## 3 The Cerenkov FEL Oscillator  
A well-known class of waveguides is characterised by having open boundaries and the possible presence of an electromagnetic (e.m.) field outside the boundary; these guides support propagating modes called \"surface waves\" [13]. Dielectric films deposited over conducting plates are devices that fall into the aforementioned category; the surface waves, in this case, present an exponential behaviour, normal to the dielectric surface, that decays when moving away. Such a waveguide category is the relevant one for the realisation of the Cerenkov FEL device because the field-particle interaction occurs in the region of the space where the \"evanescent\" e.m. field is present. These propagating devices present field distributions associable with transverse electric (TE) or transverse magnetic (TM) modes. The case of TM modes is more favourable for the FEL use due to the presence of a longitudinal electric field component related to them. The vector potential for the TM modes of a single slab geometry waveguide is expressed by [9; 13]  
\[\left\{ \begin{array}{lcl}A_{x}&=&A_{0}\frac{\xi k}{\omega_{0}}e^{-qx}\sin( kz)e^{-i\omega t}\\ A_{y}&=&0\\ A_{z}&=&A_{0}\frac{\varepsilon}{\omega}e^{-qx}\sin(kz)e^{-i\omega t}\end{array} \right. \tag{4}\]  
where \(k\) indicates the longitudinal momentum, while \(q\) is the transverse momentum in vacuum and \(p\) is the transverse momentum in the dielectric of thickness \(d\). All these parameters are linked by two coupled \"dispersion relations\" that can be deduced by the field continuity along the vacuum-dielectric interface and dielectric-conductor interface:  
\[\left\{ \begin{array}{lcl}qd&=&\frac{pd}{\varepsilon}\tan(pd)\\ (pd)^{2}+(qd)^{2}&=&d^{2}\left(\frac{\omega}{\varepsilon}\right)^{2}( \varepsilon-1)\end{array} \right.\ \ \text{and}\ \left\{ \begin{array}{lcl}p&=& \sqrt{\varepsilon\left(\frac{\omega}{\varepsilon}\right)^{2}-k^{2}}\\ q&=&\sqrt{k^{2}-\left(\frac{\omega}{\varepsilon}\right)^{2}}\end{array} \right. \tag{5}\]  
In order to obtain the electric and magnetic fields from Equation (4), we can adopt the Coulomb gauge (\(\mathbf{E}\ =\ -\partial\mathbf{A}/\partial t\); \(\mathbf{B}\ =\ \
abla\times\mathbf{A}\)) which is useful for evaluating the fields in absence of charges and currents, as for the free modes in a guiding structure. Neglecting the coupling between the electrons and the transverse electric field \(E_{x}\) (due to the small values of the electrons' velocities in the \(x\) direction), we obtain for the longitudinal field:  
\[E_{z}\ =\ -cA_{0}e^{-qx}\sin(kz)e^{-i\omega t} \tag{6}\]  
In order to achieve the maximum field intensity, it is necessary to reach the saturation regime inside the Cerenkov waveguide resonator. The dynamics of the emission process in Cerenkov FEL has been established in the past as far as the spontaneous emission process [9; 14] is concerned and with regard to the stimulated emission mechanism and the gain coefficient calculation [14;15]. Let us, therefore, consider the case discussed in Section 2 while discussing Equation (2); the relevant parameters are summarised in Table 1.  
Table 1 reports the parameters for both a single slab geometry device [9] and for the double slab one [16] adequate for an operation around \(\lambda_{\text{C}}\) = 250 \(\upmu\)m. The behaviour of the spontaneous emission for a single-electron beam with a transverse size of \(\sigma_{x}\) = 1 mm (r.m.s. ), as a function of frequency and beam centroid distance from the dielectric surface, is reported in Figure 2a). The same analysis for the double slab geometry and a beam size of \(\sigma_{x}\) = 1 cm (r.m.s.) is reported in Figure 2b):  
The one-dimensional stimulated emission process, until saturation, can be studied by means of semi-analytical techniques [17] (three-dimensional theory can be found in [18]).