Optical pulse propagation in nonlinear photonic crystals

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Abstract
We present a formalism for optical pulse propagation in nonlinear photonic crystals of arbitrary dimensionality. Using a multiple-scale analysis, we derive the dynamical nonlinear Schrödinger equation obeyed by the envelope function modulating an underlying Bloch function. Effective coefficients appear in that equation characterizing the effects of Kerr nonlinearity, linear gain or loss, and material dispersion. They depend on how the underlying Bloch function “samples” these effects in the photonic crystal, and require for their calculation a specification of these effects throughout the photonic crystal, and the calculated bandstructure of the photonic crystals in the linear, nondispersive limit. We show that wave packets from different bands can experience significantly modified effective material properties.