Magnetohydrodynamics (MHD) plays a fundamental role in a wide range of astrophysical phenomena, such as cooling of white dwarf stars^[1], amplification of magnetic fields (-fields) in supernova remnants^[2], formation of solar flares^[3], filamentary structures on the sun^[4], young stellar objects ^[5], accretion disks^[6], and magnetic reconnection^{[7, 8]}. The dynamics and stability of such astrophysical phenomena are governed by interactions between plasmas and magnetic fields, and these interactions have therefore been widely studied in laboratory experiments. Growth or suppression of hydrodynamic instabilities has been observed in plasmas in external -fields^[9–12]. Moreover, in the context of high-energy-density physics (HEDP), the hydrodynamical behavior of plasmas in intense -fields is of great relevance to the development of -assisted inertial confinement fusion^{[13, 14]} and fast ignition^[15–20].

In laboratory experiments, a flux density in the kilotesla region is needed to reveal the effect of -fields under the conditions of a laser-ablated high-energy-density plasma. There are several ways to generate such strong -fields with high-power lasers, including the important Biermann battery effect. In laser–plasma interaction, a toroidal -field is generated along the surface of the expanded plasma, where the gradients of electron density and temperature are noncollinear^[21–23]:

 (1)

The use of a capacitor–coil (CC) target to generate a controllable -field was proposed and realized on the Lekko VIII laser system by Daido et al.^[30] in 1986. Courtois et al.^[31] used a similar mechanism with a Helmholtz coil on the Vulcan laser at the Central Laser Facility in the United Kingdom and obtained a -field of a few tesla. In 2013, Fujioka et al.^[32] redesigned the CC target and used it on the Gekko XII high-power laser facility. A kilotesla level -field was achieved with a laser energy of 1.5 kJ in 1 ns^[32]. As shown in Figure 1, the target is composed of two main parts: a capacitor and a connection coil. The capacitor is formed from two metallic disks placed parallel to each other. There is a hole in the front disk that allows the laser to propagate through and focus on the rear disk. The two disks are connected by the connection coil, which generates a-field when the discharge current flows through it.

In the past few years, laser-driven coils have been applied in many laboratories with similar targets. Some typical results with nanosecond high-power lasers are listed in Table 1, together with the results from Daido et al. and Courtois et al. for comparison. The targets used are illustrated in Figure 2. Pei et al.^[33] used a double-U-turn Helmholtz coil instead of a single coil to produce a counter -field between the coils for magnetic reconnection research. Santos et al.^[35] modified the angle of the coil and the size of the disks to match the setting of the LULI laser facility. A U-shaped foil target was used on Omega by Goyon et al.^[37]. In contrast to the targets described above, Zhu et al.^[38] used a simpler design with only one single open-ended coil.

The main mechanism of laser-driven -field generation involves two steps: establishment of an electrical potential and, as a consequence, generation of a -field by the driven current.

First, when a laser pulse irradiates a disk, some electrons are heated. Suprathermal electrons escape from the disk surface, and a potential barrier is created^{[39, 40]}. This barrier potential pulls back a fraction of the escaping electrons and, at the same time, induces a return current inside the target to neutralize the charge. In comparison with a single open-ended coil^[38], the front disk on the CC target captures a proportion of the escaping electrons, thereby enhancing the potential.

The initial potential is dominated by the effect of the irradiating laser, although it is also affected by the disk material. To increase the initial potential, it is necessary to increase the number of escaping electrons. The number of escaping electrons, and thus their total charge, is determined by both the suprathermal electron temperature and the barrier potential . Considering a simple model and assuming a static barrier potential of , only electrons with an energy greater than can escape from the target surface. Assuming an energy distribution for the suprathermal electrons with an exponential high-energy tail of the form

 (2)

 (3)

 (4)

With the establishment of an initial potential between the two disks, the target behaves toward the potential like a resistor–inductor () electrical circuit. The time evolution of the current can be treated using an model:

 (5)

The -field, generated by the potential and the current in the coil , is controlled by both the incident laser and the shape of the target. For example, from a comparison of the results of Daido et al.^[30] and Fujioka et al.^[32], it can be seen that although similar target geometries were applied, the strength of the -field was increased over 25 times by increasing the laser energy by a factor of 15. Furthermore, although the use of a Helmholtz coil by Courtois et al.^[31] dramatically increased the inductance and thus decreased the peak -field, the duration of the field was increased compared with the case of a single coil, and much better spatial uniformity was achieved. Thus, it is necessary to consider not only the peak strength of the -field, but also its temporal evolution and spatial distribution.

As well as those mentioned in the above discussion, some other models have been proposed for the study of laser-driven -fields^[41–43].

A rapidly changing -field can be measured in many ways. One traditional method uses a pick-up coil, also known as a B-dot probe. The basic idea is to utilize Faraday’s law to measure -field fluctuations ^[44]. However, B-dot measurements usually suffer from strong electromagnetic pulse noise from laser–plasma interactions, and it is therefore necessary to adopt techniques that can distinguish between the signal from the magnetic coil and that from laser–plasma interactions. First of all, for a single B-dot probe, the probe should consist of a differentially twisted pair so that symmetric signals then indicate a -field, while asymmetric signals might be caused by heating of the probe by high-energy particles or X-rays. Figure 4 shows a typical signal obtained by Zhu et al.^[38], in which two symmetric pulses from the differential pair can be clearly seen. Other approaches to alleviating the noise problem involve placing multiple B-dot probes at different positions or using a multidimensional B-dot probe that detects the -field along three orthogonal axes at the same point^[45]. Since the -field generated from a laser-driven coil is strongly directional, while the noise from laser–plasma interactions is relatively isotropic, the -field from the coil can be extracted by comparing signals at different points or from different directions. One disadvantage of the B-dot probe concerns bandwidth. For a -field driven by a picosecond or femtosecond laser, the rise time can be at a picosecond scale, so the bandwidth of the B-dot probe and that of the oscilloscope need to be very high to evaluate the peak of the -field, and this is very hard to achieve.

