Laboratory generation of strong magnetic fields opens new frontiers in plasma and beam physics, astro- and solar-physics, materials science, and atomic and molecular physics. Although kilotesla magnetic fields have already been produced by magnetic flux compression using an imploding metal tube or plasma shell, accessibility at multiple points and better controlled shapes of the field are desirable. Here we have generated kilotesla magnetic fields using a capacitor-coil target, in which two nickel disks are connected by a U-turn coil. A magnetic flux density of 1.5 kT was measured using the Faraday effect 650 μm away from the coil, when the capacitor was driven by two beams from the GEKKO-XII laser (at 1 kJ (total), 1.3 ns, 0.53 or 1 μm, and 5 × 1016 W/cm2).
Laboratory generation of strong, shaped magnetic fields offers a new experimental test bed to study plasma and beam physics1, astro-2 and solar-physics3,4, materials science5, and atomic and molecular physics6. In fast-ignition laser fusion research, collimation of a relativistic electron beam by an axial magnetic field is a key scheme to increase the coupling efficiency between the laser and the core7. Magnetic field reconnection3,4 and collisionless shock generation1 in a plasma subject to a strong magnetic field are also current research objectives.
Magnetic field is generated spontaneously in a laser-produced plasma8, and several kilotesla field9,10 has been measured in a relativistically intense laser-plasma interaction experiment in small spatial and temporal scales. Kilotesla fields have been produced by magnetic flux compression using imploding metal tubes11 and plasma shells12,13,14,15. Up to 4 kT has been generated by compressing a 6-T seed magnetic field at the OMEGA laser facility. This magnetic field was used in a central-ignition inertial fusion experiment13. The temperature of the hot spot increases by embedding the magnetic field seed in the fusion fuel because the strong field inhibits electron heat conduction to the surrounding cold fuel.
Although the spontaneous magnetic field generation and the flux compression scheme are useful, others techniques are necessary because multipoint accessibility and controlled shape of the field are required for several applications. We have demonstrated that kT magnetic flux densities can be produced using a capacitor-coil target16.
Figure 1 shows photographs of a capacitor-coil target in side and front views. Two nickel disks are connected by a U-turn coil. Kilojoule, nanosecond laser pulses are focused onto the first disk through a hole in the second disk. A plasma is generated at the first disk, and suprathermal hot electrons with temperatures exceeding 10 keV are emitted from the plasma corona17. The hot electrons stream down the electron density gradient ahead of the expanding plasma plume and impact the second disk. The second disk acquires a negative charge, and a large electrical potential develops between the disks. That potential difference drives a current in the U-turn coil. A strong magnetic field pulse is generated in the coil. A previous study of this scheme18 indicated that 100-T magnetic fields could be created. In those experiments, a pick-up coil probe was used to detect the flux density at the coil. However, such probes are susceptible to the electromagnetic noise generated by the laser-plasma interactions.
The Faraday effect is a magneto-optical phenomena, producing a rotation of the plane of polarization that is linearly proportional to the component of the magnetic field in the direction of propagation. The angle of rotation θ is proportional to the length L of the optical path in the medium, the Verdet constant V of the medium, and the magnetic flux density H according to θ = L V H.
The average value of the flux density within a Faraday medium (B@measure in Table 1) is simply calculated as The thickness (100 or 500 μm) of the Faraday medium is comparable to the distance between the coil and the cylinder. Nonuniformity of the magnetic field in the medium is not negligible. Peak normalized profile of the magnetic flux density along the path of probe, shown in Fig. 2, was calculated with the initial shape of the U-turn coil, where x is a coordinate along the path of probe.
Magnetic flux density at the arbitrary position, H (x), is calculated as Here xF and xR are the positions of the front and rear surfaces of the Faraday medium. In Table 1, magnetic flux densities at 850 μm away from the coil are listed for the comparison.
Figure 3 shows a schematic of the magnetic flux density measurement with Faraday effect19. The horizontally polarized second harmonic of a Q-switched Nd:YAG laser (at a wavelength λ of 0.532 μm) was used as the probe light. The probe light pulse is coincident with the GEKKO laser. Fused silica was used as the Faraday medium with a Verdet constant of 298 ± 12 deg/T m at 0.532 μm. The fused silica has a cylindrical shape whose diameter and length are 600 μm and 100 or 500 μm, respectively. The rear surface is located 600 μm away from the coil, therefore the center of the Faraday medium is 650 or 850 μm away from the coil for the 100 or 500 μm thick medium, respectively.
