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 <@xk n xk = :

…m†

@xk

1

…m†

@xk ˆ ; @k

;

…1:3:5†

incorporating the frequency-dependent refractive index defined as   …m† ˆ1‡ n2 xk

D  E N a x…m† k e0 : N D  …m† E a xk 1 3e0

…1:3:6†

1.3 Media Corrections 25

Equation (1.3.6) represents a quantum analogue of the LorentzLorenz equation, (1.1.25). Here hai is the average molecular polarisability of the homogeneous host, based on an electronically isotropic medium, and N is the number density of host molecules. In the majority of applications to quantum and nonlinear optical phenomena it is only the photon-like branches of the dispersion curves that are of interest and the m index in the above expressions can generally be left implicit. Again, as we shall be …k† concerned with photonic regions it is legitimate to engage ak y…k† …k† y…k† and ak in place of Pk and Pk . In keeping with this policy, and to retain conceptual linkage with the vacuum formulation, we shall from here on revert to a description in terms of photons, although it is to be understood that the quanta involved in condensed phase interactions are strictly speaking optical branch polaritons. The case of vacuum propagation can then be viewed as a special case of the more general formalism. For example if the refractive index is set to unity in equation (1.3.2), the group velocity is simply c, the polariton operators become identical to those representing the annihilation and creation of vacuum photons, and the expression reduces directly to the fundamental electric field operator of equation (1.2.11), multiplied by e0. At the end of this chapter we now have a formalism in place to fully represent both linear and nonlinear optical interactions in molecular systems. For condensed phase applications, the theory duly accounts for refractive and dissipative modifications of electromagnetic fields associated with the dispersive electronic properties of the medium. The theory is not based on any phenomenological ad hoc approach, but on the founding principles of molecular quantum electrodynamics. As a result the necessary local field corrections naturally emerge from the detailed form of the auxiliary field operators – obviating the need to encompass such features indirectly in terms of macroscopic bulk susceptibilities, the device commonly employed in the semiclassical theory. As emerges in the following, the QED approach is also amenable to the consideration of quantum optical issues, and it facilitates the complete embodiment of higher orders of multipole interaction. A theory cast in these terms allows legitimate application of the photon concept to advance the description of nonlinear optical processes in molecular systems.

27

2

Perturbation Theory Double, double toil and trouble Macbeth IV.i.10, William Shakespeare

With the foundations laid in the previous chapter, the next stage of development is to establish the mode of engagement of the quantum field operators in nonlinear optical applications. For reasons associated with the time-energy uncertainty principle each conversion of input photons to an optical harmonic photon is essentially instantaneous (see Fig. 2.1). However not all the optical input becomes converted, and the key issue is the rate at which output radiation is generated. Thus, our aim must be to understand how the quantum amplitudes for the conversion processes, and ultimately the measurable rates of signal production, emerge from the theory. To this end we need to be able to predict the state of the radiation-matter system, given its initial conditions, at any later time; to answer such a question we look to approximate methods. As is well known, exact quantum mechanical calculations are impossible in all but the very simplest systems. Here we employ time-dependent perturbation theory to obtain approximate solutions for the time development. The success of this method depends on the general validity of the underlying assumption that the material system is only slightly perturbed – which broadly means that its molecular constituents retain their integrity (and also that the radiation propagating through their midst is adequately described in terms of medium-dressed photons). These assumptions are generally valid as the optical radiation, even from coherent laser sources, is associated with fields much weaker than the intramolecular Coulombic fields. If however the apOptical Harmonics in Molecular Systems: Quantum Electrodynamical Theory. David L. Andrews, Philip Allcock Copyright © 2002 WILEY-VCH Verlag GmbH, Weinheim ISBN: 3-527-40317-5

28

2 Perturbation Theory

Fig. 2.1 In second harmonic generation, the lifetime of an excited state jui displaced by an energy DE from a two-photon resonance cannot exceed h=4DE.

plied and internal fields are of the same order of magnitude, as when advanced ultra-intense pulsed lasers are deployed (irradiances in the region or in excess of 1015 W m–2), then alternative calculational methods must be employed. However at such levels of irradiance other, more destructive, interactions occur which fall outside the subject area of this book.

2.1

Time-Dependent Perturbation Theory With the full Hamiltonian given by equation (1.2.1), the timeevolution of the system wavefunction w is determined by the time-dependent Schrödinger equation: i h

@w…t† ˆ Hw…t† : @t

…2:1:1†

As outlined above, high precision solutions for w…t†, incorporating media effects, can be derived on the assumption that the coupling between matter and radiation is treated as a perturbation on the eigenstates of H0 , where X0 Hmol …n† ; …2:1:2† H0 ˆ Hbath ‡ n

thereby defining:

2.1 Time-Dependent Perturbation Theory

Hbath ˆ Hrad ‡

X 00

…Hmol …n† ‡ Hint …n†† :

…2:1:3†

n

In equation (2.1.2) the prime on the summation denotes a restriction to those molecules whose transitions are engaged either directly or indirectly in the sought optical response, which we shall refer to as ‘optically prominent’. The double prime in equation (2.1.3) denotes a sum excluding those molecules. The basis eigenstates of H0 thus comprise products of the Hamiltonian eigenstates for the optically prominent molecules and the dressed photon eigenstates of Hbath . The full Hamiltonian is thus H ˆ H0 ‡ Hint ;

…2:1:4†

with the interaction operator Hint given by X0 Hint …n† : Hint ˆ

…2:1:5†

n

It is this operator that effects a perturbation on the eigenstates of H0 and so leads to transitions between system states – in the present context, the conversion of pump radiation to optical harmonics. Given a complete matter-radiation system in an eigenstate jw…0†i of H0 at time t ˆ 0, the system wavefunction at any later time t is expressible as h†U…t; 0†jw…0†i ; jw…t†i ˆ exp… iH0 t=

…2:1:6†

defining a unitary time-evolution operator U…t; 0† for the temporal development of the system in the time interval …0; t†. Substitution of (2.1.6) into equation (2.1.1) leads to an exact result for U…t; 0†, expressible as the following series expansion: U…t; 0† ˆ 1 ‡

1 X

Z t Zt1 …i h†

mˆ1

m

Ztm

1




t0

t0

t0

~int …t2 † . . . H ~int …tm †dt1 dt2 . . . dtm ;  H~int …t1 †H

…2:1:7†

~int …t† is the interaction representation of the operator rewhere H sponsible for the coupling between light and matter, given by

29

30

2 Perturbation Theory

~int …t† ˆ eiH0 t=h Hint e H ˆ

X0

iH0 t= h

ˆ

X0

eiH0 t=h Hint …n†e

iH0 t= h

n

~int …n; t† : H

…2:1:8†

n

In the electric dipole approximation, individual interaction terms in the sum have the following expression: ~int …n; t† ˆ H

~ n ; t† ; e0 1 l…n†
d…R

…2:1:9†

where at a general position r the microscopic electric displace~ t†; in the interaction representation may conment operator, d…r; veniently be expressed as a sum of two parts: ~… † …r; t† ; ~ t† ˆ d ~…‡† …r; t† ‡ d d…r;

…2:1:10†

8 91=2 0  …m†  1 2 …m† …m† = < ‡ 2 n x X hmg xk e0 k ~…‡† …r; t† ˆi @ A   d …m† ; : 3 k;k;m 2cVn xk …k† …k†

 ek ak ei…k
r ~… † …r; t† ˆ d

…m†

xk t†

;

…2:1:11†

8 91=2 0  …m†  1 2 …m† …m† = ‡ 2 n x X<  hmg xk e0 k @ A   i …m† ; : 3 k;k;m 2cVn xk …k† y…k†

 ek ak e

…m†

i…k
r xk t†

:

…2:1:12†

In this form, the time-even and space-odd character of the electric displacement (see chapter 1.2) is again apparent. Equally, a corresponding development of the magnetic field reveals its time-odd and space-even character. In developing the quantum amplitude for a particular optical process it is necessary to determine matrix elements of the timeevolution operator; to this end it is expedient to invoke an expansion in terms of operators rather than the embedded time integrals of equation (2.1.7). The method of resolvent operators, which in fact affords a framework for both perturbative and nonperturbative analysis (Cohen-Tannoudji et al. 1992; Mukamel 1995; Dávila Romero 1999), proceeds through the introduction of a retarded Green’s function K‡ …t† ˆ U…t; 0†h…t†, together with its

2.1 Time-Dependent Perturbation Theory

advanced counterpart K …t† ˆ U…t; 0†h… t†, where h…t† is the Heaviside function. The introduction of K+ and K– allows extension of the temporal domain for the integrand of the evolution operator to infinity, enabling expression as U…t; 0† ˆ K‡ …t†

K …t† ;

…2:1:13†

with i K …t† ˆ eiEf t=h 2p

Z‡1 eiEt=h G …E†dE

…2:1:14†

1

and where Ef denotes the final system energy (which equates exactly to the initial energy). Here, the retarded and advanced propagators, G‡ …E† and G …E† respectively, are Fourier transforms of the corresponding Green’s functions 1 G …E† ˆ ih

Z‡1 eiEs=h K …s†ds ˆ lim …E g!0‡

1

H  ig†

1

:

…2:1:15†

In the implementation of equation (2.1.7) through its above development only the retarded function is retained, as is usual in optical applications; it alone generates causal solutions for positive time intervals. The next stage of the development involves determination of the matrix elements of the time-evolution operator, using equations (2.1.13)–(2.1.15). To this end it is convenient to define a set of sub-systems, each comprising one of the optically prominent molecules n and the bath. Introducing and expanding the corresponding resolvent operator, Tsub…n† (Löwdin 1965), we have Tsub…n† …z† ˆ …z ‡ …z

H0

Hint …n††

1

ˆ …z

H0 † 1 Hint …n†…z

H0 †

H0 †

1

1

‡ …z H0 † 1 Hint …n†…z H0 † 1 Hint …n†…z 1 X ‰T0 …z†Hint …n†Šp T0 …z† ; ˆ pˆ0

where z is a complex argument and

H0 †

1

...

…2:1:16†

31

32

2 Perturbation Theory

T0 …z† ˆ …z

H0 †

1

:

…2:1:17†

Through use of this result the requisite quantum amplitudes, i.e. matrix elements of the time-evolution operator, can be determined. Specifically, for a process associated with an initial system state jii and a final system state jf i we obtain a time-dependent quantum amplitude cfi …t† which can be evaluated from the equation I 1 iEf t=h e e izt=h hf jTsub…n† …z†jiidz cfi …t† ˆ hf jU…t; 0†jii ˆ 2pi 1 I 1 iEf t=h X e e izt=h hf j‰T0 …z†Hint …n†Šp T0 …z†jiidz ; ˆ 2pi pˆ0 …2:1:18† where the contour for integration encompasses the real axis. 1) Using the method of residues to evaluate the contour integral, and discarding optical-frequency oscillations, we obtain cfi …t† ˆ dfi

…Ef

Ei †

1

X

 Mfi …n† ei…Ef

Ei †t= h

 1 ;

…2:1:19†

n

where Mfi is termed the matrix element for the process (see below). The leading Kronecker delta in equation (2.1.19) trivially denotes the non-process in which the initial and final states of the entire system are identical, and can be discarded in applications to any real process. Using the relation eix 1 ˆ sin x2 eix=2 the subsequent contribution leads to a time-independent rate equation expressed by the familiar Golden Rule (Fermi 1950, Schiff 1968, Yariv 1982) 2   X 2p Mfi …n† d…Ef Cˆ h 

Ei † ;

…2:1:20†

n

where use is made of the long-time limit sinc2 ‰…Ef Ei †t=2 hŠ ! …2p h=t†d…Ef Ei †. Equation (2.1.20) is cast in terms of a Dirac delta function (Dirac 1958) which signifies overall energy conservation, and a transition matrix whose elements are 1) Contributions associated with advanced solutions disappear at this point.

2.2 Time-Orderings and State Sequences

Mfi …n† ˆ hfsub…n† jHint …n† ‡ Hint …n†Tsub…n† Hint …n†jisub…n† i 1 X Hint …n†‰T0 Hint …n†Šp ˆ hfsub…n† jHint …n† ‡ pˆ0

 T0 Hint …n†jisub…n† i ;

…2:1:21†

and where the resolvent operators T0 and Tsub…n† are evaluated for z ˆ E0.

2.2

Time-Orderings and State Sequences In evaluating the matrix elements for each n, it is immediately evident that every other molecule yields a vanishing contribution to the summands on the right of equation (2.1.21). Hence, by invoking the completeness relation for the sub-system states, the matrix elements Mfi can succinctly be expressed as Mfi ˆ hf jHint jii ‡

r …1† …2† …2†

‡ r

‡

X hf jHint jr …1† ihr …1† jHint jii …Ei

Er …1† †

X hf jHint jr ihr jHint jr …1† ihr …1† jHint jii …Ei Er …2† †…Ei Er …1† † …2† …1† ;r

X r …3† ;r …2† ;r

hf jHint jr …3† ihr …3† jHint jr …2† ihr …2† jHint jr …1† ihr …1† jHint jii ‡ ... …Ei Er …3† †…Ei Er …2† †…Ei Er …1† † …1† …2:2:1†

where all states and energies are eigenstates of H0 , with the summations over virtual intermediate states r(1), r(2), . . . taken over all such system states excluding i or f. It is worth noting that this specifically does not exclude either the initial or final molecular state from inclusion in the intermediate state summations, provided that those states are not associated in the summation with radiation states that also correspond to the initial or final conditions. This is a point which has often been misunderstood but which is quite clear in the quantum electrodynamical formulation. It is an issue of some importance in connection

33

34

2 Perturbation Theory

with the role of static moments in optical response, as will be seen in chapter 4. For consistency with the policy adopted in chapter 1 to convey concepts in familiar language, we shall from hereon again adopt a description of the (implicitly dressed) quanta of the bath as photons, also reverting to the more familiar term radiation in denoting the bath per se. In general the representation of any mphoton interaction entails terms involving m photon operators. In view of the linearity in the photon creation and annihilation operators of all the electric and magnetic multipole interactions, resulting from the structure of equations (1.3.3) and (1.3.4), this representation generates quantum amplitudes whose leading contributions come from terms involving the mth power of Hint . For most processes it is sufficient to consider only the leading non-vanishing term. Writing the system states explicitly as products of molecular and radiation states, we then have …m†

Mfi

ˆ

X

...

…1† rmol

XX …m 1† rmol

…m 1†

 hrrad 

...

…1† rrad

…m 1†

X

…m 1†

…m 1†

hfrad ; fmol jHint jrmol ; rrad

i

…m 1† rrad

…m 2†

…m 2†

; rmol jHint jrmol ; rrad

i




…1† …1† hrrad ; rmol jHint jimol ; irad i

h   i 1 Eimol Er …m 1† ‡ Eirad Er …m 1† ... mol rad h   i 1  Eimol Er …1† ‡ Eirad Er …1† 

mol

rad

…2:2:2†

In each of the (m–1) summations over the intermediate radiation …j† states jrrad i, there are only a limited number of states that can make non-vanishing contributions, as determined by the sequencing of the creation and annihilation events associated with each photon involved in the overall interaction. Each of these sequences is conventionally represented using time-ordered graphs (Ward 1965). Calculations based on this method are commonly expedited by the construction of all topologically different diagrams connecting the same initial and final states: the summa…1† …m 1† tions over the intermediate states jrrad i to jrrad i to in equation (2.2.2) are then equivalent to summations over the various timeorderings. An alternative method based on state sequences (Jenkins and Andrews 1998, Jenkins et al. 2002) enables the com-

2.2 Time-Orderings and State Sequences

plete set of interaction sequences for any process to be cast in the form of a single network. The latter method offers a more concise presentation and improved calculational expediency, particularly in the case of high-order interactions. Optical frequency doubling, or second harmonic generation (SHG) affords a simple illustration of each of these two diagrammatic methods. For clarity in developing this example, it is expedient first to write down the system states for the conversion process (although familiarity with either method in fact enables this information to be read off directly from either form of diagram). The initial and final states for the interaction are as follows: jimol ; irad i ˆ jE0 ; q…k; k†; q0 …k0 ; k0 †i ; jfmol ; frad i ˆ jE0 ; …q

…2:2:3†

2†…k; k†; …q0 ‡ 1†…k0 ; k0 †i ;

…2:2:4†

given that the molecule is initially in its ground state, and that there are initially q photons of the pump mode …k; k† and q0 photons of the harmonic mode …k0 ; k0 †. The value of q0 is left unspecified for generality; it may be zero. There are three possible orderings of photon annihilation and creation (a, b and c) that can convert the initial to the final state, each involving different virtual intermediate states. The intermediate state labels r(1) and r(2) are here re-designated as r and s, respectively: 

jrmol ; rrad i ˆ jEr ; …q jsmol ; srad i ˆ jEs ; …q

1†…k; k†; q0 …k0 ; k0 †i 2†…k; k†; q0 …k0 ; k0 †i

jrmol ; rrad i ˆ jEr ; …q jsmol ; srad i ˆ jEs ; …q

1†…k; k†; q0 …k0 ; k0 †i 1†…k; k†; …q0 ‡ 1†…k0 ; k0 †i

jrmol ; rrad i ˆ jEr ; q…k; k†; …q0 ‡ 1†…k0 ; k0 †i jsmol ; srad i ˆ jEs ; …q 1†…k; k†; …q0 ‡ 1†…k0 ; k0 †i

…2:2:5 a†  …2:2:5 b†  …2:2:5 c†

as represented by the three time-ordered diagrams of Fig. 2.2. For example in Fig. 2.2 (a), the sequence of interactions is as follows. Annihilation of one photon from the pump mode is associated by the transition of a molecule, initially in its ground state j0i, to a virtual state jri. With the annihilation of a second pump photon the molecule proceeds to a state jsi. The sequence concludes with the emission of a second harmonic photon, the mol-

35

36

2 Perturbation Theory

Fig. 2.2 The three time-ordered diagrams for SHG. Pump photons of wave-vector k and polarisation k impinge on molecular world line from the left and the harmonic …k0 ; k0 † leaves from the right. It is assumed that the initial and final state is the molecular ground state j0i; intermediate states are labelled jri and jsi respectively.

ecule returning to its ground state. Figure 2.2 (b) and (c) represent the two other sequences in which emission of the harmonic photon precedes either one or both of the pump photon interactions. It is important to emphasise that no single time-ordered diagram represents a physically distinguishable process; these diagrams are ultimately calculational aids based on the approximations of perturbation theory. As was established earlier, the photon creation and annihilation events which take place collectively at each molecule appear simultaneous, as far as real experimental measurements with finite time resolution are concerned. However the time-energy uncertainty relation does permit shortlived intermediate states which are not properly energy-conserving. This helps explain why it is necessary to include the diagrams corresponding to extremely counterintuitive time-sequences such as that of Fig. 2.2 (c). Nonetheless, all possible interaction sequences must be included in calculating any matrix element. The state sequence diagram of Fig. 2.3 accommodates all three routes between the initial and final state, those states being represented by boxes on the far left and far right of the diagram respectively. The intervening columns represent intermediate system states jr …1† i and jr …2† i, connected by links which represent

2.2 Time-Orderings and State Sequences

Fig. 2.3 State sequence diagram for SHG. The initial state is represented by the solitary box in the column on the left (outlined by the dotted lines) and the final state by the corresponding the box on the right; columns with more then one box indicate the virtual states jri and jsi. An example is afforded by the uppermost route through the diagram, which corresponds identically to the time-ordering of Fig. 2.2(c).

valid operations by Hint on a preceding state. In these diagrams, lines represent interactions; vertices represent states (compare time-ordered diagrams, where the converse applies). However the state sequence diagram for any given process represents the entire perturbative construct of the quantum amplitude, each path through it corresponding to one time-ordering, whereas no single time-ordered diagram conveys the same complete information content. For example the lowest path in Fig. 2.3 relates to the time-ordering labelled (a) in Fig. 2.2. In general the state sequence representation presents considerable advantages, especially in applications to high- order optical interactions, as will become evident in later chapters.

37

39

3

Radiation Constructs We shall all be changed, in a moment, in the twinkling of an eye 1 Corinthians 15:51

We are now in a position to make progress towards writing the quantum amplitude for any optical process of interest, in a form suitable for determination of the signal intensity through the Golden Rule. To proceed with the development, it is useful to identify within the quantum amplitude those elements that relate solely to radiation and molecular properties respectively. It is understood that any molecular response is parametrically dependent on the properties of the radiation, most obviously through its frequency-dependence. Equally, properties of the radiation within the medium depend on the material nature of the latter. Establishing a separation of the quantum amplitude as a product of radiation and molecular tensors facilitates investigation of the salient properties of each. This is an important step in our calculational procedure – and one that differs markedly from the traditional semiclassical approach, where local field corrections are carried by molecular polarisabilities as outlined in chapter 1. Here, however, local field corrections are cast as an integral component of the radiation tensors, emerging from the form of the quantum field operators (1.3.2) and (1.3.3). This reflects a more fundamental approach to dealing with the concept of displacement fields. Initially we identify a position-independent radiation tensor by extracting optical phase factors. We highlight some of the calculational methods associated with development of the radiation terms and we more briefly address the modifications required for an optical input from a pulsed laser source. 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

40

3 Radiation Constructs

3.1

Radiation Tensor Construction Consider an m-photon process involving modes of radiation …k1 ; k1 †; . . . ; …km ; km †, all potentially different, initially containing q1 ; . . . ; qm photons respectively. Equation (2.2.2) may be cast as the following tensor product: …m†

…m†

Mfi f…k1 ; k1 †. . . …km ; km †g ˆ… 1†m 1 e0 m ai1 ...im …x1 ; . . . ; xm †qi1 ...im  f…k1 ; k1 †; . . . ; …km ; km †g

…3:1:1†

Specifically, this equation entails the m-fold tensor contraction 1) of a…m† , a rank m nonlinear polarisability tensor containing all the molecular variables (with parametric dependence on the optical frequencies), with q…m† , a tensor constructed solely from radiation parameters including the local field corrections. For conciseness in both tensors, reference to a specific molecule is at present left implicit – in other words the molecular tensor refers to an unspecified molecule n at whose location the radiation tensor is evaluated. For n-harmonic generation the tensor is written as a…m† … xm ; x1 ; . . . ; x1 †, with the argument xm indicating creation of a single photon of frequency xm  nx1, and the following n arguments xi, indicating the annihilation of n ˆ …m 1† photons with frequency x1. Note that equation (3.1.1) incorporates a sign pre-factor introduced so that, in the structure of the molecular tensors to emerge in chapter 4, each denominator term is associated with a positive energy difference, assuming that all intermediate states are above the ground molecular state energy. For our present purpose we defer the explicit expressions for the molecular tensors a…m† (which, it is to be recalled, accommodate contributions from the various state sequences or time-orderings – see section 2.2) and focus on identifying the structure of the radiation tensor q…m†. In the electric dipole approximation q…m† is explicitly given by the following outer product of m matrix elements of the electric displacement operator: 1) 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).

3.1 Radiation Tensor Construction 41

q…m† f…k1 ; k1 †; . . . ; …km ; km †g ˆ h…q1 ‡ 1†…k1 ; k1 †jd? jq1 …k1 ; k1 †i . . .  h…qm

1†…km ; km †jd? jqm …km ; km †i :

…3:1:2†

Incorporation of the matrix elements of equation (3.1.1) into the rate equation (2.1.20) entails summation, over all molecules in the system, of the tensor product in the former. To this end it is expedient to isolate the one part of the above radiation tensor that is molecule-specific. This relates to the fact that q…m† is a tensor field, sensitive to the position of the molecule at which it is evaluated through the implicit position-dependence of d? – recall the phase factors of equations (1.3.2) and (1.3.3). The tensor representing the radiation field for the complete optical process at molecule n may be written, irrespective of the order or nature of the multipolar interactions involved, in the following general form: q…m† …n† ˆ q 0…m† exp…iDk
Rn † :

…3:1:3†

Here q 0…m† is a position-independent radiation tensor, Rn is the molecular position vector relative to an arbitrary fixed origin, and Dk the wave-vector mismatch for the process as defined by Dk ˆ

m X

gr kr ;

…3:1:4†

r

with gr taking the value of +1 or –1 for each photon absorbed or emitted, respectively. For instance the radiation tensor for frequency doubling carries the phase factor exp‰i…2k k0 †
Rn Š, and hence Dk ˆ …2k k0 †. Such features are responsible for the pivotal issue of coherence. Combining the above results it transpires that, for any optical process, the rate may be written in a general form as 2 N 2p X m …n† 0 iDk
Rn e0 ai1 i2 ...im qi1 i2 ...im e Cˆ d…Ei h n 

Ef † :

…3:1:5†

As the tensor contraction is explicitly highlighted through the repeated indices, the superscript designating the mth ordering is

42

3 Radiation Constructs

dropped in the above equation. Nonetheless, to highlight the sum over all molecules within the closed system we utilise the now vacant superscript to explicitly label each molecular response tensor a. Equation (3.1.5) underpins the following theoretical development; the precise structure of the molecular and radiation tensors it involves will depend on the detailed nature of the optical process to be modelled. As we have shown, radiative features that are of interest are cast in the tensor q…m† introduced in equation (3.1.2). For any mth order process the components of this tensor are explicitly given by a position-independent expression, which then follows from equation (3.1.3): 2

q0i1 ...im

! 3 0…m† 2 n …x † ‡ 2 0 0 k 4 ˆ … i†…m n† ekk0 5 0…m† 3 2cVn…xk0 † m iˆ1 i 2 !1=2 ! 3 …m† …m† …m† n Y X q hmg xk e0 n2 …xk † ‡ 2 k 5 ek  in 4 …m† 3 2cVn…xk † m jˆ1 m Yn

0…m† 0…m† X q0  he0 mg xk0

!1=2

j

…3:1:6† where there are m–n and n photons emitted into and absorbed from the ith and jth modes respectively 2) within the quantisation volume V. Later, in formulating the nonlinear response in real systems, we show how this equation is adapted specifically for molecular harmonics. If initially there are no photons of a particular mode present, as in spontaneous emission processes, q0 is equal to zero and the photon creation operator acts on the vacuum state. We can introduce a symbol s to represent a polarisation tensor defined as, si1 ...im f…k1 ; k1 †; . . . ; …km ; km †g ˆ …~ekk11 †i1 . . . …~ekkmm †im :

…3:1:7†

On the right of this equation the arguments, k and k, associated with each polarisation vector, are dropped, for brevity, since the index labels provide the necessary associations. The tilde designa2) 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.

3.1 Radiation Tensor Construction 43

tion signifies that for each photon that is annihilated in the interaction process the appropriate vector is precisely the unit polarisation vector ~ e ˆ e; for created photons ~ e ˆ e, the overbar representing complex conjugation. We note, for future reference, that it is customary when two or more photons belong to the same radiation mode (as for example is the case for the input in SHG) to place the corresponding indices of the s tensor in parentheses to signify permutational symmetry. ^ deWith a right-handed Cartesian set of unit vectors …^i; ^j; k† fined for each mode …k; k†, plane polarisations have real unit vectors expressible as e ˆ cos h^i ‡ sin h^j. The complex representation of polarisation vectors also admits radiation states associated with chirality. For example circular polarisations of left or right handedness have ^j components retarded by a phase factor of exp…ip=2† with respect to the ^i component, or advanced by exp… ip=2†, respectively  p  …3:1:8† e…L† ˆ 1= 2 ^i ‡ i^j ;  p e…R† ˆ 1= 2 ^i

 i^j :

…3:1:9†

The above expressions for plane and circular polarisation vectors are special cases of the general result that allows the representation of states with arbitrary plane, circular or elliptical polarisation to be delineated from their linear combination represented through; e…h; y; f† ˆ eiy cos h^i ‡ eif sin h^j :

…3:1:10†

where the arguments y and f refer specifically to the phase angles associated with the ^i and ^j components respectively. For example, special cases arise from the arguments as follows;

…h; y; f† ` !

8 > > > > > > > < > > > > > > > :

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 <

9 !1=2   = mg 0 x0k0 e0 h n2 …x0k0 † ‡ 2 0 1=2 0 0 ik
r ˆ i ‡ 1† e e …q i : ; 3 2cVn…x0k0 † ( )    h mg xk e0 1=2 n2 …xk † ‡ 2  i …q 1†1=2 ej eik
r 3 2cVn…xk † ( )    h mg xk e0 1=2 n2 …xk † ‡ 2 1=2 ik
r q ek e  i 3 2cVn…xk † !1=2  2    2 h mg 0 x0k0 e0 n2 …x0k0 † ‡ 2 hmg xk e0  n …xk † ‡ 2 ˆi 3 3 2cVn…xk † 2cVn…x0k0 †  f…q0 ‡ 1†…q

0 1=2 0 ei ej ek ei…2k k †
r

1†qg

…4:2:2†

From the right-hand side of equation (4.2.2), it is clear that qijk is symmetric with respect to exchange of the indices j and k, i.e. qijk ˆ qikj . This feature of the tensor is conveniently represented by placing brackets around the indices j and k, i.e. qi…jk† .

63

64

4 Molecular Properties

Now consider the result of this index symmetry in the radiation tensor on the matrix element, in the specific case of frequency doubling given by Mfi ˆ

e0 3 bijk … 2x; x; x†qi…jk† :

…4:2:3†

Because of the summation over the indices i, j and k involved in the explicit contraction of the tensors b and q, as written above, it becomes evident that only the jk symmetric part of the molecular tensor can contribute to the process. This is exhibited by first writing the molecular tensor as a sum of two parts; bi…jk† … 2x; x; x†, which is jk symmetric, and b i‰jkŠ … 2x; x; x† which is jk antisymmetric, i.e. changes sign on exchange of the indices j and k: bijk … 2x; x; x† ˆ b i…jk† … 2x; x; x† ‡ b i‰jkŠ … 2x; x; x† ;

…4:2:4†

where; n o bi…jk† … 2x; x; x† ˆ 12 bijk … 2x; x; x† ‡ b ikj … 2x; x; x† ; …4:2:5† n bi‰jkŠ … 2x; x; x† ˆ 12 bijk … 2x; x; x†

o bikj … 2x; x; x† : …4:2:6†

Thus, from equation (4.2.3), we have Mfi ˆ

e0 3 bi…jk† … 2x; x; x†qi…jk†

e0 3 bi‰jkŠ … 2x; x; x†qi…jk† ; …4:2:7†

where the second term must be zero because it is equal to its own negative. (Since in equation (4.2.7) the indices j and k are repeated and therefore summed, they are dummy indices and can be interchanged throughout; in so doing the first term is unchanged but the second reverses its sign.) Consequently it is permissible to re-express (4.2.7) as Mfi ˆ

e0 3 bi…jk† … 2x; x; x†qi…jk† ;

…4:2:8†

4.2 Symmetry

which demonstrates that only the correspondingly index-symmetric part of the molecular tensor plays any part in the process. The fact that any j,k antisymmetric tensor contracts with qi…jk† to give a vanishing result incidentally means that it is always possible to add an arbitrary tensor Ui‰jkŠ to bijk without affecting the derived value of the observable. This freedom is useful in certain types of calculation, as for example in simplifying the two-level hyperpolarisability result in section 4.3. In considering higher orders of harmonic emission, construction of the appropriately index-symmetric molecular response tensor for a process with three or more identical input photons follows along exactly similar lines. For example, for frequency tripling, we have six permutations of the three indices relating to the annihilated photons, and hence we may write. Mfi ˆ

e0 4 ci…jkl† … 3x; x; x; x†qi…jkl† ;

…4:2:9†

ci…jkl† ˆ 16fcijkl ‡ cijlk ‡ cikjl ‡ ciklj ‡ ciljk ‡ cilkj g

…4:2:10†

where

represents a sum involving all permutations of the indices j, k and l. Before moving on, it may be remarked that the six terms comprising the index-symmetric hyperpolarisability tensor defined by equations (4.1.6) and (4.2.5) could be represented by six time-ordered diagrams, each corresponding to one of the permutations of the indices i, j and k. This construction shows that the tensor has the same structure as that of the response tensor for the more general case of sum-frequency generation, except for a factor of 1/2 to avoid over-counting the frequency permutations. The inclusion of this factor appears to indicate a discontinuity between descriptions of the two phenomena, since frequencydoubling is a special case of frequency summing where the absorbed photons have equal energy. However, consideration of photon numbers removes the apparent problem, since it may be argued that when the two modes involved in frequency summing become coincident in frequency, the result is a doubling of their mode occupation number. A more detailed analysis to se-

65

66

4 Molecular Properties

cure the correct continuity necessarily entails consideration of the degree of coherence between the modes involved. We now focus on issues concerning the number of independent tensor components. Consider first, for example, the symmetrised form of the hyperpolarisability tensor represented by equation (4.2.5). Its components must satisfy identities such as bz…xy† ˆ bz…yx† , whether or not the molecule it characterises possesses the threefold or higher axis of rotational symmetry necessary to confer degeneracy on the x and y axes (see below). Hence, at most 18 of the 27 tensor components can be independent. In general for nharmonic processes, where m ˆ n ‡ 1, the tensor can have at most 3…n ‡ 1†…n ‡ 2†=2 independent components. The structural symmetry inherent in most molecules nonetheless generally reduces this number. It is also important to note that, in certain cases of harmonic emission, a more completely index-symmetric form of the associated molecular tensor is implicated. As will be shown in a later chapter, any harmonic generated in a macroscopically isotropic medium from a plane-polarised beam produces photons with the same polarisation as the input. In such a case the radiation tensor qijk becomes fully index-symmetric, and arguments similar to those given above show that only components of the fully indexsymmetric hyperpolarisability tensor, b…ijk† , can be involved. This does not mean that the tensor itself is inherently fully index-symmetric, but that such experiments (plane-polarised irradiation of an isotropic system) cannot determine any part of the tensor associated with index antisymmetry. This naturally leads us to the issue of approximate index symmetry. Any nonlinear process involving only static fields is correctly described in terms of a response tensor that is always fully index-symmetric. When optical frequencies are involved, the differences between the energy denominators of the various terms in the tensor remove this symmetry. (The only exception, though an important one, is the linear polarisability. Since this entails the product of two transition moments connecting the same pair of states, permutational index symmetry is guaranteed even in the optical frequency region.) In general, if the photon frequencies fall substantially below any transition frequencies of the material, all energy denominators lose their optical frequency components. In effect the tensor becomes fully index-symmetric. Un-

4.2 Symmetry

der such conditions the tensor for an n-harmonic process has only 12 …n ‡ 2†…n ‡ 3† independent components, e.g. 10 in the case of frequency doubling. This assumption, normally referred to as Kleinman symmetry (Kleinman 1962), is widely exploited – especially in calculations of the nonlinear optical efficiency of new materials. Nonetheless complete index symmetry is unwarranted in many applications (Wagnière 1986, Wortmann et al. 1993). We now turn to the implications of physical symmetry. This plays a significant role in determining the nullity or equality between various components of any rank m molecular response tensor (3m components in the case of dipole coupling). First we establish a simple existence condition for the m-photon optical response tensor a…m† – i.e. a condition for it not to be null. For any parametric process, the initial and final molecular states are identical and normally carry the full ground state symmetry of the molecule (in the case of a crystalline solid, that of the unit cell). Since each term in the explicit expression (4.1.1) contains a product of m transition moments, the tensor can be non-vanishing only if the totally symmetric representation of the appropriate point group or space group is spanned by the product of m translations (in the case of electric dipole, E1, coupling). Where the molecule (or crystal) possesses a centre of symmetry, the condition can only be met in the generation of odd harmonics, where m is even and the product of translations has gerade (even) symmetry. For the same reason free atoms cannot support the production of even harmonics, save under conditions that disrupt atomic symmetry. More detailed symmetry information can be determined by pursuing a group theoretical development. To this end we first consider point group representations of the optical response tensor (Jerphagnon 1970, Andrews 1990, Lalanne et al. 1996). In general, any set of 3-dimensional tensor components affords a basis for reducible representations of any molecular or crystallographic point group. For the optical response tensors of interest here, these reducible representations map into irreducible parts as shown in Tab. 4.1 (based on the transformation properties associated with E1 coupling) in which D(j±) designates an irreducible representation of weight j and either even or odd parity. Here the second harmonic tensor b… 2x; x; x† is represented as

67

68

4 Molecular Properties Tab. 4.1 Group theoretical representations and number of independent components i of second, third and fourth harmonic polarisabilities.

