A Novel Numerical Approach for the Analysis of 2D Photonic Crystals: The Cell Method

Abstract

In the study of the properties of photonic band gap materials is important to find if some bands of frequencies exist (photonic band gap) where the propagation of the electromagnetic waves is prohibited in all the directions. In order to find these frequencies, that is to solve an eigenvalues problem, a lot of methods have been proposed up to now. The Plane Wave Expansion Method (PWE) [1] works with inhomogeneous media having curved shapes and deals with large not sparse hermitian matrices. The Finite Elements Method (FEM) works with inhomogeneous media having curved shapes as well and a generalized eigenvalues problem with large sparse (quasi band diagonal) matrices has to be solved. The Finite Difference Method (FD) works with inhomogeneous media without curved shapes and deals with large sparse (quasi band diagonal) hermitian matrices and, for the structure of these matrices, the eigenvalues problem can be solved in a very efficient way [2]. The Cell Method (CM) [3], with the microcell interpolation scheme [4], works with inhomogeneous media having curved shapes, like FEM, and deals with large sparse (quasi band diagonal) hermitian matrices in non generalized eigenvalues problems, like FD. Moreover, by means of an appropriate numbering of the nodes, it is possible to solve the eigenvalues problem in the same efficient way like FD.