Coupled-mode equations for quadratically nonlinear deep gratings

Using the method of multiple scales, we present a systematic derivation of the coupled-mode equations for the propagation of light in quadratically nonlinear deep gratings. We show that the resulting coupled-mode equations for deep gratings differ from those for shallow gratings in that the coefficients have different values, and that additional nonlinear terms appear. Our equations reduce to well known results, in the appropriate shallow grating limit.