2 Simulation results

2.1 3D EPOCH simulations

We conducted 3D PIC simulations using the open-source fully relativistic code EPOCH^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{39 ]}. In the simulations, the overall computational domain and a single unit cell are both cubic with sizes of $16\ \unicode{x3bc}\text{m}\times 16\ \unicode{x3bc}\text{m}\times 16$ μm and $20\ \text{nm}\times 20\ \text{nm}\times 20$ nm, respectively. Initially, a cubic aluminum target with the size of $14\ \unicode{x3bc}\text{m}\times 14\ \unicode{x3bc}\text{m}\times 14\ \unicode{x3bc}\text{m}$ is set at the center, which has a cylindrical cavity with a radius of ${R}_0=5$ μm, and its axis overlaps the $z$ -axis. The length of the target along the z-axis is chosen to be ${L}_0=7$ μm. The target is composed of a cold and charge-neutral plasma with ion and electron densities of ${n}_{\mathrm{i}0}=5.0\times {10}^{22}$ cm ${}^{-3}$ and ${n}_{\mathrm{e}0}={Zn}_{\mathrm{i}0}=6.5\times {10}^{23}$ cm ${}^{-3}$ , respectively, with the ionization state $Z=13$ . Note that, in the following simulations, we assume the fully ionized states of the target materials, that is, $Z=6$ for carbon and $Z=13$ for aluminum, which are well validated under the applied laser intensities ${I}_{\mathrm{L}}\gtrsim {10}^{20}$ W cm ${}^{-2}$ . In addition, because the characteristic electron energies of MTI are of the order of the mega-electron volt (MeV), the plasma is postulated to be collisionless. For a solid cell, 10 and 20 pseudo particles, which respectively correspond to ions and electrons, are assigned. A seed magnetic field ${B}_0=6$ kT parallel to the $z$ -axis is uniformly set over the entire domain. Owing to limitations in our computational ability, the above cell size and particle numbers assigned to a solid cell are substantially coarser than those treated in 2D simulations, as discussed below.

The four faces of the target parallel to the $z$ -axis are normally irradiated by four laser pulses simultaneously. The pulses are polarized such that the laser electric fields are perpendicular to the seed magnetic field. The applied laser pulses are spatially uniform (planar waves) with a Gaussian shape in time, a laser wavelength ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm, a peak intensity ${I}_{\mathrm{L}}=3.0\times {10}^{21}$ W cm ${}^{-2}$ and a pulse duration ${\tau}_{\mathrm{L}}=50$ fs (FWHM: full width at half maximum). Figures 2(a) and 2(b) show the perspective views of the normalized ion density ${n}_{\mathrm{i}}/{n}_{\mathrm{i}0}$ and the $z$ -component of the magnetic field ${B}_{{z}}$ , respectively, when the peak value ${B}_{{z}}=1.03$ MT is achieved at $t\simeq 200$ fs on the central axis. The figures show a quarter of the full target volume where the two centrally orthogonal surfaces are on the left and their magnified views are on the right. Comparing the 2D and 3D results indicates that the overall plasma behavior and the achieved key physical quantities, such as the maximum magnetic field and current, agree with each other. In fact, ${B}_{\mathrm{max}}=1.2$ MT is obtained in the 2D EPOCH simulation, as shown below, instead of ${B}_{\mathrm{max}}=1.0$ MT in the 3D simulations.

Figure 2

Perspective views of the normalized ion density ${n}_{\mathrm{i}}/{n}_{\mathrm{i}0}$ and the $z$ -component of the magnetic field ${B}_{{z}}$ , respectively, observed at $t\sim 200$ fs, which is obtained by a 3D EPOCH simulation. A cubic aluminum target with a size of $14\ \unicode{x3bc}\text{m}\times 14\ \unicode{x3bc}\text{m}\times 14\ \unicode{x3bc}\text{m}$ is set at the center, which has a cylindrical cavity with a radius of ${R}_0=5\kern0.1em$ μm and an axis overlapping the $z$ -axis. The seed magnetic field ${B}_0=6$ kT parallel to the $z$ -axis is uniformly set over the entire domain. The four faces of the target parallel to the $z$ -axis are normally irradiated by uniform laser pulses simultaneously, which are characterized by ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm, ${I}_{\mathrm{L}}=3\times {10}^{21}$ W cm ${}^{-2}$ and ${\tau}_{\mathrm{L}}=50$ fs.

