Spheroidal Wave Functions in Electromagnetic Theory / Edition 1

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Overview

Spheroidal coordinates and wave functions are employed in applications such as antenna analysis and design, microwave integrated circuit design, electromagnetic compatibility and interference, and wireless communications. This is the first modern treatment of the subject complete with computer calculations that allow rigorous solutions to problems.
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Product Details

Meet the Author

LE-WEI LI, PhD, is Deputy Director of the Antenna and Scattering Laboratory and Electromagnetics Research Group at the National University of Singapore. He is a senior member of the IEEE, an editorial board member of Journal of Electromagnetic Waves and Applications, and the author of Dyadic Green's Functions in Inhomogeneous Media and Electromagnetic Theory of Complex Media.
XIAO-KANG KANG, PhD, is a research engineer in the Department of Electrical and Computer Engineering at the National University of Singapore.
MOOK-SENG LEONG, PhD, is a professor in the Department of Electrical and Computer Engineering at the National University of Singapore.
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Table of Contents

Preface v
Acknowledgments vii
1 Introduction 1
1.1 Overview 1
1.2 EM Scattering by Spheroids 4
1.3 Spheroidal Antenna 5
1.4 EM Radiation in Dielectric Spheroids 7
1.5 Oblate Spheroidal Models 8
1.6 Spheroidal Cavity System 9
1.7 Spheroidal Harmonics and Mathematica Software 10
2 Spheroidal Coordinates and Wave Functions 13
2.1 Spheroidal Coordinate Systems 13
2.2 Spheroidal Scalar Wave Functions 17
2.3 Spheroidal Angular Harmonics 18
2.3.1 Series Representation in Terms of Associated Legendre Functions 18
2.3.2 Power Series Representation 20
2.4 Eigenvalues [lambda subscript mn] and Expansion Coefficients d[superscript mn subscript r] 22
2.4.1 Case I: 23
2.4.2 Case II: 26
2.5 Spheroidal Radial Harmonics 27
2.5.1 Series Representation in Terms of Spherical Bessel Functions 27
2.5.2 Proportional Relations of Angular and Radial Functions 29
2.5.3 Power and Legendre Functional Series Representations 30
2.6 Derivatives of Spheroidal Functions 35
2.6.1 Derivatives of Angular Functions 35
2.6.2 Derivatives of Radial Functions 35
2.7 Numerical Calculations and Discussion 36
2.7.1 Mathematica Source Codes 36
2.7.2 Geometrical Features of Spheroidal Functions 37
2.7.3 Tabulated Numerical Data: New Results and Comparison 37
2.8 Spheroidal Vector Wave Functions 44
3 Dyadic Green's Functions in Spheroidal Systems 61
3.1 Dyadic Green's Functions 61
3.2 Fundamental Formulation 63
3.3 Unbounded Dyadic Green's Functions 66
3.3.1 Method of Separation of Variables 66
3.3.2 Unbounded Scalar Green's Function 67
3.3.3 Appropriate Spheroidal Vector Wave Functions for Construction of DGFs 68
3.3.4 Unbounded Green's Dyadics 69
3.4 Scattering Green's Dyadics 70
3.4.1 Scattering Green's Dyadics in the Inner Region (f = 1) 71
3.4.2 Scattering Green's Dyadics in the Intermediate Regions (2 [less than or equal] f [less than or equal] N - 1) 71
3.4.3 Scattering Green's Dyadics in the Outer Region (f = N) 72
3.5 Determination of Scattering Coefficients 73
3.5.1 Nonorthogonality and Functional Expansion 73
3.5.2 Matrix Equation Systems 76
3.6 Convergence of the Solution 86
4 EM Scattering by a Conducting Spheroid 89
4.1 Geometry of the Problem 89
4.2 Incident and Scattered Fields 89
4.3 Transformation of Incident Fields to Scattered Fields 92
4.3.1 Imposing the Boundary Conditions 92
