematic illustra- tion of a laser transition between two singlet manifolds with slightly different configura- tions, as indicated by their horizontal offset. Vi brationa] and rotational energy levels are represented by thick and thin lines, respec- tively. Laser So o Singlet states Triplet states associated with neighboring lattice atoms. Noncrystalline solids, such as glasses and plastics, have orderly structures like those of crystals, but over a short range. Three principal types of binding occur in ordinary solids: ionic, covalent, and metal- lic. Ionic solids (such as CaF 2 ) comprise a crystalline array of positive and negative ions with spherically symmetric closed shells. Since there are no free electrons to carry current, these materials are insulators. They are generally transparent in the visible region of the spectrum since their bandgaps usually lie in the ultraviolet (see Fig. 5.5-1). Covalent solids, like covalently bound molecules, consist of atoms bound by shared valence electrons. They are often insulators and can be transparent (such as diamond) or opaque (such as graphite) in the visible region. Covalent solids can be semiconductors (such GaAs), which are opaque in the visible and transparent in the infrared (see Fig. 5.5-1). Metallic solids have valence electrons that are all shared by all of the positive ions, and move in their combined potential. The ability of the elec- trons to wander at will through metallic crystals is responsible for their high electrical conductivity. Metals strongly reflect light and are opaque in the visible. It is instructive to examine how the energy levels of an isolated atom are modified as it comes into close contact with neighboring atoms in the course of forming a crystal lattice. Isolated atoms and molecules (e.g., those in gases) exhibit discrete energy levels (see Figs. 13.1-1-13.1-5). Each individual atom in a collection of such identical isolated atoms has an identical set of discrete energy levels. As these atoms are brought into proximity to form a solid, exchange interactions (arising from the quantum-mechanical requirement of indistinguishability for identical particles), along with the presence of fields of varying strengths from neighboring atoms, become in- creasingly important. The initially sharp energy levels associated with the valence electrons of isolated atoms gradually broaden into collections of numerous densely spaced energy levels that form energy bands. This process is illustrated in Fig. 13.1- 6, where electron energy levels are illustrated schematically for two isolated atoms (a), for a molecule containing two such atoms (b), and for a rudimentary ID lattice comprising five such atoms (c). The lowest-lying energy levels remain sharp because the electrons in the inner subshells are shielded from the influence of nearby atoms while the sharp energy levels associated with the outer atomic electrons become bands as the atoms enter into close proximity. This picture is elaborated in Fig. 13.1-7, where we schematically compare the en-
492 CHAPTER 13 PHOTONS AND ATOMS (a) -v -y-- (b) Figure 13.1-6 Schematic energy levels for: (a) two isolated atoms; (b) the same two atoms having formed a diatomic molecule; and (c) five identical atoms having formed a rudimentary 1 D crystal. (c) ergy levels of an isolated atom and three different kinds of solids comprising lattices of such atoms: a metal, a semiconductor, and an insulator. The lowest-lying energy levels of these solids, denoted by the electron configurations Is, 2s, and 2p, resemble those of the isolated atom because the inner electrons are shielded from interatomic forces. In contrast, the discrete higher energies of the atomic valence electrons, denoted 38 and 3p, are split into densely packed energy bands in the solids. The lowest-lying unoccupied, or partially occupied, energy band is called the conduction band while the highest-lying fully occupied energy band is known as the valence band. These two bands are separated by a forbidden band, with an energy extent E 9 known as the bandgap energy. As with electrons in individual atoms, the Pauli exclusion principle applies to the electrons in solids so that the lowest-lying energy bands are occupied firs 1. Vacuum level 3p 3s - - --, 2p >-. Q) s::: pJ 2s Is Isolated atom Figure 13.1-7 Broadening of the discrete energy levels of an isolated atom into energy bands when atoms in close proximity form a solid. Fully occupied bands are darkly shaded, unoccupied bands are lightly shaded, and partially occupied bands are both lightly and darkly shaded The forbidden band is shown as white. Typical values of the conductivity a for metals, semiconductors, and insulators are 10 8 (!1-m)-l, 10- 4 -10 5 (!1-m)-l, and 10- 10 (!1-m)-l, respectively, at room temperature. Metal Semiconductor Insulator Metals have a partially occupied conduction band at all temperatures (light and shaded region in Fig. 13.1-7). The availability of many unoccupied states in this band is responsible for their high electrical conductivity (see Sec. 5.5D). Metals comprise the great preponderance of elements in the periodic table (see Fig. 13.1-3). Semimetals, on the other hand, have overlapping valence and conduction bands. Intrinsic semiconductors have an occupied valence band (dark shading) and an un- occupied conduction band (light shading) at T == 0° K. Since there are no available free
13.1 ENERGY LEVELS 493 states in the valence band, and no electrons in the conduction band, the conductivity of an ideal intrinsic semiconductor is zero at T == 0 0 K. As the temperature of the semiconductor rises above absolute zero, an increasing number of electrons from the valence band gain sufficient thermal energy to enter the conduction band, and thereby contribute to the conductivity of the material. Insulators also have a fully occupied valence band and an unoccupied conduction band. They are distinguished from semiconductors by their larger bandgap energy (typ- ically E 9 > 3 e V). As an example, the bandgap energy for silicon (a semiconductor) is Eg 1.1 eV whereas that for diamond (an insulator) is Eg 5.5 eV. Fewer electrons in insulators have the requisite thermal energy to surmount the bandgap energy and contribute to the conductivity of the material. It should be pointed out, however, that issues such as the degree of band overlap also play a role in determining whether a material is a metal, semiconductor, or insulator. Solid-state materials playa host of important roles in photonics. Metals are highly reflective in the visible and infrared regions of the spectrum; they are often used to fabricate optical components such as mirrors. Many types of laser amplifiers and lasers are based on doped dielectric materials. Doped semiconductors, in the form of p-n junctions, heterostructures, and quantum wells, are also widely used as active laser media and detectors. We proceed to examine the energy levels of some representative doped dielectric media and semiconductor materials used as active laser media. Doped Dielectric Media Ionic or covalent solids that are insulating and transparent in a particular region of the spectrum are called transparent dielectric media. Provided they have suitable optical, thermal, and mechanical properties, such materials often serve as hosts for active laser ions. The optical properties of these host materials were considered in Sec. 5.5C in the context of the Lorentz oscillator model. Although this model is classical, it is adequate for characterizing the host material since the material is transparent in the wavelength region of interest (its resonances lie outside that region). The resonances of the dopant ions, in contrast, must be established via quantum-mechanical calculations or, as is more often the case in practice, empirically. Transition-metal and lanthanide-metal ions are the most common dopants. The extent to which the energy levels of the active laser ions are affected by the host medium is determined principally by how well their optically active electrons are shielded from neighboring lattice atoms. It will become clear that the energy lev- els of transition-metal ions are substantially modified by crystal-field effects whereas those of lanthanide-metal (rare-earth) ions are scarcely affected. By way of example, we consider the energy levels of four well-known laser systems: Cr 3 +:AI 2 0 3 (ruby), Cr 3 +:BeAI 2 04 (alexandrite), Nd3+:Y3AIs012 (Nd 3 +:YAG), and Nd 3 +:glass. Transition-metal dOfant ions. The most commonly used transition-metal dopant ions for lasers are Cr + and Ti 3 +. The electron configurations and term symbols for these elements, and their trivalent ions, are provided in Table 13.1-1. Ni 2 + and C0 2 + are also often used as dopants. We examine the energy levels of two dielectric media doped with Cr 3 +, namely ruby and alexandrite (Fig. 13.1-8). Ruby is celebrated because it was used to make the first laser, whereas alexandrite has received considerable attention because its output is tunable over a range of wavelengths. The energy levels of Ti:sapphire, an important transition-ion laser material, will be considered in Sec. 15.3A. Ruby (Cr 3 +:AI 2 0 3 ) is chromium aluminum oxide. It is a dielectric medium with refractive index n 1.76 that is composed principally of sapphire (A1 2 0 3 , also known as aluminum oxide, alumina, and corundum), in which a small fraction of
494 CHAPTER 13 PHOTONS AND ATOMS Table 13.1-1 Important transition-metal and lanthanide-metal (rare-earth) dopants for solid-state lasers: Electron configurations a and term symbols. Atom Ion Atomic Number Z Element Configuration Term Ion Configurati on Term Transition metals 22 Ti 3d 2 48 2 3P2 Ti 3 + 3d 1 2D 3 / 2 24 Cr 3d 5 48 1 78 3 Cr 3 + 3d 3 4P3/2 Lanthanide metals 60 Nd 4f4 68 2 5]4 Nd 3 + 4f3 4]9/2 68 Er 4f12 68 2 3H 6 Er 3 + 4fll 4]15/2 70 Yb 4f14 68 2 ISO Yb 3 + 4f13 2P7/2 aBy convention, the electron configurations for filled subs hells are omitted; this includes those for the 58 2 5p 6 filled subshells in the n = 5 shell of the lanthanides. the A13+ ions (rv 0.05%) are replaced by Cr 3 + ions. Alexandrite (Cr 3 +:BeAI 2 0 4 ) is formed by doping a small amount of chromium oxide (rv 0.1 %) into a chrysobery I host (BeA1 2 0 4 ). This material has a refractive index that is close to that of ruby, n 1.74; however chrysoberyl is biaxial whereas sapphire is uniaxial. Cr 3 +: Al 2 0 3 (Ruby) -4 4F 1 - 3 ,,-..... ,,-..... > > 4F 2 _ Q) Q) '-' '-' - 2 OJJ OJJ $.-< $.-< 2£ Q) Q) w 694-nm - 1 laser 4A2 0 Cr 3 +: BeAl 2 0 4 (Alexandrite) - - 4T 2 - 2£ tunable laser 680-nm ! - laser 4A "" -'" 2 4 3 2 1 o Figure 13.1-8 Selected energy levels and energy bands for Cr 3 + :A1 2 0 3 (ruby) and Cr 3 + :BeAI 2 0 4 (alexandrite). The red arrows represent laser transitions. Each laser emits light at a characteristic fixed wavelength. However alexandrite lases over a substantial range of additional wavelengths. The dark- to-light shading of the lower laser band in alexandrite indicates a decrease in its relative occupancy. Since the 3d electrons of the Cr 3 + ions in both materials are exposed to neigh- boring ions, the energy levels of these materials are determined in large part by the surrounding crystal fields and therefore depend substantially on the host material. In particular, each chromium ion is surrounded by oxygen atoms in a configuration that subjects it to a significant spatially varying potential. Best represented in the context of crystal-field theory (or ligand-field theory), this potential, along with that of the Cr 3 + nucleus, determines the energy levels of ruby and alexandrite via the Schrodinger equation. As a consequence, the energy levels of transition-metal ions in a dielectric host are generally designated by group-theoretical symbols rather than term symbols
13.1 ENERGY LEVELS 495 The resultant energies are a mixture of discrete levels and energy bands, some of which are shown, along with their group-theoretical symbols, in Fig. 13.1-8. The energy levels of the two materials are quite distinct even though they share the same dopant. In particular, the 4A 2 energy band in alexandrite comprises a collection of vibronic states that result from coupling between the electronic energy levels and the lattice vibrations of the crystal; indeed, these states are not unlike those of a dye molecule (see Fig. 13.1-5). Consequently, alexandrite lases over a substantial range of wavelengths that is not available in ruby (see Sees. 14.3E and 15.3D). Nevertheless, both materials lase at particular characteristic wavelengths that are not too far apart (694 and 680 nm, for ruby and alexandrite, respectively). Lanthanide-metal dopant ions. The lanthanides, comprising the series from 58Ce to 71 Lu, reside in row 6 of the periodic table (see Fig. 13.1-3). These elements are often called rare earths because they were long ago thought to be rare (they are not). Successive lanthanide elements are constructed by adding electrons to the 4f subshell, which lies within the filled 58 2 5 p 6 and 68 2 subshells. The lanthanides usually exist as trivalent cations; the configuration of their valence electrons takes the form 4fu, with u varying from 1 (Ce 3 +) to 14 (Lu 3 +). Nd 3 +, Er 3 +, and Yb 3 + are particularly important dopants for laser amplifiers and oscillators. Nd 3 +:glass and Er3+:silica fiber are widely used as laser amlifiers, as will be highlighted in Sees. 14.3B and 14.3C, respectively. Nd 3 +:YAG, Nd +:YV0 4 , and Yb 3 +:silica fiber often serve as laser oscillators, as discussed in Sec. 15.3A. Among the other lanthanides, Tm 3 + and H03+ are also extensively used as active laser ions. 3.0 3.0 Nd 3 +: Y AG Nd 3 +: Glass 2.5 2.5 - 2.0 2.0 --- > Q) '-' 4F 3n T 1.064-jLm laser 41 1ln 41 91 1.5 --- > Q) '-' OJ) Q) c W b/J Q) c [.iJ 4F 3n T 1.053-jLm laser 41un 4 19 / 2 1.5 1.0 1.0 0.5 0.5 o o Figure 13.1-9 Selected energy levels of Nd 3 + in YAG and Nd 3 + in glass. The arrows indicate the principal near-infrared laser transition, which has a wavelength 1.064 /-LID in YAG and 1.053 /-LID in phosphate glass. The energy-level fine structure differs in the two materials (see, e.g., Fig. 15.3-2), but this cannot be resolved in the figure. The behavior of trivalent lanthanide ions in a dielectric host and in isolation is rather similar. This results from the fact that the 4f electrons are well shielded from external effects of the lattice by the filled 58 and 5p subshells (see Table 13.1-1). This is in sharp contrast to the behavior of transition-metal ions. Thus, unlike ruby and alexandrite, rare-earth-ion energy levels are essentially independent of the host material. This is illustrated in Fig. 13.1-9 for two hosts that are quite different: Nd 3 + in YAG and Nd 3 + in glass. The principal near-infrared laser transition in the two materials, corresponding
496 CHAPTER 13 PHOTONS AND ATOMS to the energy differences between the 4F 3 / 2 and 41 11 / 2 levels, are remarkably close to each other: 1.064 /-lID for Nd 3 +:YAG and 1.053 /-lID for Nd 3 +:glass. Although the fine- structure manifolds resulting from crystal-field splittings differ in the two materials, this feature plays a relatively minor role and cannot be resolved in the figure. Actinide-metal dopant ions. The actinides, which are sometimes also gathered un- der the rubric of the rare earths, are constructed by incrementing the number of elec- trons in the 5 f subshell, which lies deep within the filled 78 2 subshell. The chemical behavior of the actinides is similar to that of their lanthanide homologs (see Fig. 13.1- 3). Although lasers have been constructed from actinide-metal ions in dielectric hosts (e.g., U 3 +:CaF 2 ), these efforts are generally impeded by the radioactivity of these elements. Semiconductors Semiconductors find widespread use in photonics. They are used as sources such as light-emitting diodes and laser diodes, and as detectors, and play many other important roles as well. We provide a brief introduction to the energy levels of bulk semiconduc- tors, quantum wells, quantum wires, and quantum dots. A more extensive exposition relating to the energy levels of semiconductors is provided in Chapter 16. Bulk semiconductors. The binary semiconductor GaAs was early on found to be useful in photonics. This material takes the form of a zincblende structure comprising two face-centered-cubic lattices, one of Ga atoms and the other of As atoms, displaced from each other by the length of a body diagonal (Fig. 13.1-10). Four molecules of GaAs are present in the conventional cell, which is a cube. Each atom is surrounded by four atoms of the opposite type, equally spaced and located at the corners of a regular tetrahedron. Semiconductors have many closely spaced allowed electron energy levels that take the form of bands, as displayed in Fig. 13.1-10 for GaAs. The bandgap energy E 9' which is the energy separating the valence and conduction bands, is 1.42 e V at room temperature. The Ga and As (3d) core levels are quite sharp, as is apparent GaAs 5 Conduction band Eg - 1 Laser 1.42 eV T o Valence band ",-..., > Q.) '-' -5 O/J $.-< Q.) -10 Ga -20 - 30 -40 -50 . . . Core levels As . . . GGa I) As Figure 13.1-10 The semiconductor GaAs takes the form of a zincblende crystal struc- ture comprising two face-centered-cubic lat- tices, one of Ga and the other of As. The higher energy levels are closely spaced and form bands. The zero of energy is (arbitrarily) defined at the top edge of the valence band. The GaAs laser diode operates on the elec- tron transition between the conduction and valence bands, in the near-infrared region of the spectrum (see Chapter 17).
13.1 ENERGY LEVELS 497 in Fig. 13.1-10. The valence band of GaAs is formed from the 48 and 4p levels (as illustrated schematically in Fig. 13.1-7). The properties of semiconductors are examined in greater detail in Chapter 16. Quantum wells. Crystal-growth techniques, such as molecular-beam epitaxy and vapor-phase epitaxy, can be used to grow materials with specially designed band struc- tures. In semiconductor quantum-well structures, the energy bandgap is engineered to vary with position in a specified manner, leading to materials with unique electronic and optical properties. An example is the multiquantum-well structure illustrated in Fig. 13.1-11. It consists of ultrathin (2- to 15- nm) layers of GaAs alternating with thin (20-nm) layers of AIGaAs. The bandgap of the GaAs is smaller than that of the AIGaAs. For motion perpendicular to the layer, the allowed energy levels for electrons in the conduction band, and for holes in the valence band, are discrete and well separated, like those of the square-well potential in quantum mechanics (see Exercise 16.1-5); the lowest energy levels are shown schematically in each of the quantum wells. The AIGaAs barrier regions can also be made ultrathin « 1 nm), in which case the electrons in adjacent wells can readily couple to each other via quantum-mechanical tunneling and the discrete energy levels broaden into miniature bands called minibands. The material is then called a superlattice structure because the minibands arise from a lattice that is "super to" (i.e., larger than) the spacing of the natural atomic lattice structure. <l) Conduction band GaAs AIGaAs Valence band o 20 40 60 80 100 120 Distance (nm) Figure 13.1-11 Quantized energies in a single-crystal AIGaAsjGaAs multiquantum-well struc- ture. The well widths can be periodic or arbitrary (as shown). Quantum wires. A semiconductor material that takes the form of a thin wire sur- rounded by a material of wider bandgap is known as a quantum wire. The wire acts as a potential well that narrowly confines electrons (and holes) in the two lateral directions but not in the direction along the axis of the wire. Quantum wires are readily made from 111- V and II-VI materials, such as InP and CdSe, respectively; they can have rectangular or circular cross section. N anotubes and nanowires, fabricated from a vast array of materials, can behave as quantum wires. In particular, carbon nanotubes, cylindrical carbon molecules with diameters of one or a few nm, display remarkable properties. The carbon molecules organize themselves into thin hollow ropes held together by van der Waals forces. Single- or multi walled nanotubes exhibit unique optical, mechanical, and electrical properties. They can behave as semiconductors or highly conductive metals, depending on their precise structure. There are a multitude of uses for carbon nanotubes in photonics, ranging from filaments for incandescent light sources to photovoltaic detectors.
498 CHAPTER 13 PHOTONS AND ATOMS Quantum dots. Also known as nanocrystals and quantum boxes, quantum dots are semiconductor particles whose dimensions usually range from 1 nm to 10 nm, but can extend to several /-lm. They can be fabricated from many different kinds of semiconductors and in many geometrical shapes (e.g., cubes, spheres, and pyramids), and are usually embedded in larger-bandgap semiconductor materials or in glasses and polymers. They are sometimes fabricated as disk-shaped structures using molecular- beam epitaxy or chemical-vapor deposition, in which case the electrons are restricted to motion in a plane, exhibiting 2D atomic-like shell structures not unlike those associated with the toroidal resonators considered in Sec. 10.4B. They can also be created by electron-beam lithography, where a pattern is etched onto a semiconductor chip and conducting metal is deposited onto the pattern. Quantum dots can be readily grown in a beaker using wet chemistry; colloidal nanocrystals are supplied in liquid suspension or dispersed in a plastic composite. The sizes of quantum dots, and thus the number of atoms they contain, can be varied over a broad range. The number of electrons can be as small as just a few or as large as millions; a 10-nm cube of GaAs contains some 40,000 atoms. All electrons belong to the dot as a whole; the energy levels are those of its excitons, namely the electron-hole pairs generated within, and confined to, the dot. As with atoms, a series of sharp energy levels results from tight electron confinement; indeed, quantum dots are often referred to as artificial atoms. Unlike atoms, however, a quantum dot fabricated from a given material has the unusual property that its energy levels are strongly dependent on its size. The color of light elicited from a CdSe quantum dot by photoexcitation, for example, can be grad- ually tuned from the red region of the spectrum for a 5-nm-diameter dot, to the violet region for a l.5-nm-diameter dot; the trend is illustrated in Fig. 13.1-12. The emitted Figure 13.1-12 Photoluminescence from colloidal CdSe quantum dots (with oleyl amine surface capping molecules) dispersed in n- hexane, in response to ultraviolet excitation at Ao = 365 nm. Quantum confinement effects al- low the emission color to be tuned with particle size (courtesy Dong-Kyun Seo, Arizona State University). photon energy increases as the dot size decreases because greater energy is required to confine the semiconductor excitation to a smaller volume. The photoexcitation wave- length is arbitrary, as long as it is to the blue of the emission wavelength. Quantum dots fabricated from InP luminesce in the near infrared, whereas those fabricated from InAs emit across the 1300-1600-nm silica-fiber-based telecommunications window. Photoexcited Si quantum dots also emit over a broad spectral range, extending from the infrared to the visible (see Sec. 17.4A). Quantum dots can also been fabricated from organic compounds. Quantum dots overcoated with a semiconductor of higher-bandgap are known as "core-shell quantum dots," whereas those overcoated with multiple semiconductors of alternating higher and lower bandgaps are known as "quantum-well-quantum dots." Such overcoatings can substantially improve the tunability and photoluminescence efficiency of the nanostructure. Ordered arrangements of quantum dots, known as quantum-dot solids, can be grown by a number of methods, including the self- assembly of spherical nanocrystals into a close-packed configuration. In the same way that tunneling can occur in multiquantum-well superlattices, so too can it occur in quantum-dot solids known as nanocrystal superlattices.
13.2 OCCUPATION OF ENERGY LEVELS 499 Quantum-dot structures are sometimes large enough to be connected to electrodes, in which case they can serve as miniature photonic devices. By constructing arrays of quantum dots of different sizes in specially designed configurations, they can sustain currents and operate over broad, or specially chosen, wavelength ranges. Quantum dots are useful as spectral tags in biological, commercial, and military applications, as well as in the detection of counterfeiting. They also find use in a broad array of applications such as efficient lasers, broadband light-emitting diodes, single-photon sources, mem- ory elements, photodetectors, solar cells, flat-panel displays, and absorbers in materials where it is desirable to filter out ultraviolet light. 13.2 OCCUPATION OF ENERGY LEVELS As indicated earlier, each atom or molecule in a collection continuously undergoes ran- dom transitions among its different energy levels. These transitions are characterized by the rules of statistical physics. Temperature is the principal determinant of both the average behavior and the fluctuations in energy-level occupancy. A. Boltzmann Distribution Consider a collection of distinguishable objects, such as atoms or molecules that form a dilute gas. Each atom is in one of its allowed energy levels E 1, E 2, . . .. If the system is in thermal equilibrium at temperature T (i.e., the atoms are kept in contact with a heat bath maintained at temperature T and their motion reaches a steady state in which the fluctuations are, on average, invariant to time), the probability P ( Em) that an arbitrary atom is in energy level Em is given by the Boltzmann distribution P(Ern) ex exp(-Em/kT), m == 1,2,3, . . . , (13.2-1) where k is the Boltzmann constant. The coefficient of proportionality is chosen such that Lm P( Em) == 1. The occupation probability P( Em) is an exponentially decreas- ing function of Em, as displayed in Fig. 13.2-1. \ Em Em \ \ \ \ \ E3 E 2 E3 E 2 '- '- El El " "- ........ ........ " '/////////////// / Energy levels Occupation P(Em) Figure 13.2-1 The Boltzmann distribution provides the probability that energy level Em of an arbitrary atom is occupied; it is an exponentially decreasing function of Em. The origin of Boltzmann distribution can be understood by considering a system of many identical entities that share a fixed total energy E. The entities are isolated
500 CHAPTER 13 PHOTONS AND ATOMS from their surroundings but are in thermal equilibrium, exchanging energy among themselves via a bath at temperature T. The divisions of energy are taken to be dis- tinguishable if they involve different energy states, and aU possible divisions of the total energy are assumed to occur with equal probability. If one of the entities takes a large share of the total energy, less is available for the remaining constituents so there are fewer possible divisions. Consequently, large energies are less probable than small energies. A quantitative description is provided by considering two entities. Because they are independent, the probability of finding one with energy E 1 and the other with energy E 2 is the product P ( E 1) P ( E 2). If the sum of the energies of the two entities is fixed at the value E 1 + E 2 , then P( E 1 )P( E 1 ) must be a function of (E1 + E 2 ), which uniquely specifies an exponential function. The equipartition energy kT for the two degrees of freedom associated with a harmonic mode leads directly to the Boltzmann distribution. Consider the Boltzmann distribution in the context of a large number of atoms N. If N m is the number of atoms occupying energy level Em, the fraction Nm/N P(Em). If N 1 atoms occupy level 1 and N 2 atoms occupy a higher level 2, the population ratio IS, on average, N 2 = cx p ( - E2 - E1 ) . N 1 kT (13.2-2) This quantity depends on the temperature T. At T == 0° K, all atoms are in the lowest energy level (ground state). As the temperature increases the populations of the higher energy levels grow. Under equilibrium conditions, the average population of a given energy level is always greater than that of a higher-lying level. This condition need not hold under nonequilibrium conditions, however, when a higher energy level can have a greater average population than a lower energy level. This state of affairs, known as a population inversion, provides the basis for laser action (see Chapters 14 and 15). It was assumed in the foregoing that there is a unique way in which an atom can find itself in one of its energy levels. It is sometimes the case, however, that several different states can correspond to the same energy (e.g., different states of angular momentum). To account for such degeneracies, (13.2-2) can be written in a more general form: N 2 = 92 cx p ( - E2 - E1 ) . N 1 91 kT (13.2-3) The degeneracy parameters 92 and 91 represent the numbers of states corresponding to the energy levels E 2 and E 1, respectively. B. Fermi-Dirac Distribution Quantum systems with overlapping wavefunctions, such as multielectron atoms and semiconductors, are subject to the Pauli exclusion principle. A state may then be occupied by at most one electron; the number of electrons N m in state m is either o or 1. The probability of occupancy of a state of energy E is then described by the Fermi-Dirac distribution (or Fermi function), f(E) = exp[(E - f)/kT] + 1 ' (13.2-4 ) where E f is known as the Fermi energy. This quantity has a maximum value of unity, indicating that the state of energy level E is definitely occupied. It decreases monotonically with increasing E, assuming a value of at the Fermi energy E == E f.
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 501 E Boltzmann P(Em) Ef Fermi-Dirac f(E) o 1/2 1 Figure 13.2-2 The Fermi-Dirac distribu- tion f (E) is well approximated by the Boltz- mann distribution P(Em) when E» Ef. It is important to recognize that f (E) is neither a probability density function nor a probability distribution function, but rather a distribution (sequence) of probabilities for different values of E, each of which stretches between 0 and 1. Nevertheless, when E » E f (and E » kT) the Fermi function behaves like the Boltzmann probability distribution, P(E) ex exp(-EjkT), (13.2-5) as is evident from (13.2-4). The Fermi-Dirac and Boltzmann distributions are com- pared in Fig. 13.2-2. Since in general E » E f for atomic electrons in outer subshells, energy levels involving optical transitions are often characterized by the Boltzmann distribution. The Fermi function is discussed in further detail in Chapter 16. 13.3 INTERACTIONS OF PHOTONS WITH ATOMS A. Interaction of Single-Mode Light with an Atom As is known from atomic theory, an atom may emit (create) or absorb (annihilate) a photon by undergoing downward or upward transitions between its energy levels, conserving energy in the process. The laws that govern these processes are described in this section. The interactions of photons with electrons and holes in semiconductors is considered in Sec. 16.2. Interaction Between an Atom and an Electromagnetic Mode Consider the energy levels E 1 and E 2 of an atom placed in an optical resonator of volume V that can sustain a number of electromagnetic modes. We are particularly interested in the interaction between the atom and the photons of a prescribed radiation mode of frequency v vo, where hvo == E 2 - E 1, since photons of this energy match the atomic energy-level difference. Such interactions are formally studied by the use of quantum electrodynamics. The key results are presented below, without proof. Three forms of interaction are possible - spontaneous emission, absorption, and stimulated emISSIon. Spontaneous Emission If the atom is initially in the upper energy level, it may decay spontaneously to the lower energy level and release its energy in the form of a photon (Fig. 13.3-1). The photon energy hv is added to the energy of the electromagnetic mode. The process is
502 CHAPTER 13 PHOTONS AND ATOMS called spontaneous emission because the transition is independent of the number of photons that may already be in the mode. 2 hv Figure 13.3-1 Spontaneous emission of a pho- ton into the mode of frequency v by an atomic transition from energy lvel 2 to energy level ]. The photon energy hv E 2 - El. 1 In a cavity of volume V, the probability density (per second), or rate, for this spontaneous transition depends on v in a way that characterizes that atomic transition, c Psp == V a(v). (13.3-1) Spontaneous Emission into a Prescribed Mode The function a(v) is a function of v centered about the atomic resonance frequency Vo; it is known as the transition cross section. The significance of this desination will become apparent subsequently, but it is clear that it has dimensions of cm (since Psp has dimensions of S-I). In principle, a(v) can be determined from the Schrodinger equation; the calculations are usually sufficiently complex, however, that a(v) is usu- ally determined experimentally. Equation (13.3-1) applies separately to every mode, with a cross section given by a == a max cos 2 B, (13.3-2) where B is the angle between the dipole moment of the atom and the field direction of the mode; the maximum cross section a max is attained when the dipole moment and field are aligned. N(t) 1 Ps p t Figure 13.3-2 Spontaneous emission into a single mode results in an exponential de- crease of the number of excited atoms with time constant 1/ Psp. N(O) The term "probability density" signifies that the probability of an emission taking place in an incremental time interval between t and t + 6.t is simply Psp 6.t. Because it is a probability density, Psp can have a numerical value greater than 1 S-l, although, of course, Psp6.t must always be smaller than 1. Thus, if there are a large number N of such atoms, a fraction of approximately 6.N == (Psp 6.t)N atoms will undergo the transition within the time interval 6.t. We can therefore write dN / dt == -Psp N, so that the number of atoms N(t) == N(O) exp( -Pspt) decays exponentially with time constant l/psp, as illustrated in Fig. 13.3-2.
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 503 Absorption If the atom is initially in the lower energy level and the radiation mode contains a photon, the photon may be annihilated and the atom concomitantly raised to the upper energy level (Fig. ] 3.3-3). This process, which is induced by the photon, is called absorption. It can occur only when the mode contains a photon. 2 1 I . Figure 13.3-3 Absorption of a photon of en- ergy hv leads to an upward transition of the atom from energy level 1 to energy level 2. hv The probability density for the absorption of a photon from a given mode of fre- quency v, in a cavity of volume V, is governed by the same law that governs sponta- neous emission into that mode, C Pab == V a (v). (13.3-3) However, if there are n photons in the mode, the probability density that the atom absorbs one photon is n times greater since the events are mutually exclusive, i.e., C P ab == n - a(v). V ( 13 .3 -4 ) Absorption of One Photon from a Mode with n Photons Stimulated Emission Finally, if the atom is in the upper energy level and the mode contains a photon, the atom may be induced to emit another photon into the same mode. This process is known as stimulated emission. It is the inverse of absorption. The presence of a photon in a mode of specified frequency, propagation direction, and polarization stimulates the emission of a duplicate ("clone") photon with precisely the same characteristics as the original (Fig. 13.3-4). This photon amplification process is the phenomenon that underlies the operation of laser amplifiers and lasers, as will be elucidated in subsequent chapters. 2 hv hv hv Figure 13.3-4 Stimulated emission is a process whereby a photon of energy hv stimulates the atom to emit a clone photon as it undergoes a downward transition. 1 The probability density Pst that this process occurs in a cavity of volume V is governed by the same law that governs spontaneous emission and absorption: C Pst == V a(v). (13.3-5)
504 CHAPTER 13 PHOTONS AND ATOMS If the mode originally carries n photons, the probability density that the atom is stim- ulated to emit an additional photon is, as in the case of absorption, c Pst == n V a(v). (13.3-6) Stimulated Emission of One Photon into a Mode with n Photons Since Pst == P ab , we make use of a common notation, Wi, to represent the probability density of stimulated emission and absorption. Inasmuch as spontaneous emission is present in addition to stimulated emission, the overall probability density that the atom emits a photon into the mode is given by Psp + Pst == (n + l)(c/V)a(v). From a quantum electrodynamic point of view, spontaneous emission may be regarded as stimulated emission induced by the zero- point fluctuations associated with the mode (see Sec. 12.1A). Because the zero-point energy is not applicable for absorption, Pab is proportional to n rather than to (n + 1). The three possible interactions between an atom and a cavity radiation mode (spon- taneous emission, absorption, and stimulated emission) obey the fundamental relations set forth above. These should be regarded as the laws that govern photon-atom inter- actions, supplementing the rules of photon optics provided in Chapter 12. We now proceed to discuss the character and consequences of these rather simple relations in some detail. The Lineshape Function The transition cross section a(v) characterizes the interaction of the atom with the radiation. Its area, s = l°Oa(v) dv, (13.3-7) which has units of cm 2 -Hz, is called the transition strength or oscillator strength and represents the strength of the interaction. Its shape governs the relative magni- tude of the interaction with photons of different frequencies. The shape (profile) of a(v) is readily separated from its strength by defining a normalized function g(v) == a(v)/ S, known as the lineshape function, which has units of Hz-l and unity area: Jo oo g(v) dv == 1. The transition cross section can then be written in terms of its strength and profile as a(v) == Sg(v). (13.3-8) The lineshape function g(v) is centered about the resonance frequency Yo, where a(v) is largest, and drops sharply as v deviates from Yo. Transitions are therefore most likely for photons of frequency v Yo. The width of the function g(v) is known as the transition linewidth. The linewidth v is usually defined as the full width of the function g(v) at half its maximum value (FWHM) (see Sec. A.2 of Appendix A). Since the area of g(v) is unity, its width is inversely proportional to its central value: v ex l/g(vo). (13.3-9) It is also useful to define a peak transition cross section at the resonance frequency, ao == a(vo). The function a(v) is then characterized by its height ao, width v, area S, and profile g(v), as illustrated in Fig. 13.3-5.
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 505 a(v) Area = S ao g(v) vo v vo v Figure 13.3-5 The transition cross section a(v) and the lineshape function g(v). B. Spontaneous Emission Total Spontaneous Emission into All Modes Equation (13.3-1) provides the probability density Psp for spontaneous emission into a specific mode of frequency v (without regard to whether the mode contains photons). As indicated in (10.3-10), the density of modes for a three-dimensional cavity increases quadratically as M(v) == 81TV 2 / c 3 . This quantity approximates the number of modes of frequency v, per unit volume of the cavity per unit bandwidth, provided that the number of modes is sufficiently large so that a continuous approximation for their number may be used. An atom may spontaneously emit one photon of frequency v into any of these modes, as shown schematically in Fig. 13.3-6. . Atom '//////////////// Optical modes Figure 13.3-6 An atom may spontaneously emit a photon into anyone (but only one) of the many optical modes with frequencies v Yo. The probability density for spontaneous emission into all modes is therefore given by the probability density for spontaneous emission into each mode, weighted by the modal density. Since modes at each frequency have an isotropic distribution of directions, each with two polarizations, we must determine the average transition cross section a (v). If B is the angle between the dipole moment of the atom and the field direction, (13.3-2) leads to a (v) == la max , (13.3-10) since (cos 2 B) == l, where (.) represents an average in 3D space. The overall spontaneous-emission probability density therefore becomes P sp = 1 00 [ a (v)] [VM(v)] dv = c l °Oa (v) M(v) dv. (13.3-11)
506 CHAPTER 13 PHOTONS AND ATOMS Because the function a (v) is sharply peaked, it is narrow in comparison with the quadratic function M(v). Since a (v) is centered about vo, M(v) is essentially constant with a value M(vo), and can thus be removed from the integral. The probability density of spontaneous emission of one photon into any mode is therefore given by - 81T S P sp == M(vo) cS == ' where A == c/vo is the wavelength of the light in the medium and S == Io oo a (v) dv. We define a time constant t sp , known as the spontaneous lifetime of the 2 1 transition, such that l/t sp P sp == M(vo) cS . Thus, (13.3-] 2) 1 P sp == - , t sp (13.3-13) Spontaneous Emission of One Photon into Any Mode which is independent of the cavity volume V. This permits us to express S as _ A 2 S== 81Tt sp , (13.3-14) which enables the transition strength to be determined from an experimental measure- ment of the spontaneous lifetime t sp . Equation (13.3-14) is useful because an analytical calculation of S would require intimate knowledge about the quantum-mechanical behavior of the system, which is not always available. Typically, t sp 10- 8 s for atomic transitions such as the first excited state of atomic hydrogen; however, t sp can vary over a large range, from subpicoseconds to minutes. EXERCISE 13.3-1 Frequency of Spontaneously Emitted Photons. Show that the probability density for an excited atom spontaneously emitting a photon of frequency between v and v + dv is Pp(v) dv == (l/isp)g(v) dv. Explain why the spectrum of spontaneous emission from an atom is proportional to its lineshape function g(v) after a large number of photons have been emitted. Relation Between Transition Cross Section and Spontaneous Lifetime Using (13.3-14) and the relation a (v) == Sg (v) shows that the average transition cross section is related to the spontaneous lifetime and the lineshape function via (13.3-15) Average Transition Cross Section The average transition cross section at the central frequency vo is therefore A 2 a (v) == g(v). 81Tt sp A 2 a o a (vo) == g(vo). 81Tt sp (13.3-16)
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 507 Because g(vo) is inversely proportional to v, in accordance with (13.3-9), the peak transition cross section a o is inversely proportional to the linewidth v, for a given value of t sp . The transition cross section a for stimulated emission into a particular mode, and the associated peak transition cross section ao, obey relations identical to those provided in (] 3.3-15) and (13.3-16), except that the effective spontaneous emission time is reduced by virtue of (13.3-2) and (13.3-10). For simplicity, we shall henceforth not always distinguish between t sp for spontaneous emission and its effec- tive value for stimulated emission. c. Stimulated Emission and Absorption Transitions Induced by Monochromatic Light We now consider the interaction of single-mode light with an atom when a stream of photons impinges on it, rather than when it is in a resonator of volume V as considered above. Let monochromatic light of frequency v, intensity I, and mean photon-flux density (photons/cm 2 -s) fjJ= hv (13.3-17) interact with an atom whose resonance frequency is yo. We wish to determine the probability densities for stimulated emission and absorption, Wi P ab == Pst, in this configuration. The number of photons n involved in the interaction process is determined by constructing a volume in the form of a cylinder of base area A, height c, and volume V == cA. The axis of the cylinder is parallel to k, the direction of propagation of the light. The photon flux that crosses the cylinder base is <I> == cj;A (photons/s). Because photons travel at the speed of light c, all of the photons within the volume of the cylinder cross its base within one second. It follows that, at any time, the cylinder contains n == cj;A == cj; V / c photons so that c cj;==n- V. (13.3-18) To determine Wi, we substitute (13.3-18) into (13.3-4) to obtain Wi == cj; a(v). (13.3-19) It is apparent that a (v) is the coefficient of proportionality between the probability density of an induced transition and the photon-flux density. The name "transition cross section" is thus apt: cj; is the photon flux per cm 2 , a(v) is the effective cross-sectional area of the atom (cm 2 ), and cj; a(v) is the photon flux "captured" by the atom for the purpose of absorption or stimulated emission. Whereas the spontaneous emission rate is enhanced by the many modes into which an atom can decay, stimulated emission involves decay only into modes that contain photons. Its rate is enhanced by the possible presence of a large number of photons in few modes. Transitions Induced by Broadband Light Consider now an atom in a cavity of volume V containing multimode polychromatic light of spectral energy density (}( v ) (energy per unit bandwidth per unit volume) that
508 CHAPTER 13 PHOTONS AND ATOMS is broadband in comparison with the atomic linewidth. The average number of photons in the frequency band from v to v + dv is {]( v) V dv j hv; each of these has a probability density (cjV)a(v) of initiating an atomic transition, so that the overall probability of absorption or stimulated emission is Wi = roo e(v)V [ a(v) ] dv. Jo hv V (13.3-20) Since the radiation is broadband, the function {]( v) varies slowly in comparison with the sharply peaked transition cross section a(v). We can therefore replace {](v)jhv under the integral with {]( vo) j hvo, which leads to W . - {](vo) 1 00 ( ) d - {](vo) S - c avv- c. hvo 0 hvo (13.3-21) Using (13.3-14), we therefore have A 3 Wi == {](vo), 81T ht sp (13.3-22) where A == cjvo is the wavelength in the medium at the central frequency vo. Defining _ A 3 n == 81Th {](vo), (13.3-23) which represents the mean number of photons per mode, allows us to write (13.3-22) in the convenient form n Wi == - . t sp (13.3-24 ) The interpretation of n as the mean number of photons per mode follows from the form of the ratio Wi A 3 {](vo) 1 Psp 81Tht sp M(vo)cS {] ( vo ) . hvoM(vo) , (13.3-25) the quantity {]( vo) j hvo represents the mean number of photons per unit volume in the vicinity of the frequency Vo while M(vo) is the number of modes per unit volume in the vicinity of Vo. The probability density Wi is thus a factor of n greater than that for spontaneous emission, since each mode contains an average of n photons. Einstein Coefficients Although Einstein did not have knowledge of (13.3-22), he carried out an important analysis of the energy exchange between atoms and radiation that permitted him to obtain general expressions for the probability densities of spontaneous and stimulated transitions. Assuming that the atoms interacted with broadband radiation of spectral energy density {]( v ), under conditions of thermal equilibrium, he obtained the follow- . . Ing expressIons:
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 509 P sp == A Wi == Iffig( Vo). (13.3-26) (13.3-27) Einstein's Postulates The constants A and Iffi are known as the Einstein A and Iffi coefficients. Comparison with (13.3-13) and (13.3-22) reveals that the A and Iffi coefficients are A== t sp A 3 Iffi== 87f ht sp (13.3-28) (13.3-29) which are associated with spontaneous and stimulated transitions, respectively. The ratio is given by Iffi A 3 A 87fh . (13.3-30) The relation between the A. and Iffi coefficients is a result of the microscopic (rather than macroscopic) probability laws of interaction between an atom and the photons of each mode. We shall present an analysis similar to that provided by Einstein in Sec. 13.4. EXAMPLE 13.3-1. Comparison Between Rates of Spontaneous and Stimulated Emission. Whereas the rate of spontaneous emission for an atom in the upper state is constant at A == 1/tsp, the rate of stimulated emission in the presence of broadband light, ]ffiQ(vo), is proportional to the spectral energy density of the light, Q(vo). The two rates are equal when Q(vo) == A/]ffi == 87r h I A 2 ; for larger values of the spectral energy density, the rate of stimulated emission exceeds that of spontaneous emission. If A == 1 J1m, for example, A/]ffi == 1.66 x 10- 14 J 1m3 -Hz. This corresponds to an intensity spectral density cQ(vo) :::::: 5 x 10- 6 W 1m 2 -Hz in free space. Thus, for a linewidth /j,v == 10 7 Hz, the optical intensity at which the stimulated emission rate equals the spontaneous emission rate is 50 W 1m2 or 5 m W I em 2 . Summary An atomic transition may be considered in terms of its resonance frequency Vo == (E 2 - E1)/h, spontaneous lifetime t sp , and lineshape function g(v), which has linewidth l::1v. The average transition cross section is _ A 2 a (v) == Sg(v) == 8 g(v). 7ft sp (13-3-15)
510 CHAPTER 13 PHOTONS AND ATOMS Spontaneous Emission . If the atom is in the upper level and in a cavity of volume V, the probability density (per second) of emitting spontaneously into one prescribed mode of frequency v is C Psp == V a(V). (13-3-1) . The probability density of spontaneous emission into any of the available modes is 87rS P sp == V 1 (13-3-13) t sp . The probability density of emitting into modes lying only in the frequency band between v and v + dv is P sp dv == (l/tsp)g(v) dv. Stimulated Emission and Absorption . If the atom in the cavity is in the upper level and a radiation mode contains n photons of frequency v, the probability density of emitting a photon into that mode is C Wi == n V a(v). (13-3-6) If the atom is instead in the lower level, and a mode contains n photons, the probability of absorption of a photon from that mode is also given by (13.3-6). . If instead of being in a cavity, the atom is illuminated by a monochromatic beam of light of frequency v, with mean photon-flux density cP (photons per second per unit area), the probability density of stimulated emission (if the atom is in the upper level) or absorption (if the atom is in the lower level) i Wi == cP a(v). (13-3-19) . If the light illuminating the atom is polychromatic, but narrowband in com- parison with the atomic linewidth, and has a mean photon-flux spectral den- sity cPv (photons per second per unit area per unit frequency), the probability density of stimulated emission/absorption is Wi == J cPv a(v) dv. (13.3-31) . If the light illuminating the atom has a spectral energy density g(v) that is broadband in comparison with the atomic linewidth, the probability density of stimulated emission/absorption is Wi == :IEg(vo), ( 13-3-27) where :IE == A 3 / 87r ht sp is the Einstein JB coefficient. In all of these formulas, C == Co / n is the velocity of light and A == Ao / n is the wavelength of light in the atomic medium, and n is the refractive index.
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 511 D. Line Broadening Because the lineshape function g(v) plays an important role in atom-photon interac- tions, we devote this subsection to a brief discussion of its origin. The same lineshape function applies for spontaneous emission, absorption, and stimulated emission. Lifetime Broadening Atoms can undergo transitions between energy levels by both radiative and nonradia- tive processes. Radiative transitions result in photon absorption and emission. Non- radiative transitions permit energy transfer by mechanisms such as lattice vibrations, inelastic collisions among the constituent atoms, and inelastic collisions with the walls of the vessel. Each atomic energy level has a lifetime T, which is the inverse of the rate at which its population decays, radiatively or nonradiatively, to all lower levels. The lifetime T2 of energy level 2 shown in Fig. 13.3-1 represents the inverse of the rate at which the population of that level decays to level 1 and to all other lower energy levels (none of which are shown in the figure), by either radiative or nonradiative means. Since 1/ t sp is the radiative decay rate from level 2 to levell, the overall decay rate 1/T2 must be greater, i.e., I/T2 > l/t sp , corresponding to a shorter decay time, T2 < t sp . The lifetime Tl of level 1 is defined similarly. Clearly, if level 1 is the lowest allowed energy level (the ground state), Tl == 00. Lifetime broadening is, in essence, a Fourier transform effect. The lifetime T of an energy level is related to the time uncertainty of the occupation of that level. As shown in Sec. A.l of Appendix A, the Fourier transform of an exponentially decaying harmonic function of time e- t / 2T ej27rvot, which has an energy that decays as e- t / T , with time constant T, is proportional to 1/[1 + j47r(v - VO)T]. The full width at half- maximum (FWHM) of the absolute square of this Lorentzian function of frequency is l::1v == 1/27rT. This spectral uncertainty corresponds to an energy uncertainty l::1E == hl::1v == h/27rT. An energy level with lifetime T therefore has an energy spread l::1E == h/27rT, provided that we can model the decay process as a simple exponential. In this picture, spontaneous emission can be viewed in terms of a damped harmonic oscillator, which generates an exponentially decaying harmonic function, as embodied in the Lorentz oscillator model presented in Sec. 5.5C. Thus, if the energy spreads of levels 1 and 2 are l::1E 1 == h/27rTI and l::1E 2 == h/27rT2, respectively, the spread in the energy difference corresponding to the transi- tion between the two levels is h ( l 1 ) hI l::1E == l::1E 1 + l::1E 2 == - - + - == - - , 27r Tl T2 27r T ( 13.3-32) where T is the transition lifetime and T- 1 == (T 1 1 + T21). The corresponding spread of the transition frequency, which is called the lifetime-broadening linewidth, is therefore l::1v == ( + ) . 27r Tl T2 (13.3-33) Lifetime-Broadening Linewidth (E 2 - E 1 )/h, and the lineshape This spread is centered about the frequency va function has a Lorentzian profile: g(v) = b.v /2n (v - vO)2 + (l::1v /2)2 . (13.3-34) Lorentzian Lineshape Function
512 CHAPTER 13 PHOTONS AND ATOMS More generally, the lifetime broadening associated with an atom or a collection of atoms may be modeled as follows. Each of the photons emitted in a transition represents a wavepacket of central frequency Vo (the transition resonance frequency), with an exponentially decaying envelope of decay time 27, which corresponds to an energy decay time equal to the transition lifetime 7. As illustrated in Fig. 13.3-7, the radiated light is taken to be a sequence of such wavepackets emitted at random times. As discussed in Example 11.1-1, this corresponds to random (partially coherent) light with a spectral intensity that is described precisely by the Lorentzian function given in (13.3-34), with v == 1/27r7. a a --+-1 27 a a a a g(v) a 111111.. III/IU.I.II, .1.11111 .. i III,U,III,I 111111111111. I' , , ' , , ' " , , , 'l'n T ' , 'IT" I , , ' " , , , 'I'I'P , 'II fl' , i j I , ' , , ' ' t o Vo v Figure 13.3-7 Wavepacket emissions at random times from a lifetime-broadened atomic system with transition lifetime 7. The light emitted has a Lorentzian spectral intensity of width l/ == 1/21f7. The value of the Lorentzian lineshape function at the central frequency Vo is g(vo) == 2/ 7r v, so that the peak transition cross section, given by (13.3-16), becomes A 2 1 ao == - . 27r 27rtspv (13.3-35) The largest transition cross section occurs under ideal conditions when the decay is entirely radiative so that 72 == t sp and 1/71 == 0 (which is the case when level 1 is the ground state from which no decay is possible). Then v == 1/27rt sp so that A 2 ao == 27r ' (13.3-36) indicating that the peak cross section is of the order of one square wavelength. When level 1 is not the ground state, or when nonradiative transitions are significant, v can be » l/t sp in which case a o can be significantly smaller than A 2 /27r. For example, for optical transitions in the range A == 0.1 to 10 /-L m , A 2 /27r 10- 11 to 10- 7 cm 2 , whereas typical observed values of ao fall in the range 10- 20 to 10- 11 cm 2 (see Table 14.3-1). Collision Broadening Collisions in which energy is exchanged, called inelastic collisions, result in transitions between atomic energy levels. This affects the decay rates and lifetimes of all levels involved and modifies the linewidth of the radiated field considered above. Collisions that do not involve an exchange of energy, called elastic collisions, also modify the linewidth of the radiated field. Elastic collisions impart random phase shifts to the wavefunction associated with the energy level, which in turn results in a random phase shift of the radiated field at each collision time. As illustrated in Fig. 13.3-8, a sine wave whose phase is modified by a random shift at random times (collision times) exhibits spectral broadening. The spectrum of such a randomly dephased function can
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 513 . t Figure 13.3-8 A sine wave interrupted at the rate feol by random phase jumps has a Lorentzian spectrum of width v = feol/7r. t Collision times I I I I . be determined using the theory of random processes. The result is again Lorentzian, with a width v == feol/7r, where feol is the collision rate (mean number of collisions per second).t The Lorentzian lineshape function that accommodates lifetime and col- lision broadening has an overalllinewidth that is the sum of the individuallinewidths, 1 ( 1 1 ) v == - - + - + 2feol . 27r 71 72 (13.3-37) Inhomogeneous Broadening Lifetime broadening and collision broadening are examples of homogeneous broad- ening, in which all of the atoms of a medium are taken to be identical and to have identical lineshape functions. Under some conditions, however, the different atoms constituting a medium have different lineshape functions or different center frequen- cies. [n that case we can define an average lineshape function 9 ( v) == (g j3 ( v ) ) , (13.3-38) where (.) represents an average with respect to the variable (3, which is used to label those atoms with lineshape function 9 (3 (v). The average lineshape function is obtained by weighting g{3(v) by the fraction of the atomic population endowed with the property (3, as pictured in Fig. 13.3-9. Vo Figure 13.3-9 The average lineshape function for an inhomogeneously broadened collection of atoms. v A particular inhomogeneous broadening mechanism is Doppler broadening. As a result of the Doppler effect, an atom moving with velocity v along a given direction exhibits a spectrum that is shifted by the frequency :::l:: ( v / c) Vo when viewed along that direction, where Vo is its central frequency. The shift is in the direction of higher frequency (+ sign) if the atom is moving toward the observer, and in the direction of t See, e.g., A. E. Siegman, Lasers, University Science, 1986, Sec. 3.2.
514 CHAPTER 13 PHOTONS AND ATOMS lower frequency (- sign) if it is moving away. For an arbitrary direction of observation, the frequency shift is ::l::(vlI/c)vo, where VII is the component of velocity parallel to the direction of observation. Since a collection of atoms in a gas exhibits a distribution of velocities, as depicted in Fig. 13.3-10, the light they emit exhibits a range of frequen- cies, which results in Doppler broadening. Direction l1li( . of observatIon 2 3 1 Figure 13.3-10 The frequency radiated by an atom depends on the direction of atomic motion relative to the direction of observa- tion. Radiation from atom I has a higher frequency than that from atoms 3 and 4. Radiation from atom 2 has a lower frequency. 4 For Doppler broadening, the velocity V therefore plays the role of the parameter /3 and g (v) == (gv(v)). As illustrated in Fig. 13.3-11, if p(v) dv is the probability that the velocity of a given atom lies between V and V + dv, the overall inhomogeneous Doppler-broadened lineshape function is g (v)= l:g(v-vo )p(V)dV. (J 3.3-39) ---+-1 Vo , , , , I I I I I I I I I , , , , , " g(v-vo*) o Velocity v o Vo v Figure 13.3-11 Velocity distribution and average lineshape function for a Doppler-broadened atomic system. EXERCISE 13.3-2 Doppler-Broadened Lineshape Function. (a) A collection of atoms in a gas has a component of velocity v along a particular direction that obeys the Gaussian probability density function p(v) = (Jv ex p ( - ;(J ) , where aJ == kT / M and M is the atomic mass. If each atom has a Lorentzian naturallineshape function of width v and central frequency Yo, derive an expression for the average lineshape function 9 (v). (13.3-40)
13.3 INTERACTIONS OF PHOTONS WITH ATOMS 515 (b) Show that if v « voav / c, 9 (v) may be approximated by the Gaussian lineshape function _ 1 [ (v - vo) 2 ] 9 (v) == V2ii aD exp - 2aE ' (13.3-41) where (TD = Vo : = J . The full-width half-maximum (FWHM) Doppler linewidth VD is then VD == V81n 2 aD :::::: 2.35 aD. (13.3-42) (] 3.3-43) (c) Compute the Doppler linewidth for the Ao == 632.8 nnl transition in Ne and for the Ao == 10.6 J1ill transition in CO 2 at room temperature, assuming that v « voav / c. These transitions are used in He-Ne and CO 2 gas lasers, respectively. (d) Show that the maximum value of the transition cross section for the Gaussian lineshape function in (13.3-41) is (To = ). 2 J 41n 2 1 0.94 ). 2 1 . 87r 7r t sp VD 87r t sp VD Compare with (13.3-35) for the Lorentzian lineshape function. (13.3-44 ) Many atom-photon interactions exhibit broadening that is intermediate between purely homogeneous and purely inhomogeneous. Such mixed broadening can be mod- eled by an intermediate lineshape function such as the Voight profile. *E. Enhanced Spontaneous Emission All of the results presented thus far in Sec. 13.3 are predicated on the assumption that v » bv, i.e., that the atomic linewidth v is far greater than the width of an electro- magnetic mode bv. This condition is usually, but not always, obeyed. In the opposite limit, when the atomic linewidth is far smaller than the width of an electromagnetic mode (Fig. 13.3-12), an enhancement of the spontaneous emission probability density can be achieved, particularly in high-Q microcavities, as we proceed to demonstrate. The enhancement of spontaneous emission is desirable for the operation of certain photon sources, as discussed in Sec. ] 7.4. p(v) 2Q 7r V q v 8VF Figure 13.3-12 Spontaneous emISSIon from an atom with normalized lineshape function g(v) into a broader normalized Lorentzian cavity mode p(v). The lineshape- function and cavity-mode center frequencies are designated by Vo and v q , respectively, while their widths are specified by v and 8v. We consider the case where Vo == v q and v « 8v. Vq v g(V) V Vo v Consider the spontaneous emission of an atom with resonance frequency Vo into an electromagnetic mode with center frequency v q == Vo in the regime v « bv, as
516 CHAPTER 13 PHOTONS AND ATOMS portrayed in Fig. 13.3-12. In accordance with (13.3-11), when the dipole moment of the atom is aligned with the field direction of the mode, the probability density for spontaneous emission into a single cavity mode p(v) is given by 1 00 C C 3A 2 1 00 ps;ax == V amax(V) p(v) dv V 8 p(vo) g(v) dv, o n4 p 0 (13.3-45) since amax(v) == 3 a (v) and a (v) == A 2 g(v)j8nt sp , as provided in (13.3-10) and (13.3-15), respectively. Inasmuch as the lineshape function g(v) is normalized, and the height of the normalized Lorentzian lineshape function of the cavity mode is 2Q j nv q , where Q == vqjflv, we obtain P max _ 1 3CA 2 2Q _ 1 3 A 3 Q sp ------.- . t sp 8nV nV q t sp 4n V (13.3-46) The net result is an enhancement of the spontaneous emission probability density relative to that in free space by a quantity known as the Purcell factor: pmax 3 A 3 sp Q Psp == 4n 2 V . ( 13.3-47) Purcell Factor The Purcell factor in (13.3-47) exhibits the following features: . The factor of 3 is a result of the alignment of the dipole moment of the atom and the field direction of the mode. . The quantity A 3 jV, which is the ratio of the cubed wavelength to the cavity volume, is substantially enhanced in a microcavity. . A high value of Q, i.e., a sharp cavity mode, enhances the Purcell factor; however, as Q increases, flv == vqjQ decreases, so that ultimately the condition v « flv is violated. As Vo deviates from v q , the height of the cavity mode at Vo becomes smaller and the en- hancement of spontaneous emission ultimately becomes a suppression of spontaneous emISSIon. *F. Laser Cooling and Trapping of Atoms It is often desirable to slow neutral atoms (laser cooling) and to trap them in a confined region of space (atom trapping). Ultracold atoms offer unparalleled accuracy for atomic clocks. Laser cooling and trapping can be achieved by arranging laser beams in such a way that they selectively impart photon momentum to a beam of atoms with well-regulated velocities (see Sec. 12.1D). Cooled and trapped atoms are essential components of atom optics, a field of research concerned with the manipulation of matter waves. Structured light waves often serve as atom-optical components; as in ordinary wave optics, reflection, refraction, diffraction, interference, and scattering of the matter waves are all observed. Matter-wave interferometry promises exceptionally sensitive measurements of local gravity anomalies. Cooled and trapped atoms are also important for the production of Bose-Einstein condensates (BECs), collections of atoms that are sufficiently slow and dense that their atomic wavefunctions overlap. One of the simplest schemes for laser cooling relies on photons from a laser beam of narrow linewidth, with a center frequency tuned slightly below the atomic line center,
13.4 THERMAL LIGHT 517 interacting with a beam of atoms moving toward the laser beam. After absorption by atoms whose Doppler-shifted frequency matches the photon frequency, an atom can return to the ground state via either stimulated or spontaneous emission. If it returns by stimulated emission, the momentum of the emitted photon is the same as that of the absorbed photon, leaving the atom with no net change of momentum. If it returns by spontaneous emission, on the other hand, the direction of the photon emission is random so that repeated absorptions and emissions result in a net decrease of the atomic momentum in the direction pointing toward the laser beam. The result is a decrease in the velocity of those atoms, as shown schematically in Fig. 13.3-13. Ultimately, the change of atomic momentum (and therefore velocity) results in the atoms moving out of resonance with the laser beam, which can be compensated by sweeping the laser- beam frequency. rfJ E o ..... ro t+-o o <l) .D E ;::::S Z Velocity v Figure 13.3-13 Velocity distribution of a beam of atoms (dashed curve) and the laser- cooled distribution (solid curve). Multiple laser beams can be used to construct an optical trap in which large numbers of neutral atoms can be confined to a small volume of space. The trapped atoms can be rapidly moved about simply by redirecting the laser beam. For trapping to occur, the kinetic energy of the collection of atoms must be sufficiently low so that the atoms cannot jump out of the trap. The use of cooling and trapping techniques can lead to temperatures in the J-lK range for neutral atoms (corresponding to atomic velocities of the order of cmls), many orders of magnitude below the hundreds of mK temperatures offered by ordinary cryogenic cooling. Even lower temperatures can be attained by means of "evaporative cooling," in which the trap depth is lowered so that atoms with energies exceeding it escape, leaving behind less energetic atoms; and via ""subrecoil cooling," in which the atomic-momentum spread is driven below the single-photon recoil momentum, and maintained at that level for long periods of time by virtue of the Levy statistics of the momentum random walk. 13.4 THERMAL LIGHT Under conditions of thermal equilibrium, and in the absence of other external energy sources, a universal form of radiation known as thermal light is emitted from black- bodies (these objects are so-named because they absorb all of the light incident on them). In this section we determine the properties of thermal light by examining the interactions among a collection of photons and atoms in thermal equilibrium, in terms of the processes of spontaneous emission, absorption, and stimulated emission. We also show how the thermal light emitted from an object can be used to image it. A. Thermal Equilibrium Between Photons and Atoms A macroscopic rate-equation approach that balances spontaneous emission, absorption, and stimulated emission, under conditions of thermal equilibrium, leads to the spectral
518 CHAPTER 13 PHOTONS AND ATOMS intensity of thermal light. The point of departure for our analysis is (13.3-13) and (13.3-24), which govern spontaneous emission and induced transitions in the presence of broadband light, respectively. Consider a cavity of unit volume whose walls have a large number of atoms with two energy levels, denoted 1 and 2, that are separated by an energy difference hv. The cavity, which is at temperature T, supports broadband radi- ation. Let N 1 (t) and N 2 (t) represent the numbers of atoms per unit volume occupying energy levels 1 and 2, at time t, respectively. Since some of the atoms are initially in level 2, as ensured by the finite temperature, spontaneous emission creates radiation in the cavity. This radiation in turn can induce absorption and stimulated emission. The three processes coexist and it is assumed that steady-state (equilibrium) conditions are attained. We assume that an average of n photons occupies each of the radiation modes whose frequencies lie within the atomic linewidth, as established in (13.3-24). We first consider spontaneous emission alone. The probability that a single atom in the upper level undergoes spontaneous emission into any of the modes, within the time increment from t to t + t, is Pspt tjtsp. There are N 2 (t) such atoms so that the avrage number of emitted photons within t is N2(t)tjtsp. This is also the number of atoms that depart from level 2 during the time interval t. Hence, the (negative) rate of increase of N 2 (t) arising from spontaneous emission is obtained from the differential equation dN 2 dt N 2 ( 13.4- 1 ) t sp The solution, N 2 (t) N 2 (O) exp( -tjt sp ), is an exponentially decaying function of time, as displayed in Fig. 13.4-1. Given sufficient time, the number of atoms in the upper level N 2 decays to zero with time constant t sp . The energy is carried off by the spontaneously emitted photons. N z (t) N z (0) Figure 13.4-1 Decay of the upper-level population caused by spontaneous emission t alone. We now incorporate absorption and stimulated emISSIon, which contribute to changes in the populations. Since there are N 1 atoms capable of absorption, the rate of increase of the population of atoms in the upper energy level arising from absorption is, based on (13.3-24), dN 2 N w. n N 1 dt 1 t . sp (13.4-2) Similarly, stimulated emission gives rise to a (negative) rate of increase of atoms in the upper state, expressed as dN 2 = -N 2 W i = _ nN 2 dt t sp (13.4-3)
13.4 THERMAL LIGHT 519 The rates of atomic absorption and stimulated emission are proportional to n , the average number of photons in each mode. Combining (13.4-]), (13.4-2), and (13.4-3) to accommodate spontaneous emission, absorption, and stimulated emission together, yields the rate equation dN 2 N 2 n N 1 n N 2 - == -- + dt t sp t sp t sp (13.4-4) Rate Equation This equation ignores transitions into or out of level 2 that arise from other effects, such as interactions with other energy levels, nonradiative transitions, and external sources of excitation. Steady-state demands that dN 2 / dt == 0, which leads to N 2 N 1 n 1 + n . (13.4-5) Clearly, N 2 / N 1 < 1. If we now make use of the fact that the atoms are in thermal equilibrium, (13.2-2) dictates that their populations obey the Boltzmann distribution: z = ex p ( - E\;El ) = ex p ( - :; ). Substituting (13.4-6) into (13.4-5) leads to a mean number of photons per mode near the frequency v that is given by (13.4-6) 1 n== exp(hv / kT) - 1 . ( 13.4-7) The foregoing derivation is predicated on the interaction of two energy levels cou- pled by absorption, as well as by stimulated and spontaneous emission, at a frequency near v. The applicability of (13.4-7) is, however, far broader. This may be understood by considering a cavity whose walls are made of solid materials that possess a contin- uum of energy levels at all energy separations, and therefore an values of v. Atoms in the walls spontaneously emit into the cavity. The emitted light subsequently interacts with the atoms, giving rise to absorption and stimulated emission. If the walls are maintained at temperature T, the combined system of atoms and radiation reaches thermal equilibrium. Equation (13.4-7) is identical to (12.2-21) - the expression for the mean photon number in a mode of thermal light for which the occupation of the modal energy levels follows the distribution p( n) ex: exp( - n hv / kT). This indicates a self-consistency in our analysis. Photons interacting with atoms in thermal equilibrium at temperature T are themselves in thermal equilibrium at the same temperature T (see Sec. 12.2C). A collection of such photons is often termed a "photon gas." B. Blackbody Radiation Spectrum Based on the discussion provided in Sec. 13.4A, the average energy E of a radiation mode is simply n hv, where n is given by (13.4-7), so that E == hv exp(hv / kT) - 1 . (13.4-8) Average Energy of a Mode in Thermal Equilibrium
520 CHAPTER 13 PHOTONS AND ATOMS E °kT 10h kT h 10 kT v h Figure 13.4-2 Semilogarithmic plot of the av- erage energy E of an electromagnetic mode in thermal equilibrium at temperature T, as a func- tion of the mode frequency v. At T == 300 0 K, kT /h == 6.25 THz, which corresponds to a wavelength of 48 J1m. The dependence of E on v, which is identical to that given in (12.2-24), is portrayed in Fig. 13.4-2. Multiplying the average energy per mode E by the modal density M(v) == 81TV 2 / c 3 provided in (10.3-10) gives rise to a spectral energy density {}( v) == M (v) E (energy per unit bandwidth per unit cavity volume) that takes the form 81Thv 3 1 e(v) = c3 exp(hvjkT) - 1 . (13.4-9) Spectral Energy Density for Blackbody Radiation This formula, which is known as the blackbody radiation spectrum, is plotted in Fig. 13.4-3 as a function of frequency. Its dependence on temperature is illustrated in Fig. 13.4-4. The total power radiated by a blackbody increases steeply with tempera- ture, as T 4 , a result known as the Stefan-Boltzmann law. The spectrum of blackbody radiation played an important role in the discovery E kT Wavelength Ao (/-lID) 1 0 2 10 10- 22 10 15 10 16 Frequency v (Hz) Figure 13.4-4 Dependence of the spectral en- ergy density (}(v) on frequency, plotted on double- logarithmic coordinates for several different tem- peratures. -.. rr. E V3 10- 15 I '-' 00 M(v) v 10- 16 "-:e' ell .£ 10- 17 VJ s:: Q) 10- 18 e.o Q) E 10- 19 c; .... ..... 10- 20 0.. CI) o o g(v) v 10- 21 10- 23 00 v Figure 13.4-3 Frequency depen- dence of the energy per mode E, the density of modes M(v), and the spec- tral energy density (}(v) == M(v) E , on double-linear coordinates. 10- 24 10 12
13.4 THERMAL LIGHT 521 of the quantum (photon) nature of light (see Chapter 12). Based on classical elec- tromagnetic theory, the modal density for a three-dimensional cavity was known to be a quadratic function of v, namely M(v) == 87rV 2 /c 3 (see Sec. 10.3C). However, the law of equipartition of energy in classical statistical mechanics specified that the average energy per mode must be constant at E == kT, independent of the modal frequency. This yielded an expression for g(v), known as the Rayleigh-Jeans formula for blackbody radiation, which failed to agree with experiment. Moreover, its integral diverged. In 1900, Max Planck observed that it was possible to obtain a theoretical expression for the blackbody spectrum that agreed with experiment by quantizing the energy of each mode. Planck's calculation led to the expression for E given in (13.4-8). Indeed, the Rayleigh-Jeans formula is recovered in the limit of sma}] photon energy: for hv « kT, we have exp(hv / kT) 1 + hv / kT whereupon (13.4-8) reverts to the classical equipartition formula E kT so that g(v) 87rv 2 kT /c 3 . EXERCISE 13.4-1 Frequency of Maximum Blackbody Energy Density. Using the blackbody radiation law g(v), show that the frequency v p at which the spectral energy density is maximum satisfies the equation 3(1 - e- X ) == x, where T == hvpl kT. Find x approximately and determine v p at T == 300 0 K. Thermography The blackbody spectral energy-density formula (13.4-9) is useful for generating maps (images) of the temperature distribution of thermal objects. This is achieved by using a camera that is sensitive in the wavelength region of the object's thermal emissions (see Fig. 13.4-4). Hot objects, such as the sun, emit most strongly in the visible region, whereas objects of moderate temperature, such as the earth and humans, typically radiate in the mid-infrared region. Cold objects radiate in the far-infrared. The imag- ing of thermal objects by means of their self-radiation is known as thermography. Thermographic cameras contain an array of photodetectors sensitive in a particular region of the spectrum (see Sec. 18.5). The technique is often used in the wavelength region 0.7 /-Lm Ao 300 /-Lm, corresponding to 12° K T 5200° K. Although thermography is facilitated at higher temperatures because of the T 4 dependence of the total radiated power, the representative images in Fig. 13.4-5 illustrate the broad range of temperatures that can be accessed. Thermography is used to garner information about objects and scenes that exhibit temperature variations. Different local temperatures are typically displayed as false colors. The technique finds use in industrial applications, such as monitoring the over- 40°C 50 60 70 80 90 100 110 22°C 24 26 28 30 32 34 36 I I I I I I I I I I I I I I I I 1115 0 K 1136 0 K 1160 0 K Figure 13.4-5 Representative thermographic images in different temperature regions. (a) Industrial-systems analysis. (b) Search and rescue. (c) Cosmology.
522 CHAPTER 13 PHOTONS AND ATOMS heating of circuit boards and the evolution of oil spills. It is of assistance in search- and-rescue missions of humans and animals, even when they are concealed in dense foliage at night. Thermography is also used in clinical medicine since skin-surface temperature is a diagnostic for blood-flow blockages and tumors. Environmental ap- plications include fire-fighting and forestry. The technique is invaluable in astronomy and cosmology since it allows astronomical objects, such as cooler red stars and red giants, to be imaged in the near infrared; planets, comets, and asteroids to be seen in the mid-infrared; and central galactic regions and emissions from cold dust to be imaged in the far-infrared. 13.5 LUMINESCENCE AND LIGHT SCATTERING Thermal excitation is not the only external source of energy that can raise an atomic or molecular system to a higher energy level and result in the emission of light. Other sources of excitation, such as electron impact and sound waves, can also cause light to be emitted as the system decays back to its ground state. Excitation in the form of one or more photons can also result in the emission of light via photoluminescence. Nonthermal radiators are known as luminescent radiators and the radiation process is called luminescence. While photoluminescence involves the absorption and subsequent emission of pho- tons, light can also scatter from an atomic or molecular system in a resonant or nonres- onant manner. Various forms of linear and nonlinear scattering, such as Rayleigh and Raman scattering, respectively, play important roles in photonics. A. Forms of Luminescence The form of the luminescence is classified according to the source of excitation as indicated by the following examples (Fig. 13.5-1). Cathodoluminescence. Cathodoluminescence is light emitted from a material as a result of excitation by energetic electrons. Examples are the images at the face of a cathode-ray tube or an image intensifier, which are induced in phosphors at the screen by the electrons. Cathodoluminescence is frequently used for assaying the composition of a material since the depth of penetration into the sample can be modified by chang- ing the electron energy and different components give rise to emission at different wavelengths. Sonoluminescence. Sonoluminescence is the emission of light from a liquid induced by acoustic cavitation: the formation, growth, and collapse of bubbles in a liquid irradi- ated with high-intensity sound or ultrasound. The light consists of picosecond-duration flashes emitted when the collapsing bubbles reach minimum size. Sonoluminescence is observed from clouds of bubbles and, under certain circumstances, from isolated bubbles. It is possible to generate single-bubble sonoluminescence flashes with a stable period and position. Chemiluminescence. Chemiluminescence is the emission of light via a chemical reaction. It is observed under those relatively rare circumstances when the reaction between two or more chemicals releases sufficient energy to populate the excited state of a reaction product. Lightsticks, for example, glow when the seal between two compartments containing chemicals is broken and they are permitted to mix. The color
13.5 LUMINESCENCE AND LIGHT SCATTERING 523 of the emitted light is determined by the dye incorporated in the chemical mixture. Lightsticks are used for illumination in underwater and military environments. Bioluminescence. Bioluminescence is chemiluminescence produced by living or- ganisms such as fireflies and glowworms. It provides a means of communication, and some organisms such as fireflies synchronize their flashes. Many deep-sea marine organisms naturally produce bioluminescence, often in the blue region of the spectrum where seawater is transparent. Biologists often attach bioluminescent proteins fromjel- lyfish to the genes of other species to permit genetic expression to be tracked optically. Electroluminescence. Electroluminescence is light resulting from the application of an electric field to a material. An important example is injection electroluminescence, which occurs when electric current is injected into a forward-biased semiconductor junction such as that in a light-emitting diode (LED), as discussed in Chapter 17. The combination of injected electrons from the conduction band with holes from the valence band results in the emission of photons. Photoluminescence. Photoluminescence is light emitted by a sample following the absorption of optical photons. An example is the glow emitted by some materials after exposure to ultraviolet light. Photoluminescence, which is discussed in greater detail in the next section, is a useful tool for investigating the properties of semiconductor materials. The light is termed radioluminescence when the photons are in the X-ray or gamma-ray regIon. '1 " .. ""'- ,,- - .} ..... . ". ."' , "" (a) Cathodoluminescence (b) Sonoluminescence (c) Chemiluminescence .. (d) Bioluminescence (e) Electroluminescence if) Photoluminescence Figure 13.5-1 (a) Cathodoluminescence from a mineral sample reveals the presence of zoned calcite and saddle dolomite in boxwork breccia. The edge dimension is 1.3 mm and the electron energy is 22 keY (courtesy Charles M. Onasch, Bowling Green State University). (b) Multibubble sonoluminescence created by an ultrasonic horn immersed in liquid (courtesy Kenneth S. Suslick, University of Illinois at Urbana-Champaign). (c) Chemiluminescence from a lightstick. (d) The deep- sea scyphomedusa AtoUa vanhoeffeni (diameter 3 em) is abundant throughout the world and pro- duces bioluminescence when disturbed (courtesy Edith A. Widder, Ocean Research & Conservation Association). (e) The electric field across a pair of parallel wires held at different potentials elicits electroluminescence from a powdered material coating them. (f) Photoluminescence from colloidal CdSe quantum dots dispersed in hexane following illumination by ultraviolet light; see Fig. 13.1-12 (courtesy Dong-Kyun Seo, Arizona State University).
524 CHAPTER 13 PHOTONS AND ATOMS If the radiative transitions are spin-allowed, i.e., if they take place between two states with equal multiplicity (singlet ---t singlet or triplet ---t triplet; see Fig. 13.1-5, for example), the luminescence process is called fluorescence. Luminescence from spin- forbidden transitions (e.g., triplet ---t singlet) is called phosphorescence. Fluorescence lifetimes are usually relatively short (often 0.1 to 20 ns), so that the luminescence pho- ton is promptly emitted following excitation. This is in contrast to phosphorescence, in which the "forbidden" nature of the transition results in longer lifetimes (often 1 ms to 10 s) and therefore substantial delay between excitation and emission. The amplification of fluorescence by stimulated emission forms the basis of laser action (see Chapters 14 and 15). B. Photoluminescence Photoluminescence occurs when a system is excited to a higher energy level by absorb- ing a photon, and then spontaneously decays to a lower energy level, emitting a photon in the process. To conserve energy, the emitted photon cannot have more energy than the exciting photon. Several examples of transitions that lead to photoluminescence are depicted schematically in Fig. 13.5-2. Nonradiative downward transitions can be part of the process, as shown by the dashed lines in (b) and (c). Ultraviolet light can be converted to visible light by this mechanism. The excited electron can be stored in an intermediate state (e.g., a trap) for an extended period of time, resulting in delayed luminescence. Intermediate downward nonradiative transitions, followed by upward nonradiative transitions, can also occur, as shown in (d). In another variation on this theme, called "quantum cutting," the absorption of an ultraviolet photon is followed by the emission of two visible photons. Photoluminescence occurs in all forms of matter. + I I I I I I --.L-L (a) (b) (c) (d) Figure 13.5-2 Various forms of single-photon photoluminescence. Multiphoton Photoluminescence Photoluminescence can also occur when a system is excited to a higher energy level by the absorption of more than one photon, followed by a subsequent spontaneous decay to a lower energy level and the concomitant emission of a photon. The exciting photons can have the same, or different, energies and the emitted photon can have an energy greater than one of the exciting photons. Multiphoton fluorescence. Two or more photons of the same energy may conspire to raise the system to a higher energy level, where it undergoes photoluminescence (fluorescence), as shown schematically in Figs. 13.5-3(a) and (b). Two-photon fluo- rescence, illustrated in Fig. 13.5-3(a), is the basis of an imaging technique known as two-photon laser scanning fluorescence microscopy (TPLSM). A fluorescent probe (fluorophore), linked to specific locations in a specimen, absorbs a pair of photons that
13.5 LUMINESCENCE AND LIGHT SCATTERING 525 arrive in its vicinity, each with energy hVl, and then emits a single fluorescence photon with energy hV2 (> hVl ), which is detected. As shown in Sec. 12.2C, the probability of observing two independently arriving photons at a given position and time is the square of observing a single such photon. Thus, by virtue of (12.1-14), the two-photon absorption rate at position r and time t, along with the emitted fluorescence-photon rate, behaves as a quadratic function of the incident intensity, i.e., is proportional to 1 2 ( r, t). (c) Figure 13.5-3 (a) Two-photon fluorescence. (b) Three-photon fluorescence. (c) Upconversion fluorescence. Nonradiative relaxation is presumed to take part in the decay in all cases. Other scenarios also occur. hV l hV l hV l hV 2 hV 2 hV l hV l (a) (b) hV 2 hV 3 The advantage of TPLSM derives in large part from this quadratic dependence: a focused excitation beam results in absorption localized to the immediate vicinity of the focal point since two-photon absorption occurs preferentially where the intensity is greatest. In comparison with ordinary (single-photon) microscopy, the region from which fluorescence is observed is thus sharpened, yielding enhanced resolution; there is also a reduction of the background light arising from out-of-focus fluorescence. Yet another advantage of TPLSM in the domain of biology is the double wavelength of the excitation since longer wavelengths penetrate more deeply into biological tissue. To ensure that the peak intensity is sufficiently high to engender two-photon absorption, and that the average intensity is sufficiently low to avoid damage to delicate tissue, the excitation is often provided by a mode-locked laser that generates ultrashort (fem- tosecond) optical pulses with high peak power but low average power. Multiphoton laser scanning fluorescence microscopy operates in much the same way, except that k independent photons, rather than two, conspire to effect each absorption, so that the emitted fluorescence-photon rate varies as 1 k (r, t). Three-dimensional multi photon microlithography. A similar approach is used to fabricate micro-objects. A lens delivers high-power optical pulses to a particular lo- cation in a specially designed transparent polymeric material. The light has sufficient intensity to effect multiphoton polymerization only in the vicinity of the focal region; it reaches that region without affecting the intervening material. Moving the focal point of the lens about allows any desired three-dimensional microstructure to be written. In practice, the strong thresholding behavior of the polymerization nonlinearity increases the resolution yet further. Upconversion fluorescence. Multiphoton photoluminescence can also occur when the two photons that conspire to excite the system are of different energies, as illus- trated in Fig. 13.5-3(c). This scheme is useful for the conversion of infrared photons to visible ones. An infrared photon of low energy (hVl) teams up with an auxiliary photon (hV2) to excite a system such as a single ion, which then produces a luminescence photon at the sum energy (hV3 == hVl + hV2).
526 CHAPTER 13 PHOTONS AND ATOMS Upconversion fluorescence via sequential absorption can be observed most easily in materials containing traps that can store the electron elevated by the first photon for a time sufficient for the second photon to arrive and boost the system to its upper state. Phosphors doped with rare-earth ions such as Er 3 + are often used. In some materials, the traps can be charged to their intermediate state in minutes by exposing the material to daylight or fluorescent light, which provides the auxiliary photons of energy hV2. An infrared signal photon of energy hVl then releases an electron from the trap, and the result is the emission of a visible luminescence photon of energy h(Vl + V2). Upconversion fluorescence can also occur via a more complex process, such as collective emission from two nearby ions that have both been excited. Practical devices often take the form of a small reflective or transmissive card with an active area about 5 cm x 5 cm, known as an infrared sensor card. The upcon- verting powder is laminated between a pair of stiff transparent plastic sheets to form the card. The upconverting powder can also be dispersed in a block of polymer for three-dimensional viewing. Although the conversion efficiency is typically quite small, these devices are nevertheless useful for visually viewing the spatial distribution of an infrared beam, such as that produced by an infrared laser. The relative spectral sensitivity and emission spectral intensity of a commercially available card are shown in Fig. 13.5-4. Visible emISSIon Infrared sensitivity 400 800 1000 1200 Wavelength (nm) 1400 1600 Figure 13.5-4 Infrared spectral sen- sitivity and relative spectral intensity of the visible upconversion-fluorescence emission from a commercially available infrared sensor card. c. Light Scattering Photoluminescence, as considered in Sec. 13.5B, involves the resonant absorption of a photon via a transition between the ground state and a real excited state; the subsequent relaxation of the excited state back to the ground state results in the emission of a luminescence photon. Absorption and subsequent re-emission from the real upper state are the defining characteristics of luminescence, fluorescence, and phosphorescence. Light scattering processes can involve transitions that occur via virtual states. Since these can be nonresonant interactions, light can be scattered over a broad range of frequencies. We consider in turn three scattering processes of importance in photonics: Rayleigh, Raman, and Brillouin scattering (see Fig. 13.5-5). Scattering is inherent and unavoidable under many circumstances but it can also prove useful for providing information about the characteristics of materials and for creating useful light sources. Rayleigh scattering. Rayleigh scattering is a process whereby a material causes an incident photon to change direction. It entails an energy-conserving (elastic) interaction so that the scattered photon has the same energy as the incident photon, as schematized in Fig. 13.5-5(a). Rayleigh scattering occurs in gases, liquids, and solids. It is engen- dered by variations in a medium, such as the random refractive-index inhomogeneities
13.5 LUMINESCENCE AND LIGHT SCATTERING 527 hV I hV I hV I hvs hV I hVA hV I hVR hVR hvs hVB (a) (b) (c) (d) Figure 13.5-5 Several forms of light scattering: (a) Rayleigh, (b) Raman (Stokes), (c) Raman (anti-Stokes), and (d) Brillouin. Dashed horizontal lines indicate virtual states and therefore nonresonant scattering. in glass (see Sec. 9.3A), or by the presence of particles whose sizes are much smaller than the wavelength of light, such as electrons, atoms, molecules, and nanoparticles. The scattered intensity is proportional to v 4 , and therefore to 1 / A, where v and Ao are the frequency and wavelength of the illumination, respectively. Short wavelengths thus undergo greater scattering than long wavelengths; Rayleigh scattering is responsible for the blue color of the sky. Scattering from spherical particles larger than Ao/10 is known as Mie scattering; this process does not depend strongly on the wavelength of the illumination and is responsible for the white glare around lights in the presence of mist and fog. Raman scattering. Raman scattering is a process by means of which a photon of frequency hVl, following an interaction with a material, emerges either at a lower frequency hv s == hVl - hV R (Stokes scattering) or at a higher frequency hV A == hVl + hV R (anti-Stokes scattering), as displayed in Figs. 13.5-5(b) and (c), respectively. Raman scattering occurs in gases, liquids, and solids. Unlike Rayleigh scattering, Ra- man scattering is inelastic; the alteration of photon frequency is brought about by an exchange of energy hV R with a rotational and/or vibrational mode of a molecule or solid. In Stokes scattering, the photon imparts energy to the material system, whereas the reverse occurs in anti-Stokes scattering. The spectrum of light scattered from a material thus generally contains a Rayleigh-scattered component, at the incident frequency, together with red-shifted and blue-shifted sidebands corresponding to in- elastically scattered Stokes and anti-Stokes components, respectively. Although the sideband power is typically weak for nonresonant interactions, lying about 10- 7 below that of the incident light, Raman scattering is useful for characterizing materials. In crystalline materials, the vibrational spectrum is generally discrete and the Raman lines are narrow. Glasses, in contrast, have broad vibrational spectra that in turn give rise to broad Raman spectra. Brillouin scattering, portrayed in Fig. 13.5-5(d), is similar to Raman scattering except that the exchange of energy hV B takes place with acoustic, rather than vibrational, modes of the medium. Stimulated Raman scattering. Stimulated Raman scattering (SRS) can take place when a signal photon enters a nonlinear optical medium together with a pump photon of higher frequency (see inset in Fig. 14.3-7). The signal photon stimulates the emis- sion of a second signal photon, which is obtained by Stokes-shifting the pump photon so that its frequency precisely matches that of the input signal photon. The surplus energy of the pump photon is transferred to the vibrational modes of the medium. The process bears some similarity to stimulated emission, but the Raman interaction is a parametric third-order nonlinear optical process (see Sec. 21.3B).
528 CHAPTER 13 PHOTONS AND ATOMS Stimulated Raman scattering is useful for making optical amplifiers (see Sec. 14.3D) and lasers (see Sec. lS.3A). Raman amplification and lasing have the distinct merit that the bandwidth over which they can be realized is governed by the vibrational spectrum of the material rather than by the linewidth of a stimulated-emission transition. The vibrational spectrum of glass is particularly broad, so that a length of optical fiber can serve as an optical amplifier or laser over a bandwidth of hundreds of nm. Raman optical amplifiers and Raman fiber lasers find widespread use in dense wavelength- division-multiplexed optical fiber communication systems (see Sec. 24.3C). SRS is also useful as a spectroscopic tool since it can reveal the underlying vibra- tional characteristics of a material. The sensitivity of Raman-based spectroscopy can be enhanced by making use of coherent anti-Stokes Raman scattering (CARS), which uses two pump lasers whose frequency difference is resonant with the vibrational frequency of the material under investigation, thereby increasing the efficiency of wave mIxIng. In another important application, optical fibers can be used to generate broadband light with the help of Raman processes. A pump gives rise to Raman-scattered spon- taneous emission that is amplified via stimulated Raman scattering as the light propa- gates through the fiber. For an optical fiber of sufficient length, and a pump of sufficient strength, the resulting Raman spectrum initiates yet further Raman frequency conver- sion, resulting in the production of broadband (supercontinuum) light (see Sec. 22.5C). Stabilization of the process can be achieved by making use of a resonator. Stimulated Brillouin scattering is similar except that acoustic vibrations, rather than molecular vibrations, are involved. READING LIST Books on Atomic, Molecular, and Condensed-Matter Physics See also the reading lists in Chapters ] 4, 15, and 16. S. Haroche and J.-M. Raimond, Exploring the Quantum, Oxford University Press, 2006. M. Plischke and B. Bergersen, Equilibrium Statistical Physics, World Scientific, 3rd ed. 2006. W. Demtroder, Molecular Physics: Theoretical Principles and Experimental Methods, Wiley- VCH, 2005. C. L. Tang, Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics, Cambridge University Press, 2005. G. Liu and B. Jacquier, eds., Spectroscopic Properties of Rare Earths in Optical Materials, Springer- Verlag, 2005. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume 3, Quantum Mechanics, 1965 and Volume 1, Mainly Mechanics, Radiation, and Heat, 1963, Addison-Wesley, 2nd ed. 2005. W. A. Harrison, Elementary Electronic Structure, World Scientific, 2004. C. Cohen- Tannoudji, Atoms in Electromagnetic Fields, World Scientific, 2nd ed. 2004. C. J. Foot, Atomic Physics, Oxford University Press, 2004. C. Kittel, Elementary Statistical Physics, Wiley, 1958; Dover, reissued 2004. H. A. Lorentz, The Theory of Electrons and its Applications to the Phenolnena of Light and Radiant Heat, Teubner, 1906; Dover, reissued 2004. B. Henderson and R. H. Bartram, Crystal-Field Engineering of Solid-State Laser Materials, Cam- bridge University Press, 2000. I. N. Levine, Quantum Chemistry, Prentice Hall, 5th ed. 1999. D. M. Roundhill and J. P. Fackler, Jr., eds., Optoelectronic Properties of Inorganic Compounds, Plenum, 1999. F. Reif, Complete Statistical Physics, Volume 5, Berkeley Physics Course, McGraw-Hill, 1998.
READING LIST 529 R. C. Powell, Physics of Solid-State Laser Materials, Springer-Verlag, 1998. C. Cohen- Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Applications, Wiley, 1992, paperback ed. 1998. D. Suter, The Physics of Laser-Atom Interactions, Cambridge University Press, 1997. F.-H. Kan and F. Gan, Laser Materials, World Scientific, 1995. H. Yokoyama and K. Ujihara, eds., Spontaneous Emission and Laser Oscillation in Microcavities, CRC Press, 1995. S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics, Cambridge University Press, paper- back ed. 1995. C. A. Morrison, Crystal Fields for Transition-Metal Ions in Laser Host Materials, Springer-Verlag, 1992. D. C. Harris and M. D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy, Oxford University Press, 1978; Dover, reissued 1989. M. Born Atomic Physics, Blackie & Son, 1935, 8th ed. 1969; Dover, reissued 1989. V. S. Letokhov, ed. Laser Spectroscopy of Highly Vibrationally Excited Molecules, Adam Hilger, 1989. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, 1975; Dover, reissued 1987. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley, 2nd ed. 1985. R. G. Breene, Jr.. Theories of Spectral Line Shape, Wiley, 1981. C. Kittel and H. Kroemer, Thermal Physics, Freeman, 2nd ed. 1980. D. ter Haar, The Old Quantum Theory, Pergamon, 1967. (Contains English translations of key early papers by Planck, Einstein, Rutherford, and Bohr.) G. Herzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand Reinhold, 1966. D. L. Livesey, Atomic and Nuclear Physics, Blaisdell, 1966. J. C. Slater, Quantum Theory of Atomic Structure, Volume 1, McGraw-Hill, 1960. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, 4th ed. 1958. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Addison-Wesley, 1958. G. Herzberg, Molecular Spectra and Molecular Structure Volume 1 Spectra of Diatomic Molecules, Van Nostrand Reinhold, 2nd ed. 1950. G. Herzberg, Atomic Spectra and Atomic Structure, Prentice Hall, 1937; Dover, reissued 1944. E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1935. Books on Laser Cooling and Trapping V. Letokhov, Laser Control of Atoms and Molecules, Oxford University Press, 2007. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers: A Reprint Volume with Commentaries, World Scientific, 2006. F. Bardou, J.-P. Bouchaud A. Aspect, and C. Cohen- Tannoudji, Levy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest, Cambridge University Press, 2002. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, 1999. Books on Thermography, Luminescence, and Scattering J. R. Lakowicz, Principles of Fluorescence Spectroscopy, Springer, 3rd ed. 2006. F. R. Young, Sonoluminescence, CRC Press, 2005. A. Tsuji, M. Maeda, M. Matsumoto, L. J. Kricka, and P. E. Stanley, eds., Bioluminescence and Chemi- luminescence: Progress and Perspectives (Proceedings of the 13th International Symposium), World Scientific, 2005. O. Breitenstein and M. Langenkamp, Lock-in Thermography: Basics and Use for Functional Diag- nostics of Electronic Components, Springer-Verlag, 2003. J.-C. Krupa and N. A. Kulagin, eds., Physics of Laser Crystals, Kluwer, 2003. D. A. Long, The Raman Effect: A Unified Treatment of the Theory of Raman Scattering by Molecules, Wiley, 2002.
530 CHAPTER 13 PHOTONS AND ATOMS M. J. Damzen, V. I. Vlad, V. Babin, and A. Mocofanescu, Stilnulated Brillouin Scattering: Funda- mentals and Applications, Institute of Physics, 2002. A. D. Wheelon, Electromagnetic Scintillation. Cambridge University Press, 200 I. B. J. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chelnistry, Biology, and Physics, Wiley, 1976; Dover, reissued 2000. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Slnall Particles, Wiley, 1983; paperback ed. 1998. C. S. Johnson, Jr. and D. A. Gabriel, Laser Light Scattering, Chapter 5 of Spectroscopy ill Biochem- istry (Volume II), T. E. Bell, ed., CRC Press, 1981; Dover, reissued 1995. Articles F. W. Wise, ed., Selected Papers 011 Semiconductor Quantunl Dots, SPIE Optical Engineering Press (Milestone Series Volume 180), 2005. T. Baldacchini, C. N. LaFratta, R. A. Farrer, M. C. Teich, B. E. A. Saleh, M. J. Naughton. and J. T. Fourkas, Acrylic-Based Resin with Favorable Properties for Three-Dimensional Two-Photon Polymerization, Journal of Applied Physics, vol. 95, pp. 6072-6076, 2004. B. R. Masters, ed., Selected Papers on Multiphoton Excitation Microscopy, SPIE Optical Engineering Press (Milestone Series Volume 175), 2003. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics. vol. 6. no. 6. 2000. M. J. Weber, ed., Selected Papers on Phosphors, Light Emitting Diodes, and Scintillators: Appli- cations of Photoluminescence, Cathodolulninescence, Electrolunlinescence, and Radiolumines- cence, SPIE Optical Engineering Press (Milestone Series Volume 151), 1998. S. J. Tans, M. H. Devoret, H. Dai, A. Thess, R. E. Smalley, L. J. Geerligs, and C. Dekker, Individual Single-Wall Carbon Nanotubes as Quantum Wires, Nature, vol. 386, pp. 474--477, 1997. R. C. Ashoori, Electrons In Artifical Atoms, Nature, vol. 379, pp. 413--419, 1996. S. J. Putterman, Sonoluminescence: Sound Into Light, Scientific Anlerican, vol. 272, no. 2, pp. 46-51, 1995. M. Kerker, ed.. Selected Papers on Light Scattering, SPIE Optical Engineering Press (Milestone Series Volume 4), 1988. J. H. van Vleck and D. L. Huber, Absorption, Emission, and Linebreadths: A Semihistorical Perspec- tive, Reviews of Modern Physics, vol. 49, pp. 939-959, 1977. V. F. Weisskopf, How Light Interacts with Matter, Scientific American, vol. 219, no. 3, pp. 60-71, 1968. E. M. Purcell, Spontaneous Emission Probabilities at Radio Frequencies, Proceedings of the Amer- ican Physical Society, Cambridge, MA, April 25-27, 1946, Abstract B 10, (Physical Review, vol. 69, p. 681, June 1946). M. Goppert-Mayer, Uber Elementarakte mit zwei Quantenspriingen. Annalen der Physik. vol. 9. pp. 273-294, 1931. A. Einstein, Zur Quantentheorie der Strahlung, Physikalische Zeitschrift, vol. 18, pp. 121-] 28, 1917 [Translation: On the Quantum Theory of Radiation, in D. ter Haar, The Old Quantum Theory, Pergamon, 1967]. PROBLEMS 13.3-3 Comparison of Stimulated and Spontaneous Emission. An atom with two energy levels corresponding to a transition with characteristics: Ao = 0.7 /-Lill, t sp = 3 illS, lI = 50 GHz, and Lorentzian lineshape, is placed in a resonator of volume V = 100 cln 3 and refractive in- dex n = 1. Two radiation modes (one at the center frequency lIo and the other at lIO+lI) are excited with 1000 photons each. Determine the probability density for stimulated emission (or absorption). If N 2 such atoms are excited to energy level 2, determine the time constant for the decay of N 2 due to stimulated and spontaneous emission. How many photons (rather than 1000) should be present so that the decay rate due to stimulated emission equals that due to spontaneous emission?
PROBLEMS 531 13.3-4 Spontaneous Emission into Prescribed Modes. (a) Consider a I-J-Lm 3 cubic cavity containing a medium of refractive index n = 1. What are the mode numbers (ql, q2, q3) of the lowest- and next-higher-frequency modes (see Chapter 10)? Show that these frequencies are 260 and 367 THz. (b) Conider a single excited atom in the cavity when it contains zero photons. Let Psp1 be the probability density (S-l) that the atom spontaneously emits a photon into the (2, 1, 1) mode, and let Psp2 be the probability density that the atom spontaneously emits a photon with frequency 367 THz. Determine the ratio Psp2/ Psp1. 13.4-2 Rate Equations for Broadband Radiation. A resonator of unit volume contains atoms having two energy levels, labeled I and 2, corresponding to a transition of resonance fre- quency Vo and linewidth v. There are N 1 and N 2 atoms in the lower and upper levels, 1 and 2, respectively, and a total of n photons in each of the modes within a broad band surrounding Vo. Photons are lost from the resonator at a rate 1/ Tp as a result of imperfect reflection at the cavity walls. Assuming that there are no nonradiative transitions between levels 2 and 1, write the rate equations for N 2 and n . 13.4-3 Inhibited Spontaneous Emission. Consider a hypothetical two-dimensional blackbody radiator (e.g., a square plate of area A) in thermal equilibrium at temperature T. (a) Determine the density of modes M(v) and the spectral energy density (i.e., the energy in the frequency range between v and v + dv per unit area) of the emitted radiation f2(v) (see Sec. 10.3). (b) Find the probability density of spontaneous emission P.. p for an atom located in a cavity that permits radiation only in two dimensions. Such a cavity may be made, for example, by using photonic-crystal omnidirectional reflectors above and below a slab. 13.4-4 Comparison of Stimulated and Spontaneous Emission in Blackbody Radiation. Find the temperature of a thermal-equilibrium blackbody cavity emitting a spectral energy den- sity f2(v), when the rates of stimulated and spontaneous emission from the atoms in the cavity walls are equal at Ao = 1 J-Lill. 13.4-5 Wien's Law. Derive an expression for the spectral energy density f2.x(A) [the energy per unit volume in the wavelength region between A and A + dA is f2.x(A) dA]. Show that the wavelength Ap at which the spectral energy density is maximum satisfies the equation 5(1- e- Y ) = y, where y = he/ ApkT, demonstrating that the relationship ApT = constant (Wien's law) is satisfied. Find ApT approximately. Show that Ap -# e/v p , where v p is the frequency at which the blackbody energy density f2( v) is maximum (see Exercise 13.4-1). Explain. ] 3.4-6 Spectral Energy Density of One-Dimensional Blackbody Radiation. Consider a hypo- thetical one-dimensional blackbody radiator of length L in thermal equilibrium at tempera- ture T. (a) Determine the density of modes M(v) (number of modes per unit frequency per unit length) in one dimension. (b) Using the average density E of a mode of frequency v, determine the spectral energy density (i.e., the energy in the frequency range between v and v + dv per unit length) of the blackbody radiation f2( v). Sketch f2( v) versus v. 13.4-7 Stefan-Boltzmann Law. Use the spectral energy density for blackbody radiation provided in (13.4-9) to confirm that the total power radiated by a blackbody is proportional to T 4 , in accord with the Stefan-Boltzmann law. Determine the proportionality constant. Hint: J o oo x 3 dx/(e X - 1) = 7f4/15. * 13.5-1 Statistics of Cathodoluminescence Light. Consider a beam of electrons impinging on the phosphor of a cathode-ray tube. Let m be the mean number of electrons striking a unit area of the phosphor in unit time. If the number m of electrons arriving in a fixed time is random with a Poisson distribution and the number of photons emitted per electron is also Poisson distributed, but with mean G , find the overall distribution p(n) of the emitted cathodolu- minescence photons. The result is known as the Neyman type-A distribution. Determine expressions for the mean n and the variance a. Hint: Use conditional probability.
CHAPTER 1 LASER AMPLIFIERS 14.1 THEORY OF LASER AMPLIFICATION 535 A. Gain and Bandwidth B. Phase Shift 14.2 AMPLIFIER PUMPING 539 A. Rate Equations B. Pumping Schemes 14.3 COMMON LASER AMPLIFIERS 547 A. Ruby B. Neodymium-Doped Glass c. Erbium-Doped Silica Fiber D. Raman Fiber Amplifiers E. Tabulation of Selected Laser Transitions 14.4 AMPLIFIER NONLINEARITY 556 A. Saturated Gain in Homogeneously Broadened Media *B. Saturated Gain in In homogeneously Broadened Media * 14.5 AMPLIFIER NOISE 562 , , Charles H. Townes Nikolai G. Basov Aleksandr M. Prokhorov (born 1915) (1922-2001) (1916-2002) Townes, Basov, and Prokhorov developed the principle of Light Amplification by Stimulated Emission of Radiation (LASER). They received the Nobel Prize for this work in 1964. 532
A coherent optical amplifier is a device that increases the amplitude of an optical field while maintaining its phase. If the optical field at the input to such an amplifier is monochromatic, the output will also be monochromatic with the same frequency. The output amplitude is increased relative to the input while the phase remains unchanged or is shifted by a fixed amount. [n contrast, an incoherent optical amplifier increases the intensity of an optical wave without preserving its phase. Coherent optical amplifiers are important in a number of applications; examples in- clude the amplification of weak optical pulses such as those that have traveled through a long length of optical fiber and the production of highly intense optical pulses such as those required for laser-fusion applications. Furthermore, it is important to understand the principles underlying the operation of optical amplifiers as a prelude to the analysis of optical oscillators (lasers) in Chapter 15. The underlying principle for achieving the coherent amplification of light is light amplification by stimulated emission of radiation, known by the acronym LASER. Stimulated emission (see Sec. 13.3) allows a photon in a given mode to induce an atom whose electron is in an upper energy level to undergo a transition to a lower energy level and, in the process, to emit a clone photon into the same mode as the initial photon. A clone photon has the same frequency, direction, and polarization as the initial photon. These two photons in turn serve to stimulate the emission of two additional photons, and so on, while preserving these properties. The result is coherent light amplification. Because stimulated emission occurs only when the photon energy is nearly equal to the atomic-transition energy difference, the process is restricted to a band of frequencies determined by the atomic linewidth. Laser amplification differs in a number of respects from electronic amplification. Electronic amplifiers rely on devices in which small changes in an injected electric current or applied voltage result in large changes in the rate of flow of charge carriers, such as electrons and holes in a semiconductor field-effect transistor (PET) or bipolar junction transistor. Tuned electronic amplifiers make use of resonant circuits (e.g., a capacitor and an inductor) or resonators (metal cavities) to limit the gain of the amplifier to the band of frequencies of interest. In contrast, atomic, molecular, and solid-state laser amplifiers rely on differences in their allowed energy levels to provide the principal frequency selection. These entities act as natural resonators that select the frequency of operation and bandwidth of the device. Optical cavities (resonators) are often used to provide auxiliary frequency tuning. Light transmitted through matter in thermal equilibrium is attenuated. This is be- cause absorption by the large population of atoms in the lower energy level is more prevalent than stimulated emission by the smaller population of atoms in the upper level. An essential ingredient for achieving laser amplification is the presence of a greater number of atoms in the upper energy level than in the lower level. This is a nonequilibrium situation, as is understood from Sec. 13.2. Achieving such a population inversion requires a source of power to excite (pump) the atoms to the higher energy level, as illustrated in Fig. 14.0-1. Although the presentation throughout this chapter is couched in terms of "atoms" and "atomic levels," these appellations are to be more broadly understood as "active medium" and "laser energy levels," respectively. The properties of an ideal optical or electronic coherent amplifier are displayed schematically in Fig. 14.0-2(a). It is a linear system that increases the amplitude of the input signal by a fixed factor, the amplifier gain. A sinusoidal input leads to a sinusoidal output at the same frequency, but with larger amplitude. The gain of the ideal amplifier is constant for all frequencies within the amplifier spectral bandwidth. The amplifier may impart to the input signal a phase shift that varies linearly with frequency, corresponding to a time delay at the output with respect to the input (see Sec. B.l of Appendix B). 533
t [77<.: .:..:: : ..:. :... . .f:; I ..:.. '.= :. :.. ':..:: .:-:.. .., :.... .. .. ......... . . .. . .. .- -. .. . .... ... . . .: . ..':.. '.. : :.:.:.. .. ... .;0.. '. . .: 534 CHAPTER 14 LASER AMPLIFIERS Pump Input photons .. Input Ideal amplifier (a) . . . . . . """t Input Real amplifier (b) . . . . . . """t Atoms Output photons .. Output ! ! ! ! ! ! """t Output ! ! ! ! ! ! """t Figure 14.0-1 The laser amplifier. An external power source (the pump) excites the active medium (represented by a collection of atoms), producing a population inversion. Photons interact with the atoms. When stimulated emis- sion is more prevalent than absorption. the medium acts as a coherent amplifier. Gain // v o V / / Gain A hase -. )a \ /v. V _/ 0 Q) ....."'0 :::I :::I o... "50.. OE ro Input amplitude Q) ....."'0 :::I :::I o... "50.. OE ro Input amplitude Figure 14.0-2 (a) An ideal amplifier is linear. It increases the amplitude of a signal (whose frequencies lies within its bandwidth) by a constant gain factor, possibly introducing a linear phase shift. (b) A real amplifier typically has a gain and phase shift that are functions of frequency, as illustrated. For large values of the input, the output signal saturates and the amplifier exhibits nonl ineari ty. Real coherent amplifiers deliver a gain and phase shift that are frequency depen- dent, typically in the manner illustrated in Fig. 14.0-2(b). The gain and phase shift determine the amplifier's transfer function. For a sufficiently large input amplitude, real amplifiers generally exhibit saturation, a form of nonlinear behavior in which the output amplitude does not increase in proportion to the input amplitude. Saturation introduces harmonic components into the output, provided that the amplifier bandwidth is sufficiently broad to pass them. Real amplifiers also introduce noise, so that a random fluctuating component is present at the output, regardless of the input. An amplifier may therefore be characterized by the following features: . Gain . Bandwidth . Phase shift . Power source . Nonlinearity and gain saturation . Noise This Chapter In this chapter, we discuss the features listed above, in turn. In Sec. 14.1 the theory of laser amplification is developed, leading to expressions for the amplifier gain, spectral bandwidth, and phase shift. The mechanisms by means of which a power source pumps the active medium and achieves a population inversion are examined in Sec. 14.2. Examples of important laser amplifiers are considered in Sec. 14.3. Sees. 14.4 and 14.5 are devoted to nonlinearity and noise in the amplification process, respectively. This chapter relies on the material presented in Chapter 13, particularly Sec. 13.3.
14.1 THEORY OF LASER AMPLIFICATION 535 14.1 THEORY OF LASER AMPLIFICATION A monochromatic optical plane wave traveling in the z direction with frequency v, electric field E(z) == Re{ E(z) exp(j27fvt)}, complex amplitude E(z), intensity I(z) == IE(z) 1 2 /21], and photon-flux density e/;(z) == I(z) / hv (photons per second per unit area) will interact with an atomic medium, provided that the atoms of the medium have two energy levels whose energy difference nearly matches the photon energy hv. The numbers of atoms per unit volume in the lower and upper energy levels are denoted N 1 and N 2 , respectively. The wave is amplified with a gain coefficient --y(v) (per unit length) and undergoes a phase shift <p(v) (per unit length). We proceed to determine expressions for --y(v) and <p(v). Positive --y(v) corresponds to amplification; negative --y (v) corresponds to attenuation. A. Gain and Bandwidth Three forms of photon-atom interaction take place (see Sec. ] 3.3). If the atom is in the lower energy level, the photon may be absorbed, whereas if it is in the upper energy level, a clone photon may be emitted by the process of stimulated emission. These two processes lead to attenuation and amplification, respectively. The third form of interaction, spontaneous emission, in which an atom in the upper energy level emits a photon independently of the presence of other photons, is responsible for amplifier noise (see Sec. 14.5). The probability density (8- 1 ) that an unexcited atom absorbs a single photon is, according to (13.3-19) and (13.3-] 5), Wi == e/; a(v), (14.1-1) where a(v) == ()..2/87ft sp ) g(v) is the transition cross section at the frequency v, g(v) is the normalized lineshape function, t sp is the spontaneous lifetime, and ).. is the wavelength of light in the medium. The probability density for stimulated emission is the same as that for absorption. Gain Coefficient The average density of absorbed photons (number of photons per unit time per unit volume) is N1W i . Similarly, the average density of clone photons generated as a result of stimulated emission is N 2 W i . The net number of photons gained per second per unit volume is therefore NW i , where N == N 2 - N 1 is the population density difference. For convenience, N is simply referred to as the population difference. If N is positive, a population inversion exists, in which case the medium can act as an amplifier and the photon-flux density can increase. If it is negative, the medium acts as an attenuator and the photon-flux density decreases. If N == 0, the medium is transparent. Since the incident photons travel in the z direction, the stimulated-emission photons also travel in this direction, as illustrated in Fig. ] 4. ] -1. An external pump providing a population inversion (N > 0) then causes the photon-flux density e/;(z) to increase with z. Because emitted photons stimulate further emissions, the growth at any position z is proportional to the population at that position; e/;( z) thus increases exponentially. To demonstrate this process explicitly, consider an incremental cylinder of length dz and unit area, as shown in Fig. 14.1-1. If e/;(z) and e/;(z) + de/;(z) are the photon-flux densities entering and exiting the incremental cylinder, respectively, then de/; ( z) must be the photon-flux density emitted from within the cylinder. This incremental number of photons per unit area per unit time, de/; ( z), is simply the number of photons gained per unit time per unit volume, NW i , multiplied by the thickness of cylinder dz: de/; == NW i dz. (14.1-2)
536 CHAPTER 14 LASER AMPLIFIERS Input light . 4J Output light I o z+dz Figure 14.1-1 The photon-flux density ej; (photons/cm 2 -s) entering an incremental cylinder containing excited atoms grows to ej; + dej; after traveling a distance dz. With the help of (14.1-1), (14.1-2) can be written in the form of a differential equation, d) = (v) </J(z), (14.1-3) where ,\2 ry(v) == Na(v) == N g(v). 87rt sp ( 14.1-4 ) Gain Coefficient The coefficient ry(v) represents the net gain in the photon-flux density per unit length of the medium. The solution of (14.1- 3) is the exponentially increasing function 4;(z) == 4;(0) exp[ry(v) z]. (14.1-5) Since the optical intensity I (z) == hvcjJ( z), (14.] -5) can also be written in terms of I as I ( z) == 1(0) exp [ry ( v) z]. (14.1-6) Thus, ry( v) also represents the gain in the intensity per unit length of the medium. The amplifier gain coefficient ry (v) is seen to be proportional to the population difference N == N 2 - N 1 . Although N was taken to be positive in the example provided above, the derivation is valid whatever the sign of N. In the absence of a population inversion, N is negative (N 2 < N 1 ) and so is the gain coefficient. The medium will then attenuate (rather than amplify) light traveling in the z direction, in accordance with the exponentially decreasing function 4;(z) == 4;(0) exp[-a(v) z], where the attenuation coefficient a(v) == -ry(v) == -N a(v). A medium in thermal equilibrium therefore cannot provide laser amplification. Gain For an interaction region of total length d (see Fig. 14.1-1), the overall gain of the laser amplifier G(v) is defined as the ratio of the photon-flux density at the output to the photon-flux density at the input, G(v) == 4;( d) /4;(0), so that G ( v) == exp [ry ( v) d ] . (14.1-7) Amplifier Gain
14.1 THEORY OF LASER AMPLIFICATION 537 Bandwidth The dependence of the gain coefficient --y( v) on the frequency of the incident light v is contained in its proportionality to the lineshape function g(v), as given in (14.1-4). The latter is a function of width v centered about the atomic resonance frequency va == (E 2 - E1)/h, where E 2 and El are the atomic energies. The laser amplifier is therefore a resonant device, with a resonance frequency and bandwidth determined by the lineshape function of the atomic transition. This is because stimulated emission and absorption are governed by the atomic transition. The linewidth v is measured either in units of frequency (Hz) or in units of wavelength (nm). These linewidths are related by A == 1(co/v)1 == +(co/v2)v == (A/Co)v. Thus, a linewidth v == 1 THz at Ao == 0.6 J-Lm corresponds to A == 1.2 nm. If the lineshape function is Lorentzian, for example, (13.3-34) provides g(v) = b.v/2n (v - vO)2 + (v /2)2 . (14.1-8) The gain coefficient is then also Lorentzian with the same width, i.e., (V/2)2 ,,(v) = ,,(vo) (v _ vo)2 + (b.v /2)2 ' as illustrated in Fig. 14.1-2, where --y(vo) == N(A 2 /47r2tspv) is the gain coefficient at the central frequency va. (] 4.1-9) "i(v) 7{v o ) o o V o v Fig u re 14.1-2 Gain coefficient ,,( v) of a Lorentzian-lineshape resonant laser amplifier. EXERCISE 14.1-1 Attenuation and Gain in a Ruby Laser Amplifier. (a) Consider a ruby crystal with two energy levels separated by an energy difference corresponding to a free-space wavelength Ao == 694.3 nn1, with a Lorentzian lineshape of width /j,v == 330 GHz. The spontaneous lifetime is t sp == 3 Ins and the refractive index of ruby is n == 1.76. If N 1 + N 2 == N a == 10 22 cm- 3 , determine the population difference N == N 2 - N 1 and the attenuation coefficient at the line center a(vo) under conditions of thermal equilibrium at T == 300 0 K (so that the Boltzmann distribution discussed in Sec. 13.2 is obeyed). (b) What value should the population difference N assume to achieve a gain coefficient ''(va) == 0.5 em- 1 at the central frequency? (c) How long should the crystal be to provide an overal] gain of 4 at the central frequency when ''(va) == 0.5 em-I?
538 CHAPTER 14 LASER AMPLIFIERS B. Phase Shift Because the gain of the resonant medium is frequency dependent, the medium is dispersive (see Sec. 5.5) and a frequency-dependent phase shift must be associated with its gain. The phase shift imparted by the laser amplifier can be determined by considering the interaction of light with matter in terms of the electric field rather than the photon-flux density or the intensity, as we have done in the foregoing. We proceed with an alternative approach, in which the mathematical properties of a causal system are used to determine the phase shift. For homogeneously broadened media, the phase-shift coefficient <p(v) (phase shift per unit length of the amplifier medium) is related to the gain coefficient --y( v) by the Hilbert transform (see Sec. B.l of Appendix B), so that knowledge of --y(v) at all frequencies uniquely determines <p(v). The optical intensity and the complex amplitude of the field are related by 1 (z) == IE(z)1 2 /21]. Since 1(z) == 1(0) exp[--y(v) z] in accordance with (14.1-6), the field complex amplitude obeys the relation E ( z) == E ( 0) exp [ --y ( v) z ] exp [ - j <p (v) z] , (14.1-10) where <p(v) is the phase-shift coefficient. The field complex amplitude evaluated at z + z is therefore E(z + z) == E(O) exp[!--y(v) (z + z)] exp [-j<p(v) (z + z)] == E ( z) exp [ --y (v) z ] exp [ - j <p (v) z] E(z) [1 + --y(v) z - j<p(v) zJ ' (14.1-11) where we have made use of a Taylor-series approximation for the exponential func- tions. The incremental change in the electric field, E(z) == E(z + z) - E(z), therefore satisfies the equation E(z) t::.z = E(z)[I'(v) - jcp(v)] . (14.1-12) This incremental amplifier may be regarded as a linear system whose input and output are E(z) and E(z)/ z, respectively, and whose transfer function is H(v) == --y(v) - j<p(v). (14.1-13) Because this incremental amplifier represents a physical system, it must be causal. But the real and imaginary parts of the transfer function of a linear causal system are related by the Hilbert transform (see Sec. B.l of Appendix B). It follows that -<p(v) is the Hilbert transform of ! --y (v) so that the amplifier phase shift function is determined by its gain coefficient. A simple example is provided by the Lorentzian atomic lineshape function with narrow width v « Yo, for which the gain coefficient --y(v) is given by (14.1-9). The corresponding phase shift coefficient <p(v) is provided in (B.I-13) of Sec. B.l, v - Vo <p(v) == v --y(v). (14.1-14) Phase-Shift Coefficient (Lorentzian Lineshape)
14.2 AMPLIFIER PUMPING 539 The Lorentzian gain and phase-shift coefficients are plotted in Fig. 14.1-3 as functions of frequency. At resonance, the gain coefficient is maximum and the phase-shift coef- ficient is zero. The phase-shift coefficient is negative for frequencies below resonance and positive for frequencies above resonance. ry(v) flv v o v <p(v) vo v Figure 14.1-3 Gain coefficient ,,(v) and phase-shift coefficient <p(v) for a laser amplifier with a Lorentzian line- shape function. 14.2 AMPLIFIER PUMPING Like other amplifiers, laser amplifiers require an external source of power to provide the energy required to augment the input signal. The pump supplies this power via mechanisms that excite the electrons in the atoms, causing them to move from lower to higher atomic energy levels. To achieve amplification, the pump must provide a population inversion (N == N 2 - N 1 > 0) on the transition of interest. However, the mechanics of pumping often involves the use of ancillary energy levels. For example, the pumping of atoms from level 1 into level 2, to achieve amplification on the 2---+ 1 transition, might be most readily accomplished by pumping the atoms from level 1 into level 3 and then by relying on natural processes of decay from level 3 to populate level 2. The pumping may be achieved optically (e.g., with a flashlamp or laser), electrically (e.g., through a gas discharge, an electron beam, an ion beam, or by means of injected charge carriers), or chemically (e.g., with a flame or via a chemical reaction that leaves the products in an excited state). For continuous-wave (CW) operation, the rates of excitation and decay of the various energy levels participating in the process must be balanced to maintain a steady-state inverted population on the 2---+ 1 transition. A. Rate Equations The equations that describe the rates of change of the population densities N 1 and N 2 as a result of pumping, as well as radiative and nonradiative transitions, are called rate equations. They are not unlike the equations presented in Sec. 13.4, but selective external pumping is now part of the process so that thermal equilibrium conditions no longer prevail.
540 CHAPTER 14 LASER AMPLIFIERS Consider the schematic energy-level diagram of Fig. 14.2-1. We focus on levels 1 and 2, which have overall lifetimes 71 and 72, respectively, permitting transitions to lower levels. The lifetime of level 2 has two contributions - one associated with decay from 2 to 1 (721), and the other (720) associated with decay from 2 to all other lower levels. When several modes of decay are possible, the overall transition rate is a sum of the component transition rates. Since the rates are inversely proportional to the decay times, the reciprocals of the decay times must be added: -1 -1 -1 7 2 == 7 21 + 7 20 . (14.2-1 ) Multiple modes of decay therefore shorten the overall lifetime (i.e., they render the decay more rapid). Aside from the radiative spontaneous emission component (of time constant t sp ) in 721, a nonradiative contribution 7nr may also be present (arising, for example, from a collision of the atom with the wall of the container, thereby resulting in a depopulation), so that -1 t -I -1 7 21 == sp + 7nr . (14.2-2) If an unpumped system like that illustrated in Fig. 14.2-1 is allowed to reach steady state, the population densities N 1 and N 2 will vanish by virtue of all of the electrons having ultimately decayed to lower energy levels. CD CD I 1 1 1 7 1 Y I I I 1 I 1 7 21 : (' ) 72 1 I 1 t S p 1 I T I Y Y nr : 1 17 20 I Y Figure 14.2-1 Energy levels 1 and 2 and their decay times. Steady-state populations of levels 1 and 2 can be maintained, however, if the energy levels above level 2 are continuously excited by pumping and ultimately populate level 2, as shown in the more realistic energy-level diagram of Fig. l4.2-2. Pumping serves to bring atoms whose electrons are in levels other than 1 and 2 out of level 1 and into level 2, at rates Rl and R 2 (per unit volume per second), respectively, as shown in simplified form in Fig. 14.2-3. As a result, levels I and 2 can achieve nonzero steady-state populations. We now proceed to write the rate equations for this system both in the absence and in the presence of amplifier radiation (the radiation resonant with the 2] transition). Rate Equations in the Absence of Amplifier Radiation The rates of increase of the population densities of levels 2 and 1 arising from pumping and decay are, respectively, dN 2 = R 2 _ N 2 dt 72 dN l = -R l _ N l + N 2 . dt 71 721 (14.2-3) ( 14.2 -4 )
14.2 AMPLIFIER PUMPING 541 R 2 V@ I I R 2 . . 7 21 . . . . 72 Ry CD f_ . I . . I . . I I t 7 20 I Y 7 1 Figure 14.2-2 Energy levels I and 2, together with surrounding higher and lower energy levels, in the presence of pumping. Figure 14.2-3 Energy levels 1 and 2 and their decay times. By means of pumping, the population density of level 2 is increased at the rate R 2 while that of level I is decreased at the rate RI. Under steady-state conditions (dN 1 /dt == dN 2 /dt == 0), (14.2-3) and (14.2-4) can be solved for N] and N 2 , and the population difference N == N 2 - N 1 can be determined. The result is No == R272 ( 1 - 71 ) + R 1 7 1 , 721 (14.2-5) Steady-State Population Difference (Absence of Amplifier Radiation) where the symbol No represents the steady-state population difference N in the absence of amplifier radiation. In accordance with (14.1-4), a large gain coefficient requires a large population difference, i.e., a large positive value of No. Equation (14.2-5) shows that this may be achieved by: . Large R 1 and R 2 . . Long 72 (but t sp , which contributes to 72 through 721, must be sufficiently short so as to make the radiative transition rate large, as will be seen subsequently). . Short 7] ifR 1 < (72/721)R2. The physical reasons underlying these conditions make good sense. The upper level should be pumped strongly and decay slowly so that it retains its population. The lower level should depump strongly so that it quickly disposes of its population. Ideally, it is desirable to have 721 t sp « 720 so that 72 t sp , and 71 « t sp . Under these conditions, (] 4.2-5) simplifies to No R 2 t sp + R]71. ( 14.2-6) In the absence of depumping (R 1 == 0), or when R 1 « (t sp /71)R 2 , this result further simplifies to No R 2 t sp . (14.2-7) EXERCISE 14.2-1 Optical Pumping. Assume that RI == 0 and that R 2 is realized by exciting atoms from the ground state E == 0 to level 2 using photons of frequency E2/h absorbed with a transition probability W. Assume that
542 CHAPTER 14 LASER AMPLIFIERS T2 t sp and T1 « t sp so that in steady state N 1 0 and No R 2 t sp . If N a is the total population of levels 0, I, and 2, show that R 2 (N a - 2N o ) W, so that the population difference is No Nat sp W /(1 + 2t sp W). Rate Equations in the Presence of Amplifier Radiation The presence of radiation near the resonance frequency Vo enables transitions between levels 2 and 1 to take place via stimulated emission as well as absorption. These processes are characterized by the probability density Wi == q; a(v), as provided in (14.1-1) and illustrated in Fig. 14.2-4. The rate equations (14.2-3) and (14.2-4) must then be extended to include this source of population loss and gain in both levels: dN 2 N 2 - == R 2 - - - N 2 W. + N 1 W. dt T2 dN l N 1 N 2 - == -R 1 - - + - + N 2 Wi - N l Wi . dt Tl T21 (14.2-8) ( 14.2-9) The population density of level 2 is decreased by stimulated emission from level 2 to level 1 and increased by absorption from level 1 to level 2. The spontaneous emission contribution is contained in T21. y(j) ;yCD . . I I TZI I I W.- l I I I " t I I I I TZ Figure 14.2-4 The population densities N 1 and N 2 (cm- 3 -s- 1 ) of atoms in en- ergy levels 1 and 2 are determined by three processes: decay (at rates 1/T1 and 1/T2' respectively, which includes the effects of spontaneous emission), depumping and pumping (at rates Rl and R 2 , respectively) and absorption and stimulated emission (at rate Wi with corresponding time constant w; 1 ). I I I Y Tl I I I , Tzo Under steady-state conditions (dN I /dt == dN 2 /dt == 0), (14.2-8) and (14.2-9) are readily solved for N 1 and N 2 , and for the population difference N == N 2 - N 1 . The result is N == No 1 + Ts Wi ' (14.2-10) Steady-State Population Difference (Presence of Amplifier Radiation) where No is the steady-state population difference in the absence of amplifier radiation, given by (14.2-5). The characteristic time Ts, which is always positive since T2 < T21, is given by Ts == T2 + Tl ( 1 _ T2 ) . T21 (14.2-11) Saturation Time Constant
14.2 AMPLIFIER PUMPING 543 In the absence of amplifier radiation, Wi == 0 so that (14.2-10) provides N == No, as expected. Because Ts is positive, the steady-state population difference in the presence of amplifier radiation always has a smaller absolute value than in its absence, i.e., I NI < I No I. If the radiation is sufficiently weak so that Ts Wi « 1 (the small- signal approximation), we may take N No. As the amplifier radiation becomes stronger, Wi increases and ultimately N 0 regardless of the initial sign of No, as shown in Fig. 14.2-5. This arises because stimulated emission and absorption dominate the interaction when Wi is very large and they have equal probability densities. It is apparent that even very strong radiation cannot convert a negative population difference into a positive one, nor vice versa. The quantity Ts plays the role of a saturation time constant, as is evident from Fig. 14.2-5. N u c No 4-. :a c .9 N --D ------------------ ::; 2 0.. o p.. 0.1 Ts Ts 10 Ts w. I Figure 14.2-5 Depletion of the steady-state population difference N == N 2 - N 1 as the rate of absorption and stimulated emission Wi increases. When Wi == 1/7s, N is reduced by a factor of 2 from its value when Wi == o. o EXERCISE 14.2-2 Saturation Time Constant. Show that if t sp « 7nr (the nonradiative part of the lifetime 721 of the 2 I transition), t sp « 720, and t sp » 71, then 7s t sp . B. Pumping Schemes We now proceed to examine specific (four-level and three-level) pumping schemes that are used in practice to achieve a population inversion. The object of these arrangements is to make use of an excitation process that increases the number of atoms populated in level 2 while decreasing the number populated in level 1. Four-Level Pumping In this arrangement, shown in Fig. 14.2-6, level 1 lies above the ground state (which is designated as the lowest energy level 0). In thermal equilibrium, level 1 will be virtually unpopulated provided that El » kT, a situation that is, of course, desirable. Pumping is accomplished by making use of an energy level (or collection of energy levels) that lies above level 2; we designate this as level 3. The 32 transition has a short lifetime (decay occurs rapidly) so that there is little population accumulation in level 3. For reasons that are made clear in Prob. 14.2-4, two-level pumping is not possible so level 2 is pumped through level 3 rather than directly. Level 2 is long- lived, so that it accumulates population, whereas level 1 is short-lived so that it sheds population; a population inversion is thereby established between levels 2 and 1. All told, four energy levels are involved in the process but the optical interaction of interest takes place between levels 2 and 1.
544 CHAPTER 14 LASER AMPLIFIERS I I Rapid I I decay : 7 32 G) Short-lived level Pump R Laser W- 1 1 @ Long-lived level I I 72 I 7 21 I I I Rapid : T 1 decay : I t CD Short-lived level 720 T @ Ground state Figure 14.2-6 Energy levels and decay rates for a four-level system. The four levels are drawn from a multitude of levels (not shown). It is assumed that the rate of pumping into level 3, and out of level 0, are the same. An external source of energy (e.g., photons with frequency E3/h) pumps atoms from level 0 to level 3 at a rate R. If the decay from level 3 to level 2 is sufficiently rapid, it may be taken to be instantaneous, in which case pumping to level 3 is equivalent to pumping level 2 at the rate R 2 == R. The situation is then the same as that shown in Fig. 14.2-4 and the expressions in (14.2-10) and (14.2-11) apply. However, in this configuration atoms are neither pumped into nor out of levell, so that R 1 == O. Thus, in the absence of amplifier radiation (Wi == cjJ == 0), the steady-state population difference is given by (14.2-5) with R 1 == 0, i.e., No == RT2 ( 1- ) . T21 (14.2-12) In most four-level systems, the nonradiative decay component in the 2---41 transition is negligible (t sp « Tnr) and T20 » t sp » T1 (see Exercise 14.2-2), so that No Rt sp , Ts t sp , (14.2-13) (14.2-14) and therefore Rt sp N 1 + t sp Wi . Implicit in the preceding derivation is the assumption that the pumping rate R is independent of the population difference N == N 2 - N 1 . This is not always the case, however, because the population densities of the ground state and level 3, N g and N 3 respectively, are related to N 1 and N 2 by ( 14.2-15) N g + N 1 + N 2 + N 3 == N a , (14.2-16) where the total atomic density in the system, N a , is a constant. If the pumping involves a transition between the ground state and level 3 with transition probability W, then R == (N g - N 3 ) W. If levels 1 and 3 are short-lived, then N 1 N 3 0, whereupon N g + N 2 N a so that N g N a - N 2 N a - N. Under these conditions, the pumping rate can be approximated as R (N a - N) W, (14.2-17)
14.2 AMPLIFIER PUMPING 545 which reveals that the pumping rate is a linearly decreasing function of the population difference N and is thus clearly not independent of it. This arises because the popula- tion inversion established between levels 2 and 1 reduces the number of atoms available to be pumped. Substituting (14.2-17) into (14.2-15), and reorganizing terms, leads to N tspNaW . 1 + t sp W + t sp Wi ( 14.2-18) Finally, the population difference can be written in the generic form of (14.2-10), N == No 1 + Ts Wi ' (14.2-19) but where now No and Ts, rather than being expressed as (14.2-13) and (14.2-14), are given by N tspNa W o 1 + t sp W ( 14.2-20) and t sp Ts . 1 + t sp W (14.2-21) For weak pumping (W « l/t sp ), No tspN a W is proportional to the pumping tran- sition probability density W, and Ts t sp , so that (14.2-13) and (14.2-14) reemerge. However, as the pumping strength increases, No decreases and ultimately saturates, while Ts decreases. Three-Level Pumping A three-level pumping arrangement, in contrast, makes use of the ground state (E 1 == 0) as the lower laser level 1, as depicted in Fig. 14.2-7. Again, an auxiliary third level (designated 3) is involved and the 32 decay is rapid so that there is no buildup of population in level 3. The 3 1 decay is slow (T32 « T31) so that the pumping serves to populate level 2, the upper laser level, which is long-lived and therefore accumulates population. Atoms are pumped from level 1 to level 3 (e.g., by absorbing light at the frequency E3/h) at a rate R; fast (nonradiative) decay effectively pumps level 2 at the rate R 2 == R. The thermally excited population of level 2 is assumed to be negligible. It is not difficult to see that under rapid 3 2 decay, the three-level system displayed in Fig. 14.2-7 is a special case of the system shown in Fig. 14.2-4 (provided that R is independent of N) with the parameters Rl == R 2 == R, T1 == 00, 72 == T21. ( 14.2-22) To avoid algebraic problems in connection with the value T1 == 00, rather than substi- tuting these special values into (14.2-10) and (14.2-11), we return to the original rate equations (14.2-8) and (14.2-9). In steady state, both of these equations provide the same result: N 2 o == R - - - N 2 Wi + N 1 Wi . T21 (14.2-23)
546 CHAPTER 14 LASER AMPLIFIERS I I Rapid I decay : 732 G) Short-lived level Pump R Laser @ Long-lived level I I : 721 I I CD Ground state Figure 14.2-7 Energy levels and decay rate for a three-level system. A multitude of other energy levels exist, but they are not germane to the considerations at hand. It is assumed that the rate of pumping into level 3 is the same as the rate of pumping out of level 1. It is not possible to determine both N 1 and N 2 from a single equation relating them. However, knowledge of the total atomic density N a in the system (in levels 1,2, and 3) provides an auxiliary condition that does permit N 1 and N 2 to be determined. Since T32 is very short, level 3 retains a negligible steady-state population; all of the atoms that are raised to it immediately decay to level 2. Thus, N 1 + N 2 == N a , (14.2-24 ) which enables us to solve (14.2-23) for N 1 and N 2 and thereby to determine the population difference N == N 2 - N 1 and the saturation time Ts. The result may be cast in the usual form of (14.2-10), N == N o /(l + Ts Wi), where now No == 2RT21 - N a Ts == 2T21 . (14.2-25) ( 14.2-26) When nonradiative decay from level 2 to level 1 is negligible (t sp « Tnr), T21 may be replaced by t sp , whereupon No 2Rt sp - N a Ts 2t sp . ( 14.2-27) ( 14.2-28) It is of interest to compare these equations with the analogous results (] 4.2-13) and (14.2-14) for a four-level pumping scheme. Attaining a population inversion (N > 0 and therefore No > 0) in the three-level system requires a pumping rate R > N a /2t sp . Thus, just to make the population density N 2 equal to N 1 (i.e., No == 0) requires a substantial pump power density, given by E3Na/2tsp. The large population in the ground state (which is the lowest laser level) is an inherent obstacle to achieving a population inversion in a three-level system that is avoided in a four-level system (in which level 1 is normally empty since T1 is short). The saturation time constant Ts t sp for the four-level pumping scheme is half that for the three-level scheme. And again, as shown in Prob. 14.2-4, a steady-state population inversion cannot be achieved by means of direct optical pumping between levels 1 and 2. The dependence of the pumping rate R on the population difference N can be included in the analysis of the three-level system by writing R == (N 1 - N 3 ) W, N 3 0, and N 1 == (Na - N), from which R (Na - N)W. Substituting this in the principal equation N == (2Rt sp - N a ) / (1 + 2t sp Wi), and reorganizing terms, we can write the population difference in the usual form, N == No 1 + Ts Wi ' (14.2-29)
14.3 COMMON LASER AMPLIFIERS 547 but now with N _ N a ( t sp W - 1) 0- 1 + t sp W (14.2-30) and Ts == 2t sp 1 + t sp W . (14.2-31) As in the four-level scheme, No and Ts saturate as the pumping transition probability W increases. EXERCISE 14.2-3 Pumping Power in Three- and Four-Level Systems. (a) Determine the pumping transition probability W required to achieve a zero population difference in a three- and a four-level laser amplifier. (b) If the pumping transition probability W = 2/t sp in the three-level system, and W = 1/2t sp in the four-level system show that No = N a /3. Compare the pumping powers required to achieve this population difference. Pumping Methods As indicated earlier, pumping may be achieved by many methods, including the use of electrical, optical, and chemical means (see Sec. 13.5A for a discussion of various forms of luminescence). A number of common methods of electrical and optical pump- ing are illustrated schematically in Fig. 14.2-8. Nuclear pumping makes use of a stream of high-energy particles or gamma rays derived from a nuclear reactor or radioisotope. It is important to recognize that Rl and R 2 represent the numbers of atoms per unit time per unit volume for which pumping is successfully achieved. The pumping process can be quite inefficient. In optical pumping, for example, many of the photons supplied by the pump can fail to raise the atoms to the upper laser level and are therefore lost. 14.3 COMMON LASER AMPLIFIERS Laser amplification can take place in a great variety of materials. The energy-level diagrams for a number of representative atoms, ions, molecules, and solids that ex- hibit laser action are displayed in Sec. 13.1. Practical laser systems usually involve many interacting energy levels that influence the population densities associated with the transition of interest, N 1 and N 2 , as illustrated in Fig. 14.2-2. Nevertheless, the essential principles of laser-amplifier operation may be codified in terms of three- and four-level systems.
548 CHAPTER 14 LASER AMPLIFIERS (a) Anode .>: :..:. :..: ...: .Gs::." :.. -: ::.":"0 . . . .- ... ". .. . .. . .: ......"...... .. (d) :(] 0 Er 3 +:Silica fiber Laser Lens diode (b) " . :. :.:.. ... . 6s" .:. .":'..: ::": ... .:. ". ". . .. . .:.: ... .. .. . (e):(]) Laser-diode Lens arra y I Nd3+:YV04 rod ROd FlaShlamp (c) () Figure 14.2-8 Examples of electrical and optical pumping. (a) Direct current (dc) is often used to pump gas lasers. The current may be passed either along the laser axis, creating a longitudinal discharge, or transverse to it. (b) Radio-frequency (RF) discharge currents are also used for pumping gas lasers. (c) Xe flashlamps or Kr CW arc lamps are useful for optically pumping ruby and rare-earth solid-state lasers. (d) Semiconductor laser diodes are often used for pumping Er 3 + :silica fiber laser amplifiers. (e) An array of laser diodes is generally used to optically pump Nd 3 +:YV0 4 lasers. This is illustrated by three laser-amplifier systems: the three-level ruby laser ampli- fier, the four-level neodymium-doped glass laser amplifier, and the three-level erbium- doped silica-fiber laser amplifier. These are discussed in turn. Although most lasers operate on the basis of a four-level pumping scheme, ruby and Er 3 + -doped silica fiber are exceptions. We also consider an important amplifier that operates on the basis of stimulated Raman scattering. All of the laser amplifiers discussed here also operate as laser oscillators (see Sec. 15.3A). Most laser amplifiers are used as power amplifiers; this form of amplifier is de- signed to increase the power of a high-quality, but low-power, laser oscillator. How- ever, some laser amplifiers, such as erbium-doped silica fiber, are also used in optical fiber communication systems as in-line amplifiers (optical repeaters) and as optical preamplifiers, designed to boost a signal prior to photodetection (see Sec. 24.1 C). Semiconductor optical amplifiers are described in Sec. 17.2. Laser amplifiers are often operated in the saturation regime (Sec. 14.4). A. Ruby Ruby (Cr 3 +:A1 2 0 3 ) is sapphire (A1 2 0 3 ), in which chromium ions (Cr 3 +) replace a small percentage of the aluminum ions (see Sec. 13.1 C). Ruby is the first material in which laser action was observed (see page 567 of Chapter 15). It serves as a didactic example since this laser amplifier is rarely used today. As with most materials, laser action can take place on a variety of transitions. The energy levels pertinent to the well-known red ruby-laser transition, labeled with their group-theoretical symbols, are displayed in Fig. 14.3-1. It is a three-level system. Levell is the ground state. Level 2 comprises a pair of closely spaced discrete levels; these levels are not resolved in Fig. 14.3-1 - the lower of the two levels, known as R 1 , corresponds to the famous red laser transition at Ao == 694.3 nm. Level 3 comprises two broad bands centered about 550 nm (green) and 400 nm (violet); these absorption bands are responsible for the reddish color of the material as light passes through it. As illustrated in Fig. 14.3-2, a ruby rod may be optically pumped from level I to level 3 by surrounding it with a helical flash lamp or by enclosing it, together with a
14.3 COMMON LASER AMPLIFIERS 549 --- > (]) '-' c+: Al 2 0 3 (Ruby) - 4p 1 - G) 1,,\ 4F 2 I - 2£ @ Pump 694-nm - laser 4A CD 2 4 3 co (]) s::: 2 1 Figure 14.3-1 Relevant energy levels for the 2E 4A 2 red ruby-laser transition at 694.3 nm. The three interacting levels are indicated by encircled numbers. o linear flashlamp, within a reflecting cylinder of elliptical cross section (see Fig. l.2-3). The flashlamp emits white light, a fraction of which is absorbed by level 3, which is quite broad, resulting in the excitation of the Cr 3 + ions to level 3. Excited Cr 3 + ions rapidly decay from level 3 to level 2 (732 is of the order of ps). They remain in level 2 for a substantial time since the spontaneous lifetime for the 2] transition is relatively long (t sp 3 ms), complying with the three-level laser scheme shown in Fig. 14.2-7. Nonradiative decay is negligible (721 t sp ). The transition has a homogeneously broadened linewidth v 330 GHz that arises principally from elastic collisions with lattice phonons. The characteristics of the ruby-laser transition and the ruby-laser oscillator are provided in Tables 14.3-1 and 15.3-1, respectively. Ruby rod '\ Flashlamp """\. v \ Input photons Ouput photons Flashlamp 1: ... '--- Ruby rod citor Elliptical/ mIrror Power supply (a) (b) Figure 14.3-2 Ruby laser-amplifier configurations. (a) Geometry used for the first laser oscillator built by Maiman in 1960 (see page 567 of Chapter 15). (b) High-efficiency pumping geometry using a linear flashlamp in a reflecting elliptical cylinder. B. Neodymium-Doped Glass Nd 3 +:glass amplifiers can be made in very large sizes and can therefore be used to generate extremely powerful optical pulses, albeit at low duty cycle because of the limited thermal conductivity of glass. Glass can be fabricated with high optical quality and retains its optical finish. Moreover, it has the merit of being isotropic and readily doped in a homogeneous fashion.
550 CHAPTER 14 LASER AMPLIFIERS The four-level neodymium-doped glass-laser amplifier plays a central role at the Na- tional Ignition Facility (NIF), located at the Lawrence Livermore National Laboratory (LLNL) in Livermore, California (see Example 14.3-1). The energy levels relevant to the Ao == 1.053 J-LID transition for the particular phosphate laser glass used at the NIF are displayed in Fig. 14.3-3. Also shown in this figure is the spectrum of the Xe flashlamp light used to pump this amplifier. 3.0 Flashlamp 3.0 Nd 3 +: Glass 2.5 2.5 2.0 --- > Q) "-" 1.5 OJ) Q) c:: 1.0 I I 0.5 0.5 1.0 . Relative spectral intensity 0 G) 4 F 3 @T 1.053-JLm laser 4 CD I /ll/2 2.0 --- > Q) "-" 1.5 OJ) Q) c:: 1.0 Pump 0.5 @ 4/9/2 o Figure 14.3-3 Left: Spectral profile of the broadband Xe flashlamp emission used to pump the neodymium-doped glass laser amplifiers at the National Ignition Facility (NIF). Right: Relevant energy levels for the 4F 3 / 2 41 11 / 2 laser transition at 1.053 J1ID in neodymium-doped phosphate glass (Schott LG-770). The four interacting energy levels are indicated by encircled numbers. Level 1 has an energy that is 0.24 e V above the ground state. This is substantially larger than the thermal energy at room temperature, kT 0.026 eV, so that the thermal population of the lower laser level is negligible. Level 3 is a collection of four absorption bands, each about 30 nm wide, centered at 805, 745,585, and 520 nm; these bands are responsible for the purple color of the material when viewed in transmission. The excited ions decay rapidly from level 3 to level 2 and then remain in level 2 for a substantial time (t sp == 375 J-Ls). Since 71 is very short ( 300 ps), the energy-level structure of Nd 3 +:glass falls within the four-level laser scheme displayed in Fig. 14.2- 6. The 2---+ 1 transition is inhomogeneously broadened as a result of the amorphous nature of the glass, which presents a different environment at each ionic location. This material therefore has a large room-temperature linewidth, v 7 THz. Other features of the 2---+ 1 transition in this material are detailed in Table 14.3-1. The gain is substantially greater than that of ruby by virtue of its four-level character. The characteristics of a small Nd 3 +:glass laser oscillator are given in Table 15.3-1. EXAMPLE 14.3-1. Neodymium-Doped Glass Laser Amplifiers at the Nationallgni- tion Facility. Neodymium-doped glass amplifiers are widely used in experiments designed to achieve the ignition of controlled thermonuclear fusion, such as those carried out at the NIF and at the Laser MegaJoule (LMJ) project in France. Both the NIF and LMJ laser systems employ a design that comprises four clusters of laser amplifiers, each of which consists of 6 amplifier bundles. Each bundle in turn contains 8 amplifying laser-glass plates stacked inside a flashlamp-pumped cavity, as shown in Fig. 14.3-4. The use of square beams allows the individual laser-glass amplifiers to be tightly packed into a compact configuration, thereby reducing the size and cost of the system. Although the laser aperture is square, the laser glass plates are rectangular since they are mounted at Brewster's angle
14.3 COMMON LASER AMPLIFIERS 551 to the direction in which the beam propagates. This minimizes Fresnel reflection losses at the slab surfaces and enhances the coupling of the flashlamp pump light and the slabs. Amplifier design at the NIF calls for the generation of optical pulses with energies of 1.8 MJ in 3.5 ns corresponding to a peak power of 500 TW. To provide time for the glass to cool between shots, no more than 5 successive shots may be fired in any 24-hour period. The seed optical pulse is provided by an Yb 3 + -doped fiber laser (see Sec. 15.3A). To facilitate the thermonuclear fusion process, the output of the beamlines is tripled in frequency to 351 nm in the ultraviolet. This is achieved via a cascaded process of frequency doubling (second- harmonic generation) to 526 nm in a nonlinear optical crystal of potassium dihydrogen phosphate (KDP), followed by parametric upconversion (sum-frequency generation) of the fundamental and sec- ond harmonic in a crystal of deuterated potassium dihydrogen phosphate (DKDP) (see Chapter 21). The frequency-tripled beams are then focused to mm-sized spots before impinging on a pellet target containing tritium and deuterium. The facility occupies a building the size of a football stadium. ,- FlaShlap arrays . . . 2m I . . . . ... Nd 3 +: Glass slab (a) (b) Figure 14.3-4 (a) A bundle of amplifiers comprises eight laser-glass plates stacked inside a flashlamp-pumped cavity at the National Ignition Facility (NIF) located at Lawrence Livermore National Laboratory. Each plate, which is made of specially formulated phosphate laser glass with a neodymium doping level:::::: 2 mol% (Schott LG-770 or Hoya LHG-8), measures 46 cm x 81 cm x 4 cm. The height of the eight-amplifier bundle is :::::: 2 m. Six such bundles make up a cluster, and four clusters comprise the 192 individual beamlines at the NIF. Each beamline in turn consists of 16 separate amplification stages so the overall system contains 3072 laser-amplifier plates. (b) Top view of the linear flashlamps and amplifying Nd 3 + :glass laser plates in a bundle. c. Erbium-Doped Silica Fiber Optical fiber amplifiers (OFAs) are useful amplifying media that offer the advan- tages of single-mode guided-wave optics (see Chapter 9). In rare-earth-doped fiber amplifiers (REFAs), the signal and pump are introduced into the doped fiber, and the signal is amplified via stimulated emission provided by downward electron transitions in the pumped ions. REFAs typically make use of Er 3 + , Pr 3 +, Tm 3 +, Nd 3 +, Yb 3 +, and Ho 3 +. They operate over a broad range of wavelengths, principally in the near infrared; however, al1 of the dopants, with the exception of Er 3 + , operate efficiently in the usual telecommunications band only for host glasses other than silica. Since silica glass is the host of choice, erbium-doped fiber amplifiers (EDFAs) are widely used in optical fiber communication systems, often in concatenation (see Chapter 24). Among other attractive features, they offer high polarization-independent gain, low insertion loss, and a broad transition near A == 1550 nm (corresponding to the wavelength of minimum loss for silica optical fibers, as shown in Fig. 9.3-2).
552 CHAPTER 14 LASER AMPLIFIERS Pumping is achieved by longitudinally coupling light into the amplifying medium from strained quantum-well InGaAs laser diodes operating at Ao == 980 nm. As shown in Fig. 14.3-5, the pump light may be injected in the same direction as the signal (the forward direction), in the opposite direction from the signal (the backward direction), or in both directions (bidirectional). Although more complex, the latter configuration is preferred since it provides a relatively uniform distribution of pump power along the length of the amplifying fiber, which serves to increase efficiency and reduce noise. Double-clad fiber configurations can be used to avoid nonlinear optical effects in the fiber core when the pump power is high, as is usually the case in fiber laser oscillators (see Fig. 15.3-5). Ytterbium or thulium are sometimes used as codopants with erbium; codoping offers various benefits such as extending the wavelength of operation and permitting greater optical power to be attained by offering an energy- transfer mechanism and preventing erbium-ion clustering. Gain fiber Gain fiber Gain fiber ) (a) (b) (c) Figure 14.3-5 Longitudinal pumping of a fiber laser amplifier. The pumping may be (a) in the forward direction; (b) in the backward direction; or (c) bidirectional. Er 3 +: silica-fiber amplifiers are often pumped by strained [nGaAs laser diodes operated at Ao == 980 nm. Raman silica fiber amplifiers, discussed in Sec. 14.3D, can also be pumped by InGaAsP laser diodes operated at a wavelength about 100 nm below that desired for amplification; however, Raman fiber laser pumping IS common. As illustrated in Fig. 14.3-6, the 980-nm pumped Er 3 +: silica fiber system operates at a wavelength in the vicinity of A == 1550 nm on the 4[13/2 4[15/2 fluorescence line. It behaves as a three-level system at T == 300° K and as a four-level system when cooled to T == 77° K. Since Er 3 + is a lanthanide-metal ion, the material in which the ions are embedded plays a minima] role in determining the energy levels. The broadening is a mixture of homogeneous (phonon mediated) and inhomogeneous (arising from local field variations in the glass); this leads to a wavelength-dependent gain profile that requires gain equalization for some applications. Salutary features of the laser transition include a long excited-state spontaneous lifetime, the absence of intermediate energy levels between the ground state and the excited state, and the absence of excited-state absorption. These properties allow gains in excess of 50 dB to be achieved in EDFAs with tens of m W of pump power. As a particular example, a gain of 30 dB is obtained by launching 5 rn W of pump power at 980 nm into a roughly 50-m length of fiber containing 300 pprn Er203. The best gain efficiencies are 10 dB/mW. Moreover, signal output powers in excess of 100 W can be generated since the output power increases in proportion to the pump power. The available bandwidth is A 40 nrn, corresponding to v 5.3 THz, which accommodates the C (conventional) band that extends from about 1530 to 1565 nm (see Sec. 24.1A). The L (long) band, which stretches from about 1565 to 1625 nm, is also readily accommodated although the optimization parameters of the EDFAs are not the same in the two bands. The large gain-bandwidth product offered by these amplifiers make them highly suitable for use in wavelength-division multiplexing (WDM) systems (see Sec. 24.3). Various characteristics of the Er 3 +: silica-fiber laser amplifier are provided in Table 14.3-1. The introduction of feedback readily converts amplification into oscillation, as discussed in Sec. 15.3A.
14.3 COMMON LASER AMPLIFIERS 553 2.0 Er 3 +: Silica fiber - 1.5 -- > cu '--" 4/ 1112 G) 4/1312 1.55-JLm laser 4 CD I 1 1512 - 1.0 o Figure 14.3-6 Schematic of energy-level manifolds for the Er 3 +: silica fiber 11 13 / 2 ----7 41 15 / 2 laser transition in the vicinity of 1550 nm. The erbium-doped silica-fiber system behaves as a three-level laser at T = 300 0 K. The three interacting levels are indicated by encircled numbers. The system can also be made to behave as a four-level laser in the vicinity of 2.9 /-LID on the 11 11 / 2 ----7 41 13 / 2 transition. Ol) cu c:: Pump - 0.5 The 4[13/2 --+ 41 15 / 2 laser transition can also be directly pumped at 1.48 /-Lill by light from InGaAsP laser diodes. This quasi-two-Ievel pumping scheme is less efficient than the three-level scheme implemented at 980 nm since the gain per unit pump power is lower and the noise is higher in the latter case. However, the pump transition linewidth and saturation signal power are greater at 1.48 /-Lill so that pumping at this wavelength is sometimes used for higher power amplifiers. In some configurations, EDFAs are simultaneously pumped at both wavelengths. With respect to other rare-earth-doped fiber amplifiers, good performance can be obtained from Tm 3 + -doped fluoride or multicomponent-silicate glass REFAs operat- ing in the 1460-1530-nm S-band wavelength range, from Pr 3 + -doped REFAs in the 1300-nm region, and from Yb 3 +:silica fiber REFAs operating in the 1050-1120-nm wavelength range. Other classes of optical amplifiers in common usage are Raman fiber amplifiers (RFAs), discussed in Sec. 14.3D, and semiconductor optical amplifiers (SOAs), studied in Sec. 17.2. The relative merits of these three classes of amplifiers are examined in Sees. 17.2D and 24.1 C. Summary The Er 3 +: silica fiber amplifier is widely used by virtue of its many salutary features: . High gain . High output power . Broad bandwidth . Polarization insensitivity . High efficiency . Low insertion loss . Low noise D. Raman Fiber Amplifiers Erbium-doped fiber amplifiers, and rare-earth-doped fiber amplifiers in general, are not the only form of optical fiber amplifiers. OFAs also operate on the basis of principles other than stimulated emission. An important version of the OFA, known as the Raman fiber amplifier (RFA), relies on stimulated Raman scattering.
554 CHAPTER 14 LASER AMPLIFIERS As discussed in Sec. 13.5C, stimulated Raman scattering (SRS) occurs when a pump photon of energy hv p , together with a signal photon of lower energy hv s , enter a nonlinear optical medium such as an optical fiber. The nonresonant version of the process is illustrated in the inset in Fig. 14.3-7; the dashed horizontal line represents a virtual state. The signal photon stimulates the emission of a clone signal photon, which is obtained by Stokes-shifting the pump photon by hV R so that its energy precisely matches that of the incident signal photon. The surplus energy from the pump photon is transferred to the vibrational modes of the glass fiber. The ensuing optical amplification is similar to the stimulated emission in an erbium-doped fiber amplifier, but the bandwidth over which Raman amplification obtains is governed by the vibrational spectrum of the glass host rather than by the transition linewidth of a dopant ion. The strength of the effect, embodied in the Raman gain coefficient R .. depends on the nonlinear properties of the glass fiber and is proportional to the pump intensity Ip == P / A, where P is the pump power and A is the area of the interaction [see (21.3-15)]. C 1.0 <l) .u 0.8 o u c .@ 0.6 00 c ro 0.4 <l) .b 0.2 ro a) 0 o - J -- --- hv s hv p Ih!J.v hV R 10 20 30 40 Stokes frequency shift v R (THz) Figure 14.3-7 Stimulated Raman scattering (SRS) is schematized in the inset. Raman gain is available over a range of Stokes frequencies determined by the vibrational characteristics of the material. Silica, germanium, phosphorus, and borate glasses all have very different SRS spec- tral distributions and magnitudes. In germanium- doped silica fiber, the peak Raman gain coefficient lies at a frequency below that of the pump by 13 THz and the bandwidth v 12.5 THz. (Gain curve adapted from R. H. Stolen, C. Lee, and R. K. Jain, Development of the Stimulated Raman Spectrum in Single-Mode Silica Fibers, Journal of the Optical Society of America B.. vol. 1, pp. 652- 657, 1984, Fig. 5.) RFAs can be either distributed or lumped. In the distributed Raman fiber ampli- fier, the signal and pump are both sent through the transmission fiber, which serves as the gain medium. The lumped Raman fiber amplifier, in contrast, makes use of a short length of highly nonlinear fiber dedicated to providing gain. The core is generally made small to increase the pump intensity Ip and thereby to reduce the length of fiber required, which can be considerable. As with the EDFA, the pump light may be injected in the forward direction, in the backward direction, or bidirectionally (see Fig. 14.3-5); backward pumping is generally employed since it reduces the noise transferred from the pump to the signal. Raman fiber amplifiers can offer substantially broader bandwidths than EDFAs. As is evident in Fig. 14.3-7, the dominant peak in the Raman gain coefficient for the usual germanium-doped silica fiber is Stokes shifted from the pump frequency by approx- imately VR == 13 THz, corresponding to about 100 nm at Ao == 1550 nm. However.. phosphosilicate glass fibers offer substantially greater Stokes shifts (see Example 15.3- 1). Pumping of the RFA can be achieved by making use of polarization-diverse laser diodes, fiber lasers, or Raman fiber lasers, operated at a wavelength about 100 nm below that desired for amplification if the medium is germanium-doped silica fiber. In this material, the bandwidth over which substantial Raman gain is available is about the same as the shift, namely v 12.5 THz, again corresponding to about A 100 nrn at 1550 nm. However, combining multiple pumps of different frequencies can lead to far greater bandwidths since the Stokes shift is linked to the pump wavelength.
14.3 COMMON LASER AMPLIFIERS 555 In principle, Raman amplification can be employed over the entire region of fiber transparency. Raman fiber amplifiers offer gains approaching 20 dB. The RFA gain efficiency in germanium-doped silica fiber is 0.02 dB/mW, which is to be compared with a gain efficiency 10 dB/mW for an EDFA. Thus, the pump power for achieving useful levels of Raman gain in such a distributed amplifier is typically hundreds of m W, far greater than that required for an EDFA. In lumped Raman amplifiers, where safety risks are not of concern, pump powers in excess of ] W can be used. Unlike EDFAs, polarization-diverse pumping is required since the Raman gain is maximized when the signal and pump beams have the same polarization. Although RFA efficiencies are substantially lower than those offered by EDFAs, they can be measurably enhanced by the use of dispersion-compensating fiber and are sometimes used in conjunction with EDFAs. The use of dispersion compensation, along with the availability of high-power laser-diode arrays, make RFAs competitive as optical fiber amplifiers, particularly as the requirements for bandwidth continue to increase toward the long-distance trans- mission of many tens of THz. Summary The Raman fiber amplifier enjoys both advantages and disadvantages in compar- ison with the erbium-doped fiber amplifier: Advantages of the RFA relative to the EDFA: . Wider bandwidth . Bandwidth extendable by use of multiple pumps . Operation over a broad range of wavelengths . Arbitrary fiber host . Compatible with existing links Disadvantages of the RFA relative to the EDFA: . Smaller gain . Greater pump power and lower efficiency . Longer fiber lengths . Sensitivity to signal polarization A comparison of the performance of optical fiber amplifiers and semiconductor optical amplifiers is provided in Secs. 17.2D and 24.1 C. E. Tabulation of Selected Laser Transitions The most commonly used laser amplifiers are those discussed in Secs. 14.3B and 14.3C. However, laser amplification is also provided by gases, dyes, free-electron systems, and semiconductors. Table 14.3-1 provides the wavelengths, cross sections, spontaneous lifetimes, linewidths, and refractive indexes for a number of represen- tative laser transitions. Although the maser principle can be implemented from the microwave to the X-ray region, the entries in Table 14.3-1 highlight transitions in the visible and infrared. The values of ao, t sp , and v vary over a broad range.
556 CHAPTER 14 LASER AMPLIFIERS Table 14.3-1 Characteristics of common laser transitions. Transition Transition Spontaneous Transition Refractive Wavelength a Cross Section Lifetime Linewidth b Index Laser Medium Ao (nm) 0"0 (cm 2 ) t sp l/ n C 5 + 18.2 5 x 10- 16 12 ps I THz I 1 ArF Excimer 193 3 x 10- 16 10 ns 10 THz [ 1 Ar+ 515 3 x 10- 12 10 ns 3.5 GHz I 1 Rhodamine-6G dye 560-640 2 x 10- 16 5 ns 40 THz H/I 1.40 He-Ne 633 3 x 10- 13 150 ns 1.5 GHz I 1 Cr 3 + :A1 2 0 3 694 2 x 10- 20 3ms 330 GHz H 1.76 Cr 3 + :BeAI 2 0 4 700-820 1 x 10- 20 260 J-LS 25 THz H 1.74 Ti 3 + :A1 2 0 3 700-1050 3 x 10- 19 3.9 J-LS 100 THz H 1.76 Yb 3 +:YAG 1030 2 x 10- 20 1 ms 1 THz H 1.82 Nd 3 + :Glass (phosphate) 1053 4 x 10- 20 370 J-LS 7THz I 1.50 Nd 3 +:YAG 1064 3 x 10- 19 230 J-LS 150 GHz H 1.82 Nd 3 +:YV0 4 1064 8 x 10- 19 100 J-LS 210 GHz H 2.0 InGaAspr 1300-1600 2 x 10- 16 2.5 ns 10 THz H 3.54 Er 3 + :Silica fiber 1550 6 x 10- 21 10ms 5THz Hn 1.46 CO 2 10 600 3 X 10- 18 3s 60 MHz I 1 aThe free-space wavelength shown in the table represents the most commonly used transition in each laser medium. The He-Ne gas laser system, for example, is most often used on the red-orange line at 0.633 J-LID, but it is also extensively used at 0.543, 1.15, and 3.39 J-LID (it also has laser transitions at hundreds of other wavelengths). bYalues reported for gases such as C02 are typical for low-pressure operation (the atomic linewidth in a gas depends on its pressure because of the presence of collision broadening, which is homogeneous). H and I indicate line broadening dominated by homogeneous and inhomogeneous mechanisms, respectively. cYalues are for Ino.72GaO.28Aso.6P0.4 assuming an injected carrier concentration n = 1.8 x 10 18 cm -3 (see Examples 17.2- 1-17.2-3). 14.4 AMPLIFIER NONLINEARITY A. Saturated Gain in Homogeneously Broadened Media Gain Coefficient It has been established that the gain coefficient --y( v) of a laser medium depends on the population difference N [see (14.1-4)], which in turn is governed by the pumping level [see (14.2-15)]; that N also depends on the transition rate Wi [see (14.2-10)]; and that Wi in turn depends on the radiation photon-flux density cP [see (14.1-1)]. It follows that the gain coefficient of a laser medium is dependent on the photon-flux density that is to be amplified. This is the origin of gain saturation and laser amplifier nonlinearity, as we now show. Substituting (14.1-1) into (14.2-10) provides N == No 1 + cPlcPs(v) (14.4-1) where 1 A 2 Ts ljJ ( ) = Tsa(v) = -- g(v). s V 81T t sp (14.4-2) Saturation Photon-Flux Density
14.4 AMPLIFIER NONLINEARITY 557 This represents the dependence of the population difference N on the photon-flux den- sity cP. Now, substituting (14.4-1) into the expression for the gain coefficient (14.1-4) leads directly to the saturated gain coefficient for homogeneously broadened media: "Yo (v) I'(v) = 1 + cjJ/4Js(v) , ( 14.4-3) Saturated Gain Coefficient where A 2 "Yo (v ) == No a(v) == No g(v) . 87rt sp (14.4-4) Small-Signal Gain Coefficient The gain coefficient is a decreasing function of the photon-flux density cP, as illustrated in Fig. 14.4-1. The quantity cPs(v) == l/T s a(v) represents the photon-flux density at which the gain coefficient decreases to half its maximum value; it is therefore called the saturation photon-flux density. When Ts t sp the interpretation of cPs(v) is straightforward: roughJy one photon can be emitted during each spontaneous emission time into each transition cross-sectional area [a(v) cPs(v) t sp == 1]. '"'/( v) '"Yo (v) 1 0.5 o 10- 2 1 0- 1 1 10 c/J c/JS<v) Figure 14.4-1 Dependence of the normalized saturated gain coefficient ,(v)/,o(v) on the normalized photon- flux density cP/cPs(v). When cP equals its saturation value cPs (v), the gain coeffi- cient is reduced to half its unsaturated value. EXERCISE 14.4-1 Saturation Photon-Flux Density for Ruby. Determine the saturation photon-flux density, and the corresponding saturation intensity, for the Ao == 694.3-nm ruby laser transition at v == Yo. Use the parameters provided in Table 14.3-1. Assume that Ts 2t sp , in accordance with (14.2-28). EXERCISE 14.4-2 Spectral Broadening of a Saturated Amplifier. Consider a homogeneously broadened am- plifying medium with a Lorentzian lineshape of width /j,y [see (14.1-8)]. Show that for a photon-flux density cP, the amplifier gain coefficient ,(v) assumes a Lorentzian lineshape with width !:::,. y s == !:::,. y 1+ cP . cPs (vo) ( 14.4-5) Linewidth of Saturated Amplifier
558 CHAPTER 14 LASER AMPLIFIERS Gain coefficien0 Vo v Figure 14.4-2 Gain coefficient re- duction and bandwidth increase result- ing from saturation when 4> == 24>s(vo). This demonstrates that gain saturation is accompanied by an increase in bandwidth, corresponding to reduced frequency selectivity, as illustrated in Fig. 14.4-2. Gain Having determined the effect of saturation on the gain coefficient (gain per unit length), we embark on determining the behavior of the saturated gain for a homogeneously broadened laser amplifier of length d [Fig. 14.4-3(a)]. For simplicity, we suppress the frequency dependencies of "Y(v) and cPs(v) and use the symbols "Y and cPs instead. If the photon-flux density at position z is cP(z), then in accordance with (14.4-3) the gain coefficient at that position is also a function of z. We know from (14.1-3) that the incremental increase of photon-flux density at the position z is dcP == "YcP dz, which leads to the differential equation dcP "YocP dz 1 + cP / cP s . (14.4-6) Rewriting this equation as (1/ cP + 1/ cPs) dcP == "Yo dz, and integrating, we obtain 1 cP(z) cP(z) - cP(O) _ n <p(0) + <Ps - 'Yo Z . (14.4-7) The relation between the photon-flux densities at the input and output, cP(O) and cP( d), respectively, is therefore [In(Y) + Y] == [In(X) + X] + "Yod , ( 14.4-8) where X == cP(O) / cPs and Y == cP( d) / cPs are the input and output photon-flux densities normalized to the saturation photon-flux density, respectively. It is useful to examine the solution for the gain G == cP( d ) / cP( 0) == Y / X in two limiting cases: 1. If both X and Yare much smaller than unity (i.e., the photon-flux densities are much smaller than the saturation photon-flux density), then X and Yare negligi- ble in comparison with In(X) and In(Y), whereupon we obtain the approximate relation In(Y) In(X) + "Yod , from which Y X exp("Yod). ( 14.4-9)
14.4 AMPLIFIER NONLINEARITY 559 In this case the relation between Y and X is linear, with a gain G == Y / X exp( "Yod) [leftmost dashed curve in Fig. 14.4-3(b)]. This accords with (14.1-7), which was obtained under the small-signal approximation, valid when the gain coefficient is independent of the photon-flux density, i.e., "Y "Yo. 2. When X » 1, we can neglect In(X) in comparison with X, and In(Y) In comparison with Y, whereupon Y X + "Yo d or cP( d) cP(O) + "YocPs d Nod cP(O) + - . Ts (14.4-10) (14.4-11) Under these heavily saturated conditions, the atoms of the medium are "busy" emitting a constant photon-flux density Nod / Ts. Incoming input photons there- fore simply leak through to the output, augmented by a constant photon-flux density that is independent of the amplifier input. For intermediate values of X and Y, (14.4-8) must be solved numerically. A plot of the solution is shown as the solid curve in Fig. 14.4-3(b). The linear input-output relationship obtained for X « 1, and the saturated relationship for X » 1, are evident as limiting cases of the numerical solution. The gain G == Y / X for "Yo d == 2 is plotted in Fig. 14.4-3(c). It achieves its maximum value exp( "Yo d) for small values of the input photon-flux density (X « 1), and decreases toward unity as X 00. 12 I I Amplifier ............. 8 --e- II ;:.... y = X exp( 'Yod) , I , I , , , , , , , , , , , I. .1 I Output E 0.. E 4 o d (a) 2 4 Input X = 4;(O)/4;s (b) ---------------- exp('Yod) 6 ,-. o ............. 4 --e- c:: . (:) 2 6 1 --------------------------- o 0.01 0.1 1 Input X = </J(O)/4;s (c) 10 Figure 14.4-3 (a) A nonlinear (saturated) amplifier. (b) Relation between the normalized output photon-flux density Y == cjJ(d)/cjJs and the normalized input photon-flux density X == cjJ(O)/cjJs. For X « 1, the gain Y/ X exp( !'ad). For X » 1, we obtain Y X + !'ad. A numerical solution of (14.4-8) is indicated by the solid curve. (c) Gain as a function of the input normalized photon-flux density X in an amplifier of length d with !'ad == 2. Saturable Absorbers If the gain coefficient "Yo is negative, i.e., if the population is normal rather than inverted (No < 0), the medium provides attenuation rather than amplification. The attenuation coefficient a(v) == -"Y(v) also suffers from saturation, in accordance with the relation a(v) == ao(v)/[l + cP/cPs(v)]. This indicates that there is less absorption for large val- ues of the photon-flux density. A material exhibiting this property is called a saturable absorber.
560 CHAPTER 14 LASER AMPLIFIERS The relation between the output and input photon-flux densities, cjJ( d) and cjJ(O), for an absorber of length d is governed by (J 4.4-8) with negative '"Yo. The overall transmittance of the absorber Y / X == cjJ( d ) / cjJ( 0) is presented as a function of X == cjJ( 0) / cjJs as the solid curve of Fig. 14.4-4. The transmittance increases with increasing cjJ(O), ultimately reaching a limiting value of unity. This effect occurs because the population difference N 0, so that there is no net absorption. I « d ./ 5' 0.8 ............. "6' 0.6 II ><: -......... 0.4 Q) u is . 0.2 en C\S $-0 0 0.1 - exp( 1'00) Input Saturable absorber Output I 10 Input X = </J(O)/</Js Figure 14.4-4 The transmittance of a saturable absorber Y / X == c/J( d) / c/J(O) versus the normalized photon-flux density X == c/J(O)/c/Js, for "rod == -2. The transmittance increases with increasing input photon-flux density. *B. Saturated Gain in Inhomogeneously Broadened Media Gain Coefficient An inhomogeneously broadened medium comprises a collection of atoms with dif- ferent properties. As discussed in Sec. 13.3D, the subset of atoms labeled (3 has a homogeneously broadened lineshape function g{3(v). The overall inhomogeneous av- erage lineshape function of the medium is described by g (v) == (g{3(v)), where (.) represents an average with respect to (3. Because the small-signal gain coefficient '"Yo (v ) is proportional to g(v), as provided in (14.4-4), different subsets (3 of atoms have different gain coefficients '"Yo{3 (v). The average small-signal gain coefficient is therefore A 2 '"Y o (v) == No g (v). 87rt sp ( 14.4-12) Solving for the saturated gain coefficient is more subtle, however, because the sat- uration photon-flux density cjJs(v), being inversely proportional to g(v) as provided in (14.4-2), is itself dependent on the subset of atoms (3. An average gain coefficient may be defined by using (14.4-3) and (14.4-2), '"Y (v) == ('"Y{3(v)), ( 14.4-13 ) where '"YO{3 (v) ')'(v) = 1 + 4Y/4Ys(v) == b g{3(v) 1 + cjJa 2 g{3(v) , with b == N O (A 2 /87rt sp ) and a 2 == (A 2 /87r) ( Ts/t sp ). Evaluating the average of (14.4- 14) requires care because the average of a ratio is not equal to the ratio of the averages. ( 14.4- J 4)
14.4 AMPLIFIER NONLINEARITY 561 Doppler-Broadened Medium Although all of the atoms in a Doppler-broadened medium share a g(v) of identical shape, the center frequency of the subset {3 is shifted by an amount v{3 proportional to the velocity v (3 of the subset. If g( v) is Lorentzian with width v, (14.1-8) provides g(v) == (v /27f)/[(v - vO)2 + (v /2)2] and g{3(v) == g(v - v(3). Substituting g{3(v) into (14.4-14) provides b(v/27f) '"'/(3 ( v) = (v _ v (3 - vo) 2 + (b. v s / 2) 2 ' (14.4-15) where vs == v 1+ cP cP s ( va ) ( 14.4-16) and 2a 2 ,,\2 Ts 2 <jJ-;l(vO) = b. _ 8 - t 7f V 7f sp 7f V ,,\2 Ts == -- g(vo). 87f t sp (14.4-17) Equation (14.4-16) was obtained for the homogeneously broadened saturated amplifier considered in Exercise 14.4-2 [see (14.4-5)]. It is evident that the subset of atoms with velocity v (3 has a saturated gain coefficient 'Y{3(v) with a Lorentzian shape of width vs that increases as the photon-flux density becomes larger. The average of'Y{3(v) in (14.4-13) is readily obtained since the shifts v{3 follow a zero-mean Gaussian probability density function p (v{3) == (27fa)-1/2 exp( -vffi/2a) with standard deviation aD (see Exercise 13.3-2). Thus, 'Y (v) == ('Y{3(v)) is given by ,",/ (v) = I: '"'/(3 (v)p(v(3) dV(3. (14.4-18) If p (v{3) is much broader than 'Y{3(v) (i.e., the Doppler broadening is much wider than vs), we may regard the broad function p(v{3) as constant and remove it from the integral when evaluating 'Y (vo). Setting v == va and v{3 == 0 in the exponential provides _ bp(O) '"'/(vo) = JI + 2<jJ a 2 /7rb.v 'Yo J l + cP / cPs (va) , (14.4-19) where the average small-signal gain coefficient 'Yo is ,,\2 1 'Yo == No 8 t 7f sp J 27fa; (14.4-20) Equation (14.4-] 9) provides an expression for the average saturated gain coefficient of a Doppler broadened medium at the central frequency va, as a function of the photon-flux density cP at v == va. The gain coefficient saturates as cP increases in accor- dance with a square-root law. The gain coefficient in an inhomogeneously broadened medium therefore saturates more slowly than the gain coefficient in a homogeneously broadened medium [see (14.4-3)], as illustrated in Fig. 14.4-5.
562 CHAPTER 14 LASER AMPLIFIERS 1 7(V o ) 1"0 0.5 o 10- 2 10- 1 10 10 2 c/J c/J s Figure 14.4-5 Comparison of gain saturation in homogeneously and inho- mogeneously broadened media. Hole Burning When a large flux density of monochromatic photons at frequency VI is applied to an inhomogeneously broadened medium, the gain saturates only for those atoms whose lineshape function overlaps VI. Other atoms simply do not interact with the photons and remain unsaturated. When the saturated medium is probed by a weak monochromatic light source of varying frequency v, the profile of the gain coefficient therefore exhibits a hole centered around VI, as illustrated in Fig. 14.4-6. This phenomenon is known as hole burning. Since the gain coefficient 'Y (3 (v) of the subset of atoms with velocity v (3 has a Lorentzian shape with width vs given by (14.4-16), it follows that the width of the hole is vs. As the flux density of saturating photons at VI increases, both the depth and the width of the hole increase. E o u it: (1) o u °a c lIO A VI II Figure 14.4-6 The gain coefficient of an inhomogeneously broadened medium is locally saturated by a large flux density of monochromatic photons at frequency VI. . II *14.5 AMPLIFIER NOISE The resonant medium that provides amplification via stimulated emission also gener- ates spontaneous emission. The light arising from the latter process, which is indepen- dent of the input to the amplifier, represents a fundamental source of laser amplifier noise. Whereas the amplified signal has a specific frequency, direction, and polariza- tion, the noise associated with amplified spontaneous emission (ASE) is broadband, multidirectional, and unpolarized. As a consequence, it is possible to filter out some of this noise by following the amplifier with a narrowband optical filter, a collection aperture, and a polarizer. The probability density (per second) that an atom in the upper laser level sponta- neously emits a photon of frequency between V and v + dv is (see Exercise 13.3-1): 1 Psp(v) dv == - g(v) dv. t sp ( 14.5-1)
14.5 AMPLIFIER NOISE 563 The probability density of spontaneously emitting a photon of any frequency is, of course, P sp == l/t sp . If N 2 is the atomic density in the upper energy level, the average spontaneously emitted photon density is N 2 P sp (v). The average spontaneously emit- ted power per unit volume per unit frequency is therefore hvN2Psp(v). This power density is emitted uniformly in all directions and is equally divided between the two polarizations. If the amplifier output is collected from a solid angle dO., as illustrated in Fig. 14.5-1, and from only one of the polarizations, it contains only a fraction dO. /47f of the spontaneously emitted power. Furthermore, if a filter is used to limit the collected photons to a narrow frequency band of width B centered about the amplified signal frequency v, the number of photons added by spontaneous emission from an incremental volume of unit area and length dz is sp(v) dz, where 1 dO. sp(v) == N 2 - g(v)B- t sp 87f ( 14.5-2) is the noise photon-flux density per unit length. Input photon flux Spontaneous photon flux Filter and polarizer Output photon flux d)__ --- '\J\.MMJ'+ e.:: :: - - --- --- T- Noise photon flux Figure 14.5-1 Spontaneous emission is a source of amplifier noise. It is broadband, radiated in all directions, and unpolarized. Optics can be used at the output of the amplifier to limit the spontaneous emission noise to a narrow optical band, solid angle do', and a single polarization. In determining the noise photon-flux density contributed by the amplifier, the photon-flux density per unit length should not be simply multiplied by the length of the amplifier. This is because the spontaneous-emission noise is itself amplified by the medium; spontaneous-emission noise generated near the input end of the amplifier provides a greater contribution than noise generated near the output end. One way to accommodate the spontaneous-emission noise is to replace the differential equation governing the growth of photon-flux density (14.1-3) by dcjJ dz == --y(v)cjJ + sp (v). (14.5-3) Equation (14.5-3) incorporates the photon-flux density arising from both the amplified signal and the amplified-spontaneous-emission noise. EXERCISE 14.5-1 Amplified Spontaneous Emission (ASE). (a) Use (14.5-3) to show that, in the absence of any input signal, spontaneous emission produces a photon-flux density at the output of an unsaturated amplifier [I'(v) I'o(v)] of length d that can be expressed as <p ( d) == <Psp { exp [1'0 (v) d] - I}, where <Psp == sp (v) /1'0 (v).
564 CHAPTER 14 LASER AMPLIFIERS (b) Since both sp(v) and T'o(v) are proportional to g(v), cfJs p is independent of g(v) so that the fre- quency dependence of cfJ( d) is governed by the factor {exp[T'o(v)d] -I}. If T'o(v) is Lorentzian with width v, i.e., T'o(v) = T'o(vo)(v/2)2 /[(v -vO)2 + (V/2)2], show that the width of the factor {exp[T'o(v)d] -I} is smaller than v, i.e., that the amplification of spontaneous emission is accompanied by spectral narrowing. In the course of amplification, the photon-number statistics (see Sec. 12.2C) of the incoming light are altered. A coherent signal presented to the input of the amplifier exhibits Poisson photon-number statistics, with a variance a equal to the mean sig- nal photon number n s. The ASE photons, on the other hand, exhibit Bose-Einstein statistics with aSE == n ASE + n SE and are therefore considerably noisier than Poisson statistics. The photon-number statistics of the light after amplification, comprising both signal and spontaneous-emission contributions, obey a probability law intermediate between the two. If the counting time is short and the emerging light is linearly polar- ized, these statistics can be well approximated by the Laguerre-polynomial photon- number distribution (see Probe 14.5-3), which has a variance given by a; == n s + ( n ASE + n SE) + 2 n S n ASE . ( 14.5 -4 ) These photon-number fluctuations contain contributions from the signal and the spon- taneous emission individually, as well as a cross-term contribution. READING LIST Books See also the reading list in Chapter 15. c. Headley and G. P. Agrawal, eds., Raman Amplification in Fiber Optical Communication Systems, Elsevier, 2005. M. N. Islam, ed., Raman Amplifiers for Telecommunications, Volume 1: Physical Principles, Springer- Verlag, 2004. M. N. Islam, ed., Raman Amplifiers for Telecommunications, Volume 2: Sub-Systems and Systems, Springer-Verlag, 2004. E. Desurvire, D. Bayart, B. Desthieux, and S. Bigo, Erbium-Doped Fiber Amplifiers: Device and System Developments, Wiley, 2002. M. J. F. Digonnet, ed., Rare-Earth-Doped Fiber Lasers and Amplifiers, Marcel Dekker, 2nd ed. 2001. P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology, Academic Press, 1999. S. Sudo, Y. Ohishi, K. Fujiura, T. Kanamori, M. Yamada, and M. Shimizu, Optical Fiber Amplifiers: Materials, Devices, and Application Technologies, Artech, 1997. E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications Wiley, 1994. S. Shimada and H. Ishio, eds., Optical Amplifiers and Their Applications, Wiley, 1994. A. Bjarklev, Optical Fiber Amplifiers: Design and System Applications, Artech, 1993. Articles C. Bibeau, M. A. Rhodes, and L. J. Atherton, Innovative Technology Enables a New Architecture for the World's Largest Laser, Photonics Spectra, vol. 40, no. 6, pp. 50-60, 2006. Issue on fiber amplifiers and lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 1, 2001. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. J. H. Campbell and T. I. Suratwala, Nd-Doped Phosphate Glasses for High-Energy/High-Peak-Power Lasers, Journal of Non-Crystalline Solids, vol. 263/264, pp. 318-341, 2000.
PROBLEMS 565 T. Li and M. C. Teich, Photon Point Process for Traveling-Wave Laser Amplifiers, IEEE Journal of Quantum Electronics, vol. 29, pp. 2568-2578, 1993. M. J. Digonnet, ed., Selected Papers on Rare Earth-Doped Fiber Laser Sources and Amplifiers, SPIE Optical Engineering Press (Milestone Series Volume 37), 1992. PROBLEMS 14.1-2 Amplifier Gain and Rod Length. A commercially available ruby laser amplifier using a 15-cm-Iong rod has a small-signal gain of 12. What is the small-signal gain of a 20-cm-Iong rod? Neglect gain saturation effects. 14.1-3 Laser Amplifier Gain and Population Difference. A 15-cm-Iong rod of Nd 3 +:glass used as a laser amplifier has a total small-signal gain of 10 at Ao == 1.06 J-Lm. Use the data in Table 14.3-1 to determine the population difference N (Nd 3 + ions per em 3 ) required to achieve this gain. 14.1-4 Amplification of a Broadband Signal. The transition between two energy levels exhibits a Lorentzian lineshape of central frequency Vo == 5 X 10 14 with a linewidth v == 1 THz. The population is inverted so that the maximum gain coefficient 1'(vo) == 0.1 em-I. The medium has an additional loss coefficient as == 0.05 em-I, which is independent of v. Estimate the loss or gain encountered by a light wave in I cm if it has a uniform power spectral density centered about Vo with a bandwidth 2v. 14.2-4 The Two-Level Pumping System. Write the rate equations for a two-level system, showing that a steady-state population inversion cannot be achieved by using direct optical pumping between levels 1 and 2. 14.2-5 Two Laser Lines. Consider an atomic system with four levels: 0 (ground state), 1, 2, and 3. Two pumps are applied: between the ground state and level 3 at a rate R 3 , and between the ground state and level 2 at a rate R 2 . Population inversion can occur between levels 3 and 1 and/or between levels 2 and 1 (as in a four-level laser). Assuming that decay from level 3 to level 2 is not possible, and that decay from levels 3 and 2 to the ground state are negligible, write the rate equations for levels 1, 2, and 3 in terms of the lifetimes 71, 731, and 721. Determine the steady-state populations N 1 , N 2 , and N 3 and examine the possibility of simultaneous population inversions between levels 3 and 1, and between levels 2 and 1. Show that the presence of radiation at the 2 1 transition reduces the population difference for the 3 1 transition. 14.4-3 Significance of the Saturation Photon-Flux Density. In the general two-level atomic system of Fig. 14.2-3, 72 represents the lifetime of level 2 in the absence of stimulated emission. In the presence of stimulated emission, the rate of decay from level 2 increases and the effective lifetime decreases. Find the photon-flux density q; at which the lifetime decreases to half its value. How is that photon-flux density related to the saturation photon- flux density q;s? 14.4-4 Saturation Optical Intensity. Determine the saturation photon-flux density q;s(vo) and the corresponding saturation optical intensity 1s(vo), for the homogeneously broadened ruby and Nd 3 + :YAG laser transitions listed in Table 14.3-1. 14.4-5 Growth of the Photon-Flux Density in a Saturated Amplifier. The growth of the photon- flux density q;(z) in a laser amplifier is described by (14.4-7). Plot q;(z)/q;s versus 1'oz for q;( 0) / q;s == 0.05. Identify the onset of saturation in this amplifier. 14.4-6 Resonant Absorption of a Medium in Thermal Equilibrium. A unity refractive-index medium of volume 1 em 3 contains N a == 10 23 atoms in thermal equilibrium. The ground state is energy level 1; level 2 has energy 2.48 eV above the ground state (Ao == 0.5 J-Lm). The transition between these two levels is characterized by a spontaneous lifetime t sp == 1 ms, and a Lorentzian lineshape of width v == 1 GHz. Consider two temperatures, Tl and T 2 , such that kT l == 0.026 eV and kT 2 == 0.26 eV. (a) Determine the populations N 1 and N 2 . (b) Determine the number of photons emitted spontaneously every second. (c) Determine the attenuation coefficient of this medium at Ao == 0.5 J-Lm assuming that the
566 CHAPTER 14 LASER AMPLIFIERS incident photon flux is small. (d) Sketch the dependence of the attenuation coefficient on frequency, indicating on the sketch the important parameters. (e) Find the value of photon-flux density at which the attenuation coefficient decreases by a factor of 2 (i.e., the saturation photon-flux density). (f) Sketch the dependence of the transmitted photon-flux density cP( d) on the incident photon-flux density cP(O) for v == va and v == va + v when cP(O)/cPs « 1. 14.4-7 Gain in a Saturated Amplifying Medium. Consider a homogeneously broadened laser amplifying medium of length d == 10 em and a saturation photon flux density cPs == 4 X 10 18 photons/em 2 -so It is known that a photon-flux density at the input cP(O) == 4 X 10 15 photons/em 2 -s produces a photon-flux density at the output cP( d) == 4 X 10 16 photons/em 2 -so (a) Determine the small-signal gain of the system Go. (b) Determine the small-signal gain coefficient 1'0. (c) What is the photon-flux density at which the gain coefficient decreases by a factor of 5? (d) Determine the gain coefficient when the input photon-flux density is cP(O) == 4 X 10 19 photons/em 2 -so Under these conditions, is the gain of the system greater than, less than, or the same as the small-signal gain determined in (a)? *14.5-2 Ratio of Signal Power to ASE Power. An unsaturated laser amplifier of length d and gain coefficient I'o(v) amplifies an input signal cPs(O) of frequency v and introduces amplified spontaneous emission (ASE) at a rate sp (per unit length). The amplified signal photon- flux density is cPs( d) and the ASE at the output is cPASE. Sketch the dependence of the ratio cPs( d) / cPASE on the product of the amplifier gain coefficient and length, I'o(v)d. *14.5-3 Photon-Number Distribution for Amplified Coherent Light. A linearly polarized su- perposition of interfering thermal and coherent light serves as a suitable model for the light emerging from a laser amplifier. This superposition is known to have random energy fluctuations w that obey the noncentral-chi-square probability distribution, ( ) 1 ( w+ws ) [ 2 ] p W == - exp - fa W ASE WA SE W ASE' provided that the measurement time is sufficiently short. tHere 10 denotes the modified Bessel function of order zero, W ASE is the mean energy of the ASE, and Ws is the (constant) energy of the amplified coherent signal. (a) Calculate the mean and variance of w. (b) Use (12.2-27) and (12.2-28) to determine the photon-number mean n and variance a, confirming the validity of (14.5-4). (c) Use (12.2-26) to show that the photon-number distribution is given by ( ) _ n SE ( _ ns ) L ( _ n s/ n ASE ) pn - - n 1 exp - n - , (1 + n ASE) + 1 + n ASE 1 + n ASE where Ln represents the Laguerre polynomial n ( n ) Xk Ln (-x) == L k k! ' k=O and n sand n ASE are the mean signal and amplified-spontaneous-emission photon numbers, respectively. (d) Plotp(n) for n s/ n == 0,0.5,0.8, and 1, when n == 5, demonstrating that it reduces to the Bose-Einstein distribution for n s /71 == 0 and to the Poisson distribution for n s/ n ==1. t See, for example, B. E. A. Saleh, Photoelectron Statistics, Springer-Verlag, 1978.
C HAP T E R 15 LASERS 15.1 THEORY OF lASER OSCillATION A. Optical Amplification and Feedback B. Conditions for Laser Oscillation 15.2 CHARACTERISTICS OF THE lASER OUTPUT A. Power B. Spectral Distribution C. Spatial Distribution and Polarization D. Mode Selection 15.3 COMMON lASERS A. Solid-State Lasers B. Gas Lasers C. Other Lasers D. Tabulation of Selected Characteristics 15.4 PULSED lASERS A. Methods of Pulsing Lasers * B. Analysis of Transient Effects *C. Q-Switching D. Mode Locking 569 575 590 605 to Arthur L. Schawlow Theodore H. Maiman (1921-1999) (born 1927) In 1958 Arthur Schawlow, together with Charles Townes, showed how to extend the principle of the maser to the optical region of the spectrum. Schawlow shared the 1981 Nobel Prize with Nicolaas Bloembergen. Theodore Maiman achieved the first successful operation of the ruby laser in 1960. 567
The laser is an optical oscillator. It comprises a resonant optical amplifier whose output is fed back to the input with matching phase (Fig. 15.0-1). The oscillation process can be initiated by the presence at the amplifier input of even a small amount of noise that contains frequency components lying within the bandwidth of the amplifier. This input is amplified and the output is fed back to the input, where it undergoes further amplification. The process continues indefinitely until a large output is produced. The increase of the signal is ultimately limited by saturation of the amplifier gain, and the system reaches a steady state in which an output signal is created at the frequency of the resonant amplifier. Feedback Amplifier WVWv 'W\f\M Ouput Power supply Figure 15.0-1 An oscillator is an amplifier with positive feedback. Two conditions must be satisfied for oscillation to occur: . The amplifier gain must be greater than the loss in the feedback system so that net gain is incurred in a round trip through the feedback loop. . The total phase shift in a single round trip must be a multiple of 27r so that the feedback input phase matches the phase of the original input. If these conditions are satisfied, the system becomes unstable and oscillation begins. As the power in the oscillator grows, the amplifier gain saturates and decreases below its initial value. A stable condition is reached when the reduced gain is equal to the loss (Fig. 15.0-2). The gain then just compensates the loss so that the cycle of amplification and feedback is repeated without change and steady-state oscillation prevails. Gain Loss o Steady-state power Power Figure 15.0-2 If the initial amplifier gain is greater than the loss, oscillation may begin. As the oscillator power increases the ampli- fier saturates, causing its gain to decrease. A steady-state condition is reached when the gain just equals the loss. Because the gain and phase shift are functions of frequency, the two oscillation con- ditions are satisfied only at one (or several) frequencies, which are the resonance frequencies of the oscillator. The useful output is extracted by coupling a portion of the power out of the oscillator. In summary, an oscillator comprises: . An amplifier with a gain-saturation mechanism . A feedback system . A frequency-selection mechanism . An output coupling scheme 568
15.1 THEORY OF lASER OSCillATION 569 The laser is an optical oscillator (Fig. 15.0-3) in which the amplifier is the pumped active medium considered in Secs. 14.1 and 14.2. Gain saturation is a basic property of laser amplifiers, as discussed in Sec. 14.4. Feedback is engendered by placing the active medium in an optical resonator, which reflects the light back and forth between its mirrors, as discussed in Chapter 10. Frequency selection is jointly achieved by the resonant amplifier and the resonator, which admits only certain modes. Output coupling is accomplished by making one of the resonator mirrors partially transmitting. Mirror / Active mediu m , I Partially trans!lli tting mIrror Laser output I - d ) I Figure 15.0-3 A laser consists of an optical amplifier (comprising an active medium) placed within an optical resonator. The output is extracted through a partially transmitting mirror. Lasers take an enormous variety of forms and are used in myriad scientific and technical applications including interferometry, spectroscopy, imaging, lithography, metrology, communications, lidar (light detection and ranging), atomic cooling, and materials processing. Needless to say, they are invaluable for fundamental studies in photonics, as well as in all branches of science, engineering, and medicine. The precursor to the laser was the maser, an acronym for microwave amplification by stimulated emission of radiation. The maser jlaser principle also holds promise for waves other than electromagnetic radiation. The saser, for example, is an acoustic version of the laser that emits a beam of phonons, offering sound amplification by stimulated emission of radiation. This Chapter This chapter provides an introduction to the operation of lasers. In Sec. 15.1 the be- havior of the laser amplifier and the laser resonator are summarized, and oscillation conditions are derived. The properties of the light emitted by lasers, such as power, spectral distribution, spatial distribution, and polarization, are considered in Sec. 15.2. Common lasers are discussed in Sec. 15.3, and Sec. 15.4 is devoted to the operation of pulsed lasers. 15.1 THEORY OF lASER OSCillATION We begin this section with a summary of the properties of the two basic components of the laser - the amplifier and the resonator. Although these topics have been discussed in detail in Chapters 14 and 10, respectively, they are reviewed here for convenience. A. Optical Amplification and Feedback Laser Amplification The laser amplifier is a narrowband coherent amplifier of light. Amplification is achieved by stimulated emission from an atomic or molecular system with a transition whose population is inverted (i.e., the upper energy level is more populated than
570 CHAPTER15 LASERS the lower). The amplifier bandwidth is determined by the linewidth of the atomic transition, or by an inhomogeneous broadening mechanism such as the Doppler effect in gas lasers. The laser amplifier is a distributed-gain device characterized by its gain coefficient (gain per unit length) 'Y(v), which governs the rate at which the photon-flux density cP (or the optical intensity I == hv cP) increases. When the photon-flux density cP is small, the gain coefficient is A 2 'Yo (v) == No a(v) == No g(v), 87Tt sp (15.1-1) Small-Signal Gain Coefficient where No = equilibrium population density difference (density of atoms in the upper en- ergy state minus that in the lower state); No increases with increasing pump- ing rate a(v) = (A2/87Tt sp )g(v) == transition cross section t sp = spontaneous lifetime 9 (v) = transition lineshape A = Ao/n == wavelength in the medium, where n == refractive index As the photon-flux density increases, the amplifier enters a region of nonlinear operation. It saturates and its gain decreases. The amplification process then depletes the initial population difference No, reducing it to N == No/[l + cP/cPs(v)] for a homogeneously broadened medium, where cPs(v) = [Ts a(v)]-l = saturation photon-flux density Ts = saturation time constant, which depends on the decay times of the energy levels involved; in an ideal four-level pumping scheme, Ts t sp , whereas in an ideal three-level pumping scheme, Ts == 2t sp The gain coefficient of the saturated amplifier is therefore reduced to 'Y(v) == Na(v), so that for homogeneous broadening (15.1-2) Satu rated Gain Coefficient The laser amplification process also introduces a phase shift. When the lineshape is Lorentzian with linewidth v, g( v) == (v /27T) / [(v - vO)2 + (v /2)2], the amplifier phase shift per unit length is 'Yo (v) I'(v) = 1 + c/;Ns(v) . (15.1-3) Phase-Shift Coefficient (Lorentzian Lineshape) This phase shift is in addition to that introduced by the medium hosting the laser atoms. The gain and phase-shift coefficients for an amplifier with Lorentzian lineshape function are illustrated in Fig. 15.1-1. v - Vo cp(v) == v 'Y(v).
15.1 THEORY OF lASER OSCillATION 571 Gain coefficient -y(v) v Phase-shift coefficient <p'(v) Va V Figure 15.1-1 Spectral dependence of the gain and phase-shift coefficients for an optical amplifier with Lorentzian lineshape function. Feedback and Loss: The Optical Resonator Optical feedback is achieved by placing the active medium in an optical resonator. A Fabry-Perot resonator, comprising two mirrors separated by a distance d, contains the medium (refractive index n) in which the active atoms of the amplifier reside. Travel through the medium introduces a phase shift per unit length equal to the wavenumber k == 27rV . C ( 15 .1-4 ) Phase-Shift Coefficient The resonator also contributes to losses in the system. Absorption and scattering of light in the medium introduces a distributed loss characterized by the attenuation coefficient as (loss per unit length). In traveling a round trip through a resonator of length d, the photon-flux density is reduced by the factor 9(19(2 exp( -2a s d), where 9( 1 and 9(2 are the reftectances of the two mirrors. The overall loss in one round trip can therefore be described by a total effective distributed loss coefficient a r , where exp(-2a r d) == 9(19(2 ex p(-2a s d), (15.1-5) so that a r == as + amI + a m 2 1 1 Cl:ml = 2d In 1 1 1 Cl: m 2 = 2d In 2 ' (15.1-6) Loss Coefficient where amI and a m 2 represent the contributions of mirrors 1 and 2, respectively. The contribution from both mirrors is 1 1 Cl: m = Cl:ml + Cl: m 2 = 2d In I2 . (15.1-7) Since a r represents the total loss of energy (or number of photons) per unit length, arC
572 CHAPTER15 LASERS represents the loss of photons per second. Thus, 1 Tp == - Qr C (15.1-8) represents the photon lifetime. The resonator sustains only frequencies that correspond to a round-trip phase shift that is a multiple of 21T. For a resonator devoid of active atoms (i.e., a "cold" resonator), the round-trip phase shift is simply k2d == 21TVd I C == q21T, corresponding to modes of frequencies V q == qvp, q== 1,2,..., (15.]-9) where Vp == c/2d is the resonator mode spacing and c == coin is the speed of light in the medium (Fig. 15.1-2). The (full width at half maximum) spectral width of these resonator modes is Vp 6v- 1" (15.1-10) where 1" is the finesse of the resonator (see Sec. 1 O.lA). When the resonator losses are small and the finesse is large, 1T 1" - d == 21TTp Vp. Qr (15.1-11) I ' c I V F = 2d v -1 q l/q l/q + 1 v Figure 15.1-2 Resonator modes are separated by the frequency l/F == c/2d and have linewidths bl/ == l/F / == 1/21rTp. B. Conditions for Laser Oscillation Two conditions must be satisfied for the laser to oscillate (lase). The gain condition determines the minimum population difference, and therefore the pumping threshold, required for lasing. The phase condition determines the frequency (or frequencies) at which oscillation takes place. Gain Condition: Laser Threshold The initiation of laser oscillation requires that the small-signal gain coefficient be greater than the loss coefficient, --Yo ( v) > Qr, (15.1-12) Threshold Gain Condition
15.1 THEORY OF lASER OSCillATION 573 or, equivalently, that the gain be greater than the loss. In accordance with (IS .1-1), the small-signal gain coefficient '"Yo (v) is proportional to the equilibrium population density difference No, which in turn is known from Chapter 14 to increase with the pumping rate R. Indeed, {IS. 1-1) may be used to translate (I 5.1-12) into a condition on the population difference, i.e., No == '"Yo(v)/a(v) > Qr/a(v). Thus, No > Nt , (15.1-13) where the quantity Qr Nt = a(v) (15.1-14) is called the threshold population difference. Nt, which is proportional to Qr, deter- mines the minimum pumping rate Rt for the initiation of laser oscillation. Using (15.1-8), Qr may alternatively be written in terms of the photon lifetime, Qr == 1/ CT p , whereupon (IS .1-14) takes the form 1 N t == . CT p a(v) {IS. 1-15) The threshold population density difference is therefore directly proportional to Qr and inversely proportional to Tp. Higher loss (shorter photon lifetime) requires more vigorous pumping to achieve lasing. Finally, use of the standard formula for the transition cross section, a( v) == (A 2 /87rt sp )g(v), leads to yet another expression for the threshold population dif- ference, (15.1-16) Threshold Population Difference from which it is clear that Nt is a function of the frequency v. The threshold is lowest, and therefore lasing is most readily achieved, at the frequency where the lineshape function is greatest, i.e., at its central frequency v == yo. For a Lorentzian lineshape function, g(vo) == 2/7rv, so that the minimum population difference for oscillation at the central frequency Vo turns out to be Nt = 87r t sp A 2 C Tp g(v) , Nt = 27r 27r /:lv t sp . A 2 C T. P (IS. I -17) It is directly proportional to the linewidth v. If, furthermore, the transition is limited by lifetime broadening with a decay time t sp , v assumes the value 1/27rt sp (see Sec. 13.3D), whereupon (IS. I -17) simplifies to 27r Nt == '2 /\ CT p 27rQr A 2 . (15.1-18) This formula shows that the minimum threshold population difference required to achieve oscillation is a simple function of the wavelength A and the photon lifetime Tp. It is clear that laser oscillation becomes more difficult to achieve as the wavelength decreases. As a numerical example, if Ao == 1 J-Lm, Tp == 1 ns, and the refractive index n == 1, we obtain Nt 2.1 X 10 7 cm- 3 .
574 CHAPTER15 LASERS EXERCISE 15.1-1 Threshold of a Ruby Laser. (a) At the line center of the Ao == 694.3-nm transition, the absorption coefficient of ruby in thermal equilibrium (i.e., without pumping) at T == 300 0 K is a(vo) -'(va) 0.2 em-I. If the concentration of Cr 3 + ions responsible for the transition is N a == 1.58 X 10 19 cm- 3 , determine the transition cross section ao == a(vo). (b) A ruby laser makes use of a 10-cm-Iong ruby rod (refractive index n == 1. 76) of cross-sectional area 1 cm 2 and operates on this transition at Ao == 694.3 nm. Both of its ends are polished and coated so that each has a reflectance of 80%. Assuming that there are no scattering or other extraneous losses, determine the resonator loss coefficient a r and the resonator photon lifetime Tp. (c) As the laser is pumped, ,(vo) increases from its initial thermal equilibrium value of -0.2 cnl- I and changes sign, thereby providing gain. Determine the threshold population difference Nt for laser oscillation. Phase Condition: Laser Frequencies The second condition of oscillation requires that the phase shift imparted to a light wave completing a round trip within the resonator must be a multiple of 21T, i.e., 2kd + 2cp(v)d == 21TQ, Q== 1,2,.... (15.1-19) If the contribution arising from the active laser atoms [2cp(v)d] is small, dividing (15.1- 19) by 2d gives the cold-resonator result obtained earlier, v == v q == q( c/2d). In the presence of the active medium, when 2cp(v) d contributes, the solution of (15.1-19) gives rise to a set of oscillation frequencies v that are slightly displaced from the cold-resonator frequencies v q . It turns out that the cold-resonator modal frequencies are all pulled slightly toward the central frequency of the atomic transition, as shown below. * Frequency Pulling U sing the relation k == 21TV / c, and the phase-shift coefficient for the Lorentzian lineshape function provided in (15.1-3), the phase-shift condition (15.1-19) provides c v - Vo v+- --y(v)==v q . 21T V (15.1-20) This equation can be solved for the oscillation frequency v == v corresponding to each cold-resonator mode v q . Because the equation is nonlinear, a graphical solution is useful. The left-hand side of (J 5.1-20) is designated 'ljJ(v) and plotted in Fig. 15.1-3 (it is the sum of a straight line representing v plus the Lorentzian phase-shift coefficient shown schematically in Fig. 15.1-1). The value of v == v that makes 'ljJ(v) == v q is graphically determined. It is apparent from the figure that the cold-resonator modes v q are always frequency-pulled toward the central frequency of the resonant medium Yo. An approximate analytical solution of (15.1-20) can also be obtained. We write (15.1-20) in the form c V-Yo v==v q -- --y(v). 21T V (15.1-21)
15.2 CHARACTERISTICS OF THE LASER OUTPUT 575 'ljJ( V) V q --------------------------------------- Central frequency of atomic transition / / / / V q _ I ------------------------/:- Oscillation frequency Cold-resonator / mode v q _] v_] V o v v q v Figure 15.1-3 The left-hand side of (15.1-20), 'ljJ(v), plotted as a function of v. The frequency v for which 'ljJ(v) == v q is the solution of (15.1-20). Each "cold" resonator frequency v q corresponds to a "hot" resonator frequency v, which is shifted in the direction of the atomic resonance central frequency Va. When v == v v q , the second term of (15.1-21) is small, whereupon v may be replaced with v q without much loss of accuracy. Thus, , C v q - va ) v q == v q - - '"'((v q , 21r V (15.1-22) which is an explicit expression for the oscillation frequency v as a function of the cold- resonator frequency v q . Furthermore, under steady-state conditions, the gain equals the loss so that '"'((v q ) == Qr 1rj'J'd == (21rjc)bv, where bv is the spectral width of the cold resonator modes. Substituting this relation into (15.1-22) leads to , ( ) bv v q v q - v q - va v . (15.1-23) Laser Frequencies The cold-resonator frequency v q is therefore pulled toward the atomic resonance fre- quency va by a fraction bv j v of its original distance from the central frequency (v q - va), as shown in Fig. 15.1-4. The sharper the resonator mode (the smaller the value of by), the less significant the pulling effect. In contrast, the narrower the atomic resonance linewidth (the smaller the value of v), the more effective the pulling. 15.2 CHARACTERISTICS OF THE LASER OUTPUT A. Power Internal Photon-Flux Density A laser pumped above the threshold (No> Nt) exhibits a small-signal gain coefficient '"'(o(v) that is greater than the loss coefficient Qr, as shown in (15.1-12). Laser oscil- lation may then begin, provided that the phase condition (15.1-19) is satisfied. As the photon- flux density q; inside the resonator increases (Fig. 15.2-1), the gain coefficient '"'((v) begins to decrease in accordance with (15.1-2) for homogeneously broadened
8V* AVO JCfdJl .,,____. V q _ I ._, ,,' V q -----,' 1111 V_I v 576 CHAPTER15 LASERS )I v Amplifier gain coefficient Cold-resonator modes )I v Laser oscillation modes )I v Figure 15.1-4 The laser oscillation frequencies fall near the cold-resonator modes; they are pulled slightly toward the atomic resonance central frequency va. The diagram is illustrative and is not to scale. media. As long as the gain coefficient remains larger than the loss coefficient, the photon flux continues to grow. Qr Loss coefficient /,(v) Gain coefficient o Photon-flux density Figure 15.2-1 Determination of the steady-state laser photon-flux density cjJ. At the time of laser turn-on, cjJ == o so that ')'(v) == ')'o(v). As the oscillation builds up in time, the in- crease in cjJ causes ')'( v) to decrease through gain saturation. When')' reaches Qr, the photon-flux density ceases its growth and steady-state conditions are achieved. The smaller the loss, the greater the val ue of cjJ. Finally, when the saturated gain coefficient becomes equal to the loss coefficient (or equivalently N == Nt), the photon flux ceases its growth and the oscillation reaches steady-state conditions. The result is gain clamping at the value of the loss. The steady- state laser internal photon-flux density is therefore determined by equating the large- signal (saturated) gain coefficient to the loss coefficient --yo(v)/[l + c/J/c/Js(v)] == Qr, which provides c/J == { <ps(V) ( -r::) - 1) , 0, --Yo ( v) > Qr --Yo ( v) < Qr. (15.2-1 ) Equation (15.2-1) represents the steady-state photon-flux density arising from laser action. This is the mean number of photons per second crossing a unit area in both directions, since photons traveling in both directions contribute to the saturation pro- cess. The photon-flux density for photons traveling in a single direction is therefore c/J/2. Spontaneous emission has been neglected in this simplified treatment. Of course,
15.2 CHARACTERISTICS OF THE LASER OUTPUT 577 (15.2-1) represents the mean photon-flux density; there are random fluctuations about this mean as discussed in Sec. 12.2. Since 'Yo (v) == No a(v) and Qr == Nt a(v), (15.2-1) may be written in the form cP == { q)S(V) ( -1) , 0, No > Nt No < Nt. (15.2-2) Steady-State Internal Photon-Flux Density Below threshold, the laser photon-flux density is zero; any increase in the pumping rate is manifested as an increase in the spontaneous-emission photon flux, but there is no sustained oscillation. Above threshold, the steady-state internal laser photon-flux density is directly proportional to the initial population difference No, and therefore increases with the pumping rate R [see (14.2-13) and (14.2-27)]. If No is twice the threshold value Nt, the photon-flux density is precisely equal to the saturation value cPs(v), which is the photon-flux density at which the gain coefficient decreases to half its maximum value. Both the population difference N and the photon-flux density cP are shown as functions No in Fig. 15.2-2. N cp C Q) o u . '3 N -- - -- - - - - -- - - ...... '""' t I ::J , : 0-.;::: I 0.- : "Cj : I I I Nt No . Pumping rate >. ...... c.u; cPs -------------------------- I ..r:::><: : p..::J : p;: : I I o : o Nt 2N t No .. Pumping rate Figure 15.2-2 Steady-state values of the population difference N, and the laser internal photon- flux density cP, as functions of No (the population difference in the absence of radiation; No increases with the pumping rate R). Laser oscillation occurs when No exceeds Nt; the steady-state value of N then saturates, clamping at the value Nt Uust as T'o(v) is clamped at ar]. Above threshold, cP is proportional to No - Nt. Output Photon-Flux Density Only a portion of the steady-state internal photon-flux density determined by (15.2-2) leaves the resonator in the form of useful light. The output photon-flux density cPo is that part of the internal photon-flux density that propagates toward mirror 1 (cPj2) and is transmitted by it. If the transmittance of mirror 1 is 'J, the output photon-flux density IS q)o = 'J' q) . 2 (15.2-3) The corresponding optical intensity of the laser output 10 is I _ hv'JcP o - 2 ' ( 15 .2 -4 ) and the laser output power is Po == loA, where A is the cross-sectional area of the laser beam. These equations, together with (15.2-2), permit the output power of the laser to be explicitly calculated in terms of cPs(v), No, Nt, 'J, and A.
