A novel view of plane wave expansion method in photonic crystals

We propose a method derived from the simple plane wave expansion that can easily solve the interface problem between vacuum and a semi-infinite photonic crystal. The method is designed to find the complete set of all the eigenfunctions, propagating or evanescent, of the translation operators $\{{\bf T_R} \}$, at a fixed frequency. With these eigenfunctions and their eigenvalues, the transmitted and reflected waves can be determined. Two kinds of applications are presented for 2D photonic crystals. The first is a selection rule for determine the normal direction of the vacuum-photonic crystal interface to achieve the highest attenuation effect at a gap frequency. The second is to calculate the transmittance and reflectance for a light incident from vacuum to an semi-infinite photonic crystal. As an example we recalculate a system studied previously by K. Sakoda et al. and get results in agreement with theirs.