# Zeeman splitting observations in laser-produced magnetized blast waves

A. Triantafyllidis, J.-R. Marquès, S. Ferri, A. Calisti, Y. Benkadoum, Y. de León, A. Dearling, A. Ciardi, J. Béard, J.-M. Lagarrigue, et al.

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A. Triantafyllidis, J.-R. Marquès, S. Ferri, A. Calisti, Y. Benkadoum, et al. Zeeman splitting observations in laser-produced magnetized blast waves. Matter and Radiation at Extremes, 2025, 10 (4), 10.1063/5.0256859. hal-05095681

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# Zeeman splitting observations in laser-produced magnetized blast waves

Special Collection: **Special Issue on ICMRE2024**

A. Triantafyllidis ; J.-R. Marquès ; S. Ferri ; A. Calisti ; Y. Benkadoum; Y. De León ;
A. Dearling ; A. Ciardi ; J. Béard ; J.-M. Lagarrigue; N. Ozaki ; M. Koenig ; B. Albertazzi

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Matter Radiat. Extremes 10, 047603 (2025)

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# Zeeman splitting observations in laser-produced magnetized blast waves

Cite as: Matter Radiat. Extremes 10, 047603 (2025); doi: 10.1063/5.0256859

Submitted: 7 January 2025 • Accepted: 20 April 2025 •

Published Online: 21 May 2025

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A. Triantafyllidis,<fnref n="1" />,ª ID J.-R. Marquès,<fnref n="1" /> ID S. Ferri,<fnref n="2" /> ID A. Calisti,<fnref n="2" /> ID Y. Benkadoum,<fnref n="1" /> Y. De León,<fnref n="3" /> ID

A. Dearling,<fnref n="4" />,<fnref n="5" /> ID A. Ciardi,<fnref n="6" /> ID J. Béard,<fnref n="7" /> ID J.-M. Lagarrigue,<fnref n="7" /> N. Ozaki,<fnref n="8" />,<fnref n="9" /> ID M. Koenig,<fnref n="1" /> ID and B. Albertazzi<fnref n="1" />,ª

## AFFILIATIONS

**Note:** This paper is part of the Special Issue on the 7th International Conference on Matter and Radiation at Extremes (ICMRE2024).

ªAuthors to whom correspondence should be addressed: angelos.triantafyllidis@polytechnique.edu and bruno.albertazzi@polytechnique.edu

## ABSTRACT

We report the observation of Zeeman splitting in multiple spectral lines emitted by a laser-produced, magnetized plasma (1–3 × 10¹<fnref n="8" /> cm⁻<fnref n="3" />, 1–15 eV) in the context of a laboratory astrophysics experiment under a controlled magnetic field up to 20T. Nitrogen lines (NII) in the visible range (563–574 nm) were used to diagnose the magnetic field and plasma conditions. This was performed by coupling our data with the Stark–Zeeman line-shape code PPPB. The excellent agreement between experiment and simulations paves the way for a non-intrusive experimental platform to get time-resolved measurements of the local magnetic field in laboratory plasmas.

© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0256859

## I. INTRODUCTION

Magnetic fields play a fundamental role in modifying the dynamics and behavior of plasmas. They can, for example, smooth out density and temperature discontinuities in interstellar medium shock waves<fnref n="1" />⁻<fnref n="3" /> and alter the geometry of supernova remnants from spherical to barrel-shaped,<fnref n="4" />,<fnref n="5" /> depending on their orientation and magnitude. A recent experimental study demonstrated the drastic structural alteration of laser-produced blast waves (BWs) in the presence of an external magnetic field<fnref n="6" /> due to mechanisms only present in the context of magnetohydrodynamics (MHD).<fnref n="7" />,<fnref n="8" /> In inertial confinement fusion (ICF), magnetic fields are ubiquitous as they can be self-generated via the Biermann-battery mechanism during the laser-matter interaction.<fnref n="9" />,¹⁰ Also, they have been shown to

help achieve optimal conditions for ignition,¹¹⁻¹³ even allowing for an increase in yield owing to the inhibition of heat transport.¹⁴⁻¹⁶ Investigating these systems requires accurate probing of the local magnetic fields, otherwise incomplete or incorrect conclusions may be drawn.

