# Automated control and optimisation of laser driven ion acceleration

B. Loughran  School of Mathematics and Physics, Queen’s University Belfast,
BT7 1NN, Belfast UK  M. J. V. Streeter  School of Mathematics and Physics,
Queen’s University Belfast, BT7 1NN, Belfast UK  H. Ahmed  Central Laser
Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, UK  S. Astbury
Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX,
UK  M. Balcazar  Gérard Mourou Center for Ultrafast Optical Science,
University of Michigan, Ann Arbor, MI 48109-2099, USA  M. Borghesi  School of
Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Belfast UK  N.
Bourgeois  Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot
OX11 0QX, UK  C. B. Curry  SLAC National Accelerator Laboratory, 2575 Sand
Hill Road, Menlo Park, California, USA  Department of Electrical and Computer
Engineering, University of Alberta, Edmonton, AB, T6G1H9, Canada  S. J. D.
Dann  Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11
0QX, UK  S. DiIorio  Gérard Mourou Center for Ultrafast Optical Science,
University of Michigan, Ann Arbor, MI 48109-2099, USA  N. P. Dover  The John
Adams Institute for Accelerator Science, Imperial College London, London, SW7
2AZ, UK  T. Dzelzanis  Central Laser Facility, STFC Rutherford Appleton
Laboratory, Didcot OX11 0QX, UK  O. C. Ettlinger  The John Adams Institute for
Accelerator Science, Imperial College London, London, SW7 2AZ, UK  M. Gauthier
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park,
California, USA  L. Giuffrida  ELI Beamlines Centre, Institute of Physics,
Czech Academy of Sciences, Za Radnicí 835, 252 41 Dolní Br̆ez̆any, Czech
Republic  G. D. Glenn  SLAC National Accelerator Laboratory, 2575 Sand Hill
Road, Menlo Park, California, USA  Department of Applied Physics, Stanford
University, Stanford, California 94305, USA  S. H. Glenzer  SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California, USA  J.
S. Green  Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot
OX11 0QX, UK  R. J. Gray  Department of Physics, SUPA, University of
Strathclyde, Glasgow G4 0NG, UK  G. S. Hicks  The John Adams Institute for
Accelerator Science, Imperial College London, London, SW7 2AZ, UK  C. Hyland
School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN,
Belfast UK  V. Istokskaia  ELI Beamlines Centre, Institute of Physics, Czech
Academy of Sciences, Za Radnicí 835, 252 41 Dolní Br̆ez̆any, Czech Republic
Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic  M. King  Department of Physics,
SUPA, University of Strathclyde, Glasgow G4 0NG, UK  D. Margarone  School of
Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Belfast UK  ELI
Beamlines Centre, Institute of Physics, Czech Academy of Sciences, Za Radnicí
835, 252 41 Dolní Br̆ez̆any, Czech Republic  O. McCusker  School of
Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Belfast UK  P.
McKenna  Department of Physics, SUPA, University of Strathclyde, Glasgow G4
0NG, UK  Z. Najmudin  The John Adams Institute for Accelerator Science,
Imperial College London, London, SW7 2AZ, UK  C. Parisuaña  SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California, USA
Department of Mechanical Engineering, Stanford University, Stanford,
California 94305, USA  P. Parsons  School of Mathematics and Physics, Queen’s
University Belfast, BT7 1NN, Belfast UK  C. Spindloe  Central Laser Facility,
STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, UK  D. R. Symes  Central
Laser Facility, STFC Rutherford Appleton Laboratory, Didcot OX11 0QX, UK  A.
G. R. Thomas  Gérard Mourou Center for Ultrafast Optical Science, University
of Michigan, Ann Arbor, MI 48109-2099, USA  F. Treffert  SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California, USA
Institut für Kernphysik, Technische Universität Darmstadt, Karolinenplatz 5,
64289 Darmstadt, Germany  N. Xu  The John Adams Institute for Accelerator
Science, Imperial College London, London, SW7 2AZ, UK  C. A. J. Palmer  School
of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, Belfast UK
<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

The interaction of relativistically intense lasers with opaque targets
represents a highly non-linear, multi-dimensional parameter space. This limits
the utility of sequential 1D scanning of experimental parameters for the
optimisation of secondary radiation, although to-date this has been the
accepted methodology due to low data acquisition rates. High repetition-rate
(HRR) lasers augmented by machine learning present a valuable opportunity for
efficient source optimisation. Here, an automated, HRR-compatible system
produced high fidelity parameter scans, revealing the influence of laser
intensity on target pre-heating and proton generation. A closed-loop Bayesian
optimisation of maximum proton energy, through control of the laser wavefront
and target position, produced proton beams with equivalent maximum energy to
manually-optimized laser pulses but using only 60% of the laser energy. This
demonstration of automated optimisation of laser-driven proton beams is a
crucial step towards deeper physical insight and the construction of future
radiation sources.

