# Comments on the Physics of Microwave-Undulators
## 4 Sources of Inhomogeneous Broadening and High Gain Regime  
It is well-known that any deviation from the ideal conditions, either in the undulator field or in the electron beam qualities, induces elements of gain deterioration, which may create a drawback to the FEL performance. These effects are known as inhomogeneous broadening contributions [29; 30; 31; 32; 33]. This last term indicates the deviation from the \"natural\" line width \(\left(\left(\frac{\Delta\lambda}{\lambda}\right)_{0}=\frac{1}{2N_{u}}\right)\) and determines a line broadening with a consequent gain reduction. The impact of these effects can be quantified through the use of the so-called inhomogeneous broadening coefficients. It should be noted that in the case of the magnetic undulator, the sources of field non-homogeneity are due to the transverse MU field components. The origin of these effects, in the case of waveguide modes, are essentially the same, even though the relevant derivation is trickier. The line broadening we have mentioned can be easily calculated from the shift in resonant wavelength(\(\lambda_{r}=\frac{\lambda_{u}}{2\gamma^{2}}\left(1+\frac{K_{u}^{2}}{2}+\gamma^{2} \theta^{2}\right)\)) determined by the deviation from the nominal value of the various quantities specifying \(\lambda_{r}\). The wavelength relative shift due to e-beam energy deviation is given by \(\frac{\delta\lambda}{\lambda}\cong 2\frac{\delta\gamma}{\gamma}\), which after averaging over the energy distribution (assumed to be Gaussian) yields the following dependence on the root mean \begin{table}
\begin{tabular}{c c} \hline Beam Current [A] & 120 \\ \hline Beam Energy [MeV] & 100 \\ \hline Normalized Beam emittance [mm\(\cdot\)mrad] & 10 \\ \hline Twiss Parameter \(\beta\) [m] & 5 \\ \hline Equivalent undulator Period length \(\lambda_{u}\) [cm] & 0.77–1.3 \\ \hline Undulator parameter Strength & \(K_{u}=0.15\)–0.26 \\ \hline FEL operating wavelength & \(\lambda\cong 100\)–170 nm \\ \hline Number of Undulator Periods & 200 \\ \hline Microwave Power [MW] & 200 \\ \hline \end{tabular}
\end{table} 
Table 2: Same as Table 1 for the UV operation. Regarding the relative energy spread, we have assumed a linear scaling with the inverse of the energy. Therefore, with respect to the case of Table 1, we foresee \(\sigma_{\varepsilon}=5\times 10^{-4}\). Figure 8: Small signal gain coefficient vs. resonant wavelength for values of \(RF\)_Power_\(=200\) MW, \(\gamma=200\) and \(N_{u}=100\) (continuous line) and \(N_{u}=200\) (dotted). square relative energy spread (\(\left\langle\frac{\delta\lambda}{\lambda}\right\rangle\cong 2\sigma_{\varepsilon}\)). The inhomogeneous broadening parameter specifying the amount of gain degradation is provided by the ratio  
\[\mu_{\varepsilon}=\frac{\left\langle\frac{\delta\lambda}{\lambda}\right\rangle}{ \left(\frac{\Delta\lambda}{\lambda}\right)_{0}}=4N_{u}\sigma_{\varepsilon}. \tag{34}\]  
The corresponding gain reduction factor is \(R=1/(1+1.7\mu_{\varepsilon}^{2})\), and accordingly to be on the safe side, the following condition should be satisfied  
\[\mu_{\varepsilon}<<1\Rightarrow\sigma_{\varepsilon}<<\frac{1}{4N_{u}}, \tag{35}\]  
which becomes rather difficult to be satisfied if large equivalent periods are to be chosen. In the case of \(N_{u}\geq 100\), the MU-FEL oscillator becomes rather problematic, because of the extremely good energy quality required for a low energy Linac. The choice of \(N_{u}\) is therefore a matter of a careful optimization of the MU-FEL design and, possibly, some trade-offs. The next quantity is the equivalent undulator strength, whose deviations from nominal values comes into play from the shifts associated with the equivalent magnetic field and period (\(B_{u}\)and \(\lambda_{u}\)). We find, accordingly, that  
\[\frac{\Delta K_{u}}{K_{u}}=\frac{\Delta B_{u}}{B_{u}}+\frac{\Delta\lambda_{u} }{\lambda_{u}}, \tag{36}\]  
and therefore the deviation from the ideal conditions comes from technological and physical constraints. Let us discuss the two contributions separately.