 | Photonic crystals: towards nanoscale photonic devices / Lourtioz J.-M., Benisty H., Vincent B., Gerard J.-M., Maystre D., Tchelnokov A., Pagnoux D. - 2nd ed. - Berlin, Heidelberg: Springer-Verlag, 2008. - xiii, 514 p.: ill. - Ref.: p.467-507. - Ind.: p.509-514. - ISBN 978-3-540-78346-6
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Foreword ........................................................ V
Part I. Theoretical Models for Photonic Crystals .............. 1
Introduction to Part 1 .......................................... 3
1. Models for Infinite Crystals ................................ 5
1.1. Plane Wave Expansion .................................. 5
1.1.1. Maxwell's Equations ........................... 5
1.1.2. The Floquet-Bloch Theorem ..................... 6
1.1.3. Hermiticity of the Field Operator ............ 10
1.1.4. Simple Examples of Bloch Functions ........... 11
1.1.5. General Plane Wave Method .................... 13
1.2. Other Methods for the Calculation of the Photonic
Band Gaps of an Infinite Crystal: the KKR Method ..... 21
1.3. Photonic Band Diagram ................................ 21
1.3.1. The Irreducible Brillouin Zone ............... 22
1.3.2. Band Diagrams of One-Dimensional Crystals .... 25
1.3.3. Band Diagrams of Two-Dimensional Photonic
Crystals ..................................... 33
1.3.4. Off-Axis Propagation in One and Two-
Dimensional Photonic Crystals ................ 40
1.3.5. Band Diagrams of Three-Dimensional Photonic
Crystals ..................................... 41
1.4. Infinite Crystals with Defects ....................... 43
1.4.1. Point Defects ................................ 44
1.4.2. Coupling of Point Defects .................... 50
1.4.3. Supercell Method ............................. 52
1.4.4. Methods derived from Tight-Binding Methods
in Solid State Physics ....................... 53
1.4.5. Extended Defects ............................. 54
1.4.6. Semi-Infinite Crystals and Surface Defects ... 56
1.4.7. Density of States in Photonic Crystals with
or without Defects ........................... 59
2. Models for Finite Crystals ................................. 65
2.1. Transfer, Reflection and Transmission Matrix
Formulations ......................................... 65
2.1.1. Reflection and Transmission Matrices ......... 66
2.1.2. Pendry Method ................................ 72
2.2. Finite Difference in Time Domain (FDTD) Method ....... 80
2.2.1. Numerical Formulation of Maxwell's
Equations .................................... 80
2.2.2. Case of an Incident Pulse .................... 84
2.2.3. Absorption Region and Boundary Conditions .... 86
2.2.4. Practical Implementation and Convergence of
the FDTD Method .............................. 87
2.2.5. Examples of Results obtained for a Point
Source with the FDTD Method .................. 89
2.3. Scattering Matrix Method ............................. 91
2.4. Other Methods: Integral and Differential Methods,
Finite Element Method, Effective Medium Theory ...... 100
2.5. Numerical Codes available for the Modelling of
Photonic Crystals ................................... 104
3. Quasi-Crystals and Archimedean Tilings .................... 107
3.1. Photonic Quasi-Crystals ............................. 108
3.2. Archimedean Tilings ................................. 111
3.3. From Photonic Quasi-Crystals to the Localization
of Light ............................................ 115
4. Specific Features of Metallic Structures .................. 121
4.1. Bulk Metals: Drude Model, Skin Effect and Metallic
Losses .............................................. 121
4.1.1. Drude Model ................................. 123
4.1.2. Low-Frequency Region: Skin Effect and
Metallic Losses ............................. 124
4.1.3. From the Infrared to the Visible and UV
Regions ..................................... 125
4.2. Periodic Metallic Structures at Low Frequencies ..... 126
4.2.1. Plasmon-Like Photonic Band Gap .............. 126
4.2.2. Transmission Spectra of Metallic and
Dielectric Photonic Crystals ................ 129
4.2.3. Complete Band Gaps in Metallic Photonic
Crystals .................................... 131
4.2.4. Structures with Continuous Metallic
Elements and Structures with Discontinuous
