Nonlinear photonic crystals: IV. Nonlinear Schrodinger equation regime

We study here the nonlinear Schrodinger (NLS) equation as the first term in a sequence of approximations for an electromagnetic (EM) wave propagating according to the nonlinear Maxwell (NLMs) equations. The dielectric medium is assumed to be periodic, with a cubic nonlinearity, and with its linear background possessing inversion symmetric dispersion relations. The medium is excited by a current J producing an EM wave. The wave nonlinear evolution is analysed based on the modal decomposition and an expansion of the exact solution to the NLM into an asymptotic series with respect to three small parameters alpha, beta and rho. These parameters are introduced through the excitation current J to scale, respectively (i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the range of wavevectors involved in its modal composition, with beta(-1) scaling its spatial extension; (iii) its frequency bandwidth, with sigma(-1) scaling its time extension. We develop a consistent theory of approximations of increasing accuracy for the NLM with its first term governed by the NLS. We show that such NLS regime is the medium response to an almost monochromatic excitation current J. The developed approach not only provides rigorous estimates of the approximation accuracy of the NLM with the NLS in terms of powers of alpha, beta and rho, but it also produces a new extended NLS (ENLS) providing better approximations. Remarkably, quantitative estimates show that properly tailored ENLS can significantly improve the approximation accuracy of the NLM compared with the classical NLS equation.