Foam materials are starting to find application in laser-heated Hohlraums used to drive inertial confinement fusion implosions. Foams made using additive manufacturing (AM) techniques are now available and may have advantages over traditional chemical (aerogel) foams. Here, we present new experimental data on laser-heated AM foams. Samples of four different types of printed AM foams were heated using a single 527 nm laser beam at the Jupiter Laser Facility. The laser pulse was ∼180 J square pulse with an FWHM of 1.6 ns and a peak intensity of 3–4 × 1014 W/cm2. The foam densities ranged from 12 to 93 mg/cc (all supercritical for 527 nm light). We measured the backscattered light (power and spectrum), the transmitted light, side-on x-ray images, and the Ti K-shell emission that was used to infer the time-integrated temperature. The fraction of backscattered light was 6%–15% of the input laser energy. The pure carbon foam sample had less backscatter than a C8H9O3 foam of similar density, which was consistent with multi-fluid calculations that predicted less ion heating for the C8H9O3 foam. The level of backscatter and the thermal front speeds for the AM foams were similar to values measured for stochastic (aerogel) foams under similar conditions.

In the indirect drive concept for inertial confinement fusion, laser-heated cavities (“Hohlraums”) produce an x-ray radiation drive that implodes a deuterium-filled capsule. The Hohlraum is designed such that it provides a nearly symmetric radiation drive. Typically, helium gas fills the Hohlraum to provide a back pressure that limits the expansion of the laser-heated walls. Laser plasma instabilities (LPIs) such as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) are a concern for Hohlraums. These instabilities can reflect a substantial fraction of the incoming laser light out of the Hohlraum, thus reducing the magnitude of the drive. Filling the entire Hohlraum with foam has been proposed as one potential mitigation for laser backscatter. A low density foam not only can act to tamp the Hohlraum wall but also allows for the possibility of doping the foam with trace elements that could potentially reduce backscatter.1 For this application, the foam density would have to be ∼1 mg/cc (ne/nc = 3.3% for 351 nm laser light) and so the laser could easily propagate through it.

Excessive wall motion can change the location of the x-ray producing laser spots on the Hohlraum wall and, thus, the symmetry of the radiation onto the capsule. Foam liners have been proposed as a means to reduce Hohlraum wall motion and help control symmetry.2–4 Here, densities of 10–100 mg/cc may be optimal, and so supercritical density foams are of interest. Foams in this density range are the focus of this paper.

In order to fully exploit foams when designing Hohlraums, they must be modeled correctly. Foams are made up of solid structures separated by large voids. Depending on the type of foam, the structures can be plate-like, rod-like, or sphere-like. Most foams are made chemically using the aerogel process.5 Aerogel foams can have very small structural elements and are stochastic (the pores are not of uniform size or spacing). It has recently become possible to print foams directly using additive manufacturing (AM) techniques. A printed foam has several possible advantages over a chemical (aerogel) foam. First, it might have less batch to batch variability in the average density. Second, it might be possible to build in arbitrary spatial variations in density or dopant material that might be beneficial for controlling LPI. Figure 1 shows electron microscope images of a 30 mg/cc dicyclopentadiene (DCPD) foam and a 12 mg/cc printed AM octet truss foam. Although the foams are of comparable density, the aerogel foam is stochastic with a wide range of pore sizes and thin nanometer-scale filaments, whereas the AM foam has a regular structure with ∼5 μm pores and ∼1 μm filaments.

FIG. 1.

Electron microscope images of (a) 30 mg/cc DCPD (CH1.2) foam and (b) 12 mg/cc octet truss AM C8H9O3 foam.

FIG. 1.

Electron microscope images of (a) 30 mg/cc DCPD (CH1.2) foam and (b) 12 mg/cc octet truss AM C8H9O3 foam.

Close modal

Looking at Fig. 1, we see that the disparate scales of the solid foam elements (nm) compared to the typical problem scale (mm) mean that it would be prohibitively expensive to resolve down to the smallest scale size in a single radiation-hydrodynamics calculation. Instead, a number of simplified models have been developed to treat the foam structure without having to completely resolve it spatially. These include 1D analytical models that account for the foam element expansion using similarity solutions.6–8 Fully three dimensional calculations that treat the solid elements as pre-expanded “pixels” have also been employed with success9 although because they are still expensive, their utility is primarily in testing and improving the simpler models.

There have been a number of experiments on the interaction of lasers with aerogel foams.10–33 These experiments have shown that to correctly model the interaction of foams with light, one must include the structure of the foam. For example, several experiments have shown that laser-driven heat fronts propagating through an under-dense foam move more slowly than through an equivalent gas of the same density.11,24,29 Preferential heating of the ions (Ti > Te) in foams has been observed.13,22 Backscatter measurements in under-dense SiO2 foam have been lower than that initially predicted when the foam is modeled as a uniform density material, which has been attributed to preferential ion heating.30 Several analytical foam models have been developed, which predict ion heating in foams due to conversion of the kinetic energy of the solid heated elements when they collide.7,8 Simplified models of foams have been implemented in hydrodynamic codes.34–36 

Since low-density AM foams are relatively new, less theoretical and experimental work has been performed. In Sec. II, we describe the four types of foam samples that we made for this study. In Sec. III, we describe multifluid and single fluid calculations of 2D analogs of several of the AM foams used in this study. We estimate foam homogenization times and compare the calculations with existing analytical theories. Then, in Sec. IV, we show what are, to our knowledge, the first experimental data on laser-heated AM foams. In these experiments, we heated the AM foam targets with a single 527 nm (green) laser beam and measured the backscatter, transmitted light, side-on time-integrated x-ray images and the time-integrated temperature. The foam densities ranged from 12 to 93 mg/cc and were all at or above critical density for green light.

The foam targets in this study were made using a process called direct laser writing via two photon polymerization (DLW-TPP).37 In this process, a tightly focused, femtosecond laser pulse is scanned across a mixture of photosensitive materials and the intense laser light drives non-linear multiphoton absorption processes that, in turn, initiate local polymerization chemistries. Recently, AM foams have been used for high-energy density physics experiments.38 In this work, we used DLW-TPP to make several different types of foam structures. The first type of foam we called a “log-pile” foam. In this structure, thin parallel filaments were printed in a layer and then another layer of parallel filaments at 90° to the previous layer was printed on top of it, and so on, as shown in Fig. 2. These foams are not isotropic, and so the orientation of the filaments relative to the laser axis may matter. We built log-pile foams in two configurations relative to the laser axis. Figure 2(a) shows the “x-y” configuration. Here, the alternating filament layers are laid down in the x-y plane (referring to the coordinate axes shown in Fig. 2), and therefore, both filament axes are orthogonal to the laser direction (z axis). Figure 2(b) shows the alternate “y-z” configuration where the filaments are laid down in the y-z plane with one of the filament directions parallel to the laser axis. The polymer that is formed by the DLW-TPP process is an acrylic with a bulk density (ρs) of 1.2 g/cc. Its composition by weight is approximately 65% carbon, 5% hydrogen, and 30% oxygen. There is less than 0.05% nitrogen. Ignoring the nitrogen, the approximate atomic formula for this material is C8H9O3. The printed filaments have an elliptical cross section, but for simplicity, they can be approximated as cylindrical filaments. Using that assumption, the formula for the average density of a log-pile foam is

(1)
FIG. 2.

50–75 mg/cc log-pile foams with 0.8 μm filament diameters and filaments separated by 10 μm.

FIG. 2.

50–75 mg/cc log-pile foams with 0.8 μm filament diameters and filaments separated by 10 μm.

