## 3 Experimental Results  
### Increased Proton Energies  
The proton energy spectra were inferred from the doses deposited into the RCF stack as detailed in Ref. [21]. Figure 4a shows two proton spectra for the target thicknesses maximizing the proton cutoff energy (\(E_{\text{max}}\)) with or without a DPM. Overlaid are predictions from PIC simulations (see Section 4). The best performance in the no-DPM case was recorded with a 1.5 um thick A1 foil, yielding \(E_{\text{max}}\simeq 25\) MeV. The best DPM shot used a 250 nm (200 nm Si + 50 nm A1) thick foil, yielding \(E_{\text{max}}\simeq 35\) MeV, i.e., a \(\sim\)40% enhancement. The proton spectra plotted in Figure 4a exhibit an exponential behavior of the form \(dN/dE=(N_{p}/T_{p})\exp(-E/T_{p})\) for \(E\leq E_{\text{max}}\), where \(E\) is the proton energy, \(T_{p}\) the effective temperature and \(N_{p}\) the total proton number. The best-fitting parameters are \(N_{p}=2.1\times 10^{11}\) and \(T_{p}=3.0\) MeV without the DPM, and \(N_{p}=1.4\times 10^{11}\) and \(T_{p}=4.0\) MeV with the DPM. Note that the first RCFs were excluded from the analysis because their signal was saturated and they did not collect a fraction of the proton beam (the lateral extent of the proton beam exceeds the size of the films for these first layers, as shown in the raw RCF data below). The low-energy (\(<\)8 MeV) component of the proton spectra not captured by the RCFs was characterized by a magnetic spectrometer in a previous campaign, conducted under similar interaction conditions, but without a DPM [50]. This measurement revealed a steeper proton spectrum (\(T_{p}\simeq 1.1\) MeV)Figure 5a shows how the proton cutoff energy varies with target thickness (\(d\)). Empty and filled symbols represent the maximum and mean cutoff energies, respectively, from all shots with a given target thickness. Both quantities exhibit the same behavior, peaking around \(d=250\) nm with a DPM and at \(d=1.5\) um without. Notably, in the absence of a DPM, no detectable proton signal is observed from Al targets thinner than 800 nm. With a DPM, significant proton acceleration is achieved even in Si foils as thin as 20 nm, with a maximum proton energy of around 15 MeV. Figure 4: (**a**) Measured proton spectra, extracted from RCF data in the DPM case with a 250 nm (200 nm Si + 50 nm Al) thick target (red points, averaged over 3 shots) and in the no-DPM case with a 1.5 μm thick Al target (blue points, averaged over 2 shots). Error bars represent the minimum and maximum values from different shots with the same target thickness. Exponential fits are shown as black dotted lines (DPM case) and purple dashed-dotted lines (no-DPM case). 2D PIC simulation results for both cases are plotted as red dashed and blue solid lines, respectively. (**b**) Simulated electron spectra together with exponential fits in the 4.5–11.5 MeV range. The line styles are the same as in panel (**a**). Figure 5: Target thickness dependence of the (**a**) proton cutoff energy and (**b**) laser-to-proton energy conversion efficiency (see text for definition). For both quantities, we distinguish between the maximum (empty symbols) and mean (filled symbols) values for all shots performed with the same target thickness. Direct and DPM shots are represented by squares and dots, respectively. The target thickness for DPM shots refers to the entire Si/Al target. For the thinnest targets, the results are single-shot, hence no mean is reported. Figure 5b shows the conversion efficiency of laser energy into protons, evaluated by integrating (over \(0\leq E\leq E_{\text{max}}\)) the exponential fit of each recorded proton spectrum (see Figure 4a). We use the fit of the spectrum rather than the actual spectrum since, as discussed above, the number of low-energy protons retrieved from the first films is underestimated (the films are saturated). The laser energy considered here is the on-target value for each configuration (i.e., taking into account the energy loss induced by the DPM).