# Controlled transition to different proton acceleration regimes: near-
critical density plasmas driven by circularly polarized few cycle pulse

Shivani Choudhary ID ELI-ALPS, ELI-HU Non-Profit Ltd., Wolfgang Sandner utca
3., Szeged 6728, Hungary  Sudipta Mondal ID ELI-ALPS, ELI-HU Non-Profit Ltd.,
Wolfgang Sandner utca 3., Szeged 6728, Hungary  Daniele Margarone  ELI
Beamlines Center, Institute of Physics, Czech Academy of Sciences, Za Radnicí
835, 252-41 Dolní Břežany, Czech Republic  Subhendu Kahaly ID
<EMAIL_ADDRESS>ELI-ALPS, ELI-HU Non-Profit Ltd., Wolfgang Sandner
utca 3., Szeged 6728, Hungary  Institute of Physics, University of Szeged, Dóm
tér 9, H-6720 Szeged, Hungary

(August 27, 2024)

###### Abstract

We investigate the different facets of ion acceleration by a relativistically
intense circularly polarized laser pulse interacting with thin near-critical
density plasma targets. Our simulations establish that plasma density gradient
and laser frequency chirp can be controlled to switch the interaction from the
transparent to the opaque regimes of operation. This enables one to choose
between a Maxwellian like ion energy distribution with a cut-off energy, in
the relativistically transparent regime, or a quasi-monoenergetic spectrum, in
the opaque regime. We subsequently demonstrate that a double-layer multi-
species target configuration, can be effectively utilized for optimal
generation of quasi mono-energetic ion bunches of a desired species. We
finally demonstrate, the feasibility of generating mono-energetic proton beams
with energy peak at $\mathcal{E}\approx 20\sim 40$ MeV with a narrow energy
spread of $\Delta\mathcal{E}/\mathcal{E}\approx 18-28.6\%$ confined within a
divergence angle of $\sim 175$ millirad at a reasonable laser peak intensity
of $I_{0}\simeq 5.4\times 10^{20}\,\mathrm{W/cm^{2}}$.

##  I Introduction

The technological advancement in targetry technology and a growing interest in
near-critical density plasma Fedeli et al. (2018); Ji et al. (2018a), both in
utilizing gas-based Sylla et al. (2012); Lifschitz et al. (2014); Kahaly et
al. (2016); Henares et al. (2019) as well as solid-based Yogo et al. (2008);
Pazzaglia et al. (2020a); Martinez et al. (2020); Wang et al. (2021)
interaction, paved a new regime of particle acceleration in relativistic
intensity laser-plasma interaction. The advent of powerful ultrashort multi-
cycle Danson et al. (2019); Radier et al. (2022) and few-cycle laser
facilities worldwide Kühn et al. (2017); Charalambidis et al. (2017) along
with sophisticated beamlines Mondal et al. (2018); Margarone et al. (2018)
makes such experiments feasible in the near future. Over several decades, the
quest to achieve monoenergetic proton beams of MeV energy has influenced the
growth in the field of laser-based particle acceleration research Macchi et
al. (2013); Schreiber et al. (2016). The interaction of ultra-intense, ultra-
short pulses with the plasma target leading to the generation of relativistic
energy particles with enormous potential for applications in materials Passoni
et al. (2019) and medical sciences Bulanov and Khoroshkov (2002); Bulanov et
al. (2014); Karsch et al. (2017), fusion schemes Weng et al. (2018); Roth et
al. (2001), industrial applications Barberio et al. (2018) and as a neutron
sourceRoth et al. (2013).

A diversity of ion acceleration processes have already been identified and
demonstrated. For example, Target Normal Sheath Acceleration (TNSA) Snavely et
al. (2000); Wilks et al. (2001); Mora (2003) has been the most widely studied
and investigated ion acceleration mechanism. TNSA predominantly occurs when a
high intensity laser pulse interacts with an opaque solid density thick foil
target producing a proton beam with maximum proton energy in the range of
several tens of MeV Hegelich et al. (2006); Ogura et al. (2012). In this
process, the hot electrons generated during the laser interaction transport
through the target and exit from the rear end, creating a sheath electric
field that accelerate the protons and other ions present at the back end layer
of the target along the direction of target normal Passoni et al. (2010). The
constraint with TNSA is mainly the broad energy spectrum that makes the use of
such beams for societal application very challenging if beam conditioning
(energy selection) is not implemented. Energy selection, on the other hand,
reduces the ion beam flux on a sample (i.e. a biological one) as in for cancer
therapy Kroll et al. (2022).

An intense ultrashort laser pulse also imparts radiation pressure during the
interaction, which is a function of the reflectivity of the relativistically
exited foil target. The complete transmission of ultra-intense laser light
from the plasma surface results in zero radiation pressure, while it maximises
in the case of complete reflection. In a typical laser-plasma interaction at
relativistic intensity, the radiation pressure can reach nearly giga-bar
($10^{9}$ bar) level Chou et al. (2022); Iwata et al. (2018) and under optimal
conditions this can favourably accelerate ions. Radiation Pressure
Acceleration (RPA) has been demonstrated to operate in two different
acceleration regimes, Hole-Boring (HB) RPA Robinson et al. (2006, 2008); Iwata
et al. (2018) ($\mu$m thick targets), and Light-Sail (LS) RPA Macchi et al.
(2009a, 2010) (for nm thin targets), with distinct features in the resulting
ion spectra. RPA has its own limitations: such as LS-RPA works well only at
ultrahigh intensities ($\gg 1\times 10^{21}W/cm^{2}$) and HB-RPA requires long
pulse ($\sim$ps), high energy lasers since a relatively long time and large
energy in the pulse is required to effectively drill a hole into the target
Iwata et al. (2018) and to accelerate ions at the target front side through
RPA. The maximum proton energy from target in RPA regime observed is about
$\mathrm{48\,MeV}$ Ma et al. (2019).

Light reflection plays a crucial role in the relativistic interaction. In non-
relativistic plasma, an incident laser pulse with frequency $\omega$ lower
than the plasma frequency, $\omega_{p}=\sqrt{n_{e}e^{2}/\epsilon_{0}m_{e}}$
(where $n_{e}$ is the electron density of the target plasma, $\epsilon_{0}$ is
vacuum permittivity, $e$ and $m_{e}$ are the electron charge and mass) is
reflected. The critical plasma density,
$n_{c}=m_{e}\omega^{2}\epsilon_{0}/e^{2}$, at which the plasma frequency
equals the wave frequency marks the transparency threshold and for
$n_{e}>n_{c}$, the plasma is described as overdense. For relativistically
intense lasers, the plasma electrons are accelerated by the laser field
lowering the effective critical density by a factor of $\langle\gamma\rangle$
Sprangle et al. (1990); Fedeli et al. (2017); Iwata et al. (2018); Pazzaglia
et al. (2020a), which is the average Lorentz factor if the electrons in the
reflecting layer increasing the transparency threshold to,
$n_{c}^{rel}\approx\langle\gamma\rangle n_{c}$ Fedeli et al. (2017). Thus, the
plasma becomes relativistically underdense for, $n_{e}<n_{c}^{rel}$. This is
the relativistically induced transparency (RIT) regime that optically switches
opaque plasma to transparent, enabling light propagation Cattani et al.
(2000); Goloviznin and Schep (2000); Siminos et al. (2012). In the recent
years, several articles have reported efficient proton acceleration from the
ultra-thin nm-scale targets in this regime Fernández et al. (2017); Sahai et
al. (2013); Poole et al. (2018); Palaniyappan et al. (2012); Higginson et al.
(2018); Singh et al. (2022).

In this article, via a series of fully relativistic 2D plane wave (one
dimension in coordinates and three dimensions in velocities) particle in cell
(PIC) simulations, over a wide range of laser and target parameters, we
demonstrate that one can transit between two-different ion acceleration
mechanisms, namely, Relativistic Induced Transparency (RIT) and Radiation
Pressure Acceleration (RPA). These regimes of ion acceleration proposed in
this study do not necessarily require PW-class lasers but potentially can be
achieved using sub-PW peak power and moderate ($\sim 10^{20}W/cm^{2}$) laser
intensities. These parametric variations provide a route to effectively tune
the spectral characteristics of the generated ions from the target. We
investigate the influence of peak plasma density, target thickness, plasma
density gradient, laser temporal chirp, and laser focal spot size effects on
the ion acceleration process from the near-critical density plasma. We
identify conditions that can effectively accelerate quasi mono-energetic ions
with narrow energy spread. The observations may potentially be investigated
experimentally at ELI-ALPS Kühn et al. (2017); Charalambidis et al. (2017);
Mondal et al. (2018) and ELI beamlines Schillaci et al. (2022) in the near
future. Additional 2D-PIC simulations give a clear indication on how target
configuration (Double layer target) can provide the mono-energetic ion bunches
of the species of choice depending upon the ion acceleration mechanisms and
sheds light on the mechanism of quasi mono-energetic ion acceleration.

The work done in this article is structured as follows. We discuss the ion
acceleration from the foil near-critical density targets with step-like
density profile and bring out the distinctive features of the processes in the
transparent and the opaque regimes both in real space and the phase space, in
section II. In section III we summarise our observations on target thickness
and plasma density effects on the transition to different proton acceleration
regimes, and investigate the robustness of the correlation between the optical
transparency and the nature of proton energy spectral shape. We discuss the
influence of an exponential density ramp in front of foil target, a feature
that can be all optically controlled in experiments with a pre-pulse, in
section IV. In section V, we explore and summarise the effect of temporal
chirp on the ion spectral characteristics for different target species and
show that preferential acceleration of a chosen ion species can be done using
a double layer target in section VI. Finally, in section VII, we discuss the
effect dimensionality and of tight focusing the ion energies from the double
layer target configuration and probe the quasi mono-energetic ion acceleration
deeper.

##  II Proton acceleration in step like sub-wavelength scale near critical
density foil

Figure 1: Spatio-temporal dynamics of an intense few-cycle circularly
polarized electric field driven thin slab target with thickness in the
sub-$\lambda$ regime (in this case foil of thickness $d=0.75\lambda$). The
spatio-temporal evolution of the laser intensity (${E_{y}^{2}+E_{z}^{2}}$
presenting the incident, reflected and transmitted light), electron density
($n_{e}$) and ion density ($n_{i}$), is depicted for two different initial
plasma densities $n_{0}$ representative of the two distinct scenarios: (a)-(b)
$n_{0}$=$n_{e}$=3$n_{c}$ representing light transmission and (c)-(d)
$n_{0}$=$n_{e}$=9$n_{c}$ showing complete reflection of light. The solid
horizontal lines mark the instant $t=48\tau$ when the peak of the pulse
envelope interacts with the target surface. (b) and (d) represents a zoomed
view of the dotted boxes demarcated in (a) and (c) respectively, capturing the
distinct signatures of the ion and electron dynamics in the two different
regimes during the relativistic interaction. Both the targets are initially
overdense and reflect the laser (since $n_{0}>n_{c}$), which interferes with
the incident pulse forming the standing wave pattern in front of the target
(see the intensity fringes on the front side of the target). (a) Near the peak
of the pulse, the target becomes optically thin (dynamically underdense),
pushing out electrons from the surface, initiating complex electron dynamics
in the ion background and expanding the ion density distribution shown in (b).
(c) The target retains overdense character within the laser pulse duration
leading to in sink motion of the electron and ion density peaks shown in (d).
We have used a Gaussian laser pulse with normalized peak laser pulse amplitude
$a_{0}$ = 20 (as in eq. 1) for both cases.

The interaction of a relativistic intensity laser pulse with a semi-infinite
plasma and subsequent plasma dynamics can be modeled accurately, to a large
extent, using 1D cold fluid model Kaw and Dawson (1970). In the reflection
region (vacuum side), this theory provides the stationary solution at the
vacuum plasma boundary by balancing the ponderomotive and the electrostatic
forces. In the plasma side, the laser ponderomotive force pushes electrons
inside the plasma to a new position leaving the immobile ions behind,
consequently, a strong charge separation field is created which peaks at the
new electron position. However, at the interaction condition, when most of the
electrons escape from the target, the laser pulse starts to propagate deeper
inside, thus, the electron dynamics become more complex and the validity of
cold-fluid approximation ceases to exist. Therefore, PIC simulations are
required to understand the behavior of interaction at this regime.

The laser electric field $\mathbf{a}$ for a Gaussian laser pulse with
elliptical polarization can be defined as follows,

$\mathbf{a}(\eta)=\frac{a_{0}f(\eta)}{\sqrt{1+\epsilon^{2}}}\bigg{[}\mathrm{cos}[\phi(\eta)]\mathbf{e_{y}}+\epsilon\leavevmode\nobreak\
\mathrm{sin}[\phi(\eta)]\mathbf{e_{z}}\bigg{]}$ (1)

where, the laser electric field $\mathbf{a}(\eta)$ is in units of $m_{e}\omega
c/e$, $f(\eta)=exp[-4ln(2)\frac{\eta^{2}}{\tau_{FWHM}^{2}}]$ is the normalised
pulse envelope function, $\phi(\eta)=2\pi\eta$ is the phase function and
$\eta=t-x$ is the propagation coordinate (where the space and time coordinates
are normalised with respect to the time period $\tau$ corresponding to the
central carrier frequency and the laser central wavelength $\lambda$
respectively). $\tau_{FWHM}$ is Full-Width-at-Half-Maxima (FWHM) of the time
dependent intensity envelop and $a_{0}$ is normalized laser pulse amplitude.
Here, $m_{e}$ and $e$ are the mass and charge of the electron. $\epsilon$ is
the ellipticity parameter which can vary from -1 (left-handed circular
polarization) to 0 (linear polarization) to +1 (right-handed circular
polarization), where the sign determines the helicity and the magnitude
defines the ellipticity in general. We kept $\epsilon=-1$ for all the cases
studied under the present investigation.

