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Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces PDF

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MODER N PROBLEM S IN CONDENSE D MATTE R SCIENCE S Series editors V.M. AGRANOVIC H Moscow, USSR A.A. MARADUDI N Irvine, California, USA Advisory editorial board F. Abeles, Paris, France N. Bloembergen, Cambridge, MA, USA E. Burstein, Philadelphia, PA, USA I.L. Fabelinskii, Moscow, USSR M.D. Galanin, Moscow, USSR V.L. Ginzburg, Moscow, USSR H. Haken, Stuttgart, W. Germany R.M. Hochstrasser, Philadelphia, PA, USA LP. Ipatova, Leningrad, USSR A.A. Kaplyanskii, Leningrad, USSR L.V. Keldysh, Moscow, USSR R. Kubo, Tokyo, Japan I.M. Lifshitz, Moscow, USSR R. Loudon, Colchester, UK A.M. Prokhorov, Moscow, USSR K.K. Rebane, Tallinn, USSR NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM · NEW YORK · OXFOR D SURFAC E POLARITON S Electromagneti c Waves at Surface s and Interface s Volume editors V. M. AGRANOVIC H Moscow, USSR D. L. MILL S Irvine California, USA 1982 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK · OXFORD © North-Hollan d Publishing Company, 1982 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN 0444 86165 3 PUBLISHERS: NORTH-HOLLAN D PUBLISHING COMPANY AMSTERDAM · NEW YORK OXFORD SOLE DISTRIBUTORS FOR THE USA AND CANADA: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 Librar y of Congress Cataloging in Publication Data Main entry under title: Surface polaritons. (Modern problems in condensed matter sciences) Includes bibliographie s and indexes. 1. Polaritons. I. Agranovich, V. M. (Vladimir Moiseevich), 1929- . II. Mills, D. L. III. Series. QC174. 8. P6S93 530.Ã41 81-224% ISBN 0-444-86165-3 (Elsevier AACR2 North-Holland ) PRINTED IN THE NETHERLAND S iv Other Volumes in this Series: EXCITONS E.I. Rashba and M.D. Sturge, editors ELECTRONIC EXCITATION ENERGY TRANSFER IN CONDENSED MATTER V.M. Agranovich and M.D. Galanin SPECTROSCOPY AND EXCITATION DYNAMIC S IN CONDENSED MOLECULAR SYSTEMS V.M. Agranovich and R.M. Hochstrasser, editors Oh, how many of them there are in the fields! But each flowers in its own way — In this is the highest achievement of a flower! Matsuo Basho 1644-1694 PREFAC E TO THE SERIE S "Surf ace Polaritons" is the first volume in a series of contributed volumes and monographs on condensed matter science that the North-Holland Publishing Company is now beginning to publish. This vast area of physics is developing rapidly at the present time, and the numerous fundamental results in it define to a significant degree the face of contemporary science. This being so, it is clear that the most important results and directions for future developments can only be covered by an international group of authors working in cooperation. Both Soviet and Western scholars are taking part in the series, and each contributed volume has, correspondingly, two editors. Furthermore, it is intended that the volumes in the series will be published subsequently in Russian by the publishing house "Nauka". The idea for the series and for its present structure was born during discussions that took place in the USSR and the USA between the President of North-Holland Publishing Company, Drs. W.H. Wimmers, and the General Editors. The establishment of this series of books, which should become a distinguished encyclopedia of condensed matter science, is not the only important outcome of these discussions. A significant development is also the emergence of a rather interesting and fruitful form of collaboration among scholars from different countries. We are deeply convinced that such international collaboration in the spheres of science and art, as well as other socially useful spheres of human activity, will assist in the establishment of a climate of confidence and peace. The General Editors of the Series, V.M. Agranovich A.A. Maradudin vi PREFAC E The purpose of this volume is to present a sequence of articles which describe the basic properties of surface polaritons, methods of generating these waves in the laboratory at frequencies of interest to condensed matter physicists, and then the physics that may be learned from them. In our view, the collection provides an excellent summary of the activity that resulted from the recent interest in these modes in the solid state physics community. A surface polariton is simply an electromagnetic wave that propagates along the surface of a medium, or along the interface between two media. The strength of the electromagnetic fields associated with the wave decay in strength exponentially as one moves away from the interface into either medium, but they vary