2.1 Statistical characterisation of stochastic magnetic fields

This paper is concerned with recovering statistical properties of stochastic magnetic fields using a proton-imaging diagnostic – so we begin with a brief outline of how stochastic magnetic fields are described statistically. We consider static magnetic fields of the form

(2.1) $$\begin{eqnarray}\boldsymbol{B}(\boldsymbol{x})=\bar{\boldsymbol{B}}(\boldsymbol{x})+\unicode[STIX]{x1D6FF}\boldsymbol{B}(\boldsymbol{x}),\end{eqnarray}$$

where $\bar{\boldsymbol{B}}$ is some non-stochastic mean field varying on the global scale of the plasma, constrained to be inside a cuboid region, dimensions $l_{\bot }\times l_{\bot }\times l_{z}$ (see figure 1) and $\unicode[STIX]{x1D6FF}\boldsymbol{B}$ is a fluctuating field with correlation length $l_{B}\ll l_{z},l_{\bot }$ ( $l_{B}$ is defined precisely in § C.2). The mean and fluctuating fields are distinguished by introducing a spatial averaging operator $\langle \cdot \rangle$ , such that $\langle \bar{\boldsymbol{B}}\rangle =\bar{\boldsymbol{B}}$ , and $\langle \unicode[STIX]{x1D6FF}\boldsymbol{B}\rangle =0$ . This averaging operator can be interpreted either as an average over intermediate scales $l$ such that $l_{B}\ll l\ll l_{z},l_{\bot }$ , or some type of ensemble average. We focus on static fields only because with respect to the fast motion of the protons the evolution of magnetic fields is assumed slow (see § 4.1 for a discussion of the validity of this assumption).

We will concentrate on one particular magnetic-field statistic in this paper: the magnetic-energy spectrum. This is defined by

(2.2) $$\begin{eqnarray}E_{B}(k)\equiv \int \text{d}\unicode[STIX]{x1D6FA}\,k^{2}\langle |\unicode[STIX]{x1D6FF}\hat{\boldsymbol{B}}(\boldsymbol{k})|^{2}\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}\hat{\boldsymbol{B}}(\boldsymbol{k})$ is the three-dimensional Fourier transform of the fluctuating magnetic field, $\boldsymbol{k}$ is the wavevector, $k=|\boldsymbol{k}|$ , and the integral is over solid angles $\unicode[STIX]{x1D6FA}$ in $\boldsymbol{k}$ space. The magnetic-energy spectrum does not in general provide a complete description of a stochastic magnetic field (see Adler Reference Adler1981). However, the magnetic-energy spectrum is a suitable focus for three reasons. Firstly, it can be used to deduce both the fluctuating RMS field strength $B_{\text{rms}}\equiv \langle \unicode[STIX]{x1D6FF}\boldsymbol{B}^{2}\rangle ^{1/2}$ and magnetic-field correlation length $l_{B}$ via relations

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle B_{\text{rms}}^{2}=8\unicode[STIX]{x03C0}\int _{0}^{\infty }\text{d}k\,E_{B}(k), & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle l_{B}=\frac{\unicode[STIX]{x03C0}}{2}\frac{\displaystyle \int _{0}^{\infty }\text{d}k\,E_{B}(k)/k}{\displaystyle \int _{0}^{\infty }\text{d}k\,E_{B}(k)}. & \displaystyle\end{eqnarray}$$

The latter result is derived in § C.3. Secondly, for the special case of isotropic, non-helical Gaussian statistics the magnetic-energy spectrum is sufficient to provide a complete statistical characterisation of the stochastic magnetic field (Adler Reference Adler1981). Thirdly, even for non-Gaussian stochastic magnetic fields (as physical fields often are), in many situations the magnetic-energy spectrum has a special significance. It describes energy distribution over wavenumber scales, and so is often the focus of theoretical predictions (for example, see Gregori, Reville & Miniati Reference Gregori, Reville and Miniati2015). Throughout the rest of this paper, we therefore explore the question of whether a proton-imaging diagnostic can be used to extract the magnetic-energy spectrum.

A discussion of the mathematical characterisation of stochastic magnetic fields by an alternative mathematical object – the magnetic autocorrelation function – is given in appendix C. This is an equivalent quantity to the magnetic-energy spectrum. It is sometimes useful to invoke it instead of the spectrum: the most physically intuitive definition of the magnetic-field correlation length is in terms of magnetic autocorrelation function, and for the derivations of spectral relations (2.11) and (3.15) presented in appendices F and I respectively, the magnetic autocorrelation function enables a clearer characterisation of the precision of approximations made than does the magnetic-energy spectrum.

2.2 The plasma-image mapping

In the Introduction, we discussed the importance of the Kugland image-flux relation between the path-integrated magnetic field and image-flux distribution when seeking to determine the magnetic-energy spectrum; here we outline the form of this relation and its correspondence to the energy spectrum. The Kugland image-flux relation was originally derived from particle conservation in real space (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012). In appendix D, we present an alternative derivation from first principles using kinetic theory; any reader who wishes to understand in greater mathematical detail the origin of the Kugland image-flux relation is encouraged to explore this appendix.

Physically, magnetic forces experienced by the imaging protons lead to global restructuring of the imaging beam. For arbitrary magnetic-field configurations and imaging-beam parameters, characterising this restructuring is extremely complicated, particularly for a stochastic magnetic field. This is made clear by stating the general relationship between the proton beam distribution function and the final proton image (see § D.1). However, under some assumptions that are typically valid for proton-imaging set-ups, the complexity of such a relation is greatly reduced (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012). We assume that a number of dimensionless parameters are small, namely the size of the proton source relative to the plasma $a/r_{i}$ , the paraxial parameter

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\equiv \frac{l_{\bot }}{r_{i}}\ll 1\end{eqnarray}$$

of the proton-imaging set-up, the point-projection parameter

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}\equiv \frac{l_{z}}{r_{s}}\ll 1,\end{eqnarray}$$

and the magnitude of angular deflections $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ away from the initial proton trajectories (the magnitude of this for stochastic fields is shown subsequently in (2.13)). We also assume $r_{s}\gtrsim r_{i}$ . A full discussion of the validity of these assumptions and their significance in simplifying the general proton-imaging problem is given in § 4.1.

In the limit where the above asymptotic parameters are indeed small, the form of the beam as a two-dimensional near-planar sheet (see figure 1 and § D.3) is retained following interaction with the magnetic field. Furthermore, internal re-distribution of proton flux within the sheet is entirely determined by velocity perturbations acquired inside the plasma. More specifically, it can be shown (p. 5, equations (3), (4) of Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012, see also § D.4) that an imaging proton with initial perpendicular position $\boldsymbol{x}_{\bot 0}$ ends up with final perpendicular position on the detector

(2.7) $$\begin{eqnarray}\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})\approx \frac{r_{s}+r_{i}}{r_{i}}\boldsymbol{x}_{\bot 0}+\frac{r_{s}}{V}\boldsymbol{w}(\boldsymbol{x}_{\bot 0}).\end{eqnarray}$$

Here $V$ is the initial speed of the proton beam, assumed mono-energetic, and the perpendicular velocity deflection $\boldsymbol{w}(\boldsymbol{x}_{\bot 0})$ caused by magnetic forces of an imaging proton with initial perpendicular position $\boldsymbol{x}_{\bot 0}$ is

(2.8) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x}_{\bot 0})\approx \frac{e}{m_{p}c}\hat{\boldsymbol{z}}\times \int _{0}^{l_{z}}\text{d}s\,\boldsymbol{B}(\boldsymbol{x}(s)),\end{eqnarray}$$

where $e$ is the proton charge, $m_{p}$ the proton mass and $\boldsymbol{x}(s)$ the proton trajectory. We can view $\boldsymbol{w}(\boldsymbol{x}_{\bot 0})$ as a function of the initial perpendicular position. We will call this function the perpendicular-deflection field for the remainder of this paper. By conservation of proton flux within the imaging beam, the image-flux distribution $\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})$ is then given by the Kugland image-flux relation  (p. 5, equation (6) of Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012):

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0}))=\mathop{\sum }_{\boldsymbol{x}_{\bot }^{(s)}=\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})}\frac{\unicode[STIX]{x1D6F9}_{0}}{|\text{Det}(\unicode[STIX]{x1D735}_{\bot 0}[\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})])|},\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}_{0}$ is the initial flux distribution (assumed uniform), $\unicode[STIX]{x1D735}_{\bot 0}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{x}_{\bot 0}$ is a gradient operator with respect to the initial plasma coordinates, and the sum indicates that the total flux at any particular position on the detector can in general have contributions from protons with many different initial positions. A numerical illustration of the validity of equations (2.8) and (2.9) can be found in § D.9.

The plasma-image mapping (2.7) can be related to the magnetic-energy spectrum (2.2) via the perpendicular-deflection field (2.8). This is entirely equivalent to relating the path-integrated field to the magnetic-energy spectrum, because the former can be recovered from the perpendicular-deflection field directly by rearranging (2.8):

(2.10) $$\begin{eqnarray}\int _{0}^{l_{z}}\text{d}s\,\boldsymbol{B}_{\bot }(\boldsymbol{x}(s))=-\frac{m_{p}c}{e}\hat{\boldsymbol{z}}\times \boldsymbol{w}(\boldsymbol{x}_{\bot 0}).\end{eqnarray}$$

If the stochastic magnetic field is assumed to be statistically isotropic and homogeneous (except for some global variation in root-mean-square (RMS) field strength magnitude), with an additional assumption of zero mean magnetic field ( $\bar{\boldsymbol{B}}=0$ , $\boldsymbol{B}=\unicode[STIX]{x1D6FF}\boldsymbol{B}$ ), it can then be shown (see appendix F) that the deflection-field spectrum is related to the magnetic-energy spectrum by

(2.11) $$\begin{eqnarray}E_{B}(k)=\frac{m_{p}^{2}c^{2}}{4\unicode[STIX]{x03C0}^{2}l_{z}e^{2}}kE_{W}(k).\end{eqnarray}$$

Here the deflection-field spectrum $E_{W}(k)$ is defined by

(2.12) $$\begin{eqnarray}E_{W}(k_{\bot })\equiv \int \text{d}\unicode[STIX]{x1D703}\,k_{\bot }\langle |\hat{\boldsymbol{w}}(\boldsymbol{k}_{\bot })|^{2}\rangle ,\end{eqnarray}$$

where $\hat{\boldsymbol{w}}(\boldsymbol{k}_{\bot })$ is the two-dimensional Fourier transform of the perpendicular-deflection field, $\boldsymbol{k}_{\bot }$ the perpendicular wavevector and the integral is over the polar angle $\unicode[STIX]{x1D703}$ . Spectral relation (2.11) implies that if we can determine the perpendicular-deflection field from the image-flux distribution, then we can recover the magnetic-energy spectrum. It is important to note that the derivation of this result relies on the solenoidality of the magnetic field.

The deflection-field spectral relation (2.11) also allows for a simple calculation of the typical deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ in terms of the magnetic field and initial proton speed. More specifically, integrating (2.11) over all wavenumbers (see appendix F) gives, in the small-deflections limit,

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\equiv \frac{w_{\text{rms}}}{V}=\frac{eB_{\text{rms}}}{m_{p}cV}\sqrt{l_{z}l_{B}},\end{eqnarray}$$

where $w_{\text{rms}}\equiv \langle \boldsymbol{w}^{2}\rangle ^{1/2}$ is the RMS of the perpendicular-deflection field, and $l_{B}$ is again the correlation length. We emphasise that both $B_{\text{rms}}$ and $l_{B}$ can be calculated independently using (2.3) and (2.4) once the magnetic-energy spectrum has been derived from (2.11).

