# Enhanced Proton Acceleration from Laser Interaction with a Tailored Nanowire Target
## 3 Results and Discussions
### Sheath Filed and Proton Acceleration  
As hot electrons originating from the nanowires propagate through the target rear side, a strong longitudinal sheath field is established. This is associated with the energy of hot  
Figure 2: Simulation results at \(t=22T_{0}\). (**a**,**b**) Normalized Poynting flux (\(S_{x}\)) for the CNT (**a**) and the PNT (**b**), where forward and reflected components are labelled as red and blue, respectively. At this time both incident and reflected components of \(S_{x}\) can be observed. (**c**) Energy spectra of hot electrons in the nanowires. (**d**,**e**) Zoom-in view of Poynting flux near the target front surface. (**f**,**g**) Phase space (\(p_{x}-x\)) of electrons in the nanowires.  
electrons (\(\epsilon_{h}\sim T_{h}\)), and the Debye length of hot electrons (\(\lambda_{D}=\sqrt{\frac{\epsilon_{0}T_{h}}{n_{b}e^{2}}}\)). For a solid slab target, the sheath field can be estimated [38] as \(E_{sheath}\sim\epsilon_{h}/\lambda_{D}\propto\sqrt{n_{h}T_{h}}\), where \(n_{h},~{}T_{h}\) are the density and temperature of the hot electrons, respectively. Based on the discussion in the previous section, the energy and number of hot electrons are both increased in the CNT, it is thus expected to generate a stronger sheath field. Figure 3 exhibits the normalized longitudinal electric field at \(t=35T_{0}\). The sheath field in the CNT becomes stronger compared to that in the PNT. Furthermore, it occupies a larger area. Figure 3c shows the electric field averaged near the axis \(y=0\). The enhancement of the sheath field in the CNT is clearly observed, and its peak value is increased from 3 to above 5. As shown in Figure 3d, the electric potential created by the sheath field is obtained by integrating the electric field presented in Figure 3c, where the right boundary is set as the zero-potential point. The maximum potential in the CNT nearly doubles, meaning that the energy gain in a single proton as it propagates from the target rear side to the right boundary is estimated to double in the CNT compared to that in the PNT.  
Figure 4a-f shows the energy density (normalized by \(n_{c}m_{p}c^{2}\)) of protons at different times. One can see that the energy densities of accelerated protons are much higher in the CNT. Besides, the accelerated protons propagate faster in the CNT. These results agree well with the above discussion on the sheath field. The energy spectrum of protons at the end of the simulation is exhibited in Figure 4g. The spectra in both cases feature a typical TNSA shape. In the CNT, the cut-off energy of the protons is about 50 MeV, which is much higher than the 22 MeV for the PNT. The proportion of high-energy protons (>5 MeV) is also greatly increased, meaning that more energetic protons are generated in the CNT. To estimate the conversion efficiency from laser to protons, protons with energy greater than 1 MeV were counted, and the laser-to-proton energy conversion efficiency in the CNT was 12.6%, while this value was only 6.7% in the PNT.  
Figure 3: Longitudinal electric field (\(E_{x}\)) at \(t=35T_{0}\) near the target rear surface for the PNT (**a**) and the CNT (**b**). (**c**) Averaged \(E_{x}\) near the region \(y=0\) for both targets. The electric field is normalized by \(m_{e}\omega_{0}c/e\). (**d**) Corresponding electric potential created by the sheath field near the region \(y=0\) for both targets.