Abstract
1 Introduction
A high-intensity laser pulse interacting with a plasma on the front surface of a solid target will generate a population of hot electrons that propagates into the target. The laser to electron conversion efficiency has been found to scale as a function of intensity[
The dynamics of these hot electrons as they travel through the target and reach the rear surface is of particular interest as they can generate high-energy X-rays that can be used for radiography[
The kinetic behaviour of electrons on the rear surface has been analytically investigated by Grismayer
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Studies have shown that the refluxing electrons can influence the generation of accelerated protons on the rear surface. Several papers have shown through experiment and simulation that the maximum proton energy generated via the TNSA mechanism increases significantly as the target thickness is reduced[
The number of reflux passes of the target that the electrons undergo has been investigated by Quinn
Refluxing electrons have also been shown to have an effect on the X-ray source size. Quinn
Upon reaching the rear surface, if the hot electrons overcome the electrostatic forces at the rear surface of the target, they will escape into free space; these are known as escaping electrons. Link
Here, using the PIC code EPOCH[
2 Simulation methodology
The PIC code EPOCH was used to simulate a laser focusing onto a planar solid target. The spatial size of the simulation box was set to
EPOCH is capable of tracking macro-particles by assigning a unique identifier to each macro-particle, and this has been used previously to identify and analyse the dynamics of hot electrons in laser–solid target simulations[
The simulation was conducted multiple times for different laser intensities, from
Identifying the different electron distributions requires the use of logical checks, such as if the electron has reached the rear of the target with positive momentum. In particular, the position check can be described as a boundary that can be placed to identify the particles that pass it. The unique particle ID of each macro-particle that meets the criteria of passing any of the defined boundaries is recorded. Afterwards, each EPOCH output from each time step is analysed and the particle IDs of those that fulfil desired criteria are stored. Then, their position, momentum and kinetic energy are sorted temporally such that we can track each electron throughout the simulation.
Before exploring the refluxing and escaping electrons using the methods discussed above, we analyse the electric fields generated on the rear surface that affect both populations of electrons.
3 Electric fields
As the laser was incident at
The peak of the electric field occurs at approximately 0.45 ps into the simulations; this is similar for all the simulations. The peak electric field scales with intensity, as shown in Figure
These temporally evolving fields play an important role in the electron dynamics, restricting some of them from escaping and causing many to reflux. We have observed higher laser intensities lead to higher electric fields. This is primarily caused by the greater number of electrons due to both the increased absorption and increased input laser energy. We also expect the temperature of the hot-electron population to increase with incident laser intensity[
4 Refluxing electrons
As stated previously, refluxing electrons are those that reach the rear of target and, due to the electrostatic potential on the rear surface, are reflected into it. To analyse this, a diagnostic boundary is placed within the target. This is positioned
With the particle tracking, we have the ability to analyse each macro-particle. To analyse the cooling of the refluxing electrons further, we can record the energy before and after the electrons interact with the sheath. Figure
To show the temporal evolution of the electron energies, the difference between the initial energy and refluxed energy of each macro-particle is calculated and plotted as a function of when they pass the boundary for the first time; this is shown in Figure
In order for an electron to lose or gain energy, it must do work or work must be done upon it, respectively. In our case, work is done to/by the electric field that the hot electrons establish on the rear surface of the target. Similarly, the electrons gain energy from work being done upon them by the high-intensity laser on the front surface. The electrons that gain or lose energy must experience a net change in potential as they reflux (
The overall energy of the refluxing spectrum is ‘cooler’ than that of the initial spectrum; this is shown in Figure
Using the particle tracking and the output of the electric fields, the field strength, position and energy of each electron can be found. Using this we can show the conditions that allow for the energy loss and energy gain scenarios. Figure
Simulations at three different intensities were conducted with outputs at high temporal resolution. We see similar results to those shown in Figure
The refluxing electrons are those failing to overcome the electric fields that are present on the rear surface of the target and returning into the target. These represent 95% of the initial accelerated hot-electron population, which agrees with the capacitor model developed by Myatt
5 Escaping electrons
Intuitively, the escaping electrons are expected to be those with the highest energy since, as stated before, they need to overcome the electric fields. However, as the earliest electrons reach the rear of the target, the electric field is at a minimum. These electrons will also have low energy in comparison to the average of the entire distribution, as they were accelerated by the rising edge of the pulse. Some of these electrons should still be able to escape, as the electric field is low and the energy threshold for escaping will be lower. This is neatly explained by the capacitor model demonstrated by Link
To analyse the escaping electrons, we use the same diagnostic boundary that is inside the target for the refluxing electrons, and add a diagnostic boundary that is placed outside the target. In this case, the second boundary is placed at
The spectra for the internal and escaping electrons for intensities ranging from
As we record the electrons internally and externally, we can identify the escaping electrons when they are inside the target. The energy that the escaping electrons had when they first passed the internal boundary is shown in Figure
A map of the hot electrons with energies above 1 MeV and outside the target four time steps is shown in Figures
Individually the sheath electrons are moving fast, as they have energies greater than 1 MeV, but the population is moving slowly. By the end of the simulation at 1 ps, the ‘front’ of the plasma electrons has reached
The ballistic electrons, however, travel away from the target at almost the speed of light and have left the simulation by
Figure
As we discussed in Section
As with the refluxing electrons, we also see similar trends with the escaping electrons for the two other additional intensity simulations we have conducted. The threshold energy for all electrons to escape for the
As stated before, the so-called ballistic electrons only make up a very small percentage of the population outside the target. The majority of the electrons that are outside the target at the end of the simulation are part of the plasma expansion, as can be seen in Figure
Experimentally, the measured escaping electrons represent a much higher number; approaching 15% of the entire incident laser energy for certain conditions[
6 Conclusions
We have simulated, using 2D PIC simulations and particle tracking, the dynamics of the hot electrons and the two created populations that form upon reaching the rear surface of a solid target, the refluxing and escaping electrons. The refluxing electrons, those that return into the target after experiencing the fields on the rear surface, have been shown on average to lose energy, resulting in a cooling of the spectra. This is confirmed across three different intensities from
The escaping electron population analysed here is a so-called ‘ballistic’ population of electrons, which leave the target at near the speed of light. These electrons must overcome the electrostatic potential that is set up on the rear and, in doing so, lose a significant percentage of their initial energy. As such, the escaping electrons are shown to be the highest-energy electrons at any given time reaching the rear of the target, as they have the capacity to overcome the field. This was again demonstrated using particle tracking across the initial internal boundary and second external boundary. A time limit is set on the external boundary that allows us to isolate the ‘ballistic’ electrons from the plasma electrons that are shown to propagate away from the target at a much slower speed but with much greater number.
The effects we observe here largely depend on the electric field that is present on the rear surface of the target. To fully understand the escaping electron dynamics at the rear target surface, further studies will be required, examining the effects of the surface morphology and contaminant layer composition, as well as laser parameters such as the pulse duration and focal spot size.
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