Isochoric heating and strong blast wave formation driven by fast electrons in solid-density targets

We experimentally investigate the fast ( < 1 ps ) isochoric heating of multi-layer metallic foils and subsequent high-pressure hydrodynamics induced by energetic electrons driven by high-intensity, high-contrast laser pulses. The early-time temperature profile inside the target is measured from the streaked optical pyrometry of the target rear side. This is further characterized from benchmarked simulations of the laser-target interaction and the fast electron transport. Despite a modest laser energy ( < 1 J ), the early-time high pressures and associated gradients launch inwards a strong compression wave developing over ≳ 10 ps into a ≈ 140 Mbar blast wave, according to hydrodynamic simulations, consistent with our measurements. These experimental and numerical findings pave the way to a short-pulse-laser-based platform dedicated to high-energy-density physics studies.


Introduction
The controlled production of extreme pressures and temperatures in dense samples is of prime interest for the study of structural and dynamic properties of matter, with applications in planetary science [1,2], astrophysics [3] and inertial confinement fusion [4,5]. Currently available high-energy (∼kJ), long-pulse (∼ns) lasers can be used to shock compress materials up to pressures of~100 Mbar [6,7]. Alternatively, in this article, we explore the capability of ultra-intense (> -10 W cm 19 2 ), utra-short-pulse (~30 fs), sub-Joule energy lasers to isochorically create hot dense matter, at >100 eV temperatures and »100 Mbar pressures, as well as the possibility of driving strong shocks of relevance for laser-fusion [5,8] or high-energy astrophysical phenomena [3,7].
It is well known that ultra-intense laser pulses interacting with solids generate relativistic electron beams (REBs) that are capable of transporting tens of percent of the laser energy deep into the targets [9,10]. The subsequent slowing down of the REB, through both direct collisions and self-generated fields, results in the heating of the bulk plasma [11][12][13][14][15][16][17][18][19][20][21][22]. Depending on the laser parameters, this takes place over time scales ranging from a few tens of fs to a few ps, i.e., prior to the hydrodynamic response of the heated material. The broad energy spectrum and finite radius and divergence of the REB give rise to a temperature profile steeply decreasing from the front surface, hence launching inwards heat and compression waves possibly developing into a shock [15].
In this paper, we provide experimental evidence for blast wave generation through the interaction of a highcontrast, sub-Joule laser and a metallic-foil target. Benchmarked radiative-hydrodynamic simulations reveal that the compression wave evolves into a shock of »140 Mbar pressure (in a Cu layer embedded into Al at shallow depth) over a micrometer scale in 10 ps, prior to significant heat conduction. As the shock front moves deeper into the target, its pressure drops rather rapidly because it is not sustained by continuous energy injection, and hence obeys the Sedov-Taylor (ST) similarity solution [23,24]. It should be noted that radiative blast waves initiated by relativistic laser-plasma interaction were first experimentally investigated in [25], yet using a much longer (20 ps) and energetic (400 J) laser pulse than in our conditions. Moreover, in contrast to the simple hydrodynamic analysis presented in that work, we here provide a comprehensive numerical modeling of our data that treats all the relevant kinetic and rad-hydro phenomena.

Methods and experimental results
The experiment was carried out using the UHI100 laser (CEA/IRAMIS, Saclay, France), delivering 800 nm wavelength pulses of 0.7 J energy and 25 fs full-width-at-half-maximum (FWHM) duration. The laser was focused at q =  45 las incidence onto flat solid foils by a f 3 off-axis parabola. About 40% of the laser energy was contained into a m 4.5 m FWHM focal spot, yielding´-4.5 10 W cm 19 2 maximum intensity. The targets weré 5 5 mm 2 foils, composed of an Al rear layer of varying thickness (1, 6, 10 or m 15 m), coated on the laser side by a m 3 m thick Cu layer (used as a REB tracer by detection of a K x-ray fluorescence; results to be detailed elsewhere), and a m 1 m thick Al interaction layer ( figure 1(a)). The total thickness of the target ranged from L = 5 to m 19 m. A double plasma mirror system allowed both the amplified spontaneous emission pedestal and prepulses to be reduced below the target ionization threshold. An intensity contrast of~10 10 was measured over a 1 ns duration before the laser peak [26], ensuring that the laser pulses interacted with unperturbed solid-density targets.
