Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory

David L. Andrews, Philip Allcock Optical Harmonics in Molecular Systems Quantum Electrodynamical Theory

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

David L. Andrews, Philip Allcock

Optical Harmonics in Molecular Systems Quantum Electrodynamical Theory

David L. Andrews, Philip Allcock School of Chemical Sciences University of East Anglia Norwich, UK e-mail: [email protected] e-mail: [email protected]

n 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 details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Die Deutsche Bibliothek – CIP-Cataloguing-inPublication Data A catalogue record for this publication is available from Die Deutsche Bibliothek. © WILEY-VCH Verlag GmbH D-69469 Weinheim, 2002 All rights reserved (including those of translation in other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. printed in the Federal Republic of Germany printed on acid-free paper

Cover “The Artwork”, artwork from the cover of the Moody Blues album “Every Good Boy Deserves Favour”. Reproduction granted by DECCA Music Group, London, UK

Typesetting K+V Fotosatz GmbH, Beerfelden Printing betz-druck gmbH, Darmstadt Bookbinding J. Schäffer GmbH & Co. KG, Grünstadt ISBN

3-527-40317-5

VII

Contents Preface

IX

1 1.1 1.2 1.3

Foundations of Molecular Harmonic Emission Classical Optics 2 Quantum Electrodynamics 12 Media Corrections 20

2 2.1 2.2

Perturbation Theory 27 Time-Dependent Perturbation Theory 27 Time-Orderings and State Sequences 33

3 3.1 3.2 3.3

Radiation Constructs 39 Radiation Tensor Construction 40 Quantum Optical Considerations 44 Pump Photonics 48

4 4.1 4.2 4.3

Molecular Properties 53 Molecular Tensor Construction 53 Symmetry 62 Two-Level Systems 71

5 5.1 5.2 5.3 5.4

Coherent and Incoherent Signals 79 Regular Solids 80 Gases, Liquids and Disordered Solids 83 Macromolecules, Suspensions and Partially Ordered Solids 87 Coherence and Wave-Vector Matching 90

6 6.1 6.2 6.3

Coherent Harmonic Generation 97 Harmonic Intensities 97 Rotational Averaging and Symmetry Criteria 102 Third Harmonic Generation 107

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

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VIII

Contents

7 7.1 7.2 7.3 7.4 8 8.1 8.2 8.3 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Special Systems for Second Harmonic Generation 109 Second Harmonic Generation at Surfaces and Interfaces 109 Electric Field-Induced Second Harmonic Generation 115 Optical Coherence in Dispersed Particles 125 Six-Wave Second Harmonic Generation 131 Incoherent Elastic Light Scattering 151 General Principles 152 Second Harmonic Scattering/Hyper-Rayleigh Scattering 153 Third Harmonic Scattering 161 Hyper-Raman Scattering 163 Constructing the Signal 164 Hyperpolarisability Theory 167 Irreducible Tensors 171 Symmetry Selection Rules 174 Scheme for the Determination of Molecular Invariants 178 Reversal and Depolarisation Ratios 182 Higher Multipole Effects 183

Appendix 1: Resonance Damping 185 Appendix 2: Rotational Averaging

191

Appendix 3: Isotropic Tensors and the Euler Angle Matrix Appendix 4: Irreducible Cartesian Tensors 205 Appendix 5: Six-Wave Mixing and Secular Resonances Appendix 6: Spectroscopic Selection Rules 215 Glossary of Symbols References 229 Bibliography Index 239

237

225

209

201

IX

Preface Creation, Evolution, . . . Population, Annihilation, . . . Contemplation, Inspiration, . . . Communication, . . . Solution. One More Time to Live, John Lodge

The acquisition of colour by white light passing through any translucent coloured material, as for example sunlight through a stained glass window, is a phenomenon familiar to all. Despite the interpretation offered by our senses, it is of course a process in which light is modified only in its distribution of intensity amongst the range of colours or frequencies of which it is comprised. No photons change frequency during the transmission process – irradiate the coloured glass with monochromatic light of any one frequency and precisely the same frequency emerges, even if somewhat attenuated. That, at least, is true for the levels of intensity we commonly encounter. With the much higher intensities available from laser sources, the situation changes and some photons of entirely different hue can be produced. These are optical harmonics, whose frequencies are simple integer multiples of the input light frequency. The production of optical harmonics owes its origin to processes in which the energy of two or more photons of incident light emerges in the creation of single photons of output, the original photons lost in the process. It is easily calculated that the probability of any such process requiring the local coincidence of two or more photons is only significant at the levels of intensity associated with pulsed laser light. This is reflected in the fact that first observations of harmonic generation were made only once laser sources became available. The upsurge of interest in harmonic generation in recent years undoubtedly Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

X

Preface

owes much to the increasing availability of laser systems offering high power ultrashort pulses; the tunability of many such systems is also important in the developing spectroscopic applications. The nature of the medium, in which harmonic conversion occurs, strongly influences the effectiveness of the process and the character of the harmonic emission. In particular, laser-induced frequency doubling (or second harmonic generation) produces tightly collimated emission in non-centrosymmetric crystals – and as a result is widely used as a means of converting laser output to a different wavelength. Inorganic crystalline materials such as potassium dihydrogen phosphate, for example, are commonly used to obtain visible radiation from the output of powerful infrared lasers. The combination of multiple-stage harmonic generation with tunable dye laser conversion opens enormous windows on the spectrum of wavelengths that can be derived from fixed-wavelength laser sources. In molecular gases and liquids, harmonic conversion processes are frequently either forbidden or occur only weakly. The principle of harmonic generation in molecules has nonetheless become an area of increasing interest for a number of reasons. It transpires that many organic crystals and polymeric solids not only have usefully large nonlinear optical characteristics, but are also surprisingly robust and thermally stable. Consequently the fabrication of organic materials for laser frequency conversion has become very much a growth area, and developments in the associated theory continue apace. Although molecular gases and liquids cannot usually compete in terms of efficiency, their vibronic structure often leads to intense and essentially discrete absorption features, and can be associated with a wavelength-dependence of refractive index which can expedite harmonic conversion. Moreover such media are themselves of considerable scientific interest through the information which can be derived by studying the harmonic processes they mediate. Any process in which each output photon is created at the expense of two or more photons of incident light can be fundamentally categorised according to two basic criteria. One consideration is whether the process is elastic or inelastic, i.e. whether the emergent photons contain precisely the summed energy of the annihilated photons, or if there is some either uptake or loss

Preface

of energy by the material in which the process occurs. The term ‘parametric’ is often used to describe elastic processes, for which the susceptibility properties of the material parametrically determine the rate of conversion. Inelastic harmonic processes in which an exchange of energy does take place are accompanied by quantum transitions in the nonlinear medium, and are thus termed non-parametric. The other major factor in determining the characteristics of a harmonic process is whether or not the response is coherent or incoherent in its nature, that is to say whether the light emergent from individual points within the sample interferes constructively or in a random fashion. Only elastic conversion processes can be coherent. Coherent harmonic generation processes are associated with laser-like emission, providing intensity levels far surpassing those associated with incoherent phenomena. The key factor that determines whether a nonlinear scattering process is coherent or not is wave-vector matching. Fulfilment of this condition, which essentially requires there to be a conservation of the net photon momentum, automatically leads to coherent emission. In much of the existing literature, the theory of harmonic production is based on the conventional formulation for solids, originating in the elegant and pioneering work of Bloembergen and others. Such theories are generally cast in terms of a bulk nonlinear susceptibility, related to microscopic properties through incorporation of the appropriate Lorentz field factors. Since the majority of theoretical applications have concerned crystalline media, such an approach is entirely appropriate. In considering the much weaker harmonic processes that occur in gases and liquids, the conventional theoretical formulation is also commonly applied, account being taken of the fluid isotropy only by considering the symmetry implications for the bulk susceptibility. However such considerations do not adequately account for the effects of local fluid structure and molecular tumbling. These exert a powerful influence on the selection rules and polarisation dependence of nonlinear optical processes. Another issue over which the classical approach is open to debate concerns its common formulation of optical nonlinearity in terms of nonlinear polarisation, a concept deeply entrenched in standard nonlinear optics. Nonetheless it establishes a framework that is not directly amenable to the resolution of certain kinds of question. In

XI

XII

Preface

particular, it can obscure the difference between coherent and incoherent optical response, leading in some cases to incorrect conclusions. Moreover the polarisation formulation introduces an unnecessary and to some extent artificial differentiation between incident and emergent radiation, masking the essential time-reversibility of optical interactions at the molecular level. It is not that the polarisation concept is itself flawed, but problems can arise when it is treated as if it represented an observable, which it never does. Only by formulating theory directly in terms of the observable, usually the harmonic intensity, can a unified treatment of both coherent and incoherent contributions be developed. Only theories properly cast in terms of molecular nonlinear optical response can accommodate both parametric and non-parametric processes, which again involves a distinction between coherent and incoherent phenomena. To properly bridge the gulf between the two disciplines of chemistry and optics, which represent the molecular and photonic heritage of nonlinear optics, demands a conceptual and mathematical bridge of sufficient strength to support its progeny. At one extreme, the chemists and materials scientists whose work is increasingly directed towards the devising, synthesis and characterisation of novel nonlinear optical materials, need a framework that can accommodate and relate to their insights into the relationships between molecular quantum mechanics, structure and optical properties. At the other, laser physicists and optical engineers need a vehicle for the furtherance of theory in a form which can reveal the detailed form of the quantum optical parameters that relate to particular materials. As a theory which addresses the full remit with the equitable rigour of quantum mechanics, molecular quantum electrodynamics (QED) is undoubtedly the tool of choice for this demanding task, and it alone is the basis for the development of theory in the following pages.

Content and Structure of the Book QED is a theory ideally suited to representation of the detailed nonlinear optical response of molecules, bringing both rigour and conceptual facility. Characterised by a quantum mechanical treatment

Preface

of both the light and matter together as a closed dynamical system, a significant part of its appeal lies in the fact that only within such a framework is it proper to employ the photon concept. Many of the insights this concept offers to nonlinear optics are lost in the classical description of light-waves. The relative unfamiliarity of molecular quantum electrodynamics to many involved in nonlinear optics invites an appraisal in which its distinctive elements can be highlighted and compared to the classical approach, and this is done in Chapter 1. This chapter also establishes the ground rules for calculations based on QED. Chapter 2 develops the general form of equations for nonlinear optics, using quantum electrodynamical states as the basis for analysis in terms of time-dependent perturbation theory. Attention is focused on both the radiative and molecular parts of the quantum amplitudes. In connection with the latter, it is shown that there is a diagrammatic alternative to the time-ordered diagrams commonly deployed for the derivation of nonlinear optical response. Useful as these are, the counterpart state-sequence diagrams offer a greater calculational immediacy, and a different perspective on the various quantum amplitude contributions. Nonlinear optical response characteristically depends on more than the intensity and wavelength of the input radiation. In Chapter 3 the influence of polarisation, wave-vectors, pulse properties and quantum optical input features are examined in detail. The following Chapter 4 examines the counterpart molecular properties, which depend crucially on molecular symmetry and electronic structure. At this stage attention is given to the dependence on frequency, and in particular the resonance and near-resonance conditions frequently employed to enhance nonlinear optical behaviour. This chapter also addresses the key structure-property principles to be considered in the design of nonlinear optical compounds. Moving to bulk systems, Chapter 5 reveals how the collective response of an ensemble of molecules or other particles is formally derived from the quantum amplitudes for harmonic emission. It is shown how the salient features operate to determine the bulk nonlinear optical response in three quite different types of system: regular solids; gases, liquids and disordered solids; and lastly macromolecules, suspensions and partially ordered solids. With reference to such systems it is then demonstrated how wave-vector matching criteria operate for coherent optical output.

XIII

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Preface

Together, the first five chapters provide the QED basis for the development of nonlinear optics. Second harmonic generation is commonly used as an illustration, though development of the general framework enables a number of common features to be identified before specific interactions are discussed explicitly in subsequent chapters. The first of these, Chapter 6, deals with the coherent process of harmonic generation, with both second and third order harmonics considered in detail and the framework for higher orders also identified. Although an obvious exemplar for the development of theory, coherent second harmonic generation is usually forbidden in isotropic molecular systems, on symmetry grounds. Nonetheless second harmonics can emerge where they are not expected, and various mechanisms for this phenomenon are considered in Chapter 7. First, attention is given to the familiar case of second harmonic evolution at surfaces and interfaces, where it is often employed for the derivation of information on local molecular orientation. The role of a static electric field is also considered: it can induce harmonic emission either by creating an anisotropic environment in the case of polar fluids or melt phase polymers, or by means of electro-optical channels. Local coherence is shown to be responsible for the emergence of coherent second harmonic signals from dispersed particles, and finally a six-wave mechanism is shown to operate at high levels of laser irradiance. The following Chapter 8 addresses the much weaker, incoherent forms of second and third harmonic emission, which in the case of the second harmonic is generally referred to as either elastic second harmonic light scattering or hyper-Rayleigh scattering. Finally Chapter 9 addresses inelastic second harmonic emission accompanied by quantum transitions in the nonlinear material, normally referred to as hyper-Raman scattering. Despite the greatly reduced levels of intensity associated with interactions of this type, the spectroscopic data they can provide contain a wealth of structural information almost unmatched by any other kind of spectroscopy, particularly through the characterisation of molecular vibrations. The selection rules are discussed in detail and a scheme is demonstrated for the derivation of added information through polarisation studies of the effect. The book is pitched at graduate level. In the introductory chapters it is assumed that the reader is familiar with basic vector cal-

Preface

culus up to the level of Maxwell’s equations; also complex analysis and quantum mechanics, including operator algebra and use of the Dirac notation. A familiarity with molecular point groups and irreducible representations is required as a background for the development of Cartesian tensors in Chapter 4 and onwards.

Acknowledgements In the course of drafting over several years we had the privilege of detailed feedback from a number of individuals, whose expertise, insights and encouragement have immeasurably helped us towards the finished manuscript. Prominent amongst these are, at UEA: Alex Bittner, Nick Blake, Luciana Dávila Romero, Robert Jenkins and Brad Sherborne, all of whom have contributed to the development of the quantum electrodynamical theory and whose comments on the manuscript at various stages have been most gratefully received. We are also greatly indebted to Gediminas Juzeliu¯nas at the Institute for Theoretical Physics and Astronomy in Vilnius, Bill Meath from the Centre for Chemical Physics at the University of Western Ontario, Geoff Stedman at the Department of Physics and Astronomy, the University of Canterbury in New Zealand, and T. Thirunamachandran from the Department of Chemistry at University College London. To the last of these we extend our special thanks for copious suggestions for improvement, and the large amount of time unsparingly devoted to the meticulous reading and checking of our material. We are indebted to the staff of Wiley-VCH for their invaluable help in production matters and forbearance in awaiting delivery of the final manuscript. Finally we would very much like to thank Decca Records for permission to reproduce on our cover, artwork from the Moody Blues album ‘Every Good Boy Deserves Favour’. Norwich, April 2002

David L. Andrews Philip Allcock

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1

1

Foundations of Molecular Harmonic Emission You fare beyond the fields we know The King of Elfland’s Daughter, Edward Dunsany

The proper description of any optical medium in atomic or molecular terms entails modelling its electronic behaviour under the influence of electromagnetic radiation, generally requiring a full quantum mechanical treatment of the material. There are, however, two distinct approaches to the detailed theoretical representation of optical processes at this level: for the radiation there exists the possibility of a classical description; the other possibility is a quantum mechanical development. The former choice, representing what is usually known as a semiclassical formulation of the theory, has the strength of relating clearly to classical electrodynamics and it is successful within a limited province. The alternative procedure where both matter and radiation are treated quantum mechanically, and which gives legitimacy to the concept of the photon, is known as quantum electrodynamics (QED). Before introducing and justifying the quantum electrodynamical basis for the theory to be delineated subsequently, it is useful to begin with an outline of the corresponding semiclassical formulation. This will provide an opportunity to define terms that are common to both theories, and to draw attention to features in which the two formalisms essentially differ. It is not our intention to defend the semiclassical approach, but to give a brief overview of its methodology.

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

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1 Foundations of Molecular Harmonic Emission

1.1

Classical Optics The starting point for a semiclassical description of optical response is invariably introduction of the concept of the electric vector polarisation P. This represents a relative displacement of positive and negative charges of the medium, on application of an electric field, associated with the induction of an electric dipole moment. As such, it denotes a parameter that is not directly measurable, but in terms of which classically measurable quantities can often be cast (see for example Born and Wolf 1999). Polarisation is a concept that is singularly appropriate for describing interactions with static or spatially homogeneous electric fields, or other fields whose spatial variation is small compared to the scale of charge separation (see below). For bulk media at common electric field strengths the electric polarisation is commonly and most simply described by a constitutive equation which, cast in the SI system to be adopted throughout, is as follows: P ˆ e0 vE ;

…1:1:1†

where E is the applied electric field, v is the scalar electric susceptibility of the medium, and e0 is the vacuum permittivity; the vector polarisation of the medium P thereby represents the induced dipole per unit volume. In the SI system, v is dimensionless and P has units of C m–2. Whilst equation (1.1.1) correctly applies to an isotropic system, any reduction in full symmetry, as for example in an axial crystal, can lead to the electric susceptibility having direction dependence. This is simply a reflection of the fact that the charges within a structured system may be more easily displaced in certain directions than in others, as illustrated in Fig. 1.1. In any such case, the polarisation need not necessarily be induced in the same direction as the field; for generality equation (1.1.1) must be recast in vector form as P ˆ e0 v
E ;

…1:1:2†

with v acquiring the status of a second rank tensor. For the polarisation component in a particular direction, we have

1.1 Classical Optics 3

Fig. 1.1 Vector addition of polarisation components in a 2-dimensional anisotropic medium. The applied electric field E is resolved into Ex and Ey components. In an isotropic medium P is aligned with E.

