# Simultaneous generation and detection of energetic particle and radiation
beams from relativistic plasma mirrors driven at kHz repetition rate

Jaismeen Kaur<EMAIL_ADDRESS>Laboratoire d’Optique Appliquée,
Institut Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Marie Ouillé  Laboratoire d’Optique Appliquée, Institut
Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Ardop Engineering, Cité de la Photonique, 11 Avenue de la
Canteranne, Bât. Pléione, 33600 Pessac, France  Dan Levy  Department of
Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100,
Israel  Louis Daniault  Laboratoire d’Optique Appliquée, Institut
Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Axel Robbes  Laboratoire d’Optique Appliquée, Institut
Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Neil Zaïm  Laboratoire d’Optique Appliquée, Institut
Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Alessandro Flacco  Laboratoire d’Optique Appliquée,
Institut Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Eyal Kroupp  Department of Physics of Complex Systems,
Weizmann Institute of Science, Rehovot 76100, Israel  Victor Malka  Department
of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100,
Israel  Stefan Haessler  Laboratoire d’Optique Appliquée, Institut
Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France  Rodrigo Lopez-Martens  Laboratoire d’Optique Appliquée,
Institut Polytechnique de Paris, ENSTA-Paris, Ecole Polytechnique, CNRS, 91120
Palaiseau, France

###### Abstract

We report on the first simultaneous measurement of high-order harmonics,
relativistic electrons and low divergence proton beams generated from plasma
mirrors driven at kHz repetition rate by relativistic-intensity milliJoule-
energy femtosecond laser pulses. This setup enables detailed parametric
studies of the particle and radiation spatio-spectral beam properties for a
wide range of controlled interaction conditions, such as pulse duration and
plasma density scale length. This versatile setup should aid in further
understanding the collective laser absorption mechanisms at play during the
laser-plasma interaction and in optimizing the secondary beam properties for
potential applications.

††preprint: AIP/123-QED

##  I Introduction

When an intense femtosecond laser pulse is tightly focused on an optically
polished solid surface, it generates a thin layer of surface plasma with near-
solid density, which becomes highly reflective for the incident laser light,
and is therefore often called "plasma mirror" (PM). Progress in high-power
laser technology going back almost two decades, in particular the generation
of ultra-short laser pulses with high temporal contrast [1, 2, 3], has enabled
the exposure of PMs to peak electric field amplitudes that can drive
relativistic plasma electron motion, making PMs potentially interesting
sources of ultrashort energetic particle and radiation beams for applications.
For typical lasers emitting around a central wavelength $\lambda_{0}$ = 800
nm, this occurs for peak focused intensities $I_{0}>10^{18}\
\textrm{W/cm}^{2}\ $ or normalized vector potentials $a_{0}=\sqrt{I_{0}\
[\textrm{W/cm}^{2}]\ \lambda_{0}^{2}\ [\mu m^{2}]/1.37\times 10^{18}}>1$. In
this so-called relativistic regime, the highly nonlinear PM response to laser
light can lead to generation of high-order laser harmonics (HHG), relativistic
electron bunches and energetic ion beams.

HHG extends into the extreme ultra-violet (XUV) and even up to the X-ray [4]
spectral ranges and corresponds to a train of attosecond pulses in the time
domain [5, 6, 7]. Two main generation mechanisms have been identified in
simulations and experiments: coherent wake emission (CWE) [8] and relativistic
oscillating mirror (ROM) [9, 10]. In CWE, the laser field injects vacuum-
heated or so-called "Brunel" electrons [11] from the surface into the bulk of
the plasma, forming a propagating electron density peak which excites plasma
oscillations in its wake, finally leading to HHG. CWE is an ubiquitous
mechanism in all HHG experiments on PMs [8] and is efficient even at
intensities as low as $I\sim 10^{15}\ \textrm{W/cm}^{2}$. ROM only comes into
play at relativistic intensities, where the PM surface itself begins to
oscillate at relativistic velocities, leading to a periodic frequency
upshifting of the reflected light, resulting in HHG. In the relativistic
regime, it is important to take electron motion into more detailed account, as
in the more advanced relativistic electron spring [12] and coherent
synchrotron emission [13, 14, 15, 16] models. The relative contributions of
relativistic HHG (RHHG) and CWE to the total emission depends mainly on the
incident laser intensity [5, 16] and the plasma gradient scale length
$L_{\mathrm{g}}$ [17]. These two HHG families are easily distinguishable
experimentally through their different spectral widths, phase properties and
divergence.

