Study of laser produced plasma in a longitudinal magnetic field

Investigation of plasmas in strong magnetic fields is important for many fields of physics including astrophysics, controlled fusion, and basic plasma physics. Magnetic fields play an important role in laser-produced plasmas. Strong magnetic fields improve conditions for inertial confinement fusion (ICF).¹ Collimation of laser generated electrons by B-fields is important for the development of the fast ignition concept.^2,3 The pulsed power MagLIF fusion concept uses linear compression of a premagnetized and laser-preheated plasma.⁴ Simulations show that strong magnetic fields may increase the plasma ion temperature at laser intensities >10¹⁸ W/cm² (Ref. 5). Strong magnetic fields can enhance laser-driven generation of proton beams.⁶ Laser produced plasmas exhibit unusual dynamics in strong magnetic fields such as the formation of narrow disklike plasma around a current-carrying rod with a 200–300 T magnetic field.⁷ Disk plasma was produced by a 0.8 ns laser pulse with an intensity of 3 × 10¹⁵ W/cm². Generation of plasma jets by a nanosecond laser pulse in a longitudinal magnetic field B ∼ 20 T was described in Refs. 8 and 9. Plasma dynamics in strong magnetic fields are relevant to astronomical events. Enhanced growth of plasma instabilities in strong external magnetic fields was observed in Ref. 10. Strong magnetic fields influence high-energy-density plasma (HEDP) dynamics due to anisotropic thermal conduction even if the B-field pressure is lower than the plasma pressure.

Magnetic fields B > 100 T are generated by rod and coil loads in pulsed power generators^11,12 and in capacitor-coil targets due to the laser driven current.^10,13 The generation of B-fields and the physics of laser plasma interactions (LPIs) in a fast rising magnetic field are governed by a complex sequence of phenomena. Plasma arises in the current carrying coils and fills the internal area. Coils generate x-ray radiation and plasma that interacts with the laser irradiated target. Eddy currents produce plasma on the metal targets¹² and can initiate a surface discharge on the dielectric target due to the high rate of dB/dt. B-field-initiated plasma on the target changes the initial conditions for laser-target interaction. These effects need to be taken into account in plasma experiments with strong magnetic fields.

In this paper, the laser-plasma interaction was investigated at intensity >10¹⁸ W/cm² in the external longitudinal magnetic field B > 50 T produced by a 1 MA pulsed power generator.¹⁴ The magnetic field was measured at the position of the laser target by Faraday rotation diagnostics. The impact of eddy currents on laser irradiated targets embedded in the magnetic field was assessed through laser imaging diagnostics. Plasma dynamics in the longitudinal magnetic field were studied using two-frame shadowgraphy and interferometry. Laser diagnostics showed collimation of the plasma plume. Laser produced plasma formed a narrow dense jet 0.2–0.3 mm in diameter with a length of 3–4 mm and an electron plasma density of (0.2–1.5) × 10²⁰ cm⁻³ on the axis. The jet propagates with a velocity of 160–200 km/s. A magnetized transport was tested using the magnetohydrodynamic (MHD) approach. The electron density and temperature of the plasma during laser irradiation were measured using x-ray K-shell spectroscopy. Comparison of experimental spectra with spectral simulations showed that the electron density increases by a factor of 2 in a field of 60–70 T due to magnetic field confinement.

Experiments were performed using Zebra, a pulsed power generator which produces a 1 MA current pulse with a rising edge of ∼80 ns from 10% to 90%.^14,15 Coil loads with internal diameters of 2.5–3.5 mm and an inductance of 1–2 nH were used for generation of the longitudinal magnetic field. Loads were installed in a current return cage 8 cm in diameter. The Faraday rotation method applied in Ref. 12 was used to characterize the magnetic field in various coil loads. An accurate measurement of strong fast rising B-fields is complicated by the extreme conditions near the coil. The rate dB/dt of the magnetic field generated by the laser-driven method exceeds 100 GT/s.

We measured the magnetic field on the axis of the coil load using two-color Faraday rotation in a small glass disk placed at the supposed position of the laser irradiated target.¹² A magnetic field of 75 T was measured at a distance of 1 mm from the edge of the half turn coil and B = 150 T in the coil center with the current of 1 MA in the loads. These values are in agreement with the calculated magnetic fields. A half-turn copper coil load between the anode and cathode electrodes is shown in Fig. 1(a).

