Chapter 1. Geometrical Optics of Guided Waves in Waveguides: Stationary                                                   Optics for Modal Waves
by Masahiro Hashirnoto                                                                                                                1
1. Introduction                                                                                                                                  1
2. Historical Background                                                                                                                 2
3. Variational Principles                                                                                                                   2
    3.1 Ray Concepts                                                                                                                         3
    3.2 Hamilton’s Principle                                                                                                               5
    3.3 Maupertuis’ Principle                                                                                                            7
    3.4 Ray Tracing — Examples —                                                                                                 10
4. Geometrical Optics Fields                                                                                                           13
5. Total Reflection of Wave-Normal Rays upon a Dielectric Interface                                    21
   5.1 Total Reflection of Two-Dimensional Spherical Waves                                                  21
   5.2 Interpretation of Ray Shift for Energy Rays                                                                      25
   5.3 Total Reflection of Three-Dimensional Nonspherical Waves                                        27
   5.4 Geometrical Law of Total Reflection in Inhomogeneous Media                                     32
6. Application to the Analysis of Dielectric Tapered Waveguides                                         38
7. Conclusions and Some Remarks                                                                                                43
Appendices                                                                                                                                       44
    A. Derivation of Eq. (6)                                                                                                               44
    B. Derivation of Eqs. (15), (16) and (17) from Eqs. (10) and (14)                                           44
    C. The Ray Tracing for Wavefronts, Wave-Normal Rays and

                          Geometrical Optics Fields                                                                                                         47
    D. Derivation of Eq. (24)                                                                                                              49
    E. Derivation of Eq. (26)                                                                                                              50
    F. Derivation of Eq. (40)                                                                                                              52
    G. Paths of Beam Waves and Dynamical Waves                                                                    52

Acknowledgements 53

References 53

Chapter 2. Inverse Scattering Problems Connected with Cylindrical Bodies by Mithat Idemen 57

