Springer Series in Optical Sciences

Volume 145

Founded byH. K. V. Lotsch

Editor-in-ChiefW. T. Rhodes

Editorial BoardAli Adibi, AtlantaToshimitsu Asakura, SapporoTheodor W. Hänsch, GarchingTakeshi Kamiya, TokyoFerenc Krausz, GarchingBo A. J. Monemar, LinköpingHerbert Venghaus, BerlinHorst Weber, BerlinHarald Weinfurter, München

For further volumes:http://www.springer.com/series/624

Springer Series in Optical Sciences

The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes,Georgia Institute of Technology, USA, provides an expanding selection of research monographs in allmajor areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques,optoelectronics, quantum information, information optics, applied laser technology, industrialapplications, and other topics of contemporary interest.

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Editor-in-ChiefWilliam T. RhodesSchool of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332-0250USAe-mail: bill.rhodes@ece.gatech.edu

Editorial BoardAli AdibiSchool of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332-0250USAe-mail: adibi@ee.gatech.edu

Toshimitsu AsakuraFaculty of EngineeringHokkai-Gakuen University1-1, Minami-26, Nishi 11, Chuo-kuSapporo, Hokkaido 064-0926, Japane-mail: asakura@eli.hokkai-s-u.ac.jp

Theodor W. HänschMax-Planck-Institut für QuantenoptikHans-Kopfermann-Straße 185748 Garching, Germanye-mail: t.w.haensch@physik.uni-muenchen.de

Takeshi KamiyaMinistry of Education, Culture, Sports,Science and TechnologyNational Institution for Academic Degrees3-29-1 Otsuka Bunkyo-kuTokyo 112-0012, Japane-mail: kamiyatk@niad.ac.jp

Ferenc KrauszLudwig-Maximilians-Universität MünchenLehrstuhl für Experimentelle PhysikAm Coulombwall 185748 Garching, Germany andMax-Planck-Institut für QuantenoptikHans-Kopfermann-Straße 185748 Garching, Germanye-mail: ferenc.krausz@mpq.mpg.de

Bo A. J. MonemarDepartment of Physics and Measurement TechnologyMaterials Science DivisionLinköping University58183 Linköping, Swedene-mail: bom@ifm.liu.se

Herbert VenghausFraunhofer Institut für NachrichtentechnikHeinrich-Hertz-InstitutEinsteinufer 3710587 Berlin, Germanye-mail: venghaus@hhi.fraunhofer.de

Horst WeberOptisches InstitutTechnische Universität BerlinStraße des 17. Juni 13510623 Berlin, Germanye-mail: weber@physik.tu-berlin.de

Harald WeinfurterSektion PhysikLudwig-Maximilians-Universität MünchenSchellingstraße 4/III80799 München, Germanye-mail: harald.weinfurter@physik.uni-muenchen.de

Tom Rother • Michael Kahnert

ElectromagneticWave Scattering onNonspherical Particles

Basic Methodology and Simulations

Second Edition

123

Tom RotherInst. Methodik der FernerkundungDeutsches Zentrum für Luft- und

Raumfahrt (DLR)NeustrelitzGermany

Michael KahnertSwedish Meteorological and

Hydrological InstituteNorrköpingSweden

ISSN 0342-4111 ISSN 1556-1534 (electronic)ISBN 978-3-642-36744-1 ISBN 978-3-642-36745-8 (eBook)DOI 10.1007/978-3-642-36745-8Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013937605

� Springer-Verlag Berlin Heidelberg 2014This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publisher’s location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

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Springer is part of Springer Science+Business Media (www.springer.com)

Additional material to this book can be downloaded from http://extras.springer.com.

The photo shows a 22-degree Halo phenomenon with sun dogs. It was taken by the author inMarch 1998 at the DLR site in Neustrelitz. This phenomenon is caused by light scattering onhexagonal ice columns (the 22-degree Halo) and ice plates (the sun dogs) in Cirrus clouds

Preface to the Second Edition

Advantage has been taken in the preparation of the second edition of this book toeliminate several errors and misprints as well as to add a chapter on group theory.The importance of such group theoretical considerations has been proven in recentapplications for particles with discrete boundary symmetries. To demonstrate thisbenefit we included an additional subsection in Chap. 9 where we consider lightscattering on Chebyshev particles of higher order. The reference chapter wasupdated accordingly.