A Leopard laser was used to produce plasma embedded in the magnetic field of the coil. The laser operated at the wavelength of 1057 nm with a pulse duration of 0.4 ps, an energy of 10–12 J on the target, and an amplified spontaneous emission contrast of 10⁶.^16,17 The wave front of the laser beam was corrected by an adaptive mirror. Laser radiation was focused by a F/1.5 parabolic mirror on the target with intensity in the focal spot of 10¹⁸–10¹⁹ W/cm². An alignment system located in the vacuum chamber of the Zebra generator focused the laser beam to a spot <10 μm in diameter.¹⁴ 

Eddy currents that arise in the magnetic field of the Zebra generator with a rate of dB/dt ∼ 1 GT/s produce plasma on metal targets.¹² For this reason, LPI was studied with Si and Si-CH targets. Laser imaging diagnostics indicate that the Si and CH targets remain intact in the magnetic field before the arrival of the main laser pulse. Si targets coated by Parylene N foil were used to identify a source of Si K-shell radiation. Laser targets are placed out of the coil load to avoid the impact of the coil plasma on targets.

Laser diagnostics of plasma included two-frame shadowgraphy and a shearing air-wedge interferometer operated at the wavelength of 532 nm. The delay between the two orthogonally polarized frames was 6.5 ns or 15 ns. The duration of the diagnostic laser pulse was 150 ps. Images were taken using CCD cameras filtered by the 10 nm interference filters. Figure 1(b) shows a sketch of the configuration of the half turn coil, target, heating laser beam, and laser diagnostics.

X-ray diagnostics included x-ray and photoconductive detectors, time-integrating spectrometers with bent KAP and quartz crystals, and a time-integrating pinhole camera. A spherically bent concave quartz crystal, cut 1010 with a spectral resolution >1300 and a spatial resolution of 0.1 mm, was used in the range of 7.1–8 Å (1.55–1.75 keV).

The first experiments were carried out with Al targets and spiral stainless steel coil loads for B-field generation. In this case, the laser pulse interacted with preplasma that formed on the targets by eddy currents. Plasma on the Al target due to eddy currents in the magnetic field with a rate of 1 GT/s was observed later in Ref. 12. Later experiments were performed with Si 10 μm foil targets, 1.5 × 1.5 mm, bare or coated by 4 μm CH plastic foil. Spiral coil loads were found to generate a strong x-ray pulse and plasma debris. Figure 2 shows x-ray pinhole images of the laser produced plasma (a) without the B-field and (b) in the magnetic field of a single turn spiral load. Figure 2(a) shows a compact plasma focal spot. Figure 2(b) demonstrates generation of a plasma jet in the axial B-field and a large x-ray spot due to the short circuit discharge in the spiral load. To avoid damage to the laser target by the x-ray pulse and coil plasma, later experiments were performed with half turn coil loads. Copper half turn coils 3.5 mm in diameter generated the magnetic field on the load axis up to 75 T at a distance of 1 mm from the load edge. The laser beam was focused on the target surface with an intensity of (2–5) × 10¹⁸ W/cm².

Figure 3 shows shadowgrams of the Si target taken (a) in the reference shot without current in the load, (b) during the Zebra shot at current I = 0.73 MA and 3 ns before the Leopard laser pulse, and (c) 12 ns after the heating laser pulse. The shadowgram [Fig. 3(b)] indicates the absence of plasma due to the eddy current effect on the Si target before the Leopard laser pulse. The shadowgram [Fig. 3(c)] presents generation of the narrow plasma jet in the axial magnetic field B = 56 T after the interaction with the laser pulse. Plasma on the rear side of the target is not only shorter but also collimated. Figure 4 shows the dynamics of the formation of the plasma jet in two frames of shadowgraphy 2 ns and 8.5 ns after the laser pulse. An image of the plasma plume in the first frame in Fig. 4(a) presents a diamagnetic cavity (1) and the beginning of collimation of the laser plume. Formation of the cavity with a very small density was observed in plasma at smaller magnetic fields.^8,9 MHD simulations in Sec. III B will associate a very small magnetic field in this cavity. The second laser frame in Fig. 4(b) shows that plasma focuses at the distance of ∼1 mm and propagates as a narrow plasma jet. A schlieren effect seen in the presented shadowgrams is evidence of large plasma gradients in the jet. A tip of the plasma jet in the second frame deviates from the initial axis of the jet. This deviation may be a result of the smaller axial magnetic field at a distance of 4–5 mm from the coil. Calculations show that the longitudinal magnetic field in the coil load drops by a factor of 5–8 at this distance. A velocity of 160–200 km/s was calculated for the propagation of the jet tip. This was done using the interferograms and the two-frame shadowgrams shown in Fig. 5. The range of velocities is close to the expansion velocity of the plasma disk in the transverse magnetic field measured in Ref. 7.

