# Parametric study on ion acceleration from the interaction of ultra-high
intensity laser pulses with near-critical density gas targets

V. Ospina-Bohórquez<EMAIL_ADDRESS>CEA, DAM, DIF, F-91297 Arpajon,
France  Université Paris-Saclay, CEA, LMCE, 91690 Bruyères-le-Châtel, France
Université de Bordeaux, CNRS, CEA, CELIA (Centre Lasers Intenses et
Applications), UMR 5107, Talence, France  Universidad de Salamanca, Salamanca,
Spain  A. Debayle<EMAIL_ADDRESS>CEA, DAM, DIF, F-91297 Arpajon,
France  Université Paris-Saclay, CEA, LMCE, 91690 Bruyères-le-Châtel, France
J. J. Santos  Université de Bordeaux, CNRS, CEA, CELIA (Centre Lasers Intenses
et Applications), UMR 5107, Talence, France  L. Volpe  Universidad de
Salamanca, Salamanca, Spain  C.L.P.U. (Centro de Láseres Pulsados), Salamanca,
Spain  L. Gremillet<EMAIL_ADDRESS>CEA, DAM, DIF, F-91297 Arpajon,
France  Université Paris-Saclay, CEA, LMCE, 91690 Bruyères-le-Châtel, France

###### Abstract

We present a parametric study based on 1-D particle-in-cell (PIC) simulations
conducted with the objective of understanding the interaction of intense
lasers with near-critical non-uniform density gas targets. Specifically, we
aim to find an optimal set of experimental parameters regarding the
interaction of a $\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$I_{L}=10^{20}$ W/cm2 ($a_{0}=8.8$), $\tau_{L}=$30\text{\,}\mathrm{fs}$$ laser
pulse with a near-critical non-uniform pure nitrogen gas profile produced by a
non commercial gas nozzle. The PIC code Calder developed at CEA was used, and
both the maximum electron density and the direct laser contribution to ion
acceleration were studied. Shock formation was achieved for a peak electron
density $n_{e}$ ranging between 0.35 $n_{c}$ and 0.7 $n_{c}$. In this density
interval, the survival of a percentage of the laser pulse until the gas
density peak, while being strongly absorbed ($>$90$\%$) and creating a hot
electron population in the gas up-ramp, is singled out as a necessary
condition for shock formation. Moreover, the laser absorption must give rise
to a super ponderomotive heating of the target electrons in order to launch an
electrostatic shock inside the plasma. The direct laser effect on ion
acceleration consists in a strong initial density perturbation that enhances
charge separation while the electron pressure gradients are identified as
fundamental for shock formation. The production of a controlled and repetitive
gas profile as well as the possibility of performing measurements with
statistical meaning are highlighted as fundamental for conducting a thorough
experimental study.

##  I Introduction

The generation of energetic ion particle sources, with energies in the
10-$100\text{\,}\mathrm{MeV}$ range, is one of the major applications of
ultra-intense short-pulse lasers, made available thanks to the development of
the chirped pulse amplification (CPA) technique.Strickland and Mourou (1985)
These particle sources are employed in ultra-fast probing methods in science
and industry, such as proton radiographyBorghesi  _et al._ (2003) and time-
resolved probing of laser-created transient states.Borghesi  _et al._ (2004);
Santos  _et al._ (2015); M. Ehret  _et al._ (2017) These particles are also
used in the creation of warm dense matter,Roth  _et al._ (2009) as well as in
isochoric heating of dense plasmasPatel  _et al._ (2003) and radioisotope and
intense neutron sources production with possible medical spin offs.Fritzler
_et al._ (2003); Spencer  _et al._ (2001) Most of these applications exploit
the unique properties of laser-driven ion beams, notably their short duration,
high number density, low emittance, high laminarity and compactness compared
to conventional particle accelerators as well as the possibility of a high
repetition rate (HRR) production when working at modern fs-laser facilities.

The search for more energetic and efficient laser-driven particle sources
motivates the investigation of novel acceleration schemes involving new states
of matter, target shapes and compositions. The laser-matter interaction is
divided in two regimes depending on the transparency/opacity properties of the
irradiated (and ionized) medium. The plasma density threshold is governed by
the so-called critical density $n_{c}=1.11\times 10^{21}/\lambda_{L}^{2}\
[\mathrm{cm^{-3}}]$ where $\lambda_{L}$ is the laser wavelength in
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$. A medium providing an electron
density $n_{e}$ (upon ionization) will be opaque if $n_{e}>n_{c}=1.74\times
10^{21}\ \mathrm{cm^{-3}}$ considering a
$\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ laser. Opaque
plasmas are know as overdense plasmas ($n>>n_{c}$) while transparent ones are
given the name of underdense plasmas ($n<<n_{c}$). Solid and liquid targets
are normally opaque to the laser. However, the laser interaction with the
outer layer of the material allows for a $\approx 1-50\%$ energy transfer to
the target electrons accelerating them to $\approx$MeV energies. These fast
electrons induce strong ($\approx\mathrm{TV/m}$) charge separation fields over
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$ scales once propagated through the
target and crossing the target-vacuum interface. Ions with the higher charge-
to-mass ratios are preferentially accelerated by the rear-side sheath field
that has been created, referring normally to protons present as organic
impurities in the material’s surface. This process is called Target Normal
Sheath Acceleration (TNSA),Macchi, Borghesi, and Passoni (2013); Daido,
Nishiuchi, and Pirozhkov (2012) a term coined in the early 2000s to name the
general and long-known process of sheath-field ion acceleration arising in
non-uniform plasmas. Broad energy spectrum proton beams of a maximum energy of
$85\text{\,}\mathrm{MeV}$Wagner  _et al._ (2016) have been produced following
this scheme.

When the medium’s density drops strongly below $n_{c}$, the laser is allowed
to propagate inside the material creating non-linear wakefields capable of
accelerating electrons to relativistic velocities. At the same time, the laser
ponderomotive force expels electrons creating a channel along the propagation
axis, which expands radially. Coulomb explosion of the local ions is then
possible after the ponderomotive expulsion of the electrons which leaves the
ions to strongly experience the unscreened field arising from charge
separation. TNSA type ion acceleration can also arise from the expelled
electrons of the expanding sheath front. Moreover, the return electron current
that is formed once exiting into vacuum sustains a strong quasistatic magnetic
field in the vacuum interface which can also efficiently accelerate as well as
collimate ions.Bulanov and Esirkepov (2007); Bulanov  _et al._ (2005); Park
_et al._ (2019); Nakamura  _et al._ (2010) Multi-MeV ion beams have been
produced in this interaction regime. Krushelnick  _et al._ (1999); Kuznetsov
_et al._ (2001); Willingale  _et al._ (2006)

Fewer studies have addressed the case of near-critical plasmas, i.e. of
electron density approaching the critical density, $n_{e}\approx n_{c}$, due
to the technical difficulty of achieving such densities in a controlled and
repetitive manner. When $n_{e}\approx n_{c}$, intense laser pulses can
propagate over large distances and volumetrically heat the plasma electrons to
relativistic temperatures. The interaction of two collisionless plasma regions
with different densities, temperatures or velocities can then lead to the
formation of non-linear electrostatic shocks capable of trapping and/or
reflecting charged particles. These shocks can be triggered in several ways:
by applying a laser piston to the overcritical boundary of a uniform
overcritical plasma;Denavit (1992); Silva  _et al._ (2004) or in partially
transparent plasmas where strong electron pressure gradients, shorter than the
local Debye length $\lambda_{D}$, and/or laser-driven ion acceleration can
trigger shock formation. Near-critical plasmas are predicted to give rise to
new ion acceleration regimes combining TNSA and collisionless shock
acceleration (CSA),Fiuza  _et al._ (2012, 2013); Dieckmann  _et al._ (2013a,
b); Lécz and Andreev (2015); Liu  _et al._ (2016, 2016); Liu, Meyer-ter-Vehn,
and Ruhl (2018) as well as enhanced hot-electron production, that is, beyond
the standard ponderomotive scaling, i.e. the mean hot electron energy
$\epsilon_{h}\approx\sqrt{1+{a_{0}}^{2}/2}-1$, where
$a_{0}=eE_{0}/m_{e}c\omega_{0}$ is the normalized laser amplitude and $E_{0}$
the laser electric field in vacuum, $m_{e}$ is the electron rest mass and $c$
the speed of light in vacuum. Our Calder codeLefebvre  _et al._ (2003)
preliminary simulations predict efficient volumetric electron heating due to
phase mixing between bulk electrons and electrons trapped in laser-induced
plasma waves.Debayle  _et al._ (2017) The strong electron pressure gradients
that result in inhomogenous plasmas are able to trigger an electrostatic
shock,Moiseev and Sagdeev (1963); Forslund and Shonk (1970); Sorasio  _et al._
(2006); Cairns  _et al._ (2014) favoring and enhancing ion
acceleration.Debayle  _et al._ (2017) Furthermore, CSA is expected to produce
more peaked ion spectra in comparison to the broad spectra produced through
TNSA.Boella  _et al._ (2018) Previous experimental attempts in this direction
have resorted to the use of double-layer targets composed of a near-critical
layer (e.g. carbon nanotubes) and a thin diamond-like solid layer Bin  _et
al._ (2015, 2018); Ma  _et al._ (2019); Pazzaglia  _et al._ (2020) with
$\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ fs lasers, or
exploded foils used in double pulse ns/ps configuration with
$\lambda_{L}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ lasers.Antici  _et
al._ (2017); Pak  _et al._ (2018); Rosmej  _et al._ (2019) Moderate-density
gas jets irradiated by $\mathrm{CO_{2}}$
($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength) laser
pulses,Haberberger  _et al._ (2012); Tresca  _et al._ (2015); Tsung  _et al._
(2012) where the critical density constraint is eased, have also been employed
and the laser pedestal has been used to sculpt the gas up-ramp into a steep
density front.Chen  _et al._ (2017); Puyuelo-Valdes  _et al._ (2019) The
advent of new high-intensity HRR facilities like Apollon (France), VEGA-3
(Spain), L4 Aton (Czech Republic) or BELLA PW (USA) drives the application of
this ion acceleration scheme using high-density gas-jets, a HRR-compatible
debris-free targetry. Pioneer experiments have been conducted both in the TW
Sylla  _et al._ (2013); Ehret  _et al._ (2020) and the PWSingh  _et al._
(2020) regime with preliminar although encouraging ion acceleration results.

