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

The B-dot probe axis was positioned parallel to the target's coil axis, either in the plane of the coil at a $30\pm 0.5\;\mathrm{mm}$ distance from the coil centre and at a $33^\circ$ angle above equator ($1\mathrm{st}$ experiment), or approximately along the coil axis at $70\pm 1\;\mathrm{mm}$ distance from the coil centre, with a $10\pm 0.5\;\mathrm{mm}$ horizontal offset to clear the path for the probe beam used for Faraday rotation and shadowgraphy ($2\mathrm{nd}$ 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 $\approx 50$ and $\approx 10\;\mathrm{ps}$ respectively in the $1\mathrm{st}$ and $2\mathrm{nd}$ experiments.

Figure 2 details the analysis of a signal obtained in the $1\mathrm{st}$ 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) the signal corresponding spectrum, calculated by the fast-Fourier-transform (FFT) of the signal (thick red curve; the thin black curve is the FFT of the signal for $t\lt -50\;\mathrm{ps}$, that is the noise spectrum).

Zoom In Zoom Out Reset image size

Figures 2(c) and (d) show the temporal evolution of the B-field, on the level of a few mT. This was obtained by integration of the signal in panel (a) for different configurations of bandpass filtering: the optimized result, represented by the thicker red curves, was obtained with the minimum frequency of ${\nu }_{\mathrm{min}}=1$ MHz to cut the dc component and $1/\nu$ noise, and a maximum frequency of ${\nu }_{\mathrm{max}}=1.5$ GHz to cut the high frequency parasitic EMP emission from higher frequency ground discharge currents [30, 31] ($1\mathrm{st}$ experiment; we used ${\nu }_{\mathrm{max}}=1\mathrm{GHz}$ for the $2\mathrm{nd}$ experiment data, according to the reached spectral resolution). Thinner curves correspond to variations either on ${\nu }_{\mathrm{min}}$ (panel (c)) or on ${\nu }_{\mathrm{max}}$ (panel (d)) testing the result sensibility to these parameters: raising of ${\nu }_{\mathrm{min}}$ yields an offset of the B-field at t = 0 (start of laser irradiation), yet the amplitude of the signal is consistent for ${\nu }_{\mathrm{min}}$ up to $20$ MHz. For fixed ${\nu }_{\mathrm{min}}=1$ MHz, the B-field rise-time and maximum yield do not significantly change for ${\nu }_{\mathrm{max}}$ down to $1$ GHz. Therefore, the large spectral peak at a few hundreds MHz in panel (b) corresponds to the main B-field signal due to the target discharge through the coil-shaped wire, the second large peak around $1$ GHz determining the second B-field peak oscillation at $t\gtrsim 2$ ns (see other measurements in figure 4).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 3D magnetostatic code Radia [29] 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 values at the B-dot spatial positions. The boundary condition consisted in imposing a closed circuit with the same current in the coil and between the two disks. Figure 3(a) represents two perpendicular planar sections of the B-field map, one horizontal containing the coil axis, z = 0 (left), the other vertical corresponding to the coil plane, y = 0 (right) (see figure 1(b) for the axis orientation: the y-axis is the coil axis and the origin is here at the coil centre). The field was calculated with an injected current of $I=340\;\mathrm{kA}$ and a coil radius $a=250\;\mu {\rm{m}}$, yielding the measured B-field peak value for the Cu target (given the $\approx 10$ μm ns⁻¹ coil rod expansion velocity measured by time-resolved optical shadowgraphy, no rod expansion was considered in the magnetostatic calculations): the B-field norm is on color scale, the arrows represent the B-field vector projections on the two planes. The spatial distribution on the z = 0-plane clearly evidences, as expected, a dipole-like B-field. One can also see on the y = 0 plane that the B-field norm is quite homogenous over the space inside the coil. Poloidal fields around the coil rod and straight parts of the wire are quite strong, and they mostly determine the B-field distribution below the coil region.

For the same calculation, figure 3(b) shows the amplitude of the B-field component parallel to the coil-axis as a function of the perpendicular distance, along the line connecting the coil and the B-dot probe centres, respectively at r = 0 and $r=30\;\mathrm{mm}$ (solid red curve). The vertical dashed lines delimit the $50\;\mu {\rm{m}}$-thick coil rod position. For a measured $5\;\mathrm{mT}$ at $r=30\;\mathrm{mm}$, the extrapolated value at the coil centre is ${B}_{0}=800$ T for the capacitor-coil target, with the uncertainty range of (795, 825 T) accounting for target-to-target eventual variations of $\pm 2\;\mu {\rm{m}}$ on the separation between the two connection points of the wire circular part with the vertical rods. An over-estimated extrapolation B-field value of ${B}_{0}=20\;\mathrm{kT}$ (for a $I=8.5$ MA current) is obtained when simulating a perfect circular coil of the same radius (dashed black curve in the insert): such unrealistic value illustrates the importance of an accurate modeling of the real target, in particular the discontinuity on the coil circular part. Yet, our target production and characterization are sufficiently precise to have only a very small contribution to the result uncertainty.