Figure 1: Experimental set-up for simultaneous detection of HHG, electrons and
protons emitted from a relativistic PM prepared by a weak Pre-pulse and driven
by an intense Main pulse. (a) Reflected laser near-field, (b) HHG
spectrograph, (c) spatial electron beam profile, (d) parabolic ion tracks and
(e) ion TOF trace are the typically recorded images by the respective
diagnostics. (f) On-target Pre-pulse (grey scale) and Main pulse (color scale)
spatial beam profiles. M1: Holey mirror

In addition to HHG, PMs can also emit ultrashort relativistic electron
bunches. It is possible to access different acceleration regimes by
controlling the plasma conditions. For relatively long density gradients
$(L_{\mathrm{g}}>\lambda_{0})$, electrons are accelerated to multi-MeV
energies by a laser wake-field produced inside the plasma gradient itself, a
process which is only observed with few-optical-cycle driving pulses [18]. For
sharper density gradients $(L_{\mathrm{g}}\ll\lambda_{0})$, electrons ejected
out of the plasma surface co-propagate with the reflected laser field in
vacuum and subsequently gain large amounts of energy by vacuum laser
acceleration [19]. The energy and charge of the electron beam emitted in this
regime have been found to be correlated to RHHG emission [20, 21, 22] and
anti-correlated to CWE [23].

Most of the experimental ion acceleration studies from PM have been focused on
the beams emitted from the back-side of the target through the well-known
target normal sheath acceleration (TNSA) mechanism [24]. TNSA-based ion
sources show large beam divergences, varying from $10^{\circ}$ to $30^{\circ}$
depending on the driving laser and target parameters [25, 26]. Often,
additional collimation devices and special target geometries [27, 24], along
with complicated target fabrication and handling are required to keep the beam
collimated. However, in case of front-side ion acceleration, a different
acceleration mechanism driven by Brunel electrons has been found to be more
efficient than TNSA for the steepest plasma density gradients [28, 29].
Acceleration takes place during the time of interaction of the laser pulse
with the plasma and results in extremely low-divergence proton beams [30].
Such exceptional beam properties, useful for numerous potential applications
[31, 32] from a fairly simple target geometry warrants further exploration of
this particular acceleration regime.

Here, we report on the recent progress on HHG, electron and proton
acceleration from relativistic PMs driven at kHz repetition rate. At the heart
of this work is the simultaneous measurement of these three signals with a
controlled surface plasma density gradient. A simultaneous measurement enables
us to observe direct correlations between the signals whilst ensuring
identical interaction conditions on target. These multiple diagnostics are a
fine in-situ probe into the collective plasma dynamics and can be a powerful
tool to better understand the rich underlying nonlinear physics. Previously
such measurements have been limited to HHG and electrons [23, 21, 33, 22]. We
extend the previous work by additionally measuring the accelerated proton
beams, characterizing them spectrally as well as spatially. At constant
driving pulse energy and focused spatial beam distribution, we can vary the
driving pulse duration between 4 fs and 1.5 ps, thus scanning over a wide
range of intensities $I\sim 10^{16}-10^{19}\ \textrm{W/cm}^{2}$ and
transitioning from the sub-relativistic to the relativistic regime, while
observing all three emission types (HHG, electrons and protons)
simultaneously.

##  II Experimental Setup

###  II.1 Driving pulses

The experiments were carried out in the _Salle Noire_ laser facility at
Laboratoire d’Optique Appliquée (LOA), delivering waveform-controlled, up to
2.5 mJ on-target, high-temporal contrast ($>10^{10}@10\>$ps) laser pulses,
centered around $\lambda_{0}=780\>$nm, with tunable pulse duration from 27 fs
to sub-4 fs [34]. While maintaining the same on-target fluence (same energy
contained within the same focal volume), the laser pulses can also be
stretched up to 1.5 ps by adding group delay dispersion using an acousto-optic
programmable dispersion filter (Dazzler, Fastlite) integrated into the laser
chain. A schematic layout of the experimental set-up is illustrated in figure
1. Under vacuum, the p-polarised Main pulses are focused by an f/1.3
$28^{\circ}$ off-axis parabolic (OAP) mirror (effective focal length
$f=54.4\>$mm) onto a rotating, thick fused optical grade silica target at an
incidence angle of $\theta_{i}=55^{\circ}$, to a $\approx 1.8\ \mu$m FWHM
spot.

###  II.2 Plasma density gradient control

A spatially overlapped, time-delayed Pre-pulse is created by picking-off
$\approx 4\%$ of the Main pulse through a holey mirror, decreasing its size by
a factor $\approx 2$ in a Galilean telescope, and sending it through a delay
stage. It is collinearly recombined with the Main pulse through a second holey
mirror (M1 in figure 1) before being focused to a $13\ \mu m$ FWHM spot with
the same OAP. Its much larger size compared to the Main pulse leads to a
spatially homogeneous plasma expansion on the target surface. The plasma
density gradient scale length, $L_{\mathrm{g}}\approx
L_{0}+c_{\mathrm{s}}\tau$, is scanned by varying the lead time $\tau$ of the
Pre-pulse before the Main pulse. Here, $L_{0}$ is the scale length increase
due to the finite temporal contrast of the Main pulse. The plasma expansion
velocity $c_{\mathrm{s}}$ is measured with the spatial domain interferometer
(SDI) technique [35], detailed in section III.1.