in a wavelike fashion as one moves parallel to it. Despite the exotic name, upon which we comment in more detail below, if both media may be described in a continuum approximation by a dielectric tensor and a magnetic permeability tensor (both are frequency dependent and complex, in general), then these waves emerge as solutions of Max- well's equations applied to the pair of media. The chapters in this volume explore a large number of realizable configurations that support pro- pagation of these modes. Despite the fact that the basic properties of surface polaritons may be obtained directly from Maxwell's equations, most physicists are not famil- iar with them. This is curious, because they have been studied actively since the turn of the present century, when Sommerfeld and his students discussed them within a certain context (Joos 1934). Interest in the waves has arisen from time to time in many subfields of physics and engineering, where in each instance they have attracted the attention of specialists in the area, but not of the general physics community. Largely because of the recent burst of activity in condensed matter physics, and especially in physics of surfaces, we now appreciate the ubiquitous nature of surface polaritons. We hope this volume will show the condensed matter physics community how surface polaritons may be used to probe the near vicinity of the crystal surface or of an interface under a variety of circumstances, in other words, how they may be used to facilitate the development of the spectroscopy of surfaces. The interaction of electromagnetic waves with an interface between two vii viii Preface media is discussed in nearly all textbooks on electromagnetic theory when a derivation is given of Fresnel formulae for the amplitude of waves reflected from or transmitted through an interface. Yet surface polaritons are almost never treated (Portis 1978), even though they emerge as solu- tions of Maxwell's equations applied to the same geometry. The reason for this may possibly be appreciated by considering a plane interface between an isotropic dielectric with dielectric constant €, and vacuum. As the reader will appreciate upon examining the discussion in several of the chapters (see, for example, the opening chapter by D. N. Mirlin) for the surface polariton to exist, the requirement is that €, assumed real here for sim- plicity, be a negative number. More specifically, the inequality e ^ -1 must be satisfied for this particular example. Most textbooks on electromagnetic theory assume, either implicitly or explicitly, that the dielectric constant of matter is always positive when the interaction of electromagnetic waves with dielectrics is examined. The static dielectric constant of any insulating solid is, ignoring spatial dispersion (i.e. for wave vectors k = 0, see Agranovich and Ginzburg 1981) required to be positive, from stability considerations in thermodynamics applied to nonconducting media (Callen 1960). No such constraint applies to the finite frequency dielectric function e(d) that enters the theory of electromagnetic wave propagation. In fact, as the sequence of chapters in this volume illustrate very nicely, in a wide class of solid systems one may have negative values for e(O) over substantial ranges of frequency. Con- sider, for example, the nearly free electron picture of simple metals. Then if Ù is the electron plasma frequency, we have €(Ù)=1-Ùñ/Ù 2 (with ñ scattering of the electrons ignored), so that the inequality €(Ù)^- 1 holds for all frequencies below Ù /\/2. For aluminum, Ù /\/2 is 10.6 eV, so on ñ ñ the surface of aluminum, surface polaritons may propagate at all frequen- cies from the microwave through the visible and well into the ultraviolet. Similar statements apply to a variety of the simple metals. This example shows that surface polaritons are not strange solutions of Maxwell's equations encountered under exotic conditions, but in fact they are rather commonplace. As several chapters explain, a photon which strikes a perfectly plane, smooth interface between a dielectric and vacuum fails to "see" or interact with surface polaritons on the interface. But if a suitable prism is placed nearby, if roughness is present or a grating is ruled on the surface, one may easily "drive" the surface polariton by linear coupling to the incoming photon. A variety of other methods of excitation are also discussed in the chapters, including the use of nonlinear interactions which allow the generation of surface polaritons on perfectly smooth surfaces or interfaces. In an isotropic dielectric, we have D = Å + 4ðÑ = â(Ù)Å. For the dielec- tric