# The role of standing wave in the generation of hot electrons by femtosecond
laser beams incident on dense ionized target

R. Babjak<EMAIL_ADDRESS>Faculty of Nuclear Sciences and Physical
Engineering, Czech Technical University in Prague, Brehova 7, 115 19 Prague 1,
Czechia  Institute of Plasma Physics, Czech Academy of Sciences, Za Slovankou
1782/3, 182 00 Praha 8, Czechia  J. Psikal  Faculty of Nuclear Sciences and
Physical Engineering, Czech Technical University in Prague, Brehova 7, 115 19
Prague 1, Czechia

###### Abstract

We demonstrate the differences in hot electron absorption mechanisms dominant
in the interaction of femtosecond laser pulse with intensities
$10^{18}~{}\rm{W/cm^{2}}$ and $10^{21}~{}\rm{W/cm^{2}}$ on fully ionized
target with steep density profile and preplasma with moderate scale length (3
$\rm{\mu}$m). We show that acceleration of each electron starts at the moment
when magnetic component of standing electromagnetic wave changes its polarity
in regime without preplasma. In the presence of preplasma the stochastic
heating is the dominant absorption mechanism along with the longitudinal
electric field. It is observed, that wave’s energy is absorbed only if
standing wave is already created at the position of electron during the
interaction with the pulse with intensity $10^{18}~{}\rm{W/cm^{2}}$. In the
case with intensity $10^{21}~{}\rm{W/cm^{2}}$, part of electrons is pre-
accelerated in front of the target before the reflection and following
stochastic heating. The presence of preplasma results in electron temperatures
close to or even exceeding ponderomotive scaling. At higher intensity, the re-
injection of electrons previously repelled by incident wave’s ponderomotive
force into high-field regions is allowed if standing wave is created.

##  I Introduction

The interaction of laser pulses with intensities exceeding
$10^{18}~{}\rm{W/cm^{2}}$ with overdense thin ionized targets found many
applications in recent years, e.g. laser-driven particle acceleration Wilks
_et al._ (2001), high harmonics generation Norreys  _et al._ (1996), fast
ignition in inertial confinement fusion Craxton  _et al._ (2015) or laboratory
astrophysics Bulanov  _et al._ (2015). When electromagnetic (EM) field at such
intensities interacts with the plasma, fraction of the radiation is absorbed
via non-linear processes present during the interaction and the rest is
reflected from the overcritical electron-plasma boundary (EPB). Therefore, the
understanding of underlying radiation-matter coupling mechanisms is essential
for this research.

Energy from the EM field that is absorbed is mostly converted into the
population of so-called hot electrons which are accelerated/generated near the
target surface or in the underdense preplasma formed in front of the target.
Many theoretical and empirical models were proposed to explain such
acceleration Macchi, Borghesi, and Passoni (2013); Beg  _et al._ (1997); Kluge
_et al._ (2011); Wilks  _et al._ (1992); Denavit (1992); Chen  _et al._
(2009); Culfa  _et al._ (2014), but they differ in prediction of electron
properties such as number of accelerated particles, quasi-temperature, cut-off
energy or angular distribution depending on simulation/experimental
parameters. The acceleration mechanisms of hot electrons seem to be the most
sensitive to laser intensity, preplasma profile and angle of incidence of
laser pulse on the target.

When steep density profile is present, the acceleration is strongly influenced
by the quasi-static electric field created at the EPB Debayle  _et al._
(2013); Bulanov  _et al._ (2013) and by the electromagnetic field in the form
of standing wave created due to the interference of incident and reflected
laser pulse Kemp  _et al._ (2013); May  _et al._ (2011); Hüller  _et al._
(2019). When preplasma Esirkepov  _et al._ (2014); Hadjisolomou  _et al._
(2020) is present, the coupling is explained mostly by stochastic heating
Paradkar  _et al._ (2011); Sheng  _et al._ (2004); Sentoku  _et al._ (2002);
Chopineau  _et al._ (2019) or Direct laser acceleration (DLA) mechanism
Pukhov, Sheng, and Meyer-ter Vehn (1999); Krygier, Schumacher, and Freeman
(2014).

This work aims to provide further insight into the regime of interaction when
femtosecond laser pulse is incident normally on the target. In the case of
steep plasma density profile, we discuss in detail the ejection of electrons
into the vacuum caused by the electrostatic field created at the vicinity of
EPB and their acceleration in the standing wave created in front of the
target. When intensity of incident laser pulse is higher, we demonstrate the
effect of target surface deformation due to strong ponderomotive pressure. In
preplasma, hot electron acceleration mechanism strongly differs compared to
the acceleration in the vicinity of EPB at sharp interface and leads to
electron temperatures close to or even exceeding ponderomotive scaling. We
clearly demonstrate the necessity of standing wave for moderate length
preplasma (in order of $\mu$m ) in order to accelerate electrons and the
presence of stochastic heating and describe why longer preplasma results in
higher electron energies. For higher intensity, we show that in the presence
of standing wave, previously repelled electrons by the ponderomotive force can
be re-injected into the high-field regions. We also propose an explanation of
higher temperatures of hot electrons in the case od p-polarization compared to
s-polarization in the target with preplasma.

