# Investigation of Pre-Pulse Effects on Ultrashort-Pulse Laser Interaction with Structured Targets
## 4 Simulations  
The experiment results indicate that plasma emission spectra and proton acceleration are sensitive to the initial conditions and temporal evolution of the pre-plasma. To validate the existence of optimal pre-plasma conditions and gain insight into the process, EPOCH-PIC [44] simulations are employed. A detailed numerical simulation of the pre-plasma generation by the pre-pulse is beyond the scope of this work. Hence, we focus on the verification of the proton acceleration with different pre-plasma conditions. In order to simulate the plasma expansion with various delay times, we assumed a constant pre-plasma temperature T\({}_{\rm e}\) = 0.28 eV during the expansion period, which leads to the 2.5 \(\upmu\)m/ns expansion speed for plastic targets. Thus, in a simplified hemispherical pre-plasma model, for a fixed APP fluence, the various time delays between APP and MB will change the size of the pre-plasma cloud and the scale length of pre-plasma, while maintaining the constant number of particles in the pre-plasma with fixed expansion velocity. We consider only the optimal fluence of 0.4 J/cm\({}^{2}\) for the total number of particles that can be estimated from the femtosecond laser ablation model described by [PERSON] et al. [45]. The densities and plasma expansion are in good agreement with the experimental measurements of pre-plasma density with similar pre-pulse conditions (\(-4.7\) ns, \(10^{13}\) W/cm\({}^{2}\)[46]). Figure 3: (**a**) A typical aluminum net filter on top of the CR39 plate used in the experiments: (**l**) is the base filter (thickness, 28 \(\upmu\)m), (**l**) and (**l**) are the vertical and horizontal strips of 14 \(\upmu\)m and 22 \(\upmu\)m thickness, respectively. (**b**) CR39 with signals from \(2.5\leq\) PE \(<2.8\) MeV protons. The adjacent image is obtained by scanning the area of interest in a SEM. (**c**) Variation in proton energy with the fluence of the artificial pre-pulse (APP) at selected time delays (\(\Delta t\)) of: (i) \(-2\) ns (red square with dashed line), (ii) \(-5\) ns (blue circle with solid line), (iii) \(-8\) ns (green diamond with dash-dot line). (**d**) Variation in average and maximum proton energy with the delay of the APP in the fluence range of 0.4–0.6 J/cm\({}^{2}\). The data points are connected by Bezier curves to help discern the trend. Figure 4a shows the 2D geometry setup in EPOCH to simulate the MP laser interaction with the structured target with a localized pre-plasma. The red arrow indicates the direction of the laser. The target structure is tilted so that the MP subtends an angle of \(60^{\circ}\) with respect to the target normal. The structured target is modeled by CH\({}_{2}\) rectangular spikes identical to the structure shown in Figure 1d. The target is assumed to be pre-ionized to \(H^{+}\) and \(C^{2+}\), with a solid electron density of \(n_{e}=80\,n_{c}\), where \(n_{c}\) is the critical plasma density. We compared these results with simulations using the same target without pre-plasma, as well as with a flat, thick-foil target. The differences in the electron heating processes in the simulations from the flat target, structured target bulk, and pre-plasma are shown in Figure 4b. The electron temperature in the flat target is lower compared with that in the structured target. The simulations of the structured target, both with and without pre-plasma, showed similar bulk electron temperatures. However, the temperature of pre-plasma electrons in structured targets is much higher compared with the bulk of structured targets. This leads to the acceleration of electrons from the pre-plasma into the tips of the structured targets and stronger charge accumulation on the tips, which leads to the local field enhancement and generation of a stronger average accelerating field that is shown in Figure 4c. The figure shows that the accelerating field in the figure with a structured target is stronger and remains longer compared with that in the case without pre-plasma.