Optical pulse propagation in nonlinear photonic crystals

Navin A. R. Bhat and J.E. Sipe. University of Toronto, Toronto,
Ontario, Canada.

We present a new formalism for optical pulse propagation in nonlinear photonic crystals. Our approach uses a reformulated k dot p theory for realistic photonic band structures, and employs the method of multiple scales to derive dynamical field equations for the envelope functions modulating the underlying Bloch functions.

Depending on the strength of the nonlinearity and the proximity of neighbouring bands, we find qualitatively different dynamical behaviour of the pulse envelope functions. For example, for fairly weak nonlinearity we find the envelope function for a single band obeys a nonlinear Schrödinger equation if all other bands are remote. In the case of a stronger nonlinearity, envelope functions of wavepackets from nearby bands are found to obey coupled-mode equations. The approximations made in the multiple-scale analysis are valid for a broad class of realistic scenarios, and although we only discuss ?(3) crystals, the method can be used to describe other nonlinear systems as well as those with or without a full band gap.

Finally, we discuss solutions of these equations and give numerical results based on calculated bandstructures of actual photonic crystals.