## 2 Fluctuations of the Plasma Source and the Electron Beams  
In this section, we present a working point of the VEGA2-based electron source. We stress that a single working point is shown, despite many other operational conditions being available at the facility. This one was found to be the most stable for the type of gas nozzle and gas species/density adopted. In more detail, a conical gas jet was used, 5 mm long at the exit-plane of the nozzle, along the laser propagation axis (see Figure 2 for the schematics of the experimental setup). The gas used for the experiments was pure helium, therefore the mechanism of electron injection was the pure self-injection. Alternative injection mechanisms can be implemented at the facility, and would give different performances of the electron and relative radiation sources. Future experimental campaigns will be devoted to this aspect. Besides the scenario of exploiting and/or mixing different gases, also different nozzle designs with different lengths, different conical apertures and/or capable of sustaining different density regimes may be considered. We do not go further in this vast topic, since the data presented in this paper refer to a single operational configuration, above described. Furthermore, we present a sequence of 10 shots for each data collected in the campaign, in such a way as to observe and quantify the shot-to-shot fluctuations over a significant amount of acquired data corresponding to 10 min of operation (indeed, the working rep rate was one shot per minute, dictated by the recovery of the vacuum level after shooting the laser on the gas target). The plasma filament was studied by means of a visible camera, watching the filament from the top, with an angle of 90\({}^{\circ}\), detecting the light from the plasma, dominated by the mechanism of Thomson scattering. The experimental results are shown in Figure 3. The maximal plasma density is measured starting from the density of neutrals, which is characterized prior to the experiment on a separate interferometric station. The laser intensity is such to ionize the gas completely before the main pulse arrives on target, in such a way that the core laser energy is transferred into the plasma. The laser beam arrives on the nozzle with a height of \(\sim\)2 mm with respect to it. The ionization rate over one cycle of the laser field is given by the Landau expression for tunneling ionization [21]:  
\[w=4\omega_{a}u^{5/2}\frac{E_{a}}{|E_{0}|}e^{-\frac{2E_{a}}{3|k_{0}|}u^{3/2}} \tag{3}\]  
Figure 2: Experimental setup (X-rays detected after deflecting the electrons into the spectrometer). The quantity \(u\) takes into account for a hydrogen-like atom different from hydrogen (for which \(u=1\)). Indeed, \(u\) is the ratio between the ionization potential of a considered atom/molecule and that of the hydrogen atom/molecule. For \(He\), considering the second ionization potential, which is larger than the first, we obtain \(u=4\). The atomic frequency is constantly equal to \(\omega_{a}=4.134\times 10^{16}\) rad/s, while the atomic field is \(E_{a}=5.14\) GV/cm. For an electric field \(E_{0}\sim 2.7\) GV/cm, corresponding to a laser intensity \(10^{16}\) W/cm\({}^{2}\), within an integration time corresponding to a laser cycle \(2\pi/\omega_{0}\) (for VEGA2 \(\omega_{0}=2.35\times 10^{15}\) rad/s), \(\int_{0}^{2\pi/\omega_{0}}dtw\simeq 1\), which means that the atom is surely ionized much before interacting with the peak of the laser pulse. The peak intensity of VEGA 2 used for the experiments in this paper was \(10^{19}\) W/cm\({}^{2}\), on average, much larger than \(10^{16}\) W/cm\({}^{2}\), therefore the front tail of the laser pulse was able to fully ionize the helium gas.