Modern Electromagnetic Scattering Theory With Applications

Sergei Tretyakov, Department of Radio Science and Engineering, Aalto University, Finland.

Acknowledgements xiii

List of Abbreviations xv

1 Introduction 1

1.1 Scattering and Diffraction Theory 1

1.2 Books on Related Subjects 3

1.3 Concept and Outline of the Book 5

References 8

2 Fundamentals of Electromagnetic Scattering 11

2.1 Introduction 11

2.2 Fundamental Equations and Conditions 11

2.2.1 Maxwell’s Equations 12

2.2.2 Constitutive Relations 12

2.2.3 Time-harmonic Scattering Problems 19

2.3 Approximate Boundary Conditions 26

2.3.1 Impedance Boundary Conditions 26

2.3.2 Generalized (Higher-order) Impedance Boundary Conditions 31

2.3.3 Sheet Transition Conditions 32

2.4 Fundamental Properties of Time-harmonic Electromagnetic Fields 35

2.4.1 Energy Conservation and Uniqueness 35

2.4.2 Reciprocity 39

2.5 Basic Solutions of Maxwell’s Equations in Homogeneous Isotropic Media 42

2.5.1 Plane, Spherical, and Cylindrical Waves 43

2.5.2 Electromagnetic Potentials and Fields of External Currents 46

2.5.3 Tensor Green’s Function 50

2.5.4 E and H Modes 54

2.5.5 Fields with Translational Symmetry 58

2.6 Electromagnetic Formulation of Huygens’ Principle 61

2.6.1 Compact Scatterers 62

2.6.2 Cylindrical Scatterers 67

2.7 Problems 70

References 84

3 Far-field Scattering 87

3.1 Introduction 87

3.2 Scattering Cross Section 87

3.2.1 Monostatic and Bistatic, Backscattering and Forward-scattering Cross Sections, Differential, Total, Absorption, and Extinction Cross Sections 87

