# Comments on the Physics of Microwave-Undulators
## 5 Final Comments  
In this article, we have discussed how the general criteria for the design of an MU-FEL can be evaluated. In particular, we have included in the FEL scaling design the parameters regarding the MU performance. As already emphasized, any quantity associated with the waveguide design may be a source of troubles for the FEL performance. Regarding the control of the power inside the waveguide, it should be noted that along with the power fluctuations \(\frac{\Delta P}{P}\) discussed in the previous section, another problem is associated with the power losses in the waveguide walls. This is reflected in a reduction of the equivalent undulator strength parameter value, thus leading to a reduction of the resonant wavelength. Large changes in power may hamper the FEL operation. In order to quantify the importance of these effects, we note that the decrease of the strength parameter \(K\) along the longitudinal direction \(z\) is [11; 20]  
\[K_{u}(z)=K_{u}^{0}e^{-\beta_{att}z}, \tag{46}\]  
where \(\beta_{att}\) is the attenuation coefficient (AC). The wavelength reduction associated with a non-zero (but sufficiently small) AC is provided by  
\[\bigg{|}\frac{\Delta\lambda}{\lambda}\cong\beta_{att}\Delta z\bigg{|}. \tag{47}\]In order to avoid any harmful effect, this shift should be well contained within the SASE gain line width, keeping \(\Delta z=L_{g}\). According to Equation (47) this requires  
\[\beta_{att}L_{g}<\rho, \tag{48}\]  
namely  
\[\beta_{att}<4\pi\sqrt{3}\frac{\rho^{2}}{\lambda_{u}}. \tag{49}\]  
The effects associated with Ohmic power losses cannot be treated as a non-homogeneous broadening. It is not due to statistical effect but to the continuous variation of the decay of the waveguide power, determining the behavior reported in Equation (46). The effect of such a \"deterministic\" variation of the magnetic field can be included in the high-gain FEL integral equation [30]  
\[\partial_{\tau}a=\imath\pi g_{0}\int_{0}^{\tau}\xi e^{-n\xi+\frac{\imath\pi \mu\tau}{2}\xi(2\tau-\xi)}a(\tau-\xi)d\xi, \tag{50}\]  
where  
\[\mu_{T}=2N\frac{\Delta B}{B}\frac{K_{0}^{2}}{1+\frac{K_{0}^{2}}{2}}, \tag{51}\]  
\[\delta B=B(L_{u})-B(0).\]  
where \(\Delta B\) is the variation of the equivalent undulator field in one section of length \(L_{u}\). The possibility of counteracting this effect is tapering the waveguide transverse area in order to increase the power density and therefore the associated \(K_{u}\) values. The amount of tapering can be quantified using Equation (49), which, using typical high gain parameter, yields \(\frac{\Delta a}{a}\ll 10^{-2}\), consistent with analogous conclusions reported in ref. [10]. In our previous considerations, we have not included the effects due to the electron beam divergence, which is the source of further gain degradation effects. They are not significant for the values of emittance we have considere, but can become important if we consider the angular spread induced by the width variation of the waveguide transverse dimensions. Rewriting the results of ref. [10] using our formalism and approximation, we find  
\[\left\langle x^{\prime}\right\rangle\cong\frac{K_{u}}{\gamma}\frac{k_{u}}{ \sqrt{k_{u}^{2}+l_{c}^{-2}}}\sqrt{\frac{N_{u}\lambda_{u}}{l_{c}}}\sigma_{u}, \tag{52}\]  
where \(l_{c}\) is the characteristic coherence length of the statistical process governing the randomness of cavity thickness variations (for further comments see ref. [10]). The inhomogeneous broadening parameter characterizing the effect of the gain degradation is  
\[\mu^{\prime}\cong\frac{1}{\rho}\left[\frac{K_{u}}{1+\frac{K_{u}^{2}}{2}}\frac {k_{u}}{\sqrt{k_{u}^{2}+l_{c}^{-2}}}\sqrt{\frac{N_{u}\lambda_{u}}{l_{c}}} \sigma_{u}\right]^{2}, \tag{53}\]  
and the use of \(l_{c}\cong 0.1\) mm, \(\sigma_{u}\cong 10^{-3}\), \(\lambda_{u}=2\) cm, \(N_{u}\lambda_{u}\cong 7\) cm, \(\rho\cong 10^{-3}\) yields an apparently not harmful contribution. By keeping less optimistic parameters (\(l_{c}\cong 0.005\) mm, \(\sigma_{a}\cong 10^{-2}\)), the associated gain degradation is not insignificant.