proton generation, high repetition rate laser-target interaction, laser-driven
particle acceleration, Bayesian optimisation

##  I Introduction

Laser-target interactions have been demonstrated to provide a highly versatile
source of secondary radiation, of interest for many applications Brenner  _et
al._ (2015); Courtois  _et al._ (2013) as well as the study of fundamental
science (Jaroszynski  _et al._ , 2006; Galy  _et al._ , 2009; Zylstra  _et
al._ , 2016). Specifically, laser-driven ion accelerators Daido  _et al._
(2012); Macchi  _et al._ (2013) have desirable characteristics pertaining to
applications in medicine Bulanov  _et al._ (2002), material science Passoni
_et al._ (2019), nuclear fusion Roth  _et al._ (2001) and imaging Borghesi
_et al._ (2002, 2001); Mackinnon  _et al._ (2004). The need for stable,
reproducible beams that can be tuned presents a necessary, yet challenging,
goal towards the realisation of many of these applications Daido  _et al._
(2012).

The most extensively researched mechanism driving laser-driven proton
acceleration is sheath acceleration (often termed Target Normal Sheath
Acceleration, or TNSA)Snavely  _et al._ (2000). Here, a laser pulse is focused
to a relativistic intensity on the surface of a solid foil, ionising the
material and heating electrons to MeV temperatures. As the accelerated
electrons escape the rear target surface, they build an electrostatic sheath
field which can reach $\gtrsim 1\,$\mathrm{T}\mathrm{V}\mathrm{/}\mathrm{m}$$
and accelerate ions to $\gg 1\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ energies over
just a few microns Roth and Schollmeier (2016). Although target materials
vary, the accelerated ions are typically dominated by protons from hydrocarbon
surface contaminants which are preferentially accelerated due to their high
charge-mass ratio Allen  _et al._ (2004).

Due to their dependence on the rear surface electrostatic sheath field, the
specific characteristics of these MeV proton beams are strongly influenced by
the laser-electron energy coupling, electron transport through the bulk of the
target, and disruption to the target rear surface. Experiments have
demonstrated the dependence of the electron and proton beam on various
experimental control parameters (e.g. laser intensity and contrast or target
thickness). Of particular importance is the plasma scale length at the front
surface which affects the laser-electron coupling mechanisms Chopineau  _et
al._ (2019); McKenna  _et al._ (2008); Gray  _et al._ (2014). In preplasmas
with long scale lengths ($>100\,$\mathrm{\SIUnitSymbolMicro m}$$) the laser
beam has been observed to filament, subsequently reducing the coupling
efficiency, while for optimal scale lengths the beam can undergo relativistic
self-focussing effects that enhance laser energy coupling (McKenna  _et al._ ,
2008; Gray  _et al._ , 2014; Esirkepov  _et al._ , 2014). The amplified
stimulated emission (ASE) pedestal and prepulses, common to short pulse
lasers, can pre-heat the target and lead to significant plasma expansion
before the main pulse arrives. For a given ASE pedestal duration, an optimal
thickness exists at which this enhanced coupling and electron recirculation
Mackinnon  _et al._ (2002) will be advantageous for the acceleration process,
while for thinner targets the inward travelling shock-wave launched by the
rapid surface pre-heating can disrupt the accelerating sheath field at the
rear surface Kaluza  _et al._ (2004); Lindau  _et al._ (2005); Lundh  _et al._
(2007). Radiative heating from x-rays, generated in the focus of a prepulse
incident on the target front-side, can similarly induce rear-surface expansion
of thin targets, impacting TNSA for interaction parameters in which the ASE
induced shock may not reach the rear surface during the acceleration window
(Kaluza  _et al._ , 2004).

