Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures

Abstract

We present a detailed analysis of finite energy solitary waves in two- and three-dimensional nonlinear periodic structures exhibiting a complete photonic band gap. Solitary waves in photonic crystals with a two-dimensional (2D) square and triangular symmetry group as well as a 3D fcc symmetry group are described in terms of an effective nonlinear Dirac equation derived using the slowly varying envelope approximation for the electromagnetic field. Unlike one-dimensional Bragg solitons, the multiple symmetry points of the 2D and 3D Brillouin zones give rise to two distinct classes of solitary wave solutions. Solutions associated with a higher order symmetry point of the crystal exist for both positive and negative Kerr nonlinearities, whereas solutions associated with a twofold symmetry point occur only for positive Kerr coefficient. Using a variational method we derive the important physical features such as the size, shape, peak intensity, and total energy of the solitary waves. This is then confirmed numerically using the finite element Ritz-Galerkin method. It is shown that the initial variational method and the finite element numerical method are in good agreement. We discuss the stability of these solitary waves with respect to small perturbations. It is suggested that an analytical stability criterion for spinor fields satisfying the nonlinear Dirac type of equation may exist, similar to the well known stability criterion for solitary waves in the nonlinear Schr\"odinger equation. Our stability criterion correctly reproduces the stability conditions of other nonlinear Dirac type of equations which have been studied numerically. Our study suggests that for an ideal Kerr medium, two-dimensional solitary waves in a band gap are stable, whereas three-dimensional ones are stable only in certain regions of the gap.