## 6 Final Remarks  
This paper has treated different topics in the small signal evolution of high gain \(SAE\) devices. We have just touched on the nonlinear effects leading to saturation. It has been underscored that the relevant pattern is accompanied by mechanisms of higher order bunching leading to the process of higher order harmonic emission. Figure 10: Same as Figure 9—all slices with the same energy spread and the maximum current associated with the rear part of the bunch (\(s*=1\)). Slice evolution vs \(z\) and intensity evolution vs. \(z\). The slices are characterized by the same amount of charge and different values of the energy spread with \(\lambda_{u}=0.015\), \(\rho_{0}=0.0011\), \(\sigma_{z}=0.05\), \(\sigma_{d}=0.002\), \(\delta=0.01\) and \(I_{0}=10^{-1}\). The intensity growth does not refer to a single slice but to the relevant average. The slices have a flat distribution in \(z\), and each is characterized by an initial seed of 0.1 W. The vertical axis is in log-scale; the insets are plotted in linear scale. Figure 9: Slice evolution inside the undulator at different \(Z\); the energy spread has been randomly chosen and the current follows the bunch Gaussian distribution with \(s*=0\), \(\lambda_{u}=0.02\), \(\rho_{0}~{}=~{}0.001\), \(\mu_{\epsilon}=0.0004\) and \(\
u=0\). The initial distribution \(z=0\) of the slices is the same as in Figure 8. The vertical axis is in log-scale; the insets are plotted in linear scale. From the phenomenological point of view, saturation can be included by modifying the small signal power evolution in Equation (29) as  
\[I_{s}(\bar{z})=I_{0}\frac{A(\bar{z})}{1+\frac{I_{0}}{I_{F}}(A(\bar{z})-1)},\qquad I _{F}=\sqrt{2}\rho P_{E} \tag{63}\]  
where \(P_{E}\) is the electron beam power intensity and \(I_{0}\) is the input seed intensity. \(I_{0}\) is usually specified by the rule of thumb that a value usually \(10^{8}\) below the saturated power density is chosen. The intensity \(I_{F}\) yields the saturated power. The term at the denominator accounts for the saturation mechanisms, using a kind of logistic model, as displayed in Figure 11. The last figure needs its own mention: it shows the power growth in the deep saturated regime. Equation (63) reproduces well the evolution up the maximum and then remains fixed without following the oscillations (see Figure 12) due to the post saturation dynamics, characterized by e-beam rotation in phase-space, induced by an exchange of power between laser field and electrons. A more complete view is reported in Figure 13. The physical content can be noted as reported below. The figure exhibits the first and third harmonics, along with the associated wave packet evolution (upper and lower panels). We have assumed the growth from a coherent seed, and no spiking appears before the saturation. With the onset of the power oscillations, the optical packets start to be characterized by the appearance of side bands, which degrade the laser pulse itself in correspondence with the minima of the oscillations [61; 62; 63]. The effects of beam qualities can be also included by replacing \(A(z)\) with approximants discussed in the previous section. Figure 11: Same parameters as in Figure 10 with the inclusion of the saturation. The slices have a flat distribution in \(z\), and each is characterized by an initial seed of 0.1 W. The vertical axis is in log-scale; the insets are plotted in linear scale. The pivotal element of the discussion of this paper has been centered around the \(FEL\) high gain equation, which, including the necessary modifications, is able to describe different physical situations regarding the \(FEL\) phenomenology. The \(FEL\) high gain equation has been derived from the \(FEL\) pendulum equation after a suitable linearization. A different treatment can be, however, exploited: it employs a Hamiltonian picture and the associated Liouville equation.