Coupling between plane waves and Bloch waves in photonic crystals with negative refraction

A comprehensive analysis of the coupling coefficients between plane waves in conventional dielectric media and Bloch waves of photonic crystals with effective negative refractions is performed by the layer Korringa-Kohn-Rostoker method. Employing the infinite layers refraction operator, semi-infinite size photonic crystals are considered. Some special coupling properties are discussed. In particular, the strong angular dependence of coupling coefficients is found even for an interface between air $(n=1)$ and a photonic crystal with effective refractive index $(n_{\mathrm{eff}}=-1)$. Thus, a negative refractive index defined by the radius of a circular equal-frequency contour does not guarantee an isotropic behavior.