Effect of equation of state on laser imprinting by comparing diamond and polystyrene foils

The Rayleigh–Taylor instability (RTI) is a hydrodynamic instability caused by gravity and is generally understood as a basic phenomenon in nature.^1,2 The RTI occurs at the interface between accelerating materials with different densities. In inertial confinement fusion (ICF) targets, the RTI and related hydrodynamic mixing are the most crucial factors determining thermonuclear ignition^3,4 during the implosion timeline. Numerous studies of the RTI and related hydrodynamic instabilities have been done based on this understanding.⁵ Because the RTI makes small perturbations at the target surface grow exponentially over time, initial perturbations at the ICF target surface should be as small as possible. Such surface perturbations are mainly due to two factors: (i) the surface roughness resulting from target fabrication and (ii) imprinting due to nonuniform laser irradiation. Therefore, numerous efforts have been made to understand the physics of laser imprinting.^6–8 

In an ICF implosion, a target is irradiated with intense laser light, forming a corona plasma that very rapidly expands over the target surface. At the target-corona interface, plasma expansion exerts pressure on the target material. Because the ablation pressure is a function of laser intensity, laser imprinting is caused primarily by pressure perturbations resulting from nonuniform laser irradiation. Such pressure perturbations introduce deformations on the target surface, followed by rippled shock propagation into the target.⁹ The pressure perturbations are reduced within the coronal plasma via thermal smoothing.¹² Figure 1 shows a schematic illustration of such laser imprinting. The imprint amplitude (i.e., the perturbation of the ablation front) $δxa$ is obtained from an incompressible model^8,10,11

where $δPa$ is the pressure perturbation as given by the cloudy-day model¹² 

where ρ_s0 is the shock-compressed density (the subscript “0” denotes unperturbed quantities), $vs0$ is the velocity of the shock front, $va0$ is the fluid velocity behind the shock, k is the wavenumber of the perturbation, $δI/I0$ is the intensity modulation of the irradiation laser, and D_sb is the effective separation between the cutoff-density point and the ablation front (called the “standoff distance”).

As can be seen in Fig. 1 or from Eqs. (1) and (2), laser imprinting is a function of several parameters. Among these, the standoff distance D_sb is very important because the imprint amplitude decreases exponentially with increasing D_sb. Many previous investigations have striven to mitigate laser imprinting by increasing the standoff distance D_sb by using low-density foam ablators,^13–16 radiation with high-Z coatings,^17,18 “picket” pulse irradiation,¹⁹ or soft-x-ray pre-irradiation.²⁰ Although these methods do mitigate laser imprinting, they also increase fuel preheating and isentropicity. Moreover, at very early irradiation times, mitigating laser imprinting by increasing the standoff distance is not effective for long-wavelength perturbations nor for small standoff distance (⁠$kDsb≪1$⁠).

Some parameters of laser imprinting are particularly important during very early irradiation times $[$i.e., when $exp−kDsbt≈1$]. Equation (1) also infers that density and compressibility, which are related to the equation of state (EOS), are functions of laser-imprinting amplitude. Based on conservation of mass across a shock wave in the form $ρS0/ρ0=vs0/vs0−va0$⁠, low compressibility implies a large difference $vs0−va0$⁠, that is, a stiff material with low compressibility (=$ρs0$/$ρ0$⁠) may reduce laser imprinting. In addition, the imprint amplitude is expected to decrease as the ablator density $ρs0$ increases. The imprint amplitude δx_a can be approximated as

Thus, laser imprinting on a material is seeded by nonuniform motion of mass due to pressure perturbations (⁠$∝δI/I0·va0$⁠), which is related to material density and stiffness, as shown above. The EOS of a solid or a liquid can be represented as a sum of three components, which describe the elastic properties of the cold body, the thermal motion of the atoms (nuclei), and the thermal excitation of the electrons. Material stiffness or compressibility is related to the Hugoniot curve for a condensed substance and to the elastic pressure curves of a cold body, which are included in the EOS for a material. In particular, at very early times, material properties including the density and compressibility play an important role in laser imprinting for long-wavelength perturbations and small standoff distance (⁠$kDsb≪1$⁠). Also, material strength is expected to be essential in low-pressure conditions because compressibility in the solid state below the elastic limit is generally very small. The compressibility implicitly includes material strength, that is, Young's ratio and Poisson's ratio. Although mitigating laser imprinting by increasing the standoff distance at the very early irradiation times is not effective, thermal smoothing may be affected by material differences because the velocity of the critical-density surface is basically related to the atomic weight and atomic number.²¹ As shown below, however, the standoff distances for diamond and polystyrene (PS) are almost the same for a given laser pulse.