As for the B-dot probes distance with respect to the coil, figure 3(c) shows the calculated relative uncertainties of the B-field measurements at the probe positions: the values are plotted over the probes circular section and account for the 3D B-field gradients over the probes cylindrical volume. The estimated B-field errors, which remain under the 6% and the 10% respectively for the $1\mathrm{st}$ and $2\mathrm{nd}$ experiments (two different probe orientations and distances relative to the coil, as described above), are mainly determined by the gradients along x and z in the first case (top), and along y for the second case (bottom), as expected from the setup geometry.

The typical final results for the produced B-field as a function of time are summarized in figure 4. The solid curves are the B-dot probe results obtained with capacitor-coil targets for the B-field at the probe positions (left-hand side ordinates) and the corresponding B₀ values at the coil'(s) centre (right-hand side ordinates, as extrapolated from the Radia magnetostatic calculations). In the $1\mathrm{st}$ experiment, panel (a), the peak values of the B-field depend on the target material, yielding ${B}_{0}\approx 800$, $\approx 600$ and $\approx 150\;{\rm{T}}$ (±6%), respectively for Cu (red), Ni (green) and Al targets (cyan). In the $2\mathrm{nd}$ experiment, panel (b), the B-dot probe measurements are showing a shot-to-shot reproducible charging using Ni targets: the B-field production is synchronous with the beginning of the laser irradiation (at t = 0, within a $\pm 100\;\mathrm{ps}$ calibration uncertainty) and the rise time of $\approx 1$ ns is consistent with the duration of the laser pulse (this is the system charge time). The reproducible measurements of a peak B-field of $\approx 1\;\mathrm{mT}$ are extrapolated to the coil centre peaking at ${B}_{0}\approx 600\;{\rm{T}}$ (±10%). This is in very good agreement with the result obtained with the Ni target in the $1\mathrm{st}$ experiment, where the probe was at a different position relative to the coil.

The targets used in the $2\mathrm{nd}$ experiment had a reduced front disk hole (diameter of 1000 instead of $1750\;\mu {\rm{m}}$), seeking for capturing more non-thermal electrons. However, a comparison of the results leads to the conclusion that the hole size is not a determining parameter for the peak B-field strength, yet the second B-field peak oscillation at $t\gtrsim 2$ ns is more pronounced. The typical duration of the pulsed B-field is of a few ns, with fluctuations mostly due to the varying distance d₀. All signals present slower amplitude variations at later times (non-represented), probably corresponding to electromagnetic coupling with objects around the target (secondary targets, diagnostics, chamber walls), converging to zero at $\sim 150$ ns.

The efficiency of the coils was tested by B-dot measurements in two blank shots: grey dashed curves in figures 4(a) and (b), corresponding respectively to two Cu disks without any connecting wire (held separately), and to a Ni target with a straight wire between the disks (no coil). The dashed curves' values correspond exclusively to the measurements at the probe positions (left-hand side ordinates). One concludes that (i) the irradiated disk charging by laser–target interaction and current of escaping electrons have a negligible contribution to the integrated signals (as well as any neutralizing current from the ground through the disk stalk) unless there is a physical connection between the disks, and (ii) the contributions from currents flowing through other target segments besides the coil (in particular the straight parts of the connecting 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 for $t\lesssim 0.4$ ns were confirmed in the $2\mathrm{nd}$ 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 circular-symbols in figure 4(b). The B-field peak values could not be measured by these two diagnostics due to laser-plasma effects, as described in the next sections.

The Faraday rotation measurements, using a $9$ ns-duration probe laser at $533$ nm wavelength incident along the coil-axis, were performed with two $500\;\mu {\rm{m}}$-thick birefringent terbium gallium garnet (TGG) crystals with its centre placed at $3.5\;\mathrm{mm}$ from the coil plane: TGG1 ($0.5$ mm-wide) was centred on the coil axis and TGG2 ($1$ mm-wide) at a $1.9$ mm perpendicular offset. The TGG Verdet constant, $11.35^\circ \;{{\rm{T}}}^{-1}\;{\mathrm{mm}}^{-1}$, is 38 times higher than that of ${\mathrm{SiO}}_{2}$ used in the earlier work by Fujioka et al [28], allowing to be sensitive to weaker B-field strengths. No Faraday rotation measurements were successful with the crystals located neither closer to the coil, because of signal blackout quasi-synchronous with the laser irradiation of the target—due to very rapid crystal ionization by the hard x-rays and fast particles issuing from the interaction region—nor further away from the coil where the local B-field was too weak. The Faraday effect was measured 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 figure 1(b)). The time-resolution of $\approx 300\;\mathrm{ps}$ was limited by the camera time-resolution and by the laser jitter. The quantification of the rotation angle was previously calibrated, without B-field, by quantifying the polarization ratio of the two transmitted perpendicular-polarization signals, defined as $R={E}_{\parallel }^{2}/({E}_{\parallel }^{2}+{E}_{\perp }^{2})$, as a function of the orientation of the incident light polarization. The obtained linear fit of the experimental data was consistent with the Malus law by an average ${\chi }^{2}\approx 0.002$ adjustment. For the shots with B-field, the incident polarization was setup at $-45^\circ$ and $+45^\circ$ relative to ${E}_{\parallel }$ and ${E}_{\perp }$ respectively, yielding a ratio $R=0.50\pm 0.01$ for the situation without B-field.

