# Collisionless Shock Acceleration of protons in a plasma slab produced in a
gas jet by the collision of two laser-driven hydrodynamic shockwaves.

J.-R. Marquès<EMAIL_ADDRESS>LULI, CNRS, École
Polytechnique, CEA, Sorbonne Université, Institut Polytechnique de Paris,
F-91128 Palaiseau Cedex, France  L. Lancia  LULI, CNRS, École Polytechnique,
CEA, Sorbonne Université, Institut Polytechnique de Paris, F-91128 Palaiseau
Cedex, France  P. Loiseau  CEA, DAM, DIF, F-91297 Arpajon Cedex, France
Université Paris-Saclay, CEA, LMCE, 91680 Bruyères-le-Chatel, France  P.
Forestier-Colleoni  LULI, CNRS, École Polytechnique, CEA, Sorbonne Université,
Institut Polytechnique de Paris, F-91128 Palaiseau Cedex, France  now at
Université Paris-Saclay, CEA, LIDYL, 91191 Gif-sur-Yvette, France  M. Tarisien
CENBG, CNRS-IN2P3, Université de Bordeaux, 33175 Gradignan Cedex, France  E.
Atukpor  CENBG, CNRS-IN2P3, Université de Bordeaux, 33175 Gradignan Cedex,
France  V. Bagnoud  GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Planckstraße 1, 64291 Darmstadt, Germany  University of Darmstadt,
Schloßgartenstr. 764289 Darmstadt, Germany  C. Brabetz  GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germany  F.
Consoli  ENEA Fusion and Technologies for Nuclear Safety Department, C.R.
Frascati, Via Enrico Fermi 45, Frascati, Rome, Italy  J. Domange  CENBG, CNRS-
IN2P3, Université de Bordeaux, 33175 Gradignan Cedex, France  F. Hannachi
CENBG, CNRS-IN2P3, Université de Bordeaux, 33175 Gradignan Cedex, France  P.
Nicolaï  CELIA, Université de Bordeaux–CNRS–CEA, Talence 33405, France  M.
Salvadori  ENEA Fusion and Technologies for Nuclear Safety Department, C.R.
Frascati, Via Enrico Fermi 45, Frascati, Rome, Italy  B. Zielbauer  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291
Darmstadt, Germany

###### Abstract

We recently proposed a new technique of plasma tailoring by laser-driven
hydrodynamic shockwaves generated on both sides of a gas jet [J.-R. Marquès et
al., Phys. Plasmas 28, 023103 (2021)]. In the continuation of this numerical
work, we studied experimentally the influence of the tailoring on proton
acceleration driven by a high-intensity picosecond-laser, in three cases:
without tailoring, by tailoring only the entrance side of the ps-laser, or
both sides of the gas jet. Without tailoring the acceleration is transverse to
the laser axis, with a low-energy exponential spectrum, produced by Coulomb
explosion. When the front side of the gas jet is tailored, a forward
acceleration appears, that is significantly enhanced when both the front and
back sides of the plasma are tailored. This forward acceleration produces
higher energy protons, with a peaked spectrum, and is in good agreement with
the mechanism of Collisionless Shock Acceleration (CSA). The spatio-temporal
evolution of the plasma profile was characterized by optical shadowgraphy of a
probe beam. The refraction and absorption of this beam was simulated by post-
processing 3D hydrodynamic simulations of the plasma tailoring. Comparison
with the experimental results allowed to estimate the thickness and near-
critical density of the plasma slab produced by tailoring both sides of the
gas jet. These parameters are in good agreement with those required for CSA.

##  I Introduction

Collisionless shocks are ubiquitous in astrophysical environments Blanford and
Eichler (1987); Jones and Ellison (1991) such as the Earth’s bow shock, solar
flares, interplanetary traveling shocks, or supernova remnants (SNRs)
Caprioli, Blasi, and Amato (2011); Koyama  _et al._ (1995). They are believed
to be responsible for non-thermal particles Adriani  _et al._ (2011);
Spitkovsky (2008a) and gamma ray bursts Spitkovsky (2008b). Using scaling laws
Drake (2000), the physics of collisionless magnetized shocks in SNRs has been
investigated experimentally using laser-produced plasmas Courtois  _et al._
(2004); Woolsey  _et al._ (2001). Their capability to accelerate particles has
been demonstrated Fiuza  _et al._ (2020).

The formation of a collisionless electrostatic shock requires the creation of
a localized region of higher pressure within a plasma with electron
temperature $T_{e}$ much larger than the ion temperature $T_{i}$. As this
region of high pressure (defined as the downstream region) expands, it can
drive a shock wave into the surrounding lower-pressure plasma (defined as the
upstream region). The shock wave front can accelerate upstream ions by
reflecting them to twice the shock velocity if the shock potential is larger
than the kinetic energy of the incoming ions in the shock-rest frame.

Since they are efficient accelerators of particles, there has been a growing
interest in exploring laser-driven shocks as compact particle accelerators
Silva  _et al._ (2004); Fiuza  _et al._ (2012, 2013); Palmer  _et al._ (2011).
Energetic ion beams from compact laser-produced plasmas have potential
applications in many fields of science and medicine, such as particle physics
Bulanov  _et al._ (2005), fast ignition of fusion targets Roth  _et al._
(2001); Temporal  _et al._ (2009); Temporal, Honrubia, and Atzeni (2009),
material science Boody  _et al._ (1996), proton radiography Borghesi  _et al._
(2002); Li  _et al._ (2006); Romagnani  _et al._ (2008), radiotherapy Bulanov
_et al._ (2002); Bulanov and Khoroshkov (2002); Malka  _et al._ (2004); Karsch
_et al._ (2017), and isotope generation for medical applications Spencer  _et
al._ (2001); Lefebvre  _et al._ (2006).

Several schemes for laser-driven ion acceleration have been proposed: Target
Normal Sheath Acceleration Snavely  _et al._ (2000); Wilks  _et al._ (2001),
Radiation Pressure Acceleration Esirkepov  _et al._ (2004); Robinson  _et al._
(2008); Macchi, Veghini, and Pegoraro (2009); Qiao  _et al._ (2012), Breakout
Afterburner Acceleration Yin  _et al._ (2007); Henig  _et al._ (2009), that
use over-dense targets (solid density foils, liquid jets), or Magnetic Vortex
Acceleration Bulanov  _et al._ (2005); Nakamura  _et al._ (2010); Park  _et
al._ (2019) and Collisionless Shock Acceleration (CSA), that use near-critical
density (NCD) targets (exploded foils, gas jets). The production, at high
repetition rate, of high-energy ion beams with a narrow energy spread and a
low divergence still remains a challenge, and CSA could potentially offer
these properties. Moreover, NCD plasmas can be generated using gas jets,
offering several advantages such as to avoid the constraints of target
replacement and realignment between the consecutive shots, to avoid the target
debris that usually spoil the surrounding optics, or to allow the production
of pure proton beams (impurity free, using H2 gas).

Proton acceleration by CSA in a hydrogen gas jet was first demonstrated Palmer
_et al._ (2011); Najmudin  _et al._ (2011) using CO2 lasers, the low critical
density ($n_{c}\approx 10^{19}$ cm-3) associated with their long wavelength
($\lambda_{0}$ = 10 $\mu$m) allowing to exploit regular-pressure, mm-scale gas
jets. A proton beam of $\sim$MeV energy was produced with narrow energy spread
($\sigma\sim 4\%$) and low normalized emittance ($<$ 8 nm.rad). Until today,
the maximum proton energy produced by CSA in a gas jet Haberberger  _et al._
(2012) is 20 MeV. It was obtained from a CO2 laser composed of several
picosecond (ps) pulses, the early pulses serving to ionize the gas jet and to
steepen the plasma density profile on the front side of the target by the
radiation pressure of the laser, so that the following pulses interacted with
a step-like density profile. Despite this promising proton acceleration, the
pulse train, inherent to the laser system, varied from shot to shot, making
the interactions challenging to reproduce. Instabilities such as laser
filamentation and hosing from the leading pulses result in variable density
profiles, which in turn lead to fluctuations in the resultant ion beam. A
solution to generate a steep plasma density profile in a more stable manner is
to tailor the near-critical gas target by a hydrodynamic shockwave (HSW)
excited before the arrival of the main ps-pulse. This HSW can be excited by a
low energy laser prepulse that can be focused inside the gas jet Tresca  _et
al._ (2015), or on a solid target placed at the entrance side of the gas jet
Chen  _et al._ (2017a). In both cases it has been demonstrated that for long
density profiles ($>$ 40 $\mu$m), ion beams with broadband energy spectrum
were produced, while for shorter plasma lengths ($<$ 20 $\mu$m),
quasimonoenergetic acceleration of protons was observed.

