Molecular photonics: fundamentals and practical aspects

Molecular Photonics Kazuyuki Horie, Hideharu Ushiki, Franqoise M. Winnik Molecular Photonics Fundamentals and Practi...

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Molecular Photonics

Kazuyuki Horie, Hideharu Ushiki, Franqoise M. Winnik

Molecular Photonics Fundamentals and Practical Aspects

KODANSHA

@WILEY-VCH

Weinheim . Berlin . New York . Chichester Brisbane Singapore . Toronto

Horie, Kazuyuki ( 1.3, 1.4, 2, 5) Department of Chemistry and Biotechnology, Graduate School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Ushiki, Hideharu (Intro., 1 . I , 1.2, 4) Department of Biomechanics and Intelligent Systems, Graduate School of Bio-applications and Systems Engineering, Tokyo University of Agriculture and Technology, 3-5-8 Saiwaicho, Fuchu-shi, Tokyo 183-8509, Japan Winnik, Franqoise M. (3) Department of Chemistry, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4M 1 Numbers in parentheses refer to the chapters

This book was carefully produced. Nevertheless, authors and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural detailes or other items may inadvertently be inaccurate.

Published jointly by Kodansha Ltd., Tokyc? (Japan), WILEY-VCH Verlag GmbH, Weinheim (Federal Republic of Germany) Library of Congress Card No. applied for A cataloguc record for this book

IS

available from the British Library

Dculsche Hibliothek Cataloguing-in-Publication Data: Horie. Kazuyuki:

Molecular Photonics / K. Horie. H. Ushiki, F.M.Winnik -Wcinlieim; New York; Chichester; Brisbane; Singapore; Toronto: Wiley-VCH. 2000 ISBN 3-527-30252-2 (WILEY-VCH) ISBN 4-06-209629-3 (KODANSHA)

Copyright 0 Kodansha Ltd., Tokyo, 2000 All rights reserved. No part of this book may be reproduced in any form, by photostat. microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd.(except in the case of brief quotation for criticism or review). Printed in Japan

Preface

The relationship between light and humankind has a long history. Light is always given a positive power, while darkness is associated with illness and depression. According to the Bible God created light on the first day. The Big Bang theory also tells us that a burst of photons occurred together with the explosion of the universe. Every aspect of modem natural science points to the fact that all life on earth needs sunlight for survival. With the advent of the laser in 1960, humankind entered a new age. The world of advanced technologies gradually moved from the age of electronics to that of optoelectronics and photonics. Everyday we encounter new technologies designated by prefixes, such as “photo” and “opto.” Optical discs, optical communication, photodiodes, and photosensors are ubiquitous. Newspapers are produced electronically by processes based on photosensitive polymers. Photoresists form the basis in the manufacture of ICs and LSls. Over the last ten years, the applications o f organic compounds have been reevaluated in various microelectronics fields, not only as necessary support materials, such as dielectrics, but also as materials able to play an active role in devices and systems. Liquid crystal displays, for example, are found everywhere. New organic compounds showing interesting and improved electronic and photonic properties are reported almost every day. New light-triggered materials are designed for molecular or bioelectronic devices. The academic world however lags behind these changes. If someone wants to study the fundamentals of light science, that individual will have to attend lectures in several university departments. Lectures on optics, laser technology, and optoelectronics will have to be taken in physics and engineering physics departments. Photochemistry in contrast is taught only in chemistry and chemical technology departments, usually as part of a course on spectroscopy, physical chemistry, or organic chemistry. Lectures on organic compounds are very rarely offered in departments of physics and electronic technology. Well aware of this unfortunate situation, two of the present authors (KH and H U ) started to discuss a systematic and unified approach to photochemistry, photophysics, and optics. V

vi

Preface

Based on their experience teaching the organic chemistry of photomaterials and photophysical chemistry (KH), and the quantum theory of light (HU) in the graduate schools of their respective universities, they wrote “The Science of Hikari-functional Molecules” published in Japanese. The third author (FMW) joined the discussion during her stay in Japan and suggested the publication of an English version of the book. She wrote Chapter 3 of the English version, based on the original Japanese text, and contributed to the editing of the complete volume, text and tables. The study of the interactions of light and materials constitutes one of the fundamental subjects in natural science and technology. In Japanese, a single word, hikuri, describes all phenomena related to light. In English, two different prefixes are used: “photo-” from the Greek photos (light) and “opto-” from the Greek optos (seen). Photochemistry and photophysics deal with light-induced changes in materials and in their electronic states, topics studied mostly by chemists. Optics deals with materials-induced changes in the properties of light, a subject developed primarily by physicists. Both aspects of lightlmaterials interactions have gained equal importance in electronics and photonic materials. Their coalescence has led to the relatively new concept of molecular photonics. The underlying unity which connects all light-induced phenomena is best appreciated if the book is studied entirely and in the order presented. The introductory chapter reviews the historical background and gives a survey of current light-related research fields. In Chapter 1, the fundamentals of molecular photonics are introduced in terms of the principles of optics, the molecular field theory, the radiation field theory, and the interaction between molecular and radiation fields. The importance of a conceptual understanding of the essence of the interaction between light and materials is emphasized throughout this chapter. Chapters 2 and 3 deal with the light-induced changes in materials. The characteristics of photochemical reactions are summarized in Chapter 2, and typical processes of photophysical chemistry such as excitation energy transfer and photoinduced electron transfer are discussed in Chapter 3 . Various examples of photofunctional molecules developed by chemists are given in these chapters. Chapters 4 and 5 are dedicated to the study of the materials-induced changes in light, thus far exploited mostly by physicists. Scattering phenomena and the materials-induced changes in light under the application of an electric, magnetic or acoustic field are presented in Chapter 4. The changes in light by light irradiation, namely multiphoton absorption processes, are introduced in Chapter 5 , where nonlinear optical phenomena and coherent spectroscopy are discussed. Throughout the book, key concepts are presented in tabular form consisting of drawings, graphs, tables, or formulae describing a given concept. In each section the most important concepts are summarized in an overview table with specific explanation in the text. We believe that such visual summaries of key concepts of an entire section result in a more active understanding of new topics as they appear in the book. These one-page-size tables might be used for transparencies. The quantum theory of light developed by Einstein in the beginning of the 20th century gave a unified concept to the particle theory and the wave theory of light. However, the successive developments of the science dealing with light-induced changes in materials and the science dealing with materials-induced changes in light evolved to different streams of science. We feel that the advent of the laser has brought us to a new age, where these two separate streams begin to interact with one another and develop nonlinearly. Molecular photonics has become a truly interdisciplinary field, where both streams are intimately interwoven. This volume attempts, as much as possible, to bring a unified approach to the

Preface

vii

study of light. Because of the breadth of these fields, a detailed description of each subject could not be provided in the space allocated. The book is written for scientists, engineers, senior students and graduate students interested in light-related sciences, not only in chemistry, but also in physics, electronic technology, and biology. Any comments on the book from the readers are welcome. The authors wish to express their special thanks to Mr. Ippei Ohta of Kodansha Scientific Inc. for his encouragement and patience, without which this book would never have been published.

Kazuyuki Horie, Hideharu Ushiki, and Francoise M. Winnik September 1999 Tokyo, Japan Hamilton, Ontario, Canada

Contents

Preface

v

Introduction The Concept of Molecular Photonics

1

0.1

Light as an Electromagnetic Wave

2

0.2

The Study of Optics and Photochemical Effects: a Historical Perspective

0.3 Recognition of Photo- and Opto-Related Areas

1 Fundamentals of Molecular Photonics

4

6

9

1.1 Fundamentals of Optics 9 1.1.1 General Formula of Wave 11 1.1.2 Refraction and Reflection 12 1.1.3 Interference 13 1.1.4 Diffraction 15 1.1.5 Polarization 18 1.2 The Molecular Field Theory 20 1.2.1 The Old Quantum Theory 21 1.2.2 Atomic Orbitals 27 1.2.3 Molecular Orbitals 28 1.3 The Radiation Field Theory 30 1.3.1 Maxwell's Equations 31 1.3.2 The Electromagnetic Potential 35 1.3.3 Quantization of the Harmonic Oscillator 39 1.3.4 Quantization of the Radiation Field 40 1.4 The Interaction of the Radiation Field and the Molecular Field 42 1.4.1 Basis of the Interaction between the Radiation Field and the Molecular Field 43 1.4.2 Absorption and Emission of Light 44

IX

x

Contents

1.4.3 The Photophysical Processes 50 A. Excited Singlet Energy Transfer and Migration 53 B. Energy Transfer in the Excited Triplet State 54 C. Interaction of Excited Molecules 56 1.4.4 Photochemical Processes 56 1.4.5 Scattering Phenomena 57 1.4.6 The Laser Principle 63 References 65

2 Photochemical Reactions

67

2.1 Characteristics of Photochemical Reactions 67 2.1.1 Photochemical Reactions and Thermal Reactions 67 2.1.2 Electronically-Excited States and Reactivity 72 2.1.3 Photochemical Reactions in the Solid State 74 2.2 Photochemical Reactions and Physical Property Control 79 2.2.1 Photosensitive Polymers 79 2.2.2 Photochromism 85 2.2.3 Photoresponsive Molecules 90 2.2.4 Photochemistry and Biotechnology 93 2.2.5 Photochemical Hole Burning 94 A. Principle of Photochemical Hole Burning 94 B. Hole Profiles and Electron-Phonon Interactions 97 C. Efficiency of Hole Formation and Temperature Dependence D. Applications of Photochemical Hole Burning 102 References 102

3 Photophysical Processes

99

105

3.1 Energy Transfer and Electron Transfer Processes I05 3.1.1 Excitation Energy Transfer 105 3.1.2 Photoinduced Electron Transfer: Theoretical Background 3.1.3 Photoconductivity and Organic Photoconductors 1 12 3.1.4 Photoinduced Electron Transfer Membranes 1 17 3.2 Photophysical Molecular Probes 122 3.2.1 Luminescence Probes 122 3.2.2 Molecular Motion Probes 127 3.2.3 Microstructural Probes 136 3.3 Chemiluminescence and Electroluminescence 140 3.3.1 Chemiluminescence 140 3.3.2 Electroluminescence 145 References 148

108

Contents

4 The Interaction of Light with Materials

151

4.1 Light Scattering 15 1 4.1.1 Rayleigh Scattering 154 4.1.2 Raman Scattering I56 4.1.3 Brillouin Scattering 157 4. I .4 Optical Propagation Loss of Optical Fibers 158 4.2 Optical Effects 163 4.2.1 Electro-Optic Effects 163 4.2.2 Electro-Optic Effects in Liquid Crystals 166 4.2.3 Magneto-Optic Effects 169 4.2.4 Acousto-Optic Effects 17 1 References 175

5 The Interaction of Light with Materials I1

177

5.1 Saturation of Absorption and Multi-Photon Absorption Processes 5.1.1 Lasers and Coherent Light 177 5.1.2 Saturation Spectroscopy 178 5.1.3 Nonlinear Susceptibility 183 5.1.4 Frequency Conversion of Light 186 5.1.5 Nonlinear Optical Materials 188 5.2 Coherent Spectroscopy 193 5.2.1 Coherent Raman Spectroscopy 193 5.2.2 Photon Echo Technique 197 References 199 Index

201

177

xi

Molecular Photonics: Firndanientals and Practical Aspects Kazuyuki Horie Hideharu Ushiki 8, FranGotse M Winnik

.

Copyright Q Kodansha Ltd Tokyo. 2000


At the point of nihility without time nor space, an explosion of dense matter marked the origin of the universe. lmmediately after this accident, a fireball of radiation at unbelievably high temperature formed in a tiny volume. Many photons were produced by collisions between quarks and anti-quarks. This is the Big Bang theory of the genesis of the universe. It is not unusual for books on the concept of light to begin with a quotation taken from the Bible on the creation of the world. Indeed, the relationship between mankind and light has a long history and it has been strongly connected to all aspects of life, from antiquity to the modem ages. Man has always been careful to distinguish light from heat. Light is always given a much greater importance throughout the ages. Light is a crucial tool in many religions. Men do not speak badly of light. Note for instance that light is used in the Japanese word Goraiko describing the admiration of the sun, while heat is included in an expression describing hell, Shakunetsu-Jigoku, the scorching heat in hell. Hence it comes as no surprise that man has always been fascinated by the multiple properties of light. This book is dedicated to the study of light, its interactions with matter, and the transformations of matter triggered by light. We have chosen an integrated approach, inspired by a recent trend trying to merge the physical aspects of light, such as wave optics and field theory, and the more chemical concepts, such as photonics and quantum theory. We have not separated from each other the descriptions of the “photo” and “opto” related areas, but rather we have merged them under the single concept of “Molecular Photonics.” Rather than trying to answer the question, “What is light?” we will strive here to illustrate how we, as human beings, understand the concept of light. We will often stress that in order to understand a difficult phenomenon or theory, it is not enough to apply one’s mind, but that one’s entire being should become involved. In other words, one cannot attain an understanding of the essence of certain phenomena by a unique way or a single concept. We urge the reader to receive all the information with an open mind and to view the various topics described as a gathering of many ideas, observations, and conclusions. In natural sciences, research proceeds when scientists become part of the developments as complex unique human beings. I n order to understand the concept of “Molecular Photonics,” it is crucial for the reader to undertake a study of fundamental principles. Chapter 1 “Fundamentals of Molecular Photonics” includes four sections dedicated to optics, the molecular field theory, the radiation field theory, and the interactions between the molecular field and the radiation field. Fundamental principles are often treated in an introductory chapter, leading the reader to think that they are of little importance and that they can be understood with ease. This trend of relegating the fundamentals to a brief introduction is getting increasingly common in natural I

2


sciences. We do not follow it in this book, since we strongly believe that in order to grasp the fundamental aspects of molecular photonics, many complex abstract concepts need to be understood. Natural sciences have grown over the years by innumerable twists and turns, reflecting the commitments and personality of researchers over many centuries. Some of the difficulties in understanding the fundamental aspects of any science can be alleviated if one uses intuition together with solid reasoning. Intuition is needed to see the relationship between a virtual image, an equation, and a plot of experimental data. We are well aware of the importance of this relationship based on intuition. We explain each fundamental concept of “Molecular Photonics” through tables and figures produced by an original computer system. Unlike many natural sciences books which tend to avoid equations at all cost, we take time here to derive equations. Mathematical equations can be of tremendous help to understand abstract concepts. An equation clarifies the predictions of an entire discussion without any ambiguity. The use of equations is very important to balance the intuitive aspects of the natural sciences. Taking note of the physical meaning of equations, without worrying too much about the mathematical derivations, is a useful first step for the readers not accustomed to use them.

0.1 Light as an Electromagnetic Wave Modem research on light is based on two apparently irreconcilable observations, on the one hand light behaves as an electromagnetic wave, as demonstrated by Hertz in 1888, and, on the other hand, light has a particulate nature as described by the quantum hypothesis put forward by Einstein in 1905. In this book the study of molecular photonics starts from these two inconsistent concepts. We think that by keeping this contradiction constantly in mind, it will become possible at last to answer the question, “How do we, as human beings, understand the concept of light?” This opinion is not like that of a manager, who will try to solve the inconsistency of two thoughts by adopting forcibly an agreement suggested by an outsider. We believe that final understanding will be achieved when both opinions will be accepted intuitively. First, following Hertz’s view, we will describe light as an electromagnetic wave. Table 0.1 lists typical wavelength and energy values of the electromagnetic waves covering the entire spectral range, from radio waves to y-Rays. Also given in Table 0.1 are the inventors’ names, the physical phenomena, practical emission sources, detectors and research fields associated with each type of electromagnetic wave. The electromagnetic spectrum encompasses the range of typical light, from the near-infra red to the near ultraviolet. Therefore in order to understand the fundamental properties of light, it is mandatory to study electromagnetic waves in general. While the wave-particle dual nature of light is often discussed, hardly anyone ever mentions the wave-particle dual nature of the electromagnetic wave. Light indeed occupies a very special place in our lives. We experience it every day, such that, intuitively, we already understand many of its properties. The dual nature of light has several unexpected consequences and it has important implications in the study of the interactions of matter with light. We can discuss at length the behavior of light as an electromagnetic wave, what really matters, however, is how light interacts with molecules and atoms, the basic building blocks of matter. The light-induced electronic transitions in a molecule must be viewed in this context. In contemporary science,

3

Table 0. I Spectral Map of Electromagnetic Wave Electromagnetic Wave

I

cm

I

1

I

I

I

I

I

I

I

loo

I

I

I

1

I

I

I

I

1

I

I

I

I

I

I

300kHz 3MHz 30MHz 300h4Hz 3GHz 3OGHz 300GHz 3THz 30THz 300THz 3PHz 30PHz 300PHz 3EHz 30EHz

Fr

I

- 1

I

I

I

I

I

0.1

I

10

I

1

1

I

I

I

loo00

1000

I

I

I

1

I

124neV 124neV 124 1 eV I 24meV 124meV 12 4eV 124keV 124keV 1 24 II eV 124 j~ eV 12 4meV 1 24eV 124eV 12 4keV 12 4neV Energy

-

1

120 II J

12d 1.2mJ

1

I

I

I

28.8 j cal 286

I

123 12OmJ

I

120J 125

I

1

1

I

I

I

1

I

l2kJ

1.2kJ

I

1

I

I

120kJ

I

I

1

14 4K

I

I

I

I

1 44kK

144K

Mcrowave

(HF) (VHF) (UHF)(SHF)

(EHF)

( 1888)

< Phenomena motion of electron or atomic nucleus in electromagnetic field Emission LC circult magnetron source klystron Detector antenna and detector radio telescope

Field

Ramation

I

12GJ

1.2GJ

I

I

I

I

I

I

communication radio

television radar

electncal and electronic eng ineenng

-

Near

144MK

I4 4MK

144GK 144MK

Ultraviolet

-._VL

Ramahon

> .

Far

H.R.Hertz

144kK

I4 4kK

c .

(LF) (MF)

lJSe

12MJ

I

Infrared Radio Wave

Dlscoverer

120MJ

2.86mcal 286mcal 28.6cal 2.86kcal 286kcal 28.6Mcal 2.86Gcal cal 28 6mcal 2 86cal 286cal 28 6kcal 286Mcal 286Mcal

rmper- 144 u K I44m~ 144mK I4 4mK 1 44K ature 144 /lK

Wave Name

12MJ

7 -

.-

RsYs

x-Rsys

r-

NearVacuumSofi Hard I R I R w uv x-ray X-ray W C Rhtgen F.W.Herschel J.W.Ritter A.H.Becquerel (1800) Newton (1801) (1895) (1896)

molecular rotation vibration maser heat-source

.-.-

. r

c-

electronic transition of inner orbital

electronic transition mercury lamp mscharge

-

nuclear reaction

synchrotron orbital radmhon leSCr tube nuclear decay photocell fluorometer Geiger-Miiller photomult~plier photograph tube thermocouple tube scmbllator matenal nucrowave IR-photography photoreabon communication heat-source Iighhng stlucture aeroradar HF-heatmg matenal analysis analysis molecular chemsby and physics astrophysics photo-reachon molecular structure

hlghflergY physics structure analysis

4


this idea resulted in the birth of a new discipline dedicated to the study of “the interaction of the radiation field with the molecular field.” Central to this study is Equation (0.1) based on perturbation theory:

where H , HR, H M ,and H I are the Hamiltonian operators for the total radiation, the radiation field, the molecular field, and the interaction of the radiation field with the molecular field, respectively. The HI term is central to the study of the interaction of the radiation field with the molecular field, but one needs to understand also the H Mand HR terms, in order to extract the H I term. Chapter 1 deals with the fundamentals of optics and photonics in terms of the three operators, HR,HM, and H I .

0.2 The Study of Optics and Photochemical Effects: a Historical Perspective Historical perspectives are very important in a course of natural science. One becomes aware of facts rarely presented in a general history course. How was a discovery made? What was the purpose of a specific experiment? What was the researcher trying to achieve in the general scientific context of his time? Trends can be uncovered, which lead from one discovery to the next. We will see then that science is not a gathering of two-dimensional information but, rather, a dynamic human drama in a four-dimensional space. Table 0.2 presents a summary of the history of modern research in light. In this table we present modern developments in light research from the fundamentals of optics (1 7th century), to the electromagnetic field, the molecular field, the radiation field, the interaction of the radiation field and the molecular field, leading finally to “Molecular Photonics,” as we know it today. The understanding of light-induced phenomena grew through the controversy between the wave and particle theories, the formulation of Maxwell’s electromagnetic theory, the measurement of the speed of light, the development of the spectral measurement method, the formulation of the quantum theory, the discovery of the laser, etc. The dawn of modern research in light can be traced to the early 17th century, with the discovery of the telescope. Thereafter optics developed rapidly. Merging of research in optics and electrostatics resulted in the classical electromagnetic theory, culminating in the formulation of Maxwell’s electromagnetic theory. In 1888 Hertz discovered the electromagnetic wave and declared victory for the wave theory of light. These major events define the first period in modern light research. At the same time, however, a new technique, the spectral measurement method based on the flame reaction, was gathering momentum in research, leading to the establishment of the periodic table. The new method gave a fatal blow to the wave theory of light. It coincided with the birth of the quantum theory which took over in the 20th century. Therefore, during this first period the historical flow took many twists and turns, torn in two different directions, by the quantum theory and the wave theory. The wave theory became quantum electrodynamics, via the discoveries of the microwaves, VHF, and UHF, leading to the creation of the laser from the maser. This line of progress yielded the concept of field based on wave optics. A second line, closely linked to the quantum theory, resulted in molecular spectroscopy and excited-state chemistry, and the creation of various

5

Table 0.2 Historical Chart of Modern Light Research Classification of Age

Optics

Ramatlon Field

Molecular Field

Invention of the telescope The law of refraction The law of diffraction Discovery of the spectral Research in static electncity pnnciple and birefnngence Controversy between wave and pamcle theones for light

18th Cen-

P

Of

Research of conductor Bscovery of the electnc charge

Static Electricity Achromatic condition

Law of light absorption

tury

Coulomb’s law ige of Electromagnetics Interference of light Discovery of planzation Measurement of 19th Century

light wavelength Lge of Spectroscopic Methods

Measurement of light velocity Discovery of optical rotatory dispersion

the Old Quantum ‘Theory

Discovery of magnetic birefnngence

20th

ge of Electro-

Dwovery of IR and W Ampere’s law

Measurement of spectra

Biot-Savart law Faraday’s law of induction Derivation of Maxwell’s equations Discovery of

Stokes’ law Spectroscopic method Equation of spectral lines Zeeman effect

electromagnetic wave

Photoelectric effect

Invention of the wireless Quantum and light quantum theory

X-ray hfiaction Bohr’s theory of atom hscovery of the ultrashort wave rhscovery of meter wave Establishment of Radiotelescope wave mechaxucs Invenhon of the maser Ihscovery of Raman effect Energy transfer Invention of the laser Development of Excmer Enussion

Cen-

Age of the Laser

nonlinear optlcs by laser Dlscovery of the quantum Hole effect

Establishment of o r w c photochemsby

6


instrumental analytical techniques, via the explanation of the atomic and molecular structures. This historical flow from the appearance of the quantum theory to that of the laser can be classified as the second age in modern light research. Recently, the merging of these two separate lines which led to the discovery of the laser has opened the third period in modern research, considered by many as “light’s golden age.”

0.3 Recognition of Photo- and Opto-Related Areas In this chapter we will attempt to uncover the events, in recent history, that led to the formulation of the concept of “Molecular Photonics.” What are the research areas supported by the concept of “Molecular Photonics?” What are the relationships between the fundamental fields described previously and “Molecular Photonics?” These questions need to be answered. We will draw a picture of “Molecular Photonics” starting form several related fields (see Table 0.3). The interaction of the radiation field with the molecular field has two aspects: “the change of the medium by light” and “the change of light by the medium.” Traditionally, the former is included in the chemistry curriculum while the latter is part of the physics curriculum. The rationale behind this separation remains unclear. The light-induced transformations of a medium are of two types, photochemical reactions following the rules of organic chemistry, and photophysical processes based on quantum chemistry, chemical kinetics, and molecular spectroscopy. The interactions describing the change of light by the medium are divided further into effects related to electrodynamics, quantum mechanics, and statistical mechanics. We propose then that “Molecular Photonics” is the main concept which includes all light-related research and, consequently, we have adopted in this book the following approach. First we describe fundamental concepts of optics (Section 1 . l ) , the molecular field theory (Section 1.2), and the radiation field theory (Section 1.3). The concept of “Molecular Photonics” emerges naturally, as the meeting point of these fundamental concepts in modem research of light. The interactions of the radiation field with the molecular field are often compared with heat-induced phenomena. Thermal energy is supplied gradually to a system, in contrast, light as an energy pack gives a large amount of energy (a photon) to a molecular system in a short time. According to Table 0.1, a molecule can gain large energy, about 100,000 K in s. Heat- and light-induced phenomena temperature scale in a flash lasting no more than are as different from each other as the live of a middle class worker and that of a gambler who has won the jackpot! We all appreciate how human behavior of the rich differs from that of the middle class. One molecule gradually wastes the energy obtained by heat. The molecule struck by light immediately changes its personality. Moreover the behavior of these molecules depends on these environments. The light-induced phenomena reproduce daily events in human society on the microscopic scale. Science philosophers often argue about the relative merits of the analytical and synthetic approaches to scientific research. A synthetic viewpoint cannot be created without analysis, and conversely, the systemization and development of a theory necessitates analysis. Analysis is a scientific tool or method which becomes meaningful only if it derives from a synthetic view of phenomena probed by experiments. The proposed new concept of “Molecular Photonics” cannot escape this methodology. It may be seen as a dogma at first. But through

7 Table 0.3 Classification of Modem Light Research and Interaction between the Radiation Field and the Molecular Field Fundamentals and Background of Molecular Photonics

Molecular Field Theory

First Classification

Second Classification

Photochemcal

(n, I ) Transition ( I , I ) Transition

Old Quantum Theory Atom and Molecule Hybndlzed Orbital Molecule wth i -electron Molecular Orbital Method

Orgamc Chermstry Intermolecular Processes

Chermcal Kinetics

,

,

~

s , 1 , #>,

8 t

"

I,

I, I , I ,

, I $ $ I# 3

It

. t ,

1 I

,I I , t ,

I,

.I,

I , I I

I

Light Scattenng

I

$ % # I

Ramahon Field Theory

Photocycloaddition Reactions, Photoisomerization, Electroocyclic and Photofragmentation Reactions, etc.

Intramolecular Processes

Molecular Spectroscopy

various Phenomena

'

t

,,I , $ ,>

I

' "

,I

)I

Electrooptical Effects

Maxwell's Equation Canomcal Equation

E

3 4

1

X

U

.-

L

Absorption, Emission, Internal Coversion, Intersystem Crossing, Deactivahon, etc

-v

E x c m r , Exciplex, Energy Transfer, Electron Transfer,etc

C

v)

L

E

O

.-

3

e,

O

Rayleigh Scattenng, Raman Scattenng, Bnllouin Scattenng, etc Pockels Effect, Kerr Effect, Electrooptical Effect in Liquid Crystals,

E 3 d

3

V

Magnetooptical Effects

Quantization of Quantization of

Zeeman Effect, Faraday Effect, Voigt Effect, etc.

.3

L L

3

O

Sonooptical Fundamentals of Optics Refraction, Reflection Interference Diffraction Polanzahon

Quantum Mechanics

;

Effects

B r a g Reflection, Raman-Nath Diffraction, etc.

Saturated Absorption and

Optical Bistability, Lamb Dips, Hole Burning,

Multi-photon Absorption

Perturbation

Frequency Transformation, etc.

Coherent spectroscopy

CARS, RKES, FWN, etc.

Time-resolved Coherent spectroscopy

Photon Echoes, Stimulated Photon Echo, etc.

QuanStatishcal Mechanics

v)

V .3

v)

x

L:

a

8


its many twists and turns between analysis and synthesis, the search for understanding becomes pleasurable and full of unsuspected discoveries. It is not a monotonous journey. It changes course abruptly as soon as a conclusion has been reached, implying, maybe, that real human beings are the driving force behind the progresses of the natural sciences.


.



1.1 Fundamentals of Optics The theories of photo- and opto-related areas can be classified into three categories: the fundamentals of optics, the molecular field theory, and the radiation field theory. As we defined molecular photonics by Equation (0.1) which relates “the interaction of the radiation field with the molecular field,” it may seem sufficient to restrict our discussion to the molecular field theory and the radiation field theory. However we believe that the fundamentals of optics are also very important to understand and appreciate all the “photo” and “opto” concepts described in this book. To support this view, consider the following. ( I ) The behavior of the electromagnetic wave can be predicted in a first approximation from the simple theory of optics. This prediction contains many approximations and several conditions, but it gives us a quick, intuitive picture of the electromagnetic wave and its properties. ( 2 ) The fundamentals of optics have been developed from phenomena observed by the human senses. This is certainly not the case for the theories of the molecular field and the radiation field. We should reach a stage in our understanding of the phenomena, where we can imagine intuitively the interaction of the radiation field with the molecular field from a basic knowledge of optics. ( 3 ) Since the emergence of the laser, the fundamentals of optics are applied in areas far beyond the traditional applied physics laboratory. It is particularly important for researchers with a chemical background to gain an intuitive understanding of the laws of optics. Typically, books on optics explain the principles of the various optical experiments and instrumentation.’ 4 J The first section of this book, while dedicated to optics will be limited to a description of the general formula of wave and the phenomena of refraction, reflection, interference, diffraction, and polarization. An overview of the fundamentals of optics presented in this section is shown in Table 1 . 1 . Most books on optics begin the explanation of the wave by stating Maxwell’s equations, and this is most appropriate, no doubt. Here we tried to follow a different approach in order to set the stage for the description of the radiation theory in the next section. However in explaining optical phenomena, it is impossible to avoid the concept of wave. So, in this section, although reluctant to do so, we introduce the equation of the three-dimensional plane wave.

9

10

Table 1.1 Summary of Fundamentals of Optics CGmralized wave equation>

Wave

$ ( r , t ) = $ o e x p { i( k r - a t ) ) $ ( x , t)=$oc 0 s t(Zz/A)(x-v t ) t & ) $ ( x , t = $oc 0 s t ( 2 7 2 / A ) x-272 Y t 4 I

+

I N2

t y Z t ( x - 1 ,1,,/2)2

s i n z ( 1 .Zjsin *’N1 I n i i t 2 ) ( 1 . Z ) 2 s i n 2 ( N 1= I I t )

Z = ( z s i n&)/A I P :Intensity of incident light For s i n 2 ( 1. I I I Z ) = O , maximumvalue I ~ A x = I a s i n 2 ( I . Z ) / ( I.212 N :number of slits I , :slit width l . l l , s i n & = m A ( m 4 . k I:-) I ,I I :space of slits ueflectivity: I+

Polarized li&t

‘P

42

1

n 1 c o s & 2 - n I c0 s + I n 2 co s & , + n l c o s 4 , =s i n2(&l-42)/si n2(&l+&2) in Jcm& z ) - in I/cos4 Ij Re=[ ( n 2 / c o s ~ + ( n I/cos+ I) = t a n 2 ( &I - & 2 ) / t a n 2 &a) R.=(

I

s:perpendicular to incident plane p:parallel to incident plane cBrewster’s angle

n lt an&s,t..k..=n~

1.1 Fundamentals of Optics

II

1.1.1 General Formula of Wave The phenomena of interference and diffraction of light cannot be understood without introducing the wave concept. In fact, the wave properties of light were established precisely from these phenomena. Here, we will introduce the essential aspects of a propagating wave and the formulae needed to explain the optical effects described throughout this section. Let us first recall the one dimensional wave formula that we met for the first time during the physics classes in senior high school.

Where A, A, v, t , and @ are the amplitude, wavelength, velocity, time, and phase, respectively. This equation describes the propagation of a cosine curve [A cos(2n/A)x] along the x-axis. Introducing the frequency v = v/A, the general formula of wave can be written as:

If the phase difference @ is neglected, Eq. (1.2) becomes Eq. ( 1 . 3 ) where we introduce the wave number k = 2n/A and the angular frequency w = 2nv. This leads us to the simplest example of a three dimensional wave, the plane wave.

A plane wave exists at a given time, when all the surfaces of constant phase form a set of planes, each generally perpendicular to the propagation direction. Under these conditions Eq. (1.3) becomes Eq. ( 1.4), defining the unit vector I in the direction perpendicular to the wave plane, that is, in the direction of the wave propagation.

+ = Acos(kI.r - wt)

(1.4)

Assuming that the wave number vector k = kI, Eq. (1.4) becomes the general formula of a plane wave. As defined, the wave number vector k indicates that the direction of the vector k is the propagation direction of the plane wave. Mr, t) = A cos(kr - wt)

(1.5)

In dealing with the general wave concept, we cannot use Eq. ( I .5) as written, since the equation for the general formula of wave is a partial differential equation of the 2nd order. But it so happens that Eq. (1.5) can be rewritten to include an exponential term such as y = AeB'. If we can remember lectures on differential equations of our undergraduate mathematics course, we will soon appreciate this convenience. For example, if during an examination we forgot the method of solving partial differential equations, we could somehow get the answer by assuming various solutions including exponential terms, such as y = Aenx,y =.flx)eBx,etc. If we could not answer the problem by this technique, the differential equation to be solved was probably famous! In this case, there was no need to panic, because the other students

12

I Fundamentals of Molecular Photonics

would probably not be able to answer the difficult problem. After all no student ever had mernorized all the calculation methods. Euler’s formula (Eq. (1.6)) relates the exponential function to the trigonometric functions. el:

= cos z

+ i sin z

(1.6)

If we transform elz into a Taylor’s series as a regular function, we can prove Eq. (1.6). This lecture on analytic functions went on like this: “The polar form of z with 121 = r and arg(z) = I9 is z = reZ0.’’ Here we transform Eq. (1.5) by using the polar form. If we overlook the strictly critical study of the argument 0,we obtain the general formula of a plane wave, using the correspondence r = A and B = (kr- wt).In physics, the following equation is always used as the wave formula. This is done to take advantage of the ease with which complex exponentials can be manipulated. Only if we want to represent the actual wave must we take the real part into account.

