Introduction

In direct-drive inertial confinement fusion (ICF)1,2, a fuel capsule is irradiated directly with laser light to achieve high-density compression. The capsule consists of a cryogenic layer of deuterium and tritium (DT) frozen onto the inner surface of a spherical shell of ablator material. Surface perturbation due to nonuniform irradiation occurs on the surface of the ablator material due to irradiation nonuniformity3,4. This spatial perturbation is amplified by Rayleigh-Taylor instability during the shell acceleration phase5,6, potentially disrupting the compressed shell and causing fuel mixing7. In the direct drive scheme, surface perturbation due to nonuniform irradiation is one of the most important issues, because imprinting a perturbation on the capsule surface degrades the symmetry of the compression. The level of surface perturbation due to nonuniform irradiation depends on the ignition condition parameters (neutron yield and target areal density) in ICF experiments8. Many previous investigations have striven to mitigate surface perturbation due to nonuniform irradiation by smoothing the effective laser irradiation nonuniformity or using low-density foam ablators9,10,11,12, high-Z coatings13,14,15 or high-Z dopants16. In our previous work, it was found that the use of stiffer and denser materials reduces the surface perturbation due to nonuniform irradiation17. Among the stiff materials, diamond is a top candidate as an ablator material for direct-drive ICF targets17. The advantages of stiff materials in ICF target design can easily combined with another suppression scheme, e.g., smoothing the effective laser irradiation nonuniformity or adding high-Z material coatings. Also, in x-ray indirect drive implosions, high-density carbon (HDC) is a leading candidate as an ablator material, because of the high implosion velocity and high stagnant pressure achievable due to its high density and optimal X-ray opacity18. Indirect-drive implosions with a HDC capsule are being conducted at the National Ignition Facility (NIF)18,19,20,21,22,23.

In the case of diamond, the shock physics is more complicated compared with conventional ICF capsule materials (e.g., plastic), due to its tightly bound crystalline structure in combination with an ultrahigh melting temperature, and the existence of the phase transition to BC824,25 at ultrahigh pressure. The existence of a solid or partially melted diamond ablator during the ICF implosion phase can provide microstructures that seed hydrodynamic instabilities18,26,27. Using a strong shock that completely melts the diamond or keeping within the coexistence regime is necessary in order to suppress distortions of the shock front due to anisotropy in the sound velocity in crystals18. The anisotropy of diamond is indeed a concern for the ICF application, but recent experiments at NIF have demonstrated that this can be mitigated by using an optimized laser pulse profile21. Although the diamond with its high stiffness is becoming a candidate ablator material, brittle materials such as diamond can easily cleave due to dynamic stress along certain crystallographic planes28,29. In the case of direct-drive inertial confinement fusion, in particular, nonuniform laser irradiation would lead to local fracture on the brittle material surface. However, there has been no previous work done on material strength issues due to nonuniform irradiation so far.

In this paper, we report on the effects of material strength against dynamic stress on the surface perturbation due to spatially nonuniform laser irradiation. An understanding of the effects of solid strength under laser irradiation is also important in the basic physical process of laser material processing and the early phases of inertial confinement fusion. We carried out measurements of the areal density of perturbations due to non-uniform laser irradiation with the face-on x-ray backlighting method, which is a standard technique in hydrodynamic instability experiments4,6. In this study, we present perturbation areal-density data and their analysis, as well as calculations of perturbation areal density and amplitude with the two-dimensional hydrodynamic simulation code PINOCO-2D30. We analyzed the effects of irradiation nonuniformity and high-Z coating on the areal density of perturbations in diamond foils. All the results from the experiments and simulations suggest that material strength factors affect the surface perturbation due to nonuniform laser irradiation. In the Methods section, details of the experimental method and specifications for measurements of the areal density of perturbations are described.

Results

Measurements of perturbation areal-density growth with face-on backlighting technique

The experiments were conducted using the GEKKO- XII Nd: glass laser facility at the Institute of Laser Engineering, Osaka University31. An overview of the experimental setup and typical stacked pulse shape are shown in Fig. 1. The diamond foils were irradiated with the second harmonic laser emission (wavelength: 0.527 μm). The stacked pulse with time delays between the beams consists of one beam for the foot pulse at an intensity of ~4.0 × 1012 W/cm2, and a subsequent pulse with two beams for the main drive pulse, at an intensity of ~1014 W/cm2 (see Fig. 1(b)). Figure 2(a) shows the spatial pattern of the foot pulse at the target surface. Foot pulses had irradiation nonuniformity (perturbation intensity/average intensity ~40%), in order to generate a surface perturbation due to nonuniform irradiation. The wavelength λ of the perturbation intensity variation was λ ~ 100 μm as shown in Fig. 2(b). Perturbations were observed on the target via amplification due to Rayleigh-Taylor instability (RTI) growth using the main drive beams, because the imprinted perturbations were typically too small for detection15.

