Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

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 χ⁽²⁾ nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective χ⁽³⁾ 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.