Laser-driven platform for generation and characterization of strong quasi-static magnetic fields

Quasi-static magnetic-fields up to $800\,$T are generated in the interaction of intense laser pulses ($500\,$J, $1\,$ns, $10^{17}\,$W/cm$^2$) with capacitor-coil targets of different materials. The reproducible magnetic-field peak and rise-time, consistent with the laser pulse duration, were accurately inferred from measurements with GHz-bandwidth inductor pickup coils (B-dot probes). Results from Faraday rotation of polarized optical laser light and deflectometry of energetic proton beams are consistent with the B-dot probe measurements at the early stages of the target charging, up to $t\approx 0.35\,$ns, and then are disturbed by radiation and plasma effects. The field has a dipole-like distribution over a characteristic volume of $1\,$mm$^3$, which is coherent with theoretical expectations. These results demonstrate a very efficient conversion of the laser energy into magnetic fields, thus establishing a robust laser-driven platform for reproducible, well characterized, generation of quasi-static magnetic fields at the kT-level, as well as for magnetization and accurate probing of high-energy-density samples driven by secondary powerful laser or particle beams.


Introduction
The properties of matter on all scales (atoms, molecules, condensed matter, plasmas) are severely modified when exposed to strong magnetic fields (B-fields) [1]. The possibility of imposing a strong B-field generated by laser to a variety of samples opens interesting perspectives for laboratory studies of magnetized plasma- [2], atomic- [3] and nuclearphysics [4]. We foresee great progress on the understanding of systems of astrophysical scale [5,6,7], on the improvement of inertial fusion energy schemes [8,9,10] and for various applications of magnetically-guided particle beams [11], among many other applications.
Explosive-driven solenoid magnetic flux compressions have been used to reach the level of a few kT [12], but the approach is a harsh one for the surrounding diagnostic hardware. B-fields of the order of 5 kT have been achieved in compression experiments carried out in Z-machines [13,14], but the efficiency is small as the needed discharge energy is of several MJ. Relativistic laser interactions with dense targets and the issuing intense currents over the target surface or into the target bulk generate super-strong B-fields [15], on the range of 10 kT for current short-pulse laser parameters, but these are rather transient as they evolve on the time-scale of ∼ 10 ps. Quasi-static magneticfield production coupled to laser facilities has been explored so far by the development of capacitor-bank pulsed discharges in solenoids (magnetic pulsers) [16], but the specific physical limitations restrain the maximum generated fields to the range of ∼ 50 T.
Instead, the use of powerful lasers interacting with capacitor-coil targets -first proposed by Daido et al back in the 1980s [17,18] and recently explored to higher levels by Fujioka et al [19] -give unprecedented quasi-static (time-scale of a few ns) high B-field amplitudes for such a compact system (∼ mm) and an energy laser pulse driver of 1 kJ: they reported a B-field ≈ 1.5 kT at 0.65 mm away from the U-turn coil center, measured by Faraday rotation of the polarization of a probe laser beam in a SiO 2 sample. But the reported value would yield, according to a magnetostatic simulation of the U-shaped capacitor-coil target [20], a non realistic magnetic energy, greater than the invested laser energy. The problem probably lies in that the tabulated Verdet constants of the birefringent crystals are of questionable validity in the presence of strong and rapidly changing B-fields.
In this study, we present an accurate characterization of the B-field produced by laser-driven capacitor-coil targets, using consistently i) high-frequency pickup B-dot probes, ii) Faraday rotation of laser probe beam polarization (we used much more sensitive birefringent crystals, placed further away from the coil to detect weaker B-field strength) and iii) deflectometry of an energetic proton beam. This combination of three independent diagnostics confirms that high power laser facilities may be used for B-field production in the range of several hundred tesla with a controlled and reproducible rise time and spatial distribution.

Methods and experimental results
The experiments were conducted at the LULI pico 2000 laser facility with a 1.057 µm wavelength (1ω 0 ), 500 J, 1 ns flat-top long-pulse laser beam, focused to intensities ≈ 10 17 W/cm 2 . The targets were made of two parallel disks (3500 µm diameter, 50 µm thickness, with a hole in the front one enabling focusing of the laser pulse into the rear disk's surface), connected by a coil-shaped wire (coil radius a = 250 µm, squared rod section of 50 µm × 50 µm): see Fig. 1. The targets were made of Cu, Ni or Al. The hole diameter of the front disk was 1750 µm or 1000 µm. The target parameters were reproducible thanks to accurate laser cutting of an initial 2D metallic form. The only variable parameter was the distance between the disks, d 0 = 900 ± 200 µm, as a consequence of the manual target folding. The laser pulse irradiates the rear disk and creates the supra-thermal electrons that are escaping the potential barrier [21,22]. A fraction of them are captured by the opposite disk. The target reacts like an RL-circuit to the potential difference between the disks and the subsequent discharge current through the coil-shaped wire I produces a dipole-like B-field over a time-scale of a few ns: The amplitude of the B-field near the coil center scales from a perfect coil of radius a like B 0 ≈ µ 0 I/2a. The target geometry, distance between the disks and wire configuration, defines the short-circuiting time of the system, which is of the order of 1 -2 ns: It happens when the plasma plume ejected from the irradiated disk reaches the opposite disk or when thermally expanding conducting wires start to overlap. This time is likely to be reduced by the front disk thermal expansion induced by X-ray irradiation from the rear disk plasma. Clearly, an important advantage of a laser-driven coil-target is that it is an open system: it works like an antenna, which does not confine but emits the field, giving access to the magnetization of secondary samples.
The experimental setup is sketched in Fig. 1-b).

