# Electromagnetic Radiation, Scattering, and Diffraction

IEEE/OUP Series on Electromagnetic Wave Theory (formerly IEEE only), Series Editor: Donald G. Dudley.

1. Edition January 2022

1152 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-81051-3**

Electromagnetic Radiation, Scattering, and Diffraction

Discover a graduate-level text for students specializing in electromagnetic wave radiation, scattering, and diffraction for engineering applications

In Electromagnetic Radiation, Scattering and Diffraction, distinguished authors Drs. Prabhakar H. Pathak and Robert J. Burkholder deliver a thorough exploration of the behavior of electromagnetic fields in radiation, scattering, and guided wave environments. The book tackles its subject from first principles and includes coverage of low and high frequencies. It stresses physical interpretations of the electromagnetic wave phenomena along with their underlying mathematics.

The authors emphasize fundamental principles and provide numerous examples to illustrate the concepts contained within. Students with a limited undergraduate electromagnetic background will rapidly and systematically advance their understanding of electromagnetic wave theory until they can complete useful and important graduate-level work on electromagnetic wave problems.

Electromagnetic Radiation, Scattering and Diffraction also serves as a practical companion for students trying to simulate problems with commercial EM software and trying to better interpret their results. Readers will also benefit from the breadth and depth of topics, such as:

* Basic equations governing all electromagnetic (EM) phenomena at macroscopic scales are presented systematically. Stationary and relativistic moving boundary conditions are developed. Waves in planar multilayered isotropic and anisotropic media are analyzed.

* EM theorems are introduced and applied to a variety of useful antenna problems. Modal techniques are presented for analyzing guided wave and periodic structures. Potential theory and Green's function methods are developed to treat interior and exterior EM problems.

* Asymptotic High Frequency methods are developed for evaluating radiation Integrals to extract ray fields. Edge and surface diffracted ray fields, as well as surface, leaky and lateral wave fields are obtained. A collective ray analysis for finite conformal antenna phased arrays is developed.

* EM beams are introduced and provide useful basis functions. Integral equations and their numerical solutions via the method of moments are developed. The fast multipole method is presented. Low frequency breakdown is studied. Characteristic modes are discussed.

Perfect for graduate students studying electromagnetic theory, Electromagnetic Radiation, Scattering, and Diffraction is an invaluable resource for professional electromagnetic engineers and researchers working in this area.

Preface xxi

Acknowledgments xxv

1 Maxwell's Equations, Constitutive Relations, Wave Equation and Polarization 1

1.1 Introductory Comments 1

1.2 Maxwell's Equations 5

1.3 Constitutive Relations 10

1.4 Frequency Domain Fields 15

1.5 Kramers-Kronig Relationship 19

1.6 Vector and Scalar Wave Equations 21

1.6.1 Vector Wave Equations for EM Fields 21

1.6.2 Scalar Wave Equations for EM Fields 22

1.7 Separable Solutions of the Source Free Wave Equation in Rectangular Coordinates and for Isotropic Homogeneous Media. Plane Waves 23

1.8 Polarization of Plane Waves, Poincare Sphere and Stokes Parameters 29

1.8.1 Polarization States 29

1.8.2 General Elliptical Polarization 32

1.8.3 Decomposition of a Polarization State into Circularly Polarized Components 36

1.8.4 Poincare Sphere for Describing Polarization States 37

1.9 Phase and Group Velocity 40

1.10 Separable Solutions of the Source Free Wave Equation in Cylindrical and Spherical Coordinates and for Isotropic Homogeneous Media 44

