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

TL;DR

This paper extends Hamiltonian formalism to analyze far-off-resonant three-wave interactions in 1D photonic crystals with quadratic nonlinearity, revealing cascading effects that produce an effective cubic nonlinearity.

Contribution

It introduces a Hamiltonian approach for far-off-resonant wave interactions in periodic media and demonstrates cascading leading to effective third-order nonlinear response.

Findings

Cascading of nonresonant interactions generates effective  nonlinear response.

Normal form transformations derive coupling coefficients for Zakharov equation.

Numerical analysis confirms the feasibility of controlling wave interactions in photonic crystals.

Abstract

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 $\chi^{(2)}$ nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective $\chi^{(3)}$ 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.