The recent advent of apparatuses capable of generating strong magnetic fields coupled with high intensity laser facilities¹⁷ necessitates the development of experimental platforms to diagnose magnetic fields in the laboratory. Several approaches have been developed over the years, but all face significant limitations. B-dot probes¹⁸ give the magnetic field change by measuring the current it induces. They were used in numerous experimental setups, but other methods are preferred nowadays due to certain issues: they disturb the plasma under study, their response time needs to be

<footnote n="1">¹LULI - CNRS, CEA, Sorbonne Universités, Ecole Polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau cedex, France</footnote>

<footnote n="2">²Aix Marseille Université, CNRS, PIIM, F-13013 Marseille, France</footnote>

<footnote n="3">³École Normale Supérieure de Lyon, Lyon, France</footnote>

<footnote n="4">⁴York Plasma Institute, University of York, York YO10 5DD, United Kingdom</footnote>

<footnote n="5">⁵Blackett Laboratory, Imperial College London, London SW7 2BZ, United Kingdom</footnote>

<footnote n="6">⁶Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LUX, 75005 Paris, France</footnote>

<footnote n="7">⁷Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS, EMFL, Université Grenoble-Alpes, Université Toulouse 3, INSA Toulouse, F-31400 Toulouse, France</footnote>

<footnote n="8">⁸Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan</footnote>

<footnote n="9">⁹Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan</footnote>
a fraction of the laser pulse duration, and can return imprecise or incorrect measurements due to electromagnetic interference.¹⁹ Optical polarimetry is an option that is based on the rotation of a polarized optical probe beam passing through a magnetic field (Faraday rotation),²⁰,²¹ but it requires a large enough magnetic field strength or plasma volume for the rotation angle to be resolvable (lower limit is about 10T over 1 mm). Similarly, placing an object with a known Verdet constant in the region of interest and measuring the rotation angle is also possible, although this is a significantly intrusive diagnostic.²² Finally, proton deflectometry²³⁻²⁵ is based on the deflection of a proton beam passing through a magnetic field as a result of the Lorentz force (generated via the Target Normal Sheath Acceleration mechanism or D-T plasmas). Nevertheless, the need for an intense laser beam dedicated to this diagnostic or imploding D-T capsules restricts the applicability of this method to few laser facilities.²⁴,²⁵ Additionally, the deflection of the protons also depends on the generated electric fields, so both transverse and axial deflectometry diagnostics are needed to distinguish their respective contributions.¹⁹

Stark-Zeeman line-shape modeling has been an important method for diagnosing plasmas, but it has its own share of benefits and drawbacks. More specifically, the distinct signatures of the Zeeman and Stark effects on spectral lines allow us to infer the average magnitude of a magnetic field permeating a plasma and its electron density. Along with temperature determination from line intensity ratios, analysis of line features has proven invaluable in numerous research fields, particularly astrophysics and laboratory plasmas.²⁶⁻³⁰ Coupled to line-shape modeling, it can provide information on the plasma and magnetic field parameters by collecting the emission spectrum. Nonetheless, the magnetic field needs to be sufficiently strong to induce an observable splitting in most spectral lines. Furthermore, when Stark broadening³¹ is too pronounced (e.g., in ICF plasmas), Zeeman splitting may not be resolvable.³²

To study the Stark-Zeeman features in a laser-plasma experiment, we developed a time-resolved, high-dispersion (HD) spectrometer in the visible domain (resolution of 0.195 nm) and performed measurements on laser-produced BWs propagating in air at the LULI2000 facility. Utilizing a pulsed-power system, we generated well-characterized magnetic fields of up to 20T, perpendicular or parallel to the BWs' propagation.¹⁷ We collected emission spectra from singly ionized nitrogen (NII), capturing multiple intense radiative transitions in the low-temperature plasma (1–15 eV, ~10¹⁸ cm⁻³) which were extensively compared with simulations using the Stark-Zeeman line-shape code PPPB.³³

This manuscript is organized as follows. In Sec. II we outline the experimental setup, how the magnetic field is generated, and describe the design of the spectrometers employed. Section III is dedicated to the experimental results, while Sec. IV demonstrates the ability of the code PPPB to reliably determine the plasma conditions. Finally, we discuss the potential and some limitations of our experimental platform as a magnetic field diagnostic in Sec. V. We gather our conclusions in Sec. VI.

