# Recirculated Wave Undulators for Compact FELs
## 5 1D FEL Equations  
In the most of relevant cases \(p_{z0}\sim\gamma_{0}mc\), we obtain \(\eta=2\gamma_{0}\omega_{0}m\). Thus, the electron laws of motion can be recast into:  
\[\vec{r}_{\perp} = \frac{\lambda_{0}}{4\pi}\vec{\theta}_{0}\xi+\frac{\lambda_{0}}{4 \pi}\frac{1}{\gamma_{0}}\int\vec{a}_{0}d\xi\] \[z = \frac{\lambda_{0}}{4\pi}\Bigg{[}1-\frac{\left(1+\gamma_{0}^{2} \theta_{0}^{2}\right)}{2\gamma_{0}^{2}}\Bigg{]}\xi-\frac{\lambda_{0}}{8\pi} \frac{\vec{\theta}_{0}}{\gamma_{0}}\cdot\int\vec{a}_{0}d\xi+\frac{\lambda_{0} }{16\pi}\frac{1}{\gamma_{0}^{2}}\int a_{0}^{2}d\xi \tag{21}\]  
where \(\vec{\theta}_{0}\) is the vector representing the initial divergence of the electron. For sake of simplicity in this paper we focus on the 1D FEL model, justified by the condition \(\theta_{0}<<1\) for any electron, i.e., low-divergence beams at the interaction region. Moreover, we consider \(a_{0}<<1\) and \(\gamma_{0}>>1\). The laws of motion can be further simplified into:  
\[\vec{r}_{\perp} \simeq \frac{\lambda_{0}}{4\pi}\frac{1}{\gamma_{0}}\int\vec{a}_{0}d\xi\] \[z \simeq \frac{\lambda_{0}}{4\pi}\xi \tag{22}\]  
Given the above equations of motion, and choosing a circularly polarized vector potential, the one-dimensional FEL equations are finally (see Appendix A for the derivation):  
\[\frac{dE}{dz} = \bigg{(}\frac{ea_{0}}{4\pi\epsilon_{0}\gamma_{r}}\bigg{)}n_{e} \Big{\langle}e^{-i\phi_{j}}\Big{\rangle}\] \[\frac{d\psi_{j}}{dz} = \bigg{(}\frac{4\pi}{\lambda_{0}}\bigg{)}\delta_{j}\] \[\frac{d\delta_{j}}{dz} = \bigg{(}\frac{ea_{0}}{4\gamma_{r}^{2}mc^{2}}\bigg{)}\left[E(z)e^{ i\phi_{j}}+E^{*}(z)e^{-i\phi_{j}}\right] \tag{23}\]  
where we have introduced the ponderomotive phase \(\psi_{j}=\omega z/c+\xi_{j}-(\omega_{0}+\omega)t+\phi_{j}\), i.e., the coordinate identifying the longitudinal position of a particle inside a bucket of physical size determined by the radiation wavelength \(\lambda_{r}\sim\lambda_{0}/2\gamma_{r}^{2}\). \(\phi_{j}\) is an arbitrary initial phase, and we have defined the relative energy deviation \(\delta_{j}\) of an arbitrary \(j\)-particle with Lorentz factor \(\gamma_{j}\):  
\[\delta_{j}=\frac{\gamma_{j}-\gamma_{r}}{\gamma_{r}}=\frac{\Delta\gamma_{j}}{ \gamma_{r}} \tag{24}\]  
Figure 3: Electron trajectory in a circularly polarized wave undulator (arbitrary units).  
The particles' density of the electron bunch has been denoted as \(n_{e}\). An important quantity has been introduced as well, which is the bunching factor:  
\[b=\left\langle e^{-i\phi_{j}}\right\rangle \tag{25}\]  
where the angular brackets indicate the average over the electron ensemble composing one FEL bucket (more details in Appendix A). The strength parameter \(K\) associated with a circularly polarized wave undulator of intensity \(I_{p}\) and wavelength \(\lambda_{0}\), in practical units reads:  
\[K=a_{0}=0.85\times 10^{-5}\lambda_{0}[m]\sqrt{I_{p}(W/m^{2})} \tag{26}\]  
The choice made in this paper of using a circularly polarized wave undulator is based on the fact that doing so, the strength parameter (\(a_{0}\)) is increased with respect to the case of a linearly polarized wave undulator (\(a_{0}/\sqrt{2}\)).