Hot-electron generation in high-intensity laser–matter experiments with copper targets Open Access

The characterization of hot-electron production in the interaction of high-intensity laser radiation with an underdense plasma is of paramount importance for inertial confinement fusion (ICF), particularly in the shock ignition scenario, which employs a laser pulse with an intensity of about 10¹⁶ W/cm² to drive a strong converging shock wave in a spherical shell target.^1–3 This intensity is about two orders of magnitude above the thresholds of two parametric instabilities^4,5—stimulated Raman scattering (SRS) and two-plasmon decay (TPD)—considered as major processes producing electrons with energies exceeding 50 keV harmful for ignition of fusion reactions.^6,7

The excitation of parametric instabilities in a high-temperature plasma is characterized by the interaction parameter $Iλ02$⁠, the product of the laser intensity I and the square of the laser wavelength λ₀, which is proportional to the energy of electron oscillation in the laser field. Values of the interaction parameter of the order of 10¹⁶ (W/cm²) · μm² are difficult to reach on high-energy laser facilities operating in the UV region (e.g., 351 nm). Systematic investigation of parametric instabilities at these facilities is also limited by access time. However, such interaction regimes can be studied at medium-scale installations operating in the IR domain, because of the longer wavelength, tighter beam focusing, and easier access.

A series of experiments conducted at the PALS laser facility⁸ at laser intensities (0.3–1.2) × 10¹⁶ W/cm² was dedicated to studies of hot-electron generation under ICF-relevant conditions.^9–12 The results of these experiments are discussed in Ref. 12. The authors used composite targets made of different materials, ranging from plastic to nickel, and with a thin copper layer at the rear side, to characterize hot-electron generation using a series of complementary diagnostics. The results can be summarized as follows:

• The hot-electron energy distribution can be approximated by a Maxwellian function with temperature 25–40 keV. These electrons carry 0.6%–2% of laser energy.

• The hot-electron production is accompanied by stimulated Raman backscattering with reflectivity ranging from 0.1% to 0.8%.

• The temporal profiles of Raman backscattering and hot electrons overlap, suggesting SRS to be a principal source of hot electrons.

• The hot-electron number and energy depend weakly on the target material, even though, according to hydrodynamic simulations, the temperature of the plasma corona in the experiments increases from 3.3 keV for a plastic target to 5.4 keV for a nickel target.

• The temporal profile of hot-electron generation measured with the Kα diagnostic shows that in plastic targets, the electrons are produced about 100 ps before the laser pulse maximum, and the duration of hot-electron emission is shorter than the laser pulse.

• Numerical simulations performed with a particle-in-cell (PIC) code for the experimental interaction conditions^9,13 confirm SRS as a source of hot electrons, but predict the number of hot electrons to be larger than that measured by a factor of two to three. This difference is attributed to the simulations being performed in two spatial dimensions and to electron collisions being ignored.

The time dependence of hot-electron generation in laser–plasma interaction experiments has been studied to a much less extent. It is often implicitly assumed that the hot-electron flux follows the temporal profile of the laser pulse. But this is not always the case. Experiments with low-Z spherical targets^14,15 showed a delay of hot-electron generation by more than 100 ps with respect to the rising part of laser pulse, but no hot-electron signal was observed after the end of the laser pulse. By contrast, a significant time delay of more than 100 ps after the end of the laser pulse was suggested in the laser interaction with thin copper targets.¹⁶ However, no direct measurement of the time dependence of the Kα emission was presented, and the time delay was deduced from the spatial extension of the x-ray spectrum. More recent experiments with layered low-Z targets¹¹ showed an increased delay and duration of the hot-electron generation with increasing ablator atomic number, but in all cases it was shorter than the laser pulse duration.

The temporal evolution of the hot-electron signal was not measured in composite targets with medium- or higher-Z ablators, such as nickel, titanium, and partly aluminum, because the Cu Kα emission induced by hot electrons was too weak or obscured by bremsstrahlung emission from the hot plasma corona in the same spectral window.¹¹ This issue, however, depends on the experimental conditions. The data presented in this paper enrich those in Ref. 11, and provide new and important information about the dynamics of hot electrons in high-Z plasmas.

In this work, we present a detailed study of hot-electron generation and transport in the interaction of a laser pulse at a wavelength of 1.3 μm with thick and thin copper targets by using Kα time-resolved imaging. A significant difference is observed in the hot-electron emission between thick and thin targets. While the hot-electron flux in thick targets follows the laser temporal evolution, the hot-electron emission is significantly delayed in thin targets. That observation provides direct confirmation of the experimental results reported in Ref. 16. However, the interpretation is revised based on the current state of the art of hot-electron transport modeling and extended experimental results. In contrast to the suggestion in Ref. 16, the delay in Kα emission in the case of a 1 μm thin target is explained by the radiation preheat and an increase in hot-electron lifetime due to their recirculation in a hot expanding plasma. This finding can be important for interpreting hot-electron diagnostics in experiments relevant to ICF conditions.