The overlapped Pre-pulse (in grey scale) and Main pulse (in color-scale) beam
profiles in-focus are shown in figure 1(f). Whether at fixed lead time or
during a $\tau$-scan, the spatial overlap of these two focal spots is
extremely stable thanks to laser’s excellent pointing stability and to the use
of a retro-reflector and a low-wobble motorized translation stage in the Pre-
pulse delay line. The retro-reflector produces an elliptical polarization
state of the Pre-pulse, which is remediated by a half-wave plate that rotates
the major axis of the ellipse into the p-polarization direction.

###  II.3 Solid target and shot sequence

A rotating target positioner refreshes the target surface at kHz repetition
rate with precise control over shooting position and rotation speed,
maintaining adequate shot spacing between consecutive laser shots [36]. To
ensure identical shot-to-shot interaction conditions, the target surface is
aligned with high precision to be perpendicular to the rotation axis with the
help of a frequency-stabilized helium-neon laser in a Mach-Zehnder
interferometer [36]. The residual target fluctuations in the normal direction
are maintained $<z_{R}/5$ peak-to-valley, where $z_{R}$ is the Rayleigh range
of the driving laser. Angular fluctuations are reduced to $<80\
\mu\textrm{rad}$ peak-to-valley, limited by mechanical stress in the target’s
rotary bearing.

A slow and a high-speed mechanical shutter (Uniblitz CS45, 45 mm diameter, 14
ms opening time), block the beams in front of the OAP for longer $\sim 1$-s
durations between acquisitions and control the shot sequence length,
respectively. The target controller, made from an Arduino micro-controller
board, (i) tracks the angular and radial positions of the laser-target
interaction point and which of these positions have already been shot at, (ii)
sets the target rotation speed according to the desired shot spacing
(typically $100\ \mu$m), and (iii) triggers the shot sequence as well as the
diagnostics detailed in section II.4. For an acquisition, the target
controller triggers the slow shutter opening and sets the target rotation
speed. Once an angular position with a fresh target surface has been reached,
the fast shutter opens and 14 ms later the diagnostics are triggered. After
the desired acquisition time (typically 100 ms, thus averaging over 100
consecutive laser shots), the fast shutter closes, the target rotation slows
down and the slow shutter closes. Once a full circle of shots at a given
radius is completed, the target is translated laterally by typically $100\
\mu$m to commence a new circular shot pattern.

We exclusively use SiO2 as target material in the form of uncoated 5 inch
diameter substrates, polished to $\lambda/10$ flatness on both faces. With a
100 $\mu$m shot spacing, these targets can in principle take 1.2 million
consecutive shots per face, corresponding to 20 min of continuous operation at
1 kHz. In practice, we take $\approx 1000$ acquisitions of $\approx 100$ shots
per day of experiment, so that one target lasts approximately one month.

###  II.4 Diagnostics

This experimental set-up has been designed to simultaneously detect three
observables resulting from the laser-PM interaction, namely the emission of
HHG radiation, energetic electrons, and proton beams. This is achieved with
three dedicated diagnostics, illustrated in figure. 1b,c,d and detailed below.
These are complemented by a fourth one consisting in recording the reflected
laser beam’s near-field beam profile, as illustrated in figure 1a.

####  II.4.1 XUV spectrometer

A rectangular hole in the screen for the near-field beam profile transmits
light within the angular acceptance of the XUV spectrometer that spectrally
characterizes the HHG emission. The spectrometer consists of an aberration-
corrected, flat-field concave grating (Hitachi 001-0639, 600 groves/mm,
incidence angle $\alpha=85.3^{\circ}$, optimized for wavelenegths
$\lambda\approx 22-124\>$nm) that images in the horizontal dimension the $\sim
1\ \mu m$ size HHG source spot on the PM surface, and spectrally disperses it
onto an image plane. Beam propagation is hardly influenced in the vertical
dimension which thus provides angular resolution to the resulting
spectrograph. A single-stack micro-channel plate (MCP) coupled with a P46
phosphor screen (Photonis Scientific, APD 1 PS 97X79/32/25/8 I 40:1 P46) is
placed in the image plane. It is finally imaged using a CCD camera (12-bit PCO
Pixelfly VGA). Figure 1(a) shows a typical recorded XUV spectrograph. The MCP
is time-gated for 200 ns synchronously with the laser pulses to suppress
background signal from longer incoherent plasma emission. The P46 phosphor
screen has a decay time of $\approx 300$ ns enabling detection at kHz
repetition rate.