Pi ˆ

X

e0 vij Ej ;

…1:1:3†

j

where i and j signify Cartesian components x, y or z. Adopting here and henceforth the convention of implied summation over repeated Cartesian indices, equation (1.1.3) can more concisely be written as Pi ˆ e0 vij Ej :

…1:1:4†

Equation (1.1.4) represents the classical linear response of a medium to an applied electric field. In describing interactions with electromagnetic radiation, where the applied field varies sinusoidally with time t, for a given point r within the medium we can write E…r; t† ˆ E0 cos…k
r

xt† ;

…1:1:5†

subject to a phase correction as appropriate. Here k is the wavevector, which points in the direction of propagation and has magnitude k ˆ jkj ˆ nx=c ;

…1:1:6†

also x is the circular frequency given by x ˆ 2pm ˆ 2pc=k ;

…1:1:7†

4

1 Foundations of Molecular Harmonic Emission

where m is the frequency and k the wavelength; n is the corresponding refractive index of the medium. Clearly an electric polarisation given by equations (1.1.2) and (1.1.4) also oscillates with the same frequency as the input. The classical picture of light scattering depicts such a fluctuating dipole as the source of emergent light with the same frequency as the incident light (Rayleigh scattering). Nonetheless it must be borne in mind, in applying (1.1.2) and (1.1.4) to radiative electric fields, that the result is based on a dipolar description of charge distribution that takes no account of any spatial variation in the field. Whereas static fields are spatially homogeneous, or approximately so, the same is certainly not true for the electromagnetic fields described by (1.1.5). Only over regions of physical dimension much smaller than the wavelength is the spatial variation of optical fields negligible. In general, it is an approximation to represent any such field as effecting a simple separation of the centres of positive and negative charge. Although this dipole approximation is very often accurate, it is not invariably appropriate for bulk media at optical frequencies – nor indeed is the polarisation formalism particularly amenable to the incorporation of higher order multipoles for a more detailed depiction of charge distribution. However the dipole approximation underpins essentially the whole of classical optics – and, although higher multipoles will later be considered for completeness, it will prove most convenient to introduce them within the context of the QED treatment in the following section. Having established the classical picture of conventional light scattering, we can now consider nonlinear optical response, i.e. the response classically associated with a polarisation that is no longer exactly proportional to, but also dependent on higher powers of, the applied field. At high field strengths the classical description fails since it provides only for Rayleigh scattering at frequency x (or, through coupling with transitions within the medium, at Raman frequencies x  Dx). Thus the origin of harmonic emission is generally understood through an extension of the theory as follows. Even within the dipole approximation, equation (1.1.4) must be recognised as only an approximation based on the expectation of linear response to the applied electric field. In general, however, the response of a material may be more accurately represented in terms of a power series in E, with the right-

1.1 Classical Optics 5

hand side of (1.1.4) as the leading term. In general we write (Bloembergen 1965): h i …1† …2† …3† Pi ˆ e0 vij Ej ‡ vijk Ej Ek ‡ vijkl Ej Ek El ‡ . . . …1†

…2†

…3†

 Pi ‡ Pi ‡ Pi ‡ . . . ;

…1:1:8†

with v of equation (1.1.2) now designated as the first-order electric susceptibility v(1). The second term in (1.1.8) represents a correction due to quadratic coupling with the electric field through the second-order susceptibility, the third term a cubic response and so on: in SI units v(n) has units of (m/V)n–1. When the electric field is not too large, i.e. under the conditions that commonly apply for non-coherent illumination, the correction terms are negligible and the response is accurately given by the leading linear term. However at the high field strengths provided by intense laser sources, the higher-order terms cannot be ignored. Moreover when several optical fields of different frequency are simultaneously present, the nonlinear polarisation must be viewed as a response induced by a concerted interaction with two or more of them. Here the total electric field at an arbitrary point in a regular solid (or at a particular molecular centre), where r ˆ 0, is correctly represented as the linear superposition E…xn . . . x1 ; t† ˆE0 …x1 † cos x1 t ‡ E0 …x2 † cos x2 t ‡ . . . ‡ E0 …xn † cos xn t :

…1:1:9†

Inserting this expression into equation (1.1.8) reveals the explicit dependence of nonlinear susceptibilities on applied fields. For example, using only the first two terms of equation (1.1.9) in calculating the first nonlinear polarisation represented by equation (1.1.8), we form the sum …2†

…2†

Pi …t† ˆ vijk …x1 ; x1 †E0j …x1 †E0k …x1 † cos2 x1 t …2†

‡ vijk …x1 ; x2 †E0j …x1 †E0k …x2 † cos x1 t cos x2 t …2†

‡ vijk …x2 ; x1 †E0j …x2 †E0k …x1 † cos x2 t cos x1 t …2†

‡ vijk …x2 ; x2 †E0j …x2 †E0k …x2 † cos2 x2 t

…1:1:10†

6

1 Foundations of Molecular Harmonic Emission

upon which use of trigonometric identities immediately reveals a source of both the second harmonics of x1 and x2, alongside their sum and difference frequencies. As such, equation (1.1.10) is regarded as signifying the polarisation source of three-wave interactions (one output with two input optical waves). If however the impinging radiation is entirely monochromatic then, returning to the full response expression of equation (1.1.8) and using only the first term of (1.1.9), we obtain  …1† …2† Pi …t† ˆe0 vij E0j …x† cos xt ‡ 12 vijk Eoj …x†Eok …x†…cos 2xt ‡ 1†  …3† ‡ 14 vijkl E0j …x†Eok …x†E0l …x†…cos 3xt ‡ 3 cos xt† ‡ . . . : …1:1:11† From this it transpires that the quadratic term involving the second-order susceptibility constitutes a source at frequency 2x, higher-order optical harmonics being associated with the following terms in the series. Hence harmonic generation represents a type of light scattering that depends on nonlinear optical response to the electric field of intense (usually laser) radiation. In addressing the origin of nonlinear optical response at the molecular level, as befits the photonic formulation to be developed later, it is first helpful to establish representations at three different levels of scale. The bulk formulation introduced above is clearly relevant to a macroscopic description, and is valid over distances that substantially exceed the scale of any local field variations within the system. Reformulation at the molecular scale is clearly necessary to accommodate the properties and internal electronic structures of the constituent atoms and molecules. However a third, intermediate scale is also commonly introduced, classically termed microscopic (but which might reasonably be termed mesoscopic to avoid confusion with the molecular scale), which addresses electromagnetic fields locally modified by their electronic environment. This scale, within which internal molecular structures are not resolved, extends from a threshold established by the molecular size to a limit determined by the extent of field variations. To proceed, it is appropriate to focus on the molecular scale, and specifically on the interplay of optical fields at sites of discrete electronic integrity – whether atoms, molecules or chromo-

1.1 Classical Optics 7

phore units within molecules. To begin, the counterpart to equation (1.1.8) which denotes a semiclassical charge displacement in molecular terms is as follows: h i 1 2 3 ˆ e a d ‡ e b d d ‡ e c d d d ‡ . . . ; lind ij j ijk j k i 0 0 0 ijkl j k l

…1:1:12†

cast in components of d, the local displacement field (see below). In equation (1.1.12) lind is an induced molecular dipole moment, a is the molecular polarisability (J m2 V–2), b the hyperpolarisability (J m3 V–3), and c the second hyperpolarisability (J m4 V–4) etc. Significant differences from the macroscopic formulation to note at this point are: firstly, non-centrosymmetric molecules may additionally possess an intrinsic dipole moment in the absence of any applied field, so that the total molecular dipole should be represented as l ˆ l0 ‡ lind ;

…1:1:13†

and secondly, the power series in equation (1.1.12) is expressed not in terms of E but d, where d is the microscopic electric displacement field. This is the local electric field experienced by each molecule, as modified by the polarisation field due to the electrical influence of neighbouring molecules (in general, the surrounding medium): d ˆ e0 e ‡ p :

…1:1:14†

Here the use of lower-case symbols signifies optical response at the microscopic level. Equation (1.1.12) is thus consistent with a description that obviates corrections associated with the influence of any individual molecule upon itself. The local average of the applied electric field e can be identified exactly with the macroscopic field E and the local polarisation p in any isotropic or cubic medium is related to the macroscopic polarisation P by p ˆ P=3, a relation typically derived by considering the charge density on an imaginary sphere about a given centre (Jackson 1999). Using spherical coordinates …r; h; u†, the field at the centre of such a sphere, associated with the polarisation-induced charge density over a surface element of area r 2 sin hdhdu, is P cos h=r 2 .

8

1 Foundations of Molecular Harmonic Emission

The mean local polarisation p is given by the corresponding spherical average of P over h and u, i.e. 1 pˆ 4p

Z2p Zp P cos2 h sin hdhdu ˆ 0

P ; 3

…1:1:15†

0

leading directly to a representation of the local microscopic field through the following material equation d ˆ e0 e ‡

P : 3

…1:1:16†

It is worth noting that, because the polarisation is embodied within the displacement field, it is not possible to directly equate terms of the same order in the macroscopic and molecular scale expansions of equations (1.1.8) and (1.1.12). A procedure for correctly disentangling the intricate relationships between microscopic and macroscopic response in classical terms has been described (Bedeaux and Bloembergen 1973). Some of the key issues that arise in relating bulk optical susceptibilities to microscopic counterparts are outlined below. To consider in detail the relationships represented by equations (1.1.8) and (1.1.12) demands a construction for the molecular origins of the bulk response. To this end we begin with the following relationship, expressing the net polarisation is the ensemble sum of all molecular contributions; Pi ˆ hN…r; Rn †lind i …n†i ;

…1:1:17†

where N is the mean number of molecules per unit volume. Then equation (1.1.12) leads to the following  Pi …x† ˆ N e0 1 aij …x†he0 ej …x† ‡ pj …x†i ‡ e0 2 bijk he0 ej …x†  ‡ pj …x†ihe0 ek …x† ‡ pk …x†i ‡ . . . 
ˆ N e0 1 aij …x† e0 Ej …x† ‡ 13 Pj …x† ‡ e0 2 bijk eo Ej …x†

 …1:1:18† ‡ 13 Pj …x† e0 Ek …x† ‡ 13 Pk …x† ‡ . . . The difficulty with this result is that it is recursive; for practical purposes it needs to be re-expressed exclusively in terms of the

1.1 Classical Optics 9

applied electric field. Any approximation based on neglect of high-order powers cannot in itself help, since the aim is to compare the structure of each successive power in equations (1.1.8) and (1.1.12). A solution to the problem can be obtained by iterating equation (1.1.18) and collecting powers of E. Such an approach is helpful providing the resulting series converges, which in turn depends on the validity of the inequality e0 e > p. As emerges below, this is indeed a reasonable constraint for molecular systems where electrical properties of the bulk are interpretable in terms of local microscopic response. From the zeroth iteration, we have Pi …x†  Naij …x†Ej …x† ;

…1:1:19†

and the inequality should be satisfied provided 1 1 3Ne0 a…x†

1:

…1:1:20†

This latter condition is invariably satisfied in gases, since at electronic frequencies the ‘polarisability volume’ …a=4pe0 † is far smaller than the mean volume per molecule, N 1 . In the condensed phase where mean molecular volumes are very much smaller, the polarisability volume may more closely approach the value of N 1 , but the value of 13 Ne0 1 a…x† generally remains less than unity. Exceptions to this may be expected in regions of high dispersion (at optical frequencies within an electronic absorption band), as will become apparent on considering the detailed structure of the polarisability in chapter 4. Performing the necessary iteration of (1.1.18) produces the following results. Collecting all terms linear in the macroscopic field E, and comparing with the corresponding term in (1.1.8), leads to the relation …1†

e0 vij …x†Ej …x† ˆ Naij …x†Ej …x† ‡ 13 N 2 e0 1 aik …x†akj …x†Ej …x† ‡ 19 N 3 e0 2 ail …x†alk …x†akj …x†Ej …x† ‡ . . . : …1:1:21† In the above equation the first order bulk susceptibility is represented as a second rank tensor. When the surrounding medium is significantly anisotropic, such a tensor construction should be

10

1 Foundations of Molecular Harmonic Emission

supported throughout the ensuing development. However, notwithstanding the intrinsic or site symmetry of individual molecules responsible for the optical response, it is more common to consider their molecular surroundings as electrically isotropic. This obviates undue complexity and allows representation of the polarisability as a scalar through ha…x†i ˆ 13 dij aij …x† (for properties of the Kronecker delta, dij, see Appendix 3), signifying the rotational invariant of the second rank tensor aij. The series (1.1.21) is readily summed and gives the following exact result  v…1† …x† ˆ Ne0 1 ha…x†i 1

1 1 3Ne0 ha…x†i



1

:

…1:1:22†

If the linear dielectric response (the relative electrical permittivity of the medium) is defined by j…x† ˆ 1 ‡ v…1† …x† ;

…1:1:23†

then substitution into equation (1.1.22) reveals the equality 

1



1 1 3 Ne0 ha…x†i

1

ˆ 13 …j…x† ‡ 2† :

…1:1:24†

When associated with optical frequencies, it is convenient to relate the linear dielectric response to a frequency-dependent refractive index, n…x†, where n2 …x† < j…x†. Equation (1.1.22) then reveals how the right hand side of equation (1.1.24), commonly known as a Lorentz factor, acts as a correction to the bulk susceptibility. The Lorentz factor clearly reduces to unity in vacuo. We can also write in terms of the refractive index: Ne0 1 ha…x†i n2 …x† 1 ˆ 2 ; 3 n …x† ‡ 2

…1:1:25†

a form of result usually known as the Lorentz-Lorenz equation. As it is the nonlinear response that is our ultimate focus it is appropriate to generate higher-order terms. Following a similar procedure for terms quadratic in e (E), the iteration of (1.1.21) results in a series that again can be summed exactly, and we obtain (Shen 1984)

1.1 Classical Optics 11

 v…2† …x1 ; x2 † ˆ 1





1 1 3 Ne0 ha…x†i

 1

1

hb…x1 ; x2 †i  1  1 1 1 1 13 Ne0 1 ha…x2 †i : 3Ne0 ha…x1 †i …1:1:26†

Here, by extension of the methods used for linear response, we take the rotational invariant of the third rank tensor representing the hyperpolarisability, hbi ˆ 16 eijk bijk , in order to give a scalar representation (the defining properties of the Levi-Civita antisymmetric tensor, eijk, are given in appendix 3). Using equation (1.1.24) we thus determine the following relation: v…2† …x1 ; x2 † ˆ 13hn2 …x† ‡ 2ihb…x1 ; x2 †i13…n2 …x1 † ‡ 2†13…n2 …x2 †‡2†; …1:1:27† where once more Lorentz factors express the homogeneous dielectric influence on the higher-order susceptibility. Working within the electric dipole approximation, further application of the methods outlined above enables corresponding relations between higher order bulk and molecular response tensors to be correctly obtained in a similar fashion. The structure of the third rank result, (1.1.27), suggests the pattern for higher orders, in which nonlinear susceptibilities emerge from their molecular counterparts through correction by a product of Lorentz factors for each frequency involved in the process. However, the results for higher ranks include additional cascading terms involving products of lower-order nonlinear susceptibilities (Armstrong et al. 1962, Flytzanis 1979). Because of their complexity, it is commonplace to find these additional terms neglected, mostly without rigorous justification. A useful and critical discussion of the shortcomings of the molecular model of macroscopic polarisation has been provided (Meredith et al. 1983). For anisotropic media the polarisabilities of the individual molecules, and consequently also the bulk susceptibilities, necessarily take on a more complicated tensorial nature.

12

1 Foundations of Molecular Harmonic Emission

1.2

Quantum Electrodynamics To properly develop the photonic or quantum optical response of materials invites the application of quantum electrodynamics (QED). The defining characteristic of this theory is that it addresses every optical interaction in terms of a closed dynamical system where light and matter are treated on an equal footing, each component addressed with full quantum mechanical rigor. It is a theory whose predictions have been tested to a higher degree of precision than any other in modern physics, and which remains unchallenged by the most sophisticated experimental measurements (Kinoshita 1990). Even in the non-covariant form commonly employed for dealing with the optical interactions of conventional matter, with charges moving at non-relativistic speeds, QED accommodates important retardation features associated with the finite time of signal propagation. The success of QED in leading to the correct form of the Casimir-Polder interaction between atoms, for example, owes its origin to this intrinsic property of its formulation (Power 1964, Cohen-Tannoudji et al. 1989, Milonni 1994, Compagno et al. 1995). Indeed, it has recently been shown that even the application of properly retarded classical electrodynamics produces results of significantly different form (Barnett et al. 2000). In the subjects to be described below, retardation effects are not specifically at issue – and the advantages of a QED foundation, which we shall highlight, are entirely independent of such features. The reader may find another body of work on resonance energy transfer and cooperative absorption, in which we have described several processes where retardation is a highly significant factor. The primary references to such work can be found elsewhere in reviews of that subject area (Andrews and Allcock 1995, Juzeliu¯nas and Andrews 2000). Not surprisingly, the semiclassical theory is inconsistent with the general principles of quantum optics, allowing for example the detection of a single photon by two different detectors (Milonni 1984). Also, the semiclassical invocation of an electric polarisation as the oscillating moment of a radiating dipole, coupling with the electric field vector of the emergent radiation, generally casts the signal amplitude in the form of a sum of contributions associated with physically distinct processes – as for ex-

1.2 Quantum Electrodynamics 13

ample in equation (1.1.10) – when it is a fundamental violation of the superposition principle to sum the amplitudes of transitions between non-identical sets of initial and final radiation states. Again, even when the semiclassical polarisation formalism is extended to accommodate electric multipoles, it does not allow proper incorporation of magnetic and diamagnetic interactions. For example, in a general three-wave interaction mediated by a species whose symmetry supports E12M1 (i.e. two electric dipole, one magnetic dipole) but not E13 couplings, the magnetic dipole interaction in the former can be associated with each of the three waves, yet for obvious reasons only two are accommodated by an electric polarisation. It has indeed been remarked that outside of QED there is no formal basis for establishing the gauge transformations which underpin the multipolar description of optical interactions (Woolley 1999, 2000). The definitive molecular formulation of quantum electrodynamics established by Power (1964) and further delineated by Craig and Thirunamachandran (1984) forms the primary basis for the theory developed below (see also Dalton et al. 1996). This framework enables direct calculation of the tensor parameters involved in linear and nonlinear optical interactions, whose detailed structure naturally emerges from the derivation of results for the observable signal intensities. The starting point for such calculations is the quantum electrodynamical Hamiltonian for the dynamical system, wherein matter is conventionally described in terms of individual components with distinct electronic integrity and overall electrical neutrality. In the following we cast theory in a form suitable to address any condensed phase system of independent atoms or molecules, for example liquids, solutions, molecular crystals, or even mesoscopically more intricate structures such as membranes. The theory can also be applied to sub-units such as ions or chromophores – assuming that it is transitions in these which dominate the optical response of the medium, so that each ion or chromophore can be treated as the optical representative of a local environment which is itself electrically neutral. For simplicity the term ‘molecules’ will now be used as an umbrella term for any such distinct optical units, each identified by a label n. In multipolar form the system Hamiltonian may be represented thus:

14

1 Foundations of Molecular Harmonic Emission

H ˆ Hrad ‡

X n

Hmol …n† ‡

X

Hint …n† :

…1:2:1†

n

Here Hrad is the Hamiltonian for the radiation field in vacuo, Hmol(n) the field-free Hamiltonian for molecule n, and Hint(n) is a term representing molecular interaction with the radiation. It is worth emphasising that the tripartite simplicity of equation (1.2.1) specifically results from adoption of the multipolar form of light-matter interaction. This is based on a well-known canonical transformation from the minimal-coupling interaction (Göppert-Mayer 1931, Power and Zienau 1959, Woolley 1971, Babiker et al. 1974, Power and Thirunamachandran 1980). The procedure results in precise cancellation from the system Hamiltonian of all Coulombic terms, save those intrinsic to the internal structure of the Hamiltonian operators for the component molecules; hence no terms involving intermolecular interactions appear in (1.2.1). An important implication of developing theory from the quantum electrodynamical Hamiltonian is that neither the eigenstates of Hrad nor those of Hmol(n) are stationary states for the system described by it. Thus the presence of the radiation field modifies the form of the molecular wavefunctions, and equally the presence of matter modifies the form of the radiation wavefunctions. Since the Hamiltonian remains the same irrespective of the state of the system, then even when no light is present the coupling still effects a modification of the molecular wavefunctions. This is, for example, manifest in the occurrence of spontaneous emission (luminescence) from isolated molecules in excited states, the lifting of degeneracy between the 22S1/2 and 22P1/2 states of atomic hydrogen (the Lamb shift), also the Casimir force between conducting plates, and yet again the corrections responsible for what was once considered the ‘anomalous’ magnetic moment of the electron (Milonni 1994). We now consider the detailed nature of the terms in the QED Hamiltonian. The simplest to deal with is the middle term, which denotes a sum of the normal non-relativistic Schrödinger operators Hmol(n) for each molecule, the operator counterparts of their classical energies. These need no further elaboration. The radiation field term Hrad is the operator equivalent of the classical expression for electromagnetic energy – which, recalling the

1.2 Quantum Electrodynamics 15

relation c 2 ˆ 1=…l0 e0 † between the vacuum electric susceptibility e0 and magnetic permeability l0, is expressible as follows in terms of the transverse microscopic electric displacement operator d? …r† and corresponding magnetic induction field operator b…r†: Z  1  1 ?2 e0 d …r† ‡ l0 1 b2 …r† d3 r : …1:2:2† Hrad ˆ 2 In source-free regions d? …r† is related to the fundamental transverse electric field operator e? …r† through d? …r† ˆ e0 e? …r†; corrections due to the presence of charged matter are introduced in section 1.3. The superscript on the electric field operator designates its transverse character with respect to the direction of propagation, redundant in the case of the magnetic field as Maxwell’s equations ensure that the divergence of the magnetic field is equal to zero and therefore the field is intrinsically transverse. To further develop and also to elucidate important properties of the above electromagnetic field operators it is convenient to introduce a vector potential, one that acts as a solution to Maxwell’s equations, and through which it is possible to encapsulate both electric and magnetic field properties in a single expression.1) We concentrate first on the second-quantised form of this vector potential, cast in terms of a summation over radiation modes as follows: ?

a …r† ˆ

X k;k

 h 2Vxe0

1=2 h …k† …k† …k† y…k† ek ak eik
r ‡ ek ak e

ik
r

i

:

…1:2:3†

…k†

Here V denotes the quantisation volume, and ek is the unit polarisation vector for the radiation mode characterised by wave-vector k, with circular frequency x ˆ cjkj, and polarisation k (a label for circular, plane etc.); where it appears, an overbar denotes complex conjugation. The polarisation vector is considered a complex quantity specifically to entertain the possibility of circular or elliptical polarisations. Associated with each mode …k; k† are an adjoint pair …k† y…k† of photon annihilation and creation operators, ak and ak respec1) For explicit details on Maxwell’s equations and the vector potential, specifically

in relation to non-relativistic QED, the reader is referred to Craig and Thirunamachandran (1984).

16

1 Foundations of Molecular Harmonic Emission

tively. These operate upon eigenstates of Hrad with q…k; k† photons (q being the mode occupation number) as follows; …k†

ak jq…k; k†i ˆ q1=2 j…q

1†…k; k†i ;

…1:2:4†

y…k†

ak jq…k; k†i ˆ …q ‡ 1†1=2 j…q ‡ 1†…k; k†i ;

…1:2:5†

reducing the number of …k; k† photons by one in the former case and increasing it by one in the latter. Vacuum energy (see below) is associated with the lack of commutativity of the creation and annihilation operators for any given radiation mode. Specifically, the commutation properties are as follows: h i …k† …k0 † ak ; ak0 ˆ 0 ;

…1:2:6†

h i y…k† y…k0 † ˆ0; ak ; ak0

…1:2:7†

h i …k† y…k0 † ˆ dk;k0 dk;k0 : ak ; ak0

…1:2:8†

We note in passing that the creation and annihilation operators are not form-invariant – by which is meant that although the same symbols are used in connection with field expansions in the minimal coupling formalism, the operators themselves differ from those employed for multipolar coupling, as the radiation states on which they operate also differ when matter is present (Power and Thirunamachandran 1999). It will repay effort to examine the symmetry properties of the vector potential and consequently those of the e? …r† and b…r† fields respectively. The vector potential is self-evidently hermitian, as befits the status of the field it represents. Its parity with respect to space-inversion is odd, since the space inversion operator I reverses the sign of r, e and k. Its character with respect to time-inversion T, also of interest, is less self-evident. Leaving the creation and annihilation operators invariant this operation gives ?

a …r† ! T

X k;k

h  2Vxe0

1=2 h …k† …k† e k ak e

i… k
r†

…k† y…k†

‡ e k ak ei…

k
r†

i …1:2:9†

1.2 Quantum Electrodynamics 17

since T reverses the sign of k and induces complex conjugation. …k† …k† Then, using the relation e k ˆ ek (McKenzie and Stedman 1979, Naguleswaran 1998) we obtain the result that a? …r† is also of odd parity in time. Now using the source-free result: e…r† ˆ

@a…r† ; @t

…1:2:10†

implemented in the interaction picture where time features explicitly – compare with the later equations (2.1.10)–(2.1.12) – we obtain the following expression for the electric field operator:  X h x 1=2 h …k† …k† ik
r ek ak e e …r† ˆ i 2Ve0 k;k ?

…k† y…k†

ek ak e

ik
r

i

:

…1:2:11†

Equally, taking the curl of equation (1.2.3), as represented explicitly by b…r† ˆ r  a…r†

…1:2:12†

we have a magnetic field given by b…r† ˆ i

 X hxl 1=2 h 0

2V

k;k

…k† …k†

bk ak eik
r

…k† y…k†

bk ak e

ik
r

i

…1:2:13†

…k†

where the complex unit vector bk is defined as the vector cross product of the unit wave-vector and the electric field polarisation vector, such that ^e : bk ˆ k k …k†

…k†

…1:2:14†

Again, both the electric and magnetic fields are obviously of hermitian character. What also emerges from their derivation through equations (1.2.3), (1.2.10) and (1.2.12) is that the electric field operator is of odd parity with respect to space, and even parity with respect to time; the magnetic field operator is of even parity with respect to space and odd with respect to time. Employing the above field operator expansions enables the radiation Hamiltonian (1.2.2) to be recast in a form that more readily identifies its quantum properties, explicitly featuring the photon

18

1 Foundations of Molecular Harmonic Emission

creation and annihilation operators. Each mode contributes to the y…k† …k† …k† y…k† Hamiltonian an operator expressible as 12 …ak ak ‡ ak ak †hx, and using the commutation relation (1.2.6) we thus obtain:  X y…k† …k† hx : …1:2:15† ak ak ‡ 12  Hrad ˆ k;k

The 12  hx associated with each radiation mode, resulting from the non-commutativity of the creation and annihilation operators, is the energy associated with the familiar vacuum fluctuations – the origin of spontaneous emission, self-energy corrections and Casimir interactions (Casimir 1948). The eigenstates jq…k; k†i of Hrad are number states; states which more closely model the coherence and other properties of laser light will be introduced later. To complete the definitions of the terms in equation (1.2.1), an expression for the multipolar interaction Hamiltonian Hint(n) can be written in its entirety as follows: Z Z m…n; r†
b…r†d3 r Hint …n† ˆ e0 1 p? …n; r†
d? …r†d3 r ZZ Oij …n; r; r0 †bi …r†bj …r0 †d3 r0 d3 r0 ; …1:2:16† ‡ 12 where p? …n; r†, m…n; r† and O…n; r; r0 † are the operators for the transverse electric polarisation vector field, the magnetisation vector field and the diamagnetisation tensor field, respectively, associated with molecule n. The above expression is exact and complete. For practical implementation, however, each of the infinite domain fields p? …n; r†, m…n; r† and O…n; r; r0 † is more usefully cast in terms of a locally defined multipolar expansion (see for example Babiker et al. 1974, Power and Thirunamachandran 1980), leading to an infinite series of interaction terms. In linear response to the electromagnetic fields the leading terms are as follows; Hint …n† ˆ

e0 1 l…n†
d? …Rn † m…n†
b…Rn †

e0 1 Qij …n†ri d? j …Rn †

... :

…1:2:17†

Here l…n† is the electric dipole (E1) operator for molecule n located at position Rn, Qij(n) is the corresponding electric quadru-

1.2 Quantum Electrodynamics 19

pole (E2) operator, and m(n) the magnetic dipole (M1) operator. To this order of approximation, the quadratic response associated with the diamagnetisation does not contribute (vide infra). We also recognise in equations (1.2.16) and (1.2.17) the microscopic transverse displacement electric field, d? , whose quantum operator form will be discussed in the next section. Explicit expressions for the components of the leading molecular multipoles are as follows: X eg…n† …qg…n† Rn †i ; …1:2:18† li …n† ˆ g…n†

Qij …n† ˆ 12

X g…n†

h eg…n† …qg…n†

Rn †i …qg…n†

Rn †j

1 3jqg…n†

i Rn j2 dij ; …1:2:19†

mi …n† ˆ 12

X g…n†

eg…n† ‰…qg…n†

Rn †  q_ g…n† Ši ;

…1:2:20†

where summations are taken over each constituent particle g…n† of charge eg and for which qg…n† is the position vector 2). In passing it may be noted that the employment of a traceless form for the electric quadrupole and higher-order multipoles is consistent with the divergence-free character of the electric displacement field upon which the gradient operator, r, acts in equation (1.2.17). In general, each electric multipole (En) is time-even and carries a (–1)n signature for space inversion; the corresponding magnetic multipole (Mn) is time-odd and has (–1)n–1 space parity. Hence the time-even, space-even nature of Hint is secure. The electric dipole term in (1.2.17) normally represents the strongest coupling between matter and radiation and is sufficient for the majority of cases, in which electronic excitations are restricted to molecular regions significantly smaller than the wavelengths of the radiation engaged. The electric quadrupole and magnetic dipole terms together are then smaller by a factor typically of the order of the fine structure constant a ˆ 1=137. Here a word of caution is necessary, however. Such guides to the size 2) Note that magnetic multipole operators are formally cast in terms of position

and momentum operators, rather than position and velocity as in (1.2.20). In this sense the velocities q_ g…n† should be interpreted as mass-corrected momenta.

20

1 Foundations of Molecular Harmonic Emission

of effects represented by higher multipoles have to be interpreted with regard to transitions that are in all respects allowed. Nonetheless for molecules of sufficiently high symmetry many transitions prove to be forbidden by electric dipole coupling, so that other electric or magnetic multipoles may represent the leading order of interaction. So, although in many quantum optical calculations the detailed multipolar form of the coupling with matter is deemed largely irrelevant, the spatial and temporal symmetries may crucially depend on the multipoles involved, and so too the magnitudes of the corresponding coupling constants. This is a theme to which we shall return in chapter 4.

1.3

Media Corrections The quantum field description delineated above is cast in a form best suited for applications in which the material part of the system comprises only individual molecules or optical centres directly involved in the interactions of interest, with no other matter present. More generally in condensed phase materials, such centres are surrounded by other atoms or molecules whose electronic properties modify the fields experienced (and produced) by those optical centres. Such influences are accommodated within the operator d?, the microscopic electric displacement field. As designated, this field is of fully transverse character as a direct consequence of working in the multipolar formalism. In mattercontaining regions, d? is related to the fundamental transverse electric and microscopic polarisation fields, e? and p? respectively, by the operator equation d? ˆ e0 e? ‡ p? :

…1:3:1†

At this stage the molecular and optical properties are neatly entwined. Despite the formal similarity with its semiclassical analogue, (1.1.14), the above equation represents a point of departure from the subsequent development of that formalism. In section 1.1 the development of theory observed the common practice of the semiclassical formalism to incorporate all the ensuing mate-

1.3 Media Corrections 21

rial-induced (Lorentz) field corrections within the optical susceptibilities. In using quantum field theory, and considering all interactions to occur through the exchange of transverse photons, it is neither necessary nor desirable to modify the corresponding molecular property tensors, as the field operators are reformulated to take full account of the light propagation environment. When all matter-induced corrections are carried with the fields, the appropriately modified operators automatically accommodate the media effects. To understand how this reformulation is achieved, it is necessary to establish the local fields within a quantum electrodynamical context. As seen above, the nature of media effects relates to the fact that, since the microscopic displacement is the net field to which molecules of the medium are exposed, it corresponds to a fundamental electric field dynamically dressed by interaction with the surroundings. Magnetic fields are also modified, though in a less obvious manner. The propagation and interactions of quantised radiation in a system where such media effects are engaged are in consequence properly described in terms of ‘dressed photons’ or polaritons, for which there is again no semiclassical counterpart. The theory of dressed optical interactions, using non-covariant molecular quantum electrodynamics (Knoester and Mukamel 1989, Juzeliu¯nas and Andrews 1994, Milonni 1995, Juzeliu¯nas 1996), signalled the achievement of a rigorous theory designed to accommodate media effects in condensed phase applications. In the present context, deployment of this theory leads to modified operators for the quantum auxiliary fields d? and h? fully accounting for the influence of the medium – the fundamental fields of course remain unchanged. Explicitly position-dependent expressions for the local displacement electric and auxiliary magnetic field operators (Juzeliu¯nas 1996), correct for all microscopic interactions, are as follows: ?

d …r† ˆi

…m† X  hm…m† g xk e0 k;k;m

!1=2

…m†

2cVn…xk †

…k† …k†

‰ek Pk;m eik
r

…k† y…k†

! …m† n2 …xk † ‡ 2 3

ek Pk;m e

ik
r

Š

…1:3:2†

22

1 Foundations of Molecular Harmonic Emission

! …m† …m† …m†
1=2 X h  m x n x g k k h? …r† ˆi cV 2l 0 k;k;m h i …k† y…k† …k† …k†  bk Pk;m eik
r bk Pk;m e ik
r

…1:3:3†

where l0 is the  magnetic permeability of the vacuum  …m† is the refractive index for the polari…l0 ˆ 1=e0 c 2 † and n xk …m† ton frequency xk as defined below. To appreciate these expressions for the new auxiliary field operators, it is expedient to dwell briefly on their key features and elucidate the new symbols appearing in the above equations. Compared with the mode expansions of their fundamental field counterparts, equations (1.2.11) and (1.2.13), the most obvious difference apparent in equations (1.3.2) and (1.3.3) relates to the introduction here of additional summations over m. This index labels the branches of polariton dispersion and runs from m ˆ 1; 2; . . . M, where M ˆ Mmol ‡ 1 and Mmol is the number of molecular response frequencies. For example in a two-level molecular system characterised by a single transition frequency (the Hopfield model) there are two branches to the dispersion curve. However no limit is imposed on the number of response frequencies, so that the polariton description extends to the quasicontinua associated with vibrational, rotational and other electronic state manifolds. In the mode expansions (1.3.2) and (1.3.3) the summations over k extend to k  2p=a, where a is a characteristic intermolecular separation. Through this restriction, the auxiliary operators are properly invoked only when dealing with the propagation and interactions in condensed media of infrared, visible or ultraviolet light – where a description in terms of refractive index is entirely legitimate. Nonetheless the theory properly accommodates not only transparent but also dispersive regions where the polariton wave-vector and frequency are not linearly related, signifying resonant or near-resonant optical response. It also affords a means for the representation of photonic band-gap materials. Figure 1.2 illustrates the photonic and exciton-like regions for conventional two-, three- and multi-level systems. The index m which identifies each of the dispersion branches in the general case has to be incorporated in the definition of the polariton frequency, as given by:

Frequency

1.3 Media Corrections 23

(a)

(b) Fig. 1.2 Schematic representation of polariton dispersion: (a) illustrates the dispersion if only a single molecular frequency is present (Hopfield model); (b) represents two molecular resonances.

1 Foundations of Molecular Harmonic Emission

Frequency

24

(c) Fig. 1.2 (c) numerous resonances.

…m†

xk

ck  ˆ  …m† n xk

…1:3:4†

where several normal frequencies are associated with each value of k, again as evident in Fig. 1.2. The mode expansions (1.3.2) and (1.3.3) also feature polariton annihilation and creation opera…k† y…k† tors, Pk;m and Pk;m respectively, with similar properties to their vacuum counterparts of equations (1.2.4) and (1.2.5). Finally, we …m† introduce the group velocity mg , defined for each specific polariton mode as m…m† g

ˆc

8 …m†  …m† 9 > > > > > < > > > > > > > :

p p ; 0; 2  4p p ; 0; 4 p 2 ; 0; 0 2 …0; 0; 0†

R

circular polarisation;

L

circular polarisation;

p

plane polarisation;

s

plane polarisation: …3:1:11†

44

3 Radiation Constructs

All polarisations given by equation (3.1.11) are normalised in the sense …e
e† ˆ 1; note that …e
e† is unity only for plane polarisations.