The remainder of this paper is structured as follows. The experimental setup and diagnostics are presented in Sec. II. Section III presents the experimental results concerning the spatial distribution and temporal evolution of hot electrons using Kα emission. Results of numerical simulations with hydrodynamic and kinetic simulations concerning the generation and transport of hot electrons are presented in Sec. IV. This is followed in Sec. V by a theoretical analysis of hot-electron transport in a copper plasma, defining the conditions where the electron recirculation occurs. Section VI summarizes our findings and conclusions.

A recently published survey of experimental methods used to characterize hot electrons accompanying laser–target interaction¹¹ pointed out the importance of the complex evaluation of data obtained by diverse diagnostics. The performance, reliability, and limits of individual diagnostics should be validated using single-valued, well-defined experimental configurations. Here, we concentrate on space–time-resolved investigation of hot-electron generation in laser-irradiated massive and thin-foil Cu planar targets via imaging of hot-electron-induced Cu Kα emission and its comparison with theory and numerical simulations.

The experiment was performed using the PALS iodine laser system at its fundamental frequency (1315 nm, 0.3 ns, 600 J).⁸ A schematic of the experimental geometry is presented in Fig. 1. The laser beam was smoothed by a random phase plate and focused using an f/2 lens at normal-to-target incidence into a Gaussian focal spot with an FWHM diameter of about 100 μm. The accuracy of the target surface positioning vs the optimum focal plane was ±20 μm. The design of the imaging system was based on a well-known coincidence between the wavelength of the Cu Kα₁ emission and the 2d interplanar spacing (1.5414 Å) of the spherical quartz (422) crystal, here with a bending radius of 500 mm and a circular entrance aperture of 24 mm. The geometry of the imager was strictly determined from a combination of the paraxial mirror equation with the Bragg condition¹⁷ complemented by a fixed distance c = 927 mm between the target and the detector, which was a Hamamatsu x-ray high-dynamic-range streak camera (XSC)²⁰ positioned at one of the flanges outside the PALS interaction chamber. By numerically solving a set of these limiting factors, we determined the target-to-crystal distance a = 313 mm and crystal-to-detector distance b = 1239 mm, which correspond to a system magnification M = b/a = 3.96. The hot-electron-induced Cu Kα emission was observed at an angle of 57° to the target surface. The radiation was diffracted from the crystal and sent onto the XSC entrance slit with dimensions of 18 mm × 80 μm, providing a spatial resolution in the horizontal (meridional) direction and a temporal resolution in the vertical (sagittal) direction, respectively. The finite vertical size of the slit resulted in a temporal smearing of the measured signal by its integration over the 33 ps-wide time window. Taking into account a typical FWHM duration of the measured hot-electron generation at the level of 200 ps or more, the signal broadening introduced by this smearing was less than 3%, i.e., negligible in most cases. The central cross-section of the two-dimensional (2D) magnified image corresponding approximately to 15% of its FWHM spatial profile was aligned on the XSC slit with a precision of about 14 arcsec, which corresponds to a positioning error of 21 μm when recalculated to the target plane. The crystal was protected by a 13 μm-thick Kapton foil, and the radiation diffracted from the crystal was further filtered by a 10 μm-thick Al foil, suppressing the radiative background. Direct irradiation of the XSC slit from the target was prevented by the Pb beam block.

The XSC was triggered using a signal derived after the second amplifier of the laser chain (see Fig. 1). To eliminate experimental jitter, the absolute temporal calibration of the hot-electron-induced signal was provided by a fiducial split-off from the main laser beam after the fourth amplifier. This signal was frequency tripled and brought via an optical fiber and a delay line directly onto the XSC slit. The temporal correlation between the recorded fiducial and the hot-electron-induced Cu Kα signals was calibrated, taking into account the difference in the optical paths from target to slit c and from target to crystal to slit a + b and a delay introduced by frequency tripling the fiducial and a partial reshaping (narrowing and 100 ps shift of its maximum, as determined from the response of a fast diode inserted into the vacuum chamber) of the main laser beam in the final amplifier. The observed variable spacing between the fiducial and the time- and space-resolved image of the hot-electron action was then used to determine the temporal correlation between the laser maximum and the time-resolved signal of the hot-electron generation. The uncertainty of fitting the fiducial and Cu Kα profiles in a single XSC record was less than ±10 ps. A considerably larger error of the measured timing between the laser and Cu Kα maxima of ±43 ps was introduced by the fiducial jitter. An extended description of the imaging system, its calibration, and the reconstruction of the measured data will be published elsewhere. Here, we note that a scatter of data taken in ten laser shots with similar interaction conditions provided a standard deviation of ±45 ps of the measured delay between the laser and Cu Kα maxima.