##  II Methods and simulations parameters

In order to study hot electron generation, the 2D version of Particle-In-Cell
code Smilei Derouillat  _et al._ (2018) was used. Linearly p-polarized laser
pulse of wavelength $\lambda=800$ nm with intensities
$10^{18}~{}\rm{W/cm^{2}}$ and $10^{21}~{}\rm{W/cm^{2}}$ at maximum (i.e., with
dimensionless pulse amplitudes $a_{0}\approx 0.68$ and $a_{0}\approx 21.6$,
respectively) was incident on target with sharp density profile and on the
target with preplasma in front of the dense thin foil. The combination of
density profiles and pulse intensities resulted in 4 investigated scenarios.
For simplicity, plasma containing only electrons and protons was assumed. The
density of overdense foil was $\approx 30~{}n_{c}$ (which corresponds to fully
ionized hydrogen in the density of solid state), where $n_{c}$ is critical
density. Scale length of exponentially increasing density of preplasma
$n=n_{c}\times\exp(x/L)$ for $-14.5~{}\rm{\mu m}<x\leq 0~{}\rm{\mu m}$ was
$L=3\rm{\mu}$m, until reached the overdense part of the target with constant
maximum density located at $0~{}\rm{\mu m}<x<5~{}\rm{\mu}$m. Target with steep
density gradient contained only overdense thin foil and the above defined
preplasma was put in front of the target for another studied cases. The size
of grid cell was set to $16\times 16~{}\rm{nm}$, 50 macroparticles per cell
were used for overdense part and preplasma contained 16 macroparticles per
cell due to its lower density. Laser pulse was propagating towards the target
in x direction from the boundary located at $x_{min}$
($x_{min}=-12~{}\rm{\mu}$m for target without preplasma and
$x_{min}=-15~{}\rm{\mu}$m for target with preplasma) with the beam axis at the
position $y=0~{}\rm{\mu}$m, it had a trapezoidal temporal profile with linear
10 fs intensity growth, 40 fs constant intensity and 10 fs linear intensity
decrease. The spatial profile was Gaussian, laser beam diameter (FWHM) was set
to 4 $\rm{\mu m}$ at focus. Laser was focused into plane coordinates (0,0).
The simulation box size was set to $31\times 31\lambda^{2}$ and to $34\times
31\lambda^{2}$ for targets without/with preplasma, which means that it
contains $1488\times 1488$ and $1632\times 1488$ cells, $4.4\times 10^{7}$ and
$7.8\times 10^{7}$ macroparticles, respectively. The initial temperature of
macroparticles was 1 keV, the macroparticles were frozen before (pre-)plasma
interaction with the laser in the simulations. The cell size was set to be
about 2/3 of the skin depth in order to resolve relevant physics. Units in the
figures are either specified explicitly, or following reference values are
used for the normalization: electric field $E_{r}=m_{e}c\omega/e$, magnetic
field $B_{r}=m_{e}\omega/e$, momentum $p_{r}=m_{e}c$, force
$F_{r}=m_{e}c\omega$, where $m_{e}$ is mass of an electron, $c$ is speed of
light, $\omega$ is laser frequency, $e$ is elementary charge.

Figure 1:  The position of overdense target with preplasma in front of it.
Yellow color shows the position of test particles.

This work vastly uses particle tracking diagnostics implemented in Smilei
together with test particles. Test particles do not contribute to charge and
current densities, but the code calculates their trajectories from EM fields
at their location. Particle tracking diagnostics saved position, momenta, and
fields at the position of particle. At simulations with steep density profile,
test electrons were put at the vacuum-plasma interface into the strip of width
$3\lambda$ ($x=\left\langle 0,3\lambda\right\rangle$) irradiated by the laser
beam where hot electrons are expected to be generated. For simulations with
preplasma, test electrons are present along the entire length of preplasma in
$x$ direction. In $y$ direction, test electrons were placed inside the region
$y=\left\langle-6\lambda,6\lambda\right\rangle$ both in simulation with and
without preplasma. Their location is displayed by yellow color in Fig. 1. All
particles were tracked from the beginning of the simulation.

Hot electron population can be easily distinguished from the target bulk
electron population in energy distribution function. Hot and cold electron
temperatures can be assigned to both populations with Boltzmann energy
distribution Liseykina, Mulser, and Murakami (2015); Tikhonchuk  _et al._
(2005). Therefore, the minimal energy $E_{min}$ of hot electron population can
be defined at the place where the slope of high energy tail changes in energy
distribution function. During the post-processing, only those particles that
were inside the overdense part of the target while satisfying the condition
$E>E_{min}$ were analyzed.

##  III Steep density gradient

When laser pulse interacted with target with steep density gradient, the
interaction started at time 17.5$T$ and finished at time 40$T$, where $T$ is
laser period.

###  III.1 Intensity $10^{18}~{}\rm{W/cm^{2}}$

Hot electron acceleration mechanism observed in our simulation results can be
described as follows. In Fig. 2, the evolution of tracked electron quantities
during the acceleration is depicted. This evolution of momentum and fields
acting on particle represents qualitative behaviour common for all hot
electrons. At the beginning of each acceleration process, the growth of
momentum in a direction out of the target ($p_{x}<0$) occurs. This happens due
to the effect of electrostatic field pulling the electron out to the vacuum as
can be seen in Fig. 2 a) at time around 31.5$T$. $p_{x}$ is negative at the
same time when $E_{x}$ component of electric field is positive at the position
of particle, which results in electric component of Lorentz force acting on
electron in direction out of the target, see Fig. 2 b). Magnetic component of
Lorentz force (B) has no impact on electron being pulled out to the vacuum.
The electrostatic field on the boundary of plasma and vacuum is created
because of oscillation of electrons around immobile ions (on the time scale of
a laser period as they have much larger mass) with frequency of 2$\omega$,
where $\omega$ is the laser frequency. This ejection of electrons into the
vacuum by electrostatic field was previously described in Refs. Debayle  _et
al._ , 2013; Cai  _et al._ , 2006.

Figure 2: (a) The evolution of momentum and fields acting on an electron
during the acceleration process by normally incident laser pulse with
intensity $10^{18}~{}\rm{W/cm^{2}}$. (b) The time evolution of forces acting
on the electron. At the beginning, the electrostatic force (E) pushes the
electron to the area in front of the target and consequently the magnetic part
of Lorentz force (B) causes the acceleration inside the target.

When electron is already in the vacuum, it interacts with the standing EM wave
created in front of the target and depicted in Fig. 3. Electric component of
the standing wave $E_{y}$ causes that electron gains momentum in the direction
along the target surface. Acceleration in the $x-$direction starts at the
moment when the magnetic field perpendicular to the plane of incidence $B_{z}$
changes its sign as shown in Fig. 2 a) in time shortly before 32$T$. At that
time, the product of $v_{y}B_{z}$ is negative, so the Lorentz force acts on a
negatively charged electron in a positive direction towards the target.
Relativistic values of velocity $v_{y}$ lead to a relatively high strength of
the magnetic part of Lorentz force in perpendicular direction to the standing
wave $B_{z}$ and $E_{y}$ fields orientation, i.e., perpendicularly to the
target surface. The process of acceleration ends at the moment when particle
crosses the plasma boundary and doesn’t feel the standing wave field. The
typical trajectories of accelerated electrons can be seen in Fig. 4 a).

