## 5 Measured Electron Beam Transport in Presence of Skew Quadrupolar Components of the First Triplet  
The QUAPEVAs tunability gives a high capability to adjust the COXINEL line to the variations that the laser-plasma system undergoes during the experiments. During experiments, pointing variations with respect to the reference beam path of the LPA beam appearing can be compensated throughout the displacement of the magnetic center of the QUAPEVAs [41] using a response matrix method. Figure 15a,b compares the measurements at the undulator entrance and simulation. The RMS beam sizes (\(\sigma_{x}=0.52\) mm \(\sigma_{z}=0.64\) mm) are similar to the simulated ones (\(\sigma_{x}=0.52\) mm \(\sigma_{z}=0.65\) mm). The slight difference may arise from the uncertainty of the initial distribution. A change of 6 T/m in QUAPEVA 2 from the configuration of Figure 15c,d causes the horizontal and vertical focusing redistribution, leading to a cross shape (see Figure 15c). The characteristic cross due to the gradient of QUAPEVA 2 is well reproduced in simulation (see Figure 15d). Figure 15e,f compares measurements and simulations at the undulator exit and shows similar experimental (\(\sigma_{x}=0.77\) mm \(\sigma_{z}=0.50\) mm) and simulation RMS beam size (\(\sigma_{x}=0.74\) mm \(\sigma_{z}=0.35\) mm). Figure 16 compares measured and simulated transverse beam profiles at the undulator entrance. Similar round shapes are observed (Figure 16a,b), corresponding to a tight focus. The RMS experimental (\(\sigma_{x}=1.31\) mm \(\sigma_{x}=1.58\) mm) and simulated (\(\sigma_{x}=0.87\) mm \(\sigma_{x}=2.32\) mm) beam sizes differ slightly, probably because of an improper initial electron beam distribution. The cross shape with a tilt observed during measurements (Figure 16c,d) is well reproduced by simulations. A good agreement between measurements and simulations is also found on the beam slopes at the undulator exit (see Figure 16e,f). The cross pattern is rotated to the opposite side with respect to Figure 16c,d, because of the focusing effect of the undulator. Figure 14: Brightness (\(\circ\)) of the electron beam at the center of the undulator versus QUAPEVA 2 \(a_{2}\) skew term and horizontal (\(x\)) and vertical (\(\diamond\)) normalized emittance in log scale. Simulation using a “monochromatic” 176 \(\pm\) 0.1 MeV beam with \(\sigma^{\prime}_{z,i}=2\) mrad and \(\sigma^{\prime}_{x,i}=3.12\) mrad, a charge of 0.1 \(\pm\) 0.02 pC, \(\epsilon_{z,i}\),\(\epsilon_{x,i}\) = 0.2 mm.mrad, transported with the “slit-undulator” optics and the slit in the chicane open to 3.2 mm. Figure 16: Transverse beam shape at the undulator entrance: measurements (**a**,**c**,**e**) and simulations (**b**,**d**,**f**). Simulations done with the high skew \(a_{2}\) case (Table 2), \(\epsilon_{z,i}\),\(\epsilon_{x,j}\) = 0.2 mm.mrad. (**a**,**b**) With \(\sigma^{\prime}_{z,i}\ =\ 2\) mrad, \(\sigma^{\prime}_{x,i}\) = 4.7 mrad and the “Undulator entrance” optics. (**c**,**d**) With \(\sigma^{\prime}_{z,i}\) = 2 mrad, \(\sigma^{\prime}_{x,i}\) = 4.7 mrad and the “Undulator” optics. (**e**,**f**) With \(\sigma^{\prime}_{z,i}\) = 2.5 mrad, \(\sigma^{\prime}_{x,i}\) = 5.9 mrad and the “Undulator exit” optics. Shot-to-shot pointing for (**a**,**c**,**e**): 0.85 \(\pm\) 0.03 mrad, 0.13 \(\pm\) 0.15 mrad, and 0.07 \(\pm\) 0.02 mrad (\(-\)0.03 \(\pm\) 0.05 mrad, \(-\)0.03 \(\pm\) 0.04 mrad, 0.18 \(\pm\) 0.04 mrad) in the horizontal (vertical) direction. Figure 15: Transverse beam shape at the undulator entrance: measurements (**a**,**c**,**e**) and simulations (**b**,**d**,**f**) with the low skew \(a_{2}\) case (Table 2), \(\sigma^{\prime}_{z,i}\) = 2 mrad and \(\sigma^{\prime}_{x,i}\) = 3.12 mrad, \(\epsilon_{z,i}\),\(\epsilon_{x,i}\) = 0.2 mm.mrad. (**a**,**b**) “Undulator entrance” optics, QUAPEVA 1: 103.27 T/m, QUAPEVA 2: \(-\)115.65 T/m, QUAPEVA 3: 94.94 T/m, (**c**,**d**) QUAPEVA 1: 103.27 T/m, QUAPEVA 2: \(-\)121.99 T/m, QUAPEVA 3: 94.94 T/m.