Close modal

If we assume that the filament radius, r, is ∼0.4 μm and the distance between filaments, ls, is ∼10 μm, then we calculate an average density of ∼75 mg/cc. When the actual targets of this type were weighed, the densities for the x-y type ranged from 76 mg/cc (undoped) to 82 mg/cc (doped). The y-z configuration came out lighter at 56 mg/cc.

We also made log-pile foam targets with both the dimensions of the filaments and the spacing between the filaments scaled up by 20. The idea here was to have filaments that were substantially larger than the wavelength of the laser light [527 nm for the Jupiter Laser Facility (JLF)] so that the filaments would be ablated by the laser. The smaller diameter filaments are of the same order as the skin depth of the laser and are expected to heat up volumetrically. In principle, these targets should have had the same density as the smaller filament log-pile foams, but when they were weighed, the density came out to ∼90 mg/cc.

The third type of foam target (Fig. 3) was also of the log-pile type, but the filaments were hollow carbon tubes, and so the density was lower (∼15–30 mg/cc). The first step was to make a 3D printed polymer nanolattice log-pile structure as before. Then, the lattice is coated with nickel. Then, the sample is heated, and the polymer is removed, leaving a carbon coating on the inside of the hollow nickel tubes. Finally, nickel is removed chemically, leaving a hollow carbon filament.

FIG. 3.

16 mg/cc log-pile carbon foam with hollow filaments separated by ∼5 μm.

FIG. 3.

16 mg/cc log-pile carbon foam with hollow filaments separated by ∼5 μm.

Close modal

The fourth type of foam is based on the octet truss structure. Figure 1(b) shows an electron microscope picture of one of the manufactured octet truss foam samples. This foam is printed using the same polymer as the log-pile foams, and so its chemical formula is C8H9O3. The average density of this structure is

(2)

where r is the radius of the filament and lcell is the cell size. With a cell size of 25 μm and a filament radius of 0.4 μm, the theoretical density is 12.3 mg/cc. The actual octet truss foam targets had a measured density of ∼12 mg/cc, which is currently the lowest density AM foam that can be directly printed at our facility.

We consider a simplified model of the hollow carbon tube log-pile foam and ask how quickly the foam, when heated by a laser, will homogenize to a uniform density plasma. We approximate the log-pile as a 2D array of infinitely long 50-nm-radius carbon filaments (solid density 2.3 g/cc) whose centroids are separated by 1 μm. The average electron density is 5.5 × 1021 #/cc, which corresponds to an average mass density of 18 mg/cc. The problem of foam homogenization time has been studied extensively, both analytically35,39,40 and numerically. Here, we use the analytical expressions from the study by Gus'kov40 to estimate these times. The problem can be broken into two parts—a fast phase in which the heated filaments expand and stagnate into each other, setting up a nonuniform density field, and a slow phase in which the density perturbations are smoothed out by viscous and thermal dissipation. The fast homogenization time is just the pore size, δ, divided by the effective speed of the exploding filaments: t1δ/Veff. The effective explosion speed depends on the heating of the filaments through the laser power and the foam geometry. The state of the plasma after the fast phase is found assuming isothermal expansion and adiabatic compression.40 The slow homogenization phase depends on the ion thermal velocity and the ion–ion collision time. The slow homogenization time scales as t2ρaδ2/T5/2. Assuming a filament length, a0, of 5 μm, a filament radius, b0, of 50 nm, a laser power of 0.1 TW, and a temperature of 1.5 keV, we calculate for the 18 mg/cc carbon foam that the fast homogenization time, t1, is 1 ps, and the slow homogenization time, t2, is 70 ps. Table I shows the calculated homogenization times for the foams in these experiments, assuming an electron temperature of 1.5 keV. Except for the 10-μm filament log-pile foam, the estimated homogenization times are shorter than the laser pulse (∼2 ns). Note that the carbon foam, which has thinner filaments, is expected to homogenize significantly faster than the other foams.

TABLE I.

Calculated fast (t1) and slow (t2) homogenization times for foams in this study assuming Te = 1.5 keV.

Target typeρ (mg/cc)t1 (ps)t2 (ps)
Carbon tube log-pile 18 70 
C8H9O3 octet truss 12 16 490 
Log-pile x-y, 0.5 μ75 760 
Log-pile y-z, 0.5 μ56 690 
Log-pile y-z, 10 μ90 630 320 000 
Target typeρ (mg/cc)t1 (ps)t2 (ps)
Carbon tube log-pile 18 70 
C8H9O3 octet truss 12 16 490 
Log-pile x-y, 0.5 μ75 760 
Log-pile y-z, 0.5 μ56 690 
Log-pile y-z, 10 μ90 630 320 000 

Next, we used a multifluid code, EUCLID,41 to simulate the dynamics of an array of four heated and expanding round filaments. We simulated both pure C and two-fluid CH filaments. First, we looked at the pure carbon (single material) case, relevant to the hollow carbon tube foam. The carbon filaments had the average density (18 mg/cc) and filament geometry as described above, but the filaments were initialized with a pre-expanded Gaussian density profile, as shown in Fig. 4. In these simulations, the electron temperature was held fixed in time to simulate a laser-heated isothermal expansion (there is no laser in this problem). The ion temperature was initialized at the same temperature as the electron temperature and varies in time as the plasmas heat or cool due to drag, thermal equilibration, and work done.

FIG. 4.

Initial electron and carbon ion density for EUCLID 2D multifluid simulation.

FIG. 4.

Initial electron and carbon ion density for EUCLID 2D multifluid simulation.

Close modal

Figure 5 shows the time dependence of the velocity and electron density at the center of the 2D mesh. This is for the case where initially Te = Ti = 0.3 keV. The fluid velocities at the center of the mesh equilibrate in less than 10 ps [Fig. 5(a)]. The ions were heated by the collisions and equilibrated to a temperature of ∼2 keV by 20 ps. Figure 5(b) shows that the density oscillations at the center have still not damped out after 20 ps. The period of the sinusoidal density oscillations is approximately equal to δ/Cs, where δ is the separation between the filaments and Cs is the ion acoustic speed using the “final” ion temperature (after it has been heated by collisions). The amplitude of the damped density oscillations can be fit to a decaying exponential with a time constant of 10 ps. This implies that the plasma density should be approximately uniform by ∼50 ps. We compared these numbers with the analytical theory40 assuming T = 0.3 keV and a laser power of 0.02 TW (consistent with lower Te) and found t1 = 4 ps (reasonable) and t2 = 9.6 ns (unreasonable). However, recall that for the slow homogenization, it is the ion temperature that matters. If we assume T = 2 keV (the equilibrated ion temperature from the EUCLID calculation), then we find t2 = 84 ps, which is getting closer to the EUCLID calculation. Note that we also performed OSIRIS42 particle-in-cell simulations of colliding C and CH sheets and saw significant interpenetration and rapid damping out of density oscillations. It seems likely that interpenetration and multifluid effects could speed up the homogenization process relative to hydrodynamic estimates, as shown here.

FIG. 5.

Plots of the central point x component of the velocity of each carbon filament (a) and the central electron density (b) for the 4 filament 2D EUCLID multifluid problem initialized with Te = Ti = 0.3 keV.

FIG. 5.

Plots of the central point x component of the velocity of each carbon filament (a) and the central electron density (b) for the 4 filament 2D EUCLID multifluid problem initialized with Te = Ti = 0.3 keV.