According to how the acceleration process was described, it is possible to
assume that the standing wave structure has the most significant impact on the
acceleration process. Electrons immediately react to changes of the
perpendicular electric field $E_{y}$. It is also possible to see that $B_{z}$
and $p_{y}$ values are shifted by $\pi$ in phase and they have the same
period. Thus, the force proportional to $v_{y}B_{z}$ is always positive. Since
the electric field is harmonic in time, the phase shift between $p_{y}$ and
$E_{y}$ is $\pi/2$. The phase shift between $E_{y}$ and $B_{z}$, which is also
$\pi/2$ (see Fig. 3), causes already mentioned $\pi$ phase shift of $p_{y}$
and $B_{z}$. The action of the magnetic part of Lorentz force thus twice a
period accelerates electrons into the target.

Figure 3: Standing wave which was created after the reflection of laser pulse
with intensity $10^{18}~{}\rm{W/cm^{2}}$ from the target without preplasma at
time 33.4$T$. Fields are normalized by reference fields: electric field
$E_{r}=m_{e}c\omega/e$, magnetic field $B_{r}=m_{e}\omega/e$. It is visible
that the standing wave is shifted because of finite conductivity of the target
and consequent skin effect.

All hot electrons were accelerated in the region between the first node of the
magnetic field at distance $\approx 1/4\lambda$ in front of the target and
EPB, see Fig. 3. In Fig. 5, points on the graph show the time and the position
of electrons at the beginning of acceleration. The color gives information
about the maximal longitudinal momentum of electrons after the acceleration.
The moment of the beginning of the acceleration is here defined as the moment
when $p_{x}$ reaches its minimal value. The acceleration twice a period
typical for $\vec{j}\times\vec{B}$ heating can be observed from vertical
bunches repeating every half-period. As mentioned before, acceleration starts
at the moment when the magnetic field changes its sign. This happens at the
same time at positions of all electrons in front of the target twice a period.
That is the reason why all electrons within one bunch start to be accelerated
at the same time independently from its position.

Figure 4: Density of the target during the interaction with laser pulse with
intensity $10^{18}~{}\rm{W/cm^{2}}$ (a) and $10^{21}~{}\rm{W/cm^{2}}$ (b) with
typical electron trajectories. It can be seen that EPB is completely
unperturbed for lower intensity and electrons are accelerated immediately
after being pulled out of the target. Higher intensity pushes the EPB inside
the target and electrons oscillate chaotically before being accelerated.

Note that minimal $x$-position for each accelerated electron is below its
position depicted in Fig. 5 a) as the longitudinal momentum $p_{x}$ is still
negative in the beginning of electron longitudinal acceleration towards the
target. The position of electrons is about $x=-0.1\lambda$, whereas the
minimum position is at around $x=-0.15\lambda$ and the electrons still does
not move behind magnetic field node at $x\approx-1/4\lambda$. One can expect
that the position of an electron during the acceleration determines its
longitudinal momentum increase after the acceleration which depends on
transverse electric field $E_{y}$, transverse magnetic field $B_{z}$, and
electron acceleration time. The thickness of the acceleration region in front
of the target should be proportional only to the strength of the longitudinal
field $E_{x}$ pulling electrons out into the vacuum. In semi-analytical model
Debayle  _et al._ (2013) assuming infinite ion density gradient with immobile
ions, the strength of such $E_{x}$ field is theoretically fully determined by
laser pulse amplitude and plasma density. However, the target expands during
the interaction with main laser pulse and due to non-zero temperature of
plasma before the main pulse interaction (electron heating by laser pedestal
or by picosecond slope of the pulse rising slowly compared to an ideal
Gaussian pulse Wagner  _et al._ (2016)). Therefore, the width of the
acceleration region strongly depends on the real value of steep plasma density
gradient and evolves during the interaction (increases with decreasing ion
density gradient in time). That is also apparent in Fig. 5 a) where more than
90 percent of the electrons start to be accelerated in the $x$-direction at
position $0>x>-0.05\lambda$ at time around 24$T$ but at wider region
$0>x>-0.1\lambda$ at later time around 32$T$.

Figure 5: Times and positions when electron acceleration towards the target
started during the interaction with normally incident pulse of intensity (a)
$10^{18}~{}\rm{W/cm^{2}}$ and (b) $10^{21}~{}\rm{W/cm^{2}}$ on target with
steep density gradient.

###  III.2 Intensity $10^{21}~{}\rm{W/cm^{2}}$

When laser pulse with intensity $10^{21}~{}\rm{W/cm^{2}}$ was incident on
target with steep gradient, interaction regime significantly changed from
idealized scenario. The strong radiation pressure deformed the target and
bored a hole in it. Consequently, initially step-like interface was strongly
disrupted and interaction region consisted of local density, electric and
magnetic field disturbances. Curved shape of overdense plasma boundary caused
that reflected wave was strongly non-uniform. Therefore, the standing wave in
front of the target was far from ideal. Due to many violations in the
interaction geometry, hot electrons were accelerated and decelerated in front
of the target before they crossed the boundary of overdense foil, as can be
seen in Fig. 4 b). The impact of electric field is negligible, longitudinal
motion of electron is determined by the magnetic component of Lorentz force.

In Fig. 5 b), one can see the starting points of acceleration defined in the
same manner as in the previous case. The EPB being pushed can be observed in
the left top corner. It can be estimated from the figure that the plasma layer
was pushed with velocity $\sim 0.08c$. The value of EPB velocity agrees with
the value obtained from densities. Since electrons are mostly accelerated
close to the boundary, the starting point of acceleration moves with it.
Firstly, it moves quickly up to the moment when an equilibrium is established
between the electrostatic pressure generated by compressed electron cloud and
the radiation pressure of laser beam on the time scale of laser period Klimo
_et al._ (2008). Then, the hole boring process continues both in electron and
ion densities during the whole interaction Robinson  _et al._ (2009). This can
be seen in the figure up to the time around 31$T$ when the points are more
scattered along $x$-position due to the increasing difference in EPB
deformation along $y$-axis.

