Computational Electrodynamics: The Finite-Difference Time-Domain Method

Transcription

1 Computational Electrodynamics: The Finite-Difference Time-Domain Method Second Edition Allen Taflove Susan C. Hagness w Artech House Boston London www. artechhouse. com

2 Contents Preface to the Second Edition Preface to the First Edition xvii xxi 1 Electrodynamics Entering the 21st Century Introduction The Heritage of Military Defense Applications Frequency-Domain Solution Techniques Rise of Finite-Difference Time-Domain Methods History of FDTD Techniques for Maxwell's Equations Characteristics of FDTD and Related Space-Grid Time-Domain Techniques Classes of Algorithms Predictive Dynamic Range Scaling to Very Large Problem Sizes Examples of Applications (including Color Plate Section, pages 9-16) Radar-Guided Missile 9, High-Speed Computer Circuit-Board Module 10, Power-Distribution System for a High-Speed Computer Multichip Module 11, Microwave Amplifier 12, Cellular Telephone 13, Optical Microdisk Resonator 14, Photonic Bandgap Microcavity Laser 15, Colliding Spatial Solitons 16, Conclusions 29 References 30 2 The One-Dimensional Scalar Wave Equation Introduction Propagating-Wave Solutions Dispersion Relation Finite Differences Finite-Difference Approximation of the Scalar Wave Equation Numerical Dispersion Relation Case 1: Very Fine Sampling in Time and Space (At > 0, Ax > 0) Case 2: Magic Time-Step (с At = Ax) Case 3: Dispersive Wave Propagation Example of Calculation of Numerical Phase Velocity and Attenuation Examples of Calculations of Pulse Propagation Numerical Stability Complex-Frequency Analysis Examples of Calculations Involving Numerical Instability Summary 61 v

3 vi Computational Electrodynamics: The Finite-Difference Time-Domain Method Appendix 2A: Order of Accuracy 63 2A.1 Lax-Richtmyer Equivalence Theorem 63 2A.2 Limitations 64 References 64 Bibliography on Stability of Finite-Difference Methods 65 Problems 65 3 Introduction to Maxwell's Equations and the Yee Algorithm Introduction Maxwell's Equations in Three Dimensions Reduction to Two Dimensions TM z Mode TE z Mode Reduction to One Dimension дг-directed, z-polarized ТЕМ Mode ^-Directed, y-polarized ТЕМ Mode Equivalence to the Wave Equation in One Dimension The Yee Algorithm Basic Ideas Finite Differences and Notation Finite-Difference Expressions for Maxwell's Equations in Three Dimensions Space Region With a Continuous Variation of Material Properties Space Region With a Finite Number of Distinct Media Space Region With Nonpermeable Media Reduction to the Two-Dimensional TM Z and TE Z Modes Interpretation as Faraday's and Ampere's Laws in Integral Form Divergence-Free Nature Alternative Finite-Difference Grids Cartesian Grids Hexagonal Grids Tetradecahedron / Dual-Tetrahedron Mesh in Three Dimensions Summary 105 References 106 Problems Numerical Dispersion and Stability Introduction Derivation of the Numerical Dispersion Relation for Two-Dimensional Wave Propagation Extension to Three Dimensions Comparison With the Ideal Dispersion Case Anisotropy of the Numerical Phase Velocity Sample Values of Numerical Phase Velocity Intrinsic Grid Velocity Anisotropy Complex-Valued Numerical Wavenumbers Case 1: Numerical Wave Propagation Along the Principal Lattice Axes Case 2: Numerical Wave Propagation Along a Grid Diagonal Example of Calculation of Numerical Phase Velocity and Attenuation Example of Calculation of Wave Propagation 131

