Homogenization of a Wire Photonic Crystal: The Case of Small Volume Fraction

We consider the diffraction of a monochromatic incident electromagnetic field by a bounded obstacle made of parallel high conducting metallic fibers of finite length which are periodically disposed. Our goal is to study the asymptotics of this three-dimensional Maxwell problem as the period, the thickness of the rods, and the resistivity are simultaneously small. We solve this problem in the case where the filling ratio of fibers vanishes but their capacity remains positive. We find a limit volumic density of current which results in a nonlocal constitutive relation between the electric and displacement fields. This extends previous results obtained in the polarized case where a two-dimensional effective local equation was found with a possibly negative effective permittivity.