1 INTRODUCTION

Shock wave generation as a result of the action of high-power laser radiation pulses on matter is among most remarkable phenomena in the high energy density physics. In practice, modern investigations in this field consist primarily in providing pressure buildup in the framework of the equation of state (EOS) studies of target materials and in realization of the inertial confinement fusion (ICF) conditions. Owing to a high energy flux density of laser radiation, laser-induced X‑ray radiation, or laser-accelerated electron beam, modern laboratory experiments allowed record high pressures of quasi-stationary shock wave in a solid target to be achieved.

A traditional experimental approach consists in the direct exposure of a target to pulses of laser or laser-induced X-ray radiation. Pressures achieved using pulses of short-wave Nd laser of terawatt power amount to several dozen megabars [1–3]. A shock wave is generated due to the so-called laser ablation pressure that arises as a result of the evaporation and thermal expansion (ablation) of matter from the irradiated target surface. Another method capable of significantly increasing the shock-wave pressure consists in energy concentration in the target as a result of the impact of a macroparticle accelerated due to the laser ablation. Using this “collisional” technique (more complicated for practical implementation) and terawatt Nd laser or laser-induced X-ray pulses, record high shock-wave pressures up to several hundred megabars have been achieved [4–7]. Moreover, there were theoretical and numerical calculations that justified the possibility of generating quasi-stationary shock waves with pressures up to several gigabars in targets under the action of relativistic electron beams accelerated in the field of petawatt laser radiation pulses [8, 9].

A universal method of increasing the shock-wave pressure is based on using its passage into a matter of higher density [10]. This approach can be considered as a variant of the “collisional” method in which the shock wave plays the role of accelerated macroparticle. In the simplest scheme of experiment with laser ablation, this method is based on using a target comprising layers of different densities where the primary shock wave is generated due to the ablation of a layer with lower density. In particular, the low-density ablator of this target can represent a gas with the density above the critical value for plasma corresponding to the wavelength of laser radiation employed. However, this leads to significant complication of the technical realization of experiment, since the target must be either arranged in a gas-filled chamber with windows for diagnostics and laser beam entry or provided with walls surrounding this “gaseous” ablator and laser beam entry window.

A more appropriate material for the low-density ablator is offered by a porous substance (foam) based on light chemical elements. The physics of high-power laser radiation acting on porous materials is now extensively studied in view of the interest in fundamental phenomena of laser–plasma interaction important for some applied tasks, particularly, for the aforementioned ICF process [11, 12]. Porous materials have a number of important advantages in respect of the absorption of laser radiation and formation of the ablation pressure. One of these advantages is a highly efficient absorption of terawatt laser pulses. As established in numerous experiments, the energy fraction of first to third harmonics of Nd laser radiation absorbed within geometric transparency length of various porous materials based on light elements amounts to 80–90% for foams with both sub- and super-critical densities [13–18]. In addition, the absorption of laser radiation in a material of supercritical density (ρ > ρcr) is a premise for developing ablation pressures on a level higher than that for the laser pulse action on solids in which the radiation can only be absorbed in plasma of subcritical density [19, 20]. Thus, a porous (foam) layer plays simultaneously the roles of (i) laser radiation absorber with minimum energy losses for self-radiation and (ii) ablator ensuring shock wave generation.

The energy transfer in partly homogenized plasma of a porous material is mediated by a hydrothermal wave [13] representing a shock wave with quasi-homogeneous temperature distribution behind the front and without ionization precursor ahead, which is caused by suppressed electron heat conductivity in view of the absence of free electrons in unperturbed porous material. The velocity of hydrothermal wave Dh is lower than that of shock wave Ds in the equivalent homogeneous medium of average density and chemical composition. This is related to delay in the pressure formation behind the hydrothermal wave as determined by the time of porous material homogenization. For a terawatt laser pulse and a porous material with a density of (3–10)ρcr, the ratio Dh/Ds amounts to 0.7–0.5 [13, 14, 17, 20–25].

