Review and meta-analysis of electron temperatures from high-intensity laser–solid interactions Open Access

The interaction between a high-intensity laser (>10¹⁸ W/cm²) and a solid target has been the subject of many studies. Many of the proposed applications of high-intensity lasers, such as multiple modes of radiography from secondary sources (proton,¹ neutron,^2,3 x-rays,^4–6 and fast-ignition schemes^7–9) depend on the understanding, and sequentially the optimization, of physical processes that govern the coupling and acceleration of the front surface electrons to mega-electron volt energies. Ponderomotive acceleration,¹⁰ direct laser acceleration,^11–14 resonance absorption,¹⁵ and stochastic heating/acceleration^16–19 are a few but not all of the mechanisms that can occur simultaneously during the same short-pulse (typically 0.1–10 ps). The previously listed mechanisms will depend differently on the target and laser parameters such as the laser intensity, polarization, and incident angle. As such, the acceleration of the hot-electrons is complex and difficult to decipher. However, the resulting electron energy spectrum can often be simply described using a Boltzmann distribution with a signature electron temperature, as widely demonstrated in many studies. This convenient description allows for a simpler representation of the electron energy distribution that can be utilized by analytical expressions and scaling of secondary sources such as target normal sheath acceleration,^20,21 the generation of positrons,²² or the x-ray dose generated by Bremsstrahlung.²³ 

The field of high-intensity laser–plasma interactions is growing rapidly as high-intensity lasers are simultaneously becoming more powerful²⁴ and more-readily available. The rapid pace of technological advancement and developments in fundamental laser technology has led to a democratization of the field, with high-intensity lasers at university scale now capable of outperforming what was previously attainable only on state-of-the-art national scale systems. In particular, the higher repetition rate of newer laser systems has created a fundamental change in how we think about and perform experimental measurements; experimental data collection that used to take weeks can now be gathered in minutes.²⁵ The increasing volume of data on top of existing decades of research makes a literature study of the electron temperature long overdue.

During the last 2–3 decades, a vast number of experimental and computational studies have been conducted to better understand the electron acceleration mechanisms and the resulting electron temperature from a variety of laser and target conditions. Many attempts at “scaling laws” that relate the laser intensity and wavelength to the temperature of the electron have been derived either been derived analytically,^10,26,27 empirically fit to experimental data,²⁸ developed from parameter scans within particle-in-cell simulations,^19,29,30 or some combination of the above. All these models show a relationship between the intensity and wavelength of the laser and the hot-electron temperature. However, there are often many other laser and target parameters, such as the pre-plasma on the front surface or the pulse duration of the laser, that have been shown on multiple occasions to also exhibit strong influences on the resulting electron distribution. Models attempting to incorporate more parameters have been recently developed.^19,30 However, they have not been compared against the vast amount of data available within the field, nor do they consider these data when formulating their models.

This paper presents a literature study and a meta-analysis of existing scaling laws in regard to experimental and simulation results of the hot-electron temperature produced in laser–solid interactions. Our goal is to present the most comprehensive literature study on the subject, collating data from more than 80 independent studies. We have attempted to gather all the parameters we deemed relevant to the electron temperature from each study where available. We also include some unpublished data from experimental and simulation studies to extend the dataset, details of which are provided in the  Appendix and supplementary material. The data gathered are split into experimental measurements in Sec. III A and simulation studies in Sec. III B. The experimental section includes both direct measurements of the electrons by electron spectrometers and inferred electron temperature by x-ray measurements.

Using this large amount of gathered data, we have developed two comprehensive scaling laws. We have first taken a heuristic approach based on our physical understanding of the acceleration mechanisms. We have also employed a Bayesian inference approach previously used to predict maximum ion energies from high-intensity laser experiments,³¹ neutron yields in inertial confinement fusion experiments,³² and optimization of 2D simulations for proton acceleration.³³ We compare the performance of the predictive models developed in this paper and predictions from existing scaling laws to the gathered data. In addition, and most importantly, the data gathered here are to be released as a dataset [Hot-Electron Temperature Data Repository (HET-DR)], initially upon request but eventually open to everyone.

Models that describe or predict electron temperature from the initial conditions are often referred to as “scaling laws.” As mentioned earlier, these are especially useful when it concerns simplifying the electron distribution function for other analytical models. Here, we will briefly discuss some of the existing scaling laws.

The intensity and wavelength of the laser plays a key role in the acceleration of the electrons. The motion of an electron in an oscillating electric and magnetic field (the laser) is determined by the Lorentz force [ $F = q ( E + v × B )$]. The magnetic field component becomes relevant as the velocity of the electron becomes relativistic. This can neatly be described using the normalized vector potential, $a 0 = e E 0 / m e c ω L ≈ 0.85 I 18 λ μ 2$⁠, where I₁₈ is the laser intensity in units of 10¹⁸ W/cm² and $λ μ$ is the laser wavelength in units of micrometers. If $a 0 > 1$⁠, then the electron motion is considered relativistic. In the presence of a non-homogeneous field, the electron can experience the ponderomotive force. The maximum energy that an electron can gain from the ponderomotive force can be expressed as $E P = m e c 2 a 0 2 / 2$⁠. As the laser intensity or laser wavelength increases, so does the energy the electron can gain from the laser.