Figure 5 shows the analysis of a successful shot for the Faraday rotation effect measurement, as a function of time: transmitted ${E}_{\parallel }^{2}$ (solid black) and ${E}_{\perp }^{2}$ (dashed black), and corresponding R (solid red) are plotted for (a) TGG1 and (b) TGG2 (the plotted values correspond to averages over the respective crystals' width). The crystals' blackout times are indicated by the light grey dashed vertical lines. Panel (c) plots the corresponding local axial component of the B-field averaged over the crystals thickness and width, solid black for TGG1, dashed red for TGG2, obtained assuming that the TGG Verdet constant does not change with time due to the increase in crystal temperature. We see that the B-field inferred from the TGG2 measurements is too weak and remains at the signal noise level, of $\pm 0.03\;{\rm{T}}$. As for the TGG1 measurements, the B-field clearly rises up to the blackout time. To compare with B-dot probe measurements we selected the time ${t}_{1}\approx 0.2$ ns, indicated in both panels (a) and (c), where the averaged B-field is evaluated to $\approx 0.22\pm 0.05\;{\rm{T}}$ along the coil axis at a $3.50\pm 0.25$ mm distance from its centre. The black square symbol in figure 4(b) corresponds to the B₀ strength (right-hand side ordinate axis) extrapolated from that measurement, using the same magnetostatic code [29] and a similar protocol as for the B-dot data. In spite of the diagnostic uncertainties—related to the crystal thickness, accessible width and equivalent height of the streak slit on the crystal plane, and the B-field gradients over such crystal volume—the obtained value for the extrapolated B₀ is fairly consistent with the B-dot probe measurements.

Zoom In Zoom Out Reset image size

The proton-deflectometry technique allows to measure the B-field directly in the coil region. A proton beam was created with a short laser pulse ($50\;{\rm{J}}$ on target, $1\;\mathrm{ps}$ FWHM at $1{\omega }_{0}$) focused onto $10\;\mu {\rm{m}}$-thick Au foils at $\approx {10}^{19}\;{\rm{W}}\;{\mathrm{cm}}^{-2}$ intensity. Proton beams of $\sim 20\;\mathrm{MeV}$ maximum energy were generated by the target normal sheath acceleration (TNSA) mechanism at the foils' rear surface [32, 33], located at a distance of $5\mathrm{mm}$ from the target coil. The proton beam propagation axis was perpendicular to the coil axis, and it was 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 $42\;\mu {\rm{m}}$-pitch mesh, positioned $3$ mm before the coil (see figure 1(b)). RCF proton imprint images correspond to a magnification of 10 for the coil plane and of 25 for the mesh plane. A $195\;\mu {\rm{m}}$-Al protection film before the RCF stack limited the detection to protons of energy ${\epsilon }_{{\rm{p}}}\geqslant 5\;\mathrm{MeV}$ and electrons with energies ${\epsilon }_{{\rm{e}}}\geqslant 260\;\mathrm{keV}$. 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 $\Delta \tau$ between the main and proton-driving lasers, scanned the effects of the B-field on the proton-trajectories over the time range of $t=\Delta \tau +\mathrm{TOF}$, with TOF between 80 and $160$ ps.

