Ultra-high (>30%) coupling efficiency designs for demonstrating central hot-spot ignition on the National Ignition Facility using a Frustraum

Recent experiments at the National Ignition Facility (NIF) on studying inertial confinement fusion (ICF) are approaching the burning plasma regime where self-heating by fusion products is the dominant source.¹ Achieving ignition, defined here as an energy gain greater than unity,² calls for an additional order-of-magnitude increase in self-heating. The path toward ignition remains challenging, requiring improved symmetry control, mitigation of laser-mediated plasma instabilities, reduced levels of hydro-instability growth of fabrication features and imperfections in the target, and controlling sources of preheat that deleteriously raise the fuel adiabat. In the indirect-drive approach to ICF, laser light is converted to soft x rays in the high-Z cylindrical hohlraum wall that are then absorbed by the capsule ablator at an efficiency typically $≼$10%. Such a modest efficiency translates into a tight performance margin against modeling errors in equations of state, thermal and radiation transport, opacities, preheat spectra, nuclear cross sections, hydroinstability growth, and from assorted nonhydrodynamic phenomena where mean free path effects may play an important role.³ Any means of improving the hohlraum-to-capsule efficiency can only help the margin budget and facilitate the path to ignition. Although the benefit of higher capsule absorbed energy has been long known,² its implementation in cylindrical hohlraums has proved challenging and elusive. Preserving drive symmetry in a cylinder when the (hohlraum) case-to-capsule ratio is $≼$2 is the largest impediment because the inner beams cannot propagate to the hohlraum (or capsule) waist at late times through the relatively dense and cool plasma created by the ablator blow off stagnating against the hohlraum wall. Figure 1 shows the three factors that strongly influence drive symmetry in a cylindrical hohlraum: (1) laser-entrance-hole (LEH) closure, (2) formation of a “wall bubble” from direct illumination of the hohlraum wall by the outer laser cones,⁴ and (3) the apparent density ridge that forms between the colliding ablator and wall plasmas near the capsule waist. As the capsule size increases, the laser pulse duration must also increase to preserve the shock timing sequence. Careful shock sequencing is necessary to maintain a low fuel adiabat and to provide a high peak implosion speed of the shell for high one-dimensional (1-D) performance margin. For a given hohlraum size, increasing the duration of the laser pulse to accommodate a larger capsule leads to more time for the LEH to close and adversely affect outer-cone propagation late in the pulse. At the same time, the wall bubble has more time to grow and potentially impede the propagation of the inner cones to the hohlraum waist. Finally, stagnation of the ablator and wall plasmas can occur relatively earlier in the pulse and further thwart symmetry control with the use of a larger-than-standard capsule.

It is clear that the cylinder geometry has its limits in accommodating a larger capsule—which begs the question as to why the cylinder has been the workhorse of hohlraum studies for the past 30 years. Aside from the value of a large and extensive database to exploit and leverage, the main reason is that the outer cones with nominally 2/3 of the energy (at the NIF) are incident on the cylinder wall at a range in angles of 40°–46° (normal to the wall) and are readily absorbed. That is to say, the risk of scattering and reflection of the outer cone beams is deemed small. There is more uncertainty on the propagation and absorption of the inner beams (60°–67° to the wall normal), but their relatively small fraction of the overall energy (∼1/3) makes this less of a concern. Of course, increasing the size of the cylindrical hohlraum radius to fit larger capsules comes at the unacceptable price of reduced drive, lower peak implosion speed $υimp,$ and reduced 1-D margin (M). Margin is related to the 1-D ignition threshold factor (ITF) by M = ITF − 1, where $ITF ∼ α−1.9υimp3.58 Pa1.16Ecap,$^5,6 α is the fuel adiabat in units of the Fermi degenerate value, $υimp$ is the peak implosion speed of the shell (including fuel and unablated shell), and $Pa$ is the ablation pressure.

A recent strategy to maintain drive symmetry in a cylinder for longer times with use of a recessed pocket in the wall near the LEHs shows promise in potentially allowing larger capsules. In this geometry—termed an “I-Raum”—the outer-cone laser rays are absorbed in the pocket, and the motion of the wall bubble toward the symmetry axis is delayed to allow prolonged inner-cone deposition near the waist up to a nanosecond longer.⁷ The goal of this approach is to exploit the prolonged symmetry control for fielding a 10%–20% larger capsule than in standard cylinders. The limiting factor for this geometry is that the inner beams must still propagate through a dense and relatively cooler ablator-wall plasma above the capsule $(z≅0)$⁠, i.e., the initial waist radius of the I-Raum remains the same as in standard cylinders to maintain sufficient drive.