####  II.4.2 Electron diagnostics

The angular electron distribution is measured with a scintillating screen [37]
(Carestream Lanex fast), placed $\sim 20\>$cm from the point of interaction,
covering a small but sufficient angular range between the target normal and
the specular direction, for $30^{\circ}<\phi<50^{\circ}$, where $\phi$ is the
angle w.r.t the target normal in the plane of incidence. The front side of the
Lanex-screen is covered with a $15\ \mu m$ thick aluminium foil to block
visible light and low-energy electrons ($E_{k}<150\>$keV). Detection of low-
energy electrons is also suppressed by the substrate of the Lanex-screen. The
green light emission of the Lanex-screen is imaged by a second CCD camera
(12-bit, Pixelink PL-B957U) through an interferometric band-pass filter. The
electron energy spectrum can be measured for $\phi=38^{\circ}-40^{\circ}$ by
inserting a purpose-built electron spectrometer, comprising of a 0.5-mm
pinhole and a pair of neodymium magnets ($B_{0}\approx 80\>$mT over $\approx
20\>$mm). The magnetic field map has been measured with a Hall probe and
injected into a numerical electron trajectory calculation to calibrate the
electron energy vs. the position along the trace of deflected electrons on the
Lanex-screen.

####  II.4.3 Proton diagnostics

Accelerated protons along the target normal are spectrally characterized using
an in-house built Thomson parabola spectrometer (TPS). A $300\ \mu m$ pin-hole
placed at about 60 cm from the point of interaction selects protons emitted
along the target normal direction, which are then dispersed in energy and
charge-mass ratio by a 2 kV/cm electric and an anti-parallel 300 mT magnetic
field, created by 50 mm long electrodes and 15 mm long magnets, respectively.
Figure 1(c) shows a typical recorded TPS trace. Protons can be easily
distinguished from the heavier ions due to their unique charge-to-mass ratio.
The TPS trace is recorded with another MCP-Phosphor screen assembly imaged by
a third CCD camera (12-bit, Pixelink PL-B957U).

Alternatively, angularly resolved proton spectra can be measured with a
movable time-of-flight (TOF) detector. It consists of a 6 mm diameter MCP,
placed $\approx 37.5\ \textrm{cm}$ from the point of interaction, and mounted
on a linear translation stage moving parallel to the target surface. This TOF-
MCP has an angular acceptance of $0.9^{\circ}$, small enough to neglect the
convolution owing to the finite size of the detector. The transient MCP
current is read out using a KEYSIGHT MSOS804A 8 GHz oscilloscope capable of
individually recording all $\sim 100\>$ shots in one acquisition, when
operated in segmented memory acquisition mode. Figure 1(d) shows a typical
recorded TOF trace.

##  III Experimental Results

###  III.1 Plasma density gradient measurement

The plasma density profile on the target surface is a critical experimental
parameter, be it for HHG [38, 39, 17, 21, 22], electron acceleration [19, 21,
18, 22], or proton acceleration [30]. Its control requires a high temporal
contrast of the femtosecond driving laser, typically such that its intensity
remains $<10^{10}\>$W/cm2 until $<10\>$ps before the pulse peak. This is 2
orders of magnitude below the intensity required to start field-ionizing[40,
41] SiO2, and also excludes possible multi-photon processes. If this is the
case, a femtosecond Pre-pulse can create a preplasma with an initially quasi-
step-like density profile, with an electron density $n_{0}$ in the half-space
$x<0$, the $x$-direction being perpendicular to the target surface. The free
expansion of such a plasma can be described by a 1D isothermal (i.e. constant
electron temperature $T_{\mathrm{e}}$) model [42] yielding a time-dependent
exponential electron density profile that starts dropping at
$x=-c_{\mathrm{s}}t$ from $n_{0}$ towards vacuum like

$n(x,t)=n_{0}\exp\left[-\frac{x}{c_{\mathrm{s}}t}-1\right],$ (1)

with a gradient scale length $L_{\mathrm{g}}=c_{\mathrm{s}}t$ that linearly
increases with the so-called ion sound velocity
$c_{\mathrm{s}}=\sqrt{Zk_{\mathrm{B}}T_{\mathrm{e}}/m_{\mathrm{i}}}$, where
$Z$ is the ion charge state, $k_{\mathrm{B}}$ the Boltzmann constant, and
$m_{\mathrm{i}}$ the ion mass. This implies that, as long as the absorption
coefficient of the Pre-pulse by the target surface is intensity-independent,
so that the resulting local electron temperature $T_{\mathrm{e}}$ is
proportional to the local Pre-pulse fluence $F$, the local expansion velocity
$c_{\mathrm{s}}$ at every position is proportional to $\sqrt{F}$.

Figure 2: Schematic representation of the experimental SDI measurement. (a)
The Pre-pulse initiates the plasma expansion. (b) The probe beam, obtained by
shaping the Main pulse with a mask into a hexagonal pattern of smaller beams,
is delayed and focused onto the expanding plasma and gets reflected. At
certain Pre-pulse lead times $\tau$, the reflected near-field beam profile
recreates the pattern imposed by the mask. (c) At certain Pre-pulse lead
times, $\tau^{\prime}$, this pattern is inverted, indicating a phase shift of
$\pi$ imposed on the reflected probe beam by the expanding plasma. The images
displayed on the "screen" are actual recorded data from a typical SDI
measurement.

