#  High-power laser experiment forming a supercritical collisionless shock in
a magnetized uniform plasma at rest

R. Yamazaki  Department of Physical Sciences, Aoyama Gakuin University, 5-10-1
Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan<EMAIL_ADDRESS>(R.Y.)
Institute of Laser Engineering, Osaka University, 2-6, Yamadaoka, Suita, Osaka
565-0871, Japan.  S. Matsukiyo  Faculty of Engineering Sciences, Kyushu
University, 6-1 Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan.  T. Morita
Faculty of Engineering Sciences, Kyushu University, 6-1 Kasuga-Koen, Kasuga,
Fukuoka 816-8580, Japan.  S. J. Tanaka  Department of Physical Sciences,
Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258,
Japan.  T. Umeda  Institute for Space-Earth Environmental Research, Nagoya
University, Furo-cho, Chikusa, Nagoya, Aichi 464-8602, Japan.  K. Aihara
Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, Kanagawa 252-5258, Japan.  M. Edamoto  Interdisciplinary Graduate
School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan.  S.
Egashira  Graduate School of Science, Osaka University, 1-1 Machikane-yama,
Toyonaka, Osaka 560-0043, Japan.  R. Hatsuyama  Interdisciplinary Graduate
School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan.  T.
Higuchi  Interdisciplinary Graduate School of Engineering Sciences, Kyushu
University, Kasuga 816-8580, Japan.  T. Hihara  Graduate School of
Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan.
Y. Horie  Interdisciplinary Graduate School of Engineering Sciences, Kyushu
University, Kasuga 816-8580, Japan.  M. Hoshino  Department of Earth and
Planetary Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan.  A. Ishii  Max Planck Institute for Gravitational Physics
(Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, Germany  N.
Ishizaka  Department of Physical Sciences, Aoyama Gakuin University, 5-10-1
Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  Y. Itadani
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University,
Kasuga 816-8580, Japan.  T. Izumi  Graduate School of Science, Osaka
University, 1-1 Machikane-yama, Toyonaka, Osaka 560-0043, Japan.  S.
Kambayashi  Department of Physical Sciences, Aoyama Gakuin University, 5-10-1
Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  S. Kakuchi  Department of
Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara,
Kanagawa 252-5258, Japan.  N. Katsuki  Interdisciplinary Graduate School of
Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan.  R. Kawamura
Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, Kanagawa 252-5258, Japan.  Y. Kawamura  Department of Physical
Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa
252-5258, Japan.  S. Kisaka  Department of Physical Science, Hiroshima
University, Higashi-Hiroshima 739-8526, Japan.  T. Kojima  Interdisciplinary
Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580,
Japan.  A. Konuma  Institute for Laser Science, The University of Electro-
Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan.  R. Kumar
Graduate School of Science, Osaka University, 1-1 Machikane-yama, Toyonaka,
Osaka 560-0043, Japan.  T. Minami  Graduate School of Engineering, Osaka
University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan.  I. Miyata
Department of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe,
Sagamihara, Kanagawa 252-5258, Japan.  T. Moritaka  Fundamental Physics
Simulation Research Division, National Institute for Fusion Science, 322-6
Oroshi-cho, Toki, 509-5292, Japan.  Y. Murakami  Interdisciplinary Graduate
School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan.  K.
Nagashima  Interdisciplinary Graduate School of Engineering Sciences, Kyushu
University, Kasuga 816-8580, Japan.  Y. Nakagawa  Graduate School of Science,
Osaka University, 1-1 Machikane-yama, Toyonaka, Osaka 560-0043, Japan.  T.
Nishimoto  School of Engineering, Osaka University, 2-1, Yamadaoka, Suita,
Osaka 565-0871, Japan.  Y. Nishioka  Interdisciplinary Graduate School of
Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan.  Y. Ohira
Department of Earth and Planetary Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033, Japan.  N. Ohnishi  Department of Aerospace
Engineering, Tohoku University, 6-6 Aramaki Aza Aoba, Aoba, Sendai, Miyagi
980-8579, Japan.  M. Ota  Graduate School of Science, Osaka University, 1-1
Machikane-yama, Toyonaka, Osaka 560-0043, Japan.  N. Ozaki  Graduate School of
Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan.
T. Sano  Institute of Laser Engineering, Osaka University, 2-6, Yamadaoka,
Suita, Osaka 565-0871, Japan.  K. Sakai  Graduate School of Engineering, Osaka
University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan.  S. Sei  Department
of Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara,
Kanagawa 252-5258, Japan.  J. Shiota  Department of Physical Sciences, Aoyama
Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  Y.
Shoji  Department of Physical Sciences, Aoyama Gakuin University, 5-10-1
Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  K. Sugiyama  Department of
Physical Sciences, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara,
Kanagawa 252-5258, Japan.  D. Suzuki  Department of Physical Sciences, Aoyama
Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  M.
Takagi  Interdisciplinary Graduate School of Engineering Sciences, Kyushu
University, Kasuga 816-8580, Japan.  H. Toda  Department of Physical Sciences,
Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 252-5258,
Japan.  S. Tomita  Astronomical Institute, Tohoku University, 6-3 Aramaki,
Aoba-ku, Sendai, Miyagi 980-8578, Japan.  Frontier Research Institute for
Interdisciplinary Sciences, Tohoku University, Sendai, 980-8578, Japan  S.
Tomiya  Department of Physical Sciences, Aoyama Gakuin University, 5-10-1
Fuchinobe, Sagamihara, Kanagawa 252-5258, Japan.  H. Yoneda  Institute for
Laser Science, The University of Electro-Communications, 1-5-1 Chofugaoka,
Chofu, Tokyo 182-8585, Japan.  T. Takezaki  Department of Creative
Engineering, National Institute of Technology, Kitakyushu College, 5-20-1
Shii, Kokuraminamiku, Kitakyushu, Fukuoka, 802-0985, Japan.  Faculty of
Engineering, University of Toyama, 3190, Gofuku, Toyama 930-8555, Japan.  K.
Tomita  Faculty of Engineering Sciences, Kyushu University, 6-1 Kasuga-Koen,
Kasuga, Fukuoka 816-8580, Japan.  Division of Quantum Science and Engineering,
Hokkaido University, Sapporo 060-8628, Japan.  Y. Kuramitsu  Graduate School
of Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871,
Japan.  Y. Sakawa  Institute of Laser Engineering, Osaka University, 2-6,
Yamadaoka, Suita, Osaka 565-0871, Japan.

###### Abstract

We present a new experimental method to generate quasi-perpendicular
supercritical magnetized collisionless shocks. In our experiment, ambient
nitrogen (N) plasma is at rest and well-magnetized, and it has uniform mass
density. The plasma is pushed by laser-driven ablation aluminum (Al) plasma.
Streaked optical pyrometry and spatially resolved laser collective Thomson
scattering clarify structures of plasma density and temperatures, which are
compared with one-dimensional particle-in-cell simulations. It is indicated
that just after the laser irradiation, the Al plasma is magnetized by a self-
generated Biermann battery field, and the plasma slaps the incident N plasma.
The compressed external field in the N plasma reflects N ions, leading to
counter-streaming magnetized N flows. Namely we identify the edge of the
reflected N ions. Such interacting plasmas form a magnetized collisionless
shock.

###### pacs:

Valid PACS appear here

††preprint: ???

##  I INTRODUCTION

Collisionless shocks are ubiquitous in astrophysical objects like supernova
remnants and solar-terrestrial and laboratory plasmas Balogh and Treumann
(2013); Burgess and Scholer (2015). When the upstream low-entropy flow comes
into the shock, the kinetic energy is converted into various forms like high-
temperature ions and electrons, magnetic turbulence, and nonthermal particles.
However, despite state-of-the-art observations Bamba  _et al._ (2003);
Johlander  _et al._ (2016), particle-in-cell (PIC) simulations Scholer  _et
al._ (2003); Lembège  _et al._ (2009); Matsukiyo  _et al._ (2011); Matsumoto
_et al._ (2013); Sironi  _et al._ (2013); Umeda  _et al._ (2014); Yamazaki
_et al._ (2019), and analytical arguments Vink and Yamazaki (2014), the
detailed mechanism of the energy dissipation is not fully understood. In many
cases, the upstream plasma is magnetized, and the pre-existing and/or self-
generated magnetic fields around the shock work as a “catalyst” in the process
of kinetic energy dissipation.

The laboratory experiment using high-power lasers is another method to study
collisionless shocks. Laser-produced plasma is fast-moving and long-lived.
Therefore, it has been expected that large-scale, long-time evolution of the
plasma interaction can be seen, which is unachievable by current PIC
simulations. There have been experiments to excite several kinds of
collisionless shocks: electrostatic shocks Kuramitsu  _et al._ (2011); Morita
_et al._ (2010); Sakawa  _et al._ (2016); Yuan  _et al._ (2017), and Weibel-
mediated shocks Ross  _et al._ (2017); Li  _et al._ (2019); Fiuza  _et al._
(2020), as well as subcritical Morita  _et al._ (2013); Niemann  _et al._
(2014); Schaeffer  _et al._ (2015) and supercritical Kuramitsu  _et al._
(2016); Schaeffer  _et al._ (2017, 2019) magnetized shocks. Previous
experiments have revealed that even in the unmagnetized case, a self-generated
magnetic field (i.e., Biermann field) is crucial in the ion reflection
Huntington  _et al._ (2015), which plays an important role in the formation of
perpendicular shocks. This is also indicated by one-dimensional (1D) PIC
simulation Umeda  _et al._ (2019).

In many astrophysical magnetized collisionless shocks producing cosmic rays,
they are supercritical (Alfvén Mach number $M_{A}\gtrsim 3$) at which a part
of incoming ions are reflected upstream and gyrate back into the shock front
Sckopke  _et al._ (1990), causing two-stream instabilities to generate plasma
waves which lead to the particle scattering and acceleration. So far, due to
limited space and time, no experimental results of clear formation of such a
supercritical shock have been reported, although some authors claimed the
observation of a precursor which was expected to evolve into the shock if the
plasma interaction proceeds Li  _et al._ (2019); Schaeffer  _et al._ (2017,
2019). In this paper, we report our experiments to generate supercritical
magnetized collisionless shocks, which is compared with 1D PIC simulations.
Unlike previous experiments Schaeffer  _et al._ (2017, 2019), our method can
make a fully magnetized plasma at rest with uniform mass density Shoji  _et
al._ (2016), so that upstream plasma parameters are well determined.