The transmitted probe light is imaged by lens 1 onto the iris. The central part of the image, having a diameter of 100 μm on the Faraday medium, is selected by the iris. The image is transferred to the visible streak camera by lens 2. Heat absorption and bandpass filters between the iris and lens 2 exclude the laser harmonics and the thermal emission from the plasma. It was experimentally confirmed that the our imaging system detects only the probe light. The Wollaston prism divides the rotated light into horizontal and vertical components.
The equator of the image is selected by a slit on the streak camera cathode in Fig. 3 and the streak image is recorded on a CCD camera. The rotation angle θ is determined from the intensity ratio IH/(IH+IV) = cosθ/(cosθ + sinθ) between the horizontal IH and vertical IV components. The temporal resolution is 150 ps.
Figure 4 is a streak image of a magnetic field measurement. A 500 μm-thick cylinder was used as the Faraday medium in this shot (35869). In the reference image, for which a capacitor-coil target was not driven by the GEKKO-XII laser, only the horizontal component was present. When the capacitor-coil target was driven by two beams of the laser, a vertical component appears and the horizontal component disappears 1.5 ns after the laser irradiation. This delay in the magnetic field indicates that the second disk needs 1.5 ns to acquire hot electrons to drive current in the coil. The intensity ratio was 0.26 ± 0.07, corresponding to 71 ± 6 degrees of rotation. The average density of magnetic flux in the Faraday medium is calculated to be 480 ± 60 T from the rotation angle and a Verdet constant. It was found that the probe light was blocked soon after the magnetic field generation.
Maximum rotation angle of 162 ± 7 deg. was obtained with a 500 μm-thick fused silica cylinder in the highest intensity shot (35866). The thickness of a cylinder was changed from 500 μm to 100 μm to confirm that the rotation is occurred in the cylinder. Rotation angle of 45 ± 8 deg. was obtained with a 100 μm-thick cylinder (35870). Positions of the cylinder center (650 μm or 850 μm) were different, two laser beams were not overlapped correctly in the shot (35870), and the intensity was a half of the highest intensity. With the consideration of the above differences, the dependence of the rotation angle on the cylinder thickness supports that the rotation is occurred dominantly in the cylinder.
The transmittance of the fused silica recovered within the duration of the probe light pulse in one shot (35870). The temporal evolution of the magnetic flux density was calculated from the intensity ratio. Dynamics of a plasma between the two nickel disks was simultaneously observed from the side using an x-ray imaging system coupled to an x-ray streak camera.
Figure 5(a) plots the evolution of the magnetic flux density measured with the Faraday effect. The hatched region indicates the duration of the probe pulse black out. Figure 5(b) shows an x-ray streak image of the plasma. The zero of time is the beginning of the laser irradiation. The magnetic field develops at 1.5 ns in Fig. 5(a), and a plasma is generated at the second disk at this same time as shown in Fig. 5(b). Consequently, hot electrons with an expansion speed of 5.1 × 107 cm/s arrive at the second disk and produced a plasma. The magnetic field once disappears at 3.0 ns in Fig. 5(a), but plasma emission at the second disk increases at that time. A plasma plume with an expansion speed of 2.5 × 107 cm/s may arrive at 3.0 ns after the laser irradiation. The resulting potential difference drives the current in the plume.
We measured the magnetic flux densities by varying the intensity and wavelength of the drive laser and the thickness of the fused silica cylinder to obtain a scaling law of the flux density against laser intensity and wavelength. Table 1 and Fig. 6 summarize the maximum magnetic field obtained. Average values of magnetic flux density within the Faraday medium (B@measure) and the flux density at 850 μm away from the coil (B@850 μm) were listed in Table 1. The maximum value of the magnetic flux density measured in this experiment was 1500 ± 330 T, 650 μm away from the coil. The dotted line in Fig. 6 shows a liner line as B(IL) = 2.7 × 10−14IL. Peak density of the magnetic flux seems to be in proportional to laser intensity not product of intensity and the square of laser wavelength.
