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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=0.3a, 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=0.35a.
Fig. 1. (Left panel) Sketch of the analyzed 8-fold quasi-periodic structure. Building tiles are depicted in red. (Right panel) The gap map of the first PBG in TM polarization for the PQC versus dielectric constant. The minimum (black line) and maximum (red line) normalized frequencies of the gap as a function of the dielectric constant are shown. The inset shows the gap width to midgap ratio as a function of the dielectric constant (points) together with an interpolation fit (green line).
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.
Fig. 2. (Left panel) Sketch of the microcavity made by removing rods around the central one. (Right panel) Energy density inside the PQC (black dashed line) and inside the cavity (solid blue line). The spectral range of a complete PBG is shown. Three cavity modes are designated by Ω 1, Ω 2 and Ω 3.
Fig. 3. Field patterns of the cavity modes of PQC corresponding to the spectral range of the first PBG. Mode Ω 1 is a quadropole, mode Ω 2 is a hexapole and mode Ω 3 is a dipole. Two degenerated hexapole modes are shown in central panels. Hexapole-0 (Hexapole-90) mode is even with respect to the vertical (horizontal) plane. Colors represent electric field amplitude. Arrows show magnetic field lines. Circles show positions of rods in the structure.
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 Cn, whereas if there is also a line of reflection the appropriate group is Cnv. Further, the cavity modes exist in finite sets. Each set forms a basis for an irreducible representation of the group. The dimension 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 C8v 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 (WN) 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.
Fig. 4. (Top panel) Sketch of different PQC waveguide configurations. W1, W2 and W3 waveguides are shown. (Bottom panel) Transmission efficiency spectra for W1 (green), W2 (red) and W3 (black) waveguides. Spectral positions of the cavity modes are shown as a vertical blue lines.
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).
Fig. 5. Energy density stored in the cavity by hexapole modes. (Left panel) Two hexapole modes are completely degenerate in square patch of PQC. (Center panel) Degeneracy is lifted, when the symmetry of the system is broken by waveguides. Hexapole-0 and Hexapole-90 modes have different resonant frequencies. (Right panel) The modes overlap is partially restored in the system based on a rectangular patch of PQC. Sketches of considered structures are shown in the top panel above the corresponding energy density spectra.
Fig. 6. Field patterns are shown for the Hexapole-0 (left panel) and Hexapole-90 (center panel) modes and for their superposition (right panel) decaying into the waveguides channels. Colors represent electric field amplitude. Circles show positions of rods in the structure. In the top panel, the sign of the electric field amplitude in waveguide channels in the direct vicinity of the cavity is shown for the appropriate modes.
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.
Fig. 7. (Left panel) Transmission efficiency of the add-drop filter based on a square patch of octagonal PQC. Transmission in the main channel (black line), reflection back at the entrance of the filter (blue line), backward (red line) and forward (green line) transmission in the upper waveguide is shown. Energy density stored in the Hexapole-0 and Hexapole-90 modes are shown for comparison by dashed black and dashed red lines, respectively. Energy density spectra are normalized to their maximum value. Electric field patterns are shown for the resonance (center panel) and out of the resonant (right panel) frequencies. Light is coupled to the add-drop filter at the Input channel and propagates in backward (forward) direction in the Output-2 (Output-1) channel for the resonance (out of the resonant) frequency. Colors represent electric field amplitude. Circles show positions of rods in the structure. (Movies 946 KB, 805 KB)
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.
Fig. 8. (Left panel) Transmission efficiency of the optimized add-drop filter based on a rectangular patch of octagonal PQC. Transmission in the main channel (black line), reflection back at the entrance of the filter (blue line), backward (red line) and forward (green line) transmission in the upper waveguide is shown. Energy density stored in the Hexapole-0 and Hexapole-90 modes are shown for comparison by dashed black and dashed red, respectively. Energy density spectra are normalized to their maximum value. Electric field patterns are shown for the resonance (center panel) and out of the resonant (right panel) frequencies. Light is coupled to the add-drop filter at the Input channel and propagates in backward (forward) direction in the Output-2 (Output-1) channel for the resonant (out of the resonance) frequency. Colors represent electric field amplitude. Circles show positions of rods in the structure. (Movies 794 KB, 699 KB)
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. |
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