Heretofore, no studies have focused on how the density and compressibility of the target material affect laser imprinting. Diamond is the most probable candidate for a stiff ablator material for ICF targets because it has a small atomic number and a large elastic limit. Sophisticated studies of diamond under ultrahigh pressure have recently been conducted by using high-intensity lasers^22–24 and Z-pinch machines²⁵ and have demonstrated that diamond has both a large elastic-plastic transition pressure²⁴ and a high melting temperature,²³ thus indicating that diamond maintains low-compressibility under high pressure and high temperature.

In this paper, we report detailed results of simulations showing how material density and compressibility affect laser imprinting. The main goal of this paper is to verify the theory and the two-dimensional calculation of laser imprinting for materials of various densities and compressibilities by comparison with the experiment. Section II describes in detail the experimental conditions of measurements of areal-density perturbations caused by nonuniform irradiation. We used face-on x-ray backlighting, which is a standard technique for measuring hydrodynamic instabilities. Section III presents and analyzes the experimental data on areal-density perturbations and target trajectories. Section IV analyzes laser imprinting by using the two-dimensional hydrodynamic simulation code PINOCO-2D. We analyze the spatial perturbation amplitude due to laser imprinting and the areal-density perturbation amplitude based on comparisons with the experimental results. We also compare the basic hydrodynamic parameters determined from the laser-imprinting amplitude. All materials used in the simulation, experiment, and the theoretical model lead to the same result that density and compressibility are the key factors of laser imprinting.

The experiment was conducted at the GEKKO-XII Nd:glass laser facility at the Institute of Laser Engineering, Osaka University.²⁶ An overview of the experimental setup is shown in Fig. 2. Flat foils were irradiated with the second harmonic (λ = 0.527 μm) of a Nd:glass laser. The pulse shape was Gaussian with a full width at half maximum of 1.3 ns, using one beam for the foot pulse and two stacked beams for the main drive pulse with time delays between the pulses. A typical pulse shape is shown in Fig. 2. Each beam was smoothed by using a random-phase plate.²⁷ Because the perturbation by laser imprinting is typically very small for direct observation, perturbations of the target material were measured via amplification through the growth of the RTI caused by the drive beams.

The intensity of the foot pulse was modulated by placing a grid mask in front of the focusing lens. Figure 3(a) shows the spatial pattern and intensity distribution of the foot pulse at the target. The nonuniformity of the higher spatial-harmonic components (wavelength = 20∼50 μm) in the imprint pulse was less than 10% of the nonuniformity of the fundamental wavelength [see Fig. 3(b)]. The modulation wavelength at the target surface was 100 μm, with ∼10% intensity nonuniformity (⁠$δI/I0$⁠). The two foot-pulse intensities were chosen mainly to investigate the imprint perturbation in the elastic and plastic states of diamond. The averaged intensities I₀ of the foot pulse were ∼4 × 10¹² W/cm² (“low-foot” condition) and ∼5 × 10¹³ W/cm² (“high-foot” condition). The high-foot pulse rapidly produces a shock stress via the plastic response (>160 GPa) for diamond.²⁴ The peak intensity of the main pulse was ∼1 × 10¹⁴ W/cm².

The targets were 13-μm-thick single-crystal diamond foils (type-Ib, initial density = 3.51 g/cm³). The surface orientation of the single-crystal diamond was (100). PS foils (initial density = 1.06 g/cm³) with a thickness of 25 μm were used as a reference material because several previous RTI experimental results are available for PS^28–30 along with precise EOS data³¹ and experiment data on laser imprinting of PS.^8,10,11 The foil surfaces were coated with 0.05 μm of Al as a shield against shine-through inside the foils during very early irradiation times.

The areal-density-perturbation growth was measured by using the face-on x-ray-backlighting technique. A backlight target (Zn) was irradiated to generate ∼1.5 keV quasimonochromatic x-rays coupled with a 6-μm-thick Al filter. The temporal evolution of the transmitted x-rays from the Zn backlighter through a diamond or PS foil was imaged by a slit (10 × 50 μm²) onto the CuI photocathode of an x-ray streak camera with 26× magnification. The temporal resolution of the x-ray streak camera was ∼140 ps. The spatial resolution was measured by using a backlit gold grid image to analyze the areal-density perturbation. The resolution function of the entire diagnostics system is given by the sum of two Gaussian functions as $Rx=1/1+αexp−x2/2σ12+αexp−x2/2σ22$⁠, where α = 0.194, σ₁ = 5.753 μm, and σ₂ = 20.381 μm for this measurement.