As a daily routine measurement, this characteristic velocity $c_{\mathrm{s}}$
of the plasma expansion initiated by the Pre-pulse is determined with the SDI
technique introduced by Bocoum et al. [35]. The principle of SDI is to
interfere light reflected off the expanded plasma with light reflected off an
unperturbed part of the target surface. The light is considered to reflect on
the critical-density surface at depth $x_{\mathrm{c}}$, where the electron
density is $n_{\mathrm{c}}\cos^{2}\theta_{i}$, with
$n_{\mathrm{c}}=\omega_{0}^{2}m_{\mathrm{e}}\epsilon_{0}/e^{2}$, the laser
carrier angular frequency $\omega_{0}$, the electron mass $m_{\mathrm{e}}$,
the vacuum permittivity $\epsilon_{0}$ and the electron charge $e$. The phase
shift due to the reflection at $x=x_{\mathrm{c}}$ rather than at the initial
target-vacuum boundary at $x=0$ is given by (as in Bragg reflection)

$\phi=2\frac{\omega_{0}}{c}x_{\mathrm{c}}\cos\theta_{i}.$ (2)

Dephasing due to propagation through the low-density tail of the plasma is at
least an order of magnitude smaller and is thus neglected. The measured phase
shift as a function of the Pre-pulse lead time $\tau$ thus tracks the position
$x_{\mathrm{c}}(\tau)$ of the critical density surface. Imposing the
exponential shape of equation 1 (where time $t$ is equivalent to $\tau$) for
the plasma density profile, this corresponds to the gradient scale length

$L_{\mathrm{g}}(\tau)=\frac{x_{\mathrm{c}}(\tau)}{\ln\left[n_{0}/(n_{\mathrm{c}}\cos^{2}\theta_{i})\right]-1}.$
(3)

Experimentally, SDI is implemented by inserting a periodic transmission mask
of period $a$ (in our case a hexagonal pattern of 3 mm holes with a period
$a=4\>$mm) into the near-field of the main beam, as shown in figure 2, thus
transforming it into the probe beam for SDI. Upon focusing by the OAP, this
turns into its far-field diffraction pattern on the target surface. In the
paraxial approximation, this is given by the probe beam’s spatial Fourier-
transform, which is again a hexagonal pattern of "probe spots" (each of the
same $\approx 1.8\ \mu$m-FWHM size as the main beam focus) with a period
$d=\lambda_{0}f/a$, where $\lambda_{0}$ is the light wavelength. For
$\lambda_{0}=800\>$nm we thus obtain $d=10\>\mu$m. This means that the central
(zeroth diffraction order) probe spot overlaps with the center of the Pre-
pulse and thus with the nearly homogeneously expanding plasma with which the
Main pulse would interact, while the surrounding hexagon of (first
diffraction-order) probe spots overlaps with the wings of the 13-$\mu$m Pre-
pulse spot, where the Pre-pulse fluence $F_{1}$ has dropped to $\approx 10$%
of its peak value $F_{0}$.

The expanding plasma thus induces a phase shift $\Delta\phi=\phi_{0}-\phi_{1}$
between the central (zeroth order) and the surrounding hexagon of (first
order) probe spots. In the simplest case, the first order spots probe an
unaffected target surface, so that $\phi_{1}=0$. In our case however, the
first-order spots reflect off an expanding plasma "launched" by a finite local
Pre-pulse fluence of $F_{1}\approx 0.1F_{0}$, with the fluence $F_{0}$ at the
Pre-pulse center. Two assumptions are now required: _(i)_ Relying on the 1D
isothermal expansion model mentioned above, we can write how much smaller the
local expansion velocity at the location of the first order probe spots,
$c_{\mathrm{s}}^{1}$, will be compared to its value, $c_{\mathrm{s}}$, at the
center: $c_{\mathrm{s}}^{1}=\sqrt{F_{1}/F_{0}}c_{\mathrm{s}}$. _(ii)_ We set
the maximum plasma density $n_{0}$ to be the same everywhere. By combining
equations 2 and 3, we deduce that the gradient scale length at the Pre-pulse
center, $L_{\mathrm{g}}$, is proportional to the measurable phase shift
$\Delta\phi$:

$L_{\mathrm{g}}(\tau)=\alpha\,\Delta\phi(\tau),\\\
\mathrm{with}\>\alpha=\frac{\lambda_{0}/(4\pi\cos\theta_{i})}{\left\\{\ln\left[n_{0}/(n_{\mathrm{c}}\cos^{2}\theta_{i})\right]-1\right\\}\left(1-\sqrt{F_{1}/F_{0}}\right)}.$
(4)