Owing to the expansion of the inner wall plasma toward the central axis, the plasma density value drops two orders of magnitude lower in the middle of the cylindrical cavity. However, both the ions and electrons are rapidly compressed near the center due to the cylindrical converging effect, which reaches a few times the value of the solid density. As a result, a hollow cylindrical structure with a diameter of approximately $700$ nm forms along the $z$ -axis (Figure 2(a)). This round-shaped cross-section forms the Larmor ring^[ Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov ^{25 ]}, which is composed of an envelope of the converging and diverging ion trajectories under the effect of the seed magnetic field. The ions and electrons around the Larmor ring collectively work to generate ultrahigh magnetic fields of the MT order at the center, as confirmed in Figure 2(b). Thus, Figure 2 demonstrates that the MTI concept indeed works under the 3D configuration with practical target and laser conditions. From the 3D simulations, a target structure with ${L}_0/{R}_0\gtrsim$ 1–2 is sufficient to achieve the same degree of the maximum magnetic field as the 2D results (not described in detail in this paper).

2.2 2D EPOCH simulations

We also conducted 2D PIC simulations under similar conditions to the above 3D case. Here we employ solid aluminum as the target material. For a solid element, we set 50 and 100 pseudo particles/cell for ions and electrons, respectively. The full size of a computational box is $22\ \unicode{x3bc}\text{m}\times 22\ \unicode{x3bc}\text{m}$ , while the size of a single cell is 10 nm. This corresponds to a resolution of 100 cells/μm with a total simulation domain size of $2200\times 2200$ meshes. Although uniformity issues with respect to laser illumination and plasma implosion are crucial topics of MTI, especially in terms of the number of laser beams^[ Reference Murakami and Nishi ^{40 ]}, they are beyond the scope of this study.

The square-shaped microtube target with a coaxial circular void is placed at the center of the computational box. The initial inner radius of the microtube ${R}_0=5\kern0.1em$ μm and the minimum thickness of the wall $\Delta R=2\kern0.1em$ μm are fixed. The side lengths of the square are ${D}_0=2\left({R}_0+\Delta R\right)=14\kern0.1em$ μm. The seed magnetic field is again ${B}_0=6$ kT along the $z$ -axis. Note that ${B}_0=4$ kT produces almost the same simulation results as ${B}_0=6$ kT, while ${B}_0=2$ kT has a significantly lower performance for the magnetic field generation than ${B}_0=6$ kT (not discussed in detail in this paper). Similar to the 3D case, the four faces of the microtube are normally and simultaneously irradiated by uniform laser pulses with ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm and $\tau =50$ fs (FWHM). In the 2D simulations, the laser peak intensity ranges from ${I}_{\mathrm{L}}=3.7\times {10}^{19}$ to $2.7\times {10}^{22}$ W cm ${}^{-2}$ .

Figure 3 shows the temporal evolution of the magnetic field generation with four different laser intensities. The sign of the generated magnetic fields is positive if the generated fields are in the same direction as the seed magnetic field. At $t\sim 110$ fs, the peak of the laser pulse reaches the target center. The maximum magnetic fields generated by MTI depend on the applied laser intensity. As the laser intensity increases, the magnetic field is generated earlier and reaches a higher peak ${B}_{\mathrm{max}}$ . This is because the imploding plasma driven by hot electrons, which has a higher temperature due to heating with a higher laser intensity^[ Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov ^{25 ]}, fills the microtube cavity more quickly. As expected, all those curves show that the magnetic fields are generated in the same direction as that of the seed magnetic field.

