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Spheroidal WaveFunctions in

Electromagnetic Theory

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Spheroidal WaveFunctions in

Electromagnetic Theory

Le-Wei Li

Xiao-Kang Kang

Mook-Seng Leong

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC.

This text is printed on acid-free paper. ©

Copyright © 2002 by John Wiley & Sons, Inc., New York. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise,except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of theappropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should beaddressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York,NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ @ WILEY.COM.

For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging-in-Publication Data.

Li, Le-Wei.Spheroidal wave functions in electromagnetic theory / Le-Wei Li, Xiao-Kang Kang,

and Mook-Seng Leong.p. cm. — (Wiley series in microwave and optical engineering)

ISBN 0-471-03170-4 (cloth : alk. paper)1. Electromagnetic theory. 2. Spheroidal functions. I. Kang, Xiao-Kang. II. Leong,

Mook-Seng. I I I . Title. IV. Series.

QC670.L492001530.14'1—dc21 2001045399

Printed in the United States of America

1 0 9 8 7 6 5 4 3 2 1

Preface

Spheroidal coordinates and spheroidal wave functions have found many im-portant and practical applications in electromagnetic theory, such as antennaanalysis and design, MIC design, EMC and EMI, and radar cross sectionscalculations. There are several hot topics in computational electromagneticsrelated to spheroidal structures. For instance, rockets, aircraft noses, andguided missiles are generally considered to have spheroidal shapes. Humanheads can be modeled approximately as prolate spheroids in the calculation ofelectromagnetic interaction between a head and a cellular phone. Raindropscan be modeled as oblate spheroids in computation of the rainfall attenuationof microwave signals in line-of-sight and satellite telecommunication systems.The theory of spheroidal wave functions was developed about half a centuryago. However, applications of these functions were not widely made in var-ious research areas involving electromagnetics. Apparently, there are manyfar fewer reports in the literature on full-wave analysis of electromagneticscattering and radiation in spheroidal coordinates than those in other coor-dinate systems, for instance, cylindrical and spherical coordinates. One ofthe difficulties is the non-existence of orthogonality among spheroidal wavefunctions. The prolate and oblate spheroidal coordinates are two systems inwhich the scalar wave equations are separable but the vector wave functionsare not separable. This therefore causes the difficulty of obtaining rigoroussolutions of those vector boundary value problems. Another challenge is thevery complicated calculation of the spheroidal harmonics, especially for those

v

vi PREFACE

spheroids with large focal distances and complex dielectric properties in thehigh-frequency regions.

This book presents in detail the theory of spheroidal wave functions, ap-plies it widely to the analysis of electromagnetic fields in various spheroidalstructures, and provides a comprehensive programming code for those compu-tations. In addition to the theory to be detailed and applications to be coveredwidely, several Mathematica package source codes are developed for the cal-culation of spheroidal wave functions and for their practical applications ofthose functions in electromagnetic theory.

The following topics are covered in the this book: (1) theory of the eigen-value problem, spheroidal harmonics, and spheroidal vector wave functions;(2) electromagnetic scattering by a conducting spheroid and a dielectric-coated conducting spheroid; (3) electromagnetic radiation from a conductingspheroidal dipole antenna, a dielectric-coated conducting spheroidal dipoleantenna, and a dielectric-coated conducting spheroidal dipole antenna cov-ered with a dielectric spheroidal radome shell; (4) dyadic Green's functionsin single- and multi-layered spheroidal structures; and (5) the specific absorp-tion rate (SAR) of power due to dipole radiation into a multilayered spheroidalhuman head. A Mathematica software package for computing spheroidal func-tions is developed on the basis for purposes of comparative study and pro-vided partially on the Web site (http://www.wolfram.com/mathsource) andcompletely on the Web site (http://www.ece.nus.edu.sg/lwli).

Proposed readers of this book are those scientists, engineers, and graduatestudents in electromagnetics, applied physics, or applied mathematics whoseresearch work involves spheroidal coordinates or models. It is hoped that theapproach of this book will help readers to understand theory and techniquesused for prolate and spheroidal structures and other nonorthogonal systems.It is, then, necessary that readers take full account of the theory and the soft-ware routine packages in this book, if they wish to find further solutions ofcomplicated structures by using the attached source codes. Those scientistsand engineers interested in some of the applications, such as SAR Distribu-tions in a Spheroidal Head Model and Rain Attenuation of Oblate SpheroidalRaindrop, can also use the numerical values there as further references.

LE-WEI LIXIAO-KANG KANG

MOOK-SENG LEONG

National University of Singapore

Acknowledgments

This book is a part of research results of a Joint Research Project sponsoredby National University of Singapore and DSO National Laboratories, Singa-pore. In this connection, the authors wish to acknowledge DSO National Labsfor the financial support and especially our colleague, Mr. Yeow-Beng Gan atTemasak Laboratory at the National University of Singapore, for his strongencouragement and help. Thanks also go to our other colleagues, ProfessorPang-Shyan Kooi and Professor Tat-Soon Yeo of the Department of Electri-cal and Computer Engineering at the National University of Singapore, whoshowed their encouragement and support in one way or another. The firstauthor also wishes to express acknowledgment to Dr. Paul C. Abbott ofthe Department of Physics at University of Western Australia for his usefuldiscussions and suggestions during the preparation of the book.