##  II Experimental Setup

We used Gekko-XII HIPER Laser system (wavelength 1053 nm, pulse duration 1.3
ns, energy 690 J per beam, focal spot size 2.8 mm). An aluminum (Al) plane
target with thickness 2 mm was irradiated by four beams simultaneously,
resulting laser intensity of $3.4\times 10^{13}$W cm-2 on the target. A
schematic side view of the target and laser configuration is shown in Fig. 1.
Before the laser shot, the chamber was filled with ambient nitrogen (N) gas
with pressure $P_{N}=5$ Torr. Just before the shot, the external magnetic
field ($B_{0}=3.6$ T) was applied. The ambient gas was ionized by ionizing
photons from Al plasma, becoming magnetized plasma with N ion density
$n_{N}=3.2\times 10^{17}$cm-3. Subsequently, Al plasma (dark gray region in
Fig. 1(b)) pushed the magnetized N plasma to generate a magnetized
collisionless shock (dotted curve in Fig. 1(b)).

Figure 1:  Schematic view of our experiment (a) before and (b) after the shot.
Solid arrows represent an external magnetic field. Al plasma expands (white
arrows) and pushes magnetized N plasma to generate a collisionless shock
(dotted curve).

As shown in Fig. 2, the $z$-axis is the vertical, and the central axis of
coils is along the $y$-axis. The target chamber center (TCC) is located at
$(x,y,z)=(0,0,0)$. Separation between the target surface and TCC was 1.4 cm,
and target normal was $\hat{n}=(\cos 14^{\circ},0,-\sin 14^{\circ})$. The
external magnetic field was applied using an electromagnetic coil consisting
of four 50-turn coils connected in parallel. The inner and outer diameters of
the coils were 60 and 110 mm, respectively, and two of them were placed at
$y=\pm 25$ mm, to generate almost uniform magnetic field perpendicular to the
plasma expansion direction as shown in Figs. 2(a) and 2(c). This
electromagnetic coil was driven by a small pulse-powered circuit Edamoto  _et
al._ (2018) consisting of four capacitors ($4\times 1.5$ mF), each charged
with a voltage of 1.4 kV, resulting in a quasi-static current of 5.3 kA in
each coil and a uniform magnetic field of 3.6 T inside the coils. The time
duration of the field of approximately 100 $\mu$s is sufficiently larger than
the typical timescale of the plasma propagation, and this field is quasistatic
during the plasma expansion.

Using streaked optical pyrometry (SOP), the time evolution of the plasma self-
emission (at a wavelength of 450 nm) along the target normal
($X^{\prime}$-axis) was observed from the $y$ direction. The plasmas were also
diagnosed with collective laser Thomson scattering (TS) method Sheffield  _et
al._ (2010). A probe laser (Nd:YAG, wavelength 532 nm, energy 370 mJ in $\sim
10$ ns) with wave number $\vec{k}_{i}$ went through the plasma in the
horizontal plane, $z=0$, at an angle $45^{\circ}$ from the $x$ and $y$-axis
($p$-axis: see Fig. 2(c)). The scattered light with wave number $\vec{k}_{s}$
was detected from two directions both of which are $90^{\circ}$ from the
incident direction. As a result, one of the measurement wave number,
$\vec{k}=\vec{k}_{s}-\vec{k}_{i}$, is toward the $x$-axis, and the other
toward the $-y$ direction (Fig. 2(d)). Triple grating spectrometers were used
to achieve a good spectral resolution of $\approx 10$ and $\approx 18$ pm for
IAW-1 and IAW-2, respectively Tomita  _et al._ (2017); Morita  _et al._
(2019). We recorded the scattered light of ion feature with an intensified
charge-couple device (ICCD) with 3 ns exposure time. In this paper, we discuss
the role of self-generated field (via Biermann battery effect) in Al plasma
Umeda  _et al._ (2019). Unfortunately, there was no direct measurement of
magnetic fields in this experiment to conclusively indicate that the Biermann
field is playing an important role.

Figure 2:  (a) Bird’s eye view of experimental setup. (b) The side view of the
setup. The target normal ($X^{\prime}$-axis) is in the $x$-$z$ plane. (c) The
top view of the setup. The probe laser ($\vec{k}_{i}$: $p$-axis) for
measurements of TS ion feature (IAW) focuses at TCC and the scattered lights
are measured from two different directions IAW-1 ($\vec{k}_{s,1}$) and IAW-2
($\vec{k}_{s,2}$), and (d) the measurement wave numbers $\vec{k}_{\rm
IAW,1}=\vec{k}_{s,1}-\vec{k}_{i}$ and $\vec{k}_{\rm
IAW,2}=\vec{k}_{s,1}-\vec{k}_{i}$ are roughly longitudinal and transverse to
the flow, respectively.

Before this experiment with N gas, we had performed similar experiments but
with different ambient gas, hydrogen and helium. This was because it had been
expected that the ion gyro radius and period could be small if the gas were
fully ionized, which would help us make the field of view of our plasma
measurement smaller. The use of a simple gas would make physical
interpretation clear. Using Gekko-XII HIPER Laser system, we had various shots
with different total laser energy and intensity, changing the number of beams
and/or focal spot size. However, our TS measurements could not identify
hydrogen or helium plasma at rest in the upstream region sufficiently before
the Al piston plasma arrived. Hence, we concluded that photoionization of
hydrogen or helium gas was difficult for our experimental setup, at least for
lasers like Gekko-XII HIPER lasers. The reason is that the number of ionizing
photons for hydrogen or helium gas is too small. For example, photoionization
cross section for hydrogen atom takes maximum at the photon absorption edge
($=13.6$ eV), and above this photon energy, the cross section approximately
scales as $E_{ph}{}^{-3}$, where $E_{ph}$ is the photon energy. Typical photon
energy from target plasma just after the shot is $E_{ph}\sim$ keV in our laser
intensity range, hence the photoionization cross section becomes very small,
$\sim 10^{-23}$cm2. Then, the mean-free path of photons with energy
$E_{ph}\sim$ keV is $\sim 10^{5}(n_{g}/10^{18}{\rm cm}^{-3})^{-1}$cm, where
$n_{g}$ is the hydrogen gas density. Our system size is 1–10 cm, so that only
a small fraction of keV photons ionize the hydrogen atoms, resulting in a very
small ionization fraction. On the other hand, since the absorption edge is
much higher for nitrogen ($\approx 400$ eV, depending on charge states of N
ions), the photoionization cross section for the nitrogen atom is much larger
($\sim 10^{-19}$cm2 for $E_{ph}\sim$ keV), which makes the upstream plasma
generation much easier.

##  III Analysis of experimental data

###  III.1 Analysis of plasma self-emission

####  III.1.1 Case of $P_{N}=B_{0}=0$

First, we show the SOP result (Fig. 3(a)) of a shot without ambient gas and
external magnetic field ($P_{N}=B_{0}=0$) to clarify the properties of piston
(Al) plasma. The Al plasma weakly emits light and freely expands with density
decreasing with time.

We show in Fig. 4(a) the time evolution of SOP counts at fixed positions
$X^{\prime}=0.8$, 1.1, and 1.4 cm (TCC). After the shot, the background
intensity is on average $\approx 25$ in our unit, and it is variable because
of statistical fluctuation. This may come from stray light of HIPER lasers,
probe laser for TS measurement, and streak detector noise. After a while, the
intensity starts to increase when Al plasma arrives; for example, at
$X^{\prime}=0.8$ cm (black-dashed line in Fig. 4(a)), the intensity becomes
twice the background level ($\approx 50$) at $t\approx 10$ ns. Assuming that
the Al plasma is freely expanding, we estimate the head speed of the Al plasma
$v_{\rm Al,0}\approx 0.8$ cm$/10$ ns $=800$ km s-1 (see blue circles in Fig.
4(a)). The line “P0” in Fig. 3(a) represents $X^{\prime}=v_{\rm Al,0}t$ with a
constant velocity $v_{\rm Al,0}=800$ km s-1.

Figure 3:  SOP images for cases with (a) no ambient gas ($P_{N}=0$) and no
external magnetic field ($B_{0}=0$), (b) $P_{N}=5$ Torr and $B_{0}=0$, and (c)
$P_{N}=5$ Torr and $B_{0}=3.6$ T. The dashed line P0 in panel (a) shows a
constant velocity of 800 km s-1. The curve of constant deceleration R1 in
panel (b) is described by parameters $v_{0}=1600$ km s-1 and $t_{0}=48$ ns,
and curves P1 and P2 are represented by $v_{0}=590$ km s-1 and $t_{0}=148$ ns.
TCC is located at $X^{\prime}=1.4$ cm ($p=0$), and epochs of TS measurements
shown in Fig. 6 are shown by white circles with error bars meaning gate width.
Assuming a plane wave with normal vector along the $X^{\prime}$ direction, we
also put white squares representing positions and times of TS measurements in
Fig. 7(b) ($p=1$ mm), and white triangles where we estimate upstream plasma
parameters ($p=-2$ mm).  Figure 4:  Time evolution of self-emission intensity
(the intensity as a function of time) at fixed positions $X^{\prime}$, for
cases with (a) no ambient gas ($P_{N}=0$) and no external magnetic field
($B_{0}=0$), (b) $P_{N}=5$ Torr and $B_{0}=0$, and (c) $P_{N}=5$ Torr and
$B_{0}=3.6$ T. In panel (a), the black-dashed, black-dotted, and red-solid
curves show temporal evolutions at $X^{\prime}=0.8$, 1.1, and 1.4 cm (TCC),
respectively. Blue circles in panel (a) on each curve represent the time of
passage at each position of a trajectory P0 in the $X^{\prime}-t$ plane (Fig.
3(a)) represented by $X^{\prime}=v_{\rm Al,0}t$ with a constant velocity
$v_{\rm Al,0}=800$ km s-1. In panels (b) and (c), black-dashed, red-solid, and
black-dotted lines are for $X^{\prime}=1.3$, 1.4 (TCC), and 1.5 cm,
respectively. Blue circles in panels (b) and (c) represent the same as panel
(a) but for R1 and R2, respectively, whose functional form is given by Eq.
(1). Similarly, green triangles in panels (b) and (c) represent the time of
passage of P1 and P2, respectively.

