Computational electrodynamics : the finite-difference time-domain method

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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,

Author Notes

Allen Taflove is a professor of electrical and computer engineering at Northwestern University, Evanston, IL. He is also the author of Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House 1995).

050

Stephen D. Gedney and Allen TafloveStephen D. Gedney and Faiza LansingThomas G. Jurgens and Jeffrey G. Blaschak and Gregory W. SaewertJames G. Maloney and Morris P. KeslerJames G. Maloney and Glenn S. Smith and Eric T. Thiele and Om P. GandhiMelinda Piket-May and Bijan Houshmand and Tatsuo Itoh

Preface to the Second Edition	p. xvii
Preface to the First Edition	p. xxi
1 Electrodynamics Entering the 21st Century	p. 1
1.1 Introduction	p. 1
1.2 The Heritage of Military Defense Applications	p. 2
1.3 Frequency-Domain Solution Techniques	p. 3
1.4 Rise of Finite-Difference Time-Domain Methods	p. 3
1.5 History of FDTD Techniques for Maxwell's Equations	p. 5
1.6 Characteristics of FDTD and Related Space-Grid Time-Domain Techniques	p. 7
1.6.1 Classes of Algorithms	p. 7
1.6.2 Predictive Dynamic Range	p. 17
1.6.3 Scaling to Very Large Problem Sizes	p. 18
1.7 Examples of Applications (including Color Plate Section, pages 9-16)	p. 19
1.7.1 Radar-Guided Missile	p. 20
1.7.2 High-Speed Computer Circuit-Board Module	p. 21
1.7.3 Power-Distribution System for a High-Speed Computer Multichip Module	p. 22
1.7.4 Microwave Amplifier	p. 23
1.7.5 Cellular Telephone	p. 24
1.7.6 Optical Microdisk Resonator	p. 25
1.7.7 Photonic Bandgap Microcavity Laser	p. 27
1.7.8 Colliding Spatial Solitons	p. 28
1.8 Conclusions	p. 29
References	p. 30
2 The One-Dimensional Scalar Wave Equation	p. 35
2.1 Introduction	p. 35
2.2 Propagating-Wave Solutions	p. 35
2.3 Dispersion Relation	p. 36
2.4 Finite Differences	p. 38
2.5 Finite-Difference Approximation of the Scalar Wave Equation	p. 39
2.6 Numerical Dispersion Relation	p. 42
2.6.1 Case 1: Very Fine Sampling in Time and Space ([Delta]t [right arrow] 0, [Delta]x [right arrow] 0)	p. 43
2.6.2 Case 2: Magic Time-Step (c[Delta]t = [Delta]x)	p. 43
2.6.3 Case 3: Dispersive Wave Propagation	p. 44
2.6.4 Example of Calculation of Numerical Phase Velocity and Attenuation	p. 49
2.6.5 Examples of Calculations of Pulse Propagation	p. 51
2.7 Numerical Stability	p. 55
2.7.1 Complex-Frequency Analysis	p. 55
2.7.2 Examples of Calculations Involving Numerical Instability	p. 59
2.8 Summary	p. 61
Appendix 2A Order of Accuracy	p. 63
2A.1 Lax-Richtmyer Equivalence Theorem	p. 63
2A.2 Limitations	p. 64
References	p. 64
Bibliography on Stability of Finite-Difference Methods	p. 65
Problems	p. 65
3 Introduction to Maxwell's Equations and the Yee Algorithm	p. 67
3.1 Introduction	p. 67
3.2 Maxwell's Equations in Three Dimensions	p. 67
3.3 Reduction to Two Dimensions	p. 70
3.3.1 TM[subscript z] Mode	p. 71
3.3.2 TE[subscript z] Mode	p. 71
3.4 Reduction to One Dimension	p. 72
3.4.1 x-Directed, z-Polarized TEM Mode	p. 72
3.4.2 x-Directed, y-Polarized TEM Mode	p. 73
