## 3 Simulation Results  
From above experimental data, we conclude a saturable absorber with lower modulation depth and no reverse absorption enables the generation of CS, where a saturable absorber with larger loss and reverse absorption leads to NLP. This provides the tunability of pulse-type in one laser system by modifying the SA parameters. A simulation was then conducted to explain the influence of the SA parameters on laser performance. The simplified numerical model of cavity setup is shown in Figure 8 with a piece of Er-doped gain fiber (EDF), three segments of single-mode fiber (SMF1, SMF2, and SMF3), an optical coupler (OC), a wavelength division multiplexer (WDM), and a saturable absorber (SA). The cavity parameters in simulation are estimated from experimental data which were shown in Table 1, including the same cavity length, relative position of each device, and the same net cavity dispersion. Saturable coefficient and reverse absorption coefficient are described by the transmittance function in Equation (1). To simulate the transformation of pulse operation states in the fiber laser, coherently coupled nonlinear Schrodinger equation (NLSE) including dispersion, nonlinearity, gain, and loss was used as the master equation to describe the pulse propagation in the fiber system (Equations (2) and (3)). \[\frac{\partial A}{\partial z}=-\frac{i}{2}\beta_{2}\frac{\partial^{2}A}{ \partial t^{2}}+i\gamma|A|^{2}A+\frac{g}{2}A+\frac{g}{2\Omega_{g}^{2}}\frac{ \partial^{2}A}{\partial t^{2}} \tag{2}\]  
\[g=g_{0}\exp\left(-\frac{\int|A|^{2}dt}{E_{sat}}\right) \tag{3}\]  
where A denotes the slow varying amplitude of the pulse envelope, z is the propagation distance, t is the pulse local time. \(\beta_{2}\) and \(\gamma\) are group velocity dispersion (GVD) and fiber nonlinear coefficient. g, g\({}_{0}\), and \(\Omega\)g denote gain function, small-signal gain coefficient, and gain bandwidth in Equations (2) and (3), respectively. Such simulation can provide insights on the evolution of laser performance with SA parameters and explain the experimental pulse operation states transformation. Figure 8: Simplified numerical simulation model. Figure 7: (**a**,**b**) Long-term stabilities of CS and NLP every 1 h sampling over 6 h.  
It should be mentioned that the positions of Kelly sidebands in the optical spectrum reflect the cavity dispersion, despite that the positions of these soliton sidebands could be slightly affected by the amount of chirp acquired by the pulse [37,38]. The total cavity dispersion is estimated by the following Equation (4) with experimentally observed parameters, including the central wavelength, the first-order Kelly sideband separation, \(\Delta\lambda_{1}\) = 4.0 nm, and the soliton pulse width t = 439 fs,  
\[\Delta\lambda N=\pm N\cdot\lambda 0\sqrt{\frac{2N}{cDL}-0.0787\frac{{\lambda_{0}}^{ 2}}{{(ct)}^{2}}} \tag{4}\]  
where N is an integer, D is the average cavity dispersion parameter, L is the fiber length. The estimated total cavity dispersion is 0.03 ps nm\({}^{-1}\). The total cavity dispersion comprises the fiber dispersion and the material dispersion from ML and BL CNT-SA. Thereafter, the material dispersion from CNT polymer could be roughly deduced as:  
\[\mathrm{D_{ML\,CNT}}=-0.3\;\mathrm{D} \tag{5}\]  
Variable parameters including \(\beta_{2}\), \(g_{0}\), \(\alpha_{s}\), Isat, and \(\beta\) were set as \(-12.2\) ps\({}^{2}\)/km, \(15\) m\({}^{-1}\), \(0.15\), \(50\) W, and \(0\) W\({}^{-1}\) to simulate the CS formation process, while the initial pulse condition was set as 10 ps chirp-free pulse and the ML CNT-SA parameters were used. Stable CS can be generated in the simulation in Figure 9a. The time domain evolution, AC trace and spectrum of the generated CS are shown in Figure 9a1-a3, respectively. Increasing the value of gain coefficient g\({}_{0}\) to 20 m\({}^{-1}\) results in splitting of single pulse split into two (so-called two-bound soliton state) in Figure 9b.