## 4 Materials and Methods
### Experiments  
For the experiment reported here, the 800-nm Ti:Sapphire \"TW\" laser beamline at the Intense Laser Irradiation Laboratory of the CNR-INO in Pisa was employed. This laser beamline provides a 30 fs duration, 450 mJ energy pulses, with an \(M^{2}\) factor close to 1.5. The laser beam was focused on the TNSA target foil with a 15\({}^{\circ}\) angle of incidence using an \(f/\approx\) 1.5 Off-Axis Parabola (Thorlabs Model MPD229-M03: Gold coating--focal length 50.8 mm--90\({}^{\circ}\) off-axis angle, reflected wavefront error \(<\)\(\lambda/2\) at 633 nm). The final intensity on the target is estimated to be about 7-8 \(\times 10^{19}\) based on the optical quality of the low-cost OAP.  
The ToF measurements were performed in air using a Si PIN diode biased with a voltage of about 80 Volt as the particle detector. An oscilloscope (Lecroy-Waverunner 64 Xi, 600-MHz bandwidth, and 5-GS/s sampling rate) is used to acquire the ToF detector signal. The time \"zero\" on the ToF traces is set relative to the X-ray/fast electrons peak that is 1.00(2) ns (given a target to Si PIN detector distance of 300(5) mm) after the proton starting time which coincides on a picosecond time-scale with the laser pulse arrival time. The uncertainty on the time difference between the X-ray/fast electrons and the cut-off proton arrival time on the particle detector is limited basically by the sampling rate. As a conservative time accuracy estimate, we can assume two times the sampling rate which results in 0.2 MeV uncertainty in the 2 to 3 MeV cut-off energy range.  
For radiochromic measurements in air, the first polyester supporting layer have been removed from the EBT3 films prior to irradiation in order for the few MeV laser-accelerated protons to reach the active layer of the film. Dosimetry evaluations are performed from the net optical density of the scanned films based on the calibration reported for EBT3 film using mono-energetic 4 and 5 MeV protons [42; 43]. By performing SRIM calculations for different discrete input energy with and without the first supporting layer the uncertainty on our estimates is about 20%.  
Conceptualization, F.B. ([PERSON]), L.L., M.C, and L.A.G.; methodology, F.B. ([PERSON]), and L.L.; software, L.L., D.P. ([PERSON]), and S.K.; validation, F.B. ([PERSON]), L.L.; formal analysis, L.L., D.P. ([PERSON]), and S.K.; investigation, F.B. ([PERSON]), L.L., D.P. ([PERSON]), S.K., L.F., P.K., and F.B. ([PERSON]); resources, F.B. ([PERSON]), L.L., and M.C.; data curation, F.B. ([PERSON]), L.L., D.P. ([PERSON]), and S.K.; writing--original draft, F.B. ([PERSON]), L.L., D.P. ([PERSON]), and S.K.; writing--review and editing, F.B. ([PERSON]), L.L., D.P. ([PERSON]), and S.K.; visualization, F.B. ([PERSON]), L.L., D.P. ([PERSON]), and S.K.; supervision, F.B. ([PERSON]), L.L., M.C., and L.A.G.; project administration, F.B. ([PERSON]), L.L., and L.A.G.; funding acquisition, F.B. ([PERSON]), L.L., M.C., D.P. ([PERSON]), and L.A.G. All authors have read and agreed to the published version of the manuscript.  
This research was co-funded by Regione Toscana (FOR FSE 2014-2020) and VCS srl (Parma, Italy) with the grant \"LaserPIXE\" within the programme ARCO-CNR. This project has also received funding from the CNR-funded Italian research Network ELI-Italy (D.M. No. 631 08.08.2016).  
Not applicable.  
Data Availability Statement: Data are available from the corresponding authors upon reasonable request.  
The authors wish to thank [PERSON] of the U. O. di Fisica Sanitaria, Azienda Ospedaliero-Universitaria Pisana (Pisa, Italy) for support with the analysis of the irradiated EBT3 radiochromic films. The authors acknowledge the useful contributions of [PERSON] and [PERSON].  
The authors declare no conflict of interest.