## 2 Fluctuations of the Plasma Source and the Electron Beams
The first ionization in fact occurs even more easily than the second, because it corresponds to a lower ionization potential (please notice that the approximation of a hydrogen-like atom cannot hold true for the \(He\) atom, thus the \(u\) factor should in that case take into account for the screening effects, but this eventually does not affect the validity of our arguments). The number of photons needed for a single He atom ionization (second ionization) is approximately 35. For a laser beam radius of \(\sigma\sim 15\) um and a propagation length in the plasma of \(L\sim 2\) mm we obtain that the total number of atoms involved effectively in the interaction is \(n_{a}\pi\sigma^{2}L\simeq 3\times 10^{12}\), given a neutral density \(\sim 2\times 10^{18}\) cm\({}^{-3}\). Thus, the number of photons lost in ionization is \(35\times 3\times 10^{12}\simeq 10^{14}\). The number of photons carried by a laser pulse with peak intensity of \(l_{0}\sim 10^{19}\) W/cm\({}^{2}\), corresponding to a laser energy of \(\sim\)3 J, 30 fs (FWHM) pulse duration and focal radius \(\sigma\sim 15\) um, is of the order of \(N_{0}\sim 10^{19}\). This demonstrates that ionization is a loss channel for the laser energy with a branching ratio of \(\sim\)\(10^{-5}\), i.e., most of the laser energy is coupled into the plasma. Knowing the maximal density, we can set a scale on the left plot of Figure 3. This is performed also under another consideration: Thomson scattered light is proportional to the local electron plasma density. Even if it remains true that electrons scattering the laser light are located in a region corresponding to a local plasma density higher than the background plasma density (due to the ponderomotive action of the laser pulse-front that accumulates electrons at the head of the pulse), such density must be proportional to the background plasma density (the amount of charge blown by the laser is proportional to the amount of charge available before such interaction). The visible camera can record scattered light at \(\omega_{0}\) and \(2\omega_{0}\), the latter being produced via non-linear Thomson scattering. The amount of power scattered by the unit volume of plasma is therefore found through (for a linearly polarized laser):  
\[\frac{dP}{dV}=\frac{n_{e}l_{0}\sigma_{T}}{1+\frac{a_{0}^{2}}{2}}\Bigg{(}1+ \frac{a_{0}^{2}}{2+a_{0}^{2}}\Bigg{)} \tag{4}\]  
Figure 3: (**Left**): Reconstructed electron plasma density axial profiles from the Thomson scattering optical diagnostics. (**Right**): shot-to-shot fluctuation of the plasma filament length, assumed to be as long as the electron acceleration length; the uncertainty is determined by the resolution of the imaging system. where \(n_{e}\) is the electron plasma density seen by the laser pulse, \(\sigma_{T}\) is the classical Thomson cross-section, while the terms in \(a_{0}=eE_{0}/m\omega_{0}c\) are relativistic corrections (\(m\) is the electron mass, \(e\) is the elementary charge and \(c\) is the speed of light in vacuum). In particular, the second term within brackets is associated with the emission of the second harmonic of the laser frequency. The laser beam is guided by relativistic self-focusing in the plasma channel and its energy is depleted in generating strong plasma perturbations, i.e., the plasma wakefields. In the extremely non-linear regime of interaction \(a_{0}>>1\), the pump depletion length is \(L_{pd}=\omega_{0}^{2}c\tau/\omega_{p}^{2}\), with \(\tau\) the laser pulse length. The plasma frequency is related to the electron plasma density as \(\omega_{p}=\sqrt{n_{e}e^{2}/\varepsilon_{0}m}\), where the vacuum dielectric constant is \(\varepsilon_{0}\). In our experiment \(L_{pd}\simeq 4.1\) mm, which was larger than the average plasma filament length, meaning that the laser was only moderately depleted during the propagation. Moreover, the laser beam envelope oscillations, typical of self-guiding phenomena, could not be properly resolved by the Thomson imaging system.