Computational Electrodynamics

The Finite-Difference Time-Domain Method, Third Edition

By: Allen Taflove, Susan Hagness

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Hardcover | 30 June 2005 | Edition Number 3

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1040 Pages

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18.4 x 26.1 x 5.5

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This extensively revised and expanded third edition of the Artech House bestseller, Computational Electrodynamics: The Finite-Difference Time-Domain Method, offers engineers the most up-to-date and definitive resource on this critical method for solving Maxwell's equations. The method helps practitioners design antennas, wireless communications devices, high-speed digital and microwave circuits, and integrated optical devices with unsurpassed efficiency. There has been considerable advancement in FDTD computational technology over the past few years, and the third edition brings professionals the very latest details with entirely new chapters on important techniques, major updates on key topics, and new discussions on emerging areas such as nanophotonics. What's more, to supplement the third edition, the authors have created a Web site with solutions to problems, downloadable graphics and videos, and updates, making this new edition the ideal textbook on the subject as well.

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Preface to the Second Edition	p. xvii
Preface to the First Edition	p. xxi
Electrodynamics Entering the 21st Century	p. 1
Introduction	p. 1
The Heritage of Military Defense Applications	p. 2
Frequency-Domain Solution Techniques	p. 3
Rise of Finite-Difference Time-Domain Methods	p. 3
History of FDTD Techniques for Maxwell's Equations	p. 5
Characteristics of FDTD and Related Space-Grid Time-Domain Techniques	p. 7
Classes of Algorithms	p. 7
Predictive Dynamic Range	p. 17
Scaling to Very Large Problem Sizes	p. 18
Examples of Applications (including Color Plate Section, pages 9-16)	p. 19
Radar-Guided Missile	p. 20
High-Speed Computer Circuit-Board Module	p. 21
Power-Distribution System for a High-Speed Computer Multichip Module	p. 22
Microwave Amplifier	p. 23
Cellular Telephone	p. 24
Optical Microdisk Resonator	p. 25
Photonic Bandgap Microcavity Laser	p. 27
Colliding Spatial Solitons	p. 28
Conclusions	p. 29
References	p. 30
The One-Dimensional Scalar Wave Equation	p. 35
Introduction	p. 35
Propagating-Wave Solutions	p. 35
Dispersion Relation	p. 36
Finite Differences	p. 38
Finite-Difference Approximation of the Scalar Wave Equation	p. 39
Numerical Dispersion Relation	p. 42
Case 1: Very Fine Sampling in Time and Space ([Delta]t [right arrow] 0, [Delta]x [right arrow] 0)	p. 43
Case 2: Magic Time-Step (c[Delta]t = [Delta]x)	p. 43
Case 3: Dispersive Wave Propagation	p. 44
Example of Calculation of Numerical Phase Velocity and Attenuation	p. 49
Examples of Calculations of Pulse Propagation	p. 51
Numerical Stability	p. 55
Complex-Frequency Analysis	p. 55
Examples of Calculations Involving Numerical Instability	p. 59
Summary	p. 61
Order of Accuracy	p. 63
Lax-Richtmyer Equivalence Theorem	p. 63
Limitations	p. 64
References	p. 64
Bibliography on Stability of Finite-Difference Methods	p. 65
Problems	p. 65
Introduction to Maxwell's Equations and the Yee Algorithm	p. 67
Introduction	p. 67
