## Abstract

We present a numerical study of optical properties of an octagonal quasi-periodic
lattice of dielectric rods. We report on a complete photonic bandgap in TM
polarization up to extremely low dielectric constants of rods. The first photonic
bandgap remains open down to dielectric constant as small as *ε*=1.6
(*n*=1.26). The properties of an optical microcavity and waveguides
are examined for the system of rods with dielectric constant *ε*=5.0
(*n*=2.24) in order to design an add-drop filter. Proposed add-drop
filter is numerically characterized and further optimized for efficient operation.
The two-dimensional finite difference time domain method was exploited for numerical
calculations. We provide a numerical evidence of effective add-drop filter based on
low index material, thus opening further opportunities for application of low
refractive index materials in photonic bandgap optics.

©2005 Optical Society of America

## 1. Introduction

The coupling of evanescent waves between two waveguides and a ring or disk resonator is the essential idea behind an effective design of add-drop filters for wavelength division multiplexing (WDM) [1–4]. Such devices have been realized using dielectric waveguides, where the confinement of light is due to the total internal reflection. In the case of a small resonator size, high radiation losses are unavoidable. The smaller the refractive index contrast between waveguide core and cladding is, the higher are radiation losses for a small resonator. To decrease losses the resonator radius should be increased, which would reduce the free spectral range (FSR) of the resulting WDM system and therefore limit the quantity of channels that can be realized [1–4].

By employing photonic bandgap (PBG) materials with complete PBG, substantial miniaturizations of optical components without severe radiation losses can be achieved. In reference [5,6], an add-drop filter based on a two-dimensional (2D) photonic crystal (PC) has been suggested. Since then, the idea has been supported by several WDM filter designs [5–10]. Typically, to achieve a complete PBG, materials with refractive index substantially higher than n=2.0 should be employed. In contrast, photonic quasicrystal (PQC) is an example of PBG material, which does not require high refractive indices to obtain complete PBG.

Photonic quasicrystals are artificial dielectric inhomogeneous media, where scattering centers are located in the vertices of a quasi-periodic tiling of space [11]. PQCs have neither true periodicity nor translational symmetry, but have a quasi-periodicity that exhibits long-range order and orientational symmetry. 2D PQCs have high-order rotational and mirror symmetries. PQCs with 8-fold (octagonal) [11–15], 10-fold (decagonal or Penrose) [14,16–18] and 12-fold (dodecagonal) [19–22] symmetries have been recently studied. These studies have demonstrated that, in general, most of the PQCs have wide complete bandgaps and small threshold value of refractive index for opening a complete gap. Recently, lasing in PQCs has been reported both based on numerical and experimental observations [18,22].

In the present paper, we study optical properties of an octagonal lattice of dielectric rods based on the Ammann-Beenker tiling of space in order to design an add-drop filter. Such a PQC possesses a full PBG in TM polarization (electric field parallel to rods axes) [12] and it has been proven to be a suitable platform for effective waveguides and micro-cavities design [13]. For this PQC add-drop filter, the low-index materials common in optical telecommunications can be employed. Moreover, in contrast with PC based add-drop filters, our design does not include any additional materials, resizing of rods or inclusion of other elements, which is usual in filters based on PCs. This substantially relives fabrication requirements.

The paper is organized as follows. In Section 2 we investigate PBG formation in PQC and PBG width as a function of dielectric constant of rods. The choice of a suitable material is made there. Modes of a microcavity in the lattice with eight-fold symmetry are studied in Section 3. Section 4 is devoted to transmission of straight waveguides in octagonal PQC. Add-drop filter design and filter efficiency are presented in Section 5. Filter optimization is addressed in Section 6. A brief conclusion is given in Section 7.

## 2. Photonic bandgap

A complete PBG in TM polarization can be realized in an octagonal lattice of rods for
fairly small dielectric constants. For example, rods with dielectric constant
*ε*=2.4 (*n*=1.55) have been studied in [14]. It is worth to analyze the relative gap width
(the ratio of the bandgap width to the midgap frequency) as a function of dielectric
constant in such a system in order to determine a threshold value of dielectric
constant, which is sufficient to open a complete PBG. We have chosen the radius of rods
to be *r*=0.3*a*, where *a* - is the
lattice parameter of octagonal quasi-periodic lattice. It is equal to the side of a
square and at the same time to the side of a rhombus of an octagonal quasi-periodic
tiling (Fig. 1, left panel).