The phenomenon of shock-wave pressure buildup on the passage from low-density porous ablator into solid material has been studied in [26–28]. These experiments employed layered targets based on aluminum covered with a layer of porous C15H20O6 material (foam) with density varied in a broad range, which were irradiated by subnanosecond (400–600 ps) pulses of Nd-laser second-harmonic or I-laser third-harmonic radiation with intensity on the order of 1014 W/cm2. The laser-induced shock-wave pressure in the aluminum layer exhibited 2.5-fold increase as compared to that in the porous layer.

In the present work, the approach based on the use of a target with porous ablator has been developed so as to enhance the “collisional” effect by separating the layers of porous ablator and solid part of the target with a vacuum gap. The second section of this article presents the results of experiments in which targets of this type were exposed to nanosecond pulses of second-harmonic radiation of terawatt Nd laser. The shock-wave velocity was measured in a solid layer of aluminum, while the porous absorber represented a layer of cellulose triacetate (CTA) based foam. Laser radiation pulses had 3.5–4 ns duration (full width at half-maximum, FWHM), which was 5–6 times greater than the values used in previous experiments [26–28]. In contrast to those, the longer pulse width allowed establishing a quasi-stationary regime of maximum pressure transfer from porous ablator to solid part of the target. The third section discusses the results of numerical modeling and considers the features of shock-wave generation and propagation in the porous ablator and solid part of a target.

2 EXPERIMENTAL

Experiments were performed in the All–Russia Research Institute of Experimental Physics at the Russian Federal Nuclear Center (RFNC, Sarov, Russia) on the LUCH laser fusion facility [29] using a method of high-speed laser plasma glow monitoring. Pulses of second-harmonic radiation of Nd phosphate glass laser with a wavelength of λ = 0.53 μm had a trapezoidal shape with FWHM of 3.4–3.8 ns, leading front width of 1.5–2.2 ns, and trailing front width of about 1 ns. Laser radiation was normally incident onto the surface of a composite target comprising layers of solid aluminum and porous material separated by a vacuum gap (Fig. 1). The aluminum part had the shape of Δb = 20 μm thick base with a step of same thickness (hs) on the rear side.

Fig. 1.
figure 1

Schematic diagram of a target comprising CTA foam layer and profiled solid aluminum layer spaced by a vacuum gap.

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The porous material represented foamed cellulose triacetate (CTA) with chemical formula (C12H16O8)n and a microstructure comprising fine porous network of crosslinked fibers (Fig. 2a). The CTA based foam was obtained by method of stimulated gelation in the (CTA–chloroform)/methanol system with a 1/2 volume ratio, followed by supercritical drying for aerogel formation. Blank foam layers were prepared by casting gel-forming polymer solution of preset concentration into a mold comprising flat gasket holder with planar glass walls, followed by cooling, solvent substitution, and supercritical drying in a special setup using carbon dioxide as extractant. The foam exhibited shrinkage (15–50% of the initial volume) on drying, thus forming a vacuum gap between porous and solid layers of the target. Geometric parameters of the foam blank upon drying and removal of the mold could be varied with the aid of Alicona Infinite Focus G5 optical 3D-profilometer. Figures 2b and 2c present photographs of assembled target as viewed from the frontal (foam) and rear (profiled aluminum) sides.

Fig. 2.
figure 2

Photographs showing (a) foam microstructure and (b, c) target viewed from the frontal (foam) and rear (profiled aluminum) sides, respectively.

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Targets were prepared with porous layers of variable density from 24 to 9 mg/cm3. Foam layers had approximately the same thickness (about 250 μm) significantly exceeding the geometric transparency length in the indicated interval of densities. The vacuum gap width was about one-third of the foam layer thickness, as selected so that to ensure, on one hand, noticeable enhancement of the “collisional” effect and, on the other hand, not too high rate of aluminum layer unloading and, hence, not too short duration of the quasi-stationary stage of shock-wave propagation in aluminum.