Coupling

Tensor

Representation

E1(E12)

b SHG i…jk†

2D…1

3

b SHG …ijk† cTHG i…jkl† cTHG …ijkl† dFHG i…jklm† dFHG …ijklm†

(E1 ) 3

E1(E1 ) 4

(E1 ) E1(E14) 5

(E1 )

D

…1 †

D

…0‡†

D

…0‡†

2D…1 D

…1 †

†

 D…2

†

i  D…3

†

18

…3 †

D

…1‡†

D

…2‡†

10 …2‡†

 2D

D

D

†

†

 D…2 …3 †

D

…4‡†

D

D

†

†

30

…4‡†

 2D…3

D

…3‡†

…5 †

15  D…4

 D…5

†

45 21

bSHG; for the third harmonic cTHG  a…4† … 3x; x; x; x† and for the fourth harmonic dFHG  a…5† … 4x; x; x; x; x† – though in each case precisely the same molecular tensors are involved in the corresponding incoherent harmonic emission processes. The first entry for each tensor gives a reduction based on index symmetry intrinsic to the interaction, as is denoted by the brackets around the subscripts in bSHG i…jk†Š for example; entries in the following line are results that apply under the Kleinman assumption of full index symmetry, e.g. bSHG …ijk† . In each case the final column gives r, the number of independent tensor components. This represents an upper limit, applicable to molecules with no intrinsic symmetry. Any molecule with structural symmetry will have fewer independent components, as we show below. In general, each nonlinear response tensor is supported by a given molecule or crystal if the totally symmetric representation of the corresponding point or space group is spanned by one or more tensor components. This remains true even for the response tensors associated with molecular excited states. Provided those states are non-degenerate, the product of the initial and final state representations will generate the totally symmetric representation – and in cases where the properties of molecules in degenerate excited states are required, the theory can be extended in an obvious way. Since components that are transformed following the operation of the symmetry matrix D(0+) invariably span the totally symmetric representation it is apparent from Tab. 4.1 that, on the basis of molecular symmetry, third harmonic generation is universally allowed. Indeed this is true for

4.2 Symmetry

all odd harmonics as the corresponding polarisabilities invariably carry components of D(0+) symmetry. By contrast, in the case of SHG it transpires that only those species whose totally symmetric representation is spanned by D(1–), D(2–) or D(3–) components can support the hyperpolarisability tensor bSHG. As seen earlier this excludes all centrosymmetric species. More interestingly, the existence condition is satisfied by all polar species; in order to support a static electric dipole these species must have D(1–) components. In fact amongst the common non-polar molecular point groups, by having D(2–) and/or D(3–) components, only the following permit a non-zero bSHG: S4, C3h, D2, D3, D4, D6, D3h, D2d, T, Td (Andrews 1990). Molecules with D(3–) but not D(1–) components are commonly referred to as octupolar, because the conditions for existence of an electric octupole are precisely those which must be fulfilled to sustain D(3–) components of the hyperpolarisability. However, although widely used, use of the term in this connection, where only electric dipole coupling is involved, is potentially misleading since electric multipoles can indeed be involved in the nonlinear optical response of molecules (for examples see chapter 7). It may be noted that if Kleinman symmetry holds, bSHG carries only D(1–) and D(3–) components; making the assumption of full index symmetry masks the SHG potential of species with D4 and D6 symmetry, which have only D(2–) components. In many molecules structural symmetry establishes an equivalence between internal Cartesian directions, and hence direct relationships between some of the tensor components. Index symmetry imposes a maximum on the number of independent components for any given response tensor, a number which is further reduced in any system with two-fold or higher proper rotational symmetry. Two features are responsible for this. One is the fact that symmetry generally dictates that certain tensor components be zero; e.g. in any species with a mirror plane perpendicular to the z-axis, bSHG z…zz† ˆ 0 since it must equal its negative. Secondly, structural symmetry usually forges relationships between different components. For example, in species with a SHG threefold axis of symmetry in the xy plane, b SHG x…xx† ˆ b x…yy† . The 2 2 explanation is that, in such cases, …x ‡ y † transforms under the totally symmetric representation but x does not, and hence neither does the product x…x2 ‡ y2 †; therefore the linear combi-

69

70

4 Molecular Properties SHG nation …bSHG x…xx† ‡ b x…yy† † must be zero. For such reasons the eighteen tensor components associated with SHG generally reduce to a significantly smaller number; in the case of species with Td tetrahedral symmetry there is in fact only one independent component (xyz). Where local symmetry permits harmonic generation based exclusively on electric dipole coupling, inclusion of higher order multipolar contributions in the exact coupling equation (1.2.13) engenders additional terms that are normally negligible. If electric dipole harmonic generation is forbidden, however, which we have seen is the case for even harmonics in a centrosymmetric system, these higher order terms can become significant and operate to effect weak harmonic emission. For example, if any one of the three ungerade electric dipole (E1) interactions involved in second harmonic generation is replaced by a gerade electric quadrupole (E2) interaction, the operator product generates a D(0+) contribution as shown in Tab. 4.2. Since this invariably spans the totally symmetric representation, the corresponding response tensor is non-zero even for centrosymmetric materials. In the first row of Tab. 2 the coupling E1(E1E2) refers to a response tensor associated with one electric dipole and one electric quadrupole annihilation of a pump photon, with electric dipole emission of the harmonic. The labelling E2(E12) in the second row relates to electric dipole annihilation of both pump photons and electric quadrupolar harmonic emission. In the third row the coupling (E12E2) can refer to either case, but the corresponding representation applies only under the assumption of Kleinman index

Tab. 4.2 Representations and number of independent components i of leading higher order multipole second harmonic polarisabilities.

Coupling E1(E1E2)

Tensor

Representation

SHG b~i…jk†l b~SHG

D…0‡†  2D…1‡†  3D…2‡†  2D…3‡†  D…4‡† D

…0‡†

(E1 E2)

SHG b~…ijkl†

D

…0‡†

E1(E1M1)

b SHG ijk

D…0‡†  3D…1‡†  2D…2‡†  D…3‡†

2

E2(E1 ) 2

2

M1(E1 ) 2

(E1 M1)

…ij†…lk†

b SHG i…jk† b SHG …ijk†

…1‡†

i

D

…2‡†

D

…1‡†

2D D

…1‡†

D

 2D D

…2‡†

…3‡†

D

…2‡†

…3‡†

D

D

…4‡†

…3‡†

D

…4‡†

45 30 15 27 18 10

4.3 Two-Level Systems 71

symmetry. Similar remarks apply if one of the electric dipole couplings is alternatively replaced by a gerade magnetic dipole (M1) interaction, though here the D(0+) representation arises only when the magnetic interaction is involved in annihilation of a pump photon. Moreover the D(0+) feature is not apparent if Kleinman symmetry is assumed. An exception to the principle of higher multipoles permitting fully E1 forbidden harmonic processes is the case of an isotropic fluid, where for example the full rotational and inversion-rotational O(3) symmetry also precludes the mediation of any other multipoles in second harmonic generation within the bulk (Andrews 1980, Andrews and Blake 1988, Stedman 1993). Work by Cao and Zhu (1995) has shown how the multipolar generation of a second harmonic signal in such systems necessitates the presence of a surface, from which the harmonic can emerge as a reflected beam.

4.3

Two-Level Systems With applied radiation of suitable frequencies or frequency multiples, proximity to resonance with particular molecular transitions leads to molecular tensors dominated by certain terms. Specifically, in the summation over intermediate states one or more terms will be associated with exceptionally small energy denominators – a principle that applies to all multiphoton interactions. For calculational purposes this invites the possibility of neglecting, in the sum over molecular states, other states whose contribution is much less significant. Reducing the sum over an infinite set of molecular states to a finite set is a technique widely employed to reduce calculational complexity. In the pursuit of analytical results for near-resonance behaviour the rapid convergence of such sums often makes it defensible to further limit the sum and consider just the ground and one electronically excited state. The literature is replete with calculations based on two-level approximations to simplify the theory in addressing the optical properties of complex molecular systems.

72

4 Molecular Properties

In the case of a single resonance, optical harmonic conversion is largely driven by transitions involving just the ground state and the resonant level. Indeed, in the realm of resonant multiphoton phenomena the two-level approximation is most appropriate for harmonic emission, whereas non-parametric processes such as multiphoton absorption generally require three or more levels for their adequate representation. In the case of frequency doubling, resonance amplification can occur at either the pump or the harmonic frequency, and it is useful to separate the molecular tensor into a sum of two terms. In the first term both summations over intermediate states jri and jsi are restricted to the ground level j0i and one resonant level jui; in the second, all other possibilities are accounted for. Hence we can write BG bijk … 2x; x; x† ˆ b TLA ijk … 2x; x; x† ‡ b ijk … 2x; x; x† ;

…4:3:1†

where TLA denotes the two-level approximation and BG the other background contributions. Note that bBG does not completely exclude the states j0i and jui from every intermediate state summation which it entails; for example it accommodates contributions associated with jri ˆ jui; jsi 6ˆ fj0i; juig. Focussing on the dominant two-level term (Bloembergen and Shen 1964), analysis of the tensor structure with due regard to the proper signs for the damping corrections (and utilising the freedom to add a j,k index-antisymmetric term which vanishes on contraction with qi…jk† – see previous section) yields the following result (Wortmann et al. 1993; Andrews and Meath 1993): " bTLA ijk …

2x; x; x† ˆ

`

…E~u0

0u l0u i lj dk 2 hx†…E~u0

hx†

`

‡

0u l0u j di lk …E~u0 ‡  hx†…E~u0

hx†

`

‡‡

0u dj l0u k li

…E~u0 ‡ hx†…E~u0 ‡ 2hx†

# …4:3:2†

assuming that the electric dipole transition moments are real. Where only diagonal components arise, the tensor product b TLA
q which features in the rate equation (3.1.5) reduces to a

4.3 Two-Level Systems 73

simpler structure first identified by Oudar and Chemla (1977). In the general expression (4.3.2), the most significant feature is the ` appearance in each term of the vector parameter d , defined by `

d ˆluu

l00 ;

…4:3:3†

i.e. the vector difference between the static electric dipole moments of the resonant and ground states. Hence the two-level hyperpolarisability displays a linear dependence on the magnitude ` of d . For example, in extensively conjugated molecules this signifies that the hyperpolarisability can reasonably be assumed to have components whose magnitude is proportional to the extent of charge delocalisation. ` In order for the dependence on d to be identified it is important to include the ground state of the molecule in sums over intermediate states. This proves significant for two reasons. First, the result (4.3.2) shows that a two-level response can be supported only by molecules with permanent ground or excited state dipoles. Secondly, it is clear that there is considerable scope for the two-level response to be enhanced, either in polar molecules possessing a resonant excited state geometry appreciably different from the ground state, or in species exhibiting the characteristically strong absorption associated with a charge transfer transition. Similar features arise in the theory of multiphoton absorption (Dick and Hohlneicker 1982, Meath and Power 1988a, b) and also single-photon cooperative absorption (Andrews and Allcock 1994, Sadlej and Fowler 1995). In connection with second harmonic generation the result has added significance since most species which support b are indeed polar (see previous section). Let us consider more closely the case of a medium possessing an excited state jui close in energy to that of the emitted harmonic, 2 hx. For practical application this condition is generally more useful than resonance at the fundamental frequency, since the latter condition is likely to result in loss of pump power through conventional (single-photon) absorption. In view of its denominator structure, the first term in equation (4.3.2) clearly provides the major contribution to the nonlinear response tensor. Explicitly,

74

4 Molecular Properties `

bTLA ijk



hDx 

0u l0u i lj d k
hx ‡ hDx i 12  hcu 

i 12 hcu


;

…4:3:4†

where Dx represents the detuning from resonance; clearly the imaginary component of the second, non-resonant denominator factor in (4.3.4) is of negligible significance. Both the first and second terms of (4.3.2) contribute strongly in the case of resonance at the fundamental frequency. It has already been established that a non-polar molecule may still support b. However if ` d vanishes and the two-level contribution b TLA disappears the ‘background’ contribution b BG may persist, as for example is the case for a tetrahedrally symmetric molecule. The background tensor is consequently dominated by a term with essentially the same denominator structure as (4.3.4), i.e. the product of a nearresonant and an off-resonant term. It has been established that a transformation of the electric dipole interaction is valid for deriving the optical characteristics of molecular systems with a response dominated by two electronic states. This procedure relates to the employment of a fluctuation dipole operator (Brueckner 1955, Bishop 1994) as given by 0 ˆ Hint

e0 1 ‰l

l00 Š
d? ;

…4:3:5†

in which l is the electric dipole operator. The subtracted moment is the permanent dipole of the initial molecular state – usually the ground state. It has also been proven that utilising equation (4.3.5) for the form of the interaction operator leads to a new and expedient algorithm for calculation of the requisite nonlinear optical polarisabilities (Andrews et al. 1999). This algorithm is based on a novel interpretation of the appropriate timeordered diagrams. In establishing the form for probability amplitudes of systems primarily driven by interactions between the ground state j0i, and one other higher electronic state jui, it proves legitimate and expedient simply to recast the permanent moments, where they arise, as follows: luu ! luu

`

l00 ˆ d ;

l00 ! 0

…4:3:6†

leaving any transition dipoles (lu0 ; l0u ) unchanged. When the various time-orderings for any optical processes of interest are

4.3 Two-Level Systems 75

considered, application of this rule enables expressions involving any state sequence which at any point entail the transition j0i j0i (and thereby the ground state dipole l00) to be discarded – so long as those entailing the excited state dipole luu ` are re-interpreted to invoke d . Illustrative applications of this algorithm are described below. In every case its implementation leads directly to results whose derivation by other means is substantially more laborious (Meath and Power 1984 a, b). As an example, we again derive the two-level second harmonic tensor, this time using the algorithm outlined by equation (4.3.6). The three time-ordered diagrams as illustrated in figures 2.1 (a)–(c) are once again employed. The sequence of molecular states between the initial and the final (both ground) states runs through two virtual states, jri and jsi. In the two-level approximation each of these intermediate states is summed to represent one of two possibilities, either the ground state j0i or the excited state jui. The permissible …j0i jsi ri j0i† sequences are concisely expressible as 0000, 00u0, 0u00, 0uu0, corresponding to the dipole products l00l00l00, l00l0ulu0, l0ulu0l00, l0uluulu0, respectively. From the three time-orderings we therefore have 3 ´ 22 = 12 contributions – each a product of three ‘transition’ dipoles (one or more of which may be permanent), divided by a product of two energy factors. Application of the algorithm determines that only the state sequence 0uu0 (l0uluulu0) need be considered in a suitable reinterpretation of the three time-ordered diagrams, since each of the other possibilities generates a l00 segment. Utilising Fig. 2.1 we therefore obtain a two-level hyperpolarisability tensor exactly as expressed by equation (4.3.2). This is the simplest example of how the algorithm quickly generates results that would otherwise demand considerable algebraic manipulation. However, it is with higher order amplitudes that the method is most obviously expedient. Even in (4.3.2), the tensor ` structure obviates simple factorisation in terms of d ; higher orders have the additional complication that terms both linear and ` in higher powers of d arise. The process of third harmonic generation serves to illustrate the power of the new algorithm, and also thereby to draw out some new physics. To derive the form of the response tensor one needs to employ either four time-ordered diagrams or an equivalent state-sequence diagram; both representations are illustrated

76

4 Molecular Properties

Fig. 4.5 Third harmonic generation. (a) The four time-ordered diagrams; (b) the state-sequence diagram.

in Fig. 4.5. The state route connecting the initial and final (ground) states here runs through three virtual states, r, s and t, and the two-level approximation requires each to be either the ground or the excited state. In this case, from the four time-orderings we get a total of 4 ´ 23 = 32 contributions, each a product of four transition or permanent dipoles divided by three energy quotients. With the benefit of the algorithmic method delineated above, we can take the four time-orderings and dispense with all but two of the following state sequences; 00000, 000u0, 00u00, 00uu0, 0u000, 0u0u0, 0uu00, 0uuu0. Specifically, discarding every

4.3 Two-Level Systems 77

sequence that includes the segment 00, we retain only 0u0u0 and 0uuu0. With proper re-interpretation of these remaining cases we immediately obtain the following explicit expression comprising only eight terms, of which each successive pair results from the successive time-ordered diagrams of Fig. 4.5 (a), also corresponding to all routes through the state-sequence diagram of Fig. 4.5 (b): cijkl … 3x; x; x; x† ˆ

…E~u0 ‡

…E~u0

u0 u0 u0 lu0 i lj lk ll 3 hx†… 2 hx†…E~u0

hx†

` ` u0 lu0 i d j d k ll

3 hx†…E~u0 2hx†…E~u0 hx† u0 u0 u0 lu0 j li lk ll ‡ …E~u0 ‡ h x†… 2 hx†…E~u0 hx† ` `

u0 lu0 j d i d k ll ‡ …E~u0 ‡ h x†…E~u0 2hx†…E~u0 u0 u0 u0 lu0 j lk li ll ‡ …E~u0 ‡ h x†…2 hx†…E~u0 h x† `

hx†

`

u0 lu0 j d k d i ll ‡ …E~u0 ‡ hx†…E~u0 ‡ 2hx†…E~u0 hx† u0 u0 u0 lu0 j lk ll li ‡ …E~u0 ‡ h x†…2 hx†…E~u0 ‡ 3hx† `

`

u0 lu0 j d k d l li …4:3:7† ‡ …E~u0 ‡ h x†…E~u0 ‡ 2hx†…E~u0 ‡ 3hx†

where again it is the index-symmetrised form, here entailing all permutations jkl, that will feature in the rate observable. The transition moment lu0 is taken to be real and hence equal to l0u. One aspect of equation (4.3.7) deserving of comment is its amenability for the identification of resonances. Three-photon resonances are manifest in the first and second terms, through the hx†; two-photon resonances appearance of the factor …E~u0 3 ~ hx† feature in the second and fourth, and single-photon …Eu0 2 x† in each of the first six. Since exploitation resonances …E~u0 h of the latter kind of resonance is usually avoided in practice because of the competing linear absorption with which it is asso-

78

4 Molecular Properties

ciated, it is the two- and three-photon resonances which are of most interest. Under suitable conditions, third-harmonic generation in either of those cases is largely driven by just two of the contributions to equation (4.3.7). Other contributions, signifying minor corrections, are of much the same order of magnitude as those relating to the background involvement of other molecular energy levels. Further features are` evident when the relative magnitudes of the dipole difference jd j and the transition dipole |lu0| are considered. One striking feature is that the second, fourth, sixth and ` eighth terms all disappear if jd j ˆ 0, leaving only terms associated with virtual excitation routes. (Note that no such se` quences were manifest in the second-harmonic result. If jd j ˆ 0 then the entire expression (4.3.2) becomes zero – any process involving an odd number of photons has to entail at least one 00 or uu segment in the interaction sequence.) In the third harmonic case, in particular, both terms associated with two-photon resonances disappear – in other words there can be no two-photon resonance enhancement of third` harmonic generation under such circumstances. If, however, jd j  jlu0 j, then the even terms of equation (4.3.7) dominate the optical response – and in the case of three-photon resonance it is the second term that provides by far the leading contribution. Such considerations can play an important role in devising and implementing better strategies for the calculation of nonlinear optical response.

79

5

Coherent and Incoherent Signals . . . there are produced in light several curious and pleasant colours, by the proportionate and harmonious motions of vibrations intermingled. Robert Hooke 1)

In the first four chapters, a framework has been established for describing nonlinear optical response in terms of individual molecules and photons. It is appropriate to consider how such a description is adapted for application to the generation of optical harmonics in full-scale systems comprising a large number of molecules. One aspect concerns proper representation of the propagation characteristics of the laser input and harmonic output. Another involves consideration of the distinctive features associated with the collective response of an ensemble of molecules or scattering centres. Scattering processes occurring at individual centres are generally modified by the electrodynamic influences of the surrounding material; moreover in media where there is a distribution of molecular orientations, the molecular response may differ from molecule to molecule. Another crucial feature is the mutual influence of optical conversion processes at different centres, leading to coherent, laser-like emission. The consideration of such features occupies this chapter, which essentially marks completion of the link between microscopic (molecular) and macroscopic (bulk) theories of harmonic emission. The theoretical framework described here is nonetheless one that specifically addresses molecular systems, and is therefore cast in terms of molecular properties. Again this contrasts with the classical 1) T. Birch, A History of the Royal Society of London. Vol. 3, p. 94 (Royal Society,

London 1756). 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

80

5 Coherent and Incoherent Signals

approach which takes account of macroscopic symmetry but not local molecular structure. To begin a consideration of optical nonlinearity in a molecular ensemble, we first return to the general Fermi rate (3.1.5), the key equation for determining ensemble response, written without explicit representation of tensor indices as follows: 2 N 2p X …m† e0 m an
q0…m† eiDk
Rn d…Ei Ef † ; Cˆ h n  where the dot between tensors represents the full inner product between the rank m molecular and radiation tensors, yielding a scalar. Also Dk is the wave-vector mismatch as defined by equation (3.1.4). For application to specific processes in media, however, the above result needs further theoretical development, and at this point it is appropriate to draw a distinction between the response from molecular solids and fluids.

5.1

Regular Solids Consider the case of a regular, orientationally ordered solid such as a molecular crystal. It is traditional to represent the nonlinear optical response of such a system in terms of the summed responses of its constituent unit crystallographic cells (Zyss and Oudar 1982). This clarifies the correlation between microscopic and bulk parameters and, given the approximate nature of the theory, it has many advantages. Within the framework of molecular quantum electrodynamics, we have also established the means to fully account for polariton interactions through use of the modified second quantised radiation field operators (equations (1.3.2) and (1.3.3)). Consequently it is possible to extend traditional methods for formulating bulk response at the molecular level, whilst fully accommodating polariton interactions and local field corrections. Consider a crystal with p molecules per unit cell. Since each cell is identically oriented, the sum over N molecules in (3.1.5) can be rewritten as a double sum, first over the molecules labelled m in each cell, then over the M ˆ …N=p† cells labelled u. For this purpose each molecular position vector Rn can be writ-

5.1 Regular Solids 81

ten as a sum of Ru, a position vector for the unit cell to which it belongs, and Rm, a vector denoting the position of the molecule within the unit cell. Hence we obtain the following result: 2 p M X 2p X iDk
Ru …m† 0…m† iDk
Rm e am
q e C ˆ 2m d…Ei he0 u  m

Ef † :

…5:1:1†

It is convenient to introduce a cell response tensor, which entails a structure factor for nonlinear optical response at the microscopic level, through the defining equation …m†

a

ˆ

p X m

iDk
Rm a…m† : m e

…5:1:2†

The components of the cell tensor are explicitly expressed in terms of molecular components through the relation …m†

^ ai1 ...im ˆ

p X m

…m†

ak1 ...km li1 k1 . . . lim km eiDk
Rm ;

…5:1:3†

once again adopting the convention of implied summation over repeated indices and utilising the redundant superscript to label the molecule sum. In equation (5.1.3) lir kr denotes the direction cosine between the crystal axis ir and the molecular axis kr. Generally, the arguments of the intracellular exponential in (5.1.2) and (5.1.3) will be vanishingly small; certainly this is the case when wave-vector matching conditions are satisfied (vide infra). Even when the mismatch Dk is at a maximum, the condition Dk
Rm  0 is commonly satisfied; for example in n-harmonic processes it still imposes only the relatively weak constraint Rm  k=4pn, where k is the wavelength of the pump radiation. Then it is possible to ignore the intracellular phase factor and regard the response functions of the unit cell components as directly additive. The additivity of chromophoric contributions to the response of large polyfunctional molecules can, to some extent, be argued on a similar basis (see section 5.3). 2) Irrespective 2) The principle of local additivity is also a key feature when addressing wave-vec-

tor matched coherent response from systems with more than one electronic level populated (see chapter 6).

82

5 Coherent and Incoherent Signals

of this potential simplification, equations (5.1.1) and (5.1.2) lead to the following result: 2 M 2p X ^ …5:1:4† C ˆ 2m a…m†
q0…m† eiDk
Ru d…Ei Ef † : he0  u Since the tensors a(m) and q0…m† each have the same value for every unit cell, by definition of the term, equation (5.1.4) simplifies to 2 M 2p …m† 0…m† 2 X iDk
Ru a
q j e …5:1:5† C ˆ 2m j^ d…Ei Ef † ; u he0  and it is helpful to rewrite the result as a sum of diagonal and off-diagonal terms as follows: ( ) M X M X 2p …m† 0…m† 2 iDk
Ruu0 d…Ei Ef † ; a
q j M‡ e C ˆ 2m j^ he0  u u0 6ˆu …5:1:6† where Ruu0 ˆ Ru Ru0 . The first term in this equation represents an incoherent contribution to the rate, Cinc, which is clearly independent of any correlation in phase between processes occurring in different unit cells: C inc ˆ M

2p …m† 0…m† 2 j^ a
q j d…Ei he2m  0

Ef † :

…5:1:7†

The second term in (5.1.6) is a cross-term which depends, through the phase factor exp…iDk
Ruu0 †, on the relative displacement of each pair of unit cells with respect to the wave-vector mismatch. Clearly such a term can persist only if the phase factors lead to constructive interference; it may therefore be interpreted as representing the source of any coherent contribution, Ccoh, and can be written as C coh ˆ

2p …m† 0…m† 2 j^ a
q j …gM e2m h 0

M†d…Ei

Ef † ;

…5:1:8†

where the coherence characteristics of the process are represented by

5.2 Gases, Liquids and Disordered Solids 83

2 M X iDk
Ru gM ˆ e : u

…5:1:9†

The detailed form and evaluation of the important parameter gM, a sensitive function of wave-vector mismatch, is discussed in section 5.4.

5.2

Gases, Liquids and Disordered Solids In molecular fluids the individual molecular centres are responsible for harmonic emission, and it is crucial to take account of the tumbling motions experienced by each molecule. Here use can be made of the ergodic theorem, which states that, in a stationary ensemble of particles, the ensemble average will equate to the time average for each of its component particles, over a sufficiently long time. In the present context this means that it is legitimate to treat the rotationally averaged system response as that of an ensemble of randomly oriented centres, for which powerful symmetry constraints are known to apply. An analysis based on this approach has the further advantage of facilitating extension to solids that are fully disordered on the optical scale, such as many glasses and amorphous polymers. The results which ensue are valid for any macroscopically isotropic system where quantised molecular rotation is not an issue. In fact, nonlinear optical response is in general sufficiently weak (and, for coherent signals, nonlinearly dependent on density – see below) as to obviate useful measurements on low-density systems where quantised rotations are manifest. For media with random orientations of the molecular components responsible for harmonic emission, it is tempting to directly replace the sum over n in equation (3.1.5) with a volume integral, an approximation which would lead to a Dirac delta function in Dk, and so imply a non-vanishing rate only in the case Dk ˆ 0. Such a result would however be based on the assumption of a spherically symmetric distribution with infinite extent, whereas in practice the interaction volume is restricted by the di-

84

5 Coherent and Incoherent Signals

mensions of the laser beam waist – and as such is markedly axial. With cognisance of this we rewrite equation (3.1.5) as a sum of diagonal and off-diagonal terms as follows: 2p C ˆ 2m he0  ‡

X N n

N X N X n

0

n 6ˆn

…m†

jan

…m†

…an


q0…m† j2

 …m†
q0…m† †…an0
q0…m† †eiDk
Rnn0 d…Ei

Ef †; …5:2:1†

where the individual molecular centres are now represented by Rnn0 ˆ Rn Rn0 . The first term again represents the incoherent contribution. Denoting the appropriate average over the orientations of each molecule or centre n by angular brackets h. . .in , we have  N  2p X …m† 0…m† 2 d…Ei hC inc i ˆ 2m an
q he0 n  n

Ef † :

…5:2:2†

Although, at any given time, the response from each molecule depends on its orientation with respect to the incident light, the averaged response from all N molecules must be identical, and equation (5.2.2) may therefore be simplified to; hC inc i ˆ N

2p D …m† 0…m† 2 E d…Ei a
q h e2m 0

Ef † :

…5:2:3†

Two important features can immediately be identified in this result. One is the obvious but significant fact that the incoherent term alone depends linearly on the number of scatterers N, and therefore delineates the magnitude of the correspondingly weak signal as colligative. Secondly, the rotational average in equation (5.2.3) is taken over the modulus square of the matrix element. In both respects, these features mark significant differences from the coherent term to be examined next. The second term in equation (5.2.1) embodies any coherent contribution to the harmonic emission. If the possibility of any time-averaged correlation between molecules is to be entertained, as can arise through dipole-dipole interaction, the result must be

5.2 Gases, Liquids and Disordered Solids 85

evaluated with regard to the pair orientational distribution within the interaction volume V as follows:   2p N 2 hC coh i ˆ 2m d…Ei Ef † 2V he0  Z D   E …m† …m†  d3 Rnn0 an
q0…m† an0
q0…m† g…Rnn0 Xn Xn0 †eiDkRnn0 nn0 : …5:2:4† Here the angular brackets h. . .inn0 denote an unweighted average over orientations, and g…Rnn0 Xn Xn0 † is the angular pair correlation function, a sensitive function of fluid structure (see for example Gray and Gubbins 1984). Where such distributional averages arise in other areas of physics they are frequently dominated by nearest-neighbour interactions. However the property which is the subject of the average in equation (5.2.4) is not of this kind, since it depends on pair separation only through the phase factor. Since, in isotropic molecular fluids, orientational correlations typically persist only up to a few molecular diameters (a distance invariably much smaller than the interaction volume) the majority of the contributions to (5.2.4) derive from a region where g…Rnn0 Xn Xn0 † is independent of the orientations Xn and Xn0 . Similar remarks apply to amorphous solids. To a good approximation it is therefore possible to effect a factorisation of the integrand, enabling analytic results of general applicability to be obtained. Only where there is long-range orientational order, as in certain anisotropic media, does this approximation warrant more critical appraisal. The treatment of strongly correlated molecules has been described in detail by a number of authors, for example Kielich (1983) and Wang (1993). A strategy for macromolecules, suspensions and partially ordered solids is discussed in the next section. Where it is justifiable to perform independent orientational averaging for each molecule, we obtain the following result from the second term of (5.2.1):

86

5 Coherent and Incoherent Signals

hC coh i ˆ

N D E X 2p …m† 0…m† d…E E † a
q i f n n he2m  0 n N D E X …m† an0
q0…m† eiDk
Rnn0 ; 0 n0 6ˆn

…5:2:5†

n

reverting to the explicit summation representation. Notice that the phase factor is factorised outside the two angular averages, since the orientation of the relative displacement vector Rnn0 is independent of the orientations of the two molecules n and n0 . Since the orientational average of each matrix element is clearly molecule-independent, equation (5.2.5) can be rewritten as hC coh i ˆ

2p D …m† 0…m† E 2 a
q d…Ei he2m  0

Ef †

N X N X n

eiDk
Rnn0 ;

n0 6ˆn

…5:2:6† or hC coh i ˆ

2p D …m† 0…m† E 2 a
q d…Ei he2m  0

Ef †…gN

N† ;

…5:2:7†

where gN is as defined by equation (5.1.9) with summation here taken over molecules rather than crystal unit cells. In contrast to the case of incoherent scattering, the orientational average in (5.2.7) is taken over the matrix element before it is squared. This apparently minor difference results in markedly different behaviour. For example it is for this reason that, for the case of second harmonic generation, in effecting the average the coherent term vanishes entirely, as will become apparent. Calculation of the tensor orientational averages in equations (5.2.3) and (5.2.7) necessitates more detailed analysis of the inner product a…m†
q0…m† , which can be written as; a…m†
q0…m† ˆ ai1 ...im q0i1 ...im :

…5:2:8†

In equation (5.2.8), all components are referred to a reference frame in which the radiation tensor components are fixed, and as such the corresponding components of the molecular non-

5.3 Macromolecules, Suspensions and Partially Ordered Solids 87

linear response tensor necessarily vary with molecular orientation. It is more convenient to re-express the molecular tensor components with reference to a molecule-fixed frame, denoted by indices kr , in terms of which they too are rotation-invariant. Moreover, a suitable choice of molecular frame facilitates consideration of the implications of molecular symmetry for relationships between the tensor components. We now write a…m†
q0…m† ˆ ak1 ...km q0i1 ...im li1 k1 . . . lim km ;

…5:2:9†

where only the direction cosines lir kr vary with molecular rotation. The orientational averages in (5.2.3) and (5.2.7) are thus obtained by averaging over the direction cosine products D E a…m†
q0…m† 2 ˆ ak ...k . . . ak ...k q0 . . . q0im‡1 ...i2m hli1 k1 . . . li2m k2m i; 1 m m‡1 2m i1 ...im …5:2:10† and D E 2 …m† 0…m† 2 a
q ˆ ak1 ...km q0i1 ...im hli1 k1 . . . lim km i :

…5:2:11†

The former, incoherent case involves a rank 2m rotational average, whereas the latter, coherent case a rank m average. Explicit results for these averages, both for isotropic and anisotropic fluids, are given in appendix 2.