As shown in Fig. 4(a), the trajectories of the irradiated foils were measured separately by using side-on x-ray backlighting so as to evaluate their basic hydrodynamics based on comparison with the results of the simulations described in Sec. III. The drive-laser conditions were the same as for the areal-density-perturbation measurements. A Cu backlit target was placed in the direction perpendicular to the target-acceleration direction. The temporal evolution of the transmitted x-rays from the Cu backlighter through accelerated flat foil (diamond or PS foil) was imaged by a slit (20 × 50 μm²) onto the CuI photocathode of an x-ray streak camera with a 20× magnification. The temporal resolution of the x-ray streak camera was ∼100 ps.

The target consists of the accelerated foil target, a 50-μm-thick Be substrate, and a 50-μm-thick Al spacer, as shown in Fig. 4(a). Figure 4(b) shows an example of a raw streaked side-on image of the target acceleration trajectory. The time origin (t = 0) is when the rise of the drive pulse reaches 50% of the maximum peak intensity. In Fig. 4(b), the Be substrate and the accelerated diamond target are indicated in the backlight image. The raw image is line scanned at various observation times and integrated over a time approximately equal to the temporal resolution [see example in Fig. 4(c)]. From the x-ray spatial profile provided by raw data, we determine the center of mass of the accelerated foil target.

The experimental trajectory is compared with the results of the ILESTA-1D simulation code³² and of the two-dimensional radiation hydrodynamic code PINOCO-2D.³³ As seen in Fig. 4(d), both simulation results are consistent with the experiment result. In this paper, we use the ILESTA-1D code to evaluate the basic hydrodynamic parameters for “unperturbed” conditions (e.g., shock breakout time, onset of foil acceleration, and standoff distance). To analyze the laser imprinting under “perturbed” conditions, we use the PINOCO-2D simulation code.

Figure 5 shows raw streaked face-on backlit images for the diamond and PS targets the high-foot and low-foot irradiation conditions. The time origin (t = 0) of the main pulse is as described above. All experimental results were acquired during the foil-acceleration phase.

“Lineouts” are extracted by integrating the raw data over the temporal resolution, as indicated in Fig. 5 for t = 1.7 ns. The lineouts shown are the backlit x-ray intensity distribution and the fitted profile. The areal-density perturbations are obtained by fitting the convolution of the resolution and a sinusoidal perturbation function to the raw lineouts, taking into account the x-ray absorption coefficient (at hν = 1.53 keV), which was separately calibrated by using “cold” foils.

In the fitting procedure, second- and third-harmonic spatial components (wavelengths of 50 and 33 μm, respectively) of the areal-density perturbation due to the irradiation nonuniformity are also taken into account. The effect of including the higher harmonic components falls within the fitting error of the areal-density perturbation plots.

Figure 6(a) shows the temporal evolution of the areal-density perturbations amplified by the RTI for diamond and PS foils. Also plotted in Fig. 6(a) are the results of simulations of the three experimental configurations by the two-dimensional radiation hydrodynamic code PINOCO-2D.³³ PINOCO-2D gives the arbitrary Lagrangian Eulerian (ALE) hydrodynamic for the radiation. This code includes hydrodynamic, flux-limited Spitzer–Härm thermal conduction,³⁴ nonlocal thermal equilibrium multigroup radiation transport, quotidian equation of state, and ray-trace laser-energy deposition. For the EOS, we incorporated an elastic-plastic boundary,³⁵ a multiphase EOS,³⁶ and a table of melting curves³⁷ for diamond with the quotidian equation of state model. The results of PINOCO-2D are consistent with the experiment results for the three configurations.