As noted by previous authors (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012; Graziani et al. Reference Graziani, Tzeferacos, Lamb and Li2016), the perpendicular-deflection field has the property of being irrotational, provided the typical deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\lesssim l_{B}/l_{z}$ (this assumption is potentially more restrictive than the small-angle approximation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\ll 1$ – as discussed in § 4.1). Irrotationality of the perpendicular-deflection field follows from the solenoidality of the magnetic field:

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\bot 0}\times \boldsymbol{w}(\boldsymbol{x}_{\bot 0})=\hat{\boldsymbol{z}}\frac{e}{m_{p}c}\int _{0}^{l_{z}}\text{d}z^{\prime }\left(\frac{\unicode[STIX]{x2202}\boldsymbol{x}_{\bot }(z^{\prime })}{\unicode[STIX]{x2202}\boldsymbol{x}_{\bot 0}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }\right)\boldsymbol{\cdot }\boldsymbol{B}\approx \hat{\boldsymbol{z}}\frac{e}{m_{p}c}\int _{0}^{l_{z}}\text{d}z^{\prime }\,\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D735}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{x}$ denotes the gradient operator with respect to the beam-proton position $\boldsymbol{x}(s)$ . The approximation $\unicode[STIX]{x2202}\boldsymbol{x}_{\bot }(z^{\prime })/\unicode[STIX]{x2202}\boldsymbol{x}_{\bot 0}\approx \unicode[STIX]{x1D644}$ holds if the proton trajectories do not cross inside the plasma (for a more detailed discussion of this result, see § D.8). As a consequence, the perpendicular-deflection field can always be written as the gradient of the deflection-field potential, defined by

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0})\equiv \int _{C}\text{d}\boldsymbol{l}\boldsymbol{\cdot }\boldsymbol{w}(\tilde{\boldsymbol{x}}_{\bot 0}),\end{eqnarray}$$

where $C$ is any path from the origin to the perpendicular coordinate $\boldsymbol{x}_{\bot 0}$ and $\text{d}\boldsymbol{l}$ is an infinitesimal line element along this path. Under the same assumption, the perpendicular-deflection field is given by the magnetic field integrated along the unperturbed trajectories:

(2.16) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x}_{\bot 0})\approx \frac{e}{m_{p}c}\hat{\boldsymbol{z}}\times \int _{0}^{l_{z}}\text{d}z^{\prime }\,\boldsymbol{B}\left(\boldsymbol{x}_{\bot 0}\left(1+\frac{z^{\prime }}{r_{i}}\right),z^{\prime }\right).\end{eqnarray}$$

We observe that introducing the vector potential $\boldsymbol{A}$ satisfying $\boldsymbol{B}=\unicode[STIX]{x1D735}\times \boldsymbol{A}$ into (2.16) recovers equations (78) of Kugland et al. (Reference Kugland, Ryutov, Plechaty, Ross and Park2012). As we discuss in §§ 3.1 and 3.2, the irrotationality property of the perpendicular-deflection field is essential for attempts for reconstructing the path-integrated field from the image-flux distribution.

2.3 The contrast parameter $\unicode[STIX]{x1D707}$

Equation (2.9) is the desired Kugland image-flux relation between the image-flux distribution and the path-integrated magnetic field, via plasma-image mapping (2.7); we now explore the properties of the Kugland image-flux relation. As is clear from the appearance of the Jacobian determinant of the plasma-image mapping in (2.9), the size of Jacobian-matrix elements of the plasma-image mapping will be of significance for characterising image-flux features. These elements have physical meaning, describing the relative size of initial gradients in perpendicular velocities of protons compared to gradients resulting from deflections due to magnetic forces. Mathematically, their size is quantified by the constrast parameter $\unicode[STIX]{x1D707}$ , which we define for a proton imaging set-up applied to a stochastic magnetic field with correlation length $l_{B}$ by

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D707}\equiv \frac{d_{s}}{{\mathcal{M}}l_{B}}=\frac{r_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}}{\tilde{l}_{B}}=\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}}\frac{r_{s}}{r_{s}+r_{i}}\frac{l_{\bot }}{l_{B}},\end{eqnarray}$$

where $d_{s}\equiv r_{s}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ is the typical perpendicular displacement of a proton from its undeflected position on the detector,

(2.18) $$\begin{eqnarray}{\mathcal{M}}=\frac{r_{i}+r_{s}}{r_{i}}\end{eqnarray}$$

the image magnification, and $\tilde{l}_{B}={\mathcal{M}}l_{B}$ is the magnified correlation length.

If $r_{s}\gg r_{i}$ , equation (2.17) reduces to the definition of $\unicode[STIX]{x1D707}$ given in previous literature (see equation (11) of Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012; Graziani et al. Reference Graziani, Tzeferacos, Lamb and Li2016): $\unicode[STIX]{x1D707}=r_{i}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}/l_{B}$ .

The universal dependence of image-flux features on $\unicode[STIX]{x1D707}$ – irrespective of the particular magnetic-field structure – enables a systematic approach to the heuristic interpretation of proton-flux images. Since $\unicode[STIX]{x1D707}$ is a function of field strength via the typical proton deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ , its identification for a particular image-flux distribution is a useful way to estimate magnetic field strengths attained in experiments. Substituting definitions (2.5) and (2.13) for $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ respectively into (2.17) gives

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\frac{r_{s}r_{i}}{r_{s}+r_{i}}\frac{eB_{\text{rms}}}{m_{p}cV}\sqrt{\frac{l_{z}}{l_{B}}}.\end{eqnarray}$$

This shows that $\unicode[STIX]{x1D707}\propto B_{\text{rms}}/V$ , the same as the RMS proton deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{\text{rms}}$ . By substituting appropriate values of physical constants into (2.19) we can explicitly write the field strength associated with a particular $\unicode[STIX]{x1D707}$ as

(2.20) $$\begin{eqnarray}B_{\text{rms}}(\text{kG})\approx 250\left(\frac{{\mathcal{M}}}{{\mathcal{M}}-1}\right)\unicode[STIX]{x1D707}\left[\frac{W(\text{MeV})}{3.3\;\text{MeV}}\right]^{1/2}\left[\frac{r_{i}(\text{cm})}{1\;\text{cm}}\right]^{-1}\left[\frac{l_{B}(\text{cm})}{l_{z}(\text{cm})}\right]^{1/2}\text{kG}.\end{eqnarray}$$

When using (2.20) to make a quick order-of-magnitude estimate, an educated guess for the correlation length could be reasonably used (though the typical field strength and correlation length can be determined independently if the magnetic-energy spectrum is known). We discuss how to estimate $\unicode[STIX]{x1D707}$ from a proton image in § 3.

Comparing (2.13) and (2.19), it is clear that $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{\text{rms}}$ and $\unicode[STIX]{x1D707}$ have different dependences on the correlation scale of the field: $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}_{\text{rms}}\propto l_{B}^{1/2}$ , while $\unicode[STIX]{x1D707}\propto l_{B}^{-1/2}$ . Thus a field with smaller-scale structures will give larger values for $\unicode[STIX]{x1D707}$ , despite the typical deflection angle being reduced (a numerical example of this is given in appendix G).

Beyond qualitative estimates, whether the path-integrated field (and hence magnetic-energy spectrum for isotropic stochastic magnetic fields) can be directly extracted from experimental data – and if so, how this extraction is carried out – changes depending on $\unicode[STIX]{x1D707}$ . This change is best elucidated for stochastic magnetic fields in terms of four contrast regimes: linear ( $\unicode[STIX]{x1D707}\ll 1$ ), nonlinear injective ( $\unicode[STIX]{x1D707}$ below some critical value $\unicode[STIX]{x1D707}_{c}\sim 1$ ), caustic ( $\unicode[STIX]{x1D707}\geqslant \unicode[STIX]{x1D707}_{c}$ ) and diffusive ( $\unicode[STIX]{x1D707}\gtrsim r_{s}/(r_{s}+r_{i})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ , where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ is again the paraxial parameter). Here, $\unicode[STIX]{x1D707}_{c}$ is defined to be the smallest value of $\unicode[STIX]{x1D707}$ associated with the imaging of a particular stochastic magnetic field such that the plasma-image mapping (2.7) is not injective (one-to-one). These four contrast regimes are discussed in the next section.

3.1 Linear regime of proton imaging – $\unicode[STIX]{x1D707}\ll 1$

If $\unicode[STIX]{x1D707}$ is small, relating the magnetic field and the image flux becomes a much simpler problem. This follows from the result (derived in appendix H – see also Graziani et al. (Reference Graziani, Tzeferacos, Lamb and Li2016)) that if $\unicode[STIX]{x1D707}\ll 1$ , the plasma-image mapping (2.7) becomes, to leading order in $\unicode[STIX]{x1D707}$ ,

(3.4) $$\begin{eqnarray}\boldsymbol{x}_{\bot }^{(s)}=\frac{r_{s}+r_{i}}{r_{i}}\boldsymbol{x}_{\bot 0}[1+O(\unicode[STIX]{x1D707})],\end{eqnarray}$$

while the Kugland image-flux relation (2.9) can be rewritten as

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})}{\unicode[STIX]{x1D6F9}_{0}^{(s)}}=\frac{r_{s}r_{i}}{r_{s}+r_{i}}\frac{4\unicode[STIX]{x03C0}e}{m_{p}c^{2}V}\int _{0}^{l_{z}}j_{z}\left(\boldsymbol{x}_{\bot 0}\left(1+\frac{z^{\prime }}{r_{i}}\right),z^{\prime }\right)\text{d}z^{\prime },\end{eqnarray}$$

where $\boldsymbol{j}=c\unicode[STIX]{x1D735}\times \boldsymbol{B}/4\unicode[STIX]{x03C0}$ is the magnetohydrodynamic (MHD) current, and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})\equiv \unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})-\unicode[STIX]{x1D6F9}_{0}^{(s)}$ is the image-flux deviation from the initial mean image flux, which in turn is related to the initial flux $\unicode[STIX]{x1D6F9}_{0}$ by the image-magnification factor ${\mathcal{M}}$ , viz., $\unicode[STIX]{x1D6F9}_{0}^{(s)}=\unicode[STIX]{x1D6F9}_{0}/{\mathcal{M}}^{2}$ . It follows from (3.5) that proton-flux images in the linear regime have a simple physical interpretation: they display the undeflected path integrated $z$ -component $j_{z}$ of the MHD current (Graziani et al. Reference Graziani, Tzeferacos, Lamb and Li2016). The linear regime is therefore so called, because the magnitude of image-flux deviations is linear in the magnetic field.

Linear-regime image-flux relation (3.5) has another consequence: estimating the magnitude of the image-flux deviations compared to the mean image flux, we see that $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}\sim \unicode[STIX]{x1D707}\unicode[STIX]{x1D6F9}_{0}^{(s)}\ll \unicode[STIX]{x1D6F9}_{0}^{(s)}$ . The linear regime is therefore characterised by small relative image-flux deviations, providing a useful observational tool for recognising proton-flux images of stochastic magnetic fields in the linear regime. An example of this phenomenon is shown in figure 3(a), a flux image of a Golitsyn field with parameters tuned to have small $\unicode[STIX]{x1D707}$ : relative image-flux deviations indeed appear faint. The smallness of coordinate-grid perturbations relative to a Cartesian grid implied by (3.4) is demonstrated in figure 3(b) for a Golitsyn field with $\unicode[STIX]{x1D707}\ll 1$ .

