Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal

We propose a modified plane-wave expansion method to calculate transmittivity and reflectivity of a semi-infinite photonic crystal (PC) with interface. This method is based on an expanded completeness basis, including both the propagation and evanescence modes. We use this approach to deal with two kinds of problems: one is to determine the normal direction of the largest attenuation strength for a semi-infinite PC in the gap frequencies; the other is to calculate the transmittivity and reflectivity of a PC slab. To demonstrate the extensive utilization of our approach, we revisit the same system as studied by K. Sakoda [Phys. Rev. B 52, 8992 (1995)] and find that our results are in good agreement with ones obtained by Sakoda's paper.