EPL (Europhysics Letters)

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Europhys. Lett., 74 (2), p. 261 (2006)
DOI: 10.1209/epl/i2005-10531-2

Coupled-resonator optical waveguides in photonic crystals with Archimedean-like tilings

Yiquan Wang

Academy of Physics and Electronic Engineering, Central University for Nationalities Beijing 100081, PRC

received 3 October 2005; accepted in final form 22 February 2006
published online 29 March 2006

Abstract
Coupled-resonator optical waveguides (CROWs) constructed in photonic crystals (PCs) with Archimedean-like tilings are studied in this article. Although unequivalent sites exist in this kind of PC, the transmission properties of the CROWs constructed in PCs with Archimedean-like tilings are comparable with those constructed in periodic PCs. Because of the special structure property of this kind of PC, there are more waveguide modes in CROW constructed in PCs with Archimedean-like tilings. Furthermore, the group velocity of the transmitted electromagnetic waves in this kind of CROWs is smaller than that in its counterpart constructed in periodic PC.

PACS
42.70.Qs - Photonic bandgap materials
41.20.Jb - Electromagnetic wave propagation; radiowave propagation
42.25.Bs - Wave propagation, transmission and absorption

***

In the past decades there has been a great deal of interest in studying photonic band gap (PBG) materials --photonic crystals (PCs) [1,2]. Because the periodic dielectric structures have spectra gaps in which electromagnetic wave propagation is forbidden in all the directions, PCs open up many exciting possibilities, such as the suppression of spontaneous emission [1], the possible observation of interesting quantum interference effect [3], and the realization of strong photon localization [2]. Many potential applications including quantum electronic devices, distributed-feedback mirror, light-emitting diodes [4], high-Q cavities [5], microwave antennae substrate [6], optical waveguides [7,8] and ultrafast optical switches [9,10] have further fueled the interest in PBG research. Optical waveguides are basic components in optical communication. There are two kinds of waveguides in photonic crystals. One is line defect formed in the otherwise perfect PC [7,8]. The other is constructed with a chain of strongly confined point defects or cavities, commonly known as a coupled-resonator optical waveguides [11,12,13] (CROWs). In addition to transmitting electromagnetic signal in a given direction, CROWs can serve as strongly wavelength-dependent delay-lines [14] and wavelength demultiplexers [15,16]. They can also be used for pulse compression [14,17] and enhancement of nonlinear effects such as second-harmonic generation [18,19,20]. All the applications of CROWs are based on the prominent property of small group velocity in a wide range of wave vectors [12]. Therefore, searching for the structures which support waveguide mode with lower group velocity is very important for the application of this kind of device. Extensive study has been carried out on CROWs constructed in periodic PC. In contrast, CROWs constructed in aperiodic PCs [21,22,23,24,25,26,27] are seldom studied. In this article, CROWs constructed in PCs with Archimedean-like tilings [28,29] are studied. Simulation results indicate that their transmission properties are comparable with those constructed in periodic PCs. Because of the special structural properties of this kind of PC, there are more waveguide modes in CROWs constructed in PCs with Archimedean-like tilings than in their counterparts formed in periodic PC. Furthermore, the group velocity is smaller than that constructed in periodic PCs with the same structural parameters.


  \begin{figure}\includegraphics{82f01.eps} \end{figure}
Figure 1: Photonic crystal with Archimedean-like tiling (a) and its band gap structure (b).


  \begin{figure}\includegraphics{82f02.eps} \end{figure}
Figure 2: The CROWs constructed in PC with Archimedean-like tiling (a) and in triangular lattice periodic photonic crystal (c) as well as their corresponding transmitted energy flux, (b) and (d). The solid line and the dashed line are energy flux with and without CROW of the same structure.

The PC in which the CROW is constructed is schematically shown in fig. 1(a). It is built up with dielectric cylinders placed perpendicularly on the lattice point of an Archimedean-like tiling. The structure can also be considered as constructed with the supercell shown in the inset of fig. 1(a). In this structure, the lattice constant of the structure is $10\ensuremath{{\rm ~mm}} $ and the dielectric constants of the cylinders and their background are 8.6 and 1.0, respectively. The radius of the cylinders is $3\ensuremath{{\rm ~mm}} $. The band gap structure calculated with the finite-difference time-domain (FDTD) method [30] is shown in fig. 1(b). It is clearly seen that this structure possesses full gaps, from 7.8 to $10.7\ensuremath{{\rm ~GHz}} $ and from 14.3 to $17.4\ensuremath{{\rm ~GHz}} $, for the TM polarized electromagnetic waves within our simulation frequency region.