Maxwell's Equations in Three Dimensions	p. 67
Reduction to Two Dimensions	p. 70
TM[subscript z] Mode	p. 71
TE[subscript z] Mode	p. 71
Reduction to One Dimension	p. 72
x-Directed, z-Polarized TEM Mode	p. 72
x-Directed, y-Polarized TEM Mode	p. 73
Equivalence to the Wave Equation in One Dimension	p. 74
The Yee Algorithm	p. 75
Basic Ideas	p. 75
Finite Differences and Notation	p. 77
Finite-Difference Expressions for Maxwell's Equations in Three Dimensions	p. 80
Space Region With a Continuous Variation of Material Properties	p. 85
Space Region With a Finite Number of Distinct Media	p. 87
Space Region With Nonpermeable Media	p. 89
Reduction to the Two-Dimensional TM[subscript z] and TE[subscript z] Modes	p. 91
Interpretation as Faraday's and Ampere's Laws in Integral Form	p. 93
Divergence-Free Nature	p. 96
Alternative Finite-Difference Grids	p. 98
Cartesian Grids	p. 99
Hexagonal Girds	p. 101
Tetradecahedron/Dual-Tetrahedron Mesh in Three Dimensions	p. 104
Summary	p. 105
References	p. 106
Problems	p. 106
Numerical Dispersion and Stability	p. 109
Introduction	p. 109
Derivation of the Numerical Dispersion Relation for Two-Dimensional Wave Propagation	p. 110
Extension to Three Dimensions	p. 112
Comparison With the Ideal Dispersion Case	p. 113
Anisotropy of the Numerical Phase Velocity	p. 114
Sample Values of Numerical Phase Velocity	p. 114
Intrinsic Grid Velocity Anisotropy	p. 120
Complex-Valued Numerical Wavenumbers	p. 124
Case 1: Numerical Wave Propagation Along the Principal Lattice Axes	p. 124
Case 2: Numerical Wave Propagation Along a Grid Diagonal	p. 127
Example of Calculation of Numerical Phase Velocity and Attenuation	p. 129
Example of Calculation of Wave Propagation	p. 131
Numerical Stability	p. 133
Complex-Frequency Analysis	p. 133
Example of a Numerically Unstable Two-Dimensional FDTD Model	p. 139
Generalized Stability Problem	p. 141
Boundary Conditions	p. 141
Variable and Unstructured Meshing	p. 142
Lossy, Dispersive, Nonlinear, and Gain Materials	p. 142
Modified Yee-Based Algorithms for Improved Numerical Dispersion	p. 142
Strategy 1: Center a Specific Numerical Phase-Velocity Curve About c	p. 143
Strategy 2: Use Fourth-Order-Accurate Spatial Differences	p. 143
Strategy 3: Use Hexagonal Grids	p. 152
Strategy 4: Use Discrete Fourier Transforms to Calculate the Spatial Derivatives	p. 156
Alternating-Direction-Implicit Time-Stepping Algorithm for Operation Beyond the Courant Limit	p. 160
Numerical Formulation of the Zheng/Chen/Zhang Algorithm	p. 162
Numerical Stability	p. 169
Numerical Dispersion	p. 171
Discussion	p. 171
Summary	p. 172
References	p. 172
Problems	p. 173
Projects	p. 174
Incident Wave Source Conditions	p. 175
Introduction	p. 175
Pointwise E and H Hard Sources in One Dimension	p. 176
Pointwise E and H Hard Sources in Two Dimensions	p. 178
Green's Function for the Scalar Wave Equation in Two Dimensions	p. 178
Obtaining Comparative FDTD Data	p. 179
Results for Effective Action Radius of a Hard-Sourced Field Component	p. 180
J and M Current Sources in Three Dimensions	p. 182
Sources and Charging	p. 183
Sinusoidal Sources	p. 184
Transient (Pulse) Sources	p. 185
Intrinsic Lattice Capacitance	p. 189
Intrinsic Lattice Inductance	p. 190
Impact Upon FDTD Simulations of Lumped-Element Capacitors and Inductors	p. 191
The Plane-Wave Source Condition	p. 193
The Total-Field/Scattered-Field Technique: Ideas and One-Dimensional Formulation	p. 194