3.2

Quantum Optical Considerations In order to relate rate equations (3.1.5) to the optical conditions delivered by a given laser source, it is clearly desirable to obtain results cast in terms of physically meaningful radiation parameters, in lieu of the quantisation volume V and photon number q which feature in equation (3.1.6). The procedure for this reformulation allows consideration of pump radiation states characterised by various forms of photon statistics, leading to results appropriate for several different kinds of intensity distribution. The number states jq…k; k†i hitherto employed in the general formulation are the usual basis for the primary development of quantum electrodynamical calculations based on time-dependent perturbation theory. As eigenstates of the unperturbed radiation Hamiltonian, they represent a radiation field for which there is a precise non-fluctuating value for the number of photons. However such states do not represent laser input. One basis rather better suited to the modelling of laser radiation is the over-complete set comprised by the coherent states 3) ja…k; k†i. These, characterised for any given radiation mode by minimisation of the uncertainty in phase and occupation number (Louisell 1973, Loudon 2000), are eigenstates of the corresponding annihilation operators, satisfying the equation …k†

ak ja…k; k†i ˆ a…k; k†ja…k; k†i ;

…3:2:1†

where a…k; k† is a complex number whose modulus relates to the mean photon number hqi through y…k† …k†

hqi ˆ ha…k; k†jak ak ja…k; k†i ˆ ja…k; k†j2 :

…3:2:2†

3) For a given radiation mode, the coherent states are expressible in terms of the

corre sponding number states through jai ˆ exp

2
1 2 jaj

1 n P a

nˆ0

n!

jni.

3.2 Quantum Optical Considerations 45

Thus the quantum amplitude for a process such as n-harmonic emission, involving the annihilation of n photons from a single beam, acquires a factor of an where a coherent state is the input – in place of the fq…q 1† . . . …q n ‡ 1†g1=2 factor which arises if a number state with q photons is employed for the calculation; (see equation (1.2.4)). For the coherent input the corresponding rate factor is then fn …q† ˆ jaj2n ˆ hqin , compared to fn …q† ˆ q!=…q n†! for the number state input. It should be mentioned that the employment of coherent (or other) states requires caution since they are not eigenstates of the radiation Hamiltonian. In consequence coherent states are invariant neither to photon creation followed by annihilation nor annihilation followed by creation, and this feature is commonly overlooked. Nonetheless, in the absence of perturbations the time-evolution of a coherent state ja…k; k†i is representable as a coherent state whose a value is modified by a time-dependent phase-shift (Carmichael 1999). A different perspective is obtained by considering the photon number to be subject to fluctuations that satisfy particular types of statistical distribution (Mandel and Wolf, 1995). Suitably weighting rate equations calculated on the basis of number states allows various kinds of radiation to be modelled. Thus, for a radiation mode with mean occupation number q, for any distribution Pq …q† (defined as the time-averaged probability of finding q photons in the quantisation volume) we have q!=…q

n†! ! fn …q† ˆ

1 X

Pq …q†q!=…q

n†! :

…3:2:3†

qˆ0

For number states Pq …q† ˆ dq;q , and the result q!=…q n†! is recovered. For coherent radiation the appropriate form of photon distribution function is a Poisson curve, expressible in terms of its mean q by the relation q

Pq …q† ˆ

e qq : q!

…3:2:4†

Substitution of this result into equation (3.2.3) readily reproduces the result fn …q† ˆ qn . One further case of interest, largely from a theoretical viewpoint, is thermal or chaotic radiation, which satis-

46

3 Radiation Constructs

fies a Bose-Einstein distribution. Such radiation has the distribution function Pq …q† ˆ

qq …q ‡ 1†…q‡1†

:

…3:2:5†

In this case substitution in (3.2.3) leads to the result fn …q† ˆ n!qn . A convenient generalisation of the above is fn …q† ˆ g …n† qn ;

…3:2:6†

where g(n) is the degree of nth order coherence. For coherent light this parameter takes the value of unity for all n; for thermal light g …n† ˆ n!, and other values typify different kinds of photon distribution. The fact that g …1† ˆ 1 for all types of radiation serves as a reminder that conventional optical processes which involve the absorption or scattering of photons singly are uniquely insensitive to photon statistics, depending only on mean photon flux. Rate equations expressed in terms of quantisation volume and photon number, whether precise or fluctuating, are not directly amenable to experimental interpretation. Moreover since the quantisation volume is simply a theoretical artefact, it must invariably cancel out in any final result. However the ratio of mean photon number and quantisation volume, which represents a mean photon density, relates to the directly measurable mean irradiance. Consider a quantisation volume represented by a cube of space of side length l and volume l3 …l3  V† through which the beam passes; the cube contains on average an energy qhxk (see Fig. 3.1). This energy traverses the cube in a time l=mg (where mg is the group velocity of the photon in the medium); hence the mean irradiance I…xk † (power per unit beam cross-sectional area) is fq hxk =…l=mg †g=l2 , so that; I…xk † ˆ

q hmg xk : V

…3:2:7†

In passing we also note that the mean interval s between photon arrival times for any one molecule of physical cross-section r is directly related to I…xk † through

3.2 Quantum Optical Considerations 47

Fig. 3.1 Photon flux through a quantisation volume l3.

sˆ

hxk  : I…xk †r

…3:2:8†

In applications to non-parametric excitation and decay processes in molecular media the value of the parameter s relative to the decay lifetime affords a useful gauge of excitation efficiency. From (3.2.3), (3.2.6) and (3.2.7) we can deduce the following algorithm for replacement of a quantum electrodynamical rate factor based on number states by a more general parameter cast in terms of mean irradiance and degree of coherence;  …q

q!

  I…xk †V n …n† gxk : ! n†! hmg xk  

3:2:9†

Any other V factor will cancel out from the rate equation once the emergent radiation is also cast in measurable parameters. For the generation of radiation through any incoherent optical process, the general lack of constraint over the propagation direction for the emergent radiation indicates that each photon is created into any one of an infinite set of radiation states, subject to energy conservation. Even in coherent processes, whose wave-vector matching nature defines the principle direction of the emergent radiation and where the initial and final molecular states are necessarily identical, the general theory leads to quantum amplitudes in which the final state of the radiation field is not completely specified. As such, sums over all possible values of k0 and k0 should remain in the amplitudes of the radiation tensor q0 . However, the restrictions imposed on parametric processes by virtue of energy conservation and wave-vector matching condi-

48

3 Radiation Constructs

tions ensures that radiation is emitted into a small pencil of solid angle centered around k0 ‡ dk0 , where the prime indicates the emitted photon. In effecting the associated sums we achieve a form of result that correctly loses dependence on the quantisation volume (that is, except for processes occurring in geometrically confined microcavities where the quantisation volume retains physical significance). In the limit of a large quantisation volume the sum over k0 is conveniently replaced by an integral of the form 1X ) lim V!1 V 0

Z

d3 k0 …2p†3

k

ˆ

1 …2p†3

Z1 I

k02 dk0 dX ;

…3:2:10†

0

where the solid angle dX extends over all directions. For coherent emission into a pencil of solid angle dX ( 4 p steradians) centered around k0 it is legitimate to substitute for the sum over k0 by; 1X dX ) V 0 …2p†3

Z1

k

k02 dk0 ;

…3:2:11†

0

a prescription commonly adopted in the representation of photoemission.

3.3

Pump Photonics Whilst the procedures described in the previous section are adequate for the description of processes observed with continuouswave input, proper representation of the optical response to pulsed laser radiation requires one further modification to the theory. It is commonly thought to be difficult to represent pulses of light using quantum field theory; indeed it is impossible to do so with internal consistency if a number state basis is employed. This is because such states are associated with infinite phase uncertainty (Loudon 2000), precluding their coherent superposition as a wavepacket. However, by expressing the radiation as a prod-

3.3 Pump Photonics 49

uct of coherent states with a definite phase relationship, it is a relatively straightforward matter to model pulsed laser radiation (Andrews 1978). The physical basis for this approach is that pulses necessarily have a finite linewidth and therefore accommodate a large number of radiation modes. Thus for the pump radiation it is appropriate to construct jirad i ˆ

Y

ja…xl †i ;

…3:3:1†

l

where ja…xl †j2 ˆ ql

…3:3:2†

represents the mean number of photons in the mode labeled by the (positive or negative) integer l. For simplicity it may be assumed that each mode is associated with the same direction of propagation and polarisation, so that the frequency label uniquely identifies each component. If the central frequency is x0 and the interval between adjacent modes is x0 , then we can write xl ˆ x0 ‡ lx0 ;

…3:3:3†

where x0 ˆ pmg …x0 †=L, and mg …x0 † denotes the speed of light at frequency x0. Equation (3.3.3) serves to represent the frequency spectrum of a laser with optical cavity length L. A phase relationship between axial cavity modes, corresponding to perfect modelocking, can now be imposed by effecting the condition; 1=2

a…xl † ˆ ql e

i…xl s‡u†

…3:3:4†

with a suitable value for s (see below) and arbitrary u. When the initial state defined by (3.3.1) is made subject to this condition and employed in the calculation of quantum amplitudes as in equation (2.2.2), it leads to the representation of a pulse train described by the following temporal envelope function J…t†: J…t† ˆ

X l

…ql xl †1=2 e

ilx0 …t‡s† ;

…3:3:5†

50

3 Radiation Constructs

in which the time t arises through evaluation of the matrix ele~…‡† as given by equation (2.1.11). Choosing s ˆ p=x0 ments of d places time zero exactly at the mid-point between two successive pulses, such that J…0†  0 and the interaction is smoothly switched on. By extending these principles to a continuous frequency distribution, single pulses of radiation can be entertained by the theory through the envelope function Z …3:3:6† J…t† ˆ A…x†e ix…t‡s† dt : The net result of incorporating all these modifications in the theory of harmonic emission (or any other process entailing the annihilation of n photons from pump radiation) is that we obtain the prescription   q! …3:3:7† ! J 2n x n …q n†! for effecting the necessary modifications to the QED rate equations. For coherent state light a time-dependent irradiance Ix …t† now appears, properly defined through   hc 2  …3:3:8† J …t† : Ix …t† ˆ V To complete the reformulation of results in terms of physically meaningful parameters, and to relax unduly restrictive assumptions, attention is now given to the possibility of stimulated emission for photons generated by the optical process of interest, for example in the case of strong harmonic pumping. This leads to a matrix element containing an additional factor …q0 ‡ 1†1=2 =V 1=2, indicating that the rate becomes linearly dependent on …q0 ‡ 1†=V. When q0 is large, the rate is essentially proportional to the harmonic photon density. In the light of above remarks on the pump radiation, it is inadvisable to work in terms of q0 . Under conditions of strong emission pumping, it is better to gauge the mean number of n-harmonic photons by employment of the relation q0 ˆ n 1 …q0

q† ;

…3:3:9†

3.3 Pump Photonics 51

where q0 is the initial number of pump photons. Equation (3.3.9) may be regarded as an integrated form of the generalised Manley-Rowe relation; dfI…xk †=xk g ˆ dz

qdfI…x0k0 †=x0k0 g ; dz

…3:3:10†

where z denotes the distance propagated through the nonlinear medium (Manley and Rowe 1959). The q0 appearing in the rate equations is best interpreted as a ratio of the stimulated to the spontaneous emission rate (see for example Haken 1984). Equation (3.3.10) effectively registers energy conservation, signifying that the rate of intensity loss suffered by the optical input is exactly matched by the rate of growth in the harmonic.

53

4

Molecular Properties . . . the further reality is once more charged with mystery Behold This Dreamer, Walter De La Mare

Having established the detailed form of radiation tensors we now turn our attention towards the corresponding molecular tensors, focussing on their construction and influence on the production of optical harmonics. Several key issues concern the dispersion behaviour of response tensors, especially in connection with resonance enhancements that can occur with optical frequencies approaching those of molecular transitions. In addressing dispersion behaviour here we invoke excited state damping to allow for the incorporation of lineshape. Once its context is established we address a number of related issues. The implications of molecular symmetry are described in detail, both with regard to conditions for harmonic emission and the relationships between tensor components. Finally two-level response is explored, both for its calculational expediency and also its power to elicit the role of static dipole moments.

4.1

Molecular Tensor Construction In the electric dipole approximation the explicit result for the response tensor a…m† that mediates an m-photon process may be written;

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

54

4 Molecular Properties

a…m† ˆ

X …1†

rmol

...

XX …m 1†

rmol …m 1†

...

…1†

rrad …m 2†

X

…m 1†

hfmol jljrmol i

…m 1†

rrad

…1†

 hrmol jljrmol i . . . hrmol jljimol i h   i 1 h  . . . E~r …1† E~imol  E~r …m 1† E~imol ‡ Er …m 1† Eirad mol rad mol  i 1 ; …4:1:1† ‡ Er …1† Eirad rad

where l is the electric dipole operator, as follows from equations (2.2.2) and (3.1.1). Summations are taken over all possible intermediate states both for the molecule and radiation. The latter are accommodated by reference to the various contributing time-orderings (equivalent to pathways through state-sequence diagrams) and generally result in a set of terms. Each term differs in its frequency dependence, as determined by the structure of its energy denominator. As will be illustrated by ensuing applications, energy denominators fall into two categories, according to whether or not conditions can be found to satisfy the condition …Errad Eirad † ˆ …Ermol Eimol †. Those terms for which such conditions can be fulfilled are potentially resonant; others, for which such conditions cannot be satisfied, are termed anti-resonant. Anti-resonant terms carry energy denominators that are finite. Resonant terms need special treatment to obviate spurious infinities, to properly account for the highly significant but nonetheless finite enhancement exhibited by the response tensor under suitable frequency conditions. This is usually achieved phenomenologically, by a prescription that accounts for the finite lifetime of each energy level. The tildes appearing over the molecular energies in equation (4.1.1) represent such a modification, the addition of complex damping terms to reflect excited state lifetimes hyr : Er ! E~r ˆ Er  12i

…4:1:2†

In general molecular states will carry such damping; only in the special case of the lowest energy (ground) state is damping redundant, because that state is considered infinitely long-lived. The negative sign in equation (4.1.2) is invariably utilised in connection with potentially resonant energy denominators, and is

4.1 Molecular Tensor Construction 55

consistent with each molecular state jri acquiring, within its h†, an exponential decay Schrödinger phase factor exp… iE~r t= component. Consequently cr may be considered a sum of the inverse lifetimes associated with each line-broadening mechanism, and its deployment through the prescription (4.1.2) signifies that it represents the full width at half maximum (FWHM) linewidth near resonance. On a point of terminology, note that the structure of molecular tensors delivered by equation (4.1.1) can be broadly identified with the semiclassical form of electric polarisability, hyperpolarisability etc. that feature in the molecular field expansion equation (1.1.12). When damping is ignored, as is commonplace in many textbook treatments of theory, the correspondence can be made exact. Although the introduction of damping develops a form that differs from the semiclassical, it differs only in anti-resonant terms. Appendix 1 details the nature of these differences and resolution of the sign in equation (4.1.2). 1) In the applications of equation (4.1.1) to specific processes it is expedient, as a reminder of the characterising input and output, to list the appropriate optical frequencies as arguments of the tensor a…m†. Written with explicit reference to these arguments, the ordering of tensor subscripts in a…m† is then assumed to relate identically to the ordering of the frequencies. Thus for exam…3† ple, in expressing as aijk … x3 ; x2 ; x1 † the component of the molecular response tensor that mediates sum-frequency conversion, the indices i, j and k correspond to interactions of the photons with frequencies x3, x2 and x1, respectively. Since the molecular tensors are seldom completely index-symmetric (Wagnière 1986), it is essential to preserve an unambiguous correlation between indices and photon frequencies. In the time-ordered diagrams, each interaction vertex carries the same index for the corresponding photon in each diagram – so that the subscript ordering of the molecular interaction vertices varies from diagram to diagram. On state-sequence diagrams, it is the various pathways between the initial and final state that correspond to the index permutations, with one of the interaction indices labelling each state connection. 1) In this work, to avoid unnecessarily complicated terminology, the term ‘molecu-

lar response tensor’ is used interchangeably with the more familiar ‘polarisability’, ‘hyperpolarisability’ etc., as appropriate.

56

4 Molecular Properties

Fig. 4.1 The two time-ordered diagrams associated with elastic light scat-

tering.

Consider, as a first illustration of the methodology, derivation of a general expression for molecular polarisability. This response tensor is of primary importance since it formally quantifies the propensity for elastic light scattering, in the electric dipole approximation. The detailed structure of the polarisability is obtained from equation (4.1.1) with m ˆ 2 (one photon is annihilated and another of the same frequency created). Here there are only two time-orderings, and equally two state sequence pathways, as illustrated by the alternative depictions of Figs 4.1 and 4.2 respectively. Each involves a single intermediate state r  r …1† and generates a term whose numerator is a product of transition dipole moment components. For the first term, corresponding to Fig. 4.1 (a) and the lower pathway in Fig. 4.2, we obtain the numerator hf jli jrihrjlj jii. To calculate the corresponding energy denominator using equation (4.1.1) requires identification of the energy components for both the molecule and the radiation. When calculated for a molecule in its electronic ground state, the initial energy Eimol ˆ E0; if the radiation field consists of q photons of frequency x, Eirad ˆ qhx. The intermediate state energy, …E~rmol ‡ Errad †, is again calculated with the aid of the diagrams. In the intermediate state of the coupled system in figure 4.1 (a) one input photon has been annihilated through interaction with a molecule which is thereby promoted to an intermediate electronic state – corresponding to the state box in the centre of the lower pathway in Fig. 4.2. There-

4.1 Molecular Tensor Construction 57

Fig. 4.2 The state-sequence diagram for scattering.

fore the total intermediate state energy is the sum of the intermediate molecular energy E~rmol and the modified radiation field hx. Following equation (4.1.1), and considering Errad ˆ …q 1† only this first state sequence (labelled a) we have as one contribution to the molecular polarisability: X r

…E~r

h0jli jrihrjlj j0i E0 ‡ …q

1† hx

q hx†

ˆ

X h0jli jrihrjlj j0i : ~ E0 hx† r …Er

…4:1:3†

By evaluating in a similar manner the contribution associated with the other time-ordering – Fig. 4.1 (b), equivalent to the upper pathway in Fig. 4.2 – and then adding the result to (4.1.3), we arrive at the following polarisability expression: ( ) X h0jli jrihrjlj j0i h0jlj jrihrjli j0i …2† ‡ ; …4:1:4† aij … x; x† ˆ …E~r0 h …E~r0 ‡ hx† x† r using the standard energy difference notation E~r E0 ˆ E~r0 . In passing it is useful to obtain from equation (4.1.4) a result for the mean polarisability, whose value is required for calculational implementation of the earlier refraction equation (1.3.6). If the transition dipoles are real (as is the case for non-degenerate transitions, or by suitable choice of degenerate basis set), and the molecular environment is randomly oriented, an isotropic average can be employed (see appendix 2) and the mean polarisability hai ˆ 13 dij aij … x; x† emerges with a lineshape as given by Barron (1982):

58

4 Molecular Properties

Fig. 4.3 Dispersive variation of equation (4.1.5) with frequency; real part (dotted line), imaginary part (solid line).

