1. Introduction
К.Inoue ...................................................... 1
1.1. History of Research on Photonic Crystals ................ 1
1.2. Types of Photonic Crystals .............................. 3
1.3. Light States in a Photonic Crystal ...................... 5
1.3.1. Description of Light State in Vacuum ............. 5
1.3.2. Light State and Its Density for a Photonic
Crystal .......................................... 6
References ................................................... 7
2. Survey of Fundamental Features of Photonic Crystals
K.Inoue, K.Ohtaka ............................................ 9
2.1. One-Dimensional Photonic Crystal: Band Calculation ...... 9
2.1.1. Bloch Theorem .................................... 9
2.1.2. Scaling Property of Photonic Band Structure ..... 14
2.2. One-Dimensional Photonic Crystal: Various Concepts
and Characteristic Features of Photonic Bands .......... 15
2.2.1. First Band at k ≈ 0 ............................. 15
2.2.2. Photonic Bands for k near the BZ Boundary ....... 15
2.2.3. Tendency of Photon Localization: Dielectric
and Air Bands ................................... 16
2.2.4. Slow Group Velocity ............................. 17
2.2.5. Density of States ............................... 17
2.3. Concept of the Light Cone and Example of One-
Dimensional Off-Axis Band .............................. 19
2.4. Band Structures of Two- and Three-Dimensional
Photonic Crystals ...................................... 23
2.4.1. Examples of Two-Dimensional Photonic Band ....... 23
2.4.2. Example of Three-Dimensional Photonic Band ...... 26
2.5. How to Experimentally Explore the Band Structure ....... 26
2.6. Defect Modes ........................................... 27
2.7. Common and Fundamental Features of Photonic Band
Structure .............................................. 29
2.7.1. Existence of Photonic Band Gap .................. 29
2.7.2. Existence of Defect or Local Modes .............. 31
2.7.3. Anomalous Group Velocity ........................ 32
2.7.4. Remarkable Polarization Dependence .............. 33
2.7.5. Manifestation of Peculiar Bands ................. 34
2.8. Application of Photonic Crystals ....................... 35
References ...................................................... 37
3. Theory I: Basic Aspects of Photonic Bands
K.Ohtaka .................................................... 39
3.1. 2D or 3D Photonic Band Structure ....................... 39
3.1.1. Full Maxwell's Equations ........................ 39
3.1.2. Plane-Wave Expansion Method ..................... 42
3.2. Parity of Mirror Reflection of 2D PBS .................. 44
3.3. Light Transmission and Reflection ...................... 47
3.3.1. Transmission through a ID Photonic Crystal ...... 47
3.3.2. Transmission through a 2D or 3D Photonic
Crystal ......................................... 49
3.3.3. Diffraction ..................................... 51
3.3.4. Transmittance and Reflectance versus PBS ........ 52
3.4. Photonic Crystals of Finite Thickness .................. 54
3.4.1. Light Cone Dividing Leaky and Nonleaky
regions ......................................... 54
3.4.2. Formation of Photonic Band Structure in a
Slab ............................................ 55
3.4.3. Thick Slab of Photonic Crystal .................. 58
3.5. Whispering Gallery Modes and Mie Resonances ............ 58
3.6. Concept of Heavy Photons and Tight-Binding Bands ....... 62
References ...................................................... 63
4. Theory II: Advanced Topics of Photonic Crystals
K.Ohtaka .................................................... 65
4.1. Methods not Based on Plane-Wave Expansion .............. 65
4.1.1. Vector Korringa-Kohn-Rostoker (KKR) Method ...... 66
4.1.2. Monolayer Scattering and Bloch Theorem .......... 70
4.2. Group Theory of Photonic Crystals ...................... 74
4.2.1. Invariance agaist Lattice Symmetry
Operations ...................................... 74
4.2.2. Group of к and Basic Group-Theoretical
Properties ...................................... 77
4.2.3. Symmetry-Related Polarizations of Plane
Waves ........................................... 79
4.2.4. Application of Group Theory ..................... 81
4.3. Leaky Modes of Slab-Type Photonic Crystals ............. 84
4.3.1. DOS Formula for a Homogeneous Dielectric
Slab ............................................ 85
4.3.2. DOS Formula for a Spherical Scatterer ........... 90
4.3.3. DOS Formula for a Slab Photonic Crystal ......... 90
4.3.4. Application of DOS Formula to a PC Slab ......... 92
4.4. Layer-Doubling Method .................................. 94
4.4.1. Procedures of Layer Doubling in Light
Propagation ..................................... 94
4.4.2. Applications of Layer-Doubling Method ........... 97