Ideas	p. 194
One-Dimensional Formulation	p. 197
Two-Dimensional Formulation of the TF/SF Technique	p. 201
Consistency Conditions	p. 203
Calculation of the Incident Field	p. 207
Illustrative Example	p. 212
Three-Dimensional Formulation of the TF/SF Technique	p. 212
Consistency Conditions	p. 216
Calculation of the Incident Field	p. 221
Pure Scattered-Field Formulation	p. 224
Application to PEC Structures	p. 224
Application to Lossy Dielectric Structures	p. 225
Choice of Incident Plane-Wave Formulation	p. 227
Waveguide Source Conditions	p. 227
Pulsed Electric Field Modal Hard Source	p. 228
Total-Field/Reflected-Field Modal Formulation	p. 229
Resistive Source and Load Conditions	p. 230
Summary	p. 231
References	p. 232
Problems	p. 232
Projects	p. 232
Analytical Absorbing Boundary Conditions	p. 235
Introduction	p. 235
Bayliss-Turkel Radiation Operators	p. 237
Spherical Coordinates	p. 238
Cylindrical Coordinates	p. 241
Engquist-Majda One-Way Wave Equations	p. 244
One-Term and Two-Term Taylor Series Approximations	p. 245
Mur Finite-Difference Scheme	p. 248
Trefethen-Halpern Generalized and Higher Order ABCs	p. 251
Theoretical Reflection Coefficient Analysis	p. 253
Numerical Experiments	p. 256
Higdon Radiation Operators	p. 261
Formulation	p. 261
First Two Higdon Operators	p. 263
Discussion	p. 264
Liao Extrapolation in Space and Time	p. 265
Formulation	p. 265
Discussion	p. 267
Ramahi Complementary Operators	p. 269
Basic Idea	p. 269
Complementary Operators	p. 270
Effect of Multiple Wave Reflections	p. 271
Basis of the Concurrent Complementary Operator Method	p. 273
Illustrative FDTD Modeling Results Obtained Using the C-COM	p. 278
Summary	p. 281
References	p. 281
Problems	p. 282
Perfectly Matched Layer Absorbing Boundary Conditions	p. 285
Introduction	p. 285
Plane Wave Incident Upon a Lossy Half-Space	p. 286
Plane Wave Incident Upon Berenger's PML Medium	p. 288
Two-Dimensional TE[subscript z] Case	p. 289
Two-Dimensional TM[subscript z] Case	p. 293
Three-Dimensional Case	p. 294
Stretched-Coordinate Formulation of Berenger's PML	p. 295
An Anisotropic PML Absorbing Medium	p. 298
Perfectly Matched Uniaxial Medium	p. 298
Relationship to Berenger's Split-Field PML	p. 301
A Generalized Three-Dimensional Formulation	p. 302
Inhomogeneous Media	p. 304
Theoretical Performance of the PML	p. 305
The Continuous Space	p. 305
The Discrete Space	p. 305
Efficient Implementation of UPML in FDTD	p. 308
Derivation of the Finite-Difference Expressions	p. 308
Computer Implementation of the UPML	p. 311
Numerical Experiments With Berenger's Split-Field PML	p. 314
Outgoing Cylindrical Wave in a Two-Dimensional Open-Region Grid	p. 314
Outgoing Spherical Wave in a Three-Dimensional Open-Region Lattice	p. 316
Dispersive Wave Propagation in Metal Waveguides	p. 318
Dispersive and Multimode Wave Propagation in Dielectric Waveguides	p. 320
Numerical Experiments With UPML	p. 322
Current Source Radiating in an Unbounded Two-Dimensional Region	p. 322
Highly Elongated Domains	p. 327
Microstrip Transmission Line	p. 330
UPML Termination for Conductive Media	p. 332
Theory	p. 332
Numerical Example: Termination of a Conductive Half-Space Medium	p. 335
UPML Termination for Dispersive Media	p. 338
Theory	p. 338
Numerical Example: Reflection by a Lorentz Medium	p. 343
Summary and Conclusions	p. 343
References	p. 345
Projects	p. 347
Near-to-Far-Field Transformation	p. 349