Figure 3

Temporal evolution of the central magnetic field, obtained from 2D EPOCH simulations, under four different laser intensities ${I}_{\mathrm{L}}$ , which are labelled and applied to an aluminum target ( ${n}_{\mathrm{i}0}=5\times {10}^{22}$ cm ${}^{-3}$ , $A=27$ and $Z=13$ ). Other fixed parameters are ${R}_0=5\kern0.1em$ μm, ${D}_0=14\kern0.1em$ μm, ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm, ${\tau}_{\mathrm{L}}=50$ fs and ${B}_0=6$ kT. The target is assumed to be uniformly irradiated on the four sides. The laser peak time is $t=75$ fs. The four highlighted circles on the green curve correspond to the sampling times for the 2D patterns given in Figure 4.

Figure 4 shows snapshots of the 2D patterns for the magnetic field (upper row), the total current (middle row) and the electron density (lower row) normalized by the initial value ( ${n}_{\mathrm{e}0}={Zn}_{\mathrm{i}0}=6.5\times {10}^{23}$ cm ${}^{-3}$ ), which correspond to the four highlighted times on the green curve in Figure 3 ( ${I}_{\mathrm{L}}=3.0\times {10}^{21}$ W cm ${}^{-2}$ ). Just after the collapse of the microtube cavity at $t=130$ fs (Figure 3), spin currents run around the center form. The currents reach 0.3 PA cm ${}^{-2}$ with a characteristic diameter of $\sim 1\kern0.1em$ μm at $t=190$ fs when the maximum magnetic field ${B}_{\mathrm{max}}\simeq 1.2$ MT. These spin currents are produced mainly by electrons. This is also supported by the fact that the ring structures of the electron density correspond to the solid density, ${n}_{\mathrm{e}}\sim 0.3{n}_{\mathrm{e}0}\simeq 2\times {10}^{23}$ cm ${}^{-3}$ . Meanwhile, practically no electrons are contained in the central area, which is referred to as the Larmor hole in Ref. [Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov25].

Figure 4

Snapshots of the 2D patterns for the magnetic field (upper row), the total current vectors (middle row) and the electron density (lower row) normalized by the initial value ( ${n}_{\mathrm{e}0}={Zn}_{\mathrm{i}0}=6.5\times {10}^{23}$ cm ${}^{-3}$ ), corresponding to the four highlighted times on the green curve in Figure 3 ( ${I}_{\mathrm{L}}=3.0\times {10}^{21}$ W/cm ${}^2$ ). Generated magnetic fields are assumed to be positive if they are in the same direction as the seed magnetic field ( ${B}_0=6$ kT). Just after the collapse of the microtube cavity at around $t=130$ fs, the spin-structured plasma flow due to the seed magnetic field is formed, increasing the magnetic strength, as observed in the current patterns.

3 Model for laser scaling

The achievable magnetic field of an MTI is given as^[ Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov ^{25 ]}

(1) \begin{align}{B}_{\mathrm{max}}\kern0.1em \left[\mathrm{MT}\right]=\frac{{\left(Z/6\right)}^{3/2}}{{\left(A/12\right)}^{1/2}}\left(\frac{n_{\mathrm{i}0}}{10^{23}\kern0.1em {\mathrm{cm}}^{-3}}\right)\left(\frac{R_0}{3\,\,\unicode{x3bc}\mathrm{m}}\right)\sqrt{\frac{{\mathrm{\mathcal{E}}}_{\mathrm{av}}}{6\ \mathrm{MeV}}},\end{align}

where $A$ and ${\mathrm{\mathcal{E}}}_{\mathrm{av}}$ are the atomic mass number for the target material and the average energy of the hot electrons, respectively. The derivation of Equation (1) assumes that the electron energy distribution follows the Maxwell–Jüttner (M-J) distribution^[ Reference Jüttner ^{41 ]}, which is characterized by a single temperature ${T}_{\mathrm{e}}$ , because the ${\mathrm{\mathcal{E}}}_{\mathrm{av}}$ of interest spans the relativistic regime. For the M-J distribution ( ${\mathrm{\mathcal{E}}}_{\mathrm{av}}\gg {mc}^2$ ), the two variables are approximately related as ${\mathrm{\mathcal{E}}}_{\mathrm{av}}\simeq 3{T}_{\mathrm{e}}$ . Here, we use Equation (1) to derive the scaling in terms of the practical laser and target parameters, and then to confirm that the scaling well reproduces the simulation results.