A complementary approach to the I-Raum is the Frustraum geometry that aims to maximize the volume above the capsule for facilitating inner-beam propagation at late times. In this way, the geometry can readily accommodate a far larger capsule than the 1 mm radius standard-sized capsule. The shape of the Frustraum, formed by conjoining two right-circular cones (or “frusta”), is meant to first minimize the surface area of the cavity and then use the energy savings (compared with a standard cylinder) toward substantially increasing the waist radius. In this manner, the surface area is arranged to be similar to a cylinder but with a significantly larger volume above the capsule (near the hohlraum waist). Thus, the peak drive temperature is expected to be similar for the same nominal capsule size. Because the envisioned larger capsules will absorb over 300 kJ extra energy, this difference is made up by using smaller LEHs in the Frustraum design (3.1 mm diameter) compared with the 3.64 mm diameter LEHs fielded in shot N170601,⁸ but comparable to the 3.1 mm diameter LEHs fielded for “575”-class hohlraums in the High-foot campaign (N141106)⁹ and “544” hohlraums used early on in the low-foot campaign.² In sum, the energetics of the Frustraum employing a 1.5 mm radius capsule can be made to strongly resemble that of the standard cylinder using a 1.0 mm-scale capsule.

The main challenge with the Frustraum geometry is ensuring sufficient drive symmetry on the oversized capsule, particularly at late times when hohlraum filling has occurred. The large diameter of the Frustraum (∼9 mm) helps reduce the plasma density above the capsule and lower inverse bremsstrahlung absorption of the inner beams. But the risk of inner beam stoppage at the wall bubble driven by the outer cones remains—just as in standard cylinders. A key difference is that the relatively low incidence angle of the outers on the inclined wall of a Frustraum may lead to significant levels of specular reflection. Besides possibly leading to a shallower wall bubble, the (specularly) reflected outer-beam energy may lead to potentially favorable heating of the channel region where the inner beams largely propagate. Whether this is a significant effect or not depends on two parameters used in the modeling: (1) the electron flux limit and (2) the amount and distribution of nonspecular (diffuse) reflection at the outer-cone turning points. A low electron flux limit, e.g., FL = 0.03–0.04, effectively bottles up the thermal energy to drive large temperature and (electron) density gradients, which in turn leads to more reflection of the incident laser light. Because of the Frustraum geometry, the specular reflection of the outer cones may fortuitously lead to channel heating largely coincident with the location of the inner beams. In this sense, outer-cone specular reflection (or glint) may help to boost propagation of the inner beams and provide sufficient symmetry in a Frustraum geometry. At the same time, the inner beams are arranged to cross the Frustraum midplane (at least early in the laser pulse) and deposit their energy at a large incidence angle compared with a cylinder. In a sense, the Frustraum manages to swap the roles of the inner- and outer-beams compared with a cylinder: less glint in the inner beams and more (and potentially favorable) glint in the outer beams. Whether this physical picture holds up depends on the choice of parameters used in the modeling and key experimental benchmarking.

The use of drive multipliers in modeling cylinder hohlraum experiments on the NIF is well known and the subject of ongoing inquiry.¹⁰ One recent explanation is that more inner-cone glint and its ensuing escape through the LEHs from use of a low value of FL (∼0.03) could circumvent the need for such drive multipliers.¹¹ This low value of the flux limit is to be compared with the widely used “high flux model” that uses FL = 0.15.¹² Although nonlocal transport effects, MHD, and ion-acoustic turbulence have long been known to inhibit electron heat flux, simplified models of them applied to hohlraums have so far not given enough inhibition to match the low FL ∼ 0.03 and its resulting x-ray drive reduction. Recent work is focusing on implementing a time-dependent FL model with 0.02 up to peak power, followed thereafter by a linear transition to 0.10 over 0.5 ns.¹³ 