constant to be negative, the external electric field Å must excite a Preface ix polarization density Ñ 180° out of phase with the exciting field E. If the medium has any sharp absorption line at frequency Ù , the excitation of 0 the medium at a frequency just above Ù will result in a negative con- 0 tribution to Ñ that can be very large, if the absorption line is sharp. That this is so, quite generally is insured by the Kramers-Kronig relations. In more elementary language, if a very lightly damped harmonic oscillator is driven just above its resonance frequency, then the response not only has large amplitude, but is 180° out of phase with the driving field. The induced polarization is thus directed anti-parallel to the electric field that excites it. Hence, the condition for surface polariton propagation is realized at the smooth planar interface between vacuum and an isotropic dielectric almost always just above an absorption line, in spectral regions where narrow lines exist. Thus, just above the reststrahl frequency of simple ionic crystals, we encounter the "surface phonon polaritons" discussed by D. N. Mirlin and others. "Surface exciton polaritons" are formed just above sharp exciton absorption lines, as discussed by J. Lagois and B. Fischer. The additional modifier refers to the elementary excitation in the crystal whose contribution to the dipole moment density Ñ drives the dielectric constant negative. We are now in a position to understand why the surface waves are referred to commonly as surface polaritons, in the condensed matter physics literature. Near sharp absorption bands such as those just des- cribed, å(Ù) is a strong function of frequency both above and below Ù . 0 When å(Ù) is positive, just below Ù and sufficiently far above it, Max- 0 well's equations admit plane wave solutions in the bulk of the material. In the frequency regime where €(Ù ) displays such sharp resonant structure near an absorption line, one may describe the electromagnetic wave as a coupled mode which is an admixture of the electromagnetic field and the elementary excitation in the medium that produces the resonance in â(Ù ) (Mills and Burstein 1974). Such electromagnetic waves, when described within this picture, are commonly called polaritons in the literature of condensed matter physics. Hence, the surface electromagnetic waves of interest to the authors in the present volume are frequently referred to as surface polaritons. As remarked earlier, surface electromagnetic waves have been studied in various subfields of physics, and within each, a specialized terminology for them has evolved. This is true within condensed matter physics, as well as within physics and engineering as a whole. This generated a problem for the editors of the present volume, in that the set of authors has been drawn from a broad spectrum of areas in condensed matter physics. Not all authors used the same terminology, and after initial correspondence and conversation, it became clear that some felt the particular choice made by them had historical precedence over other possibilities. The arguments in ÷ Preface each case seem sensible, but it proved difficult to obtain agreement on the terminology among the diverse array of authors who have contributed to the volume. Thus, A. A. Maradudin prefers to use distinctly different language to discuss waves for which the retardation terms in Maxwell's equations enter in an essential fashion, and those for which these may be ignored and an electrostatic description is sufficient. He uses the term surface polariton to describe surface electromagnetic waves of the first class, and with the metal/vacuum interface in mind, the solutions found in the electrostatic limit are described as surface plasmons. In fact, for a smooth surface, the dispersion relation of the surface electromagnetic wave is continuous, with retardation important when the frequency Ù of the wave and its wave vector parallel to the surface Q\\ satisfy Ù = cQ\\ and 9 the electrostatic limit appropriate when Q\\ is so large that cQ\\ > Ù. Here c is the velocity of light in vacuum. Thus, the terminology used by A. A. Maradudin, not at all uncommon in the theoretical literature, uses different terms to denote quasi-particles obtained in different approximations. In the chapter by V. M. Agranovich, we see the term "Coulomb wave" used to denote modes that were found when retardation is disregarded. H. Raether, along with J. Sipe and G. Stegeman use the term surface plasmon to refer to the surface electromagnetic wave on metals for all values of Ù and Qj, including the region where retardation is