Firstly, we simply estimate the averaged electron energy ${\mathrm{\mathcal{E}}}_{\mathrm{av}}$ assuming that the absorbed laser energy is uniformly transferred to all the electrons contained in the target as

(2) \begin{align}{\mathrm{\mathcal{E}}}_{\mathrm{a}\mathrm{v}}=\frac{4{\eta}_{\mathrm{a}}{D}_0{\tau}_{\mathrm{L}}{I}_{\mathrm{L}}}{\left({D}_0^2-\pi {R}_0^2\right){Zn}_{\mathrm{i}0}},\end{align}

where ${\eta}_{\mathrm{a}}$ and ${D}_0$ denote the laser absorption efficiency and the initial length of each side of the square cross-section target, respectively. To derive Equation (2), we assume that the target is irradiated on its four sides by spatially uniform lasers at an intensity of ${I}_{\mathrm{L}}$ with a Gaussian pulse duration of ${\tau}_{\mathrm{L}}$ (FWHM).

A previous study summarized various experimental data on the laser absorption efficiency for the domain ${10}^{-2}\lesssim {\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\lesssim 10$ , and gave the laser absorption efficiency in the form of $0.1{\left({{\kern2pt}\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\right)}^{1/4}\lesssim {\eta}_{\mathrm{a}}\lesssim 0.8{\left({{\kern2pt}\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\right)}^{1/4}$ , where ${\widehat{I}}_{\mathrm{L}}={I}_{\mathrm{L}}/{10}^{20}$ W cm ${}^{-2}$ and ${\widehat{\lambda}}_{\mathrm{L}}={\lambda}_{\mathrm{L}}/1$ μm^[ Reference Levy, Wilks, Tabak, Libby and Baring ^{42 ]}. The complementary domains, ${\eta}_{\mathrm{a}}\lesssim 0.1{\left({{\kern2pt}\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\right)}^{1/4}$ and ${\eta}_{\mathrm{a}}\gtrsim 0.8{\left({{\kern2pt}\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\right)}^{1/4}$ correspond to the ‘forbidden’ domain^[ Reference Levy, Wilks, Tabak, Libby and Baring ^{42 ]}, in which no experimental results can be found. In principle, ${\eta}_{\mathrm{a}}$ is bounded as ${\eta}_{\mathrm{a}}\leqq 1$ . Thus, ${\eta}_{\mathrm{a}}$ can be approximately formulated in a simple form that can also be used for a higher intensity domain ${\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\gtrsim 10$ :

(3) \begin{align}{\eta}_{\mathrm{a}}\approx {\left(1+\frac{1}{\eta_0{\left({{\kern2pt}\widehat{I}}_{\mathrm{L}}{\widehat{\lambda}}_{\mathrm{L}}^2\right)}^{1/4}}\right)}^{-1},\kern1em 0.1\lesssim {\eta}_0\lesssim 0.8.\end{align}

From Equations (1)–(3), the scaling for the maximum magnetic fields is obtained in terms of the laser and target parameters as

(4) \begin{align}{B}_{\mathrm{max}}\kern0.1em \left[\mathrm{MT}\right]=0.52{\left(\frac{\eta_{\mathrm{a}}{\widehat{\tau}}_{\mathrm{L}}{\widehat{I}}_{\mathrm{L}}{\widehat{D}}_0{\widehat{R}}_0^2{\widehat{n}}_{\mathrm{i}0}{Z}^2}{\left({{\kern2pt}\widehat{D}}_0^2-\pi {\widehat{R}}_0^2\right)A}\right)}^{1/2},\end{align}

where ${\widehat{n}}_{\mathrm{i}0}={n}_{\mathrm{i}0}/{10}^{23}$ cm ${}^{-3}$ , ${\widehat{\tau}}_{\mathrm{L}}={\tau}_{\mathrm{L}}/1$ ps, ${\widehat{R}}_0={R}_0/5\kern0.1em$ μm and ${\widehat{D}}_0={D}_0/5\kern0.1em$ μm.