In this paper, we will examine collisionless electrostatic shock (CES)
formation in transparent plasmas under conditions accessible to a laser system
characterized by $I_{L}$=1.7 x $10^{20}$ W/cm2,
$\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\tau_{L}=$30\text{\,}\mathrm{fs}$$ (parameters of the VEGA-3 laser, CLPU,
Spain), and considering as a target a highly non-uniform near-critical-density
gas jet produced by the a non-commercial supersonic nozzle Puyuelo-Valdes
(2020), plotted in Fig. 1. In the case of a $\mathrm{N_{2}}$ gas, CES are
found to occur within specifically, the quite a narrow electron density
$n_{e}$ range between 0.35 $n_{c}$ and 0.7 $n_{c}$. The transmission of a
significant fraction of the laser pulse across the density peak, while being
strongly absorbed (80$\%$ \- 90$\%$) and creating a hot electron population in
the gas up-ramp, is identified as a necessary condition for CES formation. The
latter stems from the strong electron pressure gradients located in the
density down-ramp. The direct laser effect on ion acceleration is a strong
initial density perturbation that enhances charge separation while electron
pressure gradients are identified as the main cause of shock formation. The
addition of lighter helium ions to the $\mathrm{N_{2}}$ gas leads to peak in
the velocity spectra, which may constitute a signature of CES formation.
Finally, the production of a controlled and repetitive gas profile as well as
the possibility of performing measurements with statistical meaning are
highlighted as fundamental for conducting a thorough experimental study.

This paper is organized as follows. The simulation parameters and initial
conditions are described in Section II. The electron energization and plasma
heating processes are presented in Section III. The description of the
different ion acceleration mechanisms including CSA and TNSA is exposed in
Section IV. Finally, the conclusions are summarized in Section V.

##  II Simulation parameters and initial conditions

The present 1-D simulations describe the interaction of a laser pulse
characterized by a normalized field strength $a_{0}$=8.8 ($I_{L}$=1.7 x
$10^{20}$ W/cm2), a laser wavelength
$\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and a
$\tau_{L}=$30\text{\,}\mathrm{fs}$$ FWHM duration, corresponding to the VEGA-3
laser parameters. The laser is linearly polarized along the $y$-axis and it is
injected along the $x$-axis from the left-side of the simulation box. This
study is restricted to a 1-D geometry because of the large spatial
($\approx$mm) and temporal ($\approx$10\text{\,}\mathrm{ps}$$) scales
considered, and the relatively fine resolution dictated by the $n_{c}$ maximum
density of the fully ionized $\mathrm{N_{2}}$ gas profile. Field and impact
ionization together with elastic Coulomb collisions between all charged
particles were considered. Absorbing boundary conditions were applied for
fields and particles. Each cell initially contained 100 ions of each species.
The initial ion temperature was $T_{i}=$1\text{\,}\mathrm{eV}$$. The temporal
and spatial resolutions were $\Delta t$= 0.095 $\omega_{0}^{-1}$
($0.05\text{\,}\mathrm{fs}$) and $\Delta x$= 0.1 $c/\omega_{0}$
($0.016\text{\,}\mathrm{\SIUnitSymbolMicro m}$), respectively. The modeled
gaseous targets were composed of either atomic nitrogen or a mixture of 90$\%$
$\mathrm{N_{2}}$ and 10$\%$ He.

The initial gas density profile was obtained by performing fluid dynamics
simulations with the FLUENT code Fluent (2015). The reference profile was that
predicted with the S900 shock nozzle Puyuelo-Valdes (2020) and a backing
$\mathrm{N_{2}}$ gas pressure of $400\text{\,}\mathrm{bar}$, measured far
enough from the nozzle to avoid damaging it.Ehret  _et al._ (2020) This
profile, plotted in Fig. 1 (blue curve), is characterized by a very narrow
($\approx$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) density peak (up to
$n_{e,max}=9.8\ n_{c}$) sitting on broad ($\approx$1\text{\,}\mathrm{mm}$$),
quasi-exponential symmetric wings. The chosen gas is nitrogen in order to
replicate in a simpler manner the experimental conditions.

In the following and unless explicitly marked differently the density,
velocity, time, distance, mass, electron momentum, ion momentum, energy and
electric field are normalized to the following quantities, respectively:

$n_{c},c,\omega_{0}^{-1},{c}/{\omega_{0}},m_{e},m_{e}c,m_{i}c,m_{e}c^{2},m_{e}\omega_{0}c/e,$
(1)

where $\lambda_{0}=2\pi/k_{0}=2\pi c/\omega_{0}$ is taken to be equal to
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $n_{c}$ is the critical density for
a 1 $\mu$m laser wavelength equal to 1.11$\times 10^{27}\ \mathrm{m^{-3}}$,
$c$ is the velocity of light in vacuum, $\omega_{0}^{-1}$ is the corresponding
inverse laser frequency equal to $0.53\text{\,}\mathrm{fs}$, $e$ is the
electron charge and $m_{e}$ and $m_{i}$ are the electron and ion rest masses,
respectively.

Starting from the reference gas profile we investigate the laser-driven plasma
dynamics in uniformly rescaled $\mathrm{N_{2}}$ profiles with $n_{e,max}$ =
2.1, 0.7, 0.35 and 0.14 $n_{c}$. Several runs were performed to examine the
role of the sole hot electrons to drive the CES. Additional runs were
performed to examine the changes brought to the interaction by a gas mixture
of $90\%\ \mathrm{N_{2}}$ and $10\%\ $He or $90\%\ \mathrm{N_{2}}$ and $10\%\
$H, as well as the effect of interacting with a stretched laser pulse.

Table 1 summarizes the simulations’ main parameters regarding both the laser
and gas target. Note that following the normalization given in Eq. 1,
$a_{0}=8.8$ for a $\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
transforms into $a_{0,norm.}=11$ given the normalization for
$\lambda_{L}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, as written in Table
1.

Sim. number |  $n_{at,max}$ $[n_{c}]$ |  $n_{e,max}$ $[n_{c}]$ | Gas type |  Laser ON/OFF | $a_{0}$ |  $\tau_{L}$ $[\omega_{0}^{-1}]$
---|---|---|---|---|---|---
I | 1.4 | 9.8 | $\mathrm{N_{2}}$ | ON | 11 | 56
II | 0.3 | 2.1 | $\mathrm{N_{2}}$ | ON | 11 | 56
III | 0.1 | 0.7 | $\mathrm{N_{2}}$ | ON | 11 | 56
IV | 0.05 | 0.35 | $\mathrm{N_{2}}$ | ON | 11 | 56
V | 0.02 | 0.14 | $\mathrm{N_{2}}$ | ON | 11 | 56
VI | 0.1 | 0.7 | $\mathrm{N_{2}}$ | OFF ($T_{e}=a_{0}$) | $-$ | $-$
VII | 0.1 | 0.7 | $\mathrm{N_{2}}$ | ON | 6.3 | 169
VIII | 0.1 | 0.7 |  $90\%\ \mathrm{N_{2}}$+$10\%\ $He | ON | 11 | 56
IX | 0.1 | 0.7 |  $90\%\ \mathrm{N_{2}}$+$10\%\ \mathrm{H_{2}}$ | ON | 11 | 56
X | 0.1 | 0.7 | $\mathrm{N_{2}}$ | OFF ($T_{e}=a_{0}/2$) | $-$ | $-$
Table 1:  Main parameters of the 1-D simulations.

##  III Electron acceleration and plasma heating

###  III.1 Areal density needed for complete laser absorption

To discern between high and moderate/low density simulations the criterion of
complete laser absorption inside the gas profile is considered. It is
important to differentiate between these two interaction regimes since they
will trigger different electron acceleration mechanisms, as will be discussed
later on. This condition has previously been studied in Ref. Debayle  _et al._
(2017), where the minimum areal density required for complete laser absorption
was found to scale approximately as

$\sigma_{abs}=\int_{x_{0}}^{x_{abs}}dxn_{e}(x)\approx\frac{a_{0}^{2}\tau_{L}}{2\langle\gamma_{e}\rangle}$
(2)

where $\langle\gamma_{e}\rangle\approx a_{0}$ is the relativistic Lorentz
factor corresponding to the mean bulk electron energy.

