Degenerate four-wave mixing from a waveguide with guided wave pump beams

We have analyzed the problem of degenerate four-wave mixing in which a probe beam is incident from above onto a thin-film waveguide pumped by two oppositely propagating guided waves. Four-wave mixing signals are obtained both on reflection from the film surface and on transmission into the substrate supporting the film. Analytical formulae are evaluated for a number of different cases which compare nonlinearities in the film versus the substrate.