## 2 Experimental Setup  
An experiment was conducted at the HiLASE center, Czech Republic, overseen by the Institute of Physics [12], to gather a series of laser-interaction datasets. Data were obtained from PERLA 1 kHz laser at intensities of 2.2 \(\times\) 10\({}^{16}\) W/cm\({}^{2}\) with a 2 mm Cu target. In total, 24 laser shots were gathered under fixed conditions. Figure 2 shows a basic schematic of the experimental setup. The laser was fired for a \(\sim\)1.5 ps burst with 1030 nm central wavelength and maximum laser energy up to 20 mJ at a 2 mm Cu target, with a  
Figure 1: Typical energy distribution obtained from residual ion acceleration. Data gathered from an experimental campaign that was performed at HiLASE centre, intensity 2.2 \(\times\) 10\({}^{16}\) W/cm\({}^{2}\) laser shots incident on a 2 mm Cu target. Detection threshold marks (in KeV) the value at which the signal noise becomes equal to the signal received, disallowing accurate detection below it. Faraday cup detector working in the time-of-flight technique positioned normal to the target in a backwards direction at a distance 0.18 m from the target, with a laser intensity with a magnitude of \(10^{16}\) W/cm\({}^{2}\) and varying laser focal radius. This slight changing in laser focus results from the fact that laser-target collision induces a slight 'boring' effect causing the target to bend inwards, indenting and warping the surface. This additionally induces a slight, albeit largely negligible change to the intensity. The experiment was conducted within a vacuum chamber, and the detector was negatively biased (100 V) to repel incident plasma electrons that were travelling with the ion beam. As the incident laser intensity was sufficiently low, the ion acceleration mechanism that occurred in the target backward direction is within the TNSA-like regime. Due to the low laser contrast during the experiment (\(10^{-3}\)), a significant overdense preplasma plume was generated prior to the arrival of the main pulse. Thus, on the contrary to the classical TNSA, the main acceleration direction of the ions was opposite to the gradient of density of the preplasma plume rather than of the initial target, i.e., not necessarily is in the target normal direction as suggested in particle-in-cell simulations that are described in [13]. The primary component of interest in the above setup is that of the Faraday cup ion collector. While a range of ion collectors following similar methods exist, the most readily improved and prominent of these remains to be the Faraday cup due to its reliability and simplicity. A Faraday cup at its most basic level consists of a conductive open-sided cylinder connected as a node in series to a signal amplifier. Charged particles impinging on the metallic surface are absorbed or otherwise transfer that charge to the metal which evenly distributes across the cups outer surface, building a potential difference. The plate is regularly and routinely discharged allowing a current to flow across it, which is amplified by a capacitor/resistor/amplifier parallel setup (for more detail on the Faraday cup mechanics and functionality see [14]). The voltage V\({}_{\text{out}}\) that is measured in a given time is, therefore, proportional to the total number of charged particles striking the target in that time [14; 15]. The collector is additionally often biased at a negative potential to repel electrons within the incident plasma stream to prevent inaccurate reduction of the perceived charge via negative charge transfer [16]. The nature of the signal generation makes accuracy of the incident particles energy more complex, however, as both a high energy and low energy proton will induce the same inherent signal in the detector.