Phase conjugation by cascaded second-order nonlinear-optical processes

The process of degenerate four-wave mixing in a material with a nonzero second-order susceptibility is theoretically investigated. The contributions by cascaded second-order processes to the overall third-order response are calculated by solution of the coupled-wave equations, assuming plane input fields. The driven coupling fields at frequencies 2ω = ω + ω and 0 = ω − ω, created by the second-order polarization, are very far from phase matching and are therefore calculated with proper boundary conditions without the use of the slowly varying envelope approximation. As a consequence of the large phase mismatch of the generated second-harmonic wave, the dominant pathway for cascading in degenerate four-wave mixing is via the generated dc field. With LiNbO3, KNbO3, and KTP as explicit numerical examples it is shown that cascaded χ(2) processes can make a substantial contribution to the phase-conjugate signal in a material without inversion symmetry.