More quantitatively, the RMS of relative image-flux variations is small in the linear regime. Linear-regime image-flux relation (3.5) enables the following relation between $\unicode[STIX]{x1D707}$ and the relative image-flux RMS to be derived analytically under the assumption of homogeneous and isotropic magnetic-field statistics (see appendix I):

(3.6) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6F9}_{0}}\right)_{\text{rms}}\equiv \left\langle \left(\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6F9}_{0}^{(s)}}\right)^{2}\right\rangle ^{1/2}=\sqrt{\frac{\unicode[STIX]{x03C0}}{2}}\frac{r_{i}r_{s}}{r_{s}+r_{i}}\frac{eB_{\text{rms}}}{mcV}\sqrt{\frac{l_{z}}{l_{\unicode[STIX]{x1D6F9}}}}=\sqrt{\frac{\unicode[STIX]{x03C0}}{2}\frac{l_{B}}{l_{\unicode[STIX]{x1D6F9}}}}\unicode[STIX]{x1D707},\end{eqnarray}$$

where $l_{\unicode[STIX]{x1D6F9}}$ is the relative image-flux correlation length (defined in appendix I). This can be rearranged to give a formula for $\unicode[STIX]{x1D707}$ in terms of the RMS of relative image-flux deviations:

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{0}\left(\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6F9}_{0}}\right)_{\text{rms}},\end{eqnarray}$$

where $\unicode[STIX]{x1D707}_{0}\equiv \sqrt{2l_{\unicode[STIX]{x1D6F9}}/l_{B}\unicode[STIX]{x03C0}}$ depends on the particular stochastic field configuration, but is can be shown analytically that for any isotropic, homogeneous stochastic magnetic field, $\unicode[STIX]{x1D707}_{0}\leqslant 2/\unicode[STIX]{x03C0}$ (appendix I). The estimate (3.7) of $\unicode[STIX]{x1D707}$ combined with RMS magnetic-field strength estimate (2.20) leads to a simple formula for the RMS field strength in terms of the RMS of the relative image flux:

(3.8) $$\begin{eqnarray}B_{\text{rms}}(\text{kG})\approx 40\left[\frac{W(\text{MeV})}{3.3\;\text{MeV}}\right]^{1/2}\left[\frac{r_{i}(\text{cm})}{1~\text{cm}}\right]^{-1}\left[\frac{l_{z}(\text{mm})}{1~\text{mm}}\right]^{-1/2}\left[\frac{l_{B}(\text{mm})}{0.08(\text{mm})}\right]^{1/2}\left(\!\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6F9}_{0}}\right)_{\text{rms}}\;\text{kG}.\end{eqnarray}$$

As mentioned in the discussion accompanying (2.20), a reasonable approximate value for the correlation length $l_{B}$ may be taken for the purpose of using this estimate of the magnetic-field strength. $l_{B}$ can be evaluated precisely using the quantitative techniques described in the subsequent paragraphs. We note expression (3.8) only applies for small $\unicode[STIX]{x1D707}$ ; once $\unicode[STIX]{x1D707}$ approaches unity, the RMS of the relative image-flux begins to increase nonlinearly with $\unicode[STIX]{x1D707}$ (see appendix I).

In the linear regime, the path-integrated field can always be reconstructed uniquely from the image-flux distribution. This follows from the result (2.15) that the perpendicular-deflection field can be written as the gradient of the deflection-field potential $\unicode[STIX]{x1D711}$ provided $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\lesssim l_{B}/l_{z}$ :

(3.9) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x}_{\bot 0})\approx \unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0}).\end{eqnarray}$$

The assumption $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\lesssim l_{B}/l_{z}$ is indeed valid in the linear regime, because $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}l_{z}/l_{B}\sim \unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}(r_{s}+r_{i})/r_{s}\ll 1$ . It can then be shown (appendix H) that the image-flux deviation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}$ and the deflection-field potential $\unicode[STIX]{x1D711}$ are related by a Poisson equation

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\bot 0}^{2}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0})=-\unicode[STIX]{x1D6EF}(\boldsymbol{x}_{\bot 0}),\end{eqnarray}$$

where source function $\unicode[STIX]{x1D6EF}(\boldsymbol{x}_{\bot })$ is proportional to relative image-flux deviations:

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}(\boldsymbol{x}_{\bot 0})={\mathcal{M}}\frac{V}{r_{s}}\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})}{\unicode[STIX]{x1D6F9}_{0}^{(s)}}.\end{eqnarray}$$

If suitable boundary conditions are applied, for example

(3.12) $$\begin{eqnarray}\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0})=0,\end{eqnarray}$$

equation (3.10) is well posed with a unique solution for $\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D711}$ , and so is invertible (see Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012, § VB for a more detailed discussion of this). For example, the standard two-dimensional (2-D) solution to the Poisson equation (3.10) with vanishing boundary conditions at infinity is

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0})=\frac{1}{2\unicode[STIX]{x03C0}}\int \text{d}^{2}\tilde{\boldsymbol{x}}_{\bot }\log \left(\frac{l_{\bot }}{|\boldsymbol{x}_{\bot }-\tilde{\boldsymbol{x}}_{\bot 0}|}\right)\unicode[STIX]{x1D6EF}(\tilde{\boldsymbol{x}}_{\bot }).\end{eqnarray}$$

The use of $l_{\bot }$ in this expression is arbitrary, since the integral of $\unicode[STIX]{x1D6EF}(\boldsymbol{x}_{\bot })$ over the image vanishes by conservation of particles. The perpendicular-deflection field is then given by (3.9); for the case of infinite boundary conditions, we find

(3.14) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x}_{\bot 0})=-\frac{1}{2\unicode[STIX]{x03C0}}\int \text{d}^{2}\tilde{\boldsymbol{x}}_{\bot }\frac{\boldsymbol{x}_{\bot 0}-\tilde{\boldsymbol{x}}_{\bot }}{|\boldsymbol{x}_{\bot 0}-\tilde{\boldsymbol{x}}_{\bot }|^{2}}\unicode[STIX]{x1D6EF}(\tilde{\boldsymbol{x}}_{\bot }).\end{eqnarray}$$

The path-integrated magnetic field follows from (2.10). For finite regions, Poisson equation (3.10) can in principle be inverted numerically. However, Graziani et al. (Reference Graziani, Tzeferacos, Lamb and Li2016) report that such an approach when applied to reconstructing path-integrated fields from proton images quickly becomes unsuccessful for non-asymptotically small $\unicode[STIX]{x1D707}$ . For this reason, we suggest using the field-reconstruction algorithm described in § 3.2 to reconstruct the path-integrated field instead.

If the perpendicular-deflection field (and hence the path-integrated field) has been reconstructed, the magnetic-energy spectrum can be predicted using spectral relation (2.11). However, the simple form of the relation (3.6) between image flux and magnetic field in the linear regime allows for the application of statistical methods directly to proton-flux images to obtain properties of the fields creating that image. In particular, for homogeneous and isotropic magnetic-field statistics satisfying $l_{B}\ll l_{z}$ (appendix I, and Graziani et al. Reference Graziani, Tzeferacos, Lamb and Li2016), the 1-D magnetic-energy spectrum (2.2) is related to the 2-D spectrum of image-flux deviations $\hat{\unicode[STIX]{x1D702}}(k)$ by

(3.15) $$\begin{eqnarray}E_{B}(k)=\frac{1}{2\unicode[STIX]{x03C0}}\frac{m_{p}^{2}c^{2}V^{2}}{e^{2}r_{s}^{2}l_{z}}\hat{\unicode[STIX]{x1D702}}\left(\frac{r_{i}}{r_{s}+r_{i}}k\right).\end{eqnarray}$$

Here, $\hat{\unicode[STIX]{x1D702}}(k)$ is defined by

(3.16) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D702}}(k_{\bot })\equiv \frac{1}{2\unicode[STIX]{x03C0}}\int \text{d}\unicode[STIX]{x1D703}\frac{{\mathcal{M}}^{4}}{\unicode[STIX]{x1D6F9}_{0}^{2}}\langle |\hat{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}(\boldsymbol{k}_{\bot })|^{2}\rangle ,\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}(\boldsymbol{k}_{\bot })$ is the Fourier transform of the relative image-flux deviation.

To summarise, if a stochastic magnetic field with isotropic and locally homogeneous statistics is imaged in the linear regime, the path-integrated field and magnetic-energy spectrum can be reconstructed. However, we caution that unless $\unicode[STIX]{x1D707}$ is very small, distortions to results obtained using linear analysis manifest themselves, both in reconstructing perpendicular-deflection fields, and in the magnetic-energy spectrum using linear-regime flux spectral relation (3.15) (see § 3.2 for an example).

A detailed discussion of proton imaging in the linear regime – including an alternative field-reconstruction algorithm to cope with weakly nonlinear effects – is presented in Graziani et al. (Reference Graziani, Tzeferacos, Lamb and Li2016).

3.2 Nonlinear injective regime: $\unicode[STIX]{x1D707}<\unicode[STIX]{x1D707}_{c}\sim 1$

In the nonlinear injective regime, $\unicode[STIX]{x1D707}$ is sufficiently large that beam-focusing effects associated with the nonlinear term resulting from magnetic deflections in plasma-image mapping (2.7) have a non-trivial effect on the image-flux distribution. However, $\unicode[STIX]{x1D707}$ is not so great as to lead to the proton beam intersecting itself, and hence loss of injectivity of the plasma-image mapping, which first occurs at some critical $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{c}$ (shown in appendix L). The importance of nonlinearity for moderate $\unicode[STIX]{x1D707}$ and preservation of injectivity for $\unicode[STIX]{x1D707}<\unicode[STIX]{x1D707}_{c}$ are illustrated by the plasma-image-coordinate mapping for the test Golitsyn field shown in figure 3(d): the image-coordinate grid is visually distorted, but coordinate curves do not cross each other. The nonlinear injective regime can be distinguished from the linear regime simply by the presence of image-flux structures whose deviation from the mean image flux is similar in magnitude to the mean (see figure 3 c).

When injective, the plasma-image mapping tends to preserve the morphology of the proton-flux image obtained for the same magnetic field, but with small $\unicode[STIX]{x1D707}$ – so image-flux structures can still be qualitatively interpreted in terms of path-integrated MHD current structure. However, structures with positive relative image-flux tend to be narrow due to beam focusing, and those with negative relative image-flux enlarged. This phenomenon is evident in the nonlinear injective proton-flux image of the Golitsyn field, figure 3(c).

Nonlinear effects mean that a different approach for reconstructing path-integrated fields must be adopted to that expounded for the linear regime. One such approach can be found by noting that since the plasma-image mapping (2.7) is injective, the sum in Kugland image-flux relation (2.9) disappears, leaving

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0}))=\frac{\unicode[STIX]{x1D6F9}_{0}}{\text{Det}(\unicode[STIX]{x1D735}_{\bot 0}[\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})])}.\end{eqnarray}$$

Our goal is to solve (3.17) for $\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})$ given an image-flux distribution $\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0}))$ . We proceed by noting that the plasma-image mapping (2.7) can be rewritten in terms of a potential field $\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})$ :

(3.18) $$\begin{eqnarray}\boldsymbol{x}_{\bot }^{(s)}\approx \unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})\approx \unicode[STIX]{x1D735}_{\bot 0}\left[\frac{r_{s}+r_{i}}{2r_{i}}\boldsymbol{x}_{\bot 0}^{2}+\frac{r_{s}}{V}\unicode[STIX]{x1D711}(\boldsymbol{x}_{\bot 0})\right].\end{eqnarray}$$

The existence of $\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})$ follows from that of the deflection-field potential, which is defined in § 3.1, equation (2.15). Equation (3.17) can then be restated as an equation for $\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})$ :

(3.19) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0}))=\frac{\unicode[STIX]{x1D6F9}_{0}}{\text{Det}[\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})]}.\end{eqnarray}$$

Equation (3.19) is an example of a Monge–Ampère equation, which appears in numerous mathematical and physical contexts (Gangbo & McCann Reference Gangbo and McCann1996). Despite its nonlinearity, it can be shown (Brenier Reference Brenier1991) that there is a unique (up to a constant) solution for $\unicode[STIX]{x1D719}$ with Neumann boundary conditions

(3.20) $$\begin{eqnarray}\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D719}(\boldsymbol{x}_{\bot 0})=\hat{\boldsymbol{n}}\boldsymbol{\cdot }\boldsymbol{x}_{\bot 0}.\end{eqnarray}$$

One approach for establishing this result comes from the observation that the solution of the Monge–Ampère equation also solves the $L_{2}$ Monge–Kantorovich problem (Villani Reference Villani2008). This correspondence is explained in appendix K. This means that with an appropriate field-reconstruction algorithm, $\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D719}$ can be reconstructed from a given proton-flux image, which can then be used to calculate the perpendicular-deflection field

(3.21) $$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{x}_{\bot 0})=\frac{V}{r_{s}}\unicode[STIX]{x1D735}_{\bot 0}\left(\unicode[STIX]{x1D719}-\frac{r_{s}+r_{i}}{2r_{i}}\boldsymbol{x}_{\bot 0}^{2}\right).\end{eqnarray}$$

The path-integrated magnetic field can then be calculated using (2.10). There exist a number of possible algorithms for solving the Monge–Ampère equation (3.19) (Dean & Glowinski Reference Dean and Glowinski2006), and some have recently been applied to the problem of recovering path-integrated magnetic fields from proton-flux images (Kasim et al. Reference Kasim, Ceurvorst, Ratan, Sadler, Chen, Sävert, Trines, Bingham, Burrows and Kaluza2017). In § P.4, we outline one such field-reconstruction algorithm, which is both simple to implement and computationally efficient (Sulman, Williams & Russell Reference Sulman, Williams and Russell2011). Furthermore, we have recently made available an open-source PYTHON code (‘PROton-imaged B-field nonLinear Extraction Module’, or PROBLEM) which solves the inversion problem numerically using this approach. As explained in § 2.2, the reconstructed path-integrated magnetic field can be combined with deflection-field spectral relation (2.11) to deduce the magnetic-energy spectrum.

