Electromagnetic Theory for Microwaves and Optoelectronics

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1 Keqian Zhang Dejie Li Electromagnetic Theory for Microwaves and Optoelectronics Second Edition With 280 Figures and 13 Tables 4u Springer

2 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 Maxwell's Equations in Duality form Boundary Conditions General Boundary Conditions The Short-Circuit Surface The Open-Circuit Surface The 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'alembert's Equations Solution of d'alembert's Equations Complex d'alembert Equations Hertz Vectors Instantaneous Hertz Vectors Complex Hertz Vectors Duality Reciprocity 51 Problems 52

3 Introduction to Waves Sinusoidal Uniform Plane Waves Uniform Plane Waves in Lossless Simple Media Uniform Plane Waves with an Arbitrary Direction of Propagation 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 Stokes Parameters and the Poincare Sphere The Degree of Polarization Normal Reflection and Transmission of Plane Waves Normal Incidence and Reflection at a Perfect- Conductor Surface, Standing Waves Normal Incidence, Reflection and Transmission at Nonconducting Dielectric Boundary, Traveling-Standing Waves Oblique Reflection and Refraction of Plane Waves Snell's Law Oblique Incidence and Reflection at a Perfect- Conductor Surface Fresnel's Law, Reflection and Refraction Coefficients The Brewster Angle Total Reflection and the Critical Angle Decaying Fields and Slow Waves The Goos-Hanchen Shift Reflection Coefficients at Dielectric Boundary Reflection and Transmission of Plane Waves at the Boundary Between Lossless and Lossy Media Transformission of Impedance for Electromagnetic Waves Dielectric Layers and Impedance Transducers Single Dielectric Layer, The A/4 Impedance Transducer Multiple Dielectric Layer, Multi-Section Impedance Transducer Ill A Multi-Layer Coating with an Alternating Indices.. Ill Problems 114 Transmission-Line Theory and Network Theory for Electromagnetic Waves Basic Transmission Line Theory The Telegraph Equations Solution of the Telegraph Equations Standing Waves in Lossless Lines 121

4 xiii The Reflection Coefficient, Standing Wave Ratio and Impedance in a Lossless Line States of a Transmission Line Transmission-Line Charts The Smith Chart The Schimdt Chart The Carter Chart Basic Applications of the Smith Chart The Equivalent Transmission Line of Wave Systems Introduction to Network Theory 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 Matrices and the Parameters of Two-Port Networks ;2 The Network Matrices of the Reciprocal, Lossless, Source-Free and Symmetrical Two-Port Networks The Working Parameters of Two-Port Networks The Network Parameters of Some Basic Circuit Elements Impedance Transducers The Network Approach to the A/4 Anti-Reflection Coating and the A/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 171 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 Orthogonal Curvilinear Coordinate Systems Solution of Vector Helmholtz Equations in Orthogonal Curvilinear Coordinates Method of Borgnis'Potentials Method of Hertz Vectors Method of Longitudinal Components Boundary Conditions of Helmholtz's Equations Separation of Variables 199

5 xiv Contents 4.6 Electromagnetic Waves in Cylindrical Systems Solution of Helmholtz's Equations in Rectangular Coordinates Set z as u Set x or y as u 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 230 Problems Metallic Waveguides and Resonant Cavities General Characteristics of Metallic Waveguides Ideal-Waveguide Model Propagation Characteristics ^ Dispersion Relations Wave Impedance 240 \_ 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 297

6 xv Approximate Solution for the Reentrant Cavity Principle of Perturbation Cavity Wall Perturbations Material Perturbation of a Cavity Cutoff Frequency Perturbation of a Waveguide Propagation Constant Perturbation of a Waveguide. 312 Problems Dielectric Waveguides and Resonators Metallic Waveguide with Different Filling Media The Possible TE and TM Modes LSE and LSM Modes, HEM Modes Symmetrical Planar Dielectric Waveguides TM Modes." TE Modes Cutoff Condition, Guided Modes and Radiation Modes Dispersion Characteristics of Guided Modes Radiation Modes Fields in Symmetrical Planar Dielectric Waveguides The Dominant Modes in Symmetrical Planar Dielectric Waveguides The Weekly Guiding Dielectric Waveguides Dielectric Coated Conductor Plane Asymmetrical Planar Dielectric Waveguides 339 ^ TM Modes TE Modes Dispersion Characteristics of Asymmetrical Planar Dielectric Waveguide Fields in Asymmetrical Planar Dielectric Waveguides Rectangular Dielectric Waveguides Exect Solution for Rectangular Dielectric Waveguides Approximate Analytic Solution for Weekly Guiding Rectangular Dielectric Waveguides Solution for Rectangular Dielectric Waveguides by Means of Circular Harmonics 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 Fiber Dielectric-Coated Conductor Cylinder Dielectric Resonators Perfect-Magnetic-Conductor Wall Approach 387

