Electromagnetic Theory and Applications for Photonic Crystals

Table of Contents......Page 8
CONTENTS......Page 9
1.1 INTRODUCTION......Page 10
1.2.1 PRESENTATION OF THE PROBLEM AND NOTATION......Page 11
1.2.2 FOURIER–BESSEL EXPANSIONS OF THE FIELD INSIDE THE CYLINDERS......Page 13
1.2.3 FOURIER–BESSEL EXPANSIONS OF THE FIELD OUTSIDE THE CYLINDERS......Page 15
1.2.4 FIRST SET OF EQUATIONS: CAUSALITY PROPERTY FOR EACH CYLINDER......Page 19
1.2.5 SECOND SET OF EQUATIONS: INTRODUCING THE COUPLING BETWEEN CYLINDERS......Page 20
1.2.6 FINAL EQUATION......Page 23
1.3.1 INTRODUCTION......Page 24
1.3.2 SETTING OF THE PROBLEM......Page 25
1.3.3 THE METHOD OF FICTITIOUS SOURCES (MFS)......Page 26
1.3.4 IMPLEMENTATION OF THE SCATTERING MATRIX METHOD (SMM)......Page 30
1.3.5 HYBRID METHOD USING MFS AND SMM......Page 31
1.3.6 NUMERICAL EXAMPLE......Page 32
1.4 DISPERSION RELATIONS OF BLOCH MODES......Page 33
1.4.1 INFINITE STRUCTURE......Page 34
1.4.2 FINITE-SIZE PHOTONIC CRYSTALS......Page 38
1.5.1 ULTRAREFRACTION WITH DIELECTRIC PHOTONIC CRYSTALS......Page 43
1.5.2 ULTRAREFRACTION WITH METALLIC PHOTONIC CRYSTALS......Page 44
1.5.3 NEGATIVE REFRACTION BY A DIELECTRIC SLAB RIDDLED WITH GALLERIES......Page 47
1.6 CONCLUSION......Page 50
REFERENCES......Page 51
CONTENTS......Page 55
2.1 INTRODUCTION......Page 56
2.2.1 GENERAL FRAMEWORK......Page 59
2.2.2 FIELD REPRESENTATION......Page 60
2.2.3 RAYLEIGH FIELD IDENTITY......Page 63
2.2.4 FIELD COUPLING AND CONTINUITY CONDITIONS......Page 66
2.2.5 FIELD PROBLEM OF MICROSTRUCTURED OPTICAL FIBERS......Page 68
2.2.6 INFINITE PERIODIC STRUCTURES......Page 70
2.2.7 GRATINGS......Page 76
2.3.2 GUIDING MECHANISMS AND THE “FINGER DIAGRAM”......Page 81
2.3.3 IMPLEMENTATION OF THE MULTIPOLE METHOD......Page 83
2.3.4.1 Number of Modes......Page 84
2.3.4.2 Properties of the Fundamental Mode......Page 86
2.3.5 AIR-GUIDED MODES......Page 88
2.4.1 BACKGROUND......Page 90
2.4.2 2D GREEN TENSOR AND LDOS......Page 91
2.4.3 2.5D GREEN TENSOR AND LDOS......Page 94
2.5.1 BACKGROUND AND NOMENCLATURE......Page 100
2.5.2 FORMULATION OF THE EIGENVALUE PROBLEM......Page 103
2.5.3 CALCULATION OF BAND DIAGRAMS AND BAND SURFACES......Page 105
2.5.4 CLASSIFICATION OF THE EIGENVALUES......Page 108
2.5.4.1 Formal Properties of Bloch Modes......Page 110
2.5.5 DEFECT MODE MODELING OF EXTENDED PHOTONIC CRYSTAL DEVICES......Page 112
2.5.6 INTERFACING PHOTONIC CRYSTAL MEDIA AND FRESNEL COEFFICIENTS......Page 113
2.5.7 RECURSIVELY COMBINING PC STACK SEGMENTS......Page 115
2.6.1 BACKGROUND......Page 116
2.6.2 STRAIGHT PC WAVEGUIDES......Page 117
2.6.3 RESONANT FILTERS AND JUNCTIONS......Page 118
2.6.4 MACH–ZEHNDER INTERFEROMETER......Page 122
2.7 DISCUSSION AND CONCLUSIONS......Page 124
REFERENCES......Page 127
CONTENTS......Page 131
3.1 INTRODUCTION......Page 132
3.2.1 TWO-DIMENSIONAL SCATTERING......Page 134
3.2.2 TWO-DIMENSIONAL T-MATRIX OF A CIRCULAR CYLINDER......Page 136
3.2.3 THREE-DIMENSIONAL SCATTERING......Page 137
3.2.4 THREE-DIMENSIONAL T-MATRIX OF A CIRCULAR CYLINDER......Page 138