The electron density of the laser-produced Si plasma jet was reconstructed from the interferogram in Fig. 3(c). Figure 6(a) presents a map of the plasma density calculated from the 2D phase map with Abel inversion. Figure 6(b) shows three density profiles of the jet with a gradient of ∼10²¹ cm⁻⁴. A width of the jet is in the range of 0.2–0.3 mm. MHD simulations of the formation of plasma jets in the longitudinal magnetic field are presented in Sec. III B.

Simulations were carried out using the radiation hydrodynamic code HYDRA including an MHD^18–20 description on a mesh, symmetric about the propagation axis of the jet. The laser energy deposition includes the absorption at the laser turning point (at 50%), where the energy being deposited in the simulations is made to match that of the experiment. Hot electrons generated at high laser intensity are not included in a code like HYDRA. The magnetic field from the coil is generated by imposed boundary conditions and has the same spatial profile as the magnetic field in experiment shown in Fig. 3(d). The thickness of targets matches those in experiments. As the plasma ablates off the target, the magnitude of the magnetic field inside the jet is much lower than that of the initial external field, while the field at the edges of the jet grows stronger than the initial field, leading to high magnetic pressure and to collimation. Figure 7 illustrates the process of collimation by showing the density and field strength at a delay of 1 ns after the end of the pulse. The density is concentrated in the highly magnetized region of the plasma. The density structure is similar to that seen in Ref. 9. The dynamics of the magnetic field in the plasma is dominated by resistive MHD. In the simulations, the laser pulse is too short for the Biermann effect generated magnetic fields to influence the jet formation. HYDRA simulations including the electrothermal terms in Ohm's law, such as Nernst and Seebeck terms, showed only a nominal increase in the size of the diamagnetic cavity compared to the resistive MHD. Due to the induction term, the magnetic field in Fig. 7(a) is compressed at the edges close to the magnetic field envelope and the field is greatly weakened within the envelope. The magnetic field envelope that collimates the plasma exists throughout the time evolution of the jet; however, as the jet propagates, the field diffuses via resistive diffusion into the envelope until the magnetic fields at the edge of the envelope and inside the envelope are only slightly different. In Fig. 8, the jet tip extends to distances comparable to those measured in the experiments and is compared to hydro simulations of the laser on the target without external magnetic fields. In hydro simulations without external magnetic fields, the plasma expansion is much more limited. At a delay of 8.5 ns after the pulse, the jet tip has extended to ∼3.6 mm with a fluid velocity of the dense region of the jet tip at (200–400) km/s. The jet has electron densities of (10¹⁹–5 × 10²⁰) cm⁻³ and electron temperatures of (10–100) eV. The plasma jet is a low magnetic-β structure, where the magnetic pressure maintains the jet collimated with a width of (0.1–0.2) mm as it expands in the direction parallel to the magnetic field. In Fig. 5, the dashed line indicates the position of the jet tip as a function of time in simulations. The comparison of the positions of jet tips in simulations and experiments in Fig. 5 shows that HYDRA simulations for magnetized jets are in agreement with experiments in the range of 4–20 ns. A future attempt with a hybrid fluid-kinetic code may be the way to reproduce the data more accurately.

We studied spectra of the Si laser produced plasmas to find the effect of the magnetic field on the plasma temperature and density. Laser-only shots with Si targets were carried out in a separate vacuum chamber and analyzed.

Previous spectral simulations of plasma produced by the Leopard laser were performed in Ref. 21 with a nonLTE kinetic model of iron L-shell emission. The model fit measured spectra if a 3%–5% fraction of nonMaxwellian electrons with an energy of 3–5 keV were added to the spectral model. Hot electrons shift up the ionization balance and lead to qualitative changes of the satellite line emission.²² 

Experiments driven by the Leopard laser with an intensity up to 10¹⁹ W/cm² and a contrast of 10⁶ generated a nonthermal electron beam with a temperature of 1.5 MeV estimated from the continuum emission slope.²³ A low energy electron component was not measured. Simulations of LPI of the Leopard laser with Al targets at an intensity of 10¹⁹ W/cm² were performed in Ref. 23. In the model, the plasma near the target reaches the critical density during the interaction with the amplified spontaneous emission of the laser prepulse, and hence, the main laser pulse does not interact with the solid target. We estimate that the x-ray time integrating spectrometer recorded x-ray emission from the target's plasma blowoff during the nanosecond prepulse. Simulations²³ predict a two-temperature electron distribution that is generated in the laser plasma interaction region. However, only the energetic component of the electron beam flows into the target during the interaction because the low energy component is trapped by the self-generated magnetic field. Calculations of electron collisional cross-sections and rate coefficients suggest that MeV electrons impact the charge state distribution and x-ray line emission spectra of the plasma.