The cylinder hohlraums are nearly orthogonal to the Frustraum with regard to the relative roles of inner- and outer-cone glint, so is there a database to exploit that could help benchmark Frustraum modeling in the absence of dedicated experiments to date? Fortunately, recent rugby-shaped hohlraum (nonignition) experiments with oversized aluminum single shells could lend a hand here.¹⁴ These experiments were geared toward maximizing the energy absorbed by the capsule while maintaining adequate symmetry—which coincides with the goal of the Frustraum. Both the rugby and Frustraum share the same hohlraum gas-fill density (0.3 mg/cc ⁴He) and similar glancing incidence angles of the outer cones on the hohlraum wall but differ in laser pulse shape, i.e., reverse-ramp vs 3-shock stepped pulse for a high-density carbon (HDC) ablator. Two-dimensional (2-D) integrated modeling of these rugby-shaped hohlraum experiments were performed to narrow the acceptable range of flux limit and diffuse reflection fraction across a range of key experimental observables. The conclusion is that a range of FL from 0.03 to 0.04 and diffuse reflectivity at the laser turning points of 0%–10% are consistent with the rugby dataset—with no need for drive multipliers. This range of simulation parameters was then applied to integrated modeling of the Frustraum to assess energetics, drive symmetry, laser backscatter risk, and hydro-instability exposure.

From a historical perspective, the Frustraum geometry is somewhat similar to former rugbylike hohlraum ignition designs but allows the ability to field as large a capsule as possible while trying to maintain effective inner-cone beam propagation at late times. These two goals are presumptively met by sacrificing the hohlraum surface area for greater space above the capsule. In so doing, the surface area of the Frustraum is nearly 30% greater than previously reported ignition designs with rugby hohlraums.¹⁵ 

An integrated hohlraum tune that could potentially triple the capsule absorbed energy to over 0.5 MJ out of less than 1.8 MJ of requested laser energy at 3ω (0.351 μm wavelength) and deliver robust, high yield (>20 MJ) is found. The accompanying drive symmetry is nearly uniform in time with only small excursions, clearly benefitting from the (simulated) effect of outer-cone glint to the region of the channel where the inner cones propagate. A linear gain analysis of stimulated Raman scatter (SRS) and stimulated Brillouin scatter (SBS) over all cones shows peak intensity gain exponents (or simply “gains”) of less than ten. The inner-cone backscatter risk benefits favorably from the large volume above the capsule for low electron density despite the relatively long path length in a Frustraum. The outer cones have a comparatively small path length due to the tapered geometry of the Frustraum, resulting in predicted low gains and reduced risk of laser optics damage on the NIF. A gain less than ten is often associated with tolerable risk, but this is not an absolute threshold: nonlinear effects may still lead to significantly large laser reflectivity.

The capsule designs for the Frustraum use HDC as the ablator, but the envisioned shell thicknesses do not follow a strict Euler scaling with energy due to hohlraum energetics and symmetry concerns. In order to avoid late-time hohlraum plasma filling from wall and capsule ablation and the potential loss of symmetry control, the shell designs purposely use thinner ablators than a strict Euler scaling would allow–but with full knowledge that the risk of in-flight shell breakup increases. To assess this concomitant risk, a suite of highly resolved multimode simulations are performed to gauge the effects of engineering features, e.g., fill tubes and capsule support tents, and native roughness on all the capsule surfaces. The results of these simulations suggest that the designs are very robust to perturbation growth and show more margin than the standard-sized (∼1 mm-scale) capsules despite the relative thinness of the high $Ecap$ Frustraum designs.¹⁶ 