1.1.2 Refraction and Reflection When a beam of light encounters a lens or a mirror it undergoes the phenomena of refraction and reflection. If you remember the experiments done in a dark room in primary school science lesson, during which the teacher focused a narrow light beam on a lens or mirror, then you can visualize easily the two phenomena. So, intuitively, we can recite the law of reflection: “The angle between the reflected light and a line normal to the plane of reflection is equal to the angle between this line and the incident light and the incident and reflected light travel in the same plane.” Therefore it should come as no surprise that Euclid who lived in 300 B.C. already knew this fact. While we can understand visually the phenomenon of light refraction by a planar glass, it is much more difficult to grasp intuitively the meaning of Snell’s law of refraction: “The incident light, the refracted light, and a line normal to the plane of refraction are coplanar, the ratio of the sines of the angle of incidence and of the angle of refraction is constant. This ratio depends on the wavelength of the light and on the nature of the reflecting material.” Understanding this law requires that we understand not only the concept of a light beam but also the concept of the sum of wave. Let us consider a beam of incident light as it undergoes refraction on a flat plane, as shown in Table l.l(Refraction). A wave plane PP’ encounters the refractive plane at point P. The point P’reaches the plane at points Q’and Q after t seconds. If the light velocities in each medium are vI and v2, respectively, the relationship between the refractive angle and the light velocity is given by P‘Q’ = PQ’ sin

= vlt

(1.8)

PQ = PQ’sin & = v2t

(1.9)

@I

where @I and &r are the incident and the refractive angles of the light beam, respectively. If the arbitrary distance PQ’ can be neglected in Eqs. (1.8) and (1.9), we obtain Snell’s

1. I Fundamentals of Optics

13

relationship: (1.10) Let c be the light velocity in the vacuum, then the refractive index, n, is be defined by sin $, c-v2 sin & -

= n2

(1.11)

where &, is the refractive angle of the light beam in the vacuum. Using the refractive index as defined by Eq. (1.1 l), we obtain the general formula of the refraction law:

n ,sin@,= n2 sin&.

(1.12)

The refractive index of a large number of materials is known as a function of wavelength. Typical values range from 1.3-2.3 in the region of visible light. We can see our face when looking on water or a glass surface, even though they are not mirrors. In general, the phenomena of reflection and refraction occur simultaneously when a light beam encounters a medium of different index of refraction. Often, while strolling downtown we can observe young ladies peeking at their dress in a store window. We will interpret this simply to mean that they are looking at themselves in the store window and that they are seeing their image. But this is in fact a very delicate aspect of fundamental optics. The shopper will not be able to see the goods displayed in the store window when the angle of observation through the glass plane becomes small. This phenomenon is called “total reflection.” Therefore the phenomenon of reflection differs fundamentally from that of refraction. The phenomenon of reflection applies for all angles between the light beam and a line normal to the reflection plane. On the other hand, the phenomenon of refraction occurs only when the refractive angle is larger than the incident angle in a given medium. The angle of incidence that corresponds to an angle of refraction of 90” is called the “critical angle in total refraction,” @crltlcal. It can be calculated easily from Eq. (1.13): n l sin

@cr,,lcal

= n2 sin 90” = n2.

(1.13)

An unpolished diamond in our hand is not much more than an investment. An artfully polished diamond becomes an object of aesthetic value. The diamond cutting technique is ruled by the law of total reflection.

1.1.3 Interference Thomas Young’s in 1803 performed for the first time the classic experiment that demonstrates optical interference: the two-slit interference experiment, which appears as an example in many books on optics in order to explain the concept of interference. Young was not the first to report the phenomenon. It had been observed in various forms, such as Newton’s rings, Brewster interference, and Michelson interferometer. Young’s experiments however mark a point in the history of science. They led the way to the studies of Augustin Fresnel ( 1 8 16), who introduced the measurement of the wavelength of light and established

14


the development of the wave theory of light. Young’s two-slit interference experiment is indeed an historical event. Young was undoubtedly a child prodigy. He was able to read at the age of two and by the age of four he had finished the Bible. When he reported his two-slit interference experiment, he was severely criticized by the intelligentsia: “this report does not correspond to new experiments and it does nothing except to disrupt the development of science.” Young’s papers were said to be “destitute any kind of merit.” It is well known that subsequently Young abandoned his study of optics to devote himself to the study of hieroglyphs. On the other hand, Fresnel, who made the most of Young’s thoughts and built up the foundation of classical wave optics, had a sickly constitution in his childhood. He started to read when he was eight years old and his teachers said that he made very slow progress. Why do so many people want to make fast judgments? We often hear about child prodigies, but usually they show great ability for the “rigid” field of mathematics. They rarely demonstrate outstanding ability in chemistry, social psychology, or economy. After all, one needs a “great humanity” to study science. Youth is a very important time of our life, when we build within ourselves the foundation of this “great humanity.” Let us first explain Young’s experiment as it was presented to us during physics classes in high school (see Table l.l(Interference)). The usual elementary analysis involves finding the difference in phase between the two waves arriving at a given point on a screen. In the system depicted in Table 1.1, Islit is the distance between the slits S , and S2 and lscreen is the distance between the slits and the screen. The relationship between the light wavelength A and the interval of the interference fringes, Id, is given by (1.14) In the interference diagram of Table 1.1, we define the direction parallel to the screen as the x-axis and the direction perpendicular to the book plane as the y-axis. If we choose a point Q(x,y) on the screen, then the distances between the point Q(x,y) and each slit (IsI and lS2)can be calculated using Pythagoras’ theorem. In the phenomenon of interference, the light intensity is the highest for integers multiple of the wavelength and it is the weakest for half-integers multiple of the wavelength. Let m be an integral number, then the position of the bright fringes and the interfringe distance are given by Eqs. (1.15) and (1.14), respectively. (1.15) We emphasized earlier that the phenomenon of interference was considered to be the basis of the wave concept, yet all the explanations of the phenomenon we have presented so far completely ignore the wave nature of light! How strange! Remember, though, that in order to derive Eq. (1.15), we assumed that “the light intensity is the brightest for integral multiples of the wavelength.” This statement includes the fundamental character of the wave concept. The explanation for interference given previously is based on Fresnel’s interpretation of Young’s experiment, and does not consider the interference phenomenon as a symbol of the wave concept of light. In the context of a high school physics lesson, Young’s experiment is more an application of elementary geometry than an explanation of the wave concept. Actually, it is important to take note of the fact that “the light intensity is the highest for

1 . 1 Fundamentals of Optics

15

integral multiples of the wavelength” in order to understand the interference phenomenon. Let us consider the situation where a light wave lvp(rp,t ) passes through the point P and overlaps at the point 0 with another light wave q,(r,, t ) which crosses the screen through the point Q . Here, f is the time and rpis the positional vector at the point P. Using Eq. (1.7), the wave function for the synthesized light wave at the point 0 is given by (1.16) The light intensity at the point 0 is I(ro) = vo(ro,t)q0*(ro, t ) , where I)* is the complex conjugated number of I#. When the two light waves fi and qQhave different phases, the exp (-iwt) term in Eq. ( 1.16) takes an important meaning. Intuitively, averaging over time, we can understand why light fringes do not form at point 0 under these conditions. On the other hand, if the phase of light waves 1vp and q!~, is the same, the light intensity I(r)( = lC12= C*C) at point 0 is given by the following equation:

If A and B are real numbers, and defining s = rp- r, as the difference between rpand r,, then the equation giving the light intensity at the point 0 takes the typical form of the second law of cosines: /(ro) = A 2 + B2 + A B { e t k S+ e-”’} = A2 + B2 + 2AB cos ( k s ) .

(1.18)

Consequently, the light intensity at the point 0 is controlled not only by the sum of each light wave intensity ( A 2 + B2), but also by the term 2AB cos (ks).It is this term of the Eq. (1.18) that describes the formation of interference fringes. The bright parts of the fringes of intensity I = ( A + B)2are formed at ks = 2mn, and the dark parts of the fringes I = ( A - B)z are formed at ks = ( 2 m + 1)n. The occurrence of interferences is a strong indication of an undulatory motion, since, as seen from Eq. (1.16), interferences result form the superposition of waves. Adhering strictly to this idea, we have described the general wave in Section 1.1.1 as a cosine wave function. In other words, we have derived mathematically the nature of interference, assuming that the basic wave motion is described by a cosine wave function and we derived equations corresponding to the superposition of such waves. Following this train of thoughts, we are led to believe that everything has been determined by our initial assumption and that we could easily be deceived. In order to understand a system, it is not sufficient to apply a cause to effect model; one also has to ascertain that the entire system is consistent. It is always possible to formulate a hypothesis, it will only be acceptable if it agrees with all the facts and forms an integrated system. 1.1.4 Diffraction Standing still at the seashore, immediately after a disappointment in love, a young man

16


watches the waves surging back and forth against the shore. It is doubtful whether he will recover soon from broken heart, but it is indeed a picturesque scene. When the waves surging back and forth against the shore collide with a breakwater, the waves which take a roundabout way can propagate within the breakwater. This phenomenon is known as diffraction. But, why is it that the waves inside the breakwater do not have a sharp shadow? This is related to the fact that the propagation of the waves is practically continuous. In other words, the oscillation of waves never stops abruptly at geometrical boundaries. Particles would behave differently. This “smearing” of the shadow edge is closely related to the phenomenon of interference. Historically, both concepts were essential to the understanding of light-induced phenomena. However, while it is easy to understand wave diffraction at the seashore, it is much more difficult to understand intuitively the diffraction of sunlight. This problem was of great concern to 18th century scientists who initiated the revival of the wave theory of light. Fresnel combined Huygens’ theory and the interference principle. He made an impressive contribution to the field of optics by solving the diffraction puzzle through the use of the natural features of waves. He laid the ground for Kirchhoff s outline of the diffraction theory. Huygens’ principle stipulates that “a point propagated by a wave becomes the origin of secondary spherical waves spreading out in all directions.” Fresnel added the concept of interference to Huygens’ theory. He stated that the essence of a wave is such that “the complex amplitude of a wave at a point far from the wave front is superimposable to that of all the elementary waves which propagate from each point of the wave front to the observed point.” If a monochromatic spherical wave irradiated at a point Po forms a wave front u a t time t, the complex amplitude at an arbitrary point Q on the wave front is given by the following equation (Eq. (1.19)) which uses the general wave formula given in Eq. (1.7): A @(Q)= erhO. r0

(1.19)

In this equation ro is the radius of the wave plane u. Hence, if r is the distance between a point Q of the wavefront and the observed point P, then the contribution of an elemental area, do, from the elementary waves at the point P from the observed point Q is given by (1.20) where K(0) is a function dependent on the angle 67 between the QP line and a line normal to the wave front in the amplitude of the elementary waves in plane wave u. K(B) is known as the obliquity factor. Fresnel assumed that the function K(B) is maximum for B= 0, decreases with increasing Ovalue, and is 0 for B= n/2. Consequently, the process of light propagation without an obstacle is given by Eq. (1.21). Fresnel solved this integral by dividing the wave plane u into many zones symmetrical with respect to the POPaxis. (1.21) Taking K = i/Aas the obliquity factor of the zone on the wave plane unearest to the POPaxis, Fresnel was able to explain the propagation behavior of spherical waves. In other words, with this assumption, it is possible to show that the amplitude of the secondary elementary waves decays to 1IA with a phase delay of 1/4 period with respect to the primary incident wave. In

1 . 1 Fundamentals of Optics

I7

order to explain the phenomenon of diffraction, Fresnel devised a method where many concentric circles based on the functi0n.h = r + (n/2)A are drawn around the point 0 on the wave front. Zones of concentric circles on a wave plane are usually referred to as Fresnel zones. An intuitive image of the wave propagation and diffraction is shown in Table 1.1 (Diffraction). The diffraction of waves is closely dependent on the wavelength. When the distance between obstacles is larger than the wavelength, the waves seem to pass straight through the obstacles. On the other hand, when the distance between obstacles is smaller than the wavelength, then the phenomenon of diffraction appears clearly. Following this line of thought, it becomes quite easy to accept that there is a substantial shadow in the sunlight, because the sunlight wavelength is very small. The phenomenon of diffraction appears clearly when sunlight illuminates materials where distances between obstacles are very short. This elegant analysis of the process of diffraction demonstrates once more that the advancement of science never forces one to deny systematic theory. This fact should bring us peace of mind. Kirchhoff derived the Helmholtz-Kirchhoff theorem (Eq. (1.24)) from the Helmholtz equation (Eq. ( 1.22)) and Green’s theorem (Eq. (1.23)). Consequently, the Fresnel-Kirchhoff equation containing the obliquity factor is given by Eq. (1.25),

(1.22)

V24J(x,y,z)+ k 2 4 J ( x , y , z = ) 0

(1.23)

(1.24)

(1.25)

a/&

where $(x, y , z ) and Mx, y , z) are scalar, is the differential along a normal line towards the outside, and COSS= (cos 0,- cos @/2.The obliquity factor in Fresnel-Kirchhoff equation is written as: K(e)=

i

case, - cos6

A

2

--

-1 (cos$ - cos6)eikn’2 -

A

2

(1.26)

Eq. (1.26) expresses the fact that the amplitude of the second elementary waves is inversely proportional to the wavelength A and it has phase delays of n/2. We turn our attention now to the principle of diffraction gratings, where narrow slits of identical width are aligned parallel to the same space. A diagram of a diffraction grating is shown in Table 1 . I , where A, and I, are the distances between neighboring slits, the light wavelength, and the width of the slit, respectively. The equation giving the light intensity shown in Table 1 . 1 can be derived from Eq. (1.25). In this expression of the light intensity, I” is the intensity at Q = 0 and N is the number of slits. The light intensity for diffraction when sin2(ls,,tZ) = 0. With these parameters, the grating formula gratings, is maximum (IMAX)

18


can be derived (Eq. (1.27)). It gives the relationship between wavelength and angle of diffraction: (1.27) does not depend on the total number of slits Note that the position of the maximum value IMAX N . The maximum value IMAX increases markedly with decreasing slit width, Z,, in accordance with the equation shown in Table 1.1. With diffraction gratings, one can obtain light of narrow wavelength distribution without having recourse to a prism system.

1.1.5 Polarization If you were to place on this page a piece of clear calcite or glass, the printed characters viewed though the glass would appear double. Barthollinus, a doctor of medicine and professor of mathematics in Copenhagen, was the first to report this phenomenon in 1669. In his own words: “A transparent crystal recently brought to us from Iceland is one of the greatest wonders that nature has produced. As my investigation of this crystal proceeded there showed itself a wonderful and extraordinary phenomenon: objects which are looked at through the crystal do not show, as in the case of other transparent bodies, a single refracted image, but they appear double.” That this “double image” phenomenon was quite general became apparent only in the 19th century. At the time, it posed a great challenge to scientists. One has to recall that the light waves were presumed to be longitudinal, in analogy with sound waves in air. It is the double image phenomenon which forced scientists to reconsider this assumption. Malus and Brewster studied the depolarization of light in great detail. However, it is Young, working with Fresnel, who solved the puzzle. He suggested that the vibration of light might be transverse as is a wave on a string. Fresnel went on to describe in mathematical terms the transversal wave theory of light. His thoughts are the seeds of Maxwell’s electromagnetic wave theory of light. In contrast to sound waves which vibrate in the plane of propagation, light waves are transversal: they vibrate in a plane vertical with respect to the direction of propagation. Thus, the electromagnetic wave is a transversal wave vibrating in a plane vertical with respect to the direction of wave propagation. Let us consider the electric field E ( x , y , z , t ) of a monochromatic plane wave emitted from a light source. Using Eq. (1.7), the electric field E(x, y , z , t ) is given by Eq. (1.28). Here the electric fields Ei(x,y , z, t ) ( i = x, y, z ) in each direction are given by Eqs. (1.29), assuming that the propagation direction is the z-axis. (1.28) (1.29) Polarization is characterized by four constants: the amplitudes, A , and A,,, and the phases, and #,,. Several situations can be considered. If the phase difference is assumed to be # = #?& the spherical distribution of the electric field associated with the x- and y-axes at any given time is given by the relationship: Ey = (kAJAJE,, for # = 2mn and # = (2m + l)n, where m

I . 1 Fundamentals of Optics

19

is an integer. This type of wave, where the vibrating direction of light is restricted to a plane along the propagation direction is said to be plane-polarized. Another case of particular interest arises when both constituents have the same amplitude A = A, = A,, and in addition their relative phase difference @ =2(m f 1/4)n. Then the relationship of E,Z + E,.2 = A2 applies. Such a wave is referred to as circularly-polarized. Finally, in the case where A,# A , the relationship (E,/Ar)I+ (E,./A,)2= 1 applies. This is the case of an elliptically-polarized wave. Using Eqs. (1.27) and (1.28), the general polarization Eq. (1.30) can be derived. The polarization of light plays an important role in the phenomena of birefringence and reflection. This will be described in the next section. (1.30) We will define the reflectance and transmittance of a plane-polarized wave. The coordination axes are chosen as shown in Table l.l(Polarized light). The y-axis defines the direction perpendicular to the plane of the book. Polarized light vibrating in the direction of the electric field of the incident light characterizes the p (parallel) polarization or TM (transverse magnetic) mode, while polarized light vibrating in the direction perpendicular to the plane of incidence (xy-plane) characterizes the s (senkrecht) polarization or TE (transverse electric) mode. The incident, reflective, and refractive angles are and A, and the amplitudes of incident, reflective and refractive lights are A,,, A,,, ArprA,,, Alp, and A,,, respectively. Since all the light waves have the same phase at a given arbitrary point, the relationship of k,r = k,r = k,r applies. Using the continuity of waves at a boundary plane and Snell law of refraction, respectively, we obtain the relationships between the angular and amplitude parameters given by Eqs. (1.3 1 )-( 1.34), where n l and n2 are the refractive indices of each medium. (1.31) (1.32) (1.33) (1.34) From these equations and the effective cross-section of the light beam, we can derive Eqs. ( 1.35)-( 1.38), which define the reflectance and transmittance of s- and p-polarized waves.

tan (61 + A )

( n z / cos&) - (nl / cosq!J,) (n2 / cos&) + ( n l / cosq!J,)

(1.35)

(1.36)

(1.37)

20


T, =

(1.38)

From the equation giving the reflectance in Table l.l(Polarized light), we note that the R, value increases monotonously with the incident angle. On the other hand, the R , value and then the R, value increases decreases with increasing incident angle. It reaches 0 at 4erewsler, sharply with increasing angle of incidence. According to Eq. (1.35), the R, value becomes 0 for tan(#l + &) = 00. In this case the incident angle, &rewster = d2-4+ is called Brewster’s angle. From Snell law of refraction, we obtain the corresponding angle of incidence: tan

hrewster

=n2 .

n1

(1.39)

For example, the Brewster angle for the boundary plane between air (nl = 1.O) and glass (nz = 1.5) is about 56.3”.When a light beam impinges upon are a glass with the incidence angle

hrewster, the only transmitted light is the linearly p-polarized wave. Consequently, a laser beam irradiated from an outer-mirror type laser equipped with a Brewster window is linearly polarized. 1.2 The Molecular Field Theory

In this section we present the molecular field theory as it relates to the fundamentals of optics and photonics. We will stray from the discussion of optics for a moment and study the concept of photonics from the viewpoint of molecules interacting with electromagnetic waves. In the “photo-opto” areas (see Introduction -The Concept of Molecular Photonics-), it is crucial, we believe, to study at the same time the molecular field theory dealing with the concepts of photonics and the quantum and radiation field theory dealing with the concepts of wave optics. Certainly, our recognition of light itself can advance through a deep understanding of the interactions between the molecular and the radiation fields, but light as an electromagnetic wave has also a profound effect as it interacts with the outer shell electrons of atoms and molecules, as shown in Table 0. I . In the halls of chemistry departments, we often hear how difficult it is to understand quantum chemistry. It is hard to know if this situation reflects a problem on the side of the teacher or the student. Somehow it should be easier to study quantum chemistry than classical mechanics, since so many areas of study in the 20th century have been shaped by quantum theory. In other words, the essence of quantum theory has invaded our conventional ideas. For example, we say: “If there is no conflict, it has to be right.” The logic of this statement is obviously related to the formulation of quantum theory. One would think that it should be easier to understand a field based on contemporary logic than fields rooted in classical logic. Pedagogues often believe one should study history systematically from the past to the latest events. This does not mean that the concept of a field is easy to understand, but it reflects the fact that history is a human drama. If many people took this attitude, the study of quantum theory could become quite enjoyable. In the limited space of this book we cannot explain in detail the large number of equations presented in this section. We will assume that the reader has learned quantum chemistry in his

I .2 The Molecular Field Theory

21

undergraduate years. Consequently, we will focus on the essence of the molecular field theory. Table 1.2 outlines the framework we will follow throughout this chapter, from the birth of quantum theory to its maturity.

1.2.1 The Old Quantum Theory It is commonly accepted that the old quantum theory era spans from the birth of Planck’s quantum hypothesis to the formulation of Schrodinger’s equation. This section describes the old quantum theory in three parts: the failure of classical mechanics, the birth of the quantum theory, and the completion of wave mechanics.s-x’This century obviously began with the birth of quantum theory. Many researchers appeared on the scene of quantum theory at the time, but we remember mostly the contributions of four researchers: Max Planck (1901), Albert Einstein ( 1905), Niels Bohr ( 1 9 13), and de Broglie (1923). Then Schrodinger proposed the new wave equation to conclude the age of the old quantum theory. Heisenberg established matrix mechanics and formulated the uncertainty principle. Near the e n d o f the 19th century, three phenomena could not be explained by conventional ciassical mechanics: the quantitative estimation of blackbody radiation, the photoelectric effect, and the emission spectrum of hydrogen atoms. Let us first analyze the blackbody radiation problem. When a material is heated above 500”C, it glows visibly with a red color. At higher temperature, it glows orange, yellow, or even blue. We all recall being taught at school that “the temperature of the sun surface is about 6,OOO”C because its color is yellow,” and “the temperature of a blue fixed star is about 12,OOO”C.” This phenomenon is called “blackbody radiation.” The best laboratory approximation to a blackbox is a small hole in a hollow box. Any light falling on the hole from t h e outside will be absorbed. Measurements of the light emitted through the hole when the box is heated shows that the amount of light emitted and its spectral distribution depend only on the temperature of the walls of the box. Rayleigh and Jeans were the first to develop a classical theory of blackbody radiation using the following approach. They defined as their system the set of standing waves that can exist in a cavity with nodes at the walls. Then, they determined: ( I ) the wavelengths of light which satisfy the condition for standing wave within the box; (2) the mean energy of a standing wave, which takes a value of kT. Their equation (see Table 1.2, top section) agrees well with experimental data for large values of the wavelength, but not at all with the experimental curves in the short wavelengths domain. This failure was called “the ultraviolet catastrophe.” The flaw implicit in their equation was that the energy became infinite, that is,

JcDa

E ( v ) d v = 00. However since their assumptions took their roots in the

basic geometrical features of three-dimensional space, in classical thermodynamics and kinematics, few researchers at the time doubted the validity of Jeans and Rayleigh’s approach. In order to characterize a system, the energy level of each component and its distribution function are of fundamental importance. A Boltzmann distribution is assumed to determine these quantities. Rayleigh and Jean’s assumption that “the energy levels allowed in wave motion were continuous” was not challenged. Few researchers took notice of Planck’s quantum hypothesis, which could solve the blackbody radiation problem. Planck, from his youth, was interested in the law of energy equipartition and he came to the conclusion that “the energy of a system should not change continuously, but discretely.” This insight led to the resolution of the blackbody radiation problem.

Table I .2 Summary of Molecular Field Theory

The old quantum theory Failure of cllasical mechanics Rayleigh-Jeans equation

87rkT E i v j d v=-u2d

,,E

ivi d

Y

=m

-+

I l 2 Y = R H(3-7)

Maximum kinetic energy

EuAx=hY-W

Y

CS

<Emission spectra of hydrogen atom>

Rydherg constant

RH

=h(p-po)

contradiction

=

1.09677578 'x 10 ' m

-'

Birth of quantum thory

( momentum of

light quantum theory ( E = h Gchrodmger's wave equation>

Y

h - ) c A ( E =2/m2 c p 'c '

hu the wave : p =-=

) +relativity theory

wave function ( Y = A c o s ( k

+

.+ k , + k . - w :

direct transformation of particle to wave

addition of wave propeNes onto kinetic equation : ( p : momentum vector,

Schr6dinger equation

t

'+

)

4

p+#k p

4

1 -V 1

+E = c P )

4

material wave

aZv

wave equation ( y (= ~ ) 2 A Y )

at

E

A %w

E

A is-

a

at

k : wavenumber vector, o : angular momentum, m : mass ) :

HY=EY

( A : Laplacian )

Hamiltonian Atom Hydrogen atom <Energy eigenvalue and eigenfunction for hydrogen atom> polar coordinate expression of Hamiltonian

:

I*

H=

e2

a

operator for square angular momentum

separation of parameters .1V(r,B,d)=R(r)O(B)GD(d) :Lagumre's polynomial, 0 ( B ) :Legendre's poIynomiaI, GD ( 6 ) : z component of angular momentum Equality of particles uncertainty princip1e:A r A P h ) <Slater's determinant> @articles of same species cannot be distinguished : J, (I.Z)=exp(iB)J, ( Z . I ) + J , (I,?,)=*$ (2 , l) (+:Base particle.-:Fermi p a r t i c l e ) equality of particles J,l(l) * * . * J,,(n) incaseofnelectronsystem :

R (r)

-

-

,

Pauli's pnnciple . J, i = J, + Y = 0 (two electrons cannot occupy the same state) Helium atom measured value: -79.006eV (three-body problem 4 cannot be solved strictly approximation method) E=(2Z5/4)ZE I . = -74.833eV perturbation method (use repulsion term between electrons, e / r I z , as an paturbation term) E =(2Z5/15) ' E I = -77 490eV variation method (determine minimum value by using effective nucleic charge, 2 . I I ) -77.879eV (repeat the calculation with a trial function until getting self-consistent dution) SCF method +

.

23

Molecules Hydrogen molecule (Four-bodyproblem 4 cannot be solved strictly * approximation method) valence bond method (hydrogen atom 4 electron exchange hydrogen molecule) V V S =N ( Y C O NC+Y I O N For C = 0.16, equilibrium distance between nuclei: R 4 . 8 8 , bondmg energy: D =3.23eV molecular orbital method (helium atom 4 separation of nuclei --t hydrogen molecule) -+

ry

A

I

&zi ( f c o N t C Y I O N )R e 4 . 8 5 A , D . =2.65eV, S:overlap integral

--

No-

.

Secular equation <Secular equation>

(variation method with LCAO approximation: '#=

cC i@ i

)

. . . . . . . .. *. . ..H.i o. -.S .i n.E.

Hii-SiiE. H.,-S.,E

*

-

*

*

*

*

H..-S..E

Molecules w ~ t hH electrons ( H electron approximation) S,,=($l'l (for i = j , S , , = l , for i t j , S i i = O ) overlap mtegral Coulombmtegral H I , = ( @ , 'I H I $ I ) = ( Y , resonanceintegral H I I = ( @ , 'I H I @ , ) for the presence of bond between I-th and J-th atom, H , , =B , I for the absence of bond between I-th and J-th atom, H,,=O (example ethylene)

=+I

,E=a-tR

H u -and H electron system electron system HMO extended HMO semempm cal PPP CNDO non-empmcal vanous SCF

method

u

&lorn 1

m o l e c u l e atom 2

Hydrogen atom

electron

Helium atom

electron 1

nuc eus

~

Hvdroaen molecule

electron 1

-rn

nucleus A

nucleus B

24


In the 19th century, the natural way of thinking predominated: only what a human being can feel or see is believable. Although a vast body of evidence suggested the existence of atoms and molecules, only a few scientists believed in them. The general way of looking at things at that time was through the concept of the continuity of light and energy. The hypothesis of discreteness was revolutionary. Planck’s quantum hypothesis was not a casual idea; it was based on rigorous mathematical reasoning. However, it did not attract much attention and was considered merely as a far-fetched theory made up to explain experimental results. Einstein immediately appreciated Planck’s insight and adopted the concept of discreteness. As it turns out, casual ideas rarely influence basic scientific research. Basic research is a step-by-step accumulation of ideas. The second phase in the genesis o f quantum theory is dominated by Einstein’s contributions, in particular his explanation of the photoelectric effect. Hallwacks, in 1888, discovered that electrons are emitted from a piece of metal placed inside an evacuated tube upon illumination with light of sufficient short wavelength. According to classical physics, one would predict that a more intense beam of light would be able to eject electrons of greater energy. However, this was not observed experimentally. On the contrary, the following two observations were made: ( I ) electrons are not ejected unless the wavelength of light is shorter than a given threshold; (2) the maximum energy of the ejected electrons depends not on the intensity of the light but only on its wavelength, with shorter wavelengths producing electrons of greater energy. Einstein, in 1905, proposed a new interpretation of the photoelectric effect based on the hypothesis that the energy of a beam of light consists of discrete “quanta” and that each quantum has an energy determined by its frequency, E = hv, where h is Planck’s constant and v is the frequency of light.9) The energy of an electron ejected from the metal is equal to the energy of the photon minus the energy required to detach the electron from the metal, W. Hence the maximum energy to eject an electron is EMAX = hv - W = h(v - vo).The remarkable contribution to science by Einstein’s explanation of the photoelectric effect can be focused on the following facts. ( I ) The concept of quantized energy applies to phenomena related to the emission of light as well as to the absorption of light. In this, it will serve in the interpretation of the hydrogen emission spectrum, leading Bohr to formulate his description of the hydrogen atom, the third phase in the formation of quantum theory. (2) The concept of quantized energy can be pursued further to describe the motion of energy in free space as a localized wave packet. Therefore it is at the origin of quantum electrodynamics. The wave theory of light reached its climax with Hertz’s contributions around 1888. Physics was now in a state of turmoil, torn apart by the wave-particle duality caused by the quantum theory. As a result, research in photo- and opto-related areas was divided into two streams: quantum electrodynamics and molecular spectroscopy. There is no question that the three reports, “the photoelectric effect,” “the special relativity” and “the Brownian motion” published by Einstein in 1905 changed the traditional consciousness of researchers in the 20th century. The report on the “photoelectric effect” clarified an old problem in the wave theory of light by linking discussions on the nature of light and the quantum hypothesis originating in Planck’s study of specific heat. The report on “special relativity” refuted the ether theory and created an entirely new field. The report on “Brownian motion,” aimed at the very existence of atoms and molecules, created the theory

1.2 The Moleculdr Field Theory

25

of the stochastic process as a new field of study. These three reports cannot be separated. In the formative period of quantum mechanics, the concepts of existence, understandability, and causality had become key issues for physicists. Einstein fought desperately against the school of Copenhagen (logical positivism) which in an attempt to defend realism voiced very negative judgments on these new concepts. Throughout his life Einstein would never accept the concept of non-causality. He said, “This wave-particle duality of light is the main characteristic of reality, and is explained with a clever way by quantum mechanics. Many contemporary physicists feel that this interpretation is a clincher, but I fear that it is only a temporary breakthrough.” Historically, the school of Copenhagen seems to have won the controversy. But many believe that the concept of the wave-particle duality has been expanded much more through Einstein than through the school of Copenhagen. Neils Bohr appears as the dramatis personae in the third act of the quantum theory genesis. He developed a theory to account for the assignment of the bands in the emission spectrum of hydrogen. When an alternating current is applied between two metal electrodes placed at the ends of a tube charged with moist hydrogen gas, a reddish violet glow is produced. The glow consists of a small number of discreet wavelengths of light. This spectrum, which originates mostly from hydrogen atoms cannot be interpreted on the basis of conventional classical mechanics. Kydberg was able to represent the wavelengths of all the spectral lines with a single empirical formula, shown in Table 1.2. In order to explain this formula, Bohr proposed a new model (Bohr theory) with the following hypotheses: ( 1 ) light is emitted when an electron leaves an orbital of high energy to reach an orbital of lower energy, but ( 2 ) light cannot be emitted, when there is an electron in the E , orbital of the hydrogen atom. The essence of Bohr’s theory, as summarized in Table 1.2, lies in the fact that the angular momentum, I, of an electron in a circular orbit is quantized, it corresponds to the positive integer n. Thus Bohr had achieved a “translation” of the conventional classical theory into quantum theory. This explanation has the inherent advantages and disadvantages of using a graphic image of atoms as a theoretical model. Bohr defined the angular momentum 1 = A.integer as a quantum condition, where A = h / 2 n = 1 . 0 5 4 5 9 ~ 1 0 e- ~r ~g s Bohr theory can be used to derive Rydberg’s equation (Table 1.2). De Broglie enters with French elegance in the fourth act of the quantum theory genesis. Planck’s quantum hypothesis proposed the concept of discontinuity in nature. This concept was related to the corpuscular theory of light by Einstein’s explanation of the photoelectric effect. Bohr theory was successful in giving the atomic model a new, quantized image. In 1923, de Broglie was trying to find a physical justification for Bohr’s hypothesis of quantization of the angular momentum. In classical physics, the wavelength of standing waves is quantized. De Broglie thought of a way to relate this concept to Bohr’s theory. He imagined a wave which may accompany a moving particle such as an electron. First, de Broglie showed that the momentum of light could be obtained by combining special relativity and Einstein’s explanation for the photoelectric effect. In other words, he defined a relationship between the wavelength A of light as a wave and the momentum p of light as a particle, that is, P A = h. The condition that the wave associated with the electron in the atom is a standing wave is shown in Eq. (1.40):

n=--2m A

( n = 1, 2 , ...)

