# Efficient generation of axial magnetic field
by multiple laser beams with twisted pointing directions

Yin Shi  Department of Plasma Physics and Fusion Engineering,
University of Science and Technology of China, Hefei 230026, China  Alexey
Arefiev  Department of Mechanical and Aerospace Engineering,
University of California at San Diego, La Jolla, CA 92093, USA  Jue Xuan Hao
Department of Plasma Physics and Fusion Engineering,
University of Science and Technology of China, Hefei 230026, China  Jian Zheng
Department of Plasma Physics and Fusion Engineering,
University of Science and Technology of China, Hefei 230026, China
Collaborative Innovation Center of IFSA,
Shanghai Jiao Tong University, Shanghai 200240, Peoples Republic of China

###### Abstract

Strong laser-driven magnetic fields are crucial for high-energy-density
physics and laboratory astrophysics research, but generation of axial multi-kT
fields remains a challenge. The difficulty comes from the inability of a
conventional linearly polarized laser beam to induce the required azimuthal
current or, equivalently, angular momentum (AM). We show that several laser
beams can overcome this difficulty. Our three-dimensional kinetic simulations
demonstrate that a twist in their pointing directions enables them to carry
orbital AM and transfer it to the plasma, thus generating a hot electron
population carrying AM needed to sustain the magnetic field. The resulting
multi-kT field occupies a volume that is tens of thousands of cubic microns
and it persists on a ps time scale. The mechanism can be realized for a wide
range of laser intensities and pulse durations. Our scheme is well-suited for
implementation using multi-kJ PW-class lasers, because, by design, they have
multiple beamlets and because the scheme requires only linear-polarization.

Recently, magnetic field effects in high energy density physics (HEDP) have
attracted significant interest [1, 2, 3]. These can range from guiding of
relativistic electron beams [4] to affecting the shape of inertial fusion
implosions [5]. Despite significant progress, generation of sufficiently
strong and controllable macroscopic fields at the laser facilities used for
HEDP research [6, 7, 8, 9, 10, 11] remains an outstanding challenge.

Various approaches to magnetic field generation using high-power lasers have
been explored in search of an optimal mechanism and field configuration.
Initial efforts were focused on leveraging a circularly polarized (CP) laser
beam to generate an axial quasi-static plasma magnetic field [12, 13, 14]. The
emergence of capabilities to create Laguerre-Gaussian (LG) high-intensity
beams has stimulated research into generation of the axial field using such
beams as well [15, 16, 17, 18]. The strength of the plasma field is limited by
the laser’s ability to drive a strong azimuthal current, so it is insightful
to interpret the process as a transfer of the laser’s angular momentum
$\bm{L}$ to the plasma. Here
$\bm{L}=\varepsilon_{0}\int\bm{r}\times\left[\bm{E}\times\bm{B}\right]d^{3}\bm{r}$,
where $\varepsilon_{0}$ is the dielectric permittivity, $\bm{E}$ and $\bm{B}$
are the electric and magnetic fields, respectively.

Setups involving conventional linearly polarized (LP) laser beams have also
received attention, because additional optics is required to make CP or LG
beams from the conventional beams. A large-scale uniform magnetic fields can
be created by a ns laser irradiating a capacitor–coil [3, 19, 20, 21] or a
snail target [22]. This field can then be amplified inside a plasma by a high
intensity ps or sub-ps laser pulse [23, 24, 25, 26]. Relativistic electrons
generated by high-intensity laser pulses can also generate surface or bulk
azimuthal magnetic fields when streaming through a solid density target [27,
28, 29, 30], and these fields are beneficial for hot electron transport and
electron beam collimation [13, 14, 31, 32, 33, 34, 35, 36]. Applications for
longitudinal fields include guiding of relativistic electron beams [37, 3,
19], laser-driven ion acceleration [38, 39], magnetized atomic physics [40,
3], and laboratory astrophysics [41].