## II. METHODOLOGY AND EXPERIMENTAL SETUP

Any atomic system embedded in a plasma is strongly modified by the interactions with its environment, resulting in the modification of its energy levels and lifetime. When this atomic

system is embedded in a magnetized plasma, it interacts with the magnetic moment, inducing a splitting of the spectral lines into several sub-components with different energies, the Zeeman effect.³⁴ Since the energy splitting between sub-levels depends on the magnetic field strength, it can be used to measure it. In the weak-field approximation:

$$ \Delta E_B = g_I \mu_B B M_J, \qquad (1) $$

where $g_I$ is the Landé g-factor, $\mu_B$ the Bohr magneton, B the strength of the magnetic field, $M_J$ the magnetic quantum number ($M_J = J, J-1, \dots, -J$) and J the total angular momentum. Moreover, due to the interaction with the magnetic field, the emitted light is partially polarized following the selection rules for the electric dipole radiation:

$$ \Delta J = 0, \pm 1(0 \rightarrow 0), \quad \Delta M_J = \begin{cases} \pm 1, & \sigma \text{ component,} \\ 0, & \pi \text{ component.} \end{cases} \qquad (2) $$

By observing perpendicular to the magnetic field, the $\pi$ and the $\sigma$ components show a linear polarization, but along the field direction only the $\sigma$ component is visible with a circular polarization; the $\pi$ component does not appear. Therefore, by varying the orientation of the field, different features of the line spectrum can be observed.

To study this phenomenon, we performed an experiment at the LULI2000 laser facility at LULI (École Polytechnique, France). A laser beam irradiated a cylindrical graphite pin with a diameter of 500 µm placed inside the experimental chamber at TCC. This laser beam had an energy $E_0 \approx 600J$ at the 2$\omega$ frequency ($\lambda_L = 526.5$ nm), a pulse duration $\tau \approx 1.5$ ns, and was focused down to 200 µm, shaped by a hybrid phase plate,³⁵ giving a maximum intensity of $I_0 \approx 10^{15}$ W/cm². The sudden deposition of energy onto the target generated a blast wave (BW) that propagated in the magnetized ambient medium.

<figure><img src="image_13.png" /><figcaption>FIG. 1. Experimental setup at the LULI2000 facility. A laser beam was used to drive the target (graphite pin) and generate a BW in air. A 2$\omega$ probe beam was used transversely to the propagation of the BW for the interferometry diagnostic. SOP and 2D self-emission diagnostics collected the light coming from the same direction. High- and low-dispersion optical spectrometers gathered the self-emission light at 3 and 4 mm from TCC along the laser axis from the top access point. The magnetic field was perpendicular or parallel to the BW propagation axis. Part of the coil casing is removed here for better visualization.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 2. 2D self-emission snapshots of the BW plasma in the 580–850 nm range as a function of time and magnetic field. The graphite pin target is outside the field of view to the left of the images. The shaded area on the center-left image indicates the approximate volume and position probed by the optical spectrometers (150 × 150 × 50 µm³, 3 and 4 mm for the perpendicular and parallel magnetic field to the spectrometers’ line of view, respectively).</figcaption></figure>

The experimental setup is shown in Fig. 1. Two time-resolved, optical spectrometers, one with low dispersion (LD) and one with high dispersion (HD), were used to collect the plasma self-emission from a small volume (150 × 150 × 50 µm³), located 3 mm (for parallel magnetic field) or 4 mm (for perpendicular magnetic field) in front of the target, with a magnification of 4.9. The direction of observation was perpendicular the BW propagation (see Figs. 1 and 2). The signal was divided into two arms (50% of the signal each) and imaged onto the slit of the spectrometers, each coupled with a streak camera.

The LD spectrometer used a prism to collect the signal covering a range of 250 nm (425–675 nm) with a resolution of ~2 nm, giving us information on the time evolution of numerous lines. The HD signal was gathered by a Czerny–Turner spectrometer (2400 grooves/mm) with a resolution of 0.195 nm and a window range of 16 nm. The HD spectra were collected from two different parts of the plasma, focused onto two entrance slits of the spectrometer, and then overlapped onto a streak camera to increase the signal intensity. The distance between the slits was 2.7 mm at the entrance of the streak camera, corresponding to 550 µm inside the chamber (as the plasma is probed late in time, we can consider it homogeneous in $n_e$ and $T_e$, see Sec. V B for a short discussion). We changed the spectral range of the HD spectrometer multiple times to examine different lines and identify those where the Zeeman splitting was more pronounced.

In addition, multiple other diagnostics were employed to further and independently characterize the laser-produced plasma. The electron density was also retrieved using Mach–Zehnder interferometry,³⁶ streaked optical pyrometry (SOP) provided information on the propagation of the BW from its creation until ~50 ns on a shot to shot basis,³⁷ and the 2D plasma self-emission diagnostic captured snapshots of the BW (time-integrated over 120 ps), revealing its morphology and evolution.

