Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media

Fourier analysis has been successfully applied to study optical properties of photonic crystal structures, usually composed of optically isotropic media. In a commonly used formulation [D. M. Whittaker and I. S. Culshaw, Phys. Rev. B 60, 2610 (1999)], inversion symmetry of the unit cell is required. Here, we extend the treatment of Whittaker and Culshaw to structures with asymmetric unit cells that can be composed of birefringent media. As applications we consider a high-refractive-index membrane with a triangular lattice of triangular holes, where the presence of a TE-like gap at $\ensuremath{\omega}$ and of a TM-like gap at $2\ensuremath{\omega}$ is established, and a slot waveguide made of (birefringent) porous silicon, where coupling of guided modes to radiative modes is achieved through a one-dimensional periodic grating.