Experiments determined the moments of shock-wave escape from the base layer and step of aluminum. Then, the mean velocity of shock-wave propagation in the step was defined as the ratio of step thickness to the difference Δt of escape moments. Plasma glow on the rear surface of target was monitored with the aid of a streak camera (slit photochronograph) [30] which switched the electro-optical converter with an ultimate temporal resolution of 1 ps, photocathode with a working field size of h = 8 mm, microchannel-plate image brightness amplifier, and digital CCD camera with 1000 × 1000 matrix. At a sweep rate of 10 ps per screen, the photochronograph could measure time intervals between the fronts of detected pulses with an error not exceeding 15 ns. A fraction of laser pulse radiation was directed via special optic-fiber channel onto the photochronograph slit so as to ensure temporal relation between the incident pulse and laser-induced shock-wave propagation process in the target.

Table 1 presents data on the parameters of laser pulses and targets, as well as mean shock-wave velocity in the step of aluminum layer as measured in three experimental laser shots. Laser radiation intensity on the target refers to the homogeneous part of laser spot with allowance for the trasmittance of objective (measured in a series experiments). The shock-wave velocity in Al layer step is given with an error including the uncertainty of step thickness, delay of the glow emission from the Al base layer and step, and spread of the plasma glow start (including the non-simultaneity of glow onset).

Table 1. Experimental conditions and results of shock-wave velocity measurements in Al layer step (EL, laser pulse energy on the target; τ0.5, pulse FWHM; τ1, pulse leading front width; τ2, pulse trailing front width; ILm, maximum radiation intensity on the target; ρ, TCA foam density; Δf, foam layer thickness; Δgap, vacuum gap width; D, shock-wave velocity in Al layer step)
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Figure 3 shows an example of the temporal variation of plasma glow intensity emitted from Al step in experiment 3 (see Table 1), in which the time delay between glow in the Al base layer and step amounted to Δt = 0.78 ns.

Fig. 3.
figure 3

Temporal variation of the plasma glow intensity in experiment 3 (see Table 1).

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3 NUMERICAL CALCULATIONS AND DISCUSSION OF RESULTS

Numerical calculations were performed for three groups of model targets: (i) 3 targets used in experiments presented in Table 1; (ii) 3 targets with the same porous ablator layers but not containing vacuum gaps; (iii) 3 targets without vacuum gaps but with homogeneous ablator layers of the same chemical composition, density and thickness as those indicated in Table 1. The Al layer thickness was selected equal to 50 μm, which allowed the complete pattern of shock-wave propagation in the Al base layer and step to be traced.

Numerical calculations for the targets with homogeneous ablators were performed using standard version of DIANA program package [31. 32]. Calculations for the targets with porous ablators employed DIANA-P program package that included a block for computation of the interaction of laser radiation with partly homogenized plasma of porous substance. This program ensured computation of the laser radiation absorption as a result of the bulk-retardation process in a region of with dimensions determined by the geometric transparency length [33, 34]. The geometric transparency length depends on the time of plasma homogenization in ion–ion collisions [34, 35]. The equations of motion and energy employ operators of limitation of the pressure gradient and electronic heat conductivity flux, respectively, which are also the functions of time- and coordinate-dependent plasma homogenization time [19, 35]. Parameters of the porous material structure were set by selecting the average pore size δ0 (0.5 μm for all targets under consideration) and unified fractal parameter [34, 35] set at α = 0.8 corresponding to the filamentary-membrane structure. The average thickness b0 of solid elements of the foam was selected based on the relation b0 = δ0(ρ/ρs)α [34, 35], where ρ is the average foam density and ρs = 1.1 g/cm3 is the density of solid elements. This formula yields the values of b0 = 0.025, 0.013, and 0.012 μm corresponding to ρ = 0.024, 0.011, and 0.009 g/cm3, respectively. The density of homogeneous gas in the gap with the chemical composition equivalent to that of the porous ablator was selected at 10–5 g/cm3.

Table 2 presents the results of calculations that refer to average values of the shock-wave pressure and velocity in target ablators, as well as in the Al base and step. In addition, calculations were also performed using codes adopted at the RFNC, including those taking into account radiative heat transfer in the porous material and aluminum. Values obtained in these calculations are almost identical to those presented in Table 2 and are not considered separately.

