{'fraction_non_alphanumeric': 0.021421532924896104, 'fraction_numerical': 0.009168416091855534, 'mean_word_length': 4.682083739045765, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The role of standing wave in the generation of hot electrons by femtosecond laser beams incident on dense ionized target

We demonstrate the differences in hot electron absorption mechanisms dominant in the interaction of femtosecond laser pulse with intensities 1e18 W/cm2 and 1e21 W/cm2 on fully ionized target with steep density profile and preplasma with moderate scale length (3 microns). 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 1e18 W/cm2. In the case with intensity 1e21 W/cm2, 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.

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 3λ (x = 0, 3λ ) 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 = −6λ , 6λ 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 26,27 . 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 17.5T and finished at time 40T , where T is laser period.
A. Intensity 10 18 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  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 32T . 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 π 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 π/2. The phase shift between E y and B z , which is also π/2 (see Fig. 3), causes already mentioned π 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.
All hot electrons were accelerated in the region between the first node of the magnetic field at distance ≈ 1/4λ in front of the target and EPB, see The acceleration twice a period typical for j × 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 field E x pulling electrons out into the vacuum. In semi-analytical model 12 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 29 ). 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 > −0.05λ at 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. B. Intensity 10 21 W/cm 2 When laser pulse with intensity 10 21 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 where Ξ = 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 ∆ 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 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 ≈ 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 ≈ 22 for a given laser intensity. However, this threshold is essentially increased to a th ≈ 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 32 . 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 × exp(x/L) for x ≤ 0 and scale length L = 3µm was assumed in front of the overdense ionised foil located at 0 < x < 5 µm , 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 20T and stopped the interaction at time 42.5T .
A. Intensity 10 18 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.
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 To quantify the effect of mechanisms, the contribution of electric field components was ex- where γ 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 max x < 2.5m e c. This is in agreement with acceleration process described in Fig. 8a), where the combination electric and magnetic Lorentz force con-tributes 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 exponent 33   the rear side. Distances between the bunches are identical as in the case of j × 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. 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.
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 This was observed for both laser intensities, see Table I.
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 34 , 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 max x < 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 = 2λ (1.6µm) and longer preplasma scale length was set to L s = 6.25λ (5.0µm). The trend was obvious -longer preplasma resulted in higher electron energies. Maximum p x for shortest preplasma was ≈ 150m e c, 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
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 35 , 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. [36][37][38]. It can be seen in Fig. 11  repelled by the incident pulse.
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), ω 0 = 1 corresponds to the incident wave frequency and peak on the left with slightly lower frequency equal to ω R = 0.9 is the frequency of reflected wave. This creates ≈ λ /2 wide regions of non-zero averaged field with the periodic change of sign.
Further from the beam axis (|y| > 2λ ) the channel field 39,40 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.
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 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 = 0λ at time t = 15T. As can be seen in Fig. 16  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 0.9ω 0 , where ω 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. ??.

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 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 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 j × 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 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 W/cm 2