# Comments on the Physics of Microwave-Undulators
## 4 Sources of Inhomogeneous Broadening and High Gain Regime
This is true regarding the variation of the waveguide transverse size. It reflects as a variation of the transverse \(TE_{0,1}\) mode distribution not dissimilar from the dependence of the magnetic field of ordinary undulators on the transverse coordinates. Regarding the waveguide power fluctuations or variations, the assumption is only partially true, as further discussed in the concluding section. Analogous considerations hold for the high gain regime. The pivotal parameter for this operation is the so called Pierce parameter [25; 27], which in practical units is written as  
\[\rho\cong\frac{8.36\cdot 10^{-3}}{\gamma}\left[J\left[\frac{A}{m^{2}}\right] \lambda_{u}[cm](K_{u}f_{b})^{2}\right]^{\frac{1}{3}}. \tag{44}\]It fixes either the gain length (\(L_{g}=\frac{\lambda_{u}}{4\pi\sqrt{3}\rho}\)), the saturated power or the importance of the inhomogeneous broadening gain deterioration effects. We limit ourselves to the soft X-ray operation, and the \"design\" parameters are given in Table 3.  
In Figure 9, we have plotted the achievable values of the Pierce parameter for an SASE MU-FEL operation and the associated gain length (see Table 3 for the parameters used in the calculations). The parameter case we have considered is not dissimilar to those of ref. [11] (see also Figure 10), and concerning the FEL output performances, the conclusions are that in the case of short wavelength SASE FELs, the MU offers, in principle, some advantages. The most noticeable of them is the short period, hence the possibility of operating with lower e-beam energie, and reducing the overall size and cost of the device. Furthermore, the larger transverse size, not limited by the gap of the conventional undulator magnets, reduces the wake field effects, thus allowing the use of higher currents. To make the comparison more fair, we should, however, mention that in the region above 1.5 and 1.8 cm period length, magnetic undulator of superconducting type could be exploited. The relevant performances could be comparable (if not better) than those of MU-FEL s.  
Shorter magnetic undulator periods become more problematic even for the super conducting option. The MU solution for a shorter period is also not easily viable, either for the large intra-waveguide power involved or for the technological problems mentioned below. We must, however, emphasize that the achievable values of the Pierce parameter are below the threshold of \(10^{-3}\), a circumstance which imposes extremely demanding beam performances in terms of, e.g., energy spread. The condition to be satisfied is that \(\sigma_{\epsilon}<2/\rho\)[30; 34; 35]. Regarding the present paper, an energy spread better than \(10^{-4}\) should be taken into account (ref. [11] considers \(6.5\times 10^{-5}\)). By applying the rule of thumb, we can conclude that the saturation length is about \(20L_{g}\) without the energy spread effects. The relevant inclusion determines an increase by a factor  
\[R=(1+d\tilde{\mu}_{\epsilon}^{2}), \tag{45}\]  
\[\tilde{\mu}_{\epsilon}=\frac{2\sigma_{\epsilon}}{\rho},\ \ \ \ \ d\cong 0.187\frac{ \sqrt{3}}{2}.\]  
It is evident that larger values of the MU power could in principle provide better performance, but circulating power larger than 200 MW imposes the use of very high-performing waveguides with extremely large Q-values and low losses. The bottleneck of an MU-FEL device is associated with all evidence to the intrinsic low \(K\) values due either to the short period or maximum microwave achievable power (see Equations (2) and (3)). This determines lower gain than ordinary FELs operating with permanent magnet undulators. Therefore, the MU remains of interest for FEL frequencies not exceeding those of soft X-ray. The problems associated with the beam qualities do not represent the most significant part of the problem. The inhomogeneous problems associated with the power fluctuations, which are controlled within \(10^{-4}\), are a considerable source of gain degradation.