Fast collisional electron heating and relaxation in thin foils driven by a circularly polarized ultraintense short-pulse laser

The creation of well-thermalized, hot and dense plasmas is attractive for warm dense matter studies. We investigate collisionally induced energy absorption of an ultraintense and ultrashort laser pulse in a solid copper target using particle-in-cell simulations. We find that, upon irradiation by a $2\times 10^{20}~\text{W}\,\text{cm}^{-2}$ intensity, 60 fs duration, circularly polarized laser pulse, the electrons in the collisional simulation rapidly reach a well-thermalized distribution with ${\sim}3.5~\text{keV}$ temperature, while in the collisionless simulation the absorption is several orders of magnitude weaker. Circular polarization inhibits the generation of suprathermal electrons, while ensuring efficient bulk heating through inverse bremsstrahlung, a mechanism usually overlooked at relativistic laser intensity. An additional simulation, taking account of both collisional and field ionization, yields similar results: the bulk electrons are heated to ${\sim}2.5~\text{keV}$, but with a somewhat lower degree of thermalization than in the pre-set, fixed-ionization case. The collisional absorption mechanism is found to be robust against variations in the laser parameters. At fixed laser pulse energy, increasing the pulse duration rather than the intensity leads to a higher electron temperature.


Introduction
The creation of warm dense matter (WDM) or hot dense matter (HDM) in a laboratory setting is of high interest for a broad field of research disciplines such as laboratory astrophysics (Remington 2005;Bailey et al. 2007;Fujioka et al. 2009), studies of planetary interiors (Ross 1981;Knudson et al. 2008), inertial confinement fusion (Drake 2018;Le Pape et al. 2018), understanding the equations of state under such extreme conditions (Renaudin et al. 2003;Nettelmann et al. 2008) and experimental verification of high energy density (HED) atomic physics models (Hoarty et al. 2013b;Faussurier & Blancard 2019). However, in order to benchmark atomic physics models against spectroscopic data, throughout the process and what influence the ionization history has on heating. While a high Z * is desirable for the collisional heating process, the initially cold target will not be highly ionized in the beginning. Therefore, we have studied both different degrees of fixed ionization as well as the self-consistent ionization process including both field and impact ionization.
We demonstrate that the energy absorption of an intense short laser pulse in a high-Z * solid-density target is mainly due to inverse bremsstrahlung electron heating within the plasma skin layer, and that this scenario holds in a broad range of experimentally relevant parameters. The front-layer electrons are collisionally scattered into the target body where they heat the plasma bulk to keV level temperatures, enough to reach Gbar range pressures, which is well in the regime of HDM. The scattered electrons have sufficiently low energies that they primarily heat the bulk via direct collisional thermalization. Since this mechanism relies on the scattering of the electrons accelerated by the laser field against the heavy ions, it is operative regardless the polarization. Inside the plasma, where the laser field is negligible, collisions cause fast relaxation of the electron distribution to a Maxwellian.

Simulation design
We have performed one-and two-dimensional (1-D and 2-D respectively) particle-incell (PIC) simulations of laser-solid interactions with and without collisions enabled. We have used the Smilei PIC code (Derouillat et al. 2018), which has a relativistic binary collision module (Pérez et al. 2012) based on the collisional algorithm by Nanbu (1997) and Nanbu & Yonemura (1998). In the case of a collisional plasma, we have considered either a fixed degree of ionization or self-consistent modelling of the ionization processthrough field ionization and collisional impact ionization.
We ran 1-D simulations in a box of size 20 µm with a resolution of ∆x = 0.39 nm (51 200 cells). We considered both LP and CP laser pulses with wavelength λ = 800 nm, dimensionless amplitude † a 0 = 10 (intensity I = 1 2 c 0 (m e cωa 0 /e) 2 ≈ 2 × 10 20 W cm −2 , where 0 is the vacuum permittivity, m e the electron mass, e elementary charge and ω the laser frequency) and a Gaussian temporal profile with t FWHM = 60 fs full-width-athalf-maximum (FWHM) duration in the intensity. The plasma is 2.5 µm thick, starting at x = 7.5 µm with a linear density ramp over a distance of 20 nm. The plasma consists of electrons and copper ions at solid density, n Cu,0 = 48.4n c ≈ 8.4 × 10 22 cm −3 , with 400 macro-particles per cell for each species. Here, n c = 0 m e ω 2 /e 2 is the critical density associated with the laser frequency ω. The particles are initialized from Maxwell-Jüttner distributions (in three momentum dimensions) with temperatures T e,0 = 1 eV for the electrons and T i,0 = 0.1 eV for the ions.
