From photonic crystals to metamaterials: the bianisotropic response

Abstract

Metamaterials are, in general, characterized by the bianisotropic response that relates the displacement vector D and the magnetic induction B to the electric and magnetic fields E and H via the permittivity, permeability and magnetoelectric dyadics, respectively, , and . For the first time, we derived these dyadics with great generality with the photonic crystal (PC) description as the starting point. The PC can have one- (1D), two- (2D) or three-dimensional (3D) periodicity with an arbitrary Bravais lattice and arbitrary shape of the inclusions in the unit cell, and these inclusions can be dielectric or metallic. Moreover, unlike most theories of homogenization, our theory is, in principle, applicable to band-pass or optical bands, as well as to low-pass or acoustic bands. The generalized conductivity is assumed to be given at every point in the unit cell, and we relate , and analytically to . The long-wavelength limit having been taken, these dyadics depend only on the direction of the Bloch wave vector k and not on its magnitude. In the case of inversion symmetry, the bianisotropic response simplifies to and . Applying our theory to a 2D array of metallic wires, we find that the response is paramagnetic or diamagnetic, depending on whether it is the electric or the magnetic field that is parallel to the wires. For a 3D array of mutually perpendicular wires (3D crosses), the behavior is found to change from plasma-like to free-photon-like when the wires are severed, leading to a spectral region where ε and μ are both negative. Finally, our theory confirms several well-known results for particular PCs and inclusions.