3.5 Equivalence to the Wave Equation in One Dimension	p. 74
3.6 The Yee Algorithm	p. 75
3.6.1 Basic Ideas	p. 75
3.6.2 Finite Differences and Notation	p. 77
3.6.3 Finite-Difference Expressions for Maxwell's Equations in Three Dimensions	p. 80
3.6.4 Space Region With a Continuous Variation of Material Properties	p. 85
3.6.5 Space Region With a Finite Number of Distinct Media	p. 87
3.6.6 Space Region With Nonpermeable Media	p. 89
3.6.7 Reduction to the Two-Dimensional TM[subscript z] and TE[subscript z] Modes	p. 91
3.6.8 Interpretation as Faraday's and Ampere's Laws in Integral Form	p. 93
3.6.9 Divergence-Free Nature	p. 96
3.7 Alternative Finite-Difference Grids	p. 98
3.7.1 Cartesian Grids	p. 99
3.7.2 Hexagonal Girds	p. 101
3.7.3 Tetradecahedron/Dual-Tetrahedron Mesh in Three Dimensions	p. 104
3.8 Summary	p. 105
References	p. 106
Problems	p. 106
4 Numerical Dispersion and Stability	p. 109
4.1 Introduction	p. 109
4.2 Derivation of the Numerical Dispersion Relation for Two-Dimensional Wave Propagation	p. 110
4.3 Extension to Three Dimensions	p. 112
4.4 Comparison With the Ideal Dispersion Case	p. 113
4.5 Anisotropy of the Numerical Phase Velocity	p. 114
4.5.1 Sample Values of Numerical Phase Velocity	p. 114
4.5.2 Intrinsic Grid Velocity Anisotropy	p. 120
4.6 Complex-Valued Numerical Wavenumbers	p. 124
4.6.1 Case 1: Numerical Wave Propagation Along the Principal Lattice Axes	p. 124
4.6.2 Case 2: Numerical Wave Propagation Along a Grid Diagonal	p. 127
4.6.3 Example of Calculation of Numerical Phase Velocity and Attenuation	p. 129
4.6.4 Example of Calculation of Wave Propagation	p. 131
4.7 Numerical Stability	p. 133
4.7.1 Complex-Frequency Analysis	p. 133
4.7.2 Example of a Numerically Unstable Two-Dimensional FDTD Model	p. 139
4.8 Generalized Stability Problem	p. 141
4.8.1 Boundary Conditions	p. 141
4.8.2 Variable and Unstructured Meshing	p. 142
4.8.3 Lossy, Dispersive, Nonlinear, and Gain Materials	p. 142
4.9 Modified Yee-Based Algorithms for Improved Numerical Dispersion	p. 142
4.9.1 Strategy 1: Center a Specific Numerical Phase-Velocity Curve About c	p. 143
4.9.2 Strategy 2: Use Fourth-Order-Accurate Spatial Differences	p. 143
4.9.3 Strategy 3: Use Hexagonal Grids	p. 152
4.9.4 Strategy 4: Use Discrete Fourier Transforms to Calculate the Spatial Derivatives	p. 156
4.10 Alternating-Direction-Implicit Time-Stepping Algorithm for Operation Beyond the Courant Limit	p. 160
4.10.1 Numerical Formulation of the Zheng/Chen/Zhang Algorithm	p. 162
4.10.2 Numerical Stability	p. 169
4.10.3 Numerical Dispersion	p. 171
4.10.4 Discussion	p. 171
4.11 Summary	p. 172
References	p. 172
Problems	p. 173
Projects	p. 174
5 Incident Wave Source Conditions	p. 175
5.1 Introduction	p. 175
5.2 Pointwise E and H Hard Sources in One Dimension	p. 176
5.3 Pointwise E and H Hard Sources in Two Dimensions	p. 178
5.3.1 Green's Function for the Scalar Wave Equation in Two Dimensions	p. 178
5.3.2 Obtaining Comparative FDTD Data	p. 179
5.3.3 Results for Effective Action Radius of a Hard-Sourced Field Component	p. 180
5.4 J and M Current Sources in Three Dimensions	p. 182