Introduction	p. 349
Two-Dimensional Transformation, Phasor Domain	p. 350
Application of Green's Theorem	p. 351
Far-Field Limit	p. 352
Reduction to Standard Form	p. 354
Obtaining Phasor Quantities Via Discrete Fourier Transformation	p. 356
Surface Equivalence Theorem	p. 359
Extension to Three Dimensions, Phasor Domain	p. 361
Time-Domain Near-to-Far-Field Transformation	p. 366
Summary	p. 371
References	p. 372
Project	p. 372
Dispersive and Nonlinear Materials	p. 373
Introduction	p. 373
Types of Dispersions Considered	p. 374
Debye Media	p. 374
Lorentz Media	p. 375
Piecewise-Linear Recursive Convolution Method, Linear Material Case	p. 375
General Formulation of the Method	p. 375
Application to Debye Media	p. 378
Application to Lorentz Media	p. 379
Numerical Results	p. 380
Piecewise-Linear Recursive Convolution Method, Nonlinear Dispersive Material Case	p. 382
Governing Equations	p. 382
General Formulation of the Method	p. 384
FDTD Realization in One Dimension	p. 386
Numerical Results	p. 388
Auxiliary Differential Equation Method, Linear Material Case	p. 392
Formulation for Multiple Debye Poles	p. 392
Formulation for Multiple Lorentz Pole Pairs	p. 394
Numerical Results	p. 397
Auxiliary Differential Equation Method, Nonlinear Dispersive Material Case	p. 398
Formulation for Multiple Lorentz Pole Pairs, TM[subscript Z] Case	p. 398
Numerical Results for Temporal Solitons	p. 401
Numerical Results for Spatial Solitons	p. 404
Summary and Conclusions	p. 407
References	p. 408
Problems	p. 409
Projects	p. 410
Local Subcell Models of Fine Geometrical Features	p. 411
Introduction	p. 411
Basis of Contour-Path FDTD Modeling	p. 412
The Simplest Contour-Path Subcell Models	p. 413
Diagonal Split-Cell Model for PEC Surfaces	p. 413
Average Properties Model for Material Surfaces	p. 415
The Contour-Path Model of the Narrow Slot	p. 416
The Contour-Path Model of the Thin Wire	p. 420
Locally Conformal Models of Curved Surfaces	p. 424
Dey-Mittra Technique for PEC Structures	p. 424
Illustrative Results for PEC Structures	p. 427
Dey-Mittra Technique for Material Structures	p. 433
Maloney-Smith Technique for Thin Material Sheets	p. 434
Basis	p. 434
Illustrative Results	p. 438
Dispersive Surface Impedance	p. 442
Maloney-Smith Method	p. 442
Beggs Method	p. 449
Lee Method	p. 457
Relativistic Motion of PEC Boundaries	p. 461
Basis	p. 461
Illustrative Results	p. 465
Summary and Discussion	p. 468
References	p. 470
Bibliography	p. 471
Projects	p. 472
Nonorthogonal and Unstructured Grids	p. 473
Introduction	p. 473
Nonuniform Orthogonal Grids	p. 474
Locally Conformal Grids, Globally Orthogonal	p. 482
Global Curvilinear Coordinates	p. 484
Nonorthogonal Curvilinear FDTD Algorithm	p. 484
Stability Criterion	p. 490
Irregular Nonorthogonal Structured Grids	p. 493
Irregular Nonorthogonal Unstructured Grids	p. 500
Generalized Yee Algorithm	p. 500
Inhomogeneous Media	p. 506
Practical Implementation of the Generalized Yee Algorithm	p. 508
A Planar Generalized Yee Algorithm	p. 509
Time-Stepping Expressions	p. 510
Projection Operators	p. 511
Efficient Time-Stepping Implementation	p. 513
Examples of Passive-Circuit Modeling Using the Planar Generalized Yee Algorithm	p. 514
32-GHz Wilkinson Power Divider	p. 514
32-GHz Gysel Power Divider	p. 517
Signal Lines in an IBM Thermal Conduction Module	p. 518
Summary and Conclusions	p. 522
References	p. 523
Problems	p. 525
Projects	p. 527