Experiments using more widespread solid state ps-lasers ($\lambda_{0}\approx$
1 $\mu$m, $n_{c}\approx 10^{21}$ cm-3) have been performed using high-pressure
narrow gas jets tailored by a HSW Chen  _et al._ (2017b); Puyuelo-Valdes  _et
al._ (2019), or exploded $\mu$m-size solid foils Pak  _et al._ (2018);
Tochitsky  _et al._ (2020). Compared to the proof-of-principle experiments
with CO2 lasers, the number of accelerated protons was substantially higher
($\sim 10^{4}\times$) thanks to the larger $n_{c}$ and vector potential of the
laser field ($a_{0}\sim$ 4 to 25, versus $\sim 1$ on the CO2 experiments).
Proton beams with peaks at energies up to 5.5 MeV from tailored gas jets and
54 MeV from exploded foils were obtained.

Despite the large differences in terms of laser or target types, this set of
experiments showed that the shape and maximum energy of the ion distributions
strongly depend on the profile and peak value of the plasma density. Ion beams
with peaked energy distribution and similar velocity of ion species with
different charge-to-mass ratios, consistent with CSA, were observed only for
plasmas with a steep density gradient at the laser entrance side, and of NCD.
To obtain a narrow energy spread by CSA Silva  _et al._ (2004); Fiuza  _et
al._ (2012, 2013) it is crucial to have a uniform shock velocity and ion
reflection, which implies a uniform electron temperature profile, only
achieved by a quick recirculation of the heated electrons due to the space-
charge fields at the front and at the back of the target. Therefore, the
plasma width $L$ should be limited, which for a moderate Mach number shocks
($M\gtrsim 1$), reads $L_{opt}\sim\lambda_{0}(m_{i}/m_{e})^{1/2}$. Considering
a hydrogen plasma and $\lambda_{0}$ = 1 $\mu$m yields $L_{opt}\sim$ 40 $\mu$m,
and therefore relatively sharp gradients on both sides of the target.

To generate such thin NCD plasmas, we recently proposed a new tailoring method
Marquès  _et al._ (2021); Bonvalet  _et al._ (2021) based on a narrow gas jet
coupled with two parallel nanosecond (ns) heating-lasers that propagate
perpendicular to the main ps-laser. These ns-beams are focused on both sides
of the gas jet. The sudden and localized ionization and heating occuring at
their focii generates two cylindrical HSWs. The expansion of each wave has two
effects: i) the part that expands towards vacuum expels the wing of the gas
jet, ii) while the part propagating towards the center of the jet compresses
the plasma. The collision at the jet center of the two waves generates a thin
plasma slab with steep gradients. Our hydrodynamic and ion Fokker–Planck
simulations indicated that with present high-density gas jetsHenares  _et al._
(2019), this method allows the production of a thin ($\gtrsim 10$ $\mu$m)
plasma slab, with an adjustable density up to $2\times 10^{22}$ cm-3 (20
$n_{c}$ for a 1 $\mu$m laser). This tailoring scheme can be implemented at
high repetition rate, in a debris free environment, and with different types
of gas. Such thin-NCD targets attracted a lot of attention recently not only
for use in CSA or other ion acceleration mechanisms Bulanov  _et al._ (2010),
but also for brilliant gamma-ray and electron-positron pair production Liu
_et al._ (2018); Zhu  _et al._ (2016); Rosmej  _et al._ (2019).

In this paper we present experimental results on proton acceleration in a H2
gas jet laser-tailored by the technique recently proposed in our previous
numerical work Marquès  _et al._ (2021); Bonvalet  _et al._ (2021). We study
the influence of plasma tailoring on the energy and angular distributions of
the accelerated protons, for three cases: without tailoring, by tailoring only
the entrance side of the ps-beam, or both sides of the gas jet. We observed
the transition from a transverse (to the ps-beam direction) acceleration with
a low energy broad spectrum produced by Coulomb explosion, to a forward
acceleration with a higher energy peaked spectrum, in good agreement with CSA.
The spatio-temporal evolution of the plasma profile was characterized with
optical shadowgraphy of a probe laser beam. These shadowgraphy were simulated
by a post-processing (ray-tracing) of three-dimensional hydrodynamic
simulations of the plasma tailoring, and compared with the experimental
observations to estimate the size and density of the resulting plasma slab.

##  II Experimental setup

The work was performed on two laser facilities, PHELIX at GSI (Germany) and
PICO2000 at LULI (France), both delivering laser pulses at a fundamental
wavelength $\lambda_{0}\sim$ 1 $\mu$m. The plasma was generated from a narrow,
high density, supersonic jet of hydrogen gas expelled from a 250 $\mu$m output
diameter nozzle. The two parallel ns-beams for plasma tailoring were focused
300 $\mu$m from each sides of the jet center. The ps-laser beam driving the
proton acceleration was propagating at 90∘ from the ns-beams and focused at
the center of gas jet. This experimental arrangement is sketched in Ref.
[Marquès  _et al._ (2021)](Fig. 1). The output of the gas nozzle was placed
400 $\mu$m above the propagation plane of the ps and ns laser beams. The gas
density profile in this plane was $n_{atom}(r)$ (cm-3.bar-1) $\sim 6.2\times
10^{17}[e^{-(r/186)^{2}}+0.63e^{-(r/132)}]$. The backing pressure was
adjustable up to 1000 bars allowing, without laser tailoring, to reach
$n_{e}\sim n_{c}$ (the critical plasma density, $n_{c}$(cm-3) $\sim 1.11\times
10^{21}\lambda_{0}^{-2}$($\mu$m)).

The ps-beam was focused to a Gaussian-like focal spot of 10 $\mu$m FWHM (in
intensity). The temporal profile was Gaussian with a Full Width at Half
Maximum (FWHM) of 0.5 ps at GSI and 1 ps at LULI, and a contrast on the ns
scale (Amplified Stimulated Emission, ASE) of $10^{11}$ and $10^{7}$
respectively. The peak intensity was respectively $8\times 10^{19}$ and
$2\times 10^{19}$ W/cm2, corresponding to a normalized vector potential of
$a_{0}\sim$ 8 and 4. The Rayleigh length of the ps-beam was $z_{R}\gtrsim$ 250
$\mu$m, of the order or greater than the width of the gas jet. The ns-beams
had a square-like temporal profile of $\sim$ 1 ns duration. Each ns-beam was
focused to a spot of FWHM $\sim$ 67 $\mu$m (GSI) and 42 $\mu$m (LULI)
containing up to 15 J and 6 J respectively.

On both experiments the hydrodynamics evolution of the plasma was controlled
by two-dimensional shadowgraphy and interferometry of a probe laser beam of
0.5 $\mu$m wavelength propagating at 90° from the ps-beam, parallel to the ns-
beams. This probe beam was collimated, with a diameter covering the entire
interaction region. At GSI it was generated by the leakage of a dielectric
mirror in the ps-beam path, sent to a delay-line and frequency doubled,
leading to a jitter-free probe of 0.5 ps duration. Its transverse intensity
(shadowgraphy) and phase (interferometry) profiles at the exit of the plasma
was recorded on 16-bit CCD cameras. The probe beam at LULI had a Gaussian
temporal profile of FWHM $\sim$ 7 ns. The shadowgraphy and interferometry at a
given time (snap-shot) were recorded on Gated Optical Intensifier (GOI)
coupled to 16-bit CCD cameras, with an integration window of 100 ps. The
shadowgraphy profile along the ps-beam axis was imaged on the entrance slit of
a streak camera. The long duration of the probe beam allowed to follow the
propagation of the shockwaves, from their creation to their collision, and
thus to check and adjust the synchronization of HSWs collision with the
arrival of the ps-beam. The shadowgraphy and interferometry diagnostics used
the same collecting lens, with an angular aperture of $\pm$ 3.5∘ at LULI and
$\pm$ 2∘ at GSI, leading to a spatial resolution better than 10 $\mu$m. The
focal spots of the ns-beams were recorded on a 12-bit CCD, allowing before
shot to adjust them on both sides of the gas jet, as well as control their
positions, shapes and transmitted energies on shot. The position of the ps-
beam was controlled before shot on a 8-bit CCD. In addition to these ”side-
view” diagnostics (shadowgraphy, interferometry, focal spots), the interaction
region was also imaged from the ”bottom” on a 12-bit CCD. The second harmonic
light (0.5 $\mu$m) emitted by the ps-beam along its propagation in the plasma
was recorded on the ”side-view” and ”bottom-view” diagnostics, allowing to
identify the regions where the density gradients or the laser intensity are
high, or where beam refraction or self-focusing could occur. The energy
spectrum of the protons accelerated from the H2 plasma were recorded on image
plates (Fuji BAS-MS type) coupled to magnetic spectrometers positioned at 0∘,
30∘ and 70∘ from the ps-laser axis on the LULI experiment, and at 25∘, 37∘ and
53∘ on the GSI experiment.