The estimated EPB deformation at the first moment of the pressures equilibrium
Klimo  _et al._ (2008) $\Delta_{init}=\sqrt{4\epsilon_{0}I/c}/(en_{e})\approx
130~{}\rm{nm}$ and the estimated hole boring velocity Robinson  _et al._
(2009) $u_{hb}/c=\sqrt{\Xi}/(1+\sqrt{\Xi})\approx 0.06$, where
$\Xi=I/(m_{i}n_{i}c^{3})$, agrees well with our observations in the
simulation. According to these analytical estimates, one can expect less
pronounced deformation of the target by strong radiation pressure with
increasing target density (e.g., using metal foils). Note that EPB deformation
can be seen in Fig. 4 b) when the deformation exceeds the initial value
$\Delta_{init}$ due to hole boring process after the initial EPB deformation.

The width of the acceleration region is substantially larger along
$x$-direction compared with $10^{18}~{}\rm{W/cm^{2}}$ case. Although the
standing wave structure is disrupted, it is apparent that some electrons are
accelerated beyond the first node of the magnetic field from EPB in this non-
ideal standing wave. Two strips of points where electrons are started to be
accelerated are perceptible in Fig. 5 b). The strip located closer to EPB is
between EPB and the first node, the other strip is at some distance from the
first magnetic field node where the combination of instantaneous transverse
electric field $E_{y}$ and magnetic field $B_{z}$ is favourable for electron
acceleration towards the overdense plasma.

The initial density of overdense target $n_{e}\approx 30n_{c}$ is relatively
low with regard to the commonly presented threshold for relativistic
transparency $a_{th}=n_{e}/n_{c}$ which is close to the pulse amplitude
$a_{0}\approx 22$ for a given laser intensity. However, this threshold is
essentially increased to $a_{th}\approx 0.65(n_{e}/n_{c})^{2}$ for
$(n_{e}/n_{c})>>1$ in the case of non-uniform plasmas due to the action of the
ponderomotive force which pushes electrons into the plasma creating a strong
peaking of the plasma electron density Cattani  _et al._ (2000). Moreover, the
total pulse length is 60 fs in our case, which is not enough for substantial
plasma expansion and lowering the plasma density. Thus, we are still far from
relativistically induced transparency regime during the whole interaction even
with the lowest solid density foil made from hydrogen.

##  IV Effects of preplasma

In the study of the interaction of laser pulse with overdense target slightly
evaporated by the pulse pedestal, preplasma with exponentially increasing
density $n_{e}=n_{c}\times\exp(x/L)$ for $x\leq 0$ and scale length
$L=3\rm{\mu}$m was assumed in front of the overdense ionised foil located at
$0<x<5~{}\mu$m , see Fig. 1. Therefore, the distance between the simulation
box boundary where laser pulse enters and the overdense plasma had to be
increased and laser pulse started to reflect from overdense thin foil at time
20$T$ and stopped the interaction at time 42.5$T$.

###  IV.1 Intensity $10^{18}~{}\rm{W/cm^{2}}$

The presence of preplasma in front of the target significantly changed the
interaction regime. The most of electrons were accelerated in underdense
preplasma, not in the proximity of overdense plasma boundary like in the
previous case with steep gradient.

Figure 6: (a) Electrons oscillating perpendicularly in the preplasma at the
position of laser pulse with intensity $10^{18}~{}\rm{W/cm^{2}}$ propagating
towards the overdense part of the target. (b) Electrons on the right from the
reflected wave’s wavefront at $x=-6~{}\rm{\mu}$m are red, which means that
they were already accelerated towards the overdense part of the target while
those on the left are still oscillating in the field of incident wave.

Acceleration process can be described as follows. While laser pulse propagates
through the preplasma, electrons oscillate perpendicularly to the pulse
propagation direction due to the effect of oscillating electric field. When
laser wave is reflected, it starts to propagate in the direction out of the
target and the superposition of reflected and incident wave creates a non-
ideal standing wave. At the moment when reflected wave reaches the position of
oscillating electron, it is accelerated inside the target as can be seen in
Fig. 6. In a), electrons are not accelerated yet (black color of dots) because
laser pulse still propagates towards the overcritical target surface. In b),
the wavefront of reflected wave is located at around $x=-6~{}\rm{\mu m}$ and
the standing wave is already created at positions $x>-6~{}\rm{\mu m}$. The
color of electrons indicates that they are already accelerated into the
target. It can be seen that electrons on the left from the wavefront were not
accelerated yet because standing wave is just about to be formed in that
region. This clearly shows the impact of standing wave on electron
acceleration. The fact that electrons are not accelerated before the reflected
wave reaches their position is demonstrated by the maximum distance of
acceleration from the target in Fig. 7. The maximum distance of acceleration
increases with time and corresponds to the position of reflected wave’s
wavefront. The position of reflected wavefront is approximately shown by the
black line.

Figure 7: Times and positions when electron acceleration towards the target
started during the interaction with the pulse with intensity
$10^{18}~{}\rm{W/cm^{2}}$ on target with preplasma. Black line shows the
position of the wavefront of the reflected wave from overdense plasma layer.
Figure 8: Evolution of forces acting on electron in during the acceleration
and $p_{x}$ of the electron when laser with intensity
$10^{18}~{}\rm{W/cm^{2}}$ is incident on the target with preplasma. Vertical
line indicates the moment of acceleration due to our definition. In (a),
electron was accelerated by the stochastic effect of electric and magnetic
components of Lorentz force. In (b) electron was accelerated by the
longitudinal electric field. (c) The contribution of electric field components
to the acceleration at the moment of maximum $p_{x}$.

The evolution of electron momentum and Lorentz force components acting on it
are depicted in Fig. 8. The blue vertical line defines the starting point of
acceleration used in Fig. 7. It can be seen in Fig. 8 a) that the starting
point of acceleration is only a rough estimate since more $p_{x}$ minima are
present close to the vertical line, however it is sufficient for our analysis.
Fig. 8 illustrates two boundary cases of acceleration scenarios observed for
all tracked electrons.