3.2.2 Scattering Width 91

3.2.3 Backscattering from Impedance-matched Bodies 93

3.3 Scattering Matrix 95

3.3.1 Definition 95

3.3.2 Scattering Matrix in Spherical Coordinates 97

3.3.3 Scattering Matrix in the Plane of Scattering Coordinates 99

3.4 Far-field Coefficient 101

3.4.1 Integral Representations and Far-field Conditions 102

3.4.2 Reciprocity of Scattered Fields 106

3.4.3 Forward Scattering 108

3.4.4 Cylindrical Bodies 113

3.5 Scattering Regimes 120

3.5.1 Resonant-size Scatterers 120

3.5.2 Electrically Large Scatterers 121

3.6 Electrically Small Scatterers 125

3.6.1 Physics of Dipole Scattering 126

3.6.2 Dipole Scattering in Terms of Polarizability Tensors 129

3.6.3 Magneto-dielectric Ellipsoid 131

3.6.4 Rotationally Symmetric Particles 137

3.7 Problems 148

References 162

4 Planar Interfaces 165

4.1 Introduction 165

4.2 Interface of Two Homogeneous Semi-infinite Media 167

4.2.1 Reflection and Transmission Coefficients 167

4.2.2 Brewster’s Angle 173

4.2.3 Total Internal Reflection 173

4.2.4 Interfaces with Double-negative Materials 176

4.2.5 Surface Waves 177

4.2.6 Vector Solution of Reflection and Transmission Problems 179

4.3 Arbitrary Number of Planar Layers 182

4.3.1 Solution by the Method of Characteristic Matrices 182

4.3.2 Discussion and Limiting Cases 189

4.4 Reflection and Transmission of Cylindrical and Spherical Waves 195

4.4.1 Excitation by a Linear Electric Current 195

4.4.2 Excitation by an Electric Dipole 202

4.5 A Layer between Homogeneous Half-spaces 207

4.5.1 Different Half-spaces 207

4.5.2 A PEC-backed Layer 213

4.5.3 Layer Immersed in a Homogeneous Space 215

4.6 Modeling with Approximate Boundary Conditions 224

4.6.1 Accuracy of Impedance Boundary Conditions 225

4.6.2 Accuracy of Transition Boundary Conditions 229

4.6.3 Impedance-matched Surface 232

4.7 Problems 235

References 249

5 Wedges 251

5.1 Introduction 251

5.2 The Perfectly Conducting Wedge 253

5.2.1 Formulation of Boundary Value Problem 254

5.2.2 Solution by Separation of Variables 256

5.2.3 Fields and Currents at the Edge 258

5.2.4 Reduction to an Integral Form 260

5.2.5 Special Cases 262

5.2.6 Edge-diffracted and GO Components. Diffraction Coefficient 266

5.3 Scattering from a Half-plane (Solution by Factorization Method) 271

5.3.1 Statement of the Problem 271

5.3.2 Functional Equation 273

5.3.3 Factorization and Solution 274

5.3.4 Scattered Field Far from the Edge 276

5.4 The Impedance Wedge 279

5.4.1 Boundary Value Problem, Sommerfeld’s Integrals, and Functional Equations 279

5.4.2 Normal Incidence (Maliuzhinets’ Solution) 288

5.4.3 Unit Surface Impedance 297

5.4.4 Further Exactly Solvable Cases 300

5.5 High-frequency Scattering from Impenetrable Wedges 306

5.5.1 GO Components and Surface Waves 307

5.5.2 Edge-diffracted Field, Diffraction Coefficient, and Scattering Widths 310

5.5.3 Uniform Asymptotic Approximations 316

5.5.4 GTD/UTD Formulation 319

5.6 Behavior of Electromagnetic Fields at Edges 322

5.6.1 Determining the Degree of Singularity 322

5.6.2 Analytical Structure of Meixner’s Series 328

5.7 Problems 329

References 336

6 Circular Cylinders and Convex Bodies 339

6.1 Introduction 339

6.2 Perfectly Conducting Cylinders: Separation of Variables and Series Solution 340

6.2.1 Separation of Variables 342

6.2.2 Satisfying the Boundary Conditions 342

6.2.3 Scattered Fields 343

6.2.4 Numerical Examples 345

6.3 Homogeneous Cylinders under Normal Illumination 350

6.3.1 Field Equations and Boundary Conditions 350

6.3.2 Rayleigh Series Solution 351

6.3.3 Numerical Examples 352

6.4 Watson’s Transformation and High-frequency Approximations 354

6.4.1 Watson’s Transformation 355

6.4.2 Alternative Solution by Separation of Variables 358

6.4.3 High-frequency Approximations 360

6.4.4 Surface Currents in the Penumbra Region. Fock’s Functions 369

6.5 Coated and Impedance Cylinders under Oblique Illumination 375

6.5.1 PEC Cylinder with Magneto-dielectric Coating 376

6.5.2 Impedance Cylinder 383

6.6 Extension to Generally Shaped Convex Impedance Bodies 392

6.6.1 Fock’s Principle of the Local Field in the Penumbra Region 393

6.6.2 Asymptotic Solution for the Field on the Surface of Circular Impedance Cylinders under Oblique Illumination 396

6.6.3 Fock- and GTD-type Solutions for Electrically Large Convex Impedance Bodies 398

6.7 Problems 403

References 411

7 Spheres 412

7.1 Introduction 412

7.2 Exact Solution for a Multilayered Sphere 414

7.2.1 Formulation of the Problem in Terms of Debye’s Potentials 415

7.2.2 Derivation of the Series Solution 417

7.2.3 Solution for Impedance Boundary Conditions 427

7.3 Physics of Scattering from Spheres 429

7.3.1 Classification of Scattering 430

7.3.2 Spiral Waves 436

7.3.3 Debye’s Expansions for Homogeneous Spheres 438

7.3.4 Waves in Electrically Large Homogeneous Low-absorption Spheres 442

7.4 Scattered Field in the Far Zone 463

7.4.1 Far-field Coefficient, Scattering Cross Sections, and Polarization Structure. Approximations for Electrically Large Spheres 463