The 2D finite difference time domain (FDTD) method with uniaxial perfectly matched layer
(PML) boundary conditions was used in all simulations [23]. Energy density stored in the system was calculated for different values
of the dielectric constant. For this purpose the system was uniformly fed at each grid
point with a time-pulse excitation having a random phase from point to point. The pulse
was wide enough in frequency domain to cover the range of frequencies we are interested
in. Several hundreds detectors were placed at random positions in the system, which
allows us to store field components. After a sufficiently large number of calculation
steps, the field was Fourier-transformed to calculate the energy density. The value of
the normalized frequencies of the lower and the upper boundary of the first bandgap are
plotted in Fig. 1. The inset shows the relative
gap width as a function of the dielectric constant. As can be extrapolated from the
presented data, the threshold value of dielectric constant necessary for opening a
complete PBG is extremely low in the studied structures. It can be estimated to be as
small as *ε*=1.6 (*n*=1.26). For dielectric constant
*ε*=2.1 (*n*=1.45) the gap width to midgap ratio is
close to 5%, which promises that optoelectronics components based on octagonal PQC can
be realized in silica, a common telecommunication optical material. If compared with a
square periodic lattice of dielectric rods [24],
octagonal quasi-periodic lattice provides considerable reduction of the value of
threshold dielectric constant. For the square lattice of rods with radius-to-period
ratio *r*/*b*=0.25 (the widest gap), the threshold
dielectric constant required for a complete PBG in TM polarization is
*ε*=3.8 (*n*=1.95) [24], which is almost twice the value reported here^{*}. Here *b*
is a period of the lattice. In units of lattice
parameters of quasi-periodic lattice, the radius of the rods of square periodic lattice
is then *r*=0.35*a*.

For rods of dielectric constant equal to *ε*=5.0
(*n*=2.24), the relative gap width is as large as 20%, spanning from the
normalized frequency *Ω*=*a*/*λ*=0.35 to
*Ω*=0.43. Such a wide PBG leads to the strong light localization in
PQC and promises efficient design of integrated optics components based on low index
material. In the presented study, we have chosen this hypothetic material to design an
optical microcavity, waveguides and add-drop filter, leaving investigation of integrated
optics components based on silica PQC for another study. To the best of our knowledge,
this is the first report on efficient optical elements based on a complete PBG material
with such a small dielectric constant.

## 3. Microcavity

High-quality factor (high-Q) micro-cavities are the basic building blocks needed for an
add-drop filter, providing coupling between waveguide channels. We defined a microcavity
inside a square patch of PQC by removing two layers of rods around the one located at
the geometrical center of the system (Fig. 2,
left). In Fig. 2, the energy density stored
inside the cavity is compared with the energy density inside PQC without the cavity. To
calculate the energy density stored inside the cavity, uniform feeding at each grid
point of the system was used, while detectors were placed only inside the cavity on a
uniform subgrid. In Fig. 2, the energy density
spectrum of the original PQC (black dashed line) is compared with the energy density
spectrum of the PQC with cavity (blue solid line). One can immediately identify three
localized modes in the cavity spectrum with normalized frequencies equal to
*Ω*
_{1}=0.358, *Ω*
_{2}=0.407 and *Ω*
_{3}=0.420. The corresponding mode structure is shown in Fig. 3. To calculate the field distribution of an appropriate
mode, sinusoidal excitation with the mode frequency (*Ω*
_{1}, *Ω*
_{2} or *Ω*
_{3}) was used and electric and magnetic field components were plotted after the
steady state was reached. Quadrupole, hexapole and dipole modes are supported by the
cavity.