Another approach to measuring the -field involves the use of an optical diagnostic technique, such as that based on the Faraday effect. The Faraday effect is a magneto-optical phenomenon in which, when polarized light propagates through a magneto-optical medium in a -field, the plane of polarization is rotated, with the angle of rotation being proportional to the strength of the -field in the direction of propagation:

 (6)

 (7)

 (8)

Another approach applies the Cotton–Mouton effect, which occurs when a Cotton–Mouton medium is subjected to a -field^{[46, 47]}. A linearly polarized probe laser beam becomes elliptically polarized owing to the Cotton–Mouton effect, and, by measuring the ellipticity , the -field can be estimated by solving the equation

 (9)

Additional measurements of the plasma profile, especially the electron density profile , are essential for accurate evaluation of the -field strength.

With the target normal sheath acceleration (TNSA), it is easy to obtain high-energy proton beams with a cutoff energy of tens of MeV^{[48, 49]}. An alternative is to generate a monoenergetic proton beam as a product of nuclear fusion^[23]. The protons are deflected by the Lorentz force while propagating through the magnetic coil region. An example of an experimental setting used by Santos et al.^[35] is shown in Figure 6(a). The experiments were conducted at the LULI Pico 2000 laser facility with synchronized nanosecond and picosecond laser pulses. The nanosecond laser beam was focused on a CC target to generate a -field in the coil. A secondary target was positioned beside the coil, and TNSA proton beams directed toward the coil were driven by the picosecond laser. A mesh was placed between the secondary target and the coil to quantify the -field. Figure 6(b) shows an image of proton deflectometry achieved with 13 MeV protons. The deformation of the mesh shadow and a bulb shape at the coil can be clearly seen. Figure 6(c) shows the result of a three-dimensional Monte Carlo simulation of the trajectories of 13 MeV protons through the coil. By comparing the experimentally obtained proton image with the simulation results, information about the -field can be derived. Proton deflectometry allows direct measurement of the -field in the coil region, with spatial and temporal resolutions being obtained in a single shot. The spatial resolution is found by recording a two-dimensional deflection pattern with radiochromic film (RCF) stacks. Protons are stopped at different layers in the RCF, with each layer corresponding to a specific proton energy. By taking into account the time of flight between the proton source and the coil, the pattern on each RCF layer represents the -field at different times. Evaluations of the strength of a laser-driven -field using proton deflectometry have been reported previously by Li et al.^{[23, 50]} and have been applied on a coil target by Santos et al.^[35], Zhu et al.^[38], Gao et al.^[36], Law et al.^[34] and Goyon et al.^[37].

As well as protons, high-energy electrons can be employed in deflectometry measurements. The energies of laser wakefield accelerated electrons can reach hundreds of MeV and even the GeV level^[51], and, under particular conditions, a monoenergetic electron beam can be obtained^[52]. Such electron beams provide a powerful diagnostic tool for -fields. The Larmor radius for a relativistic electron in the presence of a -field is

 (10)

We present here some typical applications of laser-driven coils to generate strong -fields for laboratory studies.

Magnetic reconnection is an important process in many astrophysical phenomena, including solar flares, star formation and auroras, and there have been a number of laboratory investigations based on laser–plasma interactions^{[7, 8]}. To mimic the conditions of low- magnetic reconnection (where is the ratio of plasma pressure to magnetic pressure), it is necessary to apply a -field to a low-density plasma. Pei et al.^[33] used a double-U-turn coil for this purpose, with an antiparallel -field being generated between the two coils. Heating by the current in the coil and by X-rays from the laser ablation leads to the creation of a low-density plasma, which then flows into the -field. Low- magnetic reconnection can occur under these conditions.

A collimated relativistic electron beam (REB) is useful for many applications, but the REBs produced by intense laser interaction with a solid target are usually strongly divergent. Simulations have shown that a kilotesla external -field could effectively decrease the angle of divergence of the REB^[16]. Such collimated REBs have been experimentally investigated by Bailly-Grandvaux et al.^[20].

Matsuo et al.^[12] applied a pair of CC targets to generate a quasi-uniform -field of a few hundred tesla in a laser-produced plasma. By adding a secondary laser-ablated sample in the -field, the MHD properties of the plasma were studied. Electron thermal conduction in the external -field was found to be strongly affected, leading to velocity changes in the ablated plasma. The results also indicated that the growth of hydrodynamic perturbations was also affected by the external -field as a result of the anisotropic thermal conductivity in the ablated plasma.

The use of laser-driven coils is an important method to provide a strong -field at a sub-kilotesla to a kilotesla level. In addition to the high field strength achievable, the controllability of generation and accessibility of a secondary sample are other advantages of this scheme. The strength, space and time ranges of the -field could be further optimized depending on the desired application through appropriate choices of incident laser, target and coil shapes. This scheme could provide a new test bed for a range of laboratory applications including, but not limited to, HEDP, laboratory astrophysics, fusion research and laser particle acceleration. Many types of target have been tested under different laser conditions. There are also some other advanced designs for generation of strong -fields using short-pulse lasers, such as the snail-shaped target proposed by Korneev et al.^[53].