To summarise, like the linear regime the path-integrated magnetic field and magnetic-energy spectrum are always recoverable from an individual proton-flux image in the nonlinear injective regime. This perhaps counter-intuitive result essentially holds because of the irrotationality of the perpendicular-deflection field, discussed in § 2.2. Furthermore, the techniques used to achieve this are more widely applicable than those derivable from linear theory. However, care must be taken when applying the field-reconstruction algorithm to arbitrary proton-flux images, because the results can be misleading if the plasma-image mapping is not injective. This is described in the next section.

3.3 Caustic regime: $\unicode[STIX]{x1D707}\geqslant \unicode[STIX]{x1D707}_{c}$

For $\unicode[STIX]{x1D707}$ greater than $\unicode[STIX]{x1D707}_{c}$ , gradients in the perpendicular-deflection field are sufficiently large that the plasma-image mapping (2.7) becomes multivalued in places (see figure 3 f) – a phenomenon sometimes referred to as mesh-twisting (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012). Physically, in this regime some of the paths of imaging protons cross before reaching the detector, and hence there exist regions of image flux whose constituent protons originate from spatially disconnected initial positions. That multivaluedness of the plasma-image mapping only occurs at $\unicode[STIX]{x1D707}\geqslant \unicode[STIX]{x1D707}_{c}$ follows from the observation that such crossing requires that the determinant of the plasma-image mapping change sign. For this to happen, the magnitude of gradients of the perpendicular-deflection field must be comparable to gradients in the undeflected mapping – which by the definition of $\unicode[STIX]{x1D707}$ (2.17) is precisely the criterion of $\unicode[STIX]{x1D707}$ sufficiently large. This alone does not guarantee the existence of $\unicode[STIX]{x1D707}_{c}$ ; however, the conditions required for the absence of mesh twisting at all values of $\unicode[STIX]{x1D707}$ are typically incompatible with stochasticity (see appendix L). The particular value of $\unicode[STIX]{x1D707}_{c}$ depends on the particular stochastic field, but typically is order unity (although heuristic arguments can be given implying that it likely decreases logarithmically with the field scale $l_{B}$ ; see appendix L).

The multivaluedness of the plasma-image mapping is closely associated with caustics, defined as curves on which the determinant of the plasma-image mapping (2.7) vanishes (equation (14) of Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012):

(3.22) $$\begin{eqnarray}\text{Det}(\unicode[STIX]{x1D735}_{\bot 0}[\boldsymbol{x}_{\bot }^{(s)}(\boldsymbol{x}_{\bot 0})])=0.\end{eqnarray}$$

This is (trivially) because a sign reversal of the determinant for a continuous mapping cannot occur without the value of that determinant passing through zero. If the determinant does indeed vanish, the denominator in Kugland image-flux relation (2.9) also does, yielding a formally infinite local image-flux. In practice the local image-flux value is limited by finite resolution of proton images (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012) – however, the morphology of proton-flux images is still dominated by caustics if there are present. This is evident in the caustic-regime proton-flux image of the test Golitsyn field, shown in figure 3(e).

The relationship between caustics and magnetic fields creating them has been studied in great depth elsewhere (see § IV of Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012); for our purposes, we simply note that the width, strength and distance between caustics do not necessarily reflect equivalent properties of the magnetic field, implying that great care must be taken when qualitatively assessing magnetic structures from proton-flux images containing caustics.

Similar care must be taken when attempting quantitative analysis. In the caustic regime the reconstruction of the path-integrated field from a proton-flux image is no longer a well-posed problem: for a given proton image, there exists a number of path-integrated fields (and hence plasma-image mappings) which give the same flux distribution via the Kugland image-flux relation (2.9). Intuitively this seems reasonable, because without the injectivity constraint, the Kugland image-flux relation insufficiently determines the path-integrated field. An explicit example of many different path-integrated fields corresponding to a simple image-flux distribution is given in appendix M. In addition to this simple analytical example, the impossibility of solving the Kugland image-flux relation (2.9) for non-injective plasma-image mappings can be demonstrated numerically (and is done so in § 3.5). Attempts to reconstruct the magnetic-energy spectrum in the caustic regime are also prone to failure. In short, we emphasise that the caustic regime is much less amenable to direct analysis than either the linear, or the nonlinear injective regime.

This being the case, identifying the presence of caustics in a proton image is essential for a sensible interpretation of that image. In principle, caustics should typically be identifiable in proton-flux images from narrow curves of high proton flux. In practice, proton-flux images have a finite spatial resolution which can disguise these, making distinguishing between the nonlinear injective regime and caustic regime a non-trivial problem. The issue, and possible ways around it, is explored in § 4.3.3.

For a systematic review of caustic theory in the context of proton imaging for electromagnetic structures, see Kugland et al. (Reference Kugland, Ryutov, Plechaty, Ross and Park2012).

3.4 Diffusive regime: $\unicode[STIX]{x1D707}\gtrsim r_{s}/(r_{i}+r_{s})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\gg 1$

For large $\unicode[STIX]{x1D707}$ , the characteristic nature of the plasma-image mapping changes yet again. The position of protons in any given local region on the detector is dominated by the perpendicular-deflection field, rather than the projected position in the absence of magnetic fields. More specifically, if $\unicode[STIX]{x1D707}\gtrsim r_{s}/(r_{i}+r_{s})\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ , then proton trajectories have already crossed as they leave the plasma. This follows because the perpendicular displacement inside the plasma due to magnetic deflections $l_{z}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ is comparable to the typical size of magnetic structures $l_{B}$ : $l_{z}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\sim \unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}(r_{s}+r_{i})/r_{s}l_{B}\gtrsim l_{B}$ . As a consequence, the perpendicular-deflection field can no longer be written as the gradient of the deflection-field potential, and the magnetic field integrated along the actual proton trajectories is not equivalent to the magnetic field integrated along undeflected trajectories (in other words, equations (2.15) and (2.16) do not apply in the diffusive regime).

The diffusive regime is incompatible with the small-deflection assumption for regular fields with $l_{B}\sim l_{z}$ , since then the deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\gtrsim 1$ . Furthermore, successful imaging in such a parameter space is practically unfeasible, because it would require a prohibitively large detector. However, the diffusive regime can be reached without violating the small-angle approximation for sufficiently small-scale fields due to the opposite scalings of $\unicode[STIX]{x1D707}$ and the typical deflection angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}$ with correlation length $l_{B}$ : $\unicode[STIX]{x1D707}\propto l_{B}^{-1/2}$ , while $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\propto l_{B}^{1/2}$ .

In the diffusive regime, image-flux variations are typically much less pronounced in magnitude than seen in the caustic regime (figure 3 g, central part of image). Qualitatively, this is due to chaotic tangling of proton trajectories from spatially uncorrelated regions of the imaged field (figure 3 h). In a technical sense, a proton-flux image in the diffusive regime is still filled with caustics, the outlines of which are still visible in the central part of figure 3(g); however, since even pairs of protons with close initial positions often end up at disparate spatial locations, the proportion of image flux concentrated into an individual caustic drops as $\unicode[STIX]{x1D707}$ increases.

An alternative interpretation of the proton-flux image in the diffusive regime can be given in terms of diffusive-type models. As first shown by Dolginov & Toptygin (Reference Dolginov and Toptygin1967), the diffusion of a fast proton beam through compact, isotropic stochastic magnetic fields with correlation length $l_{B}\ll l_{z},l_{\bot }$ can be described by perpendicular velocity-diffusion coefficient

(3.23) $$\begin{eqnarray}D_{w}=\frac{Ve^{2}B_{\text{rms}}^{2}l_{B}}{4m_{p}^{2}c^{2}}.\end{eqnarray}$$

The result is valid in the small-deflections limit $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\ll 1$ , which in turn necessitates that the proton Larmor radius $\unicode[STIX]{x1D70C}_{p}$ associated with the typical RMS magnetic-field strength $B_{\text{rms}}$ satisfy $\unicode[STIX]{x1D70C}_{p}\gg l_{B}$ :

(3.24) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70C}_{p}}{l_{B}}=\frac{mcV}{eBl_{B}}\sim \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}^{-1}\sqrt{\frac{l_{i}}{l_{B}}}\gg 1.\end{eqnarray}$$

For the reader’s convenience, § O.1 presents an alternative derivation of diffusion coefficient (3.23).

The uncertainty in perpendicular velocity $\unicode[STIX]{x0394}w$ acquired due to diffusion leads to a ‘smearing effect’: under a diffusive model with coefficient $D_{w}$ given by (3.23), the image-flux distribution is related to the (here non-uniform) initial distribution by convolution

(3.25) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})=\frac{1}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{2}}\int \text{d}^{2}\tilde{\boldsymbol{x}}_{\bot }^{(s)}\,\unicode[STIX]{x1D6F9}_{0}^{(s)}(\tilde{\boldsymbol{x}}_{\bot }^{(s)})\exp \left[-\left(\frac{\boldsymbol{x}_{\bot }^{(s)}-\tilde{\boldsymbol{x}}_{\bot }^{(s)}}{\unicode[STIX]{x1D6FF}}\right)^{2}\right],\end{eqnarray}$$

where

(3.26) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=r_{s}\frac{\unicode[STIX]{x0394}w}{V}=\frac{eB_{\text{rms}}r_{s}}{m_{p}cV}\sqrt{l_{z}l_{B}}.\end{eqnarray}$$

This result is derived in § O.2. We see that a diffusive model of proton-flux images predicts that an initially uniform flux distribution will not display relative image-flux deviations if the stochastic magnetic field being imaged has a uniform envelope. For a magnetic field with a spatially varying envelope, such as the Golitsyn field shown in figure 2, (3.25) instead indicates that different sections of the initial flux distribution will be subject to different diffusion rates. For the Gaussian envelope used to create the stochastic field shown in figure 2, this results in the central region of the proton image having a reduced image flux when compared to the mean initial image flux; the central region is surrounded by a ring of greater-than-average image flux (illustrated in figure 4 b). Comparing the diffusive prediction of the relative image-flux distribution to the actual relative image-flux distribution shown figure 4(a), we see that the diffusive model fails to capture many features of the true proton-flux image: in particular, caustic structures. However, for stochastic fields with smaller-scale structures than the Golitsyn field, the diffusive model is more accurate, and caustic structures are suppressed (two examples are provided in § O.3).

Figure 4.