The REB energy deposition was investigated by imaging the optical emission from the target rear side onto an absolutely calibrated, S20-photocathode streak camera (streaked optical pyrometry, SOP). The space-time resolution was of m »20 m and »20 ps. An interferometric filter centered at  532 5 nm was used to suppress the REB-induced coherent transition radiation, peaking at multiples of the laser frequency [15,27,28]. The top image of figure 1(b) shows the signal obtained for m = L 19 m. The time origin (t = 0) coincides with the laser peak. Figure 1(c) presents the space-averaged emissivity as a function of time for m = L 5 m (solid black curve) and m = L 19 m (solid red curve). The early-time signals correspond to the prompt thermal emission induced by the REB-driven heating of the target rear side. This emission decays away (over a 100 ps time-scale) as the target surface expands and cools down [29,30]. The late-time emissivity rebound (at » t 250 ps and » t 450 ps for m = L 5 m and m = L 19 m, respectively) is related to the hydrodynamic evolution of the initially solid target. Since the high laser contrast rules out any hydrodynamic perturbation prior to the laser peak, the longtime-scale target emission is only determined by the REB-induced heating profile, as analyzed in [15]. Remarkably, the emissivity rises more significantly and steeply for m = L 19 m (15 ) than for m = L 5 m (1.5 ), corresponding in the former case to a blast wave breakout through the rear surface (see below).

Early-time emissivity and the isochoric heating by fast electrons
The early-time electron temperature, T e , reached at the target rear surface, was measured from the emissivity at =  t 0 20 ps, averaged over a m 5 m radius around the signal maximum, and assuming a Planckian-type emission. The measurements are accurate to within~6%, owing to the absolute calibration of the streaked detection system. The circles in figure 2(a) display the temperature values obtained for different target thicknesses: the plotted values and their respective error bars account for the mean and the standard deviation We performed kinetic simulations of the REB generation and transport describing the early-time isochoric target heating, followed by hydrodynamic simulations of the subsequent target evolution to better understand the experimental data. First, we used the 2D particle-in-cell (PIC) code CALDER [32] to model self-consistently the laser-plasma interaction, electron propagation and target heating. The code describes field ionization [33], as well as elastic and inelastic collisions [34]. Our Monte Carlo binary-collisional scheme includes a Lee-More-type correction [35] in the dense/cold plasma regime. Due to computational limitations, we only treated the case m = L 5 m. Given the high laser intensity contrast, the target was initialized as a steep-gradient, solid-density plasma slab. We considered an initial 10 eV temperature and an ionization state given by the Thomas-Fermi ). The laser profile was taken to be Gaussian in space and time, and its parameters reproduced the experimental values. The mesh size was m D = D = x y 0.0064 m. A fourth-order weight factor and an alternating-order interpolation scheme [36] were employed to mitigate the numerical heating intrinsic to high-density PIC calculations. The simulation was initialized with 200 particles per cell and per species (electrons, Al 3+ , Cu 5+ ions). Note that the total number of particles significantly increased with time as a result of field and impact ionization.
The REB source angle and energy distributions were extracted at a distance of m = x 0.3 m beyond the irradiated surface. Figure 3(a) shows the angle distributions for different kinetic energy ranges (as labeled), extracted at the laser peak power. For each energy range, the respective curve can be modeled by a Gaussian , where q 0 is the mean propagation angle with respect to the target surface normal and q D is the angular dispersion around that mean value. Both parameters have a very weak dependence with time. While q D diminishes with the electron energy (faster electrons are better collimated), the q 0  dependence fairly agrees with the formula proposed in [37,38], sin sin 0 l a s 1 1 (where γ is the electron Lorentz factor), assuming conservation of the transverse canonical momentum in the boosted laser frame. The most energetic electrons are emitted close to the laser propagation axis, q =  45 las , while the slowest are preferentially injected along the surface normal. Figure 3(b) shows the kinetic energy distribution of the REB (solid black curve). As previously reported from both theoretical [39,40] and experimental [20,21] studies, the low-energy part of the spectrum is best described by a decreasing power-law function. In our case, the extracted energy distribution is well fitted by for  E 200 keV (dashed red curve). The REB distribution (of energies >10 keV) carries~10% of the laser drive energy, with a mean kinetic energy of~120 keV. The relatively low value of the laser-to-REB coupling efficiency follows from the short duration and high contrast of the laser pulse. Figure 2(b) shows the electron temperature T e profiles obtained for m = L 5 m from the PIC (dashed curve) and PIC-hybrid transport (solid curve, see below) simulations. In the PIC simulation, T e is computed from the mean energy of cold electrons (with energies <8 keV) at = t 330 fs after the on-target laser peak. At a given longitudinal position, x, the temperature is averaged around the transverse position of its maximum over a m 10 m width. At this time, the REB has not yet reached complete relaxation but it has spread enough from the irradiated region that the bulk temperature profile is no longer varying significantly along the loci of the temperature maxima. The transverse position of the maximum temperature increases linearly with x, corresponding to an effective REB propagation angle of q »  25 eff with respect to the target surface normal. The temperature profile is strongly inhomogeneous, peaking around = T 900 eV e close to the target surface, and dropping to »200 eV at the Al/Cu interface ( m = x 1 m). Further away, T e decreases at a slower rate, reaching »100 eV at the target rear side ( m = x 5 m). As a result, ultra-high pressures (>100 Mbar) are generated over the target depth.