Figure 6(b) shows the averaged target densities during the foil-acceleration phase, as calculated by the ILESTA-1D code. During the acceleration phase, the areal-density perturbation δρl detected by using face-on backlighting is expressed [see Fig. 7(b)] as $δρl=δ∫ρdx=∫xr+axrxa+axaρx,ypdx−∫xrxaρx,yudx$⁠, where the x axis is perpendicular to the target surface, x_a is the position of the ablation front, x_r is the position of the rear surface, ρ(x,y) is density distribution in the target, y_p and y_u are the perturbed and unperturbed y coordinates of the transverse direction, respectively, and a(x) is the spatial perturbation amplitude for the target depth along the x axis, including the effect of feedthrough,³⁸ and is written as $ax=aaexp−kxa−x$⁠, where $aa$ is the perturbation amplitude at the ablation front. In our calculation of areal-density perturbation, the density distribution ρ(x,y) is considered as mentioned above. Also, the areal-density perturbation δρl for uniform target density ρ is approximated as $δρl≈ρaxa−axr$⁠. Thus, the areal-density perturbations δρl increase upon increasing not only the spatial amplitude a(x) but also the target density ρ. The areal-density perturbation for PS and diamond for the high-foot pulse is at the same level as in Fig. 6(a). However, the spatial-perturbation amplitude of diamond should be smaller because the density of diamond is greater than that of PS during the observation time [as in Fig. 6(b)]. Here, the time variation of the target density is due to repeated shock propagation by laser pulses and expansion of the rear surface after shock breakout. Figure 6(c) shows the temporal evolution of the spatial perturbation at the ablation front as determined by PINOCO-2D for the three experimental configurations and for the low-foot PS as a reference. Both high-foot calculations and the low-foot calculation indicate that the spatial perturbation of diamond is less than that of PS over the entire period of observation.

Figure 8 shows density-contour plots in the perturbation-wavelength range for diamond and polystyrene at 1.2 ns obtained by using PINOCO-2D. The front and rear ablation surfaces are defined to be at 1/e of the peak density. These positions describe spatial perturbations at the front and the rear ablation surfaces. The areal-density perturbations of the two-dimensional (2D) simulation are obtained from $δρl=δ∫ρdx=∫xrypxaypρx,ypdx−∫xryuxayuρx,yudx$⁠.

The basic hydrodynamic parameters (shock-breakout time, onset time of foil acceleration, and foil acceleration) must be confirmed to estimate the perturbation amplitude. The target acceleration for the high-foot diamond is slightly less than that for the PS because diamond foil is more massive. According to the 2D simulation, the accelerations of the PS and diamond due to the high-foot pulse are ∼3.9 × 10¹⁵ cm/s² and ∼2.4 × 10¹⁵ cm/s², respectively. The growth rate of the RTI of the spatial perturbation differs slightly between the three experimental configurations, as shown in Fig. 6(c). In the present work, we confirm by using side-on radiography that the experiment data for the acceleration trajectory are reproduced by the one-dimensional (1D) and 2D simulations (as in Fig. 4). The basic parameters (shock-breakout time and onset time of foil acceleration) are evaluated by simulations (Table I). In addition, the experimental areal-density perturbation may be compared with 2D simulations [Fig. 6(a)]. By understanding the basic parameters, we can discuss the perturbations due to the irradiation nonuniformity between diamond and PS (see Sec. IV B). Note that the spatial amplitude for diamond is less than that for PS just before the growth of the RTI. Conversely, the growth of the areal-density perturbation differs from that of the spatial perturbation. Note that the areal-density perturbation is not simply the target density multiplied by the spatial perturbation at the front surface because two-dimensional effects influence the spatial distribution of the density inside the foil.

Note also the difference in basic hydrodynamics between the data for the low- and high-foot diamond experiments. For the low-foot diamond data, the areal-density perturbation is smaller than for the high-foot diamond data. This is partly because the duration over which the foil is accelerated is shorter for high-foot irradiation conditions. To correlate the basic hydrodynamics, the timing of the shock breakout and the onset of foil acceleration in the ILESTA-1D simulation for the three experimental configurations are shown in Table I. Also, Fig. 6(a) shows the areal-density perturbation plots from the simulation before and after the foil acceleration (see solid and dotted curves).

On the other hand, the RTI growth rate obtained from PINOCO-2D with classical Spitzer–Härm thermal conduction is slightly greater than the experimentally obtained growth rates. This difference in growth rates is attributed to target heating by nonlocal electron thermal-energy transport, which reduces the RTI growth^30,39 because PINOCO-2D does not consider nonlocal electron thermal-energy transport in the Fokker–Planck equation. In our experiment, however, the perturbation wavelength is relatively high so that reduced RTI growth due to nonlocal electron thermal-energy transport would not be significant. Therefore, the experimental areal density almost agrees with the PINOCO-2D calculation even in the foil-acceleration phase.