####  III.1.2 Case of $P_{N}=5$ Torr and $B_{0}=0$

Second, we had a shot with ambient N pressure $P_{N}=5$ Torr but without
external magnetic field ($B_{0}=0$). The result of SOP is shown in Fig. 3(b).
The interaction between Al and N plasmas made much brighter emission than in
the case of $P_{N}=0$. The edge of the brightest part (denoted by “P1” in Fig.
3(b)) propagates with a speed $\sim 350$ km s-1 at $t=20$ ns. The more rapid
structure “R1” goes ahead of P1 with a velocity of $\sim 700$ km s-1 at $t=15$
ns. If P1 and R1 arise at the target ($X^{\prime}=0$) at $t=0$, they cannot be
explained by constant-velocity motions. Instead, assuming a quadratic function
of time, we determined their trajectories as described below.

We obtained the functional form, $X^{\prime}(t)$, of P1 as in the following.
First, we made spatial profiles of the self-emission intensity (for fixed
time) every 1 ns from 15 to 35 ns. Second, we found that the regions with SOP
count of $\approx 1000$ (yellow region in Fig. 3(b)) have large intensity
gradient, whose scale length is $\Delta x\lesssim 0.3$ mm. Hence for each
time, we found the value of $X^{\prime}$ coordinate at which SOP count equals
1000, and we regard the points $(X^{\prime},t)$ as the position of P1 at each
epoch. Third, we adopt a quadratic function of time,

$X^{\prime}(t)=v_{0}t(1-t/t_{0})~{}~{},$ (1)

where constants $v_{0}$ and $t_{0}$ are initial velocity and a break time,
respectively. Using this functional form, the points $(X^{\prime},t)$ are
fitted with least-squares method to get $v_{0}=591\pm 6$ km s-1 and
$t_{0}=150\pm 7$ ns. Here, error means statistical in the fitting. Soon we
will apply the same method to similar edge structure “P2” in the case of
$P_{N}=5$ Torr and $B_{0}=3.6$ T (Fig. 3(c)), and derive similar values of
$v_{0}$ and $t_{0}$. They coincide with each other within errors, so that we
adopt common values $v_{0}=590$ km s-1 and $t_{0}=148$ ns in dashed lines for
P1 and P2 in Figs. 3(b) and 3(c), respectively.

For the structure R1, we could not apply the same method as for P1, because
the SOP count variation (that is, the density gradient) around R1 is much less
than P1. Scale length of the gradient of R1 is $\Delta x\approx 2$ mm, so that
it is difficult to define the positions of R1. Hence fitting by eye, we
determined parameter values of $v_{0}$ and $t_{0}$. Fortunately, one can see
from Fig. 3(b) a clear boundary around $X^{\prime}\approx 1.2$–1.5 cm for
$t\approx 8$–12 ns. In Fig. 3(b), we draw the dashed line as R1 along with
this boundary, assuming the functional form given by Eq. (1) with $v_{0}=1600$
km s-1 and $t_{0}=48$ ns. It can be confirmed from Fig. 4(b), showing the time
evolution of the intensity at fixed positions $X^{\prime}=1.3$, 1.4, and 1.5
cm, that R1 is in the period of abrupt intensity increase.

As seen in Fig. 3(b), self-emission intensity is not uniform upstream of R1.
This indicates the ionization of upstream N plasma is not uniform (see § A.1
for detailed discussion).

####  III.1.3 Case of $P_{N}=5$ Torr and $B_{0}=3.6$ T

Third, we performed a shot with $P_{N}=5$ Torr and $B_{0}=3.6$ T. Clear
difference of SOP results between with and without external field cases can be
seen (Figs. 3(b) and 3(c)). A small jump (denoted by “R2” ) is identified at
$X^{\prime}\approx 1.47$ cm at $t=23$ ns at which we had a TS measurement (see
also Fig. 5). The location of the edge of the brightest region, P2, is almost
the same as P1 at any time. Separation between R2 and P2 is smaller than that
between R1 and P1 in the case of $B_{0}=0$, suggesting that ion dynamics is
changed by the external field.

As already described previously, we determined the trajectory of the structure
P2 as in the same manner for P1. It is found that P2 is explained by the
functional form given by Eq. (1) with fitted values $v_{0}=592\pm 5$ km s-1
and $t_{0}=148\pm 5$ ns, which coincide with the values for P1 within errors.
This result is naturally understood if we assume that P1 and P2 are interface
between Al and N plasmas (see § IV). Then, the ram pressure of Al plasma is so
large that the presence or absence of the magnetic pressure in the N plasma is
negligible at least in early epoch. The intensity change is sharper for
$B_{0}=3.6$ T case than $B_{0}=0$ case; however, the physical interpretation
of the observed width of the intensity gradients is difficult because there
are many possible explanations (see § A.2 for details).

For the structure R2, when we determine the values of $v_{0}$ and $t_{0}$, the
situation is similar to the case of R1 in the case of $B_{0}=3.6$ T case than
$B_{0}=0$. SOP count variation is small and scale lengths of the intensity
gradient of R2 is $\Delta x\approx 0.5$ mm, making it difficult to define the
position of R2. Nevertheless, one can identify the boundary of green region in
Fig. 3(c) for $t\approx 15$–25 ns. Assuming again the functional form given by
Eq. (1) with $v_{0}=1000$ km s-1 and $t_{0}=64$ ns, we draw the dashed line R2
in Fig. 3(c). As in the case of R1, one can see from Fig. 4(c) that around R2,
the rate of intensity increase becomes higher than before. Figure 5 shows the
spatial profiles of the self-emission at $t=22$ and 23 ns, that is, one-
dimensional slices of Fig. 3(c). Despite fluctuation via instrumental pixel
damage and statistical noise, we identify an intensity decrease from
$X^{\prime}=1.45$ to 1.5 cm at $t=23$ ns at which we performed TS measurement.
Comparing with the profile at $t=22$ ns (black-dashed line), one can see that
R2 propagates outward.

Again as seen in Fig. 3(c), self-emission intensity is inhomogeneous upstream
of R2. This fact is similar to the case of $P_{N}=5$ Torr and $B_{0}=0$. (see
§ A.1 for detailed discussion).

Figure 5:  Spatial distribution of self-emission for the case of $P_{N}=5$
Torr and $B_{0}=3.6$ T at $t=22$ ns (black-dashed line) and 23 ns (red-solid
line), that is, one-dimensional slices of Fig. 3(c) at these epochs. In the
gray-shaded regions, the detector gain becomes smaller due to pixel damage.
Blue circles on each curve represent the position of passage at each epoch of
a trajectory R2 represented by Eq. (1) with constants $v_{0}=1000$ km s-1 and
$t_{0}=64$ ns.

###  III.2 Analysis of Ion term of Collective Thomson scattering (TS) spectra

We fit TS spectra assuming the resonance with ion-acoustic waves (IAWs) in
plasmas, which have a single ion-component in Maxwelian distribution. The IAW
spectrum is fitted with a convoluted spectral density function

$\displaystyle\tilde{S}(\mathbf{k},\omega)$ $\displaystyle=$
$\displaystyle\int
S(\mathbf{k},\omega^{\prime})R(\omega^{\prime}-\omega)d\omega^{\prime},$ (2)
$\displaystyle S(\mathbf{k},\omega)$ $\displaystyle=$ $\displaystyle\frac{2\pi
Z}{k}\left|\frac{\chi_{e}}{\epsilon}\right|^{2}f_{i}(\frac{\omega}{k}),$ (3)

where $R$ is the resolution of the spectrometer evaluated from the Rayleigh
scattering and expressed as a Gaussian, $\chi_{e}$ and $\epsilon$ are electron
susceptibility and longitudinal dielectric function, respectively,
$\mathbf{k}=\mathbf{k}_{s}-\mathbf{k}_{i}$, $\omega=\omega_{s}-\omega_{i}$,
$f_{i}$ is the ion distribution function, and the charge state $Z$ is self-
consistently derived from a collisional radiative model with the FLYCHK code
Chung  _et al._ (2005). The TS scattered light intensity is proportional to
the electron density $n_{e}$, and it is determined by an absolute calibration
of the collective TS system using the following formula:

$\displaystyle
I_{T}/I_{R}=\frac{n_{e}\sigma_{T}E_{T}}{n_{N}\sigma_{R}E_{R}}\frac{S_{i}}{2\pi},$
(4)

where $I$, $\sigma$, and $E$ are the light intensity, cross section, and
incident laser energy, respectively, the subscripts “T” and “R” represent the
Thomson and Rayleigh scattering, respectively, $n_{N}$ is the nitrogen density
for Rayleigh scattering measurement, and $S_{i}$ is the total intensity of the
ion component expressed as $S_{i}=\int S(\mathbf{k},\omega)d\omega$. Errors
for plasma parameters by TS measurements are evaluated from best-fitted values
and covariance that we get from the least-squares fitting of the observed IAW
spectra.

Results are summarized in Table 1, where we assume N plasma except for the
case of rapidly moving ($V_{X^{\prime}}\approx 737$ km s-1) component that is
seen by IAW-1 at around $\lambda\approx 530.2$ nm at $t=10$ ns, $p=0$ (TCC),
and $B_{0}=0$ (Fig. 6(b)) and the case of $t=30$ ns, $p=0$ (TCC), and
$B_{0}=0$ (Fig. 6(d)). For these cases, we show in the table both results on
the assumption of N and Al plasmas.