The spatial resolution of the recorded hot-electron signal given by the XSC pixel size was 3.92 μm/px (as recalculated to the target plane), which matches the 4.1 μm FWHM broadening of the point spread function determined using a ray tracing procedure.²¹ The vertical (time-resolving) scale of the image was calibrated with respect to a variable sweep speed of the XSC (with an average value of 5.35 ps per detector pixel). The temporal smearing of the recorded signal due to the finite vertical width of the XSC slit was ∼33 ps. Raw images taken on laser-irradiated solid and thin-foil Cu targets are depicted in Fig. 2. Part of the XSC camera output showing the mutual positions of the fiducial and the Cu Kα time- and space-resolved signal produced by hot electrons is presented in Fig. 2(a). Magnified central parts of the XSC image demonstrating the spatial evolution of the laser-induced hot-electron interaction with the solid target and the thin-foil target are shown in Figs. 2(b) and 2(c), respectively. A detailed analysis and theoretical interpretation of relevant phenomena are presented in the following sections.

The novelty of our measurements consists in a combination of the well-characterized near-normal-incidence imaging crystal with the XSC detection of the hot-electron-induced radiation and the simultaneous introduction of the fiducial onto the XSC slit, which facilitates temporal correlation of the recorded signals with the laser beam incident on the target.

The time- and space-resolved data characterizing the hot-electron generation in massive Cu targets of size 5 × 5 × 50 mm³ were obtained using a laser beam with energy 498 J and FWHM pulse duration 333 ps, corresponding to an intensity of 1.9 × 10¹⁶ W/cm². Despite small errors in the precisely fitted profiles of the fiducial mark introduced on the entrance slit of the XSC (±2.2 ps) and the Cu Kα emission (±9.1 ps) [see Fig. 2(a)], the determination of the time interval between the maxima of the asymmetrically fitted hot-electron-induced signal and the laser is encumbered by laser–fiducial fluctuations. Taking into account this uncertainty, the maximum of hot-electron generation precedes by 11 ± 44 ps the laser maximum. In this particular case, however, we benefit from an occurrence of oscillations observed in descending parts of both the laser and hot-electron profiles. Assuming that the hot-electron production closely follows the side maxima in the laser intensity, the best-fit overlap of the observed laser and hot-electron oscillations provided a refined value of 8 ± 15 ps.

Figure 3(a) shows the resulting temporal correlation between the normalized laser pulse and the hot-electron-induced Cu Kα signal. The delay of the hot-electron rising edge with respect to the laser measured at half-maximum intensity profiles equals 59 ± 15 ps. This phenomenon can be interpreted qualitatively as the time needed for the creation of the optimum plasma density scale supporting hot-electron generation via one of the alternative mechanisms. The observed duration of the hot-electron emission is considerably shorter than that of the laser pulse (224 ± 9 ps and 333 ± 10 ps FWHM, respectively). Similar shortening of the Cu Kα signal with respect to the laser pulse duration has been observed in all recorded shots (the averaged values provide 251 ± 22 ps vs 332 ± 24 ps, respectively). In contrast to the shift of the rising edge of the hot-electron production, this shortened hot-electron emission can be partly affected by the gradual heating of the target material,¹¹ but in the case of massive Cu targets, this effect should be of minor importance.

The spatial extent of the hot-electron interaction with the Cu target surface is shown in Fig. 3(b). The width of 127 ± 1 μm FWHM and 219 μm at the level of e⁻² was determined from the XSC signal distribution integrated over the full duration of the hot-electron action. This observed lateral hot-electron emission extent is comparable to the FWHM value of the focused laser beam with a diameter of ∼100 μm. The evolution of the emission zone in time is discussed in Sec. IV A, and the corresponding time dependence is shown in Fig. 6(a) below.

The second set of experimental data were collected from a 1 μm-thick Cu foil irradiated by a laser pulse with energy 466 J and FWHM duration 318 ps, corresponding to the same intensity of 1.9 × 10¹⁶ W/cm². The temporal correlation of the laser profile with the measured hot-electron-induced Cu Kα emission is presented in Fig. 4. Differing from the case of the thick target, the hot-electron maximum fitted with a single asymmetric bell-shaped profile with the FWHM duration of 288 ± 18 ps is delayed by 169 ± 45 ps with respect to the laser maximum. This time delay manifests the qualitative change in hot-electron transport. As explained in Secs. IV and V, the target areal density in this case is smaller than the electron stopping. This enables recirculation of hot electrons and leads to a significant increase of their lifetime, which increases from less than one picosecond in a massive target to more than a 100 ps in an expanding foil.