We further analyze the symmetry of the cavity modes. The symmetry operations of the
microcavity form an abstract group [25]. A cavity
with rotational symmetry of order *n* maps to the point group
*C _{n}*, whereas if there is also a line of reflection the appropriate group is

*C*. Further, the cavity modes exist in finite sets. Each set forms a basis for an irreducible representation of the group. The dimension

_{nv}*d*of the irreducible representation equals the mode degeneracy. It can be shown, that the modes are either nondegenerate (

*d*=1), in which case they must support all the symmetry operations of the group, or modes come in degenerate pairs (

*d*=2), and the individual modes then support only a subgroup of the full symmetry [25].

A cavity, supporting degenerate modes, is an ideal building block for an add-drop filter. The geometrical structure of such a filter must possess a mirror symmetry with respect to the plane perpendicular to the waveguides and the coupling element must support two degenerate resonance modes, one even and one odd, with respect to the mirror plane [5,6]. These requirements are satisfied if the coupling between waveguides is provided by a single microcavity, supporting doubly degenerate modes [5,6] or by two identical coupled micro-cavities [6,10].

The microcavity based on the octagonal quasi-periodic lattice (Fig. 2, left) possesses, at least locally, 8-fold rotational and
also mirror symmetries. All symmetry operations of such a microcavity belong to
C_{8v} group. At the same time, none of the cavity modes with frequencies in
the first PBG, supports all the symmetry operation of the cavity group (Fig. 3). It means that quadrupole, hexapole and
dipole modes are doubly degenerate modes, becoming then an appropriate choice for an
add-drop filter operation. At the same time, only the hexapole mode is deeply inside the
first PBG of the structure (Fig. 2) and also
well within the spectral range of high transmission efficiency of a PQC waveguide (Fig. 4, see also Section 4). To check the symmetry
of the hexapole mode with respect to the vertical plane, we have calculated the steady
state field patterns for the monochromatic excitation formed by point sources placed in
the vertices of a hexagon for different orientations of the hexagon axes with respect to
the mirror plane of the system. The hexagon center coincides with the geometrical center
of the cavity, while its vertices are located inside the cavity. The phase difference
between the neighboring point sources is equal to π. One can distinguish two degenerate
cavity modes, as well as their superpositions [26]. One of the degenerate modes is even, while the other one is odd with
respect to the vertical plane. We label these modes as Hexapole-0 and Hexapole-90,
correspondingly (Fig. 3, central panels).

## 4. Straight waveguides

A waveguide in a PQC can be introduced in a similar fashion as in a regular PC [27], by removing one or several rows of rods. Here,
we adopt a PC’s terminology to distinguish different waveguides by the number of removed
rows. The waveguide obtained by removing one row (*N* rows) is designated
as W1 (W*N*) waveguide. In contrast with the perfect periodic lattice, in
the case of quasi-periodic structures it is not a trivial task to insert a straight
waveguide with regularly flat walls. In selecting a proper waveguide for the add-drop
filter design, we try to keep waveguide walls as flat and as regular as possible (Fig. 4), reducing a frequency dependence of
waveguide transmission efficiency [13].

In Fig. 4, transmission efficiencies are presented for three PQC waveguides, namely W1, W2 and W3. Transmission efficiencies were calculated using FDTD method. Waveguides were excited by a gaussian-shaped temporal impulse, the Fourier transform of which is broad enough to cover the frequency range of interest. Fields were monitored by input and output detectors and transmitted waves intensities were normalized by those of incident waves. The transmission efficiency of W1 and W2 waveguides displays a limited bandwidth and has basically a resonant character [13]. Irregularities of the waveguide walls effectively localize light and the waveguide acts as a coupled cavity waveguide [28,29]. This can be also directly seen from the field pattern of the waveguide modes (not shown here). In contrast with W1 and W2 waveguides, W3 waveguide is wide enough to support propagating modes. This leads to the substantial improvement of transmission efficiency over most of the PBG spectral range (Fig. 4). We associate pronounced ripples in the transmission to the Fabry-Perot resonances at the waveguide open edges.