Time-integrated spectra of Si were analyzed using the PrismSPECT code.²⁴ PrismSPECT utilizes atomic data computed with the ATBASE suite of codes.²⁵ In ATBASE, the atomic structure is computed using Hartree-Fock models and includes configuration interaction. Electron collisional excitation and ionization cross-section calculations utilize the distorted wave (DW) Born method.²⁶ In this method, the radial functions for bound electrons are Hartree-Fock wave functions. The potential function for continuum electrons is taken as the modified Hartree-plus-statistical-exchange (HX) potential which includes screening and exchange effects of the bound electrons. We note that for very high electron energies, the scattering potential may be treated as a small perturbation.

For K-shell transitions in Si, collisional excitation and ionization cross-sections have small but non-negligible values for electron energies in the MeV region. Therefore, for low plasma temperatures, where the thermal electron energies are well below the excitation or ionization thresholds (e.g., ∼1.9 keV for He-α transition), even a small fraction of energetic nonthermal electrons can have a sizable effect on atomic kinetics and spectral signatures. For example, at a temperature of 150 eV, the He-α/y ratio is increased by a factor of two if a fraction of fast electrons increases from 0% to 0.1%. When the bulk temperature increases, more thermal electrons have energies near the peak of the collisional cross-section, and thus, thermal collisional rates become larger and dominant.

Figure 9 shows the K-shell spectrum of Si plasma taken in a laser-only experiment by a convex KAP crystal with a resolution of 500. The experimental spectrum is compared with PrismSPECT spectra. The temperature, density, and fraction of hot electrons varied²⁷ to find the best fit. The sensitivity of the spectrum with hot electron energy is weak in the range of 0.6–3 MeV. The synthetic spectrum is fit with a thermal electron temperature of T_e = 220 eV, an ion density of 3 × 10¹⁹ cm³, and a 2% fraction of hot electrons with an energy of 1.5 MeV. Due to shot-to-shot variations, the typical temperature, density, and fraction varied in the range of T_e = 180–220 eV, (2–3) × 10¹⁹ cm³, and 0.5%–2% of electrons with an energy of 1.5 MeV.

The Si plasma spectra in coupled shots with the Zebra generator were investigated using a more sensitive spectrometer equipped with a spherically bent quartz 1010 crystal.²⁸ Figure 10(a) displays a typical Si spectrum recorded in the Zebra vacuum chamber. The emission of Ly-α and He-α lines and associated satellite lines is dominated by emission from the plasma heated by both the laser pulse and the prepulse. A Si K_α line is observed and interpreted to be produced by the interaction of the hot electron beam with the solid target. Figure 10(b) shows the line intensity ratio of the He-α resonance to intercombination y transition in experiments with and without the magnetic field; this ratio carries information about the electron density.^29,30 Figure 10(c) compares a ratio of intensity of the y intercombination line to the (j + k) satellites; this ratio y/(j + k) can be used to extract the electron temperature. Average values of these ratios with standard deviations ±σ are also displayed. The ratio He-α/y increases with the magnetic field from the range near average 5.6 to the range near 7.1. The ratio y/(j + k) decreases from the range near average 1.7 to the range near 1.2.

Due to the large dispersion of parameters, the experimental datasets were analyzed with a “t”-test³¹ to determine if the two sets of data with B = 0 and B = 60 T were significantly different. The independent two-sample “t”-test for unequal sample sizes and unequal variations was applied. For the y/(j + k) dataset, the statistic parameter t = 1.89 and degree of freedom df ∼ 7. Using the EXCEL statistical function, the interval of 1.89 at the probability of 0.1 was calculated. The two y/(j + k) datasets at B = 0 and B = 60 T are significantly different with a probability of 90%. The same calculations for the He-α/y datasets at B = 0 and B = 60 T give ∼70% probability that two datasets represent different average values.

PrismSPECT simulations were performed to assess the effect of the nonthermal electron beam on Si spectra. Figure 11 shows the He-α/y ratio dependence on the plasma density calculated for several electron temperatures in the range of 150–400 eV. The results both with and without the energetic electron beam show increases of He-α/y with the plasma density expected in the experiment, but with different values and trends. Without the electron beam, experimental values are comparable with calculations only if T_e > 400 eV. However, this temperature range disagrees with the estimation from K-shell spectra in Fig. 9. The diagram in Fig. 11(b) is calculated with a 0.4% fraction of fast electrons. The left part of the diagram is consistent with the experimental range of T_e = 150–200 eV. Horizontal dashed and solid lines show the experimental He-α/y values without and with the magnetic field, respectively. The increase in the He-α/y ratio from 5.6 to 7.1 implies that the plasma density increases by a factor of 2–3 in the magnetic field. The left part of the diagram represents the high ion density, which disagrees with experimental data presented in Fig. 9. In that density range, PrismSPECT simulations show strong widening of spectral lines, which is not consistent with experimental spectra.