1. Introduction 57

1.1 Definition of the Problem 57

1.2 Scope of the Present Paper 58

1.3 Notation 58

1.4 Contents 59

2. Cylindrical Bodies in an Infinite Simple Space 60

2.1 Formulation of the Problem 60

2.2 The Case of Known Orientation 61

2.2.1 Transform of the Hankel Function 63

2.2.2 Transform of the Function w(x’) 64

2.2.3 A Relation between it ŵ(v) and Measured Values of the Scattered Field 65

2.2.4 Submanifolds of M and Different Methods 69

A. Accessible Part of M₁ 69

B. Accessible Part of M₂ 73

C. Accessible Part of M₃ 75

2.2.5 Ramm Function and the Analytic Continuation of the Collected Data 77

2.3 The Case of Unknown Orientation 81

2.3.1 Solution under the Born Approximation 83

2.3.2 Determination of the Orientation through Far-Field Measurements 83

A. The Case of 84

B. The Case of 85

3. Cylindrical Bodies Buried in a Simple Half Space 86

3.1 The Case of Known Orientation 86

3.1.1 An Expression of the Green Function 88

3.1.2 A Relation between ŵ(v) and Measured Values U_D(x’) 88

A. The Case of L=L₁ 89

B. The Case of L=L₃ 92

3.2 The Case of Unknown Orientation 93

3.2.1 Solution under the Born Approximation 95

A. Determination of the Orientation through Far-Field Measurements 96

B. Determination of the Orientation through Measurements on L₃ 97

4. Cylindrical Bodies Buried in an Infinite Slab 98

4.1 The Case of Known Orientation 99

A. The Case of L=L₁ 102

B. The Case of L=L₂ 103

4.2 The Case of Unknown Orientation 105

5. Cylindrical Bodies Buried in a Circular Cylinder 110

5.1 Formulation of the Problem 110

5.2 Solution of the Problem 112

5.2.1 Proof of the Relation (5.9a) 113

5.2.2 Proof of the Relation (5.9b) 116

5.3 An Illustrative Example 118

References 121

Chapter 3. Essentials of Nonstationary and Nonlinear Electromagnetic

Field Theory

by OIeg A. Tretyakov 123

Introduction 123

1. Evolutionary Equations for EM Fields in Cavities 126

1.1 Rotary SeIf-Adjoint Operator 126

1.2 The Vortex Eigenvectors of the Operator 128

1.3 The Nonvortex Ligenvectors of the Operator 128

1.4 Normalization and Orthogonality Conditions 130

1.5 H. Weyl’s Theorem Concerning the Basis in L₂ (V) 130

1.6 Basis in the Space of Solutions 131

1.7 Projection of the EM Field upon the Basis 132

1.8 Projection of Maxwell’s Equations upon the Basis 132

1.9 Evolutionary Equations for the EM Field in the Cavity 133

1.10 Stationary and Nonstationary Oscillations in the Cavity 133

2. Evolutionary Equations for EM Fields in Waveguides 135

2.1 Geometry of the Waveguide and Constitutive Relations 136

2.2 Standard Formulation of the Boundary Value Problem 137

2.3 The Operator Form of the Boundary Value Problem 138

2.4 The Eigenvalue Problem for the Operators and 140

2.5 Weyl’s Theorem Concerning the Basis in L₂ (S) 141

2.6 The Basis in the Space of Solutions for Maxwell’s Equations 142

2.7 The Projection of EM Field upon the Basis 142

2.8 Evolutionary Equations for the EM Field in the Waveguides 143

2.9 EM Waves and Signals in Waveguides 144

References 145

Chapter 4. Some Diffraction Problems Involving Modified Wiener-Hopf

Geometries

by Kazuya Kobayashi 147

1. Introduction 147

2. Diffraction by a Finite Sinusoidal Grating 150

