## 2 The Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Relaxation Times** & **Internal Cavity Losses** & **Saturation Fluences** & **Cavity Losses** & **Modulation Depth** & **Unsaturated Round-Trip Gain** \\ \hline \(t_{L}\) = 480 \(\upmu\)s & \(\alpha_{i}L_{r}\) = 0.1 & \(F_{sat}=h\upnu/(2\sigma)\) & \(l=2(\alpha_{i}L_{r}-ln(R))\) & \(q_{0}\) = 0.046, & \(g_{0}\) = 2\(l\) to 10\(l\) \\  & & & & & 0.023, & \\ \(t_{A}\)=1.45 ps & & \(\sigma\) = 1.8\(\times 10^{-19}\) cm\({}^{2}\) & & & 0.015, \\  & & & & & 0.010, & \\  & & & & & 0.005. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Laser and saturable absorber parameters (see the text for details). Figure 2: Q-switched regime obtained for \(q_{0}=0.015\), \(g_{0}/l\) = 4. In other cases, the temporal shape does not show a sharp peak, and the absorber is bleached for a longer time, giving rise to a flat-top-like temporal pulse, like in Figure 4, obtained for \(q_{0}=0.046\) and \(g_{0}/l\) = 4. From this output, the temporal widths, the energy per pulse and the repetition rate can be deduced. Figure 5a shows the pulse durations obtained for \(q_{0}\) ranging from 0.046 to 0.005. Very short pulse durations can be achieved in all cases, with very similar values with respect to \(g_{0}/l\). The behavior is not monotonic and changes from \(q_{0}=0.01\) to \(q_{0}=0.005\). This behavior in the pulse duration is studied later in closer detail. Figure 5b shows the pulse energy obtained for the same range of \(q_{0}\) values. Very high energies are obtained, and the highest values are obtained for the highest \(q_{0}\). Like the pulse duration, the pulse energy does not depend significantly on the pump value \(g_{0}/l\) and remains practically constant for increasing pumping values from \(g_{0}=2l\) to \(g_{0}=10l\), being the unsaturated round-trip gain \(g_{0}\) higher for higher pumping power. Figure 5c shows the repetition frequency of the pulse train and is the magnitude, which changes the most, increasing with \(g_{0}/l\), and it is apparent that, unlike in the mode locking regime, the repetition rate in Q-switched operation mode is not fixed by the cavity round trip time. The lowest frequencies are obtained for the highest values of \(q_{0}\). In Figure 5, the case \(q_{0}\) = 0.046 for bilayer graphene and \(g_{0}/l\) = 2 did not reach Q-switch operation and extinguished after a couple of spikes. Figure 4: Temporal zoom in a Q-switched pulsed obtained for \(q_{0}=0.046\), \(g_{0}/l\) = 4. Figure 3: Temporal zoom in a Q-switched pulse in the train of Figure 2a obtained for \(q_{0}=~{}0.015\), \(g_{0}/l\) = 4. The behavior of the pulse duration and pulse energy has been studied in closer detail for \(g_{0}/l\) = 8 and an absorption range for the CNTs from \(g_{0}=0.005\) to \(g_{0}=0.01\) in steps of 0.001, as well as for monolayer and bilayer graphene. Figure 6a shows how the pulse duration reaches a minimum for \(g_{0}=0.01\), while the pulse energy grows monotonically with \(q_{0}\) in the whole range. According to the behavior of these two magnitudes in Figure 5a,b, which have a low dependence on \(g_{0}/l\), the same behavior is expected for other values of \(g_{0}/l\). Figure 5: Pulse duration, pulse energy and repetition frequency for different absorptions \(g_{0}\) as a function of the pump parameter \(g_{0}/l\). Figure 6: Pulse duration and pulse energy for \(g_{0}/l\) = 8 in an absorption range \(g_{0}=0.005\) to \(g_{0}=0.01\) in steps of 0.001 for CNTs, \(g_{0}=0.023\) for monolayer graphene and \(g_{0}=0.046\) for bilayer graphene. Although the Q-switched regime is present in the range studied, when \(q_{0}\) is very low (for example \(q_{0}=0.001\)), then only relaxation oscillations appear, tending to the cwstate for longer times (Figure 7).