Because of rotational symmetry of the supercell, all the cylinders located at the vertices of the central hexagon of the supercell have the same environmental conditions. If we remove a cylinder located at different vertices of the hexagon, the defects will support the same defect modes. Therefore, the photons corresponding to the defect modes can hop between the point defects if they couple together.

The CROW constructed in PC with Archimedean-like tiling as shown schematically in fig. 2(a) is studied first. In order to study the transmission property of the CROW, the multiple-scattering method [31] is used to calculate the transmitted energy flux [32] of the CROW. In our simulation, an S-polarized slit source is put at the left-hand end of the waveguide. Both the width of the slit source and the distance between the center of the slit and the left-hand end of the waveguide are $20\ensuremath{{\rm ~mm}} $. The incident direction of the electromagnetic wave is perpendicular to the edge of the samples. The outgoing energy flux, as defined in ref. [32], at the outlet of the CROW is calculated and the result is displayed in fig. 2(b).

From fig. 2(b) it is clearly seen that the CROW constructed in PC with Archimedean-like tiling has a miniband, which has the width of $1.14\ensuremath{{\rm ~GHz}} $, from 14.78 to $15.92\ensuremath{{\rm ~GHz}} $ within the photonic gap from 14.3 to $17.4\ensuremath{{\rm ~GHz}} $. As a reference for comparison, a similar CROW constructed in triangular lattice periodic PC which has the same structural parameters as that studied above is constructed and its energy flux is simulated. The structure and the simulated results are displayed in figs. 2(c) and (d), respectively. It is shown that the CROW constructed in triangular periodic PC also has a miniband from 15.28 to $16.18\ensuremath{{\rm ~GHz}} $. Its width is $0.9\ensuremath{{\rm ~GHz}} $. Therefore, the CROW constructed in PC with Archimedean-like tiling have a wider waveguide band than its counterpart constructed in periodic PC.

In addition to the wider waveguide band, the group velocity of the CROW constructed in PC with Archimedean-like tiling is smaller than that constructed in periodic PC. In order to study accurately the group velocity of CROWs, the waveguide modes of the two kinds of CROW of an infinite length is calculated with FDTD method. The result is shown in fig. 3. Comparing figs. 2 and 3, it is clearly seen that for the CROW constructed in periodic PC, the miniband in fig. 2(d) is one-to-one with the waveguide mode AB shown in fig. 3(b). However, in CROW constructed in PC with Archimedean-like tiling, the miniband in fig. 2(b) is superimposing result of the two waveguide modes A1B1 and A2B2, as shown in fig. 3(a). This may be attributed to the special structure property of this kind of PC. The two cavities within a supercell are separated by only one cylinder. However, the two cavities belonging to two adjacent supercells are separated by two cylinders. Because of the different coupling between the cavities in the CROW, the two cavities belonging to a supercell form a group. Therefore the same defect mode is supported by each of the two cavities coupled together and results in the splitting of the original defect mode. The original defect mode located in each cavity becomes two new different defect modes. Each new defect mode locates in the total volume of the two cavities and results in a waveguide mode by coupling with the corresponding defect mode located in other groups. Therefore, there are two waveguide modes in the CROW.

In the spirit of the tight-binding approximation and taking into account that interaction occurs only between adjacent defects along the waveguide [12,14], the group velocity of the waveguide modes is calculated and the results are shown in fig. 4. The maximum value of group velocity of waveguide modes A1B1 and A2B2 is 0.0149 and 0.0183 times that of light velocity in vacuum, respectively. It is quite smaller than that of the waveguide mode AB, in which the maximum value of group velocity is 0.0294 times that of light velocity in vacuum.


  \begin{figure}\includegraphics{82f03.eps} \end{figure}
Figure 3: The waveguide modes of the CROWs shown in figs. 2(a) and (c).


  \begin{figure}\includegraphics{82f04.eps} \end{figure}
Figure 4: The group velocity of waveguide modes shown in fig. 3.

