# Direct Laser Acceleration of Bethe-Heitler positrons in laser-channel
interactions

Bertrand Martinez<EMAIL_ADDRESS>Golp/Instituto de
Plasma e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa,
1049-001 Lisbon, Portugal  Robert Babjak  Golp/Instituto de Plasma e Fusão
Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon,
Portugal  Institute of Plasma Physics, Czech Academy of Sciences, Za Slovankou
1782/3, 182 00 Praha 8, Czechia  Marija Vranic  Golp/Instituto de Plasma e
Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001
Lisbon, Portugal

###### Abstract

Positron creation and acceleration is one of the major challenges for
constructing future lepton colliders. On the one hand, conventional technology
can provide a solution, but at a prohibitive cost and scale. On the other
hand, alternative, reduced-scale ideas for positron beam generation could
bring this dream closer to reality. Here we propose a novel plasma-based
positron acceleration method using a powerful laser propagating through a
dense and narrow plasma channel. A large amount of electrons is injected
within the channel during laser propagation. This electron loading creates
static fields in the plasma, enabling positrons to be guided transversely
while they directly gain energy from the laser field itself. Within this
context, we present a theoretical model to describe how the laser injects the
electrons and estimate the beam-loaded effective electron density. We validate
our theoretical predictions through Quasi-3D PIC simulations and demonstrate
the robustness of this guiding and direct laser acceleration process for
positrons. Our approach could pave the way for testing this new positron
acceleration scheme at ELI-Beamlines, showcasing unprecedentedly high average
energy gain rate of a few TeV/m. The fireball jet produced contains GeV-level
electrons, positrons, and x-rays, opening the path towards potential
laboratory astrophysics experiments using these beams.

## Introduction

Investigation into positron acceleration stands as a pivotal endeavor of
fundamental physics with profound implications not only for future colliders,
but also for laboratory astrophysics. In particular, creating jets with GeV-
level electrons, positrons and gamma-rays will mark a major milestone toward
exploring the propagation of fireball beams in a plasma medium, and thus the
physics of Gamma-Ray Bursts.

High-energy positrons can be produced in state-of-the-art radio-frequency
accelerators. A promising concept to further boost the peak energy of existing
positron beams was introduced experimentally in the 2000s [1, 2, 3, 4], and
later complemented [5, 6, 7, 8, 9]. This approach leverages a plasma stage,
capable to withstand accelerating fields of $10\,\rm GV/m$, a thousand times
larger than conventional accelerators ($10\,\rm MV/m$). These accelerating
fields can be formed in the blowout regime of plasma wakefield acceleration.
The main challenge is to guide the positrons in the direction transverse to
their propagation while maintaining a low emittance. One solution is to tailor
the conditions of the interaction to induce the formation of a dense electron
filament that focuses positrons transversely. For instance, this can be
achieved using a Laguerre-Gauss laser [10, 11], a hollow electron beam driver
[12], a non-neutral electron-positron driver [13], an externally injected
positron beam [14, 15], or with a plasma column [16]. While this research is
conducted in particle accelerators, it became recently a topic of
investigation in high-power laser facilities [17, 18, 19, 20]. With upcoming
high-power lasers (4-5 PW), positrons created by the linear Breit-Wheeler
process [21] are expected to be accelerated by direct laser acceleration [22]
or by self-generated longitudinal plasma fields [23]. With lasers above 10 PW,
it is proposed to create nonlinear Breit-Wheeler positrons and accelerate them
in vacuum [24], via direct laser acceleration [25, 26], or in a recovering
plasma wakefield [27].

The main constraint in the realm of positron acceleration research is the
limited availability of large accelerator facilities globally. Although this
issue can be mitigated by utilizing high-power lasers, which are not only more
compact but also more affordable, there is a notable scarcity of studies
focusing on this approach in the existing literature. The lack of research in
this area underscores the need for further investigation to advance our
understanding and capabilities in the domain of positron acceleration. The
development of new theoretical models and their validation through accurate
simulation data is crucial to bring forward potential experiments at high-
power laser facilities, finding applications in the domain of laboratory
astrophysics.

Figure 1: Setup for Direct Laser Acceleration (DLA) of Bethe-Heitler
positrons. The $10\,\rm PW$ ELI-Beamlines laser first interacts with a thin
Aluminum foil in the Relativistically Self-Induced Transparency regime and
creates positrons. In the second step, it co-propagates, along with some of
the positrons, in a preformed plasma channel. A dense electron filament is
progressively built up at the center of the channel via the interplay of the
two laser field components. This enables to guide positrons over a millimeter
distance while they gain energy directly in the strong laser field, with an
average accelerating energy gain rate on the order of a few TeV/m.

In this work, we investigate Direct Laser Acceleration (DLA) of Bethe-Heitler
positrons in laser-channel interactions. Since DLA of electrons was introduced
[28, 29], it has been studied theoretically in various conditions [30, 31, 32,
33, 34, 35, 36, 37, 38]. In addition, recent experiments highlight that it
holds a potential to generate relativistic ($100^{\prime}\rm s$ of MeV) and
high charge ($100^{\prime}\rm s$ of nC) electron beams [39, 40], that are
already employed to drive bright sources of secondary particles as X-rays and
neutrons [41, 42]. With DLA, electrons gain energy in the laser field itself
which amplitude is on the order of $\simeq 10\,\rm TV/m$, a thousand times
larger than in a typical plasma wakefield. For high-power lasers with a peak
intensity of $10^{21}\,\rm Wcm^{-2}$ and an adequate focal spot, electrons are
expected to reach peak energies of 10 GeV [37]. In the transverse direction,
they are confined by the self-generated plasma fields. With their opposite
charge, positrons are transversely expelled by the same fields and therefore
cannot be accelerated. Recent but yet unexplored developments suggest that
positron guiding can be achieved in DLA either using two counter-propagating
lasers in a channel [22], or by forming a dense electron filament on the laser
axis [25].

In the setup we introduce in Fig. 1, a high-power laser (as will be soon
available at ELI-Beamlines $\sim 10\,\rm PW$) interacts with a thin aluminum
foil ($240\,\rm nm$). The pulse intensity is high enough ($\sim 10^{22}\,\rm
Wcm^{-2}$) to induce gamma-ray production via nonlinear Inverse Compton
scattering [43] and positron creation through the Bethe-Heitler process [44]
in the plasma. Due to its ultra-fast heating and expansion, the target becomes
transparent to the laser in the so-called Relativistically-Self-Induced
Transparency regime [45]. After this interaction, the laser and some of the
positrons travel together in a preformed plasma channel of near-critical
density placed on the laser propagation axis. During the first part of laser
propagation in the channel, the self-generated plasma fields are established
to be defocusing for positrons and focusing for electrons [46, 47], so a
significant fraction is lost. In the meantime, electrons are injected from the
walls of the channel to its center by the interplay of the transverse electric
and transverse magnetic components of the laser field, as sketched in Fig. 1.
The amount of electrons loaded increases as the laser propagates further in
the channel, progressively damping the transverse force expelling positrons.
When the charge density of electrons loaded becomes larger than the background
ion charge density, the transverse force acting on positrons changes sign and
they can be guided. They perform low-frequency betatron oscillations in the
loaded fields of the plasma channel, but are also subject to high-frequency
oscillations in the laser field. The oscillatory motion of a given positron
can be resonantly amplified when the two frequencies match in its co-moving
frame. From this resonance, there is a large energy transfer from the laser to
the positron and it gains momentum in the laser propagation direction. This
acceleration process is the generalization of Direct Laser Acceleration for
positrons.

