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Author: Tkachenko N.V.
Tags: optics spectroscopy spectral theory optical spectroscopy experimental physics
Year: 2005
Text
Preface
This book is intended for graduate and post graduate students, and researchers planning
to start an advanced experimental work in the fast growing field of optical spectroscopy.
Optical spectroscopy methods have numerous applications in physics, chemistry, material,
biological and medical sciences. Probably two most exciting achievements of the optical
spectroscopy are single molecule detection and ultrafast time resolution. The former shifts
research work to a molecular scale and serves as the key tool in areas known as nanochem-
istry and molecular devices. The latter extends the time scale to femtoseconds making
possible direct studies of chemical reactions on the level of the chemical bond dynamics at
the atomic scale. This research field is commonly called F emtochemistry after Nobel price
winer Ahmed H. Zewail.
The progress in the optical spectroscopy was possible because of great developments
in laser physics, optics, electronics and computers. Combining together newest lasers, ad-
vanced detectors and high technology optical components a researcher can assemble rela-
tively easy a spectroscopy instrument with characteristics which were hardly achievable in
the top level laser research labs a decade ago. Naturally, the first step in this direction is
to be informed on the tools available and to be able to evaluate the benefits and limitations
imposed by different techniques. On the other hand the researchers, who are potentially in-
terested in such spectroscopy applications, are experts in the fields of their own professional
interest, such as materials science or microbiology, and may have only basic background
knowledge in optics and modem laser physics. The aim of the book is to cover the gap by
providing background information in optics and by focusing on spectroscopy methodology,
tools and instrumentations. The goal is to provide a background for quantitative estima-
tions of the applicability range of optical spectroscopy methods and to help researchers in
planning, designing and developing of new spectroscopy instruments, and, hopefully, new
spectroscopy methods.
Logically the book can be divided on two parts. The first part, Chapters 1-4, covers a
few subjects important for technical implementations of all spectroscopy instruments. This
includes optics, opto-electronics, laser physics and related topics. This part of the book
has no intention to go deep into its subj ects nor to provide a complete overview of these
broad areas. In turn, it is supposed that readers are already familiar with the subject and
the main goals of this part of the book is to remind readers of the key concepts, important
theories, principle values and available tools which are used in spectroscopy applications.
One example of such tool is a monochromator. Monochromators can be found in almost
every spectroscopy device and they are crucial for such important characteristics of the
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devices as spectrum resolution and sensitivity.
The second part of the book, Chapters 5-13, is the main part, which divided onto Chap-
ters according to the types of the optical spectroscopy methods described. Each Chapter
starts with a general description of the design principles of a particular methods. This fol-
lows by an introduction of the approaches used to estimate the most important features of
the instrument, such as spectrum and time resolution, and by discussions of what are appli-
cability ranges for this particular spectroscopy technique. In the final part of each Chapter
examples of the instruments and measurements are provided. The order of the Chapters
roughly follows the order of inventing different spectroscopy methods and one can see that
the important direction in the development of optical spectroscopy instruments was the con-
tinuous improvement of the time resolution.
It is important to notice from the very beginning that the spectroscopy techniques do not
exist for its own - they were developed for the purpose of investigation of certain objects,
e. g. natural photosynthetic centers. The application of one or another method, or com-
bination of methods is justified by the object to be studied and the problem to be solved.
Therefore one of the main goals of the book is to compare different spectroscopy techniques
and to highlight advantages and disadvantages of them in respect to the most common ap-
plication tasks.
Two last topics discussed in the book are polarization (Chapter 14) and analysis of the
measurements (Chapter 15). They are important to all spectroscopy techniques and, there-
fore, were arranged as separated Chapters. The polarization of the light is its fundamental
property. If it is ignored, it may lead to misinterpretation of the experimental results. On
the other hand, a careful accounting for the light polarization and sample anisotropy may
help to improve the quality of the measured data. Additionally, polarization and anisotropy
can be used to gain an additional information on the samples under study. For example, the
excitation energy transfer can be studied by measuring sample anisotropy. In the latter case
the actually measured data are light intensity time profiles at different polarizations of the
light. The anisotropy is calculated then out of these primary measurements. This exam-
ple highlights the importance of the accurate analysis of the experimental data. Even more
common data analysis problem for the optical spectroscopy is extraction of the lifetimes of
the intermediate states in a photochemical reaction. This is the subj ect of Chapter 15.
At last, but not at least, I would like to express my gratitude to many great scientist who
were my teaches and my colleagues. After graduation from Moscow Institute of Physics
and Technology I have joined General Physics Institute (GPI) as young researcher, where
I was guided by Prof. A. M. Prokhorov and Dr. V. V. Savranskii in my work and doctor
thesis preparation. From them I have learned a great deal about laser physics. In GPI I have
assembled my first flash-photolysis instrument together with A. Dioumaev, V. Chukharev
and A. Sharonov, young scientists at that time. Later I was invited by Prof. H. Lemmetyi-
nen to join his team at the University of Helsinki. Prof. Lemmetyinen is a great experts in
photochemistry, and that was the time to learn new optical spectroscopy methods, such as
the time correlated single photon counting. A few year later the research team moved to
Tampere University of Technology, where the instrumentation facilities of the group were
extended by building up a new femtosecond spectroscopy system to be able to carry out op-
tical spectroscopy studies in a wide time scale from steady state to femtoseconds. Educating
students for advanced research work in the spectroscopy laboratory was the motivation to
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write this book.
Most of all, I am indebted to my family, especially my wife Natalia for her love and tol-
erance of irregular hours at home, and my daughter Evgenia for her help in text preparation.
Nikolai V. Tkachenko
Tampere, Finland
December 2005
Contents
Preface
Contents
1
Introduction
1.1 Absorption
1.1.2 Light absorption in a bulk medium
1.1.2 Absorption of complex samples
1.1.3 Electronic, vibrations and rotational levels
1.1.4 Wavelength, frequency and energy
Emission
1.2.1 Black body emission ............................................
1.2.2 Two level system (Einstein's coefficients) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Fluorescence and phosphorescence
1.2
1.3
1.4
Light amplification . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optical spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optics and Optical Device
2.1 Waves......
2.1.1 Wave equation. . . . . .
2.1.2 Harmonic waves
2.1.3 Plane waves
2
2.2 Interference...........................................................
2.3
2.4
2.2.1
2.2.2
2.2.3
Michelson interferometer
Fabry-Perot interferometer .....
Interference filters and mirrors
Diffraction . . . . . . . . . . . . . . . . . . . . . . . .
Fresnel formulation . . . . . . . . . . . . . . . . . . . .
Fraunhofer diffraction (far field approximation)
Diffraction grating.
2.3.4 Monochromator...
Calculations of optical system (matrix formulation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Geometrical optics approximation
2.4.2 Beam transfer matrix. . . . . .
2.4.3 Imaging and magnification .....
2.3.1
2.3.2
2.3.3
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Lasers for Spectroscopy Applications
3.1 Laser active medium. . . . . . . . .
3.2
3
Laser resonators . . . . . . . . . . . . . . . . . .
3.3
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Resonator with active medium .
Resonator bandwidth
Longitudinal modes
Transverse modes .........
3.2.5 Stable and unstable resonators
Continuous wave lasers
Pulsed lasers .........................................................
3 4.1 Q-Switched lasers
3 4 2 Mode-locked lasers ......
Laser amplifiers. . . .
Maine types of lasers
3.6.1 Nd:Y AG lasers.
3.6.2 Ion lasers. ....
3.6.3 Excimer lasers. .
3.6.4 Dye lasers.
3.6.5 Ti:sapphire lasers
3.6.6 Semiconductor lasers ............................................
3.6.6 Other lasers used in spectroscopy applications
Non-linear optic effect in laser applications. . . . . . . . . . . . . .
3.7.1 Second harmonic
3.7.2 Third harmonic
3.7.3 Wave mixing
3.7.4 Parametric amplification and generation of the light . . . . . . . . . .
3.2.1
3.2.2
3.2.3
3.2.4
3.5
3.6
3.7
Optical measurements
4.1 Noise statistics and accuracy of measurements
41.1 Systematic error and random noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4.1.2 Noise statistics . . . . . . . . . . . . . . . . . . . . . . .
4.2
4.1.3 Statistical approach to measurements
4.1.4 Noise sources
4.1 5 Inaccuracy of indirect measurements. . . . . . . .
Photosensitive devices
4 2.1 Photodetector performance parameters
4.2.2 Photomultiplier tubes. ......
4.2.3 Semiconductor photo-detectors
4.2.4 Other photo-detectors
Measurements of the light power
Measurements of the pulse energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
4.4
4.5
Measurements of the pulse duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Direct methods. . . . . . . . . . . . . . . . . . .
4.2.4 Autocorrelators (indirect methods)
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5
Steady State Absorption Spectroscopy
5.1 Measurements of the light absorption spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Spectrophotometer schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Single channel scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Two channel scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Spectrophotometers with array detectors ..............................
5.3 Main characteristics of spectrophotometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Spectrum range .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Spectrum resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Sensitivity and absorption range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Instruments, accessories and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Spectrophotometer specifications ...................................
5.4.2 Cuvettes for absorption spectroscopy ................................
5.4.3 Application notes and examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Steady State Emission Spectroscopy
6.1 Measurement of the Emission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Fluorimeter.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Optical Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Use of Array Detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 Evaluation of the Measured Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Spectrum Correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.5 Quantum yield determination by comparison method. . . . . . . . . . . . . . . . . . . . . .
6.2.6 Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.7 Sensitivity....................................................
6.2.8 Wavelength resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Samples for emission measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Excitation-monitoring schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Cuvettes......................................................
6.3.3 Effect of the sample absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Fluorimeter specifications ................................................
6.4.1 Water Ramon scattering line as sensitivity test. . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Commercial fluorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Emission of molecular monolayer: An example
7 Flash-photolyis
7.1 Principles
7.1.1
7.1.2
7.1.3
7.1.4
7.1.5
Optical scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transient absorbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Differential absorption spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excitation schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Time resolution and signal-to-noise ratio .....................................
7.2.1 Pulsed monitoring light .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Signal averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
7.2.1 Spectrum range and spectrum resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143
7.3 Measurements of emitting samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143
7.3.1 Effect of scattering and sample emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143
7.3.2 Applications in time resolved emission spectroscopy . . . . . . . . . . . . . . . . . . . . .. 145
7.4 Flash-photolysis instruments .............................................. 145
7.4.1 Commercial instruments and components .... . . . . . . . . . . . . . . . . . . . . . . . . .. 145
7.4.2 Flash-photolysis study of an electron transfer: An example . . . . . . . . . . . . . . . . .. 146
8
Time correlated single photon counting
8.1 Principles............................................................
8.2 Excitation sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Detection subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Emission detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Electronics....................................................
8.4 Method characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Time resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Peal-up distortions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.3 S ens i ti vi ty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 .4.4 Signal coli e cti 0 n time ............................................
8.4.5 Spectrum range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.6 Comparison with direct emission decay measurements. . . . . . . . . . . . . . . . . . . . .
8.5 Measurements and data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1 Instrument response function and decay deconvolution. . . . . . . . . . . . . . . . . . . . .
8.5.2 Time resolved and decay associated spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Commercial instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Measurements of single molecule: Application example. . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Frequency domain emission spectroscopy 171
9.1 Theoretical bacl(ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 71
9.2 Measurements scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174
9.3 Frequency domain instruments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 175
9.3.1 Light source .................................................. 176
9.3.2 Detection system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176
9.4 Comparison between frequency and time domain methods . . . . . . . . . . . . . . . . . . . . . . . .. 176
10
Picosecond time resolution with streak camera
10.1 Operation principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Main characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.1
10.2.2
10.2.3
10.2.4
Time resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectrum range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SensItIvIty ...................................................
Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 O. 3 Ins trumen t e xamp Ie s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11
Pump-probe
11.1 Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1
11.1.2
11.1.3
11.1.4
Mono-color scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two color scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurements of time resolved spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Samples and sample excitation schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Laser system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Detection subsystem and sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Time resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1
11.4.2
11.4.3
11.4.4
Group velocity dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effects of sample and optics on time resolution. . . . . . . . . . . . . . . . . . . . . . . . .
Measurements of the delay spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Can it be faster? ..............................................
11.5 Sensitivity .........................................................
11.6 Application example ..................................................
11.6.1 Photo-induced charge transfer in molecular dyad. . . . . . . . . . . . . . . . . . . . . . . .
11.6.2 Pump-probe study of thin films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Emission spectroscopy with optical gating methods
12.1 Frequency up-conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Principles of up-conversion ......................................
12.1.2 Time resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.3 Evaluation of the instrument response time. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1.4 Sensitivity...................................................
12.1.5 Excitation pulse energy .........................................
12.1.6 Spectrum range ...............................................
12.1.7 Time resolved spectra ..........................................
12.1. 8 Commercial instruments and components. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Optical Kerr effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Photo-dynamics of semiconductor quantum wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Ultra-fine spectrum resolution
13.1 N aturalline width and broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Traditional optical tools for high spectrum revolution . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Lasers for fine spectrum resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Resonator limited bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.2 Amplification bandwidth and lasing threshold. . . . . . . . . . . . . . . . . . . . . . . . .
13.3.3 Mode-beating and resonator design for single mode lasers . . . . . . . . . . . . . . . .
13.4 High resolution in absorption spectroscopy ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Laser spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.2 Intra-cavity spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5 High resolution in emission spectroscopy ...................................
13.6 Spectral hole-burning. .................................................
185
185
185
188
190
193
194
200
201
202
204
206
207
210
210
210
212
217
217
217
221
223
225
227
228
229
231
231
234
237
237
239
241
241
241
243
244
244
244
245
248
xiv
Contents
14 Polarization measurements 249
14.2 Light polarization .................................................... 251
14.2 Interaction of polarized light with media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 251
14.2.1 Magic angle ................................................. 252
14.2.2 Induced anisotropy in fluorescence measurements ...................... 253
14.2.3 Anisotropy coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 255
14.3 Applikations of polarized measurements .................................... 257
14.3.1 Tools for polarized measurements .................................. 257
14.3.2 Optical schemes for polarization measurements ........................ 258
14.3.3 Measurements of energy transfer dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . .. 259
15 Analysis of the measurements 263
15.1 Indirect measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263
15.2 Spectral data analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 264
15.3 Kinetics and reaction schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
15.3.1 First order reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
15.3.2 Second order reactions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 268
15.3.3 Complex schemes for the first order reactions. . . . . . . . . . . . . . . . . . . . . . . . .. 269
15.3.4 Time resolved measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 274
15.4 Data fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 277
15.4.1 Criteria for the fit goodness ...................................... 277
15.4.2 Minimization of mean square deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 278
15.4.3 Non-linear least square fit ....................................... 280
15 4.4 Global fitting of time resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . .. 283
15.4.5 Qualitative problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 284
16 Final remark 287
A Photon counting peal-up distortions 289
B Relation between Gaussian pulse width and its spectrum 291
C Two photon absorption 293
D Fit algorithms 295
D.l Stepping algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295
D.2 Gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 296
D.3 Newtonmethod...................................................... 296
D.4 Random search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 297
E Physical properties of some solvents 299
Bibliography
301
Index
304
Chapter 1
Introduction
Optical spectroscopy studies absorption and emission of light by matter. Originally the
studies were related to the wavelength dependences of these processes, but as new methods
and directions of research were developed the scope of optical spectroscopy has enlarged.
One of the important directions of such development today is time resolved spectroscopy,
where optical methods provide superior time resolution not achievable by any other methods
available. There are also specialized areas of optical spectroscopy such as single molecule
spectroscopy and non-linear optical spectroscopy, which have also been under active devel-
opment during the past decade.
Naturally, the reason for the great attention paid to the optical spectroscopy techniques
in recent fundamental research is new exciting knowledge gained. To mention few there
are chemical reaction dynamics at single bond level, called femtochemistry after Ahmed
H. Zewail, and single molecule spectroscopy. Furthermore, non-destructive spectroscopy
methods have found numerous applications in monitoring different processes in industry
and environmental technologies.
The aim of this book is to give an overview of the modem optical spectroscopy methods,
and to introduce the principles used to evaluate quantitatively advantages and application
ranges of the methods. The author hopes that this information will help researchers and
application engineers to plan their optical spectroscopy work and to get the most of each
method used.
In this introductory chapter we will briefly look at fundamental concepts of the light
absorption and emission. The main goal, however, is to mention the most widely used con-
cepts and terms in optical spectroscopy and its applications. It is assumed that the readers
are quite familiar with the subject and a short reminding of the topic can be appropriate for
the following consideration.
1.1 Absorption
1.1.1 Light absorption in a bulk medium
Let us consider a beam propagating in an isotropic dielectric medium. One can choose a
coordinate system so that the beam is propagating along X -axis, as presented in Fig. 1.1.
1
2
Introduction
I
1(0)
:::;.
11.1 I 1(1)
I
: I1.x
->. :<- --
a x I X
Figure 1.1: Light absorption in a medium.
At a point x the light intensity is I (x). In a layer of thickness D.X the light interacts with the
medium and at the point x + D.X the light intensity is changed by value I, which means
I(x + D.x) == I(x) + D.I(x, D.x). At D.X ---+ 0 the ratio D.I D.X) is proportional to the
light intensity:
lim LI(x, Lx) = -aI(x)
.6.x---+o D.X
(1.1)
or
dI(x) = -aI(x)
dx
(1.2)
where a is the proportionality coefficient determining efficiency of the light absorption.
The minus sign on the right side of the equation is due to the fact that the light is absorbed
by the medium, i. e. the function I( x) is decreasing with increasing x, and, thus, it has a
negative slope. To verify the statement (1.1), let us consider a transparent medium which
incorporates absorbing centers, for example dye molecules in a solution. The probability,
p, to absorb a photon by a thin layer of the medium is proportional to the surface density
of the absorbing centers, s, and absorption efficiency of the centers, a, so that p == sa. If
the volume density (concentration) of the absorbing centers is n, then the surface density
of the absorbing centers in the layer of thickness D.X is s == nD.x. For a very thin layer
the absorption probability is very small positive value, and for the incident light flux I, the
decreaseinintensityis I == -Ip,or- I == Isa == Ina x. Thus, == -naI == -aI,
where a == na.
Equation (1.2) can be rearranged
dI(x) == -adx
I(x)
(1.3)
and solved
In[I(x)J == -ax + C
(1.4 )
Where the constant C is determined by the initial conditions, which is the incident light
intensity at x == 0, i. e. 1(0) == 1 0 , in this particular case. Equation (1.4) is usually
1.1. Absorption
3
converted to the form known as Lambert law 1
I(x) == 10 exp( -ax)
(1.5)
The coefficient a is called absorption coefficient and it has measure of inverse length
(e. g. cm- 1 ). If the coefficient a > 0, then the light intensity decreases along the light
propagation direction. There is no light absorption if the coefficient a == 0, i. e. I (x) ==
10 == constant. The light intensity also increases exponentially if a < o. The latter case is
called light amplification and will be discussed in Section 1.3.
When the light propagate across finite length absorbing medium, one may be interested
in portion of the light which will be absorbed or will pass the medium. Let us assume that
the thickness of the medium is l (Fig. 1.1) and its absorption coefficient is a. The light
intensity before the medium is I in == 10 == 1(0) and, according to eq. (1.5), the light
intensity after the medium is
lout == I(l) == line- al
( 1.6)
The transmittance of the sample (the relative amount of the light passing through the sample)
IS
T == lout == e- al
I in
( 1.7)
In other words, eq. (1.6) can be rewritten as
lout == linT
( 1.8)
Consequently, the absorptance of the sample (the relative amount of the light absorbed by
the sample) is
a == I in - lout == 1 - T == 1 _ e- al
I in
( 1.9)
These values, absorptance and transmittance, are usually used when one needs to calculate
intensity of the light propagating in some optical system, i. e. when the light intensity
distributions inside the optical components are out of interest.
The absorptance and transmittance are not very convenient characteristics when some-
one is interested in the optical properties of the absorbing centers, since the absorptance
depends on the thickness of the sample and the density of centers. Because of some practi-
cal and historical reasons different values are used to characterize absorption properties of
media. The absorption coefficient, a, as given by eq. (1.6), is a common specification for
the media such as glasses or fibers. 2 The absorption coefficient does not depend on optical
path and characterizes the medium itself.
lOne can note the relation 10 = exp(C), linking eqs. (1.1) and (1.4).
2These are l11aterials with fixed density of absorbing centers, therefore the saI11ple absorption depends only on
their thickness.
4
Introduction
Example 1.1: Calculation of the absorptance and transmittance of glass-like media. Ab-
sorption coefficient of a fiber can be 0.001 m- 1 , which means that 1 km of the
fiber will absorb 1 - exp( -10- 3 . 103) == 1 - exp( -1) 0.63 == 63% of the
incoming light power. A gray filter HC-3 has absorption coefficient 1 mm- 1 in
the visible part of the spectrum, thus transmittance of the filter of 2 mm thickness is
exp( -2) 0.14 == 14%.
When the light absorption is considered at a molecular level, the absorption cross-
section, a, is a more practical value to be used since it characterizes a single molecule
and does not depend on the density (concentration) of the molecules. Then, eq. (1.6) is
presented in the form
lout == I (l) == line -ncrl
(1.10)
where n is the density of the absorbing centers (e. g. dye molecules). 3
In many practical cases power of 10 is used instead of power of e. Then, eq. (1.6) is
rewritten to the form
lout == I in 10- A
(1.11)
where A is the absorbance or optical density. Naturally, the absorbance can be calculated
from the light intensities at the sample input and output
A = -log lou.t
I in
(1.12)
Comparing eqs. (1.7) and (1.12) one obtains a relation between the transmittance and ab-
sorbance
A == - log T
(1.13)
If the molar concentration of chromophores, c, is used to express the density of light
absorbing molecules, the light absorbing characteristic of the molecules (chromophores) is
expressed by molar absorption coefficient E,4 which is the proportionality coefficient in the
relation
A == E cl
(1.14)
Note: By historical tradition the measure of molar concentration is moles per liter, i. e. M =
mol x dm -3, whereas optical path, l, is counted in cm, thus the measure of the molar
absorption coefficient is M-1cm- 1 = mol- 1 dm 3 cm- 1 , i. e. one value uses lengths
measured in different units, dm and cm!
3 The density is defined as a nUl11ber of absorbing centers (or l11olecules) per unit volUl11e and, thus, it has the
SaI11e l11eaning as concentration. However, it is il11portant to rel11el11ber that concentration can also be counted in
graI11S or l110les per volul11e. Then, a coefficient is required to calculate density frOl11 concentration.
4It is also called l110lar absorptivity and extinction coefficient.
1.1. Absorption
5
The molar absorption coefficient and the cross-sections are two characteristics specifying
the same property of chromophores, their ability to absorb the light. Comparing eqs. (1.10),
(1.9) and (1.14), and recalling that c == n / N A, where N A is the Avogadro constant, one
obtains
e- ncrl == 10- scl
( 1.15)
which gives
2.303E
a t""V t""V 3.825 X 10- 19 E
NA
( 1.16)
where a is measured in cm 2 , and E in M-1cm- 1 .
Example 1.2: Calculation of absorption of chromophore solution. Absorption cross-section
of chlorophyll a at wavelength of 440 nm is a 4 A 2 or 4 x 10- 16 cm 2 . This cor-
responds to the molar absorption coefficient of E t""V 3.825: 10- 19 10 5 M-1cm- 1 .
A chlorophyll solution at concentration c == 10- 5 M == 10 tLM placed in l == 1 cm
cuvette will have absorbance A == Ecl == 1, and will absorb 1 - 10- A == 1 - 0.1 ==
0.9 == 90% of the incident light.
In summary, there are few parameters, which can be used to characterize the light ab-
sorption by the matter. The usage of the particular parameter depends on the problem on
hands. The parameters can be
. absorptance, a, or transmittance, T, which are dimensionless values and usually ex-
pressed in % (eqs. (1.7) and (1.9));
. absorbance (optical density), A, which is dimensionless parameter, eq. (1.11);
. absorption coefficient, a, which has dimensionality of inverse length, e. g. cm -1 , eq.
(1.6);
. absorbtion cross-section, a, is used to specify absorption properties of a single molecule,
for example, and is measured by the area, e. g. cm 2 , eq. (1.10);
. molar absorption coefficient, E, which is usually used to specify absorption properties
of chemical compounds and has dimensionality ofM-1cm- 1 , eq. (1.14).
It is also important to remember that all these values are wavelength dependent. One of the
primary tasks of the optical spectroscopy is determination of the wavelength dependences
of e. g. molar absorption, E (A), which is the measurement of the absorption spectrum.
1.1.2 Absorption of complex samples
Let us now consider a practically important case of a complex sample consisting of a few
layers with different absorbing properties. This can be a solution of some compound in
6
Introduction
layer 1
layer 2 layer 3
.
I
2
I
3
I
o
x
Figure 1.2: Absorption of multi layer sample.
a cuvette, for example. Typically the absorption spectrum of the compound is of interest,
but it is possible that in the wavelength range of interest the solvent and the cuvette have
absorptions of their own. To simplify the case, the sample can be presented as a sequence
of absorbing layers, as shown in Fig. 1.2. In front of the sample, behind layer 1, the light
intensity is 10. The transmittance of the first layer is T 1 , thus the light intensity after the
layer is II == loTI. The light intensity entering layer 2 is II. The transmittance of the
layer 2 is T 2 , which gives the light intensity at the interface between layer 2 and layer 3,
1 2 == IIT2 == IoTlT2. Continuing this procedure, the light intensity after the layer 3 is
13 == 12 T 3 == IoTlT2 T 3. This gives the total transmittance of the sample T == T 1 T 2 T 3 , i. e.
the product of the transmittances of the individual layers.
In a more general case, the transmittance of a complex sample is the product of the
transmittances of the individual components forming the sample
N
T== IIT i
i=l
( 1. 1 7)
where N is the number of absorbing components, layers in the above example. This is
equally applied to a mixture of dye molecules in solution if there is no intermolecular in-
teraction. If the transmittance of one dye in a cuvette at some concentration is Tl and the
transmittance of another dye in the same cuvette is T 2 , then the mixture of the dyes will
have transmittance T == Tl T 2 , under condition that the concentrations of the dyes are the
same as in the individual measurements. 5
To calculate absorbance of a complex sample one can apply eq. (1.13) to eq. (1.17) to
obtain
N
A == L Ai
i=l
( 1.18)
This is a simple and practically important result - the absorbance of a complex sample is
the sum of absorbances of its components. Returning back to the example discussed in the
5 Which l11eans, the l11ixture was not prepared by l11ixing together the two solutions, since in that case the total
concentrations of the dyes will be lower than in the non-l11ixed solutions.
1.1. Absorption
7
Table 1.1: Energies and spectral ranges of different types of transitions.
Electronic
Vibrational
Rotational
Energy, J
(2 . . . 10) X 10- 19
(2 . . .20) X 10- 20
(2 . . .20) X 10- 21
Frequency, Hz
(3. . .15) X 10 14
(3...30) X 10 13
(3...30) X 10 12
Wavelength, tLm
0.2...1
1...10
10...100
beginning of this Section we can conclude that to obtain the absorption spectrum of the
compound of our interest we need to measure the absorbance of the sample solution, then
the absorbance of the same cuvette filled with the same solvent but without the compound,
and after subtracting the latter from the former we will obtain the spectrum of the compound.
More discussion in this subject follows in the Chapter 5.
1.1.3 Electronic, vibrational and rotational levels
At a molecular level, the light absorption results in a change of the molecule state. This is
usually discussed in terms of energy levels and transitions between them since the molecules
are quantum objects. Depending on which part of molecular subsystem is involved, the
energy levels are divided onto electronic, vibrational and rotational.
The electronic levels are associated with the energy of the electron subsystem. The
transition form one electronic level to another can be considered as the transition of one
of the electrons form one orbital to another. The vibrational levels are related with the
vibrational motions of the molecules. The transitions between them have almost no effect
on the electron subsystem. The rotational levels arise from rotations of molecules as whole.
The energies of the transitions increase in order: rotational < vibrational < electronic.
This is the order of decreasing mass of the considered subsystem: molecule < atom < elec-
tron. A smaller mass results in a higher frequency of oscillations (v rv 1/ yfiii for harmonic
oscillators). Typical ranges of the transition are summarized in Table 1.1 and presented
graphically in Fig. 1.3.
The wavelength range of the visible light is 0.4-0.7 tLm. Accounting for the near infrared
region (0.7-1.5 tLm) and ultraviolet (UV) (0.2-0.4 tLm), the optical wavelength range can be
considered to be from 0.2 to 1.5 tLm. Therefore, strictly speaking, only electronic transitions
fall into the optical range. The vibrational and rotational transitions corresponds to the
infrared and far infrared wavelength regions.
However, all the levels contribute to the absorption spectrum of the molecules in the
visible and UV ranges. The electronic levels determine positions of the absorption bands
and the vibrational and rotational levels contribute to the shapes of the bands. In addition,
the shapes of the bands are affected by numerous "line broadening" mechanisms. 6
6 A few l11echanisl11s of line broadening are discussed in Section 13.1.
8
Introduction
I I
0.1, 0.2
,
,
,
UV : visible: NIR
I I
I I I I J I I I
,.0.4 ",,";, ;' 1.0
" ,';,
" ;'
I
0.1
.....;' ;'
.' (
1 10
:.
100
A,
1liliiii( I
k, cm -1 1 a 5
I
10 4
I
1000
I
100
..
I
10 15
I
10 14
I
1013
v, Hz
I
I
electronic I vibrations rotations
I
Figure 1.3: Relations between different scales used in spectroscopy: wavelength, A, wave
number, k, and frequency, v. Ultraviolet (UV), visible and near infra-red (NIR) parts of the
spectrum are indicated in an enlarged part of the wavelength scale.
Table 1.2: Energy, frequency and wavelength units used in spectroscopy.
units value at 600 nm
wavelength A nm 600
frequency c Hz 5 X 10 14
v==X
photon energy E == hv == \c J 3.3 X 10- 19
wave number k == A-I cm- 1 16 667
electron energy U == E == hv eV 2.07
e e
1.1.4 Wavelength, frequency and energy
Since the energy of a photon determines frequency (and wavelength) of the electromagnetic
wave (by famous Planck formula E == hv or E == \c in vacuum), the absorption spec-
trum of any system represents its energetic spectrum or density spectrum of states. The
wavelength, frequency and energy are equivalent measures in spectroscopy. One practical
inconvenience of this is that numerous units used to characterize one and the same parameter
- transition energy. The most frequently used values are collected in Table 1.2.
The first three rows in Table 1.2 compare wavelength, frequency and energy of photons.
The wavelength is usually measured in nanometers in spectroscopy application, and this is
probably the most used value at present. The wave number is typical measure in vibrational
(infrared) spectroscopy, and its historical unit is the reciprocal of centimeter (cm -1). Since
the wave number is directly proportional to the frequency and energy (k == == fc ), it is
conveniently used when energy or frequency dependence is presented. The last line in the
table relates the photon energy to the energy of the electron in electric field. This appears
to be a useful presentation as the effect of a photon absorption or emission is a transition of
1.2. Emission
9
an electron from one energy level (orbital) to another, and the electron access energy can be
used in some other reaction, e. g. in electron transfer from one molecule to another.
1.2 Emission
1.2.1 Black body emission
In thermodynamic equilibrium any body absorbing light must emit equal amount of energy.
This means that any body at temperature greater than absolute zero emits energy by electro-
magnetic radiation. The explanation of the spectrum of such thermal radiation was one of
the fundamental discoveries in physics a century ago. To find the radiation density in a
thermodynamically equilibrated system Max Planck proposed a quantum approach to the
problem. He postulated that each oscillation mode of a closed cavity (resonator) could only
take certain quantized energy 7
1
En == (n - -) hv
2
(1.19)
where n == 1,2,3.... The probability to find energy En is given by the Boltzmann statistics,
p(En) rv exp( - fT ). Using eq. (1.19) and Boltzmann statistics, Planck has shown that the
spectral density of the radiation is
87rv 2 hv
Pv (v, T) = 3 ( hv ) 1
c exp kT -
(1.20)
where h is the Planck constant, k is the Boltzmann constant, c is the velocity of the light in
vacuum, v is the frequency and T is the temperature. Equation (1.20) describes emission
of so-called black body, a body which has no any specific emission/absorption features and
whose emission properties are completely determined by the thermodynamics, i. e. by eq.
(1.20).
The black body can serve as an emission model of a metal surface at relatively high tem-
perature, e. g. of tungsten halogen lamp. For quantitative characterization of the radiation
density a "gray" coefficient is used, which is the ratio of the real body emission density to
that of the ideal one. Usually the coefficient is less but close to 1. For metal surfaces the
coefficient is a constant in a wide wavelength range.
In order to evaluate the emission properties of black bodies the spectral emittance can
be used. It is defined as the total power emitted per unit wavelength interval into a solid
angle 27r by an unit area of the black body, and is given by
M == Cl
A 5 (e - 1 )
(1.21 )
where Cl == 27rhc 2 == 3.74 . 10- 16 W.m 2 and C2 == ch/kB == 1.44. 10- 2 m.K. The
spectral emittances calculated for the wavelength measured in nm and emitting area in cm 2
are presented in Fig. 1.4.
7Today one would say that the resonator was treated as a quantul11 systel11 but this was a century ago!
10
Introduction
,......., .....- .....
I / "- -. T= 5000 K
I "-
I "- ... T= 4000 K
"-
I " - T= 3000 K
I . . . . . . "
S 1 "-
I "-
u , "-
"-
, "-
"-
, "-
" "-
C) , "-
u "-
, : "-
.....
I : "-
.....
-+--' I : .....
-+--' .....
...... 0.1 I : .....
S
C) I :
...... I :
ro I :
-+--' I :
u
C) I :
I:
VJ. I:
0.01 0 500 1000 1500 2000 2500 3000
wavelength, nffi
Figure 1.4: Black body spectral emittance calculated at temperatures 3000, 4000 and 5000 K
according to eq. (1.21).
The maximum of the black body emission spectrum is (Wien's law)
Amax = 2.898 X 10 6
T
(1.22)
where the wavelength (Amax) is measured in nm and temperature (T) in Kelvins (K). Thus,
the light source must have temperature of about 5000 °C to have emission maximum at the
middle of the visible spectrum (500-550 nm). The total power emitted by unit area in a
solid angle 27r is given by Stefan-Boltzmann law
P == aT 4
(1.23)
where a == 5.67 . 10- 8 W.m- 2 K- 4 is the Stefan-Boltzmann constant.
In lamps specification one can find two parameters which determine the emission inten-
sity and the spectrum shape: color temperature and emissivity. The emissivity is the ratio
of radiation emitted by the lamp to that of the ideal black body. The color temperature is
such temperature of the ideal black body at which its spectrum (coloration), as given by eq.
(1.21), is similar to the emission spectrum of the lamp. 8
Example 1.3: Tungsten lamp. The color temperature of tungsten halogen lamps ranges
from 2000 to 3200 K, and typical emissivity is rv 0.4. If a lamp works at filament
temperature 3000 K, its spectral emittance is M 0.4 x 0.16 W.cm- 2 nm- 1
0.07 W.cm- 2 nm- 1 at 600 nm. For an emitting filament area of 0.1 cm 2 the total
power emitted at 600 nm in 1 nm wavelength range is 7 m W. The total power emitted
by a tungsten lamp with filament size of 0.1 cm 2 at temperature 3000 K is P 50 W.
8 In particular, the color tel11perature determines the el11ission l11axil11ul11, as follows frOl11 Wien law.
1.2. Emission
11
The emission maximum of the lamp is Amax 970 nm, so at shorter wavelengths
the spectrum density is lower, e. g. 3 mW.nm- 1 at 500 nm and 1 mW.nm- 1 at
400 nm, and at longer is higher, e. g. 9 mW.nm- 1 at 700 nm.
Example 1.4: High pressure Xe arc lamp. A typical color temperature of the cathode area of
the electric arc is T 6000 K. The bright cathode area diameter is usually 1-2 mm,
so the surface of the emitting cathode "sphere" can be estimated to be 3 mm 2
(1 mm sphere), which gives total power of P 200 W. The emission maximum of
such arc is at Amax 480 nm. The spectrum density of the emission is 0.3 W.nm- 1
at 500 nm (close to the maximum). At shorter wavelengths the spectrum densities
are 0.27 and 0.17 W.nm- 1 at 400 and 300 nm, respectively. At longer wavelengths
the densities are 0.27 and 0.22 W.nm- 1 at 600 and 700 nm, respectively. It should
be noted, however, that this is an estimation of the plasma thermal emission only.
Real spectra of Xe lamps consist of a number of relatively sharp emission lines on
top of rather smooth black body like emission spectrum. This is due to excited states
of Xe atom and its ions.
1.2.2 Two level system (Einstein's coefficients)
The "black body" theory considers infinite number of energy states ( oscillation modes)
which are very close to each other (forming continuous spectrum). This approach ignores
any individual properties of molecules or atoms probably involved in the emission or ab-
sorption process. In other words, this theory cannot be applied to gases or diluted dye
solutions, single molecules and atoms have individual energy levels, which are well sepa-
rated from each other. In the most simplified case one can consider a molecule with only
two states M 1 and M 2 with energies El and E 2 , respectively:
E 2
B 12 r B211 A211
(1.24)
El
This molecule will only interact with photons having energy hv == E 2 - El. The possible
photo-reactions of the system are
1. photon absorption: M 1 + hv ---+ M 2 ;
2. thermal relaxation: M 2 ---+ M 1 + D.E;
3. spontaneous photon emission: M 2 ---+ M 1 + hv;
4. stimulated photon emission: M 2 + hv ---+ M 1 + 2hv.
There are two types of reactions: spontaneous and stimulated. The stimulated reactions
require a photon to occur, these are reactions (1) and (4). The spontaneous reactions do not
12
Introduction
require any external force to occur. They take place because of access energy accumulated
by the system (e. g. a molecule in excited state), and they result in relaxation to a lower
energy state by emitting a photon (reaction (3)) or releasing the access energy by some
other means, e. g. thermal relaxation, reaction (3).
The transition probabilities can be expressed using Einstein's coefficients A and B.
Coefficient A describes spontaneous relaxation. In particular, for reaction (3), i. e. transition
from state 2 to state 1,
dN 2
- == - N 2 A 21
dt
( 1.25)
where N 2 is the population of state M 2 . Coefficients B 21 and B 12 describe stimulated
absorption, reaction (1), and stimulated emission, reaction (4), respectively. Note that ab-
sorption is always stimulated.
In case of a narrow absorption and a broad stimulating radiation field, the kinetic equa-
tion for reaction (4) is
dN 2
ill == -N 2 B 21 P
( 1.26)
where P is the energy density.
Einstein has shown that
B 12 == B 21
( 1.27)
and
A 21
B 21
87r hv 3
c 3
( 1.28)
Thus, a single coefficient, for example A 21 , describes behavior of the two level system.
The relation (1.27) deserves a separate comment as it provides a very general conclu-
sion: the probability of stimulated emission is equal to the probability of absorption. This
is of practical importance for laser applications, as will be discussed in Section 1.3.
1.2.3 Fluorescence and phosphorescence
Real atoms and molecules have many energy levels and not all transitions between the levels
are allowed. For instance, depending on spin multiplicity the states are divided on singlet
states, having total spin quantum number 0, and triplet states with non-zero total spin. Tran-
sitions between triplet and singlet states are forbidden according to the spin conservation
law. In practice this means that the rate of such transition, or the transition probability, is
very low. The process in which the state of a molecule is changed from singlet to triplet
state or backward is called inter-system crossing, and is one of the subjects of organic pho-
tochemistry.
Excitation of a molecule results in transition from the lowest (ground) state to a excited
state. In most cases this is singlet-singlet transition. 9 Being excited the molecule can relax
90 ne exception is oxygen l11olecule, 02, which has un-paired electrons in the ground state, so that its ground
state is the triplet state. Accordingly, photon absorption results in triplet-triplet transition.
1.3. Light amplification
13
to the ground state by emitting a photon (reaction (3)). This emission is called fluorescence.
However, if the inter-system crossing takes place, then the lifetime of the excited state
increases dramatically and may exceed seconds. Thus formed excited triplet state may
also emit a photon. This emission process is called phosphorescence.
The emission quantum yield of both fluorescence and phosphorescence depends on the
balance between the radiative and non-radiative relaxation rates, c/J == kk knT ' where k r is
the radiative rate, and k nr is the sum of all non-radiative relaxation rates. The denominator
is the total relaxation rate of the exited state, ko == k r + k nr , so the quantum yield can
be also expressed as c/J == . Typical rates of the singlet state relaxation for organic dye
molecules are 10 7 -10 9 S -1. The quantum yield of the fluorescence varies strongly from
molecule to molecule and usually one speaks about high quantum yield when c/J > 0.1. 10
The typical values of the excited singlet states radiative rates for organic dyes are in
range 10 6 -10 8 S-l. The most essential contribution to non-radiative decay comes from
inter-system crossing, resulting in formation of a long living excited triplet state. The ra-
diative rate for the excited triplet state is much smaller than that for the excited singlet state
(as this is forbidden transition), however the quantum yield of the phosphorescence can be
relatively high since the competing intra-molecular non-radiative relaxation is also slow. 11
In case of a small single atom emitting centers, such as metal ions in glass matrix, e. g.
N d 3 + ion, the lifetime of the singlet excited state can be much longer than that of organic
dye molecules, e. g. up to milliseconds. This is due to much slower rates of non-radiative
decays, including inter-system crossing, and lower radiation rate constant.
1.3 Light amplification
Formally, light amplification is described by eq. (1.5) with negative coefficient a. Alterna-
tively, the light amplification can be written explicitly
I( x) == Ioe{3x
( 1.29)
where (3 == -a, then coefficient (3 is said to be medium amplification coefficient. Therefore,
medium which can amplify a light is also called medium with "negative" absorption. The
methods for creating media with "negative" absorption were discovered about 50 years ago.
This discovery has opened the laser era and inventors of the lasers, C. H. Townes, N. G.
Basov and A. M. Prokhorov, have received the Nobel price in year 1964.
The stimulated emission, reaction (4), is the physical mechanism of the light ampli-
fication. A photon interacts with an excited molecule. The result of the interaction is the
relaxed molecule and two photons. Importantly, two photons on the reaction output have the
same frequency, the same phase and the same propagation direction. This makes light stim-
ulated emission (amplification) different from spontaneous emission (e. g. fluorescence).
lOPor laser dyes the fluorescence quantul11 yield is typically higher than 60%, e. g. for rhodaI11ine 6G the yield
can be as high as 95% (depending on solvent).
11 At least in solutions the inter-l11olecular quenching of the excited triplet state is typically the l11ain contributor
to the non-radiative decay. As the result, the observed quantul11 yield of the phosphorescence can be very low, and
to l11easure the phosphorescence special precautions have to be taken to inhibit inter-l11olecular quenching.
14
Introduction
The latter reaction also creates new photons, but the photons are out of phase, propagate in
different directions, and, most probably, are slightly different in frequency.
According to relation (1.27), the medium can amplify light if the population of the
higher level, M 2 , is greater than that of the lower one, M 1 , i. e. N 2 > N 1 . This is
not possible in thermodynamic equilibrium since in accordance with Boltzmann statistics
N 2 / N 1 == exp( - E 2k -;-,E 1 ), thus N 2 < N 1 at T > 0 (formally, one needs to achieve a state
of the medium with negative temperature to obtain an amplification). Therefore, only a
medium in non-equilibrium state may provide conditions for the light amplification. How
this can be done and what are the benefits of light amplification will be shortly discussed in
Chapter 3.
1.4 Optical spectroscopy
The term optical spectroscopy can be attributed to any kind of optical photon interactions
with matter. Two most general classes of such interactions are absorption and emission.
Consequently, one can distinguish between absorption spectroscopy and emission spec-
troscopy. In the former case we will speak about absorption spectra and in the latter the
emission spectra will be the subject of measurements. Technically, in both cases the light
spectra have to be measured, however, the arrangement of the measurements, application
range and interpretation of the results have their specific characters and may differ signifi-
cantly.
Another important area of optical spectroscopy is the time resolved measurements.
Among modem research methods the optical spectroscopy provides the widest possible
time range of investigations, from steady state to femtoseconds. Using time resolved opti-
cal spectroscopy a great variety of reaction can be studied in physics, chemistry and biology.
To cover this diversity five time units a widely used:
milliseconds: 1 ms = 10- 3 s, e. g. in biological reactions such as ion transport;
microseconds: 1 tLs = 10- 6 s, e. g. in diffusion controlled chemical reactions in liquid
phase, triplet state reactions;
nanoseconds: 1 ns = 10- 9 s, e. g. in photochemical reaction, singlet state reactions;
picoseconds: 1 ps = 10- 12 s, e. g. in intra- and short distance intermolecular electron
transfer, energy transfer, primary reactions in natural photosynthesis;
femtoseconds: 1 fs = 10- 15 s, e. g. in molecular vibrational motion, "hot" carriers dynam-
ics, optical (electronic) vibrations.
Advantages of the optical spectroscopy methods are their non-destructive nature and possi-
bility to monitor the studied object without physical contact to it. This makes them popular
in applications such as environment monitoring and technological process control.
Chapter 2
Optics and Optical Devices
All optical spectroscopy instruments are optical devices in that they use light sources, ma-
nipulate the light and measure the light. Optics and optical devices have a long history going
back to 17th century when the wave and corpuscular light theories were developed by two
famous scientists Christian Huygens and Isaac Newton. Nowadays optics is a well devel-
oped branch of natural sciences with numerous subtopics, application fields and wide range
of instruments and tools available commercially. Giving its importance for understanding
the principles of the optical spectroscopy instruments this Chapter will discuss a few general
topics, such as interference and interferometers, diffraction and diffraction resolution limits,
monochromators, and calculation of optical systems in geometrical optics approximation.
However, this is rather fragmentary selection of optics subj ects and readers are advised to
refer to general optics text books for more complete study of the subject. 1
The photon, being a quantum object, has a controversy of wave and particle presenta-
tions. Also there are unified theories, it is common to use wave theory to discuss interference
or diffraction properties of the light, and to present photons as particles for ray tracing or
to study their interactions with matter. Accordingly, the wave presentation of light will be
discussed at first, following by its application to interference and diffraction. In the last sec-
tion we shall switch to geometrical optics to discuss calculations of beam tracing in optical
systems.
2.1 Waves
2.1.1 Wave equation
In a simple one dimension case (1 D) the wave equation is
02j 102f
ox 2 - c 2 ot 2 == 0 (2.1)
where f == f(x, t) is a function of coordinate x and time t, and c is a constant. For example
a string vibration can be described by the wave equation, then f (x, t) can be the string
1 The author used a book by Robert Guenther as a reference [1], though there are l11any other excellent text
books on l110dem optics.
15
16
Optics and Optical Devices
t = tl
U(ctl-X)
>
X
t = t2
U( ct 2- X )
Xl
X2
:::;.
X
Figure 2.1: Propagation of a pulse along the string (lD wave). The amplitude and the shape
of the pulse do not change as the pulse propagates along the string.
displacement at position x. A general solution of the equation is
f == u( ct - x) + v( ct + x)
(2.2)
Where u and v are any functions of a single parameter. These functions present two waves
propagating in opposite directions: the wave u( ct - x) propagates in direction of increasing
x, and the wave v (ct + x) in decreasing x.
An illustration of a pulse propagating along the string is presented in Fig. 2.1. Let us
assume that at time t == t l the shape of the pulse is given by a pulse-like function u(y)
which has a single maximum at Yo. Naturally, in our case the argument of the function u is
y == ctl - x, that is u == U(ctl - x). At fixed time t == tl, function U(ctl - x) depends only
on x. The position of the maximum, Xl, is given by a simple relation Yo == ctl - Xl, i. e. at
time tl the coordinate of the maximum is Xl == ctl - Yo. At time t == t 2 the shape of the
pulse is determined by the same pulse-like function u, although now it reads as u( ct2 + x).
Thus, at time t2 the maximum of the pulse is at point X2 == ct2 - YO.2 The displacement of
maximum x == X2 - Xl in time interval t == t 2 - t l is x == ctl - Yo - (ct 2 - Yo) == c t,
and the velocity of the pulse propagation is c == , so the constant c in eq. (2.1) is the
wave velocity.
In three dimensional (3D) case, the wave equation is, for any scalar field or potential
component,
\}2 U _ 8 2 U == 0
c 2 8t 2
(2.3)
where \}2 == ::2 + ::2 + :z22 is the Laplace operator and U == U (x, y, z, t) == U (r, t) is a
function of coordinates and time. This equation describes acoustic waves, for example. The
20 ne can notice that Yo = etl - Xl = et2 - X2, which l11eans that et - X is invariant of eq. (2.1).
2.1. Waves
17
electric filed is the vector field, for which the wave equation in free space is 3
2 -+
2-+ BE
\7 E - EotLo Bt 2 == 0 (2.4)
and the velocity ofthe electromagnetic waves (e. g. light) in free space is c = VI , where
Eoj1o
EO and tLo are permeability and permitivity of vacuum, respectively. In dielectric medium
the velocity is c == .
V EEo j1j1o
Solution of 3D wave eq. (2.3) is not as straightforward as for 1D case, since there is
infinite number of propagation directions. Usually a concept of the wave front is used to
solve eq. (2.3). However, the problem can be simplified considering harmonic waves.
2.1.2 Harmonic waves
Harmonic waves are practically important for spectroscopy applications, since the emission
and absorption usually occur in relatively narrow spectrum range. The photon energy de-
termines the frequency of its electromagnetic wave, thus, electro-dynamically, photons are
essentially harmonic electro-magnetic oscillations. Another term used in optics to denote
harmonic electro-magnetic oscillations is a monochromatic wave.
Harmonic oscillations are given by functions sin(wt) or cos(wt), where w == 27rv, and
v is the oscillation frequency and w is the circular frequency.4 Another useful and widely
used mathematical notation for harmonic oscillations is (Euler formula)
e iwt == cos(wt) + i sin(wt) (2.5)
For example, a harmonic wave in 1D case can be presented as
f == foei(wt-Kox)
(2.6)
where the argument of the function was changed to be dimensionless as required by sine,
cosine or exponential functions. It is also convenient to use wt - /'1;X as the argument in
mathematical presentation of harmonic waves since it shows the frequency (w) of oscilla-
tions. In terms of eq. (2.6) the velocity of the wave is c == , and parameter /'1; is called
( circular) wave number. In space the period of wave, or the wavelength, is A == 2:c , or
A == 27r . 5
Ko
2.1.3 Plane waves
Extending the equation of monochromatic wave to 3D case, one can rewrite an equation for
plane waves
U == Uoei(wt- r) (2.7)
3 The electrol11agnetic waves have two cOl11ponents: electric and l11agnetic fields. However, as follows frOl11
Maxwell equations, these two cOl11ponents are tightly related with each other, and only one of thel11 is needed to
describe cOl11pletely the electrol11agnetic wave. By convenience the electric cOl11ponent will be used here.
4The circular frequency is convenient and preferred notation here as it gives shorter form of equations.
5Conversions frOl11 the circular to linear frequency and wave nUl11ber are w = 21fV and k = 21f , respec-
tively. An equivalent presentation of harmonic wave is f = fo e 27r (vt - kx) , with the wave velocity c = and
wavelength A = k -1 , respectively.
18
Optics and Optical Devices
Where f == f( x, y, z) is the vector from the origin of the coordinate system to point with
coordinates (x, y, z), and it is the wave vector. In isotropic dielectric medium the wave
vector, it, determines the wave propagation direction and its absolute value is I itl == /'1; == ,
or it is equal to the wave number of the 1 D case considered above. One can select the
coordinate system so that, e. g., axis Z is directed along the vector it, then the projections
of the vector are /'1;x == 0, /'1;y == 0 and /'1;Z == litl == /'1;. Thus, the product itf == /'1;xX + /'1;yY +
/'1;zZ == /'1;xX == /'1;Z, and eq. (2.7) can be rewritten as U == Uoei(wt-Koz). In other words, by
proper selection of the coordinate system, the 3D plane waves can be reduced to 1D waves.
The wave given by eq. (2.7) has infinite wave front and its amplitude, U o , is a constant
in the whole space. This is not very useful (practical) model, usually we like to know how
do waves change when propagating through different media, e. g. optical system, lens for
instance. Then it is reasonable to limit the size of the wave, i. e. the value U o can not be a
constant. This can be done by rewriting eq. (2.7) as
U == U(f)ei(wt- r)
(2.8)
where U (f) is a slow function of coordinates (compared to the wavelength) and is called
wave amplitude. Substituting eq. (2.8) into eq. (2.3) one can obtain equation for the wave
amplitude, U(f), also known as Helmholtz equation
(\7 2 + /'1;2) U == 0
(2.9)
This equation is only valid when the function U(f) is much slower than function ei( r).
Then one can consider only wave amplitude distribution over the space but can omit oscil-
lating part, ei(wt- r) .6
In optics, we usually can neglect a change of the wave amplitude, U (f), at distances
compatible with the wavelength, A. In addition oscillations of the electromagnetic field at
optical frequencies are much quicker than the time resolution of measuring instruments.
Therefore, experimentally available value is power averaged over a space region which is
much greater than the wavelength and in a time interval much longer than the wave period.
The energy flow of the electromagnetic field is given by Poynting vector S == E xii.
The light intensity7 is the time average of Poynting vector I = \ ISI), and it is proportional
to the square of the electric field amplitude for the electromagnetic wave, I ex: E mpl. 8 The
intensity is the parameter which is available experimentally and commonly used to measure
the light at different points of optical systems. Therefore, in calculations of the electric field
we shall finally look for light intensity or equivalent measure describing the electromagnetic
wave.
6It is also il11portant to notice that the a111plitude in eq. (2.9) does not depend on til11e. Therefore Heh11holtz
equation describes stationary wave flow.
7Here the intensity is power density. However, the term light intensity is ill defined in itself and can be used
to refer to different forms of the light power characteristics.
8 In dielectric l11ediul11 1 = \ 181) = t 2 C E mpl.
2.2. Interference
19
2.2 Interference
In this section the interference of plane monochromatic waves will be discussed. This means
in particular that the wave front is assumed large enough to neglect its distortions at dis-
tances characteristic for the interference phenomena. Therefore the wave presentation of
eq. (2.7) will be used.
Interference is mutual influence of two or more waves producing certain characteristic
phenomena. In the case of electromagnetic waves (e. g. light) the mutual influence is
superposition of the electric fields produced by different waves (or by difference sources
of the waves). Let us consider two sources of the electromagnetic waves. If one source
produces electric field El at a point f and another source produces field E 2 at the same
point, then the total field at this point is E == El + E-;. Let us further assume that the
waves are flat and have the same frequency w, so that the corresponding wave vectors, it l
and it 2 , have the same length, i. e. I itll == I it 2 1. Then, the electric field of the first wave is
El (f, t) == El e i (wt-if 1 r) and of the second is E-; (f, t) == E-;e i (wt-F£2 r ), respectively.9
It is important to note that the orientations of the pairs of the vectors El and it l , and E 2
and it 2 cannot be arbitrary. In dielectric media the electromagnetic waves are propagating
in the direction perpendicular to the plane formed by electric and magnetic field vectors,
thus E . it == O.
For further simplification let us assume that the vectors El and E 2 have the same orien-
tation, which means that El and E 2 are perpendicular to the plane formed by the vectors it l
and ;:£2. Therefore, the vector sum can be replaced by the sum of scalar values I Ell + I E--; I.
Then, the total field created by these two waves is
E(f, t) == El e i (wt-if1 r) + E 2 e i (wt-if 2 r)
(2.10)
The experimentally available parameter is light intensity, which is proportional to the square
of the amplitude of the electric field oscillations I rv E5, where Eo is the amplitude of the
oscillations,lo and can be calculated as I == E E*, where E* is the complex conjugated
number of Ell
IE. E* = (E 1 e i (wt-K 1 r) + E 2 e i (wt- K 2 r )) (E 1 e- i (wt-K 1 r) + E 2 e- i (wt- K 2 r ))
Ei + E + E 1 E 2 (e i (ii'1-ii'2Ji' + e- i (ii'1-ii'2Ji')
E + E + 2ElE2 cos ((it l - it 2 ) f) (2.11)
The only variable term of I is 2E l E 2 cos ((it l - it 2 ) f). It gives sinusoidal variation of the
light intensity in direction it l - it 2 . The period of the modulation in this direction is 1_ 27r - I .
K:,1-K:,2
9In a general case one of the waves l11ay include a phase argul11ent, e. g. E-;e (wt-i?,l r+cp). This, however,
will not change the following consideration, so one can safely aSSUl11e here that cp = o.
10 As was noted in Section 2.1.2 and footnote 4, the power density is 1 = \ s) = E 2 C E6. However, the
coefficient E 2 C will be ol11itted in all further calculations and a sil11ple relation, 1 = E6, will be used. This will not
create any l11istake as in all cases two transitions will be l11ade: frOl11 the intensity of the individual beaI11s to the
electric field and then back to the intensity of the interference pattern.
11 If A = ae x, then (by definition) A * = ae- x. Thus, A . A * = ae x ae- x = a 2 .
20
Optics and Optical Devices
M 2
I
M3
d 2
M1
1 1 1 2
Figure 2.2: Michelson interferometer. M l is a semi-transparent mirror, and M 2 and M3 are
100% reflectors.
Thus, using scalar wavelength A == I :I == I :I instead of wave vectors and introducing the
angle between the propagation directions of the waves, a, one can obtain interference period
as L == 2 Sin a/2) .
Since E? + E > 2E l E 2 , the intensity I is never negative. When the waves have the
same amplitudes, El == E 2 == E, the intensity of the interference pattern changes from 0 to
its maximum value of 4E 2 , which is two times greater than the sum of the intensities of the
interfering waves, 2E 2 .
Example 2.1: Interference period of two monochromatic waves. In order to obtain an inter-
ference pattern of two monochromatic plane wave at A ==500 nm (green light) with
period of L == 1 mm, the angle between the wave propagation directions must be
a == 2 sin -1 ( 2'i ) t""V £ == 0.0005 radian or 0.03°.
There are many optical devices utilizing the phenomenon of wave interference. Such
devices are called interferometers. Two types of the interferometers are particularly im-
portant for spectroscopy applications and will be discussed here. These are Michelson and
Fabry-Perot interferometers.
2.2.1 Michelson interferometer
Michelson interferometer has found numerous applications and was reproduced with mul-
tiple modifications. A classic scheme of the interferometer is shown in Fig. 2.2. It consists
of three mirrors: a semi-transparent mirror M l and two reflectors M 2 and M3. If incoming
beam has intensity I and the mirror M l has reflectance R, then the intensity of the reflected
beam is Rl and the intensity of the transmitted light is (1 - R) I, respectively.12 Let us
follow the propagation of the reflected beam first. The mirror M 2 , must be adjusted so that
12TranSl11ittance of the l11irror Ml is T = 1 - R.
2.2. Interference
21
the beam reflected by the mirror M 1 is returned back by exactly the same path. Then the re-
flected beam will hit the semi-transparent mirror M l at exactly the same point as incoming
beam (I). The intensity of the light, which will cross the mirror M l , is II == R(l - R)I.
This is the first beam on the interferometer output. Now let us follow the propagation of
the beam which is transmitted by the mirror M l at the first incidence of the incoming beam
(I). Its intensity after the mirror M l is (1 - R)I. The mirror M3 must be adjusted so that
reflected beam hits the mirror M l at exactly the same point as incoming beam (I). Part
of the beam will be reflected in the same direction as previously considered beam II, and
its intensity is 1 2 == R(l - R)I. Thus, properly adjusted Michelson interferometer splits
incoming beam (I) on two beams (II and 1 2 ) of equal intensities and propagating in the
same direction.
In order to calculate the resulting output intensity of the interferometer one needs to
sum the electric fields created by two beams and find the light intensity for the resulting
field. Considering a monochromatic light and taking into account that the beams are prop-
agating in the same direction, e. g. along Z axis, the fields can be written as El (z, t) ==
E t ei(wt-KoZ-<P1) and E 2 ( z t ) == E t ei(wt-Koz-<P2) res p ectivel y where E t is the field
ou , ou , , ou
created by one of the beams (on the interferometer output). The phases cp 1 and CP2 depend
on the propagation distances of the beam from the semi-transparent mirror M l to the reflec-
tors (M 2 or M3 , respectively) and back, and can be written as CPl == 2/'1;d l and CP2 == 2/'1;d 2 ,
where the multiplier 2 is due to the fact that each beam travels twice the distance from the
semi-transparent mirror to the corresponding reflector (M2 or M 3 ). Thus, for the electric
field of the interferometer output one can write
E
E e i (wt-Koz-2Kod 1 ) + E e i (wt-Koz-2Kod 2 )
out out
Eoutei(wt-Koz) (e-2iKodl + e-2iKod2)
(2.12)
And the light intensity is
lout E . E*
E;utei(wt-Koz) (e-2iKod1 + e-2iKod2) e-i(wt-Koz) (e2iKodl + e2iKod2)
E;llt (2 + e-2iroCdl -d2) + e2iroCdl -d 2 »)
2E;ut (1 + COS 2/'1;( d l - d 2 )) (2.13)
Converting the wave number /'1; to the wavelength one obtains
2 ( 47r( d l - d2) )
lout == 2Eout 1 + cos A
(2.14)
Finally, taking into account the reflectance of the mirror M l
( 47r(dl - d 2 ) )
lout == 2IinR(1 - R) 1 + cos A
(2.15)
Thus, the output intensity depends on relative beam propagation delay d 1 d2 , and varies
from 0 to 4R(1 - R)Iin.
22
Optics and Optical Devices
M l
M 2
:::;.
- - -> E
3
--- <:
:::;.
- - -> E
2
E
--- <:
>
- - -> E
1
I
o
I
d
:::>
Z
Figure 2.3: Fabry-Perot interferometer
A straightforward application for the Michelson interferometer is direct measurement
of the wavelength of monochromatic light. By smooth changing of the distance d l (or d 2 )
and counting the interference maxima, which comes as cosine function of the distance, eq.
(2.13), one can determine the wave number as number of maxima per unit length,13 and the
wavelength as inverse of the wave number.
A short list of the Michelson interferometer applications in the optical spectroscopy
application includes:
. wavelength determination;
. measurements of the light coherence length (the interference pattern can be observed
only for coherent beams II and 1 2 );
. optics diagnostics (an optical component, e. g. a lens, can be inserted between mirrors
M 1 and M 2 and any distortions of the wavefront will be seen in distortions of the
interference pattern on the interferometer output);
. fine displacement measurements;
. optical correlators (ultra-short pulse width measurements, will be considered in Chap-
ter 4.5.2);
. Fourier transform infrared spectroscopy;
2.2.2 Fabry-Perot interferometer
Fabry-Perot interferometer is formed by a pair of mirrors aligned parallel to each other at
some (short) distance, d, as presented in Fig. 2.3. For simplicity we will consider normal
13Note that circular frequency was used in eqs. (2.10)-(2.13). In turn, eq. (2.14) was rewritten for "linear
units", and corresponding form for the wave nUl11ber is lout = 2E ut (1 + cos411"k(dl - d2)), where k is the
wave nUl11ber, and = 211" k. See also footnotes 4 and 5.
2.2. Interference
23
incidence of the light and we will suppose that incoming light intensity is 1, i. e. E in ==
ei(wt-Koz). For the further simplification, let us suppose that the mirrors have the same
reflection coefficient r for the electric field flow, thus the intensity reflection is R == r 2 . T he
corresponding transmittance for the electric field component of the wave is f == vi I - r 2 .
The interference pattern after the interferometer is formed by multiple reflections of the
incoming beam between the mirrors. The electric field created by the incoming plane wave
right before the mirror M l is E == ei(wt-Koz) == e iwt (z == 0). Right after the mirror M l the
electric field of the incident light is fe iwt , and before the mirror M 2 is fei(wt- d) (z == d).
After the mirror M 2 the field is
El == f fei(wt-Kod) == f 2 e i (wt-Kod)
(2.16)
This is the first beam participating in the interference on the interferometer output.
The part of the light, reflected by the mirror M 2 , frei(wt-Kod), returns back to the mirror
M 1, where another portion of the light is reflected in direction to the mirror M 2 . The field
of the reflected light is fr 2 e i (wt-2Kod). Part of this beam will cross mirror M 2 and form the
second beam participating in the interference,
E 2 == frrfe i (wt-3Kod) == f 2 r 2 e i (wt-3Kod) == Elr2e-i2Kod
(2.17)
This re-reflection process will continue again and again giving beams E 3 , E 4 , ... En and so
on. It is clear, that for the beam n the electric field is
En == Elr2(n-l)e-i2 d(n-l)
(2.18)
The resulting electric field after the mirror M 2 is the sum of all the beams
ex:) ex:) ex:)
Eou.t LEn = El L r2(n-l)e-i2Kd(n-l) = El L (r2e-i2Kd) n-l
n=l n=l n=l
ex:)
El '"""" ( r2 e -i2Kod ) n == El
1 - r2e-i2Kod
n=O
(2.19)
Finally, the intensity of the transmitted beam is 14
I - ( E E* ) _ (1 - r 2 )2
out - out out - (1 2 ) 2 4 2 . 2 d
- r + r SIn /'1;
(2.20)
Converting eq. (2.20) to intensity reflection, R == r 2 , and wavelength, /'1; == 2; , one obtains
I - I. (1 - R)2
out - 2n (1 _ R)2 + 4R sin 2 2 d
(2.21 )
The transmittance, T == lout / I in , of the interferometer in a narrow wavelength range is
shown in Fig. 2.4 for d == 0.01 mm and R == 0.5 and 0.9.
14Calculations of the intensity frOl11 the electric field eq. (2.19) can be found in e. g. ref. [1] p. 108.
24
Optics and Optical Devices
1 ,\ '\ '\
\
\ 1\ , \ 1 - R=O.9 1 ' \
I \ , \ I \
\ , \ , \ I \
\ , \ I \ -- R= 0.5 , \
0.8 ' \ I I , \
I I I \ I \
\ I \ , I I \
\ , \ I \ , \
\ I I , \ I \
C) I I \ I \ I \
U \ I I I I I \
0.6 \ I \ I \ I \
I I \ I I I \
-+--' \ , \ I \ I \
-+--'
.,...., \ I \ I \ I \
S \ I \ I \ I \
rfJ \ I \ I \ I I
\ I \ I I I \
roOA \ I I I \ , \
;..., I I \ I \ I \
\ I \ I \ I \
\ , \ , \ I \
\ I \ I \ , \
\ I \ I \ I \
\ I \ I \ I \
0.2 \ I \ I \ I \
\ I \ I \ I \
, / \ / \ / ,
....-... " " .... " .... --...
goo 510 520 530 540 550
wavelength, nffi
Figure 2.4: Transmittance spectrum of the Fabry-Perot interferometer of thickness d ==
0.01 mm and formed by mirrors with reflectance R == 0.5 ( dashed line) and 0.9 (solid line).
Equation (2.21) was used for the calculations.
When 2)..d == N, where N is an integer number (0,1,2, ...), sin 2 2 d == 0 and lout == l in ,
i. e. the light crosses the interferometer without any decrease in the intensity even when the
interferometer in formed by two mirrors with high reflectance. Therefore the transmittance
spectrum consists of sharp lines at wavelengths satisfying condition 2d == N A. If there is
a transmittance maximum at AO, then N == 2)..d and the next maximum will be at Al which
correspond to N - 1. Thus the spectrum distance between maxima is
A 2
Asp = d 0 A
2 - 0
A 2
t""V
2d
(2.22)
The spacing between the lines in the wavelength domain decreases as distance between the
. .
mIrrors Increases.
When 2)..d == N + , i. e. sin 2 2 d == 1, the transmittance of the interferometer has its
minimum value
(1 - R)2
l min = l in (1 + RF
(2.23)
For example, if R == 0.5, then I min t""V O.llI in , or the light rejection is higher than what
could be expected for two "independent" mirrors, R 2 == 0.25.
One of the applications of the Fabry-Perot interferometers in optical spectroscopy is the
fine spectrum resolution. Then the value Asp (eq. (2.22)) can be treated as the spectrum
range of the Fabry-Perot interferometer, meaning that if the studied light has wider spectrum
the resulting pattern will be overlapped of different spectral parts.
2.2. Interference
25
For spectroscopy applications the interferometer is placed on the way of a plane wave
front and fine tuning of the interferometer transmittance wavelength is achieved by turn-
ing slightly the interferometer. When the light incidence angle, a, is not zero (at nor-
mal incidence a == 0) eq. (2.21) can be used after substitution d == h( cos a - sin 2 a)
where h is the distance between the mirrors. 15 Thus the transmittance maxima will be at
A == (cos a - sin 2 a) == AO (cos a - sin 2 a). For a small angle a one can use approxima-
tions cos a 1 - 2 and sin 2 a a 2 , so
1
A(a) Ao(l + _( 2 )
2
(2.24)
At angle a V 2 8P the interferometer will be again transparent to light at wavelength
AO (at N + 1).
For the purpose of spectrum resolution analysis one can introduce a contrast factor F ==
(1 )2 and a dimensionless value cp == 2 d . Then, eq. (2.21) can be rewritten as
I - I. 1
out - 'Ln 1 F . 2
+ SIn cp
The half intensity bandwidth can be determined from condition lout == l in , which results
in equation
(2.25)
1 + F sin 2 cp == 2
(2.26)
or
. fT
SIn cp == V F
(2.27)
Usually, F is a big value, e. g. F == 360 at R == 0.9, and F == 1520 at R == 0.95. Therefore,
one can use approximation
(2.28)
which is the equation to be solved in order to evaluate the spectrum resolution of the inter-
ferometer.
Interferometers for the fine spectrum resolution are usually constructed so that d »
A. This means that d » ItL in the optical wavelength range. Then, considering a small
deviation of the wavelength from the wavelength of the maximum transmittance, AO, one
obtains
== 27rd t""V 27rd ( D. A ) == 27rd 27rd D.A
Ao - LA - Ao 1 + Ao Ao + A6
(2.29)
l50 ne have to account for the traveling distance between l11irrors (cos a) and for the phase shift due to the fact
that the wave front is not parallel to the l11irror surface (sin 2 a).
26
Optics and Optical Devices
where D.A is the deviation from maximum. Since AO is the wavelength of the transmittance
maximum 2:'od == 7r N (sin 2:'od == 0) and using eq. (2.28) one can write
27rd D.A IF
A 2 F
o
(2.30)
Finally, the wavelength resolution is
LA::::; A6 {1
27rd V F
(2.31 )
Example 2.2: Spectrum resolution of Fabry-Perot interferometer. Suppose the interfer-
ometer is formed by pair of mirrors with reflectance R == 0.95 and placed at a
distance d == 1 mm from each other. The contrast factor of the interferometer
is F == (1 )2 1520. At wavelength AO == 500 nm the spectrum resolu-
tion of the interferometer is LA = 2: d ::::; 10- 3 nm. The spectrum range is
.6.A sp ; = 0.125 nm. 16 Adjustment angle a V 2 8P ::::; 0.02 radian or 1.3°.
In order to increase the resolution mirrors with higher reflectance can be used, e. g.
if R == 0.98, then F == 39600, and D.A 2 x 10- 4 nm.
2.2.3 Interference filters and mirrors
All of the above considerations can be applied to any combination of parallel light reflecting
surfaces. For example, interface between two media with different refractive indexes can
be used as the mirror. Although reflection of such interface is not large,17 one can build a
system with multiple reflecting interfaces and place the reflecting surfaces at a distances sat-
isfying the best reflection or transmission conditions at a certain wavelength, thus achieving
a high reflectance or high transmittance. One of the applications of this type of structures is
the bandpass filters, when the system is designed to be transparent at a certain wavelength
range. Alternatively, the structure can be designed to achieve high reflectance in a certain
range and can be used as high reflectance mirror. This type of mirrors are commonly called
dielectric mirrors. Yet another application is anti-reflecting coating of e. g. lens surfaces.
A big advantage of these systems is that they are made of materials which do not absorb
the light - the light is either reflected or transmitted. Therefore they can work at extremely
high light power. This is practically important for laser applications where the light peak
power or pulse energy can be extremely high (see Chapter 3).
16The spectrul11 width of studied signal l11Ust be narrower than Asp, otherwise different parts of the spectrul11
will overlap each other.
17Ifthe refraction index of the mediul11 at one side of the interface is nl and at the another is n2, the reflection
frOl11 the interface is R = nl n2 . Roughly, the refractive indexes of the l11aterials transparent in the visible
n1 n2
wavelength range (400-800 nl11) vary frOl11 1.35 (cryolite, Na3AIF6) to 2.3 (titaniul11 dioxide, Ti02), thus the
l11axil11ul11 reflection frOl11 the interface is R < 0.07.
2.3. Diffraction
27
2.3 Diffraction
Restriction of the plane wave front in space results in distortion of the front. This phe-
nomenon is called diffraction. The diffraction limits the spot size to which the light can
be focused. Also the beam presentations can be used only in a limited distance where the
change in the beam diameter can be neglected. These are probably two most important
implications for optical schemes design.
To explain the diffraction Dutch scientist Christiaan Huygens has proposed a wave the-
ory in 1670. He postulated that each point on a wave front can be treated as a source of
a spherical wave called a secondary wave or wavelet. The envelope of those waves, at the
same time, is constructed by finding the tangent to the waves. The envelop is assumed to be
the new position of the wave [1].
The mathematical treatments of the diffraction were developed later with important con-
tributions by Augustin Fresnel and Joseph Fraunhofer.
2.3.1 Fresnel formulation
Using Huygens principle Fresnel has developed a theory which allows to calculate the elec-
tric field amplitudes at any point in space for a wave front defined by some limited surface
S. The theory is applied to stationary harmonic waves. The electric field at some point in
space pointed by vector ro ( observation point) is given by Fresnel integral
E(ro) == J C(rs) Es([s) ei(FZ.R)ds
IRI
s
(2.32)
where Es (rs) is the electric field at a point on surface S given by vector rs, R == ro - rs,
C (rs) is the obliquity factor,18 and integration is done over the whole surface S. This is a
general approach when calculating diffraction of any waves.
2.3.2 Fraunhofer diffraction (far field approximation)
In Fraunhofer or far field approximation the vector R is supposed to be much greater than
the size of the wave front S. To simplify the problem let us consider plane wave propagating
through an aperture, so that the surface S is a plane and it is limited in space. Then, let the
plane S be XY plane of the coordinate system. The wave is propagating in almost Z
direction and we are interested in the deviation from this direction caused by the diffraction.
The deviation is given by function E == E (/'1;x, /'1;y), since if the wave propagates in Z
direction only (as it takes place at S) then /'1;x == 0 and /'1;y == 0, i. e. E (/'1;x, /'1;y) == 0 when
/'1;x #- 0 or /'1;y #- 0, or E == Eo <5 (/'1;x, /'1;y). After this simplification, the Fresnel integral
(2.32) can be rewritten as
E(f£X,f£y) = J J E(x,y)ei(roxx+royy)dxdy
(2.33)
l8The obliquity factor depends on orientations of the surface S and observation point To. It was later derived
by Gustav Kirchhoff, therefore integral (2.32) is also called Fresnel-Kirchhoff integral.
28
Optics and Optical Devices
The most simple example of the case is a diffraction of a wave on a slit. If the slit is
oriented in Y direction, we can consider only X plane, which gives
+d
J . eiK,x d
E (k x ) == e K,xX dx == -i
/'1; x
-d
(2.34)
where 2d is the size of the slit and the amplitude of the wave at the slit is supposed to be
unity. Taking the real part of the field amplitude (2.34) one obtains
E = Re (E ( f£x )) = sin ( f£x d)
/'1; x
(2.35)
The projection of the wave vector, /'1;x, can be expressed in terms of the observation angle cp,
which is deviation from Z axis, and the wave number /'1; as /'1;x == /'1; sin cp == 2; sin cpo Thus
eq. (2.35) gives the wavelength and angular dependence of the amplitude of the diffracted
wave
sin ( 27rd in 'P )
E==
e7r n'P )
(2.36)
The light intensity is proportional to the square of the electric field amplitude,
I = E 2 = sin 2 (f£x d )
/'1;2
x
sin 2 ( 2nd in ({J )
e7r n'P f
(2.37)
Both these functions, eqs. (2.36) and (2.37), are pulse-like functions. The angular depen-
dence of the intensity is shown in Fig. 2.5. The functions have maxima at /'1;x == 0, i. e.
cp == 0, and decrease (with some oscillations) with increase or decrease of /'1;x, respectively.
The width of the "pulse" can be determined roughly from condition (which gives the first
intensity minimum, I == 0) /'1;xd == 7r or /'1;x == J or
. A
SIn cp == 2d
This result has an important consequence. A limited in space plane wave cannot propa-
gate as a unidirectional beam. It will have divergence angle given by19
(2.38)
. A
SIn cp cp -
D
(2.39)
where D is the cross-section (aperture) of the diaphragm used to form the beam from "infi-
nite" plane wave and it was assumed that the cross section of the beam is much greater than
the wavelength, D » A. Consequently, even "ideal" lens cannot collect light into a spot
smaller than certain size, which is called diffraction limit.
19Strictly speaking, the divergence angle of the light after the slit l11easured as full width at half l11axil11ul11
is cp t'..J /'D . However for a round diaphragl11 the divergence is slightly greater. Therefore relation cp B is
reasonably accurate for a rough estil11ations of the diffraction effects.
2.3. Diffraction
29
0.01
0.008
d
0.006
-+--'
.,.....
rfJ
C)
0.004
0.002
o
-30
-20
-10 0 10
angle (<p), min
20
30
Figure 2.5: Angular distribution of the diffracted light (at A == 633 nm) after 0.2 mm slit
(d == 0.1 mm) calculated using eq. (2.37).
Example 2.3: Diffraction limit of beam divergence. Suppose a D == 0.5 mm diaphragm is
used to form a beam from a plane wave front at the wavelength A == 500 nm. The
divergence angle of that beam after the diaphragm is cp £ == 10- 3 . At a distance
l == 1 m from the diaphragm the diffraction will result in the spot size of lcp 1 mm,
which is reasonably close to the size of the diaphragm. However, at 10m distance the
diffraction spot will be 1 cm in diameter, which is essentially larger than the initial
beam diameter. In other words, 0.5 mm beam keeps its cross section at distances
shorter than 1 m, but at longer distances the diffraction beam spreading becomes
essential.
Example 2.4: Diffraction limit of beam focusing. Suppose a light at A == 500 nm is focused
by a lens with focal length f == 10 cm and aperture D == 1 cm. What is the smallest
possible spot size produced by such lens? The diffraction on the lens aperture will be
at angle cp £ . This divergence gives a spot in focal plane of size d == f cp A b ==
5 tL. In the best case the spot size will be 10 times greater than the wavelength.
2.3.3 Diffraction grating
Diffraction grating is an optical component that is used to spread light into a spectrum. Typ-
ically diffraction grating is a mirror with many thousands of parallel lines, grooves, etched
on its surface. The lines must be at one and the same intervals, called grating period. A flat
incident wave front is reflected from the grating at a number of angles determined by the
30
Optics and Optical Devices
incident
wave front
diffracted
;' ;," wave front
;';'\ .
;'
;'
( )
I
Figure 2.6: Diffraction grating with period l. The first order diffraction angle appears when
the delay between reflections from the neighbor grooves is equal to one wavelength, A.
grating period and the wavelength. The diffraction angles are determined by the condition
of phase matching for the reflections form adjacent grooves. A simple illustration in Fig.
2.6 assumes a normal light incidence and one wavelength shift between the reflections. The
angular positions of the diffraction maxima are
mA
sin () == -
l
(2.40)
where l is the grating period and m is an integer number (1,2, ...), called diffraction order.
Naturally, in eq. (2.40) A must be less than one, which means that the maximum possible
diffraction order is m < *.
The diffraction angle can be also expressed in terms of grooves number, which is defined
as reciprocal of the grating period, g == l-l,
sin () == mAg
(2.41 )
The grooves number is the parameter usually found in grating specification, and measured
in number of grooves per millimeter, i. e. mm -1 .
In spectroscopy applications the gratings are used to spread the light by the wavelength,
and thus to measure the light intensity wavelength dependence. Therefore the wavelength
resolution is the parameter of interest. In a far field approximation the wavelength resolution
is determined by two factors: the wavelength divergence due to diffraction as given by eq.
(2.40) and the diffraction divergence due to the limited size of the wave front. The angular
resolution is D.() L c s e [1], where L is the length of the illuminated area (i. e. the size of
the wave front). Equation (2.40) gives D.() cos () == m A , thus, the wavelength resolution is
D.A l
t""V
T t""V mL
(2.42)
In order to improve the wavelength resolution one can increase illuminated area L, which
is done by using bigger gratings, or use higher diffraction orders.
2.3. Diffraction
31
Input t I
. d <
sht it: \ ",
, \ ,
, \ .,
\ ' '
. \ '. .
: \
, ,
: \"
'F \.
: ,t
, . \
\
: .' \
Grating
. J t d Ou ut
." / \ .'. & t o slIt
, . / \ .', ' / '
.' ',I \ ." " I :
'. / \' " " I '
.I \ ' '. / .
I', en.'\ .,,' I :
/ 'Y ' . '
\ ,', I .
I. "
/ \. ,I F '
\ ' I .
" / \' j.
, / \ ' '.
I 'I ,
, " \ I':
I " . \ / ','
I ' .
\ ' " \ I
\ I ,'.'.
" I. /
Spherical mirrors
Figure 2.7: Monochromator optical scheme. d i and do are the sizes of input and output slits,
respectively, f is the focal length of the mirrors, and cp is the angle between the incident and
diffracted beams.
Example 2.5: Spectrum resolution of diffraction grating. Typical grooves number for grat-
ings designed for the visible-ultraviolet wavelength range is 9 == 1200 mm- l , which
corresponds to grating period l == g-l 0.8 tL. Such gratings work in the first
diffraction order. If the size of the grating is L == 5 cm, the best possible spectrum
resolution of the grating at A == 500 nm is A A 1 0.008 nm.
