## 2 Description of the Simulation Models and Experimental Set up  
Description of the Model for Generation of Angular Spectra of Electrons, Protons and Photons; GEANT4 Model for Compute the Dose Map inside the Interaction Chamber  
(a) Description of TNSA Model  
The simulation model is based upon the Geant4 [25] framework. The mathematical models for sources description and data processing pipelines have been implemented in the Python language [26]. The integration between the sources description and the custom GEANT4 based application developed in C++, was performed using the Geant4 General Particle source. We used the command line scoring system for the region of interest and the output data were analyzed using another Python pipeline. The particle sources were implemented considering the target normal sheath acceleration (TNSA) mechanism and the plasma expansion model [1]. In TNSA regime, the interaction of a high power laser beam with a thin target generates on its front surface energetic electrons, which are accelerated in forward direction through the target. During their interaction with the target material, these hot electrons can generate bremsstrahlung radiation [7; 8]. The electrons, which attain the back of the target and enter in vacuum, create an electrostatic space charge sheath and thus an electric field with an intensity of about \(10^{12}\) V/m. Due to this field, the protons from the rear target surface can be accelerated up to MeV energies [1; 2; 3; 4; 5; 6]. The temperature of hot electrons, is related to the laser intensity, [1]:  
\[T_{hot}=[(1+I\ [\mathrm{W/cm^{2}}]\lambda^{2}\ [\mathrm{\mu m}]/(1.37\times 1 0^{18})]^{1/2}-1)m_{0}c^{2}, \tag{1}\]  
where \(I\) represent the laser intensity in beam focus, \(\lambda\) is laser wavelength, \(m_{0}\) electron mass, and c light velocity [1]. The number of electrons accelerated into the target, \(N_{e}\), depends on the energy fraction absorbed by hot electrons \(f=1.2\times 10^{-15}\)\(I^{0.74}\) (\(\mathrm{W/cm^{2}}\)), through the formula [1; 27]:  
\[N_{e}=f\,E_{L}/T_{hot} \tag{2}\]  
where \(E_{L}\) is the laser energy. The electron density at the rear side of the target is \(n_{e,0}=N_{d}/(c\ \tau_{L}S_{shath})\), where \(S_{shath}=\pi(r_{0}+d\times\tan\theta)^{2}\)[1], \(\tau_{L}\) is laser pulse duration, \(d\) is the target thickness, \(r_{0}\) is the radius of laser beam in focus and \(\theta\) is the half angular broadening of the hot electrons inside the target. [PERSON] calculated this formula as a function of target thickness and laser intensity [27]:  
\[n_{e,0}=[\eta E_{L}]/[c\tau_{L}\pi(r_{0}+(d\ \tan\theta/2))^{2}k_{B}T_{hot} \sim 1.5\times 10^{19}\ [r_{0}{}^{2}/(r_{0}+(d\tan\theta/2)^{2}][I_{18}{}^{7/4}]/[(1+\] \[0.73I_{18}\lambda)^{1/2}-1]\ [\mathrm{cm^{-3}}] \tag{3}\]  
where \(I_{18}\) is the laser intensity in terms of \(10^{18}\) W/cm\({}^{2}\)[27]. The energy spectrum of electrons leaving the rear side of the target is given by different formulas, depending on how large the laser intensity is:  
\[dN/dE\sim E^{2}e^{-E/T}{}_{hot} \tag{4}\]  
(relativistic Maxwellian distribution) for \(I>10^{19}\) W/cm\({}^{2}\)[7; 12] and  
\[dN/dE\sim E^{1/2}e^{-E/T}{}_{hot} \tag{5}\]  
(Maxwellian distribution) for \(I<10^{19}\) W/cm\({}^{2}\)[7; 12]. The energy spectrum of accelerated protons within the plasma expansion model is [1]:  
\[dN/dE=[n_{e,0}c_{s}t_{acc}S_{shsat}/(2 ET_{hot})^{1/2}]\ exp(-(2E/T_{hot})^{1/2}) \tag{6}\]  
where \(t_{acc}=1.3\times\tau_{L}\) while \(m_{i}\) and \(Z_{i}\) (\(Z_{i}=1\)) are referred to as protons. The energy spectrum of bremsstrahlung photons is derived from the energy spectrum of the electrons which interact with the Ag target (\(Z=49\)) and was calculated considering the temperature of hot electrons. The bremsstrahlung radiation is due to energy deposited by the hot electrons while passing through the target.