PRABHAKAR H. PATHAK, PhD, is Professor Emeritus at Ohio State University in the Department of Electrical and Computer Engineering, and the ElectroScience Lab. He is regarded as a co-developer of the Uniform Geometrical Theory of Diffraction (UTD). His research interests are in theoretical EM, and more recently in the development of ray, beam and hybrid methods for analyzing the EM fields of large conformal arrays and small antennas on large complex platforms (e.g., aircraft/spacecraft, etc.).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.

• About the Authors xviiPreface xixAcknowledgments xxiii1 Maxwell’s Equations, Constitutive Relations, Wave Equation, and Polarization 11.1 Introductory Comments 11.2 Maxwell’s Equations 51.3 Constitutive Relations 101.4 Frequency Domain Fields 151.5 Kramers-Kronig Relationship 191.6 Vector and Scalar Wave Equations 211.6.1 Vector Wave Equations for EM Fields 211.6.2 Scalar Wave Equations for EM Fields 221.7 Separable Solutions of the Source-Free Wave Equation in Rectangular Coordinates and for Isotropic Homogeneous Media. Plane Waves 231.8 Polarization of Plane Waves, Poincaré Sphere, and Stokes Parameters 291.8.1 Polarization States 291.8.2 General Elliptical Polarization 321.8.3 Decomposition of a Polarization State into Circularly Polarized Components 361.8.4 Poincaré Sphere for Describing Polarization States 371.9 Phase and Group Velocity 401.10 Separable Solutions of the Source-Free Wave Equation in Cylindrical and Spherical Coordinates and for Isotropic Homogeneous Media 441.10.1 Source-Free Cylindrical Wave Solutions 441.10.2 Source-Free Spherical Wave Solutions 48References 512 EM Boundary and Radiation Conditions 522.1 EM Field Behavior Across a Boundary Surface 522.2 Radiation Boundary Condition 602.3 Boundary Conditions at a Moving Interface 632.3.1 Nonrelativistic Moving Boundary Conditions 632.3.2 Derivation of the Nonrelativistic Field Transformations 662.3.3 EM Field Transformations Based on the Special Theory of Relativity 692.4 Constitutive Relations for a Moving Medium 84References 853 Plane Wave Propagation in Planar Layered Media 873.1 Introduction 873.2 Plane Wave Reflection from a Planar Boundary Between Two Different Media 873.2.1 Perpendicular Polarization Case 883.2.2 Parallel Polarization Case 933.2.3 Brewster Angle θ b 973.2.4 Critical Angle θ c 1003.2.5 Plane Wave Incident on a Lossy Half Space 1043.2.6 Doppler Shift for Wave Reflection from a Moving Mirror 1103.3 Reflection and Transmission of a Plane Wave Incident on a Planar Stratified Isotropic Medium Using a Transmission Matrix Approach 1123.4 Plane Waves in Anisotropic Homogeneous Media 1193.5 State Space Formulation for Waves in Planar Anisotropic Layered Media 1353.5.1 Development of State Space Based Field Equations 1353.5.2 Reflection and Transmission of Plane Waves at the Interface Between Two Anisotropic Half Spaces 1393.5.3 Transmission Type Matrix Analysis of Plane Waves in Multilayered Anisotropic Media 142References 1434 Plane Wave Spectral Representation for EM Fields 1444.1 Introduction 1444.2 PWS Development 144References 1555 Electromagnetic Potentials and Fields of Sources in Unbounded Regions 1565.1 Introduction to Vector and Scalar Potentials 1565.2 Construction of the Solution for A 1605.3 Calculation of Fields from Potentials 1655.4 Time Dependent Potentials for Sources and Fields in Unbounded Regions 1765.5 Potentials and Fields of a Moving Point Charge 1855.6 Cerenkov Radiation 1925.7 Direct Calculation of Fields of Sources in Unbounded Regions Using a Dyadic Green’s Function 1955.7.1 Fields of Sources in Unbounded, Isotropic, Homogeneous Media in Terms of a Closed Form Representation of Green’s Dyadic, G 0 1955.7.2 On the Singular Nature of G 0 (r|r ′) for Observation Points Within the Source Region 1975.7.3 Representation of the Green’s Dyadic G 0 in Terms of an Integral in the Wavenumber (k) Space 2015.7.4 Electromagnetic Radiation by a Source in a General Bianisotropic Medium Using a Green’s Dyadic G a in k-Space 208References 2096 Electromagnetic Field Theorems and Related Topics 2116.1 Conservation of Charge 2116.2 Conservation of Power 2126.3 Conservation of Momentum 2186.4 Radiation Pressure 2256.5 Duality Theorem 2356.6 Reciprocity Theorems and Conservation of Reactions 2426.6.1 The Lorentz Reciprocity Theorem 2436.6.2 Reciprocity Theorem for Bianisotropic Media 2496.7 Uniqueness Theorem 2516.8 Image Theorems 2546.9 Equivalence Theorems 2586.9.1 Volume Equivalence Theorem for EM Scattering 2586.9.2 A Surface Equivalence Theorem for EM Scattering 2606.9.3 A Surface Equivalence Theorem for Antennas 2706.10 Antenna Impedance 2786.11 Antenna Equivalent Circuit 2826.12 The Receiving Antenna Problem 2826.13 Expressions for Antenna Mutual Coupling Based on Generalized Reciprocity Theorems 2876.13.1 Circuit Form of the Reciprocity Theorem for Antenna Mutual Coupling 2876.13.2 A Mixed Circuit Field Form of a Generalized Reciprocity Theorem for Antenna Mutual Coupling 2926.13.3 A Mutual Admittance Expression for Slot Antennas 2946.13.4 Antenna Mutual Coupling, Reaction Concept, and Antenna Measurements 2966.14 Relation Between Antenna and Scattering Problems 2976.14.1 