The magnetic field was generated using a coil in a Helmholtz configuration, developed by LNCMI.¹⁷ It was oriented parallel or perpendicularly to the propagation of the laser pulse (left to right in Fig. 2) with a constant magnitude in a volume of 1 cm³. The peak strength of the field lasts for 3–5 µs with less than 2% variation in amplitude, so orders of magnitude more than the timescales of the phenomena presented here. In the data presented in this work, we used magnetic fields of 5, 10, 15 and 20T.

The strength of the applied magnetic field is the only freely controlled parameter of Eq. (1), so the choice of material and

<table><caption>TABLE I. The nitrogen lines that were simulated.</caption><thead><tr><th>Theoretical wavelength (nm)</th><th>Transition</th></tr></thead><tbody><tr><td>566.663</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>2</sub> → 2s<sup>2</sup>2p3s <sup>3</sup>P<sub>0</sub><sup>0</sup></td></tr><tr><td>567.602</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>1</sub> → 2s<sup>2</sup>2p3s <sup>3</sup>P<sub>0</sub><sup>0</sup></td></tr><tr><td>567.956</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>3</sub> → 2s<sup>2</sup>2p3s <sup>3</sup>P<sub>2</sub><sup>0</sup></td></tr><tr><td>568.621</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>1</sub> → 2s<sup>2</sup>2p3s <sup>3</sup>P<sub>1</sub><sup>0</sup></td></tr><tr><td>571.077</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>2</sub> → 2s<sup>2</sup>2p3s <sup>3</sup>P<sub>2</sub><sup>0</sup></td></tr><tr><td>574.730</td><td>2s<sup>2</sup>2p3p <sup>3</sup>D<sub>2</sub> → 2s<sup>2</sup>2p3s <sup>1</sup>P<sub>1</sub><sup>0</sup></td></tr></tbody></table>
<figure><img src="image_15.png" /><figcaption>FIG. 3. Time-resolved spectra collected by the HD spectrometer for two bandwidths (left: 493–504 nm, right: 563–573 nm). Both orientations of the magnetic field are presented: (a) and (d) at 0T, (b) and (e) at 20T (parallel to the line of sight), and, (c) and (f) at 20T (perpendicular to the line of sight). The shaded bands in (a) and (d) indicate the time range over which the signal was integrated (see Fig. 4). The Zeeman splitting is particularly apparent by comparing images (d) and (e), as shown by the arrows.</figcaption></figure>

bandwidth to observe the Zeeman splitting is crucial. First, some
lines are more sensitive to Zeeman splitting than others, so it is
necessary to find ones that our HD spectrometer can resolve. Sec-
ond, the material should have a low Z to minimize background
noise, i.e., bremsstrahlung radiation. Taking these into account,
we filled the experimental chamber with air at 8 mbar (2 ± 0.2
× 10¹⁷ cm⁻³, 300 K) and observed the nitrogen lines in the range
425–675 nm. In particular, we focused on the transitions shown in
Table I (~568 nm) because the expected splitting is the largest in the
HD observation window: for 20T, the energy shift |ΔE| ≳ 0.0014 eV

corresponds to |Δλ| ≳ 0.3 nm, i.e., more than our instrument's
resolution. Furthermore, having different magnetic field orienta-
tions allows the observation of both σ and π components or only
the σ component.

III. EXPERIMENTAL RESULTS

First, we need to know how the BW morphology evolves in
time, with respect to the magnetic field strength (0–20T) and its
orientation. To this end, the 2D self-emission diagnostic (see Fig. 2)
<figure><img src="image_16.png" /><figcaption>FIG. 4. Time-integrated lineouts from 140 to 160 ns for all values and orientations of the magnetic field: (a) 493–504 nm, (b) 563–573 nm. The dashed vertical lines correspond to the theoretical NII lines from NIST.</figcaption></figure>

shows that for $B = 0T$ (top row) the BW maintains a spherical structure throughout its evolution and expands according to the Taylor–Sedov solution.⁶,³⁸,³⁹ In the perpendicular case (middle row), the BW front dissipates due to multiple processes,⁸,⁴⁰ while in the parallel case (bottom row) it is compressed and pinched by the magnetic field. This trend intensifies as the BW evolves and the magnetic field strengthens. The underlying physics of these two cases is inherently different. An in-depth analysis of the first case is detailed in Ref. 6, whereas the second one will be featured in a future paper.