2.3.4 Monochromator
Monochromator is an optical device which works as narrow band wavelength filter with
mechanically adjustable transmission wavelength. A typical optical scheme of monochro-
mator is presented in Fig. 2.7. The incoming light crosses the input slit of size d i and then
it is collected by a spherical mirror of focal length F placed at distance F from the entrance
slit. After the mirror a flat wave front is formed and directed to the grating. The diffracted
light is collected by the second mirror and focused to the output slit of size do. Turning the
grating one can change the wavelength which will hit the output slit. In geometry presented
in Fig. 2.7, the equation of the light transmission wavelength is
l(sin a + sin( cp + a)) == mA
(2.43)
where a is the incident angle of the light on the grating and cp is the angle between the inci-
dent and diffracted beams (this angle is fixed by the instrument geometry). Using grooves
32
Optics and Optical Devices
number the same equation reads
sin a + sin( cp + a) == mAg
(2.44 )
The right side of eq. (2.44) can be simplified if the light incidence angle a is close to 0
(normal incidence of the light on the grating)
sin a + sin ( cp + a) == sin a + sin a cos cp + cos a sin cp ==
== sin a(l + cos cp) + cos a sin cp t""V a(l + cos cp) + sin cp
and
A == a ( 1 + cos cp) + sin cp
mg mg
(2.45)
This is the monochromator dispersion equation. 2o
The spectrum resolution of the monochromators is usually determined by the slit sizes,
which determines the divergence of the wave formed by the spherical mirror Da == . In
other words, the wave formed after the spherical mirror is not flat and the diffraction limit
of the grating spectrum resolution cannot be achieved in most monochromator applications.
Then, the spectrum resolution of the monochromator is
D. A == d i (1 + cos cp )
Fmg
(2.46)
As can be seen, to achieve better spectrum resolution one can
1. use smaller slits (decrease d); this is the regular method to change the monochroma-
tor resolution, however smaller entrance slit means usually that smaller amount of
light will enter the monochromator, and at slits size approaching the wavelength the
diffraction on the slit reduces the monochromator efficiency gradually;
2. use grating with higher grooves number (increase g), the wavelength is the limit for
grooves number, in the visible part of spectrum the practical limit is 1200 mm -1 ;
3. work at higher diffraction orders (increase m), then there may be overlapping diffrac-
tion orders and the diffraction order cannot be greater than m < (Ag) -1, also with
9 == 1200 mm- l the diffraction order cannot be higher than one in the visible part of
the spectrum;
4. use mirrors with longer focal distance (increase F), this increases physical dimen-
sions of the device, i. e. usually bigger monochromators have better spectrum resolu-
tion.
20Note the linear dependence of the wavelength on the angle a. The linear relation between the grating angle
and transl11ission wavelength explains why the wavelength scale is so COl11illOn for spectroscopy devices.
2.4. Calculations of optical system (matrix formulation)
33
An important monochromator parameter is its angular aperture, which is given by the ratio
, where R is radius of the mirror (mirror size). It determines the maximum deviation of
the in-coming beam from the optical axes at the entrance slit. A bigger angular aperture
allows to collect more light and is, generally, preferable. However, bigger ratio means
bigger aberrations and, therefore, may reduce the wavelength resolution.
Example 2.6: HR320 monochromator (ISA Inc.). The HR320 monochromator is an exam-
ple of a compact monochromator which can be used in many optical spectroscopy
applications. The monochromator utilizes 32 cm focal mirrors, and in the visible
wavelength range is equipped with 9 == 1200 mm- l grating. It provides the spec-
trum resolution of 0.05 nm at 0.01 mm slits. The stray light rejection at 8 nm shift
from the monochromatic wave is 10- 5 , which is typical for single grating monochro-
mators. Overall dimensions of the monochromator are 40 x 34 x 25 cm 3 .
2.4 Calculations of optical system (matrix formulation)
2.4.1 Geometrical optics approximation
In this section we will consider an approximation of geometrical optics, which deals with
essentially flat waves with slowly changing amplitudes, but instead of operating with wave
fronts deals with beam and beam trajectories. An example of the device producing a beam
can be a laser. Also one can use a lamp with collimator and a diaphragm to obtain a beam.
Three important assumption for transition from the wave light presentation to the geo-
metrical optics are:
1. all dimensions are essentially greater than the wavelength (thus only wave amplitudes
are considered),
2. the waves are essentially plane waves,
3. travel distances of the waves are much greater than the wave cross sections.
This allows to replace waves by beams and to discuss the light propagation in terms of beam
trajectories. From a point of view of calculations of the light propagation the geometrical
optics simplifies the case by omitting the time dependence, so that one needs to consider
only the light intensities at different points in space (I(x, y, z)). The second simplification
is that the wave front is much smaller than the travel distance, so that the only information
left from the wave front is the propagation direction, but the size of the wave front itself can
be neglected. 21 This means that the problem can be simplified to beam trajectory tracing
rather than solving the problem of light intensity distribution in space.
Following the strategy of the beam trajectory tracing one can further simplify the calcu-
lations by assuming paraxial approximation, which adds:
2lThe extend to which the size of the wave front can be neglected was discussed in Section 2.3 and EXaI11ple
2.3.
34
Optics and Optical Devices
PI
P 2
I
I
: / J -----
1/-j-al------.
jL-- -- '. I
r l : r2' 2
I I
I
I ZI
I
I
I Z2
I
>
z
Figure 2.8: Paraxial approximation and beam propagation in optical system.
1. cylindrical symmetry;
2. propagation at sufficiently small angles to the symmetry axis, that it is possible to use
approximation sin( a) t""V a.
In an optical system whose symmetry axis is Z, a paraxial beam at plane z (z ==constant) is
described by two parameters: the distance from the axis r and the angle it makes with the
axis a (see Fig. 2.8). As the result, the whole diversity of the wave theory, as given e. g.
by Helmholtz equation (2.9), can be solved in terms of only two parameters, r and a, in
geometrical optics paraxial approximation.
2.4.2 Beam transfer matrix
Relation between the parameters at two planes, say planes PI (z == Zl) and P 2 (z == Z2) as
shown in Fig. 2.8, is given by a linear system
r2 == Arl + Bal
a2 == Crl + Dal
(2.47)
or, in the matrix form
( : ) (
A B
C D
) ( : )
(2.48)
or
R 2 ==MR l
(2.49)
where
M=( )
(2.50)
is the beam transfer matrix (it is also called ABCD matrix), and Rl = ( : ) and R 2 =
( : ) are the beam parameter vectors.
2.4. Calculations of optical system (matrix formulation)
35
Determinant of the matrix M is unity, i. e. det(M) == AD - CB == 1, if the media
to the left of the input plane and to the right to the output plane have the same refractive
indexes. Otherwise det (M) == , where nl is the refractive index to the left of the input
plane and n2 is the refractive index to the right to the output plane.
Simplicity of eq. (2.49) can be extended to an optical system of any complexity. Let us
consider a beam propagating in a complex optical system from plane Zl to plane Z2, then
from plane Z2 to plane Z3 and so on to plane zn+ 1.
input
output
----7
----7
---+ . . . ---+
----7
(2.51 )
Zl
Z2
Zn+l
Let us suppose that the transfer matrices between the neighboring planes are known, i. e.
matrices for transfers R i + 1 == M i R i are defined. Then, starting from the last transfer matrix
at plane Zn,
Rn+l == MnRn
(2.52)
one can progress to the next left-side transfer at plane Zn-l, which is
Rn == Mn-lRn-l
(2.53)
Substituting Rn form eq. (2.53) to eq. (2.52)
Rn+l == MnMn-lRn-l
(2.54)
Applying the same routine consequently one obtains the beam transfer equation from plane
Zl to plane Zn+l
Rn+l == MnM n - l . . . MlRl
(2.55)
Note the order of matrices in the equation: the indexes increase from the right to the left.
Thus, to calculate beam propagation through an optical system, one needs to divide the
propagation path into planes so that the transfer matrices are known for the neighboring
planes and to calculate the product of the transfer matrices.
The most often used transfer matrices are collected in Table 2.1. Derivation of the most
of these matrices is straightforward and can be done as a home exercise. Naturally, the
matrices collected in Table 2.1 can be used to derive transfer matrices of more complex
optical systems.
2.4.3 Imaging and magnification
Let us consider a typical imaging system consisting of a single lens. Let the focal length be
f and the distance from the object to the lens be d. We need to find the distance from the
lens to the obj ect image, which will be denote as x. This optical system can be presented by
four principal planes with known transfer matrices as shown in Fig. 2.9. Plane 1 (PI) is the
object plane; this is input plane. Plane 2 (P 2 ) is placed right behind the lens. Thus the first
36
Optics and Optical Devices
Table 2.1: Transfer matrices of simple optical systems.
Spherical mirror of radius R
matrix
( )
( £ )
( l)
(-\ )
(_1 )
Free space of length d
Planar interface between two different media with refractive indexes nl
and n2
Parallel-sided slab of length d and refractive index n
Thin lens of focal distance f
P1
I
1 <
I
P 2 P 3
I
:;' 1
I
. : : : .- - - - - - - - - - - - - - - - - r -
. I
I
1 <
I
l - f ->'
I ". I
I
. .l
x
P 4
I
:;' 1
I
I
I
I
I
d
M 1
Z 2 Z' 3 . . . - '. . . . . .
I I . . '. .
I I
I I
M 2 M3
Z4
z
IZ 1
I
I
I
. .
Figure 2.9: Ray tracing to obtain an image (in plane P 4) of an object (in plane PI) and
application of the matrix formalism to calculate the optical system: Zl, Z2, Z3 and Z4 are the
principal planes and M l , M 2 and M3 are corresponding transfer matrices.
transfer matrix is that for the free space of length d. Plane 3 (P 3 ) is the plane right after the
lens, and the second transfer matrix is that of the thin lens. The last plane is the image plane
or output plane. Consequently, the last transfer matrix is that of the free space of length x.
The transfer matrix of the system is the product of three transfer matrices
M
M3 M 2Ml = ( ) (-\ ) ( )
( )( I} I d y )
( 1-11 d+x d; )
I--
I I
(2.56)
2.4. Calculations of optical system (matrix formulation)
37
Having the transfer matrix for the optical system we shall answer the question: what does it
mean that the plane P4 is the image plane? The property of the image is that all the beams
emitted from one point of the object will reach one and the same point at the image plane.
In terms of the transfer matrix formalism this mean that the beam distance from the axis at
the image plane, Z4, does not depend on the beam angle at the plane of object, aI, that is
B == 0 for matrix M (see eq. (2.50)). Thus, B == d + x - d; == 0, and solving this equation
one obtains a well known fommla x = (1- - ) -1. Finally, the transfer matrix from the
obj ect to the image plane is
M == ( (1 _ _ 1 1 ) -1 a )
1- 1
The value A of the transfer matrix determines image magnification. This follows from
the relation r 4 == Arl + B al == Arl. Introducing magnification factor
r4
m --- A -
- - -
rl
( d ) -l
1- -
f
(2.57)
one obtains
M= ( 2 )
(2.58)
This is the general form of the transfer matrix of any imaging system.
The angular magnification of the imaging system is given by the element D of the
matrix M, i. e. ma == a4 == -.1. Thus, angular magnification, ma, is the inverse of the
al m
magnification m, thus ma m == 1. This is important and general result. 22
Let us introduce brightness of an obj ect (or image) as light power emitted by a surface of
unit size in an unit angle. 23 If the object brightness is b, then one can calculate the brightness
of image. The unit length of the object is converted to the length m of the image. The unit
angle of the object is converted to the angle ma of the image. Thus, the brightness of the
image (accounting for the 2D case) is b i == m 2 m;b, and since mam == 1, b i == b. In other
words, ideal optical imaging system does not change brightness of the object. If the object
is magnified, the beams come to the image plane at angles smaller than those when they
leave the object. And if the image is smaller than the object, the beams are focused at the
image plane at angles greater than those when they leave the object.
In the end of this Chapter Example 2.7 discusses an estimation of the efficiency of the
light collection from a lamp to a monochromator entrance slit to obtain a monochromatic
light on the output of the system. This is a typical task in optical spectroscopy, e. g. to select
the excitation wavelength.
22 Actually, this COl11es directly frOl11 the fact that the determinant of the transfer l11atrix is unity, i. e. AD -
CB = 1. Since B = 0, AD = 1, and thus mam = 1.
23 One can note sil11ilarity with the el11ittance introduced to characterize the black body el11ission in Section
1.2.1, see eq. (1.21). The spectrul11 integral of the el11ittance is equivalent to the brightness.
38
Optics and Optical Devices
Example 2.7: Estimation of the light collection efficiency. Let us estimate a relative amount
of the light emitted by a lamp which could be passed to a monochromator. The
light is collected by a lens and focused onto the entrance slit of the monochromator.
According to the conclusion made in this Section the brightness does not depend
on the magnification of the image on the input slit. However there are two limiting
factors: 1) if the linear magnification is too big, the image of the emitting source is
bigger than the entrance slit thus a part of the focused light is lost, 2) if the light is
focused to a very small spot the angular magnification is high and a part of the light
can be lost because of limited angular aperture of the monochromator. Therefore,
let us assume that the light is focused in a such way that the image fits inside the
slits and the lens diameter is big enough to fill the monochromator angular aperture.
Thus the light is collected in solid angle 0 7r ( 2 ) 2, where F is focal length and
R is the radius of the monochromator mirror, see Section 2.3.4 and Fig. 2.7. Since
the lamp emits in solid angle 47r the relative amount of the collected light, or the
efficiency of coupling lamp with monochromator is rJc ( 2 ) 2 == /6 ( ) 2. If
the monochromator angular aperture is == 0.3, then rJc °i 9 0.006 == 0.6%.
Chapter 3
Lasers for Spectroscopy Applications
The field of lasers and laser applications is a wide and fast growing area of research and
technology. The aim of this Chapter is a short introduction and characterization of the main
types of lasers used in optical spectroscopy. For those who wish to read more on particular
subject there are many excellent books available, e. g. [2, 3].
From the point of view of spectroscopy applications lasers can be considered as the light
sources. However, lasers have unique features which made a great progress in spectroscopy
techniques during the last few decades possible. Among these outstanding features are
. generation of short and ultra-short light pulses, which extended the time resolution to
femtosecond time domain;
. emission of narrow band radiation, which enhanced spectrum resolution and opened
possibilities for the ultra-fine spectroscopes;
The laser invention was based on two discoveries: optical resonators and light amplifying
media. The latter provides one with the possibility to build up an active optical device. The
former, the resonators, allows one to manipulate the properties of the outcoming radiation
. .
In a very precIse manner.
3.1 Laser active medium
The key component of any laser is the part which amplifies the light. Amplification means
not only increase in the light intensity, i. e. in number of photons, but that the newly pro-
duced photons are indistinguishable from the original photons. This means that the propa-
gation direction, wavelength and phase of the amplified light are the same as of incoming
light. Stimulated emission is the physical process when the newly produced photon is indis-
tinguishable from the stimulating photon, and it forms the basis for the light amplification.
The medium which has the property of light amplification is called active medium.
As it was discussed in Section 1.3 light amplification is possible in a medium with in-
verse population. The inverse population is such state where for a pair of energy levels of
an atom or molecule the population of the upper energy level is higher than that of the lower
39
40
Lasers for Spectroscopy Applications
_ _ _ _ _ _ _ _ _ pump level _ _ _ _ _ _ _ _
laser level
V p
V p
Vz
drain
_ _ _ _ _ _ _ _ _ g! I?- } _ _ _ _ _ _ _ _
/
three levels
four levels
Figure 3.1: Three and four levels laser systems.
energy level. Clearly, this situation is not equilibrated thermodynamically, and needs some
artificial work to be achieved. There are two "classic" types of the energy level layouts
which can be used to create the inverse population, and, thus, to achieve the light amplifi-
cation in a medium. They are usually referred to as three and four level systems (Fig. 3.1).
In both cases the medium is "pumped" by some external light source (or excited by some
other mean) to populate a pump level. The pump level must relax fast to the laser level.
Thus the pump level has virtually zero population at any time, N p == o. This guarantees
that the population of the ground state is N g > N p , i. e. the ground and pump levels are in
a thermodynamic equilibrium with the pump radiation. As a result, the pumping radiation
at v p will empty the ground state, N g , and increase population of the laser level, N z . In
a contrast to the pump level, the laser level must have long lifetime (this is a meta-stable
level) to accumulate as much energy as possible.
In the three level scheme, the light amplification condition (or inversion) is achieved
when N z > N g . To characterize the degree of the inverse population, or inversion, one can
use D.N == N z - N g , which gives amplification coefficient {3 == D.N az, where az is the
laser transition cross-section (see eqs. (1.10) and (1.29) for comparison). Deactivation of
the laser level may take place by different decay mechanisms, for example by spontaneous
emission or non-radiative decay. The pumping rate must exceed the sum of all deactivation
rates in order to achieve inversion, which usually requires a rather high pumping rate, e. g.
a strong light source at v p for the pumping.
The four level scheme was proposed to overcome the requirement of the high pumping
rate of the three level scheme. In four level system the working transition is the transition
between the laser level and a drain level. The drain level has very short lifetime and relaxes
to the ground level quickly, so that its population remains virtually zero at any time. The
inversion for the four level system is D.N == N z - N d , and, since N d t""V 0, the inversion
is the laser level population D.N t""V N z . Thus, the four level system needs much lower
pumping rate in order to achieve the light amplification conditions.
3.2. Laser resonators
41
Ml
Active medium
- - - - t - - - - - - -1 - - - - - -
<: I ::;;.
L
M2
..:::
::;;.
Figure 3.2: Laser resonator with active medium.
3.2 Laser resonators
3.2.1 Resonator with active medium
The simplest laser resonator is formed by a pair of flat mirrors which are aligned parallel
to each other and an active medium between them, as shown in Fig. 3.2. Let us follow the
beam intensity while it propagates inside the resonator. Suppose the intensity of the beam
traveling from the mirror M1 to the active medium is I == 10. Then, after the active (am-
plifying) medium the intensity will be I == 1 0 e[3Z, where {3 is the amplification coefficient
of the medium and l is its length. 1 The amplified beam is reflected by the mirror M2, and
after the reflection its intensity is I == I o R 2 e[3Z, where R 2 is the reflection coefficient of the
mirror M2. The reflected beam will cross the active medium once again (I == IoR2e 2 [3z) and
part of the beam will be reflected by the mirror MI. Thus after one full round of the beam
inside the resonator the intensity is II == I o R l R 2 e 2 [3z. In one round the intensity changes
by the value D.l == II - 10 == I o (R l R 2 e 2 [3Z - 1), and this takes time D.t == 2L/c, where L
is the optical length of the resonator. 2 Considering a time scale much longer than the time
needed for one round trip of the light inside the resonator one can replace b..t by dt and b..l
by dl and write a differential equation
dI == c(R l R 2 e 2 [3Z - 1) I
dt 2L
(3.1 )
which is known as the laser equation. The product R l R 2 e 2 [3Z consists of two parts. One
part, Az == R 1 R 2 , is responsible for the losses of the beam intensity and another, Aa == e 2 [3Z,
is the amplification or gain factor of the light in one round (during which it passes two times
through the active medium). Therefore, eq. (3.1) can be rewritten as
dI == c(AzAa - 1) I
dt 2L
(3.2)
The term Az can be used to collect all possible losses of the light inside the resonator. For
example, one can add an interferometer Fabry-Perot inside the resonator for fine tuning of
the laser wavelength, then Az == RlR2T where T is the transmittance of the interferometer.
1 See discussion in Section 1.3.
2The optical length l11ay differ frol11 the physical distance between the l11irrors as it takes into account the
refractive index of the l11ediul11. In particular, if the active l11ediul11 fills all the space between the l11irrors and its
refractive index is n, then the optical length is L = Dn, where D is the distance between the l11irrors.
42
Lasers for Spectroscopy Applications
When AzAa == 1, the light intensity in the resonator stays at a constant level, which
means that the laser is operating in continuous wave (CW) mode. If AzAa < 1, the intensity
decreases and the laser will not operate. Therefore, the condition Az Aa > 1 determines the
lasing threshold. This condition has an obvious meaning - the lasing starts when the light
amplification by the active medium can recover all the light losses in the resonator.
In eq. (3.2) one can introduce a time constant
2L 2L
TZ == ==
c(R l R 2 e 2 {3Z - 1) c(AzAa - 1)
(3.3)
which is called the laser time constant. Using T the laser equation can be rewritten in a
simple form: == . This differential equation has exponential solution, I == Ioe- t / Tl , if
the losses, Az, and the amplification, Aa, are independent of the time and intensity 1. 3
When there is no active medium in the resonator, i. e. a == 0 or Aa == 1, the time
constant is the resonator time constant
2L
Tr ==
C(RlR2 - 1)
This time constant is the average photon lifetime inside the resonator.
(3.4)
3.2.2 Resonator bandwidth
In the case of empty resonator, or so-called passive resonator, the whole system is reduced to
interferometer Fabry-Perot (see Section 2.2.2 on page 22). The resonator time constant (eq.
(3.4)) is yet another parameter to express the bandwidth of the interferometer Fabry-Perot:
LA = A6 A6 (1- R 1 R 2 ) = Ao Ao 1 1 (3.5)
27rLn 27r V RlR2 2L c 27r V RlR2 Tr
One can notice, that AO / c == T).. is the wave period. Then, the relative resolutions of the
interferometer Fabry-Perot, or passive resonator, is
D.A
AO
1 T)..
27r V RlR2 Tr
(3.6)
This equation has a simple meaning - the resolution of the interferometer is the better the
longer the photon lifetime in the interferometer (Tr) or the more times the light inside the
interferometer can interfere with itself.
The laser is the interferometer Fabry-Perot with the active medium inside. The above
consideration can be used to estimate the laser emission bandwidth after replacing the res-
onator time constant, Tr (eq. (3.4)) by the laser time constant, TZ (eq. (3.3)). However, for
CW lasing mode AzAa == 1, which means that l/Tz == 0, or TZ ---+ 00, and D.A ---+ o. Thus,
theoretically, one can build a laser with infinitely small wavelength bandwidth. In practice,
the bandwidth limit is determined by the factors such as thermal stability of the resonator
length and acoustic noises. Carefully designed systems can provide D.A/ A < 10- 10 , i. e.
D.V < 1 kHz. A further discussion of the subj ect can be found in Section 13.3 on page 241.
3 Actually, at least the aI11plification factor, Aa, is intensity dependent. Typically an increase in the light
intensity inside the laser resonator results in a decrease in active l11ediul11 gain, Aa.
3.2. Laser resonators
43
3.2.3 Longitudinal modes
Similar to the Fabry-Perot interferometer there are certain wavelengths which will leave the
resonator whereas others will be suppressed. The operational wavelengths of a laser can be
determined from the condition
2L == N
A
(3.7)
where N in an integer number and L is the optical length of the resonator, i. e. the length
which takes into account refractive index of the components inside the resonator. 4 The
wavelengths satisfying this condition are called longitudinal modes. For instance, if at same
wavelength N == 12345, then it is called longitudinal mode 12345. Naturally, not all the
wavelengths satisfying eq. (3.7) will be emitted by the laser but only those which can be
amplified by the active medium and for which the amplification will compensate the losses
(AzAa > 1). However, it is possible (and is usual) that in the frame of the amplification
bandwidth of active medium, D.Am, a few wavelengths will satisfy condition (3.7).
Example 3.1: Longitudinal modes of Nd: YAG laser. Nd 3 + ions (active centers of the
popular solid state laser) has amplification bandwidth of D.Aa 1 nm and the
amplification maximum wavelength is AO 1063 nm. If the resonator length is
L == 20 cm (which is reasonably short, as the typical length of the laser crystals
is 5-10 cm in non power demanding applications), then the central longitudinal
mode is No == i t""V 400 000. The wavelength distance between the modes is
D.A == 0.0025 nm. 5 Thus, in the worse case, the laser emission spectrum
will consist of D.N == AAa 4000 distinct emission lines (bands), or longitudinal
modes.
The single longitudinal mode operation is an important requirement for the fine spec-
trum resolution applications and will be discussed in Section 13.3 on page 241.
3.2.4 Transverse modes
Transverse modes can be treated as diffraction of the waves in direction perpendicular to
the resonator axis. This allows a simple estimation of the size of the fundamental mode
- zero order transverse electro-magnetic mode, TEMoo. Let us consider two plane waves
propagating in a resonator of length L (Fig. 3.3). The waves are in phase at some point on
mirror MI. One wave propagates exactly along the optical axis of the resonator and another
at a small angle. The second wave travels longer distance (L') inside the resonator than the
first wave (L). When the difference in traveling distances of the waves will be half of the
4 Actually, eq. (3.7) is the equation of "standing waves". The wave at the wavelength A = 2j; will be in phase
with itself after traveling a cOl11plete round in the resonator, and interference of the waves propagating in opposite
directions will give a standing wave pattern.
2
5Under condition L » A, the distance between the nearest l110des is A r-...- L .
44
Lasers for Spectroscopy Applications
M 1 M 2
1 L' ----- 1
_------------ d
L
Figure 3.3: Estimation of the size ofTEMoo mode.
wavelength, L' - L == A/2, the waves will quench each other. The half wavelength delay at
mirror M 2 means displacement (in direction perpendicular to the optical axis) of
d 2 = L/ 2 - L 2 = (L + y - L 2 AL
(3.8)
or
d t""V ill
(3.9)
This is a rough estimation of the size of zero order transverse mode, TEMoo. 6 The TEMoo
mode results in a round emission spot with the distribution of the light intensity across
the beam close to Gaussian. The modes of higher orders consist of multiple bright spots.
The TEM nm mode gives (n + 1) x (m + 1) spots. Therefore, TEMoo laser operation is
required for the applications, where homogeneous distribution of the light across the beam
is important.
It has to be noted, that the size of the TEMoo mode, and, therefore, the laser beam
divergence
yJ 1i
(3.10)
are determined by the length of the laser resonator.
Example 3.2: Divergence ofTEM oo mode. The diameter of the output beam of L == 20 cm
long Nd:YAG laser operating in TEMoo mode is d V>:L 0.45 mm (A ==
1063 nm). Divergence of the beam is yJ VI 2.2 X 10- 3 radians or 0.1 0 .
The same laser but with 1 m long resonator will have the beam diameter on the
output d 1 mm and divergence cp 10- 3 radians or 0.04°.
6In the fraI11e of this geol11etric optic approach one can expect the next l11axil11ul11 to appear when L' - L = A.
This l110de is called the first order transverse l110de and it is denoted as TEMIO for displacel11ent in X direction or
as TEMol for displacel11ent in Y direction, respectively. However one has to be aware of the lil11itations il11posed
by using the geol11etric optics approach for treatI11ents of the diffraction phenol11ena.
3.3. Continuous wave lasers
45
3.2.5 Stable and unstable resonators
In a general case the mirrors forming laser resonator, M 1 and M2, can be spherical ones
with curvatures Rl and R 2 . Then, the relation
0 < (1- J (1- J
(3.11 )
determines conditions for a stable resonator. The stable resonator means that any beam
(inside the resonator, of couse) will tend to propagate along the optical axis of the resonator
as the number of reflections approaches infinity. The resonator formed by a pair of flat
mirrors placed parallel to each other, as shown in Fig. 3.2, is unstable resonator, meaning
that a beam which initially propagates at a small angle to the resonator axis will leave the
resonator after some finite number of passes inside the resonator.
From the practical point of view, the stable resonators have lower losses and, thus, re-
quire lower amplification for lasing as compared to unstable resonators. An advantage of
the unstable resonators is the higher losses for higher transverse modes, which helps to build
lasers operating in TEMoo mode.
3.3 Continuous wave lasers
In spectroscopy, continuous wave (CW) lasers are used as a source of the monitoring light
and for excitation in steady state measurements. Main advantage of the lasers is the narrow
wavelength band and, respectively, the high spectrum resolution (also high spectrum density
of radiation). Usually tuning range is important technical parameter for such applications,
which limits greatly the choice of lasers.
As it follows from eq. (3.2), the CW operational mode is achieved when AzAa == 1, i. e.
== O. However, one has to mention that the gain factor, Aa == e 2 (3Z, is not a constant since
{3 depends on pumping rate creating the inversion and on the intensity inside the resonator,
I, which consumes the inversion. The CW mode can be viewed as a balance between the
pumping rate of the laser level and output power.
To estimate the lasing threshold one can neglect I, and assume {3 to be a constant at con-
stant pumping rate. Then, the lasing threshold condition is Ao = 1 1 , or (3 = l In ( 1 1 ) =
l In ( R 1 1 R2 ). Since {3 = l::..N ()', where l::..N is the inversion and ()' is the cross-section of
the laser transition, the minimum inversion needed to establish lasing (or the lasing thresh-
old) is l::..N = 2T<T In ( 1 1 ), Depending on the laser construction (Ad and on the type of
the active medium this relation determines the pumping rate of the active medium when the
lasing threshold can be achieved.
3.4 Pulsed lasers
Pulsed lasers are used for excitation, and they are important parts of the spectroscopy in-
struments designed for the time resolved experiments. There are also applications were
46
Lasers for Spectroscopy Applications
MI Active medium Polarizer f U M2
- - 1- - - - - - - t - - - 0 - - G- --1
I
Pockels cell
Figure 3.4: Optical scheme of a Q-switched laser with Pockels cell. The Pockels cell is
controlled by voltage, U. M1 and M2 are the output (semi-transparent) mirror and the rear
reflector, respectively.
utilization of short light pulses is a key tool for monitoring, e. g. the pump-probe method
requires short pulses for both excitation and monitoring.
In nanosecond time domain the pulse generation can be achieved by a very fast creation
of the inverse population of the lasing level. For example, gas N 2 and excimer lasers utilize
a short (but powerful) electric discharge to create the inversion. Limiting steps determin-
ing the pulse duration are formation of the short electric pulse and formation of the laser
emission in the resonator. Typical pulse durations for these systems are 10-40 ns and the
pulse energies are 1 mJ - 1 J. Further increase in the pulse energy orland decrease in the
pulse duration is extremely difficult as it requires high pumping current (kilo Amperes and
higher) and higher density of active centers (which is difficult for gas lasers).7
3.4.1 Q-Switched lasers
The problem of the fast creation of the inversion can be eliminated by applying so-called
Q-switching method. A Q-switch, which is functionally a light shutter, is inserted into the
laser resonator. Normally, the Q-switch locks the laser beam during pumping of the active
medium. This state can be simulated by setting R 2 == 0, thus Az == O. Lasing cannot be
established even at high inversion levels, since Az Aa < 1, and deactivation of the laser
level takes place only via spontaneous decay. The pumping results in accumulation of the
laser level population, i. e. the pumping energy is stored in form of excited (to the laser
level) active centers of the medium. When a desired degree of the inversion is reached
the Q-switch is opened and the whole accumulated energy is emitted as a short pulse in
a few passes of the laser beam across the resonator. Thus, the time limiting step is the
formation of the pulse in the resonator, but the pumping of the active medium can be done
with relatively slow rate. Many solid state lasers can operate efficiently in this mode, since
the natural lifetimes of the laser levels of the active media, such as N d 3 +, are as long as
hundred microsecond.
An optical scheme of a laser utilizing a pair polarizer-Pockels cell as the Q-switch is
shown in Fig. 3.4. The Pockels cell is an electro-optical crystal controlled by the electric
7Por eXaI11ple, at V = 10 kV potential to generate an electric pulse with energy of E = 1 J and duration
t = 1 ns, the pulse current l11Ust be I = v t = 1 kA.
3.4. Pulsed lasers
47
potential applied to it. When no voltage is applied the crystal does not change polarization
of the light. At a certain potential the Pockels cell works as A/4 plate so that when the
light returns back after reflection from mirror M2 its polarization is changed to orthogonal
one. Thus, when the voltage is applied no light passes through the Q-switch and the active
medium can be pumped to accumulate inversion. When a desired level of the inversion is
reached, the voltage is dropped down, the Q-switch opens and a short light pulse is emitted.
Typical pulse width for the Q-switched lasers is 5-10 ns. Output pulse energy can be as
high as 1 J even for a compact lasers. The pulse peak power of such lasers can be higher
than 10 MW without additional pulse amplification. The pulse repetition rate is usually not
higher than a few tenths of pulses per second.
F or a higher repetition rate acousto-optic modulators are used. Then the pulse repetition
rate can be a few tenth of kilo Hertz. The pulse duration is longer (20-50 ns) as the switching
time of acousto-optic modulators is longer. Also the pulse energy is lower with one of the
limiting factors being the pumping power.
The principal lower limit for the pulse duration of the Q-switched systems comes from
the length of the resonator and the time needed for the pulse to be established (the pulse
should cross the active medium at least two-three times) and emitted. Therefore, a practical
limit for the systems of this type is a few nanoseconds (note, during one nanosecond the
light travels the distance of 30 cm only).
3.4.2 Mode-locked lasers
To generate even shorter light pulses another mode-locking methods are used. Technically
the mode-locking methods can be actived, when an externally controlled mode-locking el-
ement is installed inside the laser resonator, or passive, when the mode-locking is achieve
using some passive optical components inside the resonator, e. g. saturable absorber. In
both cases the resonator operation is arranged so that a (single) pulse is propagating inside
the resonator. Every time the pulse crosses the active medium it is amplified, and every time
it hits the output mirror a part of the pulse leaves the resonator. In a sense, this operating
mode is similar to a CW one when the active medium is pumped continuously, but instead
of continuous light a trail of pulses is generated with repetition rate determined by the laser
resonator length, so that time interval between the pulses is T == 2L / c.
A scheme of an actively mode-locked laser is presented in Fig. 3.5. An acousto-optic
modulator opens the mirror M2 for a short time once in time interval T, thus letting only
one pulse to propagate in the resonator and suppressing CW emission and other pulses. The
acousto-optic modulator can be used to generate pulses as short as few picoseconds.
The passive mode-locking can be achieved using e. g. saturable absorbers or Kerr lens
effects. An example of a passively mode-locked Ti:sapphire laser will be discussed in Sec-
tion 3.6.5. The physical dimensions,8 the media dispersion and the amplification bandwidth
are the main factors which should be taken into account and optimized to achieve femtosec-
ond pulse generation. The amplification bandwidth determines the principal pulse width
limit - shorter pulses have broader spectrum and require broader amplification bandwidth.
Roughly the pulse width limit is t 2: c;:).. , where A is the amplification bandwidth and A
8The spacial width of alps pulse is 0.3 111111 only.
48
Lasers for Spectroscopy Applications
u
MI Active medium ..1.
- - 1- - - - - - - t - - - - - - - - - - - - -[] M2
Acousto-optic I
modulator -
Figure 3.5: Optical scheme of the mode locking with acousto-optic modulator placed as
close to the reflector M2 as possible. A frequency applied to the modulator must be f ==
c/2L.
is the (central) lasing wavelength. The dispersion results in a broadening of the pulse while
it travels inside the resonator, but can be compensated by introducing the pulse compression
components to resonator.
Many laser systems can be operated in mode-locking regime, e. g. Nd 3 +, Ar or Kr
lasers. Nevertheless, in order to generate femtosecond light pulses, active media with broad
band amplification has to be used. For example, to amplify a 100 fs pulse at 600 nm the
amplification bandwidth must be A > c t == 12 nm. Dye lasers were historically the first
to generate sub-picosecond light pulses. Nowadays Ti:sapphire lasers are almost dominating
in femtosecond laser design for spectroscopy applications. 9
3.5 Laser amplifiers
Sometimes the desired properties of the laser emission cannot be achieved at once. For ex-
ample, it is possible to generate short femtosecond pulses using the mode-locking technique,
but the pulse energy is typically a few nano Joules, which is too low for most pump-probe
applications. In such case the laser system can consist of two parts: a laser oscillator, which
produces the short but relatively weak pulses, and a laser amplifier, which amplifies the
pulses to desired energy. In the same manner, the oscillator can be designed to operate
in single longitudinal mode for fine spectroscopy applications,lo and its emission can be
amplified to a necessary power level by a separate amplifier.
The laser amplifier is essentially the same active medium but without resonator. The
beam crosses the active medium and gains more energy and power, but the properties of
the beam remains unchanged. This strategy is widely used when high power or energy is
required. Typically the same active media are used for the oscillator and amplifier, but
amplifier can be bigger in dimension or pumped to high degree of inversion, to obtain
higher power and energy output characteristics. Also in some cases special measures may
be needed to keep the important oscillator emission property unchanged, as in the case of
ultra short pulse amplification (will be discussed in Section 11.2).
9Piber lasers are the fast developing systel11s which have approach the sub-picosecond pulse width already,
and can be used in applications where a nano Joule pulse energies are sufficient.
l°It is easier to achieve single longitudinal l110de operation when the laser is operated at pUl11ping level close to
the lasing threshold, when fewer l110des exceed the threshold.
3.6. Main types of lasers
49
3.6 Main types of lasers
Types of lasers the most widely used in optical spectroscopy applications will be reviewed
briefly in this section in no particular order. 11
3.6.1 Nd:YAG lasers
Nd:YAG lasers are one of the most widely used type of solid state lasers. Its active medium
is Y 3Al50l2 (YAG) crystal doped by Nd 3 + ion at concentrations up to 1 %. It can be
pumped in a wide spectrum range, 480-600 nm, using krypton arc lamps or emitting diodes,
for example. The main lasing wavelength is 1064 nm (D.A rv 1.5 nm), for which the energy
level layout is typical four level system (see Fig. 3.1). Because of high concentration of
active centers, Nd 3 +ions, the laser crystals have high amplification, {3 > 10 3 m- l and can
provide a high output power at relatively small crystal size, e. g. 500 W output power can
be achieved for 10 cm long rod.
Nd:YAG lasers can operate in both CW and pulsed mode. In CW mode the lasers
can generate from few Watts to few kilo Watts optical power. Using Q-switching method
(see Section 3.4.1) nanosecond pulses can be generated with pulse energies from 0.1 to a
few Joules and with more than 10 MW peak power. Even shorter pulse generation can be
achieved using mode locking method, usually 5-20 ps (see Section 3.4.2).
Nd 3 + ions can also be incorporated into other matrixes, such as LiYF 4 (YLF), YV0 4 ,
and some types of glasses. The latter was used to fabricate large size active medium blocks
for power and energy demanding applications. An example of such huge laser system is
Nova laser at Livermore Lab, which can produce 1 ns pulses at 100 kJ energy.
In optical spectroscopy applications the wavelength of 1064 nm is not very useful in it-
self, but by adding a non-linear optical device to the system the selection of the wavelengths
can be extended to 532, 355 and 266 nm, which are the second, third and forth harmonics,
respectively. The higher harmonics can be used for excitation, or as the pump source for
dye lasers, Ti:sapphire lasers or optical parametric oscillators.
One can distinguish between two types ofNd Q-switched lasers. When the Q-switching
is achieved by using a Pockels cell the repetition rate of the pulses is a few tenth of Hertz
at maximum, the pumping source is a flash lamp, the pulse width is a few nanoseconds and
the pulse energy can be a few Joules. Alternatively, an acousto-optic modulator can be used
to achieve Q-switching. Then the pulse repetition rate can be a few tenth of kilo Hertz, a
continuous pumping is used (by lamp or emitting diodes), the pulse width is a few tenth of
nanoseconds and the pulse energies are typically at milli Joule level.
Example 3.3: INDI Nd: YAG laser (Spectra-Physics). INDI is a relatively compact Nd:YAG
laser which can be used e. g. in flash-photolysis experiments for sample excitation
(see Chapter 7). It generates 5-8 ns pulses at 1063, 532, 355 and 266 nm with pulse
energies 450, 200, 100 and 55 mJ respectively. The pulse repetition rate is 10Hz.
11 SOl11e laser systel11s will also be discussed with additional details in Sections devoted to their applications,
such as PUl11p-probe, Section 11.2. There are also better sources of information on different types of lasers and
laser systel11s, such as refs. [2, 3].
50
Lasers for Spectroscopy Applications
The size of the laser head is roughly 73 x 15 x 18 cm 3 . The diameter of the laser
beam is < 10 mm, and the beam divergence is 10- 4 radians.
An important application of CW mode Nd 3 + lasers is the pumping source of the fem-
tosecond mode-locked Ti:sapphire lasers. Examples of the lasers designed for this kind of
applications are Verdi from Coherent and Millennia from Spetra-Physics. Both systems
use diode pumping and implement intra-cavity second harmonic generation. They provide
2-12 W output power at 532 nm in TEMoo mode with high output stability and almost
diffraction limited beam divergence.
3.6.2 Ion lasers
These are gas lasers utilizing electron impact to generate excited ions, which are the active
media of the lasers. The principal part of the ion lasers are the plasma tubes. The current
flow in the tube can be in excess of 100 A cm -2, but the overall electric efficiency is typi-
cally less than 0.1 %. The most widely used gasses are Ar and Kr. The lasing wavelengths
for Ar ions are 351.1,363.8,457.9,465.8,472.7,476.5,488,496.5, 501.7 and 514.5 nm,
and for Kr are 350.7, 356.4,406.7,476.2,521,531,568.2,647.4,676.4 and 752.5 nm. The
output power of ion lasers can be as high as 100 W, although a typical output power for the
most commercially available ion Ar lasers is about lOW. An advantage of the ion lasers is
TEMoo operational mode with almost diffraction limited beam divergence.
In optical spectroscopy applications Ar ion lasers are usually used for pumping other
lasers, such as dye or Ti:sapphire lasers. Although solid state lasers with laser diode pump-
ing have much higher total efficiency, and provide similar characteristics in terms of the
quality of the outcoming beam, and may replace the ion lasers in nearest future.
3.6.3 Excimer lasers
Excimer is an electronically excited dimer which dissociates immediately after relaxation to
the ground state. Excimer lasers are gas lasers which utilize electric discharge to generate
emitting excimers. The excimer can be e. g. Xe;, which is meta-stable in excited state, but
after emission of a photon, or relaxation to the ground state by some other means, dissociates
immediately in two Xe atoms. Therefore, the relaxation reaction scheme is Xe; ---+ Xe2 +
hv ---+ Xe + Xe. Since the molecule Xe2 dissociates very fast, the population of the lower
laser level is virtually zero and the inversion is achieved at relatively low population of
excimers, Xe;.
The emission wavelengths of most common excimer lasers are presented in Table 3.1.
The excimer lasers are pulsed lasers with typical pulse width of 20-40 ns. The pulse en-
ergies are usually in range 0.1-1 J. The excimer lasers are popular excitation source in or-
ganic photochemistry, since many compounds have absorption in the ultraviolet wavelength
range, and no other lasers can emit directly in this wavelength range. They can also be
used to pump dye lasers to extend the selection of the excitation wavelengths to the visible
wavelength range.
3.6. Main types of lasers
51
Table 3.1: Emission wavelength of excimer lasers.
Gas
ArF, mixture of Ar and NF 3
XeCI
ArCI
Xe2
F 2
XeCI, mixture of He, HCI and Xe
wavelength, nm
193.3
357
169,175
172
157
308
Table 3.2: Tuning ranges and pumping sources of some laser dyes.
Dye
Coumarin 120
Coumarin 102
Coumarin 7
Rhodamine 6G
Rhodamine B
DCM
Oxazine 1
Oxazine 750
Lasing wavelength, nm
430-465
453-507
508-540
560-615
590-645
610-700
695-762
728-802
Pumping laser
Excimer, N 2 , Kr, 3rd harmonic ofNd
Excimer, N 2 , Kr, 3rd harmonic ofNd
Excimer, N 2 , Kr, 3rd harmonic ofNd
2nd harmonic ofNd, Excimer, N 2 , Ar
2nd harmonic ofNd, Excimer, N 2 , Ar
2nd harmonic ofNd, Ar
2nd harmonic ofNd, Ar
2nd harmonic ofNd, Ar
3.6.4 Dye lasers
Organic dye molecules are active centers in dye lasers. One of the advantages of the dye
lasers is the possibility of tuning the emission wavelength. Dye solutions are excited by
another pulsed or CW laser at fixed wavelength, but the resulting emission of the dye laser
can be tuned. The tuning ranges of he most common laser dyes are presented in Table 3.2.
Another advantages of the dye lasers are high amplification factor and high density of dye
molecules in solutions, which makes manipulations by the laser emission simpler, since the
laser is not critical to new components installed inside the resonator, and helps to design
relatively compact devices.
The dye lasers can operate in both continuous and pulsed modes. In continuous mode a
relatively high pumping power, typically higher than 0.2-0.5 W, is needed, which is due to
a short lifetime of the excited state of the dye molecules, usually less than IOns, and thus
requires a high pumping rate to create an inversion.
In nanosecond pulsed mode the excitation sources are usually the second or third har-
monics of the Q-switched N d: YAG lasers or excimer lasers. The excitation pulse width is
compatible to the lifetime of the excited state and the lasing threshold can be achieved at
the pumping energy of a few milli Joules. The efficiency of the dye lasers depends on the
dye molecule and the pumping source, and can be as high as 30%, e. g. for rhodamine 6G
pumped by the second harmonic of a Nd:YAG Q-switched laser.
A scheme of a tunable dye laser is presented in Fig. 3.6. The wavelength selecting
52
Lasers for Spectroscopy Applications
pump
beam expander
EJ
L1
L2
grating
Figure 3.6: Optical scheme of a tunable dye laser. The wavelength selecting component is
a diffraction grating, which operates also as the rear mirror. Lenses L 1 and L2 form a beam
expander to reduce the power density at the grating. Output coupler of the laser is mirror
MI.
component of the laser is a diffraction grating, which works as the rear mirror at the same
time. The gratings have relatively low damage threshold, therefore a beam expander is
introduced between the dye cell and the diffraction grating to reduce the power density. 12
The active medium of the laser is a dye flow cell or a liquid jet in high power applications.
The cell is tuned at Bragg angle in respect to the lasing beam to reduce the reflection losses.
This type of lasers can be used as the excitation source in flash-photolysis applications.
Another important application of the dye lasers in optical spectroscopy was generation
and amplification of picosecond and sub-picosecond pulses. Two properties of the dye
lasers were important for this applications: a broad amplification bandwidth, which allows
to amplify as short as 100 fs pulses, and high amplification and high density of active centers
(dye molecules), so that the thickness of the active medium can be a millimeter or so, and
it does not introduce a big pulse broadening due to the dispersion of the medium. However,
recently Ti:sapphire lasers are mostly used for femtosecond pulse generation since they
provide superior characteristics for ultra short pulse generation.
3.6.5 Ti:sapphire lasers
The active centers of the Ti:sapphire laser are Ti 3 + ions incorporated into sapphire (AI 2 0 3 )
crystal. The crystal has a number of exceptional properties for laser applications. The broad
emission and amplification band, typically 680-1100 nm, is the key feature for ultra short
pulse generation, and makes them also attractive for tunable laser systems design. The
high doping level of the crystals allows to achieve high amplification in a few millimeter
thick active medium, which is important for the high speed laser application. The crystals
have high thermal conductivity and optical damage threshold, which permits high pumping
power. Also the absorption band of the Ti 3 + ions is 480-600 nm, where a number of lasers
are available for pumping.
A scheme of a femtosecond Ti:sapphire pulse generator is shown in Fig. 3.7. The gener-
12 Since the focal distances of the lenses are sensitive to the wavelength (because of the dispersion properties
of the lens l11aterials), prisl11 beat11 expanders are usually used in dye lasers designed for a broad wavelength
applications. Also prisl11s can be used as the wavelength selecting part of the laser resonator, if the a fine spectrul11
resolution is not required.
3.6. Main types of lasers
53
, compressor
M3
Ml
output
<E-
M2
M4
Figure 3.7: Optical scheme of a femtosecond Ti:sapphire pulse generator.
ator is working in passive mode-locking regime using Kerr lens effect in the sapphire crys-
ta1. 13 The crystal is placed between a pair of spherical mirrors so that at a high peak power of
the pulsed mode the distance between the mirrors is equal to the sum of the focal distances
of the mirrors. At lower lasing power of the CW mode the distance between the mirrors
is longer than the sum of focal lengths, which makes resonator unstable and increases the
losses. Thus the Kerr lens makes the pulsing energetically preferable as compared to the
CW mode, which enforces the mode-locking operation of the laser. Two prisms (PI and
P2) form a pulse compressor, which compensate the pulse broadening in the sapphire crys-
tal due to dispersion and allows generation of femtosecond pulses. 14 The Ti:sapphire crystal
is usually excited through one of the spherical mirrors (M 1 in Fig. 3.7) with pumping beam
focused on to the crystal, which allows to achieve a good match between the excitation and
lasing beams inside the crysta1.
The pulse repetition rate of the laser in mode-locking regime is determined by the pulse
propagation time inside the resonator (from mirror M3 to M4 and back). Typical pulse
frequencies are 80-100 MHz, which corresponds roughly to 1.5 m optical path between
mirrors M3 and M4. The pulse width depends on the crystal thickness and the system
adjustment and can vary from 20 to 200 fs. At pumping power of 0.5-1 W one can expect
to obtain 200-400 mW of average power on the output of the Ti:sapphire laser, which
corresponds roughly to 3 nJ pulse energy. Although the pulse energy does not look to be
high the peak power of such pulses is higher than 30 kW, which is strong enough for e. g.
second harmonic generation.
The Ti:sapphire lasers were used to generate the shortest optical pulses, e. g. 6 fs pulse
generation was reported by few groups [4, 5]. Also in optical spectroscopy applications
pulses shorter than 20 fs are not common (see discussion in Section 11.4.4). The lasers pro-
viding 100 fs pulses are commercially available from a few manufactures, and even 20 fs
13 Kerr lens is a non-linear optic effect, which can be described as instantaneous change on the refractive index
of l11atter in high electrol11agnetic field. In case of the Ti: sapphire lasers the Kerr lens is induced in the sapphire
crystal by the light pulse traveling inside the laser resonator. The induced change in the refractive index is used to
l110dulate light losses in the resonator in order to achieve l11ode-Iooking operation.
l4The dispersion effects are very iI11portant in fel11tosecond PUl11p-probe and optical gating spectroscopy appli-
cations. They are discussed in related Sections 11.4.1 and 12.1.2.
54
Lasers for Spectroscopy Applications
pulse generators can be purchased. 15 These laser systems can be used directly in picosec-
ond time correlated single photon counting (see Chapter 8) and femtosecond up-conversion
applications (see Chapter 12). Mode-locked Ti:sapphire lasers followed by Ti:sapphire am-
plifiers are typical for femtosecond pump-probe instruments (see Chapter 11, and Section
11.2 for a discussion of the laser systems in pump-probe applications).
A wide amplification band of Ti 3 + ions is an important property of the active medium
which makes possible a wide tuning range of Ti: sapphire lasers operating in CW or nanosec-
ond pulsed modes. In optical spectroscopy the fundamental (680-1100 nm) and second
harmonic (350-540 nm) of the pulsed lasers can be used for excitation in flash-photolysis
measurements.
3.6.6 Semiconductor lasers
Semiconductor light emitting diodes and laser diodes were under active development during
few past decades. Originally the emission wavelengths were in the near infrared region,
but recently new structures were developed, e. g. GaAs, GaP, AIGaAs and InP, which can
operate in the visible wavelength range. Semiconductor lasers are compact and efficient, and
can emit up to 100 W in CW mode. They are successfully replacing traditional pumping
sources such as arc lamps in modern solid state laser systems. For example, Nd:YAG and
Ti: sapphire crystals can be efficiently pumped by semiconductor laser diodes.
A disadvantage of the lasers diodes in comparison to traditional lasers, such as N d: YAG
lasers, is rather small size of the active element (compared to the emission wavelength),
and, therefore, relatively pour quality of the laser beam in terms of divergence, power dis-
tribution and spectrum content. Therefore in applications demanding the smoothest power
distribution or fine spectrum content the semiconductor lasers cannot yet compete with the
traditional laser systems.
The pulsed laser diodes are available, but the pulse energy, pulse duration and peak
power are much lower than those of solid state or dye lasers. However, a series of pulsed
laser diodes were developed for application in time correlated single photon counting method
(see Chapter 8), where high pulse energy is not required. 16
3.6.7 Other lasers used in spectroscopy applications
There are many other lasers and laser systems which can be used in optical spectroscopy
applications depending on particular requirements, e. g. wavelength and wavelength band-
width, power, pulse duration and energy and others. For example N 2 pulsed lasers (emission
wavelength 337 nm) can be used for excitation in flash-photolysis measurements in place
of excimer lasers. There are also different types of solid state lasers emitting at different
wavelengths which may meet the requirements of particular applications, e. g. Cr 3 + ions in
sapphire or alexandrite emits in 700-800 nm range and Yr 3 + in YLF emits at 2.8 tL.
15Por eXa111ple TISSA20 frOl11 CDP corp. (Moscow) generates 20 fs pulses.
16Por eXa111ple, a selection of pulsed diodes is available frOl11 PicoQuant Gl11bH. LDH series of the laser diodes
are available in the wavelength range frOl11 375 to 1546 nl11. They provide 50-120 ps pulse width with average
power about 1111W at 40 MHz repetition rate.
3.7. Non-linear optic effect in laser applications
55
There are few laboratories utilizing so-called free electron lasers. Deceleration of rel-
ativistic electrons enables one to build up a laser emitting in a wide wavelength range ex-
panded to X-rays. This are, however, the most expensive laser system considering optical
spectroscopy applications.
Another type of the lasers, which is developing fast at present, are fiber lasers. Being
compact and efficient they can produced a few Watts of CW power which can be easily
delivered to any point by the fiber. Also sub picosecond pulse generation was achieved using
fiber lasers. These are devices which may found numerous optical spectroscopy applications
in the nearest future.
3.7 Non-linear optic effect in laser applications
The second, third and forth harmonic generation mentioned above are examples of prac-
tical importance of the non-linear optics for modern laser technologies. There are many
important fundamental and technological aspects of the subject, from which the harmonic
generation, wave mixing and parametric amplification will be discussed here accounting for
their importance for laser spectroscopy applications.
A linear response of an isotropic medium to an electric field is given by relation
fj == EE
(3.12)
where fj is the electric displacement vector, E is the electric field vector and E is the medium
dielectric constant. The effect of the medium can be also presented by the polarization
vector, j5,
-+ -+ -+
D==E+P
(3.13 )
The polarization of the medium is proportional to the electric field
-+ -+
P == XE
(3.14)
where X is the medium susceptibility. Comparing these three equations one can conclude
that E == 1 + X, or X == E - 1.
Formally, a nonlinear response of the medium can be expressed as
-+ -+ -+2 -+3
P == XlE + X2 E + X3E +...
(3.15)
where Xl is the linear susceptibility (the same as one in eq. (3.14)), X2 is the second order
susceptibility, X3 is the third order and so on.
In the case of anisotropic media the dielectric constant has to be replaced by the dielec-
tric tensor
fj == EE
(3.16)
where
Exx Exy E xz
E == Eyx Eyy E yz
E zx E zy Ezz
(3.1 7)
56
Lasers for Spectroscopy Applications
Similarly, the linear susceptibility Xl is replaced by a 3 x 3 tensor. However, already X2 is
the tensor of order 3 x 6. It must take into account six square terms: E , E;, E;, ExEy,
EzEy and ExEz. Therefore, eq. (3.15) is not strictly correct and must be replaced by a
corresponding vector form, when anisotropic medium is considered.
3.7.1 Second harmonic
The second harmonic generation can be illustrated using eq. (3.15) and considering only
the second order non-linearity. If the incident light is a harmonic wave, E == Eoe iwt , then
the medium response is (using scalar presentation)
P == XlEoe iwt + X2E;ei2wt
The second term is a wave at frequency 2w. The amplitude of the wave at 2w is proportional
to the square of the electric field, thus at a small intensity of the incident light relative
intensity of the second harmonic (SH) wave is low compared to the fundamental harmonic
(given by XlEoeiwt). The response of the medium becomes non-linear at electric field
intensities compatible to the atomic fields of the medium, which are typically greater than
10 9 Vm- l .
All optically transparent media have non-linear response starting from some light in-
tensity. The practical problem, however, arises from the fact that all media have some
dispersion and dispersion is monotonic function of the frequency (in transparency window
of the medium), i. e. the refractive indexes at wand 2w are n(w) < n(2w). This means that
the SH wave, 2w, propagates slower than the fundamental wave, w.
Let us consider SH wave generated at two points at a distance d along the propagation
d . . Wh h d . . d 7fC 7fC A h 2
IrectIon. en t e Istance IS == w[n(2w)-n(w)] == w.6.n == 2.6.n ' t e waves at ware out
of phase and will quench each other. Assuming a quartz medium and base wavelength of
A == 800 nm one obtains d == 2(1.4696 1.453l1 t""V 24 tL. This means that while propagating in
the quartz the intensity of the second harmonic will increase during the traveling through the
first 24 tL of the medium. Then the intensity will decrease, since the waves at 2w generated
at the entrance and at distance of 24 tL have opposite phases and will quench each other. At
the distance of 48 tL the intensity of the SH will be close to zero. As can be seen, in isotropic
media an efficient generation of SH is impossible.
An efficient SH generation can be achieved in anisotropic media (crystals). The waves
with different polarizations may propagate with different velocities in anisotropic crystal.
This is usually discussed in terms of crystal optical axes and "ordinary" and "extraordinary"
polarizations. A two dimensional diagram in Fig. 3.8 shows dependencies of the index of
refraction on the direction of propagation in an uniaxial crystal. When the light propagates
along the X axis the propagation velocity does not depend on polarization, no == ne. When
the light propagates along Y axis the velocities of the ordinary and extraordinary polariza-
tions are different. There is no angular dependence of no, and the dependence of ne is given
by an ellipse. The index of refraction at 2w (dashed lines) is greater than that at w (solid
line). There may be an angle when no(2w) crosses ne(w), this is angle a in Fig. 3.8. The
wave with ordinary polarization at 2w will propagate with the same velocity as the wave
with extraordinary polarization at w in direction given by the angle a. This is the phase
matching condition for the efficient second harmonic generation.
3.7. Non-linear optic effect in laser applications
57
y
...- .....
/ "-
/ "-
/ '\
/ '\
/ ,
/ ,
/ / " '\ \
/ / \ \
I / ,\
1/ \\
U , co
I I
X
,
\
\\
\'
\ ' no(co)
\ '\ o 2co
"
,
, n o (2co)
,
'\ n e( co)
"
"- ,.-
..... ...-
...... n eC2co)
Figure 3.8: Second harmonic generation: illustration of the ordinary and extraordinary
beams angular synchronism.
The phase matching condition can be expressed in terms of the wave vectors (Ikl == )
-+ -+
as 2kw == k 2w , or
kw + kw == k 2w
(3.18)
This is the momentum conservation law for two photons reaction. The energy conservation
law leads to a trivial nw + nw == n( 2w) or w + w == 2w. Therefore, the second harmonic
generation can be treated as a two photons reaction.
Equation (3.18) does not tell anything about the reaction efficiency or probability. Pho-
tons, being Bosons, do not interact in vacuum (this would be the easiest way to satisfy
condition (3.18), however in vacuum X == 0). They interact via the medium. The second
harmonic generation efficiencies for pulsed lasers can be higher than 30%, e. g. for 200 mJ
pulses ofQ-switched Nd:YAG lasers (fundamental harmonic at 1063 nm).
The second harmonic generation is widely used to extend the choice of wavelengths
generated by lasers. It is most efficient for pulsed lasers but also can be used to double the
frequency of CW lasers.
3.7.2 Third harmonic
The third harmonic cannot be obtained directly from the fundamental harmonic, since prob-
ability of three photons reaction (w + w + w == 3w) is evidently much lower than that for
the two photon reaction. Therefore, the third harmonic generation is usually achieved in
two steps. At first, the second harmonic is generated (w + w == 2w). Then, photon at 2w
and photon at w yield the third harmonic photon 2w + w == 3w. It is clear, that the second
58
Lasers for Spectroscopy Applications
step requires another non-linear crystal than the first one, since momentum conservation
law (phase matching condition) in the last step is
---+ ---+ ---+
k 2w + kw == k 3w
(3.19)
and cannot be satisfied together with the momentum conservation law for the second har-
monic generation given by eq. (3.18).
3.7.3 Wave mixing
The reaction equations, such as (3.18) or (3.19), can be written in an general form as
---+ ---+ ---+
k l + k 2 == k3
(3.20)
The energy conservation law requires
WI + W2 == W3
(3.21 )
Similar to eqs. (3.18) and (3.19), eq. (3.20) needs a non-linear crystal of a special kind to
be satisfied. Also the vectors k l and k 2 may have different orientations, as shown in Fig.
3.9.
Similarly to the second harmonic generation, the
condition (3.20) cannot be satisfied if the waves have
one and the same polarization, but unlike in the case
of the second harmonic there are two possibilities
for the waves polarizations. The primary waves, at
WI and W2, may have the same polarization, then the
Figure 3.9: Wave mixing: wave vec- sum wave, at W3, must have polarization perpendic-
tor adjustment (conservation). ular to that of the primary waves. This polarization
arrangement is called type I synchronism and is sim-
ilar to the second harmonic generation (which is WI == W2 == w). Alternatively, the primary
waves may have orthogonal polarizations and polarization of the sum wave will coincide
with one of the primary waves. This situation is called type II synchronism. Importantly,
crystals providing type II synchronism cannot generate second harmonic.
The wave mixing is the phenomenon used in up-conversion method for time resolved
emission measurements (Section 12.1). The non-linear crystal (NLC) mixes together gate
pulse, e. g. WI, and the sample emission, W2, to generate the light at higher frequency, W3.
k1 'k3 '."',..
.' ? >
k ....
2
3.7.4 Parametric amplification and generation of the light
In terms of photon-photon interaction, the wave mixing can be expressed as a reaction
PI + P2 ---+ P3
(3.22)
where PI, P2 and P3 are the photons at WI, W2 and W3, respectively. Equations (3.20) and
(3.21) state that, the photon at W3 is equivalent to the sum of two another photons, at WI and
W2, respectively. Therefore, the reaction (3.22) is reversible
P3 ---+ PI + P2
(3.23)
3.7. Non-linear optic effect in laser applications
59
under the same conditions. In other words, interaction of the photon at W3 with a non-linear
crystal may result in photon "dissociation" and appearance of two photons at WI and W2.
Clearly, this is possible only under conditions specified by eqs. (3.20) and (3.21).
The reaction (3.23) is the single photon reaction, but it can be extended to 2 photon
reaction:
P3 + PI ---+ 2Pl + P2
(3.24 )
As the result of the reaction, two photons at WI are generated and the reaction can be con-
sidered as the reaction of stimulated emission of the second photon at WI For stimulated
reactions the second photon has the same frequency and phase as initial photon at WI, thus
the reaction (3.24) results in amplification of the light at WI. In such case, the wave at WI is
called signal wave, the wave at W2 is called idler wave (it is a loss of energy) and the wave
at W3 is the pump. Usually the shorter wavelength is called signal and the longer idler, also
both wavelengths can be amplified.
The optical devices utilizing non-linear wave mixing for the light amplification are
called optical parametric amplifiers (OPA). They are widely used for amplification of the
femtosecond and picosecond pulses in pump-probe spectroscopy applications as will be
discussed in Section 11.2.
Naturally, amplification can be used for light generation by placing the amplifying
medium, the non-linear crystal, in a resonator. This kind of lasers is called optical paramet-
ric oscillators (OPO). An important advantage of the parametric lasers is the wide tuning
range. The tuning is achieved by rotating the non-linear crystal so that phase matching con-
dition (3.20) is satisfied for different combination of signal and idler waves at fixed pumping
wavelength.
Nanosecond pulsed OPO are commercially available for generation in the visible and
near infrared wavelength ranges with pumping by the third harmonic of the Nd:YAG lasers
(at 354 nm) as illustrated in Example 3.4. Optical parametric oscillators (OPA) are widely
used to produce femtosecond pulses in a wide spectrum range, which are welcomed for
pump-probe spectroscopy applications, as will be discussed in Section 11.2.
Example 3.4: Optical parametric oscillator MOPO (Spectra-Physics Inc.). MOPO-HF
model provides the tuning range 450-705 nm for the signal beam and 715-1800 nm
for the idler beam with 40 mJ pulse energy at the maximum when pumper by PR0230-
10 Nd:YAG laser, providing pulse energy of 400 mJ at 355 nm (third harmonic).
Typical pulse width of the laser 5-10 ns. The output beam of the laser can be passed
to frequency doubling unit to provide additionally emission in 220-450 nm range.
Chapter 4
Optical measurements
As any other type of measurements, the optical measurements depend on detectors which
convert the parameter to be measured into some form of electrical signal. In optical mea-
surements these are photo-detectors of different kinds. Specifically in optical spectroscopy
applications, the measured signals can be very weak and analysis of the noise characteristics
of the source of the signal, e. g. emitting sample, of the detectors and following electronic
devices becomes a part of the measuring procedure. In addition, some of the light param-
eters, such as the width of a femtosecond pulse, cannot be measured directly and require
special optical instruments to be evaluated. These are subjects, which will be discussed in
this chapter.
4.1 Noise statistics and accuracy of measurements
Suppose we want to determine power density created by a light source at some distance
from the source. We have a power meter and a diaphragm with round hole. To complete
the task we need (1) to measure the diameter of the diaphragm hole and (2) to measure
the light power after the diaphragm. We like to perform the measurements as accurate
as possible. Suppose we have measured the hole diameter and obtained a value D l , and
we have measured the light power and obtained a value II. Thus, the power density is
PI == 4 D I 12 , 1 as we can conclude from these measurements. At this point we have a question:
'if 1
how reliable is the value we have just calculated? To answer the question we can repeat the
measurements. Suppose for the second attempt we have obtained close (hopefully) but
somewhat different values, D 2 and 1 2 . Then, we can conclude that the power density is
P 2 == 4 D I22 . Now we have two values, PI and P 2 , and a series of questions:
'if 2
1. Which one is closer to the real power density?
2. If we will perform the measurements third time, what will be the result?
3. What is the accuracy of the measured values?
2
1 The area of the hole is s = 7f D 4 1 , and the power density is PI = !.l = 4 D I 12 .
s 7f 1
61
62
Optical measurements
4. What is the accuracy of the calculated power density?
5. Can we really say anything about power density?
The difference between the measured values, e. g. II and 12, is not necessarily indication
of a pour performance of the measuring devices, but it is rather a consequence of the fact
that any real measurement has some uncertainty, i. e. non-zero inaccuracy. Very often this
inaccuracy is a result of the nature of the measured value. Uncertainty or inaccuracy of the
measurements is the subj ect of this section.
The measurements can be divided into direct and indirect. In the previous example the
power and the diaphragm diameter were measured directly, which means that we apply
certain method to measure the value itself. However, the value of our interest was power
density, which was calculated using the measurements of some other values (e. g. power
and diameter). Thus, it was measured indirectly. Inaccuracy of the indirect measurements
is discussed in the end of the section.
4.1.1 Systematic error and random noise
Suppose we want to measure a parameter A, say diaphragm diameter. This means that we
want to compare it with some standard and express its value in relation to this standard. For
the example considered above we have compared the diaphragm diameter with standard
length of 1 meter, which (the standard of 1 meter) can be found in The State Institute of
Standards in Paris or in similar organizations. It is very unlikely that we will have an
opportunity and real need to compare with the original standard. Most probably we will
deal with a device which was calibrated using some standard. The calibration procedure is
performed with certain accuracy. This means that during measurements the device will add
a value, D.As, to the real parameter value A, which does not depend on the object being
measured or how may measurements were carried out. At least for the principal devices
used in laboratories the calibration inaccuracy can be found in calibration certificates or in
the device specifications.
If the measurements are carried out in conditions close to the extreme, e. g. we want
to measure a very low light intensity, then the results may differ from measurement to
measurement because of the device noise, D.A dn . The measured value, A, may vary by
itself (a low intense light flux is an example and will be considered later), thus giving an
additional deviation D.An from measurement to measurement. Therefore, the measured
val ue is
Ameas == A + D.As + D.A dn + D.An
(4.1 )
When one repeats the measurements the first two summands remain the same and the last
two will change from measurement to measurement. The value D.As is present in all mea-
surements and is called systematic error. It cannot be avoided, except by using another
more precise instrument. The two last summands, D.A dn and D.An, have similar effect on
the measured value (A meas ), they vary from experiment to experiment. They are called
random errors (or random noise, or just noise). Their influence can be reduced by clever
arrangement of the measurements, for example by averaging of a few measurements. Quite
4.1. Noise statistics and accuracy of measurements
63
often the device noise, D.A dn , and the noise of measured parameter, D.An, cannot be sepa-
rated and, therefore, considered as a single random noise Ar == DA dn + DAn. Then eq.
(4.1) can be rewritten as
Ameas == A + As + Ar
(4.2)
The last term in this equation is the subject of the following Section.
4.1.2 Noise statistics
Random values and random functions
The fact that Ar is a random value (noise) means that we can say almost nothing about its
instant value. Instead, we may use probability theory to find the way to deal with the case.
If x is a discrete random value, i. e. it may be one of Xl, X2, ... X N, then the probability
function P(Xi) gives the probability to obtain value Xi in a measurement. For example, it
can be a probability to detect 1, or 2, or 3 or ... photons in a fixed time interval, say 1 s. If
X is a continuous random value, then we can introduce probability density function p( x),
which means that the probability to obtain the random value in the interval from X to x + dx
is p(x)dx. It is evident that the probability to obtain any value is 1, thus both functions
should satisfy conditions
N
L P(Xi) == 1
i=l
(4.3)
and
+ (X)
J p(x)dx = 1
-(X)
(4.4)
For a given probability distribution one may calculate an average value
N
(x) == L XiP(Xi)
i=l
(4.5)
for a discrete random values and
+ (X)
(x) = J xp(x)dx
-(X)
(4.6)
for a continuous random value.
The average of a function f(x) of a random variable x is
+ (X)
(f(x)) = J f(x)p(x)dx
(4.7)
-(X)
64
Optical measurements
Poisson distribution
Let us consider an example typical for spectroscopy applications. Suppose we have a light
source and want to measure its emission rate k, which is the average number of photons
emitted per second. We have a measuring instrument, which can count photons in a fixed
time interval. However, the photon emission is a random process and we need to deal with
probabilities. The probability to emit a photon in a short time interval dt is k . dt (this is the
definition of the rate constant, in fact). Then, the probability that n photons will be emitted
in a time interval from 0 to t + dt, P n (t + dt), is the probability that n photons were emitted
in the interval from 0 to t, P n ( t), and none in time interval from t to t + dt, which is 1 - kdt,
plus probability that (n - 1) photons were emitted in the interval from 0 to t, Pn-l (t), and
one in the interval from t to t + dt, which is kdt. 2 Thus, we can write an equation
Pn(t + dt) == P n (t)(l - kdt) + Pn-l(t)kdt
(4.8)
The equation can be rearranged as
Pn(t + dt) - Pn(t) == -kPn(t)dt + kPn-ldt
(4.9)
The right side of the equation gives dPn(t), thus dividing eq. (4.9) by dt one obtains
dPn(t)
dt == -kPn(t) + kPn-l(t)
(4.10)
This generates a chain of the equations for n == 1,2,.... The solution of the eq. (4.10) is
P ( ) _ (kt) n _ kt
n t - e
n!
(4.11 )
which is known as Poisson distribution. One can introduce dimensionless variable x == kt
and rewrite eq. (4.11) as
Pn(X) = (x)n e-x
n!
( 4.12)
This is a general form of the Poisson distribution. However, the photon counting problem
is one of great importance in spectroscopy and the presentation (4.11) has clear physical
meaning, therefore we will use presentation (4.11) rather than (4.12).
The average number of photons is given by
/ ) _ (kt)n -kt _ -kt kt(kt)n-l
\n - n ,e - e ( _ 1 ) '
o n. 0 n .
When n == 0 the term under sum is zero, thus we can start summing from n == 1. Then, we
can substitute m == n - 1, where m == 0, 1, . . . ,00, which gives
( 4.13)
00 ( kt ) m
\ n) == e -kt kt '""""
m!
o
(4.14)
2Certainly, we can neglect probabilities of el11ission of l110re than one photon in the til11e interval dt, if the
interval dt is short enough.
4.1. Noise statistics and accuracy of measurements
65
0.2
I \
I \
\
I \
I \
I
0.15 I
\
\
\
I \
I \
I \
.......
- I
....... 0.1 I
I \
ro I \
I \
0 \
H \
\
I
0.05 I
I
I
I
I
0 -
0 1 2 3 4 5 6 7 8 9 10 11 12
number of photons, n
Figure 4.1: Poisson distribution with kt == 4.
n
One can mention that the sum in eq. (4.14) is the power series of the exponent, eX == L ,
n.
then
(n) == e-ktkte kt == kt
( 4.15)
This is the result one can predict without any derivation, the average number of photons
emitted in time interval t is the product of the emission rate constant, k, and the observation
time interval, t, i. e. it is kt.
The probability distribution of the number of photons emitted in a time interval t == ,
i. e. when (n) == kt == 4, is shown in Fig. (4.1). It is interesting to note, that the probabilities
to observe 3 and 4 photons are the same
P3(4/k) (kt)34! 4 4
- - - _ 1
P4(4/k) - 3!(kt)4 - kt - 4 -
Thus, we will observe 3 photons as frequent as 4 photons. Nevertheless, the average number
of photons will be 4. This reflects the fact that the distribution is not symmetric and has a
"longer tail" at the high number of photons. It is important to learn by this example that the
most probable value and the average value can be different values.
Gaussian distribution
At greater value of kt, e. g. at longer observation time interval, the Poisson distribution
has a more symmetric shape, as illustrated in Fig. 4.2 where the value kt == 100 was used.
Figure 4.2 shows that at high kt values the Poisson distribution is very close to the Gaussian
distribution. Indeed, at kt ---+ ex) the Poisson distribution is transformed to the Gaussian
66
Optical measurements
0.04
0.03
>:
.
-
.
0.02
o
0.01
0 60
Gaussian, Xo = 100, (J'= 10
Poisson, kt= 100
80 100 120
number of photons, n
140
Figure 4.2: Poisson distribution, kt == 100, and its Gaussian approximation (eq. (4.16))
one 3
Pn(t) = G(n, t) = 1 exp ( _ (n -kkt? )
V 27rkt 2 t
The Gaussian distribution of a random value is very often found in practical measure-
ments. Almost all types of noises are well approximated by Gaussian distribution function.
A general form of the distribution is
( 4.16)
G( ) 1 ( (x - XO)2 )
X == exp-
V2Ka 2a 2
The distribution has maximum at x == Xo, and, since the function exp( -x 2 ) is symmet-
ric one, the average value is equal to the most expected one, (x) == Xo.
Both Figs. 4.1 and 4.2 show that probability to obtain "correct" number of photons (4
and 100, respectively) in the first measurement is rather low, roughly 20% for kt == 4 and
only 4% for kt == 100. We need to repeat the measurements a few times to learn more about
the value of out interest, e. g. the emission rate.
( 4.1 7)
Data averaging and square root law
For a series of M measurements resulting in values Yl, Y2, . . . , Y M the average value is
1 M
(Y) = M L Yi
i=l
( 4.18)
3 To obtain eq. (4.16) frol11 eq. (4.11) one l11ay use Stirling approxil11ation of the factorial, In n!
n + In n - n + In 27r, and neglect by sl11aller terms of n.
4.1. Noise statistics and accuracy of measurements
67
Now, we may wonder what is the average deviation of the particular measurement, Yi, from
the average value, (y). This is given by the mean square deviation
1 M
(!J.y 2 ) = (((y) - y)2) = M L((y) - Yi)2
i=l
( 4.19)
The value J ( y2) is also called standard deviation.
Independent of the distribution type it can be shown that
\ y2) == \y2) _ (y)2
( 4.20)
This equation is somewhat easier to use in practice than eq. (4.19), since one does not need
to keep results of all the measurements (Yl, Y2, . . . , Y M) but can work with two average
values, \y2) and (y)2. 4
An important property of the Poisson distribution is that mean square deviation is
00 ( kt ) n
\ n2) == ,,( (n) - n)2 e- kt == kt
n'
o .
(4.21 )
which is, for Poisson distribution the average of square deviation is equal to the the av-
erage value, \ n 2 ) == (n). This dependence is called square root law and has different
presentation forms
J ( n2)
(n)
1
J(n)
1
v1ct
( 4.22)
or
J ( n2) == J(n)
( 4.23)
The practical use of the square root low is that if in some experiment the average number
of detected photons was N, then the standard deviation for the measurement is VN. For
example, for average value of photons 100, the average deviation from measurement to
measurement, or the standard deviation is 10.
In the case of Gaussian distribution, eq. (4.17), the mean square deviation, as given by
Eq. (4.19), is \ X2) == a 2 , which is commonly called "sigma" -value.
The averaging of M measurements gives value (x) M' where the index is used to indicate
the number on measurements. Now one may ask, how accurate is the average value, (x) M'
or how close is it to the real value, i. e. to (x) 00 ?
The value (x) M is a random value with its own distribution. The probability P ((x) M)
to obtain value (x) M is the product of probability to obtain Xl, i. e. PI (x), times probability
to obtain X2, i. e. P2(X), and so on. In other words, P ((x) M) == rr l Pi(X). For a series of
identical measurements the probability functions are the same for each measurements, thus
4During the l11easurel11ents two cUl11ulative values are collected, 2:: y and 2:: y;, which are then used to
calculate the average, the average of the square value and the standard deviation.
68
Optical measurements
1
[ill =o
. .. n = 1
-.n=2
.- n=3
0.8
. 0.6
.
.1=;
ro
.1=;
o
0.4
.
.
.
0.2
,..
'"
'"
/
/
/
/
/
. '" .'
00
,..
--
...... .....-:- . - ........, == . -- .. --
, .. - - - : :. - . - . -
.. .. .. ...... ...... ...... ......
. . .
. . .
. . .
. . .
. . .. ..
,..
,..
1
234
Observation time, t
5
Figure 4.3: Time dependence of the probabilities to observe 0, 1, 2 and 3 photons exactly,
calculated for emission rate k == 1.
p ( (x) M) == pM (x). Let us suppose that the variable x has Gaussian distribution, as this is
the most common case. Then, using eq. (4.17) one obtains
P((X)M) rv
[ ( (X_XO)2 )] M ( M(X-XO)2 )
exp - == exp -
2a 2 2a 2
( (X_XO)2 )
exp -
2a 2
M
(4.24 )
which is again the Gaussian distribution, but the width of this distribution, i. e. its standard
deviation, is aM == J-M .5 Thus, averaging of M measurements improves accuracy by
factor m, which is again the square root law.
Observation time dependence of the photon counting problem
In all previous examples a constant observation time interval was considered, so the proba-
bilities to observe different numbers of photons were presented in Figs. 4.1 and 4.2. Instead,
one may vary the observation time interval, t, and query the probability to observe exactly
n photons. The probabilities for n ==0, 1, 2 and 3 photons are presented in Fig. 4.3, which
were calculated assuming k == 1 and using eq. (4.11) with appropriate n-values. It is natural
5 Equation (4.24) gives the probability dependence on x only. To obtain correct scaling factor the normalization
(4.4) have to be done, which leads to p (x) M = ff1 exp [_ M( O)2 ].
4.1. Noise statistics and accuracy of measurements
69
that at a short time intervals, t « 1, the most expected result is zero photons. The proba-
bility to observe no photons is Po (t) == e - kt. Under condition kt « 1, it can be reduced
to Po (t) 1 - kt, and one can note that the probability to observe at least one photon
is Pn>o == 1 - Po kt, which was used as definition of the emission rate. The proba-
bility to observe exactly 1 photon grows linearly from zero, reaches its maximum value of
36.8% at kt == 1 and then decreases. 6 The probability to observe exactly two photons grows
quadratically at kt « 1 and has its maximum at kt == 2, and so forth.
4.1.3 Statistical approach to measurements
Dealing with random values one must understand, that after a single measurement not much
can be said about the process under the study if no additional information is available. In
the case of photon counting problem such information is the known probability function.
Thus, even for a single measurement the uncertainty (standard deviation) can be estimated
using eq. (4.23).
Example 4.1: Uncertainty estimation in photon counting measurements. If during t == 1 ms
N == 10000 photons were detected, the standard deviation is N == VN == 100
photons, i. e. the relative error is 6 == .6.: == 0.01, and the counting rate is (1.00 =:t
0.01) x 10 7 S-l. But, if only N == 2 photons where detected, the reliability of the
rate estimation is very poor. The standard deviation is V2 1.4 counts, therefore
the counting rate was determined with accuracy (2 =:t 1.4) x 10 3 S-l.
In the example above, the accuracy can be improved by (1) increasing the time interval
of the photon counting orland (2) by repeating the measurement a few times and calculating
the average value. Following the latter option, one repeats the measurements M times and
obtains series of values Yl, Y2, . . . , YM. Then, the average value, (y) M' can be calculated
using eq. (4.18). The mean square deviation, \ y2 ) (or a 2 ), is calculated using eq. (4.19).
In the case of Poisson distribution (y) \ y2) (see square root law, eq. (4.23)), and the
standard deviation of the averaged value is \ X2) / J]\/i. For the photon counting problem
one can notice that the total measuring time is tM == Mt, and if the photons would be
counted only once, but during this time interval the final result would be J]\/i times more
accurate than for a single measurement with counting time t. In other words, the accuracy
of the final result depends on the total counting time (more exactly on the total number of
counted photons) but does not depend on the calculation procedure. Unfortunately, there
are practical and technical reasons limiting time of a single measurements, and averaging is
a usual trick to improve the results.
The averaging helps to reduce measurement uncertainty caused by the noises of the
device or by the fluctuations of the parameter being measured. In terms of eq. (4.2) it
allows to reduce random error Ar but it does not change systematic error As.
6Note that at the best til11e interval to observe exactly one photon, kt = 1, the probability of this event is only
36.8% and is equal to the probability to observe no photons.
70 Optical measurements
4.1.4 Noise sources
There are very many sources and types of noises, also three types are the most common for
measurements in the optical spectroscopy applications.
Quantum noise (or photon noise, or short noise, or Poisson noise) is one considered in de-
tail in the previous section (eqs. (4.11 )-(4.16)). In Example 4.1 the light source
has constant emission rate, however, the experimentally available value is the
number of photons measured in a limited time interval, which is the random
value by its nature. This type of noise cannot be eliminated when dealing with
quantum objects (but in some applications can be reduced to negligible level).
This type of noise can also be found in electric circuits at low current values,
when the discrete nature of current carriers (electrons) prevails thermal noise.
