Proton radiography is a widely-fielded diagnostic used to measure magnetic structures in plasma. The deflection of protons with multi-MeV kinetic energy by the magnetic fields is used to infer their path-integrated field strength. Here, the use of tomographic methods is proposed for the first time to lift the degeneracy inherent in these path-integrated measurements, allowing full reconstruction of spatially resolved magnetic field structures in three dimensions. Two techniques are proposed which improve the performance of tomographic reconstruction algorithms in cases with severely limited numbers of available probe beams, as is the case in laser-plasma interaction experiments where the probes are created by short, high-power laser pulse irradiation of secondary foil targets. The methods are equally applicable to optical probes such as shadowgraphy and interferometry [M. Kasim et al. Phys. Rev. E 95, 023306 (2017)], thereby providing a disruptive new approach to three dimensional imaging across the physical sciences and engineering disciplines.
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