From photonic crystals (via homogenization) to metamaterials

A very general mean-field theory is presented for a photonic crystal (either dielectric or metallo-dielectric) with arbitrary 3D Bravais lattice and arbitrary shape of the inclusions within the unit cell. The material properties are described by using a generalized conductivity at every point in the unit cell. After averaging over many unit cells for small Bloch wave vectors in comparison with the inverse of the lattice constant, we have derived the macroscopic response for the artificially structured material. In the most general case, such a response turns out to be bi-anisotropic, having terms associated with the permittivity, and permeability, and magnetoelectric tensors. We have derived explicit expressions for the four tensors in terms of the geometry and material parameters of the inclusions. Nevertheless, for a photonic crystal with inversion symmetry the magnetoelectric tensors in the bi-anisotropic constitutive relation vanish. In addition, we have verified that for cubic symmetry the system becomes bi-isotropic, being characterized by two frequency-dependent scalars, namely the permittivity and permeability. It is very important that, in general, the permittivity and permeability tensors are diagonal in different reference systems. The principal axes of the permeability tensor (unlike those of the permittivity tensor) depend on the direction of the wave vector. This necessitates the development of a new Crystal Optics for anisotropic photonic metamaterials.