Figure 5 shows the scaling for the maximum magnetic field ${B}_{\mathrm{max}}$ , laser energy ${E}_{\mathrm{L}}$ and laser power ${P}_{\mathrm{L}}$ in terms of the laser intensity ${I}_{\mathrm{L}}$ for a square target with ${R}_0=5\kern0.1em$ μm and ${D}_0=14$  μm and a seed magnetic field ${B}_0=6$ kT. Laser pulses with ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm, ${\tau}_{\mathrm{L}}=50$ ps are assumed to uniformly irradiate the four target surfaces. For the 2D EPOCH simulations, here we employ two different solid materials: aluminum ( ${n}_{\mathrm{i}0}=5\times {10}^{22}$ cm ${}^{-3}$ , $Z=13$ and $A=27$ ) and carbon ( ${n}_{\mathrm{i}0}=1\times {10}^{23}$ cm ${}^{-3}$ , $Z=6$ and $A=12$ ), to compare the performance of magnetic field generation. The blue and red circles denote aluminum and carbon, respectively.

Figure 5

Scaling for ${B}_{\mathrm{max}}$ , ${P}_{\mathrm{L}}$ and ${E}_{\mathrm{L}}$ in terms of ${I}_{\mathrm{L}}$ . A square target is used with parameters ${R}_0=5\kern0.1em$ μm and ${D}_0=14\kern0.1em$ μm. The laser is assumed to uniformly irradiate the four target surfaces. The fixed parameters for the 2D simulations are ${\lambda}_{\mathrm{L}}=0.8\kern0.1em$ μm, ${\tau}_{\mathrm{L}}=50$ ps and ${B}_0=6$ kT. Yellow shading denotes the model prediction, which is given in Equation (4), and the two bounding dashed curves correspond to the minimum and maximum laser absorption efficiencies ( $0.1\lesssim {\eta}_0\lesssim 0.8$ ) in Equation (3). To draw the model curve for ${P}_{\mathrm{L}}$ and ${E}_{\mathrm{L}}$ , an optimized aspect ratio $1\lesssim {L}_0/{R}_0\lesssim 2$ is postulated.

Surprisingly, the simulation results for these two materials are in close vicinity to each other at all applied laser intensities. This coherent behavior can be explained by Equation (4), because the material-dependent part of Equation (4) has similar values for the two materials (i.e., ${\widehat{n}}_{\mathrm{i}0}{Z}^2/A\simeq 3$ ). At first glance, the results appear to be independent of the material.

The yellow shading in Figure 5 denotes the model prediction, where Equation (4) is applied with the above laser and target parameters. The shading is bounded by the two dashed curves, which correspond to the minimum and maximum laser absorption efficiency ( $0.1\lesssim {\eta}_0\lesssim 0.8$ ) in Equation (3). The simulation results are well reproduced by the model prediction, especially in the range of $5\times {10}^{20}\lesssim {I}_{\mathrm{L}}\left[\mathrm{W}\kern0.1em {\mathrm{cm}}^{-2}\right]\lesssim 5\times {10}^{21}$ . It should be noted that a lower seed magnetic field ${B}_0=4$ kT gives similar simulation results to those in Figure 5 with ${B}_0=6$ kT. Moreover, the model, which is given by Equation (4), can well reproduce the EPOCH simulation results not only for the specific values of ${R}_0=5\kern0.1em$ μm and ${D}_0=14\;$ μm, but also for targets with different values of ${R}_0$ and ${D}_0$ ranging $3\ \unicode{x3bc} \mathrm{m}\lesssim {R}_0\lesssim 7\ \unicode{x3bc}$ m and $10\ \unicode{x3bc} \mathrm{m}\lesssim {D}_0\lesssim 18\ \unicode{x3bc}$ m, respectively. The scaling, ${B}_{\mathrm{max}}\propto {I}_{\mathrm{L}}^{1/2}$ , can be roughly concluded from Figure 5, which is also confirmed in Equation (4).