578 CHAPTER15 LASERS Optimization of the Output Photon-Flux Density The useful photon-flux density at the laser output diminishes the internal photon-flux density and therefore contributes to the losses of the laser oscillator. Any attempt to in- crease the fraction of photons allowed to escape from the resonator (in the expectation of increasing the useful light output) results in increased losses so that the steady-state photon-flux density inside the resonator decreases. The net result may therefore be a decrease, rather than an increase, in the useful light output. We proceed to show that there is an optical transmittance 'I (0 < 'I < 1) that maximizes the laser output intensity. The output photon-flux density cPo == T cP/2 is a product of the mirror's transmittance 'I and the internal photon-flux density cP/2. As 'I is increased, cP decreases as a result of the greater losses. At one extreme, when 'I == 0, the oscillator has the least loss (cP is maximum), but there is no laser output whatever (cPo == 0). At the other extreme, when the mirror is removed so that T == 1, the increased losses make Qr > 'Yo (v) (Nt> No), thereby preventing laser oscillation. In this case cP == 0, so that again cPo == O. The optimal value of T lies somewhere between these two extremes. To determine it, we must obtain an explicit relation between cPo and 'I. We assume that mirror 1, with a reflectance 9(1 and a transmittance 'I == 1 - 9(1, transmits the useful light. The loss coefficient Qr is written as a function of 'I by substituting in (15.1-6) the loss coefficient due to mirror 1, 1 1 1 amI = 2d In 9(1 = - 2d In(l - 'J), (15.2-5) to obtain 1 Qr == Qs + Qm2 - 2d In(l - 'I), (15.2-6) where the loss coefficient due to mirror 2 is 1 1 a m 2 = 2d In 9(2 . (15.2-7) We now use (15.2-1), (15.2-3), and (15.2-6) to obtain an equation for the transmitted photon-flux density cPo as a function of the mirror transmittance cPo = !cPs'J [ L _ IOl _ 'J) - 1] , go == 2'Yo(v)d, L == 2(Qs + Qm2)d, (15.2-8) which is plotted in Fig. 15.2-3. Note that the transmitted photon-flux density is directly related to the small-signal gain coefficient. The optical transmittance Top is found by setting the derivative of cPo with respect to 'I equal to zero. When 'I « 1 we Cdn make use of the approximation In( 1 - 'I) -'I to obtain 'loP VgO L - L. ( 15.2-9) Internal Photon-Number Density The steady-state number of photons per unit volume inside the resonator n is related to the steady-state internal photon-flux density cP (for photons traveling in both directions)
15.2 CHARACTERISTICS OF THE LASER OUTPUT 579 0.2 I--e: --0-- (]):3 N .-....... -:3 E 0.1 $.-.(]) Ocr:, Z o o 0.1 0.2 0.3 Mirror transmittance T Figure 15.2-3 Dependence of the trans- mitted steady-state photon-flux density cPo on the mirror transmittance T. For the purposes of this illustration, the gain factor go = 2')'0 d has been chosen to be 0.5 and the loss factor L = 2( as + a m 2) d is 0.02 (2%). The optical transmittance Top turns out to be 0.08. by the simple relation cP n== -. c (15.2-10) This is readily visualized by considering a cylinder of area A, length c, and volume cA (c is the velocity of light in the medium), whose axis lies parallel to the axis of the resonator. For a resonator containing n photons per unit volume, the cylinder contains cAn photons. These photons travel in both directions, parallel to the axis of the resonator, half of them crossing the base of the cylinder in each second. Since the base of the cylinder also receives an equal number of photons from the other side, however, the photon-flux density (photons per second per unit area in both directions) is cP == 2(cAn)/ A == cn, from which (15.2-10) follows. The photon-number density corresponding to the steady-state internal photon-flux density in (15.2-2) is n = ns ( : - 1) , No > Nt , (15.2-11) Steady-State Photon-Number Density where ns == cPs (v) / c is the photon-number density saturation value. Using the relations cPs(v) == [Ts a(v )]-1 , Qr == --y(v), Qr == 1/ CT p , and --y(v) == N a(v) == Nt a(v), (15.2- 11) may be written in the form ( Tp n== No-N t )-, Ts No > Nt . (15.2-12) Steady-State Photon-Number Density This relation admits a simple and direction interpretation: (No - Nt) is the population difference (per unit volume) in excess of threshold, and (No - Nt)/Ts represents the rate at which photons are generated which, by virtue of steady-state operation, is equal to the rate at which photons are lost, n/Tp. The fraction Tp/Ts is the ratio of the rate at which photons are emitted to the rate at which they are lost. Under ideal pumping conditions in a four-level laser system, (14.2-13) and (14.2- 14) provide that Ts t sp and No Rt sp , where R is the rate (s-l-cm- 3 ) at which
580 CHAPTER15 LASERS atoms are pumped. Equation (15.2-12) can thus be written as n - == R - Rt , Tp R > Rt , (15.2-13) where Rt == Nt/t sp is the threshold value of the pumping rate. Under steady-state conditions, therefore, the overall photon-density loss rate n/ Tp is precisely equal to the excess pumping rate R - Rt. Output Photon Flux and Efficiency If transmission through the laser output mirror is the only source of resonator loss (which is accounted for in T p ), and V is the volume of the active medium, (15.2-13) provides that the total output photon flux 0 (photons per second) is o == (R - Rt)V, R > Rt. ( 15.2-14) If there are loss mechanisms other than through the output laser mirror, the output photon flux can be written as o == Ile(R - Rt)V, (15.2-15) Laser Output Photon Flux where the extraction efficiency Ile is the ratio of the loss arising from the extracted useful light to all of the total losses in the resonator Qr. If the useful light exits only through mirror 1, (15.1-8) and (15.2-5) for Qr and QmI may be used to write Ile as QmI C 1 Ile = aT = 2d Tp In 9(l . If, furthermore, 'J == 1 - I « 1, (15.2-16) provides (15.2-16) Tp Ile T F 'J", (15.2-17) Extraction Efficiency where we have defined 1/ T F == c/2d, indicating that the extraction efficiency Ile can be understood in terms of the ratio of the photon lifetime to its round-trip travel time, multiplied by the mirror transmittance. The output laser power is then Po == hv o == Ilehv(R - Rt)V. (15.2-18) With the help of a few algebraic manipulations it can be confirmed that this expression accords with that obtained from (15.2-4). Losses result from other sources as well, such as inefficiency in the pumping pro- cess. Overhead functions, such as cooling and monitoring, also consume power. The power-conversion efficiency Ilc (also called the overall efficiency or wall-plug ef- ficiency) is defined at the ratio of the output optical power Po to the supplied pump power P p , Po Ilc == P . p (15.2-19) Power-Conversion Efficiency
15.2 CHARACTERISTICS OF THE LASER OUTPUT 581 Representative values of Ilc for various types of lasers are provided in Table 15.3-1. Because the laser output power increases linearly with pump power above threshold, in accordance with (15.2-18), the differential power-conversion efficiency (also called the slope efficiency) is another oft-used measure of performance: dPo Its == dP . p (15.2-20) Slope Efficiency The slope efficiency Ils is generally larger than the power-conversion efficiency Ilc- B. Spectral Distribution The spectral distribution of the generated laser light is determined both by the atomic lineshape of the active medium (including whether it is homogeneous or inhomoge- neously broadened) and by the resonator modes. This is illustrated in the two condi- tions for laser oscillations: 1. The gain condition requiring that the initial gain coefficient of the amplifier be greater than the loss coefficient ['"Yo (v) > Qr] is satisfied for all oscillation fre- quencies lying within a continuous spectral band of width B centered about the atomic resonance frequency Yo, as illustrated in Fig. 15.2-4(a). The bandwidth B increases with the atomic linewidth v and the ratio '"Yo (vo) / Qr; the precise relation depends on the shape of the function '"Yo (v)_ 2. The phase condition requires that the oscillation frequency be one of the res- onator modal frequencies v q (assuming, for simplicity, that mode pulling is neg- ligible). The FWHM linewidth of each mode is 6v VF / [Fig. 15.2-4(b)]. Gain !'o(v) tJ.v (a) Loss a --- ----- r Vo v I v F Resonator modes (b) v 1 UW Allowed .. modes vI v 2 ... v M V Figure 15.2-4 (a) Laser oscillation can occur only at frequencies for which the gain coefficient is greater than the loss coefficient (filled-in region). (b) Oscillation can occur only within bv of the resonator modal frequencies (which are represented as lines for simplicity of illustration)
582 CHAPTER15 LASERS It follows that only a finite number of oscillation frequencies (VI, V2, . . . , VAl) are possible. The number of possible laser oscillation modes is therefore B M- , VF (15.2-21) Number of Possible Laser Modes where VF == c/2d is the approximate spacing between adjacent modes However, of these it! possible modes, the number of modes that actually carry optical power depends on the nature of the atomic line broadening mechanism. It will be shown below that for an inhomogeneously broadened medium all itI modes oscillate (albeit at different powers), whereas for a homogeneously broadened medium these modes engage in some degree of competition, making it more difficult for as many modes to oscillate simultaneously. The approximate FWHM linewidth of each laser mode might be expected to be flv, but it turns out to be far smaller than this. It is limited by the so-called Schawlow- Townes linewidth, which decreases inversely as the optical power. Almost all lasers have linewidths far greater than the Schawlow- Townes limit as a result of extraneous effects such as acoustic and thermal fluctuations of the resonator mirrors, but the limit can be approached in carefully controlled experiments. EXERCISE 15.2-1 Number of Modes in a Gas Laser. A Doppler-broadened gas laser has a gain coefficient with a Gaussian spectral profile (see Sec. 13.3D and Exercise 13.3-2) given by T'o(v) == T'o(vo) exp[-(v- vO)2 /2a], where VD == (8 In 2)1/2aD is the FWHM linewidth. (a) Derive an expression for the allowed oscillation band B as a function of VD and the ratio T'o(vo)/a n where a r is the resonator loss coefficient. (b) A He-Ne laser has a Doppler linewidth VD == 1.5 GHz and a midband gain coefficient T'o(vo) == 2 x 10- 3 em-I. The length of the laser resonator is d == 100 cm, and the reflectances of the mirrors are 100% and 97% (all other resonator losses are negligible). Assuming that the refractive index n == 1, determine the number of laser modes AI. Homogeneously Broadened Medium Immediately after being turned on, all laser modes for which the initial gain is greater than the loss begin to grow [Fig. 15.2-5(a). Photon-flux densities cPl, cP2, . . . , cPlvl are created in the M modes. Modes whose frequencies lie closest to the transition central frequency Vo grow most quickly and acquire the highest photon-flux densities. These photons interact with the medium and reduce the gain by depleting the population difference. The saturated gain is ( ) _ 'Yo (v) 'Y V - M ' 1 + E j =1 cPj/cPs(Vj) (15.2-22) where cPs(Vj) is the saturation photon-flux density associated with mode j. The validity of (15.2-22) may be verified by carrying out an analysis similar to that which led to (14.4-3). The saturated gain is shown in Fig. 15.2-5(b). Because the gain coefficient is reduced uniformly, for modes sufficiently distant from the line center the loss becomes greater than the gain; these modes lose power
15.2 CHARACTERISTICS OF THE LASER OUTPUT 583 Vo V o " ro(V)/ \ I \ I \ I \ -- _. - . I VO : v " ro(V)/ \ I \ I \ I \ ----/. \---- / r(V) / '" --'" --- . I I I I I V 1 V2". V M (a) . . I II . . (b) (c) Figure 15.2-5 Growth of oscillation in an ideal homogeneously broadened medium. (a) Immedi- ately following laser turn-on, all modal frequencies VI, V2, . . . , VAl, for whih the gain coefficient exceeds the loss coefficient, begin to grow, with the central modes growing at the highest rate. (b) After a short time the gain saturates so that the central modes continue to grow while the peripheral modes, for which the loss has become greater than the gain, are attenuated and eventually vanish. (c) In the absence of spatial hole burning, only a single mode survives. while the more central modes continue to grow, albeit at a slower rate. Ultimately, only a single surviving mode (or two modes in the symmetrical case) maintains a gain equal to the loss, with the loss exceeding the gain for all other modes. Under ideal steady- state conditions, the power in this preferred mode remains stable, while laser oscillation at all other modes vanishes [Fig. 15.2-5(c)]. The surviving mode has the frequency lying closest to Vo; values of the gain for its competitors lie below the loss line. Given the frequency of the surviving mode, its photon-flux density may be determined by means of (15.2-2). In practice, however, homogeneously broadened lasers do indeed oscillate on mul- tiple modes because the different modes occupy different spatial portions of the active medium. When oscillation on the most central mode in Fig. 15.2-5 is established, the gain coefficient can stilJ exceed the loss coefficient at those locations where the standing-wave electric field of the most-central mode vanishes. This phenomenon is called spatial hole burning. It allows another mode, whose peak fields are located near the energy nulls of the central mode, the opportunity to lase as well. Inhomogeneously Broadened Medium In an inhomogeneously broadened medium, the gain ')Io (v) represents the composite envelope of gains of different species of atoms (see Sec. 13.3D), as shown in Fig. 15.2- 6. -,.... ""- Va v Figure 15.2-6 The lineshape of an inhomogeneously broadened medium is a composite of numer- ous constituent atomic lineshapes, associated with different properties or different environments The situation immediately after laser turn-on is the same as in the homogeneously broadened medium. Modes for which the gain is larger than the loss begin to grow
584 CHAPTER15 LASERS and the gain decreases. If the spacing between the modes is larger than the width v of the constituent atomic lineshape functions, different modes interact with different atoms. Atoms whose lineshapes fail to coincide with any of the modes are ignorant of the presence of photons in the resonator. Their population difference is therefore not affected and the gain they provide remains the small-signal (unsaturated) gain. Atoms whose frequencies coincide with modes deplete their inverted population and their gain saturates, creating "holes" in the gain spectral profile [Fig. 15.2-7(a)]. This process is known as spectral hole burning. The width of a spectral hole increases with the photon-flux density in accordance with the square-root law vs == v(l + cP/ cPs)1/2 obtained in (14.4-16). ;Y{V) (b) Figure 15.2-7 (a) Laser oscillation occurs in an inhomogeneously broadened medium by each mode independently burning a hole in the overall spectral gain profile. The gain provided by the medium to one mode does not influence the gain it provides to other modes. The central modes gamer contributions from more atoms, and therefore carry more photons than do the peripheral modes. (b) Spectrum of a typical inhomogeneously broadened multimode gas laser. A V q _ 1 V q V q + 1 V (a) .. v 2Cd v This process of saturation by hole burning progresses independently for the differ- ent modes until the gain is equal to the loss for each mode in steady state. Modes do not compete because they draw power from different, rather than shared, atoms. Many modes oscillate independently, with the central modes burning deeper holes and growing larger, as illustrated in Fig. 15.2-7(a). The spectrum of a typical multimode inhomogeneously broadened gas laser is shown in Fig. 15 .2-7 (b). The number of modes is typically larger than that in homogeneously broadened media since spatial hole burning generally sustains fewer modes than spectral hole burning. * Spectral Hole Burning in a Doppler-Broadened Medium The lineshape of a gas at temperature T arises from the collection of Doppler- shifted emissions from the individual atoms, which move at different velocities (see Sec. 13.3D and Exercise 13.3-2). A stationary atom interacts with radiation of frequency va. An atom moving with velocity v toward the direction of propagation of the radiation interacts with radiation of frequency va (1 + v / c), whereas an atom moving away from the direction of propagation of the radiation interacts with radiation of frequency Yo (1 - vi c). Because a radiation mode of frequency Y q travels in both directions as it bounces back and forth between the mirrors of the resonator, it interacts with atoms of two velocity classes: those traveling with velocity + v and those traveling with velocity - v, such that v q - va == ::l:vo v / c. It follows that the mode v q saturates
15.2 CHARACTERISTICS OF THE LASER OUTPUT 585 the populations of atoms on both sides of the central frequency and bums two holes in the gain profile, as shown in Fig. 15.2-8. If v q == vo, of course, only a single hole is burned in the center of the profile. 1o( v) --.....,/ , , V o Jl v Figure 15.2-8 Hole burning in a Doppler-broadened medium. A probe wave at frequency v q saturates those atomic populations with velocities V == :t:.c(vq/vo - 1) on both sides of the central frequency, burning two holes in the gain profile. V q v The steady-state power of a mode increases with the depth of the hole(s) in the gain profile. As the frequency v q moves toward Vo from either side, the depth of the holes increases, as does the power in the mode. As the modal frequency v q begins to approach vo, however, the mode begins to interact with only a single group of atoms instead of two, so that the two holes collapse into one. This decrease in the number of available active atoms when v q == Vo causes the power of the mode to decrease slightly. Thus, the power in a mode, plotted as a function of its frequency v q , takes the form of a bell-shaped curve with a central depression, known as the Lamb dip, at its center (Fig. ] 5.2-9). Gain 'Yo (v) Loss a --- r Resonator . modes c 2d ! '1 I V : q I I i V o v v Power of mode q V o v q Figure 15.2-9 Power in a single laser mode of frequency v q in a Doppler-broadened medium whose gain coefficient is centered about va. Rather than providing maximum power at v q == va, it exhibits the Lamb dip.
586 CHAPTER15 LASERS c. Spatial Distribution and Polarization Spatial Distribution The spatial distribution of the emitted laser depends on the geometry of the resonator and on the shape of the active medium. In the laser theory developed to this point we have ignored transverse spatial effects by assuming that the resonator is constructed of two parallel planar mirrors of infinite extent and that the space between them is filled with the active medium. In this idealized geometry the laser output is a plane wave propagating along the axis of the resonator. But as is evident from Chapter 10, this planar-mirror resonator is highly sensitive to misalignment. Laser resonators usually have spherical mirrors. As indicated in Sec. 10.2, the spherical-mirror resonator supports a Gaussian beam (which was studied in detail in Chapter 3). A laser using a spherical-mirror resonator may therefore give rise to an output that takes the form of a Gaussian beam. It was also shown (in Sec. 10.2D) that the spherical-mirror resonator supports a hierarchy of transverse electric and magnetic modes denoted TEl\l l ,m,q. Each pair of indexes (l, m) defines a transverse mode with an associated spatial distribution. The (0, 0) transverse mode is the Gaussian beam (Fig. 15.2-10). Modes of a higher land m form Hermite-Gaussian beams (see Sec. 3.3 and Fig. 3.3-2). For a given (l, m), the index q defines a number of longitudinal (axial) modes of the same spatial distribution but of different frequencies v q (which are always separated by the longitudinal-mode spacing Vp == c/2d, regardless of land m). The resonance frequencies of two sets of longitudinal modes belonging to two different transverse modes are, in general, displaced with respect to each other by some fraction of the mode spacing Vp [see (10.2-34 ).] x,y Sph.erical mIrror Spherical mirror Laser intensity Figure 15.2-10 The laser output for the (0,0) transverse mode of a spherical-mirror resonator takes the form of a Gaussian beam. Because of their different spatial distributions, different transverse modes undergo different gains and losses. The (0,0) Gaussian mode, for example, is the most confined about the optical axis and therefore suffers the least diffraction loss at the boundaries of the mirrors. The (1,1) mode vanishes at points on the optical axis (see Fig. 3.3- 2); thus if the laser mirror were blocked by a small central obstruction, the (1,1) mode would be completely unaffected, whereas the (0, 0) mode would suffer significant loss. Higher-order modes occupy a larger volume and therefore can have larger gain. This disparity between the losses and/or gains of different transverse modes in different geometries determines their competitive edge in contributing to the laser oscillation, as Fig. 15.2-1] illustrates. In a homogeneously broadened laser, the strongest mode tends to suppress the gain for the other modes, but spatial hole burning can permit a few longitudinal modes to oscillate. Transverse modes can have substantially different spatial distributions so that they can readily oscillate simultaneously. A mode whose energy is concentrated in a given transverse spatial region saturates the atomic gain in that region, thereby
15.2 CHARACTERISTICS OF THE LASER OUTPUT 587 Laser output TEMO,O (O,O) modes )0 v TEM 1,1 v Figure 15.2-11 The gains and losses for two transverse modes, say (0,0) and (1, 1), usually differ because of their different spatial distributions. A mode can contribute to the output if it lies in the spectral band (of width B) within which the gain coefficient exceeds the loss coefficient. The allowed longitudinal modes associated with each transverse mode are shown. burning a spatial hole there. Two transverse modes that do not spatially overlap can coexist without competition because they draw their energy from different atoms. Partial spatial overlap between different transverse modes and atomic migrations (as in gases) allow for mode competition. Lasers are often designed to operate on a single transverse mode; this is usually the (0,0) Gaussian mode because it has the smallest beam diameter and can be focused to the smallest spot size (see Chapter 3). Oscillation on higher-order modes can be desirable, on the other hand, for purposes such as generating large optical power. Polarization Each (I, m, q) mode has two degrees of freedom, corresponding to two independent orthogonal polarizations. These two polarizations are regarded as two independent modes. Because of the circular symmetry of the spherical-mirror resonator, the two polarization modes of the same I and m have the same spatial distributions. If the resonator and the active medium provide equal gains and losses for both polarizations, the laser will oscillate on the two modes simultaneously, independently, and with the same intensity. The laser output is then unpolarized (see Sec. 11.4). Unstable Resonators Although our discussion has focused on laser configurations that make use of stable resonators (see Fig. 10.2-3), the use of unstable resonators offers a number of advan- tages in the operation of high-power lasers. These include (1) a greater portion of the gain medium contributing to the laser output power as a result of the availability of a larger modal volume; (2) higher output powers attained from operation on the lowest- order transverse mode, rather than on higher-order transverse modes as in the case of stable resonators; and (3) high output power with minimal optical damage to the resonator mirrors, as a result of the use of purely reflective optics that permits the laser light to spill out around the mirror edges (this configuration also permits the optics to be water-cooled and thereby to tolerate high optical powers without damage). D. Mode Selection A multimode laser may be operated on a single mode by making use of an element inside the resonator to provide loss sufficient to prevent oscillation of the undesired modes.
588 CHAPTER15 LASERS Selection of the Laser Line An active medium with multiple transitions (atomic lines) whose populations are in- verted by the pumping mechanism will produce a multiline laser output. A particular line may be selected for oscillation by placing a prism inside the resonator, as shown schematically in Fig. 15.2-12. The prism is adjusted such that only light of the desired wavelength strikes the highly reflecting mirror at normal incidence and can therefore be reflected back to complete the feedback process. By rotating the prism, one wavelength at a time may be selected. Argon-ion lasers, as an example, often contain a rotatable prism in the resonator to allow the choice of one of six common laser lines, stretching from 488 nm in the blue to 514.5 nm in the blue-green. A prism can only be used to select a line if the other lines are well separated from it. It cannot be used, for example, to select one longitudinal mode; adjacent modes are so closely spaced that the dispersive refraction provided by the prism cannot distinguish them. High-reflectance P . mirror nsm Output mIrror /': / /, Unwanted line Figure 15.2-12 A particular atomic line may be selected by the use of a prism placed inside the resonator. A transverse mode may be selected by means of a spatial aperture of carefully chosen shape and size. Active medium I I Aperture Laser output Selection of a Transverse Mode Different transverse modes have different spatial distributions, so that an aperture of controllable shape placed inside the resonator may be used to selectively attenuate undesired modes (Fig. 15.2-12). The laser mirrors may also be designed to favor a particular transverse mode. Selection of a Polarization A polarizer may be used to convert unpolarized light into polarized light. It is ad- vantageous, however, to place the polarizer inside the resonator rather than outside it. An external polarizer wastes half the output power generated by the laser. The light transmitted by the external polarizer can also suffer from noise arising from the fluctuation of power between the two polarization modes (mode hopping). An internal polarizer creates high losses for one polarization so that oscillation in its corresponding mode never begins. The atomic gain is therefore provided totally to the surviving polarization. An internal polarizer is usually implemented with the help of Brewster windows (see Sec. 6.2 and Exercise 6.2-1), as illustrated in Fig. ] 5.2-13. SecUonofaLongffudmalMode The selection of a single longitudinal mode is also possible. The number of longitudinal modes in an inhomogeneously broadened laser (e.g., a Doppler broadened gas laser) is the number of resonator modes contained in a frequency band B within which the atomic gain is greater than the loss (see Fig. 15.2-4). There are two alternatives for operating a laser in a single longitudinal mode: 1. Increase the loss sufficiently so that only the mode with the largest gain oscillates. This means, however, that the surviving mode would itself be weak.
15.2 CHARACTERISTICS OF THE LASER OUTPUT 589 Brewster window Active medium Brewster window Polarized laser output High-rflectance mIrror Output mIrror Figure 15.2-13 The use of Brewster windows in a gas laser provides a linearly polarized laser beam. Light polarized in the plane of incidence (the TM wave) is transmitted without reflection loss through a window placed at the Brewster angle. The orthogonally polarized (TE) mode suffers reflection loss and therefore does not oscillate. 2. Increase the longitudinal-mode spacing, VF == c/2d, by reducing the resonator length. This means, however, that the length of the active medium is reduced, so that the volume of the active medium, and therefore the available laser power, is diminished. In some cases, this approach is impractical. In an argon-ion laser, for example, VD == 3.5 GHz. Thus, if B == VD and n == 1, M == vD/(c/2d), so that the resonator must be shorter than about 4.3 cm to obtain single longitudinal- mode operation. A number of techniques that make use of intracavity frequency-selective elements have been devised for altering the frequency spacing of the resonator modes: High- IT reflectance I mirror I( Etalon o -..1 '"'1-- d 1 I I > u t .1 mirror / Active medium d Resonator loss Resonator modes lIo II c 2d A ) 1 1 A II Etalon modes A I e c 2dl ) II Laser output ) II Figure 15.2-14 Longitudinal mode selection by the use of an intracavity etalon Oscillation occurs at frequencies where a mode of the resonator coincides with an etalon mode; both must, of course, lie within the spectral window where the gain of the medium exceeds the loss.
590 CHAPTER15 LASERS . An intracavity tilted etalon (Fabry-Perot resonator) whose mirror separation d 1 is much shorter (thinner) than the laser resonator may be used for mode selection (Fig. 15.2-14). Modes of the etalon have a large spacing c/2d 1 > B, so that only one etalon mode can fit within the laser amplifier bandwidth. The etalon is designed so that one of its modes coincides with the resonator longitudinal mode exhibiting the highest gain (or any other desired mode). The etalon may be fine- tuned by means of a slight rotation, by changing its temperature, or by slightly changing its width d 1 with the help of a piezoelectric (or other) transducer. The etalon is slightly tilted with respect to the resonator axis to prevent reflections from its surfaces from reaching the resonator mirrors and thereby creating unde- sired additional resonances. The etalon is usually temperature stabilized to assure frequency stability. . Multiple-mirror resonators can also be used for mode selection. Several configu- rations are illustrated in Fig. 15.2-15. Mode selection may be achieved by means of two coupled resonators of different lengths [Fig. 15.2-15(a)]. The resonator in Fig. 15.2-15(b) consists of two coupled cavities, each with its own gain - in essence, two coupled lasers. This is the configuration used for the C 3 (cleaved- coupled-cavity) semiconductor laser discussed in Chapter 17. Another technique makes use of a resonator coupled with an interferometer [Fig. 15.2-15(c)]. The theory of coupled resonators and coupled resonator/interferometers is not ad- dressed here. (a) (b) l-----i Figure 15.2-15 Longitudinal mode selec- tion by use of (a) two coupled resonators (one passive and one active); (b) two coupled active resonators; (c) a coupled resonator- interferometer. (c) 1 15.3 COMMON LASERS Laser amplification and oscillation is ubiquitous; it occurs in an enormous variety of media, including solids (crystals, glasses, fibers, powders), gases (atomic, ionic, molecular, excimeric), and liquids (organic-dye solutions). Plasmas support laser ac- tion in the extreme-ultraviolet and X -ray regions. The energy levels of an electron in a magnetic field act as an active medium for the free-electron laser. We present several examples of lasers in these various categories. Lasers have sizes that range from nm, to the size of a football field, to the ex- tent of an astronomical body. The maser principle extends over an enormous span of electromagnetic frequencies, more than 18 orders of magnitude from I GHz in the microwave to 100 PHz in the X-ray. Laser spectral linewidths reach more than 12 orders of magnitude, from Hz to THz. And lasers generate pulses with durations from fs to CW - and with peak output powers that extend some 27 orders of magnitude, from p W to PW.
15.3 COMMON LASERS 591 A. Solid-State Lasers The energy-level diagrams of several solid-state laser materials (ruby, alexandrite, Nd 3 +:YAG, and Nd 3 +:glass) were displayed in Sec. 13.1 C (see Figs. 13.1-8 and 13.1-9) and the operation of several solid-state laser amplifiers (ruby, N d 3 + :glass, and Er 3 + :silica fiber) was discussed in Sec. 14.3 (see Figs. 14.3-1, 14.3-3, and 14.3-6, respectively). The characteristics of the principal laser transitions in these, and other, active media have been summarized in Table 14.3-1. When placed in an optical resonator that provides feedback, all of these solid-state materials behave as laser oscillators. There are many varieties of solid-state lasers since dozens of transparent dielectric media are commonly used as host materials for many different kinds of active dopant ions. Crystalline hosts include oxides, garnets, fluorides, and vanadates, the most common examples of which are Al 2 0 3 (sapphire), Y 3Al5012 (yttrium aluminum garnet or YAG), Gd3Ga5012 (gadolinium gallium garnet or GGG), YLiF 4 (yttrium lithium fluoride or YLF), and YVO.t (yttrium vanadate, also known as yttrium orthovanadate). Many different glass hosts are also in wide use; these include silicate-based compositions (such as noncrystalline Si0 2 , which is fused silica) and phosphate-based compositions, which are favored for high-power and pulsed-laser applications (see Sec. 14.3B for an example). Comparing the characteristics of lasers that use crystalline and glass hosts reveals that the former typically offer narrower linewidths (and correspondingly lower laser thresholds), lower doping levels, increased resistance to solarization (darkening caused by the ultraviolet component of flashlamp light), and higher thermal conductivities. On the other hand, glass hosts have a number of distinct merits: they are isotropic, easily fabricated with high optical quality and homogeneous doping, they retain their optical finishes, and are readily grown in large sizes (see Sec. 14.3B). Because they are poor conductors of heat, however, glass lasers are principally used in systems that operate at very high powers and low duty cycles. Line-broadening behavior and level lifetimes of solid-state lasers are often controlled by the vibrational characteristics of the host medium; crystalline hosts typically give rise to homogeneous broadening whereas glass hosts lead to inhomogeneous broadening (see Sec. 13.30). The lion's share of dopant ions used as active laser media in host crystals are transition-metal and lanthanide-metal (rare-earth) ions, but actinide-metal ions are also occasionally employed (see Fig. 13.1-3). The dopant ions are generally dispersed throughout the host and act as independent radiators, much as organic-dye ions behave in a solvent. The dopant concentration typically lies in the vicinity of 1 %; however, it can be as small as 0.01% or as large as 50%, depending on the dopant, host material, and application. To minimize strain, a host material is generally chosen so that the active dopant ion is comparable in atomic size to the substituted atom. Of this vast array of combinations, the most commonly encountered solid-state lasers are Nd 3 +:yV0 4 , Nd 3 +:YAG, Yb 3 +:YAG, Ti 3 +:sapphire, Er 3 +:silica fiber, and Yb 3 +:silica fiber, and we consider these in turn. Many other important solid-state lasers also belong to the family of rare-earth-doped dielectrics. These include Er 3 +: YAG, H0 3 +:YAG, Tm 3 +:YAG, and thulium-doped optical fiber. As discussed in Sec. 13.1C, the energy levels of the rare-earth ions (but not their fine structure) are essentially independent of the host material because the 4f electrons are well shielded from the lattice by the filled 58 and 5p subshells (see Table 13.1-1). Despite the fact that it was the first material to be crafted into a laser (see page 567), ruby is rarely used. Alexandrite finds occasional use in dermatologic applications. Solid-state lasers that are optically pumped by laser diodes (or banks of laser diodes) are known as diode-pumped solid-state (OPSS) lasers. These devices convert the rel- atively broadband, multimode output of laser diodes into the narrowband, single-mode output of solid-state lasers. They are compact and highly efficient devices, and offer a substantial variety of wavelengths. Frequency doubling, tripling, and quadrupling (see
592 CHAPTER15 LASERS Chapter 21) is often used to convert the emission into visible and ultraviolet light at many more wavelengths. Solid-state lasers find wide application in industry, medicine, and research. Neodymium-Doped Yttrium Vanadate Nd 3 +:YV0 4 is a dielectric medium with refractive index n 2.0. The host material is transparent over a broad range of wavelengths from 0.3 to 2.5 /-Lm. The energy levels relevant to lasing are illustrated in Fig. 15.3-1. Optical pumping by a semiconductor laser diode at Ao == 808 nm populates the 4F s / 2 level at 1.53 e V. Banks of laser diodes deliver high pump powers, as shown schematically in Fig. 14.2-8( d). The 4F 3 / 2 4[11/2 transition is responsible for laser action at 1.064 /-Lill, which is the principal transition of this medium. However, the 4F 3 / 2 4[13/2 and 4F 3 / 2 4[9/2 transitions also support laser action at 1.34 /-Lill and at 914 nm, respectively, the latter as a quasi-three-Ievel system. This material is distinguished from neodymium- doped glass (see Fig. 13.] -9) by its higher refractive index, homogeneous broadening, and smaller transition linewidth (see Table 14.3-1). As a four-level system, the laser threshold is substantially lower than that of ruby. Frequency-doubled Nd 3 +: YV0 4 laser light at 532 nm is often used to pump the Ti:sapphire laser (see Fig. 15.3-4).lntracavity frequency doubling of the 4F 3 / 2 4[9/2 laser light generates blue light at 457 nm. 2.0 Nd 3 +:YV0 4 ...-., > (1) '-" 4F512 G) 4F312 1.064- /-Lm Pump laser 1.5 1.0 ;>. blJ 1-< (1) t: 4/ 1312 PlJ 81JJp 'V8 I}/Jj ( Nd 3 +:YV0 4 0.5 @ 4/ 91 4/ 1112 CD o 1.064 Mill (a) (b) Figure 15.3-1 (a) Selected energy levels of Nd 3 +:yV0 4 . The red arrow indicates the principal laser transition, which has a wavelength of 1.064 /-LID in the near infrared. The four interacting energy levels are indicated by encircled numbers. (b) Configuration of a Nd 3 +: YV0 4 laser with an intracavity frequency-doubling lithium-triborate (LBO) crystal that generates light at )..0/2 == 532 nm (see Sec. 21.2A). Neodymium-Doped Yttrium Aluminum Garnet Developed in the 1960s, Nd 3 +: YAG, whose energy levels are displayed in Fig. 13.1-9, is one of the most widely used of all solid-state laser materials. Because the optically active 4f electrons are shielded from the host, its energy levels are similar to those of neodymium-doped glass and neodymium-doped yttrium vanadate (see Figs. 13.1-9 and 15.3-1, respectively). N d 3 +: YAG lasers often incorporate intracavity doubling crystals,
15.3 COMMON LASERS 593 as shown in Fig. 15.3-1 for Nd 3 +:yV0 4 . Although it can be pumped by flashlamp, Nd 3 +:YAG is most conveniently pumped by a laser diode at 808 nm, providing a compact battery-powered source of near-infrared or green light. Crystals with lengths as short as a few hundred J-Lm can serve as efficient single-frequency thin-disk lasers. The most common laser line offered by Nd 3 +:YAG is at Ao == 1.06415 J-Lm in the near infrared. The fine-structure levels of the three manifolds associated with this laser transition are displayed in Fig. 15.3-2. This particular laser line arises from a transition between the upper fine-structure level in the 4P3/2 manifold at 1.4269 e V and the third- from-bottom fine-structure level in the 4[11/2 manifold at 0.2616 eV. When frequency doubled, this transition provides the familiar green emission line at 532 nm. 0.12 4/ 11 / 2 0.34 4 F 3/2 1.50 4/ 9 /2 0.10 0.32 1.48 Figure 15.3-2 Fine structure of - - the three manifolds associated with near-infrared Nd 3 +: YAG laser tran- - 0.08 - 0.30 1.46 sitions in the vicinity of 1.06 /-Lm >' (see Fig. 13.1-9): (a) ground state ;>-. - 0.06 - 0.28 1.44 41 9 / 2 ; (b) lower laser level 41 11 / 2 ; e!J v (c) upper laser level 4P3/2' The num- s:: - 0.04 - 0.26 1.42 bers of distinct levels in the three manifolds are (23+1)/2 == 5,6, and - 0.02 - 0.24 1.40 2 respectively. The specific energies depend on the host material - the 0 0 0 levels are substantially smeared in (a) (b) (c) glass hosts. The number of distinct fine-structure levels within each manifold is determined by 9/2, where 9 == 28 + 1. The quantity 9 is the degeneracy parameter and 8 is the total overall angular momentum quantum number, which is contained in the term symbol 2S+ l 'ca, as discussed in Sec. 13.1 A. Figure 15.3-2 reveals that transitions among the different fine-structure levels within the upper and lower laser manifolds offer a multitude of possible laser wavelengths that span the range from 1.052 to 1.122 J-Lm. In particular, lasing can be achieved at Ao == 1.12238 J-Lm via a transition between the lower of the two levels in the 4P3/2 manifold at 1.4165 e V and the highest of the levels in the 4[11/2 manifold at 0.3117 e V. This represents the longest wavelength that can be attained by a transition between these manifolds. When frequency doubled, this yields yellow-green light at Ao == 561 nm. Nd 3 +:YAG can also be operated as a quasi-three-Ievellaser on the 4P3/2 4[9/2 transition, generating laser light at 946 nm; intracavity frequency doubling then provides blue light at 473 nm. Specially designed photonic crystals can be used as filters to suppress oscillation on the dominant transition. Other possibilities abound since laser action can take place on many of these transitions. The principal disadvantages of Nd 3 +:YAG relative to Nd 3 +:YV0 4 are its narrower 4P5/2 absorption band (rendering it more sensitive to wavelength variations in the pump laser diode), higher threshold, lower slope efficiency, and unpolarized output. Nevertheless, Nd 3 +:YAG continues to be the workhorse of diode-pumped solid-state lasers. Ytterbium-Doped Yttrium Aluminum Garnet Yb 3 +:YAG thin-disk lasers make use of a 940-nm laser-diode pump (Fig. 15.3-3). High-efficiency absorption of the pump light is achieved by passing it through the active medium multiple times with the help of suitably designed optics. Large gain is
594 CHAPTER15 LASERS 2.0 Yb 3 +:y AG - 1.5 732 Q) F5n 1 > Q) 1.0 '-" - >. Laser-diode pump bJ) ""'" Q) Mirror t: LI.:j Pump 1. 03 O-fJ,m laser - 0.5 2p CD ' 7/2 Retro - reflector o (a) (b) Figure 15.3-3 (a) Energy levels pertinent to the ytterbium-doped YAG laser transition at Ao == 1.030 fJ,m. (b) Schematic of a single-frequency, single-mode Yb 3 +: YAG thin-disk laser. The pump light is passed through the active medium some 25 times by an optical system that includes a parabolic mirror and a retroreflector. High gain is achieved by using Yb 3 + doping levels 25%. attained by using high Yb 3 + doping levels. Since the pump wavelength Ao == 940 nm is close to the laser wavelength Ao == 1030 nm, the thermal load per pump photon is small so that little heat is generated in the crystal. Furthermore, the thin-disk configuration allows the residual heat to be removed effectively by heat-sink mounting, thereby permitting the TEMoo spatial mode to be maintained. Despite the fact that ytterbium- doped YAG is a quasi-three-Ievel system, thin-disk lasers can generate hundreds of watts of CW optical power at 1.030 J-Lm. When doubled, this laser provides a strong source of green light at 515 nm; it can therefore replace the Ar+ laser, a far more cumbersome device, in many applications. Titanium-Doped Sapphire The Ti 3 + : sapphire laser is widely used because it is tunable over a substantial range of wavelengths. Another of its merits is that it can be mode-locked to provide ultrashort pulses (see Sec. 15.40). As the crystal grows, a small fraction of the Al ions in sapphire ( 1 %) are replaced by Ti ions. Like ruby, the material is principalJy sapphire and therefore has a refractive index n 1.76. Optical pumping is usually provided by a frequency-doubled Nd 3 +:YV0 4 or Nd 3 +:YAG laser at 532 nm (see Fig. 15.3-1); by an Ar+ -ion laser or a frequency-doubled Yb 3 +:YAG laser at 515 nm (see Fig. 15.3-3); or by direct pumping with a green laser diode. Each titanium ion, which has a single 3d 1 active electron (see Table 13.1-1), is surrounded by six oxygen atoms at an octahedral site. This ion is therefore subjected to significant crystal-field and orbital interactions. As with other transition-metal ions in dielectric hosts, the titanium-doped sapphire energy levels displayed in Fig. 15.3- 4 are designated by group-theoretical, rather than by term symbols (see Sec. 13.1 C). Moreover, the electronic energy levels are strongly coupled to the lattice vibrations., resulting in broad bands of vibronic states. Stimulated emission is thus accompanied by the simultaneous emission of one or more phonons. The occupancy of the '4f 2 band follows a Boltzmann distribution so that its upper reaches are essentially unoccupied and the system behaves as a four-level laser, as shown in Fig. ] 5.3-4(a). The laser transition indicated by a red arrow in Fig. 15 .3-4( a) can be tuned over a
15.3 COMMON LASERS 595 few tens of nm by making use of a rotatable birefringent filter installed at Brewster's angle within the cavity [Fig. 15.3-4(b)], which acts as a bandpass filter for the polarized intracavity beam. Greater changes in wavelength are effected by adjusting the internal optics since the cavity group-velocity dispersion changes with wavelength. All-in-all, a broad range of wavelenfths, from 700 nm in the red to 1050 nm in the near infrared, can be accessed. The Ti + :A1 2 0 3 laser can provide 5 W of optical power when operated CW and, when mode locked, can generate a sequence of 10-fs, 50 nJ pulses, with a repetition rate of 80 MHz and a peak power 1 MW. Ti 3 +: Al 2 0 3 (Ti : Sapphire) 3.0 2.5 2£ 732 -.. G) @T > (1) '-" ;>. bJ) Tunable (1) laser c:: UJ Pump CD ;( @ 2T 2 (a) 2.0 1.5 1.0 . Pump Ti:Sapphire 0.5 o (b) Figure 15.3-4 (a) Selected energy bands of Ti 3 +:AI 2 0 3 . The red arrow indicates the principal laser transition of this vibronic system, which is tunable between 700 and 1050 nm. Dark-to- light shading in the bands indicates a decrease in relative occupancy. (b) Schematic diagram of a Ti 3 + :A1 2 0 3 mode-locked laser. The two prisms within the dashed box provide intracavity dispersion compensation. Wavelength tuning over tens of nm is achieved by means of a rotatable birefringent filter (BRF) that acts as a bandpass filter for the polarized intracavity beam; tuning over a larger range is effected by adjusting one of the prisms. The green pump light is often provided by a frequency- doubled Nd 3 +: YV0 4 laser, such as that illustrated in Fig. 15.3-1. Because of the importance of lattice vibrations in the tunability of this laser, titanium-doped sapphire is described as a phonon-terminated or vibronic laser. The alexandrite laser (see Fig. 13.1-8) also falls in this class, as does the dye laser (see Sec. 15.3C), since molecular vibrations play the same role as lattice vibrations. In general, a vibronic transition indicates a simultaneous change in the electronic and vibrational states of a system. Fiber Lasers With suitable feedback, rare-earth-doped fibers operate as highly efficient fiber lasers from the visible to the mid infrared. A simplified schematic that illustrates the use of diode-laser pumping and fiber Bragg-grating reflectors is displayed in Fig. 15.3- 5( a). Double-clad fiber configurations are widely used to avoid the nonlinear effects attendant to concentrating high pump power in a small fiber core [Fig. ] 5.3-5(b)]. Ytterbium-doped multiclad silica fiber lasers offer particularly good performance. The laser-diode pump energy is delivered to the active medium via multimode fibers that are spliced to a coil of multiclad fiber. Feedback is provided by fiber Bragg
596 CHAPTER15 LASERS Gain fiber (a) FBG »»»»» FBG )rf)))) - - (b) Outer Cladding Inner Cladding! Outer core Inner Core Figure 15.3-5 (a) Simplified schematic of a laser-diode-pumped fiber laser with fiber Bragg gratings (FBG) as reflectors. Pumping often involves multiple broad-area multi mode laser diodes whose light is coupled into the outer core of the fiber via multimode couplers, in both the forward and backward directions. A single-mode inner core fosters single-trans verse-mode oscillation. Fiber- laser operation has been achieved in many other configurations. (b) Concentric double-clad fiber configuration. Other double-clad configurations are designed to provide increased overlap between the inner core and the skew rays of the outer core (see Fig. 9.1-2). For example, the inner core may be shifted off-center (toward the edge of the outer core), or the outer core may be rectangular, hexagonal, octagonal, or D-shaped. gratings. Output powers well in excess of 1 kW CW are available in the 1070-1080- nm wavelength range, with a FWHM linewidth 1 THz. The light exits via a single- mode fiber whose core diameter is several J-Lm. Beam quality is excellent (M 2 < 1.1), as is the overall efficiency (Ilc > 25%). Linear polarization, reduced linewidth, and a wider wavelength range can be obtained at reduced power levels. On the other side of the coin, substantially greater optical power can be obtained by operating in a multimode configuration; powers up to 50 kW are available for applications such as welding, cutting, and drilling. Erbium-doped silica fiber lasers offer hundreds of watts of CW power in the 1550- 1570-nm wavelength range, with FWHM linewidths < 400 GHz. Beam quality is near diffraction-limited (M 2 < 1.1) and the overall efficiency Ilc > 10%. These diode- laser pumped devices are compact and air-cooled, and can operate with random, linear, or circular polarization. Operation over an enhanced wavelength range, from 1530 to 1620 nm, is possible when the optical power is reduced to tens of watts. At yet longer wavelengths, in the 1.8-2.1-J-Lm range, thulium-doped fiber lasers provide optical powers of 150 W with FWHM linewidths 75 GHz. Typical overall efficiencies are Ilc 5% and beam quality is excellent (M 2 < 1.05). Again, linear polarization can be obtained at reduced optical powers. When operated at 1.94 J-Lm, the thulium-laser output matches the water-absorption wavelength of soft tissue, so this laser is useful in clinical medicine. Doped fiber lasers are used in a broad range of applications that stretch from materi- als processing, to surgery, to seeding the glass laser amplifiers at the National Ignition Facility (see Sec. 14.3B). Fiber lasers can also be operated at large average powers in Q-switched and mode-locked configurations (see Sec. 15.4). Moreover, photonic- bandgap fiber lasers can be configured so that the light emerges radially from the full circumferential surface of the fiber, rather than axially; this configuration promises new forms of imaging and display. t A comparison of the performance of fiber lasers and laser diodes is provided in Sec. 17.3C. t See o. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, Surface-Emitting Fiber Lasers, Optics Express, vol. 14, pp.3929-3935,2006.
15.3 COMMON LASERS 597 Raman Fiber Lasers Raman fiber lasers (RFLs) operate on the basis of stimulated Raman scattering (SRS), a process initially considered in Sec. 13.5C and revisited in connection with Raman fiber amplifiers (RFAs) in Sec. 14.3D. Stimulated Raman scattering is illustrated in Fig. ] 4.3-7: A signal photon of energy hv s stimulates the emission of a clone signal photon that is obtained by Stokes-shifting the pump photon by the Raman vibrational energy hV R so that the energy of the clone photon precisely matches the energy of the initial signal photon. The optical gain of a RFA is governed by the Raman gain coef- ficient 'YR [see (2] .3-15)] and its bandwidth is determined by the vibrational spectrum of the glass host (see Sec. 14.3D and Fig. 14.3-7). Just as a rare-earth-doped fiber amplifier is converted into a fiber laser by the intro- duction of optical feedback, as shown in Fig. 15.3-5, so too is a Raman fiber amplifier converted into a Raman fiber laser (RFL). Fiber Bragg gratings (FBGs) serve as re- flectors and comprise the resonator, fostering oscillation at those frequencies where their reflectance is large (see Sec. 7.1 C). The oscillation frequency v is shifted from the pump frequency v p by the Stokes frequency V R , which can take on any value within the vibrational spectrum of the glass host, as illustrated in Fig. 14.3-7. A unique feature of the Raman interaction is that the Stokes shift is linked to the pump wavelength. As a consequence, the Stokes-shifted RFL oscillation frequency generated with a resonator comprising a particular pair of fiber Bragg gratings, say FBG I, can itself serve as a pump for the same fiber. As shown in Fig. 15.3-6(a), this second pump, which has reduced frequency Vpl == v p - V R , can then create a second- order Stokes-shifted oscillation, established at the frequency of maximum reflectance V p 2 of a second pair of fiber Bragg gratings, FBG2. The cascade can continue, using nested pairs of FBGs, until terminated by use of an output coupler that directs light out of the fiber at the desired frequency. Raman fiber lasers comprising multiple orders of Stokes shifts, which are known as cascaded Raman fiber lasers, thus offer a greatly expanded range of possible wavelengths. (a) hvp ---- ------- . . . hVp2 --- hv h . Figure 15.3-6 (a) Cascaded Stokes shifts of multiple orders. (b) Schematic of a Raman phosphosilicate- fiber laser. The double-clad Yb 3 + :silica fiber pump laser is itself pumped by a laser-diode array. Numbers under the FBGs represent intensity reflectances at the specified wavelengths (in nm). h .' Double-clad Yb 3 +:silica fiber Phosphosilicate fiber (b) ( c- FBG FBG v p ») ») - 1064 1064 100% 20% 1 c Pump laser _ I c FBG 1 FBG2 ») ») 1239 1484 100% 100% FBG2 FBG 1 v ») ») - 1484 1239 50% 100% Raman wavelength shifter -I The RFL is a high-power, single-mode source that is useful in many applications, including materials processing, clinical medicine, and optical fiber communications.
598 CHAPTER15 LASERS RFLs are particularly useful for pumping Raman fiber amplifiers in dense wavelength- division-multiplexed (DWDM) systems and for the remote pumping of erbium-doped fiber amplifiers. Its most attractive feature is that oscillation can be achieved over a broad range of wavelengths by suitably choosing the pump wavelength, fiber material, and fiber Bragg gratings. Indeed, under appropriate conditions the RFL can directly serve as an RFA by injecting the signal to be amplified. EXAMPLE 15.3-1. Raman Phosphosilicate-Fiber Laser. A Raman fiber laser can be constructed by using a double-clad ytterbium-doped fiber laser, which emits in the 1050-1120-nm region, as a pump for a l-km length of phosphosilicate-fiber Raman wavelength shifter, as depicted in Fig. 15.3-6(b). The Yb 3 + :silica-fiber pump laser, which is itself pumped by a laser-diode array operating in the vicinity of 960 nm, emits single-mode light that can be coupled into the single- mode Raman wavelength shifter far more efficiently than can multimode light from a laser-diode array. As a specific example, assume we wish to convert ytterbium-laser pump light at 1064 nm to a longer wavelength, say 1484 nm, so that the Raman fiber laser is suitable for pumping a Raman fiber amplifier (see Sec. 14.3D). Since we wish to translate the wavelength over a rather large range of 80 THz, we make use of phosphosilicate fiber, which has a large Stokes shift, VR 40 THz, enabling us to make the conversion using only two Stokes orders. Alternatively, we could use germanium- doped silica fiber, but this would require six Stokes orders since the Stokes shift for this material is far smaller, VR 13 THz, as is evident in Fig. 14.3-7. As shown in Fig. 15.3-6(b), a first pair of fiber Bragg gratings, FBG 1, is used to shift the 1064-nm pump light down by 40 THz in frequency to 1239 nm, while a second pair, FBG2, shifts the light down yet another 40 THz in frequency to the desired wavelength of 1484 nm. The FBG 1 pair have reflectances of 100%, while the reflectance of one member of the FGB2 pair is reduced to 50% to couple light at 1484 nm out of the RFL. A compact RFL, such as that shown in Fig. 15.3-6(b), can deliver tens of watts of CW optical power, with a bandwidth of a few nm, at any desired wavelength in the 1200- I 700-nm range. The overall efficiency of the ytterbium-doped fiber pump laser is 25% while the slope efficiency of the Raman wavelength converter is roughly 0.25 W /W. Although bulk Raman lasers were demonstrated long ago, the fiber version of this device has brought Raman technology to the fore for several reasons. Fibers offer long lengths and therefore large gains, they can support large intensities in a single-mode core, they are efficiently pumped by diode-pumped solid-state lasers" and they readily accommodate multiple fiber Bragg gratings. Stimulated Brillouin scattering can be used in an analogous way to make Brillouin fiber lasers. As a final note, we emphasize that the Raman laser operates on the basis of Raman gain rather than stimulated emission. As such, it does not make use of a population inversion and it therefore differs from the usual laser in an fundamental way. It should also be pointed out that lasing without inversion (LWI) can be achieved within the energy-level structure of a conventional laser medium by using an external optical field to create an additional path from the lower to the upper energy level, via an auxiliary energy level. Under appropriate circumstances, the presence of the two paths can result in destructive quantum interference, and consequently the elimination of absorption. Random Lasers As discussed in Secs. 15.1 and 15.2, the oscillation frequencies of conventional lasers are determined by the Fabry-Perot resonator modes together with the gain profile of the active-medium resonant transition. The output light, transmitted through a partially reflecting exit mirror, typically has a narrow spectrum, strong directionality, and a high degree of temporal and spatial coherence. Scattering from the laser medium introduces loss and is assiduously avoided. When scattering in the active medium is very strong, however, it itself can provide feedback. Random lasers operate on the basis of feedback provided by multiple scatter- ing within a disordered gain medium, which serves as a closed 3D cavity. Photons trav-
15.3 COMMON LASERS 599 eling within the medium can be viewed as executing a random walk in 3D [Fig. 15.3- 7(a)]. Because strong scattering is associated with disordered media, lasers that operate on this principle are known as random lasers. They are also called powder lasers or plasers. In distinction to conventional lasers, the radiation scattered back to any location in the active medium has a random phase. The feedback is thus incoherent and intensity-based, rather than coherent and field-based. Inasmuch as resonant feedback is absent in such random lasers, the central oscillation frequency is governed by the active-medium gain profile. Substantial gain can be achieved because of the large overall path length in the active medium engendered by the multiple scattering. The stronger the scattering, the greater the feedback and hence the lower the laser threshold. Feedback via scattering appears to play an important role in astronomical maser action, such as that observed from molecular clouds of H 2 0, OH, and SiO. Scattering is appealing for providing feedback in the X-ray region, where specular reflection is difficult to achieve. 100 nm (a) (b) Figure 15.3-7 (a) A random laser relies on incoherent and nonresonant feedback provided by multiple scattering as well as a long path length within the gain medium. In recurrent scattering (illustrated schematically as loops), the field can repeatedly retrace one or more local paths that collectively serve as a local cavity, thereby providing coherent and resonant feedback. (b) Close- packed ZnO nanocrystallites serve as both the active medium and the scattering feedback elements in a microrandom laser. The hallmark of random lasers is the absence of directionality and spatial coherence of the emitted light. Indeed, the spatial emission lasing pattern from the face of a cuvette containing the powdered active medium often resembles that of a surface- emitting LED [see Fig. 17 .1-12( a)]. However, random lasers share many properties in common with conventional lasers, including their diversity. They can be pumped optically, electrically, or by electron beam. Lasing can take place over a broad range of wavelengths, from the infrared to the ultraviolet. The sizes of active regions can stretch from microcavities with volumes of the order of 1 J-Lm 3 to macroscopic devices with cm 3 volumes. When ground into powders, conventional solid-state laser materials such as ruby, Nd 3 +:YAG, Nd 3 +:glass, Ti 3 +:sapphire, and GaAs function as random lasers. In many powders, the gain and scattering media are one-and-the-same, but this need not be the case. For example, rhodamine 6G dye molecules serve well as an active medium while Al 2 0 3 microparticles function as a scattering medium, when both are placed in a solution of methanol. Random-laser active media encompass inorganic dielectrics, polymers, liquids, dye solutions, dye-doped liquid crystals, disordered semiconductor nanostructures, and even biological tissues. If the constituent particles of the active medium are sufficiently large and regularly shaped so that they support resonator modes, they can behave instead as random col- lections of individual microlasers, each with its own emission direction. Alternatively, a local configuration of scatterers can support resonances. If the scattering is recurrent
600 CHAPTER15 LASERS [see Fig. 15.3-7(a)], and the optical amplification exceeds the losses along the return paths, the latter can serve as a cavity. The ensuing laser emission is then sharply peaked at these fortuitous cavity-mode frequencies. Since different regions of the random medium support different collections of return paths, the oscillation frequencies depend on the particular region of the material being pumped. Such lasers are called coher- ent random lasers because of their coherent feedback and spatially random cavity configurations; they are similar to collections of microlasers with random emission directions. Clusters of scatterers can be used to fabricate individual microrandom lasers, in which light is confined to a volume of the order of a cubic wavelength by strong scattering rather than by reflection. Other forms of microcavity lasers are considered in Sec. 17.4B. EXAMPLE 15.3-2. ZnO Microrandom Laser. A closely packed collection several thousand ZnO nanocrystallites [see Fig. l5.3-7(b)], each of the order of tens of nm in diameter can coalesce into a microcluster 1 /-Lm in diameter. The nanocrystallites serve as both the gain medium and the scattering feedback elements of a microrandom laser. The emission wavelength Ao 380 nm lies near the bandgap wavelength of ZnO. Because the optical confinement arises from scattering rather than from reflection at the surface of the microcluster, such microlasers need not have regular shapes and smooth surfaces. B. Gas Lasers Atomic and Ionic Lasers Atomic and ionic gas lasers, such as He-Ne, Ar+, and Kr+, produce the beautiful multicolored beams that have been a staple of optics laboratories for decades (see Table 15.3-1). The Kr+ -ion laser, in particular, produces hundreds of milliwatts of optical power at wavelengths ranging from Ao == 350 nm in the near-ultraviolet to 676 nm in the red. It can be operated simultaneously on a number of lines to produce "white laser light." Many other monoatomic species, and their ions, also serve as active laser media and operate at innumerable wavelengths in the near infrared and visible regions. Nevertheless, atomic and ionic gas lasers are now used principally for specialized applications; diode-pumped solid-state lasers and laser diodes have superior performance, can be more readily tuned, and are physically more robust. Molecular Lasers Molecular gas lasers such as the CO 2 laser (see Table 15.3-1 and Fig. 13.1-4), which lases in the vicinity of Ao == 9.6 and 10.6 /-Lm in the mid-infrared region, can produce thousands of watts of CW power with high efficiency, and has applications such as cutting, welding, scribing, engraving, and marking. A favorite in the far-infrared is methanol, which lases at Ao == 119 and 124 /-Lm as well as at myriad other wavelengths. Indeed, most molecular transitions in the infrared region can be made to lase; even simple water vapor (H 2 0) lases at many wavelengths in the far infrared, as shown in Table 15.3-1. Excimer Lasers Excimer lasers are important in the ultraviolet region of the spectrum. The term excimer, which is a contraction of the phrase "excited dimer," is a short-lived molecule that contain two atoms in an electronic excited state; the term exciplex is often used in place of excimer when the atoms are not identical. Noble-gas halides, such as XeCI, form exciplexes because the chemical behavior of an excited noble gas atom
15.3 COMMON LASERS 601 is similar to that of an alkali atom, which readily reacts with a halogen (see Fig. 13.1- 3). When the exciplex returns to the ground state, its components dissociate and the individual atoms often repel each other. The lower laser level is therefore unpopulated, providing a built-in population inversion. Examples of excimer and exciplex lasers, along with their principal wavelengths of operation, are: F 2 (153 nm), ArF (193 nm), KrF (248 nm), XeCI (308 nm), and XeF (351 nm) [see Table 15.3-1]. Laser pulses from XeCI, for example, can be generated by passing an electric discharge through a gas mixture of Xe and C1 2 . Short-wavelength light is not absorbed deeply in most materials, which renders excimer lasers useful for cutting sharply and without the production of heat. This, plus their substantial energy per pulse, makes them useful for applications such as microlithography, micromachining, photochemistry, and refractive surgery. Semiconductor-chip fabrication using light at Ao == 193 nm in the FUV is standard. Chemical Lasers Chemiluminescence, the emission of light via a chemical reaction, is observed when the reaction between two or more substances releases sufficient energy to populate the excited state of a reaction product (see Sec. 13.5A). Chemical lasers, which comprise mixtures of gases, are self-pumped in the sense that the pump energy derives from a chemical reaction in the active medium itself. The HF laser, which operates principally in the 2.7-3.1-fLm wavelength range, is perhaps the best known among this class of lasers. A mixture of H 2 and F 2 gases is subjected to an electric discharge, which results in the production of an HF molecule in an excited vibrational state, denoted HF*. This molecule emits an infrared photon and dissociates. Its components in turn react with the H 2 and F 2 gases to create other vibrationally excited molecules, creating a chain reaction of sorts. Chemical lasers can generate high power and are of interest principally for military applications. EXAMPLE 15.3-3. Deuterium Fluoride Chemical Laser at White Sands. The most notorious chemical laser, perhaps, is the U.S. Army's Mid-Infrared Advanced Chemical Laser (MIRACL), located at the White Sands Missile Range in New Mexico. This formidable device burns ethylene (C 2 H 4 ) with nitrogen trifluoride (NF 3 ). The resulting free fluorine atoms combine with injected deuterium gas to form vibrationally excited deuterium fluoride molecules, DF*. The photon emission, molecular dissociation, and creation of new vibrationally excited molecules is similar to that of the HF laser. However, the DF device lases on multiple lines in the 3.5-4.0-p,m wavelength range, which is absorbed far less by the atmosphere than the light emitted by the HF laser. This laser produces megawatt levels of CW radiation, over durations 1 minute, in the form of a beam with a diameter 4 em. c. Other Lasers Dye Lasers Organic dye lasers played a central role in photonics in years past because of their ability to be tuned over a substantial range of wavelengths. The active medium of a dye laser is generally a solution of an organic dye compound in alcohol or water, with a concentration 10- 4 M, although the dye molecules can alternatively be imbedded in a polymer, glass, or crystalline host to form a solid-state dye laser. Dye lasers typically behave as a four-level system (see Fig. 13.1-5). Polymethine dyes provide oscillation in the red and near infrared (0.7-1.5 J-Lm), xanthene dyes lase in the visible (500-700 nm), coumarin dyes lase in the blue-green (400-500 nm), and scintillator dyes lase in the ultraviolet « 400 nm). Rhodamine-6G, the quintessential example,
602 CHAPTER15 LASERS can be tuned over the wavelength range 560-640 nm. Unfortunately, dye lasers require high maintenance, in no small part because the chemical life of the dye in the solvent is relatively short. As a result, diode-pumped solid-state lasers (see Sec. 15.3A) have by-and-Iarge replaced the dye laser, except in the most specialized of applications. The diode-pumped solid-state Ti 3 +:A1 2 0 3 laser, for example, offers broader tunability than the typical dye laser in the vicinity of Ao == 800 nm, and requires little maintenance. Frequency doubling the Ti:sapphire laser leads to a useful band of tunable radiation in the vicinity of 400 nm. Tunability near 600 nm, in the gap between 400 and 800 nm, can be achieved by frequency-doubling the output of an optical parametric oscillator operating in the 1-2-Mm wavelength region (see Secs. 21.2C and 21.4C). Extreme-Ultraviolet and X-Ray Lasers Achieving laser action in the extreme-ultraviolet (EUV) and X-ray regions of the elec- tromagnetic spectrum is a challenging enterprise because of the difficulty of achieving a population inversion at these short wavelengths. According to (15.1-16), for a fixed value of t sp , the threshold population difference Nt ex l/Tp A 2 g(v) so that the thresh- old pump power density ex l/Tp A 3 g(v). When Doppler broadening prevails, as is generally the case in EUV active laser media, (13.3-42) reveals that g(v) ex A, which leads to a threshold pump power density proportional to l/Tp A 4 . It therefore becomes increasingly difficult to attain threshold as A decreases toward the EUV and X-ray regIons. Another aspect of the challenge has to do with optical components in this wave- length region, which have traditionally been difficult to construct. This is both because the absorption coefficient is large (which decreases Tp and thus further increases Nt) and the refractive index is close to unity in most materials. Nevertheless, two ap- proaches have proved successful: . Grazing-Incidence Optics. Since EUV frequencies are well above the plasma frequency, metals exhibit a refractive index that lies just below unity (see Fig. 5.5- 10 and Sec. 5.5D). Thus, total internal reflection can be achieved and metals can serve as mirrors. This is possible only at grazing incidence, however, because the small refractive-index contrast requires a large angle of incidence (the situation is analogous to that at the boundary of the core and cladding of an optical fiber). . Multilayer Optics. The construction of multilayer-optical devices is not as straightforward as in the visible region since the refractive index is close to unity in the EUV region and it does not vary appreciably from one material to another. Nevertheless, high-reflectance multilayer mirrors can be fabricated with tolerable losses by using a large numbers of layers (see Sec. 7.1). For example, multilayer mirrors comprising tens of alternating layers of Si and Mo can provide reflectances > 70% in the EUV. Multilayer optics can also be incorporated in zone-plate structures. X-ray laser action was first achieved in a dramatic experiment carried out by re- searchers at the Lawrence Livermore National Laboratory (LLNL) in 1980. An un- derground nuclear detonation was used to create X-rays, which in turn pumped the atoms in an assembly of metal rods. The X-ray laser pulse was generated before the detonation vaporized the apparatus. Nowadays, coherent EUV and X-ray radiation is typically generated via recom- bination in a hot ionized-atom plasma. A downward electron transition in a highly ionized atom produces a high-energy photon that in turn induces the emission of a clone photon from a nearby ion via stimulated emission. Pumping in such systems is typically achieved by focusing a highly intense laser beam onto a solid target. Short pump pulses are generally used to attain high intensities. Most extreme-ultraviolet
15.3 COMMON LASERS 603 lasers and soft X-ray lasers operate on the basis of this principle. Devices such as these generate amplified stimulated emission (ASE) [see Sec. 14.5] rather than radiation emerging from a resonator since it is difficult to arrange for optical feedback at these short wavelengths. Spatial coherence can be imparted by propagation (see Sec. 11.3C). An illustrative example of an EUV laser is provided by a plasma of ionized carbon. In an experiment carried out in the mid-1980s, a 10.6- Mm-wavelength CO 2 laser pulse, of 50-ns duration and 300-1 energy, was focused onto a solid carbon disk. The infrared- laser pump pulse generated sufficient heat to strip all of the electrons from some of the carbon atoms, thereby creating a plasma of ionized carbon, which was radially confined by the use of a magnetic field. The cooling of the plasma at the tennination of the pump pulse led to the capture of electrons in the n == 3 shells, and simultaneously to a dearth of electrons in the n == 2 shells because of fast radiative decay to the ground state. The net result was a collection of hydrogen-like C 5 + ions with a population inversion (see Fig. 13.1-1). As expected from (13.1-4), the decay of an electron from the n == 3 to the n == 2 shell (the 3d 2p transition has the largest cross section) will be accompanied by the emission of a photon of energy A1rLj2e4 ( 1 1 ) E = ( 41fE o)2 2/1,2 2 2 - 3 2 . (15.3-1) With Lj == 6 this corresponds to an EUV photon of energy 68 e V and wavelength Ao == 18.2 nm. In the ionized-carbon experiment, a spontaneously emitted photon (t sp 12 ps) initiated the stimulated emission of EUV photons from other ions, resulting in amplified spontaneous emission. The single-pass gain-coefficient/length product ryd was 6 so that, in accordance with (14.1-7), the gain was G e 6 . The output was a 20-ns pulse of EUV (soft X-ray) ASE with a power of 100 kW, an energy of 2 mJ, and a divergence of 5 mrad. Similar results were obtained by using a Nd 3 +:glass-Iaser pump operated at 1.06 Mm. Active media in EUV lasers of current interest are, for the most part, highly ionized atoms that are Ne-like, Ni-like, and Pd-like. Lasers have been created from dozens of such ionic species. A common pumping configuration makes use of a cylindrical lens that focuses the pump light onto the target, generating a column of plasma that serves as a length of active region. Pumping is usually provided by a Ti 3 +:sapphire laser or by the fundamental, second, or third hannonic of a Nd 3 +:glass laser (see Secs. 15.3A and 14.3B). The use of sequential pump pulses enhances the population inversion, which improves efficiency and permits laser operation in the saturated-gain regime. Delivering the main pump pulse at grazing incidence increases absorption and reduces the required pump energy. The Ni-like Ag 19 + EUV laser considered in Example 15.3- 4 illustrates the operation of this type of laser. EXAMPLE 15.3-4. Nickel-Like Silver-Ion EUV Laser. The Ni-like AgI9+ -ion EUV laser operates on the 4d ISO 4p IP I transition. Pumping with a 4-ps-duration prepulse of 1.5 J, followed 1.2 ns later by a 4-ps-duration main pump pulse of 10.5 J, yields an EUV ASE pulse of 25 j1J at Ao = 13.9 nm. The gain coefficient is 35 cm- I and the gain-coefficient/length product is 1'd 13.6. The divergence is 6 mrad. Focusing a highly intense pump laser beam onto a solid target is not the only way to attain laser action in the EUV region. A population inversion can also be achieved by directly exciting the active medium with a brief, strong electrical pulse that creates a hot plasma. The capillary-discharge X-ray laser makes use of an active medium flowing through (or coated on the inside ot) a capillary that is a few mm in diameter and tens of cm in length. Such current-pumped devices typically offer greater repetition rates and greater spatial coherence than laser-pumped devices. However, their principal merit
604 CHAPTER15 LASERS lies in their relatively small size; indeed, such lasers are informally known as "tabletop X-ray lasers." Ionization of the active medium can also be achieved directly via field effects and multiphoton processes. The production of a population inversion then results from the collisional excitation of ions initiated by the emitted electrons, rather than by ther- malized particles in the heated plasma. Such optical-field-ionization lasers produce a cold, dense collection of ionized atoms surrounded by a hot electron distribution. In principal, it is also possible to construct inner-shell photopumped lasers, in which the inner-shell electrons of neutral atoms are ionized. Although the lifetimes of inner-shell vacancies are very short as a result of fast radiative decay from higher energy levels and from Auger transitions, the techniques of ultrafast optics may well be useful for facilitating such pumping (see Chapter 22). Substantial advances have also been made in the generation of coherent X-rays by means of high-harmonic generation (HHG). High-order harmonics can be gen- erated via an extreme nonlinear interaction between intense fsec laser pulses and the molecules of a gas. If the laser field is sufficiently strong, it can ionize the gas molecule and accelerate the liberated valence electrons away from the ion. High-order harmonic light is generated as the field reverses and the oscillating electrons return to the ion. This approach typically produces ultrashort pulses in the 5-30-nm wavelength range with significant average power. Free-electron lasers (FELs) provide yet another means for generating coherent EUV and X-ray radiation (see below), but FELs are available only at large-scale synchrotron facilities. EUV and X-ray laser applications include nanolithography for semiconductor in- tegrated circuits, nanopatteming, nanoimaging, plasma diagnostics, and the dynamic imaging and holography of biological and other structures. Free-Electron Lasers The free-electron laser (FEL) makes use of a magnetic wiggler field produced by a periodic assembly of magnets of alternating polarity, known as an undulator. The active medium is a relativistic electron beam moving in the wiggler field (Fig. 15.3-8). As the electrons traverse the magnet structure, they undergo oscillations and radiate coherently. Despite the appellation "free-electron laser," the electrons are not truly free since their motion is affected by the wiggler field. U ndulator Electron beam Electron beam Figure 15.3-8 Schematic of a typical FEL in the optical region. The undulator generates a periodic transverse wiggler field. The undulator period A is a few centimeters; it contains roughly 100 periods and has a total length of a few meters. The resonator mirrors surround the undulator and have a separation d that is about twice its length. The peak magnetic field in the undulator is typically a few kilogauss. The electron beam is guided into the undulator by bending magnets. The electron-beam current ranges from a few amperes to a few kiloamperes and the electron energy can vary from a few Me V to several Ge V; a typical value is 50 MeV. The radius of the electron beam is roughly 1 mm while the optical-beam waist is about 3 mm. The electron-beam pulse duration can vary from ps to MS. The temporal structure of the radiation follows that of the electron beam.
15.4 PULSED LASERS 605 The FEL emission can be tuned over a broad range of wavelengths by modifying the electron-beam energy, the strength of the magnetic field, and the undulator period. Depending on design, PELs emit at wavelengths that stretch from the mm-wave region to the extreme ultraviolet. Since they operate in a vacuum, high peak powers can be attained without incurring material damage and encountering thermal lensing effects. The forthcoming generation of FELs will be dedicated to the production of coherent hard X-ray radiation with wavelengths as short as Ao 1 A = 0.1 nm. The first FEL of this class, the 1-km-long Linac Coherent Light Source (LCLS) at the Stanford Linear Accelerator Center (SLAC), will accelerate bunches of roughly 10 10 electrons to generate bunches of 10 12 photons in 100-fs pulses at Ao == 1.5 A. Several FELs operating at shorter wavelengths are slated to come online by 2015. D. Tabulation of Selected Characteristics In Table 15.3-1 we provide a list, in order of increasing wavelength, of representative characteristics of some well-known lasers. The broad range of transition wavelengths, overall efficiencies, and power outputs for the different lasers is noteworthy. The transition cross section, spontaneous lifetime, and atomic linewidth for a num- ber of these laser transitions are listed in Table 14.3-1. The linewidth of the laser output is generally many orders of magnitude smaller than the atomic linewidths specified in Table 14.3-1; this is because of the additional frequency selectivity imposed by the optical resonator. Some laser systems cannot sustain a continuous population inversion and therefore operate only in a pulsed mode. 15.4 PULSED LASERS It is sometimes desirable to operate lasers in a pulsed mode since the optical power can be greatly increased when the output pulse has a limited duration. Lasers can be made to emit optical pulses with durations as short as femtoseconds; the durations can be further compressed to the attosecond regime by making use of nonlinear-optical techniques. Maximum pulse-repetition rates extend from hours for some EUV lasers to more than 100 GHz. Maximum pulse energies reach from fJ to MJ, more than 20 orders of magnitude, while peak powers extend to more than 10 MW and peak intensities reach 10 TW / cm 2 . Some lasers must be operated in a pulsed mode since CW operation cannot be sustained, as is evident in Table 15.3-1. A. Methods of Pulsing Lasers The most direct method of obtaining pulsed light from a laser is to use a continuous- wave (CW) laser in conjunction with an external switch or modulator that transmits the light only during selected short time intervals. This simple method has two distinct disadvantages, however. First, the scheme is inefficient since it blocks (and therefore wastes the light) energy during the off-time of the pulse train. Second, the peak power of the pulses cannot exceed the steady power of the CW source, as illustrated in Fig. 15.4-1(a). More efficient pulsing schemes are based on turning the laser itself on and off by means of an internal modulation process, designed so that energy is stored during the off-time and released during the on-time. Energy may be stored either in the resonator, in the form of light that is periodically permitted to escape, or in the atomic system, in the form of a population inversion that is released periodically by allowing the system to oscillate. These schemes permit short laser pulses to be generated with peak powers
606 CHAPTER15 LASERS Table 15.3-1 Typical characteristics and parameters for a number of well-known lasers made of different forms of matter, a in order of increasing wavelength. Single Mode Approximate Transition (S) or CW OveralJ Output Energy- Wavelength Multimode or Efficiency Power or Level Laser Medium Ao (M) Pulsed b Ilc(%)C Energyd Diagram Ag19+ (p) 13.9 nm M Pulsed 0.0002 25 pJ C 5 + (p) 18.2 nm M Pulsed 0.0005 2mJ Fig. 13.]-1 ArF Excimer (g) 193 nm M Pulsed 1. 200 mJ KrF Excimer (g) 248 nm M Pulsed l. 500 mJ He-Cd (g) 442 nm S/M CW 0.1 100 mW Ar + (g) 515 nm S/M CW 0.05 lOW Rhodamine-6G (1) 560-640 nm S/M CW 0.005 100 mW Fig. 13.1-5 He-Ne (g) 633 nm S/M CW 0.05 ]OmW Fig. 13.1-2 Kr + (g) 647 nm S/M CW 0.01 lW Ruby (s) 694 nm M CW 0.1 5W Fig. 14.3-1 Alexandrite (s) 700-820 nm M CW 0.1 lW Fig. 13.]-8 Ti:Sapphire (s) 700-1050 nm S/M CW 0.01 5W Fig. 15.3-4 Yb 3 +:YAG (s) 1030 nm S/M CW 5. 100W Fig. 15.3-3 Nd 3 + :Glass (s) 1053 nm M Pulsed l. 50J Fig. 14.3-3 Nd 3 +:YAG (s) 1064 nm S/M CW 5. 50W Fig. 13.1-9 Nd 3 +:YV0 4 (s) 1064 nm S/M CW 10. 30W Fig. 15.3-1 Yb 3 + :Silica fiber (s) 1075 nm S/M CW 20. 1500 W Er 3 + :Silica fiber (s) 1550 nm S/M CW ] o. 100W Fig. 14.3-6 Till 3 + : Fluoride fiber (s) 1.8-2.1 /-Lm S/M CW 5. 150W He-Ne (g) 3.39 /-Lill S/M CW 0.05 20mW Fig. 13.]-2 CO 2 (g) 10.6 /-Lill S/M CW ] O. 500W Fig. 13.]-4 H 2 0 (g) 28 /-Lill S/M CW 0.02 100 mW FEL at UCSB 60 /-Lill-2.5 mm M Pulsed 0.5 5 m] H 2 0 (g) 118.7/-L ill S/M CW 0.01 50mW CH 3 0H (g) 118.9 /-Lill S/M CW 0.02 100 mW HCN (g) 336.8 /-Lill S/M CW 0.01 20mW aGas (g), solid (s), liquid (1), plasma (p). bLasers designated "cw" can, of course, be operated in a pulsed mode; lasers designated "pulsed" are usually operated in that mode. cThe power-conversion efficiency Ilc (also called the overall efficiency and wall-plug efficiency) is the ratio of output light power to input electrical power (for pulsed lasers, the ratio of output light energy to input electrical energy). Values reported have substantial uncertainty since in some cases they include the electrical power consumed for overhead functions such as cooling and monitoring. Laser diodes exhibit the highest efficiencies, readily exceeding 50%, as discussed in Sec. 17 AC. d The output power (for CW systems) and output energy per pulse (for pulsed systems) vary over a substantial range, in part because of the wide range of pulse durations; representative values are provided. far in excess of the constant power deliverable by CW lasers, as illustrated in Fig. 15.4- 1 (b). Four common methods used for the internal modulation of laser light are: gain switching, Q-switching, cavity dumping, and mode locking. These are considered in turn. Gain Switching Gain switching is a rather direct approach in which the gain is controlled by turning the laser pump on and off (Fig. 15.4-2). In the flashlamp-pumped pulsed ruby laser,
15.4 PULSED LASERS 607 Modulator Modulator IT I I TI IT I I [g]TI t Peak power Peak power Jl ---- Dx power ) t ___ Average power ) t (a) (b) Figure 15.4-1 Comparison of pulsed laser outputs achievable with (a) an external modulator, and (b) an internal modulator. for example, the pump (flashlamp) is switched on periodically for brief periods of time by a sequence of electrical pulses. During the on-times, the gain coefficient exceeds the loss coefficient and laser light is produced. Most pulsed semiconductor lasers are gain switched because it is easy to modulate the electric current used for pumping, as discussed in Chapter 17. The laser-pulse rise and fall times achievable with gain switching are determined in Sec. 15.4B. ! Pump Pump IT I I TI Gain Loss -- Laser output t Figure 15.4-2 Gain switching. Q-Switching In Q-switching, the laser output is turned off by increasing the resonator loss (spoiling the resonator quality factor Q) periodically with the help of a modulated absorber inside the resonator (Fig. 15.4-3). Thus, Q-switching is loss switching. Because the pump continues to deliver the constant power at all times, energy is stored in the atoms in the form of an accumulated population difference during the off (high-loss)-times. When the losses are reduced during the on-times, the large accumulated population difference is released, generating intense (usually short) pulses of light. An analysis of this method is provided in Sec. 15.4C. Loss --1 ,...--, r---' ,--- -. I I I I I I I I I I I I I I I I I I I I I I I I _J L L LI I [g] TI Gain IT I Modulated absorber . t Laser .--.0 output o n (l t Figure 15.4-3 Q-switching.
608 CHAPTER15 LASERS Cavity Dumping Cavity dumping is a technique based on storing photons (rather than a population difference) in the resonator during the off-times, and releasing them during the on- times. It differs from Q-switching in that the resonator loss is modulated by altering the mirror transmittance (see Fig. 15.4-4). The system operates like a bucket into which water is poured from a hose at a constant rate. After a period of time of accumulating water, the bottom of the bucket is suddenly removed so that the water is "dumped." The bucket bottom is subsequently returned and the process repeated. A constant flow of water is therefore converted into a pulsed flow. For the cavity-dumped laser, of course, the bucket represents the resonator, the water hose represents the constant pump, and the bucket bottom represents the laser output mirror. The leakage of light from the resonator, including useful light, is not permitted during the off-times. This results in negligible resonator losses, thereby increasing the optical power inside the laser resonator. Photons are stored in the resonator and cannot escape. The mirror is suddenly removed altogether (e.g., by rotating it out of alignment), increasing its transmittance to 100% during the on-times. As the accumulated photons leave the resonator, the sudden increase in the loss arrests the oscillation. The result is a strong pulse of laser light. The analysis for cavity dumping is not provided here inasmuch as it is closely related to that of Q-switching. This is because the variation of the gain and loss with time are similar, as may be seen by comparing Fig. 15.4-4 with Fig. 15.4-3. Mirror transmittance r- I TI 1 1 I r , II I-I Gain I I I I I I I I I I I I II I I Loss __I ,-__J L___ 1__-' L IT I . t Laser output t Figure 15.4-4 Cavity dumping. One of the mirrors is removed altogether to dump the stored photons as useful light. Mode Locking The three pulse-generation approaches discussed above are based on the transient dynamics of a laser medium. Mode locking differs from these approaches in that it is a dynamic steady-state process. It is the most important technique for generating trains of ultrashort laser pulses. Pulsed laser action is attained by coupling together the modes of a laser and locking their phases to each other. An example is provided by the longitudinal modes of a multimode laser, which oscillate at frequencies that are equally separated by the intermodal frequency c/2d. When the phases of these components are locked together, they behave like the Fourier components of a periodic function, and therefore form a periodic pulse train. The coupling of the modes is achieved by periodically modulating the losses inside the resonator. Mode locking is examined in Sec. 15.4D. *8. Analysis of Transient Effects An analytical description of the operation of pulsed lasers requires an understanding of the dynamics of the laser oscillation process, i.e., the time course of laser oscillation
15.4 PULSED LASERS 609 onset and termination. The steady-state solutions presented earlier in the chapter are inadequate for this purpose. The lasing process is governed by two variables: the number of photons per unit volume in the resonator, n( t), and the atomic population difference per unit volume, N(t) == N 2 (t) - N1(t); both are functions of the time t. Rate Equation for the Photon-Number Density The photon-number density n is governed by the rate equation dn n -==--+NW i . dt Tp (15.4-1) The first term represents photon loss arising from leakage from the resonator, at a rate given by the inverse photon lifetime I/Tp. The second term represents net photon gain, at a rate NW i , arising from stimulated emission and absorption. Wi == cjJa(v) == en a (v) is the probability density for induced absorption/emission. Spontaneous emis- sion is assumed to be small. With the help of the relation Nt == a r / a(v) == 1/ CT p a(v), where Nt is the threshold population difference [see (15.1-15)], we write a(v) l/cTp Nt, from which n Wi = Nt Tp . (15.4-2) Substituting this into (15.4-1) provides a simple differential equation for the photon number density n, dn n N n - == -- + -- . dt Tp Nt Tp (15.4-3) Photon-Number Rate Equation As long as N > Nt, dn/dt will be positive and n will increase. When steady state (dn/dt == 0) is reached, N == Nt. Rate Equation for the Population Difference The dynamics of the population difference N(t) depends on the pumping configu- ration. A three-level pumping scheme (see Sec. 14.2B) is analyzed here. The rate equation for the population of the upper energy level of the transition is, according to (14.2-8), dN 2 N 2 - == R - - - W. ( N 2 - N l ) dt t sp 'I, , (15.4-4 ) where it is assumed that T2 == t sp . The pumping rate R is assumed to be independent of the population difference N. Denoting the total atomic number density N 2 + N 1 by Na,sothatN 1 == (N a -N)/2andN 2 == (N a +N)/2,weobtainadifferentialequation for the population difference N == N 2 - N 1 , dN No N -==----2W i N dt t sp t sp , (15.4-5)
610 CHAPTER15 LASERS where the small-signal population difference No == 2Rt sp - N a [see (14.2-27)]. Sub- stituting the relation Wi == n/ NtTp obtained above into (15.4-5) then yields (15.4-6) Population-Difference Rate Equation (Three-Level System) The third tenn on the right-hand side of (15.4-6) is twice the second term on the right- hand side of (15.4-3), and of opposite sign. This reflects the fact that the generation of one photon by an induced transition reduces the population of level 2 by one atom while increasing the population of level 1 by one atom, thereby decreasing the popula- tion difference by two atoms. Equations (15.4-3) and (15.4-6) are coupled nonlinear differential equations whose solution determines the transient behavior of the photon number density n(t) and the population difference N(t). Setting dN/dt == 0 and dn/dt == 0 leads to N == Nt and n == (No - N t )( T p /2t sp ). These are indeed the steady-state values of Nand n obtained previously, as is evident from (15.2-12) with Ts == 2t sp , as provided by (14.2-28) for a three-level pumping scheme. dN No N N n - == - - - - 2--. dt t sp t sp Nt Tp EXERCISE 15.4-1 Population-Difference Rate Equation for a Four-Level System. Obtain the population- difference rate equation for a four-level system for which 71 « t sp . Explain the absence of the factor of 2 that appears in (15.4-6). Gain Switching Gain switching is accomplished by turning the pumping rate R on and off; this in turn is equivalent to modulating the small-signal population difference No == 2Rt sp - N a . A schematic illustration of the typical time evolution of the population difference N (t) and the photon-number density n(t), as the laser is pulsed by varying No, is provided in Fig. 15.4-5. The following regimes are evident in the process: N Ob ----- No(t) Pump N(t) Population Loss N ------ - ---------------- t I I I N Oa I I I I I I N(t) o t} i I -£ n(t . ) P . hot . on-number density Tp I I (NOb-Nt)'it -- T sp I I I I I I I I I I I I I I t 2 . t \ . t Figure 15.4-5 Variation of the population difference N(t) and the photon-number density n(t) with time, as a square pump results in No suddenly increasing from a low value N oa to a high value Nab, and then decreasing back to a low value N oa .
15.4 PULSED LASERS 611 . For t < 0, the population difference N(t) == N oa lies below the threshold Nt and oscillation cannot occur. . The pump is turned on at t == 0, which increases No from a value N oa below threshold to a value N Ob above threshold in step-function fashion. The population difference N(t) begins to increase as a result. As long as N(t) < Nt, however, the photon-number density n == O. In this region (15.4-6) therefore becomes dN/dt == (No - N)/t sp , indicating that N(t) grows exponentially toward its equilibrium value N Ob with time constant t sp . . Once N(t) crosses the threshold Nt, at t == tI, laser oscillation begins and n(t) increases. The population inversion then begins to deplete so that the rate of increase of N(t) slows. As n(t) becomes larger, the depletion becomes more effective so that N(t) begins to decay toward Nt. N(t) finally reaches Nt, at which time n( t) reaches its steady-state value. . The pump is turned off at time t == t2, which reduces No to its initial value N oa . N(t) and n(t) decay to the values N Oa and 0, respectively. The actual profile of the buildup and decay of n(t) is obtained by numerically solving (15.4-3) and (15.4-6). The precise shape of the solution depends on t sp , Tp, N b as well as on N Oa and N Ob (see Probe 15.4-4). *c. Q-Switching Q-switched laser pulsing is achieved by switching the resonator loss coefficient a r from a large value during the off-time to a small value during the on-time. This may be accomplished in any number of ways, such as by placing a modulator that periodically introduces large losses in the resonator. Since the lasing threshold population difference Nt is proportional to the resonator loss coefficient a r [see (15.1-14) and (15.1-6)], the result of switching a r is to decrease Nt from a high value N ta to a low value N tb , as illustrated in Fig. 15.4-6. In Q-switching, therefore, Nt is modulated while No remains fixed, whereas in gain switching No is modulated while Nt remains fixed (see Fig. 15.4-5). The population and photon-number densities behave as follows: N ta ,-------, I I I I ,------ Nt I I I I _J Nj t; : t j -r 1\/ n(t) D Loss No Pump N tb N(t) opultion InVerSIOn o }\2 t n(t) Photon- number density )0 t Figure 15.4-6 Operation of a Q-switched laser. Variation of the population threshold Nt (which is proportional to the resonator loss), the pump parameter No, the population difference N(t), and the photon number n(t). . At t == 0, the pump is turned on so that No follows a step function. The loss is maintained at a level that is sufficiently high (Nt == N ta > No) so that laser oscillation cannot begin. The population difference N ( t) therefore builds up (with time constant t sp ). Although the medium is now a high-gain amplifier, the loss is sufficiently large so that oscillation is prevented.
612 CHAPTER15 LASERS . At t == tl, the loss is suddenly decreased so that Nt diminishes to a value N tb < No. Oscillation therefore begins and the photon-number density rises sharply. The presence of the radiation causes a depletion of the population inversion (gain saturation) so that N(t) begins to decrease. When N(t) falls below N tb , the loss again exceeds that gain, resulting in a rapid decrease of the photon-number density (with a time constant of the order of the photon lifetime T p ). . At t == t2, the loss is reinstated, insuring the availability of a long period of population-inversion buildup to prepare for the next pulse. The process is repeated periodically so that a periodic optical pulse train is generated. We now undertake an analysis to determine the peak power, energy, duration, and shape of the optical pulse generated by a Q-switched laser in the steady pulsed state. We rely on the two basic rate equations (15.4-3) and (15.4-6) forn(t) and N(t), respec- tively, which we solve during the on-time ti to tf indicated in Fig. 15.4-6. The problem can, of course, be solved numerically. However, it simplifies sufficiently to permit an analytical solution if we assume that the first two terms of (15.4-6) are negligible. This assumption is suitable if both the pumping and the spontaneous emission are negligible in comparison with the effects of induced transitions during the short time interval from ti to t f. This approximation turns out to be reasonable if the duration of the generated optical pulse is much shorter than t sp . When this is the case, (15.4-3) and (15.4-6) become dn ( !i _ 1 ) n dt Nt Tp dN N n - == -2- -. dt Nt Tp (15.4-7) (15.4-8) These are two coupled differential equations in n( t) and N (t) with initial conditions n == 0 and N == N i at t == ti. Throughout the time interval from ti to tf, Nt is fixed at its low value N tb . Dividing (15.4-7) by (15.4-8), we obtain a single differential equation relating n and N, dn ( Nt _ 1 ) dN 2 N ' (15.4-9) which we integrate to obtain n Nt In(N) - N + constant. (15.4-10) Using the initial condition n == 0 when N == N i finally leads to 1 N 1 n -N t In- - - ( N - N. ) 2 N i 2 '1, . ( 15 .4-11 )
15.4 PULSED LASERS 613 EXAMPLE 15.4-1. Q-Switched Neodymium-Doped-YAG Microchip Laser. A slice of Nd 3 +: YAG is brought together with a saturable absorber and an intracavity frequency-doubling crystal to form a I-mm-Iong cavity. When pumped with 1 W of light from a fiber-coupled 808- nm laser diode, this microchip laser generates Q-switched optical pulses at 532-nm. Each pulse has an energy of 30 j1J and a duration of 250 ps. The repetition rate is 10 kHz and the average power is 300 mW. Pulse Power According to (15.2-10) and (15.2-3), the internal photon-flux density (comprising both directions) is given by rp == nc, whereas the external photon-flux density emerging from mirror I (which has transmittance 'J) is CPo == 'Jnc. Assuming that the photon- flux density is uniform over the cross-sectional area A of the emerging beam, the corresponding optical output power is 1 C Po == hv A CPo == "2 hv c 'J An == hv 'J 2 d V n , where V == Ad is the volume of the resonator. According to (15.2-17), if'J « 1, the fraction of the resonator loss that contributes to useful light at the output is Il e 'J( c/2d )Tp, so that we obtain (15.4-12) nV Po==Ile hv -. Tp (15.4-13) Equation (15.4-13) is easily interpreted since the factor n V /Tp is the number of pho- tons lost from the resonator per unit time. Peak Pulse Power As discussed earlier and illustrated in Fig. 15.4-6, n reaches its peak value np when N == Nt == N tb . This is corroborated by setting dn/ dt == 0 in (15.4-7), which leads immediately to N == Nt. Substituting this into (15.4-11) therefore provides 1 ( Nt Nt Nt ) np == "2 Ni 1 + N. In N. - N. . z z z (15.4-14 ) Using this result in conjunction with (15.4-12) gives the peak power c P p == hv'J 2d Vn p . When N i » Nt, as must be the case for pulses of large peak power, N t / N i « 1, whereupon (15.4-14) gives (15.4-15) npNi. (15.4-16) The peak photon-number density is then equal to one-half the initial population density difference. In this case, the peak power assumes the particularly simple form 1 C P "2 hv 'J 2d VN i . (15.4-17) Peak Pulse Power
614 CHAPTER15 LASERS Pulse Energy The pulse energy is given by l t f E == Podt, ti (15.4-18) which, in accordance with (15.4-12), can be written as C {t f C (N f dt E = hv'J 2d V 1t; n(t) dt = hv'J 2d V 1N; n(t) dN dN. (15.4-19) Inserting (15.4-8) in (15.4-19), we obtain C {N i dN E = !hv'J 2d VN t Tp 1N! N ' (15.4-20) which integrates to 1 C N i E = 2 hv 'J 2d VN t Tp In N f . (15.4-21) The final population difference N f is determined by setting n == a and N == N f in (15.4-11), which provides N i Ni-N f In N f = Nt . (15.4-22) Substituting this into (15.4-21) gives 1 C E == "2 hv 'J 2d VT p (N i - N f ). (15.4-23) Q-Switched Pulse Energy When N i » N f, E hv 'J( c/2d) V Tp N i , as expected. It remains to solve (15.4- 22) for N f . One approach is to rewrite it in the form Y exp( -Y) == X exp( -X), where X == Nil Nt and Y == Nfl Nt. Given X == Nil Nt, we can easily solve for Y numerically or by using the graph provided in Fig. 15.4-7. Xe- X Xl X 2 Y 2 Y I Figure 15.4-7 Graphical construction for determining N f from N i , where X == Nil Nt and Y == NfiN t . For X == Xl the ordinate represents the value X I exp ( - X I). Since the corresponding solution Yi obeys Y I exp( -Y I ) == Xl exp( -Xl), it must have the same value of the ordinate. X
15.4 PULSED LASERS 615 Pulse Duration A rough estimate of the pulse duration is the ratio of the pulse energy to the peak pulse power. Using (15.4-14), (15.4-15), and (15.4-23), we obtain Nil Nt - Nfl Nt Tpul se == Tp Nil Nt - In( Nil Nt) - 1 . (15.4-24) Pulse Duration When N i » Nt and N i » N f, we have Tpul se Tp. Pulse Shape The optical pulse shape, along with all of the pulse characteristics described above, can be determined by numerically integrating (15.4-7) and (15.4-8). Examples of the resulting pulse shapes are shown in Fig. 15.4-8. n(t) Nt 3 1 2 o 2 4 6 t Tp Figure 15.4-8 Typical Q-switched pulse shapes obtained from numerical integration of the approximate rate equa- tions. The photon-number density n( t) is normalized to the threshold popula- tion difference Nt = N tb and the time t is normalized to the photon lifetime Tp. The pulse narrows and achieves a higher peak value as the ratio Nil Nt increases. In the limit Nil Nt » 1, the peak value of n(t) approaches N i . EXERCISE 15.4-2 Pulsed Ruby Laser. Consider the ruby laser discussed in Exercise 15.1-1. If the laser is now Q-switched so that at the end of the pumping cycle (at t = t i in Fig. 15.4-6) the population difference N i = 6Nt, use Fig. 15.4-8 to estimate the shape of the laser pulse, its duration, peak power, and total energy. D. Mode Locking A laser can oscillate on many longitudinal modes, with frequencies that are equally separated by the Fabry-Perot intermodal spacing Vp == c/2d. Although these modes normally oscillate independently (they are then called free-running modes), external means can be used to couple them and lock their phases together. The modes can then
616 CHAPTER15 LASERS be regarded as the components of a Fourier-series expansion of a periodic function of time of period T p == 1 I Vp == 2 d I c, which constitute a periodic pulse train. This is the approach taken in Sec. 2.6B, where we considered the interference of AI monochro- matic waves with equal intensities and equally spaced frequencies. We discuss in turn the properties of a mode-locked pulse train and methods of achieving mode locking, and then provide several examples of mode-locked lasers. Properties of a Mode-Locked Pulse Train If each of the laser modes is approximated by a uniform plane wave propagating in the z direction with a velocity c == coin, we may write the total complex wavefunction of the field in the form of a sum: U(z, t) = L Aq exp [j21fv q (t - : )] , q (15.4-25) where V q == Va + qvp, q == 0, ::i:1, ::i:2, . . . (15.4-26) is the frequency of mode q, and Aq is its complex envelope. For convenience we assume that the q == 0 mode coincides with the central frequency Va of the atomic lineshape. The magnitudes IAql may be determined from knowledge of the spectral profile of the gain and the resonator loss (see Sec. 15.2B). Since the modes interact with different groups of atoms in an inhomogeneously broadened medium, their phases arg{ Aq} are random and statistically independent. Substituting (15.4-26) into (15.4-25) provides U(z, t) = A (t - : ) exp [j21fvo (t - : )] , (15.4-27) where the complex envelope A(t) is the function '"' ( jq 27rt ) A(t) = L.." Aq exp T p q (15.4-28) and T p = = 2d . Vp c (15.4-29) The complex envelope A(t) in (15.4-28) is a periodic function of the period T p , and A(t - zlc) is a periodic function of z of period cT p == 2d. If the magnitudes and phases of the complex coefficients Aq are properly chosen, A( t) may be made to take the form of periodic narrow pulses. Consider, for example, M modes (q == 0, ::i:1, . . . , ::i:S, so that M == 2S + 1), whose complex coefficients are all equal, Aq == A, q == 0, ::i:1, . . . , ::i:S. Then S ( jq 27rt ) S xS+1 _ x-s xS+ _ x-s- A(t) == A L exp == A L x q == A == A 1. _1. T p x-I X2 - X 2 q=-S q=-S (15.4-30)
15.4 PULSED LASERS 617 where x == exp(j21ftl T p ) (see Sec. 2.6B for more details). After a few algebraic manipulations, A(t) can be cast in the form A(t) = A sin(M1ftjT F ) . sin( 1ftl T p ) (15.4-31) The optical intensity is then given by I(t, z) == IA(t - zlc)12 or I(t, z) = IAI2 sin 2 [M1f(t - zjc)jT F ] , sin 2 [1f (t - z I c) I T p ] (15.4-32) which, as illustrated in Fig. 15.4-9, is a periodic function of time. - MI I I( T F ) I Intensity - 1-- t Figure 15.4-9 Intensity of the periodic pulse train resulting from the sum of Af laser modes of equal magnitudes and phases. Each pulse has a duration that is 1\11 times smaller than the period T p and a peak intensity that is AI times greater than the mean intensity. The shape of the mode-locked laser pulse train is therefore dependent on the number of modes AI, which is proportional to the atomic linewidth v. If A[ vlvp, then Tpul se == TplM II v. The pulse duration Tpul se is therefore inversely proportional to the atomic linewidth v. Because v can be quite large, very narrow mode-locked laser pulses can generated. The ratio between the peak and mean intensities is equal to the number of modes it[, which can also be quite large. The period of the pulse train is T p == 2 d I c. This is just the time for a single round trip of reflection within the resonator. Indeed, the light in a mode-locked laser can be regarded as a single narrow pulse of photons reflecting back and forth between the mirrors of the resonator (see Fig. 15.4-10). At each reflection from the output mirror, a fraction of the photons is transmitted in the fonn of a pulse of light. The transmitted pulses are separated by the distance c(2dlc) == 2d and have a spatial width dpul se == CTpul se == 2d1M. A summary of the properties of the mode-locked laser pulse train is provided in Table 15.4-1. Table 15.4-1 Characteristic properties of a mode-locked pulse train. T p == 2d T p 1 Temporal period Pulse duration T pulse == 1\11 == v c 2d 2d Spatial period Pulse length dpulse == AI Mean intensity I Peak intensity I p == fYf I
618 CHAPTER15 LASERS d pulse I Z : ! j. I 2d I -I I . d -I Figure 15.4-10 The mode-locked laser pulse reflects back and forth between the mirrors of the resonator. Each time it reaches the output mirror it transmits a short optical pulse. The transmitted pulses are separated by the distance 2d and travel with velocity c. The switch opens only when the pulse reaches it and only for the duration of the pulse. The periodic pulse train is therefore unaffected by the presence of the switch. Other wave patterns, however, suffer losses and are not permitted to oscillate. As a particular example, we consider a Nd 3 +:glass laser operating at Ao == 1.05 Mm (see Table 14.3-1). It has a refractive index n == 1.5 and a linewidth t:::..v == 7 THz. Thus, the pulse duration Tpul se == 1/ t:::..v 140 fs and the pulse length dpulse 42 Mm. If the resonator has a length d == 15 em, the mode separation is Vp == c/2d == 1 GHz, which means that M == t:::..v / Vp == 7000 modes. The peak intensity is therefore 7000 times greater than the average intensity. In media with broad linewidths, mode locking is generally more advantageous than Q-switching for obtaining short pulses. Gas lasers generally have narrow atomic linewidths, on the other hand, so that ultrashort pulses cannot be obtained by mode locking. Although the formulas provided above were derived for the special case in which the modes have equal amplitudes and phases, calculations based on more realistic behavior provide similar results. EXERCISE 15.4-3 Demonstration of Pulsing by Mode Locking. Write a computer program to plot the intensity I(t) = IA(t)/2 of a wave whose envelope A(t) is given by the sum in (15.4-28). Assume that the number of modes AI = 11 and use the following choices for the complex coefficients Aq: (a) Equal magnitudes and the same phase (this should reproduce the results of the foregoing exam- pie) . (b) Magnitudes that obey the Gaussian spectral profile I Aq I = exp [ - ( q /5) 2 ] and the same phase. (c) Equal magnitudes but with random phases (obtain the phases by using a random number gener- ator to produce a random variable uniformly distributed between 0 and 21T). Methods of Mode Locking We have found thus far that if a large number M of modes are locked in phase, they form a giant narrow pulse of photons that reflects back and forth between the mirrors of the resonator. The spatial length of the pulse is a factor of M smaller than twice the resonator length. The question that remains is how the modes can be locked together so that they have the same phase. This can be accomplished with the help of an active or passive modulator (switch) placed inside the resonator. We consider active mode locking and passive mode locking in turn. Suppose that an optical switch controlled by an external applied signal (e.g., an acousto-optic or electro-optic switch, as discussed in Chapters 19 and 20) is placed inside the resonator, which blocks the light at all times, except when the pulse is about to cross it, whereupon it opens for the duration of the pulse (Fig. 15.4-10). Since the pulse itself is permitted to pass, it is not affected by the presence of the switch and
15.4 PULSED LASERS 619 the pulse train continues uninterrupted. In the absence of phase locking, the individual modes have different phases that are determined by the random conditions at the onset of their oscillation. If the phases happen, by accident, to take on equal values, the sum of the modes will form a giant pulse that would not be affected by the presence of the switch. Any other combination of phases would form a field distribution that is totally or partially blocked by the switch, which adds to the losses of the system. Therefore, in the presence of the switch, only when the modes have equal phases can lasing occur. The laser waits for the "lucky accident" of such phases, but once the oscillations start, they continue to be locked. The problem can also be examined mathematically. An optical field must satisfy the wave equation with the boundary conditions imposed by the presence of the switch. The multimode optical field of (15.4-25) does indeed satisfy the wave equation for any combination of phases. The case of equal phases also satisfies the boundary conditions imposed by the switch; therefore, it must be a unique solution. A passive switch such as a saturable absorber may also be used to achieve mode locking. A saturable absorber (see Sec. 14.4A) is a medium whose absorption co- efficient decreases as the intensity of the light passing through it increases; it thus transmits intense pulses with relatively little absorption while absorbing weak ones. Oscillation can therefore occur only when the phases of the different modes are related to each other in such a way that they form an intense pulse that can then pass through the switch. Semiconductor saturable-absorber mirrors (SESAMs), which are saturable absorbers operating in reflection, are in widespread use; the more intense the light, the greater the reflection provided by these devices. SESAMs can accommodate wave- lengths in the range from 800 to 1600 nm, pulse durations from fs to ns, and power levels from m W to hundreds of W. Saturable absorbers can also produce Q-switched modelocking, in which the laser emits collections of modelocked pulses within a Q- switching envelope. Passive mode locking can also be implemented by means of Kerr-lens mode lock- ing, which relies on a nonlinear-optical phenomenon in which the refractive index of a material changes with optical intensity (see Sec. 21.3). A Kerr medium, such as the gain medium itself, or a material placed within the laser cavity, acts as a lens with a focal length inversely proportional to the intensity (see Exercise 21.3-2). By placing an aperture at a judicious position within the cavity, the Kerr lens reduces the area of the laser mode for high intensities so that the light passes through the aperture. Alternatively, the reduced modal area in the gain medium can be used to increase its overlap with the strongly focused pump beam, thereby increasing the effective gain. The Kerr-lens approach is inherently broadband because of the parametric nature of the process. The rapid recovery inherent in passive mode locking generally leads to shorter optical pulses than can be achieved with active mode locking. Passive and active switches are used for the mode locking of inhomogeneously and homogeneously broadened media alike. Examples of Mode-Locked Lasers Table 15.4-2 provides a list, in order of increasing observed pulse duration, of pulse durations available using various mode-locked laser media. A broad range of pulse durations is represented. The observed pulse durations, which for a given medium can vary greatly, depend on the method used to achieve mode locking and can be limited by nonlinearities and dispersion in the medium. With the ability to tune the center wavelength over the range 700-1050 nm, and with individual pulses as short as 10- fs duration, the mode-locked laser of choice is often Ti:sapphire. A commercial version of this laser readily delivers 50-nJ pulses of duration 10 fs and peak power 1 MW, at a repetition rate of 80 MHz. Substantially better performance is available in the laboratory. Further reductions in pulse duration
620 CHAPTER15 LASERS can be achieved by using pulse-compression techniques (see Sec. 22.2A). It should also be mentioned, perhaps, that the spectral bandwidth of this laser, v, can be easily constrained to provide ps-duration mode-locked pulses. Aside from their importance in photonics research, mode-locked lasers find use in many applications, including time- resolved measurements, imaging, metrology, communications, materials processing, and clinical medicine. Table 15.4-2 Typical pulse durations for a number of mode-locked lasers subject to homogeneous (H) and inhomogeneous (I) broadening. Calculated Laser Transition Pulse Duration Observed Medium Linewidth a 6.v Tpul se == 1/6.v Pulse Duration Ti 3 +:Ab 0 3 H 100 THz 10 fs 10 fs Rhodamine-6G dye H/I 40 THz 25 fs 27 fs Nd 3 + :Glass (phosphate) I 7THz 140 fs 150 fs Er 3 + :Silica fiber HII 5THz 200 fs 200 fs Nd 3 +:YAG H 150 GHz 7 ps 7 ps Ar+ I 3.5 GHz 286 ps 150 ps He-Ne I 1.5 GHz 667 ps 600 ps CO 2 I 60 MHz 16 ns 20 ns aThe transition linewidths D"v are drawn from Table 14.3-1. EXAMPLE 15.4-2. Mode-Locking in an Ytterbium-Doped Fiber Laser. A passively mode-locked ytterbium-doped silica fiber laser operated at Ao == 1070 nm produces an average power of 10 W in the form of pulses with energy 200 nJ and peak power 40 k W. This laser generates mode-locked pulses that are 5 ps in duration at a repetition rate of 50 MHz. Since 6.v == 5 THz, the pulse duration is substantially greater than the expected value, Tpul se == 1/6.v == 200 fs. The discrepancy arises because of group velocity dispersion, which imparts broadening and chirping to a pulse traveling through an optical medium (see Fig. 5.6-3). The normal dispersion in a silica fiber near Ao == 1 J-Lm (Fig. 5.6-5) can be canceled by introducing anomalous dispersion via a fiber Bragg grating or a photonic-crystal fiber, reducing the observed pulse duration to 200 fs. The power level available from a mode-locked Ti:sapphire laser is sufficient to allow harmonic generation and other nonlinear wavelength-shifting techniques to be em- ployed (see Chapter 21), thereby providing a source of mode-locked pulses at shorter wavelengths. In particular, second-harmonic generation produces pulses in the range 350-525 nm whereas third-hannonic generation reaches the range 230-350 nm. Mode- locked operation can be extended beyond Ao == 1 Mm in the infrared by making use of Raman fiber lasers or the nonlinear-optical process of parametric downconversion (see Sec. 21.4C). The Ti:sapphire mode-locked laser oscillator serves as a source for a synchronously pumped optical parametric oscillator that employs a nonlinear optical crystal such as LBO or a periodically poled crystal. This generates modelocked signal and idler outputs that offer wavelength coverage in the range 1.0-3.3 Mm.
READING LIST 621 READING LIST Books on Lasers See also the reading list in Chapter 14 and the books on optoelectronics in Chapter 17. W. Koechner, Solid-State Laser Engineering, Springer-Verlag, 6th ed. 2006. R. S. Quimby, Photonics and Lasers: An Introduction, Wiley, 2006. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, Oxford University Press, 6th ed. 2006. Y. B. Band, Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers, Wiley, 2006. M. A. Noginov, Solid-State Random Lasers, Springer-Verlag, 2005. C. Rulliere, ed., Femtosecond Laser Pulses: Principles and Experiments, Springer, 1998, 2nd ed. 2005. W. T. Silfvast, Laser Fundamentals, Cambridge University Press, 2nd ed. 2004. O. Svelto, Principles of Lasers, Springer-Verlag, 4th ed. 2004. D. Meschede, Optics, Light and Lasers: The Practical Approach to Modem Aspects of Photonics and Laser Physics, Wiley-VCH, 2004. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The Physics of Free Electron Lasers, Springer- Verlag, 2000. F. Ciocci, G. Dattoli, A. Torre, and A. Renieri, Insertion Devices for Synchrotron Radiation and Free Electron Laser, World Scientific, 2000. M. 1. Weber, Handbook of Laser Wavelengths, CRC Press, 1999. A. A. Kaminskii, Crystalline Lasers: Physical Processes and Operating Schemes, CRC Press, 1996. J. T. Verdeyen, Laser Electronics, Prentice Hall, 3rd ed. 1995. F. J. Duarte, ed., Tunable Lasers Handbook, Academic Press, 1995. H. Yokoyama and K. Ujihara, eds., Spontaneous Emission and Laser Oscillation in Microcavities, CRC Press, 1995. M. Elitzur, Astronomical Masers, Kluwer, 1992. F. P. Schafer, ed., Dye Lasers, Springer-Verlag, 3rd ed. 1990. N. G. Basov, A. S. Bashkin, V. I. Igoshin, A. N. Oraevsky, and A. A. Shcheglov, Chemical Lasers, Springer-Verlag, 1990. P. Luchini and H. Motz, Undulators and Free-electron Lasers, Oxford University Press, 1990. P. W. Milonni and J. H. Eberly, Lasers, Wiley, 1988. P. K. Cheo, ed., Handbook of Molecular Lasers, Marcel Dekker, 1987. A. E. Siegman, Lasers, University Science, 1986. K. Shimoda, Introduction to Laser Physics, Springer-Verlag, 2nd ed. 1986. C. K. Rhodes, ed., Excimer Lasers, Springer-Verlag, 2nd ed. 1984. Books and Articles on UV and X-Ray Lasers P. Jaegle, Coherent Sources of xu V Radiation: Soft X-Ray Lasers and High-Order Harmonic Gener- ation, Springer-Verlag, 2006. J. J. Rocca, H. C. Kapteyn, D. T. Atwood, M. M. Murnane, C. S. Menoni, and E. H. Anderson, Tabletop Lasers in the Extreme Ultraviolet, Optics & Photonics News, vol. 17, no. 11, pp. 30-38, 2006. D. M. Paganin, Coherent X-Ray Optics, Oxford University Press, 2006. Feature section on extreme ultraviolet coherent sources and applications, IEEE Journal of Quantum Electronics, vol. 42, no. 1, 2006. A. Weith, M. A. Larotonda, Y. Wang, B. M. Luther, D. Alessi, M. C. Marconi, J. J. Rocca, and J. Dunn, Continuous High-Repetition-Rate Operation of Collisional Soft-X-Ray Lasers With Solid Targets, Optics Letters, vol. 31, pp. 1994-1996, 2006. J. Zhang, ed., X-Ray Lasers 2004 (Institute of Physics Conference Series), Taylor & Francis, 2005. H. C. Kapteyn, M. M. Murnane, and I. P. Christov, Extreme Nonlinear Optics: Coherent X Rays from Lasers, Physics Today, vol. 58, no. 3, pp. 39-44, 2005. Y Shvyd'ko, X-Ray Optics: High-Energy-Resolution Applications, Springer-Verlag, 2004.
622 CHAPTER15 LASERS Issue on short wavelength and EUV lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 6, 2004. J. J. Rocca, J. Dunn, and S. Suckewer, eds., X-Ray Lasers 2002 (AlP Conference Proceedings 641), American Institute of Physics, 2002. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications, Cambridge University Press, 1999. I. C. E. Turcu and J. B. Dance, X-Rays From Laser Plasmas: Generation and Applications, Wiley, 1999. Issue on short wavelength lasers and applications, IEEE Journal of Selected Topics in Quantum Electronics, vol. 5, no. 6, 1999. P. Rullhusen, X. Artru, and P. Dhez, Novel Radiation Sources Using Relativistic Electrons: From Infrared to X-Rays, World Scientific, 1998. W. W. Duley, UV Lasers: Effects and Applications in Materials Science, Cambridge University Press, 1996. R. W. Waynant and M. N. Ediger, eds., Selected Papers on U VU and X-Ray Lasers, SPIE Optical Engineering Press (Milestone Series Volume 71), 1993. R. C. Elton, X-Ray Lasers, Academic Press, 1990. D. L. Matthews and M. D. Rosen, Soft X-Ray Lasers, Scientific American, vol. 259, no. 6, pp. 86-91, 1988. D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N. M. Ceglio, A. U. Hazi, H. Medecki, B. J. MacGowan, J. E. Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M. Hawryluk, R. L. Kauffman, L. D. Pleasance, G. Rambach, J. H. Scofield, G. Stone, and T. A. Weaver, Demonstration of a Soft X-Ray Amplifier, Physical Review Letters, vol. 54, pp. 110-113, 1985. S. Suckewer, C. H. Skinner, H. Milchberg, C. Keane, and D. Voorhees, Amplification of Stimulated Soft X-Ray Emission in a Confined Plasma Column, Physical Review Letters, vol. 55, pp. 1753- 1756, 1985. Articles Feature issue on fiber lasers, Journal of the Optical Society of America B, vol. 24, no. 8, 2007. A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, Acoustic Phonon Emission from a Weakly Coupled Superlattice Under Vertical Electron Trans- port: Observation of Phonon Resonance, Physical Review Letters, vol. 96, 215504, 2006. H. Cao, Review on Latest Developments in Random Lasers with Coherent Feedback, Journal of Physics A: Mathematical and General, vol. 38, pp. 10497-10535, 2005. H. Cao, Random Lasers: Development, Features and Applications, Optics & Photonics News, vol. 16, no. l,pp. 24-29, 2005. Issue on solid-state lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 11, no. 3, 2005. M. Claussen, Astronomical Masers, Science, vol. 306, pp. 235-236, 2004. R. L. Walsworth, The Maser at 50, Science, vol. 306, pp. 236-237, 2004. U. Keller, Recent Developments in Compact Ultrafast Lasers, Nature, vol. 424, pp. 831-838, 2003. V. V. Ter-Mikirtychev, ed., Selected Papers on Tunable Solid-State Lasers, SPIE Optical Engineering Press (Milestone Series Volume 173), 2002. W. B. Colson, E. D. Johnson, M. J. Kelley, and H. A. Schwettman, Putting Free-Electron Lasers to Work, Physics Today, vol. 55, no. 1, pp. 35-41, 2002. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. N. S. Kim, M. Prabhu, C. Li, J. Song, and K-i. Ueda, 1239/1484 nm Cascaded Phosphosilicate Raman Fiber Laser with CW Output Power of 1.36 W at 1484 nm Pumped by CW Vb-Doped Double-Clad Fiber Laser at 1064 nm and Spectral Continuum Generation, Optics Communica- tions, vol. 176, pp. 219-222, 2000. M. S. Feld and K. An, The Single-Atom Laser, Scientific American, vol. 279, no. 1, pp. 56-63, 1998. M. Elitzur, Masers in the Sky, Scientific American, vol. 272, no. 2, pp. 68-74, 1995. M. D. Perry and G. Mourou, Terawatt to Petawatt Subpicosecond Lasers, Science, vol. 264, pp. 917- 924, 1994.
READING LIST 623 W. T. Silfvast, ed., Selected Papers on Fundamentals of Lasers, SPIE Optical Engineering Press (Milestone Series Volume 70), 1993. M. J. Mumma, D. Buhl, G. Chin, D. Deming, F. Espenak, and T. Kostiuk, Discovery of Natural Gain Amplification in the 10 Mill CO 2 Laser Bands on Mars: A Natural Laser, Science, vol. 212, pp. 45-49, 1981. Historical J. Hecht, Beam: The Race to Make the Laser, Oxford University Press, 2005. M. Bertolotti, The History of the Laser, Taylor & Francis, 2004. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. J. Hecht, ed., Laser Pioneer Interviews, High Tech Publications, 1985. A. Kastler, Birth of the Maser and Laser, Nature, vol. 316, pp. 307-309, 1985. Special issue: "Twenty-Five Years of the Laser," Optica Acta (Journal of Modern Optics), vol. 32, no. 9/10, 1985. Centennial issue, IEEE Journal of Quantum Electronics, vol. QE-20, no. 6, 1984. C. H. Townes, Science, Technology, and Invention: Their Progress and Interactions, Proceedings of the National Academy of Sciences (USA), vol. 80, pp. 7679-7683, 1983. A. L. Schawlow, Spectroscopy in a New Light (Nobel lecture), Reviews of Modern Physics, vol. 54, pp. 697-707, 1982. D. 0' Shea and D. C. Peckham, Lasers: Selected Reprints, American Association of Physics Teachers, 1982. D. C. 0' Shea and D. C. Peckham, Resource Letter L-1: Lasers, American Journal of Physics, vol. 49, pp. 915-925, 1981. A. L. Schawlow, Maser and Laser, IEEE Transactions on Electron Devices, vol. ED-23, pp. 773-779, 1976. A. L. Schawlow, From Maser to Laser, in Impact of Basic Research on Technology, B. Kursunoglu and A. Perlmutter, eds., Plenum, 1973. w. E. Lamb, Jr., Physical Concepts in the Development of the Maser and Laser, in Impact of Basic Research on Technology, B. Kursunoglu and A. Perlmutter, eds., Plenum, 1973. A. Kastler, Optical Methods for Studying Hertzian Resonances, in Nobel Lectures in Physics, 1963- 1970, Elsevier, 1972. C. H. Townes, Production of Coherent Radiation by Atoms and Molecules, in Nobel Lectures in Physics, 1963-1970, Elsevier, 1972. N. G. Basov, Semiconductor Lasers, in Nobel Lectures in Physics, 1963-1970, Elsevier, 1972. A. M. Prokhorov, Quantum Electronics, in Nobel Lectures in Physics, 1963-1970, Elsevier, 1972. F. S. Barnes, ed., Laser Theory, IEEE Press Reprint Series, IEEE Press, 1972. R. H. Stolen, E. P. Ippen, and A. R. Tynes, Raman Oscillation in Glass Optical Waveguide, Applied Physics Letters, vol. 20, pp. 62-65, 1972. A. L. Schawlow, ed., Lasers and Light-Readings from Scientific American, Freeman, 1969. C. H. Townes, Quantum Electronics and Surprise in the Development of Technology, Science, vol. 159, pp. 699-703, 1968. v. S. Letokhov, Generation of Light by a Scattering Medium with Negative Resonance Absorption, Soviet Physics-JETP, vol. 26, pp. 835-840, 1968 [Zhurnal Eksperimental 'noi i Teoreticheskoi Fiziki, vol. 53, pp. 1442-1452, 1967]. J. Weber, ed., Lasers: Selected Reprints with Editorial Comment, Gordon and Breach, 1967. B. A. Lengyel, Evolution of Masers and Lasers, American Journal of Physics, vol. 34, pp. 903-913, 1966. C. Cohen- Tannoudji and A. Kastler, Optical Pumping, in Progress in Optics, vol. 5, E. Wolf, ed., North-Holland, 1966. W. E. Lamb, Jr., Theory of an Optical Maser, Physical Review, vol. 134, pp. A1429-A1450, 1964. A. Yariv and J. P. Gordon, The Laser, Proceedings of the IEEE, vol. 51, pp. 4-29,1963. T. H. Maiman, Stimulated Optical Radiation in Ruby, Nature, vol. 187, pp. 493-494, 1960. A. L. Schawlow and C. H. Townes, Infrared and Optical Masers, Physical Review, vol. 112, pp. 1940- 1949, 1958.
624 CHAPTER15 LASERS R. H. Dicke, Molecular Amplification and Generation Systems and Methods, U.S. Patent 2,851,652, September 9, 1958. J. P. Gordon, H. J. Zeiger, and C. H. Townes, The Maser-New Type of Microwave Amplifier, Frequency Standard, and Spectrometer, Physical Review, vol. 99, pp. 1264-1274, 1955. N. G. Basov and A. M. Prokhorov, Possible Methods of Obtaining Active Molecules for a Molec- ular Oscillator, Soviet Physics-JETP, vol. 1, pp. 184-185, 1955 [Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, vol. 28, pp. 249-250, 1955]. V. A. Fabrikant, The Emission Mechanism of Gas Discharges, Trudi Vsyesoyuznogo Elektrotekhnich- eskogo lnstituta (Reports of the All-Union Electrotechnical Institute, Moscow), vol. 41, Elektron- nie i lonnie Pribori (Electron and Ion Devices), pp. 236-296, 1940. PROBLEMS 15.2-2 Number of Longitudinal Modes. An Ar+ -ion laser has a resonator of length 100 cm. The refractive index n = 1. (a) Determine the frequency spacing Vp between the resonator modes. (b) Determine the number of longitudinal modes that the laser can sustain if the FWHM Doppler-broadened linewidth is 6.VD = 3.5 GHz and the loss coefficient is half the peak small-signal gain coefficient. (c) What would the resonator length d have to be to achieve operation on a single longitu- dinal mode? What would that length be for a CO 2 laser that has a much smaller Doppler linewidth !J.VD = 60 MHz under the same conditions? 15.2-3 Frequency Drift of the Laser Modes. A He-Ne laser has the following characteristics: (1) A resonator with 97% and 100% mirror reflectances and negligible internal losses; (2) a Doppler-broadened atomic transition with Doppler linewidth 6.VD = 1.5 GHz; and (3) a small-signal peak gain coefficient 'ro(vo) = 2.5 X 10- 3 em-I. While the laser is running, the frequencies of its longitudinal modes drift with time as a result of small thermally induced changes in the length of the resonator. Find the allowable range of resonator lengths such that the laser will always oscillate in one or two (but not more) longitudinal modes. The refractive index n = 1. 15.2-4 Mode Control Using an Etalon. A Doppler-broadened gas laser operates at 515 nm in a resonator with two mirrors separated by a distance of 50 cm. The photon lifetime is 0.33 ns. The spectral window within which oscillation can occur is of width B = 1.5 GHz. The refractive index n = 1. To select a single mode, the light is passed into an etalon (a passive Fabry-Perot resonator) whose mirrors are separated by the distance d and its finesse is . The etalon acts as a filter. Suggest suitable values of d and :J'. Is it better to place the etalon inside or outside the laser resonator? 15.2-5 Modal Powers in a Multimode Laser. A He-Ne laser operating at Ao = 632.8 nm produces 50 mW of multimode power at its output. It has an inhomogeneously broadened gain profile with a Doppler linewidth 6.VD = 1.5 G Hz and the refractive index n = 1. The resonator is 30 cm long. (a) If the maximum small-signal gain coefficient is twice the loss coefficient, determine the number of longitudinal modes of the laser. (b) If the mirrors are adjusted to maximize the intensity of the strongest mode, estimate its power. 15.2-6 Output of a Single-Mode Gas Laser. Consider a 10-cm-long gas laser operating at the center of the 600-nm line in a single longitudinal and single transverse mode. The mirror reflectances are 9(1 = 99% and 9(2 = 100%. The refractive index n = 1 and the effective area of the output beam is 1 mm 2 . The small-signal gain coefficient l'o(vo) = 0.1 em- 1 and the saturation photon-flux density CPs = 1.43 X 10 19 photonsjem 2 -so (a) Determine the distributed loss coefficients, amI and a m 2, associated with each of the mirrors separately. Assuming that as = 0, find the resonator loss coefficient a r . (b) Find the photon lifetime Tp. (c) Determine the output photon flux density CPo and the output power Po.
PROBLEMS 625 15.2- 7 Transmittance of a Laser Resonator. Monochromatic light from a tunable optical source is transmitted through the optical resonator of an unpumped gas laser. The observed trans- mittance, as a function of frequency, is shown in Fig. P 15 .2-7. 200MHz 5x 10 14 Hz Figure P15.2-7 Transmittance of a laser resonator. v (a) Determine the resonator length, the photon lifetime, and the threshold gain coefficient of the laser. Assume that the refractive index n = 1. (b) Assuming that the central frequency of the laser transition is 5 x 10 14 Hz, sketch the transmittance versus frequency if the laser is now pumped but the pumping is not sufficient for laser oscillation to occur. 15.2-8 Rate Equations in a Four-Level Laser. Consider a four-level laser with an active volume V = 1 em 3 . The population densities of the upper and lower laser levels are N 2 and N 1 and N = N 2 - N 1 . The pumping rate is such that the steady-state population difference N in the absence of the stimulated emission and absorption is No. The photon-number density is n and the photon lifetime is 7 p . Write the rate equations for N 2 , N 1 , N, and n in terms of No, the transition cross section a(v), and the times t sp , 71, 72, 721, and 7 p . Determine the steady-state values of Nand n. 15.3-1 Operation of an Ytterbium-Doped YAG Laser. Yb 3 +: YAG is a rare-earth-doped dielec- tric material that lases at Ao = 1.030 Mm on the 2F 5 / 2 2F 7 / 2 transition (see Tables 13.1-1, 14.3-1, 15.3-1, and Fig. 15.3-3). This three-level laser is usually optically pumped with an InGaAs laser diode. ( a) The pump band (level 3) has a central energy of 1. 31915 e V and a width of 0.02475 eV. Determine the wavelength of the desired laser-diode pump and the width of the absorption band in nm. (b) At the central frequency of the laser transition vo, the peak transition cross section ao a(vo) = 2 x 10- 20 cm 2 . Given that the Yb 3 + -ion doping density is set at N a = 1.4 X 10 20 em -3, determine the absorption and gain coefficients of the material at the center of the line, a(vo) -""'((vo). Assume that the material is in thermal equilibrium at T = 300 0 K (i.e., there is no pumping). (c) Consider a laser rod constructed from this material with a length of 6 cm and a diameter of 2 mm. One of its ends is polished to a reflectance of 80% (9(1 = 0.8) while the other is polished to unity reflectance (9(2 = 1.0). Assuming that there is no scattering, and that there are no other extraneous losses, determine the resonator loss coefficient a r and the resonator photon lifetime 7 p . (d) As the laser is pumped, the gain coefficient ""'((vo) increases from its initial negative value at thermal equilibrium and changes sign, thereby providing gain. Determine the threshold population difference Nt for laser oscillation. (e) Why is it advantageous to have the energy of level 3 close to that of level 2? (f) How might the operation of the laser change if yttrium vanadate (YV0 4 ) were substi- tuted for YAG (Y 3A15012) as the host materia]? 15.3-2 Threshold Population Difference for an Ar+ -Ion Laser. An Ar + -ion laser has a I-m-Iong resonator with 98% and 100% mirror reflectances. Other loss mechanisms are negligible. The atomic transition has a central wavelength Ao = 515 nm, spontaneous lifetime t sp = 10 ns, and linewidth A = 0.003 nm. The lower energy level has a very short lifetime and hence zero population. The diameter of the oscillating mode is 1 mm. Determine (a) the photon lifetime
626 CHAPTER15 LASERS (b) the threshold population difference for laser action. 15.3-3 Spontaneous Lifetime of an EUV Transition. A visible laser transition at Ao = 500 nm has a spontaneous lifetime t sp = 10 ns. Estimate the spontaneous lifetime for an EUV laser transition at Ao = 18.2 nm, assuming that the transition strength S is the same in both cases. Compare your result with that provided in Table 14.3-1. * 15.4-4 Transients in a Gain-Switched Laser. (a) Introduce the new variables X = n/Tp, Y = N / Nt, and the normalized time s = t/Tp, to demonstrate that the rate equations (J 5.4-3) and (15.4-6) take the form dX T - = -X + X} ds dY ds = a(Y o - Y) - 2XY, where a = Tp/t sp and Yo = No/Nt. (b) Write a computer program to solve these two equations for both switching on and switching off. Assume that Yo is switched from 0 to 2 to turn the laser on, and from 2 to 0 to turn it off. Assume further that an initially very small photon flux corresponding to X = 10- 5 starts the oscillation at t = O. Speculate on the possible origin of this flux. Determine the switching transient times for a = 10- 3 , 1, and 10 3 . Comment on the significance of your results. *15.4-5 Q-Switched Ruby Laser Power. A Q-switched ruby laser makes use of a 15-cm-long rod of cross-sectional area 1 cm 2 placed in a resonator of length 20 cm. The mirrors have reflectances 9(1 = 0.95 and 9( = 0.7. The Cr 3 + density is 1.58 x 10 19 atoms/cm 3 , and the transition cross section a(vo) = 2 x 10- 20 cm 2 . The laser is pumped to an initial population of 10 19 atoms/cm 3 in the upper state with negligible population in the lower state. The pump band (level 3) is centered at 450 nm and the decay from level 3 to level 2 is fast. The lifetime of level 2 is 3 ms. (a) How much pump power is required to maintain the population in level 2 at 10 19 cm- 3 ? (b) How much power is spontaneously radiated before the Q-switch is operated? (c) Determine the peak power, energy, and duration of the Q-switched pulse. * 15.4-6 Operation of a Cavity-Dumped Laser. Sketch the variation of the threshold population difference Nt (which is proportional to the loss), the population difference N(t), the internal photon number density n(t), and the external photon flux density cjJo(t), during two cycles of operation of a pulsed cavity-dumped laser. 15.4- 7 Mode Locking with Lorentzian Amplitudes. Assume that the envelopes of the modes of a mode-locked laser are A = JP (V/2)2 q (qvp)2 + (v /2)2 ' q = -00, . . . , 00, and the phases are equal. Determine expressions for the following parameters of the gener- ated pulse train: (a) Mean power (b) Peak power (c) Pulse duration (FWHM) 15.4-8 Second-Harmonic Generation. Crystals with nonlinear optical properties are often used for second-harmonic generation, as explained in Chapter 21. In this process, two photons of frequency v are converted into a single photon of frequency 2v. Assume that such a crystal is placed inside a laser resonator with an active medium providing gain at frequency v. The frequencies v and 2v correspond to two modes of the resonator. If the rate of second-harmonic conversion is (n (s-1_m- 3 ) and the rate of photon production by the laser process (net effect of stimulated emission and absorption) is n (S-1_ n1 -3), where ( and are constants, write the rate equations for the photon number densities nand n2 at the frequencies v and 2v. Assume that the photon lifetimes at v and 2v are Tp and T p 2, respectively. Determine the steady-state values of nand n2.
CHAPTER 16 SEMICONDUCTOR OPTICS 16.1 SEMICONDUCTORS 629 A. Energy Bands and Charge Carriers B. Semiconductor Materials C. Electron and Hole Concentrations D. Generation, Recombination, and Injection E. Junctions F. Heterojunctions G. Quantum-Confined Structures 16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 660 A. Photon Interactions in Bulk Semiconductors B. Band-to-Band Transitions in Bulk Semiconductors C. Absorption, Emission, and Gain in Bulk Semiconductors D. Photon Interactions in Quantum-Confined Structures E. Refractive Index o ". . . '. \' --/ ... '..1,. * ......... '.,"1 ;, '11 I .; ... J , ..,.: .... . .. 11 '\-- . :"'..,..,1 .' . l' ;...."\. "-' \ . " ' , """ ." William P. Shockley (1910-1989), seated, John Bardeen (1908-1991), center, and \Valter H. Brattain (1902-1987), right, shared the Nobel Prize in 1956 for demonstrating that semiconductor devices could be used to achieve amplification. 627
Photonics is the technology of controlling the flow of photons, much as electronics is the technology of controlling the flow of charge carriers (electrons and holes). These two technologies join together in semiconductor optoelectronics: photons generate mobile charge carriers, and charge carriers generate and control the flow of photons. Semiconductor optoelectronic devices serve as photon sources (light-emitting diodes and laser diodes), amplifiers, detectors, waveguides, modulators, sensors, and nonlin- ear optical elements. The compatibility of semiconductor optoelectronic devices and electronic devices has fostered the development of both fields. Semiconductor materials absorb and emit photons by undergoing transitions among allowed energy levels. Although the basic rules that govern these interactions are the same as those set forth in Sec. 13.3, semiconductors have a number of unique features (see Sec. 13.1 C): . A semiconductor material cannot be viewed as a collection of non interacting atoms, each with its own individual energy levels. Because of the proximity of the atoms in the crystal lattice, the energy levels belong to the system as a whole. . Collections of closely spaced energy levels form energy bands. In the absence of external excitation, at T == 0° K, these bands are either fully occupied by electrons or totally unoccupied. The lowest-lying unoccupied energy band is called the conduction band while the highest-lying fully occupied energy band is known as the valence band. These two bands are separated by a forbidden band, with bandgap energy Eg. . An external energy source (whether thermal, optical, or electronic) can impart energy to an electron in the valence band, causing it to cross the forbidden band and enter the conduction band. This transition leaves a vacancy (hole) behind in the valence band. In the inverse process, electron-hole recombination, an electron decays from the conduction band to fill an empty state in the valence band (pro- vided that one is accessible), generating a photon and/or phonons in the process. Thus, photons interact with both types of charge carriers, electrons and holes. Two processes are fundamental to the operation of almost all semiconductor opto- electronic devices: 1. The absorption of a photon can create an electron-hole pair. Mobile charge carriers resulting from the absorption of a photon alter the electrical properties of the semiconductor. This process is the basis of operation of photoconductive photodetectors. 2. The recombination of an electron and a hole can result in the emission of a photon. This process is responsible for the operation of semiconductor photon sources. Spontaneous radiative electron-hole recombination gives rise to photon generation in the light-emitting diode. Stimulated electron-hole recombination is responsible for the generation of photons in a laser diode. This Chapter The reader is expected to be familiar with the basic principles of semiconductor physics. In Sec. 16.1 we offer a review of semiconductors and their properties, particularly those that are important in semiconductor optoelectronics. Section 16.2 provides an introduction to the optical properties of semiconductors. A simplified theory of absorption, spontaneous emission, and stimulated emission, patterned on the theory of radiative atomic transitions developed in Sec. 13.3, is presented. This and the following two chapters are to be regarded as a unit. Chapter 17 deals with the operation of semiconductor sources such as the light-emitting diode and the 628
16.1 SEMICONDUCTORS 629 laser diode. Chapter 18 is devoted to semiconductor detectors. 16.1 SEMICONDUCTORS As discussed in Sec. 13.1C, a semiconductor is a crystalline or amorphous solid whose electrical conductivity is typically intermediate between that of a metal and that of an insulator. Its conductivity can be significantly altered by modifying the temperature or doping concentration of the material, or by illuminating it with light. The band structure of semiconductors, and the ability to form junctions and heterostructures, offer unique properties. Quantum-confined semiconductor structures further extend the range of available properties. Electronic semiconductor devices are principally fabricated from silicon (Si), while optoelectronic semiconductor devices often make use of ternary or quaternary semiconductor compounds such as InGaAsP and AllnGaN (see Sec. 16.1B). A. Energy Bands and Charge Carriers Energy Bands in Semiconductors The atoms comprising solid-state materials have sufficiently strong interatomic inter- actions that they cannot be treated as individual entities (see Sec. 13.1 C). Their conduc- tion electrons are not bound to individual atoms; rather, they belong to the collection of atoms as a whole. The solution of the Schr6dinger equation for the electron energy, in the periodic potential created by the collection of atoms in the crystal lattice, results in a splitting of the atomic energy levels and the formation of energy bands. Each band contains a large number of densely packed discrete energy levels that is well approximated as a continuum. As illustrated in Fig. 16.1-1, the valence and conduction bands are separated by the bandgap energy E 9' which plays an important role in determining the electrical and optical properties of the material. Si GaAs 5 5 >-.. (]) c:: Valence band . ductioh ban' t Eg 1.42 eV T 0 Conduction band ..............u... ". . ." t Eg 1.12 eV 0 T ,,-.., > <l) '-" ,,-.., > (]) '-" -5 >-.. (]) c:: Valence band -5 -10 -10 -15 -15 Fig u re 16.1-1 Energy bands in Si and GaAs. The bandgap energy E g, which separates the valence and conduction bands, is 1.12 e V for Si and 1.42 e V for GaAs at room temperature. The origin of the bandgap may be illustrated by means of the Kronig-Penney model. In this simple theory the crystal-lattice potential, a one-dimensional version of which is depicted in Fig. 16.1-2(a), is approximated by a ID periodic rectangular- barrier potential, as shown in Fig. 16.1- 2(b). The solution of the associated Schr6dinger
630 CHAPTER 16 SEMICONDUCTOR OPTICS equation (13.1-3) for this potential yields allowed energy bands with traveling-wave solutions, separated by forbidden bands with exponentially decaying solutions. It can be shown that the results are general and apply to three dimensions. This approach is similar to that used for analyzing the optics of one-dimensional periodic media, as set forth in Sec. 7.2. The traveling-wave eigenfunctions are Bloch modes with the periodicity of the crystal lattice [see (7.2-4)]. I-- a--l Figure 16.1-2 (a) Crystal-lattice potential associated with an infinite one-dimensional collection of atoms with lattice constant a. (b) Idealized rectangular-barrier potential (height V 0) used in the Kronig-Penney model. (a) ... ,(\(\(\(\(\( ... (b) . . . Electrons and Holes As discussed in Sec. 13.1C, the wavefunctions of the electrons in a semiconductor overlap so that the Pauli exclusion principle applies. This principle dictates that no two electrons may occupy the same quantum state and that the lowest available en- ergy levels fill first. Elemental semiconductors, such as Si and Ge, have four valence electrons per atom that form covalent bonds. At T == 0° K, the number of quantum states that can be accommodated in the valence band is such that it is completely filled while the conduction band is completely empty. The material cannot conduct electricity under these conditions. As the temperature increases, however, some electrons can be thermally excited from the valence band into the empty conduction band, where unoccupied states are abundant (see Fig. 16.1-3). These electrons can then act as mobile carriers, drifting through the crystal lattice under the effect of an applied electric field, and thereby contributing to the electric current. Moreover, an electron departing from the valence band leaves behind an unoccupied quantum state, which in turn allows the remaining electrons in the valence band to exchange places with each other under the influence of an external field. The collection of electrons remaining in the valence band thus under- goes motion. This can equivalently be regarded as motion, in the opposite direction, of the hole left behind by the departed electron. The hole therefore behaves as if it has a positive charge +e. The net result is that each electron excitation creates a free electron in the conduction band and a free hole in the valence band. The two charge carriers are free to drift under the effect of the applied electric field and thereby to generate an electric current. The material behaves as a semiconductor whose conductivity increases sharply with increasing temperature, as more and more mobile carriers are thermally generated. lJ.J >-.. ep <l) <l) o ;.... ..... u (]) . . . . . . . . . . . . . ... .. . .,i Conduction band ! Bandgap _ energy Eg ; ;.; ; ;.;.; ;.; ; ; ;e; ;e; ;.; ;.; ;e;.; . . . ... . . . ... .. . . . ... . . .. . . .................. · · ... · · · · · · · · · · · · · · · · · · · · Valence ........................ · · · · · · · · · · · · · · · · · · · · · · · · band ........................ ........................ ........................ Electron Hole Figure 16.1-3 Electrons in the conduction band and holes in the valence band at T > 0° K.
16.1 SEMICONDUCTORS 631 Energy-Momentum Relations In accordance with wave mechanics, the energy E and momentum p of an electron in a region of constant potential, such as free space, are related by E == p2/2mo == n 2 k 2 12mo, where p is the magnitude of the momentum, k is the magnitude of the wavevector k == pin, and mo is the electron mass (9.1 x 10- 31 kg). The E-k relation for a free electron is thus a simple parabola. EXERCISE 16.1-1 Energy-Momentum Relation for a Free Electron. (a) Consider a one-dimensional version of the time-independent Schrodinger equation set forth in (13.1-3) for a free electron (V == 0) of mass mo. Use a trial solution of the form 'ljJ(x) ex: exp( - j kx) to show that the energy-momentum relation assumes the quadratic form fi 2 k 2 E==- 2mo ' (16.1-1) so that the energy is not quantized in this ideal case. (b) The free photon, in contrast, has a linear energy-momentum relation, as provided in (12.1-10): E == pc == cfik, (16.1-2) where c is the speed of light in the medium. What is the origin and significance of this distinction? The motion of an electron in a semiconductor material is similarly governed by the Schrodinger equation, but with a potential generated by the charges in the periodic crystal lattice of the material. As discussed earlier, this construct results in allowed energy bands separated by forbidden bands, as predicted by the Kronig-Penney model. The ensuing E-k relations for electrons and holes, in the conduction and valence bands respectively, are illustrated in Fig. 16.1-4 for Si and GaAs. The energy E is a periodic function of the components (k 1 , k 2 , k 3 ) of the wavevector k, with periodicities Crrla1,7rla2,7rla3), where a1,a2,a3 are the crystal lattice constants. Figure 16.1-4 displays cross sections of this relation along two particular directions of the wavevector k. The range of k values in the interval [-7r I a, 7r I a] defines the first Brillouin zone. The energy of an electron in the conduction band thus depends not only on the magni- tude of its momentum, but also on the direction in which it is traveling in the crystal. The semiconductor E-k diagram bears some resemblance to the photonic-crystal w-K diagram (see Fig. 7.3-5). Effective Mass It can be seen from Fig. 16.1-4 that near the bottom of the conduction band, the E-k relation may be approximated by the parabola n 2 k 2 E == Ec + - , 2mc (16.1-3) where E c is the energy at the bottom of the conduction band and k is measured from the wavevector where the minimum occurs. This relation tells us that a conduction- band electron behaves in a manner similar to that of a free electron, but with a mass mc, known as the electron (conduction-band) effective mass, that differs from the free- electron mass mo. The influence of the ions of the lattice on the motion of a conduction- band electron is thus contained in the effective mass mc. This behavior is highlighted in Fig. 16.1-5.
632 CHAPTER 16 SEMICONDUCTOR OPTICS E E - -------- - EC------- t Eg= 1.12 eV :;;;;;;;;;;;;;;;;;;;;;;;;Ev .......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... . .. ......... .......... .......... .. .... ...... .... .. .. .... Eg= 1.42eV k k Ev ;;;;;;;;;;;;;;;;;;;;;;;; .......................... .......................... .......................... .......................... ......................... ......................... .......................... .......................... ... .. .. .. .... . .. .. .... ........ .. .. .. .. .. .. .. .... Si ( [Ill] . [ 100] GaAs ( [Ill] . [ 100] Figure 16.1-4 Cross section of the E-k function for Si and GaAs along two crystal directions: [111] toward the left and [100] toward the right. Eg = 1.12 e V Eg= 1.42eV k E -- - 'I' ,. ",.- T T T .... .....- T ". T .......................... .........................< .........................< .......................... .......................... .......................... .......................... .......................... .......................... k .......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... .......................... Si GaAs Figure 16.1-5 The E-k diagrams for Si and GaAs are well approximated by parabolas at the bottom of the conduction band and at the top of the valence band. Similarly, near the top of the valence band, we have h 2 k 2 E==E -- v 2 ' mv ( 16.1-4) where Ev == Ec - Eg is the energy at the top of the valence band and mv is the hole (valence-band) effective mass, as portrayed in Fig. 16.1-5. The influence of the lattice ions on the motion of a valence-band hole is captured by the effective mass mv. The effective mass depends on the crystal structure of the material and the direction of travel with respect to the lattice since the interatomic spacing varies with crystallographic direction. It also depends on the particular band under consideration. Indeed, several parabolas of different curvature often coexist near the top of the valence band; these correspond to so-called heavy holes, light holes, and holes associated with the split-off band. Typical ratios of the averaged effective masses to the mass of the free electron mo are provided in Table 16.1-1 for Si, GaAs, and GaN. Table 16.1-1 Typical values of electron and hole effective masses in selected semiconductor materials. mc/mo mv/mo Si 0.98 0.49 GaAs 0.07 0.50 GaN 0.20 0.80
16.1 SEMICONDUCTORS 633 Direct- and Indirect-Bandgap Semiconductors Semiconductors for which the conduction-band minimum energy and the valence- band maximum energy correspond to the same value of the wavenumber k (same momentum) are called direct-bandgap materials. Semiconductors for which this is not the case are known as indirect-bandgap materials. As is evident in Fig. 16.1-5, GaAs is a direct-bandgap semiconductor whereas Si is an indirect-bandgap semiconductor. The distinction is important because a transition between the bottom of the conduction band and the top of the valence band in an indirect-bandgap semiconductor must accommodate a substantial change in the momentum of the electron. It will be shown subsequently that direct-bandgap semiconductors such as GaAs are efficient photon emitters, whereas indirect-bandgap semiconductors such as Si cannot serve as efficient light emitters under ordinary circumstances. B. Semiconductor Materials Figure 16.1-6 reproduces the section of the periodic table that comprises most of the elements important in semiconductor electronics and photonics. Both elemental and compound semiconductors play crucial roles in these technologies. II III IV V VI 2 rnrn 3 /r:fJ lrJ r] r. 11 4 M1 [Gl fG r:rs; t1 J48l [491 . 49 .. ' k501 . . . .; f5il f521 5 Ltg]1ThJ 6 11 ] D Gas D Liquid [] Solid Figure 16.1-6 Section of the periodic table relating to semiconductors. Elements indicated in blue, yellow, and silver take the form of gases, liquids, and solids, respectively, at room temperature. The full periodic table is displayed in Fig. 13.1-3. We proceed to discuss elemental, binary, ternary, and quaternary semiconductors in turn, and then consider doped semiconductors. Elemental Semiconductors Silicon (Si) and gennanium (Ge) are important elemental semiconductors in column IV of the periodic table. Virtually all commercial electronic integrated circuits and devices are fabricated using Si. Both Si and Ge also find widespread use in photonics, principally as photodetectors. These materials have traditionally not been used for the fabrication of light emitters because of their indirect bandgaps. However, some forms of Si are viable as light emitters and silicon photonics has come to the fore. The basic properties of Si and Ge are provided in Table 16.1-2.
634 CHAPTER 16 SEMICONDUCTOR OPTICS Binary 111- V Semiconductors Ternary III-V Semiconductors Compounds fonned by combining an element in column III, such as aluminum (AI), gallium (Ga), or indium (In), with an element in column V, such as nitrogen (N), phosphorus (P), arsenic (As), or antimony (Sb), are important semicon- ductors in photonics. These 12 III-V compounds are listed in Table 16.1-2, along with their crystal structure (zincblende or wurtzite), bandgap type (direct or indirect), bandgap energy Eg, and bandgap wavelength >"g == hcal Eg (the free-space wavelength of a photon of energy Eg). The bandgap energies and lattice constants of these compounds are also displayed in Fig. 16.1-7. Photon sources (light-emitting diodes and lasers) N and detectors can be readily fabricated from many of these p binary compounds. The first of the binary semiconductors to find use in photonics was gallium arsenide (GaAs), which is also sometimes used as an alternative to Si for fast electronic devices and circuits. Gallium nitride (GaN) plays a central role in photonics by virtue of its near-ultraviolet bandgap wavelength; it is also important in electronics because of its ability to withstand high temperatures. AIN, which is an insulator, has the highest bandgap of all 111- V compounds and emits photons at the shortest wavelength, in the mid-ultraviolet regIon. Compounds formed from two elements of column III with one element from column V (or one from column III with two from column V) are important ternary semiconductors. (AlxGal-x)As, for example, is a compound with properties that interpolate between those of AlAs and GaAs, depending on the compositional mixing ratio x (the fraction of Ga atoms in GaAs that are replaced by Al atoms). The bandgap energy E 9 for this material varies between 1.42 e V for GaAs and 2.16 e V for AlAs, as x varies between 0 and 1 along the line connecting GaAs and AlAs in Fig. 16.1-7(a). Because this line is essentially vertical, AlxGal-xAs is lattice matched to GaAs; a layer of arbitrary composition of this material can therefore be grown on a layer of different composition without straining the lattice. Other useful 111- V ternary compounds, such as Ga(Asl-xP x), are also represented in the bandgap- energy versus lattice-constant diagram displayed in Fig. 16.1- 7(a). (InxGal-x)As is widely used for photon sources and detectors in the near-infrared region of the spectrum. Similarly, (AlxGal-x)N and (InxGal-x)N are important ternary semiconductors for photonic devices that operate in the ultraviolet, violet, blue, and green regions of the spectrum, as can be deduced from Fig. 16.1-7 (b). In the domain of electron- ics, (InxGal-x)As/InP heterojunction bipolar transistors can be switched at speeds approaching 1 THz; indeed, various 111- V compounds can be used to fabricate ultrafast transistors that emit light.
16.1 SEMICONDUCTORS 635 These compounds are formed by mixing two elements from column III with two elements from column V (or three from column III with one from column V). Quaternary semiconductors offer more flexibility for fabricating materials with desired properties than do ternary semiconductors by virtue of an additional degree of freedom. An example is provided by Inl-xGaxAsl-yPy, whose bandgap energy varies between 0.36 eV (InAs) and 2.26 eV (GaP) as the compositional mixing ratios x and y vary between 0 and 1. The lattice constant usually varies linearly with the mixing ratio (Vegard's law). The stippled area in Fig. 16.1-7(a) indicates the range of bandgap energies and lattice constants spanned by this compound. For mixing ratios x and y that satisfy y == 2.16(1 - x), Inl-xGaxAsl-yP y can be lattice matched to InP, which can therefore serve as a convenient template (substrate). This quaternary compound is used for fabricating light- emitting diodes, laser diodes, and photodetectors, particularly in the vicinity of the 1550-nm optical fiber communications wavelength (see Chapters 17, 18, and 24). Another example is provided by (AlxInyGal_x_y)P, for which GaAs serves as a template; this compound offers high-brightness emission in the red, orange, and yellow spectral regions [see shaded region in Fig. 16.1-7(a)]. Yet another important quaternary material is the III-nitride compound (AlxInyGal-x-y)N, which serves the green, blue, violet, and ultraviolet spectral regions in the same way [see Fig. 16.1-7(b)]. Convenient templates for the Ill-nitrides are sapphire and SiC. Column IV elements can also be alloyed to form compound semiconductors. The binary alloy silicon carbide (SiC), also known as carborundum, has an indirect bandgap and is useful for fabricating ultraviolet photodetectors and as a template for 111- nitride compounds. Silicon germanium (Si1-xGe x ) enjoys a variety of applications in electronics and photonics, including use as an infrared photodetector material. Ternary and quaternary column-IV semiconductor compounds include Sil-x-yGexCy and Sil-x-y-zGexCySnz, respectively. Binary II-VI materials, i.e., compounds formed from elements in column II (e.g., Zn, Cd, Hg) and column VI (e.g., S, Se, Te) of the periodic table are also useful semiconductors. This family includes ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, HgS, HgSe, and HgTe, as shown in Fig. 16.1-8. All of these materials have a zincblende structure and all are direct-bandgap semiconductors; the exceptions are HgSe and HgTe, which are semimetals with small negative bandgaps. A particular merit of ZnSe, is that it can be deposited on a GaAs substrate with a relatively low defect density since the lattice constants of the two materials are similar. Moreover, HgTe and CdTe are nearly lattice matched, so the ternary semiconductor HgxCd 1 - x Te can be grown without strain on a CdTe substrate. This material system is widely used for fabricating photon detectors, as are other II-VI compounds (see Chapter 18). Unlike the III-V alloys, the II-VI compounds are widely found in nature, but photon sources fabricated from these mate- rials currently suffer from limited lifetimes. Nevertheless, binary II-VI semiconductor materials are readily fashioned into quantum dots with tunable photoluminescence emission wavelength (see, for example, Fig. 13.1-12). Ternary IV-VI semiconductor compounds, such as PbxSnl- x Te and PbxSnl-xSe, have also been used as infrared photodetectors and laser diodes. However, these alloys have slower response times because of their large dielectric constants. They also have high thermal coefficients of expansion, so cycling between room and cryogenic temperatures can be problematic. Quaternary III-V Senliconductors
636 CHAPTER 16 SEMICONDUCTOR OPTICS Zincblende and Diamond Table 16.1-2 Selected elemental and III-V binary semiconductors along with their crystal structures, bandgap types, bandgap energies, and GOa bandgap wavelengths. -As Crystal Bandgap Bandgap Bandgap Structure a Type b EnergyC Wavelength d Material (D/ZIW) (lID) E 9 (e V) Ag (11m) Si D I 1.12 1.11 Ge D I 0.66 1.88 Wurtzite AIN W D 6.20 0.200 AlP Z I 2.45 0.506 GOa AlAs Z I 2.16 0.574 GN AISb Z I 1.58 0.785 GaN W D 3.39 0.366 GaP Z I 2.26 0.549 GaAs Z D 1.42 0.873 GaSb Z D 0.73 1.70 InN W D 0.65 1.91 InP Z D 1.35 0.919 InAs Z D 0.36 3.44 InSb Z D 0.17 7.29 aThe crystal structure listed indicates the most commonly used form of the material: D = Diamond, Z = Zincblende, W = Wurtzite, as displayed at left. The zincblende structure comprises two interpenetrating face- centered-cubic lattices, one for each element, displaced from each other by i of the body diagonal. The diamond lattice is the same as zincblende except that all atoms are identical. The Brillouin zone for these structures is illustrated in Fig. 7.3-4. The wurtzite structure consists of two hexagonal close-packed lattices, one for each element, displaced from each other along the three-fold c axis by of its length. All atoms are tetrahedrally bonded with their neighbors. b I = Indirect bandgap; D = Direct bandgap. CData are provided at T = 300 0 K. dThe bandgap wavelength )..g is related to the bandgap energy E 9 by )..g = hco / E g; when the bandgap energy is expressed in eV and the bandgap wavelength is expressed in /-Lm, this relation becomes )..g 1.24/ Eg. Doped Semiconductors The electrical and optical properties of semiconductors can be modified substantially by the controlled introduction into the material of small amounts of specially chosen impurities called dopants. The introduction of these impurities can alter the concen- tration of mobile charge carriers by many orders of magnitude. Dopants with excess valence electrons, called donors, replacing a small proportion of the normal atoms in the crystal lattice, create a predominance of mobile electrons. The material is then said to be an n-type semiconductor. Thus, atoms from column V (e.g., P or As) replacing column-IV atoms in an elemental semiconductor (e.g., Si or Ge), or atoms from column VI (e.g., Se or Te) replacing column- V atoms in a 111- V binary semiconductor (e.g., As or Sb), produce an n-type material. Similarly, a p-type semiconductor is made by using dopants with a deficiency of valence electrons, called acceptors. The result is then a predominance of mobile holes. Column IV atoms in an elemental semiconductor replaced with column-III atoms (e.g., B or In), or column-III atoms in a III-V binary semiconductor replaced with column-II atoms (e.g., Zn or Cd), yield p-type material. Column- IV atoms act as donors for column III and as acceptors for column V, and therefore can be used to produce an excess of both electrons and holes in 111- V mate- rials. Of course, the charge neutrality of the material is not altered by the introduction of dopants.
16.1 SEMICONDUCTORS 637 2.5 t'"..bJ!.' 0.5 AIN .. .. 6 I .. "--->'., 0..'. .. C\S " -.1' 0.6 o ,. I 2.0 ',..... : 5 ',. . E >' 0.7 -3 >' 0> 4 Jnl.5 0.8 ...-.::: ...c I..LJO> biJ :>.. 1.0 s::: :>.. SiC OJ.) OJ.) '- Q) '- 3 . . Si s::: :> s::: 1.0 1.2 ro 0.., ro 0.., ro OJ.) ro OJ.) 2 "'0 OJ.) "'0 s::: "'0 s::: ro . 2.0 s::: ro Ge ro 0.5 3.0 10 0 0 5.4 5.6 5.8 6.0 6.2 6.4 6.6 3.0 3.1 3.2 3.3 3.4 Lattice constant (A) Lattice constant (A) (a) (b) 3.5 0.2 E :::t 0.3 ...-.::: ...c bJ.) 0.4 ] :> 0.5 0.6 OJ.) "'0 s::: ro 1.0 InN 2.0 10 3.6 Figure 16.1-7 Bandgap energies, bandgap wavelengths, and lattice constants for Si, Ge, SiC, and 12 III-V binary compounds. Solid and dashed curves represent direct-bandgap and indirect- bandgap compositions, respectively. A material may have a direct bandgap for one mixing ratio and an indirect bandgap for a different mixing ratio. Ternary materials are represented along the line that joins two binary compounds. A quaternary compound is represented by the area formed by its binary components. (a) Inl-xGaxAsl-YP y is represented by the stippled area with vertices at InP, InAs, GaAs, and GaP, while (AlxGal- x )ylnl-YP is represented by the shaded area with vertices at AlP, InP, and GaP. Both are important quaternary compounds, the former in the near infrared and the latter in the visible. AlxGal-xAs is represented by points along the line connecting GaAs and AlAs. As x varies from 0 to 1, the point moves along the line from GaAs and AlAs. Since this line is nearly vertical, AlxGal-xAs is lattice matched to GaAs. (b) Although the III-nitride compound InxGal-xN can, in principle, be compositionally tuned to accommodate the entire visible spectrum, this material becomes increasingly difficult to grow as the composition of In becomes appreciable. InxGal-xN is principally used in the green, blue, and violet spectral regions, while AlxGal-xN and AlxlnyGal_x_yN serve the ultraviolet region. All compositions of these III-Nitride compounds are direct-bandgap semiconductors. 3.5 3.0 0.4 >' 2.5 E \ 0.5 3 \ 0> \ 0> \ 0.6 """ Figure 16.1-8 I..LJ 2.0 \ ...c Bandgap energies, bandgap :>.. \ biJ OJ.) \ 0.7 s::: wavelengths, and lattice constants for various '- \ \ s::: 1.5 \ Q) \ :> 11- VI semiconductors (HgSe and HgTe are 0.., \ \ l.0 ro ro \ J.) 1.0 \ \ 1.2 semimetals with small negative bandgaps). \ \ 0.., s::: \ \ ro ro \ \ OJ.) HgTe and CdTe are nearly lattice matched, t:C \ \ 2.0 "'0 0.5 \ \ s::: \ \ ro as evidenced by the vertical line connect- \ \ t:C \ \ \ \ 10 ing them, so that the ternary semiconductor 0.0 \ \ \ \ HgSe HgTe \ Hg x Cd 1 - x Te can be grown without strain 5.4 5.6 5.8 6.0 6.2 6.4 6.6 on a CdTe template. It is an important mid- Lattice constant (A) infrared photodetector material. Undoped semiconductors (i.e., semiconductors devoid of intentional doping) are referred to as intrinsic materials, whereas doped semiconductors are called extrinsic materials. The concentrations of mobile electrons and holes are equal in an intrinsic semiconductor, n == p == ni, where the intrinsic concentration ni grows with in- creasing temperature at an exponential rate. On the other hand, the concentration of mobile electrons in an n-type semiconductor (majority carriers) is far greater than the concentration of holes (minority carriers), i.e., n » p. The opposite is true in
638 CHAPTER 16 SEMICONDUCTOR OPTICS a p-type semiconductor, where holes are the majority carriers, and p » n. A doped semiconductor at room temperature typically has a majority-carrier concentration that is approximately equal to the doping concentration. Single-ion implantation techniques can be used to fabricate semiconductor materials in which the number of dopant atoms, and their positions, are precisely controlled. The resulting materials exhibit properties that are more deterministic than those with random numbers of dopant atoms, which is useful in certain applications. EXAMPLE 16.1-1. Donor-Electron Ionization Energy. Consider a germanium crystal of dielectric constant E/ Eo = 16 (see Table 16.2-1), doped with arsenic donor atoms. The electron effective mass me = 0.2 mo, where mo is the free electron mass. The donor electron moves in the field of the singly charged arsenic ion (As+), and has energy levels similar to those of an electron in the hydrogen atom. Choosing n = 1 and Z = 1 in (13.1-4), and replacing Eo by E, and M r by me, to accommodate the polarization density and crystal lattice of the semiconductor material, respectively, the energy of the donor electron is given by ( 1 ) 2 4 ED = _ _ mee 41TE 21i 2 . (16.1-5) Since the energy of the electron in the ground state of hydrogen is -13.6 eV (indicating that it is 13.6 eV below ionization), the energy of the arsenic donor electron is ED = -(me/mO)(Eo/E)2 x 13.6 eV -0.01 eV. The donor electron thus resides in the forbidden band, at a level 0.01 eV below the conduction band. However, since the thermal energy kT 0.026 eV at T = 300 0 K, essentially all of the donors are ionized at room temperature and the donor electrons are elevated to the conduction band. The material thus has a conduction-band donor concentration that matches the impurity concentration. Organic Semiconductors Organic semiconductors are increasingly employed in a wide variety of fields. This in- cludes photonics, where they are used to fabricate photovoltaic devices, light-emitting diodes, and displays. Although they offer neither the speed nor small size of con- ventional semiconductor structures, they can be inexpensively fabricated in the form of thin sheets, making low-cost, mechanically flexible optoelectronic components a reality. These materials come in a virtually unlimited array of variations that can be engineered to suit specific requirements and some can be printed on a suitable substrate using inkjet technology. Organic semiconductors come in two principal varieties, as illustrated schematically in Fig. 16.1-9: 1. Small organic molecules such as pentacene, which consists of five linearly joined benzene rings [Fig. 16.1-9( a)]. 2. Conjugated polymer chains such as polyacetylene, comprising hundreds or thou- sands of carbon atoms [Fig. 16.1-9(b)]. A hallmark of these amorphous materials, tenned conjugation, is their alternating single and double carbon-carbon bonds. Although the double-bond electrons shown in Figs. 16.1-9( a) and (b) are portrayed as belonging to particular atoms, these electrons are actually delocalized and shared among multiple atoms, or along a segment of poly- mer comprising roughly 10 repeat units. The molecule, or polymer segment, behaves as a single system in which the allowed electron states form bands. In its undoped state, the valence band of a conjugated polymer chain is typically full, and its conduction band empty, so that it behaves as an insulator. However, as
16.1 SEMICONDUCTORS 639 (a) r-- Electron (b) ... ... (c) ... 0- Sodium ion Figure 16.1-9 Organic semiconductors are available in two principal varieties: (a) small organic molecules such as pentacene, and (b) conjugated polymer chains such as polyacetylene. (c) Doping polyacetylene with sodium donors yields an n-type material, while doping with iodine acceptors yields a p-type material. Each line represents a bond between two carbon atoms; double lines represent double bonds. Hydrogen bonds are omitted for simplicity. A wide variety of organic molecules and polymers are used in electronics and photonics. illustrated in Fig. 16.1-9( c), dopants such as sodium and iodine act as donors and acceptors, respectively, providing n-type and p-type variants. Small organic molecules are often conductive in their pure state. C. Electron and Hole Concentrations Determining the concentration of carriers (electrons and holes) as a function of energy requires knowledge of two features, which we consider in turn: . The density of allowed energy levels (density of states) . The probability that each of these levels is occupied Density of States The quantum state of an electron in a semiconductor material is characterized by its energy E, its vector k [the magnitude of which is approximately related to E by (16.1- 3) or (16.1-4)], and its spin. The state is described by a wavefunction that satisfies certain boundary conditions. An electron near the conduction band edge may be approximately described as a particle of mass me confined to a three-dimensional cubic box (of dimension d) with perfectly reflecting walls, i.e., a three-dimensional infinite rectangular potential well. The standing-wave solutions require that the components of the vector k == (k x , ky, k z ) assume the discrete values k == (ql7r 1 d, q2 7r 1 d, q3 7r 1 d), where the respective mode numbers (ql, Q2, Q3) are positive integers. This result is a three-dimensional general- ization of the one-dimensional infinite square well (see Exercise 16.1-5). The tip of the vector k must lie on the points of a lattice whose cubic unit cell has dimension 7r / d. There are therefore (d 1 7r)3 points per unit volume in k-space. The number of states whose vectors k have magnitudes between 0 and k is determined by counting the number of points lying within the positive octant of a sphere of radius k [with volume (! )47rk 3 /3 == 7rk 3 /6]. Because of the two possible values of the electron spin, each point in k-space corresponds to two states. There are therefore approxi- mately 2(7rk 3 /6)/(7r Id)3 == (k 3 /37r 2 )d 3 such points in the volume d 3 and (k 3 /37r 2 ) points per unit volume. It follows that the number of states with electron wavenumbers between k and k + k, per unit volume, is g(k)k == [(dldk)(k 3 /37r2)]k == (k 2 /7r2)k, so that the density of states is k 2 g( k) == 2 . 7r (16.1-6) Density of States This derivation is identical to that used for counting the number of modes that can be supported in a three-dimensional electromagnetic resonator (see Sec. 10.3). In the case of electromagnetic modes there are two degrees of freedom associated with the field
640 CHAPTER 16 SEMICONDUCTOR OPTICS polarization (i.e., two photon spin values), whereas in the semiconductor case there are two spin values associated with the electron state. In resonator optics the allowed electromagnetic solutions for k were converted into allowed frequencies via the linear frequency-wavenumber relation v == ckj27r. In semiconductor physics, on the other hand, the allowed solutions for k are converted into allowed energies via the quadratic energy-wavenumber relations given in (16.1-3) and (16.1-4). If (}e(E) E represents the number of conduction-band energy levels (per unit volume) lying between E and E + E, then, because of the one-to-one correspondence between E and k governed by (16.1-3), the densities (}e( E) and (}(k) must be related by (}e (E) dE == (}( k) dk. Thus, the density of allowed energies in the conduction band is (}e (E) == (}( k) j (dE j dk). Similarly, the density of allowed energies in the valence band is (}v(E) == (}(k)j(dEjdk), where E is given by (16.1-4). The approximate quadratic E-k relations (16.1-3) and (16.1-4), which are valid near the edges of the conduction band and valence band, respectively, are used to evaluate the derivative dE j dk for each band. The result that obtains is (16.1-8) Density of States Near Band Edges The square-root relation is a result of the quadratic energy-wavenumber formulas for electrons and holes near the band edges. The dependence of the density of states on energy is illustrated in Fig. 16.1-10. It is zero at the band edge, and increases away from it at a rate that depends on the effective masses of the electrons and holes. The values of me and mv provided in Table 16.1-] are actually averaged values suitable for calculating the density of states. ( 2m e)3/2 t!c(E) = 27r 2 ti 3 J E - En E > Ec ( 2m v)3/2 t!v(E) = 27r 2 ti 3 J Ev - E, E < Ev. (16.1-7) E Ec II ---- - - - - t-------j--- - - - - - - - - - - - - - - - - - - - - - E f2c(E) Ec--------Ec Ev Ev----u-Ev f2 v< E) Density of states (a) (b) (c) Figure 16.1-10 (a) Cross section of the E-k diagram (e.g., in the direction of the k 1 component, with k 2 and k3 fixed). (b) Allowed energy levels (at all k). (c) Density of states near the edges of the conduction and valence bands. The quantity (2c( E) dE is the number of quantum states with energy between E and E +dE, per unit volume, in the conduction band. The quantity (2v (E) has an analogous interpretation for the valence band. k Probability of Occupancy In the absence of thermal excitation (at T == 0° K), all electrons occupy the lowest possible energy levels, subject to the Pauli exclusion principle. The valence band is then
16.1 SEMICONDUCTORS 641 completely filled (there are no holes) and the conduction band is completely empty (it contains no electrons). When the temperature is raised, thermal excitations raise some electrons from the valence band to the conduction band, leaving behind empty states in the valence band (holes). The laws of statistical mechanics dictate that under conditions of thermal equilibrium at temperature T, the probability that a given state of energy E is occupied by an electron is determined by the Fermi function f (E) = 1 , exp [ (E - E f ) / kT] + 1 (16.1-9) Fermi Function where k is Boltzmann's constant (at T == 300° K, kT == 0.026 e V) and E f is a constant known as the Fermi energy or Fermi level. This function is also known as the Fermi- Dirac distribution. Each energy level E is either occupied [with probability f (E)], or empty [with probability 1 - f(E)]. The probabilities f(E) and 1 - f(E) depend on the energy E in accordance with (16.1-9). The function f( E) is not itself a probability distribution, and it does not integrate to unity; rather, it is a sequence of occupation probabilities for successive energy levels. Because f(Ef) == , whatever the temperature T, the Fermi level is that energy for which the probability of occupancy (if there were an allowed state there) would be !. The Fermi function is a monotonically decreasing function of E (Fig. 16.1-11). At T == 0° K, f (E) is 0 for E > E f and 1 for E < E f. This establishes the significance of E f; it is the division between the occupied and unoccupied energy levels at T == 0° K. Since f (E) is the probability that the energy level E is occupied, 1 - f (E) is the probability that it is empty, i.e., that it is occupied by a hole if E lies in the valence band. Thus, for energy level E: f (E) == probability of occupancy by an electron 1 - f( E) == probability of occupancy by a hole (valence band). These functions are symmetric about the Fermi level. Ec T Ej----------------- Eg E - -- -.l 1 . ....... + 1'1';:-. +.....;-.."x.. .+ ... ...J'...:_.... . . ..+ . . + . . . . . . . . . . . J'. . . ...................... ..+.+..........+.+..+... .++++........++......... ..++.++..+..+++.+..+.... ......++.+.......+..+... ....+..+....+.......+... E E T= 0 K T>OK Ec - Ej Ev - -'I I I k-l- f(E) I E- v I o 0.5 - o 0.5 1 f(E) 1 f(E) Figure 16.1-11 The Fermi function f (E) is the probability that an energy level E is filled with an electron; 1 - f (E) is the probability that it is empty. In the valence band, 1 - f (E) is the probability that energy level E is occupied by a hole. At T == 0° K, f (E) == 1 for E < E f, and f (E) == 0 for E > E f; there are then no electrons in the conduction band and no holes in the valence band When E - E f » kT, f (E) exp [ - (E - E f ) / kT], so that the high-energy tail of the Fermi function in the conduction band decreases exponentially with increasing
642 CHAPTER 16 SEMICONDUCTOR OPTICS energy. The Fermi function is then proportional to the Boltzmann distribution, which describes the exponential energy dependence of the fraction of a population of atoms excited to a given energy level (see Sec. 13.2). By symmetry, when E < E f and E f - E » kT, 1 - f (E) exp [ - ( E f - E) / kT]; the probability of occupancy by holes in the valence band then decreases exponentially as the energy decreases well below the Fermi level. Thermal-Equilibrium Carrier Concentrations Let n(E) t::,.E and p(E) t::,.E be the number of electrons and holes per unit volume, respectively, with energy lying between E and E + t::,.E. The densities n(E) and p (E) can be obtained by multiplying the densities of states at energy level E by the probabilities of occupancy of the level by electrons or holes, so that n (E) == (}e ( E) f ( E) , p (E) == (}v ( E) [1 - f ( E) ] . (16.1-10) The concentrations (populations per unit volume) of electrons and holes, nand p, are then obtained from the integrals n = ('XJ n(E) dE, lEe l Ev P = -00 p(E) dE. (16.1-11) In an intrinsic (pure) semiconductor at any temperature, n == p because thermal excitations always create electrons and holes in pairs. The Fermi level must therefore be placed at an energy value such that n == p. In materials for which mv == me, the functions n(E) and p(E) are also symmetric, so that Ef must lie precisely in the middle of the bandgap (Fig. 16.1-12). In most intrinsic semiconductors, the Fermi level does indeed lie near the middle of the bandgap. Ec Ej ----------------- E ":e;' M' . . . .... . . .,... . v .:+X+++X::::A:+:( +: + + .+ + + + + + + +-: + + + + + + + + :e: + + . ++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. L........44.......44 E Ec Ej Ev Carrier concentration Figure 16.1-12 The concentrations of electrons and holes, n( E) and p( E), as a function of energy E, for an intrinsic semiconductor. The total concentrations of electrons and holes are nand p, respecti vel y. The energy-band diagrams, Fermi functions, and equilibrium concentrations of elec- trons and holes for n-type and p-type doped semiconductors are illustrated in Figs. 16.1-13 and 16.1-14, respectively. Donor electrons occupy an energy ED slightly be- low the conduction-band edge so that they are easily raised to it. If ED == 0.01 eV, for example, at room temperature (kT == 0.026 e V) most donor electrons will be thermally excited into the conduction band (see Example 16.1-1). As a result, the Fermi level [the energy at which f(Ef) == ] will lie above the middle of the bandgap. For a p-type semiconductor, the acceptor energy level lies at an energy E A just above the valence-band edge so that the Fermi level will lie below the middle of the bandgap.
16.1 SEMICONDUCTORS 643 Our attention has been directed to the mobile carriers in doped semiconductors. These materials are, of course, electrically neutral, as assured by the fixed donor and acceptor ions, so that n + N A == P + N D , where N A and N D are, respectively, the number of ionized acceptors and donors per unit volume. - EDT ---- - Donolevel ----- E :;;;;J{:;++:;)t++;;;; .++++++++++++++++++)(++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .+++++++++++++++++++++++ .++++++++++++++++++++++++ ..... ............to..a........................to.......... J E E Ef Ev n(E) Ec o 1 f(E) Carrier concentration Figure 16.1-13 Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes, n( E) and p( E), respectively, in an n-type semiconductor. E E 1- ____ Acceptoreve ____ A T :.c.; x_:..... .,,1A....., "":.":.":..... e;."":."":. . ...: ..: :e:x::.(:: . +:« + + X + + + X + + + +...; · )(;It; + + + X + + + + + + + + + ...+ X + + ::+::::::+:::::::)(:::+::: .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ E E p(E) Ef Ev o 1 f( E) Carrier concentration Figure 16.1-14 Energy-band diagram. Fermi function f(E), and concentrations of mobile electrons and holes, n( E) and p( E), respectively, in a p-type semiconductor. EXERCISE 16.1-2 Exponential Approximation of the Fermi Function. When E - Ef » kT, the Fermi function f(E) may be approximated by an exponential function. Similarly, when E f - E » kT, 1 - f(E) may be approximated by an exponential function. These conditions apply when the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times kT (at room temperature kT 0.026 eV whereas Eg == 1.12 eV in Si and 1.42 eV in GaAs). Using these approximations, which apply for both intrinsic and doped semiconductors, show that (16. 1-11) gives n = Ncex p ( - Ec:r Ej ) p = Nvex p ( _ E j ;;;, Ev ) np = NcNvex p ( - :; ), where N e == 2(21rmekT /h2)3/2 and N v == 2(21rmvkT /h 2 )3/2. Verify that if E f is closer to the conduction band and mv == me, then n > p, whereas if it is closer to the valence band, then p > n. (16.1-12) (16.1-13) (16.1-14 )
644 CHAPTER 16 SEMICONDUCTOR OPTICS Law of Mass Action Equation (16.1-14) reveals that, in thermal equilibrium, the product _ ( 27rkT ) 3 3/2 ( E g ) np - 4 h 2 (mcmv) exp - kT (16.1-15) is independent of the location of the Fermi level E f within the bandgap and the semi- conductor doping level, provided that the exponential approximation to the Fermi function is valid. The constancy of the concentration product is called the law of mass action. For an intrinsic semiconductor, n == p n i. Combining this latter relation with (16.1-14) then leads to (16.1-16) Intrinsic Carrier Concentration revealing that the intrinsic concentration of electrons and holes increases with temper- ature T at an exponential rate. The law of mass action may therefore be written in the form ni J NcN v ex p ( - 2 ) , 2 np == n i . ( 16.1-17) Law of Mass Action The values of n i for different materials vary because of differences in the bandgap energies and effective masses. The room-temperature intrinsic carrier concentrations for Si, GaAs, and GaN are provided in Table 16.1-3. Table 16.1-3 Intrinsic carrier concentrations at T = 300 0 K.a Material ni (cm- 3 ) 1.5 x 10 10 1.8 X 10 6 1.9 X 10- 10 Si GaAs GaN a Substitution of the values of me and mv provided in Table 16.1-1, and the value for E 9 given in Table 16.l-2, into (16.1-16), does not yield the listed values of n i because of the sensitivity of the formula to the precise values of the parameters. The law of mass action is useful for determining the concentrations of electrons and holes in doped semiconductors. A moderately doped n-type material, for example, has a concentration of electrons n that is essentially equal to the donor concentration N D. Using the law of mass action, the hole concentration is then p == n/ N D . Knowledge of nand p allows the Fermi level to be determined via (16.1-11). As long as the Fermi level lies within the bandgap, at an energy greater than several times kT from its edges, the approximate relations in (16.1-12) and (16.1-13) can be used to determine it direct! y. If the Fermi level lies inside the conduction (or valence) band, the material is re- ferred to as a degenerate semiconductor. In that case, the exponential approximation of the Fenni function cannot be used, so that np i=- n. The carrier concentrations must then be obtained by numerical solution. Under conditions of very heavy doping, the donor (acceptor) impurity band actually merges with the conduction (valence) band to become what is known as the band tail. This results in an effective decrease of the bandgap.
16.1 SEMICONDUCTORS 645 Quasi-Equilibrium Carrier Concentrations The occupancy probabilities and carrier concentrations considered above are applica- ble only for a semiconductor in thermal equilibrium. They are not valid when thennal equilibrium is disturbed. There are, nevertheless, situations in which the conduction- band electrons are in thennal equilibrium among themselves, as are the valence-band holes, but the electrons and holes are not in mutual thermal equilibrium. This can occur, for example, when an external electric current or photon flux induces band-to-band transitions at too high a rate for interband equilibrium to be achieved. This situation, which is known as quasi-equilibrium, arises when the relaxation (decay) times for transitions within each of the bands are much shorter than the relaxation time between the two bands. Typically, the intraband relaxation time < 10- 12 s, whereas the radiative electron-hole recombination time 10- 9 s. Under these circumstances, it is appropriate to use a separate Fermi function for each band; the two associated Fermi levels, denoted E fe and E fv, are known as quasi- Fermi levels (Fig. 16.1-15). When E fe and E fv lie well inside the conduction and valence bands, respectively, the concentration of both electrons and holes can be quite large. E E ==_E c t Eg Ev 11(.M 11;J8+. ;e;...«e;.;.1'1I;.;"..;e;-. - - - - - - -- -- -- -- -- -- - -- -- -- -- -- -- -- - - ---; -- - -- -- -- -- Ev ,.......+++.....+...lI+..++.....+.--E - Eju -., p(E) +++++++....+.+++w...... tv I +. .. + + + )( + + + + + .. + + .. . .. . :::::::+:::::::::e:::::::: I ::::::::::::::::::::::::: I ++++++++++++++++++++++++. ....... ................................................. E E -., I r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Ec I o 1 fc(E) o 1 IV< E) Carrier concentration ....... Figure 16.1-15 A semiconductor in quasi-equilibrium. The probability that a particular conduction-band energy level E is occupied by an electron is fc( E), a Fermi function with Fermi level Efc. The probability that a valence-band energy level E is occupied by a hole is 1 - Iv (E), where Iv (E) is a Fermi function with Fermi level E fv. The concentrations of electrons and holes are n(E) and p(E), respectively. Both can be large. EXERCISE 16.1-3 Determination of the Quasi-Fermi Levels Given the Electron and Hole Concentrations. (a) Given the concentrations of electrons n and holes p in a semiconductor at T = 0° K, use (16.1- 10) and (16. 1-11) to show that the quasi-Fermi levels are E fe = Ee + (37r 2 )2/3 n 2 / 3 2mc E fv = Ev - (37r2)2/3 p2/3. 2mv (16.1-18a) (16.l-18b) (b) Show that these equations are approximately applicable for an arbitrary temperature T if nand p are sufficiently large so that E fc - Ec » kT and Ev - E fv » kT, i.e., if the quasi-Fermi levels lie deep within the conduction and valence bands.
646 CHAPTER 16 SEMICONDUCTOR OPTICS D. Generation, Recombination, and Injection Generation and Recombination in Thermal Equilibrium The thermal excitation of electrons from the valence band into the conduction band results in the electron-hole generation (Fig. 16.1-16). Thennal equilibrium requires that this generation process be accompanied by a simultaneous reverse process of deex- citation. This process, called electron-hole recombination, occurs when an electron decays from the conduction band to fill a hole in the valence band (Fig. 16.1-16). The energy released by the electron may take the form of an emitted photon, in which case the process is called radiative recombination. Ec . . . . . . . . . . . . . .. . .. . Generation 1 Recombinat" Ion Ev ; ;-; ..... 11 ... ;..; ; ; +- +... ;-; ;- ; ;- -;--;: + + + +... + + + +.. + + + +-+ + + +-.+ + + . ::::(::::::::::::::::::::: ++++++++++++++++++++++++. ++.+++++++++++++++++++++. +++++.++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. Figure 16.1-16 Electron-hole generation and recombination. Nonradiative recombination can occur via a number of independent competing processes, including the transfer of energy to lattice vibrations (creating one or more phonons) or to another free electron (Auger process). Recombination may also take place at surfaces and indirectly via traps or defect centers, which are energy levels associated with impurities or defects associated with grain boundaries, dislocations, or other lattice imperfections that lie within the forbidden band. An impurity or defect state can act as a recombination center if it is capable of trapping both an electron and a hole, thereby increasing their probability of recombining (Fig. 16.1-17). Impurity- assisted recombination may be radiative or nonradiative. Ec . . . . . . . . .. . .. . Trap Ev ;;e;; ;- -; e;;;;e; ;- -; ce;;e;; ;--; --;: , + + II. · + + +-. e: + + + .-+ + · +-+ + + · ::)(:::::::+::::::::+:::: ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++++++. ++++++++++++++++++++.+++. Figure 16.1-17 Electron-hole recombina- tion via a trap. Because it takes both an electron and a hole for a recombination to occur, the rate of recombination is proportional to the product of the concentration of electrons and holes, i.e., rate of recombination == rnp, (16.1-19) where the recombination coefficient r (cm 3 /s) depends on the characteristics of the material, including its composition and defect density, and on temperature; it also depends relatively weakly on the doping level.
16.1 SEMICONDUCTORS 647 The equilibrium concentrations of electrons and holes no and Po are established when the generation and recombination rates are in balance. In the steady state, the rate of recombination must equal the rate of generation. If Go is the rate of thermal electron-hole generation at a given temperature, then, in thermal equilibrium, Go == rnopo. (16.1-20) The product of the electron and hole concentrations nopo == Go/r is approximately the same whether the material is n-type, p-type, or intrinsic. Thus, n1 == Go/r, which leads directly to the law of mass action nopo == n 1. This law is therefore seen to be a consequence of the balance between generation and recombination in thermal equilibrium. Electron-Hole Injection A semiconductor in thermal equilibrium with carrier concentrations no and Po has equal rates of generation and recombination, Go == rnopo. Now let additional electron-hole pairs be generated at a steady rate R (pairs per unit volume per unit time) by means of an external (nonthermal) injection mechanism, such as light falling on the material. A new steady state will be reached in which the concentrations are n == no + n and P == Po + p. It is clear, however, that n == P since the electrons and holes are created in pairs. Equating the new rates of generation and recombination, we obtain Go + R == rnp. Substituting Go == rnopo into (16.1-21) leads to R == r(np-nopo) == r (non + pon + n2) == rn(no+po+n), (16.1-22) . (16.1-21) which we write in the form R == n , T (16.1-23) with 1 T== r [ (no + Po) + n] . (16.1- 24) For an injection rate such that n « no + Po, (16.1-25) Excess-Carrier Recombination Lifetime In an n-type material, where no » Po, the recombination lifetime T l/rno is inversely proportional to the electron concentration. Similarly, for a p-type material where Po » no, we obtain T l/rpo. This simple formulation is not applicable when traps play an important role in the process. The parameter T may be regarded as the electron-hole recombination lifetime of the injected excess electron-hole pairs. This is readily understood by noting that the injected-carrier concentration is governed by the rate equation 1 T . r(no + Po) d(n) == R _ n dt T ' (16.1-26)
648 CHAPTER 16 SEMICONDUCTOR OPTICS which is similar to (14.2-3). In the steady state, d(n)/dt == 0 whereupon (16.1-23), which is like (14.2-13), is recovered. If the source of injection is suddenly removed (R becomes 0) at the time to, then n decays exponentially with time constant T, i.e., n(t) == n(to) exp[-(t - to)/T]. In the presence of strong injection, on the other hand, T is itself a function of n, as evident from (16.1-24), so that the rate equation is nonlinear and the decay is no longer exponential. If the injection rate R is known, the steady-state injected concentration may be detennined from n == RT, (16.1-27) permitting the total concentrations n == no + n and P == Po + n to be determined. Furthermore, if quasi-equilibrium is assumed, (16.1-11) may be used to detennine the quasi-Fermi levels. Quasi-equilibrium is not inconsistent with the balance of gener- ation and recombination assumed in the analysis above; it simply requires that the intraband equilibrium time be short in comparison with the recombination time T. This type of analysis will prove useful in developing theories of the semiconductor light-emitting diode and the semiconductor laser diode, which are based on enhancing light emission by means of carrier injection, as will become clear in Chapter 17. EXERCISE 16.1-4 Electron-Hole Pair Injection in GaAs. Assume that electron-hole pairs are injected into n- type GaAs (E g === 1.42 eV, me 0.07 mo, mv 0.50 mo) at a rate R === 10 23 /cm 3 -s. The thermal equilibrium concentration of electrons is no === 10 16 I cm 3 . If the recombination coefficient r === 10- 11 cm 3 Is and T === 300° K, determine: (a) The equilibrium concentration of holes Po. (b) The recombination lifetime T. (c) The steady-state excess concentration n. (d) The separation between the quasi-Ferrni levels E fe - E fv, assuming that T === 0° K. Internal Quantum Efficiency The internal quantum efficiency It i of a semiconductor material is defined as the ratio of the radiative electron-hole recombination coefficient to the total (radiative and nonradiative) recombination coefficient. This parameter is important because it determines the efficiency of light generation in a semiconductor material. The total rate of recombination is given by (16.1-19). If the recombination coefficient r is split into a sum of radiative and nonradiative parts, r == rr + r nr , the internal quantum efficiency is rr rr Iti == - == . r rr+r nr (16.1-28) The internal quantum efficiency may also be written in terms of the recombination lifetimes since T is inversely proportional to r [see (16.1-25)]. Defining the radiative and nonradiative lifetimes Tr and Tnr, respectively, leads to 1 1 1 -==-+-. T Tr Tnr (16.1-29)
16.1 SEMICONDUCTORS 649 The internal quantum efficiency is then r r / r == (1/ Tr ) / (1/ T ), or T Tnr Iti == - == . Tr Tr + T nr (16.1-30) Internal Quantum Efficiency The radiative recombination lifetime Tr governs the rate of photon absorption and emission, as explained in Sec. 16.2C. Its value depends on the carrier concentrations and the material parameter rr. For low to moderate injection rates, 1 Tr , rr(no + Po) (16.1-31) in accordance with (16.1-25). The nonradiative recombination lifetime is governed by a similar equation. However, if nonradiative recombination takes place via defect centers in the forbidden band, T nr is more sensitive to the concentration of these centers than to the electron and hole concentrations. Typical values for recombination coefficients and lifetimes are listed in Table 16.1- 4. Order-of-magnitude values are given for the radiative recombination coefficients rr; the radiative, nonradiative, and overall recombination lifetimes, Tr, Tnr, and T, respectively; and the internal quantum efficiencies It i. Table 16.1-4 Representative values for radiative recombination coefficients r r, recombination lifetimes, and internal quantum efficiencies Ili, for representative semiconductors. a Materia] rr (cm 3 Is) Tr Tnr T Ili Si 10- 15 10 ms 100 ns lOO ns 10- 5 GaAs 10- 10 100 ns 100 ns 50 ns 0.5 GaN b 10- 8 20 ns O. 1 ns O. 1 ns 0.005 a Assuming n-type material with a carrier concentration no = 10 17 / cm 3 and defect centers with a concentration 10 15 /cm 3 , at T = 300 0 K. b As a matter of practice, InGaN is used; this increases the internal quantum efficiency to Ili 0.3. The radiative lifetime for bulk Si is orders of magnitude longer than its overall lifetime, principally because of its indirect bandgap. This results in a small internal quantum efficiency. For GaAs and GaN, on the other hand, the decay is largely via radiative transitions (these materials have a direct bandgap), and consequently the internal quantum efficiency is large. Direct-bandgap materials are therefore useful for fabricating light-emitting structures, whereas indirect-bandgap materials generally are not. E. Junctions Juxtapositions of differently doped regions of a single semiconductor material are called homojunctions. An important example is the p-n junction, which is discussed in this section. Junctions between different semiconductor materials are called hetero- junctions. These are discussed subsequently.
650 CHAPTER 16 SEMICONDUCTOR OPTICS The p-n Junction The p-n junction is a homojunction between a p-type and an n-type semiconductor. It acts as a diode, which can serve in electronics as a rectifier, logic gate, voltage regulator (Zener diode), or tuner (varactor diode); and in optoelectronics as a light-emitting diode (LED), laser diode (LD), photodetector, or solar cell. A p-n junction consists of a p-type and an n-type section of the same semicon- ductor materials in metallurgical contact. The p-type region has an abundance of holes (majority carriers) and few mobile electrons (minority carriers); the n-type region has an abundance of mobile electrons and few holes (Fig. 16.1-18). Both charge carriers are in continuous random thennal motion in all directions. ....- ....-1....- 0 p-type l f n-type t" ;>. I I . . . . <l) s:: <l) s:: Ej ------------ 0 l- .. ... .... .... . ..... ... . . . ..... .. . . .. ... u . . ...:... ... <l) . ! I- ..... !- o u p n ------------ . Position Figure 16.1-18 Energy levels and carrier concentrations for a p-type and an n-type semiconductor before contact. n ------------ p When the two regions are brought into contact (Fig. 16.1-19), the following se- quence of events takes place: . Electrons and holes diffuse from areas of high concentration toward areas of low concentration. Thus, electrons diffuse from the n-region into the p-region, leaving behind positively charged ionized donor atoms. In the p-region the electrons re- combine with the abundant holes. Similarly, holes diffuse from the p-region into the n-region, leaving behind negatively charged ionized acceptor atoms. In the n-region the holes recombine with the abundant mobile electrons. This diffusion process does not continue indefinitely, however, because it causes a disruption of the charge balance in the two regions. . As a result, a narrow region on both sides of the junction becomes nearly depleted of mobile charge carriers. This region is called the depletion layer. It contains only the fixed charges (positive ions on the n-side and negative ions on the p- side). The thickness of the depletion layer in each region is inversely proportional to the concentration of dopants in the region. . The fixed charges create an electric field in the depletion layer that points from the n-side toward the p-side of the junction. This built-in field obstructs the diffusion of further mobile carriers through the junction region. . An equilibrium condition is established that results in a net built-in potential dif- ference V o between the two sides of the depletion layer, with the n-side exhibiting a higher potential than the p-side. . The built - in potential provides a lower potential energy for an electron on the n-side relative to the p-side. As a result, the energy bands bend as shown in Fig.
16.1 SEMICONDUCTORS 651 Depletion layer p - -- +++ : -- ++++ - - + + n -- -- +++ + --- + + 1( Electric field >--. b1) ;.... (]) (]) o ;.... .... u (]) fIJ . . . X Figure 16.1-19 A p-n junction in thermal equilibrium at T > 0° K. The depletion-layer, energy-band diagram, and concentrations (on a logarithmic scale) of mobile electrons n(x) and holes p (x) are shown as functions of the position x. The built-in potential difference Va corresponds to an energy e va, where e is the magnitude of the electron charge. . . . 6 1 .- ;... .... I- o u p(x) n(x) x-------- ----------'" 16.1-19. In thermal equilibrium there is only a single Fermi function for the entire structure so that the Fermi levels in the p- and n-regions must align. . No net current flows across the junction. The currents associated with diffusion and built-in field (drift current) cancel for both the electrons and holes. The Biased p-n Junction An externally applied potential will alter the potential difference between the p- and n- regions. This in turn will modify the flow of majority carriers, so that the junction can be used as a "gate." If the junction is forward biased by applying a positive voltage V to the p-region (Fig. 16.1-20), its potential is increased with respect to the n-region, so that an electric field is produced in a direction opposite to that of the built-in field. The presence of the external bias voltage causes a departure from equilibrium and a misalignment of the Fermi levels in the p- and n-regions, as well as in the depletion layer. The presence of two Fermi levels in the depletion layer, E fe and E fv, represents a state of quasi-equilibrium. p ( + I n >... CD 1-4 Q) s:: Q) s:: o .b u Q) · · · · ..£ ........._...._...J eV ---------- EJc E/ v --------- r _ c ..:- ...... .... ...: t e(V o - V) . . .. s:: .9 1-4..... Q)ro -8 ro s:: u8 s:: o u p(x) n(x) Excess /'). I / - - /)..; Exces s -- electron: _ L:: holes Figure 16.1-20 Energy-band diagram and carrier concentrations for a forward- biased p-n junction. x
652 CHAPTER 16 SEMICONDUCTOR OPTICS The net effect of the forward bias is to reduce the height of the potential-energy hill by an amount e V. The majority carrier current turns out to increase by an exponential factor exp( e V I kT) so that the net current becomes i == is exp( e VI kT) - is, where is is a constant. The excess majority carrier holes and electrons that enter the n- and p- regions, respectively, become minority carriers and recombine with the local majority carriers. Their concentration therefore decreases with distance from the junction as shown in Fig. 16.1-20. This process is known as minority carrier injection. If the junction is reverse biased by applying a negative voltage V to the p-region, the height of the potential-energy hill is augmented by e V. This impedes the flow of majority carriers. The corresponding current is multiplied by the exponential factor exp( e VI kT), where V is negative; i.e., it is reduced. The net result for the current is i == is exp( e V I kT) - is, so that a small current of magnitude is flows in the reverse direction when IVI » kTle. A p-n junction therefore acts as a diode with a current-voltage (i-V) characteristic i = is [ex p ( ) - 1] , (16.1-32) Ideal Diode Characteristic as illustrated in Fig. 16.1-21. The ideal diode characteristic in (16.1-32) is known as the Shockley equation. v v ) Is + + o o v (a) (b) (c) Figure 16.1-21 (a) Voltage and current in a p-n junction. (b) Circuit representation of the p-n junction diode. (c) Current-voltage characteristic of the ideal p-n junction diode. The response of a p-n junction to a dynamic (ac) applied voltage is determined by solving the set of differential equations governing the processes of electron and hole diffusion, drift (under the influence of the built-in and external electric fields), and recombination. These effects are important for determining the speed at which the diode can be operated. They may be conveniently modeled by two capacitances, a junction capacitance and diffusion capacitance, in parallel with an ideal diode. The junction capacitance accounts for the time necessary to change the fixed positive and negative charges stored in the depletion layer when the applied v oltage c hanges. The thickness l of the depletion layer turns out to be proportional to vi Vo - V ; it therefore increases under reverse-bias conditions (negative V) and decreases under forward-bias conditions (positive V). The junction capacitance C == f Ail (where A is the area of the junction) is therefore inversely proportional to vi Vo - V . The junction capacitance of a reverse-biased diode is smaller (and the RC response time is therefore shorter) than that of a forward-biased diode. The dependence of C on V is used to make voltage- variable capacitors (varactors). Minority carrier injection in a forward-biased diode is described by the diffusion capacitance, which depends on the minority carrier lifetime and the operating current.
16.1 SEMICONDUCTORS 653 The p-i-n Junction Diode A p-i-n junction diode is made by inserting a layer of intrinsic (or lightly doped) semiconductor material between a p-type region and an n-type region (Fig. 16.1-22). Because the depletion layer extends into each side of a junction by a distance inversely proportional to the doping concentration, the depletion layer of the p-i junction pene- trates deeply into the i-region. Similarly, the depletion layer of the i-n junction extends well into the i-region. As a result, the p-i-n diode can behave like a p-n junction with a depletion layer that encompasses the entire intrinsic region. The electron energy, density of fixed charges, and the electric field in a p-i-n junction diode in thermal equilibrium are illustrated in Fig. 16.1-22. One advantage of using a diode with a large depletion layer is its small junction capacitance and its consequent fast response. For this reason, p-i-n diodes are often favored over p-n diodes for use as semiconductor photodetectors. The large depletion layer also permits an increased fraction of the incident light to be captured, thereby increasing the photodetection efficiency (see Sec. 18.3B). Depletion layer / /L' X- I p . I n o ... Electric field Electron energy .,;,. .. . I .. Ec Ev G .. x Fixed-charge density G I I I !i I I I I I I I I I I I I I I I I I I \ .. x Figure 16.1-22 Electron energy, fixed-charge density, and electric field magnitude for a p-i-n junction diode in thermal equilibrium. Electric- field magnitude F. Heterojunctions Junctions between different semiconductor materials are known as heterojunctions. Optical sources and detectors make extensive use of heterojunctions in their designs; they are used not only as active regions but also as contact layers and waveguid- ing regions. The electron affinities of the materials determine the alignments of the conduction- and valence-band edges. It is often advantageous to lattice match the semi- conductor materials and to make use of graded junctions rather than abrupt ones. The juxtaposition of different semiconductors can have manifold advantages in photonics: . Junctions between materials of different bandgap create localized jumps in the energy-band diagram. A potential-energy discontinuity provides a barrier that can be useful in preventing selected charge carriers from entering regions where they are undesired. This property may be used in a p-n junction, for example, to reduce the proportion of current carried by minority carriers, and thus to increase injection efficiency (see Fig. 16.1-23).
654 CHAPTER 16 SEMICONDUCTOR OPTICS r r p 1/ n p ...._.. >. t ..- . ... .. -----. t g Eg, Eml _____ 1------ "I >. OJ) (l) c (l) c o ....... u (l) @ l . . - - - - - .. - - . . . .--_.- o Figure 16.1-23 The p-p-n double heterojunc- tion structure. The middle layer is of narrower bandgap than the outer layers. In equilibrium, the Fermi levels align so that the edge of the conduc- tion band drops sharply at the p-p junction and the edge of the valence band drops sharply at the p-n junction. The conduction- and valence-band dis- Eg3 continuities are known as band offsets. When the ! device is forward biased, these jumps act as barri- ers that confine the injected minority carriers to the region of lower bandgap. Electrons injected from the n-region, for example, are prevented from diffusing beyond the barrier at the p-p junction. Similarly, holes injected from the p-region are not permitted to diffuse beyond the energy barrier at the p-n junction. This double-heterostructure configuration therefore forces electrons and holes to occupy a narrow common region. This sub- stantially increases the efficiency of light-emitting diodes, semiconductor optical amplifiers, and laser diodes (see Chapter 17). . Discontinuities in the energy-band diagram created by two heterojunctions can be useful for confining charge carriers to a desired region of space. For example, a layer of narrow-bandgap material can be sandwiched between two layers of a wider bandgap material, as shown in the p-p-n structure illustrated in Fig. 16.1-23 (which consists of a p-p heterojunction and a p-n heterojunction). This double-heterostructure (DH) configuration is used effectively in the fabrica- tion of LEDs, semiconductor optical amplifiers, and laser diodes, as explained in Chapter 17. . Heterojunctions are useful for creating energy-band discontinuities that accelerate carriers at specific locations. The additional kinetic energy suddenly imparted to a carrier can be useful for selectively enhancing the probability of impact ionization in a multilayer avalanche photodiode (see Sec. 18.4A). . Semiconductors of different bandgap type (direct and indirect) can be used in the same device to select regions of the structure where light is emitted. Only semi- conductors of the direct-bandgap type can efficiently emit light (see Sec. 16.2). . Semiconductors of different bandgap can be used in the same device to select regions of the structure where light is absorbed. Semiconductor materials whose bandgap energy is larger than the photon energy incident on them will be trans- parent, acting as a window layer. . Heterojunctions of materials with different refractive indexes can be used to create photonic structures and optical waveguides that confine and direct photons, as discussed in Chapters 7 and 8. G. Quantum-Confined Structures Heterostructures of thin layers of semiconductor materials can be grown epitaxially, i.e., as layers of one semiconductor material over another, by using techniques such as molecular-beam epitaxy (MBE); liquid-phase epitaxy (LPE); and vapor-phase epi- taxy (YPE), of which common variants are metal-organic chemical vapor deposition (MOCYD) and hydride vapor-phase epitaxy (HYPE). Homoepitaxy is the growth of
16.1 SEMICONDUCTORS 655 materials that have the same composition as the substrate whereas heteroepitaxy is the growth of materials on a substrate of different composition, whether lattice-matched or not. MBE makes use of molecular beams of the constituent elements that are caused to impinge on an appropriately prepared substrate in a high-vacuum environment, LPE uses the cooling of a saturated solution containing the constituents in contact with the substrate, and VPE uses gases in a reactor. The compositions and dopings of the individual layers, which can be made as thin as monolayers, are determined by manipulating the arrival rates of the molecules and the temperature of the substrate surface. When the layer thickness is comparable to, or smaller than, the de Broglie wave- length of a thermalized electron, the quantized energy of an electron resident in the layer must be accommodated, in which case the energy-momentum relation for a bulk semiconductor material is no longer applicable. The de Broglie wavelength is expressed as 7\ == hip, where h is Planck's constant and p is the electron momen- tum (7\ 50 nm for GaAs). Three structures offer substantial advantages for use in photonics: quantum wells, quantum wires, and quantum dots (see Sec. I3.1C). The appropriate energy-momentum relations for these structures are derived below. Applications of these structures are deferred to Chapters 17 and 18. Quantum Wells A quantum-well structure, displayed in Fig. 16.1-24, is a double heterostructure con- sisting of an ultrathin (;S 50 nm) layer of semiconductor material whose bandgap is smaller than that of the surrounding material. An example is provided by a thin layer of GaAs surrounded by AIGaAs (see Fig. 13.1-11). The sandwich forms ID conduction- and valence-band rectangular potential wells within which electrons and holes are confined: electrons in the conduction-band well and holes in the valence-band well. A sufficiently deep potential well can be approximated as an infinite rectangular potential well (see Fig. 16.1-25). The energy levels Eq of a particle of mass m (me for electrons and mv for holes) confined to a one-dimensional infinite rectangular well of full width d are determined by solving the time-independent Schrodinger equation (13.1-3). As shown in Exercise 16.1-5, the energy levels turn out to be h 2 (q7r I d)2 E q == 2m ' q == 1,2,3,.... (16.1-33) As an example, the first three allowed energy levels of an electron in an infinitely deep GaAs well (me == 0.07 mo) of width d == 10 nm are Eq == 54, 216, and 486 meV, respectively (recall that kT == 26 me V at T == 300 0 K). The smaller the width of the well, the larger the separation between adjacent energy levels. EXERCISE 16.1-5 Energy Levels of a Quantum Well. Solve the Schrodinger equation (13.1-3) to determine the allowed energies of an electron of mass m in an infinitely deep one-dimensional rectangular potentia] well [V(x) == 0 for 0 < x < d and V(x) == 00 otherwise], confirming that Eq == fi2(q7r/d)2/2m, q == 1,2,3, . .. , as illustrated in Fig. 16.1-25(a). Compare these energies with those for the particular finite square quantum well shown in Fig. 16.1-25(b).
656 CHAPTER 16 SEMICONDUCTOR OPTICS dl r,"u1 E E 1 lE2 E...L r .. Li. t z y x (a) Eg ) x k (b) (c) Figure 16.1-24 (a) Geometry of the quantum-well structure. (b) Energy-level diagram for electrons and holes in a quantum well. (c) Cross section of the E-k relation in the direction of k 2 or k3. The energy subbands are labeled by their quantum number ql == 1,2,.. .. The E-k relation for bulk semiconductor is indicated by the dashed curves. fi2 E4 = 78.9 md . .4" . .4 . fi2 E3 = 44. md fi2 E 2 = 19.7 md fi2 E} = 4.9 md -d/2 d/2 (a) Continuum fi2 32.0 md '\. fi2 25.9 md V o E3 = 0.81 V o fi2 11.9 md fi2 3.2 md_ d / 2 E 2 = 0.37 V o El = 0.10 Va d/2 (b) Figure 16.1-25 Energy levels of (a) a one-dimensional infinite rectangular potential well, and (b) a finite square quantum well with an energy depth Va == 321i2/md 2 . However, semiconductor quantum wells are actually three-dimensional constructs. In the quantum-well structure shown in Fig. 16.1-24, electrons (and holes) are confined in the x direction to within a distance d 1 (the well thickness), but they extend over much larger dimensions (d 2 , d 3 » d 1 ) in the plane of the confining layer. Thus, in the y-z plane, they behave as if they were in bulk semiconductor. The electron energy-momentum relation is n 2 k 2 h 2 k 2 h 2 k 2 E == Ec + --1. + + ----1. , 2mc 2mc 2mc (16.1-34) where k 1 == ql7rjd 1 , k 2 == q27rjd2, k3 == Q37rjd3, and Ql, Q2, Q3 == 1,2,3,.... Since
16.1 SEMICONDUCTORS 657 d 1 « d 2 , d 3 , the parameter k 1 takes on well-separated discrete values, whereas k 2 and k3 have finely spaced discrete values that may be approximated as a continuum. It follows that the energy-momentum relation for electrons in the conduction band of a quantum well is given by fi 2 k 2 E == Ee + Eql + - , 2me ql == 1, 2, 3, . . . , (16.1- 35) where k is the magnitude of a two-dimensional k == (k 2 , k 3 ) vector in the y-z plane. Each quantum number ql corresponds to a subband whose lowest energy is Ee + Eql. Similar relations apply for the valence band. The energy-momentum relation for a bulk semiconductor is given by (16.1-3), where k is the magnitude of a three-dimensional vector k == ( k 1 , k 2 , k 3 ). The key distinction is that for the quantum well, k 1 takes on well-separated, discrete values. As a result, the density of states associated with a quantum-well structure differs from that associated with bulk material, for which the density of states is detennined from the magnitude of the three-dimensional vector with components k 1 == ql n I d, k 2 == q2 n I d, and k3 == q3n I d for d 1 == d 2 == d 3 == d. The result is g(k) == k 2 /n 2 per unit volume [see (16.1-6)], which yields the density of conduction-band states [see (16.1-7) and Fig 16.1-10] J2 m/2 t!c(E) = 7r 2 n} ..j E - Ec, E> O. (16.1-36) In a quantum-well structure the density of states is obtained from the magnitude of the two-dimensional vector (k 2 , k 3 ). For each quantum number ql the density of states is therefore g(k) == kin states per unit area in the y-z plane, and therefore klnd 1 per unit volume. The densities ge(E) and g(k) are related by ge(E) dE == g(k) dk == (klnd 1 ) dk. Finally, using the E-k relation (16.1-35) we obtain dE/dk == n 2 klm e , from which { me ge( E) == nn 2 d 1 ' 0, E > Ee + Eql ql == 1, 2, 3, . . . . (16.1-37) E < Ee + E q1 , Thus, for each quantum number ql, the density of states per unit volume is constant when E > Ee + Eql. The overall density of states is the sum of the densities for all values of ql, so that it exhibits the staircase distribution shown in Fig. 16.1-26. Each step of the staircase corresponds to a different quantum number ql and may be regarded as a subband within the conduction band (Fig. 16.1-24). The bottoms of these subbands move progressively higher for higher quantum numbers. It can be shown by substituting E == Ee + Eql in (16.1-36), and by using (16.1-33), that at E == Ee + Eql the quantum-well density of states is the same as that for the bulk material. The density of states in the valence band has a similar staircase distribution. In contrast with bulk semiconductor, the quantum-well structure exhibits a substan- tial density of states at its lowest allowed conduction-band energy level and at its highest allowed valence-band energy level. This property has an important effect on the optical characteristics of the material, as discussed in Sec. 17.2D. Multiquantum Wells and Superlattices Multilayered structures comprising alternating semiconductor materials are known as multiquantum-well (MQW) structures (see Fig. 16.1-27). They can be fabricated so that the energy bandgap varies with position in any desired way (see, e.g., Fig. 13.1-11).
658 CHAPTER 16 SEMICONDUCTOR OPTICS dl E I E g1 _J ---------------- 1 t ql= l ----- -- 1 -------- ."."" ------f-- -------Ec ---- E E 1 ------------------- ------------------- ------ ------------- , ql =2 ,,' , Bulk ...---- . x Density of states {!(E) Figure 16.1-26 Density of states for a quantum-well structure (solid curve) and for a bulk semiconductor (dashed curve). A MQW structure can have any number of layers, from just a few to hundreds. As an example, a MQW structure with 100 layers, each of thickness 10 nm and containing some 40 atomic planes, has an overall thickness 1 Mm. As discussed in Sec. 13.1C, if the energy barriers between adjacent wells are sufficiently thin so that electrons can readily tunnel through, the discrete energy levels broaden into minibands, in which case the multiquantum-well structure is referred to as a superlattice structure. The transition from MQW subbands to superlattice minibands is analogous to the transition from discrete energy levels in an atom to energy bands in a solid as the atoms are brought into closer proximity and permitted to interact (see Figs. 13.1-6 and 13.1-7). Quantum wells and superlattices can also be created by spatially varying the doping of a material, thereby creating space-charge fields that fonn potential barriers. GaAs Figure 16.1-27 A MQW structure fabricated from alternating layers of materials of different bandgaps, such as AIGaAs and GaAs. These particular materials are often used to illus- trate multiquantum-well structures because they can be lattice matched over a broad range of compositions [see Fig. 16.1- 7(a)], which minimizes the strain between the two lattices, and because of their large difference in bandgap energies [see Table 16.1-2], which provides substantial carrier confinement. Other combinations of MQW materials commonly used in pho- tonics include AUnAs /InGaAs, AUnGaP /InGaP, GaN /InGaN, and AlxGal-xN / AlyGal_yN. AIGaAs Biased Multiquantum-Well Structures The energy-band diagrams of unbiased and biased multiquantum-well and superlattice structures are schematized in Fig. 16.1-28. The electric field causes the wells to become canted and alters the energy levels. In superlattice structures, the discrete energy levels smear into minibands. Multiquantum-well structures find use in a wide variety of photonic devices, such as active regions in light-emitting diodes, semiconductor optical amplifiers, and laser diodes (see Secs. 17.1 C, 17.2D, and 17.4, respectively). They also serve as photo detectors (see Sec. 18.2C) and modulators (see Sec. 20.5).
16.1 SEMICONDUCTORS 659 (a) I . Miniband Minigap .. - - -.....111 (c) Figure 16.1-28 Energy-band diagrams of MQW and superlattice structures fabricated from alternating layers of materials with different bandgaps, such as AIGaAs and GaAs. (a) Unbiased MQW structure. (b) Biased MQW structure. (c) Biased superlattice structure with mini bands and mlnlgap. Quantum Wires A semiconductor material that takes the form of a thin wire surrounded by a material of wider bandgap is called a quantum-wire structure (Fig. 16.1-29). The wire acts as a potential well that narrowly confines electrons (and holes) in two directions, x and y. Assuming that the wire has a rectangular cross section of area d 1 d 2 , the energy- momentum relation in the conduction band is h 2 k 2 E == Ee + Eql + E q2 + - , 2me (16.1-38) where E _ ti?(ql1f/dd 2 ql - 2 ' me E _ /1,2(q2 1f / d 2)2 q2 - 2 ' me Ql,Q2==1,2,3,... (16.1-39) and k is the vector component in the z direction (along the axis of the wire). z x E E E E , , , , , , "'- "-- , , , , , , , Ec Ec --... Ec Ec Ev Ev ---___ Ev ... ... ..... Ev Bulk Quantum well Quantum wire Quantum dot Figure 16.1-29 The density of states in different confinement configurations. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in a greater number of dimensions.
