Modified nonlinear Schrödinger equation for frequency-dependent nonlinear profiles of arbitrary sign

In recent times, materials exhibiting frequency-dependent optical nonlinearities, such as nanoparticle-doped glasses and other metamaterials, have gathered significant interest. The simulation of the propagation of intense light pulses in such media, by means of the nonlinear Schrödinger equation (NLSE), poses the problem in that straightforward inclusion of a frequency-dependent nonlinearity may lead to unphysical results, namely, neither the energy nor the photon number is conserved in general. Inspired by a simple quantum-mechanical argument, we derive an energy- and photon-conserving NLSE (pcNLSE). Unlike others, our approach relies only on the knowledge of the frequency-dependent nonlinearity profile and a generalization of Miller’s rule for nonlinear susceptibility, enabling the simulation of nonlinear profiles of arbitrary frequency dependence and sign. Moreover, the proposed pcNLSE can be efficiently solved by the same numerical techniques commonly used to deal with the NLSE. Relevant simulation results supporting our theoretical approach are presented.