High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media

We make a detailed study on the dynamics of gain-assisted superluminal optical solitons in a three-state active-Raman-gain medium at room temperature. Using a method of multiple-scales we derive a high-order nonlinear Schrödinger equation with correction terms contributed from differential gain, nonlinear dispersion, delay in nonlinear refractive index, and third-order dispersion of the system. We show that for a long pulse with realistic physical parameters the high-order correction terms are small and can be taken as perturbations. However, for a shorter pulse these higher-order correction terms are significant and hence must be treated on equal footing as the terms in the nonlinear Schrödinger equation. We provide exact soliton solutions of the higher-order nonlinear Schrödinger equation and demonstrate that such solitons have still superluminal propagating velocity and can be generated at very low light intensity.