( I .40)

26


where r is the radius of the atomic orbital. When de Broglie’s equation is related to the electronic orbital of Bohr’s atomic model, Bohr’s quantum condition is obtained. Bohr’s atomic orbitals can be explained by converting the electron into a wave in an atom. The discovery of the phenomenon of diffraction of electron-beams, such as X-Rays, confirmed experimentally the wave concept of de Broglie. One might say that de Broglie’s intuitive idea was a wonderful parody flouting the French elegance. We have compared the birth of quantum theory to a drama written and acted by Planck, Einstein, Bohr, and de Broglie. Indeed they inherited a critical mind with a deep insight into the quantum theory and they pursued continuously the development of this concept. From their research in that fertile period of time, we can study the inner workings of their considerations as human beings. Taking advantage of the failure of classical mechanics, they created a new concept. In the formation process of the quantum theory, they never denied the system of classical physics. The new physical system which they constructed included classical physics and grew to such a scale that it exerted a tremendous influence upon all science fields for nearly a century. Schrodinger appeared on the stage of quantum theory, following the birth of the old quantum theory. He defined the behavior of electrons in atoms. As shown in Table 1.2, Schrodinger derived a wavefunction from the general equation of wave. Using Planck’s quantum hypothesis and de Broglie’s equation, he guessed correctly the mathematical expressions, or operators, of the kinetic equations of particle waves. In other words he defined the quantum mechanical momentum operator, that is, p+(hli)V. This was an important step in the development of the quantum theory. The Schrodinger equation is an equation describing the conservation of energy for a system written in terms of its wavefunction. The operator applied on the wavefunction is the Hamiltonian: HI&= EV.

(1.41)

In the case of a time-dependent wavefunction, the corresponding relationship becomes E+iAalat, and Schrodinger’s wave equation is given by d HI$ = ih-y.

(1.42)

at

The wavefunction characteristics of the Schrodinger equation are shown by the following. ( 1) The wavefunctions are single-valued, continuous, and finite. (2)J!+I exp(i8) corresponds to q,due to the relationship:

(3) The wavefunction also possesses a particle nature. Thus integrating over all space, we have to apply a normalization condition in order to account for the fact that there is only one particle in the entire space defined by the wavefunction. This leads to the following equation:

jlI v 12dv= 1

(dv = dxdydz).

(1.44)

The development of the quantum theory began with Planck’s quantum hypothesis and


27

ended with the formulation of the Schrodinger equation. These developments constitute what is known as the old quantum theory.

1.2.2 Atomic Orbitals The concept of atomic orbitals is based on Bohr theory. In order to understand the atomic structure in the quantum theory framework, it is important to characterize the relationship between angular momentum and quantum number. There is no doubt that to grasp the real meaning of atomic orbitals one has to comprehend the concept of the angular momentum of electrons. In the most general mechanical system, the angular momentum is given by Eq. (1.45).

I = r x p = (vp, - zp,) i + (zp, - x p z ) j+ (xp, - yp,) k

(1.45)

Introducing the corresponding relationship p + ( A l i ) V into Eq. (1.45), we obtain each component of the angular momentum as an operator,

( I .46) where the variables I, m , and n correspond to I,m,n = x,y,z (cyclic). If the characteristics of these operators are shown by the use of commutators, one obtains [Ix,4.1 = ihI;., and this value is not zero. In other words, this equation demonstrates that it is impossible to interchange the components of the angular momentum. One cannot define a state for which both components of the angular momentum have determinated values at the same time. Therefore, we must define a new operator for the angular momentum which will be characterized by the commutators [ A , B ] = A B - BA = 0. In this specific case, the proposed new operator was [ I 2 , Il = 0 and [ I 2 , I,] = 0. It was shown conveniently that there is a state in which the double power operator and the component of angular momentum have determinated values at the same time. Let us now consider the atomic orbital functions and the energy eigenvalues of the simplest atom, the hydrogen atom. The Hamiltonian of a hydrogen atom is given in Table 1.2. As it is much easier to represent the functions for atoms and molecules in a polar coordinates system than in a Cartesian system, the hydrogen atom Hamiltonian is usually given in the polar form, also shown in Table 1.2. We can see that the Hamiltonian contains a double power of two operators of the angular momentum. Fortunately in polar coordinates it is possible to separate the variables r, 0, and @. Accordingly, the wavefunction of the hydrogen atom can be separated into three functions and it can be written as follows:

The resulting wavefunctions, R(r), @@, and q@), are described by an associated Laguerre polynomial, an associated Legendre polynomial, and a z-component of the angular momentum, respectively. These functions will not be described in more detail in this section. In the case of the hydrogen atom it is possible to calculate exactly the wavefunctions and the energy eigenvalues. However, it becomes extremely difficult to calculate precisely the

28


wavefunctions for atoms other than hydrogen. If we attempt to determine these functions we are faced with mechanics problem, the so-called three-body and n-body problems. The helium atom consists of two electrons and a nucleus. Compared to the hydrogen atom, Table 1.2, the helium atom possesses an additional electron and nuclear charge. Despite those similarities, it is very difficult to solve the Schrodinger equation that describes the helium atom. In order to solve Schrodinger’s equation of many electron systems, we have to have recourse to the method proposed by Slater. Since one cannot distinguish microscopic particles from each other, one must recognize the fact that two particles of the same kind are equal in principle, and apply the relation shown in Table 1.2, using 9 equals 4 exp(iB) as the wavefunction in the Schrodinger equation. One needs to define two wavefunctions, symmetric and antisymmetric to account for the interchange of the two particles. A photon in which the spin quantum number is an integer is called a boson (symmetric wavefunction). An electron, proton, and neutron in which the spin number is a semiinteger are called fermions (antisymmetric wavefunction). By using the two-electron system postulate, the total wavefunction of an n-electron system is represented by a Slater determinant. Generally, in determinants, the interchange of the numbers i and j changes signs, that is, a determinant conserves the characteristics of the antisymmetric wavefunction. Hence, as the value of determinants is zero when vj= N,we cannot have a wavefunction (or electronic state) such that v, = v,. This result indicates that two electrons in an atom cannot occupy the same electronic state at the same time. This fact is a mathematical guarantee of the Pauli exclusion principle. Many atoms need to be analyzed in chemistry. It is a general fact that the Schrodinger equation of many atoms cannot be solved exactly. Therefore, it became necessary to introduce many approximations in quantum theory. When hearing the term “approximation,” we feel that it is a tentative approach to solve an equation which ultimately will be exactly solved exactly. This is not the case when the concept of approximation is used in quantum theory. Here the term approximation refers to the search for the best possible method to solve an equation that, by essence, cannot be solved exactly. It is crucial to assess the true meaning of the approximation concept, if we wish to pursue the analysis of complex atomic orbitals. Historically, the Schrodinger equation of the atom of helium was solved with the help of two typical approximations, the perturbation and the variation methods. In this section, we regret that it is not possible to explain the methods in any great detail. The perturbation method is applied to a problem in which the Hamiltonian operator can be separated in two terms, for example in the case of the helium atom, the term of electron-electron repulsion is handled as a perturbation term. In the variation method, the effective nuclear charge is defined as a parameter and the Schrodinger equation of the helium atom is then solved on the basis of the variational principle. Newer and more powerful approximation methods were developed together with the progress of computers. For example the Hartree-Fock self-consistent field (SFC) method was proposed as drastic refinement of the approximation methods applied in quantum theory. The essence of approximation is to obtain a value as near as possible to the real value. Approximations do not validate a given thinking. Scientists may became anxious and pray secretly, that the logic and thought process will remain valid.

1.2.3 Molecular Orbitals Let us consider the orbitals of a hydrogen molecule, the simplest of all molecules to exist


29

at ordinary temperature. The hydrogen molecule is a two-electron system, analogous to the atom of helium. But a molecule differs from an atom as it possesses, not one, but several nuclei. In the case of molecules, new problems (chemical bonds, etc.) appear on stage, compared to the wave equation of atoms. Hence, the need for approximations is much more compelling if one attempts the analysis of the electronic states in molecules which have many electrons. It should come as no surprise that many approximations have been proposed in the field of molecular orbital method~logy.~) For example, to solve the Schrodinger equation of the hydrogen molecule, one uses typically two approximation methods: the valence bond method (Heitler-London method, VB method) and the molecular orbital method (MO method). The valence bond method describes the hydrogen molecule as an entity formed by covalent bonding between encountering hydrogen atoms. The Hamiltonian operator employed in the valence bond method is shown in Table 1.2. As this method imagines that two hydrogen atoms encounter and that a hydrogen molecule is produced after formation of a covalent bond between hydrogen atoms, it is often said that this method is based on the natural way of thinking of a human being. Therefore, we feel that it is reasonable. Hence it is very easy to understand that, in the valence bond method, the content of the covalent bond is ascribed by interchange integrals. This method is very effective for solving the wave equation of a molecule consisting of few atoms, but in the case of molecules consisting of many atoms it becomes very cumbersome. As a result, new molecular orbital methods were proposed for which a molecule was set up from the start, rather than built by the encounter of atoms. The molecular orbital method imagines that a hydrogen molecule is produced by splitting the nucleus of a helium atom, as shown in Table 1.2. The configuration interaction (CI method) is one of the improvements in the molecular orbital method. By this method one can obtain a more precise approximation by linear combination of the different electronic configurations of a molecule. It is particularly useful to study the excited states of molecules. In recent years molecular orbital methods are the preferred methods to calculate the Schrodinger wave equation of molecules. The electronic states of general polyatomic molecules are calculated by applying the molecular orbital method with a variation method. The variation method which is used in this approximation involves a linear combination of atomic orbitals, as shown in Table 1.2. In fact it is a method to evaluate the minimum energy eigenvalue of Eq. (1.48).

(1.48) In other words, this calculation process allows one to solve the eigenvalue problem by considering the determinant as the secular equation shown in Table 1.2. First, we obtain the integrals H,, and S,). Second, we calculate the energy eigenvalues E after substituting the obtained integrals in the determinant. Third, we obtain the wavefunction of the Schrodinger equation by calculating the eigenvectors. We can appreciate easily that this calculation method by approximations is readily programmable. Therefore, the estimation of the values of integrals H,, and S,) becomes the heart of the approximation and the step which in fact controls the precision of an approximation. Many approximations for the estimation of these integral values have been proposed. Well-known methods in the treatment of the n-electrons orbitals of molecules are the Huckel method, the PPP (Pariser-Pan-Pople) method, and the

30


SCF (self-consistent field) method. The Hiickel, PPP, and SCF methods are set up in three steps, first simple numerical values are substituted for these integrals, then functions to introduce the ionization potential and electron affinity of the atoms are introduced in these integrals, and finally the integrals are evaluated. When dealing with both the (7- and the nelectrons of molecules, the extended Hiickel method, the CNDO (complete neglect of differential overlap) method, and the ab initio methods are set up by substituting numerical values for these integrals, adopting functions with little meaning for these integrals, and calculating these integrals. The Hiickel method is a very effective technique to study qualitative characteristics of the n-electronic states of molecules, but it cannot be used for quantitative discussions of molecular orbitals. With the advent of powerful and inexpensive personal computers, a researcher is now able to calculate molecular orbitals to his heart’s content. When studying the a-electronic states of molecules we must calculate a 20x20 determinant even for a molecule, such as an ethanol, for which we might have strong affinities. Hence we must search with all our might the minimum energy value for the secular equation, on the basis of repeated calculation of the integrals ( H , and S,) and the eigenvalues. After all, even a computer will take quite some time to calculate molecular orbitals. If you have some experience in programming, you will appreciate that it is extremely tempting to create imaginary roots when solving the eigenvalue problem for molecular orbitals. It is well known that the method chosen to coax for numerical values in a computer calculation influences the resulting eigenvalues when many iterations are required in the calculation of the integrals (Hi, and S,) and eigenvalues. Many programs are now available for the calculation of molecular orbitals. But in order to avoid bad mistakes the following should be remembered at all cost. You must above all understand the principles of quantum chemistry and if so, you will succeed in designing, by yourselves and from the beginning, the computer programs necessary for the calculation of molecular orbitals.

1.3 The Radiation Field Theory As mentioned in our introductory section, the radiation field theory is one of the fundamental elements in photo- and opto-related areas. In fact, most books on photochemistry or optics begin with a description of the radiation field. From an historical standpoint, the electromagnetic wave theory of light was established when Hertz, in 1888, succeeded in generating and detecting the electromagnetic waves, after Maxwell had completed the derivation of the electromagnetic field equations. Then, under the influence of the emerging quantum theory at the beginning of the 20th century, the electromagnetic wave theory was modified to include the quantization of the electromagnetic wave. This current opened the door to the creation of the field of quantum optics, especially in recent years via the inventions of the laser. In this section, we start by studying Maxwell’s equations of the electromagnetic field.lOJI) We will explain how to define the radiation field based on corpuscles. Undoubtedly, this is the most fundamental aspect of the concept o f light, within the context of the electromagnetic wave theory. Consequently, we will try to understand in simple terms the quantized radiation field theory, using many figures and tables. The mathematical background required to grasp the theory of the radiation field theory is summarized in Table 1.3. Note that vector analysis is the main mathematical tool necessary t o understand classical electromagnetism.i2)We will demonstrate that all the equations discussed in this section can

I . 3 The Radiation Field Theory

3I

be derived fundamentally from the equations listed in Table 1.3. I t is necessary to consider whether we should adopt the E-B nomenclature ( E : electric field, B : magnetic induction) or the E-H nomenclature (H:magnetic field). The E-H nomenclature is recommended in the case of formal discussions in the field of electromagnetic wave theory. Though both types use the same notion of electric field, the difference between the E-B and E-H styles resides in the description of the magnetic field. In the E-B style, one assumes that the magnetic flux density B arises from the electric current, based on the BiotSavart law. On the other hand, in order to understand the magnetic field in the E-H approach, it is necessary to view the “magnetic monopole” corresponding to the electric charge defined by Coulomb’s law. Therefore, the formalism complementarity is stronger in the E-H approach than in the E-B approach. Of course, the essence of the results is the same, independently of the approach selected. We will explain the outline of the radiation field theory by using the E-H style.

1.3.1 Maxwell’s Equations The entire classical electromagnetic wave theory can be accounted for by a set of expressions given by Eqs. (1.49) and (1.50) which have come to be known as Maxwell’s equations.

V x E = - - ,dB at

V X H = I + - aD -, at

V.B=O

V.D=p

(1.49)

( 1 SO)

Where B, D , E, H , I , are the magnetic flux density, the electric flux density, the electric field, the magnetic field, and the electric charge density, respectively. Introducing E and p, the dielectric constant and the magnetic permeability, we obtain the relationship (Eq. (1.5 1)) between the key parameters in classical electromagnetic wave theory,

D=EE, B=pH

(1.51)

where the suffix 0 is given for E and p in vacuum. Practically, Maxwell’s equations are usually divided into two categories, relating either to the electric or to the magnetic field. Schematic illustrations of Maxwell’s equations are shown in Table 1.4. Eq. (1.49) was deduced from Faraday’s law of the electromagnetic induction. It demonstrates intuitively that an electric current is produced by a moving magnet. In the equations, ds and dS are the vectors tangential and perpendicular to the plane, respectively. Eq. (1 S O ) was deduced from Biot-Savart law and from Ampere’s law. It describes the fact that a magnetic field is produced by an electric current. The first set of Eqs. (1.49) and (1.50) are almost symmetrical, except for an additional term for the electric charge density contained in Eq. (1.50). This point relates to the fact that the electric charge exists whereas the magnetic monopole does not exist. On the other hand, the Eqs. (1.49) and (1 S O ) , left-hand portion, can be deduced from Gauss’ law. It is well-known that the Eqs. (1.49) and (1 SO) describe, respectively, the existence of the electric charge and the non-existence of the magnetic monopole. How can there be a

32

Table 1.3 Summary of Vector Analysis

Scalar vector

D = A . (BXC).

I

I

A, Az B. By B,

Ax

cx

c, c.

Triple product

(B.C,-BVC.)

=A.(B,C.-B,C,)tA,(B,C.-B.C,)tA~

D=AX ( B X C ) = B ( A * C ) -C ( A - B ) Differential

d(AtB) - d A t-,d B -d(A.8) dA d B =-. BtA*--, dt dt dt dt dt dt dA dA dA=-+au av

Differential

( A ( t ) ,B ( t ) )

~ o t a ldifferential

(A(u,v))

GIlldient

(Scalar potential of A : ~ , ( x , y , z ) )

d(AXB) -d AxBtAxdJ dt dt dt

a

d

( V = ~axe l + - eay2 + - e ,az)

Divergence

Rotation

(vector potential of B : A ( x , Y , z ) ) B=VXA=rotA=curlA=

Differential formulae

Integral

v (6J,)=(V4 ) $ + 6(V 11)

I

el

d/dx

A.

e2 d/dy

A,

d/dz A e ,a

I

V . (6A ) = ( V6)* A t 4 ( V . A ) Vx ( 6 A ) = ( V 6 ) X A + b( V I A ) V.(ArB)=B.(VXA)-A(VlB) Vx (Ax B ) = ( B . V ) A - ( A . V ) B t A ( V . B I - 9 ( V * A ) V ( A * B ) = ( B * V ) A - ( A * VBtAx ) (VXB)-BI(VXA) VX(Vq5)=0 , V * ( V I A ) = O, V I ( V X A ) = V ( V - A ) - V * A I A d t = ( I A . d t ) e l + ( j A . d t ) el+ ( I A . d t ) e J

( A (Ax.Ay,A.))

Curvilinear Integral

Curvilinear integral of A (x, y,

Surface integral

Surface integral of A

z)

along the curve C is

I,A

*

dr

is given by I S A . n d S , where n is unit normal

vector from inside to outside of the surface S. The following equation holds for region V surrounded by a closed surface S.

Gauss'law

Green's law

Stokes' law

The following equation holds for m i o n Senclosed hy a closed curve C on the surface

Solid angle (a)

A conical body is formed by connecting a point 0 With every point of a closed curve C forming region S. Solid angle w of S against point 0 is defined by flr * , where f i s a part of surface area of a sphere with its center at point 0 and radius r, which overlaps with the conical body.

,fSV.AdV=lsA.n dS The following equations hold for region V surrounded by a closed surface S.

I,( J , V p 4 + V # * V 6 )d V = l s J d, x4 d S IsiVXAj

- ndS=fcA-

d r

1.3 The Radiation Field Theory

33

phenomenon that seems to have no origin? Generally speaking, there should be a beginning, if there is an end. There can be no starting point if there is no beginning and no end. As such an honest view on such a phenomenon makes us nervous, we will decide to create a homogeneous loop in this system. In other words, we decide that this is a closed system. Accepting this assumption makes it possible to comprehend that a phenomenon which does not have an origin can in fact occur. We understand intuitively the complementarity between the electric and the magnetic field, even though their origins differ. Let us now study the nature of the electromagnetic wave as described by Maxwell's equations. The electromagnetic wave equations (without electric charge and current) in the vacuum are: (1.52)

V xH =

aE ~, at

V . E = 0.

(1.53)

Solving these equations simultaneously with respect to E and H yields either E or H . Thus we can study the function characteristics of E or H . If we operate a vector rotation on the right-hand side expressions of Eqs. (1.52) and (1.53) and compare them with the differential formulas listed in Table 1.3 ( V x ( V x E ) = V ( V . E )- AE) we obtain Eqs. (1.54) and (1.55), where A = V?.

AE

= V ( V .E ) - V x

(

3

(V x E ) = V (0) - V x -h .-

(1.54)

(1.55)

We observe that Eqs. ( 1 S4) and (1.55) are the same as the wave functions in Table 1.2. This result indicates that the behavior of E and H in electromagnetic phenomena is described by the general wavefunction. This observation led Maxwell to suggest the existence of the electromagnetic wave. If we assume that E and H are functions of only one variable, z , Eqs. (1.52) and (1.53) can be rewritten as Eq. ( I .56): (1.56) The relationships between the various components are given by (1.57)

In these equations, we will take E, = 0 and H z = 0. This result implies that the electric field

34

Table 1.4 Summary of Radiation Field Theory Electromagnetic field equation

r o t E=--

dB

rotH= I

at

law of

+-da Dt

d i vD=p

B=rotA

Biot-Savart's law

Gauss' law

eleceomagnetic induction

Ampere's law

d $E*ds=--IB*ds dt

P c B *d s = D 0 I

divB=O

vector potential

I.fD.dS=II/pdV 0

A="JJ/--

4r

a ",a 1 : annihilation or creation operator, k : wavenumba, e L : two unit vectors of plane wave perpendicular to each other (Radiation field is an ensemble of photons with energy of f i ~, and momentum of f i k ) I

i dV


35

is normal to the magnetic field and that both the electric and the magnetic fields are waves in a plane perpendicular to the z-axis, or in other words, the electromagnetic wave is a transversal wave. Hence we can obtain v = I/(&” p,J1’’ if we solve Eqs. (1.54) and (1.55) for the components E , and H,. As we can calculate p+, = 1.1 13x lo-” m-2 s2, the velocity of the electromagnetic wave is v = 2 . 9 9 7 ~ 1 0ms-I. ~ Therefore, the velocity of the electromagnetic wave is the same as c, the velocity of light. From this result, Maxwell predicted that light also was some sort of electromagnetic wave. We will now study the characteristics of the electromagnetic wave based on Maxwell’s equations, but from the point of view of energy. As the electromagnetic waves satisfy the general wave equation described above, the solution of Eqs. (1.54) and (1.55) will take the form of Eq. ( 1.7). The energy of the electromagnetic wave is proportional to the square of the wavefunction. Therefore we define the pointing vector S:

S=ExH.

(1.58)

Calculating the divergent of S yields the following equation:

V . S = V . ( E x H ) H . (V x E ) - E , (V x H ) (1.59)

The right-hand side term is the sum of the energies of the electric and magnetic fields. It expresses the time dependence of the energy of the electromagnetic wave, hence, the parameter S represents the energy flow per unit time and area. Table 1.4 attempts to reconstitute pictorially the development of the radiation field concepts. The propagation of an electromagnetic wave, based on Maxwell’s equations is shown in the section “Electromagnetic Wave” of Table 1.4. Note that the phenomena defined by Eqs. ( 1.49) and ( 1.50) cause the electric m d magnetic components of the electromagnetic wave to propagate alternatively. The natural consequence of the characteristics of the electromagnetic wave based on Maxwell’s equations is depicted in the section “Plane Parallel Wave” of Table 1.4. Here we see that the electromagnetic wave is a transversal wave, and that the electric and magnetic fields alternately cross perpendicularly to the plane of propagation. The velocity of propagation is v = l / ( ~ ~ pin~ the ) ” vacuum. ~ As pointed out earlier the propagation rate of the electromagnetic wave is the same as that of light.

1.3.2 The Electromagnetic Potential In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as fol lows :

36


V x E + - aB =O,

V.B=O

at

1 dE = 0, V xB -c2

at

V .E = 0.

(1.60)

(1.61)

The general solution of these equations of the electromagnetic wave yields a rather complicated mathematical expression. So, by using the second equation of Eq. (1.60) and remembering the formula, V * (V x A ) = 0 given in Table 1.3, we introduce the vector potential, A(x,y,z,t),which satisfies the following equations:

B=VXA V . B = V . (V x A ) = 0.

(1.62) (1.63)

By introducing Eq. (1.62) into the first equation of Eq. (1.60) and using the relationship of Eq. (1.64), we get Eq. (1.65).

V X E = --aB = --a (V x A ) at

at

(1.64)

(1.65) Next, we introduce the scalar potential, @ (x,y,z,t), as in Eq. (1.66), by considering the formula, Vx(VQ)= 0 given in Table 1.3.

3

(1.66)

JA7 E =- V$

(1.67)

(

V XE+-

=Vx(-V$)=O

This scalar potential, @, automatically satisfies Eq. (1.65). Thus, the electric field, E , can be expressed with no more than two potential functions, A and 4. dt

Eqs. (1.62) and ( 1.67) naturally satisfy the electromagnetic wave equation (Eq. (1.60)). The properties of Eqs. (1.62) and (1.67) can be examined from a different viewpoint. We already understand that the electromagnetic potentials introduced, A and 4, lead to the physical quantities, E and B , but in practice we measure the electric field, E , or the magnetic flux density, B . That is to say, there exist innumerable pairs of functions, A and @, which lead to E and B by the above-mentioned differential operations. Hence, it is very difficult to choose a physically-reasonable pair of A and @ among all the possible pairs. For example, A and Q can be transformed to A' and Q', respectively, as in Eq. (1.68) by using an arbitrary scalar function, W(x,y,z,t). (1.68)


37

When we introduce these transformed pairs to the definitions ( 1.62) and ( 1.67), we get V x A ' = V x A - V x (VI#)= V X A = B

aA' at

v@'= --aA at

- aA

at

+ -vqJ a at

-

(1.69)

v@- v-aw at

( 1.70)

V@=E.

That is, this pair of A and q5 also leads to the same B and E . This kind of transformation of potential functions is called a gauge transformation. Thus, we can say that the electromagnetic field is invariable against the gauge transformation. Now let us return to the original story. As was already mentioned, Eqs. (1.62) and (1.67) satisfy Eq. ( 1.60). The next step is to find out the conditions for these equations to satisfy Eq. (1.61). By introducing Eqs. (1.62) and (1.67) into Eq. (1.61) and using the differential formulae listed in Table 1.3, we get the following relations. (1.71)

a V . A + A@ = 0

-

at

(1.72)

Here when we adopt the conditions of Lorentz between A and q5 given as Eq. (1.73), we can get equations for A and @ as is shown in Eq. ( 1.74). These equations have the same form as the classical wavefunction. This means that the condition of Lorentz states the condition that A and @ obey the wavefunction. (1.73)

(1.74) As Eq. (1.7 1 ) still looks complicated and troublesome for further calculation, we introduce Eq. (1.75) as the condition for A and 9. It may seem forced, but it does not introduce any contradiction.

V . A = 0,

Vq5= 0

(1.75)

This type of gauge is called the Coulomb gauge. It reflects the condition of independence between the two components of the wave, since A oscillates in directions perpendicular to the wavenumber vector from Eq. (1.74). This corresponds to the introduction of the linearly polarized light as is shown in Table 1.4. The above process of introduction of electromagnetic potentials might be felt as an opportunistic transformation of equations, loosing the strictness of logic. However, taking the

38


most convenient way is quite natural in mathematical operations and the physical interpretation of phenomena as long as it remains self-consistent. In fact, when solving mathematical problems, an answer obtained by the most convenient transformation of parameters sometimes gives us even a truly aesthetic impression. Self-consistency of the whole system is most important. Now we introduce canonical variables to the radiation field. First, let us think about the properties of the vector potential, A . As Eq. (1.74) is the general wave equation, its general solution is expected to adopt the form of Eq. (1.7), i.e., a superposition of A exp{ i(kr - u,t)}, where n = 1, 2, ..., n. As A consists of two transverse waves perpendicular to each other (namely, E and H),the wave function of A is given by (1.76) where Z = ( t t / 2 ~ ~ I / )is" ~the normalization factor, ek,,(v= 1,2) is the two unit vectors perpendicular to each other of a plane wave with the wavenumber vector, k , V is the space volume, bk,,is the expansion coefficient, and * represents the complex conjugate. The introduction of Eq. (1.76) into Eqs. (1.62) and (1.69) gives wave functions for E and B and the energy for the electromagnetic wave, E. (1.77)

(1.78)

(1.79) Next, by considering the energy, E, as a Hamiltonian operator, H , we introduce the coordinate, Qk,,,and the momentum, Pt,. This is the process of canonicalization. (1.80)

(1.81)

(1.82) The variables introduced, Qkvand Pkv, satisfy a canonical equation. Eq. (1.82) has the form of the energy for a harmonic oscillator known from high school physics. The introduction of the electromagnetic potentials, A and $, into Maxwell equations, the derivation of the wave function of A , and the canonicalization of the space consisting of this


39

vector potential, A , have shown that this space is an ensemble of harmonic oscillator^.'^^^^) So, now we arrive at the image of the light field being an ensemble of harmonic oscillators, as represented in Table I .4. The conclusion that the radiation field is an ensemble of harmonic oscillators gives us a hint towards the process of quantization of the radiation field which will be discussed in the next section.

1.3.3 Quantization of the Harmonic Oscillator Quite often in a course on quantum chemistry, the quantization of the harmonic oscillators appears after the description of the particle in a box potential. Usually, only the solution of its Schrodinger equation is given, together with its wavefunctions and energy eigenvalues. However, the essential point of the quantization of the harmonic oscillator is not at all that it is the next step for determining the box-type potential with a higher level of precision, but that it becomes a model for the second quantization. In this section we explain the quantization of the harmonic oscillator and set the stage for discussions of the second q~antization.'~) The potential of the harmonic oscillator is expressed as:

v=- m m x 2 -- -.

h - 2

2

2

(1.83)

The Schrodinger equation for the harmonic oscillator is given by (1.84)

This equation is a Hermit differential equation (Eq. (1.87), according to the transformation of parameters shown in Eqs. (1.85) and (1.86)). (1.85)

(1.86)

d2@ -

dP2

2p d4 + ( 4 - l)(J = 0

dP

(1.87)

The general solution of Eq. ( I .87) is given as a Hermit polynomial, H,@), with the condition q = I + 1 (I is an integer). Its normalized wavefunction and energy eigenvalue are given by (1.88)

(1.89) where n is zero or a positive integer. The harmonic oscillator has an energy of h v / 2 even for

40


the n = 0 state. We introduce now an important characteristic of the harmonic oscillator. In addition to the orthogonality and existence of a normalization factor in the wavefunction, I,O. (x), the harmonic oscillator has the following interesting characteristic. (1.90)

(1.91) Here we carry out quantization, i.e., we add the character of a particle as already shown when we discussed the Schrodinger equation. By using the corresponding relationship, p x = -ih dldt, we define the following two operators, a* and a, as: (1.92) The characteristics of these operators are summarized as follows. U*

= q n ( x )=

G i qn+,(XI,

(aa* -a*a)I,O,(x) =

a a * q n ( x )= (n + i)qn(x)

Vn(X)

(1.93)

(1.95)

The process of introducing these operators is called second quantization. Their eigenvalues should be numbers (n is integer). The operator, a*, which defines a process to increase the number by one, is called a creation operator, while the operator, a, which allows for a decrease by one of the number, is called an annihilation operator. In Eq. (1.89) the increase in n results in the increase in energy eigenvalue, En. Table 1.4 shows that the number of nodes increases with increasing n. Thus, it can be said that the creation and annihilation operators control the increase and decrease in the number of nodes. By applying this image to the canonical equation of the electromagnetic wave discussed in the previous section, we will carry out quantization of radiation field.

1.3.4 Quantization of the Radiation Field Now we enter the quantization of the radiation field. First, we summarize the Hamilton operator, H , and the Schrodinger equation with total wave function, Y,,, and total energy, En, for the radiation field as an ensemble of harmonic oscillators.


41

(1.97)

(1.98)

(1.99) The strategy of quantization of radiation is based on the second quantization of vector potential, A ( r , t ) , given in Eq. ( I .76). So arbitrarily introduced expansion coefficients, bk,.*and hk),, in Eq. (1.76) should be related to creation and annihilation operators. If we use the relationships of Eqs. ( 1.80) and (1.8 1) between the expansion coefficients and the operators, a* and a , and introduce these relationships into Eq. (1.79), we get Eq. (1.102) as the expression for the quantized Hamiltonian, H , of the radiation field, which corresponds to Eq. ( 1.82) supporting these relationships. The operator for the total momentum, P,is given by Eq. (1.103). (1.100)

(1.102)

(1.103) Here we introduce new symbols, bra and cket. A symbol, IN), is called a cket-vector implying a line vector, while (qis called a bra vector, implying a row vector. In normalized orthogonal systems (&) = R"jt becomes a scalar. An equation for getting energy eigenvalue from the Schrodinger equation is expressed by using bra and ckets as: I Y * H Y dV =E

I

Y * Y dV

( Y * IH I Y ) = E ( Y * I Y ) .

t)

(1.104)

Let us consider the eigenvalues of operators H and P. As Eqs. (1.102) and ( 1.103) include creation and annihilation operators, we can assume, intuitively, that their eigenvalue becomes a number. Calculations by using Eq. (1.99) result in: (1.105)

42


As the wave function of the harmonic oscillator includes a Hermit polynomial, the eigenvalue of the operator, Nkv= akv*akv, is nkvas shown in Eq. (1.94). So, the total energy, E, and the total momentum P,of the radiation field are given by

(1.106) k

v

k

v

The new operator, Nky,is related to the number of photons specified by the condition ( k , v). For example, going from the vacuum without any photons to a state with nb photons with a (k,v ) condition is described by using the creation operator as:

( I . 107) The radiation field is an ensemble of photons with an energy of hw,,(w, = clkl) and a momentum of tik. By quantizing the electromagnetic field we have been able to introduce the concept of a photon (number). The vector potential can be rewritten as Eq. ( I . 108) by using creation and annihilation operators.