Furthermore, using 1D model equations, one can obtain the threshold limit for
the self-induced transparency and to differentiate between the regions of
transparency and opacity for semi-infinite overdense plasma target in
relativistic framework Cattani et al. (2000); Goloviznin and Schep (2000). In
a more rigorous manner, using fully relativistic PIC simulations one can
obtain the percentage of laser pulse energy reflection and transmission from
the target surface. In our study we follow the second procedure, while
conceptually benefiting from comparing the results with the expectations from
the model. A constant parameter $\xi_{0}$ is taken into account which defines
the normalized surface density of the target Vshivkov et al. (1998), and is
proportional to the product of target density $(n_{e})$ and thickness $(d)$ of
the thin plasma slab and is given by,
$\xi_{0}=\pi\bigg{(}\frac{n_{e}}{n_{c}}\bigg{)}_{\omega_{0}}\frac{d}{\lambda}$.
In case of moderate intensities ($a_{0}<1$), $\xi_{0}<1$ corresponds to
transparency regime. Whereas, for ultra-intense laser pulses ($a_{0}>1$), in
the non-linear regime, the condition of transparency is achieved when
$\xi_{0}>1$ and $a_{0}\geq\xi_{0}$ (or $a_{0}/\xi_{0}\geq 1$) Vshivkov et al.
(1998).

In order to have a clear understanding about the region of transparency and
opacity, we have used 2D plane wave simulations with fully relativistic PIC
simulation code LPIC++ Lichters et al. (1997) with modifications including in
both target conditions such as an introduction an of additional layer,
exponential ramp, as well as implementing Gaussian profile and chirp function
in the laser. In order to investigate the role of peak plasma density $n_{e0}$
for a given laser central frequency and foil thickness, we consider a quasi-
neutral plasma foil, $n_{e}(x)=n_{e0}[H(x-x_{1})-H(x-x_{2})]$, where $H$ is
the Heaviside step function and $d=x_{2}-x_{1}$ is the target thickness.

A circularly polarized (CP) laser pulse with normalized amplitude $a_{0}$ = 20
($I_{0}\simeq 5.4\times 10^{20}\,\mathrm{W/cm^{2}}$), field envelope FWHM = 5
cycles is normally incident on the sharp gradient plasma slab of thickness
0.75$\lambda$ from the left side of the simulation domain. The simulation
domain is of 100$\lambda$ with 140 cells per wavelength and 1500 particles per
cell. The laser and target parameters are taken in dimensionless units. The
space and time are normalized as x/$\lambda$ and t/$\tau$ respectively, where
$\lambda=1\,\mathrm{\mu m}$ is laser wavelength and $\tau=\lambda$/c. The
electron density is normalized with respect to critical density $n_{c}$ and
fields are normalized as $eE/m_{e}\omega c\rightarrow E$. The plasma slab is
located from 35$\lambda$ $<x<$ 35.75$\lambda$ having density of 3$n_{c}$ which
is considered as nominally overdense, for the intensity of
$\sim\mathrm{5.4\times 10^{20}\,W/cm^{2}}$ and hence calculates $\xi\sim$ 7
and therefore $\frac{a_{0}}{\xi}>1$ satisfying the transparency condition. On
the other hand, $n_{e}$ = 9$n_{c}$, is a fully overdense region for the given
intensity and results in $\xi\sim$ 21.195, where the ratio
$\frac{a_{0}}{\xi}\lesssim 1$, therefore tapping into the total reflection
region Macchi et al. (2009a); Choudhary and Holkundkar (2016). Such kind of
near-critical densities are used in experiments from, foam targets Willingale
et al. (2011), cryogenic hydrogen jet target Obst et al. (2017) and cryogenic
solid hydrogen target Polz et al. (2019)

In Figure 1, we present the spatio-temporal profile of laser intensity,
electron, and ion density for the given laser and target parameters. Figure
1(a) and (c) represent the two dynamically opposite cases with nearly (a)
$\sim 90\%$ (c) $\sim 0\%$ transmission of laser pulse through the target.
Therefore, we refer to these two regions of ion acceleration, with the
nominally underdense region as in _relativistic transparency_ (a, b) and
overdense region as in _opacity_ (c, d).

When a circular polarized (CP) laser pulse is incident on the target, due to
the absence of $J\times B$ heating phenomenon within the interaction process,
in relativistically transparency region 1(a), sufficient number of electrons
escapes the target, thus leaving the target to be underdense. As the laser
pulse propagate through the target, the electrons move in the vicinity of the
laser field in both forward (target rear) and backward (target front)
direction. Whereas, ions being heavier in mass, undergoes a marginal expansion
around the target surface (from 35$\lambda$ to 35.75$\lambda$), mostly in the
forward direction as shown in Figure 1(b). The electrons which are removed
from the target ion background create a charge separation at the interface
generating a strong electrostatic field. As a result, the electrons are pulled
back to the target surface towards the ions and oscillate around the interface
of the target. A magnified illustration of this process is shown in Figure
1(b) with ion and electron density profiles. The horizontal cyan lines in
Figure 1 demarcates the time at which the peak of the laser field interacts
with the target. As seen in 1(a), as the rising part of the laser field
interacts with the target, the ponderomotive force imparted by the laser field
compresses the electron into the ion background, eventually pushing out part
of the electrons near the peak of the laser field leading to light
transmission. Afterwards rest of the electrons execute back and forth
oscillations in the ion background leading to expansion of the ions as seen in
1(b).

The interaction in the opaque regime ($n_{e}=9n_{c}$) is presented in Figure
1(c,d). In Figure 1(c), we observe a completely different electron dynamics.
Due to the interaction of CP laser pulse with overdense target, most of the
laser pulse is reflected. Therefore, all the electrons are compressed inside
the target and are piled up at the rear side of the target leading to the
formation of two electron bunches at a later time. In order to have a clear
view, we present a zoom into the spatio-temporal profile of electron and ion
density during the interaction in Figure 1(d). We observe that, the electrons
and ions position closely overlaps with each other and co-propagate together,
as shown in ref. Ji et al. (2018b). At this point we would like to note that
the partial transmission and reflection of the light field during the
interaction is well captured in the colormap presenting the intensity envelop
in Figure 1(a) and (b). A standing wave pattern is clearly visible in the
target front side, representing the interference of the incident and reflected
light fields and hence persisting only as long as the light reflection
persists during the interaction. As seen in Figure 1(a), near the cyan line,
the target becomes partially transparent, reducing thereafter the contrast in
the standing wave pattern.

Figure 2: Temporal snapshot depicting the profiles of the driver laser field
magnitude (dashed blue line representing the incident, reflected and
transmitted fields), longitudinal electric field (red solid line representing
the accelerating field), ion density (green solid line) and electron density
(black solid line) for the two scenarios: (a)-(b) at $n_{e}$ = 3$n_{c}$
(relativistically transparent regime) and (c)-(d) at $n_{e}$ = 9$n_{c}$
(overdense regime). For each case the fields and densities are plotted at two
different time instants: (a)-(c) at a time $\mathrm{t=48\,\tau}$, when the
peak of the laser pulse is interacting with the target and (b)-(d) at latter
time $\mathrm{t=70\,\tau}$, long after the driving field has ceased to
interact with the target. Near the peak of the interaction, the lower initial
density target already starts to transmit the incident laser pulse which can
be seen in the blue shaded part on the right side of the target in (a) and is
clearly captured after the interaction in the laser pulse co-propagating to
the right along with the electron bunch as seen in (b). No such transmission
can be witnessed in (c) and (d) (pay attention to the significantly small
scale used for plotting the magnitude on the right axis for the radial
electric field presented in (d)).

To discuss the underlying physics behind the ion acceleration in these two
specific regions of interest, we present the transverse and longitudinal
electric fields along with the ion and electron density profiles in Figure 2,
for the above mentioned laser intensity and target density cases. Here, we
have considered the two specific time instants, one at t=48$\tau$, when the
peak of the driver pulse interacts with the target and, other at t=70$\tau$
which we say as post-interaction. During these two time instances, we analyze
the behavior of all the field and density components.

In the relativistic transparent region, at t=48$\tau$ in Figure 2(a), the
laser pulse energy starts to propagate through the target with limited laser
energy reflected. This eventually leads to the formation of a standing wave
pattern of low contrast at the front side of the target as mentioned before.
As discussed in the context of Fig 1(a,b) due to the formation of the electron
bunches on both sides of the target, a bipolar electrostatic field (5 arb.u.)
is generated (red solid line) at t=48$\tau$ which is the signature of target
expansion on both the side. At a similar moment, a completely different
behaviour for the opaque regime (Figure 2(c)) is observed, which exhibits a
unipolar longitudinal field. Since, in this case, a substantial amount of
laser energy is reflected from the target front surface, and this leads to the
formation of standing waves with high contrast. Therefore, the electrons
receive a laser push in the forward direction in the form of radiation
pressure and move inside the target, followed by the slowly moving ions. Thus,
creating a charge separation only at the rear side of the target Guérin et al.
(1996) and hence, resulting in a unipolar electrostatic field (20 arb.u.), as
seen in the red curve in Figure 2(c).

At a post interaction time (t=70$\tau$), we observe a strong persistent
longitudinal electrostatic field in the relativistic transparent region as
shown in Figure 2(b). In this case, the laser pulse is already propagated
through the target and a fraction of the electron population is expelled from
the target to make it positively charged. This results in the ions to expand
under its coulomb repulsion. A small fraction of electrons with very high
energy nearly co-propagate with the transmitted laser pulse (can be seen at
57$\lambda$) leading to the formation of a strong electrostatic field (2
a.u.). Since, the electrons escape from the target in both directions,
therefore it results in the generation of a negative electrostatic field, at
the target front.

In the case of overdense target, as seen in Figure 2(d), at the later time
instant of $70\tau$, when complete laser pulse has interacted with the target,
the ions and electrons move nearly together forming an overlapping double peak
structure in space. This results in generation of a weak electrostatic field
$\sim$0.5 a.u. These bunches slowly expand in the target forward direction
(rear side) under radiation pressure acceleration. This shows that the laser
pulse transfers its momentum to the electron and ions and lets them evolve
under this momentum transfer Robinson et al. (2009, 2012). Similar to the cold
fluid model, the position of the maximum electrostatic field is at the minimum
of the electron density, as shown in Figure 2(c) at the peak interaction of
the pulse. At the later instants ($70\tau$), the position of the peak
electrostatic field cannot be correctly interpreted from the cold fluid model.
Since ions being mobile and electron bunches leaving and re-entering the
target gives rise to this difficulty, as mentioned in ref. Cattani et al.
(2000).

Figure 3: Temporal snapshots of ion phasespace distribution and the
corresponding energy spectra. (a)-(b) Ion phasespace distribution in the blow-
out regime ($n_{e}=3n_{c}$) of interaction at two different time instants.
(c)-(d) Ion phasespace distribution in the opaque regime ($n_{e}=9n_{c}$) of
interaction at the same time instants. At $\mathrm{t=48\,\tau}$ the laser
pulse peak is interacting with the target. $\mathrm{t=70\,\tau}$ signifies an
instant long after the driving field has ceased to interact with the target.
The color bar in (a)-(d) represents the number of macroparticles
$d^{2}N_{i}/d\beta_{i}dx$ accelerated in the laser propagation direction with
the velocity ($\beta_{i}$), normalized with speed of light in vacuum ($c$),
per unit bin in the phasespace. (e)-(f) The ion energy spectra
$dN_{i}/d\mathcal{E}$ corresponding to the interactions represented in
(a)-(d). Both the targets behave in a reflective manner until near the peak of
the laser pulse envelop reached at $\mathrm{t=48\,\tau}$. At the peak of the
driving field the foil with $n_{e}=3n_{c}$ becomes transparent, whereas the
one with $n_{e}=9n_{c}$ remains reflective during the whole relativistic
interaction. The two different behaviours, representative of the two different
regimes of operation, are captured very well in this figure. During the
reflective regime the phase space velocity distribution presented in (a),(c)
and (d) demonstrate similar folded shape and peaky features which becomes
significantly different in (b) once the interaction enters the transparent
regime. Similar behaviour unravels their signatures within the ion energy
spectra. For the reflective part of the interaction (dark-shaded curves in
(e),(d)) the ion energy spectra show distinct quasi-monoenergetic behaviour
which is retained even after the interaction is over as seen in the light-
shaded curve (f), but erodes away once target enters transparent regime even
partially as seen in the light shaded curve in (e).