In order to assess the influence of the plasma collisionality alone, we have first carried out simulations with fixed ionization degrees Z * = 11, 19, 24 and 27. Then, to ascertain the physical accuracy of these results, we have performed simulations describing both collisional and field ionization. The collisionless skin depth l s = c/[ω(n e /n c ) 1/2 ] is resolved, even for the highest ionization where l 4 A. Sundström, L. Gremillet, E. Siminos and I. Pusztai  t = 500 fs T fit = 3.5 keV T fit = 2.5 keV Figure 1. Electron energy spectra fE at times t = 150 fs (a) and t = 500 fs (b), for LP (dotted lines) and CP (solid lines), with (black lines) and without collisions (thin, green lines); also showing CP with self-consistent field and collisional ionization (blue, solid line). Two Maxwellian-Jüttner distributions are fitted to the bulk of the CP self-consistent and fixed ionization electron spectra in panel b (red dashed and dash-dotted lines respectively).
self-consistently, the ions were initialized with Z * 0 = 5, in accordance with the widely used Thomas-Fermi model (More 1983). Both field-tunnelling and electron-ion impact ionization were enabled. The self-consistent ionization simulation was only performed with CP.
We also performed one collisional 2-D simulation to check that our results are robust to multidimensional effects. This simulation uses the same CP laser and target parameters as our 1-D base case. In order to limit the computational cost at the increased dimensionality, it was performed at a reduced resolution of 640 cells per micron in both directions (∆x = ∆y = 1.56 nm), and a simulation box size of 10 µm longitudinally and 1.6 µm (2λ) transversely. Furthermore, the number of particles per cell per species was reduced to 50. A test of these resolution parameters in one dimension showed excellent agreement in electron kinetic energy spectrum of the main body of the electrons with the corresponding high-resolution, collisional 1-D simulation; however, the lower particle count led to a poorer statistics in the high-energy tail of the electron spectrum. Figure 1 compares the electron energy spectra as obtained at CP and LP (Z * = 27), with or without collisions enabled and at CP with self-consistent ionization. The spectra are recorded at two successive times (t = 150 fs and t = 500 fs) -for reference, the peak laser intensity hits the target at t ≈ 110 fs and the pulse FWHM duration is 60 fs. For both types of polarization, much higher electron energies are achieved when allowing for collisions. The collisionally enhanced absorption results in a bulk electron temperature of T e ≈ 3.5 keV at t = 500 fs for both LP and CP, determined by fitting Maxwell-Jüttner distributions to the bulk spectra (i.e., ignoring the tails). Meanwhile, the collisionless simulations only reach an electron temperature of ∼10−100 eV; these Fast electron heating and thermalization with CP laser 5 electrons are, however, far from being thermalized and only their energetic tails are visible in the figure. The fact that both CP and LP reach very similar bulk electron temperatures when collisions are enabled indicates that the laser absorption mechanism is the same in both cases.

Results and discussion
By comparing the electron spectra with the fitted Maxwellians, we conclude that the electrons have reached a degree of thermalization wherein less than 0.1% of the kinetic energy is in the high-energy tail for CP, and ∼1% for LP. The time frame for this thermalization is consistent with a rough estimate for electron-electron collisional energy transfer rates. At t = 500 fs, the deviation from Maxwellian starts at an electron energy E 50 keV. This is consistent with the ∼300 fs collisional energy relaxation time of a 50 keV electron through a 3.5 keV plasma at 2.3×10 24 cm −3 electron density (Huba 2016).