7.4.2 Electrically Small Spheres: Dipole, Quasi-static, and Resonance Approximations 471

7.4.3 PEC Spheres 479

7.4.4 Core-shell Spheres 483

7.4.5 Impedance Spheres 488

7.5 Far-field Scattering from Homogeneous Spheres 493

7.5.1 Exact Solution and Limiting Cases 494

7.5.2 Electrically Small Lossy Spheres 495

7.5.3 Electrically Small Low-absorption Spheres 499

7.5.4 Electrically Large Lossy Spheres: Relation to the Impedance Sphere and the Role of Absorption 506

7.5.5 Electrically Large Low-absorption Spheres: Light Scattering from Water Droplets 513

7.6 Metamaterial Effects in Scattering from Spheres 542

7.6.1 Small Spheres 542

7.6.2 Invisibility Cloak 546

7.7 Problems 552

References 562

8 Method of Physical Optics 565

8.1 Introduction 565

8.1.1 On Numerical Techniques for Studying Scattering from Arbitrary-shaped Bodies 565

8.1.2 PO as one of the Approximate Analytical Techniques 566

8.1.3 Structure of the Chapter 567

8.2 Principles and General Solution 567

8.2.1 Principles of PO 567

8.2.2 Derivation of PO Solutions 569

8.2.3 PO for Cylindrical Bodies 573

8.3 Transmission through Apertures 575

8.3.1 PO Solution 575

8.3.2 GO Rays and Fresnel Zones 576

8.3.3 Contribution from the Rim of the Aperture: Edge-diffracted Rays 582

8.4 Scattering from Curved Surfaces 594

8.4.1 Fock’s Reflection Formula 594

8.4.2 Application to a Spherical Segment 600

8.4.3 Reflection Formula in the Far-field Region 605

8.4.4 Diffraction by an Edge in a Non-metallic Surface 609

8.5 Advantages and Limitations of Physical Optics 615

8.6 Problems 616

References 632

9 Physical Optics Solutions of Canonical Problems 634

9.1 Introduction 634

9.2 Vertices 635

9.2.1 Vertex on an Edge of a Thin Plate 637

9.2.2 Apex of a Pyramid 641

9.2.3 Tip of an Elliptic Cone 643

9.3 Electrically Large Plates 652

9.3.1 Arbitrarily Shaped Plates 653

9.3.2 Circular Disc 658

9.3.3 Polygonal Plates 663

9.3.4 Far-field Patterns of Polygonal Plates and Apertures 667

9.4 Bodies of Revolution 671

9.4.1 PO Solution for Bodies of Revolution 672

9.4.2 Imperfectly Reflecting Bodies under Axial Illumination 675

9.4.3 PEC Bodies under Oblique Illumination 677

9.4.4 Axial Backscattering 678

9.4.5 Examples 684

9.5 Problems 689

References 712

A Definitions and Useful Relations of Vector Analysis and Differential Geometry 714

A.1 Vector Algebra 714

A.2 Vector Analysis 716

A.3 Vectors and Vector Differential Operators in Orthogonal Curvilinear Coordinates 717

A.3.1 General Orthogonal Curvilinear Coordinates 717

A.3.2 Spherical Coordinates 718

A.4 Curves and Surfaces in Space 720

A.4.1 Curves 720

A.4.2 Surfaces 720

A.5 Problems 722

References 724

B Fresnel Integral and Related Functions 725

B.1 Fresnel Integral 725

B.2 Relation to the Error Function 728

B.3 Transition Functions of Uniform Theories of Diffraction 730

B.4 Problems 731

References 732

C Principles of Complex Integration 733

C.1 Introduction 733

C.2 Deforming the Integration Contour 734

C.2.1 Basic Facts about Functions of a Complex Variable 734

C.2.2 Integrals over Infinite Contours 736

C.3 Steepest Descent Method 737

C.3.1 Steepest Descent Path 738

C.3.2 Saddle Point Contribution 739

C.3.3 Pole Singularity near the Saddle Point 741

C.3.4 Further Cases 742

C.4 Problems 743

References 745

D The Stationary Phase Method 746

D.1 Introduction 746

D.2 One-dimensional Integrals 746

D.2.1 No Stationary Points on the Integration Interval 747

D.2.2 Isolated Stationary Points 748

D.2.3 Two Coalescing Stationary Points 751

D.3 Two-dimensional Integrals 756

D.3.1 Stationary Point in the Integration Domain 756

D.3.2 Stationary Point near the Boundary of the Integration Domain 758

D.3.3 Contribution from the Boundary of the Integration Domain 760

D.3.4 Kontorovich’s Formula 763

D.3.5 Integrand Vanishing on the Boundary 765

D.3.6 Summary of the Two-dimensional Stationary-phase Method 766

D.4 Problems 766

References 768

E Asymptotic Approximations of Bessel Functions of Large Argument and Arbitrary Order 770

E.1 Introduction 770

E.1.1 Basic Definitions and Properties 770

E.1.2 Large-argument Approximations (|z| �â 1) 772

E.1.3 Content of the Appendix 775

E.2 Debye’s Asymptotic Approximations 776

E.2.1 Debye’s Method 776

E.2.2 WKB Approximation 778

E.2.3 Bessel Functions on the Complex 𝜈 Plane 791

E.3 Almost Equal Argument and Order 795

E.3.1 Approximations in Terms of Airy Functions 796

E.3.2 Approximations in Terms of Normalized Airy Functions 797

E.3.3 Zeros in the Neighborhood of the Points 𝜈 = ±z 798

References 799

Index 801

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