## 3 Energy and Energy Spread Characterization
The measured integral depth doses correspond to beam energies of 70 MeV, 100 MeV, 150 MeV, and 220 MeV. Their normalized dose profiles are shown in Figure 6. The Bragg peak positions, \(\mathrm{R_{80}}\), and distal fall-offs were analyzed and are listed in Table 3. Due to the lesser influence of energy spread on \(\mathrm{R_{80}}\), the Bragg peak and distal fall-off are utilized to match the depth-dose distribution and determine the energy spread of the proton beam. The simulated integral depth doses for each energy spread are normalized and plotted against the measured depth-dose profile for the proton energy of 220 MeV. The calculated  
Figure 5: Peak \(\mathrm{LET_{d}}\) per proton particle with varing energy spreads, ranging from 0% to 5%, for proton energies of 70 MeV, 100 MeV, 150 MeV, and 220 MeV. Figure 6: Measured integral depth-dose curves normalized to relative values for proton energies of 70 MeV, 100 MeV, 150 MeV, and 220 MeV. distal fall-off for each energy was used to match the depth-dose distribution. For example, the measured depth dose of the 220 MeV proton beam closely matches the shape of the simulated energy spread of 0.5%, as indicated in Figure 7.  
The Gamma Index (\(\gamma\)) is a widely used quantitative metric in radiotherapy for evaluating the agreement between a calculated dose distribution--typically generated by a treatment planning system (TPS)--and a measured or reference dose distribution [22]. This metric simultaneously considers both the dose difference (\(\delta D\)) and the spatial distance (\(\delta r\)) between corresponding points in the two distributions, providing a comprehensive assessment of dosimetric accuracy. In this study, the reference distribution corresponds to the measured dose, while the evaluated distribution is derived from simulation results. The \(\gamma\)-index is computed for each point in the reference distribution by identifying the minimum Euclidean distance in the evaluated distribution that satisfies both the dose and distance criteria. Mathematically, the \(\gamma\)-index at a reference point is defined as  
\[\gamma(r_{E})=\min\{\Gamma(r_{E},r_{S})\}\quad\forall\{r_{S}\} \tag{1}\]  
where  
\[\Gamma(r_{E},r_{S})=\sqrt{\frac{\Delta r^{2}(r_{E},r_{S})}{\delta r^{2}}+ \frac{\Delta D^{2}(r_{E},r_{S})}{\delta D^{2}}} \tag{2}\]  
Here, \(r_{E}\) denotes the spatial location in the experimental (measured) dose distribution, while \(r_{S}\) represents the corresponding location in the simulated distribution. The term \(\Delta r\) is the spatial distance between the evaluated point \(r_{E}\) and the reference point \(r_{S}\), and \(\Delta D\) is the dose difference between the experimental and simulated values at those points. A reference point is considered to pass the gamma test if \(\gamma<1\), indicating that at least one evaluated point lies within the acceptance ellipsoid defined by the criteria. The most commonly adopted acceptance criteria are a 3% dose difference and a 3 mm distance-to-agreement (3%/3 mm), as originally proposed by [PERSON] et al. [23]. Figure 7: The depth-dose distribution per proton particle with varying energy spread ranging from 0% to 5% for 220 MeV. \begin{table}
\begin{tabular}{l c c c} \hline
**Energy (MeV)** & **R\({}_{\text{BP}}\) (mm)** & **R\({}_{\text{80}}\) (mm)** & **W\({}_{\text{80-20}}\) (mm)** \\ \hline
70 & 39.6 & 40.5 & 1.8 \\
100 & 75.3 & 77.0 & 2.7 \\
150 & 154.3 & 157.7 & 4.3 \\
220 & 301.0 & 305.4 & 5.2 \\ \hline \end{tabular}
\end{table} 
Table 3: The Bragg peak position (R\({}_{\text{BP}}\)), intensity, and distal fall-off width (W\({}_{\text{80-20}}\)) of the proton beam were analyzed from the measured integral depth doses at energies of 70 MeV, 100 MeV, 150 MeV, and 220 MeV. By applying the \(\gamma\)-index analysis using Equations (1) and (2), with the standard criteria of \(\delta r=3\) mm and \(\delta D=3\%\), the results--illustrated in the right plot of Figure 8--demonstrate that the \(\gamma\)-index remains below 1 for all comparison points.