According to this criterion, a $I_{L}$=1.7 x $10^{20}$ W/cm2,
$\lambda_{L}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$\tau_{L}=$30\text{\,}\mathrm{fs}$$ laser pulse should be fully absorbed
before reaching the density peak in the simulations with $n_{e,max}\geq 2.1\
n_{c}$ (Sims. I-III). In the simulations with $n_{e,max}<2.1\ n_{c}$ the laser
is able to cross the entire profile. The crossed thicknesses of complete laser
absorption $x_{abs}$ in Sims. I-III are indicated as solid lines in Figs. 1
and 2, and the obtained $x_{abs}$ values are summarized in Table 2.

Sim. number |  $n_{e,max}$ $[n_{c}]$ |  $x_{abs}$ $[c/\omega_{0}]$
---|---|---
I | 9.8 | 4520
II | 2.1 | 5868
III | 0.7 | 7061
IV | 0.35 | -
V | 0.14 | -

Table 2:  $X$-coordinates where complete laser absorption is met for Sims.
I-III considering full ionization of the nitrogen gas in Eq. 2. In Sims. IV
and V the laser is able to cross the entire profile without being completely
absorbed. Figure 1: Electron density $n_{e}$ profiles input in the 1-D PIC
simulations.

Simulation I: $n_{e}=9.8\ n_{c}$(a)

Simulation II: $n_{e}=2.1\ n_{c}$(b)

Simulation III: $n_{e}=0.7\ n_{c}$(c)

Simulation IV: $n_{e}=0.35\ n_{c}$(d)

Figure 2: $(x,t)$ evolution of the laser field $E_{y}$ propagation in the (a)
S.I, (b) S.II, (c) S.III and (d) S.IV, see Table 1. The theoretical positions
for complete laser absorption deduced from Eq. 2, are marked with solid lines
and summarized in Table 2. The profile’s peak density is marked by the pink
dashed line at 6432 $c/\omega_{0}$.

The $(x,t)$ evolution of the $y$-componen of the laser field $E_{y}$ is
plotted for various maximum electron densities (9.8 $n_{c}$, 2.1 $n_{c}$, 0.7
$n_{c}$ and 0.35 $n_{c}$) in Figs. 2a-d. The colormaps are saturated to
highlight the transmitted fraction of the pulse. The density peak is located
at $x=$ 6432 $c/\omega_{0}$ and it is marked with pink dashed lines. The
positions corresponding to full laser absorption, as predicted by Eq. 2, are
plotted as solid lines. In the highest density case (Fig. 2a) most of the
laser has been absorbed or reflected at about 5000 $c/\omega_{0}$,
approximately 1000 $c/\omega_{0}$ ahead of the density peak, a location
consistent with Eq. 2. Only $0.1\%$ of the laser energy has been transmitted
and a few percent of the laser intensity ($I$ $\propto a_{0}^{2}$).

In the high density cases, above $n_{at}\geq 2.1\ n_{c}$ (Figs. 2a and b),
strong laser reflection takes place possibly as a consequence of backward
stimulated Raman scatering (BSRS) Guérin  _et al._ (1995); Guerin  _et al._
(1996); Moreau  _et al._ (2017); Moreau (2018).

Between $0.35\geq n_{e}\geq 0.7\ n_{c}$, in the moderate density regime (Figs.
2c and d), the pulse transmitted across the density peak remains intense
enough ($a_{0}\approx 3$ at $\omega_{0}t=\ 6500$ in Fig. 2c) to induce a
plasma wakefield and thus continue energizing the down-ramp bulk electrons.
Several electromagnetic solitons are created as well but are not believed to
have a strong effect on the electron acceleration, let alone the ion dynamics.

###  III.2 Stochastic electron heating

All simulations show that, early in the interaction, wakefield-type electron
acceleration takes place in the fast-ionized plasma. This fluid stage of
electron acceleration is shown in the electron $(x,p_{x})$ phase spaces of
Figs. 3a-d corresponding to Sims. I - IV, see Table 1. The wakefield matching
condition is met for $\tau_{L}c\approx 4\pi c/\omega_{pe}$.Teychenné, Bonnaud,
and Bobin (1993) Thus, reducing the gas peak density will shift the resonance
condition to a deeper region of the plasma. This is illustrated by the
similarity of the electron $(x,p_{x})$ phase spaces in Figs. 3a-d, for raising
positions in the plasma while reducing the plasma density.

Simulation I: $n_{e}=9.8\ n_{c}$(a)

Simulation II: $n_{e}=2.1\ n_{c}$(b)

Simulation III: $n_{e}=0.7\ n_{c}$(c)

Simulation IV: $n_{e}=0.35\ n_{c}$(d)

Figure 3: Electron $(x,p_{x})$ phase space for (a) S.I, (b) S. II, (c) S. III
and (d) S. IV, see Table 1, illustrating wakefield-type electron acceleration
taking place in the low density part of the gas target up-ramp.

At the beginning of the interaction, the wakefield electron energization stage
plotted in Figs. 3a-d entails a low laser energy absorption rate. The latter
translates into the gentle first section of the laser absorption curves,
plotted against time in Fig. 4a. Simulations I-IV exhibit a transition from a
laminar wakefield stage to a turbulent regime where the laser absorption rate
strongly increases. The first stage of electron energization occurs until the
times $\sim 3000,4200,5000$ and $6000\ \omega_{0}^{-1}$ for Sims. I-IV,
respectively. The steepening of the laser absorption curves coincides with the
instant where the laser reaches the electron density $n_{e}$ that satisfies
the wakefield matching condition.

At the same time, since the laser pulse is longer than the local electron
plasma wavelength $\lambda_{pe}$ ($\tau_{L}c=56\ c/\omega_{0}>\lambda_{pe}=3\
c/\omega_{0}$, in S.I), the wakefield is generated in the front part of the
laser pulse. The front of the laser pulse is then strongly depleted in the
process of transferring its energy to the wakefield.Vieira  _et al._ (2010)
This is observed in Figs. 4b-d, which correspond to the laser $E_{y}$ field of
S.III plotted at times $\omega_{0}t=$ 3990, 5201 and 6270. This so-called
optical shock increases the ponderomotive force $F_{p}$ exerted by the laser
over the target electrons, since $F_{p}$ is proportional to the gradient of
the laser intensity $F_{p}\propto-\partial_{x}\sqrt{1+p_{x}^{2}+a_{0}^{2}}\ $
Debayle  _et al._ (2017). The development of an optical shock could also be
linked to the steepening of the laser absorption curves of Fig. 4a (see Ref.
Debayle  _et al._ (2017) and references therein).

Figure 4: (a) Percentage of absorbed laser energy with respect to time for S.I
(blue solid line), S.II (orange dashed line), S.III (yellow squared-dotted
line), S.IV (dark purple thin solid line) and S.V (green thin dashed line),
see Table 1. The instant when the laser pulse arrives to the density peak
$\omega_{0}t=6450$ is marked with a vertical pink dashed line. The insets of
the figure correspond to the $E_{y}$ laser field in S.III at times (b)
$\omega_{0}t=3990$, (c) $\omega_{0}t=5201$ and (d) $\omega_{0}t=6460$, marked
with black arrows in (a).

Simulation III: $n_{e}=0.7\ n_{c}$(a)$\leftarrow$

(b)$\leftarrow$$\leftarrow$

(c)$\uparrow$

(d)

Figure 5: Electron $(x,p_{x})$ phase-space (colorscale) extracted at times
$\omega_{0}t=$ (a) 6270, (b) 6460, (c) 6840 and (d) 7030 corresponding to
S.III. The normalized electrostatic potential $-\Phi_{x}$ is superposed as
solid pink lines.

Figures 5a-d correspond to the electron $(x,p_{x})$ phase spaces of S.III at
times $\omega_{0}t=$ 3040, 6460, 6840 and 7030. The electrostatic potential
$-\Phi_{x}$ is overlaid as solid pink lines.

Fig. 5a shows the early stage of electron energization, that takes place when
the laser interacts with the low density wing of the target. The nonlinear
wakefield has trapped a bunch of electrons that co-move with it, as indicated
by the black arrow in Fig. 5a. Note that the electrostatic potential
$-\Phi_{x}$ still comes back to zero in the first oscillations.

Afterwards, at $\omega_{0}t=$ 6460 (Fig. 5b), the laser-plasma interaction
gives rise to a two-stream electron distribution resulting from the nonlinear
development of the plasma wakefield. The mentioned electron distribution is
composed by a cold electron return current and a hot forward-moving one. Note
that the laser front has fully steepened by this point, see Fig. 4d.