In the first section of this article, we derive estimates describing the
process of electron injection and how it scales with the initial plasma and
laser conditions. Leveraging this result, we present an energy scaling law for
Direct Laser Acceleration of positrons, including a resonance condition. The
second section is dedicated to Quasi-3D PIC simulations of positron
acceleration. We validate the trend predicted by our estimate for electron
beam loading, as well as our assumption of injection from the walls of the
channel. The acceleration process of positrons is analysed in detail using
individual trajectories and a characteristic feature of DLA known as a forking
structure is observed in their simulated spectrum [48, 40, 49]. Finally, we
discuss a possible path to increase the conversion efficiency from laser
energy to positrons and how DLA of positrons compares to other approaches. We
conclude with an assessment of experimental feasibility and potential
applications.

##  I Theoretical model of positron acceleration

In this section, we introduce a theoretical model to describe Direct Laser
Acceleration of positrons in a plasma channel. Our first and main result is to
provide a quantitative estimate for the charge density of the electrons loaded
by the intense laser traveling in the plasma channel. From this general result
derived as a function of the laser and plasma conditions, we bring forward a
model describing the process of Direct Laser Acceleration for positrons.

###  I.1 Injection of electrons and beam loading

We consider a laser propagating in direction $x$ and linearly polarised in
direction $y$. It has a frequency $\omega_{L}$, a wave number $k_{L}$, a peak
field amplitude $E_{L}$ and we denote $n_{c}$ the critical density associated.
It travels in a preformed plasma channel characterized by a radius denoted
$r_{c}$. We use the notation $n_{p}$ for the density in the center of the
channel and $n_{w}$ for the density in the walls of the channel. In other
words, the transverse density profile of the plasma channel is given by
$n(r)=n_{p}+(n_{w}-n_{p})(r/r_{c})^{\alpha}$, where $r$ is the radial
coordinate defined as $r^{2}=y^{2}+z^{2}$, and $\alpha$ is an integer
describing the steepness of the profile ($\alpha=2$ for a parabolic profile).
The laser duration $\tau_{L}$ is larger than the plasma period.

As the laser travels in the plasma channel, the ponderomotive force expels
electrons and a blowout cavity is formed. Behind this cavity, electrons are
pushed transversely toward the center by the electric field component of the
laser $-e\mathbf{E}$, and rotated in the laser propagation direction by its
magnetic field component $-e\mathbf{v}\times\mathbf{B}$, where $e$ is the
electron charge and $v$ the electron velocity. To evaluate the amount of
electrons loaded, we have to simplify the problem and formulate an additional
hypothesis. We justify this assumption a posteriori in our analysis of PIC
simulations performed in section II. Although the laser focusing and
defocusing dynamic is complex, we assume that its typical transverse extent
can be approximated by the blowout radius of the interaction, denoted $r_{b}$.
To derive this radius, we balance the ponderomotive force of the laser as
defined in this work [50] with the electrostatic force created in the channel,
provided by Poisson equation $\nabla E=\rho/\epsilon_{0}$ where
$\rho(r)=-e[n_{p}+(n_{w}-n_{p})(r/r_{c})^{\alpha}]$ is exactly the generic
plasma channel profile we model. We define the dimensionless laser field
amplitude $a_{L}=eE_{L}/m\omega_{L}c$, where $m$ is the electron mass and $c$
is the light velocity. In the calculation of the ponderomotive force, we
assume that the average Lorentz factor of electrons is
$\langle\gamma\rangle\simeq a_{L}$. The blowout radius is thus a solution of

$k_{p}^{2}r^{2}\left[1+\frac{1}{\alpha+1}\left(\frac{n_{w}}{n_{p}}-1\right)\left(\frac{r}{r_{c}}\right)^{\alpha}\right]=\frac{a_{L}}{2}$
(1)

where $k_{p}$ is the plasma wave number associated with the central channel
density $n_{p}$. This equation is a generalisation of the established
expression of the blowout radius from a constant density plasma
($n_{w}=n_{p}$), to a plasma with an increasing profile ($n_{w}>n_{p}$). It
accounts for arbitrary variations of the density profile and therefore has a
broad range of validity. For instance, it encompasses the case of plasma
channels with a parabolic profile ($\alpha=2$) and also the case of a plasma
tube with an infinitely steep profile ($\alpha\rightarrow\infty$ where
$r_{b}\rightarrow r_{c}$). While analytical solutions can be derived for
specific values of the exponent $\alpha\in(0,1,2,4,6)$, it is worth noting
that the expressions become cumbersome to handle analytically.

At the blowout radius $r_{b}$, the plasma density is denoted $n_{b}=n(r_{b})$.
We assume that at every laser cycle, all electrons within the local skin depth
are injected in the channel. This means that all electrons between the two
cylinders of radius $r$ and $r+\delta$ are injected, where the skin depth
$\delta$ is defined as $\delta=(c/\omega_{L})(n_{c}/n_{b})^{1/2}$. For a laser
linearly polarised in direction $y$, the electric field component of the laser
is actually not always perpendicular to the surface, so that electrons are not
injected from all radial angles in the channel (namely on a $2\pi$ angle), but
rather on a smaller opening angle in the $\pm y$ direction, that we
approximate to $\simeq 1\,\rm rad$, as illustrated in Fig. 2

Figure 2: Electrons are injected mostly along direction $y$, as the laser is
linearly polarised in this direction and propagates along the $x$ axis. In the
model, we assume electrons are injected from an angle with an opening of $\sim
1\,\rm rad$.

With this reasoning, we can obtain a rate of electrons injected per laser
period $dN/dx\simeq r_{b}n_{b}\delta$, assuming $\delta\ll r_{b}$. However, we
are interested in estimating the density of electrons loaded in the plasma
channel, denoted $n_{e}$. To derive this, we assume that at any time $t$, the
electrons loaded fill on average a cylinder of volume $\pi r_{b}^{2}v_{g}t$,
where $v_{g}$ is the group velocity of the laser and the radius is chosen to
be the blowout radius. We deduce that both the number of electrons and the
volume they fill increase linearly with time. When we derive the ratio of
these two quantities, namely the density of electrons loaded $n_{e}$, the
linear time dependence simplifies and we deduce that the average density of
electrons loaded per laser cycle is constant

$n_{e}=\frac{1}{\pi r_{b}k_{L}}(n_{b}n_{c})^{1/2}$ (2)

For a laser with $N$ cycles, this average density is increased by a factor
$N$. We stress that this expression is derived for any plasma channel with a
transverse density profile $n(r)=n_{p}+(n_{w}-n_{p})(r/r_{c})^{\alpha}$, where
$r_{c}$ is the channel’s radius, $n_{p}$ is the central channel density,
$n_{w}$ is the wall density and $\alpha$ the steepness of the profile. In Eq.
(2), the blowout radius $r_{b}$ is given for this exact profile by Eq (1), and
the density $n_{b}$ is the channel’s density at this radius, namely
$n_{b}=n(r_{b})$ for this same profile. We underline that the two quantities
$r_{b}$ and $n_{b}=n(r_{b})$ depend on the steepness of the plasma profile
$\alpha$ as $r_{b}$ is a solution of Eq. (1). Therefore, our estimate in Eq;
(2) has a broad validity range since it provides a scaling of electron beam
loading with respect to the full plasma channel profile and the laser
intensity.