Metallic Elements ........................... 132
4.3. Periodic Metallic Structures at Optical
Frequencies. Idealized Case of a Dispersive
Lossless Dielectric ................................. 134
4.4. Surface Waves ....................................... 137
4.4.1. Surface Plasmons at a Metal/Dielectric
Plane Interface ............................. 137
4.4.2. Propagation of Surface Plasmons along a
Periodically Modulated Metal/Dielectric
Interface and Local Enhancement of the
Field ....................................... 140
4.4.3. Wood's Anomalies: Phenomenological Theory ... 144
4.4.4. Photonic Band Gaps for the Propagation of
Surface Plasmons at Periodically Modulated
Metal/Dielectric Interfaces ................. 148
4.4.5. The Photon Sieve ............................ 150
4.4.6. Surface Waves in Metals at
Radiofrequencies ............................ 151
Part II. Optical Properties of Photonic Crystals ............. 157
Introduction to Part II. The 'Many Facets' of Photonic
Crystals ............................. 159
5. Control of Electromagnetic Waves .......................... 163
5.1. The Photonic Crystal Mirror ......................... 163
5.1.1. The Semi-Infinite Photonic Crystal: Mirror
or Diffraction Grating? ..................... 163
5.1.2. Specular Reflection at a Semi-Infinite
Crystal ..................................... 166
5.1.3. Finite Photonic Crystals as Semi-
Transparent Mirrors ......................... 167
5.2. Photonic Crystal Waveguides ......................... 168
5.2.1. Index Guiding and Photonic Bandgap
Guiding ..................................... 168
5.2.2. Three-Dimensional Photonic Crystal
Waveguides .................................. 170
5.2.3. Two-Dimensional Photonic Crystal
Waveguides .................................. 172
5.2.4. Density of States and Multiplicity of
Guided Modes ................................ 174
5.2.5. Coexistence of Index Guiding and Photonic
Bandgap Guiding ............................. 178
5.3. Resonators .......................................... 185
5.3.1. Localized Modes. Origin of Losses ........... 185
5.3.2. Density of States ........................... 187
5.3.3. Waveguide formed by Coupled Cavities ........ 188
5.4. Hybrid Structures with Index Guiding.
The Light Line ...................................... 190
5.4.1. Light Cone of a Uniform Waveguide ........... 190
5.4.2. Fictitious Periodicity ...................... 191
5.4.3. True One-Dimensional Periodicity ............ 191
5.4.4. Channel Waveguides in Two-Dimensional
Photonic Crystals ........................... 195
6. Refractive Properties of Photonic Crystals and
Metamaterials ............................................. 197
6.1. Phase Refractive index, Group Refractive Index and
Energy Propagation .................................. 197
6.1.1. Phase Velocity and Group Velocity ........... 197
6.1.2. Refractive Indexes and Dispersion
Diagrams .................................... 201
6.1.3. Effective Phase Index and Group Refractive
Index ....................................... 202
6.2. Refraction of Waves at the Interface between a
Periodic Medium and a Homogeneous Medium ............ 203
6.2.1. Summary of Refraction Laws in Homogeneous
Media ....................................... 203
6.2.2. Some Well-Known Anisotropic Media:
Birefringent Solid-State Crystals ........... 205
6.2.3. Construction of the Waves Transmitted in
a Photonic Crystal .......................... 206
6.3. Superprism and Negative Refraction Effects .......... 207
6.3.1. Superprism Effect ........................... 207
6.3.2. Ultra-Refraction and Negative Refraction .... 208
6.4. Metamaterials ....................................... 210
6.4.1. Simultaneous Control of the Dielectric
Permittivity and the Magnetic
Permeability ................................ 210
6.4.2. Negative Refraction in a Slab of Perfect
Left-Handed Material ........................ 212
6.4.3. Stigmatism of a Slab of Perfect Left-
Handed Material ............................. 215
6.4.4. Perfect Lens or Superlens? .................. 217
6.4.5. Fabrication of Negative Refractive Index
Metamaterials ............................... 218
6.4.6. Electromagnetic Cloaking .................... 221
7. Confinement of Light in Zero-Dimensional Microcavities .... 225