## 5. Add-drop filter

As it was reported in Section 3, the hexapole mode of the considered microcavity is a doubly degenerate mode. To check this numerically we have calculated the energy density stored in the Hexapole-0 and Hexapole-90 modes. To excite the mode of the appropriate symmetry we have arranged point sources in the vertices of a hexagon as described in Section 3. The hexagon vertex (side) faces the top of the cavity in the case of Hexapole-0 (Hexapole-90) mode. The system was fed with a time-pulse excitation at the point sources location. The field was monitored by detectors placed on a uniform subgrid inside the cavity. Calculations reveal, that the Hexapole-0 and Hexapole-90 modes are degenerate at least within the precision of our calculations (Fig. 5, left panel).

When waveguides are introduced in the system, the local symmetry of the cavity is broken, leading to the lifting of the cavity modes degeneracy. For the system with waveguides, the energy density stored in the Hexapole-0 and Hexapole-90 modes is shown in the central panel of Fig. 5. Both modes suffer a quality factor reduction and a shift of their resonant frequencies towards lower frequencies. The Hexapole-90 mode is affected stronger due to the larger overlap of its field with the waveguides (Fig. 3).

A microcavity integrated with two waveguides, as shown in the upper row of the central
panel in Fig. 5, represents a PQC add-drop
filter. To analyze how cavity modes decay into waveguides we have calculated
steady-state field patterns corresponding to the Hexapole-0 and Hexapole-90 modes as
well as their superposition. Modes were excited as described above using a sinusoidal
excitation with the frequency *Ω*=0.406, which corresponds to the maximum
overlap of the Hexapole-0 and Hexapole-90 modes. The electric field was plotted after
the steady state was reached. In the bottom panel of Fig. 6, the electric field distribution is shown for Hexapole-0 (left) and
Hexapole-90 (center) modes and their superposition (right). Both Hexapole-0 and
Hexapole-90 modes decay in all four waveguide channels but with a different relative
phase. In the top panel of Fig. 6, the sign of
the electric field amplitude in the top and bottom waveguide channels in the direct
vicinity of the cavity is shown. The Hexapole-0 (Hexapole-90) mode decays in-phase into
the left and right (top and bottom) channels and out-of-phase into the top and bottom
(left and right) channels, keeping the symmetry of the mode with respect to the mirror
plane of the system. That leads to the following phase relation between the Hexapole-0
and Hexapole-90 modes in the waveguide channels: the modes decay almost in-phase into
the top-left and bottom-right channels, while decaying almost out-of-phase into the
top-right and bottom-left channels. The electrical field pattern of the modes
superposition confirms this simple picture (Fig.
6, right panel). In fact, the superposition decays primarily into top-left and
bottom-right channels. Then, the waveguide mode at the resonance frequency
*Ω*=0.406 coming from the bottom-left channel would excite the
superposition of the Hexapole-0 and Hexapole-90 modes, and would be further canceled by
the field decaying back from the cavity into the bottom-right waveguide channel. The
field decaying into the top-left channel will form the dropped signal.

The transmission efficiency of the add-drop filter is presented in the left panel of
Fig. 7. We use the following notations for
the filter channels: bottom-left channel is an Input channel, bottom-right is an
Output-1, top-left is an Output-2 and top-right is an Output-3 channel (if the
add-functionality is considered, then this would be the insertion port of the added
signal and it would be rather labeled as an input, but it is considered an output for
the purpose of our drop-functionality analysis). Transmission efficiencies were
calculated using the FDTD method. The Input channel was excited by a time impulse, the
Fourier transform of which is broad enough to cover the frequency range of interest.
Fields were monitored by input and output detectors and transmitted waves intensities
were normalized by the ones of incident waves. Detectors were placed close to the
waveguides ends outside the add-drop filter. The spectrum at the Output-2 detector (red
solid curve) displays a pronounced peak near the frequency of the maximum overlap of the
Hexapole-0 and Hexapole-90 modes, accompanied by almost 100% drop of the transmission
efficiency in the main channel, Output-1 (black solid curve). The electric field pattern
corresponding to the resonance frequency (*Ω*=0.406) is shown in the
center panel of Fig. 7. A monochromatic source
was placed near the Input channel outside the filter. One can clearly see the energy
transfer from the main channel (bottom waveguide) to the Output-2 channel. In the right
panel of Fig. 7, the electric field pattern for
the frequency *Ω*=0.400 out of the resonance is shown. Light is guided
through the bottom waveguide from the Input channel directly to the Output-1
channel.

