## 5 Model of Acceleration Scheme at ILIL  
The control of electron bunch features for applications is a very complex task that requires an accurate knowledge of the key parameters of our experimental configuration and the identification of the main acceleration mechanism occurring in our experiment. This is indeed not a simple task in our particular experimental conditions, which are quite different from the ideal self-injection conditions as, for example, in the bubble regime [32]. In a previous experiment [33], MeV electrons were obtained with a 200 fs \begin{table}
\begin{tabular}{l l} \hline \hline
**Input parameters** & **Output parameters** \\ \hline Laser wavelength = 800 nm & n\({}_{e}\)\(=7\times 10^{19}cm^{-3}\) \\ Laser pulse duration \(\geq\) 65 fs & Plasma channel length \(\approx\) 200 \(\mu m\) \\ Laser energy \(<\) 120 mJ & Plasma channel diameter \(\leq\) 30 \(\mu m\) \\ Optics numerical aperture = F/6 & Max energy (cut-off) \(\approx\) 10 MeV \\ Calc. focal spot radius = 3 \(\mu m\) & Electron energy peak @ 5-6 MeV \\ Calc. depth of focus = 24 \(\mu m\) & e-beam divergence (He) \(<\) 3 deg \\ Meas. focal spot diam. = 10 \(\mu m\) & e-beam divergence (N\({}_{2}\)) \(\approx\) 10 deg \\ gas-jet thickness = 1.2 mm & e-beam reproducibility (N\({}_{2}\)): high \\ Pulse contrast ratio (ns) = 10\({}^{-9}\) & Bunch charge (N\({}_{2}\)) \(>\) 0.1 nC \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of input and output parameters of the laser-plasma accelerator at ILIL. Figure 3: Electron spectrum obtained with the magnetic spectrometer showing quasi monoenergetic peak between 5 and 6 MeV. The spectrum was obtained from irradiation of a N\({}_{2}\) gas-jet with a backing pressure of 45 bar. laser pulse focused at a similar intensity as the one considered here. Those measurements were modelled invoking direct laser acceleration at the channel betatron resonance as described in [34]. However, in our experiment a significantly shorter laser pulse duration was used along with a lower electron density. These differences are likely to make laser wakefield acceleration more effective than in [33], also due to the marginal overlap between the laser pulse and the trailing channel. Starting from these assumptions, we explored the possible scenario using the Particle-in-Cell (PIC) numerical code AlaDyn [35] capable of modelling a 3-dimensional laser-plasma interaction configuration. AlaDyn is a fully self-consistent, 1/2/3-dimensional, relativistic Electro-Magnetic PIC code capable of handling laser pulse(s), injected bunch(es) and a user defined plasma, featuring high order schemes in space/time, with computational moving window, stretched computational grid and Boosted Lorentz frame. This code enabled us to take into account details of our experimental configuration, including laser parameters, focusing geometry and some of the gas-jet features. According to the theory, a significant self-focusing of the laser beam may occur under such experimental conditions. Using the well known expression for relativistic self-focusing, \(P_{cr}\approx 17(\omega/\omega_{p})^{2}\) GW and taking into account the measured electron density of \(7\times 10^{19}\) cm\({}^{-3}\), we find that the critical power in our experimental conditions is \(0.4\) TW, which is below the nominal power of our laser system. This result suggests that in our experiment, conditions are satisfied for laser beam self-focusing to occur. A confirmation of this result was obtained from 3-dimensional AlaDyn calculations as summarised in the plot of Figure 4, which shows the evolution of the laser beam intensity along the propagation distance. Two different density profiles have been considered in order to take into account shot-to-shot variability and to study the sensitivity of the dynamics to the physical parameters. In both cases, according to the simulations, the self-focusing brings the peak laser beam intensity up to a value larger than \(2.7\times 10^{19}\) W/cm\({}^{2}\).