## 4 Numerical PIC Modeling
However, this more energetic electron spectrum in the no-DPM case does not translate into better ion acceleration, as shown in Figure 4a. This counterintuitive observation suggests that the stronger ion acceleration achieved with a DPM has a different origin. As discussed in the following, this could result from an enhanced \(E_{x}\) field acting on the proton bunch due to sharper density gradients at the time of the laser peak. To explain the stronger proton acceleration observed at high laser contrast, we plot in Figure 11 the evolution of longitudinal lineouts (across the center of the focal spot) of the particle (electron and proton) densities and electric (\(E_{x}\) and \(E_{y}\)) fields, respectively measured in units of \(n_{c}\) and \(E_{0}=m_{e}c\omega_{0}/e\). The \(E_{x}\) field is averaged over the laser cycle. The time origin coincides with the on-target intensity peak. For a 200 nm Si + 50 nm Al target hit by the high-contrast pulse (top row), the accelerating \(E_{x}\) field seen by the protons (the front of which is located at \(x\simeq 11\upmu\)m) reaches a maximum strength of \(\sim\)4 \(E_{0}\) just after the laser maximum (\(t=3.5\) fs). This field decays over time (down to \(\sim\)0.75 \(E_{0}\) at \(t=55\) fs), but its spatial maximum always coincides with the outer edge of the proton bunch (Figures 11b,c), where the fastest protons reside (see the ion phase-space in Figure 12b). This spatial overlap of the peak electric field and the fastest protons maximizes the acceleration efficiency. Figure 11: Simulations of a DPM shot on a 200 nm Si + 50 nm Al target (**a–c**) and a direct shot on a 1.5 \(\upmu\)m thick Al target (**d–f**). Longitudinal cuts (across the center of the focal spot at \(y=21\upmu\)m) of the electron density (red solid curves), proton density (blue dotted curves), \(E_{y}\) electric field (cyan dashed curves) and \(E_{x}\) (averaged over the laser cycle, purple dotted-dashed curves) are plotted at various times after the laser peak has struck the target (\(t=0\)). Particle densities are measured in units of \(n_{c}\) while field strengths are shown in units of \(E_{0}=m_{e}c\omega_{0}/e\simeq 4\times 10^{12}\) V m\({}^{-1}\). The target rear side is initially located at \(x=10.25\upmu\)m with the DPM and \(x=11.5\upmu\)m without it. The situation is markedly different in the absence of a DPM, for a 1.5 \(\upmu\)m thick Al target (bottom row). As shown in Figure 11d, the prepulse-driven expansion of the target backside causes a significant reduction in the \(E_{x}\) field [64] just after the laser peak (\(E_{x}\simeq 2.5\,E_{0}\)). In addition, the spatial maximum of \(E_{x}\) no longer coincides with the front of the proton bunch (see also Figure 12a). Another interesting point to notice is the significant \(E_{y}\) (\(>\)\(E_{0}\)) field generated at \(t=30\) fs after the main pulse irradiation at low laser contrast (Figures 11e and f). This transverse field is mostly positive and hence will bend the protons towards the \(y>0\) direction, i.e., towards the laser propagation axis [65]. This should contribute to the degraded beam profile seen in Figure 6a, where the high-energy protons show an elongated shape towards the laser axis. On the contrary, at high laser contrast (Figure 11b), the accelerated protons experience a negligible \(E_{y}\).