While broad trends within TNSA proton beams have been established, comparison
of experimental results from different experiments highlights variation in
measured beam parameters and is indicative of the nuanced relationship between
laser parameters and proton beam characteristics. For example, while maximum
proton energy has been observed to increase with laser intensity, the scaling
follows a ${\sim}I^{1/2}$ dependence for long ($>300\leavevmode\nobreak\
\,$\mathrm{f}\mathrm{s}$$) duration laser pulses and a linear dependence
${\sim}I$ for ultra-short ($40-150\,$\mathrm{f}\mathrm{s}$$) pulses Fuchs  _et
al._ (2005); Borghesi  _et al._ (2008); Macchi  _et al._ (2013). A number of
numerical and experimental studies have explored the impact of laser pulse
duration on maximum proton energy due to TNSA in interactions with moderate
laser contrast. These indicate an optimal pulse duration for proton
acceleration associated with a fixed laser intensity or laser energy Oishi
_et al._ (2005); Schreiber  _et al._ (2006); Mora (2003); Carrié  _et al._
(2009) (e.g. optimal duration between $100-300\,$\mathrm{f}\mathrm{s}$$ for
laser pulse energies of $1\text{\,}\mathrm{J}$). The dependence is attributed
to differences in laser energy to electron coupling efficiency and
acceleration time relative to the rear surface expansion timescale. For higher
laser contrast, a similar trend is observed for targets with thicknesses of
tens of microns Flacco  _et al._ (2010). More advanced temporal pulse shaping
(e.g. shaping of the rising and falling edge of the pulse) has been explored
recently for the interaction of high-contrast lasers with ultra-thin targets
which indicate significant enhancement of proton maximum energies over those
observed for best laser compression Tayyab  _et al._ (2018); Ziegler  _et al._
(2021).

With the proliferation of multi-Hz high power laser pulses Danson  _et al._
(2019), and the development of HRR-compatible solid-density targetry Xu  _et
al._ (2022); Treffert  _et al._ (2022); Kim  _et al._ (2016); Kraft  _et al._
(2018); Schumacher  _et al._ (2017); Fraga  _et al._ (2012); Lemos  _et al._
(2009), it is now possible to quickly obtain large datasets from laser-driven
ion acceleration experiments. This opens the possibility to perform extensive
multi-dimensional parameter scans to elucidate the interdependence of
different experimental control parameters, as well as to apply machine
learning techniques to optimise ion beam properties - within complex multi-
dimensional parameter spaces - in automated experiments Noaman-Ul-Haq  _et
al._ (2017); Shalloo  _et al._ (2020); Jalas  _et al._ (2021a) and simulations
Dolier  _et al._ (2022); Djordjević  _et al._ (2021).

Here, we describe the first experimental demonstration of real-time Bayesian
optimisation of a laser-driven ion source, using a closed-loop algorithm. The
fully automated control system operated the laser, analysed the diagnostic
results and made changes to the experiment control parameters. This enabled
rapid and efficient optimisation of the accelerator performance through
simultaneous tuning of up to six different input parameters, producing proton
beams with equivalent peak energy using $57\text{\,}\%$ of the laser energy of
the manually-optimised interaction.

##  II Experimental Setup

The experiment (see figure 1 for setup) was performed at the Gemini TA2
facility, using a Ti:Sa laser which forms part of the Central Laser Facility
at the Rutherford Appleton Laboratory. The laser pulses contained up to
$500\text{\,}\mathrm{m}\mathrm{J}$ in a transform-limited pulse duration FWHM
of $\sim 40\,$\mathrm{f}\mathrm{s}$$, with a central wavelength of
$800\text{\,}\mathrm{n}\mathrm{m}$ and a FWHM bandwidth of
$30\text{\,}\mathrm{n}\mathrm{m}$. The laser was focused to a high intensity
($I_{L}>10^{19}\,$\mathrm{W}\mathrm{c}\mathrm{m}^{2}$$) using an $f/2.5$ off-
axis parabolic mirror and interacted with the target at an angle of incidence
of 30∘ with p-polarisation. The target was Kapton tape of
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$, spooled continuously during shots
using a motorised tape drive Xu  _et al._ (2022).

Figure 1: Illustration of the experimental setup, showing the orientation of
the laser-plasma interaction and the main diagnostics. The laser was focused,
with an f/2.5 90∘ diamond-turned OAP, to
$1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$ radius focal spot containing a
median of $35\pm 3\%$ of pulse energy. The plane of the laser-plasma
interaction was monitored by imaging self-emission at
$800\text{\,}\mathrm{n}\mathrm{m}$ at $60^{\circ}$ to the laser propagation
axis.

The interaction was diagnosed using a suite of particle diagnostics including
a scintillator (EJ-440), positioned along the rear surface target normal, to
measure the proton spatial profile, two point measurements of the proton
energy spectrum using a time-of-flight (TOF) diamond detector Margarone  _et
al._ (2011) and fibre-coupled Thomson parabola spectrometer (at 3∘ to the
target normal and along the target normal axis respectively), and a 0.15 T
permanent magnet electron spectrometer in the laser-forward direction. The
near-field of the specularly reflected laser light was also measured at the
first and second harmonic of the drive laser.