As stated above, we analyzed IAW spectra on the assumption of a single
component plasma with Maxwell distribution. In our case, the distribution
function deviates from Maxwellian around the shock transition layer. However,
at present, analysis method in such a case has not been established, because
we have no matured theory of TS spectrum in that case. This issue may be a
future problem to be resolved in the community of shock experiments. Hence, we
cannot help but say that derived parameters, listed in Table 1, are just
approximate values guiding our theoretical interpretation.

####  III.2.1 Case of $P_{N}=5$ Torr and $B_{0}=0$

In Figs. 6(a) and 6(b), two peaks corresponding to the resonance with IAWs are
seen at around 532 nm. These are from upstream N plasma almost at rest. Just
before the arrival of R1 at TCC ($t=10$ ns), we identify almost at rest, cold
($T_{e}$, $T_{i}<10$ eV) N plasma (Fig. 6(a): see § B.1 for further
discussion). A few ns after the passage of R1 ($t=15$ ns), the static N plasma
was heated ($T_{e}\approx 100$ eV and $T_{i}\approx 230$ eV). We also identify
the moving plasma at $\lambda\approx 530.2$ nm from IAW-1 but not from IAW-2,
showing the ion dynamics is collisionless (see also § E.2). Assuming this
component is moving along the $X^{\prime}$-axis, we derive the bulk velocity
$V_{X^{\prime}}\approx 740$ km s-1 (see § B.2), which is roughly consistent
with the velocity of R1 at $t\approx 15$ ns.

The TS spectrum taken at $t=23$ ns ($p=0$) does not show a clear double peak
(Fig. 6(c)). If we fit the spectrum in the same way as the other epochs, so
that assuming that the observed spectrum is made by IAW in plasma with Maxwell
distribution, then we derive unnatural parameters: namely, electrons would
move toward target with bulk velocity of $\approx 1100$ km s-1. This indicates
that the plasma in this epoch and space is nonstationary and/or in highly
nonlinear regime. In any case, the spectrum seems to show high $T_{i}$
($\approx 1$ keV: see blue dotted and dot-dashed lines in Fig. 6(c)). The TS
spectrum at $t=30$ ns reveals plasma with a clear double peak with
$V_{X^{\prime}}\approx 360$ km s-1 (Fig. 6(d): see § B.3).

Figure 6:  Background-subtracted TS spectra at TCC ($p=0$) obtained at (a)
$t=10$ ns (b) 15 ns, (c) 23 ns, and (d) 30 ns, for the case of $P_{N}=5$ Torr
and $B_{0}=0$. Shaded area around the incident wavelength of 532.0 nm is
affected by stray light. Black curves are data of IAW-1, while the red curve
in panel (b) is of IAW-2. Blue solid lines in panels (b) and (d) show the
best-fit results, while the dashed line in panel (a) explains only lower-
temperature component (see § B.1 for details). Blue dashed line (dot-dashed
line) in panel (c) is theoretically expected spectrum from N plasma in
equilibrium state with parameters $T_{i}=1$ keV, $ZT_{e}=0.73$ keV,
$Zn_{e}=2.6\times 10^{19}$ cm-3, and $V_{X^{\prime}}=450$ km s-1 ($T_{i}=1.5$
keV, $ZT_{e}=1.4$ keV, $Zn_{e}=4.1\times 10^{19}$ cm-3, and
$V_{X^{\prime}}=400$ km s-1).

####  III.2.2 Case of $P_{N}=5$ Torr and $B_{0}=3.6$ T

Next, we analyzed the data of a shot with $P_{N}=5$ Torr and $B_{0}=3.6$ T.
Figure 7(a) shows the TS spectrum at $t=23$ ns, where the vertical $p$-axis
represents the position along the probe laser (see Fig. 2(c)), so that
$(x,y,z)=(-p\cos 45^{\circ},p\sin 45^{\circ},0)$. A clear double peak of the
ion-acoustic resonance is identified when $p>0$. Assuming N plasma, we fit the
spectrum at $p=1$ mm (Fig. 7(b)) and obtain $V_{X^{\prime}}=400$ km s-1 and
the ion density $n_{N}=n_{e}/Z=1.2\times 10^{18}$ cm-3 which is about 3.6
times as large as the initial upstream one, indicating the ion compression.
Note that the double-peak feature in the TS spectrum still can be seen for
$p>1$ mm, where sensitivity becomes weaker; however, the feature vanishes for
$p<0$ mm (Figs. 7(a) and 9). Such an edge feature corresponds to R2 in Fig.
3(c), and it is first observed by spatially resolved TS measurement. In
addition, as shown in Table 1, the density has a maximum at $p\approx 1$ mm
(see § B.4 for further discussion).

Table 1: Parameters determined by TS measurements (IAW-1) assuming Maxwelian distribution and FLYCHK. We assume N plasma except for the case of fast component with $V_{X^{\prime}}\approx 740$ km s-1 at $t=15\pm 1.5$ ns and $p=0$ and for the case of $t=15\pm 1.5$ ns and $p=0$.  |  | $B_{0}=0$ |  $B_{0}=3.6$ T
---|---|---|---
$t$111Time from laser irradiation. |  $p$222$p=0$ corresponds to TCC (see Fig. 2(c)). | $T_{i}$ | $T_{e}$ | $n_{e}$ | $Z$ | $V_{X^{\prime}}$ | $T_{i}$ | $T_{e}$ | $n_{e}$ | $Z$ | $V_{X^{\prime}}$
[ns] | [mm] | [eV] | [eV] | [$\times 10^{18}$cm-3] |  | [km s-1] | [eV] | [eV] | [$\times 10^{18}$cm-3] |  | [km s-1]
$10\pm 1.5$ | $-2$ |  $5.9\pm 0.4$333Assuming $T_{i}=T_{e}$. | $1.2\pm 0.4$ | 2.8 | $4.5\pm 0.5$ | $\cdots$
$10\pm 1.5$ | 0444Cold component. |  $6.9\pm 3.1$d |  $7.5\pm 0.8$d |  $1.0$d,f | 3.2d,f |  $6.0\pm 0.5$d | $\cdots$
| 0555Warm component. |  $52\pm 7$e |  $19\pm 3$e |  $1.6$e,666Fixing N ion density, $n_{N}=n_{e}/Z=3.2\times 10^{17}$cm-3. | 5.0e,f |  $9.9\pm 0.7$e | $\cdots$
$15\pm 1.5$ | 0 | $228\pm 4$ | $98\pm 2$ | $1.7\pm 0.4$ | 5.5 | $31\pm 0.8$ | $\cdots$
| 0 | $(1.0\pm 0.3)\times 10^{3}$ | — 777Unconstrained. | $0.22\pm 0.05$ | 4.0 | $737\pm 2$ | $\cdots$
| 0888Assuming Al plasma instead of N plasma. |  $(2.0\pm 0.6)\times 10^{3}$ h | — g,h |  $0.26\pm 0.06$h | 4.9h |  $737\pm 2$h | $\cdots$
$23\pm 1.5$ | 0 | N/A (see text) | $609\pm 23$ | $188\pm 6$ | $4.9\pm 2.3$ | 6.7 | $363\pm 1.5$
$23\pm 1.5$ | 0.5 | N/A (see text) | $222\pm 6$ | $171\pm 3$ | $5.5\pm 1.8$ | 6.5 | $389\pm 0.8$
$23\pm 1.5$ | 1 | N/A (see text) | $205\pm 6$ | $161\pm 3$ | $7.4\pm 2.5$ | 6.4 | $394\pm 0.8$
$23\pm 1.5$ | 1.5 | N/A (see text) | $275\pm 8$ | $148\pm 2$ | $5.7\pm 2.5$ | 6.1 | $386\pm 0.7$
$30\pm 1.5$ | 0 | $166\pm 5$ | $95\pm 2$ | $4.9\pm 0.7$ | 5.1 | $363\pm 0.6$ | $\cdots$
| 0h |  $518\pm 8$h |  $137\pm 2$h |  $3.7\pm 0.27$h | 10.3h |  $364\pm 0.4$h | $\cdots$

##  IV Discussion

We performed 1D PIC simulations with similar conditions to our experiment. For
details of the simulation set up, see Umeda et al. Umeda  _et al._ (2019). We
adopt the same parameters as those of Run 1 (and Run 2) of Umeda et al. Umeda
_et al._ (2019), but we consider two cases with different external magnetic
field strength in the N plasma, $B_{0}=0$ and $B_{0}=3.5$ T. In the Al plasma,
the magnetic field with the strength of 10 T is externally imposed. This is a
simple artifact of a self-generated Biermann field, although our 1D simulation
cannot capture the developement of the field. Note that we use real electron-
to-ion mass ratios $m_{i}/m_{e}=49572$ and 25704 for Al and N plasmas,
respectively. Our PIC simulation scheme does not incorporate the effects of
Coulomb collisions.

Below we interpret our experimental results comparing with 1D PIC simulations.
Such an argument is similar to previous experimental studies on supercritical
magnetized shocks. Note, however, that as already stated in § I, laser
experiments are possibly capable of seeing larger-scale, longer-time evolution
of plasma interaction, which is currently unachievable with 3D PIC simulations
with real mass ratio. This causes a dilemma when comparing experimental and
simulation results since we cannot accurately evaluate multi-dimensional
effects (e.g., non-plane-parallel plasma expansion resulting in adiabatic
cooling and dilution of the Biermann magnetic field, excitation of obliquely
propagating waves in highly inhomogeneous plasmas, and so on). Hence, we
should not fully rely on the numerical simulation results, and it is not
necessary that experimental results perfectly (quantitatively) match the
simulation results. Nevertheless, as described below, our 1D PIC simulation
results in both $B_{0}=0$ and 3.5 T cases are broadly consistent with
experimental data, so that we believe that our simulations already catch
essential features related on the magnetized shock formation.