This paper focuses on the self-emission spectrum of magnetized plasma and its behavior as the magnetic field is varied, specifically on the HD spectra. We start by taking a look at the Fig. 3, where typical spectra collected by the HD spectrometer for 0 and $20T$ are exhibited. The BW’s time of arrival to the region of interest is signified by bright emissions at $t \sim 25–30$ ns which varies depending on the applied magnetic field and energy of the laser pulse. Moreover, the spectrum for $B = 0T$ displays a comparatively brighter background (mainly bremsstrahlung radiation) due to the denser and hotter BW front plasma.⁶

These images clearly exhibit the Stark–Zeeman features (Fig. 3). Here, we focused on two bandwidths (493–504 and 563–573 nm), but splitting can also be observed in other emission lines, albeit not as distinctly. The time-integrated lineouts in Fig. 4 emphasize the initial broadening and subsequent line splitting from 0 to $20T$. Moreover, by assuming that the plasma is uniform at later times, we collect the signal from two points (separated by 550 µm at TCC) and overlay it to increase the signal intensity. This is the reason why two intense emission bands are seen in Fig. 3. Interestingly, the rotation of the magnetic field (parallel or perpendicular to the observation direction) allows for a distinction between the $\sigma$ and $\pi$ components, resulting in significantly altered spectra for the same field strength. Especially in Figs. 4(e) and 4(f), the emission lines are widely different due to the $\pi$ components not being observable in the parallel configuration. More specifically, when both $\sigma$ and $\pi$ components are present (perpendicular orientation), they overlap, obscuring the splitting as they fill the gaps where the splitting would otherwise be more distinct. This results in a broadening of

the peaks rather than a clear separation, as seen in the parallel case.

#### IV. PPPB SIMULATIONS

The Stark–Zeeman line-shape code PPPB³³ was used to calculate the NII emission spectra for different electron densities, temperatures and magnetic field strengths. For the line-shape calculations, 6 intense radiative transitions have been considered in the 566–575 nm range (see Table I). For such isolated lines and the range of plasma parameters considered ($n_e = 1–3 \times 10^{18}$ cm⁻³ and $T_c = 1–15$ eV), the main broadening mechanism arises from the free electrons of the plasma,³¹,⁴¹ resulting in the lines being broadened by 0.5–1 nm. In the range $B = 0–20T$, these intense radiative transitions split in more than 50 Zeeman components. Note that by calculating the fine-structure splitting due to spin–orbit coupling $\Delta\lambda_{LS}$ and comparing it to the one caused by the magnetic field strength $\Delta\lambda_B$, we can infer whether we should use the weak- or strong-field (Paschen–Back) Zeeman effect approximation. In the following, we assume that the applied magnetic field can be considered as a perturbation, so we adopt the weak-field approximation to simulate the spectra.

As it has already been demonstrated by Tessarin et al.²⁸ the Stark–Zeeman profiles can be expressed as a convolution of the Zeeman pattern and the Stark profile. Nevertheless, in the present work, we take advantage of the flexibility and the rapidity of PPPB to calculate several thousands line shapes of the NII emission spectra over a broad range of magnetic fields and plasma conditions. The atomic data needed in the simulations (energy levels, statistical weight, and line strength) were taken from the NIST Atomic Spectra Database,⁴² while the shifts and widths were taken from Mar et al.⁴¹ and references therein. According to the latter, the experimental widths for the relevant transitions lay between the calculations of Griem et al.³¹ and Dimitrijevic and Konjević.⁴³ Both theories present a slightly different temperature dependence. In the present work, we adopt the impact theory for the electrons, so that the shifts and widths are proportional to $n_e/\sqrt{T_c}$. Doppler broadening and the response of the spectrometers have also been considered, but the former does not
seem to be important due to the low velocity of the BW ($v \sim 10$ km/s for $t \sim 150$ ns).

For the studied plasma parameters, the Stark effect gives rise to a broadening that partially masks the Zeeman components. However, different features appear depending on whether the spectra are observed perpendicularly or parallel to the magnetic field orientation. All the experimental spectra were fitted with the PPPB line shapes plus a background corresponding to bremsstrahlung radiation. Since the wavelength window of the HD spectrometer is only 16 nm, a linear dependence is assumed. Using FLY-CHK,44 we find that in the energy range 2–2.5 eV (corresponding to 564–574 nm) the ratio of radiative recombination continuum radiation to bremsstrahlung radiation is ~1:5 for $T_e = 5$ eV and ~1:2 for $T_e = 3$ eV. The former has not been included in the simulated spectra.