The laser spectral phase was controlled by an acousto-optic programmable
dispersive filter (DAZZLER) and measured using a small central sample of the
compressed pulse and a SPIDER diagnostic. The laser parameters (6 wavefront
aberrations generated through Zernike polynomials, 2nd-, 3rd- and 4th-order
temporal phase, energy and polarisation) and target position relative to the
laser focus were controlled using a fully automated control and acquisition
system. This enabled data scans consisting of bursts of shots (up to 20) at
fixed input values in parameter space. Following a burst of shots, the control
code performed analysis of the measured data from the online diagnostics and
adjusted laser or target parameters for the next burst. Although the controls
enabled adjustment of the laser temporal pulse shape, the data presented here
corresponds to pulse shapes close to best compression ($\sim
40\,$\mathrm{f}\mathrm{s}$$) with variations due to the day-to-day variation
in laser tuning.

For high-energy, multi-Hz laser facilities, prolonged high-repetition rate
operation can affect the laser pulse parameters leading to degradation in peak
intensity Li  _et al._ (2017). To ensure that our setup was not subject to
these effects, measurements of laser parameters were made over periods of
extended $1\text{\,}\mathrm{H}\mathrm{z}$ operation. These measurements
concluded that the effect of prolonged HRR operation on the quality of the
temporal pulse shape was negligible, with the standard deviation of the random
fluctuation in pulse FWHM measured as $\sim\leavevmode\nobreak\
3\,$\mathrm{f}\mathrm{s}$$ in over 1400 shots.

##  III Automated grid scans

Figure 2: a) Proton and b) electron energy spectra from the rear side of the
target during an automated target position scan ($z_{T}$) with a
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Kapton tape and an on-target laser
energy of $(438\pm 32)\,$\mathrm{m}\mathrm{J}$$. c) Average proton spectra
(and standard deviation) for different $z_{T}$ positions as indicated in the
legend. The proton spectra are recorded by the time-of-flight diamond
detector. Each column of the waterfall plots is the average of the ten shots
from each burst. The scan is comprised of 31 bursts at different target
positions spaced at $7.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ intervals
along the laser propagation axis. Negative values of $z_{T}$ are when the
target plane is closer to the incoming laser pulse and $z_{T}=0$ is the target
at the best focus of the laser pulse. The red data points, connected with a
guide line, indicate the burst-averaged 95th percentile energy as well as the
standard deviation on this value across the burst.

With the automated setup, parameter scans can be readily obtained by following
a pre-programmed procedure. In doing this, the control algorithm moved through
a series of equally spaced locations, taking a number of repeat shots at each
configuration to quantify shot-to-shot fluctuations. Figure 2 shows proton and
electron spectra for a 1D scan of target position through the laser focus with
a $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Kapton tape and a pulse length of
$\tau_{\mathrm{FWHM}}=49\pm 3\,$\mathrm{f}\mathrm{s}$$. The burst averaged
95th percentile proton energy (hereafter referred to as the maximum energy)
and average electron energies are overlaid in red with the standard deviation
for each burst indicated by the error bars. The proton and electron spectra
are seen to extend to higher energies as the target position approaches the
laser best focus as would be expected due to the increasing laser intensity at
the target surface. While the electron spectra peak around the best focus,
where the laser intensity is highest, a characteristic dip in the maximum
proton energy and flux is observed. Around best focus
($\left|z_{T}\right|<25\,$\mathrm{\SIUnitSymbolMicro m}$$), a comparatively
small number of protons are still observed at high ($\approx
3.5\,$\mathrm{M}\mathrm{e}\mathrm{V}$$) energy, with the spectrum dominated by
lower (sub-MeV) energies, as seen in figure 2c. A second signal is also seen
in the TOF spectrum, appearing as a band peaking at $\approx
0.5\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ per nucleon in figure 2a. This is most
likely due to heavy ions that were accelerated to lower velocities due to
their lower charge to mass ratio. Together with the spectrally peaked proton
spectrum, this appears similar to observations of ‘buffered’ proton
acceleration for higher intensities and thinner targets Dover  _et al._
(2016), suggesting that the sheath field is dynamic during the acceleration
process.

An increase in the laser focal spot size has previously been observed to
increase the number of accelerated particles, although with a reduced maximum
energy Green  _et al._ (2010). While this is consistent with our measurements,
the strong suppression of proton flux at the highest intensity may indicate
that the acceleration process is further compromised at the highest laser
intensities by the contrast levels of our laser, with the prepulses and
amplified spontaneous emission (ASE) causing adverse pre-heating of the
target. Similar disruption has previously been attributed to rear surface
deformation by ASE-driven shock break-out, which can effectively steer a high
energy component of the proton beam emission towards the laser axis Lindau
_et al._ (2005); Lundh  _et al._ (2007), modifying the spectrum measured at a
single angular position, or the presence of a long scale length plasma on the
rear-surface which has been shown to suppress the production of ions through
TNSA in experiments and simulations Mackinnon  _et al._ (2001); Fuchs  _et
al._ (2007); Higginson  _et al._ (2021).

