Photonic band structure in one-dimensional nonlinear crystal: Analysis of harmonic stability

We consider a one-dimensional nonlinear photonic crystal consisting of an infinite set of concentrated equidistant scatterers inserted in a linear dielectric medium. Each of the scatterers is made by a very thin layer of a nonlinear medium with high refractive index that we model by a delta function. We show that the nonlinear optical exact solutions of this system form an intensity dependent band structure. To analyze the stability of these solutions we consider a modulation harmonic perturbation of these solutions whose amplitudes are slightly above the instability threshold. We demonstrate that the nonlinearity gives rise to an oscillatory instability of the solutions, which is a localized version of the well-known modulational instability of the nonlinear Schrodinger equation. We show that the linear harmonic perturbation forms as well a band structure whose allowed bands coincide for some intervals with those of the nonlinear band structure of the solutions for which case the structures are unstable whereas in the region where both the linear and nonlinear bands do not coincide, the nonlinear waves are indeed stable so that they conform spatial solitons.