##  III Interaction without plasma tailoring

Figure 1 shows two-dimensional (2D) space-resolved shadowgraphies (snapshot)
of the probe beam from the plasma generated by the interaction of the ps-beam
with the high density H2 gas jet, at different backing pressures, times, and
ps-beams (LULI and GSI). At 800 bars (Fig. 1 a) and c)) the expected peak
density on the laser axis, assuming full ionization, is $n_{e}=8\times
10^{20}$ cm-3 (0.8 $n_{c}$). Fig. 1a) shows that the plasma at $t$ = 0.2 $\pm$
0.1 ns after the ps-pulse arrival is much larger than the laser focal volume
and extends on the whole gas jet volume, on a millimeter scale, along the
laser axis ($x$) as well as transversely ($z$). Shadowgraphies of very similar
size and shape are observed at 450 bars (Fig. 1 b)), on the LULI experiment,
as well as on the GSI experiment (Fig. 1 d)) for which the probe pulse is
shorter (0.5 ps) and the pump pulse has a $10^{4}$ times better ASE temporal
contrast. The millimeter scale of these shadowgraphies is significantly larger
than the FWHM of the gas jet, $\sim$ 270 $\mu$m at $z$ = 0, which will be
discussed in section VI. The prompt and localized energy deposition of the ps-
pulse along its propagation leads to strong ponderomotive and thermal
pressures which transversely expel the plasma from the laser axis, as observed
in Fig. 1 c), 1.7 ns after the ps-pulse.

Figure 1: Probe beam shadowgraphies from the plasma generated by the
interaction of the ps-beam with the high density H2 gas jet, at different
backing pressures, different times from the ps-pulse arrival, and for the LULI
(a, b, c) or the GSI (d) ps-beam. On top of the images ($z<$ -400 $\mu$m) is
the shadow of the nozzle. The ps-beam propagates at $z=0$ from left to right
along the $x$-axis (horizontal white dashed line), and is focused (red dashed
lines) at the center of the jet ($x$ = 0, $y$=0).

Time-resolved (streak camera) shadowgraphies along one dimension in space, the
ps-beam axis ($x$, for $z=0$), are presented in Fig. 2 for two gas jet
pressures. Despite the factor two in pressure, these shadowgraphies are very
similar. For $t$ $<$ 0 the probe beam is undisturbed, indicating that the ASE
contrast is high enough to avoid ionization of the gas before the ps-pulse
arrival. For 0 $<$ $t$ $<$ 1 ns, the shadowgraphy extends symmetrically from
the focal position up to $x\sim\pm$ 400 $\mu$m, as observed on the snapshot in
Fig 1a) and b). At these edges, for the 800 bars case (Fig. 2-a)), the plasma
density is $\sim 2\times 10^{19}$ cm-3 (0.02 $n_{c}$). The bright line
observed in Fig. 2a) at $t=0$, near the gas jet center (-200 $<x<$ +100
$\mu$m), is second harmonic emission produced by the ps-pulse, usually
occurring in regions of strong laser intensity and/or density gradients. For 0
$<t<$ 1.5 ns, the longitudinal expansion of the right edge of the plasma
($x>500$ $\mu$m) can be observed. At $t\gtrsim 1$ ns the plasma expulsion
induced by the ponderomotive and thermal pressures has lowered the density on
the laser axis, and parts of probe beam start to be transmitted again. As in
Fig. 1c), at $t\sim 1.7$ ns the plasma has been expelled from the laser axis
and the probe beam is not disturbed anymore.

Figure 2: Time-resolved (streak camera) shadowgraphies along the ps-beam axis
($x$, at $z=0$), for two gas jet pressures. The ps-pulse arrives at $t$ = 0
and propagates from left to right as in Fig. 1.

Proton spectra from the LULI experiment, measured at 0∘, 30∘, and 70∘ from the
laser axis, are presented in Fig. 3 for the same laser-plasma parameters as
Fig. 1 and 2. By directly focusing the laser on the gas jet, without plasma
tailoring (no ns-beams), the acceleration along the laser axis is very weak
(if not null), is larger at 30∘, and much more efficient at 70∘, almost
perpendicularly to the laser axis. The spectrum has two components: an
exponential part at low energy, followed by a plateau, extending up to 2 MeV
at 70∘.Such an energy distribution could be the result of ”Coulomb explosion”:
ions are accelerated by electrostatic forces caused by charge separation
induced by the laser ponderomotive pressure in the underdense part of the
plasma Sarkisov  _et al._ (1999); Krushelnick  _et al._ (1999). The maximum
energy that can be gained is the relativistic ponderomotive energy
$U_{p}=m_{e}c^{2}(\gamma-1)$, where $\gamma=(1+a_{0}^{2}/2)^{1/2}$ is the
relativistic factor of the electron quiver motion in the laser field. For the
LULI experiment $a_{0}\sim 4$, leading to $U_{p}\sim 1$ MeV, too low to
explain the high energy part of the spectra. In addition, the ponderomotive
energy depends only on the laser intensity, while the maximum energies
observed at 0∘, 30∘, and 70∘ seem to increase with the plasma density. The
plateau structure, the maximum energy value and its increase with the plasma
density could be the signature of an acceleration by multiple collisionless
shocks formed at high density Wei  _et al._ (2004); Silva  _et al._ (2004).

Figure 3: Proton spectra from the LULI experiment, without plasma tailoring,
measured at 0∘, 30∘, and 70∘ from the ps-beam axis, and for different gas jet
backing pressures. The dashed lines are the noise level of each spectrum. Same
laser-plasma parameters as in Fig. 1 and 2.

##  IV Interaction with a front face tailored plasma

The 1D time-resolved (streak camera) and 2D space-resolved (GOI snapshot)
shadowgraphy of the plasma tailored by one ns-beam, at the entrance side of
the ps-beam, is presented in Fig. 4-a) and -b) respectively, for a gas
pressure of 200 bars. The ns-pulse was sent in the gas jet $\sim$ 2.7 ns
before the ps-pulse. It was propagating along the $y$ axis (perpendicular to
the image plane) and focused at $x=$ -300 $\mu$m, $z=$ 0 (red circle in Fig.
4-b)). The hydrodynamic evolution (on the ps-beam axis) of the plasma can be
followed in Fig. 4-a). For $t<$ -2.5 ns the probe beam is fully transmitted,
indicating no pre-plasma. At $t\sim$ -2.7 ns the ns-beam promptly ionizes and
heats the edge of the gas jet. A hydrodynamic shockwave is generated, which
starts to push the plasma out of the ns-beam axis. At $-2.7<t<-2.3$ ns one can
observe the plasma motion towards vacuum ($x<$ -300 $\mu$m) and towards the
jet center ($x>$ -300 $\mu$m). At later times, the plasma moving towards
vacuum becomes too teneous to induce shadowgraphy, while the HSW propagating
towards the jet center stays dense and can easily be followed. In contrast
with the ps-pulse interaction that ionizes very quickly the whole gas jet
volume (Fig. 1), the ionization by the ns-beam concerns only its focal region,
the rest of the jet staying in the gaseous state until the arrival of the HSW
(the right side of the jet stays undisturbed). The shockwave reaches the jet
center at $t=$ 0\. At that time the ps-pulse arrives, crosses the plasma
created by the HSW, and then interacts with the high-density non-ionized part
of the gas jet (at $x=$ 0), which induces strong second harmonic emission that
saturates the camera. The 2D plasma extension at that time can be observed on
Fig. 4-b). The focal spot position of the ns-beam is indicated by the red
circle. The time integration window of the GOI was $\sim$ 120 ps, centered on
the ps-pulse arrival, so that the snapshot shows the plasma created by both
the ns-beam and the ps-beam, as well as the strong second harmonic emission of
the ps-pulse. Partly hidden by this flash, a croissant-like shape can be
distinguished, which is the plasma generated by the expansion of the HSW. This
initially cylindrical wave produces a circular shadowgraphy that has evolved
to a croissant-like shape because $\sim$ half of the HSW expelled the plasma
towards vacuum (left on the image), while the part propagating towards the
center of the jet ionized and compressed this high density region. The
horizontal extension of the croissant is larger on the bottom part of the
image ($z>$ 0) because of the conical shape of the gas jet (larger far from
the nozzle exit). The blue dashed-lines in Fig. 4-b) show the edges of the
shadowgraphy obtained for the case without tailoring (ps-beam only, Fig. 1-a),
-b), -d)). On the ns-beam side the plasma edge has been pushed towards the jet
center.

Figure 4: a) Time-resolved along the ps-beam axis (streak camera) and b) 2D
space-resolved (GOI snapshot) shadowgraphy of the plasma tailored by one ns-
beam, at the entrance side of the ps-beam, for a gas jet backing pressure of
200 bars. The ns-pulse propagates along the $y$ axis (perpendicular to the
image plane) and focuses at $x=$ -300 $\mu$m, $z=$ 0 (red circle). It arrives
in the gas jet $\sim$ 2.7 ns before the ps-pulse. The ps-beam propagates at
$z=0$ from left to right along the $x$-axis (horizontal white dashed line in
b)), and is focused (red dashed lines in b)) at the center of the jet ($x$ =
0, $y$=0). The time integration window of the snapshot in b) was $\sim$ 120
ps, centered on the ps-pulse arrival.