This directly measured delay quantitatively supports conclusions drawn from spectroscopic observations of hot-electron generation at a Cu foil of the same thickness and same laser parameters,¹⁶ but the interpretation of this observation is revised. In that experiment, the distinct maxima of the Cu Kα emission were observed at a distance of 30 μm from the original foil surface, which was attributed to the delay of 250–500 ps in hot-electron generation. By contrast, detailed hydrodynamic and kinetic simulations discussed in Sec. IV show that the delay in x-ray emission is due to the extended time of recirculation of energetic electrons in an expanding burn-through target.

The spatial distribution of hot-electron emission from the foil target is shown in Fig. 5. Selected individual profiles shown in Fig. 5(a) were integrated over a temporal window of 50 ps and vertically shifted to provide better visibility. The Gaussian fits of these profiles indicate a distinct increase in their lateral widths with time, in accordance with the dashed straight lines shown in Fig. 2(b). At later times, the signal is strongly modulated and exhibits several maxima, which may be related to poor statistics of the relatively weak Kα signal. However, it can also be related to laser beam filamentation in the plasma corona.

The time-integrated profile of the spatial distribution of Kα emission is shown in Fig. 5(b). The width of 203 ± 4 μm FWHM and 352 μm at the level e⁻² is doubled compared with the focal laser spot diameter. This broadening can be attributed to retention¹⁰ and recirculation of hot electrons near the target, causing their lateral diffusion, as discussed in the following sections. A relative increase in the Kα signal and a change in its spatial distribution due to the electron refluxing effect have previously been studied, mostly for considerably higher intensities, both numerically^22–24 and experimentally.²⁵ The effect may be particularly important and has its specifics in the case of thin foils²⁶ and ICF targets.²⁷ 

The temporal evolution of the width of the emitting zone in thick and thin targets is depicted in Fig. 6. The spatial extent of the emitting area from a thick target shown in Fig. 6(a) is about 110 μm during the laser pulse and increases only slightly afterward. By contrast, the emission area from a thin target in Fig. 6(b) increases abruptly from ∼110 to ∼240 μm at the laser maximum. This is the manifestation of a fast radial spread of hot electrons. The origin of the subsequent decrease in width at ∼100 ps is yet to be understood, but at the end of the emission, the spatial extent of emission again increases to ∼200 μm.

To relate the observed temporal and spatial characteristics of Kα emission to the mechanisms of hot-electron generation, a set of numerical simulations have been performed with the radiation hydrodynamics code CHIC²⁸ and with the two PIC codes EPOCH²⁹ and SMILEI.³⁰ 

These three numerical tools make complementary contributions to the understanding of the processes of interest. The hydrodynamic code describes the whole process of laser–target interaction on the macroscopic scale, over the laser pulse duration and target volume. Figure 7(a) shows an example of a snapshot of the electron temperature distribution in the plasma plume at the maximum of the laser pulse. Nonlinear processes such as laser-stimulated scattering on plasma waves and hot-electron generation, however, are described in hydrodynamic codes qualitatively on the basis of empirical models. The hot-electron generation is described with a PIC code on the microscopic scale, in a limited plasma volume with density smaller than the critical density and over a short time of about 10 ps at several time moments of the laser pulse. These simulations characterize the source of hot electrons and assess the validity of the empirical model included in the hydrodynamic code. Numerical simulations with another PIC code address the problem of hot-electron transport in the expanding plasma produced in laser interaction with a thin copper foil. They are extended to a longer time scale and larger plasma volume, but with some model assumptions related to the physics of electron collisions.

The CHIC code accounts for the laser inverse bremsstrahlung absorption, electron and radiation energy transport, and electron–ion temperature relaxation. This code has been widely used for modeling of laser–target interaction in ICF, and it has been successfully used for interpretation of experiments.^31–33 The code also includes an option for evaluating hot-electron generation due to TPD and SRS,^34,35 for which it has been calibrated with PIC simulations near the threshold of these instabilities at a wavelength of 0.35 μm. This is not the case of our interest, where the laser intensity is two orders of magnitude above the threshold and the wavelength is larger. However, as discussed by Antonelli et al.,³³ the model of hot-electron generation can be adapted to our conditions by decreasing the hot-electron fluxes by a factor of two to four and increasing the divergence of SRS-generated electrons.

Figure 7(a) shows the spatial distribution of electron temperature at the time of the laser pulse maximum for the case of a massive target obtained with the CHIC code. The electron temperature in the plasma corona is 5.4 keV, and the width of the heated zone is about 220 μm, which is twice as large as the laser beam diameter because of lateral thermal conduction. Profiles of the density, temperature, and flow velocity are shown in Fig. 8(a) by dashed curves at time 100 ps after the laser pulse maximum. The characteristic density scale length in the corona is about 140 μm, and collisional absorption is dominant. The time-integrated absorption is 26.5%, and the SRS and TPD contributions amount to 2.1% and 1% with hot-electron effective temperatures of 40 and 80 keV, respectively. These values are consistent with observations^10,11,19 and confirm that hot-electron generation has a minor effect on target hydrodynamics under our conditions. The hot-electron temperature also agrees with the results of PIC simulations presented in Sec. IV C. The reduced Kα emission shown in Fig. 3(b) indicates that hot electrons are generated in the central zone near the laser beam axis with a small angular divergence, and their lateral spread is limited by a short mean free path in the massive target.

