We investigated the impact of the thickness of the two-dimensional triangular photonic crystal (PC) on modal propagation along a PC slab waveguide. A single line defect optical waveguide in photonic crystal slab was designed by three-dimensional finite difference time domain method, plane wave expansion and effective index methods. The thickness of the PC slab waveguide was optimized to provide modal propagation for both TE-like and TM-like polarizations within the normalized frequency band of a/λ.=0.26–0.268.
©2007 Optical Society of America
The discovery of photonic crystals has opened up new methods for controlling light, leading to exploring many novel devices. Compact photonic circuits, compatible with CMOS technology, can be realized using photonic crystals (PC) [1–3]. A PC has a periodic dielectric constant. Multiple scattering of light from this periodic structure creates a range of frequencies, called photonic band gap (PBG); in which no mode is allowed to propagate. Recently, waveguides in PC structures have been extensively investigated in view of realization of integrated optical interconnects [4–5]. An optical waveguide is constructed by introducing a defect line in a perfect PC. This defect line would induce defect modes inside the PBG . Thus, guiding takes place over the frequency range of the defect mode. A PC slab waveguide uses the effect of PC slab to confine the light in the lateral (in-plane) direction, and the refractive index contrast in the vertical (out-of-plane) direction [7–8].
By operating the PC slab waveguide within the frequency band of the guided modes, one can avoid leaky modes which produce large propagation loss . In this paper, using 3D modal analysis along with spectral index method, we have determined the guided modes for a PC slab waveguide. We are seeking a thickness of the PC slab waveguide that provides both TE-like and TM-like modal guiding. Simultaneous TE-like and TM-like guiding are desirable for some polarization processing devices such as polarizing beam splitter (PBS) and polarization mode converter [10–11]. In , the main requirement of the PC based PBS is to accommodate both TE-like and TM-like guiding.
In the existing research literature, the 3D analysis of the PC slab waveguide was mostly replaced by a 2D analysis, and the effect of the finite thickness was introduced in the dielectric constant. In fact, the effect of the finite thickness of the slab was approximated by using effective refractive index of the fundamental guided mode of the slab [12–13]. The optimal thickness was usually chosen based on the photonic band gap size so that the optimum thickness of the slab produces the largest band gap size . For example, for silicon based PC slab with r/a=0.29 and nSi=3.4, where r, a and nSi are the radius of the air holes, the lattice constant [see Fig. 1(a)] and refractive index of silicon slab, the maximum band gap size occurs for the thickness (t) of 0.6a .
In this paper, the impact of the thickness of the PC slab waveguide on the polarization (TE-like, TM-like) dependent loss has been assessed and the thickness has been optimized to result to a modal guiding for both TE-like and TM-like polarizations.
In a triangular based PC slab waveguide the guiding mechanism for TE-like and TM-like polarizations are based on the index guiding and the Bloch reflections, respectively [13, 15, 16]. The simulations have been carried out for three different thicknesses to assess the impact of the thickness on the propagation loss. In section II, the computational methods employed to design the lossless waveguide and the design methodology are introduced. In section III, 3D-FDTD simulation results of the designed waveguides are presented and analyzed. Our conclusions are presented in section IV.
2. Proposed design methodology
In this section, the design methodology is explained. In the proposed design methodology, the variable parameter is the thickness of the silicon membrane PC slab waveguide. As the first step, the modes of the PC slab are assessed to acquire the desired range of the thickness which provides wide band gap. Thus, the study is limited to the aforementioned range of the thicknesses. In part 2.2, the dispersion diagram of the single defect line PC slab waveguide for both TE-like and TM-like polarizations are obtained and compared. Assuming that the TE-like and TM-like guiding mechanisms are based on the index-guiding and PBG guiding, the effective index method and PWEM are employed to obtain the dispersion diagrams, respectively. It is observed that by changing the thickness of the PC slab waveguide, the overlap between the TE-like and TM-like guiding varies. One can find the thickness which provides the maximum overlap between the two polarizations. Finally, to verify the arguments made based on dispersion diagrams; a 3D-FDTD simulation is carried out for the entire structure within the frequency band of the band gap of the corresponding PC slab. The loss diagrams obtained by 3D-FDTD verify the arguments made in previous steps.