B-dot probing
The B-dot probe axis was positioned parallel to the target's coil axis, either in the plan of the coil at a 30 mm distance from the coil center (1 st experiment) or along the coil axis at 70 mm distance from the coil center (2 nd experiment). The B-dot probe (and associated electronics) has a 2.5 GHz acquisition bandwidth, and the chosen sampling frequency yielded signals with time resolution of ≈ 50 and ≈ 10 ps respectively in the 1 st and 2 nd experiments. Figure 2 details the analysis of a signal obtained in the 1 st experiment with a Cu target: a) A raw detected signal (attenuators excluded), proportional to the time-derivative of the B-field at the probe position, b) B-field spectrum, calculated by the Fast-Fourier-Transform (FFT) of the signal. The large spectral peak at a few hundred 100 MHz corresponds to the B-field signal due to the target discharge through the coil-shaped wire. In spite of using capillary insulator target-stalks, we still measured a higher-frequency electro-magnetic pulse (EMP) which we think to be emitted by the target neutralization current from the target holder [22]: the peak observed at 1.65 GHz fairly agrees with a simple model, taking the target rear disk and support as an antenna and yielding ν EMP = c/4(l + πd/2), where l ≈ 40 mm and d = 3.5 mm are the target stalk length and the disk diameter, respectively. The B-dot probe signal is integrated in time by imposing a bandpass frequency filter with the minimum frequency of 1 MHz to cut the DC component and 1/ν noise, and a maximum frequency of 1 GHz to cut the high frequency parasitic EMP emission from ground discharge currents [21,22]. The temporal evolution of the filtered B-field is shown in panel c). The B-field amplitude at the probe position is on the level of a few mT. The 3D magnetostatic code Radia [20] was used to simulate the spatial distribution of the B-field taking into account the target and coil geometry, and their connection wires. We used the current I as a free parameter, varying it to obtain an equality between the measured and simulated B-field value at the B-dot spatial position. The boundary condition consisted in imposing the same current in the coil and between the two disks. The simulation gave us then an estimation of the B field within the coil: The expected B-field values at the coil center, B 0 , are shown in panel c), right axis. The spatial distribution of the magnetic field, obtained for the B-field peak value with an injected current I = 340 kA, is shown in panel d), full red curve. The dashed black curve in the insert of panel d) corresponds to the case of a perfect circular coil of the same radius a = 250 µm. For a measured 5 mT at r = 30 mm, the extrapolated value of B 0 is 0.8 kT for our target. An over-estimated extrapolation value of 20 kT is obtained when simulating a perfect coil, which shows the importance of an accurate modeling of the real target.
The typical B-dot probe results are summarized in Fig. 3 (curves). Peak values of the B-field depend on the target material, yielding B 0 ≈ 800, ≈ 600 and ≈ 150 T respectively for Cu (red), Ni (green) and Al targets (cyan). The rise in the B-field is synchronous with the beginning of the laser irradiation (t = 0) within ±100 ps The thinner grey curves correspond respectively to a shot on the rear disk holding the two disks parallel at the distance d 0 = 900 µm but without any connecting wire, and to a Ni target with a straight wire between the disks (no coil); these curves cannot be extrapolated to coil center, they refer exclusively to the B-field at the probe position (left-handside ordinates). The full symbols represent results for the B-field at the coil centre, B 0 , measured by Faraday rotation about 100 ps before signal blackout (squares) and by proton-deflectometry at ∆τ = 0.25 ns (full circle). The empty circles represent B-field estimates from proton-deflectometry images obtained at later times: the discrepancy with results from the other diagnostics is explained in the text by electrostatic effects due to electron trapping near the coil.
the results lead to conclude that the hole size is not a determining parameter for the B-field strength.
The efficiency of the coils was tested by B-dot acquisitions in two blank shots: the thin grey curve in Fig. 3-a) corresponds to two Cu disks without any connecting wire (held separately), while the thin grey curve in Fig. 3-b) corresponds to a Ni target with a straight wire between the disks (no coil). One concludes that i) the laser-target interaction and the escaping high-energy electron currents have a negligible contribution to the measured signals, and ii) the contributions from currents flowing through other target segments besides the coil (in particular the straight parts of the wire) are sufficiently weak if compared to the results obtained with coil-shaped wires: the coil is the dominant source of the magnetic flux.
The B-dot results were confirmed in the 2 nd experiment by measurements of the Faraday rotation of the polarization direction of a linearly-polarized probe laser beam, and of the laser-accelerated proton deflections: respectively full square and circularsymbols in Fig. 3-b).