Thermal noise (or Johnson noise) has actually the same origin as quantum noise but at
low frequency limit. At temperature above absolute zero the space is filled by
thermal radiation, which has quantum nature, i. e. it fluctuates. When applied
to electric circuits this noise is called Johnson noise. In all circuits the voltage
noise in a spectrum range f is
(v 2 ) == 4kTRD.f
( 4.25)
and the current noise is
(I2) = 4kT L1
R
( 4.26)
where R is the circuit active resistance. The power of the Johnson noise is
p == vi (U 2 ) (12) == 4kT D.f
( 4.27)
The noise power does not depend on resistance R.
1/ f noise and generation-recombination noise have spectrum density proportional to the
inverse of the frequency (which has given the name to this type of noise). Usu-
ally it dominates in measurements which take a long time. These measurements
are said to be low frequency measurements. Depending on the measuring tech-
nique and devices used, the frequency limit for the 1/ f noise domination can
be from 10 Hz (0.1 s in time domain) or smaller.
4.1.5 Inaccuracy of indirect measurements
Let us return to the first example of this chapter dealing with the indirect measurements, but
reformulate it in a general way. The parameter of our interest, e. g. C, was calculated based
on two experimentally obtained values, e. g. A and B. We know the measurement errors
(standard deviations) A and B and we want to evaluate the error (standard deviation)
of C, i. e. C. Yet another widely used specification of the measurements accuracy is the
relative inaccuracy, which can be calculated as a == .6. A A , b == .6.: and c == .6.cf for
parameters A, Band C, respectively.
4.1. Noise statistics and accuracy of measurements
71
How can C and c be estimated from known A, B, a and b? For four arithmetic
operations the rules are simple:
Sum, C == A + B: adding errors of the values one obtains
C + C == A + A + B + B == A + B + A + B == C + A + B
thus
C == A + B
( 4.28)
For the relative inaccuracy
_ C _ A + B
c- C - A+B
( 4.29)
The latter can be expressed in terms of a and b, but the result is rather useless.
Hence, the error of the sum of values is the sum of errors.
Subtraction, C == A - B: The case is similar to the sum, one only needs to change the
sign of B. However, the error is never negative, therefore, C == A + B. From
the point of view of the calculation accuracy the subtraction is dangerous operation.
If the values A and B are close to each other the error (which is always sum of errors)
may be greater than the subtraction result, so the result may be statistically undefined.
This is clearly seen from the relative inaccuracy, which is c == .6. B (compare
with eq. (4.29)!), thus, when A ---+ B, c ---+ 00.
Multiplication, C == AB:
C+ C
(A + A)(B + B) == AB + A B + B A + A B
C +A B +B A+ A B
Let us neglect A B, which means A » A and B » B, or we assume that our
primary values, A and B, are rather accurate, then
C t""V A B + B A
( 4.30)
and
c == A B + B A == B + A == a + b
C B A
(4.31 )
Hence, the relative error of the product, c, is the sum of the relative errors of the
multiplicands, a + b.
Division, C == :
C C == A + A
+ B + B
A
B
1 + .6.A
A
1 + .6.B
B
72
Optical measurements
Assuming that A » A and B » B, similar to the multiplication case, one
obtains
c + c
A ( 1 + A ) ( 1 _ B ) t""V A + A . A _ A . B
B A B B BA BB
C+C a-C b
where the term with A B was neglected. Finally, taking into account that the
inaccuracies must be summed independent of the sign
c == C a + C b == a + b
C
( 4.32)
The result is similar to multiplication, eq. (4.31): the relative error is the sum of the
relative errors of the numerator and denominator.
All the previous derivations are true when the error is smaller than the value, i. e. A «
IAI. Under the same conditions we may derive the error which arises when a function f()
is used to calculate value C from A. That is C == f(A), and C + C == f(A + A). The
function can be approximated by linear dependence around point A
C + b.C = f(A) + df(A) b.A = C + df(A) b.A
dA dA
( 4.33)
Thus,
C == df(A) A
dA
(4.34)
In other words, the slope of the function f() at point A determines the accuracy of its result,
C.
4.2 Photosensitive devices
4.2.1 Photo detector performance parameters
Photo-detectors convert the light power into electric signal, voltage or current, which can be
recorded or measured using standard electronic devices. The photo-detectors characteristics
essential for spectroscopy applications are
. sensitivity;
. efficiency;
. spectrum range;
. time resolution.
Lesser critical but still important characteristics for the most of applications are
4.2. Photosensitive devices
73
. area of the photosensitive element;
. dynamic range;
. physical dimensions;
. power consumption and additional electronic devices needed for the operation.
Unfortunately different sets of parameters are used to specify different types of photo-
detectors. In the following list the parameters, which can be used to compare different
classes of detectors, are given with comments on their usage and meanings:
Quantum efficiency: ratio of the photons creating a photo-response, e. g. generating elec-
tron, to the total number of the incident photons. This parameter specifies efficiency
of the light conversion to the electric signal. It is an important contributor to the
sensitivity of the device but not the only one.
Sensitivity: characterizes electric response of the device (current or voltage) created by in-
cident light power. It is measured in A.W- l or V.W- l depending on the response
type, current or voltage, respectively. This parameter tells what to expect of the de-
tector output at a given incident light power. It is wavelength dependent value. 7
Noise equivalent power (NEP): specifies the minimum light power in frequency band of
1 Hz which could be detected. It is measured in W.Hz- . For example, if one
needs to measure light power in the frequency range of f == 10 kHz, i. e. with
the time resolution of T == 2;j 16 tLS, and would like to use a photodiode with
N EP == 10- 12 W.Hz- (e. g. a Si photodiode), then the minimum detectable light
power will be P == NEP x Vl == 10- 10 W = 0.1 nW. The value of minimum
detectable power is higher when the frequency response of the detector is wider (i. e.
time resolution is faster), and it is proportional to the square root of the frequency
response. 8 This is wavelength dependent value.
Detectivity: many photo-detectors, e. g. photodiodes, exhibit a noise equivalent power that
is proportional to the square root of the detector area. For these devices a detectivity
is defined as D == VA . (N E P) -1 , where A is the detector area.
Dark counting rate: for the detectors working in photon counting mode this parameter
specifies the average counting rate under no light illumination. Usually it is measured
in counts per second, i. e. s -1 .
Dark current: for photodiodes and photomultiplier tubes specifies the output current with
no incident light. The lower value is better for the same type of detector.
7In earlier literature the term responsivity was used as synonYl11 of sensitivity.
8Prol11 the eXaI11ple of the photon counting problel11, one can see that at longer collection til11e a bigger nUl11ber
of counts is achieved which results in a sl11aller relative uncertainty of the l11easurel11ents. The decrease in relative
uncertainty is proportional to the square root of the nUl11ber of counts, thus it is proportional to the square root of
the averaging til11e and inversely proportional to the square root of the detector frequency response. In a sense, this
is consequence of the square root law discussed in Section 4.1.2
74
Optical measurements
Cathode
Dynodes
Anode
,...... - -.. ,...... - - ..........
- ...... / - "/ + -, "
'- / - '- \\ / // -- \\ \\
- .;:;::-.
/-
//
/ / /-
111/»-
I I I I
U out
Ra
hv
R
1 1
R 1 R 1
1 1
k: > 1
1 U d 1
R
R
R
U c
Figure 4.4: Schematic diagram of a photomultiplier.
Time constant and frequency response: The time constant (T) specifies how fast the sig-
nal is formed on the device output when the light is switched on instantly. The
frequency response is measured with the sinusoidally modulated light, e. g. light
intensity is I (t) == 10 [1 + sin (27r 1 t) J, and specifies the highest frequency 10 (cut
off frequency) at which the photodetector responses without significant signal reduc-
tion (usually measured at the level of -3 db relative to the low frequencies response
amplitude). The frequency response is inversely proportional to the time constant,
T t""V (27r 10)-1.9
4.2.2 Photomultiplier tubes
Photomultiplier tube (PMT or just photomultiplier) is an electronic device converting the
incoming photons to current. It consists of photo-cathode and electron multiplication sys-
tem. The photo-cathode converts photons to electrons and the multiplication system am-
plifies the electric signal. A classic scheme of a photomultiplier tube is shown in Fig. 4.4.
The photomultipliers require high voltage power supplies to operate properly. The negative
high voltage is applied to the photo-cathode, U c , and divided between the dynodes forming
electron multiplication subsystem. When a photon hits the photo-cathode it generates an
electron. The electron is accelerated due to the potential between the photo-cathode and the
first dynode, so that when it hits the dynode it generates a few secondary electrons (typically
3-4 electrons). The secondary electrons are accelerated by the electric field between the first
and the second dynodes and each of them produce another 3-4 electrons. This multiplica-
tion process continues until the electrons reach the anode, where the output signal ( current)
is collected. Typical photomultiplier consists of 9-12 dynodes, and the potential required
to obtain multiplication of 3-4 at the dynodes is 100-150 V (the electrons gain energy of
100-150 e V being accelerated between the dynodes). Thus, the power supplier of the the
photomultiplier must provide voltage of 800-2000 V, at which the current multiplication
factor can be 10 6 - 10 7 .
9The proportionality coefficient between the til11e constant T and cut off frequency 10 depends on exact def-
inition of the tiI11e constant and on the response order at the detector. Given relation is valid for the first order
response and the til11e constant l11easured at the level of the signal of 1 - e- 1 rv 0.63.
4.2. Photosensitive devices
75
Photo-
cathode
Micro-channel
plate
Anode
------------------
U3
signal
(
I
I
hv I
I
I
I
'-
U2
U1
R
Figure 4.5: Micro-channel plate photomultiplier tube.
The limiting factor of the photomultiplier time resolution is the electron traveling time
from the photo-cathode to the anode, which is typically a few nanoseconds. To reduce this
time a micro-channel plate amplification system was developed. The micro-channel plates
(MCP) are thin plates. with great number of microscopic holes, channels, having diameter 6-
20 tL. The inner surface of the channels are processed to have proper electric resistance and
secondary emissive properties. When a high voltage is applied across the plate the potential
is distributed across the plate creating electric field, which can accelerate the electrons.
The micro-channel plates replace the traditional dynode systems of the photomultipliers.
A scheme of micro-channel plate photomultiplier tube is shown in Fig. 4.5. The photons are
converted to electrons by the photo-cathode, accelerated by potential U 3 - U 2 and enter the
micro-channels. In the micro-channels the electrons hit the walls and generate secondary
electrons. The secondary electrons are accelerated and hit the channel walls thus generating
new electrons. Each generation multiplies the number of electrons due to the acceleration
similarly to the multiplication process in the dynode system. Therefore, when the electrons
are collected by the anode the signal is amplified many times.
The amplification factor of single micro-channel plate is smaller than that of a dynode
system of a typical photomultiplier. To achieve amplifications similar to those of traditional
PMTs, two or three micro-channel plates are usually assembled one after another inside one
photodetector. The electron traveling distance in the micro-channel plate photomultiplier
tubes is much shorter than that in the traditional dynode systems, which allows to improve
the time resolution by almost one order of magnitude, to less than one nanosecond in real
time measurements and to tenth of picoseconds in time correlated single photon counting
mode (see Chapter 8).
The micro-channel plates can also be used as image intensifiers. In this case a phosphor
screen is used in place of the anode (Fig. 4.5). Because of micro-channel structure of the
amplifying plate the secondary electrons on its exit keep the positions of the photoelectrons
on its input. Therefore, the optical image is converted to electron pattern by the photo-
76
Optical measurements
Table 4.1: Characteristics of photo-cathodes, CPm is the peak quantum efficiency, Am is
the wavelength of peak efficiency and id is a typical dark current (the dark current is very
sensitive to the supplied voltage and temperature of the cathode).
cathode range, nm CPm,% Am,nm i d , nA
bialkali (S-22) 300-630 26 400 0.1
multi alkali (S-20) 180-800 20 480 0.2
extended red multialcali (S-25) 300-900 7 600 1
GaAs 300-920 15 700 2
Cs- Te 160-320 14 200 0.01
cathode, which is transmitted to the micro-channel plate, amplified and converted back to
an optical image by the phosphor screen. 1 0
The main application area of the photomultipliers is detection of low light intensities
in visible and ultraviolet (UV) wavelength ranges. Particularly important feature of the
photomultipliers is that they can be used in photon counting mode.
At low photon flow the output signal of a photomultiplier consists of electric pulses,
with each pulse corresponding to one detected photon. This makes possible to use pho-
tomultipliers in photon counting regime. The output of a photomultiplier is connected to
the input of a discriminator which rej ects low amplitude pulses due to thermal electrons,
and forms fixed duration and amplitude pulses for each detected photon. The pulses from
the discriminator can be directed to a counter or be used by some other electronic device.
The dark counting rate of the photomultipliers is proportional to the dark current. For the
ultraviolet-visible sensitive photomultipliers (spectral response 160-600 nm) typical dark
counting rate is 10 counts per second, and for specially selected devices it can be just a few
counts per second. The dark counts of the photomultipliers sensitive in the near infrared
region (spectral response up to 850 nm) is typically higher than 100 S-l, but can be reduced
by more than one order of magnitude by cooling down the photo-cathode to -30 . . . -40°C.
The photo-cathodes are the principal parts of the photomultipliers determining their
spectrum response and noise characteristics. An example of the photo-cathode characteris-
tics is presented in Table 4.1. Typically the red sensitive photomultipliers have higher value
of the dark current, which is due to the fact that for those photomultipliers the work function
of the photo-cathode is lower, therefore more thermal electrons can escape the surface.
The most important characteristics of the photomultipliers which can be found in their
specifications are:
wavelength range: determined by the type of photo-cathode, typically visible and UV
parts of spectrum, a few photo-cathodes can be used in the near infra-red range up
to 850 nm (S-20) and even up to 1100 nm (S-l);
peak quantum efficiency: can be up to 20°A> in visible range;
lOThe l11icro-channel plate intensifier increases the intensity of the il11age but all the specific information about
the incol11ing light, such as wavelength, polarization and beaI11 propagation direction, is lost. This l11akes if different
frOl11 the laser a111plifier, which preserves the wave properties of the aI11plified light.
4.2. Photosensitive devices
77
size of the sensitive area: can be 2 cm in diameter or even larger;
anode dark current: determines noise level, typically less than nano Ampere for the PMT
sensitive in the visible and UV range and somewhat higher than nano Ampere for the
near infrared sensitive PMT;
dark counts: dark counting rate, for PMT depends on the type of the photo-cathode. For
the cathodes sensitive in the UV and visible part of the spectrum (wavelength shorter
than 650 nm) the dark counting rate can be 10 S-l or even smaller. For the photo-
cathodes sensitive in the near infrared part of the spectrum the dark counting rate is
higher and can be >100 S-l. By cooling the photo-cathode by 30-40 °C the dark
counting rate can be reduced by more than one order in magnitude;
amplification factor: typically 10 6 ;
transient time spread: dispersion of the pulse signal propagation from photo-cathode to
anode, typically 0.1-10 ns, important for time correlated single photon counting ap-
plications, see Section 8.4.1;
response time: typically a few nanoseconds, but for MCP PMTs can be shorter than nanosec-
ond.
Main advantages of the photomultipliers are:
. high sensitivity, due to the high multiplication factor the sensitivity can be 10 4 A. W- l ;
. can be used in photon counting mode;
. good time resolution (up to 20 ps for micro-channel plate PMT);
. relatively big photo-sensitive area (a centimeter size is typical).
Disadvantages are:
. sensitivity depends on the wavelength;
. relatively big size;
. utilizes high voltage power supply;
. difficult to construct multi -channel devices.
Leading manufactures of photomultipliers, e. g. Hamamatsu Corp., are producing pho-
tomultiplier modules which incorporates the photomultiplier and the high voltage power
supply, so that the device is supplied with a low voltage only. This simplifies utilization of
the photomultipliers. Additionally, there are modules combining togerther photomultiplier
pre-amplifier, discriminator, counter and communication port, which can be used as photon
counting unit connected directly to a computer or digital controller.
78 Optical measurements
4.2.3 Semiconductor photo-detectors
The most usually used semiconductor photo-detectors are photo-resistors, photodiodes (PD),
avalanche photodiodes (APD), photo-transistors, photodiode arrays and charge coupled de-
vices (CCD). Practically important for spectroscopy applications are photodiodes, avalanche
photodiodes, photodiode arrays and CCDs.
The photodiodes are cheap, compact and easy-to-use devices, therefore they are the
most popular photo-detectors. The spectrum range and sensitivity of the photodiodes are
determined by semiconductor material. The response time of the diode is mainly determined
by the capacitance of the p-n junction. In order to improve the response time (decrease
the capacitance) p-i-n structures were developed. Another characteristic affecting the time
resolution is the size of active area - smaller areas have smaller capacitance and provide
faster response time. Therefore fast photodiodes have typically small photo-sensitive areas,
less than 1 mm for diodes with response time shorter than 1 ns.
Although there are many different types of semiconductors, three of them are the most
common in optical spectroscopy applications:
Si photodiodes are sensitive in 300-1100 nm wavelength range and have typically sensitiv-
ity up to 0.5 A.W- l at 800 nm. ll The best p-i-n photodiodes have very good
time resolution, T < 100 ps. With a special treatment the sensitivity range
can be extended to the ultraviolet part up to 190 nm. The diodes have small
dark current (typically less than 1 nA for a millimeter size diode) and good
noise equivalent power (NEP), which can be as small as 1.5 x 10- 15 W.Hz-
(S5973, Hamamatsu, diameter of active area is 0.4 mm).
Ge photodiodes can be used in the wavelength range 800-1700 nm. The devices with small
active area have good response time, shorter than nanosecond.
InGaAs photodiodes have typical sensitivity range 900-1700 nm with maximum sensitivity
at 1550 nm. The diodes have high quantum efficiency and sensitivity (typically
0.95 A.W- l at peak sensitivity). The dark current can be smaller that nano
Ampere, which provides good NEP, e. g. 2 x 10- 15 W.Hz- for G8376-1
(Hamamatsu) with active area diameter 0.04 mm.
An advantage of the photomultipliers as compared to the photodiodes is a large amplifi-
cation of the photoelectrons achieved in dynode system. The amplification can be also
achieved in specially designed diodes at certain reverse bias voltages. These devices are
called avalanche photodiodes (APD). For Si APDs the gain factor can be as high as 100.
Although this amplification is not as high as that of the photomultipliers, it improves sensi-
tivity of the diodes to the level when they can be operated in photon counting regime. 12 An
11 Ideally, one can expect each photon to be converted to an electron, in which case the sensitivity is S = hqv =
h ' where q is the electron charge. At A = 800 nm the top possible sensitivity is S rv 0.65 A.W- 1 .
12To reduce dark current to the level when dark counting rate is reasonably low the area of the APD l11USt be
sl11all. For eXaI11ple, for APD PDM 50CT SPAD detector (frOl11 Micro Photon Devices) with active area 50 J-L, the
dark counting rate is 5000 s -1 at rOOl11 tel11perature.
4.2. Photosensitive devices
79
advantage of the APDs is also a good response time, which can be shorter than nanosec-
ond. 13
Charge coupled devices (CCDs) are another class of semiconductor photo-detectors
which are widely available in the market and actively used in optical spectroscopy. Al-
though general purpose CCDs can be used in non-demanding spectroscopy applications,
there are specially designed CCD detectors for spectroscopy. These are usually state-of-art
devices, which are rather expensive but allow to detect the whole spectrum at once and can
reduce significantly the time needed for measurements. The design goals for these detectors
(as compared to general purpose CCDs) are better linearity of the response, greater dynamic
range (the ratio of the maximum non-distorted signal to the minimum detectable signal) and
lower noise level.
Typical characteristics of Si based CCDs are similar to those of the Si photodiodes. The
wavelength range is 300-1100 nm. Peak quantum efficiency is 40-90°/6, which provides
sensitivity close to 0.6 A.W- l . There are also InGaAs linear image sensing arays, which
can be used in wavelength rage 900-1700 nm.
The main advantages of the semiconductor photo-detectors are
. small size;
. ease of use and low price;
. high linearity and dynamic range;
. good time response;
. sensitive in near infra-red range.
In addition, the great advantage of CCDs and diode arrays is multi -channel detection, which
finds numerous applications in optical spectroscopy.
Disadvantages of the semiconductor detectors are
. sensitivity depends on spectrum;
. relatively low sensitivity as compared to PMT; 14
. small size of photo-sensitive area for photodiodes with fast time response.
13 A short cOl11parison of APDs and MCP photol11ultipliers in til11e correlated single photon counting applica-
tions can be found in Section 8.3.1.
l4When cOl11pared to photol11ultipliers, the diodes have higher quantul11 yield but do not provide any aI11plifi-
cation of the electric signal, therefore the sensitivity of the diodes is l11uch lower than that of the photol11ultipliers.
Avalanche photodiodes provide an a111plification of the electric response and have sensitivity il11proved by factor of
100 or even greater, but the price of these devices is typically l11uch higher than that of the photol11ultipliers. The
avalanche photodiodes can be used in photon counting l11ode.
80 Optical measurements
4.2.4 Other photo-detectors
A disadvantage of PMTs and photodiodes (and similar semiconductor detectors) is depen-
dence of the photo-response on the wavelength, i. e. the sensitivity, S, is rather sharp func-
tion of the wavelength, S == S (A). When neither high sensitivity nor high time resolution
are required thermal detectors can be used to measure light intensity. The measured param-
eter is heating produced by the incident light. There are different types of pyroelectric de-
tectors, bolometers and alike. They can operate in wide wavelength range (300-10 000 nm)
and are very useful for steady state measurements of the light power higher than 1 m Wand
the light pulse energy higher than 1 mJ.
4.3 Measurements of the light power
If the light power is rather high, different types of thermal detectors can be used ( see above).
The main advantage of these devices is the flat spectrum response and a very wide spectrum
range (300 nm - 10 tL). Typical detection limit of these detectors is 0.1 m W. Typical mea-
surement error is 10%, however carefully designed devices can provide accuracy better than
1 %. To achieve high accuracy the sensitive surface of the device should have equally good
absorption (ideally 100%) in a wide spectrum range.
If lower light power has to be measured, pho-
todiodes can be used. A typical electric circuit for
the measurements consists of the photodiode con-
nected in series with a resistor, R, as shown in Fig.
4.6. The measured value is the potential (voltage),
U out , at the resistor R, which is often called load
resistor. A bias voltage, E, is supplied to the diode
with polarity keeping the diode closed, so that the
output voltage is virtually zero (U out == 0) with-
out illumination. 15 If the incident light power is P
and the photodiode sensitivity is S, then the pho-
tocurrent is I == S P, and corresponding potential
on the resistor is U out == I R == S RP. The product,
Su == SR, can be called voltage sensitivity. For ex-
ample, with S == 0.5 A/W, and R == 10 kOhm, one
obtains Su == 5000 V/W, or 0.1 mW of light power will produce 0.5 V response, which is
easy to measure with any voltmeter.
As can be seen, the voltage response of the scheme in Fig. 4.6 is higher when the load
resistance, R, is higher. The limiting factors for increasing Rare (1) the input resistance of
the device used to measure potential U out , and (2) the diode dark current. The latter gives
some potential on the load even without illumination. For example, a typical dark current for
diodes with big sensitive area (which is important for general purpose power measurements)
can be as high as I dark 100 nA, assuming load resistance to be R == 1 MOhm, the output
potential is IdarkR == 0.1 V, which is almost certainly the value one cannot neglect.
Photodiodes (and similar devices) have high sensitivity and wide dynamic range. Ob-
vious disadvantage of the photodiodes is rather sharp dependence of the sensitivity on the
+E
p
tI=SP
R
U out = IR
Figure 4.6: Electric circuit for mea-
surements of the light power with
photodiode.
15The actual value of the output voltage is determined by the dark current of the photodiode, Id, so that
U out = RId.
4.4. Measurements of the pulse energy
81
Pulse Detector response +E
-------- - - -- pet)
/'
U d
R C U d
a) t b)
Figure 4.7: Pulse energy measurements: a) integration of the pulse intensity by a slow
photo-detector, and b) electric integration circuit for a photodiode.
wavelength. Therefore, the photodiode must be calibrated at the wavelength of light power
measurements. Nevertheless, if calibration is done carefully, photodiodes provide very high
accuracy of measurements (much better than 1 %) in a width range of the incident power
from nano to milli Watts. Another advantage of the PD is high time resolution, which
allows to reduce measurement time to microseconds or even shorter.
4.4 Measurements of the pulse energy
The same devices as for power measurements can be used for the pulse energy measure-
ments. However, the pulse duration, Tp, and the time response of the light detector, Td, must
be Td » Tp to simplify the procedure, as illustrated in Fig. 4.7 (a). Under this condition the
response of the detector is proportional to the integral of the pulse intensity
U d '" J lp(t)dt
( 4.35)
where I p ( t) is the intensity time profile of the pulse. Thus, one has to measure photodetector
output right after the pulse. In other words, the detector integrates the pulse intensity profile
and output readings of the detector are proportional to the pulse energy. The thermal de-
tectors have relatively slow response time, ranging from milliseconds to seconds. They are
usually used to measure pulses shorter than 1 ms and with the energy greater than 0.1 mJ.
The photodiodes and similar devices can be used to detect lower pulse energies. In
such case the time response of the photodiode should be slow enough to ensure integration
of the light pulse with the accuracy required for the measurements. For example, if the
pulse duration is 1 tLS and the measurements inaccuracy must be better than 1 %, then the
detector integration time must be Td > 1tLs/0.01 == lOOtLs. Most of the photodiodes have
time response much shorter then 100 tLs. The problem is easily solved by adding simple
integration circuit, as shown in Fig. 4.7 (b). In particular case, one may select R == 100 kO
and C == 10 nF, which gives Td == RC == 1 ms.
Similarly to the light intensity measurements, one has to calibrate the photodiode at the
wavelength of measurements in order to obtain absolute power value. However, calibration
82
Optical measurements
can be carried out with continuous light, which is easier from the practical point of view.
This means that the sensitivity of the photodiode is determined as S == I / P, where I is the
photo current at illumination power P. The pulse energy is the integral of the power
E = J P(t)dt
( 4.36)
The power generates photocurrent P (t) == I ( t) / S, thus
E = J l ) dt = J l(t)dt
( 4.37)
The integral of the current gives the total charge generated by the light pulse, Q == J I(t)dt.
As far as a short pulse is considered (compared to the time constant of the RC circuit) all
the charge will be collected by the capacitor C, creating voltage U d == Q / C. Thus, for the
energy one obtains
E == Q == C U d == U d
S S Sp
( 4.38)
where Sp == 6 is the energy sensitivity. In other words, we have obtained pulse energy
sensitivity coefficient, U d == SpE, for the measuring scheme presented in Fig. 4.7 (b) using
photo current sensitivity, S, and capacitance value, C.
Example 4.2: Photodiode sensitivity for pulse energy measurements. Let us consider a
pulse energy measurement scheme presented in Fig. 4.7 (b) with capacitor C ==
10 nF. A typical silicon photo diode current sensitivity at 800 nm is S == 0.5 AIW,
which gives sensitivity Sp == 6 == 5 . 10 7 V/J. Thus a pulse with energy 1 tLJ will
create 50 V voltage jump (U d , as shown in Fig. 4.7 (a)). It is clear, that with this
arrangement one can easily measure pulse energies as small as 1 nJ, which will give
50 m V response.
Somewhat limiting parameter for this method of the pulse energy measurements is the
pulse repetition rate. The pulses must not arrive faster than the relaxation time of the mea-
suring circuit, Td. If this cannot be arranged, then one may determine pulse repetition rate,
f, and measure the average power of the pulses, Pav, using one of the methods (instru-
ments) available for the power measurements. The pulse energy can be calculated then as
E - Pav
- f .
4.5 Measurements of the pulse duration
4.5.1 Direct methods
In direct pulse duration measurements one measures the time profile of the pulse and esti-
mates the pulse duration from this time resolved picture. A duration of the light pulse can
be measured using a fast enough photosensitive detector and a device which can record and
4.5. Measurements of the pulse duration
83
display the detector electric response. The detector can be a photodiode or photomultiplier.
F or photomultipliers the time resolution is typically limited by 1 ns and for photodiodes the
time resolution can be as short as 100 ps. The general purpose fast oscilloscopes have band-
width 200-500 MHz. A faster oscilloscopes (e. g. with bandwidth 5 GHz) are available
but their prices increase fast with the bandwidth. Therefore a reasonable time resolution for
the direct pulse profile measurements is roughly 1 ns. This time resolution is sufficient for
flash-photolysis measurements, but in pump-probe experiments the pulse width can be as
short as 20 fs. Such short pulse duration cannot be measured directly.
4.5.2 Autocorrelators (indirect methods)
Electronic devices has principal limits in time resolution, one of which is signal propaga-
tion delay, which makes direct pulse measurements in time scale approaching picosecond
impossible. However there are methods to generate light pulses with duration shorter by a
few orders of magnitude, e. g. 6 fs. The time profile of such light pulses cannot be mea-
sured using electronic devices. Optical methods were developed to determine duration of
the pulses in picosecond and femtosecond time domains.
A short pulse duration means that the peak power of the pulse is extremely high. For
example, 1 ps pulse with 1 tLJ energy (is 1 tLJ a huge energy?) creates power of 1 tLJ/1 ps
= 1 MW at maximum (how big must be a power station to provide such a power for a
time significantly longer than 1 ps?). When such a light pulse propagates in a matter, the
response of the matter is not linear any more (see Section 3.7 for discussion of the medium
non-linear response). This can be used for pulse duration evaluation. One of the widely
used non-linear phenomena is the second harmonic generation (SHG).
Let us consider an optical device similar to one shown in Fig. 4.8. In both cases the
incoming beam is split into two equal parts which are then combined back to form beams
propagating in one and the same direction but delayed in respect to each other. In the upper
scheme ( a) this is achieved by utilizing Michelson interferometer with mirror M3 placed
on mechanical translation line to provide variable delay time between the beams on the
interferometer output. In the lower scheme (b) the incoming pulse, I ( t), is divided into two
equal parts by mirror MI. One part is reflected by mirrors M 2 and M3 and arrives to mirror
M4 with delay t 1. Another part is reflected (twice) by two right angle reflectors, M5 and M 6 ,
and arrives to mirror M4 with delay t2. The reflector M6 is placed on mechanical translation
line and can be moved along its axis, so that direction of the beam does not change but delay,
t2, depends on the displacement of the translation line. This arrangement is called optical
delay line. After the mirror M4 the beams are propagating together and they are directed to
a non-linear optical crystal, which operates as a second harmonic generator (SHG).
The measured parameter for both schemes is the light intensity at the second harmonic,
2w, as function of the delay time. The intensity of the second harmonic is proportional to
the square of the light intensity at the entrance of the SH G
1 2w ( t) == aI;n ( t )
( 4.39)
where Iin(t) is the intensity at fundamental harmonic before the SHG, which is the sum of
84
Optical measurements
M3
I
!>
SHG Filter
PD
M 1 co I
--<:- - -
d
a)
M 2
M 2 M3
SHG fil ter
N) PD
I 0) _1:00_>0
0) t2 -->
M4 20)
: d : Delay line
. V ,
I. _ _ _ _ _ _ _ _ _ _ .1
b)
Figure 4.8: Optical autocorrelators: a) Michelson interferometer scheme and b) right angle
reflector delay line scheme.
intensities of two beams, II and 1 2 ,
Iin(t) == II (t + tl) + 1 2 (t + t2)
( 4.40)
Since we may choose zero time arbitrary, eq. (4.40) can be rewritten as
Iin(t) == II (t) + 1 2 (t + t 2 - tl) == II (t) + 1 2 (t + D.t)
where D.t is the delay between the pulses 1 and 2 and we can change it by moving the delay
line (positions of the reflectors M6 or M3 for schemes b) and a), respectively), so that
t == 2d
c
(4.41 )
where d is the position of the delay line. For the beams of equal intensities Iin(t) == I(t) +
l(t + D.t), thus
1 2 w(t)
a (I(t) + I(t + D.t))2
aI 2 (t) + aI 2 (t + D.t) + 2aI(t)I(t + D.t)
( 4.42)
4.5. Measurements of the pulse duration
85
where a is the efficiency of the second harmonic generation.
The measured value is the total pulse energy at the second harmonic (at 2w)
+ (X)
P 2w J hw(t)dt
-(X)
+ (X) + (X) + (X)
0: J l 2 (t)dt + 0: J l 2 (t + Lt)dt + 20: J l(t)l(t + Lt)dt
( 4.43)
-(X)
-(X)
-(X)
The first and the second integrals give the same results since integration is performed in
infinite limits, J 1 2 (t)dt == Po. The last integral in eq. (4.43) is autocorrelation integral of
the function I(t). It depends on parameter D.t, i. e. on delay line position,
+ (X)
Pc(Lt) = 20: J l(t)l(t + Lt)dt
( 4.44 )
-(X)
and it is the function of our interest. The value of Pc (D.t) shows how much the pulses I(t)
and I (t + D.t) overlap each other. When the delay between pulses is zero ( t == 0) the
result of integration is Pc (0) == 2P o and when the delay between the pulses is much longer
than the pulse duration, the integration gives Pc ( 00) == Pc ( - 00) == 0, since at time when
the first pulse has non-zero intensity the second pulse has zero intensity and vise versa.
The results of the measurements using devices presented in Fig. 4.8 is the autocorrela-
tion function of the input signal, therefore these devices are called autocorrelators.
If the function I (t) is a pulse, then its autocorrelation function is a pulse too. For
example, for a Gaussian pulse I ( t) == e _t 2 the autocorrelation function is
Pc( D.t)
+ (X) + (X)
J e-t2-CHf'..t)2dt= J e- V2H 2_ 2 dt
-(X) -(X)
+ (X)
6t 2 J (2t+6t)2 6t 2
e- e- 2 dt == Ce-
-(X)
( 4.45)
which is the Gaussian pulse, but it is V2 times broader than the original pulse. This is
shown in Fig. 4.9, where autocorrelation function was normalized to fit the scale.
The pulse autocorrelation function is obtained by measuring dependence of the second
harmonic intensity as function of the delay, i. e. relative position of the reflectors M3 or M6
in schemes a) and b), respectively. An estimation of the pulse width is done by assuming a
certain pulse shape. For example, for Gaussian pulse the autocorrelation function is roughly
1.4 times wider than the original pulse width.
A series of two Gaussian pulses has autocorrelation function shown in Fig. 4.10. The
autocorrelation function consists of three "pulses". The integral, eq. (4.44), has maximum
at D.t == 0 when both pulses are overlapping with each other. Two other peaks appear
86
Optical measurements
.....
.,....,
r:/).
(])
"S0.5
1--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
I
/
.-,,/
- Gaussian pulse
- - Autocorrelation
°_5
-4
-3
-2 -1 ° 1
til11e/ delay
2
3
4
5
Figure 4.9: Autocorrelation function (dashed line) of a Gaussian pulse, I
line)
e _t 2 (solid
2
;>..
.
r:/).
(])
'E
1--1
/,
I \
I \
I \
I \
I \
I \
I \
I \
I \
I \
I \
I I
I I
I \
I \
I \
I \
I \
I \
- - Autocorrelation
- Two Gaussian pulses
1.5
0.5
/-,
I \
I \
I \
I \
I \
I \
I \
I
I
I
I
I
I
/
.-,,/
°
/'
I \
I \
I \
I \
I \
I \
I \
\
\
\
\
\
\
\
,
"...
-6
-2
°
time/delay
2
6
Figure 4.10: Autocorrelation function (dashed line) of two Gaussian pulses, I == e-(t-2)2 +
e-(t+2)2 (solid line)
4.5. Measurements of the pulse duration
87
when one pulse of f (t) overlaps one pulse of f (t + D.t), but two another pulses are not
overlapping each other, which takes place at D.t == 4 and D.t == -4 for the example shown
in Fig. 4.10.
The example in Fig. 4.10 shows that the autocorrelation function and actual pulse may
have very different shape. In particular, the autocorrelation function is always symmetrical,
whereas the actual pulse can be unsymmetrical. Therefore, the autocorrelators can be used
to estimate the pulse width, but they cannot provide exact information on the pulse shape.
The time limiting factors for the pulse duration measurements using autocorrelators
are pulse broadening in optical components of the autocorrelator and mechanical accuracy
of the delay line. For thoroughly designed devices a few femtosecond pulses have been
measured, which is sufficient time resolution for optical spectroscopy applications.
Chapter 5
Steady State Absorption Spectroscopy
Measurement of absorption spectra is a very common procedure in research, monitoring,
diagnostics and other spectroscopy applications. There is a great variety of commercially
available devices for this purpose, which are commonly called spectrophotometers. In most
cases spectrophotometers are designed to be used by people without any special knowledge
or training. The user must know how to prepare the sample and how to interpret the mea-
sured spectra. All the rest can be done by the instrument. However, there are cases when
the measurements of the absorption spectra are not a trivial "one button" operation and a
good understanding of the operation principles of the spectrophotometers are required to
find the best procedure to obtain the desired information. As an example consider a mono-
layer of dye molecules deposited on an optically transparent semiconductor electrode. The
absorption of such monolayer is very weak and the instrument has to be tuned to achieve the
best sensitivity it can, which will be achieved probably by compromising some other char-
acteristics. The second problem of this case comes from the transparent electrode which
has absorption of its own. This absorption is weak but compatible to the monolayer ab-
sorption in the wavelength range of interest. Additionally, such electrode is usually a thin
layer, e. g. of indium tin oxide (ITO), deposited on a quartz or glass plate. The refractive
index of the layer differs from that of the supporting plate resulting in an interference of
the reflections from the ITO-air and ITO-substrate interfaces. The transmission spectrum
of such electrode will look like a wavy curve with the period determined by the thickness
of the layer. Typically the transmittance variation is much stronger than the absorption of
the mono-molecular layer. Nevertheless, it is possible to develop a method for the spectrum
evaluation of the molecular layer itself, as will be illustrated in the end of this chapter.
5.1 Measurements of the light absorption spectrum
The absorption as physical phenomenon of the light interaction with the matter was consid-
ered in Chapter 1.1.1. From a technical point of view, one needs to pass the light through
the sample and to measure the light intensity before, I in , and after, lout, the sample to deter-
mine the absorption (as indicated in Fig. 5.1). Then, the transmittance, T == I i : t , eq. (1.7),
or the absorbance, A == log I I n , eq. (1.12), or some other relevant value can be calculated.
out
89
90
Steady State Absorption Spectroscopy
Light
source
Sample
A, fin
.
D
Figure 5.1: Measurement of a sample absorption. The light source produce a beam at the
wavelength A and with intensity I in . The beam intensity after the sample is lout.
I
I
I
I I
p J ; l <[E --_ PM
L1 mator L2: S I : L3
ampe
I I (, ,
Light source
Detector
Meter
Figure 5.2: Optical scheme of a simple spectrophotometer. The lamp emission is collected
by the lens L 1 and focused onto the input slit of the monochromator. The lens L2 collects
the light from the monochromator output slit and forms the monitoring beam. After the
sample the light is collected by the lens L3 and focused onto the photomultiplier detector,
PM. The electric signal from the PM is registered by a meter.
To obtain the absorption spectrum the measurements must be repeated in the desired
wavelength range by tuning the light source from one wavelength to another by small steps.
What is needed to build up an instrument for this type of measurements? First of all one
needs a source of monochromatic light which can be tuned easily in a wide range. A com-
bination of a lamp, e. g. a tungsten halogen lamp, with a monochromator can do this job
fairly well. Second, a detector is required to measure the intensities, I in and lout. And
finally some optical components are needed to connect different parts together. A simple
optical scheme of such device is presented in Fig. 5.2.
As can be seen there is only one photo-detector in the scheme. To measure the light
intensity before and after the sample the measurements must be done twice: first time with-
out the sample to obtain I in and the second time with the sample to obtain lout. The light
detection part in the scheme consists of a photomultiplier tube (PM) detector and a meter.
The actually detected signal is the PM output voltage U, which is proportional to the light
intensity on the PM entrance, U == sI, where s is the voltage sensitivity of the detector. In
our two measurement series we will receive two values: without the sample U l == s(A)I in
(first measurement), and with the sample U 2 == s(A)lout (second measurement). The ratio
of these two values gives the ratio of the light intensities before and after the sample:
U 2 lout
U l I in
(5.1)
5.2. Spectrophotometer schemes
91
which is needed to calculate, transmittance, T == I I out == U U 2 , or absorbance, A == -log T.
n 1
It is important to notice that we do not need to know the photo detector sensitivity to calcu-
late the sample absorption, since in calculations we are using the ratio of the light intensities.
Therefore, the spectrum response of the detector is not important in spectrophotometer ap-
plications as long as the detector is sensitive enough in the specified wavelength range.
After obtaining the absorption at one wavelength the monochromator can be tuned to
another wavelength and the measurement procedure is repeated until the whole spectrum
is measured. A complication of this measurement routine is that the sample have to be
removed and inserted back at each wavelength. In practice an alternative approach is used.
At first the whole spectrum is measured without the sample, thus giving the spectrum U l (A).
Then the spectrum is measured with the sample to yield the spectrum U 2 (A). Finally the
transmittance
T(A) = U 2 (A)
U l (A)
(5.2 )
or absorbance
A ( A ) == 1 U l (A)
og U 2 (A)
(5.3)
is calculated out of these two spectra.
The first measured spectrum, U l (A), is commonly called base line. 1 The measurements
of the base line can be acquired once for a series of samples, as will be discussed later in
this Chapter.
5.2 Spectrophotometer schemes
5.2.1 Single channel scheme
The scheme presented in Fig. 5.2 is really the simplest one and it has a few drawbacks.
First of all, normal lenses, e. g. quartz lenses, have a dispersion (refractive index depends
on wavelength), therefore, they introduce chromatic aberrations (focal length is different at
different wavelengths). There are achromatic lenses, which can reduce chromatic aberra-
tions but cannot solve the problem completely in a wide spectrum range. Solution of this
problem is to replace all the lenses by mirrors.
The measurements are more accurate at longer signal accumulation time (at each wave-
length). On the other hand, when the accumulation time approaches second the 1/ f-noise
starts to dominate and reduces effect of long accumulation time. The problem can be solved
by modulating the monitoring light at relatively high frequency and using so-called syn-
chronous or lock-in detection system. This method also makes the instrument lesser sensi-
tive to external light since the signal is detected at the modulation frequency and unmodu-
1 Note that the shape of this spectrul11 is determined by l11any factors such as el11ission spectrul11 of the light
source, l11onochrol11ator transl11ission spectrul11 and sensitivity spectrul11 of the photo-detector. However, the shape
of this spectrul11 is not il11portant while we can calculate the ratio U 2/ U 1 with acceptable accuracy.
92
Steady State Absorption Spectroscopy
Source of monitoring light
Sample Detection
chamber part
r - - - - - - - - - - - - - - - --I
1 Monochromator 1
j 1
I S1
Sample
6
M4
Control Ii .1
unit and
display
Figure 5.3: Single channel spectrophotometer. Source of monitoring light consists of lamp
(L), collimating mirror (M 1), light chopper and monochromator, formed by input slit S 1,
mirrors M2 and M3, diffraction grating and output slit, S2. Mirrors M4 and M5 direct the
monitoring beam to the sample. After the sample the light is directed to the photomultiplier
PM by mirror M6. The signal from the PM is processed by the synchronous detector (SD)
and collected by a control unit.
lated light (or modulated at some other frequency) should not affect the detector readings. 2
An improved version of the spectrophotometer is presented in Fig. 5.3. All the lenses
are replaced by mirrors. The monitoring light is modulated by a chopper (a rotating disc
with holes). This allows to apply synchronous detection (SD) of the signals and to avoid
influence of 1/ f-noise and to reduce gradually sensitivity to external background light.
Typical measurements circle consists of two runs of the instrument. The first spectrum is
taken without any sample. This is base line measurements. It yields the spectrum U l (A), as
discussed above. Then the spectrum with the sample is measured, U 2 (A), and the absorption
is calculated using eq. (5.3) and presented to the user. The actually measured spectra (U l (A)
and U 2 (A)) are usually hidden from the user since they tell more about the device internal
features than about the user sample.
5.2.2 Two channel scheme
The remaining drawback of the considered scheme is that when measuring spectra one
after another (U l (A) and U 2 (A)) one relies on stability of the system. Thermal fluctuations
2 For eXaI11ple, if the light is l110dulated at 333 Hz, the detection systel11 is insensitive to the laboratory illul11i-
nation light, which is "blinking" at 100 Hz (double of power supply frequency, 50 Hz).
5.2. Spectrophotometer schemes
93
I
I
I
I
I
M6 I
I
Sample
chamber
Detection
part
Source of monitoring light
1- - - - - - - - - - - - - - - --I
I Monochromator I
t 1
I S1
;- - -.
M4
Control lA
u nit and
display
Dr
Figure 5.4: Two channel spectrophotometer. The source of the monitoring light consists
of a lamp (L), a collimating mirror (M1), a light chopper and a monochromator, formed
by an input slit S 1, mirrors M2 and M3, a diffraction grating and an output slit, S2. A
semi -transparent mirror M5 splits the monitoring light. The reflected beam cross the sam-
ple (sample channel) and the beam passing M5 is used in the reference channel after the
mirror M6. The light intensities in the signal and the reference channels are measured by
the photomultipliers coupled with synchronous detectors, PM 1 - SD1 and PM2 - SD2,
respectively.
will result in deviations of the spectra from one scan to another increasing inaccuracy of
the measurements. Therefore, in order to achieve highest sensitivity it is highly desired to
measure the spectra U l (A) and U 2 (A) with as short as possible delay in time. Ideally they
are to be measured simultaneously. This is achieved in two channel scheme presented in
Fig. 5.4.
After the monochromator, the monitoring beam is split by a semi-transparent mirror M5
on two parts forming a reference and sample channels. At each wavelength two values are
measured simultaneously - the signal from the reference channel, U r , and the signal from
the sample channel, Us.
Ideally the mirror M5 must have 50% transmission in the whole wavelength range. Then
the light intensity in the reference channel is equal to the light intensity before the sample
and the ratio of the light intensities can be calculated immediately I I n == U U T , assuming
out s
that all other components after the mirror M5 are identical in both channels. Unfortunately,
it is hardly possible to manufacture a mirror with exactly 50% reflectance in the wavelength
range of interest for a general purpose spectrophotometer, for instance in 200-900 nm range.
94
Steady State Absorption Spectroscopy
A simple work around is to record the response spectrum of the instrument without sample
before the measurements and to use it to correct mismatch between the channels caused by
the spectrum response of the mirror M5 and, probably, by some other components. Similar
to the single channel spectrophotometer the response spectrum is commonly called base
line, although these are differently measured spectra. Suppose without the sample the ratio
of the signals is R(A) = i \ . This is the base line spectrum. With the sample the ratio is
S(A) == ;i \ . One may notice that Ur(A) == U (A) if the monitoring light did not change.
Therefore, the transmittance of the sample can be calculated as
T(A) = Us (A) = Us (A) x U (A) = Us (A) x U (A) = S(A)
U (A) U2(A) Ur(A) Ur(A) U2(A) R(A)
(5.4 )
Comparing eqs. (5.2) and (5.4) one can see that formally the difference between the single
and double channel schemes is that in the latter case the ratio of signals, Us/U r , is used
where the signal, U, was used for the former case.
As can be seen, the actual measurements are done in two steps. This is similar to single
channel spectrophotometer, and one may wonder what is the advantage of the two channel
scheme over the single channel? To answer the question let us suppose that the intensity
of the light source has changed between the base line and sample measurements for some
reason, e. g. because of thermal instability. This will not affect results obtained with two
channel spectrophotometer since both, the reference and signal intensities are changed si-
multaneously and the ratio :.i j will not change. For the single channel spectrophotometer
the variation in the monitoring lIght intensity will give the same relative variation in the cal-
culated transmittance. For example, if after measurement of the base line the light intensity
has increased by 1 %, the calculated transmittance will be 1 % grater then the actual value.
In conclusion, the advantage of the two channel scheme over the single channel is that it is
insensitive to small fluctuations of the light source intensity.
There is also an important practical advantage of the two channel scheme. Very often
the sample can not be measured as it is. For example, if one is interested in absorption
spectrum of some dye molecule, the dye solution must prepared and placed into a cuvette.
Then the light intensity after the cuvette will be affected by the dye absorption, As (A),
(which is the only one we are looking for) plus absorption of the cuvette, Ac (A), solvent,
Asoz (A), and the light reflections from the cuvette walls, Ar (A). 3 To obtain "pure" spec-
trum of the dye, one may prepare a reference sample which should be a cuvette similar to
one used for the sample and filled with the same solvent. The cuvette with pure solvent
(reference sample) is placed in the reference channel and the cuvette with the dye is placed
in the signal channel. Let us now switch to the transmittance of the sample (dissolved dye
molecules), Ts(A), cuvette, Tc(A), solvent, Tsoz(A), and reflections, Tr(A). The light in-
tensity in the reference channel is Ir(A) rv Tc(A)Tsoz (A)Tr(A), which is, in terms of the
measured values, Ur(A) == U (A)Tc(A)Tsoz(A)Tr(A). Similarly, for the sample channel
Us (A) == U2(A)Tc(A)Tsoz(A)Tr(A)Ts(A). The finally measured value is the ratio of the
3 Reflections frOl11 the cuvette walls differs frOl11 absorption by the fact that the light intensity is not lost (it is
not absorbed, but reflected). Nevertheless, the light intensity after the cuvette wall will be lower and in the fraI11e
of this consideration we can treat it as an absorption since the reflected light is not counted in our experil11ents.
5.2. Spectrophotometer schemes
95
intensities (of the signals from the sample and reference channels)
S(A) = Us (A) = T (A) U2(A)
Ur(A) s U (A)
(5.5)
Finally, a "pure" transmittance spectrum of the sample can be obtained by dividing signal,
S(A), by the base line spectrum, R(A), Ts (A) == i . Thus, using a pair of matching
cuvettes 4 one can obtain the spectrum of the substance of interest without contribution of
the solvent and the cuvette used.
Actually, similar measurements can be done with the single channel spectrophotometer.
Instead of recording the base line with empty sample compartment, one can record the base
line with the reference sample (and thus obtain U l (A) which takes into account absorp-
tions of the cuvette, the solvent and the reflections). Then the second measurement with
the sample cuvette will give the spectrum of the dissolved molecules only. However, this is
where the practical advantage of the two channel scheme comes up - the base line of the
two channel spectrophotometer should be recorded only once, and all the further measure-
ments can be acquired with the single run of the instrument, whereas for the single channel
spectrophotometer one has to record reference spectrum every time the cuvette or solvent is
changed. Further discussion on the subject follows in Section 5.4.3.
5.2.3 Spectrophotometers with array detectors
In both schemes considered before the measurements are acquired one by one at a number
of wavelengths until the whole spectrum of interest is collected. Using an array photo-
detectors (such as CCD detector, see Section 4.2.3) one can build up a device which can
measure the whole spectrum at once, thus decreasing gradualy the measurement time. Usu-
ally such devices are build up implementing the single channel scheme with the white mon-
itoring light and the monochromator placed after the sample, as illustrated in Fig. 5.5. The
array detector is positioned in the place of the monochromator output slit. 5
The construction of two channel schemes with array detectors would require two monochro-
mators. Also implementation of the two channel scheme is a difficult task since the chan-
nels must be identical while the size of the photosensitive element of array detector is rather
small, typically tens of microns. Another drawback of spectrophotometers with array photo-
detectors is that the spectrum resolution cannot be changed easy. The spectrum resolution is
determined by the monochromator dispersion and the distance between the photosensitive
elements of the detector. The dispersion depends on the grating grooves number and the
focal length of the collimating mirror. None of them can be changed easily. This is the price
one has to pay for much shorter acquisition time.
4Pair of l11atching cuvettes are two cuvettes l11ade of the SaI11e l11aterial and having the SaI11e internal thickness
and thickness of the light transparent walls. See Section 5.4.2 for l110re discussions on the subject.
5 Strictly speaking, the l11onochrol11ator without output slit cannot be called l11onochrol11ator. This type of
device is called spectrograph.
96
Steady State Absorption Spectroscopy
Source of : Sample :
I I
monitoring light : chamber :
Detection part
: Sample
1
1
1
1
1
1
1
1
1
1
1
Monochromator
r-- ---------------1
1 1
1 1
1 1
1 grating
Control LD-
unit and
display
1
_________________J
Figure 5.5: Spectrophotometer with array defector. The light from the lamp L is collected
by the mirror M 1 and directed onto the sample. After the sample the light is focused on the
monochromator input slit and the spectrum is detected by an array detector placed in the
focal plane of the monochromator output mirror M4.
5.3 Main characteristics of spectrophotometers
5.3.1 Spectrum range
The spectrum range of the considered spectrophotometers is determined by a number of
factors. The most essential are:
. the spectrum of the source of monitoring light,
. the sensitivity spectrum of the photo-detector used, e. g. photomultiplier tube,
. the spectrum range of the wavelength selecting device, e. g. monochromator.
In the visible and near infrared spectrum the tungsten halogen lamps are usual sources of the
light for general purpose spectrophotometers. In the ultra-violet (UV) part of the spectrum
the thermal sources of the light are inefficient (see the black body emission discussion in
Section 1.2.1). Specially designed deuterium lamps are used as the sources of the monitor-
ing light in the UV range. They can be used in far UV range, but at shorter wavelengths
another problem arises - the transparency of the output window of the lamp bulb. The high
quality quartz absorbs the light at wavelengths shorter than 200 nm. Synthetic silica, sap-
phire and magnesium fluoride are materials which allow to expand the range to 180, 170
and 120 nm respectively.
5.3. Main characteristics of spectrophotometers
97
The photo-detectors were discussed in Section 4.2 on page 72. The principal UV limit
for the photomultiplier tubes (PMT) is also determined by the material of the entrance win-
dow. 6 Therefore, 190 nm seems to be also a reasonable limit from the viewpoint of the PMT
availability, also there are PMTs which can work up to 160 nm (Cs- Te photo-cathode and
synthetic silica window), or even up to 115 nm (Ce-I photo-cathode and magnesium fluoride
window). The red limit is determined by the material of the photo-cathode and for a popu-
lar S-20 photo-cathode it is roughly 840 nm. There are only few photo-cathode which can
work up to 1000 nm, and if longer wavelengths are needed another photo-detector have to
be used. For example, with a Ge photodiode the red limit can be shifted to 1.7 tL in expense
of sensitivity (as compared to the photomultiplier).
Monochromators (diffraction gratings) can be designed to operate in any optical range,
see Section 2.3 for more information on gratings and monochromators. There are, however,
a few things to keep in mind while selecting a grating for a spectrophotometer. The gratings
are designed to have the highest diffraction efficiency at a certain wavelength which is usu-
ally specified as blazed wavelength. The grating can be optimized for diffraction order other
than the first. In other words, the grooves number is important characteristic of the grating
but not the only one to be considered in design of an optical instrument. Another important
property of the gratings is that the diffraction takes place in all possible diffraction orders.
For example, if one wants to obtain the monitoring light at 300 nm and has found a suitable
grating providing 300 nm light in the first diffraction order, then, unavoidably, there will
the second order diffraction at 600 nm, the third order at 900 nm and so far, propagating
in the same direction as the first order diffraction. The efficiency of the second and higher
diffraction orders is much lower than that of the first order, but the higher order diffraction
cannot be eliminated completely. Therefore, for devices working in a wide spectrum range
the wavelength selecting monochromators are usually combined with a set of color filters
which are used to cut off the light diffracted at higher orders and which are changed during
the wavelength scan.
In conclussion, typical wavelength range of a simple spectrophotometer is 300-900 nm,
which can be provided by a single ligth source (tungsten halogen lamp) and a general pur-
pose photomultiplier tube. To extend the range further to the ultraviolet part an editional
light source have to be used. To cover the infrared part of the spectrum a combination of
detectors must be used.
5.3.2 Spectrum resolution
The spectrum resolution is determined by the wavelength selecting element, which is a
monochromator in the most cases. The wavelength resolution of the monochromator (see
eq. (2.46) in Section 2.3.4) depends on (1) the grooves number, (2) diffraction order, (3)
the focal distance of the mirrors (M2 and M3 in Figs. 5.4 and 5.3) and (4) the slits size
(S 1 and S2 in Figs. 5.4 and 5.3). The slit sizes are easy to change and this is usually the
method to change the wavelength resolution of the instrument. The three first parameters
are optimized for a certain range of applications and they are fixed for a given device.
6The efficiency of photo-cathodes in UV range is an il11portant paraI11eter to be considered, but it is not a
lil11iting factor.
98
Steady State Absorption Spectroscopy
Smaller slits give higher spectrum resolution. At the same time smaller slits result in a
lower intensity of the monitoring light. Thus, the higher wavelength resolution means the
lower monitoring light intensity and, starting from a certain limit, will result in a decrease
in the sensitivity of the spectrophotometer. In this respect the resolution and sensitivity
are connected with each other as it is discussed in the end of the following Section and in
Chapter 13 on page 237.
5.3.3 Sensitivity and absorption range
In this section we will discuss the range of possible absorption measurements. There are
few parameters use to specify light absorption by a sample (see discussion in Section 1.1.1).
They are all equivalent and can be re-calculated one to another. Therefore the absorbance
will be mainly discussed here since it is one of the most widely used and usually presented
to the user by default.
At first let us examen the lower limit of the detectable absorbance. Apparently, there are
many parameters essential for the detection of a very low absorbances, e. g. photometric
stability, reproducibility and noise. To present a common view on the problem let us discuss
it from the point of the smallest detectable signal U and corresponding smallest detectable
absorbance, A, which will be here called sensitivity.
When the light intensity values I in and lout (see Fig. 5.1) are close to each other one
can use approximation
( Iout(A) ) ( Is )
-loglO lin(A) = -loglO 1 - l in
Is Is
t""V (In 10)- t""V 2.3-
I in I in
where Is == I in - lout. It is clear that the sensitivity ( A) depends on how accurately
one can measure the light intensities I in and lout. If inaccuracy in the intensity measure-
ments is I, then the sensitivity is A t""V 2.3 . 2 . fI == 4.6 fI . Therefore, the sensitivity,
n n
A, is completely determined by the relative accuracy of the light intensity measurements.
Apparently t == .6.J , where U is the minimum resolved voltage deviation on the photo-
detector output. U depends on many factors such as stability of the light source (which
includes stability of the lamp power supply and the quantum noise of the light), the noise
of the photo-detector and the accuracy of the meter following the detector (synchronous
detection in the case of the scheme in Figs. 5.3 and 5.4). For a clever designed electronics
(detector and meter) the ratio .6.J can be smaller than 0.00002. 7 If .6.J == 0.00002, then the
sensitivity of the instrument is A t""V 0.0001. This is a typical value for a high quality gen-
eral purpose instruments. Already the value A t""V 0.0001 needs an advanced power supply
for the light source and a heavy design of the mechanical framework of the instrument to
achieve proper mechanical stability of the components in the optical part.
The light source quantum noise can be the limiting factor for the sensitivity at low
intensities of the monitoring light. From the viewpoint of quantum noise statistics, to pro-
vide inaccuracy c5 == .6.J for the intensity measurements the number of (detected) photons
A(A)
(5.6)
7Today digital technology is used ah110st everywhere. The accuracy of 0.00002 l11eans that the signal frOl11 the
photo-detector l11Ust be l11easured using 16-bit analog to digital converter (ADC) or better.
5.4. Instruments, accessories and applications
99
should be grater than N == (6) -2 == ( .6.J ) 2 (see square root law Section 4.1.2). Assuming
the photo-detector efficiency to be CPd and the detection time interval t, the minimum light
intensity is I = hv q, t = q,: t ( Zu ) 2, where hv is the photon energy. For example, if
.6.J == 0.00002, as was considered above, CPd == 0.1, which is typical for PMTs at the wave-
lengths of their maximum efficiency, t == 0.01 s, meaning 100 measurements per second,
and hv 4 X 10- 19 J (wavelength 500 nm), then the minimum intensity is I 2tLW. This
value looks to be rather low, but is it so in reality? Typical a tungsten halogen lamp emits
1-5 m W in 1 nm bandwidth in the visible spectrum range (see examples in Section 1.2.1).
This power is emitted in all directions and only few percents can be collected (by mirror M1
in Figs. 5.3 and 5.4) and focused onto the input slit of the monochromator. Part of the light
will be lost in the monochromator (slits, diffraction grating and mirrors). After all one can
expect to obtain a few micro Watts in 1 nm bandwidth at the sample. As can be seen 1 nm
wavelength resolution is close to the limit when the light intensity may affect the instrument
sensitivity. 8 Therefore most spectrophotometers have highest spectrum resolution colse to
1 nm. The best instruments can provide the resolution close to 0.1 nm without essiential
loss of sensitivity.
The sensitivity estimation has been made for a very transparent sample. The other ex-
treme is a highly absorbing sample. Then lout is close to zero but one bit greater than zero,
i. e. U corresponds now to lout. Thus sample transmittance is T == .6.J == 0.00002.
Consequently, A t""V 4.7. This is the maximum optical density which could be measured
with an ideal instrument having detection steps of .6.J == 0.00002. 9
One may notice, that the increase in the light intensity resolution,i. e. decrease in U
and 1, respectively, extends maximum measurable value of A and decreases the minimum
resolved value, A, since they both depends on .6.J . Of course, this is true if there are no
other limiting factors, such as thermal stability of the instrument base line.
5.4 Instruments, accessories and applications
5.4.1 Spectrophotometer specifications
The aim of this Section is to examen shortly typical spectrophotometer specifications in
order to illustrate the options and the working ranges for a different classes of the instru-
ments. From a great choice of the commercially available devices three are selected as
representatives for the spectrophotometes aimed at certain application ranges. There is also
no particular reason to choose these devices namely.
As an entry level spectrophotometer we will look at Unico 2100 series. This is the
instrument which is designed to be relatively cheap and suitable for non-demanding every
day applications. It implements the single channel optical scheme. As an example of a
high quality research grade instrument Shimadzu UV-3600 will the used. This is 2 channel
80 ne l11ay increase the detection til11e interval t to achieve the SaI11e nUl11ber of photons with narrower
spectrul11 resolution. However the increase in spectrul11 resolution result in l11uch faster decrease in the light
intensity at the saI11ple because of (1) narrower spectrul11, and (2) sl11aller input slit needed to increase resolution
of the l11onochrol11ator.
9The l11easurel11ents are not very accurate in this case as the next detectable transl11ittance value is T
2 U = 0.00004. So the l11inil11ul11 transl11ittance was determined with uncertainty ::!:100%.
100
Steady State Absorption Spectroscopy
spectrophotometer designed for high sensitivity and high resolution applications. Finally
Agilent 8453E UV-visible spectroscopy system will be considered as the representative of
the spectrophotometers equipped with array detector, and, thus, having the advantage of
rapid measurements.
Now let us look at the parameters listed in the specifications of the devices.
Wavelength range ofUnico system is 325-1000 nm when equipped with tungsten halogen
lamp only. Shimadzu spectrophotometer works in the widest range 185-3000 nm,
which is achieved by switching both light sources and detectors. For Agilent system
the range is 190-1100 nm.
Wavelength resolution is fixed for Unico and Agilent systems, 5 and 1 nm, respectively.
For Agilent instrument the resolution is fixed because of use of an array detector. In
case of Unico the fixed resolution (i. e. fixed monochromator slits) is a usual mea-
sure to reduce the price of the instrument. Shimadzu spectrophotometer has tunable
resolution from 0.1 to 8 nm (in UV-visible range).
Photometric range shows mainly the maximum absorbance which still can be measured
and, as it was discussed above, it depends on the smallest detectable light intensity.
For Unico it is 2.5, and for Shimadzu is 6.
Photometric noise and photometric stability are parameters responsible for the smallest
detectable absorbance. For Unico it is <0.002, for Agilent <0.0002 and for Shimadzu
<0.00005 (at 500 nm). 10
Stray light is <0.3% for Unico, <0.03% for Agilent and <0.00005°A> for Shimadzu. The
latter number is achieved by using a high performance double monochromator.
Shimadzu spectrophotometer has the best characteristics as it belongs to the top class in-
struments. With no surprice the price difference beween the Unico 2100 type devices and
Shimadzu UV-3600 type decices is more than 10 times.
5.4.2 Cuvettes for absorption spectroscopy
Typical samples in many spectroscopy applications are solutions. Naturally, to measure
solutions one needs cuvettes. The most widely used cuvettes are probably 1 cm square
cuvettes. 11 Most of the general purpose spectrophotometers have the sample compartments
designed for easy use of the cuvettes of different optical lengths. Typical set of the lengths
is 50,20, 10,5,2 and 1 mm. Fig. 5.6 presents examples of the cuvettes with lengths 50, 10
and 2 mm. External width of all cuvettes is 12.5 mm and height is 40 mm.
Commercially available cuvettes are made of different materials, the most common be-
ing optical glass and quartz. Ultra violet-visible transmittance spectra of a glass and quartz
cuvettes are shown in Fig. 5.7. As can be seen from the figure the usable wavelength range
10 As it was discussed in Section 5.3.3, typically higher photol11etric range l11eans higher sensitivity and lower
nOIse.
lIThe cross section of the cuvette is a square with 10111111 inner size and 12.5111111 external size.
5.4. Instruments, accessories and applications
101
Figure 5.6: Spectrophotometer cuvettes with length 2, 10 and 50 Inm.
100
80
0
C) 60
()
-+->
-+->
.......
S 40
r:J"J
H
-+->
20 -
-----
---------------- --
0 200
,.
I
I
I
I .
I .'
, :'
, :
,:
r
I
J
.1
:,
:,
:,
J
1
J
300
or.......................... ................................ or.... to
- quartz
-. glass
. .. polysteren
400 500 600
wavelength, nI11
700
800
Figure 5.7: Transmittance spectra of quartz, glass and polysteren cuvettes in UV-visible
wavelength range.
for the glass cuvette is limited by roughly 320 nm in UV, whereas the quartz cuvette can
be used up to 190 nn1. 12 Very cheap cuvettes are made of polysteren, which has almost
the same transparency wavelength range as glass, but the optical quality of the polysteren
cuvettes is not as good as for the glass cuvettes.
For accurate measurements a pair of cuvettes, which are made of the same material
and have exactly the same thickness (optical 1ength), are used. This pair of cuvettes is
called matching pair. When filled with the same solvent, the spectra of the cuvettes must be
identical. The pair of matching cuvettes is useful for measurements of absorption spectra of
compounds in the wavelength range where the sol vent or the cuvette itself absorb the light,
12 111 the infrared part of the spectrul11 the range is lin1ited by rough 2500 nl11, but for speciaJ types of quartz
(water free) can be extended to 3800 nl11.
102
Steady State Absorption Spectroscopy
or when the difference in absorption between two samples is the subj ect of investigation, as
discussed in the following Section. For these reasons the high quality cuvettes are usually
sold in pairs. 13
For routine measurements 1 cm cuvettes are the most common. A typical molar ab-
sorption coefficient of organic dyes is E 10 5 M-lcm- l . A reasonable absorbance of the
samples for routine measurements is A 1. Thus, typical concentrations of the samples
prepared for absorption measurements are about c j == 10- 5 M or 10 tLM, see eq.
(1.14). This is rather small concentration,14 and one can routinely use the absorption spec-
tra measurements to monitor, e. g., the course of a chemical reaction. However, one may
need to measure an absorption band with much lower molar absorption coefficient, such as
one corresponding to partially forbidden transition. Then the required concentration can be
10 mM or higher. At this concentration one may face solubility problems. The solution of
the problem is to use a thicker cuvette. For example, using a 5 cm cuvette the concentration
should be 5 times lower to achieve the same absorbance values as in 1 cm cuvette.
The 1 cm cuvettes may be unsuitable if one needs to study millimolar molar or higher
concentrations of a compound with relatively high molar absorption coefficient. For in-
stance, studying dynamic quenching of a singlet excited state of a dye molecule with the
lifetime of the excited state of IOns, one may need to prepare a sample with concentra-
tion of the quencher molecules 10 mM or higher. 15 To measure absorption of a compound
(quencher, in this case) at so high concentration a cuvettes with a shorter optical path can
be used. This is the typical application area of the 1 mm cuvettes, and still it could be too
thick. Assuming the molar absorption coefficient to be E 10 5 M-lcm- l and the required
concentration c == 0.01 M, one obtains the the absorbance in 1 mm cuvette A == Ecl == 100,
which is clearly too high to be measured. The thickness of the cuvette must be reduced by
roughly 100 folds.
Very thin cuvettes are usually made of two glass of quartz plates. One plate is per-
fectly flat and serves as a cover for another plate which has a special profile on the surface
as illustrated in Fig. 5.8. The working volume has precise thickness, typical values for
commercially available cuvettes being 1, 0.5. 0.2, 0.1, 0.01 mm.
For the example considered above, the cuvette with thickness 0.01 mm is a reasonable
choice.
5.4.3 Application notes and examples
Base line and sample measurements
Typically the first spectrum recorded after the switching on a spectrophotometer is the base
line. Usually this is done automatically during the instrument initialization, but it is possible
to record the base line later at any time. To achieve the best results, the base line measure-
13 SOl11e l11anufactures l11ark the thickness of the cuvette together with the type of l11aterial it is l11ade of in the
top of the cuvettes. For eXaI11ple, "Q 10.01" l11eans quartz cuvette with inner thickness 10.01111111.
l4To fill in the cuvette one would need 31111 of solution. For a COl11pound with 11101ar weight 500 a.u. the saI11ple
preparation will require just 0.015 I11g of the COl11pound.
l5In case of the diffusion lil11ited quenching a typical quenching constant is K 10 10 M- 1 s- 1 , and for the
lifetil11e T = 10 ns the half quenching concentration of the quencher is c q = T K = 0.01 M.
5.4. Instruments, accessories and applications
103
working
volume
buffer
volume
cover
Figure 5.8: Cross section of a cuvette with very small thickness.
Inents must be repeated from time to time, depending on long term stability of the particular
spectrophotometer and conditions in the laboratory.
Although the base line is needed for both single and double channel instruments, the
following steps depend on the type of the instrument. As an example we can consider a
series of measurements of one and the same compound in different solvents. In the case of
double channel spectrophotometer one needs to prepare pairs of samples for each solvent
- the reference sainple, which is a cuvette with pure solvent, and the sample solution in
the same solvent. Then, inserting these pairs of samples one obtains spectra of the studied
compounds in different solvents.
To obtain the same result with the single channel spectrophotometer, at first, the base line
must be recorded with the reference sample inserted into the measurement compartment (so
that the base line accounts for the cuvette and solvent properties). Then, the sample solution
is inserted and the measurement will give the desired spectrum of the compound under study
(with subtracted effect of the cuvette and solvent). As far as the goal is to measure a series
of solvents the base line must be repeated with each new solvent, when a single channel
spectrophotometer is used. Whereas there is no need to repeat the base line measurements
for two channel instrument since it has the reference channel. This is an important practical
advantage of the two channel scheine over the single channel scheme.
A simple spectrum, transmittance and absorbance
An example of a transn1ittance and absorbance spectra of a pyropheophytin a solution in
chloroform is presented in Fig. 5.9. The spectrum was measured using a double channel
spectrophotometer and a pair of quartz matching cuvettes. Comparing the spectrum of the
quartz cuvette in Fig. 5.7 with the transmittance spectrum in Fig. 5.9 we can see that the
spectrum of the cuvette was subtracted (e. g. at 700 nm the compound has no absorption
and transmittance at this wavelength is close to 100%, whereas for the cuvette it was 88%).
Two spectra shown in Fig. 5.9 are different presentations one and the same measure-
ment. They can be recalculated one to another using eq. (1.13): A == -loglO T. 16 For a
relatively small change in transmittance the spectra look like mirror image (because of the
negative sign in eq. (1.13)). In case of a greater variation of the transmittance the non-linear
16The transl11ittance in the equation is the ratio of the light intensities, thus changing frOl11 0 to 1. In the
graphical presentation, however, the percent of the ratio are commonly used.
104
Steady State Absorption Spectroscopy
100
80
400 500 600
wavelength, nm
700
60
40
20
1.2
1
0.8
0.6
0.4
0.2
OO
Figure 5.9: Transmittance, T, (top) and absorbance, A, (bottom) spectra of one and the
same sample: pyropheophytin a solution.
character of the logarithm function comes on play resulting in clear difference in the shapes
of the bands in the transmittance and absorbance plots.
Although the transmittance and absorbance spectra are equivalent presentations of the
absorption characters of the sample, they have different usage. For instance, the band shape
analysis, e. g. Lorentzian or Gaussian band fit, must be applied to absorbance spectrum (and
in a frequency domain). Whereas in analysis of the light transmission through some optical
system the transmittance spectra are more adequate.
Spectra subtraction
In terms of the absorbance spectra the measurement result of the double channel spectropho-
tometer can be presented as a difference between two spectra
A == Asampze - Are!
(5.7)
where Asampze is the absorbance of the sample installed in the sample channel and Are! is
the absorbance of the sample installed in the reference channel. For example in Fig. 5.9 the
reference was a cuvette with a solvent, thus (see eq. (1.18) and discussion on it) Are! ==
Ac + As, where Ac and As are the absorbances of the cuvette and solvent. In the sample
compartment the pyropheophytin a solution was placed, thus Asampze == Ac + As + A pheo ,
where A pheo is the absorbance of pyropheophytin a. As desired, the measurement has
yielded the spectrum ofpyropheophytin a, A == Apheo.
The spectra subtraction feature of the double channel spectrophotometer has many dif-
ferent applications, and can be used to extract the spectrum of interest from a rather complex
sample, e. g. mixture of compounds, where absorptions of different undesired components
can not be ignored, but can be subracted using a specially prepared reference sample.
Similar subtraction experiments can be carried out using the single channel spectropho-
tometers. When the reference sample is inserted for base line recording, the following
5.4. Instruments, accessories and applications
105
0.1
-. bare ITO
- ITO with monolayer
0.05
o
0.06
0.04
0.02
o
0.02
0.01
o
- glass with monolayer
- monolayer spectrum
400
500 600 700
wavelength, nm
800
Figure 5.10: Absorption spectra of ITO, porphyrin coated ITO and glass plates and calcu-
lated spectrum of the layer on top of ITO surface (see text for explanations).
sample measurements will give the subtracted spectra according to eq. (5.7). Since the
measurements are done one after another the procedure takes longer time and the result is
less accurate when compared to the double channel spectrophotometers.
Example of low absorbing layer on ITO electrode
Sometimes the information of the interest cannot be extracted from a single spectrum mea-
surements. For example, one may want to monitor a molecular monolayer formation on the
surface of a semi-transparent semiconductor electrode. For the experiments described here
the electrodes were thin layers of indium tin oxide (ITO) deposited on one side of glass
plates. The molecular layers were formed by a specially synthesized porphyrin derivative
which can be covalently attached to chemically activated ITO or glass surface. The por-
phyrin is a photo-active compound, and the modification of the ITO electrodes by such
monolayer can be used as the first step in fabrication of an organic solar cell.
The porphyrin has a characteristic absorption spectrum. Therefore the natural method to
monitor the formation of the molecular layer is the measurements of the absorption spectra
of the samples. However, there are two complications caused by (1) the light interference
and absorption in ITO layer, and (2) by the molecular layer assembled on the rear side of
the plate, which is not covered by the ITO and will not be used in further applications of the
samples. The procedure used to obtain "pure" spectrum of the porphyrin layer on the ITO
surface was described in [6] and consists of four steps:
1. The spectra of the ITO plates are recorded with empty reference channel before the
monolayer deposition, S ITO (A). An example of the spectrum is presented in Fig.
106
Steady State Absorption Spectroscopy
5.10 (the dashed line in the top plot). ITO has almost no absorption in the visible
part of the spectrum and the shape of the spectrum is mainly determined by the light
interference, which is sensitive to a small variation in the layer thickness. Therefore
the measured spectra can vary essentially from sample to sample, and a search for a
matching pair of the ITO plates is almost hopeless. Therefore the absorption spectra
of all ITO plate are recorded before using them.
2. The layers are deposited onto ITO substrates together with a few reference glass plates
in one pot reaction. The spectra are recored again (also with empty reference chan-
nel), thus giving the spectrum of the modified ITO, SmodITO(A). An example of the
spectrum of the modified ITO plate is shown in the top plot of Fig. 5.10 by the solid
line. One can notice the difference between the spectra before and after the modifica-
tion, however the quantitative evaluation of the monolayer quality is hardly possible
yet.
3. The absorption spectra of the monolayer modified reference glass plates are measured
against non-modified reference plates, Smod gZass (A). In the contrary to ITO plates,
the glass plates made of the same type of glass have the same absorption spectra.
Therefore, the measured spectra, Smod gZass (A), are the absorption spectra of two
monolayers of porphyrin since the layers are deposited on both sides of the plates.
An example of such spectra is presented in Fig. 5.10, middle plot.
4. The absorption spectrum of the monolayer on the ITO surface can be calculated now
as the difference between the modified and bare ITO spectra minus half of the spec-
trum of the glass plate, SmonoZayer (A) == Smod ITO (A) - S ITO (A) - Smod gZass (A) /2.
This spectrum is shown in the bottom plot in Fig 5.10.
In conclusion, the absorption of the porphyrin monolayer on ITO surface is 0.022 at the
maximum of the so-called Soret band at rv 420 nm. This number can be used further to
estimate the surface density of the molecules based on the value of the molar absorption
coefficient of the porphyrin molecule. The calculations indicated that almost 100% of the
ITO surface is covered by the porphyrin monolayer.
Scattered light
Ideally the samples for the absorption spectra measurements should have no scattering.
Unfortunately the real samples may not be perfect and scatter some light. Additionally,
there are cases when the scattering is a natural feature of the sample under study, e. g.
nanoparticle or micella samples. Depending on the optics after the sample and degree of the
scattering, a part of the scattered light will not reach the detector and the measured spectrum
will be affected by the scattering properties of the sample. Typically the scattering efficiency
is proportional to the inverse of the fourth power of the wavelength (Isc rv A -4), and the
effect is more pronounced at shorter wavelength. So the fast increase in absorption toward
the shorter wavelength should be treated with caution for potentially scattering samples.
The effect of scattering can be reduced by placing the sample as close to the detector as
possible to collect as much of the light as possible. Some commercial spectrophotometers
are equipped with a special sample compartment (or holder) for this purpose.
Chapter 6
Steady State Emission Spectroscopy
Measurements of emission steady state spectra are as common as the absorption spectra
measurement. They are usually routine procedures in characterization of new compounds,
monitoring chemical reactions or industrial processes. Form the point of view of photo-
physics, the absorption and emission spectra provide information on the electronic subsys-
tem of the matter. The absorption spectra show the energy spectra of absorbed photon, there
for the absorption bands corresponds to transition energies from the ground to the excited
state (M + hv ---+ M*). The emission spectra are the energy spectra of the photons emitted
during relaxation of the excited electronic subsystem to the ground state (M* ---+ M + hv).1
However, even when the emission and absorption bands correspond to the electronic tran-
sitions between the same states, the spectra may differ significantly from each other. For
example, for a dye compound in a solution the vibrational sub-levels and interaction with
the solvent are among the key factors determining the shapes and relative positions of the
absorption and emission spectra. One can consider absorption and emission to be two com-
plementary methods which, being applied together, allow to extract an additional infor-
mation on the subject under study. Also one have to keep in mind that not all transitions
observed in absorption spectroscopy can be observed in emission spectra.
The application range of the emission spectroscopy differs from that of absorption spec-
troscopy. For example, the absorption spectroscopy is very useful in determination of con-
centration of compounds in, e. g. , solution, since the measurements give the absolute value
of the sample absorbances. Measuring of absolute emission intensities is technically com-
plicated task, therefore fluorescence methods are not common for concentration determina-
tion. On the other hand the emission spectroscopy is much more sensitive, as will be shown
later in this Chapter, and one can use it to monitor a very small amount of substance. Even
single molecule can be studied by the emission spectroscopy methods. Using fluorescence
labels one can visualize a biochemical reaction at intra-cellular scale, or monitor a curing
process in a production line. Relative simplicity of the emission spectroscopy method and
its high sensitivity forms the base for a wide range of applications of the technique.
1 Indeed, the relaxation process can be rather cOl11plex one and photons can be el11itted when the electronic
subsystel11 relaxing form one excited state to SOl11e intermediate lower laying excited state. However, in 1110St cases
the higher excited states relax quickly to the lowest excited state, which has the longest lifetil11e aI110ng the excited
states. The el11ission of the longest lived excited state is observed experil11entally.
107
108
Steady State Emission Spectroscopy
Source of
excitation
light
lex
Aex
I
fern (A e ,,) =
Emission
detecti on
system
Sample
Figure 6.1: A general scheme of the measurements of the photo-induced emission.
6.1 Measurement of the Emission Spectrum
The instruments for emission spectrum measurements are commonly called fluorimeters or
spectrofluorometers. A typical instrument for the emission spectroscopy studies consists
of two relatively independent parts. First of all the emission must be initiated, excited. In
optical spectroscopy the samples are excited by the light. 2 Thus, a source of the excitation
light is the first part of any emission spectroscopy instrument. The second part of the system
should measure the emission of the sample. A general optical scheme for the emission
spectrum measurements is presented in Fig. 6.1. There is some similarity with the general
scheme for absorption measurements, Fig. 5.1. In both cases the first part of the instrument
is the source of the light. However, in absorption spectroscopy the same light from the
source (but attenuated by the sample) is measured, whereas in emission spectroscopy the
measured light is produced by the sample and has different properties than the excitation
light. To emphasize the latter the intensity of the emission is marked as function of the
emission wavelength, I em (Aem) in Fig. 6.1.
Another difference between the measurements of the absorption and emission spectra
is that the ratio of intensities was used to calculate the absorbance, therefore the spectrum
of the light source and the spectrum response of the light detector have no direct effect on
the measured spectrum. In the case of the emission spectrum measurements the excitation
intensity is a constant and the knowledge of the spectrum sensitivity of the detection part is
of prime importance to obtain correct emission spectrum.
