Two-dimensional vector-coupled-mode theory for textured planar waveguides

We develop a model to treat coupling between guided modes in planar dielectric waveguides that have been textured in two dimensions with a thin surface grating. The formulation is based on a general Green's-function technique that self-consistently determines the field in the surface grating due to the polarization there. With simplifying approximations, this formalism is cast into a two-dimensional (2D) vector-coupled-mode theory that is more computationally efficient, and that gives considerable insight into the nature of mode coupling in 2D textured structures. These models are applied, by way of example, to illustrate some interesting properties of leaky and bound modes that are coupled together by 2D periodic texture. In particular we discuss the complex photonic band structure describing the dispersion, lifetimes, and polarization properties of the resonant states associated with the textured waveguide. In our analysis we emphasize the fundamental differences between coupling in 2D textured waveguides and infinite 2D photonic crystals. We also show that the vector-coupled-mode theory agrees well with the self-consistent formulation.