The blue curve in Fig. 7(b) shows the temporal evolution of the hot-electron flux obtained with the radiation hydrodynamics code CHIC in a two-dimensional planar geometry. The generation of hot electrons is delayed by 57 ps at the rising part of the laser pulse (black curve) and closely follows the trailing edge of the laser profile. The delay can be explained by the time needed to form an extended density profile for the excitation of SRS and TPD. This delay agrees well with the measured temporal evolution of Kα emission shown by the red curve in Fig. 3(a). However, the calculated duration of hot-electron emission of 283 ps FWHM is longer than the observed 224 ± 9 ps [see Fig. 3(a)]. This difference may be related to the fact that the simulations were performed in a 2D geometry.

The density distribution in plasma obtained from a hydrodynamic simulation of laser interaction with a thin copper foil is shown in Fig. 9 at 100 ps after the laser pulse maximum. In contrast to case of the massive target, the thin foil has already been heated and expanded. Its density at that time has decreased more than 30-fold compared with the solid density near the laser axis, producing a hole of radius about 100 μm.

The profiles of the dense part of the foil at the laser axis are shown in Fig. 8(b). At a time 100 ps after the laser maximum, the foil thickness has increased to 20 μm and the areal density has decreased twofold. However, the electron temperature in this zone is still smaller than 100 eV, and so hot electrons crossing the foil can produce Kα emission detectable with our diagnostic. The foil is accelerated and moves with a velocity of 0.1 mm/ns. The foil displacement and velocity are consistent with the position of the Kα emission zone and the plasma expansion velocity reported in Ref. 16.

The plasma profiles in the plasma corona, in the zone of laser–plasma interaction where the electron density is smaller than the critical density, n_e ≲ n_cr, are shown in Fig. 8(a). Interestingly, the plasma profiles are similar and depend only weakly on the target thickness. In the foil case (solid curves), the interaction zone is moved inside by about 50 μm, but the density, temperature, and flow velocity profiles are similar. This implies that the generation of hot electrons proceeds in the same way, independently of the target thickness, and the observed difference in the Kα signal is related to their transport in the target.

There are two consequences of thin-foil heating and hole formation on the transport of energetic electrons. First, heating leads to a spectral shift of the copper Kα emission out of the detection window. Thus, the measured signal proportional to the density of cold copper atoms decreases. Second, the target areal density becomes smaller than the hot-electron stopping distance. Instead of slowing down in a few picoseconds in a massive target, a fraction of the hot electrons leave the target from the rear side, making it positively charged. However, most of the hot electrons are retained near the target. They circulate within the potential well,^10,36 diffuse laterally, and gradually lose their energy in collisions with bulk electrons. The fact that thin foils cannot stop hot electrons is confirmed experimentally in Ref. 26, where hot electrons with effective temperature 60 keV were measured at the rear side of a tantalum target of thickness 10 μm. Moreover, the density of hot electrons may increase with time as the rate of energy loss becomes smaller than the hot-electron production rate. This situation is considered further with PIC simulations in Sec. IV C.

The plasma density, flow velocity, and temperature profiles obtained in hydrodynamic simulations are taken as input for more detailed PIC simulations, performed also in a 2D Cartesian geometry. However, these kinetic simulations are limited to much smaller spatial and temporal scales than the plasma created at the target surface. The methodology of using PIC simulations to interpret laser–plasma interaction experiments is described in Ref. 13. The simulations are performed for a sufficiently long time that the hot-electron generation attains a quasi-stationary level, typically ∼10 ps. Only the data on hot electrons obtained at that quasi-stationary phase are considered for further analysis.

For studies of the generation of hot electrons on the surface of a massive copper target, we use the EPOCH code²⁹ and consider a plasma volume near the laser axis shown by the dashed white box in Fig. 7(a) of width 150 μm and length varying from 100 to 300 μm depending on the plasma density scale length. The plasma density varies from 0.1n_cr to the critical density, which is n_cr = 6 × 10²⁰ cm⁻³ for the laser wavelength considered. The target has vacuum boundaries on all sides, with the front side spanning 80λ₀, the rear side covering 20λ₀, and each transverse side extending by 40λ₀. Convolutional perfectly matched layer (CPML) boundary conditions are applied for fields, while thermal boundary conditions are assumed for particles. The flux of the forward-propagating hot electrons is absorbed and thermalized close to the rear boundary of the target in an additional layer with the critical density. This special absorption region is described in Ref. 6. It provides a return current of cold electrons and prevents recirculation of hot electrons, which is not expected to occur for a massive target, since they lose their energy in the bulk. We use 40 cells per laser wavelength, with 25 electrons and 10 ions per cell. The simulations are performed without collisions, since the inverse bremsstrahlung absorption of the incoming laser light is already accounted for in the hydrodynamic simulations. The laser transverse profile is Gaussian with 25 μm FWHM, polarization is linear in the interaction plane, there is normal incidence, and the simulation time is typically more than 10 ps.

