## 2 The Hybrid System  
The hybrid FEL system can be assumed to be composed of three main elements: (1) an electron beam accelerator; (2) a Cerenkov slow-wave guiding structure, with an adequate radiation resonator for the saturation regime accomplishment; and lastly, (3) a magnetic undulator acting as a radiator after the velocity modulation induced by an electric field associated with the radiation stored in the Cerenkov oscillator (see Figure 1).  
The first problem to face is related to the energy of the electron beam that will be common to both generating structures. As discussed in the introduction, the synchronism conditions for Cerenkov and undulator FELs are rather different and can be summarised as follows (see Refs. [9; 11]), where the subscript \(U\) and \(C\) indicate the undulator and Cerenkov, respectively:  
\[\lambda_{U}\ =\ \frac{\lambda_{w}}{2\gamma^{2}}\Big{(}1+K^{2}\Big{)}\,;\, \lambda_{C}\ =\ 2\pi\gamma d\bigg{(}\frac{\varepsilon-1}{\varepsilon}\bigg{)} \tag{1}\]  
where \(\lambda_{w}\) indicates the undulator period, \(K\) is the undulator parameter, \(d\) is the dielectric thickness of the Cerenkov guiding structure, \(\varepsilon\) is its dielectric constant and \(\gamma\) is the Lorentz parameter of the electrons. All these parameters will be more deeply described in the following sections.  
In order to obtain an efficient modulation of the electron beam, it is required that the radiation generated from the Cerenkov oscillator and from the Undulator radiator be \"correlated\"; a way to achieve this target is to obtain the condition: \(\lambda_{U}=\lambda_{C}/n\) where \(n=1\), \(2\),... specifies the harmonic number. Such a relation, with the use of Equation (1), becomes:  
\[\frac{\lambda_{w}}{2\gamma^{2}}\Big{(}1+K^{2}\Big{)}\ =\ \frac{2\pi\gamma}{n}d \bigg{(}\frac{\varepsilon-1}{\varepsilon}\bigg{)}\ \Leftrightarrow\ \gamma^{3}\ =\ \frac{\lambda_{w}}{4\pi}n\Big{(}1+K^{2}\Big{)}\frac{ \varepsilon}{d(\varepsilon-1)} \tag{2}\]  
Equation (2) gives the value of the electron energy as a function of all the parameters involved in both the Cerenkov and undulator FELs. In order to have an idea about the numbers, we can assign some reasonable values for the involved parameters in Equation (2) such as: \(\lambda_{w}\) = 2.5 cm, \(K\) = 1, \(d\) = 5 \(\upmu\)m, \(n\) = 1, \(\varepsilon\) = 5 (dielectric constant of quartz [12]) from  
Figure 1: Schematic drawing of the hybrid FEL under study. The accelerated electron beam generates radiation interacting with the Cerenkov resonator. The electric beam associated with the stored radiation modulates the electrons’ velocities in order to obtain the most efficient emission in the magnetic undulator radiator.  
which we obtain a result, for the electron energy, \(\gamma\)-10, that corresponds to a moderate relativistic energy. By inserting these values into Equation (1), we conclude that such a source would operate in the so-called far-infrared (FIR) spectral region (also called TeraHertz (THz)-region) because \(\lambda_{U}\)-\(\lambda_{C}\)-250 \(\upmu\)m.  
Another approach can be obtained by rearranging Equation (2) as follows:  
\[d\left(\frac{\varepsilon-1}{\varepsilon}\right)\ =\ \frac{\lambda_{w}}{4\pi\gamma^{3}} \left(1+K^{2}\right)n \tag{3}\]  
from which it is possible to obtain the Cerenkov parameters values starting from the undulator FEL ones. Let us suppose we have an FEL source with \(\lambda_{w}\) = 2.5 cm, \(K\) = 1, \(\gamma\) = 20 and again working on the fundamental harmonic \(n\) = 1, Equation (3) gives: \(\left[d(\varepsilon-1)/\varepsilon\right]\)-5.10\({}^{-7}\)\(\upmu\)m. Such a value may be obtained in several ways since, for instance, \(\varepsilon\) = 2 and \(d\) = 1 \(\upmu\)m. Using all these values for the hybrid FEL parameters, we eventually obtain from Equation (1) the value for the emitted wavelength from both the elements of the hybrid FEL: \(\lambda_{U}\)-\(\lambda_{C}\)-60 \(\upmu\)m.