Exterior Radiation by a Slot Aperture Antenna Configuration 2976.14.2 Exterior Radiation by a Monopole Antenna Configuration 2996.15 Radar Cross Section 3086.16 Antenna Directive Gain 3096.17 Field Decomposition Theorem 311References 3137 Modal Techniques for the Analysis of Guided Waves, Resonant Cavities, and Periodic Structures 3147.1 On Modal Analysis of Some Guided Wave Problems 3147.2 Classification of Modal Fields in Uniform Guiding Structures 3147.2.1 TEM z Guided waves 3157.3 TM z Guided Waves 3257.4 TE z Guided Waves 3287.5 Modal Expansions in Closed Uniform Waveguides 3307.5.1 TM z Modes 3317.5.2 TE z Modes 3327.5.3 Orthogonality of Modes in Closed Perfectly Conducting Uniform Waveguides 3347.6 Effect of Losses in Closed Guided Wave Structures 3377.7 Source Excited Uniform Closed Perfectly Conducting Waveguides 3387.8 An Analysis of Some Closed Metallic Waveguides 3427.8.1 Modes in a Parallel Plate Waveguide 3427.8.2 Modes in a Rectangular Waveguide 3507.8.3 Modes in a Circular Waveguide 3587.8.4 Coaxial Waveguide 3647.8.5 Obstacles and Discontinuities in Waveguides 3667.8.6 Modal Propagation Past a Slot in a Waveguide 3797.9 Closed and Open Waveguides Containing Penetrable Materials and Coatings 3837.9.1 Material-Loaded Closed PEC Waveguide 3847.9.2 Material Slab Waveguide 3887.9.3 Grounded Material Slab Waveguide 3957.9.4 The Goubau Line 3957.9.5 Circular Cylindrical Optical Fiber Waveguides 3987.10 Modal Analysis of Resonators 4007.10.1 Rectangular Waveguide Cavity Resonator 4027.10.2 Circular Waveguide Cavity Resonator 4067.10.3 Dielectric Resonators 4087.11 Excitation of Resonant Cavities 4097.12 Modal Analysis of Periodic Arrays 4117.12.1 Floquet Modal Analysis of an Infinite Planar Periodic Array of Electric Current Sources 4127.12.2 Floquet Modal Analysis of an Infinite Planar Periodic Array of Current Sources Configured in a Skewed Grid 4197.13 Higher-Order Floquet Modes and Associated Grating Lobe Circle Diagrams for Infinite Planar Periodic Arrays 4227.13.1 Grating Lobe Circle Diagrams 4227.14 On Waves Guided and Radiated by Periodic Structures 4257.15 Scattering by a Planar Periodic Array 4307.15.1 Analysis of the EM Plane Wave Scattering by an Infinite Periodic Slot Array in a Planar PEC Screen 4327.16 Finite 1-D and 2-D Periodic Array of Sources 4377.16.1 Analysis of Finite 1-D Periodic Arrays for the Case of Uniform Source Distribution and Far Zone Observation 4377.16.2 Analysis of Finite 2-D Periodic Arrays for the Case of Uniform Distribution and Far Zone Observation 4447.16.3 Floquet Modal Representation for Near and Far Fields of 1-D Nonuniform Finite Periodic Array Distributions 4467.16.4 Floquet Modal Representation for Near and Far Fields of 2-D Nonuniform Planar Periodic Finite Array Distributions 449References 4518 Green’s Functions for the Analysis of One-Dimensional Source-Excited Wave Problems 4538.1 Introduction to the Sturm-Liouville Form of Differential Equation for 1-D Wave Problems 4538.2 Formulation of the Solution to the Sturm-Liouville Problem via the 1-D Green’s Function Approach 4568.3 Conditions Under Which the Green’s Function Is Symmetric 4638.4 Construction of the Green’s Function G(x|x′) 4648.4.1 General Procedure to Obtain G(x|x′) 4648.5 Alternative Simplified Construction of G(x|x′) Valid for the Symmetric Case 4668.6 On the Existence and Uniqueness of G(x|x′) 4838.7 Eigenfunction Expansion Representation for G(x|x′) 4838.8 Delta Function Completeness Relation and the Construction of Eigenfunctions from G(x|x′) = U (x<)T (x>) ∕ W 4888.9 Explicit Representation of G(x|x′) Using Step Functions 519References 5209 Applications of One-Dimensional Green’s Function Approach for the Analysis of Single and Coupled Set of EM Source Excited Transmission Lines 5229.1 Introduction 5229.2 Analytical Formulation for a Single Transmission Line Made Up of Two Conductors 5229.3 Wave Solution for the Two Conductor Lines When There Are No Impressed Sources Distributed Anywhere Within the Line 5259.4 Wave Solution for the Case of Impressed Sources Placed Anywhere on a Two Conductor Line 5279.5 Excitation of a Two Conductor Transmission Line by an Externally Incident Electromagnetic Wave 5419.6 A Matrix Green’s Function Approach for Analyzing a Set of Coupled Transmission Lines 5439.7 Solution to the Special Case of Two Coupled Lines (N = 2) with Homogeneous Dirichlet or Neumann End Conditions 5469.8 Development of the Multiport Impedance Matrix for a Set of Coupled Transmission Lines 5519.9 Coupled Transmission Line Problems with Voltage Sources and Load Impedances at the End Terminals 552References 55310 Green’s Functions for the Analysis of Two- and Three-Dimensional Source- Excited Scalar and EM Vector Wave Problems 55410.1 Introduction 55410.2 General Formulation for Source-Excited 3-D Separable Scalar Wave Problems Using Green’s Functions 55510.3 General Procedure for Construction of Scalar 2-D and 3-D Green’s Function in Rectangular Coordinates 56610.4 General Procedure for Construction of Scalar 2-D and 3-D Green’s Functions in Cylindrical Coordinates 56910.5 General Procedure for Construction of Scalar 3-D Green’s Functions in Spherical Coordinates 57210.6 General