###  I.2 Model for Direct Laser Acceleration of positrons

In the established framework of Direct Laser Acceleration, the laser expels
electrons via its ponderomotive force, forming an ion cavity with a positive
radial electric field $E_{y}$ and an azimuthal magnetic field $B_{z}$, such
that $E_{y}-cB_{z}\propto n_{p}y$. This positive component is illustrated
schematically in Fig. 3 with the dotted blue line. The electrons loaded at the
center of the channel also create static fields. The contribution of
relativistic electrons to the transverse static fields is smaller by a factor
$1/\gamma^{2}$ as compared to the non-relativistic ones. We therefore assume
the transverse fields are the same as for a non-relativistic electron beam of
density $n_{e}$, namely $E_{y}-cB_{z}\propto-n_{e}y$, . This new and negative
contribution is represented by the dashed orange line in Fig. 3. While we
expect these electrons to be left behind the laser after some distance, new
electrons are continuously injected, keeping the overall loaded electron
density constant in the channel. The contribution of the relativistic
electrons loaded is expected to vanish as it scales with $n_{e}/\gamma^{2}$
where $\gamma$ is the Lorentz factor of the fast electrons. Accounting for the
two contributions of the ion cavity and the non-relativistic electron beam
loaded, the resulting transverse field is

$E_{y}-cB_{z}=\frac{e}{\epsilon_{0}}\frac{y}{2}(n_{p}-n_{e})$ (3)

Depending on the amount of electron beam loading, the resulting transverse
field can be either positive or negative. With a negligible electron beam
loading ($n_{e}\ll n_{p}$), the transverse force on positrons is positive and
they are expelled radially away from the channel axis, as established by
previous works [46, 47]. However, when the electron beam loading density
overcomes the background ion density ($n_{e}>n_{p}$), the transverse force on
positrons reverts its sign, becomes negative and thus acts toward a positron
guiding. The latter case is illustrated in Fig. 3 with the green line. It is
worth highlighting that our expression for the resulting transverse field, is
obtained with intuitive arguments. However, the same expression has already
been formally derived in the context of beam loading for plasma wakefield
acceleration [10, 15].

Figure 3: Transverse fields in a plasma channel as a function of the
transverse distance to the axis $y$. When the resulting field is positive,
electrons are focused and positrons defocused. On the contrary, if the
resulting field is negative, electrons are expelled and positrons are guided.
We display two contributions: the plasma ion cavity with an ion density
$n_{p}=10^{19}\,\rm cm^{-3}$ (dotted blue line) and the electron beam loading
with density $n_{e}=1.5\times 10^{19}\,\rm cm^{-3}$ (dashed orange line). The
resulting fields (green line) are negative as the net charge density is
negative.

The transverse force experienced by an electron or a positron of charge $q=\pm
e$ in a plasma channel with beam loading is deduced from Eq. (3). We denoted
it $F_{c}$ and it can be expressed as a function of an effective plasma
frequency $\omega_{\mathrm{eff}}$

$\displaystyle F_{c}$
$\displaystyle=\frac{q}{e}\frac{y}{2}m\omega_{\mathrm{eff}}^{2}$ with
$\displaystyle\omega_{\mathrm{eff}}^{2}=\omega_{p}^{2}(1-n_{e}/n_{p})$ (4)

In the limit of low electron beam loading ($n_{e}\ll n_{p}$), the effective
plasma frequency is exactly the plasma frequency and we recover the
established transverse force of an ion cavity for an electron
$F_{c}=-m\omega_{p}^{2}y/2$ [46, 47].

The laser is modeled as a plane wave propagating in direction $x$, linearly
polarised in direction $y$ as $E_{y}^{(L)}=E_{L}\cos\omega_{L}(t-x/v_{\phi})$.
It has a magnetic field component $B_{z}^{(L)}=E_{y}^{(L)}/v_{\phi}$, where
$v_{\phi}>c$ is the phase velocity of the plane wave in the plasma of density
$n_{p}$. The equation of motion and energy conservation can be written as

$\displaystyle\frac{dp_{x}}{dt}$ $\displaystyle=qv_{y}B_{z}^{(L)}$ (5)
$\displaystyle\frac{dp_{y}}{dt}$
$\displaystyle=q\left(E_{y}^{(L)}-v_{x}B_{z}^{(L)}\right)+\frac{q}{e}\frac{y}{2}m\omega_{\mathrm{eff}}^{2}$
(6) $\displaystyle mc^{2}\frac{d\gamma}{dt}$
$\displaystyle=qv_{y}\left(E_{y}^{(L)}+\frac{1}{e}\frac{y}{2}m\omega_{\mathrm{eff}}^{2}\right)$
(7)

We note that the transverse force induced by the channel fields includes the
effect of beam loading via the effective plasma frequency
$\omega_{\mathrm{eff}}$, as defined in Eq. (4). With Eq. (5) and Eq. (7) we
find that there is an invariant of motion ($dI_{0}/dt=0$) with $I_{0}$ defined
as

$I_{0}=\gamma-\frac{v_{\phi}}{c}\frac{p_{x}}{mc}-\frac{q}{e}\frac{\omega_{\mathrm{eff}}^{2}y^{2}}{4c^{2}}$
(8)

From Eq. (6) and Eq. (7), we can deduce the equation of motion for the
transverse coordinate of the particle $y$. This derivation is a generalisation
of Ref [28] and leads to

$\displaystyle\frac{d^{2}y}{dt^{2}}+\omega_{\beta}^{2}y=-q$
$\displaystyle\left[\left(\frac{1}{c}\frac{dy}{dt}\right)^{2}-\left(1-\frac{v_{x}}{v_{\phi}}\right)\right]\frac{E_{y}^{(L)}}{m\gamma}$
$\displaystyle-\frac{q}{e}\left(\frac{1}{c}\frac{dy}{dt}\right)^{2}\frac{\omega_{\mathrm{eff}}^{2}}{2\gamma}y$
(9)

where the betatron frequency $\omega_{\beta}$ is defined as

$\omega_{\beta}=\left\\{\begin{array}[]{ll}\frac{\omega_{p}}{\sqrt{2\gamma}}\sqrt{1-\frac{n_{e}}{n_{p}}}\mbox{
for }e^{-}\mbox{ if }n_{e}<n_{p}\\\
\frac{\omega_{p}}{\sqrt{2\gamma}}\sqrt{\frac{n_{e}}{n_{p}}-1}\mbox{ for
}e^{+}\mbox{ if }n_{e}>n_{p}\end{array}\right.$ (10)

This definition evidences in what conditions electrons and positrons are
allowed to oscillate resonantly. Electrons can experience a resonant
acceleration only if the beam loading remains smaller than the initial plasma
density ($n_{e}<n_{p}$). On the contrary, positrons can experience a resonant
acceleration only if the beam loading becomes larger than the initial plasma
density ($n_{e}>n_{p}$).

