Modeling of electron refluxing and TNSA fields in laser–target interactions based on analysis of Kα emission

High-intensity ultra-short powerful lasers, when focused onto the surface of a solid target, generate an intense forward-moving electron beam of energies up to MeV's.^1,2 As the beam traverses the target, it produces a flux of x-ray radiation. Upon reaching the rear surface of the target, space-charge effects prevent majority of the electrons—except the most energetic ones—from leaving the target, thus causing the electrons to reflux. The refluxing has been studied experimentally by a considerable number of authors, such as Nersisyan et al.,³ McKeever et al.,⁴ Makita et al.,⁵ Quinn et al.,⁶ and Neumayer et al.⁷ These studies have clearly established its existence and importance, in particular its influence on the radiation emission. A tightly related phenomenon is the formation of a sheath field that accelerates protons and other ions to MeV energies^8,9 through the target-normal-sheath-acceleration (TNSA) mechanism (e.g., see reviews of Badziak¹⁰ and Macchi et al.¹¹). We also note a recent study by Huang et al.¹² of hot refluxing electrons in a thin foil, with the dynamics governed by the spatiotemporal evolution of the self-generated sheath fields.

In this paper, we analyze the experimental data by Zastrau et al.¹³ who provided radial distributions of Kα emission from Ti foils of various thicknesses, as well as from a bulk Ti target, in an ultra-intense-laser–target interaction experiment. Notably, the radiation intensity from the thin, 10  $μ m$ foil was larger by factors of 1.5 and 3 than that from the 25  $μ m$ and bulk targets, respectively. It was also observed that the radial distribution of the Kα radiation from the 10 and 25  $μ m$ foils is significantly broader than that from the bulk target. Here, these observations are reproduced with the fast-electron refluxing phenomenon accounted for, and bounds on the magnitude and spatial extent of the TNSA electric fields are obtained.

The laser energy on the target is 14 J, the pulse duration is 330 fs, the irradiation intensity at the fundamental laser frequency is $5 × 10 19 W / cm 2$⁠, and the laser spot size is $8 μ m$⁠.¹³ 1D-resolved single-shot Kα spectra are collected from the front side of the target at an angle of $50 °$ to its normal. With the Abel transform applied, spectra at different radial positions are obtained. For the present study, total Kα fluxes (i.e., these spectra integrated over the Kα profile width) are used.

The relevant experimental data are presented in Fig. 1, where the radial Kα flux distributions I(r) of various target thicknesses are shown. These profiles can be roughly divided into three regions: the nearly-flat-top core (up to a few tens micrometer), the wing (up to $∼ 100 μ m$⁠) characterized by a steep decrease in the intensity, and the slowly fading out halo extending further away.

Thus, the focus of this study is on the central region (the core and the wing) that carries a more pronounced dependence on the target thickness and, therefore, provides a better sensitivity to the limited number of model parameters. Furthermore, the experimental uncertainties outside of this region become very large. It is also consistent with the previous analysis of a subset of the same data,¹⁴ which succeeded in explaining heating of the core region.

The calculations of the fast electron trajectories within the solid titanium target accounted for collisional energy loss as well as for multiple scattering. The former process was calculated using the cold target dE/dx, since due to the relatively low temperatures attained in the experiment, plasma effects on the stopping can be neglected.¹⁵ Multiple scattering was calculated using the Bethe–Molière theory¹⁶ also assuming a cold target. For every segment of the electron path, the Kα flux contribution in the direction of the detector was determined, with the dependence of the Kα production on electron energy accounted for.¹⁷ Importantly, self-absorption of Kα photons as they make their way to the detector is included in the calculations, using the mean free path for Kα absorption in titanium of about $20 μ m$⁠.¹⁸ This value, which is on the order of the foil thicknesses used in the experiment, has an important bearing on the results.

In order to derive initial properties of the forward-moving intense electron beam, the experimental data obtained for the bulk target is used, where complex phenomena of electron refluxing do not exist. The three parameters of the model, governing the electron flow, are the effective emission spot size (limiting the area over which the electrons are accelerated at the target surface), the angular spread of the electron emission, and the fast-electron temperature $T fast$⁠. Gaussian-shaped radial and angular distributions are assumed with the standard deviations of R_b and $ϕ b$⁠, respectively.

The ballistic nature of the advancing electron beam is clearly seen in the experimental work and compilation of Green et al.,¹⁹ where for the laser intensity of the experiment here discussed¹³ the diverging angle is about 30°. Effective diverging angles of 24° and 16° at $4 × 10 20$ and $10 19 W / cm 2$⁠, respectively, have recently been measured.^20,21 A recent analysis²² of bremsstrahlung data yields an incident electron angular spread of $15 ° ± 8 °$ at $2 × 10 19 W / cm 2$⁠.