Formulation for Source-Excited 3-D Separable EM Vector Wave Problems Using Dyadic Green’s Functions 57510.7 Some Specific Green’s Functions for 2-D Problems 58310.7.1 Fields of a Uniform Electric Line Source 58310.7.2 Fields of an Infinite Periodic Array of Electric Line Sources 59010.7.3 Line Source-Excited PEC Circular Cylinder Green’s Function 59110.7.4 A Cylindrical Wave Series Expansion 59610.7.5 Line Source Excitation of a PEC Wedge 59810.7.6 Line Source Excitation of a PEC Parallel Plate Waveguide 60210.7.7 The Fields of a Line Dipole Source 60610.7.8 Fields of a Magnetic Line Source on an Infinite Planar Impedance Surface 60810.7.9 Fields of a Magnetic Line Dipole Source on an Infinite Planar Impedance Surface 61210.7.10 Circumferentially Propagating Surface Fields of a Line Source Excited Impedance Circular Cylinder 61410.7.11 Analysis of Circumferentially Propagating Waves for a Line Dipole Source-Excited Impedance Circular Cylinder 61710.7.12 Fields of a Traveling Wave Line Source 61910.7.13 Traveling Wave Line Source Excitation of a PEC Wedge and a PEC Cylinder 62010.8 Examples of Some Alternative Representations of Green’s Functions for Scalar 3-D Point Source-Excited Cylinders, Wedges and Spheres 62310.8.1 3-D Scalar Point Source-Excited Circular Cylinder Green’s Function 62310.8.2 3-D Scalar Point Source Excitation of a Wedge 63010.8.3 Angularly and Radially Propagating 3-D Scalar Point Source Green’s Function for a Sphere 63210.8.4 Kontorovich–Lebedev Transform and MacDonald Based Approaches for Constructing an Angularly Propagating 3-D Point Source Scalar Wedge Green’s Function 64010.8.5 Analysis of the Fields of a Vertical Electric or Magnetic Current Point Source on a PEC Sphere 64710.9 General Procedure for Construction of EM Dyadic Green’s Functions for Source-Excited Separable Canonical Problems via Scalar Green’s Functions 65210.9.1 Summary of Procedure to Obtain the EM Fields of Arbitrarily Oriented Point Sources Exciting Canonical Separable Configurations 65310.10 Completeness of the Eigenfunction Expansion of the Dyadic Green's Function at the Source Point 665References 66911 Method of Factorization and the Wiener–Hopf Technique for Analyzing Two- Part EM Wave Problems 67011.1 The Wiener–Hopf Procedure 67011.2 The Dual Integral Equation Approach 68211.3 The Jones Method 691References 69612 Integral Equation-Based Methods for the Numerical Solution of Nonseparable EM Radiation and Scattering Problems 69712.1 Introduction 69712.2 Boundary Integral Equations 69712.2.1 The Electric Field Integral Equation (EFIE) 69912.2.2 The Magnetic Field Integral Equation (MFIE) 70012.2.3 Combined Field and Combined Source Integral Equations 70112.2.4 Impedance Boundary Condition 70212.2.5 Boundary Integral Equation for a Homogeneous Material Volume 70312.3 Volume Integral Equations 70512.4 The Numerical Solution of Integral Equations 70612.4.1 The Minimum Square-Error Method 70612.4.2 The Method of Moments (MoM) 70812.4.3 Simplification of the MoM Impedance Matrix Integrals 71012.4.4 Expansion and Testing Functions 71312.4.5 Low-Frequency Break-Down 71812.5 Iterative Solution of Large MoM Matrices 72012.5.1 Fast Iterative Solution of MoM Matrix Equations 72112.5.2 The Fast Multipole Method (FMM) 72512.5.3 Multilevel FMM and Fast Fourier Transform FMM 73012.6 Antenna Modeling with the Method of Moments 73212.7 Aperture Coupling with the Method of Moments 73412.8 Physical Optics Methods 73612.8.1 Physical Optics for a PEC Surface 73612.8.2 Iterative Physical Optics 738References 74013 Introduction to Characteristic Modes 74213.1 Introduction 74213.2 Characteristic Modes from the EFIE for a Conducting Surface 74313.2.1 Electric Field Integral Equation and Radiation Operator 74313.2.2 Eigenfunctions of the Electric Field Radiation Operator 74313.2.3 Characteristic Modes from the EFIE Impedance Matrix 74513.3 Computation of Characteristic Modes 74613.4 Solution of the EFIE Using Characteristic Modes 74813.5 Tracking Characteristic Modes with Frequency 74913.6 Antenna Excitation Using Characteristic Modes 749References 75014 Asymptotic Evaluation of Radiation and Diffraction Type Integrals for High Frequencies 75214.1 Introduction 75214.2 Steepest Descent Techniques for the Asymptotic Evaluation of Radiation Integrals 75214.2.1 Topology of the Exponent in the Integrand Containing a First-Order Saddle Point 75314.2.2 Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point in Its Integrand Which Is Free of Singularities 75614.2.3 Asymptotic Evaluation of Integrals Containing a Higher-Order Saddle Point in Its Integrand Which Is Free of Singularities 76014.2.4 Pauli–Clemmow Method (PCM) for the Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point Near a Simple Pole Singularity 76314.2.5 Van der Waerden Method (VWM) for the Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point Near a Simple Pole Singularity 77314.2.6 Relationship Between PCM and VWM Leading to a Generalized PCM (or GPC) Solution 77514.2.7 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and a Nearby Double Pole 77714.2.8 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and Two Nearby First-Order Poles 77914.2.9 An Extension of