  We demonstrate an enhancement of the plane wave expansion method treating
two-dimensional photonic crystals by applying Fourier factorization with
generally elliptic polarization bases. By studying three examples of
periodically arranged cylindrical elements, we compare our approach to the
classical Ho method in which the permittivity function is simply expanded
without changing coordinates, and to the normal vector method using a
normal-tangential polarization transform. The compared calculations clearly
show that our approach yields the best convergence properties owing to the
complete continuity of our distribution of polarization bases. The presented
methodology enables us to study more general systems such as periodic elements
with an arbitrary cross-section or devices such as photonic crystal waveguides.