Proton spectra associated with the shadowgraphy of Fig. 4 are presented in
Fig. 5. Tailoring the gas jet on the input side of the ps-beam improved the
forward acceleration. The maximum proton energy at 0∘ and 30∘ both increased,
while the transverse (70∘) acceleration became less efficient. In addition to
the thermal exponential, the spectrum at 0∘ (Fig. 3) has a peaked component
centered near 1.35 MeV. This indicates the appearance of another acceleration
mechanism: Collisionless Shock Acceleration. No acceleration (proton) was
measured without the ps-beam.

Figure 5: Proton spectra from the LULI experiment, associated with the
shadowgraphy of Fig. 4, for a plasma tailored at the entrance side of the ps-
beam.

##  V Interaction with a front and back face tailored plasma

An example of shadowgraphies of the plasma tailored on the two opposite sides
is presented in Fig. 6. The ns-beams arrive 2.5 ns before the ps-pulse. At
this time we can see in Fig. 6-a) the prompt generation of the plasmas on both
sides of the jet center, at the focal positions of the ns-beams ($x=\pm$ 300
$\mu$m), followed by the propagation of the HSWs. Only the dense parts
converging towards the jet center are visible, the parts expanding towards
vacuum are too teneous to generate a significant shadowgraphy. The central
part of the jet stays in the gaseous state at least until $t\sim$ -1.4 ns,
when the shadowgraphies of each HSW start to overlap. The waves collide at the
jet center at $t\sim$ -100 ps, leading to a shadowgraphy with a minimum width
of $\lesssim$ 300 $\mu$m, slightly before the ps-pulse arrival ($t=0$). This
width is much smaller than the one observed in Fig. 2 without plasma tailoring
($\sim$ 800 $\mu$m). As observed in Fig. 2-a) and 5-a), the ps-pulse generates
second harmonic light at its arrival in the plasma. After that time, the
collision of the HSWs has ended and the plasma starts to expand. At $t\sim$
0.6 ns the strong ponderomotive and thermal pressures driven by the ps-pulse
have expelled the plasma from the laser axis, and the probe beam is
transmitted again. The 2D space-resolved shadowgraphy of the HSWs just before
the ps-beam arrival is presented in Fig 6-b). As already observed in Fig.
4-b), the initially cylindrical expansion of each HSW from the axis of its
driving ns-beam (red circles) has led to a croissant-like plasma of high
density. The blue dashed-lines in the figure indicate the edges of the
shadowgraphy measured without plasma tailoring (Fig. 1). It shows that
tailoring the gas jet from the two opposite sides allowed to i) significantly
reduce the plasma density at the edge of the gas jet and, ii) generate a high-
density narrow ($<$ 300 $\mu$m) plasma slab along the ps-beam axis. This is
also illustrated in Fig. 7 at a slightly higher pressure (300 bars), at two
different times: before the collision of the HSWs (Fig. 7-a), and just after
their collision and the arrival of the ps-pulse (Fig. 7-b). Before the ps-
pulse arrival the plasma is visible only at the location of the HSWs, while
after its arrival the region of the gas jet where the HSWs did not travel
(bottom of the image, $z>$ 400 $\mu$m) is promptly ionized, leading in this
region to a broad shadowgraphy comparable to the one observed without plasma
tailoring (Fig. 1). On the opposite, around the ns-beam axis, where the plasma
has been expelled by the HSWs, the ps-pulse does not create any additional
perturbation of the shadowgraphy, indicating that it propagates almost freely,
up to the HSWs colliding region where it encounters a high-density plasma slab
and generates second harmonic emission (near $x\sim$ -100 $\mu$m).

Figure 6: a) Time-resolved along the ps-beam axis (streak camera) and b) 2D
space-resolved (GOI snapshot) shadowgraphy of the plasma tailored on the two
opposite sides, for a gas jet backing pressure of 200 bars. The energy in each
ns-beam is 4.5 J. The time integration window of the snapshot in b) was $\sim$
120 ps, ending just before the ps-pulse arrival. The blue dashed-lines
indicate the edges of the shadowgraphy measured without plasma tailoring (Fig.
1). Figure 7: 2D space-resolved (GOI snapshot) shadowgraphies of the plasma
tailored on the two opposite sides, for a gas jet backing pressure of 300
bars, at two different times: a) before the collision of the HSWs, and b) just
after their collision and the arrival of the ps-pulse. The blue dashed-lines
indicate the edges of the shadowgraphy measured without plasma tailoring (Fig.
1).

Examples of proton spectra obtained by tailoring the gas jet from the two
opposite sides are presented in Fig. 8. The maximum energy and the shape of
the spectra at 70∘ is quite similar to the ones obtained without tailoring
(Fig. 3). However, despite the shot-to-shot fluctuations, the tendency
observed when tailoring only the entrance side (Fig. 5) is enhanced when
tailoring the two sides: the acceleration in the forward direction is improved
and the energy distribution becomes more peaked. At 30∘ the portion of the
spectra induced by CSA moved towards higher energies and is now clearly
separated from the low energy thermal part. The maximum energy is more than
doubled, reaching 3.2 MeV.

The number of protons accelerated above 1 MeV is also significantly increased
at 0∘ and 30∘. The comparison with the single HSW case (Fig. 5) demonstrates
that the acceleration improvement is not only the result of a better
propagation of the ps-pulse at the plasma entrance. Compared to the case
without tailoring at 800 bars (Fig. 3), the number of protons accelerated at
70∘ is similar or larger despite a lower backing pressure, 200-300 bars. This
indicates that the collision of the HSWs has allowed to compensate the
initially lower plasma density (the expected total compression factor Marquès
_et al._ (2021) is $\gtrsim$ 8), and enabled to reach a value at least
equivalent to the one at 800 bars, close to $n_{c}$. It not only increases the
number of protons available in the plasma, but helps to increase the ps-pulse
absorption Wilks  _et al._ (1992); Tsung  _et al._ (2012), thus the velocity
of the collisionless shockwave and the maximum proton energies. Let us note
that the lower backing pressure used for the tailoring cases was chosen to
avoid the ns-beam refraction at the jet entrance and a better plasma expulsion
by the HSW, while preserving the high density at the jet center thanks to the
collision of the HSWs Marquès  _et al._ (2021).

Figure 8: Proton spectra from 4 laser shots on the LULI experiment, in the
case of a plasma tailored on both sides (as in Fig. 6), and a gas jet backing
pressure of 300 bars on shot 1 and 200 bars on shots 2 to 4. The dashed lines
are the noise level of each spectrum.

Very similar results were obtained on the GSI experiment. A 2D shadowgraphy at
the collision of the two HSWs is presented in Fig. 9-a). The croissant-like
shape of the HSWs is more square. This is due to the square shape of the wings
of the ns-beam focal spots. Despite this difference, the sharpness and the
width ($\sim$ 200 $\mu$m) of the plasma slab are very similar to the LULI
experiment. The proton spectrum measured at 53∘ and associated to the
interaction of the ps-pulse with this plasma slab is presented in Fig. 9-b).
Like on the LULI experiment, the spectrum had a low energy exponential
distribution without tailoring ($<$ 2 MeV), evolving to a peaked distribution
of higher energy, up to 5.5 MeV, when two-sided tailoring is applied. Compared
to the LULI experiment, the laser intensity was a factor $\sim$ 4 larger on
GSI ($a_{0}\sim$ 8 instead of 4), leading as expected to a factor $\sim$ 2
increase on the maximum energy of the protons.

Figure 9: Typical result of the GSI experiment: a) 2D space-resolved (GOI
snapshot) shadowgraphy of the plasma tailored on both sides, b) associated
proton spectrum measured at 50∘. The gas jet backing pressure was 100 bars.
The shadowgraphy is 40 ps before the ps-pulse arrival.

Table 1 summarizes the number of accelerated protons and the energy of the
proton beam for the LULI spectra of Fig. 3, 5 and 8 associated with the three
tailoring scenario. Despite the relatively strong shot-to-shot fluctuations,
it demonstrates that tailoring the plasma from the two opposite sides improves
the acceleration towards the laser direction, in terms of number of protons as
well as of energy of the proton beam. The optimum acceleration in the forward
direction also corresponds to the minimum transverse acceleration (last line
in Table 1). Only one shot is presented for the one-side case. Nevertheless,
several shots with the one-side tailoring were performed on the GSI
experiment. They confirm the evolution of the acceleration observed on the
LULI experiment between zero, one, or two sides tailoring. The shot-to-shot
fluctuations observed on both experiments result mainly from variations on the
plasma profile at the ps-pulse arrival, which can significantly affect the
quality (uniformity, angle, Mach number) of the collisionless shockwave. These
variations originate mainly from fluctuations on the dynamics of the HSWs
which are sensitive to i) the energy in the wing of the ns-beam focal spot
(see following section) and ii) the exact position of the focal spot on the
edge of the gas jet density profile. For example, from the expression of
$n_{atom}(r)$ given in section II, the density is a factor 3.6 different
between $r$ = 250 and $r$ = 350 $\mu$m. This induces an order of magnitude
difference in the inverse bremsstrahlung absorption ($K_{BI}\propto
n_{e}^{2}$) and thus on the amplitude and velocity of each HSW, leading to
variations on the time of collision as well as on the position, density, width
and shape of the final plasma slab encountered by the ps-pulse.

