Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals

Conventional plane-wave expansion (PWE) methods, which are good for calculating such properties as photonic band gaps for materials with periodic structure, are very difficult for calculating crystals with an interface. While the dispersion relation used by PWE does not restrict the wave vectors, k, to be real, the complex k are important for interface calculations. Therefore, we extended the PWE to make it possible to easily calculate the complex k both in the two-dimensional (2D) isotropic and the general three-dimensional (3D) anisotropic cases. The advantages gained include (i) evanescent modes are obtained naturally, and EPWE provides enough information for matching boundary conditions in interface problems; (ii) 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; (iii) since EPWE is extended from the PWE, it obeys the same dispersion relation, and both results will also be the same, provided PWE employs the k derived from EPWE; and (iv) because the imaginary part of k is associated with the reciprocal of the penetration depth, the shortest width of the crystal is well-defined when it is treated as a single crystal. As an illustration, we present results for 3D isotropic GaAs crossed square prisms and find a good correspondence between the results of both methods. Further, we demonstrate why the evanescent mode is one kind of Bloch mode which does not conflict with Bloch's theorem.