Close modal

We found that CH foams act significantly different from pure carbon foams. An example where Te = 0.3 keV is shown in Fig. 6. Here, there are eight fluids, 4 for hydrogen and 4 for carbon with the average electron density same as before (5.5 × 1021). The electron density oscillation period is longer, and the amplitude of the oscillations is smaller than the pure carbon “foam.” The protons heat rapidly in 5 ps to 1.5 keV, but the carbon ions slowly heat to 800 eV over 20 ps. During this time, the protons cool to 800 eV as well. In the first picosecond, the protons are accelerated to ∼4 × 107 cm/s, whereas the ions reach only ∼1.5 × 107 cm/s, after which time the carbon ions quickly relax to a single flow. The protons continue to interpenetrate the carbon until ∼10 ps. The plasma becomes a quiescent single flow by 20 ps. The final ion temperature of 800 eV in CH is less than 2 keV in pure carbon because the rapid expansion of the H ions reduces the electron pressure gradient and reduces the acceleration of the carbon ions relative to the pure carbon filament case. Because of their long mean free path, the protons rapidly spread out to become uniform in space, while the carbon ions continue to undergo large density oscillations. Thus, we would expect the octet truss C8H9O3 foam to undergo less ion heating than the pure carbon foam, which is at about the same density. Since the SBS damping rate increases with ion temperature, we would expect less SBS backscatter for the carbon foam.

FIG. 6.

Time history of the center zone ion temperature for 2D EUCLID simulation of expanding CH filaments.

FIG. 6.

Time history of the center zone ion temperature for 2D EUCLID simulation of expanding CH filaments.

Close modal

For the y-z logpile foam configuration, we can approximate the foam as a series of parallel, infinitely long CH filaments (as above) if we ignore the filaments that are parallel to the laser direction because their initial cross section to the laser is much less than that for the orthogonal filaments. For this simulation, we used the HYDRA radiation-hydrodynamics code43 to model an array of laser-heated CH filaments. The array of filaments starts out cold and is heated by a 351-nm laser with an intensity of 3 × 1014 W/cm2. The filaments are 2 μm in diameter and are spaced 16 μm apart in the laser direction and 2 μm apart in the transverse direction. The solid density of the filaments is 1.1 g/cc, and so this geometry results in an average density of 108 mg/cc. Figure 7 shows plots of the laser intensity at 2.5 and 10 ps. We include 4 filaments in the laser (z) direction. Reflecting boundary conditions are used at the top and bottom (y direction), and so the array of filaments extends infinitely in the y direction. This means that we are effectively looking at the center of a wide beam and ignoring any effects of beam spreading. At 2.5 ps, some light is getting past the gaps between the filaments although the rays refract and not much light is getting past the fourth filament. By 10 ps, the gap between the filaments closes and becomes supercritical. The light then cannot propagate past the first layer of filaments until the density drops below critical. It seems unlikely that this short burst of low-intensity light initially passing through the foam sample at the start of the laser pulse could be measured.

FIG. 7.

Snapshots of the laser intensity in units of W/cm2 for the HYDRA 4-filament calculation at (a) 2.5 and (b) 10 ps. The CH filaments extend infinitely out of the plane. Reflecting boundary conditions mean that this array of filaments extends infinitely in the up/down (y) direction.

FIG. 7.

Snapshots of the laser intensity in units of W/cm2 for the HYDRA 4-filament calculation at (a) 2.5 and (b) 10 ps. The CH filaments extend infinitely out of the plane. Reflecting boundary conditions mean that this array of filaments extends infinitely in the up/down (y) direction.

Close modal

Figure 8 compares snapshots of the electron density at 250 ps for an 8 filament run compared to a CH gas at the same density (108 mg/cc). The position of the heat front is denoted by the point where the electron density has a sharp front and goes above critical density. The fact that the front propagates more slowly in the gas is a well-known phenomenon that has been verified experimentally.24,29 Here, we can use the average heat front speed of 0.17 mm/ns extracted from the foam calculation in Fig. 8 to compare against the theoretical front speed for a laser-heated foam in mm/ns given by7 

(3)
FIG. 8.

Snapshots of ne/nc at 250 ps for (a) 108 mg/cc CH foam and (b) 108 mg/cc CH gas.

FIG. 8.

Snapshots of ne/nc at 250 ps for (a) 108 mg/cc CH foam and (b) 108 mg/cc CH gas.

Close modal

Here, ρc is the critical density of the foam at the laser frequency, ρa is the average mass density, ne/nc is the ratio of the electron density to the critical density, and I14 is the laser intensity at the heat front normalized to 1 × 1014 W/cm2. The parameter CE is related to the dimensionality of the foam structure. The theoretical values of CE (based on similarity solutions of foam element expansion) are given by7 

(4)

CA is a laser absorption coefficient of order 1. For the 2D Hydra calculation, the intensity is 3 × 1014, ne/nc is 3.6, the average mass density is 108 mg/cc, and the critical density for CH for 351-nm light is 30 mg/cc. Solving for CE, we get a value of 0.66, which is close the theoretical value for filament-like elements.

Experiments were performed at the Jupiter Laser Facility (JLF) at the Lawrence Livermore National Laboratory. To our knowledge, these are the first experiments on laser-heated AM foams. For these experiments, we used a single frequency-doubled Nd:glass laser beam operating at a wavelength of 527 nm. The foams were heated with an ∼2 ns flat-top laser pulse that had an energy of 180–190 J and a peak power of ∼0.11 TW. The phase plate we used resulted in a 200 μm FWHM spot on the target with an average peak laser intensity of ∼3–4 × 1014 W/cm2. The primary measurements are the backscattered light, the transmitted light, the temperature of the heated foam, and side-on time-integrated x-ray images.

The experimental setup is shown in Fig. 9. The SBS was measured using a full aperture backscatter system (FABS) similar to that used on the NIF.44 The SBS light reflected back through the laser is diverted through a calibrated leaky mirror to a diode and calorimeter to record the temporal pulse shape and energy, respectively. Backscatter that comes back at an angle wider than the incident beam f-number is intercepted by a Lambertian (Spectralon) scattering plate45 and imaged using a CCD camera (calibrated in situ). The backscatter measured in the calorimeter and imaged on the plate is combined to give a total amount of backscattered energy within an ∼f/3 cone and 532 ± 5 nm. Note that we did not have an SRS spectrometer for these experiments, and so we must assume that all the measured backscattered light is either SBS or unabsorbed light scattered back into the FABS system. Preshot simulations showed that calculated linear gains for SRS were ∼10× lower than calculated SBS gains, and so SRS backscatter is expected to be negligible compared to SBS.

FIG. 9.

Illustration of the Janus experimental setup. Forward and backward unabsorbed and scattered light were characterized, along with multi-keV x-ray emission with a (broadband) time-integrated x-ray pinhole camera (XRPHC) and a high-resolution Ti K-shell spectrometer (IXS).

FIG. 9.

Illustration of the Janus experimental setup. Forward and backward unabsorbed and scattered light were characterized, along with multi-keV x-ray emission with a (broadband) time-integrated x-ray pinhole camera (XRPHC) and a high-resolution Ti K-shell spectrometer (IXS).

Close modal

A portion of the scattering plate signal was collected and sent to a fiber-coupled streak camera to record spatially averaged (3 < f < 6), but time-resolved SBS. Light transmitted through the target was also imaged with an ∼f/3 calibrated scattering plate system. A portion of the light collected from the scattering plates was sent to the fiber-coupled diode; the transmitted power onto the plate was determined by the calibrated scattering plate and the temporal profile recorded by the diode.

Two of the foam targets were doped with TiO2 using an atomic layer deposition (ALD) coating technique,46 resulting in a foam target with ∼5%–10% by atomic fraction Ti (based on weighing the foam before and after the coating process). The time-integrated, 1D spatially resolved (along the laser propagation axis) Ti K-shell emission from those shots was recorded using a toroidal imaging x-ray spectrometer (IXS).47 The IXS data were fitted with spectra calculated using the atomic-kinetics code SCRAM48 in order to infer the electron temperature of the heated foam for those shots. Time-integrated x-ray pinhole images were also captured, imaging orthogonal to the laser propagation axis. The side-on x-rays were filtered by 540 μm of Be and 50 μm of Kapton to preferentially record the Ti K-shell emission at ∼5 keV.

