# Comments on the Physics of Microwave-Undulators
## 4 Sources of Inhomogeneous Broadening and High Gain Regime
We consider first the deviations associated with \(B_{u}\) and find  
\[ \begin{split}\frac{\Delta B_{u}}{B_{u}}=\frac{\Delta E_{0}}{E_{0 }}+\frac{\Delta\Bigg{(}\sqrt{1-\left(\frac{\omega_{c}}{\omega}\right)^{2}} \Bigg{)}}{1+\sqrt{1-\left(\frac{\omega_{c}}{\omega}\right)^{2}}}=\frac{\Delta E _{0}}{E_{0}}+\frac{x\Bigg{|}\left(\frac{\Delta a}{a}+\left(\frac{\Delta \lambda}{\lambda}\right)_{c}\right)\Bigg{|}}{1+\sqrt{1-x^{2}}},\\ \left(\frac{\Delta\lambda}{\lambda}\right)_{c}\equiv\text{ microwave line -- width}.\end{split}  \tag{37}\]  
The electric field deviation from the nominal value is specified in terms of the power density waveguide fluctuation  
\[ \begin{split} E_{0}&\cong\sqrt{\frac{Z_{0}P}{ab}},\\ Z_{0}&=120\pi\,\Omega\equiv\text{vacuum -- impedance}.\end{split}  \tag{38}\]  
Including the other term, due to the waveguide size and line width, we can write the first term in Equation (36) as reported in ref. [10]  
\[\frac{\Delta B_{u}}{B_{u}}\cong\bigg{[}\frac{1}{2}\frac{\Delta P}{P}+0.59 \bigg{|}\frac{\Delta a}{a}\bigg{|}+0.5\bigg{|}\frac{\Delta b}{b}\bigg{|}+9 \cdot 10^{-2}\bigg{(}\frac{\Delta\lambda}{\lambda}\bigg{)}_{c}\bigg{]}. \tag{39}\]  
Finally, regarding the equivalent undulator period shift, we have [10]  
\[\frac{\Delta\lambda_{u}}{\lambda_{u}}\cong\bigg{[}\sqrt{2}\bigg{(}\frac{ \Delta\lambda}{\lambda}\bigg{)}_{c}+\left(\sqrt{2}-1\right)\bigg{|}\frac{ \Delta a}{a}\bigg{|}\bigg{]}. \tag{40}\]Before defining the parameters, it is necessary to quantify the effects associated with the gain dilution. We remember that the FEL line shift due to \(\Delta K_{u}/K_{u}\) is [31]  
\[\left(\frac{\delta\omega}{\omega}\right)_{K_{u}}\cong\left|\frac{K_{u}^{2}}{1+ \frac{K_{u}^{2}}{2}}\left[\frac{\Delta B_{u}}{B_{u}}+\frac{\Delta\lambda_{u}}{ \lambda_{u}}\right]\right|\cong\frac{K_{u}^{2}}{1+\frac{K_{u}^{2}}{2}}\left[ \frac{\Delta P}{2P}+\sqrt{2}\frac{\Delta a}{a}+\sqrt{2}\left(\frac{\Delta \lambda}{\lambda}\right)_{c}\right]\!, \tag{41}\]  
with \(\Delta a/a\approx\Delta b/b\). These line shifts can in principle be treated as sources of inhomogeneous broadening, provided that they are statistically independent in such a way that the induced root mean square line width can be defined as  
\[\left(\frac{\delta\omega}{\omega}\right)_{K_{u},rms}=\sqrt{\left\langle\left( \frac{\delta\omega}{\omega}\right)_{Ku}^{2}\right\rangle-\left\langle\left( \frac{\delta\omega}{\omega}\right)_{Ku}\right\rangle^{2}}, \tag{42}\]  
where the averages are taken on the distributions associated with the power fluctuation distributions, with the transverse dimension waveguide uncertainties and spectral width. In order to define a useful reference quantity, we proceed as for the case of the energy spread and introduce the quantity  
\[ \begin{split}&\mu_{K_{u}}=2N_{u}\!\left(\frac{\delta\omega}{ \omega}\right)_{K_{u},rms}\cong N_{u}\xi\sqrt{\sigma_{p}^{2}+8\sigma_{a}^{2}+ 8\sigma_{c}^{2}},\\ &\sigma_{p}=\left\langle\frac{\Delta P}{P}\right\rangle_{rms},\\ &\sigma_{a}=\left\langle\frac{\Delta a}{a}\right\rangle_{rms},\\ &\sigma_{c}=\left\langle\left(\frac{\Delta\lambda}{\lambda} \right)_{c}\right\rangle_{rms}.\end{split}  \tag{43}\]  
The importance of these parameters is easily inferred if we take into account that the RF power can be controlled to \(10^{-4}\) and the mechanical errors affecting the cavity dimensions to \(10^{-3}\), while \(\left(\frac{\Delta\lambda}{\lambda}\right)_{c}\) does not contribute since the associated deviation are better than \(10^{-8}\). For the low gain regime, these effects are not fully negligible, and the most significant contributions come from the inhomogeneous broadening due to cavity dimension shift machining errors. A further comment regarding the analysis of the gain depressing terms should be added. We have treated them as inhomogeneous broadenings as a consequence of the assumption that they are associated with independent statistical distributions, which determine the quantification of the amount of their effect through the previously defined \(\mu_{K_{u}}\) parameters.