We have presented earlier, in Figure 1(a) and elaborated in Figure 2(a) for
the case of the target with peak electron density $n_{e}=3n_{c}$, it becomes
transparent near the peak of the laser field. On the other hand, in the case
of the target with peak electron density $n_{e}=9n_{c}$ (as presented in
Figure 1(c) and Figure 2(c)) the laser field fails to pass through the target
even at the peak of the field envelop. In order to investigate the ion
dynamics in more detail and to identify the differences between the regimes of
transparency and opacity we now analyse the phase space data and look into the
behaviour of the ions in above mentioned cases in Figure 3. The ion velocity
in the direction of laser incidence (along decreasing $x$), in units of $c$ is
represented by $\beta_{i}$ in the colormap, where a positive $\beta_{i}$ in
Figure 3 indicates ion velocity into the target (from left to right in Figure
1 and Figure 2). Before the peak of the laser pulse interacts with the target,
in both the cases, the target remains predominantly reflective, as is evident
from the relatively high contrast of the interference fringes in the front
side of the target (Figure 2(a,c)). From the beginning of the interaction,
upto this point in time, the electrons face the increasing Lorentz push from
the laser field and gain in energy which is self consistently transferred to
the ions via the plasma charge separation field. As is evident in Figure 3(a),
for the target which eventually becomes transparent at the peak-interaction
(48$\tau$), the ions gain in forward momentum showing a $\beta_{i}$ peak
located at $x=35.75\lambda$ (which is inside the initial target surface at
$x=35\lambda$) in the phase space velocity distribution. We note here that the
asymmetric bipolar charge separation field $E_{x}$ (the different values of
the negative and positive peaks in the red curve in Figure 2(a)) leads to the
asymmetric ion velocity distribution in Figure 3(a), being skewed towards the
laser propagation direction. Long after the interaction (at $70\tau$) in the
transparent target, as shown in the phase space distribution in Figure 3(b),
we observe that ions have expanded in both the direction (front and rear),
with a slower ion expansion at the front side of the target (having a velocity
cutoff near $\beta_{i}\sim 0.18$), than the rear side (with velocity cut off
near $\beta_{i}\sim 0.27$). In this case, since the ions undergoes the Coulomb
repulsion, it expands in both the direction (front and rear) as in Figure
1(b).

Here we note that, if the features of the ion velocity distribution are
closely linked with the regimes of transparency during the interaction, at
this point, one may expect that until $t=48\tau$ both the targets should show
qualitatively demonstrate similar ion phase space behaviour. This behaviour is
fully corroborated by the similar features being observed in the opaque
region, as shown in Figure 3(c), near the peak of the pulse. However, there
are few qualitative differences with respect to the scenario presented in
Figure 3(a). Firstly, in Figure 3(c) we observe that at $t=48\tau$, the peak
of the $\beta_{i}$ distribution is located at a higher value compared to the
previous case. Secondly, in the case of $n_{e}=9n_{c}$, the asymmetric phase
space ion velocity distribution is more skewed. Both these points can be well
understood by looking at the $E_{x}$ field profile presented in Figure 2(c),
which shows a higher value of peak field and a sharper field profile inside
the target than in the front side. Hence the ions show some expansion at the
target front, but are much better piled up within target thickness, showcasing
the build up of charge as shown in Figure 3(c). At a longer time delay after
the interaction ($t=70\tau$), for the opaque region (Figure 3(d)), we observe
two distinct ion velocity distributions, where one ion bunch accelerates from
the front surface of the target, another ion bunch accelerates from the rear
surface of the target. Eventually, both ion bunches merge to form a single ion
distribution at $x$ = 39$\lambda$.

Figure 4: Correlation between interaction regime and the nature of the
resulting ion energy spectra: step density targets. (a) Transmitted laser
energy fraction through the target with varying thickness (d/$\lambda$) and
peak electron density ($n_{e}/n_{c}$) at simulation time 70$\tau$. The 2$\%$
(black solid line) and 40$\%$ (black dashed line) transmitted energy fractions
are shown. The 40$\%$ black dashed line indicates threshold target density
($n_{th}$) for RIT for varying target thickness. Analytically predicted
threshold target density (Eq. 2) is shown with a white dashed line and is
consistent with the 40$\%$ iso-line obtained from PIC simulations. The
colorbar shows the laser pulse energy transmission coefficient
($\mathcal{E}_{t}$) as in Eq. (3). The target density $n_{e}$ = 5$n_{c}$ and
thickness $d=1.15\lambda$ are marked with horizontal and vertical grey lines
in (a). In (b) ion energy spectral map with varying target thickness is
presented for the threshold density of $n_{e}$ = 5$n_{c}$ (along the
horizontal iso-density grey dashed line in (a)). In (c) ion energy spectral
map for target thickness of $d=1.15\lambda$ is presented with variation in
target density (along the vertical iso-thickness grey dashed line in (a)).
Note that the colormap for ion energy is plotted in logarithmic scale. The
spectral map in (b) and (c) unequivocally shows that as the interaction enters
from transparency into the opaque regime (across the 2$\%$ iso-transparency
black solid curve in (a)) the accelerated ion energy spectra goes from
exponential to quasi-monoenergetic peak structures irrespective of whether it
is along the iso-density line or along the iso-thickness line. This
establishes a consistent correlation between the regime of interaction and the
nature of ion energy spectra over a wide range of parameter space. To probe
this point further we plot the ion energy spectral map along different iso-
thickness lines with varying peak target density. (d) shows the ion energy
spectral map iso-thickness lines through the red semitransparent circles in
the black curve separating the transparent and opaque regimes in (a). The
black dotted line in (b), (c) and (d) represents the cut-off ion energy in
both the regions, whereas the orange dashed contour lines on (d) marks the
peak ion energy in relativistically transparent region. The black arrows in
the ion energy spectral maps in (b), (c) and (d) demarcates the interaction
conditions corresponding to the ($n_{e}/n_{c},d/\lambda$) pairs on the black
solid curve in (a) (identifies with target conditions indicated by each circle
shown in (a)) emphasizing the transition phase between the regimes, i.e. from
transparent to opaque conditions through the 2$\%$ iso-transmission curve. A
clear correlation can be seen between $\mathcal{E}_{t}$ and. The colorbar
shown above (d) illustrates the number of ions accelerated and represents for
(b), (c) and (d). All the spectra are obtained at time, 70$\tau$.

Now we look at the nature of ion energy spectra for the ions moving in
direction of laser incidence. The ion energy spectra at a time when the peak
of the laser pulse interacts with target (t=48$\tau$) are presented in black
in Figure 3(e) and (f), for both the cases (transparency and opaque). The ion
energy first peaks and then follows a sharp cut-off, which is a typical
spectral behavior of Hole-Boring RPA Weng et al. (2012). Since, up to the time
t=48$\tau$ (peak interaction), the interaction is not yet over and
transparency is not achieved, both the target conditions are in an overdense
region, where we observed the spectral signature of the opaque region. At this
intermediate interaction time, the ion cut-off energy is higher in the opaque
region (Figure 3(f)), than in the transparent case (Figure 3(e)) due to the
higher accelerating field operating at that instant. Also, during the post-
interaction phase (70$\tau$), in the transparency region (Figure 3(e)), the
accelerated ions follow a Plataea structure with peak energy of $\sim 30$ MeV,
whereas quasi-monoenergetic behavior in opaque region (Figure 3(f)) with the
peak ion energy of $\sim$16 MeV. The peak ion energy from the transparent
region and the cut-off ion energy from the opaque region is around $\sim$30
MeV, in these particular cases. One should take into consideration that the
respective ion energy spectra presented here correspond to spatially
integrated velocity distributions at the relevant moments. Thus, in this
section we have discussed the typical features of the interaction, the ion and
electron density distributions in the real space-time domain, the relevant
accelerating fields during and after the interaction and their consequence on
the ion phase space distributions and resulting ion energy distributions for
two special cases of interest. This has helped us identify the signatures of
two different regimes of interaction. In the following we undertake a more
systematic study of the effects of different target and laser parameters on
the ion acceleration process, in the regimes of our interest trying to
decipher the generic features that would allow us to design and establish an
approach relevant for experiments.

##  III Influence of foil thickness and peak plasma density: the transition
between the acceleration regimes

In this sub-section, we investigate the effect of thickness variation in foil
target on the ion acceleration mechanisms, namely RIT and RPA. In ref. Cattani
et al. (2000), the stationary solutions for the 1D scenario are derived
assuming the validity of a cold fluid model. Such a model operates under the
approximations that, a CP monochromatic laser pulse interacts with the
overdense plasma having a step-like electron density profile in a background
of immobile ions. In the context of the cold fluid model, at this point, we
would like to define a threshold electron density of the target $(n_{th})$.
The threshold density is the maximum electron density that allows the laser
pulse to transmit through the plasma target. The scaling law for the threshold
density is given in ref. Siminos et al. (2012) in the case of a relativistic
intense laser pulse ($a_{0}>>$1) incident on semi-infinite plasma slab. A
modified expression for threshold density incorporating the effect of the
target thickness has been presented previously in a phenomenological way in
Choudhary and Holkundkar (2016) as:

$n_{th}\sim\frac{2\lambda}{9d}\bigg{(}3+\sqrt{9\sqrt{6}a_{0}-12}\bigg{)}n_{c}$
(2)

In the following we would use this expression as a model reference and we
would also check the validity of the expression during our study, through a
comparison using a large number of PIC simulations that we conduct. In this
context, as a first step, we need to define clearly what we mean by
transparent and opaque regimes over our parameter space of interaction.

In order to define the transparent and opaque regimes in a more quantitative
and consistent way we define a parameter $\mathcal{E}_{t}$, called the
transmitted energy fraction. We define the transmitted energy fraction as,

$\mathcal{E}_{t}=\frac{\int_{\eta_{i}}^{\eta_{f}}(E_{y}^{2}+E_{z}^{2})\mid_{TF}d\eta}{\int_{\eta_{0}}^{\eta_{i}}(E_{y}^{2}+E_{z}^{2})\mid_{IF}d\eta}$
(3)

where, $E_{y}$ and $E_{z}$ are the transverse components of the propagating
laser fields, $\eta_{0}$, $\eta_{i}$ and $\eta_{f}$ are space-time
representation for before (interaction has not started yet), beginning
(initiation of the interaction) and end (interaction is over) of the
interaction regions respectively, here, $\eta=t-x$ ($t$ and $x$ are in units
of $\tau$ and $\lambda$ respectively implying $c=1$), the suffixes $IF$ and
$TF$ correspond to the incident and transmitted fields respectively. In
simulation time, $\eta_{0}$ = 0 $\tau$, $\eta_{i}$ = 36$\tau$, and $\eta_{f}$
= 60$\tau$. In Eq. 3, the numerator
$\int_{\eta_{i}}^{\eta_{f}}(E_{y}^{2}+E_{z}^{2})\mid_{TF}$ represents the
transmitted fluence after the interaction with the target is over and the
denominator $\int_{\eta_{0}}^{\eta_{i}}(E_{y}^{2}+E_{z}^{2})\mid_{IF}d\eta$
indicates the incident laser fluence on target, both expressed in the same
units. Thus, here we have defined the transmitted energy fraction as the ratio
between the transmitted laser energy and the incident laser energy. This
parameter is directly calculated by post processing the PIC simulation
results. For a given target and laser parameter, $\mathcal{E}_{t}=1$ implies
total transmission and $\mathcal{E}_{t}=0$ indicates no transmission at all
implying that laser energy is either totally reflected or absorbed. In a real
interaction the value of the parameter lies between these two extremes, i.e.
$0\leq\mathcal{E}_{t}\leq 1$. Thus, using the criterion of the transmitted
energy fraction $\mathcal{E}_{t}$, of the laser pulse, we can differentiate
between the region of transparency and opacity regimes in a quantitative
manner.

The colormap in Figure 4(a) shows the variation of the transmitted energy
fraction, $\mathcal{E}_{t}$ when the interaction spans over a range of target
peak electron densities, $n_{e}/n_{c}\in[2,10]$ and a sequence of target
thicknesses, $d/\lambda\in[0.75,1.55]$ at each peak electron density. To begin
with, each target peak plasma electron density and target thickness
combination corresponds to a nominally overdense regime implying complete
light reflection, in the case of a non-relativistic laser peak intensity, from
the plasma critical density layer $n_{c}$. At our laser peak intensity
relativistic effects come into play. For the given laser parameters,
interaction on this two dimensional (2D) target parameter space clearly brings
out several features. Firstly, along any vertical (increasing target peak
electron densities along a line of constant target thickness) or horizontal
line (increasing target thicknesses along a line of constant peak electron
density) on the colormap in Figure 4(a), the target becomes more reflective.
Secondly, there is clearly demarcated opaque region on the 2D parameter space
along with a gradually increasing transparent regime. In order to define
transparency threshold we have defined an iso-line at, $\mathcal{E}_{t}=0.02$
below which we consider the target to be transparent. Thus, the black solid
iso-line marked at 2$\%$ of transmitted energy fraction, indicates the maximum
limit in threshold density ($n_{th}$) for the laser pulse to undergo
transmission or reflection from the target. The 40$\%$ black dashed iso-line
of transmitted energy fraction shows remarkable matching with the variation of
threshold density obtained under the cold fluid approximation (the white
dashed curve) using Eq. 2. Figure 4(a) shows increasing target thickness
($d/\lambda$) approximately by a factor of two, the threshold density
($n_{th}$) reduces to nearly half, i.e. from 6$n_{c}$ at 0.75$\lambda$ to
3$n_{c}$ at 1.55$\lambda$. With increasing target thickness (0.75$\lambda$ \-
1.55$\lambda$) and density (2$n_{c}$ \- 10$n_{c}$), increases the target areal
density ($\propto n_{e}d$). Therefore, the incident laser field on the target
is insufficient to remove the substantial amount of electrons from the target
to achieve RIT and thereby reflects the pulse from the surface of the thicker
target with higher electron density. This regime of ion acceleration lies
under hole-boring (HB) RPA. Thus, one can effectively control the transition
from transparency to opacity by tuning the target thickness and density.

