  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.