VWM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and a Nearby Double Pole 78314.2.10 Nonuniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Branch Point 78414.2.11 Uniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Nearby Branch Point 78914.3 Asymptotic Evaluation of Integrals with End Points 79114.3.1 Watson’s Lemma for Integrals 79214.3.2 Generalized Watson’s Lemma for Integrals 79214.3.3 Integration by Parts for Asymptotic Evaluation of a Class of Integrals 79214.4 Asymptotic Evaluation of Radiation Integrals Based on the Stationary Phase Method 79414.4.1 Stationary Phase Evaluation of 1-D Infinite Integrals 79414.4.2 Nonuniform Stationary Phase Evaluation of 1-D Integrals with End Points 79514.4.3 Uniform Stationary Phase Evaluation of 1-D Integrals with a Nearby End Point 79614.4.4 Nonuniform Stationary Phase Evaluation of 2-D Infinite Integrals 801References 81615 Physical and Geometrical Optics 81815.1 The Physical Optics (PO) Approximation for PEC Surfaces 81815.2 The Geometrical Optics (GO) Ray Field 82015.3 GO Transport Singularities 82415.4 Wavefronts, Stationary Phase, and GO 82815.5 GO Incident and Reflected Ray Fields 83215.6 Uniform GO Valid at Smooth Caustics 840References 85416 Geometrical and Integral Theories of Diffraction 85516.1 Geometrical Theory of Diffraction and Its Uniform Version (UTD) 85516.2 UTD for an Edge in an Otherwise Smooth PEC Surface 86116.3 UTD Slope Diffraction for an Edge 87216.4 An Alternative Uniform Solution (the UAT) for Edge Diffraction 87416.5 UTD Solutions for Fields of Sources in the Presence of Smooth PEC Convex Surfaces 87416.5.1 UTD Analysis of the Scattering by a Smooth, Convex Surface 87616.5.2 UTD for the Radiation by Antennas on a Smooth, Convex Surface 88516.5.3 UTD Analysis of the Surface Fields of Antennas on a Smooth, Convex Surface 90116.6 UTD for a Vertex 91316.7 UTD for Edge-Excited Surface Rays 91516.8 The Equivalent Line Current Method (ECM) 92116.8.1 Line Type ECM for Edge-Diffracted Ray Caustic Field Analysis 92216.9 Equivalent Line Current Method for Interior PEC Waveguide Problems 92616.9.1 TE y Case 92716.9.2 TM y Case 93216.10 The Physical Theory of Diffraction (PTD) 93316.10.1 PTD for Edged Bodies - A Canonical Edge Diffraction Problem in the PTD Development 93616.10.2 Details of PTD for 3-D Edged Bodies 93716.10.3 Reduction of PTD to 2-D Edged Bodies 93916.11 On the PTD for Aperture Problems 94016.12 Time-Domain Uniform Geometrical Theory of Diffraction (TD-UTD) 94016.12.1 Introductory Comments 94016.12.2 Analytic Time Transform (ATT) 94116.12.3 TD-UTD for a General PEC Curved Wedge 942References 94517 Development of Asymptotic High-Frequency Solutions to Some Canonical Problems 95117.1 Introduction 95117.2 Development of UTD Solutions for Some Canonical Wedge Diffraction Problems 95117.2.1 Scalar 2-D Line Source Excitation of a Wedge 95217.2.2 Scalar Plane Wave Excitation of a Wedge 95817.2.3 Scalar Spherical Wave Excitation of a Wedge 96017.2.4 EM Plane Wave Excitation of a PEC Wedge 96517.2.5 EM Conical Wave Excitation of a PEC Wedge 96817.2.6 EM Spherical Wave Excitation of a PEC Wedge 97117.3 Canonical Problem of Slope Diffraction by a PEC Wedge 97417.4 Development of a UTD Solution for Scattering by a Canonical 2-D PEC Circular Cylinder and Its Generalization to a Convex Cylinder 97817.4.1 Field Analysis for the Shadowed Part of the Transition Region 98217.4.2 Field Analysis for the Illuminated Part of the Transition Region 98517.5 A Collective UTD for an Efficient Ray Analysis of the Radiation by Finite Conformal Phased Arrays on Infinite PEC Circular Cylinders 99117.5.1 Finite Axial Array on a Circular PEC Cylinder 99217.5.2 Finite Circumferential Array on a Circular PEC Cylinder 99917.6 Surface, Leaky, and Lateral Waves Associated with Planar Material Boundaries 100417.6.1 Introduction 100417.6.2 The EM Fields of a Magnetic Line Source on a Uniform Planar Impedance Surface 100417.6.3 EM Surface and Leaky Wave Fields of a Uniform Line Source over a Planar Grounded Material Slab 101117.6.4 An Analysis of the Lateral Wave Phenomena Arising in the Problem of a Vertical Electric Point Current Source over a Dielectric Half Space 101917.7 Surface Wave Diffraction by a Planar, Two-Part Impedance Surface: Development of a Ray Solution 103217.7.1 TE z Case 103317.7.2 TM z Case 103617.8 Ray Solutions for Special Cases of Discontinuities in Nonconducting or Penetrable Boundaries 1038References 103918 EM Beams and Some Applications 104218.1 Introduction 104218.2 Astigmatic Gaussian Beams 104318.2.1 Paraxial Wave Equation Solutions 104318.2.2 2-D Beams 104418.2.3 3-D Astigmatic Gaussian Beams 104718.2.4 3-D Gaussian Beam from a Gaussian Aperture Distribution 104818.2.5 Reflection of Astigmatic Gaussian Beams (GBs) 105018.3 Complex Source Beams and Relation to GBs 105118.3.1 Introduction to Complex Source Beams (CSBs) 105118.3.2 Complex Source Beam from Scalar Green’s Function 105118.3.3 Representation of Arbitrary EM Fields by a CSB Expansion 105418.3.4 Edge Diffraction of an Incident CSB by a Curved Conducting Wedge 105618.4 Pulsed Complex Source Beams in the Time Domain 1061Index 1105

