Homogenization of photonic crystals

We consider the long-wavelength limit for two-dimensional photonic crystals - periodic arrangement of dielectric rods with dielectric constant /spl epsi//sub a/ embedded in a dielectric background (/spl epsi//sub b/). Using the Fourier expansion method in the low-frequency limit we develop an effective medium theory and give a rigorous proof that, in this limit, a periodic medium behaves like a homogeneous one. We derive compact analytical formulas for the effective dielectric constants of a 2D photonic crystal. These formulas are very general, namely the Bravais lattice, the cross-sectional form of cylinders, their filling fractions and the dielectric constants are all arbitrary. So is the direction of propagation of the Bloch wave - out-of-plane in general, with special attention paid to the limiting case of high dielectric contrast between the constituents of the photonic crystal.