1.10.1 Source Free Cylindrical Wave Solutions 44

1.10.2 Source Free Spherical Wave Solutions 48

References 51

2 EM Boundary and Radiation Conditions 52

2.1 EM Field Behavior Across a Boundary Surface 52

2.2 Radiation Boundary Condition 60

2.3 Boundary Conditions at a Moving Interface 63

2.3.1 Non-Relativistic Moving Boundary Conditions 63

2.3.2 Derivation of the Non-Relativistic Field Transformations 66

2.3.3 EM Field Transformations Based on the Special Theory of Relativity 69

2.4 Constitutive Relations for a Moving Medium 84

References 85

3 Plane Wave Propagation in Planar Layered Media 87

3.1 Introduction 87

3.2 Plane Wave Reflection from a Planar Boundary Between Two Different Media 87

3.2.1 Perpendicular Polarization Case 88

3.2.2 Parallel Polarization Case 93

3.2.3 Brewster Angle _b 97

3.2.4 Critical Angle _c 100

3.2.5 Plane Wave Incident on a Lossy Half Space 104

3.2.6 Doppler Shift for Wave Reflection from a Moving Mirror 110

3.3 Reflection and Transmission of a Plane Wave Incident on a Planar Stratified Isotropic Medium Using a Transmission Matrix Approach 112

3.4 Plane Waves in Anisotropic Homogeneous Media 119

3.5 State Space Formulation for Waves in Planar Anisotropic Layered Meia 135

3.5.1 Development of State Space Based Field Equations 135

3.5.2 Reflection and Transmission of Plane Waves at the Interface Between Two Anisotropic Half Spaces 139

3.5.3 Transmission Type Matrix Analysis of Plane Waves in Multilayered Anisotropic Media 142

References 143

4 Plane Wave Spectral Representation for EM Fields 144

4.1 Introduction 144

4.2 PWS Development 144

References 155

5 Electromagnetic Potentials and Fields of Sources in Unbounded Regions 156

5.1 Introduction to Vector and Scalar Potentials 156

5.2 Construction of the Solution for A 160

5.3 Calculation of Fields from Potentials 165

5.4 Time Dependent Potentials for Sources and Fields in Unbounded Regions 176

5.5 Potentials and Fields of a Moving Point Charge 185

5.6 Cerenkov Radiation 192

5.7 Direct Calculation of Fields of Sources in Unbounded Regions Using a Dyadic Green's Function 195

5.7.1 Fields of Sources in Unbounded, Isotropic, Homogeneous Media in Terms of a Closed Form Representation of Green's Dyadic, G0 195