Figure 1
figure 1

(a) Experimental setup for face-on x-ray backlighting measurement of areal-density perturbations seeded by nonuniform laser irradiation. (b) Typical laser pulse shape.

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Figure 2
figure 2

(a) Spatial pattern of foot pulse at target surface. (b) Modulation wavelength spectrum of irradiation non-uniformity.

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Examples of raw streaked backlit images of the diamond and diamond with thin Cu coating (0.1 μmt) foils for the foot pulse intensity ~4.0 × 1012 W/cm2 are shown in Fig. 3. The time origin (t = 0) was set as the time when the onset of the main drive pulse reached the half maximum, as shown in Fig. 1(b). Figure 3 also shows the lineouts for these targets. Time-integrated lineouts were obtained for the temporal resolution duration (~140 ps). The areal density of perturbations was obtained by fitting the convolutions of the resolution functions and sinusoidal perturbation functions to the raw lineouts, taking into account the x-ray absorption coefficient (at hν = 1.53 keV), which was calibrated with “cold” materials. This detail is described in the Methods section. From the lineouts for the diamond with irradiation non-uniformity ~10%, typical sinusoidal-like perturbations of the RTI growth can be seen in Fig. 3 (a). On the other hand, in Fig. 3 (b), the backlit image for the diamond with large irradiation non-uniformity of ~40% indicates a non-sinusoidal perturbation with “sharp” structure, which is different from the usual single-mode perturbation growth of the diamond observed for laser irradiation non-uniformity ~10%17. Please note that the structure on the diamond surface is referred to as “crack”-like because we observed sharp grooves from the shadowgraph measurements. When a high-Z coating was present on the diamond, no sharp structure was observed as in Fig. 3 (c); instead, the non-uniformity was close to the typical sinusoidal perturbations of the RTI growth.

Figure 3
figure 3

Raw streaked images of the backlit diamond (a) without Cu coating for irradiation nonuniformity ~10%, (b) without Cu coating for irradiation nonuniformity ~40%. (c) with Cu coating (0.1 μmt) for irradiation nonuniformity ~40%. All raw lineouts (black lines) are at about time 1.2 ns. Red lines are curve fits for each profile.

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Figure 4 presents plots of the temporal evolution of the areal density of perturbations for the diamond with laser irradiation ~40%, with the fundamental (λ: 100 μm) and second harmonic (λ: 50 μm) perturbation. In our previous work17, for diamond with irradiation nonuniformity ~10%, it was found that only the fundamental perturbation was observed, which was well-reproduced by the two-dimensional radiation hydrodynamic simulation code PINOCO-2D30. For the diamond with large irradiation non-uniformity of ~40%, however, the second harmonic component rises to the same level as the fundamental component at very early times, prior to onset of the foil acceleration: ~0.75 ns. Both fundamental and second harmonic components individually grow with their own growth rates. For the diamond with large irradiation nonuniformity of ~40%, ordinal imprinting generation and amplification are not observed, unlike previous studies of single spatial mode planar foil experiments4,6. Also shown are the results of calculation with the PINOCO-2D code, which do not reproduce the experimental result at all. The second harmonic component from the PINOCO-2D simulation is negligibly small compared with the fundamental component.

Figure 4
figure 4

Areal-density perturbation growth for diamond targets from experiments on fundamental and second harmonic components (symbols) and from the PINOCO-2D simulations (solid curves) for each experimental configuration: (red) diamond on non-uniformity ~40%, (blue) diamond with Cu coating on non-uniformity ~40%. Also shown are schematics of crack-like surface structure for the diamond foil without Cu coating.

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The temporal evolution of the areal density of perturbations for diamond with a Cu coating with irradiation non-uniformity of ~40% is also plotted in Fig. 4. The fundamental perturbation grew with time, which is in good agreement with the PINOCO-2D simulation result. The growth with time of the second harmonic perturbation is much smaller than that of the fundamental component after the foil acceleration time of ~0.5 ns (not shown here). Hence, a sinusoidal-like perturbation arises, as seen in Fig. 3(c).

Please note that the second harmonic components without Cu coating are antiphase from those with Cu coating. The second harmonic generation on the Cu-coated diamond is very typical Rayleigh-Taylor instability in the linear to non-linear growth regime, which is called bubble-spike generation. On the other hand, the perturbation shape on the diamond is not the bubble-spike shape but rather the “crack”-like structure, as illustrated in Fig. 4. This fact clearly shows that some hidden physics exists in the generation of imprinting perturbations on the diamond surface.