4.5. Origin of Band Gap Modes ............................... 99
4.5.1. Completeness of Photonic Band Solutions ........ 100
4.5.2. Formal Treatment of a Single Defect ............ 101
4.5.3. Practical Treatment of Defect Modes ............ 104
4.6. Inhomogeneous Maxwell Equations ....................... 107
4.6.1. Green's Function and Inhomogeneous Maxwell
Equations ...................................... 107
4.6.2. Applications of the Derived Formula ............ 110
4.7. Optics of Photonic Crystals ........................... 112
References ..................................................... 116
5. Two-Dimensional Photonic Crystals
K.Inoue .................................................... 119
5.1. 2D Photonic Crystal of Arrayed Fiber Type ............. 119
5.2. 2D Photonic Crystals Fabricated Based on Anodic
Porous Alumina ........................................ 122
5.3. Other Methods and 2D Photonic Crystals with a 2D
Band Gap .............................................. 125
5.4. Photonic Crystal Fibers ............................... 127
References ..................................................... 129
6. Two-Dimensional Photonic Crystal Slabs
K.Inoue, K.Asakawa, K.Ohtaka ............................... 131
6.1. The Guided Mode and the Photonic Band Gap ............. 131
6.2. Three Types of PC Slabs ............................... 133
6.3. Fabrication of Samples ................................ 136
6.4. Optical Properties .................................... 141
6.5. Non-Bulk PC Slabs ..................................... 144
6.6. Q-values of Leaky Photonic Band Modes ................. 145
6.7. Dispersion Relation and Lifetime ...................... 147
References ..................................................... 149
7. Three-Dimensional Photonic Crystals
S.Noda, T.Kawashima, S.Kawakami ............................ 151
7.1. 3D Photonic Crystal by Wafer Fusion and Alignment
and its Band Structure ................................ 152
7.2. 3D Photonic Crystals at Infrared Wavelengths .......... 155
7.3. 3D Photonic Crystals at Near-Infrared Wavelengths ..... 158
7.4. Autocloning Technology ................................ 161
7.5. Mechanisms of Autocloning ............................. 162
7.6. Features of Autocloning ............................... 165
7.6.1. Self-healing Effect ............................ 165
7.6.2. Productivity ................................... 166
7.6.3. Flexibility of Materials ....................... 166
7.6.4. Flexibility of Lattice Type .................... 166
7.6.5. Scaling Law .................................... 166
7.6.6. Lattice Modulation ............................. 167
7.7. Concept of Lattice Modulation of Photonic Crystals
and its Application ................................... 168
7.7.1. Channel Waveguide by Using Lattice
Modulation ..................................... 168
7.7.2. Optical Resonator .............................. 169
7.8. Expansion of Band Gaps ................................ 171
7.8.1. Introducing Reactive Ion Etching ............... 171
7.8.2. Autocloned Structures Supporting FBG ........... 171
References ..................................................... 172
8. Other Types of Photonic Crystals
Y.Segawa, K.Ohtaka ......................................... 175
8.1. Double-Periodic Photonic Crystals ..................... 175
8.2. Quantum-Well Photonic Crystals ........................ 178
8.3. Isotropic Band Photonic Crystals ...................... 183
8.4. Metallic Photonic Crystals ............................ 186
8.4.1. Band Structure of a Metallic Photonic
Crystal ........................................ 186
8.4.2. Transmittance and Local Field Enhancement ...... 190
8.4.3. Effect of Absorption on the Local Field
Enhancement .................................... 191
References ..................................................... 192
9. Spectroscopic Methods for Characterization
K.Inoue .................................................... 195
9.1. How to Characterize a Sample .......................... 195
9.2. Spectroscopy in the General Case ...................... 196
9.3. Spectroscopy of a PC Slab ............................. 197
9.3.1. Transmission and Reflection Spectroscopy ....... 197
9.3.2. Near-Field Spectroscopy ........................ 204
9.3.3. Resonant-Mode Spectroscopy ..................... 204
9.4. Time-of-Flight Spectroscopy Using an Ultrafast
Pulse ................................................. 205
References ..................................................... 208
10. Interaction Between Light and Matter in Photonic
Crystals
K.Inoue, K.Ohtaka, S.Noda .................................. 211
10.1.Suppression and Enhancement of Emission ............... 211
10.2.Extraction of Light from a Slab with High
Efficiency ............................................ 214
