## 4 Influences of Non-Gaussian Transverse Distributions  
### Axisymmetric Non-Gaussian Transverse Distribution  
Realistic non-Gaussian beam distributions can be simply classified into two categories: the axisymmetric and non-axisymmetric distributions [11]. Here, we first look at the witness beam acceleration with axisymmetric non-Gaussian transverse distributions. Beams with axisymmetric transverse profiles can be fitted by the super Gaussian (SG) function (or generalized Gaussian) in the form of \(f(x)\propto e^{-|x|^{p}}\), where \(p\) is the form parameter [11; 13]. The one-dimensional (1D) projection of five different transverse beam density profiles in SG distributions are shown in Figure 4. They have the same initial RMS radius \(\sigma_{r0}=\sigma_{r,ic}\) as the baseline Gaussian case (\(p=2\)) at the injection point. Using the same simulation configuration as the Gaussian case, we study how these axisymmetric SG density distributions affect the witness beam acceleration in the quasi-linear plasma wakefield. In Figure 5, the dependence of the normalized transverse beam emittance at the end of the acceleration (\(s=10\) m) on the form parameter \(p\) is shown. It is found that the evolution of the normalized emittance of different cases generally follow that of the Gaussian case, i.e., the trend shown in Figure 3. The final value of emittance depends weakly on the distribution shape. Nonetheless, the transverse emittance of all non-Gaussian cases is shown to be slightly lower than that of Gaussian beam (\(p=2\)). In order to characterise these non-Gaussian distributions, higher-order statistical parameters such as the kurtosis can be used [22; 23]. As shown in Figure 4, for cases \(p<2\), these distributions have longer tails but highly peaked core densities, which correspond to a larger kurtosis. For more rectangular distributions with \(p>2\), they have a smaller kurtosis. The kurtosis of a beam's transverse spatial profile is a good indicator of the visually-observable halo, i.e., a low-density ring surrounding the higher-density core. However, the spatial profile kurtosis will oscillate when the transverse phase-space ellipse of the beam rotates [23]. In the transverse focusing field, the transverse motions of beam particles are governed by \(\mathrm{d}p_{x}/\mathrm{d}t=-K^{2}x\) and \(\mathrm{d}x/\mathrm{d}t=p_{x}/\gamma m_{0}\), where \(K\) is the focusing strength. The collective motion of particles allows the spatial profile and the momentum profile of the beam to be coupled and mixed during the beam propagation, which then leads to oscillations of the beam's spatial profile kurtosis as well as the momentum kurtosis. The beam halo in the 2D transverse phase-space \((x,x^{\prime})\) can be described by the halo parameter \(H\) as defined by Equation (A1). For those SG spatial profiles with form parameters of \(p=0.6,1,2,4\) and \(8\), their halo parameters are calculated as 2.05, 1.68, 1.0, 0.75 and  
Figure 4: 1D transverse beam density distributions in generalized Gaussian function. \(p\) is the form parameter. All distributions have the same standard deviation \(\sigma_{r0}\). Figure 5: Dependence of the normalized transverse beam emittance \(\epsilon_{n}\) at \(s=10\) m on the form parameter \(p\) of the initial SG distributions for both charge groups (120 pC and 400 pC). 0.65, respectively, if the corresponding momentum profiles are Gaussian. According to these values and the results shown below, a halo parameter larger than 1 means that the beam has highly-peaked non-Gaussian transverse profiles in at least one dimension of the transverse phase-space, while for \(H<1\), the beam has a flat and low-density profile in at least one dimension. The evolution trend of the beam halo parameter \(H_{x}\) of the above cases during the beam propagation is shown in Figure 6a,b. The \(y\)-direction halo parameter \(H_{y}\) is essentially equal to \(H_{x}\) in the 2D cylindrical geometry.