Weakly nonparaxial effects on the propagation of (1+1)D spatial solitons in inhomogeneous Kerr media

The widely-used approach to study the beam propagation in Kerr media is based on the slowly varying envelope approximation (SVEA) which is also known as the paraxial approximation. Within this approximation, the beam evolution is described by the nonlinear Schrödinger (NLS) equation. In this paper, we extend the NLS equation by including higher-order terms to study the effects of nonparaxiality on the soliton propagation in inhomogeneous Kerr media. The result is still a one-way wave equation which means that all back-reflections are neglected. The accuracy of this approximation exceeds the standard SVEA. By performing several numerical simulations, we show that the NLS equation produces reasonably good predictions for relatively small degrees of non-paraxiality, as expected. However, in the regions where the envelope beam is changing rapidly as in the breakup of a multisoliton bound state, the nonparaxiality plays an important role.