2.1 Modes in PC slab
First we have assessed the band gap variation with the thickness of the PC slab. We start with the calculation of the band diagram of a PC slab by the plane wave expansion method (PWEM). The unit cell of the structure which is used for the calculation of the band diagram of the PC slab is shown in Fig. 1(b). The PC slab has periodicity in x-z plane and the propagation direction is in the x direction. The dominant components of the TM-like mode are defined by Hy, Ez and Ex. The dominant components of the TE-like mode are defined by Ey, Hz and Hx where the axial component has odd symmetry with respect to the (x-z) plane. On the other hand, the axial component of the TM-like mode, Ex, has even symmetry with respect to the (x-z) plane. PWEM is employed for the periodic structures in which the structures extend periodically in all dimensions. However, the PC slab has only two-dimensional periodicity. The periodicity can be artificially extended in the vertical direction by introducing a sequence of slabs separated by sufficient amount of air to maintain electromagnetic isolation. This method is called super-cell method .
Figure 2 shows the “gap-map” and the width of the band gap of the silicon triangular lattice based PC slab versus the thickness of the slab calculated by PWEM. In this structure, the radius of the air holes and the lattice constant are r=123 nm and a=410 nm (r/a=0.3). The refractive index of Si is assumed to be nSi=3.48. The simulation results give the band gap map for the TM-like modes of the PC slab. We have imposed even symmetry to Hy, Ez and Ex components in the vertical direction. Band gaps do not exist for the modes with the odd symmetry in the vertical direction, the TE-like modes. Figure 2 shows that the band stop moves toward lower frequencies as the thickness increases which is easily understandable. No band gap exists for the thick slabs (t>1.6a). For very thin slab, guided modes still exist, but are weakly guided.
2.2 PC slab waveguide with defect line
In this section, we study the propagating modes for both TE-like and TM-like polarizations along the defect line in the PC slab waveguide shown at Fig. 1(a). To determine the TE-like modes of the PC slab waveguide, the effective index method is used. Although, TE-like modes do not have band gap, they still can be guided due to the index-guiding (in-plane as well as out-of-plane directions) [15–16]. In order to determine the appropriate thickness of the slab, we have employed the effective index method in both in-plane and out-of-plane directions to obtain the cut-off frequency of the fundamental mode. To employ the effective index method, the PC slab waveguide is replaced by a conventional slab waveguide shown in Fig. 3, where nsi ,nc and n 0 are the film, cover and free space refractive indices, respectively. The width of the film layer (depicted by W 0 in Fig. 3) of the equivalent conventional slab waveguide is W 0=a. In fact, the missing row of the air holes is replaced by the film layer of the equivalent slab waveguide. The following relation approximates the cladding refractive index for the TE-like polarization for which the electric field is parallel to the air cylinders:
Thus, nc represents the average refractive index of the cladding structure. Therefore, the problem is simplified to finding the effective refractive index of the fundamental guided mode of the effective structure shown in Fig. 3 for the TE-like polarization.
For TM-like polarization, the guiding mechanism for the in-plane direction is based on Bloch reflection and in the vertical dimension is based on total internal reflection. The TM-like modes are calculated by PWEM. This calculation uses the concept of super-cell, depicted by the dashed line in Fig.4. The propagation direction of the waveguide is in the x direction. The periodic boundary conditions have been applied in all three directions.
Finally, after obtaining the cut-off frequency of the TE-like mode and the spectrum of the TM-like modes, the cut-off frequency of the fundamental TE-like mode is compared with the frequency of the defect modes (TM-like modes) in the PBG.
To verify the modal analysis performed in previous steps, 3D-FDTD simulation is carried out to obtain the transmission loss associated with the TE-like and TM-like polarizations within the frequency range of the band gap. For the 3D-FDTD simulations we have used the FullWAVE and BandSOLVE tools of package RSOFT, version 220.127.116.11.