The method to derive Eqs. (5)-(9) is strictly similar to the one already
established for DLA of electrons. Indeed, for the case of an electron ($q=-e$)
and without beam loading ($n_{e}=0$), we recover the established definition
for the invariant of motion $I_{0}$ as defined in Ref [28], the betatron
frequency $\omega_{\beta}^{2}=\omega_{p}^{2}/2\gamma$ and the equation
describing the transverse oscillations of the particle [28]. Formally, the
only change we did is to account for beam loading by substituting the plasma
frequency $\omega_{p}$ by an effective plasma frequency
$\omega_{\mathrm{eff}}$, as defined in Eq. (4). We can draw an important
conclusion from this mathematical similarity. The scaling laws for DLA of
electrons and positrons with beam loading can be generalized from the already
established ones by a simple substitution
$\omega_{p}\rightarrow\omega_{\mathrm{eff}}$, provided the resonance condition
in Eq. (10) is fulfilled. For positrons, namely in the case of high beam
loading where $n_{e}>n_{p}$, the maximum energy at resonance averaged on a
betatron period is [28, 30, 31]

$\langle\gamma\rangle=\frac{3}{4}I^{2}\left(\frac{\omega_{L}}{\omega_{\mathrm{eff}}}\right)^{2}$
(11)

In the context of the ELI-Beamlines laser pulse ($150\,\rm fs$ laser with a
$\sin^{2}$ time envelope) where our setup could be realized, we have estimated
the average energy gain rate [33]

$\frac{d\gamma}{dx}=\frac{5}{2}\frac{a_{L}}{\sqrt{I}}\frac{\omega_{\mathrm{eff}}}{\omega_{L}}$
(12)

For high laser intensities, the relativistic positrons emit radiation, lose
energy and this should be accounted for in the theory. This was already
calculated in the case of electrons and without beam loading [33]. We verified
that it can also be generalised by the substitution of the plasma frequency
$\omega_{p}$ to the effective plasma frequency $\omega_{\mathrm{eff}}$
accounting for beam loading. The main effect of radiation reaction is that the
parameter $I_{0}$ defined in Eq. (8) is not invariant anymore. The exact
evolution of $I$ cannot be derived. Its steady-state value, averaged over
several betatron periods, can however be estimated assuming the particle
reached resonance [33]

$\displaystyle I/I_{0}$
$\displaystyle=\left\\{\begin{array}[]{ll}&\left[1+3.2\times
10^{-8}\frac{\omega_{L}^{2}}{\omega_{\mathrm{eff}}^{2}}\frac{I_{0}^{4}}{\lambda_{L}[\mathrm{\mu
m}]}\right]^{-1/4}\\\ &\left[1+2.3\times
10^{-8}a_{L}\frac{I_{0}}{\lambda_{L}[\mathrm{\mu
m}]}\right]^{-1}\end{array}\right.$ (15)

where the first expression is valid when the channel fields prevail in the
radiation reaction force, and the second as the laser field dominates in the
radiation reaction force.

To conclude, we introduced an analytical model to describe Direct Laser
Acceleration of positrons in a plasma channel accounting for electron beam
loading. The main outcome is that we obtain an estimate for the electron beam
loading as a function of the initial laser and plasma channel conditions. This
is a key theoretical milestone, as it enables to better grasp the conditions
of positron acceleration and suggest for the first time a scaling law for the
peak energy positrons may reach as they experience DLA.

##  II Quasi-3D simulations of positron acceleration

In this section, we detail the results of PIC simulations of Direct Laser
Acceleration of positrons. We first describe the process of beam loading and
verify the scaling predicted by our theoretical estimate for a wide range of
laser and plasma conditions. We also discuss the validity of the assumptions
of our theoretical model in light of the simulation data. We finally bring
unambiguous proof that positrons can experience Direct Laser Acceleration in
Quasi-3D PIC simulations.

Figure 4: Electron beam loading in laser-channel interactions as a function of
the wall density of the channel. Panels a)-e) illustrate the interaction of
the laser pulse (red/blue) with the electrons of the channel (green). In
panels f)-j), we note that transverse fields are negative in the channel, so
positrons can be guided. When moving from the top row to the bottom row, we
increase the wall density of the plasma channel in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$. The data is shown
after the peak of the laser has propagated over a distance $L_{x}=200\,\rm\mu
m$ in the channel and has already propagated in the transparency regime
through the aluminum foil placed at $x=0\,\rm\mu m$. The process is robust
with respect to the wall density, so the channel could be formed with a
capillary discharge or with another laser.

###  II.1 PIC simulations with 10 PW class lasers

The setup in Fig. 1 is modeled with the Particle-In-Cell code Osiris [51]. The
laser has a wavelength of $1\,\rm\mu m$, propagates in the x direction, and is
linearly polarised in direction $y$. The electric field of this pulse has a
Gaussian temporal profile, with a duration of $150\,\rm fs$ (FWHM) and a
Gaussian transverse profile of $5.3\,\rm\mu m$ (FWHM). Its peak intensity is
$5\times 10^{22}\,\rm Wcm^{-2}$, corresponding to a normalised field amplitude
$a_{L}=200$, a peak power of $9\,\rm PW$ and an energy of $1.3\,\rm kJ$. We
initialise the thin foil as a fully ionized Aluminum plasma with a thickness
of $240\,\rm nm$. We assume it is pre-expanded by the laser pre-pulse and has
a charge density of $3\times 10^{22}\,\rm cm^{-3}$. We also initialise an
exponential pre-plasma in front of it, with a scale length of $500\,\rm nm$.
The preformed plasma channel is placed right after the thin foil. It has a
transverse profile given by $n(r)=n_{p}+(n_{w}-n_{p})(r/r_{c})^{\alpha}$,
where $n_{p}=10^{18}\,\rm cm^{-3}$ is the central density in the channel,
$n_{w}=4\times 10^{21}\,\rm cm^{-3}$ is the wall density of the channel,
$r_{c}=20\,\rm\mu m$ is the channel’s radius and $\alpha=5$ is the steepness
of the profile. The density profile at the beginning of the plasma channel is
smooth with a $10\,\rm\mu m$ linear gradient increasing from $0$ to the value
given by the transverse profile above.

A correct description of laser guiding in a plasma channel requires 3D
simulations, that are out of reach for modern supercomputers for the required
propagation distances. To overcome this issue, we perform simulations with the
Quasi-3D simulation mode of the PIC code OSIRIS [52], that correctly models 3D
pulse focusing at a cost of the same order as a 2D algorithm. In the Quasi-3D
approach, the fields are represented in cylindrical coordinates $(x,r,\theta)$
using a Fourier decomposition in angular modes [52]. The first two modes
account for the axisymmetric self-generated channel fields (mode m = 0) and
for the non-axisymmetric linearly polarized laser field (m = 1).

The simulation domain has a size of $137.5\times 80\,\rm\mu m^{2}$, with
spatial steps $dx=dr=16\,\rm nm$ and a time step of $26.6\,\rm as$. The laser
propagates on a total distance of 1 mm in the plasma channel, which
corresponds to $1.28\times 10^{5}$ time steps. We introduce $L_{x}$ as the
distance travelled by the peak of intensity in the channel. Initially, we use
16 particles per cell for electrons and ions in the channel, and 32 particles
per cell for electrons and ions in the thin foil. We limit the simulations to
the QED processes that prevail in our conditions, namely gamma-ray generation
via nonlinear Inverse Compton scattering and pair production via Bethe-
Heitler.