4 Contents vii 4.7 Numerical Stability Complex-Frequency Analysis Example of a Numerically Unstable Two-Dimensional FDTD Model Generalized Stability Problem Boundary Conditions Variable and Unstructured Meshing Lossy, Dispersive, Nonlinear, and Gain Materials Modified Yee-Based Algorithms for Improved Numerical Dispersion Strategy 1: Center a Specific Numerical Phase-Velocity Curve About с Strategy 2: Use Fourth-Order-Accurate Spatial Differences Strategy 3: Use Hexagonal Grids Strategy 4: Use Discrete Fourier Transforms to Calculate the Spatial Derivatives Alternating-Direction-Implicit Time-Stepping Algorithm for Operation Beyond the Courant Limit Numerical Formulation of the Zheng/Chen/Zhang Algorithm Numerical Stability Numerical Dispersion Discussion Summary 172 References 172 Problems 173 Projects Incident Wave Source Conditions Introduction Pointwise E and H Hard Sources in One Dimension Pointwise E and H Hard Sources in Two Dimensions Green's Function for the Scalar Wave Equation in Two Dimensions Obtaining Comparative FDTD Data Results for Effective Action Radius of a Hard-Sourced Field Component and M Current Sources in Three Dimensions Sources and Charging Sinusoidal Sources Transient (Pulse) Sources Intrinsic Lattice Capacitance Intrinsic Lattice Inductance Impact Upon FDTD Simulations of Lumped-Element Capacitors and Inductors The Plane-Wave Source Condition The Total-Field/ Scattered-Field Technique: Ideas and One-Dimensional Formulation Ideas One-Dimensional Formulation Two-Dimensional Formulation of the TF/SF Technique Consistency Conditions Calculation of the Incident Field Illustrative Example Three-Dimensional Formulation of the TF/SF Technique Consistency Conditions Calculation of the Incident Field Pure Scattered-Field Formulation 224

5 viii Computational Electrodynamics: The Finite-Difference Time-Domain Method Application to PEC Structures Application to Lossy Dielectric Structures Choice of Incident Plane-Wave Formulation Waveguide Source Conditions Pulsed Electric Field Modal Hard Source Total-Field / Reflected-Field Modal Formulation Resistive Source and Load Conditions Summary 231 References 232 Problems 232 Projects Analytical Absorbing Boundary Conditions Introduction Bayliss-Turkel Radiation Operators Spherical Coordinates Cylindrical Coordinates Engquist-Majda One-Way Wave Equations One-Term and Two-Term Taylor Series Approximations Mur Finite-Difference Scheme Trefethen-Halpern Generalized and Higher Order ABCs Theoretical Reflection Coefficient Analysis Numerical Experiments Higdon Radiation Operators Formulation First Two Higdon Operators Discussion Liao Extrapolation in Space and Time Formulation Discussion Ramahi Complementary Operators Basic Idea Complementary Operators Effect of Multiple Wave Reflections Basis of the Concurrent Complementary Operator Method Illustrative FDTD Modeling Results Obtained Using the C-COM Summary 281 References 281 Problems Perfectly Matched Layer Absorbing Boundary Conditions (Stephen D. Gedney and Allen Taflove) Introduction Plane Wave Incident Upon a Lossy Half-Space Plane Wave Incident Upon Berenger's PML Medium Two-Dimensional TE Z Case Two-Dimensional TM Z Case Three-Dimensional Case 294

6 Contents ix 7.4 Stretched-Coordinate Formulation of Berenger's PML An Anisotropie PML Absorbing Medium Perfectly Matched Uniaxial Medium Relationship to Berenger's Split-Field PML A Generalized Three-Dimensional Formulation Inhomogeneous Media Theoretical Performance of the PML The Continuous Space The Discrete Space Efficient Implementation of UPML in FDTD Derivation of the Finite-Difference Expressions Computer Implementation of the UPML Numerical Experiments With Berenger's Split-Field PML Outgoing Cylindrical Wave in a Two-Dimensional Open-Region Grid Outgoing Spherical Wave in a Three-Dimensional Open-Region Lattice Dispersive Wave Propagation in Metal Waveguides Dispersive and Multimode Wave Propagation in Dielectric Waveguides Numerical Experiments With UPML Current Source Radiating in an Unbounded Two-Dimensional Region Highly Elongated Domains Microstrip Transmission Line UPML Termination for Conductive Media Theory Numerical Example: Termination of a Conductive Half-Space Medium UPML Termination for Dispersive Media Theory Numerical Example: Reflection by a Lorentz Medium Summary and Conclusions 343 References 345 Projects Near-to-Far-Field Transformation Introduction Two-Dimensional Transformation, Phasor Domain Application of Green's Theorem Far-Field Limit Reduction to Standard Form Obtaining Phasor Quantities Via Discrete Fourier Transformation Surface Equivalence Theorem Extension to Three Dimensions, Phasor Domain Time-Domain Near-to-Far-Field Transformation Summary 371 References 372 Project Dispersive and Nonlinear Materials Introduction Types of Dispersions Considered 374

