# Monte Carlo Study of Imaging Plate Response to Laser-Driven Aluminum Ion Beams
## 5 Calculation of PSL/Ion from SRIM Data
For each IP model, the total deposited energy is calculated as follows  
\[E_{dep}=\sum_{i=1}^{l}\big{[}\frac{S(z_{i})+S(z_{i+1})}{2}\big{]}(z_{i+1}-z_{ i}),\quad(\text{Linear model}) \tag{6}\]\[E_{dcp}=\sum_{i=1}^{l}\left(-\frac{dE}{dz}\right)e^{-\frac{1}{2}(\frac{z_{i+1}+z_{ i}}{z})}dz=\sum_{i=1}^{l}-(E_{i+1}-E_{i})e^{-\frac{1}{2}(\frac{z_{i+1}+z_{i}}{z})}, \tag{7}\]  
where \(z_{i}\) is the target depth (the distance that an Al ion passes through the target), \(S(z_{i})\) is the stopping power of target at depth \(z_{i}\), \(E_{i}\) is the kinetic energy of the Al ion at \(z_{i}\), and \(l\) is the total number of steps (\(l=100\) or \(l=700\)). In the linear model, the area under the curve of stopping power as a function of target depth represents the deposited energy. We use the average of two stopping power values in each interval, and estimate the target depth of each thin layer as \(\frac{z_{i+1}+z_{i}}{2}\) in the exponential model. The average stopping powers of phosphor layer for mono-energetic Al ions are shown as functions of the target depth in Figure 3 for 130 MeV Al ions (dashed green line) and for 200 MeV Al ions (dot dashed blue line). The 50 \(\upmu\)m thick phosphor layer of BAS-TR IP is thick enough to stop all Al ions with \(E_{sim}\leq 130\) MeV, so the entire \(E_{ion}\) is absorbed within the phosphor layer. For 200 MeV Al ions, on the other hand, the stopping power gradually increases as the ions lose their kinetic energy, but the ions penetrate through the phosphor layer with significant energy remaining. Figure 4a shows the IP response, \(R(E_{ion})\), as a function of Al ion energy for our setup. Solid black circles indicate the experimental measurements of PSL/ion data. Error bars are estimated to be \(\pm 15\%\), which account for the decay uncertainty from the fading effect during the time gap (5 min), PSL stimulated by diode laser during scanning, inaccurate calibration of the PMT, and IP surface quality (roughness) [6]. The dashed black line represents the IP response calculated using the linear model, and the solid red line shows the IP response calculated using the exponential model with \(L=44\)\(\upmu\)m. The calculated IP responses with \(\pm\) one standard deviation (\(\pm 4\)\(\upmu\)m) of \(L\) are shown when \(L=48\)\(\upmu\)m (dashed red line) and \(L=40\)\(\upmu\)m (dot dashed red line). In both models, the total deposited energy increases as the kinetic energy of Al ion increases from 50 MeV to 140 MeV, but it decreases beyond 140 MeV. The IP response from the exponential model increases more gradually and it fits the experimental data better. As is described in the experimental setup section, there was an 18 \(\upmu\)m thick filter in front of our BAS-TR IP, and only ions with sufficient kinetic energy (larger than 50 MeV) arrived at the IP front surface. Since filter arrangements can differ for each experiment, beyond the computation of our specific filter conditions, we aim to produce a response model applicable to arbitrary filter arrangements. We start with the computed incident ion energy on the front phosphor layer surface obtained by subtracting the energy loss in the 18 \(\upmu\)m thick filter from the incoming \(E_{ion}\) measured with the TPS. We computed  
Figure 3: Stopping power of the 50 \(\upmu\)m thick phosphor layer of a BAS-TR IP for 130 MeV and 200 MeV Al ions as functions of the target depth. The area under the curve represents the deposited energy in the phosphor layer. Al ion energy loss in the filter with SRIM. Figure 4b shows energy loss from an 18 \(\upmu\)m thick Al filter as a function of the incident Al ion energy. The dashed green line indicates the kinetic energy loss of energetic Al ions. The solid blue line shows the incident kinetic energy of Al ions on the phosphor layer after penetrating through the filter as a function of the initial Al ion energy before the filter. Figure 4c shows the IP response as a function of the incident Al ion energy on the phosphor layer.