## 3 Particle-In-Cell Simulation Results
Consequently, the electrons at the laser front maintain the high density compression and, a significant number of protons (see the blue and black lines in Figure 3b) are mainly accelerated by RPA in a relatively cold plasma. Note that other ion acceleration mechanisms by virtue of the fast electrons [47; 48; 49] are suppressed. Figure 3: Plasma temperature (red line) averaged over \(|y|\leq 2\lambda\) at \(t=18T_{0}\) for the cases (**a**) without and (**b**) with the NCD plasma, respectively. In each plot, the black and blue lines indicate the densities of the protons \(n_{H}\) and electrons \(n_{e}\) averaged over \(|y|\leq 2\lambda\). Figure 2: Comparison between different target configurations. (**a,b**) Proton distributions (color contour) and densities averaged at \(|y|\leq 2\lambda\) (blue line) at \(t=40T_{0}\) obtained from the cases without and with the NCD plasma, respectively. In both cases, a CP laser pulse of \(a_{0}=100\) and spot size \(R=6\lambda\) is employed. (**c,d**) The corresponding proton energy-angle (\(\theta=\arctan(p_{y}/p_{x})\)) distributions (color contour) and energy spectra (red line). At the stage of laser foil interaction, a compressed electron layer with high density is pushed forward and a large amplitude charge-separated field is excited. Thanks to the NCD plasma, this charge-separated field can sustain a substantial amplitude and propagate at relatively low speed because the CP laser pulse of \(a_{0}=100\) can propagate at around this speed in the classically overdense plasma due to relativistically transparency [44]. Figure 4a shows that the electron density peak is kept for a long time. Meanwhile, the longitudinal electric field \(E_{x}\) with large amplitude can be sustained, which is illustrated by Figure 4b. In addition, this longitudinal electric field is highly asymmetrical [43; 50], which is characterized by a strong positive electric field at the front part and a relatively weak electric field with a negative value at the rear part. Such asymmetrical field structure is favorable for the ion acceleration. The blue line in Figure 4c displays the temporal evolution of the propagating speed of \(E_{x}\). The propagating speed is defined as the forward velocity of the longitudinal coordinate \(x_{f}\), where the electric field strength peak appears. At the stage of laser-foil interaction (\(t\leq 12T_{0}\)), the propagating speed of \(E_{x}\) increases as long as the foil is accelerated by the RPA. At the stage of \(t>12T_{0}\), it is clear that the propagating speed of \(E_{x}\) substantially depends on the density of the NCD plasma as the thin foil becomes transparent. In the uniform NCD plasma with \(n_{e}=2n_{c}\), the propagating speed can be estimated as \(v_{x}\simeq v_{laser}\simeq 0.88c\). The propagating speed at first remains quasi-steady at the speed of \(0.87c\), then decreases gradually due to the laser depletion. Such a large amplitude longitudinal electric field with a sub-relativistic speed helps to boost the speed of a large amount of protons further. As shown by the pink triangles in Figure 4c, the speed of the proton energy peak reaches up to \(0.91c\) after about \(40T_{0}\). A track of a typical proton from the energy peak is plotted by the black dotted line in Figure 4b, which illustrates the proton is trapped in the accelerating field peak. This typical track suggests that the protons in the energy peak are accelerated by the co-moving longitudinal electric field until they surpass the accelerating field. This means a quasi-monoenergetic proton beam can be attained in this SRPA scheme. The dense quasi-monoenergetic beam can be seen in Figure 2d, which has a FWHM energy spread \(\sim\)13% at the peak about 1.2 GeV and an opening angle \(\sim\)8.5\({}^{\circ}\). Furthermore, the number of the proton beam can also be greatly enhanced in the case with the NCD plasma. As shown in Figure 2c,d, the number of the protons above 1 GeV in the case with the NCD plasma is 4.5 times higher than the one in the case without the NCD plasma.