A pick-up coil probe was also used to detect the magnetic flux density in the lowest laser intensity shot (35477). The diameter of the pick-up coil was 5 mm and the coil was located at 100 mm from the capacitor-coil target. The pick-up signal is easily affected by electro-magnetro noise generated by laser-plasma interactions. In our experimental conditions, we obtained a meaningful signal only at the lowest intensity and a green laser irradiation.
Magnetic field spontaneously generated by laser-plasma interactions overlaps as background on a signal. A planar nickel disk without a coil was irradiated by the laser and the spontaneous field was measured with the pick-up coil. The field density generated with the U-turn coil was ten times higher than the spontaneous one, and direction of the spontaneous field is different from that of the field generated by the U-turn coil, therefore, we neglected the spontaneous field in the Faraday rotation measurement.
Figure 7 shows temporal change of the magnetic flux density measured with the pick-up coil. The field decays with a characteristic time of 28 ns that is not so far from the decay time of a closed circuit with characteristic L/R = 17 ns, where R ~ 5.4 × 10−2 Ω and L ~ 9.2 × 10−10 H were used, respectively. The deep dip around 3 ns is found in the evolution of the field. This phenomenon is found also in the Faraday rotation measurement as shown in Fig. 5, but the mechanism is not clear. Peak flux density was 33 ± 8 T, this is also shown in Fig. 6 and Table 1.
The first important discussion issue is a validity of the Faraday rotation measurement for large B and dB/dt achieved in this experiment. In our experiment, B and dB/dt reach 1.5 kT and 3 kT/ns, respectively. Faraday rotation angles for a GaP crystal at a wavelength of 632.8 nm were measured as a function of a field strength up to 400 T measured by a pick-up coil20. The rotation angle shows a linear increase with the field strength and the coefficient of 10100 deg/T m that is very close to the value measured at weak field strength (10050 deg/T m).
The Faraday rotation in a fused silica fiber was recently used in magnetic compression experiment with a laser-accelerated foil and a cavity15. The time history of the magnetic field during the compression was successfully measured, measured amplification agrees with the ratio between initial and final dimensions of the field embedded area. B and dB/dT reached 800 T and 1 kT/ns in their experiment, therefore this result reveals that the Faraday rotation measurement can be used for at least B < 800 T and dB/dt < 1 kT/ns.
Validity of the Faraday rotation measurement for the large B and dB/dt achieved in this experiment is still an open question, however, nonlinearity of the Faraday effect induced by large B and dB/dt seems not to cause a significant error in the estimation of field strength, because those two values obtained in our experiment are only two or three times larger than the experimentally validated values.
The second discussion issue is total energy of the magnetic field generated by the laser-driven capacitor coil. By comparing the measured field density and that calculated with the initial shape of the U-turn coil, current in the coil and total energy of the field were estimated to be 8.6 MA and 15 kJ. The field energy was obtained by integrating magnetic field energy (B2/μ0) at the field peak timing in the calculation space (2 × 4 × 5 mm3). The total energy is inconceivable, because 15 kJ is much larger than the laser energy (1 kJ). Explosion of the coil due to ohmic heating by a large current and consequent implosion of the field inside the coil is a candidate mechanism to explain the experimental result, but we have no experimental evidence for that. Black-out of the probe light found in the Faraday rotation measurement may be explained by the coil implosion, because the probe light is blocked when the imploding coil collides at the center of the coil. Only a small portion of the magnetic field (100 μm in diameter and 500 μm in length) was measured in the experiment, spatial distribution of the magnetic field must be measured to conclude why the strong field is generated by the laser-driven capacitor-coil target. This measurement is a future work.
One advantage of this scheme is it enables various magnetic field geometries. The GEKKO-XII facility has twelve nanosecond laser beams, and a magnetic mirror or cusp geometry can be produced by using two or four capacitor-coil targets, each driven by two or three laser beams. This geometry affords novel experiments in magnetic field reconnection, as occurs in the solar corona and has previously been studied with a self-generated magnetic field21,22,23,24,25. It can also be used to study collisionless shock26,27,28,29,30 which is a possible mechanism for high-energy particle acceleration and cosmic-ray generation. Dynamics of electron-positron plasmas can be investigated by interactions between an intense laser and a high-Z material in this strong magnetic field31,32.