7 xvi Contents Cutoff-Waveguide Approach Cutoff-Waveguide, Cutoff-Radial-Line Approach Dielectric Resonators in Microwave Circuits...:. 395 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 Disk-Loaded Waveguide with Center Coupling Hole. 407 ] Disk-Loaded Waveguide with Edge Coupling Hole Periodic Systems Floquet's Theorem and Space Harmonics The w-/3 Diagram of Period Systems The Band-Pass Character of Periodic Systems Fields in Periodic Systems 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 the Mode Coupling Co-Directional Mode Coupling Coupling Coefficient of Dielectric Waveguides Contra-Directional Mode Coupling Distributed Feedback (DFB) Structures Principle of DFB Structures DFB Transmission Resonator The Quarter-Wave Shifted DFB Resonator A Multiple-Layer Coating as a DFB Transmission Resonator 470 Problems 472

8 xvii 8 Electromagnetic Waves in Dispersive Media and Anisotropic Media Classical Theory of Dispersion and Dissipation in Material Media Ideal Gas Model for Dispersion and Dissipation 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 Velocities of Waves in Dispersive Media Phase Velocity Group Velocity Velocity of Energy Flow Signal Velocity Anisotropic Media and Their Constitutional Relations Constitutional Equations for Anisotropic Media Symmetrical Properties of the Constitutional Tensors Characteristics of Waves in Anisotropic Media Maxwell Equations and Wave Equations in Anisotropic Media Wave Vector and Poynting Vector in Anisotropic Media Eigenwaves in Anisotropic Media kdb Coordinate System Reciprocal Anisotropic Media 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 Electromagnetic Waves in Reciprocal Media Index Ellipsoid The Effective Indices of Eigenwaves Dispersion Equations for the Plane Waves in Reciprocal Media Normal Surface and Effective-Index Surface Phase Velocity and Group Velocity of the Plane Waves in Reciprocal Crystals 525

9 xviii Contents 8.8 Waves in Electron Beams Permittivity Tensor for an Electron Beam Space Charge Waves Nonreciprocal Media Stationary Plasma in a Finite Magnetic Field Saturated-Magnetized Ferrite, Gyromagnetic Media Electromagnetic Waves in Nonreciprocal Media Plane Waves in a Stationary Plasma Plane Waves in Saturated-Magnetized Ferrites Magnetostatic Waves Magnetostatic Wave Equations Magnetostatic Wave Modes 564 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 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 Medium with a Real Quadratic Index Profile Propagation in Medium with an Imaginary Quadratic Index Profile Steady-State Hermite-Gaussian Beams in Medium with a Quadratic Index Profile Optical Resonators with Curved Mirrors Gaussian Beams in Anisotropic Media 614 Problems 619

10 xix 10 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 Reflection and Refraction of Gaussian Beams on Medium Surfaces 668 Problems 671 A SI Units and Gaussian Units 673 A.I Conversion of Amounts 673 A.2 Formulas in SI (MKSA) Units and Gaussian Units 674 A.3 Prefixes and Symbols for Multiples 676 B Vector Analysis 677 B.I Vector Differential Operations 677 B.I.I General Orthogonal Coordinates 677 B.I.2 General Cylindrical Coordinates 678 B.1.3 Rectangular Coordinates 679 B.I.4 Circular Cylindrical Coordinates 679 B.I.5 Spherical Coordinates 680 B.2 Vector Formulas 680 B.2.1 Vector Algebric Formulas 680 B.2.2 Vector Differential Formulas 681

11 xx Contents B.2.3 Vector Integral Formulas 681 B.2.4 Differential Formulas for the Position Vector 682 C Bessel Functions 683 C.I Power Series Representations 683 C.2 Integral Representations 684 C.3 Approximate Expressions 684 C.3.1 Leading Terms of Power Series (Small Argument) C.3.2 Leading Terms of Asymptotic Series (Large Argument) 684 C.4 Formulas for Bessel Functions 684 C.4.1 Recurrence Formulas 684 C.4.2 Derivatives 685 C.4.3 Integrals 685 C.4.4 Wronskian 685 C.5 Spherical Bessel Functions 686 C.5.1 Bessel Functions of Order n + 1/2 686 C.5.2 Spherical Bessel Functions 686 C.5.3 Spherical Bessel Functions by S.A.Schelkunoff 686 D Legendre Functions 687 D.I Legendre Polynomials _ 687 D.2 Associate Legendre Polynomials. '. 687 D.3 Formulas for Legendre Polynomials 688 D.3.1 Recurrence Formulas 688 D.3.2 Derivatives 688 D.3.3 Integrals 688 E Matrices and Tensors 689 E.I Matrix 689 E.2 Matrix Algebra 690 E.2.1 Definitions 690 E.2.2 Matrix Algebraic Formulas 690 E.3 Matrix Functions 691 E.4 Special Matrices 692 E.5 Tensors and Vectors 693 Physical Constants 695 Smith Chart 697 Bibliography 699 Index 704

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