Discussion

From the experimental results, the 2D hydrodynamic simulation calculations do not reproduce the experimental results when the irradiation non-uniformity is large, as shown above. In order to interpret these facts, we consider the effects of materials strength on the generation of surface perturbation due to nonuniform laser irradiation. From previous shock compression experiments, the Hugoniot elastic limits (HEL) of diamond are measured to be 80.1 (±12.4), 80.7 (±5.8) and 60.4 GPa (±3.3) for the <100>, <110>, and <111> orientations, respectively32. The elastic yield strength of diamond inferred from these measurements is 75 (±20) GPa32. In our experiments, pressure perturbations due to the foot pulse produce non-uniform stress, namely tensile stress and shear stress, in the diamond foils. In the regime beyond HEL pressures, slip and fractures, which are not taken into account in the hydrodynamic simulation code, may occur primarily on crystal planes32,33. As a result, non-sinusoidal perturbation would be generated at the diamond surface. Figure 5 shows the density and pressure distribution calculated by the PINOCO-2D code near the ablation front at early irradiation times. When the irradiation nonuniformity is large (40%), a compressed area exceeding the HEL or (elastic yield strength) of diamond locally appears as shown in Fig. 5(a). On the other hand, when irradiation nonuniformity is small (10%), most of the compressed area is over the HEL (Fig. 5(b)). The experimental data clearly indicate that the spatial perturbation is generated on the ablation surface (compression region), then the areal-density perturbation grows due to the Rayleigh-Taylor instability. Thus, the presence of crack-like structure in this experiment is not explained by cavitation nor spallation that occur when the pressure is released. At the pressure over the HEL, crystal is collapsed and transformed to plastic state which is movable like a fluid. On the other hand, crystal at elastic state is difficult to move. When the irradiation nonuniformity is large (40%), there appears two regions: elastic and plastic states on the diamond surface. Therefore, the mechanical local fracture due to elastic-plastic transition is the most probable interpretation for the generation of the crack-like structure.

Figure 5
figure 5

Simulated density and pressure contour plots for diamond foils at −0.95 ns: (a) Nonuniformity ~40%. (b) Nonuniformity ~10%. (c) Nonuniformity ~40% with Cu coating.

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We have also evaluated the effect of material strength using a thin high-Z coating that smooths the effects of irradiation non-uniformity or pressure perturbation. In previous multi-mode imprint experiments13,14,15, under initial low-intensity laser irradiation, the high-Z ablation layers were observed to expand and convert the initial nonuniform laser flux into uniform x-ray radiation, which uniformly ablates and accelerates the target. As the laser pulse shifts to higher intensities, the high-Z material burns away, and the target transitions to pure direct drive13,14,15. Figure 5(c) shows the pressure distribution for the non-uniform foot pulse irradiation (40%) for the diamond with a Cu coating. It is clear that pressure perturbation can be reduced by the high-Z coating compared to that without a Cu coating. That means local fracture at the surface is suppressed due to relaxation of the pressure perturbation. Hence the influence of material strength would be suppressed by the presence of a high-Z coating. As shown in Fig. 3, the crack-like structure, which is a non-sinusoidal perturbation, disappears for the samples coated by thin high-Z layers. This also suggests that the combination of diamond and high-Z coating are effective for suppression of the material strength effects due to large irradiation non-uniformity.

In conclusion, we have investigated the solid-strength effects on surface perturbation due to nonuniform laser irradiation when using diamond as an ablator material for ICF targets. When the irradiation non-uniformity is large, a compressed area exceeding the HEL of diamond develops, showing a crack-like surface structure. The thin high-Z surface coating is effective in suppressing local fracture due to the large pressure perturbation. These findings are particularly closely related to the target physics at the early irradiation time regime in direct-drive ICF. In particular, the performance of laser implosion is sensitive to fracture of the brittle ablator due to nonuniform irradiation. The knowledge from this study is crucial to the understanding not only of ICF target physics but also laser-material interactions and laser material processing. Further investigations on more precise measurements and modeling would reveal the detailed physics of the structure on laser irradiated surface.

Methods

Target samples and laser conditions

The targets comprised single-crystal diamond foils (Type-Ib, density: 3.51 g/cm3) with a thickness of 13–16 μm17. The surface orientation of the single-crystal diamond was the (100) plane. The target foils were coated with Al of 0.05 μm thickness as a shield for shine-through inside of the foils in the very early irradiation time regime. Some of the diamond foils were surface-coated with Cu of 0.1 μm thickness in order to compare the smoothing effects on irradiation non-uniformity, which is described above in the Results and Discussion sections.