The acceleration of so-called superponderomotive electrons, electrons whose energy far exceeds the ponderomotive limit, via direct laser acceleration (DLA). This was first described by Gahn et al.¹¹ and are discussed in detail by Robinson et al.¹² and Arefiev et al.¹³ Briefly, the energy of the electron oscillates as it goes in and out of phase with the electric field of the laser. To gain more energy, the electron needs to stay in phase with the electric field for longer or somehow reduce the dephasing time. This can be done via an external field, such as a longitudinal field caused by a plasma channel, or the injection of the electron with sufficient velocity into the laser. As discussed by Arefiev et al.,¹³ the limit of the energy gained by the electrons is determined by the superluminal phase velocity within the plasma. Ultimately, many of the electron temperatures found within this paper that exceed the ponderomotive scaling are likely due to a combination of the superponderomotive mechanisms discussed in other studies.

Additional fits and scalings exist within other papers, some of which we will refer to for data comparison in Sec. IV A.

We have gathered measurements, experimental and simulated, of the electron temperature as well as the laser and target parameters that are quoted in many publications. The parameters we have concentrated on gathering are the laser wavelength, intensity, pulse duration, focal spot size, and energy, along with the pre-plasma scale length. We have also tried to gather information about the target material, thickness, and the incidence angle of the laser. We will briefly describe the method of gathering the data.

The first important consideration for gathering a reliable dataset is the accuracy of the electron temperature itself. The measurement of electron temperature in experiments often employs a permanent magnetic field, where the deflection is inversely proportional to the energy of the electrons. This measurement is performed external to the target, which is surrounded by a potential well that prevents the majority of electrons from escaping. This field grows in time and thus affects the electron distribution in non-trivial ways.^78–80 Other methods include using the x-ray emission from the solid target to infer the electron distribution. While this method will highlight the electrons inside the target, the measurement and interpretation/deconvolution, of the x-ray distribution is difficult to perform and is the topic of ongoing research. In simulations, it is typical that the entire electron energy-angular distribution is measured/recorded from inside the target. There exists no convention as to whether the electrons are measured in a time-integrated manner through a region of interest or via a snapshot of the electrons at a particular time.

As previously mentioned, the electron energy spectrum can often be described as a particle distribution characterized by temperature. This is commonly expressed by an exponential distribution in the simple form $d N / d E = A exp ( − E / T e )$⁠, where A is the amplitude of the distribution and T_e is the electron temperature. While this is not the only way of fitting a temperature to the electron distribution, some use the Maxwell–Boltzmann or Maxwell–Juttner energy distributions, an exponential distribution tends to match the Maxwell–Boltzmann well at high energy and is the most common description found in the literature and will, therefore, be used in this paper.

The temperature T_e is often found directly in the publications but must sometimes be extracted from graphs. Moreover, there are many examples of multiple electron temperatures present in a single interaction, especially in simulation studies. It is often difficult to decipher which temperature is being measured or quoted, particularly when two electron temperatures may be present. As stated by Gibbon,⁸¹ the observer may be left with making “a choice of which [temperature] to select as the more physically relevant and/or interesting.” As such, we do not attempt to investigate if there is the possibility of multiple temperatures or, in the case of multiple temperatures, which is being quoted.

Another case, highlighted by Peebles et al. in multiple experimental examples,^50,82,83 is the case when the measured electron distribution does not appear Boltzmann-like, and instead the electron spectrum is characterized using the “half-max integrated energy.” Another method presented by Sherlock⁸⁴ is to represent the electron energy distribution instead of the number distribution. This often leads to a well-defined peak that can be easily characterized. Both methods could be applied to all electron spectra to correct for spectra that do not appear Boltzmann-like. However, neither method has been universally adopted and currently the exponential distribution description is widely used.

Publications often contain fewer than six of the main laser and target parameters that are listed at the beginning of this section. This occurs mainly for experimental data, as simulated work almost always contains a detailed description of the initial laser and target conditions. In some instances, we can infer experimental laser parameters from parameter scans/scalings performed within the studies. For example, Dover et al.³⁹ performed an energy scan to change the intensity and quoted the focal spot, pulse duration and maximum laser energy. It is trivial to infer an approximate energy from such a scan with the information provided. Where applicable, we have applied such methods to attempt to fill out the datasets.

However, one of the important parameters, the plasma scale length, is rarely measured during experiments. The plasma scale length is a description of the plasma density gradient on the surface of the target prior to the main laser pulse. Its growth is dependent on the power/intensity of the laser pedestal, including pre-pulses and the rising edge. The scale length can also evolve during long-duration (>1 ps) laser interactions, as shown in recent work by Kojima et al.⁸⁵ A measurement of the pre-plasma can be conducted using an optical probe; however, this is a difficult measurement to make or incorporate on an experiment. More-novel techniques have been explored, such as measuring the deflection of the reflected laser light to infer the plasma scale length, as demonstrated by Singh et al.⁵⁷ Other than measuring the scale length experimentally, hydrodynamic simulations can be conducted using knowledge of the laser pedestal to generate a density profile. Finally, the scale length can also be calculated analytically, as demonstrated by Simpson et al.⁵⁶ For this case, it is made much easier if much of the scale length can be attributed to a significant pre-pulse.

While the laser energy, pulse duration, and focal spot are commonly quoted in each study, it is often the case that the latter two are difficult to measure on-shot. This leads to uncertainty in the final, on-shot intensity. While there are facilities that reconstruct the vacuum on-shot focal spots,^86,87 this is only an estimation with some uncertainty. Also, the focal spot is often quoted using several different metrics. By far the most common is the FWHM, but many other metrics exist, including 1/e, 1/e² or a given percentage of energy enclosed within a certain radius exist. We have recorded which metrics are used where described. Furthermore, the intensity is often quoted as the peak intensity. The spatial profile of the focal spot will naturally lead to a spatially varying intensity profile, which will in turn lead to many electron temperatures if the scaling laws are to be believed. This particular effect is considered by Chen and Mackinnon⁶⁵ when deconvolving x-ray and electron distributions. Deciphering the actual intensity and focal spot is difficult for a single study, let alone in the host of papers studied here. As such, we take the intensity quoted in each paper at face value to simplify the problem.