Comparison of relative flux image predicted by diffusive model compared with actual result in diffusive regime. (a) The 3.3 MeV proton image-flux distribution associated with magnetic field described in figure 2, with $B_{\text{rms},0}=750~\text{kG}$ . Here, relative flux refers to $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})/\unicode[STIX]{x1D6F9}_{0}^{(s)}$ , where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})\equiv \unicode[STIX]{x1D6F9}(\boldsymbol{x}_{\bot }^{(s)})-\unicode[STIX]{x1D6F9}_{0}^{(s)}$ is the flux deviation and $\unicode[STIX]{x1D6F9}_{0}^{(s)}$ the mean image flux. The same imaging parameters as those described in figure 4 are used (see figure 4 g for normalised proton-flux image). (b) Predicted relative proton-flux image assuming uniform initial flux, and diffusive model of imaging-beam evolution (3.25), with $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}(\boldsymbol{x}_{\bot })=\unicode[STIX]{x1D6FF}_{0}\exp [-4\unicode[STIX]{x1D70E}\boldsymbol{x}_{\bot }^{2}/l_{i}^{2}]$ . Here, $\unicode[STIX]{x1D6FF}_{0}$ is given by (3.26), with $B_{\text{rms}}=750~\text{kG}$ and $l_{i}=l_{z}=l_{\bot }=0.1~\text{cm}$ .

The observation of decreased intensity of image-flux deviations relative to the mean image flux is not sufficient to identify uniquely the diffusive regime – the same observation also holds for the linear regime (as discussed in § 3.1). Operationally, the diffusive regime can usually be distinguished by noting that unless fields have an extremely small spatial correlation scale $l_{B}$ , deflections of protons will be large enough to result in net loss of proton flux from an image. This loss of flux can be measured if the initial flux is known. Another, more conclusive test identifying the diffusive regime – requiring practical modification to experimental platforms – is to introduce a partial obstruction into the path of the imaging proton beam. This could include a sharp edge, a pinhole or some type of grid. In the diffusive regime, the edges associated with any of these features will appear blurred by diffusive scattering of protons due to the stochastic fields.

Quantitative analysis of proton images in the diffusive regime is much more restricted in scope that in other regimes. Whilst the plasma-image mapping (2.7) is still formally valid, in the diffusive regime, the perpendicular-deflection field is an extremely complicated object, and is not recoverable from an individual proton-flux image (for the same reason as described for the caustic regime for § 3.3). That being said, statistical information pertaining to the magnetic field can often be extracted from proton-flux images. More specifically, for sufficiently small-scale fields the diffusion coefficient $D_{w}$ (3.23) can be measured, which is turn gives an estimate of $B_{\text{rms}}^{2}l_{B}$ . This is most simply done by using one of the partial-obstruction methods mentioned above, and calculating the extent of blurring effects. It is also conceivable that numerical extraction of $D_{w}$ could be undertaken, assuming a model for the image-flux distribution of the form (3.25), and suitable boundary conditions. How this is done more precisely – and when this is a well-posed mathematical problem – are open questions which we do not answer here.

There is an extensive literature discussing diffusion due to stochastic magnetic fields in the context of cosmic rays (Parker Reference Parker1965; Dolginov & Toptygin Reference Dolginov and Toptygin1967; Hall & Sturrock Reference Hall and Sturrock1967; Jokipii Reference Jokipii1972; Blandford & Eichler Reference Blandford and Eichler1987); however, the authors of this paper are not aware of any discussion directly concerned with the diffusive scattering of a proton-imaging beam. The key positive result of this section, then, is to have derived a relation (3.25) between the image-flux distribution and the diffusive scattering of the proton beam by a stochastic magnetic field.

3.5 Numerical demonstration of field-reconstruction algorithm

To conclude this characterisation of the four contrast regimes, we illustrate the efficacy (or lack thereof) of the proposed field-reconstruction algorithm for recovering the path-integrated field numerically in each contrast regime. More specifically, we apply the field-reconstruction algorithm outlined in § P.4 (the PROBLEM code) to the proton images presented in figure 3. The reconstructed path-integrated fields normalised to the RMS of the actual magnetic field in each case are shown in figure 5; these are compared with the true path-integrated field, shown in figure 2(b). Figure 5 also presents a calculation of magnetic-energy spectrum determined using spectral relation (2.11) (blue circles) applied to the reconstructed perpendicular-deflection field in each case. For comparison, the true spectra (red line) of the Golitsyn fields are shown, along with the results of linear-regime flux spectral relation (3.15) applied directly to each image-flux distribution (purple circles).

Figure 5.

Efficacy of techniques for reconstructing magnetic field statistics directly from proton-flux images across contrast regimes. For each proton-flux image shown in figure 3, a reconstructed path-integrated magnetic field was produced by applying the field-reconstruction algorithm (see § P.4) appropriate for inverting the Monge–Ampère equation (3.19) with boundary conditions (3.20), before recovering the perpendicular-deflection field using (3.21), and finally the path-integrated field by (2.10). The true field is shown in figure 2(b). The magnetic-energy spectrum (true result shown in red) was predicted in two ways: using spectral relation (2.11) applied to the reconstructed perpendicular-deflection field (blue), and linear-regime spectral relation (3.15) applied directly to the proton-flux images (purple). (a) Path-integrated magnetic field reconstructed from linear-regime proton-flux image ( $\unicode[STIX]{x1D707}\ll 1$ ), figure 3(a). (b) Predicted magnetic energy spectra, derived from figures 3(a) and 5(a). (c) Same as (a), but for nonlinear-injective-regime proton-flux image ( $\unicode[STIX]{x1D707}<\unicode[STIX]{x1D707}_{c}$ ), figure 3(c). (d) Same as (b), but spectra derived from figures 3(c) and 5(c). (e) Same as (a), but for caustic-regime proton-flux image ( $\unicode[STIX]{x1D707}\geqslant \unicode[STIX]{x1D707}_{c}$ ), figure 3(e). (f) Same as (b), but for figures 3(e) and 5(e). (g) Same as (a), but for diffusive-regime proton-flux image ( $\unicode[STIX]{x1D707}\gtrsim 1/\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\gg 1$ ), figure 3(f). (h) Same as (b), but for figures 3(f) and 5(f).

The possibility of successful reconstruction of the path-integrated field in the linear regime (from the proton image shown in figure 3 a) is illustrated in figure 5(a); we see strong agreement with figure 2(b) in terms of both field morphology and strength in the central region. The recovery of the magnetic-energy spectrum from the same proton-flux image is illustrated in figure 5(b): both the linear-regime flux spectral relation (3.15) and deflection-field spectral relation (2.11) recover the correct power law at the energetically dominant wavenumbers. However, at higher wavenumbers, a flattening of both predicted spectra is observed. This effect is likely due to Poisson noise, and is discussed further in § 4.3.

The result of the field-reconstruction algorithm in the nonlinear injective regime – that is, the algorithm is applied to the image-flux distribution shown in figure 3(c) – is shown in figure 5(b). Similarly to the results for the linear regime, the predicted path-integrated magnetic field is a close match to the true field in terms of both its field strength and direction. Figure 5(d) shows the predicted magnetic-energy spectrum using deflection-field spectral relation (2.11); the true spectrum is recovered over a wide range of wavenumbers. We note that despite the Poisson noise being the same in proton-flux images figure 3(a,c), the spectrum predicted from the latter is less distorted at high wavenumbers. This is because the magnitude of deviations in the image flux due to magnetic fields are larger in the latter case, so the relative effect of Poisson noise is reduced.

In contrast, the predicted magnetic-energy spectrum from linear-regime flux spectral relation (3.15) does not follow the expected $k^{-11/3}$ power law, instead moving towards a $k^{-2}$ law. This distorted power law is obtained because the strong, narrow image-flux structures that appear irrespective of the underlying magnetic-energy spectrum have a characteristic ‘sharp-edge’ spectrum (Arévalo et al. Reference Arévalo, Churazov, Zhuravleva, Hernández-Monteagudo and Revnivtsev2012). We conclude that direct application to images involving order-unity relative image-flux variations of linear-regime flux spectral relation (3.15) can lead to misleading results.

Figure 5(e) shows the results of the field-reconstruction algorithm described in § 3.2 applied to the proton-flux image figure 3(e) (an image containing caustics). While the streamlines of the path-integrated perpendicular magnetic field retain a reasonable agreement, the field strength distribution does not: the typical magnitude is reduced. The mapping recovered by the field-reconstruction algorithm is instead the unique injective mapping satisfying Monge–Ampère equation (3.19) given the image-flux distribution shown in figure 3(e).

Figure 5(f) demonstrates that the magnetic-energy spectrum predicted by spectral relation (2.11) applied to the reconstructed perpendicular-deflection field is distorted to a $k^{-2}$ spectrum, much as the magnetic-energy spectrum predicted by linear-regime flux spectral relation (3.15) is distorted as nonlinearity of the plasma-image mapping becomes important.

Finally, attempts to reconstruct the path-integrated field in the diffusive regime lead to extremely inaccurate results. This failure is illustrated by the path-integrated field figure 5(g) reconstructed from the image-flux distribution shown in figure 5(f). Predicted field strengths are orders of magnitude lower than the true values, and the recovered field’s morphology resembles that of a regular field rather than of the actual stochastic one. This inaccuracy is also replicated in the predicted magnetic-energy spectrum shown in figure 3(h): the Golitsyn spectrum is again distorted to a $k^{-2}$ power law due to the caustic ring-like structure seen in figure 3(g), and the predicted spectrum at the energetically dominant wavenumbers is strongly suppressed.

A simple quantitative way to compare the quality of the predicted reconstructions for different contrast regimes is to calculate the predicted RMS magnetic-field strength from the recovered magnetic-energy spectra using (2.3). The results are shown in table 1. It is clear that the predicted values for $B_{\text{rms}}$ are close to the actual ones in the linear and nonlinear injective regimes, but $B_{\text{rms}}$ is under-predicted in the caustic and diffusive regimes. That the typical field strengths predicted by the field-reconstruction algorithm are reduced is a manifestation of the general analytical result that the reconstructed perpendicular-deflection field will always provide a lower bound on the RMS deflection-field strength (Gangbo & McCann Reference Gangbo and McCann1996), which can in turn be used to provide a lower bound on the RMS magnetic-field strength. Explicitly, it can be shown (see appendix N) that

(3.27) $$\begin{eqnarray}B_{\text{rms}}^{2}\geqslant \frac{m_{p}^{2}c^{2}}{e^{2}l_{z}^{2}}\langle (\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D711})^{2}\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{\bot 0}\unicode[STIX]{x1D711}$ is the deflection-field potential recovered from the solution to Monge–Ampère equation (3.19). Furthermore, in appendix N, it is demonstrated for the Golitsyn test field outlined in figure 2 that this lower bound property becomes very weak as $\unicode[STIX]{x1D707}$ increases further.

Table 1.

Comparison with true result of RMS magnetic-field strength predicted using magnetic-energy spectrum derived from (2.11) in different contrast regimes.

In short, this numerical example validates the claim that for proton-flux images of stochastic magnetic fields created in the linear and nonlinear injective regimes, the true path-integrated field is extractable using an appropriate field-reconstruction algorithm. Furthermore, the predicted magnetic-energy spectrum agrees well with the true result. Neither of these results are true in the caustic and diffusive regimes.

4.1 Assumptions

As mentioned in § 2.2, the applicability of plasma-image mapping (2.7) and Kugland image-flux relation (2.9) to the proton-imaging set-up depends on various assumptions: a mono-energetic, instantaneous, uniform beam from a point source, paraxialilty, point projection and small deflections. Here, we state each of these precisely, and explore their validity. In addition to the effects stated below, in deriving the stated plasma-image mapping, a range of physical processes are neglected in line with previous work on analytic models of proton imaging (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012). These are discussed in appendix B.

