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 nanochemistry 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 Femtochemistry 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, advanced detectors and high technology optical components a researcher can assemble relatively 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 interested 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 estimations 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 ^ , 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 subjects 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
vi
Preface
devices as spectrum resolution and sensitivity. The second part of the book, Chapters 5-13, is the main part, which divided onto Chapters 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 follows 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 applicability 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 continuous 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 combination 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 application 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, therefore, 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 example 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 subject 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. Lemmetyinen 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 optical 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
Preface
vii
write this book. Most of all, I am indebted to my family, especially my wife Natalia for her love and tolerance of irregular hours at home, and my daughter Evgenia for her help in text preparation. Nikolai V. Tkachenko Tampere, Finland December 2005
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 development during the past decade. Naturally, the reason for the great attention paid to the optical spectroscopy techniques in recent fundamental research is new exciting Imowledge 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 concepts 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 1.1.1
Absorption 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
Introduction
// \ 7(0)
A/
IQ)^ J
'•J^-^
^
'
O 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 /\x the light interacts with the medium and at the point x + Ax the light intensity is changed by value A / , which means I{x + Ax) = I{x) + A/(x, Ax). At Ax ^ 0 the ratio ^^^^^"^"^ is proportional to the light intensity: lim
A/(x, Ax) /\2
-al{x)
(1.1)
or dl{x) dx
-al{x)
(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 /(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 Ax is 5 = nAx. For a very thin layer the absorption probability is very small positive value, and for the incident light flux /, the A/ decrease in intensity is A / Jp, or —A/ = IscF = InaAx. Thus, Ax -nal -al, where a — na. Equation (1.2) can be rearranged dl{x) _ -adx ~J{x)-
(1.3)
and solved \n[I{x)]
-ax -\- C
(1.4)
Where the constant C is detennined by the initial conditions, which is the incident light intensity at a; = 0, i. e. 7(0) = IQ, in this particular case. Equation (1.4) is usually
1.1. Absorption
converted to the form known as Lambert law^ I{x) = Ioex.p{—ax)
(1.5)
The coefficient a is called absorption coefficient and it has measure of inverse length (e. g. cm~^). 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) = /o = constant. The light intensity also increases exponentially if a < 0. 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 / (Fig. 1.1) and its absorption coefficient is a. The light intensity before the medium is lin = IQ = /(O) and, according to eq. (1.5), the light intensity after the medium is lout = HI) = line-""'
(1.6)
The transmittance of the sample (the relative amount of the light passing through the sample) is T=^=e-^'
(1.7)
-'-in
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 ^ _ hn - lout
_ 1 _ ^ _ 1 _ ^-al
^^9^
-^in
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 someone 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 practical 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.^ The absorption coefficient does not depend on optical path and characterizes the medium itself.
^One can note the relation /Q = exp(C), linking eqs. (1.1) and (1.4). ^ These are materials with fixed density of absorbing centers, therefore the sample absorption depends only on their thickness.
Introduction
Example 1.1: Calculation of the absorptance and transmittance of glass-like media. Absorption coefficient of a fiber can be 0.001 m~^, which means that 1 km of the fiber will absorb 1 - exp(-10-2 • 10^) = 1 - e x p ( - l ) ^ 0.63 = 63% of the incoming light power. A gray filter HC-3 has absorption coefficient ^ 1 mm~^ in the visible part of the spectrum, thus transmittance of the filter of 2 mm thickness is exp(-2) ^ 0 . 1 4 = 14%. When the light absorption is considered at a molecular level, the absorption crosssection, cr, 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 Iout^I{l)^hne-''^'
(1.10)
where n is the density of the absorbing centers (e. g. dye molecules).^ In many practical cases power of 10 is used instead of power of e. Then, eq. (1.6) is rewritten to the form Iout=hnl^-^
(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 ^ =- l o g ^
(1.12)
Comparing eqs. (1.7) and (1.12) one obtains a relation between the transmittance and absorbance yl = - l o g 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,^ which is the proportionality coefficient in the relation A^ecl
(1.14)
Note: By historical tradition the measure of molar concentration is moles per liter, i. e. M = molxdm"^, whereas optical path, /, is counted in cm, thus the measure of the molar absorption coefficient is M~^cm~^ = mol~^dm^cm~^, i. e. one value uses lengths measured in different units, dm and cm! ^The density is defined as a number of absorbing centers (or molecules) per unit volume and, thus, it has the same meaning as concentration. However, it is important to remember that concentration can also be counted in grams or moles per volume. Then, a coefficient is required to calculate density fi-om concentration. ^It is also called molar absorptivity and extinction coefficient.
1.1. Absorption
The molar absorption coefficient and the cross-sections are two characteristics specifying the same property of chromophores, their abihty to absorb the hght. Comparing eqs. (1.10), (1.9) and (1.14), and recalling that c = U/NA, where NA is the Avogadro constant, one obtains
which gives ac^^^3,82bxl0-'^e
(1.16)
NA
where a is measured in cm^, and £ in M~^cm~^. Example 1.2: Calculation of absorption of chromophore solution. Absorption cross-section of chlorophyll a at wavelength of 440 nm is cr ?^ 4 A^ or 4 x 10~^^ cm^. This corresponds to the molar absorption coefficient of e c^ 3 825XIQ-I9 ^ ^^^ M~^cm~^. A chlorophyll solution at concentration c = 1 0 ~ ^ M = 1 0 pM placed in / = 1 cm cuvette will have absorbance A = ed = 1, and will absorb 1 — 10~^ = 1 — 0.1 = 0.9 = 90% of the incident light. In summary, there are few parameters, which can be used to characterize the light absorption 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 expressed 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~^, 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^, eq. (1.10); • molar absorption coefficient, e, which is usually used to specify absorption properties of chemical compounds and has dimensionality of M~^cm~^, 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, ^(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
Introduction
layer 1 ^
\
layer 2
layer 3
T \
1
^"^^ ^ ^
/ 1
T3
/ 2
^ ^ 3 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 /Q. The transmittance of the first layer is Ti, thus the light intensity after the layer is /i = /QTI. The light intensity entering layer 2 is / i . The transmittance of the layer 2 is r2, which gives the light intensity at the interface between layer 2 and layer 3, I2 = I1T2 = I0T1T2. Continuing this procedure, the light intensity after the layer 3 is /g = I^Ts = I0T1T2TS. This gives the total transmittance of the sample T = TiTsTs, 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 A^
T
UT.
(1.17)
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 interaction. If the transmittance of one dye in a cuvette at some concentration is Ti and the transmittance of another dye in the same cuvette is T2, then the mixture of the dyes will have transmittance T = T1T2, under condition that the concentrations of the dyes are the same as in the individual measurements.^ To calculate absorbance of a complex sample one can apply eq. (1.13) to eq. (1.17) to obtain N
A=J:A^
(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 ^ Which means, the mixture was not prepared by mixing together the two solutions, since in that case the total concentrations of the dyes will be lower than in the non-mixed solutions.
1.1. Absorption
Table 1.1: Energies and spectral ranges of different types of transitions.
Electronic Vibrational Rotational
Energy, J ( 2 . . . 10) x IQ-^^ (2 . . . 20) x IQ-^o (2 . . . 20) x 10"^^
Frequency, Hz (3...15) x 10^^ ( 3 . . . 30) x 10^^ (3 . . . 30) x 10^^
Wavelength, /im 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 < electron. A smaller mass results in a higher frequency of oscillations (u ^ l / \ / m 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 /xm. Accounting for the near infrared region (0.7-1.5 //m) and ultraviolet (UV) (0.2-0.4 /xm), the optical wavelength range can be considered to be from 0.2 to 1.5 fim. 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.^ ^A few mechanisms of line broadening are discussed in Section 13.1.
Introduction
UV IvisiblelNIR —\
\
OX
\
0.2 0.1
j^ I I I —
/
1
10^
v,Hz^^
10 "^ 10^^
.d.O 10
\
-^—\
k,cm"^
\—\—I
,6A
^'
100
\
\
1000
100
X, |i
10^^
electronic i vibrations i rotations
Figure 1.3: Relations between different scales used in spectroscopy: wavelength, A, wave number, /c, 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.
wavelength frequency photon energy wave number electron energy
1.1.4
A
TT ~
E e ~
hu e
units nm Hz J cm~^ eV
value at 600 nm 600 5 X 10^^ 3.3 X 10-19
16 667 2.07
Wavelength, frequency and energy
Since the energy of a photon determines frequency (and wavelength) of the electromagnetic wave (by famous Planck formula E — hu or E — ^ \n vacuum), the absorption spectrum 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^^). Since the wave number is directly proportional to the frequency and energy {k = ^ = ^ ) , 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
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 electromagnetic 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^ En = {n-]^)hv
(1.19)
where n = 1, 2, 3.... The probability to find energy E^ is given by the Boltzmann statistics, p{En) ^ e x p ( — ^ ) . Using eq. (1.19) and Boltzmann statistics, Planck has shown that the spectral density of the radiation is
p.(^,r)=
^, _____,,^, ^ "exp(|f)-l
(1.20)
where h is the Planck constant, k is the Boltzmann constant, c is the velocity of the light in vacuum, u 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 temperature, 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=^T^ V A5 (eAT — 1 j
(1-21)
where ci = 27rhc^ = 3.74 • 10"^^ W-m^ and C2 = ch/kB = 1.44 • IQ-^ m-K. The spectral emittances calculated for the wavelength measured in nm and emitting area in cm^ are presented in Fig. 1.4. ^Today one would say that the resonator was treated as a quantum system but this was a century ago!
10
Introduction
^,
/ / / / ' ;
1^
I : I : 1; 1; 1 : 1 :
o.ih: '
'
•••••.,
,
- s
1
V N
%^^ •. "S
, / / /
•
1 1 1 1 1 1
-• T== 5000K • • • T == 4000K - T == 3000 K T
^,„^^^
-
^'"^^^^-v..^
!
/
1 :
o
, , 1 , , , , 1 , , , , 1
/ /
•'
1; / 1: /
CD OH
(/5
O.OL
'0
, 1,- , /,
500
, , 1 , , , ,
1000
1500
2000
2500
3000
wavelength, nm 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) _ 2.898 X 10^
(1.22)
where the wavelength {Xmax) 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 2TT is given by Stefan-Boltzmann law
P^aT""
(1.23)
where a = 5.67 • 10~^ W-m~^K~^ is the Stefan-Boltzmann constant. In lamps specification one can find two parameters which determine the emission intensity 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.^ Example 1.3: Tungsten lamp. The color temperature of tungsten halogen lamps ranges from 2000 to 3200 K, and typical emissivity is ^ 0.4. If a lamp works at filament temperature 3000 K, its spectral emittance is M ^ 0.4 x 0.16 W-cm~^nm~^ ^ 0.07 W-cm~^nm~^ at 600 nm. For an emitting filament area of 0.1 cm^ the total power emitted at 600 nm in 1 nm wavelength range is 7 mW. The total power emitted by a tungsten lamp with filament size of 0.1 cm^ at temperature 3000 K is P ^ 50 W. ^In particular, the color temperature determines the emission maximum, as follows from Wien law.
1.2. Emission
11
The emission maximum of the lamp is Xmax ~ 970 nm, so at shorter wavelengths the spectrum density is lower, e. g. 3 mW-nm~^ at 500 nm and 1 mW-nm~^ at 400 nm, and at longer is higher, e. g. 9 mW-nm~^ 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^ (1 mm sphere), which gives total power of P !=^ 200 W. The emission maximum of such arc is at Xmax ~ 480 nm. The spectrum density of the emission is 0.3 W-nm~^ at 500 nm (close to the maximum). At shorter wavelengths the spectrum densities are 0.27 and 0.17 W-nm~^ at 400 and 300 nm, respectively. At longer wavelengths the densities are 0.27 and 0.22 W-nm"^ 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 absorption 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 separated from each other. In the most simplified case one can consider a molecule with only two states Mi and M2 with energies Ei and E2, respectively: E2
B 12
B21
(1.24)
A21 .
^1
This molecule will only interact with photons having energy hu = E2 — Ei. The possible photo-reactions of the system are 1. photon absorption: Mi -\- hu ^ M2; 2. thermal relaxation: M2 ^ Mi + AE; 3. spontaneous photon emission: M2 ^ Mi + hu; 4. stimulated photon emission: M2 -\- hu ^ Mi + 2hi/. 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, ^
= -N,A,, (1.25) at where N2 is the population of state M2. Coefficients B21 and B12 describe stimulated absorption, reaction (1), and stimulated emission, reaction (4), respectively. Note that absorption is always stimulated. In case of a narrow absorption and a broad stimulating radiation field, the kinetic equation for reaction (4) is ^
= -N2B21P at where p is the energy density. Einstein has shown that
(1.26)
Bi2 = B2i
(1.27)
^
(1.28)
and - ^
B21
c^
Thus, a single coefficient, for example ^421, describes behavior of the two level system. The relation (1.27) deserves a separate comment as it provides a very general conclusion: 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. Transitions 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 photochemistry. Excitation of a molecule results in transition from the lowest (ground) state to a excited state. In most cases this is singlet-singlet transition.^ Being excited the molecule can relax ^One exception is oxygen molecule, O2, 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, 6 = , ^\, , where kr is the radiative rate, and knr is the sum of all non-radiative relaxation rates. The denominator is the total relaxation rate of the exited state, /CQ = /Cr + knr, so the quantum yield can be also expressed as 0 = |^. Typical rates of the singlet state relaxation for organic dye molecules are 10''-10^ s~^. The quantum yield of the fluorescence varies strongly from molecule to molecule and usually one speaks about high quantum yield when (f) > O.l}^ The typical values of the excited singlet states radiative rates for organic dyes are in range 10^-10^ s~^. The most essential contribution to non-radiative decay comes from inter-system crossing, resulting in formation of a long living excited triplet state. The radiative 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.^^ In case of a small single atom emitting centers, such as metal ions in glass matrix, e. g. Nd'^+ 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. Alternatively, the light amplification can be written explicitly /(x) = /oe^"
(1.29)
where /3 = — a, then coefficient ^ 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 amplification. 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 stimulated emission (amplification) different from spontaneous emission (e. g. fluorescence). ^^For laser dyes the fluorescence quantum yield is typically higher than 60%, e. g. for rhodamine 6G the yield can be as high as 95% (depending on solvent). ^^ At least in solutions the inter-molecular quenching of the excited triplet state is typically the main contributor to the non-radiative decay. As the result, the observed quantum yield of the phosphorescence can be very low, and to measure the phosphorescence special precautions have to be taken to inhibit inter-molecular 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, M2, is greater than that of the lower one. Mi, i. e. N2 > Ni. This is not possible in thermodynamic equilibrium since in accordance with Boltzmann statistics ^^2/^^! = e x p ( - ^ ^ ^ ^ ) , thus N2 < Ni3tT > 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 spectroscopy. 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 significantly. 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 optical 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~^ s, e. g. in biological reactions such as ion transport; microseconds: 1 /iS = 10~^ s, e. g. in diffusion controlled chemical reactions in liquid phase, triplet state reactions; nanoseconds: 1 ns = 10~^ s, e. g. in photochemical reaction, singlet state reactions; picoseconds: 1 ps = 10~^^ s, e. g. in intra- and short distance intermolecular electron transfer, energy transfer, primary reactions in natural photosynthesis; femtoseconds: 1 fs = 10~^^ s, e. g. in molecular vibrational motion, "hot" carriers dynamics, optical (electronic) vibrations. Advantages of the optical spectroscopy methods are their non-destructive nature and possibility 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, manipulate 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 developed 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 subjects and readers are advised to refer to general optics text books for more complete study of the subject.^ The photon, being a quantum object, has a controversy of wave and particle presentations. 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 section 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 (ID) the wave equation is d^f
1 d^f
_
where / = / ( 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 ^The author used a book by Robert Guenther as a reference [1], though there are many other excellent text books on modem optics.
15
Optics and Optical Devices
16
t = tl
/^
u (cti-x)
X t=t2
\ Xi
X2
U (Ct2-X)
^ X
Figure 2.1: Propagation of a pulse along the string (ID 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 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 = ti the shape of the pulse is given by a pulse-like function u{y) which has a single maximum at y^. Naturally, in our case the argument of the function u is y ^ cti— X, that is li = u{cti — rr). At fixed time t = ti, function u{cti — x) depends only on X. The position of the maximum, xi, is given by a simple relation yo = cti — xi, i. e. at time tl the coordinate of the maximum is xi = cti — yo- At time t = t2 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 ^2 the maximum of the pulse is at point X2 = ct2 — yo.^ The displacement of maximum Ax = X2 — xi in time interval At = t2 — ti is Ax = cti — yo — {ch—yo) = cAt, 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, V^/7-
1 d^U C2 dt^
= 0
(2.3)
where V^ = -^ + -^ + -^ 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 ^One can notice that yo = cti — xi = ct2 — X2, which means that ct — x is invariant of eq. (2.1).
2.1. Waves
17
electric filed is the vector field, for which the wave equation infi-eespace is^ V^E-eolio^=0
(2.4)
and the velocity of the electromagnetic waves (e.g. light) in free space is c = ^ / —*—, where eo and //Q are permeability and permitivity of vacuum, respectively. In dielectric medium the velocity is c = ^ /—^—. Solution of 3D wave eq. (2.3) is not as straightforward as for ID 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 determines 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 sm{ujt) or cos((x;t), where uu = 27TU, and u is the oscillation frequency and uu is the circular frequency."^ Another useful and widely used mathematical notation for harmonic oscillations is (Euler formula) e^^^ = cos(Ljt) + i sm{u;t)
(2.5)
For example, a harmonic wave in ID case can be presented as / = /,e'(-*-«-)
(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 ut — K,X as the argument in mathematical presentation of harmonic waves since it shows the frequency (LU) of oscillations. In terms of eq. (2.6) the velocity of the wave is c = ^, and parameter K is called (circular) wave number. In space the period of wave, or the wavelength, is A = ^^^, or A =^.5 2.1.3
Plane waves
Extending the equation of monochromatic wave to 3D case, one can rewrite an equation for plane waves U = Uoe'^'''-^^^
(2.7)
^The electromagnetic waves have two components: electric and magnetic fields. However, as follows from Maxwell equations, these two components are tightly related with each other, and only one of them is needed to describe completely the electromagnetic wave. By convenience the electric component will be used here. ^The circular frequency is convenient and preferred notation here as it gives shorter form of equations. ^Conversions from the circular to linear frequency and wave number are cj = 27ny and k = 27r/i:, respectively. An equivalent presentation of harmonic wave is f = fQe^'^'^^^^~^^\ with the wave velocity c = ^ and wavelength A = k~^, respectively.
18
Optics and Optical Devices
Where r = r(x, y, z) is the vector from the origin of the coordinate system to point with coordinates (x, y, z), and R is the wave vector. In isotropic dielectric medium the wave vector, R, determines the wave propagation direction and its absolute value is |^| = n= ^, or it is equal to the wave number of the ID case considered above. One can select the coordinate system so that, e. g., axis Z is directed along the vector R, then the projections of the vector are t\jx = 0, /^^ = 0 and Thus, the product nf= nxX -\- hiyy -\i^z^ = i^xX = tiz, and eq. (2.7) can be rewritten as /7 = Uoe'^^^'^~'^^\ In other words, by proper selection of the coordinate system, the 3D plane waves can be reduced to ID waves. The wave given by eq. (2.7) has infinite wave front and its amplitude, f/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 Uo can not be a constant. This can be done by rewriting eq. (2.7) as U = U{r)e'^''^-^^^
(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 (V^ + A^^)[/ = 0
(2.9)
This equation is only valid when the function U{r) is much slower than function e^^^^\ Then one can consider only wave amplitude distribution over the space but can omit oscillating part, e^^^*-^^) .^ In optics, we usually can neglect a change of the wave amplitude, /7(r), 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 x H. The light intensity^ is the time average of Poynting vector / = (|*S| V and it is proportional to the square of the electric field amplitude for the electromagnetic wave, / oc E'^ ^.^ 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. ^It is also important to notice that the amplitude in eq. (2.9) does not depend on time. Therefore Helmholtz equation describes stationary wave flow. ^Here 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. ^In dielectric medium / = (l^^l) =
f^lmpV
2.2. Interference
2.2
19
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 distances 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 Ei at a point f and another source produces field E2 at the same point, then the total field at this point is ^ = Ei -\- E2. Let us further assume that the waves are flat and have the same frequency cj, so that the corresponding wave vectors, KI and i^2, have the same length, i. e. |/5i| = |^2|. Then, the electric field of the first wave is Ei{r, t) = ^ie^(^^-^i^) and of the second is ^2(r, t) = ^26^^^^"^^^^ respectively.^ It is important to note that the orientations of the pairs of the vectors Ei and ^1, and E2 and R2 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 ^ • ^ = 0. For further simplification let us assume that the vectors Ei and E2 have the same orientation, which means that Ei and E2 are perpendicular to the plane formed by the vectors Ri and K2. Therefore, the vector sum can be replaced by the sum of scalar values El + E, Then, the total field created by these two waves is E{r,t) = £;^e^M-^i^"') + ^^^e^^^*"^'"'^
(2.10)
The experimentally available parameter is light intensity, which is proportional to the square of the amplitude of the electric field oscillations I ^ EQ, where EQ is the amplitude of the oscillations,^^ and can be calculated as / = EE*, where E* is the complex conjugated number of ^^^ /
=
E-E*
= (Eie'^"^*-^'^^ + E2e'^'^*-^^^A (EIC-'^''^-^'^^
=
Ef^E^^
=
Ef ^Ei^2EiE2
+ £^26"'^"^*"^'^^)
E1E2 (^6^(^1-^2)7^ _^ g-i(^i-^2)r\
cos {{1^1-^2) r)
(2.11)
The only variable term of/ is 2E1E2COS ((^1 — /^2) ^)- It gives sinusoidal variation of the light intensity in direction KI — ^2- The period of the modulation in this direction is .^ ^_^^ ,. '^In a general case one of the waves may include a phase argument, e. g. £^26*^^* ^ir+c^) jj^js^ however, will not change the following consideration, so one can safely assume here that (f = 0. ^^As was noted in Section 2.1.2 and footnote 4, the power density i s / =
(s\
= ^EQ.
However, the
coefficient ^ will be omitted in all further calculations and a simple relation, I = EQ, will be used. This will not create any mistake as in all cases two transitions will be made: from the intensity of the individual beams to the electric field and then back to the intensity of the interference pattern. ^HfA = ae*^, then (by definition) A* = ae"^^. Thus, A • A"" = ae^^ae"^^ = a^.
Optics and Optical Devices
20
M,
M, Ml
I\ YY I2
Figure 2.2: Michelson interferometer. Mi is a semi-transparent mirror, and M2 and M3 are 100% reflectors.
Thus, using scalar wavelength A = ^ ^ = ^ ^ instead of wave vectors and introducing the angle between the propagation directions of the waves, a, one can obtain interference period ^^L=
2sin(a/2)-
Since Ef -\- E2 > 2E1E2, the intensity / is never negative. When the waves have the same amplitudes, Ei — E2 — E, the intensity of the interference pattern changes from 0 to its maximum value of 4£^^, which is two times greater than the sum of the intensities of the interfering waves, 2E'^. Example 2.1: Interference period of two monochromatic waves. In order to obtain an interference 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 = 2sin"^ {^) ^ ^ = 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 important 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 multiple modifications. A classic scheme of the interferometer is shown in Fig. 2.2. It consists of three mirrors: a semi-transparent mirror Mi and two reflectors M2 and M3. If incoming beam has intensity / and the mirror Mi has reflectance R, then the intensity of the reflected beam is RI and the intensity of the transmitted light is {1 — R)I, respectively.^^ Let us follow the propagation of the reflected beam first. The mirror M2, must be adjusted so that ^Transmittance of the mirror Mi isT = 1 — R.
2.2. Interference
21
the beam reflected by the mirror Mi is returned back by exactly the same path. Then the reflected beam wiU hit the semi-transparent mirror Mi at exactly the same point as incoming beam (/). The intensity of the Hght, which will cross the mirror Mi, is / i = R{1 — 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 Mi at the first incidence of the incoming beam (/). Its intensity after the mirror Mi is (1 — R)I. The mirror M3 must be adjusted so that reflected beam hits the mirror Mi at exactly the same point as incoming beam (/). Part of the beam will be reflected in the same direction as previously considered beam Ji, and its intensity is /2 = i?(l — R)L Thus, properly adjusted Michelson interferometer splits incoming beam (/) on two beams (/i and I2) 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 propagating in the same direction, e. g. along Z axis, the fields can be written as Ei{z^t) = Eoute'^^^-^^-"^'^ and E2{z,t) = Eoute'^''^~'^^~'^^\ respectively, where Eout is the field created by one of the beams (on the interferometer output). The phases (pi and (p2 depend on the propagation distances of the beam from the semi-transparent mirror Mi to the reflectors (M2 or M3, respectively) and back, and can be written as (fi = 2i
771
i{ujt — KZ — 2Kdi)
_|_ 771
i{ojt — KZ — 2Kd2)
And the light intensity is lout
=
E' E*
—
7712 i(ujt—Kz) f —2iKdi _|_ —2md2\ ^—i{cot — Kz) f 2iKdi j _ 2iKd2\ ^out^ \^ ^ ^ )^ \^ ^ ^ )
=
E'^
/^2 + e~2^'^(^i~^2)
I g2m(dl-d2)^
2El^^ (1 + cos2K(di - ^2))
(2.13)
^out
Converting the wave number H. to the wavelength one obtains
Finally, taking into account the reflectance of the mirror Mi lout = 2hnR{l -R)U^
COS ^^^1^M\
(2.15)
Thus, the output intensity depends on relative beam propagation delay ^ i ^ ^ ^ and varies from0to4i?(l-i?)/^^.
Optics and Optical Devices
22
Mi
M,
-^
Figure 2.3: Fabry-Perot interferometer
A straightforward application for the Michel son interferometer is direct measurement of the wavelength of monochromatic light. By smooth changing of the distance di(or ^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, ^^ 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 /i and I2); • optics diagnostics (an optical component, e. g. a lens, can be inserted between mirrors Ml and M2 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 Chapter 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 ^^Note 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 number is lout = 2-B^^^ (1 + cos47Tk(di — ^2)), where k is the wave number, and n = 2Tik. 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. Ein = ^i{Ljt-Kz)^ 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 i^ = r^. The corresponding transmittance for the electric field component of the wave is / = A/1 — T'^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 Mi is £^ = e^^"^*"^^^ = e^^'^ (z = 0). Right after the mirror Mi the electric field of the incident light is /e*^^ and before the mirror M2 is /e^(^^-^^) (z = d). After the mirror M2 the field is El = ffS'^'^-'^d)
^ j2^i{ut-Kd)
^2.16)
This is the first beam participating in the interference on the interferometer output. The part of the light, reflected by the mirror M2, fre^^^^~^^^, returns back to the mirror Ml, where another portion of the light is reflected in direction to the mirror M2. The field of the reflected light is ff'^^'^i^t-'^'^d) ^ p^j,^ of this beam will cross mirror M2 and form the second beam participating in the interference, E2 = /rr/e^^^^"^^"^^ = /V^e^^^*"^^^^ = E^ir^e"^^"^^
(2.17)
This re-reflection process will continue again and again giving beams Es, E4,... En and so on. It is clear, that for the beam n the electric field is En = £'^r^(^-i)e-*2^^(^-i)
^2.18)
The resulting electric field after the mirror M2 is the sum of all the beams
=
cx)
00
n=l
n=l
Si^lrV-^-'^)
cx)
n=l Si
=^_^-_,^,
(2.19)
n=0
Finally, the intensity of the transmitted beam is^^
(^-t^o*nt) = (1 .. - r2)2 1 , +4r2sin^/^(i , ..L.2.,.
(2.20)
Converting eq. (2.20) to intensity reflection, R — r^^ and wavelength, AC = ^ , one obtains (i-i?) 2 22^(l-i^)2+4i^sin A
(2.21)
The transmittance, T — lout/hn, 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. ^^Calculations of the intensity from the electric field eq. (2.19) can be found in e. g. ref. [1] p. 108.
Optics and Optical Devices
24
ir
1
1
1
1
1
1' — 7? = 0.9~T[
^\
'ii
M
- - i? = 0.5
1
'
OM
1
1
1
1
'
1
|0.6
'
1
£
1
1
1
>
1
'
1
oo
gO.4 -
1
\
0.2
\
/
\ \
1
\
/ 1 1
^
\\ \\
'' \\
''
\\ \ \
V,,, A,,, j,v, ,,,^,v^,,,
500
510
520
530
540
550
wavelength, nm Figure 2.4: Transmittance spectrum of the Fabry-Perot interferometer of thickness d = 0.01 mm and formed by mirrors with reflectance i? = 0.5 (dashed Hne) and 0.9 (soHd line). Equation (2.21) was used for the calculations.
When^ A/", where A^ is an integer number (0, 1,2,...), sin ^ = 0 and lout = ^m, 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\. If there is a transmittance maximum at AQ, then N = ^ and the next maximum will be at A i which correspond to A^ — 1. Thus the spectrum distance between maxima is AA sp
^0
2d-X
0
2d
(2.22)
The spacing between the lines in the wavelength domain decreases as distance between the mirrors increases. When ^A = 7V+ ^, i. e. sn 2 27rd A = 1, the transmittance of the interferometer has its minimum value {l + Rf
(2.23)
For example, if R = 0.5, then Imin — 0-1 l^m? or the light rejection is higher than what could be expected for two "independent" mirrors, R^ = 0.25. One of the applications of the Fabry-Perot interferometers in optical spectroscopy is the fine spectrum resolution. Then the value AAgp (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 turning slightly the interferometer. When the light incidence angle, a, is not zero (at normal incidence c^ = 0) eq. (2.21) can be used after substitution d = h{cosa — sw? a) where h is the distance between the mirrors.^^ Thus the transmittance maxima will be at A = ^ (cos a — sin^ a) = AQ(COS a — sin^ a). For a small angle a one can use approximations cos Q; ?=^ 1 — ^ and sin^ o; ?=^ o;^, so A(a)^Ao(l + ^ a ' )
(2.24)
At angle a c:^ J ^ ^^ the interferometer will be again transparent to light at wavelength Ao (at A^ + 1). For the purpose of spectrum resolution analysis one can introduce a contrast factor F = /-^^^p and a dimensionless value (p = ^ . Then, eq. (2.21) can be rewritten as •^out ^
-^inZ.
