Nonlinear Schrödinger Equation

Abstract

Firstly, based on the small-signal analysis theory, the nonlinear Schrodinger equation (NLSE) with fiber loss is solved. It is also adapted to the NLSE with the high-order dispersion terms. Furthermore, a general theory on cross-phase modulation (XPM) intensity fluctuation which adapted to all kinds of modulation formats (continuous wave, non-return-to-zero wave, and return-zero pulse wave) is presented. Secondly, by the Green function method, the NLSE is directly solved in the time domain. It does not bring any spurious effect compared with the split-step method in which the step size has to be carefully controlled. Additionally, the fourth-order dispersion coefficient of fibers can be estimated by the Green function solution of NLSE. The fourth-order dispersion coefficient varies with distance slightly and is about 0.002 ps/km, 0.003 ps/nm, and 0.00032 ps/nm for SMF, NZDSF, and DCF, respectively. In the zero-dispersion regime, the higher-order nonlinear effect (higher than self-steepening) has a strong impact on the short pulse shape, but this effect degrades rapidly with the increase of β2. Finally, based on the traveling wave solution of NLSE for ASE noise, the probability density function of ASE by solving the Fokker-Planck equation including the dispersion effect is presented.