## 4 Influences of Non-Gaussian Transverse Distributions
### Axisymmetric Non-Gaussian Transverse Distribution
However, the emittance can always be reduced by removing these particles with large betatron oscillation amplitudes. It is therefore more convenient to consider only the particles in the core. A spatial range of the core can be chosen as the matched beam radius \(\sigma_{r,ic}\) in the plasma ion column. Similarly, the momentum range for the core can be chosen as the matched transverse momentum spread, \(\sigma_{p,ic}=\epsilon_{n0}/\sigma_{r,ic}\). Considering these two factors, we define the core range of the beam in the 4D transverse space \((x,p_{x},y,p_{y})\) as where witness electrons satisfy the condition of  
\[\frac{x^{2}+y^{2}}{\sigma_{r,ic}^{2}}+\frac{p_{x}^{2}+p_{y}^{2}}{\sigma_{p,ic }^{2}}<4. \tag{2}\]  
Figure 6: Halo parameters of witness beams with SG spatial profiles. Evolution of the transverse profile halo parameter \(H_{x}\) for the (**a**) low-charge (120 pC) cases and (**b**) high-charge (400 pC) cases are presented. **(c**)** The slice distribution of the high-charge beams’ halo parameter \(H_{x}\) at the end of acceleration (\(s=10\) m). Slice bin size is \(5\mathrm{d}z\) and \(\mathrm{d}z=0.01k_{1}^{-1}\) is the longitudinal grid resolution. The longitudinal coordinates of beam slices are relative to the initial centroid position \(\zeta_{0e}\) and normalized by the initial witness bunch length \(\sigma_{ze}\). The absolute current profile \(I\) is shown by the grey dashed line. This condition allows the evolution of the charge fraction within this core range to be measured compared with an initially fully-matched beam. For the ideal case of a fully matched beam, the charge in the core should remain constant over acceleration, in a similar way as the emittance. Figure 7 shows the charge fraction \(Q_{core}/Q_{0}\) within the core range for the considered initial spatial profiles. In Figure 7a, \(Q_{core}/Q_{0}\) first sees a rapid decrease due to the initial expansion of the beam radius. Then after the reaching full phase-mixing of the transverse electron oscillations, \(Q_{core}/Q_{0}\) remains almost constant except for a minor increase. As can be seen from Figure 7b, the fraction of charge in the beam core after acceleration increases monotonically with the initial beam halo parameter \(H_{0}\) for both charge groups. As the increase of charge in the core contributes to the increase of the core brightness, these results also suggest that the beam with a large transverse profile halo parameter at injection can maintain a higher brightness after acceleration. Since the 400-pC witness beams generate stronger transverse focusing field than 120-pC beams, the final value of \(Q_{core}/Q_{0}\) is also slightly higher for 400-pC witness beams. It should be noted that the initial value of \(Q_{core}/Q_{0}\) is independent of the beam charge since they have the same distributions. Above we have discussed the scenario where the beam's transverse spatial profiles follow the super Gaussian (SG) distribution, while their momentum profiles are still Gaussian. Figure 8 presents the opposite situation where the beam's initial transverse momentum profiles exhibit different SG distributions, while the spatial profiles are Gaussian. For the two equivalent cases with \(H_{0}=1\), i.e., with Gaussian profiles in all dimensions, we get the same results in all aspects as expected. However, the impact of the SG spatial profiles and momentum profiles are not entirely the same. In Figure 8a, we can see that the beam emittance has a stronger dependency on the initial momentum profiles than on the initial spatial profiles. For the initial low-kurtosis transverse momentum profiles (\(H_{0}<1\)), these beams exhibit a larger emittance than the SG spatial profile beams with \(H_{0}<1\) as well as all cases with \(H_{0}>1\). For the initial beams with highly-peaked momentum profiles (\(H_{0}>1\)), they show a slightly lower emittance than their spatial profile counterparts. Nonetheless, Figure 8b shows that the SG momentum profiles can generate the same effect in the core-range charge fraction as the SG spatial profiles.