Note that these results compare LP and CP for the same on-target intensity, while in an experiment, the circularly polarized pulse would in practice be at a somewhat lower intensity due to losses in the conversion from LP to CP (through a quarter-wave plate). Another practical concern may be elliptical polarization, due to imperfections in the CP conversion. While the exact dependence of the fast electron generation on the ellipticity of the polarization is non-trivial, the bulk collisional absorption itself is not expected to be affected by the ellipticity, as demonstrated by the same bulk temperatures reached in the two extreme cases of CP and LP.
The electron temperatures we quote are technically calculated before the electrons have fully thermalized with the ions, which occurs over ∼ps time scales. However, due to the high degree of ionization, the electron-ion equilibrium temperature is T eq ≈ n e /(n e + n i )T e ≈ 0.96T e . Thus, energy transfer from the electrons to the ions is insignificant.
When activating self-consistent (both field and collisional) ionization, the bulk electron temperature is slightly reduced (∼2.5 keV) compared to the fixed-ionization case. The front plasma is rapidly ionized, mostly through field ionization, so that collisional absorption quickly reaches an efficiency similar to that obtained with fixed Z * = 27 (see figure 2 (b) showing that the average ionization Z * 24 at the plasma front already at t = 100 fs). The lower T e is mostly due to the energy spent on ionization -the average ionization energy from Z * = 5 to 27 is 0.9 keV. Moreover, figure 1(b) shows that, for both CP and LP, collisions cause efficient bulk electron thermalization as early as t = 500 fs. High-energy tails are found to emerge above ∼50 keV for the fixed ionization and ∼30 keV for the self-consistent ionization. Note the large range of the logarithmic f E scale, meaning that the tails are three to five orders of magnitude lower than the bulk spectra. The non-thermal tail is heavier in LP than in CP, due to the operative j×B and vacuum heating. Also, the simulation with self-consistent ionization displays a larger tail, compared to the bulk spectrum, than its counterpart with fixed Z * . The larger tail as well as an electron population at ∼3 MeV can be explained by field-ionization events in the charge-separation layer, which is exposed to stronger laser fields. As the target front electrons are being pushed back by the ponderomotive force, the ions remaining in the charge-separation layer experience the less shielded laser field which quickly ionizes them further. Since these newly freed electrons are injected into regions of stronger laser fields, they are energized similarly to vacuum heating in LP, thus resulting in a larger population of non-thermal electrons, which, as in LP, thermalize relatively slowly. Furthermore, the average ionization level is lower inside the target with self-consistent ionization, as seen in figure 2, thus reducing the efficacy of collisional thermalization. Both these effects act to give a larger high-energy tail.
The ionization level profiles (Z * ) of the self-consistent ionization simulation shown in figure 2(a) represent the local ionization levels averaged over all macro-particles in each spatial cell. Comparing the t = 150 fs (dashed-dotted line) and the t = 500 fs (dashed line) average ionization curves, we see that the target front quickly reaches a high ionization degree while the bulk is ionized more gradually. Since there are no strong electric fields inside the plasma, the ionization of the bulk must be driven by impact ionization. Thē Z * curve at t = 500 fs (dashed line) in figure 2(a) displays plateaus atZ * = 27, 19 and to some extent 11. These plateaus result from the large jumps in ionization energies between the successive outermost electron shells (e.g., betweenZ * = 11 and 12, orZ * = 19 and 20; see figure 2a). The ionization energy to reachZ * = 28 is ∼11 keV Figure 2(b) showsZ * at the target front surface (x = 7.5 µm). At time t = 85 fs (solid line), the ionization level has saturated atZ * = 19 due to the jump in ionization energy after Z * = 19. Later, at t = 100 fs (dotted line), the laser field has become strong enough to sustain field ionization beyond Z * = 19, yielding the peak inZ * near x = 7.5 µm. Apart from the laser field, the electrostatic field (E x ) induced by the laser ponderomotive force at the target front causes additional ionization. This results in theZ * peak seen around x = 7.55 µm, which moves into the plasma as the charge-separation layer is pushed forward by the laser ponderomotive force.