7 x Computational Electrodynamics: The Finite-Difference Time-Domain Method Debye Media Lorentz Media Piecewise-Linear Recursive Convolution Method, Linear Material Case General Formulation of the Method Application to Debye Media Application to Lorentz Media Numerical Results Piecewise-Linear Recursive Convolution Method, Nonlinear Dispersive Material Case Governing Equations General Formulation of the Method FDTD Realization in One Dimension Numerical Results Auxiliary Differential Equation Method, Linear Material Case Formulation for Multiple Debye Poles Formulation for Multiple Lorentz Pole Pairs Numerical Results Auxiliary Differential Equation Method, Nonlinear Dispersive Material Case Formulation for Multiple Lorentz Pole Pairs, TM Z Case Numerical Results for Temporal Solitons Numerical Results for Spatial Solitons Summary and Conclusions 407 References 408 Problems 409 Projects Local Subcell Models of Fine Geometrical Features Introduction Basis of Contour-Path FDTD Modeling The Simplest Contour-Path Subcell Models Diagonal Split-Cell Model for PEC Surfaces Average Properties Model for Material Surfaces The Contour-Path Model of the Narrow Slot The Contour-Path Model of the Thin Wire Locally Conformal Models of Curved Surfaces Dey-Mittra Technique for PEC Structures Illustrative Results for PEC Structures Dey-Mittra Technique for Material Structures Maloney-Smith Technique for Thin Material Sheets Basis Illustrative Results Dispersive Surface Impedance Maloney-Smith Method Beggs Method Lee Method Relativistic Motion of PEC Boundaries Basis Illustrative Results Summary and Discussion 468 References 470

8 Contents xi Bibliography 471 Projects Nonorthogonal and Unstructured Grids (Stephen D. Gedney and Faiza Lansing) Introduction Nonuniform Orthogonal Grids Locally Conformal Grids, Globally Orthogonal Global Curvilinear Coordinates Nonorthogonal Curvilinear FDTD Algorithm Stability Criterion Irregular Nonorthogonal Structured Grids Irregular Nonorthogonal Unstructured Grids Generalized Yee Algorithm Inhomogeneous Media Practical Implementation of the Generalized Yee Algorithm A Planar Generalized Yee Algorithm Time-Stepping Expressions Projection Operators Efficient Time-Stepping Implementation Examples of Passive-Circuit Modeling Using the Planar Generalized Yee Algorithm GHz Wilkinson Power Divider GHz Gysel Power Divider Signal Lines in an IBM Thermal Conduction Module Summary and Conclusions 522 References 523 Problems 525 Projects Bodies of Revolution (Thomas G. Jürgens, Jeffrey G. Blaschak, and Gregory W. Saewert) Introduction Field Expansion Difference Equations for Off-Axis Cells Ampere's Law Contour Path Integral to Calculate e r Ampere's Law Contour Path Integral to Calculate е ф Ampere's Law Contour Path Integral to Calculate e z Difference Equations Surface-Conforming Contour Path Integrals Difference Equations for On-Axis Cells Ampere's Law Contour Path Integral to Calculate e z on the z-axis Ampere's Law Contour Path Integral to Calculate е ф on the z-axis Faraday's Law Calculation of h r on the z-axis Numerical Stability PML Absorbing Boundary Condition BOR-FDTD Background Extension of PML to the General BOR Case Examples 558

9 xii Computational Electrodynamics: The Finite-Difference Time-Domain Method 12.7 Application to Particle Accelerator Physics Definitions and Concepts Examples Summary 566 References 566 Problems 567 Projects Analysis of Periodic Structures (James G. Maloney and Morris P. Kesler) Introduction Review of Scattering From Periodic Structures Direct Field Methods Normal Incidence Case Multiple Unit Cells for Oblique Incidence Sine-Cosine Method Angled-Update Method Introduction to the Field-Transformation Technique Multiple-Grid Approach Formulation Numerical Stability Analysis Numerical Dispersion Analysis Lossy Materials Lossy Screen Example Split-Field Method, Two Dimensions Formulation Numerical Stability Analysis Numerical Dispersion Analysis Lossy Materials Lossy Screen Example Split-Field Method, Three Dimensions Formulation Numerical Stability Analysis UPML Absorbing Boundary Condition Application of the Periodic FDTD Method Photonic В andgap S tructures Frequency-Selective Surfaces Antenna Arrays Summary and Conclusions 623 Acknowledgments 623 References 623 Projects Modeling of Antennas (James G. Maloney, Glenn S. Smith, Eric T. Thiele, and Om P. Gandhi) Introduction Formulation of the Antenna Problem Transmitting Antenna 628