Figures 5–7 show the best fits for different values and both orientations of the magnetic field using the nitrogen transitions in this wavelength range, found using a $\chi^2$-test and further optimized by manually adjusting the background value. Generally, the Zeeman structures are masked in the perpendicular observation case, whereas in the parallel one they are quite sensitive to changes of $n_e$, $T_e$ and B. The solid curves in Fig. 5 correspond to the best fits with electron densities $n_e$ ranging from 1.6 to $2 \times 10^{18}$ cm$^{-3}$, and the shaded areas show variations to the optimal values for $\pm 0.2 \times 10^{18}$ cm$^{-3}$. Concerning the electron temperature, $T_e = 5$ eV was found for all spectra with a maximum uncertainty of $\delta T_e = 1$ eV (see Fig. 6). The magnetic field was also treated as a free parameter. In the ideal case, where the emission lines are not broadened, its value can be found purely from their splitting [see Eq. (1) for the weak-field approximation]. However, when the lines are broadened or the resolution of the instrument is not sufficiently large, the magnetic field is inferred from the line shapes. The best agreement between simulation and experiment was consistently obtained when the simulated field value matched the experimental one, determined with a maximum uncertainty of $\delta B = 2T_e$ (see Fig. 7). While the uncertainties are partly due to the resolution of the HD spectrometer, the response to variations of $T_e$ and B also depends heavily on the chosen emission lines. The aforementioned uncertainties are possibly slightly larger in the perpendicular case because the Zeeman features are less clear.

In many cases, emission from the graphite target interferes with the spectra and it also needs to be considered. Specifically, the CII line transitions $2s2p(^3P_0)3p^4S_4 \rightarrow 2s2p(^3P_0)3s^4P_6^0$ (566.247 nm) and $2s2p(^3P_0)4p^4P_6 \rightarrow 2s2p(^3P_0)3d^4P_6^0$ (569.430 nm) are included in Fig. 8. The latter one does not affect the spectra, however the former significantly improves the fit in the wing (~566.5 nm), effectively reducing the discrepancy observed at that wavelength. No oxygen lines are important in this wavelength range. It is noteworthy that the simulated spectra fail to fully account for the background brightness for $\lambda \gtrsim 569$ nm, even when a constant value is added, possibly due to radiative recombination continuum radiation on top of the bremsstrahlung radiation.

# V. DISCUSSION AND PERSPECTIVES

## A. Magnetic field dynamics

Is the magnetic field compressed by the plasma due to diamagnetism or does it diffuse inside it? Does it evolve as the BW

propagates into the magnetized plasma? The first question was answered by fitting the collected spectra with PPPB simulations, and we found that the magnetic field in the post-shock region equals or is very close to the initial value for all examined cases. However, because the HD signal is weak, the time over which we need to integrate is large. As a consequence, information on the evolution of the magnetic field is lost and we cannot answer the second question.

A way to answer these questions qualitatively is by calculating the magnetic field diffusion timescale $\tau_D \sim \frac{4n_e l^2}{\eta c^2}$ or the magnetic

<figure><img src="image_17.png" /><figcaption>FIG. 5. PPPB simulations for the NII transitions overlaid on time-integrated (over 50 ns) lineouts for 5, 10 and 20T: (a) perpendicular and (b) parallel to the magnetic field orientation. Solid curves represent the values indicated in the legend, while the shaded areas correspond to simulations with $\delta n_e = \pm 0.2 \times 10^{18}$ cm$^{-3}$ relative to them (here, curves with higher intensities correspond to higher densities). The electron temperature is $T_e = 5$ eV for all simulated spectra. A linear background has been added to account for the bremsstrahlung emission of the plasma, and the spectra are normalized for each magnetic field and orientation. The dashed vertical lines correspond to the theoretical NII lines from NIST.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**FIG. 6.** Same as Fig. 5, but with a variation in electron temperature indicated by the shaded areas for 1 or 2 eV. Here, curves with lower intensities correspond to higher $T_e$. The plasma conditions for the simulated spectrum with the solid line are $n_e = 1.6 \times 10^{18} \text{ cm}^{-3}$ and $T_e = 5$ eV. A linear background has been added to account for the bremsstrahlung emission of the plasma and the spectra are normalized for each magnetic field orientation. The dashed vertical lines correspond to the theoretical NII lines from NIST.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**FIG. 7.** Same as Fig. 5, but with a variation in the magnetic field indicated by the shaded areas for 2 or 3*T*. Here, curves with lower intensities correspond to higher *B*. The plasma conditions for the simulated spectra are $n_e = 1.6 \times 10^{18} \text{ cm}^{-3}$ and $T_e = 5$ eV.</figcaption></figure>

