# Gyrotrons as High-Frequency Drivers for Undulators and High-Gradient Accelerators
## 2 Microwave Wigglers and Undulators  
The free-electron lasers (FELs) are powerful sources of coherent radiation that have demonstrated the widest frequency range (unattainable to other devices) ranging from microwaves, through terahertz radiation and infrared, to the visible spectrum, ultraviolet, and X-rays. An indispensable core component of any FEL is one or another _insertion device_ (ID), called in such a way since it is placed (inserted) into the accelerator track of the electron beam. Generally, the IDs are of two types, namely _undulators_ and _wigglers_ which are composed of periodic magnetic structures in which the electron orbits undergo wiggles or undulations [10; 11; 12]. Being structurally equivalent, these two types of IDs can be distinguished using one of their most important characteristics, namely the dimensionless K-factor defined as (in centimeter-gram-second (CGS) units) and also in more practical units [12; 13]  
\[K=\frac{eB\lambda_{u}}{2\pi mc^{2}}=0.934B[\mathrm{T}]\lambda_{u}[\mathrm{cm }]\, \tag{1}\]  
where \(e\) and \(m\) are the magnitude of the electron charge and the electron mass, respectively, \(B\) is the maximum intensity of the magnetic field, \(\lambda_{u}\) is the period of the ID which depends on the arrangement of the used magnetic dipoles (dimensions, gaps between them, etc.) that can be coils (most frequently superconducting) or permanent magnets (e.g., Halbach arrays) [13]. Depending on the value of \(K\) (also known as \"strength parameter\" or \"deflection parameter\" [11]), the IDs are usually classified as wiggler if \(K>>1\) and undulators if \(K\leq 1\) as illustrated in Figure 1. This definition (see, e.g., [10]), however, should not be taken literally because it simply reflects the fact that generally the wigglers are distinguished from the undulators by a larger value of \(K\). As pointed out in [10], there is a grey area in the distinction between undulators and wigglers when \(K\) is in the range of between 1 and 10. Some authors define the boundary between undulators and multipole wigglers as \(K<\)1 and \(K>1\), respectively, others at \(K<5\) and \(K>5\)[10]. The main difference between these IDs is the larger field amplitude (\(B\)) and the longer period (\(\lambda_{u}\)) in the wiggler case. For example, a typical undulator presented in [11] with a period of 50 mm and a peak field intensity of 0.5 T has a \(K\) value of about 2.5. On the other hand, the typical wigglers having a magnetic field in the range of 1.5-2 T and a period of 125-200 mm are characterized by \(K\) values in the range of 20-40 [11]. Moreover, a wiggler could also be operated as an undulator by reducing the field until \(K\) becomes small enough [13]. More important, however, is that the spectral characteristics of the undulator and wiggler radiation sources are different. Most notably, in a wiggler, every electron in a bunch radiates independently (incoherently), and as a result, the spectrum of generated radiation is broadband and similar to that from a bending magnet. On the other hand, in a device employing an undulator, the radiation, emitted at different places by one oscillating electron inside the undulator (by different synchronously oscillating electrons), interferes constructively leading to a relatively narrow bandwidth of the spectrum. Moreover, in the latter case, the intensity of the radiation scales with the squared number of the poles in the magnet array. Additionally, an undulator provides a higher power per frequency, since its output is generally confined to a single frequency or its harmonics differ from the wiggler which provides radiation distributed in a wider spectral range. Furthermore, the brilliance of the undulator radiation could be significantly higher because it is confined to a much narrower beam (a cone with an aperture angle \(K/\gamma\)). Despite the progress in the development of magnet-based IDs, they have some well-known problems, e.g., the impossibility to control the polarization of the radiation and to change the undulator period.