The simulated structure [Fig. 1(a)] consists of 100 rows of holes along the propagation direction (x-direction) and 11 rows of holes (including the defect row) in z-direction. However, only 50 rows out of the middle of the structure, where the electromagnetic field distribution has become stable and all evanescent modes are vanished, are selected for loss calculation. Thus, the coupling loss has been excluded from the loss calculation. The mesh sizes along the x, y and z-directions (Δx, Δy and Δz) are Δx=Δy=Δz=20.5 nm. Thus, 20×20 mesh cells describe the PC unit cell in the in-plane (x-z plane) direction. The 3D-FDTD simulations are repeated for three thicknesses, t=0.5a, 0.75a and a.
The perfectly matched layer (PML) boundary condition is applied for all three directions. Time waveforms in 3D FDTD have been chosen as a single frequency sinusoid with normalized frequency within the frequency band of the PBG and the resolution of 0.001. The spatial distribution of the incident field follows the symmetry of the mode. For example, to excite the fundamental TM-like mode, a spatial Gaussian distribution has been used as the incident field. Where as, the spatial distribution of the incident field of the next higher order TM-like mode has an odd symmetry with respect to the (x-y) plane bisecting the middle of the defect line.
To determine the transmission loss from the 3D-FDTD simulation results, the net power crossing at several cross sections (y-z planes) are calculated by integrating the Poynting vectors across each cross sections. Thus, the axial power flow can be plotted along the propagation direction, x-direction. The errors due to the FDTD simulation lead to the fluctuation of the power along the propagation direction. To eliminate this fluctuation, least square method has been employed to fit the power to a power transmission equation. For more convenience in calculation of the transmission loss in db/mm, the transmission loss equation is chosen in the form of, T ∝ 10-αx . With this model, 10α directly gives the transmission loss in db/mm, assuming that the unit of x in the transmission loss equation is in mm.
3. Results and discussions
In Fig. 5, we present the dispersion curves of the TE-like modes of the effective slab waveguide depicted in Fig. 3 as a function of the normalized in-plane k vector for 3 values of the thicknesses, t=0.5a, 0.75a and a, respectively. t is the thickness of the PC slab waveguide. Assuming:
where, is the in-plane k vector. The light line is plotted by the dashed line in this figure. For this structure with air cladding, the light line is simply the wave number divided by the refractive index of air.
The modes below the light line are guided. Above the light line corresponds to all the background modes. For t=0.5a, the highest normalized cut-off frequency of the fundamental mode (with ka/2π=0.5) is around 0.33.
Next, we proceed to calculate the TM-like modes spectrum of the structure (t=0.5a). PWEM has been employed to obtain the spectrum of the Bloch modes inside the band gap of the PC slab. The Bloch modes are TM-like. Fig. 6(a) shows the spectrum of the fundamental and the next higher-order Bloch modes (TM-like modes). The light line is represented by the dashed line in the figure. Part of the TM-like mode on the left side of the light line physically does not represent a bound mode as such. In this region, the energy is mostly dispersed into the leaky modes without any guidance.
In Fig. 6(a), the fundamental TM-like mode is placed at the normalized frequency range of 0.28–0.29. Figure 6(a) also shows that the next higher-order TM-like mode is placed over a very narrow range of frequency, 0.301–0.302. It is observed that the TM-like modes are below the normalized cut-off frequency of the TE-like polarization of the equivalent slab discussed earlier (see Fig. 5). Therefore, no TE-like guiding is expected to take place over the frequency band of the TM-like modes.