5.4.1 Sources and Charging	p. 183
5.4.2 Sinusoidal Sources	p. 184
5.4.3 Transient (Pulse) Sources	p. 185
5.4.4 Intrinsic Lattice Capacitance	p. 189
5.4.5 Intrinsic Lattice Inductance	p. 190
5.4.6 Impact Upon FDTD Simulations of Lumped-Element Capacitors and Inductors	p. 191
5.5 The Plane-Wave Source Condition	p. 193
5.6 The Total-Field/Scattered-Field Technique: Ideas and One-Dimensional Formulation	p. 194
5.6.1 Ideas	p. 194
5.6.2 One-Dimensional Formulation	p. 197
5.7 Two-Dimensional Formulation of the TF/SF Technique	p. 201
5.7.1 Consistency Conditions	p. 203
5.7.2 Calculation of the Incident Field	p. 207
5.7.3 Illustrative Example	p. 212
5.8 Three-Dimensional Formulation of the TF/SF Technique	p. 212
5.8.1 Consistency Conditions	p. 216
5.8.2 Calculation of the Incident Field	p. 221
5.9 Pure Scattered-Field Formulation	p. 224
5.9.1 Application to PEC Structures	p. 224
5.9.2 Application to Lossy Dielectric Structures	p. 225
5.9.3 Choice of Incident Plane-Wave Formulation	p. 227
5.10 Waveguide Source Conditions	p. 227
5.10.1 Pulsed Electric Field Modal Hard Source	p. 228
5.10.2 Total-Field/Reflected-Field Modal Formulation	p. 229
5.10.3 Resistive Source and Load Conditions	p. 230
5.11 Summary	p. 231
References	p. 232
Problems	p. 232
Projects	p. 232
6 Analytical Absorbing Boundary Conditions	p. 235
6.1 Introduction	p. 235
6.2 Bayliss-Turkel Radiation Operators	p. 237
6.2.1 Spherical Coordinates	p. 238
6.2.2 Cylindrical Coordinates	p. 241
6.3 Engquist-Majda One-Way Wave Equations	p. 244
6.3.1 One-Term and Two-Term Taylor Series Approximations	p. 245
6.3.2 Mur Finite-Difference Scheme	p. 248
6.3.3 Trefethen-Halpern Generalized and Higher Order ABCs	p. 251
6.3.4 Theoretical Reflection Coefficient Analysis	p. 253
6.3.5 Numerical Experiments	p. 256
6.4 Higdon Radiation Operators	p. 261
6.4.1 Formulation	p. 261
6.4.2 First Two Higdon Operators	p. 263
6.4.3 Discussion	p. 264
6.5 Liao Extrapolation in Space and Time	p. 265
6.5.1 Formulation	p. 265
6.5.2 Discussion	p. 267
6.6 Ramahi Complementary Operators	p. 269
6.6.1 Basic Idea	p. 269
6.6.2 Complementary Operators	p. 270
6.6.3 Effect of Multiple Wave Reflections	p. 271
6.6.4 Basis of the Concurrent Complementary Operator Method	p. 273
6.6.5 Illustrative FDTD Modeling Results Obtained Using the C-COM	p. 278
6.7 Summary	p. 281
References	p. 281
Problems	p. 282
7 Perfectly Matched Layer Absorbing Boundary Conditions	p. 285
7.1 Introduction	p. 285
7.2 Plane Wave Incident Upon a Lossy Half-Space	p. 286
7.3 Plane Wave Incident Upon Berenger's PML Medium	p. 288
7.3.1 Two-Dimensional TE[subscript z] Case	p. 289
7.3.2 Two-Dimensional TM[subscript z] Case	p. 293
7.3.3 Three-Dimensional Case	p. 294
7.4 Stretched-Coordinate Formulation of Berenger's PML	p. 295
7.5 An Anisotropic PML Absorbing Medium	p. 298
7.5.1 Perfectly Matched Uniaxial Medium	p. 298
7.5.2 Relationship to Berenger's Split-Field PML	p. 301
7.5.3 A Generalized Three-Dimensional Formulation	p. 302
7.5.4 Inhomogeneous Media	p. 304
7.6 Theoretical Performance of the PML	p. 305
7.6.1 The Continuous Space	p. 305
7.6.2 The Discrete Space	p. 305
