Rotational Invariance of the Effective Refractive Index in Photonic Crystals

Abstract

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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex> </formula> -fold symmetrical <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\omega-k$</tex></formula> 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\pi/n$</tex></formula> within the first Brillouin zone. For the noncircular solution, the behavior of the effective refractive index <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$N_{\rm eff}(\theta)$</tex></formula> at normal incidence shows that <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$N_{\rm eff}(0^{\circ})$</tex></formula> has an additional constant of two in the very high symmetric EFS, higher than in the low symmetric case.