~
'. o
yZ.Zj)
1 -\- F sm if The half intensity bandwidth can be determined from condition lout = \hn, which results in equation l + FsinV = 2
(2.26)
s'lmp^J-
(2.27)
or
Usually, F is a big value, e. g. F = 360 at i? = 0.9, and F = 1520 at i^ = 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 interferometer. Interferometers for the fine spectrum resolution are usually constructed so that d :^ A. This means that d >> 1/i in the optical wavelength range. Then, considering a small deviation of the wavelength from the wavelength of the maximum transmittance, AQ, one obtains ^ = 1
AQ
2nd
^ 27Td /
A A \ _ 27Td
2nd
— ^ — ( 1 + — ) = — + ^ AA — AA AQ \ Ao / Ao AQ
(2.29)
^^One have to account for the traveHng distance between mirrors (cos a) and for the phase shift due to the fact that the wave front is not parallel to the mirror surface (sin^ a).
26
Optics and Optical Devices
where A A is the deviation from maximum. Since AQ is the wavelength of the transmittance 2^^ - TTN (sin ^ = 0) and using eq. (2.28) one can write
f^AA.Tl AQ
(2.30)
V i^
Finally, the wavelength resolution is
Example 2.2: Spectrum resolution of Fabry-Perot interferometer. Suppose the interferometer 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 ^^ ^ ~ (il^)2 ~ 1 520. At wavelength AQ = 500 nm the spectrum resolution of the interferometer is A A = 2^\/T ~ ^^~^ ^^- ^^^ spectrum range is AA^p ^ ^ = 0.125 nm.i^ Adjustment angle a o^ J^^^ ~ 0.02 radian or 1.3°. In order to increase the resolution mirrors with higher reflectance can be used, e. g. ifR = 0.98, then F = 39 600, and AA ^ 2 x lO""^ 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,^^ one can build a system with multiple reflecting interfaces and place the reflecting surfaces at a distances satisfying 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). ^^The spectrum width of studied signal must be narrower than AXsp, otherwise different parts of the spectrum will overlap each other. ^^ If the refraction index of the medium at one side of the interface is n i and at the another is n2, the reflection from the interface is R = p i '^2j Roughly, the refractive indexes of the materials transparent in the visible wavelength range (400-800 nm) vary from 1.35 (cryolite, NasAlFe) to 2.3 (titanium dioxide, Ti02), thus the maximum reflection from the interface is i? < 0.07.
2.3. Diffraction
2.3
27
Diffraction
Restriction of the plane wave front in space results in distortion of the front. This phenomenon 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 theory 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 contributions by Augustin Fresnel and Joseph Fraunhofer. 2.3.1
Fresnel formulation
Using Huygens principle Fresnel has developed a theory which allows to calculate the electric 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 fo (observation point) is given by Fresnel integral Wo)= fc{fs)^^e'^^-^^ds J \R\
(2.32)
where Es{rs) is the electric field at a point on surface S given by vector fg, R = f^ — fg, C{fs) is the obliquity factor,^^ 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 5 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{hix^f^y), since if the wave propagates in Z direction only (as it takes place at S) then K.X = ^ and t^y = 0, i. e. E{i
(2.33)
^^The obliquity factor depends on orientations of the surface S and observation point fv?. 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 E{k^) = / e'^^'^dx = -i
(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{K,))
=
^^I^i!h^
(2.35)
The projection of the wave vector, KX, can be expressed in terms of the observation angle (p, which is deviation from Z axis, and the wave number /^ as K^; = K,sm(p = ^ sin (p. Thus eq. (2.35) gives the wavelength and angular dependence of the amplitude of the diffracted wave E =
ip sm I/ 27r(isin —--A ^ ^ /{ —
(2.36)
/ 27r sin ip \
\
^ J
The light intensity is proportional to the square of the electric field amplitude,
^x
( 27r sin (/? \
Both these functions, eqs. (2.36) and (2.37), are pulse-like functions. The angular dependence of the intensity is shown in Fig. 2.5. The functions have maxima at KX = 0, i. e. (/? = 0, and decrease (with some oscillations) with increase or decrease of AI:^;, respectively. The width of the "pulse" can be determined roughly from condition (which gives the first intensity minimum, I — {)) n^d — n or KX — ^ or sin(/:^=—
(2.38)
This result has an important consequence. A limited in space plane wave cannot propagate as a unidirectional beam. It will have divergence angle given by^^ ^iiiip^ip^—
(2.39)
where D is the cross-section (aperture) of the diaphragm used to form the beam from "infinite" plane wave and it was assumed that the cross section of the beam is much greater than the wavelength, Z^ >> A. Consequently, even "ideal" lens cannot collect light into a spot smaller than certain size, which is called diffraction limit. ^Strictly speaking, the divergence angle of the Hght after the sHt measured as full width at half maximum is (/? ~ 2 ^ . However for a round diaphragm the divergence is slightly greater. Therefore relation ip '^ ^ is reasonably accurate for a rough estimations of the diffraction effects.
2.3. Diffraction
29
-30
-20
-10
0
10
20
30
angle ((p), min 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 SL 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~^. At a distance / = 1 m from the diaphragm the diffraction will result in the spot size of l(f ^ 1 mm, which is reasonably close to the size of the diaphragm. However, at 10 m 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 / = 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 ^ ^ ^. This divergence gives a spot in focal plane of size d — fip'> 511. 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. Typically 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
Optics and Optical Devices
30
incident wave front
diffracted wave front
Figure 2.6: Diffraction grating with period /. 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 mX sme/ =
T
(2.40)
where / is the grating period and m is an integer number (1,2, ...), called diffraction order. Naturally, in eq. (2.40) ^ must be less than one, which means that the maximum possible diffraction order is m < j . The diffraction angle can be also expressed in terms of grooves number, which is defined as reciprocal of the grating period, g = l~^. mXg
(2.41)
The grooves number is the parameter usually found in grating specification, and measured in number of grooves per millimeter, i. e. mm~^. 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 A^ ^ LCOSO t^]' where L is the length of the illuminated area (i. e. the size of the wave front). Equation (2.40) gives A^ cos 6 = ^^^V^, thus, the wavelength resolution is AA
I 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
Grating Input slit
j
\ Output k^o slit
^- i
t
Spherical mirrors Figure 2.7: Monochromator optical scheme, di and do are the sizes of input and output slits, respectively, / is the focal length of the mirrors, and Lp is the angle between the incident and diffracted beams.
Example 2.5: Spectrum resolution of diffraction grating. Typical grooves number for gratings designed for the visible-ultraviolet wavelength range is ^ = 1200 mm~^, which corresponds to grating period I = g~^ ^ 0.8/i. 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 AX^ X^^ 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 monochromator is presented in Fig. 2.7. The incoming light crosses the input slit of size di 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 /(sina + 8m{(p -\- a)) = mX
(2.43)
where a is the incident angle of the light on the grating and (p is the angle between the incident 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((/9 -\- a) = mXg
(2.44)
The right side of eq. (2.44) can be simphfied if the hght incidence angle a is close to 0 (normal incidence of the light on the grating) sin a + sm{cp -\- a) = sin a + sin a cos (p -\- cos asimp = = sin a ( l + cos (f) + cos asimp c:^ a{l + cos (f) + sin (p and
A = J1±S^^
+ !H^
mg
(2.45)
mg
This is the monochromator dispersion equation.^^ 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 A a = ^ . 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
AA = M i ± i i 2 ^
(2.46)
Fmg 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 monochromator 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~^; 3. work at higher diffraction orders (increase m), then there may be overlapping diffraction orders and the diffraction order cannot be greater than m < {Xg)~^, also with g = 1200 mm~^ 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 dimensions of the device, i. e. usually bigger monochromators have better spectrum resolution. 20Note the linear dependence of the wavelength on the angle a. The linear relation between the grating angle and transmission wavelength explains why the wavelength scale is so common 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 example 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 g = 1200 mm~^ grating. It provides the spectrum resolution of 0.05 nm at 0.01 mm slits. The stray light rejection at 8 nm shift from the monochromatic wave is 10~^, which is typical for single grating monochromators. Overall dimensions of the monochromator are 40 x 34 x 25 cm^.
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 geometrical 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.^^ 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 calculations by assuming paraxial approximation, which adds: ^^The extend to which the size of the wave front can be neglected was discussed in Section 2.3 and Example 2.3.
34
Optics and Optical Devices
r\
ri zi
Z2
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) c::^ 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 fi*om 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
B e a m transfer matrix
Relation between the parameters at two planes, say planes P i (z = zi) and P2 (^ = ^2) as shown in Fig. 2.8, is given by a linear system r2 = Ari + Bai a2 = Cri -\- Dai
(2.47)
or, in the matrix form A C
^2
B D
Ti
(2.48)
ai
or R2 =
(2.49)
MRi
where M
A C
B D
is the beam transfer matrix (it is also called ABCD matrix), and Ri r2 01.2
are the beam parameter vectors.
(2.50)
ai
and R2 =
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 ni 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 zi to plane Z2, then from plane Z2 to plane z^ and so on to plane Zn+iinput
-^
output
I -^ I ^ . - . ^ Zl
Z2
I
-^
(2.51)
Zn+l
Let us suppose that the transfer matrices between the neighboring planes are known, i. e. matrices for transfers i?i+i = MiRi 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-i, which is Rn = Mn-lRn-l
(2.53)
Substituting R^ form eq. (2.53) to eq. (2.52) Rn+l = MnMn-lRn-1
(2.54)
Applying the same routine consequently one obtains the beam transfer equation from plane zi to plane Zn-\-i Rn+i=MnMn-i...MiRi
(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 Imown 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 / and the distance from the object to the lens be d. We need to find the distance from the lens to the object 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 (P2) is placed right behind the lens. Thus the first
Optics and Optical Devices
36
Table 2.1: Transfer matrices of simple optical systems.
Free space of length d Planar interface between two different media with refractive indexes ni a n d 712
matrix 1 d 0 1 1 0 ni 1 0 1
Parallel-sided slab of length d and refractive index n Thin lens of focal distance /
d n
1 0
1 0
^ '^^ 1
Spherical mirror of radius R
Pi
P2P3
d-/-
I
M,
V I
M.
Figure 2.9: Ray tracing to obtain an image (in plane P4) of an object (in plane Pi) and application of the matrix formalism to calculate the optical system: zi, Z2, z^ and 2:4 are the principal planes and Mi, M2and M3 are corresponding transfer matrices.
transfer matrix is that for the free space of length d. Plane 3 (P3) 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
M'.MoM^
1 d 0 1
(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, ^4, does not depend on the beam angle at the plane of object, a i , that is B = OfoYmatrix M (see eq. (2.50)). Thus, B = d-\-x — ^ = 0, and solving this equation one obtains a well known formula a:: = ( 4 — ^ ) object to the image plane is
. Finally, the transfer matrix from the
(-1)"' 'f
^
f
The value A of the transfer matrix determines image magnification. This follows from the relation r4 = Ari + Bai = Ari. Introducing magnification factor m = —= A ri
-i- -7)'
(2.57)
one obtains
°) m
(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. m^ = ^ = :^- Thus, angular magnification, rric, is the inverse of the magnification m, thus m^m = 1. This is important and general result.^^ Let us introduce brightness of an object (or image) as light power emitted by a surface of unit size in an unit angle.^^ 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 m^ of the image. Thus, the brightness of the image (accounting for the 2D case) is bi = m^ml^b, and since mam = l,bi = 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. ^^ Actually, this comes directly from the fact that the determinant of the transfer matrix is unity, i. e. AD — CB = 1. Since B = 0, AD = 1, and thus rriam = 1. ^^One can note similarity with the emittance introduced to characterize the black body emission in Section 1.2.1, see eq. (1.21). The spectrum integral of the emittance 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 f^ ?^ TT ( ^ ) , 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 i s ? 7 c ~4 | V( ^2 F)/ = 16 ^(^).If 0.09 the monochromator angular aperture is ^ = 0.3, then rjc ~ %^ ^ 0.006 = 0.6%. 16
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 produced photons are indistinguishable from the original photons. This means that the propagation 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 indistinguishable 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 inverse 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
Lasers for Spectroscopy Applications
40
. pump ley.ei laser level
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 tw^o "classic" types of the energy level layouts which can be used to create the inverse population, and, thus, to achieve the light amplification 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, Np = 0. This guarantees that the population of the ground state is Ng > Np, i. e. the ground and pump levels are in a thermodynamic equilibrium with the pump radiation. As a result, the pumping radiation at i^p will empty the ground state, Ng, and increase population of the laser level, Ni. 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 Ni > Ng. To characterize the degree of the inverse population, or inversion, one can use AN = Ni — Ng, which gives amplification coefficient (3 = ANai, where ai 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 Up 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 AA^ = Ni — N^, and, since N^ c^ 0, the inversion is the laser level population A A" c:^ Ni. Thus, the four level system needs much lower pumping rate in order to achieve the light amplification conditions.
3.2. Laser resonators
41
^1
Active medium
I I
^^
1 1
Figure 3.2: Laser resonator with active medium.
3.2 3.2.1
Laser resonators 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 Ml to the active medium is / = /Q. Then, after the active (amplifying) medium the intensity will be / = Ioe^\ where /? is the amplification coefficient of the medium and / is its length.^ The amplified beam is reflected by the mirror M2, and after the reflection its intensity is / = IoR2e^\ where R2 is the reflection coefficient of the mirror M2. The reflected beam will cross the active medium once again (/ = /oi^2e^^0 ^^^ part of the beam will be reflected by the mirror Ml. Thus after one full round of the beam inside the resonator the intensity is /i = IoRiR2e'^^K In one round the intensity changes by the value AI = h - IQ = Io{RiR2e^^^ - 1), and this takes time At = 2L/c, where L is the optical length of the resonator.^ Considering a time scale much longer than the time needed for one round trip of the light inside the resonator one can replace At by dt and A / by dl and write a differential equation dl ^ c{RiR2e^^' - 1) dt 2L
^ ^^
which is known as the laser equation. The product RiR2e^^^ consists of two parts. One part, Ai = R1R2, is responsible for the losses of the beam intensity and another, Aa = e^^\ 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 dl
c(AiAa - 1) ,
The term Ai 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 Ai = R1R2T where T is the transmittance of the interferometer. ^See discussion in Section 1.3. ^The optical length may differ from the physical distance between the mirrors as it takes into account the refractive index of the medium. In particular, if the active medium fills all the space between the mirrors and its refractive index is n, then the optical length is L = Dn, where D is the distance between the mirrors.
42
Lasers for Spectroscopy Applications
When AiAa = 1, the Hght intensity in the resonator stays at a constant level, which means that the laser is operating in continuous wave (CW) mode. IfAiAa < 1, the intensity decreases and the laser will not operate. Therefore, the condition AiAa > 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 =
^'
2-^
^
c{R,R2e^^^ - 1)
2L
c{AiAa - 1)
^ '^
which is called the laser time constant. Using r the laser equation can be rewritten in a simple form: ^ = ^ . This differential equation has exponential solution, / = I{^e~^^^\ if the losses, A^ and the amplification, Aa, are independent of the time and intensity I? When there is no active medium in the resonator, i. e. a — {) or Aa — I, the time constant is the resonator time constant 2L
^' - c{R,R2 - 1)
^^'^^
This time constant is the average photon lifetime inside the resonator.
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: AA =
^Q
=
27TL^/F
A^ 27T^/WR^
(1 - R1R2) 2L
^ ^ ^0
^ c
1
1
.3 ^x
27r^/WR^rr
One can notice, that AQ/C = rx is the wave period. Then, the relative resolutions of the interferometer Fabry-Perot, or passive resonator, is AA Ao
1
Tx
27r^/RiR2 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 (r^) 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 resonator time constant, r^ (eq. (3.4)) by the laser time constant, r/ (eq. (3.3)). However, for CW lasing mode AiAa = 1, which means that 1/TI = 0, or ri -^ cx), and AA -^ 0. 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 A A/A < 10~^^, i. e. Az/ < 1 kHz. A further discussion of the subject can be found in Section 13.3 on page 241. ^Actually, at least the amplification factor, Aa, is intensity dependent. Typically an increase in the light intensity inside the laser resonator results in a decrease in active medium gam, A a •
3.2. Laser resonators
3.2.3
43
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 -j^N
(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.^ 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 (AiAa > 1). However, it is possible (and is usual) that in the frame of the amplification bandwidth of active medium, AA^, a few wavelengths will satisfy condition (3.7). Example 3.1: Longitudinal modes of Nd:YAG laser. Nd^+ ions (active centers of the popular solid state laser) has amplification bandwidth of AA^ ~ 1 nm and the amplification maximum wavelength is AQ ~ 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 A^o = f^ — 400 000. The wavelength distance between the modes is AA = j ^ ^ 0.0025 nm.^ Thus, in the worse case, the laser emission spectrum will consist of AA^ = ^ ^ ^ 4000 distinct emission lines (bands), or longitudinal modes. The single longitudinal mode operation is an important requirement for the fine spectrum 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, TEMQO. 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 ^Actually, eq. (3.7) is the equation of "standing waves". The wave at the wavelength A = - ^ will be in phase with itself after traveling a complete round in the resonator, and interference of the waves propagating in opposite directions will give a standing wave pattern. -'Under condition L ^ A, the distance between the nearest modes is A A ~ 2^^ L-
44
Lasers for Spectroscopy Applications
Ml
M2
Figure 3.3: Estimation of the size of TEMQO mode.
wavelength, L' — L = A/2, the waves will quench each other. The half wavelength delay at mirror M2 means displacement (in direction perpendicular to the optical axis) of cf = L'^ - L ^ = ( L ^ ^
-L^C^XL
(3.8)
or d c^ \ / A L
(3.9)
This is a rough estimation of the size of zero order transverse mode, TEMoo-^ The TEMQO 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^^ mode gives (n + 1) x (m + 1) spots. Therefore, TEMQO 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 TEMQQ mode, and, therefore, the laser beam divergence
are determined by the length of the laser resonator. Example 3.2: Divergence of TEMQQ mode. The diameter of the output beam of L = 20 cm long Nd:YAG laser operating in TEMQO mode is (i ^ A/AZ ^ 0.45 mm (A = 1063 nm). Divergence of the beam \^ Lp ^ y ^ ^ 2.2 x 10~^ radians or 0.1°. The same laser but with 1 m long resonator will have the beam diameter on the output d^\ mm and divergence Lp ^ 10~^ radians or 0.04°.
^In the frame of this geometric optic approach one can expect the next maximum to appear when L' — L = A. This mode is called the first order transverse mode and it is denoted as TEMio for displacement in X direction or as TEMoi for displacement in Y direction, respectively. However one has to be aware of the limitations imposed by using the geometric optics approach for treatments of the diffraction phenomena.
3.3. Continuous wave lasers
45
3.2.5 Stable and unstable resonators In a general case the mirrors forming laser resonator, Ml and M2, can be spherical ones with curvatures Ri and R2. Then, the relation
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, require 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 TEMQO 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 AiAa = 1, i. e. ^ = 0. However, one has to mention that the gain factor, Aa = e'^^\ is not a constant since P depends on pumping rate creating the inversion and on the intensity inside the resonator, /, 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 / , and assume /? to be a constant at constant pumping rate. Then, the lasing threshold condition is Aa = ^ , or (3 = ^ In I ^j = Yi In f ^^^ 1. Since f3 = ANa, where AA^ is the inversion and a is the cross-section of the laser transition, the minimum inversion needed to establish lasing (or the lasing threshold) is AA^ = 2l^ In f ^ j . Depending on the laser construction (Ai) 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 instruments designed for the time resolved experiments. There are also applications were
Lasers for Spectroscopy Applications
46
Ml
Active medium
Polarizer
M2
QI
Pockels cell Figure 3.4: Optical scheme of a Q-switched laser with Pockels cell. The Pockels cell is controlled by voltage, U. Ml 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 N2 and excimer lasers utilize a short (but powerful) electric discharge to create the inversion. Limiting steps determining 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 10^0 ns and the pulse energies are 1 mJ - 1 J. Further increase in the pulse energy or/and 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).'^
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 R2 = 0, thus Ai = 0. Lasing cannot be established even at high inversion levels, since AiAa < 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 Nd'^+, 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 ^For example, aXV = 10 kV potential to generate an electric pulse with energy of £^ = 1 J and duration At = 1 ns, the pulse current must be / = vAt ~ ^ ^ "
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. For 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 element 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 Section 3.6.5. The physical dimensions,^ the media dispersion and the amplification bandwidth are the main factors which should be taken into account and optimized to achieve femtosecond 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 At > -^j, where A A is the amplification bandwidth and A ^The spacial width of a 1 ps pulse is 0.3 mm only.
48
Lasers for Spectroscopy Applications
Ml
Active medium
I
U M2
Acousto-optic ~ r modulator — Figure 3.5: Optical scheme of the mode locking with acousto-optic modulator placed as close to the reflector M2 as possible. Afi*equencyapplied to the modulator must be / = 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"^+, 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 AA > ^ = 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.^
3.5
Laser amplifiers
Sometimes the desired properties of the laser emission cannot be achieved at once. For example, 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,^^ 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). ^ Fiber lasers are the fast developing systems which have approach the sub-picosecond pulse width already, and can be used in applications where a nano Joule pulse energies are sufficient. ^^It is easier to achieve single longitudinal mode operation when the laser is operated at pumping level close to the lasing threshold, when fewer modes exceed the threshold.
3.6. Main types of lasers
3.6
49
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. ^^ 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 Y3AI5O12 (YAG) crystal doped by Nd^+ 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 (AA ~ 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^+ions, the laser crystals have high amplification, (3 >10^ m~^ 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. NdYAG 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'^^ ions can also be incorporated into other matrixes, such as LiYF4 (YLF), YVO4, 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 itself, 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 of Nd 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: INDINd.YAG laser (Spectra-Physics). INDI is a relatively compact NdYAG 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 10 Hz. ^ ^ Some laser systems will also be discussed with additional details in Sections devoted to their applications, such as pump-probe, Section 11.2. There are also better sources of information on different types of lasers and laser systems, such as refs. [2, 3].
50
Lasers for Spectroscopy Applications
The size of the laser head is roughly 73 x 15 x 18 cm^. The diameter of the laser beam is < 10 mm, and the beam divergence is 10~^ radians. An important application of CW mode Nd^+ lasers is the pumping source of the femtosecond 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 TEMQO 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~^, but the overall electric efficiency is typically 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 10 W. An advantage of the ion lasers is TEMQO 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 pumping 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. Xe2, 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 Xe2 -^ Xe2 + hv ^ XQ -\- 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, Xe2. 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 energies are usually in range 0.1-1 J. The excimer lasers are popular excitation source in organic 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 NF3 XeCl ArCl Xe2 F2
XeCl, mixture of He, HCl 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
3.6.4
Lasing wavelength, nm 430-465 453-507 508-540 560-615 590-645 610-700 695-762 728-802
Pumping laser Excimer, N2, Kr, 3rd harmonic of Nd Excimer, N2, Kr, 3rd harmonic of Nd Excimer, N2, Kr, 3rd harmonic of Nd 2nd harmonic of Nd, Excimer, N2, Ar 2nd harmonic of Nd, Excimer, N2, Ar 2nd harmonic of Nd, Ar 2nd harmonic of Nd, Ar 2nd harmonic of Nd, Ar
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 10 ns, and thus requires a high pumping rate to create an inversion. In nanosecond pulsed mode the excitation sources are usually the second or third harmonics of the Q-switched Nd: 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
P^^P
^..
beam expander
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 LI and L2 form a beam expander to reduce the power density at the grating. Output coupler of the laser is mirror Ml.
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. ^^ 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'^^ ions incorporated into sapphire (AI2O3) 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^+ 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^^ Since the focal distances of the lenses are sensitive to the wavelength (because of the dispersion properties of the lens materials), prism beam expanders are usually used in dye lasers designed for a broad wavelength applications. Also prisms can be used as the wavelength selecting part of the laser resonator, if the a fine spectrum resolution is not required.
3.6. Main types of lasers
53
output ^ M4 Figure 3.7: Optical scheme of a femtosecond Tiisapphire pulse generator.
ator is working in passive mode-locking regime using Kerr lens effect in the sapphire crystal. ^^ 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 crystal due to dispersion and allows generation of femtosecond pulses. ^^ The Tiisapphire crystal is usually excited through one of the spherical mirrors (Ml 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 crystal. 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 200^00 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 Tiisapphire 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 providing 100 fs pulses are commercially available from a few manufactures, and even 20 fs ^^Kerr lens is a non-linear optic effect, which can be described as instantaneous change on the refractive index of matter in high electromagnetic 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 modulate light losses in the resonator in order to achieve mode-looking operation. ^"^The dispersion effects are very important in femtosecond pump-probe and optical gating spectroscopy applications. They are discussed in related Sections 11.4.1 and 12.1.2.
54
Lasers for Spectroscopy Applications
pulse generators can be purchased. ^^ These laser systems can be used directly in picosecond 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 amplifiers 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'^+ ions is an important property of the active medium which makes possible a wide tuning range of Ti: sapphire lasers operating in CW or nanosecond 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, AlGaAs 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 Nd: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 distribution 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.^^ 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 bandwidth, power, pulse duration and energy and others. For example N2 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^+ ions in sapphire or alexandrite emits in 700-800 nm range and Yr'^^ in YLF emits at 2.8 ytx. ^^For example TISSA20 from CDP Corp. (Moscow) generates 20 fs pulses. ^^For example, a selection of pulsed diodes is available from PicoQuant GmbH. LDH series of the laser diodes are available in the wavelength range from 375 to 1546 nm. They provide 50-120 ps pulse width with average power about 1 mW 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 relativistic electrons enables one to build up a laser emitting in a wide wavelength range expanded 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 practical 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 D = eE
(3.12)
where D 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, P , 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 £ = 1 + X, or X = £ — 1. Formally, a nonlinear response of the medium can be expressed as ^ = X i ^ + X 2 ^ ' + X 3 ^ ' + --.
(3.15)
where xi is the linear susceptibility (the same as one in eq. (3.14)), X2 is the second order susceptibility, xa is the third order and so on. In the case of anisotropic media the dielectric constant has to be replaced by the dielectric tensor (3.16)
D = EE where e =
^xx
^xy
^xz
£yx
^yy
£yz
^zx
£zy
^zz
(3.17)
56
Lasers for Spectroscopy Applications
Similarly, the linear susceptibility xi 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^, Ey, E'^, E^Ey, 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 = E^^e^^^ then the medium response is (using scalar presentation)
The second term is a wave at frequency 2u. The amplitude of the wave at 2cj 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 xi^oe^^^). 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^ V m - ^ All optically transparent media have non-linear response starting from some light intensity. 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 oj and 2cj are n{u;) < n(2cj). This means that the SH wave, 2a;, propagates slower than the fundamental wave, uj. Let us consider SH wave generated at two points at a distance d along the propagation direction. When the distance is d = r .^ ^^—r^ = ^ T ^ = ^ T ^ , the waves at 2LLJ are 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 4531) — ^^ ^- ^^^^ means that while propagating in the quartz the intensity of the second harmonic will increase during the traveling through the first 24 ji of the medium. Then the intensity will decrease, since the waves at 2uj generated at the entrance and at distance of 24 // have opposite phases and will quench each other. At the distance of 48 JJL 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, UQ = n^. When the light propagates along Y axis the velocities of the ordinary and extraordinary polarizations are different. There is no angular dependence of no, and the dependence of ng is given by an ellipse. The index of refraction at 2c<; (dashed lines) is greater than that at uo (solid line). There may be an angle when no{2uj) crosses ne(u;), this is angle a in Fig. 3.8. The wave with ordinary polarization at 2uj will propagate with the same velocity as the wave with extraordinary polarization at uo 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
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 (|^| as 2k^ = k2uj,or rvijj -]- rb(jj —
rv'2LO
(3.18)
This is the momentum conservation law for two photons reaction. The energy conservation law leads to a trivial hcu -\- huo = h{2u;) or u -\- cu = 2uj. 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. Photons, 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 of Q-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 probability of three photons reaction (u; + a; + u; = 3LU) 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 (uo -\- uu = 2cu). Then, photon at 2a; and photon at uj yield the third harmonic photon 2uj -\- uj = Suo. 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 k2uj + ku; = ksuj
(3.19)
and cannot be satisfied together with the momentum conservation law for the second harmonic 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 generalfi^rmas ki + k2 = ks
(3.20)
The energy conservation law requires LJi +(x;2 = ujs
(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 ki and ^2 may have different orientations, as shown in Fig. 3.9. Similarly to the second harmonic generation, the k J ^^ - - -. 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 LUi and 002, niay have the same polarization, then the Figure 3.9: Wave mixing: wave vec- sum wave, at CJ3, must have polarization perpendictor adjustment (conservation). ular to that of the primary waves. This polarization arrangement is called type I synchronism and is similar to the second harmonic generation (which is uui = UJ2 =00). 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. cji, and the sample emission, (jj2, to generate the light at higher frequency, CJ3. 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 ps are the photons at uoi, uj2 and 6^3, respectively. Equations (3.20) and (3.21) state that, the photon at CJ3 is equivalent to the sum of two another photons, at cji and UJ2, 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 CJ3 with a non-Hnear crystal may result in photon "dissociation" and appearance of two photons at uoi and 002Clearly, 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+P1 -^'^Pi^P2
(3.24)
As the result of the reaction, two photons at uoi are generated and the reaction can be considered as the reaction of stimulated emission of the second photon at uoi For stimulated reactions the second photon has the same frequency and phase as initial photon at uoi, thus the reaction (3.24) results in amplification of the light at uoi. In such case, the wave at uoi is called signal wave, the wave at cj2 is called idler wave (it is a loss of energy) and the wave at a;3 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 parametric 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 condition (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 PRO23010 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 220^50 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 measurements 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 parameters, 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 Di, and we have measured the light power and obtained a value / i . Thus, the power density is Pi = -^^.^ as we can conclude from these measurements. At this point we have a question: 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, D2 and I2. Then, we can conclude that the power density is P2 = ~D^' Now we have two values. Pi and P2, and a series of questions: 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? ^The area of the hole is s = ^^^,
and the power density is Pi = ^ = —j^
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. / i and I2, 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 subject 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, Aylg, 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, /^Adn- 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 AA^ from measurement to measurement. Therefore, the measured value is Ameas
= ^ + A T I , + AAdn
+ AA^
(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 /\As is present in all measurements and is called systematic error. It cannot be avoided, except by using another more precise instrument. The two last summands, /^Adn and AA^, have similar effect on the measured value {Ameas), 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, AAdn, and the noise of measured parameter, A An, cannot be separated and, therefore, considered as a single random noise A^^ = AAdn + A An- Then eq. (4.1) can be rewritten as Ameas
= ^ + A A , + AAr
(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 AAr 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 xi, X2, ... XN, 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 xto 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
Y^P{xi) = l
(4.3)
/,(.)rfx = l
(4.4)
and
— CO
For a given probability distribution one may calculate an average value N
{x) = Y.x,P{x,)
(4.5)
for a discrete random values and
(.) = / M-)d.
