# Comments on the Physics of Microwave-Undulators
## 3 MU Spectral Properties and Low Gain FEL Devices  
According to our previous results, we may safely conclude that the spectrum of the radiation emitted by the electrons traversing the waveguide is fully consistent with that from electrons moving in a conventional undulator (with a periodic static magnetic field). Figure 3: (a) X-component electron velocity (numerical red, blue crosses analytical); (b) eight-like motion in the laboratory frame (blue analytical, red numerical). We have plotted indeed \(z-\sigma_{z}t\) vs. \(x\) both evaluated in the laboratory frame. The spectrum consists of a series of peaks centered at the wavelength provided by (n is the order of the harmonics)  
\[ \begin{split}\lambda_{r,n}=\frac{\lambda_{r}}{n},\\ \lambda_{r}=\frac{\lambda_{u}}{2\gamma^{2}}\bigg{(}1+\frac{K_{u}^ {2}}{2}+(\gamma\vartheta)^{2}\bigg{)},\end{split}  \tag{25}\]  
where \(\vartheta\) is the observation angle. The power radiated inside the waveguide and the relevant spectral details can be obtained using standard means. By just applying the ordinary tools [25; 27; 28] and neglecting the contributions due to higher harmonics, we find that the power emitted per unit solid angle in the forward direction by a single electron is  
\[ \begin{split}\frac{dP}{d\Omega}\cong\dot{N}_{\Omega}\hbar \omega_{r,1}\,\\ \dot{N}_{\Omega}=\frac{\alpha K_{u}^{2}}{4\pi}\frac{\omega_{r}}{1 +\frac{K_{u}^{2}}{2}},\\ \omega_{r,1}=\frac{2\pi c}{\lambda_{r}},\lambda_{r}=\frac{ \lambda_{u}}{2\gamma^{2}}\bigg{(}1+\frac{K_{u}^{2}}{2}\bigg{)},\\ \alpha=\frac{1}{4\pi\,\varepsilon_{0}}\frac{e^{2}}{\hbar c}, \end{split}  \tag{26}\]  
where \(\dot{N}_{\Omega}\) denotes the number of photons emitted per unit time and unit solid angle. The presence of the Planck constant is artificial; the process is indeed purely classical. The power integrated on the whole solid angle in terms of the waveguide parameters reads  
\[ \begin{split} P_{1}\cong\frac{\pi}{3}N_{u}\dot{N}_{\Omega}\frac{ I}{I_{A}}m_{e}c^{2},\\ N_{u}=\frac{L\lambda}{\lambda_{u}},\end{split}  \tag{27}\]  
where \(N_{u}\) is the number of periods associated with the MU, and \(I\), \(I_{A}\) are the beam and Alfven current, respectively. In the previous formula, we neglected the effects of the lemiscates type motion due to the nondipolar longitudinal oscillations. The inclusion of the associated contributions provides the emission of all harmonics all over the solid angle and one obtains (for a single electron)  
\[ \begin{split} P_{sy}=\frac{1}{12\pi^{2}}\alpha\hbar\frac{\omega_ {sy}^{2}}{\bigg{(}1+\sqrt{1-\bigg{(}\frac{\omega_{c}}{\omega}\bigg{)}^{2}} \bigg{)}^{2}},\\ \omega_{sy}=2\pi\gamma\frac{eE_{0}\bigg{(}1+\sqrt{1-\bigg{(} \frac{\omega_{c}}{\omega}\bigg{)}^{2}}\bigg{)}}{m\,c}.\end{split}  \tag{28}\]  
The previous result is interesting (albeit a natural consequence of the analysis developed so far) because it states that the total emitted power in the waveguide is equivalent to that radiated in a bending magnet with the equivalent magnetic field provided by \(B_{u}\) in Equation (1). The radiation emitted at odd harmonics (at \(\vartheta=0\)) per unit solid angle and unit frequency is specified by  
\[\frac{d^{2}I}{d\Omega d\omega}=A_{n}(\omega)F_{n}(K_{u}),\]  
\[A_{n}(\omega)\propto\left[\frac{\sin\left(\frac{\
u_{n}}{2}\right)}{\frac{V_{n} }{2}}\right]^{2},\;\
u_{n}=2\pi N_{u}\left(\frac{\omega-n\omega_{r}}{n\omega_{ r}}\right),\]  
\[F_{n}(K_{u})=\left[\frac{n\,K_{u}}{1+\frac{K_{u}^{2}}{2}}\right]^{2}f_{b,n}^{2}, \tag{29}\]  
\[f_{b,n}=J_{\frac{n+1}{2}}(n\xi_{u})-J_{\frac{n+1}{2}}(n\xi_{u})\;,\]  
\[\xi_{u}=\frac{1}{4}\frac{K_{u}^{2}}{1+\frac{K_{u}^{2}}{2}}.\]  
The quantity \(F_{n}(K_{u})\) is of crucial importance for the FEL operation on the fundamental and at higher harmonics.