Figure 1: (a) Setup for axial magnetic field generation using four linearly-
polarized Gaussian laser beams with twisted pointing directions, shown with
solid lines, and a structured target. The size of each beam is shown with a
color-coded ellipse in the emitter plane (left side of the simulation box) and
in the focus plane. (b) Projections of the two planes on to the $(y,z)$-plane.
The parameters setting up the beam orientation are defined in the text. (c)
Surface plots of the axial magnetic field $B_{x}$ after the lasers have left
the simulation box ($t=20$ fs). The green, blue, and red surfaces represent
$B_{x}/B_{0}=-0.1$, $-0.2$, and $-0.8$, where $B_{0}=13.4$ kT.

Generation of a large-volume strong magnetic field requires significant energy
that must be delivered by the laser. Multi-kJ PW-class laser systems like LFEX
[9], NIF ARC [10], and Petal [11] offer the highest energy that can be
delivered on a ps time scale. These lasers are all composed of multiple LP
beamlets. The multi-beamlet configuration is not just an essential feature of
the laser system design, but also the key to advanced laser-plasma interaction
regimes [42]. The number of multi-beamlet facilities will increase, as SG-II
UP [8] is due to be upgraded to have multiple kJ-class ps laser beams.

This Letter presents a new multi-beam approach for efficient laser-to-plasma
angular momentum (AM) transfer resulting in magnetic field generation. The
approach, illustrated in Fig. 1 for four linearly-polarized Gaussian beams, is
motivated by the capability of multi-kJ PW-class laser systems to provide
multiple beamlets [9, 10, 43, 11]. Our scheme eliminates the need for CP or LG
beams while offering a method for generating a field above $10$ kT in a
$10^{4}~{}{\upmu\mathrm{m}}^{3}$ volume. Our scheme provides a plasma that can
potentially be used for studies of astrophysical objects involving strong
magnetic fields beyond the dynamic range of previous laboratory settings [44,
45] and to mimic a rotating plasma environment in astrophysics [46, 47].

The role of the twist in the pointing direction, Fig. 1(a), can be illustrated
using geometrical optics. Each laser beam is represented by a ray directed
along the wave vector $\bm{k}_{i}$, where $i$ is the index numbering the beam.
The photon momentum in the $i$-th beam is $\bm{p}_{i}=\hbar\bm{k}_{i}$.
Consider a pair of tilted rays, $\bm{k}_{1,2}=(k_{x},k_{\perp}^{(1,2)},0)$,
that intersect the $(y,z)$-plane at $z_{1,2}=\pm D_{f}/2$ and $y_{1,2}=0$,
where $D_{f}/2$ is the beam offset. The axial AM of a photon is
$[\bm{r}\times\bm{p}]_{x}$, so the total AM of the two beams is
$L_{x}\approx-N\hbar(k_{\perp}^{1}-k_{\perp}^{2})D_{f}/2$, where $N$ is the
number of photons in each beam. The AM can be doubled by adding two rays
offset in $y$. The rays appear twisted, so it is appropriate to refer to the
calculated AM as orbital angular momentum (OAM). They carry OAM, a distinct
form of AM, even though each beam has no intrinsic AM [48]. There are
parallels to $\gamma$-ray beams carrying OAM [49, 50, 51] composed of photons
with a twisted distribution of $\bm{p}$.

To investigate the transfer of the OAM carried by four laser beams, we have
performed a series of three-dimensional (3D) particle-in-cell (PIC)
simulations using a relativistic PIC code EPOCH [52]. Each beam is a linearly
polarized Gaussian beam. The duration of each pulse is 450 fs and the peak
intensity is $2.1\times 10^{20}~{}{\mathrm{W}\mathrm{cm}^{-2}}$. Our target is
a flat foil with sub-wavelength diameter nanowires whose purpose is to
increases the interaction volume between the laser beams and the plasma
produced by the target and thus enhances the number of hot electrons [53]. The
front and rear surfaces of the foil are located at
$x_{f1}=0~{}{\upmu\mathrm{m}}$ and $x_{f2}=4~{}{\upmu\mathrm{m}}$. The spacing
between the wires is $2~{}{\upmu\mathrm{m}}$, the wire length is
$5~{}{\upmu\mathrm{m}}$, and the wire width is $0.4~{}{\upmu\mathrm{m}}$. The
entire target is initialized as a fully ionized cold carbon plasma with an
electron density of $50n_{c}$, where $n_{c}=1.8\times 10^{21}$ cm-3 is the
critical density corresponding to a laser wavelength
$\lambda=0.8~{}{\upmu\mathrm{m}}$. All simulation parameters are listed in the
Supplemental Material.