660 CHAPTER 16 SEMICONDUCTOR OPTICS Each pair of quantum numbers (ql, q2) is associated with an energy subband that has a density of states {}( k) == 1/ TI per unit length of the wire and therefore 1/ Tld 1 d 2 per unit volume. The corresponding quantum-wire density of states (per unit volume), as a function of energy, is (}e( E) == (1/ d 1 d 2 ) (/ J2 TIn) vi E - Ee - Eql - E q2 ' 0, E > Ee + Eql + E q2 otherwise, ql, q2 == 1,2,3,.... (16.1-40) These are decreasing functions of energy, as illustrated in Fig. 16.1-29. Quantum Dots In a quantum-dot structure, the electrons are narrowly confined in all three directions within a region that we take to be a box of volume d 1 d 2 d 3 . The energy is therefore quantized to E == Ee + Eql + E q2 + E q3 , (1 6. 1-41) where E _ n 2 (ql 7r jdd 2 ql - 2 ' me E _ n 2 (q2 7r jd 2 )2 q2 - 2 ' me ql , q2, q3 == 1, 2, 3, . . . . E _ n2(q37rjd 3 )2 q3 - 2 ' me (16.1-42) The allowed energy levels are discrete and well separated so that the density of states is represented by a sequence of delta functions at the allowed energies, as illustrated in Fig. 16.1-29. Quantum dots are often called artificial atoms (see Sec. 13.1C). Even though they contain enormous numbers of strongly interacting natural atoms, the dis- crete energy levels of the quantum dot can, in principle, be chosen at will by proper design. 16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS We proceed to consider some of the basic optical properties of semiconductors, with an emphasis on the processes of absorption and emission important in the operation of photonic devices. This domain of study is known as semiconductor optics. A. Photon Interactions in Bulk Semiconductors A number of mechanisms can lead to the absorption and emission of photons in bulk semiconductors. The most important of these are: . Band-to-Band (Interband) Transitions. An absorbed photon can result in an elec- tron in the valence band making an upward transition to the conduction band, thereby creating an electron-hole pair [Fig. 16.2-1(a)]. Electron-hole recombi- nation can result in the emission of a photon. Band-to-band transitions may be assisted by one or more phonons. A phonon is a quantum of the lattice vibrations associated with molecular or acoustic vibrations of the atoms in a material.
t. Eg = 1.42 eV if T..... ....... ... T ... ... ..,. ... ... .... .. .++++++++++++++++++ +++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ (a) 16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 661 I _EA 0.088 eV. __-,-___ TE g = 0.66 eV !:;;;;;;;;+;;;++++;;+;++; :;+;;;;;;;;;;;;;;;;;;;;;; I .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .+++++.++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ .++++++++++++++++++++++++ ,++++++++++++++++++++++++ (b) (c) Figure 16.2-1 Examples of absorption and emission of photons in bulk semiconductors. (a) Band- to-band transitions in GaAs can result in the absorption or emission of photons of wavelength Aa < Ag == hcal Eg == 0.87 J-Lm. (b) The absorption of a photon of wavelength AA == hcal E A == 14 J-Lm results in a valence-band to acceptor-level transition in Hg-doped Ge (Ge:Hg). (c) Free-carrier transitions within the conduction band of Ge. . Impurity-to-Band Transitions. An absorbed photon can result in a transition be- tween a donor (or acceptor) level and a band in a doped semiconductor. In a p-type material, for example, a low-energy photon can lift an electron from the valence band to the acceptor level, where it becomes trapped by an acceptor atom [Fig. 16.2-1 (b)]. A hole is created in the valence band and the acceptor atom is ionized. Or a hole may be trapped by an ionized acceptor atom; the result is that the electron decays from its acceptor level to recombine with the hole. The energy may be released radiatively (in the form of an emitted photon) or nonradiatively (in the form of phonons). The transition may also be assisted by traps in defect states, as illustrated in Fig. 16.1-17. . Free-Carrier (Intraband) Transitions. An absorbed photon can impart its energy to an electron in a given band, causing it to move higher within that band. An electron in the conduction band, for example, can absorb a photon and move to a higher energy level within the conduction band [Fig. 16.2-1 ( c)]. This is followed by thermalization, a process whereby the electron relaxes down to the bottom of the conduction band while releasing its energy in the fonn of phonons. The strength of free-carrier absorption is proportional to the carrier density; it decreases with photon energy as a power-law function. . Phonon Transitions. Long-wavelength photons can release their energy by di- rectly exciting lattice vibrations, i.e., by creating phonons. . Excitonic Transitions. The absorption of a photon can result in the formation of an exciton. This entity is much like a hydrogen atom in which a hole plays the role of the proton. The hole and electron are bound together by their mutual Coulomb interaction. A photon may be emitted as a result of the electron and hole recombining, thereby annihilating the exciton. These transitions all contribute to the overall absorption coefficient, which is dis- played in Fig. 16.2-2 for Si and GaAs, and at greater magnification in Fig. 16.2-3 for a number of semiconductor materials. For photon energies greater than the bandgap energy Eg, the absorption is dominated by band-to-band transitions that form the basis of many photonic devices. The spectral region where the material changes from being relatively transparent (hv < Eg) to strongly absorbing (hv > Eg) is known as the absorption edge. Direct-bandgap semiconductors have a more abrupt absorption edge than indirect-bandgap materials, as is apparent from Figs. 16.2-2 and 16.2-3.
662 CHAPTER 16 SEMICONDUCTOR OPTICS 7 100 10 Wavelength Ao (J-Lm) 10 1.0 0.2 10 6 - GaAs - - - Si , , I I I I I I I I L Band- to- band ..... 10 4 '(3 S <l) o u 103 C .9 ..... 10 2 <r: ,-.... Ie 10 5 u '-' d 1 0.01 0.1 1.0 Photon energy hv (e V) 10.0 Figure 16.2-2 Observed optical absorption coefficient a versus pho- ton energy and wavelength for Si and GaAs in thermal equilibrium at T == 300 0 K. The bandgap energy Egis 1.12 eV for Si and 1.42 eV for GaAs. Silicon is relatively transparent in the band Ao 1.1 to 12 Mm, whereas in- trinsic GaAs is relatively transparent in the band Ao 0.87 to 12 Mm (see Fig. 5.5-1). 10 Wavelength Ao (J-Lm) 1.0 0.9 0.8 0 7 0 6 0 5 I I J 0.4 0.3 10 5 (j 10 4 ..... s:: <l) 'u S 8 10 3 U s:: .9 e. 10 2 <r: GaN ,-.... I e u '-' 10 o 0.5 1.0 1.5 3.0 3.5 4.0 4.5 2.0 2.5 Photon energy hv (e V) Figure 16.2-3 Absorption coefficient versus photon energy and wavelength for Ge, Si, GaAs, GaN and selected other 111- V binary semiconductors at T == 300 0 K, on an expanded scale. B. Band-to-Band Transitions in Bulk Semiconductors We proceed to develop a simple theory of direct band-to-band photon absorption and emission in bulk semiconductors, ignoring the other types of transitions. Bandgap Wavelength Direct band-to-band absorption and emission can take place only at frequencies for which the photon energy hv > Eg. The minimum frequency v necessary for this to occur is v 9 == E 9 I h, so that the corresponding maximum wavelength is >"g == Co I v 9 == hcol Eg. If the bandgap energy is given in eV (rather than in J), the bandgap wavelength
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 663 Ag == hcal eEg in Mill turns out to be (16.2-1) Bandgap Wavelength Ag (J1m) and Eg (eV) The quantity Ag is known as the bandgap wavelength (or cutoff wavelength). The bandgap wavelength Ag, and its associated bandgap energy Eg, are provided in Table 16.1-2, and in Figs. 16.1-7 and 16.1-8, for a number of semiconductor materials of importance in photonics. 111- V ternary and quaternary semiconductors of different compositions span a substantial range of bandgap wavelengths, from the mid-infrared to the mid-ultraviolet, as is evident in Fig. 16.2-4. \ r-...; 1.24 /\ 9 r-...; E . 9 10 5 3 2 1.5 1.00.90.8 0.7 0.6 0.5 I I 0.4 0.3 0.2 )..g (JLm) InN I InN A1Xfu::: r ,N I : Inl_xGa x Asl_yP y InP InP AlP InAs GaAs InAs GaAs GaSb GaAs IGe ISi o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Eg(eV) Figure 16.2-4 Bandgap wavelength Ag, and corresponding bandgap energy Eg, for selected elemental and 111- V binary, ternary, and quaternary semiconductor materials. Successive rows, starting at the top, represent AlInGaN, AIGaN, InGaN, InGaAsP, AlInGaP, InGaP, GaAsP, AIGaAs, InGaAs, and GaAsSb. The shaded regions indicate compositions for which the materials are direct- bandgap semiconductors. Conditions for Absorption and Emission Electron excitation from the valence to the conduction band may be induced by the absorption of a photon of appropriate energy (hv > Eg or A < Ag). An electron-hole pair is generated [Fig. 16.2-5(a)]. This adds to the concentration of mobile charge carriers and increases the conductivity of the material. The material behaves as a photoconductor with a conductivity proportional to the photon flux. This effect is used to detect light, as discussed in Chapter 18. Electron deexcitation from the conduction to the valence band (electron-hole re- combination) may result in the spontaneous emission of a photon of energy hv > E 9 [Fig. 16.2-5(b)], or in the stimulated emission of a photon [Fig. 16.2-5(c)], provided that a photon of energy hv > Egis initially present (see Sec. 13.3). Spontaneous emission is the underlying phenomenon on which the light-emitting diode is based, as will be seen in Sec. 17.1. Stimulated emission is responsible for the operation of semiconductor optical amplifiers and laser diodes, as will be seen in Secs. 17.2, 17.3, and 17.4. The conditions under which absorption and emission take place are summarized as follows:
664 CHAPTER 16 SEMICONDUCTOR OPTICS t II II IhV Iff! : E E E 2 _____mm__m_ 1111111111111111111 ) k (a) I I I I I I I I I I I I I I I I I I I ) k (b) 1111111111111111111 ) k (c) Figure 16.2-5 (a) The absorption of a photon results in the generation of an electron-hole pair. This process is used in the photodetection of light. (b) The recombination of an electron-hole pair results in the spontaneous emission of a photon. Light-emitting diodes (LEDs) operate on this basis. (c) Electron-hole recombination can be induced by a photon. The result is the stimulated emission of an identical photon. This is the unded ying process responsible for the operation of semiconductor laser diodes. . Conservation of Energy. The absorption or emission of a photon of energy hv requires that the energies of the two states involved in the interaction (say E 1 and E 2 in the valence band and conduction band, respectively, as depicted in Fig. 16.2-5) be separated by hv. Thus, for photon emission to occur by electron- hole recombination, for example, an electron occupying an energy level E 2 must interact with a hole occupying an energy level E 1, such that energy is conserved, 1.e., E 2 - E I == hv. (16.2-2) . Conservation of Momentum. Momentum must also be conserved in the process of photon emissionl absorption, so that P2 - PI == hv I c == hi A, or k 2 - k I == 27r / A. The photon-momentum magnitude hi A is, however, very small in comparison with the range of momentum values that electrons and holes can assume. The semiconductor E-k diagram extends to values of k of the order 27r / a, where the lattice constant a is much smaller than the wavelength A, so that 27r I A « 27r I a. The momenta of the electron and the hole participating in the interaction must therefore be approximately equal. This condition, k 2 k I , is called the k- selection rule. Transitions that obey this rule are represented in the E-k diagram (Fig. 16.2-5) by vertical lines, indicating that the change in k is negligible on the scale of the diagram. . Energies and Momenta of the Electron and Hole with Which a Photon Interacts. As is apparent from Fig. 16.2-5, conservation of energy and momentum require that a photon of frequency v interact with electrons and holes of specific energies and momenta determined by the semiconductor E-k relation. Using (16.1-3) and (16.1-4) to approximate this relation for a direct-bandgap semiconductor by two parabolas, and writing Ec - Ev == Eg, (16.2-2) may be written in the form fi 2 k 2 fi 2 k 2 E 2 - E I == - + Eg + - == hv, 2mv 2mc (16.2-3) from which k 2 = 2;r (hv - Eg), (16.2-4)
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 665 where 1 1 1 -==-+-. m r mv me (16.2-5) Substituting (16.2-4) into (16.1-3) provides that the energy levels E 1 and E 2 with which the photon interacts are m r E 2 == Ee + - (hv - Eg) me m r ) E 1 == Ev - - (hv - Eg == E 2 - hv. mv ( 16.2-6) (16.2-7) In the special case when me == mv, we obtain E 2 == Ee+ (hv- Eg), as required by symmetry. . Optical Joint Density of States. We now determine the density of states g(v) with which a photon of energy hv interacts under conditions of energy and momentum conservation in a direct-bandgap semiconductor. This quantity incorporates the density of states in both the conduction and valence bands and is called the optical joint density of states. The one-to-one correspondence between E 2 and v embodied in (16.2-6) permits us to readily relate g(v) to the density of states ge(E 2 ) in the conduction band by use of the incremental relation ge(E 2 ) dE 2 == g(v) dv, from which g(v) == (dE2/dv)ge(E2)' so that hm r g(v) == - ge(E 2 ). me ( 16.2-8) Using (16.1-7) and (16.2-6), we finally obtain the number of states per unit vol- ume per unit frequency: (2mr )3/2 Q(V) = 7r!i 2 J hv - Eg , hv > Eg, ( 16.2-9) Optical Joint Density of States which is illustrated in Fig. 16.2-6. The one-to-one correspondence between E 1 and v in (16.2-7), together with gv (E 1) from (16.1-8), results in an expression for g(v) identical to (16.2-9). g{v) Eg hv Figure 16.2-6 The density of states with which a photon of energy hv interacts increases with hv - Eg in accordance with a square-root law. . Photon Emission Is Unlikely in an Indirect-Bandgap Semiconductor. Radiative electron-hole recombination is unlikely in an indirect-bandgap semiconductor. This is because transitions from near the bottom of the conduction band to near the top of the valence band (where electrons and holes, respectively, are most likely
666 CHAPTER 16 SEMICONDUCTOR OPTICS to reside) requires an exchange of momentum that cannot be accommodated by the emitted photon. Momentum may be conserved, however, by the participation of phonons in the interaction. Phonons can carry relatively large momenta but typically have small energies ( 0.01-0.1 e V; see Fig. 16.2-2), so their transitions appear horizontal on the E-k diagram (see Fig. 16.2-7). The net result is that momentum is conserved, but the k-selection rule is violated. Because phonon- assisted emission involves the participation of three bodies (electron, photon, and phonon), the probability of its occurrence is quite low. Thus, Si, which is an indirect-bandgap semiconductor, has a substantially lower radiative recombina- tion coefficient than does GaAs, which is a direct-bandgap semiconductor (see Table 16.1-4). Silicon is therefore not an efficient light emitter, whereas GaAs is. E '1 1 11 I I Figure 16.2-7 Photon emISSIon in an indirect-bandgap semiconductor. The recom- bination of an electron near the bottom of the conduction band with a hole near the top of the valence band requires the exchange of energy and momentum. The energy may be carried off by a photon, but one or more phonons are also required to conserve momentum. This type of multi particle inter- action is therefore unlikely. k . Photon Absorption Is Not Unlikely in an Indirect-Bandgap Semiconductor. Al- though photon absorption also requires energy and momentum conservation in an indirect-bandgap semiconductor, this is readily achieved by means of a two-step process (Fig. 16.2-8). The electron is first excited to a high energy level within the conduction band by a k-conserving vertical transition. It then quickly relaxes to the bottom of the conduction band by a process called thermalization in which its momentum is transferred to phonons. The generated hole behaves similarly. Since the process occurs sequentially, it does not require the simultaneous presence of three bodies and is thus not unlikely. Silicon is therefore an efficient photon detector, as is GaAs. EI I Photon I absorption hv I I II IA Thennalization I II Figure 16.2-8 Photon absorption in an indirect-bandgap semiconductor via a ver- tical (k-conserving) transition. The photon generates an excited electron in the con- duction band, leaving behind a hole in the valence band. The electron and hole then undergo fast transitions - to the lowest and highest possible levels in the conduction and valence bands, respectively, releasing their energy in the form of phonons. Since the process is sequential it is not unlikely. k c. Absorption, Emission, and Gain in Bulk Semiconductors We now proceed to determine the probability densities of a photon of energy hv being emitted or absorbed by a bulk semiconductor material in a direct band-to-band
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 667 transition. Conservation of energy and momentum, in the form of (16.2-4), (16.2- 6), and (16.2-7), determines the energies Eland E 2 , and the momentum hk, of the electrons and holes with which the photon may interact. Three factors detennine these probability densities, as discussed below: 1. Occupancy probabilities 2. Transition probabilities 3. Optical joint density of states Occupancy Probabilities The occupancy conditions for photon emission and absorption by means of transitions between the discrete energy levels E 2 and E 1 are stated as follows: Emission condition: A conduction-band state of energy E 2 is filled (with an electron) and a valence-band state of energy E 1 is empty (i.e., filled with a hole). Absorption condition: A conduction-band state of energy E 2 is empty and a valence- band state of energy E 1 is filled. The probabilities that these occupancy conditions are satisfied for various values of E 2 and E 1 are detennined from the appropriate Fermi functions fc( E) and fv (E) asso- ciated with the conduction and valence bands of a semiconductor in quasi-equilibrium. Thus, the probability fe (v) that the emission condition is satisfied for a photon of energy hv is the product of the probabilities that the upper state is filled and that the lower state is empty (these are independent events), i.e., fe(v) == fc(E 2 ) [1 - fv(E1)J. (16.2-10) The energies E 1 and E 2 are related to v by (16.2-6) and (16.2-7). Similarly, the probability fa (v) that the absorption condition is satisfied is fa (v) == [1 - fc(E 2 )] fv(E1). (16.2-11) EXERCISE 16.2-1 Requirement for the Photon Emission Rate to Exceed the Absorption Rate. (a) For a bulk semiconductor in thermal equilibrium, show that !e(v) is always smaller than !a(v) so that the rate of photon emission cannot exceed the rate of photon absorption. (b) For a semiconductor in quasi -equilibrium (E f e -I- E f v), with radiative transitions occurring between a conduction-band state of energy E 2 and a valence-band state of energy E 1 with the same value of k, show that emission is more likely than absorption if the separation between the quasi-Fermi levels is larger than the photon energy, i.e., if E fe - E fv > hv. (16.2-12) Condition for Net Emission What does this condition imply about the locations of E fe relative to Ee, and E fv relative to Ev?
668 CHAPTER 16 SEMICONDUCTOR OPTICS Transition Probabilities Satisfying the emission/absorption occupancy condition does not assure that the emission/absorption actually takes place. These processes are governed by the proba- bilistic laws of interaction between photons and atomic systems examined at length in Secs. 13.3A-13.3C (see also Exercise 13.3-1). As they relate to semiconductors, these laws are generally expressed in terms of emission into (or absorption from) a narrow band of frequencies between v and v + dv: Summary A radiative transition between two discrete energy levels Eland E 2 is charac- terized by a transition cross section a(v) == ()..2/8T1t sp )g(v), where v is the fre- quency, t sp is the spontaneous lifetime, and 9 (v) is the lineshape function [which has linewidth f1v centered about the transition frequency vo == (E 2 - E1)/h and has unity area]. In semiconductors, the radiative electron-hole recombination lifetime Tr, which was discussed in Sec. 16.1D, plays the role of t sp so that .x 2 a(v) == _ 8 g(v) . TlTr (16.2-13) . If the occupancy condition for emission is satisfied, the probability density (per unit time) for the spontaneous emission of a photon into any of the available radiation modes in the narrow frequency band between v and v+dv IS 1 Psp(v) dv == - g(v) dv. Tr (16.2-14) . If the occupancy condition for emission is satisfied and a mean spectral photon-flux density cPv (photons per unit time per unit area per unit fre- quency) at frequency v is present, the probability density (per unit time) for the stimulated emission of one photon into the narrow frequency band between v and v + dv is )..2 Wi (v) dv == 4Jv a(v) dv == q;v- g(v) dv. 8T1T r (16.2-15) . If the occupancy condition for absorption is satisfied and a mean spectral photon- flux density cPv at frequency v is present, the probability density for the absorption of one photon from the narrow frequency band between v and v + dv is also given by (16.2-15). Since each transition has a different central frequency vo, and since we are consid- ering a collection of such transitions, we explicitly label the centra] frequency of the transition by writing g(v) as gvo(v). In semiconductors the homogeneously broadened lineshape function gvo (v) associated with a pair of energy levels generally has its origin in electron-phonon collision broadening. It therefore typically exhibits a Lorentzian lineshape [see (13.3-34) and (13.3-37)] with width f1v l/TI T 2 , where the electron- phonon collision time T 2 is of the order of picoseconds. If T 2 == 1 ps, for example, then f1v == 318 GHz, corresponding to an energy width hf1v 1.3 meV. The radiative lifetime broadening of the levels is negligible in comparison with collisional broadening.
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 669 Overall Emission and Absorption Transition Rates For a pair of energy levels separated by E 2 - El == hVa, the rates of spontaneous emission, stimulated emission, and absorption of photons of energy hv (in units of photonsjs-Hz-cm 3 of the semiconductor material), at the frequency v, are obtained as follows: The appropriate transition probability density Psp(v) or Wi (v) [as provided in (16.2-14) or (16.2-15)] is multiplied by the appropriate occupation probability fe (va) or fa (va ) [as given in (16.2-10) or (16.2-11)], and by the density of states that can interact with the photon g(va) [as set forth in (16.2-9)]. The overall transition rate for all allowed frequencies is then calculated by integrating over Va. The rate of spontaneous emission at frequency v, for example, is given by fsp(V) = J [(I/Tr)gvO(v)) !e(VO) l?(vo) dvo. (16.2-16) When the collision-broadened width f1v is substantially less than the width of the product fe(va)g(va), which is the usual situation, gva(v) may be approximated by b(v - va), whereupon the transition rate simplifies to rsp(v) == (ljTr)g(v)fe(v). The rates of stimulated emission and absorption are obtained in a similar fashion, which leads to the following formulas: 1 rsp(v) == - O(v)fe(v) Tr .x 2 rst(v) == cPv- O(v)fe(v) 87rT r .x 2 rab(V) == cPv- g(v)fa(v). 87rT r (16.2-17) (16.2-18) (16.2-19) Emission and Absorption Rates These equations, together with (16.2-9)-(16.2-11), permit the rates of spontaneous emission, stimulated emission, and absorption arising from direct band-to-band transi- tions (photonsjs-Hz-cm 3 ) to be calculated in the presence of a mean spectral photon- flux density cPv (photonsjs-Hz-cm 2 ). The products g(v)fe(v) and g(v)fa(v) are anal- ogous to the products of the lineshape function and atomic number densities in the upper and lower levels, g(v)N 2 and g(v)N 1 , respectively, used in Chapters 13-15 to study emission and absorption in atomic systems. The determination of the occupancy probabilities fe(v) and fa(v) requires knowl- edge of the quasi-Fermi levels E fe and E fv. It is via the control of these two parameters (by the application of an external bias to a p-n junction, for example) that the emission and absorption rates are modified to produce semiconductor photonic devices that carry out different functions. Equation (16.2-17) is the basic result that describes the opera- tion of the light-emitting diode (LED), a semiconductor source based on spontaneous emission (see Sec. 17.1). Equation (16.2-18) is applicable to semiconductor optical amplifiers and laser diodes, which operate on the basis of stimulated emission (see Sees. 17.2-17.4). Equation (16.2-19) is appropriate for semiconductor detectors that function by means of photon absorption (see Sec. 18.1B). Spontaneous-Emission Spectral Intensity in Thermal Equilibrium A semiconductor in thermal equilibrium has only a single Fermi function so that (16.2- 10) becomes fe(v) == f(E 2 )[1 - f(El)]. If the Fermi level lies within the bandgap, away from the band edges by at least several times kT, use may be made of the
670 CHAPTER 16 SEMICONDUCTOR OPTICS exponential approximations to the Fermi functions, f ( E 2) exp [ - ( E 2 - E f ) / kT] nd 1- f(El) exp[-(E f - E1)/kT], whereuponfe(v) exp[-(E 2 - E1)/kT], I.e., fe(v) ex p ( - z; ) . (16.2-20) Substituting (16.2-9) for Q(v) and (16.2-20) for fe(v) into (16.2-17) therefore provides ( hv - E ) fsp(V) Do J hv - Eg exp - kT 9 , hv > Eg, (16.2-21) where _ (2m r )3/2 ( _ E g ) Do - 7rn2Tr exp kT (16.2-22) is a parameter that increases with temperature at an exponential rate. The spontaneous emission rate (16.2-21), which is plotted versus hv in Fig. 16.2-9, comprises two factors: a function associated with the density of states that increases as the square-root of hv - Eg, and an exponentially decreasing function of hv - Eg arising from the Fermi function. The spontaneous emission rate can be increased by augmenting fe(v). In accordance with (16.2-10), this can be achieved by purposely causing the material to depart from thermal equilibrium in such a way that fc(E2) is made large and fv(E 1 ) is made small. This assures an abundance of both electrons and holes, which is the desired condition for the operation of an LED, as discussed in Sec. 1 7 .1. r sp ( v) Figure 16.2-9 Spectral intensity of the direct band-to-band spontaneous emission rate rsp(v) (photons/s-Hz- cm 3 ) from a semiconductor in thermal equilibrium, as a function of hv. The spectrum has a low-frequency cutoff at v == E 9 / h and extends over a range hv of frequencies of approximate width 2kT/h. Eg Gain Coefficient in Quasi-Equilibrium The net gain coefficient "Yo (v) corresponding to the rates of stimulated emission and absorption in (16.2-18) and (16.2-19) is determined by taking a cylinder of unit area and incremental length dz, and assuming that a mean spectral photon-flux density is directed along its axis (see Fig. 14.1-1). If cPv(z) and cPv(z) + dcPv(z) are the mean spectral photon-flux densities entering and leaving the cylinder, respectively, dcpv(z) must be the mean spectral photon-flux density emitted from within the cylinder. The
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 671 incremental number of photons, per unit time per unit frequency per unit area, is simply the number of photons gained, per unit time per unit frequency per unit volume [rst(v) - rab(V)], multiplied by the thickness of the cylinder dz. Hence, dcPv(z) == [rst(v) - rab(V)] dz. Substituting the rates set forth in (16.2-18) and (16.2-19) leads to dq;;(z) = 8 A2 e(v) [fe(v) - fa(v)] q;v(z) = 1'o(v) q;v(z). Z 7rTr (16.2-23) The net gain coefficient is therefore .x 2 fio(V) == _ 8 g(v) fg(v), 7rTr (16.2-24) Gain Coefficient where the Fermi inversion factor fg(v) is given by fg(v) fe(v) - fa (V) == fe(E 2 ) - fv(E 1 ), (16.2-25) as may be seen from (16.2-10) and (16.2-11), with Eland E 2 related to v by (16.2- 6) and (16.2-7). Comparing (16.2-24) with (14.1-4) reveals that g(v) fg(v) in the semiconductor system plays the role of Ng(v) in the atomic system. Using (16.2-9), the gain coefficient may be cast in the fonn with fio(v) == Dl V hv - Eg fg(v), hv > Eg J2 m/2.x2 Dl == h 2 Tr (16.2-26a) (16.2-26b) The sign and spectral form of the Fermi inversion factor fg(v) are governed by the quasi-Fermi levels E fe and E fv, which in turn depend on the state of excitation of the carriers in the semiconductor. As shown in Exercise 16.2-1, this factor is positive (corresponding to a population inversion and net gain) only when E fe - E fv > hv. When the semiconductor is pumped to a sufficiently high level by means of an external source of power, this condition may be satisfied and net gain achieved, as we shall see in Sec. 17.2. This is the physics underlying the operation of semiconductor optical amplifiers and laser diodes. Absorption Coefficient in Thermal Equilibrium A semiconductor in thermal equilibrium has only a single Fermi level E f == E fe == E fv, so that 1 fc(E) = fv(E) = f(E) = exp[(E _ Ef)/kT] + 1 . (16.2-27) The factor fg(v) == fe(E 2 ) - fv(E 1 ) == f(E 2 ) - f(El) < 0, and therefore the gain coefficient fiO(V) is always negative [since E 2 > El and f(E) decreases monotonically with E]. This is true whatever the location of the Fermi level E f. Thus, a semiconduc- tor in thermal equilibrium, whether it be intrinsic or doped, always attenuates light. The attenuation (absorption) coefficient, a(v) == -fio(V), is therefore a(v) == Dl V hv - Eg [f(E 1 ) - f(E 2 )] , (16.2-28) Absorption Coefficient
672 CHAPTER 16 SEMICONDUCTOR OPTICS where E 2 and E 1 are given by (16.2-6) and (16.2-7), respectively, and D 1 is given by (16.2-26b). If E f lies within the bandgap but away from the band edges by an energy of at least several times kT, then f(E 1 ) 1 and f(E 2 ) 0 so that [f(E 1 ) - f(E 2 )] 1. In that case, the direct band-to-band contribution to the absorption coefficient is V2 c2m/2 1 a(v) Tr (hv)2 J hv - Ego (16.2-29) Equation (16.2-29) is plotted in Fig. 16.2-10 for GaAs, using the following parameters: n == 3.6, me == 0.07 mo, mv == 0.50 mo, mo == 9.1 X 10- 31 kg, a doping level such that Tr == 0.4 ns (this differs from that given in Table 16.1-4 because of the difference in doping level), Eg == 1.42 eV, and a temperature such that [f(E 1 ) - f(E 2 )] 1. As the temperature increases, f (E 1) - f ( E 2) decreases below unity and the absorption coefficient set forth in (16.2-28) is reduced. 10 4 2 Wavelength Ao (/-lm) 1 0.5 0.4 -- ..... 'S 0.5xl0 4 "S' '-" (j o 1 (hv- Eg) (eV) 2 Figure 16.2-10 Calculated absorp- tion coefficient a(v) (cm- 1 ) resulting from direct band-to-band transitions, as a function of the photon energy hv (eV) and wavelength Ao (J1m), for GaAs. This curve should be compared with the experimental result shown in Fig. 16.2-3, which includes all absorp- tion mechanisms. o -1 In accordance with (16.2-29), absorption near the band edg e in a direct-bandgap semiconductor should follow the functional form J hv - Eg. However, the sharp onset of absorption at hv == Egis an idealization. As is evident in Fig. 16.2-3, direct- bandgap semiconductors generally exhibit an exponential absorption tail, known as the Urbach tail, with a characteristic width kT that extends slightly into the for- bidden band. This is associated with thennal and static disorder in the crystal arising from several factors, including phonon-assisted absorption, randomness in the doping distribution, and variations in material composition. Absorption near the band edge in indirect-bandgap semiconductors (e.g., Ge, Si, and GaP in Fig. 16.2-3) generally follows the functional form (hv - Eg)2 rather than the square-root relation applicable for direct-bandgap semiconductors. EXERCISE 16.2-2 Wavelength of Maximum Band-to-Band Absorption. Use (16.2-29) to determine the (free- space) wavelength Ap at which the absorption coefficient of a semiconductor in thermal equilibrium is maximum. Calculate the value of Ap for GaAs. Note that this result applies only to absorption by direct band-to-band transitions.
16.2 INTERACTIONS OF PHOTONS WITH CHARGE CARRIERS 673 D. Photon Interactions in Quantum-Confined Structures Multiquantum-well and superlattice structures were considered in Sec. 16.1G. The photon interactions in these structures bear considerable resemblance to those for bulk semiconductors (see Sec. 16.2A). Several mechanisms play important roles in absorp- tion and emission in quantum-confined structures: . Interband (band-to-band) transitions . Excitonic transitions . Intersubband transitions . Miniband transitions These are illustrated in Fig. 16.2-11 and discussed below. .... Id'tr I I I iloJ I o . --- (a) (b) (c) (d) Figure 16.2-11 Photon absorption and emission in multiquantum-well structures. (a) Interband transitions. (b) Excitonic transitions. (c) Intersubband transitions. (d) Miniband transitions in a superlattice structure. . Interband Transitions. Interband emission and absorption takes place between states in the valence and conduction bands [Fig. 16.2-11 (a)], much as in bulk semiconductors. Because of quantum confinement, however, the optical joint den- sity of states (16.2-9) must be replaced by (17.2-11). lnterband transitions are responsible for the operation of MQW light-emitting diodes, superluminescent diodes, and laser diodes (see Figs. 17.1-20, 17.2-11, and 17 .4-8, respectively), as well as MQW electroabsorption modulators (see Fig. 20.5-2). . Excitonic Transitions. The ID carrier confinement associated with MQW struc- tures results in an increase in the exciton binding energy. This leads to strong excitonic transitions, even at T == 300 0 K, as schematized in Fig. 16.2-11 (b). Excitonic transitions play an important role in many quantum-confined devices, including MQW electroabsorption modulators (see Fig. 20.5-2). . Intersubband Transitions. Transitions that take place between energy levels within a single band of a MQW structure [Fig. 16.2-11(c)] are known as intersub- band transitions. Devices that operates on the basis of these intraband transi- tions include the quantum-well quantum cascade laser [see Fig. 17.4-6(a)] and the quantum-well infrared photodetector (see Fig. 18.2-3). In the latter device, the absorption of a photon causes a transition from a bound energy level to the continuum. The picosecond carrier dynamics of intersubband systems offer very large bandwidths. . Miniband Transitions. In superlattices, the discrete MQW energy levels broaden into minibands that are separated by minigaps. Such miniband transitions [Fig. 16.2-11 (d)] play a crucial role in the operation of superlattice quantum cascade lasers [see Fig. 17.4-6(b)]. Such transitions, as well as intersubband transitions, exhibit fast relaxation and large nonlinearities, and therefore offer promise for applications such as all-optical switching and demultiplexing.
674 CHAPTER 16 SEMICONDUCTOR OPTICS E. Refractive Index The ability to control the refractive index of a semiconductor is important in the design of many photonic devices, particularly those that make use of optical waveguides, integrated optics, and laser diodes. Semiconductor materials are dispersive, so that the refractive index is dependent on the wavelength. Indeed, the refractive index is related to the absorption coefficient a(v) inasmuch as the real and imaginary parts of the susceptibility must satisfy the Kramers-Kronig relations (see Sec. 5.5B and Sec. B.1 of Appendix B). The group index and refractive index for GaAs, calculated from the Sellmeier equation discussed in Sec. 5.5C, are displayed in Fig. 16.2-12. The refractive index depends on temperature and doping level. The refractive indexes of 3.8 >< Q) "0 .s 3.7 Q) :> . ""'"' U 3.6 ro Q) 3.5 , '...... ................- n .------ 3.4 1 234 Wavelength Ao (/-lill) Figure 16.2-12 Refractive index nand group index N for GaAs as a function of the wavelength Ao. The results are calculated from the Sellmeier equation provided in Ta- ble 5.5-1. selected elemental and binary bulk semiconductors, under specific conditions and near the bandgap wavelength, are provided in Table 16.2-1. Table 16.2-1 Refractive indexes of selected semi- conductor materials. a Material Refracti ve Index Elemental semiconductors Ge Si 4.0 3.5 111- V binary semiconductors AIN AlP AlAs AISb GaN GaP GaAs GaSb InN InP InAs InSb 2.2 3.0 3.2 3.8 2.5 3.3 3.6 4.0 3.0 3.5 3.8 4.2 aResults reported are for photon energies near the bandgap energy of the material (hv Eg) and at T = 300 0 K. The refractive indexes of ternary and quaternary semiconductors can be approximated by linear interpolation between the refractive indexes of their components.
READING LIST 675 READING LIST Books on Semiconductor Physics, Devices, and Optics See also the reading lists in Chapters 17 and] 8. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, Wiley, 3rd ed. 2006. S. S. Islam, Semiconductor Physics and Devices, Oxford University Press, 2006. D. A. Neamen, An Introduction to Semiconductor Devices, McGraw-Hill, 2006. S. Kasap and P. Capper, eds., Springer Handbook of Electronic and Photonic Materials, Springer- Verlag, 2006. L. J. Olafsen, R. M. Biefeld, M. C. Wanke, and A. W. Saxler, eds., Progress in Semiconductor Ma- terials V-Novel Materials and Electronic and Optoelectronic Applications, Materials Research Society Symposium Proceedings Volume 891, Materials Research Society, 2006. M. Kuball, T. H. Myers, J. M. Redwing, and T. Mukai, eds., GaN, AIN, InN and Related Materials, Materials Research Society Symposium Proceedings Volume 892, Materials Research Society, 2006. A. Moliton, Optoelectronics of Molecules and Polymers, Springer-Verlag, 2006. B. G. Streetman and S. Banerjee, Solid State Electronic Devices, Prentice Hall, 6th ed. 2005. C. F. Klingshirn, Semiconductor Optics, Springer-Verlag, 1997, 2nd ed. 2005. S. O. Kasap, Principles of Electronic Materials and Devices, McGraw-Hill, 3rd ed. 2005. W. Barford, Electronic and Optical Properties of Conjugated Polymers, Oxford University Press, 2005. P. Wiirfel, Physics of Solar Cells: From Principles to New Concepts, Wiley-VCH, 2005. C. Kittel, Introduction to Solid State Physics, Wiley, 8th ed. 2004. K. Seeger, Semiconductor Physics: An Introduction, Springer-Verlag, 9th ed. 2004. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconduc- tors, World Scientific, 4th ed. 2004. J. Singh, Electronic and Optoelectronic Properties of Semiconductor Structures, Cambridge Univer- sity Press, 2003. P. T. Landsberg, Recombination in Semiconductors, Cambridge University Press, paperback ed. 2003. Y. Toyozawa, Optical Processes in Solids, Cambridge University Press, paperback ed. 2003. P. K. Basu, Theory of Optical Processes in Semiconductors: Bulk and Microstructures, Oxford Uni- versity Press, paperback ed. 2003. K. K. Ng, Complete Guide to Semiconductor Devices, Wiley-IEEE, 2nd ed. 2002. W. Schafer and M. Wegener, Semiconductor Optics and Transport Phenomena, Springer-Verlag, 2002. S. M. Sze, Semiconductor Devices: Physics and Technology, Wiley, 2nd ed. 2001. M. Fox, Optical Properties of Solids, Oxford University Press, 2001. K. A. Jackson and W. Schroter, eds., Handbook of Semiconductor Technology, Wiley-VCH, 2000. K. F. Brennan, The Physics of Semiconductors with Applications to Optoelectronic Devices, Cam- bridge University Press, 1999. K. Hess, Advanced Theory of Semiconductor Devices, Wiley-IEEE, 1999. M. Fukuda, Optical Semiconductor Devices, Wiley, 1999. S. Adachi, Optical Properties of Crystalline and Amorphous Semiconductors: Materials and Funda- mental Principles, Kluwer, 1999. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials, Springer-Verlag, 1999. H. Morko, Nitride Semiconductors and Devices, Springer-Verlag, 1999. M. Pope and C. E. Swenberg, Electronic Processes in Organic Crystals and Polymers, Oxford Uni- versi ty Press, 2nd ed. 1999. R. K. Willardson and E. R. Weber, eds., Semiconductors and Semimetals, Volume 57, Gallium Nitride (GaN) II, J. I. Pankove and T. D. Moustakas, eds., Academic Press, 1999. R. K. Willardson and E. R. Weber, eds., Semiconductors and Semimetals, Volume 50, Gallium Nitride (CaN) I, J. I. Pankove and T. D. Moustakas, eds., Academic Press, 1998.
676 CHAPTER 16 SEMICONDUCTOR OPTICS S. R. Rotman, ed., Wide-Gap Luminescent Materials: Theory and Applications, Kluwer, 5th ed. 1997. E. F. Schubert, ed., Delta-Doping of Semiconductors, Cambridge University Press, 1996. E. F. Schubert, Doping in III-V Semiconductors, Cambridge University Press, 1993. N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics, Prentice Hall, 1993. R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley, 2nd ed. 1985. J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press, paperback ed. 1979. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers, Part A, Fundamental Principles, Aca- demic Press, 1978. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976. A. van der Ziel, Solid State Physical Electronics, Prentice Hall, 3rd ed. 1976. J. I. Pankove, Optical Processes in Semiconductors, Prentice Hall, 1971; Dover, reissued 1975. Books on Quantum-Confined Materials and Nanostructures R. Paiella, ed., Intersubband Transitions in Quantum Structures, McGraw-Hill, 2006. S. Y. Ren, Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Springer- Verlag, 2006. M. Grundmann, The Physics of Semiconductors: An Introduction Including Devices and Nanophysics, Springer- Verlag, 2006. M. J. O'Connell, ed., Carbon Nanotubes, CRC Press, 2006. L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge University Press, 2006. C. N. R. Rao and A. Govindaraj, Nanotubes and Nanowires, Royal Society of Chemistry, 2005. E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures, Alpha Science, 2005. P. N. Prasad, Nanophotonics, Wiley, 2004. H. Kalt and M. Hetterich, eds., Optics of Semiconductors and Their Nanostructures, Springer-Verlag, 2004. T. Steiner, ed., Semiconductor Nanostructures for Optoelectronic Applications, Artech, 2004. Y. Masumoto and T. Takagahara, eds., Semiconductor Quantum Dots: Physics, Spectroscopy and Applications, Springer-Verlag, 2002. P. Harrison, Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semicon- ductor Nanostructures, Wiley, 2000. F. T. Vasko and A. V. Kuznetsov, Electronic States and Optical Transitions in Semiconductor Het- erostructures, Springer-Verlag, 1999. V. Mitin, V. Kochelap, and M. A. Stroscio, Quantum Heterostructures: Microelectronics and Opto- electronics, Cambridge University Press, 1999. R. Willardson and E. R. Weber, eds., Semiconductors and Semimetals, Volume 62, Intersubband Transitions in Quantum Wells: Physics and Device Applications I, H. C. Liu and F. Capasso, eds., Academic Press, 1999. R. K. Willardson and E. R. Weber, eds., Semiconductors and Semimetals, Volume 66, Intersubband Transitions in Quantum Wells: Physics and Device Applications II, H. C. Liu and F. Capasso, eds., Academic Press, 1999. J. T. Londergan, J. P. Carini, and D. P. Murdock, Binding and Scattering in Two-Dimensional Systems: Application to Quantum Wires, Waveguides and Photonic Crystals, Springer-Verlag, 1999. S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals, Cambridge University Press, 1998. T. Ruf, Phonon Raman-Scattering in Semiconductors, Quantum Wells and Superlattices: Basic Re- sults and Applications, Springer-Verlag, 1998. H. Yokoyama and K. Ujihara, eds., Spontaneous Emission and Laser Oscillation in Microcavities, CRC Press, 1995. L. Banyai and S. W. Koch, Semiconductor Quantum Dots, World Scientific, 1993. Articles Issue on optoelectronic materials and processing and nanostructures, IEEE Journal of Selected Topics in Quantum Electronics, vol. 11, no. 6, 2005.
PROBLEMS 677 T. Shinada, S. Okamoto, T. Kobayashi, and I. Ohdomari, Enhancing Semiconductor Device Perfor- mance Using Ordered Dopant Arrays, Nature, vol. 437, pp. 1128-1131,2005. G. Malliaras and R. Friend, An Organic Electronics Primer, Physics Today, vol. 58, no. 5, pp. 53-58, 2005. G. P. Collins, Next Stretch for Plastic Electronics, Scientific American, vol. 291, no. 2, pp. 74-81, 2004. J. Wu, W. Walukiewicz, K. M. Yu, J. W. Ager III, S. X. Li, E. E. Haller, H. Lu, and W. J. Schaff, Universal Bandgap Bowing in Group-III Nitride Alloys, Solid State Communications, vol. 127, pp. 411-414,2003. Issue on optoelectronic materials and processing, IEEE Journal of Selected Topics in Quantum Elec- tronics, vol. 8, no. 4, 2002. Issue on nanostructures and quantum dots, IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, no. 5, 2002. D. Gammon and D. G. Steel, Optical Studies of Single Quantum Dots, Physics Today, vol. 55, no. 10, pp. 36-41, 2002. Issue on organics for photonics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 5, 2001. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. Issue on nanostructures and quantum dots, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 3, 2000. Issue on organic electroluminescence, IEEE Journal of Selected Topics in Quantum Electronics, vol. 4, no. I, 1998. M. Riordan and L. Hoddeson, The Origins of the pn Junction, IEEE Spectrum, vol. 34, no. 6, pp. 46- 51, 1997. D. A. B. Miller, Optoelectronic Applications of Quantum Wells, Optics & Photonics News, vol. 1, no.2,pp. 7-15,1990. S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller, Linear and Nonlinear Optical Properties of Semiconductor Quantum Wells, Advances in Physics, vol. 38, pp. 89-188, 1989. L. Esaki, A Bird's-Eye View on the Evolution of Semiconductor Superlattices and Quantum Wells, IEEE Journal of Quantum Electronics, vol. QE-22, pp. 1611-1624, 1986. PROBLEMS 16.1-6 Donor-Electron Ionization Energies and Radii. Estimate the donor electron ionization energies ED and Bohr radii Tl for the semiconductor materials listed below (see Sec. 13.1A and Example 16.1-1). Comment, in each case, on the role of thermal excitations and the use of the bulk dielectric constants in your calculations. (a) A silicon crystal, with electron effective mass mc = 0.98 mo (see Table 16.1-1) and dielectric constant E/Eo = 12.3 (see Table 16.2-1). (b) A gallium arsenide crystal, with electron effective mass mc = 0.07 mo (see Table 16.1- 1) and dielectric constant E/ Eo = 13 (see Table 16.2-1). (c) A gallium nitride crystal, with electron effective mass mc = 0.20 mo (see Table 16.1-1) and dielectric constant E/ Eo = 6.25 (see Table 16.2-1). (d) A sample of Na+ -doped polyacetylene, an n-type conjugated polymer semiconductor, with electron effective mass mc = mo and dielectric constant E/ Eo = 3. Organic light- emitting diodes operate on the basis of recombination radiation from bound excitons. 16 1-7 Fermi Level of an Intrinsic Semiconductor. Given the expressions (16.1-12) and (16.1- 13) for the thermal equilibrium carrier concentrations in the conduction and valence bands: (a) Determine an expression for the Fermi level E f of an intrinsic semiconductor and show that it falls exactly in the middle of the bandgap only when the effective mass of the electrons mc is precisely equal to the effective mass of the holes mc'
678 CHAPTER 16 SEMICONDUCTOR OPTICS (b) Determine an expression for the Fermi level of a doped semiconductor as a function of the doping level and the Fermi level determined in (a). 16.1-8 Electron-Hole Recombination Under Strong Injection. Consider electron-hole recombi- nation under conditions of strong carrier-pair injection such that the recombination lifetime can be approximated by T = 1/ r n, where r is the recombination coefficient of the material and n is the injection-generated excess carrier concentration. Assuming that the source of injection R is set to zero at t = to, find an analytical expression for n(t), demonstrating that it exhibits power-law rather than exponential behavior. 16.1-9 Bowing Parameters for Ternary Semiconductors. The lattice constant of a ternary semi- conductor alloy, say AxB1-xC, typically varies linearly with the composition x, in ac- cordance with Vegard's law. The bandgap energy Eg, on the other hand, usually varies nonlinearly with x so that a plot of bandgap energy versuslattice constant exhibits a bowed shape. This relation is usually modeled by the quadratic equation EBC(X) = Ecx + E: c (1 - x) - bx(l - x), where b is called the bowing parameter. Use the curves provided in Figs. 16.1-7 and 16.1-8 to determine the bowing parameters for AlxGa1_xAs, GaAs1-xPx, AlxGa1- x N, InxGa1-xN, Alxln1_ x N, and Hg x Cd 1 - x Te. What significance does the bowing parameter have with respect to lattice matching of the ternary compound to a substrate? * 16.1-10 Energy Levels in a GaAsl AIGaAs Quantum Well. (a) Draw the energy-band diagram of a single-crystal multiquantum-well structure of GaAs/AIGaAs to scale on the energy axis when the AIGaAs has the composition Alo.3Gao.7As. The bandgap of GaAs, Eg(GaAs), is l.42 eV; the bandgap of AIGaAs increases above that of GaAs by 12.47 me V for each 1 % increase in the Al composition. Because of the inherent characteristics of these two materials, the depth of the GaAs conduction-band quantum well is about 60% of the total conduction-plus- valence band quantum-well depths. (b) Assume that a GaAs conduction-band well has depth as determined in (a) above and precisely the same energy levels as the finite square well shown in Fig. 16.1-25(b), for which (m va d 2 /2fi2) 1/2 = 4, where va is the depth of the well. Find the total width d of the GaAs conduction-band well. The effective mass of an electron in the conduction band of GaAs is me 0.07 mo = 0.64 X 10- 31 kg. 16.2-3 Validity of the Approximation for Absorption/Emission Rates. The derivation of the rate of spontaneous emission made use of the approximation gvo (v) 8 (v - vo) in the course of evaluating the integral rsp(v) = J [ r 9 VO (V)] fe(vo) g(vo) dvo. (a) Demonstrate that this approximation is satisfactory for GaAs by plotting the functions gvo(v), !e(VO), and Q(vo) at T = 300 0 K and comparing their widths. GaAs is colli- sionally lifetime broadened with T 2 1 ps. (b) Repeat (a) for the rate of absorption in thermal equilibrium. 16.2-4 Peak Spontaneous Emission Rate in Thermal Equilibrium. (a) Determine the photon energy hv p at which the direct band-to-band spontaneous emis- sion rate from a semiconductor material in thermal equilibrium achieves its maximum value when the Fermi level lies within the bandgap and away from the band edges by at least several times kT. (b) Show that this peak rate (photons per sec per Hz per cm 3 ) is given by Do 2(m r )3/2 ( Eg ) rsp(v p ) = M:: ill = ,;e fi2 ill exp -- k . y 2e e 7r Tr T (c) What is the effect of doping on this result? (d) Assuming that Tr = 0.4 ns, me = 0.07 mo, mv = 0.50 mo, and Eg = 1.42 eV, find the peak rate in GaAs at T = 300 0 K.
PROBLEMS 679 16.2-5 Radiative Recombination Rate in Thermal Equilibrium. (a) Show that the direct band-to-band spontaneous emission rate integrated over all emis- sion frequencies (photons per sec per cm 3 ) is given by 1 ( v ) dv = D -fiT ( kT ) 3/2 = (m r )3/2 ( kT ) 3/2 ex p( - E 9 ) sp 0 2h M 2 3/21:::3 kT ' o vn n provided that the Fermi level is within the semiconductor energy gap and away from the band edges. Note: Io oo x 1 / 2 e-J-Lx dx = (-fiT /2)J1-3/2. (b) Compare this with the approximate integrated rate obtained by multiplying the peak rate obtained in Prob. 16.2-4 by the approximate frequency width 2kT / h shown in Fig. ] 6.2-9. (c ) Using (16.1-15), set the phenomenological equilibrium radiative recombination rate rrnp = rrn (photons per second per cm 3 ) introduced in Sec. 16.1D equal to the direct band-to-band result derived in (a) to obtain the expression for the radiative re- combination coefficient V2 n 3 / 2 fi3 1 rr = (me + mv)3/2 (kT)3/2 Tr . (d) Use the result in (c) to find the value of r r for GaAs at T = 300 0 K using me = 0.07 mo, mv = 0.5 mo, and Tr = 0.4 ns. Compare this with the value provided in Table 16.1-4 (rr 10- 10 cm 3 /s).
CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES 17.1 LIGHT-EMITTING DIODES A. Injection Electroluminescence 8. LED Characteristics C. Materials and Device Structures 17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS A. Gain and Bandwidth B. Pumping C. Heterostructures D. Quantum-Well Structures E. Superluminescent Diodes 17.3 LASER DIODES A. Amplification, Feedback, and Oscillation B. Power and Efficiency C. Spectral and Spatial Characteristics 17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS A. Quantum-Confined Lasers B. Microcavity Lasers C. Materials and Device Structures 682 702 716 728 -- --- - --- - - - - - - - - - - -.... - - - -...- --- -., - The operation of semiconductor laser diodes was reported nearly simultaneously in 1962 by independent research teams from the General Electric Corporation, IBM Corporation, and Lincoln Laboratory of the Massachusetts Institute of Technology., 680
Light can be emitted from a semiconductor material as a result of electron-hole re- combination. However, materials capable of emitting such light do not glow at room temperature because the concentrations of thermally excited electrons and holes are too small to produce discernible radiation. On the other hand, an external source of energy can be used to produce electron-hole pairs in sufficient numbers such that they produce large amounts of spontaneous recombination radiation, causing the material to glow or luminesce. A convenient way of achieving this is to forward bias a p-n junction, which has the effect of injecting electrons and holes into the same region of space in the vicinity of the junction; the resulting recombination radiation is then called injection electroluminescence (see Sec. 13.5). A light-emitting diode (LED) is a forward-biased p-n junction fabricated from a direct-bandgap semiconductor material that emits light via injection electrolumines- cence [Fig. 17.0-1 (a)]. If the forward voltage is increased beyond a certain value, the number of electrons and holes in the junction region can become sufficiently large so that a population inversion is achieved, whereupon stimulated emission (i.e., emis- sion induced by the presence of photons) becomes more prevalent than absorption. The junction region may then be used as a semiconductor optical amplifier (SOA) [Fig. 17.0-1(b)] or, with appropriate feedback, as a laser diode (LD) [Fig. 17.0-1(c)]. 'i , + + p n p n p n k"J , (a) (b) II (c) II Figure 17.0-1 A forward-biased semiconductor p-n junction diode operated as: (a) a light- emitting diode (LED), (b) an semiconductor optical amplifier (SOA), and (c) a laser diode (LD). Semiconductor photon sources, in the form of both LEDs and LDs, serve as highly efficient electronic-to-photonic transducers. They have become indispensable in many applications by virtue of their small size, high brightness, high efficiency, high relia- bility, ruggedness, and durability. Visible LEDs are widely used as indicator lights and in cellular phones, computers, television receivers, games, information displays, flash- lights, signage, automotive lighting, traffic signals, architectural lighting, and liquid- crystal-display backlighting. Infrared LEDs often serve as remote controls for con- sumer products such as optical mice, headphones, microphones, and keyboards. Ultra- violet LEDs are useful in applications such as water purification, surgical sterilization, equipment and personnel decontamination, and non-line-of-sight covert communica- tions. They are also useful for the detection of chemical and biological agents, many of which fluoresce at particular wavelengths when exposed to ultraviolet light. Laser diodes find extensive use in high-density optical data-storage systems such as digital- video-disc (DVD) players; long-haul optical fiber communication systems; and scan- ning, reading, and high-resolution color printing systems. They also serve as efficient optical pumping sources for optical fiber amplifiers and solid-state lasers. As a partic- ular convenience, they can be readily modulated by controlling the injected current. 681
682 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES This Chapter This chapter is devoted to the study of light-emitting diodes (Sec. 17.1), semiconductor optical amplifiers (Sec. 17.2), laser diodes (Sec. 17.3), and quantum-confined and mi- crocavity lasers (Sec. 17.4). As background, we draw broadly on the material presented in Chapter 16. The theoretical treatments of semiconductor optical amplifiers and laser- diode oscillators closely parallel the analyses of laser amplifiers and laser oscillators provided in Chapters 14 and 15, respectively. 17.1 LIGHT-EMITTING DIODES Electroluminescence, first observed in 1907, is a phenomenon in which light is emitted by a material that is subjected to an electric field (see Sec. 13.5). Injection electrolu- minescence underlies the operation of light-emitting diodes, highly efficient devices capable of emitting light of any color. LEDs have become enormously important in a number of areas of photonics. We discuss the theory of injection electroluminescence in Sec. 17.1A, the characteristics of light-emitting diodes in Sec. 17.1B, and typical materials and device structures in Sec. 17.1 C. A. Injection Electroluminescence Electroluminescence in Thermal Equilibrium Electron-hole radiative recombination results in the emission of light from a semicon- ductor material. At room temperature the concentration of thermally excited electrons and holes is so small, however, that the generated photon flux is very small. EXAMPLE 17.1-1. Photon Emission from GaAs in Thermal Equilibrium. At room temperature, the intrinsic concentration of electrons and holes in GaAs is n1 1.8 x 10 6 cm- 3 (see Table 16.1-3). Since the radiative electron-hole recombination coefficient rr 10- 10 cm 3 /s (as specified in Table 16.1-4 for certain conditions), the electroluminescence rate rrnp = rrn 324 photons/cm 3 -s, as discussed in Sec. 16.1D. Using the bandgap energy for GaAs, Eg = 1.42 eV = 1.42 x 1.6 x 10- 19 J, this emission rate corresponds to an optical power density = 324 x 1.42 x 1.6 x 10- 19 7.4 X 10- 17 W/cm 3 . A 2-J1m layer of GaAs therefore produces an intensity I 1.5 X 10- 20 W/cm 2 , which is negligible. Light emitted from a layer of GaAs thicker than about 2 J1m suffers reabsorption. If thermal equilibrium conditions are maintained, this intensity cannot be apprecia- bly increased (or decreased) by doping the material. In accordance with the law of mass action provided in (16.1-17), the product np is fixed at n 1 if the material is not too heavily doped so that the recombination rate rrnp == rrn1 depends on the doping level only through rr. An abundance of electrons and holes is required for a large recombination rate; in an n-type semiconductor n is large but p is small, whereas the converse is true in a p-type semiconductor. Electroluminescence in the Presence of Carrier Injection The photon emission rate can be appreciably increased by using external means to increase excess electron-hole pairs in the material. This may be accomplished, for example, by illuminating the material with light, but it is typically achieved by forward biasing a p-n junction diode, which serves to inject carrier pairs into the junction region. This process is illustrated in Fig. 16.1-20 and will be explained further in
17.1 LIGHT-EMITTING DIODES 683 Sec. 17.IB. The photon emission rate may be calculated from the electron-hole pair injection rate R (pairs/cm 3 -s), where R plays the role of the laser pumping rate (see Sec. 14.2). The photon flux <I> (photons per second), generated within a volume V of the semiconductor material, is directly proportional to the carrier-pair injection rate (see Fig. 17.1-1). (rate R) Figure 17.1-1 Spontaneous photon emis- sion resulting from electron-hole radiative recombination, as might occur in a forward- biased p-n junction. Injected carriers Denoting the equilibrium concentrations of electrons and holes in the absence of pumping as no and Po, respectively, we use n == no + n and P == Po + P to represent the steady-state carrier concentrations in the presence of pumping (see Sec. 16.1D). The excess electron concentration n is precisely equal to the excess hole concentration P because electrons and holes are produced in pairs. It is assumed that the excess electron-hole pairs recombine at a rate 1/ T, where T is the overall (radiative and nonradiative) electron-hole recombination time. Under steady-state conditions, the generation (pumping) rate must precisely balance the recombination (decay) rate, so that R == n/ T. Thus, the steady-state excess-carrier concentration is proportional to the pumping rate, i.e., n == RT. (17.1-1) For carrier injection rates that are sufficiently low, as explained in Sec. 16.1D, we have T l/r(no + Po) where r is the (radiative and nonradiative) recombination coefficient, so that R r n (no + Po). Only radiative recombinations generate photons, however, and the internal quantum efficiency Ili == rr/r == T /Tr, defined in (16.1-28) and (16.1-30), accounts for the fact that only a fraction of the recombinations are radiative in nature. The injection of RV carrier pairs per second therefore leads to the generation of a photon flux <I> == Il i RV photons/s, i.e., Vn <I> == Ili RV == Ili T Vn (17.1-2) Tr The interna] photon flux <I> is proportional to the carrier-pair injection rate Rand therefore to the steady-state concentration of excess electron-hole pairs n. The internal quantum efficiency Il i plays a crucial role in determining the per- formance of this electron-to-photon transducer. Direct-bandgap semiconductors are usually used to make LEDs (and laser diodes) because Ili is substantially larger than for indirect-bandgap semiconductors (e.g., at room temperature Ili 0.5 for GaAs, whereas Ili 10- 5 for Si, as shown in Table 16.1-4). The internal efficiency Ili depends on the doping, temperature, and defect concentration of the material.
684 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES EXAMPLE 17.1-2. Injection Electroluminescence Emission from GaAs. Under cer- tain conditions, T = 50 ns and Iti = 0.5 for GaAs (see Table 16.1-4), so that a steady-state excess concentration of injected electron-hole pairs n = 10 17 em -3 will give rise to a photon flux concentration Itin/T 10 24 photons/cm 3 -s. This corresponds to an optical power density 2.3 x 10 5 W/cm 3 for photons at the bandgap energy Eg = 1.42 eVe A 2-J1m-thick slab of GaAs therefore produces an optical intensity of 46 W Icm 2 , which is a factor of 10 21 greater than the thermal equilibrium value calculated in Example 17.1-1. Under these conditions the power emitted from a device of area 200 J1m x 10 J1m is 0.9 mW. Spectral Intensity of Electroluminescence Photons The spectral intensity of injection electroluminescence light may be determined by using the direct band-to-band emission theory developed in Sec. 16.2. The rate of spontaneous emission rsp(v) (number of photons per second per Hz per unit volume), as provided in (16.2-17), is 1 rsp(v) == -g(v)!e(v), Tr (17.1-3) where Tr is the radiative electron-hole recombination lifetime. The optical joint density of states for interaction with photons of frequency v, as given in (16.2-9), is (2mr )3/2 e(v) = nn 2 J hv - Eg, (1 7 . 1-4) where m r is related to the effective masses of the holes and electrons by l/mr == l/mv + l/me [as given in (16.2-5)], and Eg is the bandgap energy. The emission condition [as given in (16.2-10)] provides !e(V) == !e(E2)[1 - !v(E 1 )], (17.1-5) which is the probability that a conduction-band state of energy E 2 == Ee + m r (hv - Eg) me ( 1 7.1-6) is filled and a valence-band state of energy El == E 2 - hv ( 1 7 .1- 7) is empty, as provided in (16.2-6) and (16.2-7) and illustrated in Fig. 17.1-2. Equa- tions (17.1-6) and (17.1- 7) guarantee that energy and momentum are conserved. The Fermi functions fe(E) == l/{exp[(E - Efe)/kT] + I} and fv(E) == l/{exp[(E- E fv)/ kT] + I} that appear in (17.1-5), with quasi-Fermi levels E fe and E fv, apply to the conduction and valence bands, respectively, under conditions of quasi-equilibrium. The semiconductor parameters Eg, Tr, mv, and me, and the temperature T, deter- mine the spectral distribution rsp(v), given the quasi-Fenni levels E fe and E fv. These in turn are determined from the concentrations of electrons and holes given in (16.1-10) and (16.1-11),
17.1 LIGHT-EMITTING DIODES 685 E --------- 2 --------1 Ec -------r---- Eg Ev _______t____ ... . . . . . . . . El --------- .--.--.,....-- · hv Figure 17.1-2 The spontaneous emission of a photon resulting from the recombination of an electron of energy E 2 with a hole of energy El == E 2 - hv. The transition is represented by a vertical arrow because the momentum carried away by the photon, hv / c, is negligible on the scale of the figure. . k rOO Qc(E)fc(E) dE = n = no + b.n; lEe l Ev -00 Qv(E)[l - fv(E)] dE = P = Po + b.n. (17.1-8) The densities of states near the conduction- and valence-band edges are, respectively, as per (16.1-7) and (16.1-8), ( 2m e)3/2 Qc(E) = 2fi3 J E - Ec; 27r ( 2m v)3/2 Qv(E) = 2fi3 J Ev - E, 27r (17.1-9) where no and Po are the concentrations of electrons and holes in thermal equilibrium (in the absence of injection), and n == RT is the steady-state injected-carrier con- centration. For sufficiently weak injection, such that the Fermi levels lie within the bandgap and away from the band edges by several kT, the Fermi functions may be approximated by their exponential tails. The spontaneous photon flux (integrated over all frequencies) is then obtained from the spontaneous emission rate r sp (v) by <I> == V 1 00 r ( v ) dv = V(m r )3/2 ( kT ) 3/2 e x p( Efe - E fv - E g ) ( 17.1-10 ) sp M 2 3/23 kT' o v 7r It Tr as is readily extrapolated from Prob. 16.2-5. Increasing the pumping level R causes n to increase, which in turn moves E fe toward (or further into) the conduction band, and E fv toward (or further into) the valence band. This results in an increase in the probability fe( E 2 ) of finding the conduction-band state of energy E 2 filled with an electron, and the probability 1 - fv(El) of finding the valence-band state of energy El empty (filled with a hole). The net result is that the emission-condition probability fe(v) == fe(E 2 )[1 - fv(E 1 )] increases with R, thereby enhancing the spontaneous emission rate given in (17.1-3) and the spontaneous photon flux <I> given above. EXERCISE 17.1-1 Quasi-Fermi Levels of a Pumped Semiconductor. (a) Under ideal conditions at T == 0° K, when there is no thermal electron-hole pair generation [see Fig. 17.1-3(a)], show that the quasi-Fermi levels are related to the concentrations of injected
686 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES electron-hole pairs n by E fe = Ee + (37r 2 )2/3.-£. (n)2/3 2me (17.1-11a) E = E - ( 31T2 ) 2/3 ( n ) 2/3 fv v 2 ' mv (17.1-11b) so that E - E = E + ( 31T2 ) 2/3 ( n ) 2/3 fe fv 9 2 ' m r (17.1-11c) where n » no, Po. Under these conditions all n electrons occupy the lowest allowed energy levels in the conduction band, and all p holes occupy the highest allowed levels in the valence band. Compare with the results of Exercise 16.1-3. (b) Sketch the functions !e(v) and rsp(v) for two values of n. Given the effect of temperature on the Fermi functions, as illustrated in Fig. 17 .1-3(b), determine the effect of increasing the temperature on r sp (v). E E, Ej \ I fc(E) fA E) ------.----- ----- E fc EI ..... . . ----. E fv ..... ----:------- ----E .. fv fv(E) fv(E) . k ) k (a) (b) Figure 17.1-3 Energy bands and Fermi functions for a semiconductor in quasi-equilibrium (a) at T = 0° K, and (b) at T > 0° K. EXERCISE 17.1-2 Spectral Intensity of Injection Electroluminescence under Weak Injection. For suffi- ciently weak injection, such that Ee - E fe » kT and E fv - Ev » kT, the Fermi functions may be approximated by their exponential tails. Show that the luminescence rate can then be expressed as ( hv - E ) rsp(v) = D V hv - Eg exp - kT 9 , hv > Eg, (1 7 .1-12a) where D (2mr)3/2 ( Efe-Efv-Eg ) = exp 1Tn2 kT ( 17 .1-12b ) is an exponentially increasing function of the separation between the quasi-Fermi levels E fe - E fv. The spectral intensity of the spontaneous emission rate is shown in Fig. 17.1-4; it has precisely the same shape as the thermal-equilibrium spectral intensity shown in Fig. 16.2-9, but its magnitude is increased by the factor D / Do = exp [( E f e - E f v) / kT], which can be very large in a presence of
17.1 LIGHT-EMITTING DIODES 687 injection. In thermal equilibrium E fe = E fv, so that (16.2-21) and (16.2-22) are recovered. r sp( v) Figure 17.1-4 Spectral intensity of the direct band-to-band injection-electroluminescence rate rsp(v) (photons per second per Hz per cm 3 ), versus hv, from (17.1-12), under conditions of weak injection. hv EXERCISE 17.1-3 Electroluminescence Spectral Linewidth. (a) Show that the spectral intensity of the emitted light described by (17.1-12) attains its peak value at a frequency v p determined by hv p = Eg + kT. (17.1-13) Peak Frequency (b) Show that the full width at half-maximum (FWHM) of the spectral intensity is v 1.8 kT /h. (17.1-14) Spectral Width (Hz) The value of v for active materials made of compound semiconductors can be larger than that specified in (17.1-14) by virtue of randomness in the chemical composition; this phenomenon is known as alloy broadening. (c) Show that this width corresponds to a wavelength spread A 1.8AkT /hc, where Ap = c/v p . For kT expressed in e V and the wavelength expressed in J-Lm, demonstrate that A 1.45 A kT. (d) Calculate v and A at T = 300 0 K, for Ap = 0.8 J-Lm and Ap = 1.6 J-Lm. (17.1-15) B. LED Characteristics As is clear from the foregoing discussion, the simultaneous availability of electrons and holes substantially enhances the flux of spontaneously emitted photons from a semiconductor. Electrons are abundant in n-type material, and holes are abundant in p-type material, but the generation of copious amounts of light requires that both electrons and holes be plentiful in the same region of space. This condition may be readily achieved in the junction region of a forward-biased p-n diode (see Sec. 16.1E). As shown in Fig. 17.1-5, forward biasing causes holes from the p side and electrons from the n side to be forced into the common junction region by the process of minority carrier injection, where they recombine and emit photons. The light-emitting diode (LED) is a forward-biased p-n junction with a large ra- diative recombination rate arising from injected minority carriers. The semiconductor material is usually direct-bandgap to ensure high quantum efficiency. In this section we determine the output power, as well as the spectral and spatial distributions of the light emitted from an LED, and derive expressions for the efficiency, responsivity, and response time.
688 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Internal Photon Flux and Internal Efficiency A schematic representation of a simple p-n homojunction diode is provided in Fig. 17.1-6. An injected DC current i leads to an increase in the steady-state carrier concentrations n, which in turn result in radiative recombination in the active-region volume V. p n +V >-.. ep <l) c:: <l) c:: o .... u <l) . . .. . --- -- 1 -- E y.. ..... · \I :. .... .. Ie !! hv eV · ...... .....:". ..... ... .... 1 · ....!...:. · ,.-:,,.,. .:. ..; - - - - - -- .. . . Ejv- .. . . . Position Figure 17.1-5 Energy-band diagram of a heavily doped p-n junction that is strongly forward biased by an applied voltage V (compare with the less strongly forward-biased energy-band diagram in Fig. 16.1-20). The dashed lines represent the quasi-Fermi levels, which are separated as a result of the bias. The simultaneous abundance of electrons and holes within the junction region results in strong electron-hole radiative recombination (injection electroluminescence). Since the total number of carriers per second passing through the junction region is i / e, where e is the magnitude of the electronic charge, the carrier injection (pumping) rate (carriers per second per cm 3 ) is simply R == i/e V. (17.1-16) Equation (17.1-1) provides that n concentration RT, which results in a steady-state carrier n == (i/e)T . V ( 1 7.1-1 7) In accordance with (17.1-2), the internal photon flux <I> is then Il i RV, which, using (17.1-3), gives <I>==Ili-. e (17.1-18) Internal Photon Flux This simple and intuitively appealing formula governs the production of photons by electrons in an LED: a fraction Il i of the injected electron flux i / e (electrons per
17.1 LIGHT-EMITTING DIODES 689 p I I I I I I I /,- l-+-I Figure 17.1-6 A simple forward-biased LED. The photons are emitted spontaneously from the junction region. n + second) is converted into photon flux. The internal quantum efficiency Ili is therefore simply the ratio of the generated photon flux to the injected electron flux. The internal photon flux can be enhanced by making use of LEDs with double- heterostructure configurations (Sec. 16.1F), and, in particular, multiquantum-well (MQW) active regions (Sec. 16.1G). The benefit obtains because double heterostruc- tures engender higher carrier concentrations, which enhances radiative recombination (the radiative lifetime Tr is reduced) and thereby increases the internal quantum efficiency Ili [see (16.1-30), (16.1-31), and (17.1-18)]. To maximize Ili, the het- erostructure confinement layers should be lattice matched to the active region. Narrow quantum wells confine carriers even more tightly, further enhancing Ili. The number of quantum wells used in a device is frequently limited because of difficulties in populating all of them. To achieve good performance, it is important to make use of materials of the highest crystal quality, which minimizes defect concentrations, and to avoid the presence of surfaces to which both carrier types have access, which minimizes nonradiative recombination. Extraction Efficiency The photon flux generated in the junction is radiated uniformly in all directions; how- ever, the flux that emerges from the device depends on the direction of emission. This is readily illustrated by considering the photon flux transmitted through a planar material along three possible ray directions, denoted A, B, and C in the geometry of Fig. 17.1-7: p n II Be A B Figure 17.1-7 Not all light generated in an LED with a planar surface is able to emerge. Ray A is partly reflected. Ray B suffers more reflection. Ray C lies outside the critical angle and therefore undergoes total internal reflection, so that it is trapped in the structure.
690 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES . The photon flux traveling in the direction of ray A is attenuated by the factor III == exp( -all), (17.1-19) where a is the absorption coefficient of the n-type material and II is the distance from the junction to the surface of the device. Furthermore, for normal incidence, reflection at the semiconductor-air boundary permits only a fraction of the light, (n-1)2 4n Il.2 = 1 - (n + 1)2 = (n + 1)2 ' (17.1-20) to be transmitted, where n is the refractive index of the semiconductor material [see Fresnel's equations (6.2-15)]. For GaAs, n == 3.6, so that Il2 == 0.68. The overall transmittance for the photon flux traveling in the direction of ray A is therefore IlA == IllIl2. . The photon flux traveling in the direction of ray B has farther to travel and therefore suffers a larger absorption; it also has greater reflection losses. Thus, IlB < IlA. . The photon flux emitted along directions lying outside a cone of (critical) angle ee == sin-l(l/n), such as illustrated by ray C, suffers total internal reflection in an ideal material and is not transmitted [see (1.2-5)]. The area of the spherical cap atop this cone is A == Jc 27rr sin e r de == 27rr 2 (1 - cas e e) while the area of the entire sphere is 47rr 2 . Thus, the fraction of the emitted light that lies within the solid angle subtended by this cone is A/ 47rr 2 , so that Il.3 = !(1 - cosec) = ! (1 - \/1 - l/n 2 ) 1/4n 2 . (17.1-21) For a material with refractive index n == 3.6, as an example, only 1.9% of the total generated photon flux can be transmitted. For a parallelepiped of refractive index n > j2, the ratio of isotropically radiated light energy that can emerge, to the total generated light energy, is 3[1- (1-1/n2)1/2], as shown in Exercise 1.2- 6. However, some fraction of the photons emitted outside the critical angle can be absorbed and reemitted within this angle, so that in practice, Il3 may assume a value larger than that specified by (17.1-21). Loss and Fresnel reflection must also be incorporated for these rays. The efficiency with which the internal photons can be extracted from the LED structure is known as the extraction efficiency Il e. Antireflection coatings (see Exercise 7.1-1) can be used to reduce Fresnel reflection and thereby increase Il e. EXERCISE 17.1-4 Extraction of Light from a Planar-Surface LED. (a) Derive (17.1-21). (b) Determine the critical angles for light escaping into air from: GaAs (n = 3.6), GaN (n = 2.5), and a transparent polymer (n = 1.5). Calculate the fraction of light that can be extracted in the three cases if absorption and Fresnel reflection are ignored. (c) What is the enhancement in the fraction of extracted light that can be achieved if a planar GaAs LED is coated with a transparent polymer of refractive index n = 1.5, assuming that absorption and Fresnel reflection at the semiconductor-polymer boundary are ignored?
17.1 LIGHT-EMITTING DIODES 691 (d) Determine the polymer refractive index that would maximize the fraction of light emitted from the LED into air if absorption is ignored but Fresnel reflection at the semiconductor-polymer and polymer-air interfaces is accommodated. The extraction efficiency can be enhanced in a multitude of ways. One approach is to select a geometry that allows a greater fraction of the light to escape. A spherical dome surrounding a point source at its center, for example, permits all rays to escape, although they remain subject to Fresnel reflection. Several other geometries offer en- hanced extraction efficiencies in comparison with the parallelepiped, as illustrated in Fig. 17.1-8: hemispherical domes, cylindrical structures (which have an escape ring along the perimeter in addition to the escape cone toward the top surface), inverted cones, and truncated inverted pyramids. However, geometries that entail complex pro- cessing steps are often avoided in practice because of increased manufacturing costs. Simple planar-surface-emitting LEDs are suitable when the intended viewing angle deviates little from the normal or when the light is coupled into an optical fiber, as for telecommunications applications. I / :' ;< K:- I ;0 J V Figure 17.1-8 LED die geometries that offer enhanced extraction efficiencies relative to the parallelepiped. An alternative approach is to roughen the planar surface, or otherwise impart a texture to it. This enhances the extraction efficiency by permitting rays beyond the critical angle to escape via scattering, as illustrated in Fig. 17.1-9. Indeed, a textured surface appears automatically under certain growth conditions. p n Figure 17.1-9 An LED with a roughened planar surface permits rays beyond the crit- ical angle to escape, thereby increasing the extraction efficiency Ile. Top-emitting LEDs often make use of current-spreading layers (also referred to as window layers), which are transparent conductive semiconductor layers that spread the region of light emission beyond that surrounding the electrical contact. Current- blocking layers, which prevent current from entering the active region below the top contact, can also be used to control the light emission. The contact geometry can be designed to maximize light transmission.
692 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES A whole host of other techniques are also used to enhance the extraction efficiency. These include the use of reflective and transparent contacts, transparent substrates, and distributed Bragg reflectors (see Chapter 7) between the active layer and an absorbing substrate to reflect the light back toward the desired direction of emission. When the substrate is transparent, another favored technique is the use of flip-chip packaging, which allows light to be extracted through the substrate rather than through the top surface of the device. The LED extraction efficiency can also be enhanced by guiding light to the surface of the device via a 2D photonic crystal (see Sec. 7.3A), comprising a regular array of 100-250-nm diameter holes formed in the current-spreading layer. Yet another approach for increasing the extraction efficiency from an LED makes use of a microresonator (see Secs. 10.1B and 10.4). A pair of mirrors (e.g., distributed Bragg reflectors) confines the light within a wavelength-sized region in one dimension. As illustrated in Fig. 17.1-10, this substantially narrows the angular confinement of the light so that a large fraction of it is emitted into a resonant mode whose angular extent falls principally within the extraction cone (see Sec. 10.1). A photonic-crystal structure can also be incorporated into a microresonator LED to guide much of the residual light toward the surface of the device, thereby increasing Il e yet further. The use of microresonators for enhancing the properties of photon sources is discussed further in Sec. 17.4B. Source Extraction _!one Figure 17.1-10 A plane-parallel-mirror microresonator LED. Two closel y spaced reflectors (one at left with a reflectance of 100% and one at right with a reflectance of, say, 50%) form a wavelength-size cavity that confines the light, funneling a large portion of it into a spatial region that lies within the extraction cone. MQdified emISSIon Spatial Pattern of Emitted Light The far-field radiation pattern for light emitted into air from a planar surface-emitting LED is similar to that of a Lambertian radiator. The intensity varies as cas e, where e is the angle from the emission-plane normal; the intensity decreases to half its value at e == 60°. This pattern arises as a result of Snell's law: light rays bend away from the normal as they exit the semiconductor-air interface. f--' , "-. .-. , ," ''-.'' _ LED " ' chip Figure 17.1-11 Epoxy-encapsulated LED. Encapsulation protects the semiconductor chip, increases light extraction by reducing refractive-index mismatch, and serves as a lens to shape the beam. LEDs are often encapsulated in transparent epoxy lenses for a number of purposes (Fig. 17.1-11). Lenses of different shapes alter the emission pattern in different ways, as illustrated schematically for hemispherical and parabolic lenses in Fig. 17.1-12. Epoxy lenses can also enhance the extraction efficiency Ile. A lens with a refractive index close to that of the semiconductor optimizes the extraction of light from the semiconductor into the epoxy. The shape of the lens can then be tailored so as to
17.1 LIGHT-EMITTING DIODES 693 maximize the extraction of light at the epoxy-air interface. Epoxy materials usually have refractive indexes that are intermediate between those of semiconductors and air and, in practice, yield a factor of 2-3 enhancement in light extraction. + Junction + + (a) (b) (c) Figure 17.1-12 Radiation patterns of surface-emitting LEDs: (a) Lambertian spatial pattern in the absence of a lens; (b) spatial pattern with a hemispherical lens; (c) spatial pattern with a parabolic lens. The radiation pattern from edge-emitting LEDs and laser diodes is usually quite narrow and can often be empirically described by the function coss e, with s > 1. If s == 10, for example, the intensity decreases to half its value at e 21 0 . Output Photon Flux and External Efficiency The output photon flux <1>0 (also called the external photon flux) is related to the internal photon flux by <1>0 == lle<1> == llelli- , e (1 7 .1- 22) where the internal efficiency II i relates the internal photon flux to the injected electron flux, and the extraction efficiency II e specifies how much of the internal photon flux is transmitted out of the structure. A single quantum efficiency that accommodates both of these processes is the external efficiency II ex: llex II ell i. (1 7 .1- 23 ) External Efficiency The output photon flux in (17.1-22) can therefore be written as <1> 0 == II ex - , e (1 7.1-24) External Photon Flux so that the external efficiency llex is simply the ratio of the external photon flux <1>0 to the injected electron flux i / e. Because the pumping rate generally varies locally within the junction region, so too does the generated photon flux. The LED output optical power Po is directly related to the output photon flux since each photon has energy hv: Po == hvcp 0 == II ex hv - . e (17.1- 25) Output Power
694 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES The internal efficiency It i for LEDs ranges between 50% and just about 100%, while the extraction efficiency Ite for properly designed devices can stretch up to 50%. The external efficiency Itex of LEDs is thus typically below 50%. As discussed in Sec. 15.2A, another measure of performance is the power-conversion efficiency (or wall-plug efficiency), which is defined at the ratio of the emitted optical power Po to the applied electrical power, Po hv It c i V == It ex e V ' (17.1-26) where V is the voltage drop across the device. For hv eV, as is the case for some commonly encountered LEDs, we obtain Itc Itex. Responsivity The responsivity 9\ of an LED is defined as the ratio of the emitted optical power Po to the injected current i, i.e., 9\ == Po/i. Using (17.1-25), we obtain 9l = o = hv.ipo = IJ.ex hv . e (17.1-27) The responsivity in W/A, when Ao is expressed in Mm, is then 1.24 9l = IJ.ex . (17.1-28) LED Responsivity (W j A; Ao in J-Lill) For example, if Ao == 1.24 Mm, then 9\ == Itex W / A; if It ex were unity, the maximum optical power that could be produced by an injection current of 1 mA would be 1 m W. Thus, for Itex == at Ao == 1.24 Mm, we have 9\ == mW/mA. In accordance with (17.1-25), the LED output power Po is proportional to the injected current i. In practice, however, this relationship is valid only over a re- stricted range. For the particular device whose light-current characteristics is shown in Fig. 17.1-13, the emitted optical power is proportional to the injection (drive) current only when the latter is less than about 20 mA. In this range, the responsivity has a constant value of about 0.3 mW/mA, as detennined from the slope of the curve. For larger drive currents, saturation causes the proportionality to fail; the responsivity then declines with increasing drive current. Since Ao == 0.420 Mm for this LED, (17.1-28) reveals that it has an external efficiency Itex == 0.10. ,,-..., 12 8 "-" Figure 17.1-13 Optical power at the out- put of an LED versus injection (drive) cur- rent. This MQW InGaNjGaN LED emits in the violet region of the spectrum, at Ao == 420 nm; the device structure is exhibited in Fig. 17.1-20. QC 10 ;.... <l) 8 o 0... 6 . g. 4 ....... ::3 & 2 ::3 o
17.1 LIGHT-EMITTING DIODES 695 Spectral Distribution The spectral intensity rsp(v) of light spontaneously emitted from a semiconductor in quasi-equilibrium has been determined, as a function of the concentration of injected carriers n, in Exercises 17.1-2 and 17.1-3. This theory is applicable to the electro- luminescence light emitted from an LED in which quasi-equilibrium conditions are established by injecting current into a p-n junction. Under conditions of weak pumping, such that the quasi-Fermi levels lie within the bandgap and are at least a few kT away from the band edges, the spectral intensity achieves its peak value at the frequency v p == (Eg + kT/2)/h (see Exercise 17.1-3). In accordance with (17.1-14) and (17.1-15), the FWHM of the spectral intensity is v 1.SkT/h (v == 10 THz for T == 300 0 K), which is independent of v. When expressed in terms of wavelength, however, the width does depend on A, A 1.45 A; kT, (17.1-29) Spectral Width (J1m) where kT is specified in eV, the wavelength is specified in Mm, and Ap == c/v p . The dependence of A on A is apparent in Fig. 17.1-14, which illustrates the observed wavelength spectral intensities for a selection of LEDs operating in the ultra- violet (indicated as magenta) and visible regions of the spectrum. AIN has the largest III-nitride bandgap, producing light at 210 nm; AIGaN is typically employed in the mid and near ultraviolet; InGaN is the material of choice in the violet, blue, and green; and AllnGaP usually serves the yellow, orange, and red. Typical spectral intensities for LEDs that operate in the near infrared are displayed in Fig. PI7.1-5; these devices are generally fabricated from InGaAsP. The spectral width increases roughly as A, in accordance with (17.1-29). However, alloy broadening can result in a further increase in the spectral width, as is evident in the spectrum for the green LED. If Ap == 1 Mm at T == 300 0 K, for example, (17.1-29) provides A 36 nm. 0.2 0.3 0.4 0.5 0.6 Wavelength Ao (/Lm) 0.7 Figure 17.1-14 Spectral inten- sities versus wavelength for LEDs that operate in the ultraviolet and visible regions of the spectrum. The peak intensities are all normal- ized to the same value. Results for LEDs operating in the infrared are presented in Fig. P17.1-5. Response Time The response time of LEDs used for illumination is usually limited by the RC time constant of the device because the junction area, and therefore the capacitance, is large. The response time of communication-system LEDs, in contrast, is generally limited principally by the lifetime T of the injected minority carriers that are respon- sible for radiative recombination. For a sufficiently small injection rate R, the injec- tion/recombination process can be described by a first-order linear differential equation (see Sec. 16.1D), and therefore by the response to sinusoidal signals. An experimental determination of the highest frequency at which an LED can be effectively modulated
696 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES is easily obtained by measuring the output light power in response to sinusoidal electric currents of different frequencies. If the injected current assumes the form i == io + i l cos(Dt), where i l is sufficiently small so that the emitted optical power P varies linearly with the injected current, the emitted optical power behaves as P == Po + PI cos(Dt + cp). The associated transfer function, which is defined as H(D) == (PI/i l ) exp(jcp), assumes the form H(D) = 1 + jDT ' (17.1-30) which is characteristic of a resistor-capacitor circuit. The rise time of the LED is T (seconds) and its 3-dB bandwidth is B == 1/21rT (Hz). A larger bandwidth B is therefore attained by decreasing the rise time T, which comprises contributions from both the radiative lifetime Tr and the nonradiative lifetime T nr through the relation l/T == l/Tr + l/Tnr. However, reducing Tnr results in an undesirable reduction of the internal quantum efficiency It i == T / Tr. It may therefore be desirable to maximize the internal quantum efficiency-bandwidth product ItiB == 1/21rTr rather than the bandwidth alone. This requires a reduction of only the radiative lifetime Tr, without a reduction of Tnr, which may be achieved by careful choice of the semiconductor material and doping level. Typical rise times of LEDs are in the range I to 50 ns, corresponding to bandwidths as large as hundreds of MHz. Electronic Circuitry An LED is usually driven by a current source, as illustrated schematically in Fig. 17.1-15(a), often implemented by means of a constant-voltage source in series with a resistor, as shown in Fig. 17.1-15(b). The emitted light is readily modulated by simply modulating the injected current. Analog and digital modulation are portrayed in Figs. 17 .1-15(c) and 17 .1-15(d), respectively. The performance of LED drivers may be improved by adding circuitry that regulates bias current, matches impedance, and provides nonlinear compensation to limit the maximum current. Fluctuations in the intensity of the emitted light may be stabilized by monitoring it with a photodetector, whose output is used as a feedback signal to control the injected current. <>-1 Input signal Data Enable (a) (b) (c) (d) Figure 17.1-15 Various circuits can be used as LED drivers. These include (a) an ideal DC current source; (b) a DC current source provided by a constant-voltage source in series with a resistor; (c) transistor control of the current injected into the LED to provide analog modulation of the emitted light; and (d) transistor switching of the current injected into the LED to provide digital modulation of the emitted light. For architectural lighting applications, a number of LEDs of a particular color are typically connected in series and driven by a pulse-width-modulated (PWM) current provided by a drive transistor. The light level is determined by the average current passing through the LEDs, which in turn is governed by the duty cycle of the PWM
17.1 LIGHT-EMITTING DIODES 697 current. Banks of LEDs of different colors (e.g., red, green, and blue) are used to generate light of an arbitrary color, including white. An addressable microprocessor can be used to control the relative light levels generated by the different-color LEDs, enabling the overall color and intensity of the light to vary with time and position in an arbitrary manner. Collections of such lighting units can be concatenated into a lighting network. c. Materials and Device Structures Photonics was revolutionized in the 1950s by the growth of single-crystal binary 111- V semiconductors, compounds that do not occur in nature. Many of these alloys have direct bandgaps and therefore yield large values of the internal quantum efficiency. Photon sources fabricated from 111- V materials also offer long lifetimes, unlike those that make use of 11- VI alloys. In 1962, GaAs was the first such material to be fabricated in the form of an LED and an LD (see p. 680). Today's LED industry is built around ternary and quaternary 111- V material systems, particularly InGaAsP, AllnGaP, and AllnGaN. High-brightness light is readily gener- ated at all colors of the rainbow, from the ultraviolet to the infrared (see Figs. 17.1-14 and 17.1-16). ..; - " -::il!:.. .{.:::.;. .:t::" ........ Figure 17.1-16 LED traffic signal based on 111- V materials. '.' '; . ,:.'.': LEDs may be constructed either in surface-emitting or edge-emitting configurations (Fig. 17.1-17). The surface-emitting LED emits light from a face of the device that is parallel to the plane of the active region. The edge-emitting LED emits light from the edge of the active region. (a) (b) Figure 17.1-17 (a) Surface-emitting LED, (b) Edge-emitting LED, We proceed to provide a brief description of the principal classes of 111- V materials, along with schematic illustrations of several representative LED device structures. GaAs The first 111- V material to play an important role in photonics was GaAs. This binary direct-bandgap semiconductor was used to fabricate the first laser diode in 1962, with an emission wavelength Ao == 0.873 Mm near its bandgap wavelength Ag. The GaAs
698 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES LED was a byproduct of the development of the GaAs LD. Shortly thereafter, sev- eral other binary direct-bandgap 111- V semiconductors, grown by vapor-phase epitaxy (VPE) and liquid-phase epitaxy (LPE), were also shown to exhibit electrolumines- cence and lasing near their bandgap wavelengths: GaSb (A g == 1.70 Mm), InP (A g == 0.919 Mm), InAs (A g == 3.44 Mm), and InSb (A g == 7.29 Mm). GaAsP The bandgap wavelength of the ternary semiconductor GaAsl-xP x moves into the visible as the mole-fraction of phosphorus increases, offering emission in the red region of the spectrum [see Fig. 16.1-7(a)]. Although the bandgap changes to indirect in the red, emission in the orange, yellow, and green can nevertheless be achieved by using nitrogen-doped versions of these materials, GaAsP:N and GaP:N. The nitrogen impurities (zinc and oxygen co-dopants are sometimes used in place of nitrogen) are incorporated into the material at sharply localized positions so that they are able to accommodate the substantial momentum changes associated with indirect transitions. However, the external efficiencies of GaAsP LEDs are typically small (0.02-0.5%), in part because of a lattice-constant mismatch with the GaAs substrate. Nevertheless, LEDs made of GaAs, GaAsP, GaAsP:N, and GaP:N are inexpensive to fabricate and thus continue to be used in low-brightness applications such as remote controls for consumer appliances and indicator lamps. In GaAsP An admixture of indium reduces the bandgap of GaAsP. The quaternary semiconductor Inl-xGaxAsl-YP Y is a versatile alloy that is widely used in the near-infrared region of the spectrum. Its bandgap is compositionally tunable over a substantial range of wavelengths [0.549 Mm (GaP) < Ag < 3.44 Mm (InAs)], and lattice matching to an InP substrate can be maintained if the compositional mixing ratios x and y are judiciously chosen [see stippled area in Fig. 16.1-7(a)]. Only a portion of this range enjoys the benefit of a direct bandgap, however. InGaAsP is used to fabricate LEDs for short-haul, modest-bit-rate communications systems operating near Ao == 1330 nm (see Fig. 17.1-18). Long-haul high-bit-rate communication systems generally operate in the vicinity of Ao == 1550 nm and make use of laser diodes rather than LEDs since it is far easier to couple the highly collimated light emitted by an LD into a single- mode fiber (see Chapter 24). Low-cost InxGal-xAs LEDs are used in a broad range of consumer applications. InGaAsP active region Figure 17.1-18 Saul-Lee-Burrus-type surface- emitting InGaAsP LED designed for use in an optical fiber communication system operating at a wavelength of 1.3 J-Lm. The active region is lattice matched to the InP substrate; the device is mounted upside down in the package (flip-chip packaging) so the light emerges through the sub- strate. An integrated lens collimates the light for enhanced coupling to a fiber. The light-emitting region is a single surface for communication- systems LEDs. + Dielectric film I.. - -- "'.'-.. I InGaASP""-- . contact \. InP layer confinement layers
17.1 LIGHT-EMITTING DIODES 699 AIGaAs Just as adding phosphorus to GaAs increases its bandgap, so too does the addition of aluminum. Like GaAsl-xP x, the ternary alloy AlxGal-xAs is a direct-bandgap ma- terial in the red and near-infrared regions of the spectrum that can be compositionally tuned. Unlike GaAsP, however, it has the merit that lattice matching to GaAs is main- tained for all mole fractions of aluminum [see Fig. 16.1-7(a)], so that the material can serve as a high-brightness source in the red. Since AlxGal-xAsjGaAs multiquantum- well structures tend to suffer from nonunifonn carrier distributions in the active region, LEDs are often fabricated using a double-heterostructure configuration of the form AlxGal-xAs/AlyGal-yAs, in which the compositions of the barriers and well differ. An unattractive feature of this material is limited device lifetime associated with the oxidation and corrosion of the material over time, when the aluminum content is sufficiently high. AllnGaP The quaternary semiconductor (AlxGal-x)ylnl-YP is a direct-bandgap material over a substantial range of the near infrared and the longer reaches of the visible spectrum [see shaded area in Fig. 16.1-7 (a)]. Lattice matching to GaAs is attained for compositions in the range (AlxGal-x)O.5Ino.5P. The lattice-matched ternary compound Ino.5Gao.5P has a bandgap wavelength of 650 nm, which is useful for applications such as laser pointers and digital-video-disc (DVD) players. AllnGaP is the material of choice for high-brightness applications, such as traffic lights and signage, in the red, orange, yellow-orange (amber), and yellow. Quantum efficiency is enhanced by using wafer- bonded transparent GaP substrates in place of GaAs, multiquantum-well (MQW) ac- tive regions, and resonant-cavity (RC) configurations, which offer decreased band- width and directed emission patterns. AllnGaPlInGaP LEDs are often used in plastic- fiber communications systems that operate in the 600-650-nm region (see Fig. 17.1- 19). AlInGaP /InGaP MQW . active region ...L II ;;11 GaAs contact layer + (\ J Figure 17.1-19 Surface-emitting AlInGaPlInGaP 650-nm MQW RC LED for use in short-haul, plastic- fiber communications. A top-emitting structure is used because of the opacity of the GaAs substrate in this device. The distributed Bragg reflectors are made of AIAs/AIGaAs layers with an aluminum content that is sufficiently large so that the 650-nm light is transmitted. A lens enhances coupling of the light to a fiber. GaAs ) / '" "-.. AlInGaP substrate AlAs / AIGaAs confinement Bragg reflectors layers GaN GaN is a binary direct-bandgap semiconductor with a bandgap wavelength >"g 0.366 Mm that falls in the near-ultraviolet region of the spectrum. Although it is a relative newcomer to photonics, GaN is arguably one of the most important. It may be grown by MBE, MOCVD, or HVPE. Although electroluminescence from this material was first observed in 1971, it was not until 1992 that the first GaN p-n homojunction
700 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES LED was fabricated. The material is typically grown on a sapphire substrate, which has a substantial lattice mismatch with GaN. Unlike the arsenide and phosphide 111- V compounds, however, GaN can tolerate a large dislocation concentration so that the mismatch turns out to be of small import. GaN is the progenitor of the highly important compounds InGaN, AIGaN, and AllnGaN,just as GaAs was the progenitor oflnGaAsP and AllnGaP. InGaN The ternary semiconductor InxGal-xN is a direct-bandgap material with a bandgap wavelength that spans the region 366 nm (GaN) < Ag < 1.61 Mm (InN). However, it is difficult to grow InGaN with a large InN content since clusters of InN form during the growth process. This phenomenon is responsible for the substantial alloy broadening of the green LED spectrum portrayed in Fig. 17.1-14. InGaN is the material of choice for high-brightness LEDs in the wavelength range 366 nm < Ag < 580 nm, comprising the near-ultraviolet, violet, blue, and green regions of the spectrum [see Fig. 16.1-7 (b)]. This III-nitride alloy is thus complementary to AllnGaP, which accommodates the red, orange, and yellow regions. As with AllnGaP, the quantum efficiency is enhanced by making use of GaNlInGaN MQW structures, as illustrated in Fig. 17.1-20. The number of quantum wells is often limited to 3-5 because of limits on the ability to populate them as a result of the hole diffusion length. The substrate is usually GaN on sapphire. Performance can also be enhanced by the use of resonant-cavity devices. Yet another configuration of interest comprises arrays of quantum dots that self-assemble on growth. GaNlInGaN MQW J-., active region z z z ° "" ° "" ° < ° .50 zz z 0""0""0 :a .50.50.5 ° ° < z ° AlGaN confinement layers + Figure 17.1-20 Surface-emitting GaNjlnGaN MQW LED operating at Ao = 420 nm in the violet spectral region. The active region comprises 5-nm GaN barriers and four 2.5-nm InxGal-xN wells (for simplicity, the effects of the characteristic intrinsic electric fields of III-nitride semiconductors are not shown). The light is extracted through the substrate of GaN on sapphire, which is transparent at 420 nm. AIGaN AlxGal-xN is also a ternary III-nitride direct-bandgap semiconductor, but its bandgap wavelength falls in the range 200 nm (AIN) < Ag < 366 nm (GaN) [see Fig. 16.1-7(b)], covering the mid- and near-ultraviolet regions (200 nm < Ao < 390 nm). One of the difficulties in making devices operate efficiently at the shorter wavelengths has been the decrease in the p-type conductivity as the AIN concentration increases, which results in a dearth of holes available for recombination in the active region. Nevertheless, AIN LEDs that emit at 210 nm have been successfully operated.
17.1 LIGHT-EMITTING DIODES 701 As with InGaN, the LED quantum efficiency is enhanced by making use of double-heterostructure, quantum-well, or MQW active regions with layers of the form AlxGal-xN/AlyGal-yN. Templates of AIGaNj AINjsapphire serve as transparent lattice-matched substrates for ultraviolet AIGaN-based emitters. The use of GaN as a substrate is avoided because of its absorption at wavelengths below 366 nm. AllnGaN It is clear from the foregoing that the ternary III-nitride compounds InGaN and AIGaN are highly suitable for fabricating sources that stretch across the visible and ultraviolet regions of the spectrum. However, the quaternary semiconductor (AlxInyGal-x-y)N has the merit that it can be lattice matched to a GaN template for appropriate values of x and y [see Fig. 16.1-7(b)], thereby increasing the quantum efficiency. This lattice matching is analogous to that of AllnGaP to GaAs and of InGaAsP to InP. AllnGaN LEDs with lattice matching to a GaN substrate are used over a wavelength range ranging from 366 nm, the wavelength of GaN, to about 250 nm, the wavelength of AllnN that is lattice matched to GaN. AllnGaNjlnGaNj AllnGaN quantum wells serve as active regions. The inclusion of indium in AIGaN also yields an enhancement in internal quantum efficiency. AllnGaN can also serve as a transparent contact layer. White-Light LEDs Appropriate combinations of red, green, and blue light are perceived as white. Two principal approaches are used for generating white light from LEDs. Blue and near- ultraviolet LEDs fabricated from III-nitrides can be used to illuminate phosphors, which then generate various other colors via photoluminescence (see Sec. 13.5B and Fig. 17.1-21). Alternatively, the light generated by LEDs of different compositions can be combined to yield light of various colors. Although the internal quantum efficiency in the green is typically lower than that in other spectral regions, the latter approach can be used to produce white light with excellent color rendering. LEDs are replacing incandescent lighting in the home and workplace, and are increasingly being used in architectural venues. The LED is superior to the incandescent lightbulb by virtue of its higher wall-plug efficiency and luminous efficacy, longer lifetime, lower cost, and compact configuration. . Figure 17.1-21 White light emission from a III-nitride blue semiconductor LED containing a phosphor. . Organic LEDs Organic light-emitting diodes can be fabricated from small organic molecules or con- jugated polymer chains (see Sec. 16.1B). Arrays of pixelated organic LEDs can be fabricated in the form of thin light-emitting plastic sheets that generate diffuse light over large areas. These serve as inexpensive, flexible, rollable, high-efficiency self- luminous displays. These devices can be used in digital cameras, cellular phones, com- puter monitors, television receivers, as well as in architectural lighting. They are less complex and thinner than liquid-crystal displays (LCDs), which require backlighting. Indeed, organic light emitters can serve as the source of backlighting for LCDs. Small-molecule organic light-emitting diodes, called OLEDs, are efficient gen- erators of electroluminescence in the red, green, and blue. Two thin ( lOO-nm) organic-semiconductor films are juxtaposed to form an organic heterostructure. As
702 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES shown in Fig. 17.1-22, this structure is sandwiched between two inorganic electrodes, an anode that injects holes and one or more cathodes that inject electrons. The injected carriers are transported to the heterojunction (active region), forming bound excitons that generate spontaneous emission upon recombination. Different heterostructure ma- terials yield different recombination-radiation wavelengths, so several heterostructures can be patterned on a substrate to provide a multicolor OLED. The active region can instead be infused with a fluorescent dopant to create blue light, and phosphorescent dopants to create green and red light, thereby increasing the internal quantum efficiency by making use of both singlet and triplet excitons (see Sec. 13.5B). White organic light-emitting diodes (WOLEDs) fabricated in this manner have nearly unity internal quantum efficiency and well-balanced color rendition. Glass substrate ! Figure 17.1-22 Typical OLED structure. Cal- cium and indium tin oxide are commonly used as the cathode and transparent anode materials, re- spectively. Exciton recombination radiation emit- ted at the organic heterojunction exits through the transparent anode and glass substrate. Organic semiconductors used to fabricate OLEDs include hole-transporting TPD (triphenyl diamine deriva- tive) and electron-transporting Alq3 (aluminum tris[8-hydroxyquinoline]). Luminescent dopants can be infused into the active regions to enhance the internal quantum efficiency and to create white light. Polymer light-emitting diodes, called PLEDs, are similar in construction to OLEDs except that they typically have an n-type active region into which holes are injected by a p-type organic layer. PPV (polyphenylene vinylene) is often used for making PLEDs. These devices are generally easier to fabricate, and have greater efficiencies than OLEDs, but they offer a more limited range of colors. The desirable features of small-molecule and large-molecule polymeric organic materials can be brought together in molecules known as phosphorescent dendrimers. These are large molecular balls containing a heavy-metal ion core, such as Ir(2-phenylpyridine)3, which facilitates exciton radiative recombination; layers of branching-ring structures are bonded around it. Such hybrid devices provide bright, high-resolution, multicolor Images. 17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS The principle underlying the operation of a semiconductor optical amplifier (SOA), also known as a semiconductor laser amplifier, is the same as that for other laser amplifiers: the creation of a population inversion that renders stimulated emission more prevalent than absorption. The population inversion is usually achieved by electric- current injection in some form of a p-n junction diode; a forward bias voltage causes carrier pairs to be injected into the junction region, where they recombine by means of stimulated emission. However, the theory of the SOA is somewhat more complex than that presented in Chapter 14 for other laser amplifiers, inasmuch as the transitions take place between bands of closely spaced energy levels rather than between well-separated discrete
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 703 levels. For purposes of comparison, nevertheless, the SOA may be viewed as a four- level laser system (see Fig. 14.2-6) in which the upper two levels lie in the conduction band and the lower two levels lie in the valence band. The extension of the laser amplifier theory set forth in Chapter 14 to semiconductor structures has been provided in Chapter 16. In this section we use the results derived in Sec. 16.2 to obtain expressions for the gain and bandwidth of semiconductor optical amplifiers. We then consider pumping schemes used to attain a population inversion and highlight the benefit of using heterostructure- and quantum-well amplifier configu- rations. Finally, we briefly review the performance of semiconductor optical amplifiers and compare them with optical fiber amplifiers. The theoretical underpinnings of SOAs are the same as those for laser-diode operation, considered in Secs. 17.3 and 17.4. A. Gain and Bandwidth Light of frequency v can interact with the carriers of a semiconductor material of bandgap energy Eg via band-to-band transitions, provided that v > Eg/h. The in- cident photons may be absorbed, resulting in the generation of electron-hole pairs, or they may produce additional photons through stimulated electron-hole recombination radiation (see Fig. 17.2-1). When emission is more likely than absorption, net optical gain ensues and material can serve as a coherent optical amplifier. E E 2 E T c E g E v El (a) h hv v hv hv I I I I I I I I I I I I I I I I I I I I ) k I I I I I I I I I I I I I I II I I I ) (b) k Figure 17.2-1 (a) The absorption of a photon results in the generation of an electron-hole pair. (b) Electron-hole recombination can be induced by a photon; the result is the stimulated emission of an identical photon. Expressions for the rate of stimulated emission fst(V) and the rate of photon ab- sorption f ab (v) are provided in (16.2-18) and (16.2-19), respectively. These quantities depend on the photon-flux spectral intensity CPv; the quantum-mechanical strength of the transition for the particular material under consideration (which is implicit in the value of the electron-hole radiative recombination lifetime Tr); the optical joint density of states Q(v); and the occupancy probabilities for emission fe(v) and absorption fa(v). The optical joint density of states Q( v) is determined by the E-k relations for electrons and holes and by the conservation of energy and momentum. With the help of the parabolic approximation for the E-k relations near the conduction- and valence- band edges, it was shown in (16.2-6) and (16.2-7) that the energies of the electron and hole that interact with a photon of energy hv are E 2 == Ee + m r (hv - Eg) , El == E 2 - hv, me (17.2-1)
704 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES respectively, where me and mv are their effective masses and l/m r == lime + l/mv. The resulting optical joint density of states that interacts with a photon of energy hv was determined to be [see (16.2-9)] (2m r )3/2 e(v) = 7rn 2 .J hv - Eg, hv > Ego (17.2-2) It is apparent that Q( v) increases as the square root of photon energy above the bandgap. The occupancy probabilities fe(v) and fa(v) are determined by the pumping rate through the quasi-Fermi levels E fe and E fv. The quantity fe(v) is the probability that a conduction-band state of energy E 2 is filled with an electron and a valence-band state of energy E 1 is filled with a hole. The quantity fa (v), on the other hand, is the probability that a conduction-band state of energy E 2 is empty and a valence-band state of energy El is filled with an electron. The Fermi inversion factor [see (16.2-25)] fg(v) == fe(v) - fa(v) == fe(E 2 ) - fv(E 1 ) (17.2-3) represents the degree of population inversion. The quantity fg (v) depends on both the Fermi function for the conduction band, fe(E) == l/{exp[(E - Efe)/kT] + I}, and the Fermi function for the valence band, fv (E) == 1 I { exp[ (E - E fv) I kT] + I}. It is a function of temperature and of the quasi-Fermi levels E fe and E fv, which in turn are determined by the pumping rate. Because a complete population inversion can in principle be achieved in a semiconductor optical amplifier [fg(v) == 1], it behaves like a four-level system. The results provided above were combined in (16.2-24) to provide an expression for the net gain coefficient, '"YO (v) == [rst(v) - (ab(v)]I<pv, .A 2 '"YO (v) == - Q(v) fg(v). 81fT r (17.2-4) Gain Coefficient Comparing (17.2-4) with (14.1-4), it is apparent that the quantity Q(v)fg(v) in the semiconductor optical amplifier plays the role of Ng(v) in other laser amplifiers, and that a(v) '"YO (v ) I n. Amplifier Bandwidth In accordance with (17.2-3) and (17.2-4), a semiconductor medium provides net optical gain at the frequency v when fe(E 2 ) > fv(E 1 ). Conversely, net attenuation ensues when fe(E 2 ) < fv(E 1 ). Thus, a semiconductor material in thermal equilibrium (un- doped or doped) cannot provide net gain whatever its temperature; this is because the conduction- and valence-band Fermi levels coincide (E fe == E fv == E f ). External pumping is required to separate the Fermi levels of the two bands in order to achieve amplification. The condition fe(E 2 ) > fv(E 1 ) is equivalent to the requirement that the photon energy be smaller than the separation between the quasi-Fermi levels, i.e., hv < E fe - Efv, as demonstrated in Exercise 16.2-1. Of course, the photon energy must be larger than the bandgap energy (hv > E g) in order that laser amplification occur by means of band-to-band transitions. Thus, if the pumping rate is sufficiently large that the separation between the two quasi-Fermi levels exceeds the bandgap energy Eg, the
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 705 medium can act as an amplifier for optical frequencies in the band E g E f e - E f v -<v< h h (17.2-5) Amplifier Bandwidth For hv < E 9 the medium is transparent, whereas for hv > E fe - E fv it is an attenuator instead of an amplifier. Equation (17.2-5) demonstrates that the amplifier bandwidth increases with E fe - E fv, and therefore with pumping level. In this respect it is unlike the atomic laser amplifier, which has an unsaturated bandwidth /:::1v that is independent of pumping level (see Fig. 14.1-2). Computation of the gain properties is simplified considerably if thermal excitations can be ignored (i.e., if T == 0 K). The Fermi functions are then simply fe(E 2 ) == 1 for E 2 < E fe and 0 otherwise; fv(E 1 ) == 1 for El < E fv and 0 otherwise. In that case the Fermi inversion factor is f (v) == { +1, 9 -1 , hv < E fe - E fv otherwise. (17.2-6) Schematic plots of the functions Q( v ), f 9 (v), and the gain coefficient '"Yo (v) are pre- sented in Fig. 17.2-2, illustrating how '"Yo (v) changes sign and turns into a loss coeffi- cient when hv > E fe - E fv. The v- 2 dependence of '"Yo (v), arising from the .A 2 factor in the numerator of (17.2-4), varies sufficiently slowly that it may be ignored. Finite temperature smoothes the functions fg(v) and '"Yo (v), as shown by the dashed curves in Fig. 17.2-2. (leV) fl v ) +1 E g Efc-Eju hv '\ \ - , hv ...... -I - "Yo(v) hv Figure 17.2-2 Dependence on energy of the optical joint density of states Q(v), the Fermi inversion factor fg(v), and the gain coefficient /'o(v) at T = 0 K (solid curves) and at room temperature (dashed curves). Photons with energy between E 9 and E fe - E fv undergo laser amplification. t 0 Dependence of the Gain Coefficient on Pumping Level The gain coefficient '"Yo (v) increases both in its width and in its magnitude as the pump- ing rate R is elevated. As provided in (17.1-1), a constant pumping rate R (number
706 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES of injected excess electron-hole pairs per cm 3 per second) establishes a steady-state concentration of injected electron-hole pairs in accordance with n == P == RT, where T is the electron-hole recombination lifetime (which includes both radiative and nonradiative contributions). Knowledge of the steady-state total concentrations of electrons and holes, n == no + n and P == Po + n, respectively, permits the Fermi levels E fe and E fv to be determined via (17.1-8). Once the Fermi levels are known, the computation of the gain coefficient can proceed using (17.2-4). The dependence of '"Yo (v) on n and thereby on R, is illustrated in Example 17.2-1. EXAMPLE 17.2-1. Gain Coefficient for an InGaAsP SOA. A sample of the quaternary material Ino.72Gao.28Aso.6Po.4, with bandgap energy E 9 == 0.95 e V, is operated as a semiconductor optical amplifier at a wavelength of Ao == 1300 nm at T == 300 0 K. The sample is undoped but has residual concentrations of 2 x 10 17 cm- 3 donors and acceptors, and a radiative electron-hole recombination lifetime Tr 2.5 ns. The effective masses of the electrons and holes are me 0.06 mo and mv 0.4 mo, respectively, and the refractive index n 3.5. Given the steady-state injected- carrier concentration n (which is controlled by the injection rate R and the overall recombination time T), the gain coefficient /'0 (v) may be computed from (17.2-4) in conjunction with (17.1-8). As illustrated in Fig. 17.2-3, both the amplifier bandwidth and the peak value of the gain coefficient "Yp increase with n. The energy at which the peak occurs also increases with n, as expected from the behavior shown in Fig. 17.2-2. Furthermore, the minimum energy at which amplification occurs decreases slightly with increasing n as a result of band-tail states, which reduce the bandgap energy. At the largest value of n shown (n == 1.8 x 10 18 cm- 3 ), photons with energies falling between 0.91 and 0.97 eV undergo amplification. This corresponds to a full amplifier bandwidth of 14.5 THz, and a wavelength range of 80 nm. A more suitable measure is the bandwidth at the full-width at half maximum (FWHM) of the gain profile, also called the 3-dB gain bandwidth, which is 10 THz, corresponding to about 50 nm at Ao == 1300 nm (see Table 14.3-1 for a comparison with other laser transitions). The calculated peak gain coefficient /,p == 270 cm- 1 at this value of n is large in comparison with most atomic laser amplifiers. 200 ..--,. I 8 -3 E 200 Q) .u S Q) o u t:: . 100 ..:..:: ro Q) 0.. 300 -... 18 100 -3 'S' '1: 0 .u S Q) o u .s -100 ro o -200 o 1.0 1.5 .6.n (10 18 cm- 3 ) 2.0 (a) 0.90 0.92 0.94 0.96 hv (eV) (b) Figure 17.2-3 (a) Calculated gain coefficient /'o(v) for an InGaAsP SOA versus photon energy hv, with the injected-carrier concentration n as a parameter (T == 300 0 K). The band of frequencies over which amplification occurs (centered near 1300 nm) increases with increasing n. At the largest value of n shown, the FWHM amplifier bandwidth is 10 THz, corresponding to 0.04 eV in energy and 50 nm in wavelength. (Adapted from N. K. Dutta, Calculated Absorption, Emission, and Gain in Ino.72Gao.28Aso.6Po.4, Journal of Applied Physics, vol. 51, pp. 6095-6100, 1980, Fig. 8.) (b) Calculated peak gain coefficient /,p as a function of n. At the largest value of n, the peak gain coefficient 270 cm- 1 . (Adapted from N. K. Dutta and R. J. Nelson, The Case for Auger Recombination in In1-xGaxAsyP1-Y' Journal of Applied Physics, vol. 53, pp. 74-92, 1982, Fig. 17.)
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 707 The onset of gain saturation in semiconductor optical amplifiers is not unlike that of other homogeneously broadened laser amplifiers, as considered in Sec. 14.4. The relatively large semiconductor transition cross section (see Table 14.3-1) implies a small saturation photon-flux density [<Ps l/Tr a(v)] and therefore a reduced gain coefficient [see (14.4-2) and (14.4-3)]. This in turn limits the overall gain that an SOA can provide. In common with other optical amplifiers, SOAs suffer from amplified spontaneous emission noise (see Sec. 14.5); however, they are also affected by noise associated with temperature and carrier fluctuations. Approximate Peak Gain Coefficient The complex dependence of the gain coefficient on the injected-carrier concentra- tion makes the analysis of the semiconductor amplifier (and laser) somewhat difficult. Because of this, it is customary to adopt an empirical approach in which the peak gain coefficient "Yp is assumed to be linearly related to n for values of n near the operating point. As the example in Fig. 17 .2-3(b) illustrates, the approximation is reasonable when "Yp is large. The dependence of the peak gain coefficient "Yp on n may then be modeled by the linear relation (17.2-7) Peak Gain Coefficient (Linear Approximation) which is illustrated in Fig. 17.2-4. The parameters 0: and nT are chosen to satisfy the following limits: . When n == 0, "Yp == -0:, where 0: represents the absorption coefficient of the semiconductor in the absence of current injection. . When n == nT, "Yp == O. Thus, nT is the injected-carrier concentration at which emission and absorption just balance so that the medium is transparent. p a ( :n: - 1 ) , p s::: . u s::: S '@ C o C) s::: C/) .- C/) ro 0 OJ) ro // nT / / / -Q / n Figure 17.2-4 Peak value of the gain coeffi- cient "!p as a function of injected-carrier concen- tration n for the approximate linear model. The quantity Q represents the attenuation coefficient in the absence of injection, whereas nT represents the injected-carrier concentration at which emis- sion and absorption just balance each other. The solid portion of the straight line matches the more realistic calculation considered in the preceding subsection. / o EXAMPLE 17.2-2. Approximate Peak Gain Coefficient for an InGaAsP SOA. The peak gain coefficient "!p versus n for InGaAsP presented in Fig. 17 .2-3(b) may be approximately fit by a linear relation that takes the form of (17.2-7), with the parameters nT 1.25 x 10 18 cm- 3 and Q == 600 cm- 1 . For n == 1.4nT == 1.75 x 10 18 cm- 3 , the linear model yields a peak gain coefficient "!p == 240 cm -1. For an InGaAsP crystal of length d == 350 pm, this corresponds to a total gain of exp( ,,!pd) 4447 or 36.5 dB. In practice, this value is reduced by gain saturation, as discussed above, as well as by coupling losses, which are typically 3 to 5 dB per facet.
708 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Increasing the injected-carrier concentration from below to above the transparency value nT results in the semiconductor changing from a strong absorber of light [fg(v) < 0] into a high-gain amplifier of light [fg(v) > 0]. The very same large transition probability that makes the semiconductor a good absorber also makes it a good amplifier, as may be understood by comparing (16.2-18) and (16.2-19). B. Pumping Optical Pumping Pumping may be achieved by the use of external light, as depicted in Fig. 17.2-5, provided that its photon energy is sufficiently large (> E g). Pump photons are absorbed by the semiconductor, resulting in the generation of carrier pairs. The generated elec- trons and holes decay to the bottom of the conduction band and the top of the valence band, respectively. If the intraband relaxation time is much shorter than the interband relaxation time, as is usually the case, a steady-state population inversion between the bands may be established, as discussed in Sec. 14.2. Pump photon I . I Output signal nput sIgna 1\n1\I\A.Jo.- h t 1\I\I\I\A.Jo.- 'vvvv P oons photon 'vvvv I I I I I I I I I I I I I I I I I I I ) k Figure 17.2-5 Optical pumping of a semi- conductor optical amplifier. Current Pumping A more practical scheme for pumping a semiconductor optical amplifier is by means of electron-hole injection in a heavily doped p-n junction - a diode. As with the LED (see Sec. 17.1) the junction is forward biased so that minority carriers are injected into the junction region (electrons into the p-type region and holes into the n-type region). Figure 17.1-5 shows the energy-band diagram of a forward-biased heavily doped p-n junction. The conduction-band and valence-band quasi-Fermi levels, E fe and E fv, lie within the conduction and valence bands, respectively, and a state of quasi- equilibrium exists within the junction region. The quasi-Fermi levels are sufficiently well separated so that a population inversion is achieved and net gain may be obtained over the bandwidth E 9 < hv < E fe - E fv within the active region. The thickness l of the active region is an important parameter of the diode that is determined principally by the diffusion lengths of the minority carriers at both sides of the junction. Typical values of l for InGaAsP are 1-3 Mm. If an electric current i is injected through an area A == wd, where wand dare the width and height of the device, respectively, into a volume lA (as portrayed in Fig. 17.2-6), then the steady-state carrier injection rate is R == i / elA == J / el per second per unit volume, where J == i / A is the injected current density. The resulting injected- carrier concentration is then T T n == T R == - i == - J. elA el (17.2-8)
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 709 Output photons 1 l + p n Input photons Figure 17.2-6 Geometry of a simple semi- conductor optical amplifier. Charge carriers travel perpendicularly to the p-n junction, whereas photons travel in the plane of the junction. The injected-carrier concentration is therefore directly proportional to the injected current density so that the results shown in Figs. 17.2-3 and 17.2-4 with n as a parameter may just as well have J as a parameter. In particular, it follows from (17.2- 7) and (17.2-8) that within the linear approximation implicit in (17.2- 7), the peak gain coefficient is linearly related to the injected current density J, i.e., p a ( - 1 ). (17.2-9) Peak Gain Coefficient The transparency current density J T is given by el J T == - nT, ItiTr (17.2-10) Transparency Current Density where It i == T / Tr again represents the internal quantum efficiency. When J == 0, the peak gain coefficient "Yp == -Q becomes the attenuation co- efficient, as is apparent in Fig. 17.2-7. When J == J T , "Yp == 0 and the material is transparent and neither amplifies nor attenuates. Net gain can be achieved only when the injected current density J exceeds its transparency value J T . Note that J T is directly proportional to the junction thickness I so that a lower transparency current density J T is achieved by using a narrower active-region thickness. This is an important consideration in the design of semiconductor optical amplifiers (and lasers). EXAMPLE 17.2-3. Gain of an InGaAsP SOA. An InGaAsP semiconductor optical amplifier operates at 300 0 K and has the following parameters: Tr == 2.5 ns, Il i == 0.5, nT == 1.25 x 10 18 cm -3, and Q == 600 cm -1. The junction has thickness l == 2 pm, length d == 200 pm, and width w == 10 pm. Using (17.2-10), the current density that just makes the semiconductor transparent is J T == 3.2 X 10 4 A/cm 2 . A slightly larger current density J == 3.5 X 10 4 A/cm 2 provides a peak gain coefficient 'Yp 56 cm- 1 as is clear from (17.2-9). This gives rise to an amplifier gain G == exp(')'pd) == exp(1.12) 3. However, since the junction area A == wd == 2 x 10- 5 cm 2 , a rather large injection current i == J A == 700 mA is required to produce this current density.
710 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES s::: . s::: C) .- .- C\S 0 Q) o C) 00 s::: 00 .- 0 ...J C\S Q) o / / J T / / / / Current density J Figure 17.2-7 Peak optical gain coefficient 1p as a function of cur- rent density J for the approximate linear model. When J = J T the material is transparent and exhibits neither gain nor loss. -Q Motivation for Heterostructures If the thickness I of the active region in Example 17.2-3 were reduced from 2 Mm to, say, 0.1 Mm, the current density J T would be reduced by a factor of 20, to the more reasonable value 1600 A/cm 2 . Because proportionately less volume would have to be pumped, the amplifier could then provide the same gain with a lower injected current density. Such a reduction in the thickness of the active region poses a potential problem, however, because the diffusion lengths of the electrons and holes in InGaAsP are several Mm and the carriers would tend to diffuse out of this smaller region. However, it is possible to confine carriers to an active region whose thickness is smaller than their diffusion lengths by making use of a heterostructure device, as discussed in Sec. 17.2C. Indeed, light can simultaneously be confined in such a structure, providing an additional advantage. c. Heterostructures As is apparent from (17.2-9) and (17.2-10), the diode-laser peak amplifier gain co- efficient ryp varies inversely with the thickness I of the active region. It is therefore advantageous to use the smallest thickness possible. The active region is defined by the diffusion distances of minority carriers on both sides of the junction. The concept of the double heterostructure is to form heterojunction potential barriers on both sides of the p-n junction to provide a potential well that limits the distance over which minority carriers may diffuse. The junction barriers define a region of space within which minority carriers are confined, allowing active regions of thickness I as small as 0.1 Mm to be achieved. Yet thinner confinement regions, 0.01 Mm, can be attained by making use of quantum-well devices, as will be discussed in Sec. 17.2D. Electromagnetic confinement of the amplified optical beam can be achieved simul- taneously if the material of the active layer is selected such that its refractive index is slightly greater than that of the two surrounding layers, in which case the structure acts as an optical waveguide (see Sec. 8.2). The double-heterostructure design therefore calls for three layers of different lattice- matched materials, as illustrated in Fig. 17.2-8: Layer 1: p-type, energy gap E g1 , refractive index n1 Layer 2: p-type, energy gap E g2 , refractive index n2 Layer 3: n-type, energy gap E g3 , refractive index n3 The semiconductor materials are selected such that E g1 and Eg3 are greater than E g2 , which achieves carrier confinement, while n2 is greater than n1 and n3, which achieves light confinement. The active layer (layer 2) is made quite thin (0.1 to 0.2 Mm) to minimize the transparency current density J T and thereby to maximize the peak gain coefficient ryp. Stimulated emission takes place in the p-n junction between layers 2 and 3.
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 711 V + E T Barrier { E g ) 1 --- - f E g2 l t Eg3 t ----T eV t 11 ) 11) 112 113 Figure 17.2-8 Energy-band diagram and refractive index as functions of position for a double-heterostructure semiconductor optical amplifier. In summary, the double-heterostructure design offers the following advantages: . Increased amplifier gain, for a given injected current density, as a result of de- creased active-layer thickness, in accordance with (17.2-9) and (17.2-10). Injected minority carriers are confined within the thin active layer between the two hetero- junction barriers and are prevented from diffusing to the surrounding layers. . Increased amplifier gain resulting from the confinement of photons within the active layer as a result of its larger refractive index. The active medium acts as an optical waveguide. . Reduced loss, resulting from the inability of layers 1 and 3 to absorb the guided photons because the bandgaps of these layers, E g1 and E g3 , are larger than the photon energy (hv == E g2 < E g1 , E g2 ). Two examples of double-heterostructure semiconductor optical amplifiers follow: . In GaAsP / InP Double-Heterostructure Laser Diode Amplifier. The active layer, In1-xGaxAs1-yPy, is surrounded by layers of InP. The composition parameters x and yare selected so that the materials are lattice matched. Operation is thereby restricted to a range of values of x and y for which E g2 corresponds to the band 1.1-1.7 Mm. . GaAsjAlGaAs Double-Heterostructure Laser Diode Amplifier. The active layer (layer 2) is fabricated from GaAs (E g2 == 1.42 eV, n2 == 3.6). The surround- ing layers (1 and 3) are fabricated from AlxGa1-xAs with Eg > 1.43 eV and n < 3.6 (by 5-10%). This amplifier typically operates within the 0.82-0.88- Mm wavelength band when the AIGaAs composition parameter is in the range x == 0.35-0.5. D. Quantum-Well Structures As discussed in Sec. 17.2C, heterostructures offer a reduced thickness of the active layer within which carriers and photons are confined. This in turn provides increased amplifier gain and reduced amplifier loss. When the thickness of the active layer is reduced yet further, say to 5-10 nm (which is smaller than the de Broglie wavelength of a thermalized electron), quantum effects playa key role. Since the active layer in
712 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES a double heterostructure has a bandgap energy smaller than that of the surrounding layers, the structure then acts as a quantum well (see Sec. 16.1 G), and is referred to as a quantum-well device. The band structure and energy-momentum (E-k) relations of a quantum well are different from those of a bulk material. The conduction band is split into a number of subbands, labeled by the quantum number q == 1,2,. . ., each with its own energy- momentum relation and density of states. The bottoms of these subbands have energies Ee + Eq, where Eq == fi2(q1f /1)2 /2me, q == 1,2,..., are the energies of an electron of effective mass me in a one-dimensional quantum well of thickness I (see Figs. 16.1- 24 and 16.1-26; ql and d 1 in Chapter 16 correspond to q and I here). Each subband has a parabolic E-k relation and a constant density of states that is independent of energy. The overall density of states in the conduction band, (}e ( E), therefore assumes a staircase distribution [see (16.1-37)] with steps at energies E e + E q, q == 1, 2, . . .. The valence band has similar subbands at energies Ev - E, where E == fi2(q1f /1)2 /2mv are the energies of a hole of effective mass mv in a quantum well of thickness I. The interactions of photons with electrons and holes in a quantum well take the form of energy- and momentum-conserving transitions between the conduction and valence bands. The transitions must also conserve the quantum number q, as illustrated in Fig. 17.2-9; they obey rules similar to those that govern transitions between the conduction and valence bands in bulk semiconductors. The expressions for the transi- tion probabilities and gain coefficient in the bulk material (see Sec. 16.2) apply to the quantum-well structure if we simply replace the bandgap energy E 9 with the energy gap between the subbands, Egq == Eg + Eq + E, and use a constant density of states rather than one that varies as the square root of energy. The total gain coefficient is the sum of the gain coefficients provided by all of the subbands (q == 1, 2, . . .). E g(v) ------ -f- q= 1 E 2 t ---- ------ i Ec ---(----T-- E g2 E E 91 E' 9 j E ___L____l__ ___v - -T E2 q= 1 ______ _l___ // . , (a) )0 k (b) // / I' ! Eg E g1 , E g2 hv Figure 17.2-9 (a) E-k relations of different subbands. (b) Optical joint density of states for a quantum-well structure (staircase curve) and for a bulk semiconductor (dashed curve). The first jump occurs at energy Egl == Eg + El + E (where El and E are, respectively, the lowest energies of an electron and a hole in the quantum well). Density of States Consider transitions between the two subbands of quantum number q. To satisfy the conservation of energy and momentum, a photon of energy hv interacts with states of energies E == Ee + Eq + (mr/me)(hv - Egq) in the upper subband and E - hv
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 713 in the lower. The optical joint density of states g( v) is related to ge (E) by g( v) (dE/dv) ge(E) == (hmr/me) ge(E). It follows from (16.1-37) that { hmr me 2m r , , hv > Eg + Eq + E q g(v) == me 7rn 2 1 == hi 0, otherwise. (17.2-11) Including transitions between all subbands q == 1, 2, . . ., we arrive at a g( v) that has a staircase distribution with steps at the energy gaps between subbands of the same quantum number (Fig. 17.2-9). Gain Coefficient The gain coefficient of the device is given by the usual expression [see (16.2-24)] A 2 '"Yo (v) == _ 8 g(v) fg(v) , 7rTr (17.2-12) where the Fermi inversion factor fg(v) depends on the quasi-Fermi levels and temper- ature, and is the same for bulk and quantum-well lasers. The density of states g(v), however, differs in the two cases, as we have shown. The frequency dependences of g(v), fg(v), and their product are illustrated in Fig. 17.2-10 for quantum-well and bulk double-heterostructure configurations. The quantum-well structure has a smaller peak gain coefficient and a narrower gain profile. g(v) /' Bulk / '- /' QW / / I fgCv) Eg E9l hv +1 hv -1 'Yo(v) c:: 'Ym / . I d 0 C/:J C/:J 0 hv \ Figure 17.2-10 Density of states Q(v), Fermi inversion factor fg(v), and gain coeffi- cient 1'0 (v) in quantum-well (solid) and bulk (dashed) structures. It is assumed in the construction of Fig. 17.2-10 that only a single step of the staircase function g(v) occurs at an energy smaller than E fe - Efv. This is the case
714 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES under usual injection conditions. The maximum gain m may then be determined by substituting f 9 (v) == 1 and Q( v) == 2m r / hI in (17.2-12), which yields A 2 m r m == 2Trhi . (17.2-13) Relation Between Gain Coefficient and Current Density By increasing the injected current density J, the concentration of excess electrons and holes n is increased and, therefore, so is the separation between the quasi-Fermi levels E fe - E fv. The effect of this increase on the gain coefficient o (v) may be assessed by examining the diagrams in Fig. ] 7.2-10. For sufficiently small J there is no gain. When J is such that E fe - E fv just exceeds the gap E gl between the q == 1 subbands, the medium provides gain. The peak gain coefficient increases sharply and saturates at the value m. An increase of J increases the gain spectral width but not its peak value. If J is increased yet further, to the point where E fe - E fv exceeds the gap E g2 between the q == 2 subbands, the peak gain coefficient undergoes another jump, and so on. The gain profile can therefore be quite broad, providing the possibility of a wide tuning range for such devices. Materials and Device Structures The structure of a semiconductor optical amplifier resembles that of a laser diode operated above transparency but below the threshold of oscillation (see Sec. 17.3). Semiconductor optical amplifiers can be made to operate in any region of the optical spectrum by judiciously choosing the semiconductor material. The center wavelength, bandwidth, and gain depend both on the material and on the structure of the device. SOAs designed for optical transmission applications in the near infrared are usually fabricated from InGaAsP, InGaAs, or InP. In the 1300-1600-nm telecommunications band, achievable bandwidths are A 50 nlli, corresponding to v 10 THz at AD == 1300 nm (see Example 17.2-1). This is broader than the bandwidths offered by EDFAs and similar to those provided by RFAs (see Secs. 14.3C and 14.3D). Quantum- well SOAs offer a substantial reduction in the drive current required to achieve trans- parency but otherwise behave similarly to bulk devices. Bandwidths for quantum-dot SOAs can stretch up to nearly 200 nm, corresponding to v 25 THz at AD == 1550 nm. The gain of an SOA is usually limited to 15 dB because of gain saturation and insertion losses of 3-5 dB per facet (see Example 17.2-2). Saturation leads to inter- channel and intersymbol interference, rendering SOAs unsuitable for use in DWDM communication systems (see Sec. 24.3C). Furthermore, the short semiconductor re- combination time (see Table 14.3-1) leaves the SOA susceptible to high-frequency noise that might reside in the pumping current and optical signal, leading to noise figures 8-10 dB as opposed to 3 dB for EDFAs. It is important to note that if an SOA is to be operated as a broadband single-pass device (i.e., as a traveling-wave amplifier), the facet reflectances must be reduced to a minimum. Failure to do so can lead to multiple reflections and a gain profile that is modulated by the resonator modes; this can also result in oscillation, which, of course, obviates the possibility of controllable amplification. Techniques for reducing reflectances include the use of antireflection coatings and tilted waveguides. As a result of the issues discussed above, optical-transmission applications of SOAs have, for the most part, been limited to metropolitan optical networks where low gain suffices to overcome losses associated with multiple optical add-drop nodes. SOAs hold greater appeal for applications such as nonlinear optical elements and optical switches (see Sec. 23.3C), as well as for wavelength conversion.
17.2 SEMICONDUCTOR OPTICAL AMPLIFIERS 715 EXAMPLE 17.2-4. Waveguide Amplifiers. Multiquantum-well semiconductor optical am- plifiers can be constructed in the form of optical waveguides, providing operation in fundamental optical modes at increased output saturation powers, and employing direct butt coupling to single- mode fibers. Such devices have relatively low losses and a small optical confinement factor. As an example, a 1550-nm InGaAsP /lnP quantum-well amplifier with a length of 1 cm provides a fiber-to- fiber gain of 13 dB. t Comparison of SOAs and OFAs The semiconductor optical amplifier enjoys advantages and disadvantages with respect to optical fiber amplifiers such as the erbium-doped fiber amplifier and the Raman fiber amplifier: Advantages: . Central wavelength selectable by choice of material . Compatible with integrated optoelectronic circuits . Electrical pumping . Small size . Low cost Disadvantages: . Low gain . Low saturated output power . High noise . Substantial interchannel and intersymbol interference . Sensitivity to thermal effects from heat dissipation . Sensitivity to facet reflections . Sensitivity to signal polarization . Control of transverse-mode characteristics . High insertion loss . Incompatibility with fiber geometry On balance, the performance of the SOA is generally inferior to that of the EDFA and the RFA, and its use is generally restricted to special applications (see Sec. 24.1 C). The relative merits of EDFAs and RFAs have been considered in Sec. 14.3D. E. Superluminescent Diodes Superluminescent diodes (SLDs) are semiconductor laser diodes with sufficiently strong current injection so that stimulated emission outweighs spontaneous emission. SLDs differ from SOAs in that no optical signal is presented to the device. Rather, the emission is amplified spontaneous emission (ASE) produced by the device itself (see Sec. 14.5). An example of an SLD is the multiquantum-well InGaAsP jlnP structure displayed in Fig. 17.2-11. The optical output power of an SLD is generally greater than that of an LED but less than that of an LD (see Fig. 17.3-5); the optical spectrum is typically narrower than that of an LED but broader than that of an LD (see Fig. 17.3-7). As with the semiconductor optica] amplifier, it is important to minimize optical feedback to avoid lasing. This may be achieved in any number of ways, such as by using a stripe t See P. W. Juodawlkis, J. J. Plant, R. K. Huang, L. J. Missaggia, and J. P. Donnelly, High-Power 1.5-J-Lm InGaAsP-InP Slab-Coupled Optical Waveguide Amplifier, IEEE Photonics Technology Letters, vol. 17, pp. 279- 281, 2005.
716 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES contact that injects current only over a portion of the device, by using a tapered-stripe geometry, or by antireflection-coating or tilting the facets of the device. + InP cladding layers Figure 17.2-11 MQW InGaAsPjInP su- perluminescent diode. SLDs generate light with substantial optical power and a band- width intermediate between that of an LED and an LD. It is important to minimize feedback so that laser oscillation does not occur. One way of achieving this is to use a stripe contact that injects current only over a portion of the device. Superluminescent diodes find use in applications where the long coherence time of laser light is troublesome because of randomly occurring interferences. Examples of such applications include interferometric instrumentation such as optical coherence tomography (see Sec. 11.2B), fiber-optic gyroscopy, and certain fiber-optic sensors. Optical fiber amplifiers are also sometimes used as sources of superluminescence light. 17.3 LASER DIODES In this section we consider the general characteristics of laser diodes, which take many forms. Quantum-confined and microcavity semiconductor lasers are considered further in Sec. 17.4. A. Amplification, Feedback, and Oscillation A laser diode is a semiconductor optical amplifier that is endowed with a path for optical feedback. As discussed in the preceding section, a semiconductor optical am- plifier is a forward-biased heavily doped p-n junction fabricated from a direct-bandgap semiconductor material. The injected current is sufficiently large to provide optical gain. The optical feedback is provided by mirrors, which are usually implemented by cleaving the semiconductor material along its crystal planes. The sharp refractive index difference between the crystal and the surrounding air causes the cleaved surfaces to act as reflectors. Thus, the semiconductor crystal acts both as a gain medium and as a Fabry-Perot optical resonator, as illustrated in Fig. 17.3-1. Provided that the gain coefficient is sufficiently large, the feedback converts the optical amplifier into an optical oscillator, i.e., a laser. The device is called a laser diode or a diode laser (it is also sometimes referred to as a semiconductor injection laser). The laser diode (LD) bears considerable similarity to the light-emitting diode (LED) discussed in Sec. 17.1. In both devices, the source of energy is an electric current injected into a p-n junction. However, the light emitted from an LED is generated by spontaneous emission, whereas the light from an LD arises from stimulated emission. Laser diodes have a number of advantages with respect to other types of lasers: high power, high efficiency, small size, compatibility with electronic components, and ease of pumping and modulation by electric current injection. However, their broader bandwidths and lower coherence can be detriInental in certain applications. Laser diodes have manifold uses, as discussed subsequently.
17.3 LASER DIODES 717 Cleaved surface TI + d p n "" Area A " Cleaved surface Figure 17.3-1 A laser diode is a forward-biased p-n junction with two parallel surfaces that act as reflectors. We begin our consideration of the conditions required for laser oscillation and the properties of the emitted light with a brief summary of the basic results that describe the semiconductor optical amplifier and the optical resonator. Laser Amplification The gain coefficient '"Yo (v) of a semiconductor optical amplifier has a peak value '"Yp that is approximately proportional to the injected-carrier concentration, which in turn is proportional to the injected current density J. Thus, as provided in (17.2-9) and (17.2-10), and as illustrated in Fig. 17.2-7, p ex ( - 1) , el J T == - nT , ItiTr (17.3-1) where Tr is the radiative electron-hole recombination lifetime, It i == T / Tr is the internal quantum efficiency, l is the thickness of the active region, a is the thermal-equilibrium absorption coefficient, and nT and J T are the injected-carrier concentration and current density required to just make the semiconductor transparent. Feedback The feedback is often obtained by cleaving the crystal planes normal to the plane of the junction, or by polishing two parallel surfaces of the crystal. The active region of the p-n junction illustrated in Fig. 17.3-1 then also serves as a planar-mirror optical resonator of length d and cross-sectional area lw. Semiconductor materials typically have large refractive indexes, so that the power reflectance at the semiconductor-air interface 9(== ( n-l ) 2 n+l (17.3-2) is substantial [see (6.2-15) and Table 16.2-1]. Thus, if the gain of the medium is sufficiently large, the refractive-index discontinuity can itself serve as an adequate reflective surface and no external mirrors are necessary. For GaAs, for example, n == 3.6, so that (17.3-2) yields 9( == 0.32.
718 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Resonator Losses The principal source of resonator loss arises from the partial reflection at the surface of the crystal. This loss constitutes the transmitted useful laser light. For a resonator of length d , the reflection loss coefficient is [see (10.1-22)] am = amI + a m 2 = ;d In( 9( 1 19(2 ) (17.3-3) if the two surfaces have the same reflectances 9(1 == 9(2 == 9(, then am == (1/ d) In(I/9(). The total loss coefficient is aT == as + am, ( 17.3 -4 ) where as represents other sources of loss, including free-carrier absorption in the semi- conductor material (see Fig. 16.2-2) and scattering from optical inhomogeneities. The 9uantity as increass as the concentraion of ipurities and interfacial imrerfections In heterostructures Increase. It can attaIn values In the range 10 to 100 cm - . Of course, the term -a in the expression for the gain coefficient (17.3-1), corre- sponding to absorption in the material, also contributes substantially to the losses. This contribution is accounted for, however, in the net peak gain coefficient "Yp given by (17.3-1). This is apparent from the expression for "Yo (v) given in (16.2-24), which is proportional to I g (v) == Ie (v) - I a (v) (i.e., to stimulated emission less absorption). Another important contribution to the loss results from the spread of optical energy outside the active layer of the amplifier (in the direction perpendicular to the junction plane). This can be especially detrimental if the thickness of the active layer I is small. The light then propagates through a thin amplifying layer (the active region) surrounded by a lossy medium so that large losses are likely. This problem may be alleviated by the use of a double heterostructure (see Sec. 17.2C and Fig. 17.2-8), in which the middle layer is fabricated from a material of elevated refractive index that acts as a waveguide confining the optical energy. Losses caused by optical spread may be phenomenologically accounted for by defin- ing a confinement factor r to represent the fraction of the optical energy lying within the active region (Fig. 17.3-2). Assuming that the energy outside the active region is totally wasted, r is therefore the factor by which the gain coefficient is reduced, or equivalently, the factor by which the loss coefficient is increased. Equation (17.3-4) must therefore be modified to reflect this increase, so that 1 aT == r (a s + am). (17.3-5) There are three types of simple laser-diode structures based on the mechanism used to confine the carriers or light in the lateral direction (i.e., in the junction plane): broad- area (in which there is no mechanism for lateral confinement), gain-guided (in which lateral variations of the gain are used for confinement), and index-guided (in which lateral refractive-index variations are used for confinement). Gain Condition: Laser Threshold The laser oscillation condition is that the gain exceed the loss, "Yp > aT' as indicated in (15.1-12). The threshold gain coefficient is therefore aT. Setting "Yp == aT and J == J t
17.3 LASER DIODES 719 T l l d p n p n 1 Refractive index I I I I I I I I I I - I I I I I I I I I I x I I I I I I . x H I I I I I I I I I I I I I I I I I I A . x I I I I . x (a) . x A . x (b) Figure 17.3-2 Spatial spread of the laser light in the direction perpendicular to the plane of the junction for: (a) homostructure, and (b) heterostructure lasers. in (17.3-1) corresponds to a threshold injected current density J t given by T _ aT + a T Jt - JT , a (17.3-6) Th reshold Cu rrent Density where the transparency current density, el J T == - nT , Il i Tr (17.3-7) Transparency Current Density is the current density that just makes the medium transparent. The threshold current density is larger than the transparency current density by the factor (aT + a) / a, which is 1 when a » aT. Since the current i == J A, where A == wd is the cross-sectional area of the active region, we can define iT == JTA and it == JtA, corresponding to the currents required to achieve transparency of the medium and laser-oscillation threshold, respectively. The threshold current density J t is a key parameter in characterizing the laser-diode performance; smaller values of J t indicate superior performance. In accordance with (17.3-6) and (17.3-7), J t is minimized by maximizing the internal quantum efficiency Ili; and by minimizing the resonator loss coefficient aT' the transparency injected- carrier concentration nT, and the active-region thickness l. As l is reduced beyond a certain point, however, the loss coefficient aT becomes larger because the confinement factor r decreases [see (17.3-5)]. Consequently, J t decreases with decreasing l until it reaches a minimum value, beyond which any further reduction causes J t to increase (see Fig. 17.3-3). In double-heterostructure lasers, however, the confinement factor remains near unity for lower values of l because the active layer behaves as an optical waveguide (see Fig. 17.3-2). The result is a lower minimum value of J t , as shown in Fig. 17.3-3, and therefore superior performance. The reduction in J t is illustrated in the following examples. Because the parameters nT and a in (17.3-1) are temperature dependent, so too are the threshold current density J t and the frequency of peak gain. Temperature
720 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Homostructure .... .c- Double .- heterostructure (J) '"d ..... s::: u '"d - o ..c:: rF.J (J) ..c:: Active-layer thickness I Figure 17.3-3 Dependence of the threshold current density J t on the thickness of the active layer l. The double-heterostructure laser exhibits a lower value of J t than the homo structure laser, and therefore superior performance. The increase of J t at small values of l is a result of the reduction in confinement for thin active layers. control can be used to stabilize the laser output and to modify the output frequency. EXAMPLE 17.3-1. Threshold Current for an InGaAsP Homostructure Laser Diode. Consider an InGaAsP homostructure laser diode with the same material parameters as in Examples 17.2-1 and 17.2-2: nT = 1.25 x 10 18 cm- 3 , a = 600 cm- 1 , Tr = 2.5 ns, n = 3.5, and Ili = 0.5 at T = 300 0 K. Assume that the dimensions of the junction are d = 200 pm, W = 10 pm, and l = 2 pm. The current density necessary for transparency is then calculated to be J T = 3.2 X 10 4 A/cm 2 . We now determine the threshold current density for laser oscillation. Using (17.3-2), the surface reflectance is 1< = 0.31. The corresponding mirror loss coefficient is am = (l/d) In(l/1<) = 59 cm -1. Assuming that the loss coefficient due to other effects is also as = 59 cm -1 and that the confinement factor r 1, the total loss coefficient is then a r = 118 cm- 1 . The threshold current density is therefore J t = [(ar + a)/a] J T = [(118 + 600)/600][3.2 x 10 4 ] = 3.8 X 10 4 A/cm 2 . The corresponding threshold current it = J t wd 760 mA, which is rather high. Homostructure lasers are rarely used because of the difficulties of achieving CW operation without cooling to dissipate heat. EXAMPLE 17.3-2. Threshold Current for an InGaAsP Heterostructure Laser Diode. We turn now to an InGaAsP /lnP double-heterostructure laser diode (see Fig. 17.2-8) with the same parameters and dimensions as in Example 17.3-1 except for the active-layer thickness, which is now taken to be l = 0.1 pm instead of 2 pm. If the confinement of light is assumed to be perfect (f = 1), we may use the same values for the resonator loss coefficient a r . The transparency current density is then reduced by a factor of 20 to become J T = 1600 A/cm 2 , and the threshold current density assumes a more reasonable value of J t = 1915 A/cm 2 . The corresponding threshold current is it = 38 mA. It is this significant reduction in threshold current that made CW operation of the double- heterostructure laser diode feasible at room temperature. B. Power and Efficiency Internal Photon Flux When the laser current density is increased above its threshold value (i.e., J > J t ), the amplifier peak gain coefficient "Yp exceeds the loss coefficient aT. Stimulated emission then outweighs absorption and other resonator losses so that oscillation can begin and the photon flux <I> in the resonator can increase. As with other homogeneously broadened lasers, saturation sets in as the photon flux becomes larger and the popu- lation difference becomes depleted [see (15.1-2)]. As shown in Fig. 15.2-1, the gain coefficient then decreases until it becomes equal to the loss coefficient, whereupon steady state is reached. As with the internal photon-flux density and the internal photon-number density
17.3 LASER DIODES 721 considered for other types of lasers [see (15.2-2) and (15.2-13)], the steady-state inter- nal photon flux <I> is proportional to the difference between the pumping rate R and the threshold pumping rate Rt. Since R ex i and Rt ex it, in accordance with (17.2-8), <I> may be written as { - t <I> == Iti e ' 0, i > it i < it. (17.3-8) Steady-State Internal Photon Flux Thus, the steady-state laser internal photon flux (photons/ s generated within the active region) is equal to the electron flux (injected electrons / s) in excess of that required for threshold, multiplied by the internal quantum efficiency It i. The internal laser power above threshold is simply related to the internal photon flux <I> by the relation P == hv<I>, so that we obtain ( ) 1.24 P = Il i i-it --;:-' (17.3-9) Internal Laser Power Ao (11 m ), P (W), i (A) where .Ao is expressed in /-Lm, i in amperes, and P in watts. Output Photon Flux and Efficiency The laser output photon flux <I>o is the product of the internal photon flux <I> and the extraction efficiency It e [see (15.2-16)], which is the ratio of the loss associated with the useful light transmitted through the mirrors to the total resonator loss a r . If only the light transmitted through mirror 1 is used, then It e == amI / a r ; on the other hand, if the light transmitted through both mirrors is used, then It e == am / a r . In the latter case, if both mirrors have the same reflectance 9(, we obtain Ite == [(1/ d) In(I/9()]/ are The laser output photon flux is therefore given by - t <I>o == IteIti . e (17.3-10) Laser Output Photon Flux The proportionality between the laser output photon flux and the injected electron flux above threshold set forth in (17.3-10) is governed by a quantity known as the external differential quantum efficiency, I Ild = Ilelli . (17.3-11) External Differential Quantum Efficiency The quantity Itd thus represents the rate of change of the output photon flux with respect to the injected electron flux above threshold: d<I>o Itd == d(i/e) . (17.3-12)
722 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES The laser output power above threshold is Po == hviJ? 0 == Il d (i - it) (hv j e ), which is written more simply as ( ) 1.24 Po == Ild i-it T' o (17.3-13) Laser Output Power Ao (pm), Po (W), i (A) when Ao is expressed in Mm. This relationship is called the light-current curve. The slope of this curve above threshold is known as the differential responsivity of the laser, which is usually specified in units of W / A: 9{ == dPo == 1.24 d di Ild Ao . [Ao (MID), Po (W), i (A)] (17.3-]4) Light-current curves for two laser diodes are displayed as the solid curves in Fig. 17.3-4: (a) a gain-guided MQW InGaAsP jlnGaAsP device operating at 1550 nm; and (b) a MQW GaN jlnGaN device operating at 405 nm. The theoretical fits provided by (17.3-13) are shown as dashed curves. 28 S 24 20 o 0.. 16 c; .g 12 0.. o '5 8 0.. '5 4 o ; , , , Ideal " , , , , , , , , , I. I. 56 S 48 40 o 0.. 32 c; .g 24 0.. o '5 16 0.. '5 8 o , , Ideal " , , , I. GaN jlnGaN o 0 40 80 120 160 200 0 0 20 40 60 80 100 Drive current i (mA) Drive current i (mA) (a) (b) Figure 17.3-4 Measured (solid) and ideal (dashed) light-current curves for: (a) a gain-guided MQW InGaAsP jlnGaAsP laser diode operated at a wavelength of 1550 nm in the near infrared (the device structure is exhibited in Fig. 17.4-8); (b) a MQW GaNjlnGaN laser diode operated at a wavelength of 405 nm in the violet. Nonlinearities, which are not accounted for by the simple theory, cause the optical output power to saturate. The parameters associated with these laser diodes are readily extracted by making use of (17.3-13) and (17.3-14); their values are presented in Table 17.3-1. Although the external differential quantum efficiency Il d is nearly identical for both devices, the differential responsivity 9{d is about a factor of four greater for the GaN jlnGaN device by virtue of its shorter operating wavelength, as is readily understood from (17.3-14). Table 17.3-1 Laser-diode operating parameters extracted from the infrared and violet light-current curves displayed in Figs. 17.3-4(a) and (b), respectively. Material Ao (nm) InGaAsP jlnGaAsP GaN jlnGaN 1550 405 it (mA) 15 35 9\d (W/A) 0.26 1.0 Ild 0.33 0.33
17.3 LASER DIODES 723 The power-conversion efficiency (or wall-plug efficiency) ltc is defined as the ratio of the emitted laser light power to the electrical input power iV, where V is the forward-bias voltage applied to the diode. Since Po == ltd( i-it) (hv / e), we have ( it ) hv ltc == ltd 1 - i e V . (17.3-15) Power Conversion Efficiency For operation well above threshold, so that i » it, and for e V hv, we obtain ltc ltd. Laser diodes can exhibit power-conversion efficiencies in excess of 50%, which is well above that for other types of lasers (see Table 15.3-1). The electrical power that is not transformed into light is transformed into heat. Because laser diodes do, in fact, generate substantial amounts of heat they are usually mounted on heat sinks, which help to dissipate the heat and stabilize the temperature. EXAMPLE 17.3-3. Comparison of Efficiencies for Multiquantum-Well and Double- Heterostructure InGaAsP Laser Diodes. Consider once again Example 17.3-2 for the InGaAsP /lnP double-heterostructure laser diode with 1l.i == 0.5, am == 59 cm- 1 , a r == 118 cm- 1 , and it == 38 mA. If the light from both output faces is used, the extraction efficiency is 1l.e == am/ar == 0.5, while the external differential quantum efficiency is 1l.d == 1l.e1l.i == 0.25. At Ao == 1300 nm, the differential responsivity of this laser is 91: d == dP 0/ di == 0.24 W / A. If, for example, i == 50 mA, we have i-it == 12 mA and Po == 12 x 0.24 == 2.9 mW. Comparison of these numbers with those reported in Fig. 17.3-4(a) and Table 17.3-1 for aMQW InGaAsP/lnGaAsP laser diode operated at 1550 nm reveals that the MQW laser has a lower threshold current and a higher external differential quantum efficiency than the double-heterostructure laser, as expected. Summary There are four efficiencies associated with the laser diode: . The internal quantum efficiency It i == r r / r == T / Tn which accounts for the fact that only a fraction of the electron-hole recombinations are radiative. . The extraction efficiency lte, which accounts for the fact that only a portion of the light lost from the cavity is useful. . The external differential quantum efficiency ltd == Ilelti, which accounts for both of the above effects. . The power-conversion (wall-plug) efficiency ltc, which is the ratio of the emitted optical power to the electrical power supplied to the device. The differentia] responsivity 9{d (W / A) is also a useful measure of performance. I Comparison of LED, SLD, and LD Efficiencies and Powers It is of interest to compare the efficiencies and optical powers associated with LEDs, SLDs, and LDs. When operated below threshold, laser diodes produce spontaneous emission and behave as light-emitting diodes (see Sec. 17.1). Indeed, the presence of spontaneous emission can be discerned at low currents in LD light-current curves. The four efficiencies attendant to LD operation have been highlighted in the sum- mary above. There are also four efficiencies associated with LEDs, as discussed in Sec. 17.1. These are the internal quantum efficiency It i, which accounts for the fact
724 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES that only a fraction of the electron-hole recombinations are radiative in nature; the transmittance efficiency Il e, which accounts for the fact that only a small fraction of the light generated in the junction region can escape from the high-index medium; the external efficiency Ilex == IliIle, which accounts for both of these effects; and the power-conversion efficiency Ilc. The responsivity 9{ is also used as a measure of LED performance. There is a one-to-one correspondence between the quantities Ili, Ile, and Ilc for the LED and the LD. Furthermore, there is a correspondence between Ilex and Ild, 9{ and 9{d, and i and (i - it). The superior performance of the laser results from the fact that Ile for the LD is greater than that for the LED. This stems from the fact that the laser operates on the basis of stimulated emission, which causes the laser light to be concentrated in particular modes so that it can be more readily extracted. The net result is that a laser diode operated above threshold has a value of Il d that is larger than the value of Ilex for an LED. Superluminescent laser diodes (SLDs), which are operated with injection that is suf- ficiently strong so that stimulated emission dominates spontaneous emission, exhibit behavior intermediate between that of LEDs and LDs. As discussed in Sec. 17.2E, feedback is frustrated in these devices to avert lasing. To forge a comparison among the performance of these three classes of devices, light-current curves for a light-emitting diode, superluminescent diode, and laser diode are provided in Fig. 17.3-5. All are MQW InGaAsP /InP structures operating at a wavelength of 1600 nm. The responsivities and efficiencies of the LD are substantially greater than those of the other two devices. Moreover, as is apparent in the inset, the light-current curve of the SLD characteristically bends upward whereas the LED curve bends downward as a result of saturation. 24 E '-'" QC 20 I-< (l) 8.. 16 u "& 12 o E So 8 ==' o 28 Fig u re 17 .3-5 Light-current curves for a light-emitting diode (LED), su- perluminescent diode (SLD), and laser diode (LD). All three devices are InGaAsP /InP MQW structures operated at a wavelength of 1600 nm. The inset provides an expanded view of the LED and SLD curves. ...... LED , _ ,..': - - -' ;,.,,, 4 SLD 00 C. Spectral and Spatial Characteristics Spectral Characteristics The spectral intensity of laser light is governed by three factors, as described in Sec. 15.2B: 1. The bandwidth B over which the active medium small-signa] gain coefficient o ( v) is greater than the loss coefficient aT.
17.3 LASER DIODES 725 2. The homogeneous or inhomogeneous nature of the line-broadening mechanism (see Sec. 13.3D). 3. The resonator modes, in particular the approximate frequency spacing between the longitudinal modes Vp == c/2d, where d is the resonator length. Semiconductor laser diodes, in particular, are characterized by the following three features: 1. The spectral width of the gain coefficient is relatively large because transitions occur between two energy bands rather than between two discrete energy levels. 2. Intraband processes are very fast so that semiconductors tend to be homoge- neously broadened. Nevertheless, spatial hole burning permits the simultaneous oscillation of many longitudinal modes (see Sec. 15.2B). Spatial hole burning is particularly prevalent in short cavities in which there are few standing-wave cycles. This permits the fields of different longitudinal modes, which are dis- tributed along the resonator axis, to overlap less, thereby allowing partial spatial hole burning to occur. 3. The semiconductor resonator length d is significantly smaller than that of most other types of lasers. The frequency spacing of adjacent resonator modes Vp == c/2d is therefore relatively large. Nevertheless, many such modes can generally fit within the broad bandwidth B over which the small-signal gain exceeds the loss [the number of possible laser modes is M == B/vp, in accordance with (15.2-21)]. EXAMPLE 17.3-4. Number of Longitudinal Modes in an InGaAsP Laser Diode. An InGaAsP crystal (n == 3.5) of length d == 400 pm has resonator modes spaced by Vp == c/2d == c o /2nd 107 GHz. Near the central wavelength Ao == 1300 nm, this frequency spacing corresponds to a free-space wavelength spacing Ap, where Ap / Ao == Vp Iv, so that Ap == AoVp /v == A/2nd 0.6 nm. If the spectral width B == 1.2 THz (corresponding to a wavelength width A == 7 nm), then approximately II longitudinal modes may oscillate. A typical spectral-intensity pattern consisting of a single transverse mode and about 11 longitudinal modes is illustrated in Fig. 17.3-6. The linewidth of individual longitudinal modes is typically of the order of tens of MHz for index-guided lasers and a few GHz for gain-guided lasers. The overall spectral width of light emitted by laser diodes is greater than that of most other lasers (see Table 14.3-1). To reduce the number of modes to one, the resonator length d would have to be reduced so that B == c/2d, requiring a cavity of length d 36 pm. AF=0.6nm I I I I I I I 1290 1300 1310 Wavelength Ao (nm) Figure 17.3-6 Spectral intensity of a 1300-nm InGaAsP index-guided buried-heterostructure laser This distribution is considerably narrower, and differs in shape, from that of a Ao 1300-nm InGaAsP LED (see Fig. P 17.1-5). The number of modes decreases as the injection current increases; the mode closest to the gain maximum increases in power while the side peaks saturate. (Adapted from R. J. Nelson, R. B. Wilson, P. D. Wright, P. A. Barnes, and N. K. Dutta, CW Electrooptical Properties of InGaAsP (A == 1.3 pm) Buried-Heterostructure Lasers, IEEE Journal of Quantum Electronics, vol. QE-17, pp. 202-207, Fig. 6 @1981 IEEE.)
726 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Comparison of LED, SLD, and LD Spectra/Intensities The spectral intensities for an InGaAsP /lnP light-emitting diode, superluminescent diode, and laser diode are compared in Fig. 17.3-7. The spectral narrowing associated with stimulation emission is evident in the SLD curve, and even more so in the LD curve. LD I , I , I , I , i SLD I ' I ' I ' I ' I ' I' , Y , 'I /: , / ' : t\ I " LED / ': \ "- , I ,... """. : "........- " '. 1.4 1.5 1.6 1.7 1.8 Ao (J.Lm) Fig u re 17 .3-7 Normalized spectral intensities for a light -emitting diode (LED), superluminescent diode (SLD), and laser diode (LD). All three devices are InGaAsP /lnP structures operating at a wavelength of 1600 nm. The LED has a broad spectrum, the LD has a narrow spectrum, and the SLD lies between. Spatia/ Characteristics As with other lasers, oscillation in laser diodes takes the form of transverse and longi- tudinal modes. In Sec. 15.2C, the indexes (l, m) were used to characterize the spatial distributions in the transverse direction, while the index q was used to represent varia- tion along the direction of wave propagation or temporal behavior. In most other types of lasers, the laser beam resides totally within the active medium so that the spatial distributions of the different modes are determined by the shapes of the mirrors and their separations. For circularly symmetric systems, the transverse modes can be rep- resented in terms of Hermite-Gaussian or Laguerre-Gaussian beams (see Sec. 10.2D). However, the situation is different in semiconductor lasers since the laser beam extends outside the active layer. The transverse modes are therefore modes of the dielectric waveguide created by the different layers of the laser diode. The transverse modes can be determined by using the theory presented in Sec. 8.3 for an optical waveguide with rectangular cross section of dimensions I and w. If I / .Ao is sufficiently small, the waveguide will admit only a single mode in the transverse direction perpendicular to the junction plane. However, w is usually larger than .Ao, so that the waveguide will support several modes in the direction parallel to the plane of the junction, as illustrated in Fig. 17.3-8. Modes in the direction parallel to the junction plane are called lateral modes. The larger the ratio W / .Ao, the greater the number of lateral modes possible. Figure 17.3-8 Schematic illustration of optical-intensity spatial distributions for the laser waveguide modes (l, m) = (1, 1), (1,2), and (1,3).
17.3 LASER DIODES 727 Far-Field Radiation Pattern A laser diode with an active layer of dimensions land w emits light with far-field angular divergence Ao/l (radians) in the plane perpendicular to the junction and Ao/W in the plane parallel to the junction, as illustrated in Fig. 17.3-9. This is similar to the results for a Gaussian beam of diameter 2W o , provided in (3.1-21), for which the divergence angle is () (2/ 7r ) (Ao / 2 W o ) == Ao / 7r W o when () « 1. The angular divergence determines the far-field radiation pattern, as discussed in Sec. 4.3. Because of the small size of its active layer, the laser diode is characterized by an angular divergence larger than that of most other lasers. As an example, for l == 2 /-Lm, W == 10 /-Lm, and Ao == 800 nm, the divergence angles are calculated to be 23° and 5°. Light from a single-transverse-mode laser diode, for which w is smaller, has an even larger angular divergence. The spatial distribution of the far-field light within the radiation cone depends on the number of transverse modes and on their optical powers. The highly asymmetric elliptical distribution of laser-diode light can make collimating it tricky. Figure 17.3-9 Angular distribution of the optical beam emitted from a laser diode. Single-Mode Operation Because higher-order lateral modes have a wider spatial spread, they are less confined; their loss coefficient Qr is therefore greater than that for lower-order modes. Conse- quently, some of the highest-order modes will fail to satisfy the oscillation conditions; others will oscillate at a lower power than the fundamental (lowest-order) mode. To achieve high-power single-spatial-mode operation, the number of waveguide modes must be reduced by decreasing the dimensions of the active-layer cross section (l and w), so that it acts as a single-mode waveguide. The attendant reduction of the junction area also has the effect of reducing the threshold current. Higher-order lateral modes may be eliminated by making use of gain-guided or index-guided laser-diode configurations. Operation on a single longitudinal mode, which produces a single-frequency output, may be achieved by reducing the length d of the resonator so that the frequency spacing between adjacent longitudinal modes exceeds the spectral width of the amplifying medium. Single-mode operation may also be attained by making use of multiple-mirror resonators, as discussed in Sec. 15.2D and illustrated in Fig. 15.2-15. Another approach for achieving single-frequency operation involves the use of dis- tributed reflectors in place of the cleaved crystal surfaces that serve as lumped mirrors in the Fabry-Perot configuration. When feedback of this type is provided, the surfaces of the crystal are antireflection coated to minimize reflections. For example, frequency- selective reflectors such as Bragg gratings can be placed in the plane of the junction
728 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES [Fig. 17.3-1 O( a)]. As discussed in Secs. 2.4 Band 7.1 C, a Bragg grating reflects light when the grating period A satisfies A == q>"/2, where q is an integer. The device portrayed in Fig. 17.3-1 O( a) is called a distributed Bragg reflector laser or, more simply, a DBR laser. Alternatively, a DBR grating placed below or above the active region can also serve as a distributed reflector, as illustrated in Fig. 17.3-1 O(b). Yet another method for providing feedback makes use of a corrugation between the active and guiding layers, as shown in Fig. 17.3-10(c); this results in a periodic refrac- tive index and therefore a grating. Structures of this kind are known as distributed- feedback lasers or, for short, DFB lasers. This class of lasers offers narrow spec- tral widths and large modulation bandwidths. They are widely used as sources for optical fiber communications systems in the 1300-1600-nm wavelength range (see Sec. 17.4C). DBR grating DBR grating \. ' MQW active region MQW acti ve region (a) (b) (c) Figure 17.3-10 (a) Schematic diagram of a distributed Bragg reflector (DBR) multiquantum-well laser diode with DBR mirrors outside the active region. (b) Diagram of a distributed feedback (DFB) multiquantum-welliaser diode with a DBR structure that resides below the active region and serves as a distributed reflector. (c) Structure for a distributed feedback (DFB) muitiquantum-welliaser diode with a corrugation between the active and guiding layers that acts as a distributed reflector. 17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS A. Quantum-Confined Lasers Quantum-confined lasers, in which carriers are confined to dimensions smaller than the de Broglie wavelength of a thermalized electron ( 50 nm in GaAs), offer excel- lent performance and are the most common of all semiconductor laser diodes. Con- finement in 1, 2, and 3 dimensions corresponds to quantum-well, quantum-wire, and quantum-dot configurations, respectively, as depicted in Fig. 17.4-1. Several examples of quantum-well and multiquantum-well LDs, SLDs, LEDs, and SOAs arose earlier. As the dimensionality of a semiconductor structure decreases, the gain-coefficient curves typically increase in height and decrease in width, offering lower threshold cur- rents, higher external differential quantum efficiencies, and narrower laser linewidths. At the same time, however, the volume of the interaction region decreases with di- mensionality, which leads to reduced output power for quantum-wire and quantum-dot lasers. In this section, we discuss quantum-well, quantum-wire, and quantum-dot semi- conductor lasers in turn. We then turn to quantum-cascade lasers, which are specially designed unipolar multiquantum-well devices that generate substantial optical power in the infrared spectral region. Quantum-Well Lasers The quantum-well device portrayed in Fig. 17.4-1 (a) offers far better performance than the double-heterostructure device, as discussed in Secs. 17.2 and 17.3. The benefit accrues from the small thickness of a single quantum well, which is typically < 10
17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 729 t t + . Bragg reflectors ) f - 0 . yj r --. _u-v \ Quantum-wire active region (b) Confinement c rs Quantum-well active region (a) Quantum-dot active region (c) Figure 17.4-1 Schematic representation of quantum-confined lasers in (a) quantum-well, (b) quantum-wire, and (c) quantum-dot configurations. Charge carriers are restricted to the active region by the confinement layers and Bragg reflectors serve as mirrors. nm; this is to be compared with 100 nm for a DR laser diode and about 2 Mm for an old-fashioned homojunction laser diode. The dependences of the peak gain coefficient ryp on the current density J for SQW and bulk DR semiconductor lasers are compared in Fig. 17.4-2. The quantum-well laser has a far smaller value of J T , the current density required for transparency, although its gain saturates at a lower level. ....... s::: Q) .- C) 2'Ym S Q) o C) s::: . '"'1m OJ.) C\S Q) p... in in Current density i Figure 17.4-2 Peak gain coefficient 'Yp versus current density J for single-quantum-well (SQW) and bulk double-heterostructure (DR) semiconductor lasers. The peak gain coefficient for the SQW laser increases sharply and then saturates at multiples of the maximum gain '"'1m [see (17.2-13)]. The QW laser offers the following salutary features in comparison with its double- heterostructure counterpart: . Smaller threshold current density . Larger external differential quantum efficiency . Larger power-conversion efficiency . Narrower width of the gain coefficient . Smaller linewidth of the laser modes . Faster response allowing greater modulation frequencies . Reduced dependence on temperature Multiquantum-Well Lasers The multiquantum-well (MQW) laser (Fig. 17.4-3) offers a greater gain coefficient than the single quantum-well (SQW) laser. Indeed, the gain coefficient of a MQW laser with N -wells is N times that of each of its wells. To effect a fair comparison of the performance of the two devices, however, the pumping should be the same in both.
730 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Consider a single quantum well injected with an excess carrier density n and a peak gain coefficient ryp. In a MQW structure, each of the N wells would be injected with only n/ N carriers. Because of the nonlinear dependence of the gain on n, however, the gain coefficient of each well is rypjN, where may be smaller or greater than I, de- pending on the operating conditions. The total gain provided by the MQW laser is then N (rypj N) == ryp. It turns out that the SQW is typically superior at low current densities, whereas the MQW usually performs better at high current densities, but by a factor smaller than N. Multiquanturp.- well actIve regIon Figure 17.4-3 Schematic of the active region of a multiquantum-well laser. The confinement layers restrict charge carriers to the quantum-well region. Confinement layers Strained-Layer Lasers The introduction of strain can provide a salutatory effect on the performance of laser diodes, in spite of the fact that the notion is counterintuitive. Strained-layer lasers can have superior properties, and can operate at wavelengths other than those accessible by means of compositional tuning. Quantum-confined strained-layer lasers have been fabricated from 111- V semiconductor materials in various configurations. Rather than being lattice-matched to the confining layers, the active region is deliberately chosen to have a different lattice constant. If sufficiently thin, it can accommodate its atomic spacings to those of the surrounding layers, and in the process become strained (if the active region is too thick it will not properly accommodate and the material will contain dislocations). The InGaAs active layer in an AIGaAsjlnGaAs strained-layer quantum-well laser, for example, has a lattice constant that is significantly greater than that of its AIGaAs confining layers. The thin InGaAs layer therefore experiences a biaxial compression in the plane of the layer, while its atomic spacings are increased above their nominal values in the direction perpendicular to the layer. The compressive strain alters the band structure in three significant ways: (I) it increases the bandgap Eg; (2) it removes the degeneracy at k == 0 between the heavy and light hole bands; and (3) it makes the valence bands anisotropic so that the highest band has a light effective mass in the direction parallel to the plane of the layer while it has a heavy effective mass in the perpendicular direction. This behavior can significantly improve the performance of lasers. First, the laser wavelength is altered by virtue of the dependence of E 9 on the strain. Second, the laser threshold current density can be reduced by the presence of the strain. Achieving a population inversion requires that the separation of the quasi-Fermi levels be greater than the bandgap energy, i.e., E fc - E fv > Eg [see (16.2-12)]. The reduced hole mass allows E fv to more readily descend into the valence band, thereby permitting this condition to be satisfied at lower values of injection current. Quantum-Wire and Multiquantum-Wire Lasers Quantum wires (see Sec. 16.IG) can also serve as the active region of a semiconductor laser, as illustrated in Fig. 17.4-I(b). Multiquantum-wire lasers comprise arrays of quantum wires, as portrayed in Fig. 17.4-4. In principle, multiquantum-wire lasers offer narrower linewidths than quantum-well lasers by virtue of their tighter carrier
17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 731 confinement. However, the fabrication of 111- V quantum-wire structures lags well be- hind that of quantum-well structures, in part because of the difficulty of creating a sufficiently dense collection of wires, and hence so too does their performance. . Cladding Multiquantum-wire active region Figure 17.4-4 Schematic of the active region of a multiquantum-wire laser. Light is ordinarily emitted in all directions although laser emission can be restricted to the end faces by making use of a suitable resonator. EXAMPLE 17.4-1. Performance of Multiquantum-Wire and Quantum-Well Lasers. A collection of five i-mm-Iong, 23-nm-wide, InGaAsP active-layer quantum wires, clad with InP and spaced 80 nm apart, operates as a room-temperature CW multiquantum-wire laser at a wavelength Ao 1550 nm. The threshold current, threshold current density, external differential quantum efficiency, and power-conversion efficiency are determined to be it == 140 mA, J t == 800 A/cm 2 , Ild == 40%, and Ilc == 2%, respectively. t As a result of the small volume of the active region and the substantial optical losses, however, the performance of this multi quantum-wire laser turns out to be inferior to that of a quantum-well laser fabricated from the same chip, which has operating parameters it == 100 mA, J t == 500 A/cm 2 , Ild == 50%, and Ilc == 6%. Quantum-Dot and Multiquantum-Dot Lasers Quantum dots, also called quantum boxes and sometimes referred to as nanocrys- tals, usually take the form of cubes, spheres, or pyramids. They typically have dimen- sions in the range 1-10 nm (a lO-nm cube of GaAs contains some 40000 atoms). The carriers may be confined by cladding the dots with a semiconductor of larger bandgap or by embedding them in glass or polymer. Figure 17.4-1(c) depicts a quantum-dot laser. The energy levels of a quantum dot are those of its excitons. Although the levels are sharp as a result of tight carrier confinement, the energies depend strongly on the size of the dot. As is dramatically illustrated in Fig. 13.1-12, the photoluminescence- photon energy increases as the dot size decreases because of the greater energy required to confine the semiconductor excitation to a smaller volume (see Sec. 13.1 C). This tunability is a salutary feature of using quantum dots as an active laser medium. A collection of quantum dots that contribute in concert can give rise to a useful level of optical power. Since quantum dots often self-assemble into ordered arrangements, it is not difficult to construct a multiquantum-dot laser with an active region that contains many quantum dots, as depicted in Fig. 17.4-5. Multiquantum-dot structures offer a good deal of design flexibility; they can, for example, emit broadband light or serve as optical amplifiers. t See H. Yagi, T. Sano, K. Ohira, D. Plumwongrot, T. Maruyama, A. Haque, S. Tamura, and S. Arai, GalnAsP flnP Partially Strain-Compensated Multiple-Quantum- Wire Lasers Fabricated by Dry Etching and Regrowth Processes, Japanese Journal of Applied Physics, vol. 43, pp. 3401-3409, 2004.
732 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES Multiquantum-dot active region Cladding Figure 17.4-5 Schematic of the active re- gion of a multi quantum-dot laser, which often consists of multiple layers. Each layer con- tains self-assembled multiple quantum dots. The typical dimensions of self-assembled quantum dots fall in the 10-50 nm range. EXAMPLE 17.4-2. Quantum-Dot Silicon Photonics. The confinement of carriers in a quantum dot results in a reduction of their positional uncertainty x. Since x k > , in accordance with (A.2-6) of Appendix A, this is accompanied by a concomitant increase in the wavenumber uncertainty k. The increase in k obviates the need for a phonon to take part in radiative recombination. The use of quantum-dot structures in this context is analogous to incorporating nitrogen impurities at sharply localized positions in indirect-bandgap GaP to make GaP:N LEDs a reality (see Sec. 17.1 C). Considering quantum-confinement effects as well, the small size of the quantum dot therefore endows indirect-bandgap semiconductors with an internal quantum efficiency that is substantially greater than that of their bulk-material counterparts. Also, surface passivation enhances the radiative rate via induced surface-localized excitons. As a result, light emission from silicon nanoparticles, as well as from porous silicon and germanium, becomes practical. Efforts are underway to demonstrate lasing in silicon nanostructures. Quantum-Cascade Lasers All of the semiconductor lasers discussed to this point operate via radiative electron- hole recombination. The production of light is a two-carrier, single-photon affair: the combination of an electron in the conduction band with a hole in the valence band generates a photon. The quantum cascade laser (QCL), in contrast, makes use of only a single carrier, the electron, but each electron generates multiple photons. The QCL is therefore unipolar rather than bipolar. Quantum-cascade lasers are constructed from a concatenated series of quantum wells, designed and biased in such a way that an electron injected into the conduction band undergoes a cascade of light-emitting intersubband transitions as it transits the device. The QCL is, perhaps, the epitome of band -structure engineering. Quantum cascade lasers are generally constructed with either quantum-well active regions or superlattice active regions. As illustrated in Fig. 17 .4-6( a), the quantum- well version consists of a sequence of stages, each comprising an n-type electron injector and an intrinsic quantum-well active region. The injector contains a collection of wells of varying widths and thin barriers that form a superlattice, with an energy- level structure consisting of minibands separated by minigaps (see Sec. 13.1C). The number of states in a miniband is the same as the number of quantum wells. In the presence of bias, the electrons are injected via resonant tunneling from the bottom (ground state) of a miniband, denoted level 3, into the upper laser level in the quantum- well active region, denoted level 2. A photon of frequency v == E21/h is emitted via stimulated emission on the 2----t 1 intersubband transition, as indicated by the red arrow (see Sec. 16.2D). The electron then decays via phonon scattering to level 0, whereupon it enters the miniband in the next stage via resonant tunneling. The process is repeated in that stage and another photon is emitted. A typical QCL contains 20-100 stages, so that a substantial number of photons is generated for each electron that transits the device. Because it makes use of intersubband transitions, the operation of a quantum-well
17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 733 ... ... I Injector Acti ve I njector Acti ve Miniband 3 Miniband 3 (a) (b) Figure 17.4-6 Schematic diagram of: (a) two stages of a QCL with a quantum-well active region, and (b) two stages of a QCL with a super lattice active region. QCLs usually contain from 20 to 100 stages; the overall length of a typical device is 1.-3 mm and its width ranges from 5 to 20 11m. Quantum-cascade lasers are often made of AlInAsJlnGaAs quantum wells that are lattice-matched to an InP substrate, or of AIGaAsjGaAs quantum wells, using MBE or MOCVD. Other material systems of interest include AIAsSbjlnGaAs, AIGaNjGaN, and SiGe. QCL closely resembles that of an atomic laser. As is evident in Fig. 17 .4-6( a), level 2 is not aligned with a miniband of the succeeding stage, so that it has a relatively long life- time (72 1 ps) and therefore accumulates population. Level I, in contrast, does not sustain population since decay to level 0 takes place via a fast nonradiative transition and subsequent tunneling into the succeeding stage (71 0.1 ps). The quantum-well active region thus behaves as a four-level laser system in which a population inversion is achieved on the 2---+ 1 transition (see Sec. 14.2B). The superlattice QCL shown in Fig. 17 .4-6(b) differs from the quantum-well QCL in that stimulated emission takes place between the bottom and top of two minibands in an active region which, in this case, comprises a superlattice (see Sec. 16.2D). The laser frequency is thus established by the height of the minigap separating the two minibands. This structure is generally more suitable for generating coherent light at longer wavelengths (Ao > 10 Mm) since the alignment between the injector and active region is less critical. Moreover, higher drive currents can be used and a population inversion is more readily achieved because of fast relaxation in the lower laser-level miniband. Yet another design for the QCL active region is the so-called bound-to- continuum scheme, where the laser action involves transitions from a discrete upper state to a superlattice miniband. This design combines the efficient electron injection into the upper laser level of quantum-well QCLs with the fast depopulation of the lower laser level of superlattice QCLs, thereby reducing the threshold and increasing the power. QCLs can be operated over an enormous range of wavelengths, from 2-70 Mm in the mid and far infrared regions, by appropriate choice of well thicknesses, which in turn determine the subband and miniband energy levels (see Exercise 16.1-5). External- cavity feedback, in conjunction with a rotatable grating (see Sec. 15.2D), offers less coarse wavelength tuning over a region comprising about 10% of the center wavelength ( 1 Mm at a center wavelength of 10 Mm). Fine wavelength tuning ( 0.1 Mm at a 10-Mm center wavelength) can be achieved by changing the injection current and/ or temperature, which modify the effective refractive index - this in turn changes
734 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES the optical path length of the cavity and thus the emission wavelength. Single-mode operation is attained by endowing the device with a distributed-feedback element. Mid-infrared QCLs operate at room temperature and emit mW to hundreds ofmW of CW coherent radiation. t High external quantum efficiency and low threshold is achieved because a single carrier produces many photons and the devices can tolerate high currents since they need not be made of low-bandgap materials. QCLs can be modulated at high speeds and can be modelocked to produce optical pulses of a few ps duration. They can also be operated as THz sources, yielding tens of m W CW when cooled. QCL devices have been operated at wavelengths as long as 150 Mm in the far infrared (v == 2 THz). A variety of other QCL designs have been developed, including superlattice devices in which the injector region is eliminated; devices in which the light is guided in surface-plasmon modes so that long-wavelength operation can be achieved without the need for thick dielectric waveguides; devices that simultaneously operate at multiple wavelengths; devices that generate supercontinuum emission by endowing the active regions in different stages with different quantum-well thicknesses; and Raman-laser devices that are injection-pumped by a quantum-cascade laser integrated into the same structure. The development of the QCL, in conjunction with room-temperature mid-infrared detectors such as HgCdTe photovoltaic arrays and VOx microbolometer arrays (see Sec. 18.5), has opened the door to a plethora of new scientific, industrial, and military applications in the mid and far infrared. Since these bands correspond to vibrational- rotational transitions in molecular species (see Sec. 13.1B), applications include trace- gas analysis, chemical sensing, isotopic analysis, and infrared spectroscopy. QCLs also appear to be suitable for eye-safe mid-infrared optical wireless communications. B. Microcavity Lasers The quantum confinement considered in Sec. 17.4A relates to the confinement of carriers to a spatial region of the order of the de Broglie wavelength of an electron (for a thermalized electron in GaAs, "A 50 nm). The microcavity lasers considered in this section, in contrast, involve the confinement of photons to a spatial region of the order of the optical wavelength (Ao 1 Mm » "A). Microresonators are resonators in which one or more of the spatial dimensions is the size of a few wavelengths of light or smaller, d A. Microcavities are usually thought of as having small dimensions in all spatial directions; however, these two terms have come to be used interchangeably. Photon confinement and carrier confinement are independent features of photonic devices. It is therefore possible to have a microcavity laser whose active region is not subject to quantum confinement (e.g., a microcavity containing a simple p-n ho- mojunction active region), or a large-resonator laser whose active region is subject to quantum confinement (e.g., a quantum-cascade laser). In practice, however, most microcavity lasers make use of quantum-confined structures for their active regions. Several representative examples of microcavity lasers are discussed in Sec. 17.4C. Microresonator lasers in which the light is confined to wavelength-sized regions in various dimensions are exemplified by the micropillar, microdisk, and microsphere structures illustrated in Fig. 17.4-7. These, and other, microresonators have been de- scribed in Sees. 10.4B and 10.4C. In laser diodes with large resonators (d » A), the modes exhibit small spacings in all directions of k-space and the density of allowed resonance frequencies M(v) can be determined via a continuous approximation (see Sec. 10.3). The overall spontaneous t Although conventional lead-salt laser diodes can operate at wavelengths as long as 30 J-Lm, they suffer from a number of difficulties that do not afflict QCLs: (1) power levels are limited to the mW range, (2) emission wavelengths are tunable only over a relatively narrow band, and (3) they require cooling.
17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 735 !X o ,----- ,AtIP .....-' 2D photonic crystal Microdisk !X - , o '" , Micropillar ...........- Figure 17.4-7 Microresonator lasers confine light within wavelength-sized regions in various dimensions. The defect in the 2D photonic crystal creates a cavity that traps the light. The analogous quantum-confined structures are the quantum well, quantum wire, and quantum dot. tt .....- r Microtoroid Microsphere ) emission probability density (S-l) depends on the modal density M(v) of the frequency space into which the atom can emit, as specified by (13.3-11). In large-resonator lasers, as in free space, the modal density assumes the quadratic form M(v) == 87rV 2 jc 3 , in accordance with (10.3-10). This offers a great number of modes for spontaneous emission; however, spontaneous emission into modes other than the laser mode repre- sents wasted energy after stimulated emission is initiated in a particular mode. Indeed, for a typical edge-emitting laser diode, the fraction (3 of spontaneous emission that contributes to a given laser mode is generally minuscule ((3 10- 5 ). The current injected into a large-resonator laser at threshold is thus principally replenishing the wasted spontaneous emission rather than contributing to the stimulated emission. However, the modal density M(v) can be substantially reduced by making use of a microcavity, as discussed in Sec. 10.4. The allowed modes of microresonators can exhibit large spacings in one or more directions of k-space, so that modes can be absent over extended spectral bands. The reduction is most dramatic in microcavities that have large spacings in all directions of k-space, which results in a discrete collection of modes (see Fig. 10.4-1). The opportunity to alter the modal environment is important in connection with spontaneous emission. Placing a source in this environment inhibits spontaneous emission into modes that do not exist, redirecting it into available modes. The emission of light into particular modes of a high-Q, small-volume microcavity can be enhanced relative to emission into ordinary optical modes via the Purcell effect, as described in Sec. 13.3E. Microcavity lasers are designed to take maximum advantage of opportunities for inhibition and enhancement. The modification of the modal density offered by a microcavity can increase the spontaneous-emission coupling coefficient (3 by several orders of magnitude, thereby reducing the laser-diode threshold current it by a commensurate amount (e.g., from the mA to the /-LA domain). Although semiconductor microcavity lasers are the most prevalent, they can also be made of materials such as organic dyes, rare-earth-doped silica, and organic polymers. Summary Microcavity lasers offer a number of desirable features in comparison with their conventional counterparts: . Reduced size . Reduced laser threshold . Reduced spectral width . Reduced spatial width . Increased efficiency Their small sizes means that they operate at low powers, however.
736 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES EXAMPLE 17.4-3. Single Photons on Demand from a Quantum Dot. Single quantum dots can serve as a source of single photons on demand when excited optically or electrically. This process has been implemented in a number of configurations. For example, a single InAs quantum dot placed in the cavity of a micropillar microcavity (see Fig. 17.4-7) generates spontaneously emitted photons that are preferentially directed out of the top face of the device via fiber-optic coupling. The Purcell spontaneous-emission enhancement factor (13.3-47) is substantial since the micropillar has a small cavity volume and a high quality factor Q (see Table 10.4-1). Because the quantum dot can emit only one photon at a time, the light is antibunched and photon-number squeezed, exhibiting a count variance that lies below its mean (see Sec. 12.3B). In another implementation, the electroluminescence is observed from a single quantum dot in a p-i-n junction. Efforts are underway to increase the reliability of the photon-generation process. A reliable single-photon source will find use in quantum cryptography, among other applications. c. Materials and Device Structures Semiconductor lasers have been fabricated in a bewildering variety of forms. They operate at wavelengths that stretch from the mid-ultraviolet to the far-infrared - and at output powers that range from n W to kW (for banks of laser diodes). Essentially all semiconductor lasers in use today make use of active regions that comprise quantum- confined structures. We first consider conventional laser diodes, which typically have large resonators, and then turn to microresonator semiconductor lasers. In particular, we examine vertical-cavity surface emitting lasers (VCSELs) and photonic-crystal lasers, which are becoming increasingly important. Conventional Laser Diodes Edge-emitting laser diodes are used in an enormous variety of applications, ranging from consumer products such as DVD players and laser printers to long-haul optical fiber communications systems. They serve as highly efficient optical pumps for optical fiber amplifiers, fiber lasers, and solid-state lasers. The materials and device structures for most conventional laser diodes closely re- semble those of light-emitting diodes (see Sec. 17.1 C). Direct-bandgap ternary and quaternary materials are typically used in the near-infrared to mid-ultraviolet region because their bandgap wavelengths can be compositionally tuned. AllnGaN, AIln- GaP, InGaAs, and InGaAsP are particularly important materials, as with LEDs. Edge- emitting devices have typical lengths I 500 Mm and widths w 2 Mm. Commonly encountered wavelengths for laser diodes are 635-650 nm for laser pointers, DVDs, and short-haul plastic-fiber communications; 785 nm for CDs; 850 nm for short-reach communications; and 1300-1600 nm for long-reach communications. Other wave- lengths at which laser diodes are commonly available include 375, 405, 440, 670, and 830 nm. Although lead-salt laser diodes can operate out to wavelengths as long as 30 Mm, they suffer from various difficulties; quantum-cascade lasers are far superior in the mid- and far-infrared regions. Edge-emitting laser diodes can be operated in single spatial and longitudinal modes. However, they are often operated as multi-spatial-mode devices for high-power appli- cations, such as pumping optical fiber amplifiers, multiclad fiber lasers, and diode- pumped solid-state (DPSS) lasers (see Sec. 15.3A). Multimode edge-emitting config- urations offer optical powers in excess of 5 W for single 50-Mm-width stripe devices and powers that stretch up to the kW level for stacks of multistripe bars. They are often fiber-coupled to provide an efficient delivery system. Commonly used wavelengths for pumping solid-state lasers are 808 nm for neodymium-doped yttrium vanadate and neodymium-doped YAG; 940 nm for ytterbium-doped YAG; and 980 nm for erbium- doped silica fiber. Typical power-conversion efficiencies are Ilc 45% but values in excess of 75% have been attained.
17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 737 Ridge-waveguide and buried-heterostructure distributed-feedback lasers have demon- strated good reliability in a variety of applications, and we consider these in turn. Other types of laser diodes, including broad-area, tapered, and OBR LOs, are also widely used. Ridge-waveguide lasers. The ridge-waveguide (RW) laser diode operates in a single spatial mode. It lases over a range of wavelengths, typically in the near IR, and finds use in applications such as spectroscopy and metrology. The ridge waveguide provides weak stripe optical waveguiding and lateral confinement by restricting current injection to the active region beneath the ridge. RW laser diodes usually take the form of a Fabry-Perot structure with cleaved facets, and can provide several hundred m W of power. The 500-fLm-Iong device displayed in Fig. 17.4-8 has an active region com- prising six 7-nm-thick, compressively strained InGaAsP quantum wells sandwiched between 10-nm-thick tensile-strained InGaAsP barriers. This particular laser diode has a threshold current it == 15 mA, an external differential quantum efficiency Il d == 0.33, a differential responsivity 9{d == 0.26 W / A, and emits about 20 m W. + InGaAsP / InGaAsP MQW active region ( InGaAs \.... contact layer l .. D" I . ' lfilnc "'.., ,," --:::::::.--" - .:::::;; iii5 ' --", ,,-' - - InP !ea I J C\g substrate Figure 17.4-8 Schematic diagram of a strained-MQW InGaAsP jInGaAsP ridge- waveguide laser diode operated at 1550 nm. The light-current curve is displayed in Fig. 17.3-4(a). Buried-heterostructure distributed-feedback lasers (DFBs). As illustrated in Fig. 17.4-9, alternating p- and n-type layers allow current flow only in the vicinity of the active region in this buried-heterostructure device, thereby enforcing lateral confinement. The dielectric film provides gain guiding. The distributed feedback (OFB) component of the device makes use of a corrugated-layer grating adjacent to the active region that serves as a distributed reflector (see Sec. 17.3C). The design + InGaAsP jlnGaAsP MQW active region Figure 17.4-9 Buried-heterostructure mul tiquantum- well distri buted- feedback laser used for optical fiber communications in the 1300-1600-nm wavelength range.
738 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES of this laser is compatible with on-chip integration. Lasers such as these offer ample gain at modest current levels, and provide output powers in excess of 1 W in a single spatial mode. Typical values of the threshold current and differential responsivity are it < 10 mA and 9{d 0.4 W j A, respectively. These edge-emitting devices offer narrow spectral widths, which is critical for the efficient operation of 1300-1600-nm wavelength-division-multiplexed (WDM) communication systems. Vertical-Cavity Surface-Emitting Lasers The most common microresonator lasers are vertical-cavity surface-emitting lasers (VeSELs); these devices are designed so that the light emerges from the top face of a 1 D planar microresonator. VeSELs typically operate in the visible and near- IR regions and can be fabricated with a broad range of diameters, stretching down to 1 /-Lm. An example of a large-area VeSEL is displayed in Fig. 17.4-10. This device has a multiquantum-well GaAsjInGaAs active region and operates at a wavelength of 995 nm. The light is repeatedly reflected through the active region by highly reflective distributed Bragg reflectors (DBRs). A key feature in the operation of a high-efficiency VeSEL is the dielectric film, which localizes carrier injection and laterally confines the optical mode. + AIGaAs / GaAs Bragg reflectors Dielectric film AlAs confinement layers GaAs/lnGaAs MQW active region AIGaAs / GaAs Bragg reflectors . GaAs substrate ! (a) (h) Figure 17.4-10 (a) Schematic diagram of a large-area (320-pm diameter) multiquantum-well GaAsjlnGaAs VCSEL operating at a wavelength of 995 nm. (b) Etched mesa showing the p contact, p-type DBR, and active region. (Adapted from M. Miller, M. Grabherr, R. King, R. Jager, R. Michalzik, and K. J. Ebeling, Improved Output Performance of High-Power VCSELs, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, pp. 210-216, Fig. 2 @2001 IEEE.) The spectral intensity, optical power, and angular distribution generated by this laser are illustrated in Fig. 17.4-11. Because the thickness of the active region is only tens of nll, the single-pass gain is typically low (a fraction of 1 %). High gain coefficient and high mirror reflectivities are thus mandatory; a typical VeSEL DBR mirror contains dozens of layers. Moreover, special care must be taken to avoid heating. Thresholds of small-area VeSEL devices fall in the /-LA region and power-conversion efficiencies reach Ilc 70%. Although the active regions are usually multiquantum wells, VeSELs have also been fabricated with multiquantum-dot active regions, as illustrated in Fig. 17.4-12. VeSELs assume an enormous variety of forms, and can incorporate auxiliary features such as photonic crystals for latera] mode control, coupled cavities, and integrated modulators that extend direct-modulation speeds toward 40 Gbit/s, as portrayed in Fig. 17.4-13. Most importantly, VeSELs offer high packing densities on a wafer scale and are readily fabricated in the form of dense arrays. As an early example, an array of about
C 1.0 .U; 5 0.8 -0 "@ .b 0.6 u (l) 0.. 0.4 .::: ] 0.2 (l) 0:: o 991 17.4 QUANTUM-CONFINED AND MICROCAVITY LASERS 739 10 QC 0.8 (l) 8. 0.6 >. ."';::: 0.8 Cf) t:: (l) .5 0.6 (l) > ..g 0.4 Q) 0.2 1.0 "@ .g 0.4 0.. o 0.2 0.. o 0 2 20 993 995 997 999 Wavelength Au CJLm) Figure 17.4-11 Spectral intensity, optical power, and angular distribution of the multi quantum- well GaAsjlnGaAs VCSEL shown in Fig. 17.4-10. The threshold current it = 1.1 A for this large-area device. (Adapted from M. Miner, M. Grabherr, R. King, R. Jager, R. Michalzik, and K. J. Ebeling, Improved Output Performance of High-Power VCSELs, IEEE Journal of Selected Topics in Quantunl Electronics, vol. 7, pp. 210-216, Figs. 8, 5, and 9 @2001 IEEE.) t .-----J GaAs/ AIO + Bragg reflectors G AIGaAs Cladding layers GaAs substrate GaAs confinement layers InGaAs / GaAs MQD active region Figure 17.4-12 VCSEL with a quantum-dot active region. GaAs / InGaAs MQW _ active region AlGaAs / GaAs :!t Bragg reflectors GaAs / InGaAs "'-- + MQW structure T / -/+ Moiulator . __ . GaAUdaAs t . ' c . tive region T --- ----... tIJI!'!:::.:: - - - - ..... 4 VeSEL rwf1. . . 1 [ ">/ 1"- - -- u - _; ::friC AlAs substrate confinement layers AIGaAs / GaAs Bragg reflectors "-- - GaAs substrate AlAs confinement layers (a) + (b) Figure 17.4-13 Variations on the theme of VCSELs. (a) VCSEL with photonic crystal for lateral mode control (b) VCSEL with monolithic ally integrated electroabsorption modulator. 1 million electrically pumped tiny vertical-cavity cylindrical InGaAs quantum-well veSELs (diameter 211m, height 5.5 11m), with lasing wavelengths in the vicinity of 970 nm, was fabricated on a single l-cm 2 chip of GaAs. These particular devices had thresholds it 100 I1A, for T == 300 0 K ew operation. A scanning electron mi- crograph of a small portion of this array is displayed in Fig. 17.4-14. VeSEL arrays can be fabricated with elements that have a pre specified distribution of laser frequencies.
740 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES .' Figure 17.4-14 Scanning electron micrographs of an early array of electrically pumped vertical- cavity Ino.2Gao.8As quantum-well lasers with di- ameters between 1 and 5 pm on a GaAs chip. The microresonators comprise AIAsjGaAs Bragg reflectors. (a) AlAs has been preferentially etched away from the Bragg reflectors in these devices, highlighting the GaAs disks, which are supported by the residual AlAs at the device centers. (b) Top view of a small portion of the array. Circular out- put beams provide easy coupling to optical fibers (courtesy Jack L. Jewell, Picolight Incorporated). O!iic ". "L- . I-' ... .- - "- - .t " (a) (b) Photonic-Crystal Microcavity Lasers Microcavities consisting of defects in 2D photonic crystals, together with miniature quantum-confined sources of emission, offer the possibility of wavelength-size thresh- oldless lasers and laser arrays. Such devices have wavelengths and defect-mode radia- tion patterns that are tunable, and they offer high direct-modulation rates. The radiation source can be a quantum well or a quantum dot. Individual devices as well as coherently coupled arrays of such devices have been demonstrated, as illustrated in Figs. 17.4-15 (a) and (b), respectively. The device por- trayed in Fig. 17.4-15(a) is a single-mode, photonic-bandgap laser that operates at room temperature. t It is electrically pumped via a sub-micron-size post and has a threshold current of 260 JLA. The active region comprises six strain-compensated In- GaAsP quantum wells and lasing takes place at .Ao == 1520 nm. The structure produces 2 n W of power at a current of mA and its differential responsivity is estimated to be 9{ 10- 5 . The quality factor and modal volume are Q 2500 and V 6 X 10- 2 JLm 3 , respectively. When the emission linewidth v is smaller than the width of an electromagnetic mode 6v, spontaneous emission in high-Q microcavities can be enhanced via the Purcell effect (see Sec. 13.3E and Fig. 13.3-12). The Purcell factor for this device was determined to be (3/47r 2 )(.A 3 /V)Q 400. InP confinement layers I I t - :=- i . =- =-J - '!a -- InP ___ -:=:2 i."!!!"",!! -ea...-o-- f " - .... -.. · ./':. con mement _,.... -::: ::;;---... layers -- I !".:!!-- : --.: : l ,. .-- - InGaAs L ::::rr contact -.;:-=--= I G A ----=_-= ).. n a s layers InGaAsP/lnGaAsP --=ntact MQW ! I layers active region InP + Dielectric substrate film InGaAsP / InGaAsP MQW active region (a) (b) Figure 17.4-15 (a) InGaAsPjInGaAsP multiquantum-well photonic-crystallaser. The TnP post has a height of 1 pm and serves as an electrical contact. (b) Array of coherently coupled quantum- well photonic-crystallasers. The nW-Ievel output powers of individual devices can be substantially increased by using arrays of coupled microcavities. The device illustrated in Fig. 17 .4-15(b) is t H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, Electrically Driven Single-Cell Photonic Crystal Laser, Science, vol. 305, pp. 1444-1447 (2004).
READING LIST 741 a photonic-crystal microcavity-array laser (also called a nanocavity-array laser), with four InGaAsPlInP quantum wells, and an emission wavelength Ao == 1534 nm.:\: Each of the 9 x 9 == 81 cavities that forms the array occupies an area of 1.5 fLm 2 and the array area is 15 fLm 2 . The device is optically pumped by a pulsed 808-nm diode laser focused to a spot size similar to that of the array. The threshold peak pump power of the coupled-cavity array is 2.5 mW and the spontaneous-emission coupling coefficient (3 0.1. A single-mode peak output power of 12 fLW was observed. The laser threshold increases with the number of coupled cavities, but the efficiency of the device rises more rapidly. Very high modulation rates are possible with such devices. READING LIST Books and Articles on Laser Amplifiers and Lasers See also the reading lists in Chapters 14 and 15. Books and Articles on Semiconductor Physics, Devices, and Nanostructures See the reading list in Chapter 16. Books on LEDs and Laser Diodes E. F. Schubert, Light-Emitting Diodes, Cambridge University Press, 2nd ed. 2006. K. Mullen and U. Scherf, eds., Organic Light Emitting Devices: Synthesis, Properties and Applica- tions, Wi ley- VCH, 2006. R. R. Alfano, ed., The Supercontinuum Laser Source, Springer-Verlag, 1989, 2nd ed. 2006. Z. H. Kafafi, ed., Organic Electroluminescence, CRC Press, 2005. D. Sands, Diode Lasers, Institute of Physics, 2005. J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer-Verlag, 2005. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, 2004. T. Suhara, Semiconductor Laser Fundamentals, Marcel Dekker, 2004. C. Ye, Tunable External Cavity Diode Lasers, World Scientific, 2004. H. K. Choi, ed., Long- Wavelength Infrared Semiconductor Lasers, Wiley, 2004. H. Zappe, Laser Diode Microsystems, Springer-Verlag, 2004. M. S. Shur and A. Zukauskas, eds., UV Solid-State Light Emitters and Detectors, NATO Science Series II: Mathematics, Physics and Chemistry, Volume 144, Springer-Verlag, 2004. J. Shinar, ed., Organic Light-Emitting Devices: A Survey, Springer-Verlag, 2004. W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers, Cambridge University Press, 2003. I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources, Springer-Verlag, 2003. S. F. Yu, Analysis and Design of Vertical Cavity Suiface Emitting Lasers, Wiley, 2003. H. Ghafouri-Shiraz, Distributed Feedback Laser Diodes and Optical Tunable Filters, Wiley, 2003. H. Li and K. Iga, eds., Vertical-Cavity Suiface-Emitting Laser Devices, Springer-Verlag, 2003. E. Gehrig and O. Hess, Spatio- Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers, Springer-Verlag, 2003. V. M. Ustinov, A. E. Zhukov, A. Yu. Egorov, and N. A. Maleev, Quantum Dot Lasers, Oxford University Press, 2003. A. Zukauskas, M. S. Shur, and R. Gaska, Introduction to Solid-State Lighting, Wiley, 2002. J. Kim, S. Somani, and Y. Yamamoto, Nonclassical Light from Semiconductor Lasers and LEDs, Springer-Verlag, 2001. :j: H. Altug and 1. Vuckovic, Photonic Crystal Nanocavity Array Laser, Optics Express, vol. 13, pp. 8819-8828, 2006.
742 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES S. Nakamura, S. Pearton, and G. Fasol, The Blue Laser Diode: The Complete Story, Springer-Verlag, 2nd ed. 2000. S. Nakamura and S. F. Chichibu, eds., Introduction to Nitride Semiconductor Blue Lasers and Light Emitting Diodes, Taylor & Francis, 2000. R. Diehl, ed., High-Power Diode Lasers: Fundamentals, Technology, Applications, Springer-Verlag, 2000. E. Kapon, ed., Semiconductor Lasers, Academic Press, 1999. C. Wilmsen, H. Temkin, and L. A. Coldren, eds., Vertical-Cavity Suiface-Emitting Lasers: Design, Fabrication, Characterization, and Applications, Cambridge University Press, 1999. J. Carroll, J. Whiteaway, and D. Plumb, Distributed Feedback Semiconductor Lasers, Institution of Engineering and Technology (London), 1998. J. P. Loehr, Physics of Strained Quantum Well Lasers, Kluwer, 1998. M.-C. Amann and J. Buus, Tunable Laser Diodes, Artech, 1998. G. Morthier and P. Vankwikelberge, Handbook of Distributed Feedback Laser Diodes, Artech, 1997. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics, Springer-Verlag, 1994, corrected ed. 1997. R. K. Willardson and E. R. Weber, eds., Semiconductors and Semimetals, Volume 48, High- Brightness Light Emitting Diodes, G. B. Stringfellow and M. G. Craford, eds., Academic Press, 1997. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, Wiley, 1995. T. Ikegami, S. Sudo, and Y. Sakai, Frequency Stabilization of Semiconductor Laser Diodes, Artech, 1995. D. Botez and D. R. Scifres, eds., Diode Laser Arrays, Cambridge University Press, 1994. N. W. Carlson, Monolithic Diode-Laser Arrays, Springer-Verlag, 1994. K. Suto and J.-I. Nishizawa, Semiconductor Raman Lasers, Artech, 1994. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, Van Nostrand Reinhold, 2nd ed. 1993. M. Ohtsu, Highly Coherent Semiconductor Lasers, Artech, 1992. Y. Yamamoto, ed., Coherence, Amplification, and Quantum Effects in Semiconductor Lasers, Wiley, 1991. R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 22, Lightwave Com- munications Technology, W. T. Tsang, ed., Part B, Semiconductor Injection Lasers, I, Academic Press, 1985. R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 22, Lightwave Communications Technology, W. T. Tsang, ed., Part C, Semiconductor Injection Lasers, II and Light Emitting Diodes, Academic Press, 1985. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers, Part B, Materials and Operating Charac- teristics, Academic Press, 1978. Books on Optoelectronics A. Krier, ed., Mid-infrared Semiconductor Optoelectronics, Springer-Verlag, 2006. J. P. Dakin and R. G. W. Brown, eds., Handbook of Optoelectronics, Volumes 1 and 2, CRC Press, 2006. J.-M. Liu, Photonic Devices, Cambridge University Press, 2005. M. A. Parker, Physics of Optoelectronics, Taylor & Francis, 2005. M. Razeghi and M. Henini, eds., Optoelectronic Devices: III-Nitrides, Elsevier, 2005. J. Piprek, Semiconductor Optoelectronic Devices: Introduction to Physics and Simulation, Academic Press 2003. T. P. Pearsall, Photonics Essentials: An Introduction with Experiments, McGraw-Hill 2003. E. Rosencher and B. Vinter, Optoelectronics, Cambridge University Press, 2002. S. O. Kasap, Optoelectronics and Photonics: Principles and Practices, Prentice Hall, 2001. J. Wilson and J. F. B. Hawkes, Optoelectronics, Prentice Hall, 3rd ed. 1998. P. Bhattacharya, Semiconductor Optoelectronic Devices, Prentice Hall, 2nd ed. 1996. S. L. Chuang, Physics of Optoelectronic Devices, Wiley 1995. J. Gowar, Optical Communication Systems (Optoelectronics), Prentice Hall, 2nd ed. 1993.
READING LIST 743 Books and Articles on Silicon Photonics R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, M. J. A. de Dood, Photoluminescence Quantum Efficiency of Dense Silicon Nanocrystal Ensembles in Si0 2 , Physical Review B, vol. 73, 132302, 2006. M. Makarova, J. Vuckovic, H. Sanda, and Y. Nishi, Silicon-Based Photonic Crystal Nanocavity Light Emitters, Applied Physics Letters, vol. 89, 221101, 2006. B. Jalali, "'. Raghunathan D. Dimitropoulos, and O. Boyraz, Raman-Based Silicon Photonics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, pp. 412-421, 2006. M. Paniccia and S. Koehl, The Silicon Solution, IEEE Spectrum, vol. 42, no. 10, pp. 38-43, 2005. L. Pavesi and D. J. Lockwood, eds., Silicon Photonics, Springer-Verlag, 2004. G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction, Wiley, 2004. L. Pavesi, S. Gaponenko, and L. Dal Negro, eds., Towards the First Silicon Laser, NATO Science Series II: Mathematics, Physics and Chemistry, Volume 93, Kluwer, 2003. L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franz<), and F. Priolo, Optical Gain in Silicon Nanocrystals, Nature, vol. 408, pp. 440-444, 2000. H. Zimmermann, Integrated Silicon Optoelectronics, Springer-Verlag, 2000. Issue on silicon-based optoelectronics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 4, no. 6, 1998. Articles Issue on nanophotonics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 6, 2006. Y. Sun, N. C. Giebink, H. Kanno, B. Ma, M. E. Thompson, and S. R. Forrest, Management of Singlet and Triplet Excitons for Efficient White Organic Light-Emitting Devices, Nature, vol. 440, pp.908-912,2006. 1. Faist, Continuous-Wave, Room-Temperature Quantum Cascade Lasers, Optics & Photonics News, vol. 17, no. 5, pp. 32-36, 2006. N. Holonyak, Jr. and M. Feng, The Transistor Laser, IEEE Spectrum, vol. 43, no. 2, pp. 50-55, 2006. H. Altug, D. Englund, and J. Vuckovic, Ultrafast Photonic Crystal Nanocavity Laser, Nature Physics, vol. 2, pp. 484-488, 2006. N. Narendran, The Solid-State Lighting Revolution, Physics World, vol. 18, no. 7, pp. 25-29, 2005. P. T. Snee, Y. Chan, D. G. Nocera, and M. G. Bawendi, Whispering-Gallery-Mode Lasing from a Semiconductor NanocrystaljMicrosphere Resonator Composite, Advanced Materials, vol. 17, pp. 1131-1136,2005. Issue on semiconductor lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 11, no. 5, 2005. Issue on organic light-emitting diodes, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 1, 2004. W. E. Howard, Better Displays with Organic Films, Scientific American, vol. 290, no. 2, pp. 76-81, 2004. H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, Electrically Driven Single-Cell Photonic Crystal Laser, Science, vol. 305, pp. 1444-1447 (2004). Issue on semiconductor lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 9, no. 5, 2003. Feature issue on mid-infrared quantum-cascade lasers, IEEE Journal of Quantum Electronics, vol. 38, no. 6, 2002. Issue on high-efficiency light-emitting diodes, IEEE Journal of Selected Topics in Quantum Electron- ics, vol. 8, no. 2, 2002. F. Capasso, C. Gmachl, D. L. Sivco, and A. Y. Cho, Quantum Cascade Lasers, Physics Today, vol. 55, no. 5,pp. 34-40, 2002. M. G. Craford, N. Holonyak, Jr., and F. A. Kish, Jr., In Pursuit of the Ultimate Lamp, Scientific American, vol. 284, no. 2, pp. 62-67, 2001. Issue on semiconductor lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 2, 2001.