The excitation and emission wavelengths depend on the object under investigation but
from the viewpoint of the instrument design they are two independent wavelengths. Thus,
the measured signal, U, e. g. photo-voltage on the output of the photomultiplier tube detec-
tor, is a function of two wavelengths: excitation, Aex, and emission, Aem,
U == U(Aex, Aem)
(6.1 )
2 Photo-excitation is not the only l11ethod to generate an excited state but probably the easiest one.
6.2. Fluorimeter
109
Excitation
monochromator
Lamp
y
Control
unit and
display
Sample
_ _ M3
- - \ Emission
- - - - S3 ; monochromator
)
= = _ _ _ _ _/-=- r
- -,- ,-' - - - - -
/ /
[]: j : ;_ : - --l
/--- ----: J
/,- -
I, Aem
,
Figure 6.2: An optical scheme of a steady state fluorimeter. The emission of the lamp is col-
lected by mirror M 1 and focused into the input slit S 1 of the excitation channel monochro-
mator. Mirror M2 collects the light from the monochromator output (slit S2) and focuses it
into the sample. Emission of the sample is collected by mirror M3 and directed to the input
slit S3 of the detection monochromator. The light intensity is measured by a photomultiplier
tube (PMT).
Naturally, for the emission spectrum measurements, Aex is kept constant, thus converting
the wavelength dependence to a simple U == U(A em ). Alternatively, one can keep Aem
constant and measure dependence U == U (Aex), which is called excitation spectrum and is
considered later in Section 6.2.6. Nevertheless, both components, excitation and emission
parts, have to be accounted to evaluate the performance of the instrument, as indicated by
the function arguments in eq. (6.1).
6.2 Fluorimeter
6.2.1 Optical Scheme
A general purpose fluorimeter should provide the widest possible range of the excitation
wavelengths and should be capable of measuring the emission spectrum in the broadest
wavelength range. Combination of a lamp and monochromator can be used as the excita-
tion source, as shown in Fig. 6.2. This is a typical solution to cover a wide spectrum range
of excitation. The excitation part of fluorimeters is schematically similar to spectropho-
tometers (Figs. 5.3 or 5.4), although different types of lamps are usually used in these two
types of devices.
110
Steady State Emission Spectroscopy
When there are no demands for wide excitation spectrum range, fluorimeters can be built
up using excitation sources with a fixed wavelength or relatively narrow wavelength range.
Then the monochromator can be replaced by a set of color or interference band pass filters.
Also an emitting diode or a laser can be used in place of the lamp and monochromator. This
usually makes the system cheaper, compact and more reliable.
The excited sample emits photons in all possible directions and a carefully designed
instrument should collect as much as possible of the emission. Therefore, the detection
part of the instrument starts from the light collecting mirror M3 in Fig. 6.2. The purpose
of the instrument is to measure the spectrum of the collected light. This can be done by
a photomultiplier tube coupled with a monochromator. The photomultipliers are the most
sensitive photo-detectors in UV and visible parts of the spectrum and are a natural choice if
the emission efficiency of the samples is expected to be low.
One can notice that in spectrophotometer the monitoring light was modulated to in-
crease the accuracy of the light intensity measurements, whereas in case of the fluorimeter
modulation and synchronous detection was not used in the scheme presented in Fig. 6.2.
From the standpoint of the light intensity measurements the difference between these two
types of instruments is in the value of the intensity to be measured. The spectrophotometers
are working with relatively high light intensities and must provide a high relative accuracy
of the intensity measurements, whereas the fluorimeters are designed to detect as low as
possible intensities. For detection of a very weak photon flux the photon counting is the
best approach. 3 The difference in application of the photon counting to measure absorption
and emission spectra is illustrated in the following example.
Example 6.1: Comparison of the photon counting method for applications in emission and
absorption spectra measurements. A typical maximum counting rate of a photon
counting module is 20 MHz, e. g. 2 x 10 7 counts per second. At this rate the proba-
bility to count two incoming photons as one is relatively high and to keep response of
the module in a linear regime the acceptable counting rate is ;S 10 6 S -1. At 10 6 S-l
rate during 1 second the signal, i. e. the average number of counts, is N == 10 6
counts, and its standard deviation is N == VN == 10 3 counts (see square root law,
eq. (4.23)). Thus, the intensity is measured with relative accuracy c5 == 0.001, or
0.1 %, which is very good accuracy for the emission spectrum measurement, but be-
ing used in absorption spectra measurements provides the absorbance resolution of
0.005 (see Section 5.3.3),4 which is rather poor result in comparison to spectropho-
tometer specifications listed in Section 5.4.1. In the same conditions, if the collection
time at single wavelength is reduced to 0.01 s, the emission spectrum is still mea-
sured with acceptable accuracy of 1 %, while the absorbance resolution is dropped
down to 0.05 value, which is unacceptable in most cases.
3 Modulation-synchronous detection technique was used previously, but today the photon counting l110dules
are widely available and reasonably priced.
4 For a typical absorbance of, e. g., 1.0 the accuracy of the l11easurel11ents is 0.5%.
6.2. Fluorimeter
111
6.2.2 Use of Array Detectors
One of the drawbacks of the fluorimeter shown in Fig. 6.2 is that only the photons emit-
ted in a narrow wavelength region (transmittance band of the detection monochromator,
Aem) are detected, but the main part of the emission is rejected by the emission monochro-
mator. This can be corrected by replacing the monochromator-photomultiplier pair by
a spectrograph-CCD combination. Then the emission is measured simultaneously at all
wavelengths and the spectrum collection time can be much shorter. This improvement
comes in expense of the sensitivity, since the CCD detectors (and other similar semiconduc-
tor array detectors) have lower sensitivity than the photomultiplier tubes. Also the spectrum
resolution of spectrograph-CCD couple is determined by the spectrograph dispersion and
the distance between the photosensitive elements of the CCD and cannot be change making
these instruments less flexible.
6.2.3 Evaluation of the Measured Signal
To make an estimation of the measured signal, U (Aex, Aem), we need to evaluate (1) the
intensity of the excitation light, (2) the emission efficiency of the sample, and (3) the ef-
ficiency of the emission detection part. The intensity of the excitation light, lex, depends
on the light source (the lamp in the case of the scheme in Fig. 6.2), selected wavelength,
Aex, and wavelength band, Aex. Efficiency of the excitation (e. g. density of the excited
molecules) depends also on the sample absorptance, a( Aex). 5 The total intensity of the
.. .
emISSIon IS
Iem == 4> I ex a ( Aex)
( 6.2)
where 4> is the quantum yield of the emission and lex == Iex(Aex, Aex). The sample emits
the light all possible directions (in solid angle 47r) and only a small part of it will be collected
by the mirror M3 (Fig. 6.2) and passed to the detection monochromator. The efficiency of
the light collection depends on
1. the angular aperture of the mirror M3 and the monochromator, which is the limiting
stage in most cases; this efficiency can be defined as a part of the light which can be
collected in the case of a point-like emitting source, 1]c;
2. the length of the sample, l, and the size of the entrance slit of the monochromator,
d em , which takes into account the fact that image of the excited area of the sample
may be larger than the size of the entrance slit; for optically transparent samples this
factor is given by 1]8 t""V d em / li, 6 where li is the length of the excited area image at
the entrance slit. The size of the slit determines monochromator spectrum resolution,
d em == b Aem, where b is a constant, thus, 1]8 == b Aem/li.
5 Note, the difference between the absorptance, a = (I n - Iout)/ I n, given by eq. (1.9) and absorbance
defined by eq. (1.11).
6 It is essential for this siI11ple estil11ation that the saI11ple is optically transparent. The light intensity inside
the saI11ple decreases exponentially, I (x) = I n (1 - e- ax ). If the saI11ple has essential absorption coefficient,
ax > 1, 1110St of the light is absorbed by the front part of the saI11ple, and the el11ission intensity decreases
essentially through the sa111ple. This is discussed later in Section 6.3.3.
112
Steady State Emission Spectroscopy
E:
\U
sample
emISSIon
band
' ,..AAem
1 1 ..
1 1 monitonng
1 bandwidth
'-'
.......
t Aem
monitoring
wavelength
A
Figure 6.3: Sample emission bandwidth, As, and detection system bandwidth, Aem.
Then, one has to take into account the fact that only a narrow part of the emission spectrum
will be selected by the detection monochromator (determined by the spectrum resolution
Aem). Thus, instead of measuring the total emission intensity, the part proportional to
the band pass of the emission detection system, Aem, and inversely proportional to the
emission bandwidth of the sample, As, will be measured. This is illustrated in Fig 6.3.
Formally speaking, the emission spectrum density of the sample, iem(A), must be used
in place of the total emission intensity. Then the total emission intensity is given by the
integral Iem == J i em (A )dA, and the emission intensity within detection system band pass,
Aem, at monitoring wavelength, Aem, is iem(Aem) Aem. Therefore, for a narrow enough
detection band pass Aem, the intensity of the detector light, i. e. the light which will enter
the detector, is
I det == 17c17s17miem(Aem) Aem
(6.3)
where 17m is the efficiency of the monitoring monochromator. If i em (A) changes signifi-
cantly in frame of the detection window Aem, eq. (6.3) must be rewritten in an integral
form
00
het = 'rJc'rJs'rJm J iem(A)r(A)dA
o
(6.4 )
where r (A) is the spectrum response function of the detection channel. 7
The emission spectrum density, iem(A), depends not only on the sample properties but
also on the excitation, e. g. the emission is stronger when excitation is stronger. There-
fore it is better to replace it with parameters having better define meaning. The values
of interest are the emission quantum yield, 4>, and the shape of the emission spectrum.
To separate different factor influencing the results of the measurements let us return to
the total emission intensity, Iem, by introducing another spectrum function, cp(A, Aem)
so that Jo oo iem(A)r(A)dA == IemCP(Aem, Aem). The function CP(Aem, Aem) defines a
7 The function r(A) is normalized so that J o oo r(A)dA = Aem.
6.2. Fluorimeter
113
relative part of emission at wavelength A within the band Aem. In fact, it is the rel-
ative sample emission spectrum measured with the resolution Aem and normalized so
that J <P ( A) dA. == 1, since the integration of the spectrum should give I em. To make
the equation shorter the second argument and the wavelength subscript will be omitted
<p(Aem, Aem) = <p(A), since the function depends only on the emission wavelength.
Finally, accounting for the detector sensitivity, S, the measured signal is
U(A) == S1]c1]s1]m Aema(Aex)cjJlex<P(A)
( 6.5)
Substituting also the value of 1]s one obtains
U(A) 4J [ 17cb A m a(Aex)Iex] [17m (A)S(A)tp(A)]
(6.6)
where all the terms are grouped into three categories (from the left to the right):
1. emission quantum efficiency, cjJ, is one of the parameters of a great interest and will
be discussed later;
2. parameters, which are independent of the monitoring wavelength, but affect the signal
intensity (note that the signal is proportional to A;m);
3. parameters, which depend on monitoring wavelength.
One of the primary goals is to measure emission spectrum, however the signal, U (A), is
the product of three spectra, 1]m(A)S(A)<p(A), from which only thee spectrum <p(A) is of
interest. Hopefully, the spectra S (A) and 1]m (A.) are smooth functions of the wavelength and
if the sample spectrum <p(A) is narrow and sharp, then the measured spectrum U(A) will
reflect (more or less) the spectrum of the sample, U(A) rv <p(A). This spectrum, obtained
directly from the detector, is called uncorrected spectrum, and has to be used with a caution.
6.2.4 Spectrum Correction
The spectra S (A.) and 17m (A.) may influence the signal too much and correction must be done
in order to determine the real emission spectrum. The signal, eq. (6.6), can be presented
as U(A) == C1]m(A)S(A)<p(A), where C is a constant. If a sample with known emission
spectrum, <Pref(A), is available, one can measure the reference sample to obtain Uref(A) ==
C ref 1]m (A)S(A)<Pref (A),8 and to calculate a correction spectrum as
Scor(A) = tprej(A) = 1
Uref(A) C re f1]m(A)S(A)
(6.7)
Then the corrected spectra of unknown samples are obtained as
C
Uc(A) == U(A)Scor(A) == - c <p(A)
ref
(6.8)
8 Although G is a constant for a given sample, it depends on the sample absorption, excitation wavelength,
etc. Therefore, for the reference sample the constant is marked as G re !, to indicate the difference between the
samples and measurement conditions of the reference and studied spectra.
114
Steady State Emission Spectroscopy
After this the spectrum, Uc(A), is proportional to the sample emission spectrum, cp(A), only.
This is emission corrected spectrum.
Obviously, to perform such spectrum correction, an emission source with known, broad
and preferably smooth emission spectrum is needed. This problem is not as simple as it
probably looks at a first glance. There is no light source which would emit pure white light,
i. e. light with flat spectrum. One type of commonly used sources of the light with broad and
well known emission spectra are black bodies. For an ideal black body everything is defined
by the temperature of the body. Accurate measurements of the temperature in the range
3000-5000 K is not a very simple task either. However, there are commercially available
black bodies with known temperatures at a certain current and voltage supplied. Also gray
coefficients are specified for the devices, which tell how much the emission density of the
real surface is lower than that of the ideal black body.
The measurements of the correction spectrum are usually accomplished in factory and
the correction spectrum is supplied together with the instrument. An example of a raw and
corrected spectra is presented in Fig. 6.4. A toluene solution of tetraphenylporphyrin was
measured using Fluorolog-3 (Spex Inc.) fluorimeter equipped with a red sensitive photo-
multipliers (R928P, Hamamatsu Co.). The lower plot shows the spectrum as obtained from
the photomultiplier (raw spectrum), and the top plot is the spectrum after correction. 9 Also
both spectra have two bands, the intensity ratio of the bands is different, which is due to the
drop in the photomultiplier sensitivity toward the near infrared wavelengths.
6.2.5 Quantum yield determination by comparison method
After the correction procedure the measurements give true shape of the emission spectra,
but the absolute value of the intensity remains unknown. Therefore, independent evaluation
of the emission quantum yield, 4>, is another relatively complicated task. It can be solved if
a reference sample with known quantum yield and emitting in a wavelength range similar to
the sample under study is available. Then, the studied and reference samples are prepared
so that they have the same absorption at a certain wavelength. This wavelength is used to
excite the samples, so that the second term in eq. (6.6) is the same for both samples. The
corrected spectra are measured and the integral intensities to be used for the quantum yield
calculations are calculated:
ex)
J U(A)dA
o
4> == 4>rej ex)
J Urej(A)dA
o
( 6.9)
where 4>rej is the quantum yield of the reference sample, and U (A) and U rej (A) are the
corrected spectra of the sample and the reference, respectively.
9 The instrul11ent allows to record directly the corrected spectrul11 so that the correction is performed during
the l11easurel11ents.
6.2. Fluorimeter
115
3x10 6 corrected
6
cd 2x10
O"\
1x10 6
0 650 700 750
6x10 5
rfJ.
'24x10 5
0
U 5
0"\2x10
0 650 700 750
wavelength, nm
Figure 6.4: Emission spectrum of tetraphenylporphyrin in toluene obtained directly from
the instrument (bottom) and the same spectrum after correction (top).
6.2.6 Excitation spectrum
Let us now consider an another type of experiment - instead of scanning the emission
wavelength, the excitation wavelength will be scanned at fixed emission (monitoring) wave-
length. In terms of eq. (6.1) this is the case of U == U (Aex). The spectrum obtained by this
method is called excitation spectrum. 10
Returning back to eq. (6.5) and collecting up the values depending on the excitation
wavelength one can obtain
U(Aex) == S1]c1]s1]m Aema(Aex)4>Iex(Aex)CP
(6.10)
As it can be seen, the signal is proportional to the absorptance of the sample a (Aex). 11
Actually, a(Aex) is not absorption of the whole sample but absorption of such part of the
sample which is responsible for the emission. As an example let us consider a sample which
is a mixture of two types of molecules with probably overlapping absorption spectra. If the
first compound in the mixture has a characteristic emission band at Aem whereas the other
does not emit light at this wavelength, then the signal (emission intensity) measured by
10 In general, the dependence of a systel11 reaction, e. g. fluorescence intensity or photo-current, on the exci-
tation wavelength is called action spectrul11. FrOl11 this point of view the excitation spectra are subclass of action
spectra.
11 When absorption coefficient is relatively low it is proportional to absorbance, since according to eq. (1.13)
A = - log T = - log(l - a) a log e O.43a.
116
Steady State Emission Spectroscopy
1 r
I,'
l' \\
, '
I' '1 - emISSIon
, I
0.8 I: , -. excitation
I: ,
l' '\ n_ absorbance
, \
d I' I
, I
0.6 I'
,
I'
""' ,
I'
,
""' I'
,
I'
0.4 I,'
,'\. "
,
1/
1/
0.2 1/
1/
-'
"',
J /r
,
0 -.....;-'";;';11"....
450 650 700
Figure 6.5: Corrected emission and excitation spectra and absorbance spectrum of rho-
damine 6G in ethanol. The emission and absorption spectra were normalized to 1. The
absorbance spectrum was multiplied by 30 to fit the scale. The emission spectrum was
recorded with excitation at Aex == 520 nm. The excitation spectrum was recorded with
monitoring at Aem == 580 nm. Spectrum resolutions were 2 nm for emission and excitation
spectra and 1 nm for absorbance.
scanning excitation wavelength Aex at the emission wavelength fixed at Aem will reveal the
absorption spectrum of the first compound only.
A practical problem for the interpretation of the excitation spectra is that the signal
depends also on the intensity of the excitation light lex (Aex), which depends on the lamp
emission spectrum and on the transmittance spectrum of the excitation monochromator. The
spectrum of the excitation light is the property of the instrument and does not change, but
the calibration procedure is not simple. First of all one needs to carry out the spectrum cor-
rection of the detection part (as was described in Section 6.2.4). Then, by using a broad band
reflecting mirror in place of the sample and by simultaneous scanning both monochromators
(excitation and emission) one can obtain the spectrum of the excitation light, which can be
used to correct the emission spectrum. For the research grade fluorimeters the emission and
excitation correction spectra are measured by the manufacture and supplied together with
the instrument.
The emission correction spectrum can be used to obtain the actual shape of the absorp-
tion spectrum, but the absolute values of the absorbance of the emitting species cannot be
calculated, since the calculations require knowledge of the emission quantum yield, over-
lap volume of the excitation and monitoring beams and many other ill defined parameters.
However, excitation spectrum is a simple qualitative test which can help to elucidate the
nature of the emission.
An example of emission, excitation and absorbance spectra is presented in Figure 6.5.
The sample, rhodamine 6G in ethanol, is a typical laser dye with the emission quantum
6.2. Fluorimeter
117
yield close to unity and with a big Stokes shift ( 25 nm).12 For comparison purpose the
magnified (by 30 folds) absorbance of the sample is shown in the same figure by the fine
dashed line. A good match between the excitation and absorbance spectra can be seen, as
expected for this mono-component solution. Also a small discrepancy between the spectra
can be noted at wavelengths corresponding to the emission sharp bands of the Xe lamp used
in the fluorimeter.
6.2.7 Sensitivity
Two emission detection methods were briefly discussed in Section 6.2.1. One can measure
photo- voltage response of a photomultiplier or photodiode after the emission monochro-
mator. Alternatively, one can count photons after the monochromator. Then, the detection
part consists of a photomultiplier, discriminator and counter. The latter method allows one
to achieve better results (both in sensitivity and accuracy). Also technically it is relatively
inexpensive in the optical wavelength range as photon counting modules are available com-
mercially at reasonable prices (see Section 4.2.2). Therefore, the photon counting method
will be considered here to make an estimation of the method sensitivity.
In order to switch to the number of counts, N c , in eq. (6.6) one needs to replace the
photomultiplier sensitivity by the quantum efficiency of the photomultiplier, 1]p M (A), and
the intensity, lex, must be replaced by the photon flux, i. e. the excitation energy divided by
the photon energy 13
N c = 7]c7]m(A)7]PM(A) bD. m a(Aex)<pip(A)
(6.11)
The term Ef:-; == N ex gives the number of photons in excitation and all the other terms give
reduction in number of photons, or, in other words, they show how much is lost at different
stages of the light propagation.
All losses can be divided in to two parts: losses by the instrument, such as quantum
efficiency of the detector, 1]p M, and losses by the sample, e. g. a weak absorbance and a
low emission quantum yield (cjJ). Then eq. (6.11) is simplified to
N c == TJi1] s N ex
(6.12)
where the instrument losses are
( ( b Aem
1]i == 1]c1]m >")1]PM >..) li
(6.13)
and the sample losses are
1]s == a(Aex)cjJ<P(A) >"em
(6.14)
12 Stokes shift is the difference between the wavelengths of maximum emission intensity and maximum ab-
sorbance.
13 The number of emitted photons is proportional to the number of absorbed photons, i. e. it is proportional to
the energy of the excitation, which is the light intensity multiplied by the illumination time.
118
Steady State Emission Spectroscopy
Note, this subdivision is rather relative - the length of the excited area, li, depends on the
sample too but has been attributed to the instrument losses, TJi.
The instrument losses 1]i are relatively simple to estimate as illustrated in Example 6.2.
For a carefully designed instrument one can expect 1]i == 10- 4 . . . 10- 3 , with the strongest
contributions (smallest value in the product) coming from inability to collect all emitted
light. The sample losses vary from sample to sample very much. To make a rough estima-
tion of somewhat typical case, Example 6.3 provides calculations made for a dye solution
which has 10% quantum efficiency of the emission.
Example 6.2: Estimation of the instrument losses of a fluorimeter. Let us estimate instru-
mental losses of a fluorimeter according to eq. (6.13). An estimation of the efficiency
of sample emission collection was done in Example 2.7 on page 38, 1]c 0.01. This
value is limited by the monochromator angular aperture. The quantum efficiency
of the monochromator depends on the wavelength and in the middle of the work-
ing range can be as good as 1]m 0.5. For a good photomultiplier one can expect
the quantum efficiency to be 1]PM < 0.1. For a moderate wavelength resolution
(1-5 nm) and a reasonable size of the monochromator slits could be 1-2 mm. If
the sample is a solution in 1 cm cuvette, the factor b6.t ern can be evaluated to be
0.1-0.2. 14 Thus, one can expect the instrument losses to be 1]i 10- 4 .
Example 6.3: Estimation of the sample losses. To make a rough estimation of the sample
losses let us consider a solution of an organic dye compound with a moderate emis-
sion quantum yield. The sample losses, 1]8' consist of three parts. First of all, the
sample will probably absorb only a small part of the excitation, say a( Aex) 0.1.
The emission quantum yield for a dye molecule can be taken as cP 0.1. 15 The term
<p(A) Aem defines how many of emitted photons will fall into the detection wave-
length band. For a reasonably accurate spectrum determination <p(A) Aem 0.01,
which means roughly 100 measurements are taken for the emission band, e. g. a
50 nm wide emission band measured with 0.5 nm steps. These values give the losses
value 1]8 10- 4 .
Combining together the losses estimated in Examples 6.2 and 6.3 one obtains 1]i1]8
10- 8 . If acceptable accuracy of the measurements is 6 == 0.01 == 1 0/0, then N c == 6- 2 == 10 4
counts must be collected to provide the accuracy (see square root law on page 67). Thus,
N ex == c == 10 12 photons are required for the excitation. If excitation wavelength is
Th TIs
Aex == 400 nm, then the photon energy is hv 5 X 10- 19 J, and the excitation energy
is Eex == 0.5 pJ. According to values obtained in Examples 1.4 and 2.7 for the emission
spectrum density of an ark lamp and a monochromator light collection efficiency, respec-
tively, one can expect to obtain excitation density of lex 270 x 0.006 2 mW/nm.
14 This is roughly the ratio of the detection monochromator input slit size to the length of the excited area of
the sample.
15 For a strongly emitting dye the quantum efficiency can be as high as 90%. However, typical emission
quantum yield of the aromatic compounds is in the range 1-500/0, if there are no quenching mechanisms reducing
it to a much lower value.
6.2. Fluorimeter
119
Consequently, using 2 nm spectrum resolution, one can accomplish single wavelength mea-
surement in time interval t ac == I E 10- 4 S = 100 tLS, which is extremely short time
ex ex
considering normal expectation of how fast emission spectrum can be measured. 16
These estimations are based on rather favorable conditions for the fluorescence spectrum
measurements. With real samples quantum efficiency of the emission may be much lower
than 10%, detection wavelength may not match the best monochromator-photomultiplier
sensitivity region and many other factors may reduce the factor 1]i1]s by many orders of
magnitude. Nevertheless, the accumulation time can be increased too by many orders of
magnitude, e. g. to t ac == 1 s, without creating any practical problems. This is more than
4 orders of magnitude in reserve which can be used if one or few measurement conditions
are not in their best. This means that in extreme case the emission spectrum can be still
measured if the emission yield is only 10- 5 , or if the sample absorption coefficient is as
low as 10- 5 . The latter value is an important result to be compared with the sensitivity of
the absorption spectrum measurements.
As it was discussed in Section 5.3.3, the resolution in absorbance measurements can be
Aabs 2 x 10- 5 at its best. This corresponds to the absorption of a == 1 - 10-.6.A abs
4.6 X 10- 5 . In emission spectroscopy one can measure emission or excitation spectrum with
inaccuracy of lOA> for a sample with absorption of 10- 5 only. Thus, in fluorescence mea-
surements one can see absorption (and emission, of course) of a sample which cannot be
measured using an advanced spectrophotometer. This is why fluorescence methods are usu-
ally considered to be more sensitive than absorption when one needs to detect a compound
at a very low concentration.
The sensitivity estimations made above were not optimized for extreme cases. If the
sample under investigation is inefficient in emission orland available at a very low concen-
tration only, one can consider the following measures to improve the signal intensity:
1. Increase accumulation time, taco For example, in 4 times longer time interval the
average number of collected photons is 4 times greater and signal-to-noise ratio is
V4 == 2 times better.
2. Increase slits of the excitation monochromator, Aex. This will increase excitation
intensity, lex, which is roughly proportional to A;x at least for relatively narrow
slits. This will work until the sample absorption band is wider than the excitation
band. Clear drawback of this approach is that with the bigger slits selectivity of the
monochromator is worse and this also increase intensity of the scattered and stray
light crossing the monitoring monochromator.
3. Increase slits of the emission monochromator, Aem. The signal is proportional to
A;m' see eqs. (6.6) and (6.11). Evidently, this will result in worse wavelength
resolution and some important features of the spectrum may be lost.
16 This is a rather theoretical value since for the conditions considered the photon counting rate will be
Nc/t ac = 10 8 photons/s, which is greater than the l11axil11ul11 counting rate of the l110dem photol11ultipliers
(typically:::; 10 7 photons/s). Therefore the practical l11easurel11ents will be carried out with lower excitation inten-
sity, e. g. narrower slits of excitation l11onochrol11ator, and will take longer til11e, e. g. 1 I11S. However, the value
can be used as a reference for the further discussions.
120
Steady State Emission Spectroscopy
4. The optimum wavelength for the sample excitation is the maximum of the product
of the excitation lamp spectrum and the sample absorption spectrum, IexA. If the
sample spectrum is broad or multi -peak, then the best excitation wavelength may
differ from the sample maximum absorption wavelength.
It should be also noted that an ideal device was considered above. In particular, except of de-
sired monitoring wavelength there will be some amount of light at other wavelengths which
is stray light crossing the monochromator. The stray light is mainly due to the light scat-
tering and its relative amount is small (usually less than 10- 6 of total incoming intensity),
but it becomes important when, e. g., the emission yield of the sample is weak. From the
practical point of view this will result in some kind of background spectrum which cannot
be easily subtracted from the measured spectrum to obtain a "pure" sample spectrum (the
background spectrum is not the property of the instrument only, but depends on the light
scattering in the sample). The effect of background spectrum can be reduced by adding color
filters in excitation and emission channels to reject excitation wavelength at the entrance of
the emission monochromator. Yet another approach is to use double monochromators.
An important property of the fluorescence measurements utilizing photon counting tech-
nique is very high dynamic range and high linearity of the response. A good photomulti-
plier may have dark counting rate smaller than 10 counts/s and the maximum counting rate
greater than 2 x 10 7 counts/s, which covers variation in signal by more than six orders of
magnitude and provides linearity (accuracy) better than 0.1°;6.
6.2.8 Wavelength resolution
An increase in wavelength resolution (decrease in Aem) results in a fast decrease in the
signal intensity (eqs. (6.6) and (6.11)). Consequently, the highest possible resolution de-
pends not only on the instrument but also on the sample, e. g. on the emission quantum
yield.
Taking the values considered in Examples 6.2 and 6.3, and following discussion in Sec-
tion 6.2.7, one can conclude that in order to increase the resolution by factor 100 (so that
Aem t""V 0.01 nm) while keeping the same signal-to-noise ratio, the signal collection time
must be increased to 10 s. This follows from the square dependence of the number of counts
on Aem, therefore 100 times increase in wavelength resolution will lead to 10 4 times de-
crease in counting rate and will require 10 4 times longer measuring time to collect the same
number of counts. This accumulation time, lOs, is technically possible but not practical -
to record a spectrum in the range, e. g., 500-550 nm with step 0.01 nm would take 5000 s
or more than 1.5 hour.
Another practical problem of low signal intensity, i. e. low counting rate, is photomulti-
plier dark counting rate. 17 In 10 seconds the number of dark counts can be 1000 or higher,
which may approach the number of emission counts and give a fault result. The dark count-
ing rate of the detector can be measured separately and subtracted from the collected counts
17 A typical dark counting rate for photol11ultipliers equipped with S-20 photo-cathode (sensitive up to 850 nl11)
is 100 counts/so The dark counting rate can be reduced gradually by cooling the photol11ultiplier photo-cathode
down by 40-50 °c.
6.3. Samples for emission measurements
121
1)
excitation
2)
excitation
3)
excitation elll1SSlon
/--1 -1
I I
I
I /
I /
I / /
I /
I /
I /
/
1/
- _ 0 - :mission
--__-r
- _ 0 - -:3>
- - = = - emission
---__-r
sample
Figure 6.6: Excitation-monitoring schemes used in emission spectroscopy: 1) right angle
scheme used for solution studies, 2) right angle scheme used for thin transparent samples,
and 3) front face excitation scheme for non-transparent samples.
during the spectrum measurements, but the noise produced by the dark counts will affect
the accuracy of the measurements in any case.
There is also a practical limit in the wavelength resolution of monochromators. Assum-
ing the resolution Aem 0.01 nm, the slits size d == 10 tL (a smaller size is hardly possible
due to the diffraction limit, not to say about complexity of manufacturing such a system),
and the groover number of the grating 9 == 1200 mm- l (a typical value for a high spectrum
resolution in the visible range), and using eq. (2.46) one obtains the focal distance of light
collimating mirror F .6.,,\ d 1 m. Thus, for 0.01 nm resolution one have to build
ern9
an instrument of more than one meter size. This is possible but unacceptable for a general
purpose device.
In conclusion, a typical wavelength resolution limit for a general purpose fluorimeters
is usually limited by 1 nm. For a high quality research grade devices the resolution can be
close to 0.1 nm. However, in both cases the best spectrum resolution can be achieved when
the sample has high enough emission quantum yield.
6.3 Samples for emission measurements
6.3.1 Excitation-monitoring schemes
The mutual orientations of the excitation beam and the direction of the emission monitoring
is perpendicular to each other in Fig. 6.6 scheme 1. This is a typical geometry for light
transparent samples, such as solutions, since in this geometry the scattered excitation light
has minimum impact on the emission detection subsystem. This excitation-monitoring
scheme is called right angle scheme.
For thin optically (semi) transparent samples the same layout of excitation-monitoring
optics around the sample can be used. In this case the sample is placed at an angle of
roughly 45° to the excitation beam in such way that the reflected excitation propagates in
direction opposite to the monitoring direction (Fig. 6.6 scheme 2).
When the samples are not optically transparent a front face illumination scheme is used,
as presented in Fig. 6.6 scheme 3. In this case the sample surface is also turned so that no
excitation light is reflected in direction of the emission detection system.
122
Steady State Emission Spectroscopy
I )
I in
1
emISSIon
monitoring
area
__ = ___J_ Iabs
c:
.
cuvette
Figure 6.8: Effect of sample absorption on the intensity of the monitored emission, Iem rv
Iabs == alex.
6.3.2 Cuvettes
A typical cuvette for the steady state emission spec-
troscopy of liquid samples in shown in Fig. 6.7. This
is a 1 cm cuvette with 4 clear walls and square cross
section. The 1 cm cuvettes for absorption and emis-
sion spectroscopy are similar to each other except that
spectrophotometer cuvettes have only 2 clear walls. The
spectrophotometer cuvette holders are usually utilizing
a spring, which helps to keep the sample at exactly the
same position, but may touch the cuvette walls and may
leave some scratches on fluorimeter cuvette walls, if it
is frequently inserted in spectrophotometer. The emis- Figure 6.7: Cuvette for emis-
sion cuvette can be used to measure absorption spec- sion spectroscopy.
trum, but some attenti on should be paid to prevent the
cuvette walls from undesired contacts with the holder.
Similar to the absorption cuvettes, the emission cuvettes are usually made of glass or
quartz (see transmittance spectra in Fig. 5.7). The quartz cuvettes must be used if the
excitation wavelength is shorter than 350 nm.
6.3.3 Effect of the sample absorption
Prom eq. (6.2) one can conclude that as more light absorbed by the sample, the emission
intensity gets stronger. So it seems that by increasing the absorbance of the sample one can
increase the measured signal continuously. For liquid samples in square cuvettes measured
in right angle scheme this is correct only at rather low sample absorbances. The reason for
that is that the emission is usually collected from the middle of the sample, and the excitation
intensity in the middle of the sainple, rather than incoming beam intensity, liTo determines
the measured emission intensity, as illustrated in Fig. 6.8. The measured emission inten-
sity is proportional to the amount of absorbed excitation only inside the monitoring area,
Iem rv I abs. The length of the monitoring area depends on the spectrum resolution of the
6.4. Fluorimeter specifications
123
detection system (size of the slit S3 in Fig. 6.2) in particular, and can be much smaller than
the total length of the cuvette. The part of the sample before the monitoring area absorb
the excitation light and works as a filter decreasing the excitation intensity lex. Starting
from a certain value of the sample absorbance this filtering effect becomes stronger than
the absorption in monitoring area and leads to a decrease in the emission intensity with an
increase in the sample absorption.
The same phenomenon leads to a decrease in emission intensity outside the cuvette in
case of overlap between the emission and absorption bands. For accurate measurements in
right angle scheme the samples must be prepared at low concentrations. Usually absorbance
of 0.1 is considered to be low enough to neglect by the absorption effects on emission or
excitation spectra. Also depending on the sample under study and the task to be solved a
higher absorption can be used safely.
In case of the front face illumination scheme (Fig. 6.6, scheme 3) there is no "saturation"
effect of the sample absorption. This scheme can be used to measure samples with high
absorbance and non-transparent samples. The scheme, however, has the disadvantage of
stronger effect of the scattered light as was discussed above, and thus, is lesser suitable for
measuring samples with low emission efficiency or achieving high spectrum resolution.
6.4 Fluorimeter specifications
6.4.1 Water Raman scattering line as sensitivity test
In order to compare different fluorimeters one needs a reference sample, which is easy to
prepare and to measure with different instruments. There are many possible candidates for
the reference, but probably the most widely used is the water. This may sound strange
since pure water has no absorption or emission bands of its own in the visible and near UV
spectrum range. The measured emission is the Raman scattered band shifted relative to the
excitation frequency by roughly 3400 cm -1.18 The efficiency of the Raman scattering is
rather low, typically 10- 9 , which makes it a good test for the instrument sensitives.
An example of the water Raman line measured with a Fluorolog-3 (Spex Inc.) is pre-
sented in Fig. 6.9. The excitation wavelength was 350 nm, which is quite standard for
fluorimeter tests. At this excitation the Raman peak maximum is at 397 nm. The slits were
set to provide 5 nm resolution for both excitation and monitoring monochromators, and
the photon counting time was 1 s. For comparison with other instruments one can notice
the maximum intensity of the Raman band and the level of the background, which were
7.34 x 10 5 and 1.3 x 10 4 counts per second, respectively, for the spectrum shown in Fig
6.9. The former value tells about the intensity of the excitation, the ability of the device to
collect the emission and the efficiency of the detection system of the instrument. The latter
value shows how well the instrument can block the stray light and other non-desired effects.
18 RaI11an scattering is the result of the photons interaction with vibrational l110des of the l11atter. For excitation
frequency vph and vibrational 1110 de at frequency V v the RaI11an scattering appears at frequency VR = vph - V v .
124
Steady State Emission Spectroscopy
8xl0 5
60
440
6xl0 5
if]
------
if]
.......
:::1
54Xl05
u
""""'
2xl0 5
380
Figure 6.9: Raman line of water recorded with excitation wavelength at 350 nm.
6.4.2 Commercial fluorimeters
There is a wide range of the fluorimeters available from different manufacturers. As an
example of a simple entry level instruments one can consider Jenway Model 6200 or Quan-
taMaster QM-2/2005 (PTI Inc.) fluorimeters. Actually, these devices are not used to mea-
sure spectra - the excitation and monitoring wavelength selection is made by a set of color
and interference filters. The main applications of these instruments are detection of small
amounts of emitting compounds and monitoring specific reactions. The excitation and mon-
itoring wavelengths are set to a specific values determined by the problem on hand and
routine measurements are carried out to compare a series of samples or to track change in
time of a monitored substance. For this type of measurements the excitation and monitoring
bandwidths are much wider than those used in fluorimeters equipped with monochromators,
which makes them potentially more sensitive.
The high performance research grade fluorimeters are usually designed as a set of mod-
ules that can be combined together to provide users with the instrument fitting best to their
needs. There are few competing instruments in this category. As an example one can con-
sider SPEX Fluorolog-3 (HORIBA Jobin Yvon Ltd.), FS920 (Edinburgh Instruments Ltd.)
and QuantaMaster (PTI Inc.) series of spectrofluorometers. The main building blocks of
this instruments are light sources (typically with Xe arc lamps), monochromators, sam-
ple compartments, and detectors (photon counting photomultipliers and infrared sensitive
photodiodes). One can select single or double monochromator optimized for UV or red
wavelength range, to find detector most suitable for a certain application or build two chan-
nel instrument with one channel optimized for UV-visible and another for red-near infrared
emission detection.
As a short example let us review technical specification ofFS920 fluorimeter in a typical
general purpose configuration (similar to one presented in Fig. 6.2). The excitation source
is 450 W Xe arc lamp. Both excitation and monitoring monochromators are Czerny- Turner
type with 30 cm focal length. The spectrum resolution of mono chroma tors is 0.05-18 nm
adjustable with 0.05 nm steps (wavelength accuracy is =::!:0.2 nm). The monitoring spec-
trum range is determined by the detector and can be 185-680 nm (standard photomultiplier,
6.5. Emission of molecular monolayer: An example
125
0.04
Absorption
0.02
0
5
-5!l. 1. Ox 1 0
rfJ
@
o
U 4
"' 5.0x10
Excitation
Emission
0.0
500 600
wavelength, nm
800
Figure 6.10: Absorption, emission and excitation spectra of 10% (molar) porphyrin film.
R1527) or 16-870 nm (red sensitive photomultiplier, R955). The dark counts are < 100
and < 2000 counts per second, respectively. In a standard water Raman sensitivity test, as
described in previous section, the peak counts are > 10 6 .
6.5 Emission of molecular monolayer: An example
The sensitivity of the emission spectroscopy can be illustrated by the measurements of the
molecular layers. In this example porphyrin mono-molecular layers were deposited on a
quartz or glass support using Langmuir-Blodgett method [7], and the spectra were acquired
using a standard laboratory spectrophotometer (UV-2501PC, Shimadzu), and a fluorimeter
(Fluorolog-3, Spex Inc.). The porphyrin, pentafluorophenylporphyrin, was mixed with ma-
trix molecules, octadecylamine, and spread on a water surface to form a monolayer. The
monolayer was then transfered onto supporting plate by lifting slowly the plate through the
monolayer. The surface density of the porphyrin molecules in the layer can be adjusted by
changing the molar ratio of the porphyrin to matrix molecules in the spreading solution. In
this example 2 films were measured - one with molar concentration of 10% and another
0.1°;6. The 10°;6 concentration corresponds to density of roughly one porphyrin molecule
per 2 nm 2 , and 0.1 % to one molecule per 200 nm 2 , respectively. 19
The absorption, emission and excitation spectra of the 10% film are shown in Fig. 6.10.
Porphyrins have a strong absorption band in the blue range of the spectrum (called Soret
band) with molar absorption coefficient as high as 4 x 10 5 M-lcm l , and relatively weak
bands in the green-red part of the spectrum, called Q-bands. The strong blue band for this
porphyrin has maximum at 426 nm and it is clearly seen in absorption spectrum. One of
19 To some degree, the porphyrin can be treated as a flat square-like 11101ecule with the size of roughly 1.2 x
1.2 x 0.5 nm 3 .
126
Steady State Emission Spectroscopy
0.004 Absorption
0.002
0
-0.002
3000 Excitation Emission
rJJ
............
rJJ
2000
;:j
0
u
""' 1000
0 400 500 600 700 800
wavelength, nm
Figure 6.11: Absorption, emission and excitation spectra of 0.1 0A> (molar) porphyrin film.
the Q-bands also can be noticed at 505 nm, but its intensity is already close to the sensi-
tivity limit of the instrument. The emission efficiency of the porphyrin is 10°A> in solution.
In solid films it is usually lower, and for the present case can be estimated to be close to
5%. However, the emission spectrum can be measured with high accuracy using excitation
wavelength of 426 nm (absorption maximum) excitation slits 3 nm, emission slits 2 nm and
accumulation time 1 S.20 The maximum counting rate was higher than 10 5 , which gave
measurement accuracy better then 0.3°A>. The excitation spectrum was recorded with moni-
toring wavelength set to 662 nm (emission maximum), excitation slits 2 nm and monitoring
slits 3 nm. Clearly, excitation spectrum matches well to the absorption spectrum, but it has
much lower noise level, so two other Q-bands can be easily identified at 538 and 578 nm.
When the concentration was decreased 100 folds, the absorption spectrum did not pro-
vide any evidence of the porphyrin molecules presence in the layer, as can be seen in Fig.
6.11. One can expect the absorption of the film to be 0.0004 at the maximum of the Soret
band (426 nm), which can be resolved only by the best commercially available spectropho-
to meters (see Section 5.4.1 for the spectrophotometer specification examples). However the
emission and excitation spectra can be measured with a reasonable signal-to-noise ratio,
also the slits were increased by 1 nm to gain better signal intensities. The useful informa-
tion extracted from these two series of measurements can be concluded from the differences
in the emission and excitation spectra at different porphyrin concentrations, which indicated
that the organization of the molecules in the layer has changed significantly when the con-
centration was reduced 100 times.
From the technical point of view the measurements of the porphyrin emission and exci-
tation spectra is a relatively easy task since the excitation and monitoring wavelengths can
20 The el11ission spectrul11 differs significantly frol11 that in solutions, which is due to special organization of
this porphyrin 11101ecules in Langl11uir- Blodgett fih11s [7].
6.5. Emission of molecular monolayer: An example
127
be well separated, 426 and 662 nm in particular case. The estimated emission efficiency
of the 0.1 % sample is acjJ == (1 - 10-0.0004) x 0.05 5 x 10- 5 . At low emission in-
tensity the main problem comes from the stray light passing the excitation monochromator
and from the selectivity of the monochromator. 21 One possible solution of the problem is
to use color filters together with monochromators. A red cut off filter can be inserted in
front of the detection monochromator and a blue band pass filter can be placed between the
excitation monochromator and sample. This helps, however, only when the emission and
excitation wavelengths are well separated as the spectrum selectivity of the color filters in
not very good and there is a limited choice of filters. Another approach is to use double
monochromator in the monitoring channel. This also allows to reduce the effect of the stray
light gradually.
21 When a l11onochrol11ator resolution is set to, e. g., Aem = 2 nl11, the light at wavelength shifted by 1 nm
(half of 2 nl11) ttOl11 the l11onochrol11ator l11axil11ul11 wavelength will be attenuated by roughly two til11es. Moving
further away frol11 the l11axiI11ul11 wavelength will reduce the intensity of passing light l110re and l11ore. However
there is no sharp border at which the l11onochrol11atic light will be blocked cOl11pletely. Usually a shift by 10 . A
frol11 the excitation wavelength is enough to l11easure an el11ission spectrul11, but this depends on the saI11ple and
should be checked if signal intensities are weak.
Chapter 7
Flash-photolysis
Introduction of new revolutionary research methods leads to new fundamental discoveries
in natural sciences. In optical spectroscopy one of the directions of great progress in the
research methods development was aimed at improving the time resolution. The first signif-
icant advances in this direction have been achieved more than 50 years ago, and were well
recognized by the scientific community by awarding the Nobel Prize in Chemistry (1967)
to Manfred Eigen, Ronald George Wreyford Norrish and George Porter. The prize was
awarded "for their studies of extremely fast chemical reactions, effected by disturbing the
equilibrium by means of very short pulses of energy". In particular, George Wreyford N or-
rish and George Porter have applied short light pulses and followed the appearance of new
absorption spectra signalling formation and relaxation of new transient states. This method
is known now as flash-photolysis.
7.1 Principles
Flash-photolysis is one of the powerful tools in modem photochemistry and photophysics.
The fundamental idea of the method is to use a short light flash to disturb the system under
study and to follow the course of the photo-reaction by monitoring absorption properties of
the system. The light flash - excitation pulse - increases instantly the energy of the system
and triggers a chain of spontaneous reactions. As an example one can consider a solution of
dye molecules. In normal conditions the dye molecules are in equilibrium with the solvent
molecules which means that electronic subsystem is in its lowest energetic state, ground
state. Absorption of a photon by the molecule rises one of its electrons to a higher orbital
and increases the energy of the molecule. The excited molecule can relax back to the ground
state via a few intermediate state, such as singlet and triplet excited state, or can participate
in a reaction (photo-reaction) such as a charge transfer or isomerization. In any case the
excited state and the following transient states have their own absorption spectra and can be
monitored by measuring the absorption change at some specific wavelengths.
Naturally, the photo-reactions take place under both continuous and pulsed excitation.
However, under continuous excitation the longest-lived transient state dominates in popula-
tion hiding any faster intermediate products. The pulsed excitation and the flash-photolysis
129
130
Flash-photolysis
Pulsed excitation source
D
, ,
V V
-- -- V - [] -- -- - -
< >
.................. --- ................,.,---
PD
G: = :>
Lamp
L1
L2
sample
L3
monitoring
monochromator
detection
monochromator
Qg
Transient recorder
( oscilloscope)
Figure 7.1: Scheme of the flash-photolysis method. PD is a photo detector and L 1, L2 and
L3 are lenses.
method in particular, allows one to monitor the fast reactions as well so that the whole
reaction scheme can be recovered.
7.1.1 Optical scheme
A general optical scheme of an instrument for flash-photolysis measurements is presented
in Fig. 7.1. The scheme is somewhat similar to the single channel spectrophotometer dis-
cussed in Section 5.2.1. There is a source of the monitoring light and the light detection
system. The first additional part is the pulsed excitation source, which can be a flash lamp
but is usually a pulsed laser nova days. The second difference is that the detection system
is aimed to measure the temporal change but not the static light intensity as in case of spec-
trophotometers. For this reason the signal from the photo detector is passed to the transient
recorder (e. g. digital oscilloscope). Finally there are two monochromators in the scheme -
one is placed between the lamp and the sample to select the desired monitoring wavelength.
Another is placed between the sample and the detector. Its role is to pass the monitoring
light and to reject the excitation.
Formally speaking, two monochromators are not needed for the purpose of the moni-
toring wavelength selection. The detection monocheromator is sufficient from the point of
view of the instrument functioning. However a continuous white light illumination of the
sample may cause a damage of some photosensitive samples. Therefore, the monitoring
monochromator main role is to reduce the monitoring light intensity at the sample. One can
use color or interference filters in place of the monitoring monochromator for this purpose.
7.1. Principles
131
There are also some other measures which help to reduce the effect of the monitoring light
on the sample, as discussed later in Section 7.2.1.
The detector must have fast enough time response to provide the time resolution re-
quired for the experiments. A reasonable limit in time resolution for the flash-photolysis
method is I-IOns, as discussed in Section 7.2. This is at the top limit of general purpose
photomultipliers, which have time resolution up to a few nanoseconds. Fast semiconductor
detectors, e. g. photodiode and avalanche photodiode, can provide higher time resolution
but there are other limiting factors as will be discussed in Section 7.2.
The photodetector signal must be recorded in the time interval suitable for the studied
reaction. Fast digital oscilloscopes are usually used for this purpose. They are typically
connected to a computer (not shown in Fig. 7.1) for signal averaging and controlling the
measurements, as discussed in Section 7.2.2.
7.1.2 Transient absorbance
The aim of the flash-photolysis measurements in to record temporal change in the sample
absorbance, A( t), induced by the excitation pulse. Absorption cannot be measured directly,
therefore the monitoring light is needed, and the light intensity, I (t), is the parameter which
is going to be recorded during the experiments. The relation between the light intensities
before and after the sample is given by eq. (1.11), which can be rewritten as
I(t, A) == I in (A)10- A (t,A)
(7.1 )
where I in and I are the monitoring light intensity before and after the sample, respectively.
There is an initial absorbance of the sample which is changing in time as the result of
the photo-excitation and following relaxation. Therefore, it is convenient to present the
absorbance as a sum of two parts A(t, A) == AO(A) + A(t, A), where AO(A) is the sample
absorbance before the excitation and A( t, A) is the absorbance change due to some photo-
reaction. The value A( t, A) is called differential absorbance. 1 Then
I(t, A) I in (A)10- Ao (A)-.6.A(t,A)
I in (A) 10-Ao (A) 10-.6.A(t,A)
I O (A)10-.6.A(t,A) (7.2)
where IO(A) == I in (A)10- Ao (A) is the monitoring light intensity after the sample at some
time before the excitation. The absorbance change is
( I(t, A) )
.6.A(t, A) = -loglO lo(A)
(7.3)
or
( I(t, A) )
.6. A ( t, A) = -loglO 1 + lo (A)
(7.4)
1 Suppose the ground state absorbance of a 11101ecule is Ao (A) and the absorbance of the excited state is
Al (A), then the excitation induced change in absorbance is A(A) = Al (A) - AO(A), which is difference in
absorbances of two states, or differential absorbance.
132
Flash-photolysis
I excitation
. flash
U o
_ __ _ _ _ _ _ _ _ __ _ _ t_ _ _ __ _ _ _ __ _
t
o
(trig 0
(
Figure 7.2: A signal recorded in flash photolysis experiment. The signal before the excita-
tion is U o . The excitation flash time is assigned to t == o.
where I(t, A) == 10(A) - l(t, A).
It is important to notice, that to obtain the differential absorbance the relative change
in the monitoring light intensity must be measured, but there is no need to determine the
absolute value of the intensity.
Since the photo detector signal is proportional to the light intensity, the intensity ratio in
eq. (7.4) can be replaced by the ratio of corresponding detector signals t: == t , where
U o is the detector signal before the excitation and U is the signal changed induced by
the excitation flash. Thus, in order to calculate the differential absorbance (as a function
of time) one needs to measure the signal before the excitation, U o , which corresponds to
10 (A), then to measure the time dependence U (t) in response to the excitation flash and to
calculate the difference U (t) == U o - U (t). After that the differential absorbance can be
calculated as
A(t, A) = -loglO (1 + o(t) )
(7.5)
In fact, one can avoid separate measurements of U o . The signal recording can be started
at a short while before the excitation flash, as shown in Fig. 7.2. This recoding mode is
called pre-triggering, assuming that the excitation flash is the source of trigger signal. The
time of the excitation flash is taken as zero time, t == o. The value U o can be calculated as
the signal average before the excitation, at t < 0,
o
U o == J U(t)dt
ttrig
ttng
(7.6)
Then the signal is recalculated as U (t) == U o - U (t), and, finally, eq. (7.5) gives the
desired differential absorbance.
It is important to notice, that the calculated differential absorbance does not depend on
the sensitivity spectrum of the photo-detector, and there is no need to measure or calculate
the equilibrium state absorbance (Ao(A)).
7.1. Principles
133
a) 0.02
a
-0.02
-0.04
-0.06
-0.08
b) 0.15
0.10
0.05
"
0.00
-0.05
-0.1 0
350
o
100 200 300
time, J..ls
400
/
-
-
".
-
-
/
/
400 450
wavelength, nm
500
Figure 7.3: a) Transient absorption decays at 420 and 450 nm and b) time resolved (t == 0)
differential absorption spectrum (connected circus) of the triplet excited state of a pyropheo-
phytin a solution. For comparison the steady state absorption spectrum of the sample (nor-
malized to fit the scale) is shown in plot b) by the dashed line.
7.1.3 Differential absorption spectra
By repeating the measurements at different wavelengths one can collect a two dimensional
data array A(t, A), which can be used to draw time resolved differential absorption spec-
tra, A( t == const, A). As an example let us consider flash-photolysis studies of a solution
of pyropheophytin a 2 presented in Fig. 7.3. The sample was excited by a laser flash at
532 nm (second harmonic of a Nd:YAG laser). Plot a shows the transient absorbances at
420 and 450 nm, which were obtained as described in the previous Section, i. e. the value
of U o was calculated by averaging the data at t < 0 (before the excitation flash) and then
the transient absorptions, A, were calculated using eq (7.5).
Similar measurements were carried out in the wavelength range 360-500 nm, thus
forming all together an array of data A(t, A). The spectrum right after the excitation,
A(t == 0, A), is shown in Fig. 7.3b. This spectrum is called time resolved differential
absorption spectrum. Naturally, the differential absorption spectrum can be calculated for
any given delay after the excitation. However, for the measurements presented in Fig. 7.3
the transient absorption signals are simple mono-exponential decays and only the spectrum
at t == 0 is of practical interest. Therefore, the time resolved differential spectrum in Fig.
7.3b is the spectrum of the transient absorbances amplitudes for the particular case.
For the data presented in Fig. 7.3a the transient absorbance is negative at 420 nm and
2 Pyropheophytin a is a derivative of pheophytin a, which is one of the natural chrol11ophores involved in
photosynthesis of green plants.
134
Flash-photolysis
positive at 450 nm. An increase in absorbance means formation of a new absorbing band,
whereas a decrease in absorbance shows a disappearance of some absorption. At 420 nm the
instant decrease of the sample absorbance means that the excitation results in disappearance
of the ground state absorption of the sample. Indeed, at 420 nm pyropheophytin a has an
absorption band, as shown in Fig. 7.3b by the dashed line. Disappearance of the ground
state absorption under light illumination is called photo-bleaching.
At 450 nm pyropheophytin a has no ground state absorption, therefore at this wavelength
one can expect only increase in absorption after excitation, which is the formation of an
excited state absorption bands. In both cases (at 420 and 450 nm) the sharp change in
absorption takes place right after excitation pulse. The instant change in absorbance is
followed by a mono-exponential relaxation with the same time constant ( 0.5 ms) at both
wavelengths. The same time constant indicates that the decay of the light induced absorption
(at 450 nm) and recovery of the ground state absorption is the result of the same reaction.
In other words, in this case the excited state is relaxing to the ground state with the time
constant of 0.5 ms. The reaction scheme can be presented as P P* 0.5 illS) P, where P
and P* are the ground and excited states ofpyropheophytin a. 3 The amplitude of the instant
change in absorbance (due to excitation pulse) is
A(A) == (Aex(A) - Agr(A)) cP
(7.7)
where Aex and Agr are absorbances of the excited and ground states, respectively, and cP
is the efficiency of the excitation or the relative fraction of the excited molecules. The
shape of the A( A) spectrum does not depend on the sample excitation efficiency, and is a
characteristic of the excited state. It can be used to identify the intermediate states formed
upon the photo-excitation. An example of identification of the intermediate steps in photo
induced electron transfer is presented in the end of this Chapter, Section 7.4.2.
If the excitation efficiency is known, then the calculations of the excited state spectrum
are straightforward
A(A)
Aex(A) = Agr(A) + (P
(7.8)
However, an accurate determination of the excitation efficiency can be a difficult task. If
there is a wavelength at which the excited state has negligible absorption compared to the
ground state, the degree of bleaching at this wavelength is equal to the excitation efficiency.
In the case of pyropheophytin a the absorption of the triplet excited state is broad and
overlaps all the absorption bands of the ground state spectrum. The excitation efficiency,
could be determined from the excitation energy density and molar absorption coefficient at
the excitation wavelength, which requires some additional measurements, but in most cases
the characteristic features of the differential spectra ( A (A)) are sufficient to establish the
mechanisms of photo-reactions.
3 The excited state observed in the experil11ents is the triplet excited state. The singlet excited state is formed
upon the photo-excitation of pyropheophytin a, but the lifetil11e of the singlet excited state is roughly 5 ns and the
singlet state cannot be resolved in the til11e scale of the presented l11easurel11ents. The efficiency of the intersystel11
crossing (conversion frol11 the singlet to the triplet state) is rather high for pyropheophytin a ( 90%), resulting in
quantitative formation of the triplet excited state observed in l11icrosecond til11e scale.
7.1. Principles
135
Excitation
beam
Monitoring
beam
cuvette
Figure 7.4: Perpendicular orientation of excitation and monitoring beams (T-scheme).
Usually the transient absorption curves ( A(t)) are used to calculate lifetimes or rate
constants. In turn, the time resolved differential spectra ( A(A)) are useful for identifi-
cation of the intem1ediate states (for example see Section 7.4.2). Also an advanced data
analysis may involve global data fitting as discussed in Chapter 15.
7.1.4 Excitation schemes
In Fig. 7.1 the excitation beam is crossing the monitoring beam at right angle. This is a
typical arrangement for liquid samples, and it is also called aT-scheme. The samples are
prepared in cuvettes with 4 clean walls similar to the fluorescence measurements, e. g. 1 cm
cuvette shown in Fig. 6.7. Usually the excitation beam is expanded to cover the whole
cuvette width and monitoring beam is passed through the cuvette as close as possible to the
excitation entrance wall, as shown in Fig. 7.4. The former is done to increase the overlap
path of the excitation and monitoring beams. The latter is used to pass the monitoring beam
through the most excited part of the salnple, as the sample absorption (at the excitation
wavelength) decreases the excitation beam intensity while it propagates inside the cuvette.
This arrangement is particularly useful for samples with high absorptions (i. e. at high
concentrations) at the excitation wavelength.
For flat samples perpendicular excitation scheme is not practical. A quasi-parallel (or
quasi co-linear) excitation scheme is used to study relatively thin samples, as shown in Fig.
7.5. When this scheme is applied, a care should be taken to provide an overlap of the
monitoring beam by the excitation beam inside the sample area, i. e. the cross section of
the excitation must be big enough to cover the monitoring area through all the sample. This
scheme can also be used to study liquid samples. In such cases thin, e. g. 1 mm, cuvettes
are used. The excitation beam diameter can be a few millimeters, or even smaller for very
thin samples. This is an advantage of the quasi -parallel scheme over the perpendicular one
- to achieve the same excitation density a smaller pulse energy is needed.
136
Flash-photolysis
Sample
Excitation overlap
area
.........................
.....:-:-:::::::::::::::::::::::::::::::::::::::::::::-:-:..... /'"
Monitoring
Figure 7.5: Quasi-parallel orientation of excitation and monitoring beams.
7.1.5 Excitation
Selection of the excitation wavelength and pulse energy depends on the sample under the
study. Clearly, the excitation wavelength must be within an absorption band of the sam-
ple.At small excitation densities, an increase in the excitation pulse energy will increase
the induced absorption change, A. However, at a certain level the signal amplitude will
be saturated. This happens when all the chromophores (active centers) are excited, i. e. at
the pulse energy density E sat > hv ex / cr, where a is the absorption cross-section of the
chromophore at the excitation wavelength, Aex == . For typical organic chromophores
V exc
the saturation energy density is Esat t""V 1 . . . 10 mJ.cm- 2 . The following example presents
an estimation of the saturation density for chlorophyll a, which has molar absorption coef-
ficient E 10 5 M- 1 cm- l at 430 nm.
Example 7.1: Estimation 0.( excitation efficiency for chlorophyll a. Chlorophyll a has the
absorption cross-section a 4.6 . 10- 16 cm 2 at Soret band maximum, 430 nm
(con esponds to E 1.2 X 10 5 M-1cm- l ). Thus, the excitation saturation energy
density at 430 nm is Esat == hv exc / a t""V 11nJ/cm 2 . At this excitation density the
fraction of the excited molecules is 1 - e- l 0.63. Two times stronger excitation
density of2 mJ/cm 2 will increase the fraction of the excited molecules to 1 - e- 2
0.86, or by 27% only. Further increase of excitation density is hardly reasonable as
only 14% of molecules remains unexcited.
For quasi-parallel excitation scheme the area of the excitation spot can be as small as
0.1 cm 2 . Then the total pulse energy needed to excite most of the chromophores is just
0.1 . . . 1 mJ. This is, however, lower limit for excitation density used in practice. Firstly, it
is easier to work with bigger excitation spots. Second, the excitation wavelength may not
match the maximum absorbance wavelength of the sample or there may be other reasons
to excite the sample at a wavelength different from its maximum absorbance. 4 Finally the
compound under study may have molar absorption coefficient much lower than used in
4 As a typical eXaI11ple one l11ay want to measure the recovery of the ground state, and to choose the wavelength
of the l11axilllum absorbance for l11onitoring. Then it is reasonable to shift the excitation to another wavelength to
reduc.e the effect of the excitation flash on the detection systel11 as it was done for measurements shown in Fig. 7.3.
7.1. Principles
137
Example 7.1 (1.2 x 10 5 M-lcm- l ). Considering all the above the excitation pulse energies
in flash-photolysis experiments are usually in the range 10-100 mJ.
There is a wide choice of lasers providing such energies in nanosecond pulses but for
the most lasers the emission wavelengths are not tunable or tunable in a limited wavelength
range. Therefore the laser operational wavelength and output energy should be considered
together when selecting a laser for the flash-photolysis experiments.
An usual design approach to the excitation subsystem of the flash-photolysis instru-
ments is to combine a few lasers, one of which generates strong pulses at a fixed wavelength
and can be used as a pumping source for another tunable laser. For example one can use a
powerful laser emitting in the blue-UV wavelength range to pump a dye laser. This can be
an excimer laser (e. g. for XeCllaser the emission wavelength is 308 nm, see on page 129)
or Nd:YAG laser with second and third harmonic generators (emitting at 532 and 354 nm,
see on page 129). The pulse energies of these lasers can easily be as high as 100-500 mJ.
The secondary dye lasers are usually build up in such a way that a few different dyes can
be used to extend the choice of the lasing wavelength (see on page 51).5 The efficiency of
the dye lasers is relatively high, 10-30°A>, and the pulse energy can be as high as 10-50 mJ
for a relatively simple laboratory laser system in the wavelength range 430-700 nm. There
are also laser dyes for operation in 700-950 nm range but the efficiency and stability of this
dyes is not as good as for visible range dyes. The most difficult part of the spectrum is the
UV range where the pumping laser wavelength (308 or 354 nm) is available only, or the
second harmonic of the dye laser have to be used.
During the past decade two new solid state tunable laser systems were developed and
became available commercially. One is Ti:sapphire laser pumped by the second harmonic
of a Nd:YAG laser (see on page 52). Ti:sapphire has high efficiency, 40%, and excellent
energetic parameters. 6 The tuning range of the Ti: sapphire laser is 690-1000 nm, and with
second harmonic generation is 345-500 nm. Starting with 100-200 mJ pumping energy at
532 nm, one can expect to obtain up to 10 mJ in second harmonic of the Ti:sapphire laser
(maximum emission at about 400 nm).
Another new solution for the tunable excitation source is optical parametric oscillator
(OPO, see on page 58). Typically OPOs are pumped by the third harmonic of a Nd:YAG
laser (354 nm), and have the tuning range 440-700 nm for the signal beam and 720-1800 nm
for the idler beam. The maximum energy of the commercial OPO laser can be as high as
50 mJ in the range 480-550 nm (e. g. MOPO series, Spectra-Physics Inc.). To achieve such
high energy, the pumping energy has to be as high as 150 mJ (at 354 nm), which requires
a powerful Nd:YAG laser with pulse energy approaching 1 J at fundamental harmonic. A
disadvantage of the OPO systems is relatively high pulse-to-pulse energy deviation, which
is due to a large number of non-linear optical components involved. 7
5To switch form one dye to another, at least the dye cuvette l11USt be changed. At 1110St resonator l11irrors have
to be switched to fit to another lasing wavelength range. In the latter case the resonator adjustI11ent l11USt be checked
and tuned.
6High density of active centers (Ti) in sapphire crystals allows to obtain high output pulse energy frol11 rela-
tively sl11all crystals.
7Por eXaI11ple, if at the fundaI11ental harmonic the pulse-to-pulse variation is 1 %, at the second harmonic one
can expect stability of 2%, and at the third 3%. Essentially non-linear nature of optical paraI11etric frequency
conversion will double this value at the best. So the output pulse-to-pulse stability after OPO is 6% in an ideal
138 Flash-photolysis
7.2 Time resolution and signal-to-noise ratio
Since the method is applied to study intermediate and essentially unstable states, the time
resolution of the flash-photolysis instrument is one of the most important questions. There
are obvious time limiting parts, namely the the excitation pulse width and time resolution of
the light detection system. The former is not a real limit at present as femtosecond pulsed
lasers are available nova days. Modem photomultipliers have time resolution as good as 1 ns
and there are commercially available photodiodes with time resolution better than 100 ps.
However, 1 ns time resolution is difficult to achieve in practical measurements. The limiting
part is usually the source of the monitoring light.
There is a relation between the accuracy (or sensitivity) and time resolution of the flash-
photolysis method. Let us consider an ideal instrument which has no noises of its own, so
that only the light quantum noise will be taken into account. Suppose we would like to
measure light intensity with accuracy c5 and with time resolution T. This means, that in time
interval T the photomultiplier (PM) should detect 1/ c5 2 photons, i. e. the photon detection
rate must be (c5 2 T) -1. Accounting for the quantum yield of the PM, 4>d, and the registration
monochromator efficiency, 4>m2, the monitoring beam photon flux after the sample should
be (c524>d4>m2T) -1. Then, we have to add the sample transmittance, T, efficiency of the
monitoring monochromator, 4>ml, and losses in all the optical components (like lenses,
monochromator slits and so on), which will be called optics efficiency 4>0. 8 Finally, an
estimation for the light intensity required at the entrance of the monitoring monochromator
IS
I - hV mon
ml-
c5 2 4>d4>ml4>m24>oTT
(7.9)
In the derivation of the equation the light detector was assumed to be a photomultiplier
and the dark current of the photomultiplier was neglected. If the dark current cannot be
neglected, as in the case of infrared sensitive photomultipliers, or another type of photo-
detector is used, e. g. a photodiode, the equation must be corrected. Formally this can be
done by taking proper value of 4>d. The inverse value, l/4>d, shows how many photons
are needed to obtain a signal equal to the noise, i. e. to reach signal-to-noise ratio equal
one. Therefore 4>d can be evaluated from noise equivalent power of the detector, see Section
4.2.1.
According to this estimation to reach 1 ns time resolution with signal-to-noise ratio 100,
i. e. accuracy c5 == 0.01, the monitoring light intensity must be > 1 m W for flash-photolysis
measurements at 500 nm as show in Example 7.2. On the other hand the total emission
spectrum density of a tungsten halogen lamp is roughly 7 mW nm- l at this wavelength (see
Example 1.3 on page 10). This lamp emission is spread in all direction and less than 1 %
can be utilized by the monitoring monochromator (see Example 2.7 on page 38). Clearly,
case.
81n case of losses in lenses, the optics efficiency is just transl11ittance. However, loses due to l11isl11atch in the
l11onochrol11ator slit and beaI11 size are due to efficiency of the beaI11 collil11ation and they are l110re essential in
practice. Therefore the term "efficiency" is used here, although for lenses efficiency and transl11ittance are equal
to each other. Indeed, the transl11ittance "l11ultiplication law" l11USt be used to calculate total optics efficiency, see
Section 1.1.2, eq. (1.17).
7.2. Time resolution and signal-to-noise ratio
139
tungsten lamps cannot be used in nanosecond flash-photolysis measurements at least with
reasonable spectrum resolution. 9
Example 7.2: Monitoring beam power. Let us estimate the monitoring lamp power (I ml )
needed to achieve T == 1 ns time resolution at c5 == 0.01 == lOA> accuracy in otherwise
favorable conditions for the measurements. The quantum efficiency of a photomul-
tiplier can be 4>d == 0.1, which is close to the top value for most photo cathodes.
Efficiencies of the monochromators can be 4>ml == 0.5 and 4>m2 == 0.5. The losses in
optics include reflections from the sample cuvette walls, reflections from all lenses
(L 1, L2 and L3 in Fig. 7.1), and mismatches between the monochromator slits and
focused spots. One can hope to achieve efficiency as high as 50%, thus 4>0 == 0.5.
Finally, the sample absorbance can be assumed to be A == 0.5, which gives sample
transmittance T == 10- A 0.3. Substituting all the values in eq. (7.9) we obtain
Iml 1 mW at 500 nm. This estimation does not account for any noises of the
photo-detector and monitoring light source but the quantum noise of the photon flux,
and should be treated as absolute minimum. Real accuracy with 1 m W monitoring
intensity and 1 ns time resolution may be much worse than 1 %.
Arc lamps have higher temperature of the emitting area. At 500 nm the emission spec-
trum density ofaXe arc lamp can be as high as 0.3 W nm -1 (see Example 1.4 on page 11),
and one can expect to collect about 3 m W at the monitoring monochromator entrance in
1 nm bandwidth. Thus with arc lamps as the source of monitoring light one can approach
the nanosecond time resolution with reasonable signal-to-noise ratio.
When the time resolution of flash-photolysis method needs to be improved the following
measures can be considered:
1. to use an arc lamp, which has higher working temperature and higher spectrum den-
sity of the emission in the visible region;
2. to use a pulsed monitoring light source, which allows to increase the temperature of
the emitting body even more, as discussed in Section 7.2.1;
3. to decrease spectrum resolution and, thus, to increase the transmission bandwidth of
the monochromators, thus increasing intensity of the monitoring light;
4. to use a laser as the source of the monitoring light, which has many folds higher
spectrum density of the emission as compared to lamps.l0
High monitoring light intensity needed for the fast measurements creates yet another prob-
lem. Namely, the monitoring light can excite the sample to extend which cannot be ne-
glected any more. Example 7.3 shows how to estimate the relative population of the excited
9 Assul11ing that 1 % of el11ission is collected at the entrance slits of the l11onitoring l11onochrol11ator, the band-
width l11USt be extended to 14 nl11 to reach 1111 W l11onitoring intensity. However, taking into account rather idealis-
tic assul11ptions l11ade in EXaI11ple 7.2, real band pass l11USt be a few til11es greater, which seel11S to be unacceptable
for 1110St practical applications.
laThe lasers have disadvantage of l11uch narrower tuning range as cOl11pared to the cOl11bination laI11p-
l11onochrol11ator, not to l11ention the difference in price between these two alternatives.
140
Flash-photolysis
state and also demonstrate that if the recovery after excitation is longer than 1 ms, the mon-
itoring light intensity must be lower than 1 m W in typical experiment conditions. This is in
contradiction with requirement of the high monitoring intensity needed to achieve nanosec-
ond time resolution. To solve the problem flash-photolysis experiments are carried out with
pulsed monitoring light as discussed in Section 7.2.1.
Example 7.3: Effect of monitoring light on sample. Let us estimate the fraction of the ex-
cited molecules of a chlorophyll a solution under continuous irradiation by a mon-
itoring beam at 430 nm, which is one of absorption bands maximum. The molar
absorption coefficient of chlorophyll at this wavelength is typical for organic dyes,
E t""V 1.2 X 10 5 M-lcm- l , i. e. absorption cross section is a 4.6.10- 16 cm 2 .
The longest living excited state of chlorophyll a is a triplet state, which has lifetime
roughly TT 1 ms. 11 Let us assume that the monitoring light intensity is I == 3 m W
(to achieve ns time resolution), and monitoring beam area is s == 0.1 cm 2 , thus the
monitoring power density is P == -f == 30 mW cm- 2 .
For a simple estimation the reaction scheme is C C T C, where C and
C T are the chlorophyll ground and triplet excited states, respectively. The kinetic
equation for the case is
dN g Nt N P
- - - - a-
dt - T 9 hv
(7.10)
where N g and Nt are the number of molecules in the ground and excited states
and N g + Nt == N == const is the total number of molecules. On the right side
of eq. (7.10) the first summand is responsible for the increase of the ground state
population due to the relaxation of the excited state, and the second summand is
responsible for the decrease of the ground state population due to photo-excitation
by the monitoring beam. In steady state conditions d:!tg == 0, thus t == Nga f:v and
the ratio of populations is == aT f:v 0.03, i. e. under such conditions 3°A> of
chlorophyll molecules will be continuously in excited state.
Equation (7.9) can also be used to analyze the accuracy of the measurements, 6, (or
signal-to-noise ratio, which is 6- 1 ) after some rearrangement
6 2 == hV mon x 1 x
Iml T 4>d4>ml4>m24>0 T
(7.11 )
The first term on the right shows dependence on the sample transmittance, the monitoring
light intensity and the wavelength. The second term collects all sources of the light losses in
the instrument (a trivial conclusion is that a better instrument will give better results). The
third term tells that the increase in the time resolution, i. e. smaller T, will give a decrease
in the measurement accuracy, 6 rv T - .
11 There are l11any l11echanisl11s leading to a quenching of the triplet excited state, such as triplet-triplet annihi-
lation and quenching by oxygen 11101ecules dissolved in solution. Therefore, concentration of chlorophyll l11USt be
low enough and the solution l11USt be de-gassed to reach a l11illisecond lifetil11e of the triplet excited state.
7.2. Time resolution and signal-to-noise ratio
141
.....
.
rJJ
C)
'S
.
I excitation
flash
signal
bI)
.
;...,
o
.....
.
o
S
duty
arc
1
1 1
1 measuring 1
I. 1
1 time 1
1 window 1
1 1
1 1
1 1
1 1
1 1
.:E
bI)
.
......
i lamp
triggering
t
Figure 7.6: Utilization of a flash lamp to increase the monitoring light intensity during a
short measuring time interval.
High sensitivity of the detector is important for high time resolution. In terms of eq.
(7.9) the sensitivity is hidden under the detector quantum efficiency, 4>d. Decrease in sen-
sitivity will mean proportional decrease in time resolution. Therefore, although there are
photodiodes with time resolution much better than 1 ns, simple replacement of a relatively
slow photomultiplier by a fast photodiode may result in gradual decrease in the time reso-
lution under other equal conditions, since the sensitivity of the photodiodes is much worse
than that of the photomultipliers (as was discussed in Section 4.2).
7.2.1 Pulsed monitoring light
The monitoring light intensity is critical for the fast time resolution. The brightness of a
thermal light source, such as arc lamp, depends on the temperature of the emitting body,
e. g. plasma temperature near the lamp cathode. Obviously there is a limit after which the
lamp electrodes will be destroyed. 12 However, for a short time the arc can be overheated
by applying a short high voltage pulse. The emission temperature of the arc may increase
to 10000 K, which gives more than 10 folds increase in emission spectrum density in the
blue part of the spectrum as compared to normal continuous lamp operation with 6000 K
cathode area temperature.
In the pulsed mode the lamp is supplied continuously with a relatively low current. This
is the current needed to keep duty ark, but the brightness of the lamp at this time is at least 10
times lower than in the normal operational mode. A few microseconds before the excitation
flash a high voltage pulse is applied to the lamp. The voltage pulse has a special temporal
profile to generate the light pulse with the shape as close to rectangle pulse as possible, as
illustrated in Fig. 7.6.
12Typical color tel11perature of Xe arc laI11ps is 5500-6000 K.
142
Flash-photolysis
The pulse width can be as long as 1 ms. However, it is difficult to keep constant emission
intensity during this time with accuracy, e. g., 1°;6. On the other hand the shape of the pulse
can be well reproduced from pulse to pulse. Therefore, the measurements can be repeated
twice: first without excitation pulse, giving the monitoring pulse temporal profile in the
measuring time window, Ubg (t), and then with the excitation pulse, U (t). The ratio of the
measurements, u(t) = rK; ) , is nonnalized signal with "compensated" monitoring pulse
shape, which can be used to calculate differential absorbance using eq. (7.5) and the proce-
dure described in Section 7.1.2. This correction procedure is somewhat similar to recording
the base line in the absorption spectra measurements. A drawback of this correction is a de-
crease in signal-to-noise ratio, since the noise will be present in the measurements without
excitation (Ubg (t)) and will be added to the noise of the measurements with the excitation.
At a short time scale, approaching sub nanosecond time domain, the correction proce-
dure is usually not needed since it is possible to find a flat enough part of the monitoring
pulse where the change in the monitoring intensity inside the measurement window can be
neglected.
The pulsed monitoring light also helps to solve the problem of monitoring light effect
on the sample, e. g. see Example 7.2. Between the measurements the sample is exposed
to a minimum possible monitoring light intensity, which is needed to keep duty arc. For
measurement time window the monitoring light intensity increases hundred times or more,
providing high enough photon flux in the monitoring beam to attain nanosecond time resolu-
tion. To block totally the monitoring light intensity between the measurements a mechanical
shutter can be inserted between the sample and monitoring monochromator. 13 The shutter
control unit must be synchronized with the lamp and excitation laser control systems. Typ-
ical time needed to open a mechanical shutter is a few milliseconds, so it must be triggered
first, e. g. 10 ms before the excitation flash. The arc lamp pulse needs a few microseconds
to reach the working intensity level, and must be triggered, e. g. 10 tLs prior to the excitation
flash. 14
The flash-photolysis systems equipped with shutters and pulsed monitoring light sources
can be used to study irreversible photo-reactions, such as photo degradation. The instrument
is adjusted with a test sample. When everything is ready for the measurements the sample
is switched to the photosensitive one so that the monitoring light is opened right before the
excitation flash and has no effect on the sample.
7.2.2 Signal averaging
The signal-to-noise ratio of raw measurements, i. e. measurements of U (t) as described
in Section 7.1.2, can be improved by repeating the experiments a few times and summing
up or averaging the results. For a random noise (which is usually the case) the signal-to-
noise ratio increases as the square root of the number of averaged data, as was discussed
in Section 4.1.3. For example, averaging 4 measurements, one can improve signal-to-noise
ratio twice, averaging 100 measurement will reduce the noise level 10 times, and so on.
13 Usually the shutter is fixed on the output slit of the l11onitoring l11onochrol11ator.
l4The developl11ent til11e of the l11onitoring light pulse depends on the design of the laI11p power supply and can
differ gradually frOl11 the value l11entioned here.
7.3. Measurements of emitting samples
143
One limiting factor in improving quality of the measurements by averaging is the time
needed to repeat the measurements. Naturally, 100 measurements will take 100 folds longer
time then a single measurement. The second limit comes from the systematic errors, e. g.
linearity of the photomultiplier response, - the lower noise level does not necessary mean
higher data quality. Repeating measurements many times one also has to take care about
photo stability of the sample and stability of the instruments as whole.
Planning the experiments one should realize that averaging does not make any tricks
in sense of eq. (7.11) and the following discussion. Repeating the measurements N times
one increases the number of photons (i. e. h lrn1 in eq. (7.11)) N times, thus decreases
V rnon
c5 2 N times, i. e. improves signal-to-noise ratio VN times. In other words, eq. (7.11)
can be interpreted as the relation between the signal-to-noise ratio ( ) and the number of
monitoring photons, h Irnl , in time interval equal to the instrument time resolution, T.
V rnon
The required number of photons can be collected in a single measurement or repeating the
measurement a few times. Therefore a stronger intensity of the monitoring beam, when
possible, can be a better solution to improve the measurement results than the averaging.
7.2.3 Spectrum range and spectrum resolution
If only the usable wavelength range is considered, the flash-photolysis method is similar
to the steady state spectrum measurements discussed in Section 5.3.1. One needs a source
of monitoring light and a photo-detector, which are principal components determining the
wavelength range.
The spectrum resolution depends on the time resolution at least in nanosecond time
domain. First of all at constant spectrum density of the monitoring light source higher spec-
trum resolution means lower total intensity of the monitoring beam. Second, the monochro-
mator spectrum resolution is proportional to the size of the slits, i. e. higher resolution means
smaller slits, thus reducing the amount of light which can be passed into the monochroma-
tor at higher resolution. If both these factors are efficient, one can expect the monitoring
light intensity to be proportional to the square of the wavelength resolution, 1m rv A ,
and, thus, signal-to-noise ratio to be directly proportional to the wavelength resolution,
c5- l t""V Am, in otherwise equal conditions (see eq. (7.11)). Similarly, the time resolu-
tion is proportional to the square of the wavelength resolution, T rv A .
7.3 Measurements of emitting samples
7.3.1 Effect of scattering and sample emission
One of disturbing factors affecting flash-photolysis measurements is the scattered excitation
light, which is mixed with monitoring light and can produce fake signal. The scattering
depends on the sample and arrangement of excitation-monitoring beams. The T-scheme
(Fig. 7.4) has an advantage over the quasi-parallel scheme (Fig. 7.5) since (1) the Rayleigh
scattering is the smallest at the right angle, and (2) the T -scheme can be arranged so that
there is no surfaces illuminated by the excitation in the monitoring area, which is impossible
144
Flash-photolysis
for quasi-parallel scheme. 15 To decrease the effect of the scattered light spectral and spatial
filtering are used.
The role of the detection monochromator is the spectral filtering, i. e. to reject the
excitation flash and to pass the monitoring beam. The efficiency of the rej ection depends on
the wavelength separation of the excitation and monitoring. Naturally, when the monitoring
wavelengths approaches the excitation the rejection efficiency decreases. Typically at a few
tens of nanometers the stray light is suppressed by a factor of 10 5 . Double monochromators
can be used to increase stray light rejection. Additionally one may use color or interference
filters to reduce scattering effect further more.
Spatial filtering can be done by placing diaphragms fitting to the size of the monitoring
beam and rej ecting the scattering which propagates in all directions. The diaphragms can
be inserted in front of the lens collecting the monitoring light behind the sample (lens L3,
Fig. 7.1) and in front of the entrance slit of the detection monochromator.
In addition to scattering, the sample may emit some light by itself in response to the
excitation flash, e. g. if fluorescing molecules are studied. The sample emission differs
from the scattered light in its spectrum and lifetime. The emission spectrum is the property
of the sample and, e. g., changing the excitation wavelength will not help in this case. The
temporal profile of the scattered light follows the temporal profile of the excitation flash,
and normally is very short. Therefore the effect of scattering is usually seen as a short peak
at the excitation time, and in some cases can be just ignored. 16 In turn, the sample emission
can be rather long lived, depending on the sample, and may interfere with the time constants
of the studied phenomena.
