## 4 Effect of the QUAPEVA Skew and Initial Beam Conditions on the Electron Beam Transport
### Brightness
Hence, the peak brightness is dominated by the horizontal and vertical emittance: As the emittance rapidly grows for high \(a_{2}\), the brightness decreases in a similar rate. For the total beam, the brightness decreases by a factor of 1.4 (42.8) from \(a_{2}\) = 0 to the low \(a_{2}\) case (high \(a_{2}\) case). The total beam brightness is only substantially changed for the high \(a_{2}\) term. For the 176 MeV \"monochromatic\" beam, the brightness decreases by a factor 51 (486) from \(a_{2}\) = 0 to the low (high) \(a_{2}\) case. Even the existence of a low \(a_{2}\) term affects dramatically the slice brightness. Figure 14 shows the brightness and horizontal emittance at the undulator center for a \"monochromatic\" 176 \(\pm\) 0.1 MeV beam of 0.1 \(\pm\) 0.02 pC with the'slit-undulator' optics while changing the skew component of QUAPEVA2 and considering the others ideal. For \(a_{2}\) = 0, the emittance from the source to the center of the undulator is 0.41 mm.mrad and 0.28 mm.mrad horizontally and vertically, respectively. By increasing \(a_{2}\) to \(\pm\)0.03, the horizontal (vertical) emittance increases by a factor of 68 (116), reaching a value of 28 mm.mrad, following a linear dependence versus \(a_{2}\), as expected  
Figure 13: Simulated horizontal (**red**) and vertical (**blue**) normalized emittance of the electron beam at the undulator center for the low \(a_{2}\) skew terms (Table 2) and the optics “slit-undulator” without slit using flat-top electron beams for different energy spread with 176 MeV central energy, \(\sigma^{\prime}_{z,i}\) = 2 mrad, \(\sigma^{\prime}_{x,i}\) = 3.12 mrad, and \(\epsilon_{z,i}\epsilon_{x,i}\) = 0.2 mm.mrad. \begin{table}
\begin{tabular}{c|c c c c|c c c c} \hline  & \multicolumn{4}{c|}{**Total Beam**} & \multicolumn{4}{c}{**176 MeV “Monochromatic” Beam**} \\ \hline
**Skew Term** & \(\epsilon_{x,RMS}\) & \(\epsilon_{x,RMS}\) & **Charge** & **Brightness** & \(\epsilon_{x,RMS}\) & \(\epsilon_{x,RMS}\) & **Charge** & **Brightness** \\
**(mm.mrad)** & **(mm.mrad)** & **(\%)** & **(A/(mm.mrad))\({}^{2}\)** & **(mm.mrad)** & **(mm.mrad)** & **(\%)** & **(A/(mm.mrad))\({}^{2}\)** \\ \hline \(a_{2}\) = 0 & 103 & 30 & 2.82 & 6.20 & 0.48 & 0.24 & 0.3 & \(3.6\times 10^{6}\) \\ \hline Low \(a_{2}\) case & 119 & 35 & 2.79 & 4.56 & 4.83 & 1.14 & 0.3 & \(7.0\times 10^{4}\) \\ \hline High \(a_{2}\) case & 328 & 234 & 1.37 & 0.14 & 8.29 & 6.56 & 0.3 & \(7.4\times 10^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Horizontal and vertical normalized emittance and charge percentage arriving of the electron beam at the undulator center for \(a_{2}\) = 0, low and high \(a_{2}\) (Table 2) cases. Simulation using the low divergence initial beam distribution (Figure 2a), 176 MeV “monochromatic” beam with \(\sigma^{\prime}_{z,i}\) = 2 mrad, \(\sigma^{\prime}_{x,i}\) = 3.12, initial charge 100 pC, mrad \(\epsilon_{z,i}\),\(\epsilon_{x,i}\) = 0.2 mm.mrad, transported with the “slit-undulator” optics (slit closed to 3.2 mm corresponding to a slice of 176 \(\pm\) 7 MeV). Each point averaged over 5 repetitions of the simulation. from the second term of Equation (2). \(\sigma_{s}\) goes from \(5.3\times 10^{-3}\)\(\upmu\)m for \(a_{2}=0\) to \(6.7\times 10^{-3}\)\(\upmu\)m for \(a_{2}=\pm 0.3\). Due to the small variations of \(\sigma_{s}\) and charge, the brightness depends mainly on the emittance, so between \(a_{2}=0\) and \(a_{2}=0.01\), the brightness decreases by a factor of 9.