skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Proton imaging of stochastic magnetic fields

Abstract

Recent laser-plasma experiments report the existence of dynamically significant magnetic fields, whose statistical characterization 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 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 characterized. 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 characterization 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. Furthermore, 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.

Authors:
ORCiD logo [1];  [2];  [3];  [4];  [2];  [1];  [5]
  1. Univ. of Oxford, Oxford (United Kingdom)
  2. Univ. of Chicago, Chicago, IL (United States)
  3. Univ. of Oxford, Oxford (United Kingdom); Univ. of Chicago, Chicago, IL (United States)
  4. Univ. of Oxford, Oxford (United Kingdom); Univ. of Nevada, Reno, NV (United States)
  5. Merton College, Oxford (United Kingdom); Univ. of Oxford, Oxford (United Kingdom)
Publication Date:
Research Org.:
Univ. of Chicago, IL (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC)
OSTI Identifier:
1495713
Grant/Contract Number:  
NA0002724
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Plasma Physics
Additional Journal Information:
Journal Volume: 83; Journal Issue: 06; Journal ID: ISSN 0022-3778
Publisher:
Cambridge University Press
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; plasma diagnostics

Citation Formats

Bott, A. F. A., Graziani, C., Tzeferacos, P., White, T. G., Lamb, D. Q., Gregori, G., and Schekochihin, A. A. Proton imaging of stochastic magnetic fields. United States: N. p., 2017. Web. doi:10.1017/S0022377817000939.
Bott, A. F. A., Graziani, C., Tzeferacos, P., White, T. G., Lamb, D. Q., Gregori, G., & Schekochihin, A. A. Proton imaging of stochastic magnetic fields. United States. doi:10.1017/S0022377817000939.
Bott, A. F. A., Graziani, C., Tzeferacos, P., White, T. G., Lamb, D. Q., Gregori, G., and Schekochihin, A. A. Tue . "Proton imaging of stochastic magnetic fields". United States. doi:10.1017/S0022377817000939. https://www.osti.gov/servlets/purl/1495713.
@article{osti_1495713,
title = {Proton imaging of stochastic magnetic fields},
author = {Bott, A. F. A. and Graziani, C. and Tzeferacos, P. and White, T. G. and Lamb, D. Q. and Gregori, G. and Schekochihin, A. A.},
abstractNote = {Recent laser-plasma experiments report the existence of dynamically significant magnetic fields, whose statistical characterization 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 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 characterized. 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 characterization 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. Furthermore, 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.},
doi = {10.1017/S0022377817000939},
journal = {Journal of Plasma Physics},
number = 06,
volume = 83,
place = {United States},
year = {2017},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 12 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Principles of Plasma Physics
journal, December 1973

  • Krall, Nicholas A.; Trivelpiece, Alvin W.; Gross, Robert A.
  • American Journal of Physics, Vol. 41, Issue 12
  • DOI: 10.1119/1.1987587

Direct observation of turbulent magnetic fields in hot, dense laser produced plasmas
journal, May 2012

  • Mondal, S.; Narayanan, V.; Ding, W. J.
  • Proceedings of the National Academy of Sciences, Vol. 109, Issue 21
  • DOI: 10.1073/pnas.1200753109

Laboratory evidence of dynamo amplification of magnetic fields in a turbulent plasma
journal, February 2018


Dynamics of Self-Generated, Large Amplitude Magnetic Fields Following High-Intensity Laser Matter Interaction
journal, November 2012


Dynamics of Electric Fields Driving the Laser Acceleration of Multi-MeV Protons
journal, October 2005


A Mexican hat with holes: calculating low-resolution power spectra from data with gaps: Power spectrum of data with gaps
journal, October 2012


Source characterization and modeling development for monoenergetic-proton radiography experiments on OMEGA
journal, June 2012

  • Manuel, M. J. -E.; Zylstra, A. B.; Rinderknecht, H. G.
  • Review of Scientific Instruments, Vol. 83, Issue 6
  • DOI: 10.1063/1.4730336

The passage of energetic charged particles through interplanetary space
journal, January 1965


Electric field detection in laser-plasma interaction experiments via the proton imaging technique
journal, May 2002