Figure 5: Average density of electrons loaded $n_{e}$ in the plasma channel as
a function of the distance traveled by the peak of the laser in the channel.
This distance corresponds to the distance the peak of the laser travels from
the entrance to the plasma channel. At each position, it is computed by
averaging the electron density in the channel, as illustrated in Figs. 4a)-e).
Each curve represents a different initial condition, where the wall density is
varied in the range $n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm
cm^{-3}$ and $n_{c}\simeq 10^{21}\,\rm cm^{-3}$ is the critical density for
the laser.

###  II.2 Laser-driven electron beam loading

In this section, we illustrate how electron beam loading takes place in PIC
simulations, and how it induces fields guiding positrons. We present the
results of five simulations, where all parameters are kept constant and
defined above (see also the first row of Table 1). We only vary the wall
density of the plasma channel in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$.

Figure 6: Average density of electrons loaded in the plasma channel from PIC
simulations (circles) and our estimate from Eq. (2) (dashed line). Each circle
represents the steady-state density of electrons loaded in the channel
inferred from a time evolution as illustrated in Fig. 5. We vary the channel’s
wall density in panel a), the channel’s central density in panel b), the
channel’s radius in panel c), and the laser power in panel d). For each panel,
the plasma and laser conditions are given in Table 1.

Figure 4a) shows the laser electric field (in red/blue) and the electron
density (in green). At the front of the laser, electrons are expelled by the
ponderomotive force and form an ion cavity. We also observe that electrons are
loaded at the rear of the laser. This is possible by the interplay of the
$-e\mathbf{E}$ and $-e\mathbf{v}\times\mathbf{B}$ components. At this instant,
the measured electron charge density is already larger than the background ion
charge density. As a result, the transverse force associated with the static
fields generated enables the guiding of positrons. Indeed, Fig. 4b) shows that
in the region where electron beam loading is large, the resulting transverse
fields are negative, and therefore the transverse force on positrons is
negative, pointing toward the channel axis. When moving from Fig. 4a) down to
Fig. 4e), the wall density in the channel increases in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$. The main trend we
note is that the amount of electrons loaded increases with the wall density.
This evolution is in line with the injection process from the walls we
described in section I. The evolution of the guiding fields is seen by moving
from Fig. 4f) down to Fig. 4j). We observe the amplitude of the resulting
transverse field on positrons increases with the wall density, due to the
increased beam loading. However, this stronger guiding is achieved to the
detriment of a smaller guiding structure. Indeed, ions are loaded and
accelerated at the center of the channel for the highest wall densities
considered:$n_{w}=8\times 10^{21}\,\rm cm^{-3}$ in Fig. 4h) and $n_{w}=2\times
10^{22}\,\rm cm^{-3}$ in Fig. 4j). The reason is that in these cases, electron
beam loading, and therefore the charge separation field mediating ion
acceleration, are the largest.

After this illustration of the beam loading process, we are eager to
understand quantitatively how electron beam loading scales in realistic PIC
simulations. To achieve this purpose, our goal is to compare the amount of
electron beam loading as seen in Fig. 4a) with the theoretical estimate from
Eq. (2). The latter provides a density averaged on the channel’s volume. For
this reason, we have computed the spatial average electron beam loading
density $n_{e}$ in the plasma channel. To do this, we consider the
instantaneous electron density as in Fig. 4a), subtract the initial electron
density profile and average spatially the density below the initial channel’s
radius of $20\,\rm\mu m$. The subtraction method is used to compute the excess
charge injected from the channel walls, as they all belong to the same
numerical species. In Fig. 5, we plot the evolution of the spatial average
electron beam loading density as a function of the laser propagation distance
in the channel denoted $L_{x}$. Each curve represents one of the simulations
from Fig. 4, where the wall density varies in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$. For all cases, we
observe that the density of electrons loaded increases as the laser propagates
further in the channel and seems to saturate after $\sim 250\,\rm\mu m$ of
laser propagation, with relatively small fluctuations. The presence of a
quasi-plateau is in line with our theoretical model, which assumes a constant
steady-state value for the average loaded electron beam density. From the data
in Fig. 5, we measure the time-average loaded electron beam density (over the
interval $L_{x}>250\,\rm\mu m$) and report it in Fig. 6a) as a function of the
wall density of the PIC simulations performed. We remark that the density of
electrons loaded can increase by one order of magnitude, from $2\times
10^{19}\,\rm cm^{-3}$ to $2\times 10^{20}\,\rm cm^{-3}$ when the wall density
is increased by two orders of magnitudes from $2\times 10^{20}\,\rm cm^{-3}$
to $2\times 10^{22}\,\rm cm^{-3}$. This evolution confirms the trend already
reported in Fig. 5 that for a higher wall density, more electrons are loaded
in the channel. We underline that it is quantitatively predicted by our
theoretical estimate from Eq. 2. While this agreement constitutes a first
argument in favor of our theoretical model describing electron beam loading,
it remains to be complemented by additional simulations.

Table 1: Summary of the initial conditions we chose in the PIC simulations presented in Fig. 6. The columns represent the main parameters of the interaction: the channel’s wall density, the channel’s central density, the channel’s radius and the laser power. Each row represents a scan over one of these parameters, while the others are kept constant. | Channel’s wall density | Channel’s central density | Channel’s radius | Laser power111In all simulations, the electric field has a Gaussian temporal profile of $150\,\rm fs$ (FWHM).
---|---|---|---|---
| $n_{w}\,(10^{21}\,\rm/cm^{3})$ | $n_{p}\,(10^{21}\,\rm/cm^{3})$ | $r_{c}\,\rm(\mu m)$ | $P\,\rm(PW)$
Fig. 6a) | $0.2,\,2,\,4,\,8,\,20$ | $10^{-3}$ | $20$ |  $9$222For all these scans, the electric field has a Gaussian transverse profile of $5.3\,\rm\mu m$ (FWHM).
Fig. 6b) | $2$ | $10^{-4},\,10^{-3},\,10^{-2},\,10^{-1}$ | $10$ |  $9$22footnotemark: 2
Fig. 6c) | $2$ | $3\times 10^{-3}$ | $10,\,20,\,30,\,40,\,50,\,60$ |  $9$22footnotemark: 2
Fig. 6d) | $2$ | $10^{-3}$ | $40$ |  $0.6,\,2,\,3.5,\,5,\,6.5$333For this parametric scan, the envelope of the laser electric field has a Gaussian transverse profile of $10.6\,\rm\mu m$ (FWHM).

We ran complementary PIC simulations in order to further test the validity of
our theoretical estimate for electron beam loading. For the sake of clarity,
we summarise the main parameters for each of the 20 PIC simulations we
conducted in Table 1. We vary four characteristic quantities, one at a time.
We consider different wall and central densities for the plasma channel, as
well as various channel’s radii and laser powers in the Peta Watt range. In
Figs. 6a)-d), the overall comparison between the density of electrons loaded
measured in PIC simulations and the theoretical estimate is correct. For the
four parameters scanned, the qualitative evolution of beam loading is well-
reproduced by Eq. 2. We therefore conclude that our estimates of electron beam
loading provides the correct scaling laws as a function of the main plasma and
laser parameters.

