Without making the nondepleted pump approximation, we solve the problem of four-wave mixing (FWM) in a nondissipative ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ medium. Instead of describing the dynamics of FWM in terms of coupled wave amplitudes, we base our solution on canonical equations that describe the propagation of the field’s intensities. This structure clearly identifies the conservative exchange of energy in the FWM process. Consequently, analysis of the FWM process is reduced to a single propagation equation that describes the energy exchange between the pump and amplified waves. This equation yields an elliptic integral solution. The conversion efficiency reduces to the simple analysis of a fourth-order polynomial, which we analyze to determine the conditions for optimization. The destructive influence of the optical Kerr effect on the phase-matching condition is shown to be eliminated by proper choice of nonzero initial wave-vector mismatch, dependent on the input intensity. Furthermore, we show that the conversion efficiency for pulses is limited by the splitting of the pulses into a set of subpulses.

• Received 23 January 1991

DOI:https://doi.org/10.1103/PhysRevA.44.6036