  • Borghesi, M.; Campbell, D. H.; Schiavi, A.
  • Physics of Plasmas, Vol. 9, Issue 5
  • DOI: 10.1063/1.1459457

Energetic proton generation in ultra-intense laser–solid interactions
journal, February 2001

  • Wilks, S. C.; Langdon, A. B.; Cowan, T. E.
  • Physics of Plasmas, Vol. 8, Issue 2, p. 542-549
  • DOI: 10.1063/1.1333697

The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity
journal, December 1961


Digital Generation of Non‐Gaussian Stochastic Fields
journal, July 1988


Diffusion, Scattering, and Acceleration of Particles by Stochastic Electromagnetic Fields
journal, January 1967


Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers
journal, August 2007


Measuring E and B Fields in Laser-Produced Plasmas with Monoenergetic Proton Radiography
journal, September 2006


Proton radiography as an electromagnetic field and density perturbation diagnostic (invited)
journal, October 2004

  • Mackinnon, A. J.; Patel, P. K.; Town, R. P.
  • Review of Scientific Instruments, Vol. 75, Issue 10
  • DOI: 10.1063/1.1788893

Proton Radiography of a Laser-Driven Implosion
journal, July 2006


D3He-proton emission imaging for inertial-confinement-fusion experiments (invited)
journal, October 2004

  • Séguin, F. H.; DeCiantis, J. L.; Frenje, J. A.
  • Review of Scientific Instruments, Vol. 75, Issue 10
  • DOI: 10.1063/1.1788892

X-ray surface brightness and gas density fluctuations in the Coma cluster: X-ray surface brightness fluctuations in the Coma cluster
journal, January 2012


Implementation of an non-iterative implicit electromagnetic field solver for dense plasma simulation
journal, December 2004

  • Welch, D. R.; Rose, D. V.; Clark, R. E.
  • Computer Physics Communications, Vol. 164, Issue 1-3
  • DOI: 10.1016/j.cpc.2004.06.028

Bayesian-Based Iterative Method of Image Restoration*
journal, January 1972

  • Richardson, William Hadley
  • Journal of the Optical Society of America, Vol. 62, Issue 1
  • DOI: 10.1364/JOSA.62.000055

Plasma Physics via Computer Simulation
book, January 1991


Fast Ion Generation by High-Intensity Laser Irradiation of Solid Targets and Applications
journal, April 2006

  • Borghesi, M.; Fuchs, J.; Bulanov, S. V.
  • Fusion Science and Technology, Vol. 49, Issue 3
  • DOI: 10.13182/FST06-A1159

Radiochromic film imaging spectroscopy of laser-accelerated proton beams
journal, March 2009

  • Nürnberg, F.; Schollmeier, M.; Brambrink, E.
  • Review of Scientific Instruments, Vol. 80, Issue 3
  • DOI: 10.1063/1.3086424

The magnetic power spectrum in Faraday rotation screens
journal, April 2003


Numerical modeling of laser-driven experiments aiming to demonstrate magnetic field amplification via turbulent dynamo
journal, April 2017

  • Tzeferacos, P.; Rigby, A.; Bott, A.
  • Physics of Plasmas, Vol. 24, Issue 4
  • DOI: 10.1063/1.4978628

Proton probing measurement of electric and magnetic fields generated by ns and ps laser-matter interactions
journal, May 2008


The geometry of optimal transportation
journal, January 1996

  • Gangbo, Wilfrid; McCann, Robert J.
  • Acta Mathematica, Vol. 177, Issue 2
  • DOI: 10.1007/BF02392620

Polar factorization and monotone rearrangement of vector-valued functions
journal, June 1991


Impulsive electric fields driven by high-intensity laser matter interactions
journal, February 2007


Simulations of the Small‐Scale Turbulent Dynamo
journal, September 2004

  • Schekochihin, Alexander A.; Cowley, Steven C.; Taylor, Samuel F.
  • The Astrophysical Journal, Vol. 612, Issue 1
  • DOI: 10.1086/422547

Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type
journal, February 2006

  • Dean, E. J.; Glowinski, R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 13-16
  • DOI: 10.1016/j.cma.2005.05.023