5.7.2 On the Singular Nature of G0.r?r¨/ for Observation Points Within the Source Region 197

5.7.3 Representation of the Green's Dyadic G0 in Terms of an Integral in the Wavenumber .k/ Space 201

5.7.4 Electromagnetic Radiation by a Source in a General Bi-anisotropic Medium Using a Green's Dyadic Ga in k-Space 208

References 209

6 Electromagnetic Field Theorems and Related Topics 211

6.1 Conservation of Charge 211

6.2 Conservation of Power 212

6.3 Conservation of Momentum 218

6.4 Radiation Pressure 225

6.5 Duality Theorem 235

6.6 Reciprocity Theorems and Conservation of Reactions 242

6.6.1 The Lorentz Reciprocity Theorem 243

6.6.2 Reciprocity Theorem for Bianisotropic Media 249

6.7 Uniqueness Theorem 251

6.8 Image Theorems 254

6.9 Equivalence Theorems 258

6.9.1 Volume Equivalence Theorem for EM Scattering 258

6.9.2 A Surface Equivalence Theorem for EM Scattering 260

6.9.3 A Surface Equivalence Theorem for Antennas 270

6.10 Antenna Impedance 278

6.11 Antenna Equivalent Circuit 282

6.12 The Receiving Antenna Problem 282

6.13 Expressions for Antenna Mutual Coupling Based on Generalized Reciprocity Theorems 287

6.13.1 Circuit Form of the Reciprocity Theorem for Antenna Mutual Coupling 287

6.13.2 A Mixed Circuit Field Form of a Generalized Reciprocity Theorem for Antenna Mutual Coupling 292

6.13.3 A Mutual Admittance Expression for Slot Antennas 294

6.13.4 Antenna Mutual Coupling, Reaction Concept, and Antenna Measurements 296

6.14 Relation Between Antenna and Scattering Problems 297

6.14.1 Exterior Radiation by a Slot Aperture Antenna Configuration 297

6.14.2 Exterior Radiation by a Monopole Antenna Configuration 299

6.15 Radar Cross Section 308

6.16 Antenna Directive Gain 309

6.17 Field Decomposition Theorem 311

References 313

7 Modal Techniques for the Analysis of Guided Waves, Resonant Cavities and Periodic Structures 314

7.1 On Modal Analysis of Some Guided Wave Problems 314

7.2 Classification of Modal Fields in Uniform Guiding Structures 314

7.2.1 TEMz Guided waves 315

7.3 TMz Guided Waves 325

7.4 TEz Guided Waves 328

7.5 Modal Expansions in Closed Uniform Waveguides 330

7.5.1 TMz Modes 331

7.5.2 TEz Modes 332

7.5.3 Orthogonality of Modes in Closed Perfectly Conducting Uniform Waveguides 334

7.6 Effect of Losses in Closed Guided Wave Structures 337

7.7 Source Excited Uniform Closed Perfectly Conducting Waveguides 338

7.8 An Analysis of Some Closed Metallic Waveguides 342

7.8.1 Modes in a Parallel Plate Waveguide 342

7.8.2 Modes in a Rectangular Waveguide 350

7.8.3 Modes in a Circular Waveguide 358

7.8.4 Coaxial Waveguide 364

7.8.5 Obstacles and Discontinuities in Waveguides 366

7.8.6 Modal Propagation Past a Slot in a Waveguide 379

7.9 Closed and Open Waveguides Containing Penetrable Materials and Coatings 383

7.9.1 Material Loaded Closed PEC Waveguide 384

7.9.2 Material Slab Waveguide 388

7.9.3 Grounded Material Slab Waveguide 395

7.9.4 The Goubau Line 395

7.9.5 Circular Cylindrical Optical Fiber Waveguides 398

7.10 Modal Analysis of Resonators 400

7.10.1 Rectangular Waveguide Cavity Resonator 402

7.10.2 Circular Waveguide Cavity Resonator 406

7.10.3 Dielectric Resonators 408

7.11 Excitation of Resonant Cavities 409

7.12 Modal Analysis of Periodic Arrays 411

7.12.1 Floquet Modal Analysis of an Infinite Planar Periodic Array of Electric Current Sources 412

7.12.2 Floquet Modal Analysis of an Infinite Planar Periodic Array of Current Sources Configured in a Skewed Grid 419

7.13 Higher Order Floquet Modes and Associated Grating Lobe Circle Diagrams for Infinite Planar Periodic Arrays 422

7.13.1 Grating Lobe Circle Diagrams 422

7.14 On Waves Guided and Radiated by Periodic Structures 425

7.15 Scattering by a Planar Periodic Array 430

7.15.1 Analysis of the EM Plane Wave Scattering by an Infinite Periodic Slot Array in a Planar PEC Screen 432

7.16 Finite 1-D and 2-D Periodic Array of Sources 437

7.16.1 Analysis of Finite 1-D Periodic Arrays for the Case of Uniform Source Distribution and Far Zone Observation 437

7.16.2 Analysis of Finite 2-D Periodic Arrays for the Case of Uniform Distribution and Far Zone Observation 444

7.16.3 Floquet Modal Representation for Near and Far Fields of 1-D Non Uniform Finite Periodic Array Distributions 446

7.16.4 Floquet Modal Representation for Near and Far Fields of 2-D Non Uniform Planar Periodic Finite Array Distributions 449