(4.6)
— oo
for a continuous random value. The average of a function f{x) of a random variable x is + 00
(fix)) = I f{x)p{x)dx
(4.7)
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 /c, 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, Pn(t -\- dt), is the probability that n photons were emitted in the interval from 0 to t, Pn (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-i{t), and one in the interval from ttot-\- dt, which is kdt? Thus, we can write an equation Pn{t + dt) = Pn(t){l - kdt) + Pn-i{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
^Ml
= _kPn(t) + kPn-l{t)
This generates a chain of the equations for n = 1,2, Pn{t)='-^e-^'
(4.10) The solution of the eq. (4.10) is (4.11)
which is known as Poisson distribution. One can introduce dimensionless variable x = kt and rewrite eq. (4.11) as Pn{x) = ^ e - -
(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
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 , . . . , oo, which gives (n) = e - ^ ^ H y ^ ^
(4.14)
^Certainly, we can neglect probabilities of emission of more than one photon in the time interval dt, if the interval dt is short enough.
4.1. Noise statistics and accuracy of measurements
0.2
65
n—^—^—^—^—^—r
0.15
2 0.1 O OH
0.05
0
1
2
3
4
5
6
7
8
9
JL:lTrT-i+f-i-± 10 11 12
number of photons, n Figure 4.1: Poisson distribution with kt = 4.
One can mention that the sum in eq. (4.14) is the power series of the exponent, e^ = J2^^ then (n) = e-'^'kte'^' = /ct
(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 ^ = | , 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 Psi^/k) _ {kt)^4\ _ ^ _ 4 _ P4{4/k) ~ 3!(H)4 ~H~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 /ct ^ oo the Poisson distribution is transformed to the Gaussian
Optical measurements
66
Gaussian, XQ=100, a=10
60
80
100 120 number of photons, n
140
Figure 4.2: Poisson distribution, kt = 100, and its Gaussian approximation (eq. (4.16))
one"
(n - ktf 1 (4.16) : exp 2kt V^TTkt The Gaussian distribution of a random value is very often found in practical measurements. Almost all types of noises are well approximated by Gaussian distribution function. A general form of the distribution is Pn{t) = G{n,t)
1
G{x)
27rcr
exp
{X - XQY 2a2
(4.17)
The distribution has maximum aXx = XQ, and, since the function exp(—o;^) is symmetric one, the average value is equal to the most expected one, (x) = XQ. 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 fct = 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. Data averaging and square root law For a series of M measurements resulting in values ^1,^2, 1
^
, yM the average value is (4.18)
i=l
^To obtain eq. (4.16) from eq. (4.11) one may use Stirling approximation of the factorial, Inn! + ^ I n n — n + ^ l n 27r, and neglect by smaller 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, y^, from the average value, {y). This is given by the mean square deviation 1
^
(Ay^> = {{{y) - yf) = - Y,{{y) - y,f
(4.19)
2=1
The value y^(A^^ is also called standard deviation. Independent of the distribution type it can be shown that (A|/2) = (f)
- (yf
(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 (yi, 7/2,..., yu) but can work with two average values, (y^) and (y) ^ An important property of the Poisson distribution is that mean square deviation is An2) = ^
((n) - nf iMLe-'=* = kt
(4.21)
which is, for Poisson distribution the average of square deviation is equal to the the average value, (An^) = (n). This dependence is called square root law and has different presentation forms 11 (n)
11
(4.22)
or A / ( A ^ = -
~ s/W)
(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 ^/N. 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 (Ax^) = cr^, which is commonly called "sigma"-value. The averaging of M measurements gives value {x) ^, where the index is used to indicate the number on measurements. Now one may ask, how accurate is the average value, ( x ) ^ , or how close is it to the real value, i. e. to {x)^l The value ( x ) ^ is a random value with its own distribution. The probability p{{x)^) to obtain value {x) ^ is the product of probability to obtain xi, i. e. pi (x), times probability to obtain X2, i. e. P2{x), and so on. In other words, p {{x)j^) = Yli=i Pi{^)- ^^^ ^ series of identical measurements the probability functions are the same for each measurements, thus ^During the measurements two cumulative values are collected, J^i/i and 5^2/f, which are then used to calculate the average, the average of the square value and the standard deviation.
Optical measurements
68
I
0.8
'
1
'
1
'
1
" \
•••
n = l
H
-• n = 2 •- n = 3
^0.6
0.4
1
— n= 0
\
O
'
-
\
0.2
-\
y
,-<.-r-" o'r^^
/X^
* * •.
1
1
1
2 3 Observation time, t
1
1
*•••"""" ^^ - J
1
4
Figure 4.3: Time dependence of the probabilities to observe 0, 1, 2 and 3 photons exactly, calculated for emission rate k = 1.
~ P^{^)' 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{{^)M)
{X - XQ)^ P((^)M)
exp
2a2
{x -
exp
M
-
M{x-XQf = exp
2^2
XQY
(4.24)
2^M
which is again the Gaussian distribution, but the width of this distribution, i. e. its standard deviation, is GM = V M ' Thus, averaging of M measurements improves accuracy by factor V 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 probabilities 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 = \ and using eq. (4.11) with appropriate n-values. It is natural ^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)
4.1. Noise statistics and accuracy of measurements
69
that at a short time intervals, t
o = I — PQ ^ kt, which was used as definition of the emission rate. The probability to observe exactly 1 photon grows linearly from zero, reaches its maximum value of 36.8% 3tkt = 1 and then decreases.^ The probability to observe exactly two photons grows quadratically at /ct
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 iV = 10 000 photons were detected, the standard deviation is AA^ = ^/N = 100 photons, i. e. the relative error is ^ = ^ ^ = 0.01, and the counting rate is (1.00 ± 0.01) X 10 *" s~^. But, if only N = 2 photons where detected, the reliability of the rate estimation is very poor. The standard deviation is \/2 ^ lA counts, therefore the counting rate was determined with accuracy (2 ±1.4) x 10^ s~^. In the example above, the accuracy can be improved by (1) increasing the time interval of the photon counting or/and (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 yi,y2, • • - .VM- Then, the average value, {y)j^, can be calculated using eq. (4.18). The mean square deviation, (AT/^) (or cr^), is calculated using eq. (4.19). In the case of Poisson distribution (y) ^ (Ay^) (see square root law, eq. (4.23)), and the standard deviation of the averaged value is (Ax^) l\fM. For the photon counting problem one can notice that the total measuring time is tu = ^ ^ , and if the photons would be counted only once, but during this time interval the final result would be \ / M 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 Av4^ but it does not change systematic error AA^. ^Note that at the best time 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
4.1.4
Optical measurements
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 detail 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 A / is (y2^ = AkTRAf
(4.25)
and the current noise is (I') = ^ A /
(4.26)
where R is the circuit active resistance. The power of the Johnson noise is P = ^{U^) (p) = AkTAf
(4.27)
The noise power does not depend on resistance R. 1/f noise
4.1.5
and generation-recombination noise have spectrum density proportional to the inverse of the frequency (which has given the name to this type of noise). Usually it dominates in measurements which take a long time. These measurements are said to be low frequency measurements. Depending on the measuring technique 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.
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) AA and AB and we want to evaluate the error (standard deviation) of C, i. e. AC. Yet another widely used specification of the measurements accuracy is the relative inaccuracy, which can be calculated as Aa = ^ , Ab = ^ and Ac = ^ for parameters A, B and C, respectively.
4.1. Noise statistics and accuracy of measurements
71
How can AC and Ac be estimated from Imown A, B, Aa and A6? For four arithmetic operations the rules are simple: Sum, C = A-\- B: adding errors of the values one obtains
C^AC=A^AA^B^AB=A^B^AA^AB=C^AA^AB thus AC = AA + AB
(4.28)
For the relative inaccuracy
The latter can be expressed in terms of Aa and Ab, 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, AC = AA + AB. 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 Ac = ^ ^ ^ ^ ^ (compare with eq. (4.29)!), thus, when A^ B,Ac^ oo. Multiplication, C = AB:
C + AC
= =
(A + AA)(B + AB) = AB + AAB + BAA + AAAB C + AAB + BAA + AAAB
Let us neglect AAAB, which means A ^ AA and B ^ AB, or we assume that our primary values, A mid B, are rather accurate, then AC ^ AAB + BAA
(4.30)
and Ac =
AAB ^ BAA AB AA ^ ^^ = — - + — - = Aa + A& C B A
^^^^^ (4.31)
Hence, the relative error of the product, Ac, is the sum of the relative errors of the multiplicands, Aa + Ab. Division, C = ^:
^ + ^ ^ - -BTAB
- B •
TT^
72
Optical measurements
Assuming that A ::^ A ^ and B >> A ^ , similar to the multipHcation case, one obtains
A (^
^ + ^^
Ayl\ /
-
B['^^)['
=
C + CAa - CAb
A5\
A
A l\A
B J - B B
A
A
AB
B
B
where the term with A A A ^ was neglected. Finally, taking into account that the inaccuracies must be summed independent of the sign Ac =
= Aa + A6
(4.32)
O
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. AA <^ \A\. Under the same conditions we may derive the error which arises when a function / ( ) is used to calculate value C from A. That is C = / ( ^ ) , and C + AC = / ( A + AA). The function can be approximated by linear dependence around point A
C + AC = f{A) + ^5z^ AA = C + ^ 5 ^ AA
(4.33)
Thus, AC = ^ ^ A A
(4.34)
In other words, the slope of the function / ( ) at point A determines the accuracy of its result, C.
4.2
Photosensitive devices
4.2.1 Photodetector 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 photodetectors. 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 electron, 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 incident light power. It is measured in A-W~^ or V-W~^ depending on the response type, current or voltage, respectively. This parameter tells what to expect of the detector output at a given incident light power. It is wavelength dependent value.^ 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~2. For example, if one needs to measure light power in the frequency range of / = 10 kHz, i. e. with the time resolution of r = ^^ ^ 16 //s, and would like to use a photodiode with NEP = 10~^^ W-Hz~ 2 (e. g. a Si photodiode), then the minimum detectable light power will be P = NEP x ^f] = 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.^ 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 = \/A • {NEP)~^, 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~^. 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. ^In earlier literature the term responsivity was used as synonym of sensitivity. ^From the example of the photon counting problem, one can see that at longer collection time a bigger number of counts is achieved which results in a smaller relative uncertainty of the measurements. The decrease in relative uncertainty is proportional to the square root of the number of counts, thus it is proportional to the square root of the averaging time 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
Optical measurements
74
Dynodes
Cathode
Anode
hv
R
o Uc
R
<
R
5; u.
1 1 / , I I I /
\rHr \f\^
^A;
R
R
R
I
u. Figure 4.4: Schematic diagram of a photomuhipHer.
Time constant and frequency response: The time constant (r) specifies how fast the signal is formed on the device output when the hght is switched on instantly. The frequency response is measured with the sinusoidally modulated light, e. g. light intensity is I{t) = /o[l + sin(27r/t)], and specifies the highest frequency /o (cut off frequency) at which the photodetector responses without significant signal reduction (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^(2^/o)- -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 system. The photo-cathode converts photons to electrons and the multiplication system amplifies 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, 17^ 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 ^ electrons). The secondary electrons are accelerated by the electric field between the first and the second dynodes and each of them produce another 3 ^ electrons. This multiplication 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 ^ at the dynodes is 100-150 V (the electrons gain energy of 100-150 eV 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 1 0 ^ - 1 0 ^ . '^The proportionality coefficient between the time constant r and cut off frequency /o depends on exact definition of the time constant and on the response order at the detector. Given relation is valid for the first order response and the time constant measured at the level of the signal of 1 — e '. 0.63.
75
4.2. Photosensitive devices
Micro-channel plate
Photocathode
Anode
signal O ^3
O U2
O
(y
R
I 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 620 /x. 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 Us — U2 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, 0 ^ is the peak quantum efficiency, A^ 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 bialkali (S-22) multialkali (S-20) extended red multialcali (S-25) GaAs Cs-Te
range, nm 300-630 180-800 300-900 300-920 160-320
0m, % 26 20 7 15 14
Am,nm 400 480 600 700 200
id, nA 0.1 0.2 1 2 0.01
cathode, which is transmitted to the micro-channel plate, amplified and converted back to an optical image by the phosphor screen. ^^ 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 photomultipliers in photon counting regime. The output of a photomultiplier is connected to the input of a discriminator which rejects 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~^, 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 characteristics 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-1); peak quantum efficiency: can be up to 20% in visible range; ^^The micro-channel plate intensifier increases the intensity of the image but all the specific information about the incoming light, such as wavelength, polarization and beam propagation direction, is lost. This makes if different from the laser amplifier, which preserves the wave properties of the amplified 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~^ or even smaller. For the photocathodes sensitive in the near infrared part of the spectrum the dark counting rate is higher and can be >100 s~^. 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^; 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 applications, see Section 8.4.1; response time: typically a few nanoseconds, but for MCP PMTs can be shorter than nanosecond. Main advantages of the photomultipliers are: • high sensitivity, due to the high multiplication factor the sensitivity can be 10"^ A-W~ ^; • 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 photomultiplier 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 devices (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 sensitivity up to 0.5 A-W~^ at 800 nm.^^ The hQ^tp-i-n photodiodes have very good time resolution, r < 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~^^ W'Hz~2 (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~^ at peak sensitivity). The dark current can be smaller that nano Ampere, which provides good NEP, e. g. 2 x 10~^^ W-Hz~2 for G8376-1 (Hamamatsu) with active area diameter 0.04 mm. An advantage of the photomultipliers as compared to the photodiodes is a large amplification 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 sensitivity of the diodes to the level when they can be operated in photon counting regime. ^^ An ^ ^ Ideally, one can expect each photon to be converted to an electron, in which case the sensitivity is S = -^ = J—, where q is the electron charge. At A = 800 nm the top possible sensitivity is 5 ~ 0.65 A-W"-*^. 19
To reduce dark current to the level when dark counting rate is reasonably low the area of the APD must be small. For example, for APD PDM 50CT SPAD detector (from Micro Photon Devices) with active area 50 /x, the dark counting rate is 5000 s~
4.2. Photosensitive devices
79
advantage of the APDs is also a good response time, which can be shorter than nanosecond. ^^ Charge coupled devices (CCDs) are another class of semiconductor photo-detectors which are widely available in the market and actively used in optical spectroscopy. Although 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%, which provides sensitivity close to 0.6 A-W"^. 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;^^ • small size of photo-sensitive area for photodiodes with fast time response. ^^ A short comparison of APDs and MCP photomultipliers in time correlated single photon counting applications can be found in Section 8.3.1. ^^When compared to photomultipliers, the diodes have higher quantum yield but do not provide any amplification of the electric signal, therefore the sensitivity of the diodes is much lower than that of the photomultipliers. Avalanche photodiodes provide an amplification of the electric response and have sensitivity improved by factor of 100 or even greater, but the price of these devices is typically much higher than that of the photomultipliers. The avalanche photodiodes can be used in photon counting mode.
Optical measurements
80
4.2.4
Other photo-detectors
A disadvantage of PMTs and photodiodes (and similar semiconductor detectors) is dependence of the photo-response on the wavelength, i. e. the sensitivity, S, is rather sharp function of the wavelength, S = S{X). When neither high sensitivity nor high time resolution are required thermal detectors can be used to measure light intensity. The measured parameter is heating produced by the incident light. There are different types of pyroelectric detectors, 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 mW and 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 //). Typical detection limit of these detectors is 0.1 mW. Typical measurement 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, photodiodes can be used. A typical electric circuit for the measurements consists of the photodiode connected in series with a resistor, R, as shown in Fig. 4.6. The measured value is the potential (voltage), Uout, 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 (Uout = 0) without illumination.^^ If the incident light power is P and the photodiode sensitivity is S, then the phoFigure 4.6: Electric circuit for mea- tocurrent is / = SP, and corresponding potential surements of the light power with on the resistor is Uout = IR = SRP. The product, Su = SR, can be called voltage sensitivity. For exphotodiode. 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 i? are (1) the input resistance of the device used to measure potential Uout, 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 Idark ~ 100 nA, assuming load resistance to be i? = 1 MOhm, the output potential is I^arkR = 0.1 V, which is almost certainly the value one cannot neglect. Photodiodes (and similar devices) have high sensitivity and wide dynamic range. Obvious disadvantage of the photodiodes is rather sharp dependence of the sensitivity on the ^^The actual value of the output voltage is determined by the dark current of the photodiode, I^, so that Uout = Rid'
4.4. Measurements of the pulse energy
Pulse
81
Detector response
A U, C
U,
-^ a) 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 measurements. However, the pulse duration, r^, 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 Ud'
Ip{t)dt
(4.35)
where Ip{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 detectors 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 /is and the measurements inaccuracy must be better than 1 %, then the detector integration time must be Td > lfis/0.01 = 100/iS. Most of the photodiodes have time response much shorter then 100 //s. 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 kQ 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 5 = I/P, where / is the photocurrent at illumination power P. The pulse energy is the integral of the power E==
f P{t)dt P{t)dt
(4.36)
The power generates photocurrent P{t) = I{t)/S, thus
E = J^-^dt=^Jl{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 Ud = Q/C. Thus, for the energy one obtains
where Sp = ^ is the energy sensitivity. In other words, we have obtained pulse energy sensitivity coefficient, Ud = SpE, for the measuring scheme presented in Fig. 4.7 (b) using photocurrent sensitivity, 5, 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 photodiode current sensitivity at 800 nm is 5' = 0.5 A/W, which gives sensitivity S^ = ^ = 5 • lO*" V/J. Thus a pulse with energy 1 fii will create 50 V voltage jump {Ud, 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 mV 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 measuring circuit, Td. If this cannot be arranged, then one may determine pulse repetition rate, / , and measure the average power of the pulses, Pav, using one of the methods (instruments) available for the power measurements. The pulse energy can be calculated then as
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 estimates 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. For 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 bandwidth 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 propagation 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 measured 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 /xJ energy (is 1 /xJ a huge energy?) creates power of 1 /xJ/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 M2 and M3 and arrives to mirror M4 with delay ti. Another part is reflected (twice) by two right angle reflectors, M5 and Me, and arrives to mirror M4 with delay ^2. The reflector Me is placed on mechanical translation line and can be moved along its axis, so that direction of the beam does not change but delay, ^2, 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, 2uj, 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 SHG
hAt) = o^lfnit)
(4.39)
where lin (t) is the intensity at fundamental harmonic before the SHG, which is the sum of
Optical measurements
84
CO
SHG M.
Filter
Ml
CO
PD ^2co
^ 2co 1 r 2co^
Mo
a)
M3
M2 /
JY' CO
/
\
\
M5
/
V
^.
SHG
h
M4
Ml
A Me
fii
PD CO
^2co
^ 2co
/ Delay line
•d
V
b)
Figure 4.8: Optical autocorrelators: a) Michelson interferometer scheme and b) right angle reflector delay line scheme.
intensities of two beams, I\ and I2, (4.40)
Since we may choose zero time arbitrary, eq. (4.40) can be rewritten as
hn(t) = hit) + hit + t2 - ti) = hit) + hit + At) where At is the delay between the pulses 1 and 2 and we can change it by moving the delay line (positions of the reflectors Me or M3 for schemes b) and a), respectively), so that 2d At= — (4.41) c where d is the position of the delay line. For the beams of equal intensities linit) = /(t) + / ( t + A t ) , thus
/2cj(t)
= =
\2
a ( / ( t ) + / ( t + At))' al^ (t) + al^ (t + At) + 2a/(t)/(t + At)
(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 2cj) + CX)
P20. =
I
h^{t)dt
-00
+ 00
+00
a / I'^(t)dt ^a
[ I^{t^
+00
At)dt + 2a f I{t)I(t + At)dt
(4.43)
— CO
The first and the second integrals give the same results since integration is performed in infinite limits, / P{t)dt = PQ. The last integral in eq. (4.43) is autocorrelation integral of the function I(t). It depends on parameter At, i. e. on delay line position. +00
Pc(Ai) = 2a j I{t)I{t + At)di
(4.44)
— OO
and it is the function of our interest. The value of Pc( At) shows how much the pulses I{t) and /(t + At) overlap each other. When the delay between pulses is zero (At = 0) the result of integration is Pc(0) = 2Po and when the delay between the pulses is much longer than the pulse duration, the integration gives Pc{oo) = Pc{—oo) = 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 autocorrelation 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 /(t) = e~* the autocorrelation function is + CX)
Pe(At)
=
+CX)
/ e-*'-(*+^*)'dt =
At^
e-^
+ CX) f
/ e
/ e" ^*+7f ' - 71 \ t
(2t + A t ) 2
^—dt = Ce-^
At^
(4.45)
which is the Gaussian pulse, but it is \/2 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 Me 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 At = 0 when both pulses are overlapping with each other. Two other peaks appear
Optical measurements
86
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
ih
^0.5
-5
-
4
-
// // '/ '' // // '/ // ' / / / / / / /
\\ — Gaussian pulse \^ -- Autocorrelation \\ ^^ \^ \\ \^ \ \ \ \ \
3
-
-
2
1
0
1
2
3
4
5
time/delay
Figure 4.9: Autocorrelation function (dashed line) of a Gaussian pulse, I = e ^ (solid line)
-
2
0
2
4
time/delay
Figure 4.10: Autocorrelation function (dashed line) of two Gaussian pulses, I = e *^* ^^ + e-(^+2)' (solid line)
4.5. Measurements of the pulse duration
87
when one pulse of f{t) overlaps one pulse of / ( t + At), but two another pulses are not overlapping each other, which takes place at At = 4 and At = —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 apphcations. 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 measured 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 monolayer 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 characteristics. 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 absorption 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 considered 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, lin, and after, lout, the sample to determine the absorption (as indicated in Fig. 5.1). Then, the transmittance, T = ^f^, eq. (1.7), or the absorbance, A = log - ^ , eq. (1.12), or some other relevant value can be calculated.
Steady State Absorption Spectroscopy
90
Light source
Sample , ,
, "),
T
1
' ^« ^
1
T
-^ out
Figure 5.1: Measurement of a sample absorption. The light source produce a beam at the wavelength A and with intensity Iin. The beam intensity after the sample is loutLight source
Detector
Lamp
Meter
Figure 5.2: Optical scheme of a simple spectrophotometer. The lamp emission is collected by the lens LI 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 combination 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, Iin 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 without the sample to obtain lin 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 Ui = s{\)Iin (first measurement), and with the sample U2 = s{X)Iout (second measurement). The ratio of these two values gives the ratio of the light intensities before and after the sample: U2
-'-out -L in,
(5.1)
5.2. Spectrophotometer schemes
91
which is needed to calculate, transmittance, T = ^f^ = ^ , or absorbance, A = — logT. It is important to notice that we do not need to know the photodetector sensitivity to calculate 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 applications 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 Ui (A). Then the spectrum is measured with the sample to yield the spectrum /72(A). Finally the transmittance
or absorbance
is calculated out of these two spectra. The first measured spectrum, Ui (A), is commonly called base line } 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 aberrations 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 wavelength). On the other hand, when the accumulation time approaches second the 1//-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 synchronous or lock-in detection system. This method also makes the instrument lesser sensitive to external light since the signal is detected at the modulation frequency and unmodu^Note that the shape of this spectrum is determined by many factors such as emission spectrum of the hght source, monochromator transmission spectrum and sensitivity spectrum of the photo-detector. However, the shape of this spectrum is not important while we can calculate the ratio U2/U1 with acceptable accuracy.
Steady State Absorption Spectroscopy
92
Detection part
Source of monitoring light
Control unit and display
[ |
Figure 5.3: Single channel spectrophotometer. Source of monitoring light consists of lamp (L), collimating mirror (Ml), light chopper and monochromator, formed by input slit SI, 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.^ 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//-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 Ui (A), as discussed above. Then the spectrum with the sample is measured, U2 (A), and the absorption is calculated using eq. (5.3) and presented to the user. The actually measured spectra (Ui (A) and f/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 (Ui{X) and /72(A)) one relies on stability of the system. Thermal fluctuations ^For example, if the light is modulated at 333 Hz, the detection system is insensitive to the laboratory illumination light, which is "blinking" at 100 Hz (double of power supply frequency, 50 Hz).
5.2. Spectrophotometer schemes
93
Source of monitoring light
Detection part
SD2
Control unit and display
Figure 5.4: Two channel spectrophotometer. The source of the monitoring hght consists of a lamp (L), a collimating mirror (Ml), a light chopper and a monochromator, formed by an input slit SI, 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 sample (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, PMl - SDl 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 Ui{X) and /72(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, Ur, and the signal from the sample channel, Ug. 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 - ^ = ^ , assuming 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 7-7-0 ^ \ \
of the signals is R{X) = jjorxy ^^^^ ^^ ^^^ ^^^^ ^^^^ spectrum. With the sample the ratio is S{X) = u (X)' One may notice that Ur{X) = U^{X) if the monitoring light did not change. Therefore, the transmittance of the sample can be calculated as
^ ^
{/°(A)
[/P(A)
Ur{X)
Ur{X)
f/p(A)
R{X)
^ •'
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/Ur, 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 simultaneously and the ratio §4^) will not change. For the single channel spectrophotometer the variation in the monitoring light intensity will give the same relative variation in the calculated 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{X), (which is the only one we are looking for) plus absorption of the cuvette, -Ac(A), solvent, Asoi{X), and the light reflections from the cuvette walls, Ar{X)? To obtain "pure" spectrum 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{X), cuvette, Tc{X), solvent, Tsoi{X), and reflections, T^(A). The light intensity in the reference channel is Ir{X) ^ Tc{X)Tsoi{X)Tr{X), which is, in terms of the measured values, Ur{X) = [/^(A)Tc(A)Tso^(A)T^(A). Similarly, for the sample channel Us{X) = U^{X)Tc{X)Tsoi{X)Tr{X)Ts{X). The finally measured value is the ratio of the ^Reflectionsfi^omthe cuvette walls differs from 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 frame of this consideration we can treat it as an absorption since the reflected light is not counted in our experiments.
5.2. Spectrophotometer schemes
95
intensities (of the signals from the sample and reference channels)
Finally, a "pure" transmittance spectrum of the sample can be obtained by dividing signal, S{X), by the base line spectrum, R{X), Ts{X) = ;RW)- Thus, using a pair of matching cuvettes'^ 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 Ui{X) which takes into account absorptions 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 measurements 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 photodetectors (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. Usually such devices are build up implementing the single channel scheme with the white monitoring 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.^ The construction of two channel schemes with array detectors would require two monochromators. Also implementation of the two channel scheme is a difficult task since the channels 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 photodetectors 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. ^Pair of matching cuvettes are two cuvettes made of the same material and having the same internal thickness and thickness of the light transparent walls. See Section 5.4.2 for more discussions on the subject. ^Strictly speaking, the monochromator without output slit cannot be called monochromator. This type of device is called spectrograph.
Steady State Absorption Spectroscopy
96
Source of monitoring iigiit
Detection part
grating
Figure 5.5: Spectrophotometer with array defector. The light from the lamp L is collected by the mirror Ml 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 monitoring 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, sapphire 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 hmit for the photomuhipHer tubes (PMT) is also determined by the material of the entrance window.^ 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 popular 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 // 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 usually 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 purpose 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 (SI 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. ^The efficiency of photo-cathodes in UV range is an important parameter to be considered, but it is not a limiting 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 MJ and corresponding smallest detectable absorbance, A A, which will be here called sensitivity. When the light intensity values Iin and lout (see Fig. 5.1) are close to each other one can use approximation ^(A)
=
- l o g i o R ^ hn{X)
=-logio(l
^
(lnl0)^c^2.3^
(5.6)
where A/g = Iin — hut- It is clear that the sensitivity (A^) depends on how accurately one can measure the light intensities Iin and lout- If inaccuracy in the intensity measurements is AJ, then the sensitivity is Av4 c^ 2.3 • 2 • ^ = 4 . 6 ^ . Therefore, the sensitivity, A A, is completely determined by the relative accuracy of the light intensity measurements. Apparently j ^ = ^ , where /SU is the minimum resolved voltage deviation on the photodetector output. A ^ 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 ^ can be smaller than 0.00002.^ If i f = 0-00002, then the sensitivity of the instrument is /S.A c^ 0.0001. This is a typical value for a high quality general purpose instruments. Already the value Av4 ^ 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 provide inaccuracy 6 — ^ for the intensity measurements the number of (detected) photons ^Today digital technology is used almost everywhere. The accuracy of 0.00002 means that the signal from the photo-detector must be measured using 16-bit analog to digital converter (ADC) or better.
5.4. Instruments, accessories and applications
99
should be grater than N = (5) = ( ^ ) (see square root law Section 4.1.2). Assuming the photo-detector efficiency to be (j)d and the detection time interval At, the minimum light intensity is / = hv-^^ = -J^^ ( ^ ) ' where hv is the photon energy. For example, if ^ = 0.00002, as was considered above, (pa = 0-1, which is typical for PMTs at the wavelengths of their maximum efficiency, At = 0.01 s, meaning 100 measurements per second, and hu ^ 4 X 10~^^ J (wavelength 500 nm), then the minimum intensity is / ~ 2/iW. This value looks to be rather low, but is it so in reality? Typical a tungsten halogen lamp emits 1-5 mW 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 Ml 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.^ 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 extreme is a highly absorbing sample. Then lout is close to zero but one bit greater than zero, i. e. AU corresponds now to lout- Thus sample transmittance is T = ^ = 0.00002. Consequently, A o^ 4.7. This is the maximum optical density which could be measured with an ideal instrument having detection steps of ^ = 0.00002.^ One may notice, that the increase in the light intensity resolution,i. e. decrease in AU and A / , respectively, extends maximum measurable value of ^ and decreases the minimum resolved value, A A, since they both depends on ^ . 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 instruments. 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 ^One may increase the detection time interval At to achieve the same number of photons with narrower spectrum resolution. However the increase in spectrum resolution result in much faster decrease in the light intensity at the sample because of (1) narrower spectrum, and (2) smaller input slit needed to increase resolution of the monochromator. '^The measurements are not very accurate in this case as the next detectable transmittance value is T = jj = 0.00004. So the minimum transmittance was determined with uncertainty ±100%.
100
steady State Absorption Spectroscopy
spectrophotometer designed for high sensitivity and high resolution appHcations. 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 of Unico 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 measure 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). ^^ Stray light is <0.3% for Unico, <0.03% for Agilent and <0.00005% 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 instruments. 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. ^^ 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 being 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 ^^As it was discussed in Section 5.3.3, typically higher photometric range means higher sensitivity and lower noise. ^^The cross section of the cuvette is a square with 10 mm inner size and 12.5 mm external size.
5.4. Instruments, accessories and applications
101
Figure 5.6: Spectrophotometer cuvettes with length 2, 10 and 50 mm. lOOr
400
500
600
800
wavelength, nm 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 nm.^^ 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 length), 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 solvent or the cuvette itself absorb the light. ^^In the infrared part of the spectrum the range is Hmited by rough 2500 nm, but for special types of quartz (water free) can be extended to 3800 nm.