(1.108) This function is essentially different from Eq. (1.76). The A in Eq. (1.76) is a classical function defined at every point of time and space, while the A in Eq. ( I . 108) is an operator functioning on various state vectors in the space occupied by photons. This operator is called an operator of quantized field. The image of the quantized radiation field is depicted in Table 1.4. An ensemble of photons with the momentum of Ak makes up the substance of the radiation field. In this section, starting from Maxwell’s equations, orthogonalizing and quantizing it, we have recognized that the light viewed as an electromagnetic field is an ensemble of harmonic oscillators and furthermore an ensemble of photons with various energies. In this process the electromagnetic potential was introduced. We would like to emphasize the importance of logic in reaching the quantization of the radiation field. There exist a variety of logics, even in natural sciences. The logics based on classical physics alone may bring us to a maze of confusing causes and results. In order to understand a concept the important point is not the amount of knowledge but seeing through it by a logical development.

1.4 The Interaction of the Radiation Field and the Molecular Field In the previous sections, we have presented the three basic fields related to modern studies of light phenomena: the fundamentals of optics, the theory of the molecular field, and the theory of the radiation field. In the Introduction we stressed that the radiation and the molecular fields cannot be studied separately. Thus, the interaction between the radiation field and the molecular field constitutes the corner stone of the three basic fields in the study of light.13*14) This interplay is best presented through the interaction term of the Hamiltonian, H I , which is discussed in the next section.

1.4 The Interaction of the Radiation Field and the Molecular Field

43

1.4.1 Basis of the Interaction between the Radiation Field and the Molecular Field The classical Hamiltonian for a substance of mass, m, and electric charge, e, in an electromagnetic field takes the following general form: H = 1 ( P - eA)' 2n1 ~

+V(r).

(1.109)

The Hamiltonian, H , for polyelectronic systems (atom or molecule) is given by Eq. can be divided into three terms: HR, a term intrinsic to the radiation field, H M ,a term intrinsic to the molecular field, and H I ,a term for the interaction between the radiation and the molecular fields. The term HI is further divided into first and second order terms of the vector potential, A , for the quantified operators, p.A = A . p , using V . A = 0. ( 1 . 1 10). I t

Assuming that HIis small compared to HR and HM, we can apply the perturbation method and use the product of the eigenfunctions for HR and HM as the zero-order approximation for the eigenfunction of H . In other words, the transition between different zero-order-approximated eigenstates is deduced from the term H I . Thus, an eigenvalue is obtained by substituting A from Eq. ( 1.108) into Eq. ( 1 . 1 10) and applying specific eigenfunctions. By expressing the eigenfunction for HR as Inkl,)and that for HM as li) etc., the product for HR and HM is given by li, nk,.)or b, nkl?nki..)etc. As A includes the creation and annihilation operators, the last term in the parenthesis in Eq. ( I . 1 lo), H , , includes a term concerning one photon, H I , ,and a term for two photons, H l z .The non-zero matrix elements for H I ,and HI*are given by the following expressions.

(1.112)

44


(1.1 17)

Table I .5 should help the reader in understanding the physical meaning of these equations which show the interaction between the radiation and molecular field. Eq. ( I . 1 1 1) depicts an electronic system undergoing a transition from the initial state (state i) to the final state (state f, by absorbing a photon ( k , v). Eq. (1.1 12) shows the emission transition from the state i to the state f, known as the scattering process, where the electronic system changes from the state i to the state f by simultaneously absorbing a photon (k, v ) or (k’, v 3 and emitting a photon (k‘, v’) or (k, v). In a similar way, Eq. ( 1 . 1 15) corresponds to a two-photon absorption process and Eq. (1.1 16) corresponds to a two-photon emission process. Scattering also is a process based upon interactions between the radiation and molecular fields. Table I .5 presents other processes, such as the second harmonic generation (SHG) and the hyper-Raman scattering as three-photon processes. The coherent anti-Stokes Raman scattering (CARS) is an example of a four-photon process. For molecules or crystals having a symmetry center, the total eigenfunctions are classified into symmetric g states and antisymmetric u states. In the equation for a three-photon process in Table 1.5, if a symmetric molecule has a g state at li), Im)will be the u state and In) will be the g state, so that the matrix elements have non-zero values. But this results in an u state for If)m, which is in contradiction with the fact that If) and li) should be in the same state for three-photon processes. When a molecule or crystal has no symmetry center, there is no distinction between g and u states. So, three-photon processes can proceed only in molecules and crystals without symmetry center. The intuitive correspondence between phenomena (schemes) and equations is emphasized in Table I .5.

1.4.2 Absorption and Emission of Light In this section, starting from the equations for the interaction between the radiation and molecular fields, we derive the transition probability per unit time, or transition rate, from the li) state to the If) state. We discuss how to relate this transition rate to the observed absorption and emission spectra of molecules.

45 Table 1.5 Summary of Interaction of Light with Molecules One-photon processes absorption

I

Two-photon processes

emission

i >

'I 1

i >

6I

f >

Rayleigh scattering

Raman scattering

Two-photon absorption

( f , n k v - l l H l ,I i , n k v )

p

nk operator denoting number of photons in the field two unit vectors of plane wave, k , perpendicular to each other ( v = 1.2) momentum operator(-1 6 V , ), V volume of the space, k wavenumber

eL

Three-photon processes Second ~UIIIONC generation (SHO)

Hyper-Raman scattering

Four-photon process Coherent anti-Stokes Raman scattering (CARS)

46


The Schrodinger equation including a time factor is given by

H Y ( r , t ) = i A { d Y ( r ,t ) / a t }

(1.1 18)

with

H = H R +HM + H i = H o +HI H,Y,"(r) = Eo"Yon(r) Y ( r , t )= ~ b , , ( t ) Y o " ( r ) e x p ( - i E o "A t) /

(1.119)

where Y is the eigenfunction, E is the eigenvalue, the subscript 0 corresponds to the unperturbed state satisfying Eq. (1.1 19), and the superscript n indicates the n-th steady state. The time dependence of bf is related to the matrix element ( YJIHll Y:) by Eq. (1.120), using Eqs. (1.1 18) and (1.1 19),

9 = L X ( Y o J I H lI Yon)bnexp{i(E,'- E,")r/ dt iA

(1.120)

A}

= 1, b,(n f i) = 0) at t = 0 , the probability therefore, for a system with an initial function Yi(bi of being in the f state at a given time is given by Eq. (1. I2 1).

(1.121)

For one-photon processes the matrix element becomes Eq. (1.122), which includes the annihilation operation, ah, and the creation operator, akv*.

(1.122)

{(f ~(ek,. p , ) e " r I i > ( n lab ~ I nhl>+(f

I(eh . p l ) e - f k rI i>(ntY, latv*

I fib)

Here we introduce the dipole approximation, exp(+ikr) = 1kik.r = 1. This is a justified approximation, since the wavelength of light is much larger than the size of a molecule or an atom. By using the exchange relationships, [p,Z, r,] = -2iAp, and [H,, c r , ] = -(iA/rn@p,, we obtain:

f

,

(1.123)

1.4 The Interaction of the Radiation Field and the Molecular Field

The time factor in Eq. (1.121) can be integrated over the frequency, field, as in Eq. ( I . 124).

Wk,

47

of the radiation

c

B introducing Eqs. ( 1.122) and ( 1.123) into Eq. (1.124) and using the dipole moment P = -

e r, and the transition dipole moment Pfi = (f I P I i), we obtain the transition rate of absorption, Wab, and that of emission, We,,,.These are called Fermi’s golden rules.”) Woh

w,,

=

=

(n/ hEtY

)c

(@,I’

kv

)x

(n/ ~ E O V

kv

1 mk )nhlekv

/

Wk )(nk,. +

‘

(f I P I

- €1 - hmk

l)lekv . (f I P I i)12W ,-

+ ttw,

( I . 125)

(1.126)

Eq. ( I . 125) shows that the transition rate of absorption, Wab,is proportional to the number of photons, nke, while Eq. ( I . 126) suggests that, for the transition rate of emission, We,, there exist two terms, one which is proportional to the number of photons and one independent of the number of photons emitted. Thus, absorption and emission of light are not symmetric phenomena, as suggested already by Einstein in his kinetic formulation for absorption and emission of light. The term proportional to the number of photons in the emission process is called stimulated emission. The term independent of nh, is called spontaneous emission. The stimulated emission is the basis of the laser oscillation, as will be described in Section 1.4.5. The transition rates for absorption and emission of light are proportional to the square of transition dipole moment, P f i .This implies that, for a transition to occur, the polarized direction of the incident or the emitted light must have a component in the direction of the vector Pfi.As the dipole moment P is an inherent property of the molecular field, the transition rates reflect the characteristics of a substance. If Pfi has a finite value, the transition is allowed; if Pfi= 0, the transition is forbidden. Transitions are controlled by several selection rules. Calculating the transition dipole moment, P, yields the outlines of the absorption and emission spectra of a molecule. The features of light absorption and light emission of typical functional groups are summarized in Table 1.6. An increase in the size of a molecule, in other words, an increase of its dipole moment, generally results in an increase of a given transition probability. This is exemplified (Table 1.6) in the case of p-phenylenes, which exhibit an increase in their molar extinction coefficient, E, with increasing number of phenylene rings. The oscillator strength is often used as a theoretical quantity proportional to the intensity of light absorbed. It is related to the absorption intensity of an harmonic oscillator consisting of an electron of mass, m, and electric charge, -e. The oscillator strength, F,, for a molecule is given by Eq. (1.127), where the wavenumber 9 is given in cm-I. F,i

=

7 2mmk IP, 12= 4.703 x loZoli IPfi 1’ 3Ae

( 1.127)

The oscillator strength can be evaluated also from the observed absorption spectra, using Lambert-Beer’s law, where E is the molar extinction coefficient given in M-‘cm-I.

48 Table 1.6 Summary of Absorption and Emission Spectra Type of bansition

Light absorbing

wavelength

I

E

M A X

(nm) (h4

goup

Shift

M A X

I

cm

) by

eleckodonating substituent

c-c

loo0 C-H loo0 C=C loo00 C=c-c=c 20000 Red ( n , n ' ) c=c-c=o 2 m shift Benzene 200 Naphthalene 200 Anthracene loo00 0 0 20 ( n ,R ) 30 Blue N=N 100 Shitt N=O 660 200 1 M A x (nm): Absorption wavelength, E H A x (I4 k r (s I ): Rate constant for fluorescence (room temperature), k P

W.b.--J

27C

d

.b'IP.(ii)

&bSTiiij

d ii

56


the transition dipole moments, Paand P b . Triplet energy transfer, in contrast, occurs via an electron exchange mechanism. The interaction integral for triplet excitation energy transfer is given by Eq. (1.156). Jab

=

:I 1

($%0(6)va0(6b)-qal(fl)1/lbl(fl

)>

(1.156)

The generalized form of the transition probability for triplet-triplet energy transfer (Eq. (1.157)) was derived by Dexter, based on Eq. ( I . 156), where pa(?) is the phosphorescence spectrum and &bsT(c) is the singlet-triplet absorption spectrum. (1.157) Here again we forgo a detailed derivation of the equations, but outline only the general approach, so that at least the reader becomes familiar with the conceptual approach.

C. Interaction of Excited Molecules It can be observed in Tables 1.10 and 1.1 1 that the emission spectra of dyes attached to the backbone of polymers or of dyes in concentrated solutions are different from those of dyes in dilute solution. Under these circumstances dyes can form complexes. An excited state complex consisting of two identical molecules is called an excimer, while an excited state complex between two different molecules is called an exciplex (Tables 1.10 and 1.1 1 ). If the interaction between two different molecules occurs in the ground state, it is called a chargetransfer complex. The occurrence of charge transfer complexes can be observed from the appearance of a new band in the absorption spectrum. A typical example, the complex formed between naphthalene and 1,4-dicyanobenzene, is shown in Table 1.1 I . The key difference between charge transfer complexes and either excimers or exciplexes, is that the latter two species are formed by interaction of an excited state molecule with a ground state molecule and not by two molecules in their ground state. The formation of an excimer or an exciplex can be observed only in the emission spectrum. The absorption or fluorescence excitation spectra are not affected. Moreover, for a given molecule, the fluorescence excitation spectra corresponding to the excimer and exciplex should be identical to that corresponding to the monomer fluorescence. Before closing this section describing photophysical phenomena, we illustrate them in the case of dyes attached to a polymer chains (Table 1. 12).221 The ordinate shows the time range of the phenomena, which covers from sec to lo0 sec in usual solutions.

1.4.4 Photochemical Processes Photochemical reactions encompass all the processes occurring when, upon absorption of light, a molecule undergoes a change in its chemical structure as it returns to the electronic ground state. This definition reflects primarily historical developments. As already shown in Table 0.3, photochemistry has been under the strong influence of organic chemistry. Modem organic chemistry is extremely well systematized, a fact often overlooked by physicists. For example, each molecule or group reacts, not randomly, but according to well-defined patterns

1.4 The Interaction o f the Radiation Field and the Molecular Field

57

(Table 1.13). In order to classify photochemical reactions, one defines first the electronic transition (a-electron, n-electron, or n-electron (non-bonding electron pair)) from which a given reaction occurs. In other words, we have to start by identifying the correspondence of photochemical reactions to the electronic states of the molecular fields. Then, one determines whether a photochemical reaction proceeds via the excited singlet state (S) or the excited triplet state (T). Finally, one has to clarify the type of bond cleavages: heterolytic (ionic type) or homolytic (free-radical type). The names of the phenomena or reactions are systematically arranged as a consequence of these considerations. 1 7 . 2 3 - 2 5 ) Phenomenologically, we can distinguish the following photochemical reactions: photoisomerization, photocyclization (photoaddition), photocleavage, hydrogen abstraction, photo-concerted reaction, etc. For a photochemical reaction to occur, efficient absorption of ultraviolet or visible light is necessary, thus the photoreactive molecules should contain in their structure one of the bond types listed in Table I . 13. The characteristics of the typical photochemical reactions of C = C and C = 0 groups are summarized as follows. ( I ) C = C bond type S,(n,n*)transition: heterolysis (ionic type) photoisomerization, electrocyclic reaction, sigmatropic rearrangement, etc. Tl(n,n*)transition: homolysis (free-radical type) photoisomerization, photocyclization (addition), etc. ( 2 ) C = 0 bond type S,(n,n*) transition: heterolysis (ionic type) hydrogen abstraction, oxetane formation, etc. S,(n,n*) transition: homolysis (free-radical type) /3-scission, etc. T,(n,n*) transition: homolysis (free-radical type) a-scission, hydrogen abstraction, oxetane formation Tl(n$) transition: homolysis (free-radical type) photocyclization (addition), etc. The photochemical processes of excited benzophenone are listed in Table 1.8 to illustrate typical photochemical reactions of the carbonyl group. A detailed explanation of each reaction path is not given here, but the reader should understand that the type of reactions can be estimated to some extent by considering the localized site of the excited electron. We have shown in Section 1.4.2 that the transition probability of an electronic system can be determined from theoretical considerations. Therefore, by controlling the radiation field it becomes possible to select photochemical reactions and to steer their outcome.

1.4.5 Scattering Phenomena Scattering phenomena encompass processes where a molecular system simultaneously absorbs a photon and emits a photon (see Table 1.5). There is no time delay between the absorption and emission, both processes occur strictly simultaneously. Thus, scattering is a two-photon process. If the energy of the emitted light is equal to that of the absorbed light, the phenomenon is known as Rayleigh scattering of light. Raman scattering corresponds to the case where the two energies are different. The mechanisms of scattering processes are the

58

Table 1.10 Summary of Excirner Naphthalene derivative

sandwich-type excimer 350

400

uavelewth (rm)

450

OM&

--

17kJ.mo I-’

g rou

I n cyclohexans

U >1 c

._ u)

c

a

c

.-c

a 5

a u n U L

-= + 0

wavelength (nm)

$50

400

450

500

wavelength (nm)

550

6C

59 Table I . I I

Summary o f Charge-Transfer Complex and Exciplex

\ " I

+ A-

D+

\

D-A

in diprop I ether / m e t h y r c y c I ohexane

*

L .-

v)

c

aphthalene (f luorescence) i n t o 1 uene t h a l ne yano%enzene

0

L

c .-

(degree o

:

c

42%)

0 .v) v) ._

E

2 20

300

wave I eng t h (nm)

300

400 5 0 0 6 n 0

400

500

wave I eng t h (nm)

31 Onm

D* f I uorescence

CN ( D+A-)

*

e x c i p I ex

(DD+A-)*

t r i p l e exciplex

60 Table 1.12 Photophysical Processes in Polymer Chains Femto-seeond +on Photoahsorption (FranckCondon principle) Strongaupling type energy transfer

Pica-second region Intemediate-coupling type energy transfer Internal conversion

Nmo-second region Weak-coupling type energy transfer

(FSrster equation)

W ab-

R.bBZ.

J-fiDjeiD),.

dD lJ

Table I . 13 Classification of Photochemical Reactions Based o n Types of Chemical Bonds Type of chemical bonds

C-C

Type of transition

Classification of photochemical reactions ( U , U * )

free radical mechanism carbanion mechanism

(n,U*)

Type of cleavage ( K , R * ) (n, I * ) Homo Hetero

Ti

TI SI

SI

Calbene mechanism(C-X) TI TI photoisomenzation SI,TI TI SI photoaddition SI,TI TI Si photoconcerted reaction C=C electrocyclic reaction SI s1 cyclic addition TI T I sigmatropic rearrangement de- K methane rearrangement S I ,T I photooxidation TI(IA photoaddition 1.2-addition S I , Sz,T I SI 1.3-addition I ,4-addition SI S I ,T I photosubstitution A-type reaction SI.TI ft B-type reaction i+ aromatic radical reaction S R N I-type reaction ring intramolecular photocyclic reaction photooxidation photormangement photo-Fries rearrangement SI photoClaisem rearrangement s1 u -scission (Nomsh type I) TI TI B -scission S i P , T i P S I " . T I " S",T"' To hydrogen abstraction intermolecular photoreduction TI>SI TI C=O intramolecular hydrogen ahtraction S I , T I SI.TI (Nomsh type 11) photocyclic addtion Oxetane formation T I > S I SI.TI cyclobutane formation TI' TI" Tnn photooxidation Ti('A,) i+ non-concerted reaction C=C photorearrangement A-type reachon -c=o B-type reaction TI' TI" TI' TI" scission S I SI rearrangement S ! SI G C acyl rearrangement 1,2-rearrangement TI TI -C1.3-rearrangement S I, Tz S I . T ~ G O decarboxylahon SI s1 tatephotoisomenzahon S I # T2 71 N=N hydrogen abstraction T I TI photocyclic reachon C=N photoaddition chain scission(C-N) S I > T I SI,TI C=S hydrogen abstraction SI SI photocyclmtion SZ,TI TI Nomsh type I reaction (C-N, N-O bond) N=O Nomsh type I1 reaction hydrogen abstraction T I TI photocyclmtion TI <State> S excited singlet state, T excited tnplet state, A I excited singlet state of oxygen <Subscript> n (n, I )transition, p ( I , K )transition, I fmt excited state, 2 second excited state <Supposition> supposed transition route (not yet clear)

* *

* *

*

* *

*

*

*

62


same in both cases. Referring to the basic equations expressing the interaction between the radiation field and the molecular system, one can identify the existence of an intermediate state li) and a final state If) as a specific feature of scattering. The transition rate of the Raman scattering, W,,,,,, can be derived from Eqs. (1.1 13) and (1.114) in a way similar to that described in Section 1.4.2 for the case of the absorption and emission of light.13)The derivations yield Eq. (1.158):

Since for scattering phenomena Wk, = Wk, Eq. (1.158) indicates that the scattering intensity increases with decreasing wavelength with a A"' power law, in Eq. (1.158) the term describing the emission of light contains (nk,,, + 1). The usual Raman scattering consists only of the spontaneous emission (nk',,'= 0) and the scattering intensity is proportional to the intensity of incident light. However if a high power laser is used as the source of the incident light, the scattering intensity undergoes a large increase, since in this case nks,,s#0. This phenomenon is called induced Raman scattering. The component of molecular polarizability, (Aa,Jfi, is defined as Eq. (1.159), (1.159) where a and b denote the coordinates, x, y, or z. The scattering transition rate is proportional to the square of the molecular polarizability, which corresponds to the transition dipole moment for the case of light absorption. The molecular polarizability, A, is classically defined as the proportionality parameter of the induced dipole moment, P, when an electric field, Eo, is applied to a molecular system as given by Eq. ( 1.160). P = A'Eo

(1.160)

This quantity plays an important role in other multi-photon processes, such as twophoton absorption, second harmonic generation and hyper-Raman scattering as three-photon processes, and coherent anti-Stokes Raman scattering (CARS), a four-photon process (Table 1.5). The two-photon absorption can be treated theoretically from Eq. (1.1 15) in the same way as the Raman scattering process discussed above. Thus, the transition rate for two-photon absorption is given by Eq. (1.16 1).

w2 dnkdfik, =

8zmk'hwknh-nkv. V2

The relationship between second harmonic generation (SHG) and hyper-Raman scattering is similar to the relation between Rayleigh and Raman scatterings, in the case of two-photon

I .4 The Interaction of the Radiation Field and the Molecular Field

63

processes. Based on the classical idea that scattering is the light emitted by dipoles in the presence of an electric field, E,, the dipole moment, P,for three-photon scattering systems is expanded as:

where A and B are tensors, and B is the hyperpolarizability defined by Eq. (1.163),

BoPq

=

zl m,n

1’

(En - E, -

2hw)(Em - El - h a )

(f I P P I nxn (Em - El

I ei I m>(m I 4 I

- Am)(&, - Ef

+ ho)

’

i>

(Em - Ef

+ 2ho)(En - E f + ho)

1

(1.163) where the subscripts, 0,p, and q denote the coordinates, x, y , or z . These phenomena are induced by interaction with higher terms, resulting in a weak emission intensity. However, the advent of high-power lasers has made it possible to observe these phenomena rather easily and has resulted in the emergence of an entirely new field, nonlinear spectroscopy (Chapter 5). If some of intermediate states, Im) and In), correspond to real excited states in three- and fourphoton processes, resonant effects are expected, resulting in the enhancement of the emission intensity. The CARS four-photon process will be discussed in Section 5.2. I . Photophysical chemistry focuses primarily on the theory of the molecular system, although, strictly speaking, light absorption and emission processes are related to the interactions of the radiation field with the molecular system. Scattering processes are typical phenomena which are mainly concerned with the interaction of the radiation field with molecular systems.

1.4.6 T h e Laser Principle The word “laser” was coined as an acronym for “Light Amplification by the Stimulated Emission of Radiation.” We have shown in Section 1.4.2 that the basis of the laser oscillation is the existence in Eq. ( I . 126) of a term of stimulated emission. The laser operates on the fact that when light travels through certain substances, it can undergo a substantial increase in intensity. Under usual circumstances, the intensity of light decreases after passing through a substance, according to the Lambert-Beer law (I = 2.303 I. exp(-&cf)). However in some The is called the gain special cases, it increases, following the law: I = 2.303 loexp(&Ba,ncf). coefficient. This special phenomenon is known as negative absorption. The Boltzmann distribution ratio of the number of molecules in the ground state, No, with an energy, E ~ to , that of the molecules in the excited state, N , , with an energy, el, is given by Eq. (1.164).

( I . 164) Where go and g , are the degrees of degeneracy for each state. It is clear that in general, N , < No, since T > 0 and E , - E,, > 0. However by pumping the system with a strong light source or a discharge, it is possible to create a situation where N , > No. This situation is called a

64


population inversion or negative temperature. A light beam passing through the sample will increase, since photons in the beam trigger stimulated emission of the same wavelength. This amplification of the light beam is at the basis of the operation of 1 a ~ e r s . I ~ ) Next, let us consider the laser oscillation under pumping conditions. The interaction Hamiltonian, HI,and the wave function W(r, t), for excitable molecules placed in a radiation field of angular frequency, w, and maximum amplitude, @, are given by Eq. (1.165):

HI = -@ ,I/

cos wt

(1.165)

( I . 166) where p is the electric dipole moment of the molecule, Oi( r ) is the time-independent part of the wave function for the state i (i = 0, l), and TI( t ) is the probability amplitude of the state i. The changes in this quantity are much slower than the transition angular frequency between the two states, w,, = - co)/A.The introduction of Eqs. (1.165) and (1.166) into the timedependent Schrodinger Eq. (1.42) and subsequent derivation give the following simultaneous equations for Toand TI.

( I . 167)

(1.168) The second terms in Eqs. (1.167) and (1.168) are rapidly vibrating non-resonant terms having minor contribution on average, so they are neglected in further calculations. This is called the rotating wave approximation. By solving these simultaneous equations with the initial condition of To= T I for t = 0, we obtain: (1.169)

(1.170)

(1.171) where Z is the Rabi frequency. The transition probability (Eq. (1.172)) for a molecule to go from the 10) state at t = 0 to the 11) state at time t, is obtained from Eq. (1.170).

( I . 172) This equation represents a sinusoidal curve and at the resonant condition, w = w,,, molecules will repeatedly undergo absorption and emission with a cycle of 2nlZ.

References

65

In this section, the outlines of the interactions between the radiation field and various molecular systems have been discussed, based on the fundamentals of optics and of the characteristics of the molecular and radiation fields. The interactions of the radiation field with molecular systems are a fundamental aspect of the processes of light absorption and emission. This concept is indispensable when dealing with scattering phenomena. On the other hand, most of the discussion in photophysics and photochemistry is focused primarily on the photophysical and photochemical processes that occur after a molecule has absorbed light. So, in these cases, discussions are often limited to the electronic states of the molecular system. In cases, such as excitation energy transfer, a keen understanding of the interactions between light and materials is needed. This situation has led an artificial distinction between the physicists’ approaches, dealing with the interaction between the radiation field and the molecular system, and the chemists’ approaches towards the study of photophysical chemistry and photochemistry. This arbitrary separation has to be destroyed to understand fully the concepts of molecular photonics.

References I . W. T. Welford, Optics, Oxford University Press (1976). 2. B. E. A. Saleh and M. C. Teich, Fundamentals qfPhotonics, Wiley-Interscience, N.Y. (1991). 3. H. Ishiguro, Optics, Kyoritsushuppan, Tokyo (1953). 4. 0. S. Heavens and R. W. Ditchburn. Insight into Optics, Wiley (1991). 5. W. Heitler, The Quantum Theory ?/Radiation, Oxford University Press (1958). 6. Y. Harada, Quantum Chemistry, Shokabo, Tokyo ( I 978). 7. S. Tornonaga. Quantum Mechanics I and I / , Misuzushobo, Tokyo (1952). 8. P. A . M. Dirac, The Principles ofQuantum Mechanics, Oxford University Press (1958). 9. R. Feynrnan, The Strange Theorv ? f l i g h t und Matter, Princeton University Press (1985). 10. H. Hirakawa, Electrornagnetics, Baifukan, Tokyo ( 1 968). I I . T. Shimizu, Physics q/Electromagnetic Wave, Asakurashuppan ( 1982). 12. J. Hori, Phvsical Mathematics, Kyoritsushuppan, Tokyo ( 1969). 13. R. Loudon, The Quantum Theory ?/Light, Oxford University Press (1973). 14. Light and Molecules I and / I , ( S . Nagakura ed.), Iwanamishoten, Tokyo (1979). 15. J. B. Birks, Photoph.vsics ?/Aromatic Molecules, Wiley-Interscience, N.Y. (1970). 16. S. Yornosa. Introduction to Quantum Physics,for Photobiology, Kyoritsushuppan, Tokyo ( 1973). 17. N. J. Turro, Modern Molecular Photochemistry, BenjarninKurnmings ( 1978). I 8. J. Guillet, Po/vmer Photophvsics and Photochemistry, Cambrige University Press (1985). 19. A. Yabe, T. Taniguchi. H. Masuhara and H. Matsuda, Introduction to Ultra-Thin Organic Membrane, Baifukan, Tokyo (1989). 20. N. Mataga and T. Kubota, Molecular interaction and Electronic Spectra, Marcel Dekker, N.Y. (1970). 21. N. Mataga, Introduction to Photochemistry, Kyoritsushuppan, Tokyo ( 1975). 22. H. Ushiki and K. Horie, Handbook of Pol.ymer Science and Technology, (N. P. Cheremisinoff ed.), Vo1.4, Chapt. I , Marcel Dekker, N.Y. (1989). 23. K. Ichirnura, New Application qf Photofinctional Polymer. CMC, Tokyo (1988). 24. T. Matsuura, Organir Photochemistry, Kagakudoj in, Tokyo ( 1970). 25. J . A. Barltrop and J. D. Coyle. Excited States in Organic Chemistry, John Wiley and Sons (1975).


.



2.1 Characteristics of Photochemical Reactions 2.1.1 Photochemical Reactions and Thermal Reactions In this chapter we will examine various facets of photochemical reactions, which encompass the light-induced transformations of a chemical structure. Synthetic organic chemists are very familiar with thermal reactions, where changes in a chemical structure are initiated by the application of thermal energy. Thermal reactions proceed through a potential energy barrier in the ground state of the reactants. In contrast, photochemical reactions take place when molecules are brought in an electronically-excited state upon photoirradiation. For a photochemical reaction to occur, it is necessary to form an electronically-excited state. Therefore to be photoactive, a compound must possess in its structure light-absorbing groups with n-electrons, known as chromophores, such as olefins, carbonyls, imines, azo groups, and aromatic rings. The advent of light sources emitting in the vacuum UV region, such as excimer lasers, provides a means to study photoreactions of saturated compounds through photo-excitation of a-bonds. Einstein’) showed in 1912 that the absorption of a photon of a certain energy is necessary for a given photoreaction to occur, “the equivalence principle of photoreaction.” A corollary of this principle is that by selecting irradiation light of specific wavelength, it is possible to perform a specific photoreaction. Since the electronically-excited state possesses excess energy as a result of photon absorption, it has access to a far greater number of ground-state molecules than thermallyactivated molecules. Therefore many reactions that would not occur readily from the ground state take place upon light irradiation at room temperature or below. An example is presented in Table 2.1, where we show the ground-state and excited-state energy potential curves corresponding to the rotation of the double bond of stilbene. The energy barrier for the transto-cis isomerization of stilbene is too large for it to occur by thermal activation. The excitedstate surface does not present such barrier, and indeed room temperature photoirradiation of trans-stilbene at 245 nm readily yields the cis-form of stilbene via It*+’p*+po. The possibility of carrying out reactions at temperatures much below room temperature is one of the attractive feature of photoreactions. In the case of trans-stilbene the excited-state surface exhibits a small barrier with some activation energy leading from the It* state to a twisted phantom excited state, ‘p*. For this reason photoisomerization does not take place below 80 K. However in the case of trans-azobenzene where -CH = CH- in stilbene is replaced by -N = N-, photoisomerization by an inversion mechanism through change in hybrid orbital 67

68 Table 2.1

Processes of Photochemical Reaction

@ Characteristic features of photochemical, radiation-induced, and thermal reactions Energy s~urce light: sun light, xenon lamp, m u r y lamp, tungsten lamp, laser (rather reasonable cast) Co, X-ray (expensive) radiation: electron beam. heat: heater, thermostat (cheap) Selectivity of excitation light: Absorption energy is determined by the molecular structure. (selective) radiation: Absorption is determined by the nature of atoms in the substance. (non-selective) heat: thermal excitation (non-selective) Homogeneity of reaction light: Homogeneity depends on the optical density of the system. Homogeneous irradiation is impossible for high optical density system. radiation: The degree of transmittance depends on the radiation source. heat: Homogeneous reaction is attained if the thermal diffusion of heat evolved is adequate. Localization of the reaction light: Space selectivity of the reaction is excellent. Space selectivity of the reaction is possible with electron beam. radiation: Space selectivity of the reaction is difficult. heat: Low-temperature reaction light: possible radiation: possible heat: impossible Mechanism of reaction The reaction proceeds through electronically excited states. light: radiation: The reaction proceeds through primary ion-dissociation state. The reaction proceeds through thermal excitation of the electronically p u n d state. heat: Processes of photochemical reactions 0potentid curve for the isomerization of stilbene

I

absorption of light

c

I

0

n / 2 ? CIS rotation angle around the double bond

trans

2. I Characteristics of Photochemical Reactions

69

(sp2+sp+sp2) can occur, enabling processes such as the photoisomerization of polymer solids even at liquid helium temperature (4 K).2) The atomic configuration of the photo-excited state and the electronic state of the frontier orbital related to the reacting electron are different from those of the ground-state molecule. This leads sometimes to the formation of a photoproduct different from the product of thermal reaction. The Woodward-Hoffmann rules controlling the pericyclic reactions of conjugated dienes and trienes shown in the following section provide a well-known example of this phenomenon. Let us now consider photochemical processes as they relate to the molecular design of photofunctional molecules in materials. In Table 2.1 (upper part) the characteristics of photochemical processes are summarized and compared to those of thermal processes. Because of their high level of selectivity in the time, space, and energy domains, there is a tremendous opportunity of external control. In the time domain, with the help of short laser pulses, the reactions can be started and stopped within very short time periods, and they can be repeated with an arbitrary interval. In the space domain, a light beam can be focused onto a very small spot by using a lens and a coherent beam of laser light, leading the way to ultrafine lithography. In the energy domain, it is possible to trigger a specific reaction by controlling the wavelength of the incident light to excite one specific chromophore in a system. These capabilities are supported by recent progresses in laser technology as light sources and by the advances of the electronics technology for photon detection and data processing. The chronology of photochemical reactions can be divided into the following three stages (Table 2.1, lower part). ( I ) The formation of an electronically-excited state of a molecule by the absorption of light. (2) The primary photochemical processes performed by electronically-excited molecules. They can be divided further into photophysical processes and photochemical reaction processes. The former include luminescent processes and nonradiative deactivation. (3) The secondary or “dark” photochemical reactions, i.e., the reactions of the reactive intermediates, such as free-radicals or activated ionic species, produced in the primary photochemical processes. These processes often proceed by a chain mechanism, as in the case of photopolymerization. To achieve a thorough understanding of a photochemical reaction, it is necessary to take into account all the events happening on a molecular level from the initial stage of the absorption of light to the isolation and structural assignment of the final reaction products. As we have discussed already photophysical processes in Section 1.3, we limit this chapter to the study of the primary and secondary photochemical processes. Typical processes are listed in Table 2.2, according to the nature of the functional groups and of their chemical reactions.