Despite these fluctuations, all the improvements observed in proton spectra
are in good agreement with laser-plasma conditions becoming closer from the
main criteria for efficient CSA Fiuza  _et al._ (2012, 2013) which are i) a
near-critical plasma density to get an efficient laser absorption of the ps-
pulse and so a strong plasma heating (MeV), ii) a narrow plasma length to
favor uniform heating, leading to a uniform shock velocity and thus the
production of monoenergetic protons, iii) an exponentially decreasing output
density gradient to get a uniform sheath field which preserves the
monoenergetic distribution as protons are reflected by the shock.

Table 1: Number of protons and total energy of the spectra for $E_{k}>$ 1 MeV,
measured at 0∘, 30∘ or 70∘, and for the different configurations of plasma
tailoring. Obtained by integrating the spectra of Fig. 3, 5 and 8 between
$E_{k}$ = 1 and 3.5 MeV. These values are expressed in $10^{8}$ sr-1 for
$dN/d\Omega$, and $10^{8}$ MeV sr-1 for $dE/d\Omega$. Acceleration in the
forward direction is highlighted with colored cells.

The proton energy expected from CSA Fiuza  _et al._ (2012, 2013) is of the
order of $E_{k}$[MeV] $\sim 2M^{2}T_{e}$[MeV], where $M$ is the Mach number of
the collisionless wave, $T_{e}$[MeV] $\sim
0.078\frac{\eta}{\alpha}a_{0}\tau$(ps)$/L$(mm), $\tau$ the duration of the
laser pulse, $\eta$ its absorption efficiency, and $\alpha$ is 3/2 for non-
relativistic plasmas ($T_{e}\ll m_{e}c^{2}$) and 3 in the relativistic case.
The absorption efficiency depends on $a_{0}$ and $n_{e}$. In our conditions,
it is expected Tsung  _et al._ (2012); Wilks  _et al._ (1992); Fiuza  _et al._
(2013); Bonvalet  _et al._ (2021) to be between 0.25 and 0.5. Taking $\alpha$
= 2.2 and assuming 0.05 $<L$(mm)$<$ 0.1 (smaller than the shadowgraphy size,
see following section), $T_{e}$ is expected to be of the order of 0.35 to 1.4
MeV on both the LULI and the GSI experiments (same $a_{0}\tau$). Taking for
$M$ the minimum value required for the shock to reflect protons Sagdeev
(1966); Fiuza  _et al._ (2013), $M=M_{cr}\sim$1.6 leads to expected proton
energies of 1.8 $<E_{k}$(MeV) $<$ 7.2, in good agreement with our
measurements.

As discussed in section I, the maximum proton energy from CSA driven by a
$\lambda_{0}$ = 1 $\mu$m laser is expected Fiuza  _et al._ (2012, 2013) to
occur for a plasma length close to $L_{opt}\sim$ 40 $\mu$m. The width of the
plasma observed in figures 6, 7 and 9-a) appears to be $\sim$ 200 to 300
$\mu$m. However, the width of the shadowgraphy could significantly
overestimate those of the density profile. For example, the gas jet FWHM is
$\sim$ 270 $\mu$m, while on figures 1 and 2 the full width of the shadowgraphy
is $\sim$ 600 to 800 $\mu$m, representing the very low wing of the density
profile, 0.08 to 0.025 of the density at the jet center. The plasma slab at
the collision of the HSWs can thus be much narrower than its apparent size on
the shadowgraphy.

##  VI Plasma tailoring - comparison with numerical simulations

The CSA strongly depends on the plasma length $L$. Our previous numerical
study Marquès  _et al._ (2021) indicated that the width of the density profile
at the collision of the two HSWs was below or of the order of tens of microns
while, as previously said, the dark part of the shadowgraphy images in Fig. 6
and 9-a) has a width of the order of 200-300 $\mu$m. This dark part originates
from light rays that are so refracted by the density gradient that they are
not collected by the lens of the imaging system. This width thus depends on
the aperture of the diagnostic (3.5∘ on the LULI experiment, 2∘ on GSI) and
can be significantly larger than the FWHM of the density profile. To estimate
this difference we performed 3D hydrodynamics simulations, and post-processed
the probe beam propagation (refraction and inverse bremsstrahlung absorption)
to simulate the shadowgraphy diagnostic. As in our previous study Marquès  _et
al._ (2021), we used the radiation-hydrodynamics code TROLL Lefebvre  _et al._
(2019), but in a three-dimensional geometry, allowing to use as inputs the
experimentally measured 3D density profile of the gas jet. We also used the
experimental temporal profile of the ns-beams. Figure 10-a) shows the temporal
evolution of the density profile along the ps-beam axis. As already detailed
in Ref. Marquès  _et al._ (2021), each of the two ns-beams, focused 300 $\mu$m
from both sides of the gas jet center ($x=0$) generates a HSW. Only the part
of the cylindrical HSW that converges towards the gas jet center leads to a
high density jump, while the part traveling towards vacuum is not visible
because of its quick spread and decrease in amplitude. The initial maximum
plasma density (at $x$ = 0) is $n_{e}^{0}=2\times 10^{20}$ cm-3 ($n_{c}/5$),
as for the shot presented in Fig. 6-a). The spatial intensity profile of the
ns-beams is a Gaussian of 25 $\mu$m radius. Fig. 10-b) is the simulation of
the shadowgraphy diagnostic. Compared to Fig. 6-a), after 2.5 ns the HSWs
still have not reached the gas jet center. Also, at the HSW creation, the
experimental shadowgraphy has a large ”horizontal” shape while the simulated
one is much more localized (narrow). In the experiment, the laser intensity
profile was in fact composed not only of the narrow high intensity part, but
also of a lower intensity wing: $I(r)=3\times
10^{14}[e^{-(r/25)^{2}}+0.0172e^{-(r/110)^{2}}]$ W/cm2. Despite its lower
intensity, this large wing contained $\sim$ 25 $\%$ of the laser energy,
enough to contribute to the HSW excitation. Fig. 11 shows the evolution of the
density profile and of the associated shadowgraphy when the wing of the ns-
beams is taken into account, and for the same laser energy (4.5 J in each
beam) as the shot in Fig. 6. The simulated shadowgraphy is very similar to the
experimental one (Fig. 6-a)), reproducing the initial ”horizontal” part
followed by two large traces that converges when the two HSWs collide after
$\sim$ 2.3 ns, at the jet center. The effect of the wing of the focal spot on
the HSW evolution is also illustrated in Fig. 12, that compares the profiles
of a) the laser focal spot, b) the electron temperature, c) the electron
density for the case without (left) or with (right) the wing. Even if the
intensity is higher without the wing, because the energy deposition is more
localized, the HSWs start with a smaller diameter and further from the jet
center, avoiding them to collide. With the wing, the HSWs are generated with a
larger diameter, enabling them to reach faster the jet center. A point to
underline is that the density digging downstream the HSW seems nevertheless
more efficient without wing (more localized energy deposition), as can also be
observed comparing Fig. 10-a) and 11a). Fig. 12-c) also shows that the
simulations reproduce very well the ”croissant-like” shape observed
experimentally (Fig. 4, 6, 7 and 9). The influence of the wing is also
observed in Fig. 9-a) where the square shape of the GSI ns-beams is retrieved
in the shape of the HSWs.

Figure 10: a): Temporal evolution of the density profile along the ps-beam
axis from 3D TROLL simulations. The two ns-beams are focused 300 $\mu$m from
both sides of the gas jet center. The initial maximum plasma density (at $x$ =
0) is $n_{e}^{0}=2\times 10^{20}$ cm-3 ($n_{c}/5$). The spatial intensity
profile of the ns-beams is $I(r)=5.4\times 10^{14}e^{-(r/25)^{2}}$, and the
energy in each beam is 6 J. b) Post-processing of the shadowgraphy diagnostic
from the TROLL outputs. The cyan line-outs are density and shadowgraphy
profiles at $t$ = 2.5 ns. Figure 11: Same as Fig. 10 but for a spatial profile
of the ns-beams with a low-intensity wing: $I(r)=3\times
10^{14}[e^{-(r/25)^{2}}+0.0172e^{-(r/110)^{2}}]$ W/cm2, and 4.5 J in each
beam. The cyan line-outs are density and shadowgraphy profiles at $t$ = 2.3
ns.

