# Radio-Frequency Undulators, Cyclotron Auto Resonance Maser and Free Electron Lasers
## 1 Introduction
In Figure 2 we report \(\gamma\) vs \(\lambda_{w}\) for two different values of the FEL wavelength (\(\lambda_{r}=1\) nm and \(\lambda_{r}=5\) nm). The corresponding \(K_{w}\) values are apparently small, for a safe FEL operation (below 0.5 for the region of interest). The choice of a sufficiently large \(\rho\) (with a suitable combination of energy and current values) ensures the saturation in a reasonable saturation length. Figure 1: Comparison between Laser Wave (or optical), magnetic and RF undulators from bottom right to top right, respectively, for an output of X-ray radiation with \(\lambda_{r}=1\) nm. The figure reports an electron beam emitted from the photocathode to the “undulator”, after a suitable acceleration. The expected undulator lengths and the e-beam energies necessary to reach the SASE-FEL performances are underlined too. The key quantity of FEL dynamics is the so-called Pierce parameter [18]  
\[\rho\approx\frac{8.36\times 10^{-3}}{\gamma}\Big{(}J\cdot(K_{w}f_{b}\lambda_{w})^{ 2}\Big{)}^{1/3}, \tag{4}\]  
where \(J=1/\Sigma\), with \(\Sigma=2\pi\sigma_{x}\sigma_{y}\), is the electron beam current density, namely the current \(I\) divided by the transverse area of the electron beam (with section \(\sigma_{x,y}\)) and \(f_{b}\) the Bessel function factor (which for circular polarized wave can be taken to be 1 [15]). In Figure 3 we report the beam current density, necessary to support a Pierce parameter of around \(5\times 10^{-4}\) for two different FEL operating wavelengths, following from Equation (4). \[J=1.71\times 10^{6}\frac{(\gamma\rho)^{3}}{(K_{w}f_{b}\lambda_{w})^{2}}. \tag{5}\]  
The importance of \(\rho\) stems from the fact that it controls all the significant parameters for the SASE FEL, like the gain length specified in [18],  
\[L_{g}=\frac{\lambda_{w}}{4\pi\sqrt{3}\rho}, \tag{6}\]  
which in turns defines the saturation length \(L_{s}\), which can be quantified as [18]  
\[L_{s}\simeq 20L_{g}. \tag{7}\]  
Figure 3: Electron beam current density (A/m\({}^{2}\)) vs. the undulator wave length for \(\lambda_{r}\equiv 1\) nm (continuous line) and \(\lambda_{r}\equiv 5\) nm (dashed line). The nonideal beam qualities (finite relative energy spread and emittance) determine a dilution of the gain and a consequent increase of the saturation length. The \(\rho\) parameter is helpful to quantify these detrimental contributions. The effect of the relative energy spread on the saturation length is, e.g., negligible if [18]  
\[\sigma_{e}<\frac{\rho}{2}, \tag{8}\]  
with the chosen value of \(\rho\) we expect an undulator length (for \(\lambda_{w}=1\) mm) of about 2 m and a beam with relative energy spread around 0.25%. The request on the electron beam parameters in terms of energy and current does not appear challenging. If we consider a beam with a normalized emittance of 1 mm\(\cdot\)mrad and assume that it is focused with an average transverse section of less than 60 \(\upmu\)m (for \(\lambda_{r}\in[1,5]\) nm), along the interaction region, the request on the necessary peak current are constrained within reasonable limits (see Figures 4 and 5). \[K_{w}=8.5\times 10^{-15}\cdot\lambda_{w}[nm]\sqrt{I_{w}[W/m^{2}]}\]0 Electron beam cross section (\(\upmu\)m) vs. the undulator wavelength for \(\lambda_{r}\equiv 1\) nm (continuous red line) and \(\lambda_{r}\equiv 5\) nm (dashed blue line); assuming a normalized emittance value of 1 mm mrad with a \(\beta_{x,y}\) Twiss value of 5 m/2\(\pi\) and using the Equation 3 with the same parameters reported in \[K_{w}=8.5\times 10^{-15}\cdot\lambda_{w}[nm]\sqrt{I_{w}[W/m^{2}]}\]1  
\[K_{w}=8.5\times 10^{-15}\cdot\lambda_{w}[nm]\sqrt{I_{w}[W/m^{2}]}\]2 Electron beam current vs. the undulator wavelength for \(\lambda_{r}\equiv 1\) nm (continuous line) and \(\lambda_{r}\equiv 5\) nm (dashed line); derived from Equation (5) using the parameters values reported in Figure 4 and assuming \(\sigma_{x}=\sigma_{y}\).