10 Contents xiii Receiving Antenna Symmetry Excitation Antenna Feed Models Detailed Modeling of the Feed Simple Gap Feed Model for a Monopole Antenna Improved Simple Feed Model Near-to-Far-Field Transformations Use of Symmetry Time-Domain Near-to-Far-Field Transformation Frequency-Domain Near-Field to Far-Field Transformation Plane-Wave Source Effect of an Incremental Displacement of the Surface Currents Effect of an Incremental Time Shift Relation to Total-Field / Scattered-Field Lattice Zoning Case Study I: The Standard-Gain Horn Case Study II: The Vivaldi Slotline Array Background The Planar Element The Vivaldi Pair The Vivaldi Quad The Linear Phased Array Phased-Array Radiation Characteristics Indicated by the FDTD Modeling Active Impedance of the Phased Array Near-Field Simulations Generic 900-MHz Cellphone Handset in Free Space MHz Dipole Antenna Near a Layered Bone-Brain Half-Space MHz Dipole Antenna Near a Rectangular Brain Phantom MHz Infinitesimal Dipole Antenna Near a Spherical Brain Phantom ,900-MHz Half-Wavelength Dipole Near a Spherical Brain Phantom Selected Recent Applications Use of Photonic-Bandgap Materials Ground-Penetrating Radar Antenna-Radome Interaction Personal Wireless Communications Devices Biomedical Applications of Antennas Summary and Conclusions 697 References 697 Projects High-Speed Electronic Circuits With Active and Nonlinear Components (Melinda Piket-May, Bijan Houshmand, and Tatsuo Itoh) Introduction Basic Circuit Parameters Transmission Line Parameters Impedance S Parameters 707

11 xiv Computational Electrodynamics: The Finite-Difference Time-Domain Method 15.3 Differential Capacitance Calculation Differential Inductance Calculation Lumped Inductance Due to a Discontinuity Flux / Current Definition Fitting Z(co) or S(co) to an Equivalent Circuit Discussion: Choice of Methods Inductance of Complex Power-Distribution Systems Method Description Example: Multiplane Meshed Printed-Circuit Board Discussion Parallel Coplanar Microstrips Multilayered Interconnect Modeling Example Digital Signal Processing and Spectrum Estimation Prony's Method Autoregressive Models Pade Approximation Modeling of Lumped Circuit Elements FDTD Formulation Extended to Circuit Elements The Resistor The Resistive Voltage Source The Capacitor The Inductor The Diode The Bipolar Junction Transistor Direct Linking of FDTD and SPICE Basic Idea Norton Equivalent Circuit "Looking Into" the FDTD Space Lattice Thevenin Equivalent Circuit "Looking Into" the FDTD Space Lattice Case Study: A 6-GHz MESFET Amplifier Model Large-Signal Model Amplifier Configuration Analysis of the Circuit Without the Packaging Structure Analysis of the Circuit With the Packaging Structure Summary and Conclusions 759 Acknowledgments 760 References 761 Additional Bibliography 763 Projects Microcavity Optical Resonators Introduction Issues Related to FDTD Modeling of Optical Structures Optical Waveguides Material Dispersion and Nonlinearities Macroscopic Modeling of Optical Gain Media Theory Validation Studies Application to Vertical-Cavity Surface-Emitting Lasers Passive Studies 782

12 Contents xv Active Studies Microcavities Based on Photonic Bandgap Structures, Quasi One-Dimensional Case Microcavities Based on Photonic Bandgap Structures, Two-Dimensional Case Microcavity Ring Resonators FDTD Modeling Considerations Coupling to Straight Waveguides Coupling to Curved Waveguides Elongated Ring Designs Resonances Microcavity Disk Resonators Resonance Behavior Suppression of Higher Order Radial Whispering-Gallery Modes Additional FDTD Modeling Studies Summary and Conclusions 819 References 822 Additional Bibliography 825 Projects 826 Acronyms 827 About the Authors 831 Index 839