## 4 Electron Beam Modulation and Undulator Emission
After some algebra, we obtain:  
\[\Delta\gamma~{}=~{}\frac{eE_{02}}{m_{0}\mathrm{c}^{2}}L\Bigg{[}\frac{\sin( \frac{\omega}{c}\frac{L}{P_{e}})}{\left(\frac{\omega}{c}\frac{L}{P_{e}}\right) }\cos(\phi\frac{\lambda_{RF}}{\lambda_{C}})+\frac{\cos(\frac{\omega}{c}\frac{ L}{P_{e}})}{\left(\frac{\omega}{c}\frac{L}{P_{e}}\right)}\sin(\phi\frac{\lambda_{ RF}}{\lambda_{C}})\Bigg{]} \tag{9}\]  
The result of what has been discussed is that the energy modulation expressed by Equation (9) can be exploited to generate powerful radiation inside a magnetic undulator utilised as a radiator. The discussion after Equation (2) in Section 2 leads to a set of parameters, summarised in Table 3, for an undulator synchronous with the fundamental Cerenkov emission. Figure 4: Electron bunch and radiation pulse interaction: (**a**) phase locking; and (**b**) electrons and radiation electric field interaction. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Undulator–Radiator FEL Parameters @ \(\lambda_{U}\) = 250 \(\upmu\)m**} \\ \hline Electron & Undulator & Undulator & Number of & Waveguide & Waveguide \\ Energy & Period & Parameter & Periods & Width & Height \\ \hline \(\gamma\) = 10.0 & \(\lambda_{\mathrm{i}w}\) = 2.5 cm & =1.0 & \(N_{L}\) = 50 & \(W\) = 0.5 cm & \(H\) = 0.5 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Undulator–Radiator FEL Parameters @ \(\lambda_{U}\) = 250 \(\upmu\)m.  
The emission of radiation from a radiator by means of an ensemble of charged particles distributed in the phase space has been faced in the recent past [25; 26]. To summarise some concepts, we can say that the electron beam, after the interaction, carries a modulation of the current \(\stackrel{{\rightarrow}}{{J}}\) that can be expressed as a Fourier expansion in terms of the harmonics \(l\) of the RF. Moreover, the electromagnetic field can be expanded in terms of the transverse modes, expressed by the aggregate index \(\lambda\), of the waveguide that is needed to confine the radiation in the far infrared spectral range, and expressing the expansion coefficients as \(A_{l,\lambda}\). Finally, the power \(P_{l,\lambda}\), emitted due to the energy exchange between the electron beam and the radiation field in the magnetic field of the undulator, can be evaluated by means of the flux of the Poynting vector. Equation (10) summarises all the above in an extensive demonstration which can be found in [25] and the references therein:  
\[ \begin{array}{l}\stackrel{{\rightarrow}}{{J}}=\sum\limits_{l =1}^{\infty}\stackrel{{\rightarrow}}{{J}}_{l}\exp(-i\omega_{l}t) \ ;\ \omega_{l}\ =\ 2\pi\frac{l}{T_{RF}}\\ A_{l,\lambda}\ =\ -\frac{Z_{0}}{2\beta_{S}}\ \frac{e}{T_{RF}}\frac{KL}{\sqrt{\Sigma_{ \lambda}}}\sum\limits_{j}^{N_{e}}\frac{1}{\beta_{z,j}/\gamma_{j}}\ \frac{\sin(\theta_{\lambda,\lambda j}/2)}{\theta_{l,\lambda j}/2}\ i\ \exp \left(\frac{\theta_{l,\lambda j}}{2}+l\psi_{j}\right)\ ;\ \theta_{l,\lambda,j}\ =\ \left(\frac{ \omega_{l}}{\varepsilon P_{2,j}}-\frac{2\pi}{\lambda_{w}}-k_{\lambda}\right)\\ P_{l,\lambda}\ =\ \frac{\beta_{S}}{Z_{0}}\left|A_{l,\lambda}\right|^{2}\end{array}  \tag{10}\]  
where \(e\) is the electron charge, \(Z_{0}\) is the vacuum impedance, \(\Sigma_{\lambda}\) is the radiation transverse mode size, \(N_{e}\) is the number of electrons contained in the bunch, \(\beta_{z,j}\) is the longitudinal velocity of the j-th electron, \(\beta_{S}\) is the radiation group velocity and \(k_{\lambda}\) is the transverse mode momentum. A brief analysis of Equation (10) tells us that the expansion coefficient \(A_{l,\lambda}\) plays a significant function in the determination of the radiation formation. It is, in fact, the role of each electron in the bunch is relevant because it carries a specific energy \(\gamma_{j}\) and a specific phase \(\psi_{j}\).