Waves and Fields in Inhomogenous Media

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 One-Dimensional Planar Inhomogeneity 45

2.1.1 Derivation of the Scalar Wave Equations 45

2.1.2 Reflection from a Half-Space 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 Half-Space (VMD) 93

2.6.2 Dipole Over Half-Space (VED) 98

2.6.3 Dipole Over a Slab 101

2.6.4 Example of Uniform Asymptotic Expansion —Transmitted Wave in a Half-Space 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 Three-Layer 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—Multi-Interface Case 167

3.2.1 The Outgoing-Wave Case 167

3.2.2 The Standing-Wave Case 170

3.3 Source in a Cylindrically Layered Medium 172

3.3.1 Discrete, Angular-Wave-Number Representation 173

3.3.2 Continuum, Angular-Wave-Number 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—Multi-Interface Case 191

3.6.1 The Outgoing-Wave Case 191

3.6.2 The Standing-Wave 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 Kramers-Kronig Relation 212

4.1.2 Causality and Contour of Integration 214

4.2 The Cagniard-de Hoop Method 215

4.2.1 Line Source in Free-Space—Two-Dimensional Green's Function 216

4.2.2 Point Source in Free-Space—Three-Dimensional Green's Function 219

4.2.3 Line Source Over Half-Space—Transient Response 221

4.2.4 Dipole Over Half Space—Transient Response 224

4.3 Multi-interface 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 Finite-Difference Method 235

4.6.1 The Finite-Difference Approximation 236

4.6.2 Stability Analysis 239

4.6.3 Grid-Dispersion Error 242

4.6.4 The Yee Algorithm 244

4.7 Absorbing Boundary Conditions 246

4.7.1 Engquist-Majda Absorbing Boundary Condition 246

4.7.2 Lindman Absorbing Boundary Condition 249

4.7.3 Bayliss-Turkel 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 Self-Adjoint Problems 285

5.2.1 General Concepts 285

5.2.2 Rayleigh-Ritz Procedure—Self-Adjoint 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 Non-Self-Adjoint Problems 295

5.3.1 General Concepts 295

5.3.2 Rayleigh-Ritz Procedure—Non-Self-Adjoint 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 Three-Region Problem 360

6.6.2 The iV-Region 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 Two-Dimensional 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 Extended-Boundary-Condition 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 Two-Scatterer Solution 463

8.6.2 iV-Scatterer Solution—A Recursive Algorithm 465

8.7 Scattering by Multilayered Scatterers 469

8.7.1 One-Interface Problem 469

8.7.2 Many-Interface Problems 471

8.8 Surface Integral Equation with Finite-Element Method 475

8-9 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 Back-Projection Tomography 514

9.1.2 Radon Transforms 516

9.1.3 Diffraction Tomography 519

9.1.4 Finite-Source Effect 522

9.1.5 Nonuniqueness of the Solution 524

9.2 One-Dimensional Inverse Problems 526

9.2.1 The Method of Characteristics 526

9.2.2 Transformation to a Schrodinger-like Equation 532

9.2.3 The GePfand-Levitan Integral Equation 534

9.2.4 The Marchenko Integral Equation 541

9.2.5 The GePfand-Levitan-Marchenko Integral Equation 543

9.3 Higher-Dimensional 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