We extend a recently developed Hamiltonian formalism for nonlinear wave interaction processes in spatially periodic dielectric structures to the far-off-resonant regime and investigate numerically the three-wave resonance conditions in a one-dimensional optical medium with ${\chi }^{\left(2\right)}$ nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective ${\chi }^{\left(3\right)}$ nonlinear response in these systems. We obtain the corresponding coupling coefficients through appropriate normal form transformations that formally lead to the Zakharov equation for spatially periodic optical media.

• Received 27 July 2007

DOI:https://doi.org/10.1103/PhysRevA.77.023809