Homogenization of three-dimensional finite photonic crystals - Abstract

Using a multiscale expansion technique applied to a scattering problem, we prove that a three-dimensional finite crystal (periodic structure) behaves as if it were homogeneous, when the period becomes very small in regard with a fixed wavelength. We show that the homogeneous medium is actually anisotropic, and we give the expression of its permittivity. More precisely, we derive the tensor of permittivity from the calculus of three scalar periodic potentials, solutions of a partial differential equation of electrostatic type. Finally, we pay a special attention to the influence of the boundary of the whole crystal on the diffracted field.