# Comments on the Physics of Microwave-Undulators
## 3 MU Spectral Properties and Low Gain FEL Devices
Figure 4: (a) \(F_{n}(K_{u})=\left[\frac{n\,K_{u}}{1+\frac{K_{u}^{2}}{2}}\right]^{2}f_{b,n}^ {2}\) vs. microwave power, n = 1 (solid line), n = 3 (dotted line), n = 5 (dashed line); (b) strength parameter \(K_{u}\) vs. \(\frac{\omega_{c}}{\omega}\) for different values of the microwave power, 300 MW (solid), 200 MW (dotted), 100 MW (dashed). \[A_{n}(\omega)\propto\left[\frac{\sin\left(\frac{\
u_{n}}{2}\right)}{\frac{V_{n} }{2}}\right]^{2},\;\
u_{n}=2\pi N_{u}\left(\frac{\omega-n\omega_{r}}{n\omega_{ r}}\right),\]0 (a) Equivalent undulator period vs. \(x=\omega_{c}/\omega\); (b) resonant wavelength vs. \(\gamma\) (dot \(x=0.9\), continuous \(x=1/\sqrt{2}\)). The small signal gain coefficient evaluated with the previously listed parameters is larger than 30%, and in absence of any inhomogeneous broadening detrimental effect (see the next section), we find the maximum gain [30]\(G\cong 0.85g_{0}+0.19g_{0}^{2}\cong 29\%\). We have assumed a relative energy spread of the order of 0.1% to avoid significant gain reduction due to line inhomogeneous broadening. The amount of gain detrimental effect according to the parameters listed in Table 1 is a reduction factor \(R=0.786\), which is not particularly harmful (for further comments see below). This small energy spread requires particular care in the choice of the Linac, which can be, e.g., super conducting. The operation with an oscillator requires a low losses cavity where the round-trip intra-cavity signal grows as indicated in Figure 7, where we have exploited the following relationships \begin{table}
\begin{tabular}{c c} \hline Beam Current [A] & 120 \\ \hline Beam Energy [MeV] & 60 \\ \hline Normalized Beam emittance [mm\(\cdot\)mrad] & 5 \\ \hline Twiss Parameter \(\beta\) [m] & 5 \\ \hline Beam Relative Energy spread (\(\sigma_{z}\)) & 0.1\% \\ \hline Microwave Power [MW] & 100 \\ \hline Equivalent undulator Period length \(\lambda_{u}\) [cm] & 2 \\ \hline Undulator parameter Strength & \(K_{u}=0.277\) \\ \hline FEL operating wavelength & \(\lambda\cong 0.7\) μm \\ \hline Number of Undulator Periods & 100 \\ \hline \end{tabular}
\end{table} 
Table 1: Sub-micron MU-FEL oscillator design parameters. Figure 6: (**a**) \(g_{0}\) vs. microwave power (MW) for different values of the beam energy (dash 50 MeV, continuous 100 MeV) \(x=\frac{1}{\sqrt{2}}\); (**b**) \(g_{0}\) vs. the beam energy for different values of the microwave power (dashed line 100 MW, solid line 300 MW); (**c**) \(g_{0}\) vs. FEL resonant wavelength (\(x=1/\sqrt{2}\) and \(P=100\) MW), for the remaining parameters see Table 1.  
\[I_{r+1}=I_{0}\frac{\left[\left(1-\eta\right)\,\left(G_{M}+1\right) \right]\,^{r}}{1+\frac{I_{0}}{L}\left\{\left[\left(1-\eta\right)\,\left(G_{M}+1 \right)\right]\,^{r}-1\right\}},\] \[I_{\epsilon}=\left(\sqrt{2}+1\right)\left(\sqrt{\frac{1-\eta}{ \eta}G_{M}}-1\right)I_{S}, \tag{32}\] \[I_{s}\equiv I_{S}\left[\frac{MW}{cm^{2}}\right]=6.9312\cdot 10^{2} \frac{1}{2}\left(\frac{\gamma}{N}\right)^{4}\left(\lambda_{u}[cm]\,K^{*}f_{b} \right)^{-2}\equiv\text{saturation intensity},\]  
where \(r\) is the round-trip number, \(I_{0}\) is the seed signal, and \(\eta\) is the cavity losses. The use of a cavity with a length of 2.5 m ensures the saturation in a time provided by  
\[\tau^{*}[s]\cong 6.7\cdot 10^{-9}\frac{\ln(0.1\,I_{e}/I_{0})}{\ln[\left(1- \eta\right)\,G_{M}+1]}L_{c}[m], \tag{33}\]  
not exceeding 1.5 \(\upmu\)s. Furthermore, for a properly optimized cavity [30], the saturated output peak power(\(P_{L}\cong 1/4N\cdot P_{E}\)) is 18 MW. The average power depends on the characteristics of the RF accelerating field and will not be discussed in the present context. It is evident that the example we have presented is just aimed at underscoring that the operation of a FEL oscillator is possible. The use of less relaxed parameters (even though not far from the present technological capabilities), see Table 2, indicates the possibility of extending the operation to the UV region.