# Recirculated Wave Undulators for Compact FELs
## Appendix A Derivation of the 1D FEL Equations for the Wave Undulator  
The wave equation for the electric field \(E\) generated by a current density \(j\) is:  
\[\frac{1}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}}-\frac{\partial^{2}E}{ \partial z^{2}}=-\mu_{0}\frac{\partial j}{\partial t} \tag{1}\]  
where the electric field is assumed complex and scalar (this can still cover the case of circular polarization) and the current density is considered along the field polarization direction. With a standard procedure, assuming slowly varying envelope, a forward propagation of the electromagnetic wave and a single-mode oscillation such that \(\partial_{t}<<c\partial_{z}\), Equation (1) can be reduced to:  
\[\frac{\partial E}{\partial z}\sim\frac{i\mu_{0}c}{2\omega}e^{-i\varphi(t)} \frac{\partial j}{\partial t} \tag{2}\]  
In the same framework of assumptions above, we average the current density over a period of rapid oscillation of the radiation field, i.e., within one FEL bucket, while considering \(\gamma_{r}>>1\):  
Figure 7: The FEL radiated power versus the propagation distance inside of the wave undulator.  
\[e^{-i\psi(t)}\frac{\partial j}{\partial t}\sim\frac{i\omega^{2}}{2\pi}\int_{- \frac{\pi}{\omega}}^{\frac{\pi}{\omega}}je^{-i\psi(t)}dt=-\frac{ie\omega^{2}a_{0 }}{4\pi\gamma_{\tau}\Sigma_{e}}\int_{j=1}^{N_{e}}\int_{-\frac{\pi}{\omega}}^{ \frac{\pi}{\omega}}e^{-i\psi(t)}\delta[z-z_{j}(t)]dt\sim-\frac{ie\omega^{2}a_{0 }}{2\pi\gamma_{\tau}\Sigma_{e}}\sum_{j=1}^{N_{e}}e^{-i\psi(t_{j})} \tag{11}\]  
where for the last passage we have used the second of Equation (21). We recognize \(n_{e}=N_{e}\omega/\Sigma_{e}c\), therefore:  
\[e^{-i\psi(t)}\frac{\partial j}{\partial t}\sim-\frac{ie\omega a_{0}n_{e}ec}{2 \pi\gamma_{\tau}\gamma_{\tau}}\Big{\langle}e^{-i\psi(t_{j})}\Big{\rangle}\equiv -\frac{i\omega a_{0}n_{e}ec}{2\pi\gamma_{\tau}\gamma_{\tau}}\Big{\langle}e^{- i\psi_{j}}\Big{\rangle} \tag{12}\]  
where \(N_{e}\) is the number of electrons in the FEL bucket and the angular brackets denote an average over the particles composing the temporal slice of width \(2\pi/\omega\). Thus, the first of Equation (23) is obtained combining Equations (10) and (12). The second and third of Equation (23) are the so-called pendulum equations, which are derived as shown below. First, we set the equation for the ponderomotive phase evolution recalling that \(\omega\sim 2\gamma_{\tau}^{2}\omega_{0}\):  
\[\frac{d\psi_{j}}{dz}=\frac{\omega}{c^{2}}\frac{dz_{j}}{dt}+\frac{\omega_{0}}{c }+\frac{\omega_{0}}{c^{2}}\frac{dz_{j}}{dt}-\frac{\omega_{0}+\omega}{c}+\frac {d\phi_{j}}{dz}=(\omega+\omega_{0})\frac{\beta_{j}}{c}-\frac{\omega}{c}\sim \frac{\omega_{0}}{c}\left(1-\frac{\gamma_{\tau}^{2}}{\gamma_{j}^{2}}\right) \tag{13}\]  
Since \(\gamma_{\tau}^{2}/\gamma_{j}^{2}=(\delta_{j}+1)^{-2}\sim 1-2\delta_{j}\) for \(\delta_{j}<<1\), the ponderomotive phase obeys to:  
\[\frac{d\psi_{j}}{dz}=\frac{\omega_{0}}{c}\left(1-\frac{\gamma_{\tau}^{2}}{ \gamma_{j}^{2}}\right)\sim 2\frac{\omega_{0}}{c}\delta_{j} \tag{14}\]  
which coincides with the second of Equation (23). Concerning the energy deviation:  
\[\frac{d\delta_{j}}{dz}=\frac{1}{\gamma_{\tau}}\frac{d\gamma_{j}}{dz}=\frac{1} {\gamma_{\tau}mc^{2}}\frac{dW_{j}}{dz} \tag{15}\]  
Where, by means of the first of Equation (21), the energy exchange rate \(dW_{j}/dz\) is given by:  
\[\frac{dW_{j}}{dz}=-\frac{e}{c}\frac{d\vec{\tau}_{\perp}}{dt}\cdot Re\Big{[} \vec{E}(z)e^{i\psi_{j}}\Big{]}\sim-\frac{ea_{0}}{2\gamma_{\tau}}Re\Big{[}E(z) e^{i\psi_{j}}\Big{]}=-\frac{ea_{0}}{4\gamma_{\tau}}\Big{[}E(z)e^{i\psi_{j}}+E^{*}(z)e^{- i\psi_{j}}\Big{]} \tag{16}\]  
Finally, by combining Equation (16) with Equation (15) it is possible to obtain the last of Equation (23).