The measurement of $\Delta\phi$ is made by observing, through an
interferometric band-pass filter around $\lambda_{0}=800\>$nm, the
modification of the probe beam’s near-field profile (cf. Fig.1a). As
demonstrated in ref. [35] and illustrated in figure 2, the pattern reverses
every time the phase shift $\Delta\phi$ reaches an integer multiple of $\pi$.
The pattern reversal is easily extracted from the recorded near-field images
by tracing the "SDI contrast" defined as
$C=[S_{\mathrm{b}}(\tau)-S_{\mathrm{d}}(\tau)]/[S_{\mathrm{b}}(\tau)+S_{\mathrm{d}}(\tau)]$,
where $S_{\mathrm{b}}$ and $S_{\mathrm{d}}$ are the integrated signals over
selected image areas that are initially (at $\tau=0$) bright and dark,
respectively. As visible in the examples shown in the left column of figure 3,
the SDI contrast oscillates and its extrema mark the times, $\tau$, when the
dephasing, $\Delta\phi$, has increased by $\pi$. The resulting dephasing
curve, $\Delta\phi(\tau)$, is found to be linear (at least for the first
$\approx 20\>$ps or so), which through equation 4 corresponds to a linearly
increasing $L_{\mathrm{g}}=c_{\mathrm{s}}\tau$, as expected from the 1D
isothermal expansion model. Its slope is easily found by a fit on the linear
range of data points and yields the sought-after expansion velocity,
$c_{\mathrm{s}}$, that is then used to calibrate the gradient scale length in
experiments.

Although it only contributes logarithmically in equation 4, the choice of the
$n_{0}$-value is a significant source of uncertainty. Barrier-suppression
intensities for field ionization [40, 41] predict ion charge states for the
Pre-pulse and SDI probe pulse intensities in our experiments, which correspond
to $n_{0}\approx 100n_{\mathrm{c}}$ only. Field ionization may, however, not
be the dominant ionization mechanism. Indeed, once the target is ionized, the
laser only interacts inside the plasma skin depth, which is much shorter than
a wavelength for a strongly overdense plasma. Then, ionization proceeds
through collisional processes in the plasma bulk, possibly further increasing
the ion charge states. Estimating the plasma ionization state (which may very
well be inhomogeneous and time-dependent) is thus not straightforward. The
$c_{\mathrm{s}}$-values shown in figure 2 are all obtained with
$n_{0}=300n_{\mathrm{c}}$, corresponding to fully ionized SiO2.

Figure 3: Experimental SDI contrast curves (left, circles are raw data, lines
are smoothed data) and extracted dephasing curves curve $\Delta\phi(\tau)$
(right, circles are extracted data, lines are linear fits including points
with $\tau<20\,$ps) for SDI scans obtained for driving laser pulse durations
of 27 fs (a), 8 fs (b) and 4 fs (c).

Figure 3 shows a series of SDI measurements that answers the question whether
the plasma expansion initiated by the Pre-pulse varies with laser pulse
duration. Since neither the energy nor the spatial profile of the Pre-pulse
vary with pulse duration, its _fluence_ remains the same ($\approx 50\>$
J/cm2) and only its _intensity_ varies. The question thus comes down to
whether the Pre-pulse absorption coefficient is intensity-dependent, which
would lead to varying $T_{\mathrm{e}}$ and thus varying $c_{\mathrm{s}}$. The
telescope and wave plate traversed by the Pre-pulse add a group delay and a
third-order dispersion of $\approx 260\>$fs2 and $\approx 200\>$fs3,
respectively. As a consequence, the 27 fs pulse is stretched to 38 fs with a
peak intensity of $1.1\times 10^{15}\>$W/cm2, the 8 fs pulse is stretched to
$\approx 100\>$fs with a peak intensity of $0.5\times 10^{15}\>$W/cm2, and the
4 fs pulse is stretched to $\approx 200\>$fs, with a profile dominated by a
$\approx 40\>$fs spike with a peak intensity of $0.7\times 10^{15}\>$W/cm2. As
visible in figure 3, the three cases lead to very similar expansion
velocities, with a variation clearly below the uncertainty of the SDI method
itself. We conclude that the $c_{\mathrm{s}}$-value is independent of the Pre-
pulse duration in the range covered by our experiments. This intensity
independence corroborates the proportionality $c_{\mathrm{s}}\propto\sqrt{F}$
used in the SDI-analysis.

The slope of the $\Delta\phi(\tau)$-curves decreases for longer delays,
$\tau>20\>$ps, indicating a decreasing velocity, $c_{\mathrm{s}}$, and thus a
cooling of the plasma electrons through transfer of energy to the ions. The
$L_{\mathrm{g}}$-values relevant to (most of) our PM experiments are however
reached before these effect set in, such that the linear evolution,
$L_{\mathrm{g}}=c_{\mathrm{s}}\tau+L_{0}$, remains valid.

