## 2 Accelerator
### Beam Dynamics and FEL Concept  
The overall design goal of SwissFEL was to build a compact facility to produce FEL pulses down to 1 A wavelength, with the lowest electron beam energy suitable to drive the FEL. This is constrained by the design of the undulator where a short undulator period (\(\lambda_{u}\)) reduces the required electron beam energy (\(\gamma\)) to fulfill the resonance condition of the FEL [27] with  
\[\lambda=\frac{\lambda_{u}}{2\gamma^{2}}\left(1+\frac{K^{2}}{2}\right) \tag{1}\]  
and undulator strength \(K=0.93\cdot B[\mathrm{T}]\cdot\lambda_{u}[\mathrm{cm}]\) in order to lase at a wavelength of \(\lambda=1\,\mathrm{\SIUnitSymbolAngstrom}\). For a compact, in-vacuum undulator [28] we assumed that a period of \(15\,\mathrm{mm}\) and an undulator \(K\) value of 1.2 are feasible. The FEL resonance condition then dictates that the maximally required beam energy to be provided by the linear accelerator is 5.8 GeV. With the nominal beam energy defined, the design value of the beam emittance (\(\epsilon_{n}\)) can be estimated from the condition [29]  
\[\frac{\epsilon_{n}}{\gamma}\approx\frac{\lambda}{4\pi} \tag{2}\]  
Figure 1: SwissFEL accelerator layout. It consists of an S-band injector, a C-band linear accelerator, and two undulator lines. The details of these components are described in the following sections. to be met for all electrons to radiate into the fundamental mode of the FEL. This is, however, a rather soft limit, as larger values simply lead to a reduced coherence of the FEL output beam [3] and lower pulse energies. For the design we assumed a target emittance of 430 nm for the longitudinally central parts of the beam. The second consideration concerns the wakefields within the undulator [30,31], which alter the electrons' local mean energy, thereby shifting them away from the resonance condition of the FEL. With an undulator gap of 4.4 mm for the nominal \(K\) of 1.2 the dominant wakefields are caused by the finite conductivity of the vacuum chamber. For the chosen material, copper, the characteristic length of the resistive wall wake potential is about 7.6 \(\upmu\)m, resulting in a parabolic shape of the wakefields with a length of 20 \(\upmu\)m. The mean energy loss can be compensated with a linear taper of the undulator, which recovers 90% of the FEL performance as calculated without wakefields or taper [32]. The SwissFSEL machine parameters are listed in Table 1.  
The basic operation mode requires a bunch with charge 200 pC, which is compressed in two stages to realize a peak current of 3 kA. The overall compression scheme is based on the stipulation that the wakefields of the C-band structures remove the induced energy chirp needed for the final compression [33]. This approach avoids wasting additional radio frequency (RF) power to remove the chirp actively by off-crest acceleration. To provide shorter but more efficient FEL pulses (in the sense that the number of emitted photons per electron is increased) SwissFSEL can operate down to 10 pC bunch charge, starting with a shorter and smaller but brighter beam from the source [34]. Besides this flexibility of tuning to any bunch charge between 10 and 200 pC, SwissFSEL offers two special modes. In the first a pulse similar to the one for the nominal 200 pC mode is generated, but with a large correlated energy chirp [35]. This is achieved by overcompressing the bunch in the last compression stage. In this case the wakefields in the main linac now add up to the reversed chirp, resulting in a peak-to-peak energy chirp of 1 to 1.5% at 5.8 GeV. The quadratic dependence of the photon energy on the electron energy Equation (1) yields a chirp in the photon pulse twice as large. The other special mode consists in the full compression of a 10 pC pulse to achieve sub-femtosecond FEL pulses.