## 3 The Cerenkov FEL Oscillator
The saturation mechanism in free electron devices is quite similar to that occurring in conventional lasers, as has been derived by [PERSON] approximately sixty years ago [19] and is also present in the theory of passive saturable absorber for laser mode-locking [20]. Two different but similar approaches can be applied; the first technique starts from the application of the Ginzburg-Landau equation to FEL systems [21] and leads to the \(G_{I}(I)\) expression in Equation (7). The second method is based on the logistic equation approach [22] and ends up the \(G_{II}(I)\) expression in Equation (7):  
\[\left\{\begin{array}{l}G_{I}(I_{n})\ =\ 0.849\cdot g_{0 MAX}\frac{1-\exp\left(- \frac{\pi}{2}\frac{I_{n}}{SAT}\right)}{\frac{\pi}{2}\frac{I_{n}}{SAT}}\ \ \ ;\ \ \ \left\{ \begin{array}{l}G_{II}(n)\ =\ \frac{I(n)-I(n-1)}{I(n-1)}\\ I(n)\ =\ I_{SP}\frac{\exp\left[\delta_{0 MAX}(1-\Gamma)-\Gamma\right]\pi}{1+\left(I_{SP }/L_{j}\right)\exp\left[\left\delta_{0 MAX}(1-\Gamma)-\Gamma\right]\pi\right)-1 \right]}\\ I_{n+1}\ =\ [I_{n}+G_{I}(I_{n})I_{n}](1-\Gamma)\end{array} \right.\ ;\ \ \left\{ \begin{array}{l}G_{II}(n)\ =\ \frac{I(n)-I(n-1)}{I(n-1)}\\ I(n)\ =\ I_{SP}\frac{\exp\left[\delta_{0 MAX}(1-\Gamma)-\Gamma\right]\pi}{1+\left(I_ {SP}/L_{j}\right)\exp\left[\left\delta_{0 MAX}(1-\Gamma)-\Gamma\right]\pi\right) -1]}\\ I_{e,i}\ =\ \frac{2}{\pi}\frac{1-\Gamma}{\Gamma}\delta_{0 MAX}\Big{\{}1-\exp \left[-\frac{1.8}{1+\delta_{0 MAX}}\frac{\delta_{0 MAX}(1-\Gamma)-\Gamma}{\Gamma }\right]\Big{\}}\,I_{SAT}\end{array} \right. \tag{7}\]  
where \(N_{eq}=L/(2\gamma^{2}\lambda_{C})\)\(\sim\) 7.3 represents the equivalent value of the number of undulator periods in Cerenkov FELs; \(I_{SAT}\)-7.4\(\cdot\)10\({}^{6}\)\([W]\) expresses the saturation power value related to the gain of the device (conventionally, the intensity is used instead of power, however, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Cerenkov FEL Parameters @ \(\lambda\) = 250 \(\upmu\)m**} \\ \hline Electron & Dielectric & Dielectric & Resonator & Resonator & Double Slab \\ Energy & Thickness & Constant & Length & Width & Height \\ \hline \(\gamma\) = 10.0 & \(d\) = 50 \(\upmu\)m & \(\varepsilon\) = 5.0 & \(L\) = 35.0 cm & \(W\) = 1.0 cm & \(D\) = 2 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cerenkov FEL Parameters @ \(\lambda\) = 250 \(\upmu\)m.  
Figure 2: Spontaneous emission spectra as a function of the beam centroid distance from the dielectric surface: (**a**) single slab waveguide geometry; and (**b**) double slab waveguide geometry. in this case, the radiation transverse cross-section is a constant due to the presence of a waveguide, making power and intensity interchangeable); \(n\) indicates the round-trip number; \(I_{AV}\)\(\sim\)17,000 \(A\) indicates the Alfven current; \(\Gamma\) represent the total losses in the radiation resonator; \(I_{SP}\) expresses the spontaneous emission power of all the electrons in the bunch and integrated over the bandwidth; \(I_{e,i}\) refers to the intracavity equilibrium power, and finally, \(g_{0 MAX}\) indicates the maximum value of the small-gain, small-signal gain coefficient as a function of frequency [15]. The relevant parameters for the gain calculation are reported in Table 2.  
The behaviour of Equation (7) is reported in Figure 3a) for the gain expressions, \(G_{I}\) and \(G_{II}\), as a function of the round trip number \(n\), and in Figure 3b) for the ongoing power radiation, \(I_{n}\) and \(I(n)\), again with respect to the round trip. The main result, as can be deduced from Figure 3, is that saturation is reached after about \(n=100\) round trips considering both methods reported in Equation (7). Considering the resonator length, as indicated in Table 1, we can deduce that saturation is gained in about \(T_{SAT}\)-200 ns, which is quite a short time with respect to the conventional macro-bunch duration of electron pulses generated by RF accelerators in the S-band (\(\sim\)3 GHz), which of the order of several microseconds.