Waves and Fields in Inhomogenous MediaISBN: 9780780347496
632 pages
February 1999, WileyIEEE Press

* Analytical methods for planarly, cylindrically and spherically layered media
* Transient waves, including the Cagniardde Hoop method
* Variational methods for the scalar wave equation and the electromagnetic wave equation
* Modematching techniques for inhomogeneous media
* The Dyadic Green's function and its role in simplifying problemsolving in inhomogeneous media
* Integral equation formulations and inverse problems
* Time domain techniques for inhomogeneous media
This book will be of interest to electromagnetics and remote sensing engineers, physicists, scientists, and geophysicists. This IEEE Press reprinting of the 1990 version published by Van Nostrand Reinhold incorporates corrections and minor updating. Also in the series. Mathematical Foundations for Electromagnetic Theory by Donald G. Dudley, University of Arizona at Tucson This volume in the series lays the mathematical foundations for the study of advanced topics in electromagnetic theory. Important subjects covered include linear spaces, Green's functions, spectral expansions, electromagnetic source representations, and electromagnetic boundary value problems. 1994 Hardcover 264 pp ISBN 0780310225 IEEE Order No. PC3715 About the Series The IEEE Press Series on Electromagnetic Waves consists of new titles as well as reprints and revisions of recognized classics that maintain longterm archival significance in electromagnetic waves and applications. Designed specifically for graduate students, practicing engineers, and researchers, this series provides affordable volumes that explore electromagnetic waves and applications beyond the undergraduate level.
PREFACE xvii
ACKNOWLEDGMENTS xxi
1 PRELIMINARY BACKGROUND 1
1.1 Maxwell's Equations 1
1.1.1 Differential Representations 1
1.1.2 Integral Representations 3
1.1.3 Time Harmonic Forms 4
1.1.4 Constitutive Relations 5
1.1.5 Poynting Theorem and Lossless Conditions 6
1.1.6 Duality Principle 9
1.2 Scalar Wave Equations 9
1.2.1 Acoustic Wave Equation 10
1.2.2 Scalar Wave Equation from Electromagnetics 12
1.2.3 Cartesian Coordinates 12
1.2.4 Cylindrical Coordinates 14
1.2.5 Spherical Coordinates 16
1.3 Vector Wave Equations 17
1.3.1 Boundary Conditions 18
1.3.2 Reciprocity Theorem 20
1.3.3 Plane Wave in Homogeneous, Anisotropic Media 22
1.3.4 Green's Function 24
1.4 Huygens' Principle 29
1.4.1 Scalar Waves 29
1.4.2 Electromagnetic Waves 31
1.5 Uniqueness Theorem 32
1.5.1 Scalar Wave Equation 33
1.5.2 Vector Wave Equation 35
Exercises for Chapter 1 37
References for Chapter 1 41
Further Readings for Chapter 1 42
2 PLANARLY LAYERED MEDIA 45
2.1 OneDimensional Planar Inhomogeneity 45
2.1.1 Derivation of the Scalar Wave Equations 45
2.1.2 Reflection from a HalfSpace 48
2.1.3 Reflection and Transmission in a Multilayered Medium 49
2.1.4 Ricatti Equation for Reflection Coefficients 53
2.1.5 Specific Inhomogeneous Profiles 56
2.2 Spectral Representations of Sources 57
2.2.1 A Line Source 58
2.2.2 A Point Source 63
2.2.3 Riemann Sheets and Branch Cuts 66
2.3 A Source on Top of a Layered Medium 70
2.3.1 Electric Dipole Fields 71
2.3.2 Magnetic Dipole Fields 74
2.3.3 The Transverse Field Components 75
2.4 A Source Embedded in a Layered Medium 76
2.5 Asymptotic Expansions of Integrals 79
2.5.1 Method of Stationary Phase 79
2.5.2 Method of Steepest Descent 82
2.5.3 Uniform Asymptotic Expansions 87
2.6 Dipole Over Layered Media—Asymptotic Expansions 93
2.6.1 Dipole Over HalfSpace (VMD) 93
2.6.2 Dipole Over HalfSpace (VED) 98
2.6.3 Dipole Over a Slab 101
2.6.4 Example of Uniform Asymptotic Expansion —Transmitted Wave in a HalfSpace 106
2.6.5 Angular Spectrum Representation 110
2.7 Singularities of the Sommerfeld Integrals 111
2.7.1 Absence of Branch Points 112
2.7.2 Bounds on the Locations of Singularities 114
2.7.3 Numerical Integration of Sommerfeld Integrals 118
2.8 WKB Method 121
2.8.1 Derivation of the WKB Solution 121
2.8.2 Asymptotic Matching 124
2.9 Propagator Matrix 128
2.9.1 Derivation of the State Equation 129
2.9.2 Solution of the State Equation 129
2.9.3 Reflection from a ThreeLayer Medium 130
