Electromagnetic Theory for Microwaves and Optoelectronics

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1 Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Translated by authors With 259 Figures Springer

2 Contents 1 Basic Electromagnetic Theory Maxwell's Equations Basic Maxwell Equations Maxwell's equations in Material Media Complex Maxwell Equations Complex Permittivity and Permeability Complex Maxwell equations in Anisotropic Media Equivalent Magnetic Charge and Current Boundary Conditions General Boundary Conditions The Short-Circuit Surface The Open-Circuit Surface She Impedance Surface Wave Equations Time-Domain Wave Equations Solution to the Homogeneous Wave Equations Frequency-Domain Wave Equations Poynting's Theorem Time-Domain Poynting Theorem Frequency-Domain Poynting Theorem Poynting's Theorem for Dispersive Media Scalar and Vector Potentials Retarding Potentials: d'aiembert's Equations Solution of d'aiembert's Equations Complex d'alembert Equations Hertz Vectors Instantaneous Hertz Vectors Complex Hertz Vectors Duality Reciprocity 47 Problems 49

3 VIII Contents 2 Introduction to Waves Sinusoidal Uniform Plane Waves Uniform Plane Waves in Lossless Simple Media Plane Waves in Lossy Media: Damped Waves Polarization of Plane Waves Combination of Two Mutually Perpendicular Linearly Polarized Waves Combination of Two Opposite Circularly Polarized Waves The Jones Matrix Stokes Parameters and the Poincare Sphere The Degree of Polarization Reflection and Refraction of Plane Waves SnelPs Law Fresnel's Law, Reflection and Refraction Coefficients Reflection at a Perfectly Conductive Plane The Brewster Angle Total Reflection and the Critical Angle Decaying Fields and Slow Waves The Goos-Hänchen Shift Reflection Coefficients at Dielectric Interfaces Reflection and Transmission of Plane Waves at Interfaces Between Lossless and Lossy Media Transformation of Impedance and Reflection Coefficients Transmission-Line Simulation of Electromagnetic Waves The Telegraph Equations and Their Solutions The Reflection Coefficient, Standing Wave Ratio, and Impedance in a Lossless Line States of a Transmission Line Transmission»Line Charts The Equivalent Transmission Line of Guided-Wave Systems Network Simulation of Electromagnetic Waves Ill Network Matrix and Parameters of a Linear Multi-port Network The Network Matrices of the Reciprocal, Lossless, Source-Free Multi-port Networks Two-Port Networks The Network Parameters of Some Basic Circuit Elements Reflection and Transmission of Waves at Multi-layer Media and Impedance Transducers 136

4 Contents IX Single Dielectric Layer, The Л/4 Impedance Transducer The Double Dielectric Layer: Double-Section Impedance Transducers The Design of a Multiple Dielectric Layer or Multi-section Impedance Transducer The Small-Reflection Approach A Multi-layer Coating with an Alternating Index Problems Time-Varying Boundary-Value Problems Uniqueness Theorem for Time-Varying-Field Problems Uniqueness Theorem for the Boundary-Value Problems of Helmholtz's Equations Uniqueness Theorem for the Boundary-Value Problems with Complicated Boundaries Solution of Vector Helmholtz Equations in Orthogonal Curvilinear Coordinates Orthogonal Curvilinear Coordinate Systems Method of Borgnis'Potentials Method of Hertz Vectors Method of Longitudinal Components Boundary Conditions of Helmholtz's Equations Separation of Variables Electromagnetic Waves in Cylindrical Systems Solution of Helmholtz's Equations in Rectangular Coordinates Set 2 as и Set x от у as щ Solution of Helmholtz's Equations in Circular Cylindrical Coordinates Solution of Helmholtz's Equations in Spherical Coordinates Vector Eigenfunctions and Normal Modes Eigenvalue Problems and Orthogonal Expansions Eigenvalues for the Boundary-Value Problems of the Vector Helmholtz Equations Two-Dimensional Eigenvalues in Cylindrical Systems Vector Eigenfunctions and Normal Mode Expansion Approximate Solution of Helmholtz's Equations Variational Principle of Eigenvalues Approximate Field-Matching Conditions 208 Problems 212