102
Steady State Absorption Spectroscopy
or when the difference in absorption between two samples is the subject of investigation, as discussed in the following Section. For these reasons the high quality cuvettes are usually sold in pairs.^^ For routine measurements 1 cm cuvettes are the most common. A typical molar absorption coefficient of organic dyes is e ^ 10^ M~^cm~^. 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 ^ ^ = 1 0 ~ ^ M o r l O JJM, see eq. (1.14). This is rather small concentration,^^ and one can routinely use the absorption spectra 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 instance, studying dynamic quenching of a singlet excited state of a dye molecule with the lifetime of the excited state of 10 ns, one may need to prepare a sample with concentration of the quencher molecules 10 mM or higher. ^^ 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 £ ?^ 10^ M~^cm~^ and the required concentration c = 0.01 M, one obtains the the absorbance in 1 mm cuvette A = ed = 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 perfectly 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^^Some manufactures mark the thickness of the cuvette together with the type of material it is made of in the top of the cuvettes. For example, "Q 10.01" means quartz cuvette with inner thickness 10.01 mm. ^"^Tofillin the cuvette one would need 3 ml of solution. For a compound with molar weight 500 a.u. the sample preparation will require just 0.015 mg of the compound. ^^In case of the diffusion limited quenching a typical quenching constant is K ^ 10-'^° M~^s~^, and for the lifetime r = 10 ns the half quenching concentration of the quencher is Cq = TK = 0.01 M.
5.4. Instruments, accessories and applications
103
^°^k'"9 ^"l""^^
buffer ' volume
\
TZr
I
TIf
cuvette base
Figure 5.8: Cross section of a cuvette with very small thickness.
ments 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 sample, 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 scheme over the single channel scheme. A simple spectrum, transmittance and absorbance An example of a transmittance 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 measurement. They can be recalculated one to another using eq. (1.13): A = — logio ^-^^ ^^^ ^ 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 ^^The transmittance in the equation is the ratio of the light intensities, thus changing from 0 to 1. In the graphical presentation, however, the percent of the ratio are commonly used.
Steady State Absorption Spectroscopy
104
400
500
600
wavelength, nm
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 spectrophotometer can be presented as a difference between two spectra A = A sample
A ref
(5.7)
where Agampie is the absorbance of the sample installed in the sample channel and Aref 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) Aref = 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 Agampie = Ac-\-As-\- Apheo, where Apheo is the absorbance of pyropheophytin a. As desired, the measurement has yielded the spectrum of pyropheophytin a, A = ApheoThe spectra subtraction feature of the double channel spectrophotometer has many different 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 spectrophotometers. When the reference sample is inserted for base line recording, the following
5.4. Instruments, accessories and applications
105
• ' I ' • ' ' I ' ' ' ' I ' ' ' '
- • bare ITO — ITO with monolayer 0.05 b
0 0.06
I
I p i
I I I I I I I I I I I I I I I I I I
— glass with monolayer
H0.04
0.02
0
I I I I n'lp'iifn^ ni' I'^l I I I I I I I I I
0.02
— monolayer spectrum
: I I \ „ - - < : ,"•'• i""!' • , I ' r r - r > ^
400
500 600 700 wavelength, nm
800
Figure 5.10: Absorption spectra of ITO, porphyrin coated ITO and glass plates and calculated 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 measurements. 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 porphyrin 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 comphcations 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, SITOWAn example of the spectrum is presented in Fig.
106
Steady State Absorption Spectroscopy
5.10 (the dashed Hne 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 channel), thus giving the spectrum of the modified ITO, SmodiToW- ^ ^ 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 modification, 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 glass W- 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 glass W, 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 spectrum of the glass plate, SmonolayerW = SmodlToW-SlToW-SmodglassW/"^' 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 ^ 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 (Igc ^ A~^), 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 photophysics, the absorption and emission spectra provide information on the electronic subsystem 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-\- hiy ^ 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 + hu)} However, even when the emission and absorption bands correspond to the electronic transitions 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 complementary methods which, being applied together, allow to extract an additional information 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 spectroscopy. For example, the absorption spectroscopy is very useful in determination of concentration of compounds in, e. g. , solution, since the measurements give the absolute value of the sample absorbances. Measuring of absolute emission intensities is technically complicated task, therefore fluorescence methods are not common for concentration determination. 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. ^ Indeed, the relaxation process can be rather complex one and photons can be emitted when the electronic subsystem relaxing form one excited state to some intermediate lower laying excited state. However, in most cases the higher excited states relax quickly to the lowest excited state, which has the longest lifetime among the excited states. The emission of the longest lived excited state is observed experimentally.
107
Steady State Emission Spectroscopy
108
Source of excitation light
Emission detection 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.^ 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, lemi^em) 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 detector, is a function of two wavelengths: excitation, Xex, and emission, Agm, U — U{Xex^ Aem) Photo-excitation is not the only method to generate an excited state but probably the easiest one.
(6.1)
6.2. Fluorimeter
109
Excitation monochromator
Sample M3 S3^
Emission monochromator
^:::;
Control ( unit and display I
-
A
iJ
Figure 6.2: An optical scheme of a steady state fluorimeter. The emission of the lamp is collected by mirror Ml and focused into the input slit SI of the excitation channel monochromator. 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, Xex is kept constant, thus converting the wavelength dependence to a simple U = U{Xem)- Alternatively, one can keep Agm constant and measure dependence U = U{Xex), 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 excitation 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 spectrophotometers (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 buih 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 increase 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.^ 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^ counts per second. At this rate the probability 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 < 10^ s~^. At 10^ s~^ rate during 1 second the signal, i. e. the average number of counts, is N = 10^ counts, and its standard deviation is AA^ = ^/N = 10^ counts (see square root law, eq. (4.23)). Thus, the intensity is measured with relative accuracy S = 0.001, or 0.1%, which is very good accuracy for the emission spectrum measurement, but being used in absorption spectra measurements provides the absorbance resolution of 0.005 (see Section 5.3.3),^ which is rather poor result in comparison to spectrophotometer 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 measured with acceptable accuracy of 1%, while the absorbance resolution is dropped down to 0.05 value, which is unacceptable in most cases.
^ Modulation-synchronous detection technique was used previously, but today the photon counting modules are widely available and reasonably priced. ^ For a typical absorbance of, e. g., 1.0 the accuracy of the measurements is 0.5%.
6.2. Fluorimeter
6.2.2
111
Use of Array Detectors
One of the drawbacks of the fluorimeter shown in Fig. 6.2 is that only the photons emitted in a narrow wavelength region (transmittance band of the detection monochromator, AAem) are detected, but the main part of the emission is rejected by the emission monochromator. 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 semiconductor 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{Xex^ ^em), we need to evaluate (1) the intensity of the excitation light, (2) the emission efficiency of the sample, and (3) the efficiency 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, Xex, and wavelength band, AAe^;. Efficiency of the excitation (e. g. density of the excited molecules) depends also on the sample absorptance, a{Xex)'^ The total intensity of the emission is hm = (phxaiKx)
(6.2)
where cj) is the quantum yield of the emission and lex = hx{Xex^ AAex). The sample emits the light all possible directions (in solid angle An) 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, rjcl 2. the length of the sample, /, and the size of the entrance slit of the monochromator, dem, 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 rjs ^ dem/k,^ where k is the length of the excited area image at the entrance slit. The size of the slit determines monochromator spectrum resolution, dem = bAXem, whcrc 6 is a constant, thus, rjs = bAXem/h^ Note, the difference between the absorptance, a = (lin — Iout)/Un, given by eq. (1.9) and absorbance defined by eq. (1.11). ^ It is essential for this simple estimation that the sample is optically transparent. The light intensity inside the sample decreases exponentially, I(x) = / i n ( l — e~'^^). If the sample has essential absorption coefficient, CKo:; > 1, most of the light is absorbed by the front part of the sample, and the emission intensity decreases essentially through the sample. This is discussed later in Section 6.3.3.
Steady State Emission Spectroscopy
112
monitoring wavelength Figure 6.3: Sample emission bandwidth, A As, and detection system bandwidth, AAe
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 AAem)- Thus, instead of measuring the total emission intensity, the part proportional to the band pass of the emission detection system, AAem, and inversely proportional to the emission bandwidth of the sample, AAg, 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 lem = / iem(A)(iA, and the emission intensity within detection system band pass, AAem, at monitoring wavelength, Aem, is ^em(Aem) AAem- Therefore, for a narrow enough detection band pass AAem, the intensity of the detector light, i. e. the light which will enter the detector, is e
^det
(6.3)
where rjm is the efficiency of the monitoring monochromator. If Zem(A) changes significantly in frame of the detection window AAem, eq. (6.3) must be rewritten in an integral form
J-det
I
VcVsVm / iem(A)r(A)dA
(6.4)
where r(A) is the spectrum response function of the detection channel.^ 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. Therefore it is better to replace it with parameters having better define meaning. The values of interest are the emission quantum yield, (j), 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, lem, by introducing another spectrum function, (f{X^ AAem) so that J^ iem{>^)r{X)dX = Iem^{\em,^Km)' The function (/^(Aem, AAem) defines a ^ The function r(A) is normalized so that J^ r(X)dX = AAe
6.2. Fluorimeter
113
relative part of emission at wavelength A within the band AAgm- In fact, it is the relative sample emission spectrum measured with the resolution AXem and normaUzed so that f (p{X)dX = 1, since the integration of the spectrum should give Iem- To make the equation shorter the second argument and the wavelength subscript will be omitted (^{Xem, AAem) = ^W^ siuce the function depends only on the emission wavelength. Finally, accounting for the detector sensitivity, 5, the measured signal is U{X) = Sr]cVsVmAKmCi(Kx)(t>hx^W
(6.5)
Substituting also the value of 'qs one obtains U(X) ^ 0
\VcbAXl, ~(^{Xex)J^e:
[rjm{X)S{XMX)]
(6.6)
where all the terms are grouped into three categories (from the left to the right): 1. emission quantum efficiency, 0, 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 AA^^); 3. parameters, which depend on monitoring wavelength. One of the primary goals is to measure emission spectrum, however the signal, t/(A), is the product of three spectra, rim{X)S{X)(f{X), from which only thee spectrum (/p(A) is of interest. Hopefully, the spectra S{X) and r]m{X) are smooth functions of the wavelength and if the sample spectrum ip{X) is narrow and sharp, then the measured spectrum U{X) will reflect (more or less) the spectrum of the sample, U{X) ~ v^(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 ^(A) and rim{X) 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{X) — Cr]ni{^)S{X)ip{X), where C is a constant. If a sample with known emission spectrum, ipref{X), is available, one can measure the reference sample to obtain Uref{X) = CrefVmWS{X)(frefWy^ and to Calculate a correction spectrum as
Then the corrected spectra of unknown samples are obtained as Uc{X) = U{X)ScoriX) = j^fiX)
(6.8)
^ref
^ Although C is a constant for a given sample, it depends on the sample absoq)tion, excitation wavelength, etc. Therefore, for the reference sample the constant is marked as Cref^ 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{X), is proportional to the sample emission spectrum, (/?(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 photomultipliers (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.^ 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, 0, 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:
JU{X)dX ^ref^
0
(6.9)
JUref{X)dX where (j)ref is the quantum yield of the reference sample, and U{X) and Uref{X) are the corrected spectra of the sample and the reference, respectively. ^ The instrument allows to record directly the corrected spectrum so that the correction is performed during the measurements.
6.2. Fluorimeter
115
1
1
1
1
3xl0' d 2x10 ^"
-
ixio' n
1
1
1
1
1
/v
^
/ /
1
1
1
1
1
corrected
/\ / \
\\
^^—^
700
1
1
J
J1
\
1
1 , , , , 1 , , ,^ - ^
650
1
. j
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) wavelength. In terms of eq. (6.1) this is the case ofU = U^Xex)- The spectrum obtained by this method is called excitation spectrum.^^ Returning back to eq. (6.5) and collecting up the values depending on the excitation wavelength one can obtain U{Xex)
=
Sr]cr]sr]m^XemCi{Xex)^hx{Xex)^
(6.10)
As it can be seen, the signal is proportional to the absorptance of the sample a{Xex)-^^ Actually, a{Xex) 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 Xem whereas the other does not emit light at this wavelength, then the signal (emission intensity) measured by ^^ In general, the dependence of a system reaction, e. g. fluorescence intensity or photo-current, on the excitation wavelength is called action spectrum. From this point of view the excitation spectra are subclass of action spectra. ^^ When absorption coefficient is relatively low it is proportional to absorbance, since according to eq. (1.13) A = - l o g T = - l o g ( l -a) ^ a l o g e ? ^ 0.43a.
Steady State Emission Spectroscopy
116
emission excitation absorbance
0.8
d 0.6 0.4 0.2
0 k~-'-^'<-'
450
500
550
600
650
700
wavelength, nm Figure 6.5: Corrected emission and excitation spectra and absorbance spectrum of rhodamine 6G in ethanol. The emission and absorption spectra were normaHzed to 1. The absorbance spectrum was multipHed by 30 to fit the scale. The emission spectrum was recorded with excitation at X^x = 520 nm. The excitation spectrum was recorded with monitoring at Agm = 580 nm. Spectrum resolutions were 2 nm for emission and excitation spectra and 1 nm for absorbance.
scanning excitation wavelength Xex at the emission wavelength fixed at Agm 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 Iex{Kx), 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 correction 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 absorption 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, overlap 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).^^ 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 monochromator. 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 commercially 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, Nc, in eq. (6.6) one needs to replace the photomultiplier sensitivity by the quantum efficiency of the photomultipher, r]PMW, and the intensity, lex, must be replaced by the photon flux, i. e. the excitation energy divided by the photon energy ^^ Zj A \ 2
771
Nc = VcrjmWvPM{X)^^^a{Xex)M^)
(6.11)
The term ^^ = Nex 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, IJPM, and losses by the sample, e. g. a weak absorbance and a low emission quantum yield (0). Then eq. (6.11) is simplified to Nc = ViVsNex
(6.12)
where the instrument losses are Vi = VcVmWr]PMW—7-^
(6.13)
and the sample losses are Vs = a(Aex)0V^(A)AAem
(6.14)
^^ Stokes shift is the difference between the wavelengths of maximum emission intensity and maximum absorbance. ^^ 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, k, depends on the sample too but has been attributed to the instrument losses, rji. The instrument losses r]i are relatively simple to estimate as illustrated in Example 6.2. For a carefully designed instrument one can expect r/i = 10~^ ... 10~^, with the strongest contributions (smallest value in the product) coming from inability to collect all emitted Hght. The sample losses vary from sample to sample very much. To make a rough estimation 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 afluorimeter. Let us estimate instrumental losses of afluorimeteraccording to eq. (6.13). An estimation of the efficiency of sample emission collection was done in Example 2.7 on page 38, r]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 working range can be as good as r^^ ^ 0.5. For a good photomultipher one can expect the quantum efficiency to be r]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 ^^^^^ can be evaluated to be 0.1-0.2.^'^ Thus, one can expect the instrument losses to be r/i ^ 10~^.
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 emission quantum yield. The sample losses, r^s, consist of three parts. First of all, the sample will probably absorb only a small part of the excitation, say a{Xex) ~ 0.1. The emission quantum yield for a dye molecule can be taken as (/> ?^ 0.1.^^ The term (f{X)AXem defines how many of emitted photons will fall into the detection wavelength band. For a reasonably accurate spectrum determination v?(A) AAgm ~ 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 r]s ~ 10~^. Combining together the losses estimated in Examples 6.2 and 6.3 one obtains ijiTjs ~ 10~^. If acceptable accuracy of the measurements isS = 0.01 =1%, then Nc = 6~'^ = 10"^ counts must be collected to provide the accuracy (see square root law on page 67). Thus, ^ex = -^~- = 10^^ photons are required for the excitation. If excitation wavelength is Xex = 400 nm, then the photon energy is /IT; :^ 5 x 10~^^ J, and the excitation energy is Eex = 0.5 /xJ. 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, respectively, one can expect to obtain excitation density of lex ~ 270 x 0.006 ^ 2 mW/nm. ^^ This is roughly the ratio of the detection monochromator input slit size to the length of the excited area of the sample. ^^ 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 1-50%, 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 measurement in time interval tac = j ^ ^ ~ 10~^ s = 100 JUS, which is extremely short time considering normal expectation of how fast emission spectrum can be measured. ^^ 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 rfirfs by many orders of magnitude. Nevertheless, the accumulation time can be increased too by many orders of magnitude, e. g. to tac = 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~^, or if the sample absorption coefficient is as low as 10~^. 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 AAabs ~ 2 X 10~^ at its best. This corresponds to the absorption of a = 1 — iQ-^^abs r.. 4.6 X 10~^. In emission spectroscopy one can measure emission or excitation spectrum with inaccuracy of 1% for a sample with absorption of 10~^ only. Thus, in fluorescence measurements 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 usually 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 or/and available at a very low concentration only, one can consider the following measures to improve the signal intensity: 1. Increase accumulation time, tac- For example, in 4 times longer time interval the average number of collected photons is 4 times greater and signal-to-noise ratio is \/4 = 2 times better. 2. Increase slits of the excitation monochromator, AAe^- This will increase excitation intensity, lex, which is roughly proportional to AA^^ 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, AAem- The signal is proportional to AAg^, see eqs. (6.6) and (6.11). Evidently, this wiU result in worse wavelength resolution and some important features of the spectrum may be lost. ^^ This is a rather theoretical value since for the conditions considered the photon counting rate will be Nc/tac = 10^ photons/s, which is greater than the maximum counting rate of the modem photomultipliers (typically < lO^photons/s). Therefore the practical measurements will be carried out with lower excitation intensity, e. g. narrower slits of excitation monochromator, and will take longer time, e. g. 1 ms. 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, Iex A. 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 desired 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 scattering and its relative amount is small (usually less than 10~^ 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 technique is very high dynamic range and high linearity of the response. A good photomultiplier may have dark counting rate smaller than 10 counts/s and the maximum counting rate greater than 2 x lO'' counts/s, which covers variation in signal by more than six orders of magnitude and provides linearity (accuracy) better than 0.1%. 6.2.8
Wavelength resolution
An increase in wavelength resolution (decrease in AAem) results in a fast decrease in the signal intensity (eqs. (6.6) and (6.11)). Consequently, the highest possible resolution depends 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 Section 6.2.7, one can conclude that in order to increase the resolution by factor 100 (so that AAem ^ 0 . 0 1 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 AAem? therefore 100 times increase in wavelength resolution will lead to 10"^ times decrease in counting rate and will require 10^ times longer measuring time to collect the same number of counts. This accumulation time, 10 s, 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 photomultiplier dark counting rate.^^ 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 counting rate of the detector can be measured separately and subtracted from the collected counts ^^ A typical dark counting rate for photomultipliers equipped with S-20 photo-cathode (sensitive up to 850 nm) is 100 counts/s. The dark counting rate can be reduced gradually by cooling the photomultiplier photo-cathode down by 40-50 °C.
6.3. Samples for emission measurements
1)
excitation
2)
excitation
121
3)
excitation
^ emission A
•
sample
Figure (i.(y\ 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. Assuming the resolution AAgm ~ 0.01 nm, the slits size d = 10 // (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 g = 1200 mm~^ (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 ^ -^j^— ?^ 1 m. Thus, for 0.01 nm resolution one have to build 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 6.3.1
Samples for emission measurements 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.
Steady State Emission Spectroscopy
122
emission monitoring area
cuvette Figure 6.8: Effect of sample absorption on the intensity of the monitored emission, Jg
6.3.2
Cuvettes
A typical cuvette for the steady state emission spectroscopy 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 emission 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 emission cuvette can be used to measure absorption spec- sion spectroscopy. trum, but some attention 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
From 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 sample, rather than incoming beam intensity, lin, determines the measured emission intensity, as illustrated in Fig. 6.8. The measured emission intensity is proportional to the amount of absorbed excitation only inside the monitoring area, Iem ^ Slabs' 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 Iex- 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~^.^^ The efficiency of the Raman scattering is rather low, typically 10~^, 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 presented 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^ and 1.3 x 10^ 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. ^^ Raman scattering is the result of the photons interaction with vibrational modes of the matter. For excitation frequency Up^ and vibrational mode at frequency u^ the Raman scattering appears at frequency upi = Vp^ — ^v
Steady State Emission Spectroscopy
124
^ 4x10'
3.60
380
400 420 wavelength, nm
440
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 QuantaMaster QM-2/2005 (PTI Inc.) fluorimeters. Actually, these devices are not used to measure 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 monitoring 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 modules 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 consider 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, sample 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 channel 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 of FS920 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 Czemy-Turner type with 30 cm focal length. The spectrum resolution of monochromators is 0.05-18 nm adjustable with 0.05 nm steps (wavelength accuracy is ±0.2 nm). The monitoring spectrum range is determined by the detector and can be 185-680 nm (standard photomultiplier,
6.5. Emission of molecular monolayer: An example
125
n—^—^—^—^—\—^—^—^—^—^—
Absorption
400
500 600 wavelength, nm
700
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^.
6.5
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 matrix 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%. The 10% concentration corresponds to density of roughly one porphyrin molecule per 2 nm^, and 0.1% to one molecule per 200 nm^, respectively.^^ 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^ M~^cm^, 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 ^^ To some degree, the porphyrin can be treated as a flat square-Hke molecule with the size of roughly 1.2 x 1.2 X 0.5 nm^.
Steady State Emission Spectroscopy
126
n—^—^—^—^—\—^—^—^—^—r
0.004
^
Absorption
0.002 0 -0.002 P \I
— I\ — \ \ — I \ — I \ — \ h— — \ —\ I \ — \ \ — \ \ — \ \ — \ h— \ — \ — I — I — \ — I — I — \ — \ — h - ^
3000
400
500
600
700
800
wavelength, nm Figure 6.11: Absorption, emission and excitation spectra of 0.1%
(molar) porphyrin film.
the Q-bands also can be noticed at 505 nm, but its intensity is already close to the sensitivity limit of the instrument. The emission efficiency of the porphyrin is 10% 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.^^ The maximum counting rate was higher than 10^, which gave measurement accuracy better then 0.3%. The excitation spectrum was recorded with monitoring 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 provide 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 spectrophotometers (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 information 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 concentration was reduced 100 times. From the technical point of view the measurements of the porphyrin emission and excitation spectra is a relatively easy task since the excitation and monitoring wavelengths can ^^ The emission spectrum differs significantly from that in solutions, which is due to special organization of this porphyrin molecules in Langmuir-Blodgett films [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 a0 = (1 - IQ-^-^^^^) x 0.05 ^ 5 x 10"^ At low emission intensity the main problem comes from the stray light passing the excitation monochromator and from the selectivity of the monochromator.^^ 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.
^^ when a monochromator resolution is set to, e. g., AAem = 2 nm, the Hght at wavelength shifted by 1 nm (half of 2 nm) from the monochromator maximum wavelength will be attenuated by roughly two times. Moving further away from the maximum wavelength will reduce the intensity of passing light more and more. However there is no sharp border at which the monochromatic light will be blocked completely. Usually a shift by 10 • AA from the excitation wavelength is enough to measure an emission spectrum, but this depends on the sample 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 significant 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 Norrish 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 Imown 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 population hiding any faster intermediate products. The pulsed excitation and the flash-photolysis 129
Flash-photolysis
130
Pulsed excitation source
VV Lamp
L2
LI monitoring monochromator
sample
^^ detection monochromator Transient recorder (oscilloscope)
Figure 7.1: Scheme of the flash-photolysis method. PD is a photo detector and LI, 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 discussed 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 spectrophotometers. For this reason the signal from the photodetector 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 monitoring 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 required for the experiments. A reasonable limit in time resolution for the flash-photolysis method is 1-10 ns, 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, /(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 /(t,A)=/,,(A)10-^(^'^)
(7.1)
where lin and / 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{X) -\- AA{t, A), where Ao{X) is the sample absorbance before the excitation and AA(t, A) is the absorbance change due to some photoreaction. The value AA{t, A) is called differential absorbance.^ Then /(t,A)
=
/,,(A)10-^°(^^-^^(^'^)
= /,,(A)io-^°(^ho-^^^*'^^ =
/o(A)10-^^^''^^
(7.2)
where Io{X) = /in(A)10~'^°*^^^ is the monitoring light intensity after the sample at some time before the excitation. The absorbance change is AA{t,X) = -log,,(^^^
(7.3)
AA{t,X) = -\og,,[l
(7.4)
or +^ ^ )
^Suppose the ground state absorbance of a molecule is Ao(A) and the absorbance of the excited state is Ai(A), then the excitation induced change in absorbance is AA(A) = ^ i ( A ) — Ao(A), which is difference in absorbances of two states, or differential absorbance.
Flash-photolysis
132
b-
,
excitation
1 flash \^U{t)
TJ ^U
T 0
—
1 hrig
' >
•
^
Figure 7.2: A signal recorded in flash photolysis experiment. The signal before the excitation is [/Q- The excitation flash time is assigned to t = 0.
where A/(t, A) = /o(A) - /(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 ^ = ^ , where UQ is the detector signal before the excitation and MJ 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, /7o, which corresponds to /o(A), then to measure the time dependence U{t) in response to the excitation flash and to calculate the difference AU{t) = f/o — U{t). After that the differential absorbance can be calculated as AA(t,A) = -logio
-
^
)
(7.5)
In fact, one can avoid separate measurements of Uo. 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 = 0. The value UQ can be calculated as the signal average before the excitation, at t < 0, u
Uo
— T^trig
/ U{t)dt
(7.6)
J ttrig
Then the signal is recalculated as AU{t) = UQ — 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 (AQ{X)).
7.1. Principles
133
100
350
200 300 time, \xs
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 pyropheophytin a solution. For comparison the steady state absorption spectrum of the sample (normalized 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 AA{t^ A), which can be used to draw time resolved differential absorption spectra, AA{t = const, A). As an example let us consider flash-photolysis studies of a solution of pyropheophytin a^ 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 UQ was calculated by averaging the data at t < 0 (before the excitation flash) and then the transient absorptions, A 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 AA{t, A). The spectrum right after the excitation, AA{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 ^Pyropheophytin a is a derivative of pheophytin a, which is one of the natural chromophores 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 resuhs 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* "^^ P, where P and P* are the ground and excited states of pyropheophytin a? The amplitude of the instant change in absorbance (due to excitation pulse) is /\A{X) = {AUX)-Agr{\))^
(7.7)
where A^^x and Ag^ are absorbances of the excited and ground states, respectively, and 0 is the efficiency of the excitation or the relative fraction of the excited molecules. The shape of the /S.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 ^e.(A) = A,.(A) + ^
^
(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{\)) are sufficient to establish the mechanisms of photo-reactions. ^The excited state observed in the experiments is the triplet excited state. The singlet excited state is formed upon the photo-excitation of pyropheophytin a, but the lifetime of the singlet excited state is roughly 5 ns and the singlet state cannot be resolved in the time scale of the presented measurements. The efficiency of the intersystem crossing (conversion from the singlet to the triplet state) is rather high for pyropheophytin a (~ 90%), resulting in quantitative formation of the triplet excited state observed in microsecond time scale.
7.1. Principles
135
Excitation beam Monitoring beam overlap area
Figure 7.4: Perpendicular orientation of excitation and monitoring beams (T-scheme).
Usually the transient absorption curves (AA{t)) are used to calculate lifetimes or rate constants. In turn, the time resolved differential spectra (AA{X)) are useful for identification of the intermediate 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 a T-scheme. The samples are prepared in cuvettes with 4 clean v^alls 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 sample, 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.
Flash-photolysis
136
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 sample. At small excitation densities, an increase in the excitation pulse energy will increase the induced absorption change, AA. 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 Egat > huex/o', where a is the absorption cross-section of the chromophore at the excitation wavelength, X^x = -;f— • For typical organic chromophores the saturation energy density is Esat — 1 . . . 10 mJ-cm"^. The following example presents an estimation of the saturation density for chlorophyll a, which has molar absorption coefficient £ ^ 10^ M~^cm~^ at 430 nm. Example 7.1: Estimation of excitation efficiency for chlorophyll a. Chlorophyll a has the absorption cross-section cr ^ 4.6 10 -16 cm^ at Soret band maximum, 430 nm (corresponds to e '^ 1.2 x 10^ M~^cm~^). Thus, the excitation saturation energy density at 430 nm is Egat = huexc/o' ^ 1 mJ/cm^. At this excitation density the fraction of the excited molecules is 1 — e~^ ^ 0.63. Two times stronger excitation density of 2 mJ/cm^ will increase the fraction of the excited molecules to 1 — e~^ ^ 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^. 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.^ Finally the compound under study may have molar absorption coefficient much lower than used in ^ As a typical example one may want to measure the recovery of the ground state, and to choose the wavelength of the maximum absorbance for monitoring. Then it is reasonable to shift the excitation to another wavelength to reduce the effect of the excitation flash on the detection system as it was done for measurements shown in Fig. 7.3.
7.1. Principles
137
Example 7.1 (1.2 x 10^ M~^cm~^). 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 instruments 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 XeCl laser 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).^ The efficiency of the dye lasers is relatively high, 10-30%, 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 exceUent energetic parameters.^ 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.^ ^To switch form one dye to another, at least the dye cuvette must be changed. At most resonator mirrors have to be switched to fit to another lasing wavelength range. In the latter case the resonator adjustment must be checked and tuned. ^High density of active centers (Ti) in sapphire crystals allows to obtain high output pulse energy from relatively small crystals. ^For example, if at the fundamental 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 parametric frequency conversion will double this value at the best. So the output pulse-to-pulse stability after OPO is 6% in an ideal
138
7.2
Flash-photolysis
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 flashphotolysis 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 6 and with time resolution r. This means, that in time interval r the photomultiplier (PM) should detect 1/(5^ photons, i. e. the photon detection rate must be (^^r)~^. Accounting for the quantum yield of the PM, (pd, and the registration monochromator efficiency, (j)m2, the monitoring beam photon flux after the sample should be {S'^(f)dm2T)~^- Then, we have to add the sample transmittance, T, efficiency of the monitoring monochromator, 0 ^ i , and losses in all the optical components (like lenses, monochromator slits and so on), which will be called optics efficiency (po-^ Finally, an estimation for the light intensity required at the entrance of the monitoring monochromator is J
^
'^^rnon
. ^ g.