According to our 1D PIC simulation for $B_{0}=0$ Umeda  _et al._ (2019), the
sharp rise of the electron density $n_{e}$ occurs at an interface between N
and Al plasmas in the electron scale, so that we interpret P1 seen in SOP
(Fig. 3(b)) as this electron-scale discontinuity. This is supported by the
fact that high electron and ion temperatures are observed at $t=15$ and 23 ns,
upstream of this interface. Our radiation hydrodynamics simulation (see § C)
shows that the Biermann magnetic field in Al plasma has initial strength
$\gtrsim 100$ T, and as the plasma expands the field becomes weaker ($\approx
10$ T at $t=8$ ns around the head of the Al plasma). It is at least partially
responsible for the N ion reflection Umeda  _et al._ (2019). Collisional
coupling is also non-negligible (see § E.1 for details). Later the interface
P1 decelerates due to the interaction between Al and N plasmas, so that its
velocity decreases to $\sim 430$ km s-1 after propagating $\sim 1$ cm from
target (at $t\approx 20$ ns).

Our 1D PIC simulation for $B_{0}=0$ also showed that some Al ions penetrate
beyond the interface, being accelerated by the ponderomotive force Umeda  _et
al._ (2019). These fast Al ions might correspond to R1 seen in Fig. 3(b) (see
§ D.1 for more discussion). The fact that the initial velocity of P1
($v_{0}\approx 590$ km s-1) is smaller than $v_{\rm Al,0}$ implies that at
least some Al ions penetrate upstream beyond P1. Another possible explanation
of R1 is N ions reflected around the head of Al plasma. This might be
indicated by our experimental result that the initial velocity of R1 ($\approx
1600$ km s-1) is just as twice as the initial Al velocity ($v_{\rm
Al,0}\approx 800$ km s-1). During the propagation, such fast ions decelerate
due to the interaction with upstream N plasma at rest, leading to the heating
of incident N plasma between P1 and R1.

Figure 7:  (a) Measured TS spectrum at $t=23$ ns for $B_{0}=3.6$ T along the
probe laser axis $p$, which was obtained by IAW-1. (b) Comparison of TS
spectra at $p=1$ mm (dashed line in panel (a)) for $B_{0}=3.6$ T (red) and 0 T
(black) cases. (c-h) Results of 1D PIC simulation with $B_{0}=3.5$ T at $t=23$
ns, in which the horizontal axis is the distance from the target $X^{\prime}$.
(c) Electron phase-space plot. (d,e) Ion phase-space plots. In these panels,
blue and yellow points represent Al and N plasmas, respectively, and green
curves show the electron bulk velocity. Bottom three panels show spatial
profiles of transverse magnetic field strength (f), electron density $n_{e}$
(g), electron temperature $T_{e}$ (blue line in panel (h)), and emissivity of
the plasma self-emission $n_{e}^{2}/\sqrt{T_{e}}$ normalized by far upstream
value (red line in panel (h)).

Our analysis results of SOP and TS measurements for $B_{0}=3.6$ T case are
again broadly consistent with results of 1D PIC simulation with $B_{0}=3.5$ T
(Figs. 7(c)-(h) and Fig. 12). Observed steep emission gradient P2 in Fig. 3(c)
is associated with an interface between Al and N plasmas (see Fig. 7(c) and
red line in Fig. 7(h)). Incident N ions do not penetrate deeply into the Al
plasma because of compressed Biermann magnetic field (Fig. 7(f)) and/or
collisional coupling. On the other hand, Al ions are trapped by compressed
external magnetic field, and do not enter the N plasma (Fig. 7(d)). Incident N
ions are initially reflected at the interface between Al and N plasmas (which
is observed as P2 in Fig. 3(c)) and later reflected by the compressed external
field in the N plasma (Fig. 12). They are gyrating back due to the external
magnetic field (Fig. 7(e)), forming a shock foot. Hence, it is natural to
interpret that R2 in Fig. 3(c) corresponds to the edge of the reflected N ions
(Fig. 7(e)), which has also been confirmed by TS measurement (Fig. 7(a)). The
density $n_{e}$ takes maximum at $p\approx 1$ mm (Table 1), which is about to
be a shock overshoot. Observed $T_{i}\approx 0.2$ keV at $p\approx 1$ mm of
the reflected component is larger than incident N ions, but is smaller than
that for $B_{0}=0$ (Fig. 7(b)). This is also consistent with PIC results.
Therefore, we claim that the spatially resolved edge of the foot of a
developing magnetized collisionless shock is measured (see § B.4, § D.2, and §
E.3 for further discussion).

We estimate upstream physical quantities. We had only one shot with
$B_{0}=3.6$ T, and TS spectrum at $t=23$ ns. Hence, we simply assume the
upstream plasma parameters except for $B_{0}$ are the same as those for the
unmagnetized case. We analyze the data of IAW-1 for the case of $P_{N}=5$ Torr
and $B_{0}=0$ at $t=10$ ns and $p=-2$ mm, at which the discontinuity R1 had
not arrived yet (white triangle in Fig. 3(b)). We fit the TS spectrum to get
the N ion density $n_{N}=n_{e}/Z=(4.4\pm 1.5)\times 10^{17}$ cm-3 (see § B.1),
suggesting the upstream medium is fully ionized. Using the best-fitted values,
we get upstream quantities such as the sound speed $a_{S}\approx 11$ km s-1,
Alfvén velocity $v_{A}\approx 32$ km s-1, and ion (electron) plasma beta
$\beta_{i}\approx 0.08$ ($\beta_{e}\approx 0.2$) for $B_{0}=3.6$ T. Simply
assuming magnetohydrodynamics, we expect that the velocity of a forming shock
may be higher than that of interface between Al and N plasmas (P2: see Figs.
3(c) and 7(c)), whose typical value is $v_{ej}\approx 400$ km s-1. Then, we
expect the magnetosonic Mach number
$M_{ms}>v_{ej}/\sqrt{v_{A}^{2}+a_{S}^{2}}\approx 12$ and Alfvén Mach number
$M_{A}>v_{ej}/v_{A}\approx 13$, so that the shock will be supercritical with
ion-ion mean-free path $\lambda_{ii}=m_{i}^{2}v_{ej}^{4}/8\pi
n_{N}Z^{4}e^{4}{\rm ln}\Lambda\approx 3.5$ cm.

##  V Summary

We have conducted experiments of generating collisionless shocks propagating
into magnetized plasma at rest with uniform mass density. It is quite
difficult to identify a shock formation only from SOP due to the bright
emission enhancement at P2. However, our PIC simulations show that R2 is
composed only of N plasma, and if we analyze the TS data assuming the N
plasma, all the experimental data self-consistently show that R2 is an edge of
the foot of a forming supercritical shock in the magnetized N plasma. We are
also sure that a shock ramp and overshoot are arising between P2 and R2 as
seen in the PIC result.

##  Appendix A Notes on SOP Analysis

###  A.1 On the inhomogeneity upstream of R1 and R2

As shown in Figs. 3(b) and 3(c), self-emission intensity is not uniform both
upstream of R1 in $B_{0}=0$ case and of R2 in $B_{0}=3.6$ T case. This
indicates the ionization of upstream N plasma is not uniform. Ahead of R1 and
R2, neither Al nor reflected N plasma co-exist, so that the cause of the non-
uniformity is likely time-dependent photoionization. Around the interface
between Al and N plasmas (P1 and P2), the electron temperature is high and it
can be more than 100 eV as shown by our 1D PIC simulations (Figs. 11 and 12).
Such high-density, hot plasma emits ionizing photons, and they change the
ionization state of the upstream N plasma. When photoionized, N ions eject hot
electrons, resulting in the plasma heating. This causes further change of the
upstream ionization state. Both collisionless and collisional processes depend
on the ion charge state. However, the upstream number density of N ions (i.e.,
the upstream mass density, which determine the whole shock dynamics from ion
scale to hydrodynamical scale) is still uniform.

###  A.2 Observed width of the density gradients

In ideal plane-parallel case, width of the transition layers P1 and P2, which
are the interfaces between the Al and N plasmas in electron scale, is given by
the gyro radius of thermal electrons around the layers if Al and N interaction
is collisionless. It is estimated as $r_{g,e}=m_{e}cv/eB\sim 3\times
10^{-4}{\rm mm}\,(B/10~{}{\rm T})^{-1}(v/500~{}{\rm km}~{}{\rm s}^{-1})$. The
mean-free path of electrons via electron-electron Coulomb collision is on the
same order or less. These lengths are much smaller than the observed apparent
width $\Delta x\sim 0.3$ mm or slightly less. This discrepancy is explained by
the three-dimensional (3D) effect: geometry of the plasma self-emission in 3D
space is not plane parallel, and even the interface fluctuates during the
propagation. In such an actual case, the emission is projected onto the
detector plane whose normal vector is in the $y$ direction. As an extreme toy
case, let us consider spherically symmetric uniformly emitting sphere with
radius $\ell_{0}$ whose emissivity is given by step function, $f(r)=const$ for
$r<\ell_{0}$ and $f(r)=0$ for $r>\ell_{0}$ ($r$ is the radial coordinate in
the spherical coordinate system). When such emission is projected onto the
detector plane, the apparent spatial profile does not remain step-function-
like, but has gradual transition whose width is on the order of the curvature
radius $\ell_{0}$. More generally, in cylindrically symmetric case, the
projected emission profile is related to the emissivity in the three-
dimensional space by Abel transform. In the present case, the exact plane
parallel assumption is clearly not a good approximation, although the local
curvature radius of the interface is highly uncertain. However, the length on
the order of 0.1 mm is not unnatural. These projection effect may become
important for not only P1 and P2 but also the other emission gradients R1, R2
and P0.

There are several other possibilities to explain the observed width $\Delta x$
of P1 and P2, such as Rayleigh-Taylor instability and the inhomogeneous
expansion of the Al plasma (that is, anisotropic velocity distribution of
kinetic energy of Al plasma). As another case, when the Biermann field in Al
plasma is weak ($\lesssim 1$ T), the diffusion length of electrons in the
direction perpendicular to the Biermann field is potentially comparable to the
observed scale width $\Delta x$. At present, precise description of the
observed width is difficult since there are several physical possibilities.