The electron energy spectrum was assumed to be Maxwellian with $T fast = 1 MeV$⁠. This value is between the ponderomotive and Beg values for the condition of the present experiment and agrees with the Beg-inspired fit of Lefebvre et al.²³ In addition, a recent bremsstrahlung experiment at similar conditions gave 1.1 MeV for the fast-electron temperature.²² The back-scattered electrons, which constitute $≲ 20$% in cold titanium targets at ambient conditions,¹ are assumed not to return to the target, even though restoring electric fields could develop on the front side of the target as here discussed.

In Fig. 3, the simulated radial intensity distribution of the Kα radiation is compared to that of the experimental result of the bulk target, assuming angular beam divergences of 15° and 30°. The best fit for $15 °$ is for the standard deviation of $36 μ m$⁠, while for the $30 °$ case, it is $34 μ m$⁠. Both plots agree with the experiment within the range of experimental errors in the core part of the distribution. It is noted that the low-intensity “halo” is evident in the experimental data. As previously mentioned, in the analysis presented here this halo radiation is ignored.

The forward-moving fast electrons exit the target and are emitted into the vacuum region at the back side of the foil, where they move in the electric field of the Debye sheath formed.^10,12,24 The electrons are returned to the foil, cross it in the opposite direction, and are emitted into the vacuum region at the front of the foil, whence they are returned into the target. This process is repeated several times with the electrons refluxing at the back and front sides of the foil and diffusing in the transverse direction until they exit the region of interest.⁸ 

A temporal evolution of the electric-field magnitude is based on that given by Fuchs et al.,³¹ in which the laser parameters are very similar to those in the experiment here analyzed.¹³ Specifically, the field is assumed to be constant for the duration of the laser pulse (330 fs) and further decreasing as a Gaussian with the standard deviation $σ = 400 fs$⁠.

In order to fit the experimental transverse radial emission profiles (which are much broader than that in the bulk target), a sheath of a considerable transverse extension is needed. We have found that a super-Gaussian flat-top distribution of order m, described by $exp [ − 2 ( R / W ) m ]$⁠, gives a better fit to the experimental data than a Gaussian distribution. Here, R is the radial or transverse coordinate, and W is a measure of the width of the distribution. It was found that $W ≃ 100 μ m$ (see Sec. IV) and $m = 9$ give satisfactory results.

It is assumed that the front-side field is the same as that at the back side of the foil. We have also tested the assumption that this field is half of that at the back of the foil, with only a moderate effect on the results. Some indications on the front-to-back electric-field ratios can be gained by comparison of thin foil backward to forward proton emission accelerated by these fields. Ceccotti et al.³² find almost identical proton spectra from back and front of the target, but for ultrahigh-contrast laser pulses. In a lower contrast experiment,³³ it was found that the less abundant protons emitted from the back of the foil are of a higher average energy than the protons emitted from the front of the foil. The contrast of the laser pulse in the experiment analyzed here¹³ lies in between these values.

As discussed in Sec. III A, the forward-moving fast electrons are injected into the foils with an angular spread of 15°. The electrons were emitted from a Gaussian spot size centered at z = 0, with a radius R_b. The initial electron energy distribution is Maxwellian with the temperature of 1 MeV. These three distributions (of the initial radial position, angle, and energy of the electron) are sampled by means of the Monte Carlo method. Each of the results given in Sec. IV is based on 15 000 events. An event is terminated when the electron exceeds the value of $80 μ m$ in the z axis direction or $250 μ m$ in the transverse direction.

In Fig. 4, we present typical calculated electron tracks for 10 and 25  $μ m$ foils, with the restoring field of 0.1TV/m (hence, $l s = 10 μ m$⁠) at both sides of the target. The electrons are emitted at 15° to the z axis from the origin (z = 0 and at r = 0) with the initial kinetic energy of 1 MeV. As seen in the figure, a few strong scattering events considerably distort the smooth appearances of the tracks. It is also observed that the electron distance in the vacuum region increases with the radial distance, since the restoring field is assumed to decrease with r.

Figure 4 also helps understanding why the radiation intensity from the 10  $μ m$ foil exceeds that of the thicker 25  $μ m$ one. Although in the case of the thinner foil, the electrons spend more time in the vacuum area, the induced Kα emission from the target undergoes a significantly lesser absorption on the way to the detector (recall that the Kα photon mean free path is about $20 μ m$ in solid titanium) than in the thicker, 25  $μ m$ foil.