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 photodetector is used, e. g. a photodiode, the equation must be corrected. Formally this can be done by taking proper value of 0^. The inverse value, 1/0^, 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 (j)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 5 = 0.01, the monitoring light intensity must be > 1 mW 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~^ 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. ^In case of losses in lenses, the optics efficiency is just transmittance. However, loses due to mismatch in the monochromator slit and beam size are due to efficiency of the beam collimation and they are more essential in practice. Therefore the term "efficiency" is used here, although for lenses efficiency and transmittance are equal to each other. Indeed, the transmittance "multiplication law" must 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.^ Example 7.2: Monitoring beam power. Let us estimate the monitoring lamp power (/mi) needed to achieve r =1 ns time resolution at (5 = 0.01 = 1 % accuracy in otherwise favorable conditions for the measurements. The quantum efficiency of a photomultiplier can be (j)d = 0.1, which is close to the top value for most photo cathodes. Efficiencies of the monochromators can be 0^1 = 0.5 and 0^2 = 0.5. The losses in optics include reflections from the sample cuvette walls, reflections from all lenses (LI, 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 0o = 0-5. Finally, the sample absorbance can be assumed to be A = 0.5, which gives sample transmittance T = 10""^ ^ 0.3. Substituting all the values in eq. (7.9) we obtain /ml ~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 mW 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 spectrum density of a Xe arc lamp can be as high as 0.3 W nm~^ (see Example 1.4 on page 11), and one can expect to collect about 3 mW 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 density 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. ^^ High monitoring light intensity needed for the fast measurements creates yet another problem. Namely, the monitoring light can excite the sample to extend which cannot be neglected any more. Example 7.3 shows how to estimate the relative population of the excited ^Assuming that 1% of emission is collected at the entrance slits of the monitoring monochromator, the bandwidth must be extended to 14 nm to reach 1 mW monitoring intensity. However, taking into account rather idealistic assumptions made in Example 7.2, real band pass must be a few times greater, which seems to be unacceptable for most practical applications. ^^The lasers have disadvantage of much narrower tuning range as compared to the combination lampmonochromator, not to mention 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 monitoring light intensity must be lower than 1 mW in typical experiment conditions. This is in contradiction with requirement of the high monitoring intensity needed to achieve nanosecond 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 excited molecules of a chlorophyll a solution under continuous irradiation by a monitoring 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, £ c::^ 1.2 X 10^ M~^cm~^, i. e. absorption cross section is cr ~ 4.6 • 10~^^ cm^. The longest living excited state of chlorophyll a is a triplet state, which has lifetime roughly r^ ~ 1 ms.^^ Let us assume that the monitoring light intensity is / = 3 mW (to achieve ns time resolution), and monitoring beam area is s = 0.1 cm^, thus the monitoring power density is P = | = 30 mW cm~^. For a simple estimation the reaction scheme is C —^ C^ ^ ^ C, where C and C^ are the chlorophyll ground and triplet excited states, respectively. The kinetic equation for the case is at
T
hv
where Ng and Nf are the number of molecules in the ground and excited states and Ng -\- 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 -^ = 0, thus ^ = NgCj^ and the ratio of populations is ^ ~ ^^^ ~ 0.03, i. e. under such conditions 3% of chlorophyll molecules will be continuously in excited state. Equation (7.9) can also be used to analyze the accuracy of the measurements, S, (or signal-to-noise ratio, which is S~^) after some rearrangement 5^ = ^ x — - ^ — - x l
(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 r, will give a decrease in the measurement accuracy, S ^ T~^ . ^^ There are many mechanisms leading to a quenching of the triplet excited state, such as triplet-triplet annihilation and quenching by oxygen molecules dissolved in solution. Therefore, concentration of chlorophyll must be low enough and the solution must be de-gassed to reach a millisecond lifetime of the triplet excited state.
7.2. Time resolution and signal-to-noise ratio
141
1 excitation 1 flash ^X ^ signal
i
c
A ^—^N
s
/ ' measuring ' / time window
bD
bD _C
^
duty
1 /
O
arc
J
'
'
\ \
\ \ ^ ^ ^ ^
a lamp triggering 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, (j)d. Decrease in sensitivity 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 resolution 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. ^^ 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. ^Typical color temperature of Xe arc lamps 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%. 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) = ^ ^^L, is normalized signal with "compensated" monitoring pulse shape, which can be used to calculate differential absorbance using eq. (7.5) and the procedure 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 decrease 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 procedure 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 resolution. To block totally the monitoring light intensity between the measurements a mechanical shutter can be inserted between the sample and monitoring monochromator.^^ The shutter control unit must be synchronized with the lamp and excitation laser control systems. Typical 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 /iS 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-tonoise 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. ^^ Usually the shutter is fixed on the output slit of the monitoring monochromator. ^^The development time of the monitoring light pulse depends on the design of the lamp power supply and can differ gradually from the value mentioned 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. ^^^^ in eq. (7.11)) N times, thus decreases 6'^ N times, i. e. improves signal-to-noise ratio A/TV 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, ^^"^^ , in time interval equal to the instrument time resolution, r. 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 spectrum resolution means lower total intensity of the monitoring beam. Second, the monochromator 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 monochromator 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, / ^ ^ ^ ^ m ' and, thus, signal-to-noise ratio to be directly proportional to the wavelength resolution, 6~^ r^ AXm, in otherwise equal conditions (see eq. (7.11)). Similarly, the time resolution is proportional to the square of the wavelength resolution, r ^ AA^.
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. ^^ 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 rejection 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^. 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 rejecting 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. ^^ 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 emission, 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, Uhg{t). Then the measurements are repeated with monitoring light on, Ue{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) — Ui)g{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 nonlinear response of the detection system.
^^ 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-scheme one can place a diofragmdiaphragm in front of the cuvette to cut away the excitation beam part hitting the walls through which the monitoring beam is passed. ^^For example, if a photo-reaction of interest has time constant of 1 /is and the response time of the instrument is 5 ns, one can start measurements at 50 ns after the excitation flash. At this delay time only 5% of reaction has gone but the scattering should already have no effect.
7.4. Flash-photolysis instruments
7.3.2
145
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) ^ 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 resolved emission spectra of the sample. A spectrum calibration procedure must be performed to correct the spectrum efficiency of the monochromator and sensitivity of the photomultiplier 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 7.4.1
Flash-photolysis instruments 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), quasiparallel (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 UVvisible 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 coupled 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 Nd: YAG laser with harmonic generators is provided, i. e. the excitation wavelengths are 1064, 532, 355, 266 and 213 nm. The excitation wavelength 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 comprehensive software package to collect and analyze the data. To build up a home-made flash-photolysis instrument one can purchase all the hardware components needed for system. Photo-detectors, digital oscilloscopes, lasers and monochromators are available form a variety of companies. Pulsed arc lamps are also produced 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 assembling 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 instrument 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 porphyrin (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 flashphotolysis 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 (Ml 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
Nd:YAG
Ge-APD
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 oscillator, PD - photodiode, Si-APD - Si avalanche photodiode, Ge-APD - Ge avalanche photodiode. Ml, M2, M3 and M4 - monochromators, SI, S2, S3 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.
Flash-photolysis
148
600
800
1000
1200
Wavelength / nm Figure 7.8: Transient absorption spectra observe by 530 nm laser excitation of Ceo (0.1 X 10~^ mol dm~^) in the presence of CoOEP porphyrin (0.1 x 10""^ mol dm~^) in Ar-saturated benzonitrile. Inset: Time profile at 740 and 1080 nm. The figure was published 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 //s delay time are presented in Fig. 7.8 for a solution of CoOEp porphyrin and fullerene Ceo • 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 fis. At 1 /iS delay time one can expect to see the differential spectrum of the reactant state mainly, and at 10 /iS 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 reactant, P is the product and r is the reaction time constant. How one should select the delay times to observe the differential spectra of the reactant and product separately? 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 > r the populations [R] < [P]. For reaction with time constant r = 3 ytxs at delay time t = 1 //s, the populations are [R] = e~ 3 ^ 0.72 and [P] ^ 0.28, thus the reactant population dominates that of the product. At delay time t = 10/xs the populations are [R] = e ~ ^ ^ 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, ^C^Q, which dominates at 1 //s delay. At 10//s delay a fullerene anion radical, Cgg, band is formed at 1080 nm. These observations leaded authors to the conclusion of the photo-induced intermolecular 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.^^
^^The 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 lifetime of 1.4 ns and relaxes to the triplet state via the inter system crossing process. This reaction was not time resolved in this experiments as the excitation pulse duration was 6 ns, but it is well-known from the time resolved fluorescence measurements (using technique described in Chapter 8) and pump-probe measurements (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 domains. 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 advanced 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.^
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.^ 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 ^Time resolved fluorescence spectroscopy has numerous application. There are excellent books available devoted to the subject, one can refer to e. g. [10, 11]. ^Different types of excitation pulse sources are discussed later in Section 8.2.
151
152
Time correlated single photon counting
Excitation source
Sample
Ml
Light pulse generator
Detection subsystem
n\
laser pulse A "start" t the first photon A "stop"
(2) PD
Monochromator
(3) TAC CFD
t
^ A^ ^
PM
/
,-f U(M)
CFD +1
TAC I—I stop start I—I TAT U(M) MCA
(4) MCA
\W\\XT^ n
Figure 8.1: A scheme (on the left) and time diagram of time-correlated single photon counting (TCSPC) method. M l 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 "starf 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, [/, is determined by the delay time, At, between the laser pulse and the pulse produced by the first detected emission photon, {.Q.U — a At, 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 < Ui < Ml, channel 2io MI
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 ) ^ ^ t < n ^ . In other words, the memory of MCA collects emission time profile measured with discreet time step ofAi=^. 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 certain 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 Ml, the TAC dead time and desired delay for the measurements.^ 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 *" — 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^-10'^. ^ 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, r = 2.13 ± 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.^ 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 10 ps 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 1050 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 /is to let the excited state relax, and the repetition rate must smaller ^Typical 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 must be / ^ 0.8ct = 4.8 m. Note that studying a nanosecond decay one may need to fit the cables with accuracy better than one meter. ^It should be noted that if there is no stop pulse after the start pulse in the time interval of the measurements, the MCA skips the event. ^The excitation pulse energy estimation will be given in Section 8.4.3 and a typical values can be found in Example 8.2.
Time correlated single photon counting
154
10000
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 monoexponential fit by the solid line. The decay lifetime is r = 2.13 =b 0.03 ns and the fit weighted mean square deviation is x^ = 1.27. The plot on the bottom shows fit residuals: weighted difference between the measured data and model decay.
than 0.5 MHz.^ 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 arrangement 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.^ 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 ^At 2 lis after the excitation e~4ions ^ 0.008 = 0.8% of initially excited pyrene molecules will remain excited, which is too high value if desired measurement accuracy is 1%. One may want to increase the time interval between the pulses even further more. ^It should be noted that the ranges 500-700 and 260-350 nm, respectively, are covered by a few different dyes. Therefore, tuning from e. g. 570 nm to 650 nm can be a complicated process, which may include readjustment of the laser system.
8.2. Excitation sources
155
dye lasers are equipped with cavity dumpers in place of output coupler. The cavity dumper works like 100% 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 acoustooptic 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. Using this method the repetition rate can be reduced to any desired value, typically to 14 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 generate 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.^ 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 emission 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~^^ 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.
^The efficiency of Ti:sapphire lasers is much higher than that of the dye lasers. Therefore there is no need to accumulate pulse energy inside the cavity using cavity dumpers. The pulse pickers are installed outside the laser and they are easier to use than cavity dumpers.
156
Time correlated single photon counting
Table 8.1: Comparison of main TCSPC characteristics of a micro-channel plate photomultiplier 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
8.3
R3809U-50 25 ps 100 s-i 5s-i 1 cm 160-850 nm 20%
PDM-50 250 ps 1500 s-^ 75s-i 50/i 350-950 nm 47%
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.^ 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 microchannel plate devices. Nowadays avalanche photodiode (APD) assemblies are available for TCSPC applications. 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 little 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 multichannel analyzers (MCA) are available in different combinations and from different manufactures. The cheapest solution can be a computer board which combines all the electronic ^See Section 4.2.2 on page 74 for the description of the operation principles of MCP 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^ 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 complete devices or as separate parts which can be interconnected to form a complete instrument. For example, PicoHarp 300 stand alone TCSPC module (PicoQuant GmbH) has electrical time resolution <10 ps, maximum counting rate 10^ s~^, 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 <10 ps 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 probabiHty theory and, namely, Poisson distribution, i. e. eq. (4.11).^^ Suppose the average number of photons per excitation pulse detected by an ideal device which has no peal-up distortions is n}^ Then the probability that no photons are detected after single excitation pulse is PQ = e~^, _
2
_
the probability of one photon detection is Pi = ne~^, of two photons is P2 = ^e~^, 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 + P2, since two photons will be counted as one, while the actual number of photons is Pi + 2P2. The difference between the counted and actual number of photons is P2 and the relative mistake in photon counting is
since n ^ 1, so 1 + n 2± 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 \% ox 5 = 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 < Ttfex — 20 000 counts/s. From the point of view of quantum noise statistics the measurement accuracy of 1% is achieved when (0.01)~^ = 10 000 photons are counted. Therefore, the photon or Poisson noise will dominate while the signal intensity is lower than 10 000 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 TQ and the time per channel is At, then the probability that the detected photon will fall in to the channel of maximum intensity is r/t — —}^ The counts ^^See also Appendix A for more detailed discussion. ^^For the sake of shortness and simplicity the we will assume here an ideal detector with 100% efficiency, so that the numbers 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. ^^This is true if At < TQ and TQ is greater than the time resolution of the instrument.
8.4. Method characteristics
159
at the channel of maximum intensity will grow r]t times slower than the total counting rate. At can be considered as an additional loss term to be added to eq. (6.11), The ratio iClLHJ rjt I It =^ ^ then Nch = VcVmVPM
.
aNex
k
W
(8.2)
To
where Nch is the number of counts in the channel of maximum intensity, N^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, At, - the shorter time the lower counting rate. One may also conclude that the counting rate depends on the lifetime of the excited state TQ. This is not quite correct. As an example let us consider fluorescence decay measurements of some molecule. If the molecule has the fluorescence emission rate kern and the lifetime of the excited state is TQ, then the fluorescence quantum yield is 0 = kem'T'o-^^ Thus, ^ = kem, and eq. (8.2) can be rewritten as Nch = ricrimVPM—j-^^aNcxkem^t^
(8.3)
The emission rate constant, kem, 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, TQ, and the quantum yield of the emission, ^, 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, 0 and TQ are not independent values but their ratio is a constant, ^ = kem- The counting rate at maximum is proportional to the radiative constant, as indicated by eq. (8.3), rather than to the decay rate of the excited state, TQ"^. 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 Uc (in all channels), one can use eq. (6.12) after replacing the counts by counting rate^"^ ric = ViVsTlex
(8.4)
where Uex is the average flux of the excitation photons. In the channel of the maximum intensity the counting rate is smaller by factor rft = ^ , and the counting rate at the maximum channel is At rich = — n c = VtViVsriex
(8.5)
^0
^^In addition to radiative decay (kem) there are usually non-radiative decay channels. If the relaxation rate via non-radiative channels is knr, then the total relaxation rate of the excited state is ko = knr + kem = (TO)~^. The fluorescence quantum yield is (j) = - ^ = j , — ^ — = kem^o. ^^Formally both parts of eq. (6.12) can be divided by an observation time thus giving counting rate instead of number of counts in both sides of the equation.
160
Time correlated single photon counting
Although formany r/t can be included in to instrument losses rji, 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 conditions as were considered previously in Examples 6.2 and 6.3 (i. e. rjiTjs = 10~^) and desired photon counting rate to be Uc = 10"^ s~^, the required excitation photon flow is Hex = ^i^^ = 10^^ s~^. At the wavelength 400 nm this means the average power of 5 X 10"'' W = 0.5 /iW. At pulse repetition rate of 1 MHz this requires pulse energy of 5 X lO"^"^ J (or 10^ 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^ 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 experimental conditions: the excitation pulse repetition rate is / = 1 MHz, the average counting rate is Uc = 4000 s~^, the emission lifetime is r = 2 ns, the time resolution (time per channel) is At = 25 ps and the desired signal intensity at maximum is J^max = 10 000 counts, i. e. the measurement inaccuracy is 1%. Then, the relative number of counts in time window of At at the channel of maximum intensity is ^ , thus the counting rate at this channel is Umax = ^ c ^ , and the time required to collect Nmax couuts is t = ^^^^^^ = ^^x,^ = 200 s or 3 m 20 s. From the point of view of the TCSPC instruments, the counting rate depends on excitation 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).^^ 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. ^^ 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 discrimination 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 technologically 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? ^^ Increasing the time per channel one can reach the limit when the emission lifetime is equal to the time per channel. Further in time increase will not increase counting rate. This is, however, very unpractical experimental condition. ^^The 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 transmission of the plate changes smoothly 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. <100 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.^^ 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 8.5.1
Measurements and data analysis 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 deltapulse excitation (theoretical one) is / ( t ) , then the experimentally measured signal is given by the convolution integral t
s{t)=
f r{T)f{t-T)dT
(8.6)
^^For picosecond milli Joule pulse generation a typical approach is to start with high frequency low energy pulses, pick up pulses at lower frequency and amplify them to milli 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 exponentials, f{t) = ^a^e~^^*, then convolution integral is p
F{t)
=
-L n
=
n
/ r(T)^Oie-'='^(*-^)dT =
»=i „
X^ttz / r{r)e-^^^'-^Ur
(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, ki.. .kn, and amplitudes, a i . . . a^, 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 thefitwhich 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 r 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 r = 2.13 =b 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. ^^
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. ^^At a reasonably large number of counts the square root of counts gives an estimation for the standard deviation, see Chapter 4.1.2 for more information.
Time correlated single photon counting
164
(a) 140
1
1
1
1
1
1
1
1
l_J_
^^
120 100 80 60 40 20
/Y
1
1
1
1
1
1
1
1
1
^^ f=Ons
1 ns X^
-
^\
_ X
""•• * ^ ^ ^
ijli-^^r^-^^t^rTT
450
•00
(b)
1
1
1
1
5 ns \ / ^ > s ; 1 1 1 1 1 1 1 7";^r*f
1
1
1
1
1
1
,
100 -\
600
500 550 wavelength, nm — 440 nm - - 480 nm
s-..
' :
•••• 5 2 0 n m • - • 560 n m
10 ~"~~-^^^:^-: =
1
1
1
1
1
1
1
1
1
1
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 collect 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 biexponential model /(t, A) = A(\)e~^
+
B{\)e~^
(8.8)
where TA and TB are the lifetime of the states A and B, respectively, and A(X) and B(X) 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) and B(X) 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 emission spectrum of the sample can be obtained by integrating eq. (8.8)^^ CO
(A)
(8.9)
It should be noted that the steady state spectrum of only A is TAA{\) and only B is TBB{\), 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.^^ Similarly the choice of the detector depends on the desired emission range and time resolution. For the best time resolution the microchannel 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. ^^The integration can be easily expended to any number of exponents, /ss(A) = ^ riai{X), where r^ and ai (A) are the Hfetimes and the corresponding decay associated spectra. ^^ At present, semiconductor pulsed laser heads are available for the excitation in the red part of the spectrum 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 spectrum the prices were a few times higher. In cases when the excitation pulse energy must be higher than a few pJ a more expensive laser system must be used, which increases the price by more than one order of magnitude.
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 counting 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. r]ir]s = 10~^) 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 / = 10 MHz,^^ the total counting rate is n = frjiffs = 0.1 s~^, 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 (rji) 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.^^ 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 Department 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 published in ref. [12] will be used here, the samples for the study were hexaphenylbenzoneperylenemonoimide imbedded in a polyvinylbutyral polymer films of 10-30 nm thickness. ^^ A higher repetition rate is hardly reasonable since (1) it is already close to the dead time of TAC-MCA detection part, and (2) it must be at least few times lower than the relaxation rate of the molecule, which is 2-10 ns for most organic dyes. ^^This is not a practical measure when emitting area of the sample has a millimeter size, as the output beam in this case will have either too big diameter or divergence.
8.7. Measurements of single molecule: Application example
167
Setup Frequency Doubled Ti:Sapphire Laser
IMAGING
OLYMPUS IX-70
ACTON Polychromator
Us
LN CCD MM
Spectra
SPAD
H SPC 630
Figure 8.4: Spectroscopic setup used for the TCSPC measurements of single-molecule fluorescence decays based on a sample scanning, far-field, confocal epifluorescence microscope. The figure was published in ref. [12], kindly provided by Prof J. Hoflcens, and reproduced here by permission of The American Chemical Society (ACS). © 2001 ACS.
To prepare such films the dye concentration was 10 ^ 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 nW, 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 molecules 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~^. 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
o
T i m e / ns
k i^fi/IA At\Ml^'i\i\kr4kihAl\i
\|'^^wyvrY|v\/M fy
\v\j\N m \i/ n " v \ /
j^- A^-^
0.4
; ,!aj
o CO
0.0
(c)
(Ifllpi^jl^^
-
1 00 t 1
5
°
^ i ?
10
15
20
;
tim e / ns
1 0 [ 1
T
\
%,'
= 4 . 2 9 ns = 0.85
N
= 644
LS
', 1
^
g
1 \-
1
0
15
r^^-
1—
1-
20
T i m e / ns
Figure 8.5: Examples of single molecule fluorescence decays. Reproduced from Fig. la and Ic 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 utiHze a short hght pulse for the sample excitation. Should the excitation be necessarily a pulse? According to the signal processing 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 examined 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 (5-pulse^ excitation is f(t), then Fourier transform of function/(t),
F{u;)=
/ f{t)e''''dt
(9.1)
gives the response of the system to a harmonic wave excitation at angular (or circular) frequency uj} One can measure the response F{uj) in a wide enough frequency range and then calculate the response to the 5-pulse by applying inverse Fourier transformation to function F(u;), CX)
m
= ^
/
F{co)e-^"'dt
(9.2)
^(5-pulse or Dirac function is S(x) = 0 at a:; / 0 and S{x) = oo at a:; = 0, but f_^ S{x)dx = 1. ^Angular frequency is uj = 2TVV, where y is the ordinary frequency, i. e. the number of waves per unit time.
171
Frequency domain emission spectroscopy
172
'
1
'
excitation
\
\
/
\
\ \
/ / ^ N L _ - ^ response 1
1
'
yr
x \ ^ \
Ap/co 1 /
\
'
X"^
N//
N.
\
1
1
1
1
/
\ V
\ \
\
1
/\~^
/
m
/
/
/
/
\ \
/
1
1
time Figure 9.1: Frequency domain measurements. The response (m cos {ujt -\- (/?)) to sinusoidal wave excitation (cos {uot)) is characterized by the phase (^ and modulation m at the angular frequency uj.
Function f(t) is the time domain presentation of the system response, and function F{uj) is the frequency domain response, respectively. If function /(t) is known, then function F{uj) 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{(JO) the sample has to be exposed to excitation by a sinusoidally modulated light. Then, at each frequency two values, phase Lp 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 formalism, a spontaneous relaxation of the emission is described by the exponential function (at t > 0)
m
(9.3)
he
where r is the lifetime to be determined. The Fourier transform of the function /(t) is the complex function^ F{u) = h I e-i^'^'dt
=
fo•-
l •
(9.4) loj
The real part of the function F{LO) is FRe{Lu) = Re{F{u;)) = fo
(WT)2
+1
^Note the integration limits, function f(t) = Oatt < 0.
(9.5)
9.1. Theoretical background
173
and imaginary part is 2
FiraiLo) = Im {FiLu)) = / o ^ ^ ^ J —
(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 {cut), then the response is r{t) = FRe{uj) cos {cut) + Fim{uj) sin {cot)
(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^ r{t) — m cos {uot — if)
(9.8)
where, m is the modulation coefficient and Lp is the phase shift. This notation is used in Fig. 9.1, and m and (/? 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 = ^FU^)
+ FULO)
(9.9)
The phase shows relative delay of the response to the excitation wave, and is given by (/; = arctan § ^ ^ FReiuj)
(9.10)
For the samples with exponentially decaying emission (eq. (9.3)) the modulation and phase are m
=
fo^=^==^
(9.11)
if
=
arctan(cjr)
(9.12)
They are presented in Fig. 9.2 as the functions of frequency a; at r = 1 and /o = 1. The dependences are straightforward in this simple case. At low frequencies {UJT < 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 (uor > 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 r of a sample using the frequency domain measurements the phase of the response, cp, can be measured at frequency uj close to the inverse of the expected ^Equivalently to the cosine excitation, one can use sine function to present excitation, sin (uot), then the response is given by m sin (ujt — ip), respectively.
Frequency domain emission spectroscopy
174
0.1
1
frequency, co Figure 9.2: Phase, (f, and modulation, m, dependence on frequency, uo, for a sample with exponentially decaying emission. The emission lifetime is r = 1.
sample lifetime and calculated as r = o;"^ tan(/;.^ In practice, however, the dependences of LP 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 uj. 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.^ After the modulator the light intensity is lex (t) = /Q [1 + a cos {out)], where a is the excitation modulation coefficient (0 < a < 1). Formally, one can say that the excitation consists of the continuous excitation with intensity /Q and the sinusoidal excitation 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. ^In most practical cases the (ordinary) frequency is measured in Hertz, in which case the equation calculating the lifetime is r = (27r/)~^ tan (p, where / is the frequency. ^Typical modulation frequency range is 1-300 MHz, which is the range of radio frequencies.
9.3. Frequency domain instruments
1
V
175
\
J
1
1
Computer
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 extracts values m and ^p from the signal (see eq. (9.8)). Therefore, only the variable part of the excitation (a cos (cjt)) is important for the method.^ The reference signal for the synchronous 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-r2 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.
^Also the continuous part of the excitation is not useful for the method, but it is unavoidable, since the light intensity cannot be negative. Indeed, the higher modulation coefficient a is better for the method.
176
9.3.1
Frequency domain emission spectroscopy
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 (p (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 metliods
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 developed 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 domain 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, (/?, and modulation, 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 frequency 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 photons, 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.^ 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.^
^In typical experimental conditions 10 000 photons are collected at the channel of maximum signal intensity, which provides 1% accuracy. Additionally, it is possible to increase the number of count by e. g. by increasing the photon collection time and increase the accuracy further more. ^The stronger emission intensity means better instant signal-to-noise ratio, therefore a shorter accumulation time is required for measurements with the same accuracy.
Chapter 10
Picosecond time resolution with streak camera For traditional electronics the fundamental limit in time resolution is set by the group velocity of the light - signal cannot propagate faster than the light. Decreasing the size of electronic 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
Photocathode
Electron deflection system
screen
/
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.^ 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 / = 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 = | ^ 10^ m s~^, 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,^ 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 photocathode 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, /(A, t).
^Although the recorded trace presents the time evolution of the electron beam intensity, these are not real time measurements, which means that this principle cannot be used process signals with high speed. ^The triggering pulse is still the signal which has all the limitations imposed by the finite value of the speed of light.
10.2. Main characteristics
181
Input beam
pin hole Streak camera slit (ID image) screen (2D image)
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 picosecond. 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 compromise 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 important 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 subjected 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 photocathodes 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 200950 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^ — 10^ using dinode system and acceleration potential. In the case of streak cameras the amplification of the primary photo electrons would result in dropping 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 microchannel plate amplifiers can provide gain factor of 10^ — 10"^, but preserve the spatial electron distribution. This measure can increase sensitivity significantly making single photon detection for the most advances systems possible.^ 10.2.4
Advantages and disadvantages
From the point of view of applications in time resolved emission spectroscopy the advantages 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 being 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 dependences, i. e. can provide complete emission dynamics at single excitation pulse. ^The output electron flow after the microchannel plate still has to be converted to photons by phosphor screen and then the optical image is recorded by a CCD detector. All these operations reduce the efficiency of the streak cameras in comparison to the photomultiplier 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;^ • requires stronger excitation pulse energy, specially for the high time resolution applications. The unique feature of the TCSPC method was the ability to measure emission decays of single 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 resolution 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, computer and comprehensive set of software for spectroscopy experiments. The main features of the system are 15 ps temporal resolution, simultaneous detection of temporal and wavelength dependences (by use of polychrometer), a high sensitivity approaching single photon counting (microchannel plate amplifier) and a wide dynamic range (10000:1).
^The TCSPC method 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 potentials 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
Pump-probe
186
Sample
Figure 11.1: Scheme of mono-color pump-probe experiments. Ml is abeam splitter, M2M3 are mirrors, PD is a photo-detector.
incoming pulse is divided in two parts by a semi-transparent mirror Ml. 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 required for the detection.^ IfEp 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 = EplO~^, 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. At, of the probe pulse relative to the pump pulse. The delay can be changed by moving the reflector M4, At =
2{d - do)
(11.1)
^The photo-detector signal is proportional to the total number 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 Hght 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 photodiode output voltage, is U{At) = sE^{M)
= sEplQ-"^^^^^
(11.2)
where s is the photo-detector sensitivity. The delay At 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, AA(At), induced by the pump pulse, which can be defined as AA{At) = A{At) — AQ where AQ is the absorbance of the sample before the excitation. To obtain AA{At) one can measure the signal from the photo-detector without excitation (or before the excitation), Uo = sEplO~'^^, and perform a simple calculation U(At) = s£;plO-^°-^^(^*) --=
sSpio-^no-^^(^*)
and thus A^(A,)^-log,.(^)
(11.4)
This is repetition of eq. (7.3) derived for the flash-photolysis method. The time uncertainty of A 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 experiments is technically a simple task. Example 11.1: Estimation of the signal intensity for the mono-color pump-probe experiments. 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, repetition 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 10 W.) The laser beam is split (by the beam splitter Ml, Fig. 11.1) in proportion 9:1 (transmittance of the mirror Ml is 10%), 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 5* = 5 • 10'' V/J (i^ =100 kTt, (7=10 nF). Thus without the sample the signal will be 50 mV. The sample will absorb part of the light and the signal will be weaker, but even 5 mV 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 AA{At) at a certain delay time At we need to measure two values, the photo-voltage with and without excitation, and to calculate absorbance AA 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 calculation method of AA, 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 /x. 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 ^10^ M~^cm~^ (this is rather high value), which corresponds to the absorption cross-section a = In(lO);^ ^ 4 x 10~^^ cm^. To excite most of the molecules the pump photon density must be at least one photon per absorbing cross-section, thus the total excitation energy density (at a visible wavelength, e. g. 500 nm) must be at least E = ^ ^ 1 mJ-cm~^. In order to obtain this energy density the pump pulse (9 nJ ^ 10~^ J) must be focused into a spot of area 10~^/10~^ = 10~^ cm^, or roughly 40 /x 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 ytxJ - 1 mJ (will be discussed in Section 11.2). Then, one can build up a (relatively)
11.1. Principles
189
continuum generator sapphire LI plate L2
band pass filter
PDl
M3
M2
U,
• ^ - " - - - 1
base pulses
M 4 \ ~ ^ ^ _^ Ml
PD2
M5
L3
Sample
U,
"^^^^^
1—1
PD3 delay line
U,
Figure 11.2: Two colors pump-probe measurements. Ml, 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 Ml. The reflected beam is directed by the mirror M2 to a white continuum generator, which consists of two lenses, LI and L2, and a sapphire plate.^ The lens LI 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 Ml, 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 At value must be changed, i.e. eq. (11.1) must be rewritten as At = ^ °^~ ^. 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 PDl monitors the intensity of the probe beam. Its main role is to improve the accuracy of the
^Almost any transparent medium can be used for continuum generation under sufficiently high power density of irradiation. Widely used materials are quartz, sapphire, water and heavy water.