Instantaneous

The instanteneity of the imaging process is the result of the temporal pulse length of proton sources $t_{\text{pulse}}$ and the transit time of protons across the plasma $t_{\text{path}}$ being small compared to the evolution time of the magnetic-field configuration of interest. For fusion-produced protons, $t_{\text{pulse}}\sim 100~\text{ps}$  (Séguin et al. Reference Séguin, DeCiantis, Frenje, Kurebayashi, Li, Rygg, Chen, Berube, Schwartz and Petrasso2004), while for TSNA-produced protons, $t_{\text{pulse}}\sim 10~\text{ps}$  (Wilks et al. Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001). For both, the transit time across the plasma can be estimated as

(4.1) $$\begin{eqnarray}t_{\text{path}}\sim 40\left[\frac{l_{z}(\text{cm})}{0.1~\text{cm}}\right]\left[\frac{W(\text{MeV})}{3.3\;\text{MeV}}\right]\text{ps}.\end{eqnarray}$$

Thus, provided the plasma dynamics to be studied is on nano-second time scales, this approximation of instantaneity is a good one. This is indeed the case for many relevant experiments involving stochastic magnetic fields (Tzeferacos et al. Reference Tzeferacos, Rigby, Bott, Bell, Bingham, Casner, Cattaneo, Churazov, Emig and Fiuza2017a ,Reference Tzeferacos, Rigby, Bott, Bell, Bingham, Casner, Cattaneo, Churazov, Emig and Fiuza b ), though by no means all (for example, see Mondal et al. Reference Mondal, Narayanan, Ding, Lad, Hao, Ahmad, Wang, Sheng, Sengupta and Kaw2012). For time-varying fields, the effect of inductive electric fields is likely significant – which in turn leads to the invalidation of expression (2.8) for the perpendicular-deflection field. In addition, magnetic fields varying on time scales shorter than $t_{\text{pulse}}$ would lead to the front part of the proton beam seeing different magnetic fields to the back part of the beam; the resulting proton image would then be the superposition of the proton images for both fields.

4.1.2 Paraxality

If the distance from the proton source to the plasma is much greater than the dimensions of the plasma, $r_{i}\gg l_{\bot },l_{z}$ , we can approximate the section of the beam passing through the plasma as planar, despite the fact that proton beams generated by fusion reactions in a $\text{D}_{2}$ capsule implosion generally take the form of a uniformly expanding spherical shell (Li et al. Reference Li, Séguin, Frenje, Rygg, Petrasso, Town, Amendt, Hatchett, Landen and MacKinnon2006). For a given proton, this paraxial approximation is effectively an expansion of the position and velocity of the proton in terms of half of the paraxial parameter

(4.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}}{2}=\frac{l_{\bot }}{2r_{i}}\ll 1,\end{eqnarray}$$

the ratio of the size of the region being imaged to the radius of curvature of the beam. The factor of two appears, because the approximating plane is chosen to be exact at the centre of the beam. For typical experimental set-ups, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\approx 0.05{-}0.2\ll 1$  (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012).

Increasing the paraxial parameter $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ does not significantly change the nature of the four contrast regimes described in § 3 qualitatively, although for sufficiently large values a slight decay in proton flux towards the edges of images may be detectable. For quantitative analysis, higher-order corrections in $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ to the plasma-image mapping (2.7) can be introduced (Kugland et al. Reference Kugland, Ryutov, Plechaty, Ross and Park2012).

4.1.3 Point projection

The point-projection assumption requires that the distance $r_{s}$ from the plasma to the detector be much greater than $l_{z}$ . This means that displacements of protons from their undeflected trajectories acquired inside the plasma due to magnetic forces are negligible compared to those displacements of protons resulting from their free-streaming motion beyond the plasma with an altered deflection velocity. Mathematically, the plasma-image mapping (2.7) is effectively an expansion in

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}\equiv \frac{l_{z}}{r_{s}}\ll 1,\end{eqnarray}$$

retaining only the leading-order term in $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}$ . In actual proton-imaging set-ups as typically implemented, $l_{z}\ll r_{i}\ll r_{s}$ , so $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FD}\ll 1$ .

4.1.4 Small proton-deflection angles

To derive the expression (2.8) for the perpendicular-deflection field, the size of typical proton-velocity deflections are assumed small compared to the initial proton velocity. Since (2.8) shows that deflection velocities are perpendicular to the initial direction, that is, that $\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{v}_{0}\approx \boldsymbol{w}\boldsymbol{\cdot }\hat{\boldsymbol{z}}\approx 0$ , the deflection angle of the proton due to magnetic forces must therefore be small:

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\equiv \cos ^{-1}\left[\frac{\boldsymbol{v}_{0}\boldsymbol{\cdot }(\boldsymbol{v}_{0}+\boldsymbol{w})}{|\boldsymbol{v}_{0}||\boldsymbol{v}_{0}+\boldsymbol{w}|}\right]\approx \frac{|\boldsymbol{w}|}{V}\ll 1.\end{eqnarray}$$

The magnitude of this parameter can be practically estimated for stochastic fields imaged by protons with energy $W$ using deflection angle RMS (2.13):

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\sim \frac{eB}{m_{p}cV}\sqrt{l_{z}l_{B}}\approx \left[\frac{B(\text{MG})}{2.6\;\text{MG}}\right]\left[\frac{W(\text{MeV})}{3.3\;\text{MeV}}\right]^{-1/2}\left[\frac{l_{z}(\text{cm})l_{B}(\text{cm})}{0.01\;\text{cm}^{2}}\right]^{1/2}.\end{eqnarray}$$

The same estimate can also be used to determine whether $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\lesssim l_{B}/l_{z}$ , and hence whether the perpendicular-deflection field is irrotational. Since $l_{B}<l_{z}$ , this condition is typically more stringent than $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\ll 1$ .

It is clear that if deflections are not small, $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}\sim 1$ inevitably implies that the system is in the diffusive regime $\unicode[STIX]{x1D707}\sim r_{i}/l_{B}\gg 1$ . This has the consequence that, despite the technical lack of validity of the derivation of plasma-image mapping (2.7) in this case, proton-flux images in the large-deflection regime are likely to have many qualitatively similar features to diffusive proton-flux images – albeit combined with a significant loss of total flux from the detector.

4.2 Theoretical complications

We have claimed in the previous sections that the imaging of stochastic magnetic fields can be classified into four general regimes depending on $\unicode[STIX]{x1D707}$ , and that in two of these regimes (linear and nonlinear injective), the path-integrated field can be extracted directly from proton-flux images; under the further assumption of isotropy, this is sufficient to determine the magnetic-energy spectrum of the stochastic field. Whilst these statements are true for many stochastic fields, there are others for which the efficacy of the proposed analysis techniques must be re-considered: these are discussed in this section. Note that we distinguish between inherent constraints on the technique, which are unavoidable, and constraints due to experimental effects, which can in principle be overcome with advances in diagnostic implementation. The former are discussed in this section, the latter in § 4.3.

4.2.1 Spatially inhomogeneous and anisotropic fields

The classification into four contrast regimes that can be distinguished simply by particular characteristics of proton-flux images holds for locally homogeneous stochastic fields. However, this classification is not strictly valid for more general fields with inhomogeneous or anisotropic statistics. In particular, for an arbitrary proton-flux image, the contrast regime associated with that image cannot be uniquely identified. This is related to the non-uniqueness of the inversion problem associated with recovering path-integrated magnetic fields associated with a particular proton-flux image – introduced in § 2.3 and discussed in §§ 3.3 and 3.4. We further illustrate this phenomenon with an example consisting of two strikingly distinct path-integrated fields (incorporating different contrast regimes) giving near-identical image-flux distributions (figure 6). Despite the mathematical impossibility of distinguishing between the path-integrated fields in figure 6(a,b) without additional information, in practice there are various qualitative tests that could in principle be employed. For example, if there are multiple species of imaging protons, and the transit time and pulse length of all these species are much smaller than the evolution time scales of the system, the multiple-beam-energy comparison technique described in § 4.3.3 to find caustics could also be used to distinguish between these path-integrated magnetic fields. More specifically, the path-integrated field could be reconstructed from a 3.3 MeV proton image (such as figure 6 c), before predicting what a 15.0 MeV proton image would look like. This could be compared to the actual 15.0 MeV proton image to test the veracity of the reconstructed path-integrated field.

Figure 6.

Illustration of ill posedness of reconstruction for spatially inhomogeneous stochastic magnetic fields. (a) Path-integrated perpendicular magnetic field associated with small-scale magnetic cocoon configuration (see § J.3) with correlation scale $l_{B}=0.8~\unicode[STIX]{x03BC}\text{m}$ . A Gaussian envelope of the form (3.2) is applied to the magnetic field, with $\unicode[STIX]{x1D70E}=3$ and $B_{\text{rms},0}=2.2~\text{MG}$ . (b) Path-integrated magnetic field associated with perpendicular-deflection field reconstructed from proton-flux image (c) generated from magnetic field described in (a), assuming an injective mapping. Typical proton deflection velocities in the central region are reduced by a factor ${\sim}25$ . (c) 3.3-MeV proton-flux image created by imaging field configuration (a) with $3\times 10^{6}$ 15.0 MeV protons from point source located at $r_{i}=1~\text{cm}$ from the field configuration, and detector located on the opposing side, with $r_{s}=30~\text{cm}$ . (d) Predicted 3.3-MeV proton-flux image, assuming proton beam with same imaging parameters as (c) experienced path-integrated magnetic field (b) while traversing the plasma.

Even in the situation when the path-integrated field is uniquely extractable from its associated proton-flux image (implying that the contrast regime of the imaging diagnostic must be linear, or nonlinear injective), for a spatially inhomogeneous field, the magnetic-energy spectrum of the stochastic field is not necessarily extractable from the path-integrated field. More specifically, the derivations of equations for the magnetic-energy spectrum (deflection-field spectral relation (2.11), or linear-regime flux spectral relation (3.15)) given in appendices F and I rely on both the assumption that magnetic field statistics do not vary along the path of the proton beam, and that the statistics are the same perpendicular and parallel to the direction of proton motion. These assumptions can likely be relaxed to include the case when the field is statistically inhomogeneous, and anisotropic in the perpendicular direction (Ensslin & Vogt (Reference Ensslin and Vogt2003) demonstrate this in the context of calculating magnetic-energy spectra from Faraday rotation measurements).

4.2.2 Variable $\unicode[STIX]{x1D707}$ across scales

It is clear from its definition (2.17) that $\unicode[STIX]{x1D707}$ is scale dependent, increasing as the scale $l_{B}$ of magnetic structures decreases relative to the path length of the imaging protons. Magnetic structures imaged in a proton-imaging set-up are, therefore, in a potentially different contrast regime depending on their strength and size. For multi-scale stochastic fields, such as those with power-law spectra of the form $E_{B}(k)\propto k^{-p}$ , we can estimate $\unicode[STIX]{x1D707}$ at a particular scale $l_{B}$ . The field strength at this scale for such a power law goes as

(4.6) $$\begin{eqnarray}B\sim B_{0}\left(\frac{l_{B}}{l_{z}}\right)^{(p-1)/2},\end{eqnarray}$$

and so $\unicode[STIX]{x1D707}$ is given by

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D707}\sim \unicode[STIX]{x1D707}_{0}\left(\frac{l_{B}}{l_{z}}\right)^{p/2-1}.\end{eqnarray}$$

We see that for $p>2$ , $\unicode[STIX]{x1D707}$ decreases with scale, whereas the opposite is true for $p<2$ . We discuss the consequences of both possibilities in turn.

The case of sufficiently steep power spectra ( $p>2$ ) – of which the Golitsyn spectrum (3.1) with $p=-11/3$ is an example – is generally much easier to investigate. The largest $\unicode[STIX]{x1D707}$ is at largest scales, which coincides with both the strongest magnetic and path-integrated structures. Whether the path-integrated field and magnetic-energy spectrum can be extracted successfully is therefore not altered by the presence of smaller-scale fields.