Reynolds number $Re_m = \frac{uL}{\eta}$, where $\eta$ is the magnetic resistivity, *L* the characteristic length scale over which diffusion or compression occurs, and *u* the velocity of the plasma. If $Re_m \gtrsim 10$ or $\tau_D$ is larger than the characteristic timescale of the phenomenon, then the system can be described by ideal MHD and the magnetic field is “frozen-in” and therefore follows the compression of matter. On the other hand, if $Re_m \lesssim 1$ then resistive MHD effects are dominant and the magnetic field is allowed to diffuse inside the expanding plasma.

From our calculations, we find $Re_m \sim 1$ and $\tau_D \sim 15–100$ ns in the post-shock region and over the time range of integration. This suggests that mostly resistive MHD features may appear, indicating that almost no compression of the magnetic field is expected, as supported by the PPPB simulations.

## B. Limitations

In the context of laser-plasma experiments, line-shape modeling has seen a wide range of applications, but several limitations

must be considered. First of all, certain combinations of $n_e$ and $T_e$ can actually result in similar spectra because the line broadening is proportional to the quantity $n_e/\sqrt{T_e}$. For this reason, it is necessary to be able to independently measure one of these variables to restrict the range of available values. Mach–Zehnder interferometry can provide $n_e$ in laser-plasma experiments with good accuracy, but only if spherical or cylindrical symmetry is preserved. Otherwise, performing an inverse Abel transform to analyze 3D objects projected onto a 2D plane produces a significant error in $n_e$,36 usually a 10%–20% error needs to be considered. In addition, when phase shifts of over $2\pi$ are present in phase maps (as often happens in dense plasmas, such as in the front of BWs), reconstructing the $n_e$ map might not be possible without significant artifacts. Still, the results from Mach–Zehnder interferometry allow us to constrain the $n_e$ and $T_e$ values; we calculate $n_e \sim 1.3–1.8 \times 10^{18} \text{ cm}^{-3}$ in the post-shock region for most cases, which is consistent with the results from spectrometry. The example in Fig. 9 highlights this fact and demonstrates the importance of crosschecking the plasma conditions using multiple diagnostics.

Furthermore, a strong external magnetic field or a high resolution instrument is required to observe any significant Zeeman
<figure><img src="image_20.png" /><figcaption>**FIG. 8.** Time-integrated spectra (over 50 ns) compared to the NII, CII and combined PPPB simulations; $n_e = 1.4 \times 10^{18} \text{ cm}^{-3}$, $T_e = 5$ eV and $n_e = 1.6 \times 10^{18} \text{ cm}^{-3}$, $T_e = 5$ eV for $B_\|\|$ of (a) 10 and (b) 15 T, respectively. A linear background has been added to account for the bremsstrahlung emission of the plasma. The dashed vertical lines correspond to the theoretical NII lines from NIST.</figcaption></figure>

splitting. This issue can be mitigated by designing an experiment where a strong compression is expected (ideal MHD regime, $Re_m \gtrsim 10$). In this case, even an initial modest field will be compressed by a few times and induce a discernible splitting. Alternatively, splitting could be more easily observed in infrared spectral lines, given that they're intense enough; the wavelength shift $\Delta\lambda$ related to an energy difference $\Delta E$ being $\Delta\lambda = \lambda^2\Delta E/hc$.

An additional feature of our experimental setup is the use of two slits in the HD spectrometer, which is done to collect more signal at the cost of temporal and spatial information. As light from two different plasma volumes is collected, part of it overlaps at the time-resolved image [see Fig. 3, especially (d)]. The BW is expected to be uniform in the post-shock region and late in time, while also evolving slowly. Therefore, since we're probing volumes separated