References 451

8 Green's Functions for the Analysis of One Dimensional Source Excited Wave Problems 453

8.1 Introduction to the Sturm-Liouville Form of Differential Equation for 1-D Wave Problems 453

8.2 Formulation of the Solution to the Sturm-Liouville Problem via the 1-D Green's Function Approach 456

8.3 Conditions Under Which the Green's Function is Symmetric 463

8.4 Construction of the Green's Function G.x?x¨/ 464

8.4.1 General procedure to obtain G.x?x¨/ 464

8.5 Alternative Simplified Construction of G.x?x¨/ Valid for the Symmetric Case 466

8.6 On the Existence and Uniqueness of G.x?x¨/ 483

8.7 Eigenfunction Expansion Representation for G.x?x¨/ 483

8.8 Delta Function Completeness Relation and the Construction of Eigenfunctions from G.x?x¨/ = U.x/T .x>/_W 488

8.9 Explicit Representation of G.x?x¨/ Using Step Functions 519

References 521

9 Applications of One Dimensional Green's Function Approach for the Analysis of Single and Coupled Set of EM Source Excited Transmission Lines 522

9.1 Introduction 522

9.2 Analytical Formulation for a Single Transmission Line Made Up of Two Conductors 522

9.3 Wave Solution for the Two Conductor Lines When There Are No Impressed Sources Distributed Anywhere Within the Line 525

9.4 Wave Solution for the Case of Impressed Sources 527

9.5 Excitation of a Two Conductor Transmission Line by an Externally Incident Electromagnetic Wave 541

9.6 A Matrix Green's Function Approach for Analyzing a Set of Coupled Transmission Lines 543

9.7 Solution to the Special Case of Two Coupled Lines .N = 2/ with Homogeneous Dirichlet or Neumann End Conditions 546

9.8 Development of the Multiport Impedance Matrix for a Set of Coupled Transmission Lines 551

9.9 Coupled Transmission Line Problems with Voltage Sources and Load Impedances at the End Terminals 552

References 553

10 Green's Functions for the Analysis of Two and Three Dimensional Source Excited Scalar and EM Vector Wave Problems 554

10.1 Introduction 554

10.2 General Formulation for Source Excited 3-D Separable ScalarWave Problems Using Green's Functions 555

10.3 General Procedure for Construction of Scalar 566

10.4 General Procedure for Construction of Scalar 2-D and 3-D Green's Functions in Cylindrical Coordinates 569

10.5 General Procedure for Construction of Scalar 3-D Green's Functions in Spherical Coordinates 572

10.6 General Formulation for Source Excited 3-D Separable EM Vector Wave Problems Using Dyadic Green's Functions 575

10.7 Some Specific Green's Functions for 2-D Problems 583

10.7.1 Fields of a Uniform Electric Line Source 583

10.7.2 Fields of an Infinite Periodic Array of Electric Line Sources 590

10.7.3 Line Source Excited PEC Circular Cylinder Green's Function 591

10.7.4 A Cylindrical Wave Series Expansion for H.2/ 596

10.7.5 Line Source Excitation of a PEC Wedge 598

10.7.6 Line Source Excitation of a PEC Parallel Plate Waveguide 602

10.7.7 The Fields of a Line Dipole Source 606

10.7.8 Fields of a Magnetic Line Source on an Infinite Planar Impedance Surface 608

10.7.9 Fields of a Magnetic Line Dipole Source on an Infinite Planar Impedance Surface 612

10.7.10 Circumferentially Propagating Surface Fields of a Line Source Excited Impedance Circular Cylinder 614

10.7.11 Analysis of Circumferentially Propagating Waves for a Line Dipole Source Excited Impedance Circular Cylinder 617

10.7.12 Fields of a Traveling Wave Line Source 619

10.7.13 Traveling Wave Line Source Excitation of a PEC Wedge and a PEC Cylinder 620

10.8 Examples of Some Alternative Representations of Green's Functions for Scalar 3-D Point Source Excited Cylinders, Wedges and Spheres 623