A laser pulse contrast measurement (using an Amplitude Sequoia) showed an ASE
intensity contrast of better than $10^{-9}$ up to
$t=-20\,$\mathrm{p}\mathrm{s}$$, after which the laser-intensity in the
coherent pedestal Hooker  _et al._ (2011) increased exponentially. Individual
prepulses with a relative intensity of $10^{-6}$ were also observed between
$t=-50\,$\mathrm{p}\mathrm{s}$$ and $t=-65\,$\mathrm{p}\mathrm{s}$$. The
measured contrast, starting at $t=-150\,$\mathrm{p}\mathrm{s}$$, was used to
perform 2D cylindrical hydrodynamic modelling of the target evolution ahead of
the arrival of the peak intensity. The modelling was performed using the FLASH
code (v4.6.2). The ASE and coherent pedestal from
$t=$-150\text{\,}\mathrm{p}\mathrm{s}$$ to
$t=$-1\text{\,}\mathrm{p}\mathrm{s}$$ was coupled to the target electrons
using ray-tracing with inverse bremsstrahlung heating, and Lee-More
conductivity and heat exchange models were used. This indicated the formation
of an approximately $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ exponential
scale-length pre-plasma. For the target thicknesses used in the experiment
($\gg$ $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$), the measured prepulse was
not large enough for the generation of a shock moving quickly enough to
perturb the density step of the target rear surface. This matches previous
results Kaluza  _et al._ (2004); Lundh  _et al._ (2007) at similar interaction
conditions, which indicate that the ablation launched density shock does not
have time to affect the rear surface during the acceleration process. This dip
in signal at highest intensities is a surprising result for targets with tens
of microns thickness. It is more commonly observed for experiments using
ultra-thin targets where it is attributed to ASE shock-breakout Morrison  _et
al._ (2018). For the interaction presented here, the rear surface may be
affected by poor long-timescale contrast (before the start of the measurement
window at $t=-150\,$\mathrm{p}\mathrm{s}$$), or through x-ray heating of the
target bulk Kaluza  _et al._ (2004). Determination of the specific processes
driving the disruption of proton acceleration for this interaction requires
additional measurements of long-timescale contrast and pre-plasma scale
length, and is beyond the scope of this optimisation demonstration.

Figure 3: One dimensional scans of a) & c) target z-position $z_{T}$ and b) &
d) astigmatism $Z_{2}^{-2}$ for $12\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thickness Kapton tape and a pre-plasma laser energy of $(453\pm
40)\,$\mathrm{m}\mathrm{J}$$. The electron and proton flux are plotted in a) &
b), and the specularly reflected fundamental and second harmonics laser
signals are plotted in c) & d). All fluxes are normalised to their observed
maxima over the 2D parameter scans. Two dimensional scans of electron and
proton flux are shown in e) & f), with the average detected electron energy
and the maximum (95th percentile) proton energies shown in g) and h)
respectively. The 2D scan is a result of 143 bursts of 15 shots and the
datapoints are the mean of each individual burst.

The laser intensity can be varied by adding wavefront abberations to the
laser, which changes the focal spot shape as well as the peak intensity.
Figure 3a-d show the burst-averaged electron and proton flux as well as the
relative specular reflectivity at the fundamental and second harmonic
wavelengths for varying target plane, $z_{T}$ (figure 3a & 3c), and 45-degree
astigmatism, $Z_{2}^{-2}$ (figure 3 b & 3d). Again, a characteristic drop was
observed in the proton flux and maximum energy for best focus. Moderately
decreasing the peak intensity by either defocusing or adding astigmatism
decreased the electron flux and average energy, but maximised the proton
acceleration. At the low intensity limit, the particle acceleration drops to
zero as expected.

Evidence of disruption to the front surface during the interaction can be seen
from the sharp drop in the fundamental and second harmonic laser reflectivity
at high intensities. This matches previous observations of target reflectivity
and harmonic generation being adversely affected for high-intensity low-
contrast laser-plasma interactions as a result of the formation of a large
scale length pre-plasma Pirozhkov  _et al._ (2009); Streeter  _et al._ (2011);
Chopineau  _et al._ (2019).