The MTI process has two characteristic timescales: the laser pulse duration ${\tau}_{\mathrm{L}}$ and the hydrodynamic time scale ${\tau}_{\mathrm{H}}\sim {R}_0/{c}_{\mathrm{s}}$ , where ${c}_{\mathrm{s}}={\left({ZT}_{\mathrm{e}}/{m}_{\mathrm{i}}\right)}^{1/2}$ denotes the sound speed, at which the microtube wall expands^[ Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov ^{25 ,} Reference Grevich, Pariskaya and Pitaevskii ^{43 ]}. From Equation (2), ${c}_{\mathrm{s}}\propto {\mathrm{\mathcal{E}}}_{\mathrm{av}}^{1/2}\propto {I}_{\mathrm{L}}^{1/2}$ , and therefore ${\tau}_{\mathrm{H}}\propto {I}_{\mathrm{L}}^{-1/2}$ . To enhance the magnetic field generation in MTI, ${\tau}_{\mathrm{L}}$ and ${\tau}_{\mathrm{H}}$ must have coherent lengths. If ${\tau}_{\mathrm{L}}\ll {\tau}_{\mathrm{H}}$ , the absorbed laser energy cannot be effectively transferred to the central core, where strong spin currents and, consequently, strong magnetic fields are generated. Thus, the hydrodynamic efficiency is reduced. On the other hand, if ${\tau}_{\mathrm{L}}\gg {\tau}_{\mathrm{H}}$ , the timing for a substantial part of the laser absorption is out of phase with MTI, reducing the practical absorption efficiency. In fact, the discrepancy between the simulation results and the model prediction in Figure 5 increases for both the laser intensity regimes, ${I}_{\mathrm{L}}\lesssim {10}^{20}$ W cm ${}^{-2}$ and ${I}_{\mathrm{L}}\gtrsim {10}^{22}$ W cm ${}^{-2}$ , which respectively correspond to ${\tau}_{\mathrm{L}}\ll {\tau}_{\mathrm{H}}$ and ${\tau}_{\mathrm{L}}\gg {\tau}_{\mathrm{H}}$ . In addition, for such an ultrahigh intensity regime, ${I}_{\mathrm{L}}\gg {10}^{22}$ W cm ${}^{-2}$ , both laser-accelerated ions and electrons become highly relativistic, degrading the generation of ultrahigh spin currents and ultrahigh magnetic fields.

The interplay between the converging and diverging relativistic charged particles, which is driven by the laser-heated quasi-isothermal plasma, is responsible for the physics ofMTI^[ Reference Murakami and Basko ^{44 ]}. The most dominant physical quantity is the average electron energy ${\mathrm{\mathcal{E}}}_{\mathrm{av}}$ , which characterizes the fundamental structure around the Larmor hole, the spin current and the resultant ultrahigh magnetic field^[ Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov ^{25 ]}. The Larmor hole radius ${R}_{\mathrm{H}}$ , which corresponds to the peak magnetic field at the center, is given by

(5) \begin{align}{R}_{\mathrm{H}}\kern0.1em \left[\mathrm{nm}\right]=37\kern0.1em {\widehat{B}}_0{\widehat{R}}_0^2{\left(\frac{\left({{\kern2pt}\widehat{D}}_0^2-\pi {\widehat{R}}_0^2\right){\widehat{n}}_{\mathrm{i}0}{Z}^2}{\eta_{\mathrm{a}}{\widehat{D}}_0{\widehat{\tau}}_{\mathrm{L}}{\widehat{I}}_{\mathrm{L}}A}\right)}^{1/2},\end{align}

where ${\widehat{B}}_0={B}_0/5$ (in kT). The maximum spin current density running around the Larmor hole, ${J}_{\mathrm{H}}\propto \left({R}_0/{R}_{\mathrm{H}}\right){Zn}_{\mathrm{i}0}{c}_{\mathrm{s}}$ , is coherently given as