Let us note that, since hydrogen is very quickly ionized, a full ionization at
the beginning of the simulation was imposed and, to avoid an early gas jet
expansion, an initial temperature of 300 K was imposed. This has no effect on
the plasma evolution induced by the ns-beams. However, in the regions not
affected by these beams or by the HSWs they produce, where the medium should
still be in its gas state, its refraction index as well as the inverse
bremsstrahlung absorption of the low-intensity probe beam are strongly
overestimated. To avoid this artifact, in the post-processing of the
shadowgraphy diagnostics the ionization state was corrected using the Saha
ionization equation. The density map presented in Fig. 10-a) includes this
correction. Without this correction, the shadowgraphy at the early times is
completely dark at the jet center, while the gas is still not ionized and
should fully transmit the probe beam (as experimentally observed and mentioned
in section IV and V).

The normalized line-outs in Fig 11-a) and -b) show respectively the density
and shadowgraphy profiles at the collision time ($t=2.3$ ns). The width of the
shadowgraphy is $\sim$ 150 $\mu$m, much larger than the width of the plasma
slab, $\lesssim$ 20 $\mu$m. Applying this same factor difference on the
experimental shadowgraphies would indicate that the plasma slab at the
collision time had a thickness of the order of 30 to 40 $\mu$m, similar to the
predicted optimal value $L_{opt}$ for CSA (section I).

Figure 12: Spatial profiles of a) the laser focal spot, b) the electron
temperature, c) the electron density for the case without the wing in the
intensity profile (left, Fig. 10) or with (right, Fig. 11). a) et b) are at
0.6 ns, the maximum of the laser, c) is at 2.3 ns (time at which the HSWs
collide in the case with the wing).

From Fig. 11-a), the peak density at the collision time is $\sim 3n_{c}$. By
conservation of the number of particles, if the experimental plasma slab is of
the order or more than 2 times thicker than in the simulations, the peak
density should be lower by this same factor, of the order or less than 1.5
$n_{c}$.

##  VII Discussion

Even if the thickness of the plasma slab is close to optimum, its density
needs also to be adjusted to an optimum value which depends on the laser
intensity. To drive a strong collisionless shock and accelerate ions at high
energies, a high laser absorption is required (see section V), which implies
$n_{e}\sim\gamma n_{c}$. With $a_{0}\sim$ 4 at LULI and 8 at GSI, this leads
to $n_{e}/n_{c}\sim$ 3 and 5.7 respectively. Our previous numerical study
Bonvalet  _et al._ (2021) also showed that the maximum proton energy drops
quickly below this optimum value. The density of the plasma slab was thus
probably too low for an optimum proton acceleration. An easy way to produce
plasma slabs of higher density is to increase the backing pressure of the gas
jet, as long as the residual density in the wing of the slab does not disturb
the propagation of the ps laser beam.

The efficiency of the acceleration depends on the intensity of the ps-laser,
but also on the size of its focal spot Fiuza  _et al._ (2013). The shock width
(which is close to the laser spot size $W_{0}$) has to be large enough such
that the plasma, expanding transversely at the sound speed, does not leave the
shock region before the acceleration occurs. Assuming an isothermal expansion,
this condition yields $W_{0}\gtrsim L/M$. Taking for $M$ the minimum value
required for the shock to reflect protons, $M_{cr}=$ 1.6, and $L=$ 40 $\mu$m
leads to $W_{0}\gtrsim$ 25 $\mu$m. The focal spot on the LULI and GSI
experiments (FWHM $\sim$ 10 $\mu$m) was thus probably too small for an optimum
acceleration. With a larger focal spot the shape of the shock will be more
flat, favoring forward acceleration, at higher energy. The larger surface
might also increase the number of accelerated protons.

The shot-to-shot fluctuations of the ns-beams, in terms of position, spatial
distribution and intensity, generate spatial and temporal fluctuations on the
HSWs, and thus on the resulting plasma slab. Coupled to the temporal jitter
between the ns and ps pulses (between 100 and 200 ps in our experiment),
significant differences in laser-plasma coupling and heating can be induced,
which is thought to be at the origin of shot-to-shot fluctuations of the
proton beam. It also indicates that the condition for efficient CSA are still
not fully reached. Hosing instabilityCeurvorst  _et al._ (2016) could also
modify the direction of propagation of the ps-pulse, ultimately leading to
fluctuations in the direction of the collisionless shock and of the proton
beam. This affects more significantly the detection in the forward direction,
mainly sensitive to CSA.

The clearly peaked spectra observed with the two-sides tailoring indicate that
even if the plasma slab is relatively narrow, the density gradient at its
backside is not very sharp, avoiding TNSA, and present a ”slowly” decreasing
profile with a small and constant sheath electric field in favor of
monoenergetic acceleration Fiuza  _et al._ (2012).

##  VIII Conclusions

We recently proposed a new technique of plasma tailoring by laser-driven
hydrodynamic shockwaves generated on both sides of a gas jet. In the
continuation of this numerical work, we studied experimentally the influence
of the tailoring on proton acceleration driven by a high-intensity ps-laser,
in three cases: without tailoring, by tailoring only the entrance side of the
ps-laser, or both sides of the gas jet. Without tailoring the acceleration is
transverse to the laser axis, with a low energy exponential spectrum, produced
by Coulomb explosion. When the front side of the gas jet is tailored, a
forward acceleration appears, that is significantly enhanced when both the
front and back sides of the plasma are tailored. This forward acceleration
produces higher energy protons (up to 5.5 MeV), with a peaked spectrum, and is
in good agreement with the mechanism of Collisionless Shock Acceleration. The
total number of accelerated protons is also enhanced by this two-sides
tailoring. The spatio-temporal evolution of the plasma profile was
characterized by optical shadowgraphy of a probe beam. The refraction and
absorption of this beam was simulated by post-processing 3D hydrodynamic
simulations of the plasma tailoring. The dynamics of the hydrodynamic
shockwaves was well reproduced only if the low-intensity wing in the focal
spot of the tailoring beams was taken into account. Comparison of the
simulated shadowgraphy with the experimental ones allowed to estimate the
thickness ($\sim$ 30-40 $\mu$m) and density ($\sim$ 1.5 $n_{c}$) of the plasma
slab produced by tailoring both sides of the gas jet. These values are close
to those required to trigger the CSA. However, the shot-to-shot fluctuations
of the forward proton beam indicate that the set of criteria for an optimum
CSA was still not reached. Improving the focal spot quality of the tailoring
ns-beams, increasing the backing pressure of the gas jet as well as the focal
spot size of the ps-beam should enable to reach a more stable CSA, generating
a proton beam with a larger number of protons, at higher energy and of lower
spectral width and emittance.

###### Acknowledgements.

The authors would like to thank the LULI staff and the GSI-PHELIX staff for
their contribution. This work has received funding from the Fédération de
recherche PLAS@PAR. Results presented here are partially based on the
experiment P189, which was performed at the PHELIX infrastructure at GSI
Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany) in the context
of FAIR Phase-0. The research leading to the PHELIX-GSI results has received
funding from the European Union’s Horizon 2020 research and innovation program
under Grant Agreement No. 871124 Laserlab-Europe, and by Grant ANR-17-CE30-
0026-Pinnacle from Agence Nationale de la Recherche.

## Conflict of Interest

The authors have no conflicts to disclose.