5.3

Macromolecules, Suspensions and Partially Ordered Solids Many imperfectly regular systems embody a degree of structurally determined orientational order. Examples include polyfunctional macromolecules such as proteins, structured aggregates of molecular units exemplified by membranes, and partially ordered solids in which orientational order extends over local domains, as in certain glasses and gels. The theoretical methods appropriate for describing such systems entail a combination of methods described in the previous two sections. To focus attention on one particular physical situation, the description below will address

88

5 Coherent and Incoherent Signals

the case of a suspension of identically structured aggregates to be referred to as units. The method that follows (Andrews, Allcock and Demidov 1995) is amenable to modification, by incorporating appropriate distribution functions, for application to more complex systems in which different units have diverse structures, or contain different numbers of molecules. Consider an assembly of M randomly oriented units labelled u, each containing p discrete molecules. The rate of harmonic emission is now given by equations of similar form to (5.1.4) and (5.1.3), and explicitly we have 2 M 2p X …m† 0…m† iDk
Ru ^ au
q e C ˆ 2m d…Ei he0 u 

Ef † ;

…5:3:1†

with Ru the position vector for the origin of an arbitrary Cartesian frame of reference of fixed orientation within unit u, and where …m†

^ a…u†i1 ...im ˆ

p X n

…n†

…n†

…n†

ak1 ...km li1 k1 . . . lim km eiDk
…Rn

Ru †

:

…5:3:2†

The latter intra-unit summation accommodates the nonlinear response tensors of each molecular component, each with its own relative orientation, referred to the common Cartesian frame. The first distinctive difference between the above result and its molecular crystal counterpart (5.1.3) lies in the much greater physical extent (potentially by many orders of magnitude) of the summation in (5.3.2), as represented by the magnitude of the vector displacement jRn Ru j. Consequently it is impossible to assume satisfaction of the condition Dk
…Rn Ru †  0. The effective additivity of contributions from different molecules is limited to a range within which coherence is sustained, as discussed below. Adding the harmonic signals associated with each distinct unit, assuming that the various units have a random distribution of orientations with respect to the radiation, we obtain C ˆ C1 ‡ C2 ; where

…5:3:3†

5.3 Macromolecules, Suspensions and Partially Ordered Solids 89

hC 1 i ˆ M

2p D …m† 0…m† 2 E ^ d…Ei au
q h e2m 0

Ef † ;

…5:3:4†

and hC 2 i ˆ

2p D …m† 0…m† E 2 au
q ^ d…Ei he2m  0

Ef †…gM

M† :

…5:3:5†

The coherence factor in (5.3.5) relates only to the spatial distribution of units within the system, allowing factorisation of the unit orientational average. In this case, physically intermediate between molecular fluids and regular solids, it is no longer appropriate simply to designate the two contributions, C1 and C2, incoherent and coherent. Certainly the latter, which represents constructive interference between different units, is significant only in processes for which the wave-vector mismatch Dk is very close to zero. The intrinsic dependence on polarisation of C2 in particular may however lead to this term vanishing, as with SHG. The former rate contribution, C1, has very different attributes. Most obvious is its association with an orientational average over the modulus square of the unit response, giving different polarisation behaviour. The interference between signals associated with molecules n and n0 within each unit generates a rate contribution, determined through (5.3.2), of hC 1 i ˆ M

2p D …m† 0…m†  …m† 0…m†  iDk
Rnn0 E e d…Ei an
q an0
q h e2m 0

Ef † ; …5:3:6†

in which both the molecular components and also the vector Rnn0 have a fixed mutual orientation. Consequently the average must here be effected with regard to the orientations of these quantities relative not only to the radiation tensor q0 but also Dk, and this phase factor cannot be factorised out. On implementing the appropriate phase-weighted average (see appendix 2) it emerges that C 1 …nn0 † averages to zero for n 6ˆ n0 except where Dk  0, and under such circumstances the additivity of the individual molecular components according to equation (5.3.2) leads to a quadratic dependence of C1 on q. Hence C1 properly reflects a response that is coherent, in the sense that it arises from constructive interference

90

5 Coherent and Incoherent Signals

between signals from various molecules within each unit. The result nonetheless scales linearly with M, the number of such units.

5.4

Coherence and Wave-Vector Matching We now focus on the detailed relationship between coherence and wave-vector matching, an issue with a highly significant bearing on both the directionality and intensity of nonlinear scattering. As seen above, whilst incoherent scattering is independent of any relationship between the wave-vectors of absorbed and emitted photons, coherent scattering in all kinds of media involves a distributional sum and leads to a factor of the form 2 N X eiDk
Rn ; gN ˆ n

…5:4:1†

with …n; N† replaced by (u, M) for the case of a crystal or other type of conglomerate. Unlike the coherences which arise in connection with short-wavelength studies, for example X-ray diffraction, the gross disparity of scale between typical nearest-neighbour distances and jDkj 1 here is such that translational regularity has little bearing on the result. Even in a regular molecular crystal any ‘accidental’ coherences 3) between the responses of two molecules generally arise only for molecules separated by distances at least comparable with, and usually several orders of magnitude larger than, the optical wavelengths involved. As a result there is no significant sensitivity to translational order, in marked contrast to the strong dependence on orientational order. The convenient assumption of a completely random spatial distribution for the molecular displacement vectors Rn leads to the following simple result (Marcuse 1980) gN  N …Dk 6ˆ 0† ;

…5:4:2†

coherences between two molecules n and n0 occur when Dk
Rnn0 ˆ 0, with out Dk being zero, because the latter is orthogonal to the intermolecular displacement vector.

3) ‘Accidental’

5.4 Coherence and Wave-Vector Matching 91

gN  N 2 …Dk ˆ 0† :

…5:4:3†

The former case, (5.4.2), which applies if the value of Dk is sufficiently large for exp…iDk
Rn † to be a rapidly oscillating function within the boundaries of the interaction volume, results in a linear dependence on the number of scatterers which characterises incoherent response, as seen earlier. The latter case, equation (5.4.3), produces a quadratic dependence on the number of scatterers, characterising coherent response. As a result of this difference, processes for which it is possible to satisfy the exact wave-vector matching condition Dk ˆ 0 invariably produce much stronger signals than those which do not. Clearly, then, the issue of whether the wave-vector mismatch can be zero or not is of considerable importance. In anticipation of chapter 9, it may be pointed out that for any inelastic nonlinear process, in which optical conversion is accompanied by an exchange of energy between the radiation field and the matter, it is not generally possible to fulfil the wave-vector matching condition. Such is the case for hyper-Raman scattering, where the initial and final molecular states differ. Even if wavevector matching could be achieved for such a process, consideration of the uncorrelated quantum mechanical phase factors associated with the final state in different molecules would dictate that equation (5.4.2) still effectively applies. For inelastic processes the coherent part of the response always vanishes, and as a consequence they are invariably incoherent. By contrast, parametric or elastic scattering processes (in which the material medium returns to its original state) can always, in principle, satisfy wave-vector matching. At the simplest level, the appropriate conditions could be achieved in a dispersion-free medium with all wave-vectors involved in the process arranged collinearly. Recalling equation (1.3.4), we have kˆ

nx

k

c

o ^; n…xk † k

…5:4:4†

and from this relation it is evident that if the refractive index for all frequencies were unity, the condition Dk ˆ 0 would immediately be satisfied simply by virtue of energy conservation. Although no truly dispersion-free materials exist, it is still often possible to satis-

92

5 Coherent and Incoherent Signals

fy the wave-vector condition if the frequencies involved are such that the appropriate refractive indices match. Even if exact wavevector matching cannot be achieved, it is still often possible to operate under conditions where the value of Dk is small. Under such circumstances however, where exp…iDk
Rn † is a slowly varying function of Rn, the approximation of (5.4.2) is inappropriate – particularly since the interaction volume is neither spherically symmetric nor of infinite extent. It is best to proceed by defining a mismatch direction z, such that Dk
Rn ˆ jDkjzn , where zn denotes the displacement of molecule n along the z-axis, subsequently converting the summation over molecules in (5.4.1) to an integral between limits 0 and L, corresponding to the boundaries of the nonlinear material. Thus we have 2 X N ijDkjzn e …5:4:5† gN ˆ ; n and therefore; 2 ZL N ijDkjzn e dz : gN  L

…5:4:6†

0

Evaluation of the integral and use of the identity eix


1 1 ˆ 2iei2x sin 12x ;

…5:4:7†

leads to the result;
N 2 sin2 12jDkjL ; gN ˆ
2 1 jDkjL 2

…5:4:8†

giving a familiar sinc2 dependence. Clearly this function attains its largest value in the limit Dk ! 0, where the result reduces to that given by (5.4.3). For efficient harmonic conversion, it is important to maximise the value of gN, which essentially requires minimising the wavevector mismatch. It is expedient to ensure that the value of 1 1 1 2 jDkjL is limited to values in the interval … 2 p; 2 p†, within which gN remains above ~0.4 of its peak value. Maximum har-

5.4 Coherence and Wave-Vector Matching 93

monic emission occurs where the pump and harmonic waves propagate collinearly, we then have assuming nxk  x0k0 the relationship jDkj ˆ jnk

k0 j ˆ

nxk  n…xk † c

n…x0k0 † ;

…5:4:9†

where n…xk † and n…x0k0 † are the refractive indices for the pump and harmonic frequencies, respectively, and the integer n takes its value from the nth harmonic. One of the most important criteria for efficient harmonic production is therefore the closeness of match between these two refractive indices. For example in the earliest experiments on second harmonic generation (Franken et al. 1961), the mismatch of pump and harmonic refractive indices was such that the conversion efficiency was *10–5%; however figures of 30% or higher are now commonly feasible when good index matching is accomplished. In fluids or optically isotropic solids where refractivity is independent of propagation direction, it is impossible to guarantee wave-vector matching unless the refractive indices n…xk † and n…x0k0 † are equal. The implications of this are less serious for gases than for condensed matter, since refractive indices vary much less with frequency in the former. Although the effects of dispersion seldom enable precise index matching to be satisfied,

1 2 jDkjL

Fig. 5.1 Normalised representation of the coherence factor gN (see equation (5.4.8)).

94

5 Coherent and Incoherent Signals

it can be effected in special cases where either the pump or harmonic lies close to an absorption band and as a result the refractive index rapidly varies with frequency. This means working in a region of dispersion, carrying the obvious disadvantage of introducing possible intensity loss through absorption. In connection with structurally complex systems such as suspensions, inter- and intra-unit coherence may be subject to rather different constraints. Though often overlooked this is a simple consequence of equation (5.4.9), taking account of the fact that the refractive indices of the disperse phase and the dispersion medium may have rather different values. It is possible under particular circumstances for correlated species to exhibit off-forward coherent scattering (Allcock et al. 1996). Using the cosine rule, the magnitude of Dk can be calculated in terms of the angle h between the pump and the nth harmonic radiation vectors to give (see Fig. 5.2); jDkj2 ˆ

nx 2 k …n2 …xk † ‡ n2 …x0k0 † c

2n…xk †n…x0k0 † cos h† : …5:4:10†

Equation (5.4.10) leads to two special cases. Maximum harmonic emission clearly occurs for forward emission, where cos h ˆ 1. Then, equation (5.4.10) reduces exactly to the form given by (5.4.9), ensuring that the pump and harmonic waves propagate collinearly. For off-forward scattering the result approximates to jDkj 

nxk 1=2 n …xk †n1=2 …x0k0 † sin 12h : c

…5:4:11†

If index matching is not possible, the condition L  p=jDkj places a constraint on the distance within which harmonic conversion is an efficient process. The harmonic intensity ceases to

Fig. 5.2 Schematic of the wave-vector geometry (not to scale) for off-forward second harmonic generation.

5.4 Coherence and Wave-Vector Matching 95

grow beyond this limit known as the coherence length, lc, which typically lies between 1 and 100 mm in the case of gases but is only a few microns in any condensed phase (Loudon 2000). Observations of SHG from isotropic suspensions of centrosymmetric particles whose dimensions exceed the coherence length (Wang et al. 1996) are due to surface harmonic conversion, as described in chapter 7. Note that coherence length has a quite different meaning in the context of laser beam characterisation, where it signifies the inverse of the linewidth expressed in wavenumber units. In optically anisotropic solids, whose refractive index is dependent on the directions of beam propagation and polarisation, (the phenomenon known as birefringence it is often possible to obtain index matching by judicious choice of crystal orientation (Maker et al. 1962). In a uniaxial crystal, waves polarised with their electric vector perpendicular to the optic axis (ordinary, or owaves) have a refractive index that is independent of the direction of propagation per se. However waves polarised with a finite electric vector component along the optic axis (extraordinary, or ewaves) have a refractive index dependent on the angle between the direction of propagation and the optic axis (see for example Born and Wolf 1999). This effect can be utilised in two types of phase-matching condition. In type-I phase-matching the pump beam propagates as an o-wave and the harmonic as an e-wave; in type-II phase-matching the fundamental is split into o-wave and e-wave components, and the harmonic propagates as an e-wave with stochastic polarisation. Obtaining the optimum indexmatching condition, to maximize harmonic conversion, thus becomes a matter of choosing a suitable configuration for the crystal, depending on the precise wavelengths of light involved. Since refractive indices are also temperature-dependent, operation in a thermally constant environment is also important. Such considerations are crucially important in the design of nonlinear optical materials.

97

6

Coherent Harmonic Generation If only the whole world could feel the power of harmony Wolfgang Amadeus Mozart

In previous chapters we have established a formulation for the quantum electrodynamical theory of molecular nonlinear optics. Methods for radiation and molecular tensor construction developed in the first four chapters first led to a focus on symmetry constraints, two-level behaviour and dispersion effects, all highly relevant for harmonic generation. Then in chapter 5 it was shown how molecular response determines that of a molecular ensemble. With this formalism fully in place we now address a number of processes involving specifically coherent harmonic emission in molecular systems. As such, this chapter acts as a centre of reference for the majority of calculational procedures required in the theory. We derive explicit results for the general case, using principles that are then appropriated for application to SHG and also some less familiar harmonic processes.

6.1

Harmonic Intensities Consider a parametric process of elastic nonlinear scattering in which n photons of laser light with frequency x (wave-vector k and polarisation vector e) are converted into a single photon of frequency nx (wave-vector k0 and polarisation vector e0 ). The quantum amplitude entails tensors of rank m ˆ …n ‡ 1†. Since all n incident photons originate from the same light source or 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

98

6 Coherent Harmonic Generation

mode, full permutational symmetry holds amongst the corresponding indices of the radiation tensor (3.1.6). As a result the latter, q0 , can be cast in the form q0i1 …i2 ...in‡1 †

!1=2

! 0…m† 2 n …x † ‡ 2 0 k ˆ in 1 0…m† 3 0 2cVn…x † 0 0 k ;k ;m k ! ! n=2 n …m† …m† …m† h mg xk e0 n2 …xk † ‡ 2  …m† 3 2cVn…xk †  1=2 q!n  …q0 ‡ 1†1=2 si1 …i2 ...in‡1 † ; …6:1:1† …q n†! X

0…m†

hmg 

0…m†

xk0 e0

where q and q0 are the number of pump photons and the number of harmonic photons, respectively, in the quantisation volume V. Here the sum over m designates the polariton branch index. Assuming that no harmonic radiation is initially present, q0 ˆ 0, demands retention of the sum over the harmonic wavevector, polarisation and branch index. Note also that in the indices of the radiation tensor the first relates to the harmonic and the remaining n indices to the pump. From equation (3.1.7), s(n+1) represents a polarisation tensor defined by si1 …i2 ...in‡1 † ˆ e0i1 …k0 ; k0 †ei2 …k; k† . . . ein‡1 …k; k†  e0i1 ei2 . . . ein‡1 ; …6:1:2† the polarisation unit vectors e and e0 referring to the pump and harmonic respectively, and the parenthesis around the indices i2 . . . in‡1 signifying permutational symmetry. The arguments associated with each unit vector are suppressed for brevity. Calculation of the harmonic intensity requires determination of the rate of coherent emission which, for a system of rigidly oriented molecules, simply requires substitution of the appropriate matrix element into equation (5.1.8). For the specific case of …3† SHG we use the notation aijk … 2x; x; x†  bijk … 2x; x; x† for the molecular tensor, and equation (6.1.1) identifies the explicit expression for the radiation tensor. Here, the rate is as follows:

6.1 Harmonic Intensities 99

C coh

4     hmg xk e0 2 n2 …xk † ‡ 2 2p  q! ˆ 6 3 …q 2†! he0 2cVn…xk †  !  2 0 0 2 2 0 X  hmg xk0 e0 n …xk0 † ‡ 2 b … 2x; x; x†e e e  i j k ijk 0 † 2cVn…x 3 0 0 0 k k ;k

 d… hx0k0

2 hxk †…gM

…6:1:3†

M†

taking summation over the polariton branch m as implicit. Substituting equations (3.2.9) and (3.2.11) into (6.1.3) allows the rate of second harmonic emission into an infinitesimal solid angle dX to be expressed as  2 4 n …xk † ‡ 2 ˆ …gM M† 3 4 h…2p†2 c 2 e30 n2 …xk † ! 2 Z1 0 0 dx h  x n2 …x0k0 † ‡ 2 0 0 0 2 0 k k  k dk dk0 2cn…x0k0 † 3 …2†

dC coh

I 2 …xk †gxk dX

0

2  b ijk … 2x; x; x†e0i ej ek d… hx0k0

2hxk † ;

…6:1:4†

where dx0k0 =dk0 denotes the group velocity of the harmonic wave. Using equation (1.3.4) and exploiting properties of the delta function (Dirac 1958) we arrive at the expression dC coh ˆ

  2 …2† I2 …xk †gxk dXx3 n…x0k0 † n2 …xk † ‡ 2 4 n2 …x0k0 † ‡ 2 3 3 4 hc5 p2 e30 n2 …xk † 2  b ijk … 2x; x; x†e0i ej ek …gM M† : …6:1:5†

We retain the subscript wave-vector indexing on the circular frequencies only where the result is specifically wave-vector sensitive, as for example in the arguments of incident radiation intensity and the refractive indices 1). Elsewhere, where only the magnitude is required to calculate the final result, it is sufficient to express the frequency as simply x.

1) The refractive index is wave-vector sensitive in the case of birefringent media,

but the labelling can be dropped where isotropic materials are considered.

100

6 Coherent Harmonic Generation

The results given by equation (6.1.5) take the same form as those calculated using the more familiar vacuum electric field operators (see for example Craig and Thirunamachandran 1984), here properly modified by refractive correction factors. In general, rates calculated for any interaction using the vacuum formulation can be made to accommodate media corrections if the quantum amplitudes are modified by refractive factors fn…xk †g 1=2 ‰fn2 …xk † ‡2g=3Š or n1=2 …x0k0 †‰fn2 …x0k0 † ‡ 2g=3Š for each photon annihilated or created, respectively, where n…x† is the refractive index at the appropriate optical frequency. This is a prescription that bypasses much of the underlying physics, including the implicit photonic branch indexing; however it expedites direct calculations on real dispersive media. Developing this approach, we can identify a key prescription for tailoring the kernel of the general rate equation (3.1.5), and also the n-harmonic emission equations of chapter 5, as follows: !  3 0 …m† 0…m† 2 …nx† e n…x 0† I…xk †e0 n 0 0 k a
q d… hxk0 n hxk † ! 16p3 c 3 2cn…xk † 2  gx…n†k a…n‡1†
s…n‡1† dX  2    n …xk †‡2 2n n2 …x0k0 †‡2 2 :  3 3 …6:1:6† For later convenience, the above result can be expressed as …m† 0…m† 2 a
q d… hx0 0 k

2 n hxk † ! D…n† a…n‡1†
s…n‡1† dX ;

with D(n) a beam parameter defined by !  3 0 …nx† e n…x 0† I…xk †e0 n 0 …n† k D ˆ 2cn…xk † 16p3 c 3  2 2n  2 0 2 n …xk0 † ‡ 2 n …xk † ‡ 2  gx…n†k : 3 3

…6:1:7†

…6:1:8†

Successively bracketed factors in equation (6.1.8) signify: the cubic dependence of the conversion efficiency on the emergent harmonic frequency; the nth power dependence on the input irradi-

6.1 Harmonic Intensities 101

ance; the 2nth power dependence on Lorentz factors for the input; quadratic dependence on the Lorentz factor for the harmonic, and lastly the degree of nth order coherence of the input. In applying equation (6.1.7) it should be borne in mind that, in the case of a crystal or suspension, a(m) is taken to be the cell or unit susceptibility ^ a…m†. Conversely for any fluid system, orientational averaging has to be performed as described in chapter 5. For …3† SHG, with aijk  bijk … 2x; x; x†, we now have; dC SHG ˆ

o 2p n …2† …3† 2 D b
s dX …gM he60 

M† :

…6:1:9†

For expression in terms of experimentally useful parameters the above expression for a differential rate is better recast in terms of a radiant intensity of harmonic emission (energy per unit time per unit solid angle) using hx0 † I…k0 † ˆ …

dC dC ˆ …2 hx† : dX dX

…6:1:10†

The total rate is obtained by integration over 4p steradians, but the requisite integration can only be effected once the angular disposition of the emission is determined, as will be established later in this chapter. We now return to the general equations for the rate of coherent harmonic emission, which for a molecular crystal is given by (5.1.8), for a fluid by (5.2.6), and for a suspension or other randomly oriented assembly of ordered aggregates, equation (5.3.5). Incorporating the beam characteristics according to the prescription given by (6.1.8) the explicit result for the rate of n-harmonic generation, CnHG, in the case of a molecular fluid, is as follows dC nHG ˆ

2p 2…n‡1†

he0 



D E 2  …n‡1† …n‡1† D a
s dX …gN N† ; …n†

…6:1:11†

where a(n+1) is the molecular nonlinear response tensor given by (4.1.1), the coherence factor gN is given by (5.4.1) and s(n+1) is the corresponding polarisation tensor (6.1.2). The corresponding result for a crystalline medium is obtained by replacing the number of molecules, N, by the number of cells, M, in equation (6.2.1); also the molecular response tensor a…n‡1† is replaced by

102

6 Coherent Harmonic Generation

^ a…n‡1† , the cell tensor, and no rotational average is taken. An identical procedure is followed for a system of randomly oriented particles or domains; however here an additional contribution arises, originating from equation (5.3.4), as will be discussed in chapter 7. Generalising (6.1.10), the rate equation is recast in terms of a radiant intensity of harmonic emission, I…nHG† …k0 †, through I…nHG† …k0 † ˆ …n hx†

dC nHG : dX

…6:1:12†

Thus, assuming that the number of scattering centres is large enough to ensure that gN  N, the following general result applies: I…nHG† …k0 † ˆ

2pnxD…n† gN 2…n‡1† e0

jha…n‡1†
s…n‡1† ij2 :

…6:1:13†

In considering the directed n-harmonic delivered by a molecular fluid, a rotational average has to be taken in order to decouple the radiation and molecular frames of reference, denoted in equation (6.1.13) by the angular brackets. At this juncture, not only the intrinsic symmetry of the molecular polarisability, but also further symmetry criteria come into play, and each can render a null or near null result, to be discussed in more detail in the next section.

6.2

Rotational Averaging and Symmetry Criteria One of the most crucial features of harmonic generation is its sensitivity to symmetry criteria both on the local molecular and the macroscopic level. As shown in chapter 4, parity arguments prove that the electric dipole generation of even harmonics is generally forbidden in any atom, or in any molecule or crystal possessing a centre of symmetry. Furthermore, such processes remain unobservable even in gases and liquids composed of non-centrosymmetric molecules, except in rare cases where an altogether different mechanism involving a higher order of optical nonlinearity

6.2 Rotational Averaging and Symmetry Criteria 103

comes into play (see chapter 7). By extension of the classical electrodynamical principles normally applied to solids, it is commonly concluded that the generation of even harmonics is forbidden in fluids as they too possess macroscopic inversion symmetry. That argument obscures the fact that the coherent process is in any case forbidden as a result of random molecular orientation. In molecular fluids, the rotational averaging denoted by angular brackets imposes additional symmetry constraints. Consider first the simplest case, that of second harmonic generation with pure electric dipole (E13) coupling. The result requires evaluation of the rotational average in equation (6.1.13), which can be accomplished through separation of the frames of reference for the molecular and radiation parameters according to equation (5.2.11). Utilising equation (6.1.13) for the particular case under consideration, the intensity of the harmonic is evidently determined by the rotational average of the modulus square of the outer product of molecular and radiation tensors, represented here as ISHG …k0 † ˆ

2 …2† 4pxD…2† gN 2 4pxD gN 0 0 b
e ee ˆ b e e e hl l l i j h i j : j ik jl k km k…lm† i e60 e60 …6:2:1†

The Greek indices in equation (6.2.1) denote components referred to molecular axes, and lik is the direction cosine of the angle between the i-axis in the radiation frame and the k-axis in the molecular frame. Evaluation of the requisite third rank tensor average, using a method described in appendix 2, yields the result; ISHG …k0 † ˆ

2 pxD…2† gN 0 b e e e e e k…lm† i j k ijk klm : 9e60

…6:2:2†

The index contraction between the fully index-antisymmetric LeviCivita tensor eijk and the jk symmetric product of polarisation components ejek invokes the vector cross product of e with itself and is clearly zero. Thus macroscopic isotropy precludes the observation of a coherent second harmonic signal in the forward (k0 ) direction. Similar conclusions must be drawn when higher- order multipolar terms in the exact development (equation (1.2.12)) are con-

104

6 Coherent Harmonic Generation

sidered – such as the magnetic dipole and electric quadrupole interactions of equation (1.2.13). Take for example the possibility of a contribution to coherent SHG associated with one electric quadrupole interaction in the pump photon annihilation process (E1E2)E1. This requires evaluation of a fourth rank orientational average (for details, again see appendix 2); in this case products of Kronecker delta pairs come into play, leading to a result entailing scalar products between pairs of the vectors e0 , e, e and k. The explicit result (Andrews 1993) is transparently zero since, for emission in the direction necessary for phase-matched coherent response, k is orthogonal to both e and e0 . Analogous remarks apply when the contribution (E12)E2 involving electric quadrupole emission of the harmonic photon is considered. In the general case of n-harmonic generation, consider that the multipolar expansions of the interactions involving the harmonic and pump waves are taken to order s and t respectively. Here a multipole of order s for the harmonic means an Es or Ms interaction, e.g. s ˆ 2 for a magnetic quadrupole interaction, M2. For the pump, t is a sum of the orders involved in each of the photon annihilation interactions, for example in SHG with one electric dipole and one electric quadrupole photon annihilation, t ˆ 3. Defining p ˆ t n and q ˆ s 1, it can be shown that the harmonic intensity can be represented by the following generalisation of equation (6.2.3) (Andrews 1980; see also Kielich 1983): I…nHG† …k0 † ˆ

2p…n hx†D…n† gN 2…2n‡1†

e0 * + 2 …n‡1† …p‡q† XX Y Y ^…j† ; k  kp k0 K…n‡p‡q‡1†
w…i† p q j i …6:2:3†

where w(i) are unit vectors (such as polarisation vectors) perpendicularly disposed to the propagation direction and K is a generalised form of response tensor which accommodates any combination of multipolar couplings. In view of the rigorous symmetry to be considered, it is worth emphasising that equation (6.2.2) embodies the form of nonlinear optical response to all orders of multipole expansion – including the diamagnetic coupling term

6.2 Rotational Averaging and Symmetry Criteria 105

in (1.2.16) which, although weak, is often unjustifiably overlooked. Although the result has a necessarily complicated form in order to entertain all such couplings, its application is manageable in most cases. For example second harmonic generation with all electric or all magnetic dipole coupling gives s ˆ 1 and t ˆ 2, so that p ˆ 0 and q ˆ 0; however if one electric quadrupole participates in the interaction with the pump, we have s ˆ 1 and t ˆ 3, giving p ˆ 1 and q ˆ 0. For crystalline media, where the local response is associated with the unit cell and there is no rotational averaging, the analysis reveals that there must always be certain multipolar response tensors which are non-zero, so that generation of every harmonic is always at least weakly allowed (see for example chapter 4). With regard to fluids that are isotropic both locally and globally, rigorous analysis of equation (6.2.3), taking into account the general structure of the rotational averages, reveals that the coherent generation of all even harmonics is forbidden to all orders of multipole approximation (Andrews & Blake 1988). Similar conclusions can also be drawn on the basis of angular momentum considerations (Stedman, 1993). This conclusion has important practical implications, since it facilitates and vindicates the use of SHG as a surface-specific probe. It is again an illustration of the importance of dealing with observables; semiclassical treatments addressing such issues in terms of nonlinear polarisation fail to differentiate coherent and incoherent response. The weak response associated with the latter (to be discussed in detail in chapter 8) is not precluded by bulk isotropy; at this stage, it may simply be noted that within the bulk of an isotropic medium, any longitudinal polarisation (i.e. one which lies along the propagation vector of the pump radiation) can only produce incoherent harmonic emission. Exceptions to the forbidden character of even harmonics in fluids or otherwise microscopically disordered media can arise where there is a degree of anisotropy either naturally present or induced. As a common example of the latter, a degree of molecular order may be conferred on a polar fluid by application of a static electric field. A higher- order process entailing the coupling of harmonic conversions can also result in production of an even harmonic through a mechanism involving an even number of electric dipole interactions. Both mechanisms are fully discussed in chapter

106

6 Coherent Harmonic Generation

7. Intense pump laser radiation can directly induce a degree of local anisotropy through a number of other mechanisms; for example (i) quadratic interaction of the strong electric fields generated by the radiation (Mossberg et al. 1978, Miyazaki et al. 1981); (ii) multiphoton ionisation, inducing essentially static fields through charge separation (Malcuit et al. 1990, Marmet et al. 1991), (iii) generation of a non-uniform density gradient in laser-induced plasma (Liang et al. 1992), (iv) in glass fibres, formation of a grating or spatially periodic electric field (Österberg and Margulis 1986, Stolen and Tom 1987). However in cases where the evolution of a second harmonic is demonstrably a structural effect, there is a clear difference between the expected behaviour of disordered or isotropic solids and genuine fluids. Resuming discussion on the allowed generation of odd harmonics in fluid media, it is important to note that isotropic averaging imposes additional polarisation constraints. When the pump radiation is plane polarised, for example, odd harmonics are necessarily produced with retention of polarisation. It also transpires that the generation of any odd harmonic is forbidden if the pump beam is circularly polarised (Bloembergen 1969; Tang and Rabin 1971). This is because odd harmonics are associated with even rank rotational averages, which lead to scalar products …e
e† in the polarisation tensor. For circular polarisations of left or right handedness the scalar product of the polarisation vector with itself is zero – see equations (3.1.8) and (3.1.9) – and so the harmonic intensity necessarily vanishes. Consideration of angular momentum conservation leads to the same conclusions; the vector sum of the input and output photon angular momenta cannot be zero in the case of a circularly polarised beam. It should be noted, however, that these arguments are applicable only to coherent harmonic production; as will be shown in chapter 8, incoherent harmonic emission remains finite both for even harmonics and for circularly polarised radiation, whether for forward or non-forward emission. In passing, note that this reasoning on the engagement of circular polarisation states applies to any nonlinear optical interaction in an isotropic medium. In general, the observable has to involve linear combinations of scalars formed by contracting the polarisation tensor, i.e. a product of polarisation vector components for each photon involved, with an isotropic tensor of the same rank. Unless the polarisation tensor has a zero-weight component (see appendix

6.3 Third Harmonic Generation 107

4), corresponding in angular momentum terms to a Dl ˆ 0 process, all such scalars vanish identically and the process is forbidden. The conditions which this imposes on the polarisation tensor lead to the result that every parametric process which entails the absorption or emission of two or more circularly polarised photons of the same handedness and direction of propagation is forbidden in an isotropic medium. This is true regardless of whether or not any of these photons have the same frequency.