Similar to the scattering, the spatial filtration can reduce the effect of the sample emis-
sion, but the spectrum filtration is helpless as in most cases the problem comes from the
emission at the monitoring wavelength rather than from the stray light. If possible, the
monitoring light intensity can be increased to reduce the relative effect of the emission.
The effect of scattered excitation and sample emission can be suppressed by repeating
experiments twice. The first time the measurements are carried out with blocked monitoring
beam, giving the time profile of the emission, U bg (t). Then the measurements are repeated
with monitoring light on, U e (t), thus recording both the monitoring signal and emission.
Subtraction of the former from the latter gives "pure" monitoring beam time profile U (t) ==
Ue(t) - Ubg(t), which can be used to calculate the transient absorbance according to eq.
(7.5). This procedure may help to reduce the effect of scattering or sample emission by 10
folds or greater if excitation pulse energies are stable and there are no saturation or non-
linear response of the detection system.
15 Surfaces, e. g. cuvette walls, are potential areas of light scattering, because of possible dust, scratches,
roughness, and so on. In the case of T -schel11e one can place a diofragl11diaphragl11 in front of the cuvette to cut
away the excitation beaI11 part hitting the walls through which the l11onitoring beaI11 is passed.
16Por eXa111ple, if a photo-reaction of interest has til11e constant of 1 J1S and the response til11e of the instrul11ent
is 5 ns, one can start l11easurel11ents at 50 ns after the excitation flash. At this delay til11e only 5% of reaction has
gone but the scattering should already have no effect.
7.4. Flash-photolysis instruments
145
7.3.2 Applications in time resolved emission spectroscopy
The flash photolysis method was developed to study transient absorption of the samples.
However it is also useful for time resolved emission spectroscopy applications. To switch
from the absorption to emission measurements the monitoring light must be turned off, then
the measured signal is directly proportional to the emission intensity, I ( t) rv U (t). This is a
simple method to study time dependence of the sample emission in nanosecond and longer
time domains.
Similar to the steady state emission spectra measurements discussed in Chapter 6, the
spectrum sensitivity of the detection system must be taken into account to obtain the time re-
solved emission spectra of the sample. A spectrum calibration procedure must be performed
to correct the spectrum efficiency of the monochromator and sensitivity of the photomulti-
plier when an instrument similar to that presented in Fig. 7.1 is used.
In practice, however, another method, which is called time correlated single photon
counting and will be discussed in Chapter 8, is used to measure emission decays in nano- to
microsecond time domain. Direct measurements of the emission decays using instruments
similar flash-photolysis are used for slower decays starting from microsecond when the
method mentioned above cannot be used.
7.4 Flash-photolysis instruments
7.4.1 Commercial instruments and components
The choice of commercially available ready made flash-photolysis systems is not as wide
as of spectrophotometers or fluorimeters, since these instruments are more complex, more
expensive and demand better understanding of the studied phenomena and measurement
techniques from users. Another reason is that one of the most expensive parts of the system
is the excitation laser, and the choice of the laser is determined by the range of samples to be
studied. Different laboratories need different laser systems and there is hardly one solution
to suit everyone at least with reasonable price. Hopefully all the components needed to
build up a flash-photolysis system are available commercially, letting users familiar with the
spectroscopy instruments build their own setup relatively easy. This is a typical approach
used by many research laboratories.
However there are companies producing ready made flash-photolysis instruments. An
example of such instrument is LP920 from Edinburgh Instruments Ltd. This is a modular
system which provides some flexibility for customers to find configurations most suitable
for their needs. The monitoring light source of the instrument is a pulsed Xe lamp with
selection of filters to reduce the intensity of the light at the sample. The sample compartment
allows three types of excitation-monitoring beam arrangements: T-scheme (Fig. 7.4), quasi-
parallel (Fig. 7.5) and diffuse scattering. The latter is useful for non-transparent samples
such as powder. It is somewhat similar to emission front face illumination scheme (Fig. 6.6)
- the scattered monitoring light is collected from the side of illumination.
Detection system consists of a monochromator and a choice of detectors. In the UV-
visible part, 200-870 nm, a photomultiplier is used. The near infrared is covered by a
cooled Ge photodiode (800-1750 nm). The signal from the photo-detectors is recorded by
146
Flash-photolysis
a fast digital oscilloscope to provide instrument resolution as fast as 7 ns. A single short
absorbance sensitivity is 0.002 with photomultiplier detector.
In alternative configuration of the instrument an image intensified CCD camera is cou-
pled with a spectrograph is used to measure time resolved differential absorption spectra
directly. This is a special gated camera, which is sensitive to light during a short gate pulse,
the shortest being 7 ns, so that instead of measuring a time profile at a fixed wavelength a
time resolved spectrum is obtained in a single shot.
In excitation channel a Q-switched N d: YAG laser with harmonic generators is provided,
i. e. the excitation wavelengths are 1064,532,355,266 and 213 nm. The excitation wave-
length range can be extended by a dye laser or optical parametric oscillator.
The whole system is controlled by a PC computer, and the company sells a comprehen-
sive software package to collect and analyze the data.
To build up a home-made flash-photolysis instrument one can purchase all the hard-
ware components needed for system. Photo-detectors, digital oscilloscopes, lasers and
monochromators are available form a variety of companies. Pulsed arc lamps are also pro-
duced by a few companies, e. g. Edinburgh Instruments Ltd. and Cairn Research Ltd. To
provide a correct sequence of triggering pulses for the monitoring and laser a computer
digital time module can be used. The sample holder and a base for the system assem-
bling depends on the problem in hand and available space, also optical tables and different
spectroscopy accessories can be used for these purposes. Usually all modem advanced in-
strument now are controlled by computers, and probably the most time consuming part of
the work is development of the instrument control and data analysis software.
7.4.2 Flash-photolysis study of an electron transfer: An example
Typical applications of the flash-photolysis methods are studies of photochemical reaction
such as photoinduced electron transfer. For example, an electron transfer between a por-
phyrin (donor) and fullerene (acceptor) was investigated by group of Prof. Osamu Ito at
Tohoku University, Japan [8]. The measurements were carried out using self made flash-
photolysis instrument. A scheme of the instrument is presented in Fig. 7.7. Details of the
instrument can be found in ref. [9].
The excitation pulses were generated by the optical parametric oscillator (OPO) pumped
by the third harmonic of the Nd:YAG laser. This allows to excite selectively one or another
compound and to study photochemical reaction starting from different excited states, e. g.
excited porphyrin or fullerene. The excitation pulse duration was 6 ns (FWHM) and the
pulse energy can be as high as 20 mJ. The monitoring light could be produced either by a
pulsed Xe arc lamp or continuous Xe arc lamp. The former was utilized when the highest
time resolution was required. The detection system of both pulsed and steady monitoring
sources consisted of two channels for measurements in the visible part of the spectrum and
in the infrared part. In all cases the monochromators (M1 and M2) with filters (F) were
used to select monitoring wavelength. The detectors were silicon avalanche photodiodes
and germanium avalanche photodiodes for the visible and infrared parts of the spectrum,
respectively. All together there were 4 possible positions (holders) to insert the sample,
depending on the wavelength range and desired time resolution. The modern avalanche
photodiodes have sensitivity approaching that of the photomultipliers, but allows to extend
7.4. Flash-photolysis instruments
147
Pulsed excitation source
354 nm
Monitoring light sources
Nd:YAG
GPO
PS
Steady
lamp
o
Digital 0
oscilloscope
<::::I>
I
F c:::::::t==J
I
PC
Digital
delay unit
M3
M4
c:::==:::J
Control and measurement subsystem
Figure 7.7: Laser flash photolysis instrument at Tohoku University, Japan. Nd:YAG is the
pumping Nd:YAG laser with a third harmonic generator, OPO - optical parametric oscilla-
tor, PD - photodiode, Si-APD - Si avalanche photodiode, Ge-APD - Ge avalanche photo-
diode, M1, M2, M3 and M4 - monochromators, S 1, S2, 83 and S4 - sample holders, PPS
- pulsed lamp power supply, PS - steady lamp power supply. For further explanations see
the text on the facing page. The scheme is reproduced here with kind permission of Prof.
Osamu Ito.
the wavelength range up to 1100 nm and 1800 nm for silicon and germanium detectors
respectively (see Section 4.2.3 on page 78).
The photodiode signals were averaged by a digital oscilloscope and processed by a PC
computer. The triggering signal for the oscilloscope was produced by a photodiode (PD) to
which a small portion of the excitation was directed by a semi-transparent mirror installed
after the OPO. For the correct operation of the system the pulsed Xe lamp must be started
before the laser pulse so that the laser pulse is generated at the maximum intensity of the
monitoring light. To synchronize the laser and Xe lamp a digital delay unit programed by
the computer was used.
An example of the measurements is presented in Fig. 7.8. The transient absorption
curves were measured in a wide spectrum range, 450-1200 nm. The inset shows examples
at two wavelengths. The whole range of interest cannot be covered by a single photodiode,
since the silicon photodiodes are insensitive to the light at wavelengths longer than 1100 nm,
148
Flash-photolysis
0.30
0.25
--e- 1
-e- 10
025
020
en 015
..c
« 0.10
<:1
005
000
740 nm
Q)
g 0.20
co
..c
o 0.15
(j)
..c
0.10
0.05
1080 nm (x5)
3C *
60
o 5 10 15
Time / J.1s
C 60 .- (x5)
0.00
600 800 1 000 1200
Wavelength / nm
Figure 7.8: Transient absorption spectra observe by 530 nm laser excitation of C 60
(0.1 x 10- 3 mol dm- 3 ) in the presence of CoOEP porphyrin (0.1 x 10- 3 mol dm- 3 )
in Ar-saturated benzonitrile. Inset: Time profile at 740 and 1080 nm. The figure was pub-
lished in [8] and reproduced here by permission of the Royal Society of Chemistry (RSC).
@ 2002 RSC.
and the germanium photodiodes cannot see the light at wavelengths shorter than 800 nm.
Therefore, it was crucial to use both detectors to complete the study.
The transient absorption curves were used further to calculate the transient absorption
spectra at different delay times. As an example time resolved differential absorption spectra
at 1 and 10 tLS delay time are presented in Fig. 7.8 for a solution of CoOEp porphyrin
and fullerene C 60 . The selection of the delay times was made base on the analysis of the
transient curves which indicated a photo-reaction with time constant of roughly 3 tLS. At
1 tLs delay time one can expect to see the differential spectrum of the reactant state mainly,
and at 10 tLS delay of the product state respectively, as illustrated in Example 7.4.
Example 7.4: Let us consider a simple spontaneous reaction: R P, where R is the re-
actant, P is the product and T is the reaction time constant. How one should select
the delay times to observe the differential spectra of the reactant and product sepa-
rately? The population of the reactant decays with time as [R] == e - , assuming that
at t == 0 [R] == 1. The population of product increases with time as [P] == 1 - e - * ,
so that [R] + [P] == 1 at any time. For delay time much shorter than the reaction time
constant, t « T, the populations [R] » [P], and for t » T the populations [R] « [P].
For reaction with time constant T == 3 tLS at delay time t == 1 tLS, the populations are
[R] == e- i 0.72 and [P] 0.28, thus the reactant population dominates that of
the product. At delay time t == 10 tLS the populations are [R] == e- 13° 0.36 and
[P] 0.64, which shows domination of the product population.
7.4. Flash-photolysis instruments
149
At selected delays the characteristic absorption bands can be seen at 740 nm and 1080 nm.
The band at 740 nm corresponds to the excited triplet state of the fullerene, 3C 0' which
dominates at 1 tLs delay. At 10 tLs delay a fullerene anion radical, C 60 ' band is formed at
1080 nm. These observations leaded authors to the conclusion of the photo-induced inter-
molecular electron transfer which is triggered by the excited triplet state of the fullerene
in presence of the porphyrin and results in formation of the porphyrin cation radical and
fullerene anion radical. 17
l7The excited triplet state of the fullerene is not a direct product of the photo-excitation. The excited singlet
state is the first transient state formed upon the excitation. The singlet state has lifetil11e of 1.4 ns and relaxes to the
triplet state via the inter systel11 crossing process. This reaction was not til11e resolved in this experil11ents as the
excitation pulse duration was 6 ns, but it is well-known frol11 the til11e resolved fluorescence l11easurel11ents (using
technique described in Chapter 8) and PUl11p-probe l11easurel11ents (see Chapter 11).
Chapter 8
Time correlated single photon counting
Time correlated single photon counting (TCSPC) technique is one of the most widely used
methods for time resolved emission measurements in nano- and sub-nanosecond time do-
mains. This time scale is important e. g. in the field of organic photo-chemistry since
most of the organic chromophores have the lifetime of the excited singlet state of a few
nanoseconds. The fluorescence intensity is proportional to the population of the excited
state. Therefore the measurements of the emission decay profiles is the natural method to
monitor the excited singlet state population in various photo-chemical reactions.
Another important feature of the method is its high sensitivity. This makes it an ad-
vanced tool in monitoring excited state dynamics of a very small amounts of substance. In
extreme case one can work with single molecules, wich have numeruos applications in e. g.
nanochemistry and cell biology. 1
8.1 Principles
A scheme of the method together with corresponding time diagrams is shown in Fig. 8.1.
The sample is excited by short light pulses, which are typically generated by a picosecond
mode-locked laser. 2 A small part of the excitation light is split by a glass or quartz plate
M and directed to a fast photodiode to produce triggering pulses. The sample emission is
collected by a lens and passed to a photomultiplier (PM) coupled with a monochromator
(similar to the steady state fluorescence measurements, Fig. 6.2). The photomultiplier is
working in photon counting mode, which means that each detected photon generates an
electric pulse on the photomultiplier output.
The electric pulses from photodiode and photomultiplier are directed to the constant
fraction discriminators (CFD), which use a constant fraction of the input pulse to determine
the timing of the output pulse relative to the input signal, i. e. the timing of output pulse is
insensitive to the amplitude of the input pulse. The triggering pulses from the photodiode
and the emission photon pulses (from PM) are passed to the time-to-amplitude converter
1 TiI11e resolved fluorescence spectroscopy has nUl11erous application. There are excellent books available
devoted to the subject, one can refer to e. g. [10, 11].
2Different types of excitation pulse sources are discussed later in Section 8.2.
151
152
Time correlated single photon counting
: Excitation source
Light pulse
generator
M1
Sample
I \
I \
( 1 ) laser pulse
A "start"
>
t
the first photon
A "stop" >
t
G 'z::>
\ I
\ I
------------ f------------.
(2)
Detection
sub-
system
Monochromato
p
ill
: <
> :
; c :/i :---t UrN)
, +1
>
t
(4)
MCA
U ( 11t)
n
Figure 8.1: A scheme (on the left) and time diagram of time-correlated single photon count-
ing (TCSPC) method. M 1 is a glass of quartz beam splitter, PD is a photodiode, PM is
a photomultiplier tube, CFD is a constant fraction discriminator, TAC is a time-to-analog
converter and MCA is a multichannel analyzer.
(TAC), which is the pulse controlled generator of linearly rising voltage. The pulse from
photodiode starts the generator operation (time diagrams (1) and (3) in Fig. 8.1), therefore
it is called "start" pulse. After the start pulse the TAC output voltage increases linearly with
time (time diagram (3) in Fig. 8.1). The emission photon pulse stops the generator - it is
called "stop" pulse (time diagrams (2) and (3) in Fig. 8.1). Thus the output voltage of the
generator, U, is determined by the delay time, t, between the laser pulse and the pulse
produced by the first detected emission photon, i. e. U == a t, where a is a constant.
The output voltage of the TAC is analyzed by the multichannel analyzer (MCA). The
analyzer has a memory divided into N channels and each channel is associated with the
voltage generated by TAC, so that the channel 1 corresponds to the voltage interval 0 <
U l < U, channel 2 to U < U 2 < 2 U and channel n to (n - l) U < Un < n U.
Since the voltage is proportional to the delay between start and stop pulses (U == a t),
the MCA channels refer to time intervals: the first channel is responsible for the time delay
from 0 to t l == .6.U , the second from tl == .6.U to t 2 == 2 .6.U , and so on. At the be g innin g
a a a
all channels are zeroed, e. g. C n == o. After start-stop pulse sequence MCA determines
to which channel the voltage generated by TAC corresponds to, and increases the value
stored in that channel by one (which means that there was one another photon emitted in
the delay time corresponding to this channel). Thus, when the measurements are complete,
8.2. Excitation sources
153
each channel shows the number of photons emitted in corresponding time interval, e. g.
channel n gives the number of photons in the interval (n - 1) .6. a U < t < n .6. a u . In other
words, the memory ofMCA collects emission time profile measured with discreet time step
of t == .6.U .
a
It is important to arrange the measurements in such a way that the stop pulses are always
enter TAC with at least few tens of nanosecond delay. The delay is needed because of a cer-
tain time required for the TAC to start to operate (to generate the linearly increasing voltage)
and because one needs to see the formation of the signal from the very beginning. The latter
means that in practice the TAC should begin to operate before the very first emission photon
may reach it. This is usually achieved by introducing an artificial delay between the PM and
TAC by adding a few extra meters of cable. The length of the cable depends on the physical
components layout,e. g. light propagation distances after beam splitter M 1, the TAC dead
time and desired delay for the measurements. 3
For each excitation pulse one photon is detected at maximum. To obtain an emission
decay profile the sample must be excited many times (typical numbers of excitation pulses
are 10 7 - 10 10 ). For most of the excitation pulses there are no detected photons. The reason
for this will be explained in Section 8.4.2. The total number of detected photons is typically
in range of 10 5 -10 7 . 4
An example of the fluorescence decay measurements is shown in Fig. 8.2. Based on
this measurement the lifetime of the excited state was determined, T == 2.13 =:t 0.03 ns.
The figure also illustrates a typical semi logarithmic plot for the data presentation. This
presentation makes data easier for visual analysis in case of an exponential emission decay.
8.2 Excitation sources
The TCSPC method does not require a strong excitation light source. 5 Pulse energy of 1 pJ
can provide emission intensity close to the maximum acceptable value for samples with
reasonably high emission yield. The most important requirements to the excitation sources
are the pulse width and repetition rate. The pulse width determines the time resolution and
ideally should be shorter than lOps so that it won't be the limiting part of the instrument.
High repetition rate is preferable for fast signal collection. With modem electronics (CFD,
TAC and MCA) and photo detectors the reasonable repetition rate can be as high as 10-
50 MHz. However, it should be noted that the high repetition rate can be unacceptable if the
studied sample has relatively long relaxation time. For example, the lifetime of the excited
singlet state of pyrene in ethanol is 410 ns, therefore the time interval between the excitation
pulses must be at least 2 tLs to let the excited state relax, and the repetition rate must smaller
3Typical signal propagation velocity in a cable with 50 Ohm wave resistance is 0.8 of the velocity of light.
Therefore, to introduce t = 20 ns delay the cable 111Ust be l Rj 0.8et = 4.8 111. Note that studying a nanosecond
decay one 111ay need to fit the cables with accuracy better than one 111eter.
4It should be noted that if there is no stop pulse after the start pulse in the ti111e interval of the 111easure111ents,
the MCA skips the event.
5The excitation pulse energy esti111ation will be given in Section 8.4.3 and a typical values can be found in
EXa111ple 8.2.
154
Time correlated single photon counting
10000
1000
VJ
.......
;::j
0 100
u
10
. ;
:,:.: .'-
r ,.
r.: '.
. . .-: ;.
.n_ - ,-: f}\i t
1
o
2
4
8
10
12
6
time, TIS
-: [ ,
o
2
4
6
8
10
12
Figure 8.2: Fluorescence decay of a zinc porphyrin solution (on top). The measured decay
is shown by the dots, the instrument response by the dotted line and the result of the mono-
exponential fit by the solid line. The decay lifetime is T == 2.13 :::I::: 0.03 ns and the fit
weighted mean square deviation is X 2 == 1.27. The plot on the bottom shows fit residuals:
weighted difference between the measured data and model decay.
than 0.5 MHz. 6
Considering this non-demanding requirements on the pulse energy, nanosecond flash
lamps were usual excitation source in TCSPC instruments some time ago. The lamps are
filled with hydrogen or nitrogen at relatively low pressure. The lamp electrodes and the
pulsed power supplies are optimized to achieve as short light pulse as possible. Typical
pulse duration for such lamps is 1 ns and the pulse repetition rate can be up to 100 kHz. The
lamps have broad emission spectra therefore monochromators are usually used to select the
excitation wavelength.
With invention of mode-locked picosecond lasers the flash lamps were replaced by the
lasers to benefit in time resolution, sensitivity and data acquisition time. Typical arrange-
ment of the excitation source is a dye laser synchronously pumped by Nd:YAG of Ar ion
mode-locked laser (see Section 3.6.4). Depending of the dye solution used the dye lasers
can generate pulses in the range of 500-700 nm. The range can be extended by using a
second harmonic generator after the dye laser to 260-350 nm. 7
Typical pulse repetition rate of mode-locked lasers is around 100 MHz. This is too high
rate for most applications and electronics (TAC and MCA). To reduce the frequency the
6 At 2 J1S after the excitation e - 4;b s Rj 0.008 = 0.8% of initially excited pyrene 11101ecules will re111ain
excited, which is too high value if desired 111easure111ent accuracy is 1 %. One 111ay want to increase the ti111e interval
between the pulses even further l11ore.
7It should be noted that the ranges 500-700 and 260-350 n111, respectively, are covered by a few different dyes.
Therefore, tuning fr0111 e. g. 570 n111 to 650 n111 can be a c0111plicated process, which 111ay include readjustI11ent of
the laser syste111.
8.2. Excitation sources
155
dye lasers are equipped with cavity dumpers in place of output coupler. The cavity dumper
works like 100°;6 reflecting mirror most of time, but at certain moment it opens the cavity
and lets the laser pulse to leave the cavity. Usually this is achieved by placing an acousto-
optic modulator inside the cavity. The principal part of the modulator is a small block of
quartz of similar light transparent material, which is supplied by a short high frequency
acoustic pulses creating a diffraction grating in it. When the acoustic frequency is applied
the light beam changes the propagation direction, due to diffraction and leaves the laser
cavity. The acoustic pulse is synchronized with optical pulse traveling inside the cavity in
a such way that only one optical pulse leaves the laser after being reflected e. g. 100 times.
U sing this method the repetition rate can be reduced to any desired value, typically to 1-
4 MHz. Another advantage of cavity dumping is that when the dumper is closed there are
no losses of the pulse energy traveling inside the resonator, and the pulse energy increases
each time the pulse crosses the active medium. In a sense, the pumping energy is not lost
but accumulated while the dumper is closed, and when the dumper is open the accumulated
energy leaves the laser. Therefore at lower repetition rate the output pulse energy is higher.
Typical pulse energy of synchronously pumped dye lasers is a few nano Joules, which is
high enough to measure samples with very weak absorption and/or very low quantum yield
of emission.
Two relatively new additions to the excitation sources for TCSPC are mode-locked
Ti:sapphire and semiconductor lasers. The Ti:sapphire lasers are widely used to gener-
ate short and ultrashort light pulses. The excitation wavelength can be tuned in the range
of 760-1050 nm or 380-520 nm after the second harmonic generator. The pulse energy is
few nano Joule at fundamental harmonic and close to nano Joule at the second harmonic.
Additionally one can use the third harmonic generator to obtain excitation pulses in the UV,
260-340 nm. To reduce the repetition rate (normally around 100 MHz) an acousto-optic
pulse picker can be used. 8
The pulsed semiconductor laser diodes were available for a long time already but their
applications were limited by relatively long pulse durations and the red-near infrared emis-
sion range. Semiconductor light emitting devices which can be used as excitation sources in
TCSPC applications were developed only recently. As an example one can consider LDH
series of laser diode heads (PicoQuant GmbH). The diodes emit light at fixed wavelength in
UV and visible part of the spectrum, e. g. at 375,405,440 and 630 nm. The diodes generate
70-100 ps pulses at 40 MHz repetition rate and provide average power of 0.3-1 mW, i. e.
the pulse energy is roughly 10 pJ (i. e. 10- 11 J).
As a conclusion, for high speed and high sensitivity applications the mode-locked lasers
are usually used as excitation sources. These are Ti:sapphire lasers and dye - Nd:YAG or
Ar laser systems. For less demanding applications emitting diodes can be considered. If
nanosecond time resolution is appropriate and the samples have relatively high emission
efficiency a flash lamp coupled with monochromator can be a reasonable solution.
8The efficiency of Ti:sapphire lasers is 111uch higher than that of the dye lasers. Therefore there is no need to
accu111ulate pulse energy inside the cavity using cavity dU111pers. The pulse pickers are installed outside the laser
and they are easier to use than cavity dU111pers.
156
Time correlated single photon counting
Table 8.1: Comparison of main TCSPC characteristics of a micro-channel plate photo-
multiplier tube (R3809U-50, Hamamatsu Photonics K. K.) and an avalanche photodiode
(PDM-50, Micro Photon Devices, Italy).
TCSPC time resolution
Dark counts, non-cooled
Dark counts, cooled
Size of photo sensor
Spectrum range
Maximum quantum efficiency
R3809U-50
25 ps
1 00 s - 1
5 S-l
1cm
160-850 nm
20%
PDM-50
250 ps
1500 S-l
75 S-l
50 tL
350-950 nm
47%
8.3 Detection subsystem
8.3.1 Emission detectors
For the best time resolution micro-channel plate photomultiplier tubes (MCP PMT ) are
used to detect the emission. 9 Depending on the type of photo-cathode they can provide the
detection wavelength range from 200 nm to 900 nm without additional cooling, and up to
1000 nm if a special photomultiplier with cooled photo-cathode is used. The time resolution
of the MCP PMT devices can be as high as 25 ps in TCSPC mode (e. g. R3809U series
from Hamamatsu Photonics K. K.). The time resolution of the ordinary PMT designed for
TCSPC applications is typically 1-0.5 ps, also the price is much lower than that of micro-
channel plate devices.
Nowadays avalanche photodiode (APD) assemblies are available for TCSPC applica-
tions. The detection wavelength range can be optimized for the visible or near infrared
wavelengths thus covering 300-1100 nm range. The time resolution of the diodes is a lit-
tle lower than that for MCP PMTs, being typically> 100 ps. Two main disadvantages of
APDs as compared to PMTs are a small area of the photo-sensor and a higher value of dark
counts. A comparison of the most important features of MCP PMT and APD for TCSPC
applications is given in Table 8.1 by the example of two typical devices. Also one have to
notice that MCP PMT are more expensive solutions.
To obtain "start" pulses any fast photodiode can be used. Although the time resolution
is not important in itself, the transient time spread is smaller for high speed photodiodes. If
an emitting pulsed laser diode is used for excitation, its power supply may already have a
"start" pulse output, which can be connected directly to the CFD.
8.3.2 Electronics
The constant fraction discriminators (CFD), time-to-amplitude converters (TAC) and multi-
channel analyzers (MCA) are available in different combinations and from different manu-
factures. The cheapest solution can be a computer board which combines all the electronic
9S ee Section 4.2.2 on page 74 for the description of the operation principles ofMCP PMT.
8.4. Method characteristics
157
components needed for TCSPC measurements. An example of such board is TimeHarp 200
(PicoQuant GmbH). It has timing resolution better than 40 ps, provides 4096 channels for
decay collection and has sustained data throughput up to 2 x 10 6 counts per second. The
board is supported by a software to control the hardware, e. g. to set the discrimination
level, and to conduct the measurements.
A more advanced electronic modules for TCSPC measurements are produced as a com-
plete devices or as separate parts which can be interconnected to form a complete instru-
ment. For example, PicoHarp 300 stand alone TCSPC module (PicoQuant GmbH) has
electrical time resolution <10 ps, maximum counting rate 10 7 S-l, minimum channel width
4 ps and maximum number of channels 65536. At present these characteristics are better
than the best available from photon detectors, so the electronics was not a limiting part in
the time resolution or the method sensitivity.
8.4 Method characteristics
8.4.1 Time resolution
In most cases the time resolution of the method is determined by the photo detector, e. g.
photomultiplier. However, unlike in the direct transient measurements, the width of the
response to a short pulse is not the limiting factor. The time resolution is determined by the
transient time spread, which tells how much the timing of the response pulse varies from
one pulse to another. This characteristic of photomultipliers and other photo-detectors is
typically much shorter than the width of the response function. For example, micro-channel
plate photomultiplier tube model R3809U-50 (Hamamatsu) has the rise time 177 ps, fall
time 410 ps and the pulse width 270 ps (FWHM), whereas transient time spread of the
device is 25 ps only. Using this detector one can build up a TCSPC system with instrument
response as short as 40 ps (FWHM). Thus, utilization of the TCSPC technique allow to
improve the time resolution by almost one order of magnitude as compared to traditional
(analog) transient measurements.
Naturally, for the best time resolution the width of the sample excitation pulses must be
as short possible. Considering the best transient time spread of the photo detectors, a pulse
width of < lOps is short enough to have no effect on the final time resolution of the system.
In practice this means that the mode-locked lasers must be used to achieve the best time
resolution. However, modem semiconductor diode lasers can provide light pulses as short
as 50 ps, which is very close to the time resolution limit put by the best MCP PMTs.
8.4.2 Peal-up distortions
The TCSPC method deals with single photons. It is important that during the detection time
interval not more than one pulse is generated by the photo detector, since all pulses but the
first will be lost. The lost pulses will result in nonlinear signal distortions called peal-up
distortions. Therefore, the emission intensity must be low enough to keep the probability of
two pulses during the measurement time window negligible. This means that the probability
of the single pulse detection cannot be very high either, i. e. for the most of "start" pulses
there is no corresponding "stop" pulse at all.
158
Time correlated single photon counting
To estimate the effect of peal-up distortions we can use the probability theory and,
namely, Poisson distribution, i. e. eq. (4.11 ).10 Suppose the average number of photons
per excitation pulse detected by an ideal device which has no peal-up distortions is n. 11
Then the probability that no photons are detected after single excitation pulse is Po == e- n ,
2
the probability of one photon detection is PI == ne- n , of two photons is P 2 == e- n , and
so on. Let us neglect the probabilities to obtain more than two photons, then the number
of detected photons by a real device with peal-up distortions is PI + P 2 , since two photons
will be counted as one, while the actual number of photons is PI + 2P 2 . The difference
between the counted and actual number of photons is P 2 and the relative mistake in photon
counting is
c5 == P 2
PI + 2P 2
n
2
l+n
n
2
(8.1 )
t""V _
since n « 1, so 1 + n t""V 1. It is usually accepted that the photon counting rate should be
less than 2% of the excitation rate. Then the peal-up distortions are less than 1 %. This puts
some limits on how fast one can collect the signal using TCSPC method, as illustrated in
Example 8.1.
Example 8.1: Estimation of the maximum counting rate. If the excitation pulse repetition
rate is fex == 1 MHz, and the desired accuracy of measurements is 1 % or c5 == 0.01,
then the probability to detect a photon for a single pulse should be less than n ==
0.01 x 2 == 0.02. Thus the photon counting rate fpM < nfex == 20 000 counts/so
From the point of view of quantum noise statistics the measurement accuracy of 1 %
is achieved when (0.01) -2 == 10000 photons are counted. Therefore, the photon or
Poisson noise will dominate while the signal intensity is lower than 10000 counts.
At higher signal amplitudes the peal-up distortion may have stronger effect on the
measured data than the quantum noise. In other words, if the signal is intended to be
collected to high intensity, i. e. higher accuracy, the counting rate must be lower to
prevent the effect of peal-up distortions.
8.4.3 Sensitivity
Sensitivity of the TCSPC method can be estimated using an approach similar to one used
for the steady state fluorescence measurement in Section 6.2.7, since the photon counting
method is used in both cases. The main difference comes from the fact that the counts are
distributed over a few channels (giving a time profile of the emission). If the time constant
of the emission decay is TO and the time per channel is t, then the probability that the
detected photon will fall in to the channel of maximum intensity is 1]t t""V .6.t . 12 The counts
TO
lOSee also Appendix A for l110re detailed discussion.
11 For the sake of shortness and si111plicity the we will aSSU111e here an ideal detector with 100% efficiency, so
that the nU111bers of photons and electric pulses on the detector output are equal to each other. The final result can
be directly applied to real detectors by replacing the photons by electric pulses on the detector output.
12This is true if t < TO and TO is greater than the ti111e resolution of the instru111ent.
8.4. Method characteristics
159
at the channel of maximum intensity will grow 1]t times slower than the total counting rate.
The ratio 1]t .6.t can be considered as an additional loss term to be added to eq. (6.11),
TO
then
a A;rn t
N ch == 1]c1]rn1]PM l aN ex -cjJcp
i TO
(8.2)
where N ch is the number of counts in the channel of maximum intensity, N ex == x is the
number of excitation photons and the wavelength dependence is not shown for shortness.
Thus, in addition to the factors discusses in Section 6.2.7, the sensitivity of TCSPC method
depends on the time per channel, t, - the shorter time the lower counting rate.
One may also conclude that the counting rate depends on the lifetime of the excited
state TO. This is not quite correct. As an example let us consider fluorescence decay mea-
surements of some molecule. If the molecule has the fluorescence emission rate kern and
the lifetime of the excited state is TO, then the fluorescence quantum yield is cjJ == kernTO.13
Thus, it == kern, and eq. (8.2) can be rewritten as
TO
a A;rn A
N ch == 1]c1]rn1]PM li aNexkerntitcp
(8.3)
The emission rate constant, kern, is the fundamental property of the molecule, which does
not change while the molecule remains unchanged. For instance, if an excited state is
quenched by some photo-chemical reaction, e. g. by an energy or electron transfer reaction,
the lifetime of the excited state, TO, and the quantum yield of the emission, cjJ, will change,
but the emission rate of the excited state, kern, will remain the same. This means that the
signal (counting rate) right after the excitation will remain the same although the lifetime
will be shorter, i. e. the signal at longer delays will be weaker. Therefore, cjJ and TO are not
independent values but their ratio is a constant, it == kern. The counting rate at maximum
TO
is proportional to the radiative constant, as indicated by eq. (8.3), rather than to the decay
rate of the excited state, TO- 1 .
One another parameter related to the question of the method sensitivity is the excitation
intensity needed to collect the data in a reasonable time. If the total counting rate is nc (in
all channels), one can use eq. (6.12) after replacing the counts by counting rate 14
nc == 1]i1] s n ex
(8.4)
where nex is the average flux of the excitation photons. In the channel of the maximum in-
tensity the counting rate is smaller by factor 1]t == .6.t , and the counting rate at the maximum
TO
channel is
t
nch == -n c == 1]t1]i1] s n ex
TO
(8.5)
BIn addition to radiative decay (kern) there are usually non-radiative decay channels. If the relaxation rate via
non-radiative channels is k nr , then the total relaxation rate of the excited state is ko = k nr + kern = (TO) -1 .
The fluorescence quantu111 yield is cjJ = k k ern = k k.+k' = kern TO.
o nr ern
l4Pormally both parts of eq. (6.12) can be divided by an observation ti111e thus giving counting rate instead of
nU111ber of counts in both sides of the equation.
160
Time correlated single photon counting
Although formally 1]t can be included in to instrument losses 1]i, eqs. (8.4) and (8.5)
serve to different purposes. Equation (8.4) can be used to estimate the average excitation
power and pulse energy required to conduct the experiments, as illustrated in Example 8.2
and eq. (8.5) is useful for signal collection time estimation, as discussed in the following
Section and Example 8.3.
Example 8.2: Estimation of excitation intensity. Assuming the same measurement condi-
tions as were considered previously in Examples 6.2 and 6.3 (i. e. 1]i1]s == 10- 8 ) and
desired photon counting rate to be nc == 10 4 S -1 , the required excitation photon flow
is nex == == 10 12 S-l. At the wavelength 400 nm this means the average power
T] T]s
of 5 x 10- 7 W = 0.5 tLW. At pulse repetition rate of 1 MHz this requires pulse energy
of 5 x 10- 13 J (or 10 6 photons per pulse). This is easy available from any pulsed
laser. In other words, the method does not require a powerful excitation source.
Example 8.2 shows that TCSPC method does not require high energy of the excitation
pulses. Naturally, one can measure a sample with a very low absorption or low emission
efficiency. This will require a higher excitation pulse energy. However, a typical pulse
energy of a mode-locked laser (see Section 8.2) is 1 nJ, which is 10 4 times greater than the
energy estimated in Example 8.2, showing that the actual limit is not in the excitation source
when picosecond laser systems are used.
8.4.4 Signal collection time
The signal collection time depends on the counting rate at the channel of maximum signal
intensity and on desired accuracy of the measurements. Example 8.3 gives an estimation of
a collection time for a typical photochemical experiment of measuring fluorescence decay
of an organic dye. In the experiments of this kind the limiting factor is the maximum
acceptable counting rate at which the peal-up distortions can be neglected, i. e. this is the
case of strong sample emission.
Example 8.3: Estimation of signal collection time. Let us assume the following experi-
mental conditions: the excitation pulse repetition rate is f == 1 MHz, the average
counting rate is nc == 4000 s -1 , the emission lifetime is T == 2 ns, the time resolu-
tion (time per channel) is t == 25 ps and the desired signal intensity at maximum is
N rnax == 10 000 counts, i. e. the measurement inaccuracy is 1 %. Then, the relative
number of counts in time window of t at the channel of maximum intensity is t ,
thus the counting rate at this channel is n rnax == nc t , and the time required to
collect N rnax counts is t == N rnax == Nrnl{T == 200 s or 3 m 20 s.
nc nc t
From the point of view of the TCSPC instruments, the counting rate depends on excita-
tion source and the experimental conditions, the most important being:
1. excitation pulse energy, higher energy gives higher rate;
8.4. Method characteristics
161
2. excitation repetition rate, the counting rate is directly proportional to the repetition
rate at fixed pulse energy;
3. time per channel (time resolution), the rate is inversely proportional to the time per
channel, as given by eq. (8.5).15
A usual method for adjusting the counting rate is to change the excitation intensity, which
can be done by placing a variable gray filter between the excitation source and the sample. 16
If the sample has a very low emission intensity these three instruments settings can be
tuned in to achieve the best result. Also one has to notice that from the statistical point
of view the time per channel value does not affect the data reliability, which is mostly
determined by the total number of the collected photons. However, at low signal intensity
by increasing the time per channel, and thus compromising the time resolution, one can
improve visualization of the measurements.
8.4.5 Spectrum range
The question of spectrum range can be applied to both excitation and emission parts of
the method. In the emission part the limiting component is the detector. Micro-channel
plate photomultiplier tubes, that were considered previously, may be used in the spectrum
range from 200 nm to 900 nm without additional cooling of the photo-cathode, and up
to 1000 nm if a special photomultiplier with cooled photo-cathode is used. With silicon
avalanche photodiode detectors the wavelength range can be extended to 1100 nm (with
sensitivity maximum at 800 nm), but the time resolution of these devices are at the limit of
0.5 ns (in TCSPC mode). At longer wavelength, the photon energy decreases making dis-
crimination between photo-electrons and thermal electrons more difficult. That is why the
cooling is required for the red and infrared sensitive devices especially for photon counting
operational mode.
The excitation wavelength range is totally determined by the sources of short (preferably
picosecond) light pulses. For lasers sources the blue and UV wavelengths are technolog-
ically more difficult. Since high pulse energy is not required for TCSPC application, the
second and third harmonic generators are used to extend the excitation wavelength to the
blue and UV ranges.
8.4.6 Comparison with direct emission decay measurements
Alternatively to TCSPC method one can excite a sample with single short light pulse and
measure directly the time profile of the emission produced as was described in Section 7.3.2.
What are than advantages of the TCSPC method over direct emission decay measurements?
l5Increasing the ti111e per channel one can reach the li111it when the e111ission lifeti111e is equal to the ti111e per
channel. Further in ti111e increase will not increase counting rate. This is, however, very unpractical experi111ental
condition.
16The variable gray filter can be a quartz (or glass) plate of rectangle or round shape with absorbing layer
deposited on it in a such way that the trans111ission of the plate changes sl1100thly along long axis or around the
plate, respectively.
162
Time correlated single photon counting
First of all the TCSPC method has superior time resolution as was discussed in Section
8.4.1.
Secondly, a very low emission intensity is sufficient for TCSPC method to obtain a
very accurate emission decay profile. This has at least two important consequences. The
experiments can be carried out using short but weak excitation pulses, which extends greatly
the choice of light sources as compared to direct emission decay measurements. What is
more important, the TCSPC method can be used even when the sample cannot emit many
photons after single excitation pulse. For instance, the sample can be a single molecule so
that there will be no more than one photon emitted per each excitation pulse. Still, using
the TCSPC method one can measure the decay profile, which is impossible by any other
methods so far developed. An example of the single molecule application is discussed in
Section 8.7.
Another advantage of the TCSPC method is its high accuracy due to photon counting
technique. Routinely the lifetime of an excited state can be determined with 1 % accuracy
or better, as illustrated in Fig. 8.2.
From the technical point of view, to study sub nanosecond emission decays both TCSPC
and direct method require sub nanosecond pulsed excitation source, e. g. < 1 00 ps. However
for direct measurements the pulse energy must be a few milli Joules but the repetition rate
can be a few Hertz or even single shots. Whereas for TCSPC the pulse energy is can be in
order of magnitude of a few pico Joules, but a mega Hertz repetition frequency is needed.
The lasers with low pulse energies but high repetition rate are cheaper and easer to operate
and maintain than milli Joule pulse lasers with low repetition rate. 17 Also under the same
conditions otherwise the exposition to the excitation must be the same for both methods to
achieve the same results.
8.5 Measurements and data analysis
8.5.1 Instrument response function and decay deconvolution
Similar to the steady state fluorescence measurements TCSPC method is known for its high
accuracy, which is based on good linear response of the instrument and high signal-to-noise
ratio. Additionally, one can easily measure temporal response of the instrument by using
any scattering sample and adjusting emission monochromator to the excitation wavelength.
The instrument response can be used to increase the accuracy of data analysis further and
to obtain quantitatively accurate information on the emission lifetime even for samples with
the lifetime approaching the time resolution of the instrument.
If the measured instrument response function is r(t) and the sample response to a delta-
pulse excitation (theoretical one) is f (t), then the experimentally measured signal is given
by the convolution integral
t
s(t) = J r(T)f(t - T)dT
(8.6)
-ex)
l7Por picosecond 111illi Joule pulse generation a typical approach is to start with high frequency low energy
pulses, pick up pulses at lower frequency and aI11plify the111 to 111illi Joule energies. See Section 3.4.
8.5. Measurements and data analysis
163
which can be used for accurate data fitting.
For example, if the sample response is a sum of ex ponentia Is, f(t) == Eaie-k t, then
convolution integral is
t n
F(t) J r(T) Laie-k,(t-T)dT =
-ex) i=l
t
n
L ai J r(T)e-k,(t-T)dT
i=l
-ex)
(8.7)
or the sum of convolution integrals of the instrument response and exponents. The function
F( t) can be calculated for a given set of the decay rates, k l . . . k n , and amplitudes, al . . . an,
and used to fit the measured data s (t), thus taking into account real instrument resolution
and increasing the accuracy of the short lifetime estimations.
An example of the fit which accounts for the instrument response is presented in Fig. 8.2 on
page 154. The measured sample shows mono-exponential decay of the emission, i. e. of
the singlet excited state, I(t) == Ioe- , where T is the lifetime of the excited state. The
figure shows also the instrument response ( dotted line), which was used to fit the data. The
calculated lifetime is T == 2.13 =::i= 0.03 ns. The plot on the bottom presents the weighted
deviation of the data from the fit. The weighting factor is square root of counts. 18
8.5.2 Time resolved and decay associated spectra
If a few emitting species are formed under the sample excitation, the emission decay profile
may look different at different wavelength. The spectrum analysis can be of great help in
order to identify the species. A natural way to obtain the spectral information with TCSPC
method is to repeat the measurements at a few wavelengths sequentially. To be able to
compare signal intensities at different wavelengths one must collect the decay curves during
the same time interval at each wavelength. Additionally a correction of sensitivity must be
acquired similarly to the steady state emission spectra measurements discussed in Section
6.2.4. After that the intensities at certain delay time can be used to draw the time resolved
emission decay spectra.
As a simple illustration let us consider a hypothetic mixture of two emitting species
with partially overlapping spectra but with different emission lifetimes. The first emitting
state, state A in Fig. 8.3, has emission maximum at 480 nm and lifetime of 1 ns, and the
second, state B, has maximum at 520 nm and lifetime of 5 ns. The bandwidths are 85 nm
for both spectra and the emission rate constant of state A is two times higher than that of B.
For this sample the steady state emission spectrum (shown by the dotted line in Fig. 8.3a)
will reveal no evidence of the mixture of two emitting species since it has only one band.
The emission decays are show in Fig. 8.3b at a few wavelengths. As expected, at 440 nm
the 1 ns decay component is dominating, whereas at 560 nm the decay is much slower and
mainly determined by the 5 ns decay component.
18 At a reasonably large nU111ber of counts the square root of counts gives an esti111ation for the standard devia-
tion, see Chapter 4.1.2 for l110re information.
164
Time correlated single photon counting
(a) 140
120
100
80
60
40
20
o; -: :-. . .
450 500 550
wavelength, nm
600
(b)
100
10
.,
"".
".
"'.
.<.:. < .. .....
............."................... ..............
-............. .......
-............. ......
-............. ......
........ ......
----- .......
----
- 440 nm
-- 480 nm
.... 520 nm
.-. 560 nm
1
o
5
10
15
time, ns
Figure 8.3 : a) Simulation of the time resolved (dashed lines, delay times are indicated on
the plot), decay associated spectra (solid lines, state A and B) and a steady state emission
spectrum (dotted line); b) emission decay curves at different wavelengths.
Repeating the measurements at large enough number of wavelengths one can then col-
lect signal intensities right after the excitation pulse, i. e. at t == 0, and plot them as function
of the wavelength, which will yield the emission time resolved spectrum at t == 0 ns after
the excitation. This is the strongest dashed spectrum in Fig. 8.3a. Similarly, the intensities
at 1 ns after the excitation can be collected to draw the time resolved spectrum at 1 ns delay
time. The last time resolved spectrum was collected for 5 ns delay time. The spectra of
states A and B were selected in such way that at any time the emission band looks like a
single band, giving no hints pointing to two actual underlying spectra. However, the three
time resolved spectra in Fig. 8.3a differ from each other not only by the intensities, but
also by positions of the maxima, which shifts to the red with delay time. This is typical
indication that the emission of the sample comes from a few different states.
The actual spectra of the two emitting states can be obtained by fitting the data to bi-
exponential model
t t
I(t, A) == A(A)e - TA + B(A)e - TB
(8.8)
where T A and TB are the lifetime of the states A and B, respectively, and A(A) and B(A)
are the emission spectra of the states A and B assuming that the excited sample consists
of two independent emitting states. In a general case, when such assumption cannot be
made, A(A) and B(A) are called pre-exponential factors, or more specifically in analysis of
the time resolved emission spectra, they are called decay associated spectra (DAS). If the
8.6. Commercial instruments
165
photo-reaction scheme includes a few intermediate states and complex relations between
them, e. g. back reactions and equilibria, interpretation of the decay associated spectra can
be rather complex task. This will be discussed briefly in Section 15.3.
Independent of the interpretation of the decay associated spectra, the steady state emis-
sion spectrum of the sample can be obtained by integrating eq. (8.8)19
ex)
lss(A) = J l(t, A)dt = TAA(A) + TBB(A)
o
(8.9)
It should be noted that the steady state spectrum of only A is T AA(A) and only B is TBB(A),
so that steady state emission spectrum of A is 2.5 times lower in intensity than that of B for
the considered example. This is why the steady state spectrum of the sample in Fig. 8.3a
(dotted line) is much closer to the spectrum of B. Indeed, the difference in shapes between
the steady state and time resolved spectra is the evidence of a complex nature of the excited
state.
Some TCSPC instruments provide a direct method to measure time resolved spectra.
This is done by counting only the photons which falls in to a certain time window and store
the number in one MCA channel. The counts are collected during a fixed time interval,
after which the monitoring wavelength is changed and the measurements are repeated by
collecting the counts to the next MCA channel. In the end of the experiments the data in
MCA channels present the time resolved spectrum.
8.6 Commercial instruments
Three principal components of any TCSPC instrument are the source of the excitation light,
the photo detector and the electronics - CFD, TAC and MCA. The design goal of the most
commercially available instrument is to provide a choice of excitation sources and detectors.
This is reasonable as the excitation laser can be the most expensive part for the instruments
with picosecond time resolution, and, thus, has to be optimized for particular applications. 2o
Similarly the choice of the detector depends on the desired emission range and time reso-
lution. For the best time resolution the micro channel plate photomultipliers are used, that
alone have price higher than 10 000 euro.
A typical solution provided by a number of companies is a base system and a set of
modules for excitation and exchangeable photo detectors. For example FL920 spectrometer
(Edinburgh Instruments Ltd.) can be equipped with a flash lamp, or pulsed laser diode, or
an external pulsed laser can be used to excite the sample. Additionally the spectrometer
can be used to measure steady state emission spectra, or to collect automatically decays at
a series of wavelengths, which are used then to present the time resolved emission spectra.
19The integration can be easily expended to any nU111ber of exponents, I ss (A) = L: T a (A), where T and
a ( A) are the lifeti111es and the corresponding decay associated spectra.
20 At present, se111iconductor pulsed laser heads are available for the excitation in the red part of the spectru111
with the pulse width close to 50 ps and price of few thousands of euros. However, for the devices operating in
the blue part of the spectru111 the prices were a few ti111es higher. In cases when the excitation pulse energy 111Ust
be higher than a few pJ a l110re expensive laser syste111111ust be used, which increases the price by l110re than one
order of 111agnitude.
166
Time correlated single photon counting
From a researcher point of view a complete TCSPC instrument can be assembled from
components supplied by different manufacturers. This is probably the most flexible way to
build up a highly optimized instrument for particular applications. Also this task requires
a good understanding of the TCSPC method principles from the researcher, but it does not
require deep knowledge in optics or electronics, and can be accomplished by a person who
has some experience in using TCSPC instruments.
8.7 Measurements of single molecule: Application example
In Section 8.4.3 the conclusion has been made that the time correlated single photon count-
ing method is a very sensitive one. What does it means from the practical point of view?
Can we measure an excited state lifetime of a single molecule, for example? To answer that
question let us make an estimation of excitation and signal intensity of a single molecule.
Apparently, the losses estimation as calculated in Examples 6.2 and 6.3 (i. e. 1]i 1]8 ==
10- 8 ) are to high to expect any detectable emission from a single molecule. Assuming
that the molecule is excited by each excitation pulse, and the pulse repetition rate is f ==
10 MHz,21 the total counting rate is n == f 1]i 1]8 == 0.1 S -1, which is much lower than the
dark counting rate of any photomultiplier. However, the system can be optimized for the
lifetime measurements to reach the sensitivity required for single molecule detection.
First of all, the monitoring monochromator can be replaced by a notch filter. There will
be no spectrum selectivity and all the emitted photons will be detected independent of the
wavelength. This measure will reduce the instrument losses (1]i) by more than one order of
magnitude. Second, one molecule is a very small object and we can use an objective lens
with high numeric aperture to collect much more emission from the sample. 22 The third
point to remember is that in eq. (8.2) the sample absorbance used to estimate excitation
efficiency. Absorbance is the characteristic of bulk materials and cannot be applied to a
single molecule. Formally, it has to be replaced by the cross-section and the excitation
intensity should be replaced by power density for counting rate estimations. Then, for the
single molecule experiments molecules with high emission efficiency are selected, where as
in Example 6.3 the emission yield was taken as moderate value. After these improvements
and optimization one can expect to reduce the losses by three orders of magnitude at least
and achieve more than 100 counts per second from a single molecule.
The actual instruments for time resolved single molecule studies are highly optimized
top level TCSPC systems combined with high resolution optical microscopes. One of
such instruments is installed in Division of Photochemistry and Spectroscopy at Depart-
ment of Chemistry, Katholieke Universiteit Leuven, Belgium. A scheme of the instrument
is presented in Fig. 8.4, and as an example of the method application the results pub-
lished in ref. [12] will be used here. the samples for the study were hexaphenylbenzone-
perylenemonoimide imbedded in a polyvinylbutyral polymer films of 10-30 nm thickness.
21 A higher repetition rate is hardly reasonable since (1) it is already close to the dead ti111e of TAC-MCA
detection part, and (2) it 111USt be at least few ti111es lower than the relaxation rate of the 11101ecule, which is 2-10 ns
for 1110St organic dyes.
22This is not a practical 111easure when e111itting area of the saI11ple has a 111illi111eter size, as the output beaI11 in
this case will have either too big diaI11eter or divergence.
8.7. Measurements of single molecule: Application example
167
Setup
.
IM ntI €i
Figure 8.4: Spectroscopic setup used for the TCSPC measurements of single-molecule flu-
orescence decays based on a sample scanning, far-field, confocal epifluorescence micro-
scope. The figure was published in ref. [12], kindly provided by Prof. 1. Hofl<ens, and
reproduced here by permission of The American Chemical Society (ACS). @ 2001 ACS.
To prepare such films the dye concentration was 10- 9 M. The very thin films and very low
concentration of the dye are needed to resolve single molecules in the sample.
A cavity dumped pulsed dye laser or the second harmonic of a mode-looked Ti:sapphire
laser (presented in the figure) were used to excite samples. In case of dye laser the repetition
rate was 4.1 MHz, the pulse width was 20 ps pulses and the average power at the sample
was roughly 600 n W, which gave roughly 0.1 pJ pulse energy. To deliver excitation pulses
to the sample and to collect the emission from the sample an oil immersion objective lens
with numerical aperture 1.4 was used. The excitation and emission were separated from
each other by a dichroic mirror (DCM). A pinhole was used for spatial signal filtration and
a notch filter to reject the excitation wavelength.
The first step in the experiment acquisition is to scan the sample and to find a suitable
object (single molecule) for the investigation. In spite of the ITIolecules have been imbedded
in the rigid matrix, they changed their behavior time by time in a few minutes intervals.
Therefore the measurements were acquired in 10 s time intervals, verifying each time that
the emission properties did not change during the measurements. During 10 s up to 6000
counts could be collected providing maximum counting rate of 600 s -1. For the lifetime
analysis 71 decays were collected with total number of counts in range 2 500-60 000, which
168
Time correlated single photon counting
corresponded to 60-1300 counts in the maximum. An example of the decays is presented
in Fig. 8.5. The data were collected in 193 channels covering time widow of 27 ns, i. e.
0.14 ns per channel.
The decays fitted well in to mono-exponential model, yielding average lifetime about
4.5 ns. However a clear difference in lifetime of different measurements can be noted, e. g.
4.29 and 4.73 ns for the data presented in Fig. 8.5, which is due to the differences in the
local environments of the individual molecules.
This example shows that the time resolved single molecule spectroscopy can provide
quantitative information about individual molecules of chromophore. This makes emission
spectroscopy in general and TCSPC technique in particular an important tool in the filed of
nanochemistry.
The high sensitivity of the TCSPC method has many different application. For example
the fluorescence labels can be used to determine the distance between specific groups of
a protein molecule. The rate of energy transfer or fluoresnce quenching rate are used to
calculate the distance between labeled sites. For this type of applications the sensitivity
and high accuracy of the emission lifetime measurements are the principal feature of the
method.
8.7. Measurements of single molecule: Application example
169
4
a: 0
-4
" (a )
I I
100 I : 00
/ \ III -0 4
en : I 0 5 10 15 20
- 4
: Ilma/ns
r::::
::J
0
()
1 0 L
I
: 't = 4.73 ns
LS
X2
= 0.89
r
\
N =1601
9
1
0 5 1 0 1 5 20 25
Tim e / n s
4
a:
o
-4
100
en
-
r::::
::J
0 :
() :
1 0 :
\
\
I
\
:
1
0
" ] (C)
,,
-0 4
10
15
20
II m a Ins
. III .
c .. · ,.: "'.....
. " .
. .
. .
.. ...
. -.
. .
't = 4.29 n s . _ ..1 _ .
LS
X 2 = 0.85
r
N = 644
9
. . It I
5
1 0
1 5
20
25
Time / ns
Figure 8.5: Examples of single molecule fluorescence decays. Reproduced from Fig. 1a
and 1c of ref. [12] by permission of The American Chemical Society (ACS). @ 2001 ACS.
Chapter 9
Frequency domain emission
spectroscopy
The methods discussed in the previous Chapters utilize a short light pulse for the sample
excitation. Should the excitation be necessarily a pulse? According to the signal process-
ing theory the response of a system can be equivalently presented in time and frequency
domains. The discussed methods are examples of the time domain measurements. In the
frequency domain approach the response of the sample to a wave-type excitation is exam-
ined in a certain frequency range and used to characterize the sample. Application of this
method to emission spectroscopy is the subject of this Chapter.
9.1 Theoretical background
Suppose the response of a system to a 6-pulse 1 excitation is f (t), then Fourier transform of
function f(t),
ex)
F(w) = J j(t)eiwtdt
(9.1 )
-ex)
gives the response of the system to a harmonic wave excitation at angular (or circular)
frequency w. 2 One can measure the response F(w) in a wide enough frequency range and
then calculate the response to the 6-pulse by applying inverse Fourier transformation to
function F(w),
ex)
j(t) = J F(w)e-iwtdt
27r
(9.2)
-ex)
18-pulse or Dirac function is 8(x) = 0 at x -I- 0 and 8(x) = 00 at x = 0, but J : 8(x)dx = 1.
2 Angular frequency is w = 27rv, where v is the ordinary frequency, i. e. the nU111ber of waves per unit ti111e.
171
172
Frequency domain emission spectroscopy
-
ro
bJ)
.......
rJJ
excitation
response
time
Figure 9.1: Frequency domain measurements. The response (m cos (wt + cp)) to sinusoidal
wave excitation (cos (wt)) is characterized by the phase cp and modulation m at the angular
frequency w.
Function f ( t) is the time domain presentation of the system response, and function
F (w) is the frequency domain response, respectively. If function f ( t) is known, then func-
tion F (w) can be calculated, and vice versa.
In application to optical spectroscopy the measurements of function f ( t) were discussed
in previous Chapters. To measure F (w) the sample has to be exposed to excitation by a si-
nusoidally modulated light. Then, at each frequency two values, phase cp and modulation m,
are measured, as illustrated in Fig. 9.1, which determine the frequency response function.
To clarify the meaning of the phase and modulation let us consider a typical example
for spectroscopy applications - determination of the lifetime. Within the time domain for-
malism, a spontaneous relaxation of the emission is described by the exponential function
(att > 0)
t
f(t) == foe- T
(9.3)
where T is the lifetime to be determined. The Fourier transform of the function f (t) is the
complex function 3
00
F(w) == fo J e-J;:+iwtdt == fo . 1
1_ iw
o T
(9.4 )
The real part of the function F(w) is
T
FRe(W) = Re (F(w)) = fo ( )2
WT + 1
(9.5)
3Note the integration li111its, function j(t) = 0 at t < O.
9.1. Theoretical background
173
and imaginary part is
WT 2
Flm(W) = lm (F(w)) = fo ( )2
WT + 1
(9.6)
The meaning of the real and imaginary parts is presentation of the response as the sum
of sine and cosine waves. If the excitation function is a cosine wave, i. e. cos (wt), then the
response IS
r(t) == FRe(W) COS (wt) + Flm(W) sin (wt)
(9.7)
The sum of cosine and sine functions is shifted in time cosine function, therefore a more
convenient presentation of eq. (9.7) is 4
r(t) == m cos (wt - cp)
(9.8)
where, m is the modulation coefficient and cp is the phase shift. This notation is used in
Fig. 9.1, and m and cp are two parameters which are obtained from the frequency domain
measurements. The modulation presents the amplitude of the sample response, and is given
by
m = V F'Ae(w) + FYm(w)
(9.9)
The phase shows relative delay of the response to the excitation wave, and is given by
Flm ( W )
cp == arctan
F Re (w )
(9.1 0)
For the samples with exponentially decaying emission (eq. (9.3)) the modulation and phase
are
cp
10 T
V I + (WT)2
arctan (WT)
(9.11 )
m
(9.12)
They are presented in Fig. 9.2 as the functions of frequency W at T == 1 and 10 == 1.
The dependences are straightforward in this simple case. At low frequencies (WT < 1) the
phase is small and modulation (or signal amplitude) is high - the sample emission intensity
follows the excitation intensity almost exactly. At high frequencies (WT > 1) the phase is
approaching 90° and the modulation (signal intensity) is small - this is the case when the
excitation frequency is higher than the reaction rate of the system, therefore the response is
delayed and small.
To determine the lifetime T of a sample using the frequency domain measurements the
phase of the response, cp, can be measured at frequency W close to the inverse of the expected
4Equivalently to the cosine excitation, one can use sine function to present excitation, sin (wt), then the
response is given by m sin (wt - cp), respectively.
174
Frequency domain emission spectroscopy
1
80
0.8
rJJ
S 60
""'0.6 H
bJ)
0 C)
....... ) '"d
'-'
ro 40 9-
-
;:j
0.4 ""'
( C)
rJJ
S ro
0.2 20
8.01
0.1 1
frequency, 0)
1<9
Figure 9.2: Phase, cp, and modulation, m, dependence on frequency, w, for a sample with
exponentially decaying emission. The emission lifetime is T == 1.
sample lifetime and calculated as T == w- l tan cp.5 In practice, however, the dependences
of cp and m on excitation frequency are measured at a number of frequencies, and used to
fit the results to obtain the lifetime. So the results of measurements are typically presented
in graphs similar to one shown in Fig. 9.2.
Recently a modification of the method was developed where the excitation light is
switched on-off (square wave modulation). This involves a different type of transform
but provides essentially the same functionality of the method.
9.2 Measurements scheme
A scheme for frequency domain emission measurements is shown in Fig. 9.3. Similar
to the other emission spectroscopy instruments there is a source of excitation light, but in
the frequency domain measurements the excitation light is modulated at frequency w. In
the scheme the light from continuously emitting source, e. g. an arc lamp coupled with
a monochromator, is passed through a modulator which is supplied by the radio frequency
(RF) from a generator. 6 After the modulator the light intensity is lex ( t) == 10 [1 + a cos (wt)],
where a is the excitation modulation coefficient (0 < a < 1). Formally, one can say that the
excitation consists of the continuous excitation with intensity 10 and the sinusoidal excita-
tion with amplitude alo. The emission induced by this excitation is collected by a lens and
filtered by a monochromator. The detection part of the instrument consists of a synchronous
detector (lock-in amplifier) supplied by the reference signal from photo-detector PD1 and
the emission signal from photo-detector PD2.
5In 1110st practical cases the (ordinary) frequency is 111easured in Hertz, in which case the equation calculating
the lifeti111e is T = (27r f) -1 tan cp, where f is the frequency.
6 Typical l11odulation frequency range is 1-300 MHz, which is the range of radio frequencies.
9.3. Frequency domain instruments
175
Light
source
o
> I Modulator I
RF
generator
=
Computer
Sample
, \
, \
, \
L
\ ,
Monochromator
Synchronous
detector
Figure 9.3: Scheme for frequency domain measurements of emission decays.
The synchronous detector is sensitive to the variable part of the signal only, and ex-
tracts values m and cp from the signal (see eq. (9.8)). Therefore, only the variable part of
the excitation (a cos (wt)) is important for the method. 7 The reference signal for the syn-
chronous detector can be obtained from the generator directly, however it is better to use an
independent photo-detector to supply the synchronous detector with the exact profile of the
excitation light as the reference signal, since the modulation frequencies are usually rather
high (can be over 100 MHz), and there may be some phase (and amplitude) distortions
induced by the modulator.
The whole system is controlled by a computer which sets the modulation frequency,
collects data form the synchronous detector and calculates the lifetimes or other parameters
of interest for particular experiment.
9.3 Frequency domain instruments
The frequency domain method was known for a long time, but it is not as widely used as time
correlated single photon counting technique (see Chapter 8). There are also commercially
available instruments, e. g. Fourolog-T2 from SPEX (Instruments S. A., Inc.), which are
relatively easy to use for routine lifetime measurements. One of advantage of the method
is, that it can be easily combined with the steady state emission spectroscopy instruments,
thus increasing flexibility of the system.
7 Also the continuous part of the excitation is not useful for the 111ethod, but it is unavoidable, since the light
intensity cannot be negative. Indeed, the higher l11odulation coefficient a is better for the 111ethod.
176 Frequency domain emission spectroscopy
9.3.1 Light source
A traditional approach to light sources for the frequency domain measurements is to use
a continuous light source, such as a lamp of CW laser, and to modulate the light using an
acousto-optic or an electro-optic modulator. This is shown in the scheme of Fig. 9.3. The
higher modulation frequencies allow to study shorter lifetimes, therefore the modulation
frequency range is one of the important characteristics of the instrument. Typical frequency
range is 1-300 MHz, though it depends on the application range of the instrument.
Recently a series of semiconductor diodes and laser diodes were developed, which can
be used in frequency domain instruments. The diodes emit light at fixed wavelengths, but
allow efficient excitation light modulation in a wide frequency range. Also they allow to
construct cheaper and more compact instruments.
9.3.2 Detection system
The core of the detection system is the synchronous detector (or lock-in amplifier), which
measures the modulation m (the signal amplitude) and the phase cp (the relative signal shift
in respect to the reference). The reference frequency can be obtained from the generator
or from a separately measured excitation beam as shown in Fig. 9.1. In the latter case the
reference detector can be a high frequency photodiode of photomultiplier. The emission
from the sample is typically detected by a photomultiplier coupled with a monochromator
to provide the highest sensitivity of the instrument.
9.4 Comparison between frequency and time domain methods
Time correlated single photon counting (TCSPC, see Chapter 8) is the time domain method
which is used for emission lifetime measurements of mostly the same range of samples
as that by the frequency domain method. Therefore it is interesting to compare these two
methods to find their advantages and disadvantages, and to sharpen the application ranges
for each one.
Results presentation and analysis
Although mathematically the time and frequency domain presentation of the measurement
results are equivalent to each other, the time domain is easier to interpret as the measured
emission intensity is directly proportional to population of the emitting states, and in most
cases can be viewed as the probability of e. g. excited molecule to be still excited at the
corresponding delay after the excitation. This simplifies analysis of different kinetic models
for the data obtained with the TCSPC method. However, it was shown that for the case
of mono-exponential decay of the emission both methods provide the same accuracy of
the lifetime measurements under otherwise same conditions [13]. Also methods were de-
veloped to analyze bi-exponential and more complex decays using the frequency domain
measurements.
9.4. Comparison between frequency and time domain methods
177
Time resolution
Ultimate time resolution of the TCSPC method goes to as short values as ten picoseconds,
which is achieved by using deconvolution procedures during data fitting, thus a separate
measurement of the instrument response is required, see Section 8.5.1. The frequency do-
main data can be analyzed directly without additional measurements and provide the time
resolution better than 1 ns. The time resolution of the frequency domain method is limited
by the highest excitation modulation frequency and accuracy of the phase, cp, and modula-
tion, m, determination at high frequency limit.
On the other side of longer sample lifetimes, the TCSPC method is limited by the pulse
repetition rate which is typically 1 MHz or higher, thus limiting the longer lifetime range
by a few hundreds of nanosecond. The frequency domain measurements can be used to
determine the lifetimes longer than the inverse of the lowest modulation frequency, also the
modulation frequency range usually starts at lower than 1 MHz values.
The TCSPC method is somewhat superior in short lifetime measurements, whereas fre-
quency domain has an advantage at longer, millisecond, lifetimes.
Sensitivity and accuracy
Technically both methods have the same sensitivity in terms of the number of detected pho-
tons, since essentially the same photo-detectors (photomultipliers) can be used. There was
also a theoretical investigation of the signal-to-noise ratios for both methods under condition
of a very low emission intensity [14], which has shown potentially the same sensitivities for
both methods. In practice, however, measurements of a very low emission intensities is a
more simple task with TCSPC method as it based on photon counting. In addition, during
TCSPC measurements all the information about photons is detected, whereas in case of the
frequency domain method the measurements are repeated at a few modulation frequencies.
Another advantage of the TCSPC method is its high accuracy which is provided by
the photon counting nature of the method. 8 This make it beneficial for studies of complex
systems as compared to frequency domain method.
For the measurements of samples with strong emission the frequency domain method
has an advantage of faster data acquisition. The TCSPC method is limited to detect no more
than one photon per excitation pulse (see Section 8.4.2), whereas frequency domain can
process strong emission intensities faster and without any specific limitations. 9
8In typical experi111ental conditions 10000 photons are collected at the channel of 111axi111u111 signal intensity,
which provides 1 % accuracy. Additionally, it is possible to increase the nU111ber of count bye. g. by increasing the
photon collection ti111e and increase the accuracy further l11ore.
9The stronger e111ission intensity 111eans better instant signal-to-noise ratio, therefore a shorter accu111ulation
ti111e is required for 111easure111ents with the SaI11e accuracy.
Chapter 10
Picosecond time resolution with streak
camera
For traditional electronics the fundamental limit in time resolution is set by the group veloc-
ity of the light - signal cannot propagate faster than the light. Decreasing the size of elec-
tronic components helps to increase the operational speed of the devices since it reduces the
signal propagation distance. Approaching one picosecond time resolution in spectroscopy,
one has to deal with optical pulses which are only 0.3 millimeter long. How to handle such
"short" signal? Is there a method to overcome the group velocity limit? Electronic devices
called streak cameras do this by switching from the group velocity to the phase velocity.
10.1 Operation principles
The TCSPC method comes to the shortest time resolution of the traditional electronics with
limiting step being the signal propagation in electric circuits, e. g. amplifiers, discriminators
and so on. The propagation limit for the signals and information is the group velocity of the
light, c. The principal idea of the streak camera design is to switch from the group to phase
velocity. The phase velocity can be greater than c. For example, if a light beam is reflected
by a rotating mirror, and the screen is placed far enough from the mirror, the light spot on
the screen can move with velocity> c. In this example the velocity of the spot movement
across the screen is not the signal propagation velocity. The signal, the light, propagates in
a different direction (perpendicular to the screen).
A scheme illustrating streak camera principles is presented in Fig. 10.1. The light hits
the photo-cathode, which converts the photons to the photo-electrons. The electrons are
accelerated in an electric field and passed to a deflection system similar to deflection systems
of oscilloscope electron tubes. A high speed sweep voltage is applied to the deflection
electrodes sweeping the electron beam across a phosphor screen. When the electrons hit the
screen they are re-converted to photons. The emission intensity at each point of the screen
is proportional to the intensity of the electron beam at the moment it swept the point. The
two dimensional emission pattern of the screen is recorded by, e. g., a CCD detector. Thus
179
180
Picosecond time resolution with streak camera
Photo-
cathode
-A\
>
light signal
Electron
deflection
system
I
I
. .7
(
. .
x
accelerating
electrode
Figure 10.1: Principle scheme of streak camera photo-detector. The electron beam is shown
by the dotted line. It is swept across the screen in vertical direction from the top to bottom.
the recorded image is a trace of the beam, for which the time was converted to the spatial
coordinate along the sweeping direction. 1
For high time resolution devices, the velocity of the electron beam movement across
the screen is greater than the velocity of the electrons in the beam, and can be greater than
the velocity of the light. For example, FESCA-200 camera (Hamamatsu) has the size of
phosphor screen l == 18 mm and the fastest full screen sweep time t == 20 ps, i. e. the
electron beam moves across the screen with velocity of v == t 10 9 m s -1, which is three
times greater than the velocity of light. With this device one can achieve an outstanding time
resolution of 200 fs. On the downside, the system has relatively high (compared to the time
resolution) trigger jitter time of 20 ps,2 which makes almost impossible signal averaging.
This means that the emission of the sample must be strong enough to allow single short
measurements with acceptable signal-to-noise ratio.
The two dimensional nature of the output image can be used further more to obtain the
time resolved spectra. The streak camera input can be connected to a spectrograph output
in such a way that the wavelength spread direction of the spectrograph is perpendicular to
the time sweep direction of the streak camera, as schematically presented in Fig 10.2. The
emission beam to be measured enters the spectrograph and is spread by the wavelength on its
output (in horizontal direction in Fig. 10.2). Thus the illuminated area of the camera photo-
cathode is a stripe, e. g. in direction X. This stripe is the spectrum of the incoming light,
which is changing quickly in time. The streak camera sweep the stripe in perpendicular
direction, Y, which becomes the time axis. So the two dimensional image on the camera
output, I(x, y), is the wavelength-time presentation of the input emission, I(A, t).
1 Although the recorded trace presents the ti111e evolution of the electron beaI11 intensity, these are not real ti111e
111easure111ents, which 111eans that this principle cannot be used process signals with high speed.
2The triggering pulse is still the signal which has all the li111itations i111posed by the finite value of the speed
of light.
10.2. Main characteristics
181
Input
beam
pin hole
slit
(lD image)
,
,
,
,
,
screen
(2D image)
t
Figure 10.2: Combination of spectrograph and streak camera converts the input light into
two dimensional image with one direction presenting the wavelength and another the time
dependences of the recorded light intensity.
10.2 Main characteristics
Streak cameras are the state of art devices produced in single units. A careful calibration
should be carried out for each camera before it can be used for quantitative measurements.
The handling of the recorded images includes correction of static and dynamic distortions
of the devices. To mention a few, there are nonlinearities of the camera sweep, the distortion
in optics, the space charge effects and nonuniform sensitivities of the photo-cathodes, the
phosphor screens and the image detectors. Therefore the streak cameras are usually sold as
complete instruments accompanied by the electronic control systems, CCD image readers
and software. The latter is designed to make all the required corrections and to give the user
the most reliable results.
10.2.1 Time resolution
The time resolution is the main characteristic of streak camera devices. At present, the
fastest streak camera, FESCA-200 (Hamamatsu) has time resolution 200 fs. A range of
devices is available from different manufacturers with the time resolution of about one pi-
cosecond. It hardly is a surprise, that the higher time resolution means much higher price.
Therefore, slower, 10-20 ps, but cheaper streak cameras are probably a reasonable compro-
mise between the time resolution and the price.
It has to be noted that the time resolution is not the only important characteristic to be
considered when selecting a streak camera for spectroscopy applications. Another impor-
tant temporal specification of the cameras is the trigger jitter. If the signal is not strong
and the averaging of a few shots is unavoidable, the trigger jitter will determine the final
time resolution of the measurements. Then the time resolution hardly can be better than
182
Picosecond time resolution with streak camera
1 ps, since the trigger signal is the "normal" electric signal and subj ected to all the speed
limitations of the electronic devices.
10.2.2 Spectrum range
The spectrum range of the streak cameras is determined totally by the type of photo-
cathodes they use. The photo-cathode materials are the same for the photomultiplier tubes
(see Section 4.2.2) and streak cameras. Typically one may expect to find devices working
in 200-700 nm (bialkali photo-cathode) 200-850 nm (photo-cathode type S-20) and 200-
950 nm (S-25) ranges. Naturally, the devices with longer red wavelength range have higher
dark current.
10.2.3 Sensitivity
For the "classic" streak cameras the sensitivity is rather low when comparing them with
photomultiplier tubes. Both streak cameras and photomultiplier tubes utilize the same types
of photo-cathodes to convert photons to electrons, but photomultipliers amplify the primary
photo electrons by factor 10 5 - 10 6 using dinode system and acceleration potential. In the
case of streak cameras the amplification of the primary photo electrons would result in drop-
ping down the time resolution to that of the photomultipliers. However the electrons can
be amplified after sweeping, when the time is already "converted" to spatial pattern. This
can be done by placing a microchannel plate amplifier, similar to one used in microchannel
plate photomultipliers (see Section 4.2.2), in front of the phosphor screen. The microchan-
nel plate amplifiers can provide gain factor of 10 3 - 10 4 , but preserve the spatial electron
distribution. This measure can increase sensitivity significantly making single photon de-
tection for the most advances systems possible. 3
10.2.4 Advantages and disadvantages
From the point of view of applications in time resolved emission spectroscopy the advan-
tages and disadvantages of streak cameras can be discussed in comparison with TCSPC
method described in Chapter 8. Also it should be noted that in terms of the time resolution,
the optical gating and, in particular, the up-conversion method provides the best results be-
ing relatively inexpensive at the same time. The up-conversion technique will be discussed
in Chapter 12.
In comparison with the TCSPC method the advantages of the streak cameras are
. better time resolution, 5-10 ps for most of the systems and up to 200 fs for the most
advanced cameras;
. single flash measurements with ability to detect both the time and wavelength depen-
dences, i. e. can provide complete emission dynamics at single excitation pulse.
3 The output electron flow after the 111icrochannel plate still has to be converted to photons by phosphor screen
and then the optical i111age is recorded by a CCD detector. All these operations reduce the efficiency of the streak
Call1eraS in c0111parison to the photo111ultiplier tubes.
10.3. Instrument examples
183
Technical disadvantages of the streak cameras as compared to the TCSPC technique are
. lower sensitivity;
. lower linearity in both time and intensity, also the time scale and intensity distortions
are corrected by the image processing software, the accuracy of the measurements are
not as good as for the TCSPC method;
. lower dynamic range in time, which is limited by the size of the screen and electron
beam diameter, while for the TCSPC method the number of channels is virtually
unlimited;
. lower dynamic range in intensity;4
. requires stronger excitation pulse energy, specially for the high time resolution appli-
cations.
The unique feature of the TCSPC method was the ability to measure emission decays of sin-
gle molecules. This type of experiments cannot be done with streak cameras since they are
not designed to measure single photons. They are useful for fast measurements of samples
with relatively strong emission intensities.
10.3 Instrument examples
There are not many manufactures of the streak cameras for spectroscopy applications, since
the cameras are the instruments based on the world's top technologies. Also the camera
prices are high and there are alternatives to attain even better time resolution using different
techniques.
One of the companies which has long term experience in streak camera production is
Hamamatsu Photonics K. K. The company produces ultra fast cameras with time resolu-
tion as short as 200 fs (FESCA-200), which can be used in different applications. They
also offer a picosecond fluorescence lifetime measurement system (C4780) which is build
around a streak camera equipped with microchannel plate. This is a complete instrument
including polychrometer, streakscope head, streakscope controller, delay generator, com-
puter and comprehensive set of software for spectroscopy experiments. The main features
of the system are 15 ps temporal resolution, simultaneous detection of temporal and wave-
length dependences (by use ofpolychrometer), a high sensitivity approaching single photon
counting (micro channel plate amplifier) and a wide dynamic range (10000: 1).
4The TCSPC 111ethod avoids dealing with analog signals and counts photons directly. In that sense it has
features that are hard to beat by any other technique.
Chapter 11
Pump-probe
Modem laser can produce light pulses as short as few femtoseconds, however the time
resolution of the flash-photolysis method for transient absorption measurements is usually
limited by a few nanoseconds, which leaves six orders of magnitude gap between poten-
tials of the laser systems and practically achievable resolution. There are at least 3 factors
limiting the time resolution of the flash photolysis method. The first one is the intensity of
the monitoring light - higher time resolution demands higher power of the monitoring light.
This is the most essential in case of a lamp-monochromator couple used as the source of
monitoring light. This problem can be solved by using CW lasers generating strong enough
monochromatic light. The second limiting factor is the excitation or even damage of the
sample by the monitoring light (this is the most important factor for the samples with long
recovery time, as was shown in Example 7.3). The third factor is the time resolution of the
detection electronics. Although electronics are developing quickly, the femtosecond time
resolution is far beyond optimistic expectations.
Pump-probe method was developed to overcome these limitations in time resolution.
The new approach of the pump-probe method is to use a short light pulse for monitoring
instead of continuous light as it is done in flash-photolysis method.
11.1 Principles
The most important part for the pump-probe instruments is the source of short light pulses.
These are different types of laser systems providing pico- and femtosecond pulses at variety
of wavelengths. The laser systems for pump-probe applications will be examined briefly in
Section 11.2. To discuss the principles of the pump-probe method we will assume that the
required short light pulses are available from some external system. By short light pulses
we will assume pulses of 20-200 fs duration, since these are the most typical pulse width
values in pump-probe applications at present.
11.1.1 Mono-color scheme
A general optical scheme for pump-probe experiments is presented in Fig. 11.1. A short
185
186
Pump-probe
M2
Pump
:!l Sample PD
A
Probe
Trap
Meter
Delay f7\
line
'- - - - - - - - - - - .
pump
A >
t
, ,
: !1 t :
>
t
Figure 11.1: Scheme of mono-color pump-probe experiments. M1 is a beam splitter, M2-
M3 are mirrors, PD is a photo-detector.
incoming pulse is divided in two parts by a semi -transparent mirror MI. The reflected beam
propagates to the mirror M2 and then hits the sample. This is the excitation pulse, which is
also called pump pulse. The rest of the light after the mirror M2 is passed to a delay line,
and after the delay it also hits the sample. This pulse serves for monitoring purpose and it
is called probe pulse.
The delay line is usually formed by a right angle reflector placed on a translation line
equipped by a stepping motor. By changing position of the reflector one can change the
traveling distance of the probe pulse thus changing the delay of the probe pulse relative to
the pump pulse.
The detection part may consist of an ordinary photodiode and a meter. The measured
parameter is the energy of the probe pulse after the sample, thus no time resolution is re-
quired for the detection. 1 If Ep is the energy of the probe pulse in front of the sample (which
does not change from pulse to pulse in an ideal case), then the energy of the probe pulse
after the sample is Em == Ep 10- A , where A is the sample absorbance. Since the sample is
excited by the short pump pulse initiating some photo-reaction, the absorption of the sample
will depend on the delay, t, of the probe pulse relative to the pump pulse. The delay can
be changed by moving the reflector M4,
t == 2(d - do)
c
(11.1)
1 The photo-detector signal is proportional to the total nU111ber of photons in the probe pulse. Therefore the
detected signal is an integral value.
11.1. Principles
187
where d is the position of the reflector M4, c is the velocity of the light and do is such
position of the delay line that the pump and probe pulses arrive to the sample at one and the
same time. Thus, the measured signal, e. g. the photo diode output voltage, is
U( t) == sEm( t) == sE p 10- A (.6.t)
(11.2 )
where s is the photo-detector sensitivity. The delay t can be scanned by moving the delay
line, i. e. it is a variable parameter.