## data availability

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

## References

  * Blanford and Eichler (1987) R. Blanford and D. Eichler, “Particle acceleration at astrophysical shocks: A theory of cosmic ray origin,” Physics Reports 154, 1 (1987).
  * Jones and Ellison (1991) F. C. Jones and D. C. Ellison, “The plasma physics of shock acceleration,” Space Science Reviews 58, 259 (1991).
  * Caprioli, Blasi, and Amato (2011) D. Caprioli, P. Blasi, and E. Amato, “Non linear diffusive acceleration of heavy nuclei in supernova remnant shocks,” Astroparticle Physics 34, 447 (2011).
  * Koyama  _et al._ (1995) K. Koyama, R. Petret, E. V. GotthelfH, U. Hwangt, M. Matsuura§, M. Ozaki, and S. S. Holt, “Evidence for shock acceleration of high energy electrons in the supernova remnant sn1006,” Nature 378, 255 (1995).
  * Adriani  _et al._ (2011) Adriani, G. C. O. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, V. Bonvicini, S. Borisov, S. Bottai, A. Bruno, F. Cafagna, D. Campana, R. Carbone, P. Carlson, M. Casolino, G. Castellini, L. Consiglio, M. P. De Pascale, C. De Santis, N. De Simone, V. Di Felice, A. M. Galper, W. Gillard, L. Grishantseva, G. Jerse, A. V. Karelin, S. V. Koldashov, S. Y. Krutkov, A. N. Kvashnin, A. Leonov, V. Malakhov, V. Malvezzi, L. Marcelli, A. G. Mayorov, W. Menn, V. V. Mikhailov, E. Mocchiutti, A. Monaco, N. Mori, N. Nikonov, G. Osteria, F. Palma, P. Papini, M. Pearce, P. Picozza, C. Pizzolotto, M. Ricci, S. B. Ricciarini, L. Rossetto, R. Sarkar, M. Simon, R. Sparvoli, P. Spillantini, Y. I. Stozhkov, A. Vacchi, E. Vannuccini, G. Vasilyev, S. A. Voronov, Y. T. Yurkin, J. Wu, G. Zampa, N. Zampa, and V. G. Zverev, “Pamela measurements of cosmic ray proton and helium spectra,” Science 332, 69 (2011).
  * Spitkovsky (2008a) A. Spitkovsky, “Particle acceleration in relativistic collisionless shocks: Fermi process at last?” The Astrophysical Journal 682, L5 (2008a).
  * Spitkovsky (2008b) A. Spitkovsky, “On the structure of relativistic collisionless shocks in electron ion plasmas,” The Astrophysical Journal 673, L39 (2008b).
  * Drake (2000) R. P. Drake, “The design of laboratory experiments to produce collisionless shocks of cosmic relevance,” Physics of Plasmas 7, 4690 (2000).
  * Courtois  _et al._ (2004) C. Courtois, R. A. D. Grundy, A. D. Ash, D. M. Chambers, N. C. Woolsey, R. O. Dendy, and K. G. McClements, “Experiment on collisionless plasma interaction with applications to supernova remnant physics,” Physics of Plasmas 11, 3386 (2004).
  * Woolsey  _et al._ (2001) N. C. Woolsey, Y. Abou Ali, R. G. Evans, R. A. D. Grundy, S. J. Pestehe, P. G. Carolan, N. J. Conway, R. O. Dendy, P. Helander, K. G. McClements, J. G. Kirk, P. A. Norreys, M. M. Notley, and S. J. Rose, “Collisionless shock and supernova remnant simulations on vulcan,” Physics of Plasmas 8, 2439 (2001).
  * Fiuza  _et al._ (2020) F. Fiuza, G. F. Swadling, A. Grassi, H. G. Rinderknecht, D. P. Higginson, D. D. Ryutov, C. Bruulsema, R. P. Drake, S. Funk, S. Glenzer, G. Gregori, C. K. Li, B. B. Pollock, B. A. Remington, J. S. Ross, W. Rozmus, Y. Sakawa, A. Spitkovsky, W. S., and H.-S. Park, “Electron acceleration in laboratory-produced turbulent collisionless shocks,” Nat. Phys. 16, 916–920 (2020).
  * 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,” Physical Review Letters 92, 015002 (2004).
  * Fiuza  _et al._ (2012) F. 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).
  * Fiuza  _et al._ (2013) F. 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,” Phys. Plasmas 20, 056304 (2013).
  * Palmer  _et al._ (2011) C. A. J. Palmer, N. P. Dover, I. Pogorelsky, M. Babzien, G. I. Dudnikova, M. Ispiriyan, M. N. Polyanskiy, J. Schreiber, P. Shkolnikov, V. Yakimenko, and Z. Najmudin, “Monoenergetic Proton Beams Accelerated by a Radiation Pressure Driven Shock,” Physical Review Letters 106, 014801 (2011).
  * Bulanov  _et al._ (2005) S. Bulanov, T. Esirkepov, P. Migliozzi, F. Pegoraro, T. Tajima, and F. Terranova, “Neutrino oscillation studies with laser driven beam dump facilities,” Nuclear Instruments and Methods in Physics Research A 540, 25 (2005).
  * Roth  _et al._ (2001) M. Roth, T. E. Cowan, S. P. B. C. Key, M. H.and Hatchett, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. Campbell, M. D. Perry, and H. Powell, “Fast ignition by intense laser accelerated proton beams,” Physical Review Letters 86, 436 (2001).
  * Temporal  _et al._ (2009) M. Temporal, R. Ramis, J. J. Honrubia, and S. Atzeni, “Fast ignition induced by shocks generated by laser-accelerated proton beams,” Plasma Phys. Control. Fusion 51, 035010 (2009).
  * Temporal, Honrubia, and Atzeni (2009) M. Temporal, J. J. Honrubia, and S. Atzeni, “Numerical study of fast ignition of ablatively imploded deuterium–tritium fusion capsules by ultra-intense proton beams,” Phys. Plasmas 9, 3098 (2009).
  * Boody  _et al._ (1996) F. P. Boody, R. Hopfl, H. Hora, and J. C. Kelly, “Laser-driven ion source for reduced-cost implantation of metal ions for strong reduction of dry friction and increased durability,” Laser and Particle Beams 14, 443 (1996).
  * Borghesi  _et al._ (2002) M. Borghesi, D. H. Campbell, A. Schiavi, M. G. Haines, O. Willi, A. J. MacKinnon, P. Patel, L. A. Gizzi, M. Galimberti, R. J. Clarke, F. Pegoraro, H. Ruhl, and S. Bulanov, “Electric field detection in laser-plasma interaction experiments via the proton imaging technique,” Physics of Plasmas 9, 2214 (2002).
  * Li  _et al._ (2006) C. K. Li, F. H. Séguin, J. A. Frenje, J. R. Rygg, R. D. Petrasso, R. P. J. Town, P. A. Amendt, S. P. Hatchett, O. L. Landen, A. J. Mackinnon, P. K. Patel, V. A. Smalyuk, T. C. Sangster, and J. P. Knauer, “Measuring E and B fields in laser-produced plasmas with monoenergetic proton radiography,” Physical Review Letters 97, 135003 (2006).
  * Romagnani  _et al._ (2008) L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser-plasma experiments,” Physical Review Letters 101, 025004 (2008).
  * Bulanov  _et al._ (2002) S. V. Bulanov, T. Z. Esirkepov, V. S. Khoroshkov, A. V. Kuznetsouv, and F. Pegoraro, “Oncological hadrontherapy with laser ion accelerators,” Physical Letters A 299, 240 (2002).
  * Bulanov and Khoroshkov (2002) S. V. Bulanov and V. S. Khoroshkov, “Feasibility of using laser ion accelerators in proton therapy,” Plasma Physics Reports 28(5), 453 (2002).
  * Malka  _et al._ (2004) V. Malka, S. Fritzler, E. Lefebvre, E. d’Humières, R. Ferrand, G. Grillon, C. Albaret, S. Meyroneinc, J. P. Chambaret, A. Antonetti, and D. Hulin, “Practicability of protontherapy using compact laser systems,” Med. Phys. 31, 1587 (2004).
  * Karsch  _et al._ (2017) L. Karsch, E. Beyreuther, W. Enghardt, M. Gotz, U. Masood, U. Schramm, K. Zeil, and J. Pawelke, “Towards ion beam therapy based on laser plasma accelerators,” ACTA ONCOLOGICA 56(11), 1359 (2017).
  * Spencer  _et al._ (2001) I. Spencer, K. Ledingham, R. Singhal, T. McCanny, P. McKenna, E. Clark, K. Krushelnick, M. Zepf, F. Beg, M. Tatarakis, A. Dangor, P. Norreys, R. Clarke, R. Allott, and I. 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 (2001).
  * Lefebvre  _et al._ (2006) E. Lefebvre, E. D’Humières, S. Fritzler, and V. Malka, “Numerical simulation of isotope production for positron emission tomography with laser-accelerated ions,” J. Appl. Phys. 100, 113308 (2006).
  * Snavely  _et al._ (2000) R. A. Snavely, M. H. Key, S. P. Hatchett, T. E. Cowan, M. Roth, T. W. Phillips, M. A. Stoyer, E. A. Henry, T. C. Sangster, M. S. Singh, S. C. Wilks, A. MacKinnon, A. Offenberger, D. M. Pennington, K. Yasuike, A. B. Langdon, B. F. Lasinski, J. Johnson, M. D. Perry, and C. M., “Intense high-energy proton beams from petawatt-laser irradiation of solids,” Physical Review Letters 85, 2945 (2000).