3.3.1 TWO-DIMENSIONAL SCATTERING......Page 139
3.3.2 LATTICE SUMS AND T-MATRIX......Page 144
3.3.3 THREE-DIMENSIONAL SCATTERING......Page 147
3.4.1 REFLECTION AND TRANSMISSION MATRICES......Page 150
3.4.3 FLOQUET-MODE APPROACH FOR LAYERED IDENTICAL ARRAYS......Page 153
3.4.4 LAYERED ARRAYS EMBEDDED IN A SLAB......Page 157
3.5 THREE-DIMENSIONAL SCATTERING FROM LAYERED CROSSED-ARRAYS......Page 159
3.5.1 SCATTERING FROM A PARALLEL ARRAY OF CIRCULAR CYLINDERS......Page 160
3.5.2 UNIT CELL OF A CROSSED-ARRAY OF CIRCULAR CYLINDERS......Page 165
3.5.3 LAYERED CROSSED-ARRAYS OF CIRCULAR CYLINDERS......Page 168
3.6.1 DISPERSION EQUATION......Page 169
3.6.2 MODE FIELD ANALYSIS......Page 172
3.7.1 TWO-DIMENSIONAL SCATTERING FROM LAYERED PARALLEL ARRAYS......Page 175
3.7.2 THREE-DIMENSIONAL SCATTERING FROM LAYERED CROSSED-ARRAYS......Page 182
3.7.3 GUIDED MODES OF TWO-DIMENSIONAL PHOTONIC CRYSTAL WAVEGUIDES......Page 189
REFERENCES......Page 195
4.1 INTRODUCTION AND OVERVIEW......Page 199
4.2 INTRODUCTION TO PHOTONIC CRYSTAL SIMULATION......Page 201
4.3 BASICS OF THE MULTIPLE MULTIPOLE PROGRAM......Page 202
4.4 HANDLING PERIODIC SYMMETRIES WHILE USING PERIODIC BOUNDARY CONDITIONS......Page 206
4.5 ADVANCED MMP AND MAS EIGENVALUE SOLVERS......Page 208
4.6 COMPUTATION OF WAVEGUIDE MODES IN PHOTONIC CRYSTALS......Page 214
4.7 COMPUTATION OF WAVEGUIDE DISCONTINUITIES......Page 218
4.8 SENSITIVITY ANALYSIS OF PHOTONIC CRYSTAL DEVICES......Page 220
4.9 OPTIMIZATION BASED ON THE SENSITIVITY ANALYSIS......Page 224
4.10 ACHROMATIC 90° BEND......Page 225
4.11 FILTERING T-JUNCTION......Page 226
4.12 CONCLUSIONS AND OUTLOOK......Page 230
REFERENCES......Page 231
CONTENTS......Page 233
5.1 INTRODUCTION......Page 234
5.2.1 TM POLARIZED CASE......Page 235
5.2.2 TE POLARIZED CASE......Page 238
5.2.3 NUMERICAL EXAMPLES AND DISCUSSION......Page 239
5.3 ANALYSIS OF PHOTONIC CRYSTALS CONSISTING OF METALLIC CYLINDERS WITH ARBITRARY CROSS SECTION......Page 241
5.3.1 A SCATTERING MATRIX OF ONE SLICED LAYER WITH TM POLARIZED INCIDENCE......Page 242
5.3.2 A GLOBAL MATRIX OF THE ORIGINAL ARRAY WITH ARBITRARY CROSS SECTION......Page 244
5.3.4 S-MATRIX SOLUTION FOR TE POLARIZED INCIDENCE......Page 245
5.3.5 A MULTILAYERED STRUCTURE PROBLEM......Page 246
5.3.6 2D PHOTONIC CRYSTAL WAVEGUIDES CONSISTING OF METALLIC CYLINDERS WITH ARBITRARY CROSS SECTION......Page 247
5.3.7 NUMERICAL EXAMPLES......Page 250
5.4.1 FORMULATION......Page 263
5.5 SCATTERING ANALYSIS OF CROSSED PHOTONIC CRYSTALS CONSISTING OF ARBITRARILY SHAPED CYLINDERS......Page 268
5.5.1 REFLECTION AND TRANSMISSION MATRICES OF ONE Z-ARRAY LAYER......Page 270
5.5.2 S-MATRIX OF ONE X-ARRAY LAYER......Page 273
5.5.3 NUMERICAL EXAMPLES......Page 277
5.6 DIFFRACTION FROM A CONDUCTIVE SLAB CUT PERIODICALLY BY RECTANGULAR HOLES......Page 280
5.6.1 FORMULATION......Page 281
5.6.2 NUMERICAL EXAMPLES......Page 287
5.7 SCATTERING ANALYSIS OF A CONDUCTIVE SLAB CUT PERIODICALLY BY RECTANGULAR HOLES IN AN ARBITRARY DIRECTION......Page 291