Similar to the flash-photolysis method the parameter of interest is the absorbance change,
A( t), induced by the pump pulse, which can be defined as A( t) == A( t) - Ao
where Ao is the absorbance of the sample before the excitation. To obtain A( t) one can
measure the signal from the photo-detector without excitation (or before the excitation),
U o == sEp 10- Ao , and perform a simple calculation
U( t) == sE p 10- Ao -.6.A(.6.t) == sE p 10- Ao 10-.6.A(.6.t) == U o 10-.6.A(.6.t)
(11.3)
and thus
( U( t) )
A( t) = -loglO U o
(11.4 )
This is repetition of eq. (7.3) derived for the flash-photolysis method.
The time uncertainty of A is determined by the widths of the pump and probe pulses
and does not depend on the time resolution of the detection system. Therefore, the time
resolution of the method is determined by the laser system used to generate light pulses and
by the pulse broadening in the optics (will be discussed in Section 11.4.2), but not limited
by the light detection electronics or any other factors. From this point of view one can say
that the pump-probe method relies only on optical time resolution.
The following example illustrates that the detection of the signal in pump-probe experi-
ments is technically a simple task.
Example 11.1: Estimation of the signal intensity for the mono-color pump-probe experi-
ments. Suppose a mode-locked Ti:sapphire femtosecond laser is used to generate
light pulses for the pump-probe experiments. The pulse duration is 100 fs, repeti-
tion rate is 100 MHz and the average power is 1 W. The pulse energy of the laser is
1 W / 100 MHz = 10 nJ. (Note, this is rather high energy for the femtosecond pulse
generator; the limiting stage here is pumping laser, which usually generates less than
lOW.) The laser beam is split (by the beam splitter M 1, Fig. 11.1) in proportion 9: 1
(transmittance of the mirror M1 is 10°;6), so that 9 nJ pulses are used for the sample
excitation (pump) and 1 nJ for the monitoring (probe). In Section 4.4 we have seen
that the sensitivity of photodiode to single pulse can be S == 5. 10 7 V /J (R == 100 kO,
C == 1 0 nF). Thus without the sample the signal will be 50 m V. The sample will ab-
sorb part of the light and the signal will be weaker, but even 5 m V can be detected
fairly easy. Moreover, the integration time of the energy meter is RC == 1 ms. If the
laser pulses are used as they are, at 100 MHz repetition rate, then during the detector
integration time 100 000 pulses will arrive, and we should obtain > 1000 V at the
188
Pump-probe
photo-diode output! Certainly, in this case we must use a very deep filter in order to
keep the photo-detector working properly.
To determine A( t) at a certain delay time t we need to measure two values, the
photo-voltage with and without excitation, and to calculate absorbance A according to
eq. (11.4). The fact that the pulse shape is not resolved by the detection system, does not
affect the time resolution of the method. Moreover, if the repetition rate is high enough the
measured values are an average of many pulses, i. e. the measured value is the monitoring
power. Measuring ( average) power instead of the pulse energy does not change the calcu-
1ation method of A, so we do not need to know even how many pulses hit the sample
during the time interval of the measurements. The detection system is a really simple part
of the method. The principal part for pump-probe method is the generator of ultra short
light pulses.
Practically difficult part can be adjustment of the pump and probe spots at the sample
if a simple mode-locked laser is used as the excitation pulse source. This is illustrated in
Example 11.2 estimating the size of excitation spot to be smaller than 40 tL. It is possible to
focus light at such small spot, but this is a difficult task. Note also that the probe beam must
be smaller than the pump and must cross the pumped volume in the bulk of the sample. The
example shows, that the direct utilization of the mode-locked lasers may face some practical
problems and a higher pulse energy is required.
Example 11.2: Estimation of the excitation spot size. Let us assume the excitation pulse
energy to be 9 nJ, as was estimated in Example 11.1 for a mode-locked Ti:sapphire
generator. Let us also assume the sample to be an organic dye compound with the
molar absorption E 105 M-lcm- l (this is rather high value), which corresponds
to the absorption cross-section a == In(10) A 4 X 10- 16 cm 2 . To excite most
of the molecules the pump photon density must be at least one photon per absorb-
ing cross-section, thus the total excitation energy density (at a visible wavelength,
e. g. 500 nm) must be at least E == h: 1 mJ.cm- 2 . In order to obtain this en-
ergy density the pump pulse (9 nJ 10- 8 J) must be focused into a spot of area
10- 8 /10- 3 == 10- 5 cm 2 , or roughly 40 tL in diameter.
A drawback of the considered method is that the pump and probe pulses have the same
wavelength, that is why this method is called mono-color pump-probe. There is a variety of
methods which allow one to extend the choice of the wavelengths for both pump and probe
pulses. However they are all the extensions of scheme presented in Fig. 11.1. More complex
scheme will use some manipulations of the pump and probe pulses, but the manipulations
will be done using optical methods to preserve the pulse width and to keep relative delay
between the pump and probe stable. Then, the time resolution is determined by the width
of the pump and probe pulses at the sample.
11.1.2 Two color scheme
The pulses generated by a mode-locked laser can be amplified to relatively high energies,
10 tLJ - 1 mJ (will be discussed in Section 11.2). Then, one can build up a (relatively)
11.1. Principles
189
continuum
generator
, band
sapphire pass
plate L2 filter M3 PD 1 U,
==:: V- ::.@::.::. ::. --- ::.-- 1
L1
M
base
pulses
A
L3
: M5
I
v.;
Q-
I I
I I
M4 ::: :: :: "'- _ PD2
-- -;J
Sample - - ::: ::
------t:>-
'V
delay line
Figure 11.2: Two colors pump-probe measurements. M1, M3 and M5 are beam splitters,
M2 and M4 are mirrors, L1-L3 are lenses, PD1-PD3 are photo-detectors.
universal instrument for dynamic measurements of the photo-induced processes. An optical
scheme of one of possible arrangements is shown in Fig. 11.2. The input base pulses are
split on two beams by the mirror MI. The reflected beam is directed by the mirror M2
to a white continuum generator, which consists of two lenses, L1 and L2, and a sapphire
plate. 2 The lens L 1 focuses the beam into sapphire plate to increase power density and to
achieve continuum generation threshold. The white continuum is collected by the lens L2
and focused on to the sample. Then the continuum is filtered by a band pass filter to select
a desired wavelength, and used as the probe pulse. The part of the light, which crosses the
mirror M 1, serves as the pump. It is directed to the delay line. Unlike in the case of the
scheme shown in Fig. 11.1, the delay line controls the excitation time (relative to the probe
pulse). Since only the relative delay between the pump and probe pulses is important, this
delay arrangement will work as well as one shown in Fig. 11.1. The only difference is that
the sign of the t value must be changed, i. e. eq. (11.1) must be rewritten as t == 2(d -d) .
The lens after the delay line, L3, is used to form an excitation beam of a suitable size, so
that the excitation spot is larger than the monitoring one across the whole sample.
The scheme (Fig. 11.2) utilizes three photo-detectors, although only detector PD2 is
needed for an "ideal" instrument. The detector PD2 measures the probe beam intensity
after the sample and has the same function as one shown in Fig. 11.1. The detector PD 1
monitors the intensity of the probe beam. Its main role is to improve the accuracy of the
2 Ah110st any transparent 111ediu111 can be used for continuu111 generation under sufficiently high power density
of irradiation. Widely used 111aterials are quartz, sapphire, water and heavy water.
190
Pump-probe
measurements. The signal is calculated as
8( t) = :
( 11.5)
where U l and U 2 are the photo responses of the detectors PD1 and PD2, respectively.3 The
detector PD3 is optional and can be used to monitor the pumping energy and, in some cases,
for further improvement of the measurement accuracy.4
Similarly to the flash-photolysis and mono-color methods one can calculate the change
in the sample absorbance, A, as
( S( t) )
A( t, A) = -loglO 8(0)
(11.6)
where S (0) is the signal obtained without the pump pulse or when the probe pulse hits the
sample before the pump (i. e. unexcited sample). Naturally, scanning the delay time, t,
one can record the time profile of the transient absorption signal.
The measurements can be repeated at another wavelength using another band pass fil-
ter in the probe beam. To simplify the procedure of the monitoring wavelength tuning a
monochromator can be installed in front of the detector PD2. The monochromator must be
placed after the sample because it will increase the pulse width gradually. The pulse width
after the sample is not important.
After a series of measurements one will collect a two-dimensional data array A( t, A).
Taking the data at fixed delay time, a time resolved spectrum is obtained, A( A) == A( t ==
const, A). This simple procedure, however, can be applied in case of time resolution of a
few picosecond or in a narrow wavelength range. With picosecond or better time resolution
the light group velocity dispersion affects the delay time at different wavelengths and should
be taken into account to obtain actual time resolved spectra. The effect of the group velocity
on pump-probe measurements is the subject of Section 11.4.1.
The scheme (Fig. 11.2) allows one to monitor the time profile of the photo-induced
absorption change at any optical wavelength. However, the excitation is fixed to the wave-
length of the fundamental pulses. The problem can be solved by a further development of
the laser system as will be discussed in Section 11.2.
11.1.3 Measurements of time resolved spectra
In the previous scheme (Fig. 11.2) a white continuum was generated in the probe channel
but only a small portion of the continuum was used to probe the sample. Clearly, one can
use the whole continuum and obtain a spectrum at a certain delay time with single pump-
probe pulse series. A modified optical scheme for spectra measurements is presented in Fig.
11.3, where the continuum generator at the probe channel and the delay line at the pump
channel are not shown for shortness. The white continuum probe pulse is split in two beams
3 One can C0111pare this two photo-detectors arrange111ent with the two channel spectrophoto111eter discussed in
Section 5.2.2. The signal fr0111 the detector PD1 serves as the reference channel of the spectrophoto111eter.
4 A typical usage of the detector PD3 is to interrupt the 111easure111ents when the variation of the excitation
pulse energy exceeding SOl11e li111it.
11.1. Principles
191
continuum M 1
reference
Ll
L2
spectro-
" h ',CCD
, " grap
I
,
,
probe
signal
sample
A
B
C
CCD sensor
,
,
signal reference..
. ................... ;{ .....
:::::::::::-. <'»> ' .':::::::::::
, . . . . . .. . . . . : .. . . : .. : .. . . . .. : .. . . . .. .. . ... . : . . . . . : . . . . . . . . .. . . . .. . . ..... . . . . . .
,. ::::::::::::::""""" ':::::::::::::'
/::Ci: ation
,,' signal reference '
' ",,,,,,,,,,, ,, ,,,,,,,,7
dj) iP"
....... .......
....... .......
....... .......
......, ......
..... .....
.... ....
... ...
entrance slit
,:signal
ref. ''.
lA
,
I
,
------...
Figure 11.3: Optical scheme for time resolved spectra measurements. Ml is a semi-
transparent mirror (beam splitter), M2 is 100% mirror, L 1 and L2 are lenses. Circles show
the beams alignment at different cross-sections (turned in horizontal plane).
and both beams are directed to the sample. They cross the sample at different points and
the pump pulse covers only one of the probe spots (as shown in the circle A, Fig. 11.3).
The beam overlapped by the pump is the signal. Another beam crosses a non-excited area
of the sample and it is used as a reference. Both the signal and the reference are focused
into the input slit of the spectrograph (circle B, Fig. 11.3), where they are not overlapping
as this plane is the image of the sample plane (circle A). On the exit of the spectrograph
the spots are spread in wavelength, so that there are two colored strips (circle C, Fig. 11.3).
These two strip-like images are recorded by a CCD detector. One strip is used to calculate
the intensity spectrum of the signal Is (A), and another of the reference Ir (A). 5
In ideal case the signal and reference channels are identical and the ratio R( A) == ; i j
should give ones over the whole spectrum without excitation. Under the excitation the
ratio R(A) can be used to calculate photoinduced absorbance, A(A) == -loglO R(A).
In practice, however, this simple method is not used, since all the components (mirrors
and lenses: M1, M2, L1 and L2) are non-ideal and inaccuracy in alignment of the beams
(e. g. asymmetric beam propagation, spots non-identity on the slit and others) reduces
the measurement accuracy to an unreasonably low value. The problem is similar to one
discussed with the two channel spectrophotometer (see Section 5.2.2, eq. (5.4)) and can
5The calculations involve sU111mation of the i111age pixels across the strip (horizontal direction for i111age in
circle C, Fig. 11.3) and recalculation of the position along the strip into the wavelength. The latter is done based
on the previous \vavelength calibration of the spectrograph-CCD couple.
192
Pump-probe
. . . ...:r', ... -0.3 ps
1::r:;."'I./:'r;...: . ;,; \ ".. ".
"" ., ,...",jI#'V 1"". .. ,.. ....'" .r..:'!:".........t.....::v-.
Vi'" J';' . "" . i:.' . ....
",\. ,..... ,,'\ .... J'- '1.:J'....k-qr: """'1,
a .'" "'. \. J'" "... .
...,.......v' .,...",... ". """"...""'....'"¥"....,.,.. ...I\
.. :..,.. .. .." .. .......... ,
,.. "oJ \>,) (/ .\ .'. ................ ..
J :..' . - ,.'\.'\......J,., . 'V."' :v\JV""i;n:
. ;.;.t#----."'-d .
..". :
:. \ \ I, .:
\'. \ ',. , .f' .
. ...\ ' :: .
.. \ .
. 0 . I .
r. \ I :
.. \. -:
\\ ',' /!
\: . ,
-0.05
<j
o.
r.
, '.
- -0.6 ps
- - -0.3 ps
0.0 ps
.... 0.3 ps
.-. 0.6 ps
-.. 48 ps
-0.1
-0.15
., ,
., J
550
600
650 700
wavelength, nrn
750
Figure 11.4: Time resolved spectra of phitochlorin-fullerene donor-acceptor dyad in ben-
zonitrile. The sample was excited at 420 nm by 60 fs pulses. The probe delays are indicated
in the plot. No correction of the group velocity dispersion was performed.
be solved in the same manner. First of all the "base line" is measured without excitation
pulses: Ro(A) = i \ . Then, during the measurements with the excitation, the change in
the sample absorbance induced by the pump is calculated by
( 1 8 (A) 1 ) ( R(A) )
A(A) = -loglO lr(A) Ro(A) = -loglO Ro(A)
( 11.7)
Thus, non-ideal features of the signal and the reference are taken into account by applying
the correction procedure. Typically this procedure helps to achieve absorbance sensitivity
better than 0.001.
Similarly to the two color pump-probe method, the measurements can be carried out
with a series of delay times. Then the collected data form two dimensional array, A(A, t),
which can be used for both dynamics and spectrum analysis. Clear advantage of the method
is that the complete figure of the photo-induced change in the sample absorption, A(A, t),
can be obtained with the single scan of the delay line. This also means that the total expo-
sition of the sample to the pump is lower than that in the two color measurements. A disad-
vantage is somewhat lower sensitivity as compared to the two color pump-probe method.
An example of the time resolved spectra measured of a chlorophyll based donor-acceptor
dyad is shown in Fig. 11.4. Photo excitation of the dyad populates the singlet excited state of
the phitochlorin chromophore, which undergoes a charge separated state with time constant
shorter than 1 ps [15]. The lifetime of the charge separated state is roughly 70 ps. The char-
acteristic feature of the ground state absorption of the phitochlorin is a sharp band around
11.1. Principles
193
670 nm, which corresponds to the transition fron the ground to the first singlet excited state.
The excitation results in an almost instant decrease of the sample absorption at this wave-
length. This is seen as negative LlA and corresponds to disappearance of the sample ground
state absorption band after excitation, which is called photo-bleaching. Formation of a new
absorption band is seen as increase in optical density, such as a broad band at 580 nm.
The spectra presented in Fig. 11.4 are the time resolved differential absorption spectra.
The approach to the spectra interpretation is similar to that discussed in Section 7.1.3 for
flash photolysis method. However, there is one important feature imposed by the much
shorter time resolution of the pump-probe method as compared to the flash photolysis.
The propagation time of the probe pulse at the shorter wavelengths takes longer time
than that at the longer wavelengths in any condensed matter, such as the lenses and the
sample itself. This has an effect of non-simultaneous signal appearance at different wave-
lengths. The spectrum obtained at t == -0.6 ps (Fig. 11.4) is a horizontal line (i. e. there
is no change in the absorption yet). The spectrum at t == -0.3 ps shows a small increase
in the absorption at the blue part of the spectrum, but the main part of the spectrum does
not show any change. At t == 0 a negative absorption (photo-bleaching) starts to grow
at the middle of the measured wavelength range (this was the reason to select this delay as
"zero" delay tin1e). This non-synchronous signal appearance happens because the refraction
indexes of the lenses and the sample decrease while the wavelength increases. Therefore
the blue part of the probe propagates slower than the red part, i. e. the blue part reaches the
sample later than the red part. If "zero" delay (pump and probe hit the sample at one and the
same time) is set for the signal appearance at the middle of the studied spectrum range, then
at "zero" delay the blue part of the probe pulse reaches the sample well after the excitation
whereas the red part hits the sample before the excitation pulse. The detailed consideration
of this effect is the subject of Section 11.4.1.
11.1.4 Samples and sample excitation schemes
In Section 7.1.4 different arrangements of the
excitation and monitoring beams in respect to the
sample were discussed and applied to the flash pho-
tolysis method. In pump-probe the choice of the
pump-probe beams organizations is much more lim-
ited. The critical point is the pulse spatial width,
which is much shorter than that in the flash photol-
ysis experiments.
Let us assume that the pump and probe pulses
are propagating in the sample at an angle a to each
other, as shown in Fig. 11.5. The pulse duration T
determines the spatial width of the pulse in detection
of its propagation, d == CT. At some moment there is Figure 11.5: Overlapping of pump
an overlap area of the pump and probe pulses. This and probe pulses.
is the area where the pump and probe pulses have
the same timing. For the part of the sample which is higher than this area the probe pulse
reaches the sample after the pump pulse, and for the lower area the probe pulse crosses
probe
pump
overlap
area
- : I :
194
Pump-probe
the sample before the pump. Thus we have to limit the front of probe so that it keeps
its delay relative to the pump pulse with accuracy roughly equal to the pulse width, i. e.
D < d/ sin a == CT / sin a, or D sin a < CT. At the same time the limited size of probe
beam impose a limit on the sample thickness, which should be thin enough to keep overlap
between the pump and probe through the whole sample thickness, i. e. the thickness must
be L < D / sin a.
Example 11.3: Estimation of the maximum angle between pump and probe beams. Suppose
that the pulse widths of the pump and probe are 100 fs. Then the product D sin a
must be smaller than CT == 0.03 mm, which is the spatial width of the 100 fs pulse. If
the size of the probe beam is 1 mm at the sample, then the angle between the pump
and probe must be smaller than sin a < C;; == 0.03, or a < 1.7 degree. This is
rather small angle. It would be more practical to reduce the probe spot size to e. g.
0.3 mm to allow the angle to be 5 degrees. In the latter case the thickness of the
sample should be smaller than D / sin a 3 mm.
The above example illustrates that in the pump-probe experiments the arrangement of
the pump and probe beams must be quasi parallel (or co-linear). Typically the spot sizes at
the sample are smaller than one millimeter, and the thicknesses of samples are smaller than
a few millimeters. Bigger sport sizes or angles will result in decrease in the time resolution
as the delays between the pump and probe inside the sample are not synchronized.
Yet another difference in sample arrangements for pump-probe experiments as com-
pared to all previously discussed deals with the higher probability of the sample degrada-
tion due to much higher peak power density in pump-probe experiments. 6 Therefore flow
cells or even jets are used for liquid samples. Also rotating disk-like cuvettes can be used
to reduce the effect of sample degradation under the excitation.
11.2 Laser systems
Ultra short pulsed lasers and laser systems are the heart of any pump-probe instrument. They
determine most of the instrument characteristics, and they are the most expensive parts of
the systems. The choice of the lasers depends on the range of samples and phenomena to
be studied. Selection of one or another configuration is usually result of thorough work
planning and negotiations with laser manufactures.
There is a wide range of lasers and lasers systems which can be used as the light source
in pump-probe instrument. It is hardly possible to mention all variants of laser system
design, therefore the goal of this section is to provide a brief overview of one common
approach to the laser system design, which can be considered as an example rather than a
complete guide.
During planning of a laser system for pump-probe application, a number of characteris-
tics must be considered together, the most important being:
6Though excitation density is usually lower in the PU111p-probe experi111ents than in flash photolysis. See
Section 11.4.4 for further discussion.
11.2. Laser systems
195
pulse width - the shortest is not always the best solution, the shorter pulses have higher
pick power and can overcome the threshold of two photon excitation or other
nonlinear effects easily when compared to longer pulses, not to mention the
price, which grows almost exponentially with the pulse width decrease;
pulse energy - depends on the sample and optics around the sample, e. g. the spot size to
provide necessary excitation energy density;
wavelength ranges for excitation and probe - determined by the classes of obj ects to be
studi ed;
pulse repetition rate - important for the optimum organization of the detection system, se-
lection of the detector type and registration method.
Primary pulse generator
The starting point of all laser systems is the generator of short pulses. The pulse width at
the generator output determines the final pump and probe pulse widths. The pulse energy,
repetition rate and even the wavelength are not of the prime importance for the generator
as they can be manipulated later. A decade ago most of the generators were dye lasers
synchronously pumped by mode-locked Nd:YAG solid state or Ar ion laser. This type of
laser system was mentioned as an example of the excitation source for the time correlated
single photon counting method in Section 8.2. Typical pulse width for such system is a few
picoseconds.
Modem ultra short pulse generators are usually Ti:sapphire mode-looked lasers pumped
by CW laser. Titanium ions have a broad luminescence band in the range 700-1050 nm
which makes them an excellent material for ultra-short pulse generation. 7 Usually, to estab-
lish the mode-locking operation of Ti:sapphire lasers a Kerr lens effect (see footnote 13 on
page 53) or/and semiconductor saturable absorbers are used. The Ti:sapphire lasers gen-
erating 50-200 fs pulses are widely available from different manufactures. There are also
commercially available generators producing as short as 20 fs pulses. Carefully designed
Ti:sapphire lasers can generate as short as 6.5 fs pulses [5,4], although these are the systems
designed for laser physics research purposes rather than for routine spectroscopy measure-
ments.
Typical average power of Ti:sapphire lasers is rv 0.5 Wand the pulse repetition rate
is 80-100 MHz. Thus the pulse energy is rv 5 nJ. The lasers can be tuned in rather wide
range of 700-1000 nm. However, this wavelength range is not useful for the most of the
optical spectroscopy applications. The pulse energy is also rather low (see estimation of the
excitation spot size in Example 11.2) thus adding limitations on the method applicability
even when the wavelength match the specification.
7There are other i111portant properties of Ti:sapphire crystals beneficial for its laser application, a1110ng the111
are high density of Ti ions, i. e. high aI11plification, and good thermal conductivity. See Section 3.6 for a brief
discussion of the subject.
196
Pump-probe
Stratcher
, ,
Amplifier
' ,
I : Ti"S ·
Pulse: . :
k : (. )\ Com p ressor
pIC er : \ 7:
- - - - - - - - - - - - - - - - - - - - - - - - - -
pump
A_
I
pulse
--
Output
pulse
Figure 11.6: Amplification of femtosecond pulses uSIng stretching-amplification-
compression method.
A
> Output
blue
Input
grating 1
Figure 11.7: Illustration of pulse stretching principles.
Amplification of ultra short light pulses
The next step in the laser system development is to amplify the femtosecond pulses. This
can be done by passing the pulses through another optically pumped Ti: sapphire crystal.
There is however a problem of keeping the pulse width short during the amplification. The
short pulses have relatively wide spectra, and because of group velocity dispersion in con-
densed media, they become broader while propagating in the amplifying crystal. To achieve
the same pulse duration on the output of the amplifier as it was before the amplification, the
pulses are first stretched in a special manner, then amplified and finally compressed back to
their original width.
A scheme of a femtosecond amplifier is shown in Fig. 11.6. First of all the input pulse
is stretched. The principles of pulse stretching are illustrated in Fig. 11.7. The stretcher
is formed by two identical gratings placed parallel to each other at some distance. The
pulse hit the first grating at angle nearly parallel to its surface. Since the pulse has a certain
spectrum width (the shorter pulse the broader spectrum) diffraction takes place with some
divergence, so that the longer wavelengths (red) diffract at bigger angle than the shorter
(blue) wavelengths. After diffraction on the second grating all the wavelengths propagate
again in the same direction, but the "red" part of the pulse travels a longer distance than
the "blue" part. Thus, after the stretcher the pulse is wider in time and "colored" so than
the "blue" part is traveling at the pulse front and the "red" at its tail. This "color" property
of the stretched pulse can be used to compress it back to its original width using a similar
11.2. Laser systems
197
optical scheme with gratings. 8
The second component in the scheme is the pulse picker, which reduces the pulse rep-
etition rate and leaves only the pulses which are going to be amplified. After amplification
the repetition rate is much lower than that of the pulse generator. Among other factors the
average pump power is the limiting factor. For example, after amplification the generator
pulses e. g. 100 time at the same repetition the average pulse power will be 100 x 0.5 W =
50 W, which would require at least 300 W of average (optical) pumping power. The actual
amplification factors are much greater than 100.
The amplification takes place in an optically pumped Ti:sapphire crystal. The pulse
travels through the crystal few times to increase the amplification and to collect all the
energy stored in the crystal. Two schemes are usually used for the amplification. One type
of amplifiers is called regenerative amplifier and another is called multi -pass amplifier. In
the latter case the pulse crosses the crystal a fixed number of times, six or eight. In the
former case the crystal is placed in a confocal resonator and the pulse travels inside the
resonator until it is amplified and then it is taken away by a cavity dumper.
The last component of the amplifier is pulse compressor, which returns the pulse width
to its original value using optical schemes complementary to the pulse stretchers.
A typical total amplification factor for multi-pass amplifiers is 10 5 -10 6 , which can pro-
vide the output pulse energy higher than 1 mJ. The repetition rate depends on the pumping
source (e. g. Q-switched Nd:YAG laser) and can vary from a single shot to a few kHz. 9
Regenerative amplifiers can be pumped by CW lasers, then the repetition rate of the
amplified pulses can be up to few hundreds kHz and pulse energy up to few tens of tLJ. For
example RegA 9000 series of amplifiers (Coherent Inc.) can operate at 250 kHz and deliver
pulse energy of 4 tLJ. With pulsed pumping the repetition rate is usually 1-5 kHz and the
pulse energy can be close to 1 mJ.
White continuum generation
The 1 mJ pulse energy does not seem to be a high value. However, accounting for the
pulse width of 100 fs or even shorter, the peak power of such pulse is higher than 10 10 W
or 10 GW. This is extremely high power! Propagation of such pulse in a medium can
easily create numerous nonlinear phenomena. One of such phenomena is a white continuum
generation. When the power density of the pulse exceed a certain limit the spectrum of the
pulse becomes very broad but the pulse duration remains virtually the same. Experimentally
it is observed as a bright white spot on the output of the medium. Although the physics
behind this phenomenon is still not clear, the white continuum generation is actively used
in pump-probe spectroscopy.
The white continuum generation can be achieved with almost any transparent medium.
The generation threshold is about 10 12 W cm- 2 . A few most popular materials for the con-
81n fact, pulse stretching takes place every ti111e the light propagates in condensed 111edia, e. g. in Ti:sapphire
active crystal of the pulse generator. Therefore the pulse C0111pressors are obligatory parts of any fe111tosecond
generators. Inside laser resonators sche111es consisting of two pris111s are usually used (see Fig. 11.11). Pris111
C0111pressors have 111uch lower losses but can be used to c0111pensate relatively sl11all stretching as they have sl11aller
dispersion c0111pared to that of gratings.
91111J at e. g. 5 kHz repetition rate 111eans 5 W of average power on the a111plifier output.
198
Pump-probe
tinuum generation are quartz, sapphire, water and heavy water. Typically a few millimeters
of such materials are sufficient to obtain efficient conversion of fundamental wavelength
of Ti:sapphire laser ("'800 nm) in to a broad spectrum covering wavelengths from 300 to
1500 nm. Typically the fundamental harmonic is focused by a lens on to 10-50 tL spot
and the white light behind the medium is collimated by another lens for further utiliza-
tion. Therefore already 1 tLJ pulse can provide more than 10 times excess over the white
continuum generation threshold.
The white continuum can be directly used as the probe pulse. The advantage of the
white continuum is that the sample can be probed at virtually any optical wavelength using
schemes similar that shown in Fig. 11.2 is used. The white continuum makes possible sin-
gle shot detection of the time resolved spectrum applying scheme similar to that presented
in Fig. 11.3. However, the continuum cannot be used for excitation, at least with a ten
nanometers bandwidth, since the spectrum density of the white light is rather low.
Pump sources and parametric amplifiers
The fundamental wavelength of Ti:sapphire laser can be tuned in to the range 750-1000 nm,
and this wavelength range is directly available for the excitation (pump). It can be extended
by adding the second and third harmonic generators to 380-500 and 260-330 nm ranges,
respectively. However, it still leaves uncovered a big part of the optical spectrum. In addi-
tion, to tune Ti:sapphire laser from 760 nm to 950 nm one will have to change all the mirrors
inside the resonator as such wide wavelength range is hard to cover by one set of mirrors.
This tuning is a time consuming procedure. What are solutions if one needs to build up a
system with widest possible selection of the excitation wavelengths?
It seems natural to utilize white continuum but to amplify it at the desired pump wave-
length. As amplification medium a suitable dye can be used, although this is not a universal
solution as the typical tuning wavelength range of laser dyes is narrower than 50 nm.
In femtosecond time domain one can take an advantage of high peak power and build
up an amplifier totally relying on non-linear optic phenomena - optical parametric (light)
amplification (OPA). The parametric amplification was considered in Section 3.7.4, and
here it will be mentioned briefly in application to the pump-probe instruments.
The parametric amplifiers require high power pumping. Naturally, the fundamental
pulses can be used for this. For the Ti:sapphire lasers this means that the OPA pump wave-
length is around 800 nm. Then, the OPA signal wavelength can be changed from roughly
1100 nm to 1600 nm and the idle from 1600 to 2900 nm. This wavelengths are still not
very useful for optical spectroscopy, but can be manipulated further more. Firstly, the sec-
ond harmonic of the OPA signal covers the range 500-750 nm. Second, the OPA signal
can be mixed with the fundamental pulses of the Ti:sapphire laser to generate pulses at
sum frequency, which produce pulses in the range 465-530 nm. 10 Mixing of the idle and
fundamental pulses covers the range 535-625 nm. As the result, any wavelength in the
visible-near infrared range can be obtained by mixing fundamental and OPA signal or idle
pulses.
10 According to eq. (3.21) the wavelength of the SU111 frequency generation can be calculated as Asum =
(A f 1 + A :; ) -1 , or the inverse of SU111 wavelength is equal to the SU111 of inverse values of the fundaI11ental and
OPA signal wavelengths.
11.2. Laser systems
199
800 nm
1 mJ
0.01-5 kHz
Ti: S amplifier
800 nm
1 Ti:S generator I
..
10 nJ, 90 MHz
white continuum
probe
1100-1600 nm
signal
1 SFG I 465-530 nm
.
30 J.1J
1 SHG I 550-800 nm
.
30 J.1J
1 SHG 1 800-1450 nrn .
20 J.1J
1 SFG I 535-625 nm
.
20 J.1J
.
OPA
1600-2900 nm
idler
1 SHG I
400 nm
Figure 11.8: Scheme for laser system generating pump and probe pulses in a broad spectrum
range. WCG is the white continuum generator, OPA is the optical parametric amplifier, SHG
is the second harmonic generator, and SFG is the sum frequency generator.
The efficiencies of OPA and wave mixing are rather high with femtosecond pulses. For
example OPerA-SFG (Coherent Inc.) can provide >30 tLJ pulses at 500 nm (mixing the
OPA signal and fundamental pulses) from 1 mJ fundamental pulses.
Universal light source for pump-probe
The above discussion can be summarized in a scheme presented in Fig. 11.8. The scheme
shows the principles to obtain different wavelengths on the output rather than actual beam
propagation, e. g. OPAs are usually complete devices with their own white continuum
generators and require only fundamental harmonic on the input. The starting point of the
scheme is the Ti:sapphire generator. This is the most important part from the point of view
of the pulse width on the output of the system. Easily available pulse width is 30-200 fs.
Shorter pulses can be obtained but it is much more difficult to keep that short pulse width
in the following system components, e. g. amplifier, OPA or harmonic generators. Also
there may be some fundamental reasons for not using too short pulses as will be discussed
in Section 11.4.4.
The pulses are amplified to energy 1 mJ or so, which is sufficient for efficient white
continuum generation and optical parametric amplification. The white continuum can be
used as the probe pulse directly. The output of the OPA has to be manipulated to obtain
pulses in the visible part of the spectrum. These manipulations are the harmonic generation
and wave mixing, i. e. the sum frequency generation. The tuning of the pump wavelength
200
Pump-probe
of such system is relatively simple as it includes (1) tuning of the OP A, which is basically
angular adjustment of the non-linear crystal, and (2) adjustment of the harmonic of wave
mixing crysta1. 11 As a result the pump pulses are available at all visible wavelengths, start-
ing from 465 nm and longer, with energy of a few tens micro Joules. Additionally the
second harmonic of the fundamental pulses (400 nm) is easily available avoiding OPA. The
pump energies are sufficient to excited areas of 1 mm 2 or larger of the sample (see Example
11.2 for the spot size estimations).
To extend the pump wavelength range further to the UV one can use higher harmonic
generation (not shown in Fig. 11.8). For instance, the fourth harmonic of OPA signal can
be used to cover 300-400 nm range, also the pulse energy is a few micro Joules then.
Different components for the laser systems similar to that shown in Fig. 11.8 and com-
plete systems are available from a number of companies, to mention few are Coherent Inc.,
Quantronix Inc. and CDP Corp. (Moscow). The configuration presented in Fig. 11.8 is the
example of the pulsed laser system for the pump-probe applications. Depending on par-
ticular requirements one can find other options, probably suiting better to his or her needs.
These lasers systems are very expensive and delicate instrument, and careful planning of the
instrument and comparison of different options can save a lot of money and further efforts
in the instrument usage and maintenance.
11.3 Detection subsystem and sensitivity
From the point of view of the probe light detection the pump-probe method is similar to
the steady state absorption spectroscopy considered in Chapter 5. The detection systems of
pump-probe instruments must measure some average light intensity as accurate as possi-
ble, being insensitive to the actual pulse durations. When the pulse repetition rate is rela-
tively high ( > 1 kHz) the modulation-synchronous detection technique is usually applied
to achieve the best results. At lower repetition rate each pulse can be detected, but the mea-
sured value is proportional to the pulse energy as was discussed in Section 11.1.1 so that the
detection scheme is optimized for accurate measurements of the relative energies with and
without pump pulses but insensitive to the pulse duration.
The sensitivity of the methods can be rather high. It depends on inaccuracy of the signal
determination, S ( t). Ideally, the calculations of signal as the ratio of intensities, e. g.
eq. (11.5), should make the result, S ( t), insensitive to the pulse-to-pulse variation in the
base pulses energy. The limiting factors of the signal measurements are the total number
of detected photons, i. e. photon quantum noise, and different types of the detector noises,
e. g. thermal noise. For the best instruments measuring the transient absorbance at fixed
wavelength (like two color scheme, Fig. 11.2) the reported results indicate accuracies as
good as 10- 5 (detectable absorbance change) [16], which is close to the best accuracies
11 As it was 111entioned above, Fig. 11.8 does not show actual layout of the optical c0111ponents or beaI11
propagations in the syste111. After the non-linear crystal, serving as aI11plifying 111ediu111 of the OPA, all three
beaI11s, fundaI11ental, signal and idler, propagate in one and the SaI11e direction. The switching between different
l11odes, SHG or SFG for signal or idler, is done by installing a proper non-linear crystal in the beaI11 and finding
the angle of phase 111atching condition for desired effect. Also care should be taken to block the infrared light (of
fundaI11ental, signal and idle re111ained unused), which propagates in the SaI11e direction as visible light and can
da111age the saI11ple.
11.4. Time resolution
201
of the steady state spectrophotometers (see Section 5.4.1). For the pump-probe systems
implementing time resolved spectrum detection, e. g. Fig. 11.3, the typical sensitivity
values (detectable absorbance change) are 10- 3 - 10- 4 .
In comparison to the flash-photolysis method the sensitivity (or accuracy) of pump-
probe technique does not depend on the time resolution. Another essential difference is that
using the flash-photolysis method the whole transient absorption profile can be measured
with a single flash, whereas the pump-probe measurements have to be repeated as many
times as many time points have to be recorded at least. This is the price one has to pay in
order to improve the time resolution gradually.
11.4 Time resolution
The size of the probe spot, angle between the pump and probe, and the sample thickness can
reduce the time resolution of the pump-probe method as was discussed in Section 11.1.4.
These factors will not be considered in this Section, thus assuming that the alignment of the
pump and probe beams is perfect. Then the time resolution of the pump-probe method is
determined by the widths of the pump and probe pulses (inside the sample). Formally the
time resolution can be calculated from the width of the convolution integral also known as
correlation function
+ (X)
!c(t) = J !pump (T)!probe (T + t)dT
-(X)
(11.8)
where !pump (t) and !probe (t) are the pulse shapes of the pump and probe, respectively.12
Assuming Gaussian pulse shape, ! ( t) = exp [- :2 ], where D.t is the pulse width, the
convolution integral is a Gaussian pulse too
!c(t)
+ (X) [ 2 ]
T 2 (T + t)
exp - 2 exp - 2 dT
J [ D.t pump ] D.tprobe
-(X)
J1i tpump tprobe [ t ]
V D.t ump + D.t robe exp - (D.t ump D.t robe )
with the pulse width 13
(11.9)
D.t c = V D.t ump + D.t robe
(11.10)
l2Equation (11.8) is valid until the inverse value of the pulse fundaI11ental frequency, v f 1 , is 111uch shorter than
the pulse width, t » v fl. Otherwise the c0111plete ti111e dependence of the pulses electric field should be used.
In the optical wavelength range (v f 1 rv 2 fs) this 111eans that the pulse width 111USt be t > 10 fs.
13The ratio before the exponent in eq. (11.9) is a constant due to the areas of pulses under the integral. The
ratio affects the aI11plitude of the pulse but not its width. The pulse width is determined by the exponent argu111ent
only.
202
Pump-probe
1.54
q
'-' 1 52
:::: .
o
.,....,
.......
u
C'j
1
(l) .5
I-<
o
><
(l)
"g 1.48
1.46
200
300
400 500 600
wavelength, nm
700
800
Figure 11.9: Wavelength dependence of the index of refraction of fused quartz. The depen-
dence was plotted using data from [17].
If tpump == tprobe == t, then tc == V2 t. For example, using a laser system which
generates e. g. 100 fs pulses the time resolution of the pump-probe method cannot be better
than 140 fs.
It is important to notice that in eq. (11.1 0) tpump and tprobe are the pulse widths
inside the sample, but not the pulse widths at the entrance of the pump-probe instrument
(e. g. before the beam splitter M1 in Figs. 11.1 and 11.2). The pulses become broader
during their propagation to the sample and inside the sample. The reason for the broadening
is the group velocity dispersion and relatively wide spectrum of ultra short pulses. The
effect of pulse broadening is getting important at a picosecond pulse width and it is usually
the main reason limiting the time resolution in the femtosecond time domain.
11.4.1 Group velocity dispersion
To evaluate the effect of the group velocity dispersion let us consider a short light pulse
propagating in a fused quartz. A dispersion curve, i. e. the wavelength dependence of the
refraction index, of a fused quartz is shown in Fig. 11.9. The difference in the index of
refraction between two wavelengths Al and A2 is n12 == n(Al) - n(A2) == nl - n2. It
determines the difference in the propagation velocity of the pulses at these wavelengths. If
two pulses at Al and A2 enter a quartz plate of thickness l simultaneously, after the plate
they will propagate with the relative delay
t == n12l
c
(11.11)
For e. g. 1 cm quartz plate the delay between the pump and probe pulses can be greater
than the typical pulse width of the lasers used in pump-probe instruments, as illustrated in
Example 11.4.
11.4. Time resolution
203
Example 11.4: Relative delay between pulses in quartz medium. Suppose two pulses at
Al == 400 nm (the blue wavelength) and A2 == 600 nm (the yellow wavelength) enter
simultaneously a quartz plate of thickness l == 1 cm. The difference in refractive
index between these wavelengths is n12 == nl - n2 == 0.012 (as can be estimated
from Fig. 11.9). Thus, after the plate the delay between the pulses will be t ==
.6.n12Z == 400 fs.
c
Similarly to quartz, all other materials transparent in the visible part of the spectrum
have their own dispersions resulting in slower propagation of the blue wavelengths relative
to the red wavelengths. 14
Not only the delay between pulses at different wavelength, but also the pulse widths
depend on the medium dispersion, since the pulses have certain spectrum widths. For a
quantitative estimation of the pulse broadening let us consider a pulse with relatively narrow
spectrum so that we can assume linear dependence of the refractive index on the wavelength.
The slope of this dependence, D == , is called dispersion. If the spectrum width of the
pulse is A, the refractive index variation within the pulse spectrum is n == AD. The
propagation time delay between wavelengths corresponding to the "blue" and "red" sides
of the pulse spectrum is
t t""Vl A D
c
(11.12)
where l is the traveling distance.
For the pump-probe instrument implementing optical scheme presented in Fig. 11.2 the
components which contribute to the pulse broadening are all lenses, the sapphire plate of
the white continuum generator, the filter and the sample. Also non-linear crystals, such as
the second harmonic generators, contribute to the pulse broadening. The following example
is an illustration of the pulse broadening estimation in quartz.
Example 11.5: Estimation of pulse broadening. Let us suppose that a spectrum limited
pulse with t 1. == 50 fs width is crossing a l == 1 cm quartz plate, and the pulse
2
central wavelength is A == 400 nm (the second harmonic of the Ti:sapphire laser).
Dispersions of quartz at 400 nm is D == t""V 10- 4 nm- l . The spectrum width
of the pulse can be estimated using eq. (B.5) (see Appendix B on page 291 for the
discussion of the relation between the pulse width and spectrum width), A1. t""V
2
2 Z.6..\l
0.88 : t t""V 10 nm. Thus the expected pulse broadening is tb t""V -----2.. D 30 fs,
c 1 C
"2
and the expected pulse width after the plate is f; == t + tb 80 fs.
14In general the wavelength dependences of the refractive indexes are not l11onotonic functions. However, in
the visible and near UV wavelength ranges 1110St of the optically transparent 111aterials have l11onotonic refraction
index dependences on the wavelength si111ilar to that shown in Fig. 11.9.
204
Pump-probe
As can be seen from the example, a 50 fs pulse is almost 2 times longer after passing
through 1 cm quartz plate. The spectrum width of shorter pulses is greater, and the broad-
ening is stronger. Longer pulses have narrower spectrum and the broadening is weaker. For
example a 0.5 ps pulse may have spectrum bandwidth as narrow as 1 nm,15 and the broaden-
ing of such pulse will be just few femtoseconds, or practically indistinguishable. The effect
of pulse broadening starts to play a limiting role in time resolution when the pulse width
approach 100 fs value and it is one of the most important factors for pulses much shorter
than 100 fs.
11.4.2 Effects of sample and optics on time resolution
The pulse broadening should be taken into account during the pump-probe instrument de-
sign. Namely, there should be as little as possible condensed media in the paths of both
pump and probe pulses, i. e. there should be as little as possible lenses and filters, and their
thicknesses must be as small as possible. In particular, the lenses can be replaced by mirrors.
There are however components which do broaden the pulse but cannot be avoided. The first
such component is the sample. The light must cross the sample and it will be broaden by
the sample. The second component is the white continuum generator. There may be other
necessary parts, such as second harmonic generators, which have to be in the path of the
pump or probe beams and will contribute to the pulse broadening.
Considering the above the samples for the pump-probe experiments are usually as thin
as possible. Similarly the media for white continuum generation, non-linear crystals, lenses
and other parts which should be on the way of the beams are selected to be as thin as
possible, sometimes compromising their performance. 16
An estimation of the time resolution for two color scheme shown in Fig. 11.2, assuming
50 fs pulses on the entrance, is given in Example 11.6. The estimation shows that the time
resolution is more than two times worse than can be expected from the pulse width. Also
this is an overestimation of the time resolution as the actual scheme will most probably
include some other optical components, such as a gray filter to adjust excitation density or
a lens to form a probe spot of a suitable size, which where not taken into account but will
increase the actual pulse broadening.
Example 11.6: Estimation of time resolution. Let us consider the scheme presented in Fig.
11.2 with the following characteristics: pulse width at the entrance t 1. == 50 fs,
2
the pump wavelength is Al == 800 nm, the probe wavelength A2 == 500 nm, the
probe spectrum width A2 == 20 nm (which is the bandpass width of the filter),
thickness of the lenses L 1-L3 l L == 2 mm, sapphire plate thickness ls == 5 mm,
thickness of the beam splitter M1 is lMl == 1 mm, and the sample is a solution
placed in a quartz cuvette of thickness lsol == 2 mm and having wall thicknesses
lc == 1 mm. For this estimation we will assume the dispersion of all components to
be Dl == 2 X 10- 5 nm- l and D 2 == 6 X 10- 5 nm- l at 800 and 500 nm, respectively
(which is dispersion of quartz).
l5This is the 111ini111u111 bandwidth of 0.5 ps pulse which is due to its duration, although there 111ay be SOl11e
other reasons for a greater spectru111 width.
16Por eXa111ple, the efficiency of the second harmonic generation is higher for thicker crystals.
11.4. Time resolution
205
l
s
Figure 11.10: Pair off axis mirrors (M1 and M2) can be used to focus and then collect the
light without affecting the pulse width.
First of all let us estimate the pump pulse width in the center of the sample.
The spectrum bandwidth of the input pulse is Al t""V 0.88 : t 2 t""V 40 nm. The
2 c 1
2
total thickness of the condensed medium on the path of the pump pulse is lpump ==
lMl + lL + lc + lsol == 5 mm. The expected pump pulse width is tpump ==
lpump .6..\.1
t + c 2 Dl 50+13==63fs.
F or the pump pulse let us check the shortest possible pulse width if the spectrum
width is A2 == 20 nm. According to eq. (B.5), t 0.88 c;: 1 40 fs, thus the
2
probe pulse width is determined by the width of the input pulses but not by the band
limiting factor of the filter. 17 Therefore we will assume initial pulse width at 500 nm
to be t 1 == 50 fs and will neglect the broadening by the lens L1. The total thickness
2
of the disperse media on the path of the probe is lprobe == ls + lL + l f + lc + lsol ==
11 mm. The pulse broadening is tb == lpTo e.6..\ D 2 55 fs, and the probe pulse
width inside the sample is tprobe == t 1 + tb 105 fs.
2
Finally, according to eq. (11.10) the time resolution is I1t = V I1t ump + I1t;robe ::;::j
120 fs
There are few measures which can help to improve the time resolution when using pulses
shorter than 100 fs. First of all the lenses can be replaced by so-called off axis mirrors. Fig.
11.10 shows how the light can be focused on e. g. sample and then collected after the sample
using a pair of off axes mirrors. The mirrors must have parabolic reflecting surface so that
there would not be any distortions of the wave front.
The pulse broadening due to dispersion results in a "coloring" of the pulse - the red
part of the pulse spectrum is propagating faster than its blue part. This broadening can be
compensated (compressed) using a prism compressor which is schematically presented in
Fig. 11.11. After the first prism the pulse is spread in spectrum so that the blue and red parts
propagate at slightly different angles, but after the second prism the blue and red parts are
traveling parallel to each other. They are reflected back by a mirror (M) and after crossing
both prisms the pulse has its original cross section, but because the traveling distance is
17 A narrower filter, e. g. .6.A = 10 nm, would increase the pulse width by limiting its spectrum. The estimation
shows that there is no reasons to use filters with the bandwidth smaller than 20 nm for the given specifications.
206
Pump-probe
output
input
red
M
blue
Figure 11.11: Prism compressor. M is a mirror, and PI and P2 are prisms.
different for the blue and red parts the scheme can be adjusted to compensate the difference
in the optical paths of the red and blue parts in dispersive media such as lenses, sapphire
plates and others. In fact, the prism compressors are used inside the Ti:sapphire generators
to compensate the pulse broadening in the Ti: sapphire crystal. The prisms can be made in
such way that the beams enter and leave them at Brewster angle, so that there is no reflected
beam if the polarization of the light is in plane of the scheme. This means that there is no
light losses, and that is the reason, why the prism compressors are widely used inside laser
resonators.
11.4.3 Measurements of the delay spectrum
A relative delay of the probe pulse measured as the function of the wavelength is shown in
Fig. 11.12. The measurements were carried out using two photon excitation of benz on it rile
at 400 nm. Benzonitrile was placed a in 1 mm rotating cuvette, and was used as the solvent
to study compound of interest in the following experiments. 18 The photo-induced absorp-
tion of benzonitrile has a broad spectrum and is formed instantly after the excitation (as
illustrated by the inset in Fig. 11.12). The transient absorption spectra were collected with
the delay time steps of 100 fs. The measured data were analyzed using a global fit proce-
dure utilizing convolution with the instrument response function. Among others, the fitting
routine determines relative delay of the probe at each wavelength, which is presented in
Fig. 11.12. The dependence in Fig. 11.12 is almost inverse image of the curve in Fig. 11.9,
which is reasonable since the delay shows integral effect of the dispersions of all the optical
component on the way of the probe pulse most of which were made of quartz.
Comparing Figs. 11.9 and 11.12 one can estimate that there is roughly 4.5 cm of quartz-
like media (lenses, filters, second harmonic crystal and others) on the way of the probe
pulse. During this experiments the base pulse duration was approximately 50 fs (FWHM)
and the final time resolution was roughly 150 fs.
18Naturally, for the compound studies the excitation energy was reduced to a level when two photon solvent
excitation is not observed.
11.4. Time resolution
207
1
0.5
0
0
r.n 0
p., 0.02 0 0
>:
ro
.....,
<1 0.01
'"d -0.5
0
-1 0 1 2 3 4
0 time, ps
-1.5 0
400 500 600 700 800 1100
wavelength, run
Figure 11.12: Probe pulse relative delay as the function of wavelength measured using two
photon excitation of benzonitrile at 400 nm. Insert shows the time profile of the transient
absorption (circles) and its fit (solid line) at 695 nm.
11.4.4 Can it be faster?
In previous Sections different complications arising from the utilization of short sub pi-
cosecond pulses were discussed, but non of them puts a principle limit in the time resolu-
tion. So what is the shortest time duration which can be resolved in optical spectroscopy?
The laser pulse duration can be as short as 6 fs at fundamental harmonic of the Ti:sapphire
lasers (around 800 nm) [4, 5], which is much shorter than any time durations discussed
above. Can one achieve e. g. 6 fs time resolution in pump-probe experiments? In this
section the time resolution will be discussed from the viewpoint of the fundamental limits,
which are imposed by the samples rather than the modem state of the laser physics.
The uncertainty principle
The Gaussian pulse with duration t has spectrum width A 2: 0.88 c t (see Appendix
B). For instance, if the Ti:sapphire laser generates 50 fs pulses the spectrum of those pulses
cannot be narrower than 40 nm, and the 10 fs pulses will have roughly 200 nm spectrum
width.
On the other hand, the pulse spectrum width is the spectrum resolution of an ideal
pump-probe instrument. The relation between the bandwidth and pulse width can be ap-
plied equally the other way around - if the spectrum width is limited by A, then the time
resolution cannot be better than
A 2
t 2: 0.88 C A
(11.13)
For example, setting the spectrum resolution to 20 nm one cannot expect the time resolution
to be better than 100 fs if the measurements are carried out at 800 nm.
208
Pump-probe
The relation between the time and spectrum resolutions is very fundamental and can be
also considered on the basis of the Heisenberg uncertainty principle. The principle states
that the product of uncertainties in time and energy cannot be smaller than n/2,
n
E t > -
- 2
(11.14)
where E is the energy uncertainty, n == 2: ' and h is the Planck constant. In ultra fast
spectroscopy experiments we want to be very "certain" about (photon) time, i. e. the time
uncertainty, t, should be as small as possible. Then we do not know the energy of the
photon accurately, which can be determined with accuracy of E 2 t in the best case.
The photon energy is E == hv, and uncertainty in photon energy means uncertainty in
photon frequency, E == h v. This gives quantum mechanical formulation of eq. (11.13)
1
t v > -
- 47r
(11.15)
The relation between the frequency and wavelength uncertainties is A v Ac 2 , which
gives t > 4 . c7:A . The classic calculations (used in Appendix B) and the latter quantum
mechanics estimation give fundamentally the same result. The difference in coefficients is
due to the fact that Gaussian pulse shape was used in classic calculations (which is not the
best to obtain the smallest uncertainties product) and that the widths in Appendix B were
calculated as full width at half maximum (FWHM) values. The latter is a practical approach,
which however differs from the mathematical Gaussian pulse width by factor 1.67.
From the sample perspective, the question of spectrum resolution, or uncertainty in the
excited state energy, becomes important in sub picosecond-femtosecond time domain. For
example, the absorption spectrum width of Rhodamine 6G dye in ethanol is Ar 40 nm
and the maximum is at A 530 nm (see Fig. 6.5 on page 116). This corresponds to the pulse
width of t 30 fs. If the dye is excited by a shorter pulse, thus having wider spectrum,
only the spectrum part with Ar will be absorbed. This is equivalent to the narrowing of the
pulse spectrum by passing it through a band pass filter, therefore the efficient pulse duration
will not be smaller than t 30 fs.
The systems with broader absorption spectra can be studied with higher time resolution.
This is usually the case of the solid state physics and supramolecular photochemistry, where
the inter-chromophore interactions result in formation of new electronic states with broad
spectra. Yet another interesting example of the high time resolution applications is the
electron solvation dynamics in liquids, e. g. water. In such experiments the UV pulses
are used and excitation is essentially a multi-photon process which helps to achieve shorter
efficient excitation duration. The photo-generated electrons have a very broad transient
absorption spectrum (i. e. with high energy uncertainty) and can be probed by a short broad
band pulses. 19
The discussed problem arises from the fact, that at pulse duration of a few tens of fem-
tosecond and shorter the harmonic wave treatment of the light becomes unsatisfactory. For
19To some extend these electrons can be considered to be free electrons, or electron plasma. That is the reason
for a very broad absorption spectrum.
11.4. Time resolution
209
Ti: sapphire laser the emission wavelength of 800 nm corresponds to the wave period of
2.7 fs. What can be said about the pulse width if the electric field makes only 2 vibrations?
Naturally, at a much shorter base wavelength, e. g. at A == 50 nm, a few femtoseconds,
means tens of waves and one can start to develop systems with even shorter attosecond
pulses, but this is not the optical wavelength range any more and goes beyond the scope of
this book.
Power density
To excite the sample a certain energy density is required. In case of molecular systems
the absorption cross-section, a, is the characteristic which allows to estimate the excitation
energetics. The excitation efficiency, cp, is the function of the pulse energy density, E,
cp == 1 - exp ( - !:v a) .20 We will consider the cases of relatively low excitation densities
giving less than 50% excitation efficiencies, and can use approximation cp !:v a. This
equation has a simple meaning: !:v is the photon density and a is the area the photon must
hit to excite the molecule. For example the absorption cross section of rhodamine 6G at
540 nm is a 1.6 x 10- 16 cm 2 and a pulse with the excitation density !:v == 1.6 X 10 15
photons per cm 2 will excite 10% of the molecules.
The energy required to excite the sample does not depend on the pulse width. However
the peak power density of the excitation pulse depends on the pulse duration, P Kt .
At shorter pulses the excitation peak power is higher. The following example provides
numerical evaluations of the power densities at different pulse durations at fixed excitation
energy density.
Example 11.7: Excitation energy and power densities. Absorption cross-section of Acri-
dine orange dye at 430 nm is a 10- 16 cm 2 (molar absorption E 27000
M- l cm -1). To excite cp == 10% of molecules in an optically transparent solution
the photon excitation density must be n == 10 15 cm -2, which is E == nhv
0.5 mJ cm -2. In typical flash-photolysis experiment the excitation pulse duration
is 10 ns, thus the excitation power density is 5 x 10 4 W cm- 2 . In typical pump-
probe experiments the pulse width is 100 fs and corresponding power density is
5 x 10 9 W cm -2, which is approaching two photon excitation threshold for some
solvents, e. g. benzonitrile.
The example above illustrates that at short excitation pulses the high power may result
in multi photon excitation phenomena (for discussion of the two photon absorption see Ap-
pendix C). The latter, when takes place in the solvent, is undesired phenomenon, since
the excited solvent will add its own transient absorption signal to the total response of the
sample. Therefore very short excitation «100 fs) can be used to study systems with rela-
tively high absorption cross-sections (molar absorptions). The shorter the pulse the higher
cross-section should be to avoid multi photon excitation of the environment. 21
20Here cp is the probability to excite the molecule.
21 Multi photon excitation can be used to excite the sample. In some cases it has advantages of utilization of
longer excitation wavelength and higher spatial localization of the excited volume.
210 Pump-probe
11.5 Sensitivity
The sensitivity of the methods (both mono- and two-colors) can be rather high. It depends
on inaccuracy of the signal determination, S ( t). Ideally, calculations of the signal as
the ratio of intensities using eq. (11.5) should make the result, S ( t), insensitive to the
pulse-to-pulse variation in the energies. The limiting factors are different types of noises,
e. g. quantum noise given by the number of the detected photons and the photo-detector
thermal noise. To improve the accuracy, one can collect the signal during a relatively long
time (thus reducing quantum noise) or apply modulation-synchronous detection technique
in a way similar to one discussed with the steady state absorption measurements. These all
allow one to achieve sensitivity better than 10- 4 (in absorbance units).
A big advantage of the pump-probe method (in comparison to flash-photolysis) is that
one does not even need to resolve individual probe pulses: only the average pulse energy
at a certain position of the delay line is important for the accurate measurements. Also the
sensitivity of the pump-probe does not depend on the time resolution, since the number of
photons collected to measure the light intensity after the sample does not depend on the
widths of the pump or probe pulses.
11.6 Application example
Application of the pump-probe methods has influence on many areas of natural sciences.
One recent example is femtochemistry and the Nobel Prize in Chemistry 1999 "for his stud-
ies of the transition states of chemical reactions using femtosecond spectroscopy" awarded
to Ahmed H. Zewail. A review of the femtochemistry adopted from Zewail's Nobel Lecture
was published in The Journal of Physical Chemistry A [18].
Two examples of working pump-probe instruments and some technical aspects of the
measurements carried out with these instruments are discussed in this section.
11.6.1 Photo-induced charge transfer in molecular dyad
As the first example of pump-probe application a study of the photo-induced charge sepa-
ration in phytochlorin-fullerene dyad will be considered [15]. The study was carried out in
the Institute of Materials Chemistry at Tampere University of Technology, Finland.
The phytochlorin chromophore has strong absorption band at 420 nm. Therefore the
second harmonic of the Ti: sapphire laser can be used to excite the dyad. A scheme of
the laser system used for the study is presented in Fig. 11.13. The 50 fs pulses were
generated by the Ti:sapphire laser pumped by CW Ar laser. The Ti:sapphire laser was tuned
to operate at wavelength of 840 nm. The pulses were amplified utilizing stretcher-pulse
picker- Ti:sapphire amplifier-compressor scheme with pumping source being a Q-switched
N d: YAG laser. The repetition rate of the amplified pulses was 10Hz and the pulse energy
rv 0.3 mJ. The amplifier output beam was divided in two parts. The first part, roughly 10%,
was passed to the second harmonic generator to produce excitation pulses at 420 nm, and
the second part was focused on 4 mm sapphire plate to generate white continuum used as
the probe pulses.
11.6. Application example
211
50 fs
840 nm
90 MHz
4-5W I I
Ti:sapphire generator .. CW Ar laser
I 420 nm
SHG ..
pump
Nd:Y AG laser
Ti: sapphire amplifier
60 fs
0.5 mJ
10 Hz
10 Hz
532 nm
10 mJ
I CG I
continuum
..
probe
Figure 11.13: Scheme of the laser system used to study transient absorption of a
phytochlorin-fullerene dyad. SHG is the second harmonic generator, and CG is the white
continuum generator.
The measurement part of the instrument was similar to that shown in Fig. 11.3. To
divide the probe beam into signal and reference the collimated white continuum from the
sapphire plate was directed to a 1 cm quartz plate as shown in Fig. 11.14 (mirrors M 1 and
M2 in Fig. 11.3 are different surfaces of the quartz plate). The reflection from the front
surface was used as the signal beam and the reflection from the rear surface as the reference
respectively. The signal and reference beams have the same spectra and intensities as the
reflectance of the surfaces depend on the refractive index of the plate but does not depend
on the side the beam approaches the surface. 22
The spectra were recorded using cooled CCD
detector and correction procedure described in Sec-
tion 11.1.3 (see eq. (11.7) and comments on it)
was used to improve data quality. In a single mea-
surement 10 pulses were averaged on CCD detector
(during one second), also the energy of each pulse
was controlled by a separate photodiode. To im-
prove the data quality 10 measurements (spectra from
the CCD) were averaged at each delay time. Typi-
cal noise level of the data was about 10- 3 (in ab-
sorbance units). The time resolution of the instru-
ment was 150-200 fs depending on the solvent used.
A few raw spectra measured for the dyad in ben-
zonitrile were presented in Fig. 11.4 on page 192.
The complete series of measurements in one wave-
length range consisted of 60-65 spectra measured with 100 fs step at first 3 ps and followed
by exponentially increasing steps to cover the whole delay time of interest. For the numeri-
cal analysis of the data, the spectra were converted into differential absorption time profiles
with step A == 3 nm. These time dependences were fitted globally to a sum of exponents,
signal
reference
f
quartz
plate
Figure 11.14: Scheme of splitting
the the white continuum on two
equal signal and reference beams.
22The reference beam travels more than 2 cm in quartz and is broadened due to the quartz dispersion. However
the pulse duration of the reference is not important for the time resolution of the measurements.
212
Pump-probe
i. e. the model decay functions were j(t,A) == L:ai(A)e-t/T , were Ti are the lifetimes
and ai (A) are the corresponding pre-exponential factors. The fitting routine included con-
volution with instrument response function r (t), and the actual fit function included probe
delays t(A) caused by the group velocity dispersion (at each wavelength)
t
F ( t, A) = J r (x - /:::,. t ( A ) ) f (t - x, A) dx
( 11.16)
-CX)
so that the fit results were corrected for the dispersion and the instrument time resolution. 23
The output of the fit procedure are so-called decay component spectra ai (A) and the corre-
sponding lifetimes, Ti which are presented in Fig. 11.15 (top plot). The decay component
spectra are analogous with the emission decay associated spectra discussed in Section 8.5.2,
but applied to the time resolved absorption analysis. They can be used to reconstruct the
differential absorption spectra if the reaction scheme is established (see Section 15.3 for
discussion of the routine).
The bottom plot in Fig. 11.15 shows the differential absorption time profiles at a few
wavelengths together with the fitted curves. The strong signal at 670 nm is due to the
photo-bleaching and recovery of the ground state absorption band of the phytochlorin band
(similar to chlorophyll dyad shown in Fig. 11.4).
The photo-chemistry of the dyad was found to be rather complex. A number of inter-
mediate states are formed in a picosecond time domain. In order to identify the states and
to establish the reaction scheme, the time resolved spectra were recorded in the visible and
near infra red (not shown) parts of the spectrum, and fitted together. Four intermediate states
involved in the relaxation of the excitation were identified as locally excited phytochlorin,
locally excited fullerene, intramolecular exciplex and intramolecular charge separated state.
The excitation populates the phytochlorin excited state, P* F, which can transfer its en-
ergy to the fullerene thus forming locally excited fullerene, P* F ---+ P F*, or to form the
intramolecular exciplex, P* F ---+ (P F) *. The locally excited fullerene is relaxing by for-
mation of the exciplex too, P F* ---+ (P F) *. The exciplex is a precursor of the charge
separated state, which is formed with the time constant rv 20 ps, (P F)* ---+ p+ F-. The
lifetime of the charge separated state is roughly 70 ps. This gives a complete description of
the photochemical reactions taking place in the dyad.
For this study the time resolution of 200 fs and accurate measurements of the time
resolved spectra were essential to (1) establish the number of the intermediate state, (2)
identify the states, and (3) find the reaction passways taking place at the event of photon
absorption.
11.6.2 Pump-probe study of thin films
The second example will illustrate the high sensitivity of the pump-probe method. The
measurements were carried out in Department of Chemical Physics at Lund University,
Sweden. The studied objects were thin films of titanium dioxide (Ti0 2 ) nanocrystallines
23The spectrum of the delays, .6.t(A), was shown in Fig. 11.12.
11.6. Application example
213
0.2
0.1
0 0.0
0
-0.1
-0.2
-0.3
550
0.03
0.02
0.01
0.00
-0.01
o
o -0.02
-0.03
-0.04
-0.05
-0.06
0°.
. 0
o
Q
9
q
++
000
\
\
6.
I
\ I
\ I
6. I
\
I I
4
I
I
I
-[]- 0.32 ps
....0 8.2 ps
- - - 6.- - - 20 ps (fixed)
m.m+mm_ 69.5 ps
4-
4. I
,t.
6.
600
650
700
750
wavelength, nm
rlx.xy,y, 748 nm
'X.
X. .
x . .)(.
)!(
x
x
.)(.
x --------------
.... -i-.;!( .
",+, ....
J'(
!II
+ .-
,
[J
682 nm
661 nm
o
20
40
60
80
100
time, ps
Figure 11.15: Transient absorption decay component spectra of phytochlorin - fullerene dyad
in benzonitrile (top) and the corresponding decay curves (bottom) at 661, 682 and 748 nm.
Reproduced from ref. [15] by permission of American Chemical Society (ACS). @ ACS,
1999.
214
Pump-probe
0.0
SE
5' -0.5
o
.s
-1.0
.1.5
-150
o
1 50 300
Time [fs]
450
600
Figure 11.16: Pump-probe measurements of RuN3- Ti0 2 thin films. The fast ( 30 fs) decay
of (negative) differential absorption was attributed to stimulated emission, and the remaining
relatively long lived signal to the photo-bleached ground state absorption. The figure is
reproduced from ref. [16] by permission of American Chemical Society (ACS). @ ACS,
2002.
photo-sensitized by ruthenium complexes (RuN3) [16].24 The films had roughly one micron
thickness and were prepared on 0.06-0.08 mm thick microscope cover slips. The laser
system consisted of a Ti:sapphire laser, amplifier, and optical parametric amplifier, which
provided excitation pulses at 530 nm and probe pulse at desired wavelengths with duration
rv 30 fs.
The absorption cross section of the dye at the excitation wavelength is a == 5 x 10 -17 cm 2 .
The excitation photon density was P rv 10 14 cm -2, which gave roughly cp a P 0.5%
excitation efficiency, i. e. 0.5% of the dye molecules are excited by the single excitation
flash. This low excitation efficiency is important for this type of measurements since the
density of the dye molecules is high and interaction of two excited dyes may result in fast
excitation quenching also called exciton annihilation. 25 Therefore, it was necessary to keep
low excitation efficiency to avoid the interaction between the excited dyes.
An example of the pump-probe measurements of the films is shown in Fig. 11.16.
The figure illustrates a high sensitivity of the method - at 30 fs time resolution as small
as 10- 5 change in absorbance can be detected. 26 The fast ( 30 fs) relaxation of the signal
was attributed to the stimulated emission (see Section 1.2.2), and the longer living negative
differential absorbance to the photo-bleached ground absorption of the sample.
The difference between the stimulated emission and emission measured bye. g. fluo-
rimeter (see Chapter 6) or time correlated single photon counting instrument (see Chapter
24RuN3 is Ru( 4,4' -dicarboxy-2,2' -bipyridine)2 (NCS)2
25This phenomenon is known for antenna subsystems of natural reaction centers, polymer films and many other
molecular structures with short « 2 nm) inter-chromophore distances. The excitation can be transfered from one
molecule to another, which is called exciton (excited state) migration. And when two excitons collide, they may
annihilate.
26The vertical scale units, mOD, denote milli optical density change, i. e. 10- 3 , and the signal-to-noise ratio
is close to 100.
11.6. Application example
215
8) is that its intensity is proportional to the intensity of the probe pulse. Without probe pulse
there is no stimulated emission. The physical reason for the stimulated emission is the pop-
ulation inversion which leads to probe pulse amplification (see Section 1.3). Formally, the
light intensity after the sample is given by eq. (1.6), lout == line- al , where a is the ab-
sorption coefficient. There are two states involved in absorption or emission of a photon.
The coefficient a is proportional to the population difference between these states. If the
lower energy state has higher population the coefficient a is positive and the light intensity
is decreasing inside the sample, which is the light absorption. If the higher energy state has
higher population (the population inversion case) the coefficient a is negative and the light
intensity increases inside the sample, which is the light amplification or stimulated emission
case.
Interpreting the time resolved spectra and component spectra one has to distinguish be-
tween the absorption of transient states and stimulated emission, since the stimulated emis-
sion corresponds to the transition to the lower energy state, whereas in event of absorption
the transition to a higher energy state takes place.
The "ordinary" emission of the sample may disturb the measurements giving a kind of
background signal. However the emission intensity does not depend on the delay between
the pump and probe pulses, i. e. it has no time dependence on pump-probe measurements.
Chapter 12
Emission spectroscopy with optical
gating methods
From the example of pump-probe method one can learn that in order to achieve femtosec-
ond time resolution, which is approaching the pulse duration of modem lasers, the measure-
ments must be done using optical methods. In pump-probe instruments the optical methods
are used to generate two short pulses, pump and probe, at desired wavelengths and variable
delay between them. The probe pulse brings the information of the sample absorption and
the duration of the probe pulse determines the time resolution of the method.
In emission spectroscopy we need an optical method to probe the sample emission in
a very short time window, i. e. an optical method to cut a short time slice of the sample
emission, which can be measured then with relatively slow photo-detectors. This can be
imagined as optically controlled gate for the sample emission. There are few methods of
optical gating. The most widely used method is frequency up-conversion. It can provide
time resolution almost as good as pump-probe technique, but in most practical applications
it is much cheaper method. 1 This chapter is mainly devoted to the up-conversion technique.
Another method, which will be briefly reviewed here, utilizes optical Kerr effect.
12.1 Frequency up-conversion
12.1.1 Principles of up-conversion
An optical scheme of an instrument implementing the frequency up-conversion method is
shown in Fig. 12.1. At the entrance short laser pulses are split by mirror M1 in two parts.
The first part (reflected beam shown by the dashed line in the scheme) is used to excite
the sample and another part serves as the gate pulses. Since the typical pulse generator is
Ti:sapphire laser, which has central emission wavelength at 800 nm, it is convenient to use
the second harmonic for the excitation (e. g. the excitation wavelength is 400 nm). Then
the second harmonic generator (SHG) is installed in front of the mirror M 1 and the mirror
1 As will be discussed later, the up-conversion can work directly with Ti: sapphire generators, thus avoiding
amplifier part, which is almost unavoidable in pump-probe instruments.
217
218
Emission spectroscopy with optical gating methods
'------------------,
1 '
! Delay
! line
-----------------
SHG
vA
M2
-
,
,2v
,
Ll
& Sample
" ,,v em
from
laser
excitation pulse
A 2v
>
t
emISSIon
v
t
"a L2
Detection D NLC L3 ....
system " "
O V .e ... ------------ -------":,--/
<.- - - - - - ------.-,- -. -: - - . ' - - - -- -- - - - - - -- - - - - - - - -/ M3
r.'..... v M4 filter
gate pulse
AV
V d = V + v em
>
, : t
: & :. I I
:: -'. . : sIgna pu se
;. -',,: v
" A ' d
" : ------:::"
t
Figure 12.1: Scheme for frequency up-conversion emission measurements. SHG is the
second harmonic generator, NLC is a non-linear crystal, L1-L2 are lenses, M1-M4 are
mirrors and D is a diaphragm
M 1 is a dichroic mirror 2 reflecting the light at 400 nm (excitation, at 2v) and transmitting
the fundamental harmonic at 800 nm (gating, at v). The excitation beam is focused on to
the sample by lens L 1. The sample emission at V em is collected by the lens L2, filtered to
rej ect the excitation light and focused on to a non-linear (NL) crystal by lens L3. The gate
pulses are passed to the delay line and then directed to the same NL crystal. Both the gate
and the emission are focused on a small spot at the NL crystal by lens L3. The crystal mixes
the gate fundamental frequency, v, and the emission frequency, v em ' to generate the sum
frequency, Vd == v + V em (see Section 3.7.3). The phase matching condition for the efficient
sum frequency generation requires certain orientation of the NL crystal, which is achieved
by cutting the crystal at certain crystallographic orientation, and by angular fine tuning of
the crystal, to satisfy the phase matching condition for the gate and emission wavelengths.
The light intensity at Vd is measured by the detection system. Therefore, the frequency
of the detected signal is shifted up by value v relative to the emission frequency V em ' that
is why the method is called up-conversion. The detection system measures an average
emission intensity at wavelength corresponding to Vd, i. e. at the wavelength
Ad == (A- l +A; )-l
(12.1)
The detection system may consists of a monochromator, photomultiplier (working in photon
counting mode), discriminator and counter. Then the measured signal is the number of
counts during a fixed time interval, e. g. 10 seconds.
The key component of the scheme in providing femtosecond time resolution is the NL
crystal. There are two light pulses in front of the crystal: the emission pulse with the
2 Dichroic mirrors are mirrors which relfect the light at one wavelengths but pass the light at some other
wavelength.
12.1. Frequency up-conversion
219
time profile to be measured and the gate pulse, as presented in the second and third time
diagrams in Fig. 12.1. The signal at the sum frequency is generated only when both the
gate and the emission hit the crystal, i. e. only during the time interval when the gate pulse
and emission overlap each other. The resulting pulse (at v d) is shown in the bottom of the
time diagram. The light intensity at sum frequency, Vd, is proportional to the product of the
instant intensities of the gate and emission. Since the gate pulse intensity is constant, by
scanning the delay line (i. e. changing the delay t) one can probe the emission intensity
at different delay times, i. e. can measure the time profile of the emission with resolution
determined by the width of the gate pulse.
Now let us look at the measuring process in more formal matter. Right after the mirror
M 1 the excitation, lex (t), and gate, I 9 (t), pulses have the same timing (and the time pro-
files). The excitation pulse at the sample is delayed by the propagation time tex. It creates
an emission, which is given by function Iem(t - tex), which is
t t
lem(t) = J lex (T)D(t - T)dT = J lex(T)D(t - T)dT
(12.2)
-CX)
-CX)
where D (t) is the sample emission response to a delta-pulse excitation, i. e. D (t) == 0 at
t < 0 and D(t) > 0 at t > o. The integral is known in mathematics as convolution integral.
Actually, determination of the function D (t) is the goal of the whole study. Nevertheless,
the goal of the measurement procedure is to obtain the function Iem (t). At the NL crystal
entrance the emission is delayed by time tem and given by the function Iem(t - tex -
tem). The delay of the gate pulse is determined by the position of the delay line, td,
and by propagation delay from the mirror M1 to the delay line plus from the delay line to
the NL crystal, tg. Thus at the crystal the gate pulse is Ig(t - tg - td). Right after
the crystal the intensity of the light at the sum frequency, Vd, (the signal to be measured) is
proportional to the product of the intensities of the gate and emission (see Section 3.7.3),
that is
Id(t) == 7]sIg(t - tg - td)Iem(t - tex - tem)
(12.3 )
where 7]s is the efficiency of the NL crystal. Id(t) is a short pulse with the shape determined
by the gate pulse mainly since I em ( t) is a slow function of time as compared to I 9 ( t) (at least
for most of the measurements). The detection system measures an average pulse energy at
sum frequency Vd, e. g. counting photons at Vd. Thus the measured signal is
+CX)
U = s J ld(t)dt
(12.4)
-CX)
where s is the detector sensitivity.
One can substitute (12.2) and (12.3) to (12.4) and analyze the result. However, it is easier
to understand the principles if we will make two assumptions. First of all, let us assume the
gate pulse to be a delta-pulse Ig(t) == I g 6(t), which means that we are not working at the
instrument time-resolution limit and can neglect by the gate pulse width compared to the
220
Emission spectroscopy with optical gating methods
time duration of the studied signal, Iem (t). Since, the gate and excitation are essentially
the same pulses we can write Iex(t) == Iex6(t) and eq. (12.2) gives Iem(t) == IexD(t).
Secondly, the delays tex, tem and tg are determined by the geometry of the optical
scheme and they are constants for a given instrument. One of the goals of the instrument
design is to arrange the scheme so that tg == tex + tem. If this is done, we can count
time t form the moment of the gate pulse arrival to the NL crystal at td == 0, then eq.
(12.3) is reduced to Id(t) == 7]sIg(t - td)Iem(t). Accepting these simplifications
+ ex:>
U = s J 'f/sI g r5 (t - tltd)lem(t)dt = S'f/slglexD(tltd)
(12.5)
-ex:>
In other words, the signal is proportional to the emission decay function at the delay time
determined by the position of the delay line. Scanning the delay line one can measure the
time profile of the emission decay. 3
The efficiency of the NL crystal is proportional to the density of the light. This means
that the base and emission beams must be focused to the smallest overlapping spots. Thus,
the excitation spot must be as small as possible also, e. g. typically the lens L 1 has focal
distance of a few centimeters.
Different non-linear crystals can be used to generate sum frequency. To mention few,
there are LiI0 3 , KDP and BBO. 4 Probably the most popular crystal is BBO (j3-BaB 2 0 4 ).
BBO has good transparency range, 190-2500 nm, high damage threshold and can be used
with type I and type II phase matching conditions. In type I the polarizations of the gate
and emission are the same and corresponds to the ordinary ray polarization in the crystal
(see Section 3.7.1). To satisfy the phase matching condition the polarization of the beam
at the sum frequency has extraordinary polarization. In the type II configuration the gate
pulse has extraordinary polarization, polarization of the emission is ordinary and of the sum
frequency extraordinary. The efficiency of conversion is roughly two times higher for the
type I BBO crystals (at 800 nm) than that for type II. Therefore type I is a natural choice for
the up-conversion, also there are cases when type II phase matching has some advantages,
as will be discussed in Section 12.1.6.
An example of the emission up-conversion measurements of a molecular donor-acceptor
system is shown in Figure 12.2. The sample, phytochlorin-fullerene dyad in benzonitrile
[15], has a short lifetime of the singlet excited state, 430 :f: 40 fs, which was well resolved
in an experiment utilizing a 50 fs pulse generator, although the final time resolution of the
instrument was estimated to be roughly 120 fs (the time resolution will be discussed in
the following Section). The important property of the method is its high accuracy both
in time, which is determined by the mechanical translation unit of the delay line, and in
intensity which is the accuracy of measurements at a low intense steady state light, e. g.
3Within simplifications made above the measured signal, U, is directly proportional to the sample response,
D(i3..td), which is reasonable as we have assumed infinitely short gate pulse width, i. e. infinitely short time
resolution. If the gate pulse cannot be neglected the result, eq. (12.5), would be a double integral equation, first
accounting for the finite time excitation and second for the finite time gate pulse.
4 A good overview of the up-conversion method, which also covers different aspects of NL crystal properties
and applications, was published by J. Shah, see ref. [19].
12.1. Frequency up-conversion
221
2000 "
,
,
,
rfJ 1500 ' I
I I
I I
I I
0 I I
U I I
"'1000 I I
I I
I I
I I
500 I I
, I
I I
I I E:I
, \
\
0 0 1 2 3 4
time, ps
4
2
0
-2
-4
Figure 12.2: Emission decay of a phytochlorin-fullerene dyad in benzonitrile. The dots
present experimental data (signal intensities in counts collected during 10 seconds) mea-
sured with 25 fs steps. The solid line is the data fit (X 2 == 1.1) and the dashed line is the
instrument response function. The plot in the bottom shows weighted residuals of the fit.
provided by using the photon counting technique. As the result a half picosecond lifetime
was determined with the accuracy better than 10%.
12.1.2 Time resolution
The time resolution of the up-conversion method is determined by the same factors as that
of pump-probe method (see Section 11.4). It depends on the width of the base pulses, tp,
and pulse broadening by different optical components of the scheme, such as SHG and NL
crystals, sample and lenses. The broadening is caused by the group velocity dispersion and
is given by (similarly to eq. (11.12) in Section 11.4)
6.t n rv D(A) l6.A
c
(12.6)
where D(A) == d \)") is the dispersion at wavelength A, A is the spectrum width of the
pulse, and l is the total thickness of the dispersive material. Taking spectrum limited pulse
approximation (see Appendix B, eq. (B.5))
A 2
A 0.88
c t p
(12.7)
222
Emission spectroscopy with optical gating methods
500
400
@" 300
.......
,.......,
o
r:.rJ
(j)
200
(j)
S
.......
--
00
-
--
-
- ;. --'-':. .
. . .
-
--
-
--
100
. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50 100 150 200
fundalnental pulse width, fs
Figure 12.3: Time resolution of the up-conversion method (solid line) calculated according
to eq. (12.8) and assuming 9 == 1.6 X 10- 3 ps2cm-l (quartz at 400 nm) and l == 2 cm.
The contribution of the dispersion is shown by the dotted line, and of the fundamental pulse
width by the dashed line.
one obtains an estimation for the time resolution presented as the sum of two factors, the
base pulse width, tp, and the pulse broadening, tn,
0.88lA 2 l
ttotal t""V tp + tn t""V tp + D(A) 2 == tp + g(A)
c t p tp
(12.8)
where g(A) == 0.88 . D(A) .
In the frame of this estimation we have neglected (1) the wavelength difference between
the excitation and emission, (2) the contribution of the gate pulses to the time resolution,
and (3) limits imposed by the NL crystal, which will be discussed later in this Section. 5
Therefore, ttotal in eq. (12.8) can be treated as the lower limit for the resolution.
