## 4 Numerical PIC Modeling  
To elucidate the mechanism behind the enhanced proton acceleration with a DPM, we have carried out 2D PIC simulations with the fully relativistic and electromagnetic smilei code [63]. The numerical domain has dimensions of \(L_{x}\times L_{y}=21\times 42\,\upmu\)m\({}^{2}\) with a mesh size \(\Delta x=\Delta y=7.8\,\upmu\)m, corresponding to 128 cells per micron. The time step is \(\Delta t=0.5\Delta x/c\), where \(c\) is the speed of light. The laser beam, of wavelength \(\lambda_{0}=0.8\,\upmu\)m, is injected from the lower left side (\(x=0\)) of the box at \(45^{\circ}\) incidence angle. It is focused with \(p\)-polarization onto the front side of the target (at \(x=10\,\upmu\)m and \(y=21\,\upmu\)m). It has a Gaussian envelope with \(3\,\upmu\)m FWHM spot size and 24 fs FWHM duration. In the no-DPM case, the laser peak intensity is \(I_{0}=1.8\times 10^{21}\,\mathrm{W}\,\mathrm{cm}^{-2}\), corresponding to \(a_{0}=29.2\). The laser beam also has a 1 ps long prepulse with the same spatial profile and a constant intensity.The intensity of the prepulse was adjusted so that the simulated proton energy spectrum matches the experimental one, see Figure 4a. In the DPM case, we consider only the main 24 fs pulse, with no prepulse, but with a reduced peak intensity of \(9\times 10^{20}\,\mathrm{W}\,\mathrm{cm}^{-2}\) (corresponding to \(a_{0}=20.5\)) to take into account the energy loss associated with the DPM. In the no-DPM case, the plasma targets consist of neutral Al foils with thickness \(d=0.8\)-\(2\,\upmu\)m. In the DPM case, we use neutral Si foils with \(d=10\)-\(300\) nm, followed by a 50 nm thick neutral Al layer. All targets are coated with a 10 nm thick hydrogen contaminant layer on their backside. Due to computational limitations, their initial ion number density is capped at a sub-solid density value, i.e., \(n_{i}=60\,n_{c}\), where \(n_{c}=m_{e}e_{0}\omega_{0}^{2}/e^{2}=1.75\,\times\,10^{21}\,\mathrm{cm}^{-3}\) is the critical density at the laser wavelength \(\lambda_{0}\equiv 2\pi c/\omega_{0}\) (\(e_{0}\) is the vacuum permittivity, \(m_{e}\) the electron mass, and \(e\) the elementary charge). Electron and ion species are represented by 256 macroparticles per cell, with a fourth-order shape-function. All simulations describe field and electron impact induced ionization. Boundary conditions for both particles and fields are open along both longitudinal (\(x\)) and transverse (\(y\)) directions. The integration time of the simulations is 500 fs after the laser peak. To illustrate the coupling of the prepulse-free laser (DPM case) with two ultrathin Si foils as a function of their thickness, we display in Figure 10 the spatial distribution of the transverse electric field (\(E_{y}\)). We observe near-complete laser transmission across the 10 nm target (Figure 10a) while the 20 nm target gives rise to partial transmission and reflection (Figure 10b). This behavior aligns with our experimental findings, i.e., high laser transmission across the 10 nm foil (Figure 3(c2)) and no detectable protons for targets thinner than 20 nm (Figure 5). Figure 10: Spatial distribution of the transverse electric field \(E_{y}\) (in \(10^{13}\)\(\mathrm{V}\,\mathrm{m}^{-1}\) units), in the DPM (no-prepulse) laser configuration with a Si target of (**a**) 10 nm and (**b**) 20 nm thickness. The simulated proton energy spectra, for a 200 nm Si + 50 nm Al target (DPM case, red dashed line) and a 1.5 \(\upmu\)m Al target (no-DPM case, blue solid line), are plotted alongside the experimental data in Figure 4a. Both simulation curves, integrated over the entire simulation duration (500 fs after peak laser impact), capture well the exponential shape of the distributions as well as their cutoff energies (\(E_{\mathrm{max}}=32\) MeV and 24 MeV, respectively). Moreover, the simulated electron spectra, plotted in Figure 4b, indicate that the no-DPM configuration generates hot electrons with energies similar to the DPM case, but in much larger numbers.