###  III.2 Short and intermediate plasma density gradients

Figure 4: Simultaneously measured angularly integrated HHG spectrum (top),
electron spectrum measured for $\phi=38^{\circ}-40^{\circ}$ (middle), and
proton spectrum at $\phi=0^{\circ}$(bottom) as a function of the plasma
density gradient scale length, for different driving pulse durations (a) 29 fs
$(a_{0}=1.0)$, and (b) 4 fs $(a_{0}=2.1)$. The dotted red line marks the
photon cut-off energy for CWE-HHG.

All data reported here are acquired and integrated over 100 ms long bursts of
laser pulses at 1 kHz repetition rate. Figure 4 shows the simultaneously
measured HHG, electron and proton spectra for a range of short to intermediate
plasma density gradients, with relativistically intense main driving pulses of
(a) 29 fs ($a_{0}=1.0\ \textrm{for}\ \lambda_{0}=800\ \textrm{nm}$) and (b) 4
fs ($a_{0}=2.1\ \textrm{for }\lambda_{0}=780\ \textrm{nm}$) duration. The
gradient scale length is calculated, up to the small offset $L_{0}$, as
$L_{\mathrm{g}}-L_{0}=c_{\mathrm{s}}\tau$ (top axis) from the plasma expansion
velocity $c_{\mathrm{s}}=18\ \textrm{nm/ps}$, measured as described in section
III.1, and the Pre-pulse lead time $\tau$ (bottom axis). Each of the scans in
figure 4 have been individually normalized to its maximum value and hence the
spectral intensity cannot be directly compared. The HHG spectrum (figure 4 top
frame) is angularly integrated over the full detected range of $[-35\
\textrm{mrad},+35\ \textrm{mrad}]$. The spectral response of the MCP has not
been taken into account so as to enhance the visibility of higher harmonic
orders. We can identify the two regimes for HHG: CWE-HHG at steep gradients
$(L_{\mathrm{g}}-L_{0}<0.03\lambda_{0})$ and RHHG at more gentle gradients of
$L_{\mathrm{g}}-L_{0}\approx 0.03\lambda_{0}$– $0.2\lambda_{0}$ [21, 22]. In
the CWE regime, the intensity-dependence of the Brunel electron trajectories
leads to a temporal aperiodicity of the emitted attosecond pulse train and,
hence, to broadened harmonics in the spectral domain [8, 21]. In contrast, the
RHHG emission in the moderately relativistic regime ($a_{0}\sim 1$) is
Fourier-limited [43], corresponding to narrower spectral widths of the
harmonics as compared to the CWE-HHG, as observed in figure 4(a) (top frame)
with 29 fs driving pulse duration. However, this transition between the CWE
and RHHG regimes becomes less distinct with shorter driving pulse durations
($a_{0}>1$) since the harmonics emerge in the spectrograph from the
interference between fewer attosecond pulses. In figure 4(b), for a 4 fs
driving pulse, around an optimal $L_{\mathrm{g}}-L_{0}\approx
0.08\lambda_{0}$, the HHG spectrum extends well beyond the CWE cutoff of 30 eV
for an SiO2 target, which is definitive proof of RHHG.

The electron energy spectrum (figure 4, middle frame) is simultaneously
measured at $\phi=38^{\circ}$– $40^{\circ}$. These positions do not sample the
center of the electron beam, which is located a few degrees closer towards the
specular direction (see Fig. 1c), but they allow simultaneous detection of the
HHG emission. At the steepest gradients, where the CWE-HHG is optimum, low
electron energy and charge is measured for both driving pulse durations of 29
fs and 4 fs. This anti-correlation between the electron beam energy and charge
and CWE-HHG has also been previously observed at sub-relativistic intensities
with 30 fs driving laser pulses [23]. We measure electrons up to 1.5 MeV in
energy in the relativistic regime for $a_{0}>1$. Higher energies are expected
in the center of the electron beam. The apparent dip in the electron spectra
at the optimal gradient for RHHG is suspected to be linked to a moving
electron beam.

Figure 5: Proton energy spectrum measured with TPS at the steepest density
gradients for different driving pulse durations (a). Angular proton energy
spectrum obtained for (b) 4 fs, (c) 7 fs, and (d) 29 fs and (e) 200 fs driving
pulses measured by the moving TOF detector. $\phi=0^{\circ}$ is the target
normal direction.

Figure 4 (bottom frame) shows the proton energy spectrum simultaneously
measured with the TPS along the target normal direction ($\phi=0^{\circ}$). We
observe that for both driving pulse durations, the proton energies benefit
from the steepest plasma density gradients and are optimised for the same
gradients as CWE-HHG, suggesting that Brunel electrons play an important role
in the acceleration mechanism [29, 30]. If an optimal plasma gradient exists
for efficient acceleration of protons, it is below our resolution limit of
$L_{0}\approx 0.01\lambda_{0}$. The highest proton energies of 0.25 MeV are
obtained with 29 fs driving pulses for the steepest density gradients and drop
below the detection threshold of 0.1 MeV for $L_{\mathrm{g}}>0.1\lambda_{0}$.
The cut-off energy decreases to $\approx 0.15\,\textrm{MeV}$ for shorter 4 fs
driving pulses.