Table 2. Results of numerical model calculations of the ablation pressure in low-density absorber and the shock-wave pressure and velocity in Al base layer and step (Pabl, average pressure behind the shock-wave front in the ablator; Dabl, mean shock-wave velocity in the ablator; Pbase, pressure in Al base layer; Dbase, shock-wave velocity in Al base; Pstep, shock-wave pressure in Al step; Dstep, shock-wave velocity in Al step; *asterisk indicates calculations for model targets with homogeneous ablators)
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Analysis of the data summarized in Table 2 allows the following conclusions to be made. All targets revealed cumulative amplification of the shock-wave pressure in aluminum in comparison to that in the ablator. This effect was most pronounced in targets with a vacuum gap between the ablator and aluminum layer. The growth of pressure in the Al base layer of various targets ranges from 3.5 to 6.3 times. The values of pressure behind the shock-wave front in the ablator (almost equal to the ablation pressures) exhibit rather insignificant differences in accordance with slightly different intensities of incident laser pulses. The values of ablation pressures fall in a range of 2.8–3.5 Mbar. The ablation pressure as dependent on the laser intensity for all targets can be evaluated using the well-known scaling relation [3, 36] established for the action of laser pulses on a substance with supercritical density (ρ > ρcr). This is related to the fact that the employed ablator densities (11 and 9 mg/cm3 in model calculations No. 5–8) are only 1.2 and 1.4 times lower than ρcr, so that laser radiation after shock-wave generation would act on the substance of supercritical density. Thus, the scaling relation for ablation pressures has the following form:

$$\begin{gathered} {{P}_{{{\text{abl}}}}}[{\text{Mbar}}] \approx {{\rho }_{{{\text{cr}}}}}{{\left( {\frac{{2(\gamma - 1)}}{{3\gamma - 1}}\frac{{{{I}_{{\text{L}}}}}}{{{{\rho }_{{{\text{cr}}}}}}}} \right)}^{{2/3}}} \\ \approx 12{{\left[ {\frac{{2(\gamma - 1)}}{{3\gamma - 1}}} \right]}^{{2/3}}}{{\left( {\frac{A}{Z}} \right)}^{{1/3}}}\frac{{I_{{{\text{L}}(14)}}^{{2/3}}}}{{\lambda _{\mu }^{{2/3}}}}, \\ \end{gathered} $$
(1)

where ρcr ≈ 1.8 × 10–3 A/Z\(\lambda _{\mu }^{2}\) is the critical density (in g/cm3); A and Z are the atomic number and degree of plasma ionization, respectively; λμ is the wavelength (in microns); IL(14) is the radiation intensity (in 1014 W/cm2 units); and γ is the adiabatic exponent.

In the approximation of fully ionized plasma in the evaporated part of ablator (A/Z ≈ 2, γ = 5/3), estimations by formula (1) for λμ = 0.53 μm with IL = 0.16 × 1014 W/cm2 and IL = 0.154 × 1014 W/cm2 yield Pabl = 3.2 Mbar, and the estimation with IL = 0.11 × 1014 W/cm2 yields Pabl = 2.5 Mbar. These values are in good agreement with the results of numerical model calculations.

Approximate estimate of shock-wave pressure increase upon the passage from ablator with density ρabl into solid layer with density ρs is given by the following formula [28, 37]:

$$G \approx {{\left( {\frac{{1 + {{\beta }^{{1/2}}}}}{{1 + {{\beta }^{{1/2}}}{{\rho }_{{{\text{abl}}}}}{\text{/}}{{\rho }_{{\text{s}}}}}}} \right)}^{2}},$$

where β = (1 + γabl); γabl and γs are the adiabatic exponents in the ablator and target, respectively. For targets involving extended ablator with ρabl ≪ ρs, this estimation yields a scale of pressure increase by a factor of about 4. In the numerical calculations performed with allowance for the real EOS of aluminum in targets without vacuum gap, this factor ranges from 3.5 to 6.3, which corresponds to attaining pressures of 12, 10, and 18 Mbar in the Al base layer for targets with ablator densities of 9 mg/cm3 (calculations No. 8 and 9 in Table 2), 11 mg/cm3 (calculations No. 5 and 6), and 24 mg/cm3 (calculations No. 2 and 3), respectively. The presence of a gap leads to about 30% shock-wave pressure increase in aluminum, whereby the pressure in Al step reaches 16 Mbar in the case of ablator with density 24 mg/cm3 (calculation No. 1) and 7 Mbar for ablators with densities 11 mg/cm3 (calculation No. 4) and 9 mg/cm3 (calculation No. 7).