The plasma density profile along the laser beam axis is approximated by an exponential function, and the scale length is calculated at quarter critical density. The electron temperature and ion charge Z = 29 are constant, and the ion temperature and ion velocity are approximated by linear functions. The plasma parameters for four simulations for the massive target are given in Table I (runs A, B, C and D) together with the corresponding laser intensity and time. Run A represents the conditions at the very beginning of the interaction −350 ps before the peak of the laser pulse. Runs B, C, and D cover the time from −250 ps to −50 ps before the peak of the pulse. Both the plasma density scale length and the electron temperature increase with time. The intensity included in the table for runs A–D corresponds to the actual intensity of a laser beam focused on the position of the quarter critical density.

Two stages of laser–plasma interaction are observed in the simulation. The initial stage of ∼6 ps is influenced by a steeply rising laser intensity and non-accommodated plasma profiles. It involves the excitation of strong electron plasma waves near quarter critical density related to TPD and SRS, generating bursts of scattered radiation and energetic electrons. This transient phase is followed by a quasi-steady stage, which is representative of the experiment. Consequently, only the simulation results obtained at the second stage and averaged over a time window of several picoseconds are considered.

Figure 10 presents the spectrum of backscattered radiation time-integrated over the interval from 6 to 10 ps recorded at the front boundary of the simulation with a massive target for runs A–D presented in Table I. The spectrum is calculated from the magnetic field B_z perpendicular to the simulation plane in vacuum in front of the target. The reflectivity is calculated as an integral over the backscattered spectrum.

In the early stage of laser interaction with the target (runs A and B), the interaction takes place near quarter critical density, as can be seen from the narrow peaks in the spectrum around 0.5ω₀ and 1.5ω₀ related to the TPD. The laser reflectivity at this stage of interaction is 46%, indicating a relatively high conversion into hot electrons. At the later stage of interaction, closer to the peak intensity of the laser pulse (runs C and D), both the density scale length and the electron temperature increase, and the laser–plasma interaction is dominated by convective SRS in a plasma with a density less than quarter critical density. It is responsible for a broad peak between 0.5ω₀ and 0.75ω₀ in Fig. 10.

The reflectivity increases with time to 68% (run B), 83% (C), and 89% (D). Such relatively high values are due to the fact that collisions are not included in the simulations. For this reason, and because only a central part of the plasma is considered in these simulations, the number of hot electrons is overestimated.

The high-energy component of the electron distribution is related to the resonance interaction with SRS- and TPD-driven plasma waves. The electrons generated in runs A and B have a lower temperature of ∼30 keV, and we attribute them to the excitation of absolute SRS or TPD instabilities because the spectra of backscattered radiation in Fig. 10 are localized near 0.5ω₀ and 1.5ω₀ and related to plasma waves excited near quarter critical density.

The hot electrons produced in runs C and D have a higher effective temperature ∼45 keV, which agrees with the hydrodynamic calculation using the CHIC code and with experimental results.¹¹ Since their generation is associated with larger backscattering spectra in Fig. 10, we attribute them to convective SRS below quarter critical density. These electrons are responsible for the Kα emission in the experiment.

The transport of laser-generated hot electrons over larger temporal and spatial scales in thin targets is studied with another set of simulations using the PIC code SMILEI.³⁰ The simulation region in the (x, z) plane is shown by the dot-dashed white box in Fig. 7(a). Only half of the profile with x > 0 is simulated, assuming symmetry with respect to the laser propagation axis, to reduce the size of the simulation box and the computation time. The box size is 300 × 2680 μm², and the number of cells is 2880 × 25 600. A large box length in the x direction enables fast electron reflection and lateral transport. Absorbing boundary conditions are applied for electromagnetic fields at all boundaries except the symmetry axis, x = 0, where the boundary condition is reflective. Removing boundary conditions are used for electrons at the z boundaries, while reflective conditions are set at the x boundaries. The cell size in both dimensions is 0.1 μm, with 10 macroparticles of each species per cell. The time step is 0.12 fs, and the simulation time is 35 ps. The maximum laser intensity is 1.8 × 10¹⁶ W/cm², with a Gaussian intensity profile of 83 μm FWHM and normal incidence. The plasma profiles are taken from the hydrodynamic simulation described in Sec. IV B at a time corresponding to the maximum laser intensity, t = 0 see run E in Table I and Fig. 8.