<figure><img src="image_21.png" /><figcaption>**FIG. 9.** Time-integrated spectra (over 50 ns) compared to the NII PPPB simulations for a parallel magnetic field orientation and the conditions shown in the legend. The similarity in the line shapes for certain combinations of $n_e$ and $T_e$ necessitate the use of independent diagnostics to choose the appropriate values. The dashed vertical lines correspond to the theoretical NII lines from NIST.</figcaption></figure>

by 550 µm and integrating over a large time range, this is not a significant issue. Some possible workarounds to avoid the overlapping would be the use of gratings with fewer grooves to reduce signal loss or using larger time windows on the streak cameras, sacrificing spectral or time resolution, respectively. Furthermore, reducing the magnification and not splitting the signal into two arms (LD and HD) can double the detected number of counts. All these changes could raise the signal intensity by approximately an order of magnitude. Finally, implementing a more rigorous goodness-of-fit method, such as Bayesian analysis, could further improve the fits and reduce the reported uncertainties.

Despite these limitations, using Zeeman splitting to measure the magnetic field in similar experimental conditions offers plenty of advantages compared to other options (B-dot probes, proton deflectometry, optical polarimetry etc), if the magnetic field is strong enough. The setup described in this work, in contrast, is comparatively simpler because it mainly depends on the choice of material and a spectrometer with a resolution of $\Delta\lambda \le 0.2$ nm or better. However, the coupling of the experimental spectra with a Stark-Zeeman code, such as PPPB, is necessary to extract the relevant plasma parameters.

## VI. CONCLUSIONS

In this article, we presented observations of the Zeeman effect in laser-produced BWs. This was achieved by designing an experimental platform capable of resolving the splitting of several spectral lines along multiple magnetic field values and orientations (0–20T, parallel and perpendicular to the axis of observation). Our observations were later coupled with Stark-Zeeman line-shape simulations performed with the PPPB code, revealing the behavior of the plasma and magnetic field; the system evolves under the resistive MHD regime, as predicted by the calculation of the magnetic Reynolds
number *Re*<sub>m</sub> and the magnetic diffusion time τ<sub>D</sub>. As a result, the magnetic field is not compressed in the post-shock region of the BW.

The excellent agreement between the experimental spectra and the PPPB simulations demonstrates the capability of our platform to infer the magnetic field strength with high accuracy, as well as its suitability as a tool in laser-plasma experiments. Our results may also be used to calibrate and benchmark other line-shape codes. Finally, since precise measurements of magnetic fields are critical for achieving and optimizing ignition in ICF, our work could also advance plasma diagnostics where other techniques are impractical.<sup>32,45,46</sup>

## ACKNOWLEDGMENTS

The authors would like to thank the anonymous Referees for their insightful feedback, which significantly clarified major points of our work. Also, they would like to thank the team at LULI2000 for their work in obtaining the data presented here, and Patrick Renaudin for insightful discussions. This work was supported by grants managed by l'Agence Nationale de la Recherche under the Investissements d'Avenir programs Grant Nos. ANR-18-EURE-0014, ANR-10-LABX-0039-PALM, and ANR-22-CE30-0044. N.O. was supported by grants from Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant No. 23K20038) and JSPS Core-to-Core program (Grant No. JPSCCA20230003). This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200—EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. The involved teams have operated within the framework of the Enabling Research Project No. AWP24-ENR-IFE.02.CEA-01 "Magnetized ICF." For the purpose of open access, the authors have applied a CC-BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**A. Triantafyllidis:** Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). **J.-R. Marquès:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (supporting); Resources (equal); Software (equal); SuperVISION (equal); Validation (equal); Writing – review & editing (supporting). **S. Ferri:** Formal analysis (supporting); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Writing – original draft (supporting);

Writing – review & editing (supporting). **A. Calisti:** Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Validation (supporting); Writing – review & editing (supporting). **Y. Benkadoum:** Data curation (supporting); Investigation (supporting); Methodology (supporting); Resources (supporting); Writing – review & editing (supporting). **Y. De León:** Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Writing – review & editing (supporting). **A. Dearling:** Data curation (supporting); Investigation (supporting); Writing – review & editing (supporting). **A. Ciardi:** Investigation (supporting); Methodology (supporting); Project administration (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **J. Béard:** Data curation (supporting); Methodology (supporting); Validation (supporting); Writing – review & editing (supporting). **J.-M. Lagarrigue:** Data curation (supporting); Methodology (supporting); Validation (supporting); Writing – review & editing (supporting). **N. Ozaki:** Data curation (supporting); Supervision (supporting); Writing – review & editing (supporting). **M. Koenig:** Conceptualization (equal); Data curation (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing – review & editing (equal). **B. Albertazzi:** Conceptualization (equal); Data curation (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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