# Investigation of Pre-Pulse Effects on Ultrashort-Pulse Laser Interaction with Structured Targets
## 5 Analysis and Discussion  
The proton energy plot in Figure 3c shows the optimal fluence, which is strong enough to produce the pre-plasma with a density that will affect the main beam, without destroying the fine structure of the target. For all APP delay times, the optimum fluence is close to the optical damage threshold 0.4-0.6 J/cm\({}^{2}\). By selecting the best time delay \(\Delta t=-\)5 ns (Figure 3d), which results in the highest proton energy cutoff, we can compare experimental data with reconstructed numerical energy spectrum Figure 4d. There is a good qualitative agreement between the experimental results and numerical simulations. Both exhibit similar behavior, with a sharp rise (likely caused by the rapid expansion of the pre-plasma in the early delay times) followed by a peak that is gradually decaying. Accurately capturing the fast dynamics of the early stages of interaction between the structured target and pre-plasma in simulations is challenging due to the complex expansion dynamics and sharp density gradients. However, the results clearly show lower proton energy cutoffs for the structured target without pre-plasma, as well as lower average accelerating field, as shown in Figure 4c. In experiments, the proton energy was gradually decreasing with longer pre-pulse delay times (\(\approx\)6-12 ns); this trend was not fully reproduced by the numerical simulations. This decrease is likely due to the recombination and cooling of the pre-plasma over longer timescales, this was not included in the PIC pre-plasma model. The observed optical spectrum provides insights into plasma dynamics with low and high APP fluences and serves as a possible indicator of nonlinear effects associated with different heating mechanisms, such as J \(\times\) B heating [47] and resonance absorption [48, 49]. Optimal heating of the pre-plasma electrons should be accompanied by strong nonlinear signals, which we anticipate observing in the spectrum. The peak observed at 402 nm can be associated with the second harmonic signal, which may suggest the presence of a short sub-micron pre-plasma scale length, with a sharp vacuum-solid boundary. In such conditions, resonance absorption is the dominant laser-plasma interaction mechanism [49, 50]. The multiple peaks at wavelengths greater than 650 nm can be primarily attributed to Stimulated Raman Scattering (SRS). It is interesting to note that the TPD signals (if any) are rather weak compared with SRS-related peaks, even though TPD has a lower intensity threshold [51, 52]. This may be attributed to the fact that the angle corrected critical plasma density value \(\left(n_{c}cos^{2}(\theta)\approx 2.6\times 10^{20}\) cm\({}^{-3}\right)\) is lower than the quarter-critical density value \(\left(\frac{n_{c}}{4}\approx 4.4\times 10^{20}\) cm\({}^{-3}\right)\), where the critical plasma density \(n_{c}\approx 1.7\times 10^{21}\) cm\({}^{-3}\) for \(\lambda_{0}\approx\) 805 nm. This implies that the obliquely incident MP would propagate through the low-density pre-plasma prior to interacting with an overdense structured target. The optical spectrum data indicates the existence of an optimum pre-plasma scale length conducive to efficient laser-plasma interaction. Under optimal APP fluence conditions (case III in Figure 2b), the signature from SRS is visible, especially at the \(\Delta t\) of \(-5\) ns, which requires the plasma density lower than the quarter critical density [49]. The spectral profile under clear SRS conditions of \(\Delta t\) and APP fluence is shown in detail in Figure 2d. Apart from the broadened second harmonic signal at 402 nm and part of the SRS signal close to 719 nm (wavelengths between 725 and 925 nm are filtered out), a distinct signal is observed at approximately 493 nm. To achieve optimal conditions, it is crucial to find a balance between the effects of the pre-plasma and the structured target. If the pre-pulse fluence is too low, the pre-plasma impact will be minimal; conversely, excessive pre-pulse fluence will destroy the microstructure of the target.