70

-

Table 2.2 Typical Photochemical Reactions

0u

-Bond cleavage CC13Br

radical generation

qH3

imtiation of radical polymerization

h v

*cc13

'73

CH,-C-N=NCN

C-CHI-NZ CN

+ *Br

RCH=CH2

RCHBrCH2CCI3

yH3 YH3 RCH=CH2 , C H ~ - C +CH~-FH% tCH3-Y. CN 6N R

hu

carben generation NO2 CH3 CH3

ion pax generation I

C H3

C H3

Q Reaction of C=C double bonds

electrocyclic reaction and cycloaddition

5

O*

'hv CH3&0

CH3

A

H3C CH3

Nomsh type I reaction ( a -cleavage),via(n.

A

')

hydrogen and electron abstraction via (n, A ' )

(plar addition via S

, radical addition via T

0

hv

II

c -CH2-

-CH2-

@

CH3

~ t~H =- ~~ H

P h ' P-h

II

hv

I

I

/H

0 .I -C*

hv

i

)

0

-CH2-C

+ CH2-

Ph I *C-OH I

*CHI

CHP

-

t

CH3 I

* cI- O H CH3

nU II

-C-CH,

'

rn

Ph B -cleavage

II

Ph

C H3

0"YHNomsh type I1 reaction

I

COOHOOH

coo

CH3CHsO

rearrangement of 1.4diene

addition to double bond sigmatropic rearrangement (3) Reaction of C=O double bonds

0

13

(from cyclopropyl ketone) h u

oxetane formation

CH3COCH3+CH3RqC=CR2CH3+0

x e t a n e

CHII

CH2

@ Reaction of conjugated unsaturated esrbonylr deconjugation

type A and type B rearrangements of cyclohexadiene

0'

~ Ph Ph %

8 Reaction of B , I

o Ph~ Ph-

0'

0

~ Ph 4Ph & Ph 4 ph&

p

hPh

dp-8 R Ar

-unsaturated arbonyls

O B

O B

1,3- and 1.2- aryl rearrangement

AA D

DD A

decarbony lation

8 Reaction of Neontaining double bonds cis-trans or syn-anti isomerization

hydrogen abstraction and transfer

C-N bond cleavage and realated reactions (demtrogenatton, u -scission)

Y

Y photocyclization and photoaddition (X=Nor CH)

8Photoreaction of aromatic rings valence isomerimtion S 1 e x c i t a t i o n S2 e x c i t a t i o n

photoaddition photosubstitution

@ Porphyrins

The effect of substituent orientation is different from (type A) or the same (type B) as that in t h d reaction.

photoionizahon

h u P o r p h y r i n

photosynthesis

0

+ P o r p h y r i n ' + e

R

Ar

72


2.1.2 Electronically-Excited States and Reactivity Organic molecules can exist in two types of electronically-excited states: the excited singlet state and the excited triplet state. The lifetime of the excited singlet state is very short usually in the range of 10-y-10-7 sec. The triplet state has a much longer lifetime, sec in solution and from to lo1 sec in solid matrices. In the case ranging from lo-' to of such long lifetimes, the influence of oxygen quenching becomes important and cannot be neglected. For intermolecular photoreactions to occur in solution, triplet state reactions are preferable, since it takes some time for the reactants to encounter. However, intramolecular photoreactions and photoreactions in the solid state where the concentration of reacting partners is very high, proceed readily via short-lived excited singlet states. The fact that most of the photoreactions cited in Table 2.2 are intramolecular reactions is a direct consequence of this unique characteristic of photoreaction. Molecules containing carbonyl groups can exist in two types of excited singlet and triplet states: the nn? state corresponding to excitation of an electron of a non-bonding orbital of the oxygen atom to a n? antibonding orbital of the CO group and the nJP state which corresponds to the excitation of an electron of the norbital of the CO group to its n? antibonding orbital. Whether the lowest excited states (S, and T I ) are of the nn? nature or nn? nature is determined by the energy levels of the n and n orbitals and hence by the chemical structure of the molecule. In the case of formaldehyde (Table 2 . 3 0 ) the energy difference for the nn? excitation is smaller than that for the nJP excitation for both the singlet and the triplet states. Therefore S , and TI are of nn? nature. The CO bond is slightly elongated in the excited state, compared to the ground state. In the nrr* excited state, the dipole moment y is smaller, compared to the ground state, since an electron is transferred from the n orbital of the oxygen atom to the n? orbital of CO group. The presence of an odd electron-number in a nonbonding orbital of the 0 atom imparts a chemical reactivity similar to that of free radicals. For this reason, photoexcited carbonyl compounds easily abstract hydrogen both intra- and intermolecularly, as is shown in Table 2.2. Formaldehyde and benzophenone are well known to undergo hydrogen abstraction from the triplet state. Since the transition from T , to So is forbidden, the rate constant for phosphorescence, k,, is much smaller than that for fluorescence, kf.As a result of spin-orbital interactions the k, for 3nrr*+S0is about l-1O-'ssi, but the k, for 3nn*+So is about 103-102 s-'. For example in the case of naphthalene, which has only a nn? state, the k, calculated from phosphorescence lifetime at 77 K is 0.04 s-I, while in the case of benzophenone where the T i state is of nn? nature, the k, takes a value of 180 s-I. The effect of solvents on the absorption spectra are also different for the nn? and nn? absorptions. The polarization of the nJP excited state is larger than that of the ground state, while the dipole moment of the nrr* excited state is smaller than that of the ground state, as is shown in Table 2 . 3 0 . j ) Thus an increase in solvent polarity results in shifts to longer wavelengths (red shifts) for nn? absorptions and in shifts to shorter wavelengths (blue shifts) for nn? absorptions. The dependence of the nn? and nJP energy levels on the conjugation between a carbonyl group and an aryl group linked to the carbonyl group is shown schematically in Table 2 . 3 0 . 3 ) The energy level of the nn? state decreases with increase of the conjugation between the carbonyl and aryl groups, but the energy level of the nJP state is independent of the extent of conjugation, in a first approximation. The excited triplet energy level is lower than the excited singlet energy level for both the nn? and the nn* states. Thus, based on the combination of

73

Table 2.3 Energy Levels and Electronic Configuration of nn' and RR* (1) Excited state of formaldehyde

T

I

dc0= 1 . 3

l A

# = l . 3 D

dco= 1 . 2 2 A

l l = 2. 3D @ Schematic energy levels of tlle n n * and n n ' excited states

@) Solvent effect on the absorption spectra

of knzophenone 1

0 log

.!.0

-

2.5

-

2.n

-

E

0

I

220

I

I

2 GO

I

I

3 00

340

380

solid line in cyclohexane (non-polar solvent) dashed line in ethanol (polar solvent)

d e g r e e of c o n j u g a t i o n

R-C=O

74


the four lines drawn in Table 2 . 3 0 , the conjugation dependence of the excited energy levels can be divided into five regions. Acetone is an example for Region I where both S, and T I are of n d ' nature with a small extent of conjugation. Benzophenone lies at the border between Regions I and 11. Acetonaphthone and 9-acetylanthracene with larger aromatic rings lie in the Regions I11 and IV, respectively, showing that the T I states for these compounds are of nn* nature. For this reason, these molecules do not abstract hydrogen upon light irradiation in hydrogen-donating solvents. The selection rules for electrocyclic reactions discovered by Woodward and Hoffmann provide a beautiful demonstration that photoreactions are carried out via electronicallyexcited states.4) For intramolecular a-bond formation to proceed between the two ends of a conjugated polyene with m double bonds, the bonding interaction should be such that orbital lobes of identical signs at the terminal atoms overlap in the highest occupied molecular orbital (HOMO) of the reaction. As orbital symmetry has to be conserved during the reaction, the reaction will proceed disrotatorily when the highest occupied molecular orbital is symmetric with respect to the plane bisecting the forming a-bond, and the reaction will proceed conrotatorily when the HOMO is antisymmetric. Thermal reactions proceed in the ground-state energy levels. For a molecule with 4n (or 4n + 2) n-electrons, where n is an integer, the highest occupied molecular orbital is the 2nth (or 2n + lth) orbital from the bottom, which proves to be antisymmetric (symmetric) from a simple calculation of molecular orbital method. Therefore, thermal reactions proceed conrotatorily for 4n n-electron systems and proceed disrotatorily for 4n + 2 n-electron systems. In contrast, photochemical reactions proceed from the electronically-excited states. The symmetry of the highest occupied molecular orbital for a photoreaction is the opposite of that for thermal reaction, and hence photochemical reactions proceed disrotatorily for 4n n-electron systems and proceeds conrotatorily for 4n + 2 n-electron systems. When two different substituent groups are bound to the terminal carbon atoms, thermal and photochemical reactions yield stereochemically different products. This opens the possibility of stereospecific syntheses using a photoreaction. A few examples of electrocyclic reactions are shown in Table 2.2. This concept of suppressing the rotational direction of a thermal reaction by introducing a fixed substituent has been applied in the molecular design of photochromic molecules unable to undergo thermal backward reaction (see Section 2.2).

2.1.3 Photochemical Reactions in the Solid State Photochemical reactions proceed in the gas phase, in solution, and even in the solid state. Photochemical reactions in the solid state are especially important when considering the applications of photofunctional molecules in materials, where photophysical processes and photochemical reactions are used to trigger some change in a given physical property of a system, such as its color, solubility, or electroconductivity. When considering reactions in the solid state, it is important to draw distinctions between reactions in crystals and reactions in amorphous solids, since the controlling factors are quite different in these two situations. Reactions in the crystalline state are controlled primarily by topological factors, such as the distances and the relative orientation of potentially reactive groups. Trans-cinnamic acid undergoes photochemical dimerization when the distance between adjacent double bonds in the crystal is shorter than 4 A (Table 2 . 5 0 ) . Irradiation of 2,5-distyrylpyrazine (DSP) which has two double bonds per molecule leads to a linear high

75 Table 2.4 Symmetry Rule of Molecular Orbitals and Reactivity

A Z

ta t o r y

0 0 0 0 0 0

0 0

0 0 0 0 0 0

0 0 0 0 0 0

photoreaction

(4n+2) n -electron system conrotatory msrotatory LUMO

4n disrotatory HOMO

IT

-electron system CONOtatOry

reacting orbital

LUMO

0 0 0 0 0 0

thermal reachon

HOMO

76

crystalline

-

Molecular motion of reactive groups

- molecular motion of the medium (glass transition tempratwe, B -kensition, 7 -transition) -- free volume and conformation necessary for the reaction to occur (molecular dispersion system bound-to-polymer system)

- interaction of the reacting p u p with the medium

Solid-state reactions

macroscopically-heterogeneous system (photoabsorption at the surface, influence of oxygen hffision) - macroscopically-homogeneous but microscopically-heterogeneoussystem (coexistence of crystalline and amorphous parts microscopically phase-separated system) macro- and microscopically homogeneous system (distribution of free volume and conformation) -

amorphous

Inhomogeneity of reaction

-

Change inthemedium during reaction

-

Factors other than

2,Sdistyrylpyrazine (a) into h e r (b) and polymer ( 4

change in aggregation structure due to network formation

~

excitation energy transfer and migration electron transfer dependence of the reactivity in polymer solids I

1

I I

the photoisomerization of poly(vinyl cinnamate) (PVCrn)

I

-

-

ry

o.70br -___.

T

r e a c t i v i t y of s i t e s

B:208K

a: 2,Sdistyrylpyrazine b: dimer c. polymer

""r 0

1. polymer-polymer reaction (dithion controlled) 2: polymer side chainsmall molecule reachon (diffusion controlled) 3 lntramolecular manchain scission 4 lntramolecular reaction UI polymer side chains 5 intramolecular reaction of a small molecule wth small c n t d free volume

0.2

0.4

0.6

conversion

0.8

1 3

A: crystalline cinnamic acid B: PVCm film C: PVCm solution in dichloroethane Insert:histogram for the site reactivity distribution corresponding to curve B

2.1 Characteristics of Photochemical Reactions

77

polymer via cyclobutane formation.

This reaction, called a four-center photopolymerization, is a typical example of topochemical reactions used to prepare polymer crystal^.^) The changes in higher-order structure during the reaction are shown in Table 2.50.Various polydiacetylene crystals have also been prepared by solid-state photopolymerization of diacetylene monomer crystals, such as 1,6-dicarbazoyl-2,4-hexadiene. These syntheses have attracted considerable interest, since they can lead to organic materials of high conductivity or of nonlinear optical properties.

Q In amorphous solids, such as polymers or glassy frozen solutions, the reactive groups are located randomly in the solid. Therefore molecular motion is required for reactions to occur. This restriction in the freedom of molecular motions is the first characteristic of reactions in amorphous solids. The controlling factors for solid-state reactions are summarized in Table 2.5 for both the crystalline and the amorphous states6)For reactions necessitating the approach of reactants and/or the separation of products, the reaction rates are determined by a balance between the molecular motion of the matrix, the amount of free volume needed for the reaction to occur, and the intrinsic characteristics of the chemical reaction. A second factor to consider in the case of amorphous reactions is that the reactivity within amorphous solids is not homogenous. For example, one has to consider heterogeneity on the macroscopic scale in systems where the light is absorbed only at the solid surface or when the diffusion of oxygen from the surface affects the reactivity. Morphological heterogeneities also play a role which may be more difficult to control, especially in the case of solids where crystalline and amorphous phases coexist, of polymer blends, or of various aggregates. Moreover, for reactions carried out below the glass transition temperature, T,, of a polymer, microheterogeneity in free volume and/or conformation distribution leads to heterogeneous progress of the reaction. For this reason, the photoisomerization of photochromic molecules such as spiropyrane (Table 2.2), known as a first-order reaction in solution, proceeds very slowly and heterogeneously in solid polymers below T,. Changes in matrix structure in the course of a reaction, observed for example during photocrosslinking of polymer side chains, also lead to a change in reactivity. Reactions in the solid state can be facilitated by phenomena occurring without mass transfer, such as excitation energy migration or the displacement of reactive sites by successive chain reactions between adjacent monomers with no mass transfer.

78


The changes in quantum yield, @, for the photodimerization of poly(viny1 cinnamate) (PVCm) (1) in films are shown in Table 2 . 5 0 together with those for the same reaction carried out in solution and in the crystalline phase.')

+*-y j 0 I

o=c,

/H

"6 /C

=c

(1)

Photodimerization of crystalline cinnamic acid proceeds with a constant 0 value. Photodimerization of PVCm in solution proceeds up to 100% conversion, albeit with smaller @values because of the low local concentration of reactive groups. In PVCm film @decreases in the course of the reaction, as indicated by the curves B and B'. The reaction proceeds only to 50% conversion. This is a direct consequence of the heterogeneous distribution o f microenvironments around the reactive sites and of the change in microenvironments as a result of crosslinks formation. The temperature dependence of photochemical reactions in polymer solids can be classified according to reaction types (Table 2 . 5 0 ) . As a general rule, the mechanisms of solid-state reactions are expected to change with temperature in the following order: Stage ( I ) Chemical control: the reaction proceeds at the same rate as the corresponding solution reaction (with normal activation energy). Stage (2) Diffusion control: the reaction proceeds heterogeneously, reflecting the restricted molecular motion of the matrix polymer. Stage (3) Freezing of the reaction due to the slowing of molecular motion. In the cases of spiropyrane photoisomerization (Table 2.2) and of the Norrish type TI photo-chain scission, the crossover from stage (1) to (2) occurs around the Tgof the matrix polymer where the micro-brownian motion of the polymer main chains is suppressed. The crossover from stage (1) to (2) for the photoisomerization of azobenzene is observed at subglass transition temperatures, T, and T,, where the local motion of the side chains and the rotation of the phenyl groups are suppressed. In each situation, the crossover temperature will depend on the amount of local free volume needed for a specific reaction to occur and on the intrinsic chemical reactivity of the transformation. Thus, the reactivity at a certain temperature of a certain solid-state reaction can be estimated knowing its rate constant in solution, the critical free volume needed for the reaction to occur, and the T., T,, T, points of the polymer. Table 2 . 5 0 1 and 2 illustrate the case of intermolecular diffusion-controlled reactions, such as the reaction between free radicals and triplet quenching, of reactions that proceed very rapidly above Tg,of reactions which undergo a change in temperature dependence at T,, and of reactions which are frozen at Tfior T,. The case 3 in Table 2 . 5 0 corresponds to the Norrish type I1 reaction in the photolysis of polymers with carbonyl groups, where @for the reaction above T, is the same as in solution. It begins to decrease below Tg,and the reaction stops entirely at TfiThe case 4 is observed for example during the photo-Fries rearrangement of the side chains of poly(pheny1 acrylate). The reaction is frozen below T, corresponding to the rotation of the side chains. The case 5 relates to the photoisomerization of azobenzene (Table

2.2 Photochemical Reactions and Physical Property Control

79

2.2) and fulgides (Table 2.2) in polymers. The reaction occurs at several mobile sites even at temperatures below T , since the critical free volume (sweep volume) required for the reaction to occur is very small in this case. Azobenzene is known to photoisomerize up to 15% even at liquid helium temperature (4 K) in polycarbonate film, suggesting an inversion mechanism with the change of sp2+sp-+sp2 in hybrid orbital of the N atom.

2.2 Photochemical Reactions and Physical Property Control 2.2.1 Photosensitive Polymers Photosensitive materials take advantage of changes in physical properties due to chemical reactions induced by irradiation with ultraviolet or visible light. A typical example is the photographic film which uses the photodecompositon of silver halides to produce an image. Photosensitive materials which are not based on silver halides, the so-called non-silver-salt type photosensitive materials, include diazotype photosensitive papers, electrophotographic materials, and photosensitive polymers for letterpress printing and integrated circuit manufacturing. Photosensitive polymers are defined as polymers possessing photosensitive groups capable of undergoing crosslinking, chain scission, or other chemical reactions upon irradiation. These reactions, in turn, lead to changes in a specific physical property, such as solubility, adhesive strength, softening point, or change from liquid to solid phase and viceversa.*) Such materials may have been used in antiquity. For example, it is reported that in ancient ages in Egypt, the linen wrapping of mummies were hardened by dipping them first in a lavender-oil solution containing high-molecular-weight bitumen and then exposing them to sunlight. The technique of lithography has its origin in France in the first half of 19th century. In the early processes a natural photosensitive resin, the bitumen of Judea, was the photosensitive material. Towards the end of the 19th century the photosensitivity of diazocompounds and the photodimerization of cinnamic acid were already known. However, the modern technology of photosensitive polymers began only in 1930 with the discovery of photoresists by photocrosslinking of unsaturated ketones. In 1952, Minsk working at Eastman Kodak, U.S., reported for the first time the use of poly(viny1 cinnamate) (1) as a photosensitive polymer using the photodimerization of cinnamate groups. This polymer was prepared by reaction of poly(viny1 alcohol) with cinnamoyl chloride. The film of polymer (1) was crosslinked upon photoirradiation via intermolecular cycloaddition leading to the formation of cyclobutane rings, as shown in Table 2.2. This crosslinking reaction effectively creates a new material insoluble in common solvents. However the parts of the film which were not irradiated remain soluble in these solvents. Thus, when photoirradiation takes place in the presence of a mask, only the irradiated parts of the film remain on the substrate after washing the film with a solvent (developing). A polymer of improved reactivity, poly(/3-vinyloxyethyl cinnamate) (2) was prepared in Japan by cationic polymerization of the monomers. The superior performance of this polymer can be attributed to an increase in the local mobility of the cinnamoyl groups due to the presence of spacer chains.

80


CH2= CH- 0 -CH,CH,CI

+ NsOCO -CH-

CH

quaternaryammnimsalt

BF,OEt,

*

a

CH,=CH-O-CH&HzOCOCH=CH

+CH,-CH+

I

OCH2CH20COCH2=CH

Such photocrosslinkable polymers are used in the manufacturing processes of printed wiring boards, integrated circuits (IC), or large scale integrated circuits (LSI). They are known as negative-type photoresists. The word “resist” refers to the use of these polymers which act as protecting films against etching of silicon dioxide. Schematic diagrams of the lithography and letterpress printing processes9)are given in Table 2 . 6 0 . There is an another class of photosentitive polymers, known as positive-type photoresists. These polymers become soluble upon irradiation, either as a result of a decrease in their molecular weight or due to the formation of solvent-soluble groups. For example, onaphthoquinonediazidesulfonate (3) is insoluble in water. Upon photoirradiation, nitrogen is evolved, Wolff rearrangement leads to ketene (4), which in the presence of water is converted to 3-indenecarboxylic acid ( 6 ) which is soluble in alkaline water. When compound (3) is mixed with a phenol-formaldehyde resin soluble in alkaline water, the composite becomes insoluble. However, subsequent photoirradiation triggers the formation of indenecarboxylic acid ( 6 ) and subsequent enhancement of the resin solubility. Hence development with an aqueous solution of an organic amine generates patterns after dissolution of the irradiated parts. This system is presently the most important positive-type resist photosensitive material used in planography and LSI lithography. A transient absorption spectrum recorded after a 3p sec pulse irradiation presents a band at 350 nm (Table 2.6@), which has been attributed to the intermediate (5) produced by the reaction of ketene with water.’”)

S03R

SO,R

S03R

(3)

(4)

=

W

OH I

‘

O

H

-

Q--JcooH

In newspaper printing, photosensitive resins are used frequently during the platemaking process. The process is very simple. It relies on the photopolymerization of a soluble monomer or oligomer resulting in the formation of an insoluble polymer film. Aromatic carbonyl

81 Table 2.6 Photoresists and Their Photochemical Reactions

0characteristic curves

0 Photoresists using step-wise photoreaction.

of negativeand positive-tone photoresists

Manufactwing process of silicon semiconductor integraed circuit by using negative-tone photoresist

..-

I

Dhotores i s t

I

light

'J

'\si -negat

wafer

i ve film

u c

.- Y

(no P)

-

ocross I i nked resist

e t c h i n g o f S i 0 2 w i t h HF

.-

L L

after laser pulse irradiation

01 0

c

01

0hufaactunng process negat i ve photo-

I sensitive

J

polymer adhesive I aye r

dissolution (water, solvent)

I drying

w i t h warm wind

0.02 0.00

YY-

-0.02

.-

of nylon letterpress prinbng plates using photoinduced chain reactions

0.04

L

al

( s e v e r a l minute)

100-

@ Transient absorptron spectm

t h e ion i n j e c t i o n

I

10

of naphthoquinonediazidsulfonale

removal o f r e s i s t

I

1

0.1

i n c i d e n t I i g h t i n t e n s i t y (rnJ-crn-')

-0

o'o ..

01 0 -

c

m

.o. 02

R

L

0 cn

R

UJ

300

350

400

450

500

w a v e l e n g t h ( n m ) laser wavelengfh 266nm pulsewdth 5ns a inwater b m dioxane c m dioxane/water (911) solid hne after 2 P s dashed h e a, after 2ms, c, after 0 3ms

82


compounds such as benzoin ether (7) and benzylalkylketal (8) are common photoinitiators. They generate free-radicals via a-scission from the triplet state. The free-radicals then catalyze the polymerization of olefinic monomers. Practical compositions of photosensitive resins include various monomers and polymers, such as acrylic and methacrylic esters, acrylamide, polyamides, thermoplastic elastomers, and poly(viny1 alcohol). 0

n

The sensitivity and the resolution of photosensitive polymers can be evaluated from a characteristic curve obtained by plotting the residual thickness of the film against irradiation energy. Schematic examples of these characteristic curves are shown in Table 2 . 6 0 for the cases of crosslinking (negative) type and solubilization (positive) type resists. For negativetype resists, the sensitivity is given by the minimum irradiation energy, E,, required for gelation and insolubilization of the polymer. It corresponds to the abscissa of the rising point of the characteristic curve in Table 2 . 6 0 . The amount of irradiation photons for gelation to occur, E,, is expressed as Equation (2.1) assuming that gelation requires on average the reaction of one monomer unit per polymer chain.

E, =

~

Id AM,@

Where, @ is the quantum yield of photocrosslinking of the photosensitive group, 1 is the film thickness, d is the specific density of the polymer, A = l-lO-K' is the fraction of absorbed irradiation energy, E being the molar extinction coefficient, C the concentration of photosensitive groups in the film, and M, is the weight-average molecular weight. The highest sensitivity of negative-type photoresists can be estimated theoretically from Eq. (2. I). For example, the maximum value for @ is 2 in the absence of chain reaction and 1 is the maximum absorbance A. Setting d = 1, we obtain E, = 0.15 mJ/cm2 for a 400 nm irradiation of a lo4 cm-thick film made with a polymer of M, = los. Current negative-type photoresists usually have a sensitivity of 1-10 mJ/cm2. Therefore improvements of their sensitivity can be anticipated with the use of more performant materials. For comparison, it should be noted that mJ/cm2. the sensitivity of silver salt photography amounts only to The important factor determining resolution is the contrast, defined as the slope y = tan0 of the characteristic curves in Table 2 . 6 0 . The value of y depends on the molecular-weight distribution, MJM,,, of the polymer. The slope y increases, and hence the level of contrast increases, as MJM,, nears unity. Deformation and shrinkage of the patterns of negative-type resists have to be minimized. However, swelling of the crosslinked parts with the developing solvent often occurs, leading to a loss in resolution. The influence of swelling is less marked

2.2 Photochemical Reactions and Physical Property Control

83

for positive-type resists. Thus, in general one can obtain patterns of higher resolution with positive-type resists than with negative-type resists. Currently the refinement in the manufacturing process is such that the attainable resolution seems to increase by a factor of four every three years, leading to finer and denser semiconductor integrated circuits: from IC, LSI to super LSI. It has been said that the limit of rcsolution of photoresists will stay around 0.4 pm because of the wavelength limitation imposed by the usual light sources. It may be possible to get a pattern of a resolution as small as 0.1 pm with the UV radiation of excimer lasers. New fine-manufacturing technologies using electron beam, X-ray, and ion beams are being actively developed in order to achieve patterns with a resolution of less than 0.1 pm. Typical reactions employed in photosensitive and electron-beam sensitive polymers are summarized in Table 2.7 according to their types. Photopolymers of high-sensitivity have been designed by combination, o f photosensitizers or crystalline matrices. The chemical amplification method with onium salts, which mimics silver salt photography, is one of the most promising method of high sensitization. A well-known example is that of the poly(p-tbutoxycarbony1oxystyrene)-oniumsalt system, where a protic acid produced by UV photolysis of a triarylsulfonium salt initiates a chain reaction generating protons (Table 2 . 7 ) . The chemical change in the polymer side groups results in an increase of the solubility of the polymer in aqueous alkaline solution, providing a positive-type resist. The name “chemical amplification” emphasizes the fact that the chemical (a proton in this case) produced in the first photoreaction a c t s a s a catalyst f o r t h e next reactions. In t h e c a s e o f t h e poly(phthalaldehyde)(9)-onium salt, the protic acid produced from the onium salt by photoirradiation catalytically cleaves the main chain acetal bonds. Moreover, above the ceiling temperature o f the polymer, t h e main c h a i n scission i n i t i a t e s e f f e c t i v e depolymerization which converts the polymer entirely to monomer molecules, providing an extremely sensitive positive-type resist. Systems where the patterns are formed only by photo- or electron-beam irradiation process are called self-developing systems. Generally speaking, the chain reactions, polymerization and depolymerization, are regarded as a kind of sensitization system. Insolubilization by crosslinking is also very effective since the change of just one monomer unit per polymer chain (usually composed of more than 1O3 monomer units) triggers a change on the macroscopic scale. Nevertheless, for lithography to reach the sensitivity of silver halide photography a totally new phenomenon is required. OCH

CHO

Incremental improvements of the known processes are unlikely to yield photoresists of such high sensitivity.

84

-

Table 2.7 Typical Reactions Used in Photosensitive Polymerization and Photoresists

0C m s s h b g -CH2--CH OI - C-CH=CHa

Photodimerization

hv

bl

Crosslinking

(DN--NtNkon:

addition i m r t ion t o double bnj ' to c+ bnj

N2+

by using the azide group

-C Ha-

Radical crosslinking by using the chloromethyl group

CH-

crosslinking by dimerization

-@ '." -CHz-CH-

@

EB,

CHzCl

cross-

X-ray

linkirg

CH2

I dichromate Crosslinkq

-CH2-YHOH

yH2 C=O

+Cr6+5-CH2-;-+Cr3+-+

0

by adding a crosslinker bis-azide

I

N ~ Q - R ~ N ~ + $

Photoinitiated radical polymenzation

@

oe-ia 0 OCHJ

5FCH H 0

> N e R a ~ $ l I

o!.+

OCH

.cHs

+

-

OCHs

radical polymerization

+- - + - +

+ +-* +-*-+ D

D

A-'

A

along a polymer chain

A

<energy transfer via dipoledipole interactions>

+-- - +

<energy transfer via electron exchange>

D

D

A

(the dotted line indicates

A

the energy migration process)

@ Redox Ipotentials (SCEunits), singlet state energies ( E 9 ) and triplet state energies (E T ) of typical organic compounds compound3

naphthalene phenanhene anthracene benzophenone pdicyanobenzene 9,l Odicyanoanthracene pbenzoquinone methyl viologen (MV ' + ) chloranil tetracyanoethylene N,N,N',N .-tetramethylppheny lendamine N,Ndimethylaniline perylene PY [R~@PY)9 1 ' + tetraphenylporphine zinc(II)

E(D

+

/D)

Es

E(NA - )

0

03

eV

I .60 1.58 1.16

-2.29 -2.20 -1.93 -1.68 -1.60 -0.89 -0.51 -0.45 0.02 0.24

3.99 3.59 3.31 3.23 4.27 2.86

0.16 0.81 0.85 1.20 1.29 0.7 I

-1.35 -1.35

3.85 2.85 3.34 2.12 2.05

ET

kJ

eV

w

385 347 319 311 413 276

2.64 2.69 1.85 3.00 3.06

255 259 179 290 295

2.3 3.10 2.70

222 299 261

2.64 2.99 1.52 2.09

255 286 147 201

1.59

154

372 275 322 205 198

3 . I Energy Transfer and Electron Transfer Processes

107

function of the donor emission, v ) the extinction coefficient of the acceptor. Eq. (3.5) implies that energy transfer will occur if there is substantial overlap between the emission spectrum of the donor and the absorption spectrum of the acceptor and for sufficiently short D* to A spatial separation distances, usually in the order of 50 to 100 A. Interaction between the triplet state of the donor and the ground state of the acceptor results in energy transfer by electron exchange. The rate constant of this process, known as Dexter electron transfer mechanism,?' is described by Eq. (3.6), where K is an energy dimension constant and L is the effective Bohr radius. (3.6) From Eq. (3.6) we observe that the rate of energy transfer by electron exchange mechanism decreases exponentially with ZRIL. Thus, it will be negligibly small as R increases more than on the order of one or two molecular diameters. Hence the effective distances for Dexter energy transfer range between 10 and 15 A. Both the Fiirster and the Dexter energy transfer mechanisms require spectral overlap of the donor emission spectrum and the acceptor absorption spectrum. However, energy transfer is known to occur even in the absence of spectral overlap, resulting in effective quenching of excited states. As an example, we can cite the quenching of the fluorescence of aromatic hydrocarbons by dienes, a process which involves thermal deactivation o f an excited state encounter complex, or exciplex, between D and A (Eq. (3.7)):

D* + A+(DA)*+D

+ A.