We now investigate the correlation between interaction regime and the nature
of the ion energy spectra, in order to see whether the understanding developed
in the previous section for two specific cases, can be validated over a wider
parameter range. To elucidate this process of transition, we present the ion
energy spectral map in Figure 4(b) at a fixed electron density($n_{e}$ =
5$n_{c}$) and varying target thickness ($d/\lambda$). The target density, in
this case, is chosen in such a way that it can cover both the regions, i.e.,
transparent (RIT) and opaque (RPA), over the range of target thickness (along
the horizontal grey dashed line in Figure 4(a)). For the first half of
thickness variation i.e., for $d<$ 1.15$\lambda$, the condition satisfies RIT
where the cutoff energy reaches maximum up to 90 MeV following the energy
distribution as previously observed in Figure 3(e). On the other hand, for the
target thickness of $d>$ 1.15$\lambda$ a quasi-monoenergetic spectrum is
observed akin to that in Figure 3(f), with the constant peak ion energy peak
at 20 MeV and cut-off energy of $\sim 55\,$MeV. It is evident from Figure 4(a)
as well, that for $n_{e}=5n_{c}$ the , the iso-density line crosses the
transparency threshold curve ($\mathcal{E}_{t}=0.02$) at d = 1.15$\lambda$,
target start to become opaque for all the thicknesses $d>1.15\lambda$.
Therefore, we can refer to thickness d = 1.15$\lambda$, as a transitioning
thickness between these two regimes. This point representing the change in the
ion spectra is marked with an arrow in 4(b).

Similarly, Figure 4(c) shows the ion energy spectral map for fixed target
thickness (d = 1.15$\lambda$) while increasing the target density from
2$n_{c}$ to 10$n_{c}$. We observe similar behavior in the ion energy spectrum
as in the case of fixed density (5$n_{c}$) target shown in Figure 4(b). Since
$n_{e}=5n_{c}$ is the threshold density (marked with an arrow) for target
thickness of 1.15$\lambda$, it acts as a transitioning density point. For
$n_{e}<5n_{c}$ the target undergoes transparency, and we observe the
representative ion energy spectrum with cut-off energy increasing from 40 MeV
for near critical density target to 90 MeV for relatively higher density
target. Whereas, in case of $n_{e}>5n_{c}$ spectral features highlight quasi-
monoenergetic behavior. However, in this case, we observe the reduced cutoff
energy from 40 MeV (threshold density $n_{e}$ = 5$n_{c}$) to 25 MeV (overdense
target $n_{e}=10n_{c}$) as well as peak ion energy from 20 MeV to 15 MeV.
Since, the target densities $n_{e}>5n_{c}$ lies under opaque region, the laser
ponderomotive push is not strong enough to remove all the electrons from the
target. This results in weak charge separation field which consequently
restricts the ion energies within certain limit.

In addition, we presented the ion energy spectral map with varying target
density for four different target thicknesses in Figure 4(d) corresponding to
the thicknesses indicated with circles filled in different shades of red,
lying on the threshold density curve in Figure 4(a). These exhibits the
typical nature of energy distribution in transparent region (RIT) and quasi-
monoenergetic distribution of ion energy spectrum in reflected regime (RPA).
For each target thickness, the threshold density acts as a transitioning
point, which is consistent with the transmitted energy fraction iso-line
(2$\%$) in Figure 4(a). Additionally, we observed that the peak ion energy
(yellow dashed in transparent region and dark blue in opaque region) follows
the similar trend as of ion cut-off energies (black dotted lines) for all the
target thicknesses, over the varying target densities ($2n_{c}-10n_{c}$ ).

Therefore, the maximum ion cut-off energy at the target threshold densities
for all the target thicknesses (0.75$\lambda$, 0.95$\lambda$, 1.15$\lambda$,
1.35$\lambda$, and 1.55$\lambda$), reaches up to 90 MeV. Additionally, the
maximum ion cut-off energy was obtained in RIT domain, ranging from $\sim$40
MeV to $\sim$90 MeV, which is significantly higher than that achieved in HB-
RPA ($\sim$25 MeV to $\sim$40 MeV). In other words, the transition point
between RIT and RPA regime is the key criteria for achieving maximum ion
energy, where RIT determines the maximum ion cut-off energy and HB-RPA region
is the criteria for achieving quasi-monoenergetic peak ion energy (maximum
$\sim$20 MeV). Hence, one can benefit from the transition between different
regimes of ion acceleration, depending upon the ion energy requirement for the
given the experimental conditions.

##  IV Controlling proton acceleration: foil with plasma density gradient

Figure 5: Correlation between interaction regime and the nature of the
resulting ion energy spectra: foils with varying plasma gradient scale
lengths. (a) Transmitted laser energy fraction for the case of target varying
scale-length (L/$\lambda$) and density ($n_{e}/n_{c}$). The colorbar is the
energy transmission coefficient ($\mathcal{E}_{t}$) as defined in Eq. 3. The
black solid line corresponds to the 2$\%$ of transmitted energy fraction, and
the black dashed line is for the 40$\%$ transmitted energy fraction. (b) and
(c) show the ion energy spectral maps with varying scale-lengths for $n_{e}$ =
3$n_{c}$ (dashed white line (a)) and 9$n_{c}$ (dashed green line (a))
respectively. The black dashed contour lines in (b) and(c) demarcate the
maximum or cut-off ion energy in each case. The colorbar on top of Figure (b)
represents the number of particles accelerated per energy bin at the
corresponding energy.

Under a realistic experimental scenario, the driving laser pulse always have a
limited temporal contrast, which leads to the expansion of the target front
surface and giving rise to the plasma gradient before the peak of the main
laser is incident on the target. In case of thin targets that are essential
for applications relevant for ion acceleration with mechanisms such as hole-
boring Pukhov and Meyer-ter Vehn (1997) or relativistic transparencyFernández
et al. (2017), the target can be destroyed before the peak of the laser pulse
can interact, if the laser contrast is poor. Influence of pulse temporal
contrast on the acceleration process in the case of optically thicker targets
has been observed experimentally McKenna et al. (2006). Thus, in all these
experiments a high temporal contrast of the main interacting laser, and a
separate fine control of the target plasma density gradient Kahaly et al.
(2013) are the prerequisites. Thus the impact of plasma density gradient on
the physics under discussion cannot be overemphasized. In this section we look
into the influence of plasma density gradient on ion acceleration process.

In this study, we use a $0.75\lambda$ thick foil target with an exponential
density profile, $n(x)=n_{0}exp(-(x-35)/L)$, at the front side of the target
as an extra controlling parameter for the interaction. Where $n_{0}$ and $L$
are the peak electron density and plasma scale-length, respectively. In Figure
5(a), we vary the target peak electron density and the plasma density scale-
length ($L/\lambda$) and plot the transmitted energy fraction as a function of
these two parameters. Altering the target density gradient, significantly
affects the threshold density criteria. Similar to the case of thickness
variation for the step target density profile, here as well we sketch an iso-
line along the different scale-lengths for 2$\%$ and 40 $\%$ of light energy
transmission through the target. With increasing scale-length, the transmitted
energy fraction of the laser pulse decreases, for a fixed laser intensity.
Subsequently, the target threshold density ($n_{th}$) reduces by approximately
30$\%$, due to the increase in overall areal density of the target. Also, we
observe that the iso-lines (2$\%$ and 40$\%$) for the case of varying scale-
length (Figure 5(a)) showed slightly higher threshold density while compared
with the step-like target thickness variation case (Figure 4(a)).

Further, Figure 5(b) and (c) exhibits the spectral features of ion energy as a
function of scale-length for relativistically transparent ($n_{e}=3n_{c}$
marked as the white dashed line in Figure 5(a)) (b) and overdense
($n_{e}=9n_{c}$ marked as the green dashed line in Figure 5(a)) (c) regions
respectively. In Figure 5(b), the ion energy spectra show features similar to
that in Figure 3(e). In this case, the ion cut-off energy increases almost
linearly from $\sim$45 MeV to $\sim$80 MeV with increasing scale-length. The
peak ion energy increases from $\sim$40 MeV to $\sim$65 MeV, closely following
the behaviour of cut-off energy. On the other hand, in Figure 5(c) the ion
energy spectra, in the overdense region ($n_{e}$ = 9$n_{c}$), shows
significantly low increase in the ion cut-off energy from $\sim$30 MeV to
$\sim$40 MeV over the entire range of target scale-length. Although, the peak
ion energy is maintained nearly at $\sim$ 19 MeV with energy spread
$(\Delta\mathcal{E}/\mathcal{E})\%$ = 2.8$\%$ for sharp gradient(L$/\lambda$ =
0) and 7.4$\%$ for long gradient ($L/\lambda=0.80$), where $\mathcal{E}$ is
the peak ion energy and $\Delta\mathcal{E}$ is the FWHM of the peak energy.
This highlights the quasi-monoenergetic feature from the overdense plasma
target. In addition, for a fixed scale-length, if we scan the target densities
(2$n_{c}$ \- 10$n_{c}$), one can transit from transparency to opacity region
interchangeably in a continuous manner. Thus, changing the ion energy spectra
to quasi-monoenergetic distribution. Therefore, control of $L/\lambda$, which
can be achieved experimentally very easily by controlling the delay of a
prepulse arriving before the main pulse starts interacting with the target,
acts as an extra optimization parameter for the production of energetic
protons.

##  V Chirp control of foil dynamics on Gradient Target

Figure 6: Simplified picture elucidating the influence of laser chirp. Upper
panel: The temporal profile of a circularly polarized (a) negatively chirped
($\zeta=-5$), (b) unchirped ($\zeta=0$), and (c) positively chirped
($\zeta=+5$) laser pulse are shown. The projected transverse field components
$a_{y}(t)$ and $a_{z}(t)$ are presented in gray solid lines, plotted on the
respective transverse planes. The variation in time dependent frequency of the
laser pulse as shown in color bar ($\Omega(t)$ as defined in Eq. 5) is encoded
on the color variation across the circularly polarised laser field. The pulse
peak here is located at $t=0$. $\Omega(t)=1$ represents the carrier frequency
and -ve $t$ indicates early in the interaction. Lower panel: The colormaps
plot the ratio of the time dependent laser pulse amplitude $a(t)$ and the
parameter $\xi(t)$ (defined in Eq.6) during the first half of the interaction
(the time axes are zoomed into the range from the start of the laser pulse at
-8$\tau$ up to the peak of the pulse intensity at 0$\tau$) for the step target
case over a range of target densities for (d) negatively chirped, (e)
unchirped, and (f) positively chirped pulse scenarios. The black contour lines
are plotted at $a(t)/\xi(t)=$ 0.7 (in opaque region), 1.0 (in threshold
region), and 1.5 (in transparent region) for the fixed density case of
$n_{e}=6n_{c}$ (black horizontal solid line). As is clearly seen, if target
remains unchanged during the interaction, in (d) for the negatively chirped
case, before the peak of the laser pulse intensity, the interaction at an
initial target density of $n_{e}=6n_{c}$ enters the transparent regime of
operation. (e) The transparent regime is approached near the peak of the pulse
for $\zeta=0$ and (f) the same target meets the transparency condition after
the peak of the laser pulse for $\zeta=+5$.