An important factor in applied tasks is related to the damping of shock waves in aluminum. In the given mode of experiments with approximately the same thickness of ablators, this damping is determined by the rate of ablator unloading: growth of the unloading rate leads to increase in the degree of shock-wave damping in aluminum. With allowance of Eq. (1), the isothermal sound velocity defined as Vs ≈ (Pabl/ρabl)1/2 amounts to

$$\begin{gathered} {{V}_{{\text{s}}}}\left[ {\frac{{{\text{cm}}}}{{\text{s}}}} \right] = {{\left( {\frac{{{{\rho }_{{{\text{cr}}}}}}}{{{{\rho }_{{{\text{abl}}}}}}}} \right)}^{{1/2}}}{{\left( {\frac{{2(\gamma - 1)}}{{3\gamma - 1}}\frac{{{{I}_{{\text{L}}}}}}{{{{\rho }_{{{\text{cr}}}}}}}} \right)}^{{1/3}}} \\ \approx 3.5 \times {{10}^{6}}{{\left[ {\frac{{2(\gamma - 1)}}{{3\gamma - 1}}} \right]}^{{1/3}}}{{\left( {\frac{A}{Z}} \right)}^{{1/6}}}\frac{{I_{{L(14)}}^{{1/3}}}}{{\lambda _{\mu }^{{1/3}}\rho _{{{\text{abl}}}}^{{1/2}}}}. \\ \end{gathered} $$
(2)

According to this formula and the conditions of calculations No. 1, 2, and 3 (see Table 2) for ablators with a density of 24 mg/cm3 at laser pulse intensity IL = 0.16 × 1014 W/cm2, the sound velocity Vs is about 120 km/s. Under the conditions of calculations No. 4, 5, and 6 for the ablators with a density of 11 mg/cm3 (2.2 times lower than in calculations No. 1, 2, and 3) at slightly reduced radiation intensity of IL = 0.11 × 1014 W/cm2, the sound velocity Vs amounts to about 160 km/s. Finally, under the conditions of calculations No. 7, 8, and 9 for the ablators with a density of 9 mg/cm3 (2.7 times lower than in calculations No. 1, 2, and 3) and almost the same radiation intensity of IL = 0.15 × 1014 W/cm2, the sound velocity Vs reaches a maximum of about 200 km/s.

In the calculations for models No. 1, 2, and 3 with minimum Vs values, the rate of shock-wave damping in aluminum is also at minimum: mean velocities in the Al base layer and step are 29–30 and 23–28 km/s, respectively. Thus, the shock-wave velocity decrease in the step is about 5 km/s, i.e., 15% of the value in Al base layer. In this case, insignificant damping effect is confirmed by the following estimation. The time of ablator unloading can be evaluated as Δf/Vs ≈ 2.3 ns, during which the shock wave with average velocity 30 km/s would travel a distance of about 70 μm, so that strong damping could be expected for Al layer thicknesses much greater than 40 μm. In calculations No. 4, 5, and 6, the increase in Vs is accompanied by growth in the degree of shock-wave damping such that the wave velocities in the Al base layer and step are reduced to 22–28 and 16–18 km/s, respectively, and velocity decrease in the step (16–11 km/s) reaches about 30% of the value in Al base layer. The time of ablator unloading in these calculations is about 1.5 ns, during which the shock wave with average velocity 21 km/s would travel a distance of about 30 μm. Finally, calculations No. 7, 8, and 9 with maximum Vs value correspond to the maximum relative decrease in the shock-wave velocity: the average wave velocities in the Al base layer and step are 25 and 17 km/s, respectively, and velocity decrease in the step is about 30% of the value in Al base layer. The time of ablator unloading in these calculations is about 1.2 ns, during which the shock wave with average velocity 18 km/s would travel a distance of about 20 μm. Thus, in calculations for the ablator density about 10 mg/cm3, the characteristic damping depths of 20–30 μm turn out to be smaller than the Al layer thickness of 40 μm, which is evidence of a rather strong damping of the shock wave.

