Optical Spectroscopy: Methods and Instrumentations

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 wa