Zeeman splitting observations in laser-produced magnetized blast waves Open Access

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^1–3 and alter the geometry of supernova remnants from spherical to barrel-shaped,^4,5 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 field⁶ due to mechanisms only present in the context of magnetohydrodynamics (MHD).^7,8 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.^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 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 10 T 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 deflectometry^23–25 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 20 T, 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, $∼1018cm−3$⁠) 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.

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 E₀ ≈ 600 J at the 2ω frequency (λ_L = 526.5 nm), a pulse duration τ ≈ 1.5 ns, and was focused down to 200 μm, shaped by a hybrid phase plate,³⁵ giving a maximum intensity of I₀ ≈ 10¹⁵ W/cm². The sudden deposition of energy onto the target generated a blast wave (BW) that propagated in the magnetized ambient medium.

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 $∼2nm$⁠, 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 $∼50ns$ 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 20 T.

The strength of the applied magnetic field is the only freely controlled parameter of Eq. (1), so the choice of material and 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. Second, 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 (⁠$∼568nm$⁠) because the expected splitting is the largest in the HD observation window: for 20 T, 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 orientations allows the observation of both σ and π components or only the σ component.

First, we need to know how the BW morphology evolves in time, with respect to the magnetic field strength (0–20 T) and its orientation. To this end, the 2D self-emission diagnostic (see Fig. 2) shows that for B = 0 T (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 20 T are exhibited. The BW’s time of arrival to the region of interest is signified by bright emissions at t ∼ 25–30 ns which varies depending on the applied magnetic field and energy of the laser pulse. Moreover, the spectrum for B = 0 T 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 20 T. 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 σ and π 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 π components not being observable in the parallel configuration. More specifically, when both σ and π 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.

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 × 10¹⁸ cm⁻³ and T_e = 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 = 0–20 T, these intense radiative transitions split in more than 50 Zeeman components. Note that by calculating the fine-structure splitting due to spin–orbit coupling Δλ_LS and comparing it to the one caused by the magnetic field strength Δλ_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 $ne/Te$⁠. 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 ∼ 10 km/s for t ∼ 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 FLYCHK,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 χ²-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 × 10¹⁸ cm⁻³, and the shaded areas show variations to the optimal values for ±0.2 × 10¹⁸ cm⁻³. Concerning the electron temperature, T_e = 5 eV was found for all spectra with a maximum uncertainty of δT_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 δB = 2 T (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 $2s2p(P03)3pS44→2s2p(P03)3sP604$ (566.247 nm) and $2s2p(P03)4pP64→2s2p(P03)3dP604$ (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 $(∼566.5nm)$⁠, 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 λ ≳ 569 nm, even when a constant value is added, possibly due to radiative recombination continuum radiation on top of the bremsstrahlung radiation.

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 $τD∼4πL2ηc2$ or the magnetic Reynolds number $Rem=uLη$⁠, where η 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 ≳ 10 or τ_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 ≲ 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 ∼ 1 and τ_D ∼ 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.

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 $ne/Te$⁠. 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 2π 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 ∼ 1.3–1.8 × 10¹⁸ 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 splitting. This issue can be mitigated by designing an experiment where a strong compression is expected (ideal MHD regime, Re_m ≳ 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 Δλ related to an energy difference ΔE being Δλ = λ²ΔE/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 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 Δλ ≤ 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.

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 (0–20 T, 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_m and the magnetic diffusion time τ_D. 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.^32,45,46

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. JPJSCCA20230003). 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.

The authors have no conflicts to disclose.

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).

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