In our experiments, we measured the areal-density perturbation of the accelerated foils by amplifying the imprint perturbation due to the RTI with face-on x-ray backlighting because the imprint amplitude is generally too small to observe with conventional experimental techniques. Because we used two materials of different densities and masses, the areal-density perturbations at a given observation time should not be compared even for the same pulse shape because of the difference in a few important parameters, namely, acceleration, in-flight density, and foil thickness. Therefore, we discuss the spatial amplitude based on the results of the PINOCO-2D simulation after we verify that the experimental results for the areal-density perturbation are consistent with the results of PINOCO-2D.

The 2D density contour plots obtained from PINOCO-2D for both targets (high-foot PS and diamond) are shown in Fig. 9. The density plots show clearly that the imprint amplitude on diamond is less than that on PS. Also, the PS target is more compressed. Figure 10 shows the calculated spatial amplitude on the laser-irradiated surface as a function of time obtained from PINOCO-2D simulations. The calculations were carried out up to the shock-breakout time for each experimental configuration. The best comparison in this experiment is high foot PS and high foot diamond. The shock-compressed density ρ_s0 for high-foot diamond up to the shock-breakout time is 1.5 to 2.0 times larger than that for high-foot PS. For a denser material, the imprinted surface perturbation is smaller, as shown by Eq. (1). Furthermore, the average difference between shock velocity and fluid velocity behind the shock front (i.e., $vs0−va0$⁠) for diamond is about 1.8 times greater than that for high-foot PS.

On the other hand, the areal-density perturbation prior to the shock-breakout time (t < −0.5 ns) is slightly greater for high-foot diamond, as shown in Fig. 6(a). As illustrated in Fig. 7(a), the areal-density perturbation δρl prior to shock breakout may be approximated by $δρl=∫xaxrρx∂ax/∂xdx≈ρsδxa−δxs+ρ0δxs$⁠, where ρ_s is the compressed target density, δx_a is the spatial amplitude of the ablation front, δx_r is the shock-front amplitude, and ρ₀ is the initial density. The target density and the amplitude of both the ablation front and the shock front should be considered to calculate the area-density perturbation. For example, at t = −1.1 ns, the compressed-target density ρ_s and initial target density ρ₀ for diamond are ∼4.41 and ∼3.51 g/cm³, respectively. On the other hand, the compressed density ρ_s and initial target density ρ₀ for PS are ∼2.90 and 1.06 g/cm³, respectively. The calculated spatial amplitude δx_a on the ablation front for diamond and PS is 0.048 and 0.126 μm, respectively.

Because the shock-propagation distance is much smaller than the perturbation wavelength, the shock-front amplitude δx_s may be approximated by δx_a. Thus, the areal-density perturbation prior to shock breakout depends strongly not only on the spatial amplitude δx_a (∼δx_s) but also on the initial density ρ₀. The simulations show that the spatial amplitude at the ablation surface for high-foot PS is about two to three times greater than that for high-foot diamond. Conversely, the initial density of diamond is ∼3.5 times greater than that of PS. Therefore, the parameters are all consistent in explaining why the areal-density perturbation for high-foot diamond is slightly greater than that for high-foot PS.

As shown in Sec. I, the standoff distance is also a key parameter for determining the imprint amplitude, which should also be considered. Figures 11(a) and 11(b) show the temporal evolution of the target compressibility in the shock-compressed region and the standoff distance obtained from ILESTA-1D, respectively, for the three experimental configurations. The compressibility is ρ_s0/ρ₀, where ρ_s0 is the averaged density of the shock-compressed region and ρ₀ is the initial density. As shown in Fig. 11(a), the compressibility for PS is ∼3.4 at the shock-breakout time, whereas the compressibility for high-foot diamond is ∼1.5. For high-foot conditions, the shock pressure is about 530 GPa, at which point the diamond remains solid²⁴ but is beyond the elastic-plastic transition pressure. As shown in Fig. 11(b), the standoff distances for high-foot PS and diamond are very similar, which means that the imprint perturbation for the high-foot condition is definitively mitigated by the differences in density and compressibility, which are strongly related to the EOS between the two target materials. In our dataset, we have no low-foot PS experiment data to compare with the low-foot diamond data. The low-foot PS simulated by PINOCO-2D indicates that exactly the same correlation exists between low-foot PS and low-foot diamond as in the high-foot dataset.