The simulation software which was provided on a CD-ROM with the firstedition can now be obtained via download from Springer’s web site or on requestfrom the authors. Besides the program mieschka and the scattering database of thefirst edition, the program Tsym for non-axisymmetric particles is also included. Itsusage is described in detail in Chap. 9 of this second edition. Heikki Laitinen, KariLumme, Dan Mackowski, Michael Mishchenko, Karri Muinonen, and TimoNousiainen are gratefully acknowledged for their kind permission to includevarious Fortran routines from their respective programs in the distribution of Tsym.

Neustrelitz, Germany T. RotherNorrköping, Sweden M. Kahnert

vii

Preface to the First Edition

Scattering of electromagnetic waves on three-dimensional, dielectric structures is abasic interaction process in physics, which is also of great practical importance.Most of our visual impressions are caused not by direct but by scattered light, aseverybody can experience by looking directly at the sun. Several modern mea-surement technologies in technical and medical diagnostics are also based on thisinteraction process. Atmospheric remote sensing with lidar and radar as well asnephelometer instruments for measuring suspended particulates in a liquid or gascolloid are only a few examples where scattered electromagnetic waves provide uswith information concerning the structure and consistence of the objects underconsideration. Using the information of the elastically scattered electromagneticwave is a common ground for most of those measuring methods. The phrase‘‘elastically scattered’’ expresses the restriction that we consider such interactionprocesses only where the scattered wave possesses the same wavelength as theprimary incident wave. This book addresses this special scattering problem.

The methodology part of this book is concerned with the solution of the partialdifferential equations underlying this scattering process. These are especially thescalar Helmholtz equation and the vector-wave equation. From the mathematicalpoint of view, we are faced with the solution of boundary value problems. Thisbecomes especially simple if the boundary values are given along a constantcoordinate line in one of the coordinate systems that allow a separation of thepartial differential equation. Such problems are sometimes called ‘‘separableboundary value problems’’ in the literature. The applied method is the so-called‘‘Separation of Variables method’’. It reduces the primary partial differentialequation to a set of ordinary differential equations whose eigensolutions serveafterwards as expansion functions to approximate the sought solution appropri-ately. Scattering of light on dielectric and ideal metallic spheres was first solved bythis method in 1908 by Gustav Mie. The Mie theory, as it is called nowadays,forms the basis in many applications even today.

On the other hand, not the least due to the possibilities of modern computers,there can be observed a growing interest in modelling more and more realisticscattering scenarios which goes beyond the conventional Mie theory for sphericalscatterers. If the geometry of the scatterer differs only slightly from that of aseparable geometry simpler perturbation methods may be applied successfully. But

ix

more often the deviation from such a separable geometry is much stronger so thatwe are forced to apply more rigorous numerical methods to solve the problem.A large variety of such methods have been developed in the past. Differing inconcept and execution these methods start from the common assumption that it isno longer possible to apply the Separation of Variables method. A critical dis-cussion of this assumption is a major objective of this book. The Green functionsare of special importance for this discussion. Based on these functions, we willshow that there can be established a formalism which provides a common meth-odological background for a variety of different numerical methods. But theso-called ‘‘T-matrix’’ method is within the focus of our interest. This specialsolution method was developed by Waterman at the end of the 1960s and thebeginning of the 1970s of the twentieth century. It has been proved to be verysuccessful in many applications.

This book at hand is restricted to the relatively simple case of modellingelectromagnetic wave scattering on single, homogeneous, but nonspherical parti-cles in spherical coordinates. Nevertheless, it provides a sound basis to develop themethods for more complex situations as we have, if scattering on an ensemble ofobjects or on inhomogeneous objects (like multilayered particles, for example) isconsidered.

The basic methodological considerations of this book are complemented in thepenultimate chapter with numerical simulations of a few typical scattering sce-narios. For these simulations we have developed a specific T-matrix code whichcan be found in the enclosed CD. Its structure and usage is presented in detail. Thisprogram will enable the reader who is more interested in tangible calculations, toperform quickly his own numerical simulations, and, most important, to estimatethe accuracy and usefulness of the obtained results. This software can also be usedin university lectures to demonstrate the principal aspects of light scattering. Someparts of this book began as notes of a special lecture given by the author at theMeteorological Institute of the University Leipzig. The considerations in thepenultimate chapter should reveal some of the numerical problems which must besolved if one tries to develop a certain T-matrix code.

Naturally, this book is not the result of only my efforts. For the multitude of(partly very heavily) discussions over the years I sincerely thank my two col-leagues at the Remote Sensing Technology Institute, Dr. K. Schmidt andDr. J. Wauer. Special thanks are due to Dr. J. Wauer for his numerical work on theRayleigh hypothesis as discussed in Chap. 6, and for his work on the softwarepackage mieschka and the database. Special thanks are due also to the GermanAerospace Center (DLR) for the confidence in my work over the years, and for thefinancial as well as administrative support which cannot be taken for grantednowadays.