Many electrons were accelerated in a way similar to Fig. 8a). The negative
momentum gain is again present followed by the acceleration towards the
target, but while electron propagated through preplasma it was periodically
accelerated until it reached the overdense plasma at time approximately 37$T$.
The final momentum gain was the result of several pushes by the standing wave
and longitudinal $E_{x}$ while electron propagated towards the target. On the
other hand, some electrons had very similar $p_{x}$ evolution during the
acceleration process as electrons accelerated in the target with steep
gradient, see Fig. 8b). At time around 25$T$, electron gains momentum in the
direction away from the target and right after it is accelerated towards the
target. Here, the initial negative momentum gain occurs in preplasma and does
not mean pulling the electron out of the overdense plasma layer. The second
difference is that mostly longitudinal electric field component was
responsible for the acceleration, not the standing wave field. However, the
most of electrons were accelerated by the combination of several strong
electric field pushes similar to Fig. 8b) and by the chaotic accelerations and
decelerations similar to Fig. 8a). Typical electron trajectories are depicted
in Fig. 9

Figure 9: Typical electron trajectories during the acceleration after the
incidence of pulse with intensity $10^{18}~{}\rm{W/cm^{2}}$ on target with
preplasma.

To quantify the effect of mechanisms, the contribution of electric field
components was expressed using formula
$\gamma^{2}(t)-\gamma_{0}^{2}=-2\int_{t_{0}}^{t}E_{x}p_{x}d\tau-2\int_{t_{0}}^{t}E_{y}p_{y}d\tau-2\int_{t_{0}}^{t}E_{z}p_{z}d\tau=I_{x}+I_{y}+I_{z}$,
where $\gamma_{0}$ is the initial gamma factor of accelerated electron. The
value of integral $I_{x}$ represents the contribution longitudinal electric
field which can be present because of charge separation and laser self-
focusing. The integral $I_{y}$ represents the contribution of background
transverse field and the oscillating field of laser pulse which was
consequently bend into $x$-direction by the magnetic field. In chosen geometry
the value of $I_{z}$ can be neglected.The value of integrals was evaluated at
the moment when electron reached its maximum in $p_{x}$, see Fig. 8c).

The effect of electric and magnetic field on electron acceleration is
comparable for electrons with lower maximal momentum, namely
$p_{x}^{max}<2.5m_{e}c$. This is in agreement with acceleration process
described in Fig. 8a), where the combination electric and magnetic Lorentz
force contributes to the final momentum. Electrons with the highest energies
were accelerated mostly by the longitudinal electric field, see Fig. 8b).

The mechanism responsible for acceleration by the transverse oscillating field
is stochastic heating. The acceleration happens in the field of standing wave
and the Lyapunov exponentLichtenberg A. J. and Lieberman M. A. (1983) for
accelerated electrons is positive. The value of Lyapunov exponent was
calculated in a following way. Pairs of randomly selected hot electrons were
created and the Lypunov exponent was calculated for all pairs according to
Ref. Lichtenberg A. J. and Lieberman M. A., 1983. The initial time $t_{0}$ was
selected as a time when reflected laser wave reached the position of the
electron. Afterwards, the sequence of Lyapunov exponents was created as a
function of initial electron pair distance in phase space
$d(\vec{x_{0}},t_{0})$. The final value of Lyapunov exponent was obtained from
the sequence as a limit for $d(\vec{x_{0}},t_{0})\to 0$.

The positive value of Lyapunov coefficient demonstrates the stochasticity of
electron motion. To confirm this we performed the simulation with counter-
propagating pulses with intensity $10^{18}~{}\rm{W/cm^{2}}$, where only test
particles were present. The effect of background field does not affect the
electron motion in such simulation. The Lyapunov exponent of electron motion
in the standing wave created by the counter-propagating pulses was positive
and the acceleration was present with maximum $p_{x}$ exceeding $m_{e}c$. Only
$I_{y}$ was non-zero during this acceleration which means that electron’s
oscillations in transverse direction were turned into positive x-direction.
Based on just mentioned observation we conclude that mechanism responsible for
electron acceleration in PIC simulations is the combination of stochastic
heating and longitudinal electric field.

###  IV.2 Intensity $10^{21}~{}\rm{W/cm^{2}}$

Figure 10: Phase space of electrons after the incidence of laser pulse with
intensity $10^{21}~{}\rm{W/cm^{2}}$ on the target with preplasma. Color shows
the relative amount of electrons where numbers on colorbar corresponds to the
exponent on the logarithmic scale.  Figure 11: Positions and energies of hot
electrons during the acceleration after the incidence of laser pulse with
intensity $10^{21}~{}\rm{W/cm^{2}}$ on the target with preplasma. (a)
Travelling wave repels electrons; (b) Electrons are injected into the pulse
where standing wave is created. Electrons from population A can be marked by
the black circle. Electrons outside the circle belong to the population B.
(c)-(d) Electrons are accelerated in the field of standing wave.

Higher intensity of laser pulse brought several new phenomena into the
interaction compared to the case with intensity $10^{18}~{}\rm{W/cm^{2}}$
while the dominant impact of standing EM wave was still present. Two hot
electron populations can be distinguished in Fig. 10. The first one
(population A) is located at $x$-axis position around 10 $\rm{\mu m}$ and
another one is spread between $x=0$ and $7$ $\rm{\mu m}$ (population B). The
rough estimate is that 45$\%$ of hot electrons belong to the population A and
55$\%$ of hot electrons belong to the population B. The population A doesn’t
seem to be formed of separate electron bunches as expected for electrons
accelerated by normally incident pulse. The distances between bunches were
smoothened when electrons crossed the rear side of the target at $x=5\rm{\mu
m}$ and were clearly visible at earlier times when they were still located in
front of or inside the target. The distance between electrons in the
population B is $\lambda/2$ since they still didn’t cross the rear side.
Distances between the bunches are identical as in the case of
$\vec{j}\times\vec{B}$ heating even though electrons were not accelerated at
overdense plasma boundary but rather in preplasma.

The acceleration process can be described by Fig. 11. Four figures describe
the propagation of laser pulse in preplasma towards the overdense plasma layer
with hot electrons positions.

In a) laser pulse is propagating through preplasma. Electrons are either
pushed forward in front of the pulse or to the sides of the pulse by the
ponderomotive force. The color of electrons shows that they were not
accelerated yet because the laser pulse was still propagating towards the
target and standing wave was not created yet.

