Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability

We consider a heterogeneous magneto-dielectric photonic crystal and derive the so-called ‘homogenized Maxwell system’ via the multi-scale method and provide ad hoc proofs for the convergence of the electromagnetic field towards the homogeneous one using the notion of two-scale convergence. The homogenized medium is described by anisotropic matrices of permittivity and permeability, deduced from the resolution of two annex problems of electrostatic type on a periodic cell. Noteworthily, this asymptotic analysis also covers the case of photonic crystals with non-cuboidal periodic cells. We solve numerically the associated system of partial differential equations with a method of fictitious charges and a finite element method (FEM) in order to exhibit the matrices of effective permittivity and permeability for given magneto-dielectric periodic composites. We then compare our results in the 2D case against some Fourier expansion approach and provide duality relations in the case of magneto-dielectric checkerboards. We further compute some low-frequency eigenmodes of a photonic crystal fiber with metallic outer boundary and compare them with the eigenmodes of a corresponding effective anisotropic waveguide, thanks to the FEM. Finally, we derive the effective properties of a 3D photonic crystal both through classical homogenization (solving numerically two decoupled annex problems) and Bloch wave homogenization. In the case of spherical inclusions, the latter approach amounts to evaluating the slope of the first band around the origin on a Bloch diagram which we compute using finite edge elements.