On the contrary, for shallow spectra ( $p<2$ ), small-scale fields become extremely important for a proper understanding of proton-flux images, as well as their analysis. Firstly, the fact that $\unicode[STIX]{x1D707}$ increases with decreasing scale (in opposition to the path-integrated field) means that the dominant features in a proton image may be at much smaller scales than the dominant features in the path-integrated field. We illustrate this with an example. Figure 7 shows a typical path-integrated magnetic field resulting from a $k^{-1}$ spectrum with a wavenumber range of a few decades. Creating a proton-flux image from this path-integrated field using the plasma-image mapping (as described in § P.3), we see that the strongest features are at the smallest scales, in contrast to the path-integrated magnetic field, where larger coherent structures are evident. This can be explained physically with recourse to an interpretation of proton-flux images as projections of path-integrated MHD current structures: for shallow enough spectra, the dominant MHD current structures are at the smallest scales. For actual proton-flux images, this fine structure would typically be masked by finite-resolution effects; that being said, for stochastic field configurations, figure 7 provides another example of the inadequacy of using image-flux structure as a proxy for magnetic-field structure.

Figure 7.

Illustration of variation in $\unicode[STIX]{x1D707}$ across scales, with consequent discrepancy in structure size between path-integrated magnetic field, and proton-flux image. (a) Predicted path-integrated field for periodic stochastic Gaussian magnetic field specified on $1001^{2}$ array of spatial extent $l_{i}=l_{\bot }=l_{z}=0.1~\text{cm}$ , with power-law spectrum of the form (4.8), spectral cutoffs $k_{l}=4\unicode[STIX]{x03C0}/l_{i}$ , $k_{u}=800\unicode[STIX]{x03C0}/l_{i}$ and field RMS $B_{\text{rms}}=10~\text{kG}$ . (b) Proton-flux image generated with 3.3 MeV protons from a point source at distance $r_{i}=1~\text{cm}$ from the array, with imaging detector at distance $r_{s}=30~\text{cm}$ on the opposite side.

More quantitatively, the ability to reconstruct both path-integrated fields and magnetic-energy spectra is limited by diffusive scattering at small scales. If $\unicode[STIX]{x1D707}(l_{B})<\unicode[STIX]{x1D707}_{c}$ at all scales, for $\unicode[STIX]{x1D707}_{c}$ the critical value of $\unicode[STIX]{x1D707}$ at which the plasma-image mapping (2.7) loses injectivity, both can be recovered. However, this inequality being reversed at small scales can prevent successful reconstruction not only at those scales, but those at which injectivity is technically preserved. This is because large-scale structures are distorted by small-scale ones.

The phenomenon of smaller-scale fields preventing successful reconstruction of larger-scale ones is best illustrated by considering an example. Figure 8(a) shows a stochastic magnetic field with a magnetic-energy spectrum of the form

(4.8) $$\begin{eqnarray}E_{B}(k)=\frac{B_{\text{rms}}^{2}}{8\unicode[STIX]{x03C0}}\frac{1}{k\log k_{u}/k_{l}},\quad k\in [k_{l},k_{u}],\end{eqnarray}$$

combined with a Gaussian envelope of the form (3.2). The wavenumber range is chosen to be large – that is, $k_{l}\ll k_{u}$ – and field strength such that the proton-imaging set-up applied to the largest structures has $\unicode[STIX]{x1D707}<\unicode[STIX]{x1D707}_{c}$ . The proton-flux image (figure 8 b) has strong image-flux structures on many scales, with larger image-flux structures somewhat dispersed by smaller, caustic ones. Figure 8(c), shows the same field, but with high wavenumbers filtered out at a scale $k_{c}$ . The resulting proton-flux image (figure 8 d) now falls into the nonlinear injective regime, indicating that $\unicode[STIX]{x1D707}$ at larger length scales is indeed below the critical value. Figure 8(e) shows the path-integrated magnetic field reconstructed from figure 8(b) using the field reconstruction algorithm described in § 3.2. While the reconstruction still captures the global morphology of the field, its fine structure is altered. This difference is shown more clearly by calculating the magnetic-energy spectra, shown in figure 8(f). The magnetic-energy spectrum resulting from spectral relation (2.11) applied the reconstructed field is suppressed at high wavenumbers. Indeed, the effect is sufficiently strong to lead to spectral distortion at wavenumbers which would otherwise be correctly reconstructed if wavenumbers $k>k_{c}$ were removed. In contrast to the reconstruction of steeper power spectra, the magnetic-energy spectrum calculated from linear theory using formula (3.15) applied to the proton image shown in figure 8(b) is closer to the true result, although the associated spectral curve is still distorted.

Figure 8.

Analysing proton-flux images of stochastic magnetic fields with broad magnetic-energy spectra. A stochastic magnetic field sample, with magnetic-energy spectrum of the form (4.8), is generated in a region, with dimensions $l_{i}=l_{z}=l_{\bot }=0.1~\text{cm}$ . Field values are specified on a $301^{3}$ array (grid spacing $\unicode[STIX]{x1D6FF}x=l_{i}/301$ ). The spectral cutoffs were set to be $k_{l}=6\unicode[STIX]{x03C0}/l_{i}$ , $k_{u}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FF}x$ , and $B_{\text{rms},0}=40~\text{kG}.$ A Gaussian envelope of the form (3.2) is again applied to the magnetic field, with $\unicode[STIX]{x1D70E}=3$ . Imaging of the field is then implemented using the parameters given in figure 4. (a) Path-integrated perpendicular magnetic field as experienced by imaging 3.3 MeV protons. (b) The 3.3 MeV proton-flux image. (c) Path-integrated perpendicular magnetic field, with magnetic-field wavemodes whose wavenumber exceeds $k_{c}=30\unicode[STIX]{x03C0}/l_{i}$ removed. (d) The 3.3 MeV proton-flux image of filtered magnetic field. (e) Predicted path-integrated magnetic field found by applying field-reconstruction algorithm. (f) Magnetic-energy spectra: true result (red), plotted with prediction using deflection-field spectral relation (2.11) (blue) and linear-regime spectral relation (3.15) (purple).

In short, for shallow magnetic-energy spectra, the presence of small-scale fields leads to the suppression of small-scale image-flux structures, which in turn hides high wavenumber modes when attempting to reconstruct both path-integrated fields and magnetic-energy spectra.

4.3 Experimental complications

In addition to the theoretical restrictions placed on the use of a proton-imaging diagnostic to assess magnetic-field statistics, additional constraints arise from limitations to the implementation of the diagnostic. These are discussed more generally elsewhere, but here we outline three effects which are particularly important when attempting to extract accurately path-integrated magnetic fields and energy spectra directly from proton images: Poisson noise, initial inhomogeneities in the initial flux distribution and finite image resolution effects imposed by a finite source size.

4.3.1 Poisson noise

Poisson noise is an unavoidable effect associated with proton-flux images involving a finite number of imaging protons, and leads to spectral distortion. The finite number of protons in the imaging beam mean that locally the initial flux distribution is noisy. Since the sample is very large, these deviations are well described by Poisson statistics. Thus, if the mean flux per pixel is $N_{\unicode[STIX]{x1D6F9}}$ , then the standard deviation of fluctuations is $\sqrt{N_{\unicode[STIX]{x1D6F9}}}$ , giving signal-to-noise ratio $1/\sqrt{N_{\unicode[STIX]{x1D6F9}}}$ .

Since such noise is uncorrelated, its contribution to the two-dimensional relative image-flux spectrum is uniform. More specifically, the requirement that the integrated power spectrum of Poisson noise has variance $1/N_{\unicode[STIX]{x1D6F9}}$ leads to

(4.9) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D702}}^{(\text{PN})}\approx \left(\frac{\unicode[STIX]{x1D6FF}x}{2\unicode[STIX]{x03C0}}\right)^{2}\frac{1}{N_{\unicode[STIX]{x1D6F9}}},\end{eqnarray}$$

for $\unicode[STIX]{x1D6FF}x$ the pixel size in length units. This in turn gives predicted magnetic-energy spectrum

(4.10) $$\begin{eqnarray}E_{B}^{(\text{PN})}(k)\approx \frac{1}{(2\unicode[STIX]{x03C0})^{3}}\frac{m_{p}^{2}c^{2}V^{2}}{e^{2}r_{s}^{2}l_{z}}\unicode[STIX]{x1D6FF}x^{2}\frac{1}{N_{\unicode[STIX]{x1D6F9}}}.\end{eqnarray}$$

Thus for a multi-scale spectrum, values of the true spectrum will be dominated by the Poisson contribution if too small. This presents a potential difficulty for analysing proton-flux images in the linear regime if either the mean image flux is low, or the physical signal is weak. The problem is evident for the predicted spectra shown in figure 5(b) (purple markers): the shallowing of the spectrum as predicted by linear-regime flux spectral relation (3.15) is due to Poisson noise.

However, in practice the Poisson noise does not usually prevent successful analysis of proton-flux images. The effect on Poisson noise on the spectrum can be anticipated using (4.10), and subsequently removed (for an example in the context of X-ray image analysis, see Churazov et al. Reference Churazov, Vikhlinin, Zhuravleva, Schekochihin, Parrish, Sunyaev, Forman, Böhringer and Randall2012). Furthermore, nonlinear field reconstruction algorithms tend to reduce spectral distortion (figure 5 b, blue markers). Thirdly, the restrictions on pixel size required to manage Poisson noise are usually less stringent that those already in place due to other factors limiting resolution (Séguin et al. Reference Séguin, DeCiantis, Frenje, Kurebayashi, Li, Rygg, Chen, Berube, Schwartz and Petrasso2004). That being said, if possible it is preferable to design proton-imaging set-ups in such a way that the magnitude of Poisson noise be significantly less than image-flux features due to magnetic fields.

4.3.2 Inhomogeneity of initial flux distribution

The methods used to reconstruct magnetic-field statistics directly from proton-flux images described in §§ 3.1 and 3.2 rely on prior knowledge of the initial flux distribution. For experimental implementations of proton imaging, this initial distribution is usually assumed to be uniform. However, as discussed in in § 4.1, at the present time it is difficult in practice to achieve a completely uniform initial flux distribution. In the absence of alternative methods for determining the initial flux, there is consequently significant uncertainty over this quantity.

Unfortunately, the ramifications of an unknown initial flux distribution for successful extraction of magnetic-field statistics cannot be disregarded: in general, path-integrated fields are no longer uniquely determined. Conceptually this is because it becomes impossible to distinguish between variations in the image flux resulting from magnetic deflections, and those due to variations in the initial flux. We illustrate this phenomenon with a numerical example – figure 9. Taking two distinct path-integrated fields (figure 9 c,d), but also choosing two distinct initial flux distributions (figure 9 a,b), it can be seen that it is possible to produce a near-identical image-flux distribution (figure 9 e,f). Naive assumption of a uniform initial flux distribution when applying a field-reconstruction algorithm can therefore lead to prediction of non-physical field structures: a far from satisfactory outcome.

Figure 9.

Effect of initial flux inhomogeneity on well posedness of field-reconstruction algorithms. (a) Non-uniform initial 3.3 MeV proton image-flux distribution, with $\unicode[STIX]{x1D6F9}_{0}(\boldsymbol{x}_{\bot 0})=\unicode[STIX]{x1D6F9}_{0}/4(5{-}6x_{\bot 0}^{2}/l_{i}^{2})$ . Imaging parameters of the set-up are the same as those described in figure 4. (b) Uniform initial flux distribution. (c) Path-integrated perpendicular magnetic field experienced by beam with initial flux distribution shown in (a). The magnetic field is the same as that described in figure 2, with $B_{\text{rms},0}=15~\text{kG}$ . (d) Path-integrated perpendicular magnetic field experienced by initial flux distribution shown in (b). The magnetic field is constructed by applying the field-reconstruction algorithm to proton-flux image (c). (e)The 3.3 MeV proton-flux image resulting from initial flux distribution (a) experiencing path-integrated field (c). (f) The 3.3 MeV proton-flux image resulting from initial flux distribution (b) experiencing path-integrated field (d).

