# Gyrotrons as High-Frequency Drivers for Undulators and High-Gradient Accelerators
## 2 Microwave Wigglers and Undulators
From the electron beam dynamics point of view, it is clear that the usage of magnetic IDs is not the only possible way to provide wiggling (undulating) electron orbits and the idea to replace them with electromagnetic waves (generated by lasers, gyrotrons (and other gyro-devices like Gyro-Klystrons, Gyro-BWO, etc.)) has been conceived a long time ago [14; 15; 16; 17; 18; 19; 20; 21]. As an illustration, a schematic of an electromagnetic wiggler/undulator powdered by an RF source is shown in Figure 2. The development of the concepts of gyrotron-driven microwave (aka RF) wigglers [14; 15; 16; 17] and undulators [18; 19; 20; 21] started soon after the gyrotrons demonstrated their potential to provide powerful coherent radiation with wavelengths ranging from centimeters to millimeters. Since these first conceptual design and feasibility studies, the research on this topic continues and is focused on novel advanced concepts [22; 23; 24; 25; 26; 27; 28; 29; 30] using other radiation sources as well. Figure 1: Schematic comparison of a wiggler and an undulator (not to scale) based on periodic magnet structure. Figure 2: Schematics of microwave wiggler/undulator powered by RF radiation source. In a microwave wiggler/undulator both the transverse electric \(E_{\perp}\) and magnetic field \(H_{\perp}\) of the electromagnetic wave cause wiggling of the electrons and the \(K\) parameter takes the form (in CGS units) [23; 30; 31]  
\[K_{w}=\frac{e(E_{\perp}+H_{\perp})}{mc^{2}(h+k)}=\frac{e(E_{\perp}+H_{\perp}) \lambda_{w}}{2\pi mc^{2}}\equiv\frac{eH_{\perp}(1+Z)\lambda_{w}}{2\pi mc^{2}}, \tag{2}\]  
where the \(\lambda_{w}=\frac{2\pi}{h+k}\) is the undulator period, \(h=2\pi/\lambda_{g}\) is the propagation constant of a wave (with a wavelength \(\lambda_{g}\)) in the waveguide at angular frequency \(\omega\), and \(k=\omega/c\) is the wavenumber of light in vacuum. The wave impedance \(Z=E_{\perp}/H_{\perp}\) for the TE and TM modes is \(Z_{TE}=\frac{k}{h}=Z_{0}/\sqrt{1-\left(\omega_{c}/\omega\right)^{2}}\) and \(Z_{TM}=\frac{h}{k}=Z_{0}\sqrt{1-\left(\omega_{c}/\omega\right)^{2}}\), respectively. In these relations \(Z_{0}\) is the impedance of the free space and \(\omega_{c}\) is the cut-off frequency of the waveguide/cavity. The wave impedance \(Z\) is an important mode-specific parameter that characterizes the used electromagnetic wave. For an operation far from the cut-off (\(\omega\gg\omega_{c}\)) it is close to the impedance of the free space. In more practical units,  
\[K_{w}=8.5\times 10^{-15}.\lambda_{w}[nm]\sqrt{I_{w}\left[\frac{W}{m^{2}}\right] }\, \tag{3}\]  
where \(I_{w}\) is the microwave power density [30]. Thus, for a specified (targeted) value of \(K_{w}\), the required power density \(I_{w}\) can be evaluated using (3). For instance, for \(K_{w}=1\) and \(\lambda_{w}=\)1 cm, this gives \(I_{w}=14\) GW/cm\({}^{2}\). If the pump wave with an angular frequency \(\omega_{p}\) and the electron beam propagate towards each other, the frequency of the field oscillation \(\omega^{\prime}\) \"seen\" by the electrons will be Doppler shifted  
\[\omega^{\prime}=\omega_{p}\gamma\Big{(}1+\frac{v_{z}}{c}\Big{)}, \tag{4}\]  
where \(\gamma\) is the relativistic Lorentz factor and \(v_{z}\) is the axial velocity of the electrons. Thus, the radiation frequency in the laboratory system is  
\[\omega=\omega_{p}\gamma^{2}\Big{(}1+\frac{v_{z}}{c}\Big{)}. \tag{5}\]  
Therefore, by controlling the energy of beam electrons the frequency of generated radiation can be easily tuned in a wide frequency band. Since the factor \(\gamma^{2}\) can be quite large (recall, for example, that for an electron energy of 511 MeV, \(\gamma\) amounts to 1000) this frequency is much higher than the frequency of the pumping wave. The emission arising under the action of the pump wave can be regarded as Compton scattering of the pump wave on moving electrons [32]. That is why, sometimes such devices are called FEL-Scattertrons.