The authors would also like to acknowledge our graduate students who pro-vided assistance in the software development of Mathematica packages, someof the numerical results in Chapters 4, 5 and 6 of this book, and proofreading,including Mr. Kian-Yong Tan, Mr. Chia-Heng Quek, Mr. Phin-Juay Loh,and Mr. Zhong-Cheng Li.

We would like to extend our gratitude to the Wiley-Interscience Series Edi-tor: Professor Kai Chang at Texas A&M University for his encouragement, tothe technical reviewers: Professor Qing-Huo Liu at Duke University and Pro-fessor Jian-Ming Jin at University of Illinois at Urbana-Champaign for theirinvaluable suggestions and strong recommendations, to the Executive Editorof Wiley-Interscience and the Managing Editors of Wiley STM Production:

vii

viii ACKNOWLEDGMENTS

George J. Telecki, Andrew Prince and Angioline Loredo for their interest inthe publication and careful copy-editing of this book, and to the EditorialAssistant of Wiley-Interscience: Sara A. Paracka for her professional help.

Finally, taking this opportunity, the authors wish to acknowledge our belovedwives and children for their understanding, encouragement, and support inone way and another.

LW Li, XK Kang, and MS Leong

Contents

Preface v

Acknowledgments vii

1 Introduction 11.1 Overview 11.2 EM Scattering by Spheroids 41.3 Spheroidal Antenna 51.4 EM Radiation in Dielectric Spheroids 71.5 Oblate Spheroidal Models 81.6 Spheroidal Cavity System 91.7 Spheroidal Harmonics and Mathematica Software 10

2 Spheroidal Coordinates and Wave Functions 132.1 Spheroidal Coordinate Systems 132.2 Spheroidal Scalar Wave Functions 172.3 Spheroidal Angular Harmonics 18

2.3.1 Series Representation in Terms ofAssociated Legendre Functions 18

2.3.2 Power Series Representation 202.4 Eigenvalues mn and Expansion Coefficients dmn

r 22

ix

x CONTENTS

2.4.1 Case I: |c|2 < 1000 232.4.2 Case II: |c|2 > 1000 26

2.5 Spheroidal Radial Harmonics 272.5.1 Series Representation in Terms of

Spherical Bessel Functions 272.5.2 Proportional Relations of Angular and

Radial Functions 292.5.3 Power and Legendre Functional Series

Representations 302.6 Derivatives of Spheroidal Functions 35

2.6.1 Derivatives of Angular Functions 352.6.2 Derivatives of Radial Functions 35

2.7 Numerical Calculations and Discussion 362.7.1 Mathematica Source Codes 362.7.2 Geometrical Features of Spheroidal

Functions 372.7.3 Tabulated Numerical Data: New Results

and Comparison 372.8 Spheroidal Vector Wave Functions 44

3 Dyadic Green's Functions in Spheroidal Systems 613.1 Dyadic Green's Functions 613.2 Fundamental Formulation 633.3 Unbounded Dyadic Green's Functions 66

3.3.1 Method of Separation of Variables 663.3.2 Unbounded Scalar Green's Function 673.3.3 Appropriate Spheroidal Vector Wave

Functions for Construction of DGFs 683.3.4 Unbounded Green's Dyadics 69

3.4 Scattering Green's Dyadics 703.4.1 Scattering Green's Dyadics in the Inner

Region (f = l) 713.4-2 Scattering Green's Dyadics in the

Intermediate Regions (2 < f < N - l) 713.4-3 Scattering Green's Dyadics in the Outer

Region (f = N) 723.5 Determination of Scattering Coefficients 73

3.5.1 Nonorthogonality and FunctionalExpansion 73

3.5.2 Matrix Equation Systems 76

CONTENTS xi

3.6 Convergence of the Solution 86

4 EM Scattering by a Conducting Spheroid 894-1 Geometry of the Problem 894-2 Incident and Scattered Fields 894-3 Transformation of Incident Fields to Scattered

Fields 924-3.1 Imposing the Boundary Conditions 924.3.2 TE Polarization for Oblique Incidence 934.3.3 TM Polarization for Oblique Incidence 994.3.4 Fields at Axial Incidence 1014.3.5 TE Fields with Incidence Angle 90° 102

4-4 Far-Field Expressions 1034.5 Numerical Computation and Mathematica Source

Codes 1064.6 Results and Discussion 108

5 EM Scattering by a Coated Dielectric Spheroid 1155.1 Geometry of the Problem 1155.2 Incident, Transmitted and Scattered Fields 1175.3 Relationship between Incident and Scattered

Fields 1195.3.1 Boundary Conditions 1195.3.2 TE Polarization for Nonaxial Incidence 1195.3.3 TM Polarization for Nonaxial Incidence 1285.3.4 Fields at Axial Incidence 130

5.4 Numerical Computation and Mathematica SourceCode 130

5.5 Results and Discussion 132

6 Spheroidal Antennas 1456.1 Introduction 1456.2 Prolate Spheroidal Antenna 146

6.2.1 Antenna Geometry 1466.2.2 Maxwell's Equations for the Spheroidal

Antenna 1466.2.3 Auxiliary Scalar Wave Function 1486.2.4 Imposing the Boundary Conditions 1496.2.5 Far-Field Expressions 150