## 4 Effect of the QUAPEVA Skew and Initial Beam Conditions on the Electron Beam Transport
### Variation of Initial Divergence  
To investigate further the influence of the initial electron beam characteristics, the dependance of the transport on the \(\sigma_{z,t}^{\prime}\), while keeping the ratio between the vertical and horizontal divergence constant, is analyzed in Figure 10. For low \(a_{2}\) (Figure 10a) at the undulator entrance, when the initial divergences grows, \(\sigma_{x}\) remains similar, while \(\sigma_{z}\) increases slightly and the charge decreases. When the beam is transported at the undulator exit (Figure 10b), the behavior is similar, with a larger growth of \(\sigma_{z}\). For high \(a_{2}\) (Figure 10a) in the undulator entrance, \(\sigma_{z}\) and \(\sigma_{x}\) increase. The charge at 1.5 mrad divergence is much lower than for the low \(a_{2}\) case and reaches the same level as for 3 mrad. When the beam exits the undulator (Figure 10b), both beam sizes increase, and the charge dependence is similar to the low \(a_{2}\) case. The contribution of the skew term becomes more significant for larger initial divergence values, and thus, the beam size and resulting 2D emittance growth could be mitigated with electron beams presenting a small initial divergence.  
The horizontal emittance at the undulator center increases with the divergence (see Figure 11a,b), going from 62 mm.mrad (62 mm.mrad) for \(\sigma_{z}^{\prime}\) = 1.5 mrad to 107 mm.mrad (110 mm.mrad) for \(\sigma_{z}^{\prime}\) = 5 mrad for the low (high) \(a_{2}\) skew term. The vertical emittance decreases until \(\sigma_{z}^{\prime}\) = 3 mrad  
\begin{table}
\begin{tabular}{c|c c c|c c c} \hline \multirow{2}{*}{**Skew Term**} & \multicolumn{4}{c|}{**Low Divergence Beam**} & \multicolumn{4}{c}{**High Divergence Beam**} \\ \cline{2-7}  & \(\sigma_{x,\text{RMS}}\) **(mm)** & \(\sigma_{x,\text{RMS}}\) **(mm)** & **Beam Center Counts (a.u.)** & \(\sigma_{x,\text{RMS}}\) **(mm)** & \(\sigma_{x,\text{RMS}}\) **(mm)** & **Beam Center Counts (a.u.)** \\ \hline \multicolumn{7}{c}{Undulator entrance (“undulator-entrance” optics)} \\ \hline \(a_{2}=0\) & 0.52 & 0.80 & 1.00 & 0.63 & 1.08 & 0.20 \\ \hline Low & 0.51 & 0.72 & 1.16 & 0.60 & 1.20 & 0.20 \\ \hline High & 0.63 & 1.01 & 0.78 & 0.98 & 1.98 & 0.14 \\ \hline \multicolumn{7}{c}{Undulator exit (“undulator-exit” optics)} \\ \hline \(a_{2}=0\) & 0.95 & 0.73 & 0.76 & 0.92 & 0.99 & 0.17 \\ \hline Low & 0.95 & 0.76 & 0.75 & 0.85 & 1.01 & 0.14 \\ \hline High & 1.22 & 0.83 & 0.50 & 1.31 & 0.97 & 0.11 \\ \hline \end{tabular}
\end{table}
Table 4: Counts at the center of the beam, RMS horizontal and vertical beam size for the different skew cases and beam distributions on the screen before and after the undulator. Each point averaged over 5 repetitions of the simulation.  
Figure 10: Counts at the center of the beam (\(\square\)), horizontal (\(\circ\)), and vertical (\(\circ\)) beam RMS size versus initial divergence for the low (**void symbols**) and high (**filled symbols**) skew (Table 2) for the electron distribution of Figure 2a, Table 1, (**a**) before and (**b**) after the undulator with the “undulator-entrance” optics and “undulator-exit” optics, respectively. Each point averaged over 5 repetitions of the simulation.  
and then starts to increase by a small amount, because of particle loss on the vertical aperture of the undulator associated to the increase of the divergence.