# Self\(-\)Mode\(-\)Locked \(2-\)um GaSb\(-\)Based Optically Pumped Semiconductor Disk Laser
## 2 Experiment Design and Setup  
Self\(-\)mode\(-\)locked pulses of SDLs are realized by the nonlinear Kerr lens effect of the pump spot, and it has been demonstrated that the nonlinear refractive index n\({}_{2}\) is large enough to cause perturbations of the cavity beam profile [47; 48; 49]. The Kerr lens effect has been explored both experimentally and theoretically in solid\(-\)state crystal lasers. In most cases, Haus master equations can explain the pulse dynamics [50]. However, it cannot describe mode\(-\)locking in semiconductor lasers satisfactorily because it is based on small changes of gain and losses in laser pulse shape within one round trip. Instead, the delay differential equation (DDE) model is widely used to describe the pulse dynamics in semiconductor lasers [51; 52; 53]. According to [PERSON] [25; 26], the DDE equations can be expressed as:  
\[\frac{1}{\gamma}\frac{\partial A}{\partial t}+A(t)=\sqrt{\kappa}exp\big{[} \big{(}1-i\alpha_{g}\big{)}g(t-\tau_{r})/2-\big{(}1-i\alpha_{q}\big{)}q(t-\tau _{r})/2\big{]}A(t-\tau_{r}) \tag{1}\]  
\[\frac{\partial g}{\partial t}=\gamma_{g}[g_{0}-g(t)]-\exp(-q)[\exp(g)-1]|A(t)| ^{2} \tag{2}\]  
\[\frac{\partial q}{\partial t}=\gamma_{q}[q_{0}-q(t)]-s[1-\exp(-q)]|A(t)|^{2} \tag{3}\]  
where \(A\) is the laser field amplitude. \(g\) and \(q\) are saturable gain and loss. \(\gamma\) is the corresponding bandwidth of the spectral filtering element. The gain bandwidth of quantum wells (QWs) is usually about 10 nm, so here we set \(\gamma\) as 200. \(\kappa\) is the linear nonresonant attenuation factor per round\(-\)trip pass; in SDLs, \(\kappa\) can be expressed as \(\kappa=\sqrt{(}R_{1}R_{2})\). Based on our experimental conditions, the reflectivity of the two end mirrors of the cavity \(R_{1}=99.9\%\), \(R_{2}=95\%\), and the value of \(\kappa\) is 0.9742. \(\alpha_{g}\) and \(\alpha_{q}\) are the linewidth enhancement (self\(-\)phase modulation) factors; for simplicity, we restrict our numerical analysis to the case in \(\alpha_{g}=\alpha_{q}=0\). \(g_{0}\) and \(q_{0}\) are the small\(-\)signal gain and loss, and also the initial value \begin{table}
\begin{tabular}{c c c c c c} \hline
**Laser Gain** & \(\tau_{p}\) & \(f_{0}\) & \(\lambda_{0}\) & \(P_{out}\) & **References** \\ \hline \multirow{4}{*}{InGaSb} & 1.1 ps & 881.2 MHz & 1950 nm & 25 mW & [19] \\ \cline{2-6}  & 384 fs & 890.4 MHz & 1960 nm & 25 mW & [20] \\ \cline{1-1} \cline{2-6}  & 324 fs & 3.03 GHz & 2061 nm & 65 mW & [21] \\ \cline{1-1} \cline{2-6}  & 4.52 ps & 3.03 GHz & 2072.8 nm & 260 mW & \\ \hline \end{tabular}
\end{table} 
Table 1: Passively mode\(-\)locked GaSb\(-\)based SDLs with SESAM at 2 μm wavelength. \(\tau_{p}\) is the pulse duration. \(f_{0}\) is the pulse repetition rate. \(\lambda_{0}\) is the center wavelength of the laser. \(P_{out}\) is the average output power. of \(g\) and \(q\). \(\gamma_{g}\) and \(\gamma_{q}\) are the carrier density relaxation rates in the gain and absorbing sections; here, we set the relaxation time \({\gamma_{g}}^{-1}\) and recovery time \({\gamma_{q}}^{-1}\) as 10 ns and 10 ps, respectively. \(s\) is the ratio of saturation energy of gain and absorber. Referring to the value in Refs. [26; 52], we set \(s\) as 25. The delay parameter \(\tau_{r}\) is the cavity round trip time. All the time in the calculation was normalized. The traditional way to obtain higher single pulse energy is to reduce the pulse repetition rate and compress the pulse width. The pulse repetition rate is inversely proportional to the cavity length. A longer cavity length means a larger \(\tau_{r}\), which may have a negative  
As shown in Figure 1, the relative value of loss caused by saturated absorption will decrease with the increase of cavity length and the cavity round\(-\)trip time. An unstable jitter occurs after the loss drops to the bottom in a single pulse cycle of the long cavity length. This phenomenon began to appear when the cavity length was 400 mm, but it did not significantly affect the pulse waveform.