We next present the electron data gathered from experimental studies, followed by the data from the simulation literature study.

The experimentally gathered electron temperature data are shown as a function of the intensity times the wavelength squared, referred to henceforth squared (⁠ $I λ 2$⁠) in Fig. 1. Due to the number of data points, we have gathered, and we have decided to omit the error bars on each measurement from the figure. The temperatures gathered via x-ray techniques are highlighted by a red outline; all other data are gathered using electron spectrometers. We have gathered data from 57 sources, 20 laser systems, including unpublished work from recent OmegaEP and Texas Petawatt experiments (parameters and configurations are discussed in the supplementary material), for a total of 475 measurements. We also plot the ponderomotive¹⁰ and Pukhov²⁹ scalings. The data collected have ranges of electron temperatures between 10 keV and 30 MeV, intensities between 10¹⁷ and 10²² W/cm², pulse durations from 30fs to 38ps, laser energies from 2mJ to 2.4 kJ, and focal spot sizes from a few micrometers to 200 μm. First, however, of the 475 measurements, only 116 have all the parameters that we specified earlier in this section. This is mostly due to the infrequent documentation of the scale length of the plasma. We will briefly discuss some of the studies and trends within the experimental datasets.

As shown in Fig. 1, there is a proportionality between the temperature measured and the incident intensity of the laser. The electron temperature is measured over several orders magnitude by Dover et al.³⁹ (⁠ $10 19 − 22$ W/cm²), Chen et al.⁴¹ (⁠ $10 17 − 21$ W/cm²), and Tanimoto et al.⁵⁹ (⁠ $10 18 − 21$ W/cm²). The results by Chen et al. and Tanimoto et al. are both obtained at two different laser facilities with different parameters. However, the scaling of temperature with intensity holds across the facilities. Chen et al. studies different target materials (Al, Cu, and Ag) and noticed little difference between target material. Tanimoto et al. studied different incident target angles (0°, 26°, and 40°), which again had little effect on the scaling with intensity. The results presented by Dover et al. show that the electron temperature falls more rapidly as a function of the laser energy than it does as a function of focal spot size. This is explained by considering the effect of the electron oscillations in the laser field and the size of the focal spot. At the highest intensities, the oscillation amplitude can be larger than the focal spot radius, which restricts the maximum attainable electron energy and therefore reduces the overall electron temperature. By incorporating this effect into the ponderomotive scaling, Dover et al. are better able to match their experimental results.

While the intensity obviously plays a major—if not the most significant—role in the resulting electron temperature, many of the other parameters can also have a strong influence. Referring to Fig. 1, data taken at ∼10¹⁹ W/cm² ranges from 500 keV to 20 MeV. These measurements were made at several different laser facilities where the laser and plasma conditions varies. Furthermore, there are several measurements made at sub-to-moderately relativistic laser intensities (∼1 × 10¹⁸ W/cm²) that yield relativistic electron temperatures. To try to better represent the additional parameters, Fig. 1 is re-plotted in Fig. 2 with the color of the data symbols representing the pulse duration, laser energy, scale length and the focal spot size in (a)–(d), respectively. Interestingly, we observe the trend previously stated that many of the data points with longer pulse duration, higher energy, larger focal spot and, where measured, longer scale length tend to be clustered toward the upper left of the plot. These experiments yield much higher electron temperatures than most of the plotted models suggest.

Some studies investigate the effects of pre-formed plasma on the resulting accelerated electrons. Culfa et al.^36,88 created a scale length (2–11 μm) on the front surface of the target using a separate, much lower intensity (∼10¹² W/cm²), longer-pulse laser (5 ns) with a 17° angle of incidence to the target normal. A 2ω probe beam, perpendicular to the surface of the target, measured the extent of the plasma prior to the interaction. The high-intensity (∼10²⁰ W/cm²), 1 ps pulse arrived after the plasma formed and at an angle of 40° to the target normal. They demonstrated an increase in the observed temperature of the electrons with longer scale lengths, from ∼14 to ∼20 MeV for scale lengths of ∼1 and ∼7 μm, respectively. However, at scale lengths larger than 7 μm, the temperature falls whilst the integrated number of electrons rises. Simulations demonstrate hosing instabilities causing the laser to deflect within the plasma, reducing the electron temperature. Courtois et al.⁴ performed a similar study, but concentrating on the x-ray dose. However, they observed that the x-ray dose and temperature continued to increase as a function of the scale length of the plasma. Geometric effects, such as a cone, can confine the growth of the plasma laterally, creating a much longer scale length. Nakamura et al.⁴⁹ and Rusby et al.⁵⁴ demonstrated the relationship between long plasmas within cones and higher-than-typical electron temperatures.