The difference between the simulations is made clearer when studying the electron phase spaces shown in figure 3 †. The figure displays time sequences of the collisional distributions with LP in the top row and CP in the second row; the third row shows the self-consistent ionization CP simulation and the bottom row shows the collisionless CP distribution. In the LP simulation, high-energy electron bunches are produced at twice the laser frequency, as seen in the t = 150 fs panel (top row), while CP with fixed ionization (second row) produces a more even distribution of hot electrons since j×B and vacuum heating mechanisms are inhibited. At t = 500 fs, most of the fast electrons have thermalized in the fixed-ionization case, while there remains a significant population of high-energy electrons "swarming" around the back of the target with self-consistent ionization.
With self-consistent ionization, two populations of relatively high-energy electrons are created during the rising phase of the laser pulse. These populations originate from two successive field-ionization phases. The first one occurs early in the interaction, when the ionization of the surface plasma momentarily saturates atZ * = 19. In the t = 150 fs panel of figure 3, this population accounts for the broad momentum distribution in the target bulk, and also for the beam (with momenta p x /(m e c) ∼ −1) being reflected in the vacuum (x > 10 µm) and refluxing into the target. The second phase starts at t 90 fs, when the laser pulse gets intense enough to ionize the surface plasma beyond Z * = 19 (compare the 85 fs and 100 fs curves in figure 2b). This yields fast electrons (visible in the upper right corner of the t = 150 fs phase space) more energetic than those generated earlier, which correspond to the bump around ∼ 3 MeV in the energy spectra of figure 1. A similar field-ionization injection of fast electrons from the surface ions was seen by Kawahito & Kishimoto (2017) in a carbon plasma, although they used LP which also caused bunching of the electrons at twice the laser frequency.
We now turn our attention to the energy density achieved in these scenarios. Since the heating process is fast compared to hydrodynamical time scales, the plasma bulk has not had time to expand, and hence the bulk electrons and ions remain at solid-range density. At the same time, the electrons reach keV temperatures, resulting in high energy densities of the order of ∼10 Gbar = 10 9 J/cm 3 . In figure 4, the electron kinetic energy density P e is displayed throughout the target at times t = 150 fs (top panel) and t = 500 fs (bottom 8 A. Sundström, L. Gremillet, E. Siminos and I. Pusztai   panel). At the earlier time, the kinetic energy density is concentrated to the front of the target, while at the later stage the energy has spread out throughout most of the plasma. The energy density in the fixed-ionization simulations reaches approximately ∼10 Gbar and is mostly homogeneous in the region x = 8−9 µm.
The high-energy electrons created with LP facilitate a better spatial homogenization of the energy density than with CP. Their slow thermalization results in a more spatially homogeneous target heating, since they can recirculate several times through the plasma. In a potential application, one should therefore make a compromise between good thermalization and homogenization. Another parameter that can be used to control homogenization is target thickness; decreasing it helps for a faster homogenization of the plasma heating. However, a thinner target will also explode faster hydrodynamically, which would give a HED application a shorter time frame to operate in.
Meanwhile, the self-consistent simulation gives an exponentially decreasing energy density profile throughout the target at t = 500 fs, indicating that thermalization is taking longer. The lower temperature and electron density reached with self-consistent ionization result in approximately an order of magnitude lower energy density compared to the fixed-ionization results. However, there is still a significant region with P e > 1 Gbar in the self-consistent ionization simulation at t = 500 fs. In this case, the energy density does not homogenize as efficiently, partly due to a decreased ability of the target to thermalize fast electrons (stemming from lowerZ * ), and partly due to the inhomogeneity of the ionization profile which affects the bulk electron density profile.
As a consequence of the strong gradients in P e around the target front side, a shock wave is launched. The shock wave presents itself as a sharp jump in electron pressure, most clearly seen close to x = 8.0 µm in the t = 500 fs panel in figure 4. The details of shock formation are sensitive to the laser and target parameters, and are more clearly seen from the ion phase space, as will be addressed by a paper in preparation (Sundström et al. 2020). However, no ion reflection occurs at the shock front, which means that the shock is hydrodynamic like in its nature.