In the left panel of Fig. 7, the transmission spectrum at the Output-3 detector is shown (green line). Any signal in this channel is unwanted, there should be a complete cancellation of the signals in this port. One can clearly see close to zero transmission efficiency at the resonance frequency, surrounded by two peaks of relatively high transmission. To understand this behavior it is instructive to superimpose the energy density spectra of the Hexapole-0 and Hexapole-90 modes on the filter efficiency spectra. In Fig. 7, dashed black and dashed red lines are the energy density spectra of the Hexapole-0 and Hexapole-90 modes, respectively. They are normalized to their maximum value. The spectral position of the transmission peaks in channel Output-3 corresponds fairly well to the spectral position of the hexapole modes. At those frequencies primarily Hexapole-90 (lower frequencies) or Hexapole-0 (higher frequencies) modes are excited, which further decay in all four waveguide channels. This can also be seen from the back reflection spectra at the Input detector (blue line). The back reflection is strongly reduced at the Hexapole-0 and Hexapole-90 modes frequencies. Near the resonant frequency the superposition of the Hexapole-0 and Hexapole-90 modes is primarily excited, leading to the dropping of the signal into the Output-1 channel as described above.

## 6. Add-drop filter optimisation

The add-drop filter introduced in the previous section possesses nearly 100% dropping efficiency and the quality factor of the resonance is close to 700. In spite of that, the overall transmission is fairly small being only 15% of the incident energy. At the same time, due to lifted degeneracy of the Hexapole-0 and Hexapole-90 modes, the cross-talk with the Output-3 channel is close to 50% of the signal intensity at near-resonance frequencies. To use the proposed add-drop filter for optoelectronics applications its characteristics should be significantly improved.

One way to improve the overall transmission of the system is to use an adiabatic coupler at the Input channel to suppress the back reflection due to the impedance mismatch. The design of an appropriate coupler is a rather complex design task, while a coupler by itself will substantially increase the size of the add-drop filter. Due to the irregular shape of the PQC boundaries, the impedance matching condition at the air-PQC waveguide interface will strongly depends on the particular cut of the quasi-periodic lattice. We have performed the transmission efficiency calculations for different lattice cuts and found a substantial overall transmission improvement for the lattice cut with a plane interface (Fig. 8). The overall transmission is raised up to 50% of the incoming signal intensity in the spectral region of interest.

The decrease of the transverse dimension of the filter also serves to increase the overlap between the Hexapole-0 and Hexapole-90 modes. The energy density stored in the Hexapole-0 and Hexapole-90 modes in the case of the rectangular patch of the PQC is shown in Fig. 5 (right panel). One can see that the modes degeneracy is partially restored, leading to improvement of the filter efficiency. The filter transmission efficiency is presented in the left panel of Fig. 8. One can see close to 95% dropping efficiency in the Output-2 channel (red line), accompanied by close to zero transmission in the main channel (black line). The cross-talk to the Output-3 channel is reduced to 20% (green line). The energy density spectra superimposed on the transmission spectra confirmed the dropping mechanism as described in the previous section. The quality factor of the dropping resonance is close to 700.

The electric field patterns corresponding to the resonant frequency
(*Ω*=0.406) and out of the resonance frequency (*Ω*=0.400)
are shown in the center and right panels of Fig.
8, respectively. A monochromatic source was placed near the Input channel
outside the filter to excite the bottom waveguide. One can clearly see the energy
transfer from the main channel to the Output-2 channel in the resonance case and to the
Output-1 channel in the out of resonance case.

## 7. Conclusions

To the best of our knowledge, we have reported for the first time on an add-drop filter design based on a PQC. Formation of the PBG and design of optical waveguides and microcavity have been studied in the case of octagonal quasi-periodic lattice of rods in air. We have analysed the size and spectral position of the first PBG for the TM polarization and have shown that a complete PBG stays open for rods of a very small dielectric constant. The minimum value of the dielectric constant necessary for the complete PBG was found to be as small as ε=1.6. The systematic analysis of optical waveguides and microcavity in the octagonal PQC has been presented for dielectric material with the dielectric constant ε=5.0. It has been proved, that it is possible to design waveguides and microcavities suitable for an add-drop filter operation. We have proposed a way to integrate these optical components into an add-drop filter and performed its numerical characterization and optimisation. An optimised structure demonstrates reasonably good performance with the dropping efficiency close to 95% and the quality factor of the resonance close to 700.

