# Recirculated Wave Undulators for Compact FELs
## 1 Introduction  
Free Electron Lasers (FELs) are indeed one of the most interesting devices belonging to the realm of the radiation sources [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Lasers are presently the most diffuse and best-working radiation sources in terms of brilliance, monochromaticity, coherence, directionality and polarization. A future perspective would be that of realizing FEL facilities in the VUV-X region exploiting compact accelerators and short undulator sections. Novel electron acceleration schemes [15; 16; 17] can provide high gradient (GV/m) acceleration. Even though the latter might solve the problem of the accelerator length, it still does not solve the issue associated with the saturation length and therefore to the undulator length. Wave undulator-based FELs might provide a valid solution [18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. In the wave undulator scheme the undulator is replaced by a laser, thus the associated period is much shorter than that of conventional undulator magnet in existing FELs, paving the way to the reduction of the saturation length. Another advantage is that the electron beam energy necessary to reach the short wavelength region scales as the square root of the undulator period: a wave undulator would permit the operational beam energy to be reduced by several orders of magnitude, as well as it would lead to a reduction in the accelerator size. The conditions for FEL operation of a wave undulator have been studied [26] without too much attention to the laser and electron beams transport. More focus on the laser beam transport has been given in Ref. [27], where a compact VUV-X FEL device has been proposed consisting of an electron LINAC and a resonator cavity to recirculate the wave undulator, used for multiple interactions with the electron beam. Conversely, in this paper we focus on the electron beam transport in the recirculated undulator, both through the interaction points and in the magnetic chicane, while considering the same design for the optical cavity used in [27]. In particular we study the evolution of the longitudinal phase-space during the beam transport and during the interaction with the wave undulator, leading to the electron microbunching responsible for coherent FEL power emission. The paper first consists of a short introduction to the ring cavity design considered for the recirculated wave undulator. Then, the exact solution of the Liouville equation for the longitudinal dynamics of particle beams is derived concerning the phase-space evolution in magnetic chicanes [28], adding corrections due to the emission of synchrotron radiation. The dynamics of relativisticelectrons undergoing oscillations in the wave undulator is developed by means of the Hamilton-Jacobi formalism. The electron trajectories are obtained, from which the 1D FEL equations are derived. Simulation results based on the numerical solution of the 1D FEL equations via an in-house Python script combined with the Liouville analytic theory for the electron beam transport in the chicane are shown, in order to demonstrate the electron microbunching with the subsequent FEL emission, as well as for quantitatively characterizing the output power from the recirculated undulator under analysis. Finally, an analytic approach is presented for the wave undulator scaling laws and the FEL power evolution, showing the pros and cons of exploiting a fully analytic model for the FEL emission while comparing numerical and analytic results relative to the same scientific case.