PRABHAKAR H. PATHAK, PhD, is Professor Emeritus at Ohio State University in the Department of Electrical and Computer Engineering, and the ElectroScience Lab. He is regarded as a co-developer of the Uniform Geometrical Theory of Diffraction (UTD). His research interests are in theoretical EM, and more recently in the development of ray, beam and hybrid methods for analyzing the EM fields of large conformal arrays and small antennas on large complex platforms (e.g., aircraft/spacecraft, etc.).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.

• About the Authors xviiPreface xixAcknowledgments xxiii1 Maxwell’s Equations, Constitutive Relations, Wave Equation, and Polarization 11.1 Introductory Comments 11.2 Maxwell’s Equations 51.3 Constitutive Relations 101.4 Frequency Domain Fields 151.5 Kramers-Kronig Relationship 191.6 Vector and Scalar Wave Equations 211.6.1 Vector Wave Equations for EM Fields 211.6.2 Scalar Wave Equations for EM Fields 221.7 Separable Solutions of the Source-Free Wave Equation in Rectangular Coordinates and for Isotropic Homogeneous Media. Plane Waves 231.8 Polarization of Plane Waves, Poincaré Sphere, and Stokes Parameters 291.8.1 Polarization States 291.8.2 General Elliptical Polarization 321.8.3 Decomposition of a Polarization State into Circularly Polarized Components 361.8.4 Poincaré Sphere for Describing Polarization States 371.9 Phase and Group Velocity 401.10 Separable Solutions of the Source-Free Wave Equation in Cylindrical and Spherical Coordinates and for Isotropic Homogeneous Media 441.10.1 Source-Free Cylindrical Wave Solutions 441.10.2 Source-Free Spherical Wave Solutions 48References 512 EM Boundary and Radiation Conditions 522.1 EM Field Behavior Across a Boundary Surface 522.2 Radiation Boundary Condition 602.3 Boundary Conditions at a Moving Interface 632.3.1 Nonrelativistic Moving Boundary Conditions 632.3.2 Derivation of the Nonrelativistic Field Transformations 662.3.3 EM Field Transformations Based on the Special Theory of Relativity 692.4 Constitutive Relations for a Moving Medium 84References 853 Plane Wave Propagation in Planar Layered Media 873.1 Introduction 873.2 Plane Wave Reflection from a Planar Boundary Between Two Different Media 873.2.1 Perpendicular Polarization Case 883.2.2 Parallel Polarization Case 933.2.3 Brewster Angle θ b 973.2.4 Critical Angle θ c 1003.2.5 Plane Wave Incident on a Lossy Half Space 1043.2.6 Doppler Shift for Wave Reflection from a Moving Mirror 1103.3 Reflection and Transmission of a Plane Wave Incident on a Planar Stratified Isotropic Medium Using a Transmission Matrix Approach 1123.4 Plane Waves in Anisotropic Homogeneous Media 1193.5 State Space Formulation for Waves in Planar Anisotropic Layered Media 1353.5.1 Development of State Space Based Field Equations 1353.5.2 Reflection and Transmission of Plane Waves at the Interface Between Two Anisotropic Half Spaces 1393.5.3 Transmission Type Matrix Analysis of Plane Waves in Multilayered Anisotropic Media 142References 1434 Plane Wave Spectral Representation for EM Fields 1444.1 Introduction 1444.2 PWS Development 144References 1555 Electromagnetic Potentials and Fields of Sources in Unbounded Regions 1565.1 Introduction to Vector and Scalar Potentials 1565.2 Construction of the Solution for A 1605.3 Calculation of Fields from Potentials 1655.4 Time Dependent Potentials for Sources and Fields in Unbounded Regions 1765.5 Potentials and Fields of a Moving Point Charge 1855.6 Cerenkov Radiation 1925.7 Direct Calculation of Fields of Sources in Unbounded Regions Using a Dyadic Green’s Function 1955.7.1 Fields of Sources in Unbounded, Isotropic, Homogeneous Media in Terms of a Closed Form Representation of Green’s Dyadic, G 0 1955.7.2 On the Singular Nature of G 0 (r|r ′) for Observation Points Within the Source Region 1975.7.3 Representation of the Green’s Dyadic G 0 in Terms of an Integral in the Wavenumber (k) Space 2015.7.4 Electromagnetic Radiation by a Source in a General Bianisotropic Medium Using a Green’s Dyadic G a in k-Space 208References 2096 Electromagnetic Field Theorems and Related Topics 2116.1 Conservation of Charge 2116.2 Conservation of Power 2126.3 Conservation of Momentum 2186.4 Radiation Pressure 2256.5 Duality Theorem 2356.6 Reciprocity Theorems and Conservation of Reactions 2426.6.1 The Lorentz Reciprocity Theorem 2436.6.2 Reciprocity Theorem for Bianisotropic Media 2496.7 Uniqueness Theorem 2516.8 Image Theorems 2546.9 Equivalence Theorems 2586.9.1 Volume Equivalence Theorem for EM Scattering 2586.9.2 A Surface Equivalence Theorem for EM Scattering 2606.9.3 A Surface Equivalence Theorem for Antennas 2706.10 Antenna Impedance 2786.11 Antenna Equivalent Circuit 2826.12 The Receiving Antenna Problem 2826.13 Expressions for Antenna Mutual Coupling Based on Generalized Reciprocity Theorems 2876.13.1 Circuit Form of the Reciprocity Theorem for Antenna Mutual Coupling 2876.13.2 A Mixed Circuit Field Form of a Generalized Reciprocity Theorem for Antenna Mutual Coupling 2926.13.3 A Mutual Admittance Expression for Slot Antennas 2946.13.4 Antenna Mutual Coupling, Reaction Concept, and Antenna Measurements 2966.14 Relation Between Antenna and Scattering Problems 2976.14.1 Exterior Radiation by a Slot Aperture