7.7 Efficient Implementation of UPML in FDTD	p. 308
7.7.1 Derivation of the Finite-Difference Expressions	p. 308
7.7.2 Computer Implementation of the UPML	p. 311
7.8 Numerical Experiments With Berenger's Split-Field PML	p. 314
7.8.1 Outgoing Cylindrical Wave in a Two-Dimensional Open-Region Grid	p. 314
7.8.2 Outgoing Spherical Wave in a Three-Dimensional Open-Region Lattice	p. 316
7.8.3 Dispersive Wave Propagation in Metal Waveguides	p. 318
7.8.4 Dispersive and Multimode Wave Propagation in Dielectric Waveguides	p. 320
7.9 Numerical Experiments With UPML	p. 322
7.9.1 Current Source Radiating in an Unbounded Two-Dimensional Region	p. 322
7.9.2 Highly Elongated Domains	p. 327
7.9.3 Microstrip Transmission Line	p. 330
7.10 UPML Termination for Conductive Media	p. 332
7.10.1 Theory	p. 332
7.10.2 Numerical Example: Termination of a Conductive Half-Space Medium	p. 335
7.11 UPML Termination for Dispersive Media	p. 338
7.11.1 Theory	p. 338
7.11.2 Numerical Example: Reflection by a Lorentz Medium	p. 343
7.12 Summary and Conclusions	p. 343
References	p. 345
Projects	p. 347
8 Near-to-Far-Field Transformation	p. 349
8.1 Introduction	p. 349
8.2 Two-Dimensional Transformation, Phasor Domain	p. 350
8.2.1 Application of Green's Theorem	p. 351
8.2.2 Far-Field Limit	p. 352
8.2.3 Reduction to Standard Form	p. 354
8.3 Obtaining Phasor Quantities Via Discrete Fourier Transformation	p. 356
8.4 Surface Equivalence Theorem	p. 359
8.5 Extension to Three Dimensions, Phasor Domain	p. 361
8.6 Time-Domain Near-to-Far-Field Transformation	p. 366
8.7 Summary	p. 371
References	p. 372
Project	p. 372
9 Dispersive and Nonlinear Materials	p. 373
9.1 Introduction	p. 373
9.2 Types of Dispersions Considered	p. 374
9.2.1 Debye Media	p. 374
9.2.2 Lorentz Media	p. 375
9.3 Piecewise-Linear Recursive Convolution Method, Linear Material Case	p. 375
9.3.1 General Formulation of the Method	p. 375
9.3.2 Application to Debye Media	p. 378
9.3.3 Application to Lorentz Media	p. 379
9.3.4 Numerical Results	p. 380
9.4 Piecewise-Linear Recursive Convolution Method, Nonlinear Dispersive Material Case	p. 382
9.4.1 Governing Equations	p. 382
9.4.2 General Formulation of the Method	p. 384
9.4.3 FDTD Realization in One Dimension	p. 386
9.4.4 Numerical Results	p. 388
9.5 Auxiliary Differential Equation Method, Linear Material Case	p. 392
9.5.1 Formulation for Multiple Debye Poles	p. 392
9.5.2 Formulation for Multiple Lorentz Pole Pairs	p. 394
9.5.3 Numerical Results	p. 397
9.6 Auxiliary Differential Equation Method, Nonlinear Dispersive Material Case	p. 398
9.6.1 Formulation for Multiple Lorentz Pole Pairs, TM[subscript Z] Case	p. 398
9.6.2 Numerical Results for Temporal Solitons	p. 401
9.6.3 Numerical Results for Spatial Solitons	p. 404
9.7 Summary and Conclusions	p. 407
References	p. 408
Problems	p. 409
Projects	p. 410
10 Local Subcell Models of Fine Geometrical Features	p. 411
10.1 Introduction	p. 411
10.2 Basis of Contour-Path FDTD Modeling	p. 412
10.3 The Simplest Contour-Path Subcell Models	p. 413
10.3.1 Diagonal Split-Cell Model for PEC Surfaces	p. 413
10.3.2 Average Properties Model for Material Surfaces	p. 415