In b) the splitting of electrons into two populations can be already observed.
The population A is located closest to the overdense layer marked by the blue
vertical line. It is marked inside the black circle. This shows that the
population A is created mostly by the electrons that are pushed in front of
the pulse. All electrons that are outside the circle belong to the population
B. They are plotted either as a black unaccelerated dots on the sides of the
pulse or they are already starting to form lines perpendicular to the laser
propagation direction by being attracted to the center of the beam with high
intensity laser field.

In c) the population B is already accelerated and is located inside the
target. The blue line showing the critical density layer roughly separates
population A from population B. All electrons that were previously on the
sides of laser pulse were attracted to the region of laser pulse and they are
accelerated towards the overdense part of the target.

In d) the most of electrons from the first population already crossed the rear
side of the foil and majority of electrons from population B is propagating
inside the target while the small part still interacts with the
electromagnetic field in front of the target.

Two populations (A and B) are also easily observed in Fig. 12 where initial
positions and times of the start of acceleration are shown. The initial time
of acceleration is again defined as a time when $p_{x}$ has a minimum value.
The black arrow represents the position of laser pulse wavefront. The point
(23.5$T$,0$\rm{\lambda}$) represents the time when constant part of the laser
pulse is reflected from the overdense region. The arrow coming from this point
separates two hot electron populations from each other. On the left, straight
lines parallel to incident wave’s wavefront correspond to times when electrons
were pushed by the propagating laser pulse towards the target. Electrons
depicted on the right were not pushed by the propagating beam. Instead, they
were injected to the high-intensity part of the laser pulse from the sides and
afterwards accelerated similarly to the case describing pulse with intensity
$10^{18}~{}\rm{W/cm^{2}}$. Again, the maximum distance of acceleration at
certain time is constrained by the position of wavefront of reflected wave.
For those electrons the standing wave seems to be inevitable condition for
acceleration to occur.

Figure 12: Times and positions when electron acceleration towards the target
started during the interaction with normally incident pulse of intensity
$10^{21}~{}\rm{W/cm^{2}}$ on target with preplasma. Black line shows the
position of the wavefront of the laser wave before and after the reflection
from overdense plasma layer. Figure 13: Evolution of forces acting on an
electron during its acceleration, position and momentum of the electron when
laser with intensity $10^{21}~{}\rm{W/cm^{2}}$ is incident on the target with
preplasma. Vertical line indicates the moment of acceleration according to our
definition. (a) $x$ components; (b) $y$ components; (c) The contribution of
the electric field components at the time of $p_{x}$ maximum and the increase
of gamma factor.  Figure 14: Typical electron trajectories during their
acceleration after the incidence of pulse with intensity
$10^{21}~{}\rm{W/cm^{2}}$ on the target with preplasma. The repulsion of
electrons on the sides of laser pulse can be observed from electron density.
Electrons oscillate on the sides of laser pulse (black part of trajectory) and
they are injected into the beam axis region where the acceleration happens
afterwards (red/yellow part of trajectory).

After a look on $y$ position of typically accelerated electron from the
population B in Fig. 13 b), it can be seen that electron was moving towards
the centre of laser pulse before it was accelerated. This electron movement
along the target surface to the centre of laser pulse is caused by the $p_{y}$
momentum component oscillating in positive values before the blue vertical
line showing the starting time of acceleration used in Fig. 12. Such injection
into the high-field regions occurs because of the presence of standing wave
and plasma channel and is further analyzed in section V. After electron
injection into the high field region, it is accelerated towards the target,
see Fig. 13 a) where the electron was consecutively accelerated during 5 laser
periods while it propagated towards the target in the standing wave. Additive
impact of several pushes from the standing wave while it propagated in
preplasma allowed electron population to be heated to temperature close to
ponderomotive scaling Wilks  _et al._ (1992)
$T_{h}=((1+a_{0}^{2})^{1/2}-1)m_{e}c^{2}$, where $T_{h}$ is hot electron
temperature in eV. Namely, the temperature of hot electrons observed in the
simulation was $8.3$ MeV, using Boltzman distribution. Temperatures of hot
electrons were higher than those predicted by the Beg’s scaling Beg  _et al._
(1997). For the case of lower intensity, hot electrons even exceed the
ponderomotive scaling. The temperature of hot electrons accelerated in the
target with preplasma exceeds the temperature of electrons accelerated by the
pulse with same intensity in the target with steep density gradient. This was
observed for both laser intensities, see Table 1.

Table 1: Comparison of hot electron temperatures for different scenarios fitted using Boltzman distribution compared with ponderomotive scaling $T_{pond}=((1+a_{0}^{2})^{1/2}-1)m_{e}c^{2}$ | T[keV] | T[keV] | T[keV] |
---|---|---|---|---
I[$\rm{Wcm^{-2}}$] | steep density | preplasma | preplasma |  $\rm{T_{pond}}$[keV]
| p-polarization | p-polarization | s-polarization |
$10^{18}$ | 24 | 250 | 150 | 96
$10^{21}$ | 4800 | 8300 | 7200 | 10538

To determine the acceleration mechanism, we have calculated the contribution
of electric field components and compared the values at the moment of $p_{x}$
maximum, see Fig. 13 c). The integral $I_{x}$ corresponds to the contribution
of longitudinal electric field. This is usually not significantly present
during the normal incidence. However, the radiation pressure deformed the
overdense target surface into parabolic-like shape which focused the laser
pulse and turned the $E_{y}$ component of laser field into $x$-direction. The
relativistic self-focusing and charge separation are also present but it is
not possible to distinguish between the possible $E_{x}$ field sources because
of simulation complexity. The integral $I_{y}$ corresponds to the contribution
of standing wave and background transverse field. Since the difference between
electron temperatures for different polarization were previously observed
Stark  _et al._ (2017), we performed simulations of target with preplasma also
with s-polarized pulse. They showed that the interaction with p-polarized
pulse leads to higher electron temperature. We propose the following
explanation. When electrons are accelerated by the standing wave, they
oscillate in either positive or negative values of $p_{y}$ while magnetic
field of the standing wave transfers the momentum into $x$-direction. When the
background field of plasma channel is present, the channel electric field can
increase the mean value of $p_{y}$ oscillations. Therefore, the higher
transverse momentum can be transferred into $x-$direction by magnetic field
which results in higher electron momentum and temperature. When oscillating
electric field is oriented out of simulation plane in s-polarized pulse, the
effect of background electric field on electron energy is negligible and
results in lower electron temperatures as expected. The positive value of
Lyapunov exponent for electron trajectories indicates that the mechanism of
electron acceleration corresponding to $I_{y}$ is stochastic heating.

