# Comments on the Physics of Microwave-Undulators
## 3 MU Spectral Properties and Low Gain FEL Devices
It depends on the undulator equivalent strength parameter and therefore on the power density feeding the waveguide, according to the practical formula [10]  
\[B_{u}[T]\cong b_{u}[T]f(x),\quad b_{u}[T]\cong 1.3\cdot 10^{-4}\sqrt{\frac{P[MW] }{S[m^{2}]}},\]  
\[f(x)=(1-x^{2})^{-\frac{1}{4}}+(1-x^{2})^{\frac{1}{4}},\quad x=\frac{\omega_{c }}{\omega},\quad S=ab, \tag{30}\]  
\[K_{u}\cong 1.86b_{u}[T]a[cm]g(x)f(x),\quad g(x)=\frac{x}{1+\sqrt{1-x^{2}}}, \quad\lambda_{u}=2 ag(x).\]  
It is worth taking into account that a new parameter \((x=\omega_{c}/\omega)\) controlling the tunability of an MU-FEL comes into play. It provides the definition either of equivalent undulator period or strength parameters. The importance of this quantity will be commented on in the following. The behavior of \(K_{u}\) and of \(F_{n}(K_{u})\) vs. the microwave power is provided in Figure 4 which yields an idea of the power necessary to obtain reasonable \(K_{u}\) values for FEL operation, either in the oscillator or in SASE configurations. This quantity is indeed critical to fix the value of the small signal gain coefficient [25; 27], representing the reference value for oscillator FEL  
\[g_{0}=\frac{16\pi}{\gamma}\frac{|J|}{I_{0}}\lambda_{r}LN_{u}^{2}\xi_{u}\,f_{ b,1}^{2} \tag{31}\]  
with \(J=\hat{I}/\Sigma\) the current density, \(\hat{I}\) the e-beam peak current and \(\Sigma\) the e-beam cross section. It is worth underscoring that apart from the obvious (and not secondary) technological details the search for the relevant design and optimization criteria does not change with respect to the strategy developed in the past [29]. In the following, we limit our examples to devices operating in the IR-UV region. In Figure 5a,b, we have plotted the equivalent undulator period length vs. \(x=\omega/\omega_{c}\) and the FEL operating wavelength vs. the beam relativistic factor. It is worth stressing that large x values shift \(\lambda_{r}\) towards longer wavelengths. This fact, as we will see in the following, will have consequences on the gain itself. In Figure 6a-c, we have reported the behavior of \(g_{0}\) in terms of the microwave parameters. In particular, in Figure 6a we have plotted the small signal gain coefficient vs. the microwave power for a FEL operating in the sub-micron region for two different beam energies (50 and 100 MeV) and 120 A peak current. In Figure 6b, we have reported the gain coefficient vs. the resonant wavelength. The values of \(g_{0}\) we have evaluated with the parameters listed in Figure 6 confirm the possibility of operating an MU-FEL oscillator driven by a low energy Linac with good beam characteristics (see next section, for the necessary details) in the region ranging from near IR to UV. The operation with larger x values is favored because it ensures larger values of the strength parameter (see Figure 4b) (which remains limited to values not exceeding 0.4). The wave undulator period vs. \(\frac{\omega_{c}}{\omega_{c}}\) is shown in Figure 5a. For example, in correspondence of \(\omega=\frac{\omega_{c}}{\sqrt{2}}\), we find \(\lambda=2\) cm, which yields reasonable \(K\) values and the possibility of a compact device. Even though the architecture of an FEL oscillator device is fairly simple, further details for the necessary implementation should be specified. In particular, we need to examine the problem of the focusing, namely the necessity of including additional quadrupoles ensuring the proper electron transport (see Appendix A). Let us now consider a specific set of parameters for a microwave undulator FEL oscillator operating in the sub-micron region with the parameters reported below (wave guide dimensions and power \(a\)[cm] = 2.5, \(b\)[cm] = 1.25, \(P\)[MW] = 100).