6.3

Third Harmonic Generation To further illustrate the general methods developed in this chapter, we conclude with an application to third harmonic generation (THG). This process has obvious frequency conversion applications and is of further interest for a number of reasons. The detailed study of its frequency dependence can give important insights into the excited states that dominate nonlinear optical channels in various media, principally through observation of its two- and three-photon resonance behaviour. A key feature is that it is a phenomenon which, like Rayleigh scattering, is universally allowed on the basis of molecular or bulk phase symmetry. All molecules have a non-zero tensor a…4†  c… 3x; x; x; x†; it is a property tensor of even parity, and all such tensors have components that transform under the totally symmetric representation of the salient molecular point group. Effecting the rotational average for any fluid sample gives a harmonic intensity that vanishes only under certain unusual polarisation conditions (see below). Specifically, from equations (5.2.7) and (6.1.7) we obtain the rate, CTHG, as follows:  dC THG ˆ

 E 2 2pD…3† dXgN D …4† c
s ; he80 

…6:3:1†

where c is explicitly constructed according to the principles of chapter 4, and the polarisation tensor s…4†  e0 eee. For fluids, evaluation of the rotational average denoted by the angular brack-

108

6 Coherent Harmonic Generation

ets in (6.3.1) entails fourth rank isotropic tensors (see appendix 2) and leads to the following result: 0

ITHG …k † ˆ



 2 6pxD…3† gN 0 …e
e†…e
e† c : k…kll† 25e80

…6:3:2†

The factor …e
e† attests to the fact that in the singular case of a circularly polarised pump the intensity of third harmonic emission is zero, as also established for SHG in section 6.2. Indeed the realization of this unusual condition provides the only means of suppressing the harmonic. The cubic dependence on laser intensity I3 …xk † embodied in the beam factor D(3) indicates that THG should be a weaker process than SHG  I2 …xk †, when both are allowed. Since SHG is forbidden in gases under normal conditions, THG is usefully employed here as a means of frequency up-conversion. However, the need for focussing to obtain sufficiently high intensities imposes additional constraints on the phase-matching. In general it transpires that if the extent of the focal region is comparable to the cell length then a small wave-vector mismatch optimises the efficiency of conversion (Ward and New 1966; Bjorklund 1975). Satisfaction of the appropriate phase-matching requirements mostly necessitates working in regions of anomalous dispersion close to resonance. In this respect two-photon resonances are more helpful than one-photon resonances, as the latter can result in strong absorption of the pump radiation. For a more extensive discussion of THG the reader is referred to Reintjes (1984).

109

7

Special Systems for Second Harmonic Generation When you have eliminated the impossible, that which remains, however improbable, must be the truth. Holmes’s Maxim, Arthur Conan Doyle

As shown in previous chapters, second harmonic generation is usually precluded in isotropic fluids on symmetry grounds. Nevertheless, conditions have been discovered or engineered in which it is possible to observe and exploit, in systems with isotropic or cubic symmetry, this and other such ‘forbidden’ optical processes. This chapter develops and applies the general theory for the special systems and mechanisms responsible.

7.1

Second Harmonic Generation at Surfaces and Interfaces At the boundary of any isotropic medium, as for example at a fluid surface or an interface between immiscible liquids, local symmetry is lowered and the principle behind the exclusion of even harmonics no longer applies. Extensive use has been made of the surface-selectivity of second harmonic generation and it has become an important tool for studying the chemistry and physics of interfaces (Shen 1989, Eisenthal 1993), both with regard to surfaces and surface-adsorbed layers. Although the harmonic is often studied by reflection, a host of experimental geometries are possible; these have been systematically discussed by Mizrahi and Sipe (1988). At any two-dimensional boundary, the general expression for the intensity of n-harmonic emission can be obtained from (5.2.7). For 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

110

7 Special Systems for Second Harmonic Generation

surface applications this equation may be regarded as a first order approximation, which in principle is subject to any corrections necessary to account for orientational correlation between molecules close to the surface. Such effects can be significant in polar liquids (see for example Yang et al., 1991) though for their assimilation in the general theory, knowledge of the detailed structure near the surface is required. The major new feature in applying equation (5.2.7) to a surface or interface region, however, concerns correct interpretation of the coherence factor gN, hitherto used as an expedient approximation for gN N. This factor explicitly arises from the double sum: gN

Nˆ

N X N X n

eiDk
Rnn0 ;

…7:1:1†

0

n 6ˆn

with the summations here to be interpreted as being taken over the surface molecules. Consequently, each Rnn0 vector lies in the (X,Y) plane, where Z specifies the surface normal and X is defined by identifying (X,Z) with the (k,k0 ) scattering plane as in Fig. 7.1. For any emission where Dk ˆ nk k0 is disposed in the Z-direction, the argument of the exponential in (7.1.1) becomes zero and the factor gN N attains the value N…N 1† associated with coherent emission. This amounts to the usual rule for reflection at an angle equal to the angle of incidence. However for reflection second harmonic generation, as can be understood from Fig. 7.1, it is generally necessary to fulfil the condition,

hi hr

^ component of Fig. 7.1 Geometry for surface harmonic generation. The X …2k

k0 † must vanish for phase-matched emission.

7.1 Second Harmonic Generation at Surfaces and Interfaces

2k sin hi ˆ k0 sin hr ; …7:1:2† where hi and hr are the angles of incidence and reflection as normally defined. Consequently we obtain, in terms of the refractive index sin hi n…2x† ; ˆ n…x† sin hr

…7:1:3†

a special case of a general relationship for frequency addition first obtained by Bloembergen and Pershan (1962). The phase-matching requirement that leads to the reflection law applies to layers of molecules situated within a distance from the surface of approximately k 1. At optical frequencies, any coherent signal from layers further from the surface would disappear through destructive interference. In this connection some confusion has arisen over the issue of surface harmonic emission through different usage of the term ‘bulk’ by various authors. In particular, ambiguity arises where surface SHG is used to study adsorbed molecules and the term ‘bulk’ is used to designate the support material. For such a purpose the term substrate is preferable, since it does not imply the major part of the solid material. Such a distinction is crucial where the substrate has an isotropic structure, as we shall see below. Consider the case of second harmonic reflection from the surface of a solid, or molecules adsorbed on a solid surface. The harmonic intensity clearly depends on the number of such molecules intercepted by the pump beam, giving an overall sec2 hi factor – consistent with maximum effect when the pump skims the surface. The strength of the harmonic is also powerfully influenced by orientational features, determined by the factor b
e0 ee. Plane-polarised components of the pump and harmonic waves are conventionally defined as s- or p-polarised by the relations; es ˆ ^J ;

…7:1:4†

^; ep ˆ cos hi^I ‡ sin hr K

…7:1:5†

e0s ˆ ^J ;

…7:1:6†

e0p ˆ

^; cos hi^I ‡ sin hr K

…7:1:7†

111

112

7 Special Systems for Second Harmonic Generation

^ are unit vectors in the (X,Y,Z) directions, respecwhere …^I; ^J; K† tively; Hopf and Stegeman (1985) employ similar relations but expressed in terms of glancing angles. Substitution of these results in the expression for b
e0 ee is straightforward. For example with an s-polarised pump, and resolving the harmonic also for its s-component, we have b YYY – note that the tensor indices here refer to the surface frame. Clearly experiments with different polarisation conditions (including circular polarisations) facilitate evaluation of the various components of the response tensor, referred to the (X,Y,Z) frame. These in turn are related to the components in the molecular frame through the relation bijk ˆ bklm hlik ljl lkm i ;

…7:1:8†

where lik is the …i; k† element of the Euler angle matrix relating the molecule-based to the laboratory-fixed frame. Here the angular brackets denote a distributional, rather than an isotropic, average over molecular orientations. This reflects the fact that there will generally be a preferred, but not a fixed, orientation with respect to the surface. Nonetheless it is commonly assumed that all molecules have identical orientation on the surface, and many experimental studies exploit this assumption in a determination of surface molecular orientation. Moreover, in view of the complexity of the results, calculations are frequently based on the additional premise that adsorbed or surface molecules have response tensors dominated by axial components. The detailed theory, allowing for partial orientational ordering, is given by Mazely and Hetherington (1987 a, b). Another interesting feature of surface SHG is that it offers a means of measuring surface chirality, affording a powerful tool for the study of adsorbates such as proteins or other chiral molecules (Byers et al. 1994; Hicks et al. 1994). Consider an experiment whose fundamental frequency input is a circularly polarised beam. For left circular input its polarisation vector is given by equation (3.1.8) and for right by (3.1.9). Identifying the ^i vector with es and ^j with ep we have 1)

1) These signs for left and right circularity are consistent with the optics conven-

tion used throughout this book, opposite to those employed in the cited publications.

7.1 Second Harmonic Generation at Surfaces and Interfaces

e…L=R†

 p ˆ 1= 2 …es  iep † ;

…7:1:9†

with the upper sign for left circularity and the lower for right. The intensity of a surface harmonic is determined by jb
e0 eej2; for detection through an s-polariser the result is expressible as follows, in terms of hyperpolarisability components referred to the surface frame jb
e0 eej2 ˆ 12 cos2 hi bYXX  i cos hi bYXY

cos hi sin hi bYXZ

 i cos hi bYYX ‡bYYY  i sin hi bYYZ  i sin hi bYZY

cos hi sin hi bYZX

sin2 hi bYZZ j2 …7:1:10†

In chiral molecules the low symmetry means that some of the tensor components in (7.1.10) vanish – in a molecule of C2 symmetry, for example, the above reduces to jb
e0 eej2 ˆ 12j cos hi sin hi b YXZ  i sin hi bYYZ  i sin hi bYZY j2 ˆ sin2 hi ibY…YZ†

cos hi sin hi bYZX cos hi bY…XZ† : …7:1:11†

The result is clearly sensitive to the handedness of the input beam, and to gauge the difference in the harmonic signal on reversing the circularity of the input a differential harmonic intensity can be defined: DISHG ˆI…L ! s† I…R ! s† / Re bY…XZ† Im bY…YZ†

Re bY…YZ† Im bY…XZ† :

…7:1:12†

The result thus depends on the imaginary part of the key hyperpolarisability components. As seen in chapter 4, these become significant under near-resonance conditions – in the case of a system with an electronic level close to resonance with the harmonic frequency, the dominant part of the tensor is as given by equation (4.3.4). Under optimum conditions the relative differential intensity ‰I…L ! s† I…R ! s†Š  12 ‰I…L ! s† ‡ I…R ! s†Š can approach a value of unity; in this respect the effect is several or-

113

114

7 Special Systems for Second Harmonic Generation

ders of magnitude more sensitive than conventional circular dichroism (which entails much weaker magnetic dipole coupling). Other, more intricate orientational factors can be manifest in studies on crystal surfaces and solid adsorbates. For example the SHG signal reflected from the (111) face of a cubic crystal depends on the angle the (k, k0 ) plane makes with the C3 axis, and a plot of the harmonic intensity against this azimuthal angle nicely displays the threefold symmetry. Moreover, it provides a sensitive measure of adsorbed species (Corn 1991). As with surface-enhanced Raman scattering, there is often a significant enhancement of the SHG signal associated with molecules adsorbed on suitable metal surfaces, thought to be associated with coupling to surface plasmons and influenced by surface roughness. The detailed theory is described by Chen et al. (1983), Aktsipetrov et al. (1990) and Li and Seshadri (1991). Returning to the issues of harmonic reflection from an isotropic fluid, on considering local symmetry implications, it again transpires that for any signal associated with (E13) pure electric dipole coupling the result of effecting the requisite molecular rotational average is zero (if the molecules at the surface are isotropically oriented). This is because equation (6.2.2), which yields the factor …e  e†, still applies. However when the possible involvement of electric quadrupole coupling (E12E2) is entertained, the polarisation factor …k
e0 †…e
e† will generally remain, since k is not parallel to k0 and there is no requirement for the first scalar product to vanish. With regard to the condition e0 kk, this signal is from a classical viewpoint interpreted as radiation associated with a longitudinal nonlinear polarisation 2); however such signals only arise at or close to the surface. For similar reasons a further term arises from electric quadrupole emission, associated in this case with a factor …k0
e†…e0
e†. Such types of nonlinear response are particularly significant where centrosymmetric molecules are concerned, since involvement of an electric quadrupolar interaction affords the principal mechanism for second harmonic production. 2) This use of the term longitudinal is potentially misleading. In the Coulomb

gauge, all radiative fields are transverse, and this of course applies to both pump and harmonic photons. The descriptor longitudinal can only be appropriated when the harmonic field is referred to the direction of propagation of the pump beam. It is the essentially classical inference of a nonlinear polarisation giving rise to the harmonic field which leads to the apparent misnomer.

7.2 Electric Field-Induced Second Harmonic Generation 115

To summarise, when SHG is detected in a transmission mode there can be no coherent (phase-matched) signal from the bulk of any isotropic medium; there is an incoherent (non-directed) signal associated with pure electric dipole (E13) coupling. For SHG studied by reflection from the surface of an isotropic medium, there is a coherent signal associated with an interaction involving one electric quadrupole (E12E2). Both the surface-specific coherent (E12E2) and the non-specific incoherent (E13) signals are classically interpreted as arising from an induced longitudinal polarisation. There is no coherent signal from the bulk associated with longitudinal polarisation (Andrews 1993). In considering surface harmonic generation from fluid surfaces one other factor to bear in mind is that a degree of orientational order may be present close to the surface. Where the constituent molecules are polar, a mechanism similar to that associated with electric field-induced second harmonic generation (see following section) can then operate. Certainly this accounts for the signal observed from certain aqueous media, in which water molecules oriented through proximity to a charged interface generate a second harmonic (Ong et al. 1992). This type of mechanism may also account for other experimental results that suggest a harmonic signal associated with the bulk of an isotropic medium. The skin depth associated with this effect invites further experimental study but crucially, in such cases, the signal derives from a region that is not truly isotropic. With a sufficiently high intensity pump, the six-wave interaction discussed in section 7.4 can also produce a second harmonic signal within the isotropic system.

7.2

Electric Field-Induced Second Harmonic Generation Although SHG is generally forbidden in isotropic media, the application of a static electric field can remove the forbidden character of the process by two distinct mechanisms. In one mechanism the static field induces an electro-optical contribution to the harmonic signal at the molecular level. This contribution is formally associated with fourth-order perturbation theory involving a static coupling term and represented by time-ordered diagrams

116

7 Special Systems for Second Harmonic Generation

Fig. 7.2 One of the twelve possible time-orderings for static electric field-

induced second harmonic generation, EFISHG. The static interaction, l
Elocal , is represented by the horizontal dotted line.

of the kind shown in Fig. 7.2 3) By this mechanism SHG becomes allowed both in centrosymmetric crystals and in fluids. The second mechanism applies to fluids or disordered media consisting of polar (and thus necessarily non-centrosymmetric) molecules. Most molecules with non-zero hyperpolarisability components are of this kind. Here, the local electric field Elocal exerts a torque on each permanent molecular dipole l determined by the vector product l  Elocal . This removes bulk isotropy by inducing a degree of molecular alignment (Mayer 1968), governed under equilibrium conditions by the Boltzmann distribution. Hence in fluid media a weighted molecular rotational average has to be effected with regard to a temperature-dependent distribution of molecular orientations. In the case of fluids, the field is applied concurrently with the pump laser, a technique known as electric field-induced second harmonic generation (EFISHG). For disordered solids such as glassy polymers,

3) In the QED formalism the dipolar interaction operator for molecular coupling

with a static field emerges as a result of accommodating virtual photon propagation between the molecule and the material comprising the field ‘source’. With zero energy transferred to or from the molecule the result equates to the semiclassical form of static interaction.

7.2 Electric Field-Induced Second Harmonic Generation 117

the static field may be applied during solidification from the melt phase, a technique known as poling. This produces a solid whose inbuilt orientational order permits the observation of SHG which persists after the field is removed (Singer et al. 1986, 1987). The rate equation for SHG …n ˆ 2† in the presence of an applied static electric field Elocal can be obtained from equation (5.2.7), where m ˆ n ‡ 1 ˆ 3 for SHG, by incorporating an (E1)4 term from fourth order perturbation theory, and also introducing the Boltzmann weighting term exp…l
Elocal =kB T†. In nematic or smectic liquid crystals, or domains within cholesteric liquid crystals, where there is a preferred orientation in the absence of any applied field, a correction to account for the associated orientational potential can be included in the exponent (Meredith et al. 1982). Once again the results that ensue are strictly subject to corrections for the orientational correlation that accompanies field-induced anisotropy – see for example Guan and Wang (1993) – but that cannot properly be modelled without an analytic form for the pair potential. On the assumption that each polar molecule experiences an orienting force, determined by the applied field, the result, utilising the beam parameter equation (6.1.8), is a radiant second harmonic intensity, IEFISHG …k0 †, given by   4pxD…2† gN IEFISHG …k † ˆ e60  b… 2x; x; x†
e0 ee‡b` … 2x; 0; x; x†
e0 E ee el
Elocal =kB T
 2 local  l
E =k T B local he i 0

`

where the correction term involves the fourth-order tensor b (Kielich, 1969; Lam and Thirunamachandran, 1982; Andrews and Sherborne, 1986) with elements defined by;

118

7 Special Systems for Second Harmonic Generation `

b ijkl … 2x; 0; x; x† ˆ

( X r;s;t

‡ ‡ ‡ ‡

…E~t0

…E~t0 …E~t0

ts sr r0 l0t i lj lk ll 2hx†…E~s0 2hx†…E~r0

hx†

ts sr r0 l0t i ll lj lk 2hx†…E~s0 hx†…E~r0 hx† ts sr r0 l0t i ll lk lj 2hx†…E~s0 2hx†…E~r0 †

ts sr r0 l0t j li ll lk …E~t0 †…E~s0 2hx†…E~r0

hx†

ts sr r0 l0t j ll li lk

…E~t0 †…E~s0 ‡ hx†…E~r0 hx† ts sr r0 l0t j ll lk li ‡ …E~t0 †…E~s0 ‡ hx†…E~r0 ‡ 2hx†

ts sr r0 l0t l li lj lk …E~t0 ‡ hx†…E~s0 hx†…E~r0 ts sr r0 l0t l lj li lk ‡ …E~t0 ‡ hx†…E~s0 ‡ hx†…E~r0

‡

‡

hx† hx†

ts sr r0 l0t l lj lk li

…E~t0 ‡ hx†…E~s0 ‡ hx†…E~r0 ‡ 2hx† ts sr r0 l0t l lk lj li ‡ …E~t0 ‡ hx†…E~s0 ‡ 2hx†…E~r0 ‡ 2hx† ts sr r0 l0t l li lk lj ‡ …E~t0 ‡ hx†…E~s0 hx†…E~r0 † ts sr r0 l0t l lk li lj ‡ …E~t0 ‡ hx†…E~s0 ‡ 2hx†…E~r0†

) :

…7:2:2†

The explicit result that follows from (7.2.1) can be cast in terms of a pair-index symmetric tensor defined by `

`

`

b ij…kl† ˆ 12 …b ijkl ‡ b ijlk † :

…7:2:3†

The group theoretical representation associated with coupling of this form is 2D…0‡†  3D…1‡†  4D…2‡†  2D…3‡†  D…4‡† , accommodating a maximum of 54 independent components in the

7.2 Electric Field-Induced Second Harmonic Generation 119 Tab. 7.1 The reduced spherical Bessel functions, j0n … ic†  jn … ic†=j0 … ic†:

Low field limit, c  1

Full expression j01 j02

 i 1c

coth c



 i

j03

‡ 3c coth c
 i 1 ‡ 15 c2 coth c

j04

105 1 ‡ 45 c2 ‡ c4

1

3 c2

10 c

6 c

15 c3




coth c ‡ 105 c3

c 3

c2 15

945







 c3 i 105
c4

High field limit, c1 
3
5
c 2c 1 1 45 ‡ 945 ‡ . . . i c

‡

2c4 315† c5 945

‡ ...




...

... 

1 ‡ 3c

3 c2

6 c

‡ 15 c2

 i 1 1

10 c

‡ 45 c2

case of a symmetry-free molecule. On assuming Kleinman symmetry the representation reduces to D…0‡†  D…2‡†  D…4‡† , accounting for just 15 components (see appendix 4). Since D(0+) ` components invariably arise the tensor b is always finite, a characteristic it shares with the molecular polarisability but markedly in contrast to the hyperpolarisability b. Hence, irrespective of the molecular symmetry, a static electric field can invariably mediate SHG by this electro-optical mechanism. Evaluating the weighted rotational averages in equation (7.2.1) entails complex tensor calculus, and a synopsis of the method may be found in appendix 2. The orienting effect of the electric field is manifest through the appearance of reduced spherical Bessel functions j0n , defined by j0n … ic†  jn … ic†=j0 … ic† ;

…7:2:4†

whose imaginary argument has a modulus determined by the Boltzmann factor c ˆ l
Elocal =kB T :

…7:2:5†

The analytic form of the j0n functions is presented in Tab. 7.1, which also gives the asymptotic low-field (high temperature) and high-field (low temperature) behaviour. To put the results into perspective, at STP 4) a molecule with a dipole moment of 1 Debye experiencing a (high) local field strength of 5 ´ 108 V m–1 yields a c factor of 0.4. 4) Standard temperature and pressure.

15 c3 105 c3

 ‡ 105 c4

120

7 Special Systems for Second Harmonic Generation

The result for the harmonic intensity, for arbitrary pump, polarisation and electric field direction, is as follows:   4pxD…2† gN IEFISHG …k † ˆ 42ij01 ‰…3p1 44100e60 0

p2 †b 2 ‡ …2p2

p1 †b1

210j02 p4 b3 2lij03 ‰…5p3 2p1 p2 †…5b 4 2b2 n ` ` ‡ Elocal 14‰…3p1 p2 †b 1‡…2p2 p1 †b 2 Š `

`

`

‡42ij01 ‰…b 3 ‡ b 4 †p4 ‡…5p3

2p1

‡ 2…3p2

2p1

b1 †Š

15j02 ‰2…5p1 2p3 p2 †b 8 `

`

p2 †…b 7 ‡ 2b 5 † `

p3 †b 6 ‡ 2…3p1 `

`

`

42ij03 ‰…5b 9

‡…5p2 3p3 4p4 †b 2 Š ‡ 3j04 ‰…5p3 2p1 p2 †

`

2p2 †b 1

3p3

b3

`

b 4 †Š

o 2 ` ` ` ` ` ` ` …2b 1 ‡ b 2 ‡35b 10 10b 5 5b 6 5b 7 10b 8 †Š …7:2:6† `

where bi and b j are molecular invariants as defined in Tab. 2, and p1 to p4 are polarisation parameters determined by the detailed experimental arrangement and explicitly given by ^ local
e†…e0
e† ; p1 ˆ …E

…7:2:7†

^ local
e0 †…e
e† ; p2 ˆ …E

…7:2:8†

^ local
e0 †…E ^ local
e†2 ; p3 ˆ …E

…7:2:9†

Tab. 7.2 Molecular invariants in EFISHG. The direction of the permanent-

molecular dipole moment defines the Z-axis in the molecular frame; repeated indices k, l and m are summed. Molecular invarients in EFISHG b1 ˆ b z…kk†

b2 ˆ b k…kz†

`

`

b2 ˆ b kk…ll†

`

`

b6 ˆ b zz…kk†

`

`

b10 ˆ b zz…zz†

b1 ˆ b kl…kl† b5 ˆ b zk…kz† b9 ˆ b kz…zl† eklz

b3 ˆ b k…lz† eklz

`

`

b3 ˆ b kl…mz† eklm

`

`

b7 ˆ b kk…zz†

`

`

b4 ˆ b zzz

`

`

b4 ˆ b km…lm† eklz

`

`

`

`

b8 ˆ b kz…kz†

`

`

7.2 Electric Field-Induced Second Harmonic Generation 121

^ local
e†…e0  e†
E ^ local : p4 ˆ …E `

…7:2:10†

If the b and b tensors are assumed to be fully index-symmetric, as in the Kleinman treatment, we can obtain a significant simplification of (7.2.6) through use of the resultant equalities b1 ˆ b2 , ` ` ` ` ` ` b 1 ˆ b 2 , and b 5 ˆ b 6 ˆ b 7 ˆ b 8 , with all other invariants save ` for b 4 and b 10 vanishing identically. However, as remarked earlier, such assumptions are generally unjustified unless both the pump and harmonic frequencies are substantially displaced from any absorption features of the sample. It is appropriate to draw out salient features of the SHG dependence on experimental parameters. The first three terms of equation (7.2.6) which involve j01 , j02 and j03 are solely due to field-induced anisotropy, in other words they reflect an orientational effect which applies only to polar molecules. In the case of poled polymers the values for the spherical Bessel functions are determined by the conditions that apply in the melt phase whilst the field is present. The subsequent terms all relate to the direct electro-optical EFISHG channel, as the pre-multiplier (Elocal) signifies, and would for example be absent in the case of poled polymer films studied in the absence of an applied field. Of the electro-optical terms in equation (7.2.6) the first entails no Bessel function and is thus independent of the orientational distribution; the remaining j01 j04 terms represent modifications due to any induced anisotropy. In situations where the degree of molecular alignment is minimal, the electro-optical channel dominates the conversion process and the signal is driven with an overall quadratic dependence on the applied field strength. Consider next the polarisation dependence. The most obvious aspect of equations (7.2.7)–(7.2.10) is the fact that if the static electric field is applied parallel to the pump and harmonic propagation direction (which we can denote as the Z-direction), then all four polarisation parameters become zero, and the harmonic signal vanishes. To maximise the harmonic intensity it is normal practice to impose the static field at right-angles to the beam, say the X-direction; then it immediately follows that p4 ˆ 0. A number of possibilities can be entertained; the pump radiation can for example be plane or circularly polarised, and the emergent harmonic can be resolved for plane or circular components. The appropriate values for p1 p3 in each of these configurations, evaluated using

122

7 Special Systems for Second Harmonic Generation Tab. 7.3 Values of the EFISHG polarisation parameters for various polari-

sation configurations with the static field applied in the X-direction and the beam propagating in the Z-direction. Here (a, b) denotes a-polarisation for the pump, b-polarisation for the harmonic, with C denoting circular polarisation of a given handedness and C* the opposite handedness. (X,X) (X,Y) (X,C) (Y,X) p1 p2 p3

1 1 1

0 0 0

2–1/2 2–1/2 2–1/2

0 1 0

(Y,Y)

(Y,C)

(C,X) (C,Y) (C,C) (C,C*)

0 0 0

0 2–1/2 0

1 2

0 1 2

 12 i 0 0

2–1/2 0 8–1/2

0 0 8–1/2

(3.1.8) and (3.1.9) for circular polarisations, are given in Tab. 7.3. From the results it is evident that with a plane-polarised pump the harmonic emerges polarised in the direction of the applied static field. A more striking feature is that application of the static field supports the harmonic conversion of a circularly polarised beam. Indeed if the harmonic is resolved for its Y-polarised component under the prescribed conditions, it transpires that only such a chirally endowed beam can produce a signal. If higher order (magnetic dipole and electric quadrupole) contributions are included in the calculations, the harmonic intensity is in fact marginally sensitive to the precise helicity of the pump radiation (Lam and Thirunamachandran 1982); the fractional change of intensity on reversing the circularity of the pump beam is typically about one part in a thousand. The case of helicity inversion, where the harmonic is resolved for a component of opposite circularity to the pump, is also of some interest. Since the only non-zero polarisation parameter is p3, reference to equation (7.2.6) shows that only terms involving j02 , j03 and j04 contribute in this case; hence the harmonic emission is here uniquely identifiable with the effects of field-induced anisotropy. In the low-field limit, where c  1, the j02 term dominates; from the results in Tab. 7.1 it follows that the harmonic intensity then displays a sixth power dependence on the local electric field, with an inverse fourth power dependence on the absolute temperature. With most polarisation configurations it is possible to ignore terms above first order in c under the normal low-field conditions. The results that follow from equation (7.2.6) arise solely from the orientationally independent term and corrections originating from the linear term in the asymptotic expression for j01,

7.2 Electric Field-Induced Second Harmonic Generation 123

as given by Tab. 7.1. These results may equally be envisaged as deriving from an expansion of the weighting exponentials in equation (7.2.1) in a Taylor series, with retention of the leading non-zero terms (Chemla and Bonneville 1978, Kielich 1979). Thus the result for the harmonic intensity may be expressed as   2 4pxD…2† Elocal gN IEFISHG …k † ˆ …3p1 225e60 0

‡…2p2

`

p1 †b kl…lk† ‡ b k…ll†

`



p2 †b kk…ll† ‡ b k…kl† 

 lk 2 : kB T

ll kB T



…7:2:11†

Where the molecular tensors have full permutational index sym` metry, a common factor of …b …kkll† ‡ b…kkl† ll =kB T† emerges. The latter is essentially identifiable with the much-quoted …5c ‡ b
l=kB T† factor of the original treatment by Hauchecorne et al. (1971), and reflects a methodology that ultimately assimilates thermal effects into the nonlinear optical formulation (Wagnière 1993). A combined study of the polarisation- and temperature- or field-dependence of the harmonic intensity can provide important information on molecular nonlinear optical and electro-optical parameters. An optimum geometry is shown in Fig. 7.3, where the electric field is applied at right angles to the beam propagation direction and at 45 8 to the pump polarisation vector. The harmonic signal is then analysed for two orthogonal polarisation components. If h is the angle between the pump and harmonic polarisation vectors, two special cases arise. With h ˆ 63:48 then …3p1 p2 † ˆ 0 and we obtain leading terms … 63:48† IEFISHG …k0 † ˆ

    2 ` ` 4pxD…2† Elocal gN lk b kl…kl† ‡ 2bk…ll† b mo…mo† : 6 kB T 225e0 …7:2:12†

With h ˆ

26:68, we have …2p2

… 26:68† IEFISHG …k0 † ˆ

p1 † ˆ 0 and thus

    2 ll ` ` 4pxD…2† Elocal gN b kk…ll† ‡2b k…kl† b mo…mo† : 6 kB T 225e0 …7:2:13†

124

7 Special Systems for Second Harmonic Generation

Fig. 7.3 One possible geometry for characterisation of EFISHG. The vec-

tors e, e0 and Elocal lie in the plane perpendicular to the pump and harmonic propagation vectors.