10.3.Direct Observation of Small Group Velocity ............ 215
10.4.Laser Action: Experimental Aspects .................... 217
10.4.1. Band-Edge Laser ............................... 217
10.4.2. Vertically-Emitting Laser ..................... 219
10.4.3. Defect Laser .................................. 222
10.4.4. Other PC Lasers ............................... 222
10.5.Laser Action: Numerical Aspects ....................... 222
10.5.1. Treatment of the Onset of Lasing .............. 222
10.5.2. Lasing Involving Group-Velocity Anomaly ....... 224
10.6.Nonlinear Optical Phenomena ........................... 226
10.7.Emission of Light from a Traveling Electron ........... 228
10.7.1. SPR Spectrum and PBS .......................... 232
10.7.2. Photon Yield of PC versus Diffraction
Grating ....................................... 234
References ..................................................... 235
11. Photonic Crystal Devices
T.Baba ..................................................... 237
11.1.How to Use Photonic Crystal Properties ................ 237
11.2.Light Emitters ........................................ 239
11.2.1. Point-Defect Laser ............................ 239
11.2.2. Band-Edge Laser ............................... 242
11.2.3. VCSEL ......................................... 243
11.2.4. High Extraction Efficiency LED ................ 243
11.3.Optical Waveguides .................................... 245
11.3.1. Line-Defect Waveguide in a Photonic
Crystal Slab .................................. 245
11.3.2. Other Types of Waveguide ...................... 250
11.4.Optical Fibers ........................................ 251
11.4.1. Holey Fiber ................................... 251
11.4.2. Photonic Band-Gap Fiber ....................... 252
11.4.3. Bragg Fiber ................................... 252
11.5.Wavelength Filters .................................... 253
11.5.1. Resonant Type Filter .......................... 253
11.5.2. Diffraction Type Filter ....................... 254
11.6.Polarization Filters .................................. 256
11.7.Dispersion Compensators ............................... 256
11.8.Light Control Devices ................................. 256
11.9.Harmonic Generation ................................... 258
References ..................................................... 258
12. Application to Ultrafast Optical Planar Integrated
Circuits
K.Asakawa, K.Inoue ......................................... 261
12.1.Introduction .......................................... 261
12.2.Why Photonic Crystal-Based Ultrafast All-Optical
Switches, PC-SMZ? ..................................... 263
12.3.Fundamental Structures of 2D PC Slab Waveguides ....... 265
12.3.1. 2D PC Slab and Defect Waveguide Structure ..... 265
12.3.2. Simulation of Band Structure .................. 266
12.4.Nanofabrication Technologies for 2D PC Slab
Waveguides ............................................ 267
12.4.1. Fine EB Lithography ........................... 267
12.4.2. Fine Dry Etching .............................. 269
12.5.Fabrication and Characterization of 2D PC Slab
Defect Waveguides ..................................... 270
12.5.1. Straight and Bend Waveguides .................. 270
12.5.2. Y-branch Waveguide ............................ 272
12.5.3. Directional Coupler ........................... 273
12.5.4. Compact and Flat-Band Delay Element ........... 277
12.6.Fabrication and Characterization of Long
Waveguides ............................................ 279
12.7.Summary ............................................... 282
References ..................................................... 282
13. Photonic Crystals in the Terahertz Region
M.W.Takeda ................................................. 285
13.1.Dispersion Relation of Terahertz Waves in Photonic
Crystals............................................... 285
13.2.Direct Excitation of Localized Planar Defect Modes .... 289
13.3.Dual-Periodic Photonic Crystals ....................... 292
13.4.Control of Microwave Emission from Photonic
Crystal ............................................... 294
References ..................................................... 297
14. Perspective
S.Noda, K.Ohtaka ........................................... 299
Appendix A.
Reciprocal Lattice Vector and Discretized Wavevector
K. Ohtaka and K. Inoue ..................................... 303
1. Reciprocal Lattice Vectors and First Brillouin Zone ......... 303
2. Density of States ........................................... 305
Appendix B.
Phase Shift of Light and Density of States
K.Ohtaka ................................................... 309
Index .......................................................... 315
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