Speed of light in a 2D photonic crystal in the low-frequency limit

We consider an arbitrary 2D periodic structure in the low-frequency limit and demonstrate that it exhibits birefringence for propagation in the plane of periodicity. For polarization parallel to the cylinder axis the effective dielectric constant, eff is just the average of the constituents, as known. On the other hand, for in-plane polarization, eff is a function of the propagation direction. Numerical calculations have been performed for triangular cylinders forming a square lattice. This leads to an anisotropy as large as 20%, much in excess of anisotropy observed in natural crystals.