Bodies of Revolution	p. 529
Introduction	p. 529
Field Expansion	p. 530
Difference Equations for Off-Axis Cells	p. 530
Ampere's Law Contour Path Integral to Calculate e[subscript r]	p. 532
Ampere's Law Contour Path Integral to Calculate e[subscript phi]	p. 534
Ampere's Law Contour Path Integral to Calculate e[subscript z]	p. 536
Difference Equations	p. 539
Surface-Conforming Contour Path Integrals	p. 542
Difference Equations for On-Axis Cells	p. 544
Ampere's Law Contour Path Integral to Calculate e[subscript z] on the z-Axis	p. 544
Ampere's Law Contour Path Integral to Calculate e[subscript phi] on the z-Axis	p. 546
Faraday's Law Calculation of h[subscript r] on the z-Axis	p. 548
Numerical Stability	p. 549
PML Absorbing Boundary Condition	p. 549
BOR-FDTD Background	p. 549
Extension of PML to the General BOR Case	p. 552
Examples	p. 558
Application to Particle Accelerator Physics	p. 560
Definitions and Concepts	p. 560
Examples	p. 563
Summary	p. 566
References	p. 566
Problems	p. 567
Projects	p. 568
Analysis of Periodic Structures	p. 569
Introduction	p. 569
Review of Scattering From Periodic Structures	p. 572
Direct Field Methods	p. 575
Normal Incidence Case	p. 575
Multiple Unit Cells for Oblique Incidence	p. 577
Sine-Cosine Method	p. 579
Angled-Update Method	p. 580
Introduction to the Field-Transformation Technique	p. 584
Multiple-Grid Approach	p. 589
Formulation	p. 589
Numerical Stability Analysis	p. 591
Numerical Dispersion Analysis	p. 592
Lossy Materials	p. 593
Lossy Screen Example	p. 595
Split-Field Method, Two Dimensions	p. 596
Formulation	p. 596
Numerical Stability Analysis	p. 598
Numerical Dispersion Analysis	p. 600
Lossy Materials	p. 601
Lossy Screen Example	p. 602
Split-Field Method, Three Dimensions	p. 603
Formulation	p. 603
Numerical Stability Analysis	p. 607
UPML Absorbing Boundary Condition	p. 610
Application of the Periodic FDTD Method	p. 614
Photonic Bandgap Structures	p. 614
Frequency-Selective Surfaces	p. 616
Antenna Arrays	p. 618
Summary and Conclusions	p. 623
Acknowledgments	p. 623
References	p. 623
Projects	p. 625
Modeling of Antennas	p. 627
Introduction	p. 627
Formulation of the Antenna Problem	p. 628
Transmitting Antenna	p. 628
Receiving Antenna	p. 630
Symmetry	p. 630
Excitation	p. 632
Antenna Feed Models	p. 634
Detailed Modeling of the Feed	p. 634
Simple Gap Feed Model for a Monopole Antenna	p. 636
Improved Simple Feed Model	p. 639
Near-to-Far-Field Transformations	p. 644
Use of Symmetry	p. 645
Time-Domain Near-to-Far-Field Transformation	p. 646
Frequency-Domain Near-Field to Far-Field Transformation	p. 648
Plane-Wave Source	p. 649
Effect of an Incremental Displacement of the Surface Currents	p. 649
Effect of an Incremental Time Shift	p. 651
Relation to Total-Field/Scattered-Field Lattice Zoning	p. 652
Case Study I: The Standard-Gain Horn	p. 653
Case Study II: The Vivaldi Slotline Array	p. 659
Background	p. 659
The Planar Element	p. 660
The Vivaldi Pair	p. 665
The Vivaldi Quad	p. 665
The Linear Phased Array	p. 668
Phased-Array Radiation Characteristics Indicated by the FDTD Modeling	p. 669
Active Impedance of the Phased Array	p. 669
Near-Field Simulations	p. 675
Generic 900-MHz Cellphone Handset in Free Space	p. 675
900-MHz Dipole Antenna Near a Layered Bone-Brain Half-Space	p. 676
840-MHz Dipole Antenna Near a Rectangular Brain Phantom	p. 678
900-MHz Infinitesimal Dipole Antenna Near a Spherical Brain Phantom	p. 680