On one hand, the cold electron current contains electrons initially
accelerated by the laser wakefield and co-moving with it, see black arrows in
Figs. 5a and b. The laser-induced displacement of these electrons from their
initial position gives rise to a strong charge separation $E_{x}$ field, which
drags them back into the plasma core at a negative momentum $p_{x}\approx-25$,
as seen in Fig 5b. This charge separation field, which propagates at a
velocity close to the speed of light $c$, is as strong as it is short-lived
since the displaced electrons are rapidly pulled back into the plasma. On the
other hand, the hot forward electron current is composed of the electrons
inside the bulk of the plasma which are already undergoing a heating process
in a turbulent regime, as highlighted by the red arrow in Fig 5b. At this
point, phase mixing between these two electron currents has already started.

At $\omega_{0}t=$ 6840 (Fig. 5c), phase mixing between these high-energy
streams continues to develop and results in fast heating of the plasma
electrons. The process of breaking of the wakefield and phase mixing between
the two cited electron currents is called beam-loading.Debayle  _et al._
(2017) The latter fully develops by $\omega_{0}t=$ 7030 (Fig. 5d). Here the
cold return current is re-injected into the target at a negative momentum
$p_{x}\approx-20\approx-\Phi_{x}/2$, in agreement with the electron heating
model in near-critical plasmas developed in Ref. Debayle  _et al._ (2017). The
electrostatic potential drops strongly behind the laser pulse and oscillates
around $-\Phi=50$ during the interaction between the two electron currents.

Figures 6a and b correspond to the late stage of the electron energization of
Sims. I and III, as seen in the electron $(x,p_{x})$ phase spaces taken at
times $\omega_{0}t=$ 12350 and 12450, respectively. Figures 6c and d are zooms
of the gas down-ramp corresponding to Sims. I and III, respectively. The
electrostatic potential $-\Phi_{x}$ is overlaid as a pink solid line in all
panels.

$\downarrow$$\leftarrow$$\leftarrow$Simulation I: $n_{e}=9.8\ n_{c}$Simulation
III: $n_{e}=0.7\ n_{c}$

Figure 6: Late stages electron $(x,p_{x})$ phase-spaces (colorscale)
corresponding to (a) S.I and (b) S.III. The normalized electrostatic potential
$-\Phi_{x}$ is overlaid as solid pink lines.

In the high-density S.I (Figs. 6a and c), the majority of hot electrons are
produced at $x\approx 3000\ c/\omega_{0}$, when the wakefield resonance
condition is satisfied, as previously discussed. Since the laser is completely
absorbed at $\omega_{0}t\approx 4500$, this electron population must heat-up
all the gas left to its right. The neutralization of this hot electron
population is ensured until the point where it reaches a background density
similar to that at which it was created (i.e. at $x\approx 9000\
c/\omega_{0}$, see density profiles of Fig. 1). At this point, charge
neutralization can no longer be ensured and a charge separation $E_{x}$ field
is created. This field pulls back the hot electrons, obliging them to
recirculate around the target (see black arrow in Fig. 6a), yielding the
hotter electron distribution at the right of the density peak.

In the moderate density S.III (Figs. 6b and d), the majority of hot electrons
are created at $x\approx 5000\ c/\omega_{0}$. Hence, charge neutrality is no
longer ensured when the hot electrons reach the symmetrical point in the up-
ramp, i.e. $x\approx 7800\ c/\omega_{0}$. The electrostatic fields arising
from the strong electron density and pressure gradients cause the electrons to
recirculate back and forth (forming a vortex in the phase space) around the
maximum density, while triggering TNSA of the local ions located at the
density peak, as will be discussed in the following section.

In S.I (S.III) the transmitted percentage of the laser energy is $0.4\%$
($4.6\%$). The laser can then not be responsible for the electron acceleration
seen in the right plasma-vacuum boundary up to momenta $p_{x}>60$ in both
simulations (see black arrows in Figs. 6c and d). This high-energy electron
bunch was initially energized in the plasma wakefield, crossed the entire
target and exits now into vacuum. The formation of the return current is seen
on the fast-ionizing front of the hot electron streams.

##  IV Ion acceleration mechanisms

The subsequent ion acceleration mechanisms and their efficiency chiefly rely
on the laser absorption conditions. Three regimes of ion acceleration,
illustrated in the $(x,t)$ ion density maps of Figs. 7a-d (corresponding to
Sims. I - IV), stem from the 1-D simulations.

In the high density cases corresponding to Sims. I and II (Figs. 7a and b),
the laser is absorbed in the gas up-ramp. The hot electrons propagate through
the gas and trigger ion acoustic waves (IAWs) propagating at $\pm\,C_{s}$ with
equal amplitudes. Here $C_{s}=\sqrt{Z^{*}T_{e}/m_{i}}\approx 0.025\ c$ is the
local ion acoustic speed (with $Z^{*}\approx 5$ and $T_{e}\approx 3$, the
ionic charge and electron temperature, respectively).

For the moderate gas densities of Sims. III and IV (Figs. 7c and d), the laser
drives strong wakefields that trap a large number of electrons, rapidly turn
turbulent and heat the plasma. The strong electron pressure triggers an
electrostatic shock as illustrated in Fig. 7c and d. Background ions
reflection takes place at $t\gg\omega_{pi}^{-1}$. In S.III (S.IV) the front of
the expanding plasma travels at $v_{i,w}\approx 0.08\ c,M\approx 2$
($v_{i,w}\approx 0.13\ c,M\approx 2$), where $M=v_{x,i}/C_{s}$ is the upstream
Mach number. As one can deduce, $T_{e}$ is higher in S.IV than in S.III. These
cases correspond to the ideal laser absorption regime and are discussed in the
following Section IV.1.

Finally, for low gas densities (S.V), the laser absorption essentially remains
in a moderately non-linear wakefield regime. The electron temperature is not
strong enough to trigger a shock and only TNSA subsists.

Simulation I: $n_{e}=9.8\ n_{c}$(a)

Simulation II: $n_{e}=2.1\ n_{c}$(b)

Simulation III: $n_{e}=0.7\ n_{c}$(c)

Simulation IV: $n_{e}=0.35\ n_{c}$(d)

Figure 7: $(x,t)$ maps of the $\mathrm{N^{7+}}$ ion density centered on the
target down-ramp for the (a) Simulation I with $n_{at,max}=1.4\ n_{c}$, (b)
Simulation II with $n_{at,max}=0.3\ n_{c}$, (c) Simulation III with
$n_{at,max}=0.1\ n_{c}$ and (d) Simulation IV with $n_{at,max}=0.05\ n_{c}$,
see Table 1. The density peak is marked with a pink dashed line.

###  IV.1 Ion acceleration in moderate density simulations: collisionless
electrostatic shock formation

Figures 8a and b show the ion $(x,p_{x})$ phase spaces for the moderate
density Sims. III and IV at the shock formation time, i.e. the onset of ion
reflection. At this point, the ion velocity profile has fully steepened. In
S.III shock reflection starts at $\omega_{0}t\approx 14250$
($7.6\,\mathrm{ps}$), and in the lower density S.IV it does so at
$\omega_{0}t\approx 13870$ ($7.4\,\mathrm{ps}$). The shock formation time does
not satisfy the normally assumed proportionality $t\propto\omega_{pi}^{-1}$.
Shock reflection could be accelerated in S.IV, in respect to S.III, by
widening the density profile until achieving optimum plasma heating
conditions, i.e. the transition from a laminar to a turbulent electron
energizing regime while crossing the gas density peak (as in S.III, see
absorption curves of Fig. 4).

Simulation III: $n_{e}=0.7\ n_{c}$(a)

Simulation IV: $n_{e}=0.35\ n_{c}$(b)

Figure 8: Ion phase-spaces for the (a) simulation III with $n_{at,max}=0.1\
n_{c}$ and (b) simulation IV with $n_{at,max}=0.05\ n_{c}$ taken at the shock
formation instants when background ion reflection is about to start.

Figures 9a and b show the ion $(x,p_{x})$ phase spaces corresponding to the
last time of Sims. III and IV. CES formation was observed in both simulations
at these times.

Simulation III: $n_{e}=0.7\ n_{c}$(a)

Simulation IV: $n_{e}=0.35\ n_{c}$(b)

Figure 9: Final $\mathrm{N^{7+}}$ ion $(x,p_{x})$ phase spaces of the (a)
$n_{at,max}$ = 0.1 $n_{c}$ and (b) $n_{at,max}$ = 0.05 $n_{c}$ maximum density
simulations.