To explore the interplay between astigmatism and defocus, an automated 2D
grid-scan was performed. Burst averaged measurements of the electron and
proton flux, mean electron energy and maximum proton energy are displayed in
figure 3 e-h. The electron generation was maximised for both zero defocus and
zero astigmatism, monotonically decreasing as $z_{T}$ and $Z_{2}^{-2}$ were
increased. The proton flux and energy are approximately maximised for a ring
around the origin, indicating a threshold intensity for disrupting the proton
acceleration which can be achieved either by defocus or increasing the focal
spot size through optical aberrations. There also appears to be enhanced
proton flux for zero defocus, $z_{T}=0\,$\mathrm{\SIUnitSymbolMicro m}$$, but
with the application of significant astigmatism $Z_{2}^{-2}=\pm
1.2\,$\mathrm{\SIUnitSymbolMicro m}$$ when compared with the increased flux
achieved just through defocusing with no astigmatism. This indicates that the
proton acceleration process is not just intensity dependent, but is also
sensitive to the spatial intensity profile on-target Wilson  _et al._ (2022).

For all results in figure 3, the laser pulse had a shorter pulse duration of
$\tau_{\mathrm{FWHM}}=46\pm 4\,$\mathrm{f}\mathrm{s}$$ and was skewed with a
slower rising edge than for the results in figure 2, as can be seen in figure
4. The pulse shape appears to effect the range of $z_{T}$ over which the
proton acceleration was suppressed. For the case of figure 2, proton
acceleration was maximised at $z_{T}=\pm 30\,$\mathrm{\SIUnitSymbolMicro m}$$,
while with the slower rising edge used for figure 3a, the maximum occurs at
$z_{T}=\pm 67\,$\mathrm{\SIUnitSymbolMicro m}$$, with very low flux obtained
at $z_{T}=\pm 33\,$\mathrm{\SIUnitSymbolMicro m}$$. For a slower rising edge,
there is more time for any disruption of the target to occur prior to the
arrival of the peak of the pulse, meaning that a larger defocus is required to
maintain proton acceleration. A lower maximum proton energy of $0.8\pm
0.1\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ was observed in figure 3a compared to
$2.8\pm 0.4\,$\mathrm{M}\mathrm{e}\mathrm{V}$$ for figure 2. This shows the
benefit of using temporal pulse shaping to minimise target pre-heating, as it
allows for efficient proton acceleration at higher intensity interaction
leading to higher energy protons. Despite the very different interaction
conditions (here - moderate laser contrast and micron thick target), this
indicates a similar trend to that observed in recent experiments using high-
contrast ultra-thin targets which demonstrated the enhancement of particle
energies and numbers by modification of the pulse shape, away from nominal
pulse compression, with a steepened rising edge in comparison to the falling
edge of the pulse Ziegler  _et al._ (2021); Tayyab  _et al._ (2018).

Figure 4: Laser pulses temporal profiles as measured by the on-shot SPIDER
diagnostic for the results of the 1D scan (figure 2), 2D scan (figure 3) and
optimisation (figure 5). The integrals of the signals are set by independent
measurements of the on-target laser energy which were $(438\pm
32)\,$\mathrm{m}\mathrm{J}$$ (1D scan), $(453\pm 40)\,$\mathrm{m}\mathrm{J}$$
(2D scan) and $(258\pm 22)\,$\mathrm{m}\mathrm{J}$$ (optimisation). The
corresponding measured FWHM pulse widths were $(49\pm
3)\,$\mathrm{f}\mathrm{s}$$, $(45\pm 4)\,$\mathrm{f}\mathrm{s}$$ and $(39\pm
1)\,$\mathrm{f}\mathrm{s}$$.

The interaction parameter space is multi-dimensional. While 1D and 2D slices
of this parameter space provide valuable insight, true mapping of the
parameter space required for deeper understanding and control of sheath-
accelerated proton beams, and the location of global optima, requires the
inclusion of more dimensions. This is particularly important when individual
parameters are coupled in complex relationships as is evident in the case of
proton acceleration from figure 3. While grid scanning is feasible for mapping
up to 2D slices of parameter space with multi-Hz laser systems, for higher
numbers of dimensions it becomes prohibitively time consuming. Additionally, a
grid-scan covers regions of parameters with high signal and no signal with the
same resolution. Given that measurements in large regions of the accessible
parameter space will return no appreciable proton acceleration, this is
uneconomical in terms of target usage, debris production and laser operation.
It is therefore desirable to use more intelligent algorithms for probing and
optimisation of the beam in highly-dimensional parameter spaces.

##  IV Bayesian Optimisation

In Bayesian Optimisation (BO) experimental data is used to update a _prior_
model to more accurately fit observations, thereby obtaining a _posterior_
model. The model is then used to make predictions over the experimental
parameter space and select the parameter set for the next measurement, with
the goal of efficiently finding the optimum within the parameter space. A
commonly used type of model is Gaussian process regression (GPR) Rasmussen and
Williams (2005), which is a well suited technique for modelling multi-
dimensional experimental data. A key advantage for experimental science is
that GPR can naturally include uncertainty quantification on the data, and
also can be used to estimate the uncertainty when making predictions. BO is
widely used for the optimisation of noisy processes that are expensive to
evaluate and for which there is no adequate analytical description, as is
typically the case in complex non-linear systems.

