Extensions and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures

Abstract

Based on an envelope-function approach developed in a previous publication [de Sterke and Sipe, Phys. Rev. A 38, 5149 (1988)], we give criteria for the experimental observation of gap solitons. Moreover, the theory is extended in several respects---it is shown that in the limit of harmonic time dependence, our treatment is equivalent to the effective-mass approximation in condensed-matter physics. Further, we derive equations of motion for the fields under more general conditions than before, including third-harmonic generation by the optical nonlinearity. By use of Hamiltonian methods it is shown that third-harmonic generation in periodic structures has quite different properties than in homogeneous materials.