The foam targets were printed directly onto a plastic disk, and the disk was glued to a target stalk. They were 500 μm wide, 1000 μm tall, and the depth (in the laser direction) varied from 100 to 400 μm. The width and height of the samples were set by the laser spot size and pointing accuracy. The laser was aimed toward the top of the target to avoid clipping on the target base. The thicknesses were chosen such that the laser heat front would burn through the foam prior to the end of the experiment. Figure 10 shows an octet truss target mounted on the target stalk. The circle denotes where the laser spot was aimed.

FIG. 10.

Octet truss target mounted on target stalk—(a) front view and (b) side view.

FIG. 10.

Octet truss target mounted on target stalk—(a) front view and (b) side view.

Close modal

1. Backscattered light

The targets for which we obtained data are listed in Table II. For most of the target types, we had repeated shots, except for the pure carbon foam target. The measured time-dependent backscatter was obtained by combining the SBS calorimeter and diode data with the energy onto the Near Backscatter Imager (NBI) plate. This measurement accounts for SBS and unabsorbed (reflected backwards off target) light. The largest uncertainty in the measurement is the fraction of light onto the NBI plate. If we account for just the light recorded on the plate, then about 50% of the total backscatter is on the NBI. But the plate was not quite large enough to capture all the light, and so the signal is not zero on part of the plate boundary. If we extrapolate to account for light off the edge of the plate, then as much as 66% of the backscatter could be outside the calorimeter (which only captures light within the beam f-number). We account for this by multiplying the calorimeter measurement by 2.5 ± 0.5 to get the total backscatter.

TABLE II.

List of the target, density, thickness, and laser energy for the JLF AM foam experiments.

ShotTarget typeρ (mg/cc)Thickness (μm)Laser energy (J)
28 Octet truss 12 453 177 
29 Log-pile x-y, doped 82 170 196 
30 Carbon tube log-pile 16 297 186 
33 Log-pile y-z, 10 μ90 206 193 
34 Log-pile x-y 77 93 174 
35 Log-pile y-z 56 146 174 
36 Log-pile x-y 76 166 188 
38 Octet truss 12 442 187 
39 Log-pile x-y, doped 82 161 184 
40 Log-pile y-z, 10 μ90 211 194 
41 Log-pile y-z 56 149 193 
ShotTarget typeρ (mg/cc)Thickness (μm)Laser energy (J)
28 Octet truss 12 453 177 
29 Log-pile x-y, doped 82 170 196 
30 Carbon tube log-pile 16 297 186 
33 Log-pile y-z, 10 μ90 206 193 
34 Log-pile x-y 77 93 174 
35 Log-pile y-z 56 146 174 
36 Log-pile x-y 76 166 188 
38 Octet truss 12 442 187 
39 Log-pile x-y, doped 82 161 184 
40 Log-pile y-z, 10 μ90 211 194 
41 Log-pile y-z 56 149 193 

Figure 11 shows a plot of the backscattered energy fraction vs the foam density for all the shots. The target types are differentiated by the color of the symbols. There is no general trend with the average density. The maximum backscatter is between 9% and 15% for all the foams tested, with the highest values corresponding to the octet truss targets and the lowest values to the hollow carbon tube and the 10-μm filament log-pile foams. Recall that in Sec. III, the multifluid calculation for a CH foam predicted reduced ion heating (and, thus, reduced in Landau damping of SBS) compared to a pure carbon foam. The higher measured backscatter for C8H9O3 compared to the pure C foam at similar density is consistent with this prediction. Also note that the backscatter for the log-pile foams does not appear to depend on the foam orientation relative to the laser direction (x-y vs y-z configurations).

FIG. 11.

Fraction of backscattered light for all AM foam targets vs foam density.

FIG. 11.

Fraction of backscattered light for all AM foam targets vs foam density.

Close modal

There have been a number of previous experiments that measured the backscattered light in aerogel (stochastic) foam targets. These include experiments in which a single beam is normally incident on the foam sample, as in our experiment.10,30,32,49 They also include experiments using multiple beams at multiple incident angles.25,26 The previous experiments have used a variety of foam densities (1–50 mg/cc), materials, and laser wavelengths (351, 527, and 1054 nm). Even so, if we compare data at roughly the same laser intensity, we find that the new data for AM foams are roughly consistent with the previous aerogel foam data. For example, in the experiments by Depierreaux,25 the 351-nm laser intensity was 4 × 1014 W/cm2. The foam was 10 mg/cc C15H20O6, which has an ne/nc value of ∼0.33. They found that the SBS was <8% and the SRS < 4% in energy. 12% backscatter is about what we measured for the 12 mg/cc C8H9O3 octet truss foam at 3 × 1014 W/cm2 and ne/nc ∼ 1.

Another example is the single beam experiments by Tanaka.10 For a C18H32O16 foam at 0.33 of critical density (351 nm light), they measured ∼6% SBS backscatter through the lens at an intensity of 3 × 1014 W/cm2 (they did a range of intensities from 5 × 1013 to 1 × 1015). They did not measure light outside the f cone, and as discussed above, this can be up to twice the light measured in the lens. So their measurements are consistent with a total backscatter of 12%–18%—again, not too different from what we measured. Looking across a number of experiments, we found that the measured backscatter for aerogel and AM foams of roughly the same chemical composition and electron density and illuminated with about the same laser intensity was similar.

The SBS spectra are more interesting. The SBS spectra for three different types of log-pile foams are shown in Fig. 12. All the spectra show both red-shifted and blue-shifted features. Calculations of the linear gain spectra using plasma conditions extracted from hydrodynamic calculations of these experiments indicate appreciable gains only for positive (red-shifted) wavelength shifts. The origin of the blue-shifted part of the spectrum is still under investigation.

FIG. 12.

SBS spectra for (a) log-pile y-z foam with 10-μm filaments, (b) log-pile x-y with 0.5-μm filaments, and (c) log-pile y-z with 0.5-μm filaments. Units are in counts. Time-dependent laser power is the yellow curve.

FIG. 12.

SBS spectra for (a) log-pile y-z foam with 10-μm filaments, (b) log-pile x-y with 0.5-μm filaments, and (c) log-pile y-z with 0.5-μm filaments. Units are in counts. Time-dependent laser power is the yellow curve.

Close modal

The spectra for all the log-pile foams look similar regardless of orientation. This is true even for the y-z foam scaled up by 20 times with the large (10 μm) filaments [Fig. 12(a)]. The spectrum has a maximum wavelength shift of approximately 5 A. They all have a broad, blue-shifted feature centered at approximately −10 A. The spectra for the two lower density foams, shown in Fig. 13, do look significantly different from the others. The maximum positive wavelength shift is much larger—more than 10 A for the octet truss foam. The blue-shifted feature is centered at about −5 A. The hollow C foam is a different material (C vs C8H9O3), whereas the octet truss is simply a different geometry and lower average density than the other log-pile foams.

FIG. 13.

SBS spectra for (a) octet truss foam (shot 28) and (b) hollow C tube foam (shot 30). Time-dependent laser power is over-plotted in the yellow curve.

FIG. 13.

SBS spectra for (a) octet truss foam (shot 28) and (b) hollow C tube foam (shot 30). Time-dependent laser power is over-plotted in the yellow curve.