Antenna Configuration 2976.14.2 Exterior Radiation by a Monopole Antenna Configuration 2996.15 Radar Cross Section 3086.16 Antenna Directive Gain 3096.17 Field Decomposition Theorem 311References 3137 Modal Techniques for the Analysis of Guided Waves, Resonant Cavities, and Periodic Structures 3147.1 On Modal Analysis of Some Guided Wave Problems 3147.2 Classification of Modal Fields in Uniform Guiding Structures 3147.2.1 TEM z Guided waves 3157.3 TM z Guided Waves 3257.4 TE z Guided Waves 3287.5 Modal Expansions in Closed Uniform Waveguides 3307.5.1 TM z Modes 3317.5.2 TE z Modes 3327.5.3 Orthogonality of Modes in Closed Perfectly Conducting Uniform Waveguides 3347.6 Effect of Losses in Closed Guided Wave Structures 3377.7 Source Excited Uniform Closed Perfectly Conducting Waveguides 3387.8 An Analysis of Some Closed Metallic Waveguides 3427.8.1 Modes in a Parallel Plate Waveguide 3427.8.2 Modes in a Rectangular Waveguide 3507.8.3 Modes in a Circular Waveguide 3587.8.4 Coaxial Waveguide 3647.8.5 Obstacles and Discontinuities in Waveguides 3667.8.6 Modal Propagation Past a Slot in a Waveguide 3797.9 Closed and Open Waveguides Containing Penetrable Materials and Coatings 3837.9.1 Material-Loaded Closed PEC Waveguide 3847.9.2 Material Slab Waveguide 3887.9.3 Grounded Material Slab Waveguide 3957.9.4 The Goubau Line 3957.9.5 Circular Cylindrical Optical Fiber Waveguides 3987.10 Modal Analysis of Resonators 4007.10.1 Rectangular Waveguide Cavity Resonator 4027.10.2 Circular Waveguide Cavity Resonator 4067.10.3 Dielectric Resonators 4087.11 Excitation of Resonant Cavities 4097.12 Modal Analysis of Periodic Arrays 4117.12.1 Floquet Modal Analysis of an Infinite Planar Periodic Array of Electric Current Sources 4127.12.2 Floquet Modal Analysis of an Infinite Planar Periodic Array of Current Sources Configured in a Skewed Grid 4197.13 Higher-Order Floquet Modes and Associated Grating Lobe Circle Diagrams for Infinite Planar Periodic Arrays 4227.13.1 Grating Lobe Circle Diagrams 4227.14 On Waves Guided and Radiated by Periodic Structures 4257.15 Scattering by a Planar Periodic Array 4307.15.1 Analysis of the EM Plane Wave Scattering by an Infinite Periodic Slot Array in a Planar PEC Screen 4327.16 Finite 1-D and 2-D Periodic Array of Sources 4377.16.1 Analysis of Finite 1-D Periodic Arrays for the Case of Uniform Source Distribution and Far Zone Observation 4377.16.2 Analysis of Finite 2-D Periodic Arrays for the Case of Uniform Distribution and Far Zone Observation 4447.16.3 Floquet Modal Representation for Near and Far Fields of 1-D Nonuniform Finite Periodic Array Distributions 4467.16.4 Floquet Modal Representation for Near and Far Fields of 2-D Nonuniform Planar Periodic Finite Array Distributions 449References 4518 Green’s Functions for the Analysis of One-Dimensional Source-Excited Wave Problems 4538.1 Introduction to the Sturm-Liouville Form of Differential Equation for 1-D Wave Problems 4538.2 Formulation of the Solution to the Sturm-Liouville Problem via the 1-D Green’s Function Approach 4568.3 Conditions Under Which the Green’s Function Is Symmetric 4638.4 Construction of the Green’s Function G(x|x′) 4648.4.1 General Procedure to Obtain G(x|x′) 4648.5 Alternative Simplified Construction of G(x|x′) Valid for the Symmetric Case 4668.6 On the Existence and Uniqueness of G(x|x′) 4838.7 Eigenfunction Expansion Representation for G(x|x′) 4838.8 Delta Function Completeness Relation and the Construction of Eigenfunctions from G(x|x′) = U (x<)T (x>) ∕ W 4888.9 Explicit Representation of G(x|x′) Using Step Functions 519References 5209 Applications of One-Dimensional Green’s Function Approach for the Analysis of Single and Coupled Set of EM Source Excited Transmission Lines 5229.1 Introduction 5229.2 Analytical Formulation for a Single Transmission Line Made Up of Two Conductors 5229.3 Wave Solution for the Two Conductor Lines When There Are No Impressed Sources Distributed Anywhere Within the Line 5259.4 Wave Solution for the Case of Impressed Sources Placed Anywhere on a Two Conductor Line 5279.5 Excitation of a Two Conductor Transmission Line by an Externally Incident Electromagnetic Wave 5419.6 A Matrix Green’s Function Approach for Analyzing a Set of Coupled Transmission Lines 5439.7 Solution to the Special Case of Two Coupled Lines (N = 2) with Homogeneous Dirichlet or Neumann End Conditions 5469.8 Development of the Multiport Impedance Matrix for a Set of Coupled Transmission Lines 5519.9 Coupled Transmission Line Problems with Voltage Sources and Load Impedances at the End Terminals 552References 55310 Green’s Functions for the Analysis of Two- and Three-Dimensional Source- Excited Scalar and EM Vector Wave Problems 55410.1 Introduction 55410.2 General Formulation for Source-Excited 3-D Separable Scalar Wave Problems Using Green’s Functions 55510.3 General Procedure for Construction of Scalar 2-D and 3-D Green’s Function in Rectangular Coordinates 56610.4 General Procedure for Construction of Scalar 2-D and 3-D Green’s Functions in Cylindrical Coordinates 56910.5 General Procedure for Construction of Scalar 3-D Green’s Functions in Spherical Coordinates 57210.6 General Formulation for Source-Excited 3-D Separable