As an example the dependence of the time resolution on the fundamental pulse width
is presented in Fig. 12.3 assuming that the most of the dispersive media on the way of
the excitation-emission beam are fused quartz and the wavelength is close to be 400 nm,
i. e. t""V 10- 4 nm- l (see Fig. 11.9 on page 202) and 9 t""V 1.6 X 10- 25 s 2 m- l == 1.6 x
10- 3 ps 2 cm- l , and the total thickness of the quartz is l == 2 cm. Noticeably, a very short
pulse has very wide spectrum and dispersion contribution limits the time resolution of the
method. Therefore, there is an optimum pulse width at a given thickness of the dispersive
media, which can be found from condition d 1ttal == 0,
p
t Pt == J g(A)l
(12.9)
5 The effect of the gate pulse can be added by using eq. (11.10).
12.1. Frequency up-conversion
223
For the considered example with the total thickness of the disperse materials of 2 cm,
the optimum pulse width is t Pt t""V 60 fs, and the corresponding time resolution is
ttotal t""V 120 fs. 6 Also the same time resolution can be achieved with a bit longer pulses
but optimizing the optical scheme so that a smaller total thickness of dispersive materials is
used, e. g. for 80 fs pulse width l must be < 1.5 cm to provide 120 fs resolution.
The group velocity and pulse broadening problems are common for pump-probe and
up-conversion methods, therefore the discussion in Section 11.4 can be equally applied
to the up-conversion and optical gating in general. There are however specific aspects of
the up-conversion technique which can be essential for the fast time resolution. Namely,
the phase and group velocities mismatch in the NL crystal limit the spectrum width of
the up-conversion and contribute to the pulse broadening, respectively. Both effects are
proportional to the crystal thickness - the thinner the crystal the better time resolution can
be. Also thinner crystals have lower up-conversion efficiency, so a stronger power density
of the gate pulses is required at shorter time resolution.
Numerical estimations of the time resolution limitations imposed by the crystal can be
found in ref. [19]. In practice, to achieve 200 fs time resolution in the visible wavelength
range 1 mm crystals can be used, provided that the other components of the instrument do
not limit the resolution. To increase time resolution to 100 fs, 0.5 mm thick crystals are
usually used. Also the time resolution better than 45 fs was demonstrated using 0.02 mm
thick crystal and non-collinear phase matching with 20 fs pulses at 800 nm (fundamental
harmonic) as the gate pulse and 31 fs pulses at 400 nm ( second harmonic) for the excitation
[20].
In all other respects the time resolution of the up-conversion method is limited by the
same factors as that of pump-probe technique. Consequently, similar measures can be
considered to improve the time resolution:
. to use mirrors instead of lenses, e. g. the lenses L 1 and L2 in Fig. 12.1 can be replaced
by a pair of off axis mirrors (see Fig. 11.10);
. to use thinner NL and SH G crystal, this will however reduce the efficiencies of the
crystal;
. to use thinner sample, this depends on the nature of the sample and may be limited
by practical reasons, e. g. solubility of the compound to be studied.
In general, the up-conversion method, being almost as fast as the pump-probe, provides the
best time resolution among the emission spectroscopy techniques at present.
12.1.3 Evaluation of the instrument response time
An accurate estimation of the lifetimes or other quantitative values is more reliable when the
instrument response function is involved in the data fitting procedure, as it was done for the
data presented in Fig. 12.2. However, the experimental determination of the instrument re-
sponse is not as simple task as e. g. in the case of the time-correlated single photon counting
6 Adding the contribution of the gate pulse (eq. (11.10)) one can expect the time resolution to be
V 120 2 + 60 2 135 ps.
224
Emission spectroscopy with optical gating methods
2000
1500
rfJ
......
0 1000
;::::S
0
u
500
I
00 0 0
o 0
o 00 0
00 0
o
000
o 00 00
o 0 0
o
o
o
/\.:
I :\
f :: ;
+,1\ : \:
"' : X
1 : :: \
, ::
J 1 \ tt+
:;:: ;....----------------------
+
-0.5 0 0.5 1 1.5
time, ps
I
-2
Figure 12.4: Determination of the temporal response of the up-conversion instrument. The
measured signal of the sample is shown by the circles (lifetime » 1 ns). Derivative of the
signal ("experimental" instrument response) is shown by the crosses connected by the dotted
lines. The fit of the data obtained by convolution of the exponential decay with Gaussian
pulse is shown by the solid line and corresponding Gaussian pulse by the dashed line. The
fit weighted residuals are shown on the bottom plot (X 2 == 1.01).
method (see Section 8.5.1). In the latter case, the instrument response is usually measured
by tuning the detection wavelength to the excitation wavelength and placing a scattering
object for the measurements. Using similar approach in case of the up-conversion, one can
measure the instrument response using excitation pulse in place of emission. To do this,
the filter after the sample has to be removed, so that the excitation pulses are propagating
together with the emission to the NL crystal, and the crystal and detection system have to be
tuned to measure the signal at the wavelength corresponding to sum for the gate and excita-
tion frequencies, which is the third harmonic, v + 2v == 3v, for the scheme presented in Fig.
12.1. 7 However, the response function measured with this arrangement may differ from the
response function at the wavelength of actual measurements, since the time resolution of the
up-conversion measurements depends on the dispersion, i. e. depends on the wavelength.
Qualitatively the time response of the instrument can be determined from the emission
formation of a sample with a long lifetime of the excited state and under condition that
there are no fast processes which can affect the formation of the emission. An example of
such experiment is presented in Fig. 12.4. If one can neglect the decay of the signal in the
frame of the measured time window, then the function D(t) in eq. (12.2) can be replaced
7 This is a useful arrangement at the stage of the instrument alignment. The third harmonic signal is strong,
which makes rough system adjustment relatively easy. Naturally, a care should be taken not to overload the
detector. Also the delay must be fine tuned as both pulses, gate and excitation, are short.
12.1. Frequency up-conversion
225
by a step function 8 and the time derivative ofeq. (12.2) will give dle:ft(t) == alex (t), i. e.
the excitation pulse time profile can be obtained by taking derivative of the emission time
profile. The computed derivative of the signal is shown by the crosses (+) connected by the
dotted line in Fig. 12.4. As can be seen the computed pulse has a half width of rv 200 fs,
but it is too noisy to be used for quantitative handling of the measurements.
Alternatively, one can assume the excitation pulse to be a Gaussian pulse and use it in
place of lex (T) in eq. (12.2). The experimental data can be fitted using the Gaussian pulse
width as one of the fit parameters. The result of the fit is shown by the solid line and corre-
sponding "theoretical" pulse is shown by the dashed line. The Gaussian pulse and the signal
derivative are in good agreement with each other, giving essentially the same estimation for
the instrument response function (rv 200 fs), but the former is more practical as it allows
to obtain quantitatively accurate results without additional measurements. Gaussian pulse
approximation was also used to fit the data presented in Figure 12.2.
12.1.4 Sensitivity
The sensitivity can be estimated in a manner similar to that used for the steady state emission
measurements, but taking into account the methods specific losses. Let us consider an
excitation pulse which holds N ex photons (at vex) and count the losses during the pulse
propagation and transformation. We need to know the sample absorption coefficient, a,
and emission efficiency, cjJ, and the efficiency of the emission collection, 7]c. Then, at the
entrance of the NL crystal the number of emission photons is 7]cacjJNex. The crystal mixes
the emission and gate pulse and only those photons which are emitted in the time frame of
the gate pulse, tg, will be detected. Consequently, in the best case only tg / T part of the
emitted photons will contribute to the up-conversion signal at v d == V 9 + V em ' where T is
the emission lifetime. Additionally, the efficiency of the NL crystal, 7]8' and the spectrum
overlap factor (ratio of the spectrum widths of the detection and emission bands), cp, have
to be taken into account. Finally, adding the detector quantum efficiency, 7]d, the number of
detected photons is
tg
N d == -7]c7]87]d a cjJcpN ex
T
(12.10)
In the following Example this equation is used to estimate the counting rate of an up-
conversion experiment.
Example 12.1: Estimation of the up-conversion counting rate. Let us assume the experi-
ment conditions to be similar to those used previously in the emission spectroscopy
examples (Section 6.2.7): 7]c 0.01, 7]d 0.1, a 0.1 and cjJ 0.1 (i. e. the sample
has rather high emission efficiency). A reasonable (but not the best) up-conversion
efficiency of the NL crystal can be 7]8 0.01 and the spectrum overlap factor will
be taken to be cp 0.1. 9 If the emission lifetime is T 2 ns (typical for dye
8 Step or 8-function is 8(t) = Oatt < Oand8(t) = 1 att > O.
9For the steady state emission spectroscopy cp 0.01 was used. For example, the emission bandwidth of
rhodamine dye is roughly 50 nm, and cp 0.01 means that one likes to measure the spectrum with 0.5 nm
226
Emission spectroscopy with optical gating methods
molecules) and the gate pulse width is tg == 100 fs, then the ratio .6.: 9 == 5 x 10- 5
and N N d 5 x 10- 5 . 0.01 . 0.01 . 0.1 . 0.1 . 0.1 . 0.1 == 5 x 10 -13. If the second
ex
harmonic of a Ti:sapphire mode-locked laser is used for the excitation, i. e. the ex-
citation wavelength is 400 nm, the excitation pulse energy is Pex 10- 10 J (0.1 nJ)
and corresponding number of photons is N ex rv 2 X 10 8 . Therefore, in average we
should detect N d 10- 4 photons per each excitation pulse, which does not look to
be a big number. However, the repetition rate of the mode-locked lasers is usually
f 100 MHz, consequently, the average photon counting rate of the detection is
fd == f Nd == 10 kHz. Thus, e. g., 10000 counts can be collected in one second. This
is reasonably short time, which allows to measure the whole emission time profile in
less than ten minutes.
The number of detected photons per excitation pulse (N d rv 2 X 10- 4 in the above
example) deserves additional comments. What will happen if there are two detected photons
per pulse? Eventually, they will be counted as one photon since the detection system has
no time resolution in time frame of the gate pulse (50-200 fs). Therefore, it is important to
keep excitation energy at a level where the probability to generate 2 "detectable" photons
per one gate pulse is negligible. Thus, the average number of detected photons per gate
pulse must be much smaller than 1. In terms of the average counting rate this means that the
counting rate must be much lower than the pulse repetition rate. For the typical repetition
rate of Ti:sapphire lasers, 100 MHz, the photon counting rate should be 1 MHz or lower.
As it was discussed for the time correlated single photon counting (TC SPC) method in
Section 8.4.3 on page 158, the term $ in eq. (12.10) is the sample radiative rate constant,
kern == $ (see footnote 13 on page 159). The signal intensity is determined by the radiative
rate N d ex kern rather than by the emission quantum yield or the lifetime of the emitting
state. The emission yield can be very low, but that does not mean that the signal will be aslo
low. If the radiative constant is high (in the above considered example kern == 2 X 10 7 S-l),
then the signal must be strong, though it can be short-living.
Also the signal intensity dependence on the pulse width is predicted to be linear by
eq. (12.10), the actual dependence on the pulse width is not that simple. By increasing
the time resolution (shorter tg) one decreases the signal intensity (sensitivity), since a
smaller number of photons will fit in to a smaller time window tg. On the other hand,
the efficiency of the NL crystal depends on the peak power of the gate pulses, therefore
parameter 7]8 depends on pulse width either. If the pulse energy remains constant, the peak
power increases with decreasing tg, thus conversion efficiency of the crystal increases.
All in all one may expect only a weak dependence of the signal intensity on the pulse width
under otherwise the same conditions. 1o
In comparison to TCSPC the up-conversion method is clearly less sensitive meaning
that more excitation photons are needed to collect the signal of the same quality (the same
resolution. In the case of the up-conversion experiments with time resolution of e. g. 200 fs the spectrum uncer-
tainty is .6.-\ 6 nm (see Section 11.4.4 and Appendix B), therefore the reasonable spectrum resolution for the
up-conversion measurements is 6 nm, which leads to cp 0.1.
lOThis is, however, a rather theoretical consideration. In practice the pulse width is not an easily adjustable
parameter. The pulse width is the property of the laser system used, and the measuring system is designed to be
optimal for that pulse width.
12.1. Frequency up-conversion
227
signal-to-noise ratio). Although eqs. (8.2) and (12.10) look very alike (they are alike, in-
deed), one has to note that in the case of TCSPC the signal is "growing" simultaneously
at all channels, while in the case of up-conversion method (or any other gating technique)
the measurements have to be repeated sequentially for each time delay. Yet another rea-
son for the lower sensitivity is the time resolution, tg, which contributes directly to the
counting rate. The time resolution of the up-conversion method is typically hundred times
shorter than that of the TCSPC technique. This improvement in the time resolution means
proportional decrease in the sensitivity.
12.1.5 Excitation pulse energy
The efficiency of the NL crystal is proportional to the peak power density of the gate pulses.
Therefore the gate and emission beams have to be focused to a small spot on the crystal. The
divergence of the beam focused to the crystal cannot be large though, since there is certain
acceptance angle at which the phase matching condition is satisfied for the wave mixing. 11
To obtain a small emission spot at the NL crystal, the emitting area of the sample aslo has
to be as small as possible. Therefore relatively short focal length lenses are used to focus
excitation on the sample. For example, using 4 cm lens (L1 in Fig. 12.1) the spot size can
be 20-40 tL in diameter.
The small excitation spot means that the excitation pulse energy should be relatively
small. A numerical estimation given in Example 12.2 shows that for organic dyes excited at
maximum of their absorption bands the pulse energy of 1 nJ can already reach the saturation
excitation density. Therefore much lower excitation pulse energies are often used in up-
conversion experiments. 12
Example 12.2: Excitation pulse energy and sample saturation. If the excitation pulses are
focused in a spot with diameter of D == 20 tL, the pulse energy required to achieve
the excitation pulse density of E == 1 mJ cm -2 is P == E D 2 0.3 nJ. At wave-
length A == 500 nm this power density means photon density of n == !:v == ;
2.5 X 10 15 cm- 2 . This photon density corresponds to the saturation density for the
chromophores with absorption cross section of a == n- l == 4 x 10- 16 cm 2 (i. e.
molar absorption E 10 5 M-lcm- l ), which is typical absorption for organic dyes.
F or compounds with higher absorption cross sections the excitation pulse energy can
be reduced without significant loss in emission intensity.
The low excitation pulse energy means that the total number of emitted photons (per
pulse) cannot be high either. The estimation in Example 12.1 is close to the maximum count-
ing rate one can expect from an organic dye solution under conditions discussed above. 13
11 The acceptance angle is inversely proportional to the crystal thickness [19]. Thin crystals have greater ac-
ceptance angle, therefore the beams can be focused to a smaller sport using lenses with shorter focal distances.
12The excitation spot can be bigger (e. g. lens L1, Fig. 12.1, with focal length of 10 cm or greater can be used),
then the excitation pulse energy also can be greater. This is typical for the up-conversion instruments using pulse
amplifiers and working at lower repetition rates.
13 Also one has to keep in mind that in conditions of Example 12.1 the total number of emitted photons per
pulse is 2 x 10 6 , and accounting for the pulse repetition rate, 100 MHz, the sample emits 2 X 10 14 photons per
second. Thus the total power emitted by the sample is roughly 0.1 m W, which will look like a bright spot.
228 Emission spectroscopy with optical gating methods
12.1.6 Spectrum range
The excitation spectrum range is determined by the pulses available from the laser source.
For a laser system consisting of Ti:sapphire generator only, the range is limited by roughly
740-1000 nm for the fundamental harmonic and 370-500 nm for the second harmonic,
respectivel y.
The choice of the excitation wavelengths can be widened greatly by using a more
complex laser system, e. g. similar to one discussed in Section 11.2. Also an important
difference in requirements for the excitation laser systems used for pump-probe and up-
conversion applications is that the latter can operate at a much lower pulse energy but de-
mands higher repetition rate. To illustrate this one can reconsider Example 12.1 with the
pulse repetition rate of 1 kHz, which is a typical rate for multi pass amplifiers, instead of
100 MHz used in the example. Then the photon counting rate is dropped to 0.2 photons per
second, which is one tenth of the dark counting rate of the best photomultipliers. This is cer-
tainly too low value to be measured with any reasonable signal-to-noise ratio in a reasonable
time.
Regenerative amplifiers operating at a few hundreds MHz pulse repetition rate, and
followed by optical parametric amplifiers are common solutions for extending the excitation
wavelength range of the up-conversion instruments.
The emission wavelength range of the up-conversion method is limited mainly by the
sensitivity range of the detection system, but this is indirect limit in a sense, that the ac-
tually detected wavelength is shifted to the blue part of the spectrum relative to the actual
sample emission range. Assuming the fundamental harmonic of the pulse generator to be
800 nm (Ti:sapphire laser) and the detection photomultiplier to be equipped by a bialkali-
type photo-cathode, with the spectrum range 200-650 nm, the emission spectrum range is
270-3470 nm. 14 Technically the measurements can be done in the wavelength range from
the ultraviolet to infrared. There are however some additional limitations.
When the emission wavelength approaches the gate wavelength, the crystal orientation
gets closer to the orientation beneficial for the second harmonic generation when type I
synchronism is used (see discussion on page 220). The second harmonic generated by the
gate pulses is much stronger than the signal and is not well separated spectrally from the
signal. This makes the up-conversion measurements at wavelengths around the gate wave-
length impossible. The effect of the gate pulse second harmonic is seen as an increased
background counting rate (the counting rate with locked sample excitation). Typically it in-
creases sharply when the difference between the emission and gate wavelengths gets smaller
than 100 nm and the gate pulses wavelength is around 800 nm. 15
Type II synchronism works with orthogonal polarizations or the gate and emission, and
there is no efficient second harmonic generation of the gate pulses. 16 Therefore type II
synchronism can be used to measure emission decays at wavelengths closer to the gate
14Por the calculations eq. (12.1) can be rearranged to give Aem = (Ad 1 - A-I) -1 .
15This depends on e. g. the angle between the gate and emission beams and spatial filtration of the signal at
sum frequency.
16There is no phase matching conditions for the second harmonic generation for type II crystals. However
some second harmonic is always generated when the peak power density is high, as it used in the up-conversion
experiments.
12.1. Frequency up-conversion
229
wavelength than that for type I. To move even closer to the gate wavelength one can use
better wavelength and spatial filtration, e. g. using double monochromator in detection and
installing diaphragms to reduce the effect of the scattered light.
The conversion of the detected signals to shorter wavelengths makes possible measure-
ments of the infrared emission with detectors sensitive in visible part of the spectrum only.
This will be illustrated in Section 12.3. Yet another important practical consequence is that
the photomultipliers with photo-cathodes having relatively high red wavelength limit can
be used. For these photo-cathodes the work function is high and, consequently, the dark
counting rate is low. As an example bialkali-type photo-cathodes were mentioned above,
which can provide dark counting rate of 2-10 counts per second.
12.1.7 Time resolved spectra
Sequential decay measurements
Since the wavelength corresponds to the energy of transition taking place in the system un-
der study, the emission decay measurements at different wavelength may provide a valuable
information on the relaxation dynamics of the excited state. 17 Switching from one detection
wavelength to another involves usually two steps. First of all the new wavelength has to
be calculated (using to eq. (12.1)) and detection wavelength must be changed accordingly.
Secondly, the NL crystal has to be tuned to match phase synchronism condition for the new
emission wavelength. 18 Then the measurements can be repeated to obtain the emission time
profile at the new wavelength.
After repeating the measurements at a number of wavelengths, the two dimensional data
array can be collected. However, extracting the time resolved spectra from such measure-
ments is not as simple procedure as it is e. g. in case of the time correlated single photon
counting method. The correction spectrum can be obtained for the detector part, but the
spectrum efficiency of sum frequency generation by the NL crystal is very sensitive to the
gate and emission spots alignment, the gate energy and different types of chromatic aber-
rations in the system. This makes the previously discussed spectrum correction procedures
(e. g. see Section 6.2.4) very unreliable in case of up-conversion method.
To extract corrected time resolved spectra from a series of decays measured using up-
conversion technique the time resolved measurements can be complemented by the steady
state emission spectrum. If Ii (t, Ai) are the decays measured at wavelengths AI, A2, . . . , AN,
and the steady state spectrum of the sample at the same excitation wavelength is s (A), then
the sensitivity correction coefficients for the decay measurements can be calculated as
[ +cx:> ] -1
Ci = S(Ai) -L Ji(t, Ai)dt
(12.11)
17Por example, dynamic Stokes shift is typical for molecules characterized by relatively large difference in
dipole moments of the excited and ground states. The shift is due to changes taking place in the nuclear subsystem
of the molecule and local environment, which are much slower than the electronic transition and typically can be
observed in pico- and sub-picosecond time domain.
18Iflenses are used to collect emission from the sample and to focus it on to the NL crystal, as it shown in Pig.
12.1, one may need to fine-tune the light focusing to compensate chromatic obberations of the optics.
230
Emission spectroscopy with optical gating methods
The integral of the emission time profile gives the value proportional to the steady state
emission intensity at this wavelength. 19 Therefore eq. ( 12.11 on the preceding page) calcu-
1ates the ratio of the measured steady state emission intensity to the steady state intensity cal-
culated from the emission time dependence, and the sensitivity corrected time-wavelength
data can be obtained as
D ( t, Al) == Ci Ii ( t, Ai)
(12.12)
This data can be used to draw the time resolved spectra (by taking slices at constant time)
at relatively long delay times. At shorter delay time the group velocity dispersion should
be taken into account to correct the delay time at different wavelengths as was discussed in
Section 11.4.1 in application to the pump-probe method.
Spectra acquisition with thin NL crystals
If the emission spectrum is not very broad, the scheme presented in Fig. 12.1 can be opti-
mized to allow recording of the time resolved spectra at each delay time. The spectral band-
width of the NL crystal 20 is inversely proportional to the crystal thickness, i. e. narrower
crystals permit up-conversion of broader spectra. Also the crystals with low dispersion have
greater conversion bandwidth. These two parameters can be optimized to gain reasonably
wide spectrum region in which the whole emission spectrum of interest will be up-converted
and measured at shorter wavelengths. For example, N. P. Ersting and co-workers [21] have
used 0.1 mm KDP type II crystal gated at 1300 nm (dispersion is lower at longer wave-
lengths) to achieve roughly 10 000 cm -1 acceptance bandwidth for the emission centered at
550 nm, which corresponds to wavelength range of 430-760 nm.
Thinner NL crystal means lower conversion efficiency under otherwise the same con-
ditions. To compensate for the conversion efficiency losses due to the crystal thickness, a
higher gate pulse energy can be used. This requires NL crystals with high damage thresh-
old. The higher pulse energy means that output pulses from Ti:sapphire generator must be
amplified. In the experiments of Ersting group the 30 fs pulses were amplified to energy
of 0.5 mJ at the wavelength of 810 nm (Ti:sapphire amplifier) and at the repetition rate of
1 kHz. Then a parametric amplifier was used to obtain 60 tLJ gate pulse at 1300 nm. The
laser system for these types of applications was discussed in Section 11.2.
For the spectrum measurements the detection system has to be modified for simultane-
ous spectrum detection at each delay time. For example, the up-converted signal can be
passed to a spectrograph and then to a CCD detector. The detected spectrum has to be
recalculated back to the corresponding emission wavelengths, using eq. (12.1) after some
rearrangement. In the instrument mentioned above, developed by Ersting group [21], a
double-prism spectrograph and back illuminated CCD camera were used. The prism spec-
19The integral gives the number of photons detected at the wavelength A . To obtain quantitatively accurate
results the measured time window must be wide enough to see the complete decay of the emission, and the dark
counts must be subtracted.
20This is the wavelength bandwidth in which the phase synchronism is satisfied for the efficient frequency
up-converSIon.
12.2. Optical Kerr effect
231
trographs have an advantage of much lower light scattering as compared to grating ones. 21
The back illuminated CCD cameras have higher sensitivity in the UV spectrum region,
which is important, since the actually detected spectra are shifted up in frequency, i. e. to
the shorter wavelengths.
12.1.8 Commercial instruments and components
In a minimum arrangement, to build up an up-conversion instrument one needs a femtosec-
ond laser, e. g. Ti:sapphire mode-locked generator, delay line, NL crystal, detection system,
e. g. a photomultiplier counting module coupled with a monochromator, and a set of optical
components such as mirrors and lenses. All these components can be purchased separately
and assembled in to a working system.
Also just a Ti:sapphire mode-locked laser can be sufficient for a wide range of applica-
tions, one can refer to Section 11.2 for discussion of the laser system generating fentosec-
ond pulses in a wide spectrum range. As compared to the pump-probe application, a higher
pulse repetition rate is preferable for the up-conversion method, but usually the same laser
system can be used for both methods.
The most widely used crystal for the frequency up-conversion is BBO (j3-barium bo-
rate). It belongs to the negative (no > ne) uniaxial group. It has high damage threshold
(> 10 10 W cm- 2 ),22 high non-linear optic characteristics (conversion efficiency roughly
five time better than for KDP) and good mechanical properties. The crystals can be ordered
to be cut at crystallographic orientation most suitable for specific applications (second har-
monic generation, frequency up-conversion, parametric amplification and so on), and with
coated surfaces to reduce reflection at specified wavelengths.
The detection system typically consists of monochromator, optionally some color re-
jecting filters and photo-detector, which must measure very low light intensities. For low
light intensity measurements photon counting photomultipliers are the most suitable de-
vices. Also modulation-synchronous detection scheme were used.
There are also commercially available up-conversion instruments. As an example one
can consider FOG 100 fluorescence up-conversion system from CDP Corp. (Moscow). This
is a complete measuring system which can be coupled with different sources of femtosecond
pulses, and includes the second harmonic generator for a convenient use with Ti:sapphire
lasers. It can provide time resolution up to 100 fs (depending on the laser and sample).
The detection system consists of a set of color filters, monochromator and photon counting
photomultiplier module.
12.2 Optical Kerr effect
The frequency up-conversion is not the only method for optical gating, although it is the
most often used one. Another optical effect, which can be used to gate short light pulses, is
21 The most essential disadvantage of the prism spectrographs and monochromators is much lower dispersion,
which is not essential for the discussed application as the size of photo-sensitive elements of the CCD camera is
small, so even that small dispersion provides an acceptable spectrum resolution.
22 Assuming e. g. 0.1 mm spot size and 100 fs pulse width, the damage threshold is achieved at 7.5 J-LJ pulse
energy, which is much greater than the pulse energies considered in Examples 12.1 and 12.2.
232
Emission spectroscopy with optical gating methods
LI
Sample
- - - - - - - - - - - - - - - - - - - - - - - - - - - I
, ,
: Kerr shutter :
, ,
, ,
Excitation
pulses
L2 : PI
Kerr cell
P2: L3
F
Gate
pulses
-- V - -..
Mono-
chromator
CCD
L4
PI
delay
line
i /' P3
Lp2
Polarizer orientations
Figure 12.5: Optical scheme of time resolved emission spectra detection using optical Kerr
effect. L 1-L4 are lenses, P 1-P3 are polarizers, and F is a filter.
the optical Kerr effect.
In general, when a strong electric field is applied to a material, the refractive index of
the material changes. If the change is linearly proportional to the electric field, the effect
is called Pockels effect. If it is proportional to the square of the field, then it is called Kerr
effect. 23 In case of the optical Kerr effect the electric field is the field of the light, so this
is the light induced change of the material refractive index. In particular, the optical Kerr
effect is responsible for self focussing of the intense laser beams in materials, which is
widely used for the self mode-locking in Ti:sapphire lasers, and also known as Kerr lens.
An optical scheme of an instrument utilizing the optical Kerr effect for the time resolved
emission decay measurements in shown in Fig. 12.5. The sample emission is passed through
the Kerr shutter, which is controlled by the gate pulses. The Kerr shutter consists of a pair
of crossed polarizers, PI and P2, and a Kerr cell. Without the gate pulse the emission does
not pass through the shutter. The gate pulse is polarized at angle of 45 degrees relative to
the polarizers PI and P2 (see polarization orientations in Fig. 12.5). When the gate pulse
hits the Kerr cell, the refractive index is changed for the light polarization of the gate pulse.
This induces anisotropy in the Kerr cell with anisotropy axis determined by the gate pulse
polarization. The anisotropic Kerr cell changes the emission polarization so that part of the
emission can pass through the polarizer P2. The refractive index changes in wide spectrum
range, therefore the Kerr shutter operates in a broad spectrum range.
The Kerr shutter opens for a very short time and emission spectrum in that short time
interval is detected by a CCD detector coupled with a monochromator. Relative delay of
the gate pulse can be controlled by a delay line,24 so that the time resolved emission spectra
23It was discovered by John Kerr in 1875.
24 Similar to other femto- picosecond methods the relative delay may differ at different wavelengths due to the
12.2. Optical Kerr effect
233
can be measured at different delay times and used to obtain the emission time profiles in a
manner similar to that discussed for the pump-probe technique.
The efficiency of the Kerr cell depend on the power density of the gate pulse and on
non-linear properties of the Kerr cell material. The lenses L2 and L4 are used to focus the
emission and gate pulse to a small spot for efficient operation of the shutter.
One of the materials with strong optical Kerr effect is CS 2 liquid. Unfortunately the
response time of CS 2 liquid is 0.8 ps, which limits its application in modem instruments.
A number of glass materials were tested and a several percent efficiency of the Kerr shutter
was demonstrated with Bi 2 0 3 doped glass [22]. Using different types of glasses the time
resolution of the method can be close to 100 fs. Better than 100 fs time resolution was
obtained with 1 mm thick fused silica used as the Kerr cell, which has relatively small
dispersion and can be used at short wavelengths [23]. Another advantage of the fused silica
is the wide operational window rage, 350-1000 nm.
Except the efficiency of the Kerr cells, critical parts of the Kerr shutter are polarizers,
namely the degree of the emission rejection, as illustrated in Example 12.3. The highest
light polarization can be achieved with prism polarizers, but the prism polarizers have high
dispersion and a small angular aperture. Therefore plastic sheet polarizers are used in in-
struments with short (femtosecond) time resolution. Disadvantage of the sheet polarizers is
relatively low polarization degree, 10 4 _10 5 . 25 As the result, the up-conversion instruments
work better with samples that have short emission lifetime. For samples with relatively long
lifetime (> 1 ns), the background signal can be as strong as the spectrum to be measured.
Example 12.3: Estimation of the signal-to-background efficiency of Kerr shutter. Let us
suppose that the lifetime of the sample emission is T == 1 ns, the measured time
window (the width of the gate pulse) is t == 0.5 ps, the efficiency of the shutter
is cp == 0.1, and the emission rej ection factor of the closed Kerr shutter (crossed
polarizers PI and P2) is 7] == 10- 5 . If the total number of collected emission photons
is N em , then the number of "background" photons, i. e. number of photons which
will reach the detector with closed shutter is N bg == 7]N em . The maximum number
of gated photons is N s == N em t cp, which is the signal (time resolved spectrum).
Thus signal-to-background ratio is N N s == .6.tcjJ == 5.
bg TTJ
Similar to all measurements in femtosecond-subpicosecond time domain the group ve-
locity dispersion affects the relative delays at different wavelengths. The dispersion correc-
tion must be carried out to obtain actual time resolved spectra. The effect of group velocity
dispersion was discussed in Sections 11.1.3 and 11.4.3 for pump-probe applications, and
it must be accounted for the emission spectra measurements with Kerr shutter in a similar
way.
group velocity dispersion effect.
25Polarization degree is the ratio of the light intensity along the polarization axis to that perpendicular to the
axis, with un-polarized light in front of the polarizer.
234
Emission spectroscopy with optical gating methods
'------------------,
1 '
! Delay
! line
-----------------
v
from
laser
QW structure
M2
\
\
\
\
\
\
\
\
\
\
yLl
\
: : : : : : : : : : : V: em : : : : ":1
L2 Sample
900 nm
Detection D NLC L3
S o y stem vem _ ----
V ----
d _-- _ _ _ __ __
<------F:-----:-:-- -- --
I I
I I
6nm
Figure 12.6: Scheme of the up-conversion instrument for measurements of non-transparent
samples (on the left) and band structure of quantum wells studied by the instrument (on the
right).
12.3 Photo-dynamics of semiconductor quantum wells
As an example of the up-conversion method application a study of quantum well carrier
relaxation dynamics will be discussed [24]. The quantum wells (QWs) have numerous
applications in quantum electronics. In frame of this study QW s were used to fabricate
semiconductor saturable absorber mirrors, which are mode-locking elements of pulsed pico-
and femtosecond lasers. For this application the recovery time of the saturable absorber, i. e.
the carrier recombination time of the QW, is the important parameter, which has to be tuned
to fit to the laser specification.
The samples were grown on GaAs substrate by molecular beam epitaxy. There were five
6 nm thick Gao.8Ino.2No.olAso.99 quantum wells, which absorb and emit light at 1080 nm.
The samples were excited at 900 nm. This excitation wavelength is somewhat shorter than
the band gap of GaAs, therefore the excitation generates carriers in conduction band, which
can be trapped by the QWs. Then, the QWs emit the light at 1080 nm as schematically
shown in Fig. 12.6. The emission intensity of the QWs is proportional to the product of the
QW population and the radiative lifetime, thus by measuring the time profile of the emission
one can monitor the population of the QWs.
Since the samples were not transparent the scheme presented in Fig. 12.1 was modified
to allow a "front face" excitation, as shown in Fig. 12.6. In this experiments the Ti:sapphire
laser was tuned to 900 nm wavelength, and the laser beam was split in two parts, so that
15% of the beam was used for the excitation (reflection from the mirror M1) and the
rest was the gate pulses beam. The excitation was focused by the lens LIon to the sample
at incident angle close to 45 degrees. The emission was collected in direction close to
the normal of the sample surface so that the reflected excitation beam did not fall into the
aperture of lens L2. In this geometry the size of the excitation spot may affect the time
resolution since the excitation-emission propagation paths are different for different points
at the sample. The focal length of the lens L 1 was 3 cm and the excitation spot size was
12.3. Photo-dynamics of semiconductor quantum wells
235
5000
1000
annealed:
"'C decay = 330 ps
4000
en
c:
::J
.ri 3000
!....
ro
....
.00 2000
c:
Q)
....
c:
unannealed:
"'C d = 31 ps
ecay
o
o
30
60 90
Time, ps
120
Figure 12.7: Photoluminescence decays of as-grown (open circles) and thermally annealed
(open squares) semiconductor quantum well sample. The solid lines shows exponential
fits of the data, which gives 31 and 330 ps lifetimes for as-grown and annealed samples,
respectively. The inset shows the same data in semi -logarithmic scale. Figure reproduced
from ref. [24] with kind permission of Springer Science and Business Media, @ 2003.
estimated to be roughly d == 30 tL. 26 Thus expected time resolution due to the non-colinear
geometry is t d CO 45° 80 fs, which is an acceptable value for the targeted time
resolution of 100-150 fs.
The detection wavelength was Ad == (A -1 + A; ) -1 == 491 nm. The detection part of
the instrument consisted of a red cut off filter (transparent at 491 nm, but rej ecting the second
harmonic of gate pulses at 450 nm) and a photon counting photomultiplier coupled with a
monochromator. For all the measurements the background counting rate was 3-4 counts
per second. 27 With excitation pulse density of rv 0.3 mJ cm -2 and the pulse repetition rate
of 91 MHz the signal counting rate was 500-700 S-l at maximum. The signal collection
time was set to 5 s, which provided 2500-3500 counts intensities of the measured signals.
The emission decays of two samples are presented in Fig. 12.7. The figure illustrates the
effect of the thermal annealing on the carrier relaxation time. The inset presents the same
data in semi -logarithmic scale, which is a convenient view for exponential decays. Varying
the conditions of the annealing (e. g. temperature and/or time) one can efficiently tune the
recovery time in the range of more than one order of magnitude from 31 to 330 ps.
From Fig. 12.7 one can notice that there is a certain time during which the emission
intensity is increasing after the sample excitation by the 50 fs laser pulses. This formation
can be seen clearly for the measurements carried our with better time resolution (smaller
time steps in this case), presented in Fig. 12.8. The signal formation is due to the carriers
26This is approximately twice of the diffraction limit for a 2 nun beam, see Section 2.3.2.
27This is the counting rate at negative delay, i. e. when the gate pulse hits the NL crystal well before the
236
Emission spectroscopy with optical gating methods
3000
ijI
"
II
II
I I
I I
I I
2000 I I
I I
r:.rJ I I
I I
1:]1
0 IQ
u I I
I I
".....r I I
I I
I I
1000 I I
t:J I
I
I
I
I
t:J
I
I
I[]
0
-0.5 a 0.5 1 1.5 2 2.5 3 3.5
tiIne, ps
Figure 12.8: Photoluminescence formation of the QWs after the 50 fs pulsed excitation
(open circles), measured response of the instrument (open squares), and calculated data fit
(solid line) and instrument response ( dashed line).
capture time, which is the time needed for carriers in conduction band to be trapped by the
QW. 28
To determine the time resolution of the instrument the reflected excitation pulses were
used instead of the emission and the measured response is shown by the open squares in Fig.
12.8. The width of the response is 100-120 fs (the measurements were acquired with 25 fs
steps). As an alternative method, the data were fitted to the exponential growth with simu-
lated Gaussian response function, which used the pulse width as one of the fit parameters.
The results of the fit are shown by the solid line, and the calculated Gaussian response pulse
by the dashed line. The measured and calculated responses are in good agreement with each
other, but the calculated response width is a little broader than the measured, being 140 fs
(FWHM). However, both values show reasonably good time resolution for the instrument
utilizing 50 fs fundamental pulses.
emISSIon.
28In this particular case the carriers are located nearby the QW, and the capture process does not involve carrier
diffusion, therefore mathematically the emission formation can be described by the exponential growth.
Chapter 13
Ultra-fine spectrum resolution
13.1 Natural line width and broadening
The shapes of the absorption and emission bands are affected by many factors and in many
cases differ significantly from a band shape of a single isolated molecule. In the visible
and near UV parts of the spectrum the absorption and emission are determined by the elec-
tron subsystem of atoms and molecules. For example, position of the absorption band of
rhodamine dye (at rv 560 nm ) is determined by the electron transitions from the highest
occupied molecular orbital (HOMO) to the lowest un-occupied molecular orbital (LUMO).
This transition has its own uncertainty, which, however, is rather small. The shape of the
absorption band depends on the molecule vibrational sub-levels and is determined also by
interaction of the molecule with the solvent. These factors result in broadening of the ab-
sorption bands.
The natural bandwidth of the electronic transition (and any other transition) is tightly
related to the lifetime of the excited state VT 1/27r, where v is the (frequency) band-
width and T is the lifetime of the excited state. 1 For the rhodamine dye the excited state
lifetime is T 3 ns and the natural bandwidth is v 5 X 10 7 Hz = 50 MHz, which
corresponds to A == Ac 2 v t""V 6 X 10- 5 nm bandwidth in the wavelength domain (or,
in wavenumbers, k == .6.; == ; rv 0.0017 cm- l ). However, at normal conditions the
absorption bandwidth is rv 20 nm (e. g. in ethanol at room temperature), which means,
that at normal conditions the absorption band shape is determined by different broadening
mechanisms rather than by the natural line shape.
Even in a gas phase one can find mechanisms resulting in broadening of the bandwidth.
Two dominating factors are collisional broadening (at high pressure) and Doppler shift. The
latter is given by a simple relation
V av
VD t""V -V o
C
(13.1)
1 For a classic oscillator the relation, VT rv 1/21f, can be directly obtained from Fourier transform of the
oscillations decay, e - kt sin (wt), where k = 1/ T. In the framework of the quantum mechanics one can notice
that Heisenberg uncertainty principle, E t rv n, gives the same result, since E = h v and n = h/21f.
237
238
Ultra-fine spectrum resolution
where V av is the average velocity of the molecules and V o is the middle band frequency. For
ideal gas the average kinetic energy, E K, of a molecule of mass m is determined by the
temperature
2
mv av
E K ==
2
== kT
2
(13.2)
Thus
V av = V3
(13.3 )
and
D..VD rv Va V 3kT
c m
(13.4 )
Example 13.1 provides an estimation of the Doppler broadening effect of a typical dye
compound.
Example 13.1: Estimation of Doppler broadening. For a molecule of m == 100 a.u. at
T == 300 K, V av t""V 800 m/s. At wavelength of 600 nm, i. e. V o == 5 X 10 14 Hz
(close to the emission maximum of rhodamine dyes), one obtains VD t""V 1.3 X
10 9 Hz == 1.3 GHz, which is A t""V 1.6 x 10- 3 nm, or k == 0.043 cm- l . This
value is greater than the natural bandwidth of the rhodamine dye, 0.0017 cm- l ,
estimated above. In general, at normal conditions, the Doppler effect will result in
essential broadening of the natural line width if the lifetime of the excited state is
longer than T > (27r v D ) -1 0.1 ns, that is, for almost all molecular systems the
Doppler broadening is greater than the natural bandwidth at room temperature.
The broadening mechanisms are usually divided into two types:
Homogeneous broadening: when the molecules (or absorbing/emitting centers) are essen-
tially equivalent and absorption/emission spectra of all the molecules are roughly the
same;
Inhomogeneous broadening: when the molecules are essentially different and the total
absorption/emission spectrum is formed by a large number of relatively narrow bands.
The distinction between the cases depends on observation time and characteristic time of
the broadening mechanism. For example, Doppler broadening can be treated as homoge-
neous broadening in spectrum measurement experiments if the time of spectrum collection
is much longer than the molecular collision time, which is certainly the case for steady state
spectroscopy. The same Doppler broadening has to be considered as inhomogeneous broad-
ening when studying emission spectrum of a gas laser. Then the characteristic time scale of
the phenomenon (frequency of the light interaction with the molecules) is the time of light
passage across the resonator, which is definitely shorter than the collision time.
13.2. Traditional optical tools for high spectrum resolution
239
In liquid phase the line broadening is caused by the interaction between the chro-
mophore and the solvent molecules. The change in chromophore-solvent state happens
with the solvent vibrational frequencies (10 12 . . . 10 14 Hz) and in the most experiments the
line broadening can be treated as homogeneous. However, using femtosecond technique one
can observe a phenomenon called hole-burning, which is observation of the inhomogeneous
broadening in the time domain shorter than the vibrational relaxation time.
As can be seen, different tasks require different spectrum resolution. In some cases one
may need resolution better than A < 10- 4 nm (or v < 10 8 Hz, or k < 0.002 cm- l )
which comes close to the natural bandwidth of a single molecule.
13.2 Traditional optical tools for high spectrum resolution
Two types of optical instruments are usually used for a fine wavelength selection: monochro-
mators (see Section 2.3.4) and Fabry-Perot interferometers (see Section 2.2.2). The monochro-
mators can be used in a wider wavelength range but have worse wavelength resolution, while
the interferometers have better spectrum resolution but can be used in a relatively narrow
spectrum range.
The spectrum resolution of monochromators was discussed in Section 2.3.4 and is given
byeq. (2.46):
D.A == d(l + COB cp)
Fmg
(13.5)
where d is the slits size, F is the focal distance of the collimating mirror (see Fig 2.7), m
is the diffraction order, 9 is the groove number and cp is the angle between the incident
and diffracted beams. Typically monochromators are equipped with adjustable slits size,
which changes the spectrum resolution of the device. The other parameters are fixed by the
design but important for the resolution. The product mg cannot be increased infinitely as it
determines diffraction angle cpo Usually the monochromators are designed to have the mg
value close to its maximum. 2 Therefore, the main difference between the monochromators
is in the focal length F - the longer the length the higher spectrum resolution.
Example 13.2: Monochromator spectrum resolution. Monochromator/spectrograph THR
1000 (ISA Inc.) has focal length F == 1 m, which provides dispersion of 0.8 nm/mm
at A ==500 nm with 1200 grooves/mm grating. The resolution of the instrument is
A ==0.008 nm ( k=0.32 cm- l ) with d == lOtLm slits. Dimensions of the device
are 1167 x 470 x 350 mm 3 .
The example above shows that even one meter focal length does not provide resolu-
tion close to the natural line width (but the resolution is already sufficient for the Doppler
broadening detection at room temperature).
2Por the normal incidence of the light sin cp = Amg (see eq. (2.44)) and for A = 400 nm mg cannot be
greater than 2000 mm -1. Typical groove number of grating used in the visible part of the spectrum is 1200 nun-I,
therefore these gratings are used for the first order diffraction, i. e. m = 1.
240
Ultra-fine spectrum resolution
Interferometer
Fabry-
Perot L2
L1
Pinhole
Monochromator
Light
source
0:
L3
..... . .
. .'
. . .
. . .
" .
.....>::..... .....>
Detector
Figure 13.1: Schematic arrangement for performing high resolution spectroscopy:
monochromator is coupled with Fabry-Perot interferometer.
The Fabry-Perot interferometers have a good spectrum resolution but transmission wave-
lengths are typically very close to each other (see Fig. 2.4 on page 24), but can be used to
improve spectrum resolution of other devices. For example, a interferometer can be coupled
with a monochromator as shown in Fig. 13.1. The thickness of the interferometer should
be thin enough to allow only one of transmission lines of the interferometer to fit into the
bandwidth of the monochromator (see eq. (2.21) and Fig. 2.4). Fine tuning of the system
can be achieved by rotating the interferometer so that the light incident angle is changed. An
estimation of the interferometer base and resolution in connection with a monochromator is
presented in Example 13.3. The wavelength resolution can be improved by using interfer-
ometer with higher contrast factor F, i. e. higher reflectance R, or by increasing resolution
of the monochromator, Am, and applying interferometer with wider base d.
Example 13.3: Use of Fabry-Perot interferometer to improve the spectrum resolution of
a monochromator. If a monochromator resolution is Am ==1 nm and the work-
ing wavelength is 500 nm, then interferometer thickness should be d < 2;:m
0.125 mm. With reflectance of R == 0.95 the contrast factor is F == (1 )2 == 1520.
Thus, the interferometer with thickness d == 0.1 mm can provide resolution (eq.
(2.31)) Ai "" 2>:d "" 10- 2 nm, which gives 100 folds improvement in the
spectrum resolution. 3
One of the obvious problems when using interferometers Fabry-Perot for wavelength
selection is the small angular acceptance, since the transmittance wavelength depends on the
light incident angle. This requires a small divergence of the measured light or a small point-
like light source, as shown in Fig. 13.1. An estimation of the acceptable beam divergence is
3The distance between the interferometer transmittance maxima is A = : . It can be expressed in the
wavenumbers k = ; = 2 2 = 2 1 d . Similarly, the bandwidth is k = (21rdH)-1. Therefore,
wavenumbers are really easier units when dealing with interferometers. For example, the distance between maxima
of d = 1 mm interferometer is 0.5 mm -1 = 5 cm -1.
13.3. Lasers for fine spectrum resolution
241
given in Example 13.4. The example shows that rather stringent requirements are imposed
on the light source to provide a high spectrum resolution in this configuration.
Example 13.4: Estimation of the acceptance angle for interferometer-monochromator cou-
pIe. If the light incident angle a is not normal, the value d must be replaced by
d COB a in eqs. (2.16)-(2.31). Thus, the transmission maximum shifts to Al == co a '
where the A is the maximum at normal incidence. The shift is Al - A Aa 2 at
small angles, i. e. COB a a, and for conditions used in Example 13.3 one obtains
a V ;' 0.005 radian 0.3 0 . If the focal length of the light collimating lens
is f == 20 cm (lens L1 in Fig. 13.1), then the light source size should be smaller than
d < af == 1 mm. This is the maximum acceptable divergence of the light to keep
the spectrum resolution of Ai == 0.01 nm.
13.3 Lasers for fine spectrum resolution
13.3.1 Resonator limited bandwidth
A typical laser resonator, two flat mirrors parallel to each other, forms a classic Fabry-Perot
interferometer. The laser radiation bandwidth is given by (eq. (3.6) on page 42)
D.A
AO
1 T)..
27r -jrl r2 T
(13.6)
where T == ( 2 al 1) is the laser time constant and T).. )..0 is the wave period. For
c r1 r2e - c
CW mode rl r2e2al == 1 and thus : == 0, i. e. the bandwidth is infinitely small. This
is consequence of placing an active medium inside the resonator, which compensates the
energy losses in the resonator and increases the lifetime of photons inside the resonator to
infinity. However, already for a passive interferometer Fabry-Perot of the size typical for
the laser resonators the bandwidth is rather small. For example, a typical length of a He-Ne
laser resonator is 40 cm and the mirror reflectances are rl == 1 and r2 == 0.95. Then, the
photon lifetime is T == ( 2£ 1) t""V 53 ns and T).. t""V 2.11 X 10- 15 s = 2.11 fs (Ao == 633 nm),
c r1 r2 -
which gives
A
Ao
T).. t""V 6.3 X 10- 9
27rT -jrl r2
or A rv 4 X 10- 6 nm, or v rv 3 MHz (at central frequency V o rv 4.7 X 10 14 Hz), or
k == v t""V 10- 4 cm- l . Therefore, already a passive resonator of such dimension has a
bandwidth narrower than the amplification bandwidth of most laser active media (including
gas lasers, which have the narrowest amplification bands).
13.3.2 Amplification bandwidth and lasing threshold
In order to establish lasing, the amplification of the active medium must exceed resonator
losses. The magnitude of amplification (or "intensity" of the amplification band) depends on
242
Ultra-fine spectrum resolution
1)
amplification line
'\
active laser mode
resonator modes
lasing threshold
A
2)
A
3)
A
Figure 13.2: Laser threshold and active lasing modes (thick lines).
the pumping rate of the lasing level. Therefore, the lasing is established when the pumping
rises the amplification higher than the losses threshold level. This is illustrated in Fig. 13.2,
where the case (1) corresponds to the beginning of the lasing, which happens when at one
of resonator modes the amplification reaches lasing threshold. In the case (1) only one
mode will be active (will be present in laser radiation). If the pumping is increased, the
amplification is stronger and relative height of the lasing threshold is lower. Then more
modes may become active. In the case (2) two modes can be emitted and in the case (3)
already 5 modes can be found in the laser emission.
The amount of actually emitted modes (cases (2) and (3) in Fig. 13.2) depends on the
type of broadening of the active medium amplification band. When the lasing is established,
it consumes some population inversion thus reducing the amplification to the threshold level.
To a first approximation, the shape of the amplification band does not change in the case of
homogeneous band broadening, but it will change in the case of inhomogeneous broaden-
ing, as shown in Fig. 13.3. Therefore, in the case of homogeneously broadening and CW
operating mode only one longitudinal mode will be emitted. On the contrary, in the case
of inhomogeneous broadening the amplification is reduced to the lasing threshold at all fre-
quencies corresponding (or close) to the laser active modes, which results in a complete
change in the shape of the amplification band (Fig. 13.3, (2)) and in a few modes emitted
together. This local reduction in the spectrum intensity is also called hole-burning.
13.3. Lasers for fine spectrum resolution
243
1) Homogeneous broadening
..' . ", amplification band
", w ithout l asing
" ft
lasing threshold
with lasing
A
2) Inhomogeneous broadening
without la sing
,'-',
lasing threshold
t t t
laser modes
A
Figure 13.3: Homogeneous and inhomogeneous broadening of the laser amplification band.
13.3.3 Mode-beating and resonator design for single mode lasers
Co-existence of several modes in laser radiation results in phenomenon called mode-beating.
The mode interference modulates the output intensity, which can be usually observed as an
almost random noise. Mode-beating has clear origin in the case of inhomogeneous laser
level broadening since the multi -mode operation is typical in such case. Unfortunately, it is
also a usual phenomenon for homogeneously broadened amplification lines. This happens
because the laser modes are standing waves. The light intensity distribution of the standing
wave along the laser optical axis is sinusoidal with period of A/2. The modes consume
inversion at maxima of the standing wave intensity and does not change inversion at min-
ima. Thus, there is a probability for less favorable modes to overcome lasing threshold and
compete with the modes having the highest amplification otherwise.
Technically, the lasers can operate in single mode regime with a very narrow band emis-
sion. However a special care should be taken to avoid multi -mode operation. There are
several methods, which allow to achieve single longitudinal mode operation:
. by using a Fabry-Perot interferometer (usually called etalon Fabry-Perot in such case)
to select a single laser mode;
244
Ultra-fine spectrum resolution
. by coupling two resonators, so that only coincident modes (hopefully one of them)
will survive;
. by using ring resonator, so that there is no standing waves and inversion is consumed
homogeneously by a single mode across the active medium;
. by using circular polarization in the case of homogeneous broadening.
13.4 High resolution in absorption spectroscopy
One can use a "classic" spectrophotometer scheme, lamp-monochromator-sample-detector
or lamp-interferometer-monochromator-sample-detector, and obtain resolution close to
0.005 nm (or 0.2 cm- l ). The monitoring light intensity will be very weak for this in-
strument (consider how much of the light can pass 10 tL entrance slit of a monochromator)
and further an increase in the spectrum resolution is hardly possible over this limit. Also
the spectrum resolution is achieved at expenses of instrument sensitivity (for example see
eq. (6.11) on page 117 and the following discussion).
13.4.1 Laser spectroscopy
One can use a tunable laser as a source of monitoring light. The scheme of the laser spec-
trophotometer can be: tunable laser-sample-detection. The intensity of monitoring light
can be very high and almost any detector can be used, e. g. a photodiode. The spectrum
resolution is determined by the laser and can be as high as 10- 6 nm (10- 5 cm- l ). Although
to achieve such resolution the mechanical stability of the laser resonator must be as good
as L < .\ L, e. g. for resonator of L == 50 cm, emission wavelength A == 500 nm
and desired resolution A == 10- 6 nm, the resonator stability must be L < 10- 6 mm,
which puts high demands on the quality of all the components and requires precise thermal
stabilization of the resonator.
One obvious disadvantage of utilization of the lasers in spectrophotometers is a narrow
tuning range. For example, rhodamine 6G dye laser can be tuned in the range of 560-
620 nm at the best. The tuning range can be changed by changing the dye and the laser
mirror, respectively. However, switching from one dye to another is a time consuming and a
complex procedure. On the other hand, measuring an absorption spectrum with resolution,
e. g. 10- 5 nm, one will hardly need to scan more than ten nanometers.
The accurate measurements require high stability of the laser power output, which is
a complex task in laser design, and, with no surprise, the laser systems with outstanding
output parameters (such as high wavelength and intensity stabilities) are quite expensive
devices.
13.4.2 Intra-cavity spectroscopy
Previously the lasers have been used as sources of the monitoring light in a manner similar
to that of lamps. However, the lasers are "open" systems and can be used differently for
the absorption measurements. Let us consider the scheme shown in Fig. 13.4. This is a
13.5. High resolution in emission spectroscopy
245
Wavelength Active
M1 selector medium Sample M2
- - [J- - - - 1- - - t - - - -0- - - t - _J u >
< z ">
<
">
L
Figure 13.4: Intra-cavity absorption measurements.
tunable laser with the sample installed inside the resonator. The measured parameter is the
laser output energy. The laser pumping power P p is spent to cover intra-cavity losses P l
and to emit the light I, that is P p == P l + I. The intra-cavity losses consist of two parts:
the losses of the laser itself Pzz and the loss due to the sample absorption Pa. At constant
pumping rate Pzz + Pa + I == P p == const, thus I == P p - Pzz - Pa. In other words,
when the absorption of the sample increases (Pa increases) the light intensity decreases. If
the sample absorption has increased from 0 to Pa then the light intensity will decrease from
P p - Pzz to P p - Pzz - Pa. Thus, measuring laser output spectrum twice, without the sample,
I(A) == P p - Al(A), and with the sample, Is(A) == P p - PZZ(A) - Pa(A), one can calculate
the power absorbed by the sample as Pa (A) == I (A) - Is (A).
It should be noted that the "monitoring" light crosses the sample many times before it
leaves the laser resonator. Therefore, compared to an ordinary spectrophotometer, the intra-
cavity scheme has an advantage of "accumulation" of the absorption by passing light many
times through the sample. This gives a gain in sensitivity. If the laser operates at conditions
close to the lasing threshold (P p - Al close to 0), then even a small change in absorption
may lead to a gradual change in the output light intensity. 4 Therefore, the main advantage
of this scheme is the higher sensitivity when operated at pumping level close to the lasing
threshold.
Unfortunately there are few clear disadvantages of the intra-cavity method. Firstly, the
measured value is the laser power output, which reflects the power losses in the sample.
However, the parameter of interest is the sample absorption, and the calculations of the
sample true absorption are not straightforward. Then, the power density inside the laser res-
onator is much higher than the laser output power and exceeds typical monitoring light in-
tensity of ordinary spectrophotometers by many orders of magnitude. Therefore the sample
must be very stable against strong irradiation. Also the price of a laser spectrophotometer
may exceed the price of an ordinary spectrophotometer by an order of magnitude.
13.5 High resolution in emission spectroscopy
Two types of experiments can be carried out in the frame of emission spectroscopy. These
are measurements of the emission spectrum and measurements of the excitation spectrum.
4In a sense, this type of measurements is similar to fluorescence measurements, the "background" signal can
be relatively low.
246
Ultra-fine spectrum resolution
The emission spectrum resolution is determined by the wavelength selectivity of the detec-
tion system. Therefore, it is limited by the wavelength selecting system used, e. g. when a
monochromator is used it can be 0.005 nm at the best.
The resolution in the excitation spectrum measurements is determined by the bandwidth
of the excitation source. Thus, one can use a tunable laser and achieve resolution as good
as 10- 6 nm. The scheme of the instrument is tunable laser-sample-detection system, or
essentially the same as one presented in Fig. 6.1.
The emission spectroscopy is a very sensitive method (see Section 6.2.7), which allows
detection of a single molecule. Single molecule time resolved emission decay measure-
ments were discussed in Section 8.7 on page 166. It was important to focus the excitation
light at a single molecule to measure fluorescence of this particular molecule. The stud-
ied molecules were embedded into polymer matrix but their spectra were homogeneously
broadened, so that spectrally they were rather similar to each other. At low temperatures,
typically at 4 K (liquid helium temperature) and lower, distortion of the molecules by the
matrix becomes small and difference in the environment between molecules can be ob-
served as inhomogeneous broadening of the spectrum. Under the conditions of inhomoge-
neous broadening single molecules can be resolved spectrally even when a large number
of molecules are covered by the excitation spot simultaneously. This is illustrated in Fig.
13.5, where the spectrum of inhomogeneously broadened band is simulated for different
number of molecules, N, in the range from 10 to 10000. The bandwidth of the (broad-
ened) spectrum was taken Vb == 200 cm -1, the bandwidth for the individual molecule
was Vm == 2 cm -1 and the central band wavenumber was v max == 20 000 cm -1 .
At relatively low number of molecules, N < Vb , the spectrum consists of a series
Vrn
of sharp well resolved lines, where each line corresponds to a particular molecule. This
is the state when all the molecules under the excitation beam can be resolved individually,
but practically nothing can be said about the band shape. At larger number of molecules,
N Vb , the absorption lines of individual molecules start to overlap with each other, and
Vrn
at N > k the shape of the broadened band can be seen clearly.
The excitation energy does not need to be very high for such experiment, as can be seen
from the following example.
Example 13.5: Estimation of the excitation intensity for single molecule detection. One can
expect to detect 1 photon per 10 000 excitations of the molecule (assuming detec-
tion efficiency 0.1, emission collection efficiency 0.01 and emission quantum yield
0.1). If the molecule has excited state lifetime of 1 tLS (which include triplet state
relaxation if it is formed) the excitation rate can be fex ==200 kHz, which means
that the molecule is excited once in 5 tLS in average, and expected counting rate is
20 counts/so The density of photons must be close to one photon per molecule cross-
section in 5 tLS time interval. For a typical dye molecule it is E rv 1 mJ/cm 2 . Thus
required power density is P == Efex == 200 W/cm 2 . For a single molecule experi-
ments the excitation spot must be as small as possible, say S == 1 tL 2 == 10-8cm 2 .
And the light intensity needed for this experiment is I == 2 tL W only.
13.5. High resolution in emission spectroscopy
247
1 N= 10
0.8
0.6
0.4
0.2
0
4 N = 100
3
2
1
0
;::::$ 15
ro N = 1000
,c 10
.
r:/1
5
0)
0
80 N= 10000
60
40
20
1 000 19500 20000 20500 21000
wavenumber, em -1
Figure 13.5: Spectra of inhomogeneously broadened samples consisted of N == 10, 100,
1000 and 10000 molecules. The bandwidth is Vb == 200 cm- l and the single molecule
line width is vm == 2 cm -1 .
The experimentally observed line widths are much smaller than that used for spec-
trum simulations in Fig. 13.5. For example vm 0.005 cm- l was reported for chlo-
rin molecules in PVB matrix at 1.7 K [25]. Such narrow lines are also called electronic
zero-phonon lines, meaning that the line is not disturbed by vibrational modes of the ma-
trix (phonons). Assuming the bandwidth to be Vb 250 cm- l (which corresponds to
A 10 nm at A == 600 nm), one can expect to resolve single molecular lines even when
N Vb == 50 000 molecules are excited at the same time.
Vrn
To resolve such narrow lines one needs a tunable laser with emission band narrower
than the line width of the studied object. Such lasers are available commercially. As an
example one can consider Matisse series (Spectra-Physics). This is passively stabilized
Ti:Sapphire ring laser which provides spectrum resolution better than 10- 3 cm- l in 700-
980 nm wavelength range (divided in three subranges). In visible part of the spectrum
tunable dye lasers can be used as the excitation sources.
248
Ultra-fine spectrum resolution
Since the excitation spectra are measured in these experiments, the detection system
does not require high spectrum resolution. The important requirement to the detection part
is high rej ection of the excitation light, which can be achieved by a combination of band pass
and cut off filters. Another important requirement is high sensitivity as the emission from
a single molecule is typically low. Therefore photomultipliers working in photon counting
mode is the usual choice for this type of measurements.
13.6 Spectral hole-burning
The spectral hole-burning is observed in the case of inhomogeneous spectrum broadening.
It can be seen as a sharp hole in the sample spectrum after illumination by a monochromatic
wave. One can distinguish between two types of hole-burning:
dynamic hole-burning when the spectrum hole is observed right after the excitation, but
recovers with time;
persistent hole-burning when the hole appears permanently after the excitation.
In the first case the reason for the hole-burning is that the excited molecule has different
absorption properties, and does not interact with the light at this wavelength during some
time. The persistent spectral hole-burning can be caused by the permanent damage of the
molecule or by some irreversible photochemical reaction, e. g. photo-isomerization.
The steady state measurements of hole-burning at cryogenic temperatures can provide
one with information about the natural line width. Another example of the hole-burning
application is investigation of mechanisms of the energy transfer and relaxation in light
harvesting subunits of natural photo-synthetic systems [26].
The dynamic hole-burning requires high light intensities. For example it is typical for
the laser active media as was discussed above in Section 13.3.2. A schematic illustration
of the hole-burning in the emission/amplification spectrum of the laser active medium is
presented in Fig. 13.3. Also dynamic hole-burning can be studied using time resolved
spectroscopy techniques.
Persistent hole-burning is one of the areas of ultra-fine spectroscopy applications. Tech-
nically it requires lasers generating very narrow band emission, as was discussed in the
previous section. There are also some potential applications of the persistent hole-burning,
e. g. for multidimensional holography [25].
Chapter 14
Polarization measurements
Polarization is important property of the light which affects the light interaction with the
matter and can result in misleading interpretation of the measurements. On the other hand,
it can be used in order to obtain more complete information on the studied obj ect. The po-
larization is one of the questions which should be thought out at the state of the experiment
planning.
14.1 Light polarization
Electromagnetic waves are known to be transverse waves, which means that the electric
field vector is orientated perpendicular to the direction of the wave propagation in vacuum.
As such, the electric field vector may have any orientation in plane perpendicular to the
propagation direction. Orientation of the electric field vector determines polarization of the
wave. A photon, being an electromagnetic wave quantum, also has polarization.
A mathematical presentation of a plane monochromatic electromagnetic wave is 1
E(r, t) = E sin (27rvt - k . r)
(14.1)
where E (r, t) is the electric field vector at a point given by vector r, v is the wave frequency,
k is the wave vector, which determines the propagation direction of the wave, and E is the
vector determining orientation and amplitude of the electromagnetic wave. The value v and
vectors k and Eo are not independent, and in dielectric medium
k.E==O
(14.2)
and
Ikl= <
(14.3)
where E is the medium dielectric constant.
1 Scalar harmonic and plane waves were considered in Sections 2.1.2 and 2.1.3, where circular frequency and
exponential presentation was used, i. e. U = Uoe (wt-i:tr).
249
250
Polarization measurements
y
z
------, Eo
Eoy
a
x
EOx X
Figure 14.1: Polarization of electromagnetic wave.
Equation (14.2) tells that the wave and the electric field vectors must be perpendicular
to each other, i. e. the electromagnetic wave is transverse wave. Equation (14.3) is the well
known relation between the wave number k = If 1 and wave frequency (in Table 1.2 the
relation is given for waves in vacuum, E == 1).
One can always choose the coordinate system so that the plane wave propagates along
one of the axis. In Fig. 14.1 the wave propagates along Z axis, therefore the electric field
vector must be in plane XY, i. e. its proj ection on axis Z is zero. Then, to specify the
electric field vector Eo in this coordinate system, only two values are needed, X projection
Eox and Y projection Eoy. If the angle between the vector Eo and X axis is a, then
Eox = Eo sin ct and Eoy = Eo cos ct, where Eo = 1 Eo I.
Experimentally accessible value is the light power, rather than electric field vector. The
light power is proportional to the square of the electric field, lo rv 1 Eo 1 2 = E5. One can
use a polarizer oriented first along X axis and then along Y axis. The measurements of the
light power will give two intensities Ix == 10 sin 2 a and Iy == 10 cos 2 a, respectively.2
It should be noted that the above considered case can be applied to linearly polarized
light only, which means that there is one axis along which the electric vector (E (r, t)) is
changing with time. The light can be circularly polarized. Circular polarization can be
presented as sum of two linearly polarized waves shifted in time
Ec(r:, t) = Eo [nx sin (27rvt - f. r:) + ny cos (27rvt - f. r:)]
(14.4)
where n x and ny are unit vectors oriented along X and Y axis, respectively. The result of
this superposition is the wave with electric vector Ec rotating in plane XY with frequency
v.
The light can be also partially polarized. All together four parameters are needed to
describe the polarization state of monochromatic plane wave. The most often used are
Stokes parameters. If there is no non-polarized light, Jones vector can be used to described
light polarization. However, in our further consideration only linear polarization will be
assumed, since this is the most practical case in the optical spectroscopy applications.
20 ne can also notice that 10 = Ix + Iy, which is just the energy conservation law.
14.2. Interaction of polarized light with media
251
The optical schemes are usually built on flat horizontal surfaces, e. g. optical tables.
Therefore, the polarization orientation is usually related to the scheme basement and de-
noted as "vertical" and "horizontal". "Horizontal" means in horizontal plane and perpen-
dicular to the beam propagation direction, whereas "vertical" is always perpendicular to the
beam propagation direction since beams are propagating in horizontal directions.
In coordinates of vertical and horizontal predefined polarization orientations light polar-
ization is characterized by polarization ratio
Iv - Ih
p==
Iv + Ih
(14.5)
where Iv and Ih are the light intensities after vertically and horizontally oriented polarizers,
respectively. However, to characterized sample emission polarization properties anisotropy
coefficient is usually used. It will be discussed in Section 14.2.3 later in this Chapter.
14.2 Interaction of polarized light with media
Polarization of a photon emitted by a molecule is determined by the molecule orientation.
The direction of the most probable photon polarization is said to be orientation of the tran-
sition dipole moment of the molecule. (N ote, it may differ from the electrostatic dipole
moment of the molecule.) Emission and absorption are transitions between the same elec-
tronic states, therefore the absorption is sensitive to the photon polarization in the same way
as the emission. Molecules can absorb photons with polarization coinciding with the orien-
tation of their transition dipole moments but do not interact with photons having polarization
perpendicular to the dipole moment orientation.
Naturally, the light polarization has great importance when anisotropic medium is stud-
ied. However, in fluorescence and other spectroscopy studies relying on the photoexcitation,
the polarization is import property even in the case of initially isotropic medium, since exci-
tation always induces some anisotropy. Even when the sample is excited by a non-polarized
light, the molecules with transition dipole moments oriented in direction of the excitation
propagation will not be excited, since they are oriented perpendicular to the electric field
orientation independent of the excitation light polarization.
Let us consider an interaction of the polarized light with randomly oriented molecules.
Let the light propagates along axis Z and to be polarized along Y axis. It is convenient to
use polar coordinate system, then the orientation of a molecule dipole moment in respect to
the light polarization is given by two angles cp and 1jJ, as shown in Fig. 14.2. If the molecule
absorption cross-section for the light polarized parallel to the transition dipole moment is
a 0, then for a molecule with the transition dipole moment at angle cp the absorption cross-
section is a 0 cos 2 cp. 3 For random distribution of the molecule orientations the absorption is
3The projection of the wave electric field E on axis Y is Ex = E cos cpo The light intensity is given by the
square of the electric field, i. e. Ix ex E = E 2 cos 2 cpo Thus, Ix = I cos 2 cp, where I is the incident light
intensity. Angle 'ljJ is not important for the cross-section calculations polarization as the electric field proj ection on
the direction of the dipole moment orientation depends on angle cp only.
252
Polarization measurements
y
x
z
Figure 14.2: Polar coordinate system.
given by averaging of all possible orientations 4
2'if 'if
(J" r = 4 J J (J" 0 cos 2 cp sin cpdcpd'ljJ
o 0
( 14.6)
Calculation of the integral for 1jJ is trivial and gives 27r. Then, for cp one obtains
'if 'if
(J" r o J cos 2 cp sin cpdcp = - o J cos 2 cpd ( cos cp )
o 0
-1
- (J"; J x 2 dx =
1
(14.7)
Which means that absorption probability of the light by randomly oriented molecules is
three times lower than that of uniformly oriented molecules in the case of the light polarized
parallel to the transition dipole moments of the molecules.
14.2.1 Magic angle
There is an angle between the transition dipole moment and the light polarization when the
absorption is equal to that of the randomly distributed molecules. The angle is called magic
angle, it is given by the condition cos 2 cp == and equal to cp == arccos 54.74 0 .
The magic angle is very useful in time resolved fluorescence and absorption studies of
liquid samples. Excitation creates an anisotropy in direction of the excitation polarization,
4 An average value of some function f ( <p, 'ljJ) in polar coordinates is given by the integral 1 =
21T 1T
4 I I f ( <p, 'ljJ) sin <pd<pd'ljJ.
o 0
14.2. Interaction of polarized light with media
253
y
Emission
J::::::-
v
" \jl
: - - - - - - - - - - ",
h
x
Z citation
Figure 14.3: Polarizations at fluorescence measurements.
meaning that the excited molecules have certain preferred orientation. Rotational diffusion
results in degradation of the anisotropy and formation of an isotropic distribution of the
dipole moments. Thus, the signal measured at light polarization parallel to the excitation
polarization will decrease due to anisotropy relaxation, and at polarization perpendicular to
the excitation will increase, if the lifetime of the excited species is longer then the rotational
time constant. If the monitoring is done at magic angle, the rotation of the molecules has no
influence on the signal and the signal decay will show the decay of the excited state only.
Another phenomenon leading to polarization change is energy transfer, which ofter ob-
served in densely packed ensembles of chromophores. The energy transfer in chlorophyll
light harvesting subsystem of natural photosynthesis reaction center is discussed in the last
Section of this Chapter discusses. This phenomenon is also observed in polymer films and
other condensed media. To avoid the energy transfer effects on the measurements one can
monitor the signal at magic angle polarization.
14.2.2 Induced anisotropy in fluorescence measurements
N ow let us consider typical arrangements of the fluorescence measurements. Let us also
assume that the dipole moment orientations are the same for excitation (absorption) and
emission, e. g. excitation populates the first singlet excited state. Suppose the excitation
beam propagates along Z axis and monitoring is carried out in direction of X axis, as
shown in Fig. 14.3. In accordance with the scheme symmetry we have to consider two
cases of the excitation polarization - one is polarization along Y axis and another along
X axis. Similarly, for the emission measurements axes Y and Z are the natural choice for
the polarization presentation. Only the mutual orientations of the excitation and monitoring
polarizations are important. By convention the orientation along axis Y is called vertical
254
Polarization measurements
and in plane X Z, i. e. perpendicular to Y, is called horizontal. 5 Experimentally available
values are emission intensities at two orthogonal polarizations, vertical and horizontal, re-
spectively. Thus we can measure four values: Ivv, Ivh, I hv and I hh , where the first index
denotes polarization of the excitation and the second shows polarization of the emission,
e. g. Ivh means emission intensity at the horizontal polarization when excited at the vertical
po lariza ti on.
If the sample is excited by the vertically polarized beam the distribution of the excited
molecules is given by cos 2 cp, if the excitation light density is much lower than the saturation
one. 6 The molecules with dipoles oriented close to the direction of axis Z can be excited
(and, thus, will emit some light), but the excitation probability is much lower than that for
the molecules with dipoles oriented along Y axis. Projection of the molecule dipole moment
on Y axis (vertical polarization) is cos cp, thus the emission intensity of the molecule at angle
cp is proportional to cos 2 cpo Consequently, Ivv can be obtained by averaging
2'if 'if
Ivv 4 J J cos 2 cp cos 2 cp sin cpdcpd'ljJ
o 0
'if +1
- J COS4 cpdcoscp = J x4dx ==
225
o -1
The projection of the dipole moment on Z axis is sin cp cos 1,LJ and Ivh is given by
( 14.8)
2'if 'if
Ivh = 4 J J sin 2 cp cos 2 cp sin cpdcp cos 2 'ljJd'ljJ
o 0
( 14.9)
2 'if
One can notice, that J cos 2 1,LJd1,LJ == 7r, and
o
'if +1
1 J 2 2 1 J 2 2 1
Iv h == -"4 cos cp (1 - cos cp) d cos cp =="4 x (1 - x ) dx == 15
o -1
(14.10)
Even so the monitoring polarization is perpendicular to the excitation polarization, the emis-
sion intensity is only 3 times lower than that of parallel polarizations of the excitation and
monitoring, i vv == 3.
vh
Similar calculations can be done for the horizontal excitation, but one can notice that due
to symmetry reason intensity I hv and I hh must be equal. 7 Thus the intensity ratio for the
5 Usually plain X Z is the horizontal plane of the optical scheme.
6The relative excitation efficiency (i. e. the relative number of excited molecules) is typically very low in
fluorescence measurements, owing to the high sensitivity of the emission spectroscopy methods. However, at
excitations approaching saturation level the distribution of the excited molecules will differ from cos 2 cp, and at
infinitely high excitations all the molecules will be excited.
7For above polarization considerations the propagation direction is not important, since the absorption and
emission efficiencies depends on the mutual orientations of the transition dipole moment and the electric field
vector. For instance, the vertical excitation is equally efficient by any beam propagating in plane X Z if it has
polarization in Y direction.
14.2. Interaction of polarized light with media
255
horizontal excitation is I I hv == 1. This is practically important result since it can be used to
hh
calibrate relative sensitivities of the detection system to horizontal and vertical polarizations
of the emission, as will be discussed in the following section.
The difference in intensities at different polarizations is due to photoinduced anisotropy
in the sample. Apparently, if the anisotropy changes during the measurement time, e. g. due
to rotational diffusion, the intensities of the signals will change too. However, if the subject
of study is the lifetime of the excited state, this change will give a fake signal, which is
not related to actual decay of the excited state. To monitor the actual decay of emission all
polarizations should be sum up, which is Itotal == Ix + I y + I z, where the subscript denotes
the polarization orientation. For the vertical excitation polarization Iy == Ivv and Iz == Ivh.
Ix was not calculated, but for the symmetry reasons it must be Ix == Iz (see footnote 7).
Thus Ix == Ivh, and the total emission intensity (for vertical excitation) can be calculated as
Itotal == Ivv + 2Ivh
(14.11)
In other words, to avoid induced anisotropy effects the vertical and horizontally polarized
emissions must be measured at vertical excitation and the non-polarized emission intensity
has to be calculated using eq. (14.11).8 Alternatively, one can measure decay at magic
angle.
14.2.3 Anisotropy coefficient
The extent of anisotropy is usually expressed in terms of anisotropy coefficient, which is
defined as
Ivv - Ivh
r==
Ivv + 2Ivh
(14.12)
The formal difference between polarization ratio, eq. (14.5), anisotropy coefficient is in
denominators. The latter relates the difference between polarization (Ivv - I vh ) to total
emission intensity (Ivv + 2I vh ), which has clear meaning for the photoinduced emission
measurements. On the contrary, the polarization ratio is used to specify the light polarization
regardless its origin.
When the excitation light has horizontal polarization, i. e. along X axis, the emission
intensities are the same for horizontal and vertical monitoring polarizations, thus
I hv == Ihh
(14.13)
The latter is very useful relation for a practical reason. The detection part of the instrument
may consists of components having different light transmissions for different polarizations,
such as monochromator. A direct comparison of the intensities measured with different
orientations of analyzing polarizer is useless as the transmission coefficients for the polar-
izations are unknown and depend on the detection wavelength. However, the experiments
80 ne can notice that with horizontal excitation the intensity Ix is not available, so the horizontal excitation
cannot be used in such case.
256
Polarization measurements
can be carried out with the horizontal excitation first and the ratio of the signals at verti-
cal and horizontal detection polarizations can be determined. This ratio is usually called
G - factor
G == I hv
Ihh
(14.14)
where I hv and Ihh are the measured intensities. By its definition, G- factor is the ratio of
sensitivities of the detection system to vertical and horizontal polarizations, and can be used
to correct the polarization measurements at vertical excitation. If the measured intensities
are I v and I h' the actual intensity ratio can be calculated as
R - Ivv _ 1 I v _ Ihh I v
- -- --
- - -
Ivh G I h I hv I h
(14.15)
which makes corrections of the detection system sensitivity to polarization. Now anisotropy
can be calculated as
1 - Ivv
Ivh
r==
1 + 2 Ivv
Ivh
1- R
1 +2R
( 14.16)
This measurement strategy can be applied to study anisotropy decays. Then two decays
have to be recorded, I v (t) and I h (t), but G- factor can be determined from static signal
intensities at the horizontal excitation.
To measure anisotropy spectrum one has to measure four intensity spectra: I hv (A),
Ihh (A) I v (A) and I h (A). The first two are needed to calculate spectrum G (A), and have
to be measured once if a series of samples is going to be studied.
The excitation induced anisotropy of an isotropic medium can vary in limits r == 0 . . . 0.4,
where the higher limit corresponds to i vv == 3. For a totally anisotropic medium the range
vh
for anisotropy coefficient values is r == -0.5...1. The lower limit is achieved when the
emission dipole moments are oriented along Z axis, i. e. I h == 1 and Iv == 0, and the higher
limit is achieved when the dipole moments oriented along Y axis, i. e. Ih == 0 and Iv == 1,
respectivel y.
For isotropic diffusional rotation anisotropy decays exponentially
r(t) == roe- 6Drt
(14.1 7)
where Dr is the rotational diffusion coefficient. Another parameter used to characterize
rotational diffusion is the rotational correlation time, which is Tc == ( 6D r) -1, thus giving
r(t) == roe- t / Tc .
The rotational diffusion coefficient of a sphere is Dr == 6 ' where 7] is the viscosity and
V is the hydrodynamic volume. This leads to the rotational correlation time of Tc == Zi . 9
9This is too rough approximation for most of practical cases. To make estimation closer to the properties of
actual molecules the correlation time is usually expressed as Tc = k ' where F is the fraction coefficient and
S is the shape factor to account for the non-spherical shapes of the molecules.
14.3. Applications of polarized measurements
257
14.3 Applications of polarized measurements
Apparently, the change in light polarization during the measurements complicates the mea-
surement procedure. On the other hand one can use them to extract additional information
of the subj ect under study. As an example of complication one can consider diffusional
rotation of the dye molecules in solutions when the fluorescence decay dynamics is stud-
ied. For a typical dye molecule, e. g. rhodamine dye, in a solvent with moderate viscosity,
e. g. propanol (7] == 1.8 cP), rotational correlation time is 0.2-0.3 ns, and the lifetime of the
fluorescence is 2-3 ns. Therefore accurate measurements of the fluorescence lifetime can
be done only when the rotational diffusion is taken into account. In this particular case one
can install a polarizer behind the sample at magic angle to prevent the excitation induced
anisotropy effects.
In the time scale essentially longer than the rotational diffusion the photoinduced anisotropy
of the sample is lost and the measurements can be carried out without regard for the light
polarization effects. This is typical situation for flash-photolysis experiments in time scale
of tenth of nanoseconds and longer. Also one have to keep in mind that diffusion correlation
time depends on both the solvent viscosity and the size of the studied objects. For example,
if studied object is suspension of relatively big particles, e. g. fragments of cellular mem-
branes, in a relatively viscous solvent, the diffusion correlation time can be as long as 1 ms,
and the flash-photolysis measurements have to be arrange to account for the anisotropy
effects.
In a very short time scale, e. g. tenth of picoseconds and shorter, the diffusion is slow and
usually can be neglected. However, by selecting a proper polarizations for the excitation and
monitoring one can improve the signal intensity. Therefore the polarizations of the beams
in pump-probe and up-conversion experiments are usually selected to be parallel to each
other, if it has been proven that the anisotropy effects can be neglected.
The polarization measurements can be used to study excitation energy transfer dynam-
ics as illustrated in Section 14.3.3. Another application of polarization measurements is
investigation solvent micro-viscosity. There are classes of compounds which were designed
and synthesized to be used as viscosity probes. These compounds have numerous practical
application, e. g. in monitoring of curing processes. 10
14.3.1 Tools for polarized measurements
There are many types of polarizers, however two main classes cover almost all options
available in practice. These are
. prism polarizers, e. g. Glan- Thompson prism;
. film polarizers, e. g. Polaroid "H -sheets".
The latter are made of stretched polymer. Also wire grid linear polarizers were developed
for application in the visible and infrared spectrum ranges. The film polarizers have rel-
atively low price and work in wide range of incident angles (typically more than =:t25°).
lOSee, for example, book by Benlard Valeur [11].
258
Polarization measurements
Sample
v
P2
v
Emission
To
detection
system
. a">'
"'.;
. 'b"
U"'-;
h
h
Figure 14.4: Scheme of anisotropy measurements: PI is excitation polarizer and P2 is
analyzing polarizer.
