Numerical simulations of femtosecond temporal solitons and spatial solitons in nonlinear optics

Experimentalists have produced all-optical switches capable of 100 femtosecond responses. Also, there are experimental observations of spatial soliton interactions. To model such effects, nonlinearities in optical materials must be included. The behavior of electromagnetic fields in nonlinear dielectrics can be determined by solving the nonlinear Maxwell´s equations. However currently, the standard method for determining the fields is to solve the nonlinear Schrodinger equation (NLSE), which is an approximation that neglects the optical carrier of the pulse. For modeling small scale engineered inhomogeneities in optical devices, on the order of 0.1 to 10 optical cycles, the assumptions in the NLSE become unjustified. In this paper, solutions are presented of calculations of the 2-D vector nonlinear Maxwell´s equations for propagating and scattering temporal and spatial solitons in material media having linear and nonlinear instantaneous and Lorentzian dispersive effects in the electric polarization. The optical carrier is retained in these calculations. A finite difference method is used to solve Maxwell´s equations and the ordinary differential equations that determine the linear and nonlinear dispersive effects