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
$\omega-k$
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
$\pi/n$
within the first Brillouin zone. For the noncircular solution, the behavior of the effective refractive index
$N_{\rm eff}(\theta)$
at normal incidence shows that
$N_{\rm eff}(0^{\circ})$
has an additional constant of two in the very high symmetric EFS, higher than in the low symmetric case.