The distorting effect of initial flux inhomogeneities, however, can to a certain extent be countered. Variations in initial flux are typically much smaller than the mean flux over solid angles ${\leqslant}1.1^{\circ }$  (Manuel et al. Reference Manuel, Zylstra, Rinderknecht, Casey, Rosenberg, Sinenian, Li, Frenje, Séguin and Petrasso2012). As a result, short-scale image-flux deviations due to deflections, particularly strong deviations from nonlinear focusing, are distinguishable from potential initial flux structures. Difficulties then only arise when considering weaker, long-scale flux variation (strong long-scale variation is inevitably associated with the appearance of focused narrow image-flux features). Therefore, to isolate genuine path-integrated magnetic field structures, we propose applying a high-pass filter to the image-flux distribution to remove long-scale variation. Field-reconstruction algorithms applied to the filtered image therefore give a reasonable estimate of fields below the scale of the filtering.

We do, however, note that long-scale path-integrated magnetic fields – if present – could in principle alter the positioning of short-scale image-flux structures, giving rise to the possibility that path-integrated fields reconstructed from filtered images are distorted compared to the true fields at that scale. The importance of this effect can be tested by comparing the results of the field reconstruction algorithm applied to the filtered image with the reconstructed path-integrated field deduced from the unfiltered image, but with filtering applied directly post-reconstruction. In any case, since long-scale variations in flux are typically not large, intuitively this effect should not be significant in practice.

4.3.3 Smearing effects due to finite source size

As mentioned in § 4.1, proton-imaging set-ups usually have a small but finite source size, placing a lower limit on the size of magnetic structures which can be imaged. This statement can be made more precise by calculating the effect of the finite source of the perpendicular velocity distribution of the beam. It can then be shown using kinetic theory (§ D.7) that when a magnetic-field configuration is imaged with a proton beam generated by a finite source, the resulting image-flux distribution $\tilde{\unicode[STIX]{x1D6F9}}(\boldsymbol{x}_{\bot }^{(s)})$ is equal to the image-flux distribution generated by a point source (the ‘unsmeared flux’ $\unicode[STIX]{x1D6F9}$ ), but convolved with a point-spread function:

(4.11) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6F9}}(\boldsymbol{x}_{\bot }^{(s)})=\int \text{d}^{2}\tilde{\boldsymbol{x}}_{\bot }^{(s)}\,\unicode[STIX]{x1D6F9}(\tilde{\boldsymbol{x}}_{\bot }^{(s)})\,S(\boldsymbol{x}_{\bot }^{(s)}-\tilde{\boldsymbol{x}}_{\bot }^{(s)}),\end{eqnarray}$$

where $S$ is given by

(4.12) $$\begin{eqnarray}S(\boldsymbol{x}_{\bot }^{(s)}-\tilde{\boldsymbol{x}}_{\bot }^{(s)})=\frac{V^{2}}{r_{s}^{2}}P\left(\frac{\tilde{\boldsymbol{x}}_{\bot }^{(s)}-\boldsymbol{x}_{\bot }^{(s)}}{r_{s}}V\right).\end{eqnarray}$$

Here, $P=P(\unicode[STIX]{x1D6FF}\boldsymbol{v}_{\bot 0})$ is the distribution function of perpendicular velocities associated with the source’s finite size; it is a function of

(4.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\boldsymbol{v}_{\bot 0}=\boldsymbol{v}_{\bot 0}-\frac{\boldsymbol{x}_{\bot 0}}{r_{i}}V,\end{eqnarray}$$

which quantifies the degree of deviation from the paraxial approximation for the initial beam-proton velocity. The form of $P$ depends on the three-dimensional emission profile of the proton source (a calculation for $P$ is the case of a uniformly emitting sphere of radius $a$ is presented in § D.7).

The limitation of image resolution by a finite source size places a number of constraints on the possibility of reconstructing the magnetic-energy spectrum. Firstly, as mentioned in § 3.3, smearing of fine image-flux structure can make the differentiation of the nonlinear injective and caustic regimes difficult. This is illustrated in figure 10. Figure 10(a,b) shows point-source proton-flux images of the same artificial test stochastic magnetic field introduced in figure 2 (with $B_{\text{rms},0}\approx 60~\text{kG}$ ), but created using two imaging proton species with different speeds: 3.3 MeV (figure 10 a) and 15.0 MeV (figure 10 b). Since $\unicode[STIX]{x1D707}$ is inversely proportional to the initial imaging proton speed, $\unicode[STIX]{x1D707}$ for the 15.0 MeV proton-flux image is lower – with the consequence that its plasma-image mapping is injective, unlike the 3.3 MeV proton plasma-image mapping. The two point-source proton-flux images are qualitatively distinct – pairs of caustics are evident in figure 10(a) – and hence can be identified as belonging to the caustic and nonlinear injective regimes respectively. Figure 10(c,d) shows the same fields imaged with protons generated from a finite spherical source. This reduces the spatial resolution of the image – and consequently, the sharp caustic structures disappear.

Figure 10.

Distinguishing the nonlinear injective and caustic contrast regimes using nonlinear field-reconstruction algorithms and multiple beam energies. (a) 3.3 MeV proton-flux image of magnetic field described in figure 2, with $B_{\text{rms},0}=64~\text{kG}$ , and same imaging parameters as those described in figure 3. (b) Same as (a), but with imaging beam composed of 15.0 MeV protons. (c) Same as (a), but now imaging implemented using a finite uniform spherical proton source with diameter $a=50~\unicode[STIX]{x03BC}\text{m}$ . (d) Same as (c), but with imaging beam composed of 15.0 MeV protons. (e) Reconstructed path-integrated magnetic field obtained by applying the field-reconstruction algorithm (described in § 3.2) to (c). (f) Same as (e), but algorithm applied to (d). (g) Predicted 3.3 MeV proton-flux image, assuming imaging proton beam experienced reconstructed path-integrated field shown in (f) as a result of traversing magnetic field (the implementation is discussed in § P.3). Under the caustic detection scheme outlined in the main text, this image is to be compared with (c). (h) Predicted 15.0 MeV proton-flux image, assuming imaging proton beam experienced path-integrated field shown in (e). This image is to be compared with (d).

To distinguish between the nonlinear injective and caustic regimes, we need more information than can be provided by a single-proton image. Fortunately, typical experimental methods for generating imaging proton beams produce at least two beam energies simultaneously, which can be imaged independently (Wilks et al. Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001; Li et al. Reference Li, Séguin, Frenje, Rygg, Petrasso, Town, Amendt, Hatchett, Landen and MacKinnon2006). If the transit and beam duration times of all proton beams are much shorter than the time scales on which dynamical evolution of the magnetic field occurs, then the proton-flux images from all beams will be of approximately the same magnetic field (see § 4.1 for a discussion of this).

Comparing proton images of the same magnetic field at different imaging beam energies is a useful technique for identifying caustics. Most simply, if proton images contain caustics, irrespective of the beam energy, then the width of strong, positive image-flux structures will typically increase with lowered beam energy; in contrast, if all images fall into the nonlinear injective regime, then the narrowest structures will occur at the lowest energies. This principle is illustrated in figure 10(c,d), where strong, positive relative image-flux features are broader for the 3.3 MeV protons than the 15.0 MeV ones – suggesting that the former contains smeared caustics.

A more robust caustic detection scheme comes from application of the field-reconstruction algorithm outlined for $\unicode[STIX]{x1D707}<\unicode[STIX]{x1D707}_{c}$ in § 3.2 on the different beam-energy proton-flux images of the same magnetic structures. If a path-integrated field reconstructed from one proton species is close to the true path-integrated field, then all imaging protons species should have seen that reconstructed field: thus the proton-flux images generated using the plasma-image mapping combined with the predicted path-integrated field should be consistent with the actual proton-flux images (this procedure is discussed in § P.3). Figure 10(g,h) shows the 3.3 MeV and 15.0 MeV proton-flux images predicted using the reconstructed perpendicular-deflection fields for 15.0 MeV (figure 10 f) and 3.3 MeV (figure 10 e) protons respectively, followed by artificial smearing with a point-spread function appropriate for a uniform finite spherical source (see appendix E). Figure 10(g) is very similar to figure 10(c), verifying the validity of the reconstructed perpendicular-deflection field shown in figure 10. However, figure 10(h) disagrees substantially with figure 10(d): for example, the width of positive image-flux structures in 10(h) is much greater than the true widths in figure 10(d). We conclude that the field-reconstruction algorithm provides an effective method for detecting the presence of caustics.

Smearing also has important consequences for accurately reconstructing both path-integrated fields and magnetic-energy spectra. More specifically, smearing reduces the strength of path-integrated fields reconstructed directly from proton-flux images by reducing the sharpness of nonlinearly focused image-flux features. It also introduces a upper wavenumber cutoff on predicted magnetic energy spectra. Both these effects are illustrated in figure 11 with a numerical example. Figure 11(a) shows a proton-flux image generated by imaging the Golitsyn field described in figure 2, along with the (near-accurate) reconstructed field (figure 11 b). Figure 11(c) shows the equivalent image-flux distribution generated by a finite source. The reconstructed morphology, while similar, is reduced in strength. The effect on spectral reconstruction in this example is more pronouced (figure 11 g): wavenumbers close to and beyond the smearing scale are suppressed in the spectrum derived from figure 11(d).

Figure 11.

Effect of finite source size on successful extraction of magnetic-field statistics. (a) ‘Unsmeared’ 3.3 MeV proton-flux image created by imaging Golitsyn field, $B_{\text{rms},0}=30~\text{kG}$ (reproduction of figure 4 c for convenience of reference). (b) Predicted path-integrated magnetic field resulting from application of field-reconstruction algorithm to figure (a). (c) ‘Smeared’ 3.3 MeV proton-flux image created by imaging same field as (a), but with a finite proton source, with diameter $a=50~\unicode[STIX]{x03BC}\text{m}$ . (d) Same as (b), but with field-reconstruction algorithm applied to (c). (e) ‘De-smeared’ 3.3 MeV proton image subsequent to application of Richardson–Lucy algorithm for ten iterations. (f) Same as (b), but with field-reconstruction algorithm applied to (e). (g) Magnetic-energy spectra: true result (red), plotted with predictions using deflection-field spectral relation (2.11) applied to (b) (blue), (d) (green) and (f) (orange). (h) Ratio of predicted to actual RMS magnetic field strength for a range of field strengths using both linear-regime flux spectral relation (3.15) (dashed) and reconstructed deflection-field spectral relation (2.11) (solid), applied to unsmeared (blue), smeared (green) and de-smeared (orange).

The impact of smearing on successful extraction of field statistics from proton-flux images can be somewhat reduced by use of a deconvolution algorithm. In particular, for a known point-spread function $S$ , there exist algorithms (such as the Richardson–Lucy algorithm) for recovering the maximum likelihood solution for the unsmeared image-flux distribution given the smeared image-flux distribution (Richardson Reference Richardson1972; Lucy Reference Lucy1974). The procedure is illustrated in figure 11(e), with associated reconstructed path-integrated field in figure 11(f). It is clear that the maximum likelihood solution for the de-smeared image-flux distribution recovered by the Richardson–Lucy algorithm does not have identical morphology to the actual unsmeared image-flux distribution. However, the morphology and strength of the reconstructed path-integrated field is much more similar; furthermore, the magnetic-energy spectrum recovered from the path-integrated field (see figure 11 g, orange markers) is a much closer match to the true result (figure 11 g, blue markers).

More quantitatively, figure 11(h) gives the ratio of the predicted RMS magnetic-field strength to the actual value for a range of field strengths, as calculated by integrating predictions of the magnetic-energy spectrum. The results from using both linear-regime flux spectral relation (3.15) and reconstructed deflection-field spectral relation (2.11) applied to unsmeared (blue), smeared (green) and de-smeared (orange) proton-flux images are shown. It is clear that for both approaches, the smeared proton-flux image leads to reductions in predictions of field strength of 10–20 % – but that these can be removed quite effectively using a deconvolution algorithm. We also note that beyond the appearance of caustics, all methods increasingly under-estimate the predicted RMS field strength; this is related to the lower-bound property of the reconstructed perpendicular-deflection fields discussed in § 3.5.

For experimental analysis with a somewhat limited resolution range, we conclude that deconvolution algorithms are helpful for deducing magnetic-energy spectra over a reasonable range of wavenumbers.