( ) 1 X jhrjlj0ij2 jhrjlj0ij2 ha… x; x†i ˆ ‡ 3 r …E~r0  hx† …E~r0 ‡ hx† ( ) 2X E~r0 2 jhrjlj0ij : ˆ 2 3 r h2 x2 E~r0

…4:1:5†

The above result is important since, in conjunction with (1.3.6), it signifies that the refractive index, n…x†, has both real and imaginary parts. The complex nature of n…x† in a typical dispersive frequency region is illustrated by Fig. 4.3. A second example of the application of equation (4.1.1), with m ˆ 3, illustrates the nonlinear molecular polarisability (or hyperpolarisability) responsible for second-harmonic generation (SHG). Here each tensor numerator comprises one of the index permutations associated with products of three transition dipole moments. Reading from the appropriate diagram, for example sr r0 using Fig. 2.2 (a), we obtain the numerator l0s i lj lk . Here again we assume that the molecule starts and finishes in its ground electronic state, simplifying the labelling with r  r …1† and sr r0 s  r …2† , and introducing the shorthand notation l0s i lj lk  h0jli jsihsjlj jrihrjlk j0i. Each denominator is a product of factors, one for each intermediate state, and again in each factor the energy of the initial state is subtracted from the (complex) intermediate state energy. In the case of Fig. 2.2 (a) we find that, for the intermediate state jsi, the molecular energy difference is

4.1 Molecular Tensor Construction 59

hx, toE~s0 …ˆ E~s E0 † and the difference in photon energies is 2 hx†. Likewise for the intermedigether giving a factor of …E~s0 2 ate state jri, the difference in molecular energies is E~r0 and the difference in photon energies is hx, giving a factor of hx†. Proceeding in a similar way for Fig. 2.2 (b) and (c) …E~r0  and summing, we obtain the following complete expression for the frequency-doubling molecular hyperpolarisability: " sr r0 XX l0s i lj lk bijk … 2x; x; x† ˆ …E~s0 2 hx†…E~r0 hx† s r ‡

sr r0 l0s j li lk …E~s0 ‡  hx†…E~r0  hx†

sr r0 l0s j lk li ‡ …E~s0 ‡  hx†…E~r0 ‡ 2 hx†

# …4:1:6†

where we have used the common hyperpolarisability nomenclature to represent the leading order of nonlinear molecular response, i.e., b… 2x; x; x†  a…3† … 2x; x; x†. Note that, although in general we would consider six permutations for such a three photon event entailed in the overall process, the indistinguishability of the two input photons reduces the number of unique time-orderings, and hence the number of terms in (4.1.6) is similarly reduced to three. By extension of the principles developed above we now construct an expression for a general nth harmonic polarisability tensor, i.e., the response tensor for a process in which a total of m ˆ …n ‡ 1† molecule-photon interactions take place. The exact structure of this tensor a…m† … nx; x; . . . ; x† can be derived by reference to the generalised time-ordered diagram shown in Fig. 4.4. This represents the successive absorption of q photons (each interaction labelled by an index ik) followed by emission of the harmonic photon (labelled in+1), followed by a further …n q† absorptions (labelled ij) before the molecule returns to its ground state. There are …n ‡ 1† topologically distinct diagrams of this type to consider, each of which can be labelled with an index q in the range 0  q  n. Hence we obtain a general result that can compactly be expressed as follows:

60

4 Molecular Properties

k',k'

Fig. 4.4 One of the (n+1) topologically distinct time-ordered diagrams for n-harmonic generation.

…m† ai1 ...in‡1 …

nx; x; . . . ; x† ˆ

...

X

qˆ0 r …1† n Q



n X X

jˆq‡1 n Q

!

…j‡1† …j† lrij r

 E~r …a† r …0† ‡…n

aˆq‡1

…q‡1† …q† lrin‡1 r

a‡1†hx

r …n†





q Q

kˆ1 q  Q bˆ1

…k‡1† …k† lrik r

E~r …b† r …0†



bhx



…4:1:7†

In writing the result in this form, both jr …0† i and jr …n‡1† i are identified with the initial molecular state j0i. A few further general remarks are in order at this stage. First note that summation over intermediate molecular states, as in equations (4.1.4) and (4.1.6) above, in principle applies not only

4.1 Molecular Tensor Construction 61

to electronic but also to vibrational levels. Although this issue initially received most attention in connection with molecular hyperpolarisabilities (Elliott and Ward 1984), it equally applies to other optical response tensors. The vibrational contributions, which were in the past largely overlooked, have been extensively studied and shown to be important in many applications (Bishop 1990, Bishop and Kirtman 1991). Secondly, in most circumstances the polarisabilities associated with nonlinear parametric processes may be regarded as properties of the ground state molecule, since it is usually the molecular ground state that constitutes the initial and final molecular level. Under normal conditions the majority of conversion events will be mediated by ground (usually S0 ) electronic state molecules, simply because of the overwhelming population of such molecules compared to those in excited states. However, other states may assume the role of the initial/final state, and their corresponding polarisabilities can be evaluated in the same way. It transpires that the polarisabilities associated with electronic excited states can exceed (or become less than) those associated with the ground state by orders of magnitude, as has been shown both in theory and experiment (Zhou et al. 1991, Heflin et al. 1992, Rodenberger et al. 1992). Thus, if an excited state acquires a significant population through strong optical pumping, the observed polarisability characteristics of the medium can be significantly enhanced – or diminished. This is an important fact in connection with harmonic generation from systems of dispersed particles, a topic to be addressed in chapter 6. For coherent parametric processes the need to employ optical frequencies in regions of dispersion, in order to satisfy wave-vector matching conditions, is a well-known experimental technique. Operating in such regions necessitates adoption of the polariton (rather than vacuum photon) formulation, as described in chapter 1. Returning to the dispersion curves in Fig. 1.2, consider for simplicity the case 1.2 (a) corresponding to a single molecular response frequency. Clearly there are three areas of interest. The diagonal curve segments represent photon-like radiation propagating through the media at transparent frequencies; the horizontal regions exhibit exciton-like molecular resonances (photons impinging on the medium at such frequencies are readily absorbed); finally level-crossing areas signify dispersive mixing of

62

4 Molecular Properties

the molecular and radiation states. Hence it is commonly necessary to operate in these latter regions for efficient frequency conversion. Whether the radiation frequency is above or below a particular molecular resonance will determine the appropriate branch index.

4.2

Symmetry In determining absolute values for components of molecular response tensors from theory it is necessary to calculate not only the energy differences that feature in the denominators but also the transition moments in the numerators of each term. Given the appropriate wavefunctions, these moments can be obtained for any species by explicit calculation, and the use of dedicated software for such purposes is now routine. However, the number and length of such calculations can often be reduced by the consideration of symmetry principles. Indeed calculation can in some cases be obviated because the results prove to be zero on symmetry grounds. Before pursuing the important connection between nonlinear polarisability and molecular symmetry, it is necessary to take account of the fact that most types of nonlinear interaction are mediated by a tensor with at least some index permutational symmetry, usually reflecting degeneracy amongst its frequency parameter set (as for example with the two input waves for SHG). For such reasons, any such tensor of rank m (i.e. m indices, each of which can stand for either x, y or z) generally has fewer than 3m independent components, even in a molecule with little or no structural symmetry. However equation (4.1.1) does not generally produce results displaying any such index symmetry. For example, the result for the hyperpolarisability, equation (4.1.6), is clearly not invariant under exchange of any of its indices i, j and k. There are two aspects to consider; one is the rigorous index symmetry that can ensue when two or more photons in a given process belong to the same radiation mode; the other is the approximate index symmetry often assumed for calculational simplicity, but which is seldom well justified. At the

4.2 Symmetry

outset, note that there are many different ways of dealing with this aspect of nonlinear optics, and the precise approach and conventions adopted should be borne in mind when applying or comparing results from different sources. Rigorous index symmetry is elicited by the coupling, in equation (3.1.1), of the molecular response tensor with the radiation tensor q defined by (3.1.2). If, in the construction of the latter, two or more of the Dirac brackets components results in factors with the same vector character, then the radiation tensor must possess a symmetry with respect to permutations of the corresponding indices. The case of frequency doubling serves to illustrate the point. Here, we have 0 0 0 qijk ˆ h…q0 ‡ 1†…k0 ; k0 †jd? i jq …k ; k †ih…q

 h…q

2†…k; k†jd? j j…q

1†…k; k†jd? k jq…k; k†i ;

1†…k; k†i …4:2:1†

where q is the number of pump photons …k; k† and q0 the number of harmonic photons of mode …k0 ; k0 † at the outset. Evaluation of the Dirac brackets, using equations (1.2.4), (1.2.5) and (1.3.2), leads to the result

qijk

8
2 are expressible as products of n/2 Kronecker deltas; isotropic tensors of odd rank n comprise products of one Levi-Civita tensor with …n 3†=2 Kronecker deltas. The elements of l display a number of other less well known properties, many of which prove useful in connection with the evaluation of rotational averages. Some of the more important relationships are as follows: eijk lik ˆ eklm ljl lkm ;

…A3:15†

eijk lik ljl ˆ eklm lkm :

…A3:16†

From these two relationships and (A3.9), respectively, emerge the following results with structural similarity to (A3.12) – (A3.14): eijk eklm lik ˆ ljl lkm

ljm lkl ;

…A3:17†

eijk eklm lik ljl ˆ 2lkm ;

…A3:18†

eijk eklm lik ljl lkm ˆ 6 :

…A3:19†

It is also readily shown that lik lil ljk ˆ ljl ; as follows from (A3.2).

…A3:20†

205

Appendix 4:

Irreducible Cartesian Tensors Any tensor may be expressed as a sum of irreducible tensors transforming under irreducible representations of the full rotation group SO(3) – or, including parity, the infinite rotation and rotation-inversion group, O(3). Irreducible tensors facilitate analysis of spectroscopic selection rules, and are traditionally treated as spherical tensors in the formalism of angular momentum formalism theory. The theoretical description of nonlinear optical processes frequently invokes vector and tensor properties most conveniently expressed in Cartesian form, however, and a development in terms of Cartesian tensors has the advantage of clearly exhibiting directional behaviour often obscured by spherical tensors. Explicit formulae for the irreducible components of Cartesian tensors beyond the well-known case of rank 2 are not widely available, although the general relation between Cartesian and spherical tensors has been established by several authors (Coope and Snider 1970, Stone 1975, 1976, Jerphagnon et al. 1978). Here we illustrate the pattern for higher order results, providing explicit expressions for rank 3. Each irreducible tensor of rank n is characterised by a weight j  n, and possesses …2j ‡ 1† independent components. The reduction of a Cartesian tensor T…n† generally results in a sum of such irreducible tensors, with some weights represented more than once and therefore distinguished by a secondary label q. Starting with the general case of a tensor without explicit index symmetry properties, we can write …j†

T…n† ˆ

Nn n X X jˆ0 qˆ1

…j;q†

T…n† ;

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

…A4:1†

206

Appendix 4: Irreducible Cartesian Tensors

where q is known as the seniority index of the irreducible tensor …j;q† …j† T…n† and Nn is the multiplicity of weight j in the reduction scheme. This multiplicity can be derived using a recursive scheme due to Gel’fand et al. (1963); the general result is explicitly given by the following formula (Mikhailov 1977): Nn…j† ˆ

X

… 1†k

k

…2n 3k …n 3

j 2†!n…n 1† ; j†!…n k†!k!

…A4:2†

where 0  k  ‰13 …n j†Š: Since each irreducible tensor has …2j ‡ 1† independent components, the total number of components in the reduction is n X …2j ‡ 1†Nn…j† ˆ 3n

…A4:3†

jˆ0

as required. The multiplicities of each weight for tensors up to rank 4 are shown in Tab. A4.1. For tensors possessing index symmetry, the

…j†

Tab. A4.1 Number of independent components, i, and multiplicity Nn of

each weight j in the irreducible representation of Cartesian tensors. …0†

…1†

…2†

Nn

…3†

i

Nn

Nn

Nn

nˆ1 Tk

3

0

1

nˆ2 Tkl T…kl†

9 6

1 1

1 0

1 1

nˆ3 Tklm Tk…lm† T…klm†

27 18 10

1 0 0

3 2 1

2 1 0

1 1 1

nˆ4 Tklmo Tkl…mo† T…kl†…mo† Tk…lmo† T……kl†…mo†† T……klmo††

81 54 36 30 21 16

3 2 2 1 2 1

6 3 1 1 0 0

6 4 3 2 2 1

3 2 1 1 0 0

…4†

Nn

1 1 1 1 1 1

Appendix 4: Irreducible Cartesian Tensors

multiplicity of most weights is diminished, as this table shows. Only alternate weights from n downwards are represented in rank n tensors with full index symmetry, and here the multiplicity of each weight is 1. The representations of tensors with partial index symmetry are obtained by coupling the appropriate non-symmetric and fully-symmetric tensors of lower rank. For example the tensor Tk…lm† has an irreducible representation obtained by coupling the results for the vector Tk (three components, weight 1) with the index-symmetric second rank tensor T…lm† (six components, weights 0 and 2). Explicit results are as follows: Rank 2 …0†

…1†

…2†

Tkl ˆ Tkl ‡ Tkl ‡ Tkl ;

…A4:4†

…0†

Tkl ˆ 13 dkl Tkk ;

…A4:5†

…1†

Tkl ˆ 12 eklm …eopm Top †  12 …Tkl …2†

Tkl ˆ 12 …Tkl ‡ Tlk †

1 3 dkl Tkk

Tlk † ;

…A4:6†

:

…A4:7†

For the index-symmetric case we have: …0†

…2†

T…kl† ˆ T…kl† ‡ T…kl† :

…A4:8†

The procedure for deriving results for higher ranks has been established by Coope et al. (1965). The explicit results for rank 3 are as follows (Jerphagnon 1970, Andrews and Thirunamachandran 1978): Rank 3 …0†

Tklm ˆ Tklm ‡

X

…1;p†

Tklm ‡

pˆa;b;c

X

…2;p†

…1a†

…A4:9†

pˆa;b

Tklm ˆ 16 eklm eopq Topq ; 1 Tklm ˆ 10 …4dkl Tqqm

…3†

Tklm ‡ Tklm ;

…A4:10† dkl Tqql

dkl Tqqk † ;

…A4:11†

207

208

Appendix 4: Irreducible Cartesian Tensors …1b†

dlm Tqkq † ;

…A4:12†

…1b†

dkm Tlqq ‡ 4dlm Tkqq † ;

…A4:13†

1 Tklm ˆ 10 … dkl Tqmq ‡ 4dkm Tqlq 1 … dkl Tmqq Tklm ˆ 10 …2a†

Tklm ˆ 16 ekls …2eqrs Tqrm ‡ 2eqrm Tqrs ‡ eqrs Tmqr ‡ eqrm Tsqr

2dms epqr Tpqr † ;

…A4:14†

…2b†

Tklm ˆ 16 elms …2eqrs Tkqr ‡ 2eqrk Tqrs ‡ eqrs Tqrk ‡ eqrk Tsqr

2dks epqr Tpqr † ;

…A4:15†

…3†

Tklm ˆ 16 …Tklm ‡ Tlkm ‡ Tkml ‡ Tlmk ‡ Tmkl ‡ Tmlk † 1 ‰dkl …Tqqm ‡ Tqmq ‡ Tmqq †km …Tqql ‡ Tqlq ‡ Tlqq † 15 ‡ d ‡ dlm …Tqqk ‡ Tqkq ‡ Tkqq †Š : …A4:16† For the case with pair index symmetry we have (Andrews, Blake and Hopkins 1988) Tk…lm† ˆ

X

…1;p†

…2†

…3†

Tk…lm† ‡ Tk…lm† ‡ Tk…lm† ;

…A4:17†

pˆc;d

where …1d†

…1a†

…1b†

…A4:18†

…2†

…2a†

…2b†

…A4:19†

Tk…lm† ˆ Tk…lm† ‡ Tk…lm† ; Tk…lm† ˆ Tk…lm† ‡ Tk…lm† : Finally, if there is full index symmetry we have …1†

…3†

…1a†

…1b†

T…klm† ˆ T…klm† ‡ T…klm† ;

…A4:20†

with …1†

…1c†

T…klm† ˆ T…klm† ‡ T…klm† ‡ T…klm† :

…A4:21†

The considerably more complex results for rank 4 are tabulated by Andrews and Ghoul (1982).