2.1 Preliminary Remark 150

2.2 Statement of the Problem 151

2.3 Wiener-Hopf Equations 154

2.4 Formal Solutions 157

2.5 Asymptotic Solutions 162

2.6 Scattered Far Field 165

2.7 Numerical Results and Discussions 170

3. Diffraction by a Parallel-Plate Waveguide Cavity 177

3.1 Preliminary Remark 177

3.2 Transformed Wave Equations 178

3.3 Field Representation in the Transform Domain 180

3.4 Simultaneous Wiener-Hopf Equations 183

3.5 Decomposition of the Wiener-Hopf Equations 185

3.6 Formal Solutions 188

3.7 Approximate Solutions 190

3.8 Scattered Field Representation 194

3.9 Numerical Results and Discussions 199

3.10 Comparison with Different Methods 205

4. Conclusions 209

Appendix A. Generalized Gamma Functions 210

Appendix B. Asymptotic Expansion of Certain Branch-Cut Integrals 216

Acknowledgements 222

References 223

Chapter 5. Some Approximate Methods Related to the Diffraction by

Strips and Slits

by A. Hamit Serbest and Alinur Büyükaksoy 229

1. Introduction 229

2. Review of Some Approximate High-Frequency Techniques

Related to Strip/Slit Problems 230

3. Application of the Triple Integral Equation Approach to the

Diffraction by a Resistive Strip Located between Two

Half-Planes with Equal Surface Impedances 233

3.1 Formulation of the Problem 233

3.2 Solution of the MWHEs 238

3.3 Analysis of the Multiply Diffracted Field 244

3.3.1 Doubly Diffracted Fields 245

3.3.2 Triply Diffracted Fields 246

4. Application of the Spectral Iteration Technique to the Diffraction

by a Resistive Strip Located between Two Half-Planes with

Different Surface Impedances 249

4.1 Doubly Diffracted Fields 249

4.2 Triply Diffracted Fields 252

5. Concluding Remarks 254

Acknowledgements 254

References 255

Chapter 6. Matrix Wiener-Hopf Factorization Methods and Applications to

Some Diffraction Problems

By Alinur Büyükaksoy and A. Hamit Serbest 257

1. Introduction 257

2. Formal Solution of a Matrix Wiener-Hopf 259

3. Wiener-Hopf-Hilbert Method 261

3.1 Essentials of the Wiener-Hopf-Hilbert Method 261

3.2 Applications of the Wiener-Hopf-Hilbert Method 265

3.2.1 Plane Wave Diffraction by the Junction of Resistive

and Soft-Hard Half Planes 265

3.2.2 High-Frequency Diffraction by a Cylindrically Curved

Surface with Different Face Impedances 270

4. Direct Factorization Methods 275

4.1 Daniele’s Factorization Method 276

4.2 Khrapkov’s Factorization Method 277

4.3 Modification of the Theory for the Exponential Behavior at Infinity 280

4.4 Applications of the Direct Factorization Methods 283

4.4.1 Plane Wave Diffraction by the Junction of Resistive and

Soft-Hard Half Planes 283

4.4.1.1 Factorization by the Daniele Method 283

4.4.1.2 Factorization by the Khrapkov Method 285

4.4.2 Diffraction by a Metal-Backed Dielectric Half Plane 287

5. Methods Based on the Weak Factorization Concept 296

5.1 Essentials of the Method 297

5.2 Plane Wave Diffraction by a Pair of Soft Half Planes 300

6. Concluding Remarks 306

Appendix. Factorization of x(a) in Terms of Maliuzhinetz Function 307

Acknowledgement 309

References 309

Chapter 7. Diffraction by an Infinite Set of Parallel Half-Planes and by an Infinite

