The Extended Plane Wave Expansion Method for 3D Anisotropic Photonic Crystals\footnote{Supported by NSF(through the Northwestern MRC)}

Conventional plane-wave expansion (PWE) methods, good for calculating such properties as photonic band gaps for materials with periodic structure, are very difficult for calculating a crystal with an interface. While the dispersion relation used by PWE does not restrict the wave vectors, {\bf k}, to be real, the complex {\bf k} are the important for interface calculations. Therefore, we modified the PWE to make it possible to easily calculate the complex {\bf k} (EPWE) both in the 2D isotropic and the general 3D anisotropic cases. Advantages gained include: (i) the frequency is initially given and regarded as a known variable, rather than as an argument, and can always be set to be a positive real number even for complex systems with real, imaginary, or complex frequency-dependent permittivity or permeability; (ii) from the complex {\bf k} results, the resonant feature of the transmittivity can be easily analyzed; (iii) since EPWE is extended from the PWE, it obeys the same dispersion relation, and their results will also be the same, when providing PWE the {\bf k} derived from EPWE; (iv) because the imaginary part of {\bf k} is associated with the reciprocal of the penetration depth, the shortest width of the crystal when it is treated as a single crystal is well-defined. As an example, we present 3D isotropic GaAs crossed square prisms and find a good correspondence between results of both methods.