The greatest uncertainty in the Frustraum design is the ability to recover adequate symmetry when the data and simulations are expected to be at odds by some degree. The model benchmarking to date is based on closely related rugby hohlraum experiments—but with important differences as already noted. Fortunately, there are two tuning levers that should provide acceptable symmetry recovery once the first dedicated Frustraum experiments are fielded. The first is the obliquity of the Frustraum wall relative to the symmetry axis. Increasing this angle provides more waist-high drive flux as needed in case the outer-cone specular reflection is not as effective as advertised in the modeling. The trade-off is that the Frustraum loses some efficiency as the wall area must necessarily increase for a fixed Frustraum length. Another risk is that higher order drive perturbations, e.g., the 4th Legendre mode, may worsen. The second potential lever in controlling lowest-order drive symmetry is to reduce the hohlraum gas fill from the nominal 0.3 mg/cc to 0.15 mg/cc or less. Recent data in cylinders at such low values of gas-fill density¹⁷ or lower¹⁸ show a tendency toward more prolate implosions than modeled, suggesting an onset of kinetic effects (or a departure from the hydrodynamic limit that underlies mainline integrated simulations). Although this anomalous behavior could be beneficial for Frustraum dynamics, the underlying physical origin would need to be identified before implementation in mainline simulations (most likely as a dedicated reduced model). Another symmetry tuning option is to significantly shorten the laser pulse to help reduce hohlraum filling at later times when loss of symmetry control is a paramount concern. One consequence of this strategy is that the fuel adiabat necessarily increases, thereby providing more margin to instability growth, hohlraum asymmetry, and unknown levels of fuel preheat—but eroding 1-D performance margin. A design is proposed that shortens the laser pulse by ∼2.5 ns to less than 7 ns while still providing moderately high yield (∼13 MJ) at a fuel adiabat of 4.6. Finally, empirically tuning the laser wavelengths between inner and outer cones could be used to effect cross-beam energy transfer (CBET) and optimize drive symmetry if needed.

It should be emphasized that despite the clear risk that outer-cone glint may not be nearly as favorable as advertised in the simulations, the upside of the Frustraum could be immense—and at relatively little cost. Essentially, tripling the capsule absorbed energy (and, in turn, the performance margin) transforms the NIF laser to act as though 6 MJ were available without investing in any optics or infrastructure upgrades. Compared with the long known option of using 2ω conversion crystals and potentially accessing ∼3MJ driver energy [as well as taking on higher laser-plasma interaction (LPI) risk $∼Iλ2,$ where I is the intensity and λ is the laser wavelength], the Frustraum option is a low-cost endeavor that could substantially improve the prospects for demonstrating ignition at 3ω over the near term.

Table I summarizes the key traits of the Frustraum as simulated compared with those over a broad range of fielded cylinder hohlraums with nominal-sized capsules. The takeaway from this comparison is that the laser-capsule coupling efficiency for similar total laser energy and peak laser power levels of the Frustraum could approach three times that of a typical cylinder hohlraum.

This paper is organized as follows: Sec. II studies the energetics, symmetry, and backscatter risk for a Frustraum driving a 0.5 MJ-class capsule and the underlying modeling assumptions used. Section III analyzes two recent rugby-shaped hohlraum experiments for benchmarking the integrated modeling used in the Frustraum design. Section IV assesses the stability properties of the 0.5 MJ-class capsule across several values of fuel adiabat. Section V looks at recovery strategies in the event that the Frustraum modeling is significantly in error, including an option for a high-adiabat ignition design that uses a laser pulse duration of less than 7 ns. Section VI looks at even higher energy capsule designs (0.6 MJ and 0.7 MJ capsule absorbed energy) to identify an approximate end point for the utility of the Frustraum for a $≼$2 MJ NIF driver energy. We conclude and summarize in Sec. VII.

The nominal Frustraum geometry is shown in Fig. 2. The shape is continuously differentiable in order to lessen the risk of corner jetting above the capsule and near the LEHs. An interior wall diameter of 9 mm and length of 12 mm are adopted to maximize the volume above the capsule while keeping the wall surface area similar to that of ongoing work on the NIF with 6.72 mm diameter cylindrical hohlraums. The wall material consists of depleted uranium (DU) for high x-ray albedo. A laser-entrance-hole diameter of 3.1 mm is used to maximize efficiency while providing sufficient laser clearances. The channel gas fill is ⁴He at a nominal density of 0.3 mg/cc. An HDC capsule design is shown in Fig. 3(a) along with the optimally chosen 3-shock laser power history in Fig. 3(b). The simulated channel radiation temperature is shown in Fig. 4, achieving a maximum value of 295 eV and remaining above 280 eV well past when the laser is off (9.2 ns). This is attributable to the choice of small LEH for energetic advantages (see below) and late time closure that helps confine the x-ray drive.