At the high ionization levels discussed in this paper, such high temperatures and densities may result in significant energy losses due to bremsstrahlung. The total bremsstrahlung emission power density can be estimated as (Huba 2016). By comparing the power density to the thermal energy density ∼ 3 2 n e T e , we arrive at a radiative time scale of the order of several ps for keV range temperatures at a density of n i = 8.4 × 10 22 cm −3 . Hence, the radiative losses from bremsstrahlung will mostly be of concern at time scales longer than those studied in this paper. However, bremsstrahlung losses cannot be completely disregarded in a WDM/HDM experiments, where current spectroscopic temporal resolution is constrained to ∼ps time scales.
Besides bremsstrahlung, line emission from relaxation of excited states may be of concern. However, at electron energies above 10 eV, electron energy losses from excitations become subdominant compared to ionization (Joshipura et al. 2006). Since energy loss from collisional ionization events is accounted for in the self-consistent ionization simulation, the temperature of T e ≈ 2.5 keV is likely not affected much by losses through line emissions.

Illustration of the collisional absorption mechanism
To illustrate the mechanism by which collisions enhance absorption, we have performed a simplified set of simulations. These are designed to generate a quasi-steady state: the laser intensity is constant after a linear ramp-up over 10 laser cycles; the ions are stationary; the plasma is 2.5 µm long and it terminates at a thermal boundary, meaning that particles which exit the boundary are reflected with momenta chosen randomly from a Maxwellian distribution at T e,0 = 10 eV for the electrons -the same at the initial temperatures. The other simulation parameters are: CP at a 0 = 10, Z * = 27 with and without collisions; resolution and other numerical parameters are as stated in § 2. The long ramp-up time has been chosen to reduce electron energization due to the laser amplitude envelope modulation (Siminos et al. 2012). Note that due to the steady state nature of this simplified simulation setting, it is hard to draw any quantitative conclusions that can be transferred to the time-varying situation.
We will now take a look at the interaction between the electrons and the laser electric field. The density of power S exerted on an electron population can be expressed as is the laser electric field -which only lies in the transverse plane - is the projection of electron velocity moment onto the transverse plane.
In a 1-D model, disregarding collisional effects, the transverse canonical momentum P ⊥ = P ⊥ − eA ⊥ is conserved, andP ⊥ = 0. Hence P ⊥ = eA ⊥ , where A ⊥ and P ⊥ are the transverse component of the magnetic vector potential and the electron momentum moment, defined analogously to V ⊥ . In quasi-steady state, A ⊥ is just rotating in the transverse plane, so the electric field is E ⊥ ≡ − ∂A ⊥ /∂t = ωA ⊥ [cos(ωt)ŷ−sin(ωt)ẑ]/ √ 2, where A ⊥ = A ⊥ (x) is the magnitude of the vector potential (necessarily transverse in 1-D). Importantly, the electric field vector is perpendicular to the vector potential and the magnitude of the electric field is E ⊥ = ωA ⊥ . We therefore expect P ⊥ and E ⊥ to be perpendicular and their magnitudes -in normalized units -to be equal, P ⊥ = E ⊥ . Figure 5 shows slices of the collisional (left) and collisionless (right) electron distri- butions in the transverse momentum plane at t = 190 fs, well after quasi-steady state has been reached, and in a thin slice 7.0−7.2 nm (one cell length) behind the immobile ion front edge of the plasma. If we were to evolve this picture in time, we would see the (negative) electric field −E ⊥ rotate clockwise, along the marked-out circular path in figure 5; the mean momentum P ⊥ would follow synchronously in this rotation. The most apparent difference between the collisional and collisionless distributions is the much larger momentum spread of the former, caused by collisional scattering of the electrons. In contrast to the collisionless case, −E ⊥ and P ⊥ are not equal in magnitude nor are they perfectly perpendicular. The missing transverse canonical momentum has been collisionally transferred to the ions, where it disappears from the simulation due to the ions being static. Note that if −E ⊥ and P ⊥ are not perfectly perpendicular in (3.1), then the absorbed power density S is non-vanishing †. We can express (3.1) as where the phase angle between −E ⊥ and V ⊥ is π/2 − α. Figure 6 displays configuration space profiles of E ⊥ , V ⊥ and S -in dimensionless units -as well as sin(α); the curves are produced from a time average over 21 time frames spanning 20 fs. In the collisionless case, we have P ⊥ (x) = E ⊥ (x) and the phase shift angle α 0 throughout the first 25 nm ≈ 8l s . Due to a finite spread in the electron transverse velocities, there will be a continuous exchange of electrons in the longitudinal direction not accounted for in the fluid description above, which induces a small deviation from α = 0 and hence S = 0. However, S changes sign at x ≈ 15 nm, beyond which the absorbed power is negative. In the collisional case, P ⊥ (x) is consistently smaller than E ⊥ (x). Furthermore, the phase shift sin(α) is much larger, which is reflected in the about two orders of magnitude larger absorbed power S than in the collisionless case.