(3.7)

The rate of energy transfer between D* and A in solution, can be determined from the Stern-Volmer equation (Eq. (3.8)). In this equation, @and 0"are the fluorescence quantum yields of D* in the presence and in the absence of A, respectively, tois the fluorescence lifetime of D* in the absence of A, and k,, the rate constant of energy transfer to be determined. Transient spectroscopy measurements yield experimental values of toand o f t, the fluorescence lifetime of D* in the presence of A. The Stern-Volmer equation, rewritten in the form (3.9), relates the fluorescence lifetime values to k,:

l i t = l/q1+ k,[A].

(3.9)

For energy transfer processes occurring in solution, the maximum value of k, is obtained when every encounter of A and D* results in energy transfer. This diffusion-controlled rate constant, kd,,, is evaluated from Smoluchowski equation (Eq. (3.10)),31where R is the radius of the reactants, D, the diffusion constant, N A , Avogadro's number, kg, Boltzmann's constant, and qothe viscosity of the solution. With these assumptions and in solvents of usual viscosity, the diffusion-controlled rate constant is of the order of magnitude of 10' to 10"' L mol-Is-'. The rate constants of excitation energy transfer are diffusion-controlled for chromophores in solution, when the process originates from the singlet state of D with large spectral overlap of the donor emission and the acceptor absorption, or from the triplet state of D, when the

108


energy level of 3D* is higher than that of jA* (3.10)

The term energy transfer is usually limited to the description of processes involving transfer of energy between two different chromophores. The term energy migration, in contrast, refers to the process of energy transfer between identical chromophores. In polymeric systems the local concentration of chromophores is high. Therefore excited chromophores and ground-state chromophores are kept in close proximity, and energy migration becomes the predominant process. The “chlorophyll antenna” acting in biological photosynthesis is a well-known example of a system where energy migration takes place. Table 3 . 1 0 presents a pictorial model of an energy harvesting system taking place in polymeric systems of photonics materials. The increase in the quantum yield of the photochemical reactions and the antenna effect are expected to occur in such a system.

3.1.2 Photoinduced Electron Transfer: Theoretical Background The fundamental mechanistic studies of quenching and sensitization in energy transfer processes summarized in the previous section were all carried out very early in the history of contemporary photochemistry. In contrast, detailed investigations of photoinduced electron transfer reactions only began in the late 1960’s. Interest in this process grew after the first report of an emission from complexes formed between perylene and N, N-dimethylaniline.41 These complexes were named “exciplexes” and postulated as originating from electron transfer between the excited singlet state of perylene and the ground state of the amine. The exciplex is defined today as a two-component excited system in which charge and electronic excitation are shared by the two components. Spectroscopic tools developed in the 1970’s were then applied to study the reaction. Most importantly they made it possible to observe directly radical ion intermediates by a combination of measurements using laser flash photolysis, electrochemical techniques and magnetic resonance spectroscopy. From then on, the scope of the photoinduced electron transfer reaction grew rapidly, especially after organic chemists discovered that it was possible to generate radical cations of electron-rich olefins and strained cyclic molecules via photosensitized electron transfer. Much progress also was made in the study of these phenomena during photoirradiation of transition metal complexes. This plethora of experimental physico-chemical data stimulated a growing interest from theoreticians, led by Marcus5’who had developed a theoretical description of the electron transfer process. His theory is now widely used to model photoinduced electron transfer reactions. It is well accepted that the same fundamental mechanism governs the initial processes in many photosensitized electron transfer reactions, whether they occur in solution during an organic synthesis or a transition metal conversion, in polymeric materials, in micelles, or in biological membranes. There are now several examples of the practical use of photoinduced electron transfer in organic photoconductors, information conversion and storage, energy conversion systems, and separation/transport devices. In many cases this unique reaction is the “key process” at work. An understanding of its detailed mechanism therefore is of importance not only for its scientific interest but also from a practical

3.1 Energy Transfer and Electron Transfer Processes

109

viewpoint. kdlf ka kP D* + A or D + A* # (D*A) or (DA*) # D ’+ A’--+reaction Lkd k-dlF k-, Jk, D+A D+A

products (3.11)

The overall processes involved in photoinduced electron transfer are outlined in Eq. (3. I I ) . In this scheme, k,, is the rate constant for unimolecular decay of the excited state, kdlr and kdd,,are diffusion-controlled rate constants, k, = k,, is the unimolecular rate constant for electron transfer, k-, is the rate constant for the backward reaction of rate constant k,, k, is the rate constant for reverse electron transfer to ground-state reactants, and kp is the rate constant for radical ion dissociation or trapping reactions in the presence of scavengers. Applying a steady-state treatment to the various intermediates in Eq. (3.1 1) one can evaluate k, (Eq. (3.12)). the overall bimolecular rate constant for quenching ofthe excited state. It is usually assumed that ka 8%), r reaches a plateau value and does not increase with increasing crosslinking density. The following interpretation was given to this observation. Micropores can form in “hard” gels. The fluorophores are preferentially bound or adsorbed to the inside surface of such micropores. The mobility of a molecule in the micropores has to be independent on the crosslinking density. Two data points, indicated by the symbols A and in Table 3 . 9 0 , were obtained with a highly crosslinked gel (DVB = 15%). The values of r are particularly small in this case. The gel was in fact prepared in a different manner than all the other gels. A non-solvent for polystyrene, hexane, was added during polymerization, resulting

I30


in the formation of a rigid macroreticular gel. It has been postulated that these gels consist of a homogeneous macroporous structure with large solvent-accessible channels inside the gels. The fluorescence anisotropy results corroborate this assumption. Another interesting aspect of the fluorescence technique is that it allows one to probe the validity of the predictions derived from theoretical models characterizing the local segmental motion modes of the polymer chain. This can be achieved for example by fluorescence depolarization measurements on solution of polymers bearing anthryl groups on the main chain. Photoprocesses, such as singlet or triplet energy transfer or excimer formation, are exceptionally fast among chromophores linked to a polymer chain. These processes are diffusion controlled reactions (molecular motion controlled reaction): a reaction occurs inevitably when the two interacting species are within a given distance and in a given orientation. Therefore it is possible to use photophysical probes to measure the rate of molecular motion and encounter. The dynamics of polymers in solution are governed by two types of motions (Table 3 . 1 0 0 ) : (1) macroscopic conformational changes of the entire macromolecule in the presence of solvent molecules ( e . g . intrapolymeric end-to-end cyclization) and (2) microconformational changes (segmental motion) consisting of rotational motions of segments of the polymer chain. Each motion has been investigated through the use of photophysical techniques. We present how intrapolymeric reactions and in particular end-to-end reactions were applied by various research groups to test the validity of theories predicting the conformation and dynamics of polymers in solution. Szwar~,~’) in 1972, was the first to explore this premise. He examined by ESR spectroscopy the dynamics of a polymethylene (PM) chain bearing naphthyl groups at both ends with the idea of determining the rate constant of polymer cyclization kintra.The approach was extended to the study of the cyclization dynamics of oligomers carrying pyrene groups at both chain ends, applying excimer formation kinetics. Since then several groups have reported experiments with longer polymer chains. Pyrene excimer formation (lifetime about 200 ns) was used by NishijimaZ8)in the case of oligomeric polymethylene chains (Py-PM-Py), by WinnikZ3)in the case of polystyrene (Py-PS-Py), polydimethylsiloxane (Py-PDMS-Py), and polyoxyethylene (Py-POE-Py) with intermediate molecular weight. Ushikiz9)examined the cyclization of di-anthrylpolystyrene (A-PS-A) with longer polymer chains ( N = 1 10-3300, N : number of monomer units) by delayed fluorescence measurements and triplet-triplet annihilation (lifetime approximately 1 ms). Values of the intramolecular cyclization rate constants measured for these polymers in several solvents and temperatures are plotted as a function of the chain length N in Table 3.100.)) The rate of local conformation changes of polymer chains has been studied by fluorescence depolarization, NMR, ESR, dielectric dispersion, and ultrasonic absorption spectroscopy. The kinetics of excimer formation within polymer substituents have been monitored for polymers carrying excimer forming aromatic side groups. The data interpretation is complicated by the fact that the rate of excimer formation is comprised of two components: (1) the rate of the internal rotation of adjacent aromatic rings and (2) the rate of excitation energy migration through the aromatic side chains. If one can separate these two effects, it is possible to evaluate the rate of main chain conformational changes from kDM,the rate constant of excimer formation. The effect of the degree of polymerization, N , on kDM,was measured for various samples of oligomeric atactic polystyrene ( N = 2-20) by pulsed picosecond radiolysis in the facilities of the Nuclear Research Center of the University of Tokyo (Table 3. The value of kDM

3.2 Photophysical Molecular Probes

131

for short oligomers (2 5 N I 8) is proportional to N , a trend which indicates that energy migration takes place over the entire molecule and excimer formation occurs by bond rotation. For oligomers of N > 8, kDMremains constant and equal to the value measured for the oligomer with N = 8. Thus, the distance of energy migration is approximately equivalent to 8 monomer units. The time taken by a single energy migration step between two adjacent sites is 30 ps. The tacticity of a polymer chain also influences the rate of excimer formation via internal rotation of adjacent substituents. Meso and racemic stereoisomers have to be distinguished when determining the time constant t for the decay of the monomer fluorescence. In the case of oligostyrene ( N = 2 4 ) , the decay of the all-racemic form follows a single exponential decay law with a time constant t = 11.0 ns for rPS2. Diastereoisomers having at least one meso structure exhibit a biexponential decay, with time constants tI< 1 ns and t2of several ns (Table 3.10@).1y)The time constants of the meso forms are much faster than those of the racemic forms, since in the meso isomer the ground state conformation (g’t) can reach the excimer conformation (tt) in a single step, while in the racemic form there is a larger steric hindrance for excimer formation from the ground state conformation. The time constant t2for trimers and tetramers does not depend on the number of meso forms if there is at least one such form present, implying that energy migration covers the entire molecule in these cases. Assuming that these trends are also valid in the case of long polymer chains, one can conclude that in isotactic PS excimer formation involves the meso forms, energy migration covers approximately 8 monomer units, and the conformational changes occur cooperatively within a few ns. The rates of excimer formation in model compounds for poly(viny1carbazole) have been determined by ps laser measurements. Two distinct rate constants for excimer formation were measured for the racemic and meso forms of either the vinylnaphthalene dimers or the vinylcarbazole dimers, with k,, of the meso forms larger by about one order of magnitude than kUMof the racemic forms. The termination reaction of free radical polymerization is a typical example of an intermacromolecular diffusion controlled reaction.3)Photophysical studies carried out in the 1980’s demonstrated for the first time that the reaction is solvent- and molecular weightdependent. The experiments involved triplet quenching of probes attached to polymer chain ends. A benzil group was linked to the end of one PS sample (PS-B) and an anthryl group was linked to the end of a second PS sample (PS-A). The quenching rate coefficient kq of the benzil phosphorescence by anthryl groups is given by Eq. (3.26), where tois the lifetime of benzil phosphorescence in the absence of anthryl and t is the benzil phosphorescence lifetime in the presence of anthryl in concentration [A]. 1

-

7

=

1

-

r,,

+ k,

[A]

(3.26)

lmportant results of these experiments are reported in Table 3 . 1 0 0 for polymer/polymer reactions and reactions between polymers and small molecules in dilute solution.31)The solvent dependence of the polymer/polymer reaction is attributed to the viscosity qo of the solvent and to coil expansion (excluded volume effect). By multiplying kq by the ratio q J T the effects of solvent viscosity and reaction temperature can be eliminated from the data and the effect of coil expansion on the reactions can be compared. In the case of a reaction between a small molecule and one polymeric chain end (e.g. benzil, B, and PS-A with N-102),

I32

Table 3.10 Chain Dynamics of Polymers in Solution

@ Schematic representation of macroconformational (left) and microconformational (right) changes in polymer chains

n

Q bpendence of the rate constant k

8

8

of intrapolymeric end-to-nd collision on the polymer chain length ( F ' :inverse of the excited state lifetime) ~

0Rate constant k

D u of excimer formation in polystyrene oligomers as a function of the degree of polymerization N

I

I

I

I 0

I02

\

I

I

103

104

n u m b e r o f atoms a l o n g p o l y m e r m a i n c h a i n n

degree of polyuerizatian

N

@ Dependence of the intermacromolecular&ffusionantroIled reaction rate constant for polystyrene samples of varymg degree of polymenzation and in several solvents

I

I 0

102

I

o3

degree o f polymeriratlon N

<Solvents>

benzene : butanone : cyclohexane :

0 ,A , 0 0 , A , El 0 , A,

reaction between PS-B and PS-A : reaction between B and PS-A : reaction between B and PS-A-PS :

0, 0 , 0 A, A, A

o , m , =

133

6)Main conformational changes for meso and racemic PS2 (E prowl end group, R : phenyl group) Tune constants of monomer singlet fluorescence of PS in cyclohexane at room teperature ’

A I and A P are prefactors of first and second components, respectively. r 7 and r 1 are relaxation times of monomer singlet fluorescence for first and second components, respectively A ,

m mm m IT

mmm m

mn m

rt

PS2 PS3 PS3 PS3

053 063 067

PS4 PS4 PS4 PS4

071 071 0 17

e x c i m e r

(ns)

As

08f01

047 037 033

06f01

09f02

___

1

029 029 083

OBfOl

07t02 IOf03

___

g r o u n d

1

s t a t e

M E S O

e-xc

i me r

R

A

C

E

M

I C

rs

(ns)

6 + I 35205 40f05 74f04 20f03 20f03 23t03 82202

134


k, takes a value of 1/2. This value is independent of chain length for longer polymers. No polymer expansion effect can be detected in this reaction. In the case of the reaction between a small molecule and a functional group located in the middle of a polymer chain (e.g. B + PS-A-PS, N > lo3),the value of k, decreases. This tendency becomes more evident as the solvent quality for the polymer decreases. The dependence of k, on N for polymedpolymer reactions in dilute solutions in a good solvent obeys Eq. (3.27):

k,

0~

N-p,

/3 = 0.3 (dilute solution in good solvents).

(3.27)

In the case of high molecular weight polymers the effect of polymer concentration on k, can be observed even when the conversion degree does not exceed 1%. Analogous experiments based on pyrene excimer formation with polystyrene samples carrying pyrene groups at both chain ends confirmed the validity of Eq. (3.27) (/3 = 0.32). For termination reactions carried out in a poor solvent for the polymer, the repulsion between polymer chains is small, as a result of favorable thermodynamic interactions, then the decrease of k, with increasing N ( N < 300) is less marked than in the case of reactions among polymers dissolved in good solvents. Entanglement effects also influence intermacromolecular reactions, resulting in a marked decrease of k, with increasing N . The magnitude of the effect is such that, for polymers of N > lo3, k, in poor solvents becomes smaller than kq in good solvents. It is noteworthy that the presence or absence of entanglements affects the solvent dependence of interpolymeric reactions in opposite directions in short oligomers and in large macromolecules. Photophysical and photochemical processes in polymer solids are extremely important in that they relate directly to the functions of photoresists and other “molecular” functional devices. These processes are influenced significantly by the molecular structure of the polymer matrix and its motion. As already discussed in Section 2.1.3, the reactivity of functional groups in polymer solids changes markedly at the glass transition temperature (T,) of the matrix. Their reactivity is also affected by the /3 transition temperature, T,, which corresponds to the relaxation of local motion modes of the main chain and by T,, the temperature corresponding to the onset of side chain rotation. These transition temperatures can be detected also by other experimental techniques, such as dynamic viscoelasticity measurements, dielectric dispersion, and NMR spectroscopy. The values obtained depend on the frequency of the measurement. Since photochemical and photophysical parameters are measures of the motion of a polymer chain, they provide means to estimate experimentally the values of T, and T,. In homogeneous solids, reactions are related to the free volume distribution. This important theoretical parameter can be discussed on the basis of photophysical processes. Recalling that excited triplet states possess much longer lifetimes than excited singlet states, it is easy to appreciate why phosphorescence is better suited than fluorescence to study molecular motions with long relaxation times, such as those taking place in solids. ~ ~ 1measured, as a function of temperature between 77 and 300 K, the Guillet et ~ 1 . have phosphorescence o f films of polystyrene, poly(methy1 methacrylate), polyethylene, poly(vinylch1oride) and poly(acrylonitri1e) containing small amounts of ketone groups or naphthyl groups. They reported their data in the form of Arrhenius plots. The plots for each polymer matrix exhibited breaks at sub-glass transition temperatures. To interpret their data Guillet et al. invoked quenching of phosphorescence by residual oxygen dissolved in the films. It is known that the diffusion coefficient of small molecules such as 0, in polymer matrices

135 Table 3. I I

Polymer Motions in the Solid Phase

0Transition temperatures of several polymer matnces and &f€usion coefficients

of reachve groups m the same matnces The values were obtained from phosphorescence decay curves of benzophenone m the polymers

T, K)

T. (C)

Tp ('c)

-100 -100

polystyrene p l y carbonate polyvmyl alcohol

-20 20,100 30

-100

100 150 85

T. K )

T. ' ( C )

Tp'(-C)

-40 40 110 poly(methy1 methacrylate) -70 20 80 poly(is0propyI methacrylate) 10 -70 poly(methy1 acrylate) and T 4 correspond to the local mohons (*) For polyacrylate mahces, T of the mam cham and of the ester side group, respechvely

4

D (ci2/s) x 10

2 x 10 3 x 10

- 1 3

-"

D (ci2/s) 3 x 10 - " 4 x 10 - I '

d i s t r i b u t i o n of f r e e volume

of f r e e v o l

Q Free volume distnbuhon in a polymer matnx, Fluctuation of the local free volume and relative rate of photoisomenzahon

t

r e l a t i v e r a t e c o n s t a n t 'fc

r"" / / 0 I

/

0

I I

ksolution

I

I

I

I I

(

!/to)

0

:

critical free volume at 4K

I t ( Vfc)

4l a r g e

0

local f r e e volume

Vf

I36


does not change at the temperature corresponding to the polymer T,. However, it is sensitive to the changes corresponding to the lower transition temperatures Tp and T,. The decay curves corresponding to the phosphorescence of benzophenone in poly(methy1 methacry late), polystyrene, polycarbonate and poly(viny1 acetate) irradiated with a nitrogen laser pulse were recorded for temperatures ranging from 80 to 453 K. In all cases the decay rates followed a single-exponential law for T > T, of the matrix polymer, but they deviated significantly from the monoexponential law for temperatures lower than the T,, but higher than either Tpor T,. This effect was attributed to a contribution of a time-dependent non-stationary state term in the diffusion-controlled rate coefficient for deactivation of the benzophenone triplet, either by quenching or by a hydrogen abstraction reaction with phenyl or ester groups of the polymer matrix. Analysis of the temperature-dependence of the phosphorescence lifetimes by means of Arrhenius plots yielded experimental transition temperature values, taken as the temperatures corresponding to discontinuities in the plots. The values are listed The diffusion constants, D, of reactive groups in polymer solids can also in Table 3.1 be obtained from photophysical measurements. It is interesting to note that at room temperature the diffusion constant of a probe in polycarbonate (PC) is higher than in PMMA and approximately equal to D in polystyrene, even though PC has the highest T, of all three polymers. This observation suggests that the probe experiences a larger free volume in PC than in PMMA, a fact which is substantiated by the macroscopic properties of PC, in particular its high impact strength, compared to PMMA or PS. There are several well known examples of photochemical reactions which exhibit a discontinuous change in rate constants and quantum yields when they are carried out in solid matrices and not in solution (see Table 2.5). The deviation from first-order kinetics for a reaction carried out in a matrix below its glass transition temperature is a characteristic feature of solid-state reactions. The kinetics of spiropyran and azobenzene photoisomerization deviate from first order when these dyes are entrapped in a solid matrix below Tg.24.34) This behavior has been attributed to the presence of a distribution of free volume within the matrix, as shown in Table 3.1 When the probe is located in sites of free volume Vr greater than the critical volume for isomerization V,,, the reaction proceeds at the same rate as in solution. For sites of Vr < V,,, the reaction is retarded, since it becomes controlled by the matrix molecular motions. At low temperature, the local molecular motions are frozen and fluctuations of local free volume become increasingly small as the temperature decreases. Consequently the fraction of sites where V f < Vfc increases.

3.2.3 Microstructural Probes Intense recent research has been devoted to explore not only the local molecular motions but also microstructure in solids by photophysical phenomena, such as non-radiative energy transfer or excited-state complex formation between chromophores. The techniques require the availability of dyes and labels known as microstructural probes.") When two polymers are mixed, the resulting blend does not possess the physical properties of the initial polymers. The new polymer composites may seem to be homogenous on a macroscopic scale, but phase separation often takes place on a microscopic scale. Since the macroscopic properties of the mixtures are dictated by their microscopic structure, it is important to be able to detect phase separation and to determine the nature of the phase


137

separated domains. Photophysical probes are helpful in monitoring phase properties in a distance scale ranging from a few A to a few tens of A. As a rule, two types of approaches are taken in the study of polymer blends. In the first approach, pioneered by Morawetz,-'') two polymers are labeled with chromophores capable of undergoing non-radiative energy transfer. One polymer is labeled with an energy donor, the other polymer is labeled with an energy acceptor. The average donor-acceptor separation in a blend of the two polymers will be much less in a one-phase system than in a system in which the polymers are segregated into separate phases. In a second spectroscopic method, excimer fluorescence is used as a molecular probe of polymer blend morphology. A critical requirement for the application of the excimer probe technique is the existence of excimer-forming sites which result when two identical aromatic rings are physically apposed in a coplanar sandwich arrangement at the equilibrium van der Waals distance. In most experiments of this type one coniponent of the blend exhibit intrinsic excimer fluorescence originating from aromatic side chains. An example of the first method is shown in Table 3 . 1 2 0 . A styreneiacrylonitrile copolymer (S-AN) was labeled with 1.45% carbazole as energy donor. It was mixed with a poly(methy1 methacrylate) (PMMA) sample labeled with 1YOanthracene as energy acceptor. was determined in 50/50 w% blends of labeled PMMA and labeled S-AN of The ratio lD/lA varying styreneiacrylonitrile content. The figure indicates that in blends of PMMA and copolymers with AN content ranging from 30 to 45 molY0, the ratio is low (extensive energy transfer takes place). This observation implies random mixing of the two polymers in these blends. This conclusion is in good agreement with results obtained from macroscopic techniques, such as T, measurements and from the optical clarity of the same polymer blends. As an example of excimer fluorescence as a molecular probe of blend miscibility we describe (see Table 3 . 1 2 0 ) experiments reported by the group of Frank.37)In this work a guest fluorescent polymer, poly(2-vinylnaphthalene) (P2VN), is introduced at a level of 0.2% in a matrix of various non-fluorescent acrylic polymers. The ratio of excimer to monomer emission for the guest P2VN is plotted vs. the host solubility parameter. The plot intensities, lu/lM, exhibits a minimum which was interpreted as corresponding to a minimum in the number of excimer-forming sites at the point where the guest solubility parameter matches that of the host. At the minimum there is maximum interpenetration of the guest polymer coils with the host polymer chains, which results in a local dilution of the aromatic rings and will reduce the number of excimer forming sites. An alternative explanation, leading to the same qualitative is that there is a decreased efficiency of energy migration in good hosts. effect on lu/lM, At temperatures slightly below T,, glassy polymers exist in a non-equilibrium state with excess entropy and free-volume. Heating a glassy polymer from a temperature slightly below the T, induces a change to a denser state closer to equilibrium. Such a change in microstructure can be observed through the use of excimer-forming spectroscopic probes. For example in the case of polystyrene, which exhibits intrinsic fluorescence, heat treatment results in an increase in excimer emission at the expense of monomer emission. For the crystalline polymer, poly(ethylene-2,6-naphthalenedicarboxylate), the ratio lD/lM also increases upon heat treatment. This trend is attributable to a large change in the degree of crystallinity of the polymer. In blends of polystyrene and poly( I -methoxy-4-vinylnaphthalene)(PMVN) two that forms excimer forms emit: the normal naphthalene excimer (D) and a second excimer (Dz) between an electronically-excited naphthalene and a ground state naphthalene. In the excimer D2 there is only partial overlap of the aromatic rings. As a result the Stokes shift of the second excimer is smaller than that of a normal excimer. The wavelength of maximum

138

Table 3.12 Microstructure Probes

@ Ratio of carbszol D to anthracene A emission intensities measured by non-radiative energy transfer experiments in blends of carbazolelabeled ply(styrene-oo-acrylonitrile) and anthracene-labeledpoly(methy1 methacry late) as a function of copolymer composition in coply-S-AN/pMMA 50/50 films. 1.2

@ Dependence of the second excimex to

monomer intensities ratio 1 D z /l Y of poly(methoxu-4-~lhthalene)eMVN) in polystyrene matrix on the reciprocal measurement temperature

.

300

1.0

,

.

100 K

.

200

I

150

annesling temperature

0:30c 0:40C

l D / lA

0.5 IDZ/IY

0.3

b

Ib

tb 3; 4b copolyner conpOSItIon ( I A N )

0 :[ 171 P M Y A = I . M0,

Q Dependence of the I o I/

:[

171

PYYA

(lO)PBzMA, (1 3)PPhMA

4

5

6

7

8

9

1

0

1mO/ Tmas 12

4

The number beside each point rcfen to the host polymer. (l)PiBoMA, (2)PiBMA, (3)pnBMA (4)PtBMA. (5)PiPMA. (6)PsBMA

(WCM4 (Il)PMMA,

3

=O.18

y

fluorescence ratio of P2VN on the solubility parameter of the host matrix

(7)pnpM4

(Ib

(9)PS (I2)PVAc

1 0 8. 0

0. 0 (ca~1/2c.-3/2)

10. 0

1 =cte 2=dtg 3=ctf

@ Emission spectra

of various

polyimides

I f

(excitation

1

WaVelength: 3SOnm)

3 0 0

4 0 0

I

5 0 0 6 0 0 wavelength (nm)

7 0 0

0


139

emission from D2 is intermediate between the monomer emission and the normal excimer emission bands. A plot of the changes in the ratio, ZD2/ZM of the intensities of the second excimer to the monomer emission as a function of the reciprocal temperature is shown in Table 3 . 1 2 0 for PWPMVN films cast at different temperature^.^^' The temperature dependence exhibited by the intensity of the second excimer fluorescence can be correlated to the polystyrene conformation at the film formation temperature. Aromatic polyimides and polyamides exhibit outstanding thermal stability and strength, two properties responsible for the industrial importance of these materials. The macroscopic properties are controlled to a large extent by the state of aggregation of the macromolecules and their relative orientation. Photophysical measurements, aimed at detecting polarization properties and the formation of excited-state charge transfer (CT) states have proven extremely useful tools to elucidate the microstructure of these materials. Information on the orientation distribution of the chains in a solid polymer sample can be gathered from polarization measurements. Upon selective excitation with linearly polarized light chromophores incorporated in the chain will emit polarized light. Analysis of the angular distribution of the polarized emitted light can, in principle, give the second and fourth moments of the orientation distribution of the probe, providing information not only on the degree of orientation but also on the orientation distribution. The experiment can be conducted with a fluorescence probe molecularly dispersed in a polymer solid. However, it is often difficult to know with certainty the relation between the orientation of the probe and that of the main polymer chains. Alternatively, the intrinsic fluorescence of the macromolecule can be monitored, as in the case of polyene-doped poly(viny1 chloride), anthryl-labeled polystyrene or polyisoprene, or high-strength fibers such as the aromatic polyamides known under the Kevlar trade mark. The fluorescence of polyimides yields information on the aggregation state of the polymer chains and on the formation of charge transfer complexes. Since the emission of the polyimide Kapton (29) is extremely weak, it had not been detected until the availability of highly

140


sensitive instrumentation. The fluorescence spectrum of Kapton has now been reported, together with the spectra of the benzophenone-(30), biphenyL(31) and (32), and bisphenol(33) polyimides. The spectra are shown in Table 3.12@,39)where the emission of (29) in the 480-600 nm wavelength domain is clearly visible after expansion of its intensity by a factor of 100. The emission spectra become red-shifted with increasing electron-withdrawing ability of the aromatic imide group and with increasing electron-donating ability of the diamine, a trend indicative of the occurrence of excited state charge transfer interactions between the aromatic moieties of the amine and the aromatic imide groups. These interactions are strongly affected by the microstructure of the polymer solid, as exemplified by the temperature studies described next. The biphenyl polyimide (32) was annealed by heating for 2 hours. On a macroscopic scale, the heat treatment results in an increase of the polymer density from 1.419 to 1.466. When excited at 350 nm, the annealed sample presented a stronger emission than before heat treatment. Moreover, upon excitation at 465 nm, a much stronger new emission appeared with a maximum at 540 nm. This result, together with the observed increase in density, suggest that the heat treatment resulted in a change in the aggregation state of the polymer chains. This change in aggregation is accompanied by a change in the steric interactions in the CT complexes. The emission from excitation at 350 nm was assigned to an exciplex-like charge transfer complex, which emits from excited states formed during the excited-state lifetime or formed by energy transfer to the exciplex sites. The emission from excitation at 465 nm was ascribed to excited complexes formed from ground-state charge transfer complexes. When films of the polyimide precursor, polyamide acid, are stretched by 50%, their morphology is also modified. Stretching the films at the polyamide acid stage prior to imidization results in an increase of the emission of the resulting polyimide when excited at 350 nm, but the emission from excitation at 465 nm is hardly affected, suggesting that these physical treatments result in different microscopic structural changes in the materials. The emission intensity of blends of polyimides gives a measure of the degree of miscibility of two polymers.40’For example in blends of the strongly fluorescent polyimide (32) and the non-fluorescent polyimide (34) added at the level of 30 mo!%, the intensity of the emission of (32) is quenched by half, indicating that the formation of intermolecular charge transfer complexes within (32) is disturbed significantly by the added polyimide. In Section 3.1.4 we described Kuhn’s early studies of energy transfer in monolayers. Their experiments are very well-known. They laid the basis for a rapidly expending area of photophysics concerned with the study of molecular assemblies such as LB-films, various monolayers, and vapor-deposited films. These photophysical techniques will become particularly important in elucidating the molecular organization in self-assembled systems.

3.3 Chemiluminescence and Electroluminescence 3.3.1 Chemiluminescence Chemiluminescence can be defined as the emission of light as a result of the generation of electronically-excited states formed by a chemical reaction rather than by the absorption of a photon.41)Chemiluminescence occurs in living species, it is then termed bioluminescence. The most famous luminescent organism is the firefly. Bioluminescence requires a lumophore,

3.3 Chemiluminescence and Electrolurninescence

141

structure (35)in the case of the North American firefly, Photinus pyralis, and enzymes which mediate the key chemiexcitation step. The enzyme which triggers the chemiluminescent reaction is called luciferase.

Modern scientific research into chemiluminescence began as early as 187 1 by an investigation of the oxidation of 2,4,5-triphenylimidazole(lophine) (36) by aqueous alkaline hydrogen peroxide. Over the years the generality of the reaction was established and many other aromatic compounds were shown to undergo chemiluminescent reactions under oxidative conditions. Examples of chemiluminescent compounds include luminol (37), lucigenin (38), and indole derivatives, such as 3-methylindole (skatole) (39). The light emitted ranges in color from yellow t o green. T h e emission spectra from well-known chemiluminescent compounds are shown in Table 3 . 1 3 0 .

In the case of indoles such as skatole or imidazoles, such as lophine, the emission of light is generated by an oxidative reaction. A sizeable fraction of the exothermicity of the reaction is converted into the electronic excitation energy of an intermediate which then emits light. It was suggested by McCapra4*)in 1966 that 1,2-dioxetanes would, on theoretical grounds, give rise to efficient population of excited states and thus exhibit chemiluminescent properties. Three years later was reported the first synthesis of a simple dioxetane, trimethyl-l,2dioxetane. Decomposition of this dioxane generates acetone in an electronic excited state and acetaldehyde, confirming McCapra’s predictions. This report stimulated intense research which eventually led to the synthesis of several remarkably stable dioxetanes shown to undergo chemiluminescence. It is interesting to note that from mechanistic studies of the chemiluminescence of synthetic dioxetanes came the suggestion that dioxetanes may in fact be involved as intermediates in other chemiluminescent reactions. For example, the mechanism of chemiluminescence of skatole is believed to involve the dioxetane intermediate (40) (see Eq. (3.28)). The chemiluminescent spectrum of skatole (39) was shown to be identical to the fluorescence spectrum of the dicarbonyl derivative (41), thus confirming the intermediacy of the dioxetane (40) which decomposes to generate (41 *).