744 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES M. H. Huang, S. Mao, H. Feick, H. Van, Y. Wu, H. Kind, E. Weber, R. Russo, and P. Yang, Room- Temperature Ultraviolet Nanowire Nanolasers, Science, vol. 292, pp. 1897-1899, 2001. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, A Quantum Dot Single-Photon Turnstile Device, Science, vol. 290, pp. 2282-2285, 2000. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. Issue on semiconductor lasers, IEEE Journal of Selected Topics in Quantum Electronics, vol. 5, no. 3, 1999. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, Two-Dimensional Photonic Band-Gap Defect Mode Laser, Science, vol. 284, pp. 1819-1821, 1999. C. J. Chang-Hasnain, VCSELs: Advances and Future Prospects, Optics & Photonics News, vol. 9, no. 5,pp. 34-39, 1998. G. R. Little, ed., Selected Papers on Fundamentals of Optoelectronics, SPIE Optical Engineering Press (Milestone Series Volume 90), 1994. J. J. Coleman, ed., Selected Papers on Semiconductor Diode Lasers, SPIE Optical Engineering Press (Milestone Series Volume 50), 1992. J. Jewell, Surface-Emitting Lasers: A New Breed, Physics World, vol. 3, no. 7, pp. 28-30, 1990. F. Capasso and S. Datta, Quantum Electron Devices, Physics Today, vol. 43, no. 2, pp. 74-82, 1990. R. H. Saul, T. P. Lee, and C. A. Burrus, Light-Emitting-Diode Device Design, in R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 22, Lightwave Communications Technology, W. T. Tsang, ed., Part C, Semiconductor Injection Lasers, II and Light Emitting Diodes, Academic Press, 1985. Historical Zh. I. Alferov, Double Heterostructure Concept and its Applications in Physics, Electronics and Technology, in G. Ekspong, ed., Nobel Lectures, Physics 1996-2000, World Scientific, 2002. H. Kroemer, Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks, in G. Ekspong, ed., Nobel Lectures, Physics 1996-2000, World Scientific, 2002. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Quantum Cascade Laser, Science, vol. 264, pp. 553-556, 1994. J. L. Jewell, A. Scherer, S. L. McCall, Y. H. Lee, S. Walker, J. P. Harbison, and L. T. Florez, Low- Threshold Electrically Pumped Vertical-Cavity Surface-Emitting Microlasers, Electronics Letters, vol. 25, pp. 1123-1124, 1989. R. D. Dupuis, An Introduction to the Development of the Semiconductor Laser, IEEE Journal of Quantum Electronics, vol. QE-23, pp. 651-657, 1987. N. G. Basov, Quantum Electronics at the P. N. Lebedev Physics Institute of the Academy of Sciences of the USSR (FIAN), Soviet Physics-Uspekhi, vol. 29, pp. 179-185, 1986 [Uspekhi Pizicheskikh Nauk, vol. 148, pp. 313-324, 1986]. J. K. Butler, ed., Semiconductor Injection Lasers, IEEE Press, 1980. E. E. Loebner, Subhistories of the Light Emitting Diode, IEEE Transactions on Electron Devices, vol. ED-23, pp. 675-699, 1976. N. G. Basov, Semiconductor Lasers, in Nobel Lectures in Physics, 1963-1970, Elsevier, 1972. R. F. Kazarinov and R. A. Suris, Amplification of Electromagnetic Waves in a Semiconductor Super- lattice, Soviet Physics-Semiconductors, vol. 5, pp. 707-709, 197]. L. Esaki and R. Tsu, Superlattice and Negative Differential Conductivity in Semiconductors, IBM Journal of Research and Development, vol. 14, pp. 61-65, 1970. J. I. Pankove and J. E. Berkeyheiser, A Light Source Modulated at Microwave Frequencies, Proceed- ings of the IRE, vol. 50, pp. 1976-1977, 1962. T. M. Quist, R. H. Rediker, R. J. Keyes, W. E. Krag, B. Lax, A. L. McWhorter, and H. J. Zeiger, Semiconductor Maser of GaAs, Applied Physics Letters, vol. 1, pp. 91-92, 1962. N. Holonyak, Jr., and S. F. Bevacqua, Coherent (Visible) Light Emission from Ga(As1-xP x) Junc- tions, Applied Physics Letters, vol. 1, pp. 82-83, 1962. M. I. Nathan, W. P. Dumke, G. Bums, F. H. Dill, Jr., and G. Lasher, Stimulated Emission of Radiation from GaAs p-n Junctions, Applied Physics Letters, vol. 1, pp. 62-64, 1962. R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. o. Carlson, Coherent Light Emission from GaAs Junctions, Physical Review Letters, vol. 9, pp. 366-368, 1962.
PROBLEMS 745 R. J. Keyes and T. M. Quist, Recombination Radiation Emitted by Gallium Arsenide, Proceedings of the IRE, vol. 50, pp. 1822-1823, 1962. N. G. Basov, O. N. Krokhin, and Yu. M. Popov, Production of Negative-Temperature States in p- n Junctions of Degenerate Semiconductors, Soviet Physics-JETP, vol. 13, pp. 1320-1321, 1961 [Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, vol. 40, pp. 1879-1880, 1961]. M. G. A. Bernard and G. Duraffourg, Laser Conditions in Semiconductors, Physica Status Solidi, vol. 1, pp. 699-703, 1961. John von Neumann, in unpublished calculations sent to Edward Teller in September 1953, showed that it was possible, in principle, to upset the equilibrium concentration of carriers in a semi- conductor and thereby obtain light amplification by stimulated emission, e.g., via the recombi- nation of electrons and holes injected into a p-n junction [see J. von Neumann, Notes on the Photon-Disequilibrium-Amplification Scheme (JvN), Sept. 16, 1953, IEEE Journal of Quantum Electronics, vol. QE-23, pp. 658-673, 1987]. H. J. Round, A Note on Carborundum, Electrical World, vol. 49, p. 309, 1907. PROBLEMS 17.1-5 LED Spectral Widths. Consider seven of the LED spectra shown in Figs. 17.1-14 and P17.1-5, namely those centered at Ao == 0.37, 0.53, 0.64, 0.91, 1.30, 1.93, and 2.25 pm. Graphically estimate the spectral widths (FWHM) in units ofnm, Hz, and eV. Compare your estimates with the results calculated from the formulas given in Exercise 17.1-3. Estimate the alloy broadening in the LED spectrum centered at Ao == 0.53 pm in units of nm, Hz, and e V. Figure P17.1-5 Spectral intensities versus wavelength for InGaAsP LEDs operating in the near-infrared region of the spectrum. The peak intensities are all normalized to the same value. The spectral width generally increases as A ' in accordance with (17.1-29). 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Wavelength Ao (/lID) 17.1-6 External Efficiency of an LED. Derive an expression for It e, the efficiency for the extrac- tion of internal unpolarized light from an LED, that includes the angular dependence of Fresnel reflection at the semiconductor-air boundary (see Sec. 6.2). 17.1- 7 Coupling Light from an LED into an Optical Fiber. Calculate the fraction of optical power emitted from an LED that is accepted by a step-index optical fiber of numerical aperture NA == 0.1 in air and core refractive index 1.46 (see Sec. 9.1). Assume that the LED has a planar surface, a refractive index n == 3.6, and an angular dependence of optical power that is proportional to cos 4 (B). Assume further that the LED is bonded to the core of the fiber and that the emission area is smaller than the fiber core. 17.2-1 Bandwidth of a Semiconductor Optical Amplifier. Use the data in Fig. 17.2-3(a) to plot the full bandwidth of the InGaAsP amplifier against the injected-carrier concentration n. Find an approximate linear formula for this bandwidth as a function of n and plot the amplifier gain coefficient versus bandwidth. 17.2-2 Peak Gain Coefficient of a Semiconductor Optical Amplifier at T == 0° K. (a) Show that the peak value IP of the gain coefficient IO(v) at T == 0° K is located at v == (E jc - Ejv)/h. (b) Obtain an analytical expression for the peak gain coefficient IP as a function of the injected-carrier concentration n at T == 0° K.
746 CHAPTER 17 SEMICONDUCTOR PHOTON SOURCES (c) Plot 'Yp versus n for an InGaAsP amplifier (Ao = 1300 nm, n = 3.5, Tr = 2.5 ns, me = 0.06 mo, mv = 0.4 mo) for values of n in the range of 1 x 10 18 to 2 X 10 18 cm -3. (d) Compare the results with the data provided in Fig. 17.2-3b. * 17 2-3 Gain Coefficient of a GaAs Semiconductor Optical Amplifier. A room-temperature (T = 300° K) p-type GaAs SOA (E g 1.40 eV, me = 0.07 mo, mv = 0.50 mo), with refractive index n = 3.6, is doped (Po = 1.2 x 1018) such that the radiative recombination lifetime Tr 2 ns. (a) Given the steady-state injected-carrier concentration n (which is controlled by the injection rate R and the overall recombination time T), use (17.2-2)-(17.2-4) to compute the gain coefficient ')'0 (v) versus the photon energy hv, assuming that T = 0° K. (b) Carry out the same calculation using a computer, assuming that T = 300° K. (c) Plot the peak gain coefficient as a function of n for both cases. (d) Determine the loss coefficient Q and the transparency concentration nT using the linear approximation model. (e) Plot the full amplifier bandwidth (in Hz, nm, and e V) as a function of n for both cases. (0 Compare your results with the gain coefficient and peak gain coefficient curves shown in Fig. PI7.2-3. --. "7 E 300 u "-" 200 Q) °u S Q) o u 100 s::: "a o --. "7 5 300 "-" ...... s::: Q) .u 200 Q) o u s::: "a 100 OJJ ro Q) P-; o 1.38 o 1040 1042 1.44 hv (eV) (a) 0.5 1.0 1.5 n (l0 18 cm- 3 ) (b) 2.0 Figure P17.2-3 Gain coefficient and peak gain coefficient of a GaAs SOA. (Adapted from M. B. Panish, Heterostructure Injection Lasers, Proceedings of the IEEE, vol. 64, pp. 1512-1540, Fig. 4 @1976 IEEE.) 17.2-4 Bandgap Reduction Arising from Band-Tail States. The bandgap reduction E 9 arising from band-tail states in InGaAsP and GaAs can be empirically expressed as Eg(eV) (-1.6 X 10- 8 ) (p1/3 + n 1 / 3 ) , where nand p are the carrier concentrations (cm- 3 ) provided by doping, carrier injection, or both. (a) For p-type InGaAsP and GaAs, determine the concentration p that reduces the bandgap by approximately 0.02 e V. (b) For undoped InGaAsP and GaAs, determine the injected-carrier density n that re- duces the bandgap by approximately 0.02 e V. Assume that n i is negligible. (c) Compute Eg+Eg and compare the result with the energy at which the gain coefficient in Fig. PI7.2-3(a) is zero on the low-frequency side. 17.2-5 Amplifier Gain and Bandwidth. GaAs has an intrinsic carrier concentration n i = 1.8 X 10 6 cm- 3 , a recombination lifetime T = 50 ns, a bandgap energy E 9 = 1.42 e V, an effective electron mass me = 0.07 mo, and an effective hole mass mv = 0.50 mo. Assume that T = 0° K.
PROBLEMS 747 (a) Determine the center frequency, bandwidth, and peak net gain within the bandwidth for a GaAs amplifier of length d = 200 pm, width w = 10 pm, and thickness l = 2 pm, when 1 mA of current is passed through the device. (b) Determine the number of voice messages that can be supported by the bandwidth determined above, given that each message occupies a bandwidth of 4 kHz. (c) Determine the bit rate that can be passed through the amplifier given that each voice channel requires 64 kbits/s. 17.2-6 Transition Cross Section. Determine the transition cross section a(v) for GaAs as a func- tion of n at T = 0° K. The probability density for stimulated emission or absorption is cpa(v), where cp is the photon-flux density. Why is the transition cross section less useful for semiconductor optical amplifiers than for other laser amplifiers? *17.2-7 Gain Profile. Consider a 1550-nm InGaAsP amplifier (n = 3.5) of the configuration shown in Fig. 17.2-6, with identical antireflection coatings on its input and output facets. Calculate the maximum reflectivity of each of the facets that can be tolerated if it is desired to maintain the variations in the gain profile arising from the frequency dependence of the Fabry-Perot transmittance to less than 10% [see (7.1-32)]. 17.3-1 Dependence of Output Power on Refractive Index. Identify the terms in the output photon flux <Po given in (] 7.3-10) that depend on the refractive index of the crystal. 17.3- 2 Longitudinal Modes. A current is injected into an InGaAsP diode of bandgap energy E 9 = 0.91 e V and refractive index n = 3.5 such that the difference in Fermi levels is E f c - E f v = 0.96 e V. If the resonator is of length d = 250 pm and has no losses, determine the maximum number of longitudinal modes that can oscillate. 17.3-3 Minimum Gain Required for Lasing. A 500-pm-Iong InGaAsP crystal operates at a wavelength where its refractive index n = 3.5. Neglecting scattering and other losses, determine the gain coefficient required to barely compensate for reflection losses at the crystal boundaries. *17.3-4 Modal Spacings with a Wavelength-Dependent Refractive Index. The frequency sepa- ration of the modes of a laser diode is complicated by the fact that the refractive index is wavelength dependent [i.e., n = n(Ao)]. A laser diode of length 430 pm oscillates at a central wavelength Ac = 650 nm. Within the emission bandwidth n( Ao) may be assumed to be linearly dependent on Ao [i.e., n(Ao) = no - a(Ao - Ac), where no = n(Ac) = 3.4 and a = dn/dAo]. (a) The separation between the laser modes with wavelength near Ac was observed to be A 0.12 nm. Explain why this does not correspond to the usual modal spacing vp = c/2d. (b) Find an estimate of a. (c) Explain the phenomenon of mode pulling in a gas laser and compare it with the effect described above in semiconductor lasers.
CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS 18.1 18.2 18.3 18.4 18.5 18.6 PHOTODETECTORS A. External and Internal Photoeffects B. General Properties PHOTOCONDUCTORS A. Intrinsic Materials B. Extrinsic Materials C. Heterostructures PHOTODIODES A. The p-n Photodiode B. The p-i-n Photodiode C. Heterostructures AVALANCHE PHOTODIODES A. Principles of Operation B. Gain and Responsivity C. Response Time D. Single-Photon Avalanche Diodes (SPADs) ARRAY DETECTORS NOISE IN PHOTODETECTORS A. Photoelectron Noise B. Gain Noise C. Circuit Noise D. Signal-to-Noise Ratio and Receiver Sensitivity E. Bit Error Rate and Receiver Sensitivity 749 758 762 767 775 777 " It -...\ " .....; ,... ,I' .. ",-, Heinrich Hertz (1857-1894) discovered the photoelectric effect in 1887; its origin was explained by Einstein in 1905. 748 Simeon Denis Poisson (1781-1840) devel- oped the fundamental probability distribution that describes photodetector noise.
A photodetector is a device that measures photon flux or optical power by converting the energy of the absorbed photons into a measurable form. Two principal classes of photodetectors are in common use, photoelectric detectors and thermal detectors: 1. The operation of photoelectric detectors is based on the photoelectric effect, also called the photoeffect. The absorption of photons by a material causes electrons to transition to higher energy levels, resulting in mobile charge car- riers. Under the effect of an electric field, these carriers move and produce a measurable electric current. The photoeffect takes two forms: external and in- ternal. The external photoeffect involves photoelectric emission, in which the photogenerated electrons escape from the material as free electrons. The internal photoeffect involves photoconductivity, in which the excited carriers remain within the material and serve to increase its conductivity. 2. Thermal detectors operate by converting photon energy into heat. As a result of the time required to effect a temperature change, thermal detectors are generally inefficient and slow in comparison with photoelectric detectors. However, recent advances in manufacturing and miniaturization have dramatically improved the performance of thermal array detectors and they are now viable contenders for imaging applications in the mid-infrared region. This Chapter This chapter is devoted to a study of various photoelectric detectors that find use in photonics. We begin in Sec. 18.1 with a discussion of the external and internal photoeffects. and we set forth several important general properties of photodetectors, including quantum efficiency, responsivity, and response time. In Secs. 18.2, 18.3, and 18.4, we direct our attention to three types of semiconductor photo detectors that rely on the internal photoeffect: photoconductors, photodiodes, and avalanche photodiodes, respectively. Array detectors, which produce electronic versions of optical images, are considered in Sec. 18.5. To assess the performance of semiconductor photo detectors in various applications, it is important to understand their noise properties, and these are set forth in Sec. 18.6. Noise in the output circuit of a photoelectric detector arises from several sources: the photon character of the light itself (photon noise), the conversion of photons to photocarriers (photoelectron noise), the generation of secondary carriers by internal amplification (gain noise), as well as receiver circuit noise. A brief discussion of the performance of analog and digital optical receivers is also provided. 18.1 PHOTODETECTORS A. External and Internal Photoeffects Photoelectron Emission If the energy of a photon illuminating a material in vacuum is sufficiently large, the excited electron can escape over the potential barrier of the surface of the material and be liberated into the vacuum as a free electron. This process, called photoelectron emission, is illustrated in Fig. 18.1-1 (a) for a metal. An incident photon of energy hv releases a free electron from within the partially filled conduction band. Energy conservation requires that electrons emitted from below the Fermi level, where they 749
750 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS are plentiful, have a maximum kinetic energy Emax == hv - w, (18.1-1) where the photoelectric work function W is the energy difference between the vacuum level and the Fermi level of the metal. Equation (18.1-1) is known as Einstein's photo- emission equation. Only if the electron initially lies at the Fermi level can it receive the maximum kinetic energy specified in (18.1-1); the removal of a deeper-lying electron requires additional energy to transport it to the Fermi level, thereby reducing the kinetic energy of the liberated electron. The lowest work function for a metal (Cs) is about 2 e V, so that optical detectors based on the external photoeffect from pure metals are useful in the visible and ultraviolet regions of the spectrum. Free 4 t elect ron I Next higher band Photon Vacuum Free f electron Photon Conduction band 'V\J\J\IL.f,- .. hv Vacuum rrr +w ____E!!l!}!. E g ! + hv W Fermilevel 1 .--4 r:.;.:._---:--.. -- Conduction band . Valence band (a) Metal (b) Semiconductor Figure 18.1-1 Photoelectric emission (a) from a metal, and (b) from an intrinsic semiconductor. The bandgap energy and electron affinity of the material are denoted E 9 and X, respectively, and W is the photoelectric work function. All three of these quantities are usually specified in e V. Photoelectric emission from an intrinsic semiconductor is portrayed schematically in Fig. 18.1-1 (b). Photoelectrons are usually released from the valence band, where electrons are plentiful. The formula analogous to (18.1-1) is Emax == hv - W == hv - (Eg + X), (18.1-2) where Eg is the bandgap energy and X is the electron affinity of the material (the energy difference between the vacuum level and the bottom of the conduction band). The energy Eg + X can be as small as 1.4 eV for certain materials (e.g., the multialkali compound NaKCsSb, which forms the basis for the so-called S-20-type photocath- ode), so that semiconductor photoemissive detectors can operate in the near infrared, as well as in the visible and ultraviolet regions of the spectrum. Furthermore, negative-electron-affinity (NEA) semiconductors have been devel- oped in which the conduction-band edge lies above the vacuum level in the bulk of the material, so that hv need only exceed Eg for photoemission to occur (a thin n- type or metallic layer deposited on p-type material can cause the bands to bend at the surface of the material so that the bottom of the conduction band does indeed lie below the vacuum level). NEA detectors, such as Cs-coated GaAs, are therefore responsive to slightly longer near-infrared wavelengths, and also exhibit improved quantum ef- ficiency and reduced dark current. Photocathodes constructed from inhomogeneous materials or oxides, such as the S-I-type photocathode, can also be used in the near infrared, but only up to wavelengths of::::;j 1 /-Lm. In their simplest form, photodetectors based on photoelectric emission take the form of vacuum tubes called vacuum photodiodes or phototubes. Electrons are emitted
18.1 PHOTODETECTORS 751 from the surface of a photoemissive material called the photocathode and travel to an electrode (anode), which is maintained at a higher electric potential. The photocathode can be opaque and operate in reflection mode [Fig. I8.I-2(a)], or semitransparent and operate in transmission mode [Fig. I8.I-2(b)]. As a result of the electron transport between the cathode and anode, a current proportional to the photon flux, known as the photocurrent, is created in the circuit. The photoemitted electrons may also create a cascade of electrons via the process of secondary emission. This occurs when the photoelectrons impact other specially placed semiconductor or cesiated-oxide surfaces in the tube, called dynodes, which are maintained at successively higher potentials. The result is an amplification of the generated photocurrent by a factor as high as 10 8 . This useful device, illustrated in Fig. I8.I-2(b), is known as a photomultiplier tube (PMT). A PMT can be used to detect and count individual photons while offering a large dynamic range; however, it is bulky and requires a high-voltage supply. An imaging device that makes use of this principle is the micro channel plate. It consists of an array of millions of capillaries (of internal diameter 10 Mm) created in a glass plate of thickness 1 mm. Both faces of the plate are coated with thin metal films that act as electrodes, across which a voltage is applied [Fig. I8.I-2(e)]. The interior walls of each capillary are coated with a material that emits secondary electrons so it behaves as a continuous dynode, multiplying the photocurrent generated at that position [Fig. 18.1- 2( d)]. The local photon flux in a faint image can therefore be converted into a substantial electron flux that can be directly measured. Furthermore, the electron flux can be reconverted into an (amplified) optical image by using a phosphor coating as the rear electrode that produces light via cathodoluminescence (see Sec. I3.5A); this combination is called an image intensifier. R L Dynodes hv (a) -v -=- (b) -v -=- Imaging /'" photocathode Capillaries -=- (c) (d) -v Figure 18.1-2 (a) Vacuum photodiode with a photocathode operated in reflection mode. (b) Electron multiplication in a photomultiplier tube with a semitransparent photocathode operated in transmission mode. ( c) Cutaway view of microchannel plate. (d) Electron multiplication in a single capillary of a microchannel plate.
752 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Photoconductivity Most modem photo detectors operate on the basis of the internal photoeffect, in which the photoexcited carriers (electrons and holes) remain within the sample. Detectors based on photoconductivity rely directly on the light-induced increase in the electrical conductivity of a material. The absorption of a photon by an intrinsic semiconductor, for example, results in the generation of a free electron excited from the valence band to the conduction band (Fig. 18.1-3). Concurrently, a hole is generated in the valence band. The application of an electric field to the material results in the transport of both electrons and holes through the material and, as a consequence, the production of an electric current in the electrical circuit. Photon hv Eleeu@D e . t'::;. ._. I hv -4 Hole .....T E g L Figure 18.1-3 Electron-hole photogeneration in a semiconductor. The semiconductor photodiode detector is a p-n junction structure that is also based on the internal photoeffect. Photons absorbed in the depletion layer generate electrons and holes, which are subjected to the local electric field within that layer. The two carriers drift in opposite directions. This transport process induces an electric current in the external circuit. Some photodetectors incorporate internal gain mechanisms so that the photocurrent can be amplified by carrier multiplication within the detector and thus make the signal more easily detectable. If the depletion-layer electric field in a photodiode is increased sufficiently by applying a large reverse bias across the junction, the electrons and holes generated may themselves acquire sufficient energy to liberate additional electrons and holes within this layer by a process called impact ionization. Devices in which this internal amplification process occurs are known as avalanche photodiodes (APDs). An APD can be used as an alternative to (or in conjunction with) a laser amplifier (see Chapters 14 and 17), in which the optical signal is amplified before detection. Each of these amplification mechanisms introduces its own form of noise, however. Semiconductor photoelectric detectors with gain therefore involve the following three basic processes: 1. Generation: Absorbed photons generate free carriers. 2. Transport: An applied electric field causes these carriers to move, resulting in the circuit current. 3. Gain: In avalanche photodiodes, large electric fields impart sufficient energy to the carriers so that they in turn free additional carriers by impact ionization. This internal amplification process enhances the responsivity of the detector. B. General Properties Certain general features are associated with all semiconductor photodetectors. Before studying the details of specific photo detectors of interest in photonics, we examine the quantum efficiency, responsivity, and response time of photoelectric detectors from a general perspective.
18.1 PHOTODETECTORS 753 Semiconductor photon detectors and semiconductor photon sources are inverse de- vices. Detectors serve to convert a photon flux at the input of the device to an electric current at its output; sources do the opposite. The same materials are often used in fabricating both types of devices (see Chapter 16). Indeed, all of the performance measures discussed in this section have their counterparts in sources (see Chapter 17). Quantum Efficiency The quantum efficiency Il (0 < Il < 1) of a photodetector is the probability that a single photon incident on the device will generate a photocarrier pair that contributes to the detector current. When many photons are incident, as is usually the case, Il becomes the flux of generated electron-hole pairs that contribute to the detector current divided by the flux of incident photons. Not all incident photons produce electron-hole pairs because not all of them are absorbed. As illustrated in Fig. 18.1-4, some of the photons are reflected at the surface of the detector while others fail to be absorbed because the material does not have suffi- cient depth (the rate of photon absorption in a semiconductor material was considered in Sec. 16.2C). Furthermore, some electron-hole pairs produced near the surface of the detector quickly recombine because of the abundance of recombination centers at surfaces, and are therefore not available to contribute to the detector current. The quantum efficiency can therefore be written as Il == (1 - 9() ( [1 - exp ( - ad)] , (18.1-3) Quantum Efficiency where 9( is the optical power reflectance at the surface, ( the fraction of electron-hole pairs that successfully contribute to the detector current, a the absorption coefficient of the material (cm- I ) discussed in Sec. 16.2C, and d the photo detector depth. Equa- tion (18.1-3) is a product of three factors: . The first factor, (1 - 9(), represents the effect of reflection at the surface of the device. Reflection can be reduced, for example, by the use of antireflection coatings. Some definitions of the quantum efficiency Il exclude reflection at the surface, which must then be considered separately. . The second factor ( is the fraction of electron-hole pairs that successfully avoid recombination at the material surface and contribute to the useful photo current. Surface recombination can be reduced by careful material growth and device design. . The third factor, fad e- ax dx/ fo oo e- ax dx == [1 - exp( -ad )], represents the fraction of the photon flux absorbed in the bulk of the material. The device should have a value of d that is sufficiently large so this factor is maximized. Of course, additional loss occurs if the light is not properly focused onto the active region of the detector. Dependence of quantum efficiency on wavelength. The quantum efficiency Il is a function of wavelength, principally because the absorption coefficient a is wavelength dependent (see Fig. 16.2-3). The characteristics of the semiconductor material thus determine the spectral window within which Il is large. For sufficiently large values of the free-space wavelength .Ao, Il is small because absorption cannot occur when .Ao > .Ag == hc o / Eg (the photon energy is then smaller than the bandgap energy and the material is transparent). The bandgap wavelength .Ag is thus the long-wavelength limit of the semiconductor material. Representative values of E 9 and .Ag are presented in Table 16.1-2 and displayed in Figs. 16.1-7 and 16.1-8 for most semiconductor materials
754 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Incident photon flux <I> Photons hv T d 1 Photosensitive region x -------- ----------- l/a 1 Reflected photon flux Transmitted photon flux Figure 18.1-4 Effect of surface reflection and incomplete absorption on the detector quantum efficiency It. x of interest in photonics. For sufficiently small values of .Ao, It also decreases because most photons are then absorbed near the surface of the device (e.g., for a == 10 4 cm- 1 , most of the light is absorbed within a distance 1/ a == 1 /-Lm). The recombination lifetime is quite short near the surface, so that the photocarriers recombine before being collected. Resonant-cavity photodetectors. The quantum efficiency It may be enhanced by constructing a detector configuration in which the light can interact with the photosen- sitive material on multiple passes. This is equivalent to increasing the photodetector depth d, which increases the absorption and reduces the transmitted photon flux. This may be achieved in practice by placing the photodetector inside a resonant cavity, which traps the light and thus increases the quantum efficiency. Responsivity The responsivity of a photo detector relates the electric current ip flowing in the de- vice circuit to the optical power P incident on it. If every photon were to generate a photocarrier pair in the device, a photon flux <I> (photons per second) would produce an electron flux <I> (electrons per second) in the photo detector circuit, corresponding to a short-circuit electric current ip == e<I>. Thus, an optical power P == hv<I> (watts) at frequency v would give rise to an electric current ip == e P / hv. However, since the fraction of photons producing detected electrons is It rather than unity, the electric current is . IteP p == Ite<I> == - P. hv ( 18.1-4 ) The proportionality factor between the electric current and the optical power, ip/ P, has units of A/W and is called the photo detector responsivity: 9\= Ile =Il' hv 1.24 (18.1-5) Photodetector Responsivity (AjW; Ao in pm) It is important to distinguish the photodetector responsivity (A/W) from the light- emitting-diode responsivity (W/A) defined in (17.1-28). The responsivity is linearly proportional to both the quantum efficiency It and the free-space wavelength .Ao, as is evident from (18.1-5) and Fig. 18.1-5. An appreciation for the order of magnitude of the responsivity is gained by setting It == 1 and .Ao == 1.24 /-Lm in (18.1-5), whereupon == 1 A/W == 1 nA/nW.
18.1 PHOTODETECTORS 755 1.2 1.0 1.0 - $ 0.8 C .;; .;;; 0.6 t:: o 0.. 0.4 0.2 >. 0.8 g Q) .u it: 0.6 E ::3 E ro 0.4 8 0.2 Figure 18.1-5 Responsivity 9{ (A/W) versus wavelength Ao, with the quantum efficiency Il as a parameter. For 11. == 1, 9{ == 1 A/W at Ao == 1.24 11m. o 0.8 1.0 1.2 1.4 Wavelength Ao (Mm) 1.6 The proportionality of 9{ to .Ao arises because the responsivity is defined on the basis of optical power, whereas most photodetectors generate currents proportional to the photon flux <I>. For a given photon flux <I> == P / hv == P.A o / hc o (corresponding to a given photodetector current i p ), the product P.A o is fixed so that an increase in .Ao requires a commensurate decrease in P, thereby leading to an increase in the responsivity. Indeed, some thermal detectors are responsive to optical power rather than to photon flux, causing 9{ to be independent of .Ao. The region over which 9{ increases with .Ao is limited, however, inasmuch the wave- length dependence of Il comes into play at both long and short wavelengths. The responsivity can also be degraded if the detector is presented with an excessively large optical power. This condition, known as detector saturation, limits the linear dynamic range of the detector, which is the range over which it responds to the incident optical power in a linear fashion. Devices with gain. The formulas presented above are predicated on the assumption that each photocarrier pair produces a charge e in the photodetector circuit. However, many devices produce a charge q in the circuit that differs from e. Such devices are said to exhibit gain. The gain G is defined as the average number of circuit electrons generated per photocarrier pair, G q/e. (18.1-6) It can be either greater than or less than unity, as will be seen subsequently. In the presence of gain, the formulas for the photocurrent and responsivity presented in (18.1-4) and (18.1-5), respectively, must be modified. Substituting q == Ge for e in these equations yields, respectively, . Il GeP zp == Il q<I> == Il Ge<I> == hv (18.1-7) Photocurrent with Gain and 9t = l\.Ge = l\.G . hv 1.24 (18.1-8) Responsivity with Gain (AjW; Ao in 11m) The device gain G is to be distinguished from the photodetector efficiency Il, which is the probability that an incident photon produces a detectable photocarrier pair. Other
756 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS useful measures of photodetector behavior, such as signal-to-noise ratio and receiver sensitivity, await discussion of detector noise properties presented in Sec. 18.6. Response Time Transit-time spread. A constant electric field E presented to a semiconductor (or metal) causes its free charge carriers to accelerate. In the course of doing so, they encounter frequent collisions with lattice ions moving about their equilibrium positions via thermal motion, as well as imperfections in the crystal lattice associated with impurity ions. These collisions cause the carriers to suffer random decelerations; the result is motion at an average velocity rather than at a constant acceleration. The mean velocity of a carrier is given by V == aTeol, where a == eE/m is the acceleration imparted by the electric field and Teol is the mean time between collisions, which serves as a relaxation time. The result is that the carrier drifts in the direction of the electric field with a mean drift velocity v == eTeoIE/m, which is conventionally written in the form v == JLE, (18.1-9) where JL == eTeol/m is the carrier mobility. The carrier motion in the photodetector creates a current in its external circuit. To determine the magnitude of the current i(t), consider an electron-hole pair generated (by photon absorption, for example) at an arbitrary position x in a semiconductor material of length w, to which a voltage V is applied, as shown in Fig. 18.1-6(a). We restrict out attention to motion in the x direction and use an energy argument. If a carrier of charge Q (a hole of charge Q == e or an electron of charge Q == -e) moves a distance dx in the time dt, under the influence of an electric field of magnitude E == V / w, the work done is -Q E dx == -Q(V / w) dx. This work must equal the energy provided by the external circuit, i(t)V dt. Thus, i(t)V dt == -Q(V/w) dx, from which i(t) == -(Q/w)(dx/dt) == -(Q/w)v(t). A carrier moving with a drift velocity v( t) in the x direction therefore creates a current in the external circuit given by Ramo's theorem: i(t) = - Q v(t). w (18.1-10) Ramo's Theorem Assuming that the hole moves with velocity Vh to the left, and the electron moves with velocity v e to the right, (18.1-10) tells us that the hole current i h == -e( -Vh)/W and the electron current ie == -( -e)ve/w, as illustrated in Fig. 18.1-6(b). Each carrier contributes to the current as long as it is moving. If the carriers continue their motion until they reach the edges of the material, the hole moves for a time X/Vh and the electron moves for a time (w - x)/v e [see Fig. 18.1-6(a)]. In semiconductors, v e is generally larger than Vh so that the fuH width of the response is X/Vh. The finite duration of the current is known as transit-time spread; it is an important limiting factor for the speed of operation of all semiconductor photodetectors. One might be inclined to argue that the charge generated in an external circuit should be 2e when a photon generates an electron-hole pair in a photodetector material, since there are two charge carriers. In fact, the charge generated is e, as is shown by calculating the total charge q induced in the external circuit as the sum of the areas under ie and i h : Vh X v e W-x ( X W-x ) q == e-- + e- == e - + == e. W Vh W v e W W (18.1-11)
18.1 PHOTODETECTORS 757 v i(t) ;(t) t V e x W evh/w eve/w (W-X)/V e x ieCt) uu____u_u_u_u__ (w-x) /V e ' [h(t) X/V h X/V h t (a) (b) Figure 18.1-6 (a) An electron-hole pair is generated at the position x. The hole drifts to the left with velocity Vh and the electron drifts to the right with velocity V e . The process terminates when the carriers reach the edges of the material. (b) Hole current ih(t), electron current ie(t), and total current i(t) induced in the circuit. The total charge induced in the circuit per carrier pair is e. This result is independent of the position x at which the electron-hole pair was created. The transit-time spread is even more severe if the electron-hole pairs are generated uniformly throughout the material, as shown in Fig. 18.] -7. For Vh < V e , the full width of the transit-time spread is then W / Vh rather than x / Vh. This occurs because uniform illumination produces carrier pairs everywhere, including at x == w, which is the point at which the holes have the farthest to travel before being able to recombine at x == o. iJz(t) ii t ) i(t) Ne(Ve+vh)lw Nevelw + . . . . . . " . . Nevhlw Nevhlw -...:- . . f .. 0 wlv h t 0 wive t o wiVe wlV h t Figure 18.1-7 Hole current i h (t), electron current ie (t), and total current i( t) induced in the circuit for electron-hole generation by N photons uniformly distributed between 0 and W (see Prob. 18.1-4). The tail in the total current results from the motion of the holes. The total current i(t) can be viewed as the impulse response function (see Appendix B, Sec. B.l) for a uniformly illuminated detector subject to transit-time spread. In summary, Ramo's theorem demonstrates that the charge delivered to the external circuit by carrier motion in the photodetector material is not provided instantaneously, but rather occupies an extended time. It is as if the motion of the charged carriers in the material pulls charge slowly from the wire on one side of the device and pushes it slowly into the wire on the other side, so that each charge passing through the external circuit is spread out in time. Ohm's law. In the presence of a uniform charge density (}, rather than a single point charge Q, the total charge in the photodetector material is (}Aw, where A is
758 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS the cross-sectional area [see Fig. 18.1-6(a)]. Equation (18.]-10) then yields i(t) -(QAwlw)v(t) == -QAv(t), so that the current density in the x direction is J(t) == -i(t)1 A == Qv(t). The well-known vector form of this equation is (J==QV. (18.1-12) Current Density Combining (18.1-12) with (18.1-9) yields J == a E, where a is the conductivity of the medium, a == Q J1 == eQ Teoll m == N e 2 Teoll m, (18.1-13) where N is the number of carriers per unit volume (see Sec. 5.5D). More generally, the conductivity is a tensor u and the vector version of this equation is Ohm's law: (J == u£ . (18.1-14) Ohm's Law For charge carried by a homogeneous conductive material with cross-sectional area A and length w, J == aE can be written as i == (aAlw)Ew == (aAlw)V == GV == V I R, where G and R are the conductance and resistance of the material, respectively. In this configuration, Ohm's law takes its beloved form V == iR . (18.1-15) RC time constant. The resistance R and capacitance C of the photodetector, along with that of its circuitry, give rise to another response time known as the RC time constant, T RC == RC. The combination of resistance and capacitance serves to integrate the current at the output of the detector, and thereby to lengthen the impulse response function. The impulse response function in the presence of transit-time and simple RC time-constant spread is determined by convolving the current i(t) displayed in Fig. 18.1-7 with the exponential function (II RC) exp( -tl RC) (see Sec. B.l). It is worthy of note that photodetectors of different types may exhibit other specific limitations on their speeds of response, which we consider on a case-by-case basis. As a final point, we mention that photo detectors of a given material and structure often exhibit a fixed gain-bandwidth product. Increasing the gain results in a decrease of the bandwidth, and vice versa. This trade-off between sensitivity and frequency response is associated with the time required for the gain process to take place. 18.2 PHOTOCONDUCTORS When photons are absorbed by a semiconductor, mobile charge carriers are generated (ideally an electron-hole pair for every absorbed photon). The electrical conductivity of the material a increases in proportion to the photon flux <I>. An electric field applied to the material by an external voltage source causes the electrons and holes to be transported. This in turn results in a measurable electric current in the circuit, as illustrated in Fig. 18.2-1 (a). Photoconductive detectors operate by registering either the photo current ip, which is proportional to the photon flux <I>, or the voltage drop across a load resistor R placed in series with the circuit.
18.2 PHOTOCONDUCTORS 759 A. Intrinsic Materials If the photon energy is greater than the bandgap of the semiconductor, photons are absorbed by virtue of band-to-band transitions. A photoconductive device may take the form of a slab or a thin film. The anode and cathode contacts are often interdigitated on the same surface of the material to maximize the light reaching the material while minimizing the transit time (see Fig. 18.2-1(b)). Light can also be admitted from the bottom of the device if the insulating substrate has a sufficiently large bandgap so that it is not absorptive. v lp Insulator Photons A I ( w -I (a) (b) Figure 18.2-1 (a) The photoconductive detector. Photogenerated carrier pairs move in response to the applied voltage V, generating a photocurrent ip proportional to the incident photon flux <I>. (b) The interdigitated electrode structure is designed to maximize the light reaching the semiconductor while minimizing the carrier transit time (thereby maximizing the bandwidth of the device). The increase in conductivity arising from a photon flux <I> (photons per second) illuminating a semiconductor volume w A (see Fig. 18.2-1) is calculated as follows. A fraction q of the incident photon flux is absorbed and gives rise to excess electron- hole pairs. The pair-production rate R (per unit volume) is thus R == q <I> / w A. If T is the excess-carrier recombination lifetime, electrons are lost at the rate n/ T where n is the electron concentration (see Chapter 16). Under steady-state conditions both rates are equal, R == 6.n/ T, so that 6.n == q T<I> / w A. The increase in the carrier concentration 6.n is accompanied by an increase in the charge density (} == e6.n, and thence, in accordance with (18.1-13), by an increase in the conductivity 6.a (} M == enM, so that A == q eT(Me + Mh) ffi ua W A 'J!, (18.2-1) where Me and Mh are the electron and hole mobilities, respectively. In accordance with (18.2-1), the increase in conductivity is proportional to the photon flux. Ohm's law (18.1-14) dictates that the photogenerated current density is given by J p == a E. Combining this with (18.2-1) and (18.1-9), which provides V e == MeE and Vh == MhE, gives J p == [qeT(v e + vh)/wA] <I>, which corresponds to an electric current ip == AJ p == [q eT( v e + Vh) / w] <I>. If Vh « v e, and the formula is cast in terms of the electron transit time across the sample Te == W / V e , we obtain ip q(T/Te)e<I>. (18.2-2) Comparison with (18.1-7) shows that the ratio T / Te in (18.2-2) corresponds to the detector gain G, for reasons we now proceed to elucidate.
760 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Gain The responsivity of a photoconductor with gain is given by (18.1-8). Simply viewed, the device exhibits internal gain because the recombination lifetime and transit time generally differ. Suppose that electrons travel faster than holes (see Fig. 18.2-1) and that the recombination lifetime is very long. As the electron and hole are transported to opposite sides of the photoconductor, the electron completes its trip sooner than the hole. The requirement of current continuity forces the external circuit to immediately provide another electron, which enters the device from the wire at the left. This new electron moves quickly toward the right, again completing its trip before the hole reaches the left edge. This process continues until the electron recombines with the hole. A single photon absorption can therefore result in an electron passing through the external circuit many times. The expected number of trips that the electron makes before the process terminates is G==TITe, (18.2-3) where T is the excess-carrier recombination lifetime and Te W Iv e is the electron transit time across the sample. The charge delivered to the circuit by a single electron- hole pair is then q == Ge > e so that the device exhibits gain. At the other extreme, the recombination lifetime may be sufficiently short such that the carriers recombine before reaching the edge of the material. This can occur if there is a ready availability of carriers of the opposite type for recombination. In that case T < Te and the gain is less than unity so that, on average, each carrier pair contributes only a fraction of the electronic charge e to the circuit. Charge is, of course, conserved and the many carrier pairs present deliver an integral number of electronic charges to the circuit. The photoconductor gain G == T I Te can therefore be interpreted as the fraction of the sample length traversed by the average excited carrier before it undergoes recombi- nation. The transit time Te is determined from the length of the device and the applied voltage via (18.1-9) and Te == wive; typical values of w == 1 mm and V e == 10 7 cm/s yield Te 10- 8 s. The recombination lifetime T can range from 10- 13 s to many seconds, depending on the photoconductor material and doping [see (16.1-24)]. Thus, G can assume a broad range of values, stretching from below unity to well above unity, depending on the parameters of the material, the size of the device, and the applied voltage. However, the gain of a photoconductor generally cannot exceed 10 6 because of the restrictions imposed by space-charge-limited current flow, impact ionization, and dielectric breakdown. Spectral Response The spectral sensitivity of photoconductors is governed principally by the wavelength dependence of Il, as discussed in Sec. 18.IB. Different semiconductors have different long-wavelength limits (see, for example, Table 16.1-2). Elemental, binary, and ternary semiconductors can be used in the photoconductive mode. Photoconductive detectors (in contrast to photoemissive detectors) operate into the infrared region on band-to- band transitions. However, operation at wavelengths beyond about 2 /-Lm generally requires that the devices be cooled to minimize the thermal excitation of electrons into the conduction band in these low-gap materials. Response Time The response time of a photoconductive detector is, of course, constrained by the transit-time and RC time-constant considerations discussed in Sec. 18.1 B. The carrier- transport response time is approximately equal to the recombination time T, so that
18.2 PHOTOCONDUCTORS 761 the carrier-transport bandwidth B is inversely proportional to T. Since the gain G is directly proportional to T in accordance with (18.2-3), increasing T serves to increase the gain, which is desirable, but concomitantly decreases the bandwidth, which is undesirable. The gain-bandwidth product G B thus turns out to be roughly independent of T; typical values of GB extend up to 10 9 . B. Extrinsic Materials Photoconductivity can be achieved at longer wavelengths by making use of doped semiconductors. Mobile charge carriers can be generated via photon absorption by dopants with energy levels lying within the forbidden gap. The process can occur in one of two ways: (1) an incident photon interacts with a bound electron at a donor site, frees it to the conduction band, and leaves behind a bound hole; or (2) an incident photon interacts with a bound hole at an acceptor site, frees it to the valence band, and leaves behind a bound electron, as illustrated in Fig. 16.2-1 (b). Donor and acceptor levels in the bandgap of doped semiconductors can have very low activation energies E A , and therefore quite substantial long-wavelength limits AA == hco/EA. These detectors must be cooled to avoid thermal excitation; liquid He at 4 0 K is often used. Representative values of E A and AA are provided in Table 18.2-1 for a number of extrinsic semiconductors. Table 18.2-1 Selected extrinsic semiconductor materials with their activation energies and long-wavelength limits. Semiconductor: Dopant Ge:Hg Ge:Cu Ge:Zn Ge:Ga Si:B E A (e V) 0.088 0.041 0.033 0.010 0.044 AA (/-lm) 14 30 38 115 23 The spectral responses of several extrinsic semiconductor materials are illustrated in Fig. 18.2-2. The responsivity increases approximately linearly with Ao, in accordance with (18.1-8), peaks slightly below the long-wavelength limit AA, and falls off rapidly beyond it. The quantum efficiency for these detectors can be quite high (e.g., Il 0.5 for Ge:Cu), although the gain may be low under usual operating conditions (e.g., G 0.03 for Ge:Hg). c- ";; "Vi t:: o 0.. r/J Q) I-; Q) .::: Q) Ge:Hg Ge:Cu Ge:Zn Ge:Ga Ge:Ga (stressed) 2 4 20 40 Wavelength Ao (Mill) 100 200 Figure 18.2-2 Relative responsivity versus wavelength Ao (pm) for five different doped-Ge extrinsic materials used as infrared photoconductive detectors.
762 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS c. Heterostructures Properly configured heterostructures can serve as useful photoconductive detectors. An example is the quantum-well infrared photodetector (QWIP). An incident infrared photon releases the electron occupying a bound energy level in a quantum-well to the continuum, thereby creating a mobile charge carrier that increases the conductivity of the material (see Fig. 18.2- 3). Figure 18.2-3 Generation of mobile charge carriers by absorption of photons in a QWIP. The device is configured such that there is a single energy level in each well, corresponding to sensitivity in a particular spectral band. The detector illustrated comprises AIGaAs barriers and n-type GaAs quantum wells, providing the electrons that occupy the energy levels. QWIPs fabricated from III-V compound semiconductors offer high responsivity from mid- to far- infrared wavelengths (Ao 4-20 JLm) and high speeds, but require cooling. The quantum-dot infrared photodetector (QDIP), a variation on this theme, can also be used for multiwavelength infrared detection via intersubband transitions. 18.3 PHOTO DIODES A. The p-n Photodiode As with photoconductors, photodiode detectors rely on photogenerated charge carri- ers for their operation. A photodiode is a p-n junction (see Sec. 16.1E) whose reverse current increases when it absorbs photons. Although p-n and p-i-n photodiodes are generally faster than photoconductors, they do not exhibit gain. Consider a reverse-biased p-n junction under illumination, as depicted in Fig. 18.3- 1. Photons are absorbed everywhere with absorption coefficient a. Whenever a photon is absorbed, an electron-hole pair is generated. But only where an electric field is present can the charge carriers be transported in a particular direction. Since a p-n junction can support an electric field only in the depletion layer, this is the region in which it is desirable to generate photocarriers. Photons ;; 2kJ 2 3 L 0 V P n I( Electric field E Figure 18.3-1 Photons illuminating an idealized reverse-biased p-n photodiode detector. The drift and diffusion regions are indicated by 1 and 2, respectively.
18.3 PHOTODIODES 763 There are, however, three possible locations where electron-hole pairs can be gen- erated: 1. Electrons and holes generated in the depletion layer (region 1) quickly drift in opposite directions under the influence of the strong electric field. Since the electric field always points in the n-tp direction, electrons move to the n side and holes to the p side. As a result, the photocurrent created in the external circuit is always in the reverse direction (from the n to the p region). Each carrier pair generates in the external circuit an electric current pulse of area e (G == 1) since recombination does not take place in the depleted region. 2. Electrons and holes generated away from the depletion layer (region 3) cannot be transported because of the absence of an electric field. They wander randomly until they are annihilated by recombination. They do not contribute a signa] to the external electric current. 3. Electron-hole pairs generated outside the depletion layer, but in its vicinity (re- gion 2), have a chance of entering the depletion layer by random diffusion. An electron coming from the p side is quickly transported across the junction and therefore contributes a charge e to the external circuit. A hole coming from the n side has a similar effect. Photodiodes have been fabricated from many of the semiconductor materials listed in Table 16.1-2, as well as from binary, ternary, and quaternary compound semiconduc- tors such as SiC, InGaAs, and InGaAsP. Devices are sometimes constructed in such a way that the light impinges normally on the p-n junction region instead of parallel to it. In that case the additional carrier diffusion current in the depletion region acts to enhance It, but this is counterbalanced by the decreased thickness of the material, which acts to reduce It. Response Time The transit time of carriers drifting across the depletion layer (w d/ V e for electrons and W d/ v h for holes) and the RC time response play a role in the response time of photodiode detectors, as discussed in Sec. I8.1B. The resulting circuit current is shown in Fig. 18.l-6(b) for an electron-hole pair generated at the position x, and in Fig. 18.1-7 for uniform electron-hole pair generation. In photodiodes there is an additional contribution to the response time arising from diffusion. Carriers generated outside the depletion layer, but sufficiently close to it, take time to diffuse into it. This is a relatively slow process in comparison with drift. The maximum times allowed for this process are the carrier lifetimes (T p for electrons in the p region and Tn for holes in the n region). The effect of diffusion time can be decreased by using a p-i-n diode, as will be seen subsequently. Nevertheless, photodiodes are generally faster than photoconductors because the strong field in the depletion region imparts a large velocity to the photogenerated carriers. Furthermore, photodiodes are not affected by many of the trapping effects associated with photoconductors. Bias As an electronic device, the photodiode has an i-V relation given by i = is [ exp ( : ) - 1] - ip , (18.3-1) as illustrated in Fig. 18.3-2. This is the usua] i-V relation of a p-njunction [see (16.1- 32)] with an added photocurrent -i p proportional to the photon flux.
764 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS <1> v 's v ip 1 + <1>=0 Ip <1»0 Figure 18.3-2 Generic photodiode and its i-V relation. There are three classical modes of photodiode operation: open-circuit (photo- voltaic), short-circuit, and reverse-biased (photoconductive). In the open-circuit mode (Fig. 18.3-3), the light generates electron-hole pairs in the depletion region. The additional electrons freed on the n side of the layer recombine with holes on the p side, and vice versa. The net result is an increase in the electric field, which produces a photovoltage across the device that increases with increasing photon flux <I>. This mode of operation is used, for example, in solar cells. The responsivity of a photovoltaic photodiode is measured in V /W rather than A/W. The short-circuit (V == 0) mode is illustrated in Fig. 18.3-4. The short-circuit current is simply the photocurrent ip. Finally, a photodiode may be operated in its reverse-biased or "photoconductive" mode, as shown in Fig. 18.3-5(a). If a series load resistor is inserted in the circuit, the operating conditions are those illustrated in Fig. 18.3-5(b). + <1> <1> o 0 0 is <1>=0 V is <1>=0 V ! ! -lpl <1>1 <1>1 -I ') p <1>2 <1>2 Figure 18.3-3 Photovoltaic operation of a photodiode. Figure 18.3-4 Short-circuit operation of a photodiode. Photodiodes are usually operated in the strongly reverse-biased mode for the fol- lowing reasons: . A strong reverse bias creates a strong electric field in the junction that increases the drift velocity of the carriers, thereby reducing transit time. . A strong reverse bias increases the width of the depletion layer, thereby reducing the junction capacitance and improving the response time. . The increased width of the depletion layer leads to a larger photosensitive area, making it easier to collect more light.
18.3 PHOTODIODES 765 <I> R L i VB VB -VB -VB V , V ! ! , , -VB/R L (a) (b) Figure 18.3-5 Reverse-biased operation of a photodiode (a) without a load resistor, and (b) with a load resistor. The operating point lies on the dashed line. B. The p-i-n Photodiode As a detector, the p-i-n photodiode has a number of advantages over the p-n pho- todiode. A p-i-n diode is a p-n junction with an intrinsic (usually lightly doped) layer sandwiched between the p and n layers (see Sec. 16.1E). It may be operated under the various bias conditions discussed in the preceding section. The energy-band diagram, charge distribution, and electric field distribution for a reverse-biased p-i-n diode are illustrated in Fig. 18.3-6. This structure serves to extend the width of the region supporting an electric field, in effect widening the depletion layer. ./ r n 0 r p Electron energy I. I. I .. I". i. ,. I lEe Ev X I Fixed-charge H i density _ ' Electric t field I \ I + I I I I I I I I I pin ) x ) x Figure 18.3-6 The p-i-n photodiode structure, energy-band diagram, charge distribution, and electric-field distribution. The device can be illuminated either perpendicularly to, or parallel to, the junction. Photodiodes with a p-i-n structure offer the following advantages: . Increasing the width of the depletion layer of the device (where the generated carriers can be transported by drift) increases the area available for capturing light. . Increasing the width of the depletion layer reduces the junction capacitance and thereby the RC time constant. On the other hand, the transit time increases with the width of the depletion layer. . Reducing the ratio between the diffusion length and the drift length of the device results in a greater proportion of the generated current being carried by the faster drift process.
766 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS The responsivity of two commercially available p-i-n photodiodes is compared with that of an ideal device (Il == 1) in Fig. 18.3-7. The maximum responsivity is at a wavelength that is shorter than the bandgap wavelength. This is because Si is an indirect-bandgap material. The photon-absorption transitions therefore typically take place from the valence-band to conduction-band states that typically lie well above the conduction-band edge (see Fig. 16.2-8). 1.0 0.2 Ideal Si photodiode 0.8 <X:: "-" :>.. . 0.6 .(/j s::: o 0.4 (1) 0::: o 0.5 1.0 Ag Wavelength Ao (/-Lm) Figure 18.3-7 Responsivity versus wavelength (pm) for ideal and com- mercially available silicon p-i-n pho- todiodes. The quantum efficiency of a carefully constructed antireflection- coated silicon device can approach unity. c. Heterostructures Heterostructure photodiodes, formed from two semiconductors of different bandgaps, can exhibit advantages over p-n junctions fabricated from a single material. A hetero- junction comprising a large-bandgap material (E g > hv), for example, can make use of its transparency to minimize optical absorption outside the depletion region. The large- bandgap material is then called a window layer. The use of different materials also offers devices with a great deal of flexibility. Several material systems are of particular interest (see Figs. 16.1-7 and 16.1-8): . AlxGal-xAs/GaAs (AIGaAs lattice matched to a GaAs substrate) is useful in the wavelength range 0.7 to 0.87 Mm. . InxGal-xAs/InP (InGaAs lattice matched to an InP substrate) can be composi- tionally tuned over the wavelength range 1300-] 600 nm, which is of interest for optical fiber communications (see Sec. 24.1D). A typical InGaAs p-i-n photode- tector operating at 1550 nm has a quantum efficiency Il 0.75 and a responsivity 9{ 0.9 A/W. . HgxCd 1 - x Te/CdTe is a material that is highly useful in the mid-infrared region of the spectrum. This is because HgTe and CdTe have nearly the same lattice parameter and can therefore be lattice matched at nearly all compositions. This material offers a compositionally tunable bandgap that operates in the wavelength range between 3 and 17 Mm. Applications include night vision, thermal imaging, and long-wavelength lightwave communications. . Quaternaries, such as Inl-xGaxAsl-yPy/InP and Gal_xAlxAsySb1_y/GaSb, which are useful over the range 0.92 to 1.7 Mm, are of interest because the fourth element provides an additional degree of freedom that allows lattice matching to be achieved for different compositionally determined values of Eg. Schottky-Barrier Photodiodes Metal-semiconductor photodiodes (also called Schottky-barrier photodiodes) are formed from metal-semiconductor heterojunctions. A thin semitransparent metallic film is used in place of the p-type (or n-type) layer in the p-n junction photodiode. The thin film is sometimes made of a metal-semiconductor alloy that behaves like a metal.
18.4 AVALANCHE PHOTODIODES 767 The Schottky-barrier structure and its energy-band diagram are shown schematically in Fig. 18.3-8. Ev Metal Semiconductor (b) (a) Figure 18.3-8 (a) Structure and (b) energy-band diagram of a Schottky-barrier photodiode formed by depositing a metal on an n-type semiconductor. These photodetectors are responsive to photon energies greater than the Schottky barrier height, hv > W - X. Schottky photodiodes can be fabricated from many materials, such as Au on n-type Si (which operates in the visible) and platinum silicide (PtSi) on p-type Si (which operates over a range of wavelengths stretching from the ultraviolet to the infrared). Schottky-barrier photodiodes are useful for a number of reasons: . Not all semiconductors can be prepared in both p-type and n-type forms; Schottky devices are of particular use in these material systems. . Semiconductors used for the detection of visible and ultraviolet light with a pho- ton energy well above the bandgap energy have a large absorption coefficient. This gives rise to substantial surface recombination and a reduction of quantum efficiency. The metal-semiconductor junction has a depletion layer present im- mediatel y at the surface, thus eliminating surface recombination. . The response speed of p-n and p-i-n junction photodiodes is in part limited by the slow diffusion current associated with photocarriers generated close to, but outside of, the depletion layer. One way of decreasing this unwanted absorption is to decrease the thickness of one of the junction layers. However, this should be achieved without substantially increasing the series resistance of the device because such an increase has the undesired effect of reducing the speed by increas- ing the RC time constant. The Schottky-barrier structure achieves this because of the low resistance of the metal. Furthermore, Schottky-barrier structures are majority-carrier devices and therefore have inherently fast responses and large operating bandwidths. Response times in the picosecond regime, corresponding to bandwidths 100 GHz, are readily available. Representative responsivity curves for several p-i-n and Schottky-barrier photodi- odes are displayed in Fig. 18.3-9. 18.4 AVALANCHE PHOTODIODES An avalanche photodiode (APD) operates by converting each detected photon into a cascade of moving carrier pairs. Weak light is then able to elicit a current that is sufficiently large so that it can be detected by the electronics following the APD. The device is configured as a strongly reverse-biased photodiode in which the junction electric field is large. The charge carriers can therefore acquire sufficient energy to excite new carriers by the process of impact ionization.
768 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS 1.0 --- < "-' , \ \ \ \ \ , , , .> . \/l s::: o 0.- \/l Q) 0.1 0.0 0.5 1.0 1.5 Wavelength,&o (/Lm) Figure 18.3-9 Responsivity 9{ versus wavelength Ao (/Lm) for a number of p-i-n (solid) and Schottky-barrier (dashed) photodiodes. For ternary and quaternary devices, the wavelength of maximum response depends on composition. Response times in the tens of ps, corresponding to bandwidths 50 GHz, are generally available. A. Principles of Operation The history of a typical electron-hole pair in the depletion region of an APD is depicted in Fig. 18.4-1. A photon is absorbed at point 1, creating an electron-hole pair (an electron in the conduction band and a hole in the valence band). The electron acceler- ates under the influence of the strong electric field, thereby increasing its energy with respect to the bottom of the conduction band. The acceleration process is constantly interrupted by random collisions with the lattice in which the electron loses some of its acquired energy. These competing processes cause the electron to reach an average saturation velocity. Should the electron be lucky and acquire an energy larger than E 9 at any time during the process, it has an opportunity to generate a second electron-hole pair by impact ionization (say at point 2). The two electrons then accelerate under the effect of the field, and each of them may be the source for a further impact ionization. The holes generated at points 1 and 2 also accelerate, moving toward the left. Each of these also has a chance of creating an impact ionization should they acquire sufficient energy, thereby generating a hole-initiated electron-hole pair (e.g., at point 3). >... ep (l) s:: (l) s:: o ;..... .... () (l) Ec Figure 18.4-1 Schematic represen- tation of the multiplication process in an APD. x
18.4 AVALANCHE PHOTODIODES 769 Ionization Coefficients The abilities of electrons and holes to impact ionize are characterized by the ionization coefficients a e and ah. These quantities represent ionization probabilities per unit length (cm- 1 ); the inverse coefficients 1/ a e and 1/ ah represent average distances be- tween consecutive ionizations. The ionization coefficients increase with the depletion- layer electric field (since it provides the acceleration) and decrease with increasing device temperature (since the increased frequency of collisions diminishes the opportu- nity a carrier has of gaining sufficient energy to ionize). The simplified theory presented below assumes that a e and ah are constants. However, it can be advantageous for the purposes of noise reduction to design devices in which the ionization coefficients depend on position and carrier history in particular ways, as discussed in Sec. 18.6B. An important parameter for characterizing the performance of an APD is the ion- ization ratio, which is defined as the ratio of the ionization coefficients, k == ah . a e (18.4-1 ) When holes do not ionize appreciably (i.e., when ah « a e so that k « 1), most of the ionization is achieved by electrons. The avalanching process then proceeds principally from left to right (i.e., from the p side to the n side of the device) in Fig. 18.4-1. It terminates some time later when all of the electrons arrive at the n side of the depletion layer. On the other hand, if electrons and holes both ionize appreciably (k 1), those holes moving to the left create electrons that move to the right, which in turn generate further holes moving to the left, in a possibly unending circulation. Although this feedback process increases the gain of the device (the total generated charge in the circuit per photocarrier pair q / e), it is nevertheless undesirable for several reasons: . It is time consuming and therefore reduces the device bandwidth . It is random and therefore increases the device noise . It can be unstable, thereby causing avalanche breakdown It is therefore desirable to fabricate APDs from materials that permit only one type of carrier (either electrons or holes) to impact ionize. If electrons have the higher ion- ization coefficient, for example, optimal behavior is achieved by injecting the electron of a photocarrier pair at the p-type edge of the depletion layer and by using a material whose value of k is as small as possible. If holes are injected, the hole of a photocarrier pair should be injected at the n-type edge of the depletion layer and k should be as large as possible. The ideal case of single-carrier multiplication is achieved when k == 0 or 00. Design As with any photodiode, the geometry of an APD should maximize photon absorption, for example by taking the form of a p-i-n structure. On the other hand, the multi- plication region should be thin to minimize the possibility of localized uncontrolled avalanches (instabilities or microplasmas) being produced by the strong electric field. Greater electric-field uniformity can be achieved in a thin region. These two conflicting requirements call for an APD design in which the absorption and multiplication regions are separate. Structures of this kind are known as separate- absorption-multiplication APD (SAM APD) devices. Their operation is most readily understood by considering a device with k 0 (e.g., Si). Photons are absorbed in a large intrinsic or lightly doped region. The photoelectrons drift across this region under the influence of a moderate electric field, and then enter a thin multiplication layer with a strong electric field where avalanching occurs. The reach-through APD structure illustrated in Fig. 18.4-2 accomplishes this. Photon absorption occurs in the
770 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS wide 7r region. Electrons drift through the 7r region into a thin p-n + junction, where they experience a sufficiently strong electric field to cause avalanching. The reverse- bias voltage applied across the device is large enough for the depletion layer to reach through the p and 7r regions into the p+ contact layer. ff' 7r m . cd rJj ..c: c u x u . t:: "'d - u . - \ J ) x Figure 18.4-2 Reach-through p+ -7f-p-n+ APD structure. The 7f region is very lightly doped p- type material The p + and n + regions are heavily doped. B. Gain and Responsivity As a prelude to determining the gain of an APD in which both kinds of carriers cause multiplication, we first consider the simpler problem of single-carrier (electron) multiplication (ah == 0, k == 0). Let Je(x) be the electric current density carried by electrons at location x, as shown in Fig. 18.4-3. Within a distance dx, on the average, the current is incremented by the factor dJe(x) == aeJe(x) dx, (18.4-2) from which we obtain the differential equation dJ e ( ) dx == aeJ e x , (18.4-3) whose solution is the exponential function J e (x) Je(w)j Je(O) is therefore Je(O) exp(aex). The gain G == G == exp(aew). (18.4-4) The electric current density increases exponentially with the product of the ionization coefficient a e and the multiplication layer width w. The result is similar to that for a laser amplifier [see (14.1- 7)]. The double-carrier multiplication problem requires knowledge of both the electron current density J e (x) and the hole current density J h (x). It is assumed that only elec- trons are injected into the multiplication region. Since hole ionizations also produce electrons, however, the growth of J e ( x) is governed by the differential equation = cxeJe(x) + cxhJh(X). (18.4-5)
18.4 AVALANCHE PHOTODIODES 771 Ji w ) I / / ./ ./ ./ " Ji x ) ,..'" ...'" --" ---- - ,. JiO) ,. I ° x w Figure 18.4-3 Exponential growth of the elec- tric current density in a single-carrier APD. As a result of charge neutrality, dJejdx == -dJhjdx, so that the sum Je(x) + Jh(x) must remain constant for all x under steady-state conditions. This is clear from Fig. 18.4-4; the total number of charge carriers crossing any plane is the same regardless of position. r I Injected C : i electron CD -c I 2 I I I I 3 I . !: .4 x @-- (I) Figure 18.4-4 Constancy of the sum of the electron and hole current densities across a plane at any x. By way of illustration, four impact ionizations and five electrons-plus-holes crossing every plane are illustrated. Since it is assumed that no holes are injected at x == W, J h (W) == 0, so that Je(x) + Jh(x) == Je(W), (18.4-6) as shown in Fig. 18.4-5. The hole current density Jh(x) can therefore be eliminated in (18.4-5) to obtain = (Ct e - Cth)Je(X) + CthJe(W). (18.4-7) This first-order differential equation is readily solved for the gain G == Je(w)j Je(O). For Qe -# Qh, the result is G == (Qe - Qh)j {Qe exp[-(Qe - Qh) w] - Qh}, from which we obtain 1-k G== exp [-(1 - k)Qew] - k . (18.4-8) APD Gain
772 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS The single-carrier multiplication result for the gain (18.4-4), with its exponential growth, is recovered when k == O. When k == 00, the gain remains unity since only electrons are injected and electrons do not multiply. For k == 1, (18.4-8) is indeterminate and the gain must be obtained directly from (18.4-7); the result is then G == 1 f (1 - D:e w). An instability is reached when D:e W == 1. The dependence of the gain on D:e W for several values of the ionization ratio k is illustrated in Fig. 18.4-6. The responsivity 9{ is obtained by using (18.4-8) in the general relation (18.1-8). JeC w ) --__ Jh(x) -- .............. ............ ..... ..... "- " , , Je(O) o W x Figure 18.4-5 Growth of the electron and hole currents as a result of avalanche multiplication. G k3=O.5 1 o 1 2 DeW Figure 18.4-6 Growth of the gain G with multiplication-layer width for several values of the ionization ratio k, assuming pure electron injection. The materials of interest are closely related to those used for p-i-n photodiodes (see Fig. 18.3-9), with the additional proviso that they should have the lowest (for electron injection) or highest (for hole injection) possible value of the ionization ratio k. Silicon APDs have ionization rates in the range k 0.1-0.2, but Si devices with k as low as 0.006 can be fabricated, providing excellent performance in the wavelength region 700-900 nm. InGaAs APDs are often used at telecommunications wavelengths (1300- 1600 nm), in spite of their higher values of k, because the t offer high responsivities (see Fig. 18.3-9) and moderate noise. Electric fields 10 V fern, corresponding to tens of volts across the device, initiate the avalanche mechanism. As the reverse-bias voltage increases, so too does the gain and dark current, as is evident in Fig. 18.4-7. EXAMPLE 18.4-1. Gain in an InGaAs APD. Because they provide internal gain, InGaAs avalanche photodiodes are widely used as photodetectors for optical fiber communication systems operating in the 1300-1600-nm telecommunications band (see Sec. 24.1D). These devices are generally operated between the punchthrough and breakdown voltages. Optimal gains are G 10 and typical dark currents are 10- 11 A. Devices fabricated from II-VI materials (e.g., HgCdTe) and IV-VI materials (e.g., PbSnTe) find use at longer wavelengths.
10- 4 10- 6 $ 10- 8 t: ::3 U 10- 10 10- 12 10 18.4 AVALANCHE PHOTODIODES 773 . : Punchthrough : voltage f r. I: I: 1 : Dark current I : /: ;I' · .;' Figure 18.4-7 Current-voltage characteristic for an InGaAs SAM APD. The device is operated at a reverse-bias voltage that lies between the punchthrough voltage and the breakdown voltage. .;' " --- -- Breakdown: voltage: 20 Reverse-bias voltage (V) 30 c. Response Time Aside from the usual transit, diffusion, and RC effects that govern the response time of photodiodes, APDs suffer from an additional multiplication time called the avalanche buildup time. The response time of a two-carrier-multiplication APD is illustrated in Fig. 18.4-8 by following the history of a photoelectron generated at the edge of the absorption region (point 1). The electron drifts with a saturation velocity V e , reaching the multiplication region (point 2) after a transit time W d/ v e . Within the multiplication region the electron also travels with a velocity v e . Through impact ionization it creates electron-hole pairs, say at points 3 and 4, generating two additional electron-hole pairs. The holes travel in the opposite direction with their saturation velocity v h. The holes can also cause impact ionizations resulting in electron-hole pairs as shown, for example, at points 5 and 6. The resulting carriers can themselves cause impact ionizations, sustaining the feedback loop. The process is terminated when the last hole leaves the multiplication region (at point 7) and crosses the drift region to point 8. The total time T required for the entire process (between points 1 and 8) is the sum of the transit times (from I to 2 and from 7 to 8) and the multiplication time denoted Tm, Wd Wd T == - + - + Tm. V e Vh (18.4-9) Because of the randomness of the multiplication process, the multiplication time T m is random. In the special case k == 0 (no hole multiplication) the maximum value of T m is readily seen from Fig. 18.4-8 to be W m w m Tm == - + - . v e Vh (18.4-10) For a large gain G, and for electron injection with 0 < k < 1, an order of magnitude of the average value of Tm is obtained by increasing the first term of (18.4-10) by the factor Gk, Tm Gkw m w m +-. v e Vh ( 18.4-11 ) A more accurate theory is rather complex.
774 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS hv Absorption region Multiplication region I ( W d -I ( wdl1 V e 1 unmmnnnm____mmm_ W m ---+I Hole current ih(t) Electron current ieCt) Tm x 3 4 5 6 7 3 4 5 6 7 T -I eVe W d + W m W d V h 1 -I 1- 8 eVil W d + W m t (a) (b) Figure 18.4-8 (a) Tracing the course of the avalanche buildup time in an APD with the help of a position-time graph. The blue lines represent electrons, and the green lines represent holes. Electrons move to the right with velocity v e and holes move to the left with velocity Vh. Electron-hole pairs are produced in the multiplication region. The carriers cease moving when they reach the edge of the material. (b) Hole current ih(t) and electron current ie(t) induced in the circuit. Each carrier pair induces a charge e in the circuit. The total induced charge q, which is the area under the ie(t) + ih(t) versus t curve, is Ge. This figure is a generalization of Fig. 18.1-6, which applies for a single electron- hole pair. EXAMPLE 18.4-2. Avalanche Buildup Time in a Silicon APD. Consider a Si APD with Wd == 50 pm, w 7n == 0.5 pm, v e == 10 7 cmls, Vh == 5 X 10 6 cmls, G == 100, and k == 0.1. Equation (18.4-10) yields T7n == 5 + 10 == 15 ps, whereupon (18.4-9) gives T == 1020 ps == 1.02 ns. On the other hand, (18.4-11) yields T7n == 60 ps, so that (18.4-9) provides T == 1065 ps == 1.07 ns. For ap-i-n photodiode with the same values of Wd, V e , and Vh, the transit time is Wd/Ve + Wd/Vh 1 ns. These results do not differ greatly because T 7n is quite low in a silicon device. D. Single-Photon Avalanche Diodes (SPADs) The ability to detect and count individual photons is important in many applications (e.g., imaging, satellite laser ranging, and deep-space laser communications). The use of photon counting mitigates against gain noise and circuit noise because the detector response is binarized. Photon counting can be achieved by using a single-photon avalanche photodetector (SPAD), also known as a Geiger-mode avalanche pho- todiode. This device is an APD biased in such a way that the arrival of a single photon precipitates avalanche breakdown, thereby creating a large current pulse that signifies the arrival of a photon. Each current pulse must be quenched to prepare for the arrival of a subsequent photon. This may be carried out either by passive or active means; the latter approach is more complex but provides a substantially greater maximum photon detection rate. Silicon SPADs operate in the visible and near-infrared spectral regions (.Ao == 400-
18.5 ARRAY DETECTORS 775 1000 nm) and offer high efficiency (Il 75%), low dark-count rates ( 75 counts/s), and sub-nanosecond timing resolution ( 100 ps). In the optical fiber communications band (.Ao == 1300-1600 nm), InGaAsjlnP heterostructures are the devices of choice, but performance is far less impressive than at shorter wavelengths: typical parameters are It 20%, dark count rate 5000 counts/s, timing resolution 500 ps. Ge and Si-Ge are also occasionally used in this region. At yet longer wavelengths (.Ao < 4 Mm), devices relying on an InAsSb absorption layer together with an AIGaAsSb multiplication layer, on a GaSb substrate, have been used. Devices fabricated from GaN and SiC have found use in the ultraviolet. SiC has the particular merit that it can tolerate high temperatures and hostile environments. In all cases, SPADs are subject to a tradeoff between efficiency and bandwidth. Photon counting can also be achieved by making use of superconducting single- photon detectors (SSPDs), which are broadband, low-noise, and fast although they require cooling. The arrival of a photon locally creates a nonsuperconducting hotspot that gives rise to a response signaling the occurrence of an event. 18.5 ARRAY DETECTORS An individual photodetector registers the photon flux striking it as a function of time. In contrast, an array containing a large number of photodetectors can simultaneously register the photon fluxes (as functions of time) from many spatial points. Array detectors therefore permit electronic versions of optical images to be formed. One type of array detector, the microchannel plate [see Fig. 18.1-2(c)], has already been discussed. Modern microelectronics technology permits the fabrication of many types of ar- rays. These contain large numbers of photo detector elements, known as pixels, that are made of photoconductors, photodiodes, avalanche photodiodes, or thermal detectors such as bolometers. A 2D array of photosensitive elements designed to record an electronic version of an image at the focal plane of an imaging system is known as a focal-plane array (FPA). In a hybrid focal-plane array, the signal collection and processing circuitry lies in a layer directly beneath the array of photosensitive elements. Two principal forms of readout circuitry are used to transport the photodetector signals: charge-coupled device (CCD) technology and complementary metal-oxide- semiconductor (CMOS) technology. Materials and Structures Array detectors take many forms, as indicated by the following examples: . Microbolometer arrays are often used in thermal imaging cameras. Incident pho- tons cause an increase in the temperature of the illuminated elements; the ac- companying change in resistance is recorded by external circuitry. These devices operate at ambient temperature and have come to the fore in recent years as their resolution and sensitivity have improved dramatically. Vanadium oxide (VOx) microbolometer arrays offer hundreds of thousands of pixels, each 25 Mm in size, and are sensitive in the mid-infrared region. These devices find extensive use in military and commercial applications. . Photoconductive arrays are typically used in the mid-infrared region. A photon whose energy is greater than the bandgap energy in a semiconductor such as InSb or HgCdTe creates an electron-hole pair that contributes to the conductivity of the material. . Arrays of extrinsic semiconductors, such as Ge:Ga, are useful for making pho- toconductive FPAs that are sensitive in the far-infrared. A photon places a donor
776 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS electron into the conduction band (or a receptor hole into the valence band)., so that it contributes to the conductivity. . Quantum-well infrared photodetectors (QWIPs) are used in megapixel focal- plane arrays. A photon provides sufficient energy to lift an electron out of a quantum well so that it contributes to the conductivity. Long-wavelength infrared (LWIR) and mid-wavelength infrared (MWIR) images are provided by GaAs/AIGaAs and GaAs/InGaAs/AIGaAs elements, respectively. Narrowband spectral filters can be used to provide hundreds of wavelength bands. . Arrays fabricated from compound-semiconductor p-i-n photodiodes, such as InGaAs and HgCdTe, are used in the visible and infrared. A photon whose energy is greater than the bandgap energy creates an electron-hole pair that contributes to the diode current. . Schottky-barrier photodiode elements fabricated from metal-semiconductor junc- tions are used in highly versatile FPA cameras. A photon whose energy is greater than the Schottky barrier creates an electron-hole pair that contributes to the diode current. PtSi can be used for imaging in many spectral regions since it is sensitive to a broad band of wavelengths stretching from the near-ultraviolet to about 6 /-Lm in the mid-infrared. In spite of the fact that it has low quantum efficiency in the infrared, PtSi is widely used since it is easily manufactured and highly stable. . Avalanche-photodiode detectors fabricated from p-n junctions with multiplica- tion regions have been crafted into array detectors. A photon whose energy is greater than the bandgap energy creates an electron-hole pair that enters a high- field semiconductor region providing gain. The resulting sub-nanosecond electri- cal pulse has an amplitude of several volts, which is sufficient to directly trigger the digital CMOS circuit, thereby obviating the need for analog-to-digital conver- SIon. . Single-photon avalanche detectors (SPADs) fabricated from reverse-biased p-n junctions make use of multiplication regions operated in the Geiger mode. A photon whose energy is greater than the bandgap energy creates an electron-hole pair that enters the high-field semiconductor region, thereby causing avalanche breakdown and the concomitant generation of a large current pulse. . Photosensitive arrays can also be operated as heterodyne detectors; conversion gain is provided by a local oscillator (see Sec. 24.5). Readout Circuitry Two principal forms of readout circuitry are used to transport the photodetector sig- nals to the camera display or output: charge-coupled device (CCD) technology and complementary metal-oxide-semiconductor (CMOS) technology: 1. CCD technology. A CCD operates by transferring the charge produced by a particular detector element to a buried CCD channel at a specified time. The charge is then sequentially transferred, via this channel, from one detector posi- tion to another until it is transported to one corner of the chip, where it is read out. Many electrode structures and clocking schemes have been developed for periodically reading out the charge accumulated by each element and generating the electronic data stream that represents the image. 2. CMOS technology. Complementary metal-oxide-semiconductor (CMOS) is a widely used manufacturing technology for fabricating electronic devices and integrated circuits. Because it consumes little power and is relatively inexpensive, this technology has spurred the mass production of FPAs. Each element in the detector is linked to several metal-oxide-semiconductor field- effect transistors (MOSFETs) that amplify and read out the detector signal. Unlike the sequential read-out used in CCDs, the detector elements in a CMOS
18.6 NOISE IN PHOTODETECTORS 777 array are individually read out. 18.6 NOISE IN PHOTODETECTORS The photodetector is responsive to photon flux (or optical power). In accordance with (18.1-4), a photon flux <I> (optical power P == hv<I» gives rise to a proportional electric current ip == It e<I> == P. However, in actuality the electric current generated in the device is a random quantity i, whose value fluctuates above and below its average value, z ip == It e<I> == P. The fluctuations of i, generally regarded as noise, are characterized by the standard deviation of the current ai, where a; == (( i - z) 2 ). For a current of zero mean (z == 0), the standard deviation reduces to the root-mean-square (rms) value of the current, ai == (i 2 ) 1/2 . A number of sources of noise are inherent in the process of photon detection: . Photon Noise. The most fundamental source of noise is associated with the ran- dom arrivals of the photons themselves, which are usually described by Poisson statistics, as discussed in Sec. 12.2. . Photoelectron Noise. In a photon detector with quantum efficiency It < 1, a single photon generates a photoelectron-hole pair with probability It and fails to do so with probability 1 - It. Because the photocarrier-generation process is random, it is a source of noise. . Gain Noise. The amplification process that provides internal gain in certain pho- todetectors, such as photoconductors and APDs, is stochastic. Each detected pho- ton generates a random number of carriers G, with an average value G. The gain fluctuations depend on the nature of the amplification mechanism. . Receiver Circuit Noise. Various components in the electrical circuitry of an optical receiver, such as resistors and transistors, contribute to receiver circuit noise. These four sources of noise are illustrated schematically in Fig. 18.6-1. The mean signal entering the detector (input optical signal) has an associated intrinsic photon noise. The photoeffect converts the photons into photoelectrons. In the process, the mean signal decreases by the factor It (the quantum efficiency). The associated photo- electron noise also decreases, but by a lesser amount than the signal; thus the signal-to- noise ratio of the photoelectron signal is lower than that of the incident photon signal. Circuit noise contributes to the detected signal. If a photo detector gain mechanism is present, it amplifies both the photoelectron signal and noise. Moreover, it introduces its own gain noise. Finally, circuit noise enters at the point of current collection. As a component in an information transmission system, an optical receiver can be characterized by the following performance measures: . The signal-to-noise ratio (SNR) of a random variable is defined as the ratio of its square-mean to its variance. Thus, the SNR of the current i is SNR == Z2 / a;, while the SNR of the photon number is SNR == n 2 / a; . . The minimum-detectable signal is defined as the mean signal that yields unity SNR. . The excess noise factor F of a random variable is defined as the ratio of its mean- square to its square-mean. Thus, the excess noise factor of the photodetector gain GisF== (G 2 )/(G)2. . The bit error rate (BER) is defined as the probability of error per bit in a digital optical receiver. . The receiver sensitivity is defined as the signal that corresponds to a prescribed value of the signal-to-noise ratio, SNR == SNRo. While the minimum-detectable
778 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Photon r ?iS , \. .. Photoelectron Circuit nOise \ / nOise , . ,.. .. ... ,. :';. .... ..:: .:.": : ":.. . ->* . ,¥-.... : Photon nOise Gain noise ,., I. i' < .. , .. Input optical signal Detected signal Input optical signal Gain Photoeffect and current collection Detected signal Photoeffect t Current collection (a) (b) Figure 18.6-1 Input and detected signals along with various sources of noise for (a) a photodetector without gain, such as a p-i-n photodiode; and (b) a photodetector with gain such as an avalanche photodiode. signal corresponds to a receiver sensitivity that provides SNR o == 1, a higher value of SNR o is often specified to ensure a given level of accuracy (e.g., SNR o == 10-10 3 corresponding to 10-30 dB). For a digital system, the receiver sensitivity is defined as the minimum optical energy (or corresponding mean number of photons) per bit required to achieve a prescribed bit error rate, which is often set at BER == 10- 9 . We begin by deriving expressions for the signal-to-noise ratio for optical detectors with these four sources of noise. Other sources of noise that we do not explicitly con- sider include background noise and dark-current noise. Background noise is photon noise associated with light from extraneous optical sources that reach the detector (these include sources other than the signal of interest, such as sunlight and starlight). Background noise is particularly deleterious in detection systems that operate in the mid- and far-infrared spectral regions because of the copious thermal radiation emitted at these wavelengths by objects at room temperature (see Fig. 13.4-4). Photodetectors also generate dark-current noise, which, as the name implies, is present even in the absence of light. Dark-current noise results from random electron-hole pairs generated thermally or by tunneling. Also ignored are leakage currents and 1/ f noise. A. Photoelectron Noise Photon Noise As described in Sec. 12.2, the photon flux associated with a fixed optical power P is inherently uncertain. The mean photon flux is <I> == P / hv (photons/s), but this quantity fluctuates randomly in accordance with a probability law that depends on the nature of the light source. The number of photons n counted in a time interval T is thus random with mean n == <I> T. For light from an ideal laser, or from a thermal source of spectral width much greater than 1/ T, the photon number obeys the Poisson probability distribution, for which cr; == n . Hence, the fluctuations associated with an average of 100 photons result in an actual number of photons that lies approximately within the range 100 ::!: 10.
18.6 NOISE IN PHOTODETECTORS 779 The photon-number signal-to-noise ratio SNR == n 2 / a; is therefore SNR == n , ( 18 .6-1 ) Photon-Number Signal-to-Noise Ratio and the minimum-detectable photon number is n == 1 photon. If the observation time T == 1 J1s and the wavelength AD == 1.24 J1m, this is equivalent to a minimum- detectable power of 0.16 pW. The receiver sensitivity for SNR o == 10 3 (30 dB) is 1000 photons. If the time interval T == 10 ns, this is equivalent to a sensitivity of 1011 photons/s or an optical power sensitivity of 16 n W at AD == 1.24 J1m. Photoelectron Noise A photon incident on a photodetector of quantum efficiency It generates a photoevent (i.e., creates a photoelectron-hole pair or liberates a photoelectron) with probability It, or fails to do so with probability 1 - It. Photoevents are assumed to be selected at random from the photon stream. An incident mean photon flux <I> (photons/s) therefore results in a mean photoelectron flux It <I> (photoelectrons/s). The number of photoelec- trons m detected in the time interval T is a random variable with mean m==Itn, (18.6-2) where n == <I> T is the mean number of incident photons in the same time interval T. If the photon number is distributed in Poisson fashion, so too is the photoelectron number, as can be ascertained by using a parallel argument to that developed in Sec. 12.2D. It follows that the photoelectron-number variance is equal to m , so that 2 - - am == m == Itn. (18.6-3) It is clear that the photoelectron noise is not additive with the photon noise. The underlying randomness inherent in the photon number, which constitutes a fundamental source of noise with which we must contend when using light to transmit a signal, therefore gives rise to a photoelectron-number signal-to-noise ratio SNR == m == It n . (18.6-4 ) Photoelectron-Number Signal-to-Noise Ratio The minimum-detectable photoelectron number is m == It n == 1 photoelectron, corre- sponding to l/It photons. The receiver sensitivity for SNR o == 10 3 is 1000 photoelec- trons or 1000/It photons. Photocurrent Noise We now examine the properties of the electric current i(t) induced in a circuit by a random photoelectron flux with mean It <I>. The treatment we provide includes the effects of photon noise, photoelectron noise, and the characteristic time response of the detector and circuitry (filtering). Every photoelectron-hole pair generates a pulse of electric current with charge (area) e and time duration Tp in the external circuit of the photodetector (Fig. 18.6-2). A photon stream incident on a photo detector therefore results in a stream of current pulses which add together to constitute the photocurrent i(t). The randomness of the photon stream is transformed into a fluctuating electric
780 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS current. If the incident photons are Poisson distributed, these fluctuations are known as shot noise. More generally, for detectors with gain C, the generated charge in each pulse is q == Ce. Photons .. .., , .".. , I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I . ., . , . ... . ) I I I I I I I I I I t Tp A: eae I I I I I I I I I I : :: : : : ::: : t I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 1 I I I I I I I - --' , " , : I I I I I I I I I I t . . ) Photoelectrons Current pulses Electric current (shot noise) i(t) ) t Figure 18.6-2 The photocurrent induced in a photodetector circuit comprises a superposition of current pulses, each associated with a detected photon. The individual pulses illustrated are exponentially decaying step functions but they can assume an arbitrary shape (see, e.g., Figs. 18.1- 6(b) and 18.1-7). Before providing an analytical derivation of the properties of the photocurrent i(t), we first consider the problem from a simplified perspective. Consider a photon flux <P incident on a photoelectric detector of quantum efficiency It. Let the random number m of photoelectrons counted within a characteristic time interval T == 1/2B (the resolution time of the circuit) generate a photo current i(t), where t is the instant of time immediately following the interval T. For rectangular current pulses of duration T, the current and photoelectron-number random variables are related by i == (e/ T)m. The photo current mean and variance are therefore given by e_ z==-m T 2 ( e ) 2 2 a i == T am' (18.6-5) (18.6-6) where m == Il <P T == Il <P /2B is the mean number of photoelectrons collected in the time interval T == 1/ 2B. Substituting a == m for the Poisson law yields the photocurrent mean and variance: z == ell <P (18.6-7) Photocurrent Mean a; == 2e'iB. (18.6-8) Photocurrent Variance It follows that the signal-to-noise ratio of the photoelectric current, SNR == Z2 / a;, is SNR == == Il <P == m . 2eB 2B ( 18.6-9) Photocurrent Signal-to-Noise Ratio
18.6 NOISE IN PHOTODETECTORS 781 The current SNR is directly proportional to the photon flux <P and inversely propor- tional to the electrical bandwidth of the circuit B. The result is identical to that for the photoelectron-number signal-to-noise ratio m , as expected, since the circuit introduces no added randomness. EXAMPLE 18.6-1. SNR and Receiver Sensitivity. For 'i == 10 nA and B == 100 MHz, CTi 0.57 nA, corresponding to a signal-to-noise ratio SNR == 310 or 25 dB. An average of 310 photoelectrons are detected in every time interval T == 1/2B == 5 ns. The minimum-detectable photon flux is <I> == 2B lIt, and the receiver sensitivity for SNR o == 10 3 is <I> == 1000 . (2B lIt) == 2 x 1011 lIt photons/so D Derivation of the Photocurrent Mean and Variance. We now proceed to prove (18.6-7) and (18.6-8) in the general case. Assume that a photoevent generated at t == 0 produces an electric pulse h(t), of area e, in the external circuit. A photoevent generated at time t 1 then produces a displaced pulse, h(t - t 1 ). Divide the time axis into incremental time intervals flt so that the probability p that a photoevent occurs within an interval is P == It <I>flt. The electric current i at time t is written as i(t) == L Xl h(t - ltlt), l ( 18.6-10) where Xi assumes the value 1 with probability p, and 0 with probability 1 - p. The variables {Xl} are independent. The mean value of Xl is 0 x (1 - p) + 1 x p == p. Its mean-square value is (Xl) == 0 2 x (1 - p) + 1 2 x P == p. The mean of the product XlX k is p2 if l i- k, and p if l == k. The mean and mean-square values of i(t) are now determined via 'i == (i) == LP h(t - ltlt) l ( 18.6-11 ) (i 2 ) == L L (XlX k ) h(t - ltlt) h(t - ktlt) l k == L Lp2 h(t -ltlt) h(t - kflt) + LPh 2 (t -ltlt). l=j:k l ( 18.6-12) Substituting p == It <I> tlt, and taking the limit tlt -+ 0 so that the summations become integrals, (18.6-11) and (18.6-12) yield, respectively, 1: = Il <f> 1= h(t) dt = ell <f> (i 2 ) = (ell <f>? + Il <f> 1=h 2 (t) dt. ( 18.6-13) ( 18.6-14 ) It follows that aT = {i 2 ) - (i? = Il <f> 1= h 2 (t) dt. (18.6-15) Defining 1 1 00 frOO h 2 (t) dt B == - h 2 (t) dt == 0 2e 2 0 2 [Jo oo h ( t) dt] 2 , we finally obtain (18.6-7) and (18.6-8). (18.6-16) . The parameter B defined by (18.6-16) represents the device/circuit bandwidth. This is readily verified by noting that the Fourier transform of h( t) is its transfer function
782 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS H(v). The area under h(t) is simply H(O) == e. In accordance with Parseval's theorem [see (A.I-7)], the area under h 2 (t) is equal to the area under the symmetric function I H ( v) 1 2 , so that roo H(v) 2 B = Jo H(O) dv. (18.6-17) The quantity B is therefore the power-equivalent spectral width of the function IH(v)1 (Le., the bandwidth of the device/circuit combination), in accordance with (A.2-10). As an example, if H(v) == 1 for -V c < v < V c and 0 elsewhere, (18.6-17) yields B == V c . These relations are applicable for all photoelectric detection devices without gain (e.g., phototubes and junction photodiodes). Use of the formulas requires knowledge of the bandwidth of the device, biasing circuit, and amplifier; B is determined by inserting the transfer function of the overall system into (18.6-17). B. Gain Noise The photocurrent mean and variance for a device with fixed (deterministic) gain G is determined by replacing e with q == Ge in (18.6-7) and (18.6-8), which leads to eGIl P "i == eGIl q> == hv a; == 2eG"iB == 2e 2 G 2 Il Bq>. (18.6-18) (18.6-19) The signal-to-noise ratio, in accordance with (18.6-9), becomes "i Ilq>- SNR == == - == m. 2eG B 2B (18.6-20) It is independent of G because the deterministic gain introduces no additional random- ness; the mean current "i and its RMS value ai are both multiplied by the same factor G. Photoelectrons . . . . . ) t G 3 G s G 1 G 2 Randomly G 4 multiplied photoelectrons ... '"t Figure 18.6-3 Each photoevent in a photodetector with gain generates a random number G l of carriers, each of which gives rise to an electrical current pulse of area eG l. The total electric current in the detector circuit i( t) is the superposition of these pulses. Electric current i(t) . I(t) ! { , t This simple result does not apply when the gain itself is random, as is the case in a photomultiplier tube, photoconductor, and avalanche photodiode. The derivation of the
18.6 NOISE IN PHOTODETECTORS 783 photocurrent mean and variance given in the previous section must then be modified. In particular, the electric current (18.6-10) should then be written as i(t) == L Xl G l h(t - lt), l ( 18.6- 21 ) where, as before, Xl takes the value 1 with probability p == Il q>t, and 0 with probabil- ity 1 - p. Included now are the independent random numbers G l representing the gain imparted to a photocarrier generated in the lth time slot, as illustrated in Fig. 18.6-3. If the random variable G l has mean value (G) == G , and a mean-square value (G 2 ), an analysis similar to that set forth in (18.6-10)-(18.6-17) yields z == eGIl q> (18.6-22) Photocu rrent Mean (Random Gain) 2 -- a. == 2eGzBF 'I, , ( 18.6-23) Photocurrent Variance (Random Gain) where the excess noise factor F is defined as (G 2 ) F = (G)2 . (18.6-24) Excess Noise Factor The excess noise factor is related to the variance of the gain ab by F == 1 + ab / (G) 2 . In the special case of deterministic gain, ab == 0 and F == 1, whereupon (18.6-23) reduces to (18.6-19). When the gain is random, ab > 0 and F > 1; both of these quantities increase with the severity of the gain fluctuations. The resulting electric current i then exhibits fluctuations that are greater than those of shot noise. In the presence of random gain, the current signal-to-noise ratio Z2 / a; becomes (18.6-25) Signal-to-Noise Ratio (Random Gain) where m is the mean number of photoelectrons collected in the time T == 1/ 2B. This is smaller than the deterministic-gain SNR by the factor F; the reduction is associated with the randomness of the gain. SNR == 2eG B F z Il <P /2B F m F' EXAMPLE 18.6-2. Excess Noise Factor for a Photomultiplier Tube. A photomultiplier tube operates on the basis of electron multiplication, via secondary electron emission at its dynodes. For a typical device, the gain randomness associated with this process yields an excess noise factor F 1.2. Since F = 1 + a/(G)2, the gain SNR = l/(F - 1) 5. If the PMT has a mean gain G = 10 8 , the standard deviation of the gain fluctuations is ac = 10 8 / V5.
784 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Excess Noise Factor for an APD When photoelectrons are injected at the edge of a uniform multiplication region in a conventional APD, the gain G of the device is given by (18.4-8). It depends on the electron ionization coefficient a e and the ionization ratio k == ah/ ae, as well as on the width of the multiplication region w. The use of a similar (but more complex) analysis, incorporating the randomness associated with the gain process, leads to an expression for the mean-square gain (G 2 ), and therefore for the excess noise factor F in (18.6- 24). This more general derivation gives rise to an expression for the mean gain G that is identical to that given in (18.4-8). Calculations shown that the excess noise factor F is then related to the mean gain and ionization ratio by F = kG + (1 - k) (2 - ) . (18.6-26) Excess Noise Factor (Conventional APD) A plot of this formula is presented in Fig. 18.6-4 with k as a parameter. 1000 k. S 100 ....... u (l) (/J .0 I (/J (/J (l) u >< 10 2 1 1 10 100 1000 - Mean gain G Figure 18.6-4 Excess noise factor F for a conventional APD with a uniform multiplication region, under electron injection, as a function of the mean gain G , for different values of the ionization ratio k. For hole injection, l/k replaces k. Equation (18.6-26) is valid when electrons are injected at the edge of the multiplication region, but both electrons and holes have the capacity to initiate impact ionizations. If only holes are injected, the same expression applies, provided that k is replaced by l/k. Gain noise is minimized by injecting the carrier with the higher ionization coefficient, and by fabricating a structure with the lowest possible value of k if electrons are injected, or the highest possible value of k if holes are injected. Thus, the ionization coefficients for the two carriers should be as different as possible. Equation (18.6- 26) is said to be valid under conditions of single-carrier-initiated double-carrier multiplication since both types of carrier have the capacity to impact ionize, even when only one type is injected. If electrons and holes are injected simultaneously, the overall result is the sum of the two partial results.
18.6 NOISE IN PHOTODETECTORS 785 The gain noise introduced by a conventional APD arises from two sources: the randomness in the locations at which ionizations occur, and the feedback process associated with the fact that both kinds of carrier can produce impact ionizations. The first of these sources of noise is present even when only one kind of carrier can multiply; it gives rise to a minimum excess noise factor F == 2 at large values of the - - mean gain G, as is apparent by setting k == 0 and letting G become large in (18.6-26). The second source of noise, the feedback process, is potentially more detrimental since it can result in a far larger increase in F. EXAMPLE 18.6-3. Excess Noise Factor for a Silicon APD. A Si APD that makes use of electron injection has a mean gain G == 100 and an ionization ratio k == 0.1. Equation (18.6-26) yields F == 11.8 so that the mean value of the detected current is increased by a factor of 100, while the signal-to-noise ratio is reduced by a factor of 11.8. In the presence of circuit noise, however, the use of an APD can serve to increase the overall SNR, as will be shown subsequently. APDs with History-Dependent Ionization Coefficients A newly generated carrier can cause an impact ionization only after traveling a suf- ficient distance through the multiplication region so that it can accumulate sufficient energy from the field. This distance is called the dead space. The ionization coeffi- cients are therefore not truly independent of location and carrier history, as assumed in the theory for the conventional APD. The dead space serves to organize the locations at which impact ionizations can occur. This in turn enhances the orderliness of the carrier- generation process and leads to a reduction in gain noise. This is particularly true when the multiplication region is very thin (w < 400 nm) and the number of multiplications is small. A further reduction in the noise can be achieved if the carrier energy is suitably controlled: . Initial-energy effects. Carriers traversing an appropriately designed field gradient before entering the multiplication region can gamer substantial kinetic energy, thereby reducing the initial dead space in the multiplication region and further regularizing the impact ionizations. . Impact-ionization threshold-energy effects. A device can be designed such that a carrier traversing the multiplication region encounters a sudden change in the ionization threshold energy as it crosses from a layer of one material into a layer of another. A carrier with insufficient energy in the first layer can result in an ionization when it enters the second layer. The localization of impact ionizations in such specially designed multilayer struc- tures yields devices with high gain, low noise, and low dark current. An example of an energy-band diagram for such a device, under reverse-bias conditions, is displayed in Fig. ] 8.6-5. Two thin multiplication layers, with relatively low threshold energy, surround a layer with higher threshold energy. Impact ionization is enhanced at the edges of the twin multiplication layers and is suppressed in the central region, which imparts energy to the carriers in transit. The materials are chosen so that hole-induced ionizations are discouraged. A theory of APD noise that accommodates these effects has been developed. t It takes the form of recurrence relations for the first and second moments, and the prob- ability distribution, of the numbers of electrons and holes. These random variables are t See M. M. Hayat, G.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, Boundary Effects on Multiplication Noise in Thin Heterostructure Avalanche Photodiodes: Theory and Experiment, IEEE Transactions on Electron Devices, vol. 49, pp. 2114-2123, 2002.
786 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS hv Figure 18.6-5 Energy-band diagram of a low-noise heterostructure APD under reverse- bias condi ti ons. deterministically related to the random gain. The recurrence relations are formulated in such a way as to accommodate dead space, as well as initial-energy and impact- ionization threshold-energy effects. Numerical solutions provide the mean gain and excess noise factor for arbitrary values of dead space and multiplication-region width. The theory properly predicts the measurements reported in Example] 8.6-4. EXAMPLE 18.6-4. Excess Noise Factor for a Very Thin GaAs APD. A thin heterostruc- ture APD similar to that displayed in Fig. 18.6-5 has a multiplication region comprising two 50-nm layers of GaAs surrounding an 85-nm layer of Alo.6Gao.4As. The measured excess noise factor is F 2.5 at a mean gain of G = 20. The theoretical excess oise factor predicted by (18.6-26) for a bulk GaAs homojunction APD (k 0.75) is F 15.5 at G = 20 (see Fig. 18.6-4). The measured noisiness of the heterostructure device is thus substantially lower than that predicted by the bulk theory, which ignores dead space as well as initial-energy and impact-ionization threshold-energy effects. These effects are evidently important for reducing gain noise and must be accommodated when modeling thin-multiplication-region APDs. Other heterostructure configu rat ions, such as a centered-well configuration, can exhibit even lower values of F at small values of G. C. Circuit Noise Yet additional noise is introduced by the electronic circuitry associated with an optical receiver. Circuit noise results from the thermal motion of charged carriers in resistors and other dissipative elements (thermal noise), and from fluctuations of charge carriers in transistors used in the receiver amplifier. Thermal Noise Thermal noise (also called Johnson noise or Nyquist noise) arises from the random motions of mobile carriers in resistive electrical materials at finite temperatures; these motions give rise to a random electric current i(t) even in the absence of an external electrical power source. The thermal electric current in a resistance R is a random function i(t) whose mean value (i(t)) == O. The variance of the current a}, which is the same as the mean-square value since the mean vanishes, increases with the temperature T. Using an argument based on statistical mechanics, which is presented in the next section, it can be shown that a resistance R at temperature T exhibits a random electric current i (t) characterized by a power spectral density (see Sec. 11.1 B) 4 hf Si(f) = Rexp(hf/kT) -1 ' (18.6-27) where f is the frequency. In the region f « kT / h, which is of principal interest since
18.6 NOISE IN PHOTODETECTORS 787 kT / h == 6.24 THz at room temperature, exp (h f / kT) 1 + h f / kT so that Si (f) 4kT / R. (18.6-28) The variance of the electric current is the integral of the power spectral density over all frequencies within the bandwidth B of the circuit, i.e., (JT = l B Si(J) df. (18.6-29) For B « kT / h, we obtain a-; 4kTB/R. (18.6-30) Thermal Noise Current Variance (Resistance R) Thus, as shown in Fig. 18.6-6, a resistor R at temperature T in a circuit of bandwidth B behaves as a noiseless resistor in parallel with a source of noise current with zero mean and an RMS value (Ji determined by (18.6-30). R (J". I Figure 18.6-6 A resistance R at temperature T is equivalent to a noiseless resistor in parallel with a noise current source with variance a; = (i 2 ) 4kT B / R, where B is the circuit bandwidth. EXAMPLE 18.6-5. Thermal Noise in a Resistor. A l-kO resistor at T = 300 0 K, in a circuit of bandwidth B = 100 MHz, exhibits an RMS thermal noise current ai 41 nA. D *Derivation of the Power Spectral Density of Thermal Noise. We derive (18.6-27) by demon- strating that the electrical power associated with the thermal noise in a resistance is identical to the electromagnetic power radiated by a one-dimensional blackbody. The factor h f / [exp ( h f / kT) - 1] in (18.6-27) is recognized as the mean energy E of an electromagnetic mode of frequency f (the symbol v is reserved for optical frequencies) in thermal equilibrium at temperature T [see (13.4-8)]. Equation (18.6-27) may therefore be written as Si(f)R = 4E. The electrical power dissipated by a noise current i passing through a resistance R is (i 2 )R = a; R, so that Si(f)R represents the electrical power density (per Hz) dissipa ted by the noise current i(t) through R. We now proceed to demonstrate that 4E is the power density radiated by a one-dimensional black- body. As discussed in Sec. 13.4B, an atomic system in thermal equilibrium with the electromagnetic modes in a cavity radiates a spectral energy density (}(v) = M(v) E , where M(v) = 87rV 2 /C 3 is the three-dimensional density of modes, and the spectral intensity density is C{}( v). Although the charge carriers in a resistor move in all directions, only motion in the direction of the circuit current flow contributes. The density of modes in a single dimension is(f) 4/c modes/m-Hz [see (10.1-10)] so that the corresponding energy density is (}(f) = M(f)E = 4E/c and the radiated power density is c{}(f) = 4 E as promised. .
788 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Circuit-Noise Parameter: Resistance-Limited and Amplifier-Limited Optical Receivers It is convenient to lump the various sources of circuit noise (thermal noise in resistors as well as noise in transistors and other circuit devices) into a single random current source i r at the receiver input that produces the same total noise at the receiver output (Fig. 18.6-7). The mean value of i r is zero while its variance a; depends on tempera- ture, receiver bandwidth, circuit parameters, and device type. <I> <I> Noiseless circuit Noisy circuit Figure 18.6-7 A noisy receiver circuit can be replaced by a noiseless receiver circuit and a single random current source with RMS value a r at its input. Furthermore, it is convenient to define a dimensionless circuit-noise parameter a r T a r a - - q - - 2Be ' ( 18.6- 3 1 ) where B is the receiver bandwidth and T == 1/2B is the receiver resolution time. Since a r is the RMS value of the noise current, a r / e is the RMS electron flux (electrons/s) arising from circuit noise, and a q == (a r / e ) T therefore represents the RMS number of circuit-noise electrons collected in the time T. The circuit-noise parameter a q is a figure of merit that characterizes the quality of the optical receiver circuit, as will become apparent in Sec. 18.6D. An optical receiver comprising a photodiode in series with a load resistor R L , followed by an amplifier, is illustrated in Fig. 18.6-8. This simple receiver is said to be resistance limited if the circuit-noise current arising from thermal noise in the load resistor substantially exceeds contributions from other sources of noise. The amplifier may then be regarded as noiseless and the circuit-noise mean-square current is simply a; == 4kT B / R L . The circuit-noise parameter defined by (18.6-31) is therefore {kT (Jq=V ' (18.6-32) which is inversely proportional to the square-root of the bandwidth B. EXAMPLE 18.6-6. Circuit-Noise Parameter. At room temperature, a resistance R L = 50 0 in a circuit of bandwidth B = 100 MHz generates a random current of RMS value a r = 0.18 /-LA. This corresponds to a circuit-noise parameter a q 5700. A receiver using a well-designed low-noise amplifier can yield a lower circuit- noise parameter than a resistance-limited receiver. Consider a receiver using an FET
18.6 NOISE IN PHOTODETECTORS 789 To noiseless amplifier R L Figure 18.6-8 Resistance-limited optical re- ceIver. amplifier. If the noise arising from the high input resistance of the amplifier can be neglected, the receiver is limited by thermal noise in the channel between the FET source and drain. With the use of an equalizer to boost the high frequencies attenuated by the capacitive input impedance of the circuit, the circuit-noise parameter at room temperature, for typical circuit component values, turns out to be VB (J"q 100 (B in Hz) . (18.6-33) Circuit-Noise Parameter (FET Amplifier Receiver) For example, if B == 100 MHz, then (J" q == 100. This is significantly smaller than the circuit-noise parameter associated with a 50-0 resistance-limited amplifier of the same bandwidth. The circuit-noise parameter (J"q increases with B because of the effect of the equalizer. t A receiver that makes use of a bipolar transistor amplifier, on the other hand, has a circuit-noise parameter (J"q that is independent of the bandwidth B over a wide range of frequencies. For bandwidths between 100 MHz and 2 GHz, (J"q is typically 500, provided that appropriate transistors are used and that they are optimally biased. D. Signal-to-Noise Ratio and Receiver Sensitivity The simplest measure of quality of reception is the signal-to-noise ratio. The SNR of the current at the input to the noiseless circuit represented in Fig. 18.6-7 is the ratio of the square of the mean current to the sum of the variances of the constituent sources of nOIse: -2 'l SNR == - 2eG"iBF + (J"; ( eGIl <P ) 2 -2 . 2e 2 G IlB<pF + (J"; (18.6-34 ) Signal-to-Noise Ratio for an Optical Receiver The first term in the denominators represents photoelectron and gain noise [see (18.6- 23)], whereas the second term represents circuit noise. For a detector without gain, G == 1 and F == 1. The noiseless circuit does not alter the signal-to-noise ratio even if it provides amplification. t For further details, see S. D. Personick, Optical Fiber Transmission Systems, Plenum, 1981, Sec. 3.4; note that the parameter a q is equivalent to Z /2 in this reference.
790 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS EXERCISE 18.6-1 Signal-to-Noise Ratio of the Resistance-Limited Optical Receiver. Assume that the opti- cal receiver shown in Fig. 18.6-8 makes use of an ideal p-i-n photodiode (q = 1) and the resistance R L = 50 0 at T = 300 0 K. The bandwidth is B = 100 MHz. At what value of photon flux <I> is the photoelectron-noise current variance equal to the resistor thermal-noise current variance? What is the corresponding optical power at Ao = 1550 nrn? It is useful to write the SNR in (18.6-34) in terms of the mean number of detected photons m in the resolution time of the receiver T == 1/ 2B, - ffi T Il<P m == Il 'J! == 2B ' (18.6-35) and the circuit noise-parameter (J"q == (J"r/2Be. The resulting expression is C 2 m 2 SNR == -2 C F m + (J" (18.6-36) Signal-to-Noise Ratio for an Optical Receiver Equation (18.6-36) has a simple interpretation. The numerator is the square of the mean number of multiplied photoelectrons detected in the receiver resolution time T == 1/2B. The denominator is the sum of the variances of the number of photoelectrons and the number of circuit-noise electrons collected in T. For a photodiode without gain C == F == 1, whereupon (18.6-36) reduces to -2 m SNR == - 2 . m+(J"q (18.6-37) Signal-to-Noise Ratio (Optical Receiver in Absence of Gain) The relative magnitudes of m and (J" determine the relative importance of photoelec- tron noise and circuit noise. The manner in which the parameter (J" q characterizes the circuit's performance as an optical receiver is now apparent. For example, if (J"q == 100, then circuit noise dominates photoelectron noise provided that the mean number of photoelectrons recorded per resolution time lies below 10000. We proceed now to examine the dependence of the SNR on photon flux <P, circuit bandwidth B, receiver circuit-noise parameter (J"q, and gain C. This will allow us to determine when the use of an APD is beneficial and will permit us to select an appropriate optical preamplifier for a given photon flux. In undertaking this parametric study, we rely on the expressions for the SNR provided in (18.6-34), (18.6-36), and (18.6-37). Dependence of the SNR on Photon Flux The dependence of the SNR on m == Il <P /2B provides an indication of how the SNR varies with the photon flux <P. Consider first a photodiode without gain, in which case (18.6-37) applies. Two limiting cases are of interest:
18.6 NOISE IN PHOTODETECTORS 791 1. Circuit-noise limit. If <P is sufficiently small, such that m « (J" (<P « 2B(J" /f}J, the photon noise is negligible and circuit noise dominates, yielding -2 m SNR 2 . (J"q (18.6-38) 2. Photon-noise limit. If the photon flux <P is sufficiently large, such that m » (J" (<1> » 2B(J"/Il), the circuit-noise term can be neglected, whereupon SNR m . (18.6-39) For small m , therefore, the SNR is proportional to m 2 and thereby to <p 2 , whereas for large m it is proportional to m and thereby to <P, as illustrated in Fig. 18.6-9. For all levels of light the SNR increases with increasing incident photon flux <P; more light improves receiver performance. SNR 10 5 m Figure 18.6-9 Signal-to-noise ratio (SNR) as a function of the mean number of photo- electrons per receiver resolution time, m == 11. <I> /2B, for a photodiode at two values of the circuit-noise parameter a q. 10 3 10 10 When the Use of an APD Provides an Advantage We now compare two receivers that are identical in all respects except that one exhibits no gain, while the other exhibits gain G together with an excess noise factor F (e.g., an APD). For sufficiently small m (or photon flux <P), circuit noise dominates. Amplifying the photocurrent above the level of the circuit noise should then improve the SNR. The APD receiver would then be superior. For sufficiently large m (or photon flux), circuit noise is negligible. Amplifying the photocurrent then introduces gain noise, thereby reducing the SNR. The photodiode receiver would then be superior. Comparing (18.6- 36) and (18.6-37) shows that the SNR of the APD receiver is greater than that of the photodiode receiver when m < cr(1-1/ G 2) / (F -1). For G » 1, the APD provides an advantage when m < (J" / (F - 1). If this condition is not satisfied, the use of an APD compromises, rather than enhances, receiver performance. When (J" q is very small, for example, it is evident from (18.6-36) that the APD SNR == m / F is inferior to the photodiode SNR == m . The SNR is plotted as a function of m for the two receivers in Fig. 18.6-10. Dependence of the SNR on APD Gain The use of an APD is beneficial for a sufficiently small photon flux, m < (J" / (F - 1). The optimal gain of the APD is determined by making use of (18.6-36): -2 SNR = -2 G m G F + (J"/ m (18.6-40)
792 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS SNR 10 5 10 3 10 10 m Figure 18.6-10 SNR versus m == 11 <I> /2B for a photodiode receiver (so lid curve) and for an APD receiver with mean gain G == 100 and excess noise factor F == 2 (dashed curve) obtained from (18.6- 36). The circuit-noise parameter (J" q == 100 in both cases. For small photon flux (circuit-noise-limited case), the APD yields a higher SNR than the photodiode. For large photon flux (photon-noise limited case), the photodiode receiver is superior to the APD receiver. The transition between the two regions occurs at m (J"/(F - 1) == 10 4 . The excess noise factor F is itself a function of G, as is clear from (18.6-26) for a thick APD. Substitution yields G 2 m SNR == -3 -2 - , kG + (1 - k)(2G - G) + a/ m (18.6-41) where k is the APD carrier ionization ratio. This expression is plotted in Fig. 18.6-11 for m == 1000 and a q == 500. For the single-carrier multiplication APD (k == 0), the SNR increases with gain and eventually saturates. For the double-carrier multiplication APD (k > 0), the SNR also increases with increasing gain, but it reaches a maximum at an optimal value of the gain, beyond which it decreases as a result of the sharp increase in gain noise. In general, there is thus an optimal choice of APD gain. SNR 10 3 10 2 10 1 1 kJ=O 10 3 G Figure 18.6-11 De pen dence of the SNR on the APD mean gain G for different ioniza- tion ratios k when m == 1000 and (J"q == 500. 10 10 2 Dependence of the SNR on Receiver Bandwidth The relation between the SNR and the bandwidth B is implicit in (18.6-34). It is governed by the dependence of the circuit-noise current variance a; on B. Consider three receivers: 1. The resistance-limited receiver exhibits a; ex B [see (18.6-30)] so that SNR ex 1/ B. (18.6-42) 2. The FET amplifier receiver obeys a q ex B 1 / 2 [see (18.6-33)] so that aT == 2eBa q ex B3/2. This indicates that the dependence of the SNR on B in (18.6-34)
18.6 NOISE IN PHOTODETECTORS 793 assumes the form SNR ex l/(B + sB 3 ), (18.6-43) where s is a constant. 3. The bipolar-transistor amplifier has a circuit-noise parameter a q that is approx- imately independent of B. Thus, a r ex B, so that (18.6-34) take the form SNR ex l/(B + s'B 2 ), (18.6-44 ) where s' is a constant. These relations are illustrated schematically in Fig. 18.6-12. The SNR always de- creases with increasing b. For sufficiently small bandwidths, all three receivers exhibit an SNR that varies as 1/ B. For large bandwidths, the SNR of the FET and bipolar transistor-amplifier receivers declines more sharply with bandwidth. SNR B. JPol ar t ral1Sistor Figure 18.6-12 Double-logarithmic plot of the dependence of SNR on bandwidth B for three types of receivers. B Receiver Sensitivity The receiver sensitivity is the minimum photon flux <1>0, with its corresponding op- tical power Po == hv<1>o and corresponding mean number of photoelectrons m o == Il fPo/2B, required to achieve a prescribed value of signal-to-noise ratio SNRo. The quantity m o can be determined by solving (18.6-36) for SNR == SNRo. We shall consider only the unity-gain receiver, leaving the more general solution as an exercise. Solving the quadratic equation (18.6-37) for m o, we obtain m o = [SNR o + J SNR?o + 4(J SNRo ] . (18.6-45) Two limiting cases emerge: Photon-noise limit (a « SNRo): m 0 == SN R o Circuit-noise limit (a » SNRo): m o == J SNR o a q . (18.6-46) (18.6-47) Receiver Sensitivity
794 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS EXAMPLE 18.6-7. Receiver Sensitivity. We assume that SNR o == 10 4 , which corresponds to an acceptable signal-to-noise ratio of 40 dB. If the receiver circuit-noise parameter a q « 50, the receiver is photon-noise limited and its sensitivity is m 0 == 10000 photoelectrons per receiver resolution time. In the more likely situation for which a q » 50, the receiver sensitivity 100 a q. If a q == 500, for example, the sensitivity is m o == 50000, which corresponds to 2B m o == 10 5 B pho- toelectrons/s. The optical power sensitivity Po == 2B m ohv lIt == 10 5 Bhv lIt is directly proportional to the bandwidth. If B == 100 MHz and It == 0.8, then at Ao == 1550 nm the receiver sensitivity is Po 1.6 J-LW. When using (18.6-45) to determine the receiver sensitivity, it should be kept in mind that the circuit-noise parameter (J" q is, in general, a function of the bandwidth B, in accordance with: Resistance-limited receiver: (J" q ex 1/ VB FET amplifier: (J" q ex VB Bipolar-transistor amplifier: (J" q independent of B For these receivers, the sensitivity m 0 depends on bandwidth B as illustrated in Fig. 18.6-13. The optimal choice of receiver therefore depends in part on the bandwidth B. o IE >. +-> : . s:: Q) V) Bipolar transistor Figure 18.6-13 Double-logarithmic plot of receiver sensitivity mo (the minimum mean number of photoelectrons per reso- lution time T == 1/2B guaranteeing a minimum signal-to-noise ratio SNRo) as a function of bandwidth B for three types of receivers. The curves approach the photon- noise limit at values of B for which a « SNR o /4. In the photon-noise limit (i.e., when circuit noise is negligible), m o == SNR o in all cases. Photon-noise limit B EXERCISE 18.6-2 Sensitivity of the APD Receiver. Derive n expression analogous to (18.6-45) for the sensitiv- ity of a receiver incorporating an APD of gain G and excess noise factor F. Show that in the limit of negligible circuit noise, the receiver sensitivity reduces to m o == F . SNRo. ( 18.6-48)
18.6 NOISE IN PHOTODETECTORS 795 E. Bit Error Rate and Receiver Sensitivity The sensitivity of an analog receiver was defined in Sec. 18.6D as the minimum power of the received light (or the corresponding photon flux) necessary to achieve a pre- scribed signal-to-noise ratio SNRo. We now turn to the sensitivity of a digital com- munications receiver. For a binary ON-OFF keying system, the sensitivity is defined as the minimum optical energy (or the corresponding mean number of photons) per bit necessary to achieve a prescribed bit error rate (BER). We first determine the sensitivity of the ideal detector and then consider the effects of circuit noise and detector gain nOIse. Sensitivity of the Ideal Optical Receiver Assume that bits" 1" and "0" of an ON-OFF keying system are represented by the pres- ence and absence of optical energy, respectively, as described in Chapter 24. During bit "1" an average of n photons is received. During bit "0" no photons are received. If the two bits are equally likely, the overall average number of photons per bit is n a == ! n . Since the actual number of detected photons is random, errors in bit identificatIon occur. For light generated by laser diodes, the probability of detecting n photons obeys ] 0 ] ] 0 100 I 1 \ \ \ \ \ \ \ \ \ Transmitted bits Received photons I I I I e: e: e: e: I f e e e e . P::: CO e : I f I ];0] 001001: Reproduced bits (a) Error (b) 10- 9 o 10 ria Figure 18.6-14 (a) Schematic illustrating errors that result from randomness in the photon number. (b) Bit error rate (BER) versus mean number of photons per bit n a in an ON-OFF keying system with an ideal receiver. the Poisson distribution p( n) == n n exp ( - n ) / n! when an average of n photons has been transmitted (see Sec. 12.2). The receiver decides that" 1" has been transmitted if it detects one or more photons. The probability PI of mistaking" 1" for "0" is therefore equal to the probability of detecting no photons, i.e., PI == P (0) == exp( - n ). When bit "0" is transmitted, there are no photons; the receiver decides correctly that bit "0" has been transmitted, so that Po == O. The bit error rate is the average of the two error probabilities, BER == !(PI + Po), from which BER == ! exp( - n ) == ! exp( -2 n a). (18.6-49) Figure 18.6-14 portrays a semilogarithmic plot of this relation. The receiver sensitivity is defined as the average number of photons per bit required to achieve a certain value of the BER. In particular, for BER == 10- 9 , a value that is often chosen, (18.6-49) provides n a 10 photons per bit. We conclude that: The receiver sensitivity (for bit error rate BER == 10- 9 ) of an optical digital communication system using an ideal receiver is 10 photons per bit.
796 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS EXERCISE 18.6-3 Effect of Quantum Efficiency and Background Noise on Receiver Sensitivity. (a) Show that for a receiver using a detector with quantum efficiency Il, but that is otherwise ideal, B ER = exp ( - 2Il n a), so that the sensi ti vi ty is 11 a = 10/ Il photons per bit, corresponding to m a = Il 11 a = 10 photoelectrons per bit. (b) Assuming that bits "I" and "0" correspond to mean photon numbers n l = n + n B and n o = n B, where n is the mean number of signal photons and n B is the mean of a Poisson-distributed background photon flux that is independent of the signal, determine an expression for the BER as a function of n and n B. Plot the BER versus n a = n for several values of n B. Determine the receiver sensitivity n a as a function of n B from this plot. [Hint: The sum of two random numbers, each with a Poisson probability distribution, is also Poisson distributed.] The ideal receiver sensitivity of 10 photons per bit is applicable only for light with a Poisson photon-number distribution. The sensitivity can be improved, in principle, by the use of photon-number-squeezed light (see Sec. 12.3B). Sensitivity of a Receiver with Circuit Noise and Gain Noise As explained in Sec. 18.6A, a photodiode transforms an average fraction q of the received photons into photoelectron-hole pairs, each of which contributes a charge e to the electric current in the external circuit. The total charge accumulated in the bit time interval T is m (units of electrons). This number is random and has a Poisson distribution with mean m == q n and variance m . Additional noise is introduced by the photodiode circuit in the form of a random electric current iT of Gaussian probability distribution with zero mean and variance a; . Within the bit time interval T, the accumulated charge q == iT Tie (units of electrons) has an RMS value a q == aT Tie. The parameter a q , called the circuit-noise parameter, depends on the receiver bandwidth B as described in Sec. 18.6C. The total accumulated charge per bit s == m + q (units of electrons) is the sum of a Poisson random variable m and an independent Gaussian random variable q. Its mean is the sum of the means, M==m==qn, (18.6-50) while its variance is the sum of the variances, (J"2 == m + a;. ( 18.6- 51 ) For m sufficiently large, the Poisson distribution may be approximated by a Gaussian so that the overall distribution may be approximated by a Gaussian distribution with mean M and variance a 2 . We adopt this approximation in the present analysis. For an avalanche photodiode (APD) of gain G, the mean number of photoelectrons is amplified by a factor G but additional noise is introduced in the amplification pro- cess. The mean of the total collected charge per bit s (units of electrons) is M == mG (18.6-52) while the variance is a 2 == m G 2 F + a2 q , (18.6-53)
18.6 NOISE IN PHOTODETECTORS 797 where F == (G 2 ) / (G)2 is the excess-noise factor of the APD (see Sec. 18.6B). The receiver measures the charge s accumulated in each bit (by use of an integrator, for example) and compares it to a prescribed threshold {J. If s > {J, bit" 1" is selected; otherwise, bit "0" is selected. The probabilities of error PI and Pa are determined by examining two Gaussian probability distributions of s that have mean /-La == 0, variance a6 == a mean /-L I == m G, variance ai == m G 2 F + a for bit "0" for bit "I". (18.6-54) The probability Pa of mistaking "0" for "1" is the integral of a Gaussian probability distribution p( s) with mean /-La and variance a6 from s == {J to s == 00. The probability PI of mistaking" 1" for "0" is the integral of a Gaussian probability distribution with mean /-LI and variance af from s == -00 to s == {J. The threshold {J is selected such that the average probability of error, BER == ! (p a + PI), is minimized. This type of analysis is the basis of the conventional theory of binary detection in the presence of Gaussian noise. If /-La and a6, and /-L I and ai, are the means and variances associated with two Gaussian variables representing bits "0" and" 1", respectively, and if ao and al are much smaller than /-LI - /-La, the bit error rate for an optimal-threshold receiver is given approximately by BER ![1 - erf(Q/J2)]. ( 18.6-55) Here Q == /-LI - /-La aa + al ( 18.6-56) and the error function erf( z) is defined as 2 (Z erf(z) Vi Jo exp( _x 2 ) dx. (18.6-57) Since a BER of 10- 9 corresponds to Q 6, we have (18.6-58) Condition for BER == 10- 9 (Gaussian Approximation) Substituting (18.6-54) into (18.6-58), defining m a == ! m as the mean number of photoelectrons detected per bit, and carrying out a bit of algebra yields /-L 1 - /-La 6 ( a a + al). m a 18F + 6a q / G . (18.6-59) Equation (18.6-59) relates the receiver sensitivity, in terms of the mean number of pho- toelectrons per bit m a required to make the BER == 10- 9 , to the receiver parameters G, F, and a q . When the APD gain is sufficiently large such that 3GF » a q , the second (circuit- noise) term on the right-hand side of (18.6-59) is negligible, whereupon m a 18F. (18.6-60) APD Receiver Sensitivity (Absence of Circuit Noise)
798 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS Table 18.6-1 Typical sensitivities (mean number of photons per bit) of several optical receivers operating at bit rates in the range of 1 Mb/s to 2.5 Gb/s. Receiver Receiver Sensitivity (photons/bit) 10 125 215 500 6000 Photon-limited ideal detector Si APD Er 3 + -doped silica-fiber preamplifierjlnGaAs p-i-n photodiode InGaAs APD p-i-n photodiode According to these calculations, a receiver that has negligible circuit noise, and makes use of a photodiode with no gain (G == 1 and F == 1), exhibits a receiver sensitivity of m a == 18 photoelectrons per bit. This result differs from the 10 photoelectrons per bit established earlier for this ideal receiver (see Exercise 18.6-3). The reason for the discrepancy is that the use of the Gaussian distribution in place of the Poisson is inappropriate for these small count numbers. Typical sensitivities of several receivers are provided in Table 18.6-1. The actual values depend on the receiver circuit-noise parameter (J" q, which in turn depends on the bit rate Bo 1/ T. READING LIST Books See also the reading lists in Chapters 16 and 17. W. Becker, Advanced Time-Correlated Single Photon Counting Techniques, Springer-Verlag, 2005. M. S. Shur and A. Zukauskas, eds., UV Solid-State Light Emitters and Detectors, NATO Science Series II: Mathematics, Physics and Chemistry, Volume 144, Springer-Verlag, 2004. M. Johnson, Photodetection and Measurement: Maximizing Performance in Optical Systems, McGraw-Hill, 2003. G. R. asche, Optical Detection Theory for Laser Applications, Wiley, 2002. M.Henini and M. Razeghi, eds., Handbook of Infrared Detection Technologies, Elsevier, 2002. A. R. Jha, Infrared Technology: Applications to Electro-Optics, Photonic Devices, and Sensors, Wiley, 2000. S. B. Howell, Handbook of CCD Astronomy, Cambridge University Press, 2000. I. S. Glass, Handbook of Infrared Astronomy, Cambridge University Press, 2000. S. Donati, Photodetectors: Devices, Circuits, and Applications, Prentice Hall, ] 999. M. A. Trishenkov, Detection of Low-Level Optical Signals: Photodetectors, Focal Plane Arrays and Systems, Springer-Verlag, 1997. K. K. Choi, The Physics of Quantum Well Infrared Photodetectors, World Scientific, 1997. E. L. Dereniak and G. D. Boreman, Infrared Detectors and Systems, Wiley, 1996. R. H. Kingston, Optical Sources, Detectors, and Systems: Fundamentals and Applications, Academic Press, 1995. G. H. Rieke, Detection of Light: From the Ultraviolet to the Submillimeter, Cambridge University Press, 1994. M. O. Manasreh, Semiconductor Quantum Wells and Superlattices for Long- Wavelength Infrared Detectors, Artech, 1993. R. H. Bube, Photoelectronic Properties of Semiconductors, Cambridge University Press, paperback ed. 1992. N. V. Joshi, Photoconductivity: Art, Science, and Technology, Marcel Dekker, ] 990. J. D. Vincent, Fundamentals of Infrared Detector Operation and Testing, Wiley, 1990. P. N. J. Dennis, Photodetectors, Springer-Verlag, 1986.
READING LIST 799 A. van der Ziel, Noise in Solid State Devices and Circuits, Wiley, 1986. R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 22, Lightwave Communications Technology, W. T. Tsang, ed., Part D, Photodetectors, Academic Press, 1985. E. L. Dereniak and D. G. Crowe, Optical Radiation Detectors, Wiley, 1984. M. J. Buckingham, Noise in Electron Devices and Systems, Wiley, 1983. R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley, 1983. R. 1. Keyes, ed., Optical and Infrared Detectors, Volume 19, Topics in Applied Physics, Springer- Verlag, 2nd ed. 1980. B. E. A. Saleh, Photoelectron Statistics, Springer-Verlag, 1978. A. Rose, Concepts in Photoconductivity and Allied Problems, Wiley, 1963; Krieger, reissued 1978. R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 12, Infrared Detec- tors II, Academic Press, 1977. R. D. Hudson, Jr. and J. W. Hudson, eds., Benchmark Papers in Optics/2: Infrared Detectors, Dow- den, Hutchinson & Ross, 1975. R. K. Willardson and A. C. Beer, eds., Semiconductors and Semimetals, Volume 5, Infrared Detec- tors, Academic Press, 1970. A. H. Sommer, Photoemissive Materials: Preparation, Properties and Uses, Wiley, 1968; Krieger, reissued 1980. Articles Special issue on photodetectors, IEEE Lasers & Electro-Optics Society News, vol. 20, no. 5, 2006. D. A. Ramirez, M. M. Hayat, G. Karve, J. C. Campbell, S. N. Torres, B. E. A. Saleh, and M. C. Teich, Detection Efficiencies and Generalized Breakdown Probabilities for Nanosecond-Gated Near In- frared Single-Photon Avalanche Photodiodes, IEEE Journal of Quantum Electronics, vol. 42, pp. 137-145, 2006. P. L. Richards and C. R. McCreight, Infrared Detectors for Astrophysics, Physics Today, vol. 58, no. 2,pp.41-47, 2005. A. Rogalski, HgCdTe Infrared Detector Material: History, Status and Outlook, Reports on Progress in Physics, vol. 68, pp. 2267-2336, 2005. J. Piotrowski and A. Rogalski, Uncooled Long Wavelength Infrared Photon Detectors, Infrared Physics and Technology, vol. 46, pp. 115-131, 2004. A. Rogalski, ed., Selected Papers on Infrared Detectors: Developments, SPIE Optical Engineering Press (Milestone Series Volume 179), 2004. Issue on photodetectors and imaging, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 4, 2004. M. G. Kang, ed., Selected Papers on CCD and CMOS Imagers, SPIE Optical Engineering Press (Milestone Series Volume 177), 2003. B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Lan- ders, Geiger-Mode Avalanche Photodiodes for Three-Dimensional Imaging, Lincoln Laboratory Journal, vol. 13, no. 2,pp. 335-350, 2002. M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, Boundary Effects on Multiplication Noise in Thin Heterostructure Avalanche Photodiodes: Theory and Experiment, IEEE Transactions on Electron Devices, vol. 49, pp. 2114-2123, 2002. M. C. Teich and B. E. A. Saleh, Branching Processes in Quantum Electronics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, pp. 1450-1457,2000. P. Yuan, S. Wang, X. Sun, X. G. Zheng, A. L. Holmes, Jr., and J. C. Campbell, Avalanche Photo- diodes with an Impact-Ionization-Engineered Multiplication Region, IEEE Photonics Technology Letters, vol. 12, pp. 1370-1372,2000. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. A. Smith, ed., Selected Papers on Photon Counting Detectors, SPIE Optical Engineering Press (Mile- stone Series Volume 143), 1998. A. Rogalski, ed., Selected Papers on Semiconductor Infrared Detectors, SPIE Optical Engineering Press (Milestone Series Volume 66), 1992. N. V. Joshi, ed., Selected Papers on Photoconductivity, SPIE Optical Engineering Press (Milestone Series Volume 56), 1992.
800 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS M. M. Hayat, B. E. A. Saleh, and M. C. Teich, Effect of Dead Space on Gain and Noise of Double- Carrier-Multiplication Avalanche Photodiodes, IEEE Transactions on Electron Devices, vol. 39, pp. 546-552, 1992. M. C. Teich, K. Matsuo, and B. E. A. Saleh, Excess Noise Factors for Conventional and Superlat- tice Avalanche Photodiodes and Photomultiplier Tubes, IEEE Journal of Quantum Electronics, vol. QE-22, pp. 1184-1193, 1986. N. Sclar, Properties of Doped Silicon and Germanium Infrared Detectors, Progress in Quantum Electronics, vol. 9, pp. 149-257, 1984. R. Chin, N. Holonyak, Jr., G. E. Stillman, J. Y. Tang, and K. Hess, Impact Ionization in Multilayered Heterojunction Structures, Electronics Letters, vol. 16, pp. 467-469, 1980. P. R. Bratt, Impurity Germanium and Silicon Infrared Detectors, in Semiconductors and Semimetals, Volume 12, Infrared Detectors II, R. K. Willardson and A. C. Beer, eds., Academic Press, 1977, pp. 33-142. H. Melchior, Demodulation and Photodetection Techniques, in F. T. Arecchi and E. o. Schulz- Dubois, eds., Laser Handbook, Volume 1, North-Holland, 1972, pp. 725-835. W. E. Spicer and F. Wooten, Photoemission and Photomultipliers, Proceedings of the IEEE, vol. 51, pp. 1119-1126, 1963. W. Shockley and J. R. Pierce, A Theory of Noise for Electron Multipliers, Proceedings of the IRE, vol. 26, pp. 321-332,1938. PROBLEMS 18.1-1 Effect of Reflectance on Quantum Efficiency. Determine the factor 1- in the expression for the quantum efficiency, under normal and 45° incidence, for an unpolarized light beam incident from air onto Si, GaAs, and InSb (see Sec. 6.2 and Table 16.2-1). 18.1-2 Responsivity. Find the maximum responsivity of an ideal (unity quantum efficiency and unity gain) semiconductor photodetector made of (a) Si; (b) GaAs; (c) InSb. 18.1- 3 Transit Time. Referring to Fig. 18.1-6, assume that a photon generates an electron-hole pair at the position x == W /3, that v e == 3Vh (in semiconductors V e is generally larger than Vh), and that the carriers recombine at the contacts. For each carrier, find the magnitudes of the currents, i h and ie, and the durations of the currents, Th and Te. Express your results in terms of e, W, and v e . Verify that the total charge induced in the circuit is e. For v e == 6 X 10 7 cm/s and W == 10 11m, sketch the time course of the currents. 18.1-4 Current Response with Uniform Illumination. Consider a semiconductor material (as in Fig. 18.1-6) exposed to an impulse of light at t == 0 that generates N electron-hole pairs uniformly distributed between 0 and w. Let the electron and hole velocities in the material be v e and Vh, respectively. Show that the hole current can be written a { Nev Nevh . - t+ h(t) == w 2 w 0, w o<t<- - - Vh elsewhere, while the electron current is { Nev; Neve ie (t) == - w2 t + w 0, and that the total current is therefore w O<t<- - - v e elsewhere, { Ne [(Vh + v e ) - (v + v;)t] , i(t) == w N eVh [ _ ] 1 w t , w w O<t<- - - v e W W -<t<-. v e - - Vh The various currents are illustrated in Fig. 18.1-7. Verify that the electrons and holes each contribute a charge N e/2 to the external circuit so that the total charge generated is N e
PROBLEMS 801 * 18.1-5 Two-Photon Detectors. Consider a beam of photons of energy hv and photon flux density cjJ (photons/cm 2 -s) incident on a semiconductor detector with bandgap hv < E 9 < 2hv, such that one photon cannot provide sufficient energy to raise an electron from the valence band to the conduction band. Nevertheless, two photons can occasionally conspire to jointly give up their energy to the electron. Assume that the current density induced in such a detector is given by J p = (cjJ2, where ( is a constant. Show that the responsivity (AIW) is given by 91 = [(/(hco)2]AP / A for the two-photon detector, where P is the optical power and A is the detector area illuminated. Explain physically the proportionality to A6 and P / A. 18.2-1 Photoconductor Circuit. A photoconductive detector is often connected in series with a load resistor R and a DC voltage source V, and the voltage V p across the load resistor is measured. If the conductance of the detector is proportional to the optical power P, sketch the dependence of V p on P. Under what conditions is this dependence linear? 18.2-2 Photoconductivity. The concentration of charge carriers in a sample of intrinsic Si is nj = 1.5 x 10 10 cm- 3 and the recombination lifetime T = 10 j1S. If the material is illuminated with light, and an optical power density of 1 mW/cm 3 at Ao = 1 j1m is absorbed by the material, determine the percentage increase in its conductivity. The quantum efficiency It = 1 2. 18.3-1 Quantum Efficiency of a Photodiode Detector. For a particular p-i-n photodiode, a pulse of light containing 6 x 10 12 incident photons at wavelength Ao = 1550 nm gives rise to, on average, 2 x 10 12 electrons collected at the terminals of the device. Determine the quantum efficiency It and the responsivity 91 of the photodiode at this wavelength. 18.4-1 Quantum Efficiency of an APD. A conventional APD with gain G = 20 operates at a wavelength Ao = 1550 nm. If its responsivity at this wavelength is 91 = 12 A1W, calculate its quantum efficiency It. What is the photocurrent at the output of the device if a photon flux c]) = 10 10 photons/s, at this same wavelength, is incident on it? 18.4-2 Gain of an APD. Show that an APD with ionization ratio k = 1, such as germanium, has a gain given by G = 1/(1- D:e w), where D:e is the electron ionization coefficient and w is the width of the multiplication layer. [Note: Equation (18.4-8) does not give a proper answer for the gain when k = 1.] ] 8.5-] Excess Noise Factor for a Single-Carrier APD. Show that an APD with pure electron injection and no hole multiplication (k = 0) has an excess noise factor F 2 for all appreciable val ues of the gain. Use (18.4-8) to show that the mean gain is then G = exp( D:e w). Calculate the responsivity of an Si APD for photons with energy equal to the bandgap energy Eg, assuming that the quantum efficiency It = 0.8 and the gain G = 70. Find the excess noise factor for a double-carrier-multiplication Si APD when k = 0.01. Compare it with the value F 2 obtained in the single-carrier-multiplication limit. *18.5-2 Gain of a Multilayer APD. Use the Bernoulli probability law to show that the mean gain of a single-carrier-multiplication multilayer APD, such as that displayed in Fig. 18.6-5, is G = (1 + p)l, where P is the probability of impact ionization at each stage and 1 is the number of stages. Show that the result reduces to that of the conventional APD when P 0 and 1 00. *18.5-3 Excess Noise Factor for a One-Stage Photomultiplier Tube. Derive an expression for the excess noise factor F of a one-stage photomultiplier tube assuming that the number of secondary emission electrons per incident primary electron is Poisson distributed with mean 8. * 18.5-4 Excess Noise Factor for a Photoconductive Detector. The gain of a photoconductive detector was shown in Sec. 18.2 to be G = T / Te, where T is the electron-hole recombination lifetime and Te is the electron transit time across the sample. Actually, G is random because T can be thought of as random. Show that an exponential probability density function for the random recombination lifetime, P( T) = (l/T) exp( -T /T), results in an excess noise factor F = 2, confirming that photoconductor generation-recombination (GR) noise degrades the SNR by a factor of 2. 18.5-5 Bandwidth of an RC Circuit. Using the definition of bandwidth provided in (18.6-16), show that a circuit of impulse response function h(t) = efT) exp( -tiT) has a bandwidth B = 1/4T. What is the bandwidth of an RC circuit? Determine the thermal noise current for a resistance R = 1 kO at T = 300 0 K connected to a capacitance C = 5 pF. 18.5-6 Signal-to-Noise Ratio of an APD Receiver. By what factor does the signal-to-noise ratio of
802 CHAPTER 18 SEMICONDUCTOR PHOTON DETECTORS a receiver using an APD of mean gain G 100 change if the ionization ratio k is increased fr om k 0.1 to 0.2? Assume that circuit noise is negligible. Show that if the m ean gain G » 1 and » 2(1 k)/k, the SNR is approximately inversely proportional to G. 18.5-7 Noise in an APD Receiver. An optical re ce iver using an APD has the following parameters: quantum efficiency Il 0.8; mean gain G 100; ionization ratio k 0.5; load resistance R L 1 kO; bandwidth B 100 kHz; dark and leakage current 1 nA. An optical signal of power 10 nW at Ao 0.87 J-lm is received. Determine the RMS values of the different noise currents, and the SNR. Assume that the dark and leakage current has a noise variance that obeys the same law as photocurrent noise and that the receiver is resistance limited. 18.5-8 Optimal Gain in an APD. A receiver using a p-i-n photodiode has a ratio of circuit-noise variance to photoelectron-noise variance of 100. If an APD with ionization ratio k 0.2 is used instead, determine the optimal mean gain for maximizing the signal-to-noise ratio and the corresponding improvement in signal-to-noise ratio. 18.5-9 Receiver Sensitivity. Determine the receiver sensitivity (Le., optical power required to achieve a SNR 103) for a photodetector of quantum efficiency Il 0.8 at Ao 1300 nm in a circuit of bandwidth B 100 MHz when there is no circuit noise. The receiver measures the electric current i. 18.5-10 Noise Comparison of Three Photodetectors. Consider three photodetectors in series with a 50-0 load resistor at 77° K (liquid nitrogen temperature) that are to be used with a I-J-lm wavelength optical system that has a bandwidth of 1 GHz: (a) a p-i-n photodiode w ith quantum efficiency Il 0.9; (b) an APD with quantum efficiency Il 0.6, gain G 100, and ionization ratio k 0; (c) a 1 0-stage photomultiplier tube (PMT) wit h quantum efficiency Il 0.3, overall mean gain G 4 10, and overall gain variance a'b G 2 / 4. (a) For each detector, find the photocurrent SNR when the detector is illuminated by a photon flux of 10 10 S-l. (b) Which devices render the signal detectable? 18.5-11 Dependence of Receiver Sensitivity on Wavelength. The receiver sensitivity of an ideal receiver (with unity quantum efficiency and no circuit noise) operating at a wavelength 870 nm is 76 dBm. What is the sensitivity at 1300 nm if the receiver is operated at the same data rate? 18.5-12 Bit Error Rates. A quantum-limited p-i-n photodiode (no noise other than photon noise) of quantum efficiency Il 1 mistakes a present Ao 870 nm optical signal of power P (bit 1) for an absent signal (bit 0) with probability 10- 10 . What is the probability of error under each of the following new conditions? (a) The wavelength is Ao 1300 nm. (b) Original conditions, but now the power is doubled. (c) Original conditions, but the efficiency is now Il 0.5. (d) Original conditions but an ideal APD with Il 1 and gain G 100 (no gain noise) is used. (e) As in (d), but the APD has an excess noise factor F 2 instead. 18.5-13 Sensitivity of an AM Receiver. A detector with responsivity 91 ( ), bandwidth B, and negligible circuit noise measures a modulated optical power P(t) Po + Ps cos(21f ft) with f < B. If Po » Ps, derive an expression for the minimum modulation power Ps that is measurable with signal-to-noise ratio SNR o 30 dB. What is the effect of the background power Po on the minimum observable signal P s ? 18.5-14 Sensitivity of a Photon-Counting Receiver. A photodetector of quantum efficiency Il 0.5 counts photoelectrons received in successive time intervals of duration T 1 tIs. Determine the receiver sensitivity (mean number of photons required to achieve SNR 103) assuming a Poisson photon-number distribution. Assuming that the wavelength of the light is Ao 870 nm, what is the corresponding optical power? If this optical power is received, what is the probability that the detector registers zero counts? * 18.5-15 A Single-Dynode Photomultiplier Thbe. Consider a photomultiplier tube with quantum efficiency Il 1 and only one dynode. Incident on the cathode is light from a hypothetical photon source that gives rise to a probability of observing n photons in the counting time
PROBLEMS 803 T 1.3 ns, which is given by p(n) , n 0, 1 0, otherwise. When one electron strikes the dynode, either two or three secondary electrons are emitted and these proceed to the anode. The gain distribution P( G) is given by P(G) 1 - 3 ' 2 - 3 ' G G 2 3 0, otherwise. Thus, it is twice as likely that three electrons are produced as two. (a) Calculate the SNR of the input photon number and compare the result with that of a Poisson photon number of the same mean. (b) Find the probability distribution for the photoelectron number p ( m) and the SNR of the photoelectron number. (c) Find the mean gain (G) and the mean-square gain (G 2 ). (d) Find the excess noise factor F. (e) Find the mean anode current 'l in a circuit of bandwidth B 1/2T. (t) Find the responsivity of this photomultiplier tube if the wavelength of the light is Ao 1550 nm. (g) Explain why (18.6-23) for a; is not applicable. . .
CHAPTER 19 ACOUSTO-OPTICS 19.1 INTERACTION OF LIGHT AND SOUND A. Bragg Diffraction *B. Coupled-Wave Theory C. Bragg Diffraction of Beams 19.2 ACOUSTO-OPTIC DEVICES A. Modulators B. Scanners C. Space Switches D. Filters, Frequency Shifters, and Isolators *19.3 ACOUSTO-OPTICS OF ANISOTROPIC MEDIA 806 819 828 ..... Ii ., .. ;, . . , \' ,,' d .. Sir William Henry Bragg (1862-1942, left) and Sir William Lawrence Bragg (1890-1971, right), a father-and-son team, were awarded the Nobel Prize in 1915 for their studies of the diffraction of light from periodic structures, such as those created by sound. 804
The refractive index of an optical medium is altered by the presence of sound. Sound therefore modifies the effect of the medium on light; i.e., sound can control light (Fig. 19.0-1). Many useful devices make use of this acousto-optic effect; these include opti- cal modulators, switches, deflectors, filters, isolators, frequency shifters, and spectrum analyzers. Sound Light ----... ----... Figure 19.0-1 Sound modifies the effect of an optical medium on light. - Medium Sound is a dynamic strain involving molecular vibrations that take the form of waves which travel at a velocity characteristic of the medium (the velocity of sound). As an example, a harmonic plane wave of compressions and rarefactions in a gas is pictured in Fig. 19.0-2. In those regions where the medium is compressed, the density is higher and the refractive index is larger; where the medium is rarefied, its density and refractive index are smaller. In solids, sound involves vibrations of the molecules about their equilibrium positions, which alter the optical polarizability and consequently the refractive index. Rarefaction Compression xt x 1-, A T 1- A IT Figure 19.0-2 Variation of the refractive index accompanying a harmonic sound wave. The pattern has a period A, the wavelength of sound, and travels with the velocity of sound. Refractive index An acoustic wave creates a perturbation of the refractive index in the form of a wave. The medium becomes a dynamic graded-index medium - an inhomogeneous medium with a time-varying stratified refractive index. The theory of acousto-optics deals with the perturbation of the refractive index caused by sound, and with the propagation of light through this perturbed time-varying inhomogeneous medium. The propagation of light in static (as opposed to time-varying) inhomogeneous (graded-index) media was discussed at several points in Chapters 1 and 2 (Sec. 1.3 and Sec. 2.4C). Since optical frequencies are much greater than acoustic frequencies, the variations of the refractive index in a medium perturbed by sound are usually very slow in comparison with an optical period. There are therefore two significantly different time scales for light and sound. As a consequence, it is possible to use an adiabatic approach in which the optical propagation problem is solved separately at every instant of time during the relatively slow course of the acoustic cycle, always treating the material as if it were a static (frozen) inhomogeneous medium. In this quasi-stationary 805
806 CHAPTER 19 ACOUSTO-OPTICS approximation, acousto-optics b"ecomes the optics of an inhomogeneous medium (usu- ally periodic) that is controlled by sound. The simplest form of interaction of light and sound is the partial reflection of an optical plane wave from the stratified parallel planes representing the refractive-index variations created by an acoustic plane wave (Fig. 19.0-3). A set of parallel reflectors separated by the wavelength of sound A will reflect light if the angle of incidence () satisfies the Bragg condition for constructive interference, . () A SIn == 2A ' ( 19.0-1 ) Bragg Condition where A is the wavelength of light in the medium (see Exercise 2.5-3). This form of light-sound interaction is known as Bragg diffraction, Bragg reflection, or Bragg scattering. The device that effects it is known as a Bragg reflector, a Bragg deflector, or a Bragg cell. xt Diffracted light Incident light 8 I ":':"-r \8 z A T Sound Transmitted light Figure 19.0-3 Bragg diffraction: an acoustic plane wave acts as a partial reflector of light (a beamsplitter) when the angle of incidence () satisfies the Bragg condition. This Chapter Bragg cells have found numerous applications in photonics. This chapter is devoted to their properties. In Sec. 19.1, a simple theory of the optics of Bragg reflectors is presented for linear, nondispersive media. Anisotropic properties of the medium and the polarized nature of light and sound are ignored. Although the theory is based on wave optics, a simple quantum interpretation of the results is provided. In Sec. 19.2, the use of Bragg cells for light modulation and scanning is discussed. Section 19.3 provides a brief introduction to anisotropic and polarization effects in acousto-optics. 19.1 INTERACTION OF LIGHT AND SOUND The effect of a scalar acoustic wave on a scalar optical wave is described in this section. We first consider optical and acoustic plane waves, and subsequently examine the interaction of optical and acoustic beams. A. Bragg Diffraction Consider an acoustic plane wave traveling in the x direction in a medium with velocity v s , frequency f, and wavelength A == v s / f. The strain (relative displacement) at
19.1 INTERACTION OF LIGHT AND SOUND 807 position x and time t is s x, t So cas flt qx, (19.1-1) where So is the amplitude, fl 27T f is the angular frequency, and q wavenumber. The acoustic intensity (W m 2 ) is 27T A is the I 1 3 S 2 s "2[!V s 0' (19.1-2) where [! is the mass density of the medium. The medium is assumed to be optically transparent and the refractive index in the absence of sound is n. The strain s x, t creates a proportional perturbation of the refractive index, analogous to the Pockels effect in (20.1-4), 1 3 "2pn s x, t , (19.1-3) /}.n x, t where P is a phenomenological dimensionless coefficient known as the photoelastic constant (or strain-optic coefficient). The minus sign indicates that positive strain (dilation) leads to a reduction of the refractive index. As a consequence, the medium has a time-varying inhomogeneous refractive index in the form of a wave n x, t n /}. no cas flt qx, (19.1-4) with amplitude A 1 3 8 uno "2pn o. (19.1-5) Substituting from (19.1-2) into (19.1-5), we find that the change of the refractive index is proportional to the square root of the acoustic intensity, /}. no (19.1-6) where ]v( p 2 n 6 [!V (19.1-7) is a material parameter representing the effectiveness of sound in altering the refractive index. The quantity M is a figure of merit for the strength of the acousto-optic effect in the material. EXAMPLE 19.1-1. Acousto-optic Effect Figure of Merit. In extra-dense flint glass {} 6.3x 10 3 kg/m 3 , V s 3.1 km/s, n 1.92, p 0.25, sothatM 1.67x 10- 14 m 2 /W. An acoustic wave of intensity 10 W /cm 2 creates a refractive-index wave of amplitude no 2.89 x 10- 5 .
808 CHAPTER 19 ACOUSTO-OPTICS Consider now an optical plane wave traveling in this medium with frequency V, angular frequency w 27rv, free-space wavelength Ao Co v, wavelength in the unperturbed medium A Ao n corresponding to a wavenumber k nw Co, and wavevector k lying in the x z plane and making an angle e with the z axis, as illus- trated in Fig. 19.1-1. x q L 2 o k () . () k r .-........... -;. ..... -JI r-' - .... -..-- . _ U_&"_ _ _ I I -..... Ir 'I I i.... . ...- L ..... - . - .. t"........ - -.. . 1. I: - -, I .::. ...- -r 'I I') - """' . , . _ L. , r - - 'L.1 'I ..'1 ____ - ..I .... -:- _ 'I n . . - - - "J - ....... . - .I - ., :..,- _ - _ r .. - ",'. - _- L r L z .... _. u L - .....-.- - '" . L - _ ...... _ - ,. --""""'---8 M - ..... . T L - - 2 _ _ .J'o_ _ ..l._ r__. _ .....,__. . . I......... .............. Linrf) Ltf,inrf) Figure 19.1-1 Reflections from layers of an inhomogeneous medium. Because the acoustic frequency f is typically much smaller than the optical fre- quency v (by at least five orders of magnitude), an adiabatic approach for studying light sound interaction may be adopted: We regard the refractive index as a static "frozen" sinusoidal function nx n no cos qx cp, (19.1-8) where cp is a fixed phase; we determine the reflected light from this inhomogeneous (graded-index) medium and track its slow variation with time by taking cp 0.t. To determine the amplitude of the reflected wave we divide the medium into in- cremental planar layers orthogonal to the x axis. The incident optical plane wave is partially reflected at each layer because of the refractive-index change. We assume that the reflectance is sufficiently small so that the transmitted light from one layer approximately maintains its original magnitude (i.e., is not depleted) as it penetrates through the following layers of the medium. If r dr dx x is the incremental complex amplitude reflectance of a layer of incremental width x at position x, the total complex amplitude reflectance for an overall length L (see Fig. 19.1-1) is the sum of all incremental reflectances, r L/2 - L /2 dx (19.1-9) The phase factor e j2kx sin () is included since the reflected wave at a position x is ad- vanced by a distance 2x sin 0, corresponding to a phase shift 2kx sin 0, relative to the reflected wave at x O. The wave numbers for the incident and reflected waves are taken to be the same. Using (19.1-8), we write dr dx dr dn dn dx dr cp, (19.1-10) where the derivative dr dn, which may be obtained from the Fresnel equations of reflection as will be shown later, is not dependent on x. We now substitute (19.1- 10) into (19.1-9), and use complex notation to write sin qx cp ej(qx-<p)
19.1 INTERACTION OF LIGHT AND SOUND 809 e-j(qx-<p) 2j, thereby obtaining 1£ 2 e j (2k sin 8-q)x dx _1£ 2 1.£ 2 e j (2k sin 8+q)x dx, (19.1-11) -1.£ 2 r where ro 1 dr (19.1-12) Performing the integrals in (19.1-11) and substituting rp nt, we obtain r r++r_, (19.1-13) where - - :!:jro sinc SIn =F q e:1::jOt \ )27r - - (19.1-14) Amplitude Reflectance r:t and sinc x sin 7rX 7rX. For reasons to become clear shortly, the terms r + and r _ are called the upshifted and downshifted reflections, respectively. The upshifted reflectance r + has its maxi- mum value when 2k sin () q, whereas the downshifted reflection is maximum when 2k sin () q. If L is sufficiently large, these maxima are sharp, so that any slight de- viation from the angles () :!: sin -1 q 2k makes the corresponding term negligible. Thus, only one of these two terms may be significant at a time, depending on the angle (). We first consider the upshifted condition, 2k sin () q, for which the downshifted reflection is negligible, and comment on the downshifted case subsequently. Bragg Condition The sinc function in (19.1-14) has its maximum value of 1.0 when its argument is zero, i.e., when q 2k sin () for upshifted reflection. This occurs when () Bp" where ()py sin -1 q 2k is the Bragg angle. Since q 27r A and k 27r A, sin (}p, A 2A. (19.1-15) Bragg Angle The Bragg angle is the angle for which the incremental reflections from planes sep- arated by an acoustic wavelength A have a phase shift of 27r so that they interfere constructively [see Exercise 2.5- 3 and (7 .1-45)]. EXAMPLE 19.1-2. Bragg Angle. An acousto-optic cell is made of flint glass in which the sound velocity is V s 3 km/s and the refractive index is n 1.95. The Bragg angle for reflection of an optical wave of free-space wavelength Ao 633 nm (A Ao/n 325 nm) from a sound wave of frequency 1 100 MHz (A V S / 1 30 Mm) is ()'B 5.4 mrad 0.31 0 . This angle is internal (i.e., inside the medium). If the cell is placed in air, ()'B corresponds to an external angle () n()p> 0.61 0 . A sound wave of 10 times greater frequency (1 1 GHz) corresponds to a Bragg angle ()'B 3.1 0 .
81 0 CHAPTER 19 ACOUSTO-OPTICS The Bragg condition can also be stated as a simple relation between the wavevectors of the sound wave and the optical waves. If q q, 0, 0 , k k sin (), 0, k cos () , and k r k sin (), 0, k cos () are the components of the wavevectors of the sound wave, the incident light wave, and the reflected light wave, respectively, the condition q 2k sin ()p> is equivalent to the vector relation k r k + q, (19.1-16) illustrated by the vector diagram in Fig. 19.1-2. Incident light Diffracted light , " , , \ , , k r - () / \ () () -------- ......-----...-._- , e , , , , , .. 27r A " , , , I ......., I , , I k , , I q : 27f _ J : A I . . , , - - - --- Sound , , , I , , T _ _... . ... .... _ _ J".-.... . Figure 19.1-2 The Bragg condition sin ()p, q/2k is equivalent to the vector relation k r k+ q. Tolerance in the Bragg Condition The dependence of the complex amplitude reflectance on the angle () is governed by the symmetric function sine q 2k sin () L 27r sine sin () sin ()p> 2L A in (19.1-14). This function reaches its peak value when () ()p> and drops sharply when () differs slightly from ()p>. When sin () sin ()p> A 2£ the sine function reaches its first zero and the reflectance vanishes (Fig. 19.1- 3). Because ()p> is usually very small, sin () (), and the reflectance vanishes at an angular deviation from the Bragg angle of approximately () ()p> A 2£. Since L is typically much greater than ,x, this is an extremely small angular width. This sharp reduction of the reflectance for slight deviations from the Bragg angle occurs as a result of the destructive interference between the incremental reflections from the sound wave. Incident Diffracted light light N -- T .- . - . - (l) ... u , s:: () ( , () ro "'-' - - u (1) I A , r;:: Q.) 2L I I I I I I I I I I I T I I I I I I ... . .- Sound 0 BB e .... ... . - ... ..... -....... -. . - . ... . ." . - -. Figure 19.1-3 Dependence of the reflectance Irl 2 on the angle (). Maximum reflection occurs at the Bragg angle ()p, sin- 1 (A/2A).
19.1 INTERACTION OF LIGHT AND SOUND 811 Doppler Shift In accordance with (19.1-14), the complex amplitude reflectance r + is proportional to exp jOt. Since the angular frequency of the incident light is w [i.e., E ex: exp jwt , the reflected wave Er r +E ex: exp j w + 0 t has angular frequency w r W + O. ( 19 .1-1 7) Doppler Shift The process of reflection is therefore accompanied by a frequency shift equal to the frequency of sound. This can almost be thought of as a Doppler shift (see Exercise 2 6- 1 and Sec. I3.3D). The incident light is reflected from surfaces that move with a velocity VS. Its Doppler-shifted angular frequency is therefore W r W 1 + 2v s sin () c , where V s sin () is the component of velocity of these surfaces in the direction of the incident and the reflected waves. Using the relations sin () A 2A, V S Af2 27r, and c AW 27r, (19.1-17) is reproduced. The Doppler shift equals the sound frequency. Because 0 « w, the frequencies of the incident and reflected waves are approx- imately equal (with an error typically smaller than 1 part in 105). The wavelengths of the two waves are therefore also approximately equal. In writing (19.1-9) we have implicitly used this assumption by using the same wavenumber k for the two waves. Also, in drawing the vector diagram in Fig. 19.1-2 it was assumed that the vectors k r and k have approximately the same length nw CO. Peak Reflectance The reflectance r + 2 is the ratio of the intensity of the reflected optical wave to that of the incident optical wave. At the Bragg angle () (), (19.1-14) gives ro 2. Substituting from (19.1-12), dr 2 1 n 2 q2 L 2 . 4 0 dn (19.1-18) An expression for the derivative dr dn may be obtained by use of the Fresnel equations (see Sec. 6.2) to determine the incremental complex amplitude reflectance r in terms of the incremental refractive-index change n between two adjacent layers. For TE (orthogonal) polarization, (6.2-8) is used with nl n + n, n2 n, ()l 90° (), and Snell's law nl sin ()l n2 sin ()2 is used to determine ()2. When terms of second order in n are neglected, the result is r n 2n sin 2 () so that dr 1 dn 2n sin 2 () · (19.1-19) Equation (6.2-9) is similarly used for the TM (parallel) polarization, yielding dr cas 2() dn 2n sin 2 () · (19.1-20) In most acousto-optic devices () is very small, so that cas 2() 1, making (19.1-19) approximately applicable to both polarizations. Substituting for ro from (19.1-19) into (19.1-18) and using the Bragg condition q 2k sin () 47rn sin () Ao , we obtain 7r 2 A 2 o L sin () 2 n6. (19.1-21)
812 CHAPTER 19 ACOUSTO-OPTICS Using (19.1-6), we conclude that the reflectance 9{ 1T 2 2A L sine 2 MIs (J 9.1-22) Reflectance is proportional to the intensity of the acoustic wave Is, to the material parameter Jv( defined in (19.1- 7) and to the square of the oblique distance L sin () of penetration of light through the acoustic wave. Substituting sine A 2A into (19.1-22), we obtain L 2 A 2 Ao (19.1-23) Thus, the reflectance is inversely proportional to A (or directly proportional to w 4 ). The dependence of the efficiency of scattering on the fourth power of the optical frequency is typical of light-scattering phenomena. The proportionality between the reflectance and the sound intensity poses a prob- lem. As the sound intensity increases, would eventually exceed unity, and the re- flected light would be more intense than the incident light! This unacceptable result is a consequence of violating the assumptions of this approximate theory. It was assumed that the incremental reflection from each layer is too small to deplete the transmitted wave which reflects from subsequent layers. Clearly, this assumption does not hold when the sound wave is intense. In reality, a saturation process occurs, ensuring that 9{ does not exceed unity. A more careful analysis (see Sec. 19.1B), in which depletion of the incident optical wave is included, leads to the following expression for the reflectance: e sin 2 , (19.1-24) where is the approximate expression (19.1-22) and e is the exact expression. This relation is illustrated in Fig. 19.1-4. Evidently, when « 1, sin , so that e 9(. 1 (l) u cd U (l) (l) 00 , / -------r----- Sound intensity Is Figure 19.1-4 Dependence of the re- flectance e of the Bragg reflector on the intensity of sound Is. When Is is small1t e , which is a linear function of Is. EXAMPLE 19.1-3. Reflectance of a Bragg Reflector. A Bragg cell is made of extra-dense flint glass with material parameter]v( 1.67 x 10- 14 m 2 /W (see Example 19.1-1). If AD 633 nm (wavelength of the He-Ne laser), the sound intensity Is 10 W jcm 2 , and the length of penetration
19.1 INTERACTION OF LIGHT AND SOUND 813 of the light through the sound is Lj sin () 1 mm, then 0.0206 and e 0.0205, so that approximately 2% of the light is reflected. If the sound intensity is increased to 100 W jcm 2 , then 0.206 and e 0.192 so that the reflectance increases to 19%. Downshifted Bragg Diffraction Another possible geometry for Bragg diffraction is that for which 2k sin () q. This is satisfied when the angle () is negative; i.e., the incident optical wave makes an acute angle with the sound wave as illustrated in Fig. 19.1- 5. In this case, the downshifted reflectance r _ in (19.1-14) has its maximum value, whereas the upshifted reflectance r + is negligible. The complex amplitude reflectance is then given by jroe- jOt . ( 19 .1- 25) r_ In this geometry, the frequency of the reflected wave is downshifted, so that W s W f2 (19.1-26) and the wavevectors of the light and sound waves satisfy the relation ks k q, (19.1-27) illustrated in Fig. 19.1-5. Equation (19.1-27) is a phase-matching condition, ensuring that the reflections of light add in phase. The frequency downshift in (19.1- 26) is consistent with the Doppler shift since the light and sound waves travel in the same directi on. x Transmitted light \ \ \ \ , , k r () () e () --- -------- , e , , , "'- "- ' 27r A " " , I " ......, I I , , k , , , I -q · 21r J :A I I . , r y A1- Incident light T -- , . . ' . .... , , I I Sound Diffracted light .. ..... ..,..... .-,... .. -. ..-..... .. .. .-.,. ... -.-...- ...... Figure 19.1-5 Geometry of downshifted reflection of light from sound. The frequency of the reflected wave is downshifted. Quantum Interpretation In accordance with the quantum theory of light (see Chapter 12), an optical wave of angular frequency wand wavevector k is viewed as a stream of photons, each of energy mu and momentum lik. An acoustic wave of angular frequency f2 and wavevector q is similarly regarded as a stream of acoustic quanta, called phonons, each of energy hf2 and momentum hq. Interaction of light and sound occurs when a photon combines with a phonon to generate a new photon of the sum energy and momentum. An incident photon of frequency wand wavevector k interacts with a phonon of frequency f2 and wavevec- tor q to generate a new photon of frequency W r and wavevector k, as illustrated in
814 CHAPTER 19 ACOUSTO-OPTICS Fig. 19.1-6. Conservation of energy and momentum require that liw r liw + fifl and fik r fik + fiq, from which the Doppler shift formula W r W + n and the Bragg condition, k r k + q, are recovered. Photon fiw Photon fiw r Phonon fin < " " Figure 19.1-6 Bragg diffraction: a photon combines with a phonon to generate a new photon of different frequency and momen- tum. *8. Coupled-Wave Theory Bragg Diffraction as a Scattering Process As described in Sec. 5.2B, light propagation through a homogeneous medium with a slowly varying inhomogeneous refractive-index perturbation n is described by the wave equation \72£ 1 8 2 £ r-....I ,-...",.; c 2 8t 2 s, (19.1-28) where 82p J-lo 8t 2 8 2 (19.1-29) s is a radiation source proportional to the second derivative of the product n£ [see (5.2-20)]. For Bragg diffraction the perturbation Lln is created by the sound wave, so that the scattering source is dependent on both the acoustic field and the optical field £, which includes both the incident and scattered fields. One approximate method of solving this scattering problem, called the first Born approximation, uses the assumption that the scattering source S is created by the incident field (rather than by the actual field). Once we know the scattering source, we can solve the wave equation for the scattered field. Assuming that the incident light is a plane wave £ Re A exp j wt k · r (19.1-30) and the perturbation caused by the acoustic wave is a plane wave Ll no cos flt q · r , (19.1-31) Lln we substitute into (19.1-29), and reorder the terms of the product n£ to obtain no n k; Re A exp j W r t k r · r + k; Re A exp j w s t ks. r , S (19.1-32) where W r W + fl, k r k + q, k r W r c; and W s w fl, ks k q, ks W s c. We thus have two sources of light of frequencies w ::I:: fl, and wavevectors k::l::q, that may emit an upshifted or downshifted Bragg-reflected plane wave. Upshifted reflection occurs if the geometry is such that the magnitude of the vector k + q equals
19.1 INTERACTION OF LIGHT AND SOUND 815 W r C W c, as can be easily seen from the vector diagram in Fig. 19.1-2. Downshifted reflection occurs if the vector k q has magnitude W s c W c, as illustrated in Fig. 19.1-5. Obviously, these two conditions may not be met simultaneously. We have thus independently proved the Bragg condition and Doppler-shift formula using a scattering approach. Equation (19.1-32) indicates that the intensity of the emit- ted light is proportional to w; w 4 , so that the efficiency of scattering is inversely proportional to the fourth power of the wavelength. This analysis can be pursued further to derive an expression for the reflectance by determining the intensity of the wave emitted by the scattering source (see Probe 19.1-2). Coupled- Wave Equations To go beyond the first Born approximation, we must include the contribution made by the scattered field to the source S. Assuming that the geometry is that of upshifted Bragg diffraction, the field £ is composed of the incident and Bragg-reflected waves: £ Re E exp jwt + Re Er exp jwrt . With the help of the relation n nocos Ot q. r , (19.1-29) gives S Re S exp jwt + Sr exp jwrt + terms of other frequencies, (19.1-33) where r, n Sr S (19.1- 34) n Com aring terms of equal frequencies on both sides of the wave equation, \72 £ 1 C f)2 £ f) 2 t S, we obtain two coupled Helmholtz equations for the incident wave and the Bragg-reflected wave, S, \72 + k2 E r r \72 + k 2 E Sr. (19.1-35) These equations, together with (19.1- 34), may be solved to determine E and Er. Consider, for example, the case of small-angle reflection (() « 1), so that the two waves travel approximately in the z direction. Assuming that k k r , the fields E and Er are described by E A exp j kz and Er Ar exp j kz , where A and Ar are slowly varying functions of z. Using the slowly varying envelope approximation (see Sec. 2.2C), \72 + k 2 Aexp jkz j2k dA dz exp jkz, (19.1-34) and (19.1-35) yield dA dz dAr dz .1 A J2'Y r (19.1-36a) .1 A J 2 'Y , (19.1- 36b ) where (19.1-37) 'Y n If the cell extends between z 0 and z d, we use the boundary condition Ar 0 0, and find that equations (19.1- 36) have the harmonic solution A 0 cas 'Y z 2 Az (19.1-38a)
816 CHAPTER 19 ACOUSTO-OPTICS Ar Z . A 0 · fiZ J SIn · 2 (] 9 .1- 3 8b ) These equations describe the rise of the reflected wave and the fall of the incident wave, as illustrated in Fig. 19.1- 7. The reflectance 9(e Ar d 2 A 0 2 is therefore given by 9(e sin 2 fi d 2 , so that 9(e sin 2 9(, where 9( fid 2 2. Using (19.1-37), reflectance in (19.1-21) with d L sin 8. Incident ... . .. 11 Reflected ,. light light .... - z - c c-'" I IA 1 2 I I I I I I t/1} I "'" "'" ..-- .. "","'" "'" t/1} o d z Figure 19.1-7 Variation of the intensity of the incident optical wave ( solid curve) and the intensity of the Bragg-reflected wave (dashed curve) as functions of the distance traveled through the acoustic wave. 1Arl2 I " I " , I ' " I -.... c. Bragg Diffraction of Beams It has been shown so far that an optical plane wave of wavevector k interacts with an acoustic plane wave of wavevector q to produce an optical plane wave of wavevector k r k+q, provided that the Bragg condition is satisfied (i.e., the angle between k and q is such that the magnitude k r k + q k 2?T A). Interaction between a beam of light and a beam of sound can be understood if the beam is regarded as a superposition of plane waves traveling in different directions, each with its own wavevector (see the introduction to Chapter 4). Diffraction of an Optical Beam from an Acoustic Plane Wave Consider an optical beam of width D interacting with an acoustic plane wave. In accordance with Fourier optics (see Sec. 4.3A), the optical beam can be decomposed into plane waves with directions occupying a cone of half-angle 88 A D . (19.1-39) There is some arbitrariness in the definition of the diameter D and the angle 88, and a multiplicative factor in (19.1-39) is taken to be ] .0. If the beam profile is rectangular of width D, the angular width from the peak to the first zero of the Fraunhofer diffraction pattern is 88 A D; for a circular beam of diameter D, 88 1.22A D; for a Gaussian beam of waist diameter D 2W o ,88 A ?TWo 2?T A D 0.64A D [see (3.1- 20)]. For simplicity, we shall use (19.1- 39). Although there is only one wavevector q, there are many wavevectors k (aU of the same length 2?T A) within a cone of angle 88. As Fig. 19.1-8 illustrates, there is only one direction of k for which the Bragg condition is satisfied. The reflected wave is then a plane wave with only one wavevector k r .
19.1 INTERACTION OF LIGHT AND SOUND 817 Incident light Diffracted light .. ....-..- - k r - b8 ' () -. - - J'II . J;: , . ....- .. . lJ - \ 8 q e-- r ---- .'. = ''', , , .. '. :'.. .";: k I ! .. 118 . . - .-. I I I I - Figure 19.1-8 Diffraction of an optical beam from an acoustic plane wave. There is only one plane-wave component of the incident light beam that satisfies the Bragg condition. The diffracted light is a plane wave. Diffraction of an Optical Beam from an Acoustic Beam Suppose now that the acoustic wave itself is a beam of width D s. If the sound fre- quency is sufficiently high so that the wavelength is much smaller than the width of the medium, sound propagates as an unguided (free-space) wave and has properties analogous to those of optical beams, with angular divergence 80s A Ds . ( 19. 1-40) This is equivalent to many plane waves with directions lying within the divergence angle. The reflection of an optical beam from this acoustic beam can be determined by finding matching pairs of optical and acoustic plane waves satisfying the Bragg condi- tion. The sum of the reflected waves constitutes the reflected optica] beam. There are many vectors k (all of the same length 21f A) and many vectors q (all of the same length 27r A); only the pairs of vectors that form an isosceles triangle contribute, as illustrated in Fig. 19.1-9. 80 \ .,' ..-- ...,.,. , . ...,-"""'- . ',n . -- t ---- --------- l . .""-- "'- . " ......... ) ""'-. "'"'- 88 "\ \ \ \ \ / Diffracted light ..,..,- Incident light -- ".. ".. . () (} 80 80s ... 88 s -,-= ,,,,..J),,..... I '. I I I I I Figure 19.1-9 Diffraction of an optical beam from a sound beam. If the acoustic-beam divergence is greater than the optical-beam divergence (80 s 80) and if the central directions of the two beams satisfy the Bragg condition, every incident optical plane wave finds an acoustic match and the reflected light beam has the same angular divergence as the incident optical beam 8e. The distribution of acoustic energy in the sound beam can thus be monitored as a function of direction, by using a probe light beam of much narrower divergence and measuring the reflected light as the angle of incidence is varied.
818 CHAPTER 19 ACOUSTO-OPTICS Diffraction of an Optical Plane Wave from a Thin Acoustic Beam: Raman Hath Diffraction Since a thin acoustic beam comprises plane waves traveling in many directions, it can diffract light at angles that are significantly different from the Bragg angle correspond- ing to the beam's principal direction. Consider, for example, the geometry in Fig. 19.1- lOin which the incident optical plane wave is perpendicular to the main direction of a thin acoustic beam. The Bragg condition is satisfied if the reflected wavevector k r makes angles :f:8, where . 8 SIn 2 A 2A. ( 19 .1-41 ) If 8 is small, sin 8 2 8 2 and A 8 . A ( 19 .1-42) The incident beam is therefore deflected into either of the two directions making angles :f:8, depending on whether the acoustic beam is traveling upward or downward. For an acoustic standing-wave beam the optical wave is deflected in both directions. : Diffracted light . - - , ' , - ',..... .- . . , - . J " .. . . k r e q k () -q k' r Incident light ..Y ..' -' .... \ e ., - - - . or _ _ t: ':. .' l () - - ... \ "'-- .- ,.. Figure 19.1-10 An optical plane wave incident nonnally on a thin-beam acoustic standing wave is partially deflected into two directions making angles ::f:A / A. The angle 8 A A is the angle by which a diffraction grating of period A deflects an incident plane wave (see Exercise 2.4-5). The thin acoustic beam in fact modulates the refractive index, creating a periodic pattern of period A confined to a thin planar layer. The medium therefore acts as a thin diffraction grating. This phase grating diffracts light also into higher diffraction orders, as illustrated in Fig. 19.1-11 (a). The higher-order diffracted waves generated by the phase grating at angles ::f:28, :f:38, . . . may also be interpreted using a quantum picture of light sound interaction. One incident photon combines with two phonons (acoustic quantum particles) to form a photon of the second-order reflected wave. Conservation of momentum requires that k r k ::f: 2q. This condition is satisfied for the geometry in Fig. 19.1-11(b). The second-order reflected light is frequently shifted to W r W :f: 20. Similar interpreta- tions apply to higher orders of diffraction. The acousto-optic interaction of light with a perpendicular thin sound beam is known as Raman Nath or Debye Sears scattering of light by sound. t t For further details, see, e.g., M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, Chapter 12.
19.2 ACOUSTO-OPTIC DEVICES 819 Ds Diffracted light J- \ e k r \0 q Incident light -.....- e --- q ...- () _ I 1"'-'J r...... - -- y y - r T - - - . ., , k . - ' - :- .'. .-.. '....:..r :..-. . . (a) (b) Figure 19.1-11 (a) A thin acoustic beam acts as a diffraction grating. (b) Conservation-of- momentum diagram for second-order acousto-optic diffraction. 19.2 ACQUSTO-OPTIC DEVICES A. Modulators The intensity of the reflected light in a Bragg cell is proportional to the intensity of sound, if the sound intensity is sufficiently weak. Using an electrically controlled acoustic transducer [Fig. 19.2-1 (a)], the intensity of the reflected light can be varied proportionally. The device can be used as a linear analog modulator of light. As the acoustic power increases, however, saturation occurs and almost total reflec- tion can be achieved (see Fig. 19.1-4). The modulator then serves as an optical switch, which, by switching the sound on and off, turns the reflected light on and off, and the transmitted light off and on, as illustrated in Fig. 19 .2-1 (b). ...c:: ..c: ...c:: b.O >. bJ) bJ) . . .. .. ...-.4 . VJ ......, VJ I:: I:: . .... C I:: VJ 0..> 0..> 0..> I:: 0..> (l) "'C 0..> "'C """"" C 0 C . 0..> . o .. I:: u . C C . I:: t 0..> t t--I t -. '. -..-,..' - - ...c:: bJ) . ---" """"" . 0..> VJ I:: o 0..> 0..> C ..... t .. "0 -..... "..-- . ..... .... ..... ..... -o.Q 0..> . VJ I:: . (l) E """"" c VJ ....-01: I:: ro...c:: b.O · ..-4 t ...... >. "'C . I:: rJ'J ::s c o 0..> cnE . ...... "'0 . VJ ::s c o 0..> cnc · ..-4 t t (a) (b) Figure 19.2-1 (a) An acousto-optic modulator. The intensity of the reflected light is proportional to the intensity of sound. (b) An acousto-optic switch. Modulation Bandwidth The bandwidth of the modulator is the maximum frequency at which it can efficiently modulate. When the amplitude of an acoustic wave of frequency fa is varied as a func- tion of time by amplitude modulation with a signal of bandwidth B, the acoustic wave is no longer a single-frequency harmonic function; it has frequency components within a band fa:l:B centered about the frequency fa (Fig. 19.2-2). How does monochromatic light interact with this multifrequency acoustic wave and what is the maximum value of B that can be handled by the acousto-optic modulator?
820 CHAPTER 19 ACOUSTO-OPTICS / I I S ::3 I-< +-' U d) 0... r./) 2B , o fo f Figure 19.2-2 The waveform of an amplitude-modulated acoustic signal and its spectrum. When both the incident optical wave and the acoustic wave are plane waves, the component of sound of frequency f corresponds to a Bragg angle, . -1'\ . -1 f'\ ,\ e == SIn - == SIn - - f 2A 2v s 2v s (assumed to be small). For a fixed angle of incidence e, an incident monochromatic optical plane wave of wavelength ,\ interacts with one and only one harmonic com- ponent of the acoustic wave, the component with frequency f satisfying (19.2-1), as illustrated in Fig. 19.2-3. The reflected wave is then monochromatic with frequency v + f. Although the acoustic wave is modulated, the reflected optical wave is not. Evidently, under this idealized condition the bandwidth of the modulator is zero! (19.2-1) 27ff Vs }===-I f L ¥ _ m 1 Vs 27rf, Vs 0 ________L____ Figure 19.2-3 Interaction of an optical plane wave with a modulated (multiple frequency) acoustic plane wave. Only one frequency component of sound reflects the light wave. The reflected wave is monochromatic and not modulated. To achieve modulation with a bandwidth B, each of the acoustic frequency com- ponents within the band fa ::l: B must interact with the incident light wave. A more tolerant situation is therefore necessary. Suppose that the incident light is a beam of width D and angular divergence be == AI D and assume that the modulated sound wave is planar. Each frequency component of sound interacts with the optical plane wave that has the matching Bragg angle (Fig. 19.2-4). The frequency band fa ::l: B is matched by an optical beam of angular divergence 8() (21f/v s )B = B. 27rj'\ V s The bandwidth of the modulator is therefore B == V s 6 e == v s ,\ D' (19.2-2) (19.2-3) or 1 B == T ' T== D , V s (19.2-4) Bandwidth
19.2 ACOUSTO-OPTIC DEVICES 821 where T is the transit time of sound across the waist of the light beam. This is an expected result since it takes time T to change the amplitude of the sound wave at all points in the light-sound interaction region, so that the maximum rate of modulation is 1/ T Hz. To increase the bandwidth of the modulator, the light beam should be focused to a small diameter. Incident light Diffracted light -........... - - T T t t 7r (to + B) i]l.fo S l t; (foB) r - ........ B - -_ - t 8B - ,B I I I Figure 19.2-4 Interaction of an optical beam of angular divergence 8e with an acoustic plane wave of frequency in the band f 0 =:t B. There are many parallel q vectors of different lengths each matching a direction of the incident light. EXERCISE 19.2-1 Parameters of Acousto-Optic Modulators. Determine the Bragg angle and the maximum bandwidth of the following acousto-optic modulators: Modulator 1 Material: Fused quartz (n == 1.46, V s == 6 km/s) Sound: Frequency f == 50 MHz Light: He-Ne laser, wavelength Ao == 633 nm, angular divergence 8e == 1 mrad Modulator 2 Material: Tellurium (n == 4.8, V s == 2.2 km/s) Sound: Frequency f == 100 MHz Light: CO 2 laser, wavelength Ao == 10.6 /Lm, and beam width D == 1 mm B. Scanners The acousto-optic cell can be used as a scanner of light. The basic idea lies in the linear relation between the angle of deflection 2() and the sound frequency f, ,\ 2() - f, V s (19.2-5) where () is assumed sufficiently small so that sin () (). By changing the sound frequency f, the deflection angle 2() can be varied. One difficulty is that () represents both the angle of reflection and the angle of incidence. To change the angle of reflection, both the angle of incidence and the sound frequency must be changed simultaneously. This may be accomplished by tilting the sound beam. Figure 19.2-5 illustrates this principle. Changing the sound frequency requires a frequency modulator (FM). Tilting the sound beam requires a sophisticated
822 CHAPTER 19 ACOUSTO-OPTICS system that uses, for example, a phased array of acoustic transducers (several acoustic transducers driven at relative phases that are selected to impart a tilt to the overalJ generated sound wave). The angle of tilt must be synchronized with the FM driver. \ \ \ \ :bt \ . --.. - .. *' -"" . I I I I I I I Figure 19.2-5 Scanning by changing the sound frequency and direction. The sound wave is tilted by use of an array of transducers driven by signals differing by a phase <po The requirement to tilt the sound beam may be alleviated if we use a sound beam with an angular divergence equal to or greater than the entire range of directions to be scanned. As the sound frequency is changed, the Bragg angle is altered and the incoming light wave selects only the acoustic plane-wave component with the matching direction. The efficiency of the system is, of course, expected to be low. We proceed to examine some of the properties of this device. Scan Angle When the sound frequency is f, the incident light wave interacts with the sound com- ponent at an angle () == (A/2v s )! and is deflected by an angle 2() == (A/v s )!, as Fig. 19.2-6 illustrates. by varying the sound frequency from fa to fa + B, the deflection angle 2() is swept over a scan angle fj.() == B. V s (19.2-6) Scan Angle This, of course, assumes that the sound beam has an equal or greater angular width 8()s == A/ Ds > fj.(). Since the scan angle is inversely proportional to the speed of sound, larger scan angles are obtained by use of materials for which the sound velocity V s is small. Number of Resolvable Spots If the optical wave itself has an angular width 8() == A/ D, and assuming that 8() « 8()s, the deflected beam also has a width 8(). The number of resolvable spots of the scanner (the number of nonoverlapping angular widths within the scanning range) is therefore N == D,.() == (A/vs)B == D B 8() A/ D v s ' (19.2-7) or
19.2 ACOUSTO-OPTIC DEVICES 823 Diffracted fo + B light /' b,.() fo , B ,..- ,..- l::1B ,.._,..-,..-,..- ,..-,..-,..-"'- ,..- ,..- ....-:: \ \ ,.._"" fo + B \ ,..- ,..- fo Incident light B { .......... bBs - t . :1 I I I I I I .... Figure 19.2-6 Scanning an optical wave by varying the frequency of a sound beam of angular divergence 80s over the frequency range fa < f < fa + B. N == TB , (19.2-8) Number of Resolvable Spots where B is the bandwidth of the FM modulator used to generate the sound and T == D /v s is the transit time of sound through the light beam (Fig. 19.2-7). 8B I .., 8B \ \ """"'" 80s - I I I I I I / / Figure 19.2-7 Resolvable spots of an acousto-optic scanner. The number of resolvable spots is therefore equal to the time-bandwidth product. This number represents the degrees of freedom of the device and is a significant indi- cator of the capability of the scanner. To increase N, a large transit time T should be used. This is the opposite of the design requirement in an acousto-optic modulator, for which the modulation bandwidth B == 1/ T is made large by selecting a small T. EXERCISE 19.2-2 Parameters of an Acousto-Optic Scanner. A fused-quartz acousto-optic scanner (v s = 6 km/s, n = 1.46) is used to scan a He-Ne laser beam (Ao = 633 nm). The sound frequency is scanned over the range 40 to 60 MHz. To what width should the laser beam be focused so that the number of resolvable points is N = 100? What is the scan angle .e? What is the effect of using a material in which sound is slower, flint glass (v s = 3.1 km/s), for example? The Acousto-Optic Scanner as a Spectrum Analyzer The proportionality between the angle of deflection and the sound frequency can be utilized to make an acoustic spectrum analyzer. A sound wave containing a spectrum
824 CHAPTER 19 ACOUSTO-OPTICS of different frequencies disperses the light in different directions with the intensity of deflected light in a given direction proportional to the power of the sound component at the corresponding frequency (Fig. 19.2-8). /3 /2 /1 1 (h (h Figure 19.2-8 Each frequency com- ponent of the sound wave deflects light in a different direction. The acousto- optic cell serves as an acoustic spectrum analyzer. /1 +/2 +/3 t c. Space Switches An acousto-optic cell can be used as a space switch (see Sec.23.3) that routes informa- tion carried by one or more optical beams to one or more selected directions. Several interconnection schemes are possible: . An acousto-optic cell in which the frequency of the acoustic wave is one of N possible values, 11,12, · · · , or IN, deflects an incident optical beam to one of N corresponding directions, (}1, (}2, . . . , or () N, as illustrated in Fig. 19.2-9. The device routes one beam to any of N directions. ()3 1 2 .... . . . /3 Figure 19.2-9 Routing an optical beam to one of N directions. By apply- ing an acoustic wave of frequency 13, for example, the optical beam is deflected by an angle 0 3 and routed to point 3. t · By using an acoustic wave comprising two frequencies, 11 and 12, simultaneously, the incident optical beam is reflected in the two corresponding directions, (}1 and (}2, simultaneously. Thus, one beam is connected to any pair of many possible directions as illustrated in Fig. 19.2-10. Similarly, by using an acoustic wave with M frequencies the incoming beam can be routed simultaneously to it! directions. An example is the acoustic spectrum analyzer for which an incoming light beam is reflected from a sound wave carrying a spectrum of it! frequencies. The light beam is routed to M points, with the intensity at each point proportional to the power of the corresponding sound-frequency component. . . . (h (}2 1 2 /1 + /2 Figure 19.2-1 0 Routing a light beam simultaneousl y to a number of direc- tions.
19.2 ACOUSTO-OPTIC DEVICES 825 . The length of the acousto-optic cell may be divided into two segments. At a certain time, an acoustic wave of frequency 11 is present in one segment and an acoustic wave of frequency 12 is present in the other. This can be accomplished by generating the acoustic wave from a frequency-shift-keyed electric signal in the form of two pulses: a pulse of frequency 11 followed by another of frequency 12, each lasting a duration T /2, where T == W /v s is the transit time of sound through the cell length W (see Fig. 19.2-11). When the leading edge of the acoustic wave reaches the end of the cell, the cell processes two incoming optical beams by deflecting the top beam to the direction ()1 corresponding to 11, and the bottom beam to the direction ()2 corresponding to 12. This is a switch that connects each of two beams to any of many possible directions. By placing more than one frequency component in each segment, each of the two beams can itself be routed simultaneously to several directions. 11 12 iiiiii iiii. ""'"'''' It: T/2. It: T/2 · I 1 2 Figure 19.2-11 Routing each of two light beams to a set of specified direc- tions. The acoustic wave is generated by a frequency-shift-keyed electric signal. . The cell may also be divided into N segments, each carrying a harmonic acoustic wave of the same frequency I but with a different amplitude. The result is a spatial light modulator (SLM) that modulates the intensities of N input beams (Fig. 19.2-12). Spatial light modulators are useful in optical signal processing (see Sec.20.IE). 1 2 Figure 19.2-12 The spatial light modulator modulates N optical beams. The acoustic wave is driven by an amplitude-modulated electric signal. N ---+-1 TIN I 111111111111111111. . I . I I IIIIII11II11 """"''''1111'1''''''1'"'1'''''' t . The most general interconnection architecture is one for which the cell is divided into L segments, each of which carries an acoustic wave with M frequencies. The device acts as a random access switch that routes each of L incoming beams to M directions simultaneously (Fig. 19.2-13).
826 CHAPTER 1 9 ACOUSTO-OPTICS 1 1 Ml M ...,.. ill. " - 2 .....J 'r J}.. 1. _. J(41'",u M i 'f WI' r .4-M.a4 . ..:...---. '!7. 17 ". L I TIL I .iI - - - 1 ---"--"-------... -- L . ,I J."., 11 ''1'1 illjIIJ' J . J Figure 19.2-13 An arbitrary- interconnection switch routes each of L incoming light beams for the random access of AI points. --.--- t Interconnection Capacity There is an upper limit to the number of interconnections that may be established by an acousto-optic device, as will be shown subsequently. If an acousto-optic cell is used to route each of L incoming optical beams to a maximum of M directions simultaneously, then product M L cannot exceed the time bandwidth product N T B, where T is the transit time through the cell and B is the bandwidth of the acoustic wave, M L < N. (19.2-9) Interconnection Capacity This upper bound on the number of interconnections is called the interconnection capacity of the device. An acousto-optic cell with L segments uses an acoustic wave composed of L seg- ments each of time duration T L. For each segment to address M independent points the acoustic wave must carry M independent frequency components per segment. For a signal of duration T L there is an inherent frequency uncertainty of L T Hz. The M frequency components must therefore be separated by at least that uncertainty. For the M components to be placed within the available bandwidth B, we must have M L T < B, from which ML < TB, and hence (19.2-9) follows. A single optical beam (L 1), for example, can be connected to any of N TB points, but each of two beams can be connected to at most N 2 points, and so on. It is a question of dividing an available time bandwidth product N T B in the form of L time segments each containing M independent frequencies. Examples of the possible choices are illustrated in the time frequency diagram in Fig. 19.2-14. f f --+-II TIN f B . . . .- .- . . . . . -. . . B BIM B -. . . t -' . J .- -. . ,- I . . .- ,- . BIN T .., T .., T ., t t t (a) Scanner (b) SLM (c) Switch Figure 19.2-14 Several examples of dividing the time-bandwidth region TB in the time- frequency diagram into N TB subdivisions (in this diagram N 20). (a) A scanner: a single time segment containing N frequency segments. (b) A spatial light modulator: N time segments each containing one frequency component. ( c) An interconnection switch: L time segments each containing M N / L frequency segments (in this diagram, N 20, M 4, and L 5).
19.2 ACOUSTO-OPTIC DEVICES 827 D. Filters, Frequency Shifters, and Isolators The acousto-optic cell is useful in a number of other applications, including filters, frequency shifters, and optical isolators. Tunable Acousto-Optic Filters The Bragg condition sin () == >"/2A relates the angle (), the acoustic wavelength A, and the optical wavelength A. If () and A are specified, reflection can occur only for a single optical wavelength >.. == 2A sin (). This wavelength-selection property can be used to filter an optical wave composed of a broad spectrum of wavelengths. The filter is tuned by changing the angle () or the sound frequency f. EXERCISE 19.2-3 Resolving Power of an Acousto-Optic Filter. Show that the spectral resolving power AI A of an acousto-optic filter equals f T, where f is the sound frequency, T the transit time, and A the minimum resolvable wavelength difference. Frequency Shifters Optical frequency shifters are useful in many applications of photonics, including optical heterodyning, optical FM modulators, and laser Doppler velocimeters. The acousto-optic cell may be used as a tunable frequency shifter since the Bragg reflected light is frequency shifted (up or down) by the frequency of sound. In a heterodyne optical receiver, a received amplitude- or phase-modulated optical signal is mixed with a coherent optical wave from a local light source, acting as a local oscillator with a different frequency. The two optical waves beat (see Sec. 2.6B) and the detected signal varies at the frequency difference. Information about the amplitude and phase of the received signal can be extracted from the detected signal (see Sec. 24.5). The acousto- optic cell offers a practical means for imparting the frequency shift required for the heterodyning process. Optical Isolators An optical isolator is a one-way optical valve often used to prevent reflected light from retracing its path back into the original light source (see Sec. 6.6C and Sec. 23.1 C). Optical isolators are sometimes used with semiconductor lasers since the reflected light can interact with the laser process and create deleterious effects (noise). The acousto-optic cell can serve as an isolator. If part of the frequency-upshifted Bragg- diffracted light is reflected onto itself by a mirror and traces its path back into the cell, as illustrated in Fig. 19.2-15, it undergoes a second Bragg diffraction accompanied by a second frequency upshift. Since the frequency of the returning light differs from that of the original light by twice the sound frequency, a filter may be used to block it. Even without a filter, the laser process may be insensitive to the frequency-shifted light. Source Filter I : . : . : . - . .. 1 _--1 /---- - ----\ - . .c Mirror - r f , , , , , , , , , , , , , , , , , , , , , , " t Figure 19.2-15 An acousto-optic isolator.
828 CHAPTER 19 ACOUSTO-OPTICS *19.3 ACOUSTO-OPTICS OF ANISOTROPIC MEDIA The scalar theory of interaction of light and sound is generalized in this section to include the anisotropic properties of the medium and the effects of polarization of light and sound. Acoustic Waves in Anisotropic Materials An acoustic wave is a wave of material strain. Strain is defined in terms of the displace- ments of the molecules relative to their equilibrium positions. If U Ul, U2, U3 is the vector of displacement of the molecules located a t position x Xl , X2, X3 , the strain indexes, i,j 1,2,3 denote the coordinates X, y, z . The element 833 {)U3 {)X3, for example, represents tensile strain (stretching) in the z direction [Fig. 19.3-1(a)], whereas 813 represents shear strain since {)Ul {)X3 is the relative movement in the X direction of two incrementally separated parallel planes normal to the z direction, as illustrated in Fig. 19 .3-1 (b). x x I I. / V # J z I Ul "J t I I t I 1 , I II , I! : / I . ",- I II UI + L\u] I f 1 I 1 t I n ti t l j .--,.. ft U I ft II I a t U3 + L\ U 3 I I n if 11 .11 rf n II tt If n U I ;P :: Z+.trz It n II I - U3 I ." " I '"!'.-':' I n .I I I " I I I I ,/ ." I .. .., - .) ... z + L\z (a) Tensile strain (b) Shear Figure 19.3-1 Displacements associated with tensile strain and shear. An acoustic wave can be longitudinal or transverse, as illustrated in the following examples. EXAMPLE 19.3-1. Longitudinal Acoustic Wave. A wave with the displacement U1 0, U2 0, U3 Aosin(Ot qz), where Ao is a constant, corresponds to a strain tensor with all components vanishing except 833 So cos(Ot qz), (19.3-1) where So qAo. This is a wave that stretches in the z direction and also travels in the l' direction. Since the vibrations are in the same direction as the wave propagation, the wave is 10ngittldinal. EXAMPLE 19.3-2. Transverse Acoustic Wave. The displacement wave, U1 Ao sin(Ot qz), U2 0, U3 0, corresponds to a strain tensor in which all components vanish except 813 831 So cos(Ot qz), ( 19.3-2) where So qAo. This wave travels in the z direction but vibrates in the x direction. It is therefore a transverse (shear) wave. The velocities of the longitudinal and transverse acoustic waves are characteristics of the medium and generally depend on the direction of propagation.
19.3 ACOUSTO-OPTICS OF ANISOTROPIC MEDIA 829 The Photoelastic Effect The optical properties of an anisotropic medium are characterized completely by the electric impermeability tensor 11 == Eo€-I (see Sec. 6.3). Given 11, we can determine the index ellipsoid and hence the refractive indexes for an optical wave traveling in an arbitrary direction with arbitrary polarization. In the presence of strain, the electric impermeability tensor is modified so that llij becomes a function of the elements of the strain tensor, llij == llij (Skl). This dependence is called the photoelastic effect. Each of the nine functions llij (s kl) may be expanded in terms of the nine variables Skl in a Taylor series. Maintaining only the linear terms, llij(Skl) llij(O) + L PijklSkl, kl i,j, l, k == 1,2,3, (19.3-3) where Pijkl == Or,ijj8s kl are constants forming a tensor of fourth rank known as the strain-optic tensor. Since both {rlij} and {SkI} are symmetrical tensors, the coefficients {Pijkl} are invariant to permutations of i and j, and to permutations of k and l. There are therefore only six instead of nine independent values for the set (i, j) and six independent values for (k, l). The pair of indexes (i, j) is usually contracted to a single index I == 1,2,...,6 (see Table 20.2-1). The indexes (k,l) are similarly contracted and denoted by the index K == 1,2,...,6. The fourth-rank tensor Pijkl is thus described by a 6 x 6 matrix PIK. Symmetry of the crystal requires that some of the coefficients PI K vanish and that certain coefficients are related. The matrix PI K of a cubic crystal, for example, has the structure PII P12 P12 0 0 0 P12 PII P12 0 0 0 PII P12 PII 0 0 0 (19.3 -4 ) PIK == 0 0 0 P44 0 0 Strain-Optic Matrix 0 0 0 0 P44 0 (Cubic Crystal) 0 0 0 0 0 P44 This matrix is also applicable for isotropic media, with the additional constraint P44 == ! (PII + P12), so that there are only two independent coefficients. EXAMPLE 19.3-3. Longitudinal Acoustic Wave in a Cubic Crystal. The longitudinal acoustic wave described in Example 19.3-1 travels along one of the axes of a cubic crystal of refractive index n. By substitution of (19.3-1) and (19.3-4) into (19.3-3) we find that the associated strain results in an impermeability tensor with elements, 1 1111 = 1122 = 2" + P12 S 0 cos(Ot - qz) n 1 1133 = 2" + Pl1 S 0 cos(Ot - qz) n - 11ij = 0, i -I- j. (19.3-5) (19.3-6) (19.3-7) Thus, the initialIy opticalIy isotropic cubic crystal becomes a uniaxial crystal with the optic axis in the direction of the acoustic wave (z direction) and with ordinary and extraordinary refractive indexes, no and ne, given by 1 1 2" = 2" + P12 S 0 cos(Ot - qz) no n (19.3-8)
830 CHAPTER 19 ACOUSTO-OPTICS 1 n 2 e 1 2 + P11 S 0 cos(flt qz). n (19.3-9) The shape of the index ellipsoid is altered periodically in time and space in the form of a wave, but the principal axes remain unchanged (see Fig. 19.3-2). Since the change of the refractive indexes is usually small, the second terms in (19.3-8) and (19.3-9) are small, so that the approximation (1 + Ll)-1/2 1 Ll/2, when ILlI « 1, may be applied to approximate (19.3-8) and (19.3-9) by no n n3p12S0 cos(flt qz) ne n n3p11So cos(flt qz). (19.3-10) (19.3-11) x Ot - qx = 0 Ot - qx = 7r /2 Ot - qx = 7r y Figure 19.3-2 A longitudinal acoustic wave traveling in the z direction in a cubic crystal alters the shape of the index ellipsoid from a sphere into an ellipsoid of revolution with dimensions varying sinusoidally with time and an axis in the z direction. EXERCISE 19.3-1 Transverse Acoustic Wave in a Cubic Crystal. The transverse acoustic wave described in Example 19.3-2 travels along one of the axes of a cubic crystal. Show that the crystal becomes biaxial with principal refractive indexes n1 n n3p44S0 cos(flt qz) (19.3-12) ( 19.3-13 ) (19.3-14) n2 n n3 n + n3p44So cos(flt qz). In Example 19.3- 3 and Exercise 19.3-1, the acoustic wave alters the index ellipsoid's principal values but not its principal directions, so that the ellipsoid maintains its orientation. Obviously, this is not always the case. Acoustic waves in other directions and polarizations relative to the crystal principal axes result in alteration of the principal refractive indexes as well as the principal axes of the crystal. Bragg Diffraction The interaction of a linearly polarized optical wave with a longitudinal or transverse acoustic wave in an anisotropic medium can be described by the same principles discussed in Sec. 19.1. The incident optical wave is reflected from the acoustic wave if the Bragg condition of constructive interference is satisfied. The analysis is more complicated, in comparison with the scalar theory, since the incident and reflected waves travel with different velocities and, consequently, the angles of reflection and incidence need not be equal. The condition for Bragg diffraction is the conservation-of-momentum (phase- matching) condition, k r k + q. (19.3-15)
19.3 ACOUSTO-OPTICS OF ANISOTROPIC MEDIA 831 The magnitudes of these wavevectors are k == (27r / Ao)n, k r == (27r / Ao)n r , and q == (27r / A), where Ao and A are the optical and acoustic wavelengths and nand n r are the refractive indexes of the incident and reflected optical waves, respectively. As illustrated in Fig. 19.3-3, if e and e r are the angles of incidence and reflection, the vector equation (19.3-15) may be replaced with two scalar equations relating the z and x components of the wavevectors in the plane of incidence: 27r 27r >'0 n r cos Or = >'0 n cos 0 27r 27r 27r >'0 n r sin Or + >'0 n sin 0 = A ' from which (19.3-16) (19.3-17) n r cas e r == n cas e A n r sin Or + n sin 0 = ;: . (19.3-18a) (19.3-18b) Given the wavelengths Ao and A, the angles e and e r may be determined by solving equations (19.3-18). Note that nand n r are generally functions of e and e r that may be determined from the index ellipsoid of the unperturbed crystal. XL z Figure 19.3-3 Conservation of momen- tum (phase-matching condition, or Bragg condition) in an anisotropic medium. I q 27r A 1 Equations (19.3-18) can be easily solved when the acoustic and optical waves are collinear, so that e == X7r /2 and e r == 7r /2. The + and - signs correspond to back and front reflections, as illustrated in Fig. 19.3-4. The conditions (19.3-18) then reduce to one condition, Ao n r x n == A. For back reflection (+ sign), A must be smaller than Ao, which is unlikely except for very-high-frequency acoustic waves. For front reflection (- sign), the incident and reflected waves must have different polarizations so that n r i=- n. 1 (19.3-19) qr k r kj kj q jk r Figure 19.3-4 Wavevector diagram for reflection of an optical wave from an acoustic wave. (a) Front reflection (b) Back reflection
832 CHAPTER 19 ACOUSTO-OPTICS READING LIST Books T.-C. Poon and T. Kim, Engineering Optics with MATLAB, World Scientific, 2006. J. P. Wolfe, Imaging Phonons: Acoustic Wave Propagation in Solids, Cambridge University Press, paperback ed. 2005. M. J. P. Musgrave, Crystal Acoustics, Holden-Day, 1970; Acoustical Society of America, 2003. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, 1984, reprinted 2003. D. Royer and E. Dieulesaint, Elastic Waves in Solids, Volume 2, Generation, Acousto-Optic Interac- tion, Applications, Springer-Verlag, 2000. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford University Press, 1957, reprinted with corrections and new material, 2001. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, 7th expanded and corrected ed. 2002, Chapter 12. A. Korpel, Acousto-Optics, Marcel Dekker, 1988, 2nd ed. 1997. N. J. Berg and J. M. Pellegrino, eds., Acousto-Optic Signal Processing, Marcel Dekker, 1983, 2nd ed. 1996. A. P. Goutzoulis and D. R. Pape, eds., Design and Fabrication of Acousto-Optic Devices, Marcel Dekker, 1994. V. E. Gusev and A. A. Karabutov, Laser Optoacoustics, American Institute of Physics, 1993. C. Scott, Field Theory of Acousto-Optic Signal Processing Devices, Artech, I 992. J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design and Applications, Wiley, 1992. F. V. Bunkin, A. A. Kolomensky, and V. G. Mikhalevich, Lasers in Acoustics, Harwood, 1991. P. K. Das and C. M. De Cusatis, Acousto-Optic Signal Processing: Fundamentals & Applications, Artech, 199]. C. S. Tsai, Guided- Wave Acoustooptics: Interactions, Devices, and Applications, Springer-Verlag, 1990. L. N. Magdich and V. Ya. Molchanov, Acoustooptic Devices and Their Applications, Gordon and Breach, 1989. M. Gottlieb, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflec- tion, Marcel Dekker, 1983. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals, Plenum, 1981. D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics, Wiley, 1979. J. Sapriel, Acousto-Optics, Wiley, 1979. M. V. Berry, The Diffraction of Light by Ultrasound, Academic Press, 1966. Articles A. Korpel, ed., Selected Papers on Acousto Optics, SPIE Optical Engineering Press (Milestone Series Volume 16), 1990. Special issue on acoustooptic signal processing, Proceedings of the IEEE, vol. 69, no. 1, 1981. PROBLEMS 19.1-1 Diffraction of Light from Various Periodic Structures. Discuss the diffraction of an optical plane wave of wavelength A from the following periodic structures, indicating in each case the geometrical configuration and the frequency shift(s): (a) An acoustic traveling wave of wavelength A. (b) An acoustic standing wave of wavelength A. (c) A graded-index transparent medium with refractive index varying sinusoidally with position (period A).
PROBLEMS 833 (d) A stratified medium made of parallel layers of two materials of different refractive indexes, alternating to form a periodic structure of period A (see Sec. 7.1 C). * 19.1- 2 Bragg Diffraction as a Scattering Process. An incident optical wave of angular frequency w, wavevector k, and complex envelope A interacts with a medium perturbed by an acoustic wave of angular frequency 0 and wavevector q, and creates a light source S described by (19.1-32). The angle () corresponds to upshifted Bragg diffraction, so that the scattering light source is S = Re{5r(r) exp(jwrt)}, where 5r(r) = -(no/n) k; Aexp( -jk r . r), W r = w + 0, and k r = k + q. This source emits a scattered field E. Assuming that the incident wave is undepleted by the acousto-optic interaction (first Born approximation, i.e., A remains approximately constant), the scattered light may be obtained by solving the Helmholtz equation \72 E + k 2 E = -5. This equation has the far-field solution (see Probe 21. 2-6) E(r) exp -jkr) r Sr(r') exp(jkr. r') dr', 7rr Jv where r is a unit vector in the direction of r, k = 27r / A, and V is the volume of the source. Use this equation to determine an expression for the reflectance of the acousto-optic cell when the Bragg condition is satisfied. Compare the result with (19.1-22). 19.1-3 Condition for Raman-Nath Diffraction. Derive an expression for the maximum width Ds of an acoustic beam of wavelength A that permits Raman-Nath diffraction of light of wavelength A (see Fig. 19.1- LO). 19.1-4 Combined Acousto-Optic and Electro-Optic Modulation. One end of a lithium niobate (LiNb0 3 ) crystal is placed inside a microwave cavity with an electromagnetic field at 3 GHz. As a result of the piezoelectric effect (the electric field creating a strain in the material), an acoustic wave is launched. Light from a He-Ne laser (Ao = 633 nm) is reflected from the acoustic wave. The refractive index is n = 2.3 and the velocity of sound is V s = 7.4 km/s. Determine the Bragg angle. Since lithium niobate is also an electro-optic material, the applied electric field modulates the refractive index, which in turn modulates the phase of the incident light. Sketch the spectrum of the reflected light. If the microwave electric field is a pulse of short duration, sketch the spectrum of the reflected light at different times indicating the contributions of the electro-optic and acousto-optic effects. 19.2-4 Acousto-Optic Modulation. Devise a system for converting a monochromatic optical wave with complex wavefunction U(t) = A exp(jwt) into a modulated wave of complex wave- function A cos(Ot) exp(jwt) by making use of an acousto-optic cell with an acoustic wave s(x, t) = 50 cos(Ot - qx). Hint: Consider the use of upshifted and downshifted Bragg reflections. 19.2-5 Frequency-Shift-Free Bragg Reflector. Design an acousto-optic system that deflects light without imparting a frequency shift. Hint: Use two Bragg cells. * 19.3-2 Front Bragg Diffraction. A transverse acoustic wave of wavelength A travels in the x direction in a uniaxial crystal with refractive indexes no and ne and optic axis in the z direction. Derive an expression for the wavelength Ao of an incident optical wave, traveling in the x direction and polarized in the z direction, that satisfies the condition of Bragg diffraction. What is the polarization of the front reflected wave? Determine A if Ao 633 nm, ne = 2.200, and no = 2.286.
