# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 1 Introduction  
The interest in Free Electron Laser (FEL), during the last decade, has moved toward devices producing X-ray beams with high brilliance matching the requirements for many applications such as nuclear materials detection [1], small-angle X-ray scattering [2], phase contrast imaging [3], macromolecular X-ray crystallography for drug discovery [4] and X-ray microscopy [5]. The Linac Coherent Light Source (LCLS) at SLAC National Accelerator Laboratory [6], FLASH at DESY Deutsches Elektronen-Synchrotron [7] with other advanced X-ray FEL [8, 9, 10, 11, 12, 13] provide, or have been designed to provide, high-brightness X-ray beams, but size, cost and operational complexity are the main drawbacks for a widespread use in small laboratories, hospitals, universities. The X-FEL technology is based on two pillars, high-energy (multi-Gev) high-quality electron beam Linacs and hundred-meters-long undulators. A high-brilliance X-ray device demands, accordingly, large facilities and therefore any progress towards \"compact\" X-ray FELs requires technological improvements in terms of high-gradient accelerators and/or short \"wavelength\" undulators. The last possibility will be considered in this paper. The availability of powerful electromagnetic sources (lasers or RF sources) offers the possibility of replacing magnetic with electromagnetic undulators. The advantage would be that of reducing both the size of the undulator and the energy of the e-beam (electron beam). In Figure 1 we sketch out the scaling of the beam energy and of the undulator length, with magnetic or RF wave undulators. The last option comprises either laser and RF solutions. The use of an electromagnetic undulator pumped by a GHz RF field allows, for the same FEL wavelength, to reduce the energy of the electron beam by several units and the length of the magnet by more than one order of magnitude. We provide below a preliminary idea of how electron beam and Radio Frequency Undulators (RFU) combine to drive an X-ray FEL Self-Amplified Spontaneous Emission (SASE) device. It is evident, as underscored below, that the price to be payed to exploit RFU, in a FEL-SASE operation, is that of employing high intensity fields. We do notspecify, for the moment, any electromagnetic source, we fix a reference RF power and the relevant operating wavelength to specify electron beam parameters suitable for FEL SASE X-ray operation. It is well known that, from the conceptual point of view, wave or magnetic undulators do not change the FEL physics and the associated design criteria. In both cases, the emission process can be traced back to Compton/Thomson backscattering and after establishing the suitable correspondence, in terms of FEL strength parameter, most of the design strategy and scaling properties work as in the case of \"ordinary\" magnetostatic undulators [14; 15]. One of the pivotal parameters ruling the electron dynamics inside the undulator is the so-called strength \(K\) which measures the amount of electron transverse momentum, acquired inside the undulator, allowing the coupling between the e-beam and a transverse co-propagating wave. The parameter \(K\) is specified  
1. for the magnetostatic case with on-axis field intensity \(B_{0}\) and period \(\lambda_{w}\), by [16] \[K_{m}=\frac{eB_{0}\lambda_{w}}{2\pi m_{e}c^{2}}\] (1)
2. for a RF/wave undulator with the microwave field power density \(I_{w}\), by [17] \[K_{w}=8.5\times 10^{-15}\cdot\lambda_{w}[nm]\sqrt{I_{w}[W/m^{2}]}\] (2)  
where the subscripts \(m,\ w\) stand for magnetic and wave, respectively, while \(e,\ m_{e}\) denote the electron charge and mass, respectively, \(\lambda_{w}\)--the RF wavelength and \(c\) is the speed of light. The electron relativistic factor \(\gamma\) is linked to the \(K\) parameter, FEL operating wavelength \(\lambda_{r}\) and undulator period by  
\[\gamma=\frac{1}{2}\sqrt{\frac{\lambda_{w}}{\lambda_{r}}(1+K^{2})}. \tag{3}\]  
being \(K=K_{w}/\sqrt{2}\), for a linearly polarized RF wave.