This work discusses a robust genetic-algorithm based hybrid optimization scheme which was applied to handle the transmission problem in planar photonic crystal waveguide bend. For test purpose, two objective functions were proposed to characterize the transmission property of bend working at a special frequency or even over an entire frequency range. Optimization results for different 90° and 120° sharp bends have shown that, with the assistant of genetic-algorithm, random sizes of the cylinders surrounding the bend corner which correspond the maximum value of test functions are able to be rapidly obtained and obvious improvement of transmission property can be observed from each optimal bend structure, especially remarkable for the case of 120-I type bend.
Photonic crystals (PCs) [1] and [2] are metamaterials based on a strong periodic modulation of the refractive index within the scale of the wavelength in one, two, or three dimensions. Due to the existence of photonic band gaps in PCs, the flow of photons can be controlled and the properties of light can be dramatically changed when it travels inside the PCs. One of the most fascinating applications of PCs is their ability to guide electromagnetic waves in narrow waveguides created by a sequence of line defects, including light propagation through extremely sharp waveguide bends with ultra-low loss transmission. This application is so important and is believed to be most promising for building low cost, highly functional optical chips where bending light path with arbitrary angles with minimum loss is highly desired [3]. In contrast to full three-dimensional (3D) PCs, two-dimensional (2D) planar PC waveguide bends are relatively easy to fabricate with conventional semiconductor processing technology. Recent studies addressed the issue of an improved design of sharp waveguide bends in 2D PCs and suggested that the transmission loss is very sensitive to the property of the band corner. Indeed, the physical mechanism which results in the transmission loss is mainly due to the impendence mismatch between the guiding mode propagating in the straight waveguide and that in bend corner. In other words, this means that if the impendence of old mode is in agreement with the new corner mode at the same frequency, there will be a coupling effect between them with no transmission loss taken place in the bend corner, or vice versa. In generally, only one of the old modes is allowed to propagate with no coupling loss, while the remaining ones are with different degrees of transmission loss or even under cut-off.
As a result, in order to improve the transmission performance in sharp waveguide bends based on 2D planar PCs, most of the designers have focused their attentions on the modification of the structure in the vicinity of the bend corner. The typical strategies include changing the refractive indexes [4], sizes [5] and [6], positions and lattice constant [7] of dielectric rods surrounding the corner, and removing or adding new rods [8], [9], [10] and [11] in the vicinity of the corner to form super defect. Unfortunately, to the present time, there is still no analytical way reported to reveal the relationships between bend corner structure and the best transmission performance. This will bring us a tough and time-consuming task to optimize all possible structure parameters in the bend corner concerning the use of a traditional step-by-step analytical routine which is only based on many usual numerical techniques, such as the finite-difference time-domain (FDTD) technique, the Green's tensor approach (GTA), the finite element method (FEM), and the multiple-scattering technique (MST).
To overcome this drawback, recently, some researchers made efforts to the use of inverse-design routine to fast searching the best design of bend corner with the help of effective global optimization tools, like topology optimization technique and genetic optimization technique. For instances, in many previous works [12], [13], [14] and [15], topology optimization technique was successfully applied in the searching of several wide-band and flat-bandwidth 2D PC waveguide bend designs and improved transmission performance in the corresponding experiments was observed, even though the final optimal geometry of bend corner is more complex and difficult for fabrication than those using traditional analytical method. In Ref. [16], the genetic algorithm (GA) in conjunction with the MST technique was firstly used in the optimization of several 2D PC waveguide bends and numerical results have verified the effectiveness of such new method. In this study, we will continue to extend the GA based inverse-design routine to the optimization of 2D PC sharp waveguide bends for not only reducing the transmission loss at a single frequency [16] but also improving the transmission performance over a wide band. More importantly, we only consider modifying the sizes of rods surrounding the corner, from a practical point of view, which will make the designs relatively simple for further fabrication.
In summary, a fast hybrid algorithm which bases on a genetic approach and the FDTD method has been applied to the problem of enhancing the transmission property through sharp bends in photonic crystal. Numerical results for four representative sharp bends have shown that, with the help of GA-assistant design technique, the improvement of transmission property can be obtained from bends working on either a fixed frequency or entire frequency range of fundamental defect waveguide mode. In particular, such improvement is quite remarkable for bend 120-I with the modification of radius of cylinders located around bend corner. It is believed that such combined optimization routine can be further easily extended to the design of low-loss air-hole type photonic crystal waveguide bends and other similar devices in the framework of photonic crystals for potential applications in high-density photonic circuits.