CHAPTER o ELECTRO-OPTICS 20.1 PRINCIPLES OF ELECTRO-OPTICS 836 A. Pockels and Kerr Effects B. Electro-Optic Modulators and Switches C. Scanners D. Directional Couplers E. Spatial Light Modulators *20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA 849 A. Pockels and Kerr Effects B. Modulators 20.3 ELECTRO-OPTICS OF LIQUID CRYSTALS 856 A. Wave Retarders and Modulators B. Spatial Light Modulators *20.4 PHOTO REFRACTIVITY 863 20.5 ELECTROABSORPTION 868 , . Friedrich Pockels (1865-1913) described the linear electro-optic effect in 1893. John Kerr (1824-1907) discovered the quadratic electro-optic effect in 1875. 834
Certain transparent materials change their optical properties when subjected to an electric field. This is a result of forces that distort the positions, orientations, or shapes of the molecules constituting the material. The electro-optic effect is a change in the refractive index that results from the application of a steady or low-frequency electric field (Fig. 20.0-1). An electric field applied to an anisotropic optical material modifies its refractive indexes and thereby the effect that it has on polarized light passing through it. Electric field im-G Light Figure 20.0-1 A steady electric field applied to an electro-optic material changes its refractive index. This in turn changes the effect of the material on light traveling through it. The electric field therefore controls the light. ---.. Electro-optic material The dependence of the refractive index on the applied electric field usually assumes one of the two following forms: . The refractive index changes in proportion to the applied electric field, an effect known as the linear electro-optic effect or Pockels effect. . The refractive index changes in proportion to the square of the applied electric field, an effect known as the quadratic electro-optic effect or Kerr effect. The change in the refractive index is typically small. Nevertheless, the phase of an optical wave propagating through an electro-optic medium can be modified sig- nificantly if the distance of travel substantially exceeds the wavelength of light. As an example, if the refractive index is increased by 10- 5 by virtue of the presence of the electric field, an optical wave propagating a distance of 10 5 wavelengths will experience an additional phase shift of 2w. Materials whose refractive index can be modified by means of an applied electric field are useful for producing electrically controllable optical devices, as indicated by the following examples: . A lens comprising a material whose refractive index can be varied is a lens of controllable focal length. . A prism whose beam-bending capability is controllable can be used as an optical scanning device. . Light transmitted through a transparent plate of controllable refractive index un- dergoes a controllable phase shift so that the plate can be used as an optical phase modulator. . An anisotropic crystal whose refractive indexes can be changed serves as a wave retarder of controllable retardation; it may be used to change the polarization properties of light. . A wave retarder placed between two crossed polarizers gives rise to transmitted light whose intensity is dependent on the phase retardation (see Sec. 6.6B). The transmittance of such a device is therefore electrically controllable so that it can be used as an optical intensity modulator or an optical switch. Controllable components such as these find substantial use in optical communications and in optical signal-processing applications. An electric field can instead modify the optical properties of a material via ab- sorption. A semiconductor material is normally optically transparent to light whose 835
836 CHAPTER 20 ELECTRO-OPTICS wavelength is longer than the bandgap wavelength (see Sec. 16.2B). However, an applied electric field can reduce the bandgap of the material, thereby facilitating ab- sorption and converting the material from transparent to opaque. This effect, known as electroabsorption, is useful for making optical modulators and switches. This Chapter We begin with a description of the electro-optic effect and the principles of electro- optic modulation and scanning. The initial presentation in Sec. 20.1 is simplified by deferring the detailed consideration of anisotropic effects to Sec. 20.2. Section 20.3 is devoted to the electro-optic properties of liquid crystals. An electric field applied to the molecules of a liquid crystal causes them to alter their orientations. This leads to changes in the optical properties of the medium, i.e., it exhibits an electro- optic effect. The molecules of a twisted nematic liquid crystal are organized in a helical pattern so that they normally act as polarization rotators. An applied electric field can be used to remove the helical pattern, thereby deactivating the polarization rotatory power of the material. Turning the electric field off results in the material regaining its original helical structure and therefore its rotatory power. Thus, the device acts as a dynamic polarization rotator. The use of additional fixed polarizers permits such a polarization rotator to serve as an intensity modulator or a switch. This behavior is the basis of most liquid-crystal display devices. The electro-optic properties of photorefractive media are considered in Sec. 20.4. These are materials in which the absorption of light creates an internal electric field, which, in turn, initiates an electro-optic effect that alters the optical properties of the medium. Thus, the optical properties of the medium are indirectly controlled by the light incident on it. Photorefractive devices therefore permit light to control light. Finally, a brief introduction to electroabsorption is provided in Sec. 20.5. 20.1 PRINCIPLES OF ELECTRO-OPTICS A. Pockels and Kerr Effects The refractive index of an electro-optic medium is a function n(E) of an applied steady (or slowly varying) electric field E. The function n(E) varies only slightly with E so that it can be expanded in a Taylor series about E == 0, n(E) == n + alE + !a2E2 + . . . , (20.1-1 ) where the coefficients of expansion are n == n(O), al == (dn/dE)IE=O, and a2 == (d 2 n/dE2)IE=O. For reasons that will become apparent below, it is conventional to write (20.1-1) in terms of two new coefficients, t == -2al/n3 and 5 == -a2/n 3 , known as the electro-optic coefficients, so that n(E) == n - !t n 3 E - !s n 3 E 2 + . . . . (20.1-2) The second- and higher-order terms of this series are typically many orders of magni- tude smaller than n. Terms higher than the third can safely be neglected. For future use it is convenient to derive an expression for the electric impermeability, 11 == Eo/E == 1/n2, of the electro-optic medium as a function of E. The parameter 11 is useful in describing the optical properties of anisotropic media (see Sec. 6.3A). The incremental change 11 == (d11/dn)n == (-2/n 3 )(-!tn 3 E - !sn 3 E2) ==
20.1 PRINCIPLES OF ELECTRO-OPTICS 837 t E + 5 E 2 , so that 11 (E) 11 + t E + 5 E 2 , (20.1-3) where 11 == 11 (0). The electro-optic coefficients t and 5 are therefore simply the coef- ficients of proportionality of the two terms of 11 with E and E 2 , respectively. This explains the seemingly odd definitions of t and 5 in (20.1-2). The values of the coefficients t and 5 depend on the direction of the applied electric field and the polarization of the light, as will be discussed in Sec. 20.2. Pockels Effect In many materials the third term of (20.1-2) is negligible in comparison with the second, whereupon n(E) n - !tn 3 E, (20.1-4 ) Pockels Effect as illustrated in Fig. 20.1-1 (a). The medium is then known as a Pockels medium (or a Pockels cell). The coefficient t is called the Pockels coefficient or the linear electro- optic coefficient. Typical values of t lie in the range 10- 12 to 10- 10 fiN (1 to 100 pm/V). For E == 10 6 Vim (10 kV applied across a cell of thickness 1 em), for example, the term !t 72 3 E in (20.1-4) is on the order of 10- 6 to 10- 4 . Changes in the refractive index induced by electric fields are indeed very small. Common crystals used as Pockels cells include NH 4 H 2 P0 4 (ADP), KH 2 P0 4 (KDP), LiNb0 3 , LiTa03, and CdTe. neE) (a) o E (b) o E Figure 20.1-1 Dependence of the refractive index on the electric field: (a) Pockels medium; (b) Kerr medium. Kerr Effect If the material is centrosymmetric, as is the case for gases, liquids, and certain crystals, n( E) must be an even symmetric function [see Fig. 20.1-1 (b)] since it must be invariant to the reversal of E. Its first derivative then vanishes, so that the coefficient t must be zero, whereupon n(E) n - !£1n 3 E 2 . (20.1-5) Kerr Effect The material is then known as a Kerr medium (or a Kerr cell). The parameter 5 is called the Kerr coefficient or the quadratic electro-optic coefficient. Typical values of 5 are 10- 18 to 10- 14 m 2 /V 2 in crystals and 10- 22 to 10- 19 m 2 N 2 in liquids. For E == 10 6
838 CHAPTER 20 ELECTRO-OPTICS Vim the term !s n 3 E 2 in (20.1-5) is on the order of 10- 6 to 10- 2 in crystals and 10- 10 to 10- 7 in liquids. B. Electro-Optic Modulators and Switches Phase Modulators A beam of light traversing a Pockels cell of length L to which an electric field E is applied undergoes a phase shift cP == n(E)koL == 27rn(E)L/ Ao, where Ao is the free- space wavelength. Using (20.1-4), we have tn 3 EL cP CPo - 7r Ao (20.1-6) where CPo == 27rnL / Ao. If the electric field is obtained by applying a voltage V across two faces of the cell separated by distance d, then E == V / d, and (20.1-6) gives cp ------------ 1 7r ---------- -- 00 V cP == CPo - 7r V 7r ' (20.1-7) Phase Modulation V 7r v where T T _ d Ao V7r - L t n 3 . (20.1-8) Half-Wave Voltage The parameter V 7r , known as the half-wave voltage, is the applied voltage at which the phase shift changes by 7r. Equation (20.1-7) expresses a linear relation between the optical phase shift and the voltage. One can therefore modulate the phase of an optical wave by varying the voltage V that is applied across a material through which the light passes. The parameter V 7r is an important characteristic of the modulator. It depends on the material properties (n and t), on the wavelength Ao, and on the aspect ratio d / L. The electric field may be applied in a direction perpendicular to the direction of light propagation (transverse modulators) or parallel thereto (longitudinal modulators), in which case d == L (Fig. 20.1-2). The value of the electro-optic coefficient t depends on the directions of propagation and the applied field since the crystal is, in general, anisotropic (as explained in Sec. 20.2). Typical values of the half-wave voltage are in the vicinity of 1 to a few kilovolts for longitudinal modulators, and hundreds of volts for transverse modulators. The speed at which an electro-optic modulator operates is limited by electrical capacitive effects and by the transit time of the light through the material. If the electric field E(t) varies significantly within the light transit time T, the traveling optical wave will be subjected to different electric fields as it traverses the crystal. The modulated
20.1 PRINCIPLES OF ELECTRO-OPTICS 839 v v v .. o.o. -- ...... .' - - ,".' - - - - ". .........-....".- . .. .¥- ......"- :...-.-.::...... - - .. .. . - -.. ... . - - --:-.." -::- » ...... . .:......-. -.'. .. ...... .. .... ... .' '. ... (a) (b) (c) Figure 20.1-2 (a) Longitudinal modulator. The electrodes may take the shape of washers or bands, or may be transparent conductors. (b) Transverse modulator. (c) Traveling-wave transverse modulator. phase at a given time t will then be proportional to the average electric field E t at times from t T to t. As a result, the transit-time-limited modulation bandwidth is 1 T. One method of reducing this time is to apply the voltage V at one end of the crystal while the electrodes serve as a transmission line, as illustrated in Fig. 20.1- 2(c). If the velocity of the traveling electrical wave matches that of the optical wave, transit time effects can, in principle, be eliminated. Commercial modulators in the forms shown in Fig. 20.1-2 generally operate at several hundred MHz, but modulation speeds of several GHz are possible. Electro-optic modulators can also be constructed as integrated-optical devices. These devices operate at higher speeds and lower voltages than do bulk devices. An optical waveguide is fabricated in an electro-optic substrate (often LiNb0 3 ) by indiffusing a material such as titanium to increase the refractive index. The electric field is applied to the waveguide using electrodes, as shown in Fig. 20.] -3. Because the configuration is transverse and the width of the waveguide is much smaller than its length (d « L), the half-wave voltage can be as small as a few volts. These modulators have been operated at speeds in excess of 1 00 GHz. Light can be conveniently coupled into, and out of, the modulator by the use of optical fibers. v · o Input light Waveguide / --- Electrodes ..... Cross section -. ..... -. ..... ...... -. ..... . --- . '. ". -- . '...... o Modulated light Figure 20.1-3 An integrated-optical phase modulator using the electro-optic effect. Dynamic Wave Retarders An anisotropic medium has two linearly polarized normal modes that propagate with different velocities, say Co nl and Co n2 (see Sec. 6.3B). If the medium exhibits the
840 CHAPTER 20 ELECTRO-OPTICS Pockels effect, then in the presence of a steady electrical field E the two refractive indexes are modified in accordance with (20.1-4), i.e., nl(E) nl - tInrE n2(E) n2 - !t2nE, (20.1-9) (20.1-10) where tl and t2 are the appropriate Pockels coefficients (anisotropic effects are exam- ined in detail in Sec. 20.2). After propagation a distance L, the two modes undergo a relative phase retardation given by r == ko[nl(E) - n2(E)]L == ko(nl - n2)L - !ko(tlnr - t2n)EL. (20.1-11) If E is obtained by applying a voltage be V between two surfaces of the medium that are separated by a distance d, (20.1-11) can be written in compact form as r ro _--] n o o V r == r o - 7r- V 7r ' (20.1-12) Phase Retardation V7f v where ro == ko(nl - n2)L is the phase retardation in the absence of the electric field and v; _ d Ao 7r - L 3 3 tl n l - t2 n 2 (20.1-13) Retardation Half-Wave Voltage is the applied voltage necessary to obtain a phase retardation 7r. Equation (20.1-12) in- dicates that the phase retardation is linearly related to the applied voltage. The medium serves as an electrically controllable dynamic wave retarder. Intensity Modulators: Use of a Phase Modulator in an Interferometer Phase delay (or retardation) alone does not affect the intensity of a light beam. How- ever, a phase modulator placed in one branch of an interferometer can function as an intensity modulator. Consider, for example, the Mach-Zehnder interferometer illus- trated in Fig. 20.1-4. If the beamsplitters divide the optical power equally, the intensity transmitted through one output port of the interferometer 10 is related to the incident intensity Ii by 10 == Ii + Ii cas <P == Ii cos 2 ( <p/2), (20.1-14) where <P == <PI - <P2 is the difference between the phase shifts encountered by light as it travels through the two branches (see Sec. 2.5A). The transmittance of the interfer- ometer is 'J' == 10/ Ii == cas 2 (<p /2). Because of the presence of the phase modulator in branch 1, according to (20.1- 7) we have <PI == <PIa - 7r V /V 7r , SO that <P is controlled by the applied voltage V in
20.1 PRINCIPLES OF ELECTRO-OPTICS 841 - 10 'I(V) - Branch 2 Branch] 0.5 r- Y7r,-t v Ii - o Figure 20.1-4 A phase modulator placed in one branch of a Mach-Zehnder interferometer can serve as an intensity modulator. The transmittance of the interferometer 'J(V) == 10/ Ii varies periodically with the applied voltage V. By operating in a limited region near point B, the device acts as a linear intensity modulator. If V is switched between points A and C, the device serves as an optical switch. accordance with the linear relation cP == CPI - CP2 == CPo - 7r V /V 7r , where the constant CPo == 'PIa - CP2 depends on the optical path difference. The transmittance of the device is therefore a function of the applied voltage V, 2 ( CPo 7r V ) 'J(V) == cos - - -- . 2 2V 7r (20.1-15) Transmittance This function is plotted in Fig. 20.1-4 for an arbitrary value of CPo. The device may be operated as a linear intensity modulator by adjusting the optical path difference so that CPo == 7r /2 and operating in the nearly linear region around 'J == 0.5. Alternatively, the optical path difference may be adjusted so that CPo is a multiple of 27r. In this case 'J(O) == 1 and 'J(V 7r ) == 0, so that the modulator switches the light on and off as V is switched between 0 and V 7r . A Mach-Zehnder intensity modulator may also be constructed in the form of an integrated-optical device. Waveguides are placed on a substrate in the geometry shown in Fig. 20.1-5. The beam splitters are implemented by the use of waveguide Y's. The optical input and output may be carried out by optical fibers. Commercially available integrated-optical modulators generally operate at speeds of a few GHz but modulation speeds exceeding 25 GHz have been achieved. Input ht V o;>S. Modulated light 10 Figure 20.1-5 An integrated-optical in- tensity modulator (or optical switch). A Mach-Zehnder interferometer and an electro- optic phase modulator are implemented using optical waveguides fabricated from a material such as LiNb0 3 . Intensity Modulators: Use of a Retarder Between Crossed Polarizers As described in Sec. 6.6B, a wave retarder (retardation r) sandwiched between two crossed polarizers, placed at 45° with respect to the retarder's axes (see Fig. 6.6- 4), has an intensity transmittance 'J == sin 2 (r / 2). If the retarder is a Pockels cell,
842 CHAPTER 20 ELECTRO-OPTICS then r is linearly dependent on the applied voltage V as provided in (20.1-12). The transmittance of the device is then a periodic function of V, .2 ( rO 7r V ) ':reV) = sm "2 - 2 V 7r ' (20.1-16) Transmittance as shown in Fig. 20.1-6. By changing V, the transmittance can be varied between 0 (shutter closed) and 1 (shutter open). The device can also be used as a linear modulator if the system is operated in the region near 'J(V) == 0.5. By selecting r 0 == 7r /2 and V « V 7r , 'J(V) == sin 2 ( 7r _ 7r ) ':reO) + d':r V = _ 7f 4 2 V 7r dV v=o 2 2 V 7r ' so that 'J(V) is a linear function with slope 7r /2V 7r representing the sensitivity of the modulator. The phase retardation ro can be adjusted either optically (by assisting the modulator with an additional phase retarder, a compensator) or electrically by adding a constant bias voltage to V. (20.1-17) . '1(V) 1 0.5 Polarizer 0 B : _hm_mu -mmu---4!'tr-t ..... (a) V7r1 v (b) Figure 20.1-6 (a) An optical intensity modulator using a Pockels cell placed between two crossed polarizers. (b) Optical transmittance versus applied voltage for an arbitrary value of r 0; for linear operation the cell is biased near the point B. In practice, the maximum transmittance of the modulator is smaller than unity because of losses caused by reflection, absorption, and scattering. Furthermore, the minimum transmittance is greater than 0 because of misalignments of the direction of propagation and the directions of polarizations relative to the crystal axes and the polarizers. The ratio between the maximum and minimum transmittances is called the extinction ratio. Ratios higher than 30 dB (1000: 1) are possible. c. Scanners An optical beam can be deflected dynamically by using a prism with an electrically controlled refractive index. The angle of deflection introduced by a prism of small apex angle a and refractive index n is f) (n - 1) a [see (1.2- 7)]. An incremental change of the refractive index n caused by an applied electric field E corresponds to an incremental change of the deflection angle, f) == an == -latn 3 E == -latn 3 V j d 2 2 ' (20.1-18)
20.1 PRINCIPLES OF ELECTRO-OPTICS 843 where V is the applied voltage and d is the prism width [Fig. 20.1-7(a)]. By varying the applied voltage V, the angle f:10 varies proportionally, so that the incident light is scanned. (a) +v \. "0 ", , , 0, . --- " ---..,.... -v · , -- j-jj D ... , , ", , I L I (b) +v · , "\ Figure 20.1-7 (a) An electro-optic prism. The deflection angle e is controlled by the applied voltage. (b) An electro-optic double prism. It is often more convenient to place triangularly shaped electrodes defining a prism on the rectangular crystal. Two, or several, prisms can be cascaded by alternating the direction of the electric field, as illustrated in Fig. 20.1- 7 (b). An important parameter that characterizes a scanner is its resolution, i.e., the number of independent spots it can scan. An optical beam of width D and wavelength Ao has an angular divergence 80 Ao D [see (4.3-7)]. To minimize that angle, the beam should be as wide as possible, ideally covering the entire width of the prism itself. For a given maximum voltage V corresponding to a scanned angle f:10, the number of independent spots is given by latn 3 V d 2 Ao D . (20.1-19) 80 Substituting a L D and V n d L Ao tn 3 , we obtain V N , 2V n (20.1-20) from which V 2NV n . This is a discouraging result. To scan N independent spots, a voltage 2N times greater than the half-wave voltage is necessary. Since V n is usually large, making a useful scanner with N » 1 requires unacceptably high voltages. More commonly used scanners therefore include mechanical and acousto-optic scanners (see Secs. 19.2B and 23.3B). The process of double refraction in anisotropic crystals (see Sec. 6.3E) introduces a lateral shift of an incident beam parallel to itself for one polarization and no shift for the other polarization. This effect can be used for switching a beam between two parallel positions by switching the polarization. A linearly polarized optical beam is transmitted first through an electro-optic wave retarder acting as a polarization rotator and then through the crystal. The rotator controls the polarization electrically, which determines whether the beam is shifted laterally, as illustrated in Fig. 20.1-8. D. Directional Couplers An important application of the electro-optic effect is in controlling the coupling be- tween two parallel waveguides in integrated-optical device. An electric field can be used to transfer the light from one waveguide to the other, so that the device serves as an electrically controlled directional coupler.
844 CHAPTER 20 ELECTRO-OPTICS Electro-optic polarization rotator Birefringent crystal Figure 20.1-8 A position switch based on electro-optic phase retar- dation and double refraction. The coupling of light between two parallel single-mode planar waveguides [Fig. 20.1-9(a)] was examined in Sec. 8.5B. It was shown that the optical powers carried by the two waveguides, PI (z) and P 2 (z), are exchanged periodically along the direction of propagation z. Two parameters govern the strength of this coupling process: the coupling coefficient e (which depends on the dimensions, wavelength, and refractive indexes), and the mismatch of the propagation constants /::::,./3 == /31 - /32 == 27r /::::"n / Ao, where /::::"n is the difference between the refractive indexes of the waveguides. If the waveguides are identical, with /::::"(3 == 0 and P 2 (0) == 0, then at a distance z == La == 7r /2e, called the transfer distance or coupling length, the power is transferred completely from waveguide 1 into waveguide 2, i.e., PI (La) == 0 and P 2 (L o ) == PI (0), as illustrated in Fig. 20.] -9(a). PI (0) Waveguide 1 Waveguide 2 P 2 (La) 'I' 1 ,-,- ,- "P2(Z) / ., ,- o ' ,- o La z (a) o J3n (3Lo (b) Figure 20.1-9 (a) Exchange of power between two parallel weakly coupled waveguides that are identical, with the same propagation constant (3. At z == 0 all of the power is in waveguide 1. At z == Lo all of the power is transferred into waveguide 2. (b) Dependence of the power-transfer ratio :r == p 2 (Lo) / PI (0) on the phase mismatch parameter (3 Lo. For a waveguide of length La and /::::"(3 #- 0, the power-transfer ratio 'J == P 2 (L o )/ PI (0) is a function of the phase mismatch [see (8.5-12a)], 7r 2 [ 1 ( /::::"(3 La ) 2 ] 'J = 4 sinc 2 2 1 + 7r ' (20.1-21) where sinc(x) sin(7rx)j(nx). Figure 20.1-9(b) illustrates this dependence. The ratio has its maximum value of unity at /::::,./3 La == 0, decreases with increasing /::::,./3 La, and vanishes when /::::,./3 La == V3 7r, at which point the optica1 power is not transferred to waveguide 2.
20.1 PRINCIPLES OF ELECTRO-OPTICS 845 A dependence of the coupled power on the phase mismatch is the key to making electrically activated directional couplers. If the mismatch /::::"(3 La is switched from 0 to J37r, the light remains in waveguide 1. Electrical control of /::::"(3 is achieved by use of the electro-optic effect. An electric field E applied to one of two, otherwise identical, waveguides alters the refractive index by /::::"n == - !n 3 r E, where r is the Pockels coefficient. This results in a phase shift /::::"(3 La == /::::"n(27r Lo/ Ao) == - (7r / Ao)n 3 r LoE. A typical electro-optic directional coupler has the geometry shown in Fig. 20.1-10. The electrodes are laid over two waveguides separated by a distance d. An applied voltage V creates an electric field E V / d in one waveguide and - V / d in the other, where d is an effective distance determined by solving the electrostatics problem (the electric-field lines go downward at one waveguide and upward at the other). The refractive index is incremented in one guide and decremented in the other. The result is a net refractive index difference 2/::::"n == -n 3 r (V/ d), corresponding to a phase mismatch !J.(3 La == -(27r / Ao)n 3 r (Lo/ d) V, which is proportional to the applied voltage V. Figure 20.1-10 An integrated electro-optic directional coupler. The voltage Va necessary to switch the optical power is that for which 1/::::"(3 La I == J37r, i.e., Va = V3 = J3 eAod La 2n 3 r 7r n 3 r' (20.1-22) where La == 7r /2e and e is the coupling coefficient. This is called the switching voltage. Since 1!J.(3 Lol == J3 7r VIVo, (20.1-21) gives 7r 2 [ 1 'J = 4 sinc 2 2 1+3( )2] . (20.1-23) Coupling Efficiency This equation (plotted in Fig. 20.] -11) governs the coupling of power as a function of the applied voltage V. An electro-optic directional coupler is characterized by its coupling length La, which is inversely proportional to the coupling coefficient e, and its switching voltage va, which is directly proportional to e. The key parameter is therefore e, which is governed by the geometry and the refractive indexes. Integrated-optic directional couplers may be fabricated, for example, by diffusing titanium into high-purity LiNb0 3 substrates. The switching voltage Va is typically less
846 CHAPTER 20 ELECTRO-OPTICS T Vo v Figure 20.1-11 Dependence of the coupling effi- ciency on the applied voltage V. When V == 0, all of the optical power is coupled from waveguide 1 into waveguide 2; when V == V o , all of the optical power remains in waveguide 1. than 10 V, and the operating speeds can exceed 10 GHz. The light beams are focused to spot sizes of a few Mm. The ends of the waveguide may be permanently attached to single-mode polarization-maintaining optical fibers (see Sec. 9.1 C). Increased band- widths can be obtained by making use of a traveling-wave version of this device. EXERCISE 20.1-1 Coupling-Efficiency Spectral Response. Equation (20.1-22) indicates that the switching voltage V o is proportion al to the wavelength. Assume that the appliecl voltage V == V o for a particular value of the wavelength Ao, so that the coupling efficiency T == 0 at Ao.lf, instead, the incident wave has wavelength Ao, plot the coupling efficiency T as a function of Ao - Ao. Assume that the coupling coefficient e and the material parameters nand t are approximately independent of wavelength. E. Spatial Light Modulators A spatial light modulator is a device that modulates the intensity of light at different positions by prescribed factors (Fig. 20.1-12). It is a planar optical element of control- lable intensity transmittance 'J(x, y). The transmitted light intensity Io(x, y) is related to the incident light intensity Ii (x, y) by the product 10 (x, y) == Ii (x, y )'J( x, y). If the incident light is uniform [i.e., Ii(x, y) is constant], the transmitted light intensity is proportional to 'J (x, y). The "image" 'J (x, y) is then imparted to the transmitted light, much like "reading" the image stored in a transparency by uniformly illuminating it in a slide projector. In a spatial light modulator, however, 'J( x, y) is controllable. In an electro-optic modulator the control is electrical. y Transmittance T(x,y) Incident light x Figure 20.1-12 The spatial light modulator. To construct a spatial light modulator using the electro-optic effect, some mecha- nism must be devised for creating an electric field E (x, y) proportional to the desired transmittance 'J( x, y) at each position. This is not easy. One approach is to place an array of transparent electrodes on small plates of electro-optic material placed between crossed polarizers and to apply on each electrode an appropriate voltage (Fig. 20.1- 13). The voltage applied to the electrode centered at the position (Xi, Yi), i == 1,2, . . .
20.1 PRINCIPLES OF ELECTRO-OPTICS 847 is made proportional to the desired value of 'J Xi, Yi (see, e.g., Fig. 20.1-6). If the number of electrodes is sufficiently large, the transmittance approximates 'J x, y . The system is in effect a parallel array of longitudinal electro-optic modulators operated as intensity modulators. However, it is not practical to address a large number of these electrodes independently; nevertheless we will see that this scheme is practical in the liquid-crystal spatia] light modulators used for display, since the required voltages are low (see Sec. 20.3B). Figure 20.1-13 An electrically addressable array of longitudinal electro-optic modulators. Optically Addressed Electro-Optic Spatial Light Modulators One method of optical1y addressing an electro-optic spatial light modulator is based on the use of a thin layer of photoconductive material to create the electric field required to operate the modulator (Fig. 20.1-14). The conductivity of a photoconductive material is proportional to the intensity of light to which it is exposed (see Sec. 18.2). When illuminated by light of intensity distribution Iw x, Y , a spatial pattern of conductance G x, Y ex Iw x, Y is created. The photoconductive layer is placed between two elec- trodes that act as a capacitor. The capacitor is initially charged and the electrical charge leakage at the position x, y is proportional to the local conductance G x, y . As a result, the charge on the capacitor is reduced in those regions where the conductance is high. The local voltage is therefore proportional to 1 G x, y and the corresponding electric field E x, y ex 1 G x, y ex 1 Iw x, y . If the transmittance 'J x, y [or the reflectance x, y ] of the modulator is proportional to the applied field, it must be inversely proportional to the initial light intensity Iw x, y . y Photoconducti ve Modulated material Electro-optic light \ / material Read light Write . Image IW<x,y) Mirror Transparent electrodes Figure 20.1-14 The electro-optic spatial light modulator uses a photoconductive ma- terial to create a spatial distribution of elec- tric field that is used to control an electro- optic material. x
848 CHAPTER 20 ELECTRO-OPTICS The Pockels Readout Optical Modulator An ingenious implementation of this principle is the Pockels readout optical modu- lator (PROM). One implementation makes use of a crystal of bismuth silicon oxide, Bi 12 Si0 20 (BSO), which has an unusual combination of optical and electrical prop- erties: (1) it exhibits the electro-optic (Pockels) effect; (2) it is photoconductive for blue light, but not for red light; and (3) it is a good insulator in the dark. The PROM (Fig. 20.1-15) comprises a thin wafer of BSO sandwiched between two transparent electrodes. The light that is to be modulated (read light) is transmitted through a polarizer, enters the BSO layer, and is reflected by a dichroic reflector, whereupon it crosses a second polarizer. The reflector reflects red light but is transparent to blue light. The PROM is operated as follows: . Priming: A large potential difference ( 4 kV) is applied to the electrodes and the capacitor is charged (with no leakage since the crystal is a good insulator in the dark). . Writing: Intense blue light of intensity distribution Iw (x, y) illuminates the crys- tal. As a result, a spatial pattern of conductance G (x, y) ex: I w (x, y) is created, the voltage across the crystal is selectively lowered, and the electric field decreases proportionally at each position, so that E (x, y) ex: 1/ G (x, y) ex: 1/ I w (x, y). As a result of the electro-optic effect, the refractive indexes of the BSO are altered, and a spatial pattern of refractive- index change n (x, y) ex: 1/ I TV (x, y) is created and stored in the crystal. . Reading: Uniform red light is used to read n(x, y) as with usual electro-optic intensity modulators [see Fig. 20.1-6(a)] with the polarizing beamsplitter playing the role of the crossed polarizers. . Erasing: The refractive-index pattern is erased by the use of a uniform flash of blue light. The crystal is again primed by applying 4 kV, and a new cycle begins. Dichroic reflector of red light Transparent electrode Transparent electrode Polarizing beamsplitter » Write light BSO (blue) Modulated light » t Incident read light (red) Figure 20.1-15 The Pockels readout optical modulator (PROM). Incoherent-to-Coherent Optical Converters In an optically addressed spatial light modulator, such as the PROM, the light used to write a spatial pattern into the modulator need not be coherent since photoconductive materials are sensitive to optical intensity. A spatial optical pattern (an image) may be written using incoherent light, and read using coherent light. This process of real- time conversion of a spatial distribution of natural incoherent light into a proportional spatial distribution of coherent light is useful in a number of optical data- and image- processing applications.
20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA 849 *20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA The basic principles and applications of electro-optics were presented in Sec. 20.1 in a simplified fashion; polarization and anisotropic effects were either ignored or introduced only generically. In this section a more complete analysis of the electro- optics of anisotropic media is presented. A brief refresher of some of the important properties of anisotropic media (see Sec. 6.3) is provided below. Crystal Optics: A Brief Rflfresher The optical properties of an anisotropic medium are characterized by a geometric construction called the index ellipsoid, L TJij XiXj - 1, J i,j==1,2,3, where TJij == TJji are elements of the impermeability tensor 11 == Eo € -1 . If the axes of the ellipsoid correspond to the principal axes of the medium, its dimensions along these axes are the principal refractive indexes nI, n2, and n3 (Fig. 20.2- 1): 2 / 2 2 / 2 2 / 2 1 Xl nl + X2 n2 + X3 n3 == . X2 Figure 20.2-1 The index ellipsoid. The coordinates (Xl, X2, X3) are the principal axes and nl, n2, n3 are the principal refractive indexes. The refractive indexes of the normal modes of a wave traveling in the direction k are na and nb. The index ellipsoid may be used to determine the polarizations and refractive indexes na and nb of the two normal modes of a wave traveling in an arbitrary direction in the anisotropic medium. This is accomplished by drawing a plane perpendicular to the direction of propagation that passes through the center of the ellipsoid. Its intersection with the ellipsoid is an ellipse whose major and minor axes have half-lengths equal to na and nb, as described in Sec. 6.3C. A. Pockels and Kerr Effects When a steady electric field E with components (El, E 2 , E 3 ) is applied to a crystal, the elements of the tensor 11 are altered. Each of the nine elements TJij becomes a function of E 1 , E 2 , and E 3 , i.e., TJij == TJij (E), so that the index ellipsoid is modified (Fig. 20.2-2). Once we know the functions TJij(E), we can determine the index ellipsoid and the optical properties for an arbitrary applied electric field E. The problem is simple in principle, but the implementation is often lengthy.
850 CHAPTER 20 ELECTRO-OPTICS Figure 20.2-2 The index ellipsoid is modified as a result of applying a steady electric field. Each of the elements llij (E) is a function of the three variables E == (E1' E 2 , E 3 ), which may be expanded in a Taylor series about E == 0, llij (E) == llij + L tijkEk + L SijklEkEl, k kl i,j,k,l == 1,2,3, (20.2-1 ) where llij == llij (0), tijk == Or]ij /oE k , Sijkl == 021lij /oEkoE l , and the derivatives are evaluated at E == o. Equation (20.2-1) is a generalization of (20.1-3), in which t is replaced by 3 3 == 27 coefficients {tij k}, and S is replaced by 3 4 == 81 coefficients { Sij kl}. The coefficients {tij k} are known as the linear electro-optic (Pockels) coef- ficients. They form a tensor of third rank. The coefficients {Sijkl} are the quadratic electro-optic (Kerr) coefficients. They form a fourth-rank tensor. Symmetry Because 11 is symmetric (llij == llji), t and S are invariant under pennutations of the indexes i and j, i.e., tijk == tjik and Sijkl == Sjikl. Also, the coefficients 5ijkl == 021lij /oEkoE l are invariant to permutations of k and I (because of the invariance to the order of differentiation), so that Sijkl == Sijlk. Because of this permutation symme- try, the nine combinations of the indexes i, j generate six instead of nine independent elements. The same reduction applies to the indexes k, I. Consequently, tijk has 6 x 3 independent elements, whereas Sijkl has 6 x 6 independent elements. It is conventional to rename the pair of indexes (i,j), i,j == 1,2,3, as a single index I == 1, 2, . . . ,6 in accordance with Table 20.2-1. The pair (k, I) is similarly replaced by an index K == 1,2, . . . ,6, in accordance with the same rule. Thus, the elements tijk and Sijkl are replaced by tlk and SIK, respectively. For example, t12k is denoted as t6k, S1231 is renamed S65, and so on. Hence, the third-rank tensor t is replaced by a 6 x 3 matrix and the fourth-rank tensor S is contracted to a 6 x 6 matrix. j\i 1 2 3 1 1 6 5 2 6 2 4 3 5 4 3 Table 20.2-1 Lookup table for the index I that represents the pair of indexes (i, j).a aThe pair (i, j) = (3,2), for example, is labeled I = 4. Crystal Symmetry The symmetry of the crystal adds more constraints to the entries of the t and 5 matrices. Some entries must be zero and others must be equal, or equal in magnitude and opposite in sign, or related by some other rule. For centrosymmetric materials, as an example, t vanishes and only the Kerr effect is exhibited. Lists of the coefficients of t and S and their symmetry relations for the 32 crystallographic point groups may be found in several of the books referenced in the reading list. Representative examples are provided in Tables 20.2-2 and 20.2-3.
20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA 851 Table 20.2-2 Pockels coefficients tlk for some representative crystal groups. 0 0 0 0 0 0 0 -t22 t13 0 0 0 0 0 0 0 t22 t13 0 0 0 0 0 0 0 0 t33 t41 0 0 t41 0 0 0 t51 0 0 t41 0 0 t41 0 t51 0 0 0 0 t41 0 0 t63 -t22 0 0 Cubic 43m Tetragonal 42m Trigonal 3m (e.g., GaAs, CdTe, InAs) (e.g., KDP, ADP) (e.g., LiNb0 3 , LiTa03) Table 20.2-3 Kerr coefficients 51 K for an isotropic medium. 511 512 512 0 0 0 512 511 512 0 0 0 512 512 511 0 0 0 544 = (511 - 512) 0 0 0 544 0 0 , 0 0 0 0 544 0 0 0 0 0 0 544 Pockels Effect The following procedure is used to determine the optical properties of an anisotropic material exhibiting the Pockels effect in the presence of an electric field E: 1. Find the principal axes and principal refractive indexes nl, n2, and n3 in the absence of E. 2. Find the coefficients {tijk} from the appropriate matrix for tlk, e.g., from Table 20.2-2, by using the rule that relates I to (i, j) provided in Table 20.2-1. 3. Detennine the elements of the impermeability tensor TJij(E) == TJij(O) + L tijkEk, k (20.2-2) where TJij (0) is a diagonal matrix with elements 1/ ni, 1 / n, and 1 / n. 4. Write the equation for the modified index ellipsoid L TJij(E) XiXj == 1. (20.2-3) 'lJ 5. Detennine the principal axes of the modified index ellipsoid by diagonaliza- tion, and find the corresponding principal refractive indexes nl (E), n2 (E), and n3(E). 6. Given the direction of light propagation, find the normal modes and their associ- ated refractive indexes from this index ellipsoid. EXAMPLE 20.2-1. Trigona/3m Crystals (LiNb0 3 and LiTa03). Trigonal 3m crystals are uniaxial (n1 = n2 = no, n3 = ne) with the matrix t provided in Table 20.2-2. Assuming that E = (0,0, E), i.e., that the electric field points along the optic axis (see Fig. 20.2-3), the modified index ellipsoid is readily shown to be ( + t 13 E) (xi + xD + ( + t33 E ) x = 1. (20.2-4 )
852 CHAPTER 20 ELECTRO-OPTICS This is an ellipsoid of revolution whose principal axes don't change when the electric field is applied. The ordinary and extraordinary indexes, no (E) and ne(E), respectively, are given by 1 1 n(E) == 2 + tl3 E (20.2-5) no 1 1 (20.2-6) n(E) == 2 + r33 E . ne Because the terms rl3E and r33E in (20.2-5) and (20.2-6) are small, we use the approximation (1 + )-1/2 1 - , valid for II « 1, to obtain no (E) no - nrI3E ne(E) ne - nr33E. (20.2- 7) (20.2-8) Note the similarity between these equations and the generic equation (20.1-4). We conclude that when an electric field is applied along the optic axis of this uniaxial crystal it remains uniaxial with the same principal axes, as shown in Fig. 20.2-3, but its refractive indexes are modified in accordance with (20.2-7) and (20.2-8). I Etl y no X2 z u VJ .&. oro . 1- J 1'1 3 E /1 2" n o t 13 Figure 20.2-3 Modification of the index ellipsoid of a trigonal 3m crystal such as LiNb0 3 resulting from the application of a steady electric field along the direction of the optic axis. .,./ x EXAMPLE 20.2-2. Tetragonal 42m Crystals (KDP and ADP). Carrying out the same process for this class of uniaxial crystals, and assuming that the electric field points along the optic axis (Fig. 20.2-4), we obtain the following equation for the index ellipsoid: X2 + X2 X2 I 2 2 + -% + 2r63ExIX2 == 1. no ne (20.2-9) The modified principal axes are obtained by rotating the coordinate system 45° about the z axis. Substituting x == (Xl + X2)/V2, x == (Xl - x2)/V2, X == X3 in (20.2-9), and relabeling the coordinate system as (Xl, X2, X3), leads to X x x n(E) + n(E) + n(E) == 1, (20.2-10) where 1 1 n(E) == n + r63 E , 1 1 == 2 - t63 E , n(E) no n3(E) == ne. (20.2-11 ) Cross-multiplying and using the Taylor-series approximation (1 + )-1/2 1 - yields
20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA 853 nl (E) no - nr63E n2(E) no + nr63E n3(E) = ne. (20.2-12) (20.2-13) (20.2-14 ) We conclude that the originally uniaxial crystal behaves as a biaxial crystal when subjected to an electric field in the direction of its optic axis, as illustrated in Fig. 20.2-4. z X3 UV) .R. oro ne /. / / / I I I I fc.-- ,- \ \ y x I I f¥ 1 3 E "2 no t63 X2 r Figure 20.2-4 Modification of the index ellipsoid resulting from the application of a steady electric field E along the direction of the optic axis of a uniaxial tetragonal 42m crystal such as KDP. EXAMPLE 20.2-3. Cubic 43m Crystals (GaAs, CdTe, and InAs). Assuming that the applied electric field points along a cubic axis of the material (taken as the z direction in Fig. 20.2-5), the index ellipsoid for these isotropic crystals (nl = n2 = n3 = n) becomes x 2 + x 2 + x 2 1 3 + 2t41Ex1X2 = 1, n (20.2-15) where r63 assumes the value t41 (see Table 20.2-2). As in Example 20.2-2, the new principal axes are rotated 45° about the z axis and the principal refractive indexes turn out to be n1(E) n - n3r41E n2(E) n + n3r41E n3(E) n. The applied field thus makes the isotropic crystal behave in a biaxial fashion (Fig. 20.2-5). Z X3 u V) ......-4 o.. n oro f Et l X2 Y x (20.2-16) (20.2-17) (20.2-18) Figure 20.2-5 Modification of the index ellipsoid as a result of applying a steady electric field E along a cubic axis of a 43m crystal such as GaAs.
854 CHAPTER 20 ELECTRO-OPTICS Cubic crystals have well-defined crystal axes but isotropic linear optical proper- ties. The imposition of a steady electric field disrupts the geometrical symmetry and leads to anisotropic optical properties, as is clear from Example 20.2-3. For initially anisotropic materials in which the applied electric field does not alter the principal axes, as in Example 20.2-1, the polarizations of the normal modes remain the same, but their associated refractive indexes become dependent on E. The medium can then be conveniently used as a phase modulator, wave retarder, or intensity modulator, in accordance with the generic theory provided in Sec. 20.1B. This principle is described further in Sec. 20.2B. Kerr Effect The optical properties of a Kerr medium can be determined by using the same proce- dure used for the Pockels medium, except that the coefficients llij(E) are given by llij(E) == llij(O) + LSijkZEkEZ. kZ (20.2-19) EXAMPLE 20.2-4. Kerr Effect in an Isotropic Medium. With a steady applied electric field E pointing along the z axis, we use the Kerr coefficients SIK in Table 20.2-3 for an isotropic medium to find the equation for the index ellipsoid, ( 1 2 ) ( 2 2 ) ( 1 2 ) 2 n 2 +SI2 E Xl +X 2 + n 2 +51l E X 3 == 1. (20.2- 20) This is the equation of an ellipsoid of revolution whose axis is the z axis, along the direction of the applied electric field. The principal refractive indexes no (E) and ne (E) are determined from 1 1 2 (20.2-21) n(E) == 2 +512E n 1 1 2 (20.2-22) n (E) == 2 + SIIE . n Since the rightmost terms in (20.2-21) and (20.2-22) are small, we again make use of the approxima- tion (1 + )-1/2 1 - to obtain no (E) n - n3s12E2 ne(E) n - n3sIIE2. (20.2-23) (20.2-24) Thus, a steady electric field E applied to an initially isotropic medium causes it to behave as a uniaxial crystal with the optic axis along the direction of the electric field. The ordinary and extraordinary indexes are quadratically decreasing functions of E. B. Modulators The principles of phase and intensity modulation using the electro-optic effect were outlined in Sec. 20.1B. Anisotropic effects were introduced only generically. Using the anisotropic theory presented in this section, the generic parameters t and s, which were used in Sec. 20.1, can now be determined for any given crystal and directions of the applied electric field and light propagation. Only Pockels modulators will be
20.2 ELECTRO-OPTICS OF ANISOTROPIC MEDIA 855 discussed, but the same approach can be applied to Kerr modulators. For simplicity, we assume that the direction of the electric field is such that the principal axes of the crystal are not altered as a result of modulation. We shall also assume that the direction of the wave relative to these axes is such that the planes of polarization of the normal modes are also not altered by the electric field. Phase Modulators A normal mode is characterized by a refractive index n(E) n - !tn 3 E, where nand t are the appropriate refractive index and Pockels coefficient, respectively, and E == V / d is the electric field obtained by applying a voltage V across a distance d. A wave traveling a distance L undergoes a phase shift V cP == CPo - 7r- V 1r (20.2-25) where CPo == 27rnLj Ao and T 7 _ d Ao V1r - Ltn 3 (20.2-26) is the half-wave voltage. The appropriate coefficients generically called nand t can be easily determined as demonstrated in the following example. EXAMPLE 20.2-5. Trigona/3m Crystals (LiNb0 3 and LiTa03). When an electric field is directed along the optic axis of this type of uniaxial crystal, the crystal remains uniaxial with the same principal axes (see Fig. 20.2-3). The principal refractive indexes are given by (20.2-7) and (20.2-8). The crystal can be used as a phase modulator in either of two configurations: Longitudinal Modulator: If a linearly polarized optical wave travels along the direction of the optic axis (parallel to the electric field), the appropriate parameters for the phase modulator are n = no, t = t13, and d = L. For LiNb0 3 , t13 = 9.6 pm/V, and no = 2.3 at Ao = 633 nm. Equation (20.2-26) then yields V 7r = 5.41 kV, the voltage necessary to change the phase by 7r. Transverse Modulator: If the wave travels in tQe x direction and is polarized in the z direction, the appropriate parameters are n = ne and t = t33. The width d is generally not equal to the length L. For LiNb0 3 at Ao = 633 nm, r33 = 30.9 pm/V, and ne = 2.2, giving a half-wave voltage V 7r = 1.9( d / L) kV. If d / L = 0.1, we obtain V 7r 190 V, which is significantly lower than the half-wave voltage for the longitudinal modulator. Intensity Modulators The difference in the dependence on the applied field of the refractive indexes of the two normal modes of a Pockels cell provides a voltage-dependent retardation, V f == f a - 7r- V 1r ' (20.2-27) where ro = 27r(nl - n2)L Ao (20.2-28) V 1r == ( d / L ) Ao 3 3 . tl n 1 - t2 n 2 (20.2-29)
856 CHAPTER 20 ELECTRO-OPTICS If the cell is placed between crossed polarizers, the system serves as an intensity modulator (see Sec. 20.1B). It is not difficult to determine the appropriate indexes nl and n2, and coefficients tl and t2, as illustrated by the following example. EXAMPLE 20.2-6. Tetragonal 42m Crystals (KDP and ADP). As described in Exam- ple 20.2-2, when an electric field is applied alo.ng the optic axis of this uniaxial crystal, it behaves as a biaxial crystal. The new principal axes are the original axes rotated by 45° about the optic axis. Assume a longitudinal modulator configuration (d / L = 1) in which the wave travels along the optic axis. The two normal modes have refractive indexes given by (20.2-12) and (20.2-13). The appropriate coefficients to be used in (20.2-29) are therefore nl = n2 = no, tl = t63, t2 = -t63, and d = L, so that r 0 = 0 and v _ Ao 7r - 2t63n . (20.2-30) For KDP at Ao = 633 nm, V 7r = 8.4 kV. EXERCISE 20.2-1 Intensity Modulation Using the Kerr Effect. Use (20.2-23) and (20.2-24) to determine an expression for the phase shift c.p and the phase retardation r in a longitudinal Kerr modulator made of an isotropic material, as functions of the applied voltage V. Derive expressions for the half-wave voltages V 7r in each case. 20.3 ELECTRO-OPTICS OF LIQUID CRYSTALS As described in Sec. 6.5, the elongated molecules of nematic liquid crystals tend to have ordered orientations that are altered when the material is subjected to mechanical or electric forces. Because of their anisotropic nature, liquid crystals can be arranged to serve as wave retarders or polarization rotators. In the presence of an electric field, their molecular orientation is modified, so that their effect on polarized light is altered. Liquid crystals can therefore be used as electrically controlled optical wave retarders, modulators, and switches. These devices are particularly useful in display technology. A. Wave Retarders and Modulators Electrical Properties of Nematic Liquid Crystals The liquid crystals used to make electro-optic devices are usually of sufficiently low conductivity that they can be regarded as ideal dielectric materials. Because of the elon- gated shape of the constituent molecules, and their ordered orientation, liquid crystals have anisotropic dielectric properties with uniaxial symmetry (see Sec. 6.3A). The electric permittivity is Ell for electric fields pointing in the direction of the molecules and E.l in the perpendicular direction. Liquid crystals for which Ell > E.l (positive uniaxial) are usually selected for electro-optic applications. When a steady (or low frequency) electric field is applied, electric dipoles are in- duced and the resultant electric forces exert torques on the molecules. The molecules rotate in a direction such that the free electrostatic energy, - E . D == - [E.l E; +
20.3 ELECTRO-OPTICS OF LIQUID CRYSTALS 857 E 1.. E + Ell E], is minimized (here, E 1 , E 2 , and E3 are components of E in the direc- tions of the principal axes). Since Ell > E1.., for a given direction of the electric field, minimum energy is achieved when the molecules are aligned with the field, so that El == E 2 == 0, E == (O,O,E), and the energy is then -EIIE2. When the alignment is complete the molecular axis points in the direction of the electric field (Fig. 20.3- 1). Evidently, a reversal of the electric field effects the same molecular rotation. An alternating field generated by an AC voltage also has the same effect. \z E Figure 20.3-1 The molecules of a positive uniaxial liquid crystal rotate and align with the applied electric field. Nematic Liquid-Crystal Retarders and Modulators A nematic liquid-crystal cell is a thin layer of nematic liquid crystal placed between two parallel glass plates and rubbed so that the molecules are parallel to each other. The material then acts as a uniaxial crystal with the optic axis parallel to the molecular orientation. For waves traveling in the z direction (perpendicular to the glass plates), the normal modes are linearly polarized in the x and y directions, (parallel and per- pendicular to the molecular directions, respectively), as illustrated in Fig. 20.3- 2( a). The refractive indexes are the extraordinary and ordinary indexes ne and no. A cell of thickness d provides a wave retardation r == 2w(n e - no)dj Ao. x y (a) Untilted state (b) Tilted state Figure 20.3-2 Molecular orientation of a liquid-crystal cell (a) in the absence of a steady electric field and (b) when a steady electric field is applied. The optic axis lies along the direction of the molecules. Now, if an electric field is applied in the z direction (by applying a voltage V across transparent conductive electrodes coated on the inside of the glass plates), the resultant electric forces tend to tilt the molecules toward alignment with the field, but the elastic forces at the surfaces of the glass plates resist this motion. When the applied electric field is sufficiently large, most of the molecules tilt toward the z axis, except those adjacent to the glass surfaces. The equilibrium tilt angle e for most molecules is a monotonically increasing function of V, which can be described byt { o e== w 1 V- "2 - 2 tan - exp (- V o ), V < V> , (20.3-1 ) t See, e.g., P.-G. de Gennes, The Physics of Liquid Crystals, Oxford University Press, 2nd ed. 1995.
858 CHAPTER 20 ELECTRO-OPTICS where V is the applied RMS voltage, a critical voltage at which the tilting process begins, and va a constant. When V - == va, e 50°; as V - increases beyond va, e approaches 90°, as indicated in Fig. 20.3-3(a). 0.5 1 2 3 (V - Vc)/vo Figure 20.3-3 (a) Dependence of the tilt angle B of the molecules toward the field direction (z axis) on the normalized RMS voltage. (b) Dependence of the normalized retardation r jr max == [n(B) - no]j(ne - no) on the normalized RMS voltage when no == 1.5, for the values of b,.n == ne - no indicated. This plot is obtained from (20.3- 1) and (20.3-2). () (a) 0 1 r r max (b) 0 o When the electric field is removed, the orientation of the molecules near the glass plates are reasserted and all of the molecules tilt back to their original orientations, in planes parallel to the plates. In a sense, the liquid-crystal material may be viewed as a liquid with memory. For a tilt angle e, the normal modes of an optical wave traveling in the z direction are polarized in the x and y directions and have refractive indexes n( e) and no, where 1 n 2 (e) cos 2 e sin 2 e n2 + 2 ' e no (20.3-2) so that the retardation becomes r == 27r[n( e) - no] d / Ao (see Sec. 6.3C). The retarda- tion achieves its maximum value r max == 27r (ne - no) d / Ao when the molecules are not tilted (e == 0), and decreases monotonically toward 0 when the tilt angle reaches 90°, as illustrated in Fig. 20.3-3(b). Note that for a tilt angle B, the direction between the optic axis and the direction of propagation is 90° - e, so that (20.3-2) differs from ( 6.3-15). The cell can be readily used as a voltage-controlled phase modulator. For an optical wave traveling in the z direction and linearly polarized in the x direction (parallel to the untilted molecular orientation), the phase shift is c.p == 27rn( e) d / Ao. For waves po- larized at 45 ° to the x axis in the x-y plane, the cell serves as a voltage-controlled wave retarder. When placed between two crossed polarizers (at ::l:45°), a half-wave retarder (r == 7r) becomes a voltage-controlled intensity modulator. Similarly, a quarter-wave retarder (r == 7r /2) placed between a mirror and a polarizer at 45° with the x axis serves as an intensity modulator, as illustrated in Fig. 20.3-4. The liquid-crystal cell is sealed between optically flat glass windows with antireflec- tion coatings. A typical thickness of the liquid crystal layer is d == 10 /-Lm and typical values of D.n == ne - no == 0.1 to 0.3. The retardation r is typically given in terms of
20.3 ELECTRO-OPTICS OF LIQUID CRYSTALS 859 Incident Polarizer light Reflected light .... c .... Figure 20.3-4 A liquid-crystal cell pro- vides a retardation r = 7r /2 in the absence of the field ("off" state), and r = 0 in the presence of the field ("on" state). After reflection from the mirror and a round trip through the crystal, the plane of polarization rotates 90° in the "off" state, so that the light is blocked. In the "on" state, there is no rotation, and the reflected light is not blocked. - Liquid- crystal cell y the retardance () == (ne - no) d, so that the retardation r == 27r {} / A.D. Retardances of several hundred nanometers are typical (e.g., a retardance of 300 nm corresponds to a retardation of 7r at A.o == 600 nm). The applied voltage usually has a square waveform with a frequency in the range between tens of Hz and a few kHz. Operation at lower frequencies tends to cause electromechanical effects that disrupt the molecular alignment and reduce the lifetime of the device. Frequencies higher than 100 Hz result in greater power consumption because of the increased conductivity. The critical voltage is typically a few volts RMS. Liquid crystals are slow. Their response time depends on the thickness of the liquid- crystal layer, the viscosity of the material, temperature, and the nature of the applied drive voltage. The rise time is of the order of tens of milliseconds if the operating voltage is near the critical voltage , but decreases to a few milliseconds at higher voltages. The decay time is insensitive to the operating voltage but can be reduced by using cells of smaller thickness. Twisted Nematic Liquid-Crystal Modulators A twisted nematic liquid-crystal cell is a thin layer of nematic liquid crystal placed between two parallel glass plates and rubbed so that the molecular orientation rotates helically about an axis normal to the plates (the axis of twist). If the angle of twist is 90°, for example, the molecules point in the x direction at one plate and in the y direction at the other [Fig. 20.3-5(a)]. Transverse layers of the material act as uniaxial crystals, with the optic axes rotating helically about the axis of twist. It was shown in Sec. 6.5 that the polarization plane of linearly polarized light traveling in the direction of the axis of twist rotates with the molecules, so that the cell acts as a polarization rotator. When an electric field is applied in the direction of the axis of twist (the z direction) the molecules tilt toward the field [Fig. 20.3-5(b)]. When the tilt is 90°, the molecules lose their twisted character (except for those adjacent to the glass surfaces), so that the polarization rotatory power is deactivated. If the electric field is removed, the orienta- tions of the layers near the glass surfaces dominate, thereby causing the molecules to return to their original twisted state, and the polarization rotatory power to be regained. Since the polarization rotatory power may be turned off and on by switching the electric field on and off, a shutter can be designed by placing a cell with 90° twist between two crossed polarizers. The system transmits the light in the absence of an electric field and blocks it when the electric field is applied, as illustrated in Fig. 20.3- 6. Operation in the reflective mode is also possible, as illustrated in Fig. 20.3-7. Here, the twist angle is 45°; a mirror is placed on one side of the cell and a polarizer on the other side. When the electric field is absent the polarization plane rotates a total of 90° upon propagation a round trip through the cell; the reflected light is therefore blocked
860 CHAPTER 20 ELECTRO-OPTICS xt xt - --- --=-- -=- "- ------ ::::- " - ::==--=-- -::.=. "'- --- - I / ----- - -- -- - I --- -- I - - -==- - -..... - - - =::::::- 1 ...:::::..- - = -- I _-==- --""':;;:::'-=-' --=- = - --=-' t-= _ _---, I f -===-......;. -- - -=-- - -=- - - .;;;;:--""::" z Z (a) Twisted state (b) Tilted (untwisted) state Figure 20.3-5 In the presence of a sufficiently large electric field, the molecules of a twisted nematic liquid crystal tilt and lose their twisted character. (a) Bright Polarizer (b) ------.......... ------ =- .......... "..... .::---=-- -=- "" --- - , / ----:;:--- - =- ==:- - - -"" I -=- -==--=-- --=- / - -=-- -==- ----....... ..=-- --=--, / --= -- - t --- --...:;:::.-::........... r=-=- =- = , ' =:5- = ---=-==- - - =- ----.c...::.. --- Dark Figure 20.3-6 A twisted nematic liquid-crystal switch. (a) When the electric field is absent, the liquid-crystal cell acts as a polarization rotator; the light is transmitted. (b) When the electric field is present, the cell's rotatory power is suspended and the light is blocked. by the polarizer. When the electric field is present, the polarization rotatory power is suspended and the reflected light is transmitted through the polarizer. Other reflective and transmissive modes of operation with different angles of twist are also possible. The twisted liquid-crystal cell placed between crossed polarizers may also be op- erated as an analog modulator. At intermediate tilt angles, there is a combination of polarization rotation and wave retardation. Analysis of the transmission of polarized light through tilted and twisted molecules is rather complex, but the overall effect is a partial intensity transmittance. There is an approximately linear range of transition between the total transmission of the fully twisted (untilted) state and zero transmission in the fully tilted (untwisted) state. However, the dynamic range is rather limited. Ferroelectric Liquid Crystals Smectic liquid crystals are organized in layers, as illustrated in Fig. 6.5-1 (b). In the smectic-C phase, the molecular orientation is tilted by an angle e with respect to the normal to the layers (the x axis), as illustrated in Fig. 20.3-8. The material has ferroelectric properties. When placed between two close glass plates the surface in-
20.3 ELECTRO-OPTICS OF LIQUID CRYSTALS 861 ,-t Polarizer Mirror Liquid- crystal cell Figure 20.3-7 A twisted nematic liquid-crystal cell with 45 0 twist angle provides a round-trip polarization rotation of 90 0 in the absence of the electric field (blocked state) and no rotation when the field is applied (unblocked state). The device serves as a switch. teractions permit only two stable states of molecular orientation at the angles :i:0, as shown in Fig. 20.3-8. When an electric field + E is applied in the z direction, a torque is produced that switches the molecular orientation into the stable state +0 [Fig. 20.3- 8(a)]. The molecules can be switched into the state -0 by use of an electric field of opposite polarity - E [Fig. 20.3-8(b)]. Thus, the cell acts as a uniaxial crystal whose optic axis may be switched between two orientations. x () Xt () () () () z Figure 20.3-8 The two states of a ferroelectric liquid-crystal cell. In the geometry of Fig. 20.3-8, the incident light is linearly polarized at an angle 0 with the x axis in the x-y plane. In the +0 state, the polarization is parallel to the optic axis and the wave travels with the extraordinary refractive index ne without retardation. In the -0 state, the polarization plane makes an angle 20 with the optic axis. If 20 == 45°, the wave undergoes a retardation r == 2w( ne - no)d / Ao, where d is the thickness of the cell and no is the ordinary refractive index. If d is selected such that r == w, the plane of polarization rotates 90°. Thus, reversing the applied electric field has the effect of rotating the plane of polarization by 90°. An intensity modulator can be made by placing the cell between two crossed polar- izers. The response time of ferroelectric liquid-crystal switches is typically < 20 J-Ls at room temperature, which is far faster than that of nematic liquid crystals. The switching voltage is typically :i:l0 V. B. Spatial Light Modulators Liquid-Crystal Displays A liquid-crystal display (LCD) is constructed by placing transparent electrodes of different patterns on the glass plates of a reflective liquid-crystal (nematic, twisted-
862 CHAPTER 20 ELECTRO-OPTICS nematic, or ferroelectric) cell. By applying voltages to selected electrodes, patterns of reflection and nonreflection are created. Figure 20.3-9 illustrates a pattern for a seven- bar display of the numbers 0 to 9. Larger numbers of electrodes may be addressed sequentially. Indeed, charge-coupled devices (CCDs) can be used for addressing liquid- crystal displays. The resolution of the device depends on the number of segments per unit area. LCDs are used in consumer items such as digital watches, pocket calculators, computer monitors, cellular phones, and television receivers. Figure 20.3-9 Electrodes of a seven-bar-segment LCD. In comparison with light-emitting diode (LED) displays, the principal advantage of LCDs is their low electrical power consumption. However, LCDs have a number of disadvantages: . They are passive devices that modulate light that is already present, rather than emitting their own light; thus they are not useful in the dark. . Nematic liquid crystals are relatively slow. . The optical efficiency is limited as a result of the use of polarizers that absorb at least 50% of unpolarized incident light. . The angle of view is limited; the contrast of the modulated light is reduced as the angle of incidence/reflectance increases. Optically Addressed Spatial Light Modulators Most LCDs are addressed electrically. However, optically addressed spatial light mod- ulators (SLM) are attractive for applications involving image and optical data process- ing. Light with an intensity distribution I w (x, y), the "write" image, is converted by an optoelectronic sensor into a distribution of electric field E (x, y), which controls the reflectance (x, y) of a liquid-crystal cell operated in the reflective mode. Another optical wave of uniform intensity is reflected from the device and creates the "read" image I (x, y) ex: (x, y). Thus, the "read" image is controlled by the "write" image (see Fig. 20.1-14). If the write image is carried by incoherent light, and the read image is formed by coherent light, the device serves as a spatial incoherent -to-coherent light converter, much like the PROM device discussed in Sec. 20.1E. Furthermore, the wavelengths of the write and read beams need not be the same. The read light may also be more intense than the write light, so that the device may serve as an image intensifier. There are several means for converting the write image Iw(x, y) into a pattern of electric field E (x, y) for application to the liquid-crystal cell. A layer of photoconduc- tive material, e..g., cadmium sulfide (CdS), placed between the electrodes of a capacitor may be used. When illuminated by the distribution Iw(x, y), the conductance G(x, y) is altered proportionally. The capacitor is discharged at each position in accordance with the local conductance, so that the resultant electric field E (x, y) ex: 1/ I w (x, y) is a negative of the original image. An alternative is the use of a sheet photo diode
20.4 PHOTOREFRACTIVITY 863 [a p-i-n photodiode of hydrogenated amorphous silicon (a-Si:H), for example]. The reverse-biased photodiode conducts in the presence of light, thereby creating a poten- tial difference proportional to the local light intensity. An example of a commercially available liquid-crystal (LC) spatial light modulator (SLM) is the Hamamatsu Parallel Aligned Spatial Light Modulator (PAL-SLM), illustrated in Fig. 20.3-10. This device uses a-Si:H as the write medium and a nematic LC with molecules in parallel alignment as a phase modulator. At each point, the impedance of the amorphous silicon layer is altered by the write light and a voltage proportional to the optical intensity is applied on the corresponding point in the LC layer. This results in rotation of the anisotropic LC molecules to align with the applied electric field. Consequently, the read light beam undergoes a proportional phase shift as it travels through the LC layer. The PAL-SLM is a continuous modulator (i.e., is not pixelated). It has a high spatial resolution, corresponding to 480 x 480 points over its active area of 2 x 2 cm 2 , and its rise (fall) time is 30 (40) ms. Transparent electrode a-Si:H \ Liquid / crystal Transparent electrode Modulated light Figure 20.3-10 Schematic of the Hamamatsu Parallel Aligned Spatial Light Modulator (PAL-SLM). This opti- cally addressed SLM has two principal layers - an amorphous silicon layer, which senses the write light intensity, and a liquid-crystal (LC) layer that serves as a reflective phase modulator of the read light. These layers are separated by a light-blocking dielectric material. The device is encased in glass substrates (not shown). )I Write light Incident readout light Light- bloCking layer \ Dielectric mIrror *20.4 PHOTOREFRACTIVITY Photorefractive materials exhibit photoconductive and electro-optic behavior, and have the ability to detect and store spatial distributions of optical intensity in the form of spatial patterns of altered refractive index. Photoinduced charges create a space-charge distribution that produces an internal electric field, that in turn alters the refractive index by means of the electro-optic effect. Ordinary photoconductive materials are often good insulators in the dark. Upon illumination, photons are absorbed, free charge carriers (electron-hole pairs) are gen- erated, and the conductivity of the material increases. When the light is removed, the process of charge photogeneration ceases, and the conductivity returns to its dark value as the excess electrons and holes recombine. Photoconductors are used as photon detectors (see Sec. 18.2). When a photorefractive material is exposed to light, free charge carriers (electrons or holes) are generated by excitation from impurity energy levels to an energy band, at a rate proportional to the optical power. This process is much like that in an extrin- sic semiconductor photoconductor (see Sec. 18.2B). These carriers then diffuse away from the positions of high intensity where they were generated, leaving behind fixed charges of the opposite sign (associated with the impurity ions). The free carriers can be trapped by ionized impurities at other locations, depositing their charge there as they
864 CHAPTER 20 ELECTRO-OPTICS recombine. The result is the creation of an inhomogeneous space-charge distribution that can remain in place for a period of time after the light is removed. This charge distribution creates an internal electric field pattern that modulates the local refractive index of the material by virtue of the (Pockels) electro-optic effect. The image may be accessed optically by monitoring the spatial pattern of the refractive index using a probe optical wave. The material can be brought back to its original state (erased) by illumination with uniform light, or by heating. Thus, the material can be used to record and store images, much like a photographic emulsion stores an image. The process is illustrated in Fig. 20.4-1 for doped lithium niobate (LiNb0 3 ). (2) Diffusion - - - --...........--.. : - - - - .... .... - - - - - ... ->--- ,"'" ,. t,_-_ d _ d_ n_-__ I (1) Photoionization (3) Recombination Fe 2 + Fe 3 + Va 1ence band (4) Space-charge formation +++++ +++++ +++++ (5) Electric field generation . x Figure 20.4-1 Energy-level diagram of LiNb0 3 illustrating the processes of photoionization, diffusion, recombination, space-charge formation, and electric-field generation. Fe 2 + impurity centers act as donors, becoming Fe 3 + when ionized, while Fe 3 + centers act as traps, becoming Fe 2 + after recombination. Important photorefractive materials include barium titanate (BaTi0 3 ), bismuth sil- icon oxide (Bi 12 Si0 20 ), lithium niobate (LiNb0 3 ), potassium niobate (KNb0 3 ), gal- lium arsenide (GaAs), and strontium barium niobate (SBN). Simplified Theory of Photorefractivity When a photorefractive material is illuminated by light of intensity I (x) that varies in the x direction, the refractive index changes by n(x). The following is a step-by-step description of the processes that mediate this effect (illustrated in Fig. 20.4-1) and a simplified set of equations that govern them: . Photogene ration. The absorption of a photon at position x raises an electron from the donor level to the conduction band. The rate of photoionization G( x) is proportional both to the optical intensity and to the number density of non ionized donors. Thus, G(x) == S (N D - Nt) I(x), (20.4-1 ) where N D is the number density of donors, Nt is the number density of ionized donors, and s is a constant known as the photoionization cross section. . Diffusion. Since I (x) is nonuniform, the number density of excited electrons n( x) is also nonuniform. As a result, electrons diffuse from locations of high concentration to locations of low concentration.
20.4 PHOTOREFRACTIVITY 865 . Recombination. The electrons recombine at a rate R( x) proportional to their number density n(x), and to the number density of ionized donors (traps) Nt, so that R(x) == 'YRn(x) Nt, (20.4-2) where 'YR is a constant. In equilibrium, the rate of recombination equals the rate of photoionization, R( x) == G( x), so that sI(x) (N D - Nt) == 'YRn(x) Nt, (20.4-3) from which n(x) = N D -+ Nt I(x). 'YR N D (20.4-4 ) . Space charge. Each photogenerated electron leaves behind a positive ionic charge. When the electron is trapped (recombines), its negative charge is deposited at a different site. As a result, a nonuniform space-charge distribution is formed. . Electric field. This nonuniform space charge generates a position-dependent elec- tric field E(x), which may be determined by observing that in steady state the drift and diffusion electric-current densities must be of equal magnitude and opposite sign, so that the total current density vanishes, i.e., dn J == e Me n ( x) E ( x) - kT Me dx == 0, (20.4-5) where Me is the electron mobility, k is Boltzmann's constant, and T is the temper- ature. Thus, E(x) = kT dn . e n(x) dx (20.4-6) . Refractive index. Since the material is electro-optic, the internal electric field E(x) locally modifies the refractive index in accordance with n(x) == -n3t E(x), (20.4- 7) where nand t are the appropriate values of refractive index and electro-optic coefficient for the material [see (20.1-4)]. The relation between the incident light intensity I (x) and the resultant refractive in- dex change n( x) may readily be obtained if we assume that the ratio (N D / Nt -1) in (20.4-4) is approximately constant, independent of x. In that case n(x) is proportional to I (x), so that (20.4-6) gives kT 1 dI E(x) = --;- I(x) dx . (20.4-8)
866 CHAPTER 20 ELECTRO-OPTICS Finally, substituting this into (20.4-7), provides an expression for the position-dependent refractive-index change as a function of intensity, 1 3 kT 1 dI n(x) == --n t - - -. 2 e l(x) dx (20.4-9) Refractive-Index Change This equation is readily generalized to two dimensions, whereupon it governs the operation of a photorefractive material as an image storage device. Many assumptions have been made to keep the foregoing theory simple: In de- riving (20.4-8) from (20.4-6) it was assumed that the ratio of number densities of unionized to ionized donors is approximately uniform, despite the spatial variation of the photoionization process. This assumption is approximately applicable when the ionization is caused by other more effective processes that are position independent in addition to the light pattern I (x). Dark conductivity and volume photovoltaic effects were neglected. Holes were ignored. It was assumed that no external electric field was applied, when in fact this can be useful in certain applications. The theory is valid only in the steady state although the time dynamics of the photorefractive process are clearly important since they determine the speed with which the photorefractive material re- sponds to the applied light. Yet in spite of all these assumptions, the simplified theory carries the essence of the behavior of photorefractive materials. EXAMPLE 20.4-1. Detection of a Sinusoidal Spatia/Intensity Pattern. Consider an intensity distribution in the form of a sinusoidal function of period A, contrast m, and mean intensity 1 0 , ( 27rx ) l(x) =10 l+mcosT ' (20.4-10) as shown in Fig. 20.4-2. Substituting this into (20.4-8) and (20.4-9), we obtain the internal electric field and refractive index distributions - sin(27rxj A) E(x) = Emax 1 + mcos(27rxj A) ' sin(27rxj A) n(x)=nmax (jA) ' 1 + mcos 27rx (20.4-11 ) where Emax = 27r(kT jeA) m and nmax n(x), respectively. n3t Emax are the maximum values of E(x) and -J,. ++ ++ ++ ++ --... --... --... ++ ++ ++ ++ ++ ++ ++ ++ -..... ..- -..... ..- -..... -..... ..- -..... ..- -..... -..... ..- -..... ..- -..... -..... ..- -..... ..- -..... Optical intensity l(x) » - - - x cfr densIty J x Fixed- r to\ charge » densIty V V V x EIctric V""" 1\ 1\ fIeld J E(x) V V \. x Index If\ 1\ 1\ change » _ L\n(x) V V \. x Nonuniform light -J,. Photo- ionization -J,. Diffusion Electric field -J,. Refractive index grating Figure 20.4-2 Response of a photorefractive material to a sinusoidal spatial light pattern
20.4 PHOTOREFRACTIVITY 867 If A = 1 Jim, m = 1, and T = 300° K, for example, Emax = 1.6 X 10 5 VIm. This internal field is equivalent to applying 1.6 kV across a crystal of l-cm width. The maximum refractive index change nmax is directly proportional to the contrast m and the electro-optic coefficient t, and inversely proportional to the spatial period A. The grating pattern .6.n(x) is totally insensitive to the uniform level of the illumination 10. When the image contrast m is small, the second term of the denominators in (20.4-11) may be neglected. The internal electric field and refractive index change are then sinusoidal patterns shifted by 90° relative to the incident light pattern, 21TX n(x) nmax sin T. These patterns are illustrated in Fig. 20.4-2. (20.4-12) Applications of the Photorefractive Effect An image I (x, y) may be stored in a photorefractive crystal in the form of a refractive- index distribution n(x, y). The image can be read by using the crystal as a spatial- phase modulator to encode the information on a uniform optical plane wave acting as a probe. Phase modulation may be converted into intensity modulation by placing the cell in an interferometer, for example. Because of their capability to record images, photorefractive materials are attractive for use in real-time holography (see Sec. 4.5 for a discussion of holography). An object wave is holographically recorded by mixing it with a reference wave, as illustrated for two plane waves in Fig. 20.4-3. The intensity of the sum of two such waves forms a sinusoidal interference pattern, which is recorded in the photorefractive crystal in the form of a refractive-index variation. The crystal then serves as a volume phase hologram (see Sec. 4.5, Fig. 4.5-10). To reconstruct the stored object wave, the crystal is illuminated with the reference wave. Acting as a volume diffraction grating, the crystal reflects the reference wave and reproduces the object wave. Wave 1 (reference) .. . 1 P ... .. .. ... . .............1 :'J e, 1- . e,C\) \.0"/ Grating Figure 20.4-3 Two-wave mixing is a form of dynamic holography. Since the recording process is relatively fast, the processes of recording and re- construction can be carried out simultaneously. The object and reference waves travel together in the medium and exchange energy via reflection from the created grating. This process is called two-wave mixing. As shown in Fig. 20.4-3 (see also Fig. 4.5-8), waves 1 and 2 interfere and form a volume grating. Wave 1 reflects from the grating and adds to wave 2; wave 2 reflects from the grating and adds to wave 1. Thus, the two waves are coupled together by the grating they create in the medium. Consequently, the transmission of wave 1 through the medium is controlled by the presence of wave 2, and vice versa. For example, wave 1 may be amplified at the expense of wave 2. The mixing of two (or more) waves also occurs in other nonlinear optical materials with light-dependent optical properties, as discussed in Chapter 21. Wave mixing has numerous applications in optical data processing (see Chapters 2] and 23), including image amplification, the removal of image aberrations, cross correlation of images, and optical interconnections.
868 CHAPTER 20 ELECTRO-OPTICS 20.5 ELECTROABSORPTION Electroabsorption is a change of the absorption characteristics of a medium in re- sponse to an externally applied electric field. In a bulk semiconductor, the application of an external electric field results in electron tunneling, which extends the absorption edge into the forbidden gap. The bandgap energy of the material is thus reduced below that provided by the band tail and the Urbach tail, so that hV2 < hVl when the field is ON, as illustrated in Fig. 20.5-1(a). This phenomenon, known as the Franz-Keldysh effect, therefore shifts the absorption spectrum to longer wavelengths [Fig. 20.5-1(b)]. The applied electric field also results in the broadening, and ultimate disappearance of, the exciton absorption peaks (see Sec. 16.2C). This effect may be used in optical electroabsorption modulators and electroab- sorption switches. In the absence of the electric field (OFF), an incident beam at the operating wavelength, which is longer than the normal bandgap wavelength, is transmitted without absorption [Fig. 20.5-1 (b)]. Upon application of the electric field (ON), however, the light is absorbed. Such modulators are often constructed in the form of a waveguide, with the electric field applied in a direction perpendicular to the direction of travel of the light beam, as shown in Fig. 20.5-1(e). In comparison with electro-optic modulators, which operate on the basis of a change of the refractive- index in response to an externally applied electric field (see Sees. 20.1B and 20.2B), electroabsorption modulators typically operate at greater speeds and at lower voltages. Since they can be integrated on a single chip with semiconductor light sources, they are convenient for use in optical fiber communication systems. They also have less chirp than directly modulated laser diodes (see Sec. 22.1). Wavelength OFF ON -r hv} ---L E (]) "[5 S (]) o u s:: .s P.. o C/) .D « OJ) ,..c I' .5 : " \ , OFF ro'(]) 'v, \ 8" : " , , , , , , , I .,1 Photon energy Modulated ;; Incident beam / Semiconductor (a) (b) (c) Figure 20.5-1 The Franz-Keldysh effect. (a) The bandgap in the absence of an external electric field (OFF) is reduced in its the presence (ON). (b) Change in the absorption spectrum caused by the presence of an electric field. The absorption peak moves toward longer wavelengths. (c) Electroabsorption modulator in a waveguide configuration. The electroabsorption effect is more pronounced in semiconductor multiquantum- well (MQW) structures (see Sees. 16.1 G and 17.2D). An electric field applied in the plane of a quantum well gives rise to behavior similar to the Franz-Keldysh effect, including a shift of the absorption edge to a longer wavelength and exciton disso- ciation. However, an electric field applied in the direction of confinement gives rise to additional phenomena, known collectively as the quantum-confined Stark effect (QCSE), as illustrated in Fig. 20.5-2: . The energy difference between the conduction- and valence-band energy levels decreases with increasing electric field (hV2 < hVl). . The band tilt causes the locations of the wavefunctions to shift toward the edges of the well.
READING LIST 869 . Exciton ionization is inhibited and exciton energy levels remain unbroadened even at high field levels, since the electron and hole remain in proximity by virtue of the confinement. OFF ON Wavelength (nm) 860 850 . 1 hV I ... 1 Incident beam Modulated ! I t beam T hV 2 1 (a) 1.43 (b) 1.44 1.45 1.46 1.47 Photon energy (e V) (c) Figure 20.5-2 (a) Energy-band diagrams of a quantum well in the absence (OFF) and presence (ON) of an external applied electric field. The field causes the interband energy difference to decrease and the wavefunctions to shift from the centers of the wells toward opposite edges. (b) Change in the absorption spectrum in an AIGaAsjGaAs multiquantum-well structure as the applied voltage (field) is increased. The exciton absorption peak moves toward longer wavelengths. (Adapted from D. A. B. Miner, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, Jr., A. C. Gossard, and W. Wiegmann, The Quantum Well Self-Electrooptic Effect Device: Optoelectronic Bistability and Oscillation, and Self-Linearized Modulation, IEEE Journal of Quantum Electronics, vol. 21, pp. 1462-1476, Fig. 1 @1985 IEEE.) (c) Schematic of a MQW electroabsorption modulator operated in a surface-normal architecture. As a result of these MQW characteristics, the wavelength shift of the absorption peak is greater, and the absorption edge is more abrupt, than in bulk semiconductors. Electroabsorption modulators based on the QCSE have excellent characteristics, in- cluding . High speeds . Large extinction ratios . Low drive voltages . Low chirp The simplest transmission implementation directs light through an intrinsic MQW structure sandwiched between p and n regions across which a voltage is applied. Switching is accomplished by simply turning the voltage on and off. A device of this sort can also be fabricated in a waveguide configuration and can be integrated with a DFB laser on a single chip. QCSE modulators and switches can also be fabricated in the form of arrays operated in a double-pass surface-normal architecture, as illustrated in Fig. 20.5-2(e). READING LIST General See also the reading lists in Chapters 6 and 21, and the books on optoelectronics in Chapter 17.
870 CHAPTER 20 ELECTRO-OPTICS M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon, 1977; Oxford University Press, paperback 2nd ed. 2001. V. G. Chigrinov, Liquid Crystal Devices: Physics and Applications, Artech, 1999. G. D. Boreman, Basic Electro-Opticsfor Electrical Engineers, SPIE Optical Engineering Press, 1998. U. Efron, ed., Spatial Light Modulator Technology: Materials, Devices, and Applications, Marcel Dekker, 1995. F. AguIla-Lopez, J. M. Cabrera, and F. AguIla-Rueda, Electrooptics: Phenomena, Materials and Applications, Academic Press, 1994. M. A. Karim, Electro-Optical Displays, Marcel Dekker, 1992. T. Tamir, ed., Guided-Wave Optoelectronics, Springer-Verlag, 1988, 2nd ed. 1990. L. D. Hutcheson, ed., Integrated Optical Circuits and COJnponents: Design and Applications, Marcel Dekker, 1987. M. Gottlieb, C. L. M. Ireland, and J. M. Ley, Electro-Optic and Acousto-Optic Scanning and Deflec- tion, Marcel Dekker, 1983. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals, Plenum, 1981. J. I. Pankove, ed., Display Devices, Volume 40, Topics in Applied Physics, Springer-Verlag, 1980. G. R. Elion and H. A. Elion, Electro-Optics Handbook, Marcel Dekker, 1979. D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics, Wiley, 1979. I. P. Kaminow, An Introduction to Electrooptic Devices, Academic Press, 1974. Photorefractive Materials J. Frejlich, Photo refractive Materials: Fundamental Concepts, Holographic Recording and Materials Characterization, Wiley, 2006. P. Gunter and J.-P. Huignard, eds., Photo refractive Materials and Their Applications. 3: Applications, Springer- Verlag, 2006. P. Gunter and J.-P. Huignard, eds., Photo refractive Materials and Their Applications. 2: Materials, Springer- Verlag, 2006. P. Gunter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications. 1: Basic Effects, Springer-Verlag, 2005. F. T. S. Yu and S. Yin, Photorefractive Optics: Materials, Properties, and Applications, Academic Press, 2000. F. M. Davidson, ed., Selected Papers on Photorefractive Materials, SPIE Optical Engineering Press (Milestone Series Volume 86), 1994. P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley, 1993. Special issue on photorefractive materials, effects, and devices, Journal of the Optical Society of America B, vol. 7, no. 12, 1990. D. M. Pepper, J. Feinberg, and N. K. Kukhtarev, The Photorefractive Effect, Scientific American, vol. 263, no. 4, pp. 62-74, 1990. J. Feinberg, Photorefractive Nonlinear Optics, Physics Today, vol. 41, no. 10, pp. 46-52,1988. B. Va. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation, Springer- Verlag, 1985. R. A. Fisher, ed., Optical Phase Conjugation, Academic Press, 1983. Articles Special issue on electrooptic materials and devices, IEEE Journal of Quantum Electronics, vo1. QE- 23, no. 12, 1987. D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, Jr., A. C. Gossard, and W. Wiegmann, The Quantum Well Self-Electrooptic Effect Device: Optoelectronic Bistability and Oscillation, and Self-Linearized Modulation, IEEE Journal of Quantum Electronics, vol. 21, pp. 1462-1476, 1985. S. H. Wemple and M. DiDomenico, Jr., Electro-Optical and Nonlinear Optical Properties of Crystals, in Applied Solid State Science: Advances in Materials and Device Research, Volume 3, pp. 263- 383, R. Wolfe, ed., Academic Press, 1972.
PROBLEMS 871 PROBLEMS 20.1-2 Response Time of a Phase Modulator. A GaAs crystal with refractive index n == 3.6 and electro-optic coefficient t == 1.6 pm/V is used as an electro-optic phase modulator operating at AD == 1.3 /-Lm in the longitudinal configuration. The crystal is 3 cm long and has a l-cm 2 cross-sectional area. Determine the half-wave voltage V 7r , the transit time of light through the crystal, and the electric capacitance of the device (the low-frequency dielectric constant of GaAs is E/Eo == 13.5). The voltage is applied using a source with 50-0 resistance. Which factor limits the speed of the device, the transit time of the light through the crystal or the response time of the electric circuit? 20.1- 3 Sensitivity of an Interferometric Electro-Optic Intensity Modulator. An integrated-optic intensity modulator using the Mach-Zehnder configuration, illustrated in Fig. 20.1-5, is used as a linear analog modulator. If the half-wave voltage is V 7r == 10 V, what is the sensitivity of the device (the incremental change of the intensity transmittance per unit incremental change of the applied voltage)? 20.1-4 An Elasto-Optic Strain Sensor. An elasto-optic material exhibits a change of the refractive index proportional to the strain. Design a strain sensor based on this effect. Consider an integrated-optical implementation. If the material is also electro-optic, consider a design based on compensating the elasto-optic and the electro-optic refractive index change, and measuring the electric field that nulls the reading of the photodetector in a Mach-Zehnder interferometer. 20.1-5 Magneto-Optic Modulators. Describe how a Faraday rotator (see Sec. 6.4B) may be used as an optical intensity modulator. *20.2-2 Silica Integrated-Optic Phase Modulator. Since bulk fused silica is centrosymmetric, it does not ordinarily exhibit the linear electro-optic (Pockels) effect. However, thermally poled silica has Pockels coefficients that are sufficiently large for use as optical modulators. Determine the phase shift introduced by a poled-silica integrated-optic phase modulator in a configuration such as that shown in Fig. 20.1-3. Assume that the electrode length is L == 25 mm, the electrode separation is d == 30 /-Lm, and the wavelength is A == 1.55/-Lm. Assume also that the optical wave is polarized in the y direction, the electric field is created by an applied voltage V == 400 V and points in the y direction, and the wave travels along the electrodes in the z direction. The material is poled in a direction such that its principal axes (Xl, X2, X3) point in the z, X, and y directions, respectively. The refractive index of the poled material is n == 1.445 and the Pockels coefficients are described by the matrix o 0 tl3 o 0 tl3 o 0 t33 o tl3 0 tl3 0 0 000 with tl3 == 0.15 pm/V. *20.2-3 Cascaded Phase Modulators. (a) A KDP crystal (t41 == 8 pm/V, t63 == 11 pm/V; no == 1.507, ne == 1.467 at AD 633 nm) is used as a longitudinal phase modulator. The orientation of the crystal axes and the applied electric field are as shown in Examples 20.2-2 and 20.2-6. Determine the half-wave voltage V 7r at AD == 633 nm. (b) An electro-optic phase modulator consists of 9 KDP crystals separated by electrodes that are biased as shown in Fig. P20.2-3. How should the plates be oriented relative to each other so that the total phase modulation is maximized? Calculate V 7r for the composite modulator. +v -- Modulated light Incident light o Figure P20.2-3
872 CHAPTER 20 ELECTRO-OPTICS *20.2-4 The "Push-Pull" Intensity Modulator. An optical intensity modulator uses two integrated electro-optic phase modulators and a 3-dB directional coupler, as shown in Fig. P20.2-4. The input wave is split into two waves of equal amplitudes, each of which is phase modulated, reflected from a mirror, phase modulated once more, and the two returning waves are added by the directional coupler to form the output wave. Derive an expression for the intensity transmittance of the device in terms of the applied voltage, wavelength. dimensions. and physical parameters of the phase modulator. Figure P20.2-4 *20.2-5 A LiNb0 3 Integrated-Optic Intensity Modulator. Design a LiNb0 3 integrated-optic intensity modulator using the Mach-Zehnder interferometer shown in Fig. 20.1-5. Select the orientation of the crystal and the polarization of the guided wave for the smallest half- wave voltage V 7r . Assume that the active region has length L == 1 mm and width d == 5 /-Lm. The wavelength is Ao == 0.85 /-Lm, the refractive indexes are no == 2.29 and ne == 2.17, and the electro-optic coefficients are t33 == 30.9, t13 == 8.6, t22 == 3.4, and t42 == 28 pm/V. *20.2-6 Double Refraction in an Electro-Optic Crystal. (a) An unpolarized He-Ne laser beam (A o == 633 nm) is transmitted through a l-cm-thick LiNb0 3 plate (ne == 2.17, no == 2.29, t33 == 30.9 pm/V, t13 == 8.6 pm/V). The beam is orthogonal to the plate and the optic axis lies in the plane of incidence of the light at 45° with the beam. The beam is double refracted (see Sec. 6.3E). Determine the lateral displacement and the retardation between the ordinary and extraordinary beams. (b) If an electric field E == 30 V /m is applied in a direction parallel to the optic axis, what is the effect on the transmitted beams? What are possible applications of this device?
C HAP T E R 1 NONLINEAR OPTICS 21.1 21.2 21.3 *21.4 *21.5 *21.6 *21.7 NONLINEAR OPTICAL MEDIA SECOND-ORDER NONLINEAR OPTICS A. Second-Harmonic Generation (SHG) and Rectification B. The Electro-Optic Effect C. Three-Wave Mixing D. Phase Matching and Tuning Curves E. Quasi-Phase Matching THIRD-ORDER NONLINEAR OPTICS A. Third-Harmonic Generation (THG) and Optical Kerr Effect B. Self-Phase Modulation (SPM), Self-Focusing, and Spatial Solitons C. Cross-Phase Modulation (XPM) D. Four-Wave Mixing (FWM) E. Optical Phase Conjugation (OPC) SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY A. Second-Harmonic Generation (SHG) B. Optical Frequency Conversion (OFC) C. Optical Parametric Amplification (OPA) and Oscillation (OPO) THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY A. Four-Wave Mixing (FWM) B. Three-Wave Mixing and Third-Harmonic Generation (THG) C. Optical Phase Conjugation (OPC) ANISOTROPIC NONLINEAR MEDIA DISPERSIVE NONLINEAR MEDIA 875 879 894 905 917 924 927 - . . '- \ -- - t- Nicolaas Bloembergen (born 1920) has carried out pio- neering studies in nonlinear optics since the early 1960s. He shared the 1981 Nobel Prize with Arthur Schawlow. . .' 873
Throughout the long history of optics, and indeed until relatively recently, it was thought that all optical media were linear. The consequences of this assumption are far-reaching: . The optical properties of materials, such as refractive index and absorption coef- ficient, are independent of light intensity. . The principle of superposition, a fundamental tenet of classical optics, is applica- ble. . The frequency of light is never altered by its passage through a medium. . Two beams of light in the same region of a medium have no effect on each other so that light cannot be used to control light. The operation of the first laser in 1960 enabled us to examine the behavior of light in optical materials at higher intensities than previously possible. Experiments carried out in the post-laser era clearly demonstrate that optical media do in fact exhibit nonlinear behavior, as exemplified by the following observations: . The refractive index, and consequently the speed of light in a nonlinear optical medium, does depend on light intensity. . The principle of superposition is violated in a nonlinear optical medium. . The frequency of light is altered as it passes through a nonlinear optical medium; the light can change from red to blue, for example. . Photons do interact within the confines of a nonlinear optical medium so that light can indeed be used to control light. The field of nonlinear optics offers a host of fascinating phenomena, many of which are also eminently useful. Nonlinear optical behavior is not observed when light travels in free space. The "nonlinearity" resides in the medium through which the light travels, rather than in the light itself. The interaction of light with light is therefore mediated by the nonlinear medium: the presence of an optical field modifies the properties of the medium, which in turn causes another optical field, or even the original field itself, to be modified. As discussed in Chapter 5, the properties of a dielectric medium through which an optical electromagnetic wave propagates are described by the relation between the polarization-density vector (r, t) and the electric-field vector £(r, t). Indeed it is use- ful to view (r, t) as the output of a system whose input is £(r, t). The mathematical relation between the vector functions (r, t) and £ (r, t), which is governed by the characteristics of the medium, defines the system. The medium is said to be nonlinear if this relation is nonlinear (see Sec. 5.2). This Chapter In Chapter 5, dielectric media were further classified with respect to their dispersive- ness, homogeneity, and isotropy (see Sec. 5.2). To focus on the principal effect of inter- est - nonlinearity - the first portion of our exposition is restricted to a medium that is nondispersive, homogeneous, and isotropic. The vectors and £ are consequently parallel at every position and time and may therefore be examined on a component-by- component basis. The theory of nonlinear optics and its applications is presented at two levels. A simplified approach is provided in Secs. 21.1-21.3. This is followed by a more detailed analysis of the same phenomena in Sec. 21.4 and Sec. 21.5. The propagation of light in media characterized by a second-order (quadratic) non- linear relation between P and c is described in Sec. 21.2 and Sec. 21.4. Applications include the frequency doubling of a monochromatic wave (second-harmonic genera- tion), the mixing of two monochromatic waves to generate a third wave at their sum or difference frequencies (frequency conversion), the use of two monochromatic waves 874
21.1 NONLINEAR OPTICAL MEDIA 875 to amplify a third wave (parametric amplification), and the incorporation of feedback in a parametric-amplification device to create an oscillator (parametric oscillation). Wave propagation in a medium with a third-order (cubic) relation between P and c is discussed in Secs. 21.3 and 21.5. Applications include third-harmonic generation, self-phase modulation, self-focusing, four-wave mixing, and phase conjugation. The behavior of anisotropic and dispersive nonlinear optical media is briefly considered in Secs. 21.6 and 21.7, respectively. Nonlinear Optics in Other Chapters A principal assumption of the treatment provided in this chapter is that the medium is passive, i.e., it does not exchange energy with the light wave(s). Waves of different frequencies may exchange energy with one another via the nonlinear property of the medium, but their total energy is conserved. This class of nonlinear phenomena are known as parametric interactions. Several nonlinear phenomena involving nonpara- metric interactions are described in other chapters of this book: . Laser interactions. The interaction of light with a medium at frequencies near the resonances of an atomic or molecular transitions involves phenomena such as absorption, and stimulated and spontaneous emission, as described in Sec. 13.3. These interactions become nonlinear when the light is sufficiently intense so that the populations of the various energy levels are significantly altered. Nonlinear optical effects are manifested in the saturation of laser amplifiers and saturable absorbers (Sec. 14.4). . Multiphoton absorption. Intense light can induce the absorption of a collection of photons whose total energy matches that of an atomic transition. For k-photon absorption, the rate of absorption is proportional to I k , where I is the optical intensity. This nonlinear-optical phenomenon is described briefly in Sec. 13.5B. . Nonlinear scattering. Nonlinear inelastic scattering involves the interaction of light with the vibrational or acoustic modes of a medium. Examples include stimulated Raman and stimulated Brillouin scattering, as described in Secs. 13.5C and 14.3D. It is also assumed throughout this chapter that the light is described by stationary continuous waves. Nonstationary nonlinear optical phenomena include: . Nonlinear optics of pulsed light. The parametric interaction of optical pulses with a nonlinear medium is described in Sec. 22.5. . Optical solitons are light pulses that travel over exceptionally long distances through nonlinear dispersive media without changing their width or shape. This nonlinear phenomenon is the result of a balance between dispersion and nonlinear self-phase modulation, as described in Sec. 22.5B. The use of solitons in optical fiber communications systems is described in Sec. 24.2E. Yet another nonlinear optical effect is optical bistability. This involves nonlinear opti- cal effects together with feedback. Applications in photonic switching are described in Sec. 23.4. 21.1 NONLINEAR OPTICAL MEDIA A linear dielectric medium is characterized by a linear relation between the polarization density and the electric field, P == EoXC, where Eo is the permittivity of free space and X is the electric susceptibility of the medium (see Sec. 5.2A). A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between P and C (see Sec. 5.2B), as illustrated in Fig. 21.1-1. The nonlinearity may be of microscopic or macroscopic origin. The polarization
876 CHAPTER 21 NONLINEAR OPTICS £ £ (a) Linear (b) Nonlinear Figure 21.1-1 The P-G relation for (a) a linear dielectric medium, and (b) a nonlinear medium. density P == Np is a product of the individual dipole moment p induced by the applied electric field e and the number density of dipole moments N. The nonlinear behavior may reside either in p or in N. The relation between p and e is linear when e is small, but becomes nonlinear when e acquires values comparable to interatomic electric fields, which are typically rv 105- 10 8 VIm. This may be understood in terms of a simple Lorentz model in which the dipole moment is p == -ex, where x is the displacement of a mass with charge -e to which an electric force -ee is applied (see Sec. 5.5C). If the restraining elastic force is proportional to the displacement (i.e., if Hooke's law is satisfied), the equilibrium displacement x is proportional to e. In that case P is proportional to e and the medium is linear. However, if the restraining force is a nonlinear function of the displacement, the equilibrium displacement x and the polarization density P are nonlinear functions of e and, consequently, the medium is nonlinear. The time dynamics of an anharmonic oscillator model describing a dielectric medium with these features is discussed in Sec. 21.7. Another possible origin of a nonlinear response of an optical material to light is the dependence of the number density N on the optical field. An example is provided by a laser medium in which the number of atoms occupying the energy levels involved in the absorption and emission of light are dependent on the intensity of the light itself (see Sec. 14.4). Since externally applied optical electric fields are typically small in comparison with characteristic interatomic or crystalline fields, even when focused laser light is used, the nonlinearity is usually weak. The relation between P and e is then approximately linear for small c, deviating only slightly from linearity as c increases (see Fig. 21.1- 1). Under these circumstances, the function that relates P to e can be expanded in a Taylor series about e == 0, P == ale + a2c2 + ia3c3 + ..., (21.1-1 ) and it suffices to use only a few terms. The coefficients aI, a2, and a3 are the first, sec- ond, and third derivatives of P with respect to e, evaluated at e == O. These coefficients are characteristic constants of the medium. The first term, which is linear, dominates at small e. Clearly, al == EoX, where X is the linear susceptibility, which is related to the dielectric constant and the refractive index of the material by n 2 == E/ Eo == 1 + X [see (5.2-11)]. The second term represents a quadratic or second-order nonlinearity, the third term represents a third-order nonlinearity, and so on. It is customary to write (21.1-1) in the form t P == Eoxe + 2de 2 + 4X(3)c 3 + ..., (21.1-2) t This nomenclature is used in a number of books, such as A. Yariv, Quantum Electronics, Wiley, 3rd ed. 1989. An alternative relation, P = Eo(X£ + X(2) £2 + X(3) £3), is used in other books, e.g., Y. R. Shen, The Principles of Nonlinear Optics, Wiley, 1984, paperback ed. 2002.
21.1 NONLINEAR OPTICAL MEDIA 877 where d == a2 and X(3) == .2 1 4 a3 are coefficients describing the strength of the second- and third-order nonlinear eftects, respectively. Equation (21.1-2) provides the essential mathematical characterization of a nonlin- ear optical medium. Material dispersion, inhomogeneity, and anisotropy have not been taken into account both for the sake of simplicity and to enable us to focus on the essential features of nonlinear optical behavior. Sections 21.6 and 21.7 are devoted to anisotropic and dispersive nonlinear media, respectively. In centro symmetric media, which have inversion symmetry so that the properties of the medium are not altered by the transformation r -r, the P-E function must have odd symmetry, so that the reversal of E results in the reversal of P without any other change. The second-order nonlinear coefficient d must then vanish, and the lowest order nonlinearity is of third order. Typical values of the second-order nonlinear coefficient d for dielectric crystals, semiconductors, and organic materials used in photonics applications lie in the range d == 10- 24 _10- 21 (C/y2 in MKS units). Typical values of the third-order nonlinear coefficient X(3) for glasses, crystals, semiconductors, semiconductor-doped glasses, and organic materials of interest in photonics are in the vicinity of X(3) == 10- 34 -10- 29 (Cm/y3 in MKS units). Biased or asymmetric quantum wells offer large nonlinearities in the mid and far infrared. EXERCISE 21.1-1 Intensity of Light Required to Elicit Nonlinear Effects. (a) Determine the light intensity (in W/cm 2 ) at which the ratio of the second term to the first term in (21.1-2) is 1 % in an ADP (NH 4 H 2 P0 4 ) crystal for which n == 1.5 and d == 6.8 X 10- 24 C /V 2 at Ao == 1.06 J-Lm. (b) Determine the light intensity at which the third term in (21.1-2) is 1 % of the first term in carbon disulfide (CS 2 ) for which n == 1.6, d == 0, and X(3) == 4.4 X 10- 32 Cm/V 3 at Ao == 694 nm. Note: In accordance with (5.4-8), the light intensity is I == 1£012/21] == (£2) /1], where 1] == 1]o/n is the impedance of the medium and 1]0 == (J-Lo/Eo)1/2 377 fl is the impedance of free space (see Sec. 5.4). The Nonlinear Wave Equation The propagation of light in a nonlinear medium is governed by the wave equation (5.2- 25), which was derived from Maxwell's equations for an arbitrary homogeneous, isotropic dielectric medium. The isotropy of the medium ensures that the vectors 1> and £ are always parallel so that they may be examined on a component-by-component basis, which provides 1 a 2 E a 2 p \J2E - 2" a 2 == J-Lo- a 2 . Co t t (21.1-3) It is convenient to write the polarization density in (21.1-2) as a sum of linear (EoXE) and nonlinear (P NL ) parts, P == EoX E + P NL , P NL == 2dE 2 + 4X(3)E 3 + .. .. (21.1-4 ) (21.1-5) Using (21.1-4), along with the relations c == coin, n 2 == 1 + x, and Co == 1/(E o J-Lo)1/2 provided in (5.2-11) and (5.2-12), allows (21.1-3) to be written as
878 CHAPTER 21 NONLINEAR OPTICS 1 8 2 c \72c - - - == -s c 2 8t 2 8 2 P NL S == -/-Lo 8t 2 . (21.1-6) (21 .1-7) Wave Equation in Nonlinear Medium It is convenient to regard (21.1-6) as a wave equation in which the term S is regarded as a source that radiates in a linear medium of refractive index n. Because P NL (and therefore S) is a nonlinear function of C, (21.1-6) is a nonlinear partial differential equation in c. This is the basic equation that underlies the theory of nonlinear optics. Two approximate approaches to solving this nonlinear wave equation can be called upon. The first is an iterative approach known as the Born approximation. This approx- imation underlies the simplified introduction to nonlinear optics presented in Secs. 21.2 and 21.3. The second approach is a coupled-wave theory in which the nonlinear wave equation is used to derive linear coupled partial differential equations that govern the interacting waves. This is the basis of the more advanced study of wave interactions in nonlinear media presented in Sec. 21.4 and Sec.21.5. Scattering Theory of Nonlinear Optics: The Born Approximation The radiation source S in (21.1-6) is a function of the field c that it, itself, radiates. To emphasize this point we write S == S ( c) and illustrate the process by the simple block diagram in Fig. 21.1-2. Suppose that an optical field Co is incident on a nonlinear medium confined to some volume as shown in the figure. This field creates a radiation source S ( co) that radiates an optical field c 1. The corresponding radiation source S ( C 1) radiates a field C2, and so on. This process suggests an iterative solution, the first step of which is known as the first Born approximation. The second Born approximation carries the process an additional step, and so on. The first Born approximation is Incident light £0 Radiated light £1 -/ Radiation source S( £0) s t -I Radiation . £ SeE) I Nonlinear medium Figure 21.1-2 The first Born approximation. An incident optical field Go creates a source S( Go), which radiates an optical field G 1. adequate when the light intensity is sufficiently weak so that the nonlinearity is small. In this approximation, light propagation through the nonlinear medium is regarded as a scattering process in which the incident field is scattered by the medium. The scattered light is determined from the incident light in two steps: 1. The incident field Co is used to determine the nonlinear polarization density P NL , from which the radiation source S ( co) is determined. 2. The radiated (scattered) field C1 is determined from the radiation source by adding the spherical waves associated with the different source points (as in the theory of diffraction discussed in Sec. 4.3).
21.2 SECOND-ORDER NONLINEAR OPTICS 879 The development presented in Sec. 21.2 and Sec. 21.3 are based on the first Born approximation. The initial field Eo is assumed to contain one or several monochromatic waves of different frequencies. The corresponding nonlinear polarization P NL is then determined using (21.1-5) and the source function S( Eo) is evaluated using (21.1-7). Since S ( Eo) is a nonlinear function, new frequencies are created. The source therefore emits an optical field E 1 with frequencies not present in the original wave Eo. This leads to numerous interesting phenomena that have been utilized to make useful nonlinear optics devices. 21.2 SECOND-ORDER NONLINEAR OPTICS In this section we examine the optical properties of a nonlinear medium in which nonlinearities of order higher than the second are negligible, so that P NL == 2dE 2 . (21.2-1) We consider an electric field E comprising one or two harmonic components and deter- mine the spectral components of P NL . In accordance with the first Born approximation, the radiation source S contains the same spectral components as P NL , and so, therefore, does the emitted (scattered) field. A. Second-Harmonic Generation (SHG) and Rectification Consider the response of this nonlinear medium to a harmonic electric field of angular frequency w (wavelength Ao == 27rc o /w) and complex amplitude E(w), E(t) == Re{E(w) exp(jwt)} == [E(w) exp(jwt) + E*(w) exp( -jwt)]. (21.2-2) The corresponding nonlinear polarization density P NL is obtained by substituting (21.2-2) into (21.2-1), PNL(t) == PNL(O) + Re{P NL (2w) exp(j2wt)} (21.2-3) where PNL(O) == d E(w)E*(w) PNL (2w) == d E 2 ( W ) . (21.2-4) (21.2-5) This process is graphically illustrated in Fig. 21.2-1. Second-Harmonic Generation (SHG) The source S(t) == -/-L08 2 P NL /8t 2 corresponding to (21.2-3) has a component at frequency 2w with complex amplitude S(2w) == 4/-Low 2 dE(w )E(w), which radiates an optical field at frequency 2w (wavelength Ao/2). Thus, the scattered optical field has a component at the second harmonic of the incident optical field. Since the amplitude of the emitted second-harmonic light is proportional to S (2w ), its intensity I (2w) is proportional to 1 S (2w ) 1 2 , which is proportional to the square of the intensity of the incident wave I(w) == IE(w)12/27] and to the square of the nonlinear coefficient d.
880 CHAPTER 21 NONLINEAR OPTICS kLzt TNL(t) ! ___u__u m _ nn_n_m__ ! uu------u----- m-71!SFK!:i" . 0 . .. · : : E t . . . . : E(t) 1 . t + /\/\/\/\/\ IVVVV\ DC Second-harmonic Figure 21.2-1 A sinusoidal electric field of angular frequency w in a second-order nonlinear optical medium creates a polarization with a component at 2w (second-harmonic) and a steady (dc) component. Since the emissions are added coherently, the intensity of the second-harmonic wave is proportional to the square of the length of the interaction volume L. The efficiency of second-harmonic generation IlsHG == I(2w)/I(w) is therefore proportional to L 2 I (w ). Since I ( w) == P / A, where P is the incident power and A is the cross-sectional area of the interaction volume, the SHG efficiency is often expressed in the form L 2 IlsHG = c 2 A P, (21.2-6) SHG Efficiency where C 2 is a constant (units ofW- 1 ) proportional to d 2 and w 2 . An expression for C 2 will be provided in (21.4-36). In accordance with (21.2-6), to maximize the SHG efficiency it is essential that the incident wave have the largest possible power P. This is accomplished by use of pulsed lasers for which the energy is confined in time to obtain large peak powers. Additionally, to maximize the ratio L 2 / A, the wave must be focused to the smallest possible area A and provide the longest possible interaction length L. If the dimensions of the nonlinear crystal are not limiting factors, the maximum value of L for a given area A is limited by beam diffraction. For example, a Gaussian beam focused to a beam width W o maintains a beam cross-sectional area A == 7r WJ over a depth of focus L == 2zo == 27rWJ/.x [see (3.1-22)] so that the ratio L 2 /A == 2L/A == 4A/A2. The beam should then be focused to the largest spot size, corresponding to the largest depth of focus. In this case, the efficiency is proportional to L. For a thin crystal, L is determined by the crystal and the beam should be focused to the smallest spot area A [see Fig. 21.2-2 (a)]. For a thick crystal, the beam should be focused to the largest spot that fits within the cross-sectional area of the crystal [see Fig. 21.2-2(b)]. L - L I · .1 I--L -I (a) (b) (c) Figure 21.2-2 Interaction volume in a (a) thin crystal, (b) thick crystal, and (c) waveguide.
21.2 SECOND-ORDER NONLINEAR OPTICS 881 Guided-wave structures offer the advantage of light confinement in a small cross- sectional area over long distances [see Fig. 21.2-2(c)]. Since A is determined by the size of the guided mode, the efficiency is proportional to £2. Optical waveguides take the form of planar or channel waveguides (Chapter 8) or fibers (Chapter 9). Although silica-glass fibers were initially ruled out for second-harmonic generation since glass is centrosymmetric (and therefore presumably has d == 0), second-harmonic generation is in fact observed in silica-glass fibers, an effect attributed to electric quadrupole and magnetic dipole interactions and to defects and color centers in the fiber core. Figure 21.2-3 illustrates several configurations for optical second-harmonic-generation in bulk materials and in waveguides, in which infrared light is converted to visible light and visible light is converted to the ultraviolet. (a) ....,.. ,..,:.:....."....,..,...<.:...,."\ Ruby laser .... ..j w 2w 694 nm (red) 347 nm (UV) KDP crystal (b) .:,: ...,-. :;. :-;---;.>'.:-.,--.-.-:-..-.-:,.;,,:-.:;.::i.; "'....:'".;;-:;; ...:.'j,'r;>N;::.-"""'-. '.' \ w Nd 3 +:y AG laser' I .- ... . - .... , m'" - "" .. ........, ..... ""_.. ... '. j 1.06 Jim (IR) (0 Ge- and P-doped silica-glass fiber 2w 530 nm (green) /' --1 J w 790nm(IR) (c) 2w 395 nm (violet) I AIGaAs laser Figure 21.2-3 Optical second-harmonic generation (a) in a bulk crystal; (b) in a glass fiber; (c) within the cavity of a laser diode. Optical Rectification The component PNL(O) in (21.2-3) corresponds to a steady (non-time-varying) polar- ization density that creates a DC potential difference across the plates of a capacitor within which the nonlinear material is placed (Fig. 21.2-4). The generation of a DC voltage as a result of an intense optical field represents optical rectification (in analogy with the conversion of a sinusoidal AC voltage into a DC voltage in an ordinary electronic rectifier). An optical pulse of several MW peak power, for example, may generate a voltage of several hundred J-L V. Light Figure 21.2-4 The transmission of an intense beam of light through a nonlinear crystal generates a DC voltage across it.
882 CHAPTER 21 NONLINEAR OPTICS B. The Electro-Optic Effect We now consider an electric field E (t) comprising a harmonic component at an optical frequency w together with a steady component (at w == 0), E(t) == E(O) + Re{E(w) exp(jwt)}. (21.2-7) We distinguish between these two components by denoting the electric field E(O) and the optical field E (w ). In fact, both components are electric fields. Substituting (21.2-7) into (21.2-1), we obtain PNL(t) == PNL(O) + Re{PNL(w) exp(jwt)} + Re{P NL (2w) exp(j2wt)}, (21.2-8) where PNL(O) == d [2E 2 (0) + IE(w)1 2 ] PNL(W) == 4dE(0)E(w) P NL ( 2w) == d E 2 ( W ) , (21.2-9a) (21.2-9b) (21.2-9c) so that the polarization density contains components at the angular frequencies 0, w, and 2w. If the optical field is substantially smaller in magnitude than the electric field, i.e., IE(w)12 « IE(0)1 2 , the second-harmonic polarization component P N L(2w) may be neglected in comparison with the components P NL (0) and P NL ( w ). This is equivalent to the linearization of P NL as a function of E, i.e., approximating it by a straight line with a slope equal to the derivative at E == E(O), as illustrated in Fig. 21.2-5. P NL y' PNL(O) ---------- -j-------- - - - - ----- I : t _1_+_________ _ _ __ I I I I I I r I I I : : I £(0) I I I E(w) 1......1: £ I.....I I.....I ...... I I ..... I I I £(0) -1 t Figure 21.2-5 Linearization of the second-order nonlinear relation P NL = 2d£2 in the presence of a strong electric field E(O) and a weak optical field E(w). Equation (21.2-9b) provides a linear relation between PNL(W) and E(w), which we write in the form PNL(W) == EotlXE(W), where X == (4d/Eo)E(0) represents an increase in the susceptibility proportional to the electric field E (0). The corresponding incremental change of the refractive index is obtained by differentiating the relation n 2 == 1 + x, to obtain 2n n == X, from which 2d n == -E(O). nEo (21.2-10) The medium is then effectively linear with a refractive index n + t6.n that is linearly controlled by the electric field E (0). The nonlinear nature of the medium creates a coupling between the electric field E (0) and the optical field E (w ), causing one to control the other, so that the nonlinear medium exhibits the linear electro-optic effect (Pockels effect) discussed in Chapter 20.
21.2 SECOND-ORDER NONLINEAR OPTICS 883 This effect is characterized by the relation fj.n == - n3tE(0), where t is the Pock- els coefficient. Comparing this formula with (21.2-10), we conclude that the Pockels coefficient t is related to the second-order nonlinear coefficient d by 4 t --d. E o n 4 (21.2-11) Although this expression reveals the common underlying origin of the Pockels effect and the medium nonlinearity, it is not consistent with experimentally observed values of t and d. This is because we have made the implicit assumption that the medium is nondispersive (i.e., that its response is insensitive to frequency). This assumption is clearly not satisfied when one of the components of the field is at the optical frequency wand the other is a steady field with zero frequency. The role of dispersion is discussed in Sec. 21.7. c. Three-Wave Mixing We now consider the case of a field E(t) comprising two harmonic components at optical frequencies WI and W2, E(t) == Re{E(wl) exp(jwIt) + E(W2) exp(jw2t)}. (21.2-12) The nonlinear component of the polarization P NL == 2dE 2 then contains components at five frequencies, 0, 2WI, 2W2, W+ == WI + W2, and w_ == WI - W2, with amplitudes PNL(O) == d [IE(WI)1 2 + IE(W2)1 2 ] PNL(2wI) == d E(WI)E(WI) PNL(2w2) == d E(W2)E(W2) PNL(W+) == 2d E(WI)E(W2) PNL(W-) == 2d E(WI)E*(W2). (21.2-13a) (21.2-13b) (21.2-13c) (21.2-13d) (21.2-13e) Thus, the second-order nonlinear medium can be used to mix two optical waves of different frequencies and generate (among other things) a third wave at the difference frequency or at the sum frequency. The former process is called frequency downcon- version whereas the latter is known as frequency up-conversion or sum-frequency generation. An example of frequency up-conversion is provided in Fig. 21.2-6: the light from two lasers with free-space wavelengths AOl == 1.06 Mm and A o 2 == 10.6 Mm enter a proustite crystal and generate a third wave with wavelength A03 == 0.96 Mm (where A;;l == Aci l + A;;2 1 ). . ...........,..,..: 0.'.:::' >::...\":....:'<.:..:.:::........:'.__.;:;'.... "\ .. Nd 3 +:YAG laser' .. WI 1.06/L ID (' W3 =wI+w2 0.96 /LID Proustite crystal C02 laser W2 .'u' .<." ".,."-- 10.6/L ID Figure 21.2-6 An example of sum-frequency generation (SFG), also called frequency up- conversion, in a nonlinear crystal.
884 CHAPTER 21 NONLINEAR OPTICS Although the incident pair of waves at frequencies WI and W2 produce polarization densities at frequencies 0, 2WI, 2W2, WI + W2, and WI - W2, all of these waves are not necessarily generated, since certain additional conditions (phase matching) must be satisfied, as explained presently. Frequency and Phase Matching If waves] and 2 are plane waves with wavevectors k i and k 2 , so that E(WI) == Al exp( -jk l . r) and E(W2) == A 2 exp( -jk 2 . r), then in accordance with (21.2-13d), P NL (W3) == 2dE(WI)E(W2) == 2dA I A 2 exp( -jk 3 . r), where and I WI + W2 = W3 I k I +k 2 =k 3 - (21.2-14) Frequency-Matching Condition (21.2-15) Phase-Matching Condition The medium therefore acts as a light source of frequency W3 == WI +W2, with a complex amplitude proportional to exp( - jk 3 . r), so that it radiates a wave of wavevector k3 == k i + k 2 , as illustrated in Fig. 21.2-7. Equation (21.2-15) can be regarded as a condition of phase matching among the wavefronts of the three waves that is analogous to the frequency-matching condition WI + W2 == W3. Since the argument of the complex wavefunction is wi - k . r, these two conditions ensure both the temporal and spatial phase matching of the three waves, which is necessary for their sustained mutual interaction over extended durations of time and regions of space. k 2 , II, .'I!:lk3 k3 Figure 21.2-7 The phase- matching condition. Three-Wave Mixing Modalities When two optical waves of angular frequencies WI and W2 travel through a second- order nonlinear optical medium they mix and produce a polarization density with components at a number of frequencies. We assume that only the component at the sum frequency W3 == WI +W2 satisfies the phase-matching condition. Other frequencies cannot be sustained by the medium since they are assumed not to satisfy the phase- matching condition. Once wave 3 is generated, it interacts with wave 1 and generates a wave at the difference frequency W2 == W3 - WI. Clearly, the phase-matching condition for this interaction is also satisfied. Waves 3 and 2 similarly combine and radiate at WI. The three waves therefore undergo mutual coupling in which each pair of waves interacts and contributes to the third wave. The process is called three-wave mixing. Two-wave mixing is not, in general, possible. Two waves of arbitrary frequencies WI and W2 cannot be coupled by the medium without the help of a third wave. Two- wave mixing can occur only in the degenerate case, W2 == 2WI, in which the second- harmonic of wave 1 contributes to wave 2; and the subharmonic W2/2 of wave 2, which is at the frequency difference W2 - WI, contributes to wave 1. Three-wave mixing is known as a parametric interaction process. It takes a variety of forms, depending on which of the three waves is provided as an input, and which are extracted as outputs, as illustrated in the following examples (see Fig. 21.2-8):
21.2 SECOND-ORDER NONLINEAR OPTICS 885 . Optical Frequency Conversion (OFC). Waves 1 and 2 are mixed in an up- converter, generating a wave at the sum frequency W3 == WI + W2. This process, also called sum-frequency generation (SFG), has already been illustrated in Fig. 21.2-6. Second-harmonic generation (SHG) is a degenerate special case of SFG. The opposite process of downconversion or frequency-difference gener- ation is realized by an interaction between waves 3 and 1 to generate wave 2, at the difference frequency W2 == W3 - WI. Up- and down-converters are used to generate coherent light at wavelengths where no adequate lasers are available, and as optical mixers in optical communication systems. . Optical Parametric Amplifier (OPA). Waves 1 and 3 interact so that wave 1 grows, and in the process an auxiliary wave 2 is created. The device operates as a coherent amplifier at frequency WI and is known as an OPA. Wave 3, called the pump, provides the required energy, whereas wave 2 is known as the idler wave. The amplified wave is called the signal. Clearly, the gain of the amplifier depends on the power of the pump. OPAs are used for the detection of weak light at wavelengths for which sensitive detectors are not available. . Optical Parametric Oscillator (OPO). With proper feedback, the parametric amplifier can operate as a parametric oscillator, in which only a pump wave is supplied. OPOs are used for the generation of coherent light and mode-locked pulse trains over a continuous range of frequencies, usually in frequency bands where there is a paucity of tunable laser sources. . Spontaneous Parametric Downconversion (SPDC). Here, the only input to the nonlinear crystal is the pump wave 3, and downconversion to the lower-frequency waves 2 and 3 is spontaneous. The frequency- and phase-matching conditions (21.2-14) and (21.2-15) lead to multiple solutions, each forming a pair of waves 1 and 2 with specific frequencies and directions. The down-converted light takes the form of a cone of multispectral light, as illustrated in Fig. 21.2-8. Further details pertaining to these parametric devices are provided in Sec. 21.4. Filter. k !: .. . .. I C "..: :¥j -:O::}Si gnal OFC Signal wI Pump w2 : I - Idler ri w2 (.:'.;..1 Amplified sig;a l »'" 1?<#/" wI OPA Signal wI Pump w3 , I I I OPO PUJ11PWO Mirror -{:J WIG! ) MIrror WI . Pump w3 ) SPDC Crystal W2 Figure 21.2-8 Optical parametric devices: optical frequency converter (OFC); optical parametric amplifier (OPA); optical parametric oscillator (OPO); spontaneous parametric down-converter (SPDC).
886 CHAPTER 21 NONLINEAR OPTICS Wave Mixing as a Photon Interaction Process The three-wave-mixing process can be viewed from a photon-optics perspective as a process of three-photon interaction in which two photons of lower frequency, WI and W2, are annihilated, and a photon of higher frequency W3 is created, as illustrated in Fig. 21.2-9(a). Alternatively, the annihilation of a photon of high frequency W3 is accompanied by the creation of two low-frequency photons, of frequencies WI and W2, as illustrated in Fig. 21.2-9(b). Since nw and ILk are the energy and momentum of a photon of frequency wand wavevector k (see Sec. 12.1), conservation of energy and momentum, in either case, requires that nwi + nw2 == tiw 3 ILk 1 + ILk 2 == ILk 3 , (21.2-16) (21.2-17) where k I , k 2 , and k3 are the wavevectors of the three photons. The frequency- and phase-matching conditions presented in (21.2-14) and (21.2-15) are therefore repro- duced. The energy diagram for the three-photon-mixing process displayed in Fig. 21.2- 9(b) bears some similarity to that for an optically pumped three-level laser, illustrated in Fig. 21.2-9(c) (see Sec. 14.2B). There are significant distinctions between the two processes, however: . One of the three transitions involved in the laser process is non-radiative. . An exchange of energy between the field and medium takes place in the laser process. . The energy levels associated with the laser process are relatively sharp and are established by the atomic or molecular system, whereas the energy levels of the parametric process are dictated by photon energy and phase-matching conditions and are tunable over wide spectral regions. , ---- - -10: \ nw l J, \ Nonradiative nw 3 nw z nw3 / \ . . Pump transItion / nw 3 'V\f\MJ\r-, nw 3 nw} nw3 Laser nw 1 hw 1 transition nw z nwZ , .. (a) (b) (c) Figure 21.2-9 Comparison of parametric processes in a second-order nonlinear medium and laser action. (a) Annihilation of two low-frequency photons and creation of one high-frequency photon. The dashed line for the upper level indicates that it is virtual. (b) Annihilation of one high- frequency photon and creation of two low-frequency photons. (c) Optically pumped 3-levellaser, a nonparametric process in which the medium participates in energy transfer. The process of wave mixing involves an energy exchange among the interacting waves. Clearly, energy must be conserved, as is assured by the frequency-matching condition, WI + W2 == W3. Photon numbers must also be conserved, consistent with the photon interaction. Consider the photon-splitting process represented in Fig. 21.2-9(b). If flcPI, flcP2, and flcP3 are the net changes in the photon fluxes (photons per second) in the course of the interaction (the flux of photons leaving minus the flux of photons entering) at frequencies WI, W2, and W3, then flcPI == flcP2 == - flcP3, so that for each of the W3 photons lost, one each of the WI and W2 photons is gained. If the three waves travel in the same direction, the z direction for example, then by taking a cylinder of unit area and incremental length z ---t 0 as the interaction
21.2 SECOND-ORDER NONLINEAR OPTICS 887 volume, we conclude that the photon flux densities cPl, cP2, cP3 (photons/s-m 2 ) of the three waves must satisfy dcPl dz dcP2 dz dcP3 -- dz (21.2-18) Photon-Number Conservation Since the wave intensities (W/m 2 ) are II (21.2-18) gives nwl cPl, 1 2 == nw2cP2, and 13 == nw3cP3, ( ) = :z ( : ) = - :z ( : ) . (21.2-19) Manley-Rowe Relation Equation (21.2-19) is known as the Manley-Rowe relation. It was derived in the con- text of wave interactions in nonlinear electronic systems. The Manley-Rowe relation can be derived using wave optics, without invoking the concept of the photon (see Exercise 21.4-2). D. Phase Matching and Tuning Curves Phase Matching in Collinear Three-Wave Mixing If the mixed three waves are collinear, i.e., they travel in the same direction, and if the medium is nondispersive, then the phase-matching condition (21.2-15) yields the scalar equation nWl / Co + nW2 / Co == nW3 / Co, which is automatically satisfied if the frequency matching condition WI + W2 == W3 is met. However, since all materials are in reality dispersive, the three waves actually travel at different velocities corresponding to different refractive indexes, nl, n2, and n3, and the frequency- and phase-matching conditions are independent: WI + W2 == W3, wlnl + W2 n 2 == W3 n 3, (21.2-20) Matching Conditions and must be simultaneously satisfied. Since this is usually not possible, birefringence, which is present in anisotropic media, is often used to compensate dispersion. For an anisotropic medium, the three refractive indexes nl, n2, and n3 are generally dependent on the polarization of the waves and their directions relative to the principal axes (see Sec. 6.3C). This offers other degrees of freedom to satisfy the matching conditions. Precise control of the refractive indexes at the three frequencies is often achieved by appropriate selection of polarization, orientation of the crystal, and in some cases by control of the temperature. In practice, the medium is often a uniaxial crystal characterized by its optic axis and frequency-dependent ordinary and extraordinary refractive indexes no (w) and ne (w). Each of the three waves can be ordinary (0) or extraordinary (e) and the process is labeled accordingly. For example, the label e-o-o indicates that waves 1, 2, and 3 are e, 0, and 0 waves, respectively. For an 0 wave, n( w) == no (w); for an e wave, n( w) == n( e, w) depends on the angle e between the direction of the wave and the optic axis of the crystal, in accordance with the relation 1 n 2 (e,w) cos 2 e sin 2 e + n(w) n(w)' (21.2-21)
888 CHAPTER 21 NONLINEAR OPTICS which is represented graphically by an ellipse [see (6.3-15) and Fig. 6.3-7]. If the polarizations of the signa] and idler waves are the same, the wave mixing is said to be Type I; if they are orthogonal, it is said to be Type II. EXAMPLE 21.2-1. Collinear Type-I Second-Harmonic Generation (SHG). For SHG, waves 1 and 2 have the same frequency (WI == W2 == w) and W3 == 2w. For Type- I mixing, waves 1 and 2 have identical polarization so that nl == n2. Therefore, from (21.2-20), the phase-matching condition is n3 == nl, i.e., the fundamental wave has the same refractive index as the second-harmonic wave. Because of dispersion, this condition cannot usually be satisfied unless the polarization of these two waves is different. For a uniaxial crystal, the process is either o-o-e or e-e-o. In either case, the direction at which the wave enters the crystal is adjusted in such a way that n3 == nl, i.e., such that birefringence compensates exactly for dispersion. no(2w) "",ne / .." no , , , , , , , , , , , , , , , , i i w 2w -- - /" /" ". "./ / / / / I \ \ \ '\. ........ " , ........ ........- , --- W Figure 21.2-10 Phase matching in e-e-o SHG. (a) Matching the index of the e wave at w with that of the 0 wave at 2w. (b) Index surfaces at w (solid curves) and 2w (dashed curves) for a uniaxial crystal. ( c) The wave is chosen to travel at an angle () with respect to the crystal optic axis, such that the extraordinary refractive index n e ((), w) of the w wave equals the ordinary refractive index no(2w) of the 2w wave. For an e-e-o process such as that illustrated in Fig. 21.2-10, the fundamental wave is extraordinary and the second-harmonic wave is ordinary, nl == n((), w) and n3 == n o (2w), so that the matching condition is: n((), w) == no(2w). This is achieved by selecting an angle () for which n((), w) == n o (2w), (21.2-22) SHG Type-I e-e-Q where n((),w) is given by (21.2-21). This is illustrated graphically in Fig. 21.2-10, which displays the ordinary and extraordinary refractive indexes (a circle and an ellipse) at w (solid curves) and at 2w (dashed curves). The angle at which phase matching is satisfied is that at which the circle at 2w intersects the ellipse at w. As an example, for KDP at a fundamental wavelength A = 694 nm, no (w) = 1.506, ne (w) = 1.466; and at A/2 == 347 nm, no(2w) = 1.534, ne(2w) = 1.490. In this case, (21.2-22) and (21.2-21) gives () == 52 0 . This is called the cut angle of the crystal. Similar equations may be written for SHG in the o-o-e configuration. In this case, for KDP at a fundamental wavelength A == 1.06 /-L m , () == 41 0 . EXAMPLE 21.2-2. Collinear Optical Parametric Oscillator (OPO). The oscillation fre- quencies of an OPO are determined from the frequency and phase matching conditions. For a Type- I o-o-e mixing configuration, WI + W2 == W3, wIno(wl) + W2 n o(W2) == W3 n ((), W3). (21.2-23) OPO Type-I Q-Q-e
21.2 SECOND-ORDER NONLINEAR OPTICS 889 For Type-II e-o-e mixing, WI + W2 = W3, WIn({}, WI) + W2 n o(W2) = W3 n ({}, W3). (21.2-24) OPO Type-II e-Q-e 1.4 E :i. 1.2 tD s::::: 1.0 > ro 0.8 1.4 E 3 1 . 2 ..c: tD Idler ] I 0 Q) . > ro 0.8 0.6 22 22.5 23 Crystal cut angle e (deg) (a) 1.6 0.6 20 40 50 30 Crystal cut angle e (deg) (b) Figure 21.2-11 Tuning curves for a collinear OPO using a BBO crystal and a 532-nm pump, which is readily obtained from a frequency doubled Nd: YAO laser (a) Type I, and (b) Type II. The functions no(w) and ne(w) are determined from the Sellmeier equation (5.5-28), and the extraordinary index n( {}, w) is determined as a function of the angle {} between the optic axis of the crystal and the direction of the waves by use of (21.2-21). For a given pump frequency W3, the solutions of (21.2-23) and (21.2-24), Wl and W2, are often plotted versus the angle {}, a plot known as the tuning curve. Examples are illustrated in Fig. 21.2-11. Phase Matching in Non-Collinear Three-Wave Mixing In the non-collinear case, the phase-matching condition k l + k 2 == k3 is equivalent to wlnlul + W2 n 2 u 2 == W3 n 3 u 3, where UI, U2, and U3 are unit vectors in the directions of propagation of the waves. The refractive indexes nl, n2, and n3 depend on the directions of the waves relative to the crystal axes, as well as the polarization and fre- quency. This vector equation is equivalent to two scalar equations so that the matching conditions become WI + W2 == W3, WI nl sin ()l == W2 n 2 sin ()2, WI nl cas ()l + W2 n 2 cas ()2 == W3 n 3, (21.2-25) where ()l and ()2 are the angles waves ] and 2 make with wave 3. The design of a 3-wave mixing device centers about the selection of directions and polarizations to satisfy these equations, as demonstrated by the following exercises and examples. EXERCISE 21.2-1 Non-Collinear Type-II Second-Harmonic Generation (SHG). Figure 21.2-12 illustrates Type-II o-e-e non-collinear SHOo An ordinary wave and an extraordinary wave, both at the fundamental frequency w, create an extraordinary second-harmonic wave at the frequency 2w. It is assumed here that the directions of propagation of the three waves and the optic axis are coplanar and the two fundamental waves and the optic axis make angles {}I, (}2, and {} with the direction of the second-harmonic wave. The refractive indexes that appear in the phase-matching equations (21.2-25) are nl = no(w), n2 = n({} + (}2, w), and n3 = n({}, 2w), i.e., no(w) sin {}I = n( {} + (}2, w) sin (}2, no(w) cas {}I + n( {} + (}2, w) cas (}2 = 2n( {}, 2w). (21.2-26) SHG Type-II Q-e-e
a90 CHAPTER 21 NONLINEAR OPTICS For a KDP crystal and a fundamental wave of wavelength 1.06JLm (Nd:Yag laser), determine the crystal orientation and the angles ()I and ()2 for efficient second-harmonic generation. . is o\'tl C _ _ --- -fj Figure 21.2-12 Non-collinear Type II second- harmonic generation. EXAMPLE 21.2-3. Spontaneous Parametric Downconversion (SPDC). In SPDC, a pump wave of frequency W3 creates pairs of waves 1 and 2, at frequencies WI and W2, and angles ()I and ()2, all satisfying the frequency- and phase-matching conditions (21.2-25). For example, in the Type-I o-o-e case, nl == no(wl), n2 == n o (w2) and n3 == n((), W3). These relations together with the Sellmeier equations for no(w) and ne (w) yield a continuum of solutions (WI, ()I), (W2, ()2) for the signal and idler waves, as illustrated by the example in Fig. 21.2-13. 6 4 WI ,-.., CrJ 2 \.-; 0.0 -0 0 '-" -2 < -4 -6 0.9 1.0 1.1 Normalized Frequency Figure 21.2-13 Tuning curves for non-collinear Type-I o-o-e spontaneous parametric downcon- version in a BBO crystal at an angle () == 33.53° for a 351.5-nm pump (from an Ar+ -ion laser). Each point in the bright area of the middle picture represents the frequency WI and angle ()] of a possible down-converted wave, and has a matching point at a complementary frequency W2 == W3 - WI with angle ()2. Frequencies are normalized to the degenerate frequency W o == W3/2. For example, the two dots shown represent a pair of down-converted waves at frequencies O.9w o and 1.1w o . Because of circular symmetry, each point is actually a ring of points all of the same frequency, but each point on a ring matches only one diametrically opposite point on the corresponding ring, as illustrated in the right graph. Tolerable Phase Mismatch and Coherence Length A slight phase mismatch k == k3 - k i - k 2 #- 0 may result in a significant reduction in the wave-mixing efficiency. If waves 1 and 2 are plane waves with wavevectors k i and k 2 , so that E(WI) == Al exp( -jk l . r) and E(W2) == A 2 exp( -jk 2 . r), then in accor- dance with (21.2-13d), P NL (W3) == 2dE(WI)E(W2) == 2dA 1 A 2 exp[-j(k 1 +k 2 ) .r] == 2dA I A 2 exp(jk. r) exp( -jk 3 . r). By virtue of (21.1-7) this creates a source with angular frequency W3, wavevector k 3 , and complex amplitude 2w5 /-La dA I A 2 exp (j k. r). It can be shown (see Prob. 21.2-6) that the intensity of the generated wave is proportional to the squared integral of the source amplitude over the interaction volume V, 2 h ex [dA 1 A 2 exp(j.6.k. r)dr (21.2-27)
21.2 SECOND-ORDER NONLINEAR OPTICS 891 Because the contributions of different points within the interaction volume are added as phasors, the position-dependent phase k . r in the phase mismatched case results in a reduction of the total intensity below the value obtained in the matched case. Consider the special case of a one-dimensional interaction volume of width L in the z direction: 13 ex I Jo£ exp(jk z)dzl 2 == L 2 sinc2(k L/27r), where k is the z component of k and sinc( x) == sin( 7rx) / (7rx). It follows that in the presence of a wavevector mismatch t6.k, 13 is reduced by the factor sinc2(kL/27r), which is unity for k == 0 and drops as k increases, reaching a value of (2/7r)2 0.4 when Ikl == 7r/L, and vanishing when Ikl == 27r/L (see Fig. 21.2-14). For a given L, the mismatch k corresponding to a prescribed efficiency reduction factor is inversely proportional to L, so that the phase-matching requirement becomes more stringent as L increases. For a given mismatch k, the length Lc == 27r /Ikl (21.2-28) Coherence Length is a measure of the maximum length within which the parametric interaction process is efficient; Lc is often called the wave-mixing coherence length. For example, for a second-harmonic generation Ikl == 2(27r / AD) In3 - nIl, where AD is the free-space wavelength of the fundamental wave and ni and n3 are the re- fractive indexes of the fundamental and the second-harmonic waves. In this case, Lc == Ao/21 n 3 - nIl is inversely proportional to In3 - nIl, which is governed by the material dispersion. For example, for I n3 - ni 1== 10- 2 , Lc == 50A. sinc2(kL/27f) 1 -47f -27f o 27f 47f kL Figure 21.2-14 The factor by which the efficiency of three-wave mixing is reduced as a result of a phase mismatch tlkL be- tween waves interacting within a distance L. The tolerance of the interaction process to the phase mismatch can be regarded as a result of the wavevector uncertainty k ex 1/ L associated with confinement of the waves within a distance L [see (A.2-6) in Appendix A]. The corresponding momentum uncertainty p == tik ex 1/ L explains the apparent violation of the law of conservation of momentum in the wave-mixing process. Phase-Matching Bandwidth As previously noted, for a finite interaction length L, a phase mismatch Ikl < 27r / L is tolerated. If exact phase matching is achieved at a set of nominal frequencies of the mixed waves, then small frequency deviations from those values may be tolerated, as long as the condition WI + W2 == W3 is perfectly satisfied. The spectral bands associated with such tolerance are established by the condition Ikl < 27r / L. As an example, in SHG we have two waves with frequencies WI == wand W3 == 2w. The mismatch k is a function k(w) of the fundamental frequency w. The device is designed for exact phase matching at a nominal fundamental frequency wo, i.e.,
892 CHAPTER 21 NONLINEAR OPTICS k(wo) == O. The bandwidth w is then established by the condition lk(wo+w)1 == 27r / L. If w is sufficiently small, we may write k(wo + w) == k' w, where k' == (d/dw)k atwo. Therefore, w == 27r/Ik'IL, from which the spectral width in Hz is v == l/lk'IL. (21.2-29) Phase-Matching Bandwidth Since k(w) == k3(2w) -2k 1 (w), the derivative k' == dk 3 (2w)/dw-2dk 1 (w)/dw == 2[dk3(2w)/d(2w) - dk1(w)/dwJ == 2[1/v3 - l/Vl], where VI and V3 are the group velocities of waves 1 and 3 at frequencies wand 2w, respectively (see Sec. 5.6). The spectral width is therefore related to the length L and the group velocity mismatch by 1 L L- 1 v == - - - - 2 v3 VI Co 1 2L IN 3 -NIl' (21.2-30) Phase-Matching Bandwidth where N 1 and N 3 are the group indexes of the material at the fundanlental and second- harmonic frequencies. It is apparent that second-harmonic generation of a broadband wave, or an ultra- narrow pulse (see Sec. 23.5), can be accomplished by use of a thin crystal (at a cost of lower conversion efficiency), and by the use of an additional design constraint, group velocity matching, V3 VI or N 3 N 1 . Phase-matching tolerance in SPDC is revealed in Fig. 21.2-13 by the thickness of the curves. E. Quasi-Phase Matching In the presence of a wavevector mismatch k, points within the interaction volume radiate with position-dependent phases k . r, so that the magnitude of the gener- ated parametric wave is significantly reduced. Since phase matching can be difficult to achieve, or can severely constrain the choice of the nonlinear coefficient or the crystal configuration that maximizes the efficiency of wave conversion, one approach is to allow a phase mismatch, but to compensate it by using a medium with position- dependent periodic nonlinearity. Such periodicity introduces an opposite phase that brings back the phases of the distributed radiation elements into better alignment. The technique is called quasi-phase matching (QPM). If the medium has a position-dependent nonlinear coefficient d(r), then (21.2-27) becomes 2 h ex Iv d(r) exp(jk . r)dr . If d(r) is a harmonic function d(r) == do exp( -jG. r), with G == k, then the phase mismatch is fully eliminated. Accordingly, the phase-matching condition (21.2-15) is replaced with (21.2-31) k 1 + k 2 + G == k3. (21.2-32) In effect, the nonlinear medium serves as a phase grating (or longitudinal Bragg grat- ing) with a wavevector G.
21.2 SECOND-ORDER NONLINEAR OPTICS 893 It is generally difficult to fabricate a medium with a continuously varying harmonic nonlinear coefficient, d(r) == do exp( -jG . r), but it is possible to fabricate simpler periodic structures, e.g., media with nonlinear coefficients of constant magnitude but periodically reversed sign. Since any periodic function can be decomposed into a superposition of harmonic functions via Fourier series, one such function can serve to correct the phase mismatch, with the others playing no role in the wave-mixing process because they introduce greater phase mismatch. QPM in Collinear Wave Mixing For collinear waves traveling in the z direction and having a phase mismatch flk, the required phase grating is of the form exp( -jGz), where G == flk. Such grating may be obtained by use of a periodic nonlinear coefficient d(z) described by the Fourier series d(z) == 2.: :- -00 d m exp( -j27rmzj A), where A is the period and {d m } are the Fourier coefficients. Any of these components may be used for phase matching. For example, for the mth harmonic, G == m27r j A == flk, so that A == m27r j flk == mLc, (21.2-33) QPM Condition i.e., the grating period A equals an integer multiple of the coherence length Lc 27r j flk. Equation (21.2-32) together with the frequency matching condition yield WI + W2 == W3, Wini + W2 n 2 + m27rcj A == W3n3. (21.2-34) QPM Tuning Curves These equations are used in lieu of (21.2-20) to determine the tuning curves and the crystal angles in the design of parametric devices. It is evident that QPM offers some flexibility in the design of desired tuning curves. QPM in a Medium with Periodically Reversed Nonlinear Coefficient The simplest periodic pattern of the nonlinear coefficient d( z) alternates between two constant values, + do and - do, at distances A j 2, as illustrated in Fig. 21.2-15. ..... .' .A. j -- .':'-::.::1<'1:- . . . . . - ... . - - - . .. t: t Jt t t t t t f+ t t < ; .! L » I I I( do d(z) z Figure 21.2-15 A nonlinear crystal with pe- riodically varying nonlinear coefficient d( z) of period A. -do The physical mechanism by which the periodic reversal of the sign of nonlinearity serves to compensate the position-dependent phase of the radiation is illustrated in Fig. 21.2-16 in the m == 1 case; i.e., the grating period A equals the coherence length Lc == 27r j flk. The improvement of the conversion efficiency afforded by QPM may be determined quantitatively as follows. In accordance with Fourier series theory, d m == (2 j m 7r ) do,
894 CHAPTER 21 NONLINEAR OPTICS z = 0 z = 7f/l1k =A/2 z = 27f/IJ.k =A .»»»»»»»»»)0)8»»»»»»0»»»»)8»»»») ) (a) Phase matched z=O (b) Phase mismatched (c) Quasi-phase matched Figure 21.2-16 Phasors of the waves radiated by incremental elements at different positions z in the nonlinear medium. (a) In the phase-matched case (6.k = 0) the phasors are all aligned and maximum conversion efficiency is attained. (b) In the presence of a phase mismatch 6.k, the phasors are misaligned and the efficiency is significantly reduced. (c) In the quasi-phase matched case, the misaligned phasors are periodically reversed by reversing the sign of the nonlinear coefficient at A/2 intervals. The conversion efficiency is partially restored. for odd m, and zero, otherwise. If phase matching is accomplished via the mth har- monic, i.e., A == mLc, then the parametric conversion efficiency is proportional to d == (2/11"m)2d. By contrast, a homogeneous medium with nonlinear coefficient do, the same length L, but with wavevector mismatch b,.k, has a conversion efficiency d sinc2(kL 1211") == d sinc 2 (L I Lc), which falls as (d/11"2) (Lcl L)2 when L » Lc. Since Lc == Aim, the improvement of conversion efficiency is a factor of 4(LI A)2, i.e., is proportional to the square of the number of periods of the periodic structure. Clearly, the use of a periodic medium can offer a significant improvement in conversion efficiency. The most challenging aspect of quasi-phase matching is the fabrication of the pe- riodic nonlinear structure. A uniform nonlinear crystal may be altered periodically by reversing the principal axis direction in alternating layers, thus creating a d coefficient with alternating sign. This may be accomplished by lithographically exposing the crystal to a periodic electric field that reverses the direction of the crystal's permanent electric polarization, a technique called poling. This approach has been applied to ferroelectric crystals such as LiTa03, KTP, and LiNb0 3 ; the latter has spawned a technology known as periodically poled lithium niobate (PPLN)]. Semiconductor crystals such as GaAs also have been used for the same purpose. 21.3 THIRD-ORDER NONLINEAR OPTICS In media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric field is reversed. The dominant nonlinearity is then of third order, P NL == 4X(3) £3 (21.3-1) (see Fig. 21.3-1) and the material is called a Kerr medium. Kerr media respond to optical fields by generating third harmonics and sums and differences of triplets of frequencies.
21.3 THIRD-ORDER NONLINEAR OPTICS 895 P NL [, Figure 21.3-1 Third-order nonlinearity. EXERCISE 21.3-1 Third-Order Nonlinear Optical Media Exhibit the Electro-Optic Effect, Kerr A monochro- matic optical field E( w) is incident on a third-order nonlinear medium in the presence of a steady electric field E(O). The optical field is much smaller than the electric field, so that IE(w) 1 2 « IE(0)1 2 . Use (21.3-1) to show that the component of P NL of frequency w is approximately given by PNL(W) 12X(3) E 2 (0)E(w). Show that this component of the polarization is equivalent to a refractive-index change n = - .5n3 E 2 (0), where .5 = - X(3). E o n 4 (21.3-2) The proportionality between the refractive-index change and the squared electric field is the Kerr (quadratic) electro-optic effect described in Sec. 20.1A, where .5 is the Kerr coefficient. A. Third-Harmonic Generation (THG) and Optical Kerr Effect Third-Harmonic Generation (THG) In accordance with (21.3-1), the response of a third-order nonlinear medium to a monochromatic optical field E(t) == Re{E(w) exp(jwt)} is a nonlinear polarization PNL(t) containing a component at frequency wand another at frequency 3w, PNL(W) == 3X(3)IE(w)12E(w) P NL (3w) == X(3) E3(w). (21.3-3a) (21.3-3b) The presence of a component of polarization at the frequency 3w indicates that third- harmonic light is generated. However, in most cases the energy conversion efficiency is low. Indeed, THG is often achieved via second-harmonic generation followed by sum-frequency generation of the fundamental and second-harmonic waves. Optical Kerr Effect The polarization component at frequency w in (21.3-3a) corresponds to an incremental change of the susceptibility X at frequency w given by Eo.6.X = p;() = 3X(3)IE(w)1 2 = 6X(3)'fJI, (21.3-4) where I == IE( w) 1 2 /21] is the optical intensity of the initial wave. Since n 2 == 1 + x' we have 2nD.n == X so this is equivalent to an incremental refractive index D.n == x/2n: 37] ( 3 ) _ n == -X I == n2 I , Eon (21.3-5)
a96 CHAPTER 21 NONLINEAR OPTICS where _ 37]0 (3) n2- 2 X . n Eo (21.3-6) Optical Kerr Coefficient Thus, the change in the refractive index is proportional to the optical intensity. The overall refractive index is therefore a linear function t of the optical intensity I, (21.3-7) Optical Kerr Effect This effect is known as the opticaJ Kerr effect because of its similarity to the electro-optic Kerr effect discussed in Sec. 20.1A, for which n is proportional to the square of the steady electric field. The optical KelT effect is a self-induced effect in which the phase velocity of the wave depends on the wave's own intensity. It is an example of nonlinear refraction. The order of magnitude of the coefficient n2 (in units of cm 2 fW) is 10- 16 to 10- 14 in glasses, 10- 14 to 10- 7 in doped glasses, 10- 10 to 10- 8 in organic materials, and 10- 10 to 10- 2 in semiconductors. It is sensitive to the operating wavelength (see Sec. 21.7) and depends on the polarization. n(I) == n + n2I. B. Self-Phase Modulation (SPM), Self-Focusing, and Spatial Solitons Self-Phase Modulation (SPM) As a result of the optical Kerr effect, an optical wave traveling in a third-order nonlinear medium undergoes self-phase modulation (SPM). The phase shift incurred by an optical beam of power P and cross-sectional area A, traveling a distance L in the medium, is c.p == -n(I)koL == 21rn(I)L/Ao == -21r(n + n2P/A)L/Ao, so that it is altered by L !lcp = - 21fn 2 AoA P, which is proportional to the optical power P. Self-phase modulation is useful in appli- cations in which light controls light. To maximize the effect, L should be large and A small. These requirements are well served by the use of optical waveguides. The optical power at which c.p == -1r is achieved is P 7r == AoA/2Ln2. A doped-glass fiber of length L == 1 m, cross-sectional area A == 10- 2 mm 2 , and n2 == 10- 10 cm 2 fW, operating at Ao == 1 Mm, for example, switches the phase by a factor of 1r at an optical power P 7r == 0.5 W. Materials with larger values of n2 can be used in centimeter-long channel waveguides to achieve a phase shift of 1r at powers of a few m W. Phase modulation may be converted into intensity modulation by employing one of the schemes used in conjunction with electro-optic modulators (see Sec. 20.1B): (1) using an interferometer (Mach-Zehnder, for example); (2) using the difference between the modulated phases of the two polarization components (birefringence) as a wave retarder placed between crossed polarizers; or (3) using an integrated-optic directional coupler (Sec. 8.5B). The result is an all-optical modulator in which a weak optical beam may be controlled by an intense optical beam. All-optical switches are discussed in Sec. 23.3C. (21.3-8) t Equation (21.3-7) is also written in the alternative form, n(I) = n+n21E12 /2, where n2 differs from (21.3- 6) by the factor 'rJ.
21.3 THIRD-ORDER NONLINEAR OPTICS 897 Self-Focusing Another interesting effect associated with self-phase modulation is self-focusing. If an intense opticaJ beam is transmitted through a thin sheet of nonlinear material exhibiting the optical Kerr effect, as illustrated in Fig. 21.3-2, the refractive-index change mimics the intensity pattern in the transverse plane. If the beam has its highest intensity at the center, for example, the maximum change of the refractive index is also at the center. The sheet then acts as a graded-index medium that imparts to the wave a nonuniform phase shift, thereby causing wavefront curvature. Under certain conditions the medium can act as a lens with a power-dependent focal length, as shown in Exercise 21.3-2. Kerr-lens focusing is useful for laser mode locking, as discussed in Sec. 15.4D. x d 4 1. n f I z Nonlinear medium Figure 21.3-2 A third-order nonlinear medium acts as a lens whose focusing power depends on the intensity of the incident beam. EXERCISE 21.3-2 Optical Kerr Lens. An optical beam traveling in the z direction is transmitted through a thin sheet of nonlinear optical material exhibiting the optical Kerr effect, n(f) = n + n2f. The sheet lies in the x-y plane and has a small thickness d so that its complex amplitude transmittance is exp( -jnkod). The beam has an approximately planar wavefront and an intensity distribution f fo[l - (x 2 + y2)/W 2 ] at points near the beam axis (x, y « W), where fo is the peak intensity and TV is the beam width. Show that the medium acts as a thin lens with a focal length that is inversely proportional to fo. Hint: A lens of focal length f has a complex amplitude transmittance proportional to exp[jk o (x 2 + y2)/2f]' as shown in (2.4-9); see also Exercise 2.4-6. Spatial Solitons When an intense optical beam travels through a substantial thickness of nonlinear homogeneous medium, instead of a thin sheet, the refractive index is altered nonuni- formly so that the medium can act as a graded-index waveguide. Thus, the beam can create its own waveguide. If the intensity of the beam has the same spatial distribution in the transverse plane as one of the modes of the waveguide that the beam itself creates, the beam propagates self-consistently without changing its spatial distribution. Under these conditions, diffraction is compensated by self-phase modulation, and the beam is confined to its self-created waveguide. Such self-guided beams are called spatial solitons. Analogous behavior occurs in the time domain when group-velocity dispersion is compensated by self-phase modulation. As discussed in Sec. 22.5B, this leads to the formation of temporal solitons, which travel without changing shape. The self-guiding of light in an 0Ftical Kerr medium is described mathematically by the Helmholtz equation, \12 E + n (I)kE == 0, where n(I) == n + n2I, ko == w / Co, and I == IEI2/27]. This is a nonlinear differential equation in E, which is simplified by writing E == A exp( - j kz), where k == nko, and assuming that the envelope
898 CHAPTER 21 NONLINEAR OPTICS A == A(x, z) varies slowly in the z direction (in comparison with the wavelength A == 27r j k) and does not vary in the y direction (see Sec. 2.2C). Using the approxi- mation (8 2 j8z 2 )[A exp( -jkz)] (-2jk8Aj8z - k 2 A) exp( -jkz), the Helmholtz equation becomes 8 2 A . 8A 2 2 2 ox 2 - 2Jk oz + ko[n (1) - n ]A = O. (21.3-9) Since the nonlinear effect is small (n21 « n), we write 2n nlAI2 n 2 n [n 2 (I) - n 2 ] == [n(I) - n] [n(I) + n] [n2 I ] [2n] == 2 == 2 1A12, 27] 7]0 (21.3-10) so that (21.3-9) becomes 0 2 A + n2 k21AI2 A = 2jk oA . 8x 2 7]0 8z (21.3-11) Equation (21.3-11) is the nonlinear Schrodinger equation. One of its solutions is A(x, z) == Ao sech ( ) exp ( -j ) . W o 4z o (21.3-12) Spatial Soliton where W o is a constant, sech(.) is the hyperbolic-secant function, Ao satisfies n2(A6j27]0) == Ij k2W cr and Zo == !kWcr == 7rWcr j A is the Rayleigh range [see (3.1- 22)]. The intensity distribution I(x,z) = IA(x,z)1 2 = A6 seCh2 ( ) 27] 27] W o (21.3-13) is independent of z and has a width W o , as illustrated in Fig. 21.3-3. The distribution in (21.3-12) is the mode of a graded-index waveguide with a refractive index n + n21 == n[l + (ljk 2 jWcr)sech2(xjWo)], so that self-consistency is assured. Since E == A exp( -jkz), the wave travels with a propagation constant k + 1j4z o == k(l + A 2 j87r2Wcr) and phase velocity cj (1 + A 2 j87r2Wcr). The velocity is smaller than c for localized beams (small W o ) but approaches c for large W 0 0 Raman Gain The nonlinear coefficient X(3) is in general complex-valued, X(3) == X) + jX}3). The self-phase modulation in (21.3-8), L 67r7]0 X(3) L c.p == 27rn2 \ A P == \ A P , Ao Eo n Ao (21.3-14) is therefore also complex. Thus, the propagation phase factor exp( - j c.p) is a combina- tion of phase shift, c.p == (67r7]ojEo)(X) jn2)(LjAoA)P, and gain exp(!,RL), with
21.3 THIRD-ORDER NONLINEAR OPTICS 899 W(z) Wo - z z t (a) (b) Figure 21.3-3 Comparison between (a) a Gaussian beam traveling in a linear medium, and (b) a spatial soliton (self-guided optical beam) traveling in a nonlinear medium. a gain coefficient given by 1271"7]0 X}3) 1 P 'YR== ' A ' Eo n /\0 (21.3-15) Raman Gain Coefficient which is proportional to the optical power P. This effect, called Raman gain, has its origin in the coupling of light to the vibrational modes of a medium, which can act as an energy source. When this gain exceeds the loss, the medium can behave as an optical amplifier (see Sec. 14.3D). With proper feedback, the Raman amplifier becomes a Raman laser (see Sec. 15.3A). The phenomenological construct of a complex nonlinear coefficient X(3) is not unlike the complex susceptibility constructed to provide loss and gain in linear media (Sec. 5.5). C. Cross-Phase Modulation (XPM) We now consider the response of a third-order nonlinear medium to an optical field comprising two monochromatic waves of angular frequencies WI and W2, E(t) == Re{ E(Wl) exp(jw1t)} + Re{ E(W2) exp(jw2t)}. On substitution in (21.3-1), the com- ponent P NL (WI) of the polarization density at frequency WI turns out to be P NL (Wl) == X(3) [3IE(Wl)12 + 6IE(W2)12] E(Wl). (21.3-16) Assuming that the two waves have the same refractive index n, this relation may be cast in the form P NL (Wl) == 2E o nt:,.nE(wl), where t:,.n == n2(1 1 + 21 2 ), (21.3-17) XPM with n2 == 37]oX(3) /E o n 2 . The quantities II == IE(Wl)1 2 /27] and 1 2 == IE(W2)1 2 /27] are the intensities of waves 1 and 2, respectively. Therefore, wave 1 travels with an effective refractive index n + t:,.n controlled by its own intensity as well as that of wave 2. Wave 2 encounters a similar effect, so that the waves are coupled. Since the phase shift encountered by wave 1 is modulated by the intensity of wave 2, this phenomenon is known as cross-phase modulation (XPM). It can result in the contamination of information between optical communication channels at neighboring frequencies, as in wavelength division multiplexing systems (WDM) (see Sec. 24.3C).
900 CHAPTER 21 NONLINEAR OPTICS As we have seen in Sec. 21.2C, two-wave mixing is not possible in a second- order nonlinear medium (except in the degenerate case). Note, however, that two-wave mixing can occur in photorefractive media, as illustrated in Fig. 20.4-3. EXERCISE 21.3-3 Optical Kerr Effect in the Presence of Three Waves. Three monochromatic waves with frequencies WI, W2, and W3 travel in a third-order nonlinear medium. Determine the complex am- plitude of the component of PNL(t) in (21.3-1) at frequency WI. Show that this wave travels with a velocity co/(n + n), where n = n2(I I + 21 2 + 21 3 ), and n2 = 31]oX(3) / Eon2, with Iq = IE(w q ) 1 2 /21], q = 1,2,3. (21.3-18) D. Four-Wave Mixing (FWM) We now examine the case of four-wave mixing (FWM) in a third-order nonlinear medium. We begin by determining the response of the medium to a superposition of three waves of angular frequencies WI, w2, and W3, with field E(t) == Re{E(wl) exp(jwIt)} + Re{E(w2) exp(jw2t)} + Re{E(w3) exp(jw3t)}. (21.3-19) It is convenient to write E ( t) as a sum of six terms E(t) == L E(wq) exp(jwqt), q =::1:: 1 , ::I:: 2 , ::I:: 3 (21.3-20) where w_ q == -w q and E( -w q ) == E* (w q ). Substituting (21.3-20) into (21.3-1), we write P NL as a sum of 6 3 == 216 terms, PNL(t) == X(3) L E(wq)E(wr)E(wz) exp[j(w q +w r +wz)t]. q, r ,l=::I:: I ,::1::2,::1::3 (21.3-21) Thus, P NL is the sum of harmonic components of frequencies WI, . . . , 3WI, . . . , 2WI :1: W2, . . . , :1:Wl :1: W2 :1: W3. The amplitude PNL(W q + W r + wz) of the component of frequency w q + W r + Wl can be determined by adding appropriate permutations of q, T, and l in (21.3-21). For example, PNL(WI + W2 - W3) involves six permutations, PNL(WI + W2 - W3) == 6X(3) E(WI)E(W2)E*(W3), (21.3-22) Equation (21.3-22) indicates that four waves of frequencies WI, W2, W3, and W4 are mixed by the medium if W4 == WI + W2 - W3, or WI+W2==W3+ W 4. (21.3-23) Frequency-Matching Condition This equation constitutes the frequency-matching condition for FWM.
21.3 THIRD-ORDER NONLINEAR OPTICS 901 Assuming that waves 1, 2, and 3 are plane waves of wavevectors k I, k 2 , and k 3 , so that E(w q ) ex exp( -jk q . r), q == 1,2,3, then (21.3-22) gives P NL (W4) ex exp( -jk l . r) exp( -jk 2 . r) exp(jk 3 . r) == exp[-j(k l + k 2 - k 3 ) . r], (21.3-24) so that wave 4 is also a plane wave with wavevector k4 == k l + k 2 - k 3 , from which k l + k 2 == k3 + k4. (21.3-25) Phase-Matching Condition Equation (21.3-25) is the phase-matching condition for FWM. Several FWM processes occur simultaneously, all satisfying the frequency and phase matching conditions. As shown before, waves 1, 2, and 3 interact and generate wave 4, in accordance with (21.3-22). Similarly, waves 3,4, and 1 interact and generate wave 2, in accordance with P NL (W2) == 6X(3) E(W3)E(W4)E*(WI), (21.3-26) and so on. The FWM process may also be interpreted as an interaction between four photons. A photon of frequency W3 and another of of frequency W4 are annihilated to create a photon of frequency WI and another of frequency W2, as illustrated in Fig. 21.3-4. Equations (21.3-23) and (21.3-25) represent conservation of energy and momentum, respectively. nw 2 nw 4 ,/ 11w 1 / '11w 3 (a) (b) Figure 21.3-4 Four-wave mixing (FWM): (a) phase-matching condition; (b) interaction of four photons. Three-Wave Mixing In the partially degenerate case for which two of the four waves have the same fre- quency, w3 == W4 wo, we have three waves with frequencies related by WI + W2 == 2wo, (21.3-27) so that the frequencies WI and W2 are symmetrically located with respect to the cen- tral frequency wo, much like the sidebands of an amplitude modulated sine wave, or the Stokes and anti-Stokes frequencies in Raman scattering. The components of the nonlinear polarization density at WI, W2, and W3 include terms of the form PNL(WI) == 3X(3) E 2 (W3)E*(W2), P NL (W2) == 3X(3) E 2 (W3)E*(WI), P NL (W3) == 6X(3) E(WI)E(W2)E*(W3). (21.3-28a) (21.3-28b) (21.3-28c)
902 CHAPTER 21 NONLINEAR OPTICS These terms are responsible for three-wave mixing, i.e., radiation at the frequency of each wave generated by mixing of the other waves. These mixing processes may be used for optical frequency conversion (OFC), optical parametric amplification (OPA) and oscillation (OPO), and spontaneous parametric downconversion (SPDC), much like three-wave mixing in second-order nonlinear media; the waves at WI, W2, and W3 may be regarded as the signal, idler, and pump of the parametric process. Note, however, that this three-wave mixing process involves four photons. For example, the annihilation of two photons at Wo and the creation of two photons at WI and W2. An example of OPA in a X(3) medium, such as a silica-glass optical fiber, is illustrated in Fig. 21.3-5. Pump Pump Signal wI Signal Signal Idler wI Wo Pump Wo Silica glass fiber wI Wo W2 Figure 21.3-5 Three-wave, four-photon optical fiber parametric amplifier (OPA). E. Optical Phase Conjugation (OPC) The frequency-matching condition (21.3-23) is satisfied when all four waves are of the same frequency: WI == W2 == W3 == W4 == W. (21.3-29) The process is then called degenerate four-wave mixing. Assuming further that two of the waves (waves 3 and 4) are uniform plane waves traveling in opposite directions, E3(r) == A3 exp( -jk 3 . r), E4(r) == A4 exp( -jk 4 . r), (21.3-30) with k4 == -k 3 , (21.3-31) and substituting (21.3-30) and (21.3-31) into (21.3-26), we see that the polarization density of wave 2 is 6X(3) A3A4Ei (r). This term corresponds to a source emitting an optical wave (wave 2) of complex amplitude E2(r) ex A3A4E(r). (21.3-32) Phase Conjugation Since A3 and A4 are constants, wave 2 is proportional to a conjugated version of wave 1. The device serves as a phase conjugator. Waves 3 and 4 are called the pump waves and waves 1 and 2 are called the probe and conjugate waves, respectively. As will be demonstrated shortly, the conjugate wave is identical to the probe wave except that it travels in the opposite direction. The phase conjugator is a special mirror that reflects the wave back onto itself without altering its wavefronts. To understand the phase conjugation process consider two simple examples:
21.3 THIRD-ORDER NONLINEAR OPTICS 903 EXAMPLE 21.3-1. Conjugate of a Plane Wave. If wave 1 is a uniform plane wave, El(r) == A1exp(-jk 1 . r), traveling in the direction k 1 , then E2(r) == Atexp(jk1 . r) is a uniform plane wave traveling in the opposite direction k 2 == -k 1 , as illustrated in Fig. 21.3- 6(b). Thus, the phase-matching condition (21.3-25) is satisfied. The medium acts as a special "mirror" that reflects the incident plane wave back onto itself, no matter what the angle of incidence. 1 ......., 2 -'-, -'" (a) (b) Figure 21.3-6 Reflection of a plane wave from (a) an ordinary mirror and (b) a phase conjugate mirror. EXAMPLE 21.3-2. Conjugate of a Spherical Wave. If wave 1 is a spherical wave centered about the origin r == 0, El(r) ex: (l/r) exp( -jkr), then wave 2 has complex amplitude E2(r) ex: (l/r) exp( +jkr). This is a spherical wave traveling backward and converging toward the origin, as illustrated in Fig. 21.3-7(b). .... .... .... .... .... .... .... _')e - -- --_. (b) Figure 21.3-7 Reflection of a spherical wave from (a) an ordinary mirror and (b) a phase conjugate mirror. Since an arbitrary probe wave may be regarded as a superposition of plane waves (see Chapter 4), each of which is reflected onto itself by the conjugator, the conjugate wave is identical to the incident wave everywhere, except for a reversed direction of propagation. The conjugate wave retraces the original wave by propagating backward, maintaining the same wavefronts. Phase conjugation is analogous to time reversal. This may be understood by examining the field of the conjugate wave G2(r, t) == Re{E 2 (r) exp(jwt)} ex Re{ Er (r) exp(jwt) }. Since the real part of a complex number equals the real part of its complex conjugate, G2(r,t) ex Re{E1(r)exp(-jwt)}. Comparing this to the field of the probe wave Gl(r, t) == Re{El(r) exp(jwt)}, we readily see that one is obtained from the other by the transformation t -t, so that the conjugate wave appears as a time-reversed version of the probe wave. The conjugate wave may carry more power than the probe wave. This can be seen by observing that the intensity of the conjugate wave (wave 2) is proportional to the product of the intensities of the pump waves 3 and 4 [see (21.3-32)]. When the powers of the pump waves are increased so that the conjugate wave (wave 2) carries more power than the probe wave (wave 1), the medium acts as an "amplifying mirror." An example of an optical setup for demonstrating phase conjugation is shown in Fig. 21.3- 8. Degenerate Four-Wave Mixing as a Form of Real- Time Holography The degenerate four-wave-mixing process is analogous to volume holography (see Sec. 4.5). Holography is a two-step process in which the interference pattern formed by the superposition of an object wave El and a reference wave E3 is recorded in a photo- graphic emulsion. Another reference wave E4 is subsequently transmitted through or reflected from the emulsion, creating the conjugate of the object wave E 2 ex E4E3Ei, or its replica E 2 ex E4EIE3' depending on the geometry [see Fig 4.5-10(a) and (b)]. The nonlinear medium permits a real-time simultaneous holographic recording and reconstruction process. This process occurs in both the Kerr medium and the
904 CHAPTER 21 NONLINEAR OPTICS .:--"t.,;.. . ,.'.-?:. ;--.....:, .., .-..'"*-. ..... '::;;,.,-::...;Q]- -::::-. \ Laser Crystal - Probe 1 - .... . . . . . ." "." .. . .... . .........,;... '. ...:'. .....,......;.... .....;-: . :.-".,'.;-;. :"'"":':-"--='.;.>',,". :':''h ,':""'"':.:"':'''-; ....'1::;.;;.. 1 Conj;gate ,:p Figure 21.3-8 An optical system for degenerate four-wave mixing using a nonlinear crystal. The pump waves 3 and 4 and the probe wave 1 are obtained from a laser using a beamsplitter and two mirrors. The conjugate wave 2 is created within the crystal. photorefractive medium (see Sec. 20.4). When four waves are mixed in a nonlinear medium, each pair of waves interferes and creates a grating, from which a third wave is reflected to produce the fourth wave. The roles of reference and object are exchanged among the four waves, so that there are two types of gratings as illustrated in Fig. 21.3-9. Consider first the process illustrated in Fig. 21.3-9(a) [see also Fig. 4.5-10(a)]. Assume that the two reference waves (denoted as waves 3 and 4) are counterpropagating plane waves. The two steps of holography are: 1. The object wave 1 is added to the reference wave 3 and the intensity of their sum is recorded in the medium in the form of a volume grating (hologram). 2. The reconstruction reference wave 4 is Bragg reflected from the grating to create the conjugate wave (wave 2). This grating is called the transmission grating. Wave 3 (reference) Wave 4 (reference) Wave 3 (reference) Wave 4 (reference) .......... .................... \ .... _,1'(}. .... 'f'I . c\,) ... \...O'() ..... 1- ,1'(}. '(}.\,e;) 'f'I . \...co{\ .- .- .- .. .. :'\ .... A1'(}. '(}.\, J .... . .... \ \...co{\. '(}.c \..O'() (a) (b) Figure 21.3-9 Four-wave mixing in a nonlinear medium. A reference and object wave interfere and create a grating from which the second reference wave reflects and produces a conjugate wave. There are two possibilities corresponding to (a) transmission and (b) reflection gratings. The second possibility, illustrated in Fig. 21.3-9(b), is for the reference wave 4 to interfere with the object wave 1 and create a grating, called the reflection grating, from which the second reference wave 3 is reflected to create the conjugate wave 2. These two gratings can exist together but they usually have different efficiencies. In summary, four-wave mixing can provide a means for real-time holography and phase conjugation, which have a number of applications in optical signal processing. Use of Phase Conjugators in Wave Restoration The ability to reflect a wave onto itself so that it retraces its path in the opposite direction suggests a number of useful applications, including the removal of wavefront
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 905 aberrations. The idea is based on the principle of reciprocity, illustrated in Fig. 21.3-10. Rays traveling through a linear optical medium from left to right follow the same path if they reverse and travel back in the opposite direction. The same principle applies to waves. . .........---------- .........--------- .........-------_. .........-------- . . Figure 21.3-10 Optical reciprocity. If the wavefront of an optical beam is distorted by an aberrating medium, the original wave can be restored by use of a conjugator that reflects the beam onto itself and transmits it once more through the same medium, as illustrated in Fig. 21.3-11. One important application is in optical resonators (see Chapter 10). If the resonator contains an aberrating medium, replacing one of the mirrors with a conjugate mirror ensures that the distortion is removed in each round trip, so that the resonator modes have undistorted wavefronts transmitted through the ordinary mirror, as illustrated in Fig. 21.3-12. . . . . . . . . . . . . . . . -'---1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . -e---1 1 Distorting medium Phase <?onjugate mIrror Mirror Distorting medium Phase conjugate mIrror Figure 21.3-11 A phase conjugate mirror reflects a distorted wave onto itself, so that when it retraces its path, the distortion is compensated. Figure 21.3-12 An optical resonator with an ordinary mirror and a phase conjugate mir- ror. *21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY A quantitative analysis of the process of three-wave mixing in a second-order nonlin- ear optical medium is provided in this section using a coupled-wave theory. Unlike the treatment provided in Sec. 21.2, all three waves are treated on equal footing. To simplify the analysis, consideration of anisotropic and dispersive effects is deferred to Sees. 21.6 and 21.6, respectively. Coupled-Wave Equations In accordance with (21.1-6) and (21.1-7), wave propagation in a second-order nonlin- ear medium is governed by the basic wave equation \72£ _ ! a 2 £ = -s C 2 8t 2 ' (21.4-1)
906 CHAPTER 21 NONLINEAR OPTICS where s _ _ 8 2 P NL - /-Lo 8t 2 (21.4-2) is regarded as a radiation source, and P NL == 2d£2 (21.4-3) is the nonlinear component of the polarization density. The field £ (t) is a superposition of three waves of angular frequencies WI, W2, and W3, with complex amplitudes E l , E 2 , and E 3 , respectively: £(t)== L Re{Eqexp(jwqt)}== L [Eqexp(jwqt)+E;exp(-jwqt)]. q=I,2,3 q=I,2,3 (21.4-4) It is convenient to rewrite (21.4-4) in the compact form £(t) == L !Eq exp(jwqt), q = :!:1,:!:2,:!:3 (21.4-5) where w_ q == -w q and E_q == E;. The corresponding polarization density obtained by substituting into (21.4-3) is a sum of 6 x 6 == 36 terms, PNL(t)==2d. L EqErexp(j(Wq+wr)tJ. q,r = :!:1,:!:2,:!:3 (21.4-6) Thus, the corresponding radiation source is S==!/-Lo d L (Wq+wr)2EqErexp(j(Wq+Wr)tJ, q,r = :!:1,:!:2,:!:3 (21.4-7) which generates a sum of harmonic components whose frequencies are sums and differences of the original frequencies WI, W2, and W3. Substituting (21.4-5) and (21.4-7) into the wave equation (21.4-1) leads to a single differential equation with many terms, each of which is a harmonic function of some frequency. If the frequencies WI, W2, and W3 are distinct, we can separate this equation into three time-independent differential equations by equating terms on both sides of (21.4-1) at each of the frequencies WI, W2, and W3, separately. The result is cast in the form of three Helmholtz equations with associated sources, C\i 72 + ki)E l == -8 1 (\7 2 + k)E2 == -8 2 (\7 2 + k)E3 == -8 3 , (21.4-8a) (21.4-8b) (21.4-8c) where 8q is the amplitude of the component of S with frequency w q and kq == nW q / Co, q == 1, 2, 3. Each of the complex amplitudes of the three waves satisfies the Helmholtz equation with a source equal to the component of S at its frequency. Under certain conditions, the source for one wave depends on the electric fields of the other two waves, so that the three waves are coupled. In the absence of nonlinearity, d == 0 so that the source term S vanishes and each of the three waves satisfies the Helmholtz equation independently of the other two, as is expected in linear optics.
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 907 If the frequencies WI, W2, and W3 are not commensurate (one frequency is not the sum or difference of the other two, and one frequency is not twice another), then the source term S does not contain any components of frequencies WI, W2, or W3. The components 8 1 , 8 2 , and 8 3 then vanish and the three waves do not interact. For the three waves to be coupled by the medium, their frequencies must be com- mensurate. Assume, for example, that one frequency is the sum of the other two, WI + W2 == W3. (21.4-9) The source S then contains components at the frequencies WI, W2, and W3. Examining the 36 terms of (21.4-7) yields 8 1 == 2/-Lowi d E3E 8 2 == 2/-Low d E3E 8 3 == 2/-LowdEIE2. (21.4-10) (21.4-11) (21.4-12) The source for wave 1 is proportional to E3E (since WI == W3 - W2), so that waves 2 and 3 together contribute to the growth of wave 1. Similarly, the source for wave 3 is proportional to E I E 2 (since W3 == WI + W2), so that waves 1 and 2 combine to amplify wave 3, and so on. The three waves are thus coupled or "mixed" by the medium in a process described by three coupled differential equations in E l , E 2 , and E 3 , (\7 2 + ki)E l == -2/-LowidE3E (\7 2 + k)E2 == -2/-LowdE3E (\7 2 + k)E3 == -2/-LowdEIE2. (21.4-13a) (21.4-13b) (21.4-13c) 3-Wave-Mixing Coupled Equations EXERCISE 21.4-1 SHG as Degenerate Three-Wave Mixing. Equations (21.4-13) are valid only when the fre- quencies WI, W2, and W3 are distinct. Consider now the degenerate case for which WI == W2 == wand W3 == 2w, so that there are two instead of three waves, with amplitudes El and E3. This corresponds to second-harmonic generation (SHG). Show that these waves satisfy the Helmholtz equation with sources 8 1 == 2P o wid E3 E ; 8 3 == Pow dEl E 1 , (21.4-14) (21.4-15) so that the coupled wave equations are (\7 2 + ki)E 1 == -2p o wid E 3 E;, (\7 2 + k)E3 == -Powd EIEI. (21.4-16a) (21.4-16b) SHG Coupled Equations Note that these equations are not obtained from the three-wave-mixing equations (21.4-13) by sub- stituting El == E 2 [the factor of 2 is absent in (2] .4-16b)].
908 CHAPTER 21 NONLINEAR OPTICS Mixing of Three Collinear Uniform Plane Waves Assume that the three waves are plane waves traveling in the z direction with complex amplitudes Eq == Aq exp( -jkqz), complex envelopes Aq, and wavenumbers kq == w q / c, q == 1,2,3. It is convenient to normalize the complex envelopes by defining the variables a q == Aq/(2TJliw q )I/2, where TJ == TJo/n is the impedance of the medium, TJo == (Mo / Eo) 1/2 is the impedance of free space, and liw q is the energy of a photon of angular frequency w q . Thus, Eq == v 2TJliwqaqexp(-jkqz), q==1,2,3, (21.4-17) and the intensities of the three waves are Iq == IEqI2/2TJ == liw q la q l 2 . The photon flux densities (photons/s-m 2 ) associated with these waves are Iq 2 cjJ q == Iiw == I a q I . q (21.4-18) The variable a q therefore represents the complex envelope of wave q, scaled such that I a q 1 2 is the photon flux density. This scaling is convenient since the process of wave mixing must be governed by photon-number conservation (see Sec. 21.2C). As a result of the interaction between the three waves, the complex envelopes a q vary with z so that a q == a q (z ). If the interaction is weak, the a q (z) vary slowly with z, so that they can be assumed approximately constant within a distance of a wavelength. This makes it possible to use the slowly varying envelope approximation wherein d 2 a q /dz 2 is neglected relative to kqdaq/dz == (27r / Aq)daq/dz and 2 da ("\7 + k)[aq exp( -jkqz)] -j2k q dz q exp( -jkqz) (21.4-19) (see Sec. 2.2C). With this approximation (21.4-13) reduce to simpler equations that are akin to the paraxial Helmholtz equations, in which the mismatch in phase is considered: da 1 . * ( . A k ) dz == -Jga3 a 2 ex p -Jti z da2 . * ( . A k ) dz == -Jga3 a l ex p -Jti z da3 dz == -jgal a 2 exp(jk z) (21.4-20a) (21.4-20b) (21.4-20c) 3-Wave-Mixing Coupled Equations where g2 == 2liw 1 W 2 W 3TJ 3 d2 (21.4-21) and k == k3 - k 2 - k 1 (21.4-22) represents the error in the phase-matching condition. The variations of aI, a2, and a3 with z are therefore governed by three coupled first-order differentia] equations (21.4- 20), which we proceed to solve under the different boundary conditions corresponding to various applications. It is useful, however, first to derive some invariants of the
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 909 wave-mixing process. These are functions of aI, a2, and a3 that are independent of z. Invariants are useful since they can be used to reduce the number of independent variables. Exercises 21.4-3 and 21.4-2 develop invariants based on conservation of energy and conservation of photons. EXERCISE 21.4-2 Photon-Number Conservation: The Manley-Rowe Relation. Using (21.4-20), show that d 2 d 2 d 2 dz lall = dz la21 = - dz la31 , (21.4-23) from which the Manley-Rowe relation (21.2-19), which was derived using photon-number conser- vation, follows. Equation (21.4-23) implies that lal1 2 + la312 and la2/ 2 + la3/ 2 are also invariants of the wave-mixing process. EXERCISE 21.4-3 Energy Conservation. Show that the sum of the intensities Iq = 12w q la q I 2 , q = 1,2,3, of the three waves governed by (21.4-20) is invariant to z, so that d dz (I I + 1 2 + 1 3 ) = O. (21.4-24) A. Second-Harmonic Generation (SHG) Second-harmonic generation (SHG) is a degenerate case of three-wave mixing in which W] == W2 == W and W3 == 2w. (21.4- 25) Two forms of interaction occur: two photons of frequency w combine to form a photon of frequency 2w (second harmonic), or one photon of frequency 2w splits into two photons, each of frequency w (degenerate parametric downconversion). The interaction of the two waves is described by the paraxial Helmholtz equations with sources. Conservation of momentum requires that 2k I == k3. (21.4-26) EXERCISE 21.4-4 Coupled-Wave Equations for SHG. Apply the slowly varying envelope approximation (21.4- 19) to the Helmholtz equations (21.4-16), which describe two collinear waves in the degenerate case, to show that da I . * ( . " k ) dz = -Jga3 a l exp -JU z da3.g . - d = -J -a I al exp(Jkz), z 2 (21.4-27a) (21.4-27b) where k = k3 - 2k l and g2 = 412w 3 7]3 d2. (21.4-28)
910 CHAPTER 21 NONLINEAR OPTICS Assuming two collinear waves with perfect phase matching (flk == 0), equations (21.4- 27) reduce to dal . * dz == -Jga3 a l da3 . 9 - == -J-alal. dz 2 (21.4-29a) (21.4-29b) SHG Coupled Equations At the input to the device (z == 0) the amplitude of the second-harmonic wave is assumed to be zero, a3(0) == 0, and that of the fundamental wave, al(O), is assumed to be real. We seek a solution for which al (z) is real everywhere. Using the energy con- servation relation ai (z) + 21 a3 (z ) 1 2 == ai (0), (21.4- 29b) gives a differential equation in a3 ( z ) , da3/dz == -j(g/2)[ai(0) - 2I a 3(Z)1 2 ], (21.4-30) whose solution may be substituted in (21.4-29a) to obtain the overall solution: a1(Z) = a1(0) sech( ga1(0)z) a3(z) = - a1(0) tanh( ga1(0)z) . Consequently, the photon flux densities CPl(Z) == lal(z)1 2 and CP3(Z) == la3(z)1 2 evolve in accordance with (21.4- 31 a) (21.4- 31 b) 2ryz CPl ( z) == CPl ( 0) sech - 2 1 2 ryz cp3(Z) == "2cpl(O) tanh 2' (21.4-32a) (21.4-32b) where ry /2 == gal (0) / yI2, i.e., ry2 == 2g2ai(0) == 2g2cpl (0) == 8 d 2 'T]3 hw 3 cpl (0) == 8d 2 'T]3 W 2 II (0). (21.4-33) Since sech2(.) + tanh2(.) == 1, cpl(Z) + 2cp3(Z) == cpl(O) is constant, indicating that at each position z, photons of wave 1 are converted to half as many photons of wave 3. The fall of cpl(Z) and the rise of cp3(Z) with z are shown in Fig. 21.4-1(b). Efficiency of SHG The efficiency of second-harmonic generation for an interaction region of length L is _ I3(L) _ hw 3 cp3(L) _ 2cp3(L) _ h 2 ryL IlsHG- h(O) - 1iw1lfh(0) - 4>1(0) -tan 2' (21.4-34) For large ryL (long cell, large input intensity, or large nonlinear parameter), the ef- ficiency approaches one. This signifies that all the input power (at frequency w) has been transformed into power at frequency 2w; all input photons of frequency w are converted into half as many photons of frequency 2w.
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 911 Fundamental <PI (0) <PI (Z) + 2<P3(Z) -t<PI (0) hw , h2w / hw Second harmonic 00 2 4 "'!z (a) (b) (c) Figure 21.4-1 Second-harmonic generation. (a) A wave of frequency w incident on a nonlinear crystal generates a wave of frequency 2w. (b) As the photon flux density <PI (z) of the fundamental wave decreases, the photon flux density <P3 (z) of the second-harmonic wave increases. Since photon numbers are conserved, the sum <PI (z) + 2<p3 (z) = <PI (0) is a constant. ( c) Two photons of frequency w combine to make one photon of frequency 2w. For small ry£ [small device length £, small nonlinear parameter d, or small input photon flux density CPl (0)], the argument of the tanh function is small and therefore the approximation tanh x x may be used. The efficiency of second-harmonic generation is then n == 1 3 (£) f'.J 11\j2 £2 == 1 9 2 £2 A. ( 0 ) == 2d 2 '113hw 3 £2 A. ( 0 ) == 2d 2 '113W 2 £2 I ( 0 ) £l,SHG 1 1 (0) f'.J 4 I 2 \fIl '/ \fIl '/ 1, (21.4- 35) so that L 2 IlSHG = C 2 A P, d 2 C 2 == 2w 2 '11 3 - '/0 3' n (21.4-36) SHG Efficiency where P == II (O)A is the incident optical power at the fundamental frequency and A is the cross-sectional area. This reproduces (21.2-6) and shows that the constant C 2 is proportional to the material parameter d 2 / n 3 , which is a figure of merit used for comparing different nonlinear materials. EXAMPLE 21.4-1. Efficiency of SHG. For a material with d 2 jn 3 = 10- 46 CjV 2 (see Ta- ble 21.6-3 for typical values of d) and a fundamental wave of wavelength 1 pm, C 2 = 38 X 10- 9 W- I = 0.038 (MW)-I. In this case, the SHG efficiency is 10% if P L 2 j A = 2.63 MW. If the aspect ratio of the interaction volume is 1000, i.e., L 2 j A = 10 6 , the required power is 2.63 W. This may be realized using L = 1 cm and A = 100 pm 2 , corresponding to a power density P j A = 2.63 X 10 6 W/cm 2 . The SHG efficiency may be improved by using higher power density, longer interaction length, or material with greater d 2 jn 3 coefficient. Phase Mismatch in SHG To study the effect of phase (or momentum) mismatch, the general equations (21.4-27) are used with tJ.k i=- O. For simplicity, we limit ourselves to the weak-coupling case for which ry L « 1. In this case, the amplitude of the fundamental wave a 1 (z) varies
912 CHAPTER 21 NONLINEAR OPTICS only slightly with z [see Fig. 21.4-1 (a)], and may be assumed approximately constant. Substituting a I (z) a I (0) in (21.4- 27b), and integrating, we obtain a3(L) = -j ahO) 1£ exp(j f).kz) dz = - C k ) ai(O)[exp(jf).kL) - 1], (21.4-37) from which CP3(L) la3(L)1 2 (gl k)2cpi(0) sin2(kLI2), where al(O) is as- sumed to be real. The efficiency of second-harmonic generation is therefore 1 3 (L) 2cP3(L) 2 L2 . 2 IJ.SHG = h(O) = 4>1(0) = C A Psmc (f).kL/27f), (21.4-38) where sinc(x) sin(nx)/(nx). The effect of phase mismatch is therefore to reduce the efficiency of second- harmonic generation by the factor sinc2(kLI2n). This confirms the previous results displayed in Fig. 21.2-14. For a given mismatch k, the process of SHG is efficient for lengths smaller than the coherence length Lc 2n 1 I k I. B. Optical Frequency Conversion (OFC) A frequency up-converter (Fig. 21.4-2) converts a wave of frequency WI into a wave of higher frequency W3 by use of an auxiliary wave at frequency W2, called the pump. A photon nw2 from the pump is added to a photon nwl from the signal to form a photon nw3 of the up-converted signal at an up-converted frequency W3 WI + W2. The conversion process is governed by the three coupled equations (21.4-20). For simplicity, assume that the three waves are phase matched (k 0) and that the pump is sufficiently strong so that its amplitude does not change appreciably within the interaction distance of interest; i.e., a2(z) a2(0) for all z between 0 and L. The three equations (21.4-20) then reduce to two, dal .'Y - -J- a 3 dz 2 da3 . 'Y dz -J 2 al' (21.4-39a) (21.4-39b) where 'Y 2ga2(0) and a2(0) is assumed real. These are simple differential equations with harmonic solutions 'Y z al (z) al (0) cos- 2 a3(z) = -jal(O) sin . (21.4-40a) (21.4-40b) The corresponding photon flux densities are 2 'Y z CPI (z) CPI (0) cos - 2 2 'Y z CP3(Z) CPI (0) sin -. 2 (21.4-41 a) (21.4-41 b)
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 913 The dependencies of the photon flux densities CPl and CP3 on z are sketched in Fig. 21.4-2(b). Photons are exchanged periodically between the two waves. In the region between z == 0 and z == 7r / 'Y, the input WI photons combine with the pump W2 photons and generate the up-converted W3 photons. Wave 1 is therefore attenuated, whereas wave 3 is amplified. In the region z == 7r / 'Y to z == 27r / 'Y, the W3 photons are more abundant; they disintegrate into WI and W2 photons, so that wave 3 is attenuated and wave 1 amplified. The process is repeated periodically as the waves travel through the medium. Signal Pump Wz <PI (O)! Signal <PI (Z) /J '}w 1 -v nw3 nwz nw3 c#'\)¥ Upconverted signal <P3(z) ,., / \ / \ I I o o 7r 37r ",!Z W Figure 21.4-2 The frequency up-converter; (a) wave mixing; (b) evolution of the photon flux densities of the input wI-wave and the up-converted W3-wave. The pump W2-wave is assumed constant; (c) photon interactions. The efficiency of up-conversion for a device of length L is 13 ( L ) W3. 2 'Y L IloFC == II (0) == WI SIn 2. For 'YL « 1, and using (21.4-21), this is approximated by 13 (L)/1 1 (0) (W3/Wl) ('YL/2)2 == (W3/ W l)g2 L 2 cp2(0) == 2wL2d27]3 1 2 (0) from which (21.4-42) L 2 l\.OFC = C 2 /i: P 2, 2 2 3 d 2 C == 2w 3 7]0 3 ' n (21.4-43) OFC Efficiency where A is the cross-sectional area and P2 == 1 2 (0)A is the pump power. This expres- sion is similar to (21.4-36) for the efficiency of second-harmonic generation. EXERCISE 21.4-5 Infrared Up-Conversion. An up-converter uses a proustite crystal (d == 1.5 x 10- 22 CjV 2 , n == 2.6, d 2 jn 3 == 1.3 x 10- 45 C 2 jV 4 ). The input wave is obtained from a CO 2 laser of wavelength 10.6 /-Lm, and the pump from a I-W Nd 3 + : YAG laser of wavelength 1.06 /-LID focused to a cross- sectional area 10- 2 mm 2 (see Fig. 21.2-6). Determine the wavelength of the up-converted wave and the efficiency of up-conversion if the waves are collinear and the interaction length is I cm.
914 CHAPTER 21 NONLINEAR OPTICS c. Optical Parametric Amplification (OPA) and Oscillation (OPO) Optical Parametric Amplifier (OPA) The OPA uses three-wave mixing in a nonlinear crystal to provide optical gain [Fig. 21.4-3(a)]. The process is governed by the same three coupled equations (21.4- 20) with the waves identified as follows. Wave 1 is the signal to be amplified; it is incident on the crystal with a small intensity II (0). Wave 3, the pump, is an intense wave that provides power to the amplifier. Wave 2, called the idler, is an auxiliary wave created by the interaction process. Pump W3 CPl (0) / / Idler / CP2(z) / / / ,;' ,;' ." nwl nw3 / nw2 ." o o 1 2 'Yz Figure 21.4-3 The optical parametric amplifier: (a) wave mixing; (b) photon flux densities of the signal and the idler (the pump photon-flux density is assumed constant); (c) photon mixing. Assuming perfect phase matching (k == 0), and an undepleted pump, a3(z) a3(0), the coupled-wave equations (21.4-20) provide da 1 . '"'j * - == -J-a dz 2 2 da2 . '"'j * dz == - J 2 aI' (21.4-44a) (21.4-44b) where '"'j == 2 9 a3 (0). If a3 (0) is real, '"'j is also real, and the differential equations have the solution '"'jz. . '"'jz al (z) == al (0) cosh - - ]a;(O) sInh- 2 2 . . 1z '"'jz a2(z) == -]a(O) sInh - + a2(0) cosh-. 2 2 (21.4-45a) (21.4-45b) If a2(0) == 0, i.e., the initial idler field is zero, then the corresponding photon flux densities are 2 '"'jz CPl ( z) == CPl (0) cosh "2 . 2 1 z CP2 (z) == CPl (0) sInh -. 2 (21.4-46a) (21.4-46b)
21.4 SECOND-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 915 Both CPI(Z) and CP2(Z) grow monotonically with z, as illustrated in Fig. 21.4-3(b). This growth saturates when sufficient energy is drawn from the pump so that the assumption of an undepleted pump no longer holds. The overall gain of an amplifier of length L is G == CPI (L ) / CPI (0) == cosh 2 ( 'Y L / 2). In the limit 'YL » 1, G == (e IL / 2 + e- IL / 2 )2 / 4 elL /4, so tha t the gain in creases exponentially with 'YL. The gain coefficient 'Y == 2ga3(O) == 2d V 2nwIw2w3173 Q3(O), from which 2 3 d 2 C == 2WIW2 170 3' n (21.4-47) OPA Gain Coefficient 'Y == 2C V I 3 (O) == 2C V P 3/ A, where P 3 == 1 3 (O)A is the pump power and A is the cross-sectional area, and C 2 is a parameter similar to that describing SHG and OFC. The interaction is tantamount to a pump photon nw3 splitting into a photon nwl that amplifies the signal, and a photon nw2 that creates the idler [Fig. 21.4-3(c)]. EXERCISE 21.4-6 Gain of an OPA. An OPA amplifies light at 2.5 pm by using a 2-cm long KTP crystal pumped by aNd: YAG laser of wavelength 1.064 pm. Determine the wavelength of the idler wave and the C coefficient in (21.4-47). Determine appropriate laser power and beam cross-sectional area such that the total amplifier gain is 3 dB. Assume that n = 1. 75 and d = 2.3 X 10- 23 C/V 2 for KTP. Optical Parametric Oscillator (OPO) A parametric oscillator is constructed by providing feedback at either or both the signal and the idler frequencies of a parametric amplifier, as illustrated in Fig. 21.4-4. In the fonner case, the oscillator is called a singly resonant oscillator (SRO); in the latter, it is called a doubly resonant oscillator (DRO). u Signal wI ) Idler Wz lu 4 nm wI Idler W z J Pump w3 Pump w3 (a) SRO (b) DRO Figure 21.4-4 The parametric oscillator generates light at frequencies WI and W2. A pump of frequency W3 = WI + W2 serves as the source of energy. (a) Singly resonant oscillator (SRO). (b) Doubly resonant oscillator (DRO). The oscillation frequencies WI and W2 of the parametric oscillator are determined by the frequency- and phase-matching conditions, WI + W2 == W3 and nlWl + n2W2 == n3w3, in the collinear case. The solution of these two equations yields WI and W2, as described in Sec. 21.2D. In addition, these frequencies must also coincide with the resonance frequencies of the resonator modes, much the same as for conventional lasers (see Sec. 15.1 B). The system therefore tends to be over-constrained, particularly in the DRO case for which both the signal and idler frequencies must coincide with resonator modes.
916 CHAPTER 21 NONLINEAR OPTICS Another condition for oscillation is that the gain of the amplifier must exceed the loss introduced by the mirrors for one round trip of propagation within the resonator. By equating the gain and the loss, expressions for the threshold amplifier gain and the corresponding threshold pump intensity may be determined, as shown below for the SRO and DRO configurations. SRO. At the threshold of oscillation, the signal's amplified and doubly reflected amplitude a1 (L) ri equals the initial amplitude a1 (0), where L is the length of the nonlinear medium and r1 is the magnitude of the amplitude reflectance of a mirror (the two mirrors are assumed identical and the phase associated with a round trip is not included since it is a multiple of 27r ). Using (21.4-45a), together with the boundary condition a2(0) == 0, we obtain ri cosh('"'(L/2) == 1, from which 9(i cosh 2 ( '"'(L/2) == 1. (21.4-48) Here, 9(1 == ri is the mirror intensity reflectance at the signal frequency. Since 9(1 is typically slightly smaller than unity, cosh 2 ( '"'(L/2) is slightly greater than unity, i.e., '"'(L/2 « 1 and the approximation cosh 2 (x) 1 + x 2 may be used. It follows that at threshold ('"'(L/2)2 (1- 9(i)/9(i. Using (21.4-47), we obtain the threshold intensity, from which the threshold power of the pump is obtained, 1 A 1 - 9(i P31threshold (0) C2 L2 9(2 ' 1 (21.4-49) SRO Threshold Pump Power where C 2 == 2W1W2 'l}d2 /n 3 and A is the cross-sectional area. For example, if L 2 / A == 10 6 , C 2 == 10- 7 W- 1 , and 9(1 == 0.9, then P 3 1threshold (0) 2.3 W. DRO. At threshold, two conditions must be satisfied: a1 (L) ri == a1 (0) and a2 (L ) r == a2 (0), where r1 and r2 are the magnitudes of the amplitude reflectances of the mirrors at the signal and idler frequencies, respectively. Substituting for a 1 (L) from (21.4-45a), and substituting for a2 (L) from (21.4-45b), and forming the conjugate, we obtain '"'(z '"'(Z (1 - 9(1) cosh -a1(0) + j9(l sinh -a;(O) == 0 2 2 - j9(2 sinh '"'(Z a1 (0) + (1 - 9(2) cosh '"'(Z a (0) == 0, 2 2 (21.4-50a) (21.4-50b) where 9(1 == ri and 9(2 == r are the intensity reflectance of the mirrors at the signal and idler frequencies, respectively. Equating the values of the ratio a 1 (0) / a 2 (0) obtained from (21.4-50a) and (21.4-50b), we obtain tanh 2 ('"'(L/2) == (1 - 9(1)(1 - 9(2)/(9(19(2). (21.4-51) Since the right-hand side of (21.4-51) is much smaller than unity, we can use the approximation tanh x x and write ('"'(L/2)2 (1 - 9(1)(1 - 9(2)/(9(19(2), from which we obtain the threshold pump power: 1 A (1 - 9(1) (1 - 9(2) P3Ithreshold(O) C 2 P 9(19(2 . (21.4-52) ORO Threshold Pump Power
21.5 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 917 The ratio of the threshold pump power for the DRO configuration, to that for the SRO configuration, as calculated from (21.4-49) and (21.4-52), is (9(1/9(2)(1 - 9(2)/(1 + 9(1). Since 9(1 1 and 9(2 1, this is approximately equal to (1 - 9(2)/2, which is a small number. Thus, the threshold power for the DRO is substantially smaller than that for the SRO. Unfortunately, DROs are more sensitive to fluctuations of the resonator length because of the requirement that the oscillation frequencies of both the signal and the idler match resonator modes. DROs therefore often have poor stability and spiky spectra. *21.5 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY A. Four-Wave Mixing (FWM) We now derive the coupled differential equations that describe FWM in a third-order nonlinear medium, using an approach similar to that employed in the three-wave mix- ing case in Sec. 21.4. Coupled-Wave Equations Four waves constituting a tota] field £(t) == L Re[E q exp(jwqt)] == L !Eq exp(jwqt) q=I,2,3,4 q=:!:I,:!:2,:!:3,:!:4 (21.5-1) travel in a medium characterized by a nonlinear density P NL == 4X(3) £3. (21.5-2) The corresponding source of radiation, S == - J-L o 8 2 P NL / 8t 2 , is therefore a sum of 8 3 == 512 terms, s == ! J-LoX(3) L (w q + w p + w r )2 EqEpEr exp[j (W q + W p + W r )t]. (21.5-3) q ,p, r=:!: 1 ,:!:2,:!:3,:!:4 Substituting (21.5-1) and (21.5-3) into the wave equation (21.4-1) and equating terms at each of the four frequencies WI, W2, W3, and W4, we obtain four Helmholtz equations with associated sources, (\7 2 + k)Eq == -Sq, q == 1,2,3,4, (21.5-4) where Sq is the amplitude of the component of S at frequency w q . For the four waves to be coupled, their frequencies must be commensurate. Con- sider, for example, the case for which the sum of two frequencies equals the sum of the other two frequencies, WI + W2 == W3 + W4, (21.5-5) and assume that these frequencies are distinct. Three waves can then combine and create a source at the fourth frequency. Using (21.5-5), terms in (21.5-3) at each of the
918 CHAPTER 21 NONLINEAR OPTICS four frequencies are 8 1 == /-LoWiX(3){6E3E4E + 3El[IE112 + 21E212 + 21E312 + 2IE412]} 8 2 == /-LoWX(3){6E3E4E + 3E2[1E212 + 21E112 + 2/E312 + 2IE412]} 8 3 == /-LoWX(3){6EIE2EJ + 3E3[IE312 + 21E212 + 21E112 + 2IE412]} 8 4 == /-LoWX(3){6EIE2E; + 3E4[IE412 + 21E112 + 21E212 + 2IE312]}. (21.5-6a) (21.5-6b) (21.5-6c) (21.5-6d) Each wave is therefore driven by a source with two components. The first component is a result of mixing of the other three waves. The first term in 8 1 , for example, is proportional to E3E4E and therefore represents the mixing of waves 2, 3, and 4 to create a source for wave 1. The second component is proportional to the complex amplitude of the wave itself. The second term of 8 1 , for example, is proportional to E l , so that it plays the role of refractive-index modulation, and therefore represents the optical Kerr effect (see Exercise 21.3-3). It is therefore convenient to separate the two contributions to these sources by defining - 2 8q == 8q + (Wq/C o ) XqEq, q== 1,2,3,4 (21.5-7) where - 2 (3) * 8 1 == 6/-LoWl X E3 E 4 E 2 - 2 (3) 8 2 == 6/-LoW2X E3E4E 8 6 2 (3) E E E * 3 == /-LoW 3 X 1 2 4 - 2 (3) 8 4 == 6/-Low4X E I E 2 E;, (21.5-8a) (21.5-8b) (21.5-8c) (21.5-8d) and Xq == 6!lX(3)(2I - Iq), Eo q == 1, 2, 3, 4. (21.5-9) Here Iq == IEqI2/2TJ are the intensities of the waves, I == II + 1 2 + 13 + 14 is the total intensity, which is constant in view of conservation of energy, and TJ is the impedance of the medium. This enables us to rewrite the Helmholtz equations (21.5-4) as 2 -2 - (\7 + kq )Eq == -8q, q==1,2,3,4, (21.5-10) where _ w q kq == nq - , Co (21.5-11 ) n == n 2 + 2nn2(2I - Iq) , (21.5-12) and _ 3TJo (3) n2 - 2 X ' Eon (21.5-13)
21.5 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 919 which matches (21.3-6). If the second term of (21.5-12) is much smaller than the first, then nq n + n2(21 - Iq) . (21.5-14) Optical Kerr Effect The Helmholtz equation for each wave is therefore modified in two ways: 1. A source representing the combined effects of the other three waves is present. This may lead to the amplification of an existing wave, or the generation of a new wave at that frequency. 2. The refractive index for each wave is altered, becoming a function of the inten- sities of the four waves. These equations are used to generate four coupled nonlinear differential equations that may be solved for the fields, or their complex envelopes, under the appropriate bound- ary conditions. This was the approach followed for second-order nonlinear processes, and will now be applied to several special cases in third-order nonlinear processes. B. Three-Wave Mixing and Third-Harmonic Generation (THG) We now consider degenerate cases for which two or three of the four waves have the same frequency. Three-Wave Mixing In the degenerate case for which two of the four waves have the same frequency W3 == W4 Wo, we have three waves with frequencies related by WI + W2 == 2wo. A coupled- wave theory of this three-wave mixing process can be formulated by identifying the radiation sources generated at the three frequencies: SI == J-L o w?X(3) {3E5E + 3E l [IE112 + 21E212 + 21 E o12J} S2 == J-LoWX(3) {3E6E; + 3E 2 [IE212 + 21E112 + 21 E o12J} So == J-L o W6X(3) {6EIE2E + 3Eo [I E oI2 + 21E112 + 21E212J} . (21.5-15a) (21.5-15b) (21.5-15c) When substituted in the Helmholtz equations (\7 2 + k;)E q == -Sq, q == 0,1, 2, the result is a set of coupled equations that can, in principle, be solved under appropriate initial conditions. For collinear waves traveling in the z direction, Eq(r) == Aq exp( -jkqz). As was done for second-order nonlinear processes, we use the slowly varying envelope ap- proximation, (\7 2 + k)[Aq exp( - jkqz)] -j2k q (dA q jdz) exp( -jkqz), and write the complex amplitudes Aq == J 2TJnw q a q , in terms of the variables a q , which are normalized such that CPq == la q l 2 are photon flux densities. The analysis is simplified by assuming that WI W2 Wo when calculating the coupling coefficients. The result is the following set of coupled equations: :l = _jg [a6 a ;exp(-j.6.kz) + al (l a ll 2 + 21a212 + 2I a oI 2 )] (21.5-16a) dd: 2 = -jg [a exp(-j.6.k z) + a2 (la212 + 21all2 + 2IaoI2)] (21.5-16b) = -jg [2ala2a exp(j.6.k z) + ao (10{)1 2 + 21all2 + 2Ia212)] , (21.5-16c)
920 CHAPTER 21 NONLINEAR OPTICS where 9 == nwo(wo/ C o )n2, (21.5-17) and k == 2ko - k 1 - k 2 (21.5-18) represents the phase-matching error. This set of nonlinear equations can be readily solved in the undepleted pump ap- proximation (I all, I a 21 « I ao I) since in this case ao ( z) is approximately constant. In the phase matched case (k == 0), (21.5-16) are approximated by two linear differen- tial equations dal ( * ) dz == -j'Y a 2 + 2al da2 . ( * ) dz == -J'Y a 1 + 2a2 , (21.5-19a) (21.5-19b) where 'Y == 9 a6 is a constant proportional to the constant pump intensity. The solution to these equations is written in terms of the initial values of the two waves: al (z) == [(1 - j'Y z ) al (0) - j'Yza2(0)] exp( - j'Yz) a2 ( z) == [- j 'Y z a ( 0) + (1 - j 'Y z ) a2 ( 0 )] exp ( - j 'Y z ) . (21.5-20a) (21.5-20b) If the initial idler amplitude is a2 (0) == 0, then the photon-flux density CPl (z ) == lal (z) 1 2 of the signal grows as CPl (z) == (1 + 'Y 2 Z2)CPl (0). The rate of growth is sensitive to the magnitude and phase of the initial idler wave. For example, if a2 (0) == re jcp al (0), then CPl (z) == [1 + (2r sin <P )'Yz + (1 + r 2 + 2r cos <P )'Y 2 z2J CPl (0), (21.5-21) which is a function of the phase difference <p that reaches its maximum value when tan<p == 2/'Yz. At small z, maximum growth occurs when <p == 7r/2. Clearly, the amplifier is a phase-sensitive amplifier. To examine the effect of pump depletion and phase mismatch, the full set of equa- tions (21.5-16) must be solved. One step in this direction is taken by writing the complex amplitudes a q == b q exp(j<pq) in terms of their magnitudes b q and phases <pq. Substituting into (21.5-16) and equating the real and imaginary parts of each equation leads to the following set of nonlinear equations in real variables: db 1 2 dz == 9 b o b 2 sin <p (21.5-22a) db 2 2 dz == gbob 1 sin <p (21.5-22b) db o . dz == -gbob 1 b 2 SIn <p (21.5-22c) = k + 9 [ 2b 6 - bi - b] + 9 [b6 b d b 2 + b6 b 2/ b l -4b 1 b 2 ] cas <p, (21.5-22d) where <p == k z + <PI + <P2 - 2<po. Two invariants can be easily identified. Consistent with conservation of optical intensity, the sum bi + b + b6 must be constant. Also,
21.5 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 921 consistent with conservation of photons, the difference bi - b must be constant (this is a version of the Manley-Rowe relation). Other invariants involving the phase cp may also be identified t and used to study the role of phase mismatch and initial amplitudes and phase difference between the signal and idler. For example, it can be readily seen from (21.5-22a) that the initial rate of growth of the signal occurs when sin cp == 0, i.e., cp == 7r /2. Third-Harmonic Generation (THG) Another degenerate special case of four-wave mixing is third-harmonic generation. Here, three of the four waves have identical frequencies, WI == W2 == W4 == w, and the fourth has the sum frequency W3 == WI + W2 + W4 == 3w . In effect, we have two waves, 1 and 3, whose amplitudes are coupled by the third-order nonlinear medium. A coupled-wave theory can be formulated using the approach followed in the four- and three-wave mixing cases. This leads to two Helmholtz equations (\7 2 + k)Eq == -8q, where 8 1 == J-l o wiX(3) {3E3EE + 3E l [IE112 + 21E312J} 8 3 == J-loWX(3) {Er + 3E 3 [IE312 + 21E112J} . (21.5-23a) (21.5-23b) These equations may be used to derive coupled equations for E l and E 3 , as was done . . In prevIous cases. EXERCISE 21.5-1 THG in the Undepleted-Pump Approximation. Assume that the fundamental and third- harmonic waves are plane waves traveling in the z direction with complex envelopes Aq, q == 1, 3. Use the slowly varying envelope approximation to write coupled differential equations for Al and A3. Show that in the undepleted pump approximation [A 3 « A 1 and Al (z) Al (0)], dd: 3 = -jgaf exp( -jllkz) , (21.5-24) where Aq == J 2T}hw q u q and tlk == 3k I - k3. Derive an expression for g. C. Optical Phase Conjugation (OPC) We now develop and solve the coupled-wave equations in the fully degenerate case for which all four waves have the same frequency WI == W2 == W3 == W4 == w. As was assumed in Sec. 21.3E, two of the waves (waves 3 and 4), called the pump waves, are plane waves propagating in opposite directions, with complex amplitudes E3 (r) == A3 exp( - j k 3. r ) and E4(r) == A4 exp( - j k 4. r ) and wavevectors related by k4 == - k3. Their intensities are assumed to be much greater than those of waves 1 and 2, so that they are approximately undepleted by the interaction process, allowing us to assume that their complex envelopes A3 and A4 are constant. The total intensity of the four waves I is then also approximately constant, I [IA312 + IA412]/2TJ. The terms 21 - II and 21 - 1 2 , which govern the effective refractive index n for waves 1 and 2 in (21.5- 14), are approximately equal to 21, and are therefore also constant, so that the optical Kerr effect amounts to a constant change of the refractive index. Its effect will therefore be ignored. t See G. Cappellini and S. Trillo, Third-Order Three-Wave Mixing in Single-Mode Fibers: Exact Solutions and Spatial Instability Effects, Journal of the Optical Society of America B, vol. 8, pp. 824-838, 1991.
922 CHAPTER 21 NONLINEAR OPTICS With these assumptions the problem is reduced to a problem of two coupled waves, ] and 2. Equations (21.5-10) and (21.5-8) give (\7 2 + k 2 )El == -E (\7 2 + k 2 )E2 == -E, (21.5-25a) (21.5-25b) where == 6/-LoW2X(3) E3E4 == 6/-Lo W2 X(3) A3 A 4 (21.5-26) and k == nw / Co, where n n + 2n21 is a constant. The four nonlinear coupled differential equations have thus been reduced to two linear coupled equations, each of which takes the form of the Helmholtz equation with a source term. The source for wave 1 is proportional to the conjugate of the complex amplitude of wave 2, and similarly for wave 2. Phase Conjugation Assume that waves 1 and 2 are also plane waves propagating in opposite directions along the z axis, as illustrated in Fig. 21.5-1, E 1 == Al exp( -jkz), E 2 == A 2 exp(jkz). (21.5-27) This assumption is consistent with the phase-matching condition since k 1 + k 2 k3 + k4. 3 v .;;....--, J ( 1 . Probe .2 Conjugate r 4 <$' , I I I A: ___ ___ _ _ _ ..... r I - At --- - Figure 21.5-1 Degenerate four-wave mix- ing. Waves 3 and 4 are intense pump waves traveling in opposite directions. Wave 1, the probe wave, and wave 2, the conjugate wave, also travel in opposite directions and have increasing amplitudes. -L o z Substituting (21.5-27) in (21.5-25) and using the slowly varying envelope approx- imation, (21.4-19), we reduce equations (21.5-25) to two first-order differential equa- tions, dA 1 . A * - == -J'Y 2 dz dA 2 . A * dz ==J'Y 1, (21.5-28a) (21.5-28b)
21.5 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE THEORY 923 where )((3) 1 == 2k == 3w'TJo n A3 A 4 (21.5-29) is a coupling coefficient whose magnitude may be written in the form 111 == 2C J 1 3 1 4 , 2 )((3) C == 3W'TJo ----=2 . n (21.5-30) Coupling Coefficient Here 13 == IA312/2'TJ and 14 == IA412/2'TJ are the intensities of the two waves and 'TJ == 7]0/ n. For simplicity, assume that A3A4 is real, so that 1 is real. The solution of (21.5- 28) is then two harmonic functions, AI(z) and A 2 (z), with a 90° phase shift between them. If the nonlinear medium extends over a distance between the planes z == - L to z == 0, as illustrated in Fig. 21.5-1, wave 1 has amplitude A I ( - L) == Ai, at the entrance plane, and wave 2 has zero amplitude at the exit plane, A 2 (0) == O. Under these boundary conditions the solution of (21.5-28) is A. Al(Z) = Z L COS/,Z cOS1 A A 2 ( z) == j 'I, sin 1 z . cos 1 L The amplitude of the reflected wave at the entrance plane, Ar == A 2 ( - L), is (21.5-31) (21.5-32) Ar == -jAT tan1L, (21.5-33) Reflected Wave Amplitude whereas the amplitude of the transmitted wave, At == Al (0), is A - Ai t - . cos 1 L (21.5-34) Transmitted Wave Amplitude Equations (21.5-33) and (21.5-34) suggest a number of applications: . The reflected wave is a conjugated version of the incident wave. The device acts as a phase conjugator (see Sec. 21.3E). . The intensity reflectance, IArI2/IAiI2 == tan 2 1L, may be smaller or greater than 1, corresponding to attenuation or gain, respectively. The medium can therefore act as a reflection amplifier (an "amplifying mirror"). . The transmittance IAtI2/IAiI2 == 1/ COS 2 1L is always greater than 1, so that the medium always acts as a transmission amplifier. . When 1 L == 7r /2, or odd multiples thereof, the reflectance and transmittance are infinite, indicating instability. The device is then an oscillator.