In order to simulate thicker targets, we used the 3D PÂRIS hybrid transport code, which employs a PIC description for the REB and a generalized Ohm's law for the background electrons [42,43]. The REB source was extracted from the PIC simulation as described above. While the two temperature profiles for m = L 5 m in figure 2(b) qualitatively agree, they present quantitative differences over the first m 4 m of the target. While the PIC results exceed (up to a factor of ∼2) the hybrid results over the front ( m 1 m thick) Al layer, they are lower (by a factor of ∼0.6) than the hybrid results in the m 3 m thick Cu layer. The discrepancy found close to the irradiated surface may point to the questionable validity of a few simplifications underlying the hybrid transport model, such as discriminating between hot and bulk electron populations and neglecting the short-scale electromagnetic (EM) modulations induced by the REB [44,45]. These simplifications may suffice to alter the REB propagation and field generation in a non-trivial way far from the laser region, notably because the EM modulations are observed in the PIC simulation to extend throughout the target, i.e., well beyond the surface through which the hot-electron distribution is extracted to serve as input for the hybrid code. Now, the lower PIC-predicted heating in the Cu layer may also stem from the different models of electrical resistivity and equations of state implemented in the collisional PIC and hybrid codes (respectively detailed in [34,43]), and, particularly, to their respective sensitivity to the non-equilibrium ( ¹ T T e i ) thermal conditions at play here. The overall consistency of these calculations, however, gives us confidence in hybrid simulations for thicker targets at early times. The temperature profiles obtained for   m L 5 19 m are plotted in figure 2(a) (solid curves). Each simulated curve matches the corresponding experimental data (circles) at x=L. Moreover, the curve for the thickest target ( m = L 19 m, thicker red curve) nicely fits all data points. This suggests that, in all considered cases, the hot electron refluxing, which is expected to decrease with L [31], weakly affects the on-axis heating of the target rear side. Our data thus gives direct access to the longitudinal heating profile in the thickest target. The very high temperatures found close to the laser-irradiated surface, on the order of -0.5 1 keV, agree with previous works conducted at similar laser intensity and power, but with 100 times higher energy [48].
