Amplitude-equation formalism for four-wave-mixing geometry with transmission gratings

An amplitude equation is derived for a four-wave-mixing geometry with nearly counterpropagating, mutually incoherent, nondiffracting pump beams, spatially overlapping in a photorefractive material with a nonlocal response. This equation extends the earlier linear two-dimensional theory to the weakly nonlinear regime. The analysis also starts from a more complete equation for the photorefractive effect, which leads to the prediction of novel effects especially apparent in the nonlinear regime. Precise predictions for the spatiotemporal behavior of the grating amplitude in the nonlinear regime are presented. The range of validity of the amplitude equation is studied. The characteristics of the instability in the nonlinear regime are analyzed through a front-selection analysis