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Proton imaging of stochastic magnetic fields

Published online by Cambridge University Press:  19 December 2017

A. F. A. Bott
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
C. Graziani
FLASH Center for Computational Science, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
P. Tzeferacos
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK FLASH Center for Computational Science, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
T. G. White
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK Department of Physics, University of Nevada, Reno, NV 89557, USA
D. Q. Lamb
FLASH Center for Computational Science, University of Chicago, 5640 S. Ellis Ave, Chicago, IL 60637, USA
G. Gregori
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
A. A. Schekochihin
Merton College, Merton Street, Oxford OX1 4JD, UK Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK


Recent laser-plasma experiments (Fox et al., Phys. Rev. Lett., vol. 111, 2013, 225002; Huntington et al., Nat. Phys., vol. 11(2), 2015, 173–176; Tzeferacos et al., Phys. Plasmas, vol. 24(4), 2017a, 041404; Tzeferacos et al., 2017b, arXiv:1702.03016 [physics.plasm-ph]) 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 (Kugland et al.Rev. Sci. Instrum. vol. 83(10), 2012, 101301) under simplifying assumptions typically valid in actual proton-imaging set-ups. We conclude that, as with 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 $\unicode[STIX]{x1D707}\equiv d_{s}/{\mathcal{M}}l_{B}$ – which quantifies the relative size of the correlation length $l_{B}$ of the stochastic field, proton displacements $d_{s}$ due to magnetic deflections and the image magnification ${\mathcal{M}}$ . For stochastic magnetic fields, we establish the existence of four contrast regimes, under which proton-flux images relate to their parent fields in a qualitatively distinct manner. These are linear, nonlinear injective, caustic and diffusive. The diffusive regime is newly identified and characterised. The nonlinear injective regime is distinguished from the caustic regime in manifesting nonlinear behaviour, but as in the linear regime, the path-integrated magnetic field experienced by the beam can be extracted uniquely. Thus, in the linear and nonlinear injective regimes we show that the magnetic-energy spectrum can be obtained under a further statistical assumption of isotropy. This is not the case in the caustic or diffusive regimes. We 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 proton-flux images.

Research Article
© Cambridge University Press 2017 

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