In these cases, the accelerated ion velocity profile continually steepens when
moving down the density gradients while the quasi-exponential density "wings"
are subject to a uniform slower acceleration in the ambipolar field
$E_{x}\approx T_{e}/L_{n}$, where $L_{n}$ is the quasi-constant density
gradient. This process bears resemblance to wavebreaking in TNSA within a pre-
expanded plasma profile,Grismayer and Mora (2006) and has been described as
the expansion of a hot plasma into a cold rarefied plasma.Perego  _et al._
(2013); Nechaev  _et al._ (2020) The steepening of the ion velocity profile
goes along with an increase in the local potential barrier up to the point of
reflecting the upstream ions, uniformly accelerated by the ambipolar field. An
ion located in front of the shock and moving at a velocity $v_{0}-v_{s}$ in
the shock frame, where $v_{s}$ is the lab-frame shock front velocity and
$v_{0}$ is the initial lab-frame ion velocity, gets reflected from the shock
front if its kinetic energy in the shock frame, $E_{k}$, is lower than the
electrostatic potential $Z^{*}e\Delta\phi$ experienced by the particle in the
shock region:

$\displaystyle E_{k}=\frac{m_{i}(v_{0}-v_{s})^{2}}{2}\ <\ Z^{*}e\Delta\phi.$
(3)

Here, $m_{i}$ is the ion mass and $Z^{*}$ the ionization degree. Shock
formation was observed in the simulations with peak electron density between
0.35 $n_{c}$ and 0.7 $n_{c}$ (Sims. III and IV). Moreover, the uniform slower
acceleration of the ions located in the target wings is due to the constant
plasma scale length $L_{n}$ of the density profile, as seen in Fig. 9, which
gives rise to a constant TNSA derived electrostatic field. The contrary would
induce a chirp in the reflected ions, making them lose their CSA
characteristic peaked spectra.Fiuza  _et al._ (2012) The CES formed in Sims.
III and IV have Mach numbers $M\approx$1.6 and $M\approx$ 2.4, respectively.
Mach numbers between $1.6\leq M\leq 3$ are characteristic of super-critical
shocks, in which ion reflection and plasma heating are the stabilizing energy
dissipation mechanisms.Fiuza  _et al._ (2012) The obtained Mach number values
agree with the CES conditions in near-critical plasmas studied by F. Fiuza et
al., Fiuza  _et al._ (2013, 2012) and M.E. Dieckmann et al.,.Dieckmann  _et
al._ (2013a, b)

Table 3:  Plasma parameters relative to the shock reflection condition in S.III ($n_{e}=0.7\ n_{c}$) for the $\mathrm{N^{7+}}$ and $\mathrm{N^{6+}}$ ion species. Ion | Time [$\omega_{0}^{-1}$] |  $m_{i}$ [$m_{e}$] |  $v_{f}$ [$c$] |  $v_{0}$ [$c$] |  $(v_{f}-v_{0})$ [$c$] |  $E_{k}$ [$m_{e}c^{2}$] | $Z^{*}$ |  $\Delta\phi$ [$m_{e}c^{2}/e$] |  $Z^{*}\Delta\phi$ [$m_{e}c^{2}/e$] |  $C_{s}$ [$c$] | M
---|---|---|---|---|---|---|---|---|---|---|---
$\mathrm{N^{7+}}$ | 13945 | 25704 | 0.109 | 0.030 | 0.079 | 79.804 | 7 | 12 | 84 | 0.048 | 1.642
$\mathrm{N^{6+}}$ | 13945 | 25704 | 0.109 | 0.020 | 0.089 | 101.34 | 6 | 12 | 72 | 0.048 | 1.850

The plasma parameters relevant for shock reflection in S.III are summarized in
Table 3. The shock reflection condition, corresponding to Eq. 3, is fulfilled
for the $\mathrm{N^{7+}}$ ionic species at $\omega_{0}t=13495$. In the case of
$\mathrm{N^{6+}}$ ions the kinetic energy exceeds the electrostatic potential
experienced by the ionic species in the shock region $E_{k}>Z^{*}e\Delta\phi$,
thus preventing electrostatic reflection, as discussed hereafter.

####  IV.1.1 Effect of the ionization degree on shock reflection

In S.III the gas down-ramp has experienced a strongly attenuated laser pulse,
and therefore has not been fully ionized. Hence, it is interesting to examine
how ionic species of different charge state interact with a CES and, in
particular, whether they can be reflected by it. Recalling the shock
reflection condition of Eq. 3, the latter is highly sensitive to the ionic
species ionization state, since it will determine the electrostatic potential
energy experienced by the incoming ions. Figs. 10a and b focus on the ion
$(x,p_{x})$ phase space down-ramp of S.III, corresponding to the
$\mathrm{N^{6+}}$ and $\mathrm{N^{7+}}$ nitrogen ionic species, respectively.
The $\mathrm{N^{6+}}$ ion species do not see a sufficiently strong
electrostatic potential and are consequently not reflected from the shock
front, located at $x=7500\ c/\omega_{0}$ in Fig. 10a. Instead, they are
slightly accelerated and finally cross the shock region. Only the fully
ionized $\mathrm{N^{7+}}$ ions experience the maximum electrostatic potential
and are reflected at twice the shock velocity, see Fig. 10b.

Simulation III: $n_{e}=0.7\ n_{c}$(a)$\mathrm{N^{6+}}$

(b)$\mathrm{N^{7+}}$

Figure 10: Influence of the ionization state in shock reflection as seen in
the ion phase spaces of the (a) $\mathrm{N^{6+}}$ and (b) $\mathrm{N^{7+}}$
ion charged species phase spaces corresponding to S.III.

####  IV.1.2 Role of laser-driven ion acceleration combined with the gas
profile in shock formation

So far, we have discussed the formation of CES in terms of the slow plasma
expansion driven by the uniformly heated bulk electrons, leaving aside the
possible influence of shorter-scale velocity of ion density perturbations
imparted by the laser. The direct laser impact on ion acceleration and shock
formation was assessed by performing a laser-free simulation with a pre-heated
electron population with $T_{e}=a_{0}=11$ and $n_{e,max}=0.7\ n_{c}$ (S.VI).
The chosen $T_{e}=a_{0}$ value approximately agrees with the mean electronic
temperature found in the original laser-on S.III, as seen in Fig. 11.

Simulation III: $n_{e}=0.7\ n_{c}$

Figure 11: Electron temperature $T_{e}$ lineout extracted from S.III at
$\omega_{0}t=15010$.

Simulation VI: laser-free(a)

Simulation III: laser-on(b)

(c)

(d)

Figure 12: (a) Final ion $(x,p_{x})$ phase spaces corresponding to (a) S.VI
and (b) S.III. Electrostatic shock barriers in (c) S.VI and (d) S.III, as seen
in the electron $(x,p_{x})$ phase spaces of the respective simulations.

Figures 12a and b show the ion $(x,p_{x})$ phase spaces corresponding to the
last time step of S.VI (with a pre-heated Maxwellian electron population) and
S.III, respectively. Figures 12c and d plot the corresponding electron
$(x,p_{x})$ phase spaces. The longitudinal electrostatic field $E_{x}$ is
overlaid as dark violet lines. The laser-free (LF) simulation (S.VI) produces
a shock that is qualitatively similar to the one triggered in the laser-on
(LO) simulation (S.III), as seen by comparing the ion $(x,p_{x})$ phase spaces
plotted in Figs. 12a and b. However, the initial plasma modulations are absent
in the laser-free simulation, which confirms their laser-driven nature.

Forcing a uniform electron temperature through the entire profile tends to
enhance TNSA in the profile’s wings. As a result, in the quasi exponentially
decreasing wings that are subject to a uniform accelerating field, the ion
velocity reaches the values $v_{i}\approx 0.05\ c$ in the LF simulation (Fig.
12a). Note that in the LO simulation $v_{i}\approx 0.02\ c$ (Fig. 12b). This
makes it harder for the fast ions accelerated from the density spike to catch
up with and eventually reflect the ions accelerated at the gas wings in the LF
simulation.

The LO simulation electron $(x,p_{x})$ phase space is strongly modulated as a
consequence of the direct-laser and wakefield-induced electron acceleration
Debayle  _et al._ (2017); Robinson  _et al._ (2011) (Fig. 12d), in comparison
with the LF simulation (Fig. 12c). Furthermore, the LO electron phase space
shows well defined electrostatic shock fronts moving both forward and
backward, as seen in Fig. 12d at $x=5000c/\omega_{0}$ and
$x=7000c/\omega_{0}$. In the LF electron $(x,p_{x})$ phase space the
electrostatic confinement in the downstream region is weaker. As a result, two
diluted electron populations centered at $x=5000c/\omega_{0}$ and
$x=8000c/\omega_{0}$ travel both forward and backward down the density
gradients, as seen in Fig. 12c.

A second LF simulation, with an initialized hot electron population with
$T_{e}=a_{0}/2$ (S.X), was performed to investigate whether a weaker target
energization could trigger an electrostatic shock in less time. We intended to
compare the shock formation time with the value obtained in the LF S.VI
initialized with $T_{e}=a_{0}$, where a shock was triggered at
$\omega_{0}t=15171$. However, by the end of S.X no shock reflection was
observed. The ion $(x,p_{x})$ phase space captured at $\omega_{0}t=19000$ is
plotted in Fig. 13. Hence, the mechanism triggering shock reflection must be
linked with the initial density, velocity or electron temperature gradients
(Fig. 11) imprinted by the laser itself.

Simulation X: $T_{e}=a_{0}/2$

Figure 13: Ion $(x,p_{x})$ phase space corresponding to the laser-free
Simulation X with $n_{at,max}=0.1\ n_{c}$ and an initialized hot electron
population with $T_{e}=a_{0}/2$.

##  V Conclusions

A parametric 1-D PIC simulations study has been conducted to derive an optimal
set of experimental parameters regarding the interaction of a near-critical
non-uniform gas profile with a $I_{L}=1.7\times 10^{20}$W/cm2 ($a_{0}=8.8$),
$\lambda_{L}=0.8\ \mu$m, $\tau_{L}=30$ fs laser pulse. We simulated realizable
experimental conditions in order to understand the different ion acceleration
mechanisms present during the interaction. At the same time, we identified a
range for electron densities in which collisionless electrostatic shock (CES)
formation occurs.

