## 2 Fluctuations of the Plasma Source and the Electron Beams
In conclusion, Equation (4) could be used to argue that the amount of light detected locally from the plasma filament was dominated by the electron plasma density more than by the laser beam evolution (unappreciable local intensity changes). In other words, the high-frequency modulations of the scattered light due to laser beam evolution in the filament could not be observed and the low-frequency modulation due to depletion was negligible. The average plasma filament length on the left of Figure 3 was calculated as:  
\[L=\frac{\int n_{e}(z)zdz}{\int n_{e}(z)dz} \tag{5}\]  
where the profiles \(n_{e}(z)\) are given on the right of Figure 3. The average value of 1.7 mm corresponds to the distance along which the laser pulse is relativistically self-guided in the plasma. Given the above assumptions, the axial profile of the accelerating field could be indirectly measured, as shown in Figure 4. In fact, the formula relating the accelerating field to the electron plasma density and laser intensity is found in the literature [14]:  
\[E=\frac{mc\sqrt{a_{0}}\omega_{p}}{e} \tag{6}\]  
On the right of Figure 4 the average gradients of acceleration are reported, in the order of 1.5 GV/cm. The axial profiles of such longitudinal plasma wakefields along the plasma \"accelerating cavity\" (indirectly reconstructed via Equation (6)) are depicted on the left of \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]0 When the beam was not magnetically deflected for energy characterization, the bunch charge was measured via a beam charge monitor after the plasma source [20]. The mean charge was around \(\sim\)370 pC, as shown in \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]1 The scaling for the bunch charge with the laser peak power \(P_{0}\) is also found in the literature in the non-linear regime of laser wakefield acceleration [14]:  
\[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]2  
where the laser wavelength is defined as \(\lambda_{0}=2\pi c/\omega_{0}\). During the experiment, the laser peak power on target was \(\sim\)100 TW, therefore we had to expect \(\sim\)400 pC per bunch according to the existing scaling laws. The experimental result was rather in agreement with the expectations. The electron beam profile at the exit of the plasma accelerator is measured by means of a Lanex scintillator screen. The shot-to-shot variation of the mean radial divergence \(\sqrt{<\theta_{ex}^{2}+\theta_{ey}^{2}>}\) (where \(\theta_{ex}\) is the electron divergence in the horizontal direction, in the same plane of laser propagation and parallel to the polarization direction, and \(\theta_{ey}\) is the vertical-divergence direction, transverse to the propagation and polarization directions) is reported on the right of \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]3 A typical electron beam profile is reported on the left of \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]4 Finally, a typical electron energy spectrum obtained by a magnetic deflector coupled to a Lanex scintillator is depicted on the left of \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]5 On the right of the same figure the shot-to-shot variations of energy and energy spreads are reported. The magnetic deflector was composed of a 10 cm long magnet, with field amplitude 1.2 T, coupled to a LANEX scintillator screen 15 cm away from the exit slit of the magnet. \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]6 (**Left**): Reconstructed accelerating wakefield axial profiles. (**Right**): shot-to-shot fluctuation of the peak of the longitudinal electric plasma wakefield; the uncertainty is determined by standard deviation of the measurements. \[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]7 Shot-to-shot fluctuations of the charge of the accelerated electron bunches.