The Extended Plane Wave Expansion Method in Three Dimensional Anisotropic Photonic Crystal

In this paper, we extend the conventional plane wave expansion method in 3D anisotropic photonic crystal to be able to calculate the complex $\mathbf{k}$ even if permittivity and permeability are complex numbers or the functions of $\omega$. There are some tricks in the derivation process, so we show the process in detail. Besides, we also provide an example for testing and explaining, and we also compare the results with the band structure derived from conventional plane wave expansion method, then we finally find that there is a good consistency between them.