Close modal

The SBS spectra have been previously measured for aerogel foams. Mariscal30 measured SBS spectra for SIO2 and Ta2O5 foams on the JLF laser. The peak intensity was 3 × 1014 W/cm2, similar to this study. For SiO2 at 15% critical density, they measured a narrow peak between 5 and 7 A. There was no signal for negative wavelength shifts. For Ta2O5 aerogel at 41% critical density, they measured a broad spectrum from 0 to 10 A. There was some low signal for negative wavelength shifts, but no peak in that area. Tanaka10 measured the spectrum for 0.33 ne/nc C18H32O16 foam at 1 × 1015 W/cm2 and also for a solid CH foil target at the same intensity. For the foam target, they found that the spectrum peaked at a red shift of 3 A. There was signal in the blue-shifted part of the spectrum, but no peak. However, for the solid CH target, they saw a double-peaked spectrum, with a second blue-shifted peak at −5 A. This suggests that perhaps in the AM foam, the initially solid filaments are contributing to the spectrum in a way that randomly oriented (and thinner) filaments in a chemical foam do not.

2. Transmitted light and time-integrated x-ray pinhole camera images

We also measured the light transmitted through the foam. Determining the time of the thermal wave breaking out from the back of the foam sample is our best measurement of the thermal propagation speed. Transmitted light was only detected for the two lowest density foams (C8H9O3 octet truss and the pure C hollow-filament foam). Figure 14 shows the input laser power, the backscatter, and the transmitted light power as a function of time for the octet truss shot with a laser energy of 187 J. The light breaks through to the back of the 0.442-μm-thick sample at 1.4 ns, and so the average front velocity through the foam is 0.316 mm/ns. If we use Eq. (1) and assume an intensity of 4 × 1014, ne/nc = 0.9, ρ/ρc = 0.9, and CE = 0.66 (from the idealized foam calculation in Sec. III B), we get a predicted front velocity of 0.78 mm/ns, which is too fast. But recall that the calculation that set the CE at 0.66 assumed an idealized 1D laser beam that was infinitely wide and did not spread. Also, there were no radial heat losses or radial mass flow. In real experiments, we find that the front velocity is always lower then this ideal 1D limit. The measured speed actually fits with a value of CE = 2.5, which is consistent with the value of 2–3 found previously for propagation through SiO2 aerogel foams.7 

FIG. 14.

Input laser power, backscattered light, and transmitted light vs time for C8H9O3 octet truss AM foam.

FIG. 14.

Input laser power, backscattered light, and transmitted light vs time for C8H9O3 octet truss AM foam.

Close modal

For the 0.293-mm thick hollow C foam sample, the light also broke through the back at 1.4 ns, and so the inferred front velocity is 0.21 mm/ns. This foam had a density of 16 mg/cc, and so ne/nc and ρ/ρc are 1.23. Using Eq. (1) again with 4 × 1014 W/cm2, we get CE = 2.5. Again, this is a similar CE compared to other foams, but compared to just the octet truss, this implies that the heat front speed is relatively more restricted in the hollow C foam.

We can also use these data to compare with other analytical models. Gus'kov6 has the following formula for the front velocity through a foam in cm/s:

(5)

Here, ρs is the solid density of the filament material in g/cc, ρa is the average density, Z is the average charge state, A is the atomic weight (g/mol), and λL is the laser wavelength in micrometers. The parameter α is related to the geometry of the foam—α = 0.5 corresponds to cylindrical filaments, while α = 1 corresponds to sheet-like foams. Using this formula, we get α = 0.91 for the octet truss foam and α = 0.88 for the pure C foam. These values of α are similar to what has been found for aerogel foams. For example, Nicolai24 found that α = 0.9 was the best fit to the measured front speed through 10 mg/cc C15H20O9 foam. Similarly, Gus'kov6 found a value of α = 0.8 to match experimental data obtained by Depierreax.25 In general, it appears that these AM foams that we tested are acting similar to stochastic aerogel foams.

For all the other higher density foams, the laser heat front did not propagate through to the far side of the sample, and so no transmitted light was detected. For these foams, all we can do is set an upper bound on the thermal velocity. For example, shot 41, a y-z foam, had a density of 56 mg/cc and was 0.149 μm thick. The pulse had 193 J and had a peak power of 0.11 TW. We can approximate this as a 1.75 ns square pulse. Assuming that the thermal front just reaches the end of the foam at the end of the pulse, this gives a maximum front velocity of 0.085 mm/ns. Using Eq. (5), we get α = 0.95.

The x-ray pinhole camera view is directly perpendicular to the laser direction (z axis). The camera looks down at the top of the target with an angle of 45° from vertical. Figure 15(a) shows the raw data for shot 28, the octet truss. The data are plotted in counts, which range from 0 to 255 (8-bit camera). Note that a few isolated pixels are saturated at 255, but the image is not saturated overall. The laser comes in from the left side in the images. The z = 0 point is placed at the centroid of the image. We then smoothed the data over 10 pixels in the z and y directions using boxcar smoothing. The smoothed image, Fig. 15(b), shows a more obvious structure. The rectangular shape of the foam block is apparent as is the fact that the heated foam block is viewed at an angle off of vertical, resulting in the apparent steepening of the gradient on the negative y side.

FIG. 15.

Side-on x-ray pinhole images of shot 28 12 mg/cc octet truss foam: (a) raw image and (b) smoothed image. The cyan line is a horizontal lineout. Laser comes in from the left. Units are counts.

FIG. 15.

Side-on x-ray pinhole images of shot 28 12 mg/cc octet truss foam: (a) raw image and (b) smoothed image. The cyan line is a horizontal lineout. Laser comes in from the left. Units are counts.

Close modal

Figure 16 shows smoothed x-ray images for other types of foam. The differences in these images are subtle. Figure 16(a) shows an image of the heated carbon foam. This foam has lower density and, hence, is thicker in the z (laser) direction (∼300 μm). So the intense part of the emission is longer in the z direction than for the higher density foams. Note that this image is the smoothest of all the images, which is consistent with the carbon foam having the shortest homogenization time (Table I). Also, the heat front burns through this foam, likely accounting for the change in the slope of the lineout at z ∼ 100 μm. Figures 16(c) and 16(d) are for log-pile foams with different filament orientations relative to the laser direction. Just as for the SBS spectra, the images are indistinguishable, implying that the AM foam orientation does not change the absorption or propagation.

FIG. 16.

Side-on x-ray pinhole camera images of (a) carbon foam, (b) 10-μm filament y-z foam, (c) 0.5-μm filament x-y foam, and (d) 0.5-μm filament y-z foam. The laser comes in from left. Units are counts.

FIG. 16.

Side-on x-ray pinhole camera images of (a) carbon foam, (b) 10-μm filament y-z foam, (c) 0.5-μm filament x-y foam, and (d) 0.5-μm filament y-z foam. The laser comes in from left. Units are counts.

Close modal

The image for the large filament foam [Fig. 16(b)] requires some explanation. A photo of this foam target is shown in Fig. 17. The front view [Fig. 17(a)] shows that the target is 600 μm wide with the vertical filaments spaced 200 μm apart. Recall that the laser spot has a diameter of 200 μm. The top view [Fig. 17(b)] shows that there are essentially two planes of filaments spaced 200 μm apart in the laser propagation direction. We would expect that early in time, some light could get past the first plane of filaments and start heating the second plane. But the calculations in Sec. III show that the gaps between the filaments would close within a few picoseconds and the first filament plane would then be supercritical. The x-ray image [Fig. 16(b)] shows a thin bright emission in the z (laser) direction that we interpret as the heating and absorption of the first plane of filaments. There seems to be no clear emission from the second plane of filaments, which would be located at Z = 200 μm in the figure. Notice that there is an inflection in the lineout at Z = 200 μm, which could be an indication of a little heating at the second plane. No transmitted light was measured, and so overall, this foam also remained supercritical to the light.

FIG. 17.

Photos of the large filament log-pile target: (a) front view and (b) top view. The laser direction is shown by the green arrow.

FIG. 17.

