Analysis of four wave mixing in slab waveguide under the undepleted pump approximation

Abstract

The four-wave mixing in a slab waveguide is investigated theoretically by using the singular perturbation technique with multiple space scales. In order to apply the perturbation technique, we first introduce the perturbation parameter concerned with the nonlinear coefficients and the multiple space scales in the propagation direction. Substituting the expansion of the field and the multiple space scales into the Maxwell’s equation, we can get the governing equations to each order of the perturbation. From the solvability conditions to have nontrivial solutions for each perturbation solutions, it can be shown that the coupled-mode equations for four waves that are asymptotically correct in the limit of weak nonlinearity are obtained. The dependence of the coupling coefficient on the waveguide width and the field distribution of the phase conjugate wave are discussed numerically. We show that the coupling coefficients have the maximum for the optimum waveguide width.