1,900-MHz Half-Wavelength Dipole Near a Spherical Brain Phantom	p. 681
Selected Recent Applications	p. 682
Use of Photonic-Bandgap Materials	p. 682
Ground-Penetrating Radar	p. 682
Antenna-Radome Interaction	p. 687
Personal Wireless Communications Devices	p. 690
Biomedical Applications of Antennas	p. 696
Summary and Conclusions	p. 697
References	p. 697
Projects	p. 701
High-Speed Electronic Circuits With Active and Nonlinear Components	p. 703
Introduction	p. 703
Basic Circuit Parameters	p. 705
Transmission Line Parameters	p. 706
Impedance	p. 706
S Parameters	p. 707
Differential Capacitance Calculation	p. 708
Differential Inductance Calculation	p. 709
Lumped Inductance Due to a Discontinuity	p. 711
Flux / Current Definition	p. 711
Fitting Z([omega]) or S([omega]) to an Equivalent Circuit	p. 712
Discussion: Choice of Methods	p. 712
Inductance of Complex Power-Distribution Systems	p. 713
Method Description	p. 713
Example: Multiplane Meshed Printed-Circuit Board	p. 715
Discussion	p. 718
Parallel Coplanar Microstrips	p. 718
Multilayered Interconnect Modeling Example	p. 720
Digital Signal Processing and Spectrum Estimation	p. 721
Prony's Method	p. 721
Autoregressive Models	p. 725
Pade Approximation	p. 729
Modeling of Lumped Circuit Elements	p. 734
FDTD Formulation Extended to Circuit Elements	p. 734
The Resistor	p. 736
The Resistive Voltage Source	p. 737
The Capacitor	p. 737
The Inductor	p. 738
The Diode	p. 740
The Bipolar Junction Transistor	p. 741
Direct Linking of FDTD and SPICE	p. 743
Basic Idea	p. 745
Norton Equivalent Circuit "Looking Into" the FDTD Space Lattice	p. 745
Thevenin Equivalent Circuit "Looking Into" the FDTD Space Lattice	p. 748
Case Study: A 6-GHz MESFET Amplifier Model	p. 750
Large-Signal Model	p. 750
Amplifier Configuration	p. 753
Analysis of the Circuit Without the Packaging Structure	p. 754
Analysis of the Circuit With the Packaging Structure	p. 756
Summary and Conclusions	p. 759
Acknowledgements	p. 760
References	p. 761
Additional Bibliography	p. 763
Projects	p. 764
Microcavity Optical Resonators	p. 767
Introduction	p. 767
Issues Related to FDTD Modeling of Optical Structures	p. 768
Optical Waveguides	p. 768
Material Dispersion and Nonlinearities	p. 772
Macroscopic Modeling of Optical Gain Media	p. 772
Theory	p. 773
Validation Studies	p. 776
Application to Vertical-Cavity Surface-Emitting Lasers	p. 780
Passive Studies	p. 782
Active Studies	p. 784
Microcavities Based on Photonic Bandgap Structures, Quasi One-Dimensional Case	p. 788
Microcavities Based on Photonic Bandgap Structures, Two-Dimensional Case	p. 793
Microcavity Ring Resonators	p. 797
FDTD Modeling Considerations	p. 797
Coupling to Straight Waveguides	p. 800
Coupling to Curved Waveguides	p. 802
Elongated Ring Designs	p. 804
Resonances	p. 806
Microcavity Disk Resonators	p. 810
Resonance Behavior	p. 811
Suppression of Higher Order Radial Whispering-Gallery Modes	p. 815
Additional FDTD Modeling Studies	p. 819
Summary and Conclusions	p. 819
References	p. 822
Additional Bibliography	p. 825
Projects	p. 826
Acronyms	p. 827
About the Authors	p. 831
Index	p. 839
Table of Contents provided by Syndetics. All Rights Reserved.

Computational Electrodynamics

The Finite-Difference Time-Domain Method, Third Edition

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Hardcover

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