The orientation of the four beams is set according to Fig. 1. Their axes
intersect a given plane perpendicular to the $x$-axis with the intersection
points forming vertices of a square. We use two planes: the emitter plane
($x_{e}=-20~{}{\upmu\mathrm{m}}$), which is the left boundary of the
simulation box, and the focus plane ($x_{f}=-16~{}{\upmu\mathrm{m}}$), which
is the plane where the beams have the smallest transverse size. The twist is
set by angle $\theta$. There is no twist for $\theta=0$, so that the axis of
each beam and the $x$-axis form a plane. We use $\phi=\arctan(-D_{p}/S)$ to
set the beam convergence, where $S$ is the distance between the emitter plane
and the focus plane and $D_{p}$ is the transverse shift of the beam axes
between the two planes, as shown in Fig. 1(b). We use $\phi=-0.27\pi$ in all
simulations.

Figure 1(c) shows the magnetic field (B-field) for $\theta=-0.28\pi$ at $t=20$
fs. We define $t=0$ fs as the time when the laser pulses leave the simulation
box. The laser-plasma interaction takes place at $t\in(-510,-60)$ fs. The
longitudinal B-field exceeds 10 kT. The volume is around
$10^{4}~{}{\upmu\mathrm{m}}^{3}$. The three surfaces show $B_{x}/B_{0}=-0.1$,
$-0.2$, and $-0.8$, where $B_{0}=13.4$ kT. Note that $B_{0}\equiv 2\pi
m_{e}c/|e|\lambda_{L}$, where $\lambda_{L}=0.8~{}{\upmu\mathrm{m}}$ is laser
wavelength in vacuum, $c$ is the speed of light, and $e$ and $m_{e}$ are the
electron charge and mass. To make its profile more clear, $B_{x}$ is averaged
temporally over a 20 fs interval and spatially using a box with stencil size
$0.4~{}{\upmu\mathrm{m}}\times 0.4~{}{\upmu\mathrm{m}}\times
0.4~{}{\upmu\mathrm{m}}$.

Owing to the approximately axisymmetric profile of $B_{x}$, we can examine its
2D distributions in Fig. 2 without missing too much information. Figure 2(a)
shows the global distribution in the $(x,y)$-plane. The nanowires are between
$x_{wire}=-5~{}{\upmu\mathrm{m}}$ and $x_{f1}=0~{}{\upmu\mathrm{m}}$. The foil
is between $x_{f1}=0~{}{\upmu\mathrm{m}}$ and $x_{f2}=4~{}{\upmu\mathrm{m}}$.
Figures 2(b)&(c) show $B_{x}$, averaged over the azimuthal angle, as a
function of $x$ and $r$ in front of and behind the target (note the different
color-scale ranges). We find that $|B_{x}|$ can be as high as $1.5B_{0}$ in
front of the wires because the lasers generate a higher concentration of hot
electrons in front of the foil. Reaching this amplitude is noteworthy because
new phenomena of laser beam transport through a plasma can arise at
$|B_{x}|\gtrsim B_{0}$ [54, 55]. Even though $B_{x}$ is weaker behind the
target, it is in the range of kT. This confirms that our scheme indeed
produces electrons carrying AM, as the lasers are unable to reach behind the
target to generate the B-field locally.

Figure 2: (a) Axial magnetic field in the $(x,y)$-plane at $t=20$ fs. (b)&(c)
Angle-averaged axial magnetic field as a function of $x$ and $r$ at $t=20$ fs.
The nanowire region is at $x_{wire}<x<x_{f1}$. The foil is at $x_{f1}\leq
x\leq x_{f2}$. (d) Time evolution of the volume-averaged magnetic field within
a box with $-15~{}{\upmu\mathrm{m}}<x<x_{wire}$, $|y|<5~{}{\upmu\mathrm{m}}$,
and $|z|<5~{}{\upmu\mathrm{m}}$.