A key factor towards triggering a CES is to have the laser shinning through
the profile’s density peak while being strongly absorbed (>90$\%$) in the gas
up-ramp, where a hot electron population is created. This was achieved with a
fourteen times reduction of the original density profile (Fig. 2). Once the
correct density is achieved, the fast-ion structure originating from TNSA
triggered in the density peak evolves into a CES during its propagation across
the plasma ramps. This shock can, given the correct conditions, reflect
background ions from the down-ramp. CES formation was achieved in the moderate
electron density $n_{e}$ range between 0.35 $n_{c}$ and 0.7 $n_{c}$ (for the
profile density peak values). IAW formation and background ion reflection,
preferentially in the direction of the IAW propagation, was seen at higher
densities. At low densities, the ion velocity profile formed by TNSA is too
shallow to overtake the ions uniformly accelerated in the quasi-exponential
target wings. Shock formation is highly sensible to the target density and to
the density gradients inherent to the gas profile. Hence, a controlled and
repetitive gas profile production is key towards controlling the laser gas
interaction.

The direct contribution of the laser to ion acceleration is that of an initial
fast and strong perturbation over the ions, which modulates both the ion and
electron phase spaces. Such modulations create and enhance density
perturbations across the profile giving rise to strong charge separation
fields inside the plasma. However, the electrostatic shock seed lays
elsewhere, with the strong electron pressure gradients, which, linked with the
profile’s density variations, create plasma slabs of different electronic
temperatures and densities. The interaction between these plasma slabs leads
to the formation of non-linear trapping structures. The latter can transition
into electrostatically-reflecting shocks while traversing the density up and
down-ramps. Fully ionized ionic species will experience a higher electrostatic
potential in the shock region an are therefore more likely to be reflected.
The initial "kick" of the laser acts as a multiplier of charge separation and
therefore of the electrostatic barriers, giving rise to stronger shock waves
than in the laser-free simulation, which will inherently travel faster and
hence reflect particles to higher velocities, see ion $(x,p_{x})$ phase spaces
of Figs. 12a and b corresponding to the laser-free and laser-on simulations,
respectively.

Since the electron pressure gradients are essential for shock formation, the
same laser interacting with a different density profile with smoother density
gradients would not be as effective as it is in this case. This advantage
clearly highlights the interest of employing shock nozzles, which are
precisely designed to produce density profiles with strong density gradients.

Given the complexity and number of the laser-matter interaction processes the
possibility of performing statistical measures in parametric schemes are
fundamental for conducting thorough experimental studies. It is then of vital
importance to work at HRR facilities, where $\approx$100 shots can be
available each day, in order to acquire good statistics. Such HRR facilities
must combine the high laser repetition rate with data flow and data
acquisition software and hardware, as well as adapted laser diagnostics and
targetry.

##  VI Acknowledgments

This work has received funding from the European Union’s Horizon 2020 research
and innovation program under grant agreement no 871124 Laserlab-Europe.

We received financial support from the French State, managed by the French
National Research Agency (ANR) in the frame of the Investments for the future
Programme IdEx Bordeaux - LAPHIA (ANR-10-IDEX-03-02).

We acknowledge Dr. J.L. Henares for providing the S900 shock nozzle design
realized by performing computer fluid dynamic simulations using the Fluent
code Ansys (2018). We acknowledge Dr. Thanh-ha Nguyen-bui for the
complementary CFD simulations performed with the ARES code Nguyen-Bui and
Duffa (2006) inputting the S900 nozzle design.

We acknowledge the support from the LIGHT S&T Graduate Program (PIA3
Investment fot the Future Program, ANR-17-EURE-0027).