Furthermore, the CROW constructed in the aperiodic PC by removing the 7 cylinders within each supercell located along the horizontal symmetry axis of the sample is studied. For the purpose of comparison, we also study the CROWs consisting of coupled hexagonal cavities H2 (seven missing cylinders) in triangular lattice PC. Their structure are schematically shown in figs. 5(a) and (c), respectively. In both kinds of CROW, the cavities are separated by one row of cylinders. Figures 5(b) and (d) show their waveguide modes. There are two modes, labeled A1B1 and A2B2, within the gap, from about 7.8 to $10.7\ensuremath{{\rm ~GHz}} $, for each kind of CROW. Below we discuss them separately.


  \begin{figure}\includegraphics{82f05.eps} \end{figure}
Figure 5: The CROWs constructed in PC with Archimedean-like tiling (a) and triangular lattice periodic PC (c) by coupling cavities H2 and their corresponding waveguide modes, (b) and (d).

The mode labeled A1B1 is a waveguide mode. Its frequency region in the CROW constructed in the aperiodic PC is from 9.32 to $10.14\ensuremath{{\rm ~GHz}} $. It is narrower than that constructed in periodic PC, where the guide mode frequency region is from 9.46 to $10.84\ensuremath{{\rm ~GHz}} $. Therefore, the group velocity, as shown in fig. 6, of electromagnetic waves transmitted in the CROW constructed in PC with Archimedean-like tiling is also smaller than that constructed in triangular lattice periodic PC.


  \begin{figure}\includegraphics{82f06.eps} \end{figure}
Figure 6: The dashed line and the solid line are the velocity of waveguide modes A1B1 transmitted in the CROW shown in figs. 5(a) and (c), respectively.


  \begin{figure}\includegraphics{82f07.eps} \end{figure}
Figure 7: The transmission spectra and the defect mode of PC with Archimedean-like tiling (a) and periodic PC (b) with a H2 cavity.

As for the mode A2B2, it is an uncoupled defect mode localized in every cavity of the CROWs. In order to understand the problem, the transmission spectra of PC with Archimedean-like tiling and triangular lattice periodic PC with only one H2 cavity as well as the distribution of electric field of their defect modes are simulated by using the multiple-scattering method. The results are displayed in fig. 7. From their transmission spectra it is clearly seen that in both cases there is only one defect mode within their gaps. The distribution of electric field of the defect modes, as shown in fig. 7, indicates that in both cases the defect mode is a dipole mode. Because of symmetry of the structures, this defect mode has a double degeneracy [33]. The two degenerated modes can be separated into the x and y dipole modes. The classification of dipole mode into x and y states arise from the polarization of the electric field in the center of the defect. The radiation patterns of the x and y dipole modes resemble those of the x and y oriented electric dipoles, respectively, positioned in the center of the defect. In our simulation, the source is placed in the left hand of the samples. Therefore, only y dipole mode can be exited. In our CROW, the electric field of the y dipole mode extends in the direction along the waveguide. Therefore, the electric fields of the y dipole modes belonging to different cavities overlap and couple together to form waveguide mode A1B1. In contrast, the electric field of x dipole mode extends in the direction perpendicular to the guiding direction. Their electric fields have no chance to overlap and interact. Therefore, they are localized in corresponding cavities and form localized mode A2B2, as shown in fig. 5.

As mentioned above, the coupled defects are formed by removing the cylinders located on the vertices of hexagon within the supercells. It is similar with the CROW constructed in triangular periodic PC. Therefore, the transmission properties of CROW constructed in the PC with Archimedean-like tiling will keep unchanged when the waveguide rotates by 60 degrees.

Inspired by the results obtained in the microwave region, we have studied the transmission properties of CROWs constructed in the aperiodic PC made from holes in a high index matrix within the frequency regions of optical communication and visible light theoretically and experimentally. Simulated and experimental results show that in the inverse system, the waveguide modes of CROWs constructed in PC with Archimedean-like tiling is measurable. They will be presented recently.

In conclusion, the CROWs constructed in PC with Archimedean-like tiling are studied. The results show that the transmission properties of CROWs constructed in the aperiodic PC are comparable with that constructed in periodic PC although unequivalent sites exist in PC with Archimedean-like tiling. Furthermore, the group velocity of electromagnetic waves transmitted in CROWs constructed in PC with Archimedean-like tiling is smaller than that of periodic PC. This is very important for the application of this kind of aperiodic PC.

$\ast\ast\ast$

This work is supported by the Chinese National Key Basic Research Special Fund (Grant No. 2001CB6104). The support of the CSTNET for computer time is gratefully acknowledged.

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