# Gyrotrons as High-Frequency Drivers for Undulators and High-Gradient Accelerators
## 3 Gyrotron-Powered Microwave Wigglers and Undulators
Several promising configurations for RF undulators powered at millimeter wavelengths and designed to produce coherent nanometer radiation from sub-GeV electron beams are analyzed and compared with one another in [23]. These schemes include a traveling-wave resonant ring, a standing wave resonator, and a resonator operating close to the cutoff. One of the most advanced recently proposed concepts is the so-called \"flying undulator\", which represents a short microwave pulse that synchronously propagates with the driving electron bunch and provides pumping of its oscillations. Profiling the parameters of a \"flying\" undulator [26] allows the realization of the non-resonant trapping regime resulting in further enhancement of the FEL efficiency. Recently, such undulators designed for the K and W bands have been studied both numerically and experimentally (\"cold tests\") [38]. Waveguide-type microwave undulator based on a helically corrugated waveguide that supports a traveling wave has been studied in [28; 33]. The use of such an approach allows partial conversion of a traveling wave mode into a backward traveling wave mode, thus increasing the effective interaction length. When operating with a TE\({}_{11}\) mode at 30.3 GHz, the undulator is predicted to realize field strength of \(\sim\)0.3 T and an undulator period of 4.95 mm when driven by 1 GW of microwave power. The authors of [28] have selected the TE\({}_{11}\) mode as the operating one because it has a strong equivalent transverse magnetic field in the waveguide axis, as well as a large flat field region in the waveguide center, although it has the disadvantage of large Ohmic losses. Other possible operating modes have also been considered. One of them is the TE\({}_{01}\) which has the lowest Ohmic losses. As already mentioned, the enormous power required by the RF undulators makes it necessary to use resonant cavities for the accumulation of microwave energy or pulse compression power amplification [33]. The filling of the cavity stored energy \(W(t)\) at time \(t\) is given by [PERSON]'s formula [39; 40] (with a few changes in the notations)  
\[W(t)=P_{0}\tau_{0}\frac{4q}{\left(1+q\right)^{2}}\left[1-\exp\left(-\frac{1+q }{2}\frac{t}{\tau_{0}}\right)\right]^{2}, \tag{6}\]  
where \(P_{0}\) is the input power produced by an microwave source (e.g., gyrotron) and coupled to the resonator with an intrinsic (Ohmic) quality factor \(Q_{0}\) (at frequency \(\omega\)) by an input port. Here, \(\tau_{0}=Q_{0}/\omega\), \(q=Q_{0}/Q_{e}\) and \(Q_{e}\) is the external Q-factor. The quantity \(q\) is therefore the coupling factor for the cavity and the input port since it is related to the fractions of the input powers that are transmitted and reflected. In a steady-state (when \(\gg\tau_{0}\)) the stored energy reaches the maximum value of \(P_{0}\tau_{0}\) if \(q=1\) (i.e., if the cavity is properly coupled). In this case, the input power fully compensates the Ohmic losses. As evident from (6), in the cases of under-coupled (\(q<1\)) and over-coupled (\(q>1\)) cavity the stored energy is smaller. The filling time of the cavity is [33]  
\[t_{tr}=\frac{Q_{0}}{\omega}\frac{2}{1+q}\ln\left(\frac{1+\chi^{1/2}}{1-\chi} \right)\,, \tag{7}\]  
where \(\chi\) is the charging factor defined as the ratio of the charged energy and its maximum value. From the above equations, it follows that cavities with high \(Q\) factors are more preferable in order to achieve maximum storage at given input power although the filling time also increases. Although in this section we were focused on the gyrotron-powered undulators it should be mentioned that their alternatives (and competitors) that are based on lasers are considered very promising. Among them are the recirculated wave undulators for compact FEL [41; 42]. The system employs a recirculated radiation pulse serving as an undulator provided by a high-power laser. The scheme uses an optical ring cavity composed of two plane mirrors, two parabolic mirrors, and one focusing lens.