10.8.1 3-D Scalar Point Source Excited Circular Cylinder Green's Function 623

10.8.2 3-D Scalar Point Source Excitation of a Wedge 630

10.8.3 Angularly and Radially Propagating 3-D Scalar Point Source Green's Function for a Sphere 632

10.8.4 Kontorovich-Lebedev Transform and MacDonald Based Approaches for Constructing an Angularly Propagating 3-D Point Source Scalar Wedge Green's Function 640

10.8.5 Analysis of the Fields of a Vertical Electric or Magnetic Current Point Source on a PEC Sphere 647

10.9 General Procedure for Construction of EM Dyadic Green's Functions for Source Excited Separable Canonical Problems via Scalar Green's Functions 652

10.9.1 Summary of Procedure to Obtain the EM Fields of Arbitrarily Oriented Point Sources Exciting Canonical Separable Configurations 653

10.10 Completeness of the Eigenfunction Expansion of 665

References 669

11 Method of Factorization and the Wiener-Hopf Technique for Analyzing Two Part EM Wave Problems 670

11.1 The Wiener-Hopf Procedure 670

11.2 The Dual Integral Equation Approach 682

11.3 The Jones Method 691

References 696

12 Integral Equation Based Methods for the Numerical Solution of Non-Separable EM Radiation and Scattering Problems 697

12.1 Introduction 697

12.2 Boundary Integral Equations 697

12.2.1 The Electric Field Integral Equation 699

12.2.2 The Magnetic Field Integral Equation 700

12.2.3 Combined Field and Combined Source Integral Equations 701

12.2.4 Impedance Boundary Condition 702

12.2.5 Boundary Integral Equation for a Homogeneous Material Volume 703

12.3 Volume Integral Equations 705

12.4 The Numerical Solution of Integral Equations 706

12.4.1 The Minimum Square-Error Method 706

12.4.2 The Method of Moments 708

12.4.3 Simplification of the MoM Impedance Matrix Integrals 710

12.4.4 Expansion and Testing Functions 713

12.4.5 Low-Frequency Break-Down 718

12.5 Iterative Solution of Large MoM Matrices 720

12.5.1 Fast Iterative Solution of MoM Matrix Equations 721

12.5.2 The Fast Multipole Method 725

12.5.3 Multi-level FMM and Fast Fourier Transform FMM 730

12.6 Antenna Modeling with the Method of Moments 732

12.7 Aperture Coupling with the Method of Moments 733

12.8 Physical Optics Methods 736

12.8.1 Physical optics for a PEC surface 736

12.8.2 Iterative physical optics 738

References 739

13 Introduction to Characteristic Modes 742

13.1 Introduction 742

13.2 Characteristic Modes from the EFIE for a Conducting Surface 743

13.2.1 Electric Field Integral Equation and Radiation Operator 743

13.2.2 Eigenfunctions of the Electric Field Radiation Operator 743

13.2.3 Characteristic Modes from the EFIE Impedance Matrix 745

13.3 Computation of Characteristic Modes 746

13.4 Solution of the EFIE using Characteristic Modes 748

13.5 Tracking Characteristic Modes with Frequency 749

13.6 Antenna Excitation using Characteristic Modes 749

References 750

14 Asymptotic Evaluation of Radiation and Diffraction Type Integrals for High Frequencies 752

14.1 Introduction 752

14.2 Steepest Descent Techniques for the Asymptotic Evaluation of Radiation Integrals 752

14.2.1 Topology of the Exponent in the Integrand Containing a First Order Saddle Point 753

14.2.2 Asymptotic Evaluation of Integrals Containing a First Order Saddle Point in its Integrand which is Free of Singularities 756

14.2.3 Asymptotic Evaluation of Integrals containing a Higher Order Saddle Point in its Integrand which is Free of Singularities 760

14.2.4 Pauli-Clemmow Method for the Asymptotic Evaluation of Integrals containing a First Order Saddle Point Near a Simple Pole Singularity 763

14.2.5 Van der Waerden Method for the Asymptotic Evaluation of Integrals containing a First Order Saddle Point Near a Simple Pole Singularity 773

