  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.