5 X Contents 4 Metallic Waveguides and Resonant Cavities General Characteristics of Metallic Waveguides Ideal-Waveguide Model Propagation Characteristics Dispersion Relations Wave Impedance Power Flow Attenuation General Characteristics of Resonant Cavities Modes and Natural Frequencies of the Resonant Cavity Losses in a Resonant Cavity: the Q Factor Waveguides and Cavities in Rectangular Coordinates Rectangular Waveguides Parallel-Plate Transmission Lines Rectangular Resonant Cavities Waveguides and Cavities in Circular Cylindrical Coordinates Sectorial Cavities Sectorial Waveguides Coaxial Lines and Coaxial Cavities Circular Waveguides and Circular Cylindrical Cavities Cylindrical Horn Waveguides and Inclined-Plate Lines Radial Transmission Lines and Radial Line Cavities Waveguides and Cavities in Spherical Coordinates Spherical Cavities Biconical Lines and Biconical Cavities Reentrant Cavities Exact Solution for the Reentrant Cavity Approximate Solution for the Reentrant Cavity Fabry-Perot Cavities Principle of Perturbation Cavity Wall Perturbations Material Perturbation of a Cavity Cutoff Frequency Perturbation of a Waveguide Propagation Constant Perturbation of a Waveguide. 285 Problems Dielectric Waveguides and Resonators Metallic Waveguide with Different Filling Media The Possible Modes LSE and LSM Modes Symmetrical Planar Dielectric Waveguides 299

6 Contents XI TM Modes ТЕ Modes Cutoff Condition, Guided Modes, and Radiation Modes Dispersion Characteristics of Guided Modes Radiation Modes Fields in Symmetrical Planar Dielectric Waveguides The Lowest Modes in Symmetrical Planar Dielectric Waveguides Dielectric Coated Conducting Plane Asymmetrical Planar Dielectric Waveguides TM Modes ТЕ Modes Dispersion Characteristics of Asymmetrical Planar Dielectric Waveguide Fields in Asymmetrical Planar Dielectric Waveguides Rectangular Dielectric Waveguides Circular Dielectric Waveguides and Optical Fibers General Solutions of Circular Dielectric Waveguides Nonmagnetic Circular Dielectric Waveguides Weakly Guiding Optical Fibers Linearly Polarized Modes in Weakly Guiding Fibers Dominant Modes in Circular Dielectric Waveguides Low-Attenuation Optical Fibers Dielectric-Coated Conducting Cylinders Dielectric Resonators Perfect-Magnetic-Wall Approach Cutoff-Waveguide Approach Cutoff-Waveguide, Cutoff-Radial-Line Approach Dielectric Resonators in Microwave Circuits 360 Problems Periodic Structures and the Coupling of Modes Characteristics of Slow Waves Dispersion Characteristics Interaction Impedance A Corrugated Conducting Surface as a Uniform System Unbounded Structure Bounded Structure A Disk-Loaded Waveguide as a Uniform System Periodic Systems Floquet's Theorem and Space Harmonics The uj-ß Diagram of Periodic Systems The Band-Pass Character of Periodic Systems Fields in Periodic Systems 381

7 6.4.5 Two Theorems on Lossless Periodic Systems The Interaction Impedance for Periodic Systems Corrugated Conducting Plane as a Periodic System Disk-Loaded Waveguide as a Periodic System The Helix The Sheath Helix The Tape Helix Coupling of Modes Coupling of Modes in Space General Solutions for Codirectional Coupling Waveguide Couplers and Switches Coupling Coefficient of Dielectric Waveguides Distributed Feedback (DFB) Structures Principle of DFB Structures DFB Transmission Resonator A Multiple Layer as a DFB Transmission Resonator The Quarter-Wave-Shifted DFB Resonator 427 Problems 430 Electromagnetic Waves in Dispersive Media Classical Theory of Dispersion in Material Media Complex Susceptibility and Complex Permittivity Kramers-Kronig Relations Complex Index of Refraction Normal and Anomalous Dispersion Complex Index for Metals Behavior at Low Frequencies, Electric Conductivity Behavior at High Frequencies, Plasma Frequency Complex propagation coefficient and Phase Velocity Group Velocity Signal Velocity Velocity of Energy Flow 450 Problems 452 Electromagnetic Waves in Anisotropic Media Anisotropic Media and Their Constitutional Relations Constitutional Equations for Anisotropic Media Symmetrical Properties of the Constitutional Tensors Governing Equations for Fields and Waves in Anisotropic Media Maxwell Equations and Wave Equations in Anisotropic Media Wave Vector and Poynting's Vector in Anisotropic Media kdb Coordinate System 460