Development of an interpretive simulation tool for the proton radiography technique
journal, March 2015

  • Levy, M. C.; Ryutov, D. D.; Wilks, S. C.
  • Review of Scientific Instruments, Vol. 86, Issue 3
  • DOI: 10.1063/1.4909536

Observation of magnetic field generation via the Weibel instability in interpenetrating plasma flows
journal, January 2015

  • Huntington, C. M.; Fiuza, F.; Ross, J. S.
  • Nature Physics, Vol. 11, Issue 2
  • DOI: 10.1038/nphys3178

Stopping of energetic light ions in elemental matter
journal, February 1999

  • Ziegler, J. F.
  • Journal of Applied Physics, Vol. 85, Issue 3
  • DOI: 10.1063/1.369844

Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation
journal, January 1996

  • Shinozuka, Masanobu; Deodatis, George
  • Applied Mechanics Reviews, Vol. 49, Issue 1
  • DOI: 10.1115/1.3101883

Invited Article: Relation between electric and magnetic field structures and their proton-beam images
journal, October 2012

  • Kugland, N. L.; Ryutov, D. D.; Plechaty, C.
  • Review of Scientific Instruments, Vol. 83, Issue 10
  • DOI: 10.1063/1.4750234

An efficient approach for the numerical solution of the Monge–Ampère equation
journal, March 2011

  • Sulman, Mohamed M.; Williams, J. F.; Russell, Robert D.
  • Applied Numerical Mathematics, Vol. 61, Issue 3
  • DOI: 10.1016/j.apnum.2010.10.006

Filamentation Instability of Counterstreaming Laser-Driven Plasmas
journal, November 2013


Particle acceleration at astrophysical shocks: A theory of cosmic ray origin
journal, October 1987


Inferring morphology and strength of magnetic fields from proton radiographs
journal, December 2017

  • Graziani, Carlo; Tzeferacos, Petros; Lamb, Donald Q.
  • Review of Scientific Instruments, Vol. 88, Issue 12
  • DOI: 10.1063/1.5013029

Fokker-Planck Equations for Charged-Particle Transport in Random Fields.
journal, February 1972

  • Jokipii, J. R.
  • The Astrophysical Journal, Vol. 172
  • DOI: 10.1086/151349

An iterative technique for the rectification of observed distributions
journal, June 1974


Measuring E and B Fields in Laser-Produced Plasmas with Monoenergetic Proton Radiography
journal, September 2006


    Works referencing / citing this record:

    Retrieving fields from proton radiography without source profiles
    journal, September 2019


    Laboratory evidence of dynamo amplification of magnetic fields in a turbulent plasma
    journal, February 2018


    Field reconstruction from proton radiography of intense laser driven magnetic reconnection
    journal, August 2019

    • Palmer, C. A. J.; Campbell, P. T.; Ma, Y.
    • Physics of Plasmas, Vol. 26, Issue 8
    • DOI: 10.1063/1.5092733

    Characterizing filamentary magnetic structures in counter-streaming plasmas by Fourier analysis of proton images
    journal, October 2019

    • Levesque, Joseph; Kuranz, Carolyn; Handy, Timothy
    • Physics of Plasmas, Vol. 26, Issue 10
    • DOI: 10.1063/1.5100728

    Distinguishing and diagnosing the spontaneous electric and magnetic fields of Weibel instability through proton radiography
    journal, December 2019

    • Du, Bao; Cai, Hong-Bo; Zhang, Wen-Shuai
    • Plasma Physics and Controlled Fusion, Vol. 62, Issue 2
    • DOI: 10.1088/1361-6587/ab57ed

    MPRAD: A Monte Carlo and ray-tracing code for the proton radiography in high-energy-density plasma experiments
    journal, December 2019

    • Lu, Yingchao; Li, Hui; Flippo, Kirk A.
    • Review of Scientific Instruments, Vol. 90, Issue 12
    • DOI: 10.1063/1.5123392

    Design of a new turbulent dynamo experiment on the OMEGA-EP
    journal, March 2019

    • Liao, Andy Sha; Li, Shengtai; Li, Hui
    • Physics of Plasmas, Vol. 26, Issue 3
    • DOI: 10.1063/1.5081062