A final note on the collisional case in figure 6 (left) is the numerical artefact that causes both E ⊥ (x) and P ⊥ (x) to level off near x = 20 nm. As the Monte Carlo collisional algorithm used in Smilei (Pérez et al. 2012) only conserves momentum statistically, a P ⊥ noise floor is generated which drives noise in E ⊥ , i.e., the base level in figure 6 (left). This effect could be alleviated by increasing the number of macro-particles. However, the absolute majority of the collisionally induced laser-energy absorption occurs in the region x = 5−15 nm and is therefore not significantly affected by the collisional noise floor.

Parameter scans
We have also performed parameter scans in order to investigate the dependencies of the collisional heating mechanism. One such scan has been in ionization, with either fixed ionization (Z * = 11, 19, 24 and 27) or self-consistent field and impact ionization. We have also conducted scans in laser intensity with a 0 ranging from 1 to 14, and pulse durations from t FWHM = 15 fs to 400 fs. The remaining parameters are as in § 2.
Let us first consider the absorbed energy from the laser. Figure 7(a) shows the kinetic energy gain by the electrons and ions (∆U ) following the laser irradiation, for a scan in a 0 (colour coded) and a scan in pulse duration (shape coded). The value displayed on the horizontal axis is the laser pulse energy J = It FWHM [π/ log(4)] 1/2 ∝ a 0 2 t FWHM , where I is the laser intensity.
In the case of a constant pulse duration, t FWHM = 60 fs (triangles), the trend scales like a power law with ∆U ∝ J 0.74 (dashed line) or ∆U ∝ a 0 1.48 , since J ∝ a 0 2 . In other words, the absorption efficiency scales as ∆U/J ∝ J −0.26 ∝ a 0 −0.52 . The a 0 = 1 point seems to deviate from the above scaling, suggesting that it is mainly valid at relativistic intensities. Thus, a 0 = 1 was excluded from the fit. This scaling is similar to the I −1/4 scaling of the normal skin effect, as described by Rozmus & Tikhonchuk (1990). Comparing the collisional mean free path, λ mfp ∼ 20 nm, to the skin depth, l s ≈ 6 nm †, it is not completely clear that the normal skin effect (λ mfp l s ) can be ruled out. However, our simulation results do not support some other scaling laws derived by Rozmus & Tikhonchuk (1990) for normal skin effect, possibly due to the nonrelativistic and simplified nature (constant intensity and semi-infinite plasma) of their analytic treatment.
The other scaling trend displayed in figure 7(a) is at constant a 0 = 10 (magenta). Here, the power law fit (dotted line) gives ∆U ∝ J 1.13 ∝ t FWHM 1.13 . In this case the absorption efficiency still has a weak positive scaling of ∆U/J ∝ t FWHM 0.13 . From the pulse duration scaling, we note that the 200 fs and 400 fs pulses begin to fall off below the scaling followed by the other data points, and they were thus also excluded from the fit. The decreasing trend with t FWHM in the long-pulse limit may be a consequence of the recirculation of hot electrons, which could lower the effective plasma collisionality in the irradiated region.