EM Vector Wave Problems Using Dyadic Green’s Functions 57510.7 Some Specific Green’s Functions for 2-D Problems 58310.7.1 Fields of a Uniform Electric Line Source 58310.7.2 Fields of an Infinite Periodic Array of Electric Line Sources 59010.7.3 Line Source-Excited PEC Circular Cylinder Green’s Function 59110.7.4 A Cylindrical Wave Series Expansion 59610.7.5 Line Source Excitation of a PEC Wedge 59810.7.6 Line Source Excitation of a PEC Parallel Plate Waveguide 60210.7.7 The Fields of a Line Dipole Source 60610.7.8 Fields of a Magnetic Line Source on an Infinite Planar Impedance Surface 60810.7.9 Fields of a Magnetic Line Dipole Source on an Infinite Planar Impedance Surface 61210.7.10 Circumferentially Propagating Surface Fields of a Line Source Excited Impedance Circular Cylinder 61410.7.11 Analysis of Circumferentially Propagating Waves for a Line Dipole Source-Excited Impedance Circular Cylinder 61710.7.12 Fields of a Traveling Wave Line Source 61910.7.13 Traveling Wave Line Source Excitation of a PEC Wedge and a PEC Cylinder 62010.8 Examples of Some Alternative Representations of Green’s Functions for Scalar 3-D Point Source-Excited Cylinders, Wedges and Spheres 62310.8.1 3-D Scalar Point Source-Excited Circular Cylinder Green’s Function 62310.8.2 3-D Scalar Point Source Excitation of a Wedge 63010.8.3 Angularly and Radially Propagating 3-D Scalar Point Source Green’s Function for a Sphere 63210.8.4 Kontorovich–Lebedev Transform and MacDonald Based Approaches for Constructing an Angularly Propagating 3-D Point Source Scalar Wedge Green’s Function 64010.8.5 Analysis of the Fields of a Vertical Electric or Magnetic Current Point Source on a PEC Sphere 64710.9 General Procedure for Construction of EM Dyadic Green’s Functions for Source-Excited Separable Canonical Problems via Scalar Green’s Functions 65210.9.1 Summary of Procedure to Obtain the EM Fields of Arbitrarily Oriented Point Sources Exciting Canonical Separable Configurations 65310.10 Completeness of the Eigenfunction Expansion of the Dyadic Green's Function at the Source Point 665References 66911 Method of Factorization and the Wiener–Hopf Technique for Analyzing Two- Part EM Wave Problems 67011.1 The Wiener–Hopf Procedure 67011.2 The Dual Integral Equation Approach 68211.3 The Jones Method 691References 69612 Integral Equation-Based Methods for the Numerical Solution of Nonseparable EM Radiation and Scattering Problems 69712.1 Introduction 69712.2 Boundary Integral Equations 69712.2.1 The Electric Field Integral Equation (EFIE) 69912.2.2 The Magnetic Field Integral Equation (MFIE) 70012.2.3 Combined Field and Combined Source Integral Equations 70112.2.4 Impedance Boundary Condition 70212.2.5 Boundary Integral Equation for a Homogeneous Material Volume 70312.3 Volume Integral Equations 70512.4 The Numerical Solution of Integral Equations 70612.4.1 The Minimum Square-Error Method 70612.4.2 The Method of Moments (MoM) 70812.4.3 Simplification of the MoM Impedance Matrix Integrals 71012.4.4 Expansion and Testing Functions 71312.4.5 Low-Frequency Break-Down 71812.5 Iterative Solution of Large MoM Matrices 72012.5.1 Fast Iterative Solution of MoM Matrix Equations 72112.5.2 The Fast Multipole Method (FMM) 72512.5.3 Multilevel FMM and Fast Fourier Transform FMM 73012.6 Antenna Modeling with the Method of Moments 73212.7 Aperture Coupling with the Method of Moments 73412.8 Physical Optics Methods 73612.8.1 Physical Optics for a PEC Surface 73612.8.2 Iterative Physical Optics 738References 74013 Introduction to Characteristic Modes 74213.1 Introduction 74213.2 Characteristic Modes from the EFIE for a Conducting Surface 74313.2.1 Electric Field Integral Equation and Radiation Operator 74313.2.2 Eigenfunctions of the Electric Field Radiation Operator 74313.2.3 Characteristic Modes from the EFIE Impedance Matrix 74513.3 Computation of Characteristic Modes 74613.4 Solution of the EFIE Using Characteristic Modes 74813.5 Tracking Characteristic Modes with Frequency 74913.6 Antenna Excitation Using Characteristic Modes 749References 75014 Asymptotic Evaluation of Radiation and Diffraction Type Integrals for High Frequencies 75214.1 Introduction 75214.2 Steepest Descent Techniques for the Asymptotic Evaluation of Radiation Integrals 75214.2.1 Topology of the Exponent in the Integrand Containing a First-Order Saddle Point 75314.2.2 Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point in Its Integrand Which Is Free of Singularities 75614.2.3 Asymptotic Evaluation of Integrals Containing a Higher-Order Saddle Point in Its Integrand Which Is Free of Singularities 76014.2.4 Pauli–Clemmow Method (PCM) for the Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point Near a Simple Pole Singularity 76314.2.5 Van der Waerden Method (VWM) for the Asymptotic Evaluation of Integrals Containing a First-Order Saddle Point Near a Simple Pole Singularity 77314.2.6 Relationship Between PCM and VWM Leading to a Generalized PCM (or GPC) Solution 77514.2.7 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and a Nearby Double Pole 77714.2.8 An Extension of PCM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and Two Nearby First-Order Poles 77914.2.9 An Extension of VWM