## 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:

- Univ. of Oxford, Oxford (United Kingdom)
- Univ. of Chicago, Chicago, IL (United States)
- Univ. of Oxford, Oxford (United Kingdom); Univ. of Chicago, Chicago, IL (United States)
- Univ. of Oxford, Oxford (United Kingdom); Univ. of Nevada, Reno, NV (United States)
- 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}

}

Works referenced in this record:

##
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