We acknowledge GENCI for providing us access to the supercomputer Irene under
the grants no. A0070506129 and no. A0080507594.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * Strickland and Mourou (1985) D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Optics Communications 56, 219–221 (1985).
  * Borghesi  _et al._ (2003) M. Borghesi, A. Schiavi, D. H. Campbell, M. G. Haines, O. Willi, A. J. Mackinnon, P. Patel, M. Galimberti, and L. A. Gizzi, “Proton imaging detection of transient electromagnetic fields in laser-plasma interactions (invited),” Review of Scientific Instruments 74, 1688–1693 (2003).
  * Borghesi  _et al._ (2004) M. Borghesi, A. J. MacKinnon, D. H. Campbell, D. G. Hicks, S. Kar, P. K. Patel, D. Price, L. Romagnani, A. Schiavi, and O. Willi, “Multi-MeV Proton Source Investigations in Ultraintense Laser-Foil Interactions,” Phys. Rev. Lett. 92, 055003 (2004).
  * Santos  _et al._ (2015) J. J. Santos, M. Bailly-Grandvaux, L. Giuffrida, P. Forestier-Colleoni, S. Fujioka, Z. Zhang, P. Korneev, R. Bouillaud, S. Dorard, D. Batani, M. Chevrot, J. E. Cross, R. Crowston, J.-L. Dubois, J. Gazave, G. Gregori, E. d’Humières, S. Hulin, K. Ishihara, S. Kojima, E. Loyez, J.-R. Marquès, A. Morace, P. Nicolaï, O. Peyrusse, A. Poyé, D. Raffestin, J. Ribolzi, M. Roth, G. Schaumann, F. Serres, V. T. Tikhonchuk, P. Vacar, and N. Woolsey, “Laser-driven platform for generation and characterization of strong quasi-static magnetic fields,” New J. Phys. 17, 083051 (2015).
  * M. Ehret  _et al._ (2017) M. Ehret _et al._ , “Picosecond-laser driven ultra-fast em fields propagating along coil targets for proton beam micro-lensing,” GSI Scientific Report 2016 GSI-2017-1, 227 (2017).
  * Roth  _et al._ (2009) M. Roth, I. Alber, V. Bagnoud, C. R. D. Brown, R. Clarke, H. Daido, J. Fernandez, K. Flippo, S. Gaillard, C. Gauthier, M. Geissel, S. Glenzer, G. Gregori, M. Günther, K. Harres, R. Heathcote, A. Kritcher, N. Kugland, S. Le Pape, B. Li, M. Makita, J. Mithen, C. Niemann, F. Nürnberg, D. Offermann, A. Otten, A. Pelka, D. Riley, G. Schaumann, M. Schollmeier, J. Schütrumpf, M. Tampo, A. Tauschwitz, and A. Tauschwitz, “Proton acceleration experiments and warm dense matter research using high power lasers,” Plasma Physics and Controlled Fusion 51, 124039 (2009).
  * Patel  _et al._ (2003) P. K. Patel, A. J. MacKinnon, M. H. Key, T. E. Cowan, M. E. Foord, M. Allen, D. F. Price, H. Ruhl, P. T. Springer, and R. Stephens, “Isochoric Heating of Solid-Density Matter with an Ultrafast Proton Beam,” Phys. Rev. Lett. 91, 125004 (2003).
  * Fritzler  _et al._ (2003) S. Fritzler, V. Malka, G. Grillon, J. P. Rousseau, F. Burgy, E. Lefebvre, E. d’Humières, P. McKenna, and K. W. D. Ledingham, “Proton beams generated with high-intensity lasers: Applications to medical isotope production,” Applied Physics Letters 83, 3039 (2003).
  * Spencer  _et al._ (2001) I. Spencer, K. W. D. Ledingham, R. P. Singhal, T. McCanny, P. McKenna, E. L. Clark, K. Krushelnick, M. Zepf, F. N. Beg, M. Tatarakis, A. E. Dangor, P. A. Norreys, R. J. Clarke, R. M. Allott, and I. N. Ross, “Laser generation of proton beams for the production of short-lived positron emitting radioisotopes,” Nuclear Instruments and Methods in Physics Research B 183, 449–458 (2001).
  * Macchi, Borghesi, and Passoni (2013) A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
  * Daido, Nishiuchi, and Pirozhkov (2012) H. Daido, M. Nishiuchi, and A. S. Pirozhkov, “Review of laser-driven ion sources and their applications,” Reports on Progress in Physics 75, 056401 (2012).
  * Wagner  _et al._ (2016) F. Wagner, O. Deppert, C. Brabetz, P. Fiala, A. Kleinschmidt, P. Poth, V. A. Schanz, A. Tebartz, B. Zielbauer, M. Roth, T. Stöhlker, and V. Bagnoud, “Maximum Proton Energy above 85 MeV from the Relativistic Interaction of Laser Pulses with Micrometer Thick CH2 Targets,” Phys. Rev. Lett. 116, 205002 (2016).
  * Bulanov and Esirkepov (2007) S. V. Bulanov and T. Z. Esirkepov, “Comment on “Collimated Multi-MeV Ion Beams from High-Intensity Laser Interactions with Underdense Plasma”,” Phys. Rev. Lett. 98, 049503 (2007).
  * Bulanov  _et al._ (2005) S. V. Bulanov, D. V. Dylov, T. Z. Esirkepov, F. F. Kamenets, and D. V. Sokolov, “Ion acceleration in a dipole vortex in a laser plasma corona,” Plasma Physics Reports 31, 369–381 (2005).
  * Park  _et al._ (2019) J. Park, S. S. Bulanov, J. Bin, Q. Ji, S. Steinke, J. L. Vay, C. G. R. Geddes, C. B. Schroeder, W. P. Leemans, T. Schenkel, and E. Esarey, “Ion acceleration in laser generated megatesla magnetic vortex,” Physics of Plasmas 26, 103108 (2019), arXiv:1904.03281 [physics.plasm-ph] .
  * Nakamura  _et al._ (2010) T. Nakamura, S. V. Bulanov, T. Z. Esirkepov, and M. Kando, “High-Energy Ions from Near-Critical Density Plasmas via Magnetic Vortex Acceleration,” Phys. Rev. Lett. 105, 135002 (2010).
  * Krushelnick  _et al._ (1999) K. Krushelnick, E. L. Clark, Z. Najmudin, M. Salvati, M. I. K. Santala, M. Tatarakis, A. E. Dangor, V. Malka, D. Neely, R. Allott, and C. Danson, “Multi-MeV Ion Production from High-Intensity Laser Interactions with Underdense Plasmas,” Phys. Rev. Lett. 83, 737–740 (1999).
  * Kuznetsov  _et al._ (2001) A. V. Kuznetsov, T. Z. Esirkepov, F. F. Kamenets, and S. V. Bulanov, “Efficiency of Ion Acceleration by a Relativistically Strong Laser Pulse in an Underdense Plasma,” Plasma Physics Reports 27, 211–220 (2001).
  * Willingale  _et al._ (2006) L. Willingale, S. P. D. Mangles, P. M. Nilson, R. J. Clarke, A. E. Dangor, M. C. Kaluza, S. Karsch, K. L. Lancaster, W. B. Mori, Z. Najmudin, J. Schreiber, A. G. R. Thomas, M. S. Wei, and K. Krushelnick, “Collimated Multi-MeV Ion Beams from High-Intensity Laser Interactions with Underdense Plasma,” Phys. Rev. Lett. 96, 245002 (2006).
  * Denavit (1992) J. Denavit, “Absorption of high-intensity subpicosecond lasers on solid density targets,” Phys. Rev. Lett. 69, 3052–3055 (1992).
  * Silva  _et al._ (2004) L. O. Silva, M. Marti, J. R. Davies, R. A. Fonseca, C. Ren, F. S. Tsung, and W. B. Mori, “Proton Shock Acceleration in Laser-Plasma Interactions,” Phys. Rev. Lett. 92, 015002 (2004).
  * Fiuza  _et al._ (2012) F. Fiuza, A. Stockem, E. Boella, R. A. Fonseca, L. O. Silva, D. Haberberger, S. Tochitsky, C. Gong, W. B. Mori, and C. Joshi, “Laser-Driven Shock Acceleration of Monoenergetic Ion Beams,” Physical review letters 109, 215001 (2012), arXiv:1206.2903 [physics.plasm-ph] .
  * Fiuza  _et al._ (2013) F. Fiuza, A. Stockem, E. Boella, R. A. Fonseca, L. O. Silva, D. Haberberger, S. Tochitsky, W. B. Mori, and C. Joshi, “Ion acceleration from laser-driven electrostatic shocks,” Physics of Plasmas 20, 056304 (2013), arXiv:1301.4262 [physics.plasm-ph] .
  * Dieckmann  _et al._ (2013a) M. E. Dieckmann, H. Ahmed, G. Sarri, D. Doria, I. Kourakis, L. Romagnani, M. Pohl, and M. Borghesi, “Parametric study of non-relativistic electrostatic shocks and the structure of their transition layer,” Physics of Plasmas 20, 042111 (2013a), arXiv:1304.6523 [physics.plasm-ph] .
  * Dieckmann  _et al._ (2013b) M. E. Dieckmann, G. Sarri, D. Doria, M. Pohl, and M. Borghesi, “Modification of the formation of high-Mach number electrostatic shock-like structures by the ion acoustic instability,” Physics of Plasmas 20, 102112 (2013b), arXiv:1310.1740 [physics.plasm-ph] .
  * Lécz and Andreev (2015) Z. Lécz and A. Andreev, “Shock wave acceleration of protons in inhomogeneous plasma interacting with ultrashort intense laser pulses,” Physics of Plasmas 22, 043103 (2015).
  * Liu  _et al._ (2016) M. Liu, S. M. Weng, Y. T. Li, D. W. Yuan, M. Chen, P. Mulser, Z. M. Sheng, M. Murakami, L. L. Yu, X. L. Zheng, and J. Zhang, “Collisionless electrostatic shock formation and ion acceleration in intense laser interactions with near critical density plasmas,” Physics of Plasmas 23, 113103 (2016), arXiv:1611.06616 [physics.plasm-ph] .
  * Liu, Meyer-ter-Vehn, and Ruhl (2018) B. Liu, J. Meyer-ter-Vehn, and H. Ruhl, “Self-trapping and acceleration of ions in laser-driven relativistically transparent plasma,” Physics of Plasmas 25, 103117 (2018), arXiv:1803.06358 [physics.plasm-ph] .
  * Lefebvre  _et al._ (2003) E. Lefebvre, N. Cochet, S. Fritzler, V. Malka, M. M. Aléonard, J. F. Chemin, S. Darbon, L. Disdier, J. Faure, A. Fedotoff, O. Landoas, G. Malka, V. Méot, P. Morel, M. Rabec LeGloahec, A. Rouyer, C. Rubbelynck, V. Tikhonchuk, R. Wrobel, P. Audebert, and C. Rousseaux, “Electron and photon production from relativistic laser plasma interactions,” Nuclear Fusion 43, 629–633 (2003).
  * Debayle  _et al._ (2017) A. Debayle, F. Mollica, B. Vauzour, Y. Wan, A. Flacco, V. Malka, X. Davoine, and L. Gremillet, “Electron heating by intense short-pulse lasers propagating through near-critical plasmas,” New J. Phys. 19, 123013 (2017).
  * Moiseev and Sagdeev (1963) S. S. Moiseev and R. Z. Sagdeev, “Collisionless shock waves in a plasma in a weak magnetic field,” Journal of Nuclear Energy 5, 43–47 (1963).
  * Forslund and Shonk (1970) D. W. Forslund and C. R. Shonk, “Formation and Structure of Electrostatic Collisionless Shocks,” Phys. Rev. Lett. 25, 1699–1702 (1970).