for Asymptotic Evaluation of an Integral Containing a First-Order Saddle Point and a Nearby Double Pole 78314.2.10 Nonuniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Branch Point 78414.2.11 Uniform Asymptotic Evaluation of an Integral Containing a Saddle Point and a Nearby Branch Point 78914.3 Asymptotic Evaluation of Integrals with End Points 79114.3.1 Watson’s Lemma for Integrals 79214.3.2 Generalized Watson’s Lemma for Integrals 79214.3.3 Integration by Parts for Asymptotic Evaluation of a Class of Integrals 79214.4 Asymptotic Evaluation of Radiation Integrals Based on the Stationary Phase Method 79414.4.1 Stationary Phase Evaluation of 1-D Infinite Integrals 79414.4.2 Nonuniform Stationary Phase Evaluation of 1-D Integrals with End Points 79514.4.3 Uniform Stationary Phase Evaluation of 1-D Integrals with a Nearby End Point 79614.4.4 Nonuniform Stationary Phase Evaluation of 2-D Infinite Integrals 801References 81615 Physical and Geometrical Optics 81815.1 The Physical Optics (PO) Approximation for PEC Surfaces 81815.2 The Geometrical Optics (GO) Ray Field 82015.3 GO Transport Singularities 82415.4 Wavefronts, Stationary Phase, and GO 82815.5 GO Incident and Reflected Ray Fields 83215.6 Uniform GO Valid at Smooth Caustics 840References 85416 Geometrical and Integral Theories of Diffraction 85516.1 Geometrical Theory of Diffraction and Its Uniform Version (UTD) 85516.2 UTD for an Edge in an Otherwise Smooth PEC Surface 86116.3 UTD Slope Diffraction for an Edge 87216.4 An Alternative Uniform Solution (the UAT) for Edge Diffraction 87416.5 UTD Solutions for Fields of Sources in the Presence of Smooth PEC Convex Surfaces 87416.5.1 UTD Analysis of the Scattering by a Smooth, Convex Surface 87616.5.2 UTD for the Radiation by Antennas on a Smooth, Convex Surface 88516.5.3 UTD Analysis of the Surface Fields of Antennas on a Smooth, Convex Surface 90116.6 UTD for a Vertex 91316.7 UTD for Edge-Excited Surface Rays 91516.8 The Equivalent Line Current Method (ECM) 92116.8.1 Line Type ECM for Edge-Diffracted Ray Caustic Field Analysis 92216.9 Equivalent Line Current Method for Interior PEC Waveguide Problems 92616.9.1 TE y Case 92716.9.2 TM y Case 93216.10 The Physical Theory of Diffraction (PTD) 93316.10.1 PTD for Edged Bodies - A Canonical Edge Diffraction Problem in the PTD Development 93616.10.2 Details of PTD for 3-D Edged Bodies 93716.10.3 Reduction of PTD to 2-D Edged Bodies 93916.11 On the PTD for Aperture Problems 94016.12 Time-Domain Uniform Geometrical Theory of Diffraction (TD-UTD) 94016.12.1 Introductory Comments 94016.12.2 Analytic Time Transform (ATT) 94116.12.3 TD-UTD for a General PEC Curved Wedge 942References 94517 Development of Asymptotic High-Frequency Solutions to Some Canonical Problems 95117.1 Introduction 95117.2 Development of UTD Solutions for Some Canonical Wedge Diffraction Problems 95117.2.1 Scalar 2-D Line Source Excitation of a Wedge 95217.2.2 Scalar Plane Wave Excitation of a Wedge 95817.2.3 Scalar Spherical Wave Excitation of a Wedge 96017.2.4 EM Plane Wave Excitation of a PEC Wedge 96517.2.5 EM Conical Wave Excitation of a PEC Wedge 96817.2.6 EM Spherical Wave Excitation of a PEC Wedge 97117.3 Canonical Problem of Slope Diffraction by a PEC Wedge 97417.4 Development of a UTD Solution for Scattering by a Canonical 2-D PEC Circular Cylinder and Its Generalization to a Convex Cylinder 97817.4.1 Field Analysis for the Shadowed Part of the Transition Region 98217.4.2 Field Analysis for the Illuminated Part of the Transition Region 98517.5 A Collective UTD for an Efficient Ray Analysis of the Radiation by Finite Conformal Phased Arrays on Infinite PEC Circular Cylinders 99117.5.1 Finite Axial Array on a Circular PEC Cylinder 99217.5.2 Finite Circumferential Array on a Circular PEC Cylinder 99917.6 Surface, Leaky, and Lateral Waves Associated with Planar Material Boundaries 100417.6.1 Introduction 100417.6.2 The EM Fields of a Magnetic Line Source on a Uniform Planar Impedance Surface 100417.6.3 EM Surface and Leaky Wave Fields of a Uniform Line Source over a Planar Grounded Material Slab 101117.6.4 An Analysis of the Lateral Wave Phenomena Arising in the Problem of a Vertical Electric Point Current Source over a Dielectric Half Space 101917.7 Surface Wave Diffraction by a Planar, Two-Part Impedance Surface: Development of a Ray Solution 103217.7.1 TE z Case 103317.7.2 TM z Case 103617.8 Ray Solutions for Special Cases of Discontinuities in Nonconducting or Penetrable Boundaries 1038References 103918 EM Beams and Some Applications 104218.1 Introduction 104218.2 Astigmatic Gaussian Beams 104318.2.1 Paraxial Wave Equation Solutions 104318.2.2 2-D Beams 104418.2.3 3-D Astigmatic Gaussian Beams 104718.2.4 3-D Gaussian Beam from a Gaussian Aperture Distribution 104818.2.5 Reflection of Astigmatic Gaussian Beams (GBs) 105018.3 Complex Source Beams and Relation to GBs 105118.3.1 Introduction to Complex Source Beams (CSBs) 105118.3.2 Complex Source Beam from Scalar Green’s Function 105118.3.3 Representation of Arbitrary EM Fields by a CSB Expansion 105418.3.4 Edge Diffraction of an Incident CSB by a Curved Conducting Wedge 105618.4 Pulsed Complex Source Beams in the Time Domain 1061Index 1105