2.9.4 Reflection from an Inhomogeneous Slab 131
2.10 Waves in Anisotropic, Layered Media 133
2.10.1 Derivation of the State Equation 133
2.10.2 Solution of the State Equation 135
2.10.3 Reflection from an Interface of Anisotropic Half Spaces 136
2.10.4 Reflection from a Slab 137
2.10.5 Geometrical Optics Series 138
Exercises for Chapter 2 140
References for Chapter 2 151
Further Readings for Chapter 2 155
3 CYLINDRICALLY AND SPHERICALLY LAYERED MEDIA 161
3.1 Cylindrically Layered Media—Single Interface Case 161
3.1.1 Vector Wave Equation in Cylindrical Coordinates 162
3.1.2 Reflection and Transmission of an Outgoing Wave 163
3.1.3 Reflection and Transmission of a Standing Wave 165
3.2 Cylindrically Layered Media—MultiInterface Case 167
3.2.1 The OutgoingWave Case 167
3.2.2 The StandingWave Case 170
3.3 Source in a Cylindrically Layered Medium 172
3.3.1 Discrete, AngularWaveNumber Representation 173
3.3.2 Continuum, AngularWaveNumber Representation 177
3.4 Propagator Matrix—Cylindrical Layers 179
3.4.1 Isotropic, Layered Media 179
3.4.2 Anisotropic, Layered Media 182
3.5 Spherically Layered Media—Single Interface Case 184
3.5.1 Vector Wave Equation in Spherical Coordinates 185
3.5.2 Reflection and Transmission of an Outgoing Wave 187
3.5.3 Reflection and Transmission of a Standing Wave 189
3.6 Spherically Layered Media—MultiInterface Case 191
3.6.1 The OutgoingWave Case 191
3.6.2 The StandingWave Case 192
3.7 Source in a Spherically Layered Medium 193
3.8 Propagator Matrix—Spherical Layers 197
Exercises for Chapter 3 199
References for Chapter 3 204
Further Readings for Chapter 3 206
4 TRANSIENTS 211
4.1 Causality of Transient Response 211
4.1.1 The KramersKronig Relation 212
4.1.2 Causality and Contour of Integration 214
4.2 The Cagniardde Hoop Method 215
4.2.1 Line Source in FreeSpace—TwoDimensional Green's Function 216
4.2.2 Point Source in FreeSpace—ThreeDimensional Green's Function 219
4.2.3 Line Source Over HalfSpace—Transient Response 221
4.2.4 Dipole Over Half Space—Transient Response 224
4.3 Multiinterface Problems 227
4.4 Direct Inversion 228
4.5 Numerical Integration of Fourier Integrals 231
4.5.1 Direct Field in a Lossy Medium—Two Diemnsional Case 232
4.5.2 Direct Field in a Lossy Medium—Three Dimensional Case 233
4.6 FiniteDifference Method 235
4.6.1 The FiniteDifference Approximation 236
4.6.2 Stability Analysis 239
4.6.3 GridDispersion Error 242
4.6.4 The Yee Algorithm 244
4.7 Absorbing Boundary Conditions 246
4.7.1 EngquistMajda Absorbing Boundary Condition 246
4.7.2 Lindman Absorbing Boundary Condition 249
4.7.3 BaylissTurkel Absorbing Boundary Condition 250
4.7.4 Liao's Absorbing Boundary Condition 251
Exercises for Chapter 4 256
References for Chapter 4 262
Further Readings for Chapter 4 265
5 VARIATIONAL METHODS 271
5.1 Review of Linear Vector Space 271
5.1.1 Inner Product Spaces 271
5.1.2 Linear Operators 274
5.1.3 Basis Functions 275
5.1.4 Parseval's Theorem 278
5.1.5 Parseval's Theorem for Complex Vectors 279
5.1.6 Solutions to Operator Equations—A Preview 280
5.1.7 The Eigenvalue Problem 284
5.2 Variational Expressions for SelfAdjoint Problems 285
5.2.1 General Concepts 285
5.2.2 RayleighRitz Procedure—SelfAdjoint Problems 288
5.2.3 Applications to Scalar Wave Equations 291
5.2.4 Applications to Vector Wave Equations 293
5.3 Variational Expressions for NonSelfAdjoint Problems 295
5.3.1 General Concepts 295
5.3.2 RayleighRitz Procedure—NonSelfAdjoint Problems 297
5.3.3 Applications to Scalar Wave Equations 298
5.3.4 Applications to Vector Wave Equations 299
5.4 Variational Expressions for Eigenvalue Problems 301
5.4.1 General Concepts 301
5.4.2 Applications to Scalar Wave Equations 303
5.4.3 Applications to Electromagnetic Problems 304
5.5 Essential and Natural Boundary Conditions 308
5.5.1 The Scalar Wave Equation Case 308
5.5.2 The Electromagnetic Case 312
Exercises for Chapter 5 315
References for Chapter 5 321
Further Readings for Chapter 5 323
6 MODE MATCHING METHOD 327
6.1 Eigenmodes of a Planarly Layered Medium 327
6.1.1 Orthogonality of Eigenmodes in a Layered Medium 328
6.1.2 Guided Modes and Radiation Modes of a Layered Medium 330
6.2 Eigenfunction Expansion of a Field 335