  * Wilks  _et al._ (2001) S. C. Wilks, A. B. Langdon, T. E. Cowan, M. Roth, M. Singh, S. Hatchett, M. H. Key, D. Pennington, A. MacKinnon, and R. A. Snavely, “Energetic proton generation in ultra-intense laser–solid interactions,” Physics of Plasmas 8, 542 (2001).
  * Esirkepov  _et al._ (2004) T. Esirkepov, M. Borghesi, S. V. Bulanov, G. Mourou, and T. Tajima, “Highly efficient relativistic-ion generation in the laser-piston regime,” Physical Review Letters 92, 175003 (2004).
  * Robinson  _et al._ (2008) A. P. L. Robinson, M. Zepf, S. Kar, R. G. Evans, and C. Bellei, “Radiation pressure acceleration of thin foils with circularly polarized laser pulses,” New J. Phys. 10, 013021 (2008).
  * Macchi, Veghini, and Pegoraro (2009) A. Macchi, S. Veghini, and F. Pegoraro, “Light sail acceleration reexamined,” Physical Review Letters 103, 085003 (2009).
  * Qiao  _et al._ (2012) B. Qiao, S. Kar, M. Geissler, P. Gibbon, M. Zepf, and M. Borghesi, “Dominance of radiation pressure in ion acceleration with linearly polarized pulses at intensities of 1021 wcm-2,” Physical Review Letters 100, 115002 (2012).
  * Yin  _et al._ (2007) L. Yin, B. J. Albright, B. M. Hegelich, K. J. Bowers, K. A. Flippo, T. J. T. Kwan, and J. C. Fernandez, “Monoenergetic ion acceleration from the laser breakout afterburner using ultrathin targets,” Physics of Plasmas 14, 056706 (2007).
  * Henig  _et al._ (2009) A. Henig, D. Kiefer, K. Markey, D. C. Gautier, K. A. Flippo, S. Letzring, R. P. Johnson, T. Shimada, L. Yin, B. J. Albright, K. J. Bowers, J. C. Fernandez, S. G. Rykovanov, H.-C. Wu, M. Zepf, D. Jung, V. K. Liechtenstein, J. Schreiber, D. Habs, and B. M. Hegelich, “Enhanced laser-driven ion acceleration in the relativistic transparency regime,” Physical Review Letters 103, 045002 (2009).
  * 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,” Physical Review Letters 105, 135002 (2010).
  * Park  _et al._ (2019) J. Park, S. S. Bulanov, J. Bin, T. Schenkel, Q. Ji, S. Steinke, J.-L. Vay, C. G. R. Geddes, C. B. Schroeder, W. P. Leemans, and E. Esarey, “Ion acceleration in laser generated megatesla magnetic vortex,” Physics of Plasmas 26, 103108 (2019).
  * Najmudin  _et al._ (2011) Z. Najmudin, C. A. J. Palmer, N. P. Dover, I. Pogorelsky, M. Babzien, A. E. Dangor, G. I. Dudnikova, P. S. Foster, J. S. Green, M. Ispiriyan, D. Neely, M. N. Polyanskiy, J. Schreiber, P. Shkolnikov, and V. Yakimenko, “Observation of impurity free monoenergetic proton beams from the interaction of a CO2 laser with a gaseous target,” Phys. Plasmas 18, 056705 (2011).
  * Haberberger  _et al._ (2012) D. Haberberger, S. Tochitsky, F. Fiuza, C. Gong, R. A. Fonseca, L. Silva, W. B. Mori, and C. Jochi, “Collisionless shocks in laser-produced plasma generate monoenergetic high-energy proton beams,” Nature Phys 8, 95 (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,” Physical Review Letters 115, 094802 (2015).
  * Chen  _et al._ (2017a) Y. Chen, M. Helle, A. Ting, D. Gordon, N. Dover, O. Ettlinger, Z. Najmudin, M. Polyanskiy, I. Pogorelsky, and M. Babzien, “Laser acceleration of protons with an optically shaped, near-critical hydrogen gas target,” AIP Conf. Proc. 1812, 090002 (2017a).
  * Chen  _et al._ (2017b) 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 $\mu$m wavelength high-intensity short-pulse laser,” Scientific Reports 7, 13505 (2017b).
  * 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,” Phys. Plasmas 26, 123109 (2019).
  * 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,” Phys. Rev. Accel. Beams 21, 103401 (2018).
  * Tochitsky  _et al._ (2020) S. Tochitsky, A. Pak, F. Fiuza, D. Haberberger, A. Lemos, N. Link, D. H. Froula, and C. Joshi, “Interpenetration and ion separation in colliding plasmas,” Physics of Plasmas 27, 083102 (2020).
  * Marquès  _et al._ (2021) J.-R. Marquès, P. Loiseau, J. Bonvalet, M. Tarisien, E. d’Humières, J. Domange, F. Hannachi, L. Lancia, O. Larroche, P. Nicolaï, P. Puyuelo-Valdes, L. Romagnani, J. Santos, and V. Tikhonchuk, “Over-critical sharp-gradient plasma slab produced by the collision of laser-induced blast-waves in a gas jet; application to high-energy proton acceleration,” Physics of Plasmas 28, 023103 (2021).
  * Bonvalet  _et al._ (2021) J. Bonvalet, P. Loiseau, J.-R. Marquès, E. Atukpor, E. d’Humières, J. Domange, P. Forestier-Colleoni, F. Hannachi, L. Lancia, D. Raffestin, M. Tarisien, V. Tikhonchuk, and P. Nicolaï, “Laser-driven collisionless shock acceleration of protons from gas jets tailored by one or two nanosecond beams,” Physics of Plasmas 28, 113102 (2021).
  * Henares  _et al._ (2019) J. L. Henares, P. Puyuelo-Valdes, F. Hannachi, T. Ceccotti, M. Ehret, F. Gobet, L. Lancia, J.-R. Marquès, J. J. Santos, M. Versteegen, and M. Tarisien, “Development of gas jet targets for laser-plasma experiments at near-critical density,” Rev. Sci. Instrum. 90, 063302 (2019).
  * Bulanov  _et al._ (2010) S. S. Bulanov, V. Y. Bychenkov, V. Chvykov, G. Kalinchenko, D. W. Litzenberg, T. Matsuoka, A. G. R. Thomas, L. Willingale, V. Yanovsky, K. Krushelnick, and A. Maksimchuk, “Generation of GeV protons from 1 PW laser interaction with near critical density targets,” Phys. Plasmas 17, 043105 (2010).
  * Liu  _et al._ (2018) J.-X. Liu, Y. Zhao, X.-P. Wang, J.-Z. Quan, T.-P. Yu, G.-B. Zhang, X.-H. Yang, Y.-Y. Ma, F.-Q. Shao, and J. Zhao, “High-flux positrons generation via two counter-propagating laser pulses irradiating near-critical-density plasmas,” Phys. Plasmas 25, 103106 (2018).
  * Zhu  _et al._ (2016) X. L. Zhu, T. P. Yu, Z. M. Sheng, Y. Yin, I. C. E. Turcu, and A. Pukhov, “Dense GeV electron–positron pairs generated by lasers in near-critical-density plasmas,” Nat. Commun. 7, 13686 (2016).
  * 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, S. G, 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 J. Phys. 21, 043044 (2019).
  * Sarkisov  _et al._ (1999) G. S. Sarkisov, V. Y. Bychenkov, V. N. Novikov, V. T. Tikhonchuk, A. Maksimchuk, S.-Y. Chen, R. Wagner, G. Mourou, and D. Umstadter, “Self-focusing, channel formation, and high-energy ion generation in interaction of an intense short laser pulse with a he jet,” Phys. Rev. E 59, 7042 (1999).
  * 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,” Physical Review Letters 83, 737 (1999).
  * Wei  _et al._ (2004) M. S. Wei, S. P. D. Mangles, Z. Najmudin, B. Walton, A. Gopal, M. Tatarakis, A. E. Dangor, E. L. Clark, R. G. Evans, S. Fritzler, R. J. Clarke, C. Hernandez-Gomez, D. Neely, W. Mori, M. Tzoufras, and K. Krushelnick, “Ion Acceleration by Collisionless Shocks in High-Intensity-Laser-Underdense-Plasma Interaction,” Physical Review Letters 93, 155003 (2004).
  * Wilks  _et al._ (1992) S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon, “Absorption of ultra-intense laser pulses.” Physical Review Letters 69, 1383 (1992).
  * 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,” J. Plasma Phys. 78, 373 (2012).
  * Sagdeev (1966) R. Z. Sagdeev, “Cooperative phenomena and shock waves in collisionless plasmas,” Rev. Plasma Phys. 4, 23 (1966).
  * Lefebvre  _et al._ (2019) E. Lefebvre, C. Bernard, S. ans Esnault, P. Gauthier, A. Grisollet, P. Hoch, L. Jacquet, G. Kluth, S. Laffite, S. Liberatore, I. Marmajou, P.-E. Masson-Laborde, O. Morice, and J.-L. Willien, “Development and validation of the TROLL radiation-hydrodynamics code for 3D hohlraum calculations,” Nucl. Fusion 59, 032010 (2019).
  * Ceurvorst  _et al._ (2016) L. Ceurvorst, N. Ratan, M. C. Levy, M. F. Kasim, J. Sadler, R. H. H. Scott, R. M. G. M. Trines, T. W. Huang, M. Skramic, M. Vranic, L. O. Silva, and P. A. Norreys, “Mitigating the hosing instability in relativistic laser-plasma interactions,” New J. Phys. 18, 053023 (2016).