924 CHAPTER 21 NONLINEAR OPTICS *21.6 ANISOTROPIC NONLINEAR MEDIA In an anisotropic medium, each of the three components of the polarization vector j> == (1\, P 2 , P 3 ) is generally a function of the three components of the electric field vector £ == (£1, £2, £3). These functions are linear for small magnitudes of £ (see Sec. 6.3) but deviate slightly from linearity as £ increases. They may therefore be expanded in a Taylor series in terms of the three components of £, just as in the scalar analysis presented in Sec. 21.1 : _" " " (3) Pi - Eo Xij£j + 2 dijk£j£k + 4 Xijkl£j£k£l j j k j kl i,j,k,l == 1,2,3. (21.6-1 ) The coefficients Xij, d;jk. and Xkl are elements of tensors that correspond to the scalar coefficients X, d, and X(3), respectively, and (21.6-1) is a vector generalization of (21.1-2). Because d ijk is proportional to 8 2 P i /8£j8£k, it is invariant to exchange of j and k. Similarly, Xkl is invariant to pennutations of j, k, and l. For lossless nondispersive media, there are additional intrinsic symmetries: Xij == Xji, as shown in Sec. 6.3A, and also d ijk and Xml are invariant to pennutations of their indexes. This full-permutation symmetry does not generally hold for dispersive nonlinear media. Exploiting the symmetry condition dijk == d ikj , elements of the tensor d ijk are usually listed as a 3 x 6 array d iJ , where the six independent combinations (j, k) == 11,22,33,23,31,12 are represented by a single index J ==1, 2, 3,4,5,6, in that order (see Table 20.2-1). For example, d 25 denotes the coefficients d 231 == d 213 . The third-order coefficients Xkl are similarly described by a 6 x 6 array X}, where the pair (i, j) is contracted into a single index I == 1,2, . . . ,6, and the pair (k, l) is contracted into K == 1, 2, . . . , 6. The structural symmetry of the crystal places additional constraints on the tensor elements dijk and Xml. When the coordinate system (1,2,3) coincides with the prin- cipal axes of the crystal, which are determined from the tensor Xij, some entries in the arrays d iJ and X} are zero, while others are equal or are related by some simple rule. Representative examples are provided in Tables 21.6-1 and 21.6-2. Values for the d iJ coefficients for a number of representative nonlinear crystals are provided in Table 21.6-3. Although cubic crystals have isotropic linear optical properties, their well-defined crystal axes (as determined by their structural symmetry) endow them with anisotropic nonlinear optical properties. Table 21.6-1 Second-order nonlinear coefficients d iJ for some representative crystal groups. [ 0 0 0 d 14 0 0 ] [ 0 0 0 d 14 0 0 ] [ 0 0 0 0 d 15 -d 22 ] o 0 0 0 d 14 0 0 0 0 0 d 14 0 -d 22 d 22 0 d 15 0 0 o 0 0 0 0 d 14 0 0 0 0 0 d 36 d 31 d 31 d 33 0 0 0 Cubic 43m Tetragonal 42m Trigonal 3m (e.g., GaAs, CdTe, InAs) (e.g., KDP, ADP) (e.g., BBO, LiNb0 3 , LiTa03) The tensors dijk and Xkl are closely related to the Pockels and Kerr tensors tijk and 5ijkl, respectively, as demonstrated in Prob. 21.6-2, and they have the same symmetries, as can be seen by comparing Tables 21.6-1 and 21.6-2, which list d iJ and X}, with
21.6 ANISOTROPIC NONLINEAR MEDIA 925 Table 21.6-2 Third-order nonlinear coefficients X} for an isotropic medium. (3) (3) (3) Xu X12 X12 (3) (3) (3) X12 XII X12 (3) (3) (3) X12 X12 XII 000 000 000 o o o (3) X44 o o o o o o (3) X44 o o o o o o (3) X44 (3) _ 1 ( (3) (3) ) , X44 -"2 XII - X 12 . Tables 20.2-2 and 20.2-3, which list tlk and 5IK for a number of crystal groups. Note, however, that d iJ is analogous to the transpose of tlk. Table 21.6-3 Representative magnitudes of second-order nonlinear optical coefficients for selected materials. a d iJ (CjV 2 ) d 22 = 2.0 X 10- 23 d 31 = 3.5 X 10- 25 d 31 = 5.9 X 10- 24 d 32 = 7.5 X 10- 24 d 33 = 3.5 X 10- 25 d 31 = 3.9 X 10- 23 d 33 = 4.1 X 10- 23 d 22 = 1.9 X 10- 23 d 31 = 4.1 X 10- 23 d 33 = 2.2 X 10- 22 d 31 = 1.1 X 10- 22 d 32 = 1.2 X 10- 22 d 31 = 2.0 X 10- 23 d 32 = 3.3 X 10- 23 d 33 = 1.3 X 10- 22 d 36 = 3.1 X 10- 24 d 36 = 4.2 X 10- 24 d 11 = 2.7 X 10- 24 d 11 = 4.3 X 10- 24 d 14 = 1.5 X 10- 21 d 11 = 5.8 X 10- 21 Crystal jJ-BaB 2 0 4 (BBO) LiB 3 0 5 (LBO) LiI0 3 LiNb0 3 KNb0 3 KTiOP0 4 (KTP) KH 2 P0 4 (KDP) NH 4 H 2 P0 4 (ADP) a-Si0 2 (quartz) KBe2B03F2 (KBBF) GaAs Te d iJ j Eo (pmjV)b 2.2 0.04 0.67 0.85 0.04 4.4 4.6 2.1 4.6 25.2 11.9 13.7 2.2 3.7 14.6 0.38 0.47 0.30 0.49 170. 650. aMost of the coefficients are as reported by D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, Springer-Verlag, 2005. Values are provided at a wavelength Ao = 1.06 /-lm except for Te, which is provided at Ao = 10.6/-lm. b The coefficients d/ Eo, specified in units of pm/V, are often used in practice. The nonlinear optical coefficients in C /V 2 (MKS units) are readily converted to pm/V by dividing d by 10- 12 Eo 8.85 X 10- 24 . Three-Wave Mixing in Anisotropic Second-Order Nonlinear Media When an optical field comprising two monochromatic linearly polarized waves of angular frequencies WI and W2, and complex amplitudes E(WI) and E(W2), travel through a second-order nonlinear crystal, the induced nonlinear polarization density vector P(W3) at frequency W3 == WI + W2 has components Pi(W3) == 2 L d ij k E j(WI)E k (W2), jk i,j, k == 1,2,3, (21.6-2)
926 CHAPTER 21 NONLINEAR OPTICS where Ej(Wl), E k (W2), and Pi(W3) are the components of these vectors along the principal axes of the crystal. This equation is a generalization of (21.2-13d). Using the contracted notation (j, k) == J, (21.6-2) may be conveniently written in the matrix form: [ : ] == 2 [ P 3 (W3) d 31 d 12 d 22 d 32 d 16 ] d 26 d 36 E 1 (Wl)E 1 (W2) E 2 (Wl)E 2 (W2) E 3 (Wl)E 3 (W2) E 2 (Wl)E 3 (W2) + E 3 (Wl)E 2 (W2) E 3 (Wl)E 1 (W2) + El (wl)E 3 (W2) E 1 (Wl)E 2 (W2) + E 2 (Wl)E 1 (W2) (21.6-3) Effective value of d. If Ej(Wl) == E(Wl) cosB 1j and E k (W2) == E(W2) COSB 2k , where B 1j and B 2k are the angles that the vectors E(Wl) and E(W2) make with the principal axes, then (21.6-2) may be written in the form Pi(W3) = 2 [dijk cosB 1j COSB2k] E(Wl)E(W2)' (21.6-4) Since the polarization density vector P(W3) is the source for wave 3, only the compo- nent P ..1 (W3) in the plane orthogonal to the wavevector k3 contributes; the component parallel to k3 cannot radiate a TEM wave. If P ..1 (W3) makes angles B 3i with the prin- cipal axes, then its magnitude is P..l(W3) == LPi(W3)cos B 3i . i (21.6-5) It follows from (21.6-4) and (21.6-5) that P..l (W3) == 2d e ff E (Wl)E(W2), (21.6-6) with an effective second-order nonlinear coefficient deff == L d ijk cas B 3 i cas B 1j cas B 2k . ijk (21.6-7) Equation (21.6-6) takes the same form as that used in the scalar formulation pro- vided in Sees. 21.2C and 21.4; deff plays the role of the coefficient d. Example 21.6-1 illustrates a direct computation of deff for a three-wave mixing configuration in an anisotropic crystal. EXAMPLE 21.6-1. Collinear Type-I Three-Wave Mixing in a KDP Crystal. In this ex- ample, we determine the effective nonlinear coefficient d eff for three collinear waves traveling in a KDP crystal at an arbitrary direction ((), cP) defined in a spherical coordinate system with the crystal optic axis pointing in the z direction, as illustrated in Fig. 21.6-1. Waves 1 and 2 are ordinary waves at frequencies WI and W2, and wave 3 is extraordinary with frequency W3 = WI + W2. Using (21.6-2) and Table 21.6-1 for crystals of 42m symmetry, such as KDP, the nonlinear components of the polarization density vector are given by PI (W3) = 2d I4 [E 2 (WI)E 3 (W2) + E 3 (WI)E 2 (W2)] P 2 (W3) = 2d I4 [E 3 (WI)E I (W2) + E I (WI)E 3 (W2)] P 3 (W3) = 2d 36 [E I (WI)E 2 (W2) + E 2 (WI)E I (W2)]. (21.6-8)
21.7 DISPERSIVE NONLINEAR MEDIA 927 In this geometry, the electric field components of waves 1 and 2 are: EI (WI) == E(WI) sin <p, EI (W2) == E(W2) sin cjJ, E 2 (WI) == -E(WI)COS<P E 3 (WI) == 0, E 2 (W2) == -E(W2) COS cjJ, E 3 (W2) == o. Therefore, based on (21.6-8), the components of the polarization density vector for wave 3 are PI(W3) == 0, P 2 (W3) == 0, P 3 (W3) == -4d 36 sin <pcos <pE(WI)E(W2). (21.6-9) In this case, the component P.l. ( W3) == - P 3 ( W3) sin (), so that d eff == - d 36 sin () sin 2cjJ. (21.6-10) This result can also be obtained by direct use of (21.6- 7) with the appropriate angles and coefficients. The effective nonlinear coefficient in (21.6-10) has its maximum magnitude d 36 if the angles are () == 90° and cjJ == 45°, as illustrated in Fig. 21.6-1. z z e y (b) Figure 21.6-1 (a) Geometry for y collinear Type-I o-o-e three-wave mixing in a uniaxial crystal whose optic axis is in the z direction. (b) Direction of propagation for achiev- ing maximum d eff . x x (a) *21.7 DISPERSIVE NONLINEAR MEDIA This section provides a brief discussion of the origin of dispersion and its effect on nonlinear optical processes. For simplicity, anisotropic effects are not included. A dispersive medium is a medium with memory (see Sec. 5.2); the polarization density P( t) resulting from an applied electric field £ (t) does not appear instantaneously. Rather, the response P ( t) at time t is a function of the applied electric field £ ( t f ) at times t f < t. When the medium is also nonlinear, the functional relation between P(t) and {£ ( t) , t f < t} is nonlinear. There are two means for describing such nonlinear dynamical systems: 1. A phenomenological integral relation between P( t) and £(t) based on a Volterra- series expansion, which is similar to a Taylor-series expansion. The coefficients of the expansion characterize the medium phenomenologically. 2. A nonlinear differential equation for P(t), with £(t) as a driving force, obtained by developing a model for the physics of the polarization process, much as the Lorentz model was developed for linear media. Integral- Transform Description of Dispersive Nonlinear Media If the deviation from linearity is small, a Volterra-series expansion may be used to describe the relation between P( t) and £ (t). The first term of the expansion is a linear
. 928 CHAPTER 21 NONLINEAR OPTICS combination of £ (t') for all t' < t, P(t) = Eo I: x(t - t')£(t') dt', (21.7-1) This describes a linear system with impulse response function Eo x( t) [see Sec. 5.2, in particular (5.2-23), and Appendix B]. The second term in the expansion is a superposition of the products c (t') £ (t") at pairs of times t' < t and t" < t, 00 P(t) = Eo J J x(2) (t - t', t - t") £( t')£(t") dt'dt", (21.7-2) -00 where x(2) (t', t") is a function of two variables that characterizes the second-order dispersive nonlinearity. The third term represents a third-order nonlinearity that can be characterized by a function x(3) (t', t", t"') and a similar triple integral relation. The linear dispersive contribution described by (21.7-1) can also be completely characterized by the response to monochromatic fields. If £(t) == Re{ E(w) exp(jwt)}, then P( t) == Re{ P(w) exp(jwt)}, where P(w) == EoX(W )E(w) and X(w) is the Fourier transform ofx(t) at v == w/27r. The medium is then characterized completely by the frequency-dependent susceptibility X(w). The second-order nonlinear contribution described by (21.7-2) is characterized by the response to a superposition of two monochromatic waves of angular frequencies WI and W2. Substituting c(t) == Re{E(wI) exp(jwIt) + E(W2) exp(jw2t)} (21.7-3) into (21.7-2), it can be shown that the polarization-density component of angular fre- quency W3 == WI + W2 has an amplitude P(W3) == 2d(W3; WI, W2) E(WI)E(W2). (21.7 -4 ) The coefficient d( W3; WI, W2) is a frequency-dependent version of the coefficient d in (21.2-13d). The relation between' this coefficient and the response function x (2) ( t' , t") is established by defining 00 X(2)(Wl,W2) = J J X(2) (t', t") exp[-j(wd + W2 t ")] dt' dt", -00 (21.7-5) which is the two-dimensional Fourier transform of X(2) (t', t") evaluated at VI == -wI/27r and V2 == -W2/27r [see (A.3-2) in Appendix A]. Substituting (21.7-3) into (21.7-2) and using (21.7-5), we obtain d( W3; WI, W2) == E o X(2) (WI, W2). (21.7 -6a) Thus, the second-order nonlinear dispersive medium is completely characterized by either of the frequency-dependent functions, X(2) (WI, W2) or d( W3; WI, W2).
21.7 DISPERSIVE NONLINEAR MEDIA 929 The degenerate case of second-harmonic generation in a second-order nonlinear medium is also readily described by substituting £(t) == Re{E(w) exp(jwt)} into (21.7-2) and using (21.7-5). The resultant polarization has a component at frequency 2w with amplitude P(2w) == d(2w; w, w) E(w)E(w), where d(2w;w,w) == !E o X(2)(w,w). (21.7 -6b ) Other d coefficients representing various wave-mixing processes may similarly be related to the two-dimensional function X(2) (WI, W2). The electro-optic effect, for ex- ample, is a result of interaction between a steady electric field (WI == 0) and an optical wave (W2 == w) to generate a polarization density at w3 == w. The pertinent coefficient for this interaction is d( w; 0, w) == 2E o X(2) (w, 0); it determines the Pockels coefficient t in accordance with (21.2-11). In a third-order nonlinear medium, an electric field comprising three harmonic functions of angular frequencies WI, W2, and W3 creates a sum-frequency polarization density with a component at angular frequency W4 == WI + W2 + W3 of amplitude P(W4) == 6X(3) (W4; WI, W2, W3) E(WI)E(W2)E(W3), (21.7-7) where the function X(3) (W4; WI, W2, W3) replaces the coefficient X(3) that describes the nondispersive case. The function X(3) (W4; WI, W2, W3) can be determined from X(3) (t ' , t", t'll) by relations similar to (21.7-6a). In short, as a consequence of dispersion, the second- and third-order nonlinear coefficients d and X(3) are dependent on the frequencies of the waves involved in the wave-mIxIng process. Differential-Equation Description of Dispersive Nonlinear Media An example of a nonlinear dynamic relation between P(t) and £(t) is provided by the differential equation d 2 P dP 2 2 2 2 dt 2 + () dt + WoP + WOEoXO bP = WOEoXO G , where 0", Wo, Xo, and b are constants. In the absence of the nonlinear term, w6 E oX o b p2, (21.7-8) reduces to (5.5-15), which is appropriate for a linear resonant dielectric medium described by the Lorentz oscillator model (see Sec. 5.5C). Each atom is then characterized by a harmonic oscillator in which an electron of mass m is subjected to an electric-field force -e£, an elastic restoring force -x, and a frictional force mO" dx / dt, w here x is the displacement of the electron from its equilibrium position and Wo == vi /m is the resonance angular frequency. The medium is then linear and dispersive with a susceptibility given by [see (5.5-18)] (21.7-8) W 2 X(w)=XO 2 2 0 +" . W o - W JWO" (21.7-9) Linear Susceptibility (Harmonic-Oscillator) When the restoring force is a nonlinear function of displacement, - x - 2x2, where and 2 are constants, the result is an anharmonic oscillator described by (21.7- 8), where b is proportional to 2. The medium is then nonlinear.
930 CHAPTER 21 NONLINEAR OPTICS EXERCISE 21.7-1 Polarization Density for an Anharmonic-Oscillator Medium. Show that for a medium containing N atoms per unit volume, each modeled as an anharmonic (nonlinear) oscillator with restraining force -K,X- K,2X2, the relation between P(t) and £( t) is the nonlinear differential equation (21.7-8), where XO = Ne2/Eomw6 and b = K,2/e 3 N 2 . Equation (21.7-8) cannot be solved exactly. However, if the nonlinear term is small, an iterative approach provides an approximate solution. Let (21.7-8) be written in the form ,c{P} == G - bP2, (21.7-10) where ,c == (w6EoXo)-1 (d 2 / dt 2 + (Jd/ dt + w6) is a linear differential operator. The iterative solution of (21.7 -10) is carried out via the following steps: 1. Find a first-order approximation PI by neglecting the nonlinear term b p2 in (21.7 -10), and solving the linear equation ,c{P 1 } G. (21.7 -11 ) 2. Use this approximate solution to determine the small nonlinear term bPi. 3. Obtain a second-order approximation by solving (21.7-10) with the term bP2 replaced by bPi. The solution of the resulting linear equation is denoted P 2 , ,c{P 2 } == £ - bPi. (21.7-12) 4. Repeat the process to obtain a third-order approximation as illustrated by the block diagram of Fig. 21.7-1. £ - Linear system EoX{ w) p bP2 P Figure 21.7-1 Block diagram represent- ing the nonlinear differential equation (21.7- 10). The linear system represented by the operator equation £{P} = £ has a transfer function EoX(W). We first examine the special case of monochromatic light, G == Re{ E(w) exp(jwt)}. In the first iteration PI == Re{Pl(w)exp(jwt)}, where P 1 (w) == EoX(w)E(w) and X(w) is given by (21.7-9). In the second iteration, the linear system is driven by a force G - bPi == Re{E(w)e jwt } - b [Re{E o x(w)e jwt }]2 == Re{E(w)e jwt } - b Re{[E o x(w)E(w)]2 e j2wt } - b IEox(w)E(w) 1 2 . Since these three terms have frequencies w, 2w, and 0, the linear system responds with susceptibilities X(w), X(2w), and X(O), respectively. The component of P 2 at
21.7 DISPERSIVE NONLINEAR MEDIA 931 frequency 2w has an amplitude P 2 (2w) == EoX(2w){ -b [EoX(w)E(w)]2}. Since P(2w) == d(2w; w, W )E(w )E(w), we conclude that d (2w; w, w) == - b E [X ( w ) ] 2 X (2w ) . (21.7-13) EXERCISE 21.7-2 Miller's Rule. For the nonlinear resonant medium described by (21.7-8), if the light comprises a superposition of two monochromatic waves of angular frequencies WI and W2, show that the second- order approximation described by (21.7-11) and (21.7-12) yields a component of polarization density at frequency W3 = WI + W2 with amplitude P2(W3) = 2d(W3; WI, w2)E(WI)E(W2), where d(W3;WI,W2) = -bEX(WI)X(W2)X(W3). (21.7-14) Miller's Rule Equation (21.7-14) is known as Miller's rule. Miller's rule states that the coefficient of second-order nonlinearity for the generation of a wave of frequency W3 == WI + W2, from two waves of frequencies WI and W2, is proportional to the product of the linear susceptibilities at the three frequencies, X (WI) X (W2) X (W3). The three frequencies must therefore lie within the optical trans- mission window of the medium (away from resonance). If these frequencies are much smaller than the resonance frequency wo, then (21.7-9) gives X( w) == Xo, and (21.7- 14) then yields d( W3; WI, W2) == -! b Ex8, which is independent of frequency. The medium is then approximately nondlspersive, and the results of the previous sections in which dispersion was neglected are applicable. Miller's rule also indicates that materials with large refractive indexes (large Xo) tend to have large d. Anisotropic Dispersive Media When both anisotropic and dispersive properties are considered, three-wave mixing in a second-order medium is described by the more general relation P i (W3) == 2 L d ijk (W3; WI, W2) E j (WI)E k (W2), jk (21.7-15) where W3 == WI + W2. The coefficients dijk are now dependent on the frequencies of the mixed waves. This relation is similar to the relation Pi(W) == Lj Xij (w)Ej(w), which describes linear media. Similarly, four-wave mixing in a third-order medium is described by Pi(W4) = 6 L X;kl(W4; WI, W2, W3) Ej (wI)E k (W2)E 1 (W3), jkl (21.7-16) where W4 == WI + W2 + W3.
932 CHAPTER 21 NONLINEAR OPTICS The frequency dependent tensor elements dijk, and X;kl obey a number of intrinsic symmetry relations that are similar to the relation xi j (w) == Xij ( -w) in linear optics: d;jk(W3; WI, W2) == djki(WI; -W2, W3) == dkij(W2;W3, -WI) (3) * ( . ) _ (3) ( . ) Xijkl W4, WI, W2, W3 - Xjkli WI, -W2, -W3, W4 (21.7-17) = X;;kl(W3; W4, -WI, -W2). (2].7-18) In these relations, the coefficient d j ki (WI; -W2, W3), for example, represents a down- conversion process in which a wave of frequency W2 and polarization k mixes with a wave of frequency W3 and polarization i and generates a wave of frequency WI == W3 - W2 and polarization j. Other coefficients can be similarly interpreted. This type of intrinsic symmetry is of course supplemented by other structura] symmetry relations that are obeyed for various classes of crystals. READING LIST Books See also the reading lists in Chapters 5, 6, 15, and 20, as well as the books on optoelectronics in Chapter 17. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 1991, 4th ed. 2006. R. Menzel, Photonics: Linear and Nonlinear Interactions of Laser Light and Matter, Springer-Verlag, 2001, 2nd ed. 2006. M. Wegener, Extreme Nonlinear Optics: An Introduction, Springer-Verlag, 2005. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey, Springer-Verlag, 2005. P. P. Banerjee, Nonlinear Optics: Theory, Numerical Modeling, and Applications, Marcel Dekker, 2004. A. Brignon and J.-P. Huignard, eds., Phase Conjugate Laser Optics, Wiley, 2004. R. W. Boyd, Nonlinear Optics, Academic Press, 1992, 2nd ed. 2003. T. Suhara and M. Fujimura, Waveguide Nonlinear-Optic Devices, Springer-Verlag, 2003. R. L. Sutherland, Handbook of Nonlinear Optics, Marcel Dekker, 2nd ed. 2003. W. P. Risk, T. R. Gosnell, and A. V. Nurmikko, Compact Blue-Green Lasers, Cambridge University Press, 2003. Y. R. Shen, The Principles of Nonlinear Optics, Wiley, 1984, paperback ed. 2002. S. Miyata and H. Sasabe, eds., Light Wave Manipulation Using Organic Nonlinear Optical Materials, CRC Press, 2000. J. Robieux, High Power Laser Interactions, Lavoisier, 2000. G. S. He and S. H. Liu, Physics of Nonlinear Optics, World Scientific, 1999. A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves, Springer-Verlag, 1999. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer-Verlag, 1991, 3rd ed. 1999. D. L. Mills, Nonlinear Optics: Basic Concepts, Springer-Verlag, 1991, 2nd ed. 1998. F. Kajzar and R. Reinisch, eds., Beam Shaping and Control with Nonlinear Optics, Plenum Press, 1998. H. S. Nalwa and S. Miyata, eds., Nonlinear Optics of Organic Molecules and Polymers, CRC Press, 1997.
READING LIST 933 N. Bloembergen, Nonlinear Optics, World Scientific, 1965, 4th ed., 1996. E. G. Sauter, Nonlinear Optics, Wiley, 1996. C. L. Tang and L. K. Cheng, Fundamentals of Optical Parametric Processes and Oscillators, Har- wood, 1995. J.- Y. Zhang, J. Y. Huang, and Y. R. Shen, Optical Parametric Generation and Amplification, Har- wood, 1995. J. Zyss, Molecular Nonlinear Optics: Materials, Physics, and Devices, Academic Press, 1994. F. A. Hopf and G. I. Stegeman, Applied Classical Electrodynamics, Volume 2, Nonlinear Optics, Wiley, 1986, reprinted 1992. P. N. Prasad and D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers, Wiley, 1991. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, 1990, paperback ed. 1991. V. S. Butylkin, A. E. Kaplan, Yu. G. Khronopulo, and E. I. Yakubovich, Resonant Nonlinear Interac- tions of Light with Matter, Springer-Verlag, 1989. M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics, Wiley, 1986. B. Y. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation, Springer- Verlag, 1985. R. A. Fisher, ed., Optical Phase Conjugation, Academic Press, 1983. H. Rabin and C. L. Tang, Quantum Electronics, Academic Press, 1975. I. P. Kaminow, An Introduction to Electrooptic Devices, Academic Press, 1974. Articles Issue on nonlinear optics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 3, 2006. C. Gmachl, O. Malis, and A. Belyanin, Optical Nonlinearities in Intersubband Transitions and Quan- tum Cascade Lasers, in Intersubband Transitions in Quantum Structures, pp. 181-235, R. Paiella, ed., McGraw-Hill, 2006. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, Multistep Parametric Processes in Nonlinear Optics, in Progress in Optics, vol. 47, pp. 1-73, E. Wolf, ed., Elsevier, 2005. Issue on nonlinear optics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 5, 2004. Issue on nonlinear optics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, no. 3, 2002. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. T. R. Gosnell, ed., Selected Papers on Upconversion Lasers, SPIE Optical Engineering Press (Mile- stone Series Volume 161), 2000. R. L. Byer, Quasi-Phasematched Nonlinear Interactions and Devices, Journal of Nonlinear Optical Physics and Materials, vol. 6, pp. 549-592, 1997. J. H. Hunt, ed., Selected Papers on Optical Parametric Oscillators and Amplifiers and Their Appli- cations, SPIE Optical Engineering Press (Milestone Series Volume 140), 1997. D. A. Roberts, Simplified Characterization of Uniaxial and Biaxial Nonlinear Optical Crystals: A Plea for Standardization of Nomenclature and Conventions, IEEE Journal of Quantum Electronics, vol. 8, pp. 2057-2074, 1992. H. E. Brandt, ed., Selected Papers on Nonlinear Optics, SPIE Optical Engineering Press (Milestone Series Volume 32), 1991. I. C. Khoo, Nonlinear Optics of Liquid Crystals, in Progress in Optics, vol. 26, E. Wolf, ed., North- Holland, 1988. D. M. Pepper, Applications of Optical Phase Conjugation, Scientific American, vol. 254, no. 1, pp. 74-83, 1986. V. V. Shkunov and B. Y. Zel'dovich, Optical Phase Conjugation, Scientific American, vol. 253, no. 6, pp. 54-59, 1985. N. Bloembergen, Nonlinear Optics and Spectroscopy (Nobel lecture), Reviews of Modern Physics, vol. 54,pp. 685-695, 1982.
934 CHAPTER 21 NONLINEAR OPTICS A. L. Mikaelian, Self-Focusing Media with Variable Index of Refraction, in Progress in Optics, vol. 17, E. Wolf, ed., North-Holland, 1980. w. Brunner and H. Paul, Theory of Optical Parametric Amplification and Oscillation, in Progress in Optics, vol. 15, E. Wolf, ed., North-Holland, 1977. R. W. Hellwarth, Third-Order Optical Susceptibilities of Liquids and Solids, Progress in Quantum Electronics, vol. 5, pp. 1-68, 1977. PROBLEMS 21.2-2 Power Exchange in Frequency Up-Conversion. A LiNb0 3 crystal of refractive index n == 2.2 is used to convert light of free-space wavelength 1.3 /-Lm into light of free-space wavelength 0.5 /-Lm, using a three-wave mixing process. The three waves are collinear plane waves traveling in the z direction. Determine the wavelength of the third wave (the pump). If the power of the 1.3-/-Lm wave drops by 1 mW within an incremental distance z, what is the power gain of the up-converted wave and the power loss or gain of the pump within the same distance? 21.2-3 Matching Conditions for Collinear Type-II SHG. Determine the angle () for a KDP crystal used in type-II second-harmonic generation at A == 1.06 /-Lm for each of the o-e- o and o-e-e configurations. Use the Sellmeier equations in Table 5.5-1 to determine the wavelength dependence of the refractive indexes. 21.2-4 Phase Matching in a Degenerate Parametric Down-Converter. A degenerate parametric down-converter uses a KDP crystal to down-convert light from 0.6 /-Lm to 1.2 /-Lill. If the two waves are collinear, what should the direction of propagation of the waves (in relation to the optic axis of the crystal) and their polarizations be so that the phase-matching condition is satisfied? KDP is a uniaxial crystal with the following refractive indexes: at Ao == 0.6 /-Lm, no == 1.509 and ne == 1.468; at Ao == 1.2 /-Lm, no == 1.490 and ne == 1.459. 21.2- 5 Matching Conditions for Three-Wave Mixing in a Dispersive Medium. The refractive index of a nonlinear medium is a function of wavelength approximated by n( Ao) no - Ao, where Ao is the free-space wavelength and no and are constants. Show that three waves of wavelengths A o l, A o 2, and A03 traveling in the same direction cannot be efficiently coupled by a second-order nonlinear effect. Is efficient coupling possible if one of the waves travels in the opposite direction? *21.2-6 Tolerance to Phase Mismatching. (a) The Helmholtz equation with a source, \72 E + k 2 E == -8, has the solution E(r) = r S(r') exp( -jkolr r'D dr', lv 47Tlr - r I where V is the volume of the source and ko == 27T / Ao. This equation can be used to determine the field emitted at a point r, given the source at all points r' within the source volume. If the source is confined to a small region centered about the origin r == 0, and r is a point sufficiently far from the source so that r' « r for all r' within the source, then Ir - r'l == (r 2 + r,2 - 2r . r')1/2 r(l - r . r' /r2) and E( ) exp( -jkor) 1 8( ' ) ( O k ---- ' ) d ' r r exp J or ° r r , 47Tr v where r is a unit vector in the direction of r. Assuming that the volume V is a cube of width L and the source is a harmonic function 8 (r) == exp ( - jks . r), show that if L » Ao, the emitted light is maximum when kor == ks and drops sharply when this condition is not met. Thus, a harmonic source of dimensions much greater than a wavelength emits a plane wave with approximately the same wavevector. (b) Use the relation in (a) and the first Born approximation to determine the scattered field, when the field incident on a second-order nonlinear medium is the sum of two waves with wavevectors k 1 and k 2 . Derive the phase-matching condition k3 == k 1 + k 2 and
PROBLEMS 935 determine the smallest magnitude of k = k3 - k 1 - k 2 at which the scattered field E vanishes. 2] .2-7 Backward SHG with QPM. Show that a periodically poled crystal may be used to generate a second-harmonic wave traveling in a direction opposite to that of the fundamental wave. Write the phase matching equation for this quasi-phase-matching process. If the equation is satisfied for the 7th-order harmonic of the periodic function, determine the ratio of the poling period to the wavelength of the fundamental wave in the medium. 21.3-4 Invariants in Four-Wave Mixing. Derive equations for energy and photon-number conser- vation (the Manley-Rowe relation) for four-wave mixing. 21.3-5 Power of a Spatial Soliton. Determine an expression for the integrated intensity of the spatial soliton described by (21.3-12) and show that it is inversely proportional to the beam width W o . 21.3-6 An Opto-Optic Phase Modulator. Design a system for modulating the phase of an optical beam of wavelength 546 nm and width W = 0.1 mm using a CS 2 Kerr cell of length L = 10 cm. The modulator is controlled by light from a pulsed laser of wavelength 694 nm. CS 2 has a refractive index n = 1.6 and a coefficient of third-order nonlinearity X(3) = 4.4 X 10- 32 Cm/y 3 . Estimate the optical power P 7r of the controlling light that is necessary for modulating the phase of the controlled light by 7r. 21.3-7 SHG in Third-Order Nonlinear Medium via a Static Electric Field. Show that SHG can occur in a third-order nonlinear medium with an applied static electric field. What physical parameters determine the efficiency of this SHG process? *21.4-7 Gain of a Parametric Amplifier. A parametric amplifier uses a 4-cm-Iong KDP crystal (n 1.49, d = 8.3 X 10- 24 C/y 2 ) to amplify light of wavelength 550 nm. The pump wavelength is 335 nm and its intensity is 10 6 W /cm 2 . Assuming that the signal, idler, and pump waves are collinear, determine the amplifier gain coefficient and the overall gain. *21.4-8 Degenerate Parametric Down-Converter. Write and solve the coupled equations that describe wave mixing in a parametric down-converter with a pump at frequency W3 = 2w and signals at WI = W2 = w. All waves travel in the z direction. Derive an expression for the photon flux densities at 2w and wand the conversion efficiency for an interaction length L. Verify energy conservation and photon conservation. *21.4-9 Threshold Pump Intensity for Parametric Oscillation. A parametric oscillator uses a 5-cm-Iong LiNb0 3 crystal with second-order nonlinear coefficient d = 4 X 10- 23 C/y 2 and refractive index n = 2.2 (assumed to be approximately constant at all frequencies of interest). The pump is obtained from a 1.06-p,m Nd:YAG laser that is frequency doubled using a second-harmonic generator. The crystal is placed in a resonator using identical mirrors with reflectances 0.98. Phase matching is satisfied when the signal and idler of the parametric amplifier are of equal frequencies. Determine the minimum pump intensity for parametric oscillation. *21.5-1 Combined SHG and SFG. Two waves of angular frequencies WI and W2, their second- harmonic waves, which have angular frequencies 2Wl and 2W2, and their sum-frequency wave, whose angular frequency is WI + W2, interact simultaneously in a second-order non- linear medium. Assuming that phase matching is satisfied for the two SHG processes, and for the SFG process, write coupled equations for this five-wave-mixing process. Solve these equations numerically and demonstrate that the presence of the second wave may suppress the SHG process for the first. *21.5-2 Coupled-Wave Equations for Degenerate Four-Wave Mixing. Consider the collinear four-wave-mixing problem in a third-order nonlinear medium, in the degenerate case W4 = W3, and WI + W2 = 2W3. Derive coupled wave equations for the amplitudes AI, A 2 , and A3 assuming that the phase matching condition is fully met. *21.6-1 Collinear Type-II Three-Wave Mixing in a BBO Crystal. Repeat the analysis carried out in Example 21.6-1 to show that the effective nonlinear coefficient d eff for Type-II o-e- e three-wave mixing for a crystal in the 3m group, such as BBO, is d eff = d 22 cas 2 () cas 3cjJ. *21.6-2 Relation Between Nonlinear Optical Coefficients and Electro-Optic Coefficients. Show that the electro-optic coefficients are related to the coefficients of optical nonlinearity by tijk = -4Eodijk/EiiEjj and fJijkl = -12EoXlz/EiiEjj. These relations are generalizations of (21.2-11) and (21.3-2), respectively. Hint: If two matrices A and B are related by B = A -1 , the incremental matrices A and B are related by B = - A -1 AA -1 .
CHAPTER ULTRAFAST OPTICS 22.1 PULSE CHARACTERISTICS 937 A. Temporal and Spectral Characteristics B. Gaussian and Chirped-Gaussian Pulses C. Spatial Characteristics 22.2 PULSE SHAPING AND COMPRESSION 946 A. Chirp Filters B. Implementations of Chirp Filters C. Pulse Compression D. Pulse Shaping 22.3 PULSE PROPAGATION IN OPTICAL FIBERS 960 A. The Optical Fiber as a Chirp Filter B. Propagation of a Gaussian Pulse in an Optical Fiber *C. Slowly Varying Envelope Diffusion Equation *0. Analogy Between Dispersion and Diffraction 22.4 ULTRAFAST LINEAR OPTICS 973 A. Ray Optics *B. Wave and Fourier Optics *C. Beam Optics 22.5 ULTRAFAST NONLINEAR OPTICS 984 A. Pulsed Parametric Processes B. Optical Solitons *C. Supercontinuum Light 22.6 PULSE DETECTION 999 A. Measu rement of Intensity B. Measu rement of Spectral Intensity C. Measurement of Phase *0. Measurement of Spectrogram .. , -- ' . In 1980, L. F. Mollenauer (left), R. H. Stolen, and J. P. Gordon (right) demonstrated the successful propagation of optical solitons in a glass fiber. 936
Interest in ultrashort optical pulses began with the invention of the laser and has been one of continuous progress toward shorter and shorter time scales. The earliest solid- state and semiconductor lasers were naturally pulsed, and the development of CW lasers required significant additional effort. The development of nanosecond pulses was followed by picosecond pulses, which ultimately led to femtosecond pulses, and more recently to attosecond pulses. However, subsequent development of shorter pulses be- came substantially more challenging. This progress has been fueled by the emergence of many important applications including communication at ultrahigh data rates and probing of ultrafast physical, chemical, and biological phenomena. These applications required either ultranarrow pulses or ultrahigh optical intensities (or field strengths). When applied to optics, the terms ultrafast and ultrashort generally describe pulses of widths in the nanosecond to femtosecond, or shorter, regimes. In electronics, how- ever, these terms refer to pulses of nanosecond to tens of picosecond widths since the ultimate speed limit of electronics is well below that of optics. A nanosecond electrical pulse has a GHz spectral width and must be guided by a broadband microwave circuit. A picosecond electrical pulse has a THz spectral width, which cannot be sustained by conventional electrical or microwave circuits. If a femtosecond electrical pulse were to be generated, it would cover a spectral band of hundreds of THz, which equals the entire frequency range extending from 0 Hz to the edge of the visible band ( 0.3 JLm). Additionally, by virtue of the uncertainty principle flEflt > h/2, such a pulse would have an energy uncertainty exceeding 1.5 e V, i.e., roughly the magnitude of the bandgap energy in typical semiconductors, which would make conventional electronics unreliable. This Chapter Ultrashort optical pulses may be generated by a combination of specially designed lasers employing various switching techniques or mode locking methods (see Sec. 15.4), but these methods are not sufficient for the generation of femtosecond pulses. The pulses generated by such lasers must be further compressed and reshaped by use of special techniques based on linear and nonlinear dispersive optical components and systems, as will be discussed in this chapter. The chapter begins with a description of the basic temporal and spectral characteris- tics of optical pulses (Sec. 22.1) and their filtering by: (1) linear dispersive optical com- ponents such as prisms and gratings (Sec. 22.2), and (2) transmission through linear dispersive media such as optical fibers (Sec. 22.3). Spatial effects are then addressed and the optics of pulsed waves with ultrawide spectral widths are examined (Sec. 22.4). Nonlinear optics of pulsed waves is subsequently addressed (Sec. 22.5), and some of the nonlinear optical phenomena that were introduced in Chapter 19 for continuous waves are generalized to pulsed waves. These include parametric wave mixing, self- phase modulation, and optical solitons (Sec. 22.5B). Finally, a number of methods of detecting ultrashort optical pulses using "slow" detectors are covered in Sec. 22.6. 22.1 PULSE CHARACTERISTICS A. Temporal and Spectral Characteristics A pulse of light is described by an optical field of finite time duration. In this chapter we use a scalar theory and represent the field components with a generic complex wavefunction U (r, t) normalized such that the optical intensity I (r, t) == I U (r, t) 1 2 937
938 CHAPTER 22 ULTRAFAST OPTICS (W/m 2 ). When we are concerned with only the temporal or spectral properties of a pulse at a fixed position r we will simply use the functions U (t) and I (t). Temporal and Spectral Representations The complex wavefunction describing an optical pulse of central frequency Vo is writ- ten in the form U(t) == A(t) exp(jwot), where A(t) is the complex envelope and Wo == 27rvo is the central angular frequency. The complex envelope itself is char- acterized by its magnitude IA(t)1 and phase cp(t) == arg{A(t)}, so that U(t) == IA(t)1 exp (j[wot + cp(t)]). The optical intensity I(t) == IU(t)1 2 == IA(t)1 2 (W/m 2 ) and the area under the intensity function J I(t)dt is the energy density (J/m 2 ). The intensity profiles of typical pulses include the Gaussian function, I (t) ex exp(-2t 2 /T 2 ) (which is examined in detail in Sec. 22.18), the Lorentzian function I(t) ex 1/(1 + t 2 /7 2 ), and the hyperbolic secant function I(t) ex sech 2 (t/7) (which appears in Sec. 22.5B in connection with optical solitons). The width of each of these pulses is proportional to the time constant 7. In the spectral domain, the pulse is described by the Fourier transform V(v) == J U(t) exp( -j27rvt)dt, which is a complex function V(v) == IV(v)1 exp[j'ljJ(v)]. The squared magnitude S( v) == 1 V (v) 1 2 is called the spectral intensity and 'ljJ (v) is the spectral phase. The function V (v) is centered at the central frequency Vo and vanishes for negative v since U(t) is a complex analytic signal (see Sec. 2.6A). The Fourier transform of the complex envelope A(v) == J A(t) exp( -j27rvt)dt == V(v - Yo) is centered at v == O. If the pulse has a narrow spectral width, then the complex envelope is a slowly varying function of time (i.e., varies slightly within an optical cycle 1/ Yo), but this is not the case for ultranarrow pulses with ultrawide spectral distributions. Figure 22.1-1 illustrates the various temporal and spectral functions that characterize an optical pulse. Wavefunction Envelope Phase <p(t) Spectral Spectral Re{ V(t)} / /IA(t)1 -- intensity S(v) phase 'ljJ(v) " \/ " , , "'-' L-JJ t t 0 Vo v (a) Temporal representation (b) Spectral representation Figure 22.1-1 Temporal and spectral representations of an optical pulse. (a) The real part of the wavefunction Re{U(t)} = IA(t)1 cos[wot+c.p(t)], the magnitude of the envelope IA(t)l, the intensity I(t), and the phase c.p(t). (b) Spectral intensity S(v) and spectral phase 'lj;(v). Temporal and Spectral Widths The temporal and spectra] widths of a pulse are the widths of the intensity I (t) IU(t)1 2 and the spectral intensity S(v) == IV(v)12, respectively, as defined by any of the measures of width set forth in Appendix A.2. Unless otherwise specified, we will use the full-width half-max (FWHM) definition and denote the temporal and spectral widths as 7FWHM and v, respectively. Because of the Fourier transform relation between U ( t) and V (1/ ), the spectral width is inversely proportional to the temporal width. The coefficient of proportionality de- pends on the pulse shape and the definition of width. This inverse relation is illustrated in Fig. 22.1-2(a) for a Gaussian pulse for which 7FWHMV == 0.44.
22.1 PULSE CHARACTERISTICS 939 /:1 v flv 100 THz 100 THz 10 THz 7J;t o 10 THz l :rCO p,. cJ> 'Q(j cO 1 THz w 1 THz Co 100 GHz 'Q(j 100 GHz 0 l' 10 GHz cJ> 10 GHz 1 GHz ] GHz 100 MHz 100 MHz 10 MHz 10 MHz (/) (/) (/) (/) T.FWHM 8 8 8 8 8 /:1), 0.. 0.. 0.. !::: 0 0 0 0 !::: !::: !::: !::: =:t (a) .-. 0 --< 0 (b) .-. 0 0 - --< a 0 ....-4 Figure 22.1-2 (a) The relation v O.44jTFWHM between the spectral width v and the temporal width TFWHM for a Gaussian pulse. (b) The corresponding width A for a pulse of central frequency Vo corresponding to the central wavelengths AO = cj Vo = 0.5 pm, 1 pm, and 1.5 /-Lm. As an example, a 10-fs pulse has a spectral width v = 44 THz, corresponding to A = 37 nm, 147 nm, and 331 nm, if the central wavelength is AO = 0.5 pm, 1 /-Lm, and 1.5 /-Lm, respectively, as indicated by the open circles in the graph. This relation is linear if v « Vo [see (22.1-1)]. The spectral intensity S(v) is often plotted as a function of the wavelength, S). (A). This conversion is obtained by use of the relation SA (A) == S( v) I dv / dA I == (C/A2)S(C/A). The spectral width v may also be converted into wavelength units. If v « vo, then the spectral width in wavelength units is approximately A IdA/ dvl v, or A 2 A v , C (22.1-1 ) Spectral Width where AO == c/vo is the wavelength corresponding to the central frequency. If v is in units of THz, AO in /-Lm, and A in nm, then A 3.3A6 v A [nm]; AO [/-Lm]; v [THz]. (22.1-2) For example, a spectral width v == 1 THz corresponds to A = 1 nm at AO = 0.55 /-Lm, and to 4 nm at AO = 1.1 /-Lm. This relation is illustrated in 22.1-2(b). For ultranarrow pulses with large v, the exact expression for A is ), = C c ),6 1I Vo - v/2 Vo + v/2 C 1- (v/2vO)2. (22.1-3) However, under these conditions, the concept of spectral width loses its significance. A 2-fs pulse, e.g., has spectral width v = 220 THz, corresponding to A = 847 nm at AO == 1 /-Lm, i.e., the spectrum is quite broad and extends from visible through infrared. Instantaneous Frequency Another descriptor of the optical pulse is the time dependence of its instantaneous frequency. The instantaneous angular frequency Wi is the derivative of the phase of U(t), and the instantaneous frequency Vi == Wi/27r, so that dcp W. == W o +- 'I, dt ' 1 dcp Vi == Vo + 27r dt . (22.1-4 ) Instantaneous Frequency
940 CHAPTER 22 ULTRAFAST OPTICS If the phase is a linear function of time, <p ( t) == 21f f t, then the instantaneous frequency Vi == Vo + f; i.e., a linearly varying phase corresponds to a fixed frequency shift. Nonlinear time dependence of the phase corresponds to time-dependent instantaneous frequency. Chirped Pulses A pulse is said to be chirped, or frequency modulated (PM), if its instantaneous frequency is time varying. If Vi is an increasing function of time at the pulse center (t == 0), i.e., <pI! == d 2 <p / dt 2 > 0, then the pulse is said to be up-chirped. If Vi is a decreasing function of time at the pulse center, i.e., <pI! < 0, it is said to be down- chirped. In particular, if the phase of an optical pulse of width 7 is a quadratic function of time <p( t) == at 2 /7 2 , where a is a constant, then <pI! == 2a/ T 2 so that the instantaneous frequency Vi == Vo + (a/ 7rT 2 )t is a linear function of time. The pulse is then said to be linearly chirped and the parameter a == <pl! 7 2 (22.1-5) Chirp Parameter is called the chirp parameter. The pulse is up-chirped if a > 0 and down-chirped if a < O. At t == T /2, the instantaneous frequency increases by a/21fT, which is of the order of magnitude of allv. Thus, the chirp parameter is indicative of the ratio between the instantaneous frequency change at the pulse half-width point and the spectral width llv. Examples of linearly chirped pulses and their instantaneous frequencies are illustrated in Fig. 22.1-3. Re{ u(t)} Re{U(t)} t t --.. N ;g ::r: 400 o ;:300 iSu 200 ......:;j VJO"' Q) tb RB --.. N ::r: 400 o ;: 300 iSu 200 ...... :;j VJO"' Q) Jj B --+- R -20 o 20 t (fs) -20 o 20 t (fs) (a) (b) Figure 22.1-3 Linearly up-chirped and down-chirped optical pulses. (a) An up-chirped pulse has an increasing instantaneous frequency. (b) A down-chirped pulse has a decreasing instantaneous frequency. In this figure, the pulse width is 20 fs and the central frequency Vo == 300 THz. The letters Rand B, which represented red and blue, are generic indicators of long and short wavelengths, respectively. If the dependence of the phase <p on time is an arbitrary nonlinear function, as in Fig. 22.1-1, then it can be approximated by a Taylor-series expansion in the vicinity of the pulse center, and the chirp coefficient a defined by (22.1-5) then represents the lowest-order chirping effect resulting from the quadratic term of the expansion.
22.1 PULSE CHARACTERISTICS 941 Time- Varying Spectrum It is often useful to trace the spectral changes of a time-varying pulse throughout its time course. Such changes are obscured in the Fourier transform, which only provides an average spectral representation of the entire signal without noting which frequen- cies occur at which times. This is particularly evident if the signal is composed of a sequence of segments each with a different spectral composition. A good example is a musical signal for which the spectral changes indicate changes of the musical score as time progresses. While the instantaneous frequency can be a measure of the time-dependent nature of the spectrum, it is not always adequate since it is based only on the phase and ignores the amplitude. A commonly used measure is based on a sliding window, or gate, that selects only one short time segment at a time, and obtains the Fourier transform of the pulse within the window duration. This is repeated at different locations of the sliding window, as illustrated in Fig. 22.1-4, and the result is plotted as a function of both frequency and time delay. The resultant 2D function is called the short-time Fourier transform. Its squared magnitude is called the spectrogram and is often plotted as a picture with the horizontal and vertical axes representing time and frequency, respec- tively, as illustrated in Fig. 22.1-4. Vet) t W(t-7}) I-- T} --1 I V(t)W(t-7}) ! i I i I Fig u re 22.1-4 The short - time "'" Fourier transform of U ( t) is constructed W(t- 72) I-- 72 -----i by a sequence of Fourier transforms of I U ( t) multiplied by a moving window V(t) W(t- 72)1i II j j II. W(t - T). The spectrogram S(v, t) is , I , , " the squared magnitude of these Fourier !. transforms. In this example, U(t) IS v (THz) S(v, t) composed of two Gaussian pulses each 150 of time constant T = 60 fs and central frequency 100 THz. The first pulse is 100 up-chirped (a = 5) and the second is down-chirped (a = -5) and has a smaller 50 amplitude. The window function W (t) I is Gaussian with time constant T = 20 0 100 200 300 t (fs) fs. If W(t) is a window function of short duration T beginning at t == 0, and if U(t) is the pulse wavefunction, then the product U ( t) W (t - T) is a segment of the pulse of duration T beginning at time T. The Fourier transform of the segment is <I>(1I, T) = J U(t)W(t - T) exp( -j27rllt)dt. (22.1-6) Short-Time Fourier Transform The function (v, t) is the short-time Fourier transform and its squared magnitude S(v, t) == I(v, t) 1 2 is the spectrogram.
942 CHAPTER 22 ULTRAFAST OPTICS B. Gaussian and Chirped-Gaussian Pulses Transform-Limited Gaussian Pulse A transform-limited Gaussian pulse has a complex envelope with constant phase and Gaussian magnitude, A( t) == Ao exp( -t 2 /7 2 ), (22.1-7) where 7 is a real time constant. The intensity I(t) == 10 exp( -2t 2 /7 2 ) is also a Gaussian function with peak value 10 == IAoI2, lie full width y127, and FWHM 7FWHM == v 21n27 == 1.187. (22.1-8) The Fourier transform of the complex envelope, A(v) ex exp (-7r 2 7 2 V 2 ), is a Gaussian function, and so is the spectral intensity S(v) ex exp [-27r 2 7 2 (V - vO)2] . The FWHM of the spectral intensity is (22.1-9) v == 0.375/7 == 0.44/7FWHM, (22.1-10) so that the product of the FWHM temporal and spectral widths is 7FWHMV == 0.44. Figure 22.1-5( a) illustrates the temporal and spectral characteristics of the transform- limited Gaussian pulse. As discussed in Appendix A.2, the transform-limited Gaussian pulse has a minimum temporal- and spectral-width product, and this is why it is called transform limited (also called Fourier-transform limited or bandwidth limited). Although the Gaussian pulse has an ideal shape that is not encountered exactly in practice, it is a useful approximation that lends itself to analytical studies. Chirped Gaussian Pulse A more general Gaussian pulse has a complex envelopeA(t) == Ao exp (-at 2 ), where a == (1 - j a) / T 2 is a complex parameter and 7 and a are real parameters, so that A(t) == Ao exp( _t 2 /7 2 ) exp(jat 2 /72). (22.1-11) The magnitude of the complex envelope is a Gaussian function I Ao I exp ( - t 2 /7 2 ) and the intensity is also Gaussian. The phase is a quadratic function cp == at 2 /7 2 so that the instantaneous frequency Vi == Vo + at / 7r7 2 is a linear function of time; i.e., the pulse is linearly chirped with chirp parameter a. The pulse is up-chirped for positive a, down-chirped for negative a, and transform-limited (unchirped) for a == O. The Fourier transform of the complex envelope A(t) == Ao exp (-at 2 ) is proportional to exp ( -7r 2 7 2 v 2 / a), which is also a Gaussian function of frequency. The spectral intensity S(v) is proportional to exp [- 27r 2 7 2 (v - vO)2/ (1 + a 2 )] , which is Gaussian w ith FW HM v == (0.375/7) V l + a 2 == (0.44/7FwHM) V l + a 2 . This is a factor of V I + a 2 greater than that of an unchirped pulse (a = 0) of the same time con stant T . The product of the FWHM temporal and spectral widths is 7FWHMV == 0.44 V l + a 2 , so that the unchirped Gaussian pulse (a = 0) has the least temporal- and spectral-width product. The spectral phase 1/J (v) ex av 2 is a quadratic function of frequency. Key equations characterizing the chirped Gaussian pulse are summarized in Ta- ble 22.1-1. Figure 22.1-5 illustrates the temporal and spectral characteristics of transform-limited and chirped Gaussian pulses.
22.1 PULSE CHARACTERISTICS 943 Table 22.1-1 Temporal and spectral properties of a chirped Gaussian pulse of peak amplitude Ao, peak intensity fo == IAo 1 2 , central frequency Yo, time constant 7, and chirp parameter a. A(t) == Ao exp[ -(1 - ja)t 2 /7 2 ] f(t) == fo exp( -2t 2 /7 2 ) Jf(t)dt == F/2 f 07 71/ e == V27 7pWHM == 1.187 cp( t) == at 2 /7 2 Complex envelope Intensity Energy density 1/ e half width FWHM width Phase (22.1-12) (22.1-13) (22.1-14) (22.1-15) (22.1-16) (22.1-17) A [ 2 2 2 ] A( ) 07 7r 7 V V == exp - . 2V 7r (1 - ja) 1 - Ja S( ) f072 [ 27r272 (v - VO)2 ] V == exp - 47r V l + a 2 1 + a 2 Fourier transform (22.1-18) Spectral intensity (22.1-19) 2 /j.V1/e == - V I + a 2 7 /j.v == 0.375 vI + a 2 == 7 1/ e half width 0.44 VI + a2 7PWHM (22.1- 20) FWHM Spectral width (22.1-21) 'ljJ(v) == -27r 2 7 2 [a/ (1 + a 2 ) ]v 2 Vi == Vo + (a/7r7 2 )t (22.1-22) (22.1-23) Spectral phase Instantaneous frequency Re{ V(t)} (a) Transform- limited pulse (b) Up-chirped pulse (c) Down-chirped pulse t 3 1.5 ] 0.75 0.6 A (/LID) I 'ljJ( v) Figure 22.1-5 Temporal and spectral profiles of three Gaussian pulses of central frequency vo= 300 THz (corresponding to a wavelength of 1 pm and a 3.3-fs optical cycle) and width 7FWHM = 5 fs (7 = 4.23 fs). (a) Transform-limited pulse; the spectral width /j.v == 88 THz ( /j.A = 73 nm). (b) Up- chirped pulse of chirp parameter a == 2; the spectral width is a factor of V I + a 2 == V5 greater than in (a), so that /j.v == 197 THz. The instantaneous frequency is a linearly increasing function of time with value vo= 300 THz at t == 0 (center of the pulse) and values Vi == Vo (1 :t: at / 7rVo 7) == 300( 1 :t: 0.497) THz at t == :t:7. The frequency is swept between 151 THz and 449 THz as t changes from -7 to +7. This corresponds to a change of the wavelength between 0.67 pm and 1.99 pm. (c) Same as in (b) but the pulse is down-chirped with chirp parameter a == -2. f t -10 o 10 t (fs)
944 CHAPTER 22 ULTRAFAST OPTICS C. Spatial Characteristics In this section we examine a few simple examples of pulsed optical waves traveling in free space, or in a linear, homogeneous, and nondisersive medium. In such media, the wavefunction U(r, t) obeys the wave equation \7 U - (1/c 2 )8 2 UI8t 2 == O. The simplest exact solutions of this equation are the pulsed plane wave and the pulsed spherical wave. We will discuss these solutions and also introduce the pulsed Gaussian beam. A more detailed study of the spatial properties of pulsed light is deferred to Sec. 22.4. Pulsed Plane Wave A pulsed plane wave traveling in the z direction has a complex wavefunction in the form U(r, t) == A(t - zlc) exp[jwo(t - zlc)], where A(t) is an arbitrary function. The corresponding intensity is I(t - zlc), where I(t) == IA(t)1 2 . If the width of I(t) is T, then the traveling pulse occupies a distance z == CT at any time and travels without change at a velocity c, as illustrated in Fig. 22.1-6. Numerical values of the pulse temporal and spatial widths in free space are: Temporal width T Spatial width CT 1 ns I ps 1 fs 1 as 30 cm 0.3 mm 0.3 /Lm 0.3 nm cl1t .. z Figure 22.1-6 The envelope of a plane- wave pulse of width T traveling in the z direction with velocity c. The pulse occupies a distance CT at any time. A pulsed plane wave traveling at an angle () with the z axis has a complex wavefunc- tion U(r, t) == A [t - (xsin() + zcos())/c] exp [-jko(xsin() + zcos())] exp(jwot) and intensity I [t - (x sin () + z cos ()) I c], where I(t) == IA(t) 1 2 . If this intensity is recorded as a function of x and z in a sequence of snapshots (each at a fixed time), then the result is as illustrated in Fig. 22.1-7(a). The bright stripe in each snapshot represents the traveling pulse at a given time. For example, a 100- fs pulse in free space appears as a stripe of width 30 /-Lm. Note that a single vertical line (fixed z) intercepting the stripe in a single snapshot (fixed t) provides a complete record of the pulse temporal profile since it records the function I ( - x sin () I c + constan t). Thus, the temporal profile may be measured by observing the spatial profile of a snapshot of the pulse. This can be utilized for pulse detection, as will be discussed in Sec. 22.5B. Pulsed Spherical Wave Another simple solution of the wave equation is the pulsed spherical wave U(r, t) == (l/r )g(t - r I c) exp[jwo(t - r I c)], where g(t) is an arbitrary function. The pulse travels in the radial directions and its wavefronts are concentric spheres, as illustrated in Fig. 22.1-7(b). At any fixed time, it occupies a spherical shell of radial width CT, where T is the width of g(t). *Paraxial Wave Modulated by Slowly Varying Pulse When the envelope of a pulsed wave varies slowly with time so that it is approximately constant within an optical cycle, it is said to have a slowly varying envelope (SVE). Because of the associated narrow spectral width, v « vo, the spatial behavior is
22.1 PULSE CHARACTERISTICS 945 approximately the same as that of a monochromatic (CW) wave at the central frequency Vo or the wavelength Ao == clvo. The wave may therefore be regarded as a quasi-CW pulsed wave. If the wave is also paraxial (see Sec. 2.2C), it may be expressed in terms of its envelope in the general form U(r, t) == A(r, t) exp( -jkoz) exp(jwot), where the envelope varies slowly with z so that it is a pp roximatel 1 constant within a distance equal to a wavelength Ao == 27r I ko; i.e., the condition 8 AI8z2 « k6A is satisfied. Since the envelope is also slowly varying in time, the approximation 8 2 AI 8t 2 « w6A is also applicable. Under such conditions, the wave equation \l2U - (II c 2 )8 2 U I 8t 2 == o leads to an approximate equation for the envelope, 2 .47r ( 8A 18A ) \l A-J- -+-- ==0 T AD 8 z c 8t ' (22.1-24 ) Paraxial SVE Equation where \l == 8 2 18x 2 + 8 2 18y2 is the transverse Laplacian operator. Equation (22.1- 24) is known as the paraxial SVE equation. For a CW wave, 8AI8t == 0 and (22.1-24) reproduces the paraxial Helmholtz equation (2.2-23). As can be seen by direct substitution, (22.1-24) is satisfied by A(p, z, t) == g(t - zlc)Ao(r), where 9 is an arbitrary function of the retarded time t - zlc and Ao(r) satisfies the paraxial Helmholtz equation \lAo - j(47rIAo)8AoI8z == 0, which is applicable in the CW case. It follows that in this approximation a paraxial wave at the wavelength Ao may be modulated by a slowly varying pulse of arbitrary shape, without altering its spatial behavior. x time x time x time - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - z (a) Plane wave (b) Spherical wave (c) Gaussian beam Figure 22.1-7 (a) Four snapshots (taken at equal time intervals) of a pulsed plane wave traveling at an angle. Each snapshot contains a single line of width CT (in the z direction), where T is the pulse width. The line moves from left to right as the pulsed wave propagates. (b) Same as (a) but for a spherical wave, ( c) Same as (a) but for a Gaussian beam. z z Pulsed Gaussian Beam One of the solutions of the paraxial Helmholtz equation is the Gaussian beam described by (3.1-5). In the pulsed quasi-CW case, the Gaussian beam is given by j Zo ( 7r p2 ) A(p, z, t) == g(t - zlc) . exp -j, . , z + J Zo AO z + J Zo (22.1-25) where g(t) is an arbitrary slowly varying function of the retarded time t - zlc and Zo is the Rayleigh range (also called the diffraction length). In this approximation, except for the retardation effect, there is no coupling between space and time; i.e., the beam
946 CHAPTER 22 ULTRAFAST OPTICS maintains its Gaussian spatial profile at all times, and the pulse maintains its initial temporal profile at all positions. Snapshots of such a beam are illustrated in Fig. 22.1- 7(c). It will be shown in Sec. 22.4 that for ultranarrow pulses, for which the SVE approx- imation is not applicable, space-time coupling can be significant, and a wave that is Gaussian in time and space in a given transverse plane becomes non-Gaussian in both time and space as it propagates in free space. 22.2 PULSE SHAPING AND COMPRESSION The temporal profile of a short optical pulse is unavoidably altered as it travels through a dispersive optical system. This occurs because the spectral components that con- stitute the pulse are attenuated and/or phase shifted by different amounts. The effect of dispersion is more dramatic for ultrashort pulses since they have greater spectral widths. Dispersive optical elements may also be designed to effect desired changes in the pulse shape, e.g., compression or stretching. In this section, we consider only temporal effects, i.e., only pulsed plane waves are considered; Sec. 22.4 deals with spatial effects in linear optical media, including diffraction and beam propagation in dispersive media. The section is also limited to linear dispersive systems; dispersion in nonlinear systems is examined in Sec. 22.5. A. Chirp Filters Linear Filtering of an Optical Pulse The transmission of an optical pulse through an arbitrary linear optical system is gener- ally described by the theory of linear systems (see Appendix B). A linear time-invariant system is characterized by a transfer function H(v), which is the factor by which the Fourier component of the input pulse at frequency v is multiplied to generate the output component at the same frequency. If U 1 (t) and U 2 (t) are the complex wavefunctions of the original and filtered pulses, respectively, then their Fourier transforms Vi (v) and V 2 (v) are related by V 2 (v) == H(v) Vi(v). (22.2-1 ) In using (22.2-1) we only need to know H(v) at frequencies within the spectral band of the pulse, which is a region of width v surrounding the central frequency va, as illustrated in Fig. 22.2-1. When v « va, it is convenient to work with the complex envelope instead of the wavefunction. Using the relation U(t) == A(t) exp(j27rvot) and the shift property of the Fourier transform, V(v) == A(v - va), where A(v) is the Fourier transform of A (t), it follows from (22.2-1) that A 2 (v - va) == H (1/ ) Al (v - va), where the subscripts 1 and 2 denote the input and output pulses, respectively. Defining the frequency difference f == v - va, we obtain A 2 (f) == H(vo + f)Al(f), or A 2 (f) == He(f)Al (f), (22.2-2) where He(f) == H(vo + f) (22.2-3) Envelope Transfer Function is called the envelope transfer function. Working with (22.2-2) is generally more convenient than working with (22.2-1), since the frequency f is typically much smaller than v. These relations are illustrated in Fig. 22.2-1.
22.2 PULSE SHAPING AND COMPRESSION 947 f J7 . c . ,y I ...... Ii -If -', i ..w. o Vo v _A_ vV VV- t Input complex wavefunction U 1 (t) Filter H(v) Output complex wavefunction U2(t) Input complex envelope A I (t) t k we(f) ::J I He(f) I o f Envelope filter He(f) = H(vo + f) Output complex envelope A 2 (t) t Figure 22.2-1 Filtering the wavefunction with a filter H(v) (upper figure) is equivalent to filtering the envelope with a filter He(f) = H(vo + f) (lower figure). The shaded area represents the spectral band of interest. The transfer functions H(v) and He(f) are complex functions, H(v) = IH(v)1 exp[-jw(v)] and He(f) == IHe(f)1 exp[-jwe(f)], where we(f) = w(vo + f) are real functions representing the phase transfer. The phase introduced by the filter often plays a more important role than the magnitude in the reshaping of pulses. Throughout this chapter we will deal with phase filters, i.e., filters for which the magnitude I H(v) I is approximately constant within the frequency range of interest. When transformed to the time domain, (22.2-2) becomes the convolution relation A 2 (t) = I: he(t - t')A 1 (t')dt', (22.2-4 ) where he ( t) is the inverse Fourier transform of He (f). The Ideal Filter An ideal filter preserves the shape of the input pulse envelope; it merely multiplies it by a constant (of magnitude < 1 for an attenuator and> 1 for an amplifier), and possibly delays it by a fixed time. The transfer function has the form He(f) == Ho exp (- j21T fTd) , (22.2-5) where Ho is a constant, G == IHol2 is the intensity reduction or gain factor, and Td is the time delay. The phase is a linear function of frequency We (f) == W 0 + 21TTdf, where Wo == arg{ Ho} is a constant phase [see Fig. 22.2-2(a)]. Using a basic Fourier- transform property (see Appendix A), the phase 21TTdf is equivalent to a time delay Td. The input and output envelopes are related by A 2 (t) == HoAl(t - Td), and the intensities are related by I2(t) == GI 1 (t - Td). For a distributed attenuator/amplifier of attenuation/gain coefficient a, velocity c, and length d, the transfer function is He(f) == exp (-ad /2) exp (- j21T f d / c) so that G == exp (-ad) and Td == d / c. A slab of ideal nondispersive material with attenuation coefficient a and refractive index n is an example of such filter, where c == co/no Here, the transfer function H(zJ) == exp (-ad/2) exp (-j(3d), where (3 = 21TV/C is the propagation constant (see Sec. 5.5A), and He(f) == exp (-ad/2)exp (-j21Tfd/c). When a and n are frequency dependent, i.e., the medium is dispersive, the filter is not ideal and the pulse shape may be significantly altered, as will be shown in Sec. 22.3.
948 CHAPTER 22 ULTRAFAST OPTICS The Chirp Filter Perhaps the most important filter in ultrafast optics is the Gaussian chirp filter, often simply called the chirp filter. It is a phase filter whose phase is a quadratic function of frequency we(f) == b7r 2 f2 [see Fig. 22.2-2(b)] so that the envelope transfer function is Gaussian, He(f) == exp (- jb7r 2 f2) , (22.2-6) Chirp-Filter Transfer Function where b is a real parameter (units of s2) called the chirp coefficient of the filter. For b > 0 the filter is said to be up-chirping, and for b < 0 it is down-chirping. The corresponding impulse-response function is the inverse Fourier transform of (22.2-6) (see Table A.2-1), which is another Gaussian function 1 he(t) = .jJifb exp(jt 2 jb). J7rb (22.2-7) Chirp-Filter Impulse-Response Function It too has a phase that is a quadratic function of time, i.e., it is a linearly chirped function, which is up-chirped for positive b and down-chirped for negative b. A cascade of two chirp filters with coefficients b 1 and b 2 is equivalent to a single chirp filter with coefficient b == b 1 + b 2 , since the transfer functions multiply. Thus, a down-chirping filter may compensate the effect of an up-chirping filter, so that the action of a chirp filter is reversible. \He(f)\----- IHe(f)l- welf ) welf ) f o f (a) Ideal filter (b) Chirp filter Figure 22.2-2 Magnitude and phase of the envelope transfer functions of (a) an ideal filter, and (b) a chirp filter (with b > 0). As can be seen by substituting (22.2-7) into (22.2-4), the pulse envelopes at the output and input of a chirp filter are related by 1 1 00 [ (t - t') 2 ] A 2 (t) = .jJifb -00 Al (t') exp j b dt'. (22.2-8) This transformation is mathematically similar to Fresnel diffraction [see (4.3-12) in Sec. 4.3B], and for a sufficiently large chirp parameter b it becomes similar to Fraun- hofer diffraction, i.e., equivalent to a Fourier transform (See Sec. 4.3A). The analogy between diffraction in space and dispersion in time, which is described by a chirp filter, is formally established in Sec. 22.3D.
22.2 PULSE SHAPING AND COMPRESSION 949 Approximation of Arbitrary Phase Filter by a Chirp Filter When the filter magnitude and phase vary slowly within the narrow spectral width of a pulse, we may assume that the magnitude is approximately constant at its central- frequency value, IH(vo + f) 1 IH(vo)1 IHol, and expand the phase function w(v) in a Taylor series centered at the frequency Yo. Retaining only the first three terms, w(vo + f) Wo + \It' f + ! W" f2, where Wo == \It(vo), W' == d\It / dvl vo , w" == d 2 \It / dv 2 l v o, we obtain H(vo + f) I Ho I exp[ - j (w 0 + \It' f + ! W" f2)]. It follows from (22.2-3) that the envelope transfer function may therefore be ap- proximated by He(f) IHol exp [-j(\It o + W' f + !w" f2)] . (22.2-9) This filter is equivalent to a cascade of an ideal filter and a chirp filter (see Fig. 22.2- 3). The ideal filter is composed of a constant multiplier Ho == IHol exp( -j\It o ), which does not alter the shape of the pulse and may be ignored, and a phase shift exp(-j27rTdf), which is equivalent to a time delay Td == W' /27r. (22.2-10) Group Delay The chirp filter has a transfer function exp ( - j b7r 2 f2) with chirp coefficient \It" b==. 27r (22.2-11 ) Chirp Coefficient .-. ..-' I H e(f) I o f Arbitrary filter Ideal filter Chirp filter Figure 22.2-3 Approximation of an arbitrary filter with slowly varying transfer function as a cascade of an ideal filter (including a time delay) and a chirp filter. \lJ elf ) - - f o f We conclude that the principal source of distortion in a dispersive system with slowly varying phase is described by a chirp filter. Examples of such systems based on angular dispersion and Bragg gratings are presented subsequently in this section. Dispersive media are also described by chirp filters, as will be shown in Sec. 22.3. A more accurate approximation of the phase filter would require the inclusion of additional terms in the Taylor-series expansion of the phase W(v). The third-order term corresponds to a phase filter exp( - j! \It'" f3), and higher-order terms can be similarly defined.
950 CHAPTER 22 ULTRAFAST OPTICS Chirp Filtering of a Transform-Limited Gaussian Pulse We now consider the effect of a chirp filter with a transfer function given by He(f) == exp (-jb7r 2 f2) and chirp coefficient b on an unchirped (transform-limited) Gaussian pulse of complex envelope Al ( t) == A 10 exp ( - t 2 /7f). Since the Fourier transform of Al (t) is Al (f) == (Al07l/2V1f) exp( -7r 2 7f f2), by virtue of (22.2-2) the filtered pulse has a complex envelope with Fourier transform A 2 (f) = Ao T exp[-7f 2 (T{ + jb)f2]. 2y7r (22.2-12) This expression may be cast as the Fourier transform of a chirped Gaussian pulse of width 72 and chirp parameter a2, which, in accordance with (22.1-18), has a Fourier transform 70 ( 7r2T,2 f2 ) A 2 (f) == A 20 2 exp _ 2. . 2 V 7r(1 - ja2) 1 - ]a2 (22.2-13) Equating the exponents in (22.2-12) and (22.2-13), we obtain 2 2. 72 71 +]b == 1 . , - ]a2 (22.2-14) and equating the amplitudes we obtain A 20 == AlO V I - ja2 71/72. Equating the real and imaginary parts of (22.2-14) leads to the expressions that relate the parameters of the output pulse to those of the input pulse: Width T2 = T1 V I + b 2 jT{, Chirp parameter a2 == b / 7f , . A AlO AmplItude 20 == . V I + jb/7f (22.2-15) (22.2-16) (22.2-17) We conclude that upon transmission through a chirp filter, an unchirped Gaussian pulse remains Gaussian and its properties are modified as follows: . The pulse width is increased by a factor V I + a == V I + b 2 /7(. For Ibl == 7f, this factor is V2. Thus, the filter begins to have a significant effect when its chip coefficient is of the order of the squared width of the original pulse. For I bl » 71 ' i.e., for large chirp coefficient or narrow original pulse, 72 I bl /71, indicating that the width of the filtered pulse is directly proportional to I bl and inversely proportional to 71, so that narrower pulses undergo greater broadening. . The initially transform-limited pulse becomes chirped with a chirp parameter a2 that is directly proportional to the filter chirp coefficient b and inversely propor- tional to the square of the original pulse width. The filtered pulse will be up- chirped if b is positive, i.e., if the filter is up-chirping, and will be down-chirped if b is negative, i.e., the filter is down-chirping. For b == 7f, the chirp parameter a2 == 1.
22.2 PULSE SHAPING AND COMPRESSION 951 . The spectral width of the pulse remains unchanged. The original pulse has a spectral widt h IJ == 0.375/T1, and the filtered pulse has an equal spectral width (0.375/T2) J I + a == 0.375/T1 == v. This is not surprising since the chirp filter is a phase filter that does not alter the spectral intensity of the original pulse. The invariance of the spectral width may als o be vi ewed as follows: The temporal width of the pulse is expanded by a factor J I + a, so that the associated spectral width must be compressed by the same factor. However, because the filtered pulse is chirped this is accompanied by a spectral broadening by the very same factor, resulting in an unchanged spectral width. The dependence of the pulse broadening ratio T2 / T1 and the chirp parameter a2 on the ratio b / Tf is illustrated in Fig. 22.2-4. '2/'1 '" 'I 72 2 t t 1 * Chirp filter b o + a2 I 13 2 c..e- :::> . - ..c u o I "'0 !:: Q) e- o .- o -5 -2 t -2 -1 o 1 2 b/ Figure 22.2-4 A chirp filter with coefficient b converts an unchirped Gaussian pulse of width /1, marked by an open circle, into a chirped Gaussian pulse of width /2 and chirp parameter a2. The pulse width increases as Ibl increases, and is greater for smaller /1 . The chirp parameter is directly proportional to b and is greater for smaller /1. Chirp Filtering of a Chirped Gaussian Pulse When a chirped Gaussian pulse is transmitted through a chirp filter, the outcome is also a chirped Gaussian pulse, with altered parameters. The pulse will be either expanded or compressed and its chirp parameter will be altered, and may under certain conditions diminish to zero so that the new pulse may become unchirped (transform limited). This compression property offers a technique for generation of picosecond and femtosecond optic a] pulses, as will be shown in subsequent sections. If the original pulse has width T1, chirp parameter aI, and complex envelope A1(t) == A 10 exp[-(I - ja1)t 2 /Tf], then upon filtering with a chirp filter He(f) == exp ( - j b7r 2 f2), the result is a chirped Gaussian pulse A 2 ( t) == A 20 exp [ - (1 - ja2)t 2 /T?], where 7,2 2 1 - ja2 T 2 1. + jb. 1 - ]a1 (22.2-18) Equating the real and imaginary parts of (22.2-18) leads to the following expressions
952 CHAPTER 22 ULTRAFAST OPTICS for the width 72 and chirp parameter a2: b b 2 72==71 1 + 2a l2"+(I+aI)4' 7 1 7 1 (22.2-19) 2 b a2 == a1 + (1 + a 1 )2". 7 1 (22.2-20) A sketch of the dependence of the pulse broadening ratio 72 171 and the chirp parameter a2 on the ratio b17? is shown in Fig. 22.2-5. To determine the value b min of the filter's chirp parameter at which the filtered pulse has its minimum width TO, we equate the derivative of 72 in (22.2-19) with respect to b to zero. The result is T1 Minimum width 70 == v I + aI ' (22.2-21) Chirp coefficient 2 a1 2 b min == - a 1 70 == - 2 7 1 . 1 +a 1 (22.2-22) Using (22.2-21) and (22.2-22) we rewrite (22.2-19) and (22.2-20) in terms of b min and 70 as follows: Width T2 = To ) 1 + (b - bin)/Tri, Chirp parameter a2 == (b - b min ) 175 . (22.2-23) (22.2-24 ) When b == b min , (22.2-23) and (22.2-24) give 72 == 70 and a2 == 0, so that the pulse is both maximally compressed and unchirped. Based on (22.2-22), if the original pulse is up-chirped (a1 > 0), then b min < 0, so that a down-chirping filter is necessary for maximal compression. If the original pulse is unchirped (al == 0), no chirp filter can compress it further, since it is already at its minimum width (b min == 0 and 70 == 71). Note that (22.2-23) and (22.2-24) are identical to (22.2-15) and (22.2-16), which were derived for the initially unchirped pulse, except that b is replaced by b - b min . Thus, the graphs in Fig. 22.2-4 are also applicable to the case of initially chirped pulse except for a shift in the horizontal direction by the value b min determined from (22.2- 22). EXAMPLE 22.2-1. Compression and Expansion of a Chirped Pulse by Use of a Chirp Filter. (a) A Gaussian pulse of width T1 and negative chirp parameter a1 = -1 is filtered by a chirp filter of coefficient b. The filtered pulse is also Gaussian and has width T2 and chirp parameter a2. In this case, the filtered pulse becomes maxim ally compressed and unchirped when b = b min = T; and the compression factor is V I + a = V2, so that the compressed pulse width TO = T1/V2. The normalized pulse width T2/TO is plotted in Fig. 22.2-5(a) versus the ratio b/T?;. For small positive values of b, the pulse is compressed and acquires positive chirp. It becomes maximally compressed (and unchirped) when biT?; = 1 (i.e., b/Tf = 0.5). As b increases further, the pulse is expanded. For negative b, the pulse is expanded and acquires additional down-chirp.
22.2 PULSE SHAPING AND COMPRESSION 953 (b) An initial1y up-chirped pulse with chirp parameter a1 == 1 is expanded with the application of an up-chirping filter (b > 0); its chirp parameter a2 > 1. Application of a down-chirping filter (b < 0) results in compression. Maximal compression is achieved at blTf; == -1 (or blTi'. == -0.5), as il1ustrated in Fig. 22.2-5(b). T2/TO 2 T2/TO 2 1 0 2 a2 0 -2 -2 -1 0 1 2 2 -2 -1 0 b/TO (a) (b) 2 o o a2 -2 2 2 b/TO Figure 22.2-5 Filtering a Gaussian pulse of width T1 and chirp parameter a1 with a chirp filter of coefficient b, which is positive/negative in the unshadedlshaded areas. The filtered pulse has width T2 and chirp parameter a2. Parameter s of the original pulse (b == 0) are marked by open circles. The minimum pulse width TO == T11 J l + ar is used for normalization. The upper graphs show the dependence of the normalized pulse width on the ratio b I T5. The lower graphs show the dependence of the chirp parameter a2 on b1T5. Two values of the original chirp parameter are shown: (a) a1 == -1, and (b) a1 == 1. Application: Chirp Pulse Amplifier The amplification of an ultrashort high-peak-power optical pulse is often limited by nonlinear effects such as saturation and self-focusing in the optical amplifier. Such limitations may be alleviated if the pulse is stretched by use of a chirp filter prior to amplification, and compressed by filtering through a second chirp filter after it has been amplified, as illustrated in Fig. 22.2-6. The first filter lowers the peak power by stretching the pulse, while maintaining its total energy. The second chirp filter, which has a chirp parameter of equal magnitude and opposite sign, compresses the pulse back to its original width. Thus, the amplification process is distributed over a longer time duration and the peak power does not exceed the amplifier limits. * -A-- Chirp filter b > 0 Amplifier Chirp filter b < 0 Figure 22.2-6 Chirp pulse amplifier.
954 CHAPTER 22 ULTRAFAST OPTICS B. Implementations of Chirp Filters Chip filters are implemented by use of dispersive optical systems. The following are some of the various origins of dispersion in optical components. . Material dispersion results from the frequency/wavelength dependence of the index of refraction and/or absorption coefficient of optical materials. . Spatial dispersion takes a variety of forms: - Angular dispersion has its origin at the frequency/wavelength dependence of the deflection angle of certain optical components. This is most pro- nounced in diffractive optical elements such as diffraction gratings and holo- graphic optical elements. Refractive elements such as prisms exhibit angular dispersion as a result of their material dispersion. - Multipath dispersion is associated with the existence of multiple paths with different optical pathlengths. An example is modal dispersion in optical waveguides, which results from the different propagation constants of the waveguide modes (Sec. 8.2). - Optical systems dominated by interferometric effects are wavelength depen- dent and therefore exhibit interferometric dispersion. For example, strat- ified media and periodic structures such as Bragg gratings have frequency- dependent reflectance and transmittance. Optical resonators have strong fre- quency selectivity, and are therefore highly dispersive. - Likewise, diffraction from small apertures is wavelength dependent and can therefore be responsible for significant changes in the profiles of short op- tical pulses; this is a form of diffractive dispersion. In general, propaga- tion through, or scattering from, spatial structures or inhomogeneities of size comparable to a wavelength contribute to this type of dispersion. Even single-mode waveguides exhibit waveguide dispersion, which is associated with the confinement of light in small structures (see Sec. 8.2). . Polarization dispersion is a result of the wavelength dependence of the anisotropic properties of optical materials, components, and systems. . Nonlinear dispersion also plays an important role in the reshaping of intense optical pulses, because of the wavelength dependence of nonlinear optical effects such as self-phase modulation and parametric interactions governed by frequency- dependent energy conservation and phase-matching conditions. Any of these dispersive effects may be used to implement the chirp filter, as demon- strated by the following examples. Angular-Dispersion Chirp Filters Optical elements that introduce angular dispersion, such as prisms and diffraction grat- ings, may function as chirp filters. A generic such element, illustrated schematically in Fig. 22.2-7(a), disperses the monochromatic components that constitute a pulsed plane-wave into different directions. Assume that the component with frequency v is directed at an angle B (v) measured from the direction of the component at the central frequency va, i.e., B(vo) == O. If £0 is the optical pathlength of the central- frequency component, then the optical pathlength of the component at frequency v is £ocosB(v), as can be seen from Fig. 22.2-7(a). The phase shift encountered by the spectral component v is 27rv w(v) == -£0 cos B(v), c (22.2- 25) and the corresponding phase filter has a transfer function H(v) == exp[-j\l1(v)].
22.2 PULSE SHAPING AND COMPRESSION 955 A pulsed beam is typically filtered by use of four identical dispersive elements arranged as shown in Fig. 22.2-7(b). One element separates the spectral components of the optical pulse into separate directions. A second inverted element brings back the rays into parallelism, as illustrated in the left block of Fig. 22.2-7 (b). The process is reversed by two identical elements in the reverse order, as illustrated in the right block of the figure. The overall system is a phase filter with 'l1(v) == (27rvjc)£ocosB(v), where £0 is the overall optical pathlength of the central-frequency component. p Original pulse v v' Filtered pulse .... ------- vo (a) (b) Chirp Filter Figure 22.2-7 (a) An optical element exhibiting angular dispersion. The component at frequency v is separated from that at the central frequency Vo by a deflection angle B (v). At the observation point Po, the path length of the central-frequency comp onent is .eo (distance PP o ). The pathlength of the component at frequency v is the distance PP 1, where PI is det ermin ed by lining up the wavefront to pass through the observation point Po. Therefore, the distance PP 1 in the triangle PP1P O is .eo cos B(v). (b) A chirp filter made with a combination of four of the elements in (a). The function B(v) depends on the dispersive element used, as will be shown in subsequent examples. Typically, B(v) is sufficiently small so that cas B(v) 1 - !B2(v) and 27rIJ [ 1 2 ] 'l1(v) -£0 1 - "2B (v) . c (22.2- 26) If B(v) is slowly varying within the pulse spectral width, then it may be approximated by a few terms of a Taylor-series expansion about the central frequency va. The deriva- tives of 'l1(v) evaluated at v == va, where B(vo) == 0, are: \]i' 27f .eo, \]i" _ 27fll .eo ( dB ) 2 . (22.2- 27) c c dv Based on (22.2-10) and (22.2-11), the filter is equivalent to a time delay T d == £0 j c and a chirp filter with chirp coefficient £0 2 b--Q \ v' 7r Ao (22.2-28) Angular-Dispersion Chirp Coefficient where Qv == dB j dv is the angular dispersion coefficient. Since b is always negative in this approximation, regardless of the sign of Qv, such filters are always down-chirping. Higher-order terms of the series expansion of the phase do, of course, introduce addi- tional pulse shaping effects. EXAMPLE 22.2-2. Prism Chirp Filter. The angle of deflection Bd(V) of a ray incident on a prism is a function of the refraction geometry and the refractive index n(v) (see Fig. 22.2-8). Since B(v) == Bd(V) - Bd(VO) the angular dispersion coefficient £Xv == dB/dv == (dBd/dn)(dn/dv). Using
956 CHAPTER 22 ULTRAFAST OPTICS the relations dn / dv = - (Ao / vo) dn / dAo = (n - N) / Vo, where N = n - Aodn / dAo is the group index of the material (see Sec. 5.6), we obtain n - N d(}d Vo dn For a thin prism with apex angle Q, the deflection angle (}d = (n - l)Q [see (1.2-7)] so that d()d /dn = Q and Qv = (22.2-29) n-N Qv = Q. Vo (22.2-30) As an example, for BK7 glass at wavelength Ao = 800 nm, n = 1.51 and N = 3.11. For a prism with Q = 15°, Qv = 1.11 X 10- 15 = 1.11 fs. For £0 = 1 cm, the chirp coefficient given by (22.2-28) is b = -5 X 10- 27 S2 - (71 fs)2. In accordance with (22.2-15) and (22.2-16), an unchirped pulse of width 71 = 50 fs transmitted through this device is broadened by a factor (1 + b 2 /7{)1/2 2.23 and becomes chirped with chirp parameter a2 = b / T = 2. Original pulse v Filtered pulse Vo Figure 22.2-8 Prism chirp filter. EXAMPLE 22.2-3. Diffraction-Grating Chirp Filter. In a diffraction grating system (Fig. 22.2-9) the angles of incidence and diffraction, (}1 and ()2, from a grating with period A are related by the diffraction condition (2.4-13). If (}2 = (}20 + (}(v), where (}20 is the angle of the central-frequency component, then for first-order diffraction, sin 0 1 + sin[02o + O(v)] = = v . (22.2-31) Taking the derivatives of both sides at v = Vo, we obtain d(} -c -A 2 o Qv = - = 2 = . dv V o A cas (}20 cA cos (}20 In the symmetrical case in which (}1 = (}20, sin (}20 = Ao/2A, and therefore 1 Ao Q =-- v Vo vi A2 - (Ao/2)2 Ao£O A b = - 7rC 2 A2 - (Ao/2)2 . For Ao = 800 nm and A = 1.6 /Lm, Qv = -2.72 x 10- 15 s = -2.72 fs. For eo -2.94 x 10- 25 = -(542 fs)2. so that Original pulse ./ / , '\ ' I --..........,.........._-_... (22.2-32) (22.2- 33 ) (22.2- 34) 10 em, b = Filtered pulse Figure 22.2-9 The diffraction grating as a down-chirping filter.
22.2 PULSE SHAPING AND COMPRESSION 957 Bragg-Grating Chirp Filters Variable-pitch (or chirped) Bragg gratings (Fig. 22.2-10) are often used as chirp fil- ters. As described in Sec. 7.1 C, a Bragg grating is a periodic structure that reflects optical waves selectively. A grating with period A reflects only waves with wave- length A satisfying the Bragg condition A == mA/2, where m is an integer; waves at other wavelengths are transmitted without change. The grating can therefore serve as a narrowband filter. If the grating has a pitch that varies with position, then each segment of the grating reflects the wave with a wavelength matching the local pitch. The reflected waves travel different distances depending on the location from which they are reflected, so that the system acts as a frequency-sensitive phase filter. If the frequency of the periodic structure varies linearly with distance, the grating is said to be linearly chirped, and it functions as a linear chirp filter. A(z) I j Figure 22.2-10 A Bragg grating with decreasing period serves as a positive chirp filter. o d z Assume that the period of a Bragg grating is a function A(z) of the position z selected such that the frequency varies linearly with z, i.e., A -l(z) == Al + z where Ao is the period at z == 0 and is a constant. To determine the effect of the grating on an optical pulse, we decompose the pulse into its spectral components and examine the effect of the grating on each component. The component of frequency v is reflected from the grating at the location z for which A == mA/2, i.e., A(z) == mA/2 == mc/2v or z == 2v /mc - 1/ Ao. That component travels a distance 2z and undergoes a phase shift W == (27rv / c)(2z) so that w == (87r /mc2)v2 + (47r / cAo) v. (22.2-35) It follows from (22.2-10) and (22.2-11) that the chirped Bragg grating is equivalent to a time delay Td == 2/ cAo and a chirp filter with chirp coefficient b == 8 m7rc2 . (22.2-36) Bragg-Grating Chirp Coefficient If > 0, i.e., the grating has an increasing frequency, as illustrated in Fig. 22.2-10, and the chirp coefficient b > 0, i.e., the filter is up-chirping. Likewise, a chirped Bragg grating with a decreasing frequency is a down-chirping filter. C. Pulse Compression A transform-limited pulse cannot be compressed by use of a chirp filter. Such a filter expands and chirps the pulse, but does not alter its spectral width. However, compres- sion may be accomplished by use of a combination of phase modulation followed by a chirp filter. The phase modulator multiplies the pulse by a time-dependent phase factor, which introduces chirp accompanied by spectral broadening but does not alter the temporal width. The chirped pulse may be subsequently compressed by use of a
958 CHAPTER 22 ULTRAFAST OPTICS chirp filter, which maintains the new spectral width while compressing the temporal width as it generates a new transform-limited compressed pulse. To compress an unchirped pulse A(t) == Aa exp( -t 2 ITf), we first convert it into a chirped pulse by multiplication with a quadratic phase factor exp(j(t 2 ), where ( is a constant, using a quadratic hase modulator (QPM). The result is a chirped pulse Al (t) == Ala exp[-(I - jal)t ITf] with chirp parameter al == (Tf. (22.2-37) If ( > 0, the pulse becomes up-chirped, and subsequent filtering with a down-chirping filter can result in compression. Alternatively, if ( < 0, the pulse becomes down- chirped and subsequent filtering with an up-chirpin g filter can r esult in c ompression. In either case, the pulse is compressed by a factor J I + ai == J I + (2T{. The system is illustrated in Fig. 22.2-11. QPM Chirp filter _jb7r 2 j2 e Transform- limited t pulse ej(p Chirped pulse Compressed transform-limited b = b min = -a}7 /(l+ai) pulse x Pulse width: 7} Chirp parameter: 0 Spectral width: I1v 71 2 al = (7} I1v. V 1 +al . - 70=7} / .J l+al ; o I1v. V 1 +al Figure 22.2-11 Compression of a transform-limited pulse by use of a quadratic phase modulator (QPM) followed by a chirp filter. If the original pulse is a chirped pulse Al (t) == Ala exp[-(l - jal)t2/Tf], then modulation by a quadratic phase exp(j(t 2 ) converts it into another chirped pulse A 2 (t) == Ala exp[-(l - ja2)t2/Tf] with the same width but with an altered chirp parameter a2 == al + (Tf. (22.2-38) Effect of QPM on Chirp Parameter Thus, a quadratic phase modulator for which the sign of ( is opposite to that of al may unchirp the initial pulse or even reverse its chirp sign. Summary The quadratic phase modulator (QPM) and the chirp filter serve dual functions. One operation is the Fourier-transform analog of the other: QPM = multiplication by a Alters spectra] width Preserves temporal width quadratic phase function Chirp = convolution with a Preserves spectral width Alters temporal width Filter quadratic phase function
22.2 PULSE SHAPING AND COMPRESSION 959 QPMs may be implemented by use of electro-optic modulators (see Sec. 20.1B), although the production of the appropriate signal exp(j(t 2 ) is not simple. Passive phase modulation occurs when intense pulses are transmitted through nonlinear media exhibiting the optical Kerr effect, as will be described in Sec. 22.5C in connection with self-phase modulation, and this effect may be used to implement QPMs. D. Pulse Shaping The pulse-shaping methods discussed so far are based on chirp filters implemented by dispersive optical components. Although chirp filters can be used for pulse stretching and compression, they cannot be used to alter the pulse shape in an arbitrary manner or to generate pulses of prescribed shape, as is often necessary in optical communi- cation and signal processing applications. General shaping of ultrafast pulses can be accomplished by use of optical frequency-to-space mapping or time-to-space mapping, together with spatial modulation, as described in this section. Frequency-fa-Space Mapping Frequency-to-space mapping of an optical pulse is achieved by means of a diffraction grating and a lens, which direct each constituent spectral component to a unique point in the lens's focal plane, as illustrated in the left side of Fig. 22.2-12. This system in effect projects the Fourier transform of the temporal profile of the pulse as a spatial pattern in the focal plane. A modulator modifies the magnitude and phase in accordance with the transfer function of the desired pulse-shaping linear filter. This is accom- plished by use of a microlithographic or holographic mask, or a programmable spatial light modulator (SLM) (see Sec. 20.3B). The inverse operation of spatial-spectral mapping is subsequently implemented by a second lens and grating, which recombine the modified spectral components to form the reshaped pulse. This amounts to an inverse Fourier transform, and the overall operation is similar to spatial filtering in Fourier optics (see Sec. 4.4B). This technique has become an established tool for general shaping of ultrafast pulses. .. f f ) SL: Lens I 1 f ) Grating II : B( v Lens /' '. > ./. Y.. J> m ! l/ x Focal plane Filtev d pulse t t Figure 22.2-12 A system for pulse shaping includes: (1) frequency-to-space mapping - a grating and a lens display the Fourier transform of the pulse as a spatial pattern in the Fourier plane; (2) modulation by a spatial light modulator (SLM); and (3) space-to-frequency mapping using a lens and a grating generating the inverse Fourier transform. The system depicted in Fig. 22.2-12 is described quantitatively as follows. If {}(v) is the deflection angle introduced by the grating at frequency v, then the Fourier component at that frequency will be focused at a position x == {}(v)f in the lens focal plane (the Fourier plane), where f is the lens focal length and the angle is assumed to be small. A mask with amplitude transmittance p( x) is therefore equivalent to a filter with transfer function H(v) == p[{}(v)fJ. If {}(v) is approximated by a linear function
960 CHAPTER 22 ULTRAFAST OPTICS of frequency, e( v) avv, where a v == de / dv is the angular dispersion coefficient of the grating [given by (22.2-32)], then the shape of the filter transfer function H(v) is a scaled version of the profile of the mask function p( x), i.e., H(v) == p(av!v). (22.2-39) In this frequency-to-space mapping, the position x in the Fourier plane corresponds to the frequency v == x / a v !, and the spectral width v extends over a width X == a v ! v. The preceding simplified analysis was based on the assumption that the original pulse is a plane wave, so that diffraction plays no role. For an original beam of finite width W in the plane of the grating, the spectral component at frequency v is deflected by an angle e(v) avv, but has an angular spread proportional to A/W == c/vW, which corresponds to a spatial spread 6x == ! Ao/W == c! /vW. This frequency- dependent spread limits the spatial resolution of the system. A mask of total width X has approximately M == X / 6x == X / (Ao! /W) independent points, where Ao is the central wavelength. The spatial spread 6x corresponds to a spectral spread 6v == (Ao! /W) / (a v !) == AO/ (a v W). This limits the spectral resolution of the pulse filtering system to M == XW / Ao! independent points. The reshaping of picosecond and femtosecond pulses has been successfully demon- strated using a number of SLM technologies, including deformable mirrors, multi- element liquid crystal modulator arrays (millisecond to submillisecond response times, high duty cycle), acousto-optic deflectors (microsecond reprogramming, low duty cy- cle), and semiconductor optoelectronic modulator arrays (nanosecond reprogramming times) . Time-fa-Space Mapping Another configuration for arbitrary pulse shaping uses a spatial light modulator (SLM) butted against a diffraction grating and followed by a 2-! lens system with an on- axis pinhole in the Fourier plane, as illustrated in Fig. 22.2-13. The grating multiplies the spectral component of frequency v, which has complex envelope Al (v), by the frequency-dependent position-dependent phase factor exp(j27r1'vx), where l' is a con- stant. The SLM modulates it by a controllable spatial pattern p( x), and the lens system functions as a spatial integrator producing an amplitude A2(v) ex Al (v) J p(x) exp(j27r"(vx) dx ex AI(V)P( -"(v), (22.2-40) where P(ZJ x ) is the spatial Fourier transform of p(x). The overall system therefore acts as a linear system with transfer function H(v) ex P( -1'v), which corresponds to an impulse response function h(t) ex p (-t/1'). (22.2-41 ) It follows that the transmittance of the SLM at the position x controls the value of the impulse response function at one-and-only-one time t == -1'X. Thus, the system serves as a direct time-to-space mapping that may be exploited to reshape or synthesize a femtosecond pulse with arbitrary temporal profile. 22.3 PULSE PROPAGATION IN OPTICAL FIBERS This section examines the propagation of an optical pulse in an extended linear dis- persive medium, such as an optical fiber, by regarding the process of propagation as
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 961 ... f -I... f p(x) . Filtered JEinm 1---- pulse pulse -II " t t > - x P 11111111' . SLM Grating Lens Figure 22.2-13 Pulse shaping based on time-to-space mapping. The system has an impulse response function h(t) that is a scaled version of the SLM transmittance function p(x). a linear filter with a transfer function governed by the frequency-dependent propa- gation constant. For pulses with a slowly varying envelope (e.g., picosecond optical pulses), the filter may be approximated by a combination of a time delay and a chirp filter. The mathematics of pulse propagation will therefore be based on the analysis in Sec. 22.2A. A differential equation describing the evolution of the pulse envelope as it travels through the medium will be derived and an analogy between this dispersion phenomenon and ordinary optical diffraction will be established. A. The Optical Fiber as a Chirp Filter The Dispersive Medium as a Filter Upon propagation in a linear lossless dispersive medium, a monochromatic plane wave of frequency v traveling a distance z in the z direction (Fig. 22.3-1) undergoes a phase shift (3(v)z, where (3(v) == 27rvn(v) / Co is the propagation constant, and n(v) is the refractive index. Propagation is therefore mathematically equivalent to multiplication by the phase factor exp[-j(3(v)z]. Since a pulsed wave of wavefunction U(z, t) is a superposition of many monochromatic waves, the phase factor H (1/) == exp [ - j (3 (v) z] is the transfer function of the linear system that represents propagation, i.e., V (z, v) == H(v)V(O, v) == exp [-j(3(v)z] V(O, v), where V(z, v) is the Fourier transform of U(z, t). For pulses with narrow spectral distribution, the complex wavefunction is writ- ten in terms of the complex envelope, U(z, t) == A(z, t) exp( -j(3oz) exp(j27rvot), where Vo is the central frequency and (30 == (3 (vo). In the Fourier domain, this trans- lates to V(z, v) == A(z, v - Yo) exp (-j(3oz), and hence the relation V(z, v) == exp [-j(3(v)] V(O, v) becomes A(z, v - Yo) == A(O, v - Yo) exp( -j[(3(v) - (3(vo)]z). In terms of the frequency difference f == v - Yo, A(z, f) == He(f)A(O, f), (22.3-1 ) where He(f) == exp { - j[(3(vo + f) - (3(vo)]z} (22.3-2) Envelope Transfer Function is the envelope transfer function. The effect of the dispersive medium on the pulse envelope is therefore modeled as a phase filter He(f) == exp[-jw(f)] with phase w(f) == [(3(vo + f) - (3(vo)]z.
962 CHAPTER 22 ULTRAFAST OPTICS I o A 1\ (0, t) I \ I I I Dispersive medium A(O, t) p HeW A(z, t) -----.. t Figure 22.3-1 Transmission of an optical pulse through a dispersive medium is equivalent to a phase filter. Approximation of a Dispersive Medium by Time Delay and Chirp Filter If the propagation constant (3(v) varies slowly within the pulse spectral width, we may use the results of the Taylor-series expansion W (f) w' f + ! W" f2, described in Sec. 22.2A, where W' and W" are the first and second derivatives of w(v) with respect to v at vo, and W(O) == O. The envelope transfer function can then be approximated by He(f) exp[ - j(w' f + ! W" f2)] == exp( - j21TTdf) exp( - jb1T 2 f2), (22.3-3) where Td == W' /21T and b == W" /21T2. It follows that the process of pulse propagation is equivalent to a combination of a time delay and a chirp filter. The factor exp(-j21TTdf) is a time delay Td == W'/21T == (1/21T)(d(3/dv)z == (d(3/dw)z == z/v, called the group delay, where v == 1/(3' == Co N (22.3-4 ) Group Velocity ((3' == d(3 / dw ) is the group velocity, and N == n - Aodn/ dAo is the group index. These parameters have been previously defined in the simplified analysis provided in Sec. 5.6. The factor exp( - jb1T 2 f2) represents a chirp filter with chirp coefficient b == W" /21T2 == (1/21T2) (d 2 (3/ dv 2 )z == 2(3" z, where (3" == d 2 (3/ dw 2 . The chirp coefficient is proportional to the distance z and is usually written in the form b = 2{3" z = Dv z, 1T (22.3-5) Chirp Coefficient where " d ( 1 ) Ad2n Dv = 21T{3 = dv v = c d>" (22.3-6) GVD Coefficient ((3" == d 2 (3/ dw 2 ) is the group velocity dispersion (GVD) coefficient. It is the derivative of the group delay per unit length with respect to the frequency v, as described previously in Sec.5.6. A medium with (3" > 0 (or Dv > 0) is said to have normal dispersion or positive
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 963 GVD, and it functions as an up-chirping chirp filter (b > 0). Conversely, a medium with (3" < 0 (or Dv < 0) is said to have anomalous dispersion or negative GVD, and it corresponds to a down-chirping filter (b < 0). EXAMPLE 22.3-1. Adjustable Chirp Filter Using Combined Angular and Material Dis- persion in a Prism. In Example 22.2-1, it was shown that when a pulsed beam is refracted by a prism, a chirping effect is introduced as a result of angular dispersion. In this example, we consider the effect of material dispersion, which was neglected before. If the central ray crossing the prism in Fig. 22.2-9 travels a distance L through the prism, then material dispersion amounts to a chirp filter with chirp coefficient b == 2{3" L == DvL/7r [see (22.3-5)]. For a prism made of BK7 glass at A = 800 nm, the dispersion coefficient Dv == 0.284 X 10- 24 s2/m so that for L = 1 cm, the chirp coefficient b == DvL/7r == +9 x 10- 28 S2 == +(30 fs)2. In Example 22.2-1, it was shown that the chirp coefficient due to angular dispersion for a thin prism with 15° apex angle is b -(71 fs)2. The total chirp coefficient is the sum of the contributions of material and angular dispersion. In this case, the net value of b is negative. The distance L can be adjusted by moving the prism in a direction orthogonal to its base, as illustrated in Fig. 22.3-2. < Figure 22.3-2 Prism chirp filter with adjustable chirp coefficient. IfIi Summary The propagation of a pulse in a dispersive medium may be approximated by two effects: a time delay associated with the group velocity v ::= 1/(3' == c o / N > and a chirp filter with chirp parameter b - 2(3" Z == Dvz / 7r proportional to , the propagation distance z. The parameters {3' and [3" are the derivatives of the . propagation constant (3 with respect to the angular frequency w, and Dv == 27r!3" is the GVD coefficient. B. Propagation of a Gaussian Pulse in an Optical Fiber Since a linear dispersive medium may be approximated by a time delay and a chirp filter, we may readily describe the propagation of a Gaussian pulse in such a medium by use of the general results of Sec. 22.2A. Transform-Limited-Gaussian Input Pulse Consider first a transform-limited (unchirped) Gaussian pulse of width TO at Z = O. At a distance z the pulse is delayed by a time Td == Z / v and is filtered by a chirp filter of chirp coefficient b == Dvz / 7r. In accordance with (22.2-15)-(22.2-17), the pulse remains Gaussian, but its width increases to T(Z) == To[l + (D/7r2T6)Z2]1/2, and it becomes chirped with chirp parameter a(z) == (Dv/7rT{;)Z and amplitude Ao(z) ==
964 CHAPTER 22 ULTRAFAST OPTICS Ao[l - j(Dv/1rT6)z]-1/2. By defining the parameter Zo such that Dv/1rT6 == l/zo, these equations may be expressed in the simpler forms provided in Table 22.3-1, which also includes an expression for the complex envelope based on (22.1-12). The magnitude of Zo is called the dispersion length and is a characteristic of the medium and the initial pulse width. The following observations emerge: Table 22.3-1 Characteristics of a Gaussian pulse traveling through a dispersive medium with group velocity v, dispersion coefficient Dv, and dispersion parameter Zo0 At Z == 0 the pulse is transform- limited with width TO, amplitude Ao, and intensity 10 == IAo 1 2 . A( ) - A {2!£ "Zo [ . 7r (t - Z/V)2 ] Z, t - 0 " exp J - " Z - JZ Dv Z - JZo TO [ (t - Z / v) 2 ] 1(z, t) == 1o exp -2 T 2 (Z) J 1(t)dt == v;/2 1 oTo T ( z) == TO vI + (z / Zo ) 2 a ( z) == Z / Zo Complex envelope (22.3-7) 7,2 7,2 Z == 7r == o Dv 2{3" Intensity (22.3-8) Energy density (22.3-9) Pulse width (22.3-10) Chirp parameter (22.3-11 ) Dispersion length I Zo I (22.3-12) !J.v == 0.375 TO Spectral width (22.3-13) . The pulse center is delayed by a time z / v; i.e., the pulse travels with the group velocity v == 1/ (3'. . The width of the pulse T(Z) has its minimum value TO at z == 0 and increases with increasing Izl, as illustrated in Fig. 22.3-3. At z == Izo\ the pulse expands by a fac- tor of V2, and at z == V3lzol its width doubles. For z » Izol, T(Z) Toz/izol == (IDvl/1rTo)z; i.e., the pulse expands linearly at a rate inversely proportional to its initial pulse width, TO. In terms of the spectral width 6.v == 0.375/ TO, the pulse width behaves as T(Z) (1/0.3751r) IDvl6.v z == 0.85IDvl6.v z, which is consistent with the fact that Dv is the pulse broadening rate per unit distance per unit spectral width (s/m-Hz). This relation may also be written in terms of the dispersion coefficient D A [ps/km-nm] as T(Z) 0.85 IDA I 6.,,\ z, which is an approximate version of (5.6-8). . The chirp parameter a( z) is 0 at z == 0, by definition, and increases linearly with the distance z, reaching a magnitude of unity at z == Izol, as illustrated in Fig. 22.3-3. The chirp sign is the same as the sign of Dv. For normal dispersion, Dv > 0 and a( z) > 0 for z > 0, meaning that the pulse is up-chirped. In the visible region, normal dispersion means that "blue" is slower than "red," which is consistent with an up-chirped pulse. The opposite occurs for anomalous dispersion. . The dispersion length I Zo I depends on the magnitude of the medium dispersion coefficient Dv and the initial pulse width TO. It is the distance at which the pulse width increases by a factor of V2 and the chirp parameter reaches a magnitude of unity. . The spectral width 6.v == 0.375/TO remains the same as the pulse travels. The spectral compression that accompanies temporal expansion of the pulse is fully compensated by an equal spectral broadening that accompanies chirping. This is
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 965 to be expected since propagation in the dispersive medium is modeled as a phase filter, which does not alter the spectral intensity. . The energy density carried by the pulse is independent of z, as one would expect in a lossless medium. Unchirped pulse width TO Up-chirped pulse width T(z) R Medium with positive aVD 3 ..............2 ,,-.., N t:::' o .............. 3 -w 2 Z/Zo 2 3 Figure 22.3-3 Propagation of an initially unchirped Gaussian pulse through a dispersive medium. The pulse remains Gaussian, but its width 7(Z) expands, and it becomes chirped with an increasing chirp parameter a ( Z ) . EXAMPLE 22.3-2. Pulse Broadening in BK7 Glass. The dispersion coefficient of BK7 glass at A= 620 nm is {3" = 1.02 X 10- 25 s2/m. For a slab of thickness 1 cm, this corresponds to a chirp coefficient b = 2{3" Z = 2.04 X 10- 23 s2/m = (4.5 pS)2. This means that when a Gaussian pulse of width 4.5 ps crosses the slab, its width expands by a factor of V2. For a shorter Gaussian pulse of time constant 70 = 100 fs and central wavelength Ao= 620 nm, the dispersion length is Izo I = 75 /21{3" 1= 0.5 mm. The pulse doubles its width upon crossing a slab of thickness J3 Zo mm = 0.87 mm. Chirped-Gaussian Input Pulse Based on (22.2-21), upon propagation through the dispersive medium a chirped- Gaussian pulse of width T1 and chirp parameter a1 at Z = 0 reaches a minimum width T1 (22.3-14 ) Minimum Width TO == J l + ai at a distance Zmin for which (Dv/7r)Zmin == b min . From (22.2-22), 7r 2 Zmin == -a1 -TO, Dv (22.3-15) which may be written in terms of the dispersion parameter Zo == 7rT6 / Dv as Zmin == -a1 Z 0. (22.3-16) Location of Minimum Width
966 CHAPTER 22 ULTRAFAST OPTICS Finally (22.2-23) and (22.2-24) translate to the following expressions for the pulse width and chirp parameter as functions of the distance z, T(Z) = TO V I + (z - Zmin)2 / z6, a(z) == (z - Zmin) / Zoo (22.3-17) (22.3-18) Equations (22.3-17) and (22.3-18) are identical to (22.3-10) and (22.3-11) for the initially unchirped case, except for a shift by a distance Zmin, which is the location of the minimum width. The expressions in Table 22.3-1 are therefore universally valid for both positive or negative Z and may be used for initially chirped pulses by placing the beginning of the medium at the location Z corresponding to the matching value of the initial chirp parameter. This is illustrated in Fig. 22.3-4, which is another plot of T(Z) and a(z) based on (22.3-10) and (22.3-11) for positive and negative values of z. As an example, for a medium with positive Zo (positive GVD, or normal dispersion), when the initial chirp parameter is al == -1, then Zmin == Zo, so that the medium begins at the position Z == - ZOo The process of pulse compression and subsequent spreading is now clear. The pulse is maximally compressed by a factor of V2 and becomes unchirped at a distance Zmin == Z00 Upon further propagation through the medium the pulse is broadened and becomes up-chirped. Down-chirped pulse width T 1 B Unchirped pulse width TO Up-chirped pulse width T( z) B I o I :Zmin I GVD ( + ) L_+- Z T(Z) / TO 2 / / T2 = bT} / -3 -2 -1 o 1 2 Z/ZO Figure 22.3-4 Propagation of an initially down-chirped Gaus- sian pulse (a1 - -1) through a medium with norma] dispersion. The pulse width 7 ( Z ) decreases from an initial value of 71 to a mInImum 70 - 71/ V2, and sub- sequently Increases. The initially negative chirp parameter increases linear! y and reverses sIgn when Z > Zmin. In this example Zmin = Zoo / o a(z) 2. o -2.... . Since the initial chirp parameter al and the dispersion coefficient Dv may be posi- tive or negative, we have a number of possibilities:
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 967 . For a medium with normal dispersion (Dv > 0) the filter is up-chirping and the parameter Zo is positive. For an initially down-chirped pulse (a1 < 0), Zmin is positive so that the pulse is indeed compressed as it travels in the positive Z direction. For an initially up-chirped pulse (a1 > 0), Zmin is negative and the pulse will not be compressed. . For a medium with anomalous dispersion (Dv < 0) the filter is down-chirping and the parameter Zo is negative. For an initially up-chirped pulse (a1 > 0), Zmin is positive so that the pulse is indeed compressed as it travels in the positive Z direction. For an initially down-chirped pulse (a1 < 0), Zmin is negative, so that the pulse will not be compressed. In summary, compression can occur if an up-chirped pulse travels through a down- chirping (anomalous) medium, or if a down-chirped pulse travels through an up- chirping (normal) medium. Pulse Compression by Use of a QPM and a Dispersive Medium As described in Sec. 22.2A, a transform-limited pulse may be compressed by use of a combination of a quadratic phase modulator (QPM) and a chirp filter. The chirp filter may be implemented by a dispersive medium, as illustrated in Fig. 22.3-5. If the width of the initial pulse is 71, then modulation by the phase factor exp (j (t 2 ) is equivalent to a chirp co efficien t a1 (7f. The spectral width of the chirped pulse is expanded by the factor V I + aI. If ( is negative, the pulse is down-chirped, and subsequent travel through a medium with positive GVD (normal dispersion) results in pulse compression to a minimum width 71 71 v I + (27{ (22.3-19) 70 v I + ai The pulse will also be compressed if ( is positive and the medium has negative GVD. Using (22.3-16) and (22.3-19), we conclude that the minimum width occurs at a distance 2 2 7r70 7r 7 1 a1 Zmin -a1 Z 0 --a1 -- 2 Dv Dv 1 + a 1 7r( 7{ -- Dv 1 + (27{ , (22.3-20) which is positive if ( and Dv have opposite signs. Ak lit GVD (+) QPM (-) T(z) 71 70 (l)..c: t/'J ... -'"0 :;j.- o a(z) (l) e- :E 0 u 0.. al z z Figure 22.3-5 Pulse compression by a quadratic phase modulator (QPM) and a medium with group velocity disper- sion (GVD).
968 CHAPTER 22 ULTRAFAST OPTICS In the limit when a1 == (7f » 1, 71 1 70 - == - a1 (71 (22.3-21) and Zmin f, where 7r f = _( Dv . (22.3-22) This distance may be regarded as the focal length of this pulse focusing system. EXAMPLE 22.3-3. Pulse Compression in a Silica-Glass Optical Fiber. (a) A Gaussian pulse of time constant T = 100 fs and central wavelength AO= 850 nm (generated, e.g., by a Ti-sapphire laser) travels through a silica-glass optical fiber. At this wavelength, silica glass has normal dispersion (positive GVD) with D).. == -200 ps/km-nm (see Fig. 5.6-5), corre- sponding to Dv == -(A/co)D).. == +4.82 x 10- 25 s2/m. If the pulse is initially un chirped, then TO = 100 fs and therefore the dispersion length is Zo == 1fT?; I Dv == 6.52 cm. At this distance the pulse expands by a factor of V2 and has a chirp coefficient a == 1. At a distance Z == 6.52 m, the pulse width increases by a factor of approximately Z I Zo == 100, becoming lOps and the chirp parameter a == 100. (b) If the initial pulse is phase modulated by a factor exp(j(t 2 ), then a1 == (T 2 . For ( == -10- 2 (fs)2 the pulse becomes down-chirped with parameter a == -1. Upon subsequent propagation through the fiber, the initial 100-fs pulse is compressed to TO =100/V2 == 77 fs at a distance Zmin == -a1Z0 == 1fT?; I Dv == 3.26 cm. Since the pulse is now narrower, it expands more rapidly upon further propagation through the fiber. At the distance Z = 6.52 m, the width increases by a factor of approximately Z I Zo 200, reaching a width of 14. 1 ps. EXERCISE 22.3-1 Dispersion Compensation in Optical Fibers. Pulse broadening in an optical fiber may be eliminated by balancing normal and anomalous dispersion. (a) An unchirped pulse of central wavelength AO == 1.55 pm and width T1 == 10 ps is transmitted through a silica-glass optical fiber. At this wavelength, silica glass has anomalous dispersion with D).. = +20 ps/km-nm. Determine the pulse width T and chirp parameter a at a distance d 1 == 100 Ian. (b) If the pulse is to be compressed back to the original width of lOps by use of another fiber of length d 2 (see Fig. 22.3-6) made of some material exhibiting normal dispersion with D).. == -100 ps/km- nm, determine d 2 . Jt t I T(z) GVD (+) R , t Jt t GVD (-) . I Tl Tl o I I a d 2 : Figure 22.3-6 Dispersion compensa- tion in optical fibers.
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 969 EXERCISE 22.3-2 Dispersion Compensation by Use of a Periodic Sequence of Phase Modulators. Pulse broadening in an optical fiber may be reduced by use of a periodic set of quadratic phase modulators spaced at a distance 2 d. Each modulator introduces a quadratic phase exp (j (t 2 ). If the dispersion coefficient ( is positive and the fiber material has negative GVD, then the pulse width and chirp parameter increase and decrease periodically as illustrated in Fig. 22.3-7. Show that the condition for this periodic pattern is (=_ 2a =_ 2djzo 7 2 7J[1+(dj z o)2] 2Dvd 7r76 [1 + (Dvdj7r7J)2] , (22.3-23) where TO and 7 are the minimum and maximum pulse widths, a is the chirp parameter, and Zo = 7r7J j Dv. I 0( 2d » I II I I II II QPM (+) GVD (-) QPM (+) GVD (-) QPM (+) GVD (-) QPM (+) ; ) z a<z l ) j --r-------r--- Z Figure 22.3-7 Dispersion compensation by use of periodic positive QPM and negative GVD. *c. Slowly Varying Envelope Diffusion Equation It was shown in Sec. 22.3A that a dispersive medium with propagation constant approx- imated by a Taylor-series expansion up to the quadratic term is equivalent to a pulse- envelope filter with transfer function He(f) == exp( -j27rzf Iv) exp( -j7rDvzf2), where v is the group velocity and Dv is the dispersion coefficient. We now show that under such conditions the envelope A(z, t) satisfies the partial differential equation (.J2 A + . 4n ( 8A + 8A ) _ 0 ( 22.3-24 ) 8t 2 J Dv 8z v 8t -. If the time delay z I v is ignored (or a coordinate system moving with the pulse velocity v is used), then (22.3-24) simplifies to 8 2 A 47r 8A ot 2 + j Dv oz = 0, (22.3-25) SVE Diffusion Equation which is recognized as the diffusion equation. D Proof of the SVE Diffusion Equation in a Dispersive Medium. The proof begins with the filter equation A(z, f) = A(O, f)He(z, f) from which A(z, f) A(O, f) exp[-j27r(zjv)f - j7r D v zf2], where A(z, f) is the Fourier transform of A(z, t). Taking the derivative with respect to z we obtain the differential equation (djdz)A(z, f) [-j27rf jv - j7rD v f 2 ]A(z, f). Forming the inverse Fourier transform of both sides with respect to f, and noting that the inverse Fourier transforms of A(z, f), j27r f A(z, f) and (j27r f)2 A(z, f) are A(z, t), 8A(z, t)8t, and 8 2 A(z, t)8t 2 , respectively, we obtain (22.3-24). .
970 CHAPTER 22 ULTRAFAST OPTICS . The impulse response function associated with the diffusion equation is 1 ( 7rt 2 ) he(t) = ..;' exp j- D ' J Dv z v Z (22.3- 26) which is identical to that of a chirp filter (22.2-7) with b == D v v7r. For an initial Gaussian distribution A(O, t) == Ao exp( -t 2 176 ), the diffusion equation is known to have a Gaussian solution, A (z, t) == Ao vi - j Zo I (z - j zo) exp [j (7r I Dv )t 2 I (z - j zo) ], where Zo == 7r76 I Dv. Accounting for the time delay, we replace t with t - z I v and reproduce (22.3-7). D *Derivation of the SVE Diffusion Equation from the Helmholtz Equation. Equation (22.3- 24) may also be directly derived from the Helmholtz equation [d 2 /dz 2 + ,82(v)]V(z, v) = o. Since U(z, t) = A(z, t) exp( -j,8oz) exp(j2nv o t), its Fourier transform is V(z, v) = A(z, v - Yo) exp( -j,8oz) where A(z, v) is the Fourier transform of A(z, t). Substituting v = Vo + f, the Helmholtz equation yields [d 2 /dz 2 + ,82(vO + f)][A(z, f) exp( -j,8oz)] = O. Using the SVE approximation d 2 /dz 2 [Aexp( -j,8ozt)] [-j2,8odA/dz - ,85A] exp( -j,8oz), Helmholtz equation becomes -j2,8odA/dz + [,82(vO + f) - ,85]A = O. For weak dispersion, ,82(vO + f) -,85 2,80 [,8(vo + f) - ,80]. As before, we approximate the propagation constant ,8(v) by a 3-term Taylor- series expansion ,8 (vo + f) ,80 + 2n f,8' + 2n 2 f2 ,8", where ,80 = ,8 (wo), ,8' = d,8 / dw I wo' and ,8" = d 2 ,8 / dw2lwo. With this, Helmholtz equation now becomes - jdA/ dz + [2n f,8' + 2n 2 f2 ,8"] A = o. Performing an inverse Fourier transform and noting that the multipliers j2n f and -4n 2 f2 are equivalent to the derivatives 8/8t and 8 2 /8t 2 , respectively, we obtain -j8A/8z - j,8'8A/8t- ,8" 8 2 AI 8t 2 = O. Finally, substituting ,8' = 1/ v and ,8" = Dv /2n, we obtain (22.3-24). . *D. Analogy Between Dispersion and Diffraction A striking mathematical similarity is observed between the SVE diffusion equation 8 2 AI 8t 2 + j (47r I Dv) 8AI8z == 0, which describes the propagation of a pulse A(z, t) in a dispersive medium (in a frame moving with velocity v, and neglecting dispersion terms higher than the quadratic term), and the paraxial Helmholtz equation \7A - j( 47r I A) 8AI8z == 0, which describes the diffraction of an optical beam A(x, y, z) through free space in the paraxial approximation. Both are diffusion equations (the former is 1D and the latter is 2D). This similarity indicates that the temporal spreading (dispersion) of a pulse as it travels through the dispersive medium obeys the same mathematical law that governs the spatial spreading (diffraction) of a beam in the transverse plane as it travels through free space, with time t playing the role of the transverse coordinate p == (x, y) and the dispersion coefficient - Dv playing the role of the wavelength A. Various features of this analogy are summarized in Table 22.3-2. The analogy between the dispersion coefficient - Dv and the wavelength A is ap- preciated more fully if time t is measured in units of distance traveled at the speed of light, ct. In these units c 2 Dv has units of distance and its role in determining the scale of pulse dispersion is quantitatively similar to the role played by the wavelength in determining the scale of diffraction. For example, if Dv = 10- 23 s2/ m , then c 2 Dv 0.9 Mm, which is equivalent to 3 fs. Another interesting analogy relates the role of a lens in altering the wavefront curvature and the role of a quadratic phase modulator (QPM) in chirping a pulse. A thin lens introduces multiplication by a phase factor exp(j 7rp 2 I Af) [see (2.4-9)], while a QPM introduces multiplication by a phase factor exp(j(t 2 ) (see Sec. 22.2C). Writing exp(j(t 2 ) == exp[j7rt 2 I( -Dvf)]' where ( == - 7rD vf, we see that the QPM
22.3 PULSE PROPAGATION IN OPTICAL FIBERS 971 Table 22.3-2 Comparison between diffraction in space (paraxial approximation) and dispersion in a dispersive medium (second-order approximation). The dispersion coefficient - Dv in pulse dispersion plays the role of the wavelength A in diffraction. The quadratic phase modulator (QPM) is analogous to a temporal lens. Diffraction Dispersion I Complex A(p, z) Complex A(z, t) envelope envelope Transverse p = J x 2 + y2 Time t coordinate Axial Axial coordinate z coordinate z Paraxial 2 .41T 8A SVE diffusion 8 2 A 41T 8A Helmholtz \7 A-J--=O (moving frame) 8t 2 + j Dv 8z = 0 equation T A 8z Wavelength A Dispersion -Dv coefficient Impulse j (7r p2 ) Impulse 1 C 7rt 2 ) response -exp -J- response exp J- function he (p) AZ AZ functi on he ( t ) yjDvz Dv z Lens exp(j 1Tp 2 / Af) QPM exp(J(t 2 ) Focal length f Focal length f = 1T /( -Dv() is equivalent to a time lens that compresses the pulse to a minimum width at z == f, where f == 7r / ( - Dv() is a focal length, confirming (22.3-22). The mathematical analogy between the temporal spreading of a Gaussian pulse in a dispersive medium (Sec. 22.3B) and the spatial spreading of a Gaussian beam in free space (Chapter 3) is summarized in Table 22.3-3. The dispersion length Zo is analogous to the diffraction length (Rayleigh range) zoo Although the latter is always positive, the former is defined such that it is positive for normal dispersion and negative for anomalous dispersion. This explains the negative sign in the parameter Zo that appears in the expression for the complex envelope of the Gaussian pulse. Table 22.3-3 Comparison between the diffraction of a Gaussian beam in free space and the dispersion of a Gaussian pulse in a dispersive medium. Gaussian Beam Gaussian Pulse I Width W(z) = W o /1 + (Z/ZO)2 Width T(Z) = To/1 + (Z/ZO)2 Diffraction Zo = 1TW5 / A Diffraction Izol = 1T75/I D vl length length Divergence eo = A/1TW O Spreading /Dvl/ 1T7 O angle rate (s/m) Wavefront 1 21T Z Chirping " 21T Z - - cp=- R(z) - A Z2 + z5 rate curvature Dv z2 + z5 Spatial W 2 (z ) z Chirp 1 z chirp a(z) = 2R(z) = Zo parameter a(z) = -cp"7 2 (Z) = - 2 Zo
972 CHAPTER 22 ULTRAFAST OPTICS Because of the mathematical analogy between spatial diffraction and temporal dis- persion, and between the lens and the quadratic phase modulator (QPM), each conven- tional optical system made of combinations of free space and lenses, has an analogous temporal system made of combinations of dispersive media and QPMs. Figure 22.3-8 lists a number of examples: Spatial Optics Temporal Optics 01) t:: ;; (l) 0.. CZ) x t T W o W TO z z -- "'-'" x 01) .5 C/'J W o :;j u 0 Z -- oJ:::) "'-'" Lens x 01) t:: .bb WI W 2 c\j E z --... "'-'" Lens x 01) t:: .>. c\j Lens Lens Lens "'-'" t T. TO z QPM (+) GVD (-) t T2 z GVD (-) QPM (+) GVD (-) t z QPM (+) QPM (+) QPM (+) GVD (-) GVD (-) E c.8 VJ t:: . :;j o x 1-1--1-1 z t z I --... "'-'" Lens QPM (+) GVD (-) GVD (-) Figure 22.3-8 Analogy of spatial optics (left column) and temporal optics (right column). The quadratic phase modulator (QPM) plays the role of the lens. The shaded areas represent the spatial width of a wave (left) and the temporal width of a pulse (right) as functions of z. In the right column, time delays are ignored and only time spread is shown. The optical pulse (right) is assumed to travel in a medium with negative GVD. The figures in the right column are also applicable for a medium with positive GVD, but in this case the QPM must be negative. . Temporal spreading of a pulse in a dispersive medium is analogous to spatial diffraction of a beam, or a wave transmitted through an aperture. . Temporal compression of a pulse by a QPM is analogous to spatial focusing a beam by a lens. For example, since a Gaussian beam is focused by a lens of focal
22.4 ULTRAFAST LINEAR OPTICS 973 length f into a width WI == (A/7rW o )f, it follows by analogy that a Gaussian pulse is compressed by a QPM into a temporal width Tl == 1/( TO == (- Dv / 7rTO) f, where f == 7r / ( - Dv() is the focal length of the QPM. Another example of the time-focusing effect of the QPM is the focusing of two separated narrow pulses at z == 0 into one single pulse at z == f. . The counterpart to single-lens imaging in conventional optics is a system using a QPM as a temporal lens that generates a magnified or minified replica of the pulse temporal profile, i.e., a temporal image (see Prob. 22.3-4). . A periodic sequence of QPMs, designed to maintain the width of a pulse, is analogous to a periodic set of relaying lenses. . The counterpart to a 2- f Fourier transform system (see Sec. 4.2) is a 2- f temporal Fourier transform system using a QPM. The system, e.g., transforms a phase mod- ulated optical pulse into an amplitude modulated pulse whose temporal profile is the Fourier transform of the original pulse. One primary difference between spatial diffraction and temporal dispersion is that the wavelength A is always positive, while its counterpart - Dv may be positive or negative. The implication of this difference may be appreciated by examining the impulse response functions in Table 22.3-2. The positivity of the wavelength A implies that a point of light must spread into a diverging phase front (a spherical wave). By analogy, in a medium with negative Dv (i.e., positive - D v ), an impulse of light spreads into a down-chirped pulse. Conversely, in a medium with positive Dv (i.e., normal dispersion), an impulse of light spreads into an up-chirped pulse. Both signs of chirp are allowed, whereas spatial diffraction permits only diverging waves. 22.4 ULTRAFAST LINEAR OPTICS The spatial and temporal characteristics of pulsed waves are inherently coupled. Spatial spreading or focusing depends on the initial temporal profile, and the temporal pulse shape is influenced by the initial spatial pattern. These effects are particularly pro- nounced for ultranarrow pulses and for optical systems exhibiting angular dispersion. Only in very special cases does a pulsed wave maintain exactly the same temporal profile as it travels (e.g., the plane wave and the spherical wave; see Sec. 22.1 C). For pulses with a slowly varying envelope, the quasi-CW approximation is applicable, and temporal and spatial changes are approximately decoupled; this approximation is not applicable, however, for ultranarrow pulses. In this section we consider propagation of ultranarrow pulsed beams in simple imag- ing systems. We begin with a simplified analysis based on ray optics and subsequently proceed with a theory based on wave optics using a Fourier-optics approach. A. Ray Optics Ray optics is based on the description of light by rays that are reflected and refracted at optical boundaries in accordance with Snell's law (Sec. 1.1). Temporal effects are included in this theory since rays are assumed to travel with a medium-dependent velocity c == co/no We used this theory in Sec. 9.3B to estimate the spreading of the time of arrival of optical rays inside an optical fiber by determining the time of travel for each of the optical paths and estimating the difference between the longest and shortest delays. If some of the components of the optical system are made of dispersive materials, then the delay introduced by these components must be based on the group velocity v == Co / N, instead of the phase velocity c == Co / n, where N == n - Adn / dA is the group
974 CHAPTER 22 ULTRAFAST OPTICS index. Estimation of the broadening of an optical pulse as it travels through an optical system is therefore an exercise in determining the difference between the longest and the shortest group delay for all of the possible optical paths. Pulse Broadening in a Single-Lens Imaging System In the single-lens imaging system illustrated in Fig. 22.4-1 an optical pulse is emitted at point PI in multiple rays that meet at the conjugate point P 2 . Each ray travels through air and glass and is delayed accordingly. If the glass material is nondispersive, then in accordance with Fermat's principle (Sec. 1.1) all rays arrive at the same time, and the pulse is not broadened. To account for the effect of dispersion, it is convenient to define the differential delay as the difference between the group delay (based on the group velocity v) and the phase delay (based on the phase velocity c). The difference between the longest and shortest differential delays then constitutes the pulse broadening. The differential delay is of course zero for the nondispersive portions of the optical path, so that one needs to be concerned only with the differential delay in the lens material. Marking each ray by its position (x, y) in the lens plane, if d (x, y) is the lens thickness at the position (x, y ), then the differential delay is 1 1 In - NI T(X, y) == - - - d(x, y) == d(x, y). c v Co (22.4-1 ) The width of the broadened pulse is the difference between the maximum and mini- mum values of T(X, y), so that d T(X, y) == In - NI- Co (22.4- 2) where d is the difference between the maximum and minimum widths of the lens. For a thin lens of focal length f, and maximum thickness do, we use (2.4-6) and (2.4- 10) and obtain d(x, y) == do - (x 2 + y2)/2R == do - (x 2 + y2)/2(n - l)f so that d == (D/2)2/2(n - l)f, where D is the lens diameter. Consequently, T == [In - NI/(n - 1)](D/2)2/2c o f, from which T== In-NI L n - 1 8F CO ' (22.4-3) Pulse Spreading where F# == f / D is the lens F-number. it t Figure 22.4-1 Pulse broadening in a single-lens imaging system is caused by material (chromatic) dispersion. As an example, for a BK7-glass lens at A = 400 nm, n == 1.53, and n- N == A dn/ dA == -0.052. If f == 30 mm and F# == 2, the pulse spreads to a width T 307 fs. In this system, the pulse broadening is a result of the differential material dispersion associated with the multiple spatial paths of the rays. Without material dispersion,
22.4 ULTRAFAST LINEAR OPTICS 975 the existence of multiple paths will not cause pulse broadening, thanks to Fermat's principle. *8. Wave and Fourier Optics The wave nature of light dictates that a monochromatic narrow optical beam spreads into a wide cone with an angle directly proportional to the wavelength and inversely proportional to the original beam width. When the beam is modulated by an ultra- short pulse with a broad spectrum, each of its wavelength components spreads into its own cone, with the short-wavelength components occupying cones of smaller an- gles. Consequently, the spectral composition of the propagated light at each point in space is altered, with the points farther from the axis having less energy at the shorter wavelengths, as illustrated in Fig. 22.4-2. At off-axis points, the spectrum is therefore shifted to a lower central frequency (red shift) and the spectral width is compressed, with an accompanied pulse broadening. This example demonstrates that the spatial and temporal characteristics of light are entwined through the very process of wave propagation, particularly when the beam is ultranarrow and the pulse is ultrashort. se v A ) I X I RIB I I I V R _ y- 8(1/) : - A I I v B { I ) Ri:\R e 8(1/) : ( I V Figure 22.4-2 Spreading of a pulsed beam. The long-wavelength components (R) spread into cones with angles greater than those of the short-wavelength components (B). This results in the suppression of the short-wavelength components at off-axis points, and hence a red shift and a compression of the spectral width accompanied by pulse broadening. Although the propagation of ultrashort light pulses through arbitrary optical sys- tems is complicated by the inherent space-time coupling, the analysis is conceptu- ally simple when the system is linear since a Fourier approach can be used to re- duce the problem to one of superposition of solutions for each of the constituent monochromatic components. An arbitrary pulsed wave U(r, t) is decomposed as a sum of monochromatic components with amplitudes given by the Fourier transform V(r, v) == J U(r, t) exp(j27rvt) dt. The propagation of each monochromatic com- ponent through the system is determined using the tools developed in Chapters 3-10, and the overall solution is subsequently obtained by superposition, i.e., by an inverse Fourier transform U(r, t) == J V(r, v) exp( -j27rvt) dv. Fourier Optics of Pulsed Waves The propagation of monochromatic light between two parallel planes (1, 2) with an arbitrary linear optical system in between is described generally by the linear transfor- mation V2(x, y, v) = J J h(x, x', y, y', v)Vi(x', y', v) dx'dy', where h is the impulse response function of the system at frequency v (see Chapter 4). For a pulsed input wavefunction U 1 (x, y, t), the output wavefunction U 2 (x, y, t) may . (22.4-4)
976 CHAPTER 22 ULTRAFAST OPTICS be readily determined by computing the Fourier transform V 2 (x, y, v), using (22.4-4) and subsequently computing an inverse Fourier transform. The impulse response function h has been determined in Chapter 4 for various opti- cal components. The results are reproduced here with the dependence on the frequency v made explicit: . Free space. In accordance with (4.1-18), a distance z of propagation through free space is equivalent, in the Fresnel approximation, to a system with impulse response function h( 1 1 ) r-v jv [ .27rv (X_XI)2+(y_yl)2 ] X,X ,y,y,v r-v - exp -J- . cz c 2z (22.4- 5) We have here ignored a factor exp( - j27rv z / c) since it represents an inconse- quential constant time delay z / c. . Aperture. Transmission through a planar aperture is equivalent to multiplication by the aperture function (unity within the aperture and 0 outside). . Lens. Transmission through a lens of focal length f is equivalent to multiplication by the quadratic phase factor ( 2 7rI J ) ( 27rv p2 ) t(x, y, /J) ::::: exp - jn---z: do exp j ---z: 2f (22.4-6) where p == V X2 + y2 is the radial distance and the focal length f is given by 1 ( 1 1 ) f = (n - 1) Rl - R 2 ' (22.4-7) where Rl and R 2 are the radii of the spherical lens [see (2.4-9) and (2.4-11)]. If the refractive index n of the lens material is wavelength dependent, then f would be dependent on the frequency v. Material dispersion results in chromatic aberration, which plays an important role in the distortion of ultrashort optical pulses. With the help of these equations, one can, in principle, determine the space-time dependence of the output wave for any input pulsed wave and for any system comprised of combinations of free-space, lenses, and apertures. Optical Fourier-Transform System ]indexFourier transform!optical Take, for example, a Fraunhofer diffraction system involving the propagation of a monochromatic wave between the front and back focal planes of a lens. This system is described by an impulse response function ( 1 jv ( . 27rv I ) h x, x , /J) ::::: cz exp - J c f xx , (22.4-8) which corresponds to a spatial Fourier transform operation for monochromatic light (see Sec. 4.2). For simplicity, we have ignored the y dependence. This system exhibits a strong temporal-spatial coupling; i.e., the temporal waveform at a fixed point in the output plane is strongly influenced by the spatial distribution of the wave in the input
22.4 ULTRAFAST LINEAR OPTICS 977 plane. Likewise, the spatial distribution in the output plane is sensitive to the temporal waveform of the input field. To illustrate this point, consider a special case for which the input wavefunction is separable in time and space, U 1 (x, t) == g(t)p(x). This may be generated by transmit- ting a pulsed plane wave of amplitude g(t) into a spatial light modulator (SLM) with frequency-independent transmittance p(x), as illustrated in Fig. 22.4-3. Substituting Vi (x, v) == G(v)p(x), where G(v) is the Fourier transform of g(t) into (22.4-8) and (22.4-4), the field in the output plane is V2(x, v) ex jvG(v)P(vx/cf), (22.4-9) where P(v x ) == J p(x) exp (j27rv x x) dx is the spatial Fourier transform of p(x). It is evident from (22.4-9) that the output field is no longer time-space separable. The temporal waveform of the field at a fixed position Xo in the output plane is given by (22.4-9), so that the transfer function of the linear system that relates U 2 (xo, t) to the input pulse 9 (t) is H(v) ex jvP(vxo/cf). (22.4-10) This temporal transfer function is a scaled version of the spatial Fourier transform of the input spatial distribution p( x). The corresponding temporal impulse response function is obtained by taking a temporal inverse Fourier transform of both sides of (22.4-10), h(t) ex p(tcf /xo), (22.4-11 ) Space-to-Time Conversion so that the value of the function h( t) at time t is controlled by the transmittance of the SLM at one-and-only-one position x == (cf / xo)t. Equivalently, the transmittance of the mask at a point x controls the value of the impulse response function of the system at one-and-only-one time t == (xo/ cf)x. The system serves as a direct space-to-time conversion, which can be used for pulse shaping. A similar pulse-shaping system using a combination of a diffraction grating and an SLM has been discussed in Sec. 22.2D. Input pulse g(t) f --pr- f, . /. ... t\o u Xo Output pulse t "1 '11', . p. ! II' p(x) SLMY X ;< t Lens Fourier plane Figure 22.4-3 A spatial Fourier transform system couples the temporal and spatial distributions of the input pulsed light. The shape of the output pulse at a fixed position is governed by the spatial distribution at the input, which is controlled by the SLM.
978 CHAPTER 22 ULTRAFAST OPTICS *c. Beam Optics The Fourier approach described in the previous section (22.4B) may be applied to the study of pulsed Gaussian beams. Consider first a Gaussian beam modulated in the plane of its waist by a pulse g(t), i.e., UI(x, y, t) == g(t) exp( _p2 jWc?) or VI (x, y, t) ex G(v) exp( _p2 jWc?), where W o is the beam radius and G(v) is the Fourier transform of g(t). At an arbitrary distance z, the spatiotemporal wavefunction is determined by use of (22.4-4) and (22.4-5), jzo(v) ( . 7rV p2 ) "V2(x,y,v) ex vG(v) . () exp -J- . () , z + ] Zo v c z + ] Zo v (22.4-12) where 7r VV;2 Zo ( v) == 0 v c (22.4-13) is the diffraction length (Rayleigh range) at frequency v. Equation (22.4-12) is the standard expression of the wavefunction of a Gaussian beam [see (3.1-5)] with the frequency dependence of the diffraction length made explicit. The beam radius and the radius of curvature given by (3.1-8) and (3.1-9) are also frequency dependent. If the spectral width is sufficiently narrow, then in accordance with the quasi-CW approximation the spatial distribution of the Gaussian beam may be approximated by its values at the central frequency v Vo, and consequently the time-space dependence is separable, as described earlier by (22.1-25). For ultranarrow (i.e., broadband) pulses, this approximation is not applicable. The temporal profile of the pulse may be determined at an arbitrary point (p, z) by evaluating the inverse Fourier transform of (22.4-12)). In general, a numerical solution IS necessary. Gaussian-Pulsed Gaussian Beam If the original wave is modulated by a Gaussian pulse g(t) == exp( -t 2 jT6)exp(j27rvot), then G(v) ex exp[-7r2T6(v - vO)2] is also Gaussian. An approximate analytical expression for V 2 (x, y, v) in the far zone [z » Zo (v) for all v] may be obtained as follows: The factor [z + jzO(v)]-l == z-I [1 + jzo(v)j z]-I in the exponent of (22.4- 12) is approximated by z-I [1 - jzo(v)jz], and the same factor in the amplitude is approximated by z-I . Using (22.4-13), we obtain the far-zone expression ( VV;2 p2 ) ( p2 ) 1;2 (x, y, v) ex jv exp [-7r2T5(v - vO)2] exp -7r 2 2 0 2 v 2 exp -j27rv- . C z 2cz (22.4-14 ) The inverse Fourier transform of (22.4-14) may now be determined. The phase factor in the exponent is equivalent to a time delay p2 j2cz. The factor jv in the amplitude is equivalent to a derivative a j at. The middle two Gaussian functions are combined into one Gaussian function of v whose inverse Fourier transform is another Gaussian function. The result may be cast in the normalized form exp[ -7r N p2 j (p2 + P6)] exp( -tj T) . U 2 (x, y, t) ex 2/ 2 1. / N exp (- J27rv p t p ) , (22.4-15) 1 + p Po + ]t p 7r TO Gaussian-Pulsed Gaussian Beam
22.4 ULTRAFAST LINEAR OPTICS 979 where t p == t - p2/2cz == t - 7rNTo(zlzo) (p 2 / 2 p6) (22.4-16) is a position-dependent delay time, Tp = TO J I + P2/P6 (22.4-17) is a position-dependent time constant, Vo v - P-1+p 2 /P6 NITo 1+p2/P6 (22.4-18) is a position-dependent central frequency, N == VOTO (22.4-19) is the number of optical cycles within the width TO of the initial pulse, and PO == 7rNW(z) == 7rNW O ZIZ o , (22.4-20) where W (z) == Woz I Zo is the far-zone beam radius for a CW wave at the central frequency Vo and Zo == 7r We? / Ao is the associated diffraction length. As a function of the normalized transverse distance pi Po and the normalized time tITO, the far-zone wavefunction is completely described by two free parameters: N and the ratio z I zoo The intensity 1 2 ( x, y, t) == I U 2 ( x, y, t) 1 2 is I ( ) exp[ - 27r N p2 I (p2 + P6)] exp( - 2t1 T;) 2 x,y,t ex 1 + 2 1 2 1 + t2 / 7r2N22. P Po p 0 (22.4-21) This is a universal function of t I TO and pi Po characterized by only one free parameter N. The spectral intensity 52 (x, y, v) == I V2 (x, y, v) 1 2 is 2 [ 2 ] [ ( ) 2 ] V 2 2 P 2 2 V - v p 52 (x, y, v) ex "2 exp -27r N 2 2 exp -27r N 2 ' P+ (22.4-22) which is a universal function of v I Vo and pi Po, characterized by the free parameter N. Based on (22.4-15)-(22.4-22), we conclude that the pulse at a point (p,z) in the far-zone has the following characteristics (see Fig. 22.4-4): . The pulse is delayed by time p2/2cz, which is the travel time between the center of the beam (0,0) and the point (p, z). . The pulse temporal profile is the product of a Gaussian function of width Tp == TO [1 + p2 I P6J 1/2 and a Lorentzian function of width 7r N TO. The width of the Gaussian function is TO at p == 0, and increases with the transverse distance p, reaching the value J2 TO at p == Po. The phase shift arctan( t p I 7r N TO) introduced by the Lorentzian function is a manifestation of the Gouy effect (see Sec. 3.1B) for pulsed Gaussian beams.
980 CHAPTER 22 ULTRAFAST OPTICS . The pulse central frequency v p depends on the transverse distance p. Starting at the value Vo on axis (p == 0), it decreases monotonically with increase of p, reaching vo/2 at p == Po. This is a consequence of the fact that long-wavelength (low-frequency) components of the pulse spread into wider cones, as illustrated in Fig. 22.4-2. For the same reason, the farther the point is from the beam axis, the smaller the spectral width and the greater the temporal width. . The initial Gaussian spatial distribution is altered dramatically as t increases. An initially single-peaked distribution builds up, is subsequently flattened, and eventually becomes double-peaked as it decays (see Fig. 22.4-4). II (p, t) 2(P'VW I 0.8 0.9 1 1.1 . vivo /2(P, t) /' B /-,,". -- ""'-- - / ">-- t -.... --............. -...........-"'" ....... 'A -- -2 Figure 22.4-4 Temporal and spatial spreading of a Gaussian beam modulated by a Gaussian pulse. Initially, the beam has radius W o and temporal width TO (left surface plot). The far-zone intensity 1 2 (p, t) is illustrated in the right surface plot. Time is normalized to the initial pulse width TO, transverse distance is normalized to Po, and the intensity has arbitrary units. At a fixed time t, 1 2 (p, t) provides a snapshot of the intensity as a function of position. It changes from a single-peaked function at t == 0 to a double-peaked function at t == TO, and eventually becomes two separate weak peaks at t == 2To and beyond. The temporal profile at a fixed position is also depicted by this surface. In the center of the beam, the pulse has its shortest width. At off-axis points the pulse is weakened, delayed, and has longer duration. The spectral intensity 8 2 (p, v) is shown (top right) as a function of the normalized frequency v / Vo at two positions, A and B, and is normalized such that the peak value is unity for each position. In this plot, N == VOTO == 5; i.e., the pulse has five optical cycles. As an example, N = 5 for a pulse of central frequency Vo = 750 THz (Ao = 400 om) and width TO = 6.67 fs. If "{;V o = 1 mm, then Zo = 7.85 m, Wozlz o = 5 mm, and Po 8 em. Focusing of a Pulsed Beam If a beam of arbitrary spatial distribution p( x, y) modulated with a pulse of arbitrary temporal shape g(t) is transmitted through a lens of focal length f followed by a distance Z of free space, then by substituting U 1 (x,y,t) == g(t)p(x,y) into (22.4-4) and (22.4-5) we obtain Y2(x, y, v) ex G(v) J J dx' dy' p(x', y') exp (- j 2V do) exp (j 2V x/ 2 ;ru/ 2 ) ( .27rV (x-x')2+(y_y')2 ) ex p - J - c 2z ' (22.4-23) where G(v) is the Fourier transform of g(t). We have here assumed that the lens has an aperture wider than the beam width.
22.4 ULTRAFAST LINEAR OPTICS 981 If the lens material is nondispersive, so that nand f are independent of v, then at points in the focal plane z == f, (22.4-23) simplifies to ( vx Vy ) ( . 27rVp2 ) V2(x,y,v)exvG(v)P ef'e! exp -J-;;- 2! ' (22.4- 24 ) where P(v x , v y ) == II dxdyp(x, y) exp [j27r (VxX + vyy)] is the spatial Fourier trans- form of p(x, y). The factor exp( - j27rvd o / c) has been ignored since it now represents a simple time delay. The wavefunction in the focal plane is the temporal inverse Fourier transform of 112 ( x, y, v), so that U 2 (x, y, t) ex J vG(v)P ( ; , ; ) exp [j27fV (t - :c ) ] dv. (22.4-25) The coupling of the temporal and spatial features of the pulsed beam is evident in (22.4- 25). In addition to the space-dependent time delay t - p 2 /2cf, the Fourier transform of the original spatial profile is scaled by the frequency-dependent factor cf/v before it is averaged over the spectral distribution of the pulse. As an example, for a Gaussian beam p(x, y) == exp (_p2 /Wc?) modulated by a Gaussian pulse g(t) == exp( -t 2 /T) exp(j27rvot), i.e., G(v) ex exp[-7r2T6(v - vO)2] and P(v x , v y ) == exp [-7r2Wc? (v; + v;)], (22.4-24) gives V2(x,y,v) ex vexp [-7f2T (v - V O )2] exp [- ( 7f:;0 ) 2 v 2 /] exp (-j27fV :c ). (22.4- 26) This expression is identical to that for the far-zone Gaussian beam (22.4-14), with z == f. Thus, the corresponding wavefunction U 2 (x,y,t) is given by (22.4-15)-(22.4- 22) with z == f. The graphs in Fig. 22.4-4 are applicable here with z == f, Zo being the diffraction length of the original (not the focused) beam, and Po == 7rNW o f /zo == NAof /W o == 7rNW, (22.4- 27) where W6 == Aof / 7r W o is the beam radius at the focal plane for a CW beam with wavelength Ao [see (3.2-15) and (3.2-17)]. As before, N == VOTO is the number of optical cycles within the initial pulse. The characteristic transverse radius Po is there- fore 7r N times greater than W6. Figure 22.4-5 is an illustration of the spatiotemporal distribution of the pulse in the focal plane. * Pulsed Beams in Dispersive Media The process of diffraction of pulsed light in a dispersive medium can be complex. If the medium is linear and homogeneous, then the Helmholtz equation [\7 2 + (32 (v)] V (r, v) == 0 describes this process for arbitrary dispersion properties, charac- terized by the propagation constant (3 (v), and for a pulse with arbitrary spatial-spectral profile V(r, v). Once V(r, v) is determined by solving this equation, the corresponding wavefunction U(r, t) may be readily determined by an inverse Fourier transform. This approach is, in principle, valid no matter how dispersive the medium or how narrow the pulse. Approximations similar to those that led independently to the paraxial Helmholtz equation, which describes beam diffraction, and the SVE equation, which describes pulse dispersion (see Table 22.3-2), may be combined to derive a partial differential equation for the envelope A(r, t) of a pulse with a narrow spectral distribution. An
982 CHAPTER 22 ULTRAFAST OPTICS f -----/ -2 2 t/70 Figure 22.4-5 Focal-plane spatiotemporal profile of the intensity of a Gaussian beam modulated by a Gaussian pulse and focused by a lens of focal length f. In this plot the initial pulse has N = 5 optical cycles and the initial beam has a diffraction length Zo » f. The difference between this and the spatiotemporal profile in Fig. 22.4-4 is attributed to the fact that here the time delay p2 j2cf = ToCrr Nj2)(f j ZO)(p2 j P6) is negligible for f « Zo at off-axis points with P < Po. approach following the same steps described in Sec. 22.2C results in the generalized paraxial wave equation: 2 8 2 A . ( 8A 1 8A ) -AoY'TA + Dv 8t 2 + J 8z + v 8t == o. (22.4-28) Generalized Paraxial Wave Equation This equation generalizes (22.1-24), which is applicable for nondispersive media (Dv == 0), as well as (2.2-23), which is applicable for the CW case for which 8 2 AI8t 2 == 8AI8t == O. D Proof of the Generalized Paraxial Wave Equation. The wavefunction and its Fourier transform are related to the envelope and its Fourier transform by U(r, t) = A(r, t) exp( -j(3oz) exp(j27rv o t) and V(r, v) = A(r, v - vo) exp( -j(3oz). The paraxial approximation, (d 2 jdz 2 )[Aexp( -j(3oz)] [-j2(3odAjdz -(35A] exp (-j(3oz), can be used to convert the Helmholtz equation to [\7 - j2(3odjdz] A + [(32(vO + f) - (36] A = o. (22.4-29) For weak dispersion, we use the approximation (32 (vo + f) - (35 2(30 [(3(vo + f) - (30] together with a 3-term Taylor-series expansion (3(vo + f) = (30 + 27r(3' f + 27r 2 (3" f2. The Helmholtz equation then becomes \7A - j2(308Aj8z + 2(30 [27rf(3' + 27r 2 f2(3"] A = o. (22.4-30) Performing an inverse Fourier transform and noting that the multipliers j27r f and -47r 2 f2 are equiv- alent to the derivatives 8 j 8t and 8 2 j 8t 2 , respectively, we obtain \7A - 2,80 [j8A/8Z + j,8' 8A/8t + ,811 8 2 Aj8t 2 ] = O. (22.4-31) Finally, substituting (3' = Ijv and (3" = Dvj27r and (30 = 27r j Ao, we obtain (22.4-28). . The paraxial SVE equation admits a space-time Gaussian solution A(x, y, z, t) == Ao - j zo' [ . 7r t - z I v ] exp -J- z - j zo' Dv z - j zo' J Zo ( 7r p2 ) exp -J- . , z + j Zo A z + J Zo (22.4- 32)
22.4 ULTRAFAST LINEAR OPTICS 983 that has a spatiotemporal Gaussian initial envelope A(x, y, 0, t) == Ao exp( -t 2 /T6) exp( _p2 /Wc?), where zb == 1fT6 / Dv and Zo == 1fWc? / A are, respectively, the dis- persion length associated with the initial pulse width TO and the diffraction length associated with the initial beam radius Woo This solution combines the diffraction of a Gaussian beam (Chapter 3) and the dispersion of a Gaussian pulse (Sec. 22.3) in a space-time separable fashion, as illustrated in Fig. 22.4-6. x --+I c7{z) CTo l _ _ _ _ _ _ _ - - - ,- - - - - - - -- - - - - - - T -- ------ --- VV(z) VVo s::: o .- u C'\j Z CS ----- T ---------------------------- ---=: ;. - . Dispersion Figure 22.4-6 Three snapshots of the spatial distribution of a pulse as it travels through a linear dispersive medium. Because of diffraction, the pulse spreads in the transverse direction x. Because of dispersion, it spreads in time (which is shown here as spatial spread in the direction of propagation z ). Since (22.4-28) and (22.4-32) are separable in time and space, we conclude that the approximations to which these equations are subject are in effect tantamount to the quasi-CW approximation described in Sec. 22.1 C. * Envelope Equation for Ultranarrow Pulsed Beam When conditions for the SVE approximation are not met (i.e., the pulse is very nar- row and the beam is very thin), then the space-time dependence is no longer sepa- rable. The differential equation that governs the pulse envelope takes a more com- plex form, although the very concept of envelope is then less meaningful. Beginning with the Helmholtz equation [\7 2 + 13 2 ( v ) ] V ( r, v) == 0 and subs titu ting V ( r, v) == A(r, V-yo) exp( -jfJoz) and v == vo+ f, we obtain [\7 + 8 2 /8z2 - j2fJ08/ 8z] A+ [fJ2(vO + f) - 135] A == O. Expanding the function [fJ2(vO + f) - 136] in a Taylor-series expansion up to the second order, we have [fJ2(vO + f) - 136] (2fJofJ')21f f + !(2fJ'2 + 213013") (21f f)2. Transforming back to the time domain and reordering terms, we obtain 8 2 A ( 8 1 8 ) ( 82 1 8 2 ) -AoV'A+Dv 8t 2 +j47r 8z + v8t A-AO 8z2 - v8t 2 A=O, (22.4-33) where v == 1/13' and Dv == 21ffJ". Equation (22.4-33) is more general than (22.4- 28) since the paraxia] approximation and the weak dispersion approximation have not been used. If 13'2 « 13013" (or Ao/v 2 « Dv) and 82A/8z2 « (41f/Ao)8A/8z, then the fourth term in (22.4-33) is negligible and (22.4-33) reproduces (22.4-28). Equation (22.4-33) may be expressed in a coordinate system moving at the pulse velocity v by using the transformation t' == t - z / v and z' == z. The result is the differential equation 8 2 A 8A ( 8 2 A 2 8 2 A ) -AoV'A + Dv 8t,2 + j47r 8z' - AD 8Z,2 - v 8t'8z' = 0, (22.4-34) which clearly exhibits spatiotemporal coupling.
984 CHAPTER 22 ULTRAFAST OPTICS 22.5 ULTRAFAST NONLINEAR OPTICS The previous sections of this chapter dealt with the propagation of optical pulses in linear media, with an emphasis on the role of group velocity dispersion (GVD) in the reshaping of short pulses. In this section, we consider the propagation of optical pulses in nonlinear media. Nonlinear effects are more frequently encountered with ultrashort pulses because of their higher intensity. Nonlinear optical phenomena were introduced in Chapter 21; in particular, three-wave mixing in media with second-order nonlin- earity and two- and four-wave mixing in media with third-order nonlinearity were considered. In this section, some of these phenomena are revisited in the context of pulsed waves. Section 22.5A deals with pulsed parametric processes, including three- wave mixing, optical rectification, and self-phase modulation; Sec. 22.5B considers optical solitons; and Sec. 22.5C is devoted to supercontinuum generation. A. Pulsed Parametric Processes Three-wave mixing in a medium with second-order nonlinearity was discussed in Sec. 21.2C for continuous waves (CW), and a coupled-wave theory was developed in Sec. 21.4. The principal conditions for wave mixing are dictated by conservation of energy and momentum. For pulsed waves with central angular frequencies WI, W2 and W3, and central wavevectors k l , k 2 , and k 3 , these conditions are: WI + W2 == W3 and k l + k 2 == k3. If dispersion effects are neglected, the CW theory is applicable to the pulsed case; i.e., the pulse is regarded as "quasi-CW" at any time during its course, and the envelopes of the three waves obey the same coupled-wave equations (21.4-20). The Walk-Off Effect If the medium exhibits first-order dispersion, but not second-order (GVD) or higher- order dispersion, then the three pulsed waves travel at their group velocities without altering their shapes (only their amplitudes are altered by the mixing process). Since these velocities are generally different, the pulses eventually separate and the paramet- ric process responsible for wave mixing ceases, a phenomenon known as the walk- off effect. Therefore, for efficient pulsed-wave mixing, an additional condition is the equality of the group velocities, VI == V2 == V3. The walk-off effect is illustrated in Fig. 22.5-1 in the degenerate case of collinear second-harmonic generation (WI == W2 == wand W3 == 2w). tt F t31----------------------------- --r- --- I SH I I I I I t I I 1 * we A N z Figure 22.5-1 A pulsed wave at the fun- damental frequency (F) and its associated second-harmonic wave (SH) separate as they travel at different velocities (in this example, the SH wave is faster). The upper graph is a space-time diagram for pulses of duration 7. The lower schematic shows three snapshots of the traveling pulses at times t 1 < t 2 < t 3 . t2 ----------------- . z It is difficult to satisfy both phase matching and group-velocity matching simulta- neously. It was shown in Sec. 21.2D and Sec. 21.4A that for a phase matching error
22.5 ULTRAFAST NONLINEAR OPTICS 985 k, second-harmonic generation diminishes significantly at a distance Lc == 27r IIkl, called the coherence length [see (21.2-28)]. For a group-velocity matching error (3' == 1 I V3 - 1 I VI, the pulses separate by a time delay (3' z == z I V3 - z I VI after traveling a distance z. When this delay equals the pulse width T, the pulses no longer overlap and the nonlinear coupling ceases. This occurs at a distance Lg == TI 1(3'1 (22.5-1 ) Walk-Off Length called the walk-off length. The shorter of the distances Lc and Lg dictates which of the two effects, phase-velocity mismatch or group-velocity mismatch, dominates. As an example, for a KDP crystal using an ordinary fundamental wave at Al = 1.06 /-Lm and an extraordinary second-harmonic wave at A3 = 0.53 /-Lm in the Type-II o-e-o configuration, the group velocity mismatch (3' == 2(1/v3 - 1/vl) 5.2 x 10- 10 s/m. For a 100-fs pulse, the walk-off length Lg == T 11(3'1 0.2 mm. *Coupled-Wave Equations for Pulsed Three-Wave Mixing The coupled-wave equations that were derived in Sec. 21.4 for CW waves may be readily generalized to pulsed waves. For collinear plane waves traveling in the z di- rection , the e lectric fields are expressed in terms of the complex envelopes as G q == Re{ J 2TJnw q aq(z, t) exp[j(wqt - (3qz)]}, q = 1,2,3, where aI, a2, and a3 are normal- Ized complex envelopes of the three pulses, and {31, (32 and (33 are the propagation constants at the centra] frequencies WI, w2, and W3. Using the slowly varying envelope approximation and a two-term Taylor-series expansion of the propagation constant (3(w) near each of the central frequencies (3(w q + n) (3q + n(3, where (3 is the derivative 8(31 8w at W q , we obtain the coupled equations: ( 8 1 8 ) . * - + -- al == -Jga3 a 2 8 z VI 8t ( 8 18 ) . * - + -- a2 == -Jga3 a 8z V2 8t 1 ( l!.- + ) a3 == -jgal a 2, 8z V3 8t (22.5-2) where v q = 1 / (3 is the group velocity of the w q wave, and 9 is a constant given by (21.4- 21). These equations are similar to the CW coupled equations (21.4-20). If the group velocities are equal, i.e., VI == V2 == V3 == V, then by use of a coordinate system moving with a velocity V, the pulsed coupled equations (22.5-2) become identical to the CW coupled equations (21.4-20), and the solutions presented in Sec. 21.4 are applicable with the variable z replaced by z - vt. If the group velocities are not equal, the solution of (22.5-2) becomes more complex. When the medium also exhibits GVD (see Prob. 22.5-2), a three-term Taylor-series expansion {3(w q + n) {3q + n(3 + n2 {3 leads to the coupled-wave equations: ( 8 1 8 . (3 8 2 ) . * 8z + VI 8t - J 2 8t2 al == -Jga3 a 2 ( 8 1 8 . (3 8 2 ) . * - + -- - J-- a2 == -Jga3 a 8 z V2 8t 2 8t 2 1 (22.5-3)
986 CHAPTER 22 ULTRAFAST OPTICS ( a 1 a . (3 a 2 ) . az + v3 at - J 2 at2 a3 == -JgaIa2. Pulsed Optical Rectification: THz Pulse Generation A pulsed wave with central frequency in the optical band and spectral width in the THz range may be down-converted into a pulse of THz radiation. In essence, the pulse is frequency shifted from the optical band to the THz band, as if it were rectified. Figure 22.5-2 is a schematic illustration of the process. Optical pulse THz pulse s Nonlinear Crystal 2 ps Figure 22.5-2 Generation of a THz pulse by down-conversion of an optical wave. When an optical pulse £(t) == Re{A(t) exp(jwot)} with slowly varying enve- lope A(t) travels through a medium with second-order nonlinear coefficient d, it in- duces a polarization density 2d£2(t), which has a term at 2wo, responsible for second- harmonic generation, and another P THz == dIA(t)1 2 (22.5-4 ) representing optical rectification (see Secs. 21.2A, 21.2C, and 21.4 B). In order to determine the appropriate phase matching conditions for this parametric process, we resort to a Fourier approach. The pulsed optical wave can be regarded as a sum of monochromatic waves with frequencies occupying a spectral band sur- rounding the central frequency woo Upon passage through the nonlinear medium, these monochromatic components are mixed in pairs, each generating a down-converted monochromatic wave at the frequency difference. In accordance with (21.2-13e), a pair of waves at the angular frequencies WI == wand W2 == W + fl generates a nonlinear polarization density PTHz(fl) == 2dE*(w)E(w + fl) at the THz frequency fl so that the sum for all the pairs is PTHz(n) = J 2dE*(w)E(w + n)dw. (22.5-5) In the time domain, this is equivalent to (22.5-4). To include nonlinear dispersion ef- fects, the nonlinear coefficient din (22.5-5) must be replaced by a frequency-dependent coefficient d(fl, w, w + fl) (see Sec. 21.7). This down-conversion process must satisfy the phase matching condition at all frequencies wand fl. This condition cannot be met exactly, and an error fj.k == k(w + fl) - k(w) - k(fl) (22.5-6) will arise. If fl « w, this relation may be written in the approximate form fj.k fldkjdw - k(fl) == fl[ljv(w) - Ijc(fl)] == [N(w) - n(fl)]fljc o , (22.5-7)
22.5 ULTRAFAST NONLINEAR OPTICS 987 where v(w) == (dk/dw)-l is the group velocity and N(w) is the group index at the optical frequency w, and c(O) and n(O) are the phase velocity and refractive index at the THz frequency O. The device must therefore be designed such that the group index at optical frequencies is equal to the phase index at THz frequencies. As was shown in Sec. 21.2D for a crystal of length L, this phase-matching error is small if L < Lc, where Lc == 27r /Ikl is the coherence length [see (21.2-28)]. To account for this effect, the factor Jo£ exp(j kz )dz == [exp(j kL) - 1]/ j k must be included within the integral of (22.5-5). Pulse Self-Phase Modulation Self-phase modulation (SPM) occurs in nonlinear media exhibiting the optical Kerr effect (see Sec. 21.3A). The phase cp introduced by this effect for a wave traveling a distance z in a medium with optical Kerr coefficient n2 is cp == -n21koz, where I is the optical intensity and ko is the wavenumber. For an optical pulse, the intensity is a function of time I ( t) and the phase is therefore time varying, cp(t) == -n21(t)koz. (22.5-8) This corresponds to a change of the instantaneous frequency [see (22.1-4)] d1 Wi == -n2 dt koz. (22.5-9) For a pulse with a simple shape, such as that shown in Fig. 22.5-3, if n2 is positive, the frequency of the trailing half of the pulse (the right half) is increased (blue shifted) since dI/dt < 0, whereas the frequency of the leading half (the left half) is reduced (red shifted) since dI/dt > O. The pulse is therefore up-chirped (i.e., its instantaneous frequency is increasing) near its center. It follows that SPM may be used to introduce chirp, and may therefore be employed for pulse shaping (see Sec. 22.2C). Optical Kerr medium (n2 > 0) t -1 r W"t W:l---- :r t R B w't w:l- :r )I t )I t Figure 22.5-3 Chirping of an optical pulse by propagation through a nonlinear optical Kerr medium. For example, a Gaussian ulse may be approximated near its center by a parabolic function,l(t) == 10 exp( -2t /7 2 ) 10[1 - 2t 2 /7 2 ], so that the time-varying compo- nent of the phase is approximately a quadratic function of time cp == 2n2 1 0kozt 2 /7 2 , corresponding to a linear chirp with chirp coefficient a == 2n210koz of the same sign as the Kerr coefficient n2. Self-phase modulation therefore introduces a quadratic phase modulation factor exp(jat 2 /7 2 ) == exp(j(t 2 ), where ( == 2n2 1 0koz/7 2 . (22.5-10)
988 CHAPTER 22 ULTRAFAST OPTICS It is convenient to write the chirp parameter introduced by SPM in the form a == Z / ZNL, ZNL == (2n2 f Oko) -1 , (22.5-11 ) SPM Chirp Parameter where IZNLI is called the nonlinear characteristic length of the Kerr medium. The phase introduced by traveling through the nonlinear material a distance 21zNLI at the peak intensity fo is unity, i.e., n2fOk02lzNLI == 1. It has been implicitly assumed in the preceding analysis that the medium is weakly dispersive so that pulse broadening is negligible; i.e., GVD is negligible in comparison with SPM. This condition is obtained if Izol » IZNLI. Analysis of pulse propagation in materials exhibiting both SPM and GVD is complex, as will be seen in the next section. The quadratic phase modulation introduced by nonlinear SPM may be used in conjunction with a linear dispersive device, such as a diffraction grating or prism module, to implement pulse compression, as described in Sec. 22.2C and illu strated in Ex ample 22.5-1. The combination results in pulse compression by a factor v I + a 2 == J I + (Z/ZNL)2. EXAMPLE 22.5-1. Pulse Compression Using Fiber SPM and Grating GVD. A 65-fs pulse of peak power Po = 300 kW at a central wavelength Ao = 620 nm is chirped by a 9-mm long silica-glass optical fiber of cross-sectional area A = 100 J-Lm 2 , as illustrated in Fig. 22.5-4. At this wavelength, n2 3.2 x 10- 20 m 2 fW so that the nonlinear characteristic length is IZNLI = 12n2IOkol-l = AoA141T"ln2lPo 0.5 mm. Since the fiber length Z = 9 mm, the chirp parameter introduc ed by t he SPM is a = Z I ZNL = 18. This corresponds to a maximum pulse compression factor V I + a 2 18, or a compressed pulse of width 3.6 fs. The fiber also introduces GVD. At 620 nm, {3" = 6 X 10- 26 s2/m, so that the dispersion length for a pulse of width TO = 65 fs is Zo = T5/2{3" = 3.5 cm. Since Zo » ZN, SPM dominates GVD. To achieve maximum compression, the grating must introduce a chirp coefficient b = [al (1 + a 2 )]T5 2.35 x 10- 28 S2 = (3.6 fs)2. U nchirped pulse )) A A " . ".fji' ,;::: ::. ':".-.'., Nonlinear optical fiber (positive SPM) Chirped pulse Diffraction grating (negative GVD) Compressed pulse Figure 22.5-4 Pulse compression by a combination of a quadratic phase modulation (QPM) (introduced by SPM) and a chirp filter. The phase modulator is implemented using an optical fiber exhibiting SPM, via the optical Kerr effect. The chirp filter is implemented using the GVD introduced by a diffraction grating. B. Optical Solitons The interplay between self-phase modulation (SPM) and group-velocity dispersion (GVD) in a medium exhibiting both the nonlinear optical Kerr effect and linear dis- persion can result in a net pulse spreading or pulse compression, depending on the magnitudes and signs of these two effects. Under certain conditions, an optical pulse of prescribed shape and intensity can travel in such a medium without ever altering its shape, as if it were traveling in an ideal linear nondispersive medium. This occurs when
22.5 ULTRAFAST NONLINEAR OPTICS 989 GVD fully compensates the effect of SPM, as illustrated in Fig. 22.5-5(c). Such pulse- like stationary waves are called solitary waves. Optical solitons are special solitary waves that are orthogonal, in the sense that when two of these waves cross one another in the medium their intensity profiles are not altered (only phase shifts are imparted as a result of the interaction), so that each wave continues to travel as an independent entity. (a) A I A I t Linear dispersive medium (negative GVD) A t (b) Nonlinear nondispersive medium (positive SPM) (c) I j t Nonlinear dispersive medium (negative GVD + positive SPM) Figure 22.5-5 (a) In a linear medium with negative GVD (anomalous dispersion), the shorter- wavelength component B has a larger group velocity and therefore travels faster than the longer- wavelength component R; this results in pulse spreading. (b) In a nonlinear medium with positive optical Kerr effect (n2 > 0), SPM introduces a negative frequency shift in the leading half of the pulse (denoted R) and a positive-frequency shift in the trailing half (denoted B). The pulse is chirped, but its shape is not altered. If the chirped wave in (b) travels in the linear dispersive medium in (a), the pulse will be compressed since the blue-shifted half catches up with the red-shifted half. (c) If the medium is both nonlinear and dispersive, the pulse can be compressed, expanded, or maintained (creating a solitary wave), depending on the magnitudes and signs of GVD and SPM. This illustration shows a solitary wave created by a balance between negative GVD and positive SPM. The soliton process may be visualized by the mechanical analog illustrated by the cartoon in Fig. 22.5-6. Here, the heavy car represents the central portion of the optical pulse. It alters the surface of the ground, assumed to be elastic, much like the intense pulse peak alters the refractive index of the medium. The fast sports car, which is analogous to the trailing side of the pulse, is slowed down by the inclination created in the surface. The slow bicycle, which is analogous to the leading side of the pulse, is accelerated by the down-sloped surface. The result of this self-sustained process is that the three members of the team travel at the same velocity, and maintain the distances separating them. .'\ OJ. .0 A1Ifa 0 111 · o. ....,..- r_6--;'" do' Figure 22.5-6 Transportation analog of the soliton. Solitons have a characteristic pulse profile and level of intensity for which the effects of SPM and GVD are balanced. For these pulses, the chirping effect of SPM perfectly compensates the natur':ll pulse expansion caused by the GVD. Any slight spreading
990 CHAPTER 22 ULTRAFAST OPTICS of the pulse enhances the compression process, and any pulse narrowing reduces the compression process, so that the pulse shape and width are maintained. Solitons can be thought of as the modes (eigenfunctions) of the nonlinear dispersive system. A mathematical analysis of this phenomenon is based on solutions of the nonlinear wave equation that governs the propagation of the pulse envelope, as described subsequently. However, we first present a simple derivation of the soliton condition. Soliton Condition A correct expression for the soliton condition is obtained by equating the sum of the phases introduced by SPM and GVD within an incremental distance z to zero. As de- scribed earlier in this section, a pulse traveling through a nonlinear medium exhibiting the optical Kerr effect undergoes SPM, which introduces a quadratic phase modulation exp(j(t 2 ), with ( == 2n2fokoZ/T6, where fo and TO are the pulse peak intensity and width, respectively, and n2 is the optical Kerr coefficient. Also, as described in Sec. 22.3, GVD in a linear dispersive medium introduces a phase shift at 2 /T6, where the chirp parameter a == z / Zo == 2(3" z / T6, (3" is the material dispersion coefficient, and I Zo I is the dispersion length (see Table 22.3-1). The condition for the pulse to travel as a soliton is that the two phase factors are equal in magnitude and opposite in sign, I.e., a ( == -2' TO (22.5-12) or equivalently, (3" kOn2 f O == -2' TO (22.5-13) Soliton Condition (Phase) or equivalently, ZNL == -zo, (22.5-14 ) Soliton Condition (Length) i.e., the GVD dispersion length equals the nonlinear characteristic length. In other words, the phase shift introduced by SPM for a propagation distance equal to twice the GVD dispersion length Izol is unity (-kOn2f02zo == 1). We may alternatively derive this condition by thinking of the medium as a periodic sequence of localized SPM elements separated by pulse spreading elements (GVD) of widths z, as illustrated in Fig. 22.5-7. The scheme is identical to the pulse relaying system described in Exercise 22.3-2. In fact (22.5-12) may be derived from (22.3-23) in the limit z ---7 o. Another expression of the soliton condition is in terms of the pulse amplitude Ao, where fo == IAoI2/2TJ and TJ is the electromagnetic impedance of the medium. The result is written in terms of the product of the pulse peak amplitude Ao and temporal width TO, AOTo == vi -(3" /ry, (22.5-15) Soliton Condition (Area) where ry== kOn2/2TJ==nn2/AoTJ (22.5-16)
22.5 ULTRAFAST NONLINEAR OPTICS 991 Ie: z .1 EJ EJ 13 T(z) SPM GVD SPM GVD SPM GVD -5 Tl ........'"d . TO o a(z) Q) e-'Q) a .- S ..c 0 U -a z z Figure 22.5-7 Simple model for a medium with negative GVD and positive SPM. is another material parameter. Note that ry and {3" are assumed to have opposite signs. Thus, the product of the peak amplitude and width AOTo is a constant determined by the ratio of the parameter (3", which describes GVD, and the parameter ry, which describes SPM. For a given material, the product AOTo is fixed, and therefore these implications follow: . The pulse peak amplitude Ao is inversely proportional to the pulse width TO. . The pulse peak power is inversely proportional to T6. . The pulse energy density J I(t)dt is inversely proportional to TO, so that a soliton of shorter duration must carry greater energy. By solving the nonlinear wave equation that governs pulse propagation in a medium exhibiting both SPM and GVD, it will be shown subsequently that one of the solutions is the soliton pulse IA(t)1 == IAol sech(t/To) (22.5-17) Soliton Envelope where sech(.) == 1/ cosh(.) is the hyperbolic-secant function illustrated in Fig. 22.5-8. This is a symmetric bell-shaped function with the following characteristics: . Peak amplitude = Ao . FWHM width of amplitude profile == 2.63 TO . Area under amplitude profile = 21T AOTo . Intensity I(t) ex IAol2 sech 2 (t/To); width TFWHM == 1.76 TO The Nonlinear Slowly Varying Envelope Wave Equation To describe the propagation of an optical pulse in a nonlinear dispersive medium exhibiting both GVD and SPM we start with the wave equation (21.1-3), [ 1 8 2 ] 82 V 2 - c ot 2 C. = /10 &2 eh + P NL ) (22.5-18) where £ (r, t) is the electric field, P L (r, t) is the linear component of the polarization density, which is governed by the medium dispersion, and P NL == 4X(3)£3 is the
992 CHAPTER 22 ULTRAFAST OPTICS T FWHM . t Figure 22.5-8 The sech function compared to a Gaussian function of the same height and width (FWHM). nonlinear component of the polarization density, which is assumed to be nondispersive. Bringing the linear term from the right-hand side to the left-hand side of (22.5-18) and rewriting the equation in the Fourier domain, we obtain [\7 2 + 13 2 (w)] £ == -/-low 2 P NL (22.5-19) where fJ(w) is the propagation constant in the linear medium and E == E(r,w) and P NL == P NL (r, w) are Fourier transfOrlTIS of £ (r, t) and P NL (r, t), respectively. In the absence of nonlinearity, (22.5-19) reproduces the Helmholtz equation (2.2-7). We consider a plane-wave optical pulse traveling in the z direction with central angular frequency Wo and central wavenumber 130 == 13 (wo) == Wo / c, £ == Re { A ( z, t) exp [j ( Wo t - 130 z] } (22.5-20) and assume that the complex envelope A is a slowly varying function of t and z (in comparison with the period 2nlw o and the wavelength 27r/fJo, respectively). Using three assumptions: (1) slowly varying envelope, (2) weak dispersion, and (3) small nonlinear effect, it will be shown that the envelope A (z, t) satisfies the differential equation: Dv a 2 A 2 . ( a 1 a ) -- +'YIAI A+J - + -- A == 0 47r at 2 az v at (22.5-21 ) Nonlinear SVE Wave Equation where v == 1/13' is the group velocity, Dv == 27r 13" is the dispersion coefficient, and 13' and 13" are the first and second derivatives of fJ(w) with respect to w at w = wo, and 'Y is given by (22.5-16). For a linear medium 'Y = 0, and the linear SVE wave equation (22.3-20) is reproduced. D *Derivation of the Nonlinear SVE Wave Equation. Beginning with the nonlinear Helmholtz equation (22.5-19) and using certain approximations we win derive the nonlinear SVE equation (22.5- 21). Substituting E == A(z, w - wo) exp( -j(3oz) and P NL == ANL(Z, W - wo) exp( -j(3oz) into (22.5-19) and defining 0 == w - wo, we obtain [ ::2 + rp(w)] [A(z,w) exp( -j.Bo z )] = -JL o W 2 ANdz,!1) exp( -j.Boz). (22.5-22) We now simplify (22.5-22) using a number of approximations: . Since w wo, the w 2 factor in the right-hand side of (22.5-22) is approximated by w5. . When the SVE approximation (d 2 jdz 2 )[Aexp(-j(3oz)] [-j2(3odAjdz - (36A] exp(-j(3oz) is applied, (22.5-22) becomes [-j2(3odjdz] A + [(32(WO + 0) - (35] A == -P o w5 A NL. (22.5-23)
22.5 ULTRAFAST NONLINEAR OPTICS 993 . Assuming weak dispersion, {32 (wo + 0) - {36 2{30 [{3 (wo + 0) - {30] . Assuming further a 3-term Taylor-series expansion (3(wo + 0) = {3o + {3'O + {3"02, (22.5-23) becomes -j2(30 + 2(30 (n(3' + n2(3") A = -lLowANL. (22.5-24) . Since P NL = 4X(3) £3, P NL contains components near the frequencies Wo and 3wo. Retaining only the term near wo, we write P NL = Re{ANL(z, t) exp[j(wot - {3oz)]}, where ANL(Z, t) is a slowly varying envelope. Using (22.5-20), it follows [see (21.3-3a)] that A NL = 3X(3) IAI2 A. (22.5- 25) Finally, we transform (22.5-24) back to the time domain, using the fact that jOA(z,O) and -02A(z,0) are equivalent to BA/Bt and B2A/Bt 2 , and using (22.5-25), we obtain the nonlinear SVE equation (22.5-21). Equation (22.5-9) may also be obtained if we assume that the nonlinear medium is approximately linear with a propagation constant (3(w) + {3, where{3 = (wo/ co)n2I. The intensity I = IAI2 /2'TJ is assumed to be sufficiently slowly varying so that it may be regarded as time independent. The Fourier analysis, which led to the differential equation (22.3-24) for the linear medium, is then simply modified by an added term proportional to (3A. This term produces the additional term 'Y IAI2 A, so that (22.5-21) is reproduced. . Nonlinear Schrodinger Equation Equation (22.5-21) must be satisfied by the complex envelope A(z, t) of a plane- wave opticaJ pulse traveling in the z direction in an extended nonlinear dispersive medium with group velocity v, dispersion parameter (3", and nonlinear coefficient 'Y. As previousJy mentioned, a solitary-wave solution is possible if (3" < 0 (i.e., the medium exhibits negative GVD) and'Y > 0 (i.e., the optical Kerr coefficient n2 > 0). It is convenient to rewrite (22.5-21) in terms of dimensionless variables by normaliz- ing the time, the distance, and the amplitude to the scales, TO, 2z0, and Ao, respectively: . TO is the pulse width . Zo === T6/ 2 (3" is the dispersion length of the linear dispersive medium for this pulse width . Ao === (-(3" /'Y)1/2 /TO is the pulse peak amplitude that satisfies the soliton condi- tion (22.5-15). U sing a retarded frame of reference, and defining the dimensionless variables, t - z/v t=== , TO z Z===- 2zo ' A 'Ij; = Ao ' (22.5- 26) the nonlinear SVE wave equation in (22.5-21) is converted to 1 8 2 1/J 2 .81/J 2 8t 2 + 11/J1 1/J + J 8z === 0, (22.5- 27) Nonlinear Schr6dinger Equation which is recognized as the nonlinear Schr6dinger equation.
994 CHAPTER 22 ULTRAFAST OPTICS Fundamental Soliton The simplest solitary-wave solution of (22.5-27) is obtained by assuming a space-time separable function in the form 'ljJ(z, t) == 'J'(t) exp[jZ(z)], where 'J'(t) and Z(z) are real functions. By direct substitution in (22.5-27) and using a separation of variables approach, this leads to two differential equations: Z' (z) == () and 'J''' (t) == 2 ({) - tJ2)'J', where () is a constant. Assuming that 'J' = 'J" = 0 at It I ---7 00, and 'J' = 1 and 'J" = 0 at t == 0 (the pulse peak), these ordinary differential equations may be solved by direct integration to yield 'J' ( t) == sech ( t) and Z ( z) == z. Therefore, 'ljJ (z, t) == sech( t) exp (j z/2) . (22.5-28) This solution is called the fundamental soliton. It corresponds to an envelope ( t - zlv ) A(z, t) = Ao sech TO exp (jz/4z o ) (22.5-29) Fundamental Soliton that travels with velocity v without altering its shape. This solution is achieved if the incident pulse at z = 0 is A(O, t) == Ao sech (tITo). (22.5-30) Higher-Order Soliton The fundamental soliton is only one of a family of solutions of the nonlinear Schr6dinger equation with solitary properties. Consistent with the initial pulse 'ljJ(0, t) == N sech(t), where N is an integer, is a solution called the N -soliton wave. Such a wave propagates as a periodic function of z with period zp == 7r 12, called the soliton period. This corresponds to a physical distance zp == 7r I Zo I == (7r I 2) T6 I I (3"1. At z = 0 the envelope A(O, t) is a hyperbolic-secant function with peak amplitude N Ao, i.e., N times greater than the fundamental soliton. As the pulse travels, it contracts initially, then splits into distinct pulses that merge subsequently and eventually reproduce the initial pulse at z = zp. This pattern is repeated periodically. As an example, the N == 2 soliton has a wavefunction 4" I}/' _ 4 cosh 3t + 3e JZ cosh t jz/2 tf/ (z, t) - e , cosh 4t + 4 cosh 2t + 3 cas 4z (22.5- 31) whose magnitude is illustrated in Fig. 22.5-9. The periodic compression and expansion of the multi-soliton wave is accounted for by a periodic imbalance between the pulse compression, which results from the chirp- ing introduced by self-phase modulation, and the pulse spreading caused by group- velocity dispersion. The initial compression has been used for generation of subpi- cosecond pulses. Soliton-Soliton Interaction When two solitons separated by some time delay are launched into the nonlinear medium their shape and time separation are altered as if they experience attractive or repulsive forces pulling them together or separating them. For example, two identical separated fundamental solitons are initially attracted as they travel through the medium and their time separation is reduced until they collapse into a single pulse, whereupon
22.5 ULTRAFAST NONLINEAR OPTICS 995 -2 0 2 Figure 22.5-9 Propagation of the fundamental soliton (N = 1) and the N = 2 soliton. they experience repulsive forces that separate them again into two pulses. The process is repeated periodically with a period Lp == 7r exp( T /2To) Zo, (22.5-32) where T is the initial center-to-center separation, TO is the width of the individual soliton and Zo is the GVD dispersion length. This can be shown by solving the nonlin- ear Schrodinger equation with the appropriate boundary condition. As an example, if T == 10TO, so that the pulses are well separated and only their tails interact, Lp 466z o is quite large. However, this effect can be significant in long optical fibers since it can set limits on fiber communication systems using solitons to represent bits, as described in Sec. 24.2E. EXAMPLE 22.5-2. Solitons in Optical Fibers. Ultrashort solitons have been generated in glass fibers at wavelengths in the anomalous dispersion regions (Ao > 1.3 pm), where the GVD is negative. They were first observed in a 700-m single-mode silica glass fiber using pulses from a mode-locked laser operating at a wavelength Ao = 1.55 pm. The pulse shape closely approximated a hyperbolic-secant function with TO = 4 ps (corresponding to TFWHM = 1.76 TO = 7 ps). At this wavelength the dispersion coefficient D).. = 16 ps/nm-krn (see Fig. 9.3-5), corresponding to {3" = Dv/21f = (-A/co)D)../21f -20 ps2/km. The refractive index n = 1.45 and the nonlinear coefficient n2 = 3.19 x 10- 20 m 2 /W, corre- sponding to l' = (1f / Ao)n2/rJ = 2.48 x 10- 16 m/y2 (where rJ = rJo/n = 260 0). The amplitude Ao = (1{3"I/1')1/2 /TO 2.25 X 10 6 Y /m, corresponding to an intensity fo = A6/2rJ 10 6 W / cm 2 . If the fiber area is 10pm 2 , this corresponds to a power of about 100 m W. The soliton period zp = 1fZo = 1fT?; /21{3" I = 1.26 km. Soliton Generation and Maintenance To excite the fundamental soliton, the input pulse must have the hyperbolic-secant profile with the exact amplitude-width product AOTO in (22.5-15). A lower value of this product will excite an ordinary optical pulse, whereas a higher value will excite the fundamental soliton, or possibly a higher-order soliton, with the remaining energy diverted into a spurious ordinary pulse. When the initial pulse has a different profile or is chirped, the resulting pulse can, under certain conditions, evolve into a fundamental or higher-order soliton after a distance equal to a few soliton periods. If the medium is lossy, the pulse power is gradually dissipated so that the nonlinear effect becomes weaker and dispersive effects take over, leading to pulse broadening and loss of the soliton nature of the pulse. In optical fibers, this problem may be addressed by use of distributed Raman amplification (see Sec. 21.3A) to overcome absorption
996 CHAPTER 22 ULTRAFAST OPTICS and scattering losses. Lumped amplification can also work if the amplifier spacing is well within the soliton period zp. Because of their unique property of maintaining their shape and width over long propagation distances, optical solitons have potential applications for the transmission of digital data through optical fibers at higher rates and for longer distances than presently possible with linear optics (see Sec. 22.1D). Optica] solitons of a few tens of picoseconds duration have been successfully transmitted through many thousands of kilometers of optical fiber. Soliton Lasers Optical-fiber lasers have also been used to generate picosecond solitons. The laser is a single-mode fiber in a ring cavity configuration (Fig. 22.5-10). The fiber is a combination of an erbium-doped fiber amplifier (see Sec. 14.3C) and an undoped fiber providing the pulse shaping and soliton action. Pulses are obtained by using a phase modulator to achieve mode locking. A totally integrated system has been developed using an InGaAsP laser-diode pump and an integrated-optic phase modulator. .' Pump -:;:; . Output Phase modulator Undoped fiber Erbium-doped (pulse shaping) fiber amplifier Figure 22.5-10 An optical-fiber soli- ton laser. Dark Solitons A dark soliton is a short-duration dip in the intensity of an otherwise continuous wave of light. Dark solitons have properties similar to the "bright" solitons described earlier, but can be generated in the normal dispersion region (Ao < 1.3 /-Lm in silica optical fibers). They exhibit robust features that may be useful for optical switching. Analogy Between Temporal and Spatial Solitons The optical solitons described in Sec. 22.5C are analogous to the spatial solitons (self- guided beams) described in Sec. 21.3B. Spatial solitons are monochrolnatic waves that are localized spatially in the transverse plane. They travel in a nonlinear medium with- out altering their spatial distribution, as a result of a balance between diffraction and spatial self-phase modulation in accordance with the nonlinear Schrodinger equation, A 8 2 A 2 . 8A - 47r 8x 2 + 'YIAI A + J 8z == 0, (22.5-33) Nonlinear Beam Diffraction where 'Y == 7rn2/ ATJo and n2 is the optical Kerr coefficient. Equation (22.5-33) is equivalent to (21.3-11). The nonlinear Schrodinger equation that describes temporal solitons in nonlinear dispersive media (22.5-25) may be rewritten in the moving frame (t ' == t-z/v, Z' == z) as Dv 8 2 A 2 .8A 47r 8t 2 + 'YIAI A + J 8z == 0, (22.5-34 ) Nonlinear Pulse Dispersion
22.5 ULTRAFAST NONLINEAR OPTICS 997 where 'Y == 7rn2/ ATJo. This is identical to (22.5-33) with time t playing the role of the transverse spatial coordinate x, and the dispersion coefficient - Dv (which gov- erns pulse dispersion) playing the role of the wavelength A (which governs beam diffraction). It is therefore evident that temporal solitons are formal analogs of spatial solitons. In fact the term soliton refers to generic solutions of the nonlinear Schrodinger equation, describing pulses that propagate without change; they may be temporal or spatial. Spatiotemporal Solitons and Light Bullets A spatiotemporal soliton is a combined temporal and spatial soliton, i.e., a pulsed beam that maintains its spatial and temporal profiles as it travels through a nonlin- ear medium exhibiting the optical Kerr effect (see Fig. 22.5-11). In this case, the temporal broadening associated with negative (anomalous) dispersion and the spatial spreading resulting from diffraction are simultaneously compensated for by self-phase modulation and self-focusing resulting from a positive nonlinear optical Kerr effect. The partial differential equation describing these three phenomena is a combination of (22.5-33) and (22.5-34), A 2 Dv 8 2 A 2 .8A --VTA+ -- +'YIAI A + J- == O. 47r 47r 8x 2 8z (22.5-35) Nonlinear Diffraction & Dispersion A necessary condition for spatiotemporal solitons is the equality of the dispersion length I Zo I == 7rTJ / I Dv I and the diffraction length Zo == 7r W / A so that TO /W o == (A/I D vl)1/2. WO, \ , . --..J CT(Z) - T W(z) t Xt ---------------------- --------------------- --- - Wo Wo · Z T ----------------------------------------;oT cTol (b) 1 Xt_____---- ------- T 1------ -+jCTO I+- (a) Figure 22.5-11 (a) Spatial and temporal spreading of a pulsed beam as a result of propagation in a linear dispersive medium. (b) A spatiotemporal soliton is a pulsed beam that maintains its spatial and temporal profiles as it propagates in a nonlinear medium. *c. Supercontinuum Light Supercontinuum light is high-brightness light with an ultrabroad continuous spectrum. Supercontinuum generation (SCG) is implemented by transmitting an ultrashort optical pulse of high peak power (a pump) through a nonlinear medium with special dispersive properties; examples are dispersion-shifted, dispersion-flattened microstruc- tured, and photonic-crystal optical fibers (PCFs). Supercontinuum light sources with spectra stretching from 400 nm to 3000 nm have been demonstrated. Several nonlinear mechanisms, including self-phase modulation (SPM), stimulated Raman scattering (SRS), four-wave mixing (FWM), and soliton self-frequency shift (SSFS), may contribute individually or jointly to SCG. These nonlinear effects are
998 CHAPTER 22 ULTRAFAST OPTICS sensitive to the sign of the medium dispersion at the central wavelength Ao of the pump pulse and to the relative location of the zero-dispersion wavelength AZD of the medium. The widest SCG spectra are obtained when Ao is close to AZD. It was the availability of nonlinear PCFs with AZD close to the wavelength of the Ti:sapphire laser that first made SCG practical. The following is a brief description of the principal nonlinear mechanisms that contribute to SCG; Fig. 22.5-12 provides schematic illustrations of these processes. Input t - ! A I A I A pulse fl _ CZ) ._ . . . Nonlinear AO AO AO fiber ...-.:::: Q .g -' rf.J 1--< Q) Q).- . . Q Q) 0 u S pectrall y broadened t light l ! SPM i "'off: SRS Q) Q) fl ... i. i:: .' .." ,. CZ) .- 1",<,;1'",," .... I I I I I I. 500 1000, 1500 A (nrn) (a) Soliton I 500 1000 , 1500 A.(nrn) (b) . . . - . - FWM FWM . 500 1000 1500 ). (nrn) (c) Figure 22.5-12 Principal nonlinear mechanisms for supercontinuum generation (SCG) via spectral broadening of an ultrashort pulse transmitted through a nonlinear dispersive fiber. (a) Self-phase modulation (SPM) combined with stimulated Raman scattering (SRS). (b) Soliton self- frequency shift (SSFS). (c) Four-wave mixing (FWM). . Self-phase modulation (SPM) is the principal mechanism for SCG in nonlinear fibers with normal dispersion (D). < 0) at the pump central wavelength Ao, since in this case solitons cannot be formed. As discussed in Sec. 22.5A, SPM results in pulse chirping, which causes spectral broad ening. A chirp coefficient a corresponds to spectral broadening by the factor V I + a 2 . For a medium of length L and optical Kerr coefficient n2, the chirp parameter is a == L / ZNL, where ZNL == (2n2 f oko)-1 is the nonlinear characteristic length and fo is the peak pulse intensity. . Stimulated Raman scattering (SRS) broadens the spectral distribution further toward the long wavelength side since it results in a frequency downshift. . When Ao is close to AZD the combined SPM/SRS broadens the spectrum into the anomalous region, creating conditions for soliton formation. Optical soli- tons generally experience a downshift of their carrier frequency, corresponding to shifts to longer wavelengths, which increases with pump power. This so-called soliton self-frequency shift (SSFS) originates from intrapulse stimulated Raman scattering (SRS). . In a microstructured fiber that has two widely separated, zero-dispersion wave- lengths with Ao lying between them, the dominant nonlinear mechanisms for spectral broadening are SPM and FWM. The SPM process broadens the pump pulse, enabling the phase-matching conditions for four-wave mixing (FWM) to be met. This generates new light at both lower and higher frequencies, corre- sponding to SCG with double-peaked spectra. With sufficient broadening, the two FWM peaks may merge into a single flat distribution.
22.6 PULSE DETECTION 999 22.6 PULSE DETECTION The measurement of an ultranarrow optical pulse is a challenging problem since the fastest available photodetector is usually too slow. Methods of addressing this problem rely primarily on the use of an ultrafast optical shutter (gate) controlled by another shorter reference pulse and a mechanism for introducing a controllable time delay between the two pulses. The measurement is repeated at different delays as the light transmitted through the gate is measured, providing an estimate of the profile of the pulse intensity I (t). To measure the pulse phase cjJ( t), interferometric techniques have been cleverly adapted in combination with nonlinear optical processes. In the spectral domain, the pulse is completely characterized by its spectral intensity S( v) and spectral phase 1jJ(v). These functions may be measured by use of optical spectrum analyzers and interferometric techniques, as will be described in this section. Another challenging aspect of ultrafast pulse detection is the fact that the optical components employed in the measurement system unavoidably alter the pulse before it is measured. Such effects must be minimized by careful system design, or compensated by appropriate post-detection signal processing methods. A. Measurement of Intensity Direct Photodetection The intensity profile of a short optical pulse may be directly measured by use of a photodetector with response time much shorter than the pulse duration. The measured photocurrent i ( t) == 9\AI ( t ) (22.6-1 ) Fast Detector is proportional to the pulse intensity I ( t), where A is the active area of the detector and 9\ is its responsivity (A/W) (see Sec. 18.1B). It is assumed here that A is sufficiently small so that the optical intensity is sampled at the position of the detector. When the detector's response time is significant, the photocurrent pulse is a broad- ened and distorted version of the optical pulse. Other measures must then be used to determine the true pulse shape. If hD(t) is the impulse response function of the detector, where I I ( t) h D (t) dt == 9\, then the photocurrent is the convolution (22.6-2) Arbitrary Detector which is a pulse of greater duration. When the response time [the width of hD(t)] is much shorter than the pulse width [the width of I(t)], then the convolution in (22.6-2) has the shape of the function with the longer duration, and the ideal relation (22.6-1) is recovered. In the other extreme, when the optical pulse duration is much shorter than the detector's response time, i(t) hD(t)A I 1(7 )d7, so that the photocurrent has the temporal profile of the detector's impulse response function, rather than the optical pulse. If the receiver circuit has a time constant 7 c longer than the short response time of the detector, then the ultimate response is i(t) 7;1 I hD(t)dt. AI I(7)d7, or i(t) = A J I(T)hD(t - T)dT, (22.6-3) Slow Detector Under such conditions, the receiver measures the area under the optical pulse, or the optical energy; the detector then lacks temporal resolution and may be modeled as an integrator. These three cases are illustrated schematically in Fig. 22.6-1. i(t) ::::: T;l9{A J I( T )dT.
1000 CHAPTER 22 ULTRAFAST OPTICS lA hD(t) L (b) (c) AA. t Figure 22.6-1 Response of a photodetector with impulse response function hD(t) to three pulses of (a) long, (b) intermediate, and (c) short duration. How might one measure the temporal profile of an ultrashort pulse of duration in the picosecond or the femtosecond regime by use of a "slow" detector with response time of a few tenths of a nanosecond at best? Measurement of Short Pulse with Slow Detector and Fast Shutter The temporal profile of a short optical pulse may be measured with a slow detector by use of a fast shutter (switch or gate). As illustrated in Fig. 22.6-2, the gate opens for only a short window of time during the course of the pulse, allowing a sample of the pulse to be detected by the slow detector. The measurement is repeated by opening the gate at different times, and a set of measured samples are used to estimate the pulse profile. Since electronically operated gates are not available at speeds in the picosecond or femtosecond range, we may use an optical gate controlled by a reference optical pulse of duration much shorter than that of the measured pulse (see Chapter 23). I(t)A Slow i(T) /t\ Gate I d etector l o i t .[1S]I(t)W(t-T) . * . 0 ;- W(t T) ! o T t let) Variable delay Wet) T Figure 22.6-2 Measurement of an optical pulse I(t) by use of an optical gate controlled by a much shorter gating pulse W ( t) . Two examples of optical gates used for measurement of ultranarrow pulses are shown in Fig. 22.6-3. We now assess the effect of the finite switching time on the measurement resolution. If W ( t) is the transmittance of the gate when initiated by a gating pulse at t == 0, then when the gating action is delayed by time T the transmitted optical pulse is I(t)W(t- T). When detected by the slow detector, the resultant photocurrent is proportional to the area under the transmitted pulse, i(t) ex J I(T)W(t - T)dT. (22.6-4 ) Under ideal conditions, the window function W(t) is a delta function 6(t) and the photocurrent is proportional to I ( T ), i.e., is a sample of the optical pulse at t == T. Otherwise, the measured photocurrent is proportional to the convolution between the optical pulse and the window function. The temporal resolution of the measurement is therefore equal to the width of the window function W (t), which is governed by the
22.6 PULSE DETECTION 1001 Vet) Kerr cell PBS SHG crystal V(t) I V r (t)1 2 Vet) U(t)Vr{t) Vr(t) Filter (a) Optical Kerr gate (b) SHG gate Figure 22.6-3 (a) An optical Kerr gate. The reference pulse intensity Ir(t) = IU r (t)1 2 alters the Kerr medium retardation. Since the test pulse is transmitted through two crossed polarizers with the Kerr medium in between, it is modulated by the gating function W(t) ex: Ir(t) = IU r (t)1 2 . (b) A second-harmonic generation (SHG) gate. The tested pulse U(t) and the gating pulse Ur(t), which have orthogonal polarization, combine in a collinear Type-II configuration (see Sec. 21.2D) and generate a pulse at the second-harmonic frequency with amplitude ex: U (t) U r (t), so that the gating function TTT(t) ex: Ur(t). gate/shutter speed. The delay T may be imparted to either the gating function W(t) or the optical pulse itself I ( t). Single-Shot Pulse versus Pulse Train The preceding method for measuring the shape of a short pulse with a slow detector can be easily implemented if a periodic train of identical pulses is available. The shutter is set at a different time delay T for each of a sequence of pulses, as illustrated in Fig. 22.6- 4, and the readings of the detector are recorded sequentially. The pulse repetition rate must, of course, be sufficiently low for the slow detector to recover before it measures a new pulse. let) I I I I :-T I I ! W(t) I I I . ( ! o I ! r 2T I I I -, I I I I I I I I I I :-+-3T t ) t Figure 22.6-4 Measurement of a pulse profile by sampling individual pulses of a pulse train at time delays 7 = mtl7, m = 0, 1,2, . . . . What if a single-shot pulse is to be measured? This may be accomplished by use of multiple detectors. Copies of the pulse are generated by a fan-out optical element and each copy is subjected to a different time delay before transmission through a gate controlled by the gating function as shown in Fig. 22.6-5. Temporal-fo-Spatial Transformation: Streak-Camera Principle The fan-out and multiple-delay concept depicted in Fig. 22.6-5 may be implemented optically by using an extended beam intercepted at an angle by a planar spatial detector (array detector, or CCD camera), as illustrated in Fig. 22.6-6. A pulsed plane wave traveling in the z direction has an intensity I (t - z / c). A wave traveling at an angle () has an intensity I (t - [x sin () + z cos ()] / c). If the beam is intercepted by a spatial detector in the plane z == 0, it detects the intensity I (t - x sin () / c), so that at the position x the pulse is delayed by time Tx == x sin () / c). Every detector element therefore has its own delay, implementing the scheme in Fig. 22.6-5. If a shutter takes a snapshot at time t == 0, the reading of the detector at x is proportional to I ( - x sin () / c).
1002 CHAPTER 22 ULTRAFAST OPTICS Bank of Array of delays Gate detectors 71 * Single-shot 72 * pulse I(t) X 73 * Gating . 1----' . Y pulse Wet) . Figure 22.6-5 Measurement of a single pulse by use of a bank of delays, a gate, and an array of detectors. Thus" the pulse shape is recorded spatially with an inverted profile scaled such that a pulse of width TO creates an image of transverse width CTo/ sin 8. A pulse of width TO =10 ps, e.g., extends over a distance CTo = 3 mm along the direction of propagation. At an angle 8 = 30° this corresponds to a width of 6 mm in the detector plane. This is the basic idea behind the streak camera. The pulsed light is reflected from a surface (a rotating drum in older technologies) and "streaked" so that rays hitting differ- ent points on the detector travel different distances and therefore experience different delays. Such position-dependent time delay may also be introduced by transmitting the beam through a glass wedge. c'l () Spatial Shutter detector n ri I B- z Figure 22.6-6 Temporal-to-spatial transforma- tion of an optical pulse by use of an oblique wave. y >' The shutter used in the system in Fig. 22.6-6 may be an optical Kerr gate or a SHG gate controlled by the gating pulse. One useful implementation of the SHG gate in such a scheme is illustrated in Fig. 22.6-7. The tested pulse and the gating pulse take the form of orthogonally polarized oblique waves at angles 8 and -8 with the z axis. Their wavefunctions are U (t - [x sin 8 + z cos 8]/ c) and U r (t - [-x sin 8 + z cos 8]/ c) so that the relative time delay is Tx (2 sin 8/ c)x at the position x. The gate is based on a non-collinear type-II SHG process. The generated wave at the second-harmonic frequency has a wavefunction proportional to the product UU r , so that the measured intensity is proportional to II r. As a result, the detected signal is proportional to the intensity autocorrelation function G I( Tx). Measurement of Intensity Autocorrelation As mentioned earlier, the basic principle for measurement of an ultrashort optical pulse 1 ( t) with a slow detector is based on the use of a shorter gating pulse W ( t) to open and close an optical gate. When no such pulse exists, the tested pulse may be compressed and used for this purpose, and in this case W ( t) is a compressed version of 1 (t). Another possibility is a squared version of the pulse, obtained for example by second- harmonic generation. The squared function 1 2 (t) is narrower than 1 (t). Higher-order nonlinear processes may also be used to generate even narrower pulses, albeit of lower intensity.
22.6 PULSE DETECTION 1003 Vet) Nonlinear crystal XL£ z Second-harmonic Spatial detector )I - uy Vet) V r(t) Figure 22.6-7 Measurement of a single-shot pulse by use of type-II SHG and time-to-space transformation (streaking). If such pulse compression is not feasible or desirable, the tested pulse may be directly used as the gating function, as illustrated schematically in Fig. 22.6-8. The photocurrent is then proportional to the intensity autocorrelation function (22.6-5) Intensity Autocorrelation Since I ( t) is a real function of finite duration, G 1 ( 7) is a symmetric function that drops from a peak value G 1(0) at 7 = 0 to zero at 7 = 00. The autocorrelation function of a pulse of arbitrary shape is generally a broader symmetric pulse. For example, a Gaussian pulse with intensity I(t) == exp( -27 2 17'!;) and width 70 has a Gaussian autocorrelation G 1 (7) ex exp(-72/76), which may be written as exp [ - 2 ( 7 I J2 70) 2], so that the width is J2 70. G1(T) = J I(t)I(t - T)dt. I(t) I:1k . 0: t I Slow G 1 ( T) /t\ detector o . T Variable delay I(t-7) 7 o 7 t Figure 22.6-8 Measurement of intensity autocorrelation. Knowledge of the autocorrelation function is generally not sufficient to determine the function itself. This can be seen by noting that the Fourier transform of G 1 ( 7 ) equals Ii (v) 1 2 , where i (v) is the Fourier transform of I ( t). Measurement of G 1 ( 7 ) permits us to determine the magnitude Ii (v) I but provides no information on its phase and hence cannot be used to completely recover the complex envelope. An exception is the case for which the pulse is symmetric, i.e., I( -t) = I(t), since in this case i(v) is real, i.e., has zero phase. If the mathematical profile of a nonsymmetric function is known, then measurement of the autocorrelation function suffices to estimate its parameters, e.g., its width.