6.2.1 Excitation of Modes due to a Line Source 335
6.2.2 The Use of Vector Notation 337
6.3 Reflection and Transmission at a Junction Discontinuity 340
6.3.1 Derivation of Reflection and Transmission Operators 341
6.3.2 The Continuum Limit Case 343
6.4 A Numerical Method to Find the Eigenmodes 346
6.5 The Cylindrically Layered Medium Case 351
6.5.1 Eigenmodes of a Cylindrically Layered Medium 351
6.5.2 Differential Equations of a Cylindrical Structure 353
6.5.3 Numerical Solution of the Eigenmodes 354
6.5.4 Eigenfunction Expansion of a Field 356
6.5.5 Reflection from a Junction Discontinuity 358
6.6 The Multiregion Problem 360
6.6.1 The ThreeRegion Problem 360
6.6.2 The iVRegion Problem 362
Exercises for Chapter 6 365
References for Chapter 6 370
Further Readings for Chapter 6 372
7 DYADIC GREEN'S FUNCTIONS 375
7.1 Dyadic Green's Function in a Homogeneous Medium 375
7.1.1 The Spatial Representation 376
7.1.2 The Singularity of the Dyadic Green's Function 378
7.1.3 The Spectral Representation 381
7.1.4 Equivalence of Spectral and Spatial Representations 384
7.2 Vector Wave Functions 387
7.2.1 Derivation of Vector Wave Functions 387
7.2.2 Orthogonality Relationships of Vector Wave Functions 388
7.2.3 Vector Wave Functions for Unbounded Media 393
7.3 Dyadic Green's Function Using Vector Wave Functions 397
7.3.1 The Integral Representations 397
7.3.2 Singularity Extraction 399
7.4 Dyadic Green's Functions for Layered Media 410
7.4.1 A General, Isotropic, Inhomogeneous Medium 410
7.4.2 Planarly Layered Media 411
7.4.3 Cylindrically Layered Media 414
7.4.4 Spherically Layered Media 416
7.4.5 Reciprocity Considerations 418
Exercises for Chapter 7 421
References for Chapter 7 424
Further Readings for Chapter 7 426
8 INTEGRAL EQUATIONS 429
8.1 Surface Integral Equations 430
8.1.1 Scalar Wave Equation 430
8.1.2 Vector Wave Equation 433
8.1.3 The Anisotropic, Inhomogeneous Medium Case 437
8.1.4 TwoDimensional Electromagnetic Case 439
8.2 Solutions by the Method of Moments 443
8.2.1 Scalar Wave Case 443
8.2.2 The Electromagnetic Case 446
8.2.3 Problem with Internal Resonances 451
8.3 ExtendedBoundaryCondition Method 453
8.3.1 The Scalar Wave Case 453
8.3.2 The Electromagnetic Wave Case 457
8.4 The Transition and Scattering Matrices 459
8.5 The Method of Rayleigh's Hypothesis 460
8.6 Scattering by Many Scatterers 463
8.6.1 TwoScatterer Solution 463
8.6.2 iVScatterer Solution—A Recursive Algorithm 465
8.7 Scattering by Multilayered Scatterers 469
8.7.1 OneInterface Problem 469
8.7.2 ManyInterface Problems 471
8.8 Surface Integral Equation with FiniteElement Method 475
89 Volume Integral Equations 479
8.9.1 Scalar Wave Case 480
8.9.2 The Electromagnetic Wave Case 481
8.9.3 Matrix Representation of the Integral Equation 483
8.10 Approximate Solutions of the Scattering Problem 484
8.10.1 Born Approximation 485
8.10.2 Rytov Approximation 487
Exercises for Chapter 8 490
References for Chapter 8 501
Further Readings for Chapter 8 505
9 INVERSE SCATTERING PROBLEMS 511
9.1 Linear Inverse Problems 511
9.1.1 BackProjection Tomography 514
9.1.2 Radon Transforms 516
9.1.3 Diffraction Tomography 519
9.1.4 FiniteSource Effect 522
9.1.5 Nonuniqueness of the Solution 524
9.2 OneDimensional Inverse Problems 526
9.2.1 The Method of Characteristics 526
9.2.2 Transformation to a Schrodingerlike Equation 532
9.2.3 The GePfandLevitan Integral Equation 534
9.2.4 The Marchenko Integral Equation 541
9.2.5 The GePfandLevitanMarchenko Integral Equation 543
9.3 HigherDimensional Inverse Problems 547
9.3.1 Distorted Born Iterative Method 548
9.3.2 Born Iterative Method 553
9.3.3 Operator Forms of the Scattering Equations 554
Exercises for Chapter 9 557
References for Chapter 9 563
Further Readings for Chapter 9 566
APPENDIX A Some Useful Mathematical Formulas 571
A.I Useful Vector Identities 571
A.2 Gradient, Divergence, Curl, and Laplacian in Rectangular, Cylindrical, Spherical, and General Orthogonal Curvilinear Coordinate Systems 571
A.3 Useful Integral Identities 573
A.4 Integral Transforms 574
APPENDIX B Review of Tensors 577
APPENDIX C Generalized Functions 583
APPENDIX D Addition Theorems 591
References for Appendices 597
Further Readings for Appendices 598
INDEX 599