The Optimization Of Band Gaps And Waveguide Bends In Two-dimensional Photonic Crystals With Finite Element Methods

In this paper,we consider the optimal design of two-dimensional photonic crystal structures,which includes a geometry optimization framework for maximizing band gaps and a bend optimization simulation for the low-loss and wide-band waveguide bends in planar two-dimensional photonic crystal structures.For each problem,the mathematical model,the computational algorithm,the optimization framework and the simulation are discussed mainly by using the finite element methods and the physical simulation software COMSOL Multiphysics.Photonic crystals have attracted a great deal of attention due to its periodically changing index of the refractive between high index region and low index region on the wavelength scale.The periodicity of the dielectric constant exhibits frequency intervals(i.e.,band gaps)in which propagating Bloch waves do not exist,and offers them unique specifications which make photonic crystals suitable for design of optical waveguide bends,cavities,mirrors,resonators,etc.The performance of photonic devices can depend strongly on their geometry.So,we first investigate the optimal design of photonic crystal structures for two-dimensional square and triangular lattice.The mathematical formulation of the band gaps optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem,which can be parameterized by the wave vector and the dielectric material.The original eigenvalue problems for computation of the band structures is discretized by using the finite element methods into a series of finite-dimensional eigenvalue problems for multiple values of the wave vector parameter.For the structures with cylinders in square or triangular lattice photonic crystals,we consider the optimization of the fill factor to max the band gaps,which numerically solved by using the finite element method and the software COMSOL Multiphysics.Since the high transmission efficiency through a sharp bend is possible in a photonic crystal waveguide,our work focuses on the Design optimization of a low-loss and wide-band sharp waveguide bend in two-dimensional photonic crystals.In this paper,for two dimensional photonic crystals involving in finite cylinders on triangle-lattice,a 120°waveguide bend and a 60° waveguide bend with low-loss and wide-band are obtained for single-mode operation.The optimal process can be divided into two steps:first,a conventional waveguide bend can be introduced by maximum photonic band gap of photonic crystals;then further optimization involves shifting the position,modifying the radii of few air holes or the geometry of few air holes.An optimization problem at a given frequency or over a frequency range needs to be solved,which depends on the field solutions obtained by using finite element methods and the optimization of photonic band gap obtained by using plane wave expansion method.With the proposed optimal method,the optimization results for 120° and 60° waveguide bends on a triangle-lattice photonic crystal defined as air holes on a Silicon background show that an obvious low-loss transmission at wavelength 1550nm can be observed and the maximum value of objective function is able to be rapidly obtained.