Figure 12 compares the ratio of intensities of the intercombination line y and satellites (j + k) without and with the nonthermal electron beam, respectively. PrismSPECT simulations without hot electrons show an increase in the y/(j + k) ratio if the plasma temperature increases as seen in Fig. 12(a). However, the inclusion of a 0.4% fraction of electrons with an energy of 1.5 MeV in the simulation model dramatically changes the trend in Fig. 12(b). Dashed and solid horizontal lines in Fig. 12(b) show the experimental y/(j + k) ratio without and with the magnetic field. The decrease in y/(j + k) from 1.7 to 1.2 corresponds to the temperature increase in 10–20 eV. According to simulations,²³ K-shell thermal spectra are, mostly, radiated by plasma near the target surface expanding during the nanosecond ASE prepulse.

Si 10 μm targets coated by 4 μm or 1 μm CH Parylene N foils were tested to identify a source of Si K-shell radiation. Laser shots with coated Si targets demonstrated that the Si lines vanish with and without the magnetic field. This means that only CH plasma was heated effectively by the laser pulse.

Analysis of experimental data with diagrams in Figs. 11(b) and 12(b) provides two sets of plasma parameters n_i(T_e) and T_e(n_i). For plasma without the magnetic field, two functions taken from y/(j + k) and He-α/y diagrams cross at the temperature T_e of 216 eV and ion density n_i of 0.85 × 10¹⁹ cm⁻³. For plasma in the magnetic field, two functions n_i(T_e) and T_e(n_i) do not cross if average experimental ratios y/(j + k) and He-α/y are used. However, functions cross if we use a 0.5σ deviation from the average value. In this case, two strips of data T_e and n_i cross at T_e = 223 eV and n_i = 2 × 10¹⁹ cm⁻³. The magnetic field B = 60–70 T results in an increase in plasma density by a factor of ∼2 and a negligible 3% increase in the plasma temperature. The increase in the plasma density is a factor of 3.3 if we apply the same 0.5σ deviation to the average value at B = 0.

Laser plasma interaction with thin Si foil targets was studied in a longitudinal B-field of 50–70 T at laser intensities of (2–5) × 10¹⁸ W/cm². The magnetic field generated by coil loads was characterized using Faraday rotation diagnostics. Two frame laser diagnostics showed no impact of eddy current on the target before the laser pulse. Shadowgraphy and interferometry showed generation of narrow dense plasma jets propagating along the longitudinal magnetic field. The plasma plume collimates and forms a jet 0.2–0.3 mm in diameter with a length of 3–4 mm and an electron plasma density of (0.2–1.5) × 10²⁰ cm⁻³ on the jet axis. The tip of the plasma jets propagated in the axial B-field with a velocity of 160–200 km/s. This type of plasma jet is relevant to the large-scale astrophysical jets.⁹ The same physical mechanism may result in the growth of instabilities on the plasma front in strong longitudinal magnetic fields. Simulations with the HYDRA MHD code show collimation of the laser plume in agreement with the plasma dynamics observed in experiments.

The effect of the magnetic field on the density and temperature of the laser produced plasma was studied using K-shell spectra and satellite lines in the Si plasma spectra. High intensity LPI with moderate contrast of the laser pulse results in heating of the plasma near the target and generation of a beam of fast nonthermal electrons. The Leopard laser generates an electron beam with MeV energy.²² The impact of the electron beam on x-ray spectra of Si plasma was analyzed using a PrismSPECT program, which includes interaction with high energy electrons. Simulations showed the strong dependence of K-shell spectra on the nonMaxwellian electron beam. Time integrated spectra of Si plasma were compared with simulations. Comparison of spectra with and without the magnetic field showed >2 times increase in the plasma density and a small increase in the plasma temperature. Spectral measurements of the enhanced plasma density are supported by the dynamics of plasma confined by the magnetic field. The laser produced plasma plume does not spread but forms a compact dense jet confined by the axial magnetic field.

The authors thank Dr. Y. Sentoku, Dr. A. Maximchuk, and Dr. S. A. Pikuz for discussions and the team at the Nevada Terawatt Facility at UNR for help with experiments. We acknowledge the support of the High Performance Computing at LLNL and Dr. M. Marinak. This work was supported by DOE Grant No. DE-SC0016500.