Owing to the specifics of PIC modeling of such a large volume, to avoid numerical heating, the maximum electron density in the dense region, n_e > 4n_cr, is artificially limited to 4n_cr, so that the ratio of the Debye radius to the spatial resolution is maintained above 1, and the plasma dynamics is correctly resolved. For the same reason, a lower limit of 2 keV is set for the electron temperature. The modeling includes collisions for electrons from the low-density region. To correctly describe their transport in the dense region, where the density has been artificially reduced, the collision frequency is proportionally increased to provide the same stopping power as expected for the target density obtained in the hydrodynamic simulations. The ions in the dense region are immobile, since their displacement and ionization are very small during the simulation time. Also, after the time moment of 1 ps since the beginning of laser–plasma interaction, when the quasi-stationary electromagnetic fields have already formed around the target, all particles except the electrons of the low-density region have been considered immobile, to prevent numerical heating and reduce the computation time.

Electromagnetic fields created around the target are presented in Fig. 12. The magnetic field distribution in Fig. 12(a) implies strong electric currents flowing along the target surface and a strong laser beam filamentation in the underdense plasma. The distribution of the electric field E_z in Fig. 12(b) reveals the charge separation field created at the front and rear edges of the target. This charge separation field is responsible for the reflection of fast electrons back into the target.

To study the effect of electron refluxing, the coordinates z and momenta p_z of macroparticles are collected during the runtime and averaged in each bin of the (z, p_z) grid. The phase plots averaged over time from the start of the simulation t = 0 up to time t₁ are shown in Fig. 13. It can be seen that fast electrons with momenta p_z ≳ m_ec are generated and ejected from the foil. Some propagate along the positive z direction toward the laser, while others propagate in the opposite direction. They are slowed down outside the target and returned to the dense region. They penetrate the target and leave it on the opposite side. The process is repeated until the fast electrons lose their energy and thermalize in collisions with cold electrons. The energetic electron excursion length exceeds 1 mm.

Electron recirculation is also studied by introducing test particles, which are subject to the fields in the simulation box but do not contribute to charge and currents and do not induce electromagnetic fields. The test particles are initialized in a small region near the laser axis, z ∈ [−20, 50] μm and x ∈ [0, 100] μm, with a total number of ∼60 000. Test particle positions, momenta, and weights are saved on each time step and analyzed a posteriori. The time dependence of the z coordinate for the selected test particles is presented in Fig. 14. Certain test particles move ∼1 mm away from the target in both directions, are reflected, and pushed back. The period of oscillations of trapped hot electrons is a few tens of picoseconds. By tracking the test particles in the energy range from 30 to 200 keV, an energy loss in one crossing of the foil of ∼3 keV is estimated.

On evaluating the number of crossings required for the electron energy to drop below 10 keV, which is the excitation energy of the copper atoms, the electron lifetime is found to be in the range 100–400 ps, which agrees with the experimentally observed value. During this time, the electrons are diffusing laterally, which is the reason for the lateral spreading of the Kα signal, seen in Fig. 2. The energetic electrons exist on a time scale ≳100 ps, oscillating around the foil, slowly losing their energy, spreading laterally, and producing Kα emission each time they cross the target.

The energy lost by energetic electrons goes into heating of particles inside the target. This energy loss is evaluated in an additional simulation where collisions inside the target are turned off. This results in higher electron energy at the end of the simulation. The difference amounts to 5% of the total laser energy delivered into the simulation box. This qualitatively agrees with, although is a few times larger than, the experimental estimate of the laser energy conversion into hot electrons.

For electrons to be able to produce the measured Cu Kα emission, they must have energy larger than 10 keV and penetrate into the target where the bulk temperature is 100 eV or less. The minimum electron energy is defined by the energy excitation of the K shell of copper. The condition of low target temperature is imposed by the spectral window of the x-ray detector, as explained in Sec. II.

Assuming the experimental value of laser energy conversion to hot electrons of 1% and the characteristic temperature of 40 keV (see Ref. 11 and Table II), the energy flux carried by these electrons is about 10¹⁴ W/cm², which corresponds to a hot-electron density n_h ∼ 10¹⁸ cm⁻³ and an electric current density j_h ≃ 2.5 GA/cm². In a massive target, this current is neutralized by the return current of bulk electrons and results in the creation of an electric field, which maintains this return current and slows down hot electrons. For the resistivity of copper of η_r ∼ 10⁻⁵ Ω · cm, the electric field is of the order of j_hη_r ∼ 25 kV/cm. So, the electric potential developed over the propagation distance of about 100 μm is less than 1 kV and has very little effect on the hot-electron transport.

The fraction of electrons with energy larger than 10 keV, f_K, is given in Table II for the electron distribution obtained in the PIC simulation. It decreases with time from ∼70% to less than 40% after the laser maximum. These losses concern essentially the electron population with lower temperature T₁. Thus, about half of the hot-electron population contributes to the excitation of copper atoms in experiments with massive targets. Their typical lifetime is the propagation time from the critical density to the cold material, which is less than 1 ps.