14.2.6 Relationship Between PCM and VWM Leading to a Generalized PCM (or GPC) Solution 775

14.2.7 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First Order Saddle Point and a Nearby Double Pole 777

14.2.8 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First Order Saddle Point and Two Nearby First Order Poles 779

14.2.9 An Extension of VWM for Asymptotic Evaluation of an Integral Containing a First Order Saddle Point and a Nearby Double Pole 783

14.2.10 Non-Uniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Branch Point 784

14.2.11 Uniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Nearby Branch Point 789

14.3 Asymptotic Evaluation of Integrals with End Points 791

14.3.1 Watson's Lemma for Integrals 792

14.3.2 Generalized Watson's Lemma for Integrals 792

14.3.3 Integration by Parts for Asymptotic Evaluation of a Class of Integrals 792

14.4 Asymptotic Evaluation of Radiation Integrals Based on the Stationary Phase Method 794

14.4.1 Stationary Phase Evaluation of 1-D Infinite Integrals 794

14.4.2 Non-Uniform Stationary Phase Evaluation of 1-D Integrals with End Points 795

14.4.3 Uniform Stationary Phase Evaluation of 1-D Integrals with a Nearby End Point 796

14.4.4 Non-Uniform Stationary Phase Evaluation of 2-D Infinite Integrals 801

References 816

15 Physical and Geometrical Optics 818

15.1 The Physical Optics (PO) Approximation for PEC Surfaces 818

15.2 The Geometrical Optics (GO) Ray Field 820

15.3 GO Transport Singularities 824

15.4 Wavefronts, Stationary Phase and GO 829

15.5 GO Incident and Reflected Ray Fields 832

15.6 Uniform GO Valid at Smooth Caustics 840

References 854

16 Geometrical and Integral Theories of Diffraction 855

16.1 Geometrical Theory of Diffraction and Its Uniform Version 855

16.2 UTD for an Edge in an Otherwise Smooth PEC Surface 861

16.3 UTD Slope Diffraction for an Edge 872

16.4 An Alternative Uniform Solution (the UAT) for Edge Diffraction 874

16.5 UTD Solutions for Fields of Sources in the Presence of Smooth PEC Convex Surfaces 874

16.5.1 UTD Analysis of the Scattering by a Smooth, Convex Surface 876

16.5.2 UTD for the Radiation by Antennas on a Smooth, Convex Surface 886

16.5.3 UTD Analysis of the Surface Fields of Antennas on a Smooth, Convex Surface 902

16.6 UTD for a Vertex 913

16.7 UTD for Edge Excited Surface Rays 916

16.8 The Equivalent Line Current Method 922

16.8.1 Line Type ECM for Edge-Diffracted Ray Caustic Field Analysis 922

16.9 Equivalent Line Current Method for Interior PEC Waveguide Problems 927

16.9.1 TEy Case 929

16.9.2 TMy Case 932

16.10 The Physical Theory of Diffraction 933

16.10.1 PTD for Edged Bodies- A Canonical Edge Diffraction Problem in the PTD Development 936

16.10.2 Details of PTD for 3-D Edged Bodies 937

16.10.3 Reduction of PTD to 2-D Edged Bodies 939

16.11 On the PTD for Aperture Problems 940

16.12 Time-Domain Uniform Geometrical Theory of Diffraction 940

16.12.1 Introductory Comments 940

16.12.2 Analytic Time Transform (ATT) 941

16.12.3 TD-UTD for a General PEC Curved Wedge 942

References 945

17 Development of Asymptotic High Frequency Solutions to Some Canonical Problems 951

17.1 Introduction 951

17.2 Development of UTD Solutions for Some Canonical Wedge Diffraction Problems 951

17.2.1 Scalar 2-D Line Source Excitation of a Wedge 952

17.2.2 Scalar Plane Wave Excitation of a Wedge 958

17.2.3 Scalar Spherical Wave Excitation of a Wedge 960

17.2.4 EM Plane Wave Excitation of a PEC Wedge 965

17.2.5 EM Conical Wave Excitation of a PEC Wedge 968

17.2.6 EM Spherical Wave Excitation of a PEC Wedge 971

17.3 Canonical Problem of Slope Diffraction by a PEC Wedge 974

17.4 Development of a UTD Solution for Scattering by a Canonical 2-D PEC Circular Cylinder and Its Generalization to a Convex Cylinder 978

