## 2 The Model  
In previous works, a model given by [PERSON] and [PERSON] [20; 21] has given good results when comparing experimental with simulated Q-switched operation using a monolayer G-SAM (graphene saturable absorber mirror) in a Nd:YLF laser [10] or predicting Q-switched operation in a short-cavity titanium:sapphire laser [19]. In this model, the intracavity power \(P\), the gain \(g\) and the density of population inversion \(q\) can be calculated with the following set of equations [20; 21]:  
\[dP/dt=1/t_{r}\left(g-l-q\right)P \tag{1}\]  
\[dg/dt=-1/t_{L}\left(\left(g-g_{0}\right)+gP/P_{L}\right) \tag{2}\]  
\[dq/dt=-1/t_{A}\left(\left(q-q_{0}\right)+qP/P_{A}\right) \tag{3}\]  
where \(t_{r}\) is the cavity round-trip time; \(t_{L}\), \(t_{A}\) the relaxation times of the laser medium and absorber, respectively; \(l\) the linear cavity losses; and \(P_{L}\), \(P_{A}\) the saturation power of the laser medium and absorber. The meaning of \(g_{0}\) and \(q_{0}\) is explained below. To compare the results obtained with graphene or carbon nanotubes as saturable absorbers deposited on a high-reflecting mirror (G-SAM or CNT-SAM), we have chosen a simple linear cavity (Figure 1), which employs of a Nd:YLF laser rod. Nd:YLF is a birefringent crystal, which has a thermal lens effect and can emit laser radiation at 1047 nm and 1053 nm with crossed polarizations, 1047 nm being the strongest line [22; 23]. In our calculations, we used the \(\lambda\) = 1047 nm emission line, and the dimensions of the rod had typical values of \(2a\) = 5 mm diameter, \(L_{r}\) = 5 cm length, refractive index 1.45. The rod was placed in a plane-mirror resonator (SAM and output mirror (OM)) with a short physical length of 100 mm, corresponding to an optical length of \(L\) = 102.25 mm. Short-length cavities give shorter pulse durations, higher output powers and higher repetition rates in the Q-Switched regime [10,19] than longer ones. The SAM was considered to have a reflectivity \(R_{\text{SAM}}\) = 0.998 without the saturable absorber on it and the output mirror \(R_{\text{OM}}\) = 0.98. A lens L (f = 10 mm) was used to focus the laser beam onto the SAM to saturate the graphene or the CNTs. With this configuration, a stability analysis can be done where the thermal effects in the laser rod were introduced with a thermal lens with a focal length f = 200 mm. This analysis reveals that the TEM\({}_{00}\) is practically collimated in the zone rod-OM and is strongly focused in the SAM. However, the geometry of the cavity has a high Fresnel number (_i.e._, N\({}_{F}=a^{2}/(L\lambda)\sim\)58, for \(a=2.5\) mm, \(L\) = 102.25 mm, \(\lambda\) = 1047 nm), and we can assume transversal pumping, so it is more realistic to think of a multimode operation and estimate larger beam sizes at the rod and at the SAM than the values given at the stability analysis, which is done for the TEM\({}_{00}\). Hence, we have done the calculations estimating \(w_{r}\sim 800\) um at the rod and \(w_{\text{SAM}}\sim 300\) um at the SAM. This estimation is based on the fact that these orders of magnitude gave good results when comparing experimental and theoretical results in a transversally-pumped Nd:YLF [10]. Internal cavity losses were taken as \(\upalpha_{i}L_{r}\) = 0.1, which is a value within the typical order of magnitude for solid-state lasers, where \(\upalpha_{i}\) is the internal absorbance and \(L_{r}\) is the length of the laser rod. With these values, the linear cavity losses can be calculated as \(l=2(\upalpha_{i}L_{r}-ln(R))\), being the average reflectivities given by \(R=\sqrt{R_{\text{SAM}}R_{\text{OM}}}\). The quantity \(q_{0}\) is the reflectivity change in the end mirror from absorbing to saturated absorber; this is the modulation depth. Carbon nanotubes have lower \(q_{0}\) values than graphene. Modulation depths are typically around 1% and below [4, 5, 6], which correspond to \(q_{0}\) = 0.01 and below. A single graphene layer absorbs \(\sim\)2.3%; the absorbance of \(n\) graphene layers is \(q_{0}=-2n\ ln(1-0.023)\).