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The dispersion and transmission characteristics of transverse electric modes of a flexible photonic crystal waveguide are investigated by numerical simulation. Calculated results indicate that for arbitrary-angle bends of this waveguide with very small curved radii no more than two wavelengths, a very high transmission (>98.5%) is observed for a broad enough bandwidth. Owing to its unique advantage of compactness and flexibility, this waveguide is expected to be applied to highly dense photonic integrated circuits after further research.
©2009 Optical Society of America
1. Introduction
In photonic crystal waveguides (PCWs) based on line defects in two-dimensional photonic crystals, a low-loss transmission through their bends with radii of curvature no more than 2–3 lattice constants can be achieved [1–5]. However, those compact PCWs are not flexible; the bending angle is restricted by the lattice type of the photonic crystals. For example, the bending angle is 90° [1,2] for in typical square lattice PCW, and it is 60° for the case of hexagonal lattice PCW [3]. There may be few other special bending angles, nevertheless, in the realization of a low-loss transmission for each special angle bends, special design is required for every individual PCW [4,5].
A flexible optical waveguide based on the omnidirectional reflection of one-dimensional photonic crystals was presented by the authors [6]. For arbitrary-angle bends of this waveguide, without requirements of any optimal design, a very high transmission (>97%) for single mode (zero-order TM mode) transmission over a wide enough frequency range is obtained. However, this TM-mode flexible waveguides (the flexible waveguides based on TM fundamental mode) has many reflection layers on each side and it is not compact enough. To achieve a high transmission through arbitrary-angle bends, not less than 10 reflection layers on each side is necessary and the bending radius is restricted to not less than 26 lattice constants of the one-dimensional photonic crystal.
The refractive index profile in the flexible waveguide plays an important role in different guide-wave mode transmission. In the case that a lower index (n0) dielectric layer is sandwiched by periodical multilayer stacks having alternative indices of n2-n1-n2-n1-n2-n1……, and n0< n1< n2, the confinement factor of the electromagnetic energy for TE modes are larger than that of TM modes [6,7], and this implies that both the number of reflection layers and the bending radius could be further reduced if TE modes is used as wave transmission instead of TM modes. In addition, according to the computed results of the omnidirectional reflection band of one-dimensional photonic crystals [8,9], the bandwidth of TE mode transmission is wider than that of TM mode one. Therefore, it can be expected that more compact waveguide structures and wider bandwidth can be achieved in TE-mode flexible waveguides (the flexible waveguides based on TE fundamental mode).
In this paper, the dispersion characteristics of TE-mode flexible waveguides with different structural parameters are studied. The transmission characteristics of their bends including 180° and randomly selected arbitrary angle bends and bend combinations are investigated, respectively. Finally, a comparison between TE-mode and TM-mode flexible waveguides is carried on, and some important conclusions are drawn.
2. Dispersion characteristics of TE-mode flexible waveguides
In the inset of Fig. 1, a schematic drawing of a flexible waveguide is shown, where a denotes the lattice constant of the one-dimensional photonic crystals (1D PCs); the refractive index for three materials are n1=1.6 (polystyrene), n2=4.6 (tellurium), n0=1 (air), and the corresponding layer’s thickness are of h1=0.75a, h2=0.25a, and h0. The rectangular area surrounded by the dotted lines in the inset is a schematic drawing of a super-cell set for the calculation of dispersion curves using plane wave expansion method [10]. It should be noticed that in the numerical simulation, 10 periodical reflection layers on each side of the defect layer should be taken. For various structural parameters h0=1.75a, 2.75a and 3.75a, the dispersion curves of TE mode are given in Fig. 1, respectively. The numerical results show that, for a frequency range located within the photonic band-gap (PBG) of the 1D PC for the case of normal incidence, 0.145–0.281 [c/a], where c is the speed of light in vacuum, when h0 is increased from 1.75a via 2.75a to 3.75a, and the zero-order even mode is gradually shifted downward; in the case of h0=1.75a or 2.75a, there is only zero-order even mode, and in case of h0=3.75a there is an additional first-order odd mode, while there is only a zero-order even mode in frequency range 0.152–0.257 [c/a]. In the paper, in the frequency range of omnidirectional reflection band for TE mode, 0.151–0.281 [c/a], which is marked in Figs. 1, the single-mode transmission characteristics of zero-order TE mode are focused on.
In Figs. 2 (a), (b), (c) and (d), the Ey field distributions at A, B, C and D points of the dispersion curves in Fig. 1 are shown respectively. It can be observed from Fig. 2, Ey at B (h0=2.75a) and C (h0=3.75a) points are totally confined within 4a on each side of the defect layers, respectively. Therefore, in following studies, h0 is set as 2.75a or 3.75a, and the width of reflection layers on each side (denoted as W) is set as 4a.
Fig. 1. The dispersion curves of TE mode flexible waveguides for different widths of defect layers (h0). The solid lines are that of even modes and the dotted line is that for the odd mode. The abscissa is of the normalized wavevector (unit: 2π/a), and the ordinate is of the normalized frequency (unit: c/a). A schematic drawing of the flexible waveguide is shown in the inset; a denotes the lattice constant of the 1D PC. n1=1.6, n2=4.6, n0=1, h1=0.75a, and h2=0.25a.