To verify the aforementioned results, 3D-FDTD simulations have been carried out for the entire structure (Fig. 1(a), t=0.5a). Time waveforms in 3D_FDTD have been chosen as a single frequency sinusoid with normalized frequency in the range of 0.265–0.34. The transmission loss (α) in db/mm has been calculated and plotted in Figs. 6(b), 7(b) and 9(b). In Fig. 6(b), and in subsequent plots, the loss is graphed along the horizontal axis to keep the format of the loss diagrams consistent with the format of the band gap diagrams used in this paper. The TE-like and TM-like modes are shown by a dashed line and a solid line, respectively. Modal propagation is seen for the normalized frequency of 0.285–0.29 and 0.3, which are very close to the modes obtained by PWEM modal analysis for TM-like mode [see Fig. 6(a)]. Here, the loss diagram for the TM-like polarization is compared with the existing research literature [17–19]. In , the out-of-plane transmission losses for suspended-membrane structures with different suspension distance above the GaAs substrate for the mode which had an even symmetry with respect to both (x-z plane) and (x-y plane) (fundamental TM-like mode according to our definition) were presented. The out-of-plane loss within the frequency band of the fundamental TM-like mode was less than 0.2 cm-1 (0.09 db/mm) caused by the remaining evanescent coupling between the waveguide mode and the radiation modes of the high index substrate . In our model, we have assumed the silicon membrane is deeply undercut; thus, there is no interaction between the waveguide guided mode and the radiation modes of the substrate. In Ref. , the disorder-induced propagation losses of fundamental TM-like mode (our definition) of the PC slab waveguide both in membrane and Silicon-on-Insulator (SOI) structures were studied. For the average uncertainty on the hole radius of Δr/a=3.2 nm (a=400 nm, r=110 nm and t=220 nm), the out-of-plane propagation loss of the membrane structure for the fundamental TM-like mode was 0.5–0.6 db/mm which was caused by the disorder induced on the radius of the air holes .
Also, Fig. 6(b) shows no TE-like modal guiding takes place in the frequency band of the fundamental and the next higher order TM-like modes, as expected. Thus, no overlap between TE-like and TM-like modal guiding exists for t=0.5a. Accordingly, in order to overlap TE-like and TM-like guiding, the cut-off frequency for TE-like mode must take place at frequencies lower than the frequency range of the Bloch modes (TM-like modes). By increasing the thickness of the PC slab waveguide, the cut-off frequency for TE-like modes moves toward lower frequencies as depicted in Fig. 5.
When we increased the thickness to 0.75a, the highest normalized cut-off frequency of the fundamental TE-like mode of the equivalent slab (Fig. 3) for κa/2π=0.5 is less than 0.25, Fig. 5, which is almost the lower edge of the band gap for TM-like wave. Thus, the cut-off frequency of the TE-like is certainly below the frequency range of the Bloch modes (TM-like modes). To obtain the characteristics of the TM-like modes of the PC slab waveguide, again PWEM has been employed. Figure 7(a) shows the band structure of the TM-like modes obtained by PWEM. The fundamental TM-like mode is placed in the frequency range of 0.255–0.27. The next higher-order TM-like mode is a very narrow band mode around the normalized frequency of 0.278. For this case, the frequency band of the TM-like modes are above the normalized cut-off frequency of the TE-like mode of the equivalent slab (Fig. 3), we expect to have both TE-like and TM-like guiding over the frequency band of the TM-like modes.
To verify these observations, a 3D-FDTD simulation was carried out for the entire structures [t=0.75a, Fig. 1(a)]. Time waveforms in 3D_FDTD have been chosen as single frequency sinusoids with normalized frequency in the range of 0.23–0.34. Figure 7(b) shows the loss per mm versus the normalized frequency for both TE-like and TM-like polarizations obtained by 3D-FDTD simulations. Modal propagation for both TE-like and TM-like polarizations over the frequency band of the fundamental TM-like mode, i.e. 0.26–0.268, is seen. According to the FDTD simulation, the next higher order TM-like mode is located at the normalized frequency band of 0.272–0.273 for which lossless TE-like propagation takes place. However, the TE-like polarization appears to experience loss at normalized frequencies higher than 0.275. The loss occurs in the in-plane direction; nevertheless, the TE-like mode is confined in the vertical direction (as per the 3D-FDTD field distribution results).
