## 4 Electron Beam Modulation and Undulator Emission  
At the end of the previous section, we saw how an intense electromagnetic pulse, that can be generated inside an optical resonator in about 200 nanoseconds, needs to be saturated. For electron macro-bunches, as long as few microseconds we will have thousands of micro-bunches available interacting with the electromagnetic field. If we analyse each of these electron micro-bunches, we recognise that, at the resonator entrance, each of them superimposes with a correspondent radiation micro-pulse as illustrated in Figure 4a). In such a situation, the electrons of the bunch, generally equally spaced in phase, will experience the oscillating longitudinal electric field \(E_{z}\) (Equation (6)) with a spatial periodicity of \(\lambda_{C}\), as showed in Figure 4b). The situation represented in Figure 4 is quite similar to that of an electron beam accelerated by a series of RF cavities such as those of a standing-wave Linac [11; 24]. The number of cavities can be estimated by calculating the number of the electric field oscillations \"contained\" within the electron micro-bunch length (\(\delta z\)-\(\tau_{c}\beta_{c}c\)) divided by the wavelength \(\lambda_{C}\): \(N_{c}\)-(\(\delta z\)/\(\lambda_{C}\)). The most widely used accelerators for FELs operate in the so-called S-band (\(\
u_{RF}\)-3 GHz) and generate micro-bunches of a typical duration of \(\tau_{c}\)-15 ps, which corresponds to a bunch length of about \(\delta z\)-5 mm. Considering a radiation wavelength of \(\lambda_{C}\)-250 \(\mu\)m, as for the example discussed in the previous sessions, we obtain \(N_{c}\)-20. It is possible, therefore, to imagine any optical cycle acting on the electrons in a way similar to that of the accelerating cavities of a Linac. We know, in fact, that electrons undergo to an energy variation that is connected to the particle-radiation relative phase. Such an energy variation will be positive for electrons close to the so-called accelerating phase and negative for those with an opposite phase. The result is a bunching of the electrons into \(N_{c}\) \"slices\". Figure 4: _Cont._  
To properly evaluate the energy variation of a single electron, let us consider that the energy exchange occurs along the time needed for the electron, to cross the resonator length \(L\). The particle, during this time \(T=L/(c\beta_{e})\), experiences a quasi-static electric field due to the fact that, even with small differences, its speed is almost synchronous with the electromagnetic pulse that travels with its group velocity \(\beta_{g}~{}=~{}d(\omega/c)/dk\) (see Equation (5)), and therefore, \(\beta_{g}\)-\(\beta_{e}\). The energy variation, for each electron, is consequently:  
\[\Delta\gamma~{}=~{}\frac{\Delta\mathrm{E}}{\mathrm{m}_{0}\mathrm{c}^{2}}~{}=~ {}\frac{\mathbf{F}\cdot\mathbf{s}}{\mathrm{m}_{0}\mathrm{c}^{2}}~{}=~{}\int \limits_{0}^{T}\frac{\mathrm{e}\mathrm{E}(t)}{\mathrm{m}_{0}\mathrm{c}^{2}} \cdot\mathbf{v}_{\mathrm{e}}\mathrm{dt}~{}=~{}\frac{\mathrm{e}\mathrm{E}_{0 2}}{\mathrm{m}_{0}\mathrm{c}^{2}}c\beta_{\mathrm{e}}\int\limits_{0}^{T}\cos( \omega t+\phi)\mathrm{dt} \tag{8}\]  
where the longitudinal electric field has been derived from Equation (6) and the constant term \(E_{02}\) contains the transverse coupling between the electron and the evanescent field in the x direction. Before getting an explicit form for Equation (8), it is worth underlining that the phase \(\varphi\) usually refers to the RF field, at an angular frequency \(\omega_{\mathrm{RF}}\), that accelerates the electron beam in the accelerator (Linac, for instance). In the present case, the phase \(\varphi\) should be referred to as the Cerenkov radiation field in the optical resonator, and therefore, ought to be \"normalised\" by means of the frequency ratio \(\omega_{C}/\omega_{RF}\): \(\phi\Rightarrow\phi(\omega_{C}/\omega_{RF})~{}=~{}\phi(\lambda_{RF}/\lambda_{C })\).