The agreement between our theoretical scaling of electron beam loading with
PIC simulations is obtained despite complex interaction conditions, the wide
range of laser-plasma parameters and nonlinear plasma processes involved. It
relies on one main hypothesis, that the electrons are extracted from the walls
of the plasma channel, at a radius approximated to be the blowout radius. In
the next section, we test the validity of this assumption directly from the
data of our PIC simulations.

###  II.3 Injection of electrons from the walls of the channel

In this section, we analyse the phase space of electrons loaded in the plasma
channel. Our goal is to investigate whether they are effectively injected from
the walls in PIC simulations, as we assumed in the theoretical modelling.

Figure 7: Electron injection as a function of the wall density of the channel.
Panels a)-e) illustrate the $(r,\mathcal{E})$ phase space of electrons. Panels
f)-j) depict the $(r_{i},\mathcal{E})$ phase space of electrons, where $r_{i}$
is the _initial_ radius of electrons when they are injected. The red curve
shows the distribution in initial radius of electrons. When moving from the
top row to the bottom row, we increase the wall density of the plasma channel
in the range $n_{w}\in(0,2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$. The
data is shown after the peak of the laser has propagated over a distance
$L_{x}=200\,\rm\mu m$ in the channel.

The phase space of electrons is displayed in Fig. 7, for simulations where we
modify the wall density of the plasma channel. The exact simulation parameters
are in the first row of Table 1, and are the same as in Fig. 4. We first focus
on one simulation, where the wall density is the smallest: $n_{w}=2\times
10^{20}\,\rm cm^{-3}$, see Figs. 7a)f). Figure 7a) shows the 2D distribution
$d^{2}N/d\mathcal{E}dr$ for all electrons in the channel. We recall that $r$
is the radial coordinate of electrons ($r^{2}=y^{2}+z^{2}$) and $\mathcal{E}$
is their kinetic energy. We observe that relativistic electrons reach energies
in the range of hundreds of MeV at the instant represented, after $200\,\rm\mu
m$ of laser propagation in the channel. This figure only gives information on
the instantaneous radius of electrons $r$, but not on their initial radius,
denoted $r_{i}$, from where they are injected in the channel. From the
simulation data, we have inferred the initial distance from the axis $r_{i}$
of each electron and computed numerically the 2D distribution
$d^{2}N/d\mathcal{E}dr_{i}$, that is plotted in Fig. 7f). We note that the
initial radius of electrons seems to coincide with the channel’s radius
$r_{i}\simeq r_{c}=20\,\rm\mu m$. To ensure this quantitatively, we integrate
the 2D distribution $d^{2}N/d\mathcal{E}dr_{i}$ to obtain a distribution in
initial radius labeled $dN/dr_{i}$, see the red curve in Fig. 7f). The peak of
the distribution $dN/dr_{i}$ is marked by a red circle and is located at a
radius $r_{i}\simeq r_{c}=20\,\rm\mu m$. This means that for these parameters
and at this instant, most of the loaded electrons are injected from the
channel wall itself. Note that we separate loaded from unloaded electrons with
the energy criteria ($>mc^{2}$), as they all belong to the same numerical
species. As we observe from Fig. 7a) down to Fig. 7e), the instantaneous phase
space of electrons indicates that they reach hundreds of MeV for all wall
densities at stake. In parallel, when moving down from Fig. 7f) to Fig. 7j),
we can observe how the injection radius of electrons varies when increasing
the wall density in the plasma channel in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$. This radius of
injection decreases from $r_{i}\simeq 20\,\rm\mu m$ down to $\simeq 14\,\rm\mu
m$ when increasing the wall density from $2\times 10^{20}\,\rm cm^{-3}$ to
$2\times 10^{22}\,\rm cm^{-3}$, as pinpointed by the red circles.

Figure 8: Injection radius $r_{i}$ of electrons loaded in the plasma channel
as a function of the distance travelled by the peak of the laser in the
channel. At each position, it is computed from the peak of the distribution in
initial radius $dN/dr_{i}$ from Fig. 7f)-j). Each curve represents a different
initial condition, where the wall density is varied in the range
$n_{w}\in(0.2,\,2,\,4,\,8,\,20)\times 10^{21}\,\rm cm^{-3}$ and $n_{c}\simeq
10^{21}\,\rm cm^{-3}$ is the critical density for the laser.

We will now investigate how these values compare with what we expect from the
analytical model. The injection radius of electrons loaded seems to fluctuate
as the laser propagates in the channel. In Fig. 8, we plot it for the
different simulations presented in Fig. 7, where we modify the wall density in
the channel. We stress that the value reported is the peak of the distribution
in of the initial radial position of injected electrons, as we computed it in
Figs. 7f)-j) (see the red distribution function and the circle). It represents
the initial radius of _most_ electrons, but not all of them. After the
acceleration stabilizes, the average $r_{i}$ of the injected electrons is
constant, which is expected as the fresh plasma always has the same plasma
profile so the injection is the same beyond the initial plasma ramp. For each
simulation, we determine this quasi-constant value by averaging the evolution
from Fig. 8 over the interval $L_{x}>250\,\rm\mu m$. The result of this
analysis is displayed in the inset of Fig. 8 along with the theoretical
estimate for the injection radius of electrons determined from Eq. (1).
Although we note the theory predicts a slightly lower numerical value than the
simulation data, the qualitative trend it provides is correct. This trend from
the theory points that for a larger wall density, the amplitude of the
perpendicular electrostatic field built to balance the ponderomotive push is
larger, and the blowout radius thus decreases. As a result, the laser is more
confined to the center of the channel, and it injects electrons at a smaller
radius than $r_{i}$ in the analytical model. The theory neglects the dynamical
evolution of the laser and the ion motion, so a perfect agreement is not
expected. Nevertheless, it seems that this model gets quite close to fully
describing the injection process.

In this section, we have confirmed the main hypothesis of our theoretical
model on electron beam loading using PIC simulations. Indeed we demonstrated
that electrons are injected from the walls of the plasma channel for various
laser and plasma conditions. The main outcome is that our model correctly
describes the injection process of electrons in the channel, and how it scales
with the plasma channel and laser profiles.

###  II.4 Evidences of Direct Laser Acceleration

We now focus on the dynamics of positrons and prove that they experience
Direct Laser Acceleration in PIC simulations. We include the analysis of
individual trajectories as well as the whole positron spectrum.

Figure 9: a) Maximum positron energy in the simulation domain as a function of
laser propagation distance in the channel with the theoretical estimates from
Eqs. (11)-(12). b) Oscillations of two types of representative positron
trajectories in the ($x-y$) plane, where $y$ is the laser polarisation
direction and $x$ the laser propagation direction. c) Evolution of the
transverse positron momentum $p_{y}$ and the transverse fields experienced by
the two types of positron trajectories. The transverse fields are the ones of
the plasma channel (mode $m=0$) and do not include the laser’s contribution
(mode $m=1$).