190
Pump-probe
measurements. The signal is calculated as S{\t)
= ^
(11.5)
where Ui and U2 are the photo responses of the detectors PDl and PD2, respectively.^ 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."^ Similarly to the flash-photolysis and mono-color methods one can calculate the change in the sample absorbance, AA, as A^(At,A) = - l o g i o ( ^ ^ )
(11.6)
where 5(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. At, one can record the time profile of the transient absorption signal. The measurements can be repeated at another wavelength using another band pass filter 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 coUect a two-dimensional data array AA(At, A). Taking the data atfixeddelay time, a time resolved spectrum is obtained, AA(A) = A ^ ( At = const, A). This simple procedure, however, can be apphed 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 wavelength 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 pumpprobe 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 ^One can compare this two photo-detectors arrangement with the two channel spectrophotometer discussed in Section 5.2.2. The signal from the detector PDl serves as the reference channel of the spectrophotometer. ^ A typical usage of the detector PD3 is to interrupt the measurements when the variation of the excitation pulse energy exceeding some limit.
11.1. Principles
191
continuum probe
signal reference'
signal reference
signal
^T
entrance slit
Figure 11.3: Optical scheme for time resolved spectra measurements. Ml is a semitransparent mirror (beam splitter), M2 is 100% mirror, LI 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 /s(A), and another of the reference /r(A).^ In ideal case the signal and reference channels are identical and the ratio R{X) = -fj^ should give ones over the whole spectrum without excitation. Under the excitation the ratio R{X) can be used to calculate photoinduced absorbance, AA{X) = —log^QR{X). In practice, however, this simple method is not used, since all the components (mirrors and lenses: Ml, M2, LI 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 ^The calculations involve summation of the image pixels across the strip (horizontal direction for image 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 wavelength calibration of the spectrograph-CCD couple.
Pump-probe
192
%\/! v. \ •
\
/ .•'\
i •
\
•:
-0.05
i- N|
<
'••
•\-
-0.1
^ '
— -0.6 ps -- -0.3 ps \ - 0.0 ps
/ [\J\ \
1 •. ff
•••• 0.3 ps
•- 0.6 ps
X-.- 48ps
-0.15 550
600
650 700 wavelength, nm
750
Figure 11.4: Time resolved spectra of phitochlorin-fullerene donor-acceptor dyad in benzonitrile. 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{\) = /ofxy- Then, during the measurements with the excitation, the change in the sample absorbance induced by the pump is calculated by A^(A) = - l o ^
10
^(A) /r(A) Ro{\)
- l o g 10
Ro[\)
(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, Ayl(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 exposition of the sample to the pump is lower than that in the two color measurements. A disadvantage 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. Photoexcitation 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 characteristic 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 wavelength. This is seen as negative A ^ 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 wavelengths. The spectrum obtained at At = —0.6 ps (Fig. 11.4) is a horizontal line (i. e. there is no change in the absorption yet). The spectrum at At = — 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 At = 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 time). 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 probe pump excitation and monitoring beams in respect to the sample were discussed and applied to the flash photolysis method. In pump-probe the choice of the pump-probe beams organizations is much more limoverlap ited. The critical point is the pulse spatial width, area which is much shorter than that in the flash photolysis 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 r determines the spatial width of the pulse in detection of its propagation, d — cr. 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
194
Pump-probe
the sample before the pump. Thus we have to hmit 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 < (i/sina = cr/sin a, or D s i n a < cr. 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 thiclmess, i. e. the thickness must be L < D / s i n 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 cr = 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 sina < ^ = 0.03, or o^ < 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 compared to all previously discussed deals with the higher probability of the sample degradation due to much higher peak power density in pump-probe experiments.^ 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 characteristics must be considered together, the most important being: ^Though excitation density is usually lower in the pump-probe experiments than in flash photolysis. See Section 11.4.4 for flarther 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 objects to be studied; pulse repetition rate - important for the optimum organization of the detection system, selection 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.^ Usually, to establish 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 generating 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 measurements. Typical average power of Ti:sapphire lasers is ^ 0.5 W and the pulse repetition rate is 80-100 MHz. Thus the pulse energy is ^ 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. ^There are other important properties of Ti:sapphire crystals beneficial for its laser application, among them are high density of Ti ions, i. e. high amplification, and good thermal conductivity. See Section 3.6 for a brief discussion of the subject.
Pump-probe
196
A,_ Input pulse
Stratcher
yx
Pulse ^> picker
/
Amplifier Ti:S
4-
Compressor
A JV Output pulse
pump Figure 11.6: Amplification of femtosecond pulses using stretching-amplificationcompression method. grating 2 Output
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 condensed 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.^ The second component in the scheme is the pulse picker, which reduces the pulse repetition 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^ —10^, which can provide 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.^ 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 //J. For example RegA 9000 series of amplifiers (Coherent Inc.) can operate at 250 kHz and deliver pulse energy of 4 //J. 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^° 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^^ W cm~^. A few most popular materials for the cousin fact, pulse stretching takes place every time the light propagates in condensed media, e. g. in Ti:sapphire active crystal of the pulse generator. Therefore the pulse compressors are obligatory parts of any femtosecond generators. Inside laser resonators schemes consisting of two prisms are usually used (see Fig. 11.11). Prism compressors have much lower losses but can be used to compensate relatively small stretching as they have smaller dispersion compared to that of gratings. ^ 1 mJ at e. g. 5 kHz repetition rate means 5 W of average power on the amplifier 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 fi spot and the white light behind the medium is collimated by another lens for further utilization. Therefore already 1 /xJ 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 single 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 addition, 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 wavelength. 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 wavelength 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 second 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.^^ 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. ^^According to eq. (3.21) the wavelength of the sum frequency generation can be calculated as Xsum = (A7 + A~- )~^, or the inverse of sum wavelength is equal to the sum of inverse values of the fundamental and OPA signal wavelengths.
11.2. Laser systems
800 nm ImJ 0.01-5 kHz
199
800 nm
Ti:S amplifier
Ti:S generator
10 nJ, 90 MHz white continuum
WCG
probe SFG
465-530 nm 30 |LiJ
1100-1600 nm signal
SHG
550-800 nm 30 fiJ
1600-2900 nm idler
SHG
800-1450 nm 20 |iJ
SFG
535-625 nm 20 jiJ
OPA
SHG
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 efiiciencies of OPA and wave mixing are rather high with femtosecond pulses. For example OPerA-SFG (Coherent Inc.) can provide >30 //J 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 OPA, which is basically angular adjustment of the non-linear crystal, and (2) adjustment of the harmonic of wave mixing crystal.^^ As a result the pump pulses are available at all visible wavelengths, starting 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^ 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 300^00 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 complete 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 particular 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 possible, being insensitive to the actual pulse durations. When the pulse repetition rate is relatively 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 measured 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, AS {At). Ideally, the calculations of signal as the ratio of intensities, e. g. eq. (11.5), should make the result, S{At), 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~^ (detectable absorbance change) [16], which is close to the best accuracies ^^As it was mentioned above, Fig. 11.8 does not show actual layout of the optical components or beam propagations in the system. After the non-linear crystal, serving as amplifying medium of the OPA, all three beams, fundamental, signal and idler, propagate in one and the same direction. The switching between different modes, SHG or SFG for signal or idler, is done by installing a proper non-linear crystal in the beam and finding the angle of phase matching condition for desired effect. Also care should be taken to block the infrared light (of fundamental, signal and idle remained unused), which propagates in the same direction as visible light and can damage the sample.
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~^ — 10~^. In comparison to the flash-photolysis method the sensitivity (or accuracy) of pumpprobe 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 + 00
fc{t)
Jpumpy
) JprobeyT
(11.8)
~r tjCLT
where fpump{t) and fprobe{t) are the pulse shapes of the pump and probe, respectively.^^ Assuming Gaussian pulse shape, f{t) = exp "At2 where At is the pulse width, the convolution integral is a Gaussian pulse too + OC
fc{t)
exp
exp
At2 '-^^pump
V^Atp Af2 ^—^^pump
,At
probe
A/2 probe
exp
{r + tf
dr
probe
(At2 pump
(11.9) + Af2 '
probe .
with the pulse width^^ Ai,
\ I ^^pump
"I"
^^probe
(11.10)
^^ Equation (11.8) is valid until the inverse value of the pulse fundamental frequency, z/7 , is much shorter than the pulse width, At ^ uj
. Otherwise the complete time dependence of the pulses electric field should be used.
In the optical wavelength range (z^7^ ~ 2 fs) this means that the pulse width must be At > 10 fs. ^^The ratio before the exponent in eq. (11.9) is a constant due to the areas of pulses under the integral. The ratio affects the amplitude of the pulse but not its width. The pulse width is determined by the exponent argument only.
Pump-probe
202
1.54
1.52
-g 1.48
1.46 200
300
400
500
600
700
800
wavelength, nm
Figure 11.9: Wavelength dependence of the index of refraction of fused quartz. The dependence was plotted using data from [17].
If Atpump = Atprobe = At, then Ate = \/2At. 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. (11.10) Atpump and At probe 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 Ml 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 Ai and A2 is An 12 = n(Ai) — n(A2) = ni —712. It determines the difference in the propagation velocity of the pulses at these wavelengths. If two pulses at Ai and A2 enter a quartz plate of thickness / simultaneously, after the plate they will propagate with the relative delay At:
Ani2l
(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 Ai = 400 nm (the blue wavelength) and A2 = 600 nm (the yellow wavelength) enter simultaneously a quartz plate of thickness I — 1 cm. The difference in refractive index between these wavelengths is Ani2 = ni — n2 = 0.012 (as can be estimated from Fig. 11.9). Thus, after the plate the delay between the pulses will be At = ^ ^ : ^ = 400 fs. 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.^"^ 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 A, the refractive index variation within the pulse spectrum is An = AXD. The propagation time delay between wavelengths corresponding to the "blue" and "red" sides of the pulse spectrum is At^—D c
(11.12)
where / 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 Ati = 5 0 fs width is crossing a / = 1 cm quartz plate, and the pulse central wavelength is A = 400 nm (the second harmonic of the Ti:sapphire laser). Dispersions of quartz at 400 nm is D = ^ c:^ 10~^ nm~^. 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), AAi c^ii Q-^^cAt
— ^^ ^^' Thus the expected pulse broadening is At^ c^ — ^ ^ ~ 30 fs, 2
and the expected pulse width after the plate is At\ = At 1 + At^ ^ 80 fs.
^^In general the wavelength dependences of the refractive indexes are not monotonic functions. However, in the visible and near UV wavelength ranges most of the optically transparent materials have monotonic refraction index dependences on the wavelength similar 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 broadening 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,^^ and the broadening 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 design. 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.^^ 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 Ati = 5 0 fs, the pump wavelength is Ai = 800 nm, the probe wavelength A2 = 500 nm, the probe spectrum width AA2 = 20 nm (which is the bandpass width of the filter), thickness of the lenses L1-L3 II — 2 mm, sapphire plate thickness /g = 5 mm, thickness of the beam splitter Ml is / M I = 1 mm, and the sample is a solution placed in a quartz cuvette of thickness Igoi = 2 mm and having wall thiclmesses /c = 1 mm. For this estimation we will assume the dispersion of all components to be Z^i = 2 X 10~^ nm~^ and D2 = 6 x 10~^ nm~^ at 800 and 500 nm, respectively (which is dispersion of quartz). ^^This is the minimum bandwidth of 0.5 ps pulse which is due to its duration, although there may be some other reasons for a greater spectrum width. ^^For example, the efficiency of the second harmonic generation is higher for thicker crystals.
205
11.4. Time resolution
Q
2 TV4 1 ^ '^^^^\
^
'
Figure 11.10: Pair off axis mirrors (Ml 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 AAi 0.88 cAti 40 nm. The total thickness of the condensed medium on the path of the pump pulse is Ip I Ml -\- II ^ h ^ hi sol = 5 mm. The expected pump pulse width is Atp. ''pump^-^
1
Ati + - ^ A 50 + 13 = 63 fs. For the pump pulse let us check the shortest possible pulse width if the spectrum width is AA2 = 20 nm. According to eq. (B.5), At ^ 0.88 cAAi 40 fs, thus the probe pulse width is determined by the width of the input pulses but not by the band limiting factor of thefilter.^^ Therefore we will assume initial pulse width at 500 nm to be At 1 = 50 fs and will neglect the broadening by the lens LI. The total thickness of the disperse media on the path of the probe is Iprobe = /s + /L + ^/ + ^c + ^hoi = —D2 ~ 55 fs, and the probe pulse 11 mm. The pulse broadening is At^ = -^—-^— width inside the sample is At probe = At 1 + At^ 105 fs. Finally, according to eq. (11.10) the time resolution is At = ^/At^^^^ + ^^^robe ' 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 ^^ A narrower filter, e. g. AA = 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
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 benzonitrile 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.^^ The photo-induced absorption 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 procedure 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 quartzlike 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.
^ ^Naturally, for the compound studies the excitation energy was reduced to a level when two photon solvent excitation is not observed.
207
11.4. Time resolution
500
600
700
800
900
1000
1100
wavelength, nm
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 picosecond pulses were discussed, but non of them puts a principle limit in the time resolution. 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 ^ 2
The Gaussian pulse with duration At has spectrum width A A > 0 - ^ 8 ^ (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 applied equally the other way around - if the spectrum width is limited by A A, then the time resolution cannot be better than At > 0.^
(11.13) cAA 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 h/2, AEAt>-
(11.14)
where A ^ is the energy uncertainty, ^ = ^ , 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, At, should be as small as possible. Then we do not know the energy of the photon accurately, which can be determined with accuracy of AE' ^ ^^ in the best case. The photon energy is E = hu, and uncertainty in photon energy means uncertainty in photon frequency, AE = hAv. This gives quantum mechanical formulation of eq. (11.13) AtAz/>—
(11.15)
47r X 2
The relation between the frequency and wavelength uncertainties is A A ^ ^^"^^ which gives At > ^ • ^ ^ . 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 AA^ ^ 40 nm and the maximum is at A ^ 530 nm (see Fig. 6.5 on page 116). This corresponds to the pulse width of At '^ 30 fs. If the dye is excited by a shorter pulse, thus having wider spectrum, only the spectrum part with AA^ 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 At ^ 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.^^ The discussed problem arises from the fact, that at pulse duration of a few tens of femtosecond and shorter the harmonic wave treatment of the light becomes unsatisfactory. For ^'^To 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, 0, is the function of the pulse energy density, E, (j) = 1 — exp (—;^cr)-^^ We will consider the cases of relatively low excitation densities giving less than 50% excitation efficiencies, and can use approximation (j) ^ -^a. This equation has a simple meaning: ^ 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 i s c r ^ l . 6 x l 0 ~ ^ ^ cm^ and a pulse with the excitation density ^ = 1.6 x 10^^ photons per cm^ 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, ^ ~ ^ • 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 Acridine orange dye at 430 nm is cr ^ 10~^^ cm^ (molar absorption e ?^ 27000 M~^ cm~^). To excite <j) =10% of molecules in an optically transparent solution the photon excitation density must be n ?^ ^ = 10^^ cm~^, which is £^ = nhu ^ 0.5 mJ cm~^. In typical flash-photolysis experiment the excitation pulse duration is 10 ns, thus the excitation power density is 5 x 10"^ W cm~^. In typical pumpprobe experiments the pulse width is 100 fs and corresponding power density is 5 X 10^ W cm~^, 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 Appendix 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 relatively high absorption cross-sections (molar absorptions). The shorter the pulse the higher cross-section should be to avoid multi photon excitation of the environment.^^ ^^Here (j) is the probability to excite the molecule. ^^ 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
11.5
Pump-probe
Sensitivity
The sensitivity of the methods (both mono- and two-colors) can be rather high. It depends on inaccuracy of the signal determination, AS (At). Ideally, calculations of the signal as the ratio of intensities using eq. (11.5) should make the result, S{At), 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~^ (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 studies 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 separation 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 Nd:YAG laser. The repetition rate of the amplified pulses was 10 Hz and the pulse energy ^ 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
Ti: sapphire generator 50 fs 840 nm 90 MHz Ti:sapphire ampUfier Nd:YAG laser
10 Hz 532 nm 10 mJ
4-5 W CW
60 fs -V 0.5 mJ 10 Hz
Ar laser
SHG
420 nm pump
CG
contmuum 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 Ml 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.^^ The spectra were recorded using cooled CCD detector and correction procedure described in Secfrom continuum generator tion 11.1.3 (see eq. (11.7) and comments on it) was used to improve data quality. In a single measurement 10 pulses were averaged on CCD detector (during one second), also the energy of each pulse was controlled by a separate photodiode. To improve the data quality 10 measurements (spectra from the CCD) were averaged at each delay time. Typiquartz cal noise level of the data was about 10~^ (in abplate sorbance units). The time resolution of the instrument was 150-200 fs depending on the solvent used. Figure 11.14: Scheme of splitting A few raw spectra measured for the dyad in benthe the white continuum on two zonitrile were presented in Fig. 11.4 on page 192. equal signal and reference beams. The complete series of measurements in one wavelength 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 numerical analysis of the data, the spectra were converted into differential absorption time profiles with step A A = 3 nm. These time dependences were fitted globally to a sum of exponents, ^^The 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 /(t, A) = ^a^(A)e~^/^% were r^ are the Hfetimes and ai{X) are the corresponding pre-exponential factors. The fitting routine included convolution with instrument response function r{t), and the actual fit function included probe delays At (A) caused by the group velocity dispersion (at each wavelength)
F{t,X)=
r{x-At{X))f{t-x,X)dx
(11.16)
so that the fit results were corrected for the dispersion and the instrument time resolution.^^ The output of the fit procedure are so-called decay component spectra ai{X) and the corresponding lifetimes, r^ 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 intermediate 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 energy to the fullerene thus forming locally excited fullerene, P*F ^ PF"^, or to form the intramolecular exciplex, P'^F ^ (PF)*. The locally excited fullerene is relaxing by formation of the exciplex too, P F * -^ (PP)*. The exciplex is a precursor of the charge separated state, which is formed with the time constant ^ 20 ps, {PF)* -^ 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 (Ti02) nanocrystallines ^The spectrum of the delays, At(A), was shown in Fig. 11.12.
11.6. Application example
213
0.2 0.1 0.0
°°OooOoo,
.a°°"^^^^^^AA,
-0.1
A
-0.2
—D
r
^ ^
• o
^i
_ A
0.32 ps 8.2 ps 20 ps (fixed) 69.5 ps
-0.3 A
1
550
'
1
650
600
1
700
'
750
wavelength, nm
0.03 •
fx''^-.,.
748 nm
0.02 0.01 0.00 "
' • V - ; •_•
1
e
• . . .
X
¥•
^
[
-0.01 Q O -0.02 • -0.03 -
'•''^'•••'~~'{..'.'.
'"'''
^^^^^^ 682 nm
•
-0.04 • -0.05 -
\i3^^^0^
661 nm
-0.06 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.
Pump-probe
214
o
-0.5
-1.0H
150
300
600
Time [fs]
Figure 11.16: Pump-probe measurements of RuN3-Ti02 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].^"^ Thefilmshad roughly one micron thickness and were prepared on 0.06-0.08 mm thick microscope cover slips. The laser system consisted of a Tiisapphire laser, amplifier, and optical parametric amplifier, which provided excitation pulses at 530 nm and probe pulse at desired wavelengths with duration - 30 fs. The absorption cross section of the dye at the excitation wavelength is cr = 5 x 10"^'' cm^. The excitation photon density was P ^ 10^^ cm~^, which gave roughly (j) ^ aP ^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.^^ 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~^ change in absorbance can be detected.^^ 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 by e. g. fluorimeter (see Chapter 6) or time correlated single photon counting instrument (see Chapter 24RUN3 is Ru(4,4'-dicarboxy-2,2'-bipyridine)2(NCS)2
^^This 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. ^^The vertical scale units, mOD, denote milli optical density change, i. e. 10~^, 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 population 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 = hn^~^\ where a is the absorption 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 between the absorption of transient states and stimulated emission, since the stimulated emission 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 femtosecond time resolution, which is approaching the pulse duration of modem lasers, the measurements 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.^ 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 Ml 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 Ml and the mirror ^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
Emission spectroscopy with optical gating methods
218
SHG Ml
Delay line
<
\
from laser
\2v M2 ^ LI
excitation pulse 2v
Sample
a^„ Detection system
^
NLC ^
L3
"^ em
M4
filter
signal pulse
v^ = v + v .
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
Ml is a dichroic mirror^ reflecting the light at 400 nm (excitation, at 21/) and transmitting the fundamental harmonic at 800 nm (gating, at z/). The excitation beam is focused on to the sample by lens LI. The sample emission at z/gm is collected by the lens L2, filtered to reject 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, u, and the emission frequency, z/gm, to generate the sum frequency, z/^ = z/ + z/gm (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 Ud is measured by the detection system. Therefore, the frequency of the detected signal is shifted up by value u relative to the emission frequency Uem, that is why the method is called up-conversion. The detection system measures an average emission intensity at wavelength corresponding to i^d, i- e. at the wavelength Ad = (A ^ + A g ^ )
(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 ^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 Ud) is shown in the bottom of the time diagram. The light intensity at sum frequency, Ud, 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 At) 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 Ml the excitation, Iex{t), and gate, Ig{t), pulses have the same timing (and the time profiles). The excitation pulse at the sample is delayed by the propagation time Atex- It creates an emission, which is given by function Iemit — Atex), which is t lemit)
=
t
I Iex{T)D{t
- T)dT =
— CX)
f Ie^{T)D{t
- T)dT
(12.2)
—OO
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 > 0. 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 lemit)- At the NL crystal entrance the emission is delayed by time At em and given by the function Iem{t — Atex — At em)' The delay of the gate pulse is determined by the position of the delay line, Atd, and by propagation delay from the mirror Ml to the delay line plus from the delay line to the NL crystal, Atg. Thus at the crystal the gate pulse is Ig{t — Atg — Atd). Right after the crystal the intensity of the light at the sum frequency, i^d, (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)
= Vslgit
- Atg
- Atd)Iem{t
- Atex
- Atem)
(12.3)
where r/g is the efficiency of the NL crystal. Id{t) is a short pulse with the shape determined by the gate pulse mainly since Iem {t) is a slow function of time as compared to Ig (t) (at least for most of the measurements). The detection system measures an average pulse energy at sum frequency Ud, e. g. counting photons at Ud- Thus the measured signal is + CXD
U = s f Id{t)dt
(12.4)
— OO
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) = IgS{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, lemit)- Since, the gate and excitation are essentially the same pulses we can write Iex{t) = Iex^{t) and eq. (12.2) gives Iem{t) = IexD{t). Secondly, the delays At ex, At em and Atg 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 Atg = At ex + At em- If this is done, we can count time t form the moment of the gate pulse arrival to the NL crystal at Atd = 0, then eq. (12.3) is reduced to Id{t) = rjslg{t — Atd)Iem{t)' Accepting these simplifications + CX)
U = S
f 7],IgS{t
- Atd)Iem{t)dt
= SVsIgleccDiAtd)
(12.5)
— CO
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.^ 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 LI has focal distance of a few centimeters. Different non-linear crystals can be used to generate sum frequency. To mention few, there are LilOs, KDP and BBO."^ Probably the most popular crystal is BBO (/3-BaB204). 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 =b 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. ^within simplifications made above the measured signal, U, is directly proportional to the sample response, D(Atd), 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. ^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 1500 o
^^1000 500
0
1
2 time, ps
3
4
4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 2 J , if \ k i i^ ii. i l U K 1 1 lU . ill /11 . 0 -2 H ' | m y fii 'ri| ^*Fl 1 11 -4 h 1 11 1 1 1 1 1 1
1
1
1
M
m WHA^TSfW mk \liW\^ 1
1
1
1
fAiJ
1
i 1
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) measured with 25 fs steps. The solid line is the data fit (x^ = 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, Atp, 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) Atn ^ D{X)
lAX
(12.6)
where D{X) = ^\^ ^ is the dispersion at wavelength A, A A is the spectrum width of the pulse, and / is the total thickness of the dispersive material. Taking spectrum limited pulse approximation (see Appendix B, eq. (B.5)) AA ^ 0.^
,4. cMr,
(12.7)
Emission spectroscopy with optical gating methods
222
500
50
100
150
200
fundamental pulse width, fs Figure 12.3: Time resolution of the up-conversion method (solid line) calculated according to eq. (12.8) and assuming g = 1.6 x 10~^ ps^cm"^ (quartz at 400 nm) and / = 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, Atp, and the pulse broadening, Atn, ^Uotal — ^tp + At^
Atp + D{X)
0.88/A^ c^AU
Atp^g{X)
I AU
(12.8)
where ^(A) = 0.88 • L>(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.^ Therefore, Attotai 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.Q. ^ c^ 10-"^ nm-i (see Fig. 11.9 on page 202) and ^ c^ 1.6 x 10"^^ s'^m-^= 1.6 x 10~^ ps^cm"^, and the total thickness of the quartz is / = 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 ^"^^f ^ = 0,
AC* = v/^(A)I ^The effect of the gate pulse can be added by using eq. (11.10).
(12.9)
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 At^^* o:^ 60 fs, and the corresponding time resolution is Attotai — 120 fs.^ 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 / 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 LI 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 SHG 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 response is not as simple task as e. g. in the case of the time-correlated single photon counting ^Adding the contribution of the gate pulse (eq. V1202 + 602 ^ 135 ps.
(11.10)) one can expect the time resolution to be
Emission spectroscopy with optical gating methods
224
2000
1
1
1
1
1
1
1
1
1
1
1
1
1
1
^
t
1500 \ j
/ ^ o ^ ro
0
o
"" o" **
° * " o*""©"* oo""—°u"o
" ^
f
§ o
V / f\ \ \
1000 r
i 1
t J
^J ij1 7
" +
A
+
\\ \ \t
'/
500
V r+
+
-
;
:
V.^^^ +
:
+ 1
-1+
1+
1
1
1
1
1
1
1
1
1
1
1
"
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^ = 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 excitation frequencies, which is the third harmonic, v^2v — 3z/, for the scheme presented in Fig. 12.1.'^ 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 ^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^ and the time derivative of eq. (12.2) will give ""J^^^^ = 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 ^ 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 corresponding "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 (r^ 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 NQX photons (at Uex) and count the losses during the pulse propagation and transformation. We need to know the sample absorption coefficient, a, and emission efficiency, 0, and the efficiency of the emission collection, r/c. Then, at the entrance of the NL crystal the number of emission photons is rjcacpNex- The crystal mixes the emission and gate pulse and only those photons which are emitted in the time frame of the gate pulse, Atg, will be detected. Consequently, in the best case only Atg/r part of the emitted photons will contribute to the up-conversion signal at Ud = i^g ^ yem, where r is the emission lifetime. Additionally, the efficiency of the NL crystal, r]s, and the spectrum overlap factor (ratio of the spectrum widths of the detection and emission bands), (/;, have to be taken into account. Finally, adding the detector quantum efficiency, 7]d, the number of detected photons is Nd = —-VcVsVda(l)(pNex (12.10) r In the following Example this equation is used to estimate the counting rate of an upconversion experiment. Example 12.1: Estimation of the up-conversion counting rate. Let us assume the experiment conditions to be similar to those used previously in the emission spectroscopy examples (Section 6.2.7): ijc ~ 0.01, rjd ^ 0.1, a ^ 0.1 and 0 ?^ 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 r/5 ^ 0.01 and the spectrum overlap factor will be taken to he (p ^ 0.1.^ If the emission lifetime is r ?^ 2 ns (typical for dye ^Step or 6'-function is 0{t) = 0 at t < 0 and 0(t) = 1 at t > 0. ^For the steady state emission spectroscopy ^p ~ 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 Atg = 100 fs, then the ratio —^ = 5 x 1 0 ^ and ^ ^ 5 X 10"^ • 0.01 • 0.01 • 0.1 • 0.1 • 0.1 • 0.1 = 5 x lO-^^. If the second harmonic of a Ti:sapphire mode-locked laser is used for the excitation, i. e. the excitation wavelength is 400 nm, the excitation pulse energy is Pex ~ 10~^^ J (0.1 nJ) and corresponding number of photons is N^x — 2 x 10^. Therefore, in average we should detect Nd ~ 10~^ 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 / ?^ 100 MHz, consequently, the average photon counting rate of the detection is fd = fNd = 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^ '^ 2 x 10~^ 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 (TCSPC) method in Section 8.4.3 on page 158, the term ^ in eq. (12.10) is the sample radiative rate constant, kem = f (see footnote 13 on page 159). The signal intensity is determined by the radiative rate A^^ oc 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 kem = 2 x 10 *" s~^), 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 At^) one decreases the signal intensity (sensitivity), since a smaller number of photons will fit in to a smaller time window Atg. On the other hand, the efficiency of the NL crystal depends on the peak power of the gate pulses, therefore parameter rjs depends on pulse width either. If the pulse energy remains constant, the peak power increases with decreasing Atg, 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.^^ 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 uncertainty is A A ~ 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 (p ^ 0.1. ^^This 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, indeed), 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 reason for the lower sensitivity is the time resolution, Atg, 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. ^^ 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 (LI in Fig. 12.1) the spot size can be 20-40 fi 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 upconversion experiments.^^ Example 12.2: Excitation pulse energy and sample saturation. If the excitation pulses are focused in a spot with diameter of D = 20 p, the pulse energy required to achieve the excitation pulse density of E' = 1 mJ cm~^ is P = E^D'^ ^ 0.3 nJ. At wavelength A = 500 nm this power density means photon density of n = -^ = ^ ^ 2.5 X 10^^ cm~^. This photon density corresponds to the saturation density for the chromophores with absorption cross section of cr = n~^ = 4 x 10~^^ cm^ (i. e. molar absorption e ^ 10^ M~^cm~^), which is typical absorption for organic dyes. For 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 counting rate one can expect from an organic dye solution under conditions discussed above.^^ ^^The acceptance angle is inversely proportional to the crystal thickness [19]. Thin crystals have greater acceptance angle, therefore the beams can be focused to a smaller sport using lenses with shorter focal distances. ^^The excitation spot can be bigger (e. g. lens LI, 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. ^^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^, and accounting for the pulse repetition rate, 100 MHz, the sample emits 2 x 10^^ photons per second. Thus the total power emitted by the sample is roughly 0.1 mW, which will look like a bright spot.