In addition, to the control over target plasma characteristics, ion
acceleration in laser-plasma interaction can greatly be optimized by tuning
the drive laser optical parameters Galow et al. (2011); Vosoughian et al.
(2015); Mackenroth et al. (2016); Choudhary and Holkundkar (2018). The
influence of laser polarization, pulse duration and peak intensity under
different target conditions have been touched upon previously Liseikina and
Macchi (2007); Macchi et al. (2009a); Macchi and Benedetti (2010); Sahai et
al. (2013); Culfa (2021); Almassarani et al. (2021) and the potential of the
relativistic transparency regime have been demonstrated experimentally
Gonzalez-Izquierdo et al. (2016); Higginson et al. (2018). Here, we introduce
a frequency chirp in the driver pulse, which is relatively simpler to control
in the experiment and investigate its efect on the acceleration process. The
frequency chirp is introduced, within the phase function
$\phi(\eta)=2\pi[\eta+g(\eta,\zeta)]$ in the expression of the laser field in
Eq. 4, where, the chirp function $g(\eta,\zeta)$, is defined in Eq. 4 below:

$\displaystyle
g(\eta,\zeta)=\bigg{(}\zeta\bigg{[}4ln(2)\frac{\eta^{2}}{\tau_{FWHM}^{2}}$
$\displaystyle+\frac{\pi^{2}\tau_{FWHM}^{2}}{4ln(2)(1+\zeta^{2})}\bigg{]}$ (4)
$\displaystyle+\frac{tan^{-1}(\zeta)}{2}\bigg{)}\frac{1}{2\pi}$

where, $\zeta$ is the chirp parameter and $\Omega(\eta)$ is the instantaneous
frequency given by,

$\displaystyle\Omega(\eta)=\frac{1}{2\pi}\frac{\partial\phi(\eta)}{\partial\eta}$
$\displaystyle=1+\frac{\partial g(\eta,\zeta)}{\partial\eta}$ (5)
$\displaystyle=1+\zeta\frac{4ln(2)}{\pi\tau_{FWHM}^{2}}\eta$

In these units, $\Omega=1$ represents the central (at the peak of the pulse)
carrier frequency of the pulse in the unchirped ($\zeta=0$) case. This
functional definition was first proposed in ref.Mackenroth et al. (2016) as a
chirped plane wave model. However, a modification is made to take advantage of
preserving peak field amplitude of the pulse during any variation in the chirp
Choudhary and Holkundkar (2018). This is done in order to ensure that when we
investigate the effect of chirp parameter $\zeta$ on the interaction, we can
keep the other laser parameters fixed. For simplifying the representation in
the following discussion we interchange $\eta$ with $t$ in the expressions
without any loss of generality.

With the chirped Gaussian pulse, defined above, we have modified the PIC code
to incorporate such a CP field and eventually perform a series of simulations
to investigate the effect of frequency chirp on ion acceleration. In Figure 6,
we have presented the (a) negatively chirped, (b) unchirped and (c) positively
chirped laser pulse, that have been used in the simulation. The time dependent
instantaneous frequency of the pulse ($\Omega(t)$) is shown with the color
axis Figure 6 (a-c). In these particular plots the peak of the laser field
envelop is located at $t=0$ and the negative value of time axis indicates
‘earlier’ in the interaction. The chirp is defined in such a way that for a
negatively (positively) chirped pulse, initially high (low) frequency
component interacts with the target followed by the low (high) frequency
component.

Before we delve into the results of fully relativistic PIC simulations, we
discuss here pedagogically the impact chirp might have by extending the
constant parameter $\xi_{0}$, introduced earlier, to an equivalent time
dependent form. Since, a chirp in the laser pulse signifies the change of
laser frequency with time, the corresponding plasma critical density also
becomes time dependent, $n_{c}(t)=(n_{c})_{\omega_{0}}\Omega(t)^{2}$. The
chirped CP plane wave pulse can be conceptually considered as a superposition
of all the monochromatic CP plane waves constructed with strengths
proportional to the instantaneous field envelops and frequencies equal to the
instantaneous frequencies at all times. Hence, the constant parameter
$\xi_{0}$, also changes and is redefined to its instantaneous form as follows,

$\xi(t)=\bigg{(}\frac{n_{e}}{n_{c}}\bigg{)}_{\Omega(t)}\frac{d\pi}{\lambda_{\Omega(t)}}=\bigg{(}\frac{n_{e}}{n_{c}}\bigg{)}_{\omega_{0}}\frac{d\pi}{\lambda}\frac{1}{\Omega(t)}=\frac{\xi_{0}}{\Omega(t)}$
(6)

Thus the ratio relevant to the condition of transparency discussed previously
in section II now becomes,
$\frac{a(t)}{\xi(t)}=\frac{a(t)\Omega(t)}{\xi_{0}}$. The colormaps in Figure 6
plot the value of this ratio for a step target with initial thickness of
$d=0.75\lambda$ and different target peak electron densities at different
instants of time. We emphasize here, that this oversimplified description does
not take into account, among many things, the dynamics of the target and takes
each time instant it treats the situation as stationary. Nonetheless, as we
would see that the discussion provides us some insight on the effect of the
sign of chirp on the interaction. The different cases for negatively chirped
($\zeta=-5$), unchirped ($\zeta=0$) and positively chirped ($\zeta=+5$) pulse
under the condition of monochromatic pulse and step-like target of finite
thickness, are depicted in Figure 6(d), (e) and (f) respectively.

The main difference between transparency and opaque regime of operation can be
distinguished by the ratio of time dependent laser pulse amplitude $a(t)$ and
the parameter $\xi(t)$. As mentioned in ref. Macchi et al. (2009a), the
condition $a_{0}/\xi_{0}>1$, indicates the region of operation in relativistic
self-induced transparency and $a_{0}/\xi_{0}\leq 1$ refers to radiation
pressure acceleration, respectively. Therefore, in Figure 6(d-f), we show the
time dependent ratio of $a(t)$ to $\xi(t)$ in colormap for varying target
density ($n_{e}/n_{c}$) and time ($t/\tau$) in order to identify the instant
of time, if any until the peak of the pulse, at which the target reaches the
condition of transparency. In our present illustration, the pulse reaches peak
amplitude at $t=0$. We consider the time up to the peak amplitude, i.e. from
$-8\tau$ to $0\tau$ to plot the ratio of $a(t)$/$\xi(t)$. Now, to understand
the region of transparency and opacity, we specifically consider the case for
a peak density of $6n_{c}$, shown with a solid horizontal black line along
with three important regions marked with contour lines at $a(t)/\xi(t)=0.7$ in
opaque region, $a(t)/\xi(t)=1.0$, transition or threshold region and
$a(t)/\xi(t)=1.5$, transparent region in Figure 6(d-f).

Before proceeding any further, we first make several observations. Firstly, in
case of negatively chirped pulse, Figure 6(d), since higher frequency
components of the pulse is interacting at the beginning ($t\leq 0$), therefore
the majority of field cycles interacts up to $t=0$. Whereas, for the
positively chirped pulse Figure 6(f) low-frequency cycles of pulse interacts
with the target initially, therefore the number of field cycles interacted up
to the peak $t\leq 0$ is much less as compared to 6(d). Secondly, according to
the ratio ($a(t)/\xi(t)$), at the peak field ($t=0\tau$) instance for
unchirped pulse (Figure 6(e)) at $n_{e}=6n_{c}$ (horizontal black line) the
transparency condition is achieved. Whereas, for negatively (Figure 6(d) and
positively chirped pulse (Figure 6(f), the transparency condition is achieved
at slightly higher target density (approximately at $n_{e}$=6.25$n_{c}$).
Thus, we note that this crossover is before $t=0\tau$ for negatively chirped
pulse and after $t=0\tau$ in positively chirped pulse.

Figure 7: Laser pulse transmitted energy fraction ($\mathcal{E}(t)$) for (a)
negatively and (b) positively chirped pulse with varying target density
($n_{e}/n_{c}$) and scale-length (L/$\lambda$). Similar to the above figures,
the iso-lines for transmitted laser energy fraction are marked at 2 $\%$
(black solid) and 40 $\%$(black dashed). The shaded squares (black square with
grey fill or the reverse) on the colormap in (a) and (b) indicate the specific
transparency conditions corresponding to the primary target electron density.
The point marked in black solid circle with grey fill in (a) and (b)
identifies the target condition where the transparency difference due to
positive and negative chirp is maximum. Ion energy spectrum from (c) the
primary layer (PL) (blue square) having peak density = $5.6n_{c}$ and scale-
length = $0.7\lambda$ and, (d) secondary layer (SL) of thickness =
$0.2\lambda$ and density = $0.1n_{c}$, for unchirped (maroon solid),
positively chirped (yellow solid) and negatively chirped (black solid) pulses.
The spectra are obtained at 70$\tau$, simulation time.

In the real interaction, for the normally incident CP laser pulse, the laser
field would dynamically propagate into the target plasma upto the critical
electron density layer before getting reflected. From the surface of
reflection the light field would penetrate upto skin depth and the
ponderomotive push of the laser would pile up electrons inside the target
leading to a dynamic in the electron density profile Macchi et al. (2009b);
Vincenti et al. (2014). For $\zeta=-5$, the higher frequency initial part
would only start piling up electrons after propagating deeper inside the
target compared to the lower frequency trailing part of the laser. On the
other hand at, $\zeta=+5$ the lower frequency leading part would start spiking
the electron density, which when optimally done, would reflect the following
higher frequency light more efficiently. Thus, unlike the simplistic picture
presented in Figure 6(d-f), in case of a longer ramp in target electron
density, the laser ponderomotive pressure can be expected to be more effective
in the case of $\zeta=+5$ in pushing the electrons out of the target and
rendering the target transparent, compared to the case when $\zeta=-5$, since
a longer ramp helps the snow ploughing effect Kahaly et al. (2013); Vincenti
et al. (2014). Thus, a priory one might expect lesser dependence on chirp in
case of step density targets. In the following, we would study this remarkable
effect of laser chirp on the previously mentioned step-like foil target having
a front density gradient akin to section IV.

In Figure 4(a) and Figure 5(a) we have already discussed the effect of target
thickness and density gradient on the transmitted energy fraction, in the case
of unchirped CP pulses. Now, we expand this idea to the chirped laser pulses.
In Figure 7(a) the transmitted pulse energy fraction for negatively chirped
pulse and (b) for positively chirped pulse over variable scale lengths and
densities are presented. In both the cases, two iso-lines are drawn at 2$\%$
(black solid) and 40$\%$ (black dashed) transmitted laser pulse energy. For a
negatively chirped pulse, in Figure 7(a), the threshold density reduces only
about $10\leavevmode\nobreak\ \%$ over the entire range of scale-length. While
comparing with Figure 5(a), the changes in threshold density with scale-length
is minimal. This is evident in transparent region, where we observe that up to
$4n_{c}$ almost entire pulse energy is transmitted through the target for all
the scale-lengths ($L=0.0\lambda$ to $0.8\lambda$). On the other hand, for the
case of positively chirped pulse, Figure 7(b), at 40$\%$ laser energy
transmission (black dashed), we see the reduction in threshold density of
about 50$\%$ for varying target scale-length. Whereas, for 2$\%$ iso-line
(black solid), we first observe the decrease of $\sim 14\%$ in target
threshold density up to $L=0.4\lambda$. Later, from $L=0.4\lambda$ to
$0.8\lambda$, the threshold density increases by nearly $33\%$. Thereby, the
gap between the 2 $\%$ and 40 $\%$ transmitted energy fraction iso-lines
widens in comparison with the negatively chirped pulse. This effect is due to
continuous compression and pilling of electrons at the rear side from the low
frequency cycles at the beginning of the interaction with the target, followed
by the high frequency cycles which pushes them out of the target surface,
thereby resulting in continuous transmission of laser energy from the target.

We have observed that, while considering the 2$\%$ iso-line one can achieve is
a higher value of threshold density in the positively chirped pulse case. This
implies that experimentally it could be possible switch the domain of
interaction from opaque to transparent regime just by controlling the chirp
parameter. One such contrasting point in the laser-matter parameter space is
indicated using the black circles with grey fill in Figures 7(a-b)
(representing peak density of 5.6$n_{c}$ and scale length as 0.7$\lambda$). In
previous sections we have observed that, over the parameter space of interest
to this work, the accelerated ions show quasi mono energetic spectral features
in the opaque regime of operation. Whereas in the transparent regime the ion
spectra have shown flat behaviour and sharp cutoffs. Nonetheless, we notice
that the bipolar accelerating field generated in the transparent case as shown
in Figure 2(a), shows a nice smooth peak profile located at the exit end of
the target ion density profile. This gives us the idea of adding a thin low
density layer at the back of our initial target, and see whether we can
benefit from this accelerating charge separation field and accelerate ions of
choice. Hence to probe if the transmitted laser pulse can allow us to
accelerate quasi mono-energetic ions we add a thin layer (thickness
0.2$\lambda$) of low-density ($n_{e}=0.1n_{c}$) Hydrogen behind the main
target of thickness 0.75$\lambda$. For the main target we use variable peak
densities (2$n_{c}$ to 10$n_{c}$) and gradient scale-lengths (0$\lambda$ to
0.8$\lambda$). This results in re-configuring of target geometry to the double
layer target. The choice of the parameters of this thin layer are made in such
a way that the electrostatic field created by the main target layer remains
unaltered. In rest, we will refer to the main target as primary layer (PL) and
additional target as secondary layer (SL).

In Figure 7(c,d) we present the ion energy spectra for the cases of unchirped
($\zeta=0$), positive ($\zeta=5$) and negative ($\zeta=-5$) chirped pulses.
The respective ion energy spectra from the PL with peak density of 5.6$n_{c}$
and scale length as 0.7$\lambda$ is shown in Figure 7(c). In Figure 7(d) we
show ion energy spectra from the SL, with above-mentioned parameters. Here, we
choose the PL parameter in such a way that approximately 20$\%$ of the laser
pulse energy propagates through the target (see the black circle with grey
fill in Figure 7(a-b). As mentioned previously, in this case, only positively
chirped pulse allows the transmission of laser pulse through the PL. For the
case of negatively chirped and unchirped pulse, the transmission coefficient
is close to zero. Previously in section II we have already established that,
in the transmission region, the ion energy spectrum shows the near Maxwellian
distribution, whereas a quasi mono-energetic feature is observed in the opaque
region. In Figure 7(c), we observe similar characteristics of Maxwellian
distribution in ion energy spectrum for positively chirped pulse and quasi-
monoenergetic features from negative and unchirped laser pulse. The cut-off
ion energy is slightly higher than 60 MeV for positively chirped pulse and
approximately 60 MeV for negatively chirped pulse. Whereas, for unchirped
pulse, the ion cut-off energy is around 40 MeV. Thus, one can utilize chirped
pulses to effectively enhance the ion cut-off energy in both the regimes
(transparency and opacity).