In the field of laser-plasma acceleration BO has been used in laser-wakefield
acceleration to optimise the generated electron and x-ray beam properties
Shalloo  _et al._ (2020); Jalas  _et al._ (2021b) and in simulated laser-
driven ion beams for maximising proton energy Dolier  _et al._ (2022). To
demonstrate its applicability in a laser-driven ion acceleration experiment we
have adapted the algorithm from Shalloo _et al._ Shalloo  _et al._ (2020).
Using the automated control of the experiment and on-line analysis of the
experimental diagnostics we were able to perform multi-dimensional
optimisation of any fitness function which outputted a scalar property of
interest, such as the maximum proton energy.

The input values passed to the model for each burst were taken from the
parameters set by the control algorithm with the exception of the plane of the
interaction, $z_{T}$. It was found that errors in positioning the tape target,
while significantly smaller than the Rayleigh range of the focusing laser
($15\text{\,}\mathrm{\SIUnitSymbolMicro m}$), were large compared to the
sensitivity of the plasma accelerator and so the target position input values
were taken from a spatially resolved measurement of the self-emission region
at the target surface, collected at $60^{\circ}$ to the front surface normal
of the tape. This was found to greatly improve the confidence of the model and
its ability to find the optimum.

Figure 5: Optimisation of the 95th percentile proton energy determined by the
rear surface time-of-flight diagnostic through the adjustment of the laser
wavefront and position of target along the laser propagation direction
($z_{T}$). The top panel shows the measured values of the proton energy
(median and median absolute difference of each burst) as a function of the
burst number (black points and error bars respectively), together with the
model predicted optimum after each burst (red line and shaded region) as well
as the final optimal value from the model (blue horizontal line). The
variation of each control parameter (given in microns) is shown in the lower
plots (black points) along with the final optimised values (blue horizontal
line), also as functions of burst number. The best individual burst is
indicated by the vertical magenta line in each plot and it can be seen that
for all parameters the experimental parameters fall very close to the optimum
value predicted by the model (e.g. they are close to the horizontal blue
line). For this data series, each burst contained 20 shots, the target was
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Kapton tape and the laser energy
was $258\pm 22\,$\mathrm{m}\mathrm{J}$$.

To demonstrate the BO algorithm, a 6D optimisation was performed to maximise
the maximum proton energy (as measured by the TOF). Five Zernike mode
coefficients - including $Z_{2}^{-2}$, $Z_{2}^{2}$ (oblique & vertical
astigmatism), $Z_{3}^{-1}$, $Z_{3}^{1}$ (vertical & horizontal coma) and
$Z_{4}^{-2}$ (oblique second astigmatism) - were used to change the spatial
phase of the laser pulse, affecting the focal spot shape and peak intensity.
In addition, the tape surface position relative to the focal plane of the
laser $z_{T}$, was varied using a linear motorised stage and an on-line self-
emission diagnostic to measure its position, as mentioned. This optimisation
started from an initially flat wavefront (optimised using a feedback loop with
a HASO wavefront sensor) and the tape target was initially positioned at the
estimated focal plane of the laser. This represents the typical starting point
for conventional optimisation of manual experiments, with the highest
intensity being assumed to be optimal.

Figure 5 shows the measured maximum proton energy along with the variation of
each input parameter as a function of burst number. By chance, one of the
initial randomly selected points produced a large enhancement in maximum
proton energy with every parameter apart from $Z_{2}^{2}$ close to its
eventual optimum value. With each additional measurement, the model gained
more knowledge of the parameter space and adjusted its prediction of the
global optimum of proton energy (red line) and its location in parameter
space. After 61 bursts, the model optimum was $(2.30\pm
0.10)\,$\mathrm{M}\mathrm{e}\mathrm{V}$$, compared to a starting point of
$(1.22\pm 0.04)\,$\mathrm{M}\mathrm{e}\mathrm{V}$$. This optimised maximum
energy is close to the value shown in figure 2 and significantly greater than
seen in figure 3, despite being limited to only
$260\text{\,}\mathrm{m}\mathrm{J}$ of on-target laser energy, compared to
$>430\,$\mathrm{m}\mathrm{J}$$ for the parameter scans.