(6) \begin{align}{J}_{\mathrm{H}}\kern0.1em \left[\mathrm{PA}\ {\mathrm{cm}}^{-2}\right]=0.63\kern0.1em \frac{\eta_{\mathrm{a}}{\widehat{D}}_0{\widehat{\tau}}_{\mathrm{L}}{\widehat{I}}_{\mathrm{L}}}{{\widehat{B}}_0{\widehat{R}}_0\left({{\kern2pt}\widehat{D}}_0^2-\pi {\widehat{R}}_0^2\right)}.\end{align}

Here it should be stressed that ${J}_{\mathrm{H}}$ does not depend on the material constants of $A$ , $Z$ and ${n}_{\mathrm{i}0}$ . Moreover, it is confirmed from Equations (4)–(6) that the product of the characteristic current ${J}_{\mathrm{H}}$ and the radius ${R}_{\mathrm{H}}$ around the Larmor hole has a coherent parametric relation with the maximum central magnetic field ${B}_{\mathrm{max}}$ , which is expressed as ${B}_{\mathrm{max}}\sim 4\pi {J}_{\mathrm{H}}{R}_{\mathrm{H}}$ , according to Ampere’s law. Applying the specific parameters used in Figure 4, Equations (5) and (6) yield the diameter of a Larmor hole as $2{R}_{\mathrm{H}}\simeq 240$ nm and the maximum spin current density ${J}_{\mathrm{H}}\simeq 0.23$ PA cm ${}^{-2}$ , respectively, where ${\eta}_{\mathrm{a}}=0.5$ is fixed as an example. These values agree well with the size of the electron-free area and maximum current density observed at $t=190$ fs in Figure 4.

The total laser power ${P}_{\mathrm{L}}$ and laser energy ${E}_{\mathrm{L}}$ are simply evaluated as

(7) \begin{align}{P}_{\mathrm{L}}=4{D}_0{L}_0{I}_{\mathrm{L}},\end{align}

(8) \begin{align}{E}_{\mathrm{L}}={P}_{\mathrm{L}}{\tau}_{\mathrm{L}},\end{align}

where ${L}_0$ denotes the target length along the microtube axis. The laser power and energy missing the target are not counted in Equations (7) and (8). Figure 5 plots ${P}_{\mathrm{L}}$ and ${E}_{\mathrm{L}}$ obtained by Equations (7) and (8), respectively, by applying the same specific numbers used for the simulations, except ${L}_0/{R}_0\simeq 1$ –2, which are inferred to be acceptable minimum lengths of microtubes to achieve similar values for the maximum magnetic fields obtained by the 2D simulations. Figure 5 also shows the model predictions for ${P}_{\mathrm{L}}$ and ${E}_{\mathrm{L}}$ , which are obtained by Equations (7) and (8), respectively, using $1\lesssim {L}_0/{R}_0\lesssim 2$ .

The 2D and 3D performances of magnetic field generation using microtube targets are similar. The results given in Figure 5 show that, for example, ${B}_{\mathrm{max}}\sim 0.5$ –0.7 MT with ${E}_{\mathrm{L}}\sim 150$ –300 J and ${P}_{\mathrm{L}}\sim 3$ –6 PW can be achieved upon uniformly irradiating the target at ${I}_{\mathrm{L}}={10}^{21}$ W cm ${}^{-2}$ . Such high-power laser performances are accessible by today’s laser technology^[ Reference Korn ^{45 –} Reference Danson, Haefner, Bromage, Butcher and Chanteloup ^{48 ]}.