While not directly related to the pre-plasma formed on the front of the solid target, experimental studies have investigated the viability of foam targets whose densities are close to the critical density. A density scan performed by Willingale et al.⁸⁹ maximized the average measured electron temperature at around a foam density of 2n_c. Compared to a flat/solid density target, the average electron temperature increases by a factor of two for optimum-density foam (4–8 MeV). However, it should be noted that the measured temperature from the foams targets varied by as much as a factor of 7, leading to temperatures as high as 21 MeV for the optimum foams. This is explained as the accelerated electrons from these foams appear to be highly directional with unstable pointing and may therefore miss the electron spectrometer opening aperture. More recently, studies by Rosmej et al.⁵² have seen even further electron acceleration enhancements by foams that are pre-ionized from heater beams. The foams used were 300–400 μm long and have an initial density of ∼0.64n_c. The temperature measured from these foams are as high as 13 MeV.

The pulse duration has been shown to play a key role in the electron temperature in several studies. Simpson et al.⁵⁶ were able to maintain a large and consistent electron temperature (>6 MeV) for a laser with pulse duration from 0.9 ps (∼10¹⁹ W/cm²) to 20.9 ps (∼9 × 10¹⁷ W/cm²). However, this is likely due to a combination of the longer pulse duration and the long scale length that was present, which was estimated to be greater than 100 μm. Peebles et al.⁵⁰ measured higher energy temperatures from longer pulses with the “intrinsic” pre-plasma on Omega-EP. Experimental work by Yogo et al. at the LFEX laser⁶³ showed that using a train of pulses of the same intensity can increase the hot-electron temperature and lead to higher proton energies and conversion efficiencies. A similar experiment was repeated at LFEX by Kojima et al.,⁸⁵ who compared a single laser pulse (1.2 ps) with a higher-intensity single pulse (1.2 ps) and a train of four pulses (4 ps total), the latter two having four times the energy of the former. The electron temperature was similar for the single high-intensity pulse and train of four pulses, and each was higher than for the single lower-intensity pulse. Simulations by Kojima et al.⁸⁵ suggest that this is due to the growth of pre-plasma and magnetic fields during the duration of the longer laser pulse. This mechanism, where the electrons are injected into the laser via the magnetic fields, is thoroughly described by Krygier et al.⁹⁰ 

A combination of longer laser pulse durations (>1 ps), larger focal spots, and higher laser energies results in much higher electron temperatures than predicted by the ponderomotive scaling at a given intensity. The previously discussed results at LFEX,^63,85 as well as results published by Williams et al.,^18,61 Mariscal et al.,⁴⁶ and Kerr et al.⁶ from NIF-ARC and Raffestin et al.⁵¹ from PETAL, all demonstrate higher-than-usual electron temperatures at a modest laser intensity. These can mostly be explained by a combination of longer pulse durations enhancing the superponderomotive electron acceleration, and larger focal spots leading to forms of stochastic acceleration. Each paper is accompanied by PIC simulations, which are briefly discussed later in Sec. III B.

Most of the data we have described to this point come from low-repetition rate laser systems. Previously unpublished data are presented here, which was taken by Mariscal and Swanson at the Colorado State University Laser using a high-repetition rate laser system and a high-repetition rate magnetic electron spectrometer. The data represent ∼100 shots (see  Appendix for details). The data are plotted at ∼2 × 10²¹ W/cm² and show a large spread in the measured electron temperature. Each of these points represent a single shot measurement at high-repetition rate. Other high-repetition experiments average the data over multiple shots. Mordovanakis et al.⁴⁷ performed an experiment using a Ti:sapphire 0.5-kHz laser system. By measuring electrons in the specular direction and averaging over 5000 laser shots, they can measure an electron distribution and fit it to a function of intensity [ $T e ∼ ( I λ 2 ) 0.65 ± 0.05$]. Another high-repetition rate experiment was conducted by Zulick et al.⁷⁷ using a 0.5-kHz system. One of few x-ray results gathered, the study used a unique method for determining the electron temperature. The laser system was a comparatively low-energy laser (2–10 mJ), with a short F-number parabola (f/1.2) and a 30 fs pulse that reached relativistic intensities (2.5–12 × 10¹⁸ W/cm²). To measure the x-ray emission, a high-purity germanium detector is employed, capable of performing measurements of a single photon to high precision. They state that between 15% and 25% of shots produce data, and it takes between 90 000 and 120 000 shots to reconstruct an x-ray spectrum. Using Monte Carlo simulations, they relate the bremsstrahlung distribution to the electron distribution $( T b = 1.34 × T e 0.92 )$ and find that the electron temperature is between the Ponderomotive scaling and the scaling measured by Mordovanakis et al.⁴⁷ 

The majority of the data discussed and shown has been gathered using electron spectrometers. However, there are some problems using this technique that are difficult to mitigate. Electrons that make it to the spectrometer must undergo a deceleration upon leaving the solid target due to the temporally and spatially evolving electrostatic fields.^78–80 The highest-energy electrons have sufficient energy to traverse the electrostatic potential and be measured by a spectrometer, but these measurements can neglect the contributions of the lower-energy electrons.

While Tanimoto et al.⁵⁹ observed little difference between the incident angle of the laser and target, work by Gray et al.⁹¹ showed that interaction at an oblique incident angle can significantly alter the observed electron energy and emission angle. Simulation work by Perez et al.⁹² was used to verify experimental data measured by Chen and Mackinnon⁶⁵ showing that irradiating a target with a preformed plasma at an angle can cause the magnetic fields of the reflected laser to bend the electrons trajectories away from the laser axis. This effect is also observed in simulations by Peebles,⁸³ who also highlights the fact that this effect is unlikely to occur when shooting at normal incidence. These are just some examples demonstrating the difficulty of relating measurements from different experiments performed on different laser systems, at different observation angles.

