Coupled-mode theory for light propagation through deep nonlinear gratings

In ordinary coupled-mode theory, the standard tool to analyze optical grating structures in both Linear and nonlinear regimes, the grating is usually assumed to be shallow. Here we generalize this theory in a systematic way to include deep gratings. We do so by expanding in the exact eigenfunctions of the linear structure (the Bloch functions) rather than simply in the forward and backward propagating modes. We show that the resulting equations for deep gratings are qualitatively similar to those for shallow ones, except that the value of some of the coefficients is different and that some additional nonlinear terms arise. We also discuss solutions to these equations and point out differences from solutions of the conventional theory.