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A plane wave is incident from medium 1 and reflected at the planar interface between media 1 and 2. Both media are isotropic and nonmagnetic (i = ft). The permittivities of the media are denoted by &, and &,. respectively. a. Fora circularly polarized incident plane wave. find the angle of incidence. in terms of &, and &,. at which the reflected wave becomes linearly polarized. State the polarization (that is, the direction of the electric field) of the reflected wave. b. When a right-hand circularly polarized plane wave is incident at the critical angle (for total reflection). determine the polarization of the reflected wave. Write down the expression of the critical angle. c. If medium 2 is replaced by a uniaxial medium with a permittivity given by 5 0 0 £=(0 ¢ 0 06's and the incident wave is a circularly polarized plane wave, find the angle of incidence at which the reflected wave becomes linearly polarized. Then, write down the expressions of the critical angles for both parallel and perpendicular polarizations. Theory and Computation of Electromagnetic Fields THEORY AND COMPUTATION OF ELECTROMAGNETIC FIELDS THEORY AND COMPUTATION OF ELECTROMAGNETIC FIELDS Second Edition JIAN-MING JIN Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Copyright © 2015 by John Wiley & Sons, Inc. 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Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 1 2015 http://www.copyright.com http://www.wiley.com/go/permissions http://www.wiley.com CONTENTS Preface xv Acknowledgments xxi PART I Electromagnetic Field Theory 1 1 Basic Electromagnetic Theory 3 1.1 Review of Vector Analysis, 3 1.1.1 Vector Operations and Integral Theorems, 4 1.1.2 Symbolic Vector Method, 6 1.1.3 Helmholtz Decomposition Theorem, 9 1.1.4 Green’s Theorems, 9 1.2 Maxwell’s Equations in Terms of Total Charges and Currents, 11 1.2.1 Maxwell’s Equations in Integral Form, 12 1.2.2 Maxwell’s Equations in Differential Form, 17 1.2.3 Current Continuity Equation, 17 1.2.4 The Lorentz Force Law, 18 1.3 Constitutive Relations, 18 1.3.1 Electric Polarization, 19 1.3.2 Magnetization, 21 1.3.3 Electric Conduction, 22 1.3.4 Classification of Media, 23 1.4 Maxwell’s Equations in Terms of Free Charges and Currents, 25 vi CONTENTS 1.5 Boundary Conditions, 27 1.6 Energy, Power, and Poynting’s Theorem, 31 1.7 Time-Harmonic Fields, 33 1.7.1 Time-Harmonic Fields, 33 1.7.2 Fourier Transforms, 35 1.7.3 Complex Power, 37 1.7.4 Complex Permittivity and Permeability, 42 References, 46 Problems, 46 2 Electromagnetic Radiation in Free Space 53 2.1 Scalar and Vector Potentials, 53 2.1.1 Static Fields, 54 2.1.2 Time-Harmonic Fields and the Lorenz Gauge Condition, 58 2.2 Solution of Vector Potentials in Free Space, 61 2.2.1 Delta Function and Green’s Function, 61 2.2.2 Green’s Function in Free Space, 62 2.2.3 Field–Source Relations in Free Space, 63 2.2.4 Why Use Auxiliary Potential Functions, 64 2.2.5 Free-Space Dyadic Green’s Functions, 66 2.3 Electromagnetic Radiation in Free Space, 69 2.3.1 Infinitesimal Electric Dipole, 69 2.3.2 Finite Electric Dipole, 72 2.3.3 Far-Field Approximation and the Sommerfeld Radiation Condition, 73 2.3.4 Circular Current Loop and Magnetic Dipole, 76 2.4 Radiation by Surface Currents and Phased Arrays, 78 2.4.1 Radiation by a Surface Current, 78 2.4.2 Radiation by a Phased Array, 81 References, 84 Problems, 85 3 Electromagnetic Theorems and Principles 89 3.1 Uniqueness Theorem, 90 3.2 Image Theory, 94 3.2.1 Basic Image Theory, 94 3.2.2 Half-Space Field–Source Relations, 99 3.3 Reciprocity Theorems, 101 3.3.1 General Reciprocity Theorem, 101 3.3.2 Lorentz Reciprocity Theorem, 102 3.3.3 Rayleigh–Carson Reciprocity Theorem, 103 3.4 Equivalence Principles, 107 3.4.1 Surface Equivalence Principle, 107 3.4.2 Application