209

Appendix 5:

Six-Wave Mixing and Secular Resonances In parametric nonlinear optical processes the sum of all the frequencies involved (appropriately signed for input or output status) is zero. In certain applications a subset of the frequencies involved also sums to zero (as for example in six-wave mixing, SWM). The secular resonances that then emerge from the perturbation theoretic formulae require careful handling to obviate unphysical divergence behaviour. In order to construct well-behaved molecular tensors for such cases it is necessary to secure a cancellation of the secular terms. To illustrate the complete process with regard to SWM, we first construct the full response tensor for …6† SWM, aSWM … x0 ; x0 ; x; x; x; x†, where generally x0 representing the frequency of scattered (or more precisely emitted) photons is given as 2x, using the standard methods introduced in chapter 4. Further the tensor is then reconstructed in a consistent form that eliminates any possible ambiguity regarding infinite response. The detailed structure of the nonlinear SWM molecular response …6† tensor, aSWM … x0 ; x0 ; x; x; x; x†, is constructed with the aid of the state-sequence diagram of Fig. 7.6. Generally six photon interactions yield a maximum 6! ˆ 720 possible permutations if each photon is uniquely defined. However, for our SWM process we initially assume that absorption takes place from a single radiation mode and that the harmonic is likewise of a single colour. This allows index symmetry to be assigned to the pump and harmonic modes respectively. Consequently the number of permutations is greatly reduced to 6!=4!2! ˆ 15, matching the unique number of pathways linking the initial and final states in Fig. 7.6. Therefore Fig. 7.6 properly represents all topologically distinct orderings of the six electric dipole interactions involved in SWM processes. We might have chosen the time-ordered diagrams, as illustrated Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

210

Appendix 5: Six-Wave Mixing and Secular Resonances

Fig. A5.1 The time-ordered diagrams for SWM. The two emergent photon modes can be reversed for fifteen other time-orderings.

in Fig. A5.1, as an alternative – both of course lead to identical tensor expressions, but the concise representation of the state-sequencing is now clearly evident (cf. Figs 7.6 and A5.1). Construction of the molecular response tensor follows from equation (4.1.1) in the usual way and is explicitly represented by:

Appendix 5: Six-Wave Mixing and Secular Resonances 211 …6†

aSWM … x0 ; x0 ; x; x; x; x† mu ut ts sr r0 mu ut ts sr r0 X l0m l0m j li lk ll lm ln i lj lk ll lm ln ˆ ‡ D1 D2 r;s;t;u;m ‡ ‡ ‡ ‡ ‡ ‡ ‡

mu ut ts sr r0 l0m j lk li ll lm ln

D3 0m mu ut ts sr r0 lj lk li lm ln ll D5 0m mu ut ts sr r0 li lj lk lm ln ll D7 0m mu ut ts sr r0 lj li lk lm ll ln D9 0m mu ut ts sr r0 lj li ll lk lm ln D11 0m mu ut ts sr r0 li ll lj lk lm ln D13 mu ut ts sr r0 l0m l l j k m li ln ll D15

‡ ‡ ‡ ‡ ‡ ‡ :

mu ut ts sr r0 l0m j lk li lm ll ln

D4 0m mu ut ts sr r0 lj li lk lm ln ll D6 0m mu ut ts sr r0 li lj lk lm ll ln D8 0m mu ut ts sr r0 li lj ll lk lm ln D10 0m mu ut ts sr r0 lj lk lm li ll lm D12 0m mu ut ts sr r0 lj lk lm ln li ll D14 …A5:1†

where the shorthand energy denominator labels, Di, are reproduced explicitly in Tab. A5.1. The hanging indices i and l are associated with photon emission whereas the jk and mn pairs result from photon absorption. Having constructed the explicit molecular response tensor, it is evident that problems appear to arise in connection with certain intermediate states. Consider a situation where the molecular intermediate state jti is represented by the molecular ground state j0i in the sum over t. When this occurs, certain energy denominators, Eti, can suffer a complete cancellation of the radiation terms to uncover an expression of the form ~00 ˆ 0. In order to circumvent these secular resonances the moE lecular response tensor has to be reconstructed in such a way as to remove the possibility of infinite response (Bishop 1994, Allcock and Andrews 1997). Such a reconstruction is demonstrated here explicitly for the SWM response tensor. Consider then a generalised six-wave interaction resulting in the emission of two output waves with differing frequencies, such that the response tensor might now be represented by

212

Appendix 5: Six-Wave Mixing and Secular Resonances

Tab. A5.1 The energy denominators, each associated with an individual route through the state sequence diagram of Figs. 7.6 or equally one for each of the timeordered diagrams illustrated in Fig. A5.1. The tilda allows for the possibility of compelx energies to fully incorporate damping.

Denominator D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13 D14 D15

~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E ~r0 …E

…6†

~s0 2 ~t0 2 ~u0 3 ~m0 ‡ 4 x†…E h hx†…E hx ‡  hx0 †…E hx ‡  hx0 †…E hx ‡  hx0 † ~s0 2 ~t0 2 ~u0 3 ~m0 ‡ 3 hx†…E  hx†…E hx ‡  hx0 †…E hx ‡  hx0 †…E hx ‡ 2 hx0 † 0 ~ 0 ~ ~ ~ hx†…Es0 2  hx†…Et0 2 hx ‡  hx †…Eu0 2 hx ‡ 2 hx †…Em0 ‡ 3 hx ‡ 2 hx 0 † 0 ~ 0 ~ ~ ~ hx†…Es0 hx ‡   hx†…Et0 2 hx ‡  hx †…Eu0 2 hx ‡ 2 hx †…Em0 ‡ 3 hx ‡ 2 hx 0 † 0 ~ 0 ~ 0 ~ 0 ~ hx †…Es0   hx ‡  hx †…Et0 2 hx ‡  hx †…Eu0 2 hx ‡ 2 hx †…Em0 ‡ 3 hx ‡ 2 hx0 † 0 ~ 0 ~ 0 ~ 0 ~ hx †…Es0   hx ‡  hx †…Et0 2 hx ‡  hx †…Eu0 3 hx ‡  hx †…Em0 ‡ 3 hx ‡ 2 hx0 † 0 ~ 0 ~ 0 ~ 0 ~ hx †…Es0   hx ‡  hx †…Et0 2 hx ‡  hx †…Eu0 3 hx ‡  hx †…Em0 ‡ 4 hx ‡ 2 hx0 † 0 0 0 ~s0 hx ‡  ~t0 2 ~u0 3 ~m0 ‡ 4 hx†…E  hx †…E hx ‡  hx †…E hx ‡  hx †…E hx ‡  hx0 † 0 0 0 ~s0 hx ‡  ~t0 2 ~u0 3 ~m0 ‡ 3 hx†…E  hx †…E hx ‡  hx †…E hx ‡  hx †…E hx ‡ 2 hx0 † ~s0 2 ~t0 3 ~u0 3 ~m0 ‡ 4 hx†…E  hx†…E hx†…E hx ‡  hx0 †…E hx ‡  hx0 † ~s0 2 ~t0 3 ~u0 3 ~m0 ‡ 3 hx†…E  hx†…E hx†…E hx ‡  hx0 †…E hx ‡ 2 hx0 † 0 ~ 0 ~ 0 ~ ~ hx†…Es0 hx ‡   hx †…Et0  hx ‡  hx †…Eu0 2 hx ‡ 2 hx †…Em0 ‡ 3 hx ‡ 2 hx0 † 0 ~ 0 ~ ~ ~ hx†…Es0 2  hx†…Et0 3 hx†…Eu0 4 hx †…Em0 ‡ 4 hx ‡  hx † ~s0 ‡ 2 ~t0  ~u0 2 ~m0 ‡ 3 hx0 †…E  hx0 †…E hx ‡ 2 hx0 †…E hx ‡ 2 hx0 †…E hx ‡ 2 hx0 † 0 ~ 0 ~ 0 ~ 0 ~ hx †…Es0   hx ‡  hx †…Et0  hx ‡ 2 hx †…Eu0 2 hx ‡ 2 hx †…Em0 ‡ 3 hx ‡ 2 hx0 †

aSWM … fx0 ‡ dg; fx0 dg; x; x; x; x†, where delta represents a small positive frequency. The total number of time-orderings for this particular SWM process is 6!=4! ˆ 30, as the permutational symmetry of indices i and l is now lost. The form of the response tensor is similar to that of equation (A5.1) with 15 additional terms arising by interchanging the two groups of indices (i, jk) and (l, mn). However, it is important to note that the modified frequencies associated with the indices i and l must now be introduced in all the appropriate denominator terms. In choosing the two groups of indices (i, jk) and (l, mn) we simply aid calculational effort later; interchanging jk and lm suffers no loss of generality as their index symmetry remains intact. In summing over the molecular intermediate state jti, in the evaluation of molecular tensor summands, secular contributions arise when jti is identified with the molecular ground state j0i. There are now eighteen individual terms that can result in the form E~t0 . It is these terms associated with secular resonance that require further analysis to identify the correct, finite response.

Appendix 5: Six-Wave Mixing and Secular Resonances 213

To tackle the problem, we have introduced in the response tensor two emergent waves as a limiting case of two-colour emission, where x0  d  2x  d ;

…A5:2†

assuming that d is a small, positive frequency tending to zero for second harmonic generation by SWM. It is then expedient to effect a further separation of the tensor into a sum of two terms, one containing all eighteen problematic secular denominators, for which jti ˆ j0i, and the other, all remaining terms, jti 6ˆ j0i n o n o …SWM† …SWM† …SWM† ‡ aijklmn : …A5:3† aijklmn ˆ aijklmn jtiˆj0i

jti6ˆj0i

The combinatorial properties of the thirty summands in the matrix element include the 18 permutations of the two groups of three indices …i; jk† and …l; mn†. A consequence of the exchange of indices …i; jk† $ …l; mn† in the sum of equation (A5.1) is that the dummy intermediate state labels suffer a similar transformation m $ s and u $ r. Exploiting this feature allows analysis to proceed in establishing common factors in the expressions to follow. Explicitly, the first term of the subset of eighteen is given by  n o…i† X 1 …SWM† ˆ a…i;jk†…l;mn† jtiˆj0i … hx0 ‡ d 2 hx† s;r;m;u " # mu u0 0s sr r0 l0m i lj lk ll lm ln  …E~r0 h x†…E~s0 2 hx†…E~u0 3 hx ‡  hx0 ‡d†…E~mo 4hx‡hx0 ‡d† …A5:4† with the temporal ordering; absorption at n and m …x†, emission l …x0 ‡ d†, absorptions k and j …x† and finally emission i …x0 d†. The first index-reversed contribution, …i; jk† $ …l; mn†, and hence the tenth term (x) in the above sum, equation (A5.3), is similarly n

o…x† …SWM† ˆ a…i;jk†…l;mn† jtiˆj0i "



…E~u0  hx†…E~m0

X s;r;m;u

… hx0

1 d

 2 hx†

# sr r0 0m mu u0 l0s l lm ln li lj lk : 2 hx†…E~r0 3 hx‡ hx0 d†…E~so 4hx‡hx0 d† …A5:5†

214

Appendix 5: Six-Wave Mixing and Secular Resonances

where each energy denominator factor is also transposed with respect to its index-reversed counterpart (cf. equation (A5.4)). The sum of contributions (i) and (x) can be expressed concisely, substituting for x0 through equation (A5.2) as n

…SWM†

a…i;jk†…l;mn†

o…i†‡…x† jtiˆj0i

" 1 ˆ …‡d† …E~r0 s;r;u;m X

…E~r0

hx†…E~s0 

mu u0 0s sr r0 l0m i lj lk ll lm ln 2 hx†…E~u0 hx ‡ d†…E~m0

2hx ‡ d† #

mu u0 0s sr r0 l0m l lm ln li lj lk

hx†…E~s0 

2 hx†…E~u0

h x

d†…E~m0

2hx

d†

…A5:6†

As the intermediate states m, u, s, r are summed over the same basis set, it is again permissible to effect the interchange …l; mn† $ …i; jk†, allowing the factorisation of the transition dipole product: n

…SWM†

a…i;jk†…l;mn†

ˆ

o…i†‡…x† jtiˆj0i

mu u0 0s sr r0 X …l0m n lm ll lk lj li † r;s;u;m

…‡d†

hx d†…E~s0 2 hx  ‰f…E~r0  x†…E~s0 2 hx†…E~u0 f…E~r0 h hx†…E~s0 2 hx†…E~u0  ‰…E~r0   …E~s0

2 hx

d†…E~u0

hx 

d†…E~u0

x†…E~m0 2hx†g h hx ‡ d†…E~m0 2hx ‡ d†g

hx†…E~m0 2hx†…E~r0 h x d† 1 d†…E~m0 2hx d†Š : …A5:7†

A common factor of d emerges from the numerator, cancelling with the factorised d from the denominator; taking the limit d ! 0 then correctly gives the finite contribution from the sum of the two terms …i† ‡ …x†. This technique can be applied to all eighteen terms (nine pairs) of equation (A5.3). The reduced set of nine summed contributions derived from equation (A5.3) reveals the correct finite form of the secular contribution to second harmonic SWM.

215

Appendix 6:

Spectroscopic Selection Rules The derivation of spectroscopic selection rules for multiphoton processes such as inelastic harmonic scattering is founded on the principles of irreducible tensor calculus described in appendix 4. In determining whether a given quantum transition is allowed or forbidden there are two aspects to consider; one concerns the implications of molecular symmetry, and the other the symmetry of the radiation field. Thus if either the nonlinear molecular response tensor a…m† or the radiation tensor q…m† vanishes, the matrix element for the transition as given by equation (3.1.1) becomes zero and the transition is forbidden. The conditions for the radiation tensor to be non-zero are dictated by the polarisation and beam propagation directions, and are analysed in the context of the experimental conditions for each nonlinear optical process. The conditions for the molecular response tensor to be non-zero are not influenced by experimental configuration, being entirely determined by the symmetry properties of the molecule and its wavefunctions. This appendix deals with these symmetry constraints. In connection with inelastic optical processes, i.e. those for which the radiation suffers a net loss or gain in energy through exchange with the molecular medium, the criterion for a…m† to be non-zero is that the product of the irreducible representations of the molecular initial state jii and final state jf i must be spanned by one or more components of the tensor. In the common case of transitions originating from, or terminating in, a totally symmetric ground state, this reduces to a requirement that one of the irreducible parts of the tensor transforms under the same representation as the excited state. It is therefore necessary to determine the transformation properties of these irreducible tenOptical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

216

Appendix 6: Spectroscopic Selection Rules

sors under the symmetry operations of the appropriate point group for any particular molecular species. This entails mapping the irreducible representations of the full three-dimensional rotation-inversion group O(3) onto the corresponding representations of point groups with lower symmetry (Kim 1999). A complete listing for all the common molecular point groups is given in Tab. A6.1 (Andrews 1990). The representation salient for infrared absorption is D(1–); for Raman scattering we have D(0+) + D(2+) and for hyper-Raman scattering 2D(1–) + D(2+) + D(3–) (in each case assuming that all photon interactions are electric dipole allowed). The table shows that, for example, in a molecule of Ih symmetry, vibrational modes of T1u, Hu, T2u and Gu character are allowed in the hyper-Raman spectrum. Table A6.2 lists the allowed weight combinations, showing for the same point group that classes 3–(1), 3–(2) and 3–(3) arise. Note that classes such as 3–(0) do not arise in hyper-Raman scattering because the molecular response tensor has no weight 0 term (as the tensor has a pair index symmetry).

Tab. A6.1 Representations of irreducible tensors in the common molecular and crystallographic point groups; even parity representations are denoted by upright characters, and odd parity by italic characters below.

Point D(0+) D(1+) group D(0–) D(1–)

D(2+) D(2–)

D(3+) D(3–)

D(4+) D(4–)

C1

A A Ag Au 3A0 ‡ 2A00 2A0 ‡ 3A00 3A ‡ 2B 3A ‡ 2B A ‡ 2E A ‡ 2E A ‡ 2B ‡ E A ‡ 2B ‡ E A ‡ E1 ‡ E2 A ‡ E1 ‡ E2 A ‡ E1 ‡ E2 A ‡ E1 ‡ E2 A ‡ 2B ‡ E 2A ‡ B ‡ E Ag ‡ 2Eg Au ‡ 2Eu

A A Ag Au 3A0 ‡ 4A00 4A0 ‡ 3A00 3A ‡ 4B 3A ‡ 4B 3A ‡ 2E 3A ‡ 2E A ‡ 2B ‡ 2E A ‡ 2B ‡ 2E A ‡ E1 ‡ 2E2 A ‡ E1 ‡ 2E2 A ‡ 2B ‡ E1 ‡ E2 A ‡ 2B ‡ E1 ‡ E2 A ‡ 2B ‡ 2E 2A ‡ B ‡ 2E 3Ag ‡ 2Eg 3Au ‡ 2Eu

A A Ag Au 5A0 ‡ 4A00 4A0 ‡ 5A00 5A ‡ 4B 5A ‡ 4B 3A ‡ 3E 3A ‡ 3E 3A ‡ 2B ‡ 2E 3A ‡ 2B ‡ 2E A ‡ 2E1 ‡ 2E2 A ‡ 2E1 ‡ 2E2 A ‡ 2B ‡ E1 ‡ 2E2 A ‡ 2B ‡ E1 ‡ 2E2 3A ‡ 2B ‡ 2E 2A ‡ 3B ‡ 2E 3Ag ‡ 3Eg 3Au ‡ 3Eu

Ci Cs C2 C3 C4 C5 C6 S4 S6

A A Ag Au A0 A00 A A A A A A A A A A A B Ag Au

A A Ag Au A0 ‡ 2A00 2A0 ‡ A00 A ‡ 2B A ‡ 2B A‡E A‡E A‡E A‡E A ‡ E1 A ‡ E1 A ‡ E1 A ‡ E1 A‡E B‡E Ag ‡ E g Au ‡ Eu

Appendix 6: Spectroscopic Selection Rules 217 Tab. A6.1 (continued)

Point D(0+) group D(0–)

D(1+) D(1–)

D(2+) D(2–)

D(3+) D(3–)

D(4+) D(4–)

S8

A ‡ E3 B ‡ E1 Ag ‡ E1g Au ‡ E1u Ag ‡ 2Bg Au ‡ 2Bu A0 ‡ E00 A00 ‡ E 0 A g ‡ Eg Au ‡ Eu A0 ‡ E001 A00 ‡ E10 Ag ‡ E1g Au ‡ E1u A2 ‡ B1 ‡ B2 A1 ‡ B1 ‡ B2 A2 ‡ E A1 ‡ E A2 ‡ E A1 ‡ E A 2 ‡ E1 A1 ‡ E1 A 2 ‡ E1 A1 ‡ E1 B1 ‡ B2 ‡ B3 B1 ‡ B2 ‡ B3 A2 ‡ E A2 ‡ E A2 ‡ E A2 ‡ E A 2 ‡ E1 A2 ‡ E1 A 2 ‡ E1 A2 ‡ E1 B1g ‡ B2g ‡ B3g B1u ‡ B2u ‡ B3u A02 ‡ E00 A002 ‡ E 0 A2g ‡ Eg A2u ‡ Eu A02 ‡ E001 A002 ‡ E10 A2g ‡ E1g A2u ‡ E1u A2 ‡ E B2 ‡ E A2g ‡ Eg A2u ‡ Eu A 2 ‡ E3 B 2 ‡ E1