228
12.1.6
Emission spectroscopy with optical gating methods
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, respectively. 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 upconversion applications is that the latter can operate at a much lower pulse energy but demands 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 certainly 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 actually 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 bialkalitype photo-cathode, with the spectrum range 200-650 nm, the emission spectrum range is 270-3470 nm.^"^ 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 wavelength 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 increases sharply when the difference between the emission and gate wavelengths gets smaller than 100 nm and the gate pulses wavelength is around 800 nm.^^ Type II synchronism works with orthogonal polarizations or the gate and emission, and there is no efficient second harmonic generation of the gate pulses.^^ Therefore type II synchronism can be used to measure emission decays at wavelengths closer to the gate ^"^For the calculations eq. (12.1) can be rearranged to give Xem = (A^ — A"-*^)"-*^. ^^This depends on e. g. the angle between the gate and emission beams and spatial filtration of the signal at sum fi-equency. ^^There 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 measurements 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 under study, the emission decay measurements at different wavelength may provide a valuable information on the relaxation dynamics of the excited state. ^^ 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. ^^ 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 measurements 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 aberrations 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 upconversion technique the time resolved measurements can be complemented by the steady state emission spectrum. If/^ (t, A^) are the decays measured at wavelengths Ai, A2,..., AAT, 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 +00
Ci = s{Xi)
/
fi{t,Xi)dt
(12.11)
^^For 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. ^^If lenses are used to collect emission from the sample and to focus it on to the NL crystal, as it shown in Fig. 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.^^ Therefore eq. ( 12.11 on the preceding page) calculates the ratio of the measured steady state emission intensity to the steady state intensity calculated from the emission time dependence, and the sensitivity corrected time-wavelength data can be obtained as D{t,Xi) = CiMt,Xi)
(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 optimized to allow recording of the time resolved spectra at each delay time. The spectral bandwidth of the NL crystal^^ 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 wavelengths) to achieve roughly 10 000 cm~^ 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 conditions. To compensate for the conversion efficiency losses due to the crystal thicloiess, a higher gate pulse energy can be used. This requires NL crystals with high damage threshold. 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 /iJ 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 simultaneous 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^^The 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. ^^This 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 hght scattering as compared to grating ones.^^ 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 femtosecond 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 applications, one can refer to Section 11.2 for discussion of the laser system generating fentosecond 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 (/5-barium borate). It belongs to the negative (rio > rie) uniaxial group. It has high damage threshold (> 10^^ W cm~^),^^ 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 harmonic 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 rejecting filters and photo-detector, which must measure very low light intensities. For low light intensity measurements photon counting photomultipliers are the most suitable devices. Also modulation-synchronous detection scheme were used. There are also commercially available up-conversion instruments. As an example one can consider FOGIOO 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 ^^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. ^^Assuming e. g. 0.1 mm spot size and 100 fs pulse width, the damage threshold is achieved at 7.5 ^iJ pulse energy, which is much greater than the pulse energies considered in Examples 12.1 and 12.2.
Emission spectroscopy with optical gating methods
232
Kerr shutter Excitation pulses
LI
Sample
PI
Kerr cell
p2
I
L3
F
MonoCCD chromator
Polarizer orientations
Figure 12.5: Optical scheme of time resolved emission spectra detection using optical Kerr effect. L1-L4 are lenses, P1-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.^^ In case of the optical Kerr effect the electric field is the field of the fight, so this is the light induced change of the material refractive index. In particular, the optical Keneffect is responsible for self focussing of the intense laser beams in materials, which is widely used for the self mode-locking in Tiisapphire 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,^"^ so that the time resolved emission spectra ^^It was discovered by John Kerr in 1875. ^^ 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 CS2 liquid. Unfortunately the response time of CS2 liquid is 0.8 ps, which limits its apphcation in modem instruments. A number of glass materials were tested and a several percent efficiency of the Kerr shutter was demonstrated with Bi203 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 instruments with short (femtosecond) time resolution. Disadvantage of the sheet polarizers is relatively low polarization degree, 10^ —10^.^^ 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 r = 1 ns, the measured time window (the width of the gate pulse) is At = 0.5 ps, the efficiency of the shutter is 0 = 0.1, and the emission rejection factor of the closed Kerr shutter (crossed polarizers PI and P2) is 77 = 10 ~^. If the total number of collected emission photons is A^em, then the number of "background" photons, i. e. number of photons which will reach the detector with closed shutter is A^^ = TjNem' The maximum number of gated photons is Ng = Nern^cj), which is the signal (time resolved spectrum). Thus signal-to-background ratio is -^ = —^ = 5. ^
°
Nbg
TTj
Similar to all measurements in femtosecond-subpicosecond time domain the group velocity dispersion affects the relative delays at different wavelengths. The dispersion correction 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. ^^Polarization 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.
Emission spectroscopy with optical gating methods
234
Delay line
Ml/
<
V
QW structure
fj-om
laser
^
~U
M2
900 nm
Detection system
^
^LC
2s7
L3
/ ^ - filter M3
L2
in— 1080 nm
LI
^i
6 nm
Sample
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 QWs were used to fabricate semiconductor saturable absorber mirrors, which are mode-locking elements of pulsed picoand 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.8lno.2No.01Aso.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 Ml) and the rest was the gate pulses beam. The excitation was focused by the lens LI on 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 LI was 3 cm and the excitation spot size was
12.3. Photo-dynamics of semiconductor quantum wells
235
5000
60 Time, ps
90
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 fi?^ Thus expected time resolution due to the non-colinear geometry is At ^ dcos^Ab ^ gg ^^^ which is an acceptable value for the targeted time resolution of 100-150 fs. The detection wavelength was \d = (A~^ + ^^m) = 4 9 1 nm. The detection part of the instrument consisted of a red cut off filter (transparent at 491 nm, but rejecting 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 ^ counts per second.^^ With excitation pulse density of ^ 0.3 mJ cm~^ and the pulse repetition rate of 91 MHz the signal counting rate was 500-700 s~^ 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 ^^This is approximately twice of the diffraction limit for a 2 mm beam, see Section 2.3.2. ^^This is the counting rate at negative delay, i. e. when the gate pulse hits the NL crystal well before the
Emission spectroscopy with optical gating methods
236
1 1 1 1 1 1 1 1 1 1 1 1 1
3000
r
11 1 1 1 1 1 1 1
11 1 1 1 1 1 1
1
•*• i' I'
o
1'
2000 o
r
11
L
1
1000 Lr
^
1
-
CKT
1 ' /P 1 'cufe>
-
t^ IX 1 AA
1 W 1 Q"
EX
'n
LmX' '"^ 1
-0.5
0
0.5
1
1
1
1.5 2 time, ps
1
, , ,1 1 1, , , ! , ,
2.5
3.5
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 simulated 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 28 In 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 electron subsystem of atoms and molecules. For example, position of the absorption band of rhodamine dye (at ^ 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 absorption bands. The natural bandwidth of the electronic transition (and any other transition) is tightly related to the lifetime of the excited state Aur ^ l/27r, where Au is the (frequency) bandwidth and T is the lifetime of the excited state.^ For the rhodamine dye the excited state lifetime is r ^3 ns and the natural bandwidth is Au ^ b x 10^ Hz = 50 MHz, which corresponds to A A = ^Au c^ 6 x 10~^ nm bandwidth in the wavelength domain (or, in wavenumbers, Ak = ^ = ^ o::^ 0.0017 cm~^). However, at normal conditions the absorption bandwidth is ^ 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 Ai^D ^ —vo c
(13.1)
^For a classic oscillator the relation, Aur ~ l/27r, can be directly obtained from Fourier transform of the oscillations decay, e~^* sin(ct;t), where k = 1/r. In the framework of the quantum mechanics one can notice that Heisenberg uncertainty principle, AEAt ^ h, gives the same result, since AE = hAu and h = h/27T.
237
238
Ultra-fine spectrum resolution
where Vav is the average velocity of the molecules and Uo is the middle band frequency. For ideal gas the average kinetic energy, EK, of a molecule of mass m is determined by the temperature EK = ^
= \kT
(13.2)
Thus
5^
(.33,
and
c yj
m
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, Vav ^800 m/s. At wavelength of 600 nm, i. e. z/^ = 5 x 10^^ Hz (close to the emission maximum of rhodamine dyes), one obtains /S.UD C^ 1.3 X 10^ Hz = 1.3 GHz, which is AA c^ 1.6 x 10"^ nm, or /\k = 0.043 c m - ^ This value is greater than the natural bandwidth of the rhodamine dye, 0.0017 cm~^, 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 r > {27vAiy]j)~^ ?^ 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 essentially 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 homogeneous 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 broadening 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 Hne broadening is caused by the interaction between the chromophore and the solvent molecules. The change in chromophore-solvent state happens with the solvent vibrational frequencies (10^^...10^^ 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 AA < 10~^ nm (or Au < 10^ Hz, or Ak < 0.002 cm~^) 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 afinewavelength selection: monochromators (see Section 2.3.4) and Fabry-Perot interferometers (see Section 2.2.2).The monochromators 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 by eq. (2.46):
AA = ^(1±^2£^
(13.5)
Fmg where d is the slits size, F is the focal distance of the collimating mirror (see Fig 2.7), m is the diffraction order, g 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 (p. Usually the monochromators are designed to have the mg value close to its maximum.^ 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 AA =0.008 nm {Ak=032 cm~^) with d = 10/xm slits. Dimensions of the device are 1167 x 470 x 350 mm^. The example above shows that even one meter focal length does not provide resolution close to the natural line width (but the resolution is already sufficient for the Doppler broadening detection at room temperature). ^For the normal incidence of the Hght siivip = Xmg (see eq. (2.44)) and for A = 400 nm mg cannot be greater than 2000 mm~ ^. Typical groove number of grating used in the visible part of the spectrum is 1200 mm~ ^, therefore these gratings are used for the first order diffraction, i. Q. m = 1.
Ultra-fine spectrum resolution
240
Interferometer FabryLight source G:''
Pinhole
Monochromator
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 wavelengths 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 interferometer with higher contrast factor F, i. e. higher reflectance R, or by increasing resolution of the monochromator, AA^, 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 AA^ =1 nm and the workA 2
ing wavelength is 500 nm, then interferometer thickness should be c? < 2 ^ A ~ — 0.125 mm. With reflectance of i? = 0.95 the contrast factor is F = ^it%2 = 1520. Thus, the interferometer with thickness d = 0.1 mm can provide resolution (eq. (2.31)) AA^ ^ ^^\/T^ ~ 10~^ ^ ^ ' which gives 100 folds improvement in the spectrum resolution.^ 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 pointlike hght source, as shown in Fig. 13.1. An estimation of the acceptable beam divergence is x^ ^The distance between the interferometer transmittance maxima is A A 2 ,. It can be expressed in the AA ^ ^ _ ^ ^ Similarly, the bandwidth is Afci = (27rd\/F)-i. Therefore, wavenumbers Ak wavenumbers are really easier units when dealing with interferometers. For example, the distance between maxima ofd=l mm interferometer is 0.5 mm~^ = 5 cm~^.
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-mono chromator couple. If the light incident angle a is not normal, the value d must be replaced by (icos a in eqs. (2.16)-(2.31). Thus, the transmission maximum shifts to Ai = ^^|^, where the A is the maximum at normal incidence. The shift is Ai — A ^ \a^ at small angles, i. e. cos a ^ a, and for conditions used in Example 13.3 one obtains a ^ J ^ ~ 0.005 radian ^ 0.3°. If the focal length of the light collimating lens is / = 20 cm (lens LI 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 AAj = 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) Ao
27r^/rir2 r
where r — —( ^h^i .. is the laser time constant and TA = — is the wave period. For CW mode rir2e^"^ = 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 ri = 1 and r2 = 0.95. Then, the photon lifetime is r = ^^^^^^^_^^ o^ 53 ns and TX C::^ 2.11 X 10"^^ s = 2.11 fs (A^ = 633 nm), which gives ^ = Ao
^ ^ ^ c . 6.3x10-^ 27rryYir2
or AA c^ A X 10~^ nm, or Az/ c^ 3 MHz (at central frequency Uo — 4.7 x 10^^ Hz), or Ak = ^ c^ 10~^ cm~^. 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
Ultra-fine spectrum resolution
242
.x ^
amplification line ^ ^ . ^
active laser mode / / lasing threshold
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 broadening, 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 frequencies 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
lasing threshold
;
' • • amplification band - without lasing '. ^ with lasing
2) Inhomogeneous broadening without lasing with lasing lasing threshold /r
t t t
%
laser modes 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 minima. 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 emission. 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~^). The monitoring light intensity will be very weak for this instrument (consider how much of the light can pass 10 // 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 spectrophotometer 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~^ nm (10~^ cm~^). Although to achieve such resolution the mechanical stability of the laser resonator must be as good as AL < ^ L , e. g. for resonator of L = 50 cm, emission wavelength A = 500 nm and desired resolution A A = 10~^ nm, the resonator stability must be AL < 10~^ 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 560620 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~^ 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
^^
Wavelength selector
\-B-
Active medium
245
. ^^
^r
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 Pp is spent to cover intra-cavity losses Pi and to emit the light / , that is Pp = Pi -\-1. The intra-cavity losses consist of two parts: the losses of the laser itself P^^ and the loss due to the sample absorption P^. At constant pumping rate Pu -\- Pa -\- I = Pp = const, thus I = Pp — Pu — 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 Pp — Pu to Pp — Pu — Pa. Thus, measuring laser output spectrum twice, without the sample, /(A) = Pp — Pu{)^), and with the sample, /s(A) = Pp — PuW — Pa (A), one can calculate the power absorbed by the sample as Pa (A) = /(A) — /s(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 intracavity 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 (Pp — Pu close to 0), then even a small change in absorption may lead to a gradual change in the output light intensity."^ 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 resonator is much higher than the laser output power and exceeds typical monitoring light intensity 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. ^In 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 detection 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~^ 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 measurements 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 studied 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 observed as inhomogeneous broadening of the spectrum. Under the conditions of inhomogeneous 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, A^, in the range from 10 to 10 000. The bandwidth of the (broadened) spectrum was taken Az/^ = 200 cm~^, the bandwidth for the individual molecule was IS.Vm = 2 cm~^ and the central band wavenumber was Vrnax = 20 000 cm~^. At relatively low number of molecules, N < ^ 7 ^ , the spectrum consists of a series 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 ^ ^7^5 the absorption lines of individual molecules start to overlap with each other, and at A^ > ^ j ^ 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 detection efficiency 0.1, emission collection efficiency 0.01 and emission quantum yield 0.1). If the molecule has excited state lifetime of 1 /iS (which include triplet state relaxation if it is formed) the excitation rate can be f^x =200 kHz, which means that the molecule is excited once in 5 //s in average, and expected counting rate is 20 counts/s. The density of photons must be close to one photon per molecule crosssection in 5 /iS time interval. For a typical dye molecule it is £" ^ 1 mJ/cm^. Thus required power density is P = Ef^x = 200 W/cm^. For a single molecule experiments the excitation spot must be as small as possible, say 5 = 1 /i^ = 10~^cm^. And the light intensity needed for this experiment is / = 2 //W only.
13.5. High resolution in emission spectroscopy
247
21000
wavenumber, cm Figure 13.5: Spectra of inhomogeneously broadened samples consisted of TV = 10, 100, 1 000 and 10 000 molecules. The bandwidth is Az/5 = 200 cm~^ and the single molecule line width is Az/^ = 2 cm~^.
The experimentally observed line widths are much smaller than that used for spectrum simulations in Fig. 13.5. For example Az/^ ^ 0.005 cm~^ was reported for chlorin 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 matrix (phonons). Assuming the bandwidth to be Az/5 ^ 250 cm~^ (which corresponds to AA ^ 10 nm at A = 600 nm), one can expect to resolve single molecular lines even when N ^ -^^ = 50 000 molecules are excited at the same time. 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~^ cm~^ in 700980 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 rejection of the excitation Hght, which can be achieved by a combination of band pass and cut off fihers. 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. Technically 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 object. The polarization 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^ E{f, t) = Eo sin hnut - k - r)
(14.1)
where E{f, t) is the electric field vector at a point given by vector f, v is the wave frequency, k is the wave vector, which determines the propagation direction of the wave, and E'o is the vector determining orientation and amplitude of the electromagnetic wave. The value v and vectors k and E^ are not independent, and in dielectric medium k- E = 0
(14.2)
u k — e— c
(14.3)
and
where e is the medium dielectric constant. ^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. [/ = C/oe**^^*~'^^^.
249
250
Polarization measurements
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 A: = ^ and wave frequency (in Table 1.2 the relation is given for waves in vacuum, s = 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 projection on axis Z is zero. Then, to specify the electric field vector EQ in this coordinate system, only two values are needed, X projection EQX and Y projection Eoy. If the angle between the vector EQ and X axis is a, then Eox = EQ sin a and Eoy = EQ COS a, where EQ = Ec] Experimentally accessible value is the light power, rather than electric field vector. The |2
= EQ. One can light power is proportional to the square of the electric field, IQ ^ \EQ use a polarizer oriented first along X axis and then along Y axis. The measurements of the light power will give two intensities Ix = IQ sin^ a and ly = /Q COS^ a, respectively.^ 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{f, 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 E,{f,t)
= Eo ix sin f 27vjyt — k - f) -\- fiy cos (27rz/t — k • f)
(14.4)
where fix and Hy 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 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. ^One can also notice that /Q = IX -\- ly, 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 denoted as "vertical" and "horizontal". "Horizontal" means in horizontal plane and perpendicular 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 polarization is characterized by polarization ratio
J^v -^ J-h
where ly 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 liglit witli 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 transition dipole moment of the molecule. (Note, it may differ from the electrostatic dipole moment of the molecule.) Emission and absorption are transitions between the same electronic 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 orientation 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 studied. 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 excitation 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 (^ and ^, as shown in Fig. 14.2. If the molecule absorption cross-section for the light polarized parallel to the transition dipole moment is (Jo, then for a molecule with the transition dipole moment at angle (p the absorption crosssection is Go cos^ (p? For random distribution of the molecule orientations the absorption is ^The projection of the wave electric field E on axis Y is Ex = E cos (/?. The light intensity is given by the square of the electric field, i. e. Ix oc El = E'^ cos2 ^. Thus, Ix = I cos^ ip, where / is the incident light intensity. Angle ip is not important for the cross-section calculations polarization as the electric field projection on the direction of the dipole moment orientation depends on angle (p only.
252
Polarization measurements
Y
A
'
X
^•..---7
Figure 14.2: Polar coordinate system.
given by averaging of all possible orientations^ 27r
TT
cr^ = -— / / (Jo cos^ (p sin (fdifdip 0
(14.6)
0
Calculation of the integral for ^p is trivial and gives 27r. Then, for cp one obtains TT
(7^
=
TT
-;^ / cos^ (p sin cpdif = —-;^ / cos^ (pd{cos (p) — J
2 7
x^(ix = —^
(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^ ^ = I ^^^ equal to cp = arccos 4= ^ 54.74°. 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, ^An average value of some function /((/?, ^ ) in polar coordinates is given by the integral / ^
/ f f{^,i^)
sin cpdipd^;.
14.2. Interaction of polarized light with media
253
Emission 5^
X
Excitation
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 observed 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
Now 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 XZ, i. e. perpendicular to Y, is called horizontal.^ Experimentally available values are emission intensities at two orthogonal polarizations, vertical and horizontal, respectively. Thus we can measure four values: I^^, lyh, hv and Ihh, where the first index denotes polarization of the excitation and the second shows polarization of the emission, e. g. lyh means emission intensity at the horizontal polarization when excited at the vertical polarization. If the sample is excited by the vertically polarized beam the distribution of the excited molecules is given by cos^ (p, if the excitation light density is much lower than the saturation one.^ 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 (f, thus the emission intensity of the molecule at angle (p is proportional to cos^ cp. Consequently, lyy can be obtained by averaging Ivv
=
~^ I I cos^ ip cos^
0
0 TV
+ 1
— — - / cos^ (pd COS if —7: 0
x^dx = -
(14.8)
-1
The projection of the dipole moment on Z axis is sin ip cos ^ and lyh is given by 27r TV
^vh —-r~ I siii^ ifcos^ (psincpdipcos^ ipdtb 47r 7 y
(14.9)
0 0 27r
One can notice, that / cos^ tpdtlj = TT, and 0 TT
+ 1
Ivh — ~li I cos^ ^ ( l ~ cos^ (p))dcos(p = 0
x^{l — x^)dx = —
(14.10)
-1
Even so the monitoring polarization is perpendicular to the excitation polarization, the emission intensity is only 3 times lower than that of parallel polarizations of the excitation and monitoring, j ^ =?>. Similar calculations can be done for the horizontal excitation, but one can notice that due to symmetry reason intensity Ihv and Ihh must be equal.^ Thus the intensity ratio for the ^Usually plain XZ is the horizontal plane of the optical scheme. ^The 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^ (f, and at infinitely high excitations all the molecules will be excited. ^For 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 XZ if it has polarization in Y direction.
14.2. Interaction of polarized light with media
255
horizontal excitation is j ^ = 1. This is practicahy important result since it can be used to 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 Itotai = Ix + ^y^h, where the subscript denotes the polarization orientation. For the vertical excitation polarization ly = lyy and /^ = lyhIx was not calculated, but for the symmetry reasons it must be Ix = h (see footnote 7). Thus Ix = Ivh, and the total emission intensity (for vertical excitation) can be calculated as hotal
= Ivv + '^Ivh
(14.1 1)
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).^ 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
The formal difference between polarization ratio, eq. (14.5), anisotropy coefficient is in denominators. The latter relates the difference between polarization {lyy — lyh) to total emission intensity (lyy -\- 2Iyh), 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 Ihv = hh
(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 polarizations are unknown and depend on the detection wavelength. However, the experiments ^One 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 vertical and horizontal detection polarizations can be determined. This ratio is usually called G-factor G=§^
(14.14)
where /^^ and /^^ 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 J*^ and J*^, the actual intensity ratio can be calculated as ^=7^ = ^ ^
= ^
^
(14.15)
which makes corrections of the detection system sensitivity to polarization. Now anisotropy can be calculated as \
Lvv_
^^'^ 1 + 24^
^
o
l^2R
(14.16)
J-vh
This measurement strategy can be applied to study anisotropy decays. Then two decays have to be recorded, I*y{t) and I^^it), 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: /^^(A), I^hW ^vvW ^^^ ^vhW- T^^ fi^^t ^^^ ^^^ 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 j ^ = 3 . For a totally anisotropic medium the range for anisotropy coefficient values i s r = —0.5...1. The lower limit is achieved when the emission dipole moments are oriented along Z axis, i. e. //^ = 1 and ly = 0, and the higher limit is achieved when the dipole moments oriented along Y axis, i. e. /^ = 0 and ly = 1, respectively. For isotropic diffusional rotation anisotropy decays exponentially r(t) = roe-^^-*
(14.17)
where Dr is the rotational diffusion coefficient. Another parameter used to characterize rotational diffusion is the rotational correlation time, which is TC = {6Dr)~ , thus giving r{t) = roe~^/^^. The rotational diffusion coefficient of a sphere is Dr = ^ ^ , where r] is the viscosity and V is the hydrodynamic volume. This leads to the rotational correlation time of TC = ^ - ^ ^This 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 = \rps •> 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
14.3
257
Applications of polarized measurements
Apparently, the change in hght polarization during the measurements complicates the measurement procedure. On the other hand one can use them to extract additional information of the subject under study. As an example of complication one can consider diffusional rotation of the dye molecules in solutions when the fluorescence decay dynamics is studied. For a typical dye molecule, e. g. rhodamine dye, in a solvent with moderate viscosity, e. g. propanol (rj = 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 membranes, 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 dynamics 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. ^^ 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 relatively low price and work in wide range of incident angles (typically more than ±25°). ^See, for example, book by Bernard Valeur [11].
258
Polarization measurements
Sample
To detection system
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 compared 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~^ at CW irradiation and 500 MW cm~^ in pulsed (10 ns) mode. Another advantage of the prism polarizers is high degree of polarization, (can be > 10^) 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 undesired 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 (PI) 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.^^ ^^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 (PI and P2 in Fig. 14.4) should have adjustment possibilities, since 4 intensities have to be measured - lyy, lyh, Ihh and /^^, as was discussed in Section 14.2.3. If the only aim of the measurements is to measure emission decay, the analyzing polarizer can be turned to magic angle, (f ^ 54.74°. 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 transfer 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 involves 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 imbedded into membrane proteins surrounding the reaction center. An excited chlorophyll (Chi) molecule can transfer the excitation energy to its neighbor by non-radiative resonance energy transfer mechanism. The neighbor transfers the energy to another neighbor and so on until the excitation energy is delivered to the reaction center. In photosystem 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 Chls the maxima are in the range 670-685 nm). However the difference is rather small and the energy transfer from P700 to the antenna Chls 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^ + Chl2 -^ Chli + Chl2, and if Chli and Chl2 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 Chli and Chl2 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 emission 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
-^
L
I
I
-I
L
600 H
600 c 400-^ 3 o O 200 H
0-J J
I
L.
J
I
L
0.4 H
0.3 H o
I 0.2-1 <
o.H 0.0-
-I—'—I—'—r 2
4 Time (ps)
6
Figure 14.5: Polarization dependent fluorescence decays of of the P700 enriched PS I reaction 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.
Ipara {t) prcscuts the decay for detection polarization parallel to the excitation polarization, and Iperp{t) shows the decay for perpendicular polarizations, respectively. The anisotropy was calculated using eq. (14.12).^^ For this measurements the time dependent anisotropy reflects mainly dynamics of the intermolecular 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 dynamics suggests that the energy can leave the reaction center. Also the anisotropy does not ^^In femtosecond up-conversion (and pump-probe) experiments the excitation and monitoring directions must be almost co-linear. Therefore the scheme presented in Fig. 14.4 cannot be used. However, the detection polarization is not changed, but the excitation polarization can be tuned (to measure Ipara and Iperp), 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 equihbrium between the antenna Chls 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 (Chi* + 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 transition dipole moments between the involved states. If the excited state has the transition dipole moment turned by e. 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 monoexponential 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 lifetime 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 = yi,y2^. • .yx and X = a;i, 0:2, • • • a;^ are two vectors, i. e. two sets of values, of length K, with direct relation between their elements: yi was obtained at xi, y2 at X2 and so on. In case of absorption spectrum, X are the wavelengths selected by operator, and Y are 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 Ym = 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^- • -PN are used to adjust the vector Ym 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 experimentally 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~^, which gives P = F~^{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^ = 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 energy. 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
Pr^i-) = r
K'^\2
(15.3)
where u is the (photon) frequency, UQ is the band position, Au is the bandwidth, and FQ 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 eV, or the wave number, traditionally counted in
15.2. Spectral data analysis
'
1 '
265
1 '
1 ' /l\ '
1 '
1
^
1
'
1
'
1
Oi
— Lorentzian - • Gaussian
0.6h
-
0.4 0.2 ^y^
^^^^^^...'-^'^ , 1 , 1 "\
1.2
/ // / /// / / / / / / y / 1\ , ^ ^
1.4
1.6
/
1
1.8
-
\ A ^. \ \ \ \ \. \ ^^^^ ,\ N ^ . , 1^ ,
1
,
-
,
1
2
,
2.2
2.4
2.6
1
2.8
,
3
V
Figure 15.1: Lorentzian and Gaussian bands calculated for I/Q = 2, Ai/ = 0.2 and FQ = 1 with eqs. (15.3) and (15.4), respectively.
cm ^, but transition to the wavelength domain requires some caution as will be discussed later. ^ 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
FG{^)
=Foexp
^0
Au
(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 Ai/1 = 2 • Av, where as for Gaussian band Az/i = 2Ai/VIog2 ?^ 2 • Az/ • 0.83 = L66 • Au. The bands are shown in Fig. 15.1 for UQ = 2, Aiy = 0.2 and FQ = 1. For Lorentzian band the intensity decreases much slower than that for Gaussian band. For example, at 2 A?/ from the maximum, the intensity of Gaussian band the intensity is 0.018 of its maximum values, and at AAv is lO"'', which can be neglected in most practical cases. For Lorentzian band shape the intensities are 0.2 and 0.059 at displacement 2Az/ and 4Az/ 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 characterized 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 ^See also Table 1.2 for other units and unit conversions.
Analysis of the measurements
266
or other model bands. In case of Gaussian band shapes the model spectrum is N
F{u)
N
ai exp
aiG{v,VQi,/^Vi)
AUi
(15.5)
where N is the number of the bands, a^ are the amplitudes of the corresponding bands is the band shape.^ The total number of model
and G (z/, Z/QZ, ^^i) = exp
parameters in eq. (15.5) is 3A^, which are band positions, Z/QZ, bandwidths, Az/^, and band intensities, ai? Conversion from frequency to wavelength domain is straightforward for absorption bands - the frequency is substituted by the wavelength, u = -I, 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 Ix and in frequency domain is /^, then the measured light intensities are Ixd\ and lydu, respectively. Since du = —j^d\^ the relation between the spectrum densities is /A
(15.6)
A2^
In particular, the Gaussian band shape in the wavelength domain is given by FG{\)
2
A0
exp A2
AA
A \
/ .
AQ-A
A
21
(15.7)
where AQ = ^ is the wavelength corresponding to the frequency of maximum in the frequency domain, A A = A Q ^ = Az/-^, and ^o is a constant. The wavelength AQ is not actually the maximum of the emission intensity in wavelength domain, it is also approaching the maximum for narrow bands, i. e. when 4 ^ ^ 0. 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~^, emission at k^ = 20 000 cm~^, and bandwidth for both spectra of A/c = 2 000 cm~^ (FWHM values are 3 330 cm~^).This corresponds to absorption maximum \a = 400 nm and bandwidth AA^ = 32 nm (FWHM value is 53 nm). For the emission band /Cecorresponds to Ag = 500 nm, but the actual emission maximum is at 495 nm. The emission bandwidth as calculated from the corresponding wavenumber values is 50 nm (FWHM value is 81 nm) which is essentially broader than the width of the ^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. •^It will be shown in Section 15.4.3 that only non-linear parameters, which are VQI and /\ui 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 2A/" in this case. ^If wavenumbers are used instead of frequencies, k = X~^, then dk = X~^dX (provided the same units are used for both values, e. g. cm~^ for wavenumber and cm for wavelength, respectively).