7.1. Microcavity Sources. Principles and Effects ......... 226
7.1.1. A Classical Effect: the Angular
Redistribution of the Spontaneous Emission
and the Example of Planar Microcavities ..... 226
7.2. Three-Dimensional Optical Confinement in Zero-
Dimensional Microcavities ........................... 245
7.2.1. Different Types of Zero-Dimensional
Microcavities ............................... 245
7.2.2. Control of the Spontaneous Emission in
Weak Coupling Regime. Some Experimental
Results ..................................... 250
7.2.3. Single-Mode Coupling of the Spontaneous
Emission .................................... 254
7.2.4. Towards Strong Coupling Regime for Solid
State 'Artificial Atoms' .................... 256
8. Nonlinear Optics with Photonic Crystals ................... 261
8.1. The Problem of Phase Matching ....................... 262
8.2. χ(1) Photonic Crystals .............................. 265
8.2.1. One-Dimensional χ(1) Photonic Crystals ...... 265
8.2.2. Two-Dimensional χ(1) Photonic Crystals ...... 273
8.3. χ(2) Photonic Crystals .............................. 274
8.3.1. One-Dimensional χ(2) Photonic
Crystals .................................... 274
8.3.2. Two-Dimensional χ(2) Photonic
Crystals .................................... 277
8.4. Photonic Crystals with Third Order
Susceptibility ...................................... 279
Part III. Fabrication, Characterization and Applications
of Photonic Bandgap Structures ...................... 283
Introduction to Part III ...................................... 285
9. Planar Integrated Optics .................................. 287
9.1. Objectives, New Devices and Challenges .............. 287
9.2. Fundamentals of Integrated Optics and
Introduction of Photonic Crystals ................... 290
9.2.1. Conventional Waveguides ..................... 290
9.2.2. Photonic Crystals in Integrated Optics ...... 295
9.3. Planar Photonic Crystals in the Substrate
Approach ............................................ 306
9.3.1. DFB and DBR Laser Diode Structures .......... 306
9.3.2. Photonic Crystals, a Strong Perturbation
for Guided Modes ............................ 307
9.3.3. Choice of the Diameter of the Holes and of
the Period of the Crystal ................... 309
9.3.4. Specific Parameters for InP- and GaAs-
Based Systems ............................... 310
9.3.5. Deep Etching ................................ 310
9.4. Membrane Waveguide Photonic Crystals ................ 311
9.4.1. Free-Standing Membranes ..................... 311
9.4.2. Reported Membranes .......................... 314
9.5. Macroporous Silicon Photonic Substrates ............. 314
9.6. Characterization Methods for Photonic Crystals in
Integrated Optics ................................... 318
9.6.1. Internal Light Source Method ................ 318
9.6.2. End-Fire Method ............................. 321
9.6.3. Wide-Band Transmission-Reflection
Spectroscopy ................................ 324
9.7. Losses of Photonic Crystal Integrated Optical
Devices ............................................. 324
9.7.1. Analysis of Losses in Planar Photonic
Crystal Waveguides .......................... 324
9.7.2. Measurement of Propagation Losses in
Straight Photonic Crystal Channel
Waveguides .................................. 326
9.7.3. Losses in the Slow-Light Regime ............. 329
9.7.4. Waveguide Bends in Photonic Crystals and
Bend Losses ................................. 329
9.7.5. Photonic Crystal Resonators and Quality
Factors ..................................... 330
9.8. Photonic Crystal Devices and Functions: Recent
Developments ........................................ 333
9.8.1. Classification of devices ................... 333
9.8.2. Coupled Resonators and Waveguides ........... 335
9.8.3. Very high-Q cavities ........................ 337
9.8.4. Other Devices and Optical Functions ......... 339
10. Microsources .............................................. 345
10.1. High-Efficiency Light-Emitting Diodes ............... 345
10.1.1. Solutions for the Extraction of Light
without Confinement ......................... 345
10.1.2. Enhanced Extraction Efficiency through
Planar Confinement .......................... 347
10.1.3. Increase of the Extraction Efficiency
using Two-Dimensional Photonic Crystals ..... 350