##  Appendix B Notes on Analysis of TS spectra

###  B.1 TS spectra at $t=10$ ns for $B_{0}=0$

We show in Fig. 8 the IAW-1 (see Fig. 2(c)) spectra at $t=10$ ns around
$\lambda=532$ nm with different positions $p=-2$, $-1$, 0, and 1 mm. The black
solid line in Fig. 8 ($p=0$, TCC) is identical to the black solid line in Fig.
3(a).

We see TS spectra from N plasma at rest. During the instrumental gate width of
3 ns, the rapid plasma component R1 (shown in Fig. 3(b)) passed around TCC.
Such rapidly moving plasma has a large Doppler shift, so that its scattered
light does not exist around $\lambda\approx 532$ nm, and may be even outside
of the whole range of wavelength coverage of our spectrometer system. (If we
assume the plasma velocity corresponding to R1 as
$V_{X^{\prime}}=dX^{\prime}/dt\approx 933$ km s-1 with $v_{0}=1600$ km s-1 and
$t_{0}=48$ ns, then the Doppler shift $\Delta\lambda\approx 2.3$ nm is
predicted.) The component R1 electrostatically interacts with the incident N
plasma, resulting in rapid heating of the N plasma. This is also suggested by
the rapid increase of self-emission at TCC (see Fig. 4(b)).

The position $p=-2$ mm was in the region at which the rapid component R1 had
not arrived yet, so we interpret that the TS spectrum at $p=-2$ mm comes from
the upstream N plasma that was unaffected by Al ejection. This is also
justified by our 1D PIC simulation (see § D.1). Then we fitted the spectrum
assuming $T_{e}=T_{i}$, and best fitted parameters are shown in Table 1. At
$p=-1$ mm, we also see such a cold component, but slightly heated compared
with the case of $p=-2$ mm. At $p=1$ mm, which becomes the downstream region
of R1 near the end of exposure of TS measurement, one can see warmer plasma
with broader separation between double peaks of IAW resonance.

The TS spectrum at $p=0$ (TCC) looks complicated. It seems to consist of the
two superimposed components (cold and warm N plasmas). First, we fit the
spectrum in the wavelength range $531.92~{}{\rm nm}<\lambda<532.05$ nm
(excluding the gray-shaded region shown in Fig. 8 that is affected by a filter
to cut the stray light) in order to explain only the cold component with
narrower separation of the double peak. Assuming $T_{i}=T_{e}$ and
$V_{X^{\prime}}=0$, we derived $T_{i}=T_{e}=9.3\pm 0.2$ eV, $n_{e}=(7.3\pm
1.6)\times 10^{17}$cm-3, and $Z=4.2$, which was represented by the blue dashed
line in Fig. 6(a). Then one can clearly see that the fit outside of the
fitting wavelength range is inadequate, which indicates the existence of
another warm component. Hence, we next fit the TS spectrum with two components
that are simply superimposed, where we assume the two components are recorded
in the instrumental gate width of 3 ns. In fitting the data, we fix the N ion
density $n_{N}=n_{e}/Z=3.2\times 10^{17}$cm-3. The best-fitted parameters are
shown in Table 1. Note that both cold and warm components are almost at rest,
so that both are highly likely N plasmas.

Figure 8:  Background-subtracted IAW-1 spectra at $t=10$ ns in the case of
$P_{N}=5$ Torr and $B_{0}=0$ for positions $p=-2$ mm (red dotted line), $-1$
mm (green dashed line), 0 (black solid line), and 1 mm (blue dot-dashed line).
The intensity in the gray-shaded region around the incident wavelength of
532.0 nm is diminished by a filter in the spectroscopic optics system to cut
the stray light. TCC corresponds to $p=0$, so that the black solid line is
identical to the data of Fig. 6(a).

###  B.2 TS spectrum at $t=15$ ns for $B_{0}=0$

When we analysed rapidly moving component for $t=15$ ns, $p=0$ and $B_{0}=0$
that was seen in the IAW-1 spectrum (Fig. 6(c)) and has a peak at
$\lambda\approx 530.2$ nm, we assumed two cases of Al and N plasmas. Table 1
shows the results. In either cases, the plasma has a bulk velocity of
$V_{X^{\prime}}\approx 740$ km s-1, and the ion temperature is on the order of
keV.

###  B.3 TS spectrum at $t=30$ ns for $B_{0}=0$

Figure 3(b) shows that at $t\approx 30$ ns, the interface P1 passes through
TCC ($X^{\prime}=1.4$ cm, $p=0$). Hence, the observed TS spectrum may either
come from Al or N plasma, so that we fit the spectrum for both cases. Indeed,
it is hard from the best-fitted values shown in Table 1 to judge which plasma
is responsible for the observed spectrum. In the case of N plasma, both the
electron density $n_{e}\approx 5\times 10^{18}$cm-3 and the ion density
$n_{N}=n_{e}/Z\approx 1\times 10^{18}$cm-3 are several times as large as
initial N plasma densities. This fact is naturally explained by 1D PIC
simulation. However, we cannot exclude the cases of Al plasma.

###  B.4 TS spectrum at $t=23$ ns for $B_{0}=3.6$ T

If R2 is the edge of reflected N ions, there should be another N ion
population at rest as seen in 1D PIC simulation at around $\lambda=532$ nm in
the TS spectrum (e.g., Schaeffer  _et al._ (2019)). Since the observed data
contains bright self-emission as a background and there is intense stray light
around the probe laser wavelength ($\lambda=532$ nm), it is difficult to
perform the spectral analysis to find such a component. Using data in the
wavelength ranges $\lambda=529.5$–530 nm and 533–534 nm, we determined the
background by fitting with a cubic function. The background-subtracted spectra
are shown in Fig. 9. One can find small excess around 532 nm. Assuming the TS
ion feature from N plasma, we fitted the excess component. For $p=1$ mm, we
obtained $Zn_{e}\approx 3\times 10^{17}$ cm-3 and $T_{i}\gtrsim 1\times
10^{2}$ eV, though the electron temperature is unconstrained since the
observed intensity is weak (red-dashed line in Fig. 10). Then, we calculate a
parameter $\alpha=1/k\lambda_{D,e}\approx 0.25(Z/3)^{-1/2}(T_{e}/100~{}{\rm
eV})^{-1/2}$, where $\lambda_{D,e}$ is the electron Debye length, and hence
the ion-term scattered light becomes dim. We also estimate a parameter
$\beta=(ZT_{e}/T_{i})^{1/2}[\alpha^{2}/(1+\alpha^{2})]^{1/2}$ that determines
the shape of the ion term Sheffield  _et al._ (2010), and we have
$\beta\approx\alpha(ZT_{e}/T_{i})^{1/2}\lesssim 0.4(T_{i}/100~{}{\rm
eV})^{-1/2}$ for small $\alpha$, so that it is less than unity as long as
$T_{i}\gtrsim 100$ eV. In such cases, the TS ion term has a single peak as
shown by the red-dashed line in Fig. 10.

In the case of the fitting result shown above (red-dashed line in Fig. 10),
the derived value of $Zn_{e}$ is about one order of magnitude smaller than
expected. If there were the same N plasma with $Zn_{e}\approx 3\times 10^{18}$
cm-3 as seen for the case of $B_{0}=0$, $t=10$ ns, and $p=-2$ mm (see Table
1), which we refer to as upstream N plasma in this paper, then the TS spectra
would have seen by the blue-dotted curve in Fig. 10. If we rely on 1D PIC
simulation results, the electron temperature alone increases to more than 100
eV around $p=1$ mm. In this case, the ion acoustic peaks in the TS spectrum
would be more separated from 532 nm, so that we would more clearly see the
component although the ion-term scattered light would be dim because the
scattering is in the noncollective regime ($\alpha\approx 0.8$). Hence, we
could not experimentally identify the coexistence of incident N plasma almost
at rest with low ion temperature and electron density of $\sim 10^{18}$cm-3.
From the experimental view point, a probable explanation of the observed
excess around 532 nm is the stray light of the probe laser.

As discussed above, in our present experiment we could not identify another N
ion population almost at rest, coexisting with the R2 component, which is
indicated in 1D PIC simulation. However, we should not directly believe the
result of 1D PIC simulations in this case. In the 2D or 3D case, plasma
interaction proceeds more rapidly, and incident N plasma becomes dilute in
velocity space. As a result, the two components are mixed with each other and
merge into a single population in the velocity space. Another possibility to
disentangle this issue is that the plasma temperatures, $T_{e}$ and $T_{i}$,
of the expected component are smaller than $\approx 6$ eV. Then, TS spectrum
from such a cold plasma has narrow peak, and in our present spectrometer
system, it is masked by a filter to cut stray light. Such a case potentially
occurs because the upstream state is inhomogeneous (see § A.1) — the less
ionized are N ions, the less hot electrons, resulting in the lower
temperatures and less electron density. There may also be the collisional
effects. Finally, we emphasize again that in our analysis of TS ion term, we
have assumed Maxwell distributions of ion and electrons and radiative-
collisional equilibrium, which are clearly violated in the shock transition
layer. The derived plasma parameters by our present analysis might be
different from actual values.

Figure 9:  Background-subtracted IAW-1 spectra at $t=23$ ns in the case of
$P_{N}=5$ Torr and $B_{0}=3.6$ T for various positions $p$. The upper panel is
for cases of $p=-2.0$, $-1.5$, $-1.0$, $-0.5$, 0.0, 0.5, and 1.0 mm, while the
lower panel for cases of $p=1.0$, 1.5, and 2.0 mm. TCC corresponds to $p=0$.
The gray-shaded region around the incident wavelength of 532.0 nm is the same
as that of Fig. 8.  Figure 10:  Enlarged view of Fig. 9 around 532 nm for
$p=1$ mm. The solid black line is background-subtracted observed data, which
is the same as red-solid lines in Fig. 9. The gray-shaded region around the
incident wavelength of 532.0 nm is the same as those of Figs. 8 and 9. The
red-dashed line in this figure represents the best-fitted result (see text for
details). The blue-dotted line is expected in the case of the same parameters
as $t=10$ ns, $P_{N}=5$ Torr, $B_{0}=0$, and $p=-2$ mm (i.e.,
$T_{i}=T_{e}=5.9$ eV, $n_{e}=1.2\times 10^{18}$cm-3, and $Z=2.8$).

