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.