C. Ascheron at Springer deserves a word of thanks for his continuous interest,support, and encouragement after the manuscript appeared via email for the firsttime suddenly in his office.

x Preface to the First Edition

My gratitude is deepest, however, to my parents, who supported me in manifoldways and pushed me into the direction of science, and to my wife Doreen,who helped me without complaint through the nearly 7 years of writing this book.She also carefully read the manuscript and wiped out a lot of needless‘‘m’’-dashes—thus demonstrating that a musician and a physicist can really benefitfrom each other.

Neustrelitz, Germany, June 2009 T. Rother

Preface to the First Edition xi

Contents

1 Scattering as a Boundary Value Problem . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Formulation of the Boundary Value Problems. . . . . . . . . . . . . . 51.3 Solving the Boundary Value Problems with the

Rayleigh Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 The Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 101.3.2 The Outer Transmission Problem . . . . . . . . . . . . . . . . . 12

2 Filling the Mathematical Tool Box . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Approximation of Functions and Fields

at the Scatterer Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Approximation by Finite Series Expansions . . . . . . . . . . 192.2.2 Best Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 The Transformation Character of the T-Matrix . . . . . . . . 24

2.3 Eigensolutions of the Scalar Helmholtz Equationin Spherical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 The Eigensolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 The Combined Summation Index . . . . . . . . . . . . . . . . . 362.3.3 Properties of the Scalar Eigensolutions . . . . . . . . . . . . . 372.3.4 Expansion of a Scalar Plane Wave . . . . . . . . . . . . . . . . 39

2.4 Eigensolutions of the Vector-Wave Equation in SphericalCoordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.1 The Vectorial Eigensolutions . . . . . . . . . . . . . . . . . . . . 422.4.2 Properties of the Vectorial Eigensolutions . . . . . . . . . . . 472.4.3 Expansion of a Linearly Polarized Plane Wave. . . . . . . . 52

2.5 Green Theorems and Green Functions Related to the ScalarBoundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5.1 The Green Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 602.5.2 The Free-Space Green Function . . . . . . . . . . . . . . . . . . 612.5.3 The Green Functions Related to the Outer Dirichlet

and Transmission Problem . . . . . . . . . . . . . . . . . . . . . . 67

xiii

2.6 Green Theorems and Green Functions Related to the VectorialBoundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.6.1 Dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.6.2 The Green Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 722.6.3 The Dyadic Free-Space Green Function. . . . . . . . . . . . . 732.6.4 The Dyadic Green Functions Related

to the Outer Dirichlet and Transmission Problem . . . . . . 77

3 First Approach to the Green Functions: The Rayleigh Method . . . 813.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.2 The Scalar Delta Distribution at the Scatterer Surface . . . . . . . . 833.3 The Scalar Green Functions Related

to the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3.1 The Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 843.3.2 The Outer Transmission Problem . . . . . . . . . . . . . . . . . 92

3.4 The Dyadic Delta Distribution at the Scatterer Surface . . . . . . . 943.5 The Dyadic Green Functions Related to the

Vector-Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.5.1 The Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 953.5.2 The Outer Transmission Problem . . . . . . . . . . . . . . . . . 102

4 Second Approach to the Green Functions:The Self-Consistent Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2 The Scalar Green Functions Related to the Helmholtz

Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.2.1 The Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 1064.2.2 The Outer Transmission Problem . . . . . . . . . . . . . . . . . 109

4.3 The Dyadic Green Functions Related to the Vector-WaveEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.3.1 The Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 1114.3.2 The Outer Transmission Problem . . . . . . . . . . . . . . . . . 113

4.4 Symmetry and Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.4.2 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Other Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 T-Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2.1 The Extended Boundary Condition Method . . . . . . . . . . 1305.2.2 Point Matching Methods . . . . . . . . . . . . . . . . . . . . . . . 136

5.3 The Method of Lines as a Special Finite-Difference Method . . . 1375.3.1 Discretization of the Scalar Helmholtz Equation

and its Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.3.2 The Limiting Behaviour of the Method of Lines. . . . . . . 146

xiv Contents

5.4 Integral Equation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.4.1 Boundary Integral Equation Method Related

to the Outer Dirichlet Problem . . . . . . . . . . . . . . . . . . . 1535.4.2 Boundary Integral Equation Method Related

to the Outer Transmission Problem . . . . . . . . . . . . . . . . 1585.4.3 Volume Integral Equation Method Related

to the Outer Transmission Problem . . . . . . . . . . . . . . . . 1605.5 Lippmann-Schwinger Equations . . . . . . . . . . . . . . . . . . . . . . . 164