Furthermore, in Figure 7(d) we show the significance of the SL in the
transparency regime. Since, only positive chirp pulse allows the laser to
transmit through the PL for a specific set of parameters as indicated in
Figure 7(c), therefore, a mono-energetic ion bunch with a peak energy at 35
MeV and cut-off energy reaching close to 45 MeV is observed. Thus, with a
secondary layer along with a target of suitable parameters, one can obtain
mono-energetic bunches in a controlled manner. On the contrary, for the case
of negatively chirped and unchirped pulse, since the PL parameters lies in the
opaque region, we observe a clear quasi-monoenergetic spectral signature in
ion spectra from PL peaking at 20 MeV (Figure 7(c)) and negligible ion
energies from the SL (Figure 7(d)).

Figure 8: The ion energy spectra from double layer target, composed of
Deuterium (PL) and thin, low-density ($\mathrm{0.2\lambda}$, $0.1n_{c}$)
Hydrogen (SL). (a,b) and (c,d) correspond to ion energy spectra from both the
layers, for primary layers with peak electron density $3n_{c}$, and $9n_{c}$
respectively, with scale-length of 0.4$\lambda$ in all the cases. The color of
the traces indicate the laser chirp conditions: positively chirped (yellow
solid), unchirped (maroon solid) and negatively chirped (black solid) cases.
The shaded squares (black square with grey fill or the reverse) on the top
left corner of each sub-figure correlates with those marked in Figure 7(a,b)
identifying the transparency conditions corresponding to the primary target
electron density.

##  VI Tuning into Quasi-Monoenergetic ion spectrum

The key aspect we observed in the section above is that, using chirped pulses
one can precisely control the laser transmission through the target with
specific properties or in other words, enhance the threshold density of the
target for the incident laser pulse. In Figure 7(d), we also observed that
using an additional layer behind the main target (double layer target) leads
to the generation of mono-energetic ion bunches in the transparency region.
Detailed modelling on the optimization of a double layer target parameter in a
different context has been proposed Pazzaglia et al. (2020b). Here we keep the
configuration of the second layer fixed and look more into the physics
aspects. More specifically in order to gain further control over ion energy
spectra, we investigate spectral features from both the layers independently
while controlling the target species and establish the proof of principle.

Towards achieving this goal, we have used a target to be composed of two
layers of different species, Deuterium in primary layer (PL) and Hydrogen in
secondary layer (SL). The target peak densities of the PL are similar to
previous section IV, $n_{e}=3n_{c}$, corresponding to transparent region and
$n_{e}=9n_{c}$, corresponding to opaque region, respectively. Also, the double
layer thickness (PL : $0.75\lambda$, SL : $0.2\lambda$) is kept similar to
Figure 7(c,d) with scale-length of $L=0.4\lambda$. The ion energy spectra are
presented in Figure 8 for unchirped, positive and negatively chirped pulses,
where (a,c) is for Deuterium target (PL) and (b,d) corresponds to Hydrogen
layer (SL). In Figure 8(a), we have observed the spectral signature similar to
Maxwellian distribution, peaking near the cut-off energy as in Figure 3(e) at
70$\tau$. For the case of negatively chirped pulse, 50$\%$ reduction in ion
cut-off energy was observed while comparing with positively chirped and
unchirped pulse. Since, for these specific set of parameters ($n_{e}$
=3$n_{c}$, d = 0.75$\lambda$, L = 0.4$\lambda$), the transmitted energy
fraction is substantially higher in negatively chirp pulse, the charge
separation field cannot sustain for longer duration to effectively accelerate
the ions. On the other hand, the number of charged particles being accelerated
are almost same for all three cases.

In Figure 8(b), we present the accelerated ion energy spectra from the
secondary layer, when the density in the PL is $3n_{c}$. Prominent mono-
energetic ion bunches with peak energy $\mathrm{\sim 22.6\,MeV}$ and energy
spread ($\Delta\mathcal{E}/\mathcal{E})\sim 28.6\%$ for the case of negatively
chirped pulse was observed. For positively chirped the peak energy and energy
spread are $\mathrm{\sim 41.7\,MeV}$ and ($\Delta\mathcal{E}/\mathcal{E})\sim
18.29\%$. And for unchirped pulse, the peak energy and energy spread are
$\mathrm{\sim 43.0\,MeV}$ and ($\Delta\mathcal{E}/\mathcal{E})\sim 19.8\%$)
respectively. The reason behind such mono-energetic behavior in ion spectra as
reconstructed from the PIC simulation results lies in the dynamics of charge
separation field. In Figure 8(c), the ion energy spectra from the
relativistically overdense primary layer (Deuterium) are shown, whereas in
Figure 8(d) the spectra are from secondary layer (Hydrogen). The spectra from
PL exhibits similar characteristics with Figure 3(f), single layer target with
the same target parameter. In Figure 8(c), it was observed that all three
cases results in the peak ion energy at the same position of almost 10 MeV,
with different cut-off energy. The maximum ion cut-off energy for positively
and negatively chirped pulses is around 30 MeV, whereas, for the case
unchirped pulse is around 20 MeV. Since, for this density, the electrostatic
field created is much lower than the case of $n_{e}=3n_{c}$. It is evident
from Figure 2(d) in sec. II for the case of unchirped pulse with single
target. This lower electrostatic field, however, remains in the vicinity of
the target surface. Since, laser pulse does not transmit through the target at
this interaction condition, therefore, low energy ion bunches were observed
from the secondary layer as shown in Figure 8(d). The maximum ion energy of
the bunches reaches upto 4 MeV only for positively chirped pulse.

Therefore, we would like to highlight that either one can generate quasi-
monoenergetic bunches from the single layer target in opaque region, or from
secondary layer in transparency region. Additionally, we get a clear
indication from the spectra about the target species being accelerated.

##  VII Ion beam quality and influence of laser focusing on the acceleration
regimes

Figure 9:  Target ion density map for the primary (Deuterium) and secondary
(Hydrogen) layers are shown in (a) for $n_{e}=3n_{c}$ (transparency regime)
and (d) $n_{e}=9n_{c}$ (reflection regime). The colorbars represent the ion
densities in log-scale for both the species in yellow-green for $D^{+}$ and
yellow-red for $H^{+}$. (b) and (e), ion energy spectrum corresponding to the
central line outs for $n_{e}=3n_{c}$ and $9n_{c}$ respectively (PL: Deuterium
layer in blue color and SL: Hydrogen layer in orange color). The angular
energy distribution of the predominant species is shown with the polar plots
for both the cases, in (c) $\mathrm{H^{+}}$, 3$n_{c}$ and (f)
$\mathrm{D^{+}}$, 9$n_{c}$. The colorbar represents the number of ions
accelerated at a certain angle with a specific energy in MeV. The ion energy
2D spatial distributions and the spectra are obtained at $t=48.806\tau$, i.e.
24$\tau$ after the peak interaction of laser pulse with the target, similar to
the condition in 1D simulations.

In order to probe the interaction deeper and to validate our previous
observations we have conducted 2D fully relativistic Particle-in-Cell
simulations using the code WarpX Vay et al. (2021). A Gaussian laser pulse
(Gaussian in both space and time) of wavelength 1$\mu m$, duration of 10.0$fs$
(FWHM of the intensity profile) as in 1D PIC simulations and beam waist
($1/e^{2}$ radius of intensity spatial profile at focus) of 8$\mu m$ is
normally incident on a double layer plasma target. The laser and target
conditions are similar to the conditions used in Figure 8, where the primary
layer (PL) target is made of Deuterium with thickness 0.75$\mu m$ and the
secondary layer (SL) is made of Hydrogen having thickness of 0.2$\mu m$. The
simulation box is 30$\mu m$ $\times$ 30$\mu m$, with the double layer target
located at the center (x=0). The simulation runs with a spatial resolution of
170 cells per wavelength and with 4 ions and 6 electrons in each dimension,
respectively. We have considered an exponential density ramp with scale-length
0.4$\mu m$ in front of the composite target. In case of PL, we have considered
two electron densities $n_{e}=3n_{c}$ and $9n_{c}$, with SL having
$n_{e}=0.1n_{c}$.

In Figure 9, we have presented the 2D PIC simulation results for the
transparency (upper panel) and the opaque regimes (lower panel) featuring the
ion density and energy spectrum from primary layer and secondary layer for
focal spot of 8$\mu m$. Figure 9 captures the time snapshot of the process,
long after the interaction is over when the laser pulse has already left the
simulation box either from the right side (post-transmission) or from the left
side (post-reflection). Figure 9(a) clearly shows that the target ions are
expanding with higher rate in laser propagation direction. The SL ions have
moved ahead of the PL ions. As is evident, due to lower mass of ions in SL,
they move faster in comparison with the ions in PL, given the same
accelerating field. Whereas, in Figure 9(d), we observed that as the laser
pulse impinges on the target, the target ions expand from both the front and
back surface. This leads to the formation of double peak structure in ion
density profile as in Figures 1(d), 2(d) and 3(d). Although, ions in SL are
lighter in mass, but they stay in close vicinity to primary layer ions. Since,
they experience reduced charge separation field at the rear end of the target.
This nature of primary layer ion density upholds with the 1D behavior, as it
was observed in Figure 1(b) and (d).

Next, the ion energy spectra from both the layers (PL and SL) for the
relativistically transparent region ($n_{e}=3n_{c}$) are presented in Figure
9(b). In this case, the ions in PL exhibit the spectral signature similar to
Maxwellain distribution, as in Figure 8(a) for the case of unchirped pulse,
with cut-off energy of around 20 MeV. Whereas, the ion spectra from the SL are
mono-energetic, peaking at $\sim$24 MeV and with energy spread
($\Delta\mathcal{E}/\mathcal{E}$) of 25.9$\%$. The ion energy (peak and cut-
off) obtained from 2D PIC are slightly lower than 1D PIC, however, 2D-PIC
simulations are able to qualitatively reproduce the characteristic of the ion
spectra. On the other hand, in Figure 9(e) the ion energy spectra from
overdense region ($n_{e}=9n_{c}$) is shown. Here, from PL we observe a quasi-
monoenergetic ion spectra similar to Figure 8(b) for unchirped pulse, with
peak energy at around 11 MeV with energy spread
($\Delta\mathcal{E}/\mathcal{E}$) of 8.5$\%$ and cut-off energy of 22 MeV.
Whereas, only few MeV ions from the secondary layer. The ion energy spectra
from the overdense region ($n_{e}$ = 9$n_{c}$) closely corroborate with the 1D
PIC simulation results as shown in Figure 8(b,d).

Peak density $n_{e}=3n_{c}$ $n_{e}=9n_{c}$ Energy (MeV) Peak Cut-off Peak Cut-
off Focalspot radius ($\mu$m) PL($\mathrm{D^{+}}$) SL($\mathrm{H^{+}}$)
PL($\mathrm{D^{+}}$) SL($\mathrm{H^{+}}$) PL($\mathrm{D^{+}}$)
SL($\mathrm{H^{+}}$) PL($\mathrm{D^{+}}$) SL($\mathrm{H^{+}}$) 4 15 13 15 20 9
0.73 21 1.45 8 28 24 28 29 11 0.21 22 0.41 12 41 38 42 45 10.7 0.21 29 0.42

Table 1: Peak and cut-off ion energy values obtained in relativistic
transparent region ($n_{e}$ = 3$n_{c}$) and overdense region ($n_{e}$ =
9$n_{c}$) from double layer target configuration (primary layer (PL) as
$\mathrm{D^{+}}$ and secondary layer (SL) as $\mathrm{H^{+}}$) with three
different driving focal spot radius ($1/e^{2}$ radius of the intensity spatial
envelop at focus) i.e. 4$\mu m$, 8$\mu m$ and 12$\mu m$. Figure 9 summarizes
the results corresponding to the case of intermediate focal spot radius of
8$\mu m$ presented in the table.

In Figure 9(c) and (f), polar ion energy spectral map is presented from the
target layers to investigate the divergence of the ion beams. Figure 9 (c)
corresponds to the $\mathrm{H^{+}}$ ion spectrum shown in 9(b). Whereas Figure
9 (f), corresponds to the $\mathrm{D^{+}}$ ion spectrum shown in 9(e). These
two cases in the simulations exhibit mono-energetic and quasi-monoenergetic
spectral characteristics. Although, the divergence of ions in Figure 9(c) is
moderately higher while compared with Figure 9(f), an interesting feature is
observed in 9(c), where ions in the central area (center of the laser focus)
accelerate with higher energy compared to the ions at the peripheral area.
Since, ions at the focus center experiences higher laser intensity and are
pushed by the driver laser at higher rate in the transparency conditions.
Therefore, only the central area of the accelerated ions in 9(c) can construct
a mono-energetic spectrum, such as shown in 9(b). Whereas, the ion energies
from the $\mathrm{D^{+}}$ target in opaque region show the low divergence
accelerated ions, due to reflection of the driving laser pulse from the
target.