Fig. 2. (a), (b), (c) and (d). Ey field distributions for operation points A, B, C and D in four dispersion curves in Fig. 1, respectively.
3. Transmission characteristics of TE-mode flexible waveguide bends
As an example, a 180° arc bend of the TE-mode flexible waveguide with a bending radius of 7a is shown in the inset of Fig. 3. The transmission spectra of the waveguide bend are calculated by the finite-difference time-domain method [11], and the results for the case of h0=2.75a and 3.75a are shown in Fig. 3 and Fig. 4, respectively. It can be observed from Fig. 3 that for the case of h0=2.75a, the frequency range of high transmission (>0.985) is 0.208–0.275 [c/a], corresponding to a relative bandwidth of 27.74%, while the frequency range for high transmission (>0.985) is 0.173–0.248 [c/a], corresponding to a relative bandwidth of 35.62% according to Fig. 4, for the case of h0=3.75a, which is larger than that for the case of h0=2.75a. Therefore, in the following studies on the transmission spectra of arbitrary-angle waveguide bends, h0=3.75a is selected. The distributions of Ey field component in the 180° arc-bend waveguide with h0=3.75a at 0.21 [c/a] is given in the inset of Fig. 4.
Fig. 3. The transmission spectrum of 180° arc waveguide bend for the case of h0=2.75a. A schematic drawing of this waveguide bend is shown in the inset; R=7a, W=4a, and other parameters are the same as those in Fig. 1.
Fig. 4. The transmission spectrum of 180° arc waveguide bend for the case of h0=3.75a. R=7a, W=4a; other parameters are the same as those in Fig. 1. Ey field profile in this waveguide bend at 0.21 [c/a] is shown in the inset.
4. Transmission characteristics of TE-mode flexible waveguide bend combinations with arbitrary angle bends
Now the transmission characteristics of waveguide bend combinations of TE-mode flexible waveguides with random and arbitrary bending angles are considered. Not losing the generality, two types of complex structures are selected as follows: a Z-type and a 7-type waveguide bend structures. The former shown in the inset of Fig. 5 is composed of two 151.05° bends with R=7a, and its transmission spectrum is shown in Fig. 5; the transmission rate is higher than 0.985 in frequency range 0.187–0.245 [c/a], corresponding to a relative bandwidth of 26.83%. The latter shown in the inset of Fig. 6 is composed of a 27.82° bend and a 117.82° bend with R=7a, and its transmission spectrum is shown in Fig. 6; the transmission rate is higher than 0.985 in frequency range 0.188–0.246 [c/a], corresponding to a relative bandwidth of 26.72%. The distributions of Ey field component in the two structures at 0.21 [c/a] are given in Figs. 7(a) and (b), respectively. For randomly selected other waveguide bends, calculation results show that similar high transmission can also be achieved so long as R and W is no less than 7a and 4a, respectively.
5. Comparison between TE-mode and TM-mode flexible waveguides
A comparison between TE-mode and TM-mode flexible waveguides is listed in Table 1, where it can be seen that to achieve a high transmittance through arbitrary-angle bends, the required values of R and W in the TE-mode waveguide are only 27% and 40% of those in the TM-mode one, respectively. As an example, for the optical communication wavelength of 1.55 microns which is located to a relative value 0.25 [c/a], R is 10.43 microns for TM mode; while if 1.55 microns is located at 0.21 [c/a], R is 2.28 microns for TE mode, which is only 22.62% of the former. Therefore, in constructing compact and highly dense optical integrated devices and systems, TE-mode waveguides have more advantages than TM-mode ones. More detailed comparison between TE-mode and TM-mode flexible waveguides can be found in Table 1.
Fig. 5. The transmission spectrum of the Z-type waveguide bend combination composed of two 151.05° bends with R=7a. A schematic drawing of this waveguide combination is shown in the inset; other parameters are the same as those in Fig. 4.
Fig. 6. The transmission spectrum of the 7-type waveguide bend combination composed of a 27.82° bend and a 117.82° bend with R=7a. A schematic drawing of this waveguide structure is shown in the inset; other parameters are the same as those in Fig. 4.
Fig. 7. The Ey field profiles at normalized frequency 0.21 [c/a] (a) in the Z-type waveguide bend combination and (b) in the 7-type waveguide bend combination.
Table 1. A Comparison with TM-mode and TE-mode flexible waveguides
6. Conclusion
Through studies on the dispersion and transmission characteristics of TE mode waves in the presented flexible photonic crystal waveguide, its compactness and flexibility for arbitrary bends is confirmed. For arbitrary-angle waveguide bends, the bending radii can be reduced to within two wavelengths and only a not wide reflection layer width of 4a are required; these bending radii and reflection layer width are only to 27% and 40% of those of TM-mode flexible waveguides, respectively. The compact flexible waveguides in flattened slab form will be studied further, and it is expected to be applied to highly dense photonic integrated circuits.
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