Andrey Osipov, Microwaves and Radar Institute, German Aerospace Center (DLR), Germany. Sergei Tretyakov, Department of Radio Science and Engineering, Aalto University, Finland.

• Preface xi Acknowledgements xiiiList of Abbreviations xv1 Introduction 11.1 Scattering and Diffraction Theory 11.2 Books on Related Subjects 31.3 Concept and Outline of the Book 5References 82 Fundamentals of Electromagnetic Scattering 112.1 Introduction 112.2 Fundamental Equations and Conditions 112.2.1 Maxwell’s Equations 122.2.2 Constitutive Relations 122.2.3 Time-harmonic Scattering Problems 192.3 Approximate Boundary Conditions 262.3.1 Impedance Boundary Conditions 262.3.2 Generalized (Higher-order) Impedance Boundary Conditions 312.3.3 Sheet Transition Conditions 322.4 Fundamental Properties of Time-harmonic Electromagnetic Fields 352.4.1 Energy Conservation and Uniqueness 352.4.2 Reciprocity 392.5 Basic Solutions of Maxwell’s Equations in Homogeneous Isotropic Media 422.5.1 Plane, Spherical, and Cylindrical Waves 432.5.2 Electromagnetic Potentials and Fields of External Currents 462.5.3 Tensor Green’s Function 502.5.4 E and H Modes 542.5.5 Fields with Translational Symmetry 582.6 Electromagnetic Formulation of Huygens’ Principle 612.6.1 Compact Scatterers 622.6.2 Cylindrical Scatterers 672.7 Problems 70References 843 Far-field Scattering 873.1 Introduction 873.2 Scattering Cross Section 873.2.1 Monostatic and Bistatic, Backscattering and Forward-scattering Cross Sections, Differential, Total, Absorption, and Extinction Cross Sections 873.2.2 Scattering Width 913.2.3 Backscattering from Impedance-matched Bodies 933.3 Scattering Matrix 953.3.1 Definition 953.3.2 Scattering Matrix in Spherical Coordinates 973.3.3 Scattering Matrix in the Plane of Scattering Coordinates 993.4 Far-field Coefficient 1013.4.1 Integral Representations and Far-field Conditions 1023.4.2 Reciprocity of Scattered Fields 1063.4.3 Forward Scattering 1083.4.4 Cylindrical Bodies 1133.5 Scattering Regimes 1203.5.1 Resonant-size Scatterers 1203.5.2 Electrically Large Scatterers 1213.6 Electrically Small Scatterers 1253.6.1 Physics of Dipole Scattering 1263.6.2 Dipole Scattering in Terms of Polarizability Tensors 1293.6.3 Magneto-dielectric Ellipsoid 1313.6.4 Rotationally Symmetric Particles 1373.7 Problems 148References 1624 Planar Interfaces 1654.1 Introduction 1654.2 Interface of Two Homogeneous Semi-infinite Media 1674.2.1 Reflection and Transmission Coefficients 1674.2.2 Brewster’s Angle 1734.2.3 Total Internal Reflection 1734.2.4 Interfaces with Double-negative Materials 1764.2.5 Surface Waves 1774.2.6 Vector Solution of Reflection and Transmission Problems 1794.3 Arbitrary Number of Planar Layers 1824.3.1 Solution by the Method of Characteristic Matrices 1824.3.2 Discussion and Limiting Cases 1894.4 Reflection and Transmission of Cylindrical and Spherical Waves 1954.4.1 Excitation by a Linear Electric Current 1954.4.2 Excitation by an Electric Dipole 2024.5 A Layer between Homogeneous Half-spaces 2074.5.1 Different Half-spaces 2074.5.2 A PEC-backed Layer 2134.5.3 Layer Immersed in a Homogeneous Space 2154.6 Modeling with Approximate Boundary Conditions 2244.6.1 Accuracy of Impedance Boundary Conditions 2254.6.2 Accuracy of Transition Boundary Conditions 2294.6.3 Impedance-matched Surface 2324.7 Problems 235References 2495 Wedges 2515.1 Introduction 2515.2 The Perfectly Conducting Wedge 2535.2.1 Formulation of Boundary Value Problem 2545.2.2 Solution by Separation of Variables 2565.2.3 Fields and Currents at the Edge 2585.2.4 Reduction to an Integral Form 2605.2.5 Special