Wave dynamics in optically modulated waveguide arrays

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulk diffraction term and an external "optical trap" is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized modes are numerically con- structed. It is shown that the equation with an induced potential is well-posed and gives rise to localized dynamically stable nonlinear modes. The derived model is of the Gross-Pitaevskii type, a nonlinear Schrödinger equation with a linear optical potential, which also models Bose-Einstein condensates in a mag- netic trap. Wave propagation in nonlinear periodic structures dis- plays unique phenomena that are absent in homogeneous media. The interplay between periodicity and nonlinearity can lead to the formation of discrete or lattice solitons, which were predicted theoretically in the context of optical wave- guide arrays f1g and then experimentally observed in f2g. Until recently, discrete solitons were considered experimen- tally in one-dimensional geometry f2g. However, by making use of the photorefractive screening nonlinearity one can "write" either one- or higher-dimensional optical waveguide arrays by interfering pairs of plane waves f3g. Indeed, such localized structures were experimentally observed in two- dimensional geometries f4g. In this paper we study wave propagation in optically modulated waveguide arrays, starting from the full time- harmonic three-dimensional Maxwell's equations. For the case where the periodic modulation along the y direction is much larger than the periodic modulation along the x direc- tion we derive, using multiscale asymptotic analysis, a semi- discrete nonlinear Schrödinger equation with the addition of bulk diffraction term and an external "optical trap." When the nonlinearity is of the defocusing type swhere in the ab- sence of modulation no finite energy solitons are knownd unstaggered localized modes are numerically constructed. The fundamental properties such as the well-posedness of the equation, existence, and the dynamical stability associ- ated with a special class of localized wave solutions, i.e., stationary wave, or ground state, are discussed. The semidis- crete model is derived from the scalar nonlinear Helmholtz equation. Below we briefly outline the justification for ne- glecting vectorial effects under certain physical assumptions. A more general and detailed study of scalar and vector semi- discrete nonlinear Schrödinger sNLSd type models will be given elsewhere. We begin by considering the three-dimensional Maxwell equations governing time-harmonic solutions of frequency v0