important. Thus, these authors use the term surface plasmon in the same manner as we employ surface polariton in the present discussion. We hope the lack of a uniform terminology will not render the volume confusing to the reader not familiar with the field; it should be clear that the same reader will encounter a similar array of terms if he or she consults the general literature, so it may be that by comparing the material in the various chapters within one set of covers, the reader can become acquain- ted with the range of descriptive terminology used in the field. The present volume confines its attention to the elucidation of the basic properties of surface polaritons, the methods of generating them in the laboratory, their use as probes of the surface or interface environment and finally to the nonlinear interactions in which they participate. These topics are covered in depth, and we are particularly pleased to see that a large volume of actual experimental data is reproduced in the figures and tables. Space limitations then require us to omit material on more specialized or less developed topics. There is no discussion of the properties of surface polaritons on magnetic crystals; one of the editors will prepare a chapter which includes discussion of such modes for another volume in the present series (Agranovich and Loudon). Surface spectroscopy and, in particular, spectroscopy of surface polari- tons continues its rapid development. In this connection, we shall mention new results that were obtained after this volume had been prepared for Preface xi publication. These concern nonlinear surface polaritons (NSP) and waves of self-induced transparency (SIT) on surfaces. Previously, when we discussed surface polaritons, surface plasmons, etc., we were concerned with solutions of linear Maxwell equations. As is known, nonlinearity appears in these equations when in the relation D = eE the dependence of e on the strength of the electric field is taken into account, and not only its dependence on the frequency and wave vector (the latter corresponds to the taking of spatial dispersion into account, see Agranovich and Ginzburg (1981)). In the simplest case, i.e., in taking the dependence of e on only the magnitude of \E\ into consideration, e = €(Ù) + OL\E\2. If a >0, this nonlinearity leads to self-focusing and to other nonlinear effects. In the papers (Tomlinson 1980, Agranovich et al. 1980, Maradudin 1981) it was established that this same nonlinearity leads to the occurrence of NSP, that those waves can exist in spectral regions in which no linear waves are formed, and that the frequency of these waves depends on the strength of the electric field at the surface. In particular, it was shown by Tomlinson (1980) and Maradudin (1981) that at the interface between linear and nonlinear isotropic media NSP exist only in s-polarization. If, however, the nonlinear medium is optically anisotropic (uniaxial), then, as shown by Agranovich et al. (1980), NSP are formed in p-polarization as well (recall that at the interface of the linear media SP are formed only in p- polarization). The frequency of NSP in p-polarization depends on the wave vector so that it is precisely in this case that NSP have nonzero group velocity Üù/dk. This feature of NSP in p-polarization is important because the waves being discussed can exist, in particular, in the transparency spectral region, according to theory. In virtue of the fact that da>/dk7*0, these waves may have considerable lengths of propagation, even along dielectric surfaces on which linear SP are usually quite rapidly damped. In connection with the aforesaid, the search for methods of exciting and detecting NSP would be of especial interest, as would be an analysis of the feasibility of employing NSP for developing new optical methods of investigating surfaces and thin films. An entirely different type of nonlinear surface waves occurs in the propagation of ultrashort pulses of intensive and resonant radiation along a surface coated with a thin film of a resonant substance. In this case, as shown by Agranovich et al (1981a), surface "27r" pulses of self-induced transparency are found to be stable. The investigation of this problem is only in an early stage (a similar effect in waveguides is discussed in the paper by Agranovich et al. (1981b)) and only the simplest situations have been studied. In particular, there is no rigorous answer to the question of whether stable surface "27r" pulses exist or not under conditions when the whole mass of the dielectric is resonant, rather than a thin film on a

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