Faraday rotation
The Faraday rotation measurements, using a 9 ns-duration probe laser at 533 nm wavelength incident along the coil-axis, were performed with two 500 µm-thick birefringent Terbium Gallium Garnet (TGG) crystals placed at 3.5 mm from the coil plan: TGG1 centered on the coil axis and TGG2 at 1.9 ± 0.5 mm perpendicular offset. The TGG Verdet constant, 11.35 • /T/m, is 38 times higher than that of SiO 2 . The Faraday effect was inspected by using a time-resolved polarimeter, constituted of a Wollaston prism to separate the two perpendicular components of the probe beam field, and a streak camera [see Fig. 1-b)]. The square symbols in Fig. 3-b) correspond to the B 0 strength (right-hand side ordinate axis) extrapolated from the measurements at the crystal's positions using the same magnetostatic code and protocol as before. The given times are ≈ 100 ps before the blackout of the probing laser signal as a consequence of rapid crystal ionization by hard X-rays and fast particles issuing from the main lasertarget interaction. No Faraday rotation measurements were successful with the crystals located closer to the coil because of signal blackout quasi-synchronous with the laser irradiation of the target. In spite of the diagnostic uncertainties (related to the size of the TGG crystals and their positions relative to the coil), the two agreeing values for the extrapolated B 0 are fairly consistent with the B-dot probe measurements.