Selective temperature experiments in these two geometries permit ` ` direct evaluation of the parameters b kk…ll† , b kl…kl† , bk…kl† ll and bk…ll† lk . Each of these parameters is readily expressible in terms of tensor components. If the permanent moment is known and taken to define the z-direction within the molecule, for example, then it is possible to evaluate bk…ll† l^k ˆ bzxx ‡ bzyy ‡ bzzz . Such determinations afford a powerful and precise means for assessing the accuracy of ab initio calculations. Other component combinations are derivable from experiments with different polarisation configurations (Andrews and Sherborne 1986). With molecules of high symmetry the number of independent components is reduced by virtue of the symmetry relationships between them (Reintjes 1984). In a molecule with a threefold or higher principal axis of symmetry, for example, bk…ll† l^k ˆ 2bzxx ‡ bzzz . However individual components are in general not directly amenable to evaluation, in contrast to the case of a rigidly oriented system. For centrosymmetric molecules, where the hyperpolarisability b vanishes, the theory simplifies considerably and in an obvious way. Here, measurements of ` both the real and imaginary parts of the b tensor are possible (Kajzar et al. 1987). Such parameters provide crucial information on molecular electronic properties, and in particular the extent of charge delocalisation (Chemla and Zyss 1987). Finally attention should be drawn to a means of enhancing the electro-optical channel for EFISHG (Shelton and Buckingham 1982). Although the low density of gases is inevitably associated with relatively weak harmonic conversion, it does offer the ameliorating prospect of coherence lengths in the millimetre region. (The coherence length lc ˆ p=jDkj is the distance over which the harmonic evolves as a coherent signal; see section 5.4). Advantage

7.3 Optical Coherence in Dispersed Particles 125

can be taken of this by arranging for the applied electric field to be spatially periodic with a period of 2lc, accomplished by placing a series of electrodes in an array such that the local field Elocal reverses sign after the distance lc. The term linear in Elocal thus acquires a phase that periodically reverses sign in the exponent of the phase factor exp…ijDkj
Rn † (see equation (7.2.5)). This allows compensation of the wave-vector mismatch so that the harmonic can continue to display coherent evolution beyond its normal coherence length. The effect proves to offer an accurate method for the measurement of refractive index dispersion in gases (Mizrahi and Shelton, 1986), and has also been shown to operate in disordered solids such as optical fibres (Kashyap 1989).

7.3

Optical Coherence in Dispersed Particles Despite the symmetry rule that precludes the generation of even harmonics in optically isotropic media, except at surfaces, a number of experimental results have indicated exceptions to the rule. As discussed in previous sections, most entail conditions resulting in a transient, local removal of isotropy, and are well understood. In this section and the next we focus on two quite different mechanisms that have recently been found to mediate SHG in systems that retain macroscopic isotropy. First we consider a mechanism relating to optical coherence in small particles in suspension, or locally ordered domains within macroscopically amorphous media. The rate equations necessary for the calculation of such coherent signals have already been introduced in chapter 5; here we consider in detail the results and their applications. The coherent generation of second harmonics from particles in suspension was first observed in the production of strongly directed SHG signals from photosynthetic bacterial membranes, randomly oriented in aqueous suspension (Song et al. 1994, Schmidt and Rayfield 1994). The paradox was resolved when it was shown that the unusually intense signal detected from such isotropic suspensions is attributable to optical coherence within the separate particles of the suspension (Andrews, Allcock and Demidov 1995). As such, the harmonic emission displays an

126

7 Special Systems for Second Harmonic Generation

amalgam of the characteristics associated with full coherence (SHG) and incoherence (hyper-Rayleigh scattering). To understand this, it is necessary to return to the development of theory in section 5.3. Consider a fluid or mesoscopically disordered material (or any other optically isotropic system) within which there exist small particles or local domains possessing a microscopically ordered structure. Examples of such systems include colloids, cell and membrane suspensions, many plastics, glasses and other modern materials. In systems of interest, each particle or domain includes a significant number of optical centres with strong optical dispersion at the frequency of the pump laser radiation or its harmonic. Let us suppose that such a system comprises M randomly oriented particles (individually denoted below by the subscript w) each composed of p discrete molecules or other optical centres (denoted by the subscript n). Within each particle there is a structurally imposed orientational correlation, i.e. a significant intrinsic rigidity, such that its net optical response can be cast in terms of an ‘effective hyperpolarisability’ tensor given by b…w†ijk ˆ

p X n

n n n iDk
…Rn b…n†klm lik ljl lkm e

Rw †

;

…7:3:1†

whose components are determined relative to a common Carten n n …ljl ; lkm † sian frame centred at Rw within the particle. The factor lik is the cosine of the angle between the space- fixed axis i(j, k) and the molecule- fixed axis k…l; m†. If the particles or ordered domains are small compared to the optical wavelengths involved, then so will be the internal distances …Rn Rw †, and thus in many circumstances – even in the complete lack of wave-vector matching …Dk ˆ 0† – the phase factor in equation (7.3.1) can often effectively be taken as unity. However, we retain its explicit form for generality. The rate of second harmonic generation by the full system of particles is expressible as a sum of two terms C1 and C2 as follows, where angular brackets denote the orientational average:

C1 ˆ

2pD…2† Mhjb…w†ijk e0i ej ek j2 i ; he60 

…7:3:2†

7.3 Optical Coherence in Dispersed Particles 127

C2 ˆ

2pD…2† …gM he60 

M†jhb…w†ijk e0i ej ek ij2 :

…7:3:3†

Intensity

In the latter, representing a coherent addition of SHG amplitudes from every optical centre in the system, care is taken to ensure that the coherence factor is explicitly represented as required when gM  M. When the particles are randomly oriented, C2 vanishes as a result of the isotropic average, as is well known. However, since the corresponding average is conducted over the modulus square in the ‘incoherent’ term, C1, this contribution persists, representing an addition of the harmonic intensities produced by different particles. As determined by equation (7.3.1), these contributions accommodate a coherent addition of signals from the various optical centres that each particle comprises. An analysis of such angularly resolved measurements of a second harmonic emerging from suspensions of the purple membrane (Allcock et al. 1996) has led to the experimental verification of this interpretation. An example of the abnormally broad angle coherent emission from such suspensions is illustrated in Fig. 7.4. The coherent addition of second harmonic signals, which can occur only in regions of local order, has interesting implications for materials strongly pumped by an ultrafast source (Andrews and Dávila Romero 2001). Here, the key feature is the relationship between the response tensors of optical centres in their ground and 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

22

Angle

67

90

Fig. 7.4 Intensity of second harmonic emission from aqueous suspension

of purple membrane as a function of angular displacement from the forward direction (Allcock et al. 1996).

128

7 Special Systems for Second Harmonic Generation

electronic excited states, under near-resonance conditions. To investigate this further, we first adopt the two-level model of chapter 4.3, returning to consider the role of other electronic levels subsequently. The hyperpolarisability for an upper state jui is readily ob00 : tained using the transformed interaction Hamiltonian Hint 00 Hint ˆ

e0 1 ‰l

luu Š
d? ;

…7:3:4†

where l without superscripts denotes the electric dipole operator. Here the algorithm given in (4.3.6) is modified by interchanging the labels 0 and u; l00 ! l00

luu ˆ

`

d ; luu ! 0

…7:3:5†

(Recall that d represents the displacement electric field operator ` and d the transition` dipole moment.) This has the effect of reversing the sign of d and also the energy difference Eu0, wherever each appears, though the hermiticity of the dipole operator ensures that for non-degenerate states the transition dipole suffers no change. It then transpires that the upper level hyperpolarisability tensor stands in the following simple relationship to its ground state counterpart: b00 ijk ˆ

buu ijk ;

…7:3:6†

where the 00 superscript serves as a reminder of the ground state designation. Similar features arise when third harmonic generation is considered. 5) Consider a system in which, prior to input of pump radiation whose conversion produces the detected harmonic, irradiation with a beam of the appropriate resonant frequency produces a significant population of the electronic level u amongst the optical centres in each particle or domain. With first order decay kinetics the probability that a certain centre n is excited at time t is given by

5) A variable sign convention for the damping (see appendix 1) would lead to be-

haviour of a marginally different analytical form from that discussed here, results such as (7.3.6) then being approximate.

7.3 Optical Coherence in Dispersed Particles 129

P…Rn ; t† ˆ P0 …Rn ; 0†e

ku …t t0 †

;

…7:3:7†

where ku is the excited state decay constant and P0 is the residual probability that the upper level is excited at time t0, the time at which the pump radiation for SHG detection is applied. Particular interest focuses on the case of ultrafast excitation of sufficient intensity to elicit the onset of saturation, where P0 > 0.5. Whilst both excited and ground state species are present the two processes previously described contribute to an effective unit hyperpolarisability given by p X   n n n iDk
…R uu n f1 P…Rn ; t†gb00 b…w†ijk ˆ …n†klm ‡P…Rn ; t†b …n†klm lik ljl lkm e

Rw †

n

ˆ

p X 

f1

n

 n n n iDk
…R n 2P…Rn ; t†gb00 …n†klm lik ljl lkm e

Rw †

;

…7:3:8†

leading to a harmonic intensity that features a characteristic decay and recovery in its temporal profile. Note that for a two-level system equation (7.3.8) is exact, as follows from use of the completeness relation for the molecular states. Suppose for simplicity that the probability of initial excitation is identical for all optical centres, removing the Rn dependence of P…Rn ; t†. The system response is characterised by a scalar factor b: bˆ

p X n

n n n 0 ik
…Rn b00 …n†klm lik ljl lkm ei ej ek e

Rw †

…7:3:9†

which will in general be a complex quantity by virtue of the damping involved in the hyperpolarisability. For simplicity assuming continuous-wave pump radiation, the intensity of harmonic emission acquires a time-dependence of bi-exponential form I…k0 † ˆ

4pxD…2† …gM e60

M†jbj2 ‰1

4e

ku t0

‡ 4e

2ku t0

Š;

…7:3:10†

where t0 ˆ t t0 and P0 …Rn †  1. The result represents a harmonic signal with a trace of the form G…t† ˆ 1 4 exp… ku t† ‡4 exp… 2ku t†, as shown in Fig. 7.5.

130

7 Special Systems for Second Harmonic Generation

Fig. 7.5 Schematic of the harmonic temporal profile G…t† from equation

(1.3.10) in arbitrary units.

The fall and recovery of the harmonic output illustrated in Fig. 7.5 is not uniquely a facet of the two-level model employed; such behaviour should be manifest in systems of considerably greater electronic complexity. Although the initial population inversion is likely to require prior pumping of higher levels, the harmonic signal will be dominated by generation at centres in either the j0i or jui state, provided that no other levels are significantly populated at that stage. Certainly other electronic levels will contribute as virtual states in evaluating the hyperpolarisabilities of the j0i and jui levels. Thus, the effect of involving other levels in the calculations is that both of the tensors b00 and buu acquire additional background contributions, such that 9 p P > n n n 0 00 iDk
…Rn Rw † > b …n†klm lik ljl lkm ei ej ek e ! b ‡ b~ > = n : …7:3:11† p P > n n n 0 uu iDk
…Rn Rw † > ^ > b …n†klm lik ljl lkm ei ej ek e ! b‡b; n

This results in a harmonic intensity with a time-dependence of the more intricate form I…k0 † ˆ

4pxD…2† …gM e60

~ M†j…b ‡ b†

…2b ‡ b~

^ b†e

ku t0 2

j : …7:3:12†

7.4 Six-Wave Second Harmonic Generation 131

Separating the real and imaginary parts of each parameter b ˆ b0 ‡ ib00 ; b~ ˆ b~0 ‡ ib~00 ; b^ ˆ b^0 ‡ ib^00 ;we thus have h 4pxD…2† I…k † ˆ …gM M† …b0 ‡ b~0 † …2b0 ‡ b~0 e60  i2  00 00 00 00 00 ku t0 ~ ~ ^ b †e ‡ …b ‡ b † …2b ‡ b : 0

b^0 †e

ku t0

i2

…7:3:13†

Of the two intensity contributions in equation (7.3.13) the first, associated with the real parts of the hyperpolarisabilities, will generally dominate – leading to a minimum in the harmonic emission at a time given by ! 2b0 ‡ b~0 b^0 0 1 : …7:3:14† tmin  ku ln b0 ‡ b~0 Since the harmonic decay and recovery owes its origin to quantum coherence between harmonic emission processes at different optical centres, the result has structural as well as kinetic implications. The local coherence, responsible for the cancellation of the harmonic signal after a characteristic delay time, is entirely dependent on structural rigidity within each particle. This need not mean that all the optical centres are identically aligned, but that they do not rotate significantly with respect to each other over the timescale for the harmonic measurements. The extent of recovery in the harmonic signal serves to register the extent of local coherence, and hence characterises the structural order.

7.4

Six-Wave Second Harmonic Generation The development of laser sources with sufficient power has allowed the observation of a number of processes associated with unusually high degrees of optical nonlinearity. In connection with frequency conversion, a key discovery has been second harmonic generation mediated by six-wave mixing (SWM). The phenomenon is significant because it affords a mechanism for the

132

7 Special Systems for Second Harmonic Generation

evolution of a coherent second harmonic signal in media where it is normally forbidden; as will be shown, the process x ‡ x ‡ x ‡ x ! 2x ‡ 2x is invariably permitted, irrespective of local or bulk symmetry (Andrews 1994). A pump-probe implementation of the process has additional value, since the output then exhibits time-dependent quantum optical interference features, which can be used to characterise system dynamics. To calculate the rate, we first construct a quantum amplitude using the general equation (2.2.2) and substitute into the Golden Rule, equation (3.1.5). The detailed structure of the nonlinear po…6† larisability aSWM … 2x; 2x; x; x; x; x†, which the process entails, can be obtained using the state-sequence diagram of Fig. 7.6 (Jenkins et al. 2002). In all there are 15 (= 6!/4!2!) pathways linking the initial and final states to be taken into consideration, comprising all topologically distinct orderings of the six electric dipole interactions involved. Utilisation of the alternative time-ordered diagrams of course leads to identical tensor expressions, but the state-sequence depiction is much more concise. (For explicit representation of the somewhat unwieldy resulting expressions see appendix 5.) At this juncture it is necessary to draw attention to the presence, and general means of dealing with, the secular resonances that arise in high orders of optical nonlinearity. Such features arise when a subset of the frequencies of the waves involved sums to zero, as occurs here with one output and two input waves. Secular resonance represents the fact that quantum amplitudes for such processes can subsume the matrix elements for lower order processes (here, conventional SHG), leading to spurious infinities if not properly addressed. The various time-orderings of the SWM interaction described here clearly include some which ‘factorise’ as a product of two sequential SHG time-orderings, apparently associated with a vanishing energy denominator. To circumvent all such ‘secular resonances’ the molecular polarisability is reconstructed, by judicious combination of terms, to remove the infinities (Bishop,1994). (Explicit details for reconstruction of the SWM polarisability tensor are given in appendix 5.) The procedure is straightforward and entails properly taking the limit as the virtual state energy approaches that of the ground state. Returning to a general formulation such that we can calculate our rate expression from first principles, we employ equation

Fig. 7.6 The state-sequence diagram representing SWM.

7.4 Six-Wave Second Harmonic Generation 133

134

7 Special Systems for Second Harmonic Generation

(5.1.8) for our current coherent harmonic process. It is necessary to construct the required radiation tensor using the general expression of equation (3.1.6) and assuming there are initially no photons in the harmonic mode, we can obtain an equation for …SWM† the rate, C coh , of coherent six-wave mixing: …SWM† C coh

8   mg xk e0 4 n2 …xk † ‡ 2 h q! : N† 3 …q 4†! 2cVn…xk † !2  4 X  hm0g x0k0 e0 n2 …x0k0 † ‡ 2  3 2cVn…x0k0 † 0 0

4p ˆ 12 …gN he0 

k ;k

 ja…6†
e0 e0 eeeej2 d…2hxk0

4hxk † : …7:4:1†

By utilising the result of equation (3.2.9) it is clear that the rate of SWM carries a quartic dependence on the intensity of the pump radiation, I…xk †. The result embodies the coherence factor gN to account for the phase-matching characteristics of the process, leading to the familiar sinc2 behaviour, demonstrating that the SWM process manifests coherent emission in the forward z-direction, with wave-vector matching 2Dk ˆ 4k 2k0  0 6). This specification ensures that the single mode selected from the sum in equation (7.4.1) can be specified with some degree of accuracy, and hence the need for a more general notation is no longer required. As shown below, all molecules support the SWM tensor. The issue of whether an isotropic molecular fluid can generate an observable SWM signal in isotropic fluids requires separate analysis, and calls for explicit evaluation of the ensemble average of the tensor product in equation (7.4.1) as …6†

ha…6†
e0 e0 eeeei ˆ a…kl†…mopq† e0i e0j ek el em en hlik ljl lkm llo lmp lnq i :

…7:4:2†

Applying a sixth rank rotational average (appendix 2) reveals that the rate equation entails an overall factor …e
e†, a scalar product that vanishes for circular polarisations (see equations (3.1.8) and (3.1.9)). The six-wave interaction is thus subject to the same embargo on conversion of a circularly polarised pump as the con6) Any small wave-vector mismatch will result in conical emission of the second

harmonic (Moll et al., 2002).

7.4 Six-Wave Second Harmonic Generation 135

ventional SHG process (Tang and Rabin 1971). In the case of a plane-polarised pump, ensemble averaging leads to the result C SWM ˆ

 2 8 p h n …xk † ‡ 2 …4† 4 N†g I …x † …g k x 3 560 c 6 e60 n4 …xk † N !2   4 m0g 0 x0k0 n2 …x0k0 † ‡ 2  Vn…x0k0 † 3 n 2 o n o …6† …6†  6…e
e0 †2 2 a…kl†…klmm† 2…e
e0 †2 3 a…kk†…llmm†  d…2 hx0k0

4 hxk † ;

…7:4:3†

exploiting the permutational symmetry in the first two and last four indices of the nonlinear response tensor. Equation (7.4.3) illustrates the fact that there need not be full retention of polarisation in the emitted harmonic. The extent of depolarisation, q, is given by  2 C SWM …e0? e† 3akk…llmm† 2akl…klmm† ; ˆ  qˆ C SWM …e0 ke† akk…llmm† ‡ 4akl…klmm†

…7:4:4†

giving values within the range 0  q  9, the lower limit signifying complete retention of polarisation. If full permutational (Kleinman) index symmetry applies to the nonlinear response tensor, equation (7.4.4) reduces to the result q ˆ 1=25. Departure of the degree of depolarisation from this value thus registers the extent of invalidity of the Kleinman assumption. Before further developing the theory to a form suited for more extensive experimental applications, we outline why SWM is a mechanism allowed for all possible molecular symmetries. From inspection of the index symmetry in the radiation tensor, it is clear that a harmonic signal can derive only from that part of the sixth …6† rank polarisability aSWM which is symmetric with respect to permutation of the four indices relating to the absorbed pump photons, and also between the two indices relating to the two harmonic photons. Under group theoretical operations of the infinite rotation and rotation-inversion group O(3) the reducible representation of the tensor emerges as 2D…0‡†  D…1‡†  4D…2‡†  2D…3‡†  3D…4‡†  D…5‡†  D…6‡† , accommodating a maximum of 90 inde-

136

7 Special Systems for Second Harmonic Generation

pendent components in the case of a molecule lacking any intrinsic symmetry. If full index symmetry is assumed (Kleinman 1962), the representation becomes D…0‡†  D…2‡†  D…4‡†  D…6‡† , accounting for just 28 independent components. In either case a crucial facet of the result is its incorporation of a D…0‡† component, which invariably spans the totally symmetric representation of any point or space group. Thus six-wave second harmonic production entails a nonlinear polarisability that never vanishes for symmetry reasons; for example it permits the process to be supported in centrosymmetric molecules or solids. The six-wave mechanism will of course be insignificant in media where the normal SHG process is allowed, since it derives from three orders higher in perturbation theory. A number of studies on the generation of optically induced harmonics from isotropic suspensions of organic dyes have led to the characterisation of SWM mechanisms (see for example Nunzi et al. 1994, Fiorini et al. 1994, 1996, Hands et al. 1998, 2000, Lin et al. 1999, 2000). In most such studies, it has been found expedient to induce production of the harmonic by seeding. This requires that samples be pumped not only at the fundamental frequency but also with a probe beam at the harmonic frequency, thereby allowing stimulated emission to enhance the interaction. The seeding of molecular harmonics owes its conceptual origins to experiments where optical second harmonics were first observed in condensed isotropic systems. Although a second harmonic was first observed to develop in glass fibres only after long exposures to laser light (Sasaki and Ohmori 1981), through an associated photo-induced structural change, it was later shown that by introducing a low intensity seeding beam at the harmonic frequency the onset of the harmonic could become essentially instantaneous (Stolen and Tom 1987). It was at this time that the proposal of an SWM mechanism was first made (Baranova and Zel‘dovich 1987, Zel‘dovich and Kapitskii 1990). The experimental set-up is usually based on three distinct beams impinging on the sample. Of these, two are counterpropagating fundamental beams of frequency x, and the third at frequency of 2x stimulates second-harmonic emission into a mode satisfying the wave-vector matching condition. For such interactions the number of state sequence pathways (or time-ordered diagrams) is increased to 180 (= 6!/2!2!), indicating a reduction in

7.4 Six-Wave Second Harmonic Generation 137

C

Fig. 7.7 Schematic of a typical SWM experimental set-up. Counter-propagat-

ing fundamental beams of mode m1 ˆ …k1 ; k1 † and m2 ˆ … k1 ; k2 †, each contribute two photons while the stimulating beam m3 ˆ …k2 ; k3 † and singal beam m4 ˆ … k2 ; k4 † each gain one photon. The laboratory axes are illustrated and other symbols represent: A – aperture, BS – beam splitter, C – chopper, F – bandpass filter, P – polariser, M – mirror and WP – wave plate.

permutational symmetry amongst the radiation and molecular polarisability indices. In the schematic experimental geometry illustrated in Fig. 7.7, four radiation modes are assigned as mi = (ki, ki), (acknowledging that k1 = –k2 and that k4 is expected to be emitted in an opposite sense to k3). Recognising the index symmetry in the molecular response tensor we can write

138

7 Special Systems for Second Harmonic Generation …SWM† C coh

4   2p I…xk1 † 2 n2 …xk1 † ‡ 2 …2† …2† ˆ 6 …gN N†g1 g2 2cn…xk1 † 3 e0 ! 4  2  2 I…x0k3 † I…x k1 † n …x k1 † ‡ 2  2cn…x k1 † 3 2cn…x0k3 † !2   n2 …x0k3 † ‡ 2 X mg4 xk4  2cVn…xk4 † 3 k4  2 2 2 n …xk4 † ‡ 2 …6† …4† …3† …2† …2† …1† e  aij…kl†…mn† ei ej ek el e…1† m n 3  d… hx0k4 ‡  hx0k3

2 hxk1

2hx

k1 † :

…7:4:5†

Notice the dependence of the signal generation rate on the intensity of the seeding harmonic beam at x0k3 , the magnitude of which is jx0k3 j  j2xk1 j  2x. On converting the sum over k4 to an integral, exploiting properties of the delta function and assuming a single induced polarisation state of the emitted harmonic we obtain an intensity given by I…k4 † ˆ

x4 16e60 c 8 p2 n4 …x† …2†

…2†

 …gN N†g1 I2 …xk1 †g2 I 2 …x k1 †I…x0k3 †  2 8  2 4 n …x† ‡ 2 n …2x† ‡ 2  3 3 2 …6† …4† …3† …2† …2† …1†  aij…kl†…mn† ei ej ek el e…1† e m n :

…7:4:6†

The features of primary interest in the optical response arise from the molecular polarisability-radiation tensor product. For an isotropic fluid sample we again have to perform a rotational average taken inside the modulus squared (see section 5.2). Index symmetry exists only in the kl and mn index pairs, as indicated by parentheses, and on applying a sixth rank average we calculate the response as; hI…k4 †i ˆ

6 j X Ei Ai : 105 iˆ6

…7:4:7†

7.4 Six-Wave Second Harmonic Generation 139

Here j represents in a single parameter all the molecule-independent and polarisation-independent constants; the Ei‘s represent the unique radiation polarisation products;





E1 ˆ e…1†
e…1† e…2†
e…2† e…3†
e…4† E3 ˆ e…1†
e…2† e…1†
e…3† e…2†
e…4†





E2 ˆ e…1†
e…1† e…2†
e…3† e…2†
e…4† E4 ˆ e…1†
e…2† e…1†
e…4† e…2†
e…3†





E3 ˆ e…1†
e…2† e…1†
e…2† e…3†
e…4† E5 ˆ e…1†
e…3† e…1†
e…4† e…2†
e…2† …7:4:8† and the Ai‘s, molecular response in the form 2 3 2 32 3 A1 8 5 5 4 4 5 a1 6 7 6 76 7 6 A2 7 6 5 11 4 6 6 4 76 a2 7 6 7 6 76 7 6 A 7 6 5 4 11 6 6 4 76 a 7 3 6 7 6 76 3 7 6 7ˆ6 76 7 : 6 A4 7 6 4 6 6 16 2 6 76 a4 7 6 7 6 76 7 6 A 7 6 4 6 6 2 16 6 76 a 7 5 4 5 4 54 5 5 A6 a6 5 4 4 6 6 11

…7:4:9†

Here each independent molecular invariant is explicitly defined as 9 …6† a1 ˆ aii…jj†…kk† > > > > …6† > a2 ˆ aij…ij†…kk† > > > > > …6† a3 ˆ aii…jk†…jk† = …6† > a4 ˆ aij…ik†…jk† > > > > …6† > a5 ˆ aji…ik†…jk† > > > > ; …6† a6 ˆ ajk…ii†…jk†

…7:4:10†

In this form we can easily derive expressions for the signal intensity of the harmonic for arbitrary field polarisations. By assuming a laboratory frame as illustrated in Fig. 7.7, the polarisation vectors for the four fields, characterised by their azimuth and ellipticity as defined in Fig. 7.8, are concisely expressible as

140

7 Special Systems for Second Harmonic Generation

Fig. 7.8 A general representation for an arbitrary polarised light beam, e…i†

as represented by equations (7.4.11). The definition of the azimutz hi and ^ and Y-axis ^ ellipticity yi ˆ tan 1 …b=a† is as illustrated with respect to the X^ with the Z-axis pointing into the plane of the paper.

^ e…1† ˆ …cos h1 cos y1 i sin h1 sin y1 †X e…2†

^ ‡ …sin h1 cos y1 ‡ i cos h1 sin y1 †Y ^ ˆ …cos h2 cos y2 i sin h2 sin y2 †X

e…3†

^ …sin h2 cos y2 ‡ i cos h2 sin y2 †Y ^ ˆ …cos h3 cos y3 i sin h3 sin y3 †X

e…4†

^ ^ sin dZ† …sin h3 cos y3 ‡ i cos h3 sin y3 †…cos dY ^ ˆ …cos h4 cos y4 i sin h4 sin y4 †X ^ ^ sin dZ† ‡ …sin h4 cos y4 ‡ i cos h4 sin y4 †…cos dY …7:4:11†

Signals can be intercepted with a plane polariser …y4 ˆ 0† and ^ 4 ˆ p=2† direc^ 4 ˆ 0† and Y…h collected separately along the X…h tions. In each case there remains freedom to select any set of polarisations for the applied fields. As an example of the many controlled polarisation studies possible, consider the applied fields ^ axis except for beam e(1), which all linearly polarised along the X makes an angle h1 to the others. The signal intensity then reduces from equation (7.4.7) to

7.4 Six-Wave Second Harmonic Generation 141

Fig. 7.9 The solid and dotted lines represent theoretically calculated traces

for SWM signals collected unter the polarisation conditions pertaining to equations (7.4.12) and (7.4.13), respectively. The solid trace is normalised and the dotted line scaled by a factor of 6. The results are calculated assuming that all molecular parameters A1 A6 yield equal contributions.

D E j …^x-pol† j…A1 ‡ A2 † ‡ …A3 ‡ A4 ‡ A5 ‡ A6 † cos2 h1 j2 ; ISWM ˆ 105 …7:4:12† D E 2 j 1 …^y pol† ˆ …A5 ‡ A6 † sin 2h1 cos d : …7:4:13† ISWM 2 105 ^ and Y ^ axis can be plotted as a The signal collected along the X function of angle h1, as illustrated in Fig. 7.9. By performing a number of similarly designed experiments, detailed information on the six-wave mixing tensor can be extracted. In the detailed analysis of SWM in organic dye suspensions, temporal effects are manifest which are generally consistent with a degree of molecular alignment through optical perturbations (Driscoll and Lawandy 1994). Such perturbations are physically distinct from the process of harmonic production, though engendered by the same optical input. Departures from the instantaneous SWM mechanism, allowed for all molecular symmetries, arise from time-dependent orientational symmetry breaking. We now illustrate how, in the case of a resonant seeding harmonic,

142

7 Special Systems for Second Harmonic Generation

the concept of a molecular population grating can be employed as a readily comprehensible means of eliciting the physics of harmonic evolution. This quantum optical effect does not require the invocation of photo-induced orientation. Its foundation is based on a selective absorption process within a molecular ensemble initially having an isotropic array of molecules. First it can be shown that, in the presence of the two writing beams m2 and m3, the created population grating is of the correct periodicity to efficiently generate phase-matched second harmonic photons from the probe beam m1. As a result, the m4 signal photons emerge at the second harmonic frequency and propagate in exactly the opposite direction to the seeding beam m3, as dictated by wavevector matching. Consider that the seeding pulses from modes m2 and m3 are coincident within the sample at time t ˆ 0, and at t ˆ s the pulse from the probe beam m1 arrives. The sample is absorbing at the harmonic frequency and so exhibits transition to the excited state jui. The transitions may proceed either by two-photon absorption of photons from the fundamental beam or single-photon absorption of photons from the harmonic beam. Based on the two time-orderings shown in Fig. 7.10, the quantum amplitude for the transition in a particular molecule n is thus written as …n†

…n;a†

Mfi ˆ Mfi

…n;b†

‡ Mfi

;

…7:4:14†

…n;a†

is the matrix element calculated from graph (a) of where Mfi …n;b† Fig. 7.10 and Mfi from graph (b). These can be written as

Fig. 7.10 Time-ordered diagrams associated with the formation of a molecular population grating for SHG. The two writing beams m2 and m3 populate the upper electronic state jui via two- and single-photon absorption respectively.

7.4 Six-Wave Second Harmonic Generation 143 …n;a† Mfi

 ˆ

 mg x h 2cVe0 n…x†



n2 …x† ‡ 2 3

2

…2†u0 …2† …2†

hq2 ia…ij† ei ej ei2k2
Rn ; …7:4:15†

and …n;b†

Mfi

ˆ

 i

 1=2  2 n …2x† ‡ 2 …3† ik3
Rn : hq3 i1=2 lu0 i ei e cVe0 n…2x† 3 hm0g x 

…7:4:16† Here the wave-vectors of the beams of modes m2, m3 are k2, k3 respectively (recall also that k2  k1 ), with hq2 i and hq3 i the corresponding mean photon numbers. In deriving equations (7.4.15) and (7.4.16) the state vectors describing the radiation fields are assumed to be constructed from coherent states. The rate at which the excited state is populated is determined by the Golden Rule, equation (3.1.5). From (7.4.14) it is apparent that three contributions arise (resulting from the square of each …n;a† …n;b† contribution, Mfi and Mfi , and their quantum interference): Cˆ

2p …n† 2 Mfi qf ˆ C 1 ‡ C 2 ‡ C 3 ; h 

…7:4:17†

where 2pqf C1 ˆ h 



 mg x h 2ce0 Vn…x†

2 

n2 …x† ‡ 2 3

4

2 …2† …2† hq2 i2 au0 e e …ij† i j ; …7:4:18†

(  2   2 1=2 hm0g x 2pqf hmg x  n …x† ‡ 2 i C2 ˆ ce0 Vn…2x† 3 h  2ce0 Vn…x†  2  n …2x† ‡ 2  3 ) …2† …2† …3†

u0 i…2k2 hq2 ihq3 i1=2 au0 …ij† lk ei ej ek e

k3 †
Rn

‡ c:c:

;

…7:4:19†

2 2 n2 …2x† ‡ 2 …3† hq3 i lu0 e i i : 3

…7:4:20†

and 2pqf C3 ˆ h 



hm0g x 

ce0 Vn…2x†



144

7 Special Systems for Second Harmonic Generation

Here, as the transition is to a particular electronic manifold, we utilise the appropriate density of states representation, qf. From (7.4.17)–(7.4.20) we observe that the rate at which the excited state is populated depends on the position of the molecule, through C 2 – and also on the molecular orientation, through the matrix elements. It is the C 2 term that produces the grating within the sample, leading to a position-dependent overall rate C…Rn † for population of the upper state during application of the two writing beams. The periodicity of the population grating, determined by exp…i…2k2 k3 †
Rn †  exp… i…2k1 k4 †
Rn †, is exactly that required for phase-matched second-harmonic generation from the probe beam, the signal being created in precisely the opposite direction to the harmonic pump beam. Defining the effective time for which the beams are applied as Dt, the probability that a molecule n positioned at Rn is excited immediately after the pulses have passed is P…Rn † ˆ C…Rn †Dt. The probe pulse arrives after a delay of s…> Dt† seconds, during which time any excited molecule may relax. Assuming that relaxation to the ground state occurs by simple exponential decay, then at time s the probability that n remains excited is P…Rn ; s† ˆ C…Rn †Dte

ku …s Dt†

;

…7:4:21†

where ku is the excited state decay constant. For calculational expediency we now assume that molecules are static during the interval between the writing and probe pulses, ignoring any movement (rotational or translational) that may occur during these periods. 7) The probe pulse thus encounters the spatially modulated population distribution in the sample and the harmonic is produced from it. Again two possibilities arise, as illustrated in Fig. 7.11 (in which only the dominant of three contributory time0…n;a† 0…n;b† and Mfi the matrix eleorderings is shown). Writing as Mfi ments for these processes, the amplitude for harmonic production is as follows: 0…n†

Mfi

ˆ ‰1

0…n;a†

P…Rn ; s†ŠMfi

0…n;b†

‡ P…Rn ; s†Mfi

;

…7:4:22†

7) This physically reasonable assumption may be dispensed with, but only at the

cost of substantially increased complexity – see Hands et al. (2000).

