## 3 Particle-In-Cell Simulation Results
Figure 4: Temporal evolutions of **(a)** electron density \(n_{e}\) and **(b)** longitudinal electric field \(E_{x}\). **(c)** Temporal evolution of the propagating velocities of the longitudinal electric field \(E_{x}\) (blue line) and the velocities (pink triangles) of the proton energy peak. Trajectory of a typical proton is plotted by black dotted line in **(b)**. The simulation parameters are the same as those in Figure 2.  
Our analysis in Section 2 shows that the density \(n_{NCD}\) of the NCD plasma affects the acceleration efficiency of protons. We implement a series of simulations with \(n_{NCD}\) ranging from \(0.6n_{c}\) to \(4n_{c}\). We find that only when \(n_{NCD}>1.2n_{c}\), the quasi-monoenergetic proton beam can be acquired. Figure 5a presents the dependency of the proton peak energy \(E_{i,peak}\) and the corresponding energy spread \(\eta\) on the density \(n_{NCD}\) when the quasi-monoenergetic proton beam is generated. The peak energy \(E_{i,peak}\) goes up as \(n_{NCD}\) drops and the laser propagating speed grows. In particular, the protons cannot be accelerated efficiently at \(n_{NCD}>3.5n_{c}\), as the protons gain a velocity around \(v_{p}\simeq 0.73c\) at the stage of laser-foil interaction, which approaches or even exceeds the propagating speed of laser pulse and the longitudinal electric field. Note that the energy spread has an optimum value around \(n_{NCD}=2n_{c}\), with which the laser propagating speed is not too high or too low for the protons to easily catch up with the accelerating field and also to be further accelerated in the NCD plasma. Then, we investigated the scaling law between the proton peak energy \(E_{i,peak}\) and the laser amplitude \(a_{0}\), as presented by red triangles in Figure 5b. The peak energy increases rapidly as the laser amplitude ranging from \(a_{0}=50\) to \(a_{0}=400\). It indicates that this SRPA scheme works well at the laser intensity \(I_{0}\simeq 2.74\times 10^{22}\) W cm\({}^{-2}\) and it can be extended to a larger laser intensity range. At \(a_{0}\geq 400\), the protons reach up to a high speed in the laser-foil interaction, while the propagating speed of the laser front and accelerating field are obviously limited in the NCD plasma. In this case, the proton acceleration in the NCD plasma is not efficient. We also check the QED effects including photon generation via nonlinear Compton scattering and the induced radiation reaction (RR), as the laser intensity at and above \(10^{23}\) W cm\({}^{-2}\) (\(a_{0}\geq 200\)) is taken [51]. One can see from the blue stars in Figure 5b that there are no significant differences in the proton peak energies when the QED module is switched on or off. In our SRPA scheme, the high energy electrons, especially those electrons around the laser propagating axis, are mainly forward propagating along with the CP laser field. In this case, the photon generation is suppressed due to the small nonlinear quantum parameter \(\chi_{e}\)[52; 53]. In the typical simulation with \(a_{0}=300\), the generated photons cover about 3% of the total laser energy. Note that we increase the foil thickness \(L\) with the enhancement of the laser amplitude in Figure 5b. For example, we take the thickness \(L\) to \(0.245\lambda\) for \(a_{0}=300\) and the simulation results are presented in Figures 6 and 7. In Figure 6, one can see that the laser front moves together with the electron, proton, and carbon layers. Therefore, both the protons and ions are efficiently accelerated. From Figure 7a, one can see that the maximum momentum \(p_{x}\) of protons increases to \(4m_{i}c\) at the interaction time \(t=50T_{0}\). The corresponding peak energy \(E_{i}\) of accelerated protons is about 2.5 GeV, as shown by the red line of Figure 7b. The black line of Figure 7b displays that high energy carbon ions, reaching up to 2.3 GeV/nucleus, can also be produced at the same time.