Coupling between the signal and idler paraxial differential equations is solved for a diffracted undepleted pump. The solutions are inherently numerical. However, we present one analytic solution, corresponding to negligible diffraction, which agrees with the simulation. Specifically, on input we consider : a sinusodial pump amplitude grating; a plane wave for the signal; and a small amount of energy in the idler.
P. R. Peterson,
"Modeling Optical Parametric Amplifiers", Proc. SPIE 1045, Modeling and Simulation of Laser Systems, (31 May 1989); doi: 10.1117/12.951311; https://doi.org/10.1117/12.951311