Positron acceleration is depicted in Fig. 9. We consider the simulation with
parameters described in Sec. II.1, where the central channel density is
$n_{p}=10^{18}\,\rm cm^{-3}$, the wall density is $n_{w}=4\times 10^{21}\,\rm
cm^{-3}$, the channel’s radius is $r_{c}=20\,\rm\mu m$ and the laser power
$9\,\rm PW$. The maximum positron energy in the simulation domain is plotted
in Fig. 9a), as a function of the laser propagation distance in the channel.
We see the instantaneous energy gain rate is as high as $\simeq 15\,\rm TeV/m$
in the first $100\,\rm\mu m$ of interaction. This transient value is unusually
large, even for plasma-based accelerators. This can be explained by the fact
that the laser field amplitude reaches $\simeq 10^{23}\,\rm Wcm^{-2}$ due to
strong self-focusing at the entrance of the channel. The maximum energy
achieved for positrons is $3.4\,\rm GeV$ after 1 millimeter of propagation in
the channel, corresponding to an average energy gain rate of $3.4\,\rm TeV/m$.
We note that both the peak energy and the instantaneous energy gain rate are
well-estimated by our analytical scaling laws from Eq. (11) and Eq. (12),
where we account for radiation reaction with Eq. (15). However, this agreement
is challenging to obtain solely from theoretical assumptions as the scaling
law has a non-linear dependence with respect to three parameters: the laser
intensity, the electron beam loading and the initial transverse position of
the accelerated positrons. Since it is not yet possible to infer these
quantities from theory, we have used the values measured in the simulation.
For instance, the agreement on the energy gain rate is obtained on the first
$100\,\rm\mu m$ of propagation for a measured laser intensity of $10^{23}\,\rm
Wcm^{-2}$ and a measured beam loading of $n_{e}=4\times 10^{19}\,\rm cm^{-3}$.
The maximum energy is reached after $500\,\rm\mu m$ of propagation, when the
measured laser intensity has reduced to $2\times 10^{22}\,\rm Wcm^{-2}$ and
the beam loading has grown larger to $n_{e}=10^{20}\,\rm cm^{-3}$. This
steady-state value of electron beam loading of $n_{e}=10^{20}\,\rm cm^{-3}$ is
very close to the theoretical estimate of Eq. (2) that provides $n_{e}=9\times
10^{19}\,\rm cm^{-3}$, as seen in Fig. 6a). Finally, both agreements with
theory are obtained considering that positrons are injected with an initial
radius of $5.3\,\rm\mu m$, namely the FWHM of the electric field and that the
phase velocity is equal to the speed of light.

In Quasi-3D PIC simulations, resonant positrons oscillate as expected from the
theoretical description. Figure 9b) details how the transverse coordinate $y$
of high-energy positrons evolves as a function of their propagation in the
channel, where $y$ is the laser polarisation direction. First they experience
fast oscillations in the laser field for a propagation distance
$L_{x}<100\,\rm\mu m$. Beyond this limit, the frequency of the oscillations
decreases and we measure a period of $\simeq 0.8\,\rm mm$. Looking at the
first type of trajectory indexed by 1 in Fig. 9b), we observe positrons are
guided along the axis and confined to the center of the channel ($y\leq
10\,\rm\mu m$) for as long as 1 millimeter. At $L_{x}=200\,\rm\mu m$ of
propagation, the positrons turn toward the center of the channel instead of
being expelled. This behavior is explained in Fig. 9c), where we represent the
transverse momentum of a representative positron $p_{y}$ and the quasi-static
(mode $m=0$) transverse channel fields it experiences $E_{y}-cB_{z}$. We note
the transverse momentum decreases rapidly from $p_{y}/mc\simeq 120$ to
$p_{y}/mc\simeq-120$ in a distance of $150\,\rm\mu m$. This momentum decrease
is quantitatively compatible with the negative self-generated fields from the
plasma channel with an amplitude that reaches $E_{y}-cB_{z}\simeq 1\,\rm
TV/m$. We can distinguish these quasi-static fields from the laser field
thanks to the azimuthal mode decomposition of our simulations. At a later
propagation distance ($L_{x}\simeq 600\,\rm\mu m$), we see in Fig. 9b) the
same positrons are focused again toward the laser propagation axis. This can
be explained as their momenta increase due to the now positive transverse
channel fields. For the sake of the example, we also added another type of
trajectory labeled "2" that oscillates resonantly. The transverse momentum of
this other test-positron changes sign, and this can be as well explained by
the strong transverse plasma fields.

After this analysis of representative trajectories, we consider the hundred
most energetic positrons and prove they experience Direct Laser Acceleration.
To this purpose, we illustrate the accumulated work performed by different
field components on positrons, over the whole propagation of the laser pulse
in the channel ($1\,\rm mm$). The energy gain $W$ is decomposed in two
directions $W_{y/x}=\int\mathbf{v}\cdot\mathbf{E}_{y/x}\,dt$, where $y$ is the
laser polarisation direction and $x$ the laser propagation direction. We also
separate the contribution of the channel fields (mode $m=0$) and the laser
(mode $m=1$). A first look at Fig. 10 shows the gain from the laser field in
its polarisation direction ($W_{y}$ for mode $m=1$ as blue circles) prevails
over the gain in plasma wakefields ($W_{x}$ for mode $m=0$ as red squares).
Despite their smaller contribution, the role of plasma wakefields is non-
negligible. In addition, the contribution of the longitudinal laser field
component ($W_{x}$ for mode $m=1$ as blue circles) is also of comparable
amplitude. We conclude that a complete modeling of Direct Laser Acceleration
should include these contributions. This effort is already discussed [53, 54]
at lower laser intensities ($\sim 10^{19}\,\rm Wcm^{-2}$), but is not yet
generalized for the intensities we consider ($\sim 10^{22}\,\rm Wcm^{-2}$).

Figure 10: Energy gain of positrons experiencing Direct Laser Acceleration
accumulated on 1 mm of laser propagation in the channel. The energy gains $W$
represent the accumulated work in the channel fields ($m=0$, red squares) and
the laser fields ($m=1$, blue circles) after a propagation distance of $1\,\rm
mm$ in the plasma channel. We recall that the laser propagation direction is
$x$, and the polarisation direction is $y$.  Figure 11: Phase space
$(y-\mathcal{E})$ of positrons. It is represented just after their creation in
the thin foil in panel a), after $500\,\rm\mu m$ of laser propagation in panel
b) and after $910\,\rm\mu m$ of laser propagation in panel c).