Figure 5(a) shows the proton energy spectrum measured with the TPS for the
steepest plasma density gradient and longer driving pulses obtained by adding
positive group delay dispersion. We observe an optimum between 100 fs and 300
fs, yielding $\approx 0.45\ \textrm{MeV}$ protons. In this acceleration
regime, protons are accelerated only during the the interaction time of the
pulse, hence they benefit from a longer driving pulse duration, despite the
corresponding drop in laser intensity [29, 30]. Figure 5(b)-(e) shows the
angle-resolved proton energy spectrum measured with the moving TOF detector
for four different driving pulse durations, 4 fs (b), 7 fs (c), 29 fs (d) and
chirped 200 fs (e), at the steepest plasma gradient. In all cases, we observe
a very low $\leq 4^{\circ}$ FWHM divergence of the proton beam. This is
significantly lower than the $16^{\circ}$ FWHM reported earlier [28], which
may be the result of a higher temporal contrast ratio of the driving laser in
our case.

###  III.3 Long density gradients regime

Figure 6: Electron spectrum measured at $\phi=40^{\circ}$ for (a) 7 fs and (b)
4 fs driving laser pulse duration and varying plasma density scale length.

For long density gradients $(L_{\mathrm{g}}\geq\lambda_{0})$, RHHG disappears
due to the onset of chaotic electron dynamics in the near-critical-density
layer [21] of the plasma density gradient. However, before getting reflected,
the laser propagates through the sub-critical tail of the plasma density
profile. In the zone of density $\sim n_{\mathrm{c}}/10$, few-cycle laser
pulses $<10\ \textrm{fs}$ are able to resonantly excite a strong plasma
wakefield. Electrons can be injected into this laser wakefield by ionization
and then get accelerated to relativistic energies. The plasma wavefront is
rotated by the density gradient and electrons are pushed away from the
specular direction [18]. Figure 6 shows the electron spectra measured in this
regime for gradient scale lengths up to $\approx 10\lambda_{0}$, reaching up
to 2.6 MeV. Note that the gradient scale lengths given by the top axis result
from the same 18 nm/ps expansion velocity, as determined for the initial
expansion phase with Pre-pulse lead times $<20\>$ps. As discussed in
connection with figure 3, we know that the expansion slows down considerably
at longer times, so that the true gradient scale lengths are certainly shorter
than those indicated in Figure 6.

##  IV Summary

We have reported the first simultaneous measurements of HHG, electron beams
and proton beams from a laser-driven PM. The kHz laser repetition rate allowed
exploring a wide range of laser and plasma parameters. We have presented
spectral measurements of HHG, electrons and protons as a function of
controlled plasma density scale lengths ranging from
$0.01\lambda_{0}-0.2\lambda_{0}$ for three different driving pulse durations:
29 fs $(a_{0}=1.0)$, 7 fs $(a_{0}=2.0)$, and 4 fs $(a_{0}=2.1)$. Such
simultaneous measurements vastly expand the level of characterization of
relativistic PMs. While ensuring identical interaction conditions on target,
we observe direct correlations between the CWE-HHG and Brunel-dominated
accelerated proton beams, and the anti-correlated emission of relativistic
electron beams. We also observe direct correlation between the RHHG and the
emitted electron beams. At extremely long density gradients, only with few-
optical-cycle driving laser pulses, we measure up to 2.5 MeV electrons
accelerated by the laser wake-field formed in the gradient.

We also measured proton spectra for a wide range of driving pulse durations,
from 4 fs up to 1.5 ps, at the steepest plasma density gradient, and find an
optimum pulse duration range for proton yield, between 100 fs and 400 fs,
under such conditions. These novel measurements of low-divergence
$(<4^{\circ}\ \textrm{FWHM})$ font-side-accelerated proton beams with a
controlled density gradient hold intrinsic value of their own. The exceptional
beam properties, along with the possibility of scaling with the driving laser
energy to multi-MeV proton energy levels [30], are promising for applications
in the field of medicine, for example to produce radio-isotopes [32].

To conclude, this work is a decisive step towards a highly detailed
characterization of fundamental PM dynamics and the optimal generation of
energetic particle and radiation beams exploiting the high repetition rate of
our laser system.

Additionally, we have also implemented carrier-envelope phase control, with
feedback at the full repetition rate of the laser system. This should enable
the generation of isolated intense attosecond pulses [44] with high conversion
efficiencies [45], by controlling the collective plasma dynamics at a the sub-
light-cycle level.

###### Acknowledgements.

This work has been supported by the Agence Nationale pour la Recherche
(ANR-14-CE32-0011-03 APERO), the European Research Council (ERC ExCoMet
694596) and the European Union’s Horizon 2020 research and innovation program
(LASERLAB-Europe under grant agreements 654148 and 871124).

## Data Availability Statement

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

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