From the point of view of practical ICF target design, the high-density ablator should be thinner to maintain the acceleration. Therefore, the spatial imprint amplitude should be compared between the same-mass conditions at the shock breakout times. In our case, because 13-μm-thick diamond foil is equivalent to 43-μm-thick PS foil, the spatial amplitude should be compared with these two conditions at their shock-breakout times. As seen in Fig. 9, the spatial amplitude of both targets (high-foot diamond and high-foot PS) starts to saturate prior to shock breakout because of the standoff distance. From PINOCO-2D, the spatial imprint amplitude for high-foot PS (43 μm) at shock breakout is also approximately 30% of that for high-foot diamond (13 μm) at its shock breakout time (results not shown here).

For low-foot diamond, the standoff distance is much less than that for high-foot diamond, as shown in Fig. 11(b). However, the imprint amplitude for low-foot diamond is the lowest at very early times. The shocked pressure for low-foot diamond is about 110 GPa, which means that the diamond pressure remains below the elastic-plastic transition pressure. As shown in Fig. 11(a), the compressibility for low-foot diamond is very small because of its material strength, whereas the pressure on high-foot diamond exceeds the elastic-plastic transition pressure. This implies that the reduction is effective because of the lower compressibility below the elastic-plastic transition pressure even though the standoff distance is smaller for low-foot diamond [see Fig. 11(b)]. The imprint amplitude for low-foot diamond overtakes that for high-foot diamond at about −1.0 ns because the standoff distance increases more rapidly with time for the high-foot conditions. Note that the 2D simulation result is consistent for low- and high-foot diamond although the code does not include a precise model of material strength below the elastic limit. This is attributed to the pressure perturbation (imprint amplitude) being so small that effects such as shear stress are negligible. Thus, the analysis of imprinting with one-dimensional compressibility is valid also for the solid phase.

Previous work on high-density carbon (HDC) reports that solid or partially melted conditions might provoke distortions of the shock front due to anisotropy in the sound velocity in crystals.^40,41 Thus, microstructures would be generated below the fully melted condition, which constitute a possible seed for hydrodynamic instability. In the first shock of over 6 Mbar, the effect of microstructures is at an acceptable level for HDC.⁴⁰ However, a strong shock precludes a low-fuel adiabat and high compression during ICF. Although the anisotropy of HDC is indeed a concern for ICF, recent experiments at the National Ignition Facility have suggested that this can be mitigated by using the laser pulse profile to allow low adiabats and high compression.⁴² In this study, there is little influence of the microstructures due to grains and the anisotropic sound velocity in the crystal because the target is single-crystal diamond. Actually, the results of the two-dimensional hydrodynamic simulation are consistent with the experimental results. Therefore, the density and compressibility, that is, the material EOS, are prominent factors for laser imprinting in many differences between diamond and PS.

The diamond ablator may reduce laser imprinting, but the effect of feed-through to the inner shell surface might be large in use of thinner diamond capsules. The implosion performance of diamond capsules in direct-drive ICF experiments should therefore be discussed in future works.

In conclusion, we show herein how density and compressibility affect laser imprinting by using diamond as a candidate stiff-ablator material for ICF targets. The effect of density and compressibility is verified both by using 2D hydrodynamic simulations and by experiments. Laser imprinting on high-foot PS is compared with the same on high-foot diamond, and the same is done for high- versus low-foot diamond. For high-foot conditions (both diamond and PS), the difference in imprinting amplitude is mainly due to the difference in density and compressibility. Conversely, the difference in imprinting amplitude for low-foot diamond is influenced by material strength (elastic-plastic transition) and by the standoff distance. The 2D simulation well reproduces the measurements of areal-density perturbation, which reveals the effect of density and compressibility on laser imprinting. The advantage of low-compressibility materials in ICF target design is that such a scheme may be combined with another suppression scheme by enhancing the standoff distance, as was proposed in previous works. For example, the mitigation method involving high-Z material coatings is easily coupled with the present scheme. By combining these schemes, both short- and long-wavelength laser imprinting may be mitigated. The physics of material stiffness affecting laser irradiation is also very important, not only for ICF-target design but also for general questions on laser-matter interactions for laser processing, laser peening, and other applications.

This work was performed under a joint research project of the Institute of Laser Engineering, Osaka University. This work was also performed with the support and under the auspices of the NIFS Collaboration Research program (NIFS10KUGK044). The authors would like to acknowledge the dedicated technical support of the staff at the GEKKO-XII facility for laser operation, target fabrication, and plasma diagnostics. This work was partly supported by the Japan Society for Promotion of Science, KAKENHI Grant No. 23340175.