5.5.1 The Scalar Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.5.2 The Dyadic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6 The Rayleigh Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.2 Plane Wave Scattering on Periodic Gratings . . . . . . . . . . . . . . . 174

6.2.1 Conventional Formulation of the Scattering Problem. . . . 1746.2.2 Formulation in Terms of Green Functions . . . . . . . . . . . 1776.2.3 T-Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.2.4 Rayleigh’s Hypothesis According to Petit, Cadilhac,

and Millar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.2.5 Rayleigh’s Hypothesis According to Lippmann,

and a Corresponding Boundary Integral Solution . . . . . . 1886.3 Numerical Near-Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1916.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Physical Basics of Electromagnetic Wave Scattering . . . . . . . . . . . 2037.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.2 The Electromagnetic Scattering Problems . . . . . . . . . . . . . . . . 204

7.2.1 Maxwell’s Equations and Boundary Conditions . . . . . . . 2047.2.2 Vector Wave Equations of the Electromagnetic Fields . . . 2087.2.3 Helmholtz Equation of the Debye Potentials

and Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2147.3 The Far-Field and the Scattering Quantities . . . . . . . . . . . . . . . 221

7.3.1 The Far-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.3.2 Definition of Scattering Quantities . . . . . . . . . . . . . . . . 230

7.4 Scalability of the Scattering Problem . . . . . . . . . . . . . . . . . . . . 237

8 Scattering on Particles with Discrete Symmetries . . . . . . . . . . . . . 2418.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

8.1.1 Symmetry Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 2418.2 Explicit Commutation Relations of the T-Matrix . . . . . . . . . . . . 251

8.2.1 Unitary Representations of Symmetry Operations . . . . . . 2518.2.2 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . 2618.2.3 Simplification of Scalar Products . . . . . . . . . . . . . . . . . 263

Contents xv

8.3 Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2658.3.1 Groups and Generators . . . . . . . . . . . . . . . . . . . . . . . . 2658.3.2 Conjugate Elements and Classes . . . . . . . . . . . . . . . . . . 2688.3.3 Linear Representations of a Group . . . . . . . . . . . . . . . . 269

8.4 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2708.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2738.4.3 The Great Orthogonality Theorem. . . . . . . . . . . . . . . . . 2778.4.4 Projection Operators into the Invariant Subspaces . . . . . . 2778.4.5 Construction of the Basis Transformation

from the Reducible to the Irreducible Basis . . . . . . . . . . 280

9 Numerical Simulations of Scattering Experiments . . . . . . . . . . . . . 2879.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.2 Numerical Simulations with mieschka . . . . . . . . . . . . . . . . . . . 288

9.2.1 Single Particles in Fixed Orientations . . . . . . . . . . . . . . 2889.2.2 Single Particles in Random Orientation . . . . . . . . . . . . . 3029.2.3 Database for Spheroidal Particles and Size Averaging . . . 3139.2.4 Morphology Dependent Resonances . . . . . . . . . . . . . . . 315

9.3 High-Order Chebyshev Particles . . . . . . . . . . . . . . . . . . . . . . . 3199.3.1 Perturbation Approach of the T-Matrix . . . . . . . . . . . . . 3209.3.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

9.4 Description of Program mieschka . . . . . . . . . . . . . . . . . . . . . . 3249.4.1 Convergence Strategy Used in mieschka . . . . . . . . . . . . 3259.4.2 Functionality and Usage of mieschka. . . . . . . . . . . . . . . 3299.4.3 Content of the Software Package . . . . . . . . . . . . . . . . . 335

9.5 Description of Program Tsym . . . . . . . . . . . . . . . . . . . . . . . . . 3369.5.1 Compilation and Input Parameters. . . . . . . . . . . . . . . . . 3379.5.2 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 3399.5.3 Result Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3409.5.4 Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

10 Recommended Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34510.1 Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34510.2 Mathematical Aspects of Scattering . . . . . . . . . . . . . . . . . . . . 34610.3 Green Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34710.4 T-Matrix Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34810.5 Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34910.6 Integral Equation Methods and Singularities . . . . . . . . . . . . . . 35010.7 Rayleigh’s Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35110.8 Electromagnetic Wave Theory. . . . . . . . . . . . . . . . . . . . . . . . 35210.9 Scattering of Electromagnetic Waves and Applications. . . . . . . 352

xvi Contents

10.10 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35410.11 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Contents xvii