The optimum found involved a shift of $70\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ from the initial position (best focus), as well as the addition of
significant wavefront aberrations compared to the initially flat wavefront.
The focal spot fluence distribution at the target plane was calculated from
on-shot measurement of the near-field phase and fluence profiles using a HASO
wavefront sensor. The resulting intensity maps are shown in figure 6 for
$Z_{2}^{-2}=-1.2,0,1.2$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at zero
defocus and for the optimised wavefront found through the optimisation. The
peak intensity was $I_{0}=5\times
10^{19}\,$\mathrm{W}\mathrm{c}\mathrm{m}^{-2}$$ for the case of a flat
wavefront at the focal plane. Each of the modified pulses had a similar peak
intensity of $I_{0}\approx 3\times
10^{19}\,$\mathrm{W}\mathrm{c}\mathrm{m}^{-2}$$, with the spot shape appearing
close to an ellipse for the optimised pulse case. From analysis of the
previous parameter scans it is inferred that this intensity distribution was
optimal maintaining a maximum possible intensity, while limiting disruption to
the rear target surface.

Figure 6: Reconstructed laser intensity profiles at
$z_{T}=0\,$\mathrm{\SIUnitSymbolMicro m}$$ for a)
$Z_{2}^{-2}=-1.2\,$\mathrm{\SIUnitSymbolMicro m}$$, b)
$Z_{2}^{-2}=0.0\,$\mathrm{\SIUnitSymbolMicro m}$$ c)
$Z_{2}^{-2}=1.2\,$\mathrm{\SIUnitSymbolMicro m}$$ and d) for the optimal pulse
(burst 53) from the optimisation shown in figure 5. The peak intensity of each
focus was (2.7, 5.1, 2.9 and 3.2) $\times 10^{19}$ Wcm-2 respectively.

The reason for the focal spot shape found in this optimisation would require
expensive 2-3D numerical simulations investigating how subtle changes in
wavefront affect the various energy transfer processes and plasma dynamics in
sheath acceleration. This would be valuable for understanding how to further
optimise laser-driven proton acceleration, but is beyond the scope of this
paper which is focused on the utility of online optimisation for proton beam
parameter optimisation as a particle source for applications as well as its
use to identify interesting regions for deeper study within a complex non-
linear system.

##  V Conclusion

In conclusion, we have demonstrated the automation and optimisation of laser-
driven proton acceleration from a solid tape drive. The ability to take high
fidelity parameter scans in one or two dimensions will be of great benefit to
the field in understanding what is a highly complex and dynamic interaction.
Bayesian optimisation of the generated proton beam was demonstrated, by using
real-time analysis of experimental diagnostics to create a closed-loop system
with limited human intervention. This can find optima that would elude manual
optimisation or single parameter scans, due to the complex interplay between
the large number of control parameters. In the case of this low-contrast
interaction, the temporal pulse shape was shown to play an important role in
determining the intensity threshold at which the proton acceleration process
was disrupted. Including temporal and spatial pulse shaping simultaneously in
the optimisation process may lead to further improvement.

Automated Bayesian optimisation can quickly find regimes of stable and
optimised operation without requiring the constant attention of laser-plasma
experts. It is anticipated that this development will be essential for
efficient utilisation of laser-driven ion acceleration for its many
applications in future user facilities Schillaci  _et al._ (2022). Bayesian
optimisation could also be extremely valuable for optimising radiation
pressure acceleration for which laser-plasma instabilities Palmer  _et al._
(2012); Sgattoni  _et al._ (2015) typically limit the acceleration process, as
well as for optimising enhanced acceleration through relativistic
transparency, which has already demonstrated highly desirable near-
$100\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ proton energies Higginson  _et
al._ (2018) and for which target evolution plays a central role. Fine tuning
of the laser parameters may be able to mitigate these instabilities or further
tailor the target evolution respectively, significantly enhancing the
accelerated proton beam.

## Data Availability Statement

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

## Acknowledgements

We acknowledge support from the UK STFC grants ST/V001639/1 with the XFEL
Physical Sciences Hub and ST/P002021/1, the UK EPSRC grants EP/V049577/1 and
EP/R006202/1, as well as the U.S. DOE Office of Science, Fusion Energy
Sciences under FWP No. 100182, and in part by the National Science Foundation
under Grant No. 1632708 and Award No. PHY – 1903414. M.J.V.S. acknowledges
support from the Royal Society URF-R1221874. G.D.G. acknowledges support from
the DOE NNSA SSGF program under DE-NA0003960. A.G.R.T acknowledges support
from the US DOE grant DE-SC0016804. D.M. acknowledges support from the project
‘Advanced research using high-intensity laser-produced photons and particles
(CZ.02.1.01/0.0/0.0/16_019/0000789)’ from European Regional Development Fund
(ADONIS)

Special thanks goes to the staff at the Central Laser Facility who provided
laser operational support, mechanical and electrical support, computational
and administrative support throughout the experiment.

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