Photos of the large filament log-pile target: (a) front view and (b) top view. The laser direction is shown by the green arrow.

Close modal

3. Temperature inference from IXS

We used high-resolution (∼1 eV) Ti K-shell spectra recorded by the IXS to estimate the time-integrated electron temperature for two 82 mg/cc Ti-doped log-pile x-y foam samples. Due to fabrication constraints, we were not able to obtain Ti-doped versions of the other types of foam. A series of SCRAM calculations were performed to infer plasma electron temperatures by finding the best fit to the spectra. The two shots (29 and 39) were nominally repeated shots, with slight differences in the laser energy and target thickness. Shown in Fig. 18 are the space-integrated IXS data [normalized to the inter-combination line (y)] along with spectra from the corresponding SCRAM inferred electron temperatures (assumed electron density was ∼quarter-critical, convolved with a 1 eV FWHM Gaussian response function). Compared to the rest of the spectral features, the resonance line (w) at 4750 eV is anticipated to be more sensitive to optical depth effects;50 as such, it was excluded from the fit. While a single temperature, optically thin calculation is often incapable of capturing all the spectral features typically observed in laser-produced plasmas,21 we find that temperatures of ∼1.1–1.2 keV reasonably capture most of the optically thin features from these shots.

FIG. 18.

Time- and space-integrated Ti He-a complex spectra from the IXS (black) and inferred electron temperatures from SCRAM (shaded area). The resonance line (w) is not included in the fit due to anticipated optical-depth effects.

FIG. 18.

Time- and space-integrated Ti He-a complex spectra from the IXS (black) and inferred electron temperatures from SCRAM (shaded area). The resonance line (w) is not included in the fit due to anticipated optical-depth effects.

Close modal

We estimate the average temperature of the heated foam plasma at the end of the laser pulse by equating the absorbed laser energy (EL) to the energy of the cylindrical plasma column,

(6)

Here, V is the plasma column volume, Z is the average ionization, A is the average atomic number (g/mol), Av is Avogadro's number (6.022 × 1023 particles/mol), k is Boltzmann's constant, Ti is the ion temperature, and Te is the electron temperature. We assume an absorbed laser energy of 170 J (average of the laser energy of shots 29 and 39 with 10% backscatter subtracted). For a fully ionized C8H9O3 plasma, Z = 4.05 and A = 8.1. We use the time integrated x-ray image of this foam, Fig. 16(c), to approximate the heated plasma as a cylinder 350 μm long and 250 μm in diameter. Using these experimental values and assuming Te = Ti, Eq. (6) gives Te = 1.4 keV—slightly higher than the measured temperature. However, we know that the ion temperature in a laser-heated foam is predicted to be higher than electron temperature.8 If we take the electron temperature as the average of the measured for the two shots (1.15 keV), we can solve Eq. (6) for Ti and get Ti = 2.3 keV. Having an ion temperature twice as high as the electron temperature is not unreasonable. For example, a previous study13 with Agar (C12H18O9) foam used spectral measurements to infer ion temperatures 1.5–2 times larger than electron temperatures. Again, it appears that the structured foams are acting similar to the unstructured foams in this regard.

We present what are, to our knowledge, the first experiments on laser-heated structured AM foams. These were done at the Jupiter Laser Facility using a single 527 nm laser beam at 3 × 1014 W/cm2. The laser pulses were approximately square with an energy of ∼180 J and an FWHM of ∼1.6 ns. Four different types of AM foams were tested, ranging in density from 12 to 90 mg/cc. These densities are near critical or supercritical to 527 nm light. We measured the backscattered light (power and spectrum), the transmitted light, side-on x-ray images, and the time-integrated temperature.

We estimated the homogenization times for these foams using existing theory.40 The pure carbon foam, which has thin hollow filaments, is predicted to homogenize on a faster timescale than the other foams, and this was seen in the x-ray image data. All the foams were predicted to homogenize on a timescale shorter than the laser pulse except for the foam with 10-μm thick filaments. We compared the analytical homogenization times with calculations of idealized 2D filamentary foam structures using the multi-fluid EUCLID code41 and found pretty good agreement as long as we accounted for the expected higher ion temperatures in the formulas. But the EUCLID calculations generally predicted slightly faster homogenization times than the hydrodynamic theory, which we attribute to interpenetration and multifluid effects. The EUCLID calculations also showed that multi-species effects could be important for determining the ion heating. In a CH foam, the rapidly expanding H ions reduce the electron pressure gradient, which results in less acceleration of the carbon ions and lower ion temperatures when these ions collide and convert their kinetic energy to heat. We also simulated an idealized, laser-heated filamentary foam using the HYDRA43 code. We confirmed that the heat front in such a system propagates slower than through a uniform plasma of the same average density. The calculated heat front speed was consistent with an analytical model7 that assumes a quasi-1D laser (no beam spreading, radial heat transport, and edge effects) and cylindrical solid elements.

The experimentally measured SBS backscattered light has 6%–15% of the input laser energy. This is consistent with experiments with various stochastic (aerogel) foams at similar laser intensity. We found that the backscatter for the C8H9O3 foam was higher than that for the pure carbon foam at similar density, which is consistent with the prediction that the multi-species foam would have lower ion temperatures and, thus, lower damping and higher amounts of SBS. The amount of backscatter and the details of the SBS spectra were largely independent of the orientation of the AM foam relative to the laser axis. We also found little difference in the backscatter of log-pile foams of similar density in which the filament diameter was scaled by a factor of 20. The spectra have a prominent, blue-shifted feature that is not understood. This feature is generally not present in the published backscatter spectra for stochastic foams.

We used the breakout time of the transmitted light through the two lowest density foams to estimate the average heat front speed through those foams. These heat front speeds were consistent with the Gus'kov model6 using values of the foam geometric parameter, α, of 0.88–0.91. They were also consistent with the Belyaev model7 for values of the fitting coefficient, CE ∼ 2.5. These were consistent with published front speeds for stochastic foams, which have α ∼ 0.8–0.9 (Refs. 6 and 24) and CE ∼ 2–3.7 The time-averaged electron temperature of the 82 mg/cc Ti-doped foam was determined by fitting the measured x-ray spectrum. The inferred temperature of ∼1.1–1.2 keV is consistent with an approximate model of a heated cylindrical plasma column and is also consistent with an ion temperature >2 keV. These experiments provide a useful dataset to compare against various published foam models and other models that are currently under development.

The authors would like to thank the staff at the Jupiter Laser Facility for help in performing these experiments. This work was funded under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC5207NA27344 with funding support from the Laboratory Directed Research and Development Program under tracking code 17-ERD-118. This document was prepared as an account of the work sponsored by an agency of the United States Government. Neither the United States Government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of the authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