Figure 2(d) shows the time evolution of the average magnetic field strength
$\langle B_{x}\rangle$ in a box with
$-15~{}{\upmu\mathrm{m}}<x<-5~{}{\upmu\mathrm{m}}$,
$|y|<5~{}{\upmu\mathrm{m}}$, and $|z|<5~{}{\upmu\mathrm{m}}$. The ps time
scale is comparable to that in Ref. [56], but the region containing the
magnetic field moves axially outward (away from the target). In terms of the
energy content within a region with $|y|$, $|z|<15~{}{\upmu\mathrm{m}}$, we
find that the energy in the magnetic field ($\varepsilon_{B}=\int
B_{x}^{2}/(2\mu_{0})dV\approx 3.0$ J) is much smaller than the kinetic energy
of electrons ($\varepsilon_{e}\approx 40.0$ J). The energy of the four beams
is $\varepsilon_{laser}\approx 580$ J. The energy conversion efficiency from
laser to hot electrons and from hot electrons to the magnetic field are both
around 10%. The overall conversion efficiency is two orders of magnitude
higher than that for a laser-driven coil in Ref. [2].

Figure 3: (a) Angle-averaged azimuthal current density $j_{\theta}$ at $t=20$
fs as a function of $x$ and $r$. (b), (c), and (d) $j_{\theta}$ in the
$(y,z)$-plane at three different locations with $x=x_{b}$, $x_{c}$, and
$x_{d}$. Note the significant difference in color-scales between the three
panels introduced to improve visibility. The current density is normalized to
$j_{0}=-|e|cn_{c}=-8.25\times 10^{16}$ A/m2.

To confirm that the twist angle $\theta$ rather than the polarization is the
key parameter, we performed a simulation without the twist ($\theta=0$) and a
simulation with an opposite twist to the original direction
($\theta=0.28\pi$). We found that no axial magnetic field is generated without
the twist and that $B_{x}$ reverses its direction when we reverse the twist.
The azimuthal B-field is generated in all three cases due to the ubiquitous
axial current driven by the laser pulses. We also performed a simulation with
the original setup but randomly selected direction of the E-field polarization
in each laser beam. The angle-averaged $B_{x}$ is similar to the $B_{x}$ in
Fig. 2, confirming that laser polarization has only a secondary effect on the
magnetic field generation in our setup.

In the remainder of this letter we focus on the region in front of the target.
We start with an analysis of the azimuthal current density $j_{\theta}$ that
is thought to be responsible for the axial magnetic field generation. Figure
3(a) shows $j_{\theta}$, averaged over the azimuthal angle, in the
$(x,r)$-plane at $t=20$ fs. The direction of $j_{\theta}$ alternates in the
nanowire region [$x\in(-5,0)~{}{\upmu\mathrm{m}}$]. The underlying cause is
the presence of strong nonuniformities in the ion density associated with the
original nanowires.

Transverse distributions of $j_{\theta}$ in the $(y,z)$-plane at different $x$
positions ($x_{b}$, $x_{c}$, $x_{d}$) are shown in Fig. 3(b-d). These
positions are marked by dashed lines in Fig. 3(a). In agreement with Fig.
3(a), $|j_{\theta}|$ is the biggest at $x_{d}$ and the smallest at $x_{b}$. To
perform an order of magnitude estimation for the maximum value of $|B_{x}|$,
we assume that $j_{\theta}$ is uniform inside a cylinder of radius $R$ and
length $\Delta x$. Then the Biot-Savart law [57] yields

$\displaystyle\max|B_{x}|$ $\displaystyle\approx$
$\displaystyle\dfrac{\mu_{0}}{2}\int_{0}^{R}\int_{-\Delta x}^{\Delta
x}\dfrac{|j_{\theta}|r^{2}\mathrm{d}x\mathrm{d}r}{(r^{2}+x^{2})^{3/2}}$ (1)
$\displaystyle=$ $\displaystyle\mu_{0}|j_{\theta}|\Delta
x\operatorname{arsinh}(R/\Delta x),$

where $\mu_{0}=1.26\times 10^{-6}$ H/m is permeability in vacuum. According to
Fig. 3(a), we can set $R\approx\Delta x\approx 5~{}{\upmu\mathrm{m}}$. In Fig.
3(d), the current density reaches $|j_{\theta}|\approx 0.05|j_{0}|$, where
$j_{0}\equiv-|e|cn_{c}$. Using this value, we obtain $\max|B_{x}|\sim 20$ kT,
which is close to the peak magnetic field, $B_{x}\sim 1.5B_{0}$, in Fig. 2(b).