  * Sorasio  _et al._ (2006) G. Sorasio, M. Marti, R. Fonseca, and L. O. Silva, “Very High Mach-Number Electrostatic Shocks in Collisionless Plasmas,” Phys. Rev. Lett. 96, 045005 (2006), arXiv:physics/0512231 [physics.plasm-ph] .
  * Cairns  _et al._ (2014) R. A. Cairns, R. Bingham, P. Norreys, and R. Trines, “Laminar shocks in high power laser plasma interactions,” Physics of Plasmas 21, 022112 (2014), arXiv:1402.2804 [physics.plasm-ph] .
  * Boella  _et al._ (2018) E. Boella, F. Fiúza, A. S. Novo, R. Fonseca, and L. Silva, “Ion acceleration in electrostatic collisionless shock: on the optimal density profile for quasi-monoenergetic beams,” Plasma Physics and Controlled Fusion 60, 035010 (2018).
  * Bin  _et al._ (2015) J. H. Bin, W. J. Ma, H. Y. Wang, M. J. V. Streeter, C. Kreuzer, D. Kiefer, M. Yeung, S. Cousens, P. S. Foster, B. Dromey, X. Q. Yan, R. Ramis, J. Meyer-ter-Vehn, M. Zepf, and J. Schreiber, “Ion Acceleration Using Relativistic Pulse Shaping in Near-Critical-Density Plasmas,” Phys. Rev. Lett. 115, 064801 (2015).
  * Bin  _et al._ (2018) J. H. Bin, M. Yeung, Z. Gong, H. Y. Wang, C. Kreuzer, M. L. Zhou, M. J. V. Streeter, P. S. Foster, S. Cousens, B. Dromey, J. Meyer-ter-Vehn, M. Zepf, and J. Schreiber, “Enhanced Laser-Driven Ion Acceleration by Superponderomotive Electrons Generated from Near-Critical-Density Plasma,” Phys. Rev. Lett. 120, 074801 (2018), arXiv:1710.09855 [physics.plasm-ph] .
  * Ma  _et al._ (2019) W. J. Ma, I. J. Kim, J. Q. Yu, I. W. Choi, P. K. Singh, H. W. Lee, J. H. Sung, S. K. Lee, C. Lin, Q. Liao, J. G. Zhu, H. Y. Lu, B. Liu, H. Y. Wang, R. F. Xu, X. T. He, J. E. Chen, M. Zepf, J. Schreiber, X. Q. Yan, and C. H. Nam, “Laser Acceleration of Highly Energetic Carbon Ions Using a Double-Layer Target Composed of Slightly Underdense Plasma and Ultrathin Foil,” Phys. Rev. Lett. 122, 014803 (2019), arXiv:1801.10338 [physics.plasm-ph] .
  * Pazzaglia  _et al._ (2020) A. Pazzaglia, L. Fedeli, A. Formenti, A. Maffini, and M. Passoni, “A theoretical model of laser-driven ion acceleration from near-critical double-layer targets,” Communications Physics 3, 133 (2020).
  * Antici  _et al._ (2017) P. Antici, E. Boella, S. N. Chen, D. S. Andrews, M. Barberio, J. Böker, F. Cardelli, J. L. Feugeas, M. Glesser, P. Nicolaï, L. Romagnani, M. Scisciò, M. Starodubtsev, O. Willi, J. C. Kieffer, V. Tikhonchuk, H. Pépin, L. O. Silva, E. d. Humières, and J. Fuchs, “Acceleration of collimated 45 MeV protons by collisionless shocks driven in low-density, large-scale gradient plasmas by a 1e20 W/cm2, 1 mum laser,” Scientific Reports 7, 16463 (2017), arXiv:1708.02539 [physics.plasm-ph] .
  * Pak  _et al._ (2018) A. Pak, S. Kerr, N. Lemos, A. Link, P. Patel, F. Albert, L. Divol, B. B. Pollock, D. Haberberger, D. Froula, M. Gauthier, S. H. Glenzer, A. Longman, L. Manzoor, R. Fedosejevs, S. Tochitsky, C. Joshi, and F. Fiuza, “Collisionless shock acceleration of narrow energy spread ion beams from mixed species plasmas using 1 $\mu$ m lasers,” Physical Review Accelerators and Beams 21, 103401 (2018), arXiv:1810.08190 [physics.plasm-ph] .
  * Rosmej  _et al._ (2019) O. N. Rosmej, N. E. Andreev, S. Zaehter, N. Zahn, P. Christ, B. Borm, T. Radon, A. Sokolov, L. P. Pugachev, D. Khaghani, F. Horst, N. G. Borisenko, G. Sklizkov, and V. G. Pimenov, “Interaction of relativistically intense laser pulses with long-scale near critical plasmas for optimization of laser based sources of MeV electrons and gamma-rays,” New Journal of Physics 21, 043044 (2019).
  * Haberberger  _et al._ (2012) D. Haberberger, S. Tochitsky, F. Fiuza, C. Gong, R. A. Fonseca, L. O. Silva, W. B. Mori, and C. Joshi, “Collisionless shocks in laser-produced plasma generate monoenergetic high-energy proton beams,” Nature Physics 8, 95–99 (2012).
  * Tresca  _et al._ (2015) O. Tresca, N. P. Dover, N. Cook, C. Maharjan, M. N. Polyanskiy, Z. Najmudin, P. Shkolnikov, and I. Pogorelsky, “Spectral Modification of Shock Accelerated Ions Using a Hydrodynamically Shaped Gas Target,” Phys. Rev. Lett. 115, 094802 (2015), arXiv:1503.06180 [physics.plasm-ph] .
  * Tsung  _et al._ (2012) F. Tsung, S. Y. Tochitsky, D. J. Haberberger, W. B. Mori, and C. Joshi, “CO2 Laser acceleration of forward directed MeV proton beams in a gas target at critical plasma density,” Journal of Plasma Physics 78, 373–382 (2012).
  * Chen  _et al._ (2017) S. N. Chen, M. Vranic, T. Gangolf, E. Boella, P. Antici, M. Bailly-Grandvaux, P. Loiseau, H. Pépin, G. Revet, J. J. Santos, A. M. Schroer, M. Starodubtsev, O. Willi, L. O. Silva, E. d’Humières, and J. Fuchs, “Collimated protons accelerated from an overdense gas jet irradiated by a 1 um wavelength high-intensity short-pulse laser,” Scientific Reports 7, 13505 (2017), arXiv:1708.02492 [physics.plasm-ph] .
  * Puyuelo-Valdes  _et al._ (2019) P. Puyuelo-Valdes, J. L. Henares, F. Hannachi, T. Ceccotti, J. Domange, M. Ehret, E. d’Humieres, L. Lancia, J. R. Marquès, X. Ribeyre, J. J. Santos, V. Tikhonchuk, and M. Tarisien, “Proton acceleration by collisionless shocks using a supersonic H2 gas-jet target and high-power infrared laser pulses,” Physics of Plasmas 26, 123109 (2019).
  * Sylla  _et al._ (2013) F. Sylla, A. Flacco, S. Kahaly, M. Veltcheva, A. Lifschitz, V. Malka, E. d’Humières, I. Andriyash, and V. Tikhonchuk, “Short Intense Laser Pulse Collapse in Near-Critical Plasma,” Phys. Rev. Lett. 110, 085001 (2013), arXiv:1206.6329 [physics.plasm-ph] .
  * Ehret  _et al._ (2020) M. Ehret, C. Salgado-Lopez, V. Ospina-Bohorquez, J. A. Perez-Hernandez, M. Huault, M. de Marco, J. I. Apinaniz, F. Hannachi, D. De Luis, J. Hernandez Toro, D. Arana, C. Mendez, O. Varela, A. Debayle, L. Gremillet, T. H. Nguyen-Bui, E. Olivier, G. Revet, N. D. Bukharskii, H. Larreur, J. Caron, C. Vlachos, T. Ceccotti, D. Raffestin, P. Nicolai, J. L. Feugeas, M. Roth, X. Vaisseau, G. Gatti, L. Volpe, and J. J. Santos, “Ion acceleration by an ultrashort laser pulse interacting with a near-critical-density gas jet,” arXiv e-prints , arXiv:2012.09455 (2020), arXiv:2012.09455 [physics.plasm-ph] .
  * Singh  _et al._ (2020) P. K. Singh, V. B. Pathak, J. H. Shin, I. W. Choi, K. Nakajima, S. K. Lee, J. H. Sung, H. W. Lee, Y. J. Rhee, C. Aniculaesei, _et al._ , “Electrostatic shock acceleration of ions in near-critical-density plasma driven by a femtosecond petawatt laser,” Scientific reports 10, 1–10 (2020).
  * Puyuelo-Valdes (2020) P. Puyuelo-Valdes,  _Laser-driven ion acceleration with high-density gas-jet targets and application to elemental analysis_ , Theses, Université de Bordeaux ; Institut national de la recherche scientifique (Québec, province) (2020).
  * Fluent (2015) A. Fluent, “Ansys fluent,” Academic Research. Release 14 (2015).
  * Guérin  _et al._ (1995) S. Guérin, G. Laval, P. Mora, J. Adam, A. Heron, and A. Bendib, “Modulational and raman instabilities in the relativistic regime,” Physics of Plasmas 2, 2807–2814 (1995).
  * Guerin  _et al._ (1996) S. Guerin, P. Mora, J. Adam, A. Héron, and G. Laval, “Propagation of ultraintense laser pulses through overdense plasma layers,” Physics of Plasmas 3, 2693–2701 (1996).
  * Moreau  _et al._ (2017) J. Moreau, E. d’Humières, R. Nuter, and V. Tikhonchuk, “Stimulated raman scattering in the relativistic regime in near-critical plasmas,” Physical Review E 95, 013208 (2017).
  * Moreau (2018) J. Moreau,  _Interaction d’une impulsion laser intense avec un plasma sous dense dans le régime relativiste_ , Ph.D. thesis, Université de Bordeaux (2018).
  * Teychenné, Bonnaud, and Bobin (1993) D. Teychenné, G. Bonnaud, and J.-L. Bobin, “Wave-breaking limit to the wake-field effect in an underdense plasma,” Physical Review E 48, R3248–R3251 (1993).
  * Vieira  _et al._ (2010) J. Vieira, F. Fiúza, L. O. Silva, M. Tzoufras, and W. B. Mori, “Onset of self-steepening of intense laser pulses in plasmas,” New Journal of Physics 12, 045025 (2010).
  * Grismayer and Mora (2006) T. Grismayer and P. Mora, “Influence of a finite initial ion density gradient on plasma expansion into a vacuum,” Physics of Plasmas 13, 032103 (2006).
  * Perego  _et al._ (2013) M. Perego, P. D. Howell, M. D. Gunzburger, J. R. Ockendon, and J. E. Allen, “The expansion of a collisionless plasma into a plasma of lower density,” Physics of Plasmas 20, 052101 (2013).
  * Nechaev  _et al._ (2020) A. A. Nechaev, M. A. Garasev, A. N. Stepanov, and V. V. Kocharovsky, “Formation of a Density Bump in a Collisionless Electrostatic Shock Wave During Expansion of a Hot Dense Plasma into a Cold Rarefied One,” Plasma Physics Reports 46, 765–783 (2020).
  * Robinson  _et al._ (2011) A. Robinson, R. Trines, J. Polz, and M. Kaluza, “Absorption of circularly polarized laser pulses in near-critical plasmas,” Plasma Physics and Controlled Fusion 53, 065019 (2011).
  * Ansys (2018) Ansys, “Ansys® academic research mechanical, release 18.1,” ANSYS (2018).
  * Nguyen-Bui and Duffa (2006) N.-H. Nguyen-Bui and G. Duffa, “Modeling and simulation of turbulent reacting flow around a hypersonic space probe,” Journal of spacecraft and rockets 43, 919–923 (2006).