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

1.
S.
Wilks
,
W.
Kruer
,
J.
Denavit
,
K.
Estabrook
,
D.
Hinkel
,
D.
Kalantar
,
A.
Langdon
,
B.
MacGowan
,
D.
Montgomery
, and
E.
Williams
,
Phys. Rev. Lett.
74
(
25
),
5048
5051
(
1995
).
2.
K.
Baker
,
C.
Thomas
,
T.
Baumann
,
R.
Berger
,
M.
Biener
,
D.
Callahan
,
P.
Celliers
,
F.
Eisner
,
S.
Felker
,
A.
Hamza
,
D.
Hinkel
,
H.
Huang
,
O.
Jones
,
N.
Landen
,
J.
Milovich
,
J.
Moody
,
A.
Nikroo
,
R.
Olson
, and
D.
Strozzi
, in
APS DPP
(
APS
,
Savannah, GA
,
2015
), Vol.
60
.
3.
S.
Bhandarkar
,
T.
Baumann
,
N.
Alfonso
,
C.
Thomas
,
K.
Baker
,
A.
Moore
,
C.
Larson
,
D.
Bennett
,
J.
Sain
, and
A.
Nikroo
,
Fusion Sci. Technol.
73
(
2
),
194
209
(
2018
).
4.
A.
Moore
,
N.
Meezan
,
C.
Thomas
,
S.
Bhandarkar
,
L.
Divol
,
N.
Izumi
,
A.
Nikroo
,
T.
Baumann
,
M.
Rubery
,
J.
Williams
,
N.
Alfonso
,
O.
Landen
,
W.
Hsing
, and
J.
Moody
,
Phys. Plasmas
27
,
082706
(
2020
).
5.
R.
Pekala
,
J. Mater. Sci.
24
(
9
),
3221
3227
(
1989
).
6.
S.
Gus'kov
,
J.
Limpouch
,
P.
Nicolai
, and
V.
Tikhonchuk
,
Phys. Plasmas
18
(
10
),
103114
(
2011
).
7.
M.
Belyaev
,
R.
Berger
,
O.
Jones
,
S.
Langer
, and
D.
Mariscal
,
Phys. Plasmas
25
(
12
),
123109
(
2018
).
8.
S.
Gus'kov
and
V.
Rozanov
,
Quantum Electron.
27
(
8
),
696
701
(
1997
).
9.
J.
Milovich
,
O.
Jones
,
R.
Berger
,
G.
Kemp
,
J.
Oakdale
,
J.
Biener
,
M.
Belyaev
,
D.
Mariscal
,
S.
Langer
,
P.
Sterne
, and
M.
Stadermann
,
Plasma Phys. Controlled Fusion
(published online).
10.
K. A.
Tanaka
,
B.
Boswell
,
R. S.
Craxton
,
L. M.
Goldman
,
F.
Guglielmi
,
W.
Seka
,
R. W.
Short
, and
J. M.
Soures
,
Phys. Fluids
28
(
9
),
2910
2914
(
1985
).
11.
J.
Koch
,
K.
Estabrook
,
J.
Bauer
,
C.
Back
,
L.
Klein
,
A.
Rubenchik
,
E.
Hsieh
,
R.
Cook
,
B.
MacGowan
,
J.
Moody
,
J.
Moreno
,
D.
Kalantar
, and
R.
Lee
,
Phys. Plasmas
2
(
10
),
3820
3831
(
1995
).
12.
A. E.
Bugrov
,
S. Y.
Guskov
,
V. B.
Rozanov
,
I. N.
Burdonskii
,
V. V.
Gavrilov
,
A. Y.
Goltsov
,
E. V.
Zhuzhukalo
,
N. G.
Kovalskii
,
M. I.
Pergament
, and
V. M.
Petryakov
,
J. Exp. Theor. Phys.
84
(
3
),
497
505
(
1997
).
13.
V.
Gavrilov
,
A.
Gol'tsov
,
N.
Koval'skii
,
S.
Koptyaev
,
A.
Magunov
,
T.
Pikuz
,
I.
Skobelev
, and
A.
Faenov
,
Quantum Electron.
31
(
12
),
1071
1074
(
2001
).
14.
C.
Constantin
,
C.
Back
,
K.
Fournier
,
G.
Gregori
,
O.
Landen
,
S.
Glenzer
,
E.
Dewald
, and
M.
Miller
,
Phys. Plasmas
12
(
6
),
063104
(
2005
).
15.
K.
Fournier
,
C.
Constantin
,
J.
Poco
,
M.
Miller
,
C.
Back
,
L.
Suter
,
J.
Satcher
,
J.
Davis
, and
J.
Grun
,
Phys. Rev. Lett.
92
(
16
),
165005
(
2004
).
16.
K.
Fournier
,
J.
Satcher
,
M.
May
,
J.
Poco
,
C.
Sorce
,
J.
Colvin
,
S.
Hansen
,
S.
MacLaren
,
S.
Moon
,
J.
Davis
,
F.
Girard
,
B.
Villette
,
M.
Primout
,
D.
Babonneau
,
C.
Coverdale
, and
D.
Beutler
,
Phys. Plasmas
16
(
5
),
052703
(
2009
).
17.
F.
Girard
,
M.
Primout
,
B.
Villette
,
D.
Brebion
,
H.
Nishimura
, and
K.
Fournier
,
High Energy Density Phys.
7
(
4
),
285
287
(
2011
).
18.
F.
Perez
,
J.
Kay
,
J.
Patterson
,
J.
Kane
,
B.
Villette
,
F.
Girard
,
C.
Reverdin
,
M.
May
,
J.
Emig
,
C.
Sorce
,
J.
Colvin
,
S.
Gammon
,
J.
Jaquez
,
J.
Satcher
, and
K.
Fournier
,
Phys. Plasmas
19
(
8
),
083101
(
2012
).
19.
F.
Perez
,
J.
Patterson
,
M.
May
,
J.
Colvin
,
M.
Biener
,
A.
Wittstock
,
S.
Kucheyev
,
S.
Charnvanichborikarn
,
J.
Satcher
,
S.
Gammon
,
J.
Poco
,
S.
Fujioka
,
Z.
Zhang
,
K.
Ishihara
,
N.
Tanaka
,
T.
Ikenouchi
,
H.
Nishimura
, and
K.
Fournier
,
Phys. Plasmas
21
(
2
),
023102
(
2014
).
20.
F.
Perez
,
J.
Colvin
,
M.
May
,
S.
Charnvanichborikarn
,
S.
Kucheyev
,
T.
Felter
, and
K.
Fournier
,
Phys. Plasmas
22
(
11
),
113112
(
2015
).
21.
M.
May
,
G.
Kemp
,
J.
Colvin
,
D.
Liedahl
,
P.
Poole
,
D.
Thorn
,
K.
Widmann
,
R.
Benjamin
,
M.
Barrios
, and
B.
Blue
,
Phys. Plasmas
26
(
6
),
063105
(
2019
).
22.
M.
Primout
,
D.
Babonneau
,
L.
Jacquet
,
F.
Gilleron
,
O.
Peyrusse
,
K.
Fournier
,
R.
Marrs
,
M.
May
,
R.
Heeter
, and
R.
Wallace
,
High Energy Density Phys.
18
,
55
66
(
2016
).
23.
M.
Tanabe
,
H.
Nishimura
,
S.
Fujioka
,
K.
Nagai
,
N.
Yamamoto
,
Z.
Gu
,
C.
Pan
,
F.
Girard
,
M.
Primout
,
B.
Villette
,
D.
Brebion
,
K.
Fournier
,
A.
Fujishima
, and
K.
Mima
,
Appl. Phys. Lett.
93
(
5
),
051505
(
2008
).
24.
P.
Nicolai
,
M.
Olazabal-Loume
,
S.
Fujioka
,
A.
Sunahara
,
N.
Borisenko
,
S.
Gus'kov
,
A.
Orekov
,
M.
Grech
,
G.
Riazuelo
,
C.
Labaune
,
J.
Velechovsky
, and
V.
Tikhonchuk
,
Phys. Plasmas
19
(
11
),
113105
(
2012
).
25.
S.
Depierreux
,
C.
Labaune
,
D. T.
Michel
,
C.
Stenz
