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Complex Fourier factorization techniques in calculation of modes of discontinuous optical structures with an arbitrary cross-section

We demonstrate an improvement of the plane wave expansion method treating two-dimensional photonic crystals by applying Fourier factorization with generally elliptic polarization bases. By studying one example 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.