Combining the two above scalings yields the approximate scaling ∆Ũ = 0.23 mJ/µm 2 × a 0 1.48 t FWHM 100 fs in the (a 0 , t FWHM ) plane. Figure 7(b) shows this scaling to agree well with the full set of data, including data points where both a 0 and t FWHM are varied. The observed scaling does break down at the low-a 0 or long-duration limits. However, the range of validity stretches over two orders of magnitude in pulse energy and a similar range in absorbed energy, and the laser parameters captured by this power law are experimentally feasible and relevant to isochoric heating experiments. Collisional effects in general decrease at higher particle energies and the absorption happens through collisional scattering of the laser-driven electrons in the skin layer. Therefore, if the laser field (a 0 ) is increased, so that the electrons in the skin layer reach higher energies, then the efficiency of the collisional absorption should decrease. We also report on a scan in (fixed) ionization degree Z * . Although this parameter cannot be controlled independently in experiments, this scan aims to provide insight into the target collisionality, which scales as (Z * ) 2 -ignoring other effects, e.g. individual particle energy. However, by varying Z * while keeping the ion density n Cu,0 fixed, we inevitably also change the electron density n e,0 = Z * n Cu,0 , which may introduce other density-related effects. Nevertheless, the electron density stays highly overcritical -the lowest electron density in this scan is n e,0 = 532.4n c for Z * = 11. Figure 8  Due to the accompanying changes in electron density, the average absorbed energy per electron ∆E, also shown in figure 8 (black triangles, right axis), happens to decrease by approximately 15 % from Z * = 11 to 27. The decrease in ∆E with Z * may seem surprising † The skin depth adjusted for collisions has been inferred from figure 6 (left). if the dominant heating mechanism is collisional. However, this might be due to other density effects, such as the increased skin depth at lower electron density which allows a deeper laser penetration and thus a stronger laser-to-electron coupling efficiency. When we examine the electron energy spectra (not shown) in this scan, the Maxwellian-fitted bulk electron temperatures are all T e = 3.5±0.1 keV.
Besides just the pure amount of energy absorbed from the laser, we are also interested in how well thermalized the plasma is. As a measure of that, figure 9 shows the fraction of electron kinetic energy in the non-thermal electrons. This is calculated by the fraction of the energy in the high-energy tail to the total electron kinetic energy, is a Maxwell-Jüttner distribution fitted to the bulk of the electron energy spectrum f E (E) (as shown in figure 1) and T e is the temperature inferred from the fit. Figure 9 shows scans in fixed ionization degree Z * (a), laser amplitude a 0 (b) and pulse duration t FWHM (c). The displayed values are taken 200 fs after the end of the laser pulse. Due to a varying heat transport speed, the fraction of non-thermal electron energy is only taken in the region in which the bulk electron temperature is no longer increasing. This should still give a representative estimate of the non-thermal fraction, since the fast electrons have already recirculated by the chosen time, see the t = 250 fs panels of figure 3. The precise values in figure 9 are sensitive to the choice of time and region to include, thus these results are only qualitative. Nevertheless, the general trends shown here are still representative of the observed situation -importantly, the relation between LP and CP is robust. Figure 9(a) shows that there is a trend toward lower non-thermal fraction at higher ionization levels, which is consistent with the faster thermalization expected at high Z * . This trend also suggests that the higher absorbed energy per electron at lower Z * (figure 8) is linked to a relative increase in the non-thermal population.
Regarding the scans in the laser amplitude and duration in figure 9(b) and 9(c), respectively, no clear trend appears to emerge among the CP laser pulses (marked by ×). Then there are the self-consistent ionization (star) and LP (downward triangle) simulations: both have approximately one order of magnitude higher fraction of energy in non-thermal electrons than the equivalent (fixed-ionization, CP) counterpart. The higher fraction of non-thermal energy with LP stems from the j×B and vacuum heating mechanism. The higher non-thermal energy fraction with a self-consistent ionization process is discussed in conjunction with its phase-space distribution in figure 3.
