Rotational Invariance of the Effective Refractive Index in Photonic Crystals

The rotational invariance of the effective refractive index in the 2-D photonic crystal is discussed. It is proven that, in the case of the n -fold symmetrical ω-k equifrequency surface (EFS), there are two solutions that show the invariance of the effective refractive index after rotating a multiple of the basic repeated angle 2 π/n within the first Brillouin zone. For the noncircular solution, the behavior of the effective refractive index N_eff(θ) at normal incidence shows that N_eff(0^°) has an additional constant of two in the very high symmetric EFS, higher than in the low symmetric case.