To quantify the rotating effect of the plasma, we computed the density of the
axial AM for electrons and ions. Due to the significant difference in mass,
the ratio of the axial AM absorption between electrons and ions is
$\eta_{ei}\approx 0.01$. We can estimate the AM density of hot electrons using
the azimuthal current density. We write the AM density of electrons as
$L_{xe}\approx r\gamma_{a}m_{e}n_{e}v_{\theta}$, where $\gamma_{a}$ is the
relativistic gamma-factor, $n_{e}$ is the number density, and $v_{\theta}$ is
the effective azimuthal velocity. We set $v_{\theta}\approx-
j_{\theta}/|e|n_{e}$ to find that $L_{xe}\approx
r\gamma_{a}m_{e}j_{\theta}/|e|$. For $r=w_{0}$ and
$\gamma_{a}\approx\sqrt{1+a_{0}^{2}}\approx 10$, we have $L_{xe}\approx 1.6$
kg/m-s. Using the electron density from simulations, $n_{e}\approx 10^{27}$
m-3, we find that the rotating velocity is around $v_{\theta}\approx 0.1c$.
Our setup produces a rotating plasma environment with electron density and
rotation velocity two orders of magnitude higher than an LG beam in [16].

The OAM transfer from the laser beams to the electrons can be determined using
the conservation of AM [12, 15, 56]. The OAM of absorbed laser photons is
transferred to electrons and ions, with the electron fraction equal to
$\eta_{ei}$. Then, based on the photon absorption, the axial AM density of the
electrons is roughly

$\displaystyle L_{ex}(x,r)$ $\displaystyle\approx$
$\displaystyle\frac{\eta\eta_{ei}}{x_{\kappa}}\frac{0.75\tau_{g}I_{0}}{c}\sin(\phi)\sin(\theta)D_{xr},$
$\displaystyle D_{xr}$ $\displaystyle=$ $\displaystyle
re^{x/x_{\kappa}-2(r-D_{f}/2)^{2}/w_{0}^{2}},$ (2)

where $I_{0}$ is the peak intensity of the incident laser pulses and
$\tau_{g}$ is their duration. For simplicity, we assume that the absorption
coefficient $f_{abs}$ of the laser intensity over the axial distance is
$f_{abs}=f_{0}\exp(x/x_{\kappa})$. We find from the simulation that
$\eta=\int_{x=x_{e}}^{x=0}f_{abs}dx\approx x_{\kappa}f_{0}\approx 0.1$
($x_{\kappa}\approx 3~{}{\upmu\mathrm{m}}$). We use
$r=D_{f}/2=6~{}{\upmu\mathrm{m}}$ and $x=0~{}{\upmu\mathrm{m}}$ to find the
peak AM density of the electrons, $L_{ex}\approx 2.4$ kg/m-s. This result is
on the same scale as the peak AM density ($\approx$ 6.9 kg/m-s) in our
simulations. It is also close to the result ($\approx$ 1.6 kg/m-s) calculated
using $j_{\theta}$ in Fig. 3. The peak AM density in the simulation exceeds
our model’s prediction, which may be due to the locally positive AM density in
the nanowire region. According to Eq. (Efficient generation of axial magnetic
field  by multiple laser beams with twisted pointing directions), the axial
B-field can be controlled by changing the sign of twist angle $\theta$, which
has been confirmed in the Supplemental Material. Our model ignores the
dependence of $f_{abs}$ on such parameters like $I_{0}$, $\phi$, and $\theta$,
but the actual absorption mechanism may be more complex [58, 59, 60]. Using
$L_{ex}$, we can obtain the azimuthal current density and the associated axial
B-field,