Strip Grating: Comparison of Different Methods

by Ernst Luneburg 317

Abstract 317

1. Introduction 318

2. Diffraction of Plane Waves by Parallel Half-Planes 322

2.1 The Single Half-Plane 322

2.2 Two and Three Half-Planes 324

2.3 Infinite Set of Soft/Soft Parallel Half-Planes 325

2.3.1 The Wiener-Hopf Method 326

2.3.2 The Riemann-Hilbert Method 330

2.3.3 The Mode Matching Technique 337

2.4 Infinite Set of Soft/Hard Parallel Half-Planes 339

2.4.1 The Wiener-Hopf Method 340

2.4.2 The Riemann-Hilbert Method 345

2.4.3 The Mode Matching Technique 349

2.5 Infinite Set of Hard and Soft Parallel Half-Planes 350

3. Diffraction by an Infinite Strip Grating 351

3.1 Statement of the Problem 351

3.2 The Normal Incidence Case 352

3.2.1 The Wiener-Hopf Method 353

3.2.2 Remarks 355

3.3 The Arbitrary Incidence Case 355

3.3.1 The Wiener-Hopf Method 355

3.3.2 The Riemann-Hilbert Method 357

3.3.3 The Generalized Mode Matching Technique 362

3.3.4 Some Numerical/Graphical Results 364

4. Conclusions 367

References 367

Chapter 8. Wavefront and Complex Resonance Descriptions in Time Transient

EM Responses by Simple Geometries

by Hiroshi Shirai 373

1. Introduction 373

2. Bilateral Relation between Complex Resonances and Wavefronts 374

2.1 Resonance Equation for Deriving Complex Resonances 377

3. Example 1: Transient Response by a Dielectric Cylinder 380

3.1 Direct Contribution 382

3.2 SEM Formulation 383

3.3 WEM Formulation 385

3.4 Numerical Results and Discussion 387

4. Example 2: Transient Response by a Dielectric Sphere 395

4.1 Direct Contribution 398

4.2 SEM Formulation 398

4.3 WEM Formulation 400

4.4 Numerical Examples and Discussion 401

5. Conclusions 407

Appendix A. Field Formulation by High Frequency Asymptotic Rays 410

Appendix B. Finite filbert Transforms 413

Acknowledgement 415

References 415

Chapter 9. Green’s Function-Dual Series Approach in Wave Scattering by

Combined Resonant Scatterers

by Alexander I. Nosich 419

1. Introduction 419

1.1 Historical Background 419

1.2 On the Extension of the Range of Applications 420

2. Mathematical Foundations of the Method 421

2.1 About the RUP in a Complex Plane 421

2.2 Solution of an RHP Typical for Electromagnetics 423

2.3 Exact Solution of Canonical Dual Series Equations 424

3. Free-Space H-Wave Scattering by an Open Circular Screen 426

3.1 Formulation of the Problem 426

3.2 Derivation and Regularization of Dual Series Equations 428

3.3 Calculation of Near Field and Surface Current 430

3.4 Far-Field Scattering Characteristics 432

3.5 Low-Frequency Asymptotic Solution 435

4. Scattering by Screens in Stratified Dielectric Medium 436

4.1 On the Modified Condition of Radiation 436

4.2 Screen near a Plane Dielectric Interface 438

4.3 Impedance-Plane Surface Wave Scattering from

a Screen-Shaped Inhomogeneity 445

4.4 Mode Conversion and Scattering due to Screens in Dielectric-Slab Waveguide 450

5. Scattering by Screens near Infinite Periodic Grating 458

5.1 Formulation of the Problem and Derivation of Basic Equations 458

5.2 Diffraction by a Screen near a Plane Strip Grating 461

6. Conclusion 466

Acknowledgements 466

References 466

Chapter 10. Numerical-Analytical Approach for the Solution to the Wave

Scattering by Polygonal Cylinders and Flat Strip Structures

by Eldar I. Veliev and Vladimir V. Veremey 471

1. Introduction 471

2. Solution to the Problem of H-Polarized Electromagnetic Wave

Scattering by a Polygonal Cylinder 473

2.1 Statement of the Problem: Derivation of a System of Dual Integral Equations 473

2.2 Solution of a Class of Dual Integral Equations with the

Kernel in Trigonometric Functions 478

2.3 Reduction of the Problem of H-Polarized Wave Scattering

by Polygonal Cylinders to the SLAB 483

3. H-Polarized Plane Wave Scattering by a Finite Number of Flat Strips 488

4. Wave Scattering by a Flat Strip 492

5. H-Polarized Plane Wave Scattering by a Rectangular Cylinder 497

6. E-Polarized Plane Wave Scattering by a Regular Polygonal Cylinder 508

7. Conclusion 511

References 511

Chapter 11. An Introduction to the Yasuura Method

by Yoichi Okuno 515

1. Introduction 515

2. Scattering by an Obstacle with a Smooth Cross Section 516

2.1 Formulation of the Problem 516

2.2 Modal Functions and an Approximate Solution 518

2.3 Scattered Field Calculation: E Wave Case 519

2.3.1 The Conventional Yasuura Method in the E Wave Case 519

2.3.2 The Yasuura Method with a Smoothing Procedure in the E Wave Case 520

2.4 Scattered Field Calculation: H Wave Case 522

2.4.1 The Conventional Yasuura Method in the H Wave Case 522

2.4.2 The Yasuura Method with a Smoothing Procedure in the H Wave Case 523

2.5 The Significance of the Smoothing Procedure 524

2.5.1 Examination from the Fourier Analysis 524

2.5.2 A Comment on Convergence Rate 525

2.6 Current Density Calculation: E Wave Case 526

2.6.1 The ACYM in the E Wave Case 527

2.6.2 The AYMSP in the E Wave Case 528

2.7 Current Density Calculation: H Wave Case 531

2.7.1 The ACYM in the H Wave Case 532

2.7.2 The AYMSP in the H Wave Case 532

2.8 Necessity of the Methods for the Current Density 533

3. Scattering by an Obstacle with an Edged Cross Section 534

3.1 Formulation of the Problem and Statement of Distinctive Features 534

3.2 Modal Functions and an Approximate Solution 536

3.3 Scattered Field Calculation: E Wave Case 536

3.4 Scattered Field Calculation: H Wave Case 538

3.5 Current Density Calculation: E Wave Case 540

3.6 Current Density Calculation: H Wave Case 544

4. Method of Numerical Analysis 547

4.1 Scattered Field Calculation 547

4.1.1 A Numerical Method Based on the CYM 547

4.1.2 A Numerical Method Based on the YMSP or YMSSP 550

4.2 Current Density Calculation 552

4.2.1 A Numerical Method Based on the ACYM 552

4.2.2 A Numerical Method Based on the AYMSP or AYMSSP 553

5. Conclusion 554

Appendix A 555

Appendix B 556

Appendix C 557

Appendix D 557

Appendix E 558

Appendix F 560

Appendix G 561

Appendix H 561

Acknowledgements 562

References 562

Index 567