We have verified that resistive effects play a major role in the target heating along the REB propagation axis, at least up to m 10 m depth, consistently with a REB current density >´-5 10 A cm

Late-time emissivity and blast-wave formation
The evolution of the target following the fast REB-induced heating, which accounts for the late-time SOP signals of figures 1(b) and (c), was simulated using the 2D axisymmetric Lagrangian radiation-hydrodynamic code CHIC [46,47]. The ionization and the opacity data are tabulated, assuming a local thermodynamic equilibrium LTE or a non-LTE depending on the local plasma parameters at each time step. The radiative transport is computed assuming that the radiation field is quasi-stationary and weakly anisotropic (multigroup diffusion). The code makes use of the QEOS  Figure 4 displays results from the rad-hydro simulation for m = L 19 m. Due to the high pressures (>300 Mbar) induced on axis over the front Al layer, the target surface rapidly expands into vacuum and a rarefaction wave is launched inwards. Plasma expansion also occurs at the rear side, but at a slower velocity because of the lower local temperature (»20 eV). A compression wave is driven off axis by the oblique pressure gradients, but not on axis due to the fast rarefaction wave. In contrast to standard ns laser-driven shock formation [50], the compression wave that forms here is not preceded by a thermal wave precursor. As seen in figure 4(a), a shock has already built up at » t 12 ps at a depth of m »0.3 m beyond the Al/Cu interface and at a transverse position m ¢  y 5 m. At the shock front in the Cu layer, the pressure reaches~140 Mbar (i.e., about twice the upstream pressure value) and the density is raised to  25 ps, so that no shock is launched longitudinally. Along an oblique direction, by contrast, the upstream temperatures are much lower (~1 eV) and the pressure gradient is strong enough to generate a shock in t » shock 10 ps ( t »  250 rar ps). Figures 4(b) and (c) show the oblique shock propagating in the thicker, rear-side Al layer at later times: the shock-front peak pressure decreases from » 30 Mbar at 125 ps ( figure 4(b)) to »8 Mbar at 440 ps ( figure 4(c)). The corresponding compression factor of the Al layer drops from 3.7 to 2.6. Figure 4  x (density (top) and pressure (bottom)): (a) = t 12 ps: the shock is formed off-axis and propagates into the Cu layer; (b) = density. In the Cu layer, the shock pressure is found to decrease with time ast 0.5 , remaining in excess of 50 Mbar till » t 70 ps. Its sudden drop (to~15 Mbar) follows from the shock transmission into the rear Al layer. It reaches a maximum of » 30 Mbar in Al at » t 140 ps, and then decreases ast 1.3 . The decreasing trends of the shock-front pressure in each of the two materials are fairly well predicted by the ST model (solid curves) [24], P sf where r sf is the density at the shock-front and g = 5 3 ad is the specific heats ratio.
The blast wave breaks out through the rear surface at 440 ps (figure 4(c)), at a transverse distance of m10 m, from which ensues a rapid increase in the optical emissivity of the rear surface (solid red curve in figure 1(c)). For a direct comparison with the experimental data, the simulated emissivity was convolved by a m 20 m FWHM Gaussian function in the target rear plane modeling the response function of the SOP detector. The resulting synthetic emissivity (bottom image in figure 1(b) and dashed red curve in figure 1(c)) matches well the experimental signal, both in terms of chronometry and strength. The discrepancy between the measured rear-side emissivity and the hydrodynamic simulation at > t 0.6 ns is probably related to the fact that the synthetic emission is integrated over p 2 sr, while the collection of the experimental radiation is limited to the small aperture of the optical imaging system, which looks at the rear target surface from a q =  45 obs angle (as depicted in figure 1(a)). The experimental emissivity remains high after 0.6 ns because of oblique shock propagation and off-axis shock breakout (in the direction of the collecting lens). Under our experimental conditions, the shock breakout could be clearly detected only for m = L 19 m due to the relatively low rear-side temperatures at early times, limiting the rear rarefaction wave and plasma expansion into vacuum. We checked, by turning off the radiative effects in the hydro simulations, that the strength and chronometry of the shock propagation did not vary significantly. By contrast, the late-time rebound in the emissivity for m = L 5 m (solid black curve in figure 1(c)) is relatively weaker, and not so well reproduced by the simulation (dashed black curve).

Conclusions
In summary, by means of absolutely calibrated streaked optical pyrometry, we have accurately characterized the isochoric heating of metal foils by fast electrons driven by a 0.7 J, high-intensity (´-4.5 10 W cm 19 2 ), highcontrast short-pulse laser. Despite a modest laser driver, our measurements, supported by kinetic transport simulations, indicate that electron temperatures above 100 eV are reached up to m 5 m depths. Rad-hydro simulations show that the steep electron temperature and pressure gradients formed at shallow depths trigger ã 140 Mbar blast wave. Its breakout through the target rear side accounts for the rise in the emissivity observed in the m 19 m thick target. The measured breakout chronometry and strength are consistent with rad-hydro simulations, and therefore provide quantitative-albeit indirect-signatures of the efficiency of REB energy deposition as a function of target depth. Our results stand out from previous works by much reduced laser energy and pulse duration, well-controlled interaction conditions, and a quantitative description of the full range of physical processes leading to blast-wave formation. Finally, our experimental data and numerical simulations highlight the great potential of high-contrast, few-femtosecond laser pulses as compact tools for the study of ultrahigh-pressure hydrodynamics, of fundamental interest to inertial confinement fusion [5], highenergy astrophysical phenomena [3,51], as well as for the creation of transient (~10 ps) warm dense matter states.