Artificial Ferro-Magnetic Anisotropy: Homogenization of 3D Finite Photonic Crystals

Abstract

In this paper, we attempt to replace heterogeneous ferro-magnetic photonic crystals by homogeneous structures with anisotropic matrices of permittivity and permeability, both deduced from the resolution of annex problems of electrostatic type on a periodic cell. The asymptotic analysis relies on the multi-scale method which is a tool in the theory of homogenization with rapidly oscillating coefficients [2]. We note a singular perturbation for the divergence of the electromagnetic field associated to scaled permittivity ε(x/η) and permeability μ(x/η), which are periodic functions of period η≪1. We establish the sharp convergence of the oscillating field towards the homogenized one via the notion of two-scale convergence [1]. We finally solve numerically the associated system of partial differential equations with a Finite Element Method in order to exhibit the matrices of effective permittivity and permeability for given 2D ferro-magnetic periodic composites.