Validity of the effective-medium approximation of photonic crystals

A definition of the effective permittivity and permeability is proposed for two-dimensional photonic crystals. Expression for these effective parameters is deduced from the combination of the dispersion law with the formula for the reflection coefficient of a semi-infinite photonic crystal in the single-mode approximation. From these expressions, it is shown that the effective permittivity and permeability take purely real values when the truncation plane of the photonic crystal contains an inversion center of the infinite crystal. In addition, these effective parameters are demonstrated to be continuous and bounded when the crystal mode turns from propagating to evanescent only if the eigenmode fields fulfill specific symmetry conditions. Finally, the validity of the single-mode approximation and the effective-medium model of photonic crystals is investigated. In particular, arguments suggesting that this model fails for bands exhibiting negative refraction are presented.