## 4 Effect of the QUAPEVA Skew and Initial Beam Conditions on the Electron Beam Transport
### Brightness  
Figure 12 shows the transversal beam shape versus \(a_{2}\) at the undulator center. For \(a_{2}=0\), the beam (see Figure 12a) is focused (beam horizontal size 0.67 mm and vertical size 0.37 mm RMS). For the low \(a_{2}\), the beam (see Figure 12b) is slightly defocused (beam horizontal size 0.73 mm and vertical size 0.38 mm RMS). For the high \(a_{2}\), the beam (see Figure 12c) exhibits a vertical and horizontal focusing redistribution per energy of the beam and a small tilt. The total beam emittance at the center of the undulator (see Table 5) increases by a small amount from \(a_{2}=0\) (\(\epsilon_{x}=103\) mm.mrad \(\epsilon_{z}=30\) mm.mrad) to low \(a_{2}\) (\(\epsilon_{x}=119\) mm.mrad \(\epsilon_{z}=35\) mm.mrad), with a factor 1.2 (1.16) horizontally (vertically), while keeping the charge at 2.82% of the initial beam one, which corresponds to a \(\pm 7\) MeV slice around the reference energy thanks to the slit that cuts the useless electron energies. For the high \(a_{2}\), the increase is more important, the horizontal (vertical) emittance gets larger by a factor of 2.73 (6.71) from the low \(a_{2}\) case, achieving \(\epsilon_{x}=328\) mm.mrad \(\epsilon_{z}=234\) mm.mrad, and the charge decreases to 1.4% of the initial beam one. Table 5 shows the emittance and charge for a \"monochromatic\" 176 \(\pm\) 0.1 MeV slice of a flat top beam of central energy 176 MeV and \(\sigma_{y}=\pm 5\)%, at the center of the undulator. The emittance increases quickly with the presence of the \(a_{2}\) skew term. From \(a_{2}=0\) to low \(a_{2}\), an increase of 10 (4.7) of the horizontal (vertical) emittance takes place, and for the high \(a_{2}\) skew term, an additional increase by a factor 1.7 (5.7) horizontally (vertically) occurs, attaining an emittance value of 8.29 mm.mrad horizontally and 6.56 mm.mrad vertically from the 0.48 mm.mrad and 0.24 mm.mrad value of emittance for the \(a_{2}=0\) case. The emittance of the \"monochromatic\" 176 MeV is 2 orders of magnitude lower than the total beam emittance, showing that the chromatic effects dominate the total beam emittance behavior. Figure 11: Vertical (**blue**) and horizontal (**red**) normalized emittance versus the initial vertical divergence at the undulator center for the low (**a**) and high (**b**) skew case (Table 2) with the low divergence electron distribution (see Figure 2a, Table 1) and the “undulator” optics. Figure 12: Simulated transversal beam shape at the undulator center for (**a**) \(a_{2}=0\), the (**b**) low and (**c**) high \(a_{2}\) (Table 2). ‘Slit-undulator’ optics (slit open to 3.2 mm corresponding to a slice of 176 \(\pm\) 7 MeV). Initial low divergence beam distribution of Figure 2a with \(\sigma^{\prime}_{z,j}=2\) mrad, \(\sigma^{\prime}_{x,j}=3.12\) mrad \(\epsilon_{z,j}\),\(\epsilon_{x,j}=0.2\) mm.mrad. Figure 13 shows the horizontal and vertical emittance at the undulator center of an initial flat-top energy distribution electron beam versus energy spread. From the monochromatic beam (\(\sigma_{\gamma}\) = 0) to =5% energy spread, the emittance has a linear dependence versus energy spread, as shown in the chromatic term (see Equation (2)), with a slope of 12.4 mm.mrad/% (2.9 mm.mrad/%) in the horizontal (vertical) plane. \(\epsilon_{x}\) (\(\epsilon_{z}\)) increases to 63.8 mm.mrad (15.3 mm.mrad) for a \(\sigma_{\gamma}\) = 5% spread. For \(\sigma_{\gamma}\) larger than 1%, the chromatic emittance starts to dominate. For \(\sigma_{\gamma}\) higher than 5%, \(\epsilon_{x}\) starts to converge around 74 mm.mrad and \(\epsilon_{z}\) to increase faster, because the lower and higher energies vertically defocus at the undulator and are thus cut by its physical aperture. Table 5 shows the brightness calculated with Equation (1) for the total beam (with \(\sigma_{s}\) of \(980\pm 100\)\(\mu\)m) and the \"monochromatic\" beam (with \(\sigma_{s}\) of \(5.4\pm 0.2\)\(\mu\)m).