## 2 The Model
Hence, with carbon nanotubes and mono- and multi-layer graphene, we can cover a large range in the parameter \(q_{0}\). All graphene or nanotube layers were taken as uniform within the spot size of the beam at the SAM. Other parameters, like relaxation times \(t_{A}\) for the absorber, saturation powers \(P_{A}\) and saturation fluences \(F_{A}\), are more similar between graphene and carbon nanotubes. \(t_{A}=1.45\) ps for graphene [9], while \(t_{A}\) ranges from 0.75 ps to 1 ps for carbon nanotubes [4,6]. For graphene, \(F_{sat}\) is 14.5 \(\upmu\)J/cm\({}^{2}\)[9], while \(F_{sat}\) is 10 \(\upmu\)J/cm\({}^{2}\) for carbon nanotubes [4,6]. Therefore, it seems reasonable to choose for all parameters their values in graphene and to study just the influence of \(q_{0}\). Figure 1: Cavity design for a transversally-pumped Nd:YLF laser. \(L\): cavity length, \(L_{r}\): laser rod length, L: focusing lens, OM: output mirror, SAM: saturable absorber mirror. Saturation powers for the absorber or the laser medium \(P_{sat}=P_{A},P_{L}\) were calculated from the saturation fluence \(F_{sat}\). We used \(P_{sat}=F_{sat}\pi w^{2}/\tau\), with \(\tau\) the relaxation time of the laser medium (\(t_{L}=480\)\(\upmu\)s [22]) and of the absorber (\(t_{A}=1.45\) ps [9]), \(w\) the beam waists at the laser rod (\(w_{r}\)) or at the SAM (\(w_{\mathbf{SAM}}\)), which, as mentioned, were estimated considering multimode operation. The saturation fluence for Nd:YLF is given by \(F_{sat}=h\upnu/(2\sigma)\), with \(h\) the Planck's constant, \(\upnu\) the laser emission frequency and \(\sigma\) the emission cross-section (\(\sigma\) = 1.8 \(\times\)\(10^{-19}\) cm\({}^{2}\) for the 1047-nm line of Nd:YLF [22]). \(g_{0}\) is the unsaturated round-trip gain, and this parameter grows with increasing pumping intensity. We gave \(g_{0}\) values ranging from 2- to 10-times the linear losses \(l\), in steps of 1. By solving Equations (1) to (3) using a 4 th order Runge-Kutta method, we have studied the Q-switched regime (pulse duration, pulse energy and repetition frequency of the Q-switch train) as a function of \(g_{0}/l\), for different values of \(q_{0}\). The values chosen were \(q_{0}=\) 0.046 (two layers of graphene), 0.023 (one-layer of graphene) and 0.015, 0.01, 0.005 (representative values for CNTs) The integration time was always set long enough so that the stable regime (constant peak power) was reached. Tables 1 and 2 summarize the value of the parameters used in the calculations. Figure 2 shows the pulses obtained by integration of Equations (1) to (3) in the case \(q_{0}\) = 0.015, \(g_{0}/l\) = 4, where it can be seen how the regime is stationary only after half a millisecond approximately. The lower the value of \(q_{0}\), the longer the integration time needed to reach the stationary Q-switched operation for low \(q_{0}/l\) values (for example, 5 ms, for \(q_{0}\) = 0.005 and \(g_{0}/l\) = 2). Figure 3 is a temporal zoom on the last pulse of this train. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Laser Rod** & **Laser Rod** & **Refractive Index** & **Cavity Length** & \begin{tabular}{c} **Output Mirror** \\ **Reflectivity** \\ \end{tabular} & **SAM Reflectivity** & \
\begin{tabular}{c} **Beam Waist** \\ **at Laser Rod** \\ \end{tabular} &
\begin{tabular}{c} **Beam Waist** \\ **at SAM** \\ \end{tabular} \\ \hline \(2a\) = 5 mm & \(L_{r}\) = 5 cm & 1.45 & L = 100 mm & \(R_{OM}\) = 0.98 & \(R_{SAM}\) = 0.998 & \(\omega_{r}\) = 800 \(\upmu\)m & \(\omega_{SAM}\) = 300 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cavity parameters (see the text for details).