The effect of longitudinal electric field and stochastic heating on electron
acceleration is comparable for electrons with lower maximum momenta
($p_{x}^{max}<120~{}m_{e}c$). On the other side, stochastic heating is the
dominant acceleration mechanism for the most energetic electrons. Typical
electron trajectories are depicted in Fig. 14. At the beginning, electrons are
located on the sides of the laser pulse because they were repelled by the
ponderomotive force. After some time they are injected into the beam center
where they are accelerated by the mechanisms described above. The injection
mechanism is explained in the following section.

The influence of preplasma scale length on electron energies was examined by
two additional simulations with p-polarized pulse. Shorter preplasma had a
scale length $L_{s}=2\lambda~{}(1.6\rm{\mu m})$ and longer preplasma scale
length was set to $L_{s}=6.25\lambda~{}(5.0\rm{\mu m})$. The trend was obvious
- longer preplasma resulted in higher electron energies. Maximum $p_{x}$ for
shortest preplasma was $\approx 150m_{e}c$, maximum $p_{x}$ for moderate
preplasma ($3.75\lambda,3.0\rm{\mu m}$) was $\approx 200m_{e}c$ and longest
preplasma target resulted in electrons with almost $300m_{e}c$.

Higher electron energies in the case of moderate preplasma were the result of
longer interaction length of electrons with the standing wave. The longer
interaction length is a known property of stochastic heating Paradkar  _et
al._ (2011). Since in the case of moderate preplasma the maximum possible
distance of interaction with the standing wave was already achieved (defined
by the length of laser pulse), the result of increase in electron energy for
the longest examined preplasma is different. When laser pulse propagated in
preplasma, the modulation of laser pulse amplitude started to be developed,
which created the lower amplitude region in the center. Electrons were
captured inside this region in the center of propagating laser pulse and they
were pre-accelerated even before the laser reflection. The pre-acceleration of
electrons before the stochastic heating led to the higher electron energies.
Similar phenomena was observed in Ref.Hadjisolomou  _et al._ , 2020. It caused
the higher energies of accelerated electrons, even though the capturing
mechanism is assigned to different effect in the reference.

Interesting property of $p_{x}$ evolution for the majority of hot electrons
was the negative momentum values just before the acceleration into the target.
This was however the same feature of all electrons being accelerated in four
distinct interaction regimes described.

##  V Standing wave electron injection

Figure 15:  (a) The magnetic field component of the standing wave for pulse
with intensity $10^{21}~{}\rm{W/cm^{2}}$ at $y=1.5\lambda$ and its Fourier
transform. $\omega=1$ corresponds to the incident wave and shifted peak of
$\omega=0.9$ corresponds to the reflected wave. (b) Magnetic field in front of
the overdense target averaged over single laser period in times between 27$T$
and 28$T$. (c) Electric field in front of the overdense target averaged over
the same laser period.

Ponderomotive force is well known for repelling particles from high field
regions of laser pulses down the gradient. However, the nature of
ponderomotive force in standing wave changes. It was previously shown Kaplan
and Pokrovsky (2005), that ponderomotive force by the standing wave in
relativistic regime can cause high-field regions to be attracting electrons.
It was also shown that particles inside the standing wave exhibit complicated
chaotic oscillations in Refs. Bauer, Mulser, and Steeb, 1995; Lehmann and
Spatschek, 2012; Sheng  _et al._ , 2002. It can be seen in Fig. 11 b) that
only electrons at the position where standing wave was already present (marked
by black circle) were injected to the beam axis. The most of electrons outside
the circle is still on the sides of the beam, repelled by the incident pulse.

Figure 16: (a)Positions of electrons at time instant when laser pulses of
intensity $10^{21}~{}\rm{W/cm^{2}}$ were propagating towards each other.
Electrons are repelled by the ponderomotive force from the region where pulses
are located. (b)Positions of electrons after laser pulses overlapped and
created a standing wave present at $x=[-5,5]~{}\lambda$. Electrons are no
longer repelled from the high-field regions and oscillate inside the standing
wave.

To investigate closer the injection mechanism in standing wave into its high-
field regions, we performed time-averaging of electromagnetic fields over one
laser period shown in Fig. 15 b) and c). The region of non-zero averaged field
can be observed in front of the target. Close to the beam axis, the field
created due to the frequency shift of reflected pulse is present. Overdense
plasma surface is pushed by the ponderomotive pressure which causes the red
shift of reflected wave. This can be seen in the Fourier transform of standing
wave. In Fig. 15a), $\omega_{0}=1$ corresponds to the incident wave frequency
and peak on the left with slightly lower frequency equal to $\omega_{R}=0.9$
is the frequency of reflected wave. This creates $\approx\lambda/2$ wide
regions of non-zero averaged field with the periodic change of sign.

Further from the beam axis ($|y|>2\lambda$) the channel field Khudik  _et al._
(2016); Vranic, Fonseca, and Silva (2018) dominates. It is created by the
charge separation and acts on the repelled electrons in the direction towards
the beam center. The channel field is also present close to the beam axis and
the resulting time-averaged field is a superposition of the channel field and
the field created by the red-shifted reflected pulse.

Figure 17: Example phase space trajectories of electrons in the field of
standing EM wave. Red dots correspond to the initial time of simulation and
blue dotd correspond to the time 15$T$ when pulses started to overlap. (a)-(d)
were obtained for counter-propagating pulses with identical frequencies,
(e)-(f) for the pulses with shifted frequencies.