5.8 CONCLUSION......Page 299
REFERENCES......Page 300
CONTENTS......Page 303
6.1 INTRODUCTION......Page 304
6.2 BASIC THEORY......Page 306
6.2.2 MAXWELL’S EQUATIONS IN MATRIX NOTATION......Page 307
6.2.3 GENERALIZED TRANSMISSION LINE EQUATIONS IN GENERAL ORTHOGONAL COORDINATES......Page 308
6.2.4 DISCRETIZATION......Page 309
6.2.5 ABSORBING BOUNDARY CONDITIONS......Page 315
6.2.7 EIGENVALUE AND MODAL MATRICES......Page 316
6.3.1 TRANSFORMATION THROUGH WAVEGUIDE SECTIONS......Page 317
6.3.3 TRANSFORMATION AT METALIZED WAVEGUIDE INTERFACES......Page 318
6.3.4 STEPS OF THE ANALYSIS PROCEDURE......Page 319
6.4.1 INTRODUCTION......Page 320
6.4.2 SYMMETRIC PERIODIC STRUCTURES......Page 321
6.4.3 PROPAGATION CONSTANTS OF FLOQUET MODES......Page 322
6.4.4 WAVE IMPEDANCE/ADMITTANCE OF FLOQUET MODES......Page 323
6.4.5 EFFICIENT CALCULATION OF THE FLOQUET MODE PARAMETERS......Page 324
6.4.6 CONCATENATION OF N PERIODS......Page 326
6.5.1 PROPAGATION ALGORITHM......Page 327
6.5.2 EIGENMODE ALGORITHM......Page 331
6.5.3 LINES OF VARYING LENGTH......Page 333
6.6.1 PORT RELATIONS......Page 336
6.6.2 MAIN DIAGONAL SUBMATRICES......Page 338
6.6.3 OFF-DIAGONAL SUBMATRICES: COUPLING TO OTHER PORTS......Page 340
6.7.1 BAND STRUCTURE CALCULATIONS......Page 344
6.7.2 PC WAVEGUIDE DEVICES......Page 347
6.7.3 BRAGG GRATINGS......Page 350
6.8 CONCLUSION......Page 355
REFERENCES......Page 356
CONTENTS......Page 359
7.1 INTRODUCTION......Page 360
7.2.1 FORMULAE FOR 2D WAVEGUIDE PROBLEMS......Page 363
7.2.2 THE FDFD METHOD WITH PMLS......Page 366
7.2.3 NUMERICAL TREATMENT AT THE DIELECTRIC INTERFACE......Page 371
7.3 MODAL ANALYSIS OF PHOTONIC CRYSTAL FIBERS......Page 373
7.3.1 HOLEY FIBERS......Page 374
7.3.2 TWO-CORE PCFS......Page 380
7.3.3 HONEYCOMB PCFS......Page 381
7.3.4 HOLLOW PCFS......Page 384
7.4.1 THE FDFD FORMULAE......Page 387
7.4.1.1 The TE Mode......Page 388
7.4.1.2 The TM Mode......Page 390
7.4.2 PERIODIC BOUNDARY CONDITIONS FOR 2D PCS......Page 391
7.5.1 PCS WITH SQUARE LATTICES......Page 393
7.5.2 PCS WITH TRIANGULAR LATTICES......Page 394
7.5.3 FINGER PLOTS OF 2D PCS......Page 396
7.6 MODAL ANALYSIS OF PLANAR PC WAVEGUIDES......Page 399
7.7 CONCLUSION......Page 403
REFERENCES......Page 405
8.1 INTRODUCTION......Page 409
8.2.1 FORMULATION OF PROBLEM......Page 410
8.2.1.1 Transformation of Maxwell’s Equations......Page 411
8.2.1.2 An FDTD Method Based on MD-WDFs......Page 414
8.2.2 NUMERICAL DISPERSION AND NUMERICAL ERRORS......Page 419
8.3 PHOTONIC CRYSTAL STRAIGHT WAVEGUIDE......Page 423
8.3.1 NUMERICAL RESULTS......Page 424
8.3.2 FLOQUET MODE......Page 425
8.3.3 DISPERSION RELATION......Page 428
8.4.1 DIRECTIONAL COUPLER......Page 431
8.4.2.1 Pillar Type Photonic Crystal Bent Waveguide......Page 435
8.4.2.2 Air-Hole Type Photonic Crystal Bent Waveguide......Page 444
8.5 WAVELENGTH MULTI/DEMULTIPLEXER......Page 446
8.5.1 DESIGN PARAMETERS......Page 447
8.5.2 NUMERICAL RESULTS......Page 448
8.6 CONCLUSION......Page 449
REFERENCES......Page 451