# Comments on the Physics of Microwave-Undulators
## 2 Microwave Field Equation
The equation ruling the y-component is obtained by the use of the same procedure. and indeed we obtain  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]1  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]2  
Inserting (9) in the previous equation, we are left with  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]3  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]4  
and, limiting ourselves to the integration of the non-oscillating part only, we obtain  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]5  
and  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]6  
Thus, the motion along the y-direction is accordingly characterized by a parabolic drift. We have checked the consistency of our approximation by a full integration of Equation (6) and a comparison with the analytical treatment. In Figure 2 we have reported the behavior of the energy vs. time and the x,y-trajectories for the case of the parameters reported in the captions. The particles are injected around the center of a waveguide with sides \(X=2.5\) cm, \(Y=5\) cm. The energy is essentially conserved, and the transverse dynamics reflect the superposition of different oscillating components  
\[B_{y} =\frac{E_{0}}{c}\sin(\frac{\pi\,y}{a})\sqrt{1-\left(\frac{\omega _{c}}{\omega}\right)^{2}}\sin(\omega t+k_{g}z),\]7 (**a**) Comparison between numerical and analytical solution for the x-motion component; (**b**) same as (**a**) for the y-position (simulation parameters). It is evident that the comparison between analytical and numerical solutions is excellent; however, the injection at the point we have specified is not useful for our purposes. The electron trajectory undergoes significant deviations in the transverse directions which, if not properly corrected, may cause the loss of the beam itself. We therefore consider the particle injection in such a way that (initially) \(\dfrac{\pi y_{0}}{a}\cong\dfrac{\pi}{2}\). Under such a condition, the magnetic field component along the \(z-direction\) is initially vanishing and remains such if the \(y\) component has no significant drift during the particle transit inside the waveguide. Equation (6) eventually reduces to  
\[ \begin{split} m_{e}\gamma\dfrac{d\beta_{x}}{dt}&=- \dfrac{eE_{0}}{c}\Big{[}1+\dfrac{\omega_{g}}{\omega}\Big{]}\sin(k_{u}z),\\ m_{e}\gamma\dfrac{d\beta_{y}}{dt}&=0,\\ m_{e}\gamma\dfrac{d\beta_{z}}{dt}&=e\dfrac{E_{0}}{c }\dfrac{\omega_{g}}{\omega_{c}}\beta_{x}\sin(k_{u}z).\end{split}  \tag{16}\]  
By further pushing our approximations, we can write the first of Equation (9) as  
\[\ddot{x}=-c\dfrac{K_{u}k_{u}}{\gamma}z\sin(k_{u}z). \tag{17}\]  
In conclusion, the results we have obtained are indeed consistent with the integration of the equations of motion of a (ultra)-relativistic electron moving in a magnetostatic undulator, whose vector field distribution is  
\[\vec{B}\equiv(0,B_{u}\sin(k_{u}z),0), \tag{18}\]  
provided that the equivalent magnetic field and period be those given, respectively, in the second and first of Equation (1). The integration of Equation (17) at the lowest order in \(K_{u}/\gamma\) (or simply specializing Equations (9), (14) and (15)), yields  
\[\dot{x}(t)=\dot{x}_{0}-\dfrac{cK_{u}}{\gamma}(1-\cos(\omega_{u}t)). \tag{19}\]  
After a further integration, we find  
\[ \begin{split} x(t)=\dot{x}_{0}t-\dfrac{cK_{u}}{\gamma}\bigg{(}t- \dfrac{1}{\omega_{u}}\cos(\omega_{u}t)\bigg{)}=\bigg{(}\dot{x}_{0}-\dfrac{cK_{ u}}{\gamma}\bigg{)}+A\cos(\omega_{u}t),\\ A=\dfrac{cK_{u}}{\gamma\omega_{u}}=\dfrac{K_{u}}{\gamma}\dfrac{ \lambda_{u}}{2\pi}.\end{split}