In this section, we describe the optical pulse propagation in a fiber laser using the NLS equation for the pulse envelope $\psi (z,T)$ in the presence of mode locking with the use of an SA, including the gain dispersion, losses for the cavity and fiber, gain, group-velocity dispersion (GVD), self-phase modulation (SPM), and two-photon absorption (TPA). This equation can be written as^{5}Display Formula
$\u2202\psi \u2202z+i2(igT22+\beta 2)\u22022\psi \u2202T2=[i\gamma +12(\delta SApsatCNT\u2212gpsatEr\u2212\alpha 2)]|\psi |2\psi +12(g\u2212\alpha \u2212\delta SA)\psi ,$(1)
where $\psi (z,T)$ is the amplitude of the optical pulse, $T$ is the time, $z$ is the propagation distance, $\alpha $ is a coefficient that takes into account material losses in the cavity, $\delta SA$ is the SA parameter, $\alpha 2$ is the TPA parameter, $\gamma $ is the SPM parameter, $\beta 2$ is the second-order dispersion coefficient, $psatEr$ is the saturation power of the gain medium ($Er+3$), $psatCNT$ is the saturation power of SA (CNTs), and $gT22$ is a frequency-dependent gain dispersion factor. We assume a chirped pulse given by Display Formula$\psi (z,T)=\chi (z,T)+i\mu (z,T),$(2)
where Display Formula$\chi (z,T)=\xi \u2009sech(\sigma T)cos{kz\u2212c\u2009log[cosh(\sigma T)]},$(3)
Display Formula$\mu (z,T)=\xi \u2009sech(\sigma T)sin{kz\u2212c\u2009log[cosh(\sigma T)]},$(4)
where $\xi $, $\sigma $, $k$, and $c$ are four arbitrary parameters representing the amplitude, width, wave number, and chirp of the pulse, respectively.