# Gyrotrons as High-Frequency Drivers for Undulators and High-Gradient Accelerators
## 3 Gyrotron-Powered Microwave Wigglers and Undulators  
The principles of gyrotron-powered electromagnetic wigglers for FELs have been proposed and discussed in the pioneering paper by [PERSON] et al. [14] a long time ago. Analogous concepts have been developed also in [36]. A generic arrangement of an FEL with a gyrotron-driven wiggler/undulator is shown in Figure 3.  
On the basis of a theoretical analysis [14], it has been shown that the usage of short wavelength electromagnetic wigglers in waveguides and resonant cavities can significantly reduce the required electron beam accelerating voltages and in such a way would allow building compact FEL's. The gain calculations have been performed for both the low- and high-gain Compton regime and take into account the effects of emittance, transverse wiggler gradient, and thermal spread of the electron velocities. This paper gives also an insight into the scaling laws for the gain and the required electromagnetic wiggler field power. Moreover, both traveling wave and standing wave resonator fields have been considered as possible electromagnetic wigglers. In the former case, TE\({}_{\rm 1n}\) (\(n\) = 2, 3, 4... ) modes in a cylindrical waveguide have been examined. They have fields on the axis that can be approximated by plane waves. Such modes are well suited for use as wigglers with high voltage axial electron beams. Moreover, high-power gyrotrons can produce significant power when operating in these modes. Alternatively, an elect  
Figure 3: Schematic of an FEL with gyrotron-powered microwave wiggler/undulator (different components not to scale). in a resonator can be used in order to achieve high values of the electromagnetic field by storage of the gyrotron power in a single high-Q cavity mode TE\({}_{\text{1 nl}}\) (l being the axial mode index). The resulting standing wave wiggler field can be represented as the sum of two oppositely propagating traveling waves of equal amplitude. This results in two separate FEL resonance conditions corresponding to the two different components of the motion due to the forward and backward waves. Since the low-frequency resonances are undesirable, special design solutions need to be used to prevent oscillations on them. In the case of a standing wave wiggler, several different design schemes are possible using a conventional gyrotron (with a waveguide cavity) or a quasi-optical gyrotron (with an open-mirror resonator) [14]. Moreover, the power source may be separated from the wiggler cavity, powering the latter through an external coupling. As pointed out in the pioneering paper [14], in this case, \"high-power microwave or millimeter wave circulators or isolators would be necessary to prevent reflection of power back to the source. Although these components may be feasible for lower frequency EM wigglers (\(\lambda\)\(\sim\)0.5-1 cm), higher frequency wigglers are likely to require an alternate concept.\" So far, this severe problem is not solved for the high-frequency gyrotrons and the only available means is the careful coupling and matching the wave-beam with the RF undulator so that to minimize the reflections. One of the first configurations that have been proposed in [14] is that of an FEL's operating in the infrared region using moderate energy (<10 MeV electron beams) with electromagnetic wigglers powered by millimeter-wavelength gyrotron. In this scheme, both the gyrotron and the FEL interaction take place in the same closed cavity which, however, has two sections. The gyrotron interaction occurs in the first region where there is the necessary axial magnetic field (needed for its operation) and where the wave number of the excited mode is almost entirely in the transverse direction (i.e., close to cut-off). The radiation produced in the gyrotron section is trapped inside the high-Q cavity and acts as a wiggler for the FEL interaction there. The electron beam accelerated to the required energy enters the cavity through tapered openings (which are cut-off to the trapped mode) or through slots in the resonator walls.