to Scattering by a Conducting Object, 109 3.4.3 Application to Scattering by a Dielectric Object, 114 3.4.4 Volume Equivalence Principle, 116 CONTENTS vii 3.5 Duality Principle, 120 3.6 Aperture Radiation and Scattering, 121 3.6.1 Equivalent Problems, 121 3.6.2 Babinet’s Principle, 124 3.6.3 Complementary Antennas, 127 References, 128 Problems, 129 4 Transmission Lines and Plane Waves 135 4.1 Transmission Line Theory, 135 4.1.1 Governing Differential Equations and General Solutions, 135 4.1.2 Reflection and Transmission, 138 4.1.3 Green’s Function and Eigenfunction Expansion, 140 4.2 Wave Equations and General Solutions, 144 4.2.1 Wave Equations and Solution by Separation of Variables, 144 4.2.2 Characteristics of a Plane Wave, 146 4.2.3 Wave Velocities and Attenuation, 147 4.2.4 Linear, Circular, and Elliptical Polarizations, 151 4.2.5 Wave Propagation in Metamaterials, 154 4.3 Plane Waves Generated by a Current Sheet, 156 4.4 Reflection and Transmission, 159 4.4.1 Reflection and Transmission at Normal Incidence, 159 4.4.2 Reflection and Transmission at Oblique Incidence, 161 4.4.3 Total Transmission and Total Reflection, 164 4.4.4 Transmission into a Left-Handed Medium, 168 4.4.5 Plane Waves Versus Transmission Lines, 170 4.5 Plane Waves in Anisotropic and Bi-Isotropic Media, 174 4.5.1 Plane Waves in Uniaxial Media, 174 4.5.2 Plane Waves in Gyrotropic Media, 179 4.5.3 Plane Waves in Chiral Media, 183 References, 190 Problems, 191 5 Fields and Waves in Rectangular Coordinates 199 5.1 Uniform Waveguides, 199 5.1.1 General Analysis, 200 5.1.2 General Characteristics, 204 5.1.3 Uniform Rectangular Waveguide, 208 5.1.4 Losses in Waveguides and Attenuation Constant, 215 5.2 Uniform Cavities, 220 5.2.1 General Theory, 221 5.2.2 Rectangular Cavity, 223 5.2.3 Material and Geometry Perturbations, 226 5.3 Partially Filled Waveguides and Dielectric Slab Waveguides, 229 5.3.1 General Theory, 229 5.3.2 Partially Filled Rectangular Waveguide, 231 5.3.3 Dielectric Slab Waveguide on a Ground Plane, 236 viii CONTENTS 5.4 Field Excitation in Waveguides, 241 5.4.1 Excitation by Planar Surface Currents, 242 5.4.2 Excitation by General Volumetric Currents, 243 5.5 Fields in Planar Layered Media, 245 5.5.1 Spectral Green’s Function and Sommerfeld Identity, 245 5.5.2 Vertical Electric Dipole above a Layered Medium, 247 5.5.3 Horizontal Electric Dipole above a Layered Medium, 249 5.5.4 Dipoles on a Grounded Dielectric Slab, 251 References, 257 Problems, 257 6 Fields and Waves in Cylindrical Coordinates 261 6.1 Solution of Wave Equation, 261 6.1.1 Solution by Separation of Variables, 262 6.1.2 Cylindrical Wave Functions, 263 6.2 Circular and Coaxial Waveguides and Cavities, 266 6.2.1 Circular Waveguide, 267 6.2.2 Coaxial Waveguide, 273 6.2.3 Cylindrical Cavity, 276 6.3 Circular Dielectric Waveguide, 279 6.3.1 Analysis of Hybrid Modes, 279 6.3.2 Characteristics of Hybrid Modes, 283 6.4 Wave Transformation and Scattering Analysis, 287 6.4.1 Wave Transformation, 288 6.4.2 Scattering by a Circular Conducting Cylinder, 289 6.4.3 Scattering by a Circular Dielectric Cylinder, 293 6.4.4 Scattering by a Circular Multilayer Dielectric Cylinder, 296 6.5 Radiation by Infinitely Long Currents, 300 6.5.1 Line Current Radiation in Free Space, 300 6.5.2 Radiation by a Cylindrical Surface Current, 304 6.5.3 Radiation in the Presence of a Circular Conducting Cylinder, 306 6.5.4 Radiation in the Presence of a Conducting Wedge, 309 6.5.5 Radiation by a Finite Current, 312 References, 319 Problems, 320 7 Fields and Waves in Spherical Coordinates 325 7.1 Solution of Wave Equation, 325 7.1.1 Solution by Separation of Variables, 325 7.1.2 Spherical Wave Functions, 328 7.1.3 TEr and TMr Modes, 329 7.2 Spherical Cavity, 331 7.3 Biconical Antenna, 335 7.3.1 Infinitely Long Model, 335 7.3.2 Finite Biconical Antenna