While we have discussed some of the difficulties in measuring the electron spectra experimentally, PIC simulations can offer an alternative method to study experimental observations with well-defined input parameters. Most importantly, PIC simulations can measure the energies of the electrons prior to reaching the rear surface of the target. In Sec. III B, we present literature review of PIC simulations and the observed electron temperatures.

Computing power has rapidly progressed over the past few decades, as has the accessibility of PIC simulation software, which is an incredibly valuable tool in the understanding of laser-plasma interactions. We have gathered simulations from numerous sources and have conducted some additional simulations ourselves. While many stand-alone simulations studies exist, many are conducted and published alongside experimental data to help understand the measurements. As such, some of the publications and results from the experimental section have partnering results in this section. Most of the papers list the core parameters that we looked for in the experimental data. However, there are additional simulation setup parameters that are vitally important and are often not listed such as the cell size/resolution of the simulation, particles per cell (ions or electrons), the maximum or minimum densities, or the initial temperature of the ions and electrons. We have, again, tried to gather this additional information.

There are many different options for PIC codes that currently exist, and it is important that given similar inputs that they yield similar outputs. Smith et al.⁹⁸ conducted a code comparison between three PIC codes: EPOCH,¹⁰¹ LSP,^102,103 and WarpX.¹⁰⁴ Conducting each simulation with the same input parameters, they observe strong agreement between all three codes, including the electron temperature/spectra. Although this study does not include all the PIC codes that are discussed in the following paragraphs, this finding reassures us that the choice of PIC codes should not heavily influence the results.

The gathered simulated electron temperatures as a function of the laser intensity are plotted in Fig. 3. We show 26 datasets obtained from 11 different PIC codes, representing almost 8000 individual points of data. We again see that within a given study, for example, Xu et al.¹⁰⁰ Miller et al.³⁰ or Pukhov et al.,²⁹ the electron temperature increases with the laser intensity. However, much like the experimental data, the electron temperature can also be strongly influenced by other parameters.

As stated earlier, the study by Pukhov et al.²⁹ highlights the relationship between the intensity and the scaling length. At short scale lengths, the simulated electron temperature conforms to the ponderomotive scaling, but at longer scale lengths the electron temperature is much higher. The effect of the scale length on the temperature is also investigated by Culfa et al.⁸⁸ to explain their experimental results. As mentioned earlier, Culfa et al. experimentally observed the electron temperature increase up to a scale length of 7 μm and then the temperature begins to decrease. This trend is only reproduced with 2D PIC simulations, since 1D simulations do not capture the hosing instability or the effects of the laser refracting before the critical surface. The off-normal angle of incidence likely exacerbates the effect of the reflection in this case.

Many simulation studies have shown the importance of the pulse duration on the electron temperature. Peebles et al.⁸² showed that changes in the scale length (from 5 to 10 μm) had little effect on the temperature of the electrons when the pulse duration was relatively short (∼150 fs). However, when the simulations were repeated using a longer pulse (1 ps), the longer scale length resulted in a hot-electron temperature increase in more than a factor of two. Kemp and Wilks,¹⁴ using 1D simulations with intensities between 10¹⁷ and 5 × 10¹⁸ W/cm², showed that both longer plasma lengths and longer pulse durations can lead to hotter electron temperatures. However, this study used a uniform plasma density that was 1% of the critical density, which is atypical experimentally. However, a simple density profile can allow for isolation of the physical mechanism involved in the acceleration process and the parameters scan within the study (long-pulses and plasmas lengths) are rare amongst the gathered datasets. Studies with lower densities, which are not included here, tend to investigate wake-field, self-modulated, DLA, or a combination of these acceleration mechanisms (we refer to Ref. 105 and references therein). Sorokovikova et al.⁹⁹ demonstrated the generation of a similar uniform plasma density from an initial scale length density profile due to the ponderomotive pressure of the incoming laser. The density plateau or shelf has a potential that traps the plasma electrons within. The electrons, if injected into the laser field with the optimum conditions, will accelerate to superponderomotive energies. The length/extent of the plasma plateau, for a constant intensity of 10²⁰ W/cm² and initial scale length of 10 μm, was shown to be proportional to the laser pulse duration (1–10 ps) and is demonstrated in both 1D and 2D. A ponderomotive-like temperature of 2.7 MeV was observed, but the tail of the electron distribution (T_e = 11–35 MeV) scaled proportionally to the pulse duration (1–10 ps). However, at the shortest pulse duration of 1 ps, the tail of distribution is superponderomotive, which is likely due to the relatively long initial scale length.

Simulations by Shen et al.¹⁹ use a homogeneous underdense plasma to represent experiments using foam targets.⁵² 3D simulations are designed such that the intensity of the laser is many orders of magnitude higher than the self-focusing limit, resulting in breakup into many small filaments. Some electrons can traverse several of the filaments that form in the plasma, gaining energy stochastically as they do. Since the foam targets are several hundreds of micrometers thick, the electrons can again be accelerated to superponderomotive energies.