17.4.1 Field Analysis for the Shadowed Part of the Transition Region 982

17.4.2 Field Analysis for the Illuminated Part of the Transition Region 985

17.5 A Collective UTD for an Efficient Ray Analysis of the Radiation by Finite Conformal Phased Arrays on Infinite PEC Circular Cylinders 991

17.5.1 Finite Axial Array on a Circular PEC Cylinder 992

17.5.2 Finite Circumferential Array on a Circular PEC Cylinder 998

17.6 Surface, Leaky, and Lateral Waves Associated with Planar Material Boundaries 1004

17.6.1 Introduction 1004

17.6.2 The EM Fields of a Magnetic Line Source on a Uniform Planar Impedance Surface 1004

17.6.3 EM Surface and Leaky Wave Fields of a Uniform Line Source over a Planar Grounded Material Slab 1012

17.6.4 An Analysis of the Lateral Wave Phenomena Arising in the Problem of a Vertical Electric Point Current Source over a Dielectric Half Space 1020

17.7 Surface Wave Diffraction by a Planar, Two-Part Impedance Surface: Development of a Ray Solution 1032

17.7.1 TEz Case 1032

17.7.2 TMz Case 1036

17.8 Ray Solutions for Special Cases of Discontinuities in Nonconducting or Penetrable Boundaries 1038

References 1040

18 EM Beams and Some Applications 1042

18.1 Introduction 1042

18.2 Astigmatic Gaussian Beams 1043

18.2.1 Paraxial Wave Equation Solutions 1043

18.2.2 2D Beams 1044

18.2.3 3D Astigmatic Gaussian Beams 1047

18.2.4 3D Gaussian Beam from a Gaussian Aperture Distribution 1048

18.2.5 Reflection of Astigmatic Gaussian Beams 1050

18.3 Complex Source Beams and Relation to GBs 1051

18.3.1 Introduction to Complex Source Beams 1051

18.3.2 Complex Source Beam from Scalar Green's Function 1051

18.3.3 Representation of arbitrary EM fields by a CSB expansion 1054

18.3.4 Edge Diffraction of an Incident CSB by a Curved Conducting Wedge 1056

18.4 Pulsed Complex Source Beams in the Time Domain 1061

References 1063

A Coordinate Systems, Vectors and Dyadics 1065

B The Total Time Derivative of a Time Varying Flux Density Integrated Over a Moving Surface 1072

C The Delta Function 1075

D Transverse Fields in Terms of Axial Field Components for TMz and TEz Waves Guided Along z 1078

E Two Di_erent Representations for Partial Poisson Sum Formulas and Their Equivalence 1080

F Derivation of 1-D Green's Second Identity 1082

G Green's Second Identity for 3-D Scalar, Vector and Vector-DyadicWave Fields 1083

H Formal Decomposition and Factorization Formulas 1086

I On the Transition Function F 1089

J On the Branch Cuts Commonly Encountered in the Evaluation of Spectral Wave Integrals 1092

K On the Steepest Descent Path (SDP) for Spectral Wave Integrals 1096

L Parameters Used in the Uniform GO Solution for the Lit and Shadow Sides of a Smooth Caustic 1099

M Asymptotic Approximations of Hankel Functions for Large Argument and Various Orders 1101

ROBERT J. BURKHOLDER, PhD, is a Research Professor Emeritus at Ohio State University in the Department of Electrical and Computer Engineering, and the ElectroScience Lab. He has over 30 years of experience in theoretical and numerical modeling methods for realistic EM radiation, propagation, and scattering applications.