## 4 Effect of the QUAPEVA Skew and Initial Beam Conditions on the Electron Beam Transport
### Variation of Skew Quadrupole Component  
In the case of a low initial divergence electron beam (see Figure 2a, Table 1) for an ideal quadrupole (\(a_{2}=0\)), the RMS beam size is 0.52 \(\pm\) 0.02 mm and 0.80 \(\pm\) 0.10 mm horizontallly and vertically, respectively (see Figure 9a). When a small skew quadrupole term (Table 2) is added (Figure 9b), the beam remains well centered and focused (RMS beam size of 0.52 \(\pm\) 0.05 mm and 0.72 \(\pm\) 0.08 mm horizontallylly and vertically, respectively). For high \(a_{2}\) (Table 2) (see Figure 9c), the beam exhibits a tilted cross shape, resulting from the change of gradient angle (see Equation (3)).  
For a low value of \(a_{2}\), the beam size at the undulator entrance (see Table 4) is slightly smaller horizontally and larger vertically than for an ideal quadrupole (\(a_{2}=0\)). A substantial decrease in intensity occurs when \(a_{2}\) is enhanced to the high \(a_{2}\) value at the undulator entrance. After the undulator, the intensity and beam sizes (Table 4) for null and low \(a_{2}\) are close. Figure 9g-l presents simulations using an input beam with a divergence three times larger and smaller charge density for the energy of interest (high divergence beam case of Figure 2b and Table 1). For \(a_{2}=0\) (see Figure 9g), the beam is well focused. For a slight increase of \(a_{2}\) (Table 2), the beam remains similarly focused. A larger increase of \(a_{2}\) (Table 2) (see Figure 9i) leads to a tilted cross. Figure 9j-l shows a reduction of intensity resulting from electron beam loss along the reduced vertical aperture. The transport becomes much less efficient for higher initial divergence. The corresponding intensities and beam sizes for the different cases are shown in Table 4. The beam size for larger \(a_{2}\) at the undulator entrance is higher than in the low divergence case. However, the size is closer to those of the null and low \(a_{2}\) cases at the undulator exit. For a low divergence beam, the intensity drops to 0.8 (0.5) at the screen before (after) the undulator for  
Figure 9: Electron beam transversal shape and vertical–horizontal position per energy at the screens before (**a–c,g–i**) and after (**d–f,j–l**) the undulator for low (**b,e,h,k**), high (**c,f,i,l**), and null (**a,d,g,j**) \(a_{2}\) terms (values of Table 2). (**a–f**) Low divergence electron beam distribution (Figure 2a) with \(\sigma^{\prime}_{z,i}=2\) mrad, \(\sigma^{\prime}_{x,i}=3.12\) mrad, and \(\epsilon_{x,i},\epsilon_{z,i}=0.2\) mm.mrad (see Table 1). (**g–l**) High divergence electron beam distribution (Figure 2b) with an initial \(\sigma^{\prime}_{z,i}=5\) mrad, \(\sigma^{\prime}_{x,i}=11.75\) mrad, and \(\epsilon_{x,i},\epsilon_{z,i}=0.2\) mm.mrad (see Table 1). Transport with the “undulator entrance” optics and “undulator exit” optics.  
the high \(a_{2}\) value. In the high divergence case, the highest intensity achieved is 0.2 (0.14) for the low (high) \(a_{2}\) case and decreases until 0.14 (0.11) at the undulator exit. For a high divergence beam (see Figure 2), the charge arriving to the undulator is a small fraction of the initial one.