A 2D Pseudo-Spectral Approach of Photonic Crystal Slabs

Abstract

We consider an L1 periodic dielectric slab which is characterized by the dielectric function a 2D model for photonic crystals. We assume that there is no variation in ydirection, with fields varying time-harmonically according to exp j ωt . In order to solve electromagnetic wave propagation in such structures, we diagonalize the Maxwell’s equations with respect to the z coordinate. As demonstrated in this paper, diagonalized forms greatly facilitate the implementation of the finite difference method. The L1 periodicity of the fields suggests expansions in terms of spatially harmonic functions. However, contrary to the commonlyused Bloch inhomogeneous plane waves, we utilize expansions of the form. For the determination of the coefficient functions ψn z we employ a sophisticated, yet, easy-to-apply implementation of a finite difference discretization scheme in the zdirection which permits virtually arbitrary L1 periodic profile functions. It will be demonstrated that the proposed hybridization of the plane-wave decomposition and the finite difference method leads to a robust and flexible method of analysis with a wide range of applications. As an example, we consider TE-polarized electromagnetic waves which propagate in the assumed dielectric slab along the x axis.