Detailed temporal dynamics of the laser-induced shock-wave propagation is illustrated in Figs. 4, 5, and 6 showing pressure profiles calculated at various moments of time using models No. 1, 2, and 3 (Table 2) for targets with different ablators of same average density (24 mg/cm3):

Fig. 4.
figure 4

Pressure profiles obtained in calculation No. 1 (Table 2) for a model target containing 50 μm thick Al layer spaced by 87 μm thick vacuum gap from 275 μm thick porous layer with average density 24 mg/cm3 irradiated by laser pulse of 0.16 × 1014 W/cm2 intensity, as calculated at various moments of time t (ns): 3 (1); 3.5 (2); 4.2 (3); 4.4 (4); 4.75 (5); 5.45 (6).

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Fig. 5.
figure 5

Pressure profiles obtained in calculation No. 2 (Table 2) for a model target containing 50 μm thick Al layer and 275 μm thick porous ablator with average density 24 mg/cm3 (without gap) irradiated by laser pulse of 0.16 × 1014 W/cm2 intensity, as calculated at various moments of time t (ns): 3 (1); 3.5 (2); 4.1 (3); 5.0 (4).

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Fig. 6.
figure 6

Pressure profiles obtained in calculation No. 3 (Table 2) for a model target containing 50 μm thick Al layer and 275 μm thick homogeneous ablator with average density 24 mg/cm3 (without gap) irradiated by laser pulse of 0.16 × 1014 W/cm2 intensity, as calculated at various moments of time t (ns): 2.5 (1); 2.8 (2); 3.0 (3); 3.1 (4); 3.6 (5); 4.4 (6).

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– target with porous ablator and vacuum gap (model No. 1);

– target with porous ablator without vacuum gap (model No. 2);

– target with equivalent homogeneous ablator without gap (model No. 3).

In all calculations, the shock wave was generated by identical laser pulses with the same intensity (0.16 × 1014 W/cm2). In calculation No. 3 for the model target containing homogeneous ablator without gap, the shock wave passes via 275 μm thick ablator to reach the boundary of aluminum within tabl ≈ 2.8 ns (Fig. 6), so that the average wave velocity in the homogeneous ablator is Dabl ≈ 1.0 × 107 cm/s. At the moment of maximum laser pulse intensity (t = 2.2 ns), the shock-wave velocity amounts to 1.2 × 107 cm/s. Note that, for the ablation pressure of 3.5 Mbar, estimation of the shock-wave velocity by formula

$$D \approx {{[(\gamma + 1){{P}_{{{\text{abl}}}}}{\text{/}}2{{\rho }_{{{\text{abl}}}}}]}^{{1/2}}}$$

yields a close value of Dabl ≈ 1.4 × 107 cm/s.

In the porous ablator, the shock wave propagates (Figs. 4 and 5) much slower than in the equivalent homogeneous ablator due to the process of plasma homogenization. In the target without vacuum gap, the shock wave travels via ablator of the same thickness and reaches the boundary of aluminum for tabl ≈ 3.5 ns, so that the average velocity is about 7 × 106 cm/s (Fig. 5). The time of plasma homogenization can be estimated using the following equation [34]:

$$\begin{gathered} {{\tau }_{{\text{h}}}}\,[{\text{s}}] \approx \frac{{{{\delta }_{0}}}}{{{{V}_{{\text{i}}}}}}\left[ {1 + \frac{{{{\delta }_{0}}}}{{{{\lambda }_{{{\text{ii}}}}}}}} \right] \approx 3.3 \times {{10}^{{ - 12}}}\frac{{{{A}^{{1/2}}}{{\delta }_{0}}}}{{{{{(Z + 1)}}^{{1/2}}}{{T}^{{1/2}}}}} \\ \times \left[ {1 + 7.3\frac{{{{Z}^{4}}{{{(Z + 1)}}^{{1/2}}}{{\delta }_{0}}\rho }}{{A{{T}^{2}}}}} \right], \\ \end{gathered} $$
(3)

where δ0 is the average pore size in microns, Vi is the thermal ion velocity, λii is the ion–ion collision length, T [keV] is the plasma temperature, and ρ [g/cm3] is the average density of porous ablator.