10.2. Ridge-Type Waveguide Lasers confined by Photonics
Crystals ............................................ 352
10.3. Bulk Photonic Crystal Band Edge Lasers .............. 355
10.4. Photonic crystal VCSELs ............................. 358
10.5. Microcavity Lasers .................................. 360
10.6. Potential Interest of Single-Photon Sources ......... 364
11. Photonic Crystal Fibres ................................... 371
11.1. Another Implementation of Periodic Structures ....... 371
11.2. Fabrication of Microstructured Optical Fibres ....... 372
11.3. Solid-Core Microstructured Optical Fibres ........... 375
11.3.1. Confinement Losses and Second Mode
Transition .................................. 375
11.3.2. Attenuation and Bend Loss ................... 378
11.3.3. Chromatic Dispersion Properties ............. 378
11.3.4. Main Applications of Solid-Core
Microstructured Optical Fibres .............. 380
11.4. True Photonic Crystal Fibres (PCF) .................. 382
11.4.1. Photonic Bandgap Cladding ................... 382
11.4.2. Losses of Photonic Crystal Fibres with
Finite Cladding ............................. 385
11.4.3. Photonic Crystal Fibres with Optimised
Structures .................................. 387
11.4.4. Main Applications of Photonic Crystal
Fibres ...................................... 389
12. Three-Dimensional Structures in Optics .................... 393
12.1. Geometrical Configurations proposed for Three-
Dimensional Structures .............................. 394
12.1.1. Structures with Omnidirectional Photonic
Band Gaps ................................... 394
12.1.2. Incomplete Band Gap Three-Dimensional
Structures .................................. 397
12.2. Examples of Fabrication Processes and Realizations
of Three-Dimensional Photonic Crystals in the
Optical Region ...................................... 399
12.2.1. Complete Band Gap Structures ................ 399
12.3. Metallic Three-Dimensional Photonic Crystals in
the Optical Region .................................. 410
12.4. Three-Dimensional Photonic Crystals and Light
Emitters ............................................ 412
13. Microwave and Terahertz Antennas and Circuits ............. 413
13.1. Photonic Crystal Antennas ........................... 414
13.1.1. Photonic-Crystal Antenna Substrates ......... 415
13.1.2. Photonic-Crystal Antenna Mirrors ............ 418
13.1.3. Photonic Crystal Antenna Radomes or
Superstrates ................................ 422
13.2. Controllable Structures and Metamaterials ........... 424
13.2.1. Principles and Characteristics of
Electrically Controllable Photonic
Crystals .................................... 424
13.2.2. Electrically Controllable Photonic Crystal
Antennas .................................... 426
13.2.3. Antennas and Metamaterials .................. 429
13.3. Microwave Circuits and Ultra-Compact Photonic
Crystals ............................................ 430
13.3.1. Ultra-Compact Photonic Crystals ............. 430
13.3.2. Microwave Filters and Waveguides realised
from Ultra-Compact Photonic Crystals ........ 433
13.4. From Microwaves to Terahertz Waves .................. 435
13.5. From Microwaves to Optics ........................... 436
13.5.1. Impedance Matching of Photonic Waveguides ... 437
13.5.2. Photonic Crystal THz Imaging System ......... 439
13.5.3. 'Microwave Inspired' Nanostructures and
Nanodevices ................................. 440
Conclusion and Perspectives ................................... 443
Appendices .................................................... 447
Appendix A. Scattering Matrix Method: Determination of the
Field for a Finite Two-Dimensional Crystal
formed by Dielectric Rods ......................... 449
A.l. Incident Field ............................... 449
A.2. Field inside the Rods ........................ 449
A.3. Field in the Vicinity of a Rod ............... 452
Appendix B. Magneto-Photonic Cystals .......................... 459
Appendix C. Stigmatism of a Slab of Perfect Left-Handed
Material: Integral for the Total Field ............ 463
References .................................................... 467
Index ......................................................... 509
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