Figure 6(a) shows a typical image of the RCF layer corresponding to the imprint of protons of energy ${\epsilon }_{{\rm{p}}}=13\pm 1$ MeV, obtained in a shot with $\Delta \tau =0.25$ ns. The corresponding probing time is $t\approx 0.35$ ns. The mesh-shadow deformations are detected for distances till $\sim 500\;\mu {\rm{m}}$ from the coil centre (the given spatial scale corresponds to the plane of the coil centre, perpendicular to the proton beam axis, where the unperturbed mesh shadow has a pitch of $105\;\mu {\rm{m}}$). However, the most outstanding feature is the centred bulb region void of any proton imprint due to the very strong B-field. Figure 6(b) shows the result of a Monte-Carlo simulation of the trajectories of randomly injected protons within the energy range of the experimental signal in (a). 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 centre ${B}_{0}=95$ T. Figure 6(c) shows the map of the experimental mesh vertical deformations (measured from the film corresponding to figure 4(a)), fairly indicating a dipole-like B-field spatial distribution, with a typical length scale of $1$ mm. In figure 6(d) the horizontal lineout of the experimental vertical deformation map at $z=0\pm 105\;\mu {\rm{m}}$ (circles) is compared to the corresponding synthetic deformations for different strengths of the B-field (curves), leading to the evaluation of ${B}_{0}=95\pm 20$ T at the corresponding probing time, $t\approx 0.35$ ns. This value, solid red circle in figure 4(b) (right-hand side ordinates), agrees with the evaluation from the B-dot measurements.

Zoom In Zoom Out Reset image size

Figure 7(a) shows the RCF images obtained for ${\epsilon }_{{\rm{p}}}=13\pm 1$ MeV protons with different delays $\Delta \tau$ 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 $t\gt 0.35$ ns, following the same protocol as before, would hold the B₀ values and uncertainties represented by the small orange circles in figure 4(b): the B₀ decreasing behaviour 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. Figure 7(b) shows images from RCF layers corresponding to ${\epsilon }_{{\rm{p}}}=16.8\pm 0.8$ MeV protons, for the same first two delays $\Delta \tau$: for each delay, the usual mesh imprint and void-bulb proton signatures are smaller, consistently with the higher proton energy. Yet, we detect a second particle species producing a large circular halo on both images, superposed to the 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, and relatively homogeneously deposing energy over the successive RCF layers. Accounting for their spectrum (inferred from 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 between 3 and $5$ MeV. Comparing images from different shots (see figure 7(b)), we observe that the halo size increases for increasing $\Delta \tau$.

Zoom In Zoom Out Reset image size

The signature of the relativistic electrons evolves in opposition to the proton signature. This can be explained by a monotonous increase, over the ns time scale of the main laser irradiation, of magnetized plasma electrons in regions of strong B-field around or at the vicinity of the coil. The main-laser interaction with the target rear disk creates a plasma $3$ mm below the coil. As determined by x-ray spectrometry along with a comparison with calculated atomic spectra, its temperature is of the order of $T\approx 1.5\pm 0.1$ keV, with a high-energy electron component of ${T}_{{\rm{e}}}\approx 40\pm 5$ keV. A fraction of these electrons can stream upwards to the coil-wire region in less than $30$ ps, where they can be magnetized: their Larmor radius is $\lt 10\;\mu {\rm{m}}$ (${B}_{0}\approx 100\;{\rm{T}}$ is achieved at $t\approx 300\;\mathrm{ps}$ (see figure 4)), much smaller than the typical size of the strong B-field region, $\sim 2a=500\;\mu {\rm{m}}$. The progressively increasing charge of magnetized electrons at the coil vicinity can produce an electrostatic effect sufficient important to influence the deflection trajectories of the probing particles, with increasing focusing or defocusing contributions respectively for the TNSA protons and for the relativistic electrons issuing from the short laser pulse interaction with the Au foil. A trapped electron charge as small as $5\times {10}^{-7}\;{\rm{C}}$ yields an electrostatic potential at a distance $r\sim a$ from the coil centre already comparable to the $20$ MeV proton maximum energy. This charge corresponds to a density of magnetized electrons ${n}_{{\rm{e}}}^{\mathrm{mag}}\sim 5\times {10}^{16}\;{\mathrm{cm}}^{3}$ in a sphere of radius a, which is about 5% of the density of the 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, ${\omega }_{\mathrm{ce}}\gtrsim {\omega }_{\mathrm{pe}}$ (assuring that the potential stays localized at the coil vicinity), determines a maximum ${n}_{{\rm{e}}}^{\mathrm{mag}}\lesssim {10}^{17}\;{\mathrm{cm}}^{-3}$ (for ${B}_{0}\sim 100\;{\rm{T}}$), 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 ${p}_{{\rm{e}}}={n}_{{\rm{e}}}^{\mathrm{mag}}{T}_{{\rm{e}}}\sim 6\times {10}^{7}\;$ Pa, which remains much smaller than the density of magnetic energy ${p}_{B}={B}_{0}^{2}/2{\mu }_{0}\sim 4\times {10}^{9}\;$ Pa, for ${B}_{0}=100$ T. (ii) Diamagnetic currents of the trapped electrons ${j}_{1}\sim {{en}}_{{\rm{e}}}^{\mathrm{mag}}{v}_{{\rm{e}}}$, yield a magnetic field over the characteristic length a of ${B}_{1}\sim {\mu }_{0}{j}_{1}a\sim 10$ T, eventually opposed but much smaller than the B₀ inferred by the other diagnostics.