In order to examine the influence of standing wave on electron dynamics, we
performed the simulation only with test particles interacting with two
counter-propagating laser pulses with intensity $10^{21}~{}\rm{W/cm^{2}}$ and
identical temporal and spatial profile like in the previous simulations. Such
approach allows us to observe particle dynamics without effects of collective
plasma behavior. Test particles were evenly spread across the simulation box
and laser pulses crossed at $x=0\lambda$ at time $t=15\rm{T}$. As can be seen
in Fig. 16 a), while pulses propagate towards each other, ponderomotive force
pushes electrons in front of the pulse and on the sides in agreement with our
PIC simulation of pulse interacting with preplasma. The vertical column of
particles around the blue line represents the region between two counter-
propagating pulses. Electrons in this region were either pushed by the pulses
or they were initially placed there. Those electrons are also the source of
electron population A in section IV.2. Fig. 16 b) shows positions of electrons
after beams crossed. The wavefront of the laser pulse propagating from the
right is located approximately at $x=-5\lambda$, and the wavefront of the
pulse propagating from the left is located approximately at $x=5\lambda$. The
region where particles are present coincides with the area where standing wave
was already created due to the overlap of both pulses. This means that pushing
electrons down the wave packet gradient is not the dominant ponderomotive
force behavior and particles are allowed to oscillate inside the high-
intensity region in the standing wave at such intensities. In later times when
pulses propagate further, the region where particles are not repelled
increases. This clearly demonstrates that the creation of standing wave by the
reflected laser pulse had strong impact on the injection of electrons into the
standing wave in the case of laser pulse with intensity
$10^{21}~{}\rm{W/cm^{2}}$ incident on target with preplasma.

Observation of electron trajectories in phase space shows that standing EM
wave allows electrons to oscillate in the high-field regions instead of being
reflected by the ponderomotive force. Trajectories vary greatly, however
several patterns can be observed, see Fig. 17. Examples a)-d) correspond to
electrons interacting with pulses with same frequency and e)-)f correspond to
pulses with shifted frequencies. In Fig. 17 a), $x-p_{x}$ trajectory
oscillates around the position of $B_{z}$ field antinode and the node of
$E_{y}$ field of standing wave. In $y-p_{y}$ trajectory, oscillation in
positive $p_{y}$ values is visible while electron moves in the direction
towards the beam axis at $y=0$. This is in agreement with trajectory obtained
from PIC simulation in Fig. 13. In Fig. 17 b), it can be seen how electron
skips from one $B_{z}$ antinode to another in $x-p_{x}$ phase space. In
$y-p_{y}$, electron oscillates in negative $p_{y}$ values and moves at first
in direction towards the beam axis and after it continues in oscillation and
heads towards lower pulse intensity regions. In Fig. 17 c), electron at first
oscillates around $B_{z}$ antinode in $x$-direction and afterwards is
accelerated to momenta exceeding values of oscillation in the laser field. The
$p_{y}$ momentum oscillates in positive values and after some time starts to
oscillate in negative values. The moment when oscillation sign changes is
identical to the time when electron starts to be accelerated in $x$-direction.
In Fig. 17 d), the immediate acceleration by the standing wave to high
energies without oscillation around $E_{y}$ node is present.

The case in Fig. 17 e) and f) demonstrates the impact of reflected wave’s
shifted frequency. The frequency of the pulse propagating from the left was
set to $0.9\omega_{0}$, where $\omega_{0}$ is the frequency of the pulse
propagating from the right boundary. The shift of oscillation center in
$x$-direction is the only observed difference compared to the situation when
pulses with identical frequencies couter-propagate.

The discussion above allows us to describe the electron injection in the
following way. At first, electrons are repelled by the propagating laser pulse
far from the beam axis. The channel field pushing electrons towards the beam
center is simultaneously created. However, ponderomotive force overcomes the
effect of channel field and keeps electrons on the sides. After the reflection
of incident laser wave the standing wave is created and electrons are no
longer repelled by the ponderomotive force. Instead, they are oscillating
around the magnetic field antinodes. This allows electrons to be injected by
the channel field closer to the beam axis where the motion is dominated by the
standing wave. There, electrons are either accelerated in $x$-direction or
they oscillate around the $B$ field antinode while they are injected further
in transverse direction as shown in Fig. LABEL:fig:18-0-xpx.

##  VI Conclusion

We have demonstrated, that hot electron acceleration mechanisms strongly
differ depending on the laser intensity and density profile. For the case of
intensity $10^{18}~{}\rm{W/cm^{2}}$ and steep plasma gradient, we showed that
electron is injected into the vacuum by the electrostatic field where it
interacts with the standing wave and starts to be accelerated back into the
plasma at times, when magnetic field changes its polarity twice a laser
period. The increase of intensity to $10^{21}~{}\rm{W/cm^{2}}$ caused the
deformation of overdense plasma boundary, which resulted in chaotic
oscillations of electrons before the acceleration. Without the presence of
preplasma, electrons were accelerated at vacuum-plasma boundary because of
$\vec{j}\times\vec{B}$ heating.

The presence of preplasma caused, that all electrons were accelerated in the
underdense region, which resulted in significantly higher hot electron
temperatures. Significant impact of pulse polarization on electron temperature
was explained by the background field of plasma channel. For intensity
$10^{18}~{}\rm{W/cm^{2}}$, irreversible energy gain was not present until the
moment, when reflected wave reached the position of an electron. Afterwards,
electrons were accelerated towards the overdense part of the target while the
stochastic acceleration along with longitudinal electric field present in
preplasma were dominant in both regimes. When laser pulse with intensity
$10^{21}~{}\rm{W/cm^{2}}$ was incident on the target, the first population of
hot electrons was pre-accelerated by the front of laser pulse. The second
population of hot electrons was initially repelled from the high-field regions
by the ponderomotive force of incident wave. However, when standing wave was
created, they were injected back into the beam axis region along the magnetic
field antinode and accelerated afterwards. This injection is a result of
relativistic electron dynamics in the field of standing electromagnetic wave
with the combination of plasma channel field. It was demonstrated, that longer
preplasma scale length results in higher electron energies.

###### Acknowledgements.

This work was partially supported by CTU project “Research on optical
(nano)structures and laser plasma” No. SGS19/192/OHK4/3T/14. Computational
resources were supplied by the project "e-Infrastruktura CZ" (e-INFRA
LM2018140) provided within the program Projects of Large Research, Development
and Innovations Infrastructures. J. P. also acknowledges the support and
computational resources obtained from European Regional Development Fund -
Project CAAS ("Center of Advanced Applied Sciences") No.
CZ.02.1.01/0.0/0.0/16$\\_$019/0000778. Fruitful discussions with M. Vranic
(IPFN, Instituto Superior Técnico) and D. Maslarova (IPP, The Czech Academy of
Sciences) are gratefully acknowledged.

## Data Availability Statement

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

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