Mutual action of self-phase modulation and cross-phase modulation on the parametric four-photon mixing. Exact analytical solutions in the form of Jacobi functions

The purpose of this work is to investigate the influence of the self-phase modulation and cross-phase modulation on the parametric interaction between pump signal and idler at few coherent lengths. The coherent length of four-photon mixing process can be quite long, up to 500 m to 1 km, if the spectral separation between the signal waves is relatively small. The problem with generation of new frequencies on distances less than one coherent length in parametric four-photon processes, without including the Raman scatter was solved only in approximation for a fixed electric field or fixed intensity of the pump wave. The overall task which includes the change in intensity and the phase of the pump wave has not been studied in detail in the literature up to now. The main purpose of the present work is to solve the more general problem: the mutual action of all chi((3)) nonlinear processes on the parametric four-photon mixing.