Cases 2625.2.6 Edge-diffracted and GO Components. Diffraction Coefficient 2665.3 Scattering from a Half-plane (Solution by Factorization Method) 2715.3.1 Statement of the Problem 2715.3.2 Functional Equation 2735.3.3 Factorization and Solution 2745.3.4 Scattered Field Far from the Edge 2765.4 The Impedance Wedge 2795.4.1 Boundary Value Problem, Sommerfeld’s Integrals, and Functional Equations 2795.4.2 Normal Incidence (Maliuzhinets’ Solution) 2885.4.3 Unit Surface Impedance 2975.4.4 Further Exactly Solvable Cases 3005.5 High-frequency Scattering from Impenetrable Wedges 3065.5.1 GO Components and Surface Waves 3075.5.2 Edge-diffracted Field, Diffraction Coefficient, and Scattering Widths 3105.5.3 Uniform Asymptotic Approximations 3165.5.4 GTD/UTD Formulation 3195.6 Behavior of Electromagnetic Fields at Edges 3225.6.1 Determining the Degree of Singularity 3225.6.2 Analytical Structure of Meixner’s Series 3285.7 Problems 329References 3366 Circular Cylinders and Convex Bodies 3396.1 Introduction 3396.2 Perfectly Conducting Cylinders: Separation of Variables and Series Solution 3406.2.1 Separation of Variables 3426.2.2 Satisfying the Boundary Conditions 3426.2.3 Scattered Fields 3436.2.4 Numerical Examples 3456.3 Homogeneous Cylinders under Normal Illumination 3506.3.1 Field Equations and Boundary Conditions 3506.3.2 Rayleigh Series Solution 3516.3.3 Numerical Examples 3526.4 Watson’s Transformation and High-frequency Approximations 3546.4.1 Watson’s Transformation 3556.4.2 Alternative Solution by Separation of Variables 3586.4.3 High-frequency Approximations 3606.4.4 Surface Currents in the Penumbra Region. Fock’s Functions 3696.5 Coated and Impedance Cylinders under Oblique Illumination 3756.5.1 PEC Cylinder with Magneto-dielectric Coating 3766.5.2 Impedance Cylinder 3836.6 Extension to Generally Shaped Convex Impedance Bodies 3926.6.1 Fock’s Principle of the Local Field in the Penumbra Region 3936.6.2 Asymptotic Solution for the Field on the Surface of Circular Impedance Cylinders under Oblique Illumination 3966.6.3 Fock- and GTD-type Solutions for Electrically Large Convex Impedance Bodies 3986.7 Problems 403References 4117 Spheres 4127.1 Introduction 4127.2 Exact Solution for a Multilayered Sphere 4147.2.1 Formulation of the Problem in Terms of Debye’s Potentials 4157.2.2 Derivation of the Series Solution 4177.2.3 Solution for Impedance Boundary Conditions 4277.3 Physics of Scattering from Spheres 4297.3.1 Classification of Scattering 4307.3.2 Spiral Waves 4367.3.3 Debye’s Expansions for Homogeneous Spheres 4387.3.4 Waves in Electrically Large Homogeneous Low-absorption Spheres 4427.4 Scattered Field in the Far Zone 4637.4.1 Far-field Coefficient, Scattering Cross Sections, and Polarization Structure. Approximations for Electrically Large Spheres 4637.4.2 Electrically Small Spheres: Dipole, Quasi-static, and Resonance Approximations 4717.4.3 PEC Spheres 4797.4.4 Core-shell Spheres 4837.4.5 Impedance Spheres 4887.5 Far-field Scattering from Homogeneous Spheres 4937.5.1 Exact Solution and Limiting Cases 4947.5.2 Electrically Small Lossy Spheres 4957.5.3 Electrically Small Low-absorption Spheres 4997.5.4 Electrically Large Lossy Spheres: Relation to the Impedance Sphere and the Role of Absorption 5067.5.5 Electrically Large Low-absorption Spheres: Light Scattering from Water Droplets 5137.6 Metamaterial Effects in Scattering from Spheres 5427.6.1 Small Spheres 5427.6.2 Invisibility Cloak 5467.7 Problems 552References 5628 Method of Physical Optics 5658.1 Introduction 5658.1.1 On Numerical Techniques for Studying Scattering