10.4 The Contour-Path Model of the Narrow Slot	p. 416
10.5 The Contour-Path Model of the Thin Wire	p. 420
10.6 Locally Conformal Models of Curved Surfaces	p. 424
10.6.1 Dey-Mittra Technique for PEC Structures	p. 424
10.6.2 Illustrative Results for PEC Structures	p. 427
10.6.3 Dey-Mittra Technique for Material Structures	p. 433
10.7 Maloney-Smith Technique for Thin Material Sheets	p. 434
10.7.1 Basis	p. 434
10.7.2 Illustrative Results	p. 438
10.8 Dispersive Surface Impedance	p. 442
10.8.1 Maloney-Smith Method	p. 442
10.8.2 Beggs Method	p. 449
10.8.3 Lee Method	p. 457
10.9 Relativistic Motion of PEC Boundaries	p. 461
10.9.1 Basis	p. 461
10.9.2 Illustrative Results	p. 465
10.10 Summary and Discussion	p. 468
References	p. 470
Bibliography	p. 471
Projects	p. 472
11 Nonorthogonal and Unstructured Grids	p. 473
11.1 Introduction	p. 473
11.2 Nonuniform Orthogonal Grids	p. 474
11.3 Locally Conformal Grids, Globally Orthogonal	p. 482
11.4 Global Curvilinear Coordinates	p. 484
11.4.1 Nonorthogonal Curvilinear FDTD Algorithm	p. 484
11.4.2 Stability Criterion	p. 490
11.5 Irregular Nonorthogonal Structured Grids	p. 493
11.6 Irregular Nonorthogonal Unstructured Grids	p. 500
11.6.1 Generalized Yee Algorithm	p. 500
11.6.2 Inhomogeneous Media	p. 506
11.6.3 Practical Implementation of the Generalized Yee Algorithm	p. 508
11.7 A Planar Generalized Yee Algorithm	p. 509
11.7.1 Time-Stepping Expressions	p. 510
11.7.2 Projection Operators	p. 511
11.7.3 Efficient Time-Stepping Implementation	p. 513
11.8 Examples of Passive-Circuit Modeling Using the Planar Generalized Yee Algorithm	p. 514
11.8.1 32-GHz Wilkinson Power Divider	p. 514
11.8.2 32-GHz Gysel Power Divider	p. 517
11.8.3 Signal Lines in an IBM Thermal Conduction Module	p. 518
11.9 Summary and Conclusions	p. 522
References	p. 523
Problems	p. 525
Projects	p. 527
12 Bodies of Revolution	p. 529
12.1 Introduction	p. 529
12.2 Field Expansion	p. 530
12.3 Difference Equations for Off-Axis Cells	p. 530
12.3.1 Ampere's Law Contour Path Integral to Calculate e[subscript r]	p. 532
12.3.2 Ampere's Law Contour Path Integral to Calculate e[subscript phi]	p. 534
12.3.3 Ampere's Law Contour Path Integral to Calculate e[subscript z]	p. 536
12.3.4 Difference Equations	p. 539
12.3.5 Surface-Conforming Contour Path Integrals	p. 542
12.4 Difference Equations for On-Axis Cells	p. 544
12.4.1 Ampere's Law Contour Path Integral to Calculate e[subscript z] on the z-Axis	p. 544
12.4.2 Ampere's Law Contour Path Integral to Calculate e[subscript phi] on the z-Axis	p. 546
12.4.3 Faraday's Law Calculation of h[subscript r] on the z-Axis	p. 548
12.5 Numerical Stability	p. 549
12.6 PML Absorbing Boundary Condition	p. 549
12.6.1 BOR-FDTD Background	p. 549
12.6.2 Extension of PML to the General BOR Case	p. 552
12.6.3 Examples	p. 558
12.7 Application to Particle Accelerator Physics	p. 560
12.7.1 Definitions and Concepts	p. 560
12.7.2 Examples	p. 563
12.8 Summary	p. 566
References	p. 566
Problems	p. 567
Projects	p. 568
13 Analysis of Periodic Structures	p. 569
13.1 Introduction	p. 569
13.2 Review of Scattering From Periodic Structures	p. 572