A ‡ E2 ‡ E3 B ‡ E1 ‡ E2 Ag ‡ E1g ‡ E2g Au ‡ E1u ‡ E2u 3Ag ‡ 2Bg 3Au ‡ 2Bu A0 ‡ E0 ‡ E00 A00 ‡ E 0 ‡ E 00 Ag ‡ 2Bg ‡ Eg Au ‡ 2Bu ‡ Eu A0 ‡ E001 ‡ E02 A00 ‡ E10 ‡ E200 Ag ‡ E1g ‡ E2g Au ‡ E1u ‡ E2u 2A1 ‡ A2 ‡ B1 ‡ B2 A1 ‡ 2A2 ‡ B1 ‡ B2 A1 ‡ 2E A2 ‡ 2E A1 ‡ B1 ‡ B2 ‡ E A2 ‡ B1 ‡ B2 ‡ E A1 ‡ E 1 ‡ E 2 A2 ‡ E1 ‡ E2 A1 ‡ E 1 ‡ E 2 A2 ‡ E1 ‡ E2 2A ‡ B1 ‡ B2 ‡ B3 2A ‡ B1 ‡ B2 ‡ B3 A1 ‡ 2E A1 ‡ 2E A1 ‡ B1 ‡ B2 ‡ E A1 ‡ B1 ‡ B2 ‡ E A1 ‡ E 1 ‡ E 2 A1 ‡ E1 ‡ E2 A1 ‡ E 1 ‡ E 2 A1 ‡ E1 ‡ E2 2Ag ‡ B1g ‡ B2g ‡ B3g 2Au ‡ B1u ‡ B2u ‡ B3u A01 ‡ E0 ‡ E00 A001 ‡ E 0 ‡ E 00 A1g ‡ B1g ‡ B2g ‡ Eg A1u ‡ B1u ‡ B2u ‡ Eu A01 ‡ E001 ‡ E02 A001 ‡ E10 ‡ E200 A1g ‡ E1g ‡ E2g A1u ‡ E1u ‡ E2u A1 ‡ B1 ‡ B2 ‡ E A1 ‡ A2 ‡ B1 ‡ E A1g ‡ 2Eg A1u ‡ 2Eu A1 ‡ E 2 ‡ E 3 B1 ‡ E1 ‡ E2

A ‡ E1 ‡ E2 ‡ E3 B ‡ E1 ‡ E2 ‡ E3 Ag ‡ E1g ‡ 2E2g Au ‡ E1u ‡ 2E2u 3Ag ‡ 4Bg 3Au ‡ 4Bu A0 ‡ 2A00 ‡ E0 ‡ E00 2A0 ‡ A00 ‡ E 0 ‡ E 00 Ag ‡ 2Bg ‡ 2Eg Au ‡ 2Bu ‡ 2Eu A0 ‡ E001 ‡ E02 ‡ E002 A00 ‡ E10 ‡ E20 ‡ E200 Ag ‡ 2Bg ‡ E1g ‡ E2g Au ‡ 2Bu ‡ E1u ‡ E2u A1 ‡ 2A2 ‡ 2B1 ‡ 2B2 2A1 ‡ A2 ‡ 2B1 ‡ 2B2 A1 ‡ 2A2 ‡ 2E 2A1 ‡ A2 ‡ 2E A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ B1 ‡ B2 ‡ 2E A2 ‡ E1 ‡ 2E2 A1 ‡ E1 ‡ 2E2 A2 ‡ B1 ‡ B2 ‡ E1 ‡ E2 A1 ‡ B1 ‡ B2 ‡ E1 ‡ E2 A ‡ 2B1 ‡ 2B2 ‡ 2B3 A ‡ 2B1 ‡ 2B2 ‡ 2B3 A1 ‡ 2A2 ‡ 2E A1 ‡ 2A2 ‡ 2E A2 ‡ B1 ‡ B2 ‡ 2E A2 ‡ B1 ‡ B2 ‡ 2E A2 ‡ E1 ‡ 2E2 A2 ‡ E1 ‡ 2E2 A2 ‡ B1 ‡ B2 ‡ E1 ‡ E2 A2 ‡ B1 ‡ B2 ‡ E1 ‡ E2 Ag ‡ 2B1g ‡ 2B2g ‡ 2B3g Au ‡ 2B1u ‡ 2B2u ‡ 2B3u A001 ‡ A02 ‡ A002 ‡ E0 ‡ E00 A01 ‡ A02 ‡ A002 ‡ E 0 ‡ E 00 A2g ‡ B1g ‡ B2g ‡ 2Eg A2u ‡ B1u ‡ B2u ‡ 2Eu A02 ‡ E001 ‡ E02 ‡ E002 A002 ‡ E10 ‡ E20 ‡ E200 A2g ‡ B1g ‡ B2g ‡ E1g ‡ E2g A2u ‡ B1u ‡ B2u ‡ E1u ‡ E2u A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ A2 ‡ B2 ‡ 2E A1g ‡ 2A2g ‡ 2Eg A1u ‡ 2A2u ‡ 2Eu A2 ‡ E1 ‡ E2 ‡ E3 B2 ‡ E1 ‡ E2 ‡ E3

A ‡ 2B ‡ E1 ‡ E2 ‡ E3 2A ‡ B ‡ E1 ‡ E2 ‡ E3 Ag ‡ 2E1g ‡ 2E2g Au ‡ 2E1u ‡ 2E2u 5Ag ‡ 4Bg 5Au ‡ 4Bu A0 ‡ 2A00 ‡ 2E0 ‡ E00 2A0 ‡ A00 ‡ E 0 ‡ 2E 00 3Ag ‡ 2Bg ‡ 2Eg 3Au ‡ 2Bu ‡ 2Eu A0 ‡ E01 ‡ E001 ‡ E02 ‡ E002 A00 ‡ E10 ‡ E100 ‡ E20 ‡ E200 Ag ‡ 2Bg ‡ E1g ‡ 2E2g Au ‡ 2Bu ‡ E1u ‡ 2E2u 3A1 ‡ 2A2 ‡ 2B1 ‡ 2B2 2A1 ‡ 3A2 ‡ 2B1 ‡ 2B2 2A1 ‡ A2 ‡ 3E A1 ‡ 2A2 ‡ 3E 2A1 ‡ A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ 2A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ 2E1 ‡ 2E2 A2 ‡ 2E1 ‡ 2E2 A1 ‡ B1 ‡ B2 ‡ E1 ‡ 2E2 A2 ‡ B1 ‡ B2 ‡ E1 ‡ 2E2 3A ‡ 2B1 ‡ 2B2 ‡ 2B3 3A ‡ 2B1 ‡ 2B2 ‡ 2B3 2A1 ‡ A2 ‡ 3E 2A1 ‡ A2 ‡ 3E 2A1 ‡ A2 ‡ B1 ‡ B2 ‡ 2E 2A1 ‡ A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ 2E1 ‡ 2E2 A1 ‡ 2E1 ‡ 2E2 A1 ‡ B1 ‡ B2 ‡ E1 ‡ 2E2 A1 ‡ B1 ‡ B2 ‡ E1 ‡ 2E2 3Ag ‡ 2B1g ‡ 2B2g ‡ 2B3g 3Au ‡ 2B1u ‡ 2B2u ‡ 2B3u A01 ‡ A001 ‡ A002 ‡ 2E0 ‡ E00 A01 ‡ A001 ‡ A02 ‡ E 0 ‡ 2E 00 2A1g ‡ A2g ‡ B1g ‡ B2g ‡ 2Eg 2A1u ‡ A2u ‡ B1u ‡ B2u ‡ 2Eu A01 ‡ E01 ‡ E001 ‡ E02 ‡ E002 A001 ‡ E10 ‡ E100 ‡ E20 ‡ E200 A1g ‡ B1g ‡ B2g ‡ E1g ‡ 2E2g A1u ‡ B1u ‡ B2u ‡ E1u ‡ 2E2u 2A1 ‡ A2 ‡ B1 ‡ B2 ‡ 2E A1 ‡ A2 ‡ 2B1 ‡ B2 ‡ 2E 2A1g ‡ A2g ‡ 3Eg 2A1u ‡ A2u ‡ 3Eu A1 ‡ B1 ‡ B2 ‡ E1 ‡ E2 ‡ E3 A1 ‡ A2 ‡ B1 ‡ E1 ‡ E2 ‡ E3

S10 C2h C3h C4h C5h C6h C2v C3v C4v C 5v C6v D2 D3 D4 D5 D6 D2h D3h D4h D5h D6h D2d D3d D4d

A B Ag Au Ag Au A0 A00 Ag Au A0 A00 Ag Au A1 A2 A1 A2 A1 A2 A1 A2 A1 A2 A A A1 A1 A1 A1 A1 A1 A1 A1 Ag Au A01 A001 A1g A1u A01 A001 A1g A1u A1 B1 A1g A1u A1 B1

218

Appendix 6: Spectroscopic Selection Rules

Tab. A6.1 (continued)

Point D(0+) group D(0–)

D(1+) D(1–)

D(2+) D(2–)

D(3+) D(3–)

D(4+) D(4–)

D5d

A2g ‡ E1g A2u ‡ E1u A 2 ‡ E5 B 2 ‡ E1 R ‡P R‡ ‡ P Rg ‡ P g R‡ u ‡ Pu T T Tg Tu T1 T2 T1 T1 T1g T1u T1 T1 T1g T1u

A1g ‡ E1g ‡ E2g A1u ‡ E1u ‡ E2u A1 ‡ E 2 ‡ E 5 B1 ‡ E1 ‡ E4 R‡ ‡ P ‡ D R ‡P ‡D R‡ g ‡ Pg ‡ Dg R u ‡ P u ‡ Du E‡T E‡T Eg ‡ T g E u ‡ Tu E ‡ T2 E ‡ T1 E ‡ T2 E ‡ T2 Eg ‡ T2g Eu ‡ T2u H H Hg Hu

A2g ‡ E1g ‡ 2E2g A2u ‡ E1u ‡ 2E2u A2 ‡ E2 ‡ E3 ‡ E5 B2 ‡ E1 ‡ E3 ‡ E4 R ‡P‡D‡U R‡ ‡ P ‡ D ‡ U Rg ‡ Pg ‡ Dg ‡ Ug R‡ u ‡ P u ‡ Du ‡ Uu A ‡ 2T A ‡ 2T Ag ‡ 2Tg Au ‡ 2Tu A 2 ‡ T1 ‡ T2 A1 ‡ T1 ‡ T2 A 2 ‡ T1 ‡ T2 A2 ‡ T1 ‡ T2 A2g ‡ T1g ‡ T2g A2u ‡ T1u ‡ T2u T2 ‡ G T2 ‡ G T2g ‡ Gg T2u ‡ Gu

A1g ‡ 2E1g ‡ 2E2g A1u ‡ 2E1u ‡ 2E2u A1 ‡ E 2 ‡ E 3 ‡ E 4 ‡ E 5 B1 ‡ E1 ‡ E2 ‡ E3 ‡ E4 R‡ ‡ P ‡ D ‡ U ‡ C R ‡P ‡D‡U‡C R‡ g ‡ Pg ‡ Dg ‡ Ug ‡ Cg R u ‡ P u ‡ Du ‡ Uu ‡ C u A ‡ E ‡ 2T A ‡ E ‡ 2T Ag ‡ Eg ‡ 2Tg Au ‡ Eu ‡ 2Tu A1 ‡ E ‡ T 1 ‡ T 2 A2 ‡ E ‡ T1 ‡ T2 A1 ‡ E ‡ T 1 ‡ T 2 A1 ‡ E ‡ T1 ‡ T2 A1g ‡ Eg ‡ T1g ‡ T2g A1u ‡ Eu ‡ T1u ‡ T2u G‡H G‡H Gg ‡ Hg Gu ‡ Hu

D6d C1v D1h T Th Td O Oh I Ih

A1g A1u A1 B1 R‡ R R‡ g Ru A A Ag Au A1 A2 A1 A1 A1g A1u A A Ag Au

Appendix 6: Spectroscopic Selection Rules 219 Tab. A6.2 Weights allowed under the irreducible representations (irreps) of the common molecular and crystallographic point groups in response tensors up to rank 4; ± signs indicate tensor parity.

Point group

Irrep

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + –

C1 Ci

A Ag Au A0 A00 A B A E A B E A E1 E2 A B E1 E2 A B E Ag Eg Au Eu A B E1 E2 E3 Ag E1g E2g Au E1u E2u Ag Bg Au Bu A0 E0 A00 E 00

…1† (1)

Cs C2 C3 C4

C5

C6

S4

S6

S8

S10

C2h

C3h

(1)

(1) (1) (1) (1) (1) (1) (1)

(1) (1) (1) (1) (1) (1) (1) (1)

(1) (1) (1)

(1) (1) (1)

(1)

(1)

(1)

(1)

(1) (1) (1) (1)

(1) (1)

(012) (012) (012) (12) (012) (12) (012) (12) (012) (2) (12) (012) (12) (2) (012) (12) (2) (012) (2) (12) (012) (12)

(1) (1) (1)

(012) (12) (012) (012) (12) (012) (12) (012) (2) (12) (012) (12) (2) (012) (12) (2) (2) (012) (12)

(012)

(2) (12) (012) (12) (2)

(1) (1) (1) (1) (1) (1) (1)

(3) (23) (123) (0123) (123) (23)

(12)

(0123) (123) (23) (3)

(0123) (123)

(12) (012) (2)

(01234) (1234) (01234) (1234) (01234) (1234) (01234) (234) (1234) (01234) (1234) (234) (01234) (34) (1234) (234) (01234) (234) (1234) (01234) (1234)

(01234) (4) (34) (234) (1234) (01234) (1234) (234)

(0123) (123) (23)

(012) (12) (012) (2)

(0123) (123) (0123) (0123) (123) (0123) (123) (0123) (23) (123) (0123) (123) (23) (0123) (3) (123) (23) (23) (0123) (123)

(01234) (01234)

(0123) (123)

(012) (12) (2)

(1) (1) (1) (1)

(0123) (123) (0123) (123) (0123) (123) (0123) (23) (123) (0123) (123) (23) (0123) (3) (123) (23) (0123) (23) (123) (0123) (123)

(0123)

(0123) (012) (12) (2)

(012) (12)

(1)

(0123) (0123)

(012) (12)

(1) (1)

(1)

(012)

(0123) (23) (3) (123)

(01234) (01234) (1234) (01234) (01234) (1234) (01234) (1234) (01234) (234) (1234) (01234) (1234) (234) (01234) (34) (1234) (234) (234) (01234) (1234)

(01234) (1234) (4) (01234) (1234) (234) (34)

(01234) (1234) (234) (01234) (1234)

(0123) (123) (3) (123) (0123) (23)

(01234) (234) (34) (1234)

(01234) (1234) (34) (1234) (01234) (234)

220

Appendix 6: Spectroscopic Selection Rules

Tab. A6.2 (continued)

Point group

Irrep

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + –

C4h

Ag Bg Eg Au Bu Eu A0 E10 E20 A00 E100 E200 Ag Bg E1g E2g Au Bu E1u E2u A1 A2 B1 B2 A1 A2 E A1 A2 B1 B2 E A1 A2 E1 E2 A1 A2 B1 B2 E1 E2

(1)

C5h

C6h

C2m

C3m

C4m

C5m

C6m

(012) (2) (12)

(1) (1)

(012) (2) (12)

(1) (1)

(0123) (23) (123)

(012) (1) (1)

(0123) (23) (012)

(12) (2)

(1)

(012)

(1)

(12) (2) (1)

(012)

(1) (02) (12) (12) (12) (02) (1) (12) (02) (1) (2) (2) (12) (02) (1) (12) (2) (02) (1)

(12) (2) (12) (02) (12) (12) (1) (02) (12) (1) (02) (2) (2) (12) (1) (02) (12) (2) (1) (02)

(12) (2)

(12) (2)

(1) (1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

(1)

(1) (1) (1)

(1) (1) (1) (1)

(1)

(1)

(1)

(0123) (23) (123)

(12) (2)

(1)

(01234) (234) (1234)

(123) (3) (0123) (3) (123) (23)

(023) (123) (123) (123) (023) (13) (123) (02) (13) (23) (23) (123) (02) (13) (123) (23) (02) (13) (3) (3) (123) (23)

(123) (3) (0123) (23)

(0123) (3) (123) (23) (123) (023) (123) (123) (13) (023) (123) (13) (02) (23) (23) (123) (13) (02) (123) (23) (13) (02) (3) (3) (123) (23)

(01234) (234) (1234) (01234) (4) (234) (1234) (34) (01234) (34) (1234) (234)

(0234) (1234) (1234) (1234) (0234) (134) (1234) (024) (134) (234) (234) (1234) (024) (13) (1234) (234) (024) (13) (34) (34) (1234) (234)

(1234) (34) (01234) (4) (234)

(01234) (34) (1234) (234) (1234) (0234) (1234) (1234) (134) (0234) (1234) (134) (024) (234) (234) (1234) (13) (024) (1234) (234) (13) (024) (34) (34) (1234) (234)

Appendix 6: Spectroscopic Selection Rules 221 Tab. A6.2 (continued)

Point group

Irrep

D2

A B1 B2 B3 A1 A2 E A1 A2 B1 B2 E A1 A2 E1 E2 A1 A2 B1 B2 E1 E2 Ag B1g B2g B3g Au B1u B2u B3u A01 A02 E0 A001 A002 E 00 A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu

D3

D4

D5

D6

D2h

D3h

D4h

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + – (1) (1) (1)

(1) (1) (1)

(1) (1)

(1) (1)

(1)

(1)

(1)

(1)

(1) (1)

(1) (1)

(1)

(1)

(1)

(1)

(1) (1) (1)

(02) (12) (12) (12) (02) (1) (12) (02) (1) (2) (2) (12) (02) (1) (12) (2) (02) (1)

(02) (12) (12) (12) (02) (1) (12) (02) (1) (2) (2) (12) (02) (1) (12) (2) (02) (1)

(12) (2) (02) (12) (12) (12)

(12) (2)

(02) (12) (12) (12)

(1) (1) (1) (1) (1)

(02) (1) (2)

(1) (1)

(12) (02) (1) (2) (2) (12)

(1)

(1) (1)

(1)

(023) (123) (123) (123) (023) (13) (123) (02) (13) (23) (23) (123) (02) (13) (123) (23) (02) (13) (3) (3) (123) (23) (023) (123) (123) (123)

(12) (02) (1) (2)

(02) (1) (2) (2) (12)

(02) (13) (23) (3) (3) (123) (02) (13) (23) (23) (123)

(023) (123) (123) (123) (023) (13) (123) (02) (13) (23) (23) (123) (02) (13) (123) (23) (02) (13) (3) (3) (123) (23)

(023) (123) (123) (123) (3) (3) (123) (02) (13) (23)

(02) (13) (23) (23) (123)

