Proton imaging of stochastic magnetic fields - Bott, A; Graziani, C; Tzeferacos, P; White, P; Lamb, D; Gregori, G; Schekochihin, A (Cambridge University Press, 2017)
Recent laser-plasma experiments [1, 2, 3, 4] report the existence of dynamically significant magnetic fields, whose statistical characterisation is essential for a complete understanding of the physical processes these experiments are attempting to investigate. In this paper, we show how a proton imaging diagnostic can be used to determine a range of relevant magnetic field statistics, including the magnetic-energy spectrum. To achieve this goal, we explore the properties of an analytic relation between a stochastic magnetic field and the image-flux distribution created upon imaging that field. This ‘Kugland image-flux relation’ was previously derived [5] under simplifying assumptions typically valid in actual proton-imaging set-ups. We conclude that, as in the case of regular electromagnetic fields, features of the beam’s final image-flux distribution often display a universal character determined by a single, field-scale dependent parameter – the contrast parameter µ ≡ ds/MlB – which quantifies the relative size of the correlation length lB of the stochastic field, proton displacements ds due to magnetic deflections, and the image magnification M. For stochastic magnetic fields, we establish the existence of four contrast regimes – linear, nonlinear injective, caustic and diffusive – under which proton-flux images relate to their parent fields in a qualitatively distinct manner. As a consequence, it is demonstrated that in the linear or nonlinear injective regimes, the path-integrated magnetic field experienced by the beam can be extracted uniquely, as can the magnetic-energy spectrum under a further statistical assumption of isotropy. This is no longer the case in the caustic or diffusive regimes. We also discuss complications to the contrast-regime characterisation arising for inhomogeneous, multi-scale stochastic fields, which can encompass many contrast regimes, as well as limitations currently placed by experimental capabilities on one’s ability to extract magnetic field statistics. The results presented in this paper are of consequence in providing a comprehensive description of proton images of stochastic magnetic fields, with applications for improved analysis of individual proton-flux images, or for optimising implementation of proton-imaging diagnostics on future laser-plasma experiments.