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Photonic Crystals and Light Localization
in the 21 st Century

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NATO Science Series

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Series C: Mathematical and Physical Sciences - Vol. 563

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(defined by kc ) can propagate in the guide. For frequencies less than the
cut-off frequency, kz is imaginary, and such modes (evanescent modes) can-
not propagate in the waveguide [39]. Figure 7(b) compares the parallel-plate
waveguide model dispersion relations [obtained from Eq. (6], dotted lines)
with the dispersion relations obtained from the transmission phase mea-
surements [using Eq. (3], solid lines). As can be seen from the plots, the
results are in good agreement for different separation widths of the guide,
except for the higher frequency regions of the waveguide. This discrepancy
is mainly related to the inaccurate reflection phase information (due to
experimental limitations) at higher incidence angles, e > 70°.

6. Tight-binding description of the localized coupled-cavity modes
in photonic crystals

As we have demonstrated in previous sections, by introducing a defect
into the photonic crystals, it is possible to create highly localized defect
modes within the photonic band gap, which is analogous to the localized
impurity states in a semiconductor [15]. Although the modes of each cav-
ity were tightly confined at the defect sites, overlap between the nearest-
neighbor modes is enough to provide the propagation of photons via hop-
ping [Fig. 8(a)]. This picture can be considered as the classical wave analog
of the tight-binding (TB) method in solid state physics [18,40-43]. Recently,
the TB scheme was also successfully used for various photonic structures.
Waveguiding along the impurity chains in photonic insulators [40], waveg-
uiding through coupled resonators [43], and one-dimensional superstruc-
ture gratings [41] were theoretically investigated by using TB formalism.
Lidorikis et al. tested the TB model by comparing the ab initio results
of two-dimensional PBe structures with and without defects [42]. They
obtained the TB parameters by an excellent fitting to ab initio results.
Splitting of the coherent coupling of whispering gallery mode in quartz
polystyrene spheres was reported and explained within the TB photon pic-
ture [44]. The optical modes in the micrometer-sized semiconductor coupled
cavities were investigated by Bayer et al. [45].

We used the 3D layer-by-Iayer dielectric photonic crystals. The defec-
tive unit cells were created by removing a single rod from a single layer
of the cell, where each cell consists of 4 layers having the symmetry of a
face-centered tetragonal structure. The experimental set-up consists of a
HP 8510C network analyzer and microwave horn antennas to measure the
transmission-amplitude and transmission-phase properties of various defect
structures built around photonic crystals [Fig. 8(b)]. The electric field po-
larization vector of the incident EM wave e was parallel to the rods of the
defect layer for all measurements.

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Figure 8. (a) Schematics of propagation of photons by hopping between the coupled
evanescent defect modes. The overlap of the defect modes is large enough to provide
propagation of the EM waves along tightly confined cavity modes. (b) The experimental
setup for measuring the transmission characteristics of the coupled defect structures in
three-dimensional photonic crystals. The electric field polarization is directed along the
removed rods.

By using the aforementioned experimental setup, we first measured the
transmission amplitude through a crystal with a single defective unit cell.
This resulted in a localized defect mode within the PBG which is analo-
gous to acceptor impurity state in semiconductor physics [15]. The defect
mode occurred at a resonance frequency of n = 12.150 GHz with a Q-
factor (quality factor, defined as center frequency divided by the peak's
full width at half-maximum) of rv 1000 [Fig. 9(a)]. Next, we measured the
transmission through the crystal that contains two consecutive single rods
removed unit cells. We observed that the mode in the previous case split
into two resonance modes at frequencies Wl = 11.831 GHz and Wl = 12.402
GHz [Fig. 9(b)]. The intercavity distance for this structure was a = 1.28

Page 594


Light Emitting Diodes (LED), 93,119,
Lithography, 33,83,131

x-ray, 32
Localization, 489

length,37, 423,452
of light, 2, 52,191,389,417,489
plasmon, 329
photon, 52
scaling theory of, 476
threshold, 52, 447, 455

Losses, 20, 117
Liquid Crystals, 55
Luminescence, 125,257

Maxwell-Garnett theory, 199,459
Maxwell's equations, 26,161,208,329,
Mean free path

transport, 202

cylinders, 36, 335
photonic band gaps, 35, 329, 342, 373

Microscopy, 150
Confocal, 244

Microlaser, 133, 398
Microwaves, 25, 30, 155
Microcavities,7, 175
Mie resonance, 104, 458
Mie scattering, 2
Mirrors, 133
Mobility edge, 477

localized, 8, 182,422
transversed electric (TJ, 106, 149
transversed magnetic (TM)' 2,106,149

Multilayer film, 173

Nonlinear, 377, 567, 577, 589

microcavity,7, 175
filters, 1, 25
localization, 52

Opto-electronic devices, 117

Perturbation, 535
Phase velocity, 61

diffusion, 458
localization, 191
mobility edge, 455

band gaps, 25
circuit, 10
crystal fiber, 305
crystals, 1,25,41,83,93,117,173

Plane Wave Method, 58, 70,194
Polarization effects, 108, 145
Polymer, 221, 229, 405
Porous silicon, 143
Poynting vector, 165

Quality factor, 9, 30, 282

Random laser, 389,405,417,435
Random media, 69,417,489,510
Reflection coefficient, 11, 193,210,223,
Resonators, 25, 360, 437

cavity, 102,327
mode, 102
tunneling, 181, 186

Scalar wave
approximation, 26
equation, 69
gaps, 69

cross section, 448
mean free path, 202, 447, 454
multiple, 60, 202, 447
Mie, 204, 458

Second harmonic generation, 577, 589
Self-energy, 477
Self-assembly, 204, 229
Self-organization, 204, 231, 239, 374
Silica spheres, 264
Spontaneous emission, 24, 41, 191, 198,

band structure, 522
plasmons, 567
states, 9

TE modes, 106, 149

Hamiltonian, 294
model, 298, 476

TMmodes, 2, 106, 149
Transfer matrix techniques, 34
Transmission coefficient, 11, 75, 86, 109,

Page 595

mean free path, 202
velocity, 447

Ultrafast spectroscopy, 196
Ultrasound, 60

phase, 61
group, 61, 196,213

VVaveguides, 78,105,129,175,181,290,

bends, 2, 95, 126, 136
splitters, 2
crossings 2, 8
tapered, 16

Wavelength division multiplexing


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