Due to the development of multi-kilojoule, longer-pulse (>1 ps) laser systems such as LFEX, PETAL and NIF-ARC, recent simulations studies attempt to capture the unique physics observed on experimental campaigns-namely, electron temperatures that far exceed the ponderomotive or Pukhov scaling. For example, simulations presented by Raffestin et al.⁵¹ and Williams et al.⁶¹ show filaments that form in the underdense plasma in front of the target. Unlike similar studies, the large focal spot at these facilities causes many filaments of a large lateral extent such that electrons expelled from one filament can be accelerated in another (similar to the findings of Shen et al.). Higher electron temperatures are also shown by Williams et al.¹⁸ when simulating a focusing cone. Additional acceleration occurs when the cone wall is present to reflect the laser light, causing an interference pattern within the plasma that can increase the likelihood that electrons dephase.⁹⁷ Iwata et al.⁹⁵ demonstrated a clear link between the pulse duration and the maximum observed electron temperature within the simulation. Both a single Gaussian pulse with a FWHM of 1.5 ps and another pulse with the same rising and falling profile plus an additional 1.5-ps flat-top yielded maximum temperatures of 0.46 and 1.43 MeV, respectively. This is similar to the corresponding experimental measurement by Yogo et al.⁶³ However, one of the reasons Iwata et al. observed higher-temperature electrons is due to recirculating electrons within the thin (5 μm) target; similar to the effects observed by Gray et al.⁹³ using a larger focal spot. Kojima et al.⁸⁵ simulated a similar laser configuration as Iwata et al. but with a much thicker target. Experimentally, Kojima et al. used a 1-mm gold cube, of which they simulated the first 25 μm and allow relativistic electrons (>511 keV) to escape the simulation box. Despite this, electrons were still accelerated to much higher energies and temperatures when the pulse duration was longer. They argue that the significant magnetic fields can trigger injection of the electrons, akin to the process outlined by Krygier et al.,⁹⁰ yielding high electron energies.

The largest dataset given to us was that supplied by Djordjevic et al.¹⁰⁶ Due to the vast dataset, we have chosen to plot the data with transparency in Figs. 3–5, and as smaller points in Fig. 6. This dataset of 1D and 2D simulations was created to train machine learning models and perform transfer learning in order to reproduce physical results, such as ion energies and electron spectra, at much lower computations costs. With such a large dataset, many different inputs and outputs, can be related to one another. Some of the parameter space investigated in this study differs in some significant ways to other papers discussed in this literature review. As such, we have only selected some of the data using the following criteria. We have set a lower density threshold of 10 n_c, as some of the study investigated low-density targets with emphasis on ion acceleration. We have also sub-selected data with scale lengths greater than 0.8 μm (equal to the laser wavelength). Many of the simulations are conducted with scale lengths on the order of nanometers, but this often causes the target to behave like a plasma-mirror with high reflectivity and therefore low absorption and acceleration. To reduce the ambiguity of this swath of parameter space, we have also sub-selected data with scale lengths greater than 0.8 μm (equal to the laser wavelength) to account for this. The majority of the data lays well within the bounds of other data. However, there is a selection of data, particularly those with either longer scale lengths or pulse durations, that appear to the upper left of the simulation distribution, much higher than the rest of the data.

Despite the number of simulation datasets gathered here, there are some absences and limitations. There are very few simulations that are conducted in 3D. It is a known effect that increased intensity from relativistic-self-focusing and filamentation will be underestimated in 2D compared to 3D. Also, the magnetic field dynamics are not captured fully in 2D. Simulations performed in 1D omit even more physical processes.¹⁰⁷ The changes that these geometrical effects may have on the electron temperature would require further study.

We have re-plotted the simulation data in Fig. 4 with the laser/target parameters colored. We have not included the energy plot, as most studies do not quote an input laser energy. Similar to the experimental data in Fig. 2, the data with longer pulse durations and longer scale lengths achieve the highest electron temperatures at a given laser intensity.

The intensity of the incident laser clearly has a strong influence on the hot-electron temperature. However, as shown in the literature study, the intensity alone cannot explain the differences observed in the experimental and simulation data at the same intensity. Many studies show both the pulse duration and scale length of the plasma also influence the hot-electron temperature. There are two existing scaling laws that consider the previously mentioned parameters: the scaling presented by Shen et al.,¹⁹ shown in Eq. (4) and the scaling presented by Miller et al.³⁰ shown in Eq. (5). However, these two models are formulated using datasets of limited size. With the current dataset we have gathered here, we can develop a model that encompasses a much larger dataset while considering the intensity, wavelength, pulse duration, and scale length.

We wish to determine the temperature dependence on both the scale length and pulse duration, and we begin by considering them multiplied together to simplify the model. To remove the influence of the laser intensity, we can divide the temperature by one of the many existing scaling laws that takes intensity into account. We choose to divide the observed temperature data by the predicted ponderomotive temperature [Eq. (1)] to obtain the ponderomotive multiplier, P_m, is defined as $P m = T e / T pond$⁠, where T_e is the measured temperature.

There are some physical limitations to the fitted model in Eq. (6). The model suggests that the electron temperature increases with the scale length without limit. However, when the scale length becomes very long, such as observed by Culfa et al.,³⁶ hosing instabilities can severely inhibit electron acceleration. Similarly, the model suggests that increasing the pulse duration will continually increase the electron temperature which is shown by Kemp and Wilks, is shown to be false. Regardless of these shortcomings, the scaling fits well with the data shown in Fig. 5.

While this yields a simple scaling, it would be more insightful if we could consider each parameter separately. We consider the methodology used by Takagi et al.,³¹ who used Bayesian inference to consider the multiple-parameter problem of ion acceleration. We similarly apply the Bayesian inference using PyMC library. As the hot-electron temperature is a key parameter in ion acceleration, Takagi et al. were also able to apply the Bayesian inference to the hot-electron temperature. While they only consider the pulse duration and the intensity in their model they demonstrate that the hot-electron temperature is most-strongly dependent on the intensity, to a power of 0.41. This is close to the square-root proportionality that many of the scaling laws suggest.