7.4 Six-Wave Second Harmonic Generation 145

Fig. 7.11 The two dominant time-ordered diagrams (a) and (b) each represent one of the two possible SHG routes of the time-dependent SHG originating from either the ground or first electronic excited states j0i and jui respectively.

employing the same superposition principle used earlier in (7.3.8) and here reflecting a statistical weighting of the appropriate quantum amplitudes. The two components involved in the harmonic generation process are identical in terms of the photonics, differing only in their molecular mediation, and so we have 0…n;a† Mfi

2  2 n …x† ‡ 2 ˆ hq1 i 3 !1=2 ! X hm0g x0k4  n2 …x0k4 † ‡ 2  2ce0 Vn…x0k4 † 3 k ;k  i

4



mg x h 2ce0 Vn…x†

4

…4† …1† …1† i…2k1 k4 †
Rn b00 i…jk† ei ej ek e 0…n;b†

;

…7:4:23†

which is identical, save for replaand an expression for Mfi 00 uu cing bi…jk† by b i…jk†. Here, hq1 i is the mean number of photons in the coherent state mode m1 and the direction and polarisation of the harmonic are for the moment unspecified. As shown previously, a two-level model would require that the hyperpolarisability of the upper level is precisely minus that of the ground state, as in equation (7.3.6). In most SWM systems of experimental interest the two-level model is too restrictive; however the upper

146

7 Special Systems for Second Harmonic Generation

and lower level hyperpolarisabilities will usually be substantially different, an important feature in the following analysis. To continue, the full matrix element for the grating mechanism for SHG is given by 0…n† Mfi

!1=2 2   2 X h m0g x0k4 hmg x  n …x† ‡ 2 ˆ i hq1 i 3 2ce0 Vn…x0k4 † 2ce0 Vn…x† k4 ;k4 ! n2 …x0k4 † ‡ 2 …4† …1† …1† ei ej ek  3 o Xn …4† …1† …1† i…2k1 k4 †
Rn  b00 ‡ P…R ; s†Db ; n i…jk† ei ej ek e i…jk† n

…7:4:24† where the difference in the hyperpolarisabilities between the upper and lower states has been written Dbi…jk† ˆ buu i…jk†

b00 i…jk† :

…7:4:25†

The rate of production of SHG from the ensemble as given by the Golden Rule is: 2 2p X 0…n† Mfi d…Ef Cˆ h n 

Ei † ;

…7:4:26†

where the delta function will be used to specify the density of states for the second (reading) process. Taking an orientational average and effecting the usual split into incoherent (single site) and coherent (multi-site interference) terms, we have * + 2p X 0…n† 2 X 0…n† 0…n0 † d…Ef Mfi Mfi hCi ˆ Mfi ‡ h  0 n

Ei † :

…7:4:27†

n6ˆn

The dominant contribution to SHG is hence the coherent term, Ccoh hC coh i ˆ

2p XD 0…n† ED 0…n† E Mfi Mfi d…Ei h  0 n6ˆn

Ef † ;

…7:4:28†

7.4 Six-Wave Second Harmonic Generation 147

assuming that molecules are orientationally uncorrelated. For any particular molecule one can simply employ equation (7.4.24), excluding the sum over n. Returning to that equation and effecting the orientational average leads to disappearance of the first term within the parenthesis, as is usual for SHG in isotropic media. The second term, however, contains ‘hidden’ orientational factors through P…Rn ; s†, as a result of which the average is nonzero. From (7.4.21) we have D C…Rn †Dte

ku …s Dt†

E D E Db i…jk† ˆ …C 1 ‡ C 2 ‡ C 3 †Dbi…jk† Dte

ku …s Dt†

…7:4:29† As noted earlier, of the three contributory terms it is C2 which is responsible for the observed signal as it alone exhibits the necessary phase-matching on substitution in equation (7.4.24). We thus ignore the other terms in (7.4.29) – as justified by experimental observations that no SHG signal emerges if either of the writing beams m2 or m3 is blocked. From the C2 term in equation (7.4.29) we now find D C…Rn †Dte

ku …s Dt†

Db i…jk†

E

2   2 2pqf hmg x n …x† ‡ 2 ˆi 3 h  2ce0 Vn…x†   1=2  2 0 hmg x  n …2x† ‡ 2  ce0 Vn…2x† 3  hq2 ihq3 i1=2 Dn u0 …2† …2† …3† i…2k2  au0 …lm† ln el em en e E o ‡ c:c: Dbi…jk† Dte ku …s Dt†

k3 †
Rn

…7:4:30† with the exponential explicitly exhibiting the phase-matching condition (and the complex conjugate term accounting for the fact that SHG can be produced from a fundamental beam propagating in the opposite direction, as also observed experimentally). The phase-matched, orientationally averaged matrix element is hence

:

148

7 Special Systems for Second Harmonic Generation 0…n† hMfi i

4   2  2 1=2 hm0g x 2pqf hmg x  n …x† ‡ 2 ˆ ce0 Vn…2x† 3 h  2ce0 Vn…x† !1=2  2  X hm0g x0k4 n …2x† ‡ 2 1=2  hq1 ihq2 ihq3 i 3 2ce0 Vn…x0k4 † k4 ;k4 ! n2 …x0k4 † ‡ 2  3 D E …2† …2† …3† …4† …1† …1† u0 tku …s Dt†  au0 l Db ; i…jk† el em en ei ej ek Dte …lm† n …7:4:31†

which is necessarily position-independent – so that the phasematching double sum in equation (7.4.28) can be evaluated for the ensemble of N molecules as N…N 1†  N 2 for large N. The resulting rate of SHG production is: 8 4  2 mg x h n …x† ‡ 2 …NDt† hC coh i ˆ 3 2ce0 Vn…x† h3  2   2 0 hmg x  n …2x† ‡ 2  hq1 i2 hq2 i2 hq3 i ce0 Vn…2x† 3 ! !2 X hm0g x0k4  n2 …x0k4 † ‡ 2 d… hx0k4 ‡ 2hx†  0 † 2ce Vn…x 3 0 k4 k4 ;k4 D 2 E …2† …2† …3† …4† …1† …1† u0 2ku …s Dt† e  au0 l Db e e e e e i…jk† …lm† n m n i j l k e 8p3 q2f



2

…7:4:32† Converting the sum over k4 into an integral and assuming a single polarisation state we can recast the result in terms of mean intensities of the beams. Noting that the magnitude of the harmonic is forced to be jxk4 j  2x by the delta function, the final expression for the intensity of coherent SHG from the grating may be written as

7.4 Six-Wave Second Harmonic Generation 149

I…k4 † ˆ

q2f x4 …NDt†2

I2 …xk1 †I 2 …xk2 †I…x0k3 † 2 c 8 e60 n4 …x† h  2 8  2 4 n …x† ‡ 2 n …2x† ‡ 2  3 3 D 2 E u0 u0 …2† …3† …4† …1† …1†  a…lm† ln Dbi…jk† el e…2† e e e e m n i j k e

2ku …s Dt†

:

…7:4:33† Equation (7.4.33) exhibits a quadratic dependence on the intensities of the two fundamental beams and a linear dependence on the harmonic writing beam intensity. It also carries a quadratic dependence on the sample density I…k4 † / N 2, characteristic of coherent processes. The polarisation dependence is exactly that found for the case of coincident pulses, equation (7.4.6), save for its dependence on different molecular tensors. However the exponential decay in time exhibits the grating-induced origin of the harmonic signal (Hands et al. 2000).

151

8

Incoherent Elastic Light Scattering Wherefore they are no longer twain, but one flesh Matthew 19:6, Authorised Version

Elastic harmonic light scattering is a process of harmonic up-conversion that is incoherent in nature, and thus results in essentially non-directed harmonic emission (Kielich 1968). As in the coherent process of harmonic generation it may be represented at the microscopic level as involving the conversion of n (two or more) photons of laser light with frequency x (wave-vector k and polarisation vector e) into a single photon of frequency nx (wave-vector k0 and polarisation vector e0 ). It is nonetheless considerably weaker than harmonic generation by virtue of the fact that the wave-vector matching condition discussed in chapter 5 is not satisfied. Although a characteristic of the incoherent conversion process is harmonic emission over 4p steradians, the resultant intensity distribution is generally anisotropic and contrasts markedly with the laser-like output associated with coherent response. In this chapter attention is focussed principally on the second harmonic process known as either elastic second harmonic scattering (SHS) or hyper-Rayleigh scattering. The third harmonic case is also examined, more briefly. Inelastic harmonic scattering, which shares many of the features to be discussed below, is the subject of the concluding chapter 9. Before proceeding with the development of theory it may be noted that there are alternative schools of thought that represent all incoherent light scattering processes in fluids (both linear and nonlinear) as statistical phenomena, specifically associated with temporal fluctuations in molecular orientation. In terms of observables, results of broadly similar form emerge from such theories. However our formalism properly addresses all such systems – and 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

152

8 Incoherent Elastic Light Scattering

indeed reveals that even an isolated molecule at rest can give rise to incoherent (non wave-vector matched) light scattering.

8.1

General Principles In order to develop the theory of incoherent harmonic light scattering it is necessary to return to the general formulation of chapter 5. The basic rate equation for incoherent nonlinear response in a molecular fluid is given by equation (5.2.3). From this result, the explicit equation for the rate, Cinc, of n-harmonic scattering in a fluid medium is 2p D C inc ˆ N 2m ja…n‡1† … nx; x; . . . ; x†
q0…n‡1† f…k0 ; k0 †; he0  E …8:1:1†  …k; k†; . . . ; …k; k†gj2 d…Ei Ef † Here a…n‡1† … nx; x; . . . ; x† is the same nonlinear response tensor as appears in the theory of coherent n-harmonic generation, with components as defined by equation (4.1.7). As noted in the last chapter, for even n such tensors only have non-zero components for non-centrosymmetric molecules. The corresponding result for a molecular crystal, which follows from equation (5.1.7), differs from (8.1.1) only in that; (i) the number of unit cells M appears in place of N; (ii) the unit cell response tensor ^a…n‡1† replaces a…n‡1† , and (iii) the angular brackets denoting orientational averaging are no longer required. In this case it can be expected that emission in the forward direction will be dominated by the much stronger coherent signal, if the crystal symmetry permits. Under such circumstances the incoherent signal will nonetheless be detectable in other directions. Using methods similar to those employed for the representation of coherent emission, it can readily be shown that the above result for the rate of nonlinear scattering can be translated into an expression for the harmonic radiant intensity, InHS. The result emerges as follows: InHS …k0 † ˆ

2pnxD…n† N 2…n‡1† e0

hja…n‡1† … nx; x; . . . ; x†
s…n‡1† j2 i;

…8:1:2†

8.2 Second Harmonic Scattering/Hyper-Rayleigh Scattering 153

where again the beam parameter D(n) is as defined by equation (6.1.8) and the polarisation tensor s is constructed from the outer product of polarisation vectors for all photons involved in the process, s ˆ e0 e . . . e, as described by equation (6.1.2). Equation (8.1.2) differs crucially from the corresponding result (6.2.3) for coherent response in the positioning of the angular brackets denoting the rotational average required for application to fluid media. For incoherent processes the rotational average is performed on the modulus square of the inner product of the nonlinear susceptibility and polarisation tensors; consequently a rank (2n+2) rotational average is entailed. The linear dependence on the number of scatterers N in (8.1.2), as opposed to the coherence factor in (6.2.3), is also significant – it is this difference which accounts for the very much weaker signal associated with the incoherent process.

8.2

Second Harmonic Scattering/Hyper-Rayleigh Scattering From equation (8.1.2), the general expression for the intensity of elastic second harmonic (hyper-Rayleigh) scattering is the starting point for the development of a result for the angular disposition of the harmonic: ISHS …k0 † ˆ

4pxD…2† N hja… 2x; x; x†
e0 eej2 i ; e60

…8:2:1†

or, explicitly in terms of the tensor and vector components; ISHS …k0 † ˆ

4pxD…2† N hbi…jk† … 2x; x; x†e0i ej ek bl…mn† e60

… 2x; x; x†e0l em en i ;

…8:2:2†

with the index-symmetric nonlinear polarisability bi(jk) as defined by (4.2.5). Without the large angular brackets the result (8.2.2) is directly applicable to media comprising scatterers with a fixed orientation – the response tensor and the number of particles are interpreted in terms of cell quantities in the case of crystalline media, as discussed earlier. The fact that the hyperpolarisa-

154

8 Incoherent Elastic Light Scattering

bility vanishes for centrosymmetric crystals means that the appearance of any hyper-Rayleigh signal can be used as indicator of disorder in such crystals (Vogt and Neumann 1978). In the case of fluid media, application of equation (8.2.2) necessarily requires evaluation of a sixth rank rotational average. As shown in appendix 2, this tensor average comprises products of Kronecker delta triplets, one triplet referred to the laboratory frame and the other to the molecular frame. The laboratory frame deltas contract with the polarisation vector components to give scalar products amongst e, e, e0 and e0 . It thus transpires that the full result for the SHS intensity is as follows: ISHS …k0 † ˆ

4pxD…2† N f‰30…e0
e†…e
e0 †…e
e† 105e60 12…e
e0 †…e0
e†

12…e0
e†…e0
e†

10…e
e†…e
e† ‡ 8Šbk…kl† bl…mm†

‡ ‰ 12…e0
e†…e
e0 †…e
e† ‡ 16…e0
e†…e0
e† ‡ 2…e
e0 †…e0
e† ‡ 4…e
e†…e
e† 6Šbk…kl† bm…lm† ‡ ‰ 10…e0
e†…e
e0 †…e
e† ‡ 4…e0
e†…e0
e† ‡ 4…e
e0 †…e0
e† ‡ 8…e
e†…e
e†

5Šbk…ll† bk…mm†

‡ ‰8…e0
e†…e
e0 †…e
e† 6…e0
e†…e0
e† 6…e
e0 †…e0
e† 5…e
e†…e
e† ‡ 11Šb k…lm† bk…lm† ‡ ‰ 12…e0
e†…e
e0 †…e
e† ‡ 2…e0
e†…e0
e† ‡ 16…e
e0 †…e0
e† ‡ 4…e
e†…e
e†

6Šb k…lm† bl…km† g ; …8:2:3†

where use has been made of the symmetry in the last two indices of the nonlinear susceptibility tensor, whose components are for simplicity assumed to be real. The five independent quadratic parameters of the form b . . . b . . . appearing in (8.2.3) are distinct and independent molecular parameters that completely determine the angular and polarisation behaviour of second harmonic scattering. 1) They were first identified and discussed by Bersohn et al. (1966), and were the subject of a detailed analysis by Kielich and Bancewicz (1990). 1) Under the assumption of Kleinman index symmetry implicit in early treatments,

the first three molecular invariants become equivalent, bk…kl† bl…mm† ˆ bk…kl† bm…lm† ˆ bk…ll† bk…mm† , as do the latter two, bk…lm† bk…lm† ˆ bk…lm† bl…km† and results are cast in terms of only two parameters.

8.2 Second Harmonic Scattering/Hyper-Rayleigh Scattering 155

The polarisation vector scalar products in (8.2.3) are generally non-zero, except that …e
e† is zero when the pump is circularly polarised, as follows from their evaluation using equations (3.1.8) and (3.1.9). For all other cases the non-zero components ensure that a finite emission is guaranteed. The result faithfully represents the detailed polarisation- and angle-dependence of incoherent second harmonic emission (Strizhevski and Klimenko 1968, Kielich and Kozierowski, 1974; Andrews and Thirunamachandran 1978). Since emission can occur in a non-forward direction, it is associated with what would classically be regarded as a longitudinal polarisation of the medium at the harmonic frequency. Specifically, the intensity of the associated harmonic intensity, produced at right angles to a plane-polarised pump beam, immediately follows from (8.2.3) with ekk0 and …e
e† ˆ 1: ISHS …k0 † ˆ

4pxD…2† N ‰ 2b k…kl† bl…mm† 105e60 ‡ 6bk…lm† bk…lm†

2b k…kl† bm…lm† ‡ 3bk…ll† b k…mm†

2bk…lm† bl…km† Š :

…8:2:4†

A degree of confusion in the literature over the possible role of higher multipoles in harmonic emission from fluids has tended to obscure the fact that such longitudinal polarisations arise even in the electric dipole approximation, and are invariably associated with incoherent emission. One of the key features in the resolution of the debate has been a shift away from description in terms of the classical nonlinear polarisation, a concept that is not amenable to eliciting the distinction between coherent and incoherent emission. When plane-polarised radiation is employed, a salient feature of harmonic scattering is its degree of polarisation, conventionally expressed in terms of a depolarisation ratio, qSHS ? , defined by …?!k†

ˆ qSHS ?

ISHS …k0 † …?!?†

ISHS

…k0 †

;

…8:2:5† …k!l†

where we adopt the notation ISHS …k0 † to denote the experimental configuration with incident and scattered photons having polarisations k e and l e0 , respectively. The explicit representations || and ? in equation (8.2.5) denote polarisations lying in, and per-

156

8 Incoherent Elastic Light Scattering

pendicular to, the scattering plane which contains the vectors k and k0 . The scattering geometry is neatly illustrated in Fig. 8.1. On considering the angular disposition of the polarisation vectors it is evident that the result given by equation (8.2.5) is indeis given by pendent of the scattering angle h. Explicitly, qSHS ? qSHS ˆ ? 2bk…kl† bl…mm† 2b k…kl† bm…lm† ‡ 3bk…ll† b k…mm† ‡ 6bk…lm† b k…lm† 2b k…lm† bl…km† 4bk…kl† bl…mm† ‡ 4bk…kl† bm…lm† ‡ b k…ll† b k…mm† ‡ 2bk…lm† b k…lm† ‡ 4bk…lm† bl…km† …8:2:6†

In special cases where both pump and harmonic frequencies are far removed from any resonances, the Kleinman approximation can be applied and it becomes legitimate to assume full index symmetry for the molecular tensor. Then we obtain the following, simpler result: ˆ qSHS ?

4b…klm† b…klm† b …kkl† b…mml† : 6b…klm† b…klm† ‡ 9b…kkl† b…mml†

…8:2:7†

Relations equivalent to (8.2.6) and (8.2.7), cast in terms of irreducible tensor invariants (see appendix 4) have been variously obtained by Maker (1970), Andrews and Thirunamachandran (1978) and also Kielich and Bancewicz (1990). Such expressions are amenable to the incorporation of rotational line structure (see also Bancewicz et al. 1973). Moreover they greatly facilitate the derivation of the selection rules for inelastic second harmonic (hyper-Raman) scattering, as will be shown in chapter 9. Second harmonic scattering of circularly polarised light generally leads to a degree of inversion of its helicity and with this in mind we define the circular polarisations using the scattering geometry illustrated in Fig. 8.1. Here the in-plane vector ||e was ^ from a right-handed triad, as does the chosen such that ^e and k || 0 primed vector e , such that for circular polarisations we define e…L† ˆ p12 …k e ‡ i? e† ;

…8:2:8†

e…R† ˆ p12 …k e

…8:2:9†

i? e† ;

8.2 Second Harmonic Scattering/Hyper-Rayleigh Scattering 157

1808–h

Fig. 8.1 An illustration of the scattering geometry with respect to nHS

experiments.

consistent with (3.1.8) and (3.1.9). The effect, which has a complicated dependence on the scattering angle h ˆ cos 1 …k
k0 †, is most acute in either the forward- or back-scattering geometries (Kielich and Kozierowski 1974), and is traditionally measured in terms of a reversal ratio, R…h†, defined as …C!C†

R…h† ˆ

ISHS

…C!C†

…k0 †

ISHS …k0 †

;

…8:2:10†

where C denotes a circular polarisation of either left or right handedness, and C* the opposite handedness. The reversal ratio acquires a value of unity for right-angled scattering; for the commonly studied forward-scattering geometry we obtain R…0† ˆ 4bk…kl† bl…mm† 4b k…kl† bm…lm† bk…ll† b k…mm† ‡ 5bk…lm† bk…lm† ‡ 10b k…lm† bl…km† ; 4bk…kl† bl…mm† ‡ 10b k…kl† bm…lm† bk…ll† b k…mm† ‡ 5bk…lm† b k…lm† 4b k…lm† bl…km†

(8.2.11) or, in the case of Kleinman symmetry R…0† ˆ

15b…klm† b…klm† 9b…kkl† bk…mml† : b …klm† b…klm† ‡ 5b …kkl† bk…mml†

…8:2:12†

The corresponding result for back-scattering can simply be recovered by use of the general relation R…p

h† ˆ R 1 …h† :

…8:2:13†

This relation, (8.2.13), essentially signifies that the angular momentum of a circularly polarised photon propagating in one di-

158

8 Incoherent Elastic Light Scattering

rection is equivalent to that of a photon of opposite circularity travelling in the opposite direction. To derive the maximum information from second harmonic scattering, it is necessary to perform a complete polarisation study, following methods first proposed by McClain (1972). Optimally, this entails performing five separate experiments, with different scattering geometries and polarisations, such that the harmonic intensities given by equation (8.2.3) (replacing the argument k0 with the corresponding scattering angle h, as defined in Fig. 8.1), form a linearly independent system. These equations can then be solved for the five independent molecular invariants they involve. One such combination of experiments is as follows: …C!C†

ISHS …0† ˆ I1 ; …C!C†

ISHS

…8:2:14†

…0† ˆ I2 ;

…8:2:15†

…C!k†

…8:2:16†

…?!k†

…8:2:17†

ISHS …p=2† ˆ I3 ; ISHS …p=2† ˆ I4 ; …?!?†

ISHS

…p=2† ˆ I5 :

…8:2:18†

From (8.2.3), and with the aid of Cramer’s rule for solving systems of linear equations, it transpires that the molecular invariants are uniquely expressible in terms of I1 . . . I5 through   1 4pxD…2† N … 2I1 bk…ll† bk…mm† ˆ 3 e60

2I2

2I3 ‡ 6I4 ‡ 3I5 † ; …8:2:19†

bk…lm† bk…lm†

  1 4pxD…2† N ˆ3 …I1 ‡ I2 ‡ I3 ‡ 2I4 ‡ I5 †; e60

bk…lm† bl…km†

  1 4pxD…2† N ˆ3 … I1 ‡ 4I2 e60

I3

…8:2:20†

2I4 ‡ 4I5 † ; …8:2:21†

8.2 Second Harmonic Scattering/Hyper-Rayleigh Scattering 159

bk…kl† b l…mm†

  1 4pxD…2† N ˆ3 … 3I1 e60

3I2 ‡ 2I3

I4 ‡ 7I5 †; …8:2:22†

  1 4pxD…2† N …4I1 bk…kl† b m…lm† ˆ 3 e60

I2

I3

2I4 ‡ 4I5 † : …8:2:23†

Such results, which permit determination of the relative magnitudes of the molecular parameters, can be employed to rigorously test the results of ab initio calculations and also the degree of validity of the Kleinman approximation. One interesting feature of SHS concerns the polarisation of the harmonic when the pump radiation is unpolarised, u. As with its optically linear analogue, Rayleigh scattering, the harmonic process itself produces a degree of polarisation in the scattered light. It may be shown by application of equation (8.2.3) that the harmonic intensity is an unusual linear combination of the intensities produced by the two orthogonal pump polarisations ||, ^, and circular pump polarisations of opposite helicity L, R. For the harmonic component of polarisation l …u!l†

…k!l†

…?!l†

…L!l†

…R!l†

ISHS …k0 † ˆ 18 ‰2ISHS …k0 † ‡ 2ISHS …k0 † ‡ ISHS …k0 † ‡ ISHS …k0 †Š; …8:2:24† which holds for any scattering angle (Andrews and Thirunamachandran, 1977 b). The result should be compared to the following expressions that are well known to hold for Rayleigh scattering; …u!l†

IRS

…k!l†

…k0 † ˆ 12 ‰IRS

…?!l†

…k0 † ‡ IRS

…L!l†

…k0 †Š ˆ 12 ‰IRS

…R!l†

…k0 † ‡ IRS

…k0 †Š:

…8:2:25† The relations (8.2.25) do not hold for second harmonic scattering. The reason is that in SHS the possibility of the two incident photons having different polarisations has to be entertained. Consequently the relationship

160

8 Incoherent Elastic Light Scattering

qu …p=2† ˆ

2q? …1 ‡ q? †

…8:2:26†

for right-angled Rayleigh scattering also has no counterpart in SHS. In fact, for right-angled scattering, the result for qSHS is as u follows qSHS u …p=2† ˆ bk…ll† b k…mm† ‡ 23bk…lm† bk…lm† 10b k…lm† bl…km† ‡ 4bk…kl† b l…mm† 10bk…kl† bm…lm† 3bk…ll† b k…mm† ‡ 13bk…lm† bk…lm† ‡ 5b k…lm† bl…km† 2bk…kl† b l…mm† ‡ 5bk…kl† bm…lm†

(8.2.27) or again, where Kleinman index symmetry holds in the hyperpolarisability tensor: qSHS u …p=2† ˆ

13b…klm† b …klm† 5b…kkl† b…mml† : 18b…klm† b …klm† ‡ 6b…kkl† b…mml†

…8:2:28†

The general result for scattering at an angle h, in either the Rayleigh or the hyper-Rayleigh case, is correctly given by qu …h† ˆ qu …p=2† ‡ ‰1

qu …p=2†Š cos2 h :

…8:2:29†

Whilst the style of analysis presented above is cast directly in terms of harmonic intensities and intensity ratios for specific resolved polarisation components, formulations in terms of Stokes vectors are also given in the literature. These vectors (Stokes 1852), comprising four parameters that facilitate the depiction of classical radiation states with arbitrary (generally elliptical) polarisation, are widely employed in the theory of conventional linear light scattering and associated spectroscopy (Kliger et al. 1990). The Stokes vector for the emitted light is normally related by a (4 ´ 4) Mueller matrix to the corresponding vector for the incidence, a formalism which has been neatly extended by Shi et al. (1993) to accommodate nonlinear scattering. An earlier and more extensive treatment of second harmonic scattering in terms of Stokes parameters by Kozierowski and Kielich (1984) incorporates both polarised and unpolarised light, giving results for the

8.3 Third Harmonic Scattering 161

depolarisation ratio, reversal ratio and degree of circularity of the harmonic. In fluid media comprising centrosymmetric molecules, second harmonic scattering can still occur, even though much more weakly, through the involvement of fourth rank molecular response tensors. One facilitating mechanism involves the application of a static electric field and entails the EFISHG tensor b  a…4† … 2x; 0; x; x† defined by (7.2.2). Even in the absence of any such applied field, the involvement of electric quadrupolar ~ 2x; x; x† (see chapter 4) can interactions, as in the tensor b… produce a signal. In this case the calculation calls for evaluation of an eighth rank tensor isotropic average; details are given by Kielich et al. (1974). The generation of local anisotropy through molecular orientational correlation can also play a role in supporting SHS, in fluids comprising centrosymmetric species (Kielich et al. 1971, Kielich and Kozierowski 1974). The theory underlying the operation of this mechanism has been presented largely from the perspective of a molecular statistical treatment; in terms of the methodology employed here, the harmonic scattering can be understood on the basis of interactions between neighbouring molecules effectively reducing their inversion symmetry, leading to finite values for their hyperpolarisabilities.

8.3

Third Harmonic Scattering Most of the work on elastic harmonic scattering has concerned the second harmonic process. This undoubtedly reflects the fact that the signal associated with incoherent third harmonic scattering (THS) is exceptionally weak and therefore experimentally problematic, it also perhaps attests to the fact that the theory is exceptionally intricate. Equation (8.1.2) provides the basis for calculation of the third harmonic intensity which, for the case of fluid media, involves a rotational average requiring the implementation of eighth rank tensor calculus. Again assuming that the molecular tensor c… 3x; x; x; x† is real, the result is expressible in terms of seven independent molecular parameters. Suffice it here to give the result, ITHS …k0 †, for the intensity of planepolarised components (Kielich and Kozierowski 1970)

162

8 Incoherent Elastic Light Scattering …P!P† ITHS

  6pxD…3† N ˆ f‰16ck…lmo† ck…lmo† 630e80 ‡ 24ck…lmm† ck…loo† 6ck…klm† co…lmo†

6ck…lmo† cl…kmo†

3ck…lmm† cl…koo†

12ck…lmm† co…klo†

3ck…kll† cm…moo† Š

0 2

‡ 3…e
e † ‰ 4ck…lmo† ck…lmo† ‡ 6ck…lmo† cl…kmo† 6ck…lmm† ck…loo† ‡ 3ck…lmm† cl…koo† ‡ 12ck…lmm† co…klo† ‡ 6ck…klm† co…lmo† ‡ 3ck…kll† cm…moo† Šg :

…8:3:1†

From this result the depolarisation ratio as given by (8.2.5) follows: ˆ ‰16ck…lmo† ck…lmo† qTHS ? 3ck…lmm† cl…koo†

6ck…lmo† cl…kmo† ‡ 24ck…lmm† ck…loo† 12ck…lmm† co…klo†

6ck…klm† co…lmo†

3ck…kll† cm…moo† Š  ‰4ck…lmo† ck…lmo† ‡ 12ck…lmo† cl…kmo† ‡ 6ck…lmm† ck…loo† ‡ 6ck…lmm† cl…koo† ‡ 24ck…lmm† co…klo† ‡ 12ck…klm† co…lmo† ‡ 6ck…kll† cm…moo† Š

1

:

…8:3:2†

Symmetry aspects of these results have been considered in detail by Kozierowski (1970), through the simplifying assumption of Kleinman index symmetry. Results for the scattering of circular polarisations have also been derived by Kozierowski (1974).

163

9

Hyper-Raman Scattering To stretch the octave. . . The Decadent to His Soul, Richard Le Gaillienne

The hyper-Raman effect is an incoherent process of inelastic second harmonic scattering. It entails the pairwise annihilation of pump photons, frequency x, and release of photons with frequency x0  2x, accompanied by transitions in the medium. In this respect it is unlike all the processes described in earlier chapters, for it involves an overall exchange of energy between the radiative and material parts of the dynamical system. The term hyper-Raman signifies that this three-photon interaction may be regarded as a nonlinear analogue of Raman scattering. One of the features the two processes share is that molecules may undergo either upwards (Stokes) or downward (anti-Stokes) transitions, as illustrated in Fig. 9.1. In the spectrum of harmonically scattered light, the hyper-Raman effect generally produces a number of lines, each associated with one of the allowed molecular transitions (Terhune et al. 1965). These lines exhibit shifts to both the low and high frequency sides of the harmonic 2x. Since the selection rules are in general markedly different from those that apply to conventional (twophoton) Raman scattering (or one-photon absorption) processes, hyper-Raman scattering is of considerable spectroscopic utility. Most experimental studies involve analysis of the Stokes (negative shift) features associated with upward transitions from the ground state. The anti-Stokes part of the spectrum carries essentially the same information content but is weaker – as only molecules that are initially in an excited state can undergo anti-Stokes transitions, and under most circumstances the population of such molecules is small compared to the ground state. 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

164

9 Hyper-Raman Scattering

hx hx0  hx

hx

hx

hx0 

Fig. 9.1 A schematic of hyper-Raman Stokes (a) and anti-Stokes (b) in-

elastic scattering. Solid horizontal lines represent a manifold of molecular states and the dotted lines virtual states.