142

Table 3.13 Chemiluminescence

@ Emission spectra induced by oxidative reaction of typical chemiluminescent compounds

(1) lophme KOH-C H 6 OH-0 (2) skatole KOH-DMSO-H z 0-0 z (3) lucigenin Na z CO s -H z 0-H a 0 a (4) lucigenin KOH-C H I OH-pyridine-H a 0 z (lucigenin.2.8 x 10 M)

I CL

(5) luminol

wavelength (nm)

(a) concerted reaction

@ Mechanism and energy diagram of the thermal decomposition of dioxetanes

(b) reaction

0-0 U

involving a biradical

-p' '7 -

0

0"

II

II

(c) energy levels

CH3 -C

I I

I

- C - CH3

3CH3

I

C'

CH3CH3

hv

HR

CH3*

II

0 CH3

V

v v

'/

CH3

0 structure of common chemiluminescent organic compounds

HO luciferin(Ph0totimus Pyralis)

A skatole

triphenylimidazole (lophine)

luminol

m Y+ CH3

No;

lucigenin

3.3 Chemiluminescence and Electroluminescence

(39)

143

(40)

(3.28)

The thermal decomposition of a dioxetane leading to a dicarbonyl group in its excited state occurs by a concerted mechanism following Woodward-Hofmann rules (see Table 3.120). As the dioxetane is a four-atom ring, its thermal cleavage should bring one carbonyl group in its excited state in order to maintain orbital symmetry during the reaction. The enthalpy change of the thermal decomposition of dioxetane, AHR,is very large. In the case of the tetramethyl-l,2-dioxetaneshown in Table 3 . 1 2 0 it takes a value of 272 kJ mol-I. The activation energy, E,, of the reaction was estimated to take a value of 105 kJ mol-I. The sum AHR+ E, (377 kJ mol-I) is greater than the energy required to populate either the excited singlet state of acetone (E, = 35 1 kJ mol-I) or its triplet state (ET= 326 kJ mol-I). Thus, from the thermodynamic standpoint, the generation of either excited state is possible by thermal decomposition of the dioxetane. In the case of strongly chemiluminescent compounds, the fluorescent group contains either strongly electron-donating groups or carries nitrogen atoms. In this situation the thermal decomposition occurs via a nllc state and not a nllc state, involving a singlet excited state. On the other hand in the case of simple dioxetanes, the thermal decomposition is believed to involve a two-step mechanism via the intermediate formation of a diradical. The chemiluminescent compound luminol is also well-known for its use in criminal investigations, in particular to trace illicit drugs. In alkaline solution luminol undergoes the chemical reaction shown in Eq. (3.29). The reaction occurs in high yield. It generates 3amino-l,2-phthalic acid as the dianion (42) and a nitrogen molecule, together with a strong green emission from the reactive intermediate shown in Eq. (3.29). The emission spectrum is characteristic of the fluorescence of the dianion (42). In recent years the chemiluminescence of isoluminol obtained by deprotonation of luminol has been used in the quantitative analysis of peracids formed from lipids in living organisms. These intermediates are believed to be involved in the phenomenon of aging. Thus chemiluminescence is an important tool in unraveling the complex mechanisms involved in the aging process.

144


(37)

(3.29)

Bioluminescence is a process of extremely high quantum yield. Typical values range from 0.88 in the case of the firefly to 0.28 for the marine firefly. A notable feature of the firefly bioluminescence process is the requirement for ATP, the result being the formation of luciferyl adenylate (35a). This reaction predisposes the luciferin to attack by base and oxygen, and provides the leaving group which results in the formation of the dioxetanone (43) in the presence of an enzyme, luciferase. The thermolysis of the dioxetanone is believed to occur via intramolecular electron transfer according to a mechanism known as CIEEL (chemically initiated electron exchange l u m i n e ~ c e n c e ) . The ~ ~ ) excited singlet state (44) of the decomposition product emits light with high quantum yield. If an electron donating aromatic hydrocarbon (ArH, Eq. (3.31)) is present during the thermal decomposition of the dioxetane, the excited state formation is generated by a highly efficient electron transfer. In the case of the firefly luciferin, the electron transfer is believed to take place intramolecularly. For intermolecular CIEEL processes, sensitized chemiluminescence is achieved by adding a strongly fluorescent aromatic hydrocarbon. H

H

[oa:$(y"]* +COr

(44) +ArH

- go

(44)+hv+COa

+ A r H ' d

(3.31) 0

+COr+ArH'

Recently a topic related to bioluminescence has sparked a lot of interest. Researchers attempted to introduce the firefly luciferase gene into killifish eggs. Just before the birth of

3.3 Chemiluminescence and Electroluminescence

145

the killifish, there was a report about the success of the gene transfer. The killifish egg was fertilized by injecting the firefly luciferase gene into its nucleus, which is about 100 microns in diameter. The condition of the killifish was watched very closely a few days before and after hatching. A weak luminescence was observed around the eyes of the fish and in its muscular system. If the luciferase gene had been inserted in the killifish chromosomes with other genes, different parts of the fish should have become bioluminescent. We hope to see soon a luminescent killifish swimming gracefully and emitting flashes of light from various parts of its colorful body.

3.3.2 Electroluminescence Electroluminescence occurs when a system is excited by an electric field to a higher energy level and then spontaneously decays to a lower energy level by emitting a photon. An important example is injection electroluminescence which occurs when electric current is injected into a semiconductor junction diode. As injected electrons drop from the conduction band to the valence band, they emit photons. Light-emitting diodes (LED’S) are well-known devices which function on this principle. Recently there has been intense research in electroluminescence devices for various types of information processing technologies. A special interest centers on the replacement of cathode-ray tubes (CRT’s) by flat displays. The electroluminescence phenomenon provides a route to such devices which use solid organic dyes to produce luminescent devices of exceptionally high resolution and brightness. The electroluminescent devices commonly available today consist of inorganic materials, such as ZnS thin films doped with manganese or other rare earth ions as luminescent centers. The objective is to obtain a full color luminescent system based on the three fundamental colors o f light (red, green, and blue). In recent years the use o f organic molecules a s electroluminescent centers has been investigated in great detail. We introduce here the principles which underlie the electroluminescence of organic molecules. The present status will be presented as well. Upon application of an electric field on strongly fluorescent organic solids, holes are injected from the cathode and electrons from the anode. Holes and electrons migrate through the solid. Upon encounter, they recombine to generate singlet excitons of the fluorescent molecule. If the electrodes are transparent, the exciton fluorescence can be observed. The electroluminescence of organic solids was discovered in the early 1960’s in single crystals of dyes such as anthracene. In later years research has focused on the preparation of electroluminescent films which are easily fabricated and processed. The successful preparation of flat electroluminescent devices of very high luminous intensity (> 1000 cd m-2) and low operational voltage (40 V) was announced in the late 1980’s. Since then the interest in electroluminescent organic devices has grown markedly. From a practical aspect the most important requirement placed on electroluminescent devices is that the electron/hole recombination must occur with high efficiency in order to produce a strong luminescent signal. Diagrams o f several multilayer organic electroluminescent devices are shown in Table 3 . 1 4 0 . The first structure (3.140(a)) proposed by an Eastman-Kodak is a double layer composed of a hole transporting layer and a layer responsible for both electron transport and luminescence. The second structure proposed by a Kyushu University is shown in 3.14@(b). It is also a double layer, but in this

I46

Table 3.14 Electrolurninescence

0 Diagrams of mdtihyed eIectroluminescent devices (b)

M g A g

M g A g electron transport layer

luminescent layer hole transport layer

luminescent layer

I TO

ITO

M

M g A g electron transport layer luminescent layer hole transport layer

g :

C u (120nm)

1un inescent

I TO

(a), @),

(d): two layers

(c): three

PSD NSD

300

400

layers

PD

500 600 700 wavelength (nm)

800

147 Table 3. I 5 Chemical Structures of Compounds Used in Carrier Transport Layers and Luminescent Layers of Organic Electroluminescent Devices (1) Hole transport layer triphenylamine derivatives (TAD)

luminescent triphenylamine derivatives ' (NSD) (52Onm)

luminescent triphenylamine derivatives ' (PSD) (46Onm)

(2) Electron transport layer oxadmole denvatives (PBD)

t-Bu&&jpQ)

c H,

w

aminoquinolinol derivatives ' (54Onm)

pennone denvatives * (PD) (620nm)

t-Bu

0 distyrylbenzene derivatives '

(480nm)

(3) Luminescent layer

DCMl '

(606nm)

(The superscript * indicates the use of

the dyes in luminescent layers. The number in brackets corresponds to the wavelength of maximum luminescence.)

coumarin540 *

C=CH-CH=C

(52Onm)

tetraphenylbutadiene ' (43om

148


case it is the hole-transport layer that functions also as luminescent layer, the second layer being solely an electron transporting layer. A triple layer structure (3.140(c)) consisting of an electron-transport layer, a luminescent layer, and a hole transport layer was proposed later by the Kyushu University group. In both cases electrons are formed in the luminescent layer and combined, in a highly efficient process, with holes near the organic interface. A transparent I T 0 (Indium Tin Oxide) layer was chosen as the anode which injects holes into the hole transport layer. A vacuum deposited layer of magnesium, gold or aluminum was used as the cathode for electron injection. Other requirements are placed on electroluminescent devices in addition to supporting efficient hole and electron transport: ( I ) the exciton energy in the transport layer has to be higher than the exciton energy in the luminescent layer; (2) the formation of molecular complexes between the fluorescent layers should be precluded; (3) the luminescent layer has to possess a high fluorescence quantum yield; and (4) the film should be processed readily. In Table 3.15 are shown the structures of the main compounds used in the carrier and electroluminescent layers. Various types of triphenylamines are common hole transport materials. Oxadiazoles are used for electron-transport. For simultaneous electron transport and luminescence, the Eastman-Kodak group employed large aluminum quinolinol complexes; while the Kyushu University group selected perinones and NSD, a luminescent triphenylamine derivative. In the former case, Coumarin-540 or DCMl was added to the aluminum quinolinol complex in order to increase the electroluminescence intensity and to produce desired shifts in the fluorescence wavelength of the dye molecules. In the latter case the combination of NSD for simultaneous hole transport and luminescence and an oxadiazole derivative (PBD) in the electron transport layer resulted in luminescent emission of much higher intensity (by a factor as large as lo4)than that of devices consisting only of NSD. Note that it is difficult to obtain inorganic electron luminescent molecules that are highly efficient in the blue. In contrast phenyl substituted dienes are outstanding materials for blue luminescence. They are used in the triple layer structure depicted in Table 3.140(c). This device exhibits a luminous energy of approximately 700 cd m-2 with an operating voltage of 10 V and a current density of 100 mA cm-2. Electroluminescence spectra for the three fundamental colors are shown in Table 3 . 1 4 0 , for a device consisting of PSD (blue emission), NSD (green emission) and PD (red emission). The luminescence emission maxima of the most important dyes are included in Table 3.15. There is still a major drawback associated with the use o f organic thin films in electroluminescent devices. They are sensitive to ambient oxygen and water, hence the actual operating voltage required to generate a given electric current is significantly higher than the operating voltage o f inorganic devices. A systematic experimental study o f the electroluminescence of organic dyes is required, with special attention to the detailed mechanism of the luminescence and degradation processes.

References I . Th. Forster, Ann. Phys., 2, 55 (1948). 2. D. L. Dexter, J. Chem. Phys., 21, 836 (1953). 3. I . Mita and K. Horie, J. Macromol. Sci. -Rev.Macromol. Chem. Phys., C27( I ) , 91 (1987). 4. H. Leonhardt and A. Weller, Ber. Bunsenges. Phys. Chem., 67,791 (1963). 5. R . A. Marcus, J. Chem. Phys., 24, 966 (1956); Rev. Phys. Chem., 15, 155 (1964). 6. D. Rehm and A. Weller, Isr. J. Chem., 8 , 259 (1970).

References 7. 8. 9. 10. 1 1.

12. 13.

14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

I49

N. Mataga, T. Asahi, Y. Kanda, T. Okada and T. Kakitani, Chem. Phys., 127, 249 (1988). J. R. Miller, L. T. Calcaterra and G. L. Closs, J. Am. Chem. Soc., 106, 3047 (1984). T. Kakitani and N. Mataga. J. Phys. Chem., 90, 993 (1986); 91, 6277 (1987). M. Tachiya and S. Murata, J. Phys. Chem., 96, 8441 (1992). C. F. Carlson, US Patent, 2. 22 I, 176 ( 1938). L. Onsager, fhys. Rev., 54.554 (1938); P. M. Borsenberger, L. E. Contois and D. C. Hoesterey, J. Chem. Phys., 68, 637 (1978). H. Kokado, in Organic Electron Transfer Process, Chem. Rev. Ser. Vol. 2, (Chem. SOC.Jpn. ed.), p. 191, 194, Gakkaishuppan Center, Tokyo (1988). L. Stryer, Biochemistry. 3rd Ed., W. H. Freeman and Co. (1988). J . J. Grimaldi, S. Boileau and J . M. Lehn, Nuture, 265, 230 (1977). H. Kuhn and D. Mobius. Angew. Chem. Internut. Ed. Engl., 10, 620 (1971). M. Fujihira, Oyo-Butsuri(Appl. Ph.vs.). 57, 1892 (1988). Pho/ophysicul and Photochemical Tools in Polymer Sciences, (M. A. Winnik ed.), NATO AS1 Ser. Vol. 182, Reidel (1986). H. Itagaki, K. Horie and 1. Mita, Progr. Polym. Sci., 15, 361 (1990). H. Ushiki and K. Horie, in Handbook o/Polsmer Science und Technology, (N. P. Cheremisinoff ed.), Vol. 4, Chapt. I , Marcel Dekker, N.Y. (1989). G. Weber and D. J . R. Laurence, Biochem. J., 56, 31 (1954). E. M. Kosower, H. Dodiuk, K. Tanizawa, M. Offolenghi and N. Orbach, J. Am. Chem. Soc., 97, 2167 (1975). M. A. Winnik, Chem. Rev., 81. 491 (1981); Arc. Chem. Res., 18, 73 (1985). K. Horie and I . Mita. Adv. Polym. Sci., 88, 77 (1989). G. Webber, Adv. Protein Chem., 8, 415 (1953). K. Horie, 1. Mita. J. Kawabata, S. Nakahama, A. Hirao and N. Yamazaki, Pol-vm. J . , 12, 319 (1980). K. Shimada, Y. Shimozato and M. Szwarc, J. Am. Chem. Soc., 97, 5834 (1975). T . Kanaya, K. Goshiki, M. Yamamoto and Y. Nishijima, J. Am. Chem. Soc.. 104, 3580 (1982). K. Horie, W. Schnabel, I . Mita and H. Ushiki, Macromol., 14, 1422 (1981). H. Itagaki, K. Horie, I . Mita, M. Washio, S. Tagawa and Y. Tabata, J. Chem. Phys., 79, 3996 (1983). K. Horie and I. Mita, Mucromol., 11, I175 (1978). A. C. Somersall, E. Dan and J. E. Guillet, Mucromol., 7, 233 (1974). K. Horie, K. Morishita and I . Mita, Macromol., 17, 1746 (1984). G. Smets, Ad\'. Polym. Sci., 50, 17 (1983). T. Naito, K. Horie and I. Mita, Mucromol., 24, 2910 (1991). H. Morawetz, Sciences, 203, 405 (1979); Pol.vm. Eng. Sci., 23, 689 (1983). M. A . Gashgari and C. W. Frank, Mucromol., 15, 1558 (1981). H. Itagaki, K. Horie and 1. Mita, Eur. Pol-vm. J . , 19, 1201 (1983). M. Hasegawa, I. Mita, M. Kochi and R. Yokota, J. Pol.vm. Sci., Pol~vm.Lett., 27, 263 (1989); Eur. Pol,vm. J . , 25, 349 ( 1989). M. Hasegawa, M. Kochi, I. Mita and R. Yokota, Polymer, 32, 3225 (1991). K. Maeda, Kagaku-no-Ryoiki(FieldofChemistry),29, 640 (1975); N. J . Turro, P. Lechtken, N. E. Schore, G. Schuster, H. C. Steinmeter and A. Yekta, Arc. Chem. Res., 7, 97 (1974). F. McCapra, Endeavour, 32, 139 (1973). J. K. Koo and G. 9. Schuster, J. Am. Chem. Soc., 99, 6107 (1977). C. W. Tang and S. A. VanSlyke, Appl. Phys. Leu., 51.913 (1987). C. Adachi. T . Tsutsui and S. Saito, Appl. Phys. Letr., 5 5 , 1489 (1989); T. Tsutsui and S. Saito, in Inrrinsically Conducting Polymers: An Emerging Technology, (M. Aldissi ed.), p. 123-134, Kluwer, Amsterdam (1993).


.



In the Introduction, we presented the fundamental hypothesis of this book on “The Concept of Molecular Photonics.” We proposed that, in order to understand truly the “nature of light,” it is crucial to study in detail how light interacts with materials. In Chapter 1 “Fundamentals of Molecular Photonics,” we emphasized that materials are best described by the molecular field model. Then, in Chapters 2 and 3, we described in concrete terms the mechanisms and products of photochemical reactions, energy transfer processes, and electron transfer processes. These were treated as “Photochemical Reactions” and “Photophysical Processes,” respectively. We gave examples of practical applications of the phenomena, but we were careful to restrict these two chapters to aspects relevant to the interaction of light and materials. In other words, we examined the changes which occur in materials as they enter in contact with light. The time has come now to remember that photo- and opto-phenomena consist not only “of light-induced changes in materials” but also of “materials-induced changes in the light.” Any action carries with it, a specific “re-action’’ (recall Table 0.3). The “light-induced changes in materials” were classified into photophysical and photochemical reaction processes. The “materials-induced changes in the light” will be classified as of inner perturbation type or outer perturbation type, depending on whether the changes in light propagation are induced by the action of light or by the perturbation of materials not due to light. Therefore the inner perturbation type includes various phenomena of nonlinear optics, since, as you may recall, the electric field associated with light has many high-order terms. The main processes of the outer perturbation type can be ascribed to fluctuations of the medium refractive index (light scattering phenomena, acousto-optic effects, etc.) or to the interaction between electromagnetic fields (electro-optic effects, magnetooptic effects, etc.).This chapter is concerned with optical effects of the outer perturbation type.

4.1 Light Scattering Light scattering phenomena were mentioned already in Chapter 1 (Section 1.4.5). They were described as representative of the interaction of light with material from the viewpoint of the particle theory of light. They were defined as processes where a molecular system, simultaneously absorbs and emits a photon from the viewpoint of the wave theory of light, Rayleigh and Brillouin scatterings are explained in terms of “light scattered by fluctuations of material density.” In this section, we will explain Rayleigh, Raman, and Brillouin scatterings as representative examples of the changes in light propagation as a light beam 151

152 Table 4.1 Summary of Light Scattering Phenomena

Iliscovery

Raman scattemg Raman( 1928)

Rayleigh scattering Raylei& 187 1)

Brillouin scattemg theory: Brillouin(l922)

exDeriment: Gross(1930)

Fundamental interaction

> origin

Density fluctuation in a substance Brownian motion in solution Zimm dot(0olvmers)

+

Scatternu intensity I f, = ( 2 3 ~ / 3 c 3 loo' )

KC/I (q)=(l/M)

I.=!i(n2V/r2Ai ' )

x,I ( A a & ) 1'

I +<s2%2 /3]+2A2 C q= (4rll) s i n (a/t)

[LTP. ( P 1 E /a

t I

b

I o :intensity of incident light o :2 k ( Y 0 - Y I I )

Cconcentration by weight periodic change in the onemtation vector stripe dielectric contants: t A > e 11 electric conductivity: u 11 > u 1 Dynamic scattering (DSh4) effect due to turbulent flow occurring when the ionic power caused by the electric field exceeds the elasticity limit of the liquid crystals ( e 1- E I I ) ( O I / U , l ) ) ( n t I l / V ) ( E I / € ,1- ( I l / U l f ) K, :shear torque coefficient, v :viscosity, flat display: RCA Co. (1%8) Guest-host effect the reorientation of LC molecules induces an orientation change in the dye molecules dielectric constant: & 1 # e 1 1 , optical valve: Brit(RCA Co.)(1971) Accumulation effect generarion of scattering center nematic-cholesteric mixed LC system Optical axis deformation shape change of the optical axis in the LC layer Banana-type molecule induces piezoelectric effect Nematic-isotropic phase transition change in nematic-isotropic phase-hamition temperature dielectric constant: e 1 > e 11 Optical axis rotation <Schiekel:1971> change in the molecular axis by anisotropy of the dielectric constant dielectric constant: E A # t 11 , flat display Screw structure realignmemt <Schadt:1971> aligns LC molecules perpendicular to the elecrode surface &electric constant: E I > E 1 1 , flat display:Schadt(l971) Helix pitch change change in optical properties due to a change in herix pitch increase in pitch: The electric field is parallel to the helix axis. decrease in pitch The electric field is pe~pendicularto the helix axis. color display panel:Fergason(WH Co.)(1%4) Helix axis rotation Grandjean texture (disordered) focal conic texture (ordered) dielectric constant: E L # E 11 electronic photography:Haas(Xex Co.)(1%8) flat display:Wysocki(RCACo.)(1971) Cholesteric-nematic phase. transition <Meyer: 1969> change from negative uniaxiality to positive uniaxiality The electric field is parallel to the helix axis. +

Optically positive uniaxiality

(n.> n o )

Low viscosity

.

n :refractive index along the exbaordinary axis (molecular or stretching axis)

n

:refractive index along the ordinmy axis @erpendicular to n direction)

Cholestenc liquid crystal(Ch)

Optically negative uniaxiality

(n.< n.)

+

Helix structure

4.2 Optical Effects

169

has a bad defect which is hard to change. An “ambiguous” person indeed is very much like an “intricately controlled apparatus.”

4.2.3 Magneto-Optic Effects The propagation of light through materials can be influenced by the application of a magnetic field. The effect was discovered by Michael Faraday in 1845 who observed that the plane of vibration of linearly polarized light incident on a piece of glass rotates when a strong magnetic field is applied in the propagation direction. Since then, other magnetooptic effects have been discovered besides the Faraday effect, such as the magneto-optic Kerr effect, the Cotton-Mouton effect, the Voigt effect, and the magnetic double refraction. Hence there is also the Zeeman effect on the microscopic scale (atomic and molecular fields). A summary of these effects is shown in Table 4.9. Turning our attention first to the Faraday effect, the angle 6 (in minutes of arc) through which the plane of vibration rotates is given by the empiric expression: B= VLH

(4.34)

where V is a factor of proportionality, known as Verdet’s constant, L is the optical propagation distance traversed in the lines of magnetic force, and H is the intensity of the magnetic field. The theoretical treatment of the Faraday effect involves the quantum theory of dispersion. The separation of electronic energy levels depending on the orbital and spin angular moments of atoms and molecules may be formed similarly to the case of the Zeeman effect. If the linearly polarized light propagates in a direction parallel to the lines of magnetic force, the absorbed light will separate into the two components of circularly polarized light: one line absorbs righthanded circularly polarized light and the other line absorbs left-handed circularly polarized light. In this case, the dependence of the refractive index for each absorption line ( n , and n,) on the light angular frequency w is the same, but a rotation angle B of the linear polarization plane is induced only by the difference between the angular frequencies of the Zeeman components, that is, n, - n, # 0 and the relationship between the rotation angle of the linear polarization plane and the refractive index is given by the following equation:

(4.35) Hence the dielectric polarization, P,, induced by a magnetic flux density B,(O) and a photoelectric field, E,(w), is given by

(4.36) where 6,“is a cyclic tensor. On the other hand, a phase gap, 6, between the linear polarizations parallel (I,,) and perpendicular (IL) to the magnetic field may be induced when a transparent medium, placed in a magnetic field which can cause a Zeeman effect, is irradiated with light propagating perpendicularly to the lines of magnetic force. This phenomenon is known as the CottonMouton effect (Voigt effect). The phase gap is given by Eq. (4.37),

I70 Effect

Faraday effect

( I 845)

Magnetic Ken effect (1888)

ZIXmFUl

Table 4.9 Magneto-Optic Effects Phenomenon and Illustration Rotation of the polarized plane when linearly polarized light propagates in a transparent medium in a direction parallel to the magnetic field rotation angle: O=(oL / Z c ) ( n , - n , ) -*H magnetic w :angular frequency of light field e Iect r i c L :medium thickness n I , n , :refractive index for left or right-handed circularly polarized light, rotation angle for weak magnetic field: O=VLH rl w ):Verdet constant. magnetic rotation dispersion L Application: optical isolator, magneto-optical switch Rotation of the polarized plane of reflective light when linearly polarized light is reflected by a magnetic material or a material in magnetic field

h electric field

longltudinal transverse effect Kerr effect Kerr effect

Application: observation of magnetic domain, read-out of magneto-optical memory Splitting of the energy level of a nucleus, an electron, an atom, or a molecule in a magnetic field by the interaction of the degenerated magnetic momentum with the external magnetic field

effect centripetal force of the circular orbital:

Fo=mw'r (1896)

CononMouton effect

(1907) Wigt effect) Magnetic birefringence

H=0

force induced by the motion of the electron across the magnetic field: FH=He ( w + A w ) splitting by the magnetic field. A w = e H / 2 rn

Generation of a phase difference due to the difference in refractive indices of the linearly polarized light parallel and perpendicular to the magnetic field when light propasates in a bansparent medium H magnetlc field in the direction perpendicular to the magnetic field phase difference: 6 = ( Z n L / e )

( n l - n t ~ )

A

6 = I C L H 2

C : Cotton-Mouton constant App1ication:polarizer

6 :phase dlfference

L

electron a-A0

4.2 Optical Effects

6 = ACLW

17 I

(4.37)

where 13. is the wavelength of the light, L is the optical path distance, and C is the CottonMouton constant. This phenomenon can be ascribed to a difference between the two refractive indices (n,,and n l ) based on the linear polarization planes parallel and perpendicular to the magnetic field. Here the refractive index n,, of the linear polarization plane parallel to the lines of magnetic force is independent of the intensity of the magnetic field, but the refractive index nL = ( n ,+ n,)/2 of the linear polarization depends on it. In this case, the phase gap is given by the following expression:

6

=

~

2nL (nl - n,, ). e

(4.38)

Hence the dielectric polarization, P,, induced by the magnetic flux density, E,(O), and the photoelectric field E,( w ) is given by (4.39) When a magnetic compound is irradiated with linearly polarized light, the polarization of the reflected light changes. This phenomenon is known as the magnetic Kerr effect. According to Table 4.9, the magnetic Kerr effect can be divided into polar, vertical, and horizontal Kerr effects, according to various combinations between the direction of the lines of the magnetic force and the directions of the incident and reflected light waves. The reflected light based on the polar and vertical Kerr effects changes into an elliptical polarization, and the axis of the ellipse rotates. On the other hand, in the horizontal Kerr effect, it is the reflection factor of the linear polarization components, I,,, which differs from that of the other components II. There are a number of practical applications of the magneto-optic effects. For example, optical modulators and magneto-optic switches in fiber optics communication systems are well-known devices based on the Faraday effect. In an optical fiber, when some light is reflected, it mixes with the normal light signals. Consequently normal fiber optics communication fails. Therefore the reflected returned light has to be shielded. This can be done by optical insulators which consist of two mutually perpendicular polarizers and a crystal of a Faraday rotor garnet. The apparatus where a Faraday rotor is interposed between a polarizer and a polarization insulator chip constitutes a magneto-optic switch. The magnetic Kerr effect is used in read-out processes in magneto-optic disks. You will be able to understand its principles with the help of Table 4.9.

4.2.4 Acousto-Optic Effects Electro-optic and magneto-optic phenomena contain terms of nonlinear optics effects (see Eqs. (4.32), (4.33), (4.36), and (4.39)). On the other hand, acousto-optic effects which arise from a periodical density fluctuation of the medium, analogous to the Brillouin scattering phenomenon, do not contain terms of nonlinear optics, as a general rule.5)The perturbation of light propagation by sonic waves differs from that induced by electric and magnetic fields. As the electric susceptibility, xe,is a function of the density of the medium, it will be influenced by the periodical density fluctuation induced in a medium by sound waves.

172


Consequently, we can understand that sound waves change the refractive index of a medium according to Eq. (4.31). In a piezoelectric medium sound waves can induce a new electric field. This phenomenon is very complex because the induced electric field relates to a change in the refractive index of the medium. In general, the acousto-optic effects are classified into two phenomena depending on the frequency of the sound waves, the Raman-Nath diffraction and the Bragg diffraction (see Table 4.10). When a medium placed in low frequency sound waves is irradiated with a narrow light beam perpendicular to the direction of propagation of the sound waves, its flux suffers bending scan. This phenomenon can be ascribed to the occurrence of a gradient of refractive index induced in the medium by the stationary sound wave. On the other hand, when a medium placed in high frequency sound waves is irradiated with a wide light flux perpendicular to the propagation direction of the sound wave, we can observe a diffraction phenomenon. This phenomenon is known as the Raman-Nath diffraction. It can be used to create phase lattices. In other words, the phase velocity of light decreases in the parts of the medium of high density and the light perpendicular to the direction of sound wave propagation (parallel to the sound wave plane) forms a periodical wave based on the density distribution induced by the sound wave in the medium. Hence, the light is diffracted in various directions after it passes through this modulation region. The degree of diffraction decreases with increasing sound wave frequency. If a medium placed in a sound wave of very high frequency is irradiated with light of wavelength A with an incident angle 0 to the direction of the sound wave propagation, we can observe Bragg diffraction. This phenomenon is ascribed to the existence of interference light waves in a particular direction after the incident light waves are diffracted successively by many sound wave planes (frequency: f).The condition for Bragg diffraction is given by

2d sine = m g A

(4.40)

where d is the wavelength of the sound wave and m g is an integer. In this case, the diffracted light of m g is the Doppler light shifted by me$ Sono-optic modulators which can polarize and modulate laser light are practical applications of these phenomenon. Many devices function according to the optical effects described in this chapter. For example, light modulators, polarizers, optical shutters are based on electro-optic effects, light modulators, optical insulators, optical switches, polarizers, etc., use magneto-optic effects. Light modulators and optical switches also use sono-optic effects. The principles behind processes used in optical insulator, magneto-optic switches, and reading-out in magneto-optic disks are depicted in Table 4.1 1. Optical insulators are used to block the reflected returning light in fiber optics communication systems. The framework of this apparatus consists of a Faraday rotor and polarizers. It takes advantage of the rotation of the plane of linear polarization when a polarized light beams travels through a Faraday rotor. The same principle is applied in magneto-optic switches. If the direction of the magnetic field applied in a Faraday rotor is changed, the direction of the light path can be momentarily changed through a polarization insulator chip. Thus this apparatus has the ability to act as a rapid response switch. The read-out process in magneto-optic disks is based on the magnetic Kerr effect. When the linearly polarized light is reflected by magnetic material, the polarization plane rotates. This accounts for the on-off (0 or 1) information storage in magnetic disks, as the polarization plane of the reflected light changes, depending on the direction of

173 Table 4.10 Acousto-Optic Effects Effects

Phenomenon and Illusutation

of the acoustic wave m a m d u m Bending scan

of light

condtion L 2

Relationship between polarizability and phase velocity and intensity of light Phase velocity of light wave in nonmagnetic body

~ Z E alp

~

E

a t-

at

4

o c v =~- = - = c R e ~ k n

~

Lamb-Beer rule

Intensity of light wave in non-magnehc body

I I = k * ( S)=-& o cn IE i r j 1'

c =(E

fi 0 )

I

2

'*

light velocity in vacuum, n extmchon coefficient, < S > h e average of Poyntmg vector

Absorphon of light by molecule F e n ' s golden rule

W.b : transition rate, P I Einstein's A coefficient

I

: transition dipole moment, p (

n2+2

3

P f 1 + 3e z , r ,

,

Afl=

E

3Z E

): energy density of light per unit volume

n a 6 C 3

I Pfl I lWfI3

n : refractive index Frequency dependence of refractive index Kramers-Kroningequation

Scattering phenomena and dielectnc polanzation

dielectnc polamation ( mduced by light wth amplihde,

E , and frequency, o )

P = A ( t )E ( t ) dA = A , E ( r ) exp(-iut)+-E dQ dA A ( t ) =Aa+-Q

( r Qo [e~pi-i(utw~)t~+exp~-I(o-oa)t)]

aQ

total scattered light mtensity per unit volume o Iaut=NUqIIn total cross sechon of Rnman scattenng

Dipole moment

P

( t)

Q generallzedcoordmate

scattenng cross sechon, m mcident. out scattered

~

~

5.1 Saturation of Absorption and Multi-Photon Absorption Processes

I85

L

(5.9)

x,,"],

In this equation, A is the polarizability, B is the hyperpolarizability, and x,lr(2), x,,J3)are the 1 st, 2nd, and 3rd order susceptibilities, respectively. These are tensors, where i, j , k , I correspond to the space coordinates (x, y , z ) and crystal axes. The refractivity is related to XI/"). The occurrence of birefringence in anisotropic media is a direct consequence of the fact that x,,('Jis a tensor. Susceptibilities of order higher than one are called nonlinear susceptibilities. The nonlinear refractive index, n2, and the nonlinear absorption coefficient, a2,both depend on the intensity of light, I. They are defined by Eqs. (5.10) and (5.1 l),

n = no + n21

(5.10)

a = a , + a21

(5.1 1 )

where the subscript 0 refers to the linear case. The 3rd order nonlinear susceptibility, k3), is related to the nonlinear refractive index, n2, by n2 =-.

Re[X" ] cn

(5.12)

E"

In general, an odd-number-order nonlinear polarization cannot be observed for systems which Table 5 . 5 Nonlinear Optical Phenomena Susceptibility

x

x x

(2)

( 2 w ,w

(21 ( W .