1004 CHAPTER 22 ULTRAFAST OPTICS B. Measurement of Spectral Intensity Optical Spectrum Analyzer The spectral intensity S(v) == IA(v)12 of an optical pulse of complex envelope A(t) may be measured by use of an optical spectrum analyzer. The analyzer is simply a bank of spectral filters tuned to a set of frequencies/wavelengths. If a bank of "slow" detectors is used to detect the energy in each of the spectral components, then the result of measurement is the spectral intensity S( v). It is not generally possible to retrieve a function A(t) from the magnitude of its Fourier transform A(v) in the absence of phase information. An exception is the case of a symmetric pulse, whose Fourier transform is real. An optical implementation is shown in Fig. 22.6-9. (v) Vet) Fourier V(v) 1.1 2 S - transform Spatial . V. ! detector Optical spectrum analyzer [0 (a) (b) Figure 22.6-9 Measurement of spectral intensity with an optical spectrum analyzer. (a) System. (b) Optical implementation using prisms. Interferometric Spectrum Analyzer The spectral intensity S(v) of an optical pulse may also be measured by use of an interferometer (Fig. 22.6-10). Recall from Sec. 11.2 that the Michelson interferometer may be used as a Fourier-transform spectrometer. When a pulsed optical beam of complex wavefunction U ( t) is split into two beams by use of a beamsplitter and one beam is delayed by time T with respect to the other, the result is an optical field [U(t) + U( t - T)] with intensity IU(t) + U( t - T) 1 2 . When detected with a "slow detector," the result is a function of the optical delay, RU(T) = J IU(t) + U(t - T)1 2 dt = J IU(t)1 2 dt + J IU(t - T)1 2 dt + Re J U*(t)U(t - T)dt Substituting U(t) == A(t) exp(j27rv o t), we obtain RU(T) == GA(O) + Re {GA(T) exp( -j27rVOT)} == GA(O) + IGA(T)I cos [27rVOT - arg {GA(T)}], (22.6-6) where GA(T) = J A*(t)A(t - T)dt (22.6-7) is the autocorrelation function of the complex envelope. The function G A ( T) equals the inverse Fourier transform of the spectral intensity S(v) == IA(v) 1 2 . The measurement
22.6 PULSE DETECTION 1005 Ru( T) is a fringe pattern of visibility IG A (T) 1/ G A(O). The scheme permits us to deter- mine G A ( T) through careful analysis of the visibility and location of the fringes. The interferometer therefore provides the same information as the conventional spectrum anal yzer. Slow detector U(t) -* RefiT) I -+--. o T U(t) t Variable delay T U(t-T) (a) CTt (b) t Figure 22.6-10 Interferometric measurement of the pulse spectral intensity. The interferogram is used to determine the autocorrelation function of the pulse envelope G A ( T) whose Fourier transform is the spectral intensity. c. Measurement of Phase Full characterization of an optical pulse involves measurement of the complex enve- lope, i.e., the magnitude and phase of the wavefunction U(t) == JT(i) exp[j27rvat + <p( t)], o r equivalently the magnitude and phase of its Fourier transform V(v) == V S(v) exp[j'ljJ(v)]. The techniques presented in Sec. 22.6A provide measurement of the intensity I(t), but no information on the phase <p(t). Those presented in Sec. 22.6B provide measurement of the spectral intensity S(v), with no information on the spectral phase 'ljJ(v). Under certain conditions, a complex function may be fully determined from knowledge of its magnitude and the magnitude of its Fourier transform, i.e., from I(t) and S(v). This section introduces other measurements that are directly sensitive to the phase <p( t) or the spectral phase 'ljJ (v). Phase measurement is often based on interferometry since the intensity at the output of an interferometer is highly sensitive to the difference between the phases of the interfering waves. Spectral Interferometry A conventional method for measuring phase is heterodyning, which is a form of "time- domain" interferometry (see Sec. 2.6B). The pulse U(t) == JT(i) exp[j27rvat + <p(t)] is mixed with a known reference pulse Ur(t) == JT(i) exp[j27rv r t + <Pr(t)] with a different central frequency V r == Va + I. The intensity of the sum IU(t) + U r (t)1 2 == I(t) + Ir(t) + 2 V I (t)Ir(t) cos [27r It + <Pr(t) - <p(t)] (22.6-8) is an interferogram with a beat frequency I (fringes per second) equal to the difference between the central frequencies, and a time-varying phase <Pr(t) - <p(t), which may be readily extracted from the interferogram. For ultrafast pulses, however, the detector is slow and all temporal features of the interferogram are washed out, so that the technique of heterodyning, or temporal interferometry, is not applicable. Interferometry does work, however, if performed in the Fourier domain, and is then known as spectral interferometry. The pulse U(t) is delayed by a fixed time T and added to a known reference pulse U r ( t) of the same frequency. The Fourier transform
1006 CHAPTER 22 ULTRAFAST OPTICS of the sum U (t - T) + U r (t) is then measured with a slow detector, creating an interferogram, as ill ustrated in Fig. 22.6-11. If th e Fouri er transforms of U (t) and U r (t) are V(v) == J S(v) exp [j(v)] and v;. (v ) == J Sr(v) exp [jr(v )], respectively, then the spectral interferometer measures the interferogram: IV(v)e-j27rTV + v;. (v) 1 2 == S(v) + Sr(V) +2 J S(v)Sr(v) COS [27rTV + r(V) - (V)], (22.6-9) which is a fringe pattern (in frequency) with visibility determined by the spectral intensity S(v) and fringe locations governed by the phase difference (v) -r(v). The measurement therefore yields full information on V (v) and hence on U ( t). The duality between temporal and spectral interferometry may be seen by noting that (22.6-8) and (22.6-9) are identical in form, with t and v playing dual roles, and the delay T playing the role of the frequency difference f. The main difficulty with spectral interferometry is the need for a known reference pulse. I / U(t) U(t-T) Fourier ... + ..... T , transform - - ) - Ur(t) Detector Fixed delay I \ o v Figure 22.6-11 A spectral interferometer generates an interferogram in the Fourier domain. Self-Referenced Spectral Interferometry The tested pulse cannot be used as its own reference since the phase term in (22.6- 9) vanishes if r(v) == (v). One method of addressing this problem is to use a frequency-shifted version of the tested pulse as a reference, i.e., Vr(v)= V(v + f). The result is an interferogram, IV(v)e-j27rVT + V( v + f)12 == S( v) + S(v + f) +2 J S(v)S(v + f) cos [27rVT + (v) - (v + f)], (22.6-10) and the system is illustrated schematically in Fig. 22.6-12. From this Fourier-domain interferogram, the phase difference (v + f) - ( v) may be estimated. If the frequency shift f is small, the phase difference may be used as an approximation of the derivative d/dv, which may be integrated to provide the phase (v). Nonlinear Interferometry As was shown earlier, the conventional (time-domain) interferometer, which mea- sures the area under the function I U ( t) + U (t - T) 1 2 , provides ful] information on the spectral intensity, but no information on the spectral phase. One approach for extracting or verifying phase information from an interferometer is to transform the sum U ( t) + U (t - T) by a squaring operation [U ( t) + U (t - T)] 2 prior to detection, i.e., extract the area under the function I [U ( t) + U (t - T)] 21 2 . This leads to R)(T) = J I [U(t) + U(t - T)]21 2 dt, (22.6-11 )
22.6 PULSE DETECTION 1007 - U(t) U(t-T) Fourier ..... + ..... T - transform - , J - Detector +1 U(t) e j27rft Fixed delay o v Frequency shift Figure 22.6-12 Self-referenced spectral interferometer. an operation illustrated by the block diagram in Fig. 22.6-13. The squaring operation may be readily implemented by a process of optical second-harmonic generation in a nonlinear optical crystal. Variable delay * R( T) / \ / U(t) >t U(t) ( . )2 \ SHG Slow detector o T T U(t - T) Figure 22.6-13 Nonlinear interferometer. To show that the new function R) ( 7) contains phase information, we substitute U(t) == A(t)exp(j27rvot) into (22.6-11), and separate terms with frequencies 0, Vo and 2vo to obtain RB)(7) ex C O (7) + 4Re {Cl(7)ej27rVoT} + 2Re {C2(7)ej47rVOT} (22.6-12) where CO(T) = J 1 2 (t)dt + J I 2 (t - T)dt + 4 J I(t)I(t - T)dt == 2G 1 (O) +4G 1 (7), (22.6-13) C 1 (T) = J A*(t)A(t - T) [I(t) + I(t - T)] dt, (22.6-14) C 2 (T) = J [A*(t)A(t - T)]2dt, (22.6-15) and G I( 7) is the intensity autocorrelation function given by (22.6-5). The function R) ( 7) is therefore the sum of three terms: a nonoscillatory term Co ( 7) and two oscillatory terms at frequencies Vo and 2vo. These terms may be separated by Fourier analysis of R) ( 7 ). The first term depends on the intensity autocorrelation function G I ( T) and has no phase dependence. The two other terms depend on both the pulse intensity and phase. The overall function is bounded by an upper envelope with max- imum value RB)(O) == 16 J 1 2 (t)dt == 16G 1 (O) and a lower envelope with minimum
1008 CHAPTER 22 ULTRAFAST OPTICS value RB) (0) == O. Its asymptotic value is RB) ( 00) == Co ( 00) == 2 J [2 (t) dt == 2G ](0). The ratio RB) (00) / RB) (0) therefore changes from a peak value of 8 at 7 = o to an asymptotic value of unity at 7 = 00. As an example, for a linearly chirped Gaussian pulse with time constant 70 and chirp parameter a, C O (7) == 2G](0) [1 + 2exp(-7 2 /76)] C 1 (7) == 2G](0)exp [-(3+a2)72/47] cos(a7 2 / 27 6), C 2 (7) == G](O)exp [-(1 +a 2 )7 2 /76]. (22.6-16) (22.6-17) (22.6-18) The normalized function RB) ( 7) / R) ( 00) is plotted in Fig. 22.6-14 for three values of the chirp parameter a. It is evident that the profile of the interferogram, particularly the point at which the oscillatory terms vanish, is highly sensitive to a, and can therefore be used to estimate a from experimental data. 8 8 8 , \ , I , a=O , a=4 a=8 , , 6 6 , 6 , , , I , , , I I , I 4 I 4 4 , I , I , I I 2 2 2 -- \ I 0 7/70 0 2 7/70 0 2 7/70 -2 -1 0 2 -2 -I 0 -2 -1 0 Figure 22.6-14 Normalized intensity autocorrelation function RHG ( (0) Co ( 00 ) == 2 J [2(t)dt as a function of the normalized time delay 7/70 for a chirped Gaussian pulse with three values of chirp parameter a. There is no general procedure for estimating the pulse phase from the measurement of RB) ( 7 ). However, this measurement can be used to verify known models for pulse amplitude and phase or estimate unknown parameters. Nonlinear Interferometry with Nonlinear Detectors. In an alternative implemen- tation of the nonlinear interferometer depicted in Fig. 22.6-13, the squaring opera- tion is carried out by the detector itself. This is accomplished by use of a detector based on two-photon absorption, e.g., a photodiode with a bandgap energy greater than the photon energy, but smaller than twice the photon energy. In such a detector, the photocurrent is proportional to the square of the intensity (since it absorbs pairs of photons). As a result, the nonlinear interferometer measures the function R2)(T) = J IU(t) + U(t - T)1 4 dt, (22.6-19) which, like (22.6-11), contains information on the pulse distribution and width.
22.6 PULSE DETECTION 1009 *D. Measurement of Spectrogram As mentioned in Sec. 22.1 A, the spectrogram of an optical pulse U ( t) is a time- frequency representation equal to the squared magnitude of the Fourier transform of the pulse as seen through a moving window or gating function W ( t) : S(v, T) = 1<I>(v, T)12; <I> (v, T) = J U(t)W(t - T) exp( -j27rvt)dt. (22.6-20) The spectrogram may be measured by transmitting the pulse U(t) through an optical gate controlled by a time-delayed gating function W (t - T), and measuring the spec- trum of the product U(t)W(t - T) with a spectrum analyzer at each time delay T, as depicted schematically in Fig. 22.6-15. An optical implementation relies on a moving mirror to introduce the time delay, an optical spectrum analyzer such as that shown in Fig. 22.6-9, and an appropriate optical gate. The technique is known as frequency- resolved optical gating (FROG). v Gate Vet) Vet) W(t-7) Fourier S(7, .... X I--- - transform Variable - delay J Detector Wet) 7 W(t-7) v) 7 Figure 22.6-15 Measurement of the spectrogram S( v, 7) by frequency-resolved optical gating (FROG). In the absence of a sufficiently short gating function W (t), the pulse U (t) itself, or another related pulse, may be used for this purpose. The relation between W ( t) and U ( t) depends on the nature of the used optical gate, as illustrated by the following examples: . For a second-harmonic generation (SHG) gate (see Fig. 22.6-7) with input waves U (t) and U (t - T) at the fundamental frequency, the wave at the second-harmonic frequency is proportional to the product U ( t) U (t - T), so that W ( t) ex U ( t) and <I> (v, T) = J U(t)U(t - T) exp( -j27rvt)dt. (22.6-21) The time-frequency function in (22.6-21) is known as the Wigner Distribu- tion Function. The overall optical system that implements the block diagram in Fig. 22.6-15 is depicted in Fig. 22.6-16(a) and the system is known as the SHG-FROG. This system is suitable for single-shot measurement, as discussed earlier. . For a polarization-based optical Kerr gate [Fig. 22.6-3(a)], W(t) is proportional to the pulse intensity I(t) so that W(t) ex I(t) == IU(t) 1 2 and <I> (v, T) = J U(t)IU(t - T)1 2 exp( -j27rvt)dt. (22.6-22) When this gate is used to implement the block diagram in Fig. 22.6-15 the system becomes the polarization-gated FROG (PG- FROG) illustrated in Fig. 22.6-16(b).
1010 CHAPTER 22 ULTRAFAST OPTICS U(t) U(t) SHG crystal Spectrum analyzer U(t)U(t-T) U(t-T) t CT SHG gate L>[] (a) SHG-FROG U(t) Kerr cell (b) PG-FROG Spectrum analyzer U(t) I U(t-'t) 1 2 t CT Optical Kerr gate L>[] Figure 22.6-16 Two implementations of frequency-resolved optical gating (FROG). (a) SHG- FROG. (b) PG-FROG. Other nonlinear optical configurations have been devised, including a gate based on third-harmonic generation, which corresponds to the gating function W(t) ex U 2 (t), and a gate based on self-diffraction, which corresponds to W (t) ex [U* (t)] 2 . Estimation of the Pulse Wavefunction from the Spectrogram In any of its many variations, the spectrogram S(v, T) is a 2D "picture" that may be used to characterize the optical pulse or display signatures of its key features. It may also be used to estimate the pulse complex wavefunction U ( t), magnitude and phase. The estimation of U(t) from the measured spectrogram S(v, T) is not straightfor- ward. A general expression for S(v, T), as measured by any of the previously men- tioned gating systems, may be written in the form S(v, T) = 1<I>(v, T)1 2 ; <I> (v, T) = J g(t, T) exp( -j27rvt)dt, where g(t, T) == U(t)W(t-T) and W(t) is related to U(t); for example, W(t) == U(t) for the SHG-FROG, and W(t) == IU(t)1 2 for the PG-FROG. If the complex function <I> (v, T) were known, then U ( t) may be readily estimated as follows. By taking the inverse Fourier transform of <I> (v, T) with respect to v at each T, we obtain (22.6-23) g(t, T) = J <I> (v, T) exp(j27rvt)dv. Knowing g(t, T) == U(t)W(t - T), the wavefunction U(t) may be computed by integration over T, (22.6-24) J g(t, T)dT = J U(t)W(t - T)dT = U(t) J W(t - T)dT ex: U(t). (22.6-25) The proportionality constant equals the area of the window function, which is un- known.
READING LIST 1011 The problem of estimating <I> (v, 7) from the measured S(v, T) == 1<I>(v, 7)1 2 is a "missing-phase problem." Many algorithms have been devised for solving this and similar phase problems. One iterative approach follows the steps illustrated by the diagram: s 1if>1 if> v rf arg{ if>} r--" if> 1. Beginning with the measured spectrogram S(v, 7), the magnitude I <I> (v, 7)1 [S( V, 7)] 1/2 is detennined. Using some initial guess for the missing phase arg { <I> ( v, 7) }, the previous procedure [inverse Fourier transform <I> ( v, 7) with respect to v and integrate over 7] is used to estimate U ( t) up to an unknown proportionali ty constant. 2. Knowing U(t), <I> (v, 7) is computed and a new estimate of the unknown phase arg{ <I> (v, 7)} is determined and used in combination with the measured magni- tude I <I> ( v, 7) I to obtain a new and better estimate of U ( t ) . 3. The process is repeated until it converges to a pulse wavefunction U(t) that is consistent with the measured spectrogram. An example is shown in Fig. 22.6-17. 0 0.5 1 » 0 0.5 . 1 7r (!) s::: V':i (!) <.p( t) C>j 0.5 1 .-.. .. I E ...s::: -1 0.5 . p., E 0.5 0 -. E -3 -3 .....;: ..-:::: 0 7r -40 40 t (fs) 0.4 .q 1 7r 0.4 , V':i (!) s::: V':i C>j ...... (!) ...s::: , .5 0... 0.5 r....Y!.>.l,,-, o t) \-; (!) t) 0... 0.3 (!) en 0.3 o I 7r . . - I - . -40 -20 0 20 40 0.6 0.8 1 -40 -20 0 20 40 t (fs) >. (11 m ) t (fs) (a) (b) (c) Figure 22.6-17 (a) Measured spectrogram SA (.x, T) of a 2.5-cycle 4.5-fs pulse by SHG-FROG. (b) Estimated temporal and spectral characteristics of the pulse. (c) SHG-FROG spectrogram computed from the pulse in (b) is approximately the same as the measurement in (a). (Adapted from A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, IEEE Journal of Quantum Electronics, vol. 35, pp. 459- 478, Figs. 17(a), 17(b), and 18 @] 999 IEEE; R. Trebino, ed., Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, Kluwer, 2000, figure on associated CD-ROM.) READING LIST General See also the reading list in Chapter 21. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, Elsevier, 2nd ed. 2006. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 1991, 4th ed. 2006. P. Gibbon, Short Pulse Laser Interactions with Matter: An Introduction, Imperial College Press (London), 2005.
1012 CHAPTER 22 ULTRAFAST OPTICS C. Rulliere, ed., Femtosecond Laser Pulses: Principles and Experiments, Springer-Verlag, 2nd ed. 2005. M. Uesaka, ed., Femtosecond Beam Science, Imperial College Press (London), 2005. F. X. Kartner, ed., Few-Cycle Laser Pulse Generation and Its Applications, Springer-Verlag, 2004. A. A. Andreev, Generation and Application of Ultrahigh Laser Fields, Nova Science, 2002. R. Trebino, ed., Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses, Kluwer, 2000. T. Kamiya, F. Saito, O. Wada, and H. Yajima, eds., Femtosecond Technology: From Basic Research to Application Prospects, Springer-Verlag, 1999. A. B. Shvartsburg, Time-Domain Optics of Ultrashort Waveforms, Oxford University Press, 1996. T. Sueta and T. Okoshi, eds., Ultrafast and Ultra-Parallel Optoelectronics, Wiley, 1996. R. Trebino and I. A. Walmsley, eds., Generation, amplification, and measurement of ultrashort laser pulses, SPIE Proceedings, vol. 2116, 1994. W. Kaiser, ed., Ultrashort Laser Pulses: Generation and Applications, Springer-Verlag, 1993. A. B. Shvartsburg, Non-Linear Pulses in Integrated and Waveguide Optics, Oxford University Press, 1993. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses, American Institute of Physics, 1992. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers, Harwood, 1989. W. Rudolph and B. Wilhelmi, Light Pulse Compression, Harwood, 1989. J. Herrmann and B. Wilhelmi, Lasersfor Ultrashort Light Pulses, North-Holland, 1987. Books 011 Solitons L. Mollenauer and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, Academic Press, 2006. T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, 2006. B. A. Malomed, Soliton Management in Periodic Systelns, Springer-Verlag, 2006. J. R. Taylor, ed., Optical Solitons: Theory and Experiment, Cambridge University Press, 1992, reprinted 2005. A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers, Springer-Verlag, 3rd ed. 2003. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, 2003. N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Chapman & Hall, 1997. P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, 1989, reprinted 1993. P. J. Olver and D. H. Sattinger, eds., Solitons in Physics, Mathematics, and Nonlinear Optics, Springer-Verlag, 1990. R. K. Dodd, J. C. Elbeck, J. D. Gibson, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, 1982, reprinted 1984. G. L. Lamb, Jr., Elements of Soliton Theory, Wiley, 1980. K. Lonngren and A. Scott, eds., Solitons in Action, Academic Press, 1978. Articles Issue on ultrafast science and technology, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 2, 2006. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Wester- walbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, Direct Measurement of Light Waves, Science, vol. 305, pp. 1267-1270,2004. G. A. Mourou and D. Umstadter, Extreme Light, Scientific American, vol. 286, no. 5, pp. 81-86, 2002. Issue on ultrafast phenomena and their applications, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 4, 2001.
PROBLEMS 1 013 D. E. Leaird and A. M. Weiner, Femtosecond Direct Space-to- Time Pulse Shaping, IEEE Journal of Quantum Electronics, vol. 37, pp. 494-504, 2001. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, Second-Harmonic Generation Frequency- Resolved Optical Gating in the Single-Cycle Regime, IEEE Journal of Quantum Electronics, vol. 35,pp. 459-478, ]999. I. A. Walmsley, Measuring Ultrafast Optical Pulses Using Spectral Interferometry, Optics & Photon- ics News, vol. 10, no. 4, pp. 29-33, 1999. A. M. Weiner, Femtosecond Fourier Optics: Shaping and Processing of Ultrashort Optical Pulses, in International Trends in Optics and Photonics, T. Asakura, ed., Springer-Verlag, 1999, pp. 233- 246. Issue on ultrafast optics, IEEE Journal of Selected Topics in Quantum Electronics, vol. 4, no. 2, 1998. M. Segev and G. I. A. Stegeman, Self-Trapping of Optical Beams: Spatial Solitons, Physics Today, vol. 51, no. 8, pp. 42-48, 1998. V. Binjrajka, C.-C. Chang, A. W. R. Emanuel, D. E. Leaird, and A. M. Weiner, Pulse Shaping of Incoherent Light by Use of a Liquid-Crystal Modulator Array, Optics Letters, vol. 21, pp. 1756- 1758, 1996. M. M. Wefers, K. A. Nelson, and A. M. Weiner, Multi-Dimensional Femtosecond Pulse Shaping, in Ultrafast Phenomena X, P. F. Barbara, J. G. Fujimoto, W. H. Knox, and W. Zinth, eds., Springer- Verlag, 1996,pp. ]59-160. A. M. Weiner, Femtosecond Optical Pulse Shaping and Processing, Progress in Quantum Electronics, vol. 19, pp. ]61-238, ]995. H. A. Haus, Optical Fiber Solitons, Their Properties and Uses, Proceedings of the IEEE, vol. 81, pp. 970-983, 1993. R. Trebino and D. J. Kane, Using Phase Retrieval to Measure the Intensity and Phase of Ultrashort Pulses: Frequency-Resolved Optical Gating, Journal of the Optical Society of America A, vol. 10, pp. 1101-1111, 1993. M. Kempe and W. Rudolph, Femtosecond Pulses in the Focal Region of Lenses, Physical Review A, vol.48,pp.4721-4729,1993. M. Kempe and W. Rudolph, The Impact of Chromatic and Spherical Aberration on the Focusing of Ultrashort Light Pulses by Lenses, Optics Letters, vol. 18, pp. 137-139, 1993. M. Kempe, W. Rudolph, U. Stamm, and B. Wilhelmi, Spatial and Temporal Transformation of Femtosecond Laser Pulses by Lenses and Lens Systems, Journal of the Optical Society of America B, vol. 9, pp. 1158-1165, 1992. A. M. Weiner, Dark Optical Solitons, in Optical Solitons: Theory and Experiment, J. R. Taylor, ed., Cambridge University Press, 1992, pp. 378-408. T. R. Gosnell and A. J. Taylor, eds., Selected Papers on Ultrafast Laser Technology, SPIE Optical Engineering Press (Milestone Series Volume 44), 1991. I. Christov, Generation and Propagation of Ultrashort Optical Pulses, in Progress in Optics, vol. 29, pp. 199-291, E. Wolf, ed., Elsevier, 1991. K. E. Oughstun, Pulse Propagation in a Linear, Causally Dispersive Medium, Proceedings of the IEEE, vol. 79, pp. 1379-1490, 1991. T. E. Bell, Light That Acts Like Natural Bits, IEEE Spectrum, vol. 27, no. 8, pp. 56-57, 1990. B. H. Kolner and M. Nazarathy, Temporal Imaging with a Time Lens, Optics Letters, vol. 14, pp. 630- 632, 1989. A. H. Zewail, Laser Femtochemistry, Science, vol. 242, pp. 1645-1653, 1988. H. A. Haus and M. N. Islam, Theory of the Soliton Laser, IEEE Journal of Quantum Electronics, vol. 21,pp. ]]72-1188,1985. PROBLEMS 22.1-1 Superposition of Two Gaussian Pulses. A transform-limited Gaussian pulse is added to a chirped Gaussian pulse of chirp parameter a and otherwise identical parameters. Derive
1014 CHAPTER 22 ULTRAFAST OPTICS expressions for the intensity, phase, spectral intensity, spectral phase, and chirp parameter of the superposition pulse. 22.1-2 The Hyperbolic-Secant Pulse. A pulse has a complex envelope sech(t/7), where sech(.) = 1/ cosh(.) and 7 is a time constant. Show that the width of the intensity function 7FWHM = 1.767, the spectral intensity S(v) = sech 2 (7r 2 7v), and the FWHM spectral width v = 0.993/7. Compare to the Gaussian pulse. (See Fourier transform table in Appendix A). 22.2-1 Thick-Prism Chirp Filter. A thick prislll is used as a chirp filter. The angle of incidence is selected to satisfy the Brewster condition in order to minimize the reflection loss. The apex angle 0: is selected such that the incident ray and the central deflected ray are symmetric with respect to the prism. Under these two conditions, show that the angle of deflection ()d satisfies the condition d()d/dn = -2, and the chirp coefficient is given by b -4(n - N) 2 fo)..o /7rC 2 . Show that, all parameters equal, the chirp coefficient is greater than that in the thin prism (see Example 22.2-2) by a factor of 4/0: 2 . 22.2-2 Bragg-Grating Chirp Filter. Design a Bragg-grating chirp filter for pulses of central fre- quency va = 300 Hz (wavelength of 1 J-Lm) and FWHM 7FWHM = 0.44 ps. The filter is to have a chirp coefficient b = (2 pS)2. Specify the dimensions of the grating and the maximum and minimum pitch of its periodic structure to ensure that all spectral components of the pulse are reflected by the grating. 22.3- 3 Propagation of a Rectangular Pulse through an Optical Fiber. A rectangular pulse of width 7 travels through an optical fiber, which is modeled as a chirp filter with chirp parameter b = Dvz/7r [see (22.3-5)]. Show that at a sufficiently long distance z, the pulse changes its shape from a rectangular function to a sinc function. Derive an expression for the new pulse width. 22.3-4 Temporal Imaging with a "Time Lens." An optical pulse of width 71 and arbitrary shape travels a distance d I through a fiber with positive GVD, whereupon it is modulated by a phase factor exp (j(,t 2 ) and subsequently travels a distance d 2 through a fiber of the same material. The width of the final pulse is 72. Assuming that d I and d 2 are much longer than the dispersion length Zo of the fiber, show that the new pulse will be a delayed replica of the original pulse with time magnification 72/71 = d 2 / d I if the condition 1/ d I + 1/ d 2 = 1/ f is satisfied, where f = -7r / (, Dv is the focal length of the phase modulator for this medium «(, is negative and f is positive). This means that the system is equivalent to a temporal imaging system. 22.5-] Mixing of Chirped Waves and Chirp Amplification. (a) Three pulsed collinear plane waves with central angular frequencies WI, W2, and W3 = WI + W2 are mixed in a second- order nonlinear medium with nonlinear coefficient d. The medium is dispersive and has indexes of refraction nI, n2, and n3 and group velocities VI, V2, and V3 at the three centra] frequencies. The three pulses are chirped with chirp parameters aI, a2, and a3. What should be the relation between aI, a2, and a3 for efficient 3-wave mixing. Hint: Assume that energy conservation and momentum conservation (phase matching) relations are satisfied at all instants of time. (b) Demonstrate that the chirp parameter of the signal and/or the idler may be greater than that of the pump. Discuss possible applications of this "chirp amplification" process. *22.5-2 Pulsed Three-Wave Mixing in a Medium with GVD. Derive the 3-wave-mixing coupled- wave equations (22.5-3) for a medium with GVD. You may use the following procedure. Begin with the Helmholtz equation with a source equal to the Fourier transform of S = J-L 0 8 2 P NL /8t 2 , where P NL = 2d£ 2 . Express the field £ as a superposition of three waves with distinct central frequencies and slowly varying envelopes, and convert the Helmholtz equation into three separate equations at the three frequencies. Simplify these equations using the SVE approximation, weak dispersion, and a 3-term Taylor-series expansion of the propagation coefficient. Use an inverse Fourier transform to convert the equations back to the time domain. 22.5- 3 Dependence of Soliton Charactaristics on G VD. Compare the characteristics of two fun- damental solitons of equal energy traveling in two extended media (e.g., optical fibers) with GVD coefficient D). = 20 ps/km-nm and D). = 10 ps/km-nm, but otherwise identical optical properties (same refractive index and same Kerr coefficient n2). Compare the soliton widths, peak amplitude, area under amplitude profile, and soliton distance. 22.5-4 Solitons in Optical Fiber. Show that the product of the peak intensity and dispersion length
PROBLEMS 1015 for the fundamental soliton is a constant, 1olzol == Ao/47rn2. For a silica glass fiber with Kerr coefficient n2 = 3.19 x 10- 2o m 2 /W determine the peak intensity 10 for a dispersion distance I Zo I == 30 km. 22.6-1 Measurement of a Gaussian Pulse. A Gaussian transform-limited optical pulse of 50-fs width (FWHM) and central frquency corresponding to an 800-nm wavelength is measured by use of the intensity correlator illustrated in Fig. 22.6-8. (a) Determine the shape and FWHM width of the measured autocorrelation function. (b) It has been suggested that the measurement would be improved if one of the pulses, say the one traveling in the upper branch, is deliberately stretched by passage through a silica glass fiber. What should the length of the fiber be, if the pulse is to be stretched by a factor of 5? (Silica glass has a dispersion coefficient D A == -110 ps/km-nm at 800-nm.) What would be the width of the new correlation function after the insertion of the fiber? (c) If this idea is applied to the nonlinear interferometer shown in Fig. 22.6-13, and the fiber is also placed in the upper branch, describe possible merits and problems with this idea as a tool for pulse measurement. 22.6-2 Interferometer with Two-Photon Absorbing Detector. An interferometer using a two- photon absorber as a detector measures the function in (22.6-19). Compare this interfer- ometer with a nonlinear interferometer using a second-harmonic generator followed by a conventional detector, which measures (22.6-11). Expand (22.6-19) in a form similar to that in (22.6-12) and compare the different terms.
CHAPTER 3 OPTICAL INTERCONNECTS AND SWITCHES 23.1 OPTICAL INTERCONNECTS A. Free-Space Refractive and Diffractive Interconnects B. Guided-Wave Interconnects C. Nonreciprocal Optical Interconnects D. Optical Interconnects in Microelectronics 23.2 PASSIVE OPTICAL ROUTERS A. Wavelength-Based Routers B. Polarization-, Phase-, and Intensity-Based Routers 23.3 PHOTONIC SWITCHES A. Architectures of Space Switches B. Implementations of Optical Space Switches C. All-Optical Space Switches D. Wavelength-Domain Switches E. Time-Domain Switches F. Packet Switches 23.4 OPTICAL GATES A. Bistable Systems B. Principle of Optical Bistability C. Bistable Optical Devices 1018 1030 1038 1058 , .,r'" . , ' \ .L ,.." " The development of optical interconnects and photonic switches began in earnest in the 1980s under the aegis of Bell Laboratories, an organization created by AT&T in 1925. Bell Laboratories became part of Lucent Technologies in 1996 and was subsequently merged into Alcatel in 2006. 1016
Interconnections and switches are essential components of distributed systems, such as communication, information processing and computing systems. The emergence of optical fibers as the favored technology for communication systems and networks has stimulated the development of a variety of photonic switches, and the introduction of wavelength division multiplexing (WDM) (see Sec. 24.4B) has added a new dimension into the switching fabric, which has motivated the development of special wavelength- based photonic switches. On the other hand, several decades of research in digital optical computing have not yielded commercial products that are competitive with electronic computers. Yet, as a byproduct of this effort, a number of technologies for optical logic gates have been developed, and an important role for optical interconnects in electronic computer systems has emerged. An optical beam is characterized by several attributes - position (space), direction, wavelength (or frequency), intensity, phase (for coherent waves), polarization, time (for optical pulses), or a code (based on a sequence of optical pulses), as illustrated schematically in Fig. 23.0-1. One of these attributes, e.g., intensity, may be modulated and used to transport a signal between two points. Another attribute, e.g., wavelength, may be used to mark different signals carried on the same beam, a process known as multiplexing. A wavelength-sensitive optical router is necessary to separate (demul- tiplex) the different signals. Switches are used to direct an optical signal from one point to one of several possible destinations. These are called space-domain switches, or simply space switches. Alternatively, a switch may transfer a signal from one time slot (or one wavelength channel) to another. These are called time-domain switches (or wavelength-domain switches). Switches controlled by an address included in each packet of incoming data are called packet switches. Position Time Intensity Phase Figure 23.0-1 Attributes of an optical beam that may be used for modulation, multiplexing, routing and switching. This Chapter This chapter introduces the basic principles of optical interconnects, passive optical routers, and photonic switches (see Fig. 23.0-2). Many of the fundamental principles of photonics that have been introduced in earlier chapters (Fourier optics and holography, guided-wave optics, electro-optics, semiconductor optics, acousto-optics, nonlinear optics, and ultrafast optics) find use here. Figure 23.0-2 A system directing each of iti optical beams entering AI input ports to one or several of N output ports. If the connections are fixed and independent of the nature of the incoming beams, the system is referred to as an interconnect. If an optical attribute of the input beams, such as wavelength, dictates the output ports to which they are directed, the system is called a passive routing element. If the connections are reconfigurable, based on an external control signal, the system is called a space switch. 1 2 3 1 2 3 M N Control Section 23.1 covers optical interconnects via free-space, planar lightwave circuits, and optical fibers. In these interconnects, the input beams are directed to prescribed output ports regardless of their attributes or the information they carry. 1017
1018 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Passive optical routers are described in Sec. 23.2. Here, each input optical beam is directed to one or more output ports based on beam attributes such as wavelength, polarization, or intensity. For example, different wavelength components in a sin- gle beam may be routed to separate ports. The device then serves as a wavelength- division demultiplexer. The inverse of this operation, wherein beams with different wavelengths are combined into a single beam, is called wavelength-division multiplex- ing (WDM). Such operations are important in modern optical communication systems (see Sec. 24.4B). Photonic space switches are described in Sec. 23.3. The simplest example of these switches is an ON-OFF switch that connects or disconnects two ports (i.e., transmits or blocks the beam), or a switch that selectively directs a beam to one of two possible locations, regardless of the beam's data content or attributes. An introduction to the general types and properties of these switches is followed by a brief overview of the different technologies used for their implementation, including electro-optic, semicon- ductor laser amplifiers, liquid crystals, microelectromechanical systems, and all-optical devices. Time-division switches and packet switches are also described. Section 23.4 is devoted to optical logic gates based on bistable optical devices. These are switches with memory, i.e., systems for which the output takes one of two (or several) values, depending on both the current value and previous history of the input. 23.1 OPTICAL INTERCONNECTS Digital signal-processing and computing systems contain large numbers of intercon- nected gates, switches, and memory elements. In electronic systems the interconnec- tions are made by use of conducting wires, coaxial cables, or conducting channels within semiconductor integrated circuits. Photonic interconnections may similarly be realized by use of optical waveguides with integrated-optic or fiber-optic couplers. Free-space light beams may also be used for interconnections, wherein beams are directed by microlenses or diffractive optical elements. This option is not available in electronic systems since electron beams must be in vacuum and cannot cross one another without mutual repulsion. Figure 23.1-1 illustrates a number of configurations of interconnects (also called couplers). Each input port is connected to one or many output ports, and vice-versa. For example, in the fan-out or the T-coupler configuration, the input port is connected to each of the output ports. In the star coupler or the directional coupler, each input port is connected to each and every output port. Interconnection Matrix The diagrams shown in Fig. 23.1-1 are only schematic connectivity diagrams that do not specify the quantitative relations between the optical fields or intensities at the connected ports. For linear coherent optical interconnects, the optical field U;o) at the fth output port (f == 1, 2, . . . , N) is related to the optical fields U,g) at the input ports, m == 1, 2, . . . , M, by a superposition: M U(o) == '" TfJ U(i) f m m' m=l (23.1-1 ) where the weights {T fm } are complex numbers defining an interconnection matrix T. For example, the 2 x 2 3-dB coupler in Fig. 23 .1-1 (c) is described by an interconnection
(a) Shift/ Banyan Reversal! crossover (b) T-coupler Fan-out (multicast) (c) 3-dB coupler Star coupler 23.1 OPTICAL INTERCONNECTS 1019 Crossover Perfect shuffle Fan-in Projection Figure 23.1-1 Examples of interconnects. (a) One-to-one. (b) One-to-many or many-to-one. (c) Many-to-many. matrix 1 [ 1 j ] T= J2 j l' (23.1-2) identical to that of an ideal beamsplitter (see Sec. 7.1A). For this device, the optical power incoming from one beam (in the absence of the other) is divided equally be- tween the two outgoing beams. Other interconnects may be similarly described. The interconnection matrix of a cascaded interconnect may be determined by use of matrix multiplication, as described in Sec. 7.1 A. Since the light is assumed to be coherent, the phase relation between the incoming beams and the phases introduced by the elements of the interconnection device play important roles. Indeed, interferometric effects are often used to redistribute the in- coming power among the output ports in prescribed manners. If the light is incoherent, then the intensity (and hence the power) at each output port is a weighted superposition of the intensities (powers) at the input ports (see Sec. 11.3B): M p(o) == '""" I T: 1 2 p(i) f fm m. m=l (23.1-3) For example, the powers at the output and input ports of a 3-dB coupler are related by a matrix whose elements are all equal to . Key performance specifications of practical couplers include the following power ratios, usually expressed in dB [== -10 log(l/ratio )]: . The insertion loss describes the port-to-port power transmittance, ideally 0 dB for a lossless path. . For a coupler distributing power among multiple output ports, the splitting ratio is the ratio of the power at one output port to the power at all output ports. For example, for an ideal 3-dB coupler, the splitting ratio is 3 dB. . The crosstalk is the ratio of the undesired power received at an output port to the input power directed to another output port(s). . The excess loss is the ratio of the total output power to the total input power.
1020 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Nonreciprocal Interconnects: Isolators and Circulators The designation of the ports of an interconnect as input or output ports implies a specific direction of transmission - from input to output (from left to right in the examples in Fig. 23.] -1). Certain interconnects are reciprocal, i.e., if the transmission is directed instead from the output ports to the input ports, the interconnection matrix remains the same. Otherwise, the interconnect is nonreciprocal. Isolators. The simplest example of nonreciprocal interconnects is a 1 x 1 unidirec- tionallink that transmits in only one direction, as illustrated in Fig. 23.1-2(a). This is often implemented by use of an optical isolator, much like a diode or a one-way valve (see Sec. 6.4B). The performance of an isolator is specified by the insertion loss [power transmittance in the forward direction (dB)] and the reverse isolation [power transmittance in the reverse direction (dB)]. Multiport Nonreciprocal Interconnects. The input/output designation is not appli- cable when a port plays a dual role, as transmitter and receiver. The interconnect is then designated simply by the number of ports. Figure 23.1-2(b) and (c) are example of 3- port interconnects using unidirectional links. These interconnects are used in duplex (two-way) communication systems, as depicted in the 4-port interconnect in Fig. 23.1- 2(e). In another 4-port system, shown in Fig. 23.1-2(d), the connections between the left and right ports are in the parallel configuration in the forward direction (left to right), and in the cross configuration in the backward direction (right to left). (a) :__ P1- 3 (b) , .« : .......---- (c) ::: (d) 1 4 · F 2 . (e) . gr ---- Figure 23.1-2 (a) 2-port unidirectional link (isolator). (b) and (c) 3-port interconnect using two unidirectional links. (d) Bidirectional (duplex) communication. Circulators. Another example of nonreciprocal interconnects is the optical circula- tor. This is an interconnect with three or more ports connected by unidirectional links pointing in the same direction. As illustrated in Fig. 23.1-3, the 4-port circulator is equivalent to the interconnect in Fig. 23.1- 2( d). Circulators find many applications in communication systems and networks. An example of the use of circulators in optical add-drop multiplexers (OADM) is described in Sec. 23.2A (see Fig. 23.2-3). ,- -2 3 : - ;:: 4 ,)' ' 0 ' , , , , , , I I = ,(h' Figure 23.1-3 4-port circulator. The two configurations are equivalent.
23.1 OPTICAL INTERCONNECTS 1021 A. Free-Space Refractive and Diffractive Interconnects Conventional optical components (mirrors, lenses, prisms, etc.) are used routinely in optical systems as interconnects. One example is an imaging system in which a lens is used to connect points of the object and image planes. To appreciate the order of magnitude of the density of such interconnections, note that in a well-designed imaging system as many as 1000 x 1000 independent points per mm 2 in the object plane are connected optically by means of the lens to a corresponding 1000 x 1000 points per mm 2 in the image plane. For this to be implemented electrically, a million noninter- secting and properly insulated conducting channels per mm 2 would be required! Standard optical components may be used to implement special interconnects, such as shift, reversal, crossover, shuffle, fan-in, fan-out, star coupling, and projection, as Fig. 23.1-4 illustrates (see also Fig. 23.1-1). They can be miniaturized by use of micro- optics components, such as miniature beamsplitters, lenses, graded-index rods, prisms, filters, and gratings. Such components are also compatible with optical fibers, which are often used for light transmission. e--f\... , Reversal Reversal Crossover Perfect shuffle Directional coupler Fan-out Fan-in Star coupler Projection Figure 23.1-4 Examples of simple optical interconnects created by conventional optical compo- nents: A prism bends parallel optical rays preferentially and establishes an ordered interconnection map corresponding to a reversal or crossover. Two appropriately oriented prisms perform a perfect- shuffle - an operation used in sorting algorithms and in the fast Fourier transform (FFT). A lens establishes a fan-in, a fan-out, or a reversal. A beamsplitter together with two lenses creates a directional coupler. A glass rod serves as a star coupler. An astigmatic optical system, such as a cylindrical lens, implements a projection by connecting points of each row in the input plane to one point in the output plane. Arbitrary optical interconnection maps require the design of custom optical com- ponents that may be quite complex and impractical. However, computer-generated holograms made of a large number of segments of phase gratings of different spatial frequencies and orientations have been used successfully to create high-density optical interconnections. A phase grating is a thin optical element whose complex amplitude transmittance is a two-dimensional periodic function with unit amplitude. The simplest phase grating has complex amplitude transmittance t(x, y) == exp[-j21T(V x X + vyY)], where V x and v y are the spatial frequencies in the x and y directions; they determine the period and orientation of the grating. It was shown in Secs. 2.4B and 4.1A that when a coherent optical beam of wavelength A is transmitted through this grating, it undergoes a phase shift, causing it to tilt by angles sin- 1 AV x AV x and sin- I AV y AV y , where AV x « 1 and AV y « 1, as illustrated in Fig. 23.1-5. By varying the spatial frequencies V x and v y (i.e., the periodicity and orientation of the grating) the tilt angles are altered. As described in Sec. 4.1A, this principle may be used to make an arbitrary intercon- nection map by use of a phase grating made of a collection of segments of gratings of
1022 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES } 1/ l/y t Figure 23.1-5 Bending of an opticaJ wave as a result of transmission through a phase grating. The deflection angles, assumed to be small, depend on the spatial frequency and orientation of the grating. y different spatial frequencies. Optical beams transmitted through the different segments undergo different tilts, in accordance with the desired interconnection map (Fig. 23.1- 6). Figure 23.1-6 Holographic interconnec- tion map created by an array of phase gratings of different periodicities and orientations. If the grating segment located at position (x, y) has frequencies V x == V x (x, y) and v y == vy(x, y), the angles of tilt are approximately AV x and AV y , and the beam hits the output plane at the point (x', y') satisfying x' - x d AV x , y' - Y d AV y , (23.1-4) where d is the distance between the hologram and the output plane and all angles are assumed to be small. Given the desired relation between (x', y') and (x, y), i.e., the interconnection map, the necessary spatial frequencies V x and v y may be determined at each position using (23.1-4). Holographic interconnection devices are capable of establishing one-to-many or many-to-one interconnections (i.e., connecting one point to many points, or vice versa). In Fig. 23.1-6, for example, the center grating element is a superposition of two harmonic functions so that its complex amplitude transmittance t(x, y) == exp(-j27r(V x lX+ VylY)] +exp(-j27r(Vx2X+ V y 2Y)]; the incident beam is split equally into two components, one tilted at angles (AVxl, AVyl) and the other at (AV x 2, AV y 2), where all angles are small. Weighted interconnections may be realized by assigning different weights to the different gratings. Arbitrary interconnections may therefore be created by appropriate selection of the grating spatial frequencies at each point of the hologram. EXERCISE 23.1-1 Interconnection Capacity. The space-bandwidth product of a square hologram of size a x a is the product (Ba)2, where B is the highest spatial frequency (lines/mm) that may be printed on the hologram. Show that if the hologram is used to direct each of L incoming beams to Ai directions, the product AI L cannot exceed (B a ) 2 ,
23.1 OPTICAL INTERCONNECTS 1023 1 2 it! L < (Bd)2. L Hint: Use an analysis similar to that presented in Sec. 19.2C in connection with acousto-optic inter- connection devices [see (19.2-9)]. What is the maximum number of interconnections per mm 2 if the highest spatial frequency is 1000 lines/mm and if every point in the input plane is connected to every point in the output plane? In the limit in which the grating elements have infinitesimal areas, we have a contin- uous (instead of discrete) interconnection map: a geometric coordinate transformation rule that transforms each point (x, y) in the input plane into a corresponding point of the output plane (x', y'). If the desired transformation is defined by the two continuous functions x' == 'l/Jx (x, y), y' == 'l/Jy ( x, y), (23.1-5) the grating frequencies must vary continuously with x and y as in a frequency- modulated (FM) signal. (See Fig. 23.1-7.) Assuming that the grating has a transmit- tance t(x, y) == exp[-jcp(x, y)], the associated local (or instantaneous) frequencies are given by 8cp 27TV x == 8x ' 8cp 27rv y = ay . (23.1-6) (This is analogous to the instantaneous frequency of an FM signal.) Substituting into (23.1-4), we obtain 'l/Jx(x, y) - x d A 8cp -- 27T 8x ' 'l/Jy(x, y) - Y d A 8cp -- 27T 8y . (23.1-7) These two partial differential equations may be solved to determine the grating phase function cp(x, y). e-j<p(x,y) I( d Figure 23.1-7 Diffraction from a phase hologram as a continuous interconnection system.
1024 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES EXAMPLE 23.1-1. Fan-In Map. Suppose that all points (x,y) in the input plane are to be steered to the point (x', y') == (0,0) in the output plane, so that a fan-in interconnection map is created. Substituting 'l/;x(x,y) == 'l/;y(x,y) == 0 in (23.1-7) and solving the two partial differential equations, we obtain cp(x,y) == -7r(x 2 + y2)/Ad. Not surprisingly, this is exactly the phase shift introduced by a lens of focal length d (see Sec. 2.4B). EXERCISE 23.1-2 The Logarithmic Map. Show that the logarithmic coordinate transformation x' == 'l/; x ( x, y) == In x y' == 'l/;y (x, y) == In y (23.1-8) (23.1-9) is realized by a hologram with the phase function ) 27r ( 1 2 1 2 ) cp (x, y == A d x In x - x - "2 x + y In y - y - "2 Y . (23.1-10) Once the appropriate phase <p( x, y) is decided, the optical element is fabricated by using the techniques of computer-generated holography. This approach allows a complex function exp[-j<p(x, y)] to be encoded with the help of a binary function taking only two values, 1 and 0, or 1 and -1, for example. This is similar to encoding an image by use of black dots whose size or density vary in proportionality to the local gray value of the image (an example is the halftone process used for printing images in newspapers). With the help of a computer, the binary image is printed on a mask (a transparency) that plays the role of the hologram. The binary image may also be printed by etching grooves in a substrate, which modulate the phase of an incident coherent wave, a technology known as surface-relief holography. References discussing computer-generated holography are provided in the reading list. Dynamic (reconfigurable) interconnections may be constructed using acousto-optic devices or magneto-optic devices. But the number of interconnection points is much smaller than is achievable by use of holographic gratings. Dynamic holographic in- terconnections may be achieved by use of nonlinear optical processes, such as four- wave mixing in photorefractive materials. Two waves interfere to create a grating from which a third wave is reflected. The angle between the two waves determines the spatial frequency of the grating, which determines the tilt of the reflected wave (Secs. 20.4 and 21.3E). B. Guided-Wave Interconnects Optical interconnects are implemented in planar lightwave circuits (PLCs) by pattern- ing optical waveguides in LiNb0 3 or silicon substrates (see Sec. 8.5B), much like metal wires in electronic printed circuits or integrated circuits. Examples are illustrated in Fig. 23.1-8 and combinations and cascades of these basic interconnects can be used to create more complex interconnects. Waveguide couplers are used to distribute optical power in prescribed amounts. The coupler shown in Fig. 23.1-8(b), for example, is described by an interconnection matrix (see Sec. 8.5B) T == [ cos eL -j sin eL - j sin e L ] cos eL ' (23.1-11 )
23.1 OPTICAL INTERCONNECTS 1025 (a) T-coupler (b) 3-dB coupler Figure 23.1-8 Integrated-optic devices implementing some of the interconnections in Fig. 23.1-1. where e is the coupling coefficient and L is the interaction length. The incoming power in input port 1 is therefore divided among output ports 1 and 2 by factors cos 2 eL and sin 2 eL, respectively. For eL == 7r / 4, the power is divided equally and the coupler becomes a 3-dB coupler. Applications of optica] fiber technology, particularly in telecommunication, have stimulated the development of many fiber-optic interconnects. The examples shown in Fig. 23.1-9 parallel those shown in Fig. 23.1-8 for planar integrated optics, and the fiber couplers shown in Fig. 23.1-9(b) are described by the same interconnection matrix (23.1-11). ---.. Double-core fiber j.. = -cE( Splitter or TAP Double-core fiber '=-=== C? Combiner (a) T-coupler (b) 3-dB coupler !d (d) Star coupler Figure 23.1-9 Fiber-optic couplers implementing some of the interconnections in Fig. 23.1-1. (a) Double-core fiber used as a T-coupler, splitter, or combiner. (b) 3-dB coupler made of two fused fibers and another using two GRIN-rod lenses separated by a beamsplitter film. (c) Fan-in or fan-out. (d) Star coupler using fused fibers and another using a mixing rod, a slab of glass through which light from one fiber is dispersed to reach all other fibers. c. Nonreciprocal Optical Interconnects Optical implementations of nonreciprocal interconnects are based primarily on the Faraday rotator. As was demonstrated in Sec. 6.4B), an optical isolator may be imple- mented by use of a 45° Faraday rotator sandwiched between two polarizers oriented at 45° from one another. Linearly polarized light is transmitted in the forward direction and b]ocked in the reverse direction. It was also shown in Sec. 6.4B) that a combination of a 45° Faraday rotator followed by a half-wave retarder is a useful nonreciprocal device. The state of polarization of a forward-traveling linearly polarized light, with the plane of polarization oriented at 22.5° with the fast axis of the retarder, is not altered. But the plane of polarization of the backward wave is rotated by 90°. The device may therefore be used together with polarizing beam splitters to implement nonreciprocal interconnects as illustrated by the example in Fig. 23.1-10(a) and also optical circulators as shown in Fig. 23.1-10 (b).
1026 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Faraday rotator (0) CD PBS Retarder Faraday rotator (b) PBS Retarder PBS Figure 23.1-10 (a) Implementation of the 3-port nonreciprocal interconnect in Fig.23.1-2(b) by use of a polarizing beamsplitter (PBS) together with the Faraday rotator and half-wave retarder combination. Light travels from port 1 to port 2, and from port 2 to port 3. (b) Optica] circulator implementing the 4-port 1-2-3--4-1 interconnect shown in Fig. 23.1-3 by use of two PBSs, and a Faraday-rotator and half-wave-retarder combination. D. Optical Interconnects in Microelectronics The possibility of using optical interconnects in place of conventional electrical in- terconnects in microelectronics and computer systems has led to substantial research and development effort for several decades. With the successful use of fiber optics for computer-to-computer communication (in local area networks, for example; see Sec. 24.4), systems employing optical fibers for processor-to-processor, backplane- to-backplane, and board-to-board communications have been developed. Examples of board-to-board interconnects using optical fibers are illustrated in Fig. 23.1-11 (a). Such short-reach fiber optical communication links can operate at data rates much greater than electrical links and the technology is well established, as described in Chapter 24. Board-to-board free-space optical interconnects have also been prototyped, as illustrated by the example in Fig. 23.1-11 (b). Each board has a transmitter-receiver optoelectronic chip, including light sources, e.g., a vertical-cavity surface-emitting laser (VCSEL) array and a photodetector array, along with their associated circuitry. Circuit board (a) Circuit board : : ---=- . . ..- -- Fibers -, . -._._.-...... ,/' " Laser OptoelectronIc d " d " h " 10 e transmItter c IR Lens . y Lens Photodiode "....." Optical beam --- ...._.,......_.... ,," - . Circuit board (b) Circuit board Figure 23.1-11 Board-to-board interconnect using (a) a fiber-optic array; (b) a free-space micro- optic link. Prototypes for chip-to-chip communication via optical interconnects have also been
23.1 OPTICAL INTERCONNECTS 1027 developed. In the example depicted in Fig. 23.1-12, each chip is interfaced with a transmitter-receiver optoelectronic chip, and the two optoelectronic chips are con- nected via a planar dielectric waveguide. Free-space optical links via reflecting mirrors have also been considered. .. EleC(rO nc ......... ., d < -F-" ..."0 ......-:- ------ :_- .... . C lu ...,- "... Board -- / Mirror Waveguide Mirror Optoelectronic chip Figure 23.1-12 Chip-to-chip inter- connect and optical waveguide link. The use of ultra-short-reach optical interconnects within an electronic chip is clearly more challenging. Intra-chip optical interconnects have been motivated by advances in high-speed high-density microelectronic circuitry and the emergence of parallel processing architectures, which have created communication bottlenecks making in- terconnections a major problem. In very-large-scale integrated circuits (VLSI), inter- connects occupy a large portion of the available chip area. To minimize the effect of interconnection time delays, which are becoming as long as, or even longer than, gate delays, considerable design effort is being devoted to the equalization of interconnect lengths. Optical interconnects have the potential for alleviating some of these problems, but their commercial viability has not been established. Rationale for Chip Optical Interconnects Optical interconnects offer a number of advantages for inter- and intra-chip intercon- nects, stemming principally from the short wavelength of light and the corresponding high frequency (e.g., 20-50 THz), which is substantially greater than the bandwidth of transmitted data. Electronic interconnects use baseband signals at relatively much lower frequencies (e.g., in the GHz regime). . Density. The most dense set of interference-free interconnects uses unguided beams, each with a small width and a small divergence angle, limited only by diffraction (the product of the width and the angle of a narrow beam is of the order of a wavelength, which is small at optical frequencies). Since such beams can intersect (pass through one another) without mutual interference (assuming that the medium is linear), they can be used in a three-dimensional configuration to create interconnects with densities unmatchable by electrical wires. Light may also be guided in planar or quasi-planar low-loss dielectric waveguides of widths as small as a wavelength. They can be packed densely with minimal crosstalk. Electrical interconnects, on the other hand, must use metallic conductors, such as strip lines, which serve as transmission lines or waveguides for the electromagnetic waves associated with the oscillating electric charges. Metallic conductors introduce losses and cannot be packed tightly since they become susceptible to electromagnetic interference if they come in close proximity. . Bandwidth. The bandwidth of an electronic strip line of length £ and cross- sectional area A placed above a ground plane is proportional to the ratio AI £2. This can be seen by noting that in a line limited by RC effects, the resistance R ex £1 A and the capacitance C ex £, so that the time constant RC ex £2 / A. A similar argument applies to lines limited by LC effects. The bandwidth is therefore determined by the aspect ratio £1 VA and cannot be altered by miniaturizing the device or making it bigger. Optical interconnects do not suffer from this aspect ratio limit since bandwidth is governed by other physical effects
1028 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES and is generally greater. Additionally, in optical interconnects, the maximum bandwidth of the data carried by each connection is not affected by the density of proximate interconnects. Said differently, the crosstalk between neighboring lines is not influenced by increase of the data rate. This stems from the small ratio of the bandwidth to the carrier frequency of the modulated light. This is not the case in electronic interconnects for which the density of electronic interconnects must be reduced sharply at high modulation frequencies in order to eliminate capacitive and inductive coupling between proximate interconnects. Optical interconnects therefore have greater density-bandwidth product, in comparison with electronic interconnects. . Delay. Photons travel at a speed of 0.3 mmJps in free space and 0.86 mmJps in silicon. The corresponding propagation time delay is 3.3 ps/mm and 9.4 ps/mm, respectively. Propagation delays of electrical signals in striplines fab- ricated on ceramics and polyimides are approximately 10.2 and 6.8 ps/mm, re- spectively. Delay is therefore not in itself an issue. However, whereas the velocity of light is independent of the number of interconnections branching from an opti- cal interconnect, in electronic transmission lines the velocity is inversely propor- tional to the capacitance per unit length so that it depends on the total capacitive "load"; the propagation delay time therefore increases with increase of the fan- outs. Optics offers a greater flexibility of fan-out and fan-in interconnections, limited only by the available optical power. . Power. To avoid reflections, electrical interconnects must be terminated with their matched impedance. This usually requires a larger expenditure of power. In opti- cal interconnects, reflection can be significantly reduced by use of antireflection coating, and power requirements are limited by the sensitivity of photodetectors and the efficiencies of the electrical-to-optical and optical-to-electrical conver- sions as well as the power transmission efficiency of the routing elements. Implementation of Chip Optical Interconnects Intrachip optical interconnects are ultrashort optical communication links connecting points within the chip. Each link comprises three components: an electronic-optical transducer (transmitter) modulated by the electric signal at a point within the chip, an optical beam, and an optical-electronic transducer (receiver) feeding the signal to another point in the same chip. The optical beams may be guided in silicon-based waveguides and may be routed by an external device, such as a hologram, as illustrated in the in Fig. 23.1-l3(a). A special case of the external-routing configuration is a one-way interconnect between one, or several, external point(s) and points on the chip. This system is simpler to fabricate since it requires no on-chip transmitters. One ueful application is optical clock distribution. In this case, a signal from an external clock modulates an external light source that broadcasts the signal to multiple photodetectors on the chip using a reflection hologram, as illustrated in Fig. 23.1-13(b). This ensures accurate synchro- nization of high-speed synchronous circuits and alleviates the problem of clock skew that results from differential time delays. The hologram may, of course, be eliminated and the light "broadcast" directly to all points on the chip. This creates a robust system that is insensitive to misalignment, but the power efficiency is low since a larger portion of the optical power is wasted. Ideally, all three components of the optical link should be monolithically integrated with the chip silicon substrate and be compatible with CMOS (complementary metal oxide semiconductor) technology. Silicon photodiodes can be readily embedded in silicon chips, and silicon-on-insulator (SOl) optical waveguides (see Sec. 8.3) may be used as optical connections, although the real-estate for such guides may not be available on the chip. The main difficulty lies in the transmitters since light sources
23.1 OPTICAL INTERCONNECTS 1029 Hologram -rf(5?/:'" ,., ___ __......... __._nnun_____.____....__.... //' / /, 'j'/' ' Hologram --......, ----"' . ,- t .....:::-=:y, - . 67 " ,. Photodetector Silicon chip ; -,., .. 6 Laser source I . .{(:_u. ......:., ,- ...., ,d .'. " .. __ Detectors , .,._<- <of' Silicon chip (a) (b) Figure 23.1-13 (a) Interconnects between on-chip sources and detectors via an external reflection hologram used as a routing element. (b) One-way interconnects directing clock pulses from an external light source to photodetectors in a silicon chip. cannot be efficiently made in silicon because it is an indirect-bandgap material (see Sec. 16.1 D). Efficient light-emitting materials, such as AIGaAs/GaAs, grown on silicon substrates by heteroepitaxy are not sufficiently reliable because of the lattice-parameter and therma]-expansion mismatch between the two materials. Other ideas for direct generation of light in silicon by use of photonic-crystal structures remain in the domain of ongoing research. Another approach to addressing the transmitter problem is to replace the optical sources with electro-optic modulators illuminated by an external light source and mod- ulated by the local electric signals within the chip. However, all-silicon modulators remain either too large or two slow for optical interconnect applications. One practical approach for addressing the mismatch between the compound- semiconductor optoelectronic technology, which is used to fabricate optical sources and modulators, and the CMOS silicon technology, which is the basis of modem electronics, is hybrid integration. This approach is based on bonding separately fabricated optoelectronic and electronic chips. A hybrid-integration process known as flip-chip bonding can integrate thousands of optoelectronic devices on a single silicon chip with lateral alignment better than one micron. Using this packaging technology, light sources or optical modulators fabricated in two-dimensional arrays in a surface-norma] architecture may be bonded to a silicon chip, as illustrated schematically in Fig. 23.1-14(a). An example is an 850-nm GaAs-based VCSEL array operated at rates exceeding 10 Gb/s and using low-voltage drives (approximately 1 V). The same configuration may be used for arrays of electro-optic modulators based on semiconductor electroabsorption (see Sec. 20.5). These arrays of sources or modulators may be used in the external-routing configuration shown in Fig. 23.1- 13(a). Figure 23.1-14(b) illustrates another configuration for on-chip routing using SOl waveguides with an InP light source bonded to the silicon chip. . <:",: :'3 -,. AIGaAs source Light or modulator n MQWi p *.__<.' ;..'1'.:,/' ....:t:-=.;:.. . -.. / [ .---""'! - .....--- _ _...>..m.. ,Silicon CMOS chi. (a) ..:[( -;.:-:-- . ..si_..." - f <>v> ./ . _t< ..::. f" (b) Figure 23.1-14 Hybrid integration. The transmitters are mounted in optoelectronic chips integrated with the silicon chip using flip-chip bonding. (a) An AIGaAs optical source, or modulator, bonded to a silicon chip in a surface normal architecture. (b) An InP light source bonded to a silicon chip; the light is coupled into an on-chip silicon-on-insulator (SOl) ridge waveguide.
1030 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES 23.2 PASSIVE OPTICAL ROUTERS Light beams may be routed on the basis of physical attributes such as wavelength, intensity, phase, polarization, or time. As illustrated in the example in Fig. 23.2-I(a), the component with attribute Xl, in each beam, is routed to the fth output port, where f == 1,2,..., N, and N is the number of output ports, which equals the number of attributes. The following are some examples: . A demultiplexer (DEMUX) is a 1 x N attribute-based router that sorts the components with attributes Xl, X 2 , . . . , X N in a single input beam and directs them to separate output ports, as shown in Fig. 23.2-I(b). The DEMUX may be implemented by use of a broadcast-and-select operation - a 1 x N fan-out interconnect, which broadcasts copies the incoming beam to all output ports, is followed by a bank of filters that pass through components of selected attributes and reject aU others. . The multiplexer (MUX) is the inverse of the DEMUX. As illustrated in Fig. 23.2- l(e), input beams with distinct optical attributes Xl, X 2 ,..., X N are combined into a single beam, which can be subsequently separated by use of a demultiplexer. Multiplexing and demultiplexing based on wavelength, frequency, and time, are used extensively in optical communication systems. . The optical add-drop multiplexer (OADM), shown in Fig. 23.2-I(d), is another important routing device used in communication networks. Here, a demultiplexer sorts components of different attributes, separates the component of a selected at- tribute, say X 2 , drops its data content and adds instead new data, and subsequently combines all components into a single beam by use of a multiplexer. XN X3,X N X1,X N XI.X2. ... XN . X2. X 3 (a) (b) DEMUX (c) MUX (d) OADM Figure 23.2-1 (a) Attribute-based routing. (b) Demultiplexer (DEMUX). (c) Multiplexer (MUX). (d) Add- drop multiplexer (ADM). A. Wavelength-Based Routers Wavelength-based routers are commonly used in wavelength-division multiplexing (WDM) optical fiber networks. As described in Sec. 24.3C, these systems use multiple wavelength channels in the same optical fiber. They employ routers that combine the channels at the fiber input and separates them at the output using wavelength-based routers called wavelength-division multiplexers and wavelength-division demulti- plexers, respectively. Implementations of Wavelength-Division Multiplexers/Demultiplexers The following techniques, illustrated in Fig. 23.2-2, are used for wavelength-division demultiplexing. . An angularly dispersive optical component separates the components of different wavelengths within a single optical beam into separate optical beams. The sim- plest optical components exhibiting angular dispersion are the prism [Fig. 23.2- 2(a)] and the diffraction grating [Fig. 23.2-2(b)]. The angular dispersion of a prism is limited by the rate of change of the refractive index with respect to the wavelength, dn / dA, which is usually not sufficiently large to adequately separate
23.2 PASSIVE OPTICAL ROUTERS 1031 components of slightly different wavelengths. Prisms made of photonic-crystal materials (see Chapter 7), called superprisms, can have two to three orders of magnitude greater dispersive power. Diffraction gratings (Sec. 2.4B) have angular dispersion stronger than ordinary prisms. They are capable of resolving wave- length differences corresponding to a few GHz. . Wavelength separation may also be implemented by use of a bank of filters tuned to the different wavelengths. The incoming light is broadcast to the different filters, with each filter transmitting a single wavelength channel and blocking all others. Alternatively, the beam may be directed through a sequence of filters with narrow spectral width, such as dielectric interference thin-film filters (TFF), each of which transmits one wavelength and reflects all others to the next filter, as illustrated in Fig. 23.2-2(c). A GRIN rod is used to guide the rays between the filters. . In a similar implementation, the wavelength dependence of the reflectance of a fiber Bragg grating (FBG) (Sec. 7.1 C) is exploited to separate wavelength components; the component at the Bragg wavelength (AB == A/2, where A is the grating period), is reflected and all other components are transmitted. Multiple Bragg gratings are used to separate multiple wavelengths [Fig. 23.2-2(d)]. . In yet another implementation, a sequence of microring-resonator filters, each tuned to one wavelength, is used [Fig. 23.2-2(e)]. . Other implementations use interferometers such as the Mach-zehnder interfer- ometer and the waveguide grating routers, as will be described subsequently. Al + A2 + A3 Al + A2 + A3 Al A2 A3 - Al A2 A3 AI Az A3 (a) Prism (b) Diffraction gratings Al + A2 + A3 GRIN rod Al Al + A2 + A3 Al + A3 A2 () 15}))))) n})})) FBG Al + A2 + A3 A3 t . A2 A3 (c) Thin-film filters (d) Fiber Bragg grating (FBG) (e) Microring resonators Figure 23.2-2 Wavelength-division demultiplexers. (a) Prism. (b) Diffraction grating with a lens or graded-index (GRIN) rod. (c) Dielectric interference thin-film filters (TFF). (d) Fiber Bragg grating (FBG). (e) Microring resonator filter. Optical Add-Drop Multiplexer (OADM). The optical add-drop multiplexer (OADM) extracts data from and adds data to selected wavelength channels of a multi-channel optical beam. Individual wavelength channel may be accessed by use of a demultiplexer followed by a multiplexer, as in the layout in Fig. 23.2-1. The data are extracted (dropped) from selected channels using detectors, and new data are added to selected channels by use of modulated optical sources. In another implementation, the selected wavelength channel is separated from the other channels by means of a wavelength-sensitive optical component. Examples of OADM
1032 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES based on this layout and using fiber Bragg grating (FBG) and multiple microring resonators are illustrated in Fig. 23.2-3 and Fig. 23.2-3, respectively. Circulator Circulator .-\1,.-\2,A3 Al,.-\2,.-\3 .-\ 1 , .-\3 .-\1, A2, A3 ----... Al Al Add Figure 23.2-3 An optical add-drop multiplexer (OADM) uses a fiber Bragg grating (FBG) to reflect the dropped wavelength component At, and a circulator directs it to a detector. Other components (A2 and A3) are passed through. Another circulator is used to add light modulated by new data at AI. The FBG reflects back any backward-propagating light at AI. Input ),2, ),3 ),2, ),3 Output J J ) Figure 23.2-4 An optical add-drop multiplexer ),1 (OADM) uses multiple microring resonators to extract channel Al from a multichannel input beam and drop it to a detector. Other components (A2 and A3) are passed through. New data at Al are selected by the filter and added to the output beam. Multiple microring resonators have greater wave- Add ),1 ),1 Drop length selectivity (i.e., narrower spectral width ----------- ) and greater rejection ratio) than single microring . resonators. The Mach-Zehnder Interferometer as a Demultiplexer Since interferometers are sensitive to the wavelength they are suitable for wavelength- division routing. For example, the integrated-optic Mach-Zehnder interferometer (MZI) shown in Fig. 23.2-5 may be used as a two-wavelength demultiplexer. To direct the components of wavelength Al and A2 to different output ports, the pathlength difference d is selected such that the phase difference cjJ == 27rd I A is an even multiple of 7r at Al and an odd multiple of 7r at A2; i.e., fld == qIAI/2 and fld == q2A2/2, where qi is an even integer and q2 is an odd integer. ),1 PI ! I\/ do + tld ) ),1, ),2 ),2 P2 t ), do \/V\L I ) 0 tld tld tld ), 4 2 Figure 23.2-5 Wavelength-division routing (demultiplexing) by use of an integrated-optic Mach- Zehnder interferometer. The resolution of the routing device, i.e., the closest wavelengths that can be sepa- rated, is determined by writing II Al -II A2 == (qi - q2) 1 2fld , and taking IqI - q21 == 1
23.2 PASSIVE OPTICAL ROUTERS 1033 so that III Al - II A21 I VI - v21 is therefore 1/2/:)"d. The corresponding frequency difference /:),.v == c /:),. v == 2/:)" d . (23.2-1 ) For example, if d == 1 mm and n == 1.5, then v == 100 GHz. Smaller separations /:),.v require proportionally longer pathlength differences /:),. d The spectra] sensitivity of the MZI router may be determined by writing its inter- connection matrix: T = G i] [exp[-j27f(O + d)/'\] a ] [ 1 j ] exp(-j27rd o IA) j 1 ' (23.2-2) where do and do + d are the pathlengths of the interferometer branches, and the first and third matrices in this matrix product represent 3-dB couplers. For an input field of unit power at input port 2, the power received at ports 1 and 2 are PI == 1 T 211 2 and P 2 == IT 221 2 , respectively, so that PI == cos 2 (7r /:)"d I A), P 2 == sin2(7rdl A). (23.2-3) These powers are plotted in Fig. 23.2-5 as functions of A. It is clear from this de- pendence that the smaller the ratio AI d, the more rapidly these functions alternate between 0 and l, i.e., the greater the possibility for demultiplexing closely spaced wavelengths. Multiple MZIs may be cascaded to separate more than two wavelengths. For exam- ple, four wavelengths may be separated in a two-step process, as illustrated in Fig. 23.2- 5(b). The first MZI separates the odd-numbered from the even-numbered wavelengths, and subsequent MZIs do finer wavelength separations. AI, A2 , A3, A4 Al A3 . A2 Figure 23.2-6 Wavelength-division rout- ing (demultiplexing) by use of integrated- optic cascaded Mach-Zehnder interferome- ters. A4 Waveguide Grating Routers (WGR) Other interferometric configurations may be used to provide greater wavelength selec- tivity. For example, multipath interferometers are highly selective to wavelength since they exhibit sharp resonance. Such interferometers may be custom designed in planar waveguides, and multiple interferometers may be configured to provide wavelength routing of a large number of wavelengths in devices with many input and output ports. The principle is to configure each connection between an input port and output port as an independent multipath interferometer that transmits only specific wavelengths. Since the multipath interferometer is similar to the diffraction-grating spectrometer, the router is known as the waveguide grating router (WGR). These devices are also called arrayed waveguides (AWG).
1034 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES The multipath interferometer. Before we consider the operation of the WGR, we first review the properties of the multipath interferometer (see Sec. 2.5B). An L-path interferometer is a connection with L optical paths whose length increases progres- sively and linearly so that adjacent paths have exactly the same pathlength difference fld. The wave received at the output port is the sum of L waves of equal amplitudes and equal phase difference cjJ == 27r fl d / A at wavelength A. The power transmittance, T = sin 2 (Lc.p /2) = sin 2 (L7f b.d / >..) sin 2 ( <p /2) sin 2 ( 7r fl d / A) , (23.2-4) is a periodic function of cjJ with sharp peaks occurring when cjJ equals integer multiples of 27r (see Fig. 2.5-7). The dependence of T on A is not periodic, but comprises sharp peaks at A == d and integer fractions thereof, as illustrated in Fig 23.2-7. The larger the number of paths L, the sharper the peaks. do + L/::).d Q) t) s::: ro +-' +-' .1"'"'1 S fJ) s::: ro /::).d/3 /::).d/2 /::).d A Figure 23.2-7 Wavelength dependence of the transmittance of a multi path interferometer. The WGR as a wavelength-division demultiplexer. A waveguide-grating router (WGR) may be used as a 1 x N wavelength-based router that directs each of N wavelength components, AI, A2, . . . , AN, at the input port to one of the N output ports, as shown in Fig. 23.2-8. There are N multipath interferometers, one for each of the output ports. Each interferometer has a unique pathlength difference d selected such that only a specific wavelength is transmitted. This is accomplished if the connections leading to the mth output port are designed to have a pathlength difference fld m that is an integer multiple of Am, but not an integer multiple of the other wavelengths. The design is simpler if the wavelengths AI, A2, . . . , AN are distributed uniformly as a decreasing sequence, Am == Ao - mA, where flA is the wavelength channel separation and Ao == Al - flA. A necessary condition of operation of the demultiplexer IS: dm == Am == Ao - mA, m == 1,2, . . . , N, (23.2-5) i.e., the pathlength difference for the connections to the mth output port decreases linearly with m. The other condition is that dm is not equal to an integer multiple of AI! for all f =I- m. This condition is automatically satisfied if the the shortest wavelength AN is greater than one half of the longest wavelength AI, as depicted in Fig. 23.2-8. In the implementation shown in Fig. 23.2-8, each pathlength between the input port and an output port is the sum of the waveguide length and the distances traveled in the star couplers. The waveguide lengths may be selected to increase progressively by a fixed length fld w . For a star coupler with circular boundaries, the pathlength difference may be approximated by a linearly decreasing function of m, so that dm == dw + (da - m db), (23.2-6) where fld a and db are constants dependent on the geometry of the couplers. The condition in (23.2-5) can therefore be satisfied if fld w + da == Ao and db ==
23.2 PASSIVE OPTICAL ROUTERS 1035 L Al+A2+ ...AN . - g ---- ---...,.. - A AN Am !1 d 1 n_ nn_._ - - - - - - - - - - - - - - - - - - - - - - - - - - n__ !1d2 - nnnn_nnnnnnn_nnnnn_ AN A2 Al 2AN A 2 · A2 I Al !1dN - .nnnnnnn_n_.n_ . Star coupler l\I Star coupler Figure 23.2-8 Wavelength-division demultiplexing by use of a wave-grating router (WGR). LlA. The resolution of the wavelength demultiplexer, i.e., the minimum wavelength separation LlA, is therefore limited by the minimum value of the geometrical factor db. The WGR as an N x N wavelength router. The WGR may also be used as a more general N x N wavelength router. The connections between the .eth input port and the mth output port fonn a multipath interferometer with pathlength difference dRm == Aoo- (.e+m )LlA, which decreases linearly with both.e and m (Aoo and A are constants dependent on the geometry of the WGR). Light is transmitted between these ports if the wavelength ARm equals dRm, i.e., ARm == AOO - (.e + m)A, .e, m == 1,2,..., N. I (23.2-7) WGR Equation Equation (23.2-7) is a generalization of (23.2-6). Although the WGR does not imple- ment an arbitrary wavelength routing, it can offer solutions to certain routing problems such as simultaneous wavelength multiplexing operations. B. Polarization-, Phase-, and Intensity-Based Routers Polarization-Based Routing The simplest example of passive optical routing is based on polarization. In polarization- division demultiplexing the parallel and orthogonal polarization components of an optical beam are separated by use of a polarizing beamsplitter (PBS), as illustrated in Fig. 23.2-9. Polarization-based multiplexing is achieved by use of the PBS as a beam combiner (with light traveling from right to left instead of left to right). PI Figure 23.2-9 Polarization-division routing using a polarizing beamsplitter (PBS). For beams traveling from left to right, the prism is a demultiplexer. For beams traveling from right to left, it is a multiplexer. P2 Phase-Based Routing Another simple example of passive optical routing is based on phase. Here a sequence of optical pulses with phases 0 or 7r are to be sorted based on phase and routed to two output ports. This may be accomplished by use of a simple interferometer, as shown in Fig.23.2-10.
1036 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Bif;= 0 7r 0 7r 7r 0 Input ll.D.illl1lIL -- t Output 1 t OutPu t2 t ll.D.illl1lIL t t Reference Figure 23.2-10 Phase-division routing. Intensity-Based Routing A light beam with time-varying intensity may be routed into separate beams based on the intensity. For example, a light beam carrying a sequence of pulses with two inten- sities, as illustrated in Fig. 23.2-11, is separated into a beam with the high-intensity pulses and another with the low-intensity pulses. This demultiplexing operation re- quires the use of nonlinear optical elements. It is often implemented by converting the intensity variation into a phase change by use of an optical Kerr cell (see Sec. 21.3A), as described next. Nonlinear Mach-Zehnder interferometer (MZI). The nonlinear MZI is a con- ventional MZI with a nonlinear optical element, such as a Kerr cell, placed in one of the interferometer branches. The cell introduces a phase shift proportional to the light intensity. The system is adjusted such that the phase difference between the interferometer branches is an odd multiple of 7r for one intensity, and an even multiple of 7r for the other. This diverts the stream of pulses into two ports, one with the high- intensity pulses and the other with the low-intensity pulses, as illustrated in Fig. 23.2- 1 1 (a). The interferometer may also be implemented in optical fibers, as illustrated in Fig. 23.2-11 (b). It,h lln.nllnIG npu1 t (a) _JllIL ili t l n c=:J t : ::: hfUl-n (b) Figure 23.2-11 Intensity-based 1 x 2 router using a Mach-Zehnder interferometer with a nonlinear Kerr medium implemented in (a) bulk optics and (b) fiber optics. Nonlinear asymmetric Sagnac interferometer. An intensity-based 1 x 2 router using a nonlinear fiber Sagnac interferometer is illustrated in Fig. 23.2-12. In this interferometer, light enters from fiber 1 and is split into a clockwise wave and a coun- terclockwise wave. If the optical pathlengths of these waves are identical, constructive interference occurs and the light propagates back into fiber 1 and is directed to output port 1, so that the device acts as a mirror. This occurs if the fiber is linear, or if the fiber is nonlinear and the intensities of the two waves are equal. However, if the coupler feeding the interferometer loop is not symmetric, then the intensities in the two paths are not equal so that the phase shifts introduced via the optical Kerr effect are generally different. When the phase difference is 7r, destructive interference ensues and light is
23.2 PASSIVE OPTICAL ROUTERS 1037 diverted into fiber 2 and output port 2. Since the phase difference is proportional to the intensity of the incident wave, the system acts as a 1 x 2 self-controlled intensity- division router (a demultiplexer). Asymmetry between the clockwise and counterclockwise waves in the Sagnac inter- ferometer may also be introduced by placing an erbium-doped fiber amplifier (EDFA) at an asymmetric location within the loop. This amplifies one of the interfering waves during the first half of its trip around loop, so that it travels more than one half of a round trip at a high intensity. The other wave is amplified in the second half-round- trip and travels a shorter distance at high intensity and therefore encounters a smaller nonlinear phase shift. The system is known as the nonlinear optical loop mirror (NOLM). fiR n Output 1 t .- llnnIlni It t Put2 t Fiber 2 Fiber loop Figure 23.2-12 Intensity-based 1 x 2 router using a nonlinear Sagnac inter- ferometer serving as a nonlinear optical loop mirror (NOLM). Nonlinear directional coupler (NLDC). A waveguide or fiber-optic directional cou- pler made of a Kerr material can also serve as an intensity-based router, as illustrated in Fig. 23.2-13. If the intensity of an input pulse is low, the medium is linear and the light is coupled from one guide to the other periodically as it travels (see Fig. 8.3-4). If the coupler's length equals the transfer distance Lo, the light is transferred completely from the input waveguide to the other waveguide. For pulses with large intensity the propagation constants are altered by the Kerr effect, creating an intensity-dependent phase mismatch that varies with the distance. Propagation then obeys the nonlinear coupled equations dd: 1 = -jeexp(jb..j3z)a2(z) - h'lall2al dG2 . ( . ) ( ) . 1 1 2 dz == -Je exp -Jt1{3z Gl z - Jry G2 G2, (23.2-8) (23.2-9) which are generalizations of the linear coupled equations (8.5-4) for the linear direc- tional coupler. Here, e == 7r /2Lo is the coupling coefficient and ry is proportional to the optical Kerr coefficient n2 [see Sec. 21.3A and Eq. (22.5-16)]. The system can be designed such that the high-intensity pulses exit the coupler from the same waveguide, i.e., are separated from the low-intensity pulses. Soliton directional coupler. The NLDC router may suffer from pulse breakup. Since the intensity of an optical pulse varies during its time course, so does the refractive index and the corresponding propagation constant in the nonlinear medium. Different fractions of the pulse power therefore cross between the two fibers, and this leads to pulse reshaping and possibly breakup. This does not occur in a fiber-optic NLDC [Fig. 23.2-13(b)] if the pulse is an optical soliton (see Sec. 22.5B). Because the non- linear phase shift of an optical soliton is constant over the pulse's envelope, the soliton pulse remains intact as it is routed between the coupled fibers. Another advantage of operating the NLDC in the soliton mode is that the transition between the output ports is a much sharper function of the input pulse power.
1038 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES (a) It (b) gI ... Double-core fiber . I UnIL t lt 12t Figure 23.2-13 Intensity-based 1 x 2 router using a directional coupler made of nonlinear optical material. The device may be implemented in (a) integrated-optic, and (b) fiber-optic technology. 23.3 PHOTONIC SWITCHES A. Architectures of Space Switches A switch is a device that establishes and releases connections among transmission paths in a communication or signal-processing system. A control unit processes the commands for connections and sends a control signal to operate the switch in the desired manner. While interconnects always operate on the incoming signals in the same manner, switches are controllable, active, or reconfigurable interconnects that are modified by an external command. Examples of switches are shown in Fig. 23.3-1. 1 1 1 ... ...... .. :)<:' 2 2 ",' ........., 2 1 1 Control Control Control 3 . 4 , 4 1 2 I N 1 - --- --- ---- --- --- 2 --- --- ---- I I - I ---..--- --- ---- I I I I ! --- - I I I I I I i I : I - : I I I I I N I I I I --_!.._-_! t Control 4 (a) (b) (c) (d) (e) Figure 23.3-1 (a) 1 x 1 switch connects or disconnects two lines. It is an ON-OFF switch. (b) 1 x 2 switch connects one line to either of two lines. (c) 2 x 2 crossbar switch connects two lines to two lines. It has two configurations: the bar state and the cross state, and may be regarded as a controllable directional coupler. (d) 1 x N switch connects one line to one of N lines. (e) N x N crossbar switch connects N lines to N lines. Any input line can always be connected to a free (unconnected) output line without blocking (i.e., without conflict). A 1 X 1 switch can be used as an elementary unit from which switches of larger sizes can be built. An N x N crosspoint-matrix (crossbar) switch, for example, may be constructed by using an array of N 2 1 x 1 switches organized at the points of an N x N matrix to connect or disconnect each of the N input lines to a free output line [see Fig. 23.3-1(e)]. The mth input reaches al] elementary switches of the mth row, while the lth output is connected to outputs of all elementary switches of the lth column. A connection is made between the mth input and the lth output by activating the (m, l) 1 x 1 switch. Examples are shown in Fig. 23.3-2. An N x N switch may also be built by use of 2 x 2 switches. Examples are shown in Fig. 23.3-3.
23.3 PHOTONIC SWITCHES 1039 (a) 2 3 (b) 1 2 3 2 3 Figure 23.3-2 (a) A 1 x 3 switch made of three 1 x 1 switches. (b) A 3 x 3 switch made of nine 1 x 1 switches in a broadcast-and-select configuration. 1. 2 2 3 3 2 2 4 4 3 3 5 5 4 4 6 6 (a) 7 7 8 8 (b) Figure 23.3-3 (a) A 4 x 4 switch made of five 2 x 2 switches. Input line 1 is connected to output line 3, for example, if switches A and C are in the cross state and switch E is in the bar state. (b) An 8 x 8 switch made of 28 2 x 2 switches. Switch Characteristics A switch is characterized by the following parameters: . Size (number of input and output lines) and direction(s), i.e., whether data can be transferred in one or two directions. . Switching time (time necessary for the switch to be reconfigured). . Propagation delay time (time taken by the signal to cross the switch). . Throughput (maximum data rate that can flow through the switch ). . Switching energy (energy needed to activate and deactivate the switch). . Power dissipation (energy dissipated per second in the process of switching). . Insertion loss (drop in signal power introduced by the connection). . Crosstalk (undesired power leakage to other lines). . Blocking probability. Probability that a connection cannot be established because of a conflict with another connection. . Physical dimensions. This is important when large arrays of switches are built. B. Implementations of Optical Space Switches Optoelectronic Switches Electronic switches have evolved steadily since the early years of telephony, generally tracking the steady advances in microelectronics. Nanoscale CMOS (complementary- symmetry metal-oxide-semiconductor) electronic gates can now operate at switch- ing times less than 0.1 ns and with switching energies smaller than 1 fJ. Advanced MOSFET (metal-oxide-semiconductor field-effect transistor) gates can be switched at subpicosecond times. Electronic chips for crossbar switching with large number of ports (e.g., 128 x 128) are readily available. It is therefore natural to use these devices for optical switching. But this requires optical-to-electrical conversion at the input of
1040 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES the switch and electrical-to-optical conversion at its output, as illustrated schematically in (Fig. 23.3-4). .. Optoelectronic .r- transmitter chip , / f ' ". /- .. _., Electronic crossbar .-- switch / . = Electronic control ",:,f: ;" ",,-' ,- ."),,.,, . Outgoing Fibers Figure 23.3-4 An optoelectronic crossbar switch. Incoming optical signals carried by optical fibers are detected by an array of photodetectors on an optoelectronic chip, switched using an electronic crossbar switch, and regenerated using an array of light sources (e.g., VCSELs) feeding outgoing optical fibers. Since the optical/electrical/optical conversions that are necessary for the operation of optoelectronic switches introduce unnecessary time delays and power loss, it is desirable to develop "transparent" photonic switches that operate directly on the optical signals by use of mechanical, electrical, acoustical, magnetic, or thermal effects, as described in this section. Some Basic Configurations for Optical Switches The most elementary optical switches are the optical scanner and the modulator. A scanner that deflects an optical beam into one of N possible directions is a 1 x N switch [ Fig. 23.3-5(a)]. An optical modulator operated in the ON-OFF mode serves as a 1 x 1 switch. Modulation may be direct, relying on some physical effect that transmits or blocks the light, or inteiferometric, using for example an optical phase modulator placed in one arm of an interferometer, which converts phase modulation into intensity modulation [Fig. 23.3-5(b)]. Another elementary optical switch is a directional coupler operated as a 2 x 2 switch. This may be implemented by use of an interferometer with a phase modulator in one or both arms [Fig. 23.3-5(c)]. The two branches of the interferometer may also represent two orthogonal polarization components, and the phase modulator is then a wave retarder that introduces a relative phase shift between the two polarizations. t Output I ...N ""'""'". , ...."'" --- ---- ::------ $:::----- t "'......:-........ , ..... .... , ..... Control "... 2 Input Input 1 -+- -+- Control Input 2 t (a) (b) (c) Figure 23.3-5 (a) An optical scanner as a 1 x N switch. (b) An interferometer with a phase modulator as a 1 x 1 switch. ( c) An interferometer with a phase modulator as a 2 x 2 switch. Elementary optical switches may be combined or cascaded in free space or in planar waveguide technology to make switches of higher dimensions. For example, as illus- trated in Fig. 23.3-6, a planar array of 16 optical modulators, each serving as a 1 x 1 switch, may be configured in an optical system operating as a 4 x 4 crossbar switch in the broadcast-and-select configuration.
4 "\ 4x4 modulators 23.3 PHOTONIC SWITCHES 1041 Figure 23.3-6 A 4 x 4 crossbar switch. Each of the 16 elements is a 1 x 1 switch transmitting or blocking light depending on a control signal. Light from the input mth point, m = 1, 2, 3, 4, is broadcast to all switches in the mth column. Light from all switches of the lth row is directed to the lth output point, I = 1,2,3,4. The system is an implementation of the 4 x 4 switch depicted in Fig. 23.3-I(e). Modulation and deflection of light can be achieved by the use of mechanical, electro-mechanical, electrical, acoustic, magnetic, thermal, or optical control; the switches are then called optomechanical (or mechano-optic), microelectromechanical systems (MEMS), electro-optic, acousto-optic, magneto-optic, or thermo-optic. The remainder of this section provides brief outlines of these technologies. All-optical, or opto-optic, switches are described in Sec. 23.3C. The switching times of these devices are compared in the following diagram: r Thermo- MEMS ..' '; Mechano- Magneto-: i LC SOA Acousto-<- All-Optical Electro- Electronics I 1 ns 1 fs I 1 ps I 1 JLs I 1 illS Mechano-Optic Switches A mechano-optic (or optomechanical) 1 x N switch (a scanner) may be implemented by use of a moving (rotating or alternating) mirror, prism, or holographic grating that deflects a light beam to a set of directions (Fig. 23.3-7). An optical fiber can be connected to any of a number of other optical fibers by mechanically moving the input fiber to align with the selected output fiber using a mechanism such as that illustrated in Fig. 23.3-7(c). Piezoelectric elements may be used for faster mechanical action. Prism cb 1 --_/ / J. 1 --+= w Figure 23.3-7 Deflecting light into different directions using (a) a rotating mirror or prism; (b) a rotating holographic disk. Each sector of the holographic disk contains a grating whose orientation and period determine a scanning plane and scanning angle of the deflected light. (c) An optical fiber attached to a rotating wheel is aligned with one of a number of optical fibers attached to a fixed wheel. The fibers are placed in V-grooves. An index-matching liquid is used for better optical coupling. Microelectromechanical systems (MEMS) are miniaturized systems powered by electrostatic actuators and fabricated in large arrays using processes similar to those
1042 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES of microelectronics. Switching speeds range between 10 ms and 10 MS. A cross-bar switch, for example, may be implemented by use of a set of MEMS popup mirrors, as shown in Fig. 23.3-8(a), or a set of moving mirrors, as shown in Fig. 23.3-8(b). The 4 Control 1 Popup mirror (a) (b) Figure 23.3-8 (a) MEMS popup-mirror switch. (b) MEMS moving-mirror switch. major limitation of opto-mechanical switches is their relatively slow response (switch- ing times are in the millisecond regime). Their major advantages are low insertion loss and low crosstalk. Electro-Optic Switches As discussed in Sec. 20.1, electro-optic materials alter their refractive indexes in the presence of an electric field. They may be used as electrically controlled phase mod- ulators or wave retarders. When placed in one arm of an interferometer, or between two crossed polarizers, the electro-optic cell serves as an electrically controlled light modulator or a 1 x 1 (on-oft) switch (see Sec. 20.1B). Since it is difficult to make large arrays of switches using bulk crystals, the most promising technology for electro-optic switching is integrated optics (see Chapter 8 and Sec. 20.1). Integrated-optic waveguides are fabricated using electro-optic dielectric substrates, such as lithium niobate LiNb0 3 , with strips of slightly higher refractive in- dex at the locations of the waveguides, created by diffusing titanium into the substrate. An example of a 1 x 1 switch using an integrated-optic Mach-Zehnder interferome- ter (MZI) is described in Sec. 20.1B and shown in Fig. 23.3-9(a). An example of a 2 x 2 switch is the directional coupler discussed in Sec. 20.1D and illustrated in Fig. 23.3- 9(b). Two waveguides in close proximity are optically coupled; the refractive index is altered by applying an electric field adjusted so that the optical power either remains in the same waveguide or is transferred to the other waveguide [Fig. 23.3-9(c)]. These switches operate at a few volts with speeds that can exceed 20 GHz. Figure 23.3-9 (a) A 1 x 1 switch using an integrated-optic Mach-Zehnder interferometer. (b) A 2 x 2 switch using an integrated-optic Mach-Zehnder interferometer. ( c) A 2 x 2 switch using an integrated-optic directional coupler.
23.3 PHOTONIC SWITCHES 1043 An N x N integrated-optic switch can be built by use of a combination of 2 x 2 switches. A 4 x 4 switch is implemented by use of five 2 x 2 switches connected as in Fig. 23.3-3(a). This configuration can be fabricated on a single substrate in the geometry shown in Fig. 23.3-10. Lithium niobate electro-optic switches of size up to 32 x 32 have been fabricated. 4 Figure 23.3-10 An integrated-optical 4 x 4 switch using five directional couplers A, B, C, D, and E on a single substrate. The limit on the number of switches per unit area is governed by the relatively large physical dimensions of each directional coupler and the planar nature of the interconnections within the chip. To reduce the dimensions and increase the packing density of switches, intersecting (instead of parallel) waveguides are used. Because of the rectangular nature of integrated-optics technology, it is difficult to obtain efficient coupling to cylindrical waveguides (e.g., optical fibers). Relatively large insertion losses are encountered, especially when a single-mode fiber is con- nected to an integrated-optic switch. Because the coupling coefficient is polarization dependent, the polarization of the guided light must be properly selected. This imposes a restriction requiring that the input and output connecting fibers must be polarization maintaining (see Sec. 9.2B). Elaborate schemes are required to make polarization- independent switches. Semiconductor Photonic Switches Semiconductor devices exhibit a number of electronic and optical properties that can be exploited for fast optical switching. As described in Sec. 20.5, electroabsorption, which is based on the Franz-Keldysh effect, and the quantum confined Stark effect (QCSE) in multiple quantum well (MQW) structures are used to control the absorption of light at wavelengths near the bandgap wavelength by application of an electric field. These electrically controlled optical modulators are used as 1 x 1 switches operated at high speeds with switching times shorter than 20 ps. They can be fabricated in large arrays, in the surface-norma] configuration, bonded to silicon substrates, as illustrated schematically in Fig. 23.3-11. It ".-- / / - . v"':it'.;:-- - ..---- Figure 23.3-11 An array of MQW switches based on the QCSE in the surface- normal configuration. '_M'''_'.'' . Another important device used in optical switching is the semiconductor optical amplifier (SOA). Since the SOA may be rapidly turned on and off by applying and removing the injected electric current (see Sec. 17.2A), it can be used as a fast 1 x 1 switch. Switching times in the nanosecond regime have been reported. In the absence of gain (i.e., the device is in the OFF state), the device acts as a strong absorber and in the presence of gain (i.e., the device is in the ON state) it becomes an amplifier, so that
1044 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES very large extinction ratios (more than 40 dB) are obtained. SOA switches operating at wavelengths of 1.55 J-Lm and 1.3 J-Lm using InGaAsPlInP double heterostructure, and also MQW structures (see Sec. 17.2D) have been demonstrated. Arrays of SOAs may be fabricated and interconnected via optical fibers, as illus- trated by the example in Fig. 23.3-12. Since SOAs provide gain, they may be added to the circuit to compensate for the large splitting losses. Hybrid and monolithic inte- gration of SOA switches with silica-based planar lightwave circuits (PLCs) have been demonstrated for small circuits, but large-scale integration remains to be a challenge. Since SOAs can function as wavelength converters, they may be used in wavelength switching (optical data carried on one wavelength are "copied" on a different wave- length). Because of their nonlinear optical properties, SOAs are also used as ultrafast all-optical switches, as discussed in Sec. 23.3C. Input 1 I I I Vt:!- Output 1 Preamplifiers SOA Switches Output 2 Postamplifiers Input 2 Figure 23.3-12 A 2 x 2 switch using four 1 x 1 SOA switches in the broadcast-and-select configuration shown in Fig. 23.3-2. Liquid-Crystal Switches Liquid crystals (LCs) provide another technology that can be used to make electrically controlled optical switches. As described in Sec. 20.3, an LC cell may be configured to act as an electrically controlled wave retarder or polarization rotator. This may be converted into intensity modulation by use of crossed polarizers. In another switching configuration, the change of the LC refractive index caused by the applied electric field is used for switching. The incoming light enters the LC at an angle via another medium with a refractive index selected such that total internal reflection occurs only when the electric field is applied. A large array of electrodes placed on a single liquid-crystal panel serves as a set of 1 x 1 switches (a digital spatial light modulator), which may be used in the broadcast-and-select configuration shown in Fig. 23.3-6 to implement an N x N crossbar switch. An alternative configuration for implementing a 2 x 2 crossbar LC switch is il- lustrated in Fig. 23.3-13. In this configuration, which is a polarization version of the Mach-Zehnder interferometer shown in Fig. 23.3-5(c), the LC cell rotates the polar- ization of the beams in the interferometer arms by 90° if the control signal is on, thus switching the connections from the bar state to the cross state. This switch is polarization independent, i.e., the beams are directed to the desired ports regardless of their polarization state. LC polarization rotator t Output 1 Figure 23.3-13 A 2 x 2 crossbar liquid- crystal switch. The two polarization components of an input beam are separated by the left polarizing beamsplitter (PBS) and recombined by the right PBS after passage through the liquid-crystal cell LC, which serves as a 7r /2 polarization rotator if the control signal is on. Without polarization rotation, the beams enter- ing at inputs 1 and 2 are directed to outputs 1 and 2, respectively, i.e., the switch is in the bar state. With polarization rotation, the beams are directed to the opposite output, ports corresponding to the cross state. PBS Output 2 Input 1 PBS Input 2 t Control
23.3 PHOTONIC SWITCHES 1045 Because of their relatively low switching speed, LC switches are used in applica- tions for which speed is not an issue, such as fault protection. switching and reconfig- urable optical add-drop multiplexing in optical fiber networks (see 24.4B). Acousto-Optic Switches Acousto-optic switches use the property of Bragg deflection of light by sound (Chap- ter 19). The power of the deflected light is controlled by the intensity of the sound. The angle of deflection is controlled by the frequency of the sound. An acousto-optic modulator is a 1 x 1 switch. An acousto-optic scanner (Fig. 23.3-14) is a 1 x N switch, where N is the number of resolvable spots of the scanner (see Sec. 19.2B). Acousto-optic cells with N == 2000 are available. If different parts of the acousto-optic cell carry sound waves of different frequencies, an N x M switch or interconnection device is obtained. Limitations on the maximum product N M achievable with acousto- optic cells have been discussed in Sec. 19.2C. Arrays of acousto-optic cells are also available. 1 1 1 2 1 2 L --..j TIL 1111,1, Inn ,I,I,I,.u""" i i I nm t -""jUt ' '" "'!II I " .,,, . '" t (a) (b) (c) Figure 23.3-14 Acousto-optic switches. (a) 1 x 2 ON-OFF switch, (b) 2 x 2 directional coupler, (c) L x !vI cross-bar switch. - - - - - - 1 2 M Magneto-Optic Switches Magneto-optic materials alter their optical properties under the influence of a magnetic field. Materials exhibiting the Faraday effect, for example, act as polarization rotators in the presence of a magnetic flux density B (see Sec. 6.4B); the rotatory power p (angle per unit length) is proportional to the component of B in the direction of propagation. When the material is placed between two crossed polarizers, the optical power transmission 'J == sin 2 e is dependent on the polarization rotation angle e == pd, where d is the thickness of the cell. The device is used as a 1 x 1 switch controlled by the magnetic field. Magneto-optic materials have recently received more attention because of their use in optical-disk recording. In these systems, however, a thermomagnetic effect is used in which the magnetization is altered by heating with a strong focused laser. Weak linearly polarized light from a laser is used for readout. The magneto-optic material is usually in the form of a film (e.g., bismuth-substituted iron garnet) grown on a nonmagnetic substrate. The magnetic field is applied by use of two intersecting conductors carrying electric current. The system operates in a binary mode by switching the direction of magnetization. Arrays of magneto-optic switches can be constructed by etching isolated cells (each of size as small as lOx 10 J-Lm) on a single film. Conductors for the electric-current drive lines are subsequently deposited using usual photolithographic techniques. Large arrays of magneto-optic switches (1024 x 1024) have become available and the tech- nology is advancing rapidly. Switching speeds of 100 ns are possible.
1046 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Thermo-Optic Switches Thermo-optic switches are generally based on the thermo-optic effect, which is the change of the refractive index caused by temperature variation of the material. The thermo-optic coefficient of silica glass, for example, is dn/ dT 10- 5 per degree (Celsius) and polymers have larger coefficients. Since this change is very small, the thermo-optic switch is often used in an interferometric configuration. Thermo-optic integrated-optic switches are fabricated in fiber-matched silica-on-Si (SOS) waveguides. An example is the Mach-Zehnder interferometer switch illustrated in Fig. 23.3-15. A thin-film metal heater deposited directly on the waveguide is used to control the temperature of the material in one of the interferometer branches. A tem- perature change T results in a phase shift (27rL/Ao)n == (27rL/Ao)(dn/dT)T, where L is the length of the heated region. For example, in a silica-based switch with L / Ao == 2 X 10 3 , the temperature change necessary to introduce a phase shift of 7r is T == 25° C. Other interferometric switching configurations based on arrayed waveguide gratings (AWG) have also been used in silica-based planar waveguides and in polymeric waveg- uides. The principal limitation of these switches, however, is long switching time, which is in the millisecond range. They are therefore more suitable for reconfiguring light paths in optical networks. Another thermo-optic switching technology is based on changing the refractive index by use of micro heaters to generate a bubble jet in a fluid. As illustrated in Fig. 23.3-16, the bubbles convert the fluid into a mirror that reflects the light beam. Fluid, refractive index n Microheater Waveguides, refractive index n Bubbles t Figure 23.3-15 Thermo-optic Mach-Zehnder interferometer switch. Figure 23.3-16 Bubble jet switch. c. All-Optical Space Switches In an all-optical (or opto-optic) switch, light controls light with the help of a nonlin- ear optical material. The control light alters some optical property of the nonlinear material, which changes some attribute of the controlled light, directing it from one port to another. For example, in a Kerr medium the refractive index is altered by the control light, which changes the phase of the controlled light, directing it from one output port of an interferometer to another. Other nonlinear interactions that can be used for switching include light sensitive retardation or absorption coefficient, and optical soliton collision, which is accompanied by time delay or frequency shift. Nonlinear Mach-Zehnder Interferometer (MZI) Switch A Mach-Zehnder interferometer (MZI) with a nonlinear optical element in one of its branches [Fig. 23.3-17)(a)] may be used as an all-optical 1 x 2 switch directing an optical beam at one of its two input ports to either of its output ports. The switch is
23.3 PHOTONIC SWITCHES 1047 controlled by an optical beam illuminating the nonlinear element. In the absence of the control beam, the interferometer is balanced such that the input light is directed to one of the output ports. When the control beam is applied, it induces a change in the refractive index n, which in turns creates an incremental phase shift of 7r so that the input beam is directed to the other output port. The interferometer may be implemented using bulk optics [Fig. 23.3-17(a)] or fiber optics [Fig. 23.3-17(a)]. This 1 x 2 switch may, of course, be used as a 1 x ION-OFF switch by simply ignoring one of the output ports. Input Output 1 Input Output 1 ----.. ----.. ----.. Control Nonlinear element ----.. Output 2 Control t (a) (b) Figure 23.3-17 (a) An all-optical 1 x 2 switch using a Mach-Zehnder interferometer with an optical Kerr cell. (b) A fiber-optic Mach-Zehnder interferometer. Ultrafast Nonlinear Asymmetric Sagnac Interferometer Switch interferometric configurations to achieve ultrafast optical switching, Switching speeds of several hundred Gb/s have been demonstrated despite the relatively slow carrier recovery process in SOAs. The switching speed of a nonlinear all-optical switch is limited by the response time of the nonlinear optical effect. This typically includes a short rise time at the onset of the control optical pulse, and a longer decay time following the pulse removal. The switching speed of a nonlinear MZI using a Kerr cell, for example, is limited by the response time of the Kerr effect. Much greater switching speeds, limited by the short rise time of the nonlinear effect, may be accomplished by use of an ingenious interferometric configuration for which both branches of the interferometer include the same nonlinear element, and the light pulse that is to be switched crosses it at different times. This is readily implemented by a fiber Sagnac interferometer with a nonlinear optical element placed at an asymmetric location within the fiber loop, as illustrated in Fig. 23.3-18. When an input optical pulse enters the loop from fiber 1, it is split by a symmetric coupler into a clockwise pulse and a counterclockwise pulse of equal amplitudes. If the two pulses encounter the same phase shift as they make their round-trip path around the loop, they recombine and return into the same fiber and leave out of output port 1. If they undergo phase shifts differing by 7r, they recombine and emerge into the other fiber and leave out of output port 2. These are the two states of a 1 x 2 switch. The nonlinear element is controlled by a short control optical pulse, which changes its refractive index by n. This change builds up with a short rise time 7i and decays with a much longer relaxation time Tr. Since the nonlinear element is placed at an offset location within the fiber loop, the two pulses cross it at different times, 71 and 72. If both pulses cross the nonlinear element when it is active, i.e., in the presence of the full change n, they undergo the same phase shift and the recombined pulse is received in fiber 1. This also occurs if both pulses cross the nonlinear element when it is inactive. However, if one pulse crosses when the nonlinear element is active and the other when
1048 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES it is inactive, they undergo different phase shifts, and if the phase difference is 7r, the pulse emerges in fiber 2, and out of output port 2. The switching action is therefore governed by the time difference 71 - 72, which is proportional to the distance of the nonlinear element from the mid-point of the fiber loop. If 71 - 72 is slightly greater than the rise time 7i, the switching action can be controlled with precision limited by the rise time, instead of the full response time 7r. Femtosecond switching times have been reported, so that the switch can be operated at terahertz speeds. This switch has been used for time-division demultiplexing and is known as the Terahertz Optical Asymmetric Demultiplexer (TOAD).t -+- Nonlinear element t j -I :"'''''ln ' , , ' '... 71 72 t IflI Output 1 ---+' Input /" 71 r-......" l:1n n:n ', 71 72 t Ifl2 Output 2 e. . '--- Fiber 2 Control r-......" !:In ' , , ' ...... 7[ 72 t IflI Figure 23.3-18 An all-optical fiber nonlinear asymmetric Sagnac interferometer used as a 1 x 2 switch. The switch is controlled by an optical pulse, which initiates a refractive index change f)"n in a nonlinear element placed at an offset location within the interferometer loop. The input pulse coming from fiber 1 is split into a clockwise pulse and a counterclockwise pulse that traverse the nonlinear element at different times. The switch changes the connection from output port 1 to output port 2 if one of these pulses arrives just before, and the other just after the onset of f)"n. This results in a phase difference of 7r and a diversion of the output pulse to output port 2. Nonlinear Optical Retardation Switch An all-optical switch may be based on the nonlinear Kerr effect in an anisotropic medium. The application of a control optical pulse creates different changes in the principal refractive indexes so that the medium may be used as an optically controlled wave retarder. When the medium (e.g., a crystal or an optical fiber) is placed between two crossed polarizers, as illustrated in Fig. 23.3-19, it functions as an on- off switch. When the retardation is 0, the light is blocked and the switch is in the OFF state. For a retardation of 7r, the light is transmitted and the switch is in the ON state. Soliton Switches Optical solitons are ultrashort pulses that propagate in nonlinear dispersive optical fibers without spreading (see Sec. 22.5B). A 1 x 2 all-optical switch may be realized by use of one optical soliton to control the routing of another into one of two output ports. The interaction between the two solitons may take the form of a collision or a recombination into one vector soliton. In either case, some optical property of the input soliton is altered by the interaction, and the changed property is used to effect the routing. Sub-picoseond switching speeds with switching energy in tens of pJ have been implemented in soliton technology. t See J. P. Sokoloff, P. R. Prucnal, I. Glesk, and M. Kane, A Terahertz Optical Asymmetric Demultiplexer (TOAD), IEEE Photonics Technology Letters, vol. 5, pp. 787-790, 1993.
23.3 PHOTONIC SWITCHES 1049 Polarizer Input . Input Output Output Control Anisotropic Kerr medium Polarizer Control (a) (b) Figure 23.3-19 An anisotropic nonlinear Kerr medium serving as an all-optical switch: (a) a crystal, and (b) an anisotropic (birefringent) optical fiber. In the presence of the control light, the medium introduces a phase retardation 7r, so that the polarization of the linearly polarized input light rotates 90° and is transmitted by the output polarizer. In the absence of the control light, the medium introduces no retardation and the light is blocked by the polarizer. The filter is used to block the control light, which has a different wavelength. Soliton-collision switching. If two solitons with slightly different frequencies, and hence slightly different group velocities, collide, i.e., pass through one another, the arrival time and the phase of each soliton are altered. One of the pulses serves as the control pulse, and the other as the signal pulse. Either the time delay or the phase shift that accompanies collision with the control pulse is used to route the signal pulse. Time-based routing is implemented by use of an optical gate that opens during a prescribed time window. Phase-based routing is effected by use of an interferometer. Vector-soliton switching. A vector soliton comprises two orthogonally polarized optical pulses copropagating through a nonlinear birefringent fiber. Since both pulses must be present for the vector soliton to form, the system may be used as an optical switch with one pulse serving to control the other. Two pulses with orthogonal polarization travel in a birefringent fiber at slightly different group velocities and therefore separate in time, a phenomenon known as walk- off (see Sec. 22.5A). If the fiber is also nonlinear, cross-phase modulation (XPM) (see Sec. 21.3C) results in a frequency upshift in one pulse and a frequency downshift in the other. Because of group velocity dispersion (GVD), these shifts are accompanied by a change in the group velocities. When the group velocity difference due to birefringence is exactly compensated by that due to GVD (via XPM), the two pulses travel jointly, as a single vector soliton, a phenomenon also known as soliton trapping. As illustrated in Fig. 23.3-20, a 1 x 1 soliton switch is implemented by using one of two orthogonally polarized pulses as the control pulse, and the other as the signal to be transmitted or blocked. If the two pulses have the same wavelength A, then when they travel through the nonlinear birefringent fiber they form a vector soliton whose components have shifted wavelengths A ::l: 6A. One of these components is selected by a filter and constitutes the output of the switch. In the absence of the control pulse, the vector soliton is not formed and the wavelength is not shifted, so that the light is blocked by the filter. IIV h Control Ie Slow J\ Fast ( r SOli lk A Acf*8A Birefringent nonlinear fiber Figure 23.3-20 A fiber-optic all-optical switch using vector solitons.
1050 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Fundamental Limits on All-Optical Switches Minimum values of the switching energy E and the switching time T of all-optical switches are governed by the following fundamental physical limits. Photon-number fluctuations. The minimum energy needed for switching is in prin- ciple one photon. However, since there is an inherent randomness in the number of photons emitted by a laser or light-emitting diode, a larger mean number of photons must be used to guarantee that the switching action almost always occurs whenever desired. For these light sources and under certain conditions (see Sec. 12.2C) the number of photons arriving within a fixed time interval is a Poisson-distributed random number n with probability distribution p(n) == n n exp( - n ) In!, where n is the mean number of photons. If n == 21 photons, the probability that zero photons are delivered is p(O) == e- 21 10- 9 . An average of 21 photons is therefore the minimum number that guarantees delivery of at least one photon, with an average of 1 error every 10 9 trials. The corresponding energy is E == 21hv. For light of wavelength Ao == 1 J-Lm, E == 21 x 1.24 26 eV == 4.2 aJ. This is regarded as a lower bound on the switching energy; it should be noted, however, that this is a practical bound rather than a fundamental limit, inasmuch as photon-number-squeezed light (see Sec. 12.3B) may in principle be used. To be on the less optimistic side, a minimum of 100 photons may be used as a reference. This corresponds to a minimum switching energy of 20 aJ at Ao == 1 J-Lm. Note that, at optical frequencies, hv is much greater than the thermal unit of energy kT at room temperature (kT == 0.026 eV at T == 300 0 K). Energy-time uncertainty. Another fundamental quantum principle is the energy- time uncertainty relation (J E(JT > hi 47r [see (12.1-16)]. The product of the minimum switching energy E and the minimum switching time T must therefore be greater than hi 47r (i.e., E > hi 47r T == hv 147rv T). This bound on energy is smaller than the energy of a photon hv by a factor 47rV T. Since the switching time T is not smaller than the duration of an optical cycle 1 I v, the term 47rv T is always greater than unity. Because E is chosen to be greater than the energy of one photon, hv, it follows that the energy-time uncertainty condition is always satisfied. Switching time. The only fundamental limit on the minimum switching time arises from energy-time uncertainty. In fact, optical pulses of a few femtoseconds (a few optical cycles) are readily generated. Such speeds cannot be attained by semiconductor electronic switches (and are also beyond the present capabilities of Josephson de- vices). Subpicosecond switching speeds have been demonstrated in a number of optical switching devices. Switching energies can also, in principle, be much smaller than in semiconductor electronics. Size. Limits on the size of photonic switches are governed by diffraction effects, which make it difficult to couple optical power to and from devices with dimensions smaller than a wavelength of light. Practical limitations. The primary limitation on all-optical switching is a result of the weakness of the nonlinear effects in currently available materials, which makes the required switching energy rather large. Another important practical limit is related to the difficulty of thermal transfer of the heat generated by the switching process. This limitation is particularly severe when the switching is performed repetitively. If a minimum switching energy E is used in each switching operation, a total energy E IT is used every second. For very short switching times this power can be quite large. The
23.3 PHOTONIC SWITCHES 1051 maximum rate at which the dissipated power must be removed sets a limit, making the combination of very short switching times and very high switching energies untenable. Note, however, that thennal effects are less restrictive if the device is operated at less than the maximum repetition rate; i.e., the energy of one switching operation has more than a bit time to be dissipated. D. Wavelength-Domain Switches The switches described so far are space-domain switches, i.e., they establish transmis- sion paths that route optical beams between specific physical positions (the input and output ports of the switch). Their wavelength-domain logical counterparts are called wavelength-domain switches, as illustrated by the following examples. EXAMPLE 23.3-1. Reconfigurable Wavelength Selector. An example of an optical de- vice that uses a combination of passive wavelength routers and space switches is the wavelength selector illustrated in Fig. 23.3-21. This switch selects one or more wavelengths from an incoming beam with N wavelengths. It uses a demultiplexer to separate the N wavelength components, a set of NIx 1 switches to select the desired wavelengths, and a multiplexer to reconstitute the output beam as shown in Fig. 23.3-21. The overall system is a combination of a passive wavelength routers and space switches. A1. A 2' ... AN A L A2. ... AN DEMUX Ix 1 switches MUX Figure 23.3-21 A reconfigurable wavelength selector. EXAMPLE 23.3-2. Reconfigurable Optical Add-Drop Multiplexer (ROADM). The ROAD is a reconfigurable OADM with the option to add, drop, or pass-through, as illustrated in Fig. 23.3-22. It uses a demultiplexer (DEMUX), a multiplexer (MUX), as well as a 1 x 2 switch and a 2 x 1 switch per add-drop channel. A},A2' ... AN DEMUX Figure 23.3-22 Reconfigurable optical add-drop multiplexer (ROADM). EXAMPLE 23.3-3. Wavelength-Channel Interchange (WCI). The WCI switch, also called the A switch, routes data between wavelength channels in the same optical beam. An N x N WCI switch may be implemented by mapping the wavelength channels to the space domain using a demultiplexer, converting the wavelengths using a bank of N wavelength converters (WCs), and recombining the channels into a single beam by use of an N x 1 coupler, as shown in Fig. 23.3-23. A wavelength converter changes the wavelength of a beam without altering the data, i.e., "copies" the data from one wavelength channel to another.
1052 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES - Al .-\2 A3 . .. AN DEMUX - - I - I - A I ---cJ---e AN _ aJ I _ I _ - I I - I I - A2 ---cJ---e A3 I I I I I I I I I I I I I I I I I I I I A3 ---cJ---e AI - - I I - - ...l_ L...L..J I L...L..J I LJ AN --c=J---. A2 - I - I - WCs MUX Al A2 A3 ... AN Figure 23.3-23 Implementation of a wavelength-channel interchange (WCI). In this example, data in wavelength channel 2 (green) of the input beam, for example, are routed to wavelength channel 3 (yellow) of the output beam. Data bits are depicted as colored and white squares. This switch is implemented by use of a wavelength demultiplexer to separate and directs the wavelength channels to a bank of wavelength converters. A fan-in N x 1 coupler recombines the switched channels into a single beam. Multidimensional Space- Wavelength Switches The previous examples of wavelength-domain switches involve a single optical beam with multiple wavelength channels. Switching may also be applied to multichannel multiple beams. Consider, for example, the switching of N beams, each with one of N wavelength channels. The switch redistributes the wavelength channels among the beams. Two implementations are shown in Fig. 23.3-24. The first implementation uses a broadcast -and-select router to redirect the wave- length channels to different ports. This is accomplished by means of a star coupler that broadcasts the contents of all N beams to each of a set of wavelength filters, each tuned to a single wavelength channel [see Fig. 23.3-24(a)]. Finally, for further processing, the wavelengths of the switched channels are converted to the original wavelengths (without change of their data content), by use of a bank of wavelength converters (WCs). The second implementation uses two sets of WCs with a wavelength grating router (WGR) in-between, as shown in Fig. 23.3-24(b). The first WC converts the wavelengths to values that satisfy the WGR equation (23.2-7) for the appropriate destinations. The WGR switch is more efficient than the broadcast-and-select switch since the lat- ter wastes considerable power at the filters. However, the broadcast -and-select switch has the advantage of being reconfigurable. )q A2 A3 Al A2 A3 Al A2 A3 AN AN AN AN (a) Star Tunable WCs coupler filters WCs (b) WCs Figure 23.3-24 (a) Broadcast-and-select space-wavelength switch. (b) WGR space-wavelength switch. Implementations of Wavelength Converters A wavelength converter (WC) transfers data carried by an optical beam at some wave- length to a different wavelength. The wavelengths often represent channels of a WDM fiber communication system (see Sec. 24.3C) and their separation is not large since they lie in the same band. Wavelength converters are implemented by use of nonlinear optical devices, parametric or nonparametric.
23.3 PHOTONIC SWITCHES 1053 In nonparametric WCs, the intensity of the first beam, which is modulated by the data, alters an optical property of a medium, such as gain coefficient, absorption coeffi- cient, or refractive index of a semiconductor, in proportionality to the intensity, so that the data is "written" into the medium. When a second beam of different wavelength is transmitted through the medium it is modulated by the altered property, so that the data are "read" by, and transferred to, the second beam. As depicted in Fig. 23.3-25(a), the gain of a saturated semiconductor optical am- plifier (SOA) is a decreasing function of the intensity. When the original intensity- modulated beam is transmitted through, the gain is modulated as an inverted function, and so is the intensity of the read beam. The process is called cross-gain modulation (XGM). In an unsaturated SOA, the refractive index is modulated by the write beam since it is dependent on the carrier density. The read beam is therefore phase modulated. The process is known as cross-phase modulation (XPM). An interferometer is necessary to convert phase modulation into intensity modulation, as shown in Fig. 23.3-25(b). In a WC based on parametric interaction, beams of different wavelengths are cou- pled via the nonlinear effect (see Chapter 21). For example, in a second-order nonlinear medium a wave of frequency WI may be downconverted to a frequency W2 == W3 -WI by use of an auxiliary wave of frequency W3. The amplitude of the downconverted wave is related to that of the original wave, so that the data embedded in the magnitude or the phase of the original wave are transferred to the downconverted wave. The main difficulty of this three-wave mixing process is that if the frequencies WI and W2 are close, the frequency W3 of the auxiliary wave must be approximately twice as large. If only waves of approximately equal frequencies are to be used, cascaded nonlinear parametric processes may be implemented. The first process is a second- harmonic generation (SHG) process in which WI is converted to 2WI, and the second is a three-wave mixing process of downconversion generating a wave of frequency W2 == 2W3 - WI. All three waves now have approximately the same frequency. Alternatively, a four-wave mixing process in a third-order nonlinearity, such as an optical fiber [see Fig. 23.3-25(c)]. As described in Sec. 21.3, this process involves the mixing of four- waves of frequencies satisfying the relation WI + W2 == W3 + W4. In the partially- degenerate case W3 == W4 == WO, so that W2 == 2wo - WI. c 'a o Intensity II , lflIlflflflJU w2 W2 MZI W2 JUUUULlUl WO JUUUULlUl "-.=' , WI JUUULJ1..JUl SO A (a) XGM W2 WI X(3) medium W2 JUUUULlUl WI JUUUULlUl (b) XPM (c) FWM Figure 23.3-25 Wavelength conversion. Data is transferred from a beam of frequency WI to a beam of frequency W2. (a) Cross-gain modulation (XGM) in a semiconductor optical amplifier. (b) Cross-phase modulation in a semiconductor. Phase modulation of the converted beam is transformed into intensity modulation by use of a Mach-Zehnder interferometer (MZI). (c) Partially-degenerate four-wave mixing (FWM) in a third-order nonlinear medium using an auxiliary wave of frequency Wo = (WI + W2). E. Time-Domain Switches The time-domain switch routes signals between time slots (see Fig. 23.3-26). In digi- tal communication systems, a signal is divided into a sequence of time frames of equal
1054 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES duration, each divided into N time slots where the data reside. An example of a time- domain switch is the time-slot interchange (TSI) switch, which transfers the data at the fth time slot of each frame to the mth time slot of the same frame. This corresponds to the wavelength-channel interchange (WCI) switch described in the previous section. 1 1 I I 2 2 ....... - - 3 3 ------- 1 2 3 N . 1 2 3 t t N (a) Space switch (b) Time switch Figure 23.3-26 Correspondence between time- and space-domain switches. (a) Space-domain switch. In the shown example, data in line 2 are routed to line 3. (b) Time-domain switch implementing a time-slot interchange (TSI). In the shown example, data in time slot 2 are routed to time-slot 3 in each frame. Two-dimensional space-time switches employ a combination of time-domain and space-domain switches. The switch connects a set of input lines, each carrying a digital signal composed of a sequence of time frames, to a similar set of output lines. Data in each time slot in each input line are transferred to one, or several, time slots in one or several output lines, in accordance with some rule. An example is the time-space-time (TST) switch, which is made of a cascade of a time-slot interchange (TSI), a space switch, and another TSI, as shown in Fig. 23.3-27. I I I I I I I I I I I I I I I I I I . I I I I I I I I I I I I I I I I I . I I I I I I I I I I I I I I I I I . I I I I I I I I I I I I I I I I I I I I I I I I I I . I I I I I I I I TSI TSI I I I I I I I I I I I I r I I I I I . I I I I I I I I · ( T . . h IIDe SWltc . ( Space switch · ( T " . h IIDe SWitc . ( Figure 23.3-27 Time-Space- Time (TST) switch. Time-Division Multiplexing and Demultiplexing A simple example of space-time switches is the time-division demultiplexer. It has one input line and N output lines, where N is the number of time slots in each frame. The switch routes data in the fth time slot of the input line to the fth time slot of the fth output line; f == 1, 2, . . . , N. The process is repeated periodically in all frames. This switch is therefore equivalent to a time-to-space mapping. In the time-division demultiplexer shown in Fig. 23.3-28, for example, there are N == 4 time slots per frame. The slots have data in the form of pulses of various heights. The switch directs the first pulse to the first output port, and the second pulse to the second output port, and so on. Such a switch may be constructed by use of a 1 x N space switch connecting the input port sequentially to one of its four output ports. The inverse of a time-division demultiplexer, called a time-division multiplexer (TDM), interleaves pulses in N separate ports into a single sequence of pulses. This inverse operation may be visualized in Fig. 23.3-28 with the input and output ports exchanging roles; i.e., the pulses travel from right to left, instead of left to right. The 1 x N time-division demultiplexer may be implemented by use of NIx ION-OFF .
23.3 PHOTONIC SWITCHES 1055 switches, as illustrated in Fig. 23.3-2(a), turned on and off sequentially with control pulses from a clock. F rame 2 . 1 10 : L 2 o I 3 1 r-I: : 4 i D I : n1 : 2 : 101 1 3 ! LIT- 1 1 4 : : ....., t I ( T . I F rame I t F rame 1 fr ame 2 Figure 23.3-28 Time-division demultiplexing with N == 4. Optical Time-Division Multiplexing (TOM) An optical implementation of the TDM is illustrated in Fig. 23.3-29(a). Copies of the input beam are transmitted through a set of NIx 1 optical switches controlled by a set of optical pulses from a clock delayed by multiples of the time delay T / N, where T is the frame period. In another implementation, copies of the input beam are delayed successively by multiples of T / N so that the N input pulses are synchronized in time but separated in space; the 1 x 1 switches are controlled by the same clock signal, as illustrated in Fig. 23.3-29(b). The system is similar to that used to detect the temporal profile of an optical pulse (see Fig. 22.6-5 in Sec. 22.6A). Optical delays may be implemented by use of optical fibers (approximately 5 ns/m for silica-glass fibers). The 1 x 1 switches may be implemented optically using an all-optical nonlinear interferometric switch. An example is the Terahertz Optical Asymmetric Demultiplexer (TOAD) described in Sec. 23.3C. 1 x 1 switches Optical delays 1 x 1 switches .. -_. . II Star coupler Star coupler (b) (a) Figure 23.3-29 Implementation of time-division demultiplexing by use of a set of optical time delays and 1 x 1 optical switches. In this illustration, N == 4.
1056 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Optical Time-Slot Interchange (TSI) The TSI switch (Fig. 23.3-26) is a time-domain switch that interchanges the data in the time slots of each frame. Its implementation may be based on a combination of space switches and space-time switches. The configuration shown in Fig. 23.3-30, for example, uses a time-division demultiplexer that routes the time slots to separate lines (time-to-space mapping); time delays are introduced to synchronize the pulses in one time slot of duration T / N before entering an N x N cross-connect (a space switch). Another set of time delays is then introduced to restore the pulses to their original time slots, and a time-division multiplexer is subsequently used to bring these time slots to a single line (space-to-time mapping). 3T/4 1 . 1 2 3 4 i t DEMUX MUX t Time-to-space Delays Space Switch Delays Space-to-time Figure 23.3-30 Time-slot interchange (TSI). Optical Programable Time Delays and Buffers Controllable time delays are essential components in time-domain switching. Buffers are memory elements used to temporarily store data or to compensate for differences in the data flow rates. As seen in Fig. 23.3-29 and Fig. 23.3-30, such delays are introduced by use of opticaJ fibers of appropriate length ( 5 ns/m). Programable delays may be implemented by allowing the optical pulses to circulate for a programable number of cycles in a fiber loop. As illustrated in Fig. 23.3-31, this is accomplished by use of a crossbar switch that permits the pulse to enter the loop at the desired time and releases it after a selected number of cycles. (0) Delay T (0) Dela y T (0) Delay T K :r-------( ---------t t+ T t+ 2T t+mT Figure 23.3-31 Programable delay line using a fiber loop and a crossbar switch. At time t = 0, the switch is in the cross state so that the optical pulse is admitted into the loop. At time t = T, the pulse returns back to the input port of the switch, which is then put in the bar state, so that the pulse undergoes another round trip with an additional delay T. At time t = mT, the pulse is released by changing the switch to the cross state. F. Packet Switches The switches presented so far in this section are relational switches that establish mappings between input and output ports depending on the state of the switch, which ,.
23.3 PHOTONIC SWITCHES 1057 is controlled by external signals that are not dependent on the data entering the input ports. This type of switching is called circuit switching. In a different type of switches, called packet switches, the switch configuration is set up according to destination information contained in the input data themselves. The data are organized in packets, each with a header containing the address of the packet's destination, as illustrated in Fig. 23.3-32. The packet switch contains a header recognition unit that reads the ad- dresses and sends a control signal that sets the switch to the appropriate configuration. 3 _,__- __,_- N ..-..- I __,_- ._.,,- -.--- 2 ---..- ..-..- Switch Header .... I Payload - Packet t Header recognition Figure 23.3-32 Packets and packet switches. A header address recognition system may use a bank of correlators that correlate the bit sequence representing the address of the incoming packet with the bit se- quences representing each of the possible addresses in a lookup table, and identifies the address with the highest correlation. For example, if the address of the incoming packet is the bit sequence (aI, a2, . . . , aN) and that of one of the addresses in the table is (b l , b 2 , . . . , b N ), the correlation is the sum al b l + a2b2 + . . . , aNb N . Since the bits of the incoming header arrive sequentially in time, implementation of the correlation operation requires the use of delays, multipliers, and an adder. One optical implementation uses an optical fiber with N fiber Bragg grating (FBG) reflectors placed at equal distances, as shown in Fig. 23.3-33. The reflectors have reflectance (b l , b 2 , . . . , b N) and serve as the multipliers. The round-trip delays introduced by the fiber segments bring the bits of the incoming header in synchrony so that they add up to yield the correlation sum. _<== ,.Jw.. JLJ1fL...fi..JL a] aN -;- llllW.UU.l FBG bI (( «( FBG b2 n . . . ci I' .) .T' I , -\ (((( ((( FBG .'.; (<< ((( ( ==:I" FBG bN Figure 23.3-33 Optical correlator for recognition of header address. A packet switch may also be implemented sequentially by use of a set of elementary 2 x 2 switches each routing the incoming packet to its upper or lower output port depending on one bit in the header address. For example, if the bit is 1 or 0, the switch routes the data to the upper or lower port, respectively. In other systems, the 2 x 2 switch sorts its two incoming packets and directs the packet with the greater address number to the lower output port and the other packet to the upper output port. For example, the 8 x 8 three-stage switch illustrated in Fig. 23.3-34, called Banyan switch, employs twelve 2 x 2 self-routing switches. The address of each packet is expressed as a binary number (Xl, X2, X3). Routing in the first stage is based on the most significant bit Xl, and routing in stages 2 and 3 is based on bits X2 and X3,
1058 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES respectively. In each case, if the bit is 1, the packet is routed to the lower output port; otherwise, it goes to the upper port. The switch is configured in such a way that after three stages, the packet arrives at its desired destination. However, it is not difficult to show that a conflict may arise when two packets are to be routed to the same output port of a 2 x 2 switch. More complex configurations have been devised in order to avoid such internal blocking. For example, networks using combinations of sorting and routing units can completely avoid internal blocking. o Stage 1 Stage 2 Stage 3 o = (000) 1 = (001) 2=(010) 3 = (011) 4 = 100) 5 = (101) 6 = (110) ------.. r ..:.:.:... :., 1 llO =><= -- " , 2 " ..--" 6 = (110) 7 7=(111) Figure 23.3-34 An 8 x 8 three-stage Banyan switch. A packet incoming at input port number 2 with a header address number 6 is directed to its destination, output port number 6, after passage through three 2 x 2 self-routing switches. Since the address is represented by the binary number 6=(110), the packet is directed to the (lower, lower, upper) ports of these switches, respectively, following the path marked by the dashed line, and ultimately reaches output port 6=(110). Contention occurs when packets of different input ports are simultaneously des- tined to the same output port. Methods for contention resolution include routing the conflicting packet via a different path or delaying it to a different time period by use of a buffer. In the optical domain, the packet may also be converted to a different wavelength and transmitted along a different wavelength channel. Optical buffers and wavelength converters were described in Sec. 23.3D. 23.4 OPTICAL GATES Highly sophisticated digital electronic systems (e.g., a digital computer) contain a large number of interconnected basic units: switches, gates, and memory elements (flip- flops). This section introduces bistable optical devices, which offer possibilities for optical gates and flip-flops. A. Bistable Systems A bistable (or two-state) system has an output that can take only one of two distinct stable values, no matter what input is applied. Switching between these values may be achieved by a temporary change of the level of the input. In the system illustrated in Fig. 23.4-1, for example, the output takes its low value for small inputs and its high value for large inputs. When an increasing input exceeds a certain critical value (threshold) f)2, the output jumps from the low to the high value. When the input is subsequently decreased, the output jumps back to the lower value when another critical value f)1 < f)2 is crossed, so that the input-output relation forms a hysteresis loop. There is an intermediate range of input values (between f)1 and f)2) for which low or high outputs are possible, depending on the history of the input. Within this range, the
23.4 OPTICAL GATES 1 059 ...... ;:3 0... ...... ;:3 o .J "' ,. ... I 'l91 I 'l92 Input Figure 23.4-1 Input-output hysteresis relation for a bistable system. system acts like a seesaw. If the output is low, a large positive input spike flips it to high. A large negative input spike flips it back to low. The system has a "flip-flop" behavior; its state depends on its history (whether the last spike was positive or negative; Fig. 23.4-2). 2 ..... :::s 0.- ..... :::s o ..... :::s 0.- ..... :::s o 1 :3 t Input 1 2 3 t 2 3 t Figure 23.4-2 Flip-flopping of a bistable system. At time 1 the output is low. A positive input pulse at time 2 flips the system from low to high. The output remains in the high state until a negative pulse at time 3 flips it back to the low state. The system acts as a latching switch or a memory element. Bistable devices are important in digital electronics and are a basic building block of computer systems. They are used as switches, logic gates, and memory elements. The device parameters may be adjusted so that the two critical values (the thresholds 'l9 1 and 'l9 2 ) coalesce into a single value 'l9. The result is a single-threshold steep S-shaped nonlinear output-input relation. When biased appropriately the device can have large differential gain and can be used as an amplifier, like a transistor. It can also be used as a thresholding element in which the output switches between two values as the input exceeds a threshold, as a pulse shaper, or as a limiter (Fig. 23.4-3). A stable threshold and stable bias are necessary for these operations. Bistable devices are also used as logic elements. The binary data are represented by pulses that are added and their sum used as input to the bistable device. With an appropriate choice of the pulse heights in relation to the threshold (see Fig. 23.4-4), the device can be made to switch to high only when both pulses are present, so that it acts as an AND gate. The AND logic gate is a digital device with two binary inputs and one binary output. Both inputs must be in the" 1" state for the output to be in the "1" state. Otherwise, the output is in the "0" state. Logic gates may be used as switches. For example, by using one of the inputs to an AND gate, as the control, the gate becomes 1 x 1 ON-OFF switch.
1060 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES ..... ::3 0... ..... ::3 o Output mm__m__m_ 6 _ A__A__m_ : ---\1-\1-- t ..... ::3 0... ..... ::3 o Output --m_un- m - - -- -- - -- -- -- -- --- Input t {) Input ::i : 0... . : (a) t, (b) t Figure 23.4-3 The bistable device as (a) an amplifier or (b) a thresholding device, pulse shaper, or limi ter. ..... ::3 0... C - Output 10 Output 00010 D Input Ii = II + 12 t 0+0 --------------- ------------- 0+1 -------...------... ------------- 0+0 --------------- 1 + 1 --------------- ]+0 /1 =D- 10 12 AND Figure 23.4-4 The bistable device as an AND logic gate. The input Ii = II + 1 2 , where II and 1 2 are pulses representing the binary data. The output 10 is high if and only if both inputs are present. t /1 --- --- 1"2 --- - --- ---. B. Principle of Optical Bistability Two features are required for making a bistable device: nonlinearity and feedback. An electronic bistable (flip-flop) circuit is made by connecting the output of each of two transistors to the input of the other (see any textbook on digital electronics). An optical bistable system is realized by use of a nonlinear optical element whose output beam is used in a feedback system to control the transmission of light through the element itself. Consider the generic optical system illustrated in Fig. 23.4-5. By means of feedback the output intensity 10 is somehow made to control the transmittance 'I of the system, so that 'I is some nonlinear function 'I == 'J(I o ). Since 10 == 'IIi, Ii '1(1 0 ) /0 10 Ii = ']'(1 0 ) (23.4-1) Input-Output Relation for a Bistable System Figure 23.4-5 An optical system whose transmittance T is a function of its output 10. If 'J(Io) is a nonmonotonic function, such as the bell-shaped function shown in Fig. 23.4-6(a), Ii will also be a nonmonotonic function of 1 0 , as illustrated in Fig. 23.4-6(b). Consequently, 10 must be a multivalued function of Ii; i.e., there are some values of Ii with more than one corresponding value of 1 0 , as illustrated in Fig. 23.4-6(c).
23.4 OPTICAL GATES 1 061 The system therefore exhibits bistable behavior. For small inputs (Ii < 'l9 1 ) or large inputs (Ii > 'l9 2 ), each input value has a single corresponding output value. In the intermediate range, 'l9 1 < Ii < 'l9 2 , however, each input value corresponds to three pos- sible output values. The upper and lower values are stable, but the intermediate value [the line joining points 1 and 2 in Fig. 23.4-6(c)] is unstable. Any slight perturbation added to the input forces the output to either the upper or the lower branch. Starting from small input values and increasing the input, when the threshold 'l9 2 is exceeded the output jumps to the upper state without passing through the unstable intermediate state. When the input is subsequently decreased, it follows the upper branch until it reaches 'l9 1 whereupon it jumps to the lower state, as illustrated in Fig. 23.4-7. 'Ii ,/ 3 Io/// 10 12 I. I 2 I. I 72 '1"(/0 ) I} 10 1 0 /72 II II 12 10 (a) (b) (c) Figure 23.4-6 (a) Transmittance 'T(Io) versus output 10. (b) Input Ii = Iol'T(Io) versus output 10. For 10 < a or 10 > b, 'T(Io) = 'II and Ii = Iol'Tl is a linear relation with slope 1/'Tl. At the intermediate value of 10 for which 'I has its maximum value 'T 2 (point 2), Ii dips below the line Ii = Ioj'!l and touches the lower line Ii = Iol'T2 at point 2. (c) The output 10 versus the input Ii is obtained simply by replotting the curve in (b) with the axes exchanged. (The diagram is rotated 90° in a counterclockwise direction and mirror imaged about the vertical axis.) 10 'l91 'l92 I. I Figure 23.4-7 Output versus input of the bistable device shown in Fig. 23.4-5. The dashed line represents an unstable state. The instability of the intermediate state may be seen by considering point P in Fig. 23.4-7. A small increase of the output 10 causes a sharp increase of the transmittance 'J(1 0 ) since the slope of 'J(10) is positive and large [see Fig. 23.4-6(a) and note that P lies on the line joining points 1 and 2]. This in turn results in further increase of 'J(I o ), which increases 10 even more. The result is a transition to the upper stable state. Similarly, a small decrease in 10 causes a transition to the lower stable state. The nonlinear bell-shaped function 'J(10) was used only for illustration. Many other nonlinear functions exhibit bistability (and possibly multi stability, with more than two stable values of the output for a single value of the input).
1062 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES EXERCISE 23.4-1 Examples of Nonlinear Functions Exhibiting Bistability. Use a computer to plot the rela- tion between 10 and Ii == Io/'T(Io), for each of the following functions: (a) 'T(x) == l/[(x - 1)2 + a 2 ]. (b) 'T ( x) == 1/ [1 + a 2 sin 2 (x + B)]. (c) 'T(x) == + c os(x + B). (d) 'T(x) == sinc 2 v a 2 + x 2 . (e) 'T(x) == (x + 1)2/(x + a)2. Select appropriate values for the constants a and B to generate a bistable relation. The functions in (b) to (e) apply to bistable systems that will be discussed subsequently. c. Bistable Optical Devices Numerous schemes can be used for the optical implementation of the foregoing basic principle. Two types of nonlinear optical elements can be used (Fig. 23.4-8): 1. Dispersive nonlinear elements, for which the refractive index n is a function of the optical intensity. 2. Dissipative nonlinear elements, for which the absorption coefficient a is a func- tion of the optical intensity. The optical element is placed within an optical system and the output light intensity 10 controls the system's transmittance in accordance with some nonlinear function T(I o ). Ii Ii (a) (b) Figure 23.4-8 (a) Dispersive bistable optical system. The transmittance 'T is a function of the refractive index n, which is controlled by the output intensity 10. (b) Dissipative bistable optical system. The transmittance 'T is a function of the absorption coefficient Q, which is controlled by the output intensity 10. Dispersive Nonlinear Elements A number of optical systems can be devised whose transmittance T is a nonmonotonic function of an intensity-dependent refractive index n == n(Io). Examples are inter- ferometers, such as the Mach-Zehnder and the Fabry-Perot etalon, with a medium exhibiting the optical Kerr effect, n == no + n2 I o, (23.4-2) where no and n2 are constants.
23.4 OPTICAL GATES 1 063 In the Mach-Zehnder inteiferometer, the nonlinear medium is placed in one branch, as illustrated in Fig. 23.4-9. The power transmittance of the system is (see Sec. 2.5A) 1 1 ( d ) 'J = 2 + 2 cas 21T .Ao n + <Po , (23.4-3) where d is the length of the active medium, Ao the free-space wavelength, and CPo a constant. Substituting from (23.4-2), we obtain 1 1 ( d ) 'J(Io) = 2 + 2 cas 21T .Ao n2 I o + <p , (23.4-4) where cP == CPo + (27rd / Ao)no is another constant. As Fig. 23.4-9 shows, this is a nonlinear function comprising a periodic repetition of the generic bell-shaped function used earlier to demonstrate bistability [see Fig. 23.4-6(a)]. I. I T(1o) 10 >"0 -I n2 d 10 Figure 23.4-9 A Mach-Zehnder interferometer with a nonlinear medium of refractive index n controlled by the transmitted intensity 10 via the optical Kerr effect. In a Fabry-Perot etalon with mirror separation d, the intensity transmittance is (see Sec. 2.5B) 'J == 'J max 1 + (21'/7r)2 sin 2 ((27rd/Ao)n + cpo]' (23.4-5) where 'J max, 1', and CPo are constants and Ao is the free-space wavelength. Substituting for n from (23.4-2) gives 'J(I) = 'J rnax o 1 + (21' / 7r)2 sin 2 [(27rd / Ao)n2 I o + cp] , (23.4-6) where cp is another constant. As illustrated in Fig. 23.4-10, this function is a periodic sequence of sharply peaked bell-shaped functions. The system is therefore bistable.
1064 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES I. 1 7(/0) 10 I I -+I 2n2d )I 10 Figure 23.4-10 A Fabry-Perot interferometer containing a medium of refractive index n controlled by the transmitted light intensity 10. Intrinsic Bistable Optical Devices The optical feedback required for bistability can be internal instead of external. The system shown in Fig. 23.4-11, for example, uses a resonator with an optically nonlinear medium whose refractive index n is controlled by the internal light intensity I within the resonator, instead of the output light intensity 10. Since 10 == 'To I, where 'To is the transmittance of the output mirror, the action of the internal intensity I has the same effect as that of the external intensity 1 0 , except for a constant factor. If the medium exhibits the optical Kerr effect, for example, the refractive index is a linear function of the optical intensity n == no + n21 and the transmittance of the Fabry-Perot etalon is 'J(1 0 ) = 'J rnax . 1 + (21' /1r)2 sin 2 [(21rd / Ao)n2Io/'To + <p] (23.4-7) Thus the device operates as a self-tuning system. 1 II( d .1 T I I n I 10 -- , 7(1 0 ) AO To -+-1 10 2n2d Figure 23.4-11 Intrinsic bistable device. The internal light intensity 1 controls the active medium and therefore the overall transmittance of the system 'T. Dissipative Nonlinear Elements A dissipative nonlinear material has an absorption coefficient that is dependent on the optical intensity I. The saturable absorber discussed in Sec. 14.4A is an example in which the absorption coefficient is a nonlinear function of I, ao a == 1 + I / Is ' (23.4-8) where ao is the small-signal absorption coefficient and Is is the saturation intensity. If the absorber is placed inside a Fabry-Perot etalon of length d that is tuned for peak transmission (Fig. 23.4-12), then 'T == 'T 1 (1 - 9{e- ad )2 ' (23.4-9)
23.4 OPTICAL GATES 1 065 where == V I2, I and 2 are the mirror reflectances, and 'II is a constant (see Secs. 2.5B and 10.IA for details). If ad « 1, i.e., the medium is optically thin, e- ad 1 - ad, and 'I 'II . [1 - (1 - ad)]2 (23.4-10) Because a is a nonlinear function of I, 'I is also a nonlinear function of I. Using the relation I == Io/'J o and (23.4-8) and (23.4-10), [ ] 2 10 + ISI 'J(10) = 'J 2 10 + (1 + a)1s1 ' (23.4-11) where 'J 2 == 'J I /(l - )2, a == aod/(l - ), and ISI == Is'J o . For certain values of a, the system is bistable [recall Exercise 23.4-1, example (e)]. Ii H 10 Mirror Saturable absorber Figure 23.4-12 A bistable device consisting of a saturable absorber in a resonator. Suppose now that the saturable absorber is replaced by an amplifying medium with saturable gain 1"0 1" == 1 + I / Is . (23.4-12) The system is nothing but an optical amplifier with feedback, i.e., a laser. If exp( 1"0 d) < 1, the laser is below threshold; but when exp(1"od) > 1, the system becomes unstable and we have laser oscillation. Lasers do exhibit bistable behavior. However, the theory of these phenomena is beyond the scope of this book. In some sense, the dispersive bistable optical system is the nonlinear-index-of- refraction (instead of nonlinear-gain) analog of the laser. Materials Optical bistability has been observed in a number of materials exhibiting the optical Kerr effect (e.g., sodium vapor, carbon disulfide, and nitrobenzene). The coefficient of nonlinearity n2 for these materials is very small. A long path length d is therefore required, and consequently the response time is large (nanosecond regime). The power requirement for switching is also high. Semiconductors, such as GaAs, InSb, InAs, and CdS, exhibit a strong optical nonlin- earity due to excitonic effects at wavelengths near the edges of the bandgap. A bistable device may simply be made of a layer of the semiconductor material with two parallel partially reflecting faces acting as the mirrors of a Fabry-Perot etalon (Fig. 23.4-13). Because of the large nonlinearity, the layer can be thin, allowing for a smaller response time. GaAs switches based on this effect have been the most successful. Switch-on times of a few picoseconds have been measured, but the switch-off time, which is domi- nated by relatively slow carrier recombination, is much longer (a few nanoseconds). A switch-off time of 200 ps has been achieved by the use of specially prepared samples
1066 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES Cleaved surface Cleaved surfaces Ii 10 Semiconductor material Figure 23.4-13 A thin layer of semiconductor with two parallel reflecting surfaces can serve as a bistable device. in which surface recombination is enhanced. The switching energy is 1 to 10 pI. It is possible, in principle, to reduce the switching energy to the femtojoule regime. InAs and InSb have longer switch-off times (up to 200 ns). However, they can be speeded up at the expense of an increase of the switching energy. Semiconductor multiquantum- well structures (see Secs. 16.1 G and 17.2D) have also been shown to exhibit bistability, and so have organic materials. The key condition for the usefulness of bistable optical devices is the capability to make them in large arrays. Arrays of bistable elements can be placed on a single chip with the individual pixels defined by the light beams. Alternative]y, reactive ion etching may be used to define the pixels. An array of 100 x 100 pixels on a l-cm 2 GaAs chip is possible with existing technology. The main difficulty is heat dissipation. If the switching energy E == 1 pJ, and the switching time T == 100 ps, then for N == 10 4 pixels/cm 2 the heat load is NE/T == 100 W /cm 2 . This is manageable with good thermal engineering. The device can perform 10 14 bit operations per second, which is larfie in comparison with electronic supercomputers (which operate at a rate of about 10 0 bit operations per second). Hybrid Bistable Optical Devices The bistable optical systems discussed so far are all-optical. Hybrid electrical/optical bistable systems in which electrical fields are involved have also been devised, as illustrated by the four examples shown Fig. 23.4-14. In the first example [Fig. 23 .4-14( a)], a Pockels cell is placed inside a Fabry-Perot etalon; the output light is detected using a photodetector, and a voltage proportional to the detected optical intensity is applied to the cell, so that its refractive index variation is proportional to the output intensity. Using LiNb0 3 as the electro-optic material, 1-ns switching times have been achieved with 1-J-LW switching power and 1-fl switching energy. The second example [Fig. 23.4-14(b)], which is an integrated optical version of the first, has also been implemented. In the third example [Fig. 23 .4-14( c)], an electro-optic modulator employing a Pock- els cell wave retarder is placed between two crossed polarizers; see Sec. 20.1B. Again the output light intensity 10 is detected and a proportional voltage V is applied to the cell. The transmittance of the modulator is a nonlinear function of V, T == sin 2 (r 0/2 - 7r V /2 V 7r ) , (23.4-13) where ro and V 7r are constants. Because V is proportional to 1 0 , T(lo) is a nonmono-. tonic function and the system exhibits bistability. In the fourth example [Fig. 23.4-14(d)], an integrated-optical directional coupler is used The input light Ii enters from one waveguide and the output 10 leaves from the other waveguide; the ratio T == 10/ Ii is the coupling efficiency (see Sec. 20.1D). Using (20.1-23) yields 'J= ()2sinc2 [ h/ 1 + 3 (VjV o )2] , (23.4-14)
Mirror V H ,---, 1-1 n (a) Pockels cell Mirror Polarizer (c) 23.4 OPTICAL GATES 1 067 (d) Beamsplitter Figure 23.4-14 (a) A Fabry-Perot interferometer containing an electro-optic medium (Pockels cell). The output optical power is detected and a proportional electric field is applied to the medium to change its refractive index, thereby changing the transmittance of the interferometer. (b) An integrated-optical implementation. where V is the applied voltage and Va is a constant. A bistable system is created by making V proportional to the output intensity 10 [see Exercise 23.4-1, example (d)]. Spatial light modulators (SLMs) may also be used to construct arrays of bistable elements (Fig.23 .4-15). For example, in an optically addressed liquid-crystal SLM (see Sec. 20.3B), the reflectance of each element is a nonlinear function of the intensity of light illuminating its write side. By using feedback, the write intensity is proportional to the intensity 10 of the beam that is reflected from the element itself, i.e., == (10), and 10 == 1i(10), so that bistable behavior is exhibited. Different points on the surface of the device can be addressed separately, so that the modulator serves as an array of bistable optical elements. Typical switching times are in tens of milliseconds and switching powers are less than 1 J-L W. , 10 Figure 23.4-15 An optically addressed spatial light modulator operates as an array of bistable optical elements. The reflectance of the "read" side (right) of the valve at each position is a function 9( == 9((1 0 ) of the intensity 10 at the "write" side (left). The electro-optical properties of semiconductors offer many possibilities for mak- ing bistable optical devices. As mentioned earlier, the laser amplifier is an important example in which the nonlinearity is inherent in the saturation of the amplifier gain. InGaAsP laser-diode amplifiers have been operated as bistable switches with optical switching energy less than 1 fJ, and switching time less than 1 ns. Self-Electro-Optic-Effect Device Another electro-optic semiconductor device that exhibits bistability is the self-electro- optic-effect device (SEED). The SEED is a p-i-n photodiode with a heterostructure
1068 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES multiquantum-well (MQW) semiconductor in the intrinsic region [Fig. 23.4-16(a)]. The diode is reverse-biased so that a large electric field is created in the MQW. By virtue of the quantum confined Stark effect (QCSE) (see Sec. 20.5), the optical ab- sorption coefficient is a nonlinear function a(V) of the voltage V across the MQW [Fig. 23 .4-16(b)]. Consequently, the optical transmittance 'J(V) is a nonlinear function of V that mirrors a(V) [Fig. 23.4-16(c)]. Bistable behavior is exhibited in the SEED as a result of the feedback mechanism introduced by the photodiode electrical circuit, which makes the voltage V depen- dent on the incident optical power Pi. This occurs since the absorbed light creates a proportional photocurrent ip == 9\(V) Pi that flows into the external circuit causing a drop in V. Here, 9\(V) is the responsivity, which is proportional to the absorption coefficient a (V). For example, if the circuit uses an external voltage source Va with a load resistance R L in series, then V == Va - ipRL == Va - R L 9\(V)Pi. The device is therefore described by two equations: Po == 'J(V)P i p. _ 1 va - V Z - 9\(V) R L ' (23.4-15) (23.4-16) where 9\(V) is proportional to a(V), 'J(V) mirrors a(V), and a(V) is the nonlinear function shown in Fig. 23.4-16(b). These two equations define a parametric relation between the input and output optical powers, which exhibits bistability, as shown schematically in Fig. 23.4-16(d). The resistor in the driver circuit may also be replaced by another electronic device such as a field effect transistor (FET) or another SEED. Since the QCSE is strongly dependent on the wavelength [see Fig. 20.5-2(b)], the bistable characteristics of the SEED are wavelength dependent. P i n $' c"-:" .9 a ....... ....... e- a (b) o .- Q... rJJ U ..D ...... p. Po '""" (l) I - 0 - - U 0 - P-. Voltage V ....... ::s 0.. MQW II $' ....... ::s t::' 0 (l) I VI U (c) c Input Power Pi ro ....... ....... 's rJJ (d) Va c ro (a) Voltage V Figure 23.4-16 (a) The self-electro-ptic-effect device (SEED) is a reverse biased MQW photodiode with optically controllable transmittance. (b) Dependence of the absorption coefficient on the voltage V via the QCSE. This relation is obtained from Fig. 20.5-2(b). (c) Dependence of the optical transmittance on the voltage V. (d) Bistable relation between the input and output optical power. The SEED operates without a resonator since the feedback is created in the electrical circuit by the optically generated photocurrent. But it is not exactly an all-optical device since it involves electrical processes within the material and the circuit, and requires an external voltage source. SEED devices can be fabricated in arrays operating at moderately high speeds (switching times of tens of ns) and very low power.
READING LIST 1069 READING LIST Books T. S. EI-Bawab, Optical Switching, Springer-Verlag, 2006. L. Pavesi and G. Guillot, eds., Optical Interconnects: The Silicon Approach, Springer-Verlag, 2006. H. Ukita, Micromechanical Photonics, Springer-Verlag, 2006. W. Kabacinsk, Nonblocking Electronic and Photonic Switching Fabrics, Springer-Verlag, 2005. S. Kawai, ed., Handbook of Optical Interconnects, CRC Press/Taylor & Francis, 2005. N. Gehani, Bell Labs: Life in the Crown Jewel, Silicon Press, 2003. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, Morgan Kaufmann, 2nd ed. 2002, Chapter 14. H. T. Mouftah and J. M. H. Elmirghani, eds., Photonic Switching Technology: Systems and Networks, IEEE Press, 1999. C. S. Tocci and H. J. Caulfield, eds., Optical Interconnection: Foundations and Applications, Artech, 1994. H. S. Hinton, An Introduction to Photonic Switching Fabrics, Plenum, 1993. J. E. Midwinter, ed., Photonics in Switching, Volume 1, Background and Components, Academic Press, 1993. J. E. Midwinter, ed., Photonics in Switching, Volume 2, Systems, Academic Press, 1993. M. N. Islam, Ultrafast Fiber Switching Devices and Systems, Cambridge University Press, 1992. A. D. McAulay, Optical Computer Architectures: the Application of Optical Concepts to Next Gen- eration Computers, Wiley, 1991. R. Arrathoon, ed., Optical Computing: Digital and Symbolic, Marcel Dekker, 1989. T. K. Gustafson and P. W. Smith, eds., Photonic Switching, Springer-Verlag, 1988. H. M. Gibbs, Optical Bistability: Controlling Light with Light, Academic, 1985. C. M. Bowden, M. Cifton, and H. R. Roble, eds., Optical Bistability, Plenum, 1981. Articles N. Holonyak, Jr. and M. Feng, The Transistor Laser, IEEE Spectrum, vol. 43, no. 2, pp. 50-55, 2006. I. Glesk, B. C. Wang, L. Xu, V. Baby, and P. R. Prucnal, Ultra-Fast All-Optical Switching in Optical Networks, in Progress in Optics, vol. 45, pp. 53-] 17, E. Wolf, ed., Elsevier, 2003. D. Huang, T. Sze, A. Landin, R. Lytel, and H. L. Davidson, Optical Interconnects: Out of the Box Forever?, IEEE Journal of Selected Topics in Quantum Electronics, vol. 9, pp. 614-623, 2003. G. A. Keeler, B. E. Nelson, D. Agarwal, C. Debaes, N. C. Helman, A. Bhatnagar, and D. A. B. Miller, The Benefits of Ultrashort Optical Pulses in Optically Interconnected Systems, IEEE Journal of Selected Topics in Quantum Electronics, vol. 9, pp. 477--485, 2003. M. J. Potasek, All-Optical Switching for High Bandwidth Optical Network, Optical Networks Mag- azine, vol. 3, no. 6, pp. 30--43, 2002. L. Y. Lin and E. L. Goldstein, Opportunities and Challenges for MEMS in Lightwave Communica- tions, IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, pp. 163-172, 2002. Special issue on arrayed grating routers/WDM MUX/DEMUXs and related applications/uses, IEEE Journal of Selected Topics in Quantum Electronics, vol. 8, no. 6, 2002. M. Forbes, J. Gourlay, and M. Desmulliez, Optically Interconnected Electronic Chips: A Tutorial and Review of the Technology, Electronics & Communication Engineering Journal, vol. 13, pp. 22]- 232, 2001. D. A. B. Miller, Rationale and Challenges for Optical Interconnects to Electronic Chips, Proceedings of the IEEE, vol. 88, pp. 728-749, 2000. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. Special issue on optical interconnections for digital systems, Proceedings of the IEEE, vol. 88, no. 6, 2000. S. Yao, B. Mukherjee, and S. Dixit, Advances in Photonic Packet Switching: An Overview, IEEE Communications Magazine, vol. 38, no. 2, pp. 84-94, 2000. D. J. Bishop and V. A. Aksyuk, Optical MEMS Answer High-Speed Networking Requirements, Electronic Design, pp. 85-99, April 5, 1999.
1070 CHAPTER 23 OPTICAL INTERCONNECTS AND SWITCHES A. V. Krishnamoorthy, L. M. F. Chirovsky, W. S. Hobson, R. E. Leibenguth, S. P. Hui, C. J. Zydzik, K. W. Goossen, J. D. Wynn, B. J. Tseng, J. Lopata, J. A. Walker, J. E. Cunningham, and L. A. D' Asaro, Vertical-Cavity Suiface-Emitting Lasers Flip-Chip Bonded to Gigabit-per-Second CMOS Circuits, IEEE Photonics Technology Letters, vol. ] 1, no. 1, pp. 128-130, 1999. Issue on smart photonic components, interconnects, and processing, IEEE Journal of Selected Topics in Quantum Electronics, vol. 5, no. 2, 1999. D. A. B. Miller, Physical Reasons for Optical Interconnection, International Journal uf Optoelec- tronics, vol. 11, pp. 155-168,1997. A. V. Krishnamoorthy and D. A. B. Miller, Scaling Optoelectronics-VLSI Circuits into the 21st Century: A Technology Roadmap, IEEE Journal of Selected Topics in Quantum Electronics, vol. 2, no. 1, pp. 55-76, 1996. A. Marrakchi, ed., Selected Papers on Photonic Switching, SPIE Optical Engineering Press (Mile- stone Series Volume 121), 1996. C. S. Tsai, Integrated Acoustooptic and Magnetooptic Devices for Optical Information Processing, Proceedings of the IEEE, vol. 84, pp. 853-869, 1996. R. Reinisch and G. Vitrant, Optical Bistability, Progress in Quantum Electronics, vol. 18, pp. 1-38, 1994. R. F. Kalman, L. G. Kazovsky, and J. W. Goodman, Space Division Switches Based on Semiconduc- tor Optical Amplifiers, IEEE Photonics Technology Letters, vol. 4, pp. 1048-105], 1992. D. A. B. Miller, Quantum-Well Self-Electro-Optic Effect Devices, Optical and Quantum Electronics, vol. 22, pp. S61-S98, 1990. G. I. Stegeman and E. M. Wright, All-Optical Waveguide Switching, Optical and Quantum Electron- ics, vol. 22, pp. 95-122, 1990. Issue on optical interconnects, Applied Optics: Information Processing, vol. 29, no. 8, 1990. Issue on optical interconnections and networks, SPIE Proceedings, vol. 1281, 1990. Issue on nonlinear optical materials and devices for photonic switching, SPIE Proceedings, vol. 1216, 1990. Issue on optical interconnects in the computer environment, SPIE Proceedings, vol. 1178, 1990. Y. Silberberg, Photonic Switching Devices, Optics News, vol. 15, no. 2, pp. 7-12, 1989. Issue on photonic switching, IEEE Journal of Selected Areas in Communicatiuns, vol. 6, no. 7, 1988. J. E. Midwinter, Digital Optics, Smart Interconnect or Optical Logic? Part 1, Physics in Technology, vol. 19, pp. 101-]08, 1988. J. E. Midwinter, Digital Optics, Smart Interconnect or Optical Logic? Part 2, Physics in Technology, vol. 19, pp. 153-] 65, 1988. D. H. Hartman, Digital High Speed Interconnects: A Study of the Optical Alternative, Optical Engi- neering, vol. 25, pp. 1086-1102, 1986. P. R. Haugen, S. Rychnovsky, A. Husain, and L. D. Hutcheson, Optical Interconnects for High Speed Computing, Optical Engineering, vol. 25, pp. 1076-1085, 1986. A. A. Sawchuk and B. K. Jenkins, Dynamic Optical Interconnections for Parallel Processors, SPIE Proceedings, vol. 625, pp. 143-153, 1986. S. F. Su, L. Jou, and J. Lenart, A Review on Classification of Optical Switching Systems, IEEE Communications Magazine, vol. 24, no. 5, pp. 50-55, 1986. D. A. B. Miller, D. S. Chemla, T. C. Damen, T. H. Wood, C. A. Burrus, Jr., A. C. Gossard, and w. Wiegmann, The Quantum Well Self-Electrooptic Effect Device: Optoelectronic Bistability and Oscillation, and Self-Linearized Modulation, IEEE Journal of Quantum Electronics, vol. 21, pp. 1462-1476, 1985. J. W. Goodman, F. I. Leonberger, S. Y. Kung, and R. A. Athale, Optical Interconnections for VLSI Systems, Proceedings of the IEEE, vol. 72, pp. 850-866, 1984. L. A. Lugiato, Theory of Optical Bistability, in Progress in Optics, vol. 21, E. Wolf, ed., North- Holland, 1984. P. W. Smith, Applications of All-Optical Switching and Logic, Philosophical Transactions of the Royal Society of London, vol. A313, pp. 349-355, 1984. P. W. Smith and W. J. Tomlinson, Bistable Optical Devices Promise Subpicosecond Switching, IEEE Spectrum, vol. 18, no. 6, pp. 26-33, 1981.
PROBLEMS 1 071 L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Optical Data Processing and Filtering Systems, IRE Transactions on Information Theory, vol. IT-6, pp. 386-400, 1960. PROBLEMS 23. 1- 3 Interconnection Hologram for a Conformal Map. Design a hologram to realize the geo- metric transformation defined by x' == 'l/Jx (x, y) == In -J x 2 + y2 y' == 'l/Jy(x, y) == tan- l y . x This is a Cartesian-to-polar transformation followed by a logarithmic transformation of the polar coordinate r == (x 2 + y2)1/2. Determine an expression for the phase function cp(x, y) of the hologram required. 23.2-1 Cascaded MZI MUXlDEMUX. Three Mach-Zehnder interferometers (MZIs) are cas- caded as shown in Fig. 23.2-6 to multiplex or demultiplex four wavelength channels with wavelength separation A == 0.2 nm and central wavelength 1550 nm. Determine the necessary pathlength differences d in each interferometer if the refractive index is n == 2.3. 23.2-2 WGR DEMUX. A wavelength grating router (WGR) (see Fig. 23.2-8) is used to demulti- plex four wavelength channels with with wavelength separation A == 0.2 nm and central wavelength 1550 nm. Determine the pathlength difference parameter db that must be introduced by the star coupler if its refractive index is n == 2.3. 23.2-3 WGR as a 2 x 2 Wavelength Router. A WGR is configured as a 2 x 2 wavelength router. Input port 1 has two wavelength channels, Al and A2, and input port 2 has two wavelength channels, A3 and A4. Design a router that transposes the input wavelengths among the two output ports, i.e., directs the Al and A3 channels to output port 1, and the A2 and A4 channels to output port 2. Write the routing conditions in terms of the four optical pathlength differences dll, dI2, d21 and d22, of the multipath interferometers connecting each of the input ports to each of the output ports. 23.3-1 Power Loss and Crosstalk. A 4 x 4 switch may be implemented by use of five 2 x 2 switches. If each of these switches introduces a power loss of 0.5 dB and a crosstalk of -30 dB, determine the worst case power loss and crosstalk for the 4 x 4 switch. 23.3-2 MZI Crossbar Switch. An electro-optic Mach-Zehnder interferometer is used as a crossbar switch. The application of a voltage V == V n on the electro-optic material in one arm of the interferometer introduces a phase shift of 7r. If the switch is set in the bar state when V == 0, what must the applied voltage V be to change the switch to the cross state. Determine the crosstalk (in dB) caused by a 1 % error in that applied voltage. 23.3-3 TSI Switch. As shown in Fig. 23.3-30, the time-slot interchange (TSI) switch may be im- plemented by a five step process: time-to-space routing, time delays, space switching, time delays, and space-to-time routing. Construct another implementation using the programable delay lines shown in Fig. 23.3-31. 23.4-2 Optical Logic. Figure 23.4-4 illustrates how a nonlinear thresholding optical device may be used to make an AND gate. Show how a similar system may be used to make NAND, OR, and NOR gates. Is it possible to make an XOR (exclusive OR)? Can the same system be used to obtain the OR of N binary inputs? 23.4-3 Bistable Interferometer. A crystal exhibiting the optical Kerr effect is placed in one of the arms of the Mach-Zehnder interferometer. The transmitted intensity 10 is fed back and illuminates the crystal. Show that the intensity transmittance of the system is 10/ Ii == (Io) == + cos(7rIo/In + cp), where In and cp are constants. Assuming that cp == 0, sketch 10 versus Ii and derive an expression for the maximum differential gain dIo / dI i .
CHAPTER 4 OPTICAL FIBER COMMUNICATIONS 24.1 FIBER-OPTIC COMPONENTS 1074 A. Optical Fibers B. Sources for Optical Transmitters C. Optical Amplifiers D. Detectors for Optical Receivers 24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1084 A. Evolution of Optical Fiber Communication Systems B. Performance of Optical Fiber Systems C. Attenuation- and Dispersion-Limited Systems D. Attenuation and Dispersion Compensation and Management E. Soliton Optical Communications 24.3 MODULATION AND MULTIPLEXING 1101 A. Modulation B. Multiplexing C. Wavelength-Division Multiplexing (WDM) 24.4 FIBER-OPTIC NETWORKS 1106 A. Network Topologies and Multiple Access B. Wavelength-Division Multiplexing (WDM) Networks 24.5 COHERENT OPTICAL COMMUNICATIONS 1112 ., N ..;It t .f- .,' t, < ... 4,'", , ..... \ . ". $-. ., :. .'1 ,-. \ ,Y , ': r .... . . .f ,. . ...._ '. .. llo. - ""..c. ,.. . ." '" ,r"" ........ 0 ..- i- -' :'&. .- .. -. zC' --. .-, , .AtI... r? "If"'. >;A' v --"" \ ".. " '\ . l,\' '\ 'j .' '" ....' '\\'1i . '::J: "..: ...,......... ow r........ ...... tf ::..... .. . I' ' .. .--.. . 1"" < ,:C'!!5. . ., '... "(' om .... ........... ......... .,ri ¥" , ,. Intercontinental optical fiber communications network. 1072
Until the mid-1970s virtually all communication systems relied on the transmission of information over electrical cables or have made use of radio-frequency and microwave electromagnetic radiation propagating in free space. It would appear that the use of light would have been a more natural choice for communications since, unlike electric- ity and radio waves, it did not have to be discovered. The reasons for the delay in the development of this technology are twofold: the difficulty of producing a light source that could be rapidly switched on and off and therefore could encode information at a high rate, and the fact that light is easily obstructed by opaque objects such as clouds, fog, smoke, and haze. Unlike radio-frequency and microwave radiation, light is rarely suitable for free-space communication. Lightwave communication has come into its own, however, and indeed it is now the preferred technology in many applications including the transmission of data, voice, video, and telemetry in short-distance communication and local-area networks, as well as long-haul communication and internet traffic. Lightwave technology affords the user enormous transmission capacity, distant spacings of repeaters, immunity from electro- magnetic interference, and relative ease of installation. Lightwave transmission is the only technology capable of meeting the vast and exponentially increasing demands of global communication, and its is now reaching individual dwellings via fiber to the home (FTTH) broadband systems. The spectacular successes of lightwave communication have their roots in two crit- ical photonic inventions: the development of the light-emitting diode (LED) and the development of the low-loss optical fiber as a light conduit. Suitable detectors of light have been available for some time, although their performance has been improved dramatically in recent years. Interest in optical communications was initially stirred by the invention of the laser in the early 1960s. However, the first generation of optical fiber communication systems made use of LED sources and indeed many present local- area communication systems continue to do so. Nevertheless, most lightwave commu- nication systems (such as long-haul single-mode optical fiber systems and short-haul free-space systems) do benefit from the large optical power, narrow linewidth, and high directivity provided by the laser. This Chapter This chapter is an introduction to optical fiber communication systems and networks. A point-to-point communication link comprises three basic elements, as illustrated in Fig. 24.0-l: a compact light source modulated by the electrical signal, a low-Ioss/low- dispersion optical fiber, and a photodetector converting the optical signal back into an electrical signal. These optical components have been discussed in detail in Chap- ters 17,9, and 18, respectively. Optical amplifiers have also proved themselves to be very valuable in fiber systems; these devices are discussed in Chapter 14. To make this chapter self-contained, Sec. 24.1 provides an abbreviated summary of the pertinent properties of fibers, sources, detectors, and amplifiers, examining their role in the context of the overall design, operation, and performance of an optical communication link. Other optical accessories such as splices, connectors, couplers, switches, and multiplexing devices are also essential to the successful operation of fiber links and networks; the principles of some of these devices are described in Chapter 23 and in other parts of this book. Section 24.2 introduces the basic design principles applicable to long-distance dig- ital and analog optical fiber communication systems using intensity modulation. The maximum fiber length that can be used to transmit data, at a given rate and with a prescribed level of performance, is determined. Performance deteriorates if the data rate exceeds the fiber bandwidth, or if the received power is smaller than the receiver 1073
1 074 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Signal- :t!:-- - CO) ""''r" " - - 0/ E Signal CO) Receiver Transmitter ,....._"'.','.'","',,., t J 1,1 Source Figure 24.0-1 An optical fiber communication system. An electrical signal is converted into an optical signal (EtO) by modulating an optical source. The optical signal is transmitted through the fiber to the receiver. At the receiver, the optical signal is converted back into an electrical signal by use of a detector and demodulator (DIE). For long fibers, optical amplifiers (OA) may be used to boost the weakened optical signal. Alternatively, several optical links may be cascaded to form a longer link by use of an intermediate process of electrical amplification and signal regeneration between adjacent links. Such units are called regenerators or repeaters. Fiber Fiber sensitivity (so that the signal cannot be distinguished from noise). This is followed, in Sec. 24.3, by an introduction to modulation and multiplexing systems used in optical fiber communications. Fiber-optic networks are communication links connecting multiple users that are distributed in some geographic area, and controlled by a set of routers and switches. Section 24.4 provides an introduction to such networks, including wavelength- division multiplexing (WDM) networks. Coherent optical communication systems, which are introduced in Sec. 24.5, use light not as a source of controllable power but rather as an electromagnetic wave of controllable amplitude, phase, or frequency. These systems are the natural extension to higher frequencies of conventional radio and microwave communications. They provide substantial gains in receiver sensitivity, permitting greater spacings between repeaters and increased data rates. 24.1 FIBER-OPTIC COMPONENTS A. Optical Fibers An optical fiber is a cylindrical dielectric waveguide made of low-loss materials, usu- ally fused silica glass of high chemical purity. In the simplest optical fiber, the step- index fiber, the core of the waveguide has a constant refractive index slightly higher than that of the cladding (the outer medium) so that light is guided along the fiber axis by total internal reflection. As described in Chapter 9, the transmission of light through the fiber may be studied by examining the trajectories of rays within the core. In accordance with a more complete analysis based on electromagnetic theory, light travels in the fiber in the form of modes; each is a wave with a distinct spatial distribution, polarization, propagation constant, group velocity, and attenuation coefficient. There is, however, a correspon- dence between each mode and a ray that bounces within the core in a distinct trajectory. The step-index fiber is characterized by its core radius a, the refractive indexes of the core and cladding, nl and n2, and the fractional refractive index change == (nl - n2)/nl, which is usually very small ( == 0.001-0.02). Light rays making angles with the fiber axis smaller than the complement of the critical angle, () c == cos-1(n2/nl)' are guided within the core by multiple total internal reflections at the core-cladding boundary. The angle ()c in the fiber corresponds to an acceptance angle
24.1 FIBER-OPTIC COMPONENTS 1075 () a == sin -1 (NA) for rays incident from air into the fiber, where NA = sinB a = J ni - n n 1 vI2K (24.1-1 ) Numerical Aperture is the numerical aperture. Multimode Fibers (MMF) The number of guided modes AI is governed by the V parameter, V == 27r(a/ Ao) NA, where a/ Ao is the ratio of the core radius a to the wavelength Ao. In a fiber with V » 1, there are a large number of modes, AI V 2 /2. Since the modes travel with different group velocities, this results in pulse spreading, which increases linearly with the fiber length, an effect called modal dispersion. When an impulse of light travels a distance L in the fiber, it arrives as a sequence of pulses centered at the mode delay times, as illustrated in Fig. 24.1-1. The composite pulse has an approximate RMS width afiber -L, 2C1 (24.1-2) Response Time (Step-Index MMF) where C1 == co/n1. It is therefore more desirable to use fibers with small . For example, if n1 == 1.46 and == 0.01, the response time per km /2C1 24 ns/km. For a 100-km fiber, an impulse spreads to a width of 2.4 j1S. Modal dispersion can also be reduced by use of graded-index (GRIN) fibers. In such fibers the refractive index of the core varies gradually from a maximum value n1 on the fiber axis to a minimum value n2 at the core-cladding boundary. Rays follow curved trajectories, with paths shorter than those in the step-index fiber. The axial ray travels the shortest distance at the smallest phase velocity (largest refractive index), whereas oblique rays travel longer distances at higher phase velocities (smaller refractive indexes), so that the delay times are approximately equalized. If the fiber is graded optimally (using an approximately parabolic profile), then the pulse spreading rate (ps/km) is equal to that of the equivalent step-index fiber multiplied by a factor of /2. For example, for == 0.01, the pulse spread is reduced by a factor of 500. This factor, however, is usually not fully met in practical graded-index fibers because of the difficulty of achieving ideal index profiles. Single-Mode Fibers (SMF) When the core radius a and the numerical aperture NA of a step-index fiber are suffi- ciently small so that V < 2.405, only a single mode is allowed and the fiber is called a single-mode fiber (SMF). One advantage of using an SMF is the elimination of pulse spreading caused by modal dispersion. Pulse spreading occurs, nevertheless, since the initial pulse has a finite spectrallinewidth and the group velocities (and therefore the delay times) are wavelength dependent. This effect is called chromatic dispersion. There are two origins of chromatic dispersion: material dispersion, which results from the dependence of the refractive index on the wavelength, and waveguide dis- persion, which is a consequence of the dependence of the group velocity of the mode on the ratio between the core radius and the wavelength. Material dispersion is usually larger than waveguide dispersion.
1076 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS A short optical pulse of spectral width a;\ spreads to a temporal width afiber == IDla.x L , (24.1-3) Response Time (SMF) proportional to the propagation distance L (km) and to the source linewidth a.x (nm), where D is the dispersion coefficient (ps/km-nm). The parameter D involves a combi- nation of material and waveguide dispersion. For weakly guiding fibers ( « 1), D may be separated into a sum D.x + Dw of the material and waveguide contributions. As an example, an SMF with a light source of spectrallinewidth a.x == 1 om (from a typical single-mode laser) and a fiber dispersion coefficient D == 1 ps/km-nm (for operation near Ao == 1300 nm with minimal waveguide dispersion), the response time given by (24.1-3) is aT / L == 1 ps/km. A fiber of length 100 km has a 100- ps response time. The geometries, refractive-index profiles, and pulse broadening in multimode step- index and graded-index fibers and in single-mode fibers are schematically compared in Fig. 24.1-1. MMF: Step-Index n 1 Impulse-response h(t) aT (a) - t MMF: GRIN nl (b) t SMF nl 1- ------ ------ n2 I o -.. aT (c) I t Figure 24.1-1 (a) Step-index multimode fibers (MMF): relatively large core diameter; uniform refractive indexes in the core and cladding; large pulse spreading due to modal dispersion. (b) Graded- index (GRIN) MMF: refractive index of the core is graded; there are fewer modes; pulse broadening due to modal dispersion is reduced. (c) Single-mode fibers (SMF): small core diameter; no modal dispersion; pulse broadening is due only to material and waveguide dispersion. Material Attenuation and Dispersion The wavelength dependence of the attenuation coefficients of fused-silica-glass fibers is illustrated in Fig. 24.1-2. As the wavelength increases beyond the visible band, the attenuation drops to a minimum of approximately 0.3 dB /km at Ao == 1300 nm, increases slightly at 1.4 Mm because of OH-ion absorption, and then drops again to its absolute minimum of 0.16 dB/km at Ao == 1550 nm, beyond which it rises sharply. Fibers with suppressed OH absorption have been recently developed. The wavelength dependence of the dispersion coefficient D.x of fused silica glass is also illustrated in Fig. 24.1-2. It changes from negative values at short wavelengths to positive values at long wavelengths, and is zero at Ao 1312 nm. In a medium with negative dispersion, shorter-wavelength components of a pulse are slower than longer- wavelength components. This is known as normal dispersion. The opposite (called
24.1 FIBER-OPTIC COMPONENTS 1077 anomalous dispersion) occurs in a medium exhibiting positive dispersion coefficient (see Sec. 5.6). Although the sign of the dispersion coefficient does not affect the pulse- broadening rate, the sign can play an important role in pulse propagation through media consisting of cascades of materials with different dispersion sign, as described in Sec. 24.2D (see also Sec. 22.3). ..... I:: Q) .- u .- E u 1::-- o ._ '"d ..... '-" ro t) I:: Q) ..... ..... < 3 Frequency (THz) 240 230 220 210 200 190 180 0.3 0.1 40 o E sc L U ..... 1::---... . 8 0 u s= .- I 8 Q) -40 0__ U rJJ I:: 0.. -80 0'-" .- ..-.:::: Q -120 Q) 0.. C/) a -160 d \0. a \0 -.::t a lr) lr) lr) M\O N r-- lr) lr) \0 \0 - 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 Wavelength Ao (nm) Figure 24.1-2 Wavelength dependence of the minimum attenuation a and the material dispersion coefficient D). of silica-glass fibers. The dashed line represents the attenuation of silica-glass fibers with suppressed OH absorption. Three spectral bands (shaded) are noted: the band centered at 870 nm, which was used in earlier systems has a = 1.5 dB/km and D). = -80 pslkm-nm; the 0 (original) band centered at 1310 nm, for which a = 0.3 dB /km and dispersion is minimal; and the C (conventional) band centered at 1550 nm, for which attenuation is minimal (a = 0.16 dB/km) and D). = +80 ps/km-nm. Three additional bands are used in wavelength division multiplexing systems (WDM): E = extended, S = Short, L = Long; and U = Ultra-long. Dispersion-Modified Fibers (DSF) As described in Sec. 9.3B, advanced designs of single-mode fibers use graded-index cores with special refractive index profiles selected such that the overall chromatic dispersion coefficient D has desired values at certain wavelengths, or wavelength de- pendence that is useful in fiber communication systems, as in the following examples: . In dispersion-shifted fibers (DSF), D vanishes at Ao = 1550 nm, where at- tenuation is minimum, rather than at 1312 nm. In non-zero dispersion-shifted fibers (NZ-DSF), D is significantly reduced in the 1500-1600 nm window, but is not zero. A small amount of dispersion can be useful in alleviating nonlinear distortions encountered by narrow intense pulses. The wavelength dependence of D in DSF and NZ-DSF fibers is illustrated in Fig. 24.1-3 [see Fig. 9.3-6(a)]. . In dispersion-flattened fibers, D vanishes at two wavelengths and is reduced at intermediate wavelengths [see Fig. 9.3-6(b)]. . In dispersion compensating fibers (DCF), D is proportional to that of the con- ventional step-index fiber over an extended wavelength band, but has the opposite
1078 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS sign. A short fiber with a reversed large dispersion coefficient can be used to com- pensate the pulse spreading introduced in long conventional fibers [see Fig. 9.3- 6( c) ] . 20 Frequency (THz) 230 220 210 200 190 180 ,. Conventional , , , , , .' ..... s:: Q) .u -- 10 S E Q) s:: o I U E . s:: 0 o V':J .- 0.. V':J "-" ;..... &Q . -10 o ". ." o s ..C L u Figure 24.1-3 Wavelength depen- dence of the chromatic dispersion co- efficient D of a conventional fiber and examples of a dispersion-shifted fiber (DSF) and non-zero-dispersion shifted fiber (NZ-DSF). The desig- nations 0.653 and 0.655 are spec- ifications of the ITU (International Telecommunications Union). NZ-DSF -20 Other dispersion-modified fibers include the holey fibers and the photonic-crystal fibers (PCF), described in Sec. 9.4. In these fibers, chromatic dispersion is dominated by waveguide dispersion, which is strongly dependent on the geometry of the holes. Dispersion flattening over broad wavelength ranges can be achieved as can dispersion shifting to wavelengths lower than the zero-material-dispersion wavelength. A holey fiber may be designed to operate as a single-mode waveguide over a broad range of wavelengths (endlessly single-mode fibers). In fibers with a hollow core and a cladding with holes arranged in a periodic structure, light is guided in the core by reflection from the surrounding photonic-crystal cladding. Since the light travels in the hollow core, it suffers lower losses and reduced nonlinear effects. Polarization-Mode Dispersion Another form of pulse spreading, known as polarization-mode dispersion (PM D), is caused by random anisotropic changes in the fiber introduced by environmental factors along its length. Random variations in the magnitude and orientation of the birefringence introduce differential delays between the two polarization modes, and as described in Sec. 9.3B, the average RMS value of the pulse broadening associated with PMD is proportional to the square root of the fiber length: apMD == DpMDVL, I (24.1-4) Polarization-Mode Dispersion where D pMD is a dispersion parameter typically ranging from 0.1 to 1 psi vkm . PMD becomes important at high data rates when other forms of dispersion are compensated. Nonlinear Optical Effects Silica-glass fibers exhibit two kinds of optical nonlinear effects - third-order non- linearity, which underlies the optical Kerr effect, and nonlinear inelastic scattering, which includes stimulated Raman and Brillouin scattering. When high-power optical pulses are transmitted through single-mode fibers, which have small cross-sectional area, the optical intensity may be sufficiently high for these nonlinear interactions to occur, causing a number of deleterious effects that damage the signal integrity in communication systems:
24.1 FIBER-OPTIC COMPONENTS 1079 . Self-phase modulation (SPM) is a form of nonlinear dispersion caused by the opti- cal Kerr effect (the slight dependence of the refractive index, and hence the phase velocity, on the optical intensity, as described in Sec. 21.3A). Since different segments of the optical pulse travel at different velocities, pulse spreading ensues (see Sec. 22.3B). The optical Kerr effect may also result in crosstalk between counter-propagating waves in two-way communication systems. . Cross-phase modulation (XPM) results from nonlinear wave mixing wherein the phase velocity of a wave at one wavelength depends on the intensities of waves at other wavelengths traveling simultaneously in the same fiber (see Sec. 21.3C). In wavelength-division multiplexing (WDM) systems, XPM can cause serious crosstalk between the different channels. . Four-wave mixing (FWM) is also associated with third-order nonlinear effects (see Sec. 21.3D). It causes crosstalk between four waves of different wavelengths traveling simultaneously in the same fiber since the waves may exchange energy. This introduces an intensity-dependent gain/loss into channels of a WDM system. . Stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) are inelastic scattering processes involving interactions between light and molecular or acoustic vibrations of the medium. In these processes, two optical waves of dif- ferent wavelengths interact via a molecular vibration mode (SRS) or an acoustic wave (SBS) (see Secs. 13.5C, 14.3D, and 15.3A). Such interactions also lead to undesirable crosstalk between channels of a WDM system. The nonlinear properties of fibers may also be harnessed for useful applications in communication systems. Nonlinear dispersion via SPM may be adjusted to compensate for chromatic dispersion in the fiber. The result is the spreadless pulses known as optical solitons (see Sec. 22.5B). Nonlinear interactions can also be used to provide useful gain via FWM or SRS. Optical Raman Amplifiers are described in Sec. 14.3D. B. Sources for Optical Transmitters The basic requirements for the light sources used in optical communication systems depend on the nature of the intended application (long-haul communication, local-area network, etc.). The principal features are: . Power. The source power must be sufficient so that, after transmission through the fiber, the received signal is detectable with the required accuracy. . Speed. It must be possible to modulate the source power at the rate desired for imparting information. . Linewidth. The source must have a narrow spectrallinewidth so that the effect of chromatic dispersion in the fiber is minimized. . Noise. Random fluctuations in the source power must be avoided, particularly for coherent communication systems. . Other features. Other important features include ruggedness, insensitivity to en- vironmental changes such as temperature, reliability, low cost, and long lifetime. Both light-emitting diodes (LEDs) and laser diodes (LDs) are used as sources in optical fiber communication systems. These devices are discussed in Chapter 17. Light-emitting diodes are fabricated in two basic structures: surface emitting and edge emitting. Suiface-emitting diodes have the advantages of ruggedness, reliability, lower cost, long lifetime, and simplicity of design. The basic limitation attendant to their use is their relatively broad linewidth, which can exceed 100 nm in the 1300- 1600-nm band (see Fig. P17.1-5). When operated at maximum power, modulation frequencies up to 100 Mb/s are possible, but higher speeds (up to 500 Mb/s) can be attained at reduced powers. The edge-emitting diode has a structure similar to that of a laser diode without a feedback mechanism. It produces more power output with
1080 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS relatively narrower spectrallinewidth, at the expense of increased complexity. Laser diodes have the advantages of high power (tens of W), high speeds (many Gb/s), narrow spectrallinewidths (tens of MHz), and ease of coupling into single- mode fiber. However, they are sensitive to temperature variations. Multimode laser diodes suffer from partition noise, which is a random distribution of the laser power among the modes. When subjected to chromatic dispersion in the fiber, this leads to random intensity fluctuations and reshaping of the transmitted pulses. Laser diodes also suffer from frequency chirping, which is a change of the laser frequency as the optical power is modulated. Chirping results from modifications of the refractive index that accompany changes of the charge-carrier concentrations as the injected current is altered. As discussed in Chapter 17, AllnGaP jlnGaP LEDs are often used in inexpen- sive plastic-fiber communications systems that operate in the 600-650-nm region (see Fig. 17.1-19). More common, however, is In I-x GaxASI-YP y, a versatile alloy that is widely used for the fabrication of both LEDs and LDs in the near-infrared region of the spectrum. It offers a direct bandgap that is compositionally tunable over a substantial range of wavelengths, and lattice matching to an InP substrate can be maintained. InGaAsP is used to fabricate LEDs for short-haul, modest-bit-rate communications systems operating at Ao = 1.3 J-Lm (see Fig. 17.1-18). Long-haul high-bit-rate commu- nication systems generally operate at Ao == 1.55 J-Lm and make use of laser diodes rather than LEDs since it is far easier to couple the highly collimated light emitted by an LD into a single-mode fiber. The most common laser-diode configuration is the distributed feedback (DFB) laser (Fig. 24.1-4). As discussed in Sec. 17.4C, this device makes use of a corrugated- layer grating adjacent to the active region, which acts as a distributed reflector that substitutes for the mirrors of a Fabry-Perot laser. This design is compatible with on- chip integration. These edge-emitting lasers offer narrow spectral widths, which is critical for the efficient operation of 1.3 and 1.55 J-Lm wavelength-division-multiplexed (WDM) optical communication systems. n-InP InGaAsP jlnGaAsP MQW active region Figure 24.1-4 Buried-heterostructure multiquantum-well distributed- feedback laser used for optical fiber communications. The dielectric film provides gain guiding while the alternating p and n-type layers p-InP allow current flow only in the vicinity of the active region. Lasers such as these offer ample gain at modest current levels, and Fiber provide output powers of 1 W or more in a single spatial mode. Typical values . of the threshold current and differential responsivity are it < 10 mA and 91d 0.4 W / A, respectively. InP substrate c. Optical Amplifiers Optical amplifiers are indispensable components in modem long-haul optical fiber communication systems. They find use as postamplifiers (power amplifiers), line am- plifiers, and preamplifiers. As shown in Fig. 24.1-5, power amplifiers augment the optical power before light is launched into an optical fiber, line amplifiers serve as repeaters to boost the signal in the course of transmission (see Fig. 24.0-1), and pream- plifiers provide gain before photodetection.
24.1 FIBER-OPTIC COMPONENTS 1081 (a) Transmitter -0 ' CO) Fiber (b) Transmute t_ o CO) '->e_ (I) __ tR&et< Fiber {-<.:.... Fiber -.- -. . -:". "':m*"= ."."." .". :::.:: TransmItter -0 (t) (c) Fiber .. .. ; : n _):ReCelr: Figure 24.1-5 Optical fiber amplifiers are used in three configurations in an optical fiber communication system: (a) postamplifiers; (b) line amplifiers; and (c) preamplifiers. We consider three kinds of optical amplifiers: . Optical fiber amplifiers (OFAs). These include erbium-doped fiber amplifiers (EDFAs) (Sec. 14.3C), rare-earth-doped fiber amplifiers (REFAs) (Sec. 14.3C), and Raman fiber amplifiers (RFAs) (Sec. 14.3D). . Semiconductor optical amplifiers (SO As ) (Sec. 17.2). . Optical parametric amplifiers (OPAs) (Sec. 21.4C). With the exception of the OPA, all of the optical amplifiers listed above are nonpara- metric devices inasmuch as they rely on an exchange of energy between the field and the amplifying medium (see introduction to Chapter 21). EDFAs and RFAs turn out to be the most suitable amplifiers for optical fiber communications, as discussed below. Optical Fiber Amplifiers (OFAs) OFAs comprise three varieties: EDFAs, REFAs, and RFAs: Erbium-doped fiber amplifiers (EDFAs). Erbium-doped fiber amplifiers (EDFAs), which were the first OFAs to be developed, are widely used in optical fiber communi- cation systems. As discussed in Sec. 14.3C, they offer high polarization-independent gain, high output power, low insertion loss, low noise, and a broad transition near A == 1.55 J-Lm (corresponding to the wavelength of minimum loss for silica optical fibers, as shown in Fig. 24.1-2). Pumping is achieved by longitudinally coupling light into the amplifying medium, usually from strained quantum-well InGaAs laser diodes operating at Ao == 980 nm. The pump light may be injected in the forward or backward direction, or bidirectionally. Gains in excess of 50 dB can be achieved in EDFAs with tens of m W of pump power, and signal output powers in excess of 100 Ware readily generated. The available bandwidth is LlA 40 nm, corresponding to f1v 5.3 THz, which accommodates the C band. The L band is also readily covered although the optimization parameters of the EDFAs are not the same in the two bands. The large gain and bandwidth offered by these amplifiers make them highly suitable for use in wavelength-division multiplexing (WDM) systems (see Sec. 24.3C). Rare-earth-doped fiber amplifiers (REF As). Several ions other than Er 3 + (e.g., Pr 3 +, Tm 3 +, and Nd 3 +) are useful for making rare-earth-doped fiber amplifiers (RE- FAs) that cover the OIE/S/U bands (see Fig. 24.2-3). REFAs can therefore be used to extend the amplification bandwidth well beyond the 60-nm (7.5- THz) bandwidth achievable by using individually optimized EDFAs in the C and L bands. Unfortu- nately, however, REFAs other than Er 3 + function far better with fluoride and tellurite
1082 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS glasses than with silica glass. This materials challenge is not easily surmounted, for a number of reasons: (1) silica-glass fiber has been widely used and its manufacturing technology is entrenched; (2) each type of REFA requires its own fiber-glass matrix; (3) the splicing of different glasses is not straightforward; and (4) each type of REFA requires its own laser-diode pump at the appropriate wavelength. Nevertheless, by mixing and matching REFAs of Er 3 + and Tm 3 +, the available bandwidth A can be increased from 60 nm to 150 nm, corresponding to v 18.8 THz at 1550 nm. Raman fiber amplifiers (RFAs). Raman fiber amplifiers operate on the basis of stimulated Raman scattering (see Sec. 13.5C). As discussed in Sec. 14.3D, there are two standard RFA configurations: (1) distributed RFAs where the signal and pump are both sent through the transmission fiber, which serves as the gain medium, and (2) lumped RFAs in which a short length of highly nonlinear fiber serves as the amplifier and provides gain. As with the EDFA, pumping can be in the forward or backward direction, or bidirectional. The bandwidth over which Raman gain is available in silica fiber is about 100 nm (corresponding to about 12.5 THz at 1550 nm) so RFAs typically offer greater band- widths than EDFAs. Furthermore, multiple pumps of different frequencies can be com- bined to provide yet greater bandwidths; indeed, Raman amplification can, in principle, be employed over the entire region of fiber transparency. The gain of an RFA, which is 20 dB is substantially lower than that of an EDFA, as is the efficiency, but they can be increased by making use of dispersion-compensating fiber. Also, polarization-diverse pumping is required. The relative merits of EDFAs and RFAs have been considered in Sec. 14.3D. In spite of the shortcomings of RFAs in comparison with EDFAs, their wider bandwidths (extendable by using multiple pumps), arbitrary wavelength of operation, and compatibility with existing systems make them increasingly competitive as OFAs. Semiconductor Optical Amplifiers (SOAs) Semiconductor optical amplifiers (SOAs) (see Sec. 17.3) can be made to operate in any region of the optical spectrum by judiciously choosing the semiconductor material. They are compact and compatible with integrated optoelectronic circuits, particularly as postamplifiers or preamplifiers on an integrated optoelectronic circuit, and can be electrically pumped. SOAs designed for optical transmission applications in the near infrared are usually fabricated from InGaAsP, InGaAs, or InP. In the 1.3-1.6-j1m com- munications band, achievable bandwidths using quantum-well SOAs are A 50 nm, corresponding to v 6.5 THz at Ao == 1550 nm, although quantum-dot SOAs can provide nearly 200 nm of bandwidth. However, because of their low gain ( 15 dB), optical transmission applications have principally been limited to metropolitan optical networks where low gain suffices to overcome losses associated with multiple optical add-drop nodes. As discussed in Sec. 17.2D, SOAs have a number of disadvantages with respect to OFAs: they are incompatible with fiber geometry, exhibit substantial interchannel and intersymbol interference, high noise, sensitivity to temperature, and residual sensitivity to signal polarization. As a result, they hold greater appeal for applications such as all- optical switching in optical networks (see Chapter 23) and wavelength conversion than as linear optical amplifiers. Optical Parametric Amplifiers (OPAs) The optical parametric amplifier (OPA) discussed in Secs. 21.2C and 21.4C has the merit that it offers substantial gain and broadband tunability over an extended spectral region that stretches from the infrared to the visible. However, it has a number of features that limit its deployment in wavelength-division-multiplexed (WDM) applica- tions:
24.1 FIBER-OPTIC COMPONENTS 1083 . The WDM signals must be phase matched to the pump, which requires dispersion flattening. . Large-scale WDM implementation with equal spacing of channels is impeded by the presence of four-wave mixing. . This amplifier is sensitive to signal polarization so that polarization-multiplexed pumping is required. We conclude that SOAs and OPAs are less useful than OF As in optical fiber com- munication systems. D. Detectors for Optical Receivers A comprehensive discussion of semiconductor photon detectors is provided in Chapter 18. Two types of detectors are commonly used in optical communication systems: the p-i-n photodiode and the avalanche photodiode (APD). The APD has the advantage of providing gain before the first electronic amplification stage in the receiver, thereby reducing the detrimental effects of circuit noise. However, the gain mechanism itself introduces noise and has a finite response time, which may reduce the bandwidth of the receiver. Furthermore, APDs require greater voltage and more complex circuitry to compensate for their sensitivity to temperature fluctuations. The signal-to-noise ratio and the sensitivity of receivers using p-i-n photodiodes and APDs are discussed in Sec. 18.6. Detectors in the 870-nm Wavelength Range Silicon p-i-n photodiodes and APDs are used at these wavelengths. In state-of-the- art preamplifiers, silicon APDs enjoy a 10-to-15-dB sensitivity advantage over silicon p-i-n photodiodes because their internal gain makes the noise of the preamplifier relatively less important. Detectors in the 1300-1600-nm Wavelength Range Silicon cannot be used in this wavelength region because it is transparent (see Fig. 5.5- 1); this is because its bandgap wavelength lies below the wavelength of the light (A g < Ao). Rather, InGaAs and Ge p-i-n photodiodes are used, but InGaAs is preferred because of its smaller dark noise and greater thermal stability. Typical InGaAs p-i- n photodiodes have quantum efficiencies Il 0.8, responsivities 91: 1 A/W (see Fig. 18.3-9), bandwidths 10 GHz into 500, and dark currents 0.1 nA. Waveguide structures offer larger bandwidths. InGaAs APDs are widely used. Like all narrow-bandgap materials, however, In- GaAs suffers from large tunneling leakage currents when subjected to strong electric fields. This problem is mitigated by making use of a heterostructure with a small- bandgap material for the absorption region and a larger-bandgap material for the mul- tiplication region (SAM APD). Figure 24.1-6 illustrates a variation on this theme: a separate-absorption-grading-multiplication (SAGM APD), in which the absorption takes place in InGaAs and the multiplication in InP. The InGaAsP grading layer pro- vides a smooth transition between the two regions. Since holes multiply in this device, the salient ionization ratio is Ilk (see Fig. 18.6- 4). For InP, Ilk 0.5 when the mean gain G == 10, so the gain noise is substantially greater than that in Si. Nevertheless, these devices work well; they typically have efficiencies Il 0.8, responsivities 91: 10, bandwidths 10 GHz, gain-bandwidth products 100 GHz, and dark currents 0.1 nA.
1084 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS InP Substrate InGaAsP I Gra:i: I" InGaAs \ Absorption InP Multiplication APD Figure 24.1-6 Structure of a separate- absorption-grading-multiplication (SAGM) APD. Fiber 24.2 OPTICAL FIBER COMMUNICATION SYSTEMS The simplest communication system is a point-to-point link. The information is carried by a signal - a physical variable (electrical, electromagnetic, optical, etc.) modulated at one point and observed at the other. To transmit more than one signal simultaneously through the same link, the signals must be marked by some distinct attribute (e.g., time, frequency, or wavelength), or identified by some distinct code. The scheme is called multiplexing. In an optical fiber communication system, the link is an optical fiber through which a light wave modulated by the signal is transmitted. The modulated physical variable carrying the information may be the optical intensity, amplitude, frequency, phase, or polarization. The simplest example is the intensity-modulation communication system illustrated in Fig. 24.2-1. The simplest example of optical multiplexing is wavelength- division multiplexing (WDM), in which multiple signals are transmitted through the same fiber at distinct optical wavelengths, as illustrated in Fig. 24.2-2. . t Transmitter Receiver (a) Fiber t Transmitter 1 0 1 0 0 1 1 0 1 101001101 Receiver IVl, iX'J\ . t (b) Fiber Figure 24.2-1 Optical fiber communication systems using intensity-modulation. (a) Analog system: the power of the light source is proportional to the signal, which is a continuous function of time representing, e.g., an audio or video waveform. (b) Digital ON-OFF keying: bits"]" and "0" are represented, respectively, by the presence and absence of an optical pulse. . . ....... . .. . :. J ,. ' . . .. . . . ..::: . :". -i J I I I - Al AI" Ii D 0 - j. :. - AN- ".,,:,%f. ) 9te Figure 24.2-2 Wavelength- division multiplexing (WDM) AN One measure of the performance of an analog communication system is the band- width B (Hz). It is the maximum frequency at which modulated optical power may be t
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1085 transmitted through the link such that the received signal is detectable with a prescribed signal-to-noise ratio. The bandwidth is determined by the response time of the overall communication channel and by the attenuation and the noise level at the receiver. Similarly, a measure of the performance of a digital communication system is the maximum bit rate Bo (bits per second, or b/s) at which bits of the received signal are discernible with error rate not exceeding a prescribed value. This data rate is deter- mined by the attenuation and the pulse spreading introduced by the system, and also by the noise level at the receiver. The following bit rates represent optical carrier (OC) levels defined by the Synchronous Optical Network (SONET), which is a standard for optical telecommunication technology: Table 24.2-1 Approximate bit rates of the SONET standard. OC-I OC-3 OC-12 OC-24 OC-48 OC-192 OC-768 52 Mb/s 156 Mb/s 622 Mb/s 1.25 ObIs 2.5 ObIs 10 ObIs 40 ObIs This section begins with an overview of the evolution of optical fiber communica- tion systems followed by a quantitative analysis of the performance limits of simple digital and analog systems using intensity modulation. A. Evolution of Optical Fiber Communication Systems As illustrated in Fig. 24.1-2, the minimum attenuation in silica glass occurs at 1550 nm, whereas the minimum material dispersion occurs at 1312 nm. The choice between these two wavelengths depends on the relative importance of power loss versus pulse spreading, as explained in Sec. 24.2B. However, the availability of an appropriate light source is also a factor. First-generation optical fiber communication systems operated at 870 nm (the wavelength of AIGaAs light-emitting diodes and laser diodes), where both attenuation and material dispersion are relatively high. More advanced systems operate at 1300 and 1550 nm. The various operating wavelengths, materials and types of fibers, light sources, detectors, and amplifiers that may be used for building an optical link offer many possible combinations, some of which are summarized in Fig. 24.2-3. Progress in the implementation of optical fiber systems has historically followed a path toward longer wavelengths, from multimode fibers (MMF) to single-mode fibers (SMF), from light-emitting diodes (LEDs) to laser diodes (LDs), from p-i-n (PIN) photodiodes to avalanche photodiodes (APDs), and from semiconductor optical amplifiers (SOAs) to optical fiber amplifiers (OFAs). Appropriate materials for the longer wavelengths (e.g., quaternary sources and detectors) had to be developed to make this progress possible. The evolution of fiber components and systems has been motivated by a desire to increase the transmission bit rate Bo (bits/s or b/s) and the length L (km) of the communication link (the repeater spacing); the product LBo (km-b/s) has been used as a measure of progress. The following seven systems describe this evolution and Fig. 24.2-4 depicts the increase in LBo over the years. The first three systems, which are often referred to as the first three generations of optical fiber systems, have achieved a 1000-fold increase in LBo from 1974 to the 1990s. These technologies are used as examples in the discussion of system performance in Sec. 24.2B. Subsequent progress has extended these basic systems in a number of directions, leading to an increase of LBo by another five orders of magnitude from 1990 to 2005. This tenfold increase every four years has been called the "optical Moore's law."
1086 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS - I I 0 E S C L U Wavelength I I I I I I I I I Ao (nm) 800 -900 1000 1100 1200 1300 1400 1500 1600 1700 Source: LED LASER Detector: p-i-n APD SOA Amplifier: OFA AIGaAs I -, I InGaAsP I Si I I Ge -' ; I InGaAs --.- -- InGaAsP I I EDFA I REF A RFA I AIGaAs I Fiber: MMF: SI / GRIN SMF >- , i Silica glass Figure 24.2-3 Types and materials of optical sources, detectors, amplifiers, and fibers used at various wavelengths. The first generations of optical links operated at wavelengths near 870 nm, 1310 nm, and 1550 nm. System 1: Multimodefiber (MMF) at 870 nm. This is the early technology of the 1970s. Fibers are either step-index or graded-index. The light source is either an LED or a laser (initially GaAs and subsequently AIGaAs). Both siliconp-i- nand APD photodiodes are used. The performance of this system is limited by the fiber's high attenuation and modal dispersion. A typical intercity communication link of this era operated at Bo = 100 Mb/s, with a repeater spacing L = 10 km, i.e., LBo == 1 km-Gb/s. System 2: Single-mode fiber (SMF) at 1310 nm. The move to single-mode fibers and a wavelength region where material dispersion is minimal led to a substantial improvement in performance, limited by fiber attenuation. InGaAsP lasers are used with either InGaAs p-i-n or APD photodetectors (Ge APDs are also sometimes used). A typical long-haul link in this class operated at OC- 12 (622 Mb/s) with repeater spacing L = 40 km and LBo 25 km-Gb/s. System 3: Single-mode fiber (SMF) at 1550 nm. At this wavelength the fiber has its lowest attenuation. Performance is limited by material dispersion, which is reduced by the use of low-chirp single-frequency distributed-feedback (DFB) lasers (InGaAsP). Subsequent use of dispersion-shifted fibers (DSF) has alleviated the dispersion problem and boosted the performance. An example of this system is a long-haul terrestrial or undersea link operating at 2.5 Gb/s (OC-48) over a distance L = 100 km, for which LBo 250 km- Gb/s. Advances in transmitters and receivers have boosted this system to 10 Gb/s (OC-192), bringing LBo to one km- Tb/s. System 4: Coherent system. As described in Sec. 24.5, rather than measuring the intensity of the signal light directly by a photodetector, a coherent system makes use of coherent detection, in which light from a local source (called the local oscillator) is mixed with the signal light at the detector. The use of coherent detection enhances the receiver sensitivity thus allowing greater communication distances; however, this comes at the expense of increased complexity. As a result, the commercial implementation of coherent systems has lagged behind that of direct-detection systems, particularly as a result of the emergence of optical fiber amplifiers.
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1087 Systenl5: Link with optical amplifiers. The advent of semiconductor and optical fiber amplifiers (see Sec. 24.1 C) has had a dramatic impact on the performance of optical fiber communication systems. Placed periodically along the fiber, these amplifiers compensate attenuation and therefore extend the distance between electronic repeaters. An example is the transpacific TPC-5 system, which operates at bit rates of up to Bo = 10 Gb/s with distances up to L = 20,000 km, i.e., LBo = 200 km-Tb/s. System 6: Optical soliton system. Solitons are short (typically 1 to 50 ps) optical pulses that can travel through long optical fibers without changing the shape of their pulse envelope. As discussed in Sec. 22.5B, the effects of fiber dispersion and nonlinear self-phase modulation (arising, for example, from the optical Kerr effect) precisely cancel each other, so that the pulses act as if they were traveling through a linear nondispersive medium. Erbium-doped fiber amplifiers are effectively used in conjunction with soliton transmission to overcome absorption and scattering losses. Experimental systems have been operated at 10 Gb/s over fiber lengths in excess of 12,000 km (LBo = 1200 km-Tb/s). System 7: Wavelength-division multiplexing (WDM). The introduction of WDM has provided a significant increase in the capacity of the system by the use of multiple wavelengths (channels) transmitted through the same fiber. Broad- band optical amplifiers are used to provide simultaneous amplifications for all channels. An example is the TPC-6 system for which Bo = 100 Gb/s, L = 9,000 km, and LBo = 900 km-Tb/s. In combination with dispersion- managed transmission and forward error correction, rates of 5-10 Tbit/s per fiber over distances of 10 000 km are now possible. 10 9 10 8 -.. 10 7 C/) - ..D Co? 10 6 8 10 5 2 10 4 ro 10 3 I (!) g 10 2 ro C/) 6 10 Soliton ##.. //" .# ,,// ' @OFA Q) 1550 nm /'l SMF (DSF) //' .t.. ..- SF Laser /" .- / .# . // ."'" . , #"'" l .t .t : @ Coherent . ///Q) WDM 1 1970 1975 1980 1985 1990 1995 2000 2005 2010 Figure 24.2-4 The history of optical fiber communication systems compnses continuous improvement of the bit rate distance product LBo. B. Performance of Optical Fiber Systems The first step in assessing the performance of a fiber communication system is to come up with a mathematical model describing the effect of the various system components, principally the optical fiber, on the modulated signal. This permits us to estimate the
1088 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS shape of received distorted signal, and hence determine the signal-to-noise ratio in analog systems and the expected bit error rate in digital systems. In most applications, the fiber may be treated as a linear system described by an impulse response function h(t) or its Fourier transform, the transfer function H(f), where f is the modulation frequency. Three important parameters characterize these functions: . Power transmission. This is the fraction of steady (unmodulated) input optical power received at the output. It equals the transfer function H (f) at f == O. Since H(f) is the Fourier transform of h(t), H(O) == J h(t) dt is the area under h(t). For a fiber of length L and attenuation coefficient ex (dB/km), H(O) == exp[-aL], where a 0.23 ex is the attenuation coefficient in units of km - 1 . Localized power losses at couplers may also be included in ex in distributed units of dB /km. . The response time aT is the width of h ( t). It determines the temporal spreading of optical pulses and therefore sets the maximum data rate that can be used in digital systems. The response time is proportional to the fiber length. For example, in a single-mode fiber, aT == IDla.xL, where a.x (nm) is the source linewidth and D (ps/km-nm) is the dispersion coefficient (ps/km-nm). . The bandwidth a f (Hz) is the width of the transfer function 1 H (f) I. In an analog system, the bandwidth determines the maximum frequency at which the input power may be modulated and successfully detected by the receiver. Since H(f) and h ( t) are related by a Fourier transfonn, the bandwidth a f is inversely pro- portional to the response time aT. The coefficient of proportionality depends on the actual profile of h(t) (see Appendix A, Sec. A.2). Here, we use the relation a f == 1/27Ta T for the purpose of illustration. The maximum fiber length that can be used to transmit a signal with a desired performance level is set by the following principal impairments introduced by the system: . Attenuation results in an exponential drop of the optical power as a function of distance [Fig. 24.2-5(a)]. At a distance for which the received power becomes smaller than the receiver sensitivity (the minimum power required by the re- ceiver), the system's performance becomes unacceptable. . Dispersion results in an increase of the width of the optical pulses that represent data bits in a digital system as a function of distance [Fig. 24.2-5(b)]. When the width exceeds the bit interval, adjacent pulses overlap, resulting in intersym- bol interference (ISI), which introduces undesirable errors. In an analog system, dispersion washes out high-frequency components of the modulated signal and reduces the system's bandwidth. $-; Q) o p... ..c::: " . ,; +oJ .' .... ..' .,.."......I. ,. ...., , ,.,...,. "........ ". . ''" .,..... '.'. .... .,.,.. ,......." '"d . "W'w,_."""""'__"'..,...'"'''''''' -.. ..,..-.... -- --"'., . Q) rfJ "3 Bit time __. l!!!!!Y... _ _ _ _ _ __ Distance (a) Attenuation Distance (b) Dispersion Figure 24.2-5 (a) Dependence of the optical power on the distance. (b) Dependence of the pulse width on the distance. The maximum length of the optical link is set by either (a) attenuation, when the received power drops below the receiver sensitivity, or (b) dispersion, when the pulse width exceeds the bit time.