from Arbitrary-shaped Bodies 5658.1.2 PO as one of the Approximate Analytical Techniques 5668.1.3 Structure of the Chapter 5678.2 Principles and General Solution 5678.2.1 Principles of PO 5678.2.2 Derivation of PO Solutions 5698.2.3 PO for Cylindrical Bodies 5738.3 Transmission through Apertures 5758.3.1 PO Solution 5758.3.2 GO Rays and Fresnel Zones 5768.3.3 Contribution from the Rim of the Aperture: Edge-diffracted Rays 5828.4 Scattering from Curved Surfaces 5948.4.1 Fock’s Reflection Formula 5948.4.2 Application to a Spherical Segment 6008.4.3 Reflection Formula in the Far-field Region 6058.4.4 Diffraction by an Edge in a Non-metallic Surface 6098.5 Advantages and Limitations of Physical Optics 6158.6 Problems 616References 6329 Physical Optics Solutions of Canonical Problems 6349.1 Introduction 6349.2 Vertices 6359.2.1 Vertex on an Edge of a Thin Plate 6379.2.2 Apex of a Pyramid 6419.2.3 Tip of an Elliptic Cone 6439.3 Electrically Large Plates 6529.3.1 Arbitrarily Shaped Plates 6539.3.2 Circular Disc 6589.3.3 Polygonal Plates 6639.3.4 Far-field Patterns of Polygonal Plates and Apertures 6679.4 Bodies of Revolution 6719.4.1 PO Solution for Bodies of Revolution 6729.4.2 Imperfectly Reflecting Bodies under Axial Illumination 6759.4.3 PEC Bodies under Oblique Illumination 6779.4.4 Axial Backscattering 6789.4.5 Examples 6849.5 Problems 689References 712A Definitions and Useful Relations of Vector Analysis and Differential Geometry 714A.1 Vector Algebra 714A.2 Vector Analysis 716A.3 Vectors and Vector Differential Operators in Orthogonal Curvilinear Coordinates 717A.3.1 General Orthogonal Curvilinear Coordinates 717A.3.2 Spherical Coordinates 718A.4 Curves and Surfaces in Space 720A.4.1 Curves 720A.4.2 Surfaces 720A.5 Problems 722References 724B Fresnel Integral and Related Functions 725B.1 Fresnel Integral 725B.2 Relation to the Error Function 728B.3 Transition Functions of Uniform Theories of Diffraction 730B.4 Problems 731References 732C Principles of Complex Integration 733C.1 Introduction 733C.2 Deforming the Integration Contour 734C.2.1 Basic Facts about Functions of a Complex Variable 734C.2.2 Integrals over Infinite Contours 736C.3 Steepest Descent Method 737C.3.1 Steepest Descent Path 738C.3.2 Saddle Point Contribution 739C.3.3 Pole Singularity near the Saddle Point 741C.3.4 Further Cases 742C.4 Problems 743References 745D The Stationary Phase Method 746D.1 Introduction 746D.2 One-dimensional Integrals 746D.2.1 No Stationary Points on the Integration Interval 747D.2.2 Isolated Stationary Points 748D.2.3 Two Coalescing Stationary Points 751D.3 Two-dimensional Integrals 756D.3.1 Stationary Point in the Integration Domain 756D.3.2 Stationary Point near the Boundary of the Integration Domain 758D.3.3 Contribution from the Boundary of the Integration Domain 760D.3.4 Kontorovich’s Formula 763D.3.5 Integrand Vanishing on the Boundary 765D.3.6 Summary of the Two-dimensional Stationary-phase Method 766D.4 Problems 766References 768E Asymptotic Approximations of Bessel Functions of Large Argument and Arbitrary Order 770E.1 Introduction 770E.1.1 Basic Definitions and Properties 770E.1.2 Large-argument Approximations (|z| â 1) 772E.1.3 Content of the Appendix 775E.2 Debye’s Asymptotic Approximations 776E.2.1 Debye’s Method 776E.2.2 WKB Approximation 778E.2.3 Bessel Functions on the Complex 𝜈 Plane 791E.3 Almost Equal Argument and Order 795E.3.1 Approximations in Terms of Airy Functions 796E.3.2 Approximations in Terms of Normalized Airy Functions 797E.3.3 Zeros in the Neighborhood of the Points 𝜈 = ±z 798References 799Index 801