##  Appendix C Estimate of the Biermann Battery Magnetic field

We performed 2D radiation hydrodynamics simulation Ohnishi (2012) to have
electron pressure and density of Al plasma ejected by laser irradiation. Then,
in order to estimate the strength of the Biermann magnetic field in the Al
plasma, we solve, as a post process, the induction equation with Biermann term

$\displaystyle\frac{\partial\bm{B}}{\partial t}$ $\displaystyle=$
$\displaystyle\bm{\nabla}\times(\bm{v}\times\bm{B})+c\frac{\bm{\nabla}p_{e}\times\bm{\nabla}n_{e}}{n^{2}_{e}e}~{}~{},$
(5)

to get the evolution of the field. It is found that until 4 ns from the shot,
the magnetic field strength in the Al plasma is more than 100 T at the point
where the electron density is equal to the critical density of $1.0\times
10^{21}$ cm-3. Note that even in this case, the plasma beta is not less than
unity, so that the generated magnetic field does not alter the overall
dynamics of Al plasma. After that, the plasma expands and the field strength
becomes small (e.g., $\approx 10$ T at $t\approx 8$ ns around the head of Al
plasma); however, it is still strong enough to reflect N ions into Al plasma.
Our simulation result on the field generation is quantitatively consistent
with previous numerical studies Schoeffler  _et al._ (2014, 2016); Fox  _et
al._ (2018).

##  Appendix D Details of 1D PIC simulation results

###  D.1 Result for $B_{0}=0$ T

This case is identical to Run 2 of Umeda et al. Umeda  _et al._ (2019).
Results for $t\approx 14$ ns was presented in Fig. 4 of Umeda et al. Umeda
_et al._ (2019), which shows the electron-scale tangential discontinuity at
$X^{\prime}=0.62$ cm. The electron density rapidly increases there. We
interpret this discontinuity as corresponding to the structure “P1” found in
SOP.

In Fig. 11 we show the results for $t\approx 10$ ns and $t\approx 23$ ns, the
latter of which can be directly compared with Figs. 7(c)–7(h). A part of Al
ions penetrates into incident N plasma after accelerated by ponderomotive
force at the interface (P1) between Al and N plasma. At $t\approx 10$ ns, it
reaches the position $X^{\prime}\approx 0.8$ cm. As shown in the electron
phase space plots (top panels of Fig. 11), a small fraction ($\lesssim 1$%) of
electrons originally associated with Al ions, shown by blue dots in the right-
side region of P1, that is, electrons injected at the left boundary
$X^{\prime}=0$, go across the interface P1. (In the phase space plots, deeper
blue means less number of particles, while light blue or nearly white regions
indicate high particle density.) Nitrogen ions are being reflected by the
Biermann magnetic field in Al plasma; however, its edge is still in the Al
plasma (that is, in the left region of the interface P1 at $X^{\prime}\approx
0.45$ cm). Interaction between the penetrating Al plasma and incident N plasma
is responsible for plasma heating and enhancement of the plasma self-emission
upstream of P1 ($X^{\prime}\approx 0.45$–0.8 cm). Hence, we interpret the tip
of penetrating Al plasma corresponds to the structure “R1” found in SOP in the
early epoch. In our 1D PIC simulation, the region around TCC ($X^{\prime}=1.4$
cm) is unaffected by the penetrating Al plasma. Namely, the position $p=-2$ mm
(definition of $p$-axis is shown in Fig. 2(c)) corresponds to
$X^{\prime}\approx 1.7$ cm, hence it is expected that the observed IAW-1 TS
data at $t=10$ ns and $p=-2$ mm is appropriate to estimate the parameters of
initial upstream N plasma.

At $t\approx 23$ ns, penetrating Al plasma reaches the position
$X^{\prime}\approx 1.9$ cm. Nitrogen ions are reflected by Biermann magnetic
field in Al plasma, and the edge of the reflected component arrives at
$X^{\prime}\approx 1.6$ cm. Before reflected N ions come back, incident N
plasma is heated just after the passage of Al ions by electrostatic
interaction ($X^{\prime}\approx 1.6$–1.9 cm), and electron temperature $T_{e}$
becomes larger. When we calculate the spatial profile of the intensity of the
plasma self-emission (assuming the bremsstrahlung emission, $\propto
n_{e}^{2}/\sqrt{T_{e}}$), we find that it starts to increase at the edge of
the reflected N ions at $X^{\prime}\approx 1.6$ cm. Hence, we interpret this
edge corresponds to the structure “R1” found in SOP in later epoch.

Our 1D PIC simulation used typical parameters of Al and N plasmas around TCC
($X^{\prime}=1.4$ cm) at late epoch (e.g., $t\simeq 20$–30 ns) Umeda  _et al._
(2019). The high-speed flow at 1600 km s-1, labeled as “R1”, is not seen in
the 1D PIC simulation. On the other hand, parameters of Al plasma and N plasma
around the target in the experiment are quite different from those of the 1D
PIC simulation. For example, Biermann magnetic field in Al plasma is estimated
as $\gtrsim 100$ T at the initial state of the experiment by radiation
hydrodynamics simulation. Such a strong magnetic field may generate a high-
speed flow at 1600 km s-1 in 2D/3D system. Then, the position of the emission
edge “R1” found in SOP may be much farther from the target. Nevertheless, it
can be said that our PIC simulation qualitatively well explains our
experimental results.

Figure 11:  Results of 1D PIC simulation with $B_{0}=0$ T at $t\approx 10$ ns
(left) and $t\approx 23$ ns (right). Format is the same as Figs. 7(c)–7(h).
The horizontal axis is the distance from the target $X^{\prime}$. From top to
bottom, panels of the first row are electron phase-space plots, and those of
the second and third rows are ion phase-space plots. In these panels, blue and
yellow points represent Al and N plasmas, respectively, and green curves show
the electron bulk velocity. Panels of the fourth and fifth rows show the
transverse magnetic field strength $B_{y}$ and electron density $n_{e}$,
respectively. The bottom panels represent electron temperature $T_{e}$ (blue
line) and emissivity of the plasma self-emission $n_{e}^{2}/\sqrt{T_{e}}$
normalized by far upstream value (red line). Note that the electron density
$n_{e}$ is displayed with a logarithmic scale. The bottom panel represents
electron temperature $T_{e}$ (blue line) and emissivity of the plasma self-
emission $n_{e}^{2}/\sqrt{T_{e}}$ normalized by far upstream value (red line).
Note that to enlarge the small variation around a structure “R1”, we take the
range of the plasma self-emission $n_{e}^{2}/\sqrt{T_{e}}$ up to 20 (left
panel) or 10 (right panel), which are different from plots in Figs. 7(h) and
Fig. 12.

###  D.2 Result for $B_{0}=3.5$ T

The results at $t\approx 5$, 10, and 15 ns are shown in Fig. 12. After $t=10$
ns, incoming N ions are reflected at the region of strong magnetic field that
is the compressed external field (center and right panels of Fig. 12). The
reflection point is then the upstream side of the interface between Al and N
plasmas (structure “P2” shown in Fig. 12, and in Figs. 3(c) and 7(c)). Hence,
after $t=10$ ns, incoming N ions are reflected in the N plasma. These
reflected N ions form the structure “R2” (see Figs. 3(c), 7(a), and 7(c)). The
N ions reflected at $t\approx 10$ ns are responsible for “R2” at $t=23$ ns. In
earlier epoch ($t\approx 5$ ns: left panels of Fig. 12), incoming N ions
penetrates into Al plasma and they are reflected by the Biermann field in the
Al plasma (left panels of Fig. 12). However, such ions have already come back
to N plasma before $t=23$ ns, and have turned again by the external field,
going back toward the downstream region (it has been passed 0.7 times the ion
gyro period). Hence, they are not at the edge R2 when $t=23$ ns, but at around
$X^{\prime}=0.9$–1.0 cm.

Figure 12:  Results of 1D PIC simulation with $B_{0}=3.5$ T at $t\approx 5$ ns
(left), $t\approx 10$ ns (center), and $t\approx 15$ ns (right). Format is the
same as Figs. 7(c)–7(h). The horizontal axis is the distance from the target
$X^{\prime}$. From top to bottom, panels of the first row are electron phase-
space plots, and those of the second and third rows are ion phase-space plots.
In these panels, blue and yellow points represent Al and N plasmas,
respectively, and green curves show the electron bulk velocity. Panels of the
fourth and fifth rows show the transverse magnetic field strength $B_{y}$ and
electron density $n_{e}$, respectively. The bottom panels represent electron
temperature $T_{e}$ (blue line) and emissivity of the plasma self-emission
$n_{e}^{2}/\sqrt{T_{e}}$ normalized by far upstream value (red line). Note
that the electron density $n_{e}$ is displayed with a logarithmic scale.