To comprehend the behavior of the TE-like mode, the dispersion diagram of the laterally even TE-like mode is computed and presented in Fig. 8 (depicted by the solid dotted line). The accurate dispersion analysis of the TE-like mode is based on the spatial fourier transform (SFT) of the electromagnetic field distribution in the PC slab waveguide along the propagation direction at any point on the normal plane, the plane normal to the propagation direction (plane y-z) . The electromagnetic field distribution is determined using 3D-FDTD. The peaks of the SFT spectrum describe the propagating modes of the structure. These peaks are independent of the location, (y,z) and the electromagnetic filed components. The spectrum of field component has one peak in the regions where no photonic crystal slab mode exists. This peak is associated with the guided TE-like mode. On the other hand, in regions where photonic crystal slab modes exist, other peaks associated with the photonic ctysal slab modes are also observed. The dispersion diagram of the TE-like mode is also computed employing PWEM in regions where no photonic crystal mode exist. The results achieved by both methods are similar.
The TE-like guiding takes place below all photonic crystal modes depicted by the gray area in Fig. 8. This mode is the index-guiding mode that does not see the periodic structure, but the effective material with the refractive index less that nSi. There is no upper limit for this mode since it does not see the periodic PC. This mode is folded back at the zone boundary of the first Brillouin zone (κa/2π=0.5) [15, 16]. The lower limit of the mode takes place approximately at the normalized frequency of a/λ=0.2. This modes cross the region where the PC modes exist at a/λ=0.27 which determines the upper limit since it leak the energy into the PC. Thus, in loss diagram, the upper limit is observed for the TE-like mode.
The simulation results for two different thicknesses suggest that for t=0.75a, lossless propagation for TM-like polarization over the normalized frequency band of 0.26–0.268 and 0.272–0.273 is achieved in which TE-like guiding is also lossless. In fact, by increasing the thickness to 0.75a, the overlap between TE-like and TM-like modal guiding has been increased in the sense that the cut-off frequency for TE-like modes has been pushed down below the frequency band of the fundamental TM-like mode.
To investigate, whether the overlap between the frequency bands of the TE-like and TM-like modes improves with increasing the thickness of the PC slab wave, the thickness is increased to t=a. The normalized cut-off frequency for t=a for the TE-like mode is around 0.15 according to Fig. 5. The fundamental TM-like mode obtained by PWEM is placed in the normalized frequency range of 0.243–0.26 and the next higher-order TM-like mode is a narrow band mode placed approximately at the normalized frequency of 0.266 [Fig. 9(a)]. Hence, for this case the TM-like modes are located above the normalized cut-off frequency of the TE-like mode of the equivalent slab (Fig. 3). Thus, modal TE-like and TM-like guiding is expected to be seen within the frequency band of the TM-like modes. Furthermore, the higher order Bloch mode (TM-like mode) is also pushed down inside the PBG.
3D-FDTD simulation results for both TE-like and TM-like polarizations are shown in Fig 9(b). 3D-FDTD simulation results for the TM-like mode, shown by the solid line in Fig 9(b), show a lossless propagation over the frequency range of 0.255–0.258 and 0.264–0.265. The higher order mode is also observed within the band gap. The simulation results for the TE-like mode show that the TE-like guiding is lossless up to the normalized frequency of 0.23. Then, above this frequency the propagation is not lossless any more. The dispersion diagram for the laterally even TE-like mode, Fig. 10, indicates that at frequencies above a/λ=0.23 the TE-like mode depicted by dotted solid line in the figure falls into the gray region and leaks the energy to PC slab modes. Therefore, the TE-like polarization is not guided over the frequency range of the fundamental TM-like mode and the next higher-order TM-like mode. Therefore, thicker slabs do not necessarily provide wide band lossless propagation for both TE-like and TM-like polarizations.
The simulation results suggest that both TE-like and TM-like modal guiding in PC slab waveguide can be achieved by optimizing the thickness of the PC slab waveguide. For a PC slab waveguide to accommodate both TE-like and TM-like modal guiding, it must be thick enough to ensure that the TM-like mode lies above the cut-off frequency of the TE-like mode. A rough estimation can be obtained by using effective index method. For example the cut-off frequency of the TE-like mode for t=0.5a lies above the TM-like mode frequency; thus, no overlap between TE-like and TM-like guiding is observed. On the other hand, for thick slabs where the cut-off frequency is well below the frequency band of the band gap, the TE-like wave has already become lossy at the frequency band of the TM-like mode and lost its energy to PC slab modes. Therefore, no overlap between the modal guiding of the two polarizations exists. 3D-FDTD simulation results show that t=0.75a is a good choice for maintaining lossless TE-like and TM-like guiding; since, the upper cut-off frequencies for both TE-like and TM-like waves are almost the same.