Astronomers are interested in the x-ray spectrum of pulsars in the presence of megatesla magnetic fields2. Intensity dips2 are observed in the continuum spectrum. One explanation for these dips is Compton scattering in a Landau-quantized plasma. When the cyclotron radius of an electron in a strong magnetic field is small compared to its de Broglie wavelength and the allowed energy levels are discretized, then absorption bands appear. An energy gap equal to 0.1 eV at 1 kT arises in the extreme ultraviolet range.
Two laser beams at GEKKO-XII were used to drive the capacitor-coil target. The maximum total laser energy, minimum laser spot diameter, pulse duration, and wavelengths were 1000 J, 50 μm, 1.2 ns, and 1.053 or 0.53 μm, respectively.
The capacitor-coil target was designed based on previous work16,18. The target consists of two 50-μm-thick nickel disks and a U-turn coil made of 50 μm × 50 μm nickel rod. The diameter of the disk is 3568 μm and the diameter of the hole in the first disk is 1784 μm, to prevent the GEKKO beams from hitting the periphery of the hole. The separation between the two disks is 780 μm, limited by the fraction of the hot electrons that do not reach the second disk due to the E × B drift in the self-generated magnetic field around the plasma. The radius of curvature of the U-turn coil is 500 μm. A glass stalk, used to support the target in the vacuum chamber, is connected to the first disk.
The probe laser had multiple peaks in time-domain due to multi-mode operation of a laser oscillator without a single-mode seeding light, envelope of the probe pulse had 7 ns of full width at half maximum. The probe beams was shaped with a 1-mm-diameter aperture, the shaped probe beam passed through the Faraday medium. Energy of the probe beam was adjusted by inserting several filters in the laser path to the acceptable energy for a visible streak camera.
Verdet constant at 532 nm
The Verdet constant depends on the wavelength of the probe. It was calculated for 532 nm light by a model. The Verdet constant V (in rad T−1 m−1) depends on the wavelength (in nm) according to33 where e, m, and c are the electron charge, electron mass, and speed of light. Here γ is the magnetooptical anomaly equal to 0.692. The dispersion in the refractive index is34 Consequently the Verdet constant at 532 nm is calculated to be 287 deg/T m.
By another method, the Verdet constant at 532 nm was determined to be 310 deg/T m by interpolation of data35,36 listed in Table 2. fitted to the power law V = 298 ± 12 deg/T m is used in the this study as a Verdet constant of a fused silica at 0.532 μm.
Error in estimation of magnetic flux density from faraday rotation
Intensity ratio between horizontal and vertical components of the rotated light (IH/(IH + IV) has fluctuation of ± 0.07 of a typical value. This fluctuation was considered in the calculation of the rotation angle from the intensity ratio. Uncertainly of Verdet constant of a fused silica at 0.532 μm light (V = 298 ± 12 deg./T m) was considered in the calculation of the magnetic flux density from the rotation angle. The upper and lower values of the flux density were calculated by using V = 286 and 310 deg./T m from the rotation angle, respectively. Consequent error was in the range from ± 12 to ± 31% of the flux density.
The authors thank the technical support staff at the GEKKO-XII facility for laser operation, target fabrication, and plasma diagnostics. The authors acknowledge Dr. N. Hasegawa (JAEA) and Dr. K. Kondo (TIT) for valuable contributions to this experiment. The authors also thank Prof. J. Honrubia (UPM), Prof. F. Wang (NAO at CAS), Dr. T. Enoto (RIKEN), Dr. D. Salzmann (WI), and Prof. H. Takabe (ILE) for fruitful discussions. This research was supported by the Japanese Ministry of Education, Science, Sports, and Culture (MEXT), by Special Education and Research Expenses for ‘Laboratory Astrophysics with High-Power Laser’ and a Grant-in-Aid for Young Scientists (A) for ‘Extreme Magnetic Field Generation for Quantum Beam Control and Laboratory X-ray Astronomy (No. 24684044),’ Collaboration Research program by NIFS (NIFS12KUGK057 and NIFS11KUGK054) and by the Institute of Laser Engineering at Osaka University (under contract ‘Laboratory X-ray Astrophysics with Strong Magnetic Field’).