Electromagnetic Radiation, Scattering, and Diffraction

About the Authors xix 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