In Eq. (3), the first term describes the time of primary pore filling and the second term refers to the density leveling by means of ion–ion collisions. For δ0 = 0.5 μm, ρ = 0.024 g/cm3, T = 1 keV, and Z = 1, estimation of the plasma homogenization time yields about 1.5 ns. A similarity relationship for the velocity of shock-wave propagation in partly homogenized plasma of a porous substance is given by the following expression [25]:

$${{D}_{{\text{h}}}}\left[ {\frac{{{\text{cm}}}}{{\text{s}}}} \right] \approx 9.7 \times {{10}^{4}}\frac{{{{A}^{{2/3}}}I_{{{\text{L}}(14)}}^{{1/3}}}}{{{{Z}^{{2/3}}}\lambda _{\mu }^{{4/3}}{{\rho }^{\alpha }}\rho _{s}^{{1 - \alpha }}}}.$$
(4)

For the fully ionized plasma in laser-irradiated porous target with parameters IL = 0.16 × 1014 W/cm2, λ = 0.53 μm, ρ = 24 mg/cm3, ρs = 1.1 g/cm3, and α = 0.8, estimation of the shock-wave velocity by formula (4) yields a Dh value about 5 × 106 cm/s. This estimation evidences, approximately to the same extent as the numerical calculations, much slower propagation of the laser-induced shock wave in a porous ablator as compared to the homogeneous one. In the case of a target with vacuum gap (Fig. 4), the formation of pressure on the Al layer surface equal to the laser-wave pressure in the ablator is due to plasma percolation via the vacuum gap, acceleration behind the shock-wave front, and retardation in the Al layer for a time of tg ≈ 0.5 ns. For this reason, a powerful shock wave in Al layer in the target with vacuum gap is formed at a time of tabl + tg ≈ 3.45 ns. Simple estimation of the time of plasma percolation via 87 μm thick gap for the ablator substance moving behind the shock-wave front at a velocity of 1.3 × 107 cm/s gives a value close to tg ≈ 0.6 ns.

In the targets without gaps (Figs. 5 and 6), the pressures at which shock waves enter from the porous and homogeneous ablators into aluminum have close values about 18 Mbar and then slightly decrease within the first 10 μm of shock-wave passage in aluminum. The decrease grows but not significantly (to 14 Mbar) over the next 10 μm and then drop almost by half (to 7 Mbar) within the next 20 μm to attain a quasi-stationary regime of propagation at a pressure close to that in the ablator. In the target with porous ablator and vacuum gap (Fig. 4), the pressure at which the shock wave enters aluminum is about 26 Mbar, does not decrease in the first 10 μm of propagation, and then slightly decreases (to 15–17 Mbar) within the next 20-μm path. The pressure level of about 10 Mbar is retained even upon traveling over 40 μm. The rate of shock wave damping is rather correctly described by estimations presented above during the analysis of data summarized in Table 2.

Figure 7 shows calculated temporal profiles of hydrodynamic efficiency η of laser energy transfer to the shock wave, defined as the ratio of the kinetic energy of aluminum with density above initial to the laser energy absorbed in a target by the given moment of time. Maximum η values reach 1.28% at t = 3.3 ns for the target containing porous ablator with vacuum gap, 1.42% at t = 3.8 ns for the target containing same porous ablator without gap, and 1.82% at t = 4.5 ns for the target with homogeneous ablator of same equivalent density without gap. The higher hydrodynamic efficiency observed for the target with homogeneous ablator is due to a lower energy absorbed in this ablator because the shock wave propagates without delay for the homogenization of plasma.

Fig. 7.
figure 7

Temporal variation of hydrodynamic efficiency η (ratio of the η kinetic energy of aluminum with density above initial to the absorbed laser energy) in targets with (1) 24 g/cm3 average density porous ablator with vacuum gap, (2) same porous ablator without gap, and (3) homogeneous ablator of same equivalent density without gap.