## Acknowledgments

This work was partially supported by the EU-IST project APPTech IST-2000-29321 and the German BMBF project PCOC 01 BK 253. DNC also acknowledges the partial support of the DFG Research Unit 557. AVL acknowledge the partial support by Danish Technical Research Council via PIPE project. CMST acknowledges the support of the Science Foundation of Ireland.

## Footnotes

^{*} | When the manuscript of the presented paper had been submitted, a following preprint
by A. Matthews et al. was issued (arXiv:physics/0501072, January 14,
2005), which reports on a comparably low threshold dielectric constant of 1.73,
required for a complete PBG in TM polarization in a triangular lattice of dielectric
rods. |

## References and links

**1. **D. Rafizadeh,
et al., “Waveguide coupled AlGaAs/GaAs microcavity ring and disk
resonators with high finesse and 21.6-nm free spectral range,”
Opt. Lett. **22**, 1244–1246
(1997). [CrossRef] [PubMed]

**2. **S.
C. Hagness, et al., “FDTD microcavity
simulations: design and experimental realization of waveguide coupled single-mode
ring and whispering-gallery-mode disk resonators,” J.
Lightwave Technol. **15**, 2154–2165
(1997). [CrossRef]

**3. **B.
E. Little, et al., “Ultra-compact
Si-SiO2 microring resonator optical channel dropping filters,”
IEEE Photon. Technol. Lett. **10**, 549–551
(1998). [CrossRef]

**4. **D. J.
W. Klunder, et al., “Experimental and
numerical study of SiON microresonators with air and polymer
cladding,” J. Lightwave Technol. **21**, 1099–1110
(2003). [CrossRef]

**5. **S. Fan, P.
R. Villeneuve, J. D. Joannopoulos, and H.
A. Haus, “Channel drop tunneling
through localized states,” Phys. Rev. Lett. **80**, 960 (1998). [CrossRef]

**6. **S. Fan, P.
R. Villeneuve, J. D. Joannopoulos, and H.
A. Haus, “Channel drop filters in
photonic crystals,” Opt. Express **3**, 4–11 (1998),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4. [CrossRef] [PubMed]

**7. **S. S. Oh, C.-S. Kee, J.-E. Kim, H.
Y. Park, T. I. Kim, I. Park, and H. Lim,
“Duplexer using microwave photonic band gap
structure,” Appl. Phys. Lett. **76**, 2301–2303
(2000). [CrossRef]

**8. **M. Bayindir and E. Ozbay,
“Dropping of electromagnetic waves through localized modes in
three-dimensional photonic bandgap structures,” Appl. Phys
Lett. **81**, 4514–4516
(2002). [CrossRef]

**9. **T. Asano, B.S. Song, Y. Tanaka, and S. Noda,
“Investigation of a channel-add/drop filtering device using
acceptor-type point defects in a two-dimensional photonic crystal
slab,” Appl. Phys Lett. **83**, 407 (2003). [CrossRef]

**10. **M. Qiu and B. Jaskorzynska,
“Design of a channel drop filter in a two-dimensional triangular
photonic crystal,” Appl. Phys. Lett. **83**, 1074–1076
(2003). [CrossRef]

**11. **J.-B. Suck, M. Schreiber, and P. Häussler,
eds., *Quasicrystals* (Springer,
Berlin, 2002).

**12. **Y. S. Chan, C.
T. Chang, and Z.
Y. Liu, “Photonic band gaps in two
dimensional photonic quasicrystals,” Phys. Rev.
Lett. **80**, 956–959
(1998). [CrossRef]

**13. **S. S. M. Cheng, L.
M. Li, C. T. Chan, and Z.
Q. Zhang, “Defect and transmission
properties of two-dimensional quasiperiodic photonic band-gap
systems,” Phys. Rev. B **59**, 4091–4098
(1999). [CrossRef]

