Rigorous derivation of coupled mode equations for short, high-intensity grating-coupled, co-propagating pulses

We present a rigorous derivation of coupled mode equations for the grating-assisted coupling of short, co-propagating pulses. Since this is a co-propagating geometry, the required grating period is much longer than the wavelength of the light and is typically 0.1-10 mm. This can exceed the spatial extent of the pulses substantially, and the validity of the usual assumptions that lead to coupled mode theory is therefore not guaranteed. Here we show rigorously that the standard nonlinear coupled mode equations can be used to describe these experiments and we give expressions for the parameters.