4.3.2 TE Polarization for Oblique Incidence 93
4.3.3 TM Polarization for Oblique Incidence 99
4.3.4 Fields at Axial Incidence 101
4.3.5 TE Fields with Incidence Angle 90[degree] 102
4.4 Far-Field Expressions 103
4.5 Numerical Computation and Mathematica Source Codes 106
4.6 Results and Discussion 108
5 EM Scattering by a Coated Dielectric Spheroid 115
5.1 Geometry of the Problem 115
5.2 Incident, Transmitted and Scattered Fields 117
5.3 Relationship between Incident and Scattered Fields 119
5.3.1 Boundary Conditions 119
5.3.2 TE Polarization for Nonaxial Incidence 119
5.3.3 TM Polarization for Nonaxial Incidence 128
5.3.4 Fields at Axial Incidence 130
5.4 Numerical Computation and Mathematica Source Code 130
5.5 Results and Discussion 132
6 Spheroidal Antennas 145
6.1 Introduction 145
6.2 Prolate Spheroidal Antenna 146
6.2.1 Antenna Geometry 146
6.2.2 Maxwell's Equations for the Spheroidal Antenna 146
6.2.3 Auxiliary Scalar Wave Function 148
6.2.4 Imposing the Boundary Conditions 149
6.2.5 Far-Field Expressions 150
6.2.6 Numerical Computations and Mathematica Code 150
6.2.7 Results and Discussion 151
6.3 Dielectric-coated Prolate Spheroidal Antenna 152
6.3.1 Coated Dielectric Antenna Geometry 152
6.3.2 Obtaining the Auxiliary Wave Functions 158
6.3.3 Imposing the Boundary Conditions 161
6.3.4 Numerical Computations 162
6.3.5 Mathematica Code 163
6.3.6 Results and Discussion 165
6.4 Prolate Spheroidal Antenna enclosed in a Confocal Radome 168
6.4.1 Geometry of the Antenna with Radome 168
6.4.2 Obtaining the Auxiliary Wave Functions 174
6.4.3 Imposing the Boundary Conditions 174
6.4.4 Numerical Computations 176
6.4.5 Mathematica Code 177
6.4.6 Results and Discussion 179
7 SAR Distributions in a Spheroidal Head Model 191
7.1 Introduction 191
7.2 Multilayered Prolate Spheroidal Head Model 192
7.3 Formulation of the Problem 194
7.3.1 Expansions of EM Fields Using Spheroidal Wave Functions 194
7.3.2 EM Boundary Conditions for Multispheroidal Interfaces 195
7.3.3 Specific Absorption Rate 195
7.4 Numerical Computation 196
7.5 Results and Discussion 197
7.6 Effects on Wire Antennas Due to the Presence of the Multilayered Spheroid 209
7.7 Numerical Results and Discussion 218
8 Analysis of Rainfall Attenuation Using Oblate Raindrops 227
8.1 Introduction 227
8.1.1 Rainfall Attenuation 227
8.1.2 Raindrop Models in Different Sizes 228
8.1.3 Oblate Spheroidal Raindrops 229
8.2 Problem Formulation 230
8.2.1 Geometry of the Problem 230
8.2.2 Definition of the EM Field 230
8.2.3 Boundary Conditions and Solution of Unknowns 234
8.2.4 Total Cross Section 237
8.3 Size Parameters of Raindrops 238
8.3.1 Radius-Independent Oblate Spheroid Raindrop 238
8.3.2 Radius-Dependent Oblate Spheroid Raindrop 238
8.4 Numerical Calculation and Results 239
9 EM Eigenfrequencies in a Spheroidal Cavity 245
9.1 Introduction 245
9.2 Theory and Formulation 246
9.2.1 Background Theory 246
9.2.2 Derivation 247
9.3 Numerical Results for TE Modes 249
9.3.1 Numerical Calculation 249
9.3.2 Results and Comparison 250
9.4 Numerical Results for TM Modes 252
9.4.1 Numerical Calculation 252
9.4.2 Results and Comparison 252
9.5 Discussion 254
Appendix A Expressions of Spheroidal Vector Wave Functions 255
Appendix B Intermediates I[superscript mn subscript t,e](c) in Closed Form 263
B.1 The Case where m [greater than or equal] 1 264
B.2 The Case where m = 0 269
Appendix C U[superscript q(i),t] and V[superscript q(i),t] Used in the Matrix Equation System 273
References 277
Index 292
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