8 Contents XIII 8.3 Reciprocal Dielectric Crystals Isotropic Crystals Uniaxial Crystals Biaxial Crystals Electromagnetic Waves in Uniaxial Crystals General Expressions Plane Waves Propagating in the Direction of the Optical Axis Plane Waves Propagating in the Direction Perpendicular to the Optical Axis Plane Waves Propagating in an Arbitrary Direction General Formalisms of EM Waves in Reciprocal Media Index Ellipsoid Dispersion Equations for the Plane Waves in Reciprocal Media Normal and Effective-Index Surfaces Phase Velocity and Group Velocity of the Plane Waves in Reciprocal Crystals Waves in Electron Beams Permittivity Tensor for an Electron Beam Space Charge Waves Nonreciprocal Media Stationary Plasma in a Finite Magnetic Field Ferrite in a Finite Magnetic Field, Gyromagnetic Media Electromagnetic Waves in Nonreciprocal Media Plane Waves in a Stationary Plasma in a Finite Magnetic Field Plane Waves in Saturated-Magnetized Ferrites Magnetostatic Waves Magnetostatic Wave Equations Magnetostatic Wave Modes 523 Problems Gaussian Beams Fundamental Gaussian Beams Characteristics of Gaussian Beams Condition of Paraxial Approximation Beam Radius, Curvature Radius of Phase Front, and Half Far-Field Divergence Angle Phase Velocity Electric and Magnetic Fields in Gaussian Beams Energy Density and Power Flow Transformation of Gaussian Beams The q Parameter and Its Transformation 543

9 XIV Contents ABCD Law and Its Applications Transformation Through a Non-thin Lens Elliptic Gaussian Beams Higher-Order Modes of Gaussian Beams Hermite-Gaussian Beams Laguerre-Gaussian Beams Gaussian Beams in Quadratic Index Media The General Solution Propagation in Media with a Real Quadratic Index Profile Propagation in Medium with an Imaginary Quadratic Index Profile Steady-State Hermite-Gaussian Beams in Media with a Quadratic Index Profile Optical Resonators with Curved Mirrors Gaussian Beams in Anisotropic Media 572 Problems Scalar Diffraction Theory Kirchhoff's Diffraction Theory Kirchhoff Integral Theorem Fresnel-Kirchhoff Diffraction Formula Rayleigh-Sommerfeld Diffraction Formula Fraunhofer and Fresnel Diffraction Diffraction Formulas for Spherical Waves Fraunhofer Diffraction at Circular Apertures Fresnel Diffraction at Circular Apertures Diffraction of Gaussian Beams Fraunhofer Diffraction of Gaussian Beams Fresnel Diffraction of Gaussian Beams Diffraction of Plane Waves in Anisotropic Media Fraunhofer Diffraction at Square Apertures Fraunhofer Diffraction at Circular Apertures Fresnel Diffraction at Circular Apertures Refraction of Gaussian Beams in Anisotropic Media Eigenwave Expansions of Electromagnetic Fields Eigenmode Expansion in a Rectangular Coordinate System Eigenmode Expansion in a Cylindrical Coordinate System Eigenmode Expansion in Inhomogeneous Media Eigenmode Expansion in Anisotropic Media Eigenmode Expansion in Inhomogeneous and Anisotropic Media 623

10 Contents XV Reflection and Refraction of Gaussian Beams on Medium Surfaces 625 Problems 628 A SI Units and Gaussian Units 631 A.l Conversion of Amounts 631 A.2 Conversion of Formulas 632 В Vector Analysis 633 B.l Vector Differential Operations 633 B.l.l General Orthogonal Coordinates 633 B.l.2 General Cylindrical Coordinates 634 B.1.3 Rectangular Coordinates 635 B.l.4 Circular Cylindrical Coordinates 635 B.l.5 Spherical Coordinates 636 B.2 Vector Formulas 636 B.2.1 Vector Algebric Formulas 636 B.2.2 Vector Differential Formulas 637 B.2.3 Vector Integral Formulas 637 B.2.4 Differential Formulas for the Position Vector 638 С Bessel Functions 639 C.l Power Series Representations 639 C.2 Integral Representations 640 C.3 Approximate Expressions 640 C.3.1 Leading Terms of Power Series C.3.2 (Small-Argument Approximation) 640 Leading Terms of Asymptotic Series (Large-Argument Approximation) 640 C.4 Formulas for Bessel Functions 640 C.4.1 Recurrence Formulas 640 C.4.2 Derivatives 641 C.4.3 Integrals 641 C.4.4 Wronskian 641 C.5 Spherical Bessel Functions 642 C.5.1 Bessel Functions of Order n + 1/2 642 C.5.2 Spherical Bessel Functions 642 C.5.3 Spherical Bessel Functions Used by Schelkunoff D Legendre Functions 643 D.l Legendre Polynomials 643 D.2 Associate Legendre Polynomials 643 D.3 Formulas for Legendre Polynomials 644 D.3.1 Recurrence Formulas 644 D.3.2 Derivatives 644

11 XVI Contents D.3.3 Integrals 644 E Matrices and Tensors 645 E.l Matrix 645 E.2 Matrix Algebra 646 E.2.1 Definitions 646 E.2.2 Matrix Algebraic Formulas 646 E.3 Matrix Functions 647 E.4 Special Matrices 648 E.5 Tensors and Vectors 649 Physical Constants and Smith Chart 651 Bibliography 653 Index 659

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