(0234) (1234) (1234) (1234) (0234) (134) (1234) (024) (134) (234) (234) (1234) (024) (13) (1234) (234) (024) (13) (34) (34) (1234) (234) (0234) (1234) (1234) (1234)

(024) (13) (234) (34) (34) (1234) (024) (134) (234) (234) (1234)

(0234) (1234) (1234) (1234) (0234) (134) (1234) (024) (134) (234) (234) (1234) (024) (13) (1234) (234) (024) (13) (34) (34) (1234) (234)

(0234) (1234) (1234) (1234) (34) (34) (1234) (024) (13) (234)

(024) (134) (234) (234) (1234)

222

Appendix 6: Spectroscopic Selection Rules

Tab. A6.2 (continued)

Point group

Irrep

D5h

A01 A02 E10 E20 A001 A002 E100 E200 A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u A1 A2 B1 B2 E A1g A2g Eg A1u A2u Eu A1 A2 B1 B2 E1 E2 E3 A1g A2g E1g E2g A1u A2u E1u E2u

D6h

D2d

D3d

D4d

D5d

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + – (02) (1)

(1) (1)

(02) (13) (12)

(2) (02) (1)

(1) (1)

(23)

(12) (2) (02) (1)

(1)

(1)

(12) (2) (02) (1)

(1)

(1)

(1)

(1)

(1) (1)

(1) (1)

(02) (1) (2) (2) (12) (02) (1) (12)

(12) (2) (2) (2) (02) (1) (12)

(02) (1) (1) (1) (2) (12) (02) (1) (12) (2)

(1) (1) (1)

(1) (1)

(02) (13) (23) (23) (123) (023) (13) (123)

(02) (1) (12)

(1) (1) (1)

(123) (3) (02) (13) (3) (3) (123) (23)

(123) (3) (02) (13) (23)

(02) (13) (3) (3) (123) (23) (23) (23) (02) (13) (123)

(02) (1) (12) (2)

(1234) (34) (024) (13) (34) (34) (1234) (234)

(024) (134) (234) (234) (1234) (0234) (134) (1234)

(023) (13) (123) (02) (13)

(02) (1) (12) (2)

(024) (13) (4) (234)

(3) (23) (123) (02) (13) (123) (23)

(02) (13) (123) (23) (3)

(02) (13) (123) (23)

(024) (13) (4) (4) (34) (234) (1234) (024) (13) (1234) (234)

(1234) (34) (024) (13) (4) (234)

(024) (13) (34) (34) (1234) (234) (234) (234) (024) (134) (1234)

(0234) (134) (1234) (4) (4) (024) (13) (1234) (234) (34)

(024) (13) (1234) (234)

Appendix 6: Spectroscopic Selection Rules 223 Tab. A6.2 (continued)

Point group

Irrep

D6d

A1 A2 B1 B2 E1 E2 E3 E4 E5 R‡ R P D U C R‡ g Rg Pg Dg Ug Cg R‡ u Ru Pu Du Uu Cu A E T Ag Eg Tg Au Eu Tu A1 A2 E T1 T2 A1 A2 E T1 T2

C1m

D1h

T

Th

Td

O

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + – (02) (1)

(1)

(02) (13) (02) (1) (12)

(1) (1) (2)

(02) (13) (123) (23) (3)

(2) (1) (1) (1) (1)

(1)

(12) (02) (1) (12) (2)

(02) (1) (12) (2)

(1) (1)

(1)

(1)

(1)

(0) (2) (12) (0) (2) (12)

(0) (2) (12)

(0)

(1)

(1)

(1)

(3) (23) (13) (02) (123) (23) (3)

(02) (13) (123) (23) (3)

(2) (1) (2) (0)

(0) (2) (2) (1) (0)

(2) (1) (2)

(2) (1) (2)

(234) (34) (4) (1234) (024) (13) (1234) (234) (34) (4) (024) (13) (1234) (234) (34) (4)

(13) (02) (123) (23) (3) (03) (2) (123) (03) (2) (123)

(0) (2) (12)

(1)

(1)

(123) (02) (13) (123) (23) (3)

(1) (02) (12) (2)

(1)

(1)

(1) (02) (12) (2)

(024) (13)

(0) (3) (2) (13) (23) (0) (3) (2) (13) (23)

(03) (2) (123)

(03) (2) (123) (3) (0) (2) (23) (13) (0) (3) (2) (13) (23)

(034) (24) (1234) (034) (24) (1234)

(04) (3) (24) (134) (234) (04) (3) (24) (134) (234)

(024) (13) (1234) (4) (34) (234) (13) (024) (1234) (234) (34) (4)

(13) (024) (1234) (234) (34) (4) (034) (24) (1234)

(034) (24) (1234) (3) (04) (24) (234) (134) (04) (3) (24) (134) (234)

224

Appendix 6: Spectroscopic Selection Rules

Tab. A6.2 (continued)

Point group

Irrep

Oh

A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u A T1 T2 G H Ag T1g T2g Gg Hg Au T1u T2u Gu Hu

I

Ih

Rank 1 Rank 1 Rank 2 Rank 2 Rank 3 Rank 3 Rank 4 Rank 4 + – + – + – + – (0)

(0) (3) (2) (13) (23)

(2) (1) (2)

(1)

(0)

(1)

(1)

(1)

(0) (1)

(2) (0) (1)

(1)

(2) (1) (2) (0) (1)

(2)

(2) (1)

(0) (1)

(2)

(0) (1) (3) (3) (2) (0) (1) (3) (3) (2)

(04) (3) (24) (134) (234) (0) (3) (2) (13) (23) (0) (1) (3) (3) (2)

(0) (1) (3) (3) (2)

(0) (1) (3) (34) (24) (0) (1) (3) (34) (24)

(04) (3) (24) (134) (234) (0) (1) (3) (34) (24)

(0) (1) (3) (34) (24)

225

Glossary of Symbols Latin Script Symbols a? …r† …k† ak y…k† ak b…r† …k† bk c d d? …r† ~? …r; t† d `

d e e? …r† e …k† ek E E0 Elocal Ex Er E~r g(n) h  h? …r† H

Vector potential operator Photon annihilation operator with arguments k, k Photon creation operator with arguments k, k Magnetic induction field operator Magnetic field polarisation vectors with arguments, k, k Speed of light in vacuo Classical microscopic electric displacement field Transverse displacement electric field operator Time- dependent transverse displacement electric field operator Dipole moment difference Classical microscopic electric field Transverse electric field operator Charge on an electron Electric field polarisation vectors, with arguments k, k Classical applied electric field Amplitude of the classical electric field Local electric field xth order electric multipole, where x is an integer Energy of molecular state with label r Molecular state energy with a finite lifetime hcr E~r ˆ Er 12 i nth order degree of coherence Planck’s constant divided by 2p Auxiliary magnetic field operator Hamiltonian operator

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

226

Glossary of Symbols

H0 Hmol Hrad Hint i ^i ^I I…x† InHG …k0 † ^j ^J J(t) k k kB ^ k ^ K lim km m m…n; r† m…n† Mx M Mfi n N n…x† O…n; r; r0 † p P p? …n; r† …k† Pk;x y…k†

Pk;x q

Unperturbed Hamiltonian operator Molecular Hamiltonian operator Radiation Hamiltonian operator Interaction Hamiltonian p operator Complex number 1 Cartesian radiation frame unit vector Cartesian surface harmonic reference frame unit vector Mean irradiance Radiant nth harmonic intensity Cartesian radiation frame unit vector Cartesian surface harmonic reference frame unit vector Temporal radiation envelope function Radiation wave-vector Magnitude of radiation wave-vector Boltzmann’s constant Cartesian radiation frame unit vector Cartesian surface harmonic reference frame unit vector Direction cosines between differing Cartesian reference frames An mth order system, where m is an integer Magnetisation vector field operator Molecular magnetic dipole operator xth order magnetic multipole, where x is an integer Number of cells within a crystal Matrix element (quantum amplitude) linking the initial and final states Label for the nth harmonic, where n is an integer Total number of molecules in a closed system Frequency-dependent refractive index Diamagnetisation tensor field operator Classical microscopic polarisation vector Classical macroscopic polarisation vector Transverse electric polarisation vector operator Polariton annihilation operator with arguments k, x; k Polariton creation operator with arguments k, x; k Photon mode occupation number

Glossary of Symbols

Q…n† r Rn s(m) t T u U(t,0) V ^ ^ Y; ^ Z X;

Molecular electric quadrupole operator Position vector Position of molecule e mth order radiation tensor Time Temperature Subscript label for a unit cell Time-evolution operator Quantisation volume Cartesian laboratory reference frame unit vectors

Italic Script Symbols D…n† I m n N P T

Beam parameter for an nth harmonic process Space inversion operator Polariton branch index Classical frequency-independent refractive index Mean number of molecules per unit volume General probability function Time-inversion operator

Greek Script Symbols a…n† a ja…k; k†i b c C d…x† dij eijk e0 gn j k l0 l

General nth order molecular polarisability tensor 2nd rank molecular polarisability tensor Coherent radiation state vector for the mode (k, k) 3rd rank molecular polarisability tensor 4th rank molecular polarisability tensor Transition rate Dirac delta function Kronecker delta tensor Levi-Civita antisymmetric tensor Electric permittivity of free space Coherence factor Relative permittivity Mode polarisation: Radiation wavelength Magnetic permeability of free space Classical molecular dipole

227

228

Glossary of Symbols

l0 lind l…n† m mg n q…m† q0…m† v w…t† v…n† x xk X

Classical static molecular dipole Classical induced molecular dipole Molecular electric dipole operator Radiation frequency Radiation group velocity Molecular label General mth rank radiation tensor Position-independent mth rank radiation tensor Classical electrical susceptibility (scalar) Time-dependent wavefunction Classical nth order electrical susceptibility tensor Circular frequency …x ˆ 2pm† Wave-vector- dependent circular frequency Solid angle (steradians)

229

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Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

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The following list will guide the reader to some of the more useful general texts available; although the list is categorised under several headings, many of the books will prove useful in more than one area.

Classical Optics and Electrodynamics Born, M. and E., Wolf, 1999, Principles of Optics, 7th edn, (University Press, Cambridge). Hopf, F. A. and G. I. Stegeman, 1985, Applied Classical Electrodynamics Vol. 1 (Wiley,New York). Jackson, J. D., 1999, Classical Electrodynamics, 3rd edn (Wiley, New York). Landau, L. D. and E. M. Lifshitz, 1960, Electrodynamics in Continuous Media (Pergamon, New York). Wilson, J. and J. F. B. Hawkes, 1983, Optoelectronics: An Introduction (Prentice-Hall, London).

Quantum Electrodynamics Berestetskii, V. B., E. M. Lifshitz and L. P. Pitaevskii, 1982, Quantum Electrodynamics (Pergamon, Oxford).

Cohen-Tannoudji, C., J. DupontRoc and G. Grynberg, 1989, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, New York). Cohen-Tannoudji, C., J. DupontRoc and G. Grynberg, 1992, Atom-Photon Interactions (Wiley, London). Compagno, G., R. Passante and F. Persico, 1995, Atom-Field Interactions and Dressed Atoms (University Press, Cambridge). Craig, D. P. and T. Thirunamachandran, 1984, Molecular Quantum Electrodynamics (Academic, London). Healy, W. P., 1982, Non-Relativistic Quantum Electrodynamics (Academic, New York). Milonni, P. W., 1994, The Quantum Vacuum. An Introduction to Quantum Electrodynamics (Academic, San Diego). Power, E. A., 1964, Introductory Quantum Electrodynamics (Longmans, London).

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Quantum and Nonlinear Optics Akhmanov, S. A. and R. V. Khokhlov, 1972, Problems of Nonlinear Optics (Gordon and Breach, New York). Bloembergen, N., 1965, Nonlinear Optics (Benjamin, New York). Boyd, R. W., 1992, Nonlinear Optics (Academic, Boston). Butcher, P. N. and D. Cotter, 1990, The Elements of Nonlinear Optics (University Press, Cambridge). Haken, H., 1984, Laser Theory (Springer, Berlin). Hanna, D. C., M. A. Yuratich and D. Cotter, 1979, Nonlinear Optics of Free Atoms and Molecules (Springer, Berlin). Lalanne, J. R., A. Ducasse and S. Kielich, 1996, Laser-Molecule Interaction. Laser Physics and Nonlinear Optics (Wiley, New York). Loudon, R., 2000, The Quantum Theory of Light, 3rd edn (Clarendon, Oxford). Louisell, W. H., 1973, Quantum Statistical Properties of Radiation (Wiley, New York). Mandel, L. and E. Wolf, 1995, Optical Coherence and Quantum Optics (University Press, Cambridge). Marcuse, D., 1980, Principles of Quantum Electronics (Academic, New York). Mukamel, S., 1995, Principles of Nonlinear Optical Spectroscopy, (Oxford University Press, Oxford). Nussenzveig, H. M., 1973, Introduction to Quantum Optics (Gordon and Breach, London). Reintjes, J. F., 1984, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, Orlando).

Shen, Y. R., 1984, The Principles of Nonlinear Optics (Wiley, New York). Yariv, A., 1989, Quantum Electronics (Wiley, New York) 3rd edition.

Molecular Materials for Nonlinear Optics Chemla, D. S. and J. Zyss, 1987, Nonlinear Optical Properties of Organic Molecules and Crystals Vols 1 and 2 (Academic, Orlando). Hann, R. A. and D. Bloor, 1989, Organic Materials for Nonlinear Optics (Royal Society of Chemistry, London). Kuzyk, M. G. and C. W. Dirk, 1998, Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials (Marcel Dekker, New York). Marder, S. R., J. E. Sohn and G. D. Stucky, 1991, Materials for Nonlinear Optics: Chemical Perspectives (American Chemical Society, Washington). Moloney, J. V., 1998, Nonlinear Optical Materials (The IMA Volumes in Mathematics and Its Applications/101) ed. (Springer, Berlin) Prasad, P. N. and D. J. Williams, 1991, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, New York). Wagnière, G. H., 1993, Linear and Nonlinear Optical Properties of Molecules (VCH, Zürich). Zyss, J., 1994, Molecular Nonlinear Optics: Materials, Physics and Devices, ed. (Academic, San Diego).

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Index a aggregates 87, 126 angular momentum 106, 157, 182 annihilation and creation operator 15, 24 auxiliary fields 21

b birefringence 95 Boltzmann weighting 117, 165, 196 Born-Oppenheimer approximation 167 Bose-Eistein distribution 46

c chiral molecules 112, 176 ff., 183 circular polarisation 43, 106, 112, 121, 134, 155, 156, 179 ff., 183 coherence lengh 95 coherent states 44 complete polarisation study 158 ff., 178 ff. completeness relation 169 convention implied summation 3

d damping 54, 185 ff. degrees of coherence 46 depolarisation 135, 155, 160, 162, 182 dipole approximation 4 dipole, induced 2, 7 dispersion 185

dispersion behaviour 53 ff., 93 down-conversion, degenerate 187

e electric dipole operator 18 electric displacement operator 15, 20 ff. electric field operator 17 electric field-induced second harmonic generation 161 electric polarisation 2 electric quadrupole interaction 70 ff., 104, 114, 122, 161 electric quadrupole operator 18 electrical permittivity 10 elliptical polarisation 43, 140 ergodic theorem 83, 191 Euler angle matrix 201

f fluctuation dipole operator 74 fourth harmonic generation 68 Franck-Condon overlap 168

g Golden Rule 32 group velocity 24

h Hopfield model 22 ff. hyperpolarisability 7, 59 hyper-Raman scattering 163 ff., 216

Optical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

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Index

hyper-Rayleigh scattering 170

151 ff.,

i implied summation, convention 3 index permutational symmetry 62 ff., 206 induced dipole 2, 7 irradiance 46 irreducible Cartesian tensors 205 isotropic tensors 201 ff.

k Kleinman symmetry 67, 71, 119, 135, 154 ff., 157, 160 Kronecker delta 201

l Levi-Civita antisymmetric tensor 202 lifetime 28 line-broadening 55 longitudinal polarisation 114, 155 Lonrentz factor 10 Lorentz-Lorenz equation 10, 25

m macromolecules 87, 126 magnetic dipole interaction 13, 71, 122, 184 magnetic dipole operator 19 magnetic induction operator 15, 17 magnetic permeability 22 Manley-Rowe relation 51 membranes 87, 125 ff. mode-locking 49 multiple coupling 104 multipolar Hamiltonian 13 multipole coupling 18

p parity 17, 19, 30, 67, 176 permittivity, electrical 10 ff. perturbation theory 27 ff.

phase-matching, types I and II 95 photon arrival times 47 photonic band-gap materials 22 plane polarisation 43 Poisson distribution 45 polarisability 185 polar molecular 115 ff. polar molecules 7, 69, 73 ff., 196 polarisability 7, 25, 56 ff. polarisability volume 9 polarisation, electric 2 ff. polaritons 21 ff. poling 117 population grating 142 ff. pulsed laser radiation 48

q quantisation volume 46 ff.

r radiant intensity 101 Rayleigh scattering 4, 159 refractive index 4, 10, 24, 58 ff. resonance 53 ff., 72 ff., 77, 113, 128, 185 ff. reversal ratio 157 ff., 182 ff. rotational averaging 191 ff. rotational symmetry 66, 69, 114, 124

s scalar susceptibility 2 second harmonic scattering 151 ff. secular resonances 123, 209 ff. selection rules, spectroscopic 215 ff. semiclassical approach 1 ff. seniority index 206 six-wave mixing 131 ff., 209 ff. space-inversion 16 ff., 30 spherical Bessel functions 119 f. state sequence diagrams 34 ff. structure factor 81 surface chirality 112 ff. surface second harmonic generation 71, 109 ff.

Index

susceptibility – linear 2 ff. – non linear 5 – scalar 2 symmetry, molecular 176 ff.

time-reserval 16 ff., 30, 186 two-level system 71 ff., 128 ff. two-photon absorption 142 66 ff.,

u unpolarised light

159, 180

t

v

thermallight 46 third harmonic generation 65, 68, 75 ff., 107 ff., 128, 161 time reversal 187 time-dependent Schrödinger equation 28 time-energy uncertainty principle 27, 36 time-evolution operator 29 ff. time-ordered diagrams 35

vacuum energy 16, 18 vector potential 15 – operator 15 ff. vibrational levels 61, 164 ff., 216

w wave-vector 3, 22 ff. wave-vector matching 111, 134, 151 weight, tensor 205

81, 90 ff.,

241