The positron spectrum simulated contains an unambiguous proof they experience
DLA. We show the $(y,\mathcal{E})$ phase space of positrons in Fig. 11 at
three instants, where $y$ is the laser polarisation direction. At the
beginning of the interaction $L_{x}=40\,\rm\mu m$ in Fig. 11a), the laser has
just travelled through the thin foil and enters the channel. At this point,
positrons reach a maximum energy of $1.5\,\rm GeV$ and it is achieved by the
positrons located on the laser propagation axis $y=0$. This gain is solely due
to positrons oscillating in the laser field as it goes through the thin foil
in the Relativistically Self-Induced Transparency regime. At a later
propagation distance of $L_{x}=500\,\rm\mu m$ in Fig. 11b), the maximum energy
is now $2.5\,\rm GeV$. It is reached by positrons that are not on the laser
propagation axis ($y=0$), but slightly off-axis for a transverse position
$y\simeq 5\,\rm\mu m$. This behavior comes from the amplitude of the betatron
oscillations of the positrons of $\simeq 5-10\,\rm\mu m$, as seen in Fig. 9b).
This so-called forking structure has already been observed experimentally for
electrons and is used as an evidence of the DLA mechanism [48]. Our
simulations confirm that the same type of evidence can be expected for
positrons. We also highlight that the central beam energy is $\sim 1\,\rm
GeV$. The final inset in Fig. 11c) displays the positron spectrum close to the
end of the plasma channel for $L_{x}=910\,\rm\mu m$. The maximum energies are
higher $\simeq 3\,\rm GeV$, while the central beam energy kept increasing to
$1.5\,\rm GeV$. However, this increase is done to the detriment of beam
charge. Indeed, positrons are continuously lost over the propagation of the
laser in the channel due to an imperfect guiding. We have quantified the loss
of positrons for these laser and plasma parameters. The Tab. 2 provides the
initial laser energy for ELI-Beamlines at 9 PW, the total energy of all the
positrons created in the foil, and the energy of the positrons and electrons
in the beam after 1 mm of propagation. The $1.3\,\rm kJ$ laser produces
$2\times 10^{6}$ positrons in total in the thin foil, with a total energy of
$42\,\rm\mu J$ at the time of creation. After 1 mm of propagation in the
plasma channel, there are $10^{4}$ positrons left in the beam, for a total
energy of $3\,\rm\mu J$ (these represent about $0,5\leavevmode\nobreak\ \%$ of
initial positrons, so their initial energy content was $\sim
0.2\leavevmode\nobreak\ \rm\mu J$, while the rest was obtained through
acceleration). One can define an energy conversion efficiency from the laser
to the positrons created in the foil $10^{-6}\%$, which may be improved by
optimizing the laser and foil parameters in the future. The total energy
contained in the positron beam after 1 mm represents $\simeq 7\%$ of the
energy converted to BH positrons within the target.

Table 2: Energy of the incident laser (column 1), of all the positrons created in the foil (column 2), of the positrons in the beam (column 3) and of the electrons in the beam (column 4). The energies of electrons and positrons in the beam are given after 1 mm of laser propagation in the plasma channel. | Laser |  $e^{+}$ (foil) |  $e^{+}$ (beam) |  $e^{-}$ (beam)
---|---|---|---|---
Energy | $1.3\,\rm kJ$ | $42\,\rm\mu J$ | $3\,\rm\mu J$ | $4\,\rm J$
Efficiency ($\%$) | $100$ | $3.2\times 10^{-6}$ | $3.0\times 10^{-7}$ | $0.3$

To conclude this section, we brought evidence that Direct Laser Acceleration
of positrons can take place using Quasi-3D PIC simulations. This confirmation
is done through the analysis of single trajectories and also the full positron
spectrum.

##  III Discussion

We will now compare our approach with existing studies documented in the
literature. While post-acceleration configurations offer better positron
charge and beam quality, they lack compactness as they require a kilometer-
long accelerator [6, 7, 8, 9, 12, 10, 55, 56, 57, 5, 58, 16, 59, 60]. The
distinct advantage of our method lies in its ability to generate and
accelerate positrons to a GeV level with a single 10 PW laser pulse. Other
schemes utilizing multi-PW lasers either solely concentrate on pair creation
without acceleration [61, 62, 63] or employ a 2D geometry that hinders
accurate quantitative predictions [64, 65, 24, 22]. Despite this observation,
it is worth underlining a recent experiment related to the creation of a high-
quality positron beam [66], marking a milestone toward laser-driven positron
acceleration in a plasma wakefield. Our Quasi-3D approach enables us to obtain
precise quantitative estimates for our positron generation and acceleration
scheme. We remark that DLA of positrons was observed in one numerical work
[22]. However, we bring an original contribution by introducing a theoretical
framework describing this process, along with validation against PIC
simulations.

We have also assessed the experimental feasibility of our setup. We considered
nominal parameters for the ELI laser pulse [17], with a total energy of
$1.3\,\rm kJ$ and a peak power of $9\,\rm PW$. There is no particular
constraint of the thin Aluminum foil, except that it should be thin enough
($\simeq 240\,\rm nm$) to enable a significant laser transmission. The
preformed plasma can be produced with commercially available plasma jet [67],
with a density of $10^{21}\,\rm cm^{-3}$ and a length that can be as short as
$400\,\rm\mu m$. For instance, a dense and narrow plasma channel was formed in
an experiment related to ion acceleration in near-critical-density plasmas
with 0.1-1 PW-class lasers [68]. Although the channel is not fully
characterized, its wall density is on the order of $10^{20}\,\rm cm^{-3}$, its
length is $800\,\rm\mu m$ and its radius is $150\,\rm\mu m$. These numbers are
comparable to what we are requiring for our setup, paving the way for a future
experiment. In the context of ELI-Beamlines, the channel could be formed with
the 1 PW laser system (HALPS), while the positron acceleration could be done
with the 10 PW system (ATON).

## Conclusions

To conclude, we have investigated Direct Laser Acceleration of Bethe-Heitler
positrons in laser-channel interactions. In this setup, positrons are created
in a thin aluminum foil and accelerated in a preformed plasma channel by the
same high-power laser pulse (10 PW).

We first build a theoretical framework of electron beam loading during the
propagation of a high-power laser in a dense plasma channel. Our main
assumption is that electrons are injected from the walls of the plasma channel
to its center via the interplay of the two laser field components. In light of
this, we derived the density of electrons loaded as a function of the diverse
parameters of the plasma channel and the laser power. In addition, we use this
result to introduce a theoretical framework for DLA of positrons. The main
outcome is that we obtain a quantitative estimate of electron beam loading as
a function of the initial laser and plasma conditions, thus gaining a better
understanding of the exact acceleration conditions of positrons.

We later confirm the assumptions and predictions of this theoretical model
with Quasi-3D PIC simulations. In particular, our estimate for electron beam
loading predicts the correct scaling measured from Quasi-3D PIC simulations
where we vary a large range of laser and plasma conditions. We have also
checked that electrons are actually injected from the walls of the plasma
channel directly in PIC simulations, as assumed in our model. Finally, we
analyzed single trajectories of positrons and their full spectrum and found
several pieces of evidence they experience DLA.

This work directly contributes to advance our theoretical understanding of
plasma-based positron acceleration. It also contributes to lay the bases of a
potential experiment at ELI-Beamlines to test a new positron acceleration
process. As a potential prospect, the jet produced may be used to mimic the
propagation of astrophysical jets from Gamma Ray Bursts in the laboratory
frame, as it contains three species found in fireball jets: electrons,
positrons and x-rays.

###### Acknowledgements.

We acknowledge the support of the Portuguese Science Foundation (FCT) Grant
No. CEECIND/01906/2018, PTDC/FIS-PLA/3800/2021 DOI: 10.54499/PTDC/FIS-
PLA/3800/2021 and UI/BD/151560/2021 DOI:10.54499/UI/BD/151560/2021 We
acknowledge PRACE for awarding us access to MareNostrum at BSC (Spain),
EuroHPC for awarding us access to LUMI-C at CSC (Finland) and FCT for awarding
us access to Oblivion at University of Evora (Portugal).

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