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1089 . Noise added by optical components, such as optical amplifiers, and by random propagation effects, such as polarization mode dispersion, introduces additional errors. . Nonlinear distortion associated with intense optical pulses results in the cross mixing of spectral components, and the introduction of interference between mul- tiplexed signals in wavelength-division multiplexing (WDM) systems. The communication system is more sensitive to transmission impairments at high bit rates (or high modulation frequencies) because of the following effects: . For a fixed average power, a higher bit rate corresponds to fewer photons per bit, and therefore to greater photon noise. Other noise sources in the receiver also become more important at high data rate. The receiver sensitivity is therefore an increasing function of the bit rate [Fig. 24.2-6( a)]. . A higher bit rate corresponds to shorter pulses [Fig. 24.2-6(b)] with broader spec- tra and greater dispersion. Such pulses undergo greater broadening, which leads to greater intersymbol interference (ISI). . For a fixed optical energy per bit, a higher bit rate (shorter bit time) requires greater optical power [Fig. 24.2-6(c)], which evokes nonlinear interactions lead- ing to nonlinear IS!. >. ....... : ....... .f';; c Q) rf1 I-; Q) > . a:) u Q) ':, :.II' " / ", " t> " q/"", I-; Q) . 8. '_ _ _ _1}_ ! _ _ ,0:: _ _ _ _ _ _ _ _ _1 ro Q) 0.. Q) rf1 ........ ::3 Bit rate (a) (b) (c) Figure 24.2-6 Effect of bit rate on (a) receiver sensitivity, (b) pulse width at the receiver, and (c) peak power. At higher bit rate, the communication system is more sensitive to attenuation, dispersion, and nonlinear effects. Bit rate Bit rate As we will see in the remainder of this section, the design of a long-haul high-bit- rate optical fiber communication link involves the selection of fibers with the lowest attenuation and/or dispersion, and careful power and pulse width budgeting, while guarding against the deleterious nonlinear effects associated with ultra-intense pulses. Bit-Error Rates The performance of a digital communication system is measured by the probability of error per bit, which is referred to as the bit error rate (BER). For an ON-OFF keying system, such as that shown in Fig. 24.2-1, bits "1" and "0" are represented, respectively, by the presence and absence of an optical pulse. If PI is the probability of mistaking" l" for "0," and Po is the probability of mistaking "0" for" 1," and if the two bits are equally likely to be transmitted, then BER == PI + po. A typical acceptable BER is 10- 9 (i.e., an average of one error every 10 9 bits). Errors occur as a result of noise in the received signal, or due to pulse spreading into neighboring bits, which results in intersymbol interference. Figure 24.2-7 shows an example of random realizations of the pulse corresponding to bit" 1," superimposed with random realizations of the signal received from possible neighboring pulses when
1090 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS the bit is "0." This diagram is known as the eye diagram. The more open the "eye," the more distinguishable are the" 1" and "0" bits and the less the likelihood of error. . t t t Figure 24.2-7 Closing of the eye diagram (left to right) as a result of noise and pulse broadening. Receiver Sensitivity The sensitivity of a digital optical receiver is defined as the minimum number of photons (or the corresponding optical energy) per bit necessary to guarantee that the rate of error (BER) is smaller than a prescribed rate (e.g., 10- 9 ). Errors occur because of the randomness of the number of photoelectrons detected during each bit, as well as the noise in the receiver circuit itself. The sensitivity of receivers using various photodetectors is discussed in Sec. 18.6E. For example, when the light source is a stabilized laser, the detector has unity quantum efficiency, and the receiver circuit is noise-free, then an average of at least n o == 10 photons per bit is required to ensure that BER < 10- 9 . Therefore, the sensitivity of the ideal receiver is 10 photons/bit. This means that bit" 1" should carry an average of at least 20 photons, since bit "0" carries no photons. In the presence of other forms of noise, a larger number of photons is required. A sensitivity of n o photons corresponds to an optical energy hv n o per bit and an optical power Pr == (hv n o)/(l/Bo), Pr == hv n oBo, (24.2-1 ) which is proportional to the bit rate Bo. As the bit rate increases, a higher optical power is required to maintain the number of photons/bit (and therefore the BER) constant. It is shown in Sec. 18.6E that when circuit noise is important, the receiver sensitivity n o depends on the receiver bandwidth (i.e., on the data rate Bo). This behavior com- plicates the design problem. For simplicity, we shall assume in the following analysis that the receiver sensitivity (photons per bit) is independent of Bo. c. Attenuation- and Dispersion-Limited Systems In this section, we examine the performance limits imposed by attenuation and dis- persion on a digital intensity modulation ON-OFF keying (OOK) system. Nonlinear effects are ignored and the fiber transmission system itself is assumed to introduce no nOIse. Consider an optical fiber link operated as a digital communication system at a data rate of Bo bits/s over a distance of L (km). The source has power Ps (m W), wavelength A (nm), and spectral width a).. (nm). The fiber has attenuation coefficient ex (dB/km) and chromatic dispersion coefficient D).. (ps/km-ns). The receiver has a sensitivity of n o (photons per bit), corresponding to power sensitivity Pr == (hc/A) n oBo (mW), which must be received for the system to operate at an acceptable error rate. The performance limits are established by determining the maximum distance L over which the link can transmit Bo bits/s without exceeding the prescribed bit-error
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1091 rate. Clearly, L decreases with increase of Bo. Alternatively, we may determine the maximum bit rate Bo a link of length L can transmit with an error rate not exceeding the allowable limit. The maximum bit-rate-distance product LBo serves as a single number that describes the capability of the link. We shall determine the typical depen- dence of L on Bo, and derive expressions for the maximum bit-rate-distance product LBo for various types of fibers. Two conditions must be satisfied for acceptable operation of the link: 1. The received power must be at least equal to the receiver power sensitivity Pr. This condition is met by preparing a power budget from which the maximum fiber length is determined. A margin of 6 dB above Pr is usually specified. 2. The width of the received pulses must not significantly exceed the bit time in- terval1/ Bo, or else adjacent pulses overlap and cause intersymbol inteiference, which increases the error rates. This condition is met by preparing a budget for the pulse spreading resulting from the transmitter, the receiver, and various forms of dispersion in the fiber. If the bit rate Bo is fixed and the link length L is increased, two situations leading to performance degradation may occur: the received power becomes smaller than the receiver power sensitivity Pr, or the received pulses become wider than the bit time 1/ Bo. If the former situation occurs first, the link is said to be attenuation limited. If the latter occurs first, the link is said to be dispersion limited. Attenuation-Limited Performance: Power Budget Attenuation-limited performance is assessed by preparing a power budget. Since fiber attenuation is measured in dB units, it is convenient to also measure power in dB units. Using 1 mWas a reference, dBm units are defined by P == 10 10glO P, Pin mW; P in dBm. (24.2-2) For example, P == 0.1 mW, 1 mW, and 10 mW correspond to P == -10 dBm, 0 dBm, and 10 dBm, respectively. In these logarithmic units, power losses are additive. If Psis the power of the source (dBm), (X is the fiber loss (dB /km), Pc is the splicing and coupling loss (dB), and L is the maximum fiber length such that the power delivered to the receiver is the receiver sensitivity P r (dBm), then Ps - Pc - Pm - (XL == Pr (dB units), (24.2-3) where Pm is a safety margin. The optical power is plotted schematically in Fig. 24.2-8 as a function of the distance from the transmitter. The receiver power sensitivity Pr == 10 10glO Pr (dBm) is obtained from (24.2-1), n ohvBo Pr == 10 log 3 dBm. 10- (24.2-4 ) Thus, Pr increases logarithmically with Bo, and the power budget must be adjusted for each Bo as illustrated in Fig. 24.2-9. The maximum length of the link is obtained by substituting (24.2-4) into (24.2-3), 1 ( n ohvBo ) L==- Ps-Pc-Pm-1010g 3 ' (X 10- (24.2-5)
1092 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Connector Connector Connector Fiber Fiber Receiver Ps _____________________c_E______________________---- --- 8 '"d '-' ;...., (l.) o 0... . 0... o nn } Receiver sensitivity Margin Pm { P r j;:',:", :;{-:;y.: :<':;::',-:::=;;;,';;:::;;;::",;:C0.Z'/=G.-:::::,::::;'::.:1.G:;;;;;;;V:;;:::Z,i7;;;:;;'.,:;,T2:Z ,.J;;;T;;:,7",::\=,7y;:,7>::,::.::::.';- :,'::- -,:-::';:7,::,,:: ,::. .-::;- -t: . o L z Figure 24.2-8 Power budget of an optical link. -10 E -20 '-" o -------------------f------------- aL ! Source power Ps I 10M 100M 100M Bit rate Bo (b/s) Figure 24.2-9 Power budget as a function of bit rate Bo. As Bo increases, the power Pr required at the receiver increases (so that the energy per bit remains constant), and the maximum length L decreases. -60 - 70 ... lOOk Pc + Pm ... .... \..."\? r ... ... ... \\.J ...... e{\ ... .... e '2) ... ... ece\ ...............t (]) -30 o 0... ..- -40 ro u .R -50 o I IG I lOG 1M from which 10 L == Lo - - log Bo, ex (24.2-6) Attenuation-Limited Fiber where Lo == [ps - Pc - Pm - 30 -1010g( n ohv)]/exo The length drops with increase of Bo at a logarithmic rate with slope 10/ ex. Figure 24.2-10 is a plot of this relation for the operating wavelengths 870, 1300, and 1550 nm. Dispersion-Limited Performance: Time Budget When a pulse representing a data bit is generated by the transmitter, propagated through the fiber, and detected by the receiver, it loses power and gains width. The final pulse width a r depends on the original pulse width as, the response time of the transmitter atx, the response time of the fiber aT' which results from various forms of dispersion, and the response time of the receiver a rx . The actual shape of the receiver pulse may be determined by convolving the original pulse profile with the impulse response
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1093 1000 ]' '-' Coaxial' , , , , cable 100 (l.) U t:: ro ....... U"J 6 10 lOOk 1M 10M 100M lG lOG Bit rate BO (b/s) Figure 24.2-10 Maximum fiber length L as a function of bit rate Bo under attenuation-limited conditions for a fused silica-glass fiber operating at wavelengths Ao == 870, 1300 and 1550 nm assuming fiber attenuation coefficients ex == 2.5, 0.35, and 0.16 dB /km, respectively; source power Ps == 1 mW (P s == 0 dBm); receiver sensitivity n o == 300 photons/bit for receivers operating at 870 and 1300 nm and n o == 1000 for the receiver operating at 1550 nm; and Pc == Pm == O. For comparison, the L-B o relation for a typical coaxial cable is also shown. functions of the transmitter, the fiber, and the receiver (assuming that all systems are linear). If all functions are Gaussian, the square of the width of the final pulse equals the sum of the squares of the widths of all constituent functions, so that 2 2 2 a o == as + a sys , (24.2-7) where 2 222 a sys == a tx + a rx + aT' (24.2-8) and a sys is the width of the response function of the communication system (trans- mitter + fiber + receiver). These relations are used in practical design even though the response functions are not Gaussian. A principal design condition for the communication link ensures that the width of the received pulse does not exceed a prescribed fraction of the bit period T == 1/ Bo, in order avoid intersymboJ interference (ISI). A time budget must be prepared (Fig. 24.2- 11) to ensure that this condition is met. The choice of that fraction is arbitrary and a number of ad hoc values are used. For example, some designers require that the system's response time a sys does not exceed 70 % of the bit period for non-return- to-zero (NRZ) pulses and 35 % for return-to-zero (RZ) pulses (see Fig. 24.3-4 in Sec. 24.3A for definitions of these modulation formats). For a given receiver and transmitter, the design of the link centers around determin- ing the maximum fiber length L. Since the only length-dependent contribution to a sys comes from the fiber aT' in the following analysis we will adopt a design condition that the maximum allowed value of aT be 25 % of the bit-time interval T, i.e., 1 1 (}" T = 4 T = 4Bo . (24.2-9) The choice of the factor is clearly arbitrary and serves only for comparison of the different types of fibers. We now consider the distance versus bit-rate relations that arise from this condition for the various dispersion-limited cases mentioned in Sec. 24.1 A. The results are plotted in Fig. 24.2-12.
1094 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS - I r Fiber 1 Transmitter Receiver Limit .::'.-:;_ __ __ _....-..;:=::;..:.:::.:.:.::.:::--_ 2 a fiber 2 a tx a; 0 L z Figure 24.2-11 Budget for the pulse temporal width. . Multimodefiber (MMF). For multimode fiber, the width of the received pulse after propagation a distance L is dominated by modal dispersion. For step-index fibers, (24.1- 2) and (24.2-9) result in the L- Bo relation Cl LBo == 2 . (24.2-10) Step-Index MMF where Cl == colnl is the speed of light in the core materia] and == (nl - n2) I nl is the fiber fractional index difference. In a graded-index fiber of optimal (approximately parabolic) refractive index profile, the pulse width is smaller by a factor 2/, and LBo is greater by the same factor. For nl == 1.46 and == 0.01, the bit-rate-distance product LBo 10 km-Mb/s for step-index fibers and LBo 2 km-Gb/s for graded index fibers. . Single-ll1ode fiber (SMF). Assuming that pulse broadening in a single-mode fiber results from material dispersion only (i.e., neglecting waveguide dispersion), then for a source of linewidth a A the width of the received pulse is given by (24.1-3), so that 1 LBo = 4J D xl a .x ' (24.2-11 ) SMF where DAis the dispersion coefficient of the fiber material. For operation near Ao == 1300 nm, IDA I may be as small as 1 ps/km-nm. Assuming that a).. == 1 nm (the linewidth of a single-mode laser), the bit-rate-distance product LBo 250 km-Gb/s. For operation near Ao == 1550 nm, D A == 17 ps/km-nm, and for the same source spectral width a).. == 1 nm, LBo 15 km-Gb/s. . Single-mode fiber with transform-limited pulses. To reduce chromatic dispersion, the spectral linewidth a).. of the source must be small. Spectral widths that are a small fraction of 1 nm are obtained with single-frequency lasers and external modulators. However, an extremely narrow spectral width is incompatible with an extremely short pulse because of the Fourier transform relation between the spectral and temporal distributions. As described in Sec. A.2 of Appendix A,
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1095 1000 --- 100 E '-' u t:: ro ..... rfJ o / Graded-index Step-index / 1.55 11m 10 \ \ \ \ \ \(f) \ \ \ \ \ \ \ \ Transform- \ \ limi ted \ \ \ \ MMF 1M 10M I 100M \ \ \ \ \ NZ-DSF \ \/ \ \ \ \ \ \ \ \ \ \ \ 1 lOOk lG lOG 100G IT Bit rate BO (b/s) Figure 24.2-12 Dispersion-limited maximum fiber length L as a function of the bit rate Bo for multi mode fibers (MMF) and single-mode fibers (SMF). Six lines are shown (left to right): (a) MMF, step-index (nl = 1.46, = 0.01), LBo = 10 km-Mb/s; (b) MMF, graded-index with parabolic profile (nl = 1.46, = 0.01), LBo = 2 km-Gb/s; (c) SMF limited by material dispersion, operating at 1550 nm with D A = 17 ps/km-nm and a A = 1 nm, BoL 15 km-Gb/s; (d) SMF limited by material dispersion, operating at 1300 nm with IDA/ = 1 ps/km-nm and a A = 1 nm, BoL = 250 km- Gb/s; (e) SMF with transform limited pulses operating at 1550 nm with IDA I = 17 ps/km-nm; (f) same as (e) with non-zero dispersion-shifted fiber (NZ-DSF) with chromatic dispersion coefficient D A = -! ps/km-nm. pulses with the least product of temporal and spectral widths have a Gaussian profile. Such transform-limited pulses therefore suffer the least dispersion. A transform-limited Gaussian pulse of width TO and complex envelope exp( -t 2 / T6) has a Gaussian spectral intensity of width (FWHM) a v == 0.375/TO (see Sec. 22.1B). This corresponds to a A == '8Ao/8vlav == (A/c)av == 0.375A/CTO. If the pulse has a width equal to half a bit period, i.e., TO == T /2 == 1/2Bo, then A 2 a A == 0.75Bo, Co (24.2-12) which is directly proportional to the bit rate Bo. For example, for Ao == 1550 nm and Bo == 10 Gb/s, a A == 0.06 nm. As described in Sec. 22.3B, when a transform- limited Gaussian pulse of width TO travels through a dispersive medium with dispersion coefficient Dv it is broadened by a factor of J2 at the characteristic distance Zo == 7rT6 / Dv. At this distance a pulse of initial width TO == T /2 stretches by a time (J2 - 1) T /2 0.21 T. We may therefore take Zo as the maximum accertable length L of the communication link. Using the relations L == Zo == 7rTo / Dv, TO == T /2 == 1/2Bo, and Dv == DAA/ CO we obtain the distance bit-rate relation 2 7r Co LBo = 4 ID>J ' (24.2-13) SMF Transform-Limited Pulse
1096 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS The maximum distance L is therefore inversely proportional to B, i.e., drops more rapidly with the data rate than in previous cases. Also, the product BoL is inversely proportional to the data rate Bo. Figure 24.2-12 shows the L - Bo rela- tion for AD == 1550 nm and D).. == 17 ps/km-nm. For example, at Bo = 10 Gb/s, L = 64 km, but at Bo = 40 Gb/s, L drops to 4 km. The use of transform-limited pulses therefore extends the dispersion-limited bit rate bounds substantially, although that rate drops more rapidly with further increase of bit rate. . Single-mode dispersion-compensated fiber with transform-limited pulses. With single-mode fibers and transform-limited optical pulses, the maximum fiber dis- tance for a given bit rate reaches its highest value, limited only by the dispersion coefficient. This coefficient can be reduced by use of dispersion-shifted fibers (DSF). As shown in Fig. 24.2-12, reduction of D from l7 ps/km-nm to 4 ps/km- nm, by use of DSF, increases the maximum length from 64 km to 272 krn at 10 Gb/s. However, DSF fibers come with slightly higher attenuation coefficient. Combined Attenuation- and Dispersion-Limited Performance The attenuation-limited and dispersion-limited distance-bit-rate relations are com- bined in Fig. 24.2-13 by superposing Figs. 24.2-10 and 24.2-12 and selecting the smaller of the attenuation- or dispersion-limited distances. These relations describe the performance of generations of optical fibers operating at AD = 870 nm (multimode), at 1300 nm, and 1550 nm (single-mode). Several simplifying assumptions and arbitrary choices have been used to create this chart, and the values obtained should therefore be regarded only as indications of the order of magnitude of the relative performance of the different types of fibers. Nevertheless, a number of important conclusions can be made: 1000 __ 100 e '-' u s::: ro ..... V'J Q 10 MMF , , , , , , , \ Transform- , limited , , , , , , \ NZ-DSF , 'v , , , , , , 100M IG Bit rate Bo (b/s) Figure 24.2-13 Maximum distance L versus bit rate Bo for six cases of fibers. This graph is obtained by superposing the graphs in Figs. 24.2-10 and 24.2-12. Each line represents the maximum distance L of the link at each rate Bo that satisfies both the attenuation and dispersion limits, i.e., guarantees the reception of the required power and pulse width at the receiver. 1 lOOk 1M 10M lOG 100G IT . At low bit rates, the fiber link is generally attenuation limited; L drops with Bo logarithmically. At high bit rates, the link is dispersion limited and L is inversely
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1097 proportional to Bo for optical pulses limited by the source linewidth and inversely proportional to B6 for transform-limited. . For high-data-rate long-haul communication links, single-mode fibers are essen- tial. The choice between the 1300-nm and the 1550-nm wavelengths is not ob- vious since, for conventional fibers, chromatic dispersion is smallest at 1300 nm while the attenuation is smallest at 1550 nm. This explains the cross-over of the L - Bo lines at these wavelengths. . By use of dispersion-shifted fibers (DSF), it is possible to reduce the overall chro- matic dispersion coefficient at 1550 nm, making operation at 1550 nm generally superior to operation at 1300 nm. Performance of Analog Communication System As in digital optical fiber communication links, the performance of an analog link is limited by the fiber attenuation and/or dispersion. Because of fiber attenuation the received signal is weakened and may not be discernible from noise. Because of fiber dispersion the transmission bandwidth a f == l/27ra T is limited so that high-frequency signal components are attenuated more than low-frequency components, resulting in signal degradation. Both of these deleterious effects increase with the increase of the fiber length L. The received optical power drops exponentially with L, whereas the fiber bandwidth is inversely proportional to L. Nonlinear effects do not playa role in analog systems since the power is distributed and not concentrated in narrow pulses. The maximum allowable length of the analog fiber link is determined by ensuring that two conditions are met: . The fiber attenuation must be sufficiently small so that the received power is greater than the receiver power sensitivity P r. . The fiber bandwidth a f must be greater than the spectral width B of the transmit- ted signal. As discussed in Sec. 18.6, the sensitivity of an analog optical receiver is the smallest optical power necessary for the signal-to-noise ratio (SNR) of the photocurrent to exceed a prescribed value SNRo. For an ideal receiver (with unity quantum efficiency and no circuit noise) SNR == n == (P/hv)/2B, where B is the receiver bandwidth, P the optical power (watts), and n the average number of photons received in a time intervall /2fl, regarded as the resolution time of the system. If SNR o is the minimum allowed signal-to-noise ratio, the receiver sensitivity becomes n o == SNR o photons per resolution time and the corresponding power Pr == hv n o(2B). (24.2-14 ) This is identical to the expression (24.2-1) for the power sensitivity of the digital receiver if the resolution time 1/2B of the analog system is equated with the bit time 1/ Bo of the digital system. Because of the equivalence between (24.2-14) and (24.2-1) and because of the applicability of the power budget equation (24.2-3) to analog systems as well, the L- Bo relations determined earlier for the binary digital system are applicable to the analog system, with Bo replaced by 2B, provided that the acceptable performance of the analog system is SNR o == 10. As an example, a l-km fiber link capable of transmitting digital data at a rate of 2 Obis with a BER not exceeding 10- 9 can also be used to transmit analog data of bandwidth 1 GHz with a signal-to-noise ratio of at least 10. In analog systems, however, the required signal-to-noise ratio is usually much greater than 10, so that the receiver sensitivity must be much greater than 10 photons per resolution time. For high-quality audio and video signals, for example, a 60-dB signal-to-noise ratio is often required. This corresponds to SNR o == 10 6 , or n o ==
1098 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS 10 6 photons per resolution time. Additional design considerations are particularly important in analog systems. For example, the nonlinear response of the light source and photodetector cause additional signal degradation and place restrictions on the dynamic range of the transmitted waveform. D. Attenuation and Dispersion Compensation and Management Attenuation Compensation The performance of attenuation-limited fiber communication systems may be signifi- cantly enhanced by use of optical fiber amplifiers placed at appropriate distances within the fiber link, as illustrated in Fig. 24.2-14. Amplifiers elevate the diminished optical power, so that the received power remains above the receiver sensitivity for longer links. This process is ultimately limited by the noise introduced by the amplifiers themselves since, unlike repeaters, optical amplifiers do not regenerate the exact digital signal. However, before this limit is reached, dispersion often takes over and the system becomes dispersion limited. Dispersion compensation is therefore indispensable in long-haul optical fiber communication systems using optical amplifiers. Fiber Fiber ....... 1 Fiber . :... :.' :. -0 --- . z Figure 24.2-14 Compensation of attenuation by use of optical fiber amplifiers. Dispersion Compensation The pulse spreading introduced by propagation through an optical fiber of length Land dispersion coefficient D)., may be reversed by use of another fiber, called a dispersion- compensating fiber, with dispersion coefficient D of opposite sign and length L' selected such that the magnitudes of the dispersion introduced by the two fibers are equal, i.e., DL' == - D).,L. (24.2-15) The pulse spreading and compression introduced by an alternating sequence of such fibers is illustrated in Fig. 24.2-15. The compensating fiber is often relatively short and its dispersion coefficient must therefore be high. Since dispersion in conventional fibers is positive for wavelengths above 1310 nm, the dispersion compensating fiber must have negative dispersion in this band. This can be afforded by use of dispersion- shifted fibers (DSF). Other optical components may be used in place of DCFs. As described in Sec. 22.2, the propagation of an optical pulse through a dispersive medium is equivalent to a quadratic chirp filter, which is a phase-only filter with a phase proportional to the square of the frequency. A fiber of length L and dispersion coefficient D)., is a quadratic chirp filter with chirping coefficient b == D)., L. The effect of this filter may be com- pletely eliminated by use of an inverse compensation filter - another quadratic chirp filter with chirping coefficient of equal magnitude and opposite sign, b' == -b. The dispersion compensating fiber plays such a role, but other optical components, such as gratings and interferometers, may also be used (see Sec. 22.2).
24.2 OPTICAL FIBER COMMUNICATION SYSTEMS 1099 D)..J.+) I DJ-)dJl D)..J.+) I DJ-)d II( L . I >II( L ' . I II( L . I 0( L'---! , " , , I I I I g -- II I: ....-1 8 I I . i D)..L : D L' i :: -- ' Q I I I I Z I I I I I I I ,J:::: I I I I .:c; I I ....-4 I I I I I I I I Q) I I I I I I ;:j I I I I D)..J.+) --. z Figure 24.2-15 Dispersion compensation by use of fiber segments of opposite dispersion. The compensation filter may be placed at the transmitter end of the link, thus prec- ompensating the dispersion that is subsequently introduced by the fiber. Alternatively, it may be placed at the receiver end, thus postcompensating the broadened pulses immediately before they are detected. More commonly, multiple compensation filters are placed periodically within the link, providing distributed compensation. Under linear propagation conditions, the actual locations of the compensation filters is not important. However, in order to avoid the deleterious nonlinear effects, compensation filters are placed such that short pulses are avoided over extended distances within the fiber. Broadband Dispersion Compensation: Dispersion Management For broadband communication systems, such as WDM, the condition for dispersion compensation, (24.2-15), must be satisfied at an wavelengths within the spectral band; i.e., the error e).. == DL' - D)..L must be zero everywhere. Since the dispersion coefficients are wavelength dependent, this condition is difficult to satisfy. Fig. 24.2-16 illustrates a situation for which e).. == 0 at a wavelength Al in the middle of the band, wherein the compensation is perfect, a positive e).. at a wavelength A2 corresponding to a net positive dispersion, and a negative e.x at another wavelength A3 with net negative dispersion. . . . . . . . I . » Z D).. ;' A "'""' e s:: --- 00 0.- '-" s:: o .00 0 $-i Q) 0.- 00 a " z -""" Q I o """ A , Figure 24.2-16 Perfect dispersion compensation at AI, and imperfect dispersion compensation with net positive and negative dispersion at A2 and A3, respectively. The error e).. vanishes if the slopes of D).. and D are equal. If both D.x and D are approximately linear functions of A with the same slope, and if e).. == 0 at the central wavelength AI, then e.x 0 everywhere. The design of a
1100 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS dispersion compensation filter with the appropriate value of the dispersion coefficient and the slope of its wavelength dependence is known as dispersion management. E. Soliton Optical Communications The ultimate dispersion compensation occurs naturally in optical solitons. These non- spreading pulses have an intensity that is sufficiently high for the nonlinear optical properties of the fiber to play a principal role in their formation. As described in Sec. 22.5B, optical solitons are pulses for which nonlinear dispersion (the dependence of the phase velocity on the intensity via the optical Kerr effect) completely com- pensates linear chromatic dispersion - the net result is that the pulse travels without altering its width or shape. The gain provided by a fiber amplifier can also be used to compensate for fiber attenuation so that the pulse maintains its peak intensity and continues to travel as a soliton. As expressed in (22.5-13), for a pulse of width TO, peak intensity 1 0 , and free-space wavelength Ao at its central frequency, the condition for soliton formation is 27r - {3" ,n2 1 0 ==, .1\0 TO (24.2-16) Soliton Condition where n2 is the Kerr coefficient and -(3" == (A/27rco)D).. is proportional to the dispersion coefficient D A. The intensity profile of the soliton is described by 1 (t) == 10 sech 2 (tITO), which is a bell-shaped function of FWHM 1.76 TO. In a digital optical communication system, a soliton of width TO much smaller than the bit interval T represents bit "1," while bit "0" is represented by the absence of a soliton. This is necessarily a retum-to-zero (RZ) modulation format. Soliton communication systems are neither attenuation-limited nor dispersion- limited. Instead, they are limited by nonlinear intersymbol interference that occurs as a result of the nonlinear interaction between the tails of solitons representing neighboring bits. For example, when two identical solitons separated by the bit interval T travel a sufficiently long distance through the same fiber, they eventually collapse and merge into a single pulse, which subsequently separates back into the original two pulses. As mentioned in Sec. 22.5B, this process is repeated periodically with a period [see (22.5-32)] Lp == 7re r / 2 Zo, (24.2-17) where r == T / TO is the ratio of the separation to the soliton width and 2zo == -T6 / (3" == 27rCoT6 / AD).. is the fiber dispersion distance. The period Lp increases exponentially with the ratio r. If r » 1, i.e., if the bit interval T is much greater than the soliton width TO, then Lp can be made much longer than ZOo If the fiber length L is much smaller than Lp, then the interaction between neighboring bits is minimal. For a fixed ratio r, the condition L « Lp may be written in terms of the bit rate Bo == 1/ T as 2 7r 2 C o e r / 2 LBo « A 2 D . o ).. r (24.2-18) This places a limit on the ultimate distances and transmission rates allowed. EXAMPLE 24.2-1. Soliton Communication System. A soliton communication system transmits data at 10 Gb/s through a single-mode dispersion-shifted fiber at Ao == 1550 nrn using
24.3 MODULATION AND MULTIPLEXING 1101 10-ps (FWHM) soliton pulses. At this wavelength, the dispersion coefficient D.;\ = 1 ps/nm-km and the nonlinear coefficient n2 = 2.6 x 10- 20 m 2 /W. The fiber effective cross-sectional area is A eff = 60 J-Lm 2 . We proceed to determine the source optical power and the maximum length of the link. The 10-ps FWHM pulse width corresponds to a time constant TO = 10/1.76 = 5.7 ps. To satisfy the soliton condition (24.2-16), the peak intensity is 10 = 3.75 X 10 8 W/m 2 , corresponding to a peak power 10Aeff = 22.5 m W, which must be delivered by the source. The fiber dispersion distance 2zo = (21rc o / A)Tg / D.;\ 25 km. Since the bit interval T = 1/ Bo = 100 ps, the ratio r = T /TO = 17.6, and the interaction period given in (24.2-17) is Lp 2.1 X 10 4 Zo. The fiber length must be much shorter than this length. In this example, (24.2-18) provides LB6 « 26 (Tb/s)2 km. 24.3 MODULATION AND MULTIPLEXING A. Modulation Optical communication systems are classified in accordance with the optical variable modulated by the transmitted signal. Two principal types are used: field modulation and intensity modulation. Field modulation. The field of a monochromatic optical wave serves as a sinusoidal carrier of very high frequency (200 THz at Ao == 1500 nm, for example). In amplitude modulation (AM), phase modulation (PM), and frequency modulation (FM) systems, the amplitude, phase, or frequency is varied in proportionality to the signal (Fig. 24.3- 1). Because of the extremely high frequency of the optical carrier, a very wide spectral band is available, and large amounts of information can, in principle, be transmitted. Although modulation of the optical field is an obvious extension of conventional radio AM 1 Si g nal I j ,,, j i , itl'I.. j , , j i , itl I , , , , , , , , ' 'A' , , , , , , , " t · - Modulator - PM t & , iL, & , jLI & , ""1 11 "'11"'" t Optical field Ii j i , 1 , i 1111 i , 1 i " , i """'''''''''''''' t Figure 24.3-1 Amplitude and frequency modulation of the optical field. and microwave communication systems to the optical band, it is rather difficult to implement, for several reasons: . It requires a source whose amplitude, frequency, and phase are stable and free from fluctuations, i.e., a highly coherent laser. . Direct modulation of the phase or frequency of the laser is usually difficult to implement. An external modulator using the electro-optic effect, for example, may be necessary. . Because of the assumed high degree of coherence of the source, multimode fibers exhibit large modal noise; a single-mode fiber is therefore necessary. . Unless a polarization-maintaining fiber is used, a mechanism for monitoring and controlling the polarization is needed. . The receiver must be capable of measuring the magnitude and phase of the optical field. This is usually accomplished by use of a heterodyne detection system.
1102 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Because of the requirement of coherence, optical communication systems using field modulation are called coherent communication systems. These systems are discussed in Sec. 24.5. Intensity modulation. In an intensity modulation system, the optical intensity (or power) is proportional to the signal, or a coded version thereof, as illustrated in Fig. 24.3-2). The majority of commercial fiber communication systems at present use intensity modulation. The power of the source is modulated by varying the injected current in an LED or a laser diode. The fiber may be single-mode or multimode and the optical power received is measured by use of a direct-detection receiver. The high- frequency optical field oscillations play no role in the modulation and demodulation; only power is varied at the transmitter and detected at the receiver. However, the wavelength of light may be used to identify different signals traveling through the same link - a process known as wavelength division multiplexing (WDM). t Signal .-/ ""--/ '-.. )t Optical intensity Modulator . t Figure 24.3-2 Intensity modula- tion. Modulation format. Once the modulation variable is chosen (intensity, frequency, or phase), any of the conventional modulation formats (analog, pulse, or digital) can be used. An important example is pulse code modulation (PCM). In PCM the analog signal is sampled periodically at an appropriate rate and the samples are quantized to a discrete finite number of levels, each of which is binary coded and transmitted in the form of a sequence of binary bits, "1" and "0," represented by pulses transmitted within the time interval between two adjacent samples (Fig. 24.3-3). --- VJ :E 00 '-' VJ ;> \0 tr) (".I )I t PCM signal 64 kbits/s !1 0 0 1 0 1 1 0 10 1 1 0 1 1 0 I! 1 0 1 0 1 0 1 O! 1 1 0 0 1 0 lIt Figure 24.3-3 An example of PCM. A 4-kHz voice signal is sampled at a rate of 8 x 10 3 samples per second. Each sample is quantized to 2 8 == 256 levels and represented by 8 bits, so that the signal is a sequence of bits transmitted at a rate of 64 kb/s. . . . . . . I I I I I Signal samples (8000/s) 125 f.1s If intensity modulation is adopted, each bit is represented by the presence or absence of a pulse of light. This type of modulation is called ON-OFF keying (OOK). For frequency or phase modulation, the bits are represented by two values of frequency or phase. The modulation is then known as frequency shift keying (FSK) or phase shift keying (PSK). These modulation schemes are illustrated in Fig. 24.3-4. It is also possible to modulate the intensity of light with a harmonic function serving as a subcarrier whose amplitude, frequency, or phase is modulated by the signal (in the AM, FM, PM, FSK, or PSK format).
24.3 MODULATION AND MULTIPLEXING 1103 W 1 1 0 1 0 0 Signal OOK Modulator ) (NRZ) I I t (a) t Optical intensity OOK ) I t (RZ) t I I I 4 I 0 1 0 Signal 11111111!!' "111111111111111 ,,! FSK Modulator (b) Optical field ,I 1111111 1 ' , , 'I"" II ! 1111 " I" , , , , t I I I I I I I I I I ii" ""I "''" ""I PSK t , , , 'I , , , d , , 'I' , , 'I , , , I t I Figure 24.3-4 Examples of binary modulation of light: (a) ON-OFF keying intensity modulation (OOK/IM); (b) frequency shift keying (FSK) and phase shift keying (PSK) field modulation. B. Multiplexing Multiplexing enables the transmission and retrieval of more than one signal through the same communication link, as illustrated in Fig. 24.3-5. This is accomplished by marking each signal with a distinct physical label or a code that may be identified at the receiver. SIL ro S2 · ..... ro L o. : . M U · X S 1 + S2 + ... SN Sl · " 'I S2 ro " s 8 Figure 24.3-5 Transmission of N signals through the same channel by use of a multi- plexer (MUX) and demultiplexer (DMUX). SN. There are three standard multiplexing systems: frequency-division multiplex- ing, (FDM), time-division multiplexing (TDM), and code-division multiplexing (CDM). FDM. In FDM, carriers of distinct frequencies are modulated by the different signals. At the receiver, the signals are identified by the use of filters tuned to the carrier frequencies, as illustrated in Fig. 24.3-6( a). Signal 1 2 N Signal 1 2 ... N 1 2 ... N 1 2 ... N Frame 1 Frame 2 Frame 3 time (b) TDM II 12 (a) FDM IN frequency Figure 24.3-6 (a) In frequency-division multiplexing (FDM), a spectral band centered about a distinct frequency is allocated to each signal. (b) In time-division multiplexing, a sequence of time slots is allocated to each signal. The time slots of different signals are interleaved. TDM. In TDM, data is transmitted in a sequence of time frames, each with a set of time slots allocated to bits or bytes of the different signals, as illustrated in Fig. 24.3- 6(b). These bits must be synchronized to the same clock. At the receiver, each signal is identified by its location within the frame. An example of a hierarchical TDM system is the T-system illustrated in Fig. 24.3-7.
11 04 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS 4 kb/s %,. M Tl U X ,1.544 '" Mb/s i M -! T2 U . X 6.312 Mb/s ! M T3 ? ..U. .., X . 44.736 Mb/s M T4 ,U . X 274.176 Mb/s x24 x4 x6 x7 Figure 24.3-7 The T-system. A set of 24 4-kb/s signals are multiplexed by a TDM generating a T 1 composite signal at 1.544 Mb/s. Four such signals are multiplexed to generate a T2 signal, and so on. CDM. In CDM, each signal is assigned a code (or key) in the form of a unique function of time defined within the bit period. The code can be a sequence of one/zero bits at a much higher rate than that of the original data. Codes of different signals must be uncorrelated (orthogonal) so that they can be separated at the receiver by use of a correlator. In one encoding scheme, shown in Fig. 24.3-8, each bit "1" of the original data is replaced with the code sequence. Each receiver correlates its own code with the received signal. This locks it to only the bits associated with its own code, disregarding all other bits. I I Address !n nn n n nm Code : t I Oigina l , 1 sIgnal ----' I I I Encoded I signal : I I o 1 ) t t Figure 24.3-8 CDM encoding. Multiplexing may be electronic or optical. In electronic multiplexing, the signals are multiplexed (FDM or TDM) to generate a composite electronic signal that is used to modulate the light source in any of the optical modulation schemes discussed in Sec. 24.3A. For example, an FDM electronic signal may be generated by use of a set of carrier frequencies, called "subcarriers," to modulate the intensity of the light source (1M modulation). At the receiver, the light is detected and the demultiplexing is accomplished by use of electronic filters. Another example is a TDM electronic signal, such as the T4 signal shown in Fig. 24.3-7, used to intensity modulate the light source; the demultiplexing of the detected signal is accomplished electronically. In optical multiplexing, the labels distinguishing the multiplexed signals are optical in nature. For example, in optical FDM, different optical frequencies are used as the carriers of the various signals. These frequencies are separated at the receiver by use of optical filters. When the frequencies of the carriers in optical FDM are widely spaced (say, greater than 20 GHz) this form of optical FDM has become known as wavelength-division multiplexing (WDM). WDM systems are popular since they can be used to expand the capacity of an existing fiber network without laying more fiber. c. Wavelength-Division Multiplexing (WDM) A WDM system uses light sources of different wavelengths, each intensity modulated by a different electrical signal. The modulated light beams are mixed into the fiber
24.3 MODULATION AND MULTIPLEXING 1105 using an optical multiplexer (OMUX). Demultiplexing is implemented at the receiver end by use of an optical demultiplexer (ODMUX), which separates the different wave- lengths and directs them to different detectors. Optical multiplexers and demultiplexers are described in Sec. 23.2A. The electronic signal associated with each wavelength is often an electronically multiplexed set of other signals, and electronic demultiplexing is then necessary at the receiver end. The overall system is illustrated in Fig. 24.3-9. ro ..... ro o E/O; Ale A2 0 0 M U : e X AN. <.... Al ).2 .. AN O/E; · ro ..... .0 )( . . A I' A2' ".. </?qAN Figure 24.3-9 Wavelength division multiplexing (WDM). The spectral bands used in modem optical fiber communication systems are shown in Fig. 24.3-10, along with the attenuation of silica-glass fibers. WDM systems use any combination of wavelengths within these bands. The spacing between the wavelengths of the different channels must be sufficiently greater than the spectral widths of the modulated light in each channel, which is determined by the linewidth of the light sources, and also by the spectral widths of the data carried by the channel. The channel spacing must also be sufficiently large to permit optical multiplexing and demultiplex- ing with minimal crosstalk. 100 GHz 1231 40 : 1111111 ... III 180 Frequency (THz) 230 E O . 5 r:Q SOA 220 210 0.3 "u !+:: 4-< 80.2 s::: .9 0.1 s::: < 35nm o c L u VJ r--- \0 ...... 1600 1700 Wavelength >"0 (nm) -+ Figure 24.3-10 A 40-channel WDM system in the C spectral band, where fiber attenuation is minimal. The channel spacing is 100 GHz. 1200 1300 1400 1500 WDM systems are classified into two categories, coarse and dense, depending on the number of channels and the channel spacing. Coarse WDM (CWDM) systems use a few channels with widely spaced wave- lengths (20 nm or more). An example is a system with two wavelengths, one at 1310 nm and another at 1550 nm. CWDM is used in cable television networks, wherein different wavelengths are used for the downstream and upstream signals. The Ethernet LX-4 physical layer standard is another example in which four wavelengths near 1310 nm
1106 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS are used, each carrying a 3.125 Gb/s data stream. Metropolitan networks use CWDM systems with a 20 nm wavelength spacing. Dense WDM (DWDM) systems have a large number of channels (generally more than 8) with closely spaced wavelengths. At a wavelength of 1550 nm in the C band, a frequency spacing f1v == 200 GHz corresponds to a wavelength spacing LlA == (A/Co)f1v == 1.6 nm. DWDM systems use channel spacings as small as 50 GHz or even 25 GHz, corresponding to wavelength spacings of 0.4 nm and 0.2 nm, respec- tively. As shown in Fig. 24.3-10, the width of the C band is 35 nm, or approximately 4.4 THz. This can accommodate 40 channels with a 100-GHz (0.8 nm) spacing. DWDM tends to be used at a higher level (and higher data rates) in the communications hierar- chy, for example, on the Internet backbone. Design of DWDM systems is significantly more difficult than CWDM because the lasers need to be significantly more stable, and precision temperature control is often required to prevent wavelength drift. 24.4 FIBER-OPTIC NETWORKS A communication network comprises a set of communication links connecting mul- tiple users (terminals) distributed within some geographical area. Messages or data may be passed from one terminal to another by transmission through one or several links along paths controlled by routers and switches. A local-area network (LAN), for example, connects terminals such as computers, printers, video monitors, or faxing and copying machines in a restricted region such as a building, a campus, or a manufac- turing plant. Larger networks include the telephone network, the global Telex network, and the Internet. The network may use electrical cables, optical fibers, or satellite links. Fiber-optic networks use fiber-optic links together with electronic or optical routers and switches (see Chapter 23). A. Network Topologies and Multiple Access A network of N nodes may be constructed by use of a dedicated point-to-point link between each node and all other nodes. This requires N(N - 1) duplex (i.e., bidi- rectional) point-to-point links, and employs 2N(N - 1) transmitters and 2N(N - 1) receIvers. Topologies using fewer point -to-point links, and fewer transmitters and receivers, include the bus, the star, and the ring topologies, illustrated in Fig. 24.4-1, as well as a system for accessing the shared links. In these networks, only N transmitters and ", -v 'V 't, (c) Bus . /#... Wi :+ - .> (a) Star (b) Ring (d) Mesh Figure 24.4-1 Network topologies: (a) star, (b) bus, (c) ring, and (d) mesh. N receivers are necessary. In the star network, each node is connected to all other nodes via the star coupler at the center of the network; power transmitted by one node
24.4 FIBER-OPTIC NETWORKS 1107 is distributed equally among the other nodes. In the bus and ring networks, the fiber passes through the nodes, and data may be extracted from, or added to, the optical signal by any node. Since light transmitted by each node travels different distances to different nodes, the receivers must be able to process received power at various levels, i.e., must have a large dynamic range. A more general configuration is the mesh network. Several networks of the same or different topologies are often connected to create a larger network, as illustrated by the example in Fig. 24.4-2. Figure 24.4-2 A network comprised of ring and bus subnetworks connected by digital cross- connects (XC) at central offices. Backbone ring networks carry heavier traffic and feed access networks. Interface The interface between the terminal and the fiber network at each node includes a receiver, a transmitter and an electronic add-drop multiplexer (ADM), as illustrated in Fig. 24.4-3(a). The receiver detects the optical signal, and the ADM extracts data and adds new data that modulate a source and transmit a new optical signal through another fiber. Such interface is said to be opaque since the light is detected and regenerated at the node. A transparent interface is coupled to the fiber network optically, as illustrated in Fig. 24.4-3(b). Optical directional couplers are described in Sec. 23.1 and Sec. 23.3. An optical interface to a bidirectional (duplex) fiber uses two directional couplers to transmit and receive in either direction, as shown in Fig. 24.4-3(c). t t- - I t MUX ! . t pIE t - . 0/ - -.. (t ' 3::; -.. ....,,',...,...,.,...:'.. - ;" .;:",;,; · :t .'1 t I DMUX ",,".. t E/o . t ", _;. 01 E t "lJf;, .. 'm (a) .... U ... (b) . r r DM (c) . .. r . Figure 24.4-3 Interface between the node and the fiber network. (a) Opaque interface. The signal is converted from optical to electronic (OlE) and the ADM extracts data and adds new data, which is used to generate a new optical signal (EtO). (b) Optically coupled (transparent) interface using a directional coupler. (c) Optically coupled interface to a duplex fiber using two directional couplers.
1108 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Multiple Access The signals transmitted by the network nodes share the same fiber (the medium). To avoid confusion, a scheme for multiple access or medium access is necessary. Time- domain, frequency-domain, and code-domain multiple access systems are in use: . Time-division multiple access (TDMA) is similar to time-division multiplexing (TDM), which is used in conventional point-to-point communication systems (see Sec. 24.3B). The nodes send their data through the shared medium during interleaved time slots. Buffers may be used to store data until the appropriate time. Since it is not possible to synchronize the timing of all nodes, guard times separating consecutive slots are necessary. . Frequency-division multiple access (FDMA) is similar to frequency-division multiplexing (FDM) (see Sec. 24.3B). Here, the nodes send their data through the shared medium in preassigned spectral bands, and there is no need to synchronize the bit clocks of the input signals. In optical networks, FDMA is called wavelength-division multiple access (WDMA) and is the counterpart to wavelength-division multiplexing (WDM). . Code-division multiple access (CDMA) is similar to code-division multiplexing (CDM) (see Sec. 24.3B). In CDMA, each node is preassigned a unique address code. Data transmitted by a node is encoded with the address code of the desti- nation node. Each node correlates its own address code with the incoming signal. This locks it to only the bits associated with its own address, disregarding all other bits. The data come in a sequence of packets each with the address of its destination (see Sec. 23.3F). Synchronous Optical Network (SONET) SONET [and its international version, the Synchronous Digital Hierarchy (SDH)] is a TDM standard for transmission over optical fibers. It addresses the difficulty of time- division multiplexing of signals with slightly different clock rates by embedding these signals within frames of longer duration. The pay load (the signal bits) are allowed to float within the frames, but the frames are perfectly synchronous. SONET provides a hierarchy of multiplexed signals in which the basic unit, known as the STS-I signal or the optical carrier-1 (OC-l), transports data at 51.84 Mb/s. Combining N such signals generates the OC-N signal, which has an N times greater rate, as listed in Table 24.4- 1. For example, OC-192 and OC-768 operate at approximately 10 Gb/s and 40 Gb/s, respectively. Table 24.4-1 Transmission rates (Mb/s) in the STS hierarchy used in the SONET network. OC-1 51.84 OC-3 155.52 OC-12 622.08 OC-24 1,244.16 OC-48 2,488.32 OC-192 9,995.33 OC-768 39,813.12
24.4 FIBER-OPTIC NETWORKS 1109 EXAMPLE 24.4-1. Ring Network. An example of a fiber-optic 4-node ring network operating at different data rates is shown in Fig. 24.4-4. Each of the four nodes transmits data to the other three nodes at either the OC-12 ( 622 Mb/s) or the OC-24 ( 1.24 Gb/s) rate, as shown. The fiber segment connecting nodes 1 and 2 carries the heaviest traffic at a combined rate OC-12+0C- 12+0C-24 = OC-48 ( 2.5 Gb/s). The 2-3 and 3-4 segments carry lighter combined traffic at the OC- 24 rate. ;J . t L CD x>""' @"j: Figure 24.4-4 A 4-node ring network. Q) B. Wavelength-Division Multiplexing (WDM) Networks A wavelength-division multiplexing (WDM) fiber-optic network uses coarse or dense WDM for communication along its links and WDMA for medium access. The nodes are connected in some topology (e.g., star, ring, bus, or mesh), and each node transmits into one or several wavelength channels and receives from one or several wavelength channels. The existence of multiple wavelength channels for each physical connection adds another dimension to the network and offers some flexibility, at the expense of some complexity. Broadcast-and-Select WDM Network The simplest WDM network is the broadcast-and-select network. Each node trans- mits at a unique fixed wavelength and broadcasts its transmission to all other nodes via a passive optical couplers. The receiver in each node selects the one wavelength addressed to it by use of a tunable filter. As an example, in the 5-node network shown in Fig. 24.4-5(a), nodes 1,2, . . . ,5 transmit at wavelengths AI, A2, . . . , As, respectively. An optical star coupler broadcasts each transmission to all other nodes. In the shown state, for example, node 1 is tuned to channel As; nodes 2, 3, and 4 are tuned to channel AI; and node 5 is tuned to channel A2. As illustrated in the equivalent connection diagram in Fig. 24.4-5(b), node 2 transmits to node 5, node 5 transmits to node 1, and node 1 multicasts its transmission to nodes 2, 3, and 4. (b) Figure 24.4-5 A WDM broadcast-and- select network (a) and its equivalent logical connections (b). In another example, shown in Fig. 24.4-6(a), the receiver of each node is tuned to
1110 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS the wavelength transmitted by its next neighbor. Thus, the network, which has a star physical topology, is equivalent to a ring logical topology, as illustrated in Fig. 24.4-6 (b). ;... A \2 ....... i.... A2.4 / / / A3 (a) A2 2 Al A3 A4 (b) Figure 24.4-6 A WDM network in a star physical topology (a) is equivalent to the ring logical topology (b). (b) The network changes its state, i.e., the wavelengths to which each node is tuned, as desired. Dynamic coordination is required in order to avoid conflict and collisions. 4 3 Multi-Hop Broadcast-and-Select WDM Network The requirement that each of the nodes of the broadcast-and-select network be capable of selectively detecting any of the wavelengths transmitted by the other nodes can be demanding. This requirement is alleviated in a multi-hop network, for which each node is allocated two different wavelength channels for transmission and only two different channels for reception. At any time, a node may transmit at one of its two allocated wavelengths and may receive by tuning to one its two allocated wavelengths. The channels are allocated to the nodes in such a way that a node may access any other node by following either a single-hop (i.e., direct) connection or a two-hop connection via an intermediate node. For example, in the network shown in Fig. 24.4-7 (a), node 2 can transmit to node 1 directly via channel A3. Although node 1 cannot transmit to node 2 directly, since they share no common wavelength, this transmission may occur in two hops: node 1 transmits to node 3 on the Al channel, and node 3 subsequently transmits to node 2 on the A6 channel. This configuration is therefore called the multi- hop broadcast-and-select network. (a) (c) Figure 24.4-7 (a) A WDM multi-hop broadcast-and-select network. (b) A two-hop connection from node 1 to node 2 via node 3. (c) Logical topology of the network. The broadcast-and-select configuration, single-hop or multi-hop, is not suitable for networks with large number of nodes. Since the power transmitted by each node must reach all other nodes, the system becomes inefficient for a large number of nodes. Also the number of channels used, which must equal or exceed the number of nodes, becomes prohibitive for large networks.
24.4 FIBER-OPTIC NETWORKS 1111 Wavelength-Routed Networks In a wavelength-routed network, a pair of nodes communicates by use of one of the wavelength channels following some connection path. Another pair of nodes may use the same wavelength channel if their connection path does not share a common link with the path of the first pair. For example, in the network shown in Fig. 24.4-8( a), nodes 1 and 2 communicate on channel AI, and so do nodes 2 and 3. However, nodes 1 and 3 must use a different wavelength A2 if they use the path connecting them via node 2. Similarly, nodes 4 and 1 communicate via a third channel A3 since their path contains links that use the Al and A2 channels. In this network, each link carries one or more wavelengths (but not necessarily all of the wavelengths, as is the case in the broadcast-and-select network). For example, the link between nodes 4 and 5 carries traffic at three wavelength channels, but each of the other four links carry only two channels. Also, each node transmits and receives data at one or more wavelengths. For example, node 5 receives data from node 4 at Al and from node 3 at A2; it transmits data to node 1 at AI; data carried by channel A3 pass through this node without being detected. The logical connections in this network are shown in Fig. 24.4-8(b). (a) . Ai-_-- A, A}. A3 :- -=:- Al -'.' - ]- "' \; t 1 TL"' (b) (c) Figure 24.4-8 (a) A 5-node 3-channel wavelength-routed ring network. ( b) Its logical topology. (c) An optical-add-drop multiplexer (OADM) used at node 5. The key component in a wavelength-routed WDM network is the optical add-drop multiplexer (OADM) (see Sec. 23.2A). Each node has an OADM that extracts (drops) data from certain wavelength channels on the incoming fiber, adds data to certain channels on the outgoing fiber, and lets data on certain channels of the incoming fiber pass through without change to the outgoing fiber. An OADM is made of an optical demultiplexer (ODMUX), an add-drop multiplexer (ADM), and an optical multiplexer (OMUX). As an example, the OADM used in node 5 of the network in Fig. 24.4-8(a) is shown in Fig. 24.4-8(c). Agile networks use reconfigurable OADMs (denoted by the acronym ROADM). Wavelength-routed networks with configurations other than the ring configuration have nodes with multiple incoming and outgoing fibers. At these nodes, more complex routers are necessary. For example, a node with two incoming and two outgoing fibers, as shown in Fig. 24.4-9, employs an optical cross-connect (OX C) that receives data from selected incoming fibers/channels, adds data to selected outgoing fibers/channels, and routes data on selected incoming channels to selected outgoing channels. The oxe uses multidimensional space-domain wavelength-domain switches and ADMs (see Sec. 23.3D). A wavelength-routed network also uses a hub node which uses a server to process data at all wavelength channels.
1112 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS ':. I 1 "": : A} ; ; . :'X. : SWitch : . & Al ADM A2 .. A} m A4 ,;; t l ..-:;: ----.. ----.. A 1 ,4A 2 ,4A},4A 4 A 1 ,4A 2 ,4A},4A 4 Al .: :. ,. A2 0 M A3 U A4 X t ..-:;: ] 1 ' 1 " j;' '" 1 { f''A- Figure 24.4-9 An optical cross-connect (OXC) at a node with two incoming and two outgoing fibers, each with four wavelength channels. .. .. Add till t Drop r .. "t !. ! ! . EXAMPLE 24.4-2. WDM Upgrade of a Ring. Network. A 4-node wavelength-routed WDM ring network operates at 3 channels of wavelength AI, A2, and A3 at the rates shown in Fig. 24.4-10. This network is an upgraded version of the network in Fig. 24.4- 4. Nodes 1 and 3 access the wavelengths Al and A2; node 4 accesses the wavelengths Al and A3; and node 2 accesses all three wavelengths. In the upgraded network, the nodes communicate at twice the rates of the original network, but the highest rate in any of the WDM channels does not exceed that of the original network. The fiber segment connecting nodes 1 and 2 carries the heaviest traffic at a combined OC-96 rate ( 5 ObIs), but the highest rate at any given wavelength is OC-48 ( 2.5 Gb/s). CD > ,X l' 'x2 ,X l' 'x3 ,X I" 'x2" 'x3 ,X l' 'x2 Q) m "" Figure 24.4-10 A schematic of a 4-node 3-channel WDM ring network. 24.5 COHERENT OPTICAL COMMUNICATIONS Coherent optical communication systems use field modulation (amplitude, phase, or frequency) instead of intensity modulation. They employ coherent light sources, single- mode fibers, and heterodyne receivers. In this section we examine the principles of operation of these systems, determine their performance advantage, and briefly discuss the requirements on the components of the system. Heterodyne and Homodyne Receivers Photodetectors are responsive to the photon flux and, as such, are insensitive to the optical phase. It is possible, however, to measure the complex amplitude (both mag- nitude and phase) of the signal optical field by mixing it with a coherent reference optical field of stable phase, called the local oscillator, and detecting the superposition using a photodetector, as illustrated in Fig. 24.5-1. As a result of interference (beating)
24.5 COHERENT OPTICAL COMMUNICATIONS 1113 between the two fields, the detected electric current contains information about both the amplitude and phase of the signal field. This detection technique is called optical heterodyning, optical mixing, pho- tomixing, light beating (see Sec. 2.6B), or coherent optical detection (as opposed to direct detection. The coherent optical receiver is the optical equivalent of a superheterodyne radio receiver. The signal and local-oscillator waves usually have different frequencies (v s and VL). When V s == VL the detector is said to be a homodyne detector. Let G s == Re{A s exp(j27rv s t)} be the signal optical field, with As == IAsl exp(j'Ps) its complex amplitude and V s its frequency. The magnitude IAs I or the phase 'Ps are modulated with the signal at a rate much slower than v s. The local oscillator field is described similarly by GL, A L , VL, and 'PL. The two fields are mixed using a beam- splitter or an optica] coupler, as illustrated in Fig. 24.5-1. If the incident fields are perfectly parallel plane waves and have precisely the same polarization, the spatial dependence may be suppressed and the total field is the sum of the two constituent fields G == G s + GL. Taking the absolute square of the sum of the complex waves, we obtain IAs exp(j27rV s t) + A L exp(j27rVLt) 1 2 == {As{2 + {A L {2 + 2{As{{AL{ cos[27r(v s - VL)t + (C{Js - C{JL)]. (24.5-1) Since the intensities Is, I L , and I are proportional to the absolute-square values of the complex amplitudes, I == Is + I L + 2 J IsIL cOS[27rVIt + ('Ps - 'PL)], (24.5-2) where VI quency). V s - VL is the difference frequency (also called the intermediate fre- Signal: v s Beamsplitter Photodetector t ected L£J signal: vI Photodetector Si gvs ' rr tl ::: lel Local ----.- J : , - oscillator: vL Coupler ----.- ----.- Local oscillator: vL t (a) (b) Figure 24.5-1 Optical heterodyne detection. A signa] wave of frequency V s is mixed with a local oscillator wave of frequency VL using (a) a beamsplitter, and (b) an optical coupler. The photocurrent varies at the frequency difference VI == V s - VL. The optical power P collected by the photodetector is the product of the intensity and the detector area, so that P == Ps + P L + 2 J P sPL cOS[27rVIt + ('Ps - 'PL)], (24.5- 3) where Ps and P L are the powers of the signal and the local oscillator beams, respec- tively. The third term of (24.5-3) varies with time at the difference frequency VI with a phase 'Ps - 'PL. If the signal and local oscillator beams are close in frequency, their difference VI can be many orders of magnitude smaller than the individual frequencies. Misalignment between the directions of the two waves reduces or washes out the interference term [the third term of (24.5-3)], since the phase 'Ps - 'PL then varies
1114 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS sinusoidally with position within the area of the detector. As is readily understood from Fig. 2.5-4, this can be avoided by keeping the angle () between the wavefronts sufficiently small, such that () « AI a where a is the size of the photodetector aperture. The photocurrent i generated in a semiconductor photon detector is proportional to the incident photon flux <I> (see Sec. 18.1B). When VI is much smaller than V s and VL, the superposed light is quasi-monochromatic and the total photon flux <I> == PI h v is proportional to the optical power, where v == (vs + VL). The mean photocurrent is therefore'i == Ile<I> == (Ilelh v )P, where e is the electron charge and Il the detector's quantum efficiency, so that 'i == 'is + 'iL + 2 V'is1:L cos [21TvIt + ('Ps - 'P L)] , (24.5-4) where 1:s == IlePslh v and'iL == IlePLlh v are the photocurrents generated by the signal and local oscillator individually. The local oscillator is usually much stronger than the signal, so that the first term in (24.5-4) is negligible and 'i ';::j 'i L + 2 V 'is 'i L cos [ 21T VI t + ('P s - 'P L ) ] . (24.5-5) Photomixing Current The time dependence of the detected current 'i is sketched in Fig. 24.5- 2( a). The second term in (24.5-5), which oscillates at the difference frequency VI, carries the useful information. With knowledge of'iL and 'PL, the amplitude and phase of this term can be determined, and 'is and 'Ps estimated, from which the intensity and phase (and hence the complex amplitude) of the measured optical signal can be inferred. The information-containing signal variables 1:s or 'Ps are usually slowly varying functions of time in comparison with the difference frequency VI, so they act as slow modulations of the amplitude and the phase of the harmonic function 2yffi COS(21TVIt - 'PL). This amplitude- and phase-modulated current can be demodulated by drawing on the conventional techniques used in AM and PM radio receivers. (a) Uf IL _! ---+I Ijvf "- ..... (b) ___u_- t i L J t t Figure 24.5-2 (a) Photocurrent generated by the heterodyne detector. The envelope and phase of the time-varying component carries complete information about the complex amplitude of the optical field representing the signal. (b) Photocurrent generated by the homodyne detector. From a photon-optics point of view, this process can be understood in terms of the detection of polychromatic (two-frequency) photons (see Prob. 12.1-11). The homodyne system is a special case of the heterodyne system for which V s == VL and VI == O. The demodulation process is different. A phase-locked loop is used to lock the phase of the local oscillator so that 'P L == 0 and (24.5-5) yields 'i == 'iL + 2 V'is'iL COS 'Ps. (24.5-6) Amplitude and phase modulation is achieved by varying 1:s and 'Ps, respectively.
24.5 COHERENT OPTICAL COMMUNICATIONS 1115 Advantages of Heterodyne/Homodyne Receivers In comparison with the direct-detection receiver, the heterodyne receiver has the fol- lowing advantages: . It is capable of measuring the optical phase and frequency. . It permits the use of wavelength-division multiplexing (WDM) with smaller chan- nel spacing ( 100 MHz). In conventional direct-detection systems the channel spacing is of the order of 100 GHz. . It permits the use of electronic equalization to compensate for pulse broadening in the fiber. Pulse broadening is a result of the dephasing of the different wave- length/frequency components because of differences in group velocities. Since the receiver monitors the phase, this dephasing may be removed by proper electronic filtering. . By use of a strong reference field, the heterodyne receiver has an inherent noise- less gain conversion factor that effectively amplifies the signal above the circuit noise level. . It provides a 3-dB SNR advantage over even the noiseless direct-detection re- ceiver, as shown in Sec. 24.5. . It is insensitive to unwanted background light with which the local oscillator does not mix. Heterodyning is one of the few ways of attaining photon-noise-limited detection in the infrared, where background noise is so prevalent. The cost of these advantages is an increase in the system's complexity since hetero- dyning requires a stable local oscillator, an optical coupler in which the mixed fields are precisely aligned, and complex circuits for phase locking. Coherent Systems An essential condition for the proper mixing of the local oscillator field and the re- ceived optical field is that they must be locked in phase, be parallel, and have the same polarization in order to permit interference to take place. This places stringent requirements on the two lasers and on the fiber. The lasers must be single-frequency and have minimal phase and intensity fluctuations. The local oscillator is phase-locked to the received optical field by means of a control system that adjusts the phase and fre- quency of the local oscillator adaptively (using a phase-locked loop). The fiber must be single-mode (to avoid modal noise). The fiber must also be polarization-maintaining, or the receiver must contain an adaptive polarization-compensation system. A schematic diagram of a coherent optical fiber communication system using two lasers and phase modulation is shown in Fig. 24.5-3. The local oscillator field is mixed with the received optical field using an optical directional coupler. One branch of the coupler output contains the sum of the two optical fields and the other branch contains the difference. Using (24.5-4), the detected currents 'l:l:: == 'ls + 'lL ::t: 2 V'ls'lL COS[21TV[t + (CPs - CPL)] (24.5- 7) are subtracted electronically, yielding 4 V'ls'lL COS[21TV[t+ (CPs -CPL)], which is demod- ulated to recover the message. This type of coherent receiver is known as a balanced mixer. It has the advantage of canceling out intensity fluctuations of the local oscillator. A number of coherent optical fiber communication systems have been implemented at Ao == 1550 nm (where fiber attenuation is minimal) with bit-rate-distance products matching theoretical expectations. One example is provided by a system operating at a bit rate 4 Obis. A DFB laser with a I5-MHz CW linewidth was directly modulated in an FSK signal format. The local oscillator was a tunable DBR laser (see Sec. 17.3C). This system exhibited a receiver sensitivity 190 photonslbit and was used for transmission over a 160-km length of fiber.
1116 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Phase modulator Polarization controller :. ::, 0 Coupler Balanced mixer Phase I_ ! Aplifier de . tector . :. . . .. ReceIved 'I'; si g nal - - Transmitted signal --.. !llt DFB laser Single-mode fiber .. f I( Tunable DFB laser .: Frequency .. lock Figure 24.5-3 Coherent optical fiber communication system. * Performance of Analog Coherent Communication Systems Heterodyne detection is necessary whenever the phase of the optical field is to be measured. However, heterodyne detection can also be useful for measuring the optical intensity since it provides gain through the presence of the strong local oscillator. As such, it offers an alternative to both optical amplification (see Chapter 14 and Sec. 17.2) and APD amplification (see Sec. 18.4). This can provide a signal-to-noise ratio advantage over direct detection, as we show in this section. The mean photocurrent 'i generated by a photodiode is accompanied by noise of varIance 2 2 - B 2 ai == e'l + a r , (24.5-8) where B is the receiver's bandwidth; the first term is due to photon noise and the second represents circuit noise (see Sec. 18.6). The intensity of the local oscillator can be made sufficiently large so that even if the signal is weak, the total current 'i is such that the circuit noise a; is negligible in comparison with the photon noise 2e'iB. Assuming that 'iL 'is and 2e'i L B » a;, we use (24.5-5) and approximate (24.5-8) by 'i 'iL + 2 V'is'iL cos[27rv]t + (<.ps - <.pL)] a; 2e'i L B. (24.5-9a) (24.5-9b) In the case of amplitude modulation, the signal is represented by the RMS value of the sinusoidal waveform in (24.5-9a), with the phase ignored. The electrical signal power is therefore! [2 y1'is'iLJ 2 == 2'is'iL and the noise power is a; == 2e'i L B, so that the power signal-to-noise ratio is 2'is'iL 'is SNR == == -. 2e'i L B eB (24.5-10) If m == 'i/2Be is the mean number of photoelectrons counted in the resolution time interval T == 1/2B, then SNR == 2 m . (24.5-11 ) Signal-to-Noise Ratio Heterodyne Receiver In comparison, the SNR of the direct-detection photodiode receiver measuring the
24.5 COHERENT OPTICAL COMMUNICATIONS 1117 same signal current 'is without the benefit of heterodyning is -2 'l SNR == s 2e'isB + a; -2 m m + a ' (24.5-12) where a == (a r /2Be)2 is the circuit-noise parameter discussed in Sec. 18.6C. The principal advantage of the heterodyne system is now apparent. For strong light or low circuit noise ( m » a), the direct-detection result is SNR == m . The heterodyne receiver, which yields SNR == 2 m , offers a factor-of-2 improvement (3-dB advantage). But for weak light (or large circuit noise) the advantage can be even more substantial, since the heterodyne receiver has SNR == 2 m , whereas the SNR of the direct-detection receiver is reduced by circuit noise to SNR == m / (1 + a / m ) . The performance of a direct-detection avalanche photodiode receiver is also inferior to that of a heterodyne photodiode receiver. In accordance with (18.6-36), the SNR obtained when the APD gain is sufficiently large to overcome circuit noise is m SNR == F ' (24.5-13 ) where F is the APD excess noise factor (F > 1). Therefore, even a noiseless APD receiver (F == 1) is a factor of 2 inferior to the heterodyne receiver. * Performance of Digital Coherent Communication Systems In this section the performance and sensitivity of a digital coherent communication system are determined in the cases of amplitude and phase modulation. ON-OFF keying (OOK) homodyne system. Consider an ON-OFF keying (OOK) system transmitting data at a rate Bo bits/s and using a homo dyne receiver. Bits "1" and "0" are represented by the presence and absence of the signal 'is during the bit time T == 1/ Bo, respectively. Assuming that <Ps == <PL == 0 and VI == V s - VL == 0, the measured current has the following means and variances obtained from (24.5-9a) and (24.5-9b ): mean 111 'iL + 2 V'iL'is , variance ar 2'iLeB for bit "1" mean 110 'iL, variance a6 2'iLeB for bit "0." (24.5-14 ) The receiver bandwidth B == Bo/2 since the bit time T == 1/ Bo is the sampling time 1/2B for a signal of bandwidth B. The performance of the binary communication system under the Gaussian approx- imation has been discussed in Sec. sec22-4. The bit error rate is given by (18.6-55), where Q = /11 - /10 = J 1:s = vm a1 + ao 2eB ' (24.5-15) and m == 'is/2eB is the mean number of detected photoelectrons in bit 1. For a bit error rate BER == 10- 9 , Q 6 and therefore m == 36, corresponding to a receiver sensitivity m a == ! m == 18 photoelectrons per bit (averaged over both bits).
1118 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS Table 24.5-1 Receiver sensitivity for different receivers and modulation systems under ideal conditions (photons per bit). OOK PSK FSK 10 Homodyne 18 9 Heterodyne 36 18 36 Direct Detection Phase-shift-keying (PSK) homodyne system. Here, bits "I" and "0" are repre- sented by a phase shift CPs == 0 and 7r, respectively. Assuming that cP L == 0, the means and variances of the photocurrent for bits" I " and "0" are, from (24.5-9), mean Ml == 'lL + 2 y1'lL'ls , variance ai == 2e'lLB for bit "]" mean Uo == 'lL - 2 y1'lL'ls , variance a6 == 2e'lLB for bit "0" and therefore Q = /11 - /10 = 2 V 'is = 2 vrn . al + ao 2eB (24.5-16) For a BER == 10- 9 , Q == 6, from which m == 9. Since each of the two bits must carry an average of nine photoelectrons in this case, the average number of photoelectrons per bit is ma == m == 9. It follows that the receiver sensitivity is 9 photoelectronslbit. The PSK homodyne receiver is twice as sensitive as the OOK homodyne receiver because it requires half the number of photoelectrons. Comparison. The sensitivity of the heterodyne digital receiver can be determined by following a similar analysis. Table 24.5-1 lists the receiver sensitivities of several digital modulation systems, assuming It == 1. Although it appears that the direct- detection OOK system has about the same performance as the best coherent system (homodyne PSK), in practice this is not so. In the homodyne system, circuit noise is overcome, whereas in the direct-detection system, circuit noise cannot be ignored, unless an APD is used. When an APD is used in a direct-detection receiver, circuit noise is overcome, but the APD gain noise raises the receiver sensitivity from 10 to at least 10F, where F is the excess-noise factor. Direct-detection systems would have performance comparable to coherent-detection systems if a perfect APD with F == 1 (no excess noise) were available. READING LIST Books on Optical Fiber Communications See also the reading lists in Chapters 8, 9, 17, 18, 22, and 23. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 1991, 4th ed. 2006. S. Ramachandran, Fiber-Based Dispersion Compensation, Springer-Verlag, 2006. J. N. Damask, Polarization Optics in Telecommunications, Springer-Verlag, 2005. K.-P. Ho, Phase-Modulated Optical Communication Systems, Springer-Verlag, 2005. J. C. Palais, Fiber Optic Communications, Prentice Hall, 5th ed. 2005. A. Galtarossa and C. R. Menyuk, eds., Polarization Mode Dispersion, Springer-Verlag, 2005. C.-F. Lin, Optical Components for Communications: Principles and Applications, Springer-Verlag, 2004.
READING LIST 1119 H. Kolimbiris, Fiber Optics Communications, Prentice Hall, 2004. T. Schneider, Nonlinear Optics in Telecommunications, Springer-Verlag, 2004. M. Cvijetic, Optical Transmission Systems Engineering, Artech, 2004. G. P. Agrawal, Fiber-Optic Communication Systems, Wiley, 3rd ed. 2002. I. P. Kaminow and T. Li, eds., Optical Fiber Telecommunications IVA: Components, Academic Press, 2002. I. P. Kaminow and T. Li, eds., Optical Fiber Telecommunications IVB: Systems and Impairments, Academic Press, 2002. C. DeCusatis, ed., Fiber Optic Data Communication: Technological Trends and Advances, Academic Press, 2002. C. DeCusatis, Handbook of Fiber Optic Data Communication, Academic Press, 2nd ed. 2002. R. L. Freeman, Fiber-Optics Systems for Telecommunications, Wiley, 2002. E. W. Van Stryland and M. Bass, eds., Fiber Optics Handbook: Fiber, Devices, and Systems for Optical Communications, McGraw-Hill, 2002. N. Grote and H. Venghaus, eds., Fibre Optic Communication Devices, Springer-Verlag, 2001. D. K. Mynbaev and L. L. Scheiner, Fiber-Optic Communications Technology, Prentice Hall, 2001. G. Mahlke and P. Gassing, Fiber Optic Cables: Fundamentals, Cable Design, System Planning, Wi ley- V CH, 4th revised and enlarged ed. 2001. G. Keiser, Optical Fiber Communications, McGraw-Hill, 3rd ed. 2000. R. Sabella and P. Lugli, High Speed Optical Communications, Kluwer, 1999. G. Guekos, ed., Photonic Devices for Telecommunications: How to Model and Measure, Springer- Verlag, 1999. H. J. R. Dutton, Understanding Optical Communications, Prentice Hall, 1999. G. Lachs, Fiber Optic Communications: Systems, Analysis, and Enhancements, McGraw-Hill, 1998. I. P. Kaminow and T. L. Koch, eds., Optical Fiber Telecommunications IlIA: Components, Academic Press, 1997. I. P. Kaminow and T. L. Koch, eds., Optical Fiber Telecommunications IIIB: Systems and Impair- ments, Academic Press, 1997. S. B. Alexander, Optical Communication Receiver Design, SPIE Optical Engineering Press, 1997. L. Kazovsky, S. Benedetto, and A. E. Willner, Optical Fiber Communication Systems, Artech, 1996. R. M. Gagliardi and S. Karp, Optical Communications, Wiley, 1976, 2nd ed. 1995. S. Ryu, Coherent Lightwave Communication Systems, Artech, 1995. S. Shimada, ed., Coherent Lightwave Communications Technology, Chapman & Hall, 1995. J. M. Senior, Optical Fiber Communications: Principles and Practice, Prentice Hall, 2nd ed. 1992. J. E. Midwinter, Optical Fibers for Transmission, Wiley, 1979; Krieger, reissued 1992. S. E. Miller and I. P. Kaminow, eds., Optical Fiber Telecommunications II, Academic Press, 1988. C. K. Kao, Optical Fibre, Institution of Electrical Engineers, 1988. S. D. Personick, Fiber Optics: Technology and Applications, Plenum, 1985. C. K. Kao, Optical Fiber Systems, McGraw-Hill, 1982. S. E. Miller and A. G. Chynoweth, eds., Optical Fiber Telecommunications, Academic Press, 1979. B. E. A. Saleh, Photoelectron Statistics with Applications to Spectroscopy and Optical Communica- tion, Springer-Verlag, 1978. Books on Fiber-Optic Networks P. R. Prucnal, ed., Optical Code Division Multiple Access: Fundamentals and Applications, Taylor & Francis/CRC Press, 2006. B. Mukherjee, Optical WDM Networks, Springer-Verlag, 2006. E. Desurvire, Wiley Survival Guide in Global Telecommunications: Broadband Access, Optical Com- ponents and Networks, and Cryptography, Wiley, 2004. E. Desurvire, Wiley Survival Guide in Global Telecommunications: Signaling Principles, Protocols, and Wireless Systems, Wiley, 2004. G. Bernstein, B. Rajagopalan, and D. Saha, Optical Network Control: Architecture, Protocols, and Standards, Addison-Wesley, 2004.
1120 CHAPTER 24 OPTICAL FIBER COMMUNICATIONS D. Greenfield, The Essential Guide to Optical Networks, Prentice Hall, 2002. R. J. Bates, Optical Switching and Networking Handbook, McGraw-Hill, 2001. A. Jukan, QoS-Based Wavelength Routing in Multi-Service WDM Networks, Springer-Verlag, 2001. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective, Morgan Kaufmann, 2nd. ed. 2002. M. T. Fatehi and M. Wilson, Optical Networking with WDM, McGraw-Hill, 2002. C. D. Chaffee, Building the Global Fiber Optics Superhighway, Plenum, 2001. T. E. Stem and K. Bala, Multiwavelength Optical Networks: A Layered Approach, Prentice Hall, 1999. A. Bononi, Optical Networking, Springer-Verlag, 1999. P. E. Green, Fiber Optic Networks, Prentice Hall, 1993. Articles Issue on optical communications, IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 4, 2006. Issue on optical communications, IEEE Journal of Selected Topics in Quantum Electronics, vol. 10, no. 2, 2004. Millennium issue, IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 6, 2000. D. L. Begley, ed., Selected Papers on Free-Space Laser Communications 11, SPIE Optical Engineer- ing Press (Milestone Series Volume 100), 1994. E. G. Rawson, ed., Selected Papers on Fiber Optic Local Area Networks, SPIE Optical Engineering Press (Milestone Series Volume 91), 1994. L. D. Hutcheson and S. C. Mettler, eds., Selected Papers on Fiber Optic Communications, SPIE Optical Engineering Press (Milestone Series Volume 88), 1993. S. F. Jacobs, Optical Heterodyne (Coherent) Detection, American Journal of Physics, vol. 56, pp.235-245, 1988. P. R. Prucnal, M. A. Santoro, and T. R. Fan, Spread Spectrum Fiber Optic Local Area Network Using Optical Processing, IEEE Journal of Lightwave Technology, vol. LT-4, pp. 547-554, 1986. M. C. Teich, Laser Heterodyning, Optica Acta (Journal of Modern Optics), vol. 32, pp. 1015-1021, 1985. PROBLEMS 24.1-1 Optical Fiber Systems. Discuss the validity of each of the following statements and indi- cate the conditions under which your conclusion is applicable. (a) The wavelength.A o = 1300 nm is preferred to .Ao = 870 nm for all optical fiber commu- nication systems. (b) The wavelength .Ao = 1550 nm is preferred to .Ao = 1300 nm for all optical fiber communication systems. (c) Single-mode fibers are superior to multimode fibers because they have lower attenuation coefficients. (d) There is no pulse spreading at .Ao 1312 nm in silica-glass fibers. (e) Compound semiconductor devices are required for optical fiber communication sys- terns. (f) APDs are noisier than p-i-n photodiodes and are therefore not useful for optical fiber systems. 24.1-2 Components for Optical Fiber Systems. The design of an optical fiber communication system involves many choices of sources, fibers, amplifiers, and detectors, some of which are shown in Fig. 24.2-3. Make appropriate choices for each of the applications listed below. More than one answer may be correct. Some choices, however, may be incompatible. (a) A transoceanic cable carrying data at a 2.5 Gb/s rate with 100-km repeater spacings. (b) A 1-m cable transmitting analog data from a sensor at 1 kHz.
PROBLEMS 1121 (c) A link for a computer local-area network operating at 500 Mb/s. (d) A l-km data link operating at 100 Mb/s with ::i::50°C temperature variations. 24.2-1 Performance of a Plastic Fiber Link. A short-distance low-data-rate communication sys- tem uses a plastic fiber with attenuation coefficient 0.5 dB/m, an LED generating 1 m W at a wavelength of 870 nm, and a photodiode with receiver sensitivity 20 dBm. Assuming a power loss of 3 dB each at the input and output couplers, determine the maximum length of the link. Assume that the data rate is sufficiently low so that dispersion effects play no role. 24.2-2 Maximum Length of Attenuation-Limited System. An optical fiber communication link is designed for operation at 10 Mb/s. The source is a 100-pW LED operating at 870 nm and the fiber has an attenuation coefficient of 3.5 dB Ikm. The fiber is made of I-Ian segments and connectors between segments have a loss of 1 dB each. Input and output couplers each introduce a loss of 2 dB. The safety margin is 6 dB. Two receivers are available, a Si p-i-n photodiode receiver with sensitivity 5000 photons per bit, and a Si APD with sensitivity 125 photons per bit. Determine the receiver sensitivity Pr (dBm units) and the maximum length of the link for each receiver. 24.2-3 Maximum Data Rate of Attenuation-Limited System. A 50-km optical fiber link is operated at a wavelength of 1550 nm. The source is a 2-mW InGaAsP laser and the fiber has attenuation coefficient of 0.2 dB Ikm. Connectors and couplers introduce a total loss of 8 dB and the safety margin is 6 dB. The receiver is an InGaAs APD with a sensitivity of 1 000 photons per bit for a bit error rate of 1 0- 9 . Determine the maximum data rate that can be used assuming an attenuation-limited system. If the required error rate is 10- 11 , what is the maximum data rate? 24.2-4 Maximum Length of an Analog Link. An optical fiber communication link uses intensity modulation to transmit data at a bandwidth B 10 MHz with a signal-to-noise ratio of 40 dB. The source is a Ao 870 nm light-emitting diode producing 100 pW of average power with a maximum modulation index of 0.5. The fiber is a multimode step-index fiber with an atte n uation coefficient of 2.5 dB Ikm. The detector is an avalanche photodiode with mean gain G 100, excess noise factor F 5, and responsivity 0.5 (excluding the gain). Assuming that the circuit noise is negligible, use the theory presented in Sec. 18.6D to calculate the optical power sensitivity of the receiver and the attenuation-limited maximum length L of the fiber. 24.2-5 Time Budget for Dispersion-Limited System. A 100-km single-mode fiber link operating at a wavelength of 1550 nm. The source is an InGaAsP laser diode of spectral width 0.2 nm and response time 20 ps. The fiber has a dispersion coefficient 17 ps/km-nm. The receiver uses an InGaAs APD and has a response time of 0.1 ns. Determine the maximum data rate based on the criterion that the response time of the fiber does not exceed 25 % of the bit time. Also, determine the maximum data rate using the criterion that the response time of the overall system does not exceed 70 % of the bit time. If the a dispersion-shifted fiber is used instead, so that the dispersion coefficient is reduced to 1 ps/km-nm, what are the maximum data rates under the two criteria? 24.3-1 Number ofWDM Channels. How many WDM channels fit in the C band (1530-1565 nm) and in the 0 band (1260-1360 nm) if the channel spacing is 75 GHz? 24.3-2 Number of Nodes in a Broadcast-and-Select WDM Network. The maximum number of nodes N that can be used by a broadcast-and-select WDM network is often limited by the available optical power. Determine N for a local area network using an optical star coupler connected to each of the nodes by a fiber of 2-km length, 0.3 dB/km attenuation coefficient, and 1 dB of connector loss. The star coupler distributes the power equally among its outputs and introduces an additional loss of 3 dB. Each node uses a I-mW optical source, the receiver sensitivity is -35 dBm, and a 5-dB safety margin is assumed. 24.3-3 Wavelength-Routed WDM Ring Network. Consider a 4-node 6-channel WDM network. Each node uses an add-drop multiplexer to transmit or receive at any of three different wavelengths assigned to it, but passes through the other three wavelengths. For example, node I may add or drop data at channels At, A2, or A3, but passes through data at A4, As, and A6. Allocate sets of three add-drop channels to each of the nodes 2, 3, and 4, such that any node on the ring may communicate with any of the other nodes. The idea is that each node must have one add-drop channel common with each of the other three nodes, but this channel must not be common with nodes in-between.
APPENDIX This appendix provides a brief review of the Fourier transform, and its properties, for functions of one and two variables. A.1 ONE-DIMENSIONAL FOURIER TRANSFORM The harmonic function F exp j27rllt plays an important role in science and engineer- ing. It has frequency 1I and complex amplitude F. Its real part F cos 27rllt + arg F is a cosine function with amplitude F and phase arg F . The variable t usually represents time; the frequency 1I has units of cycles/s or Hz. The harmonic function is regarded as a building block from which other functions may be obtained by a simple superposition. In accordance with the Fourier theorem, a complex-valued function f t , satisfying some rather unrestrictive conditions, may be decomposed as a superposition integral of harmonic functions of different frequencies and complex amplitudes, CX) f t F 1I exp j27rvt dll. (A. 1-1 ) Inverse Fourier Transform -CX) The component with frequency v has a complex amplitude F v given by CX) Fv f t exp j27rvt dt. (A.1-2) Fourier Transform -CX) F v is termed the Fourier transform of f t , and f t is the inverse Fourier trans- form of F v . The functions f t and F v form a Fourier transform pair; if one is known, the other may be determined. In this book we adopt the convention that exp j27rllt is a harmonic function with positive frequency, whereas exp j27rvt represents negative frequency. The opposite convention is used by some authors who define the Fourier transform in (A. 1-2) with a positive sign in the exponent, and use a negative sign in the exponent of the inverse Fourier transform (A. 1-1 ). In communication theory, the functions f t and F 1I represent a signal, with f t its time-domain representation and F 1I its frequency-domain representation. The squared-absolute value f t 2 is called the signal power, and F v 2 is the energy 112
A.1 ONE-DIMENSIONAL FOURIER TRANSFORM 1123 spectral density. If F v 2 extends over a wide frequency range, the signal is said to have a wide bandwidth. Properties of the Fourier Transform Some important properties of the Fourier transform are provided below. These proper- ties can be proved by direct application of the definitions (A. I-I) and (A. 1-2) (see any of the books in the reading list). . Linearity. The Fourier transform of the sum of two functions is the sum of their Fourier transforms. . Scaling. If I t has a Fourier transform F v , and T is a real scaling factor, then I t T has a Fourier transform T F TV . This means that if I t is scaled by a factor T, its Fourier transform is scaled by a factor 1 T. For example, if T > 1, then I t T is a stretched version of It, whereas F TV is a compressed version of F V . The Fourier transform of I t is F v. . Time Translation. If I t has a Fourier transform F v , the Fourier transform of I t T is exp j27rVT F v . Thus delay by time T is equivalent to multiplica- tion of the Fourier transform by a phase factor exp j27rVT. . Frequency Translation. If F v is the Fourier transform of It, the Fourier trans- form of I t exp j27rvat is F v Va. Thus multiplication by a harmonic func- tion of frequency Va is equivalent to shifting the Fourier transform to a higher frequency Va. . Symmetry. If I t is real, then F v has Hermitian symmetry [i.e., F v F* v ]. If I t is real and symmetric, then F v is also real and symmetric. . Convolution Theorem. If the Fourier transforms of 11 t and 12 tare F 1 v and F 2 v , respectively, the inverse Fourier transform of the product Fv F 1 V F 2 V (A.1-3) . IS 00 I t 11 T 12 t T dT. (A. 1-4) Convolution -00 The operation defined in (A. 1-4) is known as the convolution of 11 t with 12 t . Convolution in the time domain is therefore equivalent to multiplication in the Fourier domain. . Correlation Theorem. The correlation between two complex functions is defined as 00 I t I; T 12 t + T dT. (A.1-5) Correlation -00 The Fourier transforms of 11 t , 12 t , and 1 t are related by Fv F{ V F 2 V . (A.1-6)
1124 APPENDIX A FOURIER TRANSFORM . Parseval's Theorem. The signal energy, which is the integral of the signal power f t 2, equals the integral of the energy spectral density F v 2, so that 00 ex) f t 2 dt F v 2 dv. (A. 1-7) Parseval's Theorem -00 -00 Examples The Fourier transforms of some important functions used in this book are listed in Table A.2-1. By use of the properties of linearity, scaling, delay, and frequency trans- lation, the Fourier transforms of other functions may be readily obtained. In this table: width centered about t o. . 8 t is the impulse function (Dirac delta function), defined as 8 t lima oo a rect at . It is the limit of a rectangular pulse of unit area as its width approaches zero (so that its height approaches infinity). . sine t sin 7rt 1ft is a symmetric function with a peak value of 1.0 at t 0 and zeros at t :I:: 1, :1::2, . . .. A.2 TIME DURATION AND SPECTRAL WIDTH It is often useful to have a measure of the width of a function. The width of a function of time f t is its time duration and the width of its Fourier transform F v is its spectral width (or bandwidth). Since there is no unique definition for the width, a plethora of definitions are in use. All definitions, however, share the property that the spectral width is inversely proportional to the temporal width, in accordance with the scaling property of the Fourier transform. The following definitions are used at different places in this book. The Root-Mean-Square Width The root-mean-square (rms) width at of a nonnegative rea] function f t is defined by a 2 t , - where t (A.2-1) - If f t represents a mass distribution (t representing position), then t represents the centroid and at the radius of gyration. If f t is a probability density function, these . quantities represent the mean and standard deviation, respectively. As an example, the Gaussianfunction f t exp t 2 2a; has an S width at. Its Fourier transform is given by F v 1 27r a v exp v 2 2a ' where a v 1 27rat (A.2-2) is the S spectral width.
A.2 TIME DURATION AND SPECTRAL WIDTH 1125 Table A.2-1 Selected functions and their Fourier transforms. Function I(t) F(v) - -- --- Uniform J o 1 8(v) I o 1J . -- - -- Impulse 8(t) 1 I o lJ o t Rectangular reet (t) sinc (v) I -I 0 -1/2 0 1/ 2 t lJ Exponential a exp (-Itl) 2 1 +(21f1l)2 -I 0 I t -I 0 1 lJ Gaussian exp (-7ft 2 ) exp (-7TV 2 ) -1 0 ] t -1 0 1 l/ Hyperbolic secant sech (7ft) sech (7flJ) -I 0 ) t -I 0 1 v Chirpb exp (j7ft 2 ) e j1f/4 exp (j1ft 2 ) 101 r 1 0 1 v M = 2S+ 1 Impulses -) 0 J S 1:.8(t - m) r m=-S sin (M7fv) sin (7rlJ) lJ Comb . . . . 00 1:.8(t - m) 00 1:.8(v - m) . . . . m = -00 m = -00 -I 0 1 v -I 0 1 r aThe double-sided exponential function is shown. The Fourier transform of the si ngle-sided exponential, J(t) exp( t) with t > 0, is F(v) 1/[1 + j21rv]. Its magnitude is 1/ -J 1 + (21rv)2. bThe functions cos(1rt 2 ) and cos(1rV 2 ) are shown. The function sin(1rt 2 ) is shown in Fig. 4.3-6. This definition is not appropriate for functions with negative or complex values. For such functions the RMS width of the squared-absolute value f t 2 is used, a 2 t 00 t -ex) l 2 f t 2 dt f t 2 dt , where - t ex) -ex) oo f t 2 dt We call this version of at the power-rms width. With the help of the Schwarz inequality, it can be shown that the product of the power RMS widths of an arbitrary function f t and its Fourier transform F v must
1126 APPENDIX A FOURIER TRANSFORM be greater than 1 47r, 1 (A.2-3) Duration Bandwidth Reciprocity Relation where the spectral width a 1/ is defined by , where iJ 00 II F II 2 dll -00 CX) · F II 2 dll -00 00 II iJ 2 F II 2 dll a 2 y -00 00 F II 2 dll -00 Thus the time duration and the spectral width cannot simultaneously be made ar- bitrarily small. The Gaussian function f t exp t 2 4a; , for example, has a power-rms width at. Its Fourier transform is also a Gaussian function, F II 1 2 7r a 1/ exp lI 2 4a , with power-rms width a y 1 47rat . (A.2-4) Since ata1/ 1 47r, the Gaussian function has the minimum permissible value of the duration bandwidth product. In terms of the angular frequency w 27rll, 1 (A.2-5) If the variables t and w, which usually describe time and angular frequency (rad s), are replaced with the position variable x and the spatial angular frequency k rad m , respectively, then (A.2-5) translates to 1 a xak > · 2 (A.2-6) In quantum mechanics, the position x of a particle is described by the wavefunction 'ljJ x , and the wavenumber k is described by a function <p k which is the Fourier transform of 'ljJ x . The uncertainties of x and k are the S widths of the probability densities 'ljJ x 2 and <p k 2, respectively, so that a x and ak are interpreted as the uncertainties of position and wavenumber. Since the particle momentum is p hk (where n h 21r and h is Planck's constant), the position-momentum uncertainty product satisfies the inequality n (A.2- 7) Heisenberg Uncertainty Relation which is known as the Heisenberg uncertainty relation. The Power-Equivalent Width The power-equivalent width of a signal f t is the signal energy divided by the peak signal power. If f t has its peak value at t 0, for example, then the power-equivalent
A.2 TIME DURATION AND SPECTRAL WIDTH 1127 width is -00 2 2 dt. (A.2-8) 7 00 f t fO The double-sided exponential function f t exp t 7, for example, has a power-equivalent width 7, as does the Gaussian function f t exp 7rt 2 27 2 . This definition is used in Sec. 11.1, where the coherence time of light is defined as the power-equivalent width of the complex degree of temporal coherence. The power-equivalent spectral width is similarly defined by 13 00 F v 2 2 dv. FO (A.2-9) -ex) If f t is real, so that F v 2 is symmetric, and if it has its peak value at v 0, the power-equivalent spectral width is usually defined as the positive -frequency width, 00 F v 2 o (A.2-10) B In the case F v 7 1 + j27rV7 , for example, B 1 4T . (A.2-11) This definition is used in Sec. 18.6A to describe the bandwidth of photodetector circuits susceptible to photon and circuit noise (see also Problem 18. 5- 5). may be written in the form dt, (A.2-10) B 1 2T' (A.2-12) where 2 dt T (A.2-13) is yet another definition of the time duration [the s q uare of the area under f t divided The 1/e-, Half-Maximum, and 3-dB Widths Another type of measure of the width of a function is its duration at a prescribed fraction of its maximum value (1 2, 1 2, 1 e, or 1 e 2 , as examples). Either the half- width or the fuII width on both sides of the peak is used. Two commonly encountered measures are the fuII-width at half-maximum ( HM) and the half-width at 1 2- maximum, cdlled the 3-dB width. The following are three important examples: . The exponential function f t exp t 7 for t > 0 and f t 0 for t < 0, which describes the response of a number of electrical and optical systems, has a
1128 APPENDIX A FOURIER TRANSFORM 1 e-maximum width tl/e T. The magnitude of its Fourier transform F v T 1 + j27rVT has a 3-dB width (half-width at 1 2-maximum) V3-dB 1 27rT. (A.2-14) . The double-sided exponential function f t exp t T has a half-width at 1 e-maximum tl/e T. Its Fourier transform F v 2T 1 + 27rVT 2 , known as the Lorentzian distribution, has a full-width at half-maximum VFWHM 1 (A.2-15) , 7r7 and is usually written in the form F v v 27r v 2 + v 2 2 where v VFWHM. The Lorentzian distribution describes the spectrum of certain light emissions (see Sec. I3.3D). . The G au ssian function f t exp t 2 2T 2 has a full-w id th at 1 e-maximum width at 1 e-maximum Vl/e 2 (A.2-16) 7rT and a full-width at half-maximum VFWHM 21n2 , (A.2-17) 7r7 so that VFWHM In 2 Vl/e 0.833 Vl/e (A.2-18) The Gaussian function is also used to describe the spectrum of certain light emis- sions (see Sec. I3.3D) as well as to describe the spatial distribution of light beams (see Sec. 3.1). A.3 TWO-DIMENSIONAL FOURIER TRANSFORM We now consider a function of two variables f x, y . If x and y represent the coordi- nates of a point in a two-dimensional space, then f x, y represents a spatial pattern (e.g., the optical field in a given plane). The harmonic function F exp j27r VxX + vyy is regarded as a building block from which other functions may be composed by superposition. The variables V x and v y represent spatial frequencies in x and y directions, respectively. Since x and y have units of length mm), V x and v y have units of cycles/mm, or lines/mm. Examples of two-dimensional harmonic functions are illustrated in Fig. A.3-I. The Fourier theorem may be generalized to functions of
A.3 TWO-DIMENSIONAL FOURIER TRANSFORM 1129 y y y n )--- x --.... x x .. (a) (b) (c) Figure A.3-1 The real part IFI cos[21rllxx + 21rllyY + arg{ F}] of a two-dimensional harmonic function: (a) lIx 0; (b) lIy 0; (c) arbitrary case. For this illustration we have assumed that arg{ F} 0 so that dark and white points represent positive and negative values of the function, respectively. two variables. A function I x, y may be decomposed as a superposition integral of harmonic functions of x and y, ex) I x,y F v x , V y exp j27r VxX + Vyy dv x dv y (A.3-1) Inverse Fourier Transform -ex) where the coefficients F v x , v y are determined by use of the two-dimensional Fourier transform ex) F v x , v y f x, y exp j27r VxX + vyy dx dYe (A.3-2) Fourier Transform -00 Our definitions of the two- and one-dimensional Fourier transforms, (A.3-2) and (A.I-2) respectively, differ in the sign of the exponent. The choice of this sign is, of course, arbitrary, as long as opposite signs are used in the Fourier and inverse Fourier transforms. In this book we have adopted the convention that exp j27rvt has positive temporal frequency v., whereas exp j27r VxX + vyy has positive spatial frequencies V x and v y . We have elected to use different signs in the spatial (two-dimensional) and temporal (one-dimensional) cases in order to simplify the notation used in Chap. 4 (Fourier optics), in which the traveling wave exp +j27rvt exp j kxx + kyy + kzz has temporal and spatial dependences with opposite signs. Properties The two-dimensional Fourier transform has many properties that are obvious general- izations of those of the one-dimensional Fourier transform, and others that are unique to the two-dimensional case: . Convolution Theorem. If I x, y is the two-dimensional convolutions of two func- tions 11 x, y and 12 x, y with Fourier transforms F 1 v X , v y and F 2 v X , v y ,
1130 APPENDIX A FOURIER TRANSFORM respectively, so that 00 f x,y I I I f 1 X , Y 2 X x' , y y' dx' d y' , (A.3-3) -00 the the Fourier transform of f x, y is F v x , v y Fl v x , v y F 2 v X , v y · (A.3-4) Thus, as in the one-dimensional case, convolution in the space domain is equiva- lent to multiplication in the Fourier domain. . Separable Functions. If I x, y Ix x I y y is the product of one function of x and another of y, then its two-dimensional Fourier transform is a product of one function of V x and another of v y . The two-dimensional Fourier transform of I x, y is then related to the product of the one-dimensional Fourier transforms of Ix x and fy y by F v x , v y Fx V x Fy v y . For example, the Fourier transform of 6 x Xo 6 y Yo, which represents an impulse located at Xo, Yo , is the harmonic functio n exp j27r VxXo + vyYo ; and the Fourier tr an sform of the and so on. . Circularly Symlnetric Functions. The Fourier transform of a circularly symmetric function is also circularly symmetric. For example, the Fourier transform of I x,y 1 , x 2 + y2 < 1 0, otherwise, (A.3-5) denoted by the symbol circ x, y and known as the cire function, is F v x , v y J 1 27rV P v p 2 + 2 V x V y , (A.3-6) , V p where J 1 is the Bessel function of order 1. These functions are illustrated in Fig. A.3-2. j(x,y) F(vx, Vy) . 1 .. ,::a = / l II 1 . I ,/ A 1 T / I - - , - - -- - I - - 1 y A - ,. () - .l - V y ........ 1 / 0.61 x Vx (a) (b) Figure A.3-2 (a) The circ function and (b) its two-dimensional Fourier transform.
READING LIST 1131 READING LIST E. Kamen, Introduction to Signals and Systems, Macmillan, ] 987, 2nd ed. ] 990. A. Gabel and R. A. Roberts, Signals and Linear Systems, Wiley, 1973, 3rd ed. 1987. C. D. McGillem and G. R. Cooper, Continuous and Discrete Signal and System Analysis, Oxford University Press, 3rd ed. 1991. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, Prentice Hall, 1983, 2nd ed. 1997 . R. N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, 3rd ed. 2000. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, Wiley, 1978. L. E. Franks, Signal Theory, Prentice Hall, 1969, revised ed. 1981. A. Papoulis, Systems and Transforms with Applications in Optics, McGraw-Hill, 1968; Krieger, reissued 1986. A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, 1962, reprinted 1987.
APPENDIX This appendix provides a review of the basic characteristics of one- and two-dimensional linear systems. B.1 ONE-DIMENSIONAL LINEAR SYSTEMS Consider a system whose input and output are the functions 11 t and 12 t , respec- tively. An example is a harmonic oscillator driven by a time-varying force 11 t that responds by undergoing a displacement 12 t . The system is characterized by a rule that relates the output to the input. In general, the rule may take the form of a differential equation, an integra] transform, or a simple mathematical operation such as 12 t log 11 t · Linear Systems A system is said to be linear if it satisfies the principle of superposition, i.e., if its response to the sum of any two inputs is the sum of its responses to each of the inputs separately. The output at time t is, in general, a weighted superposition of the input contributions at different times T, 00 12 t h t; T 11 T dT, (B. I - 1 ) -ex) where h t; T is a weighting function representing the contribution of the input at time T to the output at time t. If the input is an impulse at T, so that 11 t t T, then (B. I-I) gives 12 t h t T . Thus h t; T is the impulse-response function of the system (also known as the Green's Function). Linear Shift-Invariant Systems A linear system is said to be time-invariant or shift-invariant if, when its input is shifted in time, its output shifts by an equal time, but otherwise remains the same. The impulse-response function is then a function of the time difference h t; T h t T. Under these conditions (B.] -I) becomes 00 12 t h t T 11 T dT. (B.I-2) -ex) Thus the output 12 t is the convolution of the input 11 function h t [see (A.I-4)]. If 11 t t , then 12 t then 12 t h t T, as illustrated in Fig. B.I-I. 1132 t with the impulse-response h t ; and if lIt t T,
B.1 ONE-DIMENSIONAL LINEAR SYSTEMS 1133 h(t) /l(t - T) o T Input Systel11 h(t) Output t o T t Figure 8.1-1 Response of a linear shift-invariant system to impulses. The Transfer Function In accordance with the convolution theorem discussed in Appendix A, the Fourier transforms Fl v , F 2 V , and H v , of 11 t , 12 t , and h t , respectively, are related by F 2 V H V F 1 V . (B.1-3) If the input 11 t is a harmonic function F 1 v exp j27rvt , the output 12 t H V F 1 v exp j27rvt is also a harmonic function of the same frequency but with a modified complex amplitude F 2 v F 1 V H v , as illustrated in Fig. B.I-2. The multiplicative factor H v is known as the system's transfer function. The transfer function is the Fourier transform of the impulse-response function. Equation (B. 1- 3) is the key to the usefulness of Fourier methods in the analysis of linear shift- invariant systems. To determine the output of a system for an arbitrary input, we simply decompose the input into its harmonic components, multiply the complex amplitude of each harmonic function by the transfer function at the appropriate frequency, and superpose the resultant harmonic functions. Examples . Ideal system: H v 1 and h t t ; the output is a replica of the input. . Ideal system with delay: H v exp j27rVT and h t t T; the output is a replica of the input delayed by time T. . System with exponential response: H v T 1 + j27rVT and h t e- t / r for t > 0, and h t 0, otherwise; this represents the response of a system described by a first-order linear differential equation, e.g., that representing an R-C circuit with time constant T. An impulse at the input results in an exponentially decaying response. . Chirped system: H v exp j7rV 2 and h t e- j7r / 4 exp j7rt 2 ; the system distorts the input by imparting to it a phase shift proportional to v 2 . An input impulse generates an output in the form of a chirped signal, i.e., a harmonic function whose instantaneous frequency (the derivative of the phase) increases linearly with time. This system describes the propagation of optical pulses through media with a frequency-dependent phase velocity (see Sec. 5.6). It also describes changes in the spatial distribution of light waves as they propagate through free space (see Sec. 4.1C). Linear Shift-Invariant Causal Systems The impulse response function h t of a linear shift-invariant causal system must vanish for t < 0, since the system's response cannot begin before the application of the input. The function h t is therefore not symmetric and its Fourier transform, the
1134 APPENDIX 8 LINEAR SYSTEMS Input System H(v) Output t t e j27rvt H( v) e j27rvt Figure 8.1-2 Response of a linear shift-invariant system to a harmonic function. transfer function H v , must be complex. It can be shown t that if h t 0 for t < 0, then the real and imaginary parts of H v , denoted H' v and HI! v respectively, are related by H' v 1 00 HI! S ds 7r -00 S v 1 00 H' s ds, 7r -00 V S (B.I-4) H" v (B.I-5) Hilbert Transform where the Cauchy principal values of the integrals are to be evaluated, i.e., 00 v- 00 lim > 0 + -00 v+ , > o. -00 Functions that satisfy (B .1-4) and (B .1- 5) are said to form a Hilbert transform pair, HI! v being the Hilbert transform of H' v . If the impulse response function h t is also real, its Fourier transform must be symmetric, H v H* v . The real part H' v then has even symmetry, and the imaginary part HI! v has odd symmetry. The integrals in (B.I-4) and (B.I-5) may then be rewritten as integrals over the interval 0, 00 . The resultant equations are known as the Kramers Kronig relations H'v 2 00 S HI! S 2 2 ds 7r 0 S v 2 00 V H' s 2 2 ds. 7r 0 V S (B.I-6) HI! V (B.1-7) Kramers Kronig Relations In summary, the Hilbert-transform relations, or the Kramers Kronig relations, relate the real and imaginary parts of the transfer function of a linear shift-invariant causal system, so that if one part is known at all frequencies, the other part may be determined. Example: The Harmonic Oscillator The linear system described by the differential equation d 2 d 2 12 t II t (B.I-8) t See, e.g., L. E. Franks, Signal Theory, Prentice Hall, 1969, revised ed. 1981.
B.2 TWO-DIMENSIONAL LINEAR SYSTEMS 1135 describes a harmonic oscillator with displacement 12 t under an applied force 11 t , where Wa is the resonance angular frequency and a is a coefficient representing damp- ing effects. The transfer function H v of this system may be obtained by substituting 11 t exp j27rvt and 12 t H v exp j21fvt in (B.I-8), which yields 1 27r 2 v: 2 o 1 v 2 + jvv' (B.I-9) Hv where Va Wa 27r is the resonance frequency, and v a 27r. The real and imagi- nary parts of H v are therefore vJ v 2 v 2 2 + vv 2 vv v 2 2 + vv 2 (B .1-1 0) 1 27r 2 vJ 1 27r 2 v5 H'v H" v (B .1-11 ) Since the system is causal, H' v and H" v satisfy the Kramers Kronig relations. When Va » v, H' v and H" v are narrow functions centered about Va. For v Va, v5 v 2 2va Va v so that (B. 1-1 0) and (B .1-11) may be approximated by 1 21f 2 va v 4va v 2 + v 2 2 (B. 1-12) H" v H'v 2 v Vo H " v v. (B. 1- 13) The transfer function of the harmonic-oscillator system is used in Chaps. 5 and 14 to describe dielectric and atomic systems. Equation (B .1-12) has a Lorentzian form. B.2 TWO-DIMENSIONAL LINEAR SYSTEMS A two-dimensional system relates two two-dimensional functions 11 x, y and 12 x, y , called the input and output functions. These functions may, for example, represent optical fields at two parallel planes, with x, y representing position variables; the system comprises the free space and optical components that lie between the two planes. The concepts of linearity and shift invariance defined in the one-dimensional case are easily generalized to the two-dimensional case. The output 12 x, Y of a linear system is related to its input II x, y by a superposition integral ex) 12 X, Y h X Y . x , Y ' f x , Y ' dx ' dy ' , , " 1 , (B.2-I) -00 where h x, y; x' y' is a weighting function that represents the effect of the input at the point x' , y' on the output at the point x, y . The function h x, y; x', y' is the impulse-response function of the system (also known as the point-spread function). The system is said to be shift-invariant (or isoplanatic) if shifting its input in some direction shifts the output by the same distance and in the same direction without otherwise altering it (see Fig. B.2-1). The impulse response function is then a function
1136 APPENDIX 8 LINEAR SYSTEMS of position differences h x, y; x', y' h x x', y y'. Equation (B.2-1) then becomes the two-dimensional convolution of h x, y with 11 x, y : 00 12 x, Y hx I x,y I I I , d ' d ' y 1 x,y X y. (B.2-2) -00 Applying the two-dimensional convolution theorem discussed in Sec. A.3 of Ap- pendix A, we obtain F 2 lJ x , lJ y H lJ x , lJ y F 1 lJ x , lJ y , (B.2-3) where F 2 lJ x , v , H lJ x , lJ y , and F 1 lJ x , lJ y are the Fourier transforms of 12 x, y , h x, y , and 11 x, y , respectively. y y ... . Input System h(x, y) Output \ ) .., ) . x x II (x,y) 12 (x,y) Figure 8.2-1 Response of a two-dimensional linear shift-invariant system to hannonic functions. A harmonic input of complex amplitude F 1 lJ x , lJ y therefore produces a harmonic output of the same spatial frequency but with complex amplitude F 2 lJ x , lJ y H lJ x , lJ y F 1 lJ x , lJ y , as illustrated in Fig. B.2-2. The multiplicative factor H lJ x , lJ y is the system's transfer function. The transfer function is the Fourier transform of the impulse-response function. Either of these functions characterizes the system completely and enables us to determine the output corresponding to an arbitrary input. y y " --... x Input Output System . .... X H(vx, Vy) e-j27r(vxx+ VyY) H(v x ' v y ) e-j27r(vxx + vyY) Figure 8.2-2 Response of a two-dimensional linear shift-invariant system to harmonic functions. In summary, a two-dimensional linear shift-invariant system is characterized by its impulse-response function h x, y or its transfer function H lJ x , lJ y . For exam- ple, a system with h x, y circ x Ps, y Ps smears each point of the input into a patch in the form of a cir c le of radius Ps. It has a transfer function H lJ x , lJ y 2. The system severely attenuates spatial frequencies higher than 0.61 Ps lines/mm. READING LIST See the reading list in Appendix A.
APPENDIX This appendix is a brief overview of the modes of a linear system that is described explicitly by an input-output relation in the form of a matrix or an integral operation, or implicitly by a linear partial differential equation. Consider first a linear system described by an explicit input-output relation char- acterized by a linear operator /:.; that operates on an input vector X to generate the corresponding output vector x L y y /:.;X. (C. I-I) The vector X may be an array of complex numbers represented by a column matrix, or a complex function of one or several variables. The modes of such a system are the special inputs that are unaltered (except for a multiplicative constant) upon passage through the system, i.e., AqXq X q L /:.; X q AqXq, (C.] -2) Eigenvalue Problem where q is an index labeling the mode. The vector X q is called an eigenvector. The multiplication constant Aq, called the eigenvalue, is generally a complex number. The condition in (C.1-2) is known as the eigenvalue problem. Consider second a linear dynamical system whose state is described by N contin- uous variables constituting a vector X t . The evolution of any of the N variables of this N-dimensional vector is, in general, dependent on the all N variables. However, the same system may be described in a new coordinate system such that the N new variables evolve independently, so that the system is decomposed into N independent one-dimensional systems. These decoupled variables are the modes of the system. Consider third a linear system described implicitly by a linear partial differential equation that may be cast in the form in (C.1-2), where /:.; is a differential operator and X is a complex function of one, or several, variables. In this case, the modes are simply solutions of the differential equation and the eigenvectors are called the eigenfunctions. The notion of input and output is not meaningful in this case. In this appendix, we describe a number of applications of modal analysis in pho- tonics. But first we recall briefly a few geometrical concepts from linear algebra. Associated with each pair of vectors X and Y is a complex scalar X, Y called the inner product. The square root of the inner product of a vector X by itself, X, X , is called the norm of X and is a measure of its "length." The inner product of two vectors of unit norm can be thought of as cosine the "angle" between them. Two vectors are said to be orthogonal if their i nner product is zero. If t h e ve cto rs are arrays of complex on. 1137
1138 APPENDIX C MODES OF LINEAR SYSTEMS The following are two classes of operators /:.; for which the solutions of the eigen- value problem have special properties. Hermitian Operators. Hermitian operators are defined by the property X, /:.; Y /:.;X, Y , i.e., the inner product is the same if the operator is applied to either of two vectors. The eigenvalues of a Hermitian operator are real and the eigenvectors are orthogonal. Further, the eigenvectors o f a Hermitian operator obey the variational energy. It states that the eigenvector Xl with the lowest eigenvalue minimizes Ev; the eigenvector X 2 with the next lowest eigenvalue minimizes Ev, subject to the condition that it is orthogonal to Xl, and so on. Unitary Operators. Passive, lossless physical systems are described by unitary op- erators, which are defined by the norm-preserving property /:.;X, /:.;X X, X . An example is the operation of "rotation." The eigenvalues of unitary operators are unimodular ( Aq 1), i.e., they represent a pure phase. 1) Modes of a Discrete Linear System A discrete linear system is described by a matrix relation Y MX, where the input vector X is a set of N complex numbers Xl, X 2 ,. . . , X N arranged in a column matrix, M is an N x N matrix that represents the linear system, and Y, the output vector is also a column matrix of dimension N. The modes are those input vectors that remain parallel to themselves upon transmission through the system, so that the matrix . equatIon MX q AqXq (C.1-3) is obeyed. Thus, the modes of the system are the eigenvectors X q of the matrix M, and the scalars Aq are the corresponding eigenvalues, as determined by solving the algebraic equation det M AI 0, where I is the identity matrix. There are N such modes, labeled by the index q 1,2, . . . , N. The special case of binary systems (N 2) is particularly important in optics. In a binary system, each vector is a pair of complex numbers Xl, X 2 arranged in a column matrix X. The system is characterized by a 2 x 2 square matrix M whose elements are denoted A, B, C, and D. The relation Y MX signifies Yi 112 A B C D Xl X 2 · The eigenvalues are determined by solving the algebraic equation A A D A BC 0 for the two eigenvalues Al and A2. The following are examples of optical systems described by binary linear systems: Application: Polarization Matrix Optics. In polarization matrix optics (Sec. 6.1B), the vector Xl, X 2 represents components of the input electric field in two orthogo- nal directions (the Jones vector), and Y I , Y 2 similarly represents the output electric field. The matrix M is the Jones matrix of the system. In this case, the modes are the polarization states that are maintained as light is transmitted through the system. Application: Ray Matrix Optics. In geometrical paraxial optics (Sec. 1.4), the po- sition and angle of an optical ray are described by a vector Xl, X 2 , and the effect of optical components, such as lenses and mirrors, is described by a matrix M, called the ABCD matrix. For a closed optical system, such as a resonator, the modes are ray positions and angles that self-reproduce after a round trip, so that they are confined within the resonator.
APPENDIX C MODES OF LINEAR SYSTEMS 1139 Application: Multilayer Matrix Optics. In multilayer matrix optics (Sec. 7.1 A) light is reflected and refracted at each boundary, so that there are forward- and backward-traveling waves at each plane, with amplitudes described by a vector X Xl, X 2 . A system containing a set of boundaries between an input and an output plane is described by a transmission matrix M. The modes of such a system are the vectors that self reproduce upon transmission through the system, so that if the system is replicated periodically, as in ID photonic crystals (Sec. 7.2), the propagation modes are the modes of the system M. 2) Modes of a Continuous System Described by an Integral Operator Linear systems represented by integral operators are discussed in Appendix B. Con- sider, for example, a function of time f t , such as an optical pulse or a broadband optical field, transmitted through a linear time-invariant system, such as an optical filter. The system is described by the convolution operation, ex) 9 t h t T f T dT. (C. 1-4) -00 In this system, vectors X and Yare the functions f t and 9 t , and the operator /:.; is an integral operator. The modes of this system are the harmonic functions exp j27rvt . This is evident since the input function exp j27rvt generates another harmonic output function H v exp j27rvt , where H v is the Fourier transform of h t . In this case, there is a continuum of modes with continuous eigenvalues H v . Here, the index q is the frequency v, which takes continuous values. Another example is a linear shift-invariant system that operates on a two-dimensional (2D) function f x, y of the position x, y , as described in (B .2-1), ex) 9 x,y h X x', y y' f x', y' dx ' dy'. (C. 1-5) -00 The eigenfunctions are 2D harmonic functions exp j27r VxX + vyy , and the eigenval- ues are H v x , v y , the 2D Fourier transform of h x, y . Again, there is a continuum of eigenfunctions, labeled by the spatial frequencies v x , v y . Translational Symmetry and Harmonic Modes. It is not surprising that the har- monic functions are the eigenmodes of a shift-invariant system. Because the harmonic function is invariant to time shift, i.e. remains a harmonic function if translated in time, it is the eigenfunction of the time-invariant (stationary) linear system. Likewise, since 2D harmonic functions are invariant to translation in the plane, they are the eigenfunctions of a space-invariant (homogeneous) linear system. If the linear system is not space-invariant, i.e., does not enjoy translational symme- try, then it is represented (in the 2D case) by the more general linear operation: 00 9 x,y h I' f ' , d ' d ' x, y; x , y x , y x y, (C.I-6) -ex) In this case, the eigenfunctions are not necessarily harmonic functions. They can be determined by solving the eigenvalue problem in (C.I-2), which now takes the form of an integral equation 00 h "f I , dx ' dy ' x, y; x , Y J q X , Y Aqfq x, y, q 1,2,.... (C. 1- 7) -00
1140 APPENDIX C MODES OF LINEAR SYSTEMS The functions fq x, y and the constants Aq are the eigenfunctions and eigenvalues of the system, respectively, and the index q labels a discrete set of modes. Application: Optical Resonator Modes. An example (discussed in Sec. 10.2E) is light traveling between two parallel mirrors of a laser resonator. The distributions of the optical field in the transverse plane at the beginning and at the end of a single round trip are the input and output to the system. The modes of the resonator are those field distributions that maintain their shape after one round trip. The kernel h x, y; x', y' in (C.I-7) represents propagation in free space and reflection from the first mirror, followed by backward free-space propagation and reflection from the second mirror. Clearly, the presence of curved mirrors, or mirrors of finite extent, makes this system shift-variant. If the mirrors are spherical and are assumed to modulate the incoming light by a phase factor that is a quadratic function of the radial distance, then the res- onator modes are Hermite Gaussian functions of x and y.. In the presence of apertures, (C.I- 7) can only be solved numerically (see Sec. 10.2E). 3) Modes of a System Described by Ordinary Differential Equations The dynamics of certain physical systems are described by a set of coupled ordinary differential equations. For example, the dynamics of N coupled oscillators are de- scribed by N differential equations written in the matrix form: X I X2 X3 .... .... iCJ .. X MX, (C.I-8) where X is a column matrix with components Xl, X 2 ,. . . , XN , X d 2 X dt 2 , and M is an N x N matrix with time-independent coefficients, so that the system is time invariant. Time invariance requires that the modes be harmonic functions of the form exp jwt , i.e., the vector X t X 0 exp jwt . Substituting in (C.I-8), we obtain MX w 2 X, (C.I-9) This equation is in the form of a discrete-system eigenvalue problem. Its eigenvalues provide the resonance frequencies WI, W2, . . . , W N of the modes, and its eigenvectors are called the normal modes. All components of the eigenvector X q of mode q oscil- late at the same resonance frequency w q , without altering their relative amplitudes or phases. In this sense, the modes are stationary solutions that are decoupled from one another. 4) Modes of a System Described by a Partial Differential Equation Fields and waves are described by partial differential equations such as Maxwell's equations, which describe the dynamics of the electric and magnetic fields in a di- electric medium, and the Schrodinger equation, which describes the dynamics of the wavefunction of a particle subject to some potential. If these physical systems are sta- tionary, i.e., the dielectric medium and the potential distribution are time independent, then each mode must be a harmonic function of time exp jwt with some frequency w. The wave equation is therefore converted into the generalized Helmholtz equation w 2 \7 x 11 r V x H 2 H, (C. I-I 0) Co where 11 r Eo E r is the electric impermeability of the dielectric medium [see (7.0-2)]. Likewise, the Schrodinger equation yields the time-independent Schrodinger . equatIon: fi2 V 2 + V r 'ljJ r 2m E'ljJ r , (C. I-II)
APPENDIX C MODES OF LINEAR SYSTEMS 1141 where E wand V r is the potential distribution [see (13.1-3)]. Each of these equations is now in the form of the eigenvalue problem (C.1-2), where /:.; is a Hermitian differential operator characterized by the functions 11 r or V r . The eigenvalues, which are real, provide the frequencies w q of the modes (and hence the corresponding energies Eq in the case of Schrodinger equation). The eigenfunctions are the spatial distributions of the electromagnetic fields (or the wavefunctions) for each mode. Note that the field (or the wavefunction) of the qth mode evolve with time as exp jwqt at all positions, so that each mode is stationary, as required. Modes of Fiel ave in a Homogeneous Medium with Boundary Conditions. If the dielectric medium is homogeneous, i.e., 11 r is constant, then the system is shift- invariant. To be consistent with this translational symmetry, the modes of the electro- magnetic system must be harmonic functions of position, i.e., plane waves. Similarly, if the potential V r is constant, then the modes are plane waves wavefunctions, so that the particle is equally likely to be found anywhere. In other situations, 11 r and V r are constant within a finite region bounded by a surface that imposes certain boundary conditions. For example, electromagnetic modes of a cavity resonator with perfectly conducting surfaces can be obtained by requiring that the parallel components of the electric field vanish at the surface. For a rectangular resonator, the modes are harmonic functions of position standing waves oscillat- ing in unison [see sec. 10.3C]. Likewise, the modes of a particle in a quantum box (dot) are obtained by requiring that the wavefunction vanishes at the boundaries (see sec. 16.1 G). In yet another geometry, a homogeneous dielectric medium may be bounded in one direction, e.g., by two parallel planar mirrors. Here, the boundary conditions corre- spond to a discrete set of standing waves in the direction orthogonal to the mirrors (transverse direction), with traveling waves in the parallel (axial) direction, so that the modes travel in this optical waveguide as harmonic functions in the axial direction, without altering their transverse distributions (see Sec. 8.1). If /3q is the propagation constant of mode q, then the eigenvalue is the phase factors exp j {3qZ . Modes of Fields aves in a Periodic Medium. As evident from the previous exam- ples, the modes of a system described by a partial differential equation are dictated by the spatial distribution of the medium, i.e., the functions 11 r or V r . If this function is constant, the modes must be invariant to an arbitrary translation. If it is periodic, then the modes must be invariant to translation by a period. This type of translational symmetry requires that the modes be Bloch waves (see Sec. 7.2A). For example, if the medium is homogeneous in the x and y directions but periodic in the Z direction, a Bloch mode has the form of a harmonic function exp j K z , modulated by a periodic standing wave PK z with period equal to that of the medium; the dependence on x and y is, of course, harmonic. For a given value of K, the frequencies of the modes and the shapes of the corresponding standing waves PK z depend on the shape of the periodic function 11 r or V r . This type of translational symmetry results in a spectrum of eigenvalues (and hence frequencies W or energies E w) in the form of bands separated by bandgaps within which no modes are allowed. Thus, an electron in a periodic potential distribution exhibits the well-known band structure of solids (see Sec. 13.1 C). Likewise, an optical field in a periodic dielectric medium, i.e., a photonic crystal, exhibits a similar band structure with photonic bandgaps (see Sec. 7.2 and Sec. 7.3).
.. . Roman Symbols and Acronyms a Radius of an aperture or fiber [m]; also, Radius of a circle [nl]; also, Lattice constant [m]; also, Chirp parameter for an optical pulse a Amplitude (magnitude) of an optical wave; also, Normalized complex amplitude of an optical field (I al 2 photon flux density) a Normalized complex amplitude of an optical field in a cavity (lal 2 photon number) a Acceleration of a carrier [m · S-2] Complex envelope of a monochromatic plane wave; also, Pulse amplitude Complex envelope of a monochromatic wave Fourier transform of the complex envelope of an optical pulse Complex vector envelope of a monochromatic plane wave; also, Vector potential [V. s. m -1 ] Complex envelope of a polychromatic (e.g., pulsed) wave Complex envelope of an optical pulse Area [m 2 ]; also, Element of the ABCD ray-transfer matrix; also, Element of the ABCD wave-transfer matrix Coherence area [m 2 ] Einstein A coefficient [S-I] Alternating current Add-drop multiplexer Amplitude modulation Avalanche photodiode Amplified spontaneous emission Arrayed waveguide gratings A A(r) A(v) A A(r, t) A(t) A Ac A AC ADM AM APD ASE AWG b Radius of a circle [m]; also, Chirp coefficient [S2] B Magnetic flux-density complex amplitude [Wb · m-2]; also, Bandwidth [Hz] Bo Bit rate [bits · S-1 ] Magnetic flux density [Wb · m- 2 ]; also, Power-equivalent spectral width [Hz] B Element of the ABCD ray-transfer matrix; also, Element of the ABCD wave-transfer matrix ]ffi Einstein B coefficient [m 3 · J- 1 · S-2] BEC Bose-Einstein condensate BER Bit error rate BGR Bragg grating reflector BRF Birefringent filter BSO Bismuth silicon oxide C Speed of light; Phase velocity [m · S-I] Co Speed of light in free space [m · S-I] 1142
SYMBOLS AND UNITS 1143 C == Electrical capacitance [F] C ( .) == Fresnel integral e == Coupling coefficient in a directional coupler [m -1 ] C == Element of the ABCD ray-transfer matrix; also, Element of the ABCD wave-transfer matrix CARS == Coherent anti-Stokes Raman Scattering CCD == Charge-coupled device CD == Compact-disc CDM == Code-division multiplexing CDMA == Code-division multiple access CLSM == Confocal laser-scanning microscopy CMOS == Complementary metal-oxide-semiconductor CW == Continuous-wave CWDM == Coarse wavelength-division multiplexing d == Differential dr == Incremental volume [m 3 ] ds == Incremental length [m] d == Coefficient of second-order optical nonlinearity [C . y-2] d ijk == Element of the second-order optical nonlinearity tensor [C . y-2] d iJ == Element of the second-order optical nonlinearity tensor (contracted indexes) [C . y-2] d( W3; WI, W2) == Coefficient of second-order optical nonlinearity (dispersive medium) [C . V- 2 ] d == Distance, Length [m] d p == Penetration depth [m] dpulse == Length of a modelocked optical pulse [m] d s == Length along a small dimension [m] D == Diameter [m]; also, Electric flux-density complex amplitude [C . m- 2 ] Dw == Waveguide dispersion coefficient [8 . m- 2 ] D x, D-y == Lateral widths [m] D A == Material dispersion coefficient [8 . m- 2 ] Dv == Material dispersion coefficient [S2 . m- I ] 1) == Electric flux density [C . m- 2 ] D == Element of the ABCD ray-transfer matrix; also, Element of the ABCD wave-transfer matrix DBR == Distributed Bragg reflector DC == Direct current DCF == Dispersion compensating fiber DEMUX == Demultiplexer DFB == Distributed-feedback DFF == Dispersion-flattened fiber DGD == Differential group delay DH == Double-heterostructure DKDP == Deuterated potassium dihydrogen phosphate DMUX == Demultiplexer DPSS == Diode-pumped solid-state DRO == Doubly resonant oscillator DSF == Dispersion-shifted fiber DVD == Digital-video-disc DWDM == Dense wavelength-division multiplexing
1144 SYMBOLS AND UNITS e == Magnitude of electron charge [C] ex == Unit vector in the x direction E == Electric-field complex amplitude [V . m- 1 ]; also, Steady or slowly varying electric field [V . m- 1 ] £ == Electric field [V . m -1 ] E == Energy [J] E A == Acceptor energy level [J] E e == Energy at the bottom of the conduction band [J] ED == Donor energy level [J] E f == Fermi energy [J] E fe == Quasi-Fermi energy for the conduction band [J] E fv == Quasi-Fermi energy for the valence band [J] E 9 == Bandgap energy [J] E v == Energy at the top of the valence band [J] Ev == Energy spectral density [J . HZ-I] EDFA == Erbium-doped fiber amplifier EIT == Electromagnetically induced transparency E/O == Electronic to optical EUV == Extreme-ultraviolet I == Focal length of a lens [m]; also, Frequency [Hz] I (E) == Fermi function la == Probability that absorption condition is satisfied le(E) == Fermi function for the conduction band leol == Collision rate [S-I] Ie == Probability that emission condition is satisfied I g == Fermi inversion factor Iv (E) == Fermi function for the valence band f == Focal length [m] f == Frequency of sound [Hz]; also, Modulation frequency [Hz] F == Excess-noise factor of an avalanche photodiode F # == F-number of a lens == Finesse of a resonator; also, Force [kg. m . S-2] FBG == Fiber Bragg grating FDM == Frequency-division multiplexing FDMA == Frequency-division multiple access FET == Field-effect transistor FEL == Free-electron laser FFT == Fast Fourier transform FIR == Far infrared FM == Frequency-modulated FON == Fiber-optic network FPA == Focal-plane array FROG == Frequency-resolved optical gating FSK == Frequency shift keying FUV == Far ultraviolet FWHM == Full width at half maximum FWM == Four-wave mixing 9 == Resonator g-parameter
SYMBOLS AND UNITS 1145 g(r1, r2) = Normalized mutual intensity g(r1, r2, T) = Complex degree of coherence g(v) = Lineshape function of a transition [Hz-I] 9 ( T) = Complex degree of temporal coherence go = Gain factor gvo (v) = Electron-photon collisionally broadened lineshape function in a semiconductor [Hz-I] 9 = Coupling coefficient in a parametric interaction [m- 3 ] 9 = Degeneracy parameter G = Gain of an amplifier; also, Gain of a photodetector; also, Conductance [0- 1 ] G(rl,r2) = Mutual intensity [W. m- 2 ] G(r1, r2, T) = Mutual coherence function [W . m- 2 ] G(v) = Gain of an optical amplifier G(T) = Temporal coherence function [W . m- 2 ] G = Coherency matrix [W . m -2]; also, Gyration vector of an optically active medium G = Rate of photoionization in a photorefractive material [m- 3 . S-l] Go = Rate of thermal electron-hole generation in a semiconductor [m- 3 . S-I] G = Reciprocal-lattice vector [m -1 ] <G n (.) = Hermite-Gaussian functions GR = Generation-recombination GRIN = Graded-index GVD = Group velocity dispersion h = Complex round-trip amplitude attenuation factor in a resonator; also, Planck's constant [J .s] h( t) = Impulse response function of a linear system h( x, y) = Impulse response function of a two-dimensional linear system h D ( t) = Detector impulse response function fi = h/21r [J . s] H = Magnetic-field complex amplitude [A . m -1 ] 1{ = Magnetic field [A . m -1 ] H (,) = Transfer function of a linear system H' (v) = Real part of the transfer function of a linear system H" (v) = Imaginary part of the transfer function of a linear system H(v x , v y ) = Transfer function of a two-dimensional linear system He (f) = Envelope transfer function of a linear system S)1) (.) = Hankel function of the first kind of order f lliI n (.) = Hermite polynomials HHG = High-harmonic generation HVPE = Hydride vapor-phase epitaxy i = Electric current [A]; also, Integer ie = Electron current [A] ih = Hole current [A] ip = Photoelectric current [A] is = Reverse current in a semiconductor n diode [A] it = Threshold current of a laser diode [A] iT = Transparency current for a laser-diode amplifier [A] I = Optical intensity [W . m- 2 ] I(t) = Intensity of an optical pulse [W . m- 2 ] Is = Saturation optical intensity of an amplifier or an absorber [W. m -2]; also, Acoustic intensity [W . m- 2 ]
1146 SYMBOLS AND UNITS Iv = Spectral intensity [W . m- 2 . HZ-I] Io ( .) = Modified Bessel function of order zero J = Fourier transform of intensity profile; also, Moment of inertia [kg. m 2 ] 1M = Intensity modulation IR = Infrared ISI = Intersymbol interference j = A ; also, Integer J = Electric current density [A . m - 2 ] J e = Electron current density [A . m- 2 ] J h = Hole current density [A . m -2] J£ ( .) = Bessel function of the first kind of order f J p = Photoelectric current density [A . m- 2 ] J t = Threshold current density of a laser diode [A . m- 2 ] J T = Transparency current density for a laser-diode amplifier [A . m- 2 ] " J = Jones vector a = Total angular-momentum quantum number (J = Electric current density vector [A . m- 2 ] k = Wavenumber [m -1]; also, Integer ko = Fr ee-space wavenumber [m -1 ] k T = v k + k = Transverse component of the wavevector [m-I] k x , ky = Components of the wavevector in the x and y directions [m- I ]; also, Spatial angular fre- quencies in the x and y directions [rad . m- I ] ko = Central wavenumber [m -1 ] k = Wavevector [m-I] kg = Grating wavevector [m -1 ] k = Ionization ratio for an avalanche photodiode k = Boltzmann's constant [J . K- I ] Km ( .) = Modified Bessel function of the second kind of order m K = Bloch wavenumber [m-I] K = Bloch wavevector [m -1 ] KDP = Potassium dihydrogen phosphate l = Length [m]; also, Integer le = Coherence length [m] f = Azimuthal quantum number fo = Optical pathlength of the central frequency component of a pulse L = Length [m]; also, Electrical inductance [H]; also, Loss factor; also, Integer Le = Coherence length in a parametric interaction [m] Lo = 7r /2e = Coupling length (transfer distance) in a directional coupler [m] /:.; = Linear operator £ = Orbital angular momentum quantum number 2S+I£a = Term symbol for angular-momentum quantum numbers with LS coupling L = Angular momentum [J . s] IL ( .) = Laguerre polynomial of order l and index m LAN = Local-area network LANL = Los Alamos National Laboratory LASER = Light amplification by stimulated emission of radiation LBO = Lithium-triborate
SYMBOLS AND UNITS 1147 LC = Liquid-crystal LCD = Liquid-crystal display LCP = Left-circularly polarized LD = Laser diode LED = Light-emitting diode LHS = Left-hand side LLNL = Lawrence Livermore National Laboratory LMA = Large mode-area LP = Linearly polarized LPE = Liquid-phase epitaxy LWI = Lasing without inversion LWIR = Long-wavelength infrared m = Mass of a particle [kg]; also, Integer; also, Contrast or modulation depth me = Effective mass of a conduction-band electron [kg] m r = Reduced mass of an electron-hole pair in a semiconductor [kg] mv = Effective mass of a valence-band hole [kg] m = mo = Free electron mass [kg] m = Photon number; also, Photoelectron number m = Magnetic quantum number M = Magnification in an image system; also, Number of modes; also, Integer ]v{ = Magnetization density [A . m- 1 ]; also, Number of modes of thermal light; also, Figure of merit for the acousto-optic effect [m 2 . W- 1 ] M = Mass of an atom or molecule [kg] M r = Reduced mass of an atom or molecule [kg] M(v) = Density of modes in a resonator or cavity [m- 3 . Hz- 1 for a 3D resonator; m- 1 . Hz- 1 for a 1- D resonator] M = Ray-transfer matrix; also, Wave-transfer matrix M 2 = Factor indicating deviation of optical-beam profile from Gaussian form MBE = Molecular-beam epitaxy MEMS = Microelectromechanical systems MIM = Metal-insulator-metal MIR = Mid infrared MIRACL = Mid-infrared advanced chemical laser MKS = Meter jkilogramj second unit system MMF = Multimode fiber . MOCVD = Metal-organic chemical vapor deposition MQD = Multiquantum dots MQW = Multiquantum-well MOSFET = Metal-oxide-semiconductor field-effect transistor MUV = Mid ultraviolet MUX = Multiplexer MWIR = Mid-wavelength infrared MZI = Mach-Zehnder interferometer n = Refractive index; also, Integer n( r) = Refractive index of an inhomogeneous medium n( ()) = Refractive index of the extraordinary wave with its wavevector at an angle () with respect to the optic axis of a uniaxial crystal ne = Extraordinary refractive index
1148 SYMBOLS AND UNITS no = Ordinary refractive index n2 = Optical Kerr coefficient (nonlinear refractive index) [m 2 . W- 1 ] n = Photon-number density [m- 3 ] ns = Saturation photon-number density [m- 3 ] n = Photon number n = Principal quantum number n = Concentration of electrons in a semiconductor [m- 3 ] n i = Concentration of electrons/holes in an intrinsic semiconductor [m -3] no = Equilibrium concentration of electrons in a semiconductor [m- 3 ] N = Group index; also, Integer; also, Number of atoms; also, Number of resolvable spots of a scanner N F = Fresnel number N = Number density [m- 3 ]; also, N = N 2 - N 1 = Population density difference between energy levels 2 and 1 [m- 3 ] N a = Atomic number density [m- 3 ] N A = Number density of ionized acceptor atoms in a semiconductor [m- 3 ] N D = Number of density of ionized donor atoms in a semiconductor [m- 3 ] Nt = Laser threshold population difference [m- 3 ] No = Steady-state population difference in the absence of amplifier radiation [m- 3 ] NA = Numerical aperture NEA = Negative-electron-affinity NIF = National Ignition Facility NIR = Near infrared NLDC = Nonlinear directional coupler NOLM = Nonlinear optical loop mirror NRZ = Non-retum-to-zero NUV = Near ultraviolet NZ-DSF = Non-zero-dispersion shifted fiber OA = Optical amplifier OADM = Optical add-drop multiplexer OC = Optical carrier OCT = Optical coherence tomography ODMUX = Optical demultiplexer OlE = Optical to electronic OFA = Optical fiber amplifier OFC = Optical frequency conversion OLED = Organic light-emitting diode OMUX = Optical multiplexer OOK = ON-OFF keying OPA = Optical parametric amplifier OPC = Optical phase conjugation OPO = Optical parametric oscillator OXC = Optical cross-connect p = Probability; also, Momentum [kg. ill. S-I]; also, Grade profile parameter of a graded-index fiber p(n) = Probability of n events p( x, y) = Aperture function or pupil function Pab = Probability density for absorption (mode containing one photon) [S-I]
SYMBOLS AND UNITS 1149 Psp == Probability density for spontaneous emission (into one mode) [8- 1 ] Pst == Probability density for stimulated emission (mode containing one photon) [8- 1 ] P == Dipole moment [C . m] p == Normalized electric-field quadrature component p == Photoelastic constant (strain-optic coefficient) Pij kl == Element of the strain-optic tensor PI K == Element of the strain-optic tensor (contracted indexes) P == Concentration of holes in a semiconductor [m -3] Po == Equilibrium concentration of holes in a semiconductor [m -3] P == Electric polarization-density complex amplitude [C . m -2] P(v x , V y ) == Fourier transform of the aperture function p( x, y) P ab == Probability density for absorption (mode containing many photons) [8- 1 ] P NL == Complex amplitude of the nonlinear component of the polarization density [C . m- 2 ] P sp == Probability density for spontaneous emission (into any mode) [8- 1 ] Pst == Probability density for stimulated emission (mode containing many photons) [8- 1 ] P (.) == Microsphere-resonator adjoint Legendre function P == Electric polarization density [C . m- 2 ] P L == Linear component of the polarization density [C . m- 2 ] P NL == Nonlinear component of the polarization density [C . m- 2 ] P == Optical power [W] P p == Pump power [W] P l/ == Power spectral density [W . HZ-I] P 7f == Half-wave optical power in a Kerr medium [W] JtD == Degree of polarization P == Optical power [dBm] PAL-SLM == Parallel aligned spatial light modulator PBG == Photonic bandgap PBS == Polarizing beamsplitter PCF == Photonic-crystal fiber PCM == Pulse code modulation PG-FROG == Polarization-gated frequency-resolved optical gating PIN == p-type-intrinsic-n-type PLC == Planar lightwave circuit PLED == Polymer light-emitting diode PM == Phase modulation PMD == Polarization mode dispersion PMT == Photomultiplier tube PPLN == Periodically poled lithium niobate PROM == Pockels readout optical modulator PSK == Phase shift keying PWM == Pulse-width modulation q == Electric charge [C]; also, Wavenumber of an acoustic wave [m-1]; also, Integer (mode index, diffraction order, quantum number) q(z) == Complex Gaussian-beam parameter [m] q == Wavevector of an acoustic wave [m -1 ] Q == Electric charge [C]; also, Quality factor of an optical resonator QCSE == Quantum confined Stark effect QD == Quantum dot QDIP == Quantum-dot infrared photodetector
1150 SYMBOLS AND UNITS QED = Quantum electrodynamics QPM = Quasi-phase matching; also, Quadratic phase modulator QWIP = Quantum-well infrared photo detector r = Radial distance in spherical coordinates [m]; also, Radial distance in a cylindrical coordinate system [m] r n = Radii of allowed electron orbits in a Bohr atom [m] rl = Bohr radius of the ground state of hydrogen [m] r = Position vector [m] r = Unit vector in radial direction in spherical coordinates r = Complex amplitude reflectance Irl = Magnitude of round-trip amplitude attenuation factor in a resonator r(v) = Rate of photon emission/absorption from a semiconductor [8- 1 . m- 3 . HZ-I] t = Linear electro-optic (Pockels) coefficient [m . V-I]; also, Rotational quantum number tijk = Element of the linear electro-optic tensor [m . V-I] tlk = Element of the linear electro-optic tensor (contracted indexes) [m. V-I] r = Electron-hole recombination coefficient [m 3 .8- 1 ] r nr = Nonradiative electron-hole recombination coefficient [m 3 .8- 1 ] rr = Radiative electron-hole recombination coefficient [m 3 .8- 1 ] R = Radius of curvature [m]; also, Electrical resistance [0] R(z) = Radius of curvature of a Gaussian beam [m] Ro = Radius of cylinder in which a meridional ray is confined [m] R( ()) = Jones matrix for coordinate rotation by an angle () 1( = Intensity or power reflectance R = Pumping rate [8- 1 . m- 3 ]; also, Recombination rate in a semiconductor [8- 1 . m- 3 ]; also, Electron-hole injection rate in a semiconductor [8- 1 . m- 3 ] Rt = Laser threshold pumping rate [8- 1 . m- 3 ] R = Lattice vector [m] 9\ = Responsivity of a photon source [W.A -1]; also, Responsivity of a photon detector [A. W- 1 ] 9\d = Differential responsivity of a laser diode [W . A-I] JR n £ (r) = Hydrogen-atom associated Laguerre function of order f and index n RC = Resonant-cavity RC = Resistor-capacitor combination RCP = Right-circularly polarized REFA = Rare-earth-doped fiber amplifiers RF = Radio-frequency RFA = Raman fiber amplifier RFL = Raman fiber laser RHS = Right-hand side RMS = Root-mean square ROADM = Reconfigurable optical add-drop multiplexer RW = Ridge-waveguide RZ = Retum-to-zero s = Length or distance [m] s(x, t) = Strain wavefunction Sij = Element of the strain tensor s( r 1 , r2, v) = Normalized cross-spectral density £j = Quadratic electro-optic (Kerr) coefficient [m 2 . V- 2 ]; also, Spin quantum number Sijkl = Element of the quadratic electro-optic tensor [m 2 . V- 2 ]
SYMBOLS AND UNITS 1151 SIK == Element of the quadratic electro-optic tensor (contracted indexes) [m 2 . y-2] S == Transition strength (oscillator strength) [m 2 . Hz] S(r, t) == Complex amplitude for a radiation source [Y . m- 3 ] S ( .) == Fresnel integral S == Source of optical radiation created by an incident field [Y . m -3]; also, Spin angular- momentum quantum number S == Poynting vector [W . m - 2 ] S(r) == Eikonal [m] S( rl , r2, v) == Cross-spectral density [W . m -2 . HZ-I] S(v) == Spectral intensity of an optical wave or pulse [W. m- 2 . HZ-I]; also, Power spectral density [W . m - 2 . Hz-l ] S(v, t) == Spectrogram of an optical pulse [S(v, t) == I <I> (v, t)12] S == Scattering matrix S == Photon-spin angular momentum [J . s] S [] == Stokes parameters SAGM == Separate-absorption-grading-multiplication SAM == Separate-absorption-multiplication SBN == Strontium barium niobate SBS == Stimulated Brillouin scattering SCG == Supercontinuum generation SDH == Synchronous digital hierarchy SEED == Self-electro-optic-effect device SESAM == Semiconductor saturable-absorber mirror SFG == Sum-frequency generation SH == Second-harmonic SHG == Second-harmonic generation SHG-FROG == Second-harmonic generation frequency-resolved optical gating SLAC == Stanford Linear Accelerator Center SLD == Superluminescent diode SLM == Spatial light modulator SMF == Single-mode fiber SNOM == Scanning near-field optical microscopy SNR == Signal-to-noise ratio SOA == Semiconductor optical amplifier SOl == Silicon-on-insulater SONET == Synchronous optical network SOS == silica-on-silicon SPAD == Single-photon avalanche diode SPDC == Spontaneous parametric downconversion SPM == Self-phase modulation SPP == Surface plasmon polariton SQW == Single-quantum-well SRO == Singly resonant oscillator SRS == Stimulated Raman scattering SSFS == Soliton self-frequency shift SSPD == Superconducting single-photon detector SVE == Slowly varying envelope SXR == Soft X-ray t == Time [s]
1152 SYMBOLS AND UNITS t sp == Spontaneous lifetime [8] t == Complex amplitude transmittance; also, Normalized time for an optical pulse [8] T == Temperature [K] T == Jones matrix == Intensity or power transmittance; also, Power-transfer or power-transmission ratio T == Transit time [8]; also, Counting time [8]; also, Switching time [8]; also, Bit time interval [8]; also, Resolution time (T == 1/2B where B == Bandwidth) [8]; also, Period of a wave (T == l/v where v == frequency) [8] T p == l/vp == Inverse of Fabry-Perot resonator-mode frequency spacing (T p == 2d/c) [8]; also, Period of a mode-locked laser pulse train [8] T 2 == Electron-phonon collision time [8] TDM == Time-division multiplexing TDMA == Time-division multiple access TE == Transverse electric TEM == Transverse electromagnetic TFF == Thin-film filter TGa == Terbium gallium garnet THG == Third-harmonic generation TM == Transverse magnetic TOAD == Terahertz optical asymmetric demultiplexer TPLSM == Two-photon laser scanning fluorescence microscopy TSI == Time-slot interchange TST == Time-space-time u == Displacement [m] u(r, t) == Wavefunction of an optical wave ii == Unit vector u == Number of electrons in a subshell U ( r) == Complex amplitude of a monochromatic optical wave U (r, t) == Complex wavefunction of an optical wave U (t) == Complex wavefunction of an optical pulse li(x) == Unit step function UV == Ultraviolet v == Group velocity of a wave [m . 8- 1 ] v( r, v) == Fourier transform of the wavefunction of an optical wave V s == Velocity of sound [m . 8- 1 ] V == Velocity of an atom or object [m . 8- 1 ] V e == Velocity of an electron [m . 8- 1 ] Vh == Velocity of a hole [m . 8- 1 ] V == Velocity vector of a charge carrier [m. 8- 1 ] tJ == Vibrational quantum number V == Volume [m 3 ]; also, Modal volume [m 3 ]; also, Voltage [V] V (r, v) == Fourier transform of the complex wavefunction of an optical wave V (v) == Fourier transform of the complex wavefunction of an optical pulse V c == Critical voltage for a liquid-crystal cell [V] V 7r == Half-wave voltage of an electro-optic retarder or modulator [V] Va == Built-in potential difference in a p-n junction [V]; also, Switching voltage of a directional coupler [V] V == Visibility
SYMBOLS AND UNITS 1153 W == Verdet constant [min. Oersted -1 . em -1 ] V == Fiber V parameter V (r) == Potential energy [J] V == Abbe number of a dispersive medium VCSEL == Vertical-cavity surface emitting laser VLSI == Very-large-scale integration VOx == Vanadium Oxide VPE == Vapor-phase epitaxy VUV == Vacuum ultraviolet w == Width [m] w == Integrated photon flux (integrated optical power in units of photon number) W d == Width of the absorption region in an avalanche photodiode [m] w m == Width of the multiplication region in an avalanche photodiode [m] W == Time-averaged electromagnetic energy density [J . m- 3 ] W(t) == Window function for short-time Fourier transform W(z) == Width (radius) of a Gaussian beam at axial distance z from the beam center [m] W o == Waist radius of a Gaussian beam [m] W == Electromagnetic energy density [J . m- 3 ] W == Probability density for absorption of pump light [S-I] Wi == Probability density for absorption and stimulated emission [S-I] W == Photoelectric work function [J] WC == Wavelength converter WCI == Wavelength-channel interchange WDM == Wavelength-division multiplexing WDMA == Wavelength-division multiple access WGM == Whispering-gallery mode WGR == Waveguide grating router WKB == Wentzel-Kramers-Brillouin WOLED == White organic light-emitting diode x == Position coordinate; displacement [m] x == Unit vector in the x direction x(t) == Inverse Fourier transform of the susceptibility of a dispersive medium X(v) x == Normalized electric-field quadrature component X == Normalized photon-flux density at the input to an optical amplifier "" X == Input vector to a linear system X ( u) == Real function associated with the Hermite-Gaussian beam X(2) (WI, W2) == Second-order nonlinear susceptibility X ( .) == Normalized rate of change of radial distribution in the core of a step-index fiber XC == Cross-connect XGM == Cross-gain modulation XPM == Cross-phase modulation XUV == Extreme ultraviolet y == Position coordinate [m] Y == Normalized photon-flux density at the output of an optical amplifier "" Y == Output vector from a linear system ( v) == Real function associated with the Hermite-Gaussian beam
1154 SYMBOLS AND UNITS Y(.) = Normalized rate of change of radial distribution in the cladding of a step-index fiber YIG = Yttrium iron garnet z = Position coordinate (Cartesian or cylindrical coordinates) [m] Zo = Rayleigh range of a Gaussian beam [m]; also, Rayleigh range of a Gaussian pulse traveling in a dispersive medium [m] z = Normalized distance for an optical pulse [m] Z = Atomic number Z( .) = Real function associated with the Hermite-Gaussian beam Greek Symbols a = Attenuation or absorption coefficient [m -1 ]; also, Apex angle of a prism; also, Twist coef- ficient of a twisted nematic liquid crystal [m- 1 ] a e = Electron ionization coefficient in a semiconductor [m -1 ] ah = Hole ionization coefficient in a semiconductor [m -1 ] am = Loss coefficient of a resonator attributed to a mirror [m -1 ] a r = Effective overall distributed loss coefficient [m -1 ] as = Loss coefficient of a laser medium [m- 1 ] ap = Mean value of p for a coherent state ax = Mean value of x for a coherent state a v = Angular dispersion coefficient [HZ-I] ex = Attenuation coefficient of an optical fiber [dB Ikm] /3 = k z = Propagation constant [m-l]; also, Phase-retardation coefficient of a twisted nematic liquid crystal [m -1 ] /3' = First derivative of /3 with respect to w [m- 1 . s] /3" = Second derivative of /3 with respect to w [m- 1 .8 2 ] /3(v) = Propagation constant in a dispersive medium [m- 1 ] /30 = /3(vo) = Propagation constant at the central frequency Vo [m- 1 ] (3 = Spontaneous-emission coupling coefficient '"'/ = Gain coefficient [m -1]; also, Coupling coefficient in a parametric device [nl-l]; also, Non- linear coefficient in soliton theory; also, Lateral decay coefficient in a waveguide [nl- 1 ]; also, Magnetogyration coefficient [m 2 . Wb- 1 ] ,",/(v) = Gain coefficient of an optical amplifier [m- 1 ] '"'/p = Peak gain coefficient of a laser-diode amplifier [m- 1 ] '"'/o(v) = Small-signal gain coefficient of an optical amplifier [m- 1 ] r = Retardation; also, Confinement factor 8 (.) = Delta function or impulse function 8x = Increment of x 8 v = Spectral width of a resonator mode [Hz] = Thickness of a thin optical component [m]; also, Fractional refractive-index change in an optical fiber or waveguide x = Increment of x n = Concentration of excess electron-hole pairs [m- 3 ] nT = Concentration of injected carriers for a semiconductor optical amplifier at transparency [m- 3 ] v = Spectral width or linewidth [Hz] ve = liTe = spectral width [Hz] VD = Doppler linewidth [Hz]
SYMBOLS AND UNITS 1155 VFWHM == Full-width-at-half-maximum spectral width [Hz] v s == Linewidth of a saturated amplifier [Hz] E == Electric permittivity of a medium [F 1m]; also, Focusing error [m- 1 ] Eij == Component of the electric permittivity tensor [F 1m] Ee == Effective electric permittivity [F 1m] Eo == Electric permittivity of free space [F 1m] € == Electric permittivity tensor [F 1m] (( z) == Excess axial phase of a Gaussian beam 'r} == Impedance of a dielectric medium [0] 'r}o == Impedance of free space [0] 11 == Electric impermeability llij == Component of the electric impermeability tensor T) == Electric impermeability tensor 11. == Photodetector quantum efficiency 11.c == Power-conversion efficiency (also overalJ efficiency, walJ-plug efficiency) 11. d == External differential quantum efficiency 11. e == Extraction efficiency; also, Transmission efficiency 11. ex == External efficiency 11. i == Internal quantum efficiency 11. s == Differential power-conversion efficiency; also, Slope efficiency () == Angle; also, Twist angle in a liquid crystal; also, Deflection angle of a prism () == 90° - () = Complement of angle () () a == Acceptance angle ()B == Brewster angle ()13 == Bragg angle ()c == Critical angle () c == Complementary critical angle () d == Deflection angle of a prism () s == Angle subtended by source ()o == Divergence angle of a Gaussian beam /'.. e == Unit vector in polar direction in spherical coordinates {) == Threshold 8£m(()) == Hydrogen-atom associated Legendre function K == Elastic constant of a harmonic oscillator [J . m- 2 ] A == Wavelength [m] AA == Acceptor long-wavelength limit [m] Ac == Cutoff wavelength [m] Ap == Wavelength spacing of adjacent Fabry-Perot resonator modes [m] Ag == Bandgap wavelength (long-wavelength limit) of a semiconductor [m] Ao == Free-space wavelength [m] Ap == Wavelength of maximum blackbody energy density [m] Aq == Eigenvalues of an eigenvalue problem AO == Central wavelength [m] A == de Broglie wavelength [m]
1156 SYMBOLS AND UNITS A == Spatial period of a grating or periodic structure [m]; also, Wavelength of an acoustic wave [m] J-l == magnetic permeability [H. m- 1 ]; also, Carrier mobility in a semiconductor [m 2 .8- 1 . V-I] J-le == Electron mobility [m 2 . 8- 1 . V-I] J-lh == Hole mobilit¥ [m 2 . 8- 1 . V-I] J-lo == Magnetic permeability of free space [H . m- 1 ] v == Frequency [Hz] v c == Cutoff frequency [Hz] Vp == Frequency spacing of adjacent Fabry-Perot resonator modes; free spectral range of a Fabry- Perot spectrometer [Hz] v p == Frequency of maximum blackbody energy density [Hz] V s == Spatial bandwidth of an imaging system [m- 1 ] v q == Frequency of mode q [Hz] V x , v y == Spatial frequencies in the x and y directions [m- 1 ] V13 == Bragg frequency [Hz] VA == Anti-Stokes-shift frequency [Hz] Va == Brillouin frequency [Hz] VR == Raman frequency [Hz] Vs == Stokes-shift frequency [Hz] v p == Radial component of the spatial frequency: v p == J v + v; [m -1 ] Vo == Central frequency [Hz] == Coupling coefficient in four-wave mixing sp(v) == Amplifier noise photon-flux density per unit length [m- 3 .8- 1 ] P == R otatory p ower of an optically active medium [m- 1 ]; also, Resistivity [0 . m]; also, P == J x 2 + y2 == Radial distance in a cylindrical coordinate system [m] Pc == Coherence distance [m] ps == Radius of the Airy disk [m]; also, Radius of the blur spot of an imaging system [m] {! == Mass density of a medium [kg. m- 3 ]; also, Charge density [C . m- 3 ] {!( k) == Wavenumber density of states [m -2] (!(v) == Spectral energy density [J. m- 3 . HZ-I]; also, Optical joint density of states [m- 3 . Hz-I] (!c(E) == Density of states near the conduction band edge [m- 3 . J-l in a bulk semiconductor] (!v(E) == Density of states near the valence band edge [m- 3 . J-l in a bulk semiconductor] p(v) == Normalized Lorentzian cavity mode [Hz-I] a == Conductivity [0- 1 . m- 1 ] a 5 == Pauli spin matrix a(v) == Transition cross section [m 2 ] a max == Maximum transition cross section [m 2 ] a q == Circuit-noise parameter a x == Standard deviation of a random variable x; RMS width of a function of x ao == a(vo) == Transition cross section at the central frequency Vo [m2] (J" == Damping coefficient of a harmonic oscillator [8- 1 ] (J == Conductivity tensor [0- 1 . m- 1 ] T == Lifetime [8]; also, Decay time [8]; also, Relaxation time [8]; also, Width of a function of time [8]; also, Excess-carrier electron-hole recombination lifetime in a semiconductor [8]
SYMBOLS AND UNITS 1157 Tc == Coherence time [8] Teol == Mean time between collisions [8] Td == Delay time [8] Te == Electron transit time [8] Th == Hole transit time [8] T m == Multiplication time in an avalanche photodiode [8] T nr == N onradiative electron-hole recombination lifetime [8] T p == Resonator photon lifetime [8] Tpul se == Duration of a modelocked optical pulse [8] Tr == Radiative electron-hole recombination lifetime [8] T RC == RC time constant [8] Ts == Saturation time constant of a laser transition [8] T21 == Lifetime of a transition between energy levels 2 and 1 [8] cp == Angle in a cylindrical coordinate system; also, Photon flux density [m- 2 .8- 1 ] cp(p) == Momentum wavefunction [8 1 / 2 . kg- 1 / 2 . m- 1 / 2 ] CPv == Spectral photon flux density [m- 2 . 8- 1 . HZ-I] CPs(v) == Saturation photon-flux density [m- 2 .8- 1 ] -- <I> == Unit vector in azimuthal direction in spherical coordinates cP == Phase difference cp(t) == Complex envelope phase of an optical pulse CPo == Phase shift from reflection at a resonator mirror cp(v) == Phase-shift coefficient of an optical amplifier [m- 1 ] <I> == Photon flux [8- 1 ] <I>(v, t) == Short-time Fourier transform <I>m (cp) == Hydrogen-atom harmonic function <I>v == Spectral photon flux [8- 1 . HZ-1 ] x == Electric susceptibility; also, Electron affinity [J] X' == Real part of the electric susceptibility X X" == Imaginary part of the electric susceptibility X X(v) == Electric susceptibility of a dispersive medium X ij == Component of the electric susceptibility tensor X(3) == Coefficient of third-order optical nonlinearity [C . m . y-3] xL == Element of the third-order optical nonlinearity tensor [C . m . y-3] xY2 == Element of the third-order optical nonlinearity tensor (contracted indexes) [C . m . y-3] X == Polarization-ellipse angle of ellipticity X == Electric susceptibility tensor 'ljJ == Normalized amplitude of an optical pulse 'ljJ (t) == Spectral phase of an optical pulse 'ljJ(x) == Particle position wavefunction [m- 1 / 2 ] 'ljJ(r, t) == Particle wavefunction [m- 3 / 2 .8- 1 / 2 ] 1.1-> == Polarization-ellipse orientation of major axis w(t:) == Nonlinear polarization density [C . m- 2 ] we (f) == Envelope transfer function phase W == Angular frequency [rad .8- 1 ] Wp, == Bragg angular frequency [rad . 8- 1 ]
1158 SYMBOLS AND UNITS W p == Plasma frequency [rad . 8- 1 ] Wo == Central angular frequency [rad . 8- 1 ] n == Angular frequency of an acoustic wave [rad . 8- 1 ]; also, Angular frequency of a harmonic electric signal [rad . 8- 1 ]; also, Solid angle Mathematical Symbols 8 == Partial differential \7 == Gradient operator \7 . == Divergence operator \7 x == Curl operator \72 == Laplacian operator (\7 2 == 8 2 /8x 2 + 8 2 /8y2 + 8 2 /8Z2 in Cartesian coordinates) \7 == Transverse Laplacian operator (\7 == 8 2 /8x 2 + 8 2 /8y2 in Cartesian coordinates) x == (x) == Mean of the quantity x
AUTHORS Bahaa E. A. Saleh has been Professor and Chairman of the Department of Electrical and Computer Engineering at Boston University since 1994. He serves as Deputy Director of the Cen- ter for Subsurface Sensing and Imaging Systems, an Engineer- ing Research Center supported by the National Science Foun- dation. He received the B.S. degree from Cairo University in 1966 and the Ph.D. degree from the Johns Hopkins University in 1971, both in electrical engineering. He has held faculty and research positions at the University of Santa Catarina in Brazil, Kuwait University, the Max-Planck-Institut fur biophysikalis- che Chemie in Gottingen, the University of California at Berke- ley, the European Molecular Biology Laboratory in Heidelberg, and Columbia Univer- sity in New York. He was a faculty member at the University of Wisconsin-Madison from 1977 to 1994, and Chairman of the Department of Electrical and Computer Engineering from 1990 to 1994. His research contributions cover a broad spectrum of topics in optics and photonics including statistical and quantum optics, optical communications and signal process- ing, nonlinear optics, photodetectors, digital image processing, and vision. He is Co- Director of the Quantum Imaging Laboratory and a member of the Boston University Photonics Center. He is the author of Photoelectron Statistics (Springer-Verlag, 1978) and the co-author of the first edition of Fundamentals of Photonics (Wiley, 1991). He has published chapters in ten books and authored or coauthored more than 250 papers in technical journals. He holds a number of patents. Saleh served as Topical Editor of the Journal of the Optical Society of America A from 1980 to 1990 and as Editor-in-Chief from 1991 to 1997. Among his professional activities, he was a Member and Chair of the Board of Editors, Member and Chair of the Publications Council, and Member of the Board of Directors of the Optical Society of America (OSA). He also served as a Member of the Board of Editors of the Journal of the European Optical Society B: Quantum Optics and as Vice-President of the International Commission for Optics (ICO). He was Series Editor of the Adam Hilger Series in Optics and Optoelectronics of the Institute of Physics in the UK and has been Editor of the Wiley Series in Pure and Applied Optics. Saleh is a Fellow of the Institute of Electrical and Electronics Engineers, the Optical Society of America, and the Guggenheim Foundation. He is the recipient of the 1999 OSA Esther Hoffman Beller Award for outstanding contributions to optical science and engineering education, and the 2004 SPIE BACUS award for his contributions to photomask technology. He is a member of Phi Beta Kappa, Sigma Xi, and Tau Beta Pi. 1159
1160 AUTHORS Malvin Carl Teich received the S.B. degree in physics from the Massachusetts Institute of Technology in 1961, the M.S. de- gree in electrical engineering from Stanford University in 1962, and the Ph.D. degree from Cornell University in 1966. His first 1 professional affiliation was with MIT's Lincoln Laboratory in Lexington, Massachusetts. He joined the faculty at Columbia University in 1967, where he served as a member of the Electri- cal Engineering Department (as Chairman from 1978 to 1980), the Applied Physics Department, the Columbia Radiation Lab- oratory, and the Fowler Memorial Laboratory at the Columbia College of Physicians & Surgeons. In 1995 Teich became Professor Emeritus of Engineering Science and Applied Physics at Columbia. He concurrently became a faculty member at Boston University, where he is now teaching and pursuing his research interests with appointments in the Departments of Electrical and Computer Engineering, Physics, and Biomedical Engineering. He is Co-Director of the Quantum Imaging Laboratory and a Member of the Photonics Center, the Center for Adaptive Systems, the Hearing Research Center, and the Program in Neuroscience. During periods of sabbatical leave, he has served as a visiting faculty member at the University of Colorado at Boulder and at the University of California at San Diego. He frequently serves as an expert in patent-infringement and trade-secret litigation cases. Teich is a Fellow of the Institute of Electrical and Electronics Engineers, the Optical Society of America, the American Physical Society, the American Association for the Advancement of Science, and the Acoustical Society of America. He is a member of Sigma Xi and Tau Beta Pi. In 1969 he received the IEEE Browder J. Thompson Memorial Prize for his paper "Infrared Heterodyne Detection." He was awarded a Guggenheim Fellowship in 1973. In 1992 he was honored with the Memorial Gold Medal of Palacky University in the Czech Republic, and in 1997 he received the IEEE Morris E. Leeds Award. He has authored or coauthored some 350 journal articleslbook chapters and holds six patents. He is the co-author of the first edition of Fundamentals of Photonics (Wiley, 1991) and of Fractal-Based Point Processes (Wiley, 2005, with S. B. Lowen). Among his professional activities, he served as a member of the Editorial Advisory Panel for the journa] Optics Letters, as a Member of the Editorial Board of the Journal of Visual Communication and Image Representation, and as Deputy Editor for the journal Quantum Optics. He is currently a Member of the Editorial Board of the journal Jemna Mechanika a Optika and a Distinguished Lecturer of the IEEE Engineering in Medicine and Biology Society.
INDEX ABCD law, 92-94 matrix, 29 Absorption, 170-173,503,507-511 coefficient, 17 J fiber, 349 Acousto-optics,804-833 Add-drop multiplexer, optical (OADM), 1031, 1111 Add-drop multiplexer, reconfigurable optical (ROADM),1051 Airy's formulas, 249 Alcatel, 1016 Amplified spontaneous emission, 562, 603, 707, 715 Amplifier chirp pulse, 953 laser, 532-566, 569 optical fiber (OFA), 551-555, 1081 optical parametric (OPA), 1082 phase sensitive, 920 semiconductor optical (SOA), 702-716, 1082 Anisotropic media acousto-optic, 828-831 Anti-glare screen, 241 Anti-reflection coating, 252 Array detectors CCD, 776 CMOS, 776 materials, 775 readout circuitry, 776 structures, 775 Arrayed waveguide, see Waveguide grating router Atoms, 482-531 absorption, 503, 507-511 electron configuration, 486 energy levels, 483 hydrogen, 485 interaction with electromagnetic mode, 501 interaction with photons, 501-517 laser cooling, 516 laser trapping, 516 lineshape function, 504 manifold, 486 multielectron, 486 Pauli exclusion principle, 486 periodic table, 488 spontaneous emission, 50], 505-507 stimulated emission, 503, 507-5] I subshell, 486 term symbol, 486 Attenuation coefficient, 171 fiber, 348 Avalanche photodiodes, 767-775 advantages of, 791 buildup time, 774 excess noise factor, 783, 784 gain, 770 gain noise, 782 initial-energy effects, 785 ionization coefficients, 769 ionization ratio, 769 low-noise, 785 principle of operation, 768 reach-through device, 769 response time, 773 responsivity, 770 separate-absorption- grading - mul tiplication (SAGM), 1083 separate-absorption-multiplication, 769 single-photon, 774 thin, 785 threshold energy effects, 785 Bandgap wavelength, 662 Bardeen,John,627 Basov, Nikolai G., 532 Beam optics, 74-99 Beam, optical, 74-99 Bessel, 98 Bessel-Gaussian, 98, 170 donut, 97 Gaussian, 75-94 Hermite-Gaussian, 94-97 Laguerre-Gaussian, 97 quality, 85 vector, 169 Beamsplitter, 11 polarizing, 235 Beat frequency, 70 Beating, light, 70 1161
1162 INDEX Bell Laboratories, 1016 Bessel beam, 98 Bessel-Gaussian beam, 98 Biaxial crystal, 217 Bioluminescence, 523 Bistable optical devices, 1058-1068 Bit error rate, 777,1089 Blackbody spectrum, 519 Stefan-Boltzmann law, 520 Bloch modes, 265-267, 269,283 phase, 269 wavenumber, 266 Bloch, Felix, 243 Bloembergen, Nicolaas, 873 Bohr, Niels, 482 Boltzmann probability distribution, 465, 499 Born approximation, 814, 878 Born postulate, 484 Born, Max, 403 Bose-Einstein distribution, 466 Boundary conditions, 154 Bragg angle, 64, 258, 809 condition, 258, 809 diffraction, 806, 814 frequency, 258 reflection, 64 Bragg grating, 257-264, 602 beams, 816 chirp filter, 957 fiber, 596, 598, 1031 total reflection, 261 waveguide, 311 Bragg, William Henry, 804 Bragg, William Lawrence, 804 Brattain, Walter H., 627 Brewster angle, 212 window, 213 Brillouin zone, 266, 267, 271, 631 irreducible, 283 Bullet, light, 997 Cascaded-component matrices, 27 Cathodoluminescence, 522 Cavity resonator, 392-398 Characteristic equation fiber, 334 Chemiluminescence, 522 Chirp coefficient, 948 function, 1125 parameter, 940 pulse amplifier, 953 Chirp filter, 946-957 angular-dispersion, 954 Bragg-grating, 957 grating, 956 prism, 955 circ function, 1130 Circuit noise, 786 thermal noise, 786 Circular dichroism, 242 Circulator, optical, 239, 1020, 1025 Coherence, 403-443 area, 415 complex degree of temporal coherence, 408 cross-spectral density, 416 image formation, 429-435 interference, 419-427 length, 890 longitudinal, 417 mutual coherence function, 413 mutual intensity, 414, 428 power spectral density, 409 propagation, 427-435 quasi-monochromatic light, 424 spatial, 413, 423 spectral width, 411 temporal, 406, 420 temporal coherence function, 407 time, 408 visibility, 420, 423 Young's interference experiment, 423 Coherency matrix, 437 Coherent anti-Stokes Raman scattering (CARS), 528 Coherent optical amplifier, 533 Coherent optical communications, 11 l2-1118 Comb function, 1125 Complex q parameter, 76 Complex amplitude, 43, 75 Complex analytic signal, 67 Complex degree of temporal coherence, 408 Complex envelope, 44 Complex representation, 42, 67 Complex wavefunction, 42, 68 Component matrices, 26 Conductive medium, 181 Conductivity, 181 Confinement factor waveguide, 305 Conical refraction, 242 Constitutive relation, 156 Converter incoherent -to-coherent, 848 wavelength, 1052 Convolution, 1123 Convolution theorem, 1129 Coordinate transformation polarization, 207 Correlation, 1123 Coupled waveguides, 315 Coupled-mode theory waveguide, 316 Coupled-wave theory acousto-optics, 814
four-wave mixing, 917 pulsed three-wave mixing, 985 second-harmonic generation, 909 three-wave mixing, 905 Coupler directional, 843 grating, 315 input to waveguide, 313, 314 prism, 315 waveguide, 313-320 Coupling efficiency, 845 Critical angle, 211 Cross-connect, optical (OX C), 1111 Cross-phase modulation (XPM), 899, 1079 Crystal biaxial, 217 structure, 282 symmetry, 850 uniaxial, 217 Crystal-field theory, 494 Cutoff frequency waveguide, 302 Degree of polarization, 439 Detectors, see Photodetectors Dichroism, 235 circular, 242 Dielectric boundary reflection from, 50 refraction at, 50 Dielectric constant, 157 Dielectric medium doped, 493 Diffraction, 121-127 analogy with dispersion, 970 Bragg, 806, 814 Fraunhofer, 122 Fresnel, 124 grating, 56 Raman-Nath,818 Diffusion equation, 969, 970 Diode lasers, see Laser diodes Directional coupler, 843 soliton, 1037 Dispersion, 173-190 analogy with diffraction, 970 chromatic, 355 coefficient, 185 fiber, 351-359 group velocity (GVD), 185 material, 352 material and modal, 354 measures of, 175 modal, 351 multi-resonance medium, 189 nonlinear, 359 normaJ and anomalous, 187 polarization mode, 356 pulse propagation, 184-187 INDEX 1163 waveguide, 354 Dispersion compensating fiber (DCF), 356 Dispersion relation anisotropic material, 221 fiber, 334 photonic crystal, 270, 273, 284 waveguide, 295, 306 Dispersion-flattened fiber, 356 Dispersion-shifted fiber, 355 Doppler effect, 70 radar, optical, 70 shift, 811 Double refraction, 225, 242 Doubly negative materials, 191 Drude model, 182 Efficiency overall, 580 power-conversion, 580 wall-plug, 580 Eigenfunction, 1137 Eigenvalue, 1137 Eigenvalue problem, 1137 Eigenvector, 1137 Eikonal, 49 Eikonal equation, 23, 49, 342 Einstein A and ]ffi coefficients, 509 Einstein, Albert, 445, 482 Electric flux density, 153 Electro-optic effect, 882 Kerr, 895 Electro-optics, 834-872 anisotropic media, 849-856 Electroabsorption,868-869 Electroluminescence,523 Electromagnetic optics, 150-196 constitutive relation, 156 material equation, 163, 216 relation to scalar wave optics, 169 Electromagnetic wave anisotropic medium, 160 conductive medium, 181 dispersive medium, 160, 164,173,184-187 energy, 155 in dielectric medium, 156-169 in free space, 152-155 inhomogeneous medium, 158, 164 intensity, 155, 162 magnetic material, 190 momentum, 155 monochromatic, 162-169 negative-index material, 192 nonlinear medium, 161 power, 155, 162 resonant medium, 176 transverse (TEM), 165 Energy
1164 INDEX electromagnetic, 155, 162 optical, 41 Energy levels azimuthal quantum number, 485 Bohr atom, 485 Boltzmann Distribution, 499 C 5 + , 485 degeneracy parameter, 593 Fermi-Dirac distribution, 500 hydrogen, 485 magnetic quantum number, 485 manifold, 593 multielectron atoms, 486 occupation of, 499 principal quantum number, 485 spin quantum number, 486 Evanescent wave waveguide, 304 Excess noise factor, 777, 783, 784 Extinction coefficient, 171 waveguide, 304 skewed ray, 328 step-index, 327-330, 332-340 Fiber Bragg grating (FBG), 596, 598, 620, 1031 Fiber optics, 325-364, 1074 Filter acousto-optic, 827 Finesse, 65, 254, 372 Fluorescence, 524 Four-wave mixing (FWM), 900, 917, 1079 degenerate, 902, 903 Fourier optics, 102-137 periodic media, 274-277 pulsed waves, 975 Fourier transform, 1122-1131 one-dimensional, 1122-1128 optical, 116, 118 table of selected functions, 1125 two-dimensional, 1128-1130 Fourier, Jean-Baptiste Joseph, 102 Fourier-transform spectroscopy, 421 Franz-Keldysh effect, 868 Fraunhofer approximation, 116 diffraction, 122 Fraunhofer, Josef von, 102 Frequency light, 39 Frequency conversion, optical (OFC), 885, 912 Frequency shifter acousto-optic, 827 Frequency-resolved optical gating (FROG), 1009 Fresnel approximation, 46, 112 diffraction, 124 zone plate, 110 Fresnel equations, 211 Fresnel, Augustin Jean, 197 Fabry-Perot etalon, 254-257 Faraday effect, 230-231 rotator, 238 Fast light, 188 Fermat, Pierre de, 1 Fermi energy, 500 function, 500, 640 inversion factor, 671 level, 641 Fermi-Dirac distribution, 500, 641 Fiber, 325-364 V parameter, 334 absorption, 349 attenuation, 348 characteristic equation, 334 differential group delay (DGD), 357 dispersion, 351-359 dispersion compensating (DCF), 356 dispersion relation, 334 dispersion- flattened, 356 dispersion-shifted, 355 effective-index, 361 graded-index, 330-331 group velocity, 339, 347 holey, 244, 359-362 meridional ray, 328 modes, 334, 344 numerical aperture, 329 optimal index profile, 347 photonic-crystal, 359-362 polarization-maintaining, 341 propagation constant, 338, 346 quasi-plane wave, 342 response time, 352 single-mode, 340 Gabor, Dennis, 102 Gain coefficient saturated, 570 small-signal, 535, 570 Gates, optical, 1058-1068 Gauss, Carl Friedrich, 74 Gaussian chirped pulse, 942 function, 1125, ] 128 pulse, 942 pulsed beam, 945 Gaussian beam, 48, 75-94 M 2 factor, 85 ABCD law, 92 collimation of, 90 complex q parameter, 76 complex amplitude, 75 complex envelope, 76 confocal parameter, 80 depth of focus, 80 divergence angle, 80
expansion, 90 focusing, 88 Gouy effect, 81 in spherical-mirror resonator, 381 intensity, 77 phase, 81 power, 78 properties, 77-86 quality, 85 Rayleigh range, 76 reflection from spherical mirror, 91 relaying, 89 shaping, 88 transmission through free space, 93 transmission through lens, 86 transmission through thin optical compo- nent, 93 vector, 168 wavefront, 81 width, 79 General Electric Corporation, 680 Geometrical optics, see Ray optics Glass BK7, 956, 963, 965,974 silica, 968 Goos-Hanchen effect waveguide, 307 Goos-Hanchen shift, 241 Gordon, J. P., 936 Gouy effect, 81 Graded-index fiber, 21, 330-331 optics, 17-22 slab, 19 Green's function, 1132 Group velocity, 184 fiber, 339, 347 photonic crystal, 272 waveguide, 296, 306 Group velocity dispersion (GVD), 185 Guided waves, 289-324 Guided-wave optics, 289-324 Gyration vector, 229 Hagen-Rubens relation, 215 Harmonic oscillator, 1134 quantum theory, 471 Heisenberg uncertainty relation, 473, 1126 Helmholtz equation, 43, 332 generalized, 245, 1140 paraxial, 48 Hermite polynomials, 95 Hermite-Gaussian beam, 94-97 complex amplitude, 96 intensity, 97 Hermite-Gaussian function, 96 Hermitian operator, 1138 Hertz, Heinrich, 748 Heterodyne receiver, 1112 INDEX 1165 Heterodyning optical, 70 lIeterostructures photoconductors, 762 photodiode, 766 semiconductor optical amplifiers, 710 lIigh-harmonic generation, 604 Hilbert transform, 1134 Holey fiber, 359-362 Holography, 138-145 apparatus, 142 computer-generated, 1024 Fourier-transform, 141 off-axis, 140 rainbow, 144 real-time, 903 spatial filter, 141 volume, 143 Homodyne receiver, 1112 Huygens, Christiaan, 38 Huygens-Fresnel principle, 115 Hyperbolic secant function, 1125 IBM Corporation, 680 Image formation, 9 Imaging equation, 55 incoherent illumination, 429 partially coherent illumination, 429 partially coherent light, 435 single-lens, 55 Imaging system, 127-137 4- f, 129 imaging equation, 429 impulse response function, 429 near-field, 136 point spread function, 429 single-lens, 128, 132 two-point resolution, 431 Impedance, 165 Impermeability tensor, 218 Impulse response function, 1132, 1135 free space, 114 single-lens imaging system, 128, 133 Incoherent optical amplifier, 533 Index ellipsoid, 218, 220, 849 Infrared frequencies, 39 sensor card, 526 wavelengths, 39 Inner product, 1137 Instantaneous frequency, 939 Intensity electromagnetic, 155, 162 optical, 41, 68 partially coherent light, 405 polychromatic wave, 68 Interconnection capacity, 826
1166 INDEX Interconnection matrix, 1018 Interconnects, optical, 1018-1029 chip, 1027 diffractive, 1021 free-space, 1021 guided-wave, 1024 in microelectronics, 1026 nonreciprocal, 1020, 1025 Interference, 58-66 effect of spatial coherence, 423 effect of temporal coherence, 420 infinite number of waves, 64 light from extended source, 424 multiple waves, 62, 71 partially coherent light, 419-427 photon, 454 plane wave and spherical wave, 61 two oblique plane waves, 61 two partially coherent waves, 419 two spherical waves, 62 _ two waves, 58, 70 visibility, 73 Interferometer, 59-66 Mach-Zehnder, 59, 1032, 1046 Michelson, 59 Michelson stellar, 435 multipath, 1034 nonlinear, 1006 Sagnac,59,1047 self-referenced spectral, 1006 single-photon, 455 spectral, 1005 Young's double-pinhole, 423 Internal quantum efficiency, 648 Ionization coefficients, 769 history -dependent, 785 Ionization ratio, 769 Isolator, optical, 238, 1020, 1025 acousto-optic, 827 Isoplanatic system, 430 argon-fluoride excimer, 556, 605 argon-ion, 556, 605, 620 atomic, 600 Brillouin fiber, 598 capillary-discharge, 603 carbon dioxide, 556, 605, 620 carbon plasma, 556, 603, 605 cascaded Raman fiber, 597 cavity dumping, 608 chemical, 601 coherent random, 600 diode-pumped solid-state, 591 double-clad fiber, 595 dye, 601 efficiency, 580 erbium-doped silica fiber, 556, 596, 605, 620 examples of, 590-605 excimer, 600 exciplex, 600 extreme-ultraviolet, 602 fiber, 595 free-electron, 604, 605 frequency pulling, 574 gain clamping, 576 gain switching, 606, 610 gas, 600-601 He-Cd, 605 He-Ne, 556, 605, 620 homogeneous broadening, 582 hydrogen cyanide, 605 InGaAsP, 556 inhomogeneous broadening, 583 inner-shell photopumped, 604 internal photon-flux density, 575 internal photon-number density, 578 intracavity tilted etalon, 590 ionic, 600 krypton-fluoride excimer, 605 krypton-ion, 605 lasing without inversion, 598 line selection, 588 longitudinal-mode selection, 588 loss coefficient, 571 methanol, 605 microrandom, 600 mode locking, 608, 615-620 molecular, 600 multiclad fiber, 595 multiple-mirror resonator, 590 neodymium-doped glass, 556, 605, 618, 620 neodymium-doped YAG, 556, 592, 605, 620 neodymium-doped yttrium vanadate, 556, 592,605 number of modes, 582 optical-field-ionization, 604 oscillation frequencies, 574, 575 John, Sajeev, 243 Jones matrix, 205 J ones vector, 203 k surface, 223 Kao, Charles, 325 Kerr coefficient, 837 effect, 837, 854 electro-optic effect, 895 medium, 894 Kerr, John, 834 Kramers-Kronig relations, 175, 1134 Laguerre-Gaussian beam, 97 Laser, 567-626 active mode locking, 618 alexandrite, 556, 605
output characteristics, 575-590 output photon flux, 580 output photon-flux density, 577 passive mode locking, 618 phonon-terminated, 595 photon lifetime, 572 photonic-bandgap fiber, 596 plasers, 599 polarization, 587 polarization selection, 588 powder, 599 pulsed, 605-620 Q-switching, 607, 611 quantum cascade, 732 Raman fiber, 597 Raman phosphosilicate fiber, 598 random, 598 rate equations, 609 rhodamine-6G dye, 556, 605, 620 rub556,567,574,591,605,615 silver-plasma, 603, 605 soft X-ray, 602 solid state, 591-600 solid-state dye, 601 spatial distribution, 586 spatial hole burning, 583 spectral distribution, 581 spectral hole burning, 584 tabulation of selected, 605 theory of oscillation, 569-575 thin disk, 593 threshold, 572 threshold population difference, 573 thulium-doped fluoride fiber, 596, 605 titanium-doped sapphire, 556, 594, 605, 619,620 transient effects, 608-620 transverse-mode selection, 588 unstable resonators, 587 vibronic, 595 water vapor, 605 X-ray, 602 ytterbium-doped silica fiber, 551, 595, 605, 620 ytterbium-doped YAG, 556, 593, 605 Laser amplifier, 532-566, 569 amplified spontaneous emission, 562, 603, 707, 715 attenuation coefficient, 536 bandwidth, 537 continuous-wave operation, 539 Doppler-broadened medium, 561 double-clad fiber, 552 erbium-doped silica fiber, 551 examples of, 547-556 four versus three-level pumping, 546, 547 four-level pumping, 543 gain, 536 gain coefficient, saturated, 570 INDEX 1167 gain coefficient, small-signal, 535, 570 hole burning, 562 homogeneously broadened media, 556-560 in-line amplifiers, 548 inhomogeneously broadened media, 560- 562 Lorentzian phase-shift coefficient, 538 National Ignition Facility, 550 neodymium-doped glass, 549 noise, 562-564 nonlinearity, 556-562 optical fiber, 551, 553 optical pumping, 541 phase-shift coefficient, 538 photon-number statistics, 564 population difference, 541, 542 population inversion, 535, 539 power amplifiers, 548 preamplifiers, 548 pumping, 539-547 pumping dependent on population differ- ence, 544, 546 pumping methods, 547 Raman fiber, 553 rare-earth-doped fiber, 551 rate equations, 539-543 rates and decay times, 540 ruby, 548 saturated gain, 558 saturated gain coefficient, 556, 560 saturation photon-flux density, 557 saturation time constant, 543 small-signal approximation, 543 spontaneous-emission noise, 562 steady state, 539 tabulation of selected, 555-556 three-level pumping, 545 two-level pumping, 543 Laser diodes, 716-728 buried-heterostructure, 737 comparison with light-emitting diodes, 723, 726 comparison with superluminescent diodes, 723, 726 device structures, 736-741 differential responsivity, 722 distributed- feedback, 737 efficiency, 721 external differential quantum efficiency, 721 far-field radiation pattern, 727 gain condition, 718 internal photon flux, 720 light-current curve, 722 materials, 736-741 multi quantum-dot lasers, 731 muitiquantum-welliasers, 729 multiquantum-wire lasers, 730 output photon flux, 721
1168 INDEX photonic-crystal microcavity, 740 power output, 721 power-conversion efficiency, 723 quantum-dot lasers, 731 quantum-well lasers, 728 quantum-wire lasers, 730 ridge-waveguide, 737 single-mode operation, 727 spatial characteristics, 726 spectral characteristics, 724 strained-layer, 730 threshold, 718 vertical-cavity surface-emitting, 738 Lawrence Livermore National Laboratory, 550, 551,602 Layered media, 246-264 off-axis wave, 252 Lens, 13 double-convex, 55 imaging, 55 planoconvex, 55 thin, 54, 86 Ligand field theory, 494 Light guides, 15 Light line, 273 Light trapping, 16 Light-emitting diodes (LEDs), 682-702 characteristics of, 687-697 comparison with laser diodes, 723, 726 comparison with superluminescent diodes, 723, 726 device structures, 697-702 die geometries, 691 electronic circuitry, 696 external efficiency, 693 extraction efficiency, 689 internal efficiency, 688 internal photon flux, 688 materials, 697-702 organic, 701 output photon flux, 693 photonic-crystal, 692 polymer, 702 response time, 695 responsivity, 694 roughened surface, 691 spatial pattern, 692 spectral distribution, 695 white-light, 701 Line broadening, 511-515 collision, 512 Doppler, 514 inhomogeneous, 513 lifetime, 511 Linear system, 1132-1136 causal, 1133 impulse-response function, 1135 isoplanatic, 1135 modes, 1137-1141 one-dimensional, 1132-1135 point-spread function, 1135 shift-invariant, 1132, 1135 time-invariant, 1132 transfer function, 1133, 1136 two-dimensional, 1135-1136 Lineshape function, 504 Liquid crystals, 232-234 cholesteric, 232 display, 861 electro-optics, 856-863 ferroelectric, 860 modulator, 859 nematic, 232, 856 optical properties, 233 smectic,232 twisted nematic, 232, 859 Lorentz model, 176 Lorentzian function, 178 Lucent Technologies, 1016 Luminescence, 522-526 bioluminescence, 523 cathodoluminescence, 522 chemiluminescence, 522 electroluminescence,523 fluorescence, 524 multiphoton fluorescence, 524 phosphorescence, 524 photoluminescence, 523-526 sonoluminescence, 522 upconversion fluorescence, 525 Magnetic flux density, 153 Magnetization density, 154 Magneto-optics, 230-231 Magnetogyration coefficient, 231 Maiman, Theodore H., 567 Mandel's formula, 468 Manley-Rowe relation, 887, 909, 92], 935 Maser astronomical, 599 Material equation, 163,216,229,231 Matrix method polarization optics, 203-209 Matrix optics, 24-34 Bragg grating, 259 layered media, 246-264, 1139 periodic media, 268-274 polarization, 1138 ray transfer, 1138 Maxwell's equations boundary conditions, 154 in a medium, 153, 162 in free space, 152 Maxwell, James Clerk, 150 Metal plasma frequency, 602 Metamaterials, 191, 322 Michelson stellar interferometer, 435
Microcavity microdisk,396 micropillar, 396 microsphere, 397 microtoroid, 396 photonic-crystal, 399 rectangular, 395 Microcavity lasers, 734-741 Microresonator, see Microcavity Microscopy near-field, 136 scanning near-field optical (SNOM), 137 Miller's rule, 931 Miniband, 497, 658, 659, 673, 733 quantum cascade laser, 733 Minimum-detectable signal, 777 Mirror, 6 planar, 50 spherical, 91 MIT Lincoln Laboratory, 680 Mixing optical, 70 Mode locking, 615-620 active, 618 examples, 619 Kerr-lens, 619 methods, 618 passive, 618 properties, 616 saturable absorber, 619 semiconductor saturable-absorber mirrors, 619 Modes discrete linear system, 1138 homogeneous medium, 1141 integral operator, ] 139 linear system, 1137-1141 ordinary differential equations, 1140 partial differential equation, 1140 periodic medium, 1141 resonator, 1140 waveguide, 295, 300 Modulation field, 1] 01 frequency shift keying (FSK), 1102 intensity, 1102 ON-OFF keying (OOK), 1102, 1117 phase shift keying (PSK), 1102, 1118 Modulator acousto-optic, 819 electro-optic, 838 electroabsorption, 868 intensity, 840, 841, 855 interferometric, 840 liquid-crystal, 859 Mach-Zehnder, 840 phase, 838, 855 Pockels readout optical (PROM), 848 quadratic phase (QPM), 958 INDEX 1169 Molecule, 488-490 dye, 490 rotating diatomic, 489 vibrating diatomic, 489 vibrating triatomic, 489 Mollenauer, L. F., 936 Momentum electromagnetic, 155 Multiphoton fluorescence, 524 multiphoton micro lithography, 525 multi photon microscopy, 525 two-photon microscopy, 524 Multiphoton microlithography, 525 Multiphoton microscopy, 525 Multiple access code-division (CDMA), 1108 frequency-division (FDMA), 1108 time-division (TDMA), 1108 Multiplexing code-division (CDM), 1104 frequency-division (FDM), 1103 time-division (TDM), 1055, 1103 wavelength-division (WDM), 1030, 1104 Multiquantum well, 657 Multiquantum-dot lasers, 731 Muitiquantum-welliasers, 729 Multiquantum-wire lasers, 730 Mutual coherence function, 413 Mutual intensity, 414, 428 Nano-optics, 322 Nanophotonics, 137 Negative-index materials, 192 Network bus, 1106 interface., 1107 local-area (LAN), 1106 mesh, 1106 ring, 1106, 1109 star, 11 06 topologies, 1106 WDM, 1109 Network, fiber-optic, 1106-1112 Newton, Sir Isaac, 1 Noise background, 778 bit error rate, 777 circuit, 786 dark-current, 778 excess noise factor, 777 gain, 782, 785 minimum-detectable signal, 777 photocurrent, 779 photodetector, 777-798 photoelectron, 779 photon, 778 receiver sensitivity, 778 signal-to-noise ratio, 777 Nonlinear coefficient, 876, 924, 926
1170 INDEX Nonlinear optical loop mirror (NOLM), 1037 Nonlinear optics, 873-932 anisotropic dispersive medium, 931 anisotropic medium, 924 Born approximation, 878 coherence length, 890 coupled-wave equations, 905, 917 cross-phase modulation (XPM), 899 degenerate four-wave mixing, 902 dispersive medium, 927 doubly resonant oscillator (DRO), 916 electro-optic effect, 882 four-wave mixing (FWM), 900,917 high-harmonic generation, 604 Kerr medium, 894 Manley-Rowe relation, 887, 909 nonlinear Schrodinger equation, 898 optical frequency conversion (OFC), 912 optical Kerr effect, 895 optical Kerr lens, 897 optical parametric amplifier (OPA), 885, 914 optical parametric oscillator (OPO), 885, 915 optical phase conjugation (OPC), 902, 921 optical rectification, 881 parametric interactions, 886 periodically poled materials, 893 phase matching, 884, 887 phase mismatch, 911 phase-sensitive amplifier, 920 polarization density, 875-877 quasi-phase matching (QPM), 892 Raman gain, 898 real-time holography, 903 scattering theory, 878 second-harmonic generation (SHG), 551, 879,909 second-harmonic generation efficiency, 880,910 second-order, 879-894, 905-917 self-focusing, 897 self-phase modulation (SPM), 896 singly resonant oscillator (SRO), 916 solitons, spatial, 897 spontaneous parametric downconversion (SPDC), 885 sum-frequency generation, 551 third-harmonic generation (THG), 895, 921 third-order, 894-905, 917-923 three-wave mixing, 883, 90], 908, 919, 925 tuning curves, 887 wave equation, 877 Nonreciprocal polarization devices, 238 Normal modes anisotropic crystal, 218, 220 optically active medium, 229 polarization system, 208 Normalsurlace,223 Numerical aperture, ] 6, 329 Ohm's law, 757 Omnidirectional reflection, 278 Optic axis, 217 Optical activity, 228-230 Optical coherence tomography (OCT), 422 Optical component, 50-57 diffraction grating, 56 graded-index, 57 lens, 54, 55 mirror, 50 prism, 53 transmission through, 5] transparent plate, 52 Optical fiber amplifier (OFA), 551-555 comparison with semiconductor optical amplifier, 715 Optical fiber communications, 1072-] 1 ] 8 amplifiers, ] 080 analog, ] 097 attenuation compensation, ] 098 attenuation-limited, ] 091 components, 1074-] 083 dispersion compensation, 1098 dispersion management, 1099 dispersion-limited, ] 092 modulation, 1101-1] 02 multiplexing, 1 ] 03-] ] 06 networks, 1106-1112 nonlinear effects, ] 078 power budget, 1091 receivers, 1083 soliton, 1100 systems, 1084-1101 time budget, ] 092 transmitters, 1079 Optical indicatrix, see Index ellipsoid Optical Kerr effect, 895, 900 Optical Kerr lens, 897 Optical phase conjugation (OPC), 902, 92] Organic light-emitting diodes (OLEOs), 70] Organic semiconductors, 638 Oscillator optical parametric (OPO), 885, 915 Paraboloidal wave, 46 Parametric amplifier, optical (OPA), 885, 9] 4 Parametric oscillator, optical (OPO), 885, 915 Paraxial Helmholtz equation, 48 Paraxial rays, 7 Paraxial wave, 47 Parseval's theorem, ] 124 Partial coherence, see Coherence Partial polarization, 4340 coherency matrix, 437 degree of polarization, 439 Poincare sphere, 437 Stokes parameters, 437
INDEX 1171 unpolarized light, 438 Pauli exclusion principle, 486 Periodic media, 280, 282 Periodic optical system, 29-34 Periodic table elements, 488 semiconductors, 633 Periodically poled materials, 893 Permeability, magnetic, 153 Permittivity, electric, 153 effective, 181 tensor, 160 Phase matching, 276, 884, 887, 889 Phase modulator, quadratic (QPM), 958 Phase velocity, 45, 184 photonic crystal, 272 Phase-sensitive amplifier, 920 Phase-shift coefficient, 570 Phosphorescence, 524 Photoconductors, 752, 758-762 extrinsic materials, 761 gain, 760 intrinsic materials, 759 response ti me, 760 spectral response, 760 Photocurrent noise, 779 Photodetectors array detectors, 775 avalanche photodiodes, 767-775 circuit noise, 786 external photoeffect, 749 gain in, 755 general properties, 752-758 internal photoeffect, 749 noise in, 777-798 photoconductors, 752, 758-762 photodiodes, 762-767 photoelectric emission, 749 photoelectric work function, 749 quantum efficiency, 753 quantum-well infrared detector, 762 Ramo's theorem, 756 receiver sensitivity, 789 resonant cavity, 754 response time, 756 responsivity,754 signal-to-noise ratio, 789 thermal, 749 Photodiodes, 762-767 p-i-n junction, 765 p-n junction, 762 Avalanche, 767-775 bias, 763 heterostructure, 766 response time, 763 Schottky-barrier, 766 Photoelastic constant, 807 Photoelastic effect, 829 Photoelectron noise, 779 Photoluminescence, 523-526 Photon, 4458 energy, 448 in Gaussian beam, 451 in Mach-Zehnder interferometer, 455 in Young's interferometer, 454 interference, 454 momentum, 452, 453 orbital angular momentum, 454 polarization, 448 position, 451 spin, 454 time, 456 transmission through beamsplitter, 452 transmission through polarizer, 450 Photon detectors, see Photodetectors Photon lifetime, 572 Photon optics, 444-476 Photon statistics, 458--476 Bose-Einstein distribution, 466 coherent light, 463 exponential probability density function, 468 Laguerre-polynomial distribution, 564 Mandel's formula, 468 mean number of photon, 461 mean photon flux, 459 Poisson distribution, 463 random partitioning, 469 signal-to-noise ratio, 465 spectral densities of photon flux, 461 sub-Poisson, 475 thermal light, 465 vacuum state, 474 variance, 464, 467 Photon stream, 458--476 Photonic bandgap (PBG), 270, 273, 284 Photonic crystals, 265-288 band structure, 270, 273, 284 bandgap, 270,273,284 Bloch modes, 265 dispersion relation, 273 fi ber, 359-362 group velocity, 272 holes and poles, 286 holes on a diamond lattice, 285 microcavity lasers, 740 omnidirectional reflection, 278 one-dimensional, 265-279 phase velocity, 272 point defects, 286 projected dispersion diagram, 273 three-dimensional, 282-286 two-dimensional, 280-282 waveguide, 311 Woodpile, 285 Yablanovite, 285 Photorefractivity, 863-867 Planar boundaries 9
1172 INDEX Planck's constant, 448 Planck,ax,445,446 Plane wave, 44, 165 Plasma frequency, 182 Plasmonics, 321-322 Pockels coefficient, 837 effect, 837, 851 readout optical modulator (PRO), 848 Pockels, Friedrich, 834 Poincare sphere, 202, 437 Point-spread function, 429, 1135 Poisson, Simeon Denis, 748 Polarization circular, 201, 439 ellipse, 199 linear, 200, 439 matrix representation, 203-209 partial, 436-440 rotator, 207 TE, 211 T, 212 unpolarized light, 438 Polarization density, 154 Polarization mode dispersion (PD), 356 Polarization optics, 197-242 Polarization-maintaining fiber, 341 Polarization-mode dispersion, 1078 Polarizer, 205, 208 Polarizing beamsplitter, 235 Polychromatic light, 66-72 Polymer light-emitting diodes (PLEDs), 702 Power, 41, 155, 162 Power spectral density, 409 Poynting theorem, 155 Poynting vector, 155 Principal axes, 217 Principal refractive indexes, 217 Prism, 11, 53 electro-optic, 842 Prokhorov, Aleksandr ., 532 Propagation partially coherent light, 427-435 Pulse characteristics, 937 chirped Gaussian, 942 compression, 957, 967 detection, 999-1011 Gaussian, 942 Gaussian beam, 945 linear filtering, 946 plane wave, 944 propagation in dispersive media, 184-187 propagation in fiber, 960-973 self-phase modulation, 987 shaping and compression, 946-960 slowly varying, 944 spherical wave, 944 transform-limited, 942 Pulsed light, 66-72 Pupil function, 129, 130, 132 generalized, 133 Purcell factor, 515, 735, 740 Quadric representation, 217 Quality factor microresonator,395 microsphere resonator, 398 resonator, 376 Quantum cascade laser, 732 Quantum dot, 498, 660 artificial atom, 498 Quantum efficiency photodetector,753 Quantum number azimuthal, 485 magnetic, 485 principal, 485 spin, 486 Quantum state, 471-476 coherent, 474 photon-number-squeezed,475 quadrature-squeezed, 474 thermal, 465 twin beam, 476 Quantum well, 497,655 infrared detector, 762 semiconductor optical amplifiers, 711-715 Quantum wire, 497, 659 Quantum-confined lasers, 728-741 structures, 654-660 Quantum-dot lasers, 731 Quantum- well infrared detector (QWIP), 762 lasers, 728 Quantum-wire lasers, 730 Quarter-wave film, 252 Quasi-phase matching (QP), 892 Radiation pressure, 453 Raman cascaded fiber laser, 597 distributed fiber amplifier, 554 fiber amplifier, 553, 598 fiber laser, 597 gain, 898 lumped fiber amplifier, 554 phospho silicate fiber laser, 598 scattering, 527 stimulated Raman scattering, 527 stimulated scattering, 554, 597 Stokes shift, 554, 597 Raman-Nath diffraction, 818 Ramo's theorem, 756 Random light, see Coherence Rate equation photon-number, 609
INDEX 1173 population-difference, 609 Ray equation, 17 Ray optics, 1-37 beamsplitter, 11 cascaded-component matrices, 27 Eikonal equation, 23 external refraction, 9 graded-index fibers, 21 graded-index optics, 17-22 graded- index slab, 19 Hero's principle, 4 homogeneous medium, 4 internal refraction, 9 lenses, 13 light guides, 15 light trapping, 16 matrices of simple components, 26 matrix optics, 24-34 mirror, 6 mirror reflection, 5 optical components, 6-17 paraxial rays, 7 periodic optical system, 29-34 planar boundaries, 9 postulates, 3-6 prism, 11 ray equation, 17 ray-transfer matrix, 24 reflection and refraction, 5 relation to wave optics, 49 Snell's law, 6 spherical boundaries, 12 total internal reflection, 10 Ray-transfer matrix, 24 Rayleigh range, 76 Rayleigh scattering, 349 Rayleigh, Lord (John William Strott), 74 Reach-through APD, 769 Receiver sensitivity, 778, 789, 793 bit error rate, 795-798 Receiver, heterodyne, 1112 Receiver, homodyne, 1112 Reciprocal lattice, 280 Reciprocal system, 250 Rectangular function, 1125 Rectification, optical, 881, 986 Reflection, 50, 209-215 external, 211, 212 internal, 211, 213 omnidirectional, 278 phase shift, 212 TE polarization, 211 TM polarization, 212 total internal, 211, 241 Refraction, 50, 209-215 conical, 242 double, 225, 242 TE polarization, 211 TM polarization, 212 Refractive index, 158, 172 Resonance frequencies, 377, 385, 391 Resonant medium, 176 Resonator 9 parameters, 379 bow-tie, 370 cavity, 392-398 circular, 391 concentric, 380 confinement condition, 379, 380, 384 confocal, 380, 385 diffraction loss, 388 energy per mode, 467 finesse, 65, 372 frequency spacing, 373 loss coefficient, 374 losses, 371 microring, 1031 modes, 367-371, 381, 1140 multiple microring, 1032 photon lifetime, 375 photonic-crystal, 399 planar-mirror, 367-377 quality factor, 376 rectangular, see Cavity resonator resonance frequencies, 377, 385, 391 ring, 370 spectral width, 371 spherical-mirror, 378-389 stability, 380 three-dimensional, see Cavity resonator traveling-wave, 370 two-dimensional, 390-392 Resonator modes axial, 387 density, 371, 393, 396 Hermite-Gaussian, 386 off-axis, 377 planar-mirror resonator, 367-371 spherical resonator, 381 transverse, 387 whispering-gallery (WGM), 391 Resonator optics, 365-402, 571 multiple-scattering feedback, 598 Response time photodetector, 756 Responsivity photodetector, 754 Retarder, 206, 236 half-wave, 207 quarter-wave, 207 Rotator Faraday, 238 polarization, 233, 237 Rotatory power, optically active medium, 230 Router, passive optical, 1030-1037 intensity-based, 1036 Mach-Zehnder interferometer, 1032
1174 INDEX optical add-drop multiplexer (OADM), 1031 polarization-based, 1035 waveguide grating, 1033 wavelength-based, 1030 wavelength-division multiplexer, 1030 Russell, Philip St John, 325 Saturable absorber, 559 Scanner, 842 acousto-optic, 821 holographic, 109 Scattering, 526-528 anti-Stokes, 527 Brillouin, 527 coherent anti-Stokes Raman scattering (CARS), 528 Raman, 527 Rayleigh, 526 stimulated Brillouin, 528 stimulated Raman, 527 Stokes, 527 Scattering matrix, 247 relation to wave-transfer matrix, 248 Schawlow, Arthur, 873 Schawlow, Arthur L., 567 Schrodinger equation nonlinear, 898, 993 time dependent, 484 time independent, 484 sech( .) pulse, 991 Second-harmonic generation (SHG), 879, 909 efficiency, 880, 910 phase mismatch, 911 Self-focusing, 897 Self-phase modulation (SPM), 896, 1079 Self-phase modulation (SPM), pulse, 987 Sellmeier equation, 179, 674 Semiconductor laser amplifiers, see Semiconduc- tor optical amplifiers Semiconductor optical amplifiers (SOAs), 702- 716 bandwidth, 703, 704 comparison with optical fiber amplifiers, 715 gain, 703 gain coefficient, 705 heterostructures, 710 pumping, 708 quantum-well, 711-715 superluminescent diodes, 715 waveguide, 715 Semiconductors, 4999 n-type,636 p-i-n junction, 653 p-n junction, 650 p-type,636 absorption in, 663, 666, 671 alloy broadening, 687 amplifiers, 702-7] 6 bandgap wavelength, 662 Brillouin zone, 631 bulk, 496 carrier concentrations, 642, 645 carrier injection, 647 carrier mobility, 756 characteristics of LEDs, 687-697 density of states, 639 direct-bandgap, 633 doped, 636 effective mass, 631 electroluminescence from, 682-687 electrons and holes in, 630 emission from, 663, 666 energy bands, 629 energy-momentum relations, 631 Fermi function, 640 Fermi inversion factor, 67] gain in, 666 gain in quasi-equilibrium, 670 generation and recombination, 646 heterojunction,653 II-VI materials, 635 111- V materials, 633-635 indirect-bandgap, 633 internal quantum efficiency, 648 intrinsic, 636 IV-VI materials, 635 Kronig-Penney model, 629 laser diode, 716-728 law of mass action, 644 light-emitting diodes (LEDs), 682-702 microcavity lasers, 734-741 mini bands, 497, 658, 659, 673, 733 multiquantum well, 657 nanocrystal, 498 occupancy probability, 640 optics of, 660-679 optics of bulk, 660-672 optics of quantum-confined structures, 673 organic, 638 periodic table of, 633 photodetectors, 748-803 photon sources, 680-747 properties of, 627-660 quantum box, 498 quantum cascade laser, 732, 733 quantum dot, 498, 660 quantum well, 497, 655 quantum wire, 497, 659 quantum-confined lasers, 728-741 quantum-confined structures, 654-660 recombination coefficient, 646 recombination lifetime, 647 refractive index of, 674 saturable-absorber mirrors, 619 spontaneous emission from, 669 superlattice, 657
superlattice structures, 497, 658, 659, 673, 733 transition probabilities, 668 Separate-absorption-multiplication APD, 769 Shift-variant system, 430 Shockley, William P., 627 Signal-to-noise ratio, 777, 789 dependence on APD gain, 791 dependence on photon flux, 790 dependence on receiver bandwidth, 792 Single-mode fiber, 340 waveguide, 302 Single-photon avalanche photodiode (SPAD), 774 Skin depth, 182 Slow light, 188 Snell's law, 6 Solid, 490-499 actinide-metal doped, 496 covalent, 491 doped dielectric, 493 ionic, 491 lanthanide-metal doped, 495 metallic, 491 rare-earth doped, 495 ruby, 493 transition-metal doped, 493 Solitary wave, 989 Soliton, 988-997, 1048 condition, 990 dark, 996 directional coupler, 1037 fundamental, 994 generation, 995 higher-order, 994 interaction, 994 laser, 996 optical fiber communications, 1100 period, 994 spatial, 897 spatiotemporal,997 temporal and spatial, 996 Sonoluminescence, 522 Spatial filter, 131 holographic, 141 Spatial frequency, 103 Spatial harmonic function, 105 Spatial light modulator (SLM), 847, 848 acousto-optic, 825 electro-optic, 846 liquid-crystal, 861 Spatial spectral analysis, 106 Spectrogram, 941, 1009 Spectrum analyzer acousto-optic, 823 Speed of light in a medium, 40, 157 in free space, 40, 153 INDEX 1175 Spherical boundary, 12 Spherical mirror, 91 Spherical wave, 45, 166 Spontaneous emission, 501, 505-507 enhanced, 515 Purcell factor, 515 Spontaneous parametric downconversion (SPDC), 885 Stark effect, quantum-confined (QCSE), 868 Statistical optics, 403-443 Step-index fiber, 327-330, 332-340 Stimulated Brillouin scattering (SBS), 528, 1079 Stimulated emission, 503, 507-511 Stimulated Raman scattering (SRS), 527, 1079 Stokes parameters, 202, 204, 437 Stolen, R. H., 936 Strain-optic tensor, 829 Supercontinuum light, 997 Superlattice structure, 497, 657-659, 673, 733 quantum cascade laser, 733 Superluminescent diodes (SLDs), 715 comparison with laser diodes, 723, 726 comparison with light-emitting diodes, 723, 726 Superposition principle of, 41 Superprism, 1031 Surface plasmon polariton (SPP), 322 Susceptibility, electric, 156 resonant medium, 177 tensor, 160 Switch acousto-optic, 824 electro-optic, 838 electroabsorption, 868 quantum-confined Stark effect (QCSE), 868 waveguide, 319 Switch, photonic, 1038-1058 acousto-optic, 1045 all-optical, 1046 architectures, 1038 characteristics, 1039 electro-optic, 1042 fundamental limits, 1050 implementations, 1039 liquid-crystal, 1044 magneto-optic, 1045 mechano-optic, 1041 multidimensional space-wavelength, 1052 nonlinear Mach-Zehnder interferometer, 1046 nonlinear optical retardation, 1048 nonlinear Sagnac interferometer, 1047 optoelectronic, 1039 packet, 1056 semiconductor, 1043 soliton, 1048 space, 1038
1176 INDEX thermo-optic, 1046 time-division multiplexing, 1054 time-domain, 1053 time-slot interchange, 1056 wavelength-domain, 1051 Synchronous optical network (SONET), 1108 Tail band, 644 Fermi, 641 Urbach, 672 Tensor, 216 Term symbol, 486 Thermal light, 517-522 blackbody spectrum, 519 thermography, 521 Thermography, 521 Third-harmonic generation (THG), 895, 921 Three-wave mixing, 883, 901, 908, 919, 925 THz generation, 986 Time- varying spectrum, 941 TOAD, 1048 Total reflection Bragg grating, 261 Townes, Charles H., 532 Transfer function, 1133, 1136 free space, 111 ray-optics imaging system, 130 single-lens imaging system, 135 Translational symmetry, 1139 Transmission matrix see wave-transfer matrix, 247 Transmittance, complex amplitude diffraction grating, 56 graded-index plate, 57 optical component, 51 plate of varying thickness, 52 prism, 53 thin lens, 54 transparent plate, 52 Transverse electromagnetic (TEM) Wave, 165 Tuning curves, 887 Two-photon microscopy, 524 Two-point resolution, 431 Tyndall,John,289 Ultraviolet frequencies, 39 wavelengths, 39 Uncertainty field quadrature components, 473 position-momentum, 473 time--energy, 456 Undulator,604 Uniaxial crystal, 217 Unitary operator, 1138 Unpolarized light, 438 Vacuum state, 474 Van Cittert-Zemike theorem, 433 Variational principle, 1138 Vector beam, 169 Vector potential, 166 Velocity group, 184,272,296,306,339,347 phase, 45, 272 Verdet constant, 231 Vertical-cavity surface-emitting lasers (VCSELs 738 Visibility, 73, 420, 423 Visible frequencies, 39 wavelengths, 39 Walk-off effect, 984 Wave complex amplitude, 43 complex analytic signal, 67 complex envelope, 44 complex representation, 42, 67 complex wavefunction, 42, 68 equation, 40 monochromatic, 41-49 paraboloidal, 46 paraxial, 47 plane, 44 pulsed-plane, 69 spherical, 45 Wave equation, 153 Wave optics, 38-73 postulates of, 40-41 relation to ray optics, 49 Wave retarder dynamic, 839 Wave-transfer matrix, 246 relation to scattering matrix, 248 Wavefront optical, 44 Waveguide asymmetric planar, 308 Bragg-grating, 311 channel, 310 confinement factor, 305 coupled-mode theory, 316 coupling, 313-320 cutoff frequency, 302 dispersion relation, 295, 306 evanescent wave, 304 extinction coefficient, 304 field distribution, 303 GaAs/AIGaAs,311 Goos-Hanchen effect, 307 group velocity, 296, 306 metal, 321-322 modes, 291, 300 number of modes, 295 periodic, 320 photonic-crystal, 311
INDEX 1177 planar-dielectric, 299-308 planar-mirror, 291-299 plasmonic,321-322 propagation constant, 293 rectangular dielectric, 309 rectangular mirror, 308 silica-on-silicon, 311 silicon-on-insulator, 311 single mode, 302 switch, 319 Ti:LiNb0 3 ,311 two-dimensional, 308 Waveguide-grating router, 1033 Wavelength infrared, 39 light, 39, 44 ultraviolet, 39 visible, 39 X-ray, 39 Wavelength converter, 1052 Wavelength-channel interchange, 1051 Wavelength-division multiplexer, 1030 Wavevector, 44 WDM network, 1109 broadcast -and-select, 1109 multi-hop, 1110 wavelength-routed, 1111 White organic light-emitting diode (WOLED), 702 Width, 1124-1128 1/ e-, 1127 3-dB, 1127 full-width at half-maximum (FWHM), 1127 power-equivalent, 1126 root-mean-square, 1124 Wiener-Khinchin theorem, 410, 442 Wiggler field, 604 Wigner distribution function, 1009 Wolf, Emil, 403 X-ray frequencies, 39 laser, 602 wavelengths, 39 Yablanovite, 285 Yablonovitch, Eli, 243 Young's interference experiment, 423, 454 Young, Thomas, 38 Zero-point energy, 448 Zone plate, Fresnel, 110
USEFUL CONSTANTS Speed of light in free space Co 2.9979 x 10 8 mls Planck's constant h 6.6261 x 10- 34 J. s Permittivity of free space fo 8.8542 x 10- 12 F 1m Electron charge e 1.6022 x 10- 19 C Permeability of free space 110 1.2566 x 10- 6 Him Electron mass mo 9.1094 x 10- 31 kg I mpedance of free space '1]0 376.73 n Boltzmann's constant k 1.3807 x 10- 23 J/oK PREFIXES FOR UNITS 10- 18 10- 15 10- 12 10- 9 lo-{j 10- 3 10 3 10 6 10 9 10 12 10 15 10 18 I I I I I I I I atto femto plCO nano micro milli kilo mega glga tera peta exa (a) (f) (p) (n) (/I) (m) (k) (M) (G) (T) (P) (E) PHOTONS Energy E=hv eV Frequency v J cm- I 10 Example: A photon of frequency 1/= 300THz has free-space wavelength >"0= 1 pm and energy E= 1.99 x 10- 19 J= 1.24eV= l()4cm- 1 OPTICAL PULSES Pulse width FEI\II'OSH 'O )) OPTICS 10 fs 100 fs T Pu I se length CT 100 THz 10 THz I THz 100 GHz 10 GHz 1 GHz 100 MHz STRUCTURES N \NO-OP I"ICS 1\11( 'I O-()It I"ICS IU'I h. (WTICS I IA I Inm I 10 nm I 100nm I l/lm I 10 /lm I 100 pm I I mm I lcm