Envelope-function approach for the electrodynamics of nonlinear periodic structures

Abstract

An envelope-function approach is used to give a theoretical description of the electromagnetic properties of nonlinear periodic structures. This method, in which the electric field is separated into slow and fast spatial components, shows that the slow field component satisfies the nonlinear Schr\"odinger equation. The well-known soliton solutions of this equation provide a theoretical description of the gap solitons found by Chen and Mills [Phys. Rev. Lett. 58, 160 (1987)] in their numerical studies of these structures. The more general solutions of the nonlinear Schr\"odinger equation provide a framework for understanding the properties of finite nonlinear periodic stacks. Our method allows us to find these solutions analytically.