## 2 Emittance Growth Issues on LPA
### Experimental Test Bench: The COXINEL Line  
The influence of the skew components of the first quadrupoles was tested on the LPA-based COXINEL line [72; 73; 74; 67], designed for FEL. For such a purpose, the relative energy spread (RMS) \(\sigma_{\gamma}\) over one coherent length should be less than \(\sigma_{\gamma}<\rho\), with \(\rho\) as the Pierce parameter [75]. This requirement becomes an issue since relative energy spreads of percentage level have been measured for LPA beams [13], so until more improvements are made in LPA systems, these beams have to be first modified to satisfy the FEL condition and transported to the undulator. Mitigations rely in decompressing the beam in a magnetic chicane [76; 77; 78; 41; 44] enabling us to reduce the slice energy spread to acceptable levels for FEL, or by using a transverse gradient undulator (TGU) [79; 80] that generates a linear transverse dependence of the undulator field associated to dispersion, introducinga transverse displacement with the energy according to the undulator resonance condition for a particular undulator gradient. The TGU minimizes the energy spread effect, leading to an improvement of FEL gain and radiation power.  
Figure 1 shows the sketch of the COXINEL line. Electrons were produced and accelerated by a Titanium:Sapphire laser focused on a gas jet. A triplet of permanent magnets quadrupoles with tunable gradient (QUAPEVA [57; 58]) positioned 5 cm from the source handles the bit of mrad initial divergence of the electron beam via a strong magnetic field, insuring a proper beam focusing. Then, a demixing chicane consisting of four electro-magnetic dipoles accompanied with a removable and adjustable slit placed in the middle stretches the beam longitudinally, sorts electrons by energy, and then selects the energy of interest via the slit. A second set of four electromagnetic quadrupoles (EMQs) enables a beam to focus into a cryo-ready undulator of 107 periods. In addition, scintillating screens enable measuring the beam size and shape at different locations along the line [67]. The coordinate system shown in Figure 1 uses x as the horizontal direction, z as the vertical one, and s as the longitudinal one.  
A laser system at \"Salle Jaune\" of Laboratoire d'optique Appliquee was used for the LPA source. The titanium:sapphire (Ti:Sa)-based laser system provides a linearly polarized light in the horizontal direction, 800 nm, 30 fs (FWHM), 30 TW pulse to the interaction chamber where it is focused into a spot of 20 \(\upmu\)m (FWHM) in a supersonic jet with a millimeter scale nozzle of gas mixture composed of 99% helium and 1% of nitrogen. LPA is operated in the ionization injection configuration for its robustness and relatively high charge. The electron beam generated by LPA is firstly characterized with a removable electron spectrometer, located at 355 mm from the gas jet. The spectrometer uses a permanent magnet dipole of 1.1 T field and 100 mm long to horizontally disperse the beam, a lancex screen, and a CCD camera with a spectral resolution of 2.7% and 3.8% between 50 MeV and 280 MeV. The electron beam is then monitored with a first beam imager positioned 64 cm from the LPA source and mounted on a motorized stage. The imager is composed of a lancx scintillating screen protected by a 75 \(\upmu\)m black ionized aluminum foil, a pair of lenses, and a CCD camera (scA640-70 gm Basler scout with a resolution of 150 \(\upmu\)m). The ratio \(r=\frac{\sigma_{x}}{\sigma_{z}}\) of the horizontal beam size \(\sigma_{x}\) to the vertical one \(\sigma_{z}\) measured on the imager is assumed to be equal to the horizontal to vertical divergence ratio. From the electron spectrometer, the vertical divergence is measured, and thus, the horizontal divergence. The COXINEL test experiment enables providing different values of the initial electron beam divergences and of the skew angle of the first strong focusing particles.