Propagation analysis of forward degenerate four-wave mixing

We discuss the propagation problem associated with the forward four-wave degenerate mixing induced by the third-order non-linear susceptibility
\chi_{(-\omega; \omega, \omega, -\omega)}^{(3)} = \chi_{1}^{(3)} + i\chi_{2}^{(3)}
. Exact analytical solutions are presented of two-photon resonances, evidencing the influence of the Kerr effect on the efficiency of the process. Around a two-photon resonance, we solve the problem of the nearly parallel propagation of two beams undergoing auto and induced two-photon absorption (TPA). The resonant interaction is studied in detail and analytical solutions are provided with operational accuracy for current experimental conditions. We show by the way that in the two-beam two-photon experiments, when the phase matching condition is fulfilled, although TPA is governed by
\chi_{2}^{(3)}
, the real part of
\chi^{(3)}
must be considered in a substantial way. All the above solutions are put in a form simple enough that the numerical computations can be performed on a programmable calculator.