In Sec. IV, we developed two new scaling laws, Eqs. (6) and (7). We now seek to compare the scaling laws from the literature and the ones we have formulated here. The ponderomotive [Eq. (1)] and Pukhov scalings [Eq. (3)] are chosen as well-established scalings, and newer models developed by Shen et al.¹⁹ [Eq. (4)] and Miller et al.³⁰ [Eq. (5)]. The predicted electron temperature from each model is plotted as function of the observed temperature for the simulated (blue) and experimental (yellow) data in Fig. 6. The red line represents where the observed and predicted temperatures are equal; above (below) the line the model is over- (under-) predicting the temperature.

There are many way to represent the data and quantify the quality of the models. We choose as metrics the mean (⁠ $⟨ r ⟩ = ∑ r i / N$⁠) and standard deviation (⁠ $σ r = ∑ r i 2 / N$⁠) of the residuals for each scaling. In computing these metrics, we work with the $log 10$ of the data, which aids in reducing the weight of the higher-energy data. The mean determines if the model is generally over- or under-predicting on average. The standard deviation of the residuals serves as an overall goodness-of-fit metric that quantifies the spread of the error in the model about zero. A perfect model would replicate the observed data exactly, yielding zero for both metrics. A good model, therefore, should have low values for both numbers simultaneously. To visually represent the qualities of each model, we have plotted histograms of the experimental and simulation residuals in Fig. 7. The experimental data for the ponderomotive and Pukhov scalings in Figs. 7(b) and 7(d) are plotted in two histograms, which represent data with the scale length given (darker) and all points in the dataset (lighter). Both distributions in each cases have spread; therefore, it is fair to include all the data when plotting the scatter in Fig. 6 and calculating the mean and standard deviation.

The ponderomotive and Pukhov scalings generally under- and over-predict the electron temperature, respectively. However, this is to be expected since the ponderomotive model is best-suited to shorter scale lengths and the Pukhov model is fit to long-scale length interactions. Each of these two models has a relatively large spread in the residual for both the simulation and experimental data. The average standard deviation for each model between simulation and experiment is 0.47 for the ponderomotive scaling and 0.70 for the Pukhov scaling.

The four newer models all perform reasonably well for both datasets. Note that because these models require knowledge of the pulse duration and scale length, the experimental dataset against which these models are compared is significantly reduced. However, incorporating the dependence on pulse duration and scale length results in much better agreement with the data in the field. There are simulation data where the models in Figs. 6(d)–6(f) are under predicting the electron temperature where the majority come from the larger dataset. These data points all appear to have a short scale length (∼1 μm) and a long pulse duration (>1 ps). These simulations may exhibit similar behavior as those presented by Kojima et al.⁸⁵ who observed the scale length evolve during the pulse. Using a shorter initial scale length than the later longer scale length in these models would under predict the electron temperature, which is consistent with what is seen.

We list the four models in order from the lowest to highest average standard deviation for simulation and experiment: Bayesian scaling (0.23), simulation empirical scaling (0.24), Miller scaling (0.26), and Shen scaling (0.28). It remains unclear whether the simulation or experimental data are more important in determining the accuracy of the models for reasons explained previously. While experiments better indicate reality, the measurement uncertainty is greater both in parameter estimation and data collection.

The hot-electron temperature is crucial in understanding and optimizing many potential applications of laser–solid and laser–plasma interactions. We hope that the large data collection and analysis performed in this literature review will provide insight and be of aid to the field. As mentioned, this collected data are to be released as a dataset (HET-DR). The hundreds of experimentally measured and simulated electron temperatures shown in Figs. 1 and 3 show that the intensity of the laser alone is unable to explain the large variation in electron temperature. However, by investigating some of these parameters, which is enabled by the quantity of the data gathered, we can observe trends in the data. We have included recent models in our comparison, such as the scaling developed by Shen et al.¹⁹ and Miller et al.,³⁰ whose models encompass more laser parameters. Additionally, we have developed new models using the data gathered in this study. Each model that incorporates an additional dependence on the scale length and pulse duration performs better than previous models that rely only on the intensity of the laser. The scatter that remains between these models and the data, experimental and simulations, is to be expected as there are many acceleration mechanisms (e.g., ponderomotive force acceleration, betatron resonance acceleration, stochastic acceleration, wakefield/electrostatic acceleration, resonance absorption) that can influence the final results and parameters we have not investigated such as target angle or measurement angle.

There are also outstanding issues that are not investigated in this paper. For example, the focal spot of the laser can affect the electron temperature, as shown in the studies by Dover et al.³⁹ and Gray et al.⁹³ The scale length can develop substantially during long-duration interactions, which was highlighted by Kojima et al.⁸⁵ In this regime, the pulse duration and the scale length are intrinsically linked. The approach by Miller et al. was to measure the scale length at the peak intensity during the simulations. However, this method cannot be applied to experimental measurements.

Another complication is that multiple temperature distributions may develop from multiple acceleration mechanisms that can occur simultaneously in a single interaction. As stated, when multiple temperatures may be present we have not tried to discern the particular temperature that each study has observed. However, an important step would be to understand the energy partition between each distribution when multiple temperatures are present. This understanding could aid in deducing the importance of various acceleration mechanisms in the production of secondary sources.

