Scattering-method calculation of propagation modes in two-dimensional photonic crystals of finite thickness

We present a scattering-method calculation of electromagnetic wave propagation modes inside dielectric waveguides of finite thickness. The algorithm is based on an expansion into propagation eigenmodes within the layer with appropriate wave matching at the boundaries. It is applicable to waveguide structures made of layers of dielectric material where the structure of each layer is periodic in two dimensions but is translationally invariant in the third dimension. The method is useful for extracting such information as the effect of the finite thickness on the band structure, the influence of cladding layers and the substrate on dispersion, and the escape rates of radiative modes. As a demonstration, we present the dispersion relations for an ${\mathrm{Al}}_{0.1}{\mathrm{Ga}}_{0.9}\mathrm{As}$ photonic-crystal core layer with a triangular lattice of cylindrical perforations, either placed on top of a uniform $\alpha -{\mathrm{Al}}_2{\mathrm{O}}_3$ substrate or suspended freely in the air.