The radial distribution of the Kα emission for the 25, 10, and 5  $μ m$ foil targets and their ratios to the bulk Kα yield are presented here.

The electron temperature of 1 MeV and the initial angular electron spread of 15° are used, as inferred from the bulk Kα distribution (see Sec. III A). The three other parameters that influence the results are varied: the size of the initial radial distribution of the “hot” electrons (R_b), the magnitude of the restoring electric field (E₀), and its radial extension (W).

Analysis of the 25 and 10  $μ m$ targets suggests a nearly identical set of the parameters providing best fit to the experimental results. In particular, $R b = 60 μ m$ is inferred. This is larger than $30 − 40 μ m$ inferred for the bulk emission (see Fig. 3). The difference is attributed to the different conditions formed at the front surface of the bulk and foil targets. It is obtained that the value of W must be at least $125 μ m$⁠. $W = 150 μ m$ gives the best fit, while larger yet values degrade the agreement slightly. This value significantly exceeds the estimate based on a single pass of the fast electrons,³¹ even when allowing for a rapid traversal expansion of the sheath field.²⁵ As for E₀, the simulations suggest the values of a few TV/m, but in general, the results depend relatively weakly on the value of E₀ in excess of $∼ 1 TV / m$ and nearly reach saturation at $∼ 5 TV / m$⁠. The value and the geometry of the sheath electric field are comparable to those found in other studies. For example, while studying the bremsstrahlung emission, Chen and Sawada³⁴ found electric fields of up to 10 TV/m extending to radii of up to $500 μ m$⁠. Higher yet fields of 15 TV/m were also reported.¹² Lateral expansion of the electric field was also dealt with by McKenna et al.,³⁵ finding fields millimeters from the laser focus. In another study,²⁹ an order-of-magnitude lower electric field of $∼ 0.5 TV / m$ was inferred; however, the irradiation intensity in that experiment, $10 18 W / cm 2$⁠, was also significantly weaker.

A typical simulated radial Kα distribution and its comparison with the experimental data are shown in Fig. 5(a). Figure 5(b) shows a detailed study of the 10  $μ m$ to bulk Kα intensity ratios as a function of the radius. The results are given for different restoring fields E₀ with standard electron energy deposition (solid lines) as well as for the enhanced resistive stopping assumed (dashed lines). For the latter, the stopping is multiplied by a factor of 6 to account for increase in the stopping due to the intense return currents induced in the target, as it has previously been discussed.¹⁴ The difference between the two approaches is very minor.³⁶ The reason for studying the intensity ratios as a function of R is to exclude the effect of the halo, which is observed in the experimental data, but not accounted for in the present theoretical modeling (see Fig. 3). In Fig. 5(b), the region where the halo begins to play an important role is indicated by the vertical dashed line. It is seen that except for the halo region, the agreement between the model and the experiment is very good for an electric field of $≳ 5 TV / m$ assumed. We note that the range of the experimental uncertainties, indicated by the gray area in Fig. 5(b), is calculated assuming the worst-case scenario, i.e., that the error bars in Fig. 1 are all systematic and, thus, are accumulated by the integration in Eq. (1). It is believed that the true uncertainties are smaller.

A similar analysis of the 25  $μ m$-foil results is given in Fig. 6. It is of interest to note that the Kα flux in the 10  $μ m$ case is about 1.5 times larger than that of the thicker 25  $μ m$ foil. This non-intuitive result is due to the Kα self-absorption effect within the target, as explained above in discussion of Fig. 4.

Thus, for both 10 and 25  $μ m$ targets, the electric field of a few $TV / m$ extending over $∼ 100 μ m$ radially is inferred. Contrary to the thicker targets, the 5  $μ m$ foil is very different, as seen in Fig. 7: a lower intensity and a significantly narrower distribution is evident. In these respects, it resembles the bulk target. Indeed, the radial distribution of the hot electrons with $R b = 36 μ m$—the same as in the case of the bulk target—appears to explain the observed data best. The halo radiation is also very like that of the bulk target (while less prominent in the two other foil cases). Furthermore, the weaker intensity indicates a significantly lower, by two orders of magnitude, field, extending to a smaller area. It was suggested¹³ that the distinct properties of the 5  $μ m$ target were due to a specific preplasma formation, since this specific foil was coated with 250 nm of copper. Alternatively, one can attribute these observations to a significant distortion of the target surfaces by a shock wave launched before the main laser pulse,³⁷ whereas in the case of the thicker targets, the TNSA sheath is formed before the slow moving shock wave reaches the rear side of the targets.

