Metamaterials and the homogenization of composite materials

Metamaterials and photonic crystals are often distinguished by the requirement that the former have features fine compared with the wavelength, and thus be homogenizable. We discuss the assumption implicit in this: that the metamaterial microstructure have well defined effective dielectric constant and magnetic permeability (concentrating on the former case). We illustrate subtleties in the calculation of effective dielectric constant in the particular case of square prisms composed of dielectric arranged in a square array in a host medium of another dielectric constant, commenting on poles, zeros, essential singularities and branch cuts of the effective dielectric constant function.