15.3. Kinetics and reaction schemes
15000
20000
25000 -1
267
30000
wavenumber, cm
350 400
450 500 550 wavelength, n m
600 65C
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. Additionally, 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 measurements 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 + hiy, where kr is the radiative decay rate constant. It can decay non-radiatively to k
the ground state, M* —^ M, where kn is the non-radiative decay rate constant, or it can k
relax to some intermediate state, M* -^ P, where kp is the reaction rate constant. ^ If the population of the excited state is Uex than the value Uexkrdt is the number of photons ^One 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 or in form of differential equation
^
= - „ „ l . - , , A , - „ „ ^ = -*.„„
(,5.8,
where the negative signs in the right side of the equation are due to the fact that relaxation decreases the population, and ko = kr ^ kn + kp is the total relaxation rate. The solution of eq. (15.8) is ne.(t) = ce-^°^
(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(O), one obtains nex(t)=ne.(0)e-^°^
(15.10)
Noteworthy, in time resolved spectroscopy measurements, which follows the population riexit),^ an exponential decay will be observed with time constant r = /CQ"^, i. e. the individual reaction rate constants kr, kn and kp cannot be obtained form the decay measurements. To extract individual rates from k^ additional experiments must be carried out. For example, emission quantum yield is ^^
^^
(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 kr = ^em^o- Similarly, if the quantum yield (j)p of the photo product P was determined, than the rate constant of the reaction M* ^ P is kp = (j)pkp. 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 ^
= -k2nl
(15.12)
^These 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 /c2 is the second order rate constant. The measure of the second order rate constant is the inverse measure of population multiphed by the inverse of time, e. g. if population is measured in number of molecules per cubic centimeter, cm""^, and time in seconds, then the second rate constant has units of s~^cm^. The solution of the equation is ne.{t) = — ^
(15.13)
where C2 is the integration constant which can be determined from the reaction initial conditions. Under assumption of a very short pulse excitation at t = 0, which generates Uexi^) excited states, the constant is C2 = [^ecc(0)]~^ and eq. (15.13) can be rewritten as
^-W = ^ S ^ T X T
(1^-14)
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 ^2^ea: ^ ^0, wherc /CQ 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 population 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 [i^ex{t)]~^ = k2t -\- C2, the dependence must be a straight line with the slope given by the second order rate. 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, ... A4, and the reaction rate constants are kij, 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 usually much smaller.'^ 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 Ai is -j^
= -(/^12 + ki3 + h4)ni + k2in2 + ksiUs + ^^41^4
(15.15)
^In 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.
Analysis of the measurements
270
The terms on the right side with negative sign are responsible for the decrease of the state Ai population, e. g. reaction Ai —^ A2, and positive for the increase, e. g. reaction Similar equations can be written for the other state, but from four equations only three are independent, since the total population should not change, i. e. Til -\- n2 -\- n^ -\- n4 = riQ = const
(15.16)
where no is the total population, e. g. total number of molecules. Therefore the system is described by three differential equations of type (15.15). The general solution of the system of three linear differential equations is the sum of three exponential terms ni{t) = Cue -kit + Ci2e -kot + Cise' -kst
Figure 15.3: General four state reaction scheme. (15.17)
where ki, /c2 and k^ are the rate constants and cn, Q2 and Q3 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 ^ i and A2 are the only emitting states in the scheme, the total emission of the sample as the function of time is I{t) = krini{t) + kr2Ti2{t), where kri and kr2 are the radiative rate constants of the states Ai and A2, 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 A^ — 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 TV — 1 exponents, although the number of intrinsic rate constants is at most A^(A^ — 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 Ai
ki2
A
Ao
k2
A,
(15.18)
where Ai is the primary excited state, e. g. singlet excited state, A^ is the final ground state and A2 is the intermediate state. The reaction scheme is described by a system of two differential equations dni dt dn2 dt
-ki2ni ki2ni
(15.19)
-k23n2
In matrix form this equation can be rewritten as d dt
N =
where N
(15.20)
K-N ni
is the population vector, and K
^2
-^12
0
kl2
-k23
is the charac-
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 -ki2-(3 0 = 0, which are /3i = ki2 and P2 = k23, inthispardet ki2 -k23 - (3 fel2
ticular case.^ The eigen-vectors are Vi
and V2
1 between the pre-exponential factors, so that the solution has form
They give relation
(15.21) where ci and C2 are the constants determined by the initial conditions. For the linear reaction scheme (15.18) the solution is ni{t)
n2{t)
=
cier.f^-f^i2t ^ Cl
fel2
-
k23-ki2
-ki2t
_
_ C2e
-k23t
(15.22)
Assuming the sample was excited at t = 0, and the excitation populates state Ai, we can take the initial condition to be n i ( 0 ) = 1 and 712(0) = 0.^ Then the constants are Ci = 1 ^In general, eigen-values are solutions of equation (A — X- I)x = 0, where / is the unity matrix. Please refer to mathematics book for further information of matrix algebra and linear differential equation solutions. ^The physical meaning of the values n i and 712 is the probability for system to be in states Ai and A2, respectively. Alternatively they can be interpreted as some reduced concentrations. If someone needs to operate with actual concentrations, then the initial conditions are n i ( 0 ) = nex and 712(0) = 0, where riex 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 •
Analysis of the measurements
272
^12 ~ ^ ' ^23 ~ ^
-A
I 0.5
-•
A.
OH
O
I
I I
I I I I
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.
fel2 -kr.
and C2 ni(t) n2(t)
-. Finally, the populations of the states Ai and A2 are -ki2t -ki2t -ki2
_
-k23t
)
(15.23)
Naturally, population of state As (which is the ground state for examples considered above) can be obtained from eq. (15.16), i. e. ria = 1 — ni — n2, in particular case. The first obvious result is that decay of state Ai does not depend on the rate constant ^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 A2 state forms and decays with rate constants ki2 and /C23, 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 ki2 > /C23, this is intuitively clear situation when A2 state decays slower than its precursor state Ai, i. e. reaction Ai -^ A2 is faster than A2 ^ As. When reaction A2 -^ As is faster than Ai ^ A2, which is ki2 < /C23, 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 A2 populations are the same in both cases, but the population values are five times smaller for the inverse kinetics. For the final state As the population time profiles are just the same for both cases and only state Ai shows clearly the difference between the normal and inverse kinetics. The relation between the intrinsic rate constants ki2 and /C23 determines the boarder between the normal and inverse kinetics, but what happens when ki2 = A^23? This is so-
273
15.3. Kinetics and reaction schemes
called degenerated case. For the state Ai the solution is still ni ^ e"^^^* por A2 solution is 712 ^ te~^i2* 10 j\^Q 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-i^i < ki^ij^i one has to consider inverse kinetics. Also the time profile of the first state depends only on ki2, for the second on ki2 and k2z, 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 Ai ki3 A^ and modifying the first equation in system (15.19) ^ = —(/ci2 + kis)ni, which gives eigenvalues (observable rates) ki2 + /cis and /C23 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.^^ Reaction scheme with equilibrium Another practically important case is the reaction scheme with equilibrium ki2
Ai ^ A2
A.
(15.24)
It can be solved using the same mathematical approach as one described above for the linear scheme.^^ The characteristic matrix for this scheme is K
-ki2 ki2
(15.25)
-k2i - k23
The eigen-values (or observable rates) are —(3i = {ki2 + /C21 + kis -\- ks)/2 and —(32 = {ki2 + k2i + ki3 - ks)/2, where kg = ^{ki2 + /C21 + /^is)^ - ^ki2k2i, and the eigenvectors are 1 1 Vi and V2 2fei2 2/ei2 -ks-\-k2i-\-k23—ki2
ks-\-k2i-\-k23-ki2
If the initial conditions are supposed to be the same as in the previous example, ni (0) 1 and 712(0) = 0, the solutions for the state populations are ni(t) n2(t)
+ C2e'P2t k^M
(15.26)
where ci = ^s+kr2-k2i-k23 ^^^ ^^ = ks-k^2+k2i+k23 ^ P()j. equilibrium reaction the decay of the state Ai is bi-exponential. Because of the reaction A2 -^ Ai the decay of the ^^ According to the scheme, state Ai does not "know" what happens with state A2, since all the reactions occur in one direction. ^ ^ The eigen-vectors are also slightly different, giving lower pre-exponential factors in expression for n2 (t). ^^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 (^i) depends on the reactions of state A2. 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 (—Pi and —/?2) does not allow to calculate intrinsic rates (/ci2, /C21 and ^^23), but the experiments can be organized in such way that all three intrinsic rate constants will be calculated. For instance, if Ai 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 ^ ^ be fitted to bi-exponential model, which is I{t) = aie ^1 + a2e ^2, where n and r2 are thefittime constants and ai and a2 are the corresponding pre-exponential factors. Comparing this with the first equation in (15.26) one can see that ri = —/5f ^ and r2 = —/52^^. The absolute values of the pre-exponential factors are not very useful, but their ratio is r = — = — = ^g+^i2-fe2i-fc23 xhus, for three unknown rate constants there are three ^2
C2
ks-ki2-\-k21-\-k23
'
equations, which can be solved. After some manipulations (see supporting information in ref. [28]) one can obtain ki2 =
^+r.,+ 1
k23= fei =
_ {h2riT2y' r^^ + r^^ - ki2 - fes
(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 reactions 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 significantly. 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 relaxes 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 photochemistry 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) = hi7jkjrnj{t)
(15.28)
where kir is the radiative rate constant and hi7j 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 population 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{X)nj{t), where Sj{X) is the emission spectrum of the state Aj. Naturally, for the total emission intensity Ij = J^ ij{X)dX, thus J^ Sj{X)dX = hujkjr. 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{X,t) = Y,Sj{X)n,{t)
(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) = Xl/=i Cjie~^^^}^ thus M
i{x,t) = j2
M
N
^CjiSj{X)
1=1
e
-kit
Y,bie-^'^
(15.30)
1=1
In other words, the emission time profile at wavelength A is the sum of exponents. The rates, ki, 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, Sj, of the intermediate states (at A) and eigen-vectors, Cji. Transient absorption models If absorbance of state Aj is aj{X) and its relative population is rij, than the absorbance of that state at time t is aj{X)nj{t). Instead of absorbance and relative population one can ^^In general A^ = M , and A^ 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, Cj (A) and molar concentrations Cj {t) to calculate sample absorption. This will not change the following consideration after replacing aj by e^, and Uj by Cj, respectively. The actually measured value in transient absorption experiments is the change in absorbance 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 n^(t),^^ then the transient absorption is N-l
Aa{X,t) = Y^ aj{X)nj{t) -ag{X)nj{t)
(15.31)
j=i
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
Aa{X,t) = Yl ^My~^'^
(15.32)
j=i
where hj (A) are the pre-exponential factors similar (but not equal) to those in square brackets 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 photoproduct the absorption cross-section (or molar absorption coefficient) is know at some wavelength, 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 For 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 instrument time resolution are much shorter than the reaction time of the sample. This is not ^^In 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 account to improve thefitaccuracy. The instrument time response can be measured, see e. g. Section 8.5.1, and used to correct the decay model accordingly^^ W= / r{r)f{t-r)dT
(15.33)
where r(t) is the instrument response function, ^^ f{t) is the decay model for the sample measured with ideal instrument, and fc{t) is the decay model measured with the real instrument 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 Now that we have the experimental data and mathematical models for the measurements, we canfitthe 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
^' = ^f2(yi-MP)?
(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 y^^f^^n,
(15.35)
where n^ 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 z + 1 are independent values. Combining eqs. (15.34) and (15.35) one obtains 1 ^ a2 = - ^ n , 2
(15.36)
z=l
^^In Section 8.5.1 the method accounting for the instrument response was discussed in apphcation to time correlated single photon counting technique. Therefore, eq. (15.33) is a copy of eq. (8.6). ^^Measurements 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 = ^ X^i=i ^i = 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 cr^ criterion shows the noise mean square deviation. The (7^ criterion is not the only one. For example one can use absolute values of the deviations aabs = J2i=i \fi~yi\- However, the cr^ 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. ^^ Application of the a^ criterion with eq. (15.34) requires stationary noise statistics. However, there are many important cases where the noise is not stationary. In spectroscopy such case is the photon counting. Uncertainty for y counts is ^/y, 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^ = J2w,{MP)-y^f
(15.37)
2=1
In the case of fluorescence measurements or any other photon counting experiments, the weight factor is Wi = ^ = . Then, uncertainty of measurement i, which is ^/yi, is multiplied by ^ = , resulting in stationary noise with mean square deviation cr^ = 1.^^ This simple modification allows one to increase applicability of the cr^ criterion. From the point of mean square deviation criterion, the best approximation is the one which gives the smallest cr^. 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^-value. 15.4.2
Minimization of mean square deviation
Formally eq. (15.37) is a function of parameters P and minimization of a^ can be done by solving equations dpi
= 0
= 0 (15.38) = 0 ^^Practically important for spectroscopy application photon counting statistic approaches the normal distribution 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 likelihood criterion of fit goodness [12]. ^^In this case the criterion is commonly called x^, meaning that Wi = —^^.
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)=pi
(15.39)
The function is just a constant (so number of parameters is A^ = 1). There is a single parameter which leads to a single equation for the minimization problem Q
2
Q
M
^^ i = l
^^
M
i=l
and after rearrangement M
M
pi'^^i = Yl ^^y^ 2=1
(15.41)
2=1
Which gives M
P^ =
'^
(15.42)
2=1
To shorten the equation we can use an average 1 ^ ^=ME^^ M .
(15.43)
2=1
instead of keeping summation. Then eq. (15.42) has a simple and clear form n =^ (15.44) w In the case of stationary noise, i. Q. Wi = 1 for all i, one obtains pi = y, i. 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^p2X
(15.45)
This gives a system of two equations
[ if: = af7E2^i^^(^i+^2x,-^,)2 =
2Y.fi^w^{pi^p2x^-y^) = 0
I W^ = W^T.f=i'^iiPi + V2Xi - yif = '2J2fLiUJi{Pi^P2Xi-yi)xi = 0 (15.46)
280
Analysis of the measurements
Using average values instead of summing J piw \ piwx
-\- P2WX = wy -\- p2wx'^ = wyx
(15.47)
Determinant of the system is det = w ' wx'^ — wx^
(15.48)
Clearly, determinant must be non-zero, i.e. w - wx^ — wx'^ ^ 0, or system (15.47) cannot be solved. If det ^ 0, parameters are _ Pi
^
wy-wx^ —wx-wyx — W-WX'^—WX'^ w-wyx — wx-wy —
/^2
9"
("15 49) ^ ' ^
0
w-wx-^—wx"^
In the case of stationary noise (wi = 1) the parameters for the linear approximation are ll
:
^ i f
(15.50)
In the same manner one can derive equations to calculate parameters P for any polynomial, F = pi^p2X-\- ...pNX^~^. Differentiation (eq. (15.38)) gives a system of equations Piw + . . . + PNWX^~^
=
wy
—
onoi^N-1
(15.51) Piwx + . . . +
2N-2 PNWX"^^^ ~^
= wyx
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 PQ and calculates the corresponding mean square deviation a^ using eq. (15.37). Then, a new parameter vector Pi is generated according to the selectedfitalgorithm, and corresponding af is calculated. The values a^ and af are compared and this information is used to generate the next parameter vector, P3. This procedure is repeated until somefittermination criterion is fulfilled. These repeating circles are called iterations. The most essential difference between thefitalgorithms 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 algorithm. However, no onefitalgorithm can guarantee that the found parameters correspond
15.4. Data fitting
281
to the global minimum of cr^, which is in contrast with the polynomial function approximation 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 = cr{P), according to eq. (15.37).^^ The function cr{P) may have a few local minima, which are such values of P = Pi that a (Pi) < a {Pi + 6) for any S in some range ± A P , i. e. \5\ < AP . From all possible minima of function cr{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, PQ, 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 minimum. 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 measurements, 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
/(t) = ^ a , e - ^ ^ - ^
(15.52)
i=i
where M is the number of the exponential terms, kj are the rates and aj are the corresponding pre-exponential factors. The values kj and aj are the fit parameters, so formally P = (/ci,..., /CM, 0^1, •.., <^m), 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
/(P,x) = ^ a , 7 i , ( P ^ x )
(15.53)
where P ' is the subset of parameter vector P , i. e. the complete parameter vector is P = {P\ a i , . . . , a ^ ) , and Uj {P\ x) are the functions of P ' and x, which are rates kj and time t in the case of exponential fitting. The coefficients aj are linear model parameters, in contrast to the subset P\ which are non-linear parameters.^^ The linear parameters aj are also called ^^For a given set of experimental data the values X and Y are constants, unlike the parameters P which can be changed in certain limits. ^^The 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.^^ To find the minimum of the mean square deviation at fixed P', one can solve a system o
2
of equations obtained from conditions | ^ = 0, which are similar to eqs. (15.38) with aj in place ofpj, f
M
N
N
Y, ^jYl WiUi{P\Xi)Uj{P\Xi) I j=0 i=l
=
Yl WiUi{P\Xi) i=l (15.54)
M
N
N
Y, 0.3 E WiUM{P',Xi)Uj{P',Xi) j=0
=
Y
2=1
WiUM{P\Xi)
i=l
This is system of linear equations which can be written in matrix form as (7 • yl = D, N
where C is M x M matrix with elements Cji = Y WiUi{P\ Xi)Uj{P', Xi) and vector D N
has elements di = Y 'WiUi{P' ^xi). The formal solution of the equation is A =
C~^D,
i=l
which can be formalized for any dimensionality of the system, i. e. for any M?^ 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 (kj 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-^^
(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))^\na-kt
= A-kt
(15.56)
^^In the case of exponential decay, Ui = e~^^* at t > 0, and Ui = 0 at t < 0. Thus function Ui has maximum value of 1 at t = 0. ^^ Formally, the system has a solution if matrix C has non-zero determinant, det C 7^ 0, which requires that the functions Uj () are orthogonal or independent functions, i. e. no one of functions Uj () 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 Hnear function of time. Thus one can minimize K
(J^ = Y.w,{A - kU - In2/0'
(15.57)
i=l
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 = 6 + ae~^^) makes result of logarithm application a non-linear function.^^ 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.^^ 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 F = ^ (A, t). As it was discussed in Section 15.3.4, a common model is the sum of exponents of type
fiX,t) = Y,bj{X)e-''^'
(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, kj, 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 cr^-value, which is calculated as average of af-values at all wavelengths
"'.'jY.'' ' ' - L
<'55i>)
1=
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 ^^Semi-logarithmic scales are very common to present emission decays even when decays are multiexponential, 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 Figs. 8.2 and 12.7 (inset). ^^For 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
Eir-f ni ol =
'-=\
(15.60)
Ewhere rii is the noise level of the /-th decay.^^ On the output of the global fit one will have the rates kj and corresponding spectra, hj{\). In emission spectroscopy these spectra are called decay associated spectra (see Section 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 hj (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 J^ ^j(A), since e^ = 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 hj{\) is the linear reaction 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 obtained as sums of spectra hj{\) starting from the intermediate state index, i. e. for state L
j = J the spectrum is sj{X) = J2 ^jW-^^ 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 ^^In 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. ^^The errors of these calculations are of the order -^"^^, e. g. if formation rate for some intermediate state is 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^ 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'^ values between biexponential 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^ 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 2 0 ^ 0 % difference in cr^-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 Imow the spectrum range of measurements, expected magnitude of the signal, excitation wavelengths, time scale of the phenomena and other similar parameters. 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.^ 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 pumpprobe and up-conversion experiments to measure transient absorption and emission decays in femto- to nanosecond time domain, respectively.^ 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 ^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]. ^The laser system considered in Section 11.2, Fig. 11.8 is an example of such instrument. Also the upconversion 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.^ 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.
^This 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 At,^ there is a probability that two or more photons will hit the detector during time interval At and, thus, will be counted as one photon.^ 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 At if the average photon flux is n, is given by the Poisson distribution, i. e. eq. (4.11), (nAt) N lAt N iV! The average number of counted photons in time interval At is
(A.1)
oo
(A.2) 2=1
whereas actual number of photons oo
(A.3)
Nact = XI *-P* = ^^* i=0
Substituting P^ from eq. (A.l) to eq. (A.2) one obtains Nr
E (nAt)
aAt
i=l
{nAty
-nAt
At y^ i^^t) -nAt
-nAt
7I
{^nAt
i) = i
i+E
{nAty
-nAt
(A.4)
.i=0 ^Dead time is the time interval during which the next photon cannot be detected. ^Strictly speaking, the term detected photons, rather than the photons entering the photo-detector, has to be used. However, an ideal detector 100% 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 probabihty to obtain any number of photons but 0. The difference between the actual number of photons Nad, eq. (A.3), and the detected number of photons Nc, eq. (A.4) is A7V = Nact - iVe = nAt - 1 + e"^"^^
(A.5)
Expending exponent as power series AN = ^
^
- ^ - ^
+ ...
(A.6)
one obtains the relative error due to finite time resolution
^
AN
1 ^^
(5 = - — = -nAt Nact
2
(uAtf - ^ — ^ + ...
(A.7)
6
For example, if the desired accuracy of the measurements is ^ = 0.01 and the detector dead time is At = 50 ns, then the photon counting rate must be lower than n = 26At = 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
m = /nAt1
(B.l)
where the term -7=^ is due to normalization J_^ fif)dt = 1. A usual measure of the pulse or bandwidth is the full width at half maximum (FWHM) value, A t i . The relation between At and At 1 is given by equation
exp
[I^H)
(B.2)
Ai2
or Ail = 2 A i V - l n 0 . 5 = 2 V l n 2 A i « 1.67Ai. 2
To obtain the spectrum of the pulse one needs to calculate the Fourier integral +00
F{uj)
1 v/7rAt
^dt
-^At^uj^
(B.3)
where uj is the circular frequency, which can be converted to normal frequency as 27r/ = uj. Thus, the spectrum of a Gaussian pulse is a Gaussian band F{f) = e~^ ^^ ^ with the bandwidth of A / = (7rAt)"\ In eq. (B.3) the optical frequency of the pulse, fo = f, was omitted, which can be justified if/o >> At~^, 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 (At < 20 fs) the high frequency "filHng" of the pulse should be taken into account in eq. (B.l). In frame of the relatively narrow spectrum approximation, the wavelength bandwidth is A 2
A A = ^f^, and substituting the pulse width one obtains AA = ^
(B.4)
or using FWHM values for the pulse and bandwidth, respectively, A A . = 1 ^ . ^ « 0 . 8 8 ^ 2
TT
cAti
cAti
2
2
(B.5)
For example, the spectrum width of a At i = 5 0 fs pulse (At ^ 30 fs) at A = 800 nm (Ti:sapphire laser) is AAi ^ 0-88^^^ ~ 38 nm. This corresponds to the case of so-called 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 Hght intensity the probability of absorption of two and more photon at once increases 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 /. If/in is the light intensity before a sample, then after the sample the intensity is [29] _ (1 - Rfe--' ""*" ' " l + f / . „ ( l - i ? ) ( l - e - 0
.ru ^""^
where R is the reflectivity of the sample, a is the linear absorption coefficient (the one introduced 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 -r-
J- in,
1 + Plh This can be used to define two photon absorptance (in a manner similar to the linear absorptance, see eq. (1.9) in Section 1.1.1) 0^2ph —
J
Naturally, the two photon absorptance depends on the incident intensity. At low values of (511in the two photon absorptance is proportional to the input light intensity lin, 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 T a b l e d . 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 7^^ = -^ • j ^ ^ ?^ 0.2 x 10^ W cm~^ at 264 nm (the third harmonic 293
294
Two photon absorption
Table C.l: Two photon absorption coefficients, /?, of some materials measured at wavelengths A. material fused silica water methanol hexane chloroform
f3x 1 0 i \ c m w - i 2.4 <0.13 49 34 57 95
wavelength, nm 264 355 264 264 264 264
reference [30] [29] [30] [30] [30] [30]
of Ti:sapphire laser). Assuming a spot size typical for pump-probe experiments d = 1 mm, i. e. spot area s = ^(P ^ 10~^ mm^, and the excitation pulse width At = 100 fs, the pulse energy, at which the two photon absorption will be 1%, is E = hnsAt ^ 0.2 /xJ, 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 called data fit. General fit algorithms can be divided on two steps repeated sequentially:
1.
For some approximation P J 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. C o m m o n problems of different minimization methods are guesses of an initial approximation, p 0 (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 A p = A p l . . . A p N , are defined and actively used.
D.1
Stepping algorithm
This is probably the simplest fitting algorithm. A new approximation is generated by changing only one of parameters, say parameter Pk, and keeping the other parameters unchanged. The parameter pk is scanned with the step Apk until minimum of cr2 is found. ~ When optimization o f p k is complete the next parameter is subjected to the same procedure. When all 1Typically, 0.2 is computed for a parameter vector P and for the same vector but with Pk + APk substituted in place ofpk. If o-2(pk) > o'2(pk + Apk), then Pk is increased sequentially by steps Apk (i.e. testing 0.2(pk + 2Apk), a2(pk + 3Apk ) . . . . ) while o-2 is decreasing. If o-2(pk) < 0.2(pk + Apk), then Pk is decreased sequentially (i.e. testing o-2(Pk - APk ), o-2(Pk - 2 A p k ) . . . . ) while o-2 is decreasing. 295
296
Fit algorithms
the parameters are optimized, the steps A 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 direction of the most quick decrease in a 2, and testing sequentially points (in P space) in that direction. This can be done by comparing (72 for pairs of parameter vectors when only one parameter is changed. To calculate i component of 0 .2 gradient one needs to calculate (72 for P = P l , . . . , P i , . . . P N and (7~2 for P = Pl, . .,p~. +. A .p , . PN, then/-component of the gradient is si-
0(72 Opi
(7~ _ (72 Ap
(D.1)
When all the gradient components are determined the new parameter vector p j + l can be computed based on the previous approximation PJ and gradient vector S = Sx, s 2 , . . . , SN as
(D.2)
p~+l = p j _ s i A p
or in vector form f j + l = p j q_ S A p . If parameter p j + l gives better (smaller) (72 value, the next point in the same direction is tested, i. e. pj+2 = p j + l + S A p = PJ + 2 S A p . The procedure is repeated until minimum of (72 (in direction given by the maximum slope of (72(p), or by vector S) is found. Formally 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 parameters 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
N e w t o n method
Further improvement can be done by presenting (72 as power series of parameters P N
(72(p) = (7~ + E
002"
N
=~pi (Poi - Pi) + E
i=1
N
~
02(72
OpiOp~ (p°i - P~)(Pok - Pk) + . . .
(D.3)
i=1 k=l
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
(72(p) = (702+ Z
N
E
i=1 k = l
Aik(poi - Pi)(Pok -- Pk)
(D.4)
D.4. Random search
297
This is polynomial of the second order of N variables. It has minimum of cr~ at point Po : Pol,. • •, Polv. 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 perform a series of experiments by testing different parameter vectors (say Pi 4- Api ) and obtaining corresponding ~7z. This set of data can be used for mean square minimization by solving equations similar to eqs. (I 5.5 l) and obtaining parameters P0 = pol,. •., P0N. 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
R a n d o m 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 common spectroscopy solvents: A,,.~ is the ultraviolet cutoff wavelength (absorbance 1.0 in I cm cuvette), ~ is the refractive index, e is the dielectric constant, ~l is the viscosity, T,, is the melting point, and T~, is the boiling point (data from [17, 31])
Solvent acetone acetonitrile benzene benzonitrile chloroform cyclohexane 1,2-dichloroethane diethyl ether dimethyl sulfoxide ethanol glycerol hexane methanol
1-propanol pyridine tetrahydrofuran toluene water
k,,~., nm 330 190 280 245 210
226 218 265 210 207 210 210 210 330
220 286 190
~ 1.3591 1.3460 1.5011 .5289 .4459
.4262 .4448 .3527 .4170 .3611 .4746 .3749 .3284 .3840 .5067 .4052 .4960 1.333
e 21.0 36.6 2.284 25.9 4.72 2.023 10.4
l/ . 1():~, N s m 2 0.302 0.341 0.65 1.3 0.54 0.98 0.78
4.22
0.22
47.24 25.3 46.5 1.890 33.0 20.8 13.26 7.52 2.385 80.2
1.1 1.03 945 0.31 0.58 1.8 0.87 0.46 0.55 0.89
299
T,,,, °C -94 44 5.5 -12.7 6::I.6 6.6 -35.7 116.3 18.5 -114 18 -9,5.4 97.7 -127 -41.6 -108.5 -94.9 0
T~, °C 56 81.6 80 191 61.1 80.7 83.5 34.5 189 78.3 290 68.7 64.7 97.2 115.2 65 110.6 100
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Index convolution integral, 162, 277 correction spectrum, 113, 114 cuvette, 100, 122 matching pair, 95, 101 CW lasers, 45 CW mode, 42
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
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
band limited pulse, 291 base line, 91, 92, 94 BBO, 220, 231 black body, 9 spectral emittance, 9 Brewster angle, 206 cavity 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 discriminator, 151 continous wave mode, 42 contrast factor, 25
eigen-value, 271 304
Index
Einstein coefficients, 12 electronic levels, 7 emission corrected spectrum, 114 emissivity, 10 equilibrium, 273 Euler formula, 17 excitation spectrum, 109, 115 exciton annihilation, 214 extinction coefficient, 4 extraordinary polarization, 56, 220 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 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 harmonic waves, 17 Helmholtz equation, 18 hole-burning, 242 homogeneous broadening, 238, 242 idler wave, 59 image intensifier, 75 indirect measurements, 263 indirect problem, 264
305
inhomogeneous broadening, 238,242 instrument response function, 162, 206, 212,223,277, 284 interference, 19 interferometer, 20 intermediate state, 269 mtersystem crossing, 12 intrinsic rate constant, 269 inverse kinetics, 272 reverse population, 39, 46 reverse problem, 264 reversion, 40 iteration, 280 Johnson noise, 70 Kerr effect, 232 Kerr lens, 53, 195, 232 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 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
306
monochromator, 31, 91, 95, 97, 109, 146 dispersion equation, 32 multichannel analyzer, 152 Nd:YAG laser, 49, 51, 57, 146, 154, 195, 210 NEE 73 noise equivalent power, 73 non-correlated noise, 277 non-linear parameter, 281 off axis mirror, 205,223 optical density, 4 optical parametric amplifier, 59, 198 optical parametric oscillator, 59, 146 ordinary polarization, 56, 220 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 compressor, 205 probability
Index
density function, 63 function, 63 pump-probe, 83, 185,293 mono-color scheme, 185 two-color scheme, 188 Q-switching, 46 qualitative problem, 285 quantum efficiency, 73 quantum noise, 70 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 saturable absorber, 195 second harmonic, 167, 198,203, 210, 217, 228, 235 generation, 50, 53, 56, 83, 154, 231 generator, 83 second order rection, 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, 111, 191 spectrophotometer, 89
Index
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, 117 stratcher, 196 streak camera, 179 susceptibility, 55 synchronous detection, 92,200, 210 detector, 174 systematic error, 62 T-scheme, 135 TEM, 43 mode, 44, 45, 50 thermal noise, 70 thermal relaxation, 11 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, 172 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
307
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