,
P.
Nicolai
,
M.
Grech
,
G.
Riazuelo
,
S.
Weber
,
C.
Riconda
,
V. T.
Tikhonchuk
,
P.
Loiseau
,
N. G.
Borisenko
,
W.
Nazarov
,
S.
Huller
,
D.
Pesme
,
M.
Casanova
,
J.
Limpouch
,
C.
Meyer
,
P.
Di-Nicola
,
R.
Wrobel
,
E.
Alozy
,
P.
Romary
,
G.
Thiell
,
G.
Soullie
,
C.
Reverdin
, and
B.
Villette
,
Phys. Rev. Lett.
102
(
19
),
195005
(
2009
).
26.
B.
Delorme
,
M.
Olazabal-Loume
,
A.
Casner
,
P.
Nicolai
,
D. T.
Michel
,
G.
Riazuelo
,
N.
Borisenko
,
J.
Breil
,
S.
Fujioka
,
M.
Grech
,
A.
Orekhov
,
W.
Seka
,
A.
Sunahara
,
D. H.
Froula
,
V.
Goncharov
, and
V. T.
Tikhonchuk
,
Phys. Plasmas
23
(
4
),
042701
(
2016
).
27.
M.
Desselberger
,
M.
Jones
,
J.
Edwards
,
M.
Dunne
, and
O.
Willi
,
Phys. Rev. Lett.
74
(
15
),
2961
2964
(
1995
).
28.
R.
Watt
,
J.
Duke
,
C.
Fontes
,
P.
Gobby
,
R.
Hollis
,
R.
Kopp
,
R.
Mason
,
D.
Wilson
,
C.
Verdon
,
T.
Boehly
,
J.
Knauer
,
D.
Meyerhofer
,
V.
Smalyuk
,
R.
Town
,
A.
Iwase
, and
O.
Willi
,
Phys. Rev. Lett.
81
(
21
),
4644
4647
(
1998
).
29.
J.
Colvin
,
H.
Matsukuma
,
K.
Brown
,
J.
Davis
,
G.
Kemp
,
K.
Koga
,
N.
Tanaka
,
A.
Yogo
,
Z.
Zhang
,
H.
Nishimura
, and
K.
Fournier
,
Phys. Plasmas
25
(
3
),
032702
(
2018
).
30.
D.
Mariscal
,
O.
Jones
,
R.
Berger
,
S.
Patankar
,
K.
Baker
,
T.
Baumann
,
M.
Biener
,
C.
Goyon
,
B.
Pollock
,
J.
Moody
, and
D.
Strozzi
,
Phys. Plasmas
28
,
013106
(
2021
).
31.
M.
Cipriani
,
S.
Gus'kov
,
R.
De Angelis
,
F.
Consoli
,
A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
, and
G.
Di Giorgio
,
Phys. Plasmas
25
(
9
),
092704
(
2018
).
32.
N. G.
Borisenko
,
Y. A.
Merkul'ev
,
A. S.
Orekhov
,
S.
Chaurasia
,
S.
Tripathi
,
D. S.
Munda
,
L. J.
Dhareshwar
,
V. G.
Pimenov
, and
E. E.
Sheveleva
,
Plasma Phys. Rep.
39
(
8
),
668
673
(
2013
).
33.
A.
Caruso
,
C.
Strangio
,
S. Y.
Gus'kov
, and
V. B.
Rozanov
,
Laser Part. Beams
18
(
1
),
25
34
(
2000
).
34.
J.
Velechovsky
,
J.
Limpouch
,
R.
Liska
, and
V.
Tikhonchuk
,
Plasma Phys. Controlled Fusion
58
(
9
),
095004
(
2016
).
35.
M.
Cipriani
,
S.
Gus'kov
,
R.
De Angelis
,
F.
Consoli
,
A.
Rupasov
,
P.
Andreoli
,
G.
Cristofari
,
G. D.
Giorgio
, and
F.
Ingenito
,
Laser Part. Beams
36
(
1
),
121
128
(
2018
).
36.
M.
Belyaev
,
R.
Berger
,
O.
Jones
,
S.
Langer
,
D.
Mariscal
,
J.
Milovich
, and
B.
Winjum
,
Phys. Plasmas
27
(
11
),
112710
(
2020
).
37.
J.
Fischer
and
M.
Wegener
,
Laser Photonics Rev.
7
(
1
),
22
44
(
2013
).
38.
J.
Oakdale
,
R.
Smith
,
J.
Forien
,
W.
Smith
,
S.
Ali
,
L.
Aji
,
T.
Willey
,
J.
Ye
,
A.
van Buuren
,
M.
Worthington
,
S.
Prisbrey
,
H.
Park
,
P.
Amendt
,
T.
Baumann
, and
J.
Biener
,
Adv. Funct. Mater.
27
(
43
),
1702425
(
2017
).
39.
S.
Gus'kov
,
V.
Rozanov
, and
N.
Zmitrenko
,
JETP Lett.
66
(
8
),
555
(
1997
).
40.
S.
Gus'kov
,
J. Russ. Laser Res.
26
(
4
),
312
327
(
2005
).
41.
D.
Ghosh
,
T.
Chapman
,
R.
Berger
,
A.
Dimits
, and
J.
Banks
,
Comput. Fluids
186
,
38
57
(
2019
).
42.
R.
Fonseca
,
L.
Silva
,
F.
Tsung
,
V.
Decyk
,
W.
Lu
,
C.
Ren
,
W.
Mori
,
S.
Deng
,
S.
Lee
,
T.
Katsouleas
, and
J.
Adam
, in
Proceedings of the International Conference on Computational Science—ICCS 2002
, edited by
P.
Sloot
,
C.
Tan
,
J.
Dongarra
, and
A.
Hoekstra
(Springer-Verlag, Berlin Heidelberg,
2002
), Vol.
2331
, pp.
342
351
.
43.
M.
Marinak
,
G.
Kerbel
,
N.
Gentile
,
O.
Jones
,
D.
Munro
,
S.
Pollaine
,
T.
Dittrich
, and
S.
Haan
,
Phys. Plasmas
8
(
5
),
2275
2280
(
2001
).
44.
D.
Froula
,
D.
Bower
,
M.
Chrisp
,
S.
Grace
,
J.
Kamperschroer
,
T.
Kelleher
,
R.
Kirkwood
,
B.
MacGowan
,
T.
McCarville
,
N.
Sewall
,
F.
Shimamoto
,
S.
Shiromizu
,
B.
Young
, and
S.
Glenzer
,
Rev. Sci. Instrum.
75
(
10
),
4168
4170
(
2004
).
45.
G.
Georgiev
and
J.
Butler
,
Appl. Opt.
46
(
32
),
7892
7899
(
2007
).
46.
S.
Ghosal
,
T.
Baumann
,
J.
King
,
S.
Kucheyev
,
Y.
Wang
,
M.
Worsley
,
J.
Biener
,
S.
Bent
, and
A.
Hamza
,
Chem. Mater.
21
(
9
),
1989
1992
(
2009
).
47.
E.
Gamboa
,
D.
Montgomery
,
I.
Hall
, and
R.
Drake
,
J. Instrum.
6
,
P04004
(
2011
).
48.
S.
Hansen
,
J.
Bauche
,
C.
Bauche-Arnoult
, and
M.
Gu
,
High Energy Density Phys.
3
(
1–2
),
109
114
(
2007
).
49.
A.
Bugrov
,
S.
Gus'kov
,
V.
Rozanov
,
I.
Burdonskii
,
V.
Gavrilov
,
A.
Gol'tsov
,
E.
Zhuzhukalo
,
N.
Koval'skii
,
V.
Kondrashov
,
M.
Pergament
,
V.
Petryakov
, and
S.
Tsoi
,
J. Exp. Theor. Phys.
88
(
3
),
441
448
(
1999
).
50.
G.
Perez-Callejo
,
L.
Jarrott
,
D.
Liedahl
,
E.
Marley
,
G.
Kemp
,
R.
Heeter
,
J.
Emig
,
M.
Foord
,
K.
Widmann
,
J.
Jaquez
,
H.
Huang
,
S.
Rose
,
J.
Wark
, and
M.
Schneider
,
Phys. Plasmas
26
(
6
),
063302
(
2019
).