9.1

Constructing the Signal In order to accommodate both Stokes and anti-Stokes transitions in the theory to be developed below, for generality we denote the initial molecular state by jii and the final state by jf i, such that the condition of energy conservation is expressed through the relation h…2xk Efi ˆ 

x0k0 † :

…9:1:1†

Whereas the transitions jf i jii can in principle be of various types – electronic, vibrational or rotational – most attention has

9.1 Constructing the Signal 165

focussed on the vibrational case for which jEfi j ˆ hxm , with xm a molecular or lattice vibrational frequency. The derivation of a result for the intensity, IHR(k'), of hyperRaman scattering exactly parallels the treatment of second harmonic light scattering, discussed in chapter 8. Starting from the general rate equation for fluid media, (5.2.3), it leads to the following expression for the scattered intensity: IHR …k0 † ˆ

4pxD…2† Ni fi hjb … x0k0 ; xk ; xk †
e0 eej2 i ; e60

…9:1:2†

where Ni is the number of molecules in the initial state and b fi is the corresponding molecular response tensor. Here e and e' are the pump and emergent photon polarisation vectors and D(2) the pump beam parameter defined by equation (6.1.8). Under equilibrium conditions at temperature T, anti-Stokes transitions are associated with a value of Ni that is smaller, compared to their Stokes counterparts, by the Boltzmann factor exp(–Efi/kBT). The solid state counterpart to equation (9.1.2), in principle derivable from (5.1.7), can be complicated by coupling between vibrations in molecules at different sites. Accommodating a delocalised phonon structure, the theory exhibits the facility for hyperRaman scattering to result in the excitation of lattice vibrations, a topic that falls outside our province. Where the vibrations of discrete molecules are concerned, and no more than one molecule per unit cell undergoes transition (a condition fully consistent with the intrinsic weakness of the effect), then employment of equation (5.1.7) is permissible and leads to the result IHR ˆ

4pxk D…2† M ^fi hjb … x0k0 ; xk ; xk †
e0 eej2 i ; e60

…9:1:3†

where M is the number of unit cells and b^fi is the cell response tensor as defined by (5.1.2). The similarity in form between the above result (9.1.3) and its elastic counterpart, equation (8.2.1), leads to a result that can similarly be expressed in terms of five molecular invariants. However in order to elicit the influence of selection rules, further development is required (see section 9.3). From equation (2.2.2), we use the relation (9.1.1) to recast energy denominators that emerge expressed in terms of the scat-

166

9 Hyper-Raman Scattering

tered frequency in favour of the pump frequency. We then obtain the following general expression for the hyper-Raman tensor:

bijk … x0k0 ; xk ; xk † ˆ fi

X r;s

‡

fs

…E~si

li lsrj lrik 2hx†…E~ri

hx†

fs lj lsri lrik

…E~sf ‡  hx†…E~ri fs

hx†

lj lsrk lrii ‡ ~rf ‡ 2hx† …E~sf ‡  hx†…E

 :

…9:1:4†

The structure of the numerator in each term reflects a progression of the overall transition through two virtual intermediate states, f s r i. As in all such second harmonic processes, it is only the partially index-symmetric part of the tensor, fi

fi

fi

bi…jk† ˆ 12…bijk ‡ bikj † ;

…9:1:5†

that contributes to the signal as given by equation (9.1.3). Careful attention needs to be given to the detailed nature of the states in equation (9.1.4). As noted in chapter 4, these designations should not be regarded as pertaining to electronic levels alone. As the nature of hyper-Raman scattering necessitates consideration of vibrational form in the initial and final states for the process, these should also be taken into account in the sums over intermediate states. Hence the transition dipole moments which feature in the response tensor must be interpreted as vibronic, rather than purely electronic. It is now well established that vibrational contributions play an important role in molecular hyperpolarisabilities (Bishop, 1990, Bishop and Kirtman 1991), and at this juncture it is pertinent to consider the detailed relationship between hyper-Raman and hyperpolarisability tensors. In many respects this relationship, first considered in detail by Long and Stanton (1970), is similar to the one between conventional Raman and polarisability tensors first developed in the historic ‘Polarisability Theory’ (Placzek 1934).

9.2 Hyperpolarisability Theory

9.2

Hyperpolarisability Theory To begin, it is necessary to develop in more explicit detail the basic structure of the molecular state vectors. For most purposes it is sufficient to employ the Born-Oppenheimer approximation, based on the premise that both the nuclear and electronic motions respectively are separable. The intricate theory required for describing more specialised cases of hyper-Raman scattering entailing vibronic coupling has been dealt with by Chung and Ziegler (1988) for the case of resonance, and by Neddersen et al. (1989) for off-resonance scattering. For our purposes concerning distinctively molecular response, we may express relevant quantum states in the form jrijR…r†i, representing a factorisation of the molecular wavefunctions in the form wr …q; Q†wr;R …Q†. Here q and Q denote the complete set of electronic and nuclear coordinates, respectively; r labels the electronic state and R the nuclear state. The parametric dependence of the electronic wavefunction on the nuclear coordinates reflects the fact that the electrons are responsive to the Coulomb field generated by the nuclei; the labelling of the nuclear wavefunction with both electronic and nuclear quantum numbers indicates that different electronic states generally have different potential energy surfaces, so that the forces and equilibrium configurations which govern each type of nuclear motion also differ. The state vectors involved in the hyper-Raman process, hitherto labelled i, r, s and f, may explicitly be represented as j0ijI…0†i; jrijR…r†i; jsijS…s†i and j0ijF…0†i; with corresponding energies …E0 ‡ EI…0† †; …Er ‡ ER…r† †; …Es ‡ ES…s† †, and …E0 ‡ EF…0† † respectively. Notice that, in analogy with Raman scattering, the initial and final electronic state remains unchanged overall. The hyperRaman tensor (9.1.4) thus emerges with an explicit structure given by:

167

168

9 Hyper-Raman Scattering 0 bFI ijk … x ; x; x† ˆ

X X r;s R…r†;S…s†



 ‡

sr r0 hF…0†jl0s i jS…s†ihS…s†jlj jR…r†ihR…r†jlk jI…0†i …E~s0 ‡ E~S…s†I…0† 2hx†…E~r0 ‡ E~R…r†I…0† hx†

sr r0 hF…0†jl0s j jS…s†ihS…s†jli jR…r†ihR…r†jlk jI…0†i …E~s0 ‡ E~S…s†F…0† ‡ hx†…E~r0 ‡ E~R…r†I…0† hx†

sr r0 hF…0†jl0s j jS…s†ihS…s†jlk jR…r†ihR…r†jli jI…0†i ‡ …E~s0 ‡ E~S…s†F…0† ‡ hx†…E~r0 ‡ E~R…r†F…0† ‡ 2hx†



…9:2:1† where lab are specifically electronic transition dipole moments. Since the electronic wavefunctions are parametrically dependent on nuclear configuration, these transition moments are, in an Natomic molecule, functions of the 3N nuclear coordinates. This dependence is more usefully expressed in linear combinations of these coordinates comprising 3 molecular position coordinates, 3 orientation coordinates and (3N–6) vibrational normal mode coordinates (linear molecules having one less orientational and one extra vibrational degree of freedom). For a free molecule in the gas phase the nuclear wavefunction is generally separable as a product of rotational and vibrational parts. However in most gasphase hyper-Raman studies the rotational structure is not resolved. (For a notable exception see Campbell and Ziegler 1993.) Equally, for molecules in the liquid phase, translational and rotational motions are not associated with quantised energy levels because those motions are strongly and stochastically perturbed by the molecular environment. In both cases it is legitimate for the rotational and translational motions to be accommodated as statistical features. Through application of the usual ensemble averaging procedure discussed in appendix 2, we obtain results corresponding to the intensities of hyper-Raman lines effectively integrated over any rotational fine structure. The transition dipole moments that feature in equation (9.2.1) now invite closer attention. For any vibronic transition of general form jnijN…n†i jmijM…m†i, involving a mode of vibration with coordinate Qj, the electric transition moment can be written in a Taylor series expansion involving a single variable, contributions

9.2 Hyperpolarisability Theory

from all other nuclear motions vanishing through the linear independence of normal coordinates. Thus we have qlnm hN…n†jlnm …Qi †jM…m†i ˆ hN…n†jlnm …Qe † ‡ …Qj Qe † qQj Qe ‡ . . . ; jM…m†i ˆ lnm …Qe †hN…n†jM…m†i qlnm hN…n†j…Qj Qe †jM…m†i‡ . . . ; ‡ qQ j

Qe

…9:2:2† where Qe denotes the equilibrium configuration. For transitions between vibrational levels of different electronic states, where m 6ˆ n, the first term in (9.2.2) involving the Franck-Condon overlap integral hNn jMm i generally dominates. Moreover unless there is a substantial difference in form between the potential energy surfaces associated with the two electronic states, the result is vanishingly small unless the vibrational levels M and N differ by generally no more than a few quanta. For transitions within the vibrational manifold of one electronic state, where m ˆ n, the overlap integral in the first term of (9.2.2) vanishes unless M ˆ N, and in the second term the transition integral, represented through hN…n†j…Qj Qe †jM…m†i, vanishes unless N ˆ M  1: Assuming that the fundamental laser frequency x and its harmonic 2x are far removed from any potential resonances, the contributions of the vibrational energy terms E~R…r†I…0† ; E~R…r†F…0† ; E~S…s†I…0† and E~S…s†F…0† in the denominators of the hyper-Raman tensor (9.2.1) will mostly be negligible, since they entail at most only a few vibrational quanta. Consequently, it is possible to effect closure1) over R(r) and S(s), giving to a good approximation the result bFI ijk … 2x; x; x†  hF…0†jb ijk … 2x; x; x†jI…0†i ;

…9:2:3†

where bijk … x0 ; x; x† is the electronic hyperpolarisability, as previously defined by equation (4.1.6) with the summations over r and s specifically effected over electronic states. The electronic wavefunctions in terms of which this hyperpolarisability is cast

1) Closure is effected using the quantum mechanical completeness relation,

P

Y

jYihYj ˆ 1:

169

170

9 Hyper-Raman Scattering

are nonetheless still sensitive to the nuclear geometry, so that (9.2.3) can be expanded in a Taylor series resulting from, and similar in form to (9.2.2): hF…0†jbijk …Qj †jI…0†i ˆ hF…0†jbijk …Qe †‡…Qj

qb ijk Qe † qQj Qe

‡. . . ; jI…0†i ˆ bijk …Qe †hF…0†jI…0†i qbijk hF…0†j…Qj Qe †jI…0†i‡. . . ; …9:2:4† ‡ qQ j Qe

We now consider the circumstances under which each term in (9.2.4) plays the major role. Since the Dirac bracket in the first term relates to the overlap of vibrational wavefunctions belonging to the same (ground) electronic state, it vanishes unless F…0† ˆ I…0†. Provided the molecule has a non-zero hyperpolarisability in its equilibrium nuclear configuration, the first term of (9.2.4) thus provides the dominant contribution for elastic second harmonic scattering, as discussed in chapter 8. It is the second term of equation (9.2.4) that represents the leading contribution for inelastic harmonic scattering, which accounts for the fact that the hyper-Raman effect is still weaker than the inherently weak hyper-Rayleigh process. It is from this term that the detailed selection rules arise which govern the appearance or nonappearance of vibrational transitions in the hyper-Raman spectrum. Several important facets are immediately apparent. For any particular vibrational mode to be manifest in a hyper-Raman spectrum, the molecular hyperpolarisability has to change during vibration as the molecule passes through its equilibrium configuration – or else the first derivative …qbijk =Qj † ˆ 0, resulting in a null result. This is a direct counterpart to the rule that operates in conventional Raman scattering, where it is a change in molecular polarisability that counts. However, notice that although a centre of symmetry in the equilibrium geometry of a molecule will make bijk …Qe † vanish, such a molecule may have a non-zero value for …qb ijk =qQj †. Physically, this may be understood as originating from an effective lowering of the molecular symmetry in the course of non-totally symmetric nuclear vibrations. Thus molecules that exhibit no hyper-Rayleigh scattering may nonetheless exhibit hyper-Raman features. This behaviour contrasts with Raman

9.3 Irreducible Tensors

scattering, where the Rayleigh line signifying elastic scattering is always present, and invariably dominates the vibrationally shifted Raman lines. In the case of centrosymmetric molecules, hyper-Raman active vibrations will necessarily be of odd parity, ungerade character (see below). Ab initio wavefunction studies can be used to predict hyper-Raman spectra with a good degree of qualitative agreement to experiment (see for example Jayatilaka et al. 1992). It will also be observed that the hF…0†j…Qj Qe †jI…0†i factor in (9.2.4) leads to the usual vibrational selection rule F ˆ I  1. It is for this reason that the displacement …2x x0 † of any vibrational hyper-Raman frequency from the second harmonic can be identified with the corresponding vibrational frequency. Higher order terms in (9.2.4), which account for the possibility of overtones and combination bands, are usually negligible; however for a full treatment, interested readers may refer to Petrov (1985).

9.3

Irreducible Tensors To proceed towards detailed analysis of the selection rules governing hyper-Raman scattering, it is helpful to recast the molecular response tensor in terms of its various irreducible parts. In this connection, since the rate equations are expressible in terms of the index-symmetrised tensor (9.1.5), it is important to take account of the permutational symmetry. From this feature it follows that the tensor has a maximum of eighteen independent components, and can be written as a sum of four irreducible tensors, i.e. tensors whose components form bases for irreducible representations of the molecular point group. Two of these irreducible tensors are of weight 1 (each with 3 components), transforming under the symmetry operations of the molecular point group as polar vectors. Also represented are one weight 2 pseudo-tensor (5 components), transforming as a polar second rank antisymmetric tensor, and one weight 3 tensor (7 components) with the transformation properties of a traceless, fully index-symmetric third rank tensor (see appendices 4 and 6). Thus we can write …1A†

…1B†

…2†

…3†

0 bFI i…jk† … x ; x; x† ˆ b i…jk† ‡ b i…jk† ‡ b i…jk† ‡ b i…jk† ;

…9:3:1†

171

172

9 Hyper-Raman Scattering

where explicitly we have X

…1i†

1 bi…jk† ˆ10 ‰dij …3b l…lk†

b…k…ll† † ‡ dik …3bl…lj†

bj…ll† †

iˆA;B

2djk …b l…li†

2bi…ll† †Š

…9:3:2†

  …2† bi…jk† ˆ 16 eijn …elmn bl…mk† ‡ elmk bl…mn† † ‡ eikn …elmn bl…mj† ‡ elmj bl…mn† † …9:3:3† …3†

bi…jk† ˆ 13…bi…jk† ‡ bj…ki† ‡ b k…ij† †   1 15 dij …2b l…lk† ‡b k…ll† †‡ dik …2b l…lj† ‡ b j…ll† †‡djk …2b l…li† ‡b i…ll† † : …9:3:4† The decomposition of equation (9.3.2) into the two independent weight-1 terms here labelled 1A and 1B is not unique, but a convenient division is as follows …1A†

1 …3dij bl…lk† ‡ 3dik bl…lj† bi…jk† ˆ 10 …1B†

1 bi…jk† ˆ 10 … dij bk…ll†

2djk bl…li† † ;

…9:3:5†

dik bj…ll† ‡ 4djk bi…ll† † ;

…9:3:6†

as these satisfy the following contraction relations: …1A†

dij bi…jk†

…1A†

dik b i…jk†

…1A†

djk b i…jk†

…1B†

dij bi…jk†

…1B†

dik bi…jk†

…1B†

djk b i…jk†

9 ˆ dij b i…jk† > > > > > ˆ dik bi…jk† > > > > > = ˆ0 > ˆ0 > > > > > > ˆ0 > > > ; ˆ djk bi…jk†

…9:3:7†

…1A†

…1B†

This particular choice for bi…jk† and bi…jk† leads to the simplest possible form for the ensuing scattering intensities. The similarity in structure noted earlier, between equation (9.1.3) and the corresponding intensity expression for elastic second harmonic scattering, (8.2.1), correctly suggests that the hyper-Raman result can be cast in terms of five molecular invariants, as in the explicit result for the elastic case, (8.2.3). To elicit

9.3 Irreducible Tensors

the influence of selection rules the result is nonetheless best recast in terms of a different linearly independent set, based on the above irreducible tensor decomposition

…1A† …1B†

…9:3:8†

…1A† …1A†

…9:3:9†

…1B† …1B†

…9:3:10†

FI bFI i…ij† b j…kk† ˆ b i…ij† b j…kk† ; FI bFI i…ij† b k…kj† ˆ b i…ij† b k…kj† ; FI bFI i…jj† b i…kk† ˆ b i…jj† b i…kk† ; …1A† …1B†

…1A† …1A†

…1B† …1B†

…2†

…2†

FI 1 bFI i…jk† b i…jk† ˆ 5… 2b i…ij† b j…kk† ‡ 3b i…ij† b k…kj† ‡ 2b i…jj† b i…kk† ‡ 5b i…jk† b i…jk† …3†

…3†

‡ 5bi…jk† bi…jk† † ; …1A† …1B†

…9:3:11† …1A† …1A†

FI 1 bFI i…jk† b j…ik† ˆ 10…6b i…ij† b j…kk† ‡ b i…ij† b k…kj† …3†

…1B† …1B†

bi…jj† bi…kk†

…2†

…2†

5b i…jk† bi…jk†

…3†

‡ 10b i…jk† bi…jk† † :

…9:3:12†

Using the above relationships, the following expression emerges for the intensity, IHRS, of hyper-Raman scattering: IHRS …k0 † ˆ

4pxD…2† Ni  14‰7…e0
e†…e
e0 †…e
e† 105e60

3…e0
e†…e0
e†

…1A† …1B†

2…e
e†…e
e†Šbi…ij† bj…kk† ‡ 7‰ 6…e0
e†…e
e0 †…e
e† …1A† …1A†

‡ 9…e0
e†…e0
e† ‡ …e
e†…e
e†Šbi…ij† b k…kj† ‡ 7‰ 4…e0
e†…e
e0 †…e
e† ‡ …e0
e†…e0
e† …1B† …1B†

‡ 4…e
e†…e
e†Šbi…jj† bi…kk† ‡ 35‰2…e0
e†…e
e0 †…e
e† …2†

…2†

…e0
e†…e0
e† 2…e
e0 †…e0
e† …e
e†…e
e†‡2Š bi…jk† bi…jk† ‡ 5‰ 4…e0
e†…e
e0 †…e
e†

4…e0
e†…e0
e† ‡ 10…e
e0 †…e0
e† …3† …3† …e
e†…e
e† ‡ 5Š bi…jk† b i…jk† : …9:3:13†

Using this general result to interpret data, one can obtain a very detailed insight into the molecular vibrations and hence the nuclear configuration of the scattering system.

173

174

9 Hyper-Raman Scattering

9.4

Symmetry Selection Rules The symmetry selection rules for hyper-Raman scattering can be derived by considering the integral associated with the transition jm00 ije0 i jm0 ije0 i he0 jhm0 jbi…jk† jm00 ije0 i ;

…9:4:1†

where je0 i designates the ground electronic state. We confine attention to fundamental vibrational transitions, i.e. those with m0 m00 ˆ 1. For such transitions the product of the vibrational wavefunctions transforms as the corresponding normal mode coordinate. For the Dirac bracket, equation (9.4.1), not to vanish identically bi(jk) must have at least one component belonging to the same irreducible representation as the normal mode. Implementing this condition, the corresponding selection rules are easily obtained by examining the symmetry properties of the weight 1, 2 and 3 components of the b tensor under the operations of the various molecular point groups. Explicit expressions for these components are given in Tab. 9.1, and Tab. 9.2 lists the irreducible representations spanned by these components for molecules of point group symmetry D5h, D6h, D4d, D6d, Oh, Ih and D?h. The corresponding representations for point groups of lower symmetry are easily obtained from correlation tables, and are explicitly given in appendix 6. Using these results the vibrational modes for a molecule of given symmetry can be classified according to the weights of the b tensor with the same irreducible representation. This scheme facilitates comparison between the selection rules for hyper-Raman scattering and those applying to the more familiar forms of vibrational spectroscopy, i.e. infrared absorption and the Raman effect. Before we discuss the details of the classification we note two general features of the selection rules. First, all infrared active modes are also hyper-Raman active; the weight 1 component of the b tensor has precisely the same transformation properties as the electric dipole moment. Secondly, for centrosymmetric molecules, since only ungerade modes are hyper-Raman allowed, all such modes are necessarily Raman inactive. However, for noncentrosymmetric systems vibrational modes with the same sym-

9.4 Symmetry Selection Rules 175 Tab. 9.1

Irreducible components of the b tensor b1  b xxx ‡ b yyx ‡ b zzx b2  b xxy ‡ b yyy ‡ bzzy b3  b xxz ‡ byyz ‡ bzzz b4  b xxx ‡ b xyy ‡ bxzz b5  b yxx ‡ byyy ‡ b yzz b6  b zxx ‡ b zyy ‡ b zzz

b 7  b xyz b yzx q b 8  13 …byzx q b 9  13 …byyz q b10  13 …byzz q b11  13 …bzzx

b12  b13  b14  b15  b16  b17  b18 

q

2 3 …b xyz

1 2 …b xxx 1 2 …b yyy

q

ü ý þ

ü

set A ï

ï ý weight 1 ü ï ý set B ï þ þ

2bzxy ‡ b xyz † bzyy ‡ bzxx

bxxz †

b zzy ‡ b xxy

byxx †

b xzz ‡ b xyy

byyx †

ü ï ï ý weight 2 ï ï þ

‡ byzx ‡ bzxy † 2byyx

bxyy †

2b xxy

b yxx †

1 6 …2b xxz

q

‡ bzxx

2b yyz

b zyy †

1 10 …2b zzz

2bxxz

2byyz

bzxx

1 60 …8b zzx

‡ 4bxzz

3b xxx

2b yyx

1 60 …8b zzy

‡ 4b yzz

3b yyy

q q

2b xxy

ü ï ï ï ï ý weight 3 bzyy † ï ï bxyy † ï ï b yxx † þ

176

9 Hyper-Raman Scattering

metry as a weight 1 or 2 component of the b tensor can be Raman active.2) Although this is a necessary requirement, it is not sufficient because identical weight components of the Raman and hyper-Raman tensors, in general, transform differently under reflection. For example, Tab. 9.1 shows that in the point group D4d only a vibration of E2 symmetry can be both Raman and hyperRaman active. However, in the pure rotation (chiral) groups Cn, Dn, T, O and I any vibration belonging to the same representation as a weight 2 component of the b tensor is also Raman active. Examination of Tab. 9.2 shows that hyper-Raman active vibrational modes may be classified into six types (Andrews and Thirunamachandran 1978, Andrews 1990). The classification3) is based on a structure detailed in appendix 6: Class 3–(123): For modes belonging to this class the b tensor has contributions of weights 1, 2 and 3. These modes are not only hyper-Raman but also infrared active. Raman activity is neither universally allowed nor forbidden; each case can be decided using the results give in Tab. 9.2 and appendix 6. Examples are the Raman active m3(E) antisymmetric stretching and m4(E) bending modes of ammonia (C3m), and the Raman inactive m7(B1u) bending mode of ethylene (D2h). Class 3–(23): For these modes the b tensor has contributions of weight 2 and 3 alone. Since weight 1 is forbidden, such modes are invariably infrared inactive. Again, Raman activity may or may not be present. Examples are the E'' modes (such as the m13 methylene rocking mode) of cyclopropane (D3h) and the Raman inactive m4(Au) torsional mode of ethylene (D2h). Class 3–(13): These modes have b contributions of weights 1 and 3 alone. They are always infrared active and Raman inactive. An example is the m3(A2u) out-of-plane bending mode in xenon tetrafluoride (D4h). 2) Irreducible tensor decomposition of the Raman scattering tensor, assuming the

index-symmetric form appropriate for non-resonant applications, yields one weight 0 and one weight 2 contribution. 3) This classification scheme has also found applications in multiphoton absorption spectroscopy (Galasso 1992).

D5h D6h D4d D6d Oh Ih D1h

Group a (weight 0) A01 A1g A1 A1 A1g A Pg

l (weight 1)

A002 +E01 A2u+E1u B2+E1 B2+E1 T1u T P1u‡ Q u ‡ u ‡ g

Raman

Infrared

A01 +E01 +E001 A1g+E1g+E2g A1+E2+E3 A1+E2+E5 Eg+T2g H Pg‡ Q g ‡ g ‡Dg

a (weight 2) 2A002 +2E01 2A2u+2E1u 2B2+2E1 2B2+2E1 2T1u 2TP 1u Q 2 ‡ u ‡2 u

b (weight 1)

Hyper-Raman

A01 +E01 +E001 A1u+E1u+E2u B1+E1+E2 B1+E1+E4 Eu+T2u H Pu‡ Q u ‡ u ‡ Du

b (weight 2)

A002 +E01 +E02 +E002 A2u+B1u+B2u+E1u+E2u B2+E1+E2+E3 B2+E1+E3+E4 A2u+T1u+T2u E +Gu 2uQ Pu+T ‡ ‡ u u ‡Du ‡ Uu

b (weight 3)

Tab. 9.2 Infrared, Raman and hyper-Raman activity: Irreducible representations of dipole l, polarisability a and hyperpolarisability b.

9.4 Symmetry Selection Rules 177

178

9 Hyper-Raman Scattering

Class 3–(3): With b contributions of weight 3 alone, these modes are invariably both infrared and Raman inactive. An example is the m6(B1u) hydrogen stretch in benzene (D6h). Class 3–(2): Here the b tensor has only weight 2 contributions. These modes are infrared inactive, but can be Raman active. For example, the Raman active m2(E) deformation modes of methane (Td) belong to this class, as does the Raman inactive m4 …A001 † methylene twisting mode of cyclopropane (D3h). Class 3–(1): The final class contains modes which have only weight 1 b contributions. These modes are infrared active but always Raman inactive. As is evident from Tab. 9.2, molecules belonging to the common point groups do not have modes of this type. A rare example is a T1u vibration in species of Ih icosahedral symmetry, such as buckminsterfullerene, C60.

9.5

Scheme for the Determination of Molecular Invariants The expression for the hyper-Raman intensity, equation (9.3.13), contains five independent molecular invariants, namely the different types of b products. From this result one can determine a value for each of these from a set of five experimental observations, suitably devised to produce linearly independent results. Determination of the invariants which are zero and non-zero for each line in the spectrum directly allows symmetry classification of the corresponding vibrational mode, using the scheme introduced in the previous section. Suitable experimental configurations are conveniently described in terms of the propagation and polarisation vectors of the incident and scattered photons. As shown in the previous Fig. 8.1, ? e and ? e' are equivalent unit ^k ^0 plane, h is the scattering angle, and cirvectors normal to the k cular polarisations are consistent with (3.1.8) and (3.1.9). Using a …k!l† similar notation IHR …h† to denote the experimental configuration with incident and scattered photons having polarisations ke and le' respectively, as used in chapter 8, a convenient set of five intensity measurements is as follows:

9.5 Scheme for the Determination of Molecular Invariants 179 …?!?†

…p=2† ˆ I1

…9:5:1†

…p=2† ˆ I2

…9:5:2†

…p=2† ˆ I3

…9:5:3†

IHR

…?!jj†

IHR

…R!jj†

IHR

…R!R†

IHR

…R!L†

IHR

…0† ˆ I4

…9:5:4†

…0† ˆ I5 :

…9:5:5†

This particular set includes the two pairs of measurements required for the calculation of the reversal ratio and depolarisation ratio (see section 9.6). Explicit expressions for I1 to I5 follow directly from equation (9.3.13): I1 ˆ

I2 ˆ

4pxD…2† Ni …1A† …1B† …1A† …1A† …1B† …1B† 28bi…ij† bj…kk† ‡ 28bi…ij† bk…kj† ‡ 7b i…jj† bi…kk† 525e60 …3† …3†
…9:5:6† ‡ 30bi…jk† bi…jk† ; 4pxD…2† Ni 525e60

…1A† …1B†

…1A† …1A†

…1B† …1B†

28bi…ij† bj…kk† ‡ 7b i…ij† bk…kj† ‡ 28bi…jj† bi…kk†

…2† …2† …3† …3†
‡ 35bi…jk† bi…jk† ‡ 20bi…jk† bi…jk† ;

…9:5:7†

I3 ˆ

4pxD…2† Ni …2† …2† …3† …3†
14b i…jk† bi…jk† ‡ 5bi…jk† bi…jk† ; 6 105e0

…9:5:8†

I4 ˆ

4pxD…2† Ni 525e60

…1A† …1B†

…1A† …1A†

…1B† …1B†

42bi…ij† bj…kk† ‡ 63bi…ij† bk…kj† ‡ 7bi…jj† bi…kk†

…2† …2† …3† …3†
‡ 35bi…jk† bi…jk† ‡ 5b i…jk† bi…jk† ;

I5 ˆ

…9:5:9†

4pxD…2† Ni …3† …3†
bi…jk† bi…jk† : 7e60

…9:5:10†

These form a regular system of equations, which can be solved, using Cramer’s rule, to give …1A† …1B† bi…ij† bj…kk†

  3 e60 ˆ …7I1 2 4pxD…2† Ni

I2 ‡ 2I3

3I4

3I5 † ; …9:5:11†

180

9 Hyper-Raman Scattering …1A† …1A†

bi…ij† bk…kj† ˆ

…1B† …1B†

  3 e60 …4I1 2 4pxD…2† Ni 

bi…jj† bi…kk† ˆ 3

2I2

I3 ‡ 4I4

I5 † ; …9:5:12†



e60 …3I1 ‡ 6I2 ‡ 2I3 4pxD…2† Ni

2I4

2I5 † ; …9:5:13†

…2† …2† bi…jk† b i…jk†

…3†

…3†

bi…jk† b i…jk†

  5 e60 ˆ …3I3 2 4pxD…2† Ni   e60 ˆ7 …I5 † : 4pxD…2† Ni

I5 † ;

…9:5:14† …9:5:15†

Thus, the assignment of each spectral line to a vibrational mode of a particular class is achievable by digital processing of the spectra obtained from these five experiments. For this purpose it suffices to determine the relative intensities, in each of the five independent spectra, of lines with the same frequency shift from the harmonic. Processing the linear combinations, equations (9.5.11)–(9.5.15), generates new spectra corresponding to each of the five b tensor products. The appearance or non-appearance of a line at a particular frequency in the computed spectra then provides for assignment of the corresponding vibrational mode to one of the six hyper-Raman classes. Tables 9.3–9.5 present a complete set of intensity results for forward, backward and right-angled scattering respectively (u denoting unpolarised light), in terms of I1–I5, equations (9.5.6)(9.5.10), and I6–I11 expressible as linear combinations of I1–I5: I6 ˆ 14 …2I3 ‡ I4 ‡ I5 † ;

…9:5:16†

I7 ˆ 14 …I4 ‡ I5 † ;

…9:5:17†

I8 ˆ 21 …I1 ‡ I2 † ;

…9:5:18†

I9 ˆ 18 …2I1 ‡ 2I2 ‡ I4 ‡ I5 † ;

…9:5:19†

I10 ˆ 14 …2I2 ‡ I3 † ;

…9:5:20†

1 I11 ˆ 16 …2I1 ‡ 6I2 ‡ 2I3 ‡ I4 ‡ I5 † :

…9:5:21†

9.5 Scheme for the Determination of Molecular Invariants 181 Tab. 9.3 Intensities of hyper-Raman forward scattering (h = 0).

jj ! ?! L! R! u!

jj

?

L

R

I1 I2 I7 I7 I9

I2 I1 I7 I7 I9

I8 I8 I4 I5 I9

I8 I8 I5 I4 I9

Tab. 9.4 Intensities of hyper-Raman backward scattering (h = p).

jj ! ?! L! R! u!

jj

?

L

R

I1 I2 I7 I7 I9

I2 I1 I7 I7 I9

I8 I8 I5 I4 I9

I8 I8 I4 I5 I9

Tab. 9.5 Intensities of hyper-Raman scattering at right angles (h = p/2, 3p/2).

jj ! ?! L! R! u!

jj

?

L

R

I2 I2 I3 I3 I10

I2 I1 I7 I7 I9

I2 I8 I6 I6 I11

I2 I8 I6 I6 I11

Except for I1, I3 and I5 all expressions contain contributions from weights 1, 2 and 3. Consequently, the appearance of a line in the spectra for the above scattering configurations does not provide a firm basis for classifying the corresponding vibrational mode. Linear processing of the results associated with I1–I5 remains the most expedient means of symmetry classification.

182

9 Hyper-Raman Scattering

9.6

Reversal and Depolarisation Ratios The reversal ratio, generally dependent on the scattering angle h, is here defined as for Raman and hyper-Rayleigh scattering: R…h† ˆ

I …R!L† …h† : I …R!R† …h†

…9:6:1†

Once more using equation (9.3.13) as the kernel we obtain the following expression:  R…h† ˆ 7…1 2 cos h ‡ cos2 h† …1A† …1B† …1A† …1A† …1B† …1B†
 6bi…ij† bj…kk† ‡ 9bi…ij† bk…kj† ‡ bi…jj† bi…kk† ‡ 35…5

2 cos h

…2†

…2†

3 cos2 h†b i…jk† bi…jk†

…3† …3† ‡ 10…13 ‡ 14 cos h ‡ 3 cos2 h†bi…jk† bi…jk†  …1A† …1B† …1A† …1A† 6bi…ij† bj…kk† ‡ 9bi…ij† bk…kj†  7…1 ‡ 2 cos h ‡ cos2 h† …1B† …1B†
…2† …2† ‡ bi…jj† bi…kk† ‡ 35…5 ‡ 2 cos h 3 cos2 h†bi…jk† bi…jk† …3† …3† 1 ‡ 10…13 14 cos h ‡ 3 cos2 h†bi…jk† bi…jk† : …9:6:2†

For forward scattering, h = 0, the result equates to I5/I4. For vibrational modes belonging to class 3–(3) the result for the reversal ratio is R(0) = 15; this result relates to the highest achievable reversal of circularity in the emergent harmonic, consistent with an exchange of 3 units of angular momentum through the weight 3 vibrational transition of the scatterer. Class 3–(2) vibrational modes cannot mediate the uptake of 3 angular momentum units, and hence R(0) = 0, corresponding to complete retention of circularity. For other classes, all of which involve both weight 3 and other weights, values of the reversal ratio are dependent on the appropriate molecular invariants. As also for Raman scattering, the reversal ratio for right angled scattering is unity; here there is no preferred circularity in the harmonic emission (signalling the irrelevance of orthogonal angular momentum components), and the result again satisfies the condition (8.2.13). The depolarisation ratio is defined as for hyper-Rayleigh scattering (equation (8.2.5)). For the particular case of hyper-Raman

9.7 Higher Multipole Effects

scattering considered here, the depolarisation ratio is given by I2/ I1. For class 3–(3) vibrational modes the value of q? is 2/3, whereas for class 3–(2) modes the ratio is infinite. The latter behaviour is unknown in Raman scattering except under resonance conditions. These two classes are of special interest since they correspond to modes that are active in hyper-Raman scattering but not in infrared absorption; it was originally thought that all such modes have identical reversal and depolarisation ratios (Cyvin et al. 1965). The other class of infrared inactive modes, class 3–(23), exhibits reversal and depolarisation ratios intermediate between the class 3–(3) and 3–(2) values.

