## 4 Conclusions  
It is worth pointing out that our stable radiation pressure acceleration (SRPA) scheme could provide a reference for the ion acceleration experiments with PW-class laser systems recently available and 10-PW-class laser systems available in the near future [54; 55]. A NCD plasma behind the solid thin foil might be obtained by the ionization of an ultra low-density plastic foam [56]. Alternatively, such an NCD plasma might be achieved with the help of a secondary nanosecond laser pulse at a lower intensity [57].  
To summarize, we have investigated, by analysis and PIC simulation, a possible stable radiation pressure acceleration (SRPA), when a multi-petawatt CP laser pulse irradiates a combination target of a \(CH_{2}\) thin foil and a large-scale NCD plasma. With a suitable density range, the NCD plasma can not only significantly suppress strong electron heating in the laser-foil interaction, but also decrease the propagating speed of the laser pulse and the induced acceleration field to allow the ions to catch up. In this way, the ions can nearly co-move with the acceleration field in a relatively cold environment and, therefore, they can be efficiently accelerated. By simple analysis, a rough range of suitable NCD plasma density has been given and verified by our PIC simulation. Our simulations have shown that collimated quasi-monoenergetic proton beams with peak energy of 360 MeV and 1.2 GeV can be generated by a pulse of \(a_{0}=50\) (\(6.85\times 10^{21}\) W cm\({}^{-2}\)) with a \(n_{NCD}=3.5n_{c}\) plasma and a pulse of \(a_{0}=100\) (\(2.74\times 10^{22}\) W cm\({}^{-2}\)) with a \(n_{NCD}=2n_{c}\) plasma, respectively. The SRPA  
Figure 6: (**a**) The electron density (with the colorbar in a log scale) at \(t=35T_{0}\), where the laser field along the axis \(y=0\) is plotted by the red dashed line. (**b**) The densities of protons (\(n_{H}\)) and carbon ions(\(n_{C^{+}}\)) with the colorbar in a log scale, where the longitudinal electric field \(E_{x}\) along the axis \(y=0\) is plotted by the green dotted line.  
Figure 7: (**a**) Phase space \(p_{x}-x\) of protons at \(t=30T_{0}\), \(t=40T_{0}\), and \(t=50T_{0}\), respectively. (**b**) Spectra of protons (red line) and carbon ions (black line) at \(t=50T_{0}\). Here, the laser and target parameters are the same as those in Figure 6.  
scheme can efficiently work with the laser intensity ranging from \(6.85\times 10^{21}\) W cm\({}^{-2}\) to \(4.38\times 10^{23}\) W cm\({}^{-2}\) and the QED effects have a slight influence on this scheme.  
Investigation, M.L. and J.-X.G.; Supervision, W.-M.W. and Y.-T.L.; Writing--original draft, M.L. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Not applicable.  
This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant Nos. XDA25050300, XDA25010300), the National Key R&D Program of China (Grant No. 2018 YFA0404801), the National Natural Science Foundation of China (Grant Nos. 11827807) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2020 MS138). The authors declare no conflict of interest.