13.3 Direct Field Methods	p. 575
13.3.1 Normal Incidence Case	p. 575
13.3.2 Multiple Unit Cells for Oblique Incidence	p. 577
13.3.3 Sine-Cosine Method	p. 579
13.3.4 Angled-Update Method	p. 580
13.4 Introduction to the Field-Transformation Technique	p. 584
13.5 Multiple-Grid Approach	p. 589
13.5.1 Formulation	p. 589
13.5.2 Numerical Stability Analysis	p. 591
13.5.3 Numerical Dispersion Analysis	p. 592
13.5.4 Lossy Materials	p. 593
13.5.5 Lossy Screen Example	p. 595
13.6 Split-Field Method, Two Dimensions	p. 596
13.6.1 Formulation	p. 596
13.6.2 Numerical Stability Analysis	p. 598
13.6.3 Numerical Dispersion Analysis	p. 600
13.6.4 Lossy Materials	p. 601
13.6.5 Lossy Screen Example	p. 602
13.7 Split-Field Method, Three Dimensions	p. 603
13.7.1 Formulation	p. 603
13.7.2 Numerical Stability Analysis	p. 607
13.7.3 UPML Absorbing Boundary Condition	p. 610
13.8 Application of the Periodic FDTD Method	p. 614
13.8.1 Photonic Bandgap Structures	p. 614
13.8.2 Frequency-Selective Surfaces	p. 616
13.8.3 Antenna Arrays	p. 618
13.9 Summary and Conclusions	p. 623
Acknowledgments	p. 623
References	p. 623
Projects	p. 625
14 Modeling of Antennas	p. 627
14.1 Introduction	p. 627
14.2 Formulation of the Antenna Problem	p. 628
14.2.1 Transmitting Antenna	p. 628
14.2.2 Receiving Antenna	p. 630
14.2.3 Symmetry	p. 630
14.2.4 Excitation	p. 632
14.3 Antenna Feed Models	p. 634
14.3.1 Detailed Modeling of the Feed	p. 634
14.3.2 Simple Gap Feed Model for a Monopole Antenna	p. 636
14.3.3 Improved Simple Feed Model	p. 639
14.4 Near-to-Far-Field Transformations	p. 644
14.4.1 Use of Symmetry	p. 645
14.4.2 Time-Domain Near-to-Far-Field Transformation	p. 646
14.4.3 Frequency-Domain Near-Field to Far-Field Transformation	p. 648
14.5 Plane-Wave Source	p. 649
14.5.1 Effect of an Incremental Displacement of the Surface Currents	p. 649
14.5.2 Effect of an Incremental Time Shift	p. 651
14.5.3 Relation to Total-Field/Scattered-Field Lattice Zoning	p. 652
14.6 Case Study I: The Standard-Gain Horn	p. 653
14.7 Case Study II: The Vivaldi Slotline Array	p. 659
14.7.1 Background	p. 659
14.7.2 The Planar Element	p. 660
14.7.3 The Vivaldi Pair	p. 665
14.7.4 The Vivaldi Quad	p. 665
14.7.5 The Linear Phased Array	p. 668
14.7.6 Phased-Array Radiation Characteristics Indicated by the FDTD Modeling	p. 669
14.7.7 Active Impedance of the Phased Array	p. 669
14.8 Near-Field Simulations	p. 675
14.8.1 Generic 900-MHz Cellphone Handset in Free Space	p. 675
14.8.2 900-MHz Dipole Antenna Near a Layered Bone-Brain Half-Space	p. 676
14.8.3 840-MHz Dipole Antenna Near a Rectangular Brain Phantom	p. 678
14.8.4 900-MHz Infinitesimal Dipole Antenna Near a Spherical Brain Phantom	p. 680
14.8.5 1,900-MHz Half-Wavelength Dipole Near a Spherical Brain Phantom	p. 681
14.9 Selected Recent Applications	p. 682
14.9.1 Use of Photonic-Bandgap Materials	p. 682
14.9.2 Ground-Penetrating Radar	p. 682
14.9.3 Antenna-Radome Interaction	p. 687
14.9.4 Personal Wireless Communications Devices	p. 690
14.9.5 Biomedical Applications of Antennas	p. 696
14.10 Summary and Conclusions	p. 697
References	p. 697
Projects	p. 701
