## Abstract

Enhanced coupled-nonlinear equations are introduced for full compensation of cross-phase modulation and partial compensation of four-wave mixing (FWM) via split-step digital backward propagation. Compared to full FWM compensation, the new backward propagation equations provide a significant reduction in the number of steps required in the split-step method. This increased step size together with more relaxed sampling requirements reduced the computational load by more than a factor of 20.

© 2009 Optical Society of America

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### Equations (10)

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(1)
$$E=\sum _{m=1}^{N}{\hat{E}}_{m}\mathrm{exp}(im\mathrm{\Delta}\omega t),$$
(2)
$$-\frac{\partial E}{\partial z}+\frac{\alpha}{2}E+\frac{i{\beta}_{2}}{2}\frac{{\partial}^{2}E}{\partial {t}^{2}}-\frac{{\beta}_{3}}{6}\frac{{\partial}^{3}E}{\partial {t}^{3}}+i\gamma |E{|}^{2}E=0,$$
(3)
$$-\frac{\partial {\hat{E}}_{m}}{\partial z}+\frac{\alpha}{2}{\hat{E}}_{m}+\sum _{p=1}^{3}{K}_{pm}\frac{{\partial}^{p}{\hat{E}}_{m}}{\partial {t}^{p}}+i\gamma (2\sum _{q=1}^{N}|{\hat{E}}_{q}{|}^{2}-|{\hat{E}}_{m}{|}^{2}){\hat{E}}_{m}=0,$$
(4)
$${K}_{1m}=m{\beta}_{2}\mathrm{\Delta}\omega -\frac{1}{2}{m}^{2}{\beta}_{3}\mathrm{\Delta}{\omega}^{2},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{K}_{2m}=\frac{1}{2}i{\beta}_{2}-\frac{1}{2}m{\beta}_{3}\mathrm{\Delta}\omega ,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{K}_{3m}=-\frac{1}{6}{\beta}_{3}.$$
(5)
$${S}_{\mathrm{T}\text{-}\mathrm{NLSE}}=2\frac{N\mathrm{\Delta}f}{B},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}\text{samples}/\text{symbol},$$
(6)
$$-\frac{\partial {\hat{E}}_{m}}{\partial z}+\frac{\alpha}{2}{\hat{E}}_{m}+\sum _{p=1}^{3}{K}_{pm}\frac{{\partial}^{p}{\hat{E}}_{m}}{\partial {t}^{p}}+i\gamma (2\sum _{q=1}^{N}|{\hat{E}}_{q}{|}^{2}-|{\hat{E}}_{m}{|}^{2}){\hat{E}}_{m}+{F}_{2m}+{F}_{4m}=0,$$
(7)
$${F}_{2m}=2{\hat{E}}_{m+1}{\hat{E}}_{m-1}{\hat{E}}_{m}^{*},$$
(8)
$${F}_{4m}={\hat{E}}_{m+1}^{2}{\hat{E}}_{m+2}^{*}+{\hat{E}}_{m-1}^{2}{\hat{E}}_{m-2}^{*}+2{\hat{E}}_{m-1}{\hat{E}}_{m+1}{\hat{E}}_{m+1}^{*}\phantom{\rule{0ex}{0ex}}+2{\hat{E}}_{m+1}{\hat{E}}_{m-2}{\hat{E}}_{m-1}^{*}+2{\hat{E}}_{m+2}{\hat{E}}_{m-2}{\hat{E}}_{m}^{*}.$$
(9)
$${\hat{E}}_{m}^{j+1}={\hat{E}}_{m}^{j}\mathrm{exp}(i\gamma |{\hat{E}}_{m}^{j}{|}^{2}h)+h({F}_{2m}^{j}+{F}_{4m}^{j}).$$
(10)
$${H}_{m}(\omega )=\mathrm{exp}\left[\right(i{\beta}_{2}\frac{(\omega -m\mathrm{\Delta}\omega {)}^{2}}{2}+i{\beta}_{3}\frac{(\omega -m\mathrm{\Delta}\omega {)}^{3}}{6}\left)h\right].$$