For the sake of completeness and to verify the dimensional effect in our
simulation, we have studied the 2D effect on ion acceleration. In 2D
simulations, not only the dimension of the target changes to 2D-plane but also
the driving laser pulse waist is taken into account, therefore focal spot
plays a crucial role in determining the maximum or cut-off and peak ion
energies from the targets. We have summarized the ion energy cut-off and peak
values in Table 1 for the above mentioned target species, density and
thickness with different laser focal spot sizes. We observed that in case of
tight focusing i.e. 4$\mu m$, the ion energies (peak and cut-off) obtained
from PL and SL in transparency region ($n_{e}$ = 3$n_{c}$) is almost half than
that from focal spot size of 8$\mu m$. Similarly, if we further increase the
focal spot size three-fold i.e. 12$\mu m$, the ion energies are also enhanced
almost three times, when compared with focal spot size of 4$\mu m$ and almost
1.5times higher than focal spot size of 8$\mu m$. Here, we would like to
specify that the laser intensity is kept constant through, thus effectively
means an increase in the laser energy. On the other hand, the ion energies
from the opaque region ($n_{e}$ = 9$n_{c}$) does not show significant changes
with the variation in focal spot size. This is due to the completely
reflection of the laser pulse irrespective to the focal spot sizes, therefore
the electron dynamics and charge separation field remain almost identical for
different focal spot sizes. When we compare these ion energies obtained from
the 1D PIC simulation as in Figure 8, are approximately in range for 12$\mu m$
focal spot size. Since, larger focal spot size (spot size $\geq$ 12$\mu m$)
mimics the laser conditions as of 1D simulation near the axis.

##  VIII Concluding Remarks

In this article, we have demonstrated that by tuning accessible sub-PW class
laser parameters and target conditions, one can harness the transition between
two-different acceleration mechanisms, namely Relativistic Induced
Transparency (RIT) and Radiation Pressure Acceleration (RPA) to directly
administer the ion bunch spectral features to a large extent. This study
enables us to optimize the features of the accelerated ion beams by scanning
the target parameters (density, thickness and geometry), but also adding
additional flexibility via controlling the laser frequency chirp and prepulses
(to create a suitable preplasma). We have demonstrated that the interaction
conditions for each of the ion acceleration mechanisms of our interest (RIT
and RPA) can be identified by analyzing the transmission energy coefficient
($\mathcal{E}_{t}$) with varying target and laser parameters.

Due to very different charge particle dynamics in RIT and in RPA region, the
charge separation field generated in RIT is substantially stronger when
compared with the field generated in the opaque region leading to considerably
different spectral features generated in RIT and RPA, Maxwellian-like ion
spectra in the RIT and quasi-mono-energetic in the RPA. We have provided a
thorough prescription on how to go from one regime of operation to another and
effectively generate quasimonoenergetic ion spectrun. Furthermore, we have
established the proof of principal that any chosen ion species accelerated to
a quasimonoenergetic energy spectrum under conditions realizable and easily
controllable in experiments, using a double layer configuration. 2D PIC
simulations clearly unveil the ion acceleration process from different target
layer and establish the ion beam quality. In addition, the focal spot
dimension plays an important role and the ion energy spectral behavior with
larger focal spot corroborate with 1D PIC simulation result. With the large
number of physical parameters involved, this numerical study is important for
the future experimental verification and further extension towards
multidisciplinary applications of laser driven ion beams Mondal et al. (2018);
Margarone et al. (2018). The possibility of controlling transition between
different ion acceleration mechanisms, within the same experimental setup, can
pave the way to optimize the ion energies and spectral features depending on
region of operation. In addition the possibility of controlled experiments in
this direction might open the doors to study the correlation of ion
acceleration with proposed cross disciplinary studies like strong field plasma
quantum opticsLamprou et al. (2021).

## Acknowledgements

This research has been supported by the IMPULSE project which receives funding
from the European Union Framework Programme for Research and Innovation
Horizon 2020 under grant agreement No 871161. ELI-ALPS is supported by the
European Union and co-financed by the European Regional Development Fund
(ERDF) (GINOP-2.3.6-15-2015-00001). S.K. and S.M. acknowledges Project No.
2020-1.2.4-TÉT-IPARI-2021-00018, which has been implemented with support
provided by the National Research, Development and Innovation Office of
Hungary, and financed under the 2020-1.2.4-TET-IPARI-CN funding scheme. S.C.
and S.K. acknowledge the High Performance Computation (HPC) facility/service
at ELI-ALPS.