The present modeling allows for calculating the temporal shape of the Kα radiation pulse, which is important in the context of using laser–matter interactions as intense ultra-fast radiation sources.^38,39 The temporal shapes of the Kα pulses for different targets are shown in Fig. 8. The bulk case is the easiest to understand: while the fast electrons are supplied by the laser and propagate inside the target, creating K-shell holes along their paths, the number of Kα photons increases. As long as the electrons emitted at t = 0 do not deepen beyond the Kα absorption length, i.e., during the first ∼70 fs, the growth is linear. Afterward, the growth rate slows down and the intensity approaches “saturation.” At the end of the laser pulse (⁠ $t = 300 fs$⁠), the supply of the new fast electrons ends, while those previously emitted enter deeper yet target regions from which the number of $K α$ photons reaching the observer decays exponentially. The shape resembles that of the capacitor voltage in a charge–discharge cycle, with the FWHM determined by the laser-pulse duration. The particle-in-cell simulations of Reich et al.,⁴⁰ when extrapolated to the irradiation intensities relevant for the present study, indicate that 90% of the Kα pulse energy is emitted over about 1–2 ps, reasonably close to our result.

The shape of the Kα pulse in the case of the 25  $μ m$ target closely matches that of the bulk target for the first ∼100 fs, including the beginning of the saturation phase. At this point, however, the initially injected electrons reach the rear target side and begin their way back. As a result, the radiation intensity returns to the linear growth. The electrons continue bouncing off the restoring fields at both sides of the target (see Fig. 4), forming the clearly seen “ripples” with the period of $2 d / c ≈ 170 fs$⁠, decaying in time due to the electron scattering and energy spread. As the laser pulse switches off, the radiation intensity begins to decrease due to an increasing fraction of fast electrons overcoming the weakening field and leaving the target forever.

The case of the 10  $μ m$ target is similar, except that, due to the smaller thickness, the photon attenuation is much weaker, while the bouncing is 2.5 times faster. As a result, the ripples are nearly invisible. Finally, due to the much weaker fields formed at the surfaces of the 5  $μ m$ target, the fast electrons spend a significant time, about 300 fs, in the vacuum before returning to the foil. This results in a reduced Kα yield compared to the other foil targets. Evidently, the duration of the Kα pulse in the case of foil targets (about 1 ps) strongly depends on the electric-field evolution assumed in the model.³¹ An experimental investigation of the Kα duration by Nilson et al.⁴¹ determined a few times longer pulses. However, limited-mass targets were used in that study, which might be responsible for extended electron rebouncing times.

The modeling predicts a near uniformity of the Kα emissivity inside the target along the z direction. However, a larger x-ray flux near the front target surface was reported, for example, by Seely et al.⁴² and Langhoff et al.⁴³ Novel tomographic experiments⁴⁴ are expected to provide accurate data on the z-dependence of the Kα emissivity.

A model simulating Kα emission resulting from the laser-induced electron beam interacting with planar foil and bulk titanium targets is constructed. The motion of the fast electrons inside the targets is calculated allowing for multiple scattering and collisional energy loss, while outside, it is governed by electric fields. For the latter, the TNSA theory is used as a framework for the present modeling, with the electron refluxing being key to the quantitative determination of the Kα intensity ratios. Previously reported experimental results, where various targets were exposed to a $5 × 10 19 W / cm 2$ laser radiation, are analyzed. An electric field of $≳ 1 TV / m$⁠, radially extending over $∼ 100 μ m$⁠, is inferred when the 10 and 25  $μ m$-thick targets were used. However, in the case of the thinner 5  $μ m$ foil, a much weaker field appears to be formed, probably due to a strong deformation of the target by the shock wave. Notably, the effective size of the fast-electron-beam source in the 5  $μ m$-foil case is the same as that of the bulk target (⁠ $R b ≈ 35 μ m$⁠), whereas for the thicker foils, it is about twice larger. In addition, we simulate temporal profiles of the Kα radiation for all targets used in the experiment.

This work was supported in part by the German–Israeli Foundation for Scientific Research and Development (GIF).

The authors have no conflicts to disclose.

Eran Nardi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Evgeny Stambulchik: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Zeev Zinamon: Conceptualization (supporting); Formal analysis (supporting); Investigation (equal); Methodology (equal); Writing – original draft (supporting). Yitzhak Maron: Conceptualization (supporting); Funding acquisition (lead); Project administration (lead); Writing – review & editing (supporting).

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