By contrast, in the laser interaction with a thin foil, the rate of hot-electron production is approximately the same as in the massive target, since they are generated under similar interaction conditions, but electrons lose less energy when crossing the expanding foil because of a smaller areal density. The hot electrons do not produce Kα emission, because all the target material within the laser focal spot is heated to a temperature above 100 eV. For a target areal density of less than 1 mg/cm², all electrons with energy larger than ∼13 keV cross the foil. This value agrees with the results of PIC simulations in Sec. IV D, where an electron with an initial energy of about 100 keV needs to cross the foil about 30 times before thermalization.

The electron excursion time increases strongly with increasing electron energy from 0.3 ps for 20 keV to 5.5 ps for 200 keV. At the same time, the collisional energy loss, according to Eq. (4), decreases from 6 to 1.6 keV over the same interval. Consequently, the time of electron slowing down increases strongly with electron energy. It takes only 1.6 ps to reduce an electron’s energy from 30 to 20 keV, but this time increases to 200 ps for a 200 keV electron. These estimates are in agreement with the numerical simulations shown in Figs. 13 and 14. Consequently, the lifetime of electrons with energies exceeding 100–150 keV is on a 100 ps scale, and these electrons are responsible for the delayed Kα emission observed in the experiment.

During the recirculation time, electrons diffuse laterally from the hot plasma into cold material, excite copper atoms, and thus produce Kα emission. This is in agreement with the rapid increase in the size of the emission zone shown in Fig. 6(b). Moreover, since the rate of generation of hot electrons in laser interaction with foil remains the same as with a massive target and their lifetime increases, their density increases proportionally after the maximum laser pulse, in agreement with the observation presented in Fig. 4. The overall number of emitted K_α photons is the same for the massive and thin targets, because the total number of generated electrons and their energy are approximately the same. However, the number of detected photons is smaller in the experiments with thin foils for two reasons: first, the electrons are stopped in a plasma with a temperature larger than 100 eV where the energy of emitted photons is outside the detector’s spectral window; second, the number of electrons contributing to the delayed Kα emission is relatively small because their energies exceed four to five times the hot-electron temperature.

We have presented a set of precise measurements of the space and time evolution of energetic electrons created in the interaction of intense laser pulses with massive and thin copper targets. In the massive targets, the hot-electron lifetime is very short, and the Kα emission signal follows the temporal evolution of the laser intensity. The observed delay in the Kα emission and increase in the size of the emission zone in experiments with foil targets is explained by an increase in the lifetime of hot electrons in the expanding target and their temporal accumulation. This scenario is confirmed by the PIC simulations described in Secs. IV C and IV D. The spatial displacement of Kα emission in the laser interaction with thin foils is in agreement with the result reported previously,¹⁶ but the physical origin of this effect has now been clarified. It does not relate to a delay in hot-electron generation, but rather to a strongly enhanced hot-electron lifetime, due to their recirculation in expanding plasma.

By comparison of experimental data with detailed numerical simulations, we have demonstrated that this is related to the condition where the foil thickness is smaller than the stopping range of laser-produced hot electrons. This makes possible the following physical effects:

• The hot electrons and x-ray radiation from the plasma corona preheat the foil, which results in the spectral shift of Kα emission out of the detector’s acceptance range.

• Electrons capable of crossing the foil enter the recirculation regime. This is manifested in a significant increase in their lifetime and lateral diffusion.

• An increase in the lifetime of hot electrons results in an increase in their density, which also contributes to the delay in the maximum of Kα emission.

These effects of increased lifetime and population of hot electrons could be important in applications related to laser-driven secondary sources of x-ray radiation and charged particles and in evaluation of hot-electron preheating in inertial confinement fusion.

We acknowledge partial funding via EUROfusion Enabling research Project No. AWP21-ENR-01-CEA-02 “Advancing Shock Ignition for Direct-Drive Inertial Fusion,” within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200—EUROfusion). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Commission. The authors give thanks to the Czech Ministry of Education, Youth and Sports (CMEYS) for funding the operation of the PALS facility (Grant No. LM2023068) and to the PALS staff and visiting guest scientists for assistance in performing the experiments. Our special thanks go to Philipp Korneev for his valuable contribution to interpretation of experimental results. O. Klimo acknowledges the EuroHPC Joint Undertaking for awarding access to Karolina at IT4Innovations (VŠB-TU), Czechia under Project No. EHPC-REG-2023R02-006 (DD-23-157), and the Ministry of Education, Youth and Sports of the Czech Republic through e-INFRA CZ (Grant No. ID:90140).

The authors have no conflicts to disclose.

O. Renner: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). O. Klimo: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Krus: Data curation (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). P. Nicolaï: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Poletaeva: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). N. Bukharskii: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). V. T. Tikhonchuk: Conceptualization (lead); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead).

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