Phenomenon

. W )

w , 0 )

(2'

( W 3 . O I , W 2 )

x

(2)

( w 2 , w 3 , w

x

(3)

( 3w .w,w ,

1

(3)

(3)

(

,

-

w 4 ,

0 ,w

w

I ,

w

,w ) 2 ,

w+w

Pockels effect

w + o

optlcal rmxlng

w,fw2

Thrd harmomc generation (THG)

W )

( w , - w ,w , w )

(31 ( W

Second harmonic generation (SHG)

Parametnc amplification

)

x ' 3 ' ( o , o , o , w )

x x x

Change in frequency

w

3 )

W I

w

+w +w

+

+

--t

-

2 w w w3 w 2 + 0 3

3 w

Kerr effect

w+o+o

Optical bistability

w+w-w

+

w

Phase conjugation

w + w - w

+

w

W l t W 2 t 0 . 3

+

Four wave mixing (FWM)

-*

w

w,

186

5 The Interaction of Light with Materials I1

have an inversion symmetry (see Section 1.4.1 ). Various classes of nonlinear optical phenomena are listed in Table 5.5: linear susceptibility controlled phenomena, including refraction and absorption; second harmonic generation (SHG), optical mixing, and parametric amplification are related to the 2nd order nonlinear susceptibility; the 3rd order nonlinear susceptibility controls phenomena, such as third harmonic generation (THG), optical bistability, phase conjugation, and four-wave mixing.

5.1.4 Frequency Conversion of Light Coherence and monochromaticity are two important characteristics of the laser light. Laser light radiation of nonlinear optical materials triggers the emission of light, either of . ~ ) us consider the mechanism of light different wavelength or of different c o h e r e n ~ e . ~Let emission in the case of nonlinear polarization. The electric field of the light, E , emitted in a situation of nonlinear polarization is expressed by the following nonlinear Maxwell equation: (5.13) where b is the magnetic permeability in the vacuum and E = ~ ~ +( 21 ' ) )is the dielectric constant of the material. PNLdenotes the nonlinear dielectric polarization with forced oscillation given by pNL = pNLe"Wi'-kNiX).

(5.14)

Assuming that the amplitude of the light electric field changes during propagation in the x direction, then the electric field of the light, E , emitted by the forced oscillation of the nonlinear dielectric polarization is given by

E = E(X)e'{~4dt'-~(%tIrl,

(5.15)

Introducing Eqs. (5.14) and (5.15) into Eq. (5.13) and using Eq. (5.16) lead to Eq. (5.17),

(5.17) Here, assuming that the amplitude remains constant during a propagation distance equal to the wavelength (see Eq. (5.18)), we get Eq. (5.19). (5.18)

(5.19) When PNL is independent ofx, Eq. (5.19) is easily converted to

187 Table 5.6 Frequency Transformation o f Light

s-ry

The madiation of light wth frequency, 0 I , onto a matenal sometimes generates nonlimar polanzed wave, leadmg to the

emission of light wth different wavelength, o m The case of w = m o I w I 4- w 2 = w 3 is called optical mxing or frequency upanversion

is called optical harmonic generatlon, the case of

Equation of optical harmoruc generatlon

< Intensity of optical harmonic generation >

I

:

I sample path length, I c coherent length, n decreases, If phase matchmg condition A k =O does not hold, I depends on the incident angle qj . ie ,I m = 2 second harmonic generation (SHG) rn = 3 third h a r m o ~ cgeneration (THG) < Transformation efficiency of SHG >

fi

0

I

Intensity of mcident light,

permeability of vacuum, e

o : permittivity

I

refrachve mdex

of vacuum, A : cross section of beam, d : nonlinear optical constant

Apparatus for harmonic generation measurements

< Apparatus for THG > mirror

S HG

photomu I t ip I i e r

I 1-

w,

V

<
w 2 /

/’

/

#3

188

5 The Interaction of Light with Materials 11

(5.20) 2

where Ak denotes the difference between the wavenumber of the nonlinear polarization, k N L , and the wavenumber of the newly emitted light, k(wL). Accordingly, the intensity of the newly emitted light, I, is obtained from the relationship, I IE(r)12,as shown in Table 5.6. In the case where Ak = 0, the light intensity, I, is proportional to the square of the sample length, 1. When Ak f 0, the light intensity, I , reaches a maximum for Akll2 = nl2. The sample length in this case, I, = nlAk, is known as the coherence length. Light emitted from nonlinear dielectric polarization is of highest intensity for Ak = 0, emphasizing the importance of phase matching. The equations related to second harmonic generation as well as an example of a third harmonic generation experimental set-up are shown in Table 5.6. When two laser beams ( E l , wlr k ( w l )and E2, q,k ( q ) ) are introduced into a nonlinear optical medium, the nonlinear dielectric polarization, PNL, exhibits the following forced oscillations,

-

-

pNL

erl(w * o z N - I k ( o ,

)+k(@Z))Xl

(5.21)

which describe the generation of two new light beams with angular frequencies, wlf% and wavenumbers, k ( w l f q ) ,a phenomenon known as optical mixing. In the case where w1= q and m,., = 2 w 1 ,this phenomenon corresponds to second harmonic generation. Radiation of laser light (w,, k,) through a nonlinear dielectric medium induces two kinds of nonlinear dielectric polarizations (m= wI - w3 and w3 = wI- w z ) , a phenomenon called optical parametric oscillation which can be considered as the reverse of optical mixing. The energy and phase matching are given by

The induction of parametric oscillation in a resonator yields optical parametric amplification. Thus irradiation of nonlinear optical media with coherent light induces nonlinear dielectric polarization and emission of frequency-converted light. Second harmonic generation (SHG) and third harmonic generation (THG) are typical examples of this effect. They are used for example to generate laser light emitting in the ultraviolet and visible spectral domains.

5.1.5 Nonlinear Optical Materials Nonlinear optical materials are defined as materials which can convert the wavelength of incident light or which exhibit changes in their refractive indices as a result of changes in the intensity of the incident light or the application of an external electric field.4) They have important applications in laser light frequency conversion devices and in various optical switches. Moreover their availability has generated an entirely new research effort towards the development of optical computers. Since, as shown in the previous section, nonlinear optical phenomena originate from the nonlinear susceptibility (see Eq. (5.9)), in order to be useful in

5. I Saturation of Absorption and Multi-Photon Absorption Processes

189

practical applications nonlinear optical materials must have large nonlinear susceptibilities, f ) . Additional demands are placed on these materials, for example they should have low optical absorption, a fast response time, and it should be possible to process them into films without loss of activity. In the early days of nonlinear optics, inorganic crystals, such as KDP and LiNb03, were used almost exclusively. In the 1970’s, it was observed that several organic compounds in the powder form possess nonlinear optical properties. This discovery has stimulated intensive research into the development of organic nonlinear optical materials. Nonlinear polarization of organic compounds primarily involves n-electrons and nelectrons not directly involved in chemical bonds, since an external field affects these electrons much more than a-electrons directly participating in chemical bonds. Many molecules with conjugated n-electrons, however, show centrosymmetry and hence do not exhibit 2nd-order nonlinear optical effects, as discussed earlier. There have been various approaches to break the centrosymmetry of organic molecules, such as (1) the introduction of bulky substituents, (2) the introduction of chirality, and ( 3 ) the control of dipole moments. Empirical experimental approaches, together with molecular orbital calculations, by the Pariser-Parr-Pople (PPP) method, or the complete neglect of differential overlap (CNDO) method, have led to the following design rules. In order to exhibit a large second-order molecular polarizability, j3, organic materials should possess an extensively conjugated nelectron system exhibiting (1) a large transient dipole moment between the ground state and the excited state, and (2) a large difference between the permanent dipole moments of the ground state and the excited state (charge transfer structure). This is achieved in molecules which consist of a n-conjugated system carrying electron-donating and electron-withdrawing substituents. Nonlinear polarization characteristics of centrosymmetric molecules modified by the introduction of substituent groups are expressed in the following manner. The dielectric polarization consists of odd-number-order terms as in Eq. (5.23).

d2’and

P = A.E + C.EEE + ...

(5.23)

Substituents on an aromatic ring induce a distortion in the n-electron system, mostly via mesomeric effects. These induce changes in the light electric field and, consequently, influence the dielectric polarization P . In general terms, P can be expressed by Eq. (5.24).

P = A . ( E + E,)+ C.(E + E,) ( E + E,) ( E + Ek) + ... = (A’E,+ C.E,E,Ek+ ...) + ( A + 3C*E,E,+ ...)E + (3C.E,+ ...)EE + C.EEE + ... = P M + aE + PEE + yEEE + ... (5.24) Where p Mis the induced dipole moment due to the mesomeric effect (E,= pM/A).The secondorder molecular polarizability, j3, is given by Eq. (5.25). (5.25) A s the molecular polarizabilities, A and C , are inherent properties of unsubstituted centrosymmetric molecules, the j3 value of the corresponding substituted derivative is

I90

5 The Interaction of Light with Materials 11

proportional to the dipole moment induced by the mesomeric effect. Quantum mechanical expressions of the molecular polarizability can be derived from Eq. ( 1.163) which describe the hyperpolarizability of scattering phenomena, discussed in Chapter 1. The second-order molecular polarizability is given by the following equations:

(5.26)

(5.27) where r i denotes the components of the transient dipole moments associated with the ground state, g, and the excited state, n, and Arni(= rnk- r g i )is the difference between the permanent dipole moments of the ground state and the excited state. In practical systems, the second-order molecular polarizability can be approximated by the fourth term of Eq. (5.27). Assuming here a two-level system where the permanent and transient dipole moments have the same direction (the x-axis), then the second-order molecular polarizability is expressed by Eq. (5.28), (5.28)

which shows that significant second-order molecular polarizability can be achieved when the values of Ar; and rg$ are large. This corresponds qualitatively to the results of quantum mechanical calculation.

5.1 Saturation of Absorption and Multi-Photon Absorption Processes

191

The second-order molecular polarizability, /3, and the third-order nonlinear susceptibility, have been measured for many compounds (see Table 5.7). Note that the value of nitroaniline, where the centrosymmetric benzene ring carries an electron withdrawing nitrogroup and an electron-donating amino-group, is larger than that of monosubstituted benzenes. In the early days, optical nonlinearity of organic materials was measured usually with powder samples, mainly because it is very difficult to isolate organic compounds in the form of molecular crystals. In the case of centrosymmetric crystal lattices, macroscopic secondorder nonlinear optical characteristics are not detected. Molecular crystals are organized assemblies of individual molecules held together by intermolecular forces. Their macroscopic nonlinear optical constants are estimated as the sum of the molecular polarizability of individual molecules. Thus, neglecting intermolecular interactions in the crystal, the nonlinear optical constant, dllK,is expressed by

f ) ,

(5.29)

A=-

nI2 + 2

3

(5.30)

where N is the number of molecules per unit volume, Z is the number of molecules per unit cell, and s designates a molecule. The suffixes, I, J, K denote the dielectric main axes of the crystal, i , j , k are the molecular coordinates,f; is the Lorentzian factor, and n, is the ordinary refractive index. Experimental values of nonlinear optical constants for molecular crystals of organic nonlinear optical materials are given in Table 5.7. The control of the orientation of nonlinear optical materials assemblies is the subject of intensive study. Major advances in orientational control have been reported recently. Particularly attractive are the techniques of Langmuir-Blodgett film deposition, the vapor-deposition of noncentrosymmetric films, and the electric-field poling of dye-dispersions in films. Conjugated polymers possess a large off-resonant nonlinear response with an ultrafast relaxation time. However, it is extremely difficult to prepare noncentrosymmetric polymers with a large n-conjugated system. Noncentrosymmetry is not a requisite for third-order nonlinear optical properties. Polyacetylene, which has a main-chain n-conjugated electronic system, was prepared in the early 1970's by Wegner. Since then, many conjugated polymers have been prepared and shown to exhibit third-order nonlinearity. Values of third-order of conjugated polymers are given in Table 5.7. The conditions nonlinear susceptibility, y3), for obtaining large j y 3 ]are similar to those necessary for getting large second-order molecular polarizability. Odd-number-order dielectric polarization can be measured even for symmetric structures. Technical applications rely on processable materials which can be developed into thin films. Few conjugated polymers satisfy these requirements. A large number of interdisciplinary research groups are active in this area, in Europe, North America, and Japan. Light travels faster in materials than electrons in metals. Optical information processing has tremendous advantages over electronic information processing, since it can transfer more information simultaneously and with less noise than the current systems. The opportunities open to the techniques have been demonstrated in optical telecommunication systems. Future applications presently under evaluation are in the area of optical computing and optical switching.

192 Table 5.7 Nonlinear Optical Materials optical materials

ADP KDP SiO z LiIO s

L

W

3

KNhOS AIGaAdAIAs-MQW As 1 S s Glass Nitrobenzene Fluorobenzene Aniline o-Fluoronitrobenzene m-Fluoronitrobenzene p-Fluoronitrobenzene o-Nitroaniline m-Nitroaniline

nonlinear optical constant n ( o ) n(2w) dlHxlO-a(e.su)

refractive index

1.5067 1.4942 1.5341

1.4816 1.4708 1.5470

1.719 2.1544 2.1196

1.750 2.2325 2.2029

1.26 1.04 0.80.1.20

2nd-order 3rd-order molecular nonlinear polarkability susceptibility j 3 ~ 1 O - ~ ~ ( e s u~)( ~ ' x l O - ' ' ( e s u )

(d s e ) (d s o ) (d I I )

2.8 (1.90 p m)

8.59 (d s a ) 91 (d s ) X SiO 61 (d 3 s ) X SiO z 35000 720

1.630

1.700

p-Nitroaniline 4-Nitro-trans-stilbene 4-Amino-trans-stilbene

39 (d s I ) X SiO 41 (d s s ) X SiO

P

2.2 1.I6 1.1 -1.75 -1.64 -2.14 10.2 6.0

(1.97 p m) (1.90 p m)

Oiquid) Oiquid) Oiquid) (liquid) (liquid) (liquid) (acetone) (acetone)

P

16.7 (dioxane) 29 (benzene) 12 (benzene)

4-.hho-4'-nitro

-trans-stilbene 3-Methyl-4-nitropyridin - 1-oxide

260

m-Aminophenol

1.663~ 1.829y 1.6252 1.562

1.750~ 1.997~ 1.6602 1.589

2-methyl-4-nitroaniline

I .8

2.2

N-(4-Nitrophenyl) -L-prolinol Urea( A 4 , 6 3 3 p m)

1.68

1.77

AzobenzeneiPMA Polyacetylene : E II c Elc Poly@phenylenevinylce)

Poly(methy1phenylsilane)

1.48~ 1.48~ 1.5%

(acetone)

13.5(d I 4 ) X KDP 1.5(d I ) X SiO I 6.7(d s s ) X SiO z 500 (d I I ) X SiO z 40 ( d s ~ ) X S i O g 164 ( d 1 I ) X S i O z 61 (d Z I ) X SiO z 3.6(d

I 4

)

320 32000 300 3200 150

X SiO z or X KDP should be multiplied by the d I u value of SiO or KDP. The x,y,z in r e h t i v e index values denote the direction of axis. Wavelengths in parentheses are those of incident laser light, and is 1.064 p m for the case of no indication.

(1.90 LL (1.90 p (1.90 p (1.85 p

m) m) m) m)

5.2 Coherent Spectroscopy

193

5.2 Coherent Spectroscopy The emergence of lasers has set the stage for new types of spectroscopy. As we have seen, laser light triggers phenomena quite different from those created by conventional light sources. Coherence is certainly the most useful property of laser light when one considers the design of new spectroscopic measurements. Several techniques of measurements with coherent light are outlined in this

5.2.1 Coherent Raman Spectroscopy In Chapter 1 we described conventional scattering processes, Rayleigh scattering and Raman scattering. These scattering processes involve the simultaneous absorption and emission of a photon by a molecule. When a sample is exposed to three laser beams of different angular frequencies, w,, q,w3,third-order nonlinear dielectric polarization can be induced, leading to the emission of coherent light with an angular frequency, w4( = IwI - w2 + w31).In a sample having a Raman shift of 00,the induced Raman resonance will occur with = Iw, - w21 or ol, = lo3- 4, where w , corresponds to the angular a frequency shift frequency of the incident laser light, y to that of the Stokes scattering light, and w4 to that of the new emission. Coherent anti-Stokes Raman spectroscopy (CARS) corresponds to the case of ~3 = w, and wI > q.Coherent Stokes Raman spectroscopy (CSRS) corresponds to the case of w3= wIand w , < y.Raman-induced Kerr-effect Spectroscopy (RIKES) is based on the processes relating to the situation where y = q.Four-wave mixing (FWM) occurs when wI# w2# w3,and degenerated four-wave mixing (DFWM) if wI= q = q.Experimentally, interference fringes are induced in a sample by two pumping laser beams. The light due to these interference fringes is observed by the probing beam. Four-wave mixing is described by four coupled wave equations. Because three waves mix to generate a fourth, the nonlinear polarization associated with four-wave mixing is proportional to a product of three fields. Four-wave mixing can be represented schematically as follows: where CARS corresponds to the case where wI= w3 and RIKES to the case intensity of the emitted light, 14,is given by the following general equation:

0,

= -q. The

(5.32) where (2”)is the third-order nonlinear susceptibility, I , and I2 the intensities of the two pumping beams, I3 the intensity of probing beam, fB the interaction length of the three beams, and n the refractive index of the sample. Since light produced by higher-order dielectric polarization induced by coherent beams is also coherent, the matching of the wave vectors is a necessary condition for four-wave mixing (Eq. (5.31)). Thus, the emission of the signal light exhibits an angular dependence. The definition, theoretical treatment, and experimental set-up of CARS, RIKES, and FWM are summarized in Tables 5.8, 5.9, and 5.10, respectively. The set-ups differ slightly for each technique, to satisfy the specific conditions of energy and wave vector matching. In

194 Table 5.8 Coherent Anti-Stokes Raman Spectroscopy (CARS)

summary

Coherent anti-Stokes Raman spectroscopy is a prevailing nonlinear optical mixing technique, where optical mixing of light

beams with frequencies, w light, w

2

I

and w z generates light with new frequency (

w 3 =2 w

- w z ).

I

Here, w

is incident laser

I

corresponds to Stokes scattering light, and w s is the signal light of anti-Stokes line. The case of w

I

<w

called coherent Stokes Raman spectrosco~(CSRS). The characteristics of CARS me excellent directivity of emitted light

and hgh sensitivity by 5

-

z is

9 orders of magnitudes compared to usual Raman scattering. Equation

< Intensity of CARS: Solid state > The CARS intensity is expressed as a function of dif€erence frequency ( w I - w z ). In solid states, CARS spectra are

x

dispersive due to the large contribution of

I

( 0 1 - 0 2 )

w

o

-1

I

xl,kl'31 2=

and the interference of contributions from

and

x

.

I X"R+%Bl

: frequency for Stokes shift, a

' : tensor factor for Raman interaction,

P:differenceindensitymatrixinstationarystate : nonlinear susceptibility for nonresonant term,

x

x

N : number density of the system,

(

~

1

1

-

p ~ n ),

D : combination factor,

T 2 : dephasing time (relaxation time to equilibrium state of mn-diagonal elements in density matrix) Principle and apparatus apparstus for CARS measurements

scheme of energy levels

I

w

a r on-ion Saser

c r

L i t t row

1 mirror

.

prism spectrometer matching of wavenumber vector k.

k,

matching condition of wavenumber vector: I k a - 2 k t

mirror

I + k I Linearly polanzed pumping light tnduces refractive mdex amsotropy in the sample, and linearly polanzed probmg light changed to elliptically planzed light

n 2 nonlmear refractive index ( n = n o 1 I intensity of pumping light, y,

(

’’

+n

2

I )

,

1 A 4

3rd-order nonlinear susceptibility

interaction length between two beams,

phase mfference angle of elliptmlly polanzed light,

Apparatus RIKES (Raman Induced Kerr Effect Spectroscopy)

OHD-RIKES(0ptical Heterodyne Detected Raman Induced Kerr Effect Spectroscopy)

el I i p t ical l y polarized light I

p r o b i n g laser

E~~~

] €0

IS

I96

Table 5.10 Four Wave Mixing (FWM) SummarY

Four-wave mixing is a method where an intRference fnnge genmted by two pump+ beams provides diffraction p t m g

which difiracts probing beam This method eliminates back-ground noise such as in CARS due to non-resonant electronic transition or in RIKES due to strain-induced birefringence. The method hap an advantage of obsewing all kinds of nonlinear

effects regardless of their mechanisms. Principle and equation matching of wavenumk vector

scheme of energy levels

'4 f > @oxcar configuration)

i >

. .

s

I

k2

-

k

k

!2

'

=k

kl

k3

< Diffraction intensity >

I ,, I

: intensity of pumping beams,

2

1

I

I

: interaction length between two beams,

y,

'

: intensity of probing )

beam

: 3rd-order nonllnear susceptibility

APplram

DFWM (Degenerated Four Wave Mixing) path length

variable

,

+ k 3- k 2

5.2 Coherent Spectroscopy

197

RIKES, dichroism and birefringence proportional to the intense incident pumping beam are produced, resulting in a change in the polarized probing beam. The phase change of the linearly polarized probing light to an elliptically polarized light, A@, and the nonlinear refractive index, n2, are given by Eqs. (5.33) and (5.34), respectively,

n2 = Re[x"'] cn2Eo ~

(5.34)

where ZI is the intensity of the pumping beam, I, is the interaction length, and 2')is the third-order nonlinear susceptibility. As the signal produced by four-wave mixing is coherent, CARS is about 5-9 times more sensitive than conventional Raman scattering. Moreover, the signal can be detected by a photomultiplier tube placed at a certain distance of the sample unlike the case of conventional spectroscopy. These features make CARS a particularly attractive technique to measure concentrations and temperatures of atoms or molecules in combustion fields. The measurement of the third-order nonlinear susceptibility, k'),is carried out by using four-wave mixing based on Eq. (5.32). FWM and DFWM have become important techniques in the evaluation of nonlinear optical materials. High-resolution coherent spectroscopy has improved by two orders of magnitudes the accuracy of the rotational and vibrational constants of molecules, Coupling of photons with optical phonons in anisotropic media induces the formation of polaritons, which show a strong dependence on the wave vector. This wave vector dependence is measured by changing the angle between incident beams. CARS and FWM play important roles in spectroscopy in the momentum space or the k space.

5.2.2 Photon Echo Technique All the examples treated thus far have been restricted to the case of continuous-wave operation. In this section we will consider time-dependent, or transient, effects. Let us imagine a sample exposed to two consecutive laser pulses separated by a delay time, fd. There are cases where an echo pulse is emitted after a time delay f d (see Section 2.2.4). To observe this photon echo the delay time must be such that f d < 3T2, where T2 is the dephasing time. Photon echo is one of several transient coherent spectroscopy techniques. It is analogous to the transient spectroscopy in pulsed NMR spectroscopy. For example, consider a system interacting with an external field. When the external field is suddenly removed, the system does not return to thermal equilibrium at once, but rather it emits an attenuating electromagnetic wave. This phenomenon of free induced relaxation is similar to the sound echo one can hear for a while after a bell has been struck. When exposed to two consecutive laser pulses, a medium emits light at a dephasing time. This dephasing time is related to the homogeneous line width of the absorption. The time dependence of dielectric polarization after a very short pulse irradiation is given by Eq. (5.35).

I98

Table 5 . I 1 Photon Echo summary Photon echo is a phenomenon that, when two successive light pulses with time interval, sample emits echo pulse at time, 2

t

1

t d ,are given to a sample, the t4 < 3 TP

, after the first pulse. Photon echo can be obsmed in the. condition of

Principle

I

lat DUIea

2nd nulea

t

3

< Decay of echo intensity >

t

z=

t

t d+L 2

u : average value of the rate change due to collision

U : rate of elastic scattering

T 2 : &phasing time (relaxation time to equilibrium state of non-diagonal elements in density matrix) Appemtus

F fit dye laser

variable l i g h t delay

mirror

. -

acoustooot i c

\I

d&&

I i f ier

j h a l f wave Iengt h plate photomultiplier

References

I99

The photon echo technique is outlined in Table 5.1 1. Its principle is briefly explained here. The Bloch vectors of the molecules in a medium are arranged instantaneously in the direction of the e, axis by the first laser pulse ( t = 0). During the dephasing process they begin to rotate with different azimuthal rates as a result of the inhomogeneity of the system. After the dephasing time, T,, the distribution of these vectors has become random. If a second laser pulse irradiation is applied to the system at t = tdrthe Bloch vectors rotate by n radians about the e , axis. They will be arranged again in the same direction on the e2axis after a delay time td from the second pulse irradiation, leading to macroscopic dielectric polarization and the emission of an echo pulse. The amplitude of the echo pulse changes as a function of the delay time between the first and second pulses, according to Eq. (5.36): 212

I 1(2t,) 1- e-K

-e

Lkdl2',,~

-

(5.36)

7

where U is the rate of elastic scattering, u the average value of the rate change due to collision, and T, the dephasing time. The time t, is given by the following relationship between the delay time, td and the pulse duration, t,: t, =

td

t, +--.

(5.37)

2

The dephasing time, T2, can be measured by the photon echo technique or determined from the homogeneous width of saturation spectroscopy, which is a Fourier transform of the former, as easily seen from Eq. (5.35). When a sample is irradiated with three consecutive laser pulses at times, 0, 1,. and t3, an echo pulse is emitted at time, t2 + f 3 . This is called stimulated photon echo. Several additional echo techniques have been proposed. In this chapter, the modulation of light by light has been sketched out. We stressed the importance of the laser discovery, which gave ready access to extremely intense coherent light. All the phenomena we described require coherent light. The electric field associated with the intense and phase-matched laser light induces higher-order dielectric polarization. This distortion of the transient dipole moment forms a new electric field and emits light. The advent of lasers has promoted new areas in the fields of photochemistry, photophysics, and optics. The laser has become the driving force towards the third golden age of photo- and opto-studies.

References 1. T. Shimazu, Laser und its Application, Sangyoushuppan, Tokyo (1969). 2 . Applications 9fLasers to Chemical Problems, (T. R. Evans ed.), John Wiley and Sons (1982). 3 . Laserfor Spectroscopic Chemicul Anu!,.sis, (T. Shimazu ed.). Gakkaishuppan Center, Tokyo ( 1986). 4. S. Umegaki, Orgunic Nonlinear Optical Materids, Bunshinshuppan, Tokyo ( 1990). 5. M . D. Levenson and S. S. Kano, Introduction to Nonlinear Laser Spectroscopv. Academic Press, N . Y . ( 1982).


.


Index

chlorophyll pigment 1 17 cholesteric liquid crystals 166 circularly-polarized wave 19 1 1-cis-retinal 90 coherent anti-Stokes Raman scattering 62 coherent anti-Stokes Raman spectroscopy 193 coherent light 177 coherent Raman spectroscopy 193 Cotton-Mouton effect 169 Coulomb gauge 37 creation operator 40 critical angle in total refraction 13

A

absorption 44 acousto-optic effect 17 1 amplitude 1I annihilation operator 40 ANS 123 anthracene 50 aromatic polyimide 139 N-arylaminonaphthalene sulfonate atomic orbitals 27 azobenzene 67,85 azobenzene bis(crown ether) 92

123

D

B benzophenone 50, 57 birefringence 19 birefringent 163 bis(triphenylimidazo1e) 85 blackbody radiation 2I Bloch vector 199 Bragg diffraction 172 Brewster’s angle 20 Brillouin scattering I57

Debye-Waller factor 97 degenerated four-wave mixing 193 dephasing time 96, 199 Dexter energy transfer mechanism 107 DFWM 193 diarylethene 88 diffraction I5 diffusion control 78 N,N-dime t h y lam i nonap h t h a 1ene su I fona t e 129 dioxetane 143 dipole approximation 46 dipole-dipole interactions 105 distribution of free volume 136 2,s-distyrylpyrazine 74 DNS 129 Doppler-free spectroscopy 180 Doppler shift I80

C

canonical equation 38 CARS 62, 193 charge generation layer 1 I3 charge-transfer complex 56 charge transfer fluorescence 139 I I3 charge transport layer chemical amplification 83 chemical control 78 chemically initiated electron exchange luminescence chemiluminescence

E

144 140

eigenfunction eigenvalue 20 I

46 46

202 Index

electric field 36 electroluminescence 145 electromagnetic theory 4 electromagnetic wave 2 electron exchange mechanism 56 electronically-excited state 67 electron transfer 108 electro-optic effect 163 electro-optic Ken effect 163 elliptically-polarized wave 19 emission 44 end-to-end cyclization I30 energy migration 108 energy transfer 53, 105, 108 equivalence principle of photoreaction excimer 50, 56 excimer formation 130 exciplex 50, 56 excluded volume effect 13 1 F Faraday effect 169 Faraday rotor 172 Fermi’s golden rules 47 fluorescence 50 fluorescence anisotropy ratio 127 fluorescence depolarization 127 fluorescence probe 123 Forster energy transfer mechanism 107 Forster equation 54 four-wave mixing 193 Franz-Keldysh effect 163 freezing of the reaction 78 Fresnel-Kirchhoff equation 17 fulgide 88 FWM 193 G gain coefficient 63 gauge transformation 37 graded-index type 160

H Hamiltonian 26, 43 history of light research 4 hole burning 94, 180 homogeneous width 96 Hiickel method 29 Huygens’ theory 16 hydrophobic probe I23

hyperpolarizability 63 hyper-Raman scattering 62 1

3-indenecarboxylic acid 80 interference 13 intermacromolecular diffusion controlled reaction 131 intersystem crossing 50 inverted region I 11 J

Julilidone malonitrile

127

67 Lambdip 180 Lambert-Beer’s law 47 Langmuir-Blodgett films 120 laser 63, 177 light 2 light scattering 15 1 linearly polarized light 20 liquid crystals 92, 166 lithography 79 local conformation change 130 low-energy excitation mode 97 luminescence probe 122 luminol 141 M magnetic flux density 36 magneto-optic disk 172 magneto-optic effect 169 magneto-optic Kerr effect 169 magneto-optic switch 172 Marcus theory 1 11 Maxwell’s equations 3I microenvironment probe 123 microstructural probe I36 microstructure probe 123 migration 53 molar extinction coefficient 47 molecular devices 120 molecular field theory 20 molecular motion probe 123, 127 molecular orbitals 28 molecular photonics 6 molecular polarizability 62 molecular rotor 127

Index 203

N

naphthoquinonediazidesulfonate 80 nematic liquid crystals I66 nonlinear dielectric polarization 186 nonlinear optical material 188 nonlinear optical phenomenon 186 nonlinear refractive index 185 nonlinear susceptibility 183 nonradiative transition 50 normal region 111 Norrish type 11 reaction 78 0 old quantum theory 21 optical fibers 158 optical insulator 172 optical propagation loss 158 optics 9 oscillator strength 47

P phantom excited state 67 I1 phase phenol-formaldehyde resin 80 phonon frequency 97 phonon side hole 97 phosphorescence 50 phosphorescence depolarization 129 phosphorescence of benzophenone 136 photochemical hole burning 94 photochemical reaction 67 s in the solid state 74 photochromism 85 photoconductivity 112 24 photoelectric effect photon echo 102, 197 photon-gated PHB 102 photophysical processes 50 - in polymer solids 134 photopolymerization 69 photoresist 80 photoresponsive polymer 90 photosensitive polymer 79 photosynthesis 117 1 18 photosystem I photosystem i I 117 phthalocyanine 99 plane-polarized wave 19 Pockels effect 163 ~

polarization 18 polydiacetylene 77 polymer blends 137 polymeric liquid crystals 92 polymeric optical fibers 158 poly(pheny1 acrylate) 78 poly(N-vinylcarbazole) 1 13 poly(viny1 cinnamate) 78 poly(pvinyloxyethy1 cinnamate) population inversion 64 porphine 99 p polarization 19 primary photochemical process PVK 113 N-(1-pyridinio) amidates 93

79

69

Q quantization of the angular momentum quantization of the harmonic oscillators quantization of the radiation field 40 quantum theory 4 quartz optical fibers 158

25 39

R radiation field theory 30 Raman-induced Kerr-effect spectroscopy 193 Raman-Nath diffraction 172 Raman scattering 57, 156 Rayleigh scattering 57, 154 reactions in crystals 74 reflection 12 refraction 12 refractive index 13 resolution 82 Rhodopsin 90 RlKES 193 rotational diffusion constant 129

S salicilydeneaniline 85 saturation spectroscopy 178 scalar potential 35 scattering phenomenon 57 Schrodinger equation 26, 46 secondary photochemical reaction 69 second harmonic generation 62, 188 second-order molecular polarizability 189 second quantization 40 sensitivity 82

204 Index

smectic liquid crystals 166 Smoluchowski equation 107 Snell’s relationship 12 spironaphthoxazine 89 spiropyran 85 s polarization 19 Stark effect 102, 163 nlr* state 72 n?T+ state 72 stepped-index type 160 Stem-Volmer equation 107 stilbene 67

T TEmode 19 termination of free radical polymerization 131 tetraphenylporphine 99 thermal reaction 67 thioindigo 85 third harmonic generation 188 TICT 126 time-dependent wavefunction 26 time-domain optical memories 102 TM mode 19 TNF 113 TNS 123 total reflection 13 6

transient hole burning 96 transient spectral hole burning 180 transition dipole moment 47 transition probability 44 - of the excitation energy transfer 54 transition rate 44 triarylmethane leuco-form 92 2,4,7-trinitrofluorenone 1 13 twisted intramolecular charge transfer 126 two-photon absorption 62 V

vector potential 35 velocity 11 Voigt effect 169 W

wave formula 11 wavelength 11 wave number vector 11 Woodward and Hoffmann selection rules

Y

Young’s experiment

14

Z

Zeeman effect

I69

74

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