**14. **M. Hase, H. Miyazaki, M. Egashira, N. Shinya, K.
M. Kojima, and S. Uchida,
“Isotropic photonic band gap and anisotropic structures in
transmission spectra of two-dimensional fivefold and eightfold symmetric
quasiperiodic photonic crystals”, Phys. Rev. B **66**, 214205 (2002). [CrossRef]

**15. **K. Wang, S. David, A. Chelnokov, and J.-M. Lourtioz,
“Photonic band gaps in quasicrystal-related approximant
structures,” J. Mod. Optics **50**, 2095–2105 (2003)

**16. **M. A. Kaliteevski, S. Brand, R.
A. Abram, T. F. Krauss, R. De La
Rue, and P. Millar,
“Two-dimensional Penrose-tiled photonic quasicrystals; diffraction
of light and fractal density of modes,” J. Mod.
Opt. **47**, 1771–1778 (2000).

**17. **M. A. Kaliteevski, S. Brand, R.
A. Abram, T. F. Krauss, P. Millar, and R.
M. De La Rue, “Diffraction and
transmission of light in low-refractive index Penrose-tiled photonic
quasicrystals,” J. Phys. Cond. Matt. **13**, 10459–10470
(2001). [CrossRef]

**18. **M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa,
“Lasing Action due to the Two-Dimensional Quasiperiodicity of
Photonic Quasicrystals with a Penrose Lattice,” Phys. Rev.
Lett. **92**, pp.123906. [PubMed]

**19. **M. E. Zoorob, M. D.
B. Charlton, G. J. Parker, J.
J. Baumerg, and M.
C. Netti, “Complete photonic bandgaps
in 12-fold symmetric quasicrystals,” Nature **404**, 740–743
(2000). [CrossRef] [PubMed]

**20. **X. Zhang, Z.
Q. Zhang, and C.
T. Chang, “Absolute photonic band
gaps in 12-fold symmetric photonic quasicrystals,” Phys.
Rev. B. **63**, 081105-1 to 081105-5
(2001). [CrossRef]

**21. **Y.W. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang,
“Localized modes in defect-free dodecagonal quasiperiodic photonic
crystals,” Phys. Rev. B **68**, 165106 (2003). [CrossRef]

**22. **K. Nozaki and T. Baba,
“Quasiperiodic photonic crystal microcavity
lasers,” Appl. Phys. Lett. **84**, 4875–4877
(2004). [CrossRef]

**23. **A. Lavrinenko, P.
I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M.
H. Chong, “Comprehensive FDTD
modelling of photonic crystal waveguide components,” Opt.
Express **12**, 234–248 (2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-234 [CrossRef] [PubMed]

**24. **M. J. A. de
Dood, E. Snoeks, A. Moroz, and A. Polman,
“Design and optimization of 2D photonic crystal waveguides based on
silicon,” Opt. Quantum Electr. **34**, 145–159
(2002). [CrossRef]

**25. **M. J. Steel, T.
P. White, C. M. de
Sterke, R.
C. McPhedran, and L.
C. Botten, “Symmetry and degeneracy in
microstructured optical fibers,” Opt. Lett. **26**, 488–490
(2001). [CrossRef]

**26. **S. Guo and S. Albin,
“Numerical techniques for excitation and analysis of defect modes
in photonic crystals,” Opt. Express **11**, 1080–1089 (2003),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1080. [CrossRef] [PubMed]

**27. **J. D. Joannopoulos, R.
D. Meade, and J.
N. Winn, *Photonic Crystals: Molding the
Flow of Light*, (Princeton Univ. Press,
1995).

**28. **A. Yariv, Y. Xu, R.
K. Lee, and A. Scherer,
“Coupled-resonator optical waveguide: a proposal and
analysis,” Opt. Lett. **24**, 711–713
(1999). [CrossRef]

**29. **M. Bayindir, B. Temelkuran, and E. Ozbay,
“Tight-binding description of the coupled defect modes in three
dimensional photonic crystals,” Phys. Rev. Lett. **84**, 2140–2143
(2000). [CrossRef] [PubMed]