# A New Line for Laser-Driven Light Ions Acceleration and Related TNSA Studies
## 4 Results: Numerical Investigation of TNSA Regime  
Numerical simulations were carried out using the ALaDyn code [18] following the prospects outlined in the table of Table 1. We considered a laser pulse with a duration of \(\tau\) = 25-40 fs full width half maximum (FWHM) and a Gaussian focal spot \(D_{f}\) = 7.5 \(\upmu\)m, with an energy per pulse in the range of \(E_{L}\) = 0.4-6 J. The corresponding peak intensity ranges from 2 \(\times\) 10\({}^{19}\) to 5 \(\times\) 10\({}^{20}\) and the normalized intensity a\({}_{0}\) = 0.85(\(l^{2}/10^{18}\) W/cm\({}^{2}\))\({}^{1/2}\) ranges from 3 to 15. Some simulations with higher intensity were also considered. The lower intensity value corresponds to the front-end laser operation, which has been extensively explored [15; 19; 20] and was found to be fully consistent with the above scaling. In the simulations, the laser impinges with different angles \(\theta\) = 0\({}^{\circ}\), 10\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\) on a fully ionized, CH\({}_{2}\) slab with a thickness \(l_{\rm th}\) = 2-2.5 \(\upmu\)m and electronic density \(n_{e}\) = 100\(n_{c}\) = 1.74 \(\times\) 10\({}^{23}\) cm\({}^{-3}\). No effects originating from pre-plasma due to finite pre-pulse are taken into account. The first outcome of Particle In Cell (PIC) investigation summarized in Figure 3 shows a significant dependence of proton acceleration on the laser incidence angle, but only at lower intensities a\({}_{0}\) < 5. In this range, going from incidence \(\theta\) = 0\({}^{\circ}\) to \(\theta\) = 30\({}^{\circ}\), the proton cut-off energy increases by 40%. At higher intensities, a\({}_{0}\) > 10, this factor drops to only a few percentage points.  
In PIC simulations, systematic parametric scans are necessarily restricted to 2D geometry. The main difficulty to extract useful quantitative information on proton acceleration is that 2D TNSA models give a logarithmic increase with time of the \(E_{m}(t)\) energy. The choice of a definite time at which to stop the simulation to evaluate the final proton energy is therefore intrinsically arbitrary. To overcome this limitation, a method has been introduced [21] that allows a unique proton energy, the so-called asymptotic \(E\) value, to be extracted using only the information on the growth rate of energy in a finite time range. The results reported in Figure 3 were consistent with this approach, which enabled a more detailed description of the intensity and angular dependence of TNSA cut-off energy.  
Figure 3: Calculated protons cut-off energy (MeV) at different laser intensities with strength parameter a\({}_{0}\) = 3–20. Data on the red line all refer to laser pulse duration \(\tau\) = 30 fs, incidence angle \(\theta\) = 10\({}^{\circ}\), and target thickness \(l_{\rm th}\) = 2 \(\upmu\)m. Different data at a\({}_{0}\) = 3, \(Emax\) = [1.2, 1.4, 1.82] MeV refer to different incidence angles (\(\theta\) = 0–30\({}^{\circ}\)). The lower value at a\({}_{0}\) = 10 refers to a larger \(l_{\rm th}\) = 2.5 target thickness, whereas the lower value at a\({}_{0}\) = 15 refers to a lower \(\tau\) = 25 fs pulse duration. TNSA: Target Normal Sheath Acceleration.