Soliton self-frequency shift in non-uniform fiber tapers: analytical description through an improved moment method

We develop an improved moment method derived from the generalized nonlinear Schr\"{o}dinger equation to model soliton propagation in optical fibers. We account for the full Raman gain spectrum of the material and derive a system of coupled differential equations describing the evolution of the five moments of the optical pulse, valid for arbitrary soliton durations. We further simplify the equations by casting them into a moving frequency frame that follows the central frequency of the Raman-shifting soliton. We show that for short solitons there exists a non-negligible Raman-induced chirp, which contributes to slowing down the soliton self-frequency shift. By comparing with the numerical solution of the generalized nonlinear Schr\"{o}dinger equation, the improved moment method is shown to accurately represent soliton self-frequency shift under higher order dispersion, self-steepening and pulse chirp. Numerical examples are presented for a dispersion-shifted fused silica fiber and a ZBLAN non-uniform fiber taper. The latter demonstrates enhanced soliton self-frequency shift through axial dispersion and nonlinearity engineering along the taper length.