## 3 Beam Dynamics of a Gaussian Electron Bunch  
Figure 2 shows the 2D colourmaps (\(y=0\)) of the plasma wakefields for the case with a low-charge (120 pC) witness beam. The initial charge distribution of the witness beam is Gaussian in both the transverse and longitudinal directions. It can be seen that the longitudinal wakefield \(E_{z}\) is nearly constant in the vicinity of the witness beam as shown in Figure 2a. Its average value also remains almost unchanged due to a very small amount of dephasing over the 10-m propagation distance. Therefore, the average beam energy increases linearly. However, since the accelerating gradient is not fully constant along the whole witness beam, it leads to a finite energy spread after acceleration, as shown in Figure 3a. Figure 3: Evolution of the Gaussian witness beam characteristics as a function of the beam propagation distance \(s\). (**a**) The average beam energy (blue lines) and the relative energy spread \(\sigma_{\gamma_{e}}/\langle\gamma_{e}\rangle\) (black lines). \(\langle\gamma_{e}\rangle\) is average Lorentz gamma factor that represents the beam energy. (**b**) The RMS transverse size \(\sigma_{r}\) (blue lines) and the normalized projected emittance \(\varepsilon_{n}\) (black lines). Results of both the low-charge (120 pC, denoted by dash-dotted lines) and high-charge (400 pC, denoted by solid lines) cases are shown. Figure 2: Plasma wakefields for the low-charge (120 pC) case in colourmaps. (**a**) The longitudinal wakefield \(E_{z}\). The 1D lineout shows the on-axis value of \(E_{z}\), whose value range is the same as the right colorbar. (**b**) The transverse wakefield \(W_{x}=E_{x}-cB_{y}\). The dashed line and the dotted line show the value at \(r=\sigma_{re}\) and \(r=2\sigma_{re}\), respectively. The initial loading position \(\xi_{0e}\) (or the longitudinal centroid) of the witness electron bunch, shown by vertical grey dash-dotted lines in both plots. The transverse wakefield \(W_{x}\) is shown in Figure 2b, which is focusing for the witness electrons. However, as shown in Figures 1 and 2b, the transverse wakefield varies continuously in \(\xi\)-direction before the bubble area. The plasma bubble with linear focusing force only covers the rear part of the witness beam. This happens since the electron bunch takes time to expel the plasma electrons from its propagation axis, with the exact dynamics subject to the beam charge distribution. Therefore, for a moving beam, the plasma bubble trails behind its low density head. At the head of the witness bunch where the plasma electron blow-out is incomplete, the witness electrons are exposed to the relatively weak plasma wakefield with non-linearity in both directions. The initial witness radius is matched to the strong focusing fields in the bubble, which causes the head of the witness beam, where the fields are weaker, to diverge. As the focusing field varies in \(\xi\), the radius of each beam slice oscillates at different local betatron frequencies. The phase difference of oscillations between different slices of the beam then leads to an increase in the projected (whole beam) radius and normalised emittance, as shown in Figure 3. Nonlinear focusing fields may also contribute to the emittance growth. The emittance and radius of the witness beam continue to grow over the first few tens of centimetres until the full phase-mixing. After reaching their maximum values, the normalised emittance remains essentially constant over the remaining acceleration length, while the RMS radius decreases due to adiabatic damping [21], following the scaling law \(\sigma_{r}\propto\gamma_{e}^{-1/4}\). Here \(\gamma_{e}\) is the Lorentz gamma factor of the witness beam. As a solution for the incomplete blowout of the plasma electrons at the head of the witness, one can increase the charge of the witness beam to a higher value [9], e.g., 400 pC.