Proton-deflectometry
The proton-deflectometry technique allows to inspect the B-field directly in the coil. A proton beam was created with a short and intense laser pulse (50 J on target, 1 ps FWHM at 1ω 0 ) focused into 10 µm thickness Au foils. Proton beams of ∼ 20 MeV maximum energy were generated by the Target Normal Sheath Acceleration (TNSA) mechanism at the foils rear surface [23], located at a distance of 5 mm from the target coil. The proton propagation axis was perpendicular to the coil axis, and they were detected 45 mm away from the coil by a 15-layers radiochromic film (RCF) stack. The proton deflections due to the B-field were quantified with the help of a 42 µm-pitch mesh, positioned 3 mm before the coil [see Fig. 1-b)]. RCF proton imprint images correspond to a magnification of 10 for the coil plan and of 25 for the mesh plan. A 195 µm-Al protection film before the RCF stack limited the detection to protons of energy p ≥ 5 MeV and electrons with energies e ≥ 260 keV. The proton imprint signal on each RCF layer corresponds to a narrow energy range of their spectrum, due to the Bragg peak energy absorption. Accounting for the time-of-flight (TOF) between the proton source and the coil, each shot, with a chosen delay ∆τ between the main and proton-driving lasers, scanned the effects of the B-field on the proton-trajectories over the time range of t = ∆τ + TOF, with TOF between 80 and 160 ps.  to the plan of the coil center, perpendicular to the proton beam axis, where the unperturbed mesh shadow has a pitch of 105 µm). However, the most outstanding feature is the centered bulb region void of any proton imprint due to the very strong B-field. Figure 4-b) shows the result of a Monte-Carlo simulation of the trajectories of randomly injected protons within the energy range of the experimental signal. Both the mesh-shadow deformations and bulb size compare very well with the experimental image when imposing a coil current I = 40 kA, yielding the B-field in the coil center B 0 = 95 T. This value agrees with the B-dot measurements [full circular-symbol in Fig. 3-b)]. Figure 4-c) shows the map of the measured mesh deformations, indicating a dipole-like B-field spatial distribution, with a typical length scale of 1 mm. Figure 5-a) recapitulates the RCF images obtained for p = 13±1 MeV protons with different delays ∆τ between the laser pulses. Surprisingly, the bulb size and the overall mesh imprint deformations decrease with time: protons of the same energy undergo smaller deflections if injected at later times. The measurement of the void bulb size and mesh-imprint deformations for t > 0.35 ns, following the same protocol as before, would hold the B 0 values represented by the void circular blue symbols in Fig. 3-b): the B 0 decreasing behavior as a function of time is in contradiction with the B-dot probe measurements, and does not agree with the laser-charging process up to t = 1 ns. The proton-trajectories must be then perturbed by some plasma effect, summing up with the magnetic force. Figure 5-b) shows images from RCF layers corresponding to p = 16.8 ± 0.8 MeV protons, for the same first two delays ∆τ : we detect a second particle species, producing an large circular halos on both images, superposed to the usual mesh imprint and bulb proton signature. For each shot, such a halo is clearly visible with the same size and shape over the successive last six layers of the RCF stack, identifying it as the signature of relativistic electrons, accelerated at the Au-foil front surface by the short laser pulse. Accounting for their spectrum (inspected by Particle-in-Cell simulations of the short pulse laser interaction), the foil potential barrier and the Al-filtering of the RCF stack, the halo signal should correspond to electrons with energies approximately between 3 and 5 MeV. Comparing images from different shots [see Fig. 5-b)], we observe that the halo size increases for increasing time delay ∆τ . This can be explained by a monotonous increase, over the ns time scale of the main laser irradiation, of magnetized (trapped) density of plasma electrons in regions of strong B-field around or at the vicinity of the coil. The plasma is created by the laser interaction with the target, 3 mm below the coil. This results in the increasing focusing and defocusing electrostatic effects as a function of the delay ∆τ observed respectively for the TNSA protons and for the relativistic electrons issuing from the short laser pulse interaction with the Au foil.
As inspected by X-ray spectrometry along with a comparison with calculated atomic spectra, the temperature of the main laser-produced plasma is of the order of T ≈ 1.5 ± 0.1 keV, with a high-energy electron component of T e ≈ 40 ± 5 keV which eventually take less than 30 ps to reach the coil. Their Larmor radius is < 10 µm at the coil center [B 0 ≈ 100 T is achieved at t ≈ 300 ps (cf. Fig. 3)], much smaller than the typical size of the strong B-field region, ∼ 2a = 500 µm. A trapped electron charge as small as 5 × 10 −7 C yields an electrostatic potential at a distance r ∼ a from the coil center already comparable to the 20 MeV proton maximum energy. This charge corresponds to a density of magnetized electrons n mag e ∼ 5 × 10 16 cm 3 in a sphere of radius a, which is about 5% of the density of the expected supra-thermal electrons expected to stream through the coil region. This would be enough to electrostatically balance the main magnetic force on the probing TNSA protons and relativistic electrons. The magnetization condition for the main plasma electrons, ω ce ω pe (assuring that the potential stays localized at the coil vicinity), determines a maximum n mag e 10 17 cm 3 (for B 0 ∼ 100 T), consistent with the previous estimation.
Other plasma effects can influence proton-deflectometry measurements, but their effect is expected to be negligible compared to the electrostatic one: i) The pressure of the magnetized electrons in the coil region p e = n mag e T e ∼ 6×10 7 Pa, which remains much smaller than the density of magnetic energy p B = B 2 0 /2µ 0 ∼ 4 × 10 9 Pa, for B 0 = 100 T. ii) Diamagnetic currents of the trapped electrons j 1 ∼ en mag e v e , yield a magnetic field over the characteristic length a of B 1 ∼ µ 0 j 1 a ∼ 10 T, eventually opposed but much smaller than the B 0 inspected by the other diagnostics.

Conclusions
In conclusion, the results obtained in our experiments were obtained simultaneously by three independent diagnostics, yielding consistent results, and show a reproducible quasi-static B-field generation by laser interaction with capacitor-coil targets, with a typical few ns duration and 1 mm 3 -volume, yielding peak strengths of several hundred Tesla, depending on the target material: the dipole-like magnetic field total energy is of the order of 8%, 4.5% and 0.3% of the invested laser pulse energy, respectively for Cu, Ni and Al targets. The observed differences with the different target materials may be attributed to i) the different resistive behavior at low temperature, though the current rise time and consequent wire heating is identically rapid, and, more likely, to ii) the plasma temperature yielding different short-circuiting times: this will be object of further investigations. The correct extrapolation of the B-field amplitude at the center of the coils from distant measurements (at a few mm for Faraday rotation, and at a few cm for the pick-up coil probes) is possible by an accurate coding of the target shape and magnetostatic computation of the current intensity looping in the capacitor-coil targets. Electrostatic plasma effects, likely due to electron magnetization around the coil region, sum up with the main B-field effect to explain decreasing proton deflections for t > 0.4 ns. While the typical mm 3 -volume and ns-duration of the produced B-field pulses are small compared to the parameters achieved in the large-scale experiments mentioned in the introduction, they are characterized by an unprecedentedly high conversion efficiency (approaching the range of 10%) of the driver energy into magnetic energy. Moreover, this all-optical technique lends itself to the magnetization and accurate probing of highenergy-density samples driven by secondary powerful laser or particle beams: Given the ≈ 10 µm/ns expansion velocity of the coil rod, inspected by time-resolved optical shadowgraphy, the strong B-field region is accessible for several ns.