There are many avenues for further research regarding this literature study. There is likely more data that is currently available from the high-intensity laser community that is not included here. Adding those data points to this dataset could enhance the models developed in this paper. As a result, the authors have decided to release the datasets presented here to allow for further refinement of the models. Finally, with the advent of higher-repetition-rate high-intensity laser system, larger experimental datasets are expected within the foreseeable future. However, as this paper makes clear, it is vital that as many experimental parameters are measured as possible (e.g., plasma scale length, focal spot size, laser energy, etc.). This is particularly helpful when comparing data across different experiments or to simulations.

See the supplementary material for the data gathered for this work. The data is in the form of an excel file and contains the metrics we have found in each publication. We have also provided links to the publications. The data is as accurate as possible, often extracted from graphs, and in few occurrences, calculated based on the other metrics provided.

I would like to acknowledge the fellow authors of the paper who have helped with gathering and interpretation of the datasets. I would like to thank Herbie Smith, Ghassan Zeraouli, and Paul Campbell for assisting greatly with gathering the Texas Petawatt electron data (experimental procedure briefly described in the  Appendix). I would also extend an acknowledgment to Marco Garten and Axel Huebl who upon hearing of this endeavor, decided to help gather additional data for the dataset. Mariscal and Swanson acknowledge that the experimental campaign described below was granted through the LaserNetUS proposal K146 Mariscal CSU-ALEPH 2022 & DOE Office of Science, Fusion Energy Sciences under Contract No. DE- SC0019076: the LaserNetUS initiative at CSU Advanced Beam Laboratory Experiments, LDRD 20-ERD-048.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and funded by the LLNL LDRD program under tracking code 19-SI-002 and the Office of Fusion Energy Sciences under Award No. DE-SC0021057. IM release No. LLNL-JRNL-858859.

The authors have no conflicts to disclose.

D. R. Rusby: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). A. J. Kemp: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Methodology (equal); Writing – review & editing (equal). S. C. Wilks: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Methodology (equal); Writing – review & editing (equal). K. G. Miller: Conceptualization (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). M. Sherlock: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Methodology (equal); Writing – review & editing (equal). H. Chen: Conceptualization (supporting); Data curation (supporting); Investigation (supporting). R. A. Simpson: Formal analysis (equal); Methodology (equal). D. A. Mariscal: Data curation (equal); Investigation (equal). K. Swanson: Data curation (equal); Investigation (equal). B. Z. Djordjevic: Data curation (equal); Formal analysis (equal); Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). A. Link: Investigation (supporting); Methodology (supporting). G. J. Williams: Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). A. J. MacKinnon: Funding acquisition (equal); Project administration (equal).

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

Some of the experimental and simulation data presented here are from unpublished datasets. We provide brief details of each in this  Appendix to provide context.

The work by Mariscal and Swanson took place at the Colorado State Laser ALEPH. Using an F/2 parabola, the spot radius was approximately 2.5 μm. The laser energy delivered to target was 8.2 ± 2.3 J in approximately 45 fs. This produced an on-target intensity of 1.87 ± 0.5 × 10²¹ W/cm². The laser is incident at 22.5° onto an array of 9 μm Al targets that were aligned prior to each shot to ensure best focus. The electrons are recorded on a newly developed high-repetition-rate magnetic electron spectrometer.¹⁰⁸ Measurements on the Omega-EP laser were conducted by the authors of this paper. Shots were performed at Omega-EP over several shot days using several different wire configurations as well as flat targets. The laser conditions were varied, but the general setup is as follows. We have conducted several shots at 1, 4, and 10 ps where the laser energy at each pulse duration was 300, 300, and 900 J, respectively. The wavelength of the Omega-EP laser is 1.053 μm. An F/3 off-axis parabola was used; however the focal spot had 80% of the energy enclosed in a radius of 15.7 ± 1.7 μm. The targets were a 1-mm thick, 1-mm diameter gold puck. To characterize the escaping electrons and positrons, we utilized the electron, positron, proton spectrometer (EPPS), as described by Chen et al.,¹⁰⁹ which was positioned on the laser axis.

The measurements labeled Texas Petawatt (2021) were also conducted by the authors using the Texas Petawatt laser system at the University of Texas in Austin. Using the F/3 parabola, the laser delivers 125 J in 145 fs in a focal radius of 4.5 μm. The laser wavelength is 1.054 μm. A 10-μm copper target was irradiated at 10°. Pulse duration and focal spot scans were performed to vary the laser intensity. The measurements were also performed using the EPPS, which was placed ∼8° away from the laser axis.

Simulations were conducted for this review. We conducted 21 simulations in 2D using EPOCH. We varied the scale length (3, 5, and 8 μm) and the intensity (⁠ $10 18 − 20$ W/cm²) while keeping the pulse duration at 300 fs FWHM. We used a 10 μm FWHM focal spot for all simulations. These simulations were performed with a 50 × 50 μm grid with 1000 cells in each direction with a resolution of 50 nm. The target (Copper) started at x = 0 and extended past the end of the simulation box. The scale length in front of the target was described by a double exponent, where the long scale length (which was varied) was used from near-vacuum to the critical density, and a scale length of 0.5 μm is used from 1 to 50 times the critical density. Both electrons and copper ions used 20 particles per cell. The elections had an initial temperature of 50 eV. The electron spectrum is recorded within the 15 μm of target before the electrons leave the simulation box. The extracted temperatures are shown in Fig. 3 as red diamonds. We observe from the simulations that the scale length has an impact on the resulting temperature, which compounds with the intensity. For example, simulations conducted at 1 × 10¹⁹ W/cm² have electron temperatures of 1.75, 2, and 2.5 MeV for 3, 5, and 8 μm scale lengths, respectively. These data fit well when plotted against the fitted model shown in Fig. 5.