## 2 Results and Discussion  
### Magnetic Beamline Design  
The MBL is specifically designed to achieve a few MeV proton beam of about a 10-mm diameter collimated over a few centimeters in length when propagating in air. The MBL consists of magnetic quadrupoles comprising standard neodymium-based commercial permanent magnets of 25 \(\times\) 12 \(\times\) 4 mm\({}^{3}\) dimensions that are embedded in a soft iron supporting cage, as shown schematically in Figure 1a. The simple design and components of the developed permanent quadrupoles results in a compact and cost-effective beamline. Figure 1: Design of the magnetic beamline: (**a**) Schematic of the quadrupoles, the green shows the four permanent magnets, the brown shows the supporting soft-iron structure, dimensions in mm; (**b**) magnetic field in the quadrupole measured at position x = 0 close to the permanent magnet surface: Measured values (black points), numerically reconstructed field (blue line), supergaussian fit (blue dashed line), truncated ideal quadrupole field (red line); (**c**) vertical cross section of the MBL simulation (Q\({}_{i}\) are the quadrupoles and D\({}_{i}\) the virtual detectors used in the simulation); and (**d**) horizontal cross section of the MBL simulation. The transport of the laser-accelerated protons through the MBL is studied using the Monte Carlo GEANT4 toolkit [37] (see Methods for details). An analytical function for the magnetic field of the quadrupoles is used in the GEANT4 simulations. Such a function is obtained by fitting the magnetic field numerically, reconstructed from actual measurements, of which an example is reported in Figure 1b (see Materials and Methods for details). In Figure 1b, the truncated ideal quadrupole magnetic field is also reported as a reference for comparison. The aim of the simulation is to find a configuration of the MBL that efficiently transfers the few MeV divergent proton beam from the TNSA source to a collimated beam in air. The proton beam exits the vacuum chamber through a Kapton window of a 10-mm diameter and 13-\(\upmu\)m thickness, which sustains a vacuum to the 10\({}^{-4}\) mbar level. The compact MBL set-up comprises six quadrupole magnets placed with alternating field orientation and gaps of 5 mm, 15 mm, 5 mm, 45 mm, and 35 mm between each other starting from the TNSA source side. The total length of the MBL is 255 mm. The first quadrupole is placed at 12.5 mm from the TNSA source and the overall distance between the Kapton window and the TNSA source is measured to be 290(5) mm. In Figure 1c,d the schematics of the two orthogonal transverse cross sections of the MBL are shown along with the simulated proton beam trajectories (see Materials and Methods). The characteristics of the proton beam transported in air is evaluated in details by performing simulations at an initial proton energy in narrow ranges. In Figure 2a, the particle distribution and the final energy obtained in air at 1 cm after the Kapton window are reported for initial energy in the ranges 2.4 to 2.6 MeV, 2.9 to 3.1 MeV, and 3.4 to 3.6 MeV. To compare with the experimental measurements, the time-of-flight (ToF) of the protons as a function of the initial energy is also evaluated. The graph reported in Figure 2b shows the ToF as well as the final energy considering the transport through the MBL, the Kapton window, and 1 cm of air at ambient conditions. Figure 2: Proton beam characteristics at 1 cm after Kapton windows. (**a**) Proton particles distribution and final energy at various initial energy ranges calculated over an area of 40 mm \(\times\) 40 mm centred on the MBL axis at position (20 mm, 20 mm); the highlighted numbers represents the values on the MBL axis; (**b**) graph of the ToF as function of the initial proton energy, with final energy also indicated. The properties of the proton beam when propagating in air are analyzed using the software SRIM [38].