15 High-Speed Electronic Circuits With Active and Nonlinear Components	p. 703
15.1 Introduction	p. 703
15.2 Basic Circuit Parameters	p. 705
15.2.1 Transmission Line Parameters	p. 706
15.2.2 Impedance	p. 706
15.2.3 S Parameters	p. 707
15.3 Differential Capacitance Calculation	p. 708
15.4 Differential Inductance Calculation	p. 709
15.5 Lumped Inductance Due to a Discontinuity	p. 711
15.5.1 Flux / Current Definition	p. 711
15.5.2 Fitting Z([omega]) or S([omega]) to an Equivalent Circuit	p. 712
15.5.3 Discussion: Choice of Methods	p. 712
15.6 Inductance of Complex Power-Distribution Systems	p. 713
15.6.1 Method Description	p. 713
15.6.2 Example: Multiplane Meshed Printed-Circuit Board	p. 715
15.6.3 Discussion	p. 718
15.7 Parallel Coplanar Microstrips	p. 718
15.8 Multilayered Interconnect Modeling Example	p. 720
15.9 Digital Signal Processing and Spectrum Estimation	p. 721
15.9.1 Prony's Method	p. 721
15.9.2 Autoregressive Models	p. 725
15.9.3 Pade Approximation	p. 729
15.10 Modeling of Lumped Circuit Elements	p. 734
15.10.1 FDTD Formulation Extended to Circuit Elements	p. 734
15.10.2 The Resistor	p. 736
15.10.3 The Resistive Voltage Source	p. 737
15.10.4 The Capacitor	p. 737
15.10.5 The Inductor	p. 738
15.10.6 The Diode	p. 740
15.10.7 The Bipolar Junction Transistor	p. 741
15.11 Direct Linking of FDTD and SPICE	p. 743
15.11.1 Basic Idea	p. 745
15.11.2 Norton Equivalent Circuit "Looking Into" the FDTD Space Lattice	p. 745
15.11.3 Thevenin Equivalent Circuit "Looking Into" the FDTD Space Lattice	p. 748
15.12 Case Study: A 6-GHz MESFET Amplifier Model	p. 750
15.12.1 Large-Signal Model	p. 750
15.12.2 Amplifier Configuration	p. 753
15.12.3 Analysis of the Circuit Without the Packaging Structure	p. 754
15.12.4 Analysis of the Circuit With the Packaging Structure	p. 756
15.13 Summary and Conclusions	p. 759
Acknowledgements	p. 760
References	p. 761
Additional Bibliography	p. 763
Projects	p. 764
16 Microcavity Optical Resonators	p. 767
16.1 Introduction	p. 767
16.2 Issues Related to FDTD Modeling of Optical Structures	p. 768
16.2.1 Optical Waveguides	p. 768
16.2.2 Material Dispersion and Nonlinearities	p. 772
16.3 Macroscopic Modeling of Optical Gain Media	p. 772
16.3.1 Theory	p. 773
16.3.2 Validation Studies	p. 776
16.4 Application to Vertical-Cavity Surface-Emitting Lasers	p. 780
16.4.1 Passive Studies	p. 782
16.4.2 Active Studies	p. 784
16.5 Microcavities Based on Photonic Bandgap Structures, Quasi One-Dimensional Case	p. 788
16.6 Microcavities Based on Photonic Bandgap Structures, Two-Dimensional Case	p. 793
16.7 Microcavity Ring Resonators	p. 797
16.7.1 FDTD Modeling Considerations	p. 797
16.7.2 Coupling to Straight Waveguides	p. 800
16.7.3 Coupling to Curved Waveguides	p. 802
16.7.4 Elongated Ring Designs	p. 804
16.7.5 Resonances	p. 806
16.8 Microcavity Disk Resonators	p. 810
16.8.1 Resonance Behavior	p. 811
16.8.2 Suppression of Higher Order Radial Whispering-Gallery Modes	p. 815
16.8.3 Additional FDTD Modeling Studies	p. 819
16.9 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

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