## References

  * Fedeli et al. (2018) L. Fedeli, A. Formenti, L. Cialfi, A. Pazzaglia, and M. Passoni, Scientific Reports 8, 3834 (2018), ISSN 2045-2322, URL http://www.nature.com/articles/s41598-018-22147-6.
  * Ji et al. (2018a) L. Ji, B. Shen, and X. Zhang, New Journal of Physics 20, 053043 (2018a), ISSN 1367-2630, URL https://iopscience.iop.org/article/10.1088/1367-2630/aac24b.
  * Sylla et al. (2012) F. Sylla, M. Veltcheva, S. Kahaly, A. Flacco, and V. Malka, Review of Scientific Instruments 83, 033507 (2012), ISSN 0034-6748, URL http://scitation.aip.org/content/aip/journal/rsi/83/3/10.1063/1.3697859http://aip.scitation.org/doi/10.1063/1.3697859.
  * Lifschitz et al. (2014) A. Lifschitz, F. Sylla, S. Kahaly, A. Flacco, M. Veltcheva, G. Sanchez-Arriaga, E. Lefebvre, and V. Malka, New Journal of Physics 16, 033031 (2014), URL https://doi.org/10.1088/1367-2630/16/3/033031.
  * Kahaly et al. (2016) S. Kahaly, F. Sylla, A. Lifschitz, A. Flacco, M. Veltcheva, and V. Malka, Scientific Reports 6 (2016), URL https://doi.org/10.1038/srep31647.
  * 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, et al., Review of Scientific Instruments 90, 063302 (2019), ISSN 0034-6748, URL http://aip.scitation.org/doi/10.1063/1.5093613.
  * Yogo et al. (2008) A. Yogo, H. Daido, S. V. Bulanov, K. Nemoto, Y. Oishi, T. Nayuki, T. Fujii, K. Ogura, S. Orimo, A. Sagisaka, et al., Physical Review E 77, 016401 (2008), ISSN 1539-3755, URL https://link.aps.org/doi/10.1103/PhysRevE.77.016401.
  * Pazzaglia et al. (2020a) A. Pazzaglia, L. Fedeli, A. Formenti, A. Maffini, and M. Passoni, Communications Physics 3 (2020a), URL https://doi.org/10.1038/s42005-020-00400-7.
  * Martinez et al. (2020) B. Martinez, E. D’Humières, and L. Gremillet, Physical Review Research 2, 043341 (2020), ISSN 2643-1564, eprint 2006.16603, URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.043341.
  * Wang et al. (2021) P. Wang, G. Qi, Z. Pan, D. Kong, Y. Shou, J. Liu, Z. Cao, Z. Mei, S. Xu, Z. Liu, et al., High Power Laser Science and Engineering 9, e29 (2021), ISSN 2095-4719, URL https://www.cambridge.org/core/product/identifier/S2095471921000189/type/journal{_}article.
  * Danson et al. (2019) C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-C. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas, L. A. Gizzi, J. Hein, D. I. Hillier, et al., High Power Laser Science and Engineering 7, e54 (2019), ISSN 2095-4719, URL https://www.cambridge.org/core/product/identifier/S2095471919000367/type/journal{_}article.
  * Radier et al. (2022) C. Radier, O. Chalus, M. Charbonneau, S. Thambirajah, G. Deschamps, S. David, J. Barbe, E. Etter, G. Matras, S. Ricaud, et al., High Power Laser Science and Engineering 10 (2022), URL https://doi.org/10.1017/hpl.2022.11.
  * Kühn et al. (2017) S. Kühn, M. Dumergue, S. Kahaly, S. Mondal, M. Füle, T. Csizmadia, B. Farkas, B. Major, Z. Várallyay, E. Cormier, et al., Journal of Physics B: Atomic, Molecular and Optical Physics 50, 132002 (2017), ISSN 0953-4075, URL http://stacks.iop.org/0953-4075/50/i=13/a=132002?key=crossref.3201ce77c8a33bfcf55456334c5ce7d6https://iopscience.iop.org/article/10.1088/1361-6455/aa6ee8.
  * Charalambidis et al. (2017) D. Charalambidis, V. Chikán, E. Cormier, P. Dombi, J. A. Fülöp, C. Janáky, S. , M. Kalashnikov, C. Kamperidis, S. Kühn, et al., in _Progress in Ultrafast Intense Laser Science XIII, Springer Series in Chemical Physics_ , edited by K. Yamanouch (Springer, Cham, 2017), pp. 181–218, URL https://doi.org/10.1007/978-3-319-64840-8_10.
  * Mondal et al. (2018) S. Mondal, M. Shirozhan, N. Ahmed, M. Bocoum, F. Boehle, A. Vernier, S. Haessler, R. Lopez-Martens, F. Sylla, C. Sire, et al., Journal of the Optical Society of America B 35, A93 (2018), ISSN 0740-3224, URL https://www.osapublishing.org/abstract.cfm?URI=josab-35-5-A93.
  * Margarone et al. (2018) D. Margarone, G. A. P. Cirrone, G. Cuttone, A. Amico, L. Andò, M. Borghesi, S. S. Bulanov, S. V. Bulanov, D. Chatain, A. Fajstavr, et al., Quantum Beam Science 2 (2018), ISSN 2412-382X, URL https://www.mdpi.com/2412-382X/2/2/8.
  * Macchi et al. (2013) A. Macchi, M. Borghesi, and M. Passoni, Rev. Mod. Phys. 85, 751 (2013), URL http://link.aps.org/doi/10.1103/RevModPhys.85.751.
  * Schreiber et al. (2016) J. Schreiber, P. R. Bolton, and K. Parodi, Review of Scientific Instruments 87, 071101 (2016), eprint https://aip.scitation.org/doi/pdf/10.1063/1.4959198, URL https://aip.scitation.org/doi/abs/10.1063/1.4959198.
  * Passoni et al. (2019) M. Passoni, F. M. Arioli, L. Cialfi, D. Dellasega, L. Fedeli, A. Formenti, A. C. Giovannelli, A. Maffini, F. Mirani, A. Pazzaglia, et al., Plasma Physics and Controlled Fusion 62, 014022 (2019), URL https://doi.org/10.1088%2F1361-6587%2Fab56c9.
  * Bulanov and Khoroshkov (2002) S. Bulanov and V. Khoroshkov, Plasma Phys. Rep. 28, 453 (2002), ISSN 1063-780X.
  * Bulanov et al. (2014) S. Bulanov, J. Wilkens, T. Esirkepov, G. Korn, G. Kraft, S. Kraft, M. Molls, and V. Khoroshkov, Physics-Uspekhi 57, 1149 (2014).
  * Karsch et al. (2017) L. Karsch, E. Beyreuther, W. Enghardt, M. Gotz, U. Masood, U. Schramm, K. Zeil, and J. Pawelke, Acta Oncologica 56, 1359 (2017), URL https://doi.org/10.1080/0284186X.2017.1355111.
  * Weng et al. (2018) S. Weng, Z. Sheng, M. Murakami, M. Chen, M. Liu, H. Wang, T. Yuan, and J. Zhang, Matter Radiat. Extremes 3, 28 (2018), ISSN 2468-080X, heavy-ion-driven fusion and HEDP (I), URL http://www.sciencedirect.com/science/article/pii/S2468080X17300456.
  * Roth et al. (2001) M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, et al., Phys. Rev. Lett. 86, 436 (2001).
  * Barberio et al. (2018) M. Barberio, M. Scisció, S. Valliéres, F. Cardelli, S. N. Chen, G. Famulari, T. Gangolf, G. Revet, A. Schiavi, M. Senzacqua, et al., Nat. Commun. 9, 372 (2018), URL https://doi.org/10.1038/s41467-017-02675-x.
  * Roth et al. (2013) M. Roth, D. Jung, K. Falk, N. Guler, O. Deppert, M. Devlin, A. Favalli, J. Fernández, D. Gautier, M. Geissel, et al., Phys. Rev. Lett. 110, 044802 (2013), URL http://link.aps.org/doi/10.1103/PhysRevLett.110.044802.
  * 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, et al., Phys. Rev. Lett. 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, Phys. Plasmas, 8, 542 (2001).
  * Mora (2003) P. Mora, Phys. Rev. Lett. 90, 185002 (2003), URL https://link.aps.org/doi/10.1103/PhysRevLett.90.185002.
  * Hegelich et al. (2006) B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letzring, M. Paffett, H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernández, Nature 439, 441 (2006), URL http://dx.doi.org/10.1038/nature04400.
  * Ogura et al. (2012) K. Ogura, M. Nishiuchi, A. S. Pirozhkov, T. Tanimoto, A. Sagisaka, T. Z. Esirkepov, M. Kando, T. Shizuma, T. Hayakawa, H. Kiriyama, et al., Opt. Lett. 37, 2868 (2012), URL http://opg.optica.org/ol/abstract.cfm?URI=ol-37-14-2868.
  * Passoni et al. (2010) M. Passoni, L. Bertagna, and A. Zani, New J. Phys. 12, 045012 (2010).
  * Kroll et al. (2022) F. Kroll, F.-E. Brack, C. Bernert, S. Bock, E. Bodenstein, K. Brüchner, T. E. Cowan, L. Gaus, R. Gebhardt, U. Helbig, et al., Nature Physics 18, 316 (2022), URL https://doi.org/10.1038/s41567-022-01520-3.
  * Chou et al. (2022) H.-G. J. Chou, A. Grassi, S. H. Glenzer, and F. Fiuza, Physical Review Research 4 (2022), URL https://doi.org/10.1103/physrevresearch.4.l022056.
  * Iwata et al. (2018) N. Iwata, S. Kojima, Y. Sentoku, M. Hata, and K. Mima, Nature Communications 9, 623 (2018), URL https://doi.org/10.1038/s41467-018-02829-5.
  * Robinson et al. (2006) A. P. L. Robinson, A. R. Bell, and R. J. Kingham, Phys. Rev. Lett. 96, 035005 (2006), URL https://link.aps.org/doi/10.1103/PhysRevLett.96.035005.
  * Robinson et al. (2008) A. P. L. Robinson, M. Zepf, S. Kar, R. G. Evans, and C. Bellei, New J. Phys. 10, 013021 (2008), URL http://stacks.iop.org/1367-2630/10/i=1/a=013021.
  * Macchi et al. (2009a) A. Macchi, S. Veghini, and F. Pegoraro, Phys. Rev. Lett. 103, 085003 (2009a), URL http://link.aps.org/doi/10.1103/PhysRevLett.103.085003.
  * Macchi et al. (2010) A. Macchi, S. Veghini, T. V. Liseykina, and F. Pegoraro, New J. Phys. 12, 045013 (2010), URL http://stacks.iop.org/1367-2630/12/i=4/a=045013.
  * Ma et al. (2019) W. J. Ma, I. J. Kim, J. Q. Yu, I. W. Choi, P. K. Singh, H. W. Lee, J. H. Sung, S. K. Lee, C. Lin, Q. Liao, et al., Phys. Rev. Lett. 122, 014803 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.122.014803.
  * Sprangle et al. (1990) P. Sprangle, E. Esarey, and A. Ting, Phys. Rev. Lett. 64, 2011 (1990), URL https://link.aps.org/doi/10.1103/PhysRevLett.64.2011.
  * Fedeli et al. (2017) L. Fedeli, A. Formenti, C. E. Bottani, and M. Passoni, The European Physical Journal D 71 (2017), URL https://doi.org/10.1140/epjd/e2017-80222-7.
  * Cattani et al. (2000) F. Cattani, A. Kim, D. Anderson, and M. Lisak, Phys. Rev. E 62, 1234 (2000), URL http://link.aps.org/doi/10.1103/PhysRevE.62.1234.
  * Goloviznin and Schep (2000) V. V. Goloviznin and T. J. Schep, Phys. Plasmas 7, 1564 (2000).
  * Siminos et al. (2012) E. Siminos, M. Grech, S. Skupin, T. Schlegel, and V. T. Tikhonchuk, Phys. Rev. E 86, 056404 (2012), URL https://link.aps.org/doi/10.1103/PhysRevE.86.056404.
  * Fernández et al. (2017) J. C. Fernández, D. C. Gautier, C. Huang, S. Palaniyappan, B. J. Albright, W. Bang, G. Dyer, A. Favalli, J. F. Hunter, J. Mendez, et al., Phys. Plasmas 24, 056702 (2017), URL https://doi.org/10.1063/1.4983991.
  * Sahai et al. (2013) A. A. Sahai, F. S. Tsung, A. R. Tableman, W. B. Mori, and T. C. Katsouleas, Phys. Rev. E 88, 043105 (2013), URL https://link.aps.org/doi/10.1103/PhysRevE.88.043105.
  * Poole et al. (2018) P. L. Poole, L. Obst, G. E. Cochran, J. Metzkes, H.-P. Schlenvoigt, I. Prencipe, T. Kluge, T. Cowan, U. Schramm, D. W. Schumacher, et al., New J. Phys. 20, 013019 (2018), URL http://stacks.iop.org/1367-2630/20/i=1/a=013019.
  * Palaniyappan et al. (2012) S. Palaniyappan, B. M. Hegelich, H.-C. Wu, D. Jung, D. C. Gautier, L. Yin, B. J. Albright, R. P. Johnson, T. Shimada, S. Letzring, et al., Nat. Phys. 8, 763 EP (2012), article, URL http://dx.doi.org/10.1038/nphys2390.
  * Higginson et al. (2018) A. Higginson, R. J. Gray, M. King, R. J. Dance, S. D. R. Williamson, N. M. H. Butler, R. Wilson, R. Capdessus, C. Armstrong, J. S. Green, et al., Nat. Commun. 9, 724 (2018), URL https://doi.org/10.1038/s41467-018-03063-9.
  * Singh et al. (2022) P. K. Singh, F. Y. Li, C. K. Huang, A. Moreau, R. Hollinger, A. Junghans, A. Favalli, C. Calvi, S. Wang, Y. Wang, et al., Nature Communications 13 (2022), ISSN 20411723, eprint 2110.13942.
  * Schillaci et al. (2022) F. Schillaci, L. Giuffrida, M. Tryus, F. Grepl, S. Stancek, A. Velyhan, V. Istokskaia, T. Levato, G. Petringa, G. A. P. Cirrone, et al., Quantum Beam Science 6, 30 (2022), URL https://doi.org/10.3390/qubs6040030.
  * Kaw and Dawson (1970) P. Kaw and J. Dawson, Phys. Fluids 13, 472 (1970), URL http://scitation.aip.org/content/aip/journal/pof1/13/2/10.1063/1.1692942.
  * Vshivkov et al. (1998) V. A. Vshivkov, N. M. Naumova, F. Pegoraro, and S. V. Bulanov, Physics of Plasmas 5, 2727 (1998), eprint https://doi.org/10.1063/1.872961, URL https://doi.org/10.1063/1.872961.
  * Lichters et al. (1997) R. Lichters, R. E. W. Pfund, and J. Meyer-Ter-Vehn, MPQ Report 225 (1997), URL http://www.lichters.net/download.html.
  * Choudhary and Holkundkar (2016) S. Choudhary and A. R. Holkundkar, Eur. Phys. J. D 70, 234 (2016), URL https://doi.org/10.1140/epjd/e2016-70391-2.
  * Willingale et al. (2011) L. Willingale, P. M. Nilson, A. G. R. Thomas, S. S. Bulanov, A. Maksimchuk, W. Nazarov, T. C. Sangster, C. Stoeckl, and K. Krushelnick, Physics of Plasmas 18, 056706 (2011), eprint https://doi.org/10.1063/1.3563438, URL https://doi.org/10.1063/1.3563438.
  * Obst et al. (2017) L. Obst, S. Göde, M. Rehwald, F. Brack, J. Branco, S. Bock, M. Bussmann, T. E. Cowan, C. B. Curry, F. Fiuza, et al., Scientific Reports 7, 10248 (2017), URL https://doi.org/10.1038/s41598-017-10589-3.
  * Polz et al. (2019) J. Polz, A. P. L. Robinson, A. Kalinin, G. A. Becker, R. A. C. Fraga, M. Hellwing, M. Hornung, S. Keppler, A. Kessler, D. Klöpfel, et al., Scientific Reports 9, 16534 (2019), URL https://doi.org/10.1038/s41598-019-52919-7.
  * Ji et al. (2018b) L. Ji, B. Shen, and X. Zhang, New Journal of Physics 20, 053043 (2018b), URL https://doi.org/10.1088/1367-2630/aac24b.
  * Guérin et al. (1996) S. Guérin, P. Mora, J. C. Adam, A. Héron, and G. Laval, Physics of Plasmas 3, 2693 (1996), URL https://doi.org/10.1063/1.871526.
  * Robinson et al. (2009) A. Robinson, P. Gibbon, M. M Zepf, S. Kar, R. Evans, and B. C., Plasma Phys. Control. Fusion 51, 024004 (2009).
  * Robinson et al. (2012) A. P. L. Robinson, R. M. G. M. Trines, N. P. Dover, and Z. Najmudin, Plasma Phys. Control. Fusion 54, 115001 (2012), URL http://stacks.iop.org/0741-3335/54/i=11/a=115001.
  * Weng et al. (2012) S. M. Weng, M. Murakami, P. Mulser, and Z. M. Sheng, New Journal of Physics 14, 063026 (2012), URL https://doi.org/10.1088%2F1367-2630%2F14%2F6%2F063026.
  * Pukhov and Meyer-ter Vehn (1997) A. Pukhov and J. Meyer-ter Vehn, Physical review letters 79, 2686 (1997).
  * Fernández et al. (2017) J. C. Fernández, D. Cort Gautier, C. Huang, S. Palaniyappan, B. J. Albright, W. Bang, G. Dyer, A. Favalli, J. F. Hunter, J. Mendez, et al., Physics of plasmas 24, 056702 (2017).
  * McKenna et al. (2006) P. McKenna, F. Lindau, O. Lundh, D. Neely, A. Persson, and C.-G. Wahlström, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, 711 (2006), URL https://doi.org/10.1098/rsta.2005.1733.
  * Kahaly et al. (2013) S. Kahaly, S. Monchocé, H. Vincenti, T. Dzelzainis, B. Dromey, M. Zepf, P. Martin, and F. Quéré, Phys. Rev. Lett. 110, 175001 (2013), URL https://link.aps.org/doi/10.1103/PhysRevLett.110.175001.
  * Galow et al. (2011) B. J. Galow, Y. I. Salamin, T. V. Liseykina, Z. Harman, and C. H. Keitel, Phys. Rev. Lett. 107, 185002 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.185002.
  * Vosoughian et al. (2015) H. Vosoughian, Z. Riazi, H. Afarideh, and E. Yazdani, Phys. Plasmas 22, 073110 (2015), URL https://doi.org/10.1063/1.4926817.
  * Mackenroth et al. (2016) F. Mackenroth, A. Gonoskov, and M. Marklund, Phys. Rev. Lett. 117, 104801 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.117.104801.
  * Choudhary and Holkundkar (2018) S. Choudhary and A. R. Holkundkar, Phys. Plasmas 25, 103111 (2018), eprint https://doi.org/10.1063/1.5039918, URL https://doi.org/10.1063/1.5039918.
  * Liseikina and Macchi (2007) T. V. Liseikina and A. Macchi, Applied Physics Letters 91, 171502 (2007), URL https://doi.org/10.1063/1.2803318.
  * Macchi and Benedetti (2010) A. Macchi and C. Benedetti, Nucl. Instrum. Methods Phys. Res. A 620, 41 (2010), ISSN 0168-9002, URL http://www.sciencedirect.com/science/article/pii/S0168900210001257.
  * Culfa (2021) O. Culfa, The European Physical Journal D 75 (2021), URL https://doi.org/10.1140/epjd/s10053-021-00208-2.
  * Almassarani et al. (2021) M. Almassarani, S. Meng, B. Beleites, F. Ronneberger, G. G. Paulus, and A. Gopal, Plasma 4, 670 (2021), URL https://doi.org/10.3390/plasma4040034.
  * Gonzalez-Izquierdo et al. (2016) B. Gonzalez-Izquierdo, M. King, R. J. Gray, R. Wilson, R. J. Dance, H. Powell, D. A. Maclellan, J. McCreadie, N. M. H. Butler, S. Hawkes, et al., Nat. Commun. 7, 12891 (2016), URL http://dx.doi.org/10.1038/ncomms12891.
  * Macchi et al. (2009b) A. Macchi, T. V. Liseikina, S. Tuveri, and S. Veghini, Comptes Rendus Physique 10, 207 (2009b), URL https://doi.org/10.1016/j.crhy.2009.03.002.
  * Vincenti et al. (2014) H. Vincenti, S. Monchocé, S. Kahaly, G. Bonnaud, P. Martin, and F. Quéré, Nature Communications 5, 3403 (2014).
  * Pazzaglia et al. (2020b) A. Pazzaglia, L. Fedeli, A. Formenti, A. Maffini, and M. Passoni, Communications Physics 3, 133 (2020b), URL https://doi.org/10.1038/s42005-020-00400-7.
  * Vay et al. (2021) J.-L. Vay, A. Huebl, A. Almgren, L. D. Amorim, J. Bell, L. Fedeli, L. Ge, K. Gott, D. P. Grote, M. Hogan, et al., Physics of Plasmas 28, 023105 (2021), eprint https://doi.org/10.1063/5.0028512, URL https://doi.org/10.1063/5.0028512.
  * Lamprou et al. (2021) T. Lamprou, R. Lopez-Martens, S. Haessler, I. Liontos, S. Kahaly, J. Rivera-Dean, P. Stammer, E. Pisanty, M. F. Ciappina, M. Lewenstein, et al., Photonics 8, 192 (2021), URL https://doi.org/10.3390/photonics8060192.