##  Appendix E Effects of Coulomb collisions

In this paper, we have mainly considered collisionless processes. However,
collisional effects are not negligible in some cases. Our current 1D PIC
simulations do not capture the role of Coulomb collisions. The mean-free path
of electron-electron Coulomb collision is smaller than any other scales in
most cases with typical parameters, so that electrons are hydrodynamically
coupled. Below, we discuss effects of ion-ion collisions for structures P1,
P2, R1, and R2. In the following, we adopt an approximate formula of the mean-
free path of ions with mass $m_{1}$, charge $Z_{1}e$, and initial velocity
$v_{1}$, running into the plasma with ion mass $m_{2}$, charge $Z_{2}e$,
electron density $n_{e2}$, ion density $n_{i2}$, electron temperature
$T_{e2}$, and ion temperature $T_{i2}$, which is given by $\lambda_{ii}=\mu
m_{1}v_{r}{}^{4}/4\pi n_{2i}Z_{1}{}^{2}Z_{2}{}^{2}e^{4}\ln\Lambda$, where
$\mu=m_{1}m_{2}/(m_{1}+m_{2})$ and the mean relative velocity
$v_{r}=\sqrt{v_{1}{}^{2}+(8kT_{i2}/\pi m_{2})}$. The Coulomb logarithm is
calculated with $\Lambda=\min\\{\lambda_{e2},\lambda_{i2}\\}/b_{L}$, where
$b_{L}=Z_{1}Z_{2}e^{2}/\mu v_{r}$, and $\lambda_{e2}=\sqrt{kT_{e2}/4\pi
n_{e2}e^{2}}$ and $\lambda_{i2}=\sqrt{kT_{i2}/4\pi Z_{2}n_{e2}e^{2}}$ are
electron and ion Debye lengths, respectively Spitzer (1965); Pauls  _et al._
(1995).

###  E.1 Structure P1 and P2

Our claim is that N ions penetrating into Al plasma is reflected by Biermann
magnetic field in the Al plasma. Although the Al plasma parameters are
uncertain at times and positions of our interest, we set $T_{e}=T_{i}=100$ eV,
$Z_{\rm Al}=9$, the ion density $n_{\rm Al}=1\times 10^{18}$cm-3, and the
magnetic field strength $B_{\rm Al}=10$ T as typical values. A trajectory of
P1 and P2, $X^{\prime}(t)$, gives the velocity, $dX^{\prime}/dt$, of 510 km
s-1 at $t=10$ ns, and of 406 km s-1 at $t=23$ ns, so we set the relative
velocity $v_{r}=500$ km s-1 as a fiducial value. The mean-free path of N ions
with $Z_{\rm N}=3$ in the Al plasma is then $\lambda_{ii}\approx 0.5~{}{\rm
cm}\,(3/Z_{\rm N})^{2}(9/Z_{\rm Al})^{2}(8.7/\ln\Lambda)v_{r,500}{}^{4}/n_{\rm
Al,18}$, and gyro radius of the N ions is given by $r_{g,{\rm N}}=0.24~{}{\rm
cm}\,(3/Z_{\rm N})v_{r,500}(B_{\rm Al}/10~{}{\rm T})^{-1}$, where
$v_{r,500}=v_{r}/500~{}{\rm km}~{}{\rm s}^{-1}$ and $n_{\rm Al,18}=n_{\rm
Al}/10^{18}{\rm cm}^{-3}$, so that their ratio is $\lambda_{ii}/r_{g,{\rm
N}}=2.2(3/Z_{\rm N})(9/Z_{\rm Al})^{2}(8.7/\ln\Lambda)(B_{\rm Al}/10~{}{\rm
T})v_{r,500}{}^{3}/n_{\rm Al,18}$. When we only change temperatures to
$T_{e}=T_{i}=10$ eV, then we get $\lambda_{ii}/r_{g,{\rm N}}=2.6$. If $Z_{\rm
N}=6$ with the other parameters being fiducial, the ratio becomes
$\lambda_{ii}/r_{g,{\rm N}}=1.2$. In these cases, although Coulomb collision
is marginally subdominant, it is non-negligible. However, at least, it can be
said that the Biermann field plays a crucial role in N ion reflection in the
Al plasma. Note that while the electron-electron Coulomb collision mean-free
path is much smaller than $\lambda_{ii}$ and $r_{g,{\rm N}}$, electrons around
P1 and P2 cannot generate an electric field that is able to reflect incoming N
ions.

###  E.2 Structure R1

An explanation of structure R1 is fast Al ions penetrating into initial
upstream N plasma as shown in our 1D PIC simulation. The mean-free path of the
Al ions with $Z_{\rm Al}=9$ in the N plasma ($T_{e}=T_{i}=6$ eV, $Z_{\rm
N}=3$, and the N ion density $n_{\rm N}=3.2\times 10^{17}$cm-3 corresponding
to $P_{N}=5$ Torr) is estimated as $\lambda_{ii}\approx 11~{}{\rm
cm}\,(3/Z_{\rm N})^{2}(9/Z_{\rm Al})^{2}(8.1/\ln\Lambda)v_{r,700}{}^{4}$,
where $v_{r,700}=v_{r}/700~{}{\rm km}~{}{\rm s}^{-1}$. If R1 is the head of N
ions reflected by interface P1, their mean-free path is also large
($\lambda_{ii}\gtrsim 10v_{r,700}{}^{4}$ cm for reflected ion charge number
$Z_{\rm N}=3$–6). Hence, the collisional effect is negligible for the
propagation of R1.

###  E.3 Structure R2

Our claim is that R2 at $t=23$ ns is the edge of N ions that are reflected by
compressed external magnetic field in the N plasma just upstream of P2 at
$t\approx 10$ ns. Although the parameters of the N plasma at the reflection
region are again uncertain, we set $T_{e}=200$ eV, $T_{i}=30$ eV, the electron
density $n_{e}=1\times 10^{19}$cm-3, and the strength of the compressed
external field $B_{N}=10$ T, considering the result of our 1D PIC simulation
at $t=10$ ns. The value of $Z_{\rm N}$ is also uncertain but for the worst
case, we assume that both incoming N ions and the N plasma have $Z_{\rm N}=6$.
Then we obtain the ion-ion mean-free path of incoming N ions
$\lambda_{ii}\approx 0.2~{}{\rm cm}\,(6/Z_{\rm
N})^{3}(7.5/\ln\Lambda)v_{r,500}{}^{4}/n_{e,19}$ and their gyro radius
$r_{g,{\rm N}}=0.12~{}{\rm cm}\,(6/Z_{\rm N})v_{r,500}(B_{\rm N}/10~{}{\rm
T})^{-1}$, where $v_{r,500}=v_{r}/500~{}{\rm km}~{}{\rm s}^{-1}$ and
$n_{e,19}=n_{e}/10^{19}{\rm cm}^{-3}$. Hence, the ratio of these scale length
is $\lambda_{ii}/r_{g,{\rm N}}\approx 1.3(3/Z_{\rm
N})^{2}(7.5/\ln\Lambda)(B_{\rm Al}/10~{}{\rm T})v_{r,500}{}^{3}/n_{e,19}$.
According to our PIC simulation results, the electron density $n_{e}\lesssim
10^{19}$cm-1 and the field strength of the compressed external magnetic field
$B_{N}\gtrsim 10$ T at the reflection region. Furthermore, the incoming N ions
have $Z_{N}=3$. Hence, we expect that the ratio $\lambda_{ii}/r_{g,{\rm N}}$
should be larger, and ion-ion collision is subdominant for the N ion
reflection for the origin of R2 at $t=23$ ns.

After the reflection at $t\approx 10$ ns, the reflected N ions return to the
upstream N plasma ($T_{e}=T_{i}=6$ eV, $Z_{\rm N}=3$, and the N ion density
$n_{\rm N}=3.2\times 10^{17}$cm-3) and gyrate due to the external magnetic
field. Assuming that the reflected ions have $Z_{\rm N}=6$ with velocity $400$
km s-1, which are values inferred from the TS analysis results at $t=23$ ns,
we get the ion-ion mean-free path of reflected N ions $\lambda_{ii}\approx
1.2~{}{\rm cm}$. Just after the reflection ($t\approx 10$ ns), the velocity of
the reflected ions should be larger than $400$ km s-1 as shown in 1D PIC
simulation, so that the Coulomb collision is negligible when reflected ions
move in the upstream N plasma.

Coulomb collision also changes ionization state of N ions, that is the value
of $Z_{\rm N}$. If the N plasma with $T_{e}\approx T_{i}\gtrsim 200$ eV is in
collision-ionization equilibrium (as assumed in analyzing TS IAW data), then
$Z_{\rm N}$ becomes large, e.g., $Z_{\rm N}=5$–7. However, the equilibrium
state is achieved when $n_{e}t_{p}\gtrsim 10^{11-12}$s cm-3, where $t_{p}$ is
the plasma age, that is, the elapsed time from the plasma generation Spitzer
(1965). The value of $n_{e}t_{p}$ to reach the equilibrium depends on the
initial values of $T_{e}$, $T_{i}$, $n_{e}$, and $Z_{\rm N}$. In our case,
$t_{p}$ is roughly given by the crossing time of the high-temperature
reflection region, which has typical width of $0.1$ cm, so that $t_{p}\approx
2v_{r,500}^{-1}$ ns and $n_{e}t_{p}\approx 2\times
10^{10}n_{e,19}v_{r,500}^{-1}$ s cm-3. The time evolution of $Z_{\rm N}$
should be calculated, but it is complicated. Such detailed calculation is
beyond the scope of this paper. However, for our values of $n_{e}t_{p}$, the
value of $Z_{\rm N}$ may be smaller than the value for the equilibrium state
($Z_{\rm N}=5$–7).

###### Acknowledgements.

The authors would like to acknowledge the dedicated technical support of the
staff at the Gekko-XII facility for the laser operation, target fabrication,
and plasma diagnostics. We thank A. Takamine and H. Maeda for useful
discussions. We would also like to thank anonymous referees for their careful
reading and helpful comments to improve the paper. This research was partially
supported by the joint research project of Institute of Laser Engineering,
Osaka University, JSPS KAKENHI Grants No. 18H01232, 19H01868, 17K18270,
18H01245, 18H01246, 19K14712, 17J03893, and 15H02154, the Sumitomo Foundation
for environmental research projects (203099) (S.M.), and JSPS Core-to-Core
Program B: Asia-Africa Science Plat-forms Grant No. JPJSCCB20190003 (Y.
Sakawa). Computer simulations were performed on the CIDAS supercomputer system
at the Institute for Space-Earth Environmental Research in Nagoya University
under the joint research program. R.Y. and S.J.T. deeply appreciate Aoyama
Gakuin University Research Institute for partially funding our research.

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