9.7

Higher Multipole Effects The intrinsically weak signal of hyper-Raman scattering, even when all three of the photon interactions it entails are electric dipole allowed, originates from three factors; its optical nonlinearity, incoherence, and dependence on the first derivative of the molecular response tensor, equation (9.2.4). When allowed, higher multipoles can contribute to the signal but only still more weakly, and their role is not identifiable unless associated with distinctive polarisation behaviour that affords experimental detection. To conclude and to illustrate the underlying principles, we briefly consider one such possibility: circular differential hyper-Raman scattering. In its optically linear analogue, circular differential scattering is a well established branch of vibrational Raman spectroscopy with significant biological applications (Barron 1982, 2000). Molecular vibrations in a local environment that is chiral (‘optically active’), as in most of nature, are associated with transitions that carry a chiral signature. When induced by circularly polarised light, such transitions are induced at different rates according to the circularity of the radiation. The interactions of a right-handed molecule with light of a right-handed circular polarisation are different from those it experiences with a left-handed polarisation, for example. Equally the responses of two opposite chiral forms of a given molecule differ in their interactions with light

183

184

9 Hyper-Raman Scattering

of a given handedness. When molecules are randomly oriented, or measurements are made over a spatial range that extends beyond any local ordering in a system with long-range disorder, the fractional difference in rate is small, typically 10–3 or 10–4, although now routinely detectable. The weakness of the effect reflects the fact that it has a quantum interference origin involving multipoles beyond the electric dipole. In the context of hyper-Raman emission, in constructing an equation for the intensity based on the Fermi rule, (2.1.20), the matrix element has to be regarded as a sum Cˆ

2p jMfi …E13 † ‡ Mfi …E12 E2† ‡ Mfi …E12 M1† ‡ . . . j2 d…Ef h 

Ei † : …9:7:1†

Evaluating the modulus square, the leading term comes from the interference of the E13 term with itself and the leading corrections, the interference of E13 with E12E2 and with E12M1. It is evident that the latter contributions generate chiral sensitivity. This is because both the molecular tensor and the radiation tensor for the E13 quantum amplitude, the basis for all calculations earlier in this chapter, are of odd spatial parity whereas both E12E2 and E12M1 are of even parity (see section 1.2). For any interaction where both types of behaviour are allowed, molecular wavefunctions cannot have a definite parity signature (such as the gerade or ungerade designation which applies to centrosymmetric molecules); this means that the molecule has to be chiral. Detailed results for the ensuing circular differential intensity have been given by Andrews and Thirunamachandran (1979).

185

Appendix 1:

Resonance Damping The issue of damping is highly significant when operating near to resonance, a condition which applies to an optical process when there exists a molecular state differing in energy from the initial state by an amount approaching the energy of one or more of the photons involved. The correct designation of sign for damping factors in the energy denominators of optical response tensors has been the subject of considerable debate (see for example Andrews et al. 1998, Buckingham and Fischer 2000, Stedman et al. 2001). Partly this stems from a common confusion in the literature between two entirely different forms of damping; partly it reflects attempts to impose conflicting conditions on the molecular response. The former obscurity is easily dealt with – though it forms the ground in which seeds of the latter conflict have been sown. First, note that the embodiment of photon 1) energies in the denominators of expressions such as equations (4.1.4) and (4.1.6) owes its origin to the development of the signal amplitude from equation (3.1.1). To confer analyticity on the dispersion behaviour, imaginary infinitesimals are commonly added to the photon energies in each denominator factor to displace poles from the real axis in a complex frequency plane. Each photon energy thereby acquires an infinitesimal addendum, hx ! hx ‡ is, with s ! ‡0. For example, in the polarisability equation (4.1.4), since the sign of  hx differs in the two energy denominators, this modification introduces addenda of –is and +is, respectively, to the two denominator expressions. The real and imaginary parts of a 1) Here we give a reminder that in the present context the term ‘photon’ more ac-

curately denotes a bath-quantum polariton – see chapter 1. 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

186

Appendix 1: Resonance Damping

polarisability so modified are even and odd functions of frequency, respectively – so enabling the relation a… x; x† ˆ a…x; x† to be satisfied. Nonetheless, the result has no meaning other than in the designed limit s ! ‡0. The second type of resonance modification to energy denominators, introduced in chapter 4, is designed to reflect the finite lifetime of each molecular energy level, through modification of the corresponding energy, E~r ˆ Er  12 ihcr . Adopting the negative sign systematically proves to be consistent with time-reversal symmetry, as detailed below. The result of implementing this correction in the polarisability equation (4.1.4), for example, is hcr to each energy denominator, with the the addition of 12 i same sign in each term (Barron 1982). In non-resonant processes,  hcr is typically several orders of magnitude smaller than Er , its precise value determined by the nature of the molecule and structural inhomogeneities. Near to resonance, damping serves to give a realistic lineshape to the optical response. For example, in light scattering close to an optical absorption band where  hx  Er0 , the first term of equation (4.1.4) dominates and the response acquires a Lorentzian lineshape. Historically two conventions have been used extensively in the literature for setting the sign of damping factors. Although mutually incompatible, justifications for each convention have been made by appeal to causality – a different line generally being taken in semiclassical and in fully quantum mechanical approaches. To illustrate, the two conventions with respect to SHG are as follows: in the semiclassical or variable-sign convention (vsc) the signs are chosen to be opposite for interactions preceding and following emission of the harmonic, as follows (Bloembergen 1965, Hanna et al. 1979, Shen 1984): bvsc ijk …

2x; x; x† ˆ

" XX s

r

‡ ‡

…Es0

sr r0 l0s i lj lk

2hx

1 hcs †…Er0 2 i 0s sr r0 lj li lk

…Es0 ‡  hx ‡ 12 ihcs †…Er0

hx

hx

1 hcr † 2 i

1 hcr † 2 i

#

sr r0 l0s j lk li

…Es0 ‡  hx ‡ 12 ihcs †…Er0 ‡ 2hx ‡ 12 ihcr †

:

…A1:1†

Appendix 1: Resonance Damping 187

In the fully quantum mechanical development (constant-sign convention, csc), as adopted in most of the literature on Raman and nonlinear Raman scattering, all signs of the damping are identical: " sr r0 XX l0s i lj lk csc bijk … 2x; x; x† ˆ …Es0 2 hx 12 ihcs †…Er0 hx 12 ihcr † s r ‡ ‡ :

sr r0 l0s j li lk

…Es0 ‡  hx …Es0 ‡  hx

1 hcs †…Er0 2 i sr r0 l0s j lk li 1 hcs †…Er0 ‡ 2 i

hx 2hx

1 hcr † 2 i

#

1 hcr † 2 i

…A1:2†

It needs to be understood that the status of the results (A1.1) and (A1.2) is quite different and belies their common symbology. The former expression is a semiclassical construct for a molecular second order nonlinear polarisability; the latter is the molecular factor in a quantum electrodynamical amplitude developed from (3.1.1). Neither has legitimacy outside its own province. 2) Although there has been debate over these damping corrections, it should be borne in mind that there is no difference between the conventions regarding the signing of resonant terms, and such differences as exist in anti-resonant terms do not translate into measurable differences in rates. The csc result (A1.2) is the material component of a quantum amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate down-conversion. On Hermitian conjugation H and time reversal T, each of the six terms of the symmetrised hyperpolarisability tensor bi…jk† … 2x; x; x†, equation (4.1.6), transforms into one of the six counterpart terms in bi…jk† …2x; x; x†, the degenerate down-conversion tensor. For example the last term of b i…jk† … 2x; x; x†, in the order, which logically follows from equations (A1.2) and (4.1.6), behaves as follows: 2) In Andrews et al. (1998) the quantum electrodynamical response was cast as a

quantum amplitude A to highlight its inapplicability within semiclassical equations.

188

Appendix 1: Resonance Damping sr r0 l0s k lj li

…Es0 ‡ hx …Es0 ‡ hx

HT

1 hcs †…Er0 ‡2 hx 2 i 0r rs s0 li lj lk 1 hcs †…Er0 ‡2 hx 2 i

1 hcr † 2 i

!

1 hcr † 2 i

On interchanging the dummy state sum indices r and s, the result is exactly the first term of bijk …2x; x; x† as follows by reversing the sign of x in (A1.2). The time reversal represented above is depicted in the time-ordered diagrams of Fig. A1.1. As written, the effect of the combined operation HT is valid not only for non-degenerate and time-even states (consistent with spin-paired molecular orbitals) but also for degenerate and timeodd states (Naguleswaran 1998, Andrews et al. 1998, Stedman 2001). The vsc result (A1.1) fails to satisfy such time-reversal symmetry. The requirement for temporal symmetry remains unequivocal, despite violation of time-reversal invariance by the system itself (its engagement of molecular interaction with the bath leading to state decay), specifically through the inclusion of damping. In certain processes the two conventions can lead to

k;k k;k

k0 ;k0

Fig. A1.1 The three time-ordered diagrams for degenerate down-conver-

sion, representing exact time reversal of the SHG orderings illustrated in Fig. 2.2. Consequently the emitted photons now carry the wave vector and polarisation labels, k and k, and the annihilated photon is characterised by k0, k0 .

Appendix 1: Resonance Damping 189

results with very significant differences. For example the correct constant-signing rule is necessary to uphold the principle that linear electro-optical response cannot occur in any isotropic liquid (Dávila-Romero et al. 2000). A possible source of confusion over the correct form of damping can be identified on comparing equations (A1.1) and (A1.2). In the latter, the csc result is associated with the consistently positive signs of the energy differences Es0 and Er0; in the former vsc result, damping signs vary with the variable signs of the photon energies  hx, 2 hx. If the imaginary infinitesimals discussed earlier are substituted by finite, physically meaningful damping constants, spurious results conforming to the variablesign rule emerge. Logically, vsc requires that, for any electro- or magneto-optical processes, interactions with which static fields are associated should carry no damping. However from a quantum field viewpoint, static perturbations must induce damping. In the multipolar gauge all electromagnetic interactions are mediated through the exchange of virtual photons, and a static field is not different from a time-varying field – except that, whilst causality is of course satisfied, explicit retardation features disappear in the limit of zero frequency. The damping associated with any molecular excited state is therefore not frequency-dependent; it has a characteristic magnitude, regardless of the frequency of the perturbation with which it is associated.

191

Appendix 2:

Rotational Averaging The theoretical representation of optical processes in any orientationally disordered molecular medium requires that account be taken of the distribution of orientations amongst the component molecules. By the ergodic theorem, the mean response from an ensemble of such molecules under equilibrium conditions equates to the time-averaged response from each individual molecule; thus in a system of free or partially free molecules it is required to evaluate an orientationally averaged response. For conventional light absorption and scattering processes the procedure is elementary, but for nonlinear interactions it creates calculational problems on a scale that rapidly escalates with the degree of nonlinearity. Calculation is simplest for fluids such as gases or liquids, where the distribution of molecular orientations is in general isotropic, with O(3) bulk symmetry. The application of a static electric field to such a medium may however introduce a degree of molecular alignment if the constituent molecules are polar, reducing the symmetry of the medium to C1m ; then a rotational averaging which incorporates a Boltzmann weighting factor is appropriate. A similar orientationally weighted average occurs in the description of molecules adsorbed at surfaces or liquid crystals. In this appendix, essential details of the procedure for dealing with each of these cases is provided, together with tables of the main results.

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

192

Appendix 2: Rotational Averaging

A.

Isotropic Systems Consider the following expression for the optical response of a molecule with fixed orientation with respect to the radiation. In general terms the signal A associated with a given nonlinear interaction can be expressed by A / Si1 ...in Ti1 ...in ;

…A2:1†

where S is a rank n tensor representing the radiation field and T is a generalised response tensor of the same rank. For coherent processes, n can be identified with the number of photons involved in the interaction, within the dipole approximation; S then comprises a product of the polarisation vectors for each photon, and T is a corresponding rank n molecular tensor characterising the molecular response. For incoherent processes involving m photons, the leading contribution to the signal normally comes from a term with n ˆ 2m; S is then the outer product of the appropriate rank m optical polarisation tensor and its complex conjugate, and a similar construction from the rank m molecular response tensor generates T. Since A is a scalar, the choice of reference frame i for the Cartesian axes in equation (A2.1) is arbitrary, and may suitably be chosen as a laboratory-fixed frame in terms of which the components of S are independent of molecular orientation. It is then convenient to re-express the components of the response tensor T to a molecule-fixed frame k in which they too are rotation-invariant, through the relation Ti1 ...in ˆ li1 k1 . . . lin kn Tk1 ...kn :

…A2:2†

Here lip kp denotes the direction cosine between the ^ip and k^p axes, as given by the …ip ; kp † element of the Euler angle matrix (see appendix 3). Note that the designation …ip ; kp † represents the pth set of indices …i; k†, each one of which can be either x, y or z. In the orientational average of the right-hand side of equation (A2.1), following implementation of (A2.2), only the direction cosines are dependent on molecular orientation and the result for the averaged signal can be represented by

A: Isotropic Systems 193

hAi ˆ Si1 ...in Tk1 ...kn hli1 k1 . . . lin kn i :

…A2:3†

It thus remains to evaluate the average represented by …n†

Ii1 ...in ;k1 ...kn ˆ hli1 k1 . . . lin kn i :

…A2:4†

Whilst explicit integration over the Euler angles offers the most obvious means of identifying the result of (A2.4), it is a method which in principle entails 32n distinct integrals. Despite simplifications that can be effected by exploiting the symmetry properties of the Euler matrix, the procedure remains a formidable task for any n > 2. A much simpler method makes use of the fact that the result given by (A2.4), which is a double tensor of rank n both in the laboratory and molecular frames, must be rotationinvariant in each frame. Consequently it has to be expressible as a combination of products of isotropic (weight 0) rank n tensors referred to the two frames. Knowledge of this structure enables the explicit results to be obtained by a relatively straightforward matrix inversion method, and the results up to rank 8 are known (Andrews and Thirunamachandran 1977 a, Andrews and Ghoul 1981, Andrews and Blake 1989). Results up to rank six are given below: I …0† ˆ 1 ;

…A2:5†

…1†

…A2:6†

…2†

…A2:7†

…3†

…A2:8†

Ii1 ;k1 ˆ 0 ; Ii1 i2 ;k1 k2 ˆ 13 di1 i2 dk1 k2 ; Ii1 i2 i3 ;k1 k2 k3 ˆ 16 ei1 i2 i3 ek1 k2 k3 ; 2

…4†

Ii1 i2 i3 i4 ;k1 k2 k3 k4

3T 2 di1 i2 di3 i4 4 1 6 7 6 ˆ 4 di1 i3 di2 i4 5 4 1 30 di1 i4 di2 i3 1

1 4 1

3 32 1 dk1 k2 dk3 k4 7 76 1 54 dk1 k3 dk2 k4 5 dk1 k4 dk2 k3 4 …A2:9†

194

Appendix 2: Rotational Averaging …5†

Ii1 i2 i3 i4 i5 ;k1 k2 k3 k4 k5 ˆ 2

ei1 i2 i3 di4 i5

3T 2

6 7 6 ei1 i2 i4 di3 i5 7 6 7 6e 7 6 i1 i2 i5 di3 i4 7 6 7 6 ei1 i3 i4 di2 i5 7 6 7 6e 7 4 i1 i3 i5 di2 i4 5 ei1 i4 i5 di2 i3

6 6 6 6 6 6 6 6 6 4

1 30

3

1

1

1

1

1 1 1

3 1 1

1 3 0

1 0 3

0 1 1

1 0

0 1

1 1

1 1

3 1

0

32

ek1 k2 k3 dk4 k5

3

7 76 1 76 ek1 k2 k4 dk3 k5 7 7 76 6 7 17 76 ek1 k2 k5 dk3 k4 7 7 76 6 7 17 76 ek1 k3 k4 dk2 k5 7 6 7 1 54 ek1 k3 k5 dk2 k4 7 5 ek1 k4 k5 dk2 k3 3

…A2:10†

…6†

di1 i6 di2 i5 di3 i4

3T 2 di1 i2 di3 i4 di5 i6 6d d d 7 6 6 i1 i2 i3 i5 i4 i6 7 6 6 7 6 6 di1 i2 di3 i6 di4 i5 7 6 6 7 6 6d d d 7 6 6 i1 i3 i2 i4 i5 i6 7 6 6 7 6 6 di1 i3 di2 i5 di4 i6 7 6 6 7 6 6d d d 7 6 6 i1 i3 i2 i6 i4 i5 7 6 6 7 6 6 di1 i4 di2 i3 di5 i6 7 6 6 7 6 6 7 6 6 di1 i4 di2 i5 di3 i6 7 6 6 7 6 6 di i di i di i 7 6 1 4 2 6 3 5 6 7 6 6 7 6 6 di1 i5 di2 i3 di4 i6 7 6 6 7 6 6 di i di i di i 7 6 6 15 24 367 6 6 7 6 6 di1 i5 di2 i6 di3 i4 7 6 6 7 6 6 di i di i di i 7 6 6 16 23 457 6 6 7 6 4 di1 i6 di2 i4 di3 i5 5 4

2

1 210

2 2 5 2

5 2

2 5 2 2

2 5

2 2 5

5 2

2 2

2 2 5

5 5 5

2 2 2 5 2

5 2 5 2 5

2 2

2 5

2 2

2 5 2

2 2

5 5

5 2 2

2 2

2 5

5 5 2

5 5 5

2 2 2

2 5

2 5 2

2 2 5

2 5

2 2 5

5 5 2

2 2

2 5

2 2 5 2

5 2 2 2

5 5

5 5 5

5 2 2

2 2 2

5 2

2 5 2

5 5 2

5 2 2

2 2

5 2

2 5

2 5

5 2 5

2 2 2

2 5 2

2 5

dk1 k6 dk2 k5 dk3 k4

3 32 5 dk1 k2 dk3 k4 dk5 k6 6 7 27 76 dk1 k2 dk3 k5 dk4 k6 7 7 76 6 7 27 76 dk1 k2 dk3 i6 dk4 k5 7 6 7 2 76 dk1 k3 dk2 k4 dk5 k6 7 7 7 76 6 7 5 76 dk1 k3 dk2 k5 dk4 k6 7 7 6 7 27 76 dk1 k3 dk2 k6 dk4 k5 7 7 76 6 dk1 k4 dk2 k3 dk5 k6 7 27 7 76 7 76 5 76 dk1 k4 dk2 k5 dk3 k6 7 7 76 6 dk k dk k dk k 7 27 1 4 2 6 3 5 7 76 7 76 2 76 dk1 k5 dk2 k3 dk4 k6 7 7 76 6 7 27 76 dk1 k5 dk2 k4 dk3 k6 7 7 76 5 76 dk1 k5 dk2 k6 dk3 k4 7 7 76 6 7 57 76 dk1 k6 dk2 k3 dk4 k5 7 7 76 5 54 dk1 k6 dk2 k4 dk3 k5 5 5 16 5 5 16

5 16 2 2 2 16

5 16 2 2 2 2 16 5 5 2 5 16

5 16 2 2 2 2 16 5 5 2 5 16

5 5 2

5 5 2

2 2 2

5 2 5

2 2 2

2 5

2 5 2

5 2

5 16 2 2 2 2 16 5 5 2 5 16

5 5

5 5 2

16 5 5 16

Ii1 i2 i3 i4 i5 i6 ;k1 k2 k3 k4 k5 k6 ˆ

…A2:11†

A: Isotropic Systems 195

196

Appendix 2: Rotational Averaging

B.

Partially Ordered and Phase-Weighted Systems When a degree of preferred orientation is conferred upon the molecules comprising a fluid medium, isotropy is commonly retained in the two dimensions perpendicular to the director of order. For example in a system of polar molecules, the application of a static electric field leads to each molecule experiencing a torque l  Elocal =e0 , where l is the permanent molecular dipole moment and Elocal is the local microscopic field. The corresponding interaction energy is given by DE ˆ l
Elocal =e0. Under equilibrium conditions the molecular orientational distribution is thus governed by Boltzmann statistics with a weighting factor exp… l
Elocal =e0 kB T†. In general, the mean signal associated with a nonlinear optical process in any such partially ordered medium is determined by an equation of the form hAi ˆ

hSi1 ...in Ti1 ...in eu
v i ; heu
v i

…A2:12†

where u and v are vectors referred to the laboratory and molecular frames respectively. In the specific case described above, for example, we may identify u with l and v with Elocal =e0 kT. Results of similar form are also required for the calculation of rotational averages weighted by phase factors, in which the laboratory frame vector u (commonly denoting a wave-vector mismatch) is imaginary. Proceeding as before, the right-hand side of (A2.12) is rewritten as follows hAi ˆ

Si1 ...in Tk1 ...kn hli1 k1 . . . lin kn eu
v i heu
v i

…A2:13†

The requisite averages in angular brackets, calculated by Andrews and Harlow (1984), are expressible in terms of spherical Bessel functions involving a scalar parameter c determined by; c ˆ …u
v†=…^ u
^v† ;

…A2:14†

in terms of which the weighted averages are represented by

B. Partially Ordered and Phase-Weighted Systems 197 …n†w

^ ; ^v† ˆ Ii1 ...in ;k1 ...kn …c; u

hli1 k1 . . . lin kn ec^u
^v i : hec^u
^v i

…A2:15†

In the expressions below, j0n denote reduced spherical Bessel functions defined by j0n ˆ jn =j0, as given together with their asymptotic limits in Table 7.1. For phase-weighted, rather than partially ordered averages, the imaginary nature of u yields an imaginary c and hence the spherical Bessel functions emerge in terms of trigonometric instead of hyperbolic functions. ^ ; ^v† ˆ 1 : I …0†w …c; u

…A2:16†

…1†w

^ ; ^v† ˆ ij01 … ic†^ Ii1 ;k1 …c; u ui1 ^vk1 :

…A2:17†

…2†w

^ ; ^v† ˆ 13 di1 i2 dk1 k2 ‡ 12 j01 … ic†ei1 i2 is u ^ is ek1 k2 ks ^vks Ii1 i2 ;k1 k2 …c; u 3 0 2 j2 …

^ i2 13 di1 i2 † ic†…^ ui1 u 1 dk k † :  …^vk1 ^vk2 3 12

…A2:18†

^ i3 di1 i2 u

3T 2

^ i2 u ^ i3 ic†‰^ ui 1 u ‡ dk1 k3 ^vk2 ‡ dk2 k3 ^vk1 †Š

5i 0 2 j3 … 1 ^ 5 …di1 i2 ui3

ek1 k3 ks …^vk2 ^vks

1

2

dk2 k3 ^vk1 4 #" 1 ek1 k2 ks …^vk3 ^vks 2

1

3

76 7 6 7 17 54 dk1 k3 ^vk2 5

1

dk1 k2 ^vk3

4

32

1

1

1

4

^ i2 ‡ di2 i3 u ^ i1 †Š‰^vk1 ^vk2 ^vk3 ‡ di1 i3 u

6 7 6 …3†w 6 ^ ; ^v† ˆ 16 ei1 i2 i3 ek1 k2 k3 ‡ 10i j01 … ic†6 ^ i2 7 Ii1 i2 i3 ;k1 k2 k3 …c; u 4 di1 i3 u 5 4 ^ i1 di2 i3 u " #T " ^ is 13 di3 is † ei1 i2 is …^ ui 3 u 1 0 2 j2 … ic† ^ is 13 di2 is † ui 2 u ei1 i3 is …^

2

#

1 3 dk2 ks † 1 ^ 5 …dk1 k2 vk3

1 3 dk3 ks †

…A2:19†

198 Appendix 2: Rotational Averaging

35 0 8 j4 …

5i 0 8 j3 …

^ i4 u ^ is ui2 u ei1 i3 is ‰^

1 35 …dk1 k2 dk3 k4

3T 2

6 6 6 6 6 6 6 6 6 6 6 4 3

4 11

3

3

3 3T 2

4

32

3

3

ek1 k3 ks ‰^vk2 ^vk4 ^vks

^ i2 u ^ i 4 ‡ di 1 i 4 u ^ i2 u ^ i3 ‡ di2 i3 u ^ i1 u ^ i4 ‡ di2 i4 u ^ i1 u ^ i3 ‡ ‡ di1 i3 u

1 ^ 5 …dk1 ks vk4

1 ^ 5 …dk2 ks vk4

‡ dk1 k3 dk2 k4 ‡ dk1 k4 dk2 k3 †Š

1

1

32 ek1 k2 ks ^vks dk3 k4

3

7 ‡ dk4 ks ^vk1 ‡ dk1 k4 ^vks †Š 7 5

3

76 7 7 6 17 76 ek1 k3 ks ^vks dk2 k4 7 76 7 7 6 17 76 ek1 k4 ks ^vks dk2 k3 7 76 7 7 6 17 76 ek2 k3 ks ^vks dk1 k4 7 76 7 7 6 17 54 ek2 k4 ks ^vks dk1 k3 5 3 ek3 k4 ks ^vks dk1 k2

0

‡ dk4 ks ^vk2 ‡ dk2 k4 ^vks †Š

1

3

1

1

0

1 1

1

3

0

1

1

0

1

0

1

0

1

1

3

1

1

1

3

1

1

1

3

…A2:20†

1 ^ ^ ^ 5 …dk1 ks vk2 ‡ dk2 ks vk1 ‡ dk1 k2 vks †Š 1 ^ i1 u ^ i2 † ‡ 35 …di1 i2 di3 i4 ‡ di1 i3 di2 i4 ‡ di4 i4 di2 i3 †Š di3 i4 u

76 6 3 17 54 ek2 k3 ks ‰^vk1 ^vk4 ^vks 1 3 ek3 k4 ks ‰^vk1 ^vk2 ^vks

32

1 3 d k1 k2 †

^ is di1 i2 ei3 i4 is u 3 1 3 d k3 k4 † 7 1 7 3 d k2 k4 † 7 7 1 7 3 d k2 k3 † 7 7 1 7 3 d k1 k4 † 7 7 1 7 3 d k1 k3 † 5

6 6 6 6 6 6 6 6 6 6 6 4

3T 2

‡ dk1 k3 ^vk2 ^vk4 ‡ dk1 k4 ^vk2 ^vk3 ‡ dk2 k3 ^vk1 ^vk4 ‡ dk2 k4 ^vk1 ^vk3 ‡ dk3 k4 ^vk1 ^vk2 †

1 ^ ^ 7 …di1 i2 ui3 ui4

1 ^ 5 …di1 is ui4

1 1

dk1 k2 …^vk3 ^vk4 76 6 37 76 dk1 k3 …^vk2 ^vk4 76 6 37 76 dk1 k4 …^vk2 ^vk3 76 6 37 76 dk2 k3 …^vk1 ^vk4 76 6 37 54 dk2 k4 …^vk1 ^vk3 11 dk3 k4 …^vk1 ^vk2

^ i2 ‡ di2 i4 u ^ is †Š ‡ di4 is u

3

4

3 4

3

3 3 11

3

4

3 11

3

4

3

3

3 11 3

3

3

3

3

1

4

1

^ is di3 i4 ei1 i2 is u

7 6 6 ei1 i3 is u ^ is di2 i4 7 7 6 dk1 k2 dk3 k4 1 7 6 7 6 ei1 i4 is u ^ d 76 7 i 0 is i2 i3 7 6 6 7 17 7 54 dk1 k3 dk2 k4 5 ‡ 10 j1 … ic†6 6 ei2 i3 is u ^ is di1 i4 7 7 6 4 dk1 k4 dk2 k3 7 6 7 6 ei i i u 4 2 4 s ^ is di1 i3 5 32

7 6 6 ^ i1 ‡ di1 i4 u ^ is †Š 7 ‡ di4 is u 5 4 1 1 ^ ^ ^ 1 5 …di1 is ui2 ‡ di2 is ui1 ‡ di1 i2 uis †Š

1 ^ 5 …di2 is ui4

1 3 di 1 i 2 †

7 1 7 3 di 2 i 4 † 7 7 1 7 3 di 2 i 3 † 7 7 1 7 3 di 1 i 4 † 7 7 1 7 3 di 1 i 3 † 5

1 3 di 3 i 4 †

3T 2 11

4 7 6 7 6 1 5 4 1 di1 i4 di2 i3

di1 i2 di3 i4

1 ^ ^ 7 …dk1 k2 vk3 vk4

^ i2 u ^ i3 u ^ i4 ic†‰^ ui1 u

^ i2 u ^ is ui1 u ei3 i4 is ‰^

6 ^ i4 u ^ is ic†6 ui1 u 4 ei2 i3 is ‰^

2

 ‰^vk1 ^vk2 ^vk3 ^vk4

‡

ˆ

6

1 6 30 4 di1 i3 di2 i4

^ i4 di1 i2 …^ ui3 u 6 6 di1 i3 …^ ^ i4 ui2 u 6 6 6 di1 i4 …^ ^ i3 ui2 u 6 1 0 14 j2 … ic†6 6 di i …^ ^ i4 6 2 3 ui1 u 6 6 di i …^ ^ i3 4 2 4 ui1 u ^ i2 di3 i4 …^ ui1 u

2

…4†w ^ ; ^v† Ii1 i2 i3 i4 ;k1 k2 k3 k4 …c; u

2

2

B. Partially Ordered and Phase-Weighted Systems 199

201

Appendix 3:

Isotropic Tensors and the Euler Angle Matrix The angular relationship between any two Cartesian reference frames i and k with a common origin can be represented in terms of three Euler angles …u; h; v† (see for example Zare 1988). The Euler angle matrix l comprises the direction cosines between the unit vectors in the two frames, and may be defined as follows: 2

xi

3

6 7 6 yi 7 ˆ 4 5 zi 2 cos u cos h cos v sin u sin v 6 4 cos u cos h cos v sin u sin v 2

xk

3

cos u sin h

sin u cos h cos v ‡ cos u sin v sin u cos h sin v ‡ cos u cos v sin u sin h

6 7 7 6 4 yk 5

3 sin u cos v 7 sin u sin v 5 cos h …A3:1†

zk

The rows and columns of the …3  3† matrix l defined by (A3.1) are orthogonal, i.e. lik lil ˆ dkl ;

…A3:2†

lik ljk ˆ dij ;

…A3:3†

where Latin indices i,j etc. refer to the i-frame, Greek indices k, l etc. to the k-frame, and there is implied summation over repeated indices. The index-symmetric second rank tensor dij is the Kronecker delta whose elements are zero except for dxx ˆ dyy ˆ dzz ˆ 1, and whose products with itself yield 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

202

Appendix 3: Isotropic Tensors and the Euler Angle Matrix

dij djk ˆ djk ;

…A3:4†

dij dij ˆ dii ˆ 3 :

…A3:5†

The orthogonality relationships (A3.2) and (A3.3) may be interpreted as establishing the isotropic nature of the Kronecker delta; for example (A3.2) can be written as dij lik ljl ˆ dkl :

…A3:6†

As such the Kronecker delta is an isotropic tensor, i.e. when re-expressed with reference to a new frame, its elements remain invariant. The Kronecker delta is the only isotropic tensor of rank 2. Elements of the Euler angle matrix also satisfy determinantal relations. These involve the Levi-Civita antisymmetric tensor eijk, whose components take a value of ‡1 if the indices constitute a right-handed cycle, 1 for a left-handed cycle, and zero otherwise: exyz ˆ eyzx ˆ ezxy ˆ

ezyx ˆ

eijk …non-cyclic† ˆ 0 :

eyxz ˆ

exzy ;

…A3:7† …A3:8†

In (A3.8) case two of the indices are necessarily identical, thus establishing that the Levi-Civiata tensor is traceless. This tensor, which is the isotropic tensor of rank 3, is associated with the Euler angle results lik ljl lkm eijk ˆ eklm ;

…A3:9†

lik ljl lkm eklm ˆ eijk :

…A3:10†

For the products of the Levi-Civita tensor with itself we obtain eijk elmn

dil ˆ djl dkl

eijk eimn ˆ djm dkn eijk eijn ˆ 2dkn ;

dim djm dkm

din djn ; dkn

djn dkm ;

…A3:11† …A3:12† …A3:13†

Appendix 3: Isotropic Tensors and the Euler Angle Matrix 203

eijk eijk ˆ 6 :

…A3:14†

Isotropic tensors of even rank n > 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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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).

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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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

240

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