Their main disadvantages are relatively high losses, lower degree of polarization as com-
pared to prism polarizers (contrast is < 1000) and limited working spectrum (most of them
absorb strongly in the blue and near ultraviolet ranges).
The prism polarizers have low losses, thus can work at higher light intensities. The
damage threshold can be as high as 500 W cm- 2 at CW irradiation and 500 MW cm- 2 in
pulsed (IOns) mode. Another advantage of the prism polarizers is high degree of polariza-
tion, (can be > 106) in a wide spectrum range (for Glan- Thompson polarizers 400-1700
nm). A disadvantage of the prism polarizers is a relatively narrow range of incident angles
(usually < 10°). In ultra-fast spectroscopy applications the prism polarizers are very unde-
sired because of their high dispersion, which may lead to essential pulse broadening and,
thus, loss in the time resolution.
In addition to polarizers one can use polarization rotating plates (A/4 and A/2) to change
polarization state of the light, e. g. convert linearly polarized light to circular polarized or
to turn polarization by 90° .
14.3.2 Optical schemes for polarization measurements
Polarization arrangements for typical fluorescence measurements is schematically presented
in Fig. 14.4. In steady state measurements the excitation source is usually a lamp, which
produce unpolarized light. Then, a polarizer (P 1) is installed in front of the sample to
provide excitation at desired polarization. In time resolved experiments lasers are used for
the excitation. The emission of lasers is polarized, but may have undesired orientation. To
change polarization of the excitation, half wave plates are used. Orientation of the plate
must be at 45 degrees to the beam polarization to turn the polarization by 90 degrees. 11
11 A more universal approach is to use two quarter wave plate. The first is installed at 45 degrees angle to the
laser beam polarization and convert the linear polarization to the circular one. Then, by rotating the second plate,
any polarization of the excitation can be obtained. An advantage of using half and quarter wave plates is that the
polarization is changed without energy losses.
14.3. Applications of polarized measurements
259
The analyzing polarizer (P2) is installed between the sample and detection system. Both
polarizers (P 1 and P2 in Fig. 14.4) should have adjustment possibilities, since 4 intensities
have to be measured - Ivv, Ivh, Ihh and Ihv, as was discussed in Section 14.2.3.
If the only aim of the measurements is to measure emission decay, the analyzing polar-
izer can be turned to magic angle, cp 54.74 0 . After that the measurements are nor affected
by the anisotropy change of the sample.
14.3.3 Measurements of energy transfer dynamics
As an example of polarized spectroscopy application a study of the excitation energy trans-
fer dynamics in natural photo-synthetic system will be discussed in this section [27]. In
plants and green algae the conversion of the solar energy to electro-chemical potential in-
volves the light absorption by antenna chromophores and excitation energy transfer to the
reaction center, where the excitation is used to transfer an electron from one molecule to
another. These primary processes are extremely fast but can be studied by the time resolved
spectroscopy methods.
The antenna subsystem of green plants is formed by the chlorophyll molecules imbed-
ded into membrane proteins surrounding the reaction center. An excited chlorophyll (Chi)
molecule can transfer the excitation energy to its neighbor by non-radiative resonance en-
ergy transfer mechanism. The neighbor transfers the energy to another neighbor and so on
until the excitation energy is delivered to the reaction center. In photo system I (PS I) the
reaction center consists of special chlorophyll dimer (P700), a few chlorophylls and two
quinones. Excited P700 can initiate a chain of electron transfer reactions which delivers
an electron to the quinone in 21-35 ps. Since P700 is chlorophyll dimer it differs slightly
from the antenna chlorophylls, having the absorption spectrum shifted slightly to the red
(the maximum of P700 is at 700 nm, whereas for the antenna Chis the maxima are in the
range 670-685 nm). However the difference is rather small and the energy transfer from
P700 to the antenna Chis is also possible.
The excited state of chlorophylls can be studied by measuring the fluorescence, but the
time resolved fluorescence measurements are not sensitive to the energy transfer process.
The reaction of energy transfer can be schematically presented as Chl + Chl 2 ---+ Chl l +
Chi;, and if Chi 1 and Chl 2 are two molecules with the same spectroscopic properties, as it
takes place in the antenna subsystem, the reaction will not give any change in the emission
or transient absorption spectra. Nevertheless, the molecules Chi 1 and Chl 2 are different
molecules and they may have different orientations. If so, the polarizations of the emissions
of these molecules will be different and this fact can be used to monitor the energy transfer
dynamics. The time resolved polarized fluorescence measurements of the P700 enriched
reactions centers are presented in Fig. 14.5 [27].
The experiments were carried out using the up-conversion method (see Chapter 12). To
acquire the information on emission polarization the excitation scheme must be modified
to allow rotation of the excitation polarization. In up-conversion experiments the emis-
sion is mixed with gate pulses and, since the mixing is sensitive to polarization (due to
the phase matching condition in non-linear crystal), only one polarization of the emission
is detected. Therefore in the case of up-conversion measurements it is technically easier
to manipulate the polarization of the excitation. The fluorescence decay trace marked as
260
Polarization measurements
600
o
I para(t)
(a)
600.."
5400
o
o
200
0.4
(b)
0.3
Ei:
e
0
.!2 0.2
c
«
0.1
0.0
o
2 4
Time (ps)
6
8
Figure 14.5: Polarization dependent fluorescence decays of of the P700 enriched PS I re-
action centers excited at 700 nm and monitored at 749 nm ( a), and anisotropy decay (b)
calculated from the data in (a) [27]. The solid line in (a) shows the instrument function.
Reproduced by permission of Elsevier Science S. A. @ 2000 Elsevier Science S. A.
I para (t) presents the decay for detection polarization parallel to the excitation polarization,
and I perp (t) shows the decay for perpendicular polarizations, respectively. The anisotropy
was calculated using eq. (14.12).12
For this measurements the time dependent anisotropy reflects mainly dynamics of the
intennolecular energy transfer. The maximum value of anisotropy is close to 0.4, which is
expected for the photoinduced anisotropy of isotropic sample. In Fig. 14.5 the excitation
wavelength corresponds to the absorption maximum of P700, therefore the anisotropy dy-
namics suggests that the energy can leave the reaction center. Also the anisotropy does not
l2In femtosecond up-conversion (and pump-probe) experinlents the excitation and monitoring directions must
be almost co-linear. Therefore the scheme presented in Fig. 14.4 cannot be used. However, the detection polar-
ization is not changed, but the excitation polarization can be tuned (to measure I para and I perp ), so one does not
need to measure G- factor if the excitation intensity does not change when changing polarization.
14.3. Applications of polarized measurements
261
decay to zero, which can be interpreted as an equilibrium between the antenna Chis and
P700. The scheme for this part of the reaction center dynamics is
- -
Chi * P700* A . . .
where the left side presents the energy transfer equilibrium between antenna chlorophylls
and P700 (Chl* + P700 Chi + P700*). The fast decay of the anisotropy suggested the
presence of excitation transfer with a time constant of less than 0.3 ps. The right side of the
scheme presents the first steps of the electron transfer chain (P700* + A ---+ P700+ + A -,
and so on). The electron transfer is observed as the total fluorescence intensity decay.
The important conditions for the anisotropy measurements is a small difference in tran-
sition dipole moments between the involved states. If the excited state has the transition
dipole moment turned bye. g. 45 degrees relative to that of the ground state, the degree of
the photoinduced anisotropy of such sample is expected to be very low. This may take place
when the second singlet state in excited. It is relaxing quickly to the first singlet excited
state, via internal conversion mechanism, so that the observed emission (fluorescence) is
the transition from the first excited to the ground state. Then the excitation and emission
are transitions between two different pairs of states, which may have different orientations
of the dipole moments. To avoid such effect, the samples in above example were excited at
700 nm, which populates directly the lowest singlet excited state of P700.
Chapter 15
Analysis of the measurements
Most of the measurements are carried out to obtain just a few parameters characterizing the
system under study out of hundreds or thousands of actually measured values. For example,
the aim of fluorescence decay measurements of some dye in solution can be an estimation
of lifetime of the singlet excited state. Typically one will fit the fluorescence decay to mono-
exponential model to extract the single value, lifetime, from many data points specifying the
emission decay. This kind of the experimental data analysis is the subject of this Chapter.
First a general approach to the problem will be briefly discussed. Then we will look at most
typical models used in analysis of the spectroscopy data. And finally the fit procedures will
be reviewed.
15.1 Indirect measurements
Suppose we like to know lifetime of the singlet excited state of some compound. The life-
time cannot be measured directly. However, one can measure fluorescence decay and try to
extract the lifetime from the decay curve. The latter can be done by fitting the experimental
data to a certain theoretical dependence and finding parameters (lifetime in our case) best
suiting to the measured data. This is a general problem and can be formulated in a general
manner.
The experimental data, such as absorption spectra or emission decays, are dependences
of some measured parameter, e. g. absorption, on experimentally controlled parameters,
e. g. wavelength, which can be formally presented as
Y == Y(X)
(15.1)
where Y == Yl, Y2, . . . Y K and X == Xl, X2, . . . X K are two vectors, i. e. two sets of values,
of length K, with direct relation between their elements: Yl was obtained at Xl, Y2 at X2
and so on. In case of absorption spectrum, X are the wavelengths selected by operator, and
Yare the absorbances at these wavelengths.
For the quantitative analysis we need a theoretical model of the system under study
which predicts Y values as function of X values and depends on some model parameters P
Y m == F(P, X)
(15.2)
263
264
Analysis of the measurements
For example, in analysis of an emission decay of a singlet excitaed state the model parameter
is the state lifetime.
The model parameters P == PI, P2, . . . P N are used to adjust the vector Y m to be as
similar to the experimental data Y as possible. When this adjustment procedure is complete
we will conclude that the system under study is characterized by parameters P, so the goal
is to evaluate the parameters, e. g. lifetimes.
The problem, formulated in this manner, is called indirect problem since the experimen-
tally obtained values, Y, have to be processed in a certain way in order to obtain parameters
of our interest, P. It is also called am inverse problem, since from the point of view of the
theoretical model of the studied object we know how to obtain Y values from given model
parameters P, but we want to estimate the parameters P for the given values Y, i. e. we are
looking for an inverse function F- l , which gives P == F- l (Y).
Naturally, model and real data do not coincide completely. Consequently, we are looking
for the best approximation of the measured data Y by the model values Y m == F(P, X) and
we should accept certain deviation of the model values from the experimental data. In
some cases the best approximation can be obtained by using some analytical expressions
for computations, such as linear approximation, which means that there is an analytical
solution of the inverse problem. In most cases, which are practically important, there is
no analytical solution to the inverse problem, and to find parameters of interest, P, trial
procedures are used. The aim of the trials is to fit the model to the measured data by
adjusting its parameters.
In two following Sections the most common models used in analysis of the optical
spectroscopy measurements will be discussed. There are two classes of such data - the
spectra and the time resolved measurements, emission decays and transient absorbances.
15.2 Spectral data analysis
In most cases the theoretical predictions of the absorption spectra operate with the transition
energies. The experimental data, absorption spectra, are the dependences of the absorption
on the wavelength, or wave number, or any other equivalent presentation of the photon en-
ergy. If the experimental absorption spectrum shows only one band the problem is reduced
to finding the wavelength corresponding to the absorbance maximum. If the case is not
that simple and absorption spectrum is composed of a number of bands, which are probably
overlapping each other, more complex analysis methods are required to find out transition
energies and probably some other parameters.
A complex spectrum can be usually presented as superposition of bands. For an isolated
molecule a single band has Lorentzian band shape
F v2 Fo
L(v) = (v - VO)2 + D..v 2
(15.3 )
where v is the (photon) frequency, Vo is the band position, v is the bandwidth, and Fo is
the intensity at maximum. It is important to note that the band shape is given in frequency
domain, which is directly proportional to the photon energy. The frequency can be directly
replaced by the energy, e. g. counted in e V, or the wave number, traditionally counted in
15.2. Spectral data analysis
265
1
- Lorentzian
-. Gaussian
0.8
0.6
;;;-
'-'
0.4
0.2
0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
v
Figure 15.1: Lorentzian and Gaussian bands calculated for Vo == 2, v == 0.2 and Po == 1
with eqs. (15.3) and (15.4), respectively.
cm -1, but transition to the wavelength domain requires some caution as will be discussed
later. 1
Lorentzian band shape, however, is very rarely observed in practice, since the interaction
with environment results in a broadening of the spectrum, as was discussed in Section 13.1.
Most broadening mechanisms result in Gaussian band shape
Fc(v) = Fa exp [ - ( v :o y]
(15.4 )
The bandwidth in eqs. (15.3) and (15.4) has different values. For Lorentzian band
the full width at half maximum (FWHM) is Vl == 2 . v, where as for Gaussian band
2
Vl == 2 v Vlog 2 2 . v . 0.83 == 1.66 . v. The bands are shown in Fig. 15.1
2
for Vo == 2, v == 0.2 and Po == 1. For Lorentzian band the intensity decreases much
slower than that for Gaussian band. For example, at 2 v from the maximum, the intensity
of Gaussian band the intensity is 0.018 of its maximum values, and at 4 v is 10- 7 , which
can be neglected in most practical cases. For Lorentzian band shape the intensities are 0.2
and 0.059 at displacement 2 v and 4 v from the maximum, respectively.
Pure Gaussian or Lorentzian bands are rarely observed in practice. For instance in
molecular systems the overlapping bands and vibrational sub-levels are common reasons
for relatively complex band shapes, although there are distinct bands which can be charac-
terized by certain transition energies. In practice the analysis of absorption and emission
bands can be accomplished by approximating the measured spectrum by a sum of Gaussian
1 See also Table 1.2 for other units and unit conversions.
266
Analysis of the measurements
or other model bands. In case of Gaussian band shapes the model spectrum is
F(v) = f; o'i exp [ - ( V Oi ) 2] = f; o'iG (v, VOi, Vi)
where N is the number of the bands, ai are the amplitudes of the corresponding bands
and G (v, VOi, Vi) = exp [ - ( V';: h f] is the band shape. 2 The total number of model
parameters in eq. (15.5) is 3N, which are band positions, VOi, bandwidths, Vi, and band
intensities, ai. 3
Conversion from frequency to wavelength domain is straightforward for absorption
bands - the frequency is substituted by the wavelength, v == , e. g. in eqs. (15.3) or
(15.4). However, in the case of emission spectra the actually measured intensity depends
on the bandwidth of the detection system (e. g. monochromator), which is reflection of the
fact that in emission spectra the signal intensity is the spectrum power density. Therefore,
if the spectrum density in wavelength domain is I).. and in frequency domain is Iv, then
the measured light intensities are I)..dA and Ivdv, respectively. Since dv == - {2 dA,4 the
relation between the spectrum densities is
(15.5)
c
I).. == A 2 Iv
In particular, the Gaussian band shape in the wavelength domain is given by
(15.6)
[ ( ) 2 ( ) 2 ]
Ao AO AO - A
FC(A) == A 2 exp - A A
( 15.7)
where AO == ..£. is the wavelength corresponding to the frequency of maximum in the fre-
Va
quency domain, A == AO .6.v == v ).. , and Ao is a constant. The wavelength AO is not
Va C
actually the maximum of the emission intensity in wavelength domain, it is also approach-
ing the maximum for narrow bands, i. e. when ; ---+ o.
As an example the absorption and emission spectra in frequency (wavenumber) and
wavelength domains are presented in Fig. 15.2. The bands in wavenumber scale (frequency
domain) were calculated using Gaussian bands, eq. (15.4) with absorption maximum at
ka == 25 000 cm -1, emission at ke == 20 000 cm -1, and bandwidth for both spectra of
k == 2000 cm- l (FWHM values are 3330 cm- l ).This corresponds to absorption max-
imum Aa == 400 nm and bandwidth Aa == 32 nm (FWHM value is 53 nm). For the
emission band kecorresponds to Ae == 500 nm, but the actual emission maximum is at
495 nm. The emission bandwidth as calculated from the corresponding wavenumber val-
ues is 50 nm (FWHM value is 81 nm) which is essentially broader than the width of the
2 Indeed, any other band shape can be used in place of Gaussian. In the following discussion we will not use
any specific features of the Gaussian band shape.
3It will be shown in Section 15.4.3 that only non-linear parameters, which are vo and v in this case, require
the fitting procedure, whereas the linear parameters (a ) can be computed. Therefore the number of parameters
which are subject to fit is 2N in this case.
4 If wavenumbers are used instead of frequencies, k = A-I, then dk = A - 2 dA (provided the same units are
used for both values, e. g. cm- 1 for wavenumber and cm for wavelength, respectively).
15.3. Kinetics and reaction schemes
267
1 1
Abs
0.8 0.8
d d
cod 0.6 0.6 cod
.- .-
rJ) rJ)
C) 0.4 0.4 C)
.- .-
0.2 0.2
19OOO
20000 25000
-1
vvavenurnber, crn
30000 350 400 450 500 550 600 658
wavelength, nm
Figure 15.2: Absorption and emission spectra in frequency (wavenumber) and wavelength
domains.
absorption spectrum although in the wavenumber presentation the widths are the same. Ad-
ditionally, one can notice that in the wavelength domain the bands are not symmetric, which
is the consequence of the conversion procedure.
15.3 Kinetics and reaction schemes
Time resolved measurements are very important part of spectroscopy studies since they
allow one to elucidate the mechanism of the photo-reactions. By the time resolved mea-
surements we will assume excitation of the sample by a short light pulse and measurements
of the time evolution of the emission or absorption induced by that pulse. In most cases
the excitation pulse width will be assumed to be much shorter than any photo-reaction, so
that the pulse width can be neglected. This simplifies the modeling of the sample response.
Also the methods accounting for the excitation pulse width will be discussed in the end of
this Section.
15.3.1 First order reactions
Let us consider a molecule in excited state. It can relax by emitting a photon, M*
M + hv, where k r is the radiative decay rate constant. It can decay non-radiatively to
the ground state, M* M, where k n is the non-radiative decay rate constant, or it can
relax to some intermediate state, M* P, where kp is the reaction rate constant. 5 If
the population of the excited state is nex than the value nexkrdt is the number of photons
50 ne can notice a relation between the rate constant and the reaction probability. For example, the probability
of the excited molecule to emit a photon in time interval dt is krdt.
268
Analysis of the measurements
emitted in time interval dt, which is also the number of molecules relaxing to the ground
state by emitting photons. The total number of excited molecules relaxing in time interval
dt is nexkrdt + nexkndt + nexkpdt, or in form of differential equation
dn ex
ill == -nexkr - nexkn - nexkp == -konex
(15.8)
where the negative signs in the right side of the equation are due to the fact that relaxation
decreases the population, and ko == k r + k n + kp is the total relaxation rate.
The solution of eq. (15.8) is
nex(t) == ce- kat
(15.9)
where c is a coefficient which can be determined from some initial conditions, which are
usually the populations right after the excitation. Assuming that the excitation was a very
short light pulse at time t == 0 and the initial population of the excited sate is c == nex (0),
one obtains
nex (t) == nex (O)e -kat
(15.10)
Noteworthy, in time resolved spectroscopy measurements, which follows the popula-
tion nex (t), 6 an exponential decay will be observed with time constant T == kG 1 , i. e. the
individual reaction rate constants k r , k n and kp cannot be obtained form the decay mea-
surements. To extract individual rates from ko additional experiments must be carried out.
For example, emission quantum yield is
k r
CPern = k r + k n + kp
k r
ko
(15.11)
and, if it was measured (e. g. see Section 6.2.5) in addition to the excited state decay
measurements, then the radiative rate constant can be calculated as k r == CPemkO. Similarly,
if the quantum yield CPP of the photo product P was determined, than the rate constant of
the reaction M* ---+ P is kp == cppkp.
The above consideration can be applied to any spontaneous reaction, e. g. radioactive
decay. The exponential decays are very common for the reactions of different natures and
exponential decay models can be found in many other non-spectroscopy applications.
15.3.2 Second order reactions
In photochemistry the excited state quenching is one of the processes of great importance.
In particular it may happen as the result of interaction between two excited states, e. g.
after their collision. Schematically this type of excited state quenching can be presented as
M* + M* M, and mathematically it is expressed by the differential equation of the
second order
dn ex _ -k 2
dt - 2nex
(15.12)
6These can be emission decay measurements or transient absorption measurements at wavelength specific for
the excited state under consideration.
15.3. Kinetics and reaction schemes
269
where k 2 is the second order rate constant. The measure of the second order rate constant
is the inverse measure of population multiplied by the inverse of time, e. g. if population
is measured in number of molecules per cubic centimeter, cm -3, and time in seconds, then
the second rate constant has units of s- l cm 3 .
The solution of the equation is
1
nex (t) == k
2 t + C2
where C2 is the integration constant which can be determined from the reaction initial con-
ditions. Under assumption of a very short pulse excitation at t == 0, which generates nex (0)
excited states, the constant is C2 == [n ex (0)] -1 and eq. (15.13) can be rewritten as
( ) nex(O)
n t-
ex - nex (0)k 2 t + 1
In order to be observable experimentally the quenching must be faster than the other
excited state relaxation reactions. Comparing eqs. (15.8) and (15.12) this means that
k2nex » ko, where ko is the sum of all first order relaxation rates. This case is usually
referred to as diffusion controlled reaction.
From the experimental point of view, if a measured value is proportional to the pop-
ulation of some state, as in the case of the excited state emission, a simple test for the
second order reactions is to plot the inverse of that value as a function of time. Since
[n ex (t)] -1 == k 2 t + C2, the dependence must be a straight line with the slope given by the
second order rate.
(15.13)
(15.14)
15.3.3 Complex schemes for the first order reactions
In most practically importance cases the photo-excitation initiates a chain of reactions. For
example, in photochemistry the excitation yields the excited singlet state, which may relax
first to the excited triplet state, and then to the ground state. There are also possibilities
for other types of photochemical reactions as photo-induced charge transfer or exciplex
formation. The states which are formed following the photo-excitation, and relaxing further
to some other states, e. g. to the ground state, are called intermediate states or intermediates.
In order to discuss a general case, a scheme consisting of four states with all possible
transitions between them is presented in Fig. 15.3. The states are marked AI, ... A 4 , and
the reaction rate constants are k ij , where i and j are the indexes of the reactant and product,
respectively. These rates are also called intrinsic rate constants since they are properties
of the elementary reactions. We will also assume that all the elementary reactions are the
first order reactions. This scheme can be used to present a great variety of the reactions,
and in case of photo-reactions the number of actually possible elementary reactions is usu-
ally much smaller. 7 However the conclusions, which can be made from this consideration,
deserve to be formulated in a general way.
The kinetic (differential) equation describing the population change of state A 1 is
dnl
& == -(k 12 + k 13 + k 14 )nl + k 2l n 2 + k 3l n 3 + k4ln4 (15.15)
7In the case of photo-reaction, the energy difference between most states is much greater than the thermal
energy, as between the ground and excited state. Therefore most of transitions can be excluded from the scheme
in Fig. 15.3. For example, one can neglect by probability of spontaneous transition from the ground to the excited
state.
270
Analysis of the measurements
The terms on the right side with negative sign are responsible for the decrease of the state
Al population, e. g. reaction Al A 2 , and positive for the increase, e. g. reaction
A k2l A
2 ----7 1 .
Similar equations can be written for the other
state, but from four equations only three are inde- k 12
:> A 2
pendent, since the total population should not change,
1. e.
nl + n2 + n3 + n4 == no == const (15.16)
k 14
k 32
where no is the total population, e. g. total num-
ber of molecules. Therefore the system is described
by three differential equations of type (15.15). The
general solution of the system of three linear dif-
ferential equations is the sum of three exponential
terms
k34
Figure 15.3: General four state reac-
tion scheme.
ni(t) == Cile-k1t + Ci2e-k2t + Ci3 e - k3t
( 15.1 7)
where k l , k 2 and k3 are the rate constants and Cil, Ci2 and Ci3 are the pre-exponential
factors. Three rate constants in eq. (15.17) are the functions of the intrinsic rates and
they are common for all the states in the scheme. The experimentally accessible values,
e. g. sample absorption or emission, are the linear combination or state populations. For
example, if Al and A 2 are the only emitting states in the scheme, the total emission of the
sample as the function of time is I(t) == krlnl(t) + kr2n2(t), where k rl and k r2 are the
radiative rate constants of the states Aland A 2 , respectively. Thus, in any experiment one
will observe a transient process which can be approximated by three exponents at most.
The important general conclusion from the above example of four states scheme is
. if the number of the states in the reaction scheme is N, then the temporal behavior of
any of the intermediate state can be described by the sum of at most N - 1 exponents,
with the same rate constants for all the states;
. in any time resolved experiment the signal can be approximated by the sum of at most
N - 1 exponents, although the number of intrinsic rate constants is at most N (N - 1).
The pre-exponential factors are also functions of the intrinsic rates and of initial conditions.
Analytical expressions for rates and pre-exponential factors can be obtained for any number
of the intermediate states, but already for four state in general case the expressions are bulky
and difficult to analyze. Also in practical photophysics and photochemistry even when four
states are involved in photo reaction, the number of elementary reactions to be considered
is much smaller than that shown in the general reaction scheme in Fig. 15.3. Therefore a
few simplified but more practical cases will be considered here to illustrate the application
the reaction kinetics analysis.
15.3. Kinetics and reaction schemes
271
Linear chain of reactions
Probably the simplest reaction scheme is the linear chain of reactions. As an example
one can consider a typical photochemical reaction of some dye molecule. The excitation
produce the singlet excited state, which can relax to the triplet excited state, and the latter
relaxes to the ground state. Yet another example can be a carrier relaxation considered in
Section 12.3. The excitation generates carriers in the conduction band. Then the carriers
are trapped by the quantum wells, where they finally recombine. The reaction scheme for
the both cases is
A k12 A k23 A
1 -----+ 2 -----+ 3
(15.18)
where Al is the primary excited state, e. g. singlet excited state, A3 is the final ground
state and A 2 is the intermediate state. The reaction scheme is described by a system of two
differential equations
{ irf: - k 12 n l
dt - k 12 n l -k 23 n 2
In matrix form this equation can be rewritten as
(15.19)
d
-N==K.N
dt
(15.20)
nl -k 12
where N == is the population vector, and K
n2 k 12 - k 23
teristic matrix of the equation (the matrix with rate constants in our case). Solution of
the equations can be expressed in terms of eigen-values and eigen-vectors of matrix K.
The eigen-values are the decay rate constants, and can be found as solutions of equation
det ( -k k 12 - /3 k 0 /3 ) == 0, which are /31 == -k 12 and /32 == -k 23 , in this par-
12 - 23-
1 k 12
and V 2 == k 23 -k 12 . They give relation
o 1
between the pre-exponential factors, so that the solution has form
N == Cl V l e fJ1t + C2 V 2 e fJ2t
o
is the charac-
ticular case. 8 The eigen-vectors are VI
(15.21)
where Cl and C2 are the constants determined by the initial conditions. For the linear reaction
scheme (15.18) the solution is
nl (t) cle- k12t
n2 ( t ) C k12 e- k12t + C2e-k23t
1 k 23 -k 12
(15.22)
Assuming the sample was excited at t == 0, and the excitation populates state AI, we can
take the initial condition to be nl (0) == 1 and n2 (0) == 0. 9 Then the constants are Cl == 1
8In general, eigen-values are solutions of equation (A - A . I)x = 0, where I is the unity matrix. Please refer
to mathematics book for further information of matrix algebra and linear differential equation solutions.
9The physical meaning of the values nl and n2 is the probability for system to be in states Al and A 2 ,
respectively. Altenlatively they can be interpreted as some reduced concentrations. If someone needs to operate
with actual concentrations, then the initial conditions are nl (0) = nex and n2 (0) = 0, where nex is the
concentration of the excited state at t = 0, thus all the following consideration can be converted to concentrations
by multiplying the results by nex .
272
Analysis of the measurements
1
k 12 = 5, k 23 = 1
o .
. . .
. . .
. . .
. . .
o
.-
cd
0.5
P-
o
P-
/--......
/ ......
...... .
,.
.' ......
......
......
......
......
.....
-A
1
-. A
2
...A
3
-
---
---
. . .
. . .
. .
. .
o
.-
cd
0.5
P-
o
P-
/ ..- - - - - - -
o . .
o 0.5 1
-A
1
-. A
2
...A
3
---
1.5
time
2
2.5
Figure 15.4: Populations of the states in the linear reaction scheme, eq. (15.18), calculated
for two pairs of the rate constants as indicated in the plots.
and C2 == - k k 12k . Finally, the populations of the states Al and A 2 are
23 - 12
nl (t) ==
n2 ( t) ==
e- k12t
k 1 2 ( e-k12t _ e- k23t )
k23 - k12
(15.23)
Naturally, population of state A3 (which is the ground state for examples considered above)
can be obtained from eq. (15.16), i. e. n3 == 1 - nl - n2, in particular case.
The first obvious result is that decay of state Al does not depend on the rate constant k 23 .
The second practically important conclusion is that in the case of linear reaction scheme the
rates, which are observed in the experiments, correspond to the intrinsic rate constants in the
scheme. The third not so obvious result is that A 2 state forms and decays with rate constants
k 12 and k 23 , but which one is observed as the formation or decay depends on the relation
between them. To illustrate this, two series of population time dependences are presented in
Fig. 15.4. The top plot presents the case when k 12 > k 23 , this is intuitively clear situation
when A 2 state decays slower than its precursor state AI, i. e. reaction Al ---+ A 2 is faster
than A 2 ---+ A3. When reaction A 2 ---+ A3 is faster than Al ---+ A 2 , which is k 12 < k 23 , the
case is called inverse kinetics, and the state populations are presented in the bottom plot in
Fig. 15.4. Noticeably, shapes of the time profiles of the state A 2 populations are the same
in both cases, but the population values are five times smaller for the inverse kinetics. For
the final state A3 the population time profiles are just the same for both cases and only state
Al shows clearly the difference between the normal and inverse kinetics.
The relation between the intrinsic rate constants k 12 and k 23 determines the boarder
between the normal and inverse kinetics, but what happens when k 12 == k 23 ? This is so-
15.3. Kinetics and reaction schemes
273
called degenerated case. For the state Al the solution is still nl rv e- k12t . For A 2 solution
is n2 rv te- k12t .10 The widely used exponential fitting of the spectroscopy data is not
applicable to the case, and will give unpredictable results. In practice this is, hopefully, a
rare situation.
For longer reaction chains the main conclusions made for two stage reaction remain
valid. Namely, experimentally observable rates correspond to the intrinsic rate constants in
the scheme and if at some point ki-l,i < ki,i+l one has to consider inverse kinetics. Also
the time profile of the first state depends only on k 12 , for the second on k 12 and k 23 , and so
on.
The scheme (15.18) can be an over simplification. In example of singlet-triplet-ground
state reaction the emission of the singlet state may have the quantum yield which cannot
be neglected. This can be corrected by adding one more reaction path Al A3 and
modifying the first equation in system (15.19) d:Jtl == -(k 12 + k 13 )nl, which gives eigen-
values (observable rates) k 12 + k 13 and k 23 but the general structure of the solution (eqs.
(15.23)) will be the same. Therefore most of above consideration can be used in that case. 11
Reaction scheme with equilibrium
Another practically important case is the reaction scheme with equilibrium
k12 k
Al +== A 2 A3
k21
(15.24)
It can be solved using the same mathematical approach as one described above for the linear
scheme. 12 The characteristic matrix for this scheme is
K==
-k 12
k 12
k 2l
-k 2l - k 23
(15.25)
The eigen-values (or observable rates) are -/31 (k 12 + k 2l + k 13 + ks )/2 and -/32 ==
(k 12 + k 2l + k 13 - ks)/2, where ks == yI (k 12 + k 2l + k 13 )2 - 4k 12 k 2l , and the eigen-
vectors are
1 1
VI == 2k 12 and V 2 == 2k 12
-ks+k21 +k23- k 12 ks+k21 +k23- k 12
If the initial conditions are supposed to be the same as in the previous example, nl (0) ==
1 and n2 (0) == 0, the solutions for the state populations are
nl (t) ==
n2 ( t) ==
Cl e,61 t
_ k 1 e,61 t
ks
+
+
C2 e,62 t
k 1 e,62 t
ks
(15.26)
where C == ks+k12-k21-k23 and c == ks-k12+k21 +k 23 For e q uilibrium reaction the deca y
1 2ks 2 2ks .
of the state Al is bi-exponential. Because of the reaction A 2 ---+ Al the decay of the
10 According to the scheme, state Al does not "know" what happens with state A2, since all the reactions occur
in one direction.
11 The eigen-vectors are also slightly different, giving lower pre-exponential factors in expression for n2 (t).
12 An example of this scheme application and derivation of the equations can be found e. g. in ref. [28] (see
supporting information for the derivations).
274
Analysis of the measurements
primary formed state (AI) depends on the reactions of state A 2 . This is important point for
the interpretation of the experimental data. A bi -exponential fluorescence decay of some
compound is an indication that the singlet excited state is in equilibrium with some other
intermediate state.
The second point to notice in eq. (15.26) is that the experimentally observable rates
(- /31 and - (32) does not allow to calculate intrinsic rates (k 12 , k 2l and k 23 ), but the ex-
periments can be organized in such way that all three intrinsic rate constants will be cal-
culated. For instance, if Al is the singlet excited state and its decay can be measured at
some wavelength, so that it is not affected by any other emission, the emission decay can
t t
be fitted to bi-exponential model, which is I(t) == ale --:;:;- + a2e - T2 , where 71 and 72 are
the fit time constants and al and a2 are the corresponding pre-exponential factors. Com-
paring this with the first equation in (15.26) one can see that 71 == - /31 1 and 72 == - /3:;1 .
The absolute values of the pre-exponential factors are not very useful, but their ratio is
r == Ql == C1 == k s +k12- k 21- k 23 . Thus for three unknown rate constants there are three
a2 C2 ks -k12+k21 +k 23 '
equations, which can be solved.
After some manipulations (see supporting information in ref. [28]) one can obtain
k 12 ==
k 23 ==
k 2l ==
-1 -1
rT 1 +T 2
r+l
(k 12 7 l72)-1
-1 -1 k k
7 1 + 7 2 - 12 - 23
(15.27)
The same problem can be solved for different initial conditions using the eigen-values
and eigen-vectors obtained for the reaction scheme (15.24) and eq. (15.21).
More complex schemes will result in more complex expressions for the eigen-values
and eigen-vectors. However, when the relaxation of the excitation consists of a few reac-
tions for which the reactions rate constant differ by a few orders of magnitude, a separate
consideration of the fast and slow parts of the reaction scheme can simplify the analysis sig-
nificantly. Already mentioned singlet-triplet-ground state relaxation is the example of this
sort photochemical reaction typical for many organic dye compounds. The photo-excitation
populates the singlet excited state, which usually has lifetime of a few nanoseconds and re-
laxes to the triplet excited state via inter-system crossing process. The triplet state lifetime
can be a few microseconds. Therefore, modelling the photochemistry of the singlet excited
state one can neglect the relaxation of the triplet excited state. Respectively, the photochem-
istry of the triplet state happens after complete relaxation of the singlet excited state, and
can be modelled independent of the singlet state photochemistry.
15.3.4 Time resolved measurements
In the previous sections the time evolution of the populations of different intermediate states
was discussed. However the actually measured values in the time resolved spectroscopy
experiments are the sample emission intensity and transient absorption. Thus, to proceed
with the experimental data analysis the models for the population time evolutions have to
be converted to emission intensities and absorbances of the samples.
15.3. Kinetics and reaction schemes
275
Emission decay models
The radiative rate constant is the parameter specifying how many photons per unit time is
emitted by an excited state. In notations used above the emission intensity of some state Aj
IS
Ij(t) == hvjkjrnj(t)
(15.28)
where k ir is the radiative rate constant and hVj is the average photon energy. Therefore, if
there is a wavelength at which only state Aj emits, then the emission at this wavelength is
proportional to the population of the state, and measuring the emission decay one can see
the time evolution of the population of that state. Although the absolute value of the popu-
lation is hard to obtain from the time resolved emission measurements, the time evolution in
itself is important, since it allows one to obtain the lifetimes and, probably, to calculate the
intrinsic rate constants, as in cases of the linear reaction chain and the equilibrium reaction
considered above.
In eq. (15.28) the intensity Ij is the total emission intensity, i. e. the light emitted in all
directions and at all wavelengths. In most cases we are not interested in absolute values of
the emission intensity, but the emission spectrum can be important in many practical cases.
To account for the emission spectrum the emission spectrum density can be introduced
ij(t, A) == Sj(A)nj(t), where Sj(A) is the emission spectrum of the state Aj. Naturally, for
the total emission intensity Ij == Jo oo ij (A)dA, thus Jo oo Sj (A)dA == hvjk jr . If there are few
intermediate emitting states, then to obtain the emission model of the sample we need to
sum up emissions of these states
N
i(A, t) == L Sj(A)nj(t)
j=l
(15.29)
If only spontaneous reactions are involved in the relaxation of the photoexcited sample the
population kinetics is the sum of exponents, nj (t) == E :vr 1 Cjle- kzt , 13 thus
i(A, t) = [; [ [; CjlSj (A)] e- k1t = [? le- k1t
In other words, the emission time profile at wavelength A is the sum of exponents. The
rates, k l , are the eigen-values of the corresponding characteristic matrix, i. e. the same as
for populations, and the pre-exponential factors (shown in square brackets in eq. (15.30))
are the derivatives of the emission spectrum intensities, S j, of the intermediate states (at A)
and eigen-vectors, Cjl.
(15.30)
Transient absorption models
If absorbance of state Aj is aj (A) and its relative population is nj, than the absorbance of
that state at time t is aj(A)nj(t). Instead of absorbance and relative population one can
BIn general N = M, and N is the number of intermediate states. However in practice, not all the intermediate
states may emit any light and/or for some states the number of exponents can be less than the total number of the
intermediate states.
276
Analysis of the measurements
use molar absorption coefficients, E j (A) and molar concentrations C j (t) to calculate sample
absorption. This will not change the following consideration after replacing aj by Ej, and
nj by C j , respectively.
The actually measured value in transient absorption experiments is the change in ab-
sorbance of the sample (see Sections 7.1.3). Therefore the signal is calculated relative to
the ground state absorbance, which is absorbance of the sample before the excitation. If the
ground state population is ng (t), 14 then the transient absorption is
N-l
a(A, t) == L aj(A)nj(t) - ag(A)nj(t)
j=l
(15.31)
where the summation is done over all intermediate states (and the total number of states
including the ground state is N).
For the scheme consisting of spontaneous reactions the populations are the sums of
exponents similar to eq. (15.17). Therefore the transient absorption can be also presented
as
N-l
a(A, t) == L bj(A)e-k]t
j=l
(15.32)
where b j ( A) are the pre-exponential factors similar (but not equal) to those in square brack-
ets in eq. (15.30). The number of exponents is the number of states minus one, as was
discussed in Section 15.3.3.
From the point of view of mathematical description, there is no difference between the
emission and absorbance time resolved measurements. The measured time profiles reflect
the time evolution of the intermediate state populations. If it is possible to find a wavelength
at which only one intermediate state is absorbing or emitting, then the measurements at this
wavelength will show the time evolution of this particular state.
From the practical point of view, the transient absorption measurements may have an
advantage of quantitative estimation of the yields of intermediate states. If for some photo-
product the absorption cross-section (or molar absorption coefficient) is know at some wave-
length, the signal amplitude at this wavelength can be used to calculate the population of
this state at a certain delay time.
Fits of time resolved measurements
F or the reasons discussed above the exponential fittings of the transient absorption and
emission data are very common in optical spectroscopy. The model for the fitting is given
by eqs. (15.30) or (15.32). The number of the exponential terms is the number of the
intermediate state in the relaxation of the excited state. If the relaxation is a sequential
chain of reactions, then rates in the exponents are the rate constants of the reactions.
In all previous considerations we assumed that the excitation pulse width and the in-
strument time resolution are much shorter than the reaction time of the sample. This is not
14In the scheme presented in Fig. 15.3 one of the states must be the ground state.
15.4. Data fitting
277
always true, and the instrument time resolution is the factor which has to be taken into ac-
count to improve the fit accuracy. The instrument time response can be measured, see e. g.
Section 8.5.1, and used to correct the decay model accordingly15
t
fe(t) = J r(T)f(t - T)dT
(15.33)
-CX)
where r (t) is the instrument response function,16 I ( t) is the decay model for the sample
measured with ideal instrument, and Ie (t) is the decay model measured with the real instru-
ment having response r (t). This integral is called convolution integral. Then the meaning
of the convolution integral is to present the excitation as a series of short pulses shifted in
time and with amplitudes given by r( t), and to sum up the sample responses to each pulses
which were before time t. The result of application of this procedure to fit the emission
decay data are shown in Fig. 8.2 on page 154 and Fig. 12.2 on page 221.
15.4 Data fitting
N ow that we have the experimental data and mathematical models for the measurements, we
can fit the data. As was discussed in the first Section of this Chapter, from the mathematical
point of view we need to solve an inverse problem.
15.4.1 Criteria for the fit goodness
The first step in solving the inverse problem is to find a criterion for the model goodness.
The most widely used criterion is the mean square deviation
K
2 1 2
a == K (Yi - Ii(P))
i=l
(15.34)
This equation is easy to understand if we will consider accurate measurements disturbed by
a stationary and non-correlated normal noise, so that
Yi == Ii + ni
(15.35)
where ni is the noise. Stationary means that the noise dispersion does not depend on X, i. e.
the noise statistics do not change during the measurements. Non-correlated means that the
noise values at measurements i and i + 1 are independent values. Combining eqs. (15.34)
and (15.35) one obtains
1 K
a 2 == - n
K I
i=l
(15.36)
15In Section 8.5.1 the method accounting for the instrument response was discussed in application to time
correlated single photon counting technique. Therefore, eq. (15.33) is a copy of eq. (8.6).
16Measurements of the response functions were discussed e. g. in Sections 8.5.1 and 12.1.3.
278
Analysis of the measurements
which is the noise dispersion, since n == 1< E l ni == 0 by definition. One can say, that
the aim of the fit is to separate the noise from the "real" values and to determine the noise
statistics, i. e. to calculate the noise dispersion, a. Ideally, the a 2 criterion shows the noise
mean square deviation.
The a 2 criterion is not the only one. For example one can use absolute values of the
deviations a abs == E 1 I!i - Yi I. However, the a 2 criterion is the best for the normal noise
distribution and, since most of the noises have normal distribution, this criterion is the most
widely used. 17
Application of the a 2 criterion with eq. (15.34) requires stationary noise statistics. How-
ever, there are many important cases where the noise is not stationary. In spectroscopy such
case is the photon counting. Uncertainty for Y counts is Vfj, that is, uncertainty or noise
level depends on the signal intensity. For example, in fluorescence decay measurements the
noise level decreases with decrease in emission intensities, thus it is changing with time.
Therefore eq. (15.34) cannot be applied directly. Situation can be corrected by introducing
a weight factor, Wi
K
a 2 == L Wi(!i(P) - Yi)2
i=l
(15.37)
In the case of fluorescence measurements or any other photon counting experiments, the
weight factor is Wi == . Then, uncertainty of measurement i, which is Jfii, is multiplied
vy
by , resulting in stationary noise with mean square deviation a 2 == 1. 18 This simple
vy
modification allows one to increase applicability of the a 2 criterion.
From the point of mean square deviation criterion, the best approximation is the one
which gives the smallest a 2 . Therefore, the problem can be reformulated in mathematically
strict manner - better approximation means smaller mean square deviation as it is given by
eq. (15.34) or (15.37). Thus, from the mathematical point of view solution of the inverse
problem or the problem of indirect measurement is reduced to minimization of the a 2 -value.
15.4.2 Minimization of mean square deviation
Formallyeq. (15.37) is a function of parameters P and minimization of a 2 can be done by
solving equations
8cr 2 - 0
8PI -
8cr 2
_ 8 ==0
P2
(15.38)
8cr 2
_ 8 ==0
PN
17Practically important for spectroscopy application photon counting statistic approaches the normal distri-
bution at relatively high number of counts, as was discussed in Section 4.1.2. At low number of counts, e. g.
N < 100, Poisson distribution of counts has to be taken into consideration, which leads to the maximum likeli-
hood criterion of fit goodness [12].
18In this case the criterion is commonly called X 2 , meaning that w = .;;, .
vy
15.4. Data fitting
279
In general the system (15.38) has no analytical solution. However, there are some important
cases when the system can be solved. One of them is polynomial function.
The simplest polynomial is the polynomial of zero order, which is
F(x) ==Pl
(15.39)
The function is just a constant (so number of parameters is N == 1). There is a single
parameter which leads to a single equation for the minimization problem
Ba 2 B M 2 M
- == - L Wi (PI - Yi) == 2 L Wi (PI - Yi) == 0
BPI BPI i=l i=l
(15.40)
and after rearrangement
M M
PlLWi == LWiYi
i=l i=l
(15.41)
Which gives
M
L: WiYi
i=l
M
L: Wi
i=l
(15.42)
PI ==
To shorten the equation we can use an average
1 M
0, = M L o'i
i=l
(15.43)
instead of keeping summation. Then eq. (15.42) has a simple and clear form
wy
PI == -=-
W
(15.44)
In the case of stationary noise, i. e. Wi == 1 for all i, one obtains PI == Y , 1. e. PI is the
average of all measurements, which is an obvious result, indeed.
Similarly, one can work out the linear approximation (the first order polynomial)
F(x) == PI + P2 X
(15.45)
This gives a system of two equations
{ == 8 1 L: l Wi (PI + P2 X i - Yi)2 ==
8cr 2 8 ,\:,M ( ) 2
8P2 == 8P2 Li=l Wi PI + P2 X i - Yi ==
2 L: l Wi (PI + P2 X i - Yi) == 0
2 L: l Wi (PI + P2 X i - Yi)Xi == 0
(15.46)
280
Analysis of the measurements
U sing average values instead of summing
{ PlW
PlWX
+ P2 WX
+ P2 WX2
wy
wyx
(15.47)
Determinant of the system is
det == W . wx 2 - WX 2
(15.48)
Clearly, determinant must be non-zero, i.e. W . wx 2 - WX 2 i=- 0, or system (15.47) cannot
be solved. If det i=- 0, parameters are
PI
P2
WY'WX2 -wx.WfjX
w . wx2 - wx 2
----
w.wyx-wx.wy
w . wx2 - wx 2
(15.49)
In the case of stationary noise (Wi == 1) the parameters for the linear approximation are
PI
P2
Y'X2 -x.fiX
x 2 - x 2
- --
yx-x.y
x 2 - x 2
(15.50)
In the same manner one can derive equations to calculate parameters P for any polyno-
mial, F == PI + P2X + ...PNx N -1. Differentiation (eq. (15.38)) gives a system of equations
{ - + + N 1
PlW ... PN WX-
Pl WX +... + PNWx 2N - 2
wy
(15.51)
wyxN-l
which has dimension N and looks somewhat similar to eq. (15.47). The system can be
solved analytically, although the complexity of the solutions grows fast with increasing the
order of the polynom.
15.4.3 Non-linear least square fit
If nonlinear functions are involved in the model, then, most probably, the system of eqs.
(15.38) has no general analytical solution, and the minimization problem must be solved
numerically. A short discussion of most common fitting algorithm can be found in Appendix
D. In short, one starts with some initial estimation, a guess, of the parameters Po and
calculates the corresponding mean square deviation a6 using eq. (15.37). Then, a new
parameter vector PI is generated according to the selected fit algorithm, and corresponding
ar is calculated. The values a6 and ar are compared and this information is used to generate
the next parameter vector, P 3 . This procedure is repeated until some fit termination criterion
is fulfilled. These repeating circles are called iterations. The most essential difference
between the fit algorithms is the method used to generate new parameter.
The fit termination criteria can be inability of the algorithm to obtain smaller a-value,
or too small change in a-value between iterations, or something else depending on the al-
gorithm. However, no one fit algorithm can guarantee that the found parameters correspond
15.4. Data fitting
281
to the global minimum of a 2 , which is in contrast with the polynomial function approxi-
mation discussed above, where by solving system of linear equations one obtains the best
approximation.
The mean square deviation a is the function of parameters P, which are variable values
in sense of fit routine, a == a(P), according to eq. (15.37).19 The function a(P) may have
a few local minima, which are such values of P == Pz that a(Pz) < a(Pz + (5) for any <5
in some range P, i. e. 1<51 < P . From all possible minima of function a(P) we are
looking for one with the smallest <5-value, which is called global minimum and which is the
solution of the minimization problem.
The existence of the local minima is one of the reasons why the fit algorithms may fail
to find the actual solution of the problem, the global minimum. A usual method used to
check the fit result is to run fitting a few times starting with different initial approximation,
Po, and to verify that the final results are the same (withing the fit accuracy).
There are many factors affecting the ability of the fit programs to find the global mini-
mum. The most important are the quality of the data, e. g. signal-to-noise ratio, complexity
of the model and, of couse, type of the fitting algorithm. Higher number of fit parameters
usually means higher complexity of the fit model and is more difficult to handle.
Linear and non-linear parameter
Let us look at one example of practical importance. In analysis of the time resolved mea-
surements, e. g. flash-photolysis data, the multi -exponential fitting is a usual approach (as
was discussed above in Section 15.3.4). Then, the mathematical model of the decay is
M
f(t) == L aje- kJt
j=l
(15.52)
where M is the number of the exponential terms, k j are the rates and aj are the corre-
sponding pre-exponential factors. The values k j and aj are the fit parameters, so formally
P == (k l , . . . , k M , aI, . . . , am), i. e. there are 2M fit parameters. However, one can notice
that the fit model is just a linear combination of the non-linear functions, exponents in our
example. To present this property in a general way we can write
M
f(P,x) == Lajuj(pl,x)
j=l
(15.53)
where P' is the subset of parameter vector P, i. e. the complete parameter vector is P ==
(PI, al,..., am), and Uj(P I , x) are the functions of P' and x, which are rates k j and time t
in the case of exponential fitting. The coefficients aj are linear model parameters, in contrast
to the subset pI, which are non-linear parameters. 20 The linear parameters aj are also called
19Por a given set of experimental data the values X and Yare constants, unlike the parameters P which can
be changed in certain limits.
20The resulting model function y = f (P, x) is linearly proportional to each of linear parameters.
282
Analysis of the measurements
amplitudes, e. g. for mono-exponential decay the linear parameter, pre-exponential factor,
gives the signal amplitude. 21
To find the minimum of the mean square deviation at fixed pI, one can solve a system
of equations obtained from conditions : = 0, which are similar to eqs. (15.38) with o'j
in place of Pj,
M N
E aj E WiUl (PI, Xi)Uj (PI, Xi)
j=O i=l
N
E WiUl (PI, Xi)
i=l
(15.54)
M N
E aj E WiUM(PI, Xi)Uj(PI, Xi)
j=O i=l
N
E WiUM(PI, Xi)
i=l
This is system of linear equations which can be written in matrix form as C . A == D,
N
where Cis M x M matrix with elements CjZ == E WiUZ(PI, Xi)Uj(PI, Xi) and vector D
i=l
N
has elements d z == E WiUZ (PI, Xi). The formal solution of the equation is A == C- l D,
i=l
which can be formalized for any dimensionality of the system, i. e. for any M. 22 Thus, to
find amplitudes one does not need to use fitting procedure - the amplitudes can be found
by solving system of linear equations. In other words, the number of fit parameters can be
reduced to non-linear parameters, such as the rates in exponential decay (k j in eq. (15.52)),
whereas to find linear parameters one can solve the system of linear equations and avoid
any fitting routines. This reduces the complexity of the fitting procedure and increases the
calculation speed and accuracy.
Special cases
It is beneficial to avoid fitting when it is possible, since fitting always adds some uncertainty,
e. g. by finding local minima. Sometimes it is possible to modify the model function, F (P),
so that the problem is reduced to one of the cases which can be solved analytically. For
example, mono-exponential decay,
F ( a, k) == ae - kt
(15.55)
results in transcendental equations when eq. (15.38) is applied to find minimum. However,
taking logarithm of the function one obtains
In (F ( a, k )) == In a - kt == A - kt
(15.56)
21 In the case of exponential decay, U = e - k t at t ;::: 0, and U = 0 at t < O. Thus function U has maximum
value of 1 at t = o.
22Formally, the system has a solution if matrix C has non-zero determinant, det C i=- 0, which requires that
the functions u J 0 are orthogonal or independent functions, i. e. no one of functions u J 0 can be presented as
linear combination of the other functions. For example, for exponential functions this means that the rates must be
different.
15.4. Data fitting
283
which is linear function of time. Thus one can minimize
K
a 2 == L Wi (A - kt i - In Yi)2
i=l
(15.57)
Then using eq. (15.49) after conversion Yi to In Yi, one can avoid the fitting and calculate the
lifetime (of couse, weight coefficients have to be transformed accordingly). Unfortunately,
this simple method can be applied to mono-exponential decay only. For instance, addition
of a constant level (F == b + ae- kt ) makes result of logarithm application a non-linear
function. 23
There are some other models (fit functions) which allows some simplifications of the
minimization problems. There are also some mathematical treatments, such as Fourier and
Laplace transforms, which are aimed to find analytical solutions of minimization problem
for particular decay models, e. g. exponential decays. However there is no general method
which would allow one to avoid fitting procedure in most cases of practical importance.
15.4.4 Global fitting of time resolved measurements
In Section 15.3.4 we have seen that the models of transient absorption and emission decays
are functions the wavelength and time. Also the measured data can be collected as two
dimensional array presenting both time and wavelength dependences. 24 For instance, it is
a common practice for flash-photolysis studies to measure the transient absorptions at a
series of wavelengths, so that the measured data can be presented as Y == Y (A, t). As it was
discussed in Section 15.3.4, a common model is the sum of exponents of type
f(A, t) == L b j (A)e-k]t
(15.58)
as it is given by eqs. (15.30) and (15.32).
When such time-wavelength data are available, it is beneficial to fit all of them together
rather then one decay after another. The reason fot this is that the rates, k j , do not depend
on the wavelength but doing separate fits at separate wavelengths one will eventually obtain
as many sets of the rates as the number of wavelengths used.
The global fit can be done by minimizing the global a -value, which is calculated as
average of a?-values at all wavelengths
L
2 1" 2
a g == L az
Z=l
(15.59)
where L is the number wavelengths used to collect the data. Then the same fit algorithms
can be used for "normal" single wavelength and global fits. The practically important
23Semi-logarithmic scales are very common to present emission decays even when decays are multi-
exponential, since one can easily discriminate between e. g. mono- and bi-exponential decay by drawing trend
line. An example of the semi-logarithmic plots are presented in Pigs. 8.2 and 12.7 (inset).
24Por all time resolved measurements the methods to collect time-wavelength dependences were discussed,
e. g. see Sections 7.1.3, 8.5.2, 11.1.3 and 12.1.7.
284
Analysis of the measurements
point in this approach is the noise level at different wavelengths, which should be the same
through all the spectrum. Otherwise weight coefficients must be added to equalize the noise
effect
L
'"'" ...!... a 2
L nz Z
2 Z=l
a g == L
E z
Z=l
(15.60)
where nz is the noise level of the l-th decay.25
On the output of the global fit one will have the rates k j and corresponding spectra,
b j (A). In emission spectroscopy these spectra are called decay associated spectra (see Sec-
tion 8.5.2), and in time resolved absorption spectroscopy they are usually referred to as
component spectra. Interpretation of these spectra depends on the reaction scheme selected
to model the sample, and in general can be a complex problem. However, there are few
relatively simple cases which deserve to be mentioned.
The studied sample can be a mixture of few compounds, e. g. of two isomers. If these
compounds have different fluorescence lifetimes but overlapping spectra, the rime resolved
spectroscopy can be used to obtain the emission spectra of each compound. The spectra
b j (A) are just the emission spectra of the compounds. This was discussed in Section 8.5.2 on
page 163.
The spectrum right after excitation is E b j (A), since eO == 1. This calculated spectrum
can be useful when deconvolution procedure is used to improve the time resolution of the
measurements. Then there is uncertainty in determination of the time delay for spectrum
measurements caused by the pulse width, and the measured spectra at short delay times
may differ form the calculated spectra, which gives the spectrum corrected to the instrument
response function.
Another example of relatively simple interpretation of the spectra b j (A) is the linear re-
action scheme without inversion. The spectrum of the longest living component (smallest
rate) is the spectrum of the last intermediate state. If in additions the intrinsic rates increases
gradually from the reaction to reaction, then the spectra of intermediate states can be ob-
tained as sums of spectra b j (A) starting from the intermediate state index, i. e. for state
L
j == J the spectrum is SJ(A) == E b j (A).26
j=J
15.4.5 Qualitative problems
Previously we have considered the case when the model is known and only the particular
values of the parameters have to be optimized. However, the model itself can be under
25In practice the variation in the noise level by a few times can be ignored, but if the difference is one order of
magnitude or greater then the fit procedure will be forced to optimize the most noisy data and will ignore data with
lower noise, if noise normalization was not done.
26The errors of these calculations are of the order kt+ 1 , e. g. if formation rate for some intermediate state is
J
more than ten times higher than the decay rate, then the calculated spectra will differ from actual spectra by less
than 10%.
15.4. Data fitting
285
question. This means that one may have a choice of models, for example,the first order and
the second order reactions, and wants to find out which one fits better to the measured data.
This problem is called qualitative problem.
Let us assume that a transient absorption of some system has been measured and we
would like to know how many intermediate states are formed during relaxation of the excited
state. If all the reactions are the first order reactions, then for a single intermediate state the
decay (measured signal) must be mono-exponential, for two intermediate states must be
bi-exponential and so on. Thus, if we know how many exponential terms are needed to fit
the data we can conclude about the number of intermediate states. This is a typical question
for the qualitative problem. To answer the question we can do sequential fitting of the data
using mono-exponential mode, bi -exponential and so on. If for bi -exponential fit the a 2
value is much lower than that for mono-exponential fit, say by factor of 5, then we can
say that there are at least 2 intermediate states. If the difference in a 2 values between bi-
exponential and 3-exponential fits is small, say less than 10%, we can say that there is no
statistical reason to expect more than 2 intermediate states. The answer we have at the end
is not that there are exactly 2 states, but that under experimental conditions used we have
no prove for more than 2 states. This reflects the fact that all the measurements are made
with certain accuracy. It is possible, that by improving the measurement techniques one
can find statistical evidences for 3 -exponential decay and, thus, for the presence of the third
intermediate state.
The question of what is statistically reliable decrease in a 2 value when switching from
one model to anther is answered in frame of F -statistics. It depends on the number of
experimental data available, number of fit parameters used in each model and and some
other factors. For a typical measurements in optical spectroscopy experiments the 20-40%
difference in a 2 -value can be considered as reliable improvement when testing different fit
models.
Chapter 16
Final remarks
In this book the most widely used optical spectroscopy methods were reviewed. However,
in practice the starting point of any actual research or application is object-centric. The first
step is determination of the classes of phenomena which have to be studied or examined.
The second step is finding right tools for the work to be done. In case of optical spectroscopy
instruments one needs to know the spectrum range of measurements, expected magnitude
of the signal, excitation wavelengths, time scale of the phenomena and other similar param-
eters. Based on this specification one or another method or instrument can be selected and
used.
Indeed, there are rather universal spectroscopy instruments which can be used to solve
different classes of problems. For example spectrophotometers and fluorimeters are usually
general purpose devices. Nevertheless, moving to time resolved measurements, especially
to ultra fast systems one can distinguish specializations of the instrument. In fact, all modem
pump-probe systems are unique instruments designed with special goals in mind. In many
cases the state of the pump-probe instruments as we can see them now is the result of
numerous improvements, redesigns, refittings and readjustments made in order to extend the
facilities of the system to solve yet another class of problems. 1 This is an important point -
the advanced spectroscopy instruments are in continuous development, and the best way to
keep them in good working conditions is to experiment with them and to keep improvements
In progress.
An important approach to the complex instrument design is the modular structure. For
example, the same laser system can be used to generate femtosecond pulse for pump-
probe and up-conversion experiments to measure transient absorption and emission decays
in femto- to nanosecond time domain, respectively.2 The system can be extended by adding
a streak camera to acquire emission decays and spectra in single short experiments, or by
adding a pulse picker and a time correlated single photon counting instrument to measure
1 A famous example of complex femtosecond system, which has been continuously improving during more
than decade, is the one developed by Ahmed H. Zewail group [18].
2The laser system considered in Section 11.2, Fig. 11.8 is an example of such instrument. Also the up-
conversion requires relatively high pulse repetition rate, therefore regenerative amplifier has to be used or excitation
pulses for the up-conversion experiments have to be taken directly after the Ti:sapphire generator.
287
288
Final remarks
e. g. emission decays of single molecules in pico- to nanosecond time domain. 3
One of the goals of this book was to inspire new researchers to design new and to
improve exciting spectroscopy methods and instruments, and provide them with starting
background knowledge in this way. I am happy if you, reader, will find this book useful
for you, and I wish you a new exciting discoveries in your work in the field of optical
spectroscopy.
3This application was discussed in Section 8.7 and ref. [12].
Appendix A
Photon counting peal-up distortions
Peal-up distortions are specific for photon counting techniques. For example, measuring a
continuous photon flow with photon counting detector which has dead time t, 1 there is
a probability that two or more photons will hit the detector during time interval t and,
thus, will be counted as one photon. 2 Another example is the time correlated single photon
counting method, where in the time window of the measurements only the first photon is
detected, and all the following photons are ignored.
The probability of detecting N photons in time window t if the average photon flux is
n, is given by the Poisson distribution, i. e. eq. (4.11),
P _ (n t)N -n t
N - N! e (A. 1)
The average number of counted photons in time interval t is
CX)
N c == LPi
i=l
(A.2)
whereas actual number of photons
CX)
N act == L iP i == n t
i=O
(A.3)
Substituting P N from eq. (A. 1 ) to eq. (A.2) one obtains
(n t)i -n.6.t _ -n.6.t (n t)i _ -n.6.t [ (n t)i _ ]
., e - e ., - e 1 + ., 1
z. z. z.
i=l i=l i=l
e- n .6.t [ ) (n t)i _ 1 ] == e- n .6.t (e n .6.t -1) == 1- e- n .6.t
1,=0
N c
(A.4)
1 Dead time is the time interval during which the next photon cannot be detected.
2S tr ictly speaking, the term detected photons, rather than the photons entering the photo-detector, has to be
used. However, an ideal detector 1000/0 efficiency will be assumed here for the sake of simplicity, so that there
is no difference between the photons and counts. Still the result can be used for real devices by adding detector
quantum yield to switch from counts to photons.
289
290
Photon counting peal-up distortions
which is, indeed, the probability to obtain any number of photons but o.
The difference between the actual number of photons N act , eq. (A.3), and the detected
number of photons N c , eq. (A.4) is
N == N act - N c == n t - 1 + e- n .6.t
(A.5)
Expending exponent as power series
!1N= (n!1t)2 _ (n!1t)3 +...
2 6
(A.6)
one obtains the relative error due to finite time resolution
8 = !1N = n!1t _ (n!1t)2 +...
N act 2 6
(A.7)
For example, if the desired accuracy of the measurements is c5 == 0.01 and the detector
dead time is t == 50 ns, then the photon counting rate must be lower than n == 2c5 t ==
100 kHz.
Appendix B
Relation between Gaussian pulse width
and its spectrum
Let us consider a light pulse width Gaussian shape and middle wavelength at A. We will
assume a band limited pulse, which means that pulse width is determined by its spectrum.
In other words, the pulse width is the shortest possible for a given spectrum width, or vice
versa the spectrum is the narrowest for a given pulse duration.
The envelop of the Gaussian pulse is given by function
1 t 2
f(t) == e- t2
V1i t
(B .1 )
where the term ft.6.t is due to normalization J : f(t)dt == 1. A usual measure of the
pulse or bandwidth is the full width at half maximum (FWHM) value, t 1. The relation
2
between t and tl is given by equation
2
exp
( 1lt f
t2
1
2
(B.2 )
or t 1 == 2 t v - In 0.5 == 2 vln 2 t 1.67 t.
2
To obtain the spectrum of the pulse one needs to calculate the Fourier integral
+CX)
J . 1 t 2 1 A 2 2
F ( w) = e - "wt V1i 1l t e - "" t 2 dt = e - "4 <..>.t w
-CX)
(B.3 )
where w is the circular frequency, which can be converted to normal frequency as 27r f == w.
Thus, the spectrum of a Gaussian pulse is a Gaussian band F(f) == e- n2 .6.t 2 j2 with the
bandwidth of f == (7r t) -1.
In eq. (B.3) the optical frequency of the pulse, fo == , was omitted, which can be
justified if f 0 » t -1 , i. e. in the case of relatively broad pulse (compared to the one wave
291
292
Relation between Gaussian pulse width and its spectrum
period). For shorter pulses ( t < 20 fs) the high frequency "filling" of the pulse should be
taken into account in eq. (B. 1).
In frame of the relatively narrow spectrum approximation, the wavelength bandwidth is
A == f Ac 2 , and substituting the pulse width one obtains
A == A 2
7r tc
or using FWHM values for the pulse and bandwidth, respectively,
(B. 4 )
41n2 A 2 A 2
A 1 == . 0.88
"2 7r c t 1 c t 1
"2 "2
(B. 5)
For example, the spectrum width of a tl == 50 fs pulse ( t 30 fs) at A == 800 nm
2
(Ti:sapphire laser) is Al 0.88 : t 2 38 nm. This corresponds to the case of so-called
2 c 1
"2
spectrum limited pulse width, which means the shortest possible pulses at given spectrum
width. The actual spectrum of the pulse can be broader, but cannot be narrower.
Appendix C
Two photon absorption
At high light intensity the probability of absorption of two and more photon at once in-
creases and may become an important phenomenon at experimental conditions typical for
pump-probe or up-conversion measurements. Let us consider the light propagation through
a sample of thickness l. If I in is the light intensity before a sample, then after the sample
the intensity is [29]
(1 - R)2e- aZ
lout == I in (3
1 + alin(l - R)(l - e- aZ )
(c. 1 )
where R is the reflectivity of the sample, a is the linear absorption coefficient (the one intro-
duced in Section 1.1.1) and (3 is the coefficient responsible for the two photon absorption,
or the two photon absorption coefficient. Neglecting the light reflectance by the sample
and the linear absorption of the sample one can write a simplified equation for the output
intensity in case of pure two photon absorption
I t""V I in
out t""V 1 + (3lI in
This can be used to define two photon absorptance (in a manner similar to the linear absorp-
tance, see eq. (1.9) in Section 1.1.1)
a2ph ==
I in - lout
I in
t""V
t""V
(3lI in
1 + (3lI in
1
1 + ((3lI in )-l
Naturally, the two photon absorptance depends on the incident intensity. At low values
of (3lI in the two photon absorptance is proportional to the input light intensity I in , which
means that the absorption probability is proportional to the square of intensity, as it should
be for two photon reaction. Two photon absorption coefficients ((3) of few media are listed
in Table C.1.
The two photon absorption can be observed at power density approaching giga Watts per
square centimeter values. For example, one millimeter of water will absorb 1 % of the light at
incident power density of I in == Jz . l a 0.2 X 10 9 W cm -2 at 264 nm (the third harmonic
293
294
Two photon absorption
Table C.1: Two photon absorption coefficients, /3, of some materials measured at wave-
lengths A.
material /3 x 1011, cm W- l wavelength, nm reference
fused silica 2.4 264 [30]
< 0.13 355 [29]
water 49 264 [30]
methanol 34 264 [30]
hexane 57 264 [30]
chloroform 95 264 [30]
of Ti:sapphire laser). Assuming a spot size typical for pump-probe experiments d == 1 mm,
i. e. spot area s == d2 10- 2 mm 2 , and the excitation pulse width t == 100 fs, the pulse
energy, at which the two photon absorption will be 1 %, is E == Iins t 0.2 tLJ, which is
reasonably small and easily achievable value in pump-probe experiments.
Appendix D
Fit algorithms
When eqs. (15.38) cannot be solved analytically the minimization problem is solved by
applying one of numerous iterative methods of minimization. This procedure is commonly
caned data fit.
General fit algorithms can be divided on two steps repeated sequentially:
1. For some approximation pJ the goodness of the approximation, a; , is evaluated using
eq. (15.37);
2. Base on a; values evaluation and, probably, on some other knowledge (depending
on fitting algorithm and problem under study) the next approximation is generated,
pj+l.
The steps 1 and 2 are repeated until one of fit termination criteria is reached. These criteria
can be a certain degree of goodness, or a certain number of unsuccessful attempts to improve
the goodness, or something else. The most essential difference between fitting algorithms
is the method used to generate new approximation.
Common problems of different minimization methods are guesses of an initial approxi-
mation, pO (which is not obvious in most practical cases) and ability to find global minimum
rather than one of multiple local minima. For the most of fitting algorithms deviations of
parameters, i. e. vector p == 6:..Pl . . . 6.p N, are defined and actively used.
D.I Stepping algorithm
This is probably the simplest fitting algorithm. A new approximation is generated by chang-
ing only one of parameters, say parameter Pk, and keeping the other parameters unchanged.
The parameter Pk is scanned with the step 6.Pk until minimum of a 2 is found. I When opti-
mization of Pk is complete the next parameter is subjected to the same procedure. When all
1 Typically, a 2 is computed for a parameter vector P and for the same vector but with Pk + D...Pk substituted
in place of Pk. If a 2 (Pk) > a 2 (Pk + f).Pk), then Pk is increased sequentially by steps D,.Pk (i.e. testing
(T2(Pk + 2APk), a 2 (Pk + 3L\Pk), ...) while a 2 is decreasing. If a 2 (Pk) < a 2 (Pk + L\Pk), then Pk is
decreased sequentially (i.e. testing a 2 (Pk - APk), a 2 (Pk - 2APk), . . .) while a 2 is decreasing.
295
296
Fit algorithms
the parameters are optimized, the steps p can be reduced and procedure repeated. The fit
is complete when the procedure with reasonably small steps is complete.
This method is efficient enough if the number of parameters is low, typically N ==
1 . . . 3. With higher number of parameters the efficiency of the method depends on the
scanning order of parameters. The method is very sensitive to the initial approximation and
in the case of wrong guess may not achieve global minimum at all. It is also very sensitive
to the local minima.
D.2 Gradient method
The converge rate of the stepping algorithm can be improved by determination of the direc-
tion of the most quick decrease in (T2, and testing sequentiaIly points (in P space) in that
direction. This can be done by comparing CJ2 for pairs of parameter vectors when only one
parameter is changed. To calculate i component of a 2 gradient one needs to calculate CJ2
for P == Pl, . . . , Pi, . . . P Nand al for P == PI, . . . , Pi + P, . . . P N, then i-component of
the gradient is
oCJ 2 a? - a 2
Si==- 2 (0.1)
OPi P
When all the gradient components are detennined the new parameter vector pj+l can be
computed based on the previous approximation pj and gradient vector S == SI 82, . . . , SN
as
'+1 .
== pi - Si P (0.2)
or in vector form pj+l == pj + S p. If parameter pj+1 gives better (smaller) a 2 value,
the next point in the same direction is tested, i. e. pj+2 == pj+l + S P == pj + 2SD..p.
The procedure is repeated until minimum of a 2 (in direction given by the maximum slope
of CJ2(P), or by vector S) is found. Fonnal1y this method is similar to stepping algorithm
with optimized stepping direction.
This method improves efficiency of fit and can be used with greater number of param-
eters as compared to the simple stepping algorithm. Still it depends very much on initial
approximation and can be trapped in a local minimum.
D.3 Newton method
Further improvement can be done by presenting a 2 as power series of parameters P
N () 2 N N [)2 2
a 2 (p) = a6 + L 8 a . (POi - Pi) + L L 8 . (POi - Pi) (POk - Pk) + . .. (D.3)
i=l P2 i=l k=l P2 Pk
One can note, that the second term is zero as it is follows from eqs. (15.38). Limiting the
series by the second order terms
N N
a 2 (p) == a6 + L L Aik(POi - Pi) (POk - Pk)
i=l k=I
(D.4)
D.4. Random search
297
This is polynomial of the second order of N variables. It has minimum of a5 at point
Po === POI . . . , PON. Now the problem can be reformulated - we need to find coefficients
in eq. (D.4), i. e. Aik and POi- To solve the problem we can perfonn a series of experi-
ments by testing different parameter vectors (say Pi :!: Pi) and obtaining corresponding
a 2 . This set of data can be used for mean square minimization by solving equations similar
to eqs. (15.5 I) and obtaining parameters Po == POI, . . . , PON. This set of parameters, Po, is
the next approximation in sense of this fit algorithm.
This algorithm is very efficient when the initial approximation is close to the global
minimum (when the series (D.3) can be limited by the second order terms). Still the problem
of local minima cannot be avoided with this algorithm.
D.4 Random search
There are many modifications of the random search algorithms with common approach that
the new approximation is generated using a random number generator in a certain range
around some initial approximation. A typical fit strategy is to test parameters randomly
distributed around the best already found parameter vector. If a better parameter vector is
found, it becomes the center of the new search. If a better parameter was not found during
predefined number of attempts, the search range is squeezed.
The initial search range can be broad enough and this helps to minimize the probability
of trapping the fit in a local minimum. At the same time random search is less sensitive
to the initial approximation as compared to the previously considered algorithms. Random
search can be applied successfully to the fitting problem of tenth of parameters.
The disadvantage of the random search is low converge rate.
Appendix E
Physical properties of some solvents
Physical properties of COmlTIOn spectroscopy solvents: A llv is the ultraviolet cutoff wave-
length (absorbance 1.0 in 1 crn cuvette) , n is the refractive index (- is the dielectric constant,
'1 is the viscosity, Trn is the melting point, and T'J is the boi ling point (data from [17, 3 1])
Solvent Aut', nm n f '/ . ] 0: 1, N s m - 2 Tm °C T b , 0 C
acetone 330 1.3591 21.0 0.302 -94 56
acetonitrile 190 1.3460 36.6 0.341 -c!4 81.6
benzene 280 1.5011 2.284 0.65 5.5 80
benzonitri Ie 1.5289 25.9 1.3 -]2.7 191
chJorofonn 245 1 .4459 4.72 0.54 - G:3. G 61.1
cyclohexane 210 1.4262 2.023 0.98 6.6 80.7
1,2-dichloroethane 226 1.4448 10.4 0.78 -3G.7 83.5
diethyl ether 218 1.3527 4.22 0.22 -11G.3 34.5
dimethyl sulfoxide 265 1.4170 47.24 1.1 18.5 ]89
ethanol 210 1.3611 25.3 1.03 -114 78.3
glycerol 207 1.4746 46.5 945 18 290
hexane 210 1.3749 1.890 0.31 -05.4 68.7
methanol 210 1 .3284 33.0 0.58 -07.7 64.7
I-propano 1 210 1. 3840 20.8 1.8 -127 97.2
pyridine 330 1.5067 13.26 0.87 --11.6 115.2
tetrah y dro fu ran 220 1.4052 7.52 0.46 - lOK.G 65
toluene 286 1.4960 2.385 0.55 -D4.D 110.6
water 190 1.333 80.2 0.89 0 100
299
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edition.
Index
absorbance, 4, 5,91,103
absorptance, 3, 5, 111
absorption coefficient, 3, 5
absorption cross-section, 4, 5, 188, 209,
251
acousto-optic modulator, 155
active medium, 39
amplification coefficient, 41
angular aperture, 33, 111, 233
anisotropy coefficient, 255
APD, 78
Ar ion laser, 154
autocorrelation function, 85
autocorrelator, 85
avalanche photodiode, 78, 131, 146, 156,
161
band limited pulse, 291
base line, 91, 92, 94
BBG, 220, 231
black body, 9
spectral emittance, 9
Brewster angle, 206
cavi ty dumper, 155
CCD, 78,79,95,111,179,181,191,211,
230,232
charge couple device, 78
circular frequency, 17
circular polarization, 244, 250
color temperature, 10
component spectra, 284
constant fraction discrill1inator, 151
continous wave mode, 42
contrast factor, 25
304
convolution integral, 162, 277
correction spectrum, 113, 114
cuvette, 100, 122
matching pair, 95, 101
CW lasers, 45
CW mode, 42
dark counts, 73, 120, 125, 156, 166, 228
dark current, 73
DAS, 164
dead time, 289
decay associated spectra, 164, 284
decay component spectra, 212
detectivity, 73
dichroic mirror, 167
dielectric mirrors, 26
differential
absorbance, 132,142,214
absorption spectra, 133, 146, 193,
211
differential absorbance, 131
differential absorption, 212
diffraction, 27
grating, 29, 52, 97
limit, 28
diffusion controlled reaction, 269
discriminator, 76, 77, 117, 218
dispersion, 203, 221
Doppler
broadening, 238, 239
shift, 237
dye laser, 51, 154
eigen-value,271
Index
305
Einstein coefficients, 12
electronic levels, 7
emission corrected spectrum, 114
emissivity, 10
equilibrium, 273
Euler fonnula, 17
excitation spectrum, 109, 115
exciton annihilation, 214
extinction coefficient, 4
extraordinary polarization, 56, 220
inhomogeneous broadening, 238, 242
instrument response function, 162, 206,
212,223,277,284
interference, 1 9
interferometer, 20
intennediate state, 269
intersystem crossing, 12
intrinsic rate constant, 269
inverse kinetics, 272
inverse population, 39, 46
inverse problem, 264
inversion, 40
iteration, 280
Fabry-Perot interferometer, 22, 42, 239,
240,243
first order reaction, 267
flash lamp, 154
flash-photolysis, 83
fluorescence, 13
fluorimeter, 108
frequency domain, 172
frequency response, 74
Fresnel integral, 27
front face scheme, 121
FWHM,291
Johnson noise, 70
Kerr effect, 232
Kerr lens, 53, 195,232
G- factor, 256
Gaussian
band, 265, 291
distribution, 66
pulse, 85, 201, 207, 225, 291
geometrical optics, 33
grating period, 29
gray coefficient, 9
groovers number, 30
group velocity dispersion, 190, 196, 202,
212,221,230,233
Lambert law, 3
laser
equation, 41
resonator, 41, 241, 244
time constant, 42
lasing threshold, 42, 242
light amplification, 13
linear parameter, 281
linear polarization, 250
linear reaction scheme, 271, 284
lock-in amplifier, 174
longitudinal mode, 43, 242
Lorentzian band, 264
harmonic waves, 1 7
Helmholtz equation, 18
hole-burning, 242
homogeneous broadening, 238, 242
magic angle, 252
mean square deviation, 67, 277
Michelson interferometer, 20
micro-channel plate
MCP, 75
photomultiplier tube, 75, 156, 161
mode-beating, 243
mode-locked laser, 47, 231
mode-locking, 154, 195
molar absorption coefficient, 4, 5
molar absorptivity, 4
monochromatic wave, 17
idler wave, 59
image intensifier, 75
indirect measurements, 263
indirect problem, 264
306
Index
monochromator, 31, 91, 95, 97, 109, 146
dispersion equation, 32
multichannel analyzer, 152
density function, 63
function, 63
pump-probe, 83, 185, 293
mono-color scheme, 185
two-color scheme, 188
Nd:YAG laser, 49,51,57,146,154,195,
210
NEP, 73
noise equivalent power, 73
non-correlated noise, 277
non-linear parameter, 281
Q-switching, 46
qualitative problem, 285
quantum efficiency, 73
quantum noise, 70
off axis mirror, 205, 223
optical density, 4
optical parametric amplifier, 59, 198
optical parametric oscillator, 59, 146
ordinary polarization, 56, 220
Raman scattering, 123
random
error, 62
noise, 62
search, 297
value, 63
reference channel, 93, 103, 191
resonator
bandwidth, 42
losses, 241
modes, 242
time constant, 42
responsivity, 73
right angle scheme, 121, 122
rotational correlation time, 256
rotational diffusion, 256
rotational levels, 7
parametric amplifiers, 198
paraxial approximation, 33
peal-up distortions, 157, 289
phase matching, 56, 218
phosphorescence, 13
photo-bleaching, 134, 193
photodiode, 78, 83, 117, 131, 138, 141,
145,151,186, 187,244
photomultiplier, 74, 83, 90, 108, 117, 124,
218
photon counting, 76, 110, 117, 120, 218,
226
photon noise, 70
plane wave, 17
Pockels
cell, 46, 49
effect, 232
Poisson
distribution, 64, 289
noise, 70
polarization, 47, 56, 58, 220, 228, 232,
233,249
ratio, 251
vector, 55
population inversion, 215
pre-triggering, 132
prism COIn pressor, 205
probabi lity
saturable absorber, 195
second hannonic, 167, 198, 203, 210, 217,
228,235
generation, 50,53, 56, 83, 154,231
generator, 83
second order recti on, 268
second order susceptibility, 55
semiconductor laser, 54, 155
sensitivity, 73
sigma-value, 67
signal wave, 59
singlet state, 12, 14, 151, 163,253,261
spectral hole-burning, 248
spectrofluorometer, 108
spectrograph, Ill, 191
spectrophotometer, 89
Index
307
spectrum correction, 113, 116
spontaneous
emission, 11
reaction, 11, 268
square root law, 67, 68, 73, 110, 118
standard deviation, 67
standing wave, 243
stationary noise, 277
Stefan-Boltzmann law, 10
stimulated
emission, 11, 39, 59, 214
reaction, 11
Stokes shift, 11 7
stratcher, 196
streak camera, 179
susceptibility, 55
synchronous
detection, 92, 200, 210
detector, 174
systematic error, 62
type II crystal, 220
uncertainty principle, 207, 237
uncorrected emission spectrum, 113
up-conversion, 54, 58, 182, 217, 293
vibrational levels, 7
wave mixing, 218
wave vector, 18
white continuum, 190, 210
generation, 197
generator, 189, 199
Wien law, 10
Xe arc lamp, 11, 146
zero-phonon line, 247
T-scheme, 135
TEM,43
mode, 44, 45, 50
thermal noise, 70
thermal relaxation, II
Ti:sapphire laser, 52,155,167,195,203,
210,231,232,292,294
time constant, 74
time correlated single photon counting,
151, 195,289
time domain, 1 72
time-to-amplitude converter (TAC), 152
transient time spread, 77, 157
transition dipole moment, 251
transmittance, 3, 5, 91, 94, 103
transverse mode, 43
transverse wave, 249
trigger jitter, 181
triplet states, 12
tungsten lamp, 10
two photon absorption, 293
coefficient, 293
type I crystal, 220
type I synchronism, 58