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An important factor is the higher hydrodynamic efficiency observed for the target containing porous ablator with vacuum gap in comparison to the target with same porous ablator without the gap. Taking into account that greater laser energy is absorbed in the target with gap by the moment of energy transfer to the Al layer, this result is indicative of a greater energy transferred to Al layer during shock-wave impact via the vacuum gap.

At later stages of the shock wave propagation in aluminum, the hydrodynamic efficiency of both targets attains the “ablation” level of about 1%. The maximum values of η = 1.8–2.1% are more than twice as large as those for the usual laser ablation regime of shock wave generation, whereby a laser pulse acts immediately on the surface of solid (e.g., aluminum) target [38]. It should be noted that the hydrodynamic efficiency of laser energy transfer in the ablation regime to shock-wave energy in a solid target is at least ten times lower than the efficiency of laser ablation acceleration of a thin planar layer of substance as a whole, which can reach (at the optimum dose of ablated mass about 80%) up to 38% [3, 36, 39].

Comparative scaling analysis of reported experimental data [26, 28] showed that experiments performed in the present work under equivalent conditions in respect of the laser pulse intensity and foam characteristics gave much higher values of the shock-wave velocity in aluminum. Indeed, shock waves generated by laser pulses of IL ≈ 0.5 × 1014 W/cm2 intensity and 600 ps duration in earlier experiments with foam–aluminum targets at absorber densities of 10 and 20 mg/cm3 [26, 28] had velocities of 22–24 and 22–26 km/s, respectively, at 13–18 μm depth in aluminum. Taking into account the scaling relation for ablation pressure, Pabl ∝ \(I_{{\text{L}}}^{{2/3}}\), and the laser pulse intensity of IL ≈ 0.15 × 1014 W/cm2 used in the present work, those values would correspond to 16–17 and 16–18.5 km/s, respectively. Thus, shock-wave velocities of about 25 and 29 km/s observed in aluminum at a greater depth of 20–40 μm in our experiments exceed the scaled values reported in [26, 28] by a factor of about 1.5.

There are two possible reasons for this difference. First, the much longer duration of laser pulses that ensures maximum ablation pressure achieved during the entire process of shock wave propagation in the porous ablator and aluminum layer. Second, the presence of vacuum gap between the ablator and aluminum, which ensures attaining the maximum efficiency of shock-wave energy transfer to aluminum as confirmed by the results of numerical calculations

4 CONCLUSIONS

In these experiments, nanosecond laser pulses of (1–1.5) × 1013 W/cm2 intensity acting upon planar targets comprising a layer of low-density porous ablator-absorber (10–25 mg/cm3) spaced by vacuum gap from a layer of aluminum generated shock waves with velocities 25–29 km/s in aluminum. Analysis of the experimental results with the aid of numerical calculations modeling the generation and propagation of shock waves in partly homogenized laser plasma of porous (foam) ablator showed that the main factor ensuring the attainment of high shock-wave velocities are (i) the presence of a vacuum gap that provides 1.2–1.5 fold amplification of the shock-wave pressure in aluminum and (ii) optimum duration (about 4 ns) of the laser pulse that ensures maximum ablation pressure in the porous absorber during the entire process of hock-wave propagation in the ablator and aluminum layers of target.

In view of the scaling relation for initiating ablation pressures, Pabl ∝ \(I_{L}^{{2/3}}\), a tenfold increase in laser pulse intensity and the passage to third harmonic can be expected to produce an eight-fold growth in shock-wave pressure (up to 200–250 Mbar) in the solid target and about 2.8-fold growth in shock-wave velocity (up to 8–9 × 106 cm/s), which is a quite serious result for EOS experiments with planar targets under standard conditions of pulsed laser irradiation.

Finally, it should be noted that the length of quasi-stationary shock-wave propagation in aluminum (about 10 μm) corresponds to a distance of about 30 μm in plastics (a typical sold-plastic ablator thickness in direct-ignition ICF targets [3, 40, 41]. This implies that the use of composite ablators comprising sold plastic and porous substance layers can lead not only to increase in the efficiency of laser radiation absorption and leveling of ICF target irradiation inhomogeneity, but also to increase in hydrodynamic efficiency of the absorbed laser energy transfer to the target.