## 4 Electron Beam Modulation and Undulator Emission
In particular, the electron phase \(\psi_{j}\), together with the interaction phase \(\theta_{l,\lambda,j}\) enter in a more general phase term in \(A_{l,\lambda}\) that, when summing up all of the contributions of the \(N_{e}\) electrons of the bunch, may generate constructive or destructive interactions. The maximum radiated power in the undulator is obtained when all the electrons contribute to the radiation formation with the same phase term (only depending on the harmonic number \(l\)), therefore, with an adequate distribution of the electrons in the phase-space. What has been discussed, so far, about Equation (10), refers to the longitudinal properties of the particle-wave interaction, which concur with the undulator parameter \(K\) and period \(\lambda_{w}\) and the electron longitudinal velocities \(\beta_{z,j}\). The transverse characteristics of the radiation emission are summarised by the mode aggregate index \(\lambda\). It is evident, however, that the transverse space can play a role when the electron beam emittance values could be relevant for the radiation wavelength to be generated. A deeper and exhaustive analysis can be found in [27]. With such a mathematical background, an appropriate computational code was set up for the evaluation of the coherent generation of radiation in the undulator-radiator. The code is capable of evaluating the impact of an electron bunch, with a specific distribution in the phase-space, for the radiation generation, calculating the contribution of any electron, integrated along the interaction length. The code, at the moment, does not take into account saturation effects. The effect of the energy modulation introduced by the electron-field interaction inside the Cerenkov FEL optical resonator can be assessed by comparing the emission from an electron distribution \(D\big{(}\gamma_{j},\psi_{j}\big{)}\) and the same distribution to which an energy modulation is added \(D\big{(}\gamma_{j}+\Delta\gamma\big{(}\psi_{j}\big{)}_{j},\psi_{j}\big{)}\), where the contribution \(\Delta\gamma\big{(}\psi_{j}\big{)}_{j}\) is calculated from Equation (9) for each electron. To this aim, an ad hoc electron distribution was generated in the phase-space having a uniform partitioning in energy with a total \(\Delta\gamma\) = 0.2 around \(\gamma\) = 10 (see Table 3) corresponding to a \(\sigma_{\gamma}\approx 5\cdot 10^{-3}\) r.m.s. The phase partitioning is again uniform, with an added Poissonian component, with a total \(\Delta\psi\) = 20\({}^{\circ}\) corresponding to the actual micro-bunch duration expressed as phase interval for Linac [11]. In Figure 5, we report the aforementioned electron distribution (Figure 5a) together with the corresponding power spectrum (Figure 5b) calculated with Equation (10). The total power, integrated over the whole bandwidth, results in \(P_{TOT}\)-55 W.  
As anticipated, a similar calculation can be performed for the electron distribution of Figure 5a), to which the energy modulation \(\Delta\gamma_{j}\) of Equation (9) is added for each electron of the distribution with its actual phase \(\psi_{j}\) and energy (through the velocity \(\beta_{c,j}\)). The amplitude of the electric field is the one evaluated at the end of Section 3 from the saturated intracavity radiation power \(E_{0z}\)-1.8 \(\times\) 10\({}^{7}\) [V/m]. In Figure 6, we report the energy modulated electron distribution (Figure 6a) together with the corresponding power spectrum (Figure 6b), again calculated by means of Equation (10). As can be clearly seen, the emission is strongly increased by orders of magnitude due to the modulation introduced which, moreover, is not clearly distinguishable when looking at Figures 5a and 6a. It is further worth underlining how the total bandwidth is slightly reduced with respect to the unmodulated case. The total power, integrated over the whole bandwidth, now results as \(P_{TOT}\)-1.28 \(\times\) 10\({}^{7}\) W.  
Figure 5: Emission by undulator–radiator: not the modulated case: (**a**) phase-space; and (**b**) power spectrum.