We have numerically demonstrated the impact of the thickness on the modal behavior of the PC slab waveguide. The PWEM, effective index method and 3D-FDTD simulation results suggest the thickness of the PC slab can be optimized to achieve lossless (TE-like, TM-like) wave propagation.
(TE-like, TM-like) modal propagation inside the PC slab waveguide is of great interest in many integrated optical devices. To achieve this, the thickness of the PC waveguide must be optimized to provide both TE-like and TM-like modal guiding. The initial value of the thickness can be estimated by effective index method and PWEM. The slab must be thick enough to provide TE-like modal guiding inside the PBG frequency band of the TM-like modes. However, very thick slabs introduce more defect modes inside the PBG and shift the spectrum of the TE-like modes toward frequencies lower than the frequency range of the TM-like mode. In our studies, for t=0.75a, both TE-like and TM-like modal propagation within the normalized frequency band of 0.26–0.268 has been achieved. Assuming that the normalized frequency of 0.265 corresponds to 1.55 μm, the available spectrum window of the silicon membrane PC slab waveguide for both TE-like and TM-like guiding is 1.53 –1.58 μm.
References and links
1. W. Bogaerts, R. Baets, and P. Dumon, et al., “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–421 (2005). [CrossRef]
2. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger,”Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Electron. 4, 938–947 (1998). [CrossRef]
4. M. Zelsmann, E. Picard, T. Charvolin, and E. Hadji, “Broadband optical characterization and modeling of photonic crystal waveguides for silicon optical interconnects,” J. Appl. Phys. 95, 1606–1608 (2004). [CrossRef]
6. A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, “Properties of the slab modes in photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1554–1564 (2000). [CrossRef]
7. L. C. Andreani and M. Agio, “Photonic bands and Gaps maps in a photonic crystal slab,” IEEE J. Quantum Electron. 38, 891–898 (2002). [CrossRef]
8. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
9. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000). [CrossRef]
10. T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Design of compact photonic crystal-based polarizing beam splitter,” IEEE Photon. Technol. Lett. 17, 1435–1437 (2005). [CrossRef]
11. Y. Tanaka, S. Takayama, T. Asano, and S. Noda, “Polarization mode conveter based on 2D photonic crystal slab,” LEOS 2005, 339–340 (2005).
12. A. Adibi, R. K. Lee, Y. Xu, A. Yariv, and A. Scherer, “Design of photonic crystal optical waveguides with single mode propagation in the photonic band gap,” Electron. Lett. 36, 1376–1378 (2000). [CrossRef]
13. H. Benisty, C. Weisbuch, D. Labilloy, M. Rattier, C. J. M. Smith, and T. F. Krauss, “Optical and confinement properties of two-dimensional photonic crystals,” J. Lightwave Technol. 17, 2063–2077 (1999). [CrossRef]
14. H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassange, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000). [CrossRef]
15. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000). [CrossRef]
16. T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, “Light propagation characterization of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate,” IEEE J. Quantum Electron. 38, 1743–752 (2002). [CrossRef]
17. L. C. Andreani and M. Agio, “Intrinsic diffraction losses in photonic crystal waveguides with line defects,” Appl. Phys. Lett. 82, 2011–2013 (2003). [CrossRef]
18. W. Kuang, C. Kim, A. Stapleton, W. J. Kim, and J. D. O’Brien, “Calculated out-of-plane transmission loss for photonic-crystal slab waveguides,” Opt. Lett. 28, 1781–1783 (2003). [CrossRef] [PubMed]
20. A. Jafarpour, C. M. Reinke, A. Adibi, Y. Xu, and R. K. Lee, “A new method for the calculation of the dispersion of nonperiodic photonic crystal waveguides,” J. Quantum Electron. 40, 1060–1067 (2004). [CrossRef]