$\displaystyle
B_{x}\propto\frac{j_{\theta}(x,r)}{j_{0}}\propto(\eta\eta_{ei})\frac{a_{0}^{2}}{\gamma_{a}}\frac{c\tau_{g}}{x_{\kappa}}\frac{D_{xr}}{r}\sin(\phi)\sin(\theta).$
(3)

Figure 4: Volume-averaged magnetic field $|\langle B_{x}\rangle|$ as a
function of peak laser intensity $I_{0}$ (blue asterisk markers) and laser
pulse duration $\tau_{g}$ (red square markers). The averaging is performed
within a box with $-15~{}{\upmu\mathrm{m}}<x<x_{wire}$,
$|y|<5~{}{\upmu\mathrm{m}}$, and $|z|<5~{}{\upmu\mathrm{m}}$. The dashed
curves (blue and red) show the fits based on Eq. (3).

To investigate the robustness of this mechanism to the choice of laser
parameters, we perform scans over laser peak intensity $I_{0}$ and pulse
duration $\tau_{g}$. The dependence of the volume-averaged longitudinal field
on $I_{0}$, shown in Fig. 4 with asterisk markers, matches well the dependence
given by Eq. (3) and shown with the blue dashed line. The blue dashed line is
$|\langle B_{x}\rangle|$[kT] $=0.85a_{0}^{2}/\sqrt{1+a_{0}^{2}}$. Even at
$I_{0}\approx 3\times 10^{19}$W/cm2 ($a_{0}=4$), the axial magnetic field
strength can be as high as 5 kT. The pulse duration scan, shown in Fig. 4 with
square markers, is performed for a fixed peak intensity of $I_{0}\approx
2.1\times 10^{20}$W/cm2 ($a_{0}=10$). The red dashed line, $|\langle
B_{x}\rangle|$ [kT] $=0.024\tau_{g}$[fs], has the same dependence on
$\tau_{g}$ as that given by Eq. (3). The laser pulse duration is believed to
affect the number of hot electrons and, as a result, the magnetic field
generation. For $\tau_{g}$ as small as 30 fs, we can still get a volume-
averaged magnetic field of 1.3 kT. Additional simulations with a laser
wavelength of 1.053 ${\upmu\mathrm{m}}$ produce similar results, confirming
that our scheme is applicable to both Ti:Sa and neodymium-based lasers.

In summary, we have demonstrated via 3D kinetic simulations a novel mechanism
for generating a multi-kT axial magnetic field using multiple regular laser
pulses. The twist in the pointing direction of the pulses is the key to
driving an azimuthal plasma current that sustains the magnetic field. The
twist angle is a convenient control knob for adjusting the direction and
magnitude of the axial magnetic field. The field occupies a volume that is
tens of thousands of cubic microns and it persists on a ps time scale. The
mechanism can be realized for a wide range of laser intensities and pulse
durations. Our scheme requires just regular linearly-polarized laser beams,
which makes it suitable for implementation at existing and future multi-kJ PW-
class laser facilities that, by design, have to have multiple beamlets [9, 10,
11, 42, 61], including the SG-II UP facility [8] that is expected to have
multiple kJ-class ps pulses in the near future.

## Acknowledgements

Y S acknowledges the support by USTC Research Funds of the Double First-Class
Initiative, Strategic Priority Research Program of CAS (Grant No.
XDA25010200), CAS Project for Young Scientists in Basic Research (Grant No.
YSBR060) and Newton International Fellows Alumni follow-on funding. Y S also
acknowledge Rui Yan and Robert Kingham for enthusiastic discussions. A.
Arefiev’s research was supported under the National Science Foundation–Czech
Science Foundation partnership by NSF Award No. PHY-2206777. Simulations were
performed with EPOCH (developed under UK EPSRC Grants EP/G054950/1,
EP/G056803/1, EP/G055165/1, and EP/M022463/1). The computational center of
USTC and Hefei Advanced Computing Center are acknowledged for computational
support.

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