Even a very small fraction of non-thermals may affect the interpretation of X-ray diagnostics (Rosmej 1997;Chen et al. 2009;Renner & Rosmej 2019), meaning that LP can be more intrusive than CP in WDM/HDM studies. We have also conducted simulations with a larger pre-expanded plasma (exponentially decaying density profile with a scale length of 80 nm). While not presented here, those simulations show that LP can result in up to ∼10% of the electron kinetic energy in non-thermal electrons, which would of course be even more intrusive and significantly affect the X-ray diagnostics. With CP, the pre-plasma weakens the energy absorption by about a factor of two, but the fraction of energy in fast electrons stays 1%.

Two-dimensional simulation results
Up to this point, all the results presented have been produced in 1-D simulations. However, to investigate the applicability of these results in higher dimensions, where transverse plasma modulations can arise at the target boundary and affect the bulk heating and hot-electron generation (Kemp & Divol 2016), we have performed a 2-D simulation at our baseline laser parameters (see § 2 for details). A density map of the electron density near the illuminated target surface is shown in figure 10(a). The density is shown at t = 250 fs. There is a transverse band of higher density at x ≈ 2.72 µm that represents a shock front propagating into the plasma. Notably, this shock front remains straight, with no evidence of substantial density modulations.
To more clearly demonstrate the absence of transverse instability effects, the relative transverse variation of the electron density n/ n y − 1 is displayed in figure 10(b), where · y denotes a transverse average over the full width of the simulation box in y. Any transverse density structure should therefore be clearly visible in this representation. The substantial deviations from the average density observed in the low-density (n e 0.2n 0 ) region correspond to statistical noise due to a low number of computational particles in said region. Importantly, the deviations seen in figure 10(b) have no structure to them, and the same applies for the shock, suggesting that transverse effects are inoperative in the present highly collisional case (at least within the simulated time window). Lastly, to confirm that the collisional heating behaves similarly in the 2-D and 1-D simulations, figure 10(c) shows the electron kinetic energy spectra of the corresponding 1-D and 2-D simulations at t = 250 fs. † The spectra of the 2-D and 1-D simulations are essentially the same. Although not shown here, the two spectra agree similarly well also † Due to the 2-D simulation having a smaller longitudinal box size, and thus the target front being located at x = 2.5 µm instead of x = 7.5 µm, the times of comparison for the 1-D simulation at earlier times. Since the following thermalization process is almost entirely collisional, and thus independent of dimensionality, it is safe to conclude that the energy absorption is not affected by going from one to two dimensions, under the interaction conditions considered.

Conclusions
We have performed collisional and collisionless 1-D and 2-D PIC simulations and shown that a collisional, inverse bremsstrahlung, absorption can be used for strong plasma heating in a solid-density, high-Z * material, such as copper, with ultrahigh intensity, short-pulse lasers. Using CP, the electron population quickly thermalizes to well-formed Maxwellian distributions suitable for experimental verification of HED physics models. The collisional simulations show that the target electrons are quickly heated to T e ∼ 3.5 keV bulk temperature on a ∼300 fs time scale. The target energy density reaches ∼10 Gbar, which is within the realm of ultrahigh energy density. The use of CP provides faster collisional thermalization of the electron population compared to LP, something which is valuable for experimental tests of HED atomic physics models. A test of the collisional absorption using a 2-D simulation, demonstrates the transferability of the 1-D results to higher dimensions. In contrast to previous work conducted with lower-Z * targets (Kemp & Divol 2016), the high collisionality is not favourable for driving transverse plasma modulations, resulting in the same absorption levels in two dimensions as in one dimension, at least for CP. are shifted by 15 fs later compared to the 2-D simulation, due to increased travel time for the laser pulse. The simulation time at which the 1-D spectrum is plotted is thus t = 265 fs.
We have carried out scans over laser parameters and ionization. The scans over laser settings show that the mechanism is robust to changes in the laser, over two orders of magnitude in laser energy, with lower intensity and longer pulses at the same laser pulse energy yielding better energy absorption. Also, the higher collisionality incurred from a higher ionization level improves energy absorption and electron thermalization. A more realistic simulation run with self-consistent ionization, including both impact and field ionization, reached T e ∼ 2.5 keV, confirming that collisional heating is still operational in a self-consistently ionized plasma, although its thermalization is then less complete than at fixed ionization, due to high-energy electrons generated through ionization events in strong-field regions.