The diamond foils were irradiated with the second harmonic laser emission (Wavelength = 0.527 μm) at an incident angle of 37.4°. The laser pulse consisted of one beam for the foot pulse and two beams for the main drive pulse, with time delays between the beams (Fig. 1(b)). The pulse shape was Gaussian with a 1.3 ns duration at full width at half maximum (FWHM). The laser pulse was focused on to the diamond foil to a spot with a size of ~600 μm (FWHM). The average intensity, I0, of the foot pulse was ~4 × 1012 W/cm2. The peak intensity of the main drive was ~1014 W/cm2. Intensity modulation δI of the foot pulse was introduced using a grid mask placed in front of the focusing lens, whereas the main drive was kept uniform. The spatial pattern and intensity distribution of the foot pulse on the target are shown in Fig. 2(a). The modulation wavelength on the target surface was 100 μm, with intensity non-uniformity δI/I0 ~ 10% or ~40%. The higher spatial harmonic components (wavelength: 20–50 μm) in the imprint pulse were less than 10% of the fundamental wavelength (as shown in Fig. 2(b)).

Measurements of areal-density perturbation with face-on x-ray backlighting technique

Perturbations were observed on the target via amplification due to the RTI growth using the main drive beams because the imprinted perturbations were typically too small for detection17. The areal-density perturbation growth was measured using a face-on x-ray backlighting technique. A backlight target (Zn) was irradiated to generate ~1.53 keV quasi-monochromatic x-rays coupled with a 6 μm-thick aluminum filter. Temporal evolution of the transmitted x-rays from the Zn backlight target through a diamond foil was imaged through a slit (10 × 50 μm2) onto the CuI photocathode of an x-ray streak camera. The total magnification was ~25.9×, and the temporal resolution of the x-ray streak camera was ~140 ps.

Data analysis

The spatial resolution was measured using a backlit grid image that took into account the analysis of the areal-density perturbation. The Au mesh (63.5 μm/period) was used to obtain the backlit grid image. The resolution function R(x) of the entire diagnostics system is given by the sum of two Gaussian functions as

$$R(x)=\left(\frac{1}{\sqrt{2\pi }}\right)\left[\frac{1}{({\sigma }_{1}+\alpha {\sigma }_{2})}{e}^{\left(-,\frac{{x}^{2}}{2{{\sigma }_{1}}^{2}}\right)}+\frac{\alpha }{({\sigma }_{1}+\alpha {\sigma }_{2})}{e}^{\left(-,\frac{{x}^{2}}{2{{\sigma }_{2}}^{2}}\right)}\right],$$
(1)

where α = 0.242, σ1 = 4.881 μm, and σ2 = 11.303 μm. The areal-density perturbations δ(ρl) were 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 (μ = 660.9 cm2/g) for diamond. The raw lineouts function I(x) is expressed as

$$I(x)=\int R(x-u){I}_{0}(u){e}^{-\mu \{\delta {(\rho l)}_{1}\cos (ku)+\delta {(\rho l)}_{2}\cos (2ku)\}}du,$$
(2)

where I0 is the spatial intensity distribution of the backlight X-ray source, and δ(ρl)1 and δ(ρl)2 are perturbation amplitudes of the fundamental component and the second harmonic component, respectively. k (2π/perturbation wavelength) is the wave number of the raw lineouts. Fitting by convolution, considering the fundamental and second harmonic components, was in good agreement with the line profile. The trajectories of the irradiated foils were also determined using side-on x-ray backlighting in order to evaluate their basic hydrodynamics.

Simulations were carried out using the two-dimensional (2D) radiation hydrodynamic code PINOCO-2D for comparison with the three experimental configurations. PINOCO-2D gives the arbitrary Lagrangian Eulerian (ALE) hydrodynamic for the radiation. This code includes hydrodynamic, flux-limited Spitzer-Härm thermal conduction34,35, nonlocal thermal equilibrium multigroup radiation transport, quotidian equation of state36, and ray-trace laser-energy deposition. For the EOS, we incorporated a multiphase EOS37, and a table of melting curves38 for diamond with the quotidian equation of state model. Please note that the combined EOS model does not include local fracture nor crack progress.

The areal-density perturbations δ(ρl) of the 2D simulation are obtained from \(\delta \rho l=\delta (\int \rho dx)={\int }_{{x}_{r}({y}_{p})}^{{x}_{a}({y}_{p})}\rho (x,{y}_{p})dx-\) \({\int }_{{x}_{r}({y}_{u})}^{{x}_{a}({y}_{u})}\rho (x,{y}_{u})dx.\)Here, the x axis is perpendicular to the target surface, xa is the position of the ablation front, xr is the position of the rear surface, ρ(x,y) is density distribution in the target, and yp and yu are the perturbed and unperturbed y coordinates of the transverse direction, respectively. The ablation front xa and rear front xr are defined to be at 1/e of the peak density. In our calculation of perturbation areal density, the density distribution ρ(x,y) is considered as mentioned above.