A power balance analysis supports the high values of high radiation temperature and capsule absorbed energy as follows. Equating the laser power converted into soft x rays with Marshak wall losses, capsule absorbed energy and losses out the pair of LEHs, gives

where $ηx$ is the laser conversion efficiency into drive x rays, $PL$ is the laser power in terawatts, $TR$ is the radiation temperature in hectoelectron volts (heV), $Aw$ is the interior surface area of the Frustraum, $αw$ is the wall albedo, $αHDC$ is the ablator albedo, CCR is the case-to-capsule ratio (or $Aw/AHDC$⁠, where $AHDC$ is the initial surface area of the ablator), and $fLEH$ is the ratio of LEH area to wall area. Upon using η_x = 0.75, $αw=0.95$⁠, $αHDC$=0.43 (obtained from 1-D hohlraum + capsule simulations), and the geometrical quantities CCR and $fLEH$ from Fig. 2, we obtain a peak radiation temperature of 291 eV at a laser power of 460 TW—in reasonable agreement with the simulated value of 295 eV. Next, we estimate the capsule absorbed energy $Ecap$ from

where $τL≡EL/PL$ =3.87 ns is approximately the duration of peak power, $EL$ is the laser energy (of 1.78 MJ),

and $RHDC,0$=1500 μm. Using that the peak implosion velocity of the shell $υ0$ is nearly 400 μm/ns and approximating its time dependence as $υ0t/τL2$⁠, we obtain $Ecap$=517 kJ—which is close to the one-dimensional simulated value of ∼500 kJ absorbed energy in the capsule.

In summary, a 0-D analysis of hohlraum power balance is consistent with the 2-D integrated hohlraum simulations of a Frustraum with regard to drive temperature and capsule absorbed energy. As a reminder, integrated hohlraum calculations refer to (radiation-hydrodynamic) LASNEX¹⁹ simulations that incorporate 3-D laser ray tracing, deterministic radiation transport, in-line (XSN²⁰ and DCA²¹) non-LTE opacities, equations of state and thermonuclear burn on a numerical mesh that includes the capsule, hohlraum gas fill, LEH window, and hohlraum wall. The values of capsule absorbed energy quoted in this paper are derived from first extracting a frequency-dependent source (FDS) from the integrated hohlraum simulations across the channel region, applying a pair of source multipliers to the 1-D capsule-only simulation driven by this FDS to match the onset of neutron production from first shock passage through the fuel and the subsequent velocity history of the fuel-ablator interface, summing over the total energy of the ablator vs time, and then identifying its maximum value before compressional burn begins.

The 2-D integrated hohlraum simulations employed here do not use a laser power multiplier to compensate for drive deficiencies that have characterized previous modeling of NIF experiments with a high value of electron flux limit (0.15), as required by the high flux model.¹² Instead, we use a lower value (0.03–0.04) as argued in Ref. 11 and well supported by recent rugby-shaped hohlraum experiments on the NIF—as described in detail in Sec. III. In the case that future Frustraum experiments on the NIF show the need for drive multipliers to accommodate unexpected energy losses, the designs shown here still leave some headroom in energy and power.

The next hurdle for Frustraum designs is ensuring adequate drive symmetry. One of the intended design merits of the Frustraum geometry is that the outer cones are incident on the narrow portion or throat of the Frustraum [close to a node of the Legendre polynomial $P3$(cosθ)], helping to control time-integrated (Legendre polynomial coefficient) $a4$ drive asymmetries by virtue of a large inner- to outer-ring separation.²² Figures 5(a)–5(d) shows the initial ray geometry for each of the four NIF “cones.” As the wall blow off or ablation evolves in time, the inner cones (23° and 30°) will effectively merge into a single cone near the midplane, while the outer cones (44° and 50°) will achieve first bounce closer to the symmetry axis (z) and are increasingly separated from the inner cones. The challenge with arranging the outer cones to reach the wall at a smaller angle relative to the symmetry axis (from capsule center) is ensuring adequate time-integrated $a2$ symmetry control as well. Figures 6(a) and 6(b) show the (normalized) $a2$ and $a4$ ablation pressure Legendre coefficients vs time for the above Frustraum design. Although these histories are not directly measurable, they prove useful in gauging the effects of 2-D symmetry excursions on implosion performance and setting acceptable limits in an ignition design. To achieve this high degree of lowest-order symmetry control in the simulations, the ratio of inner-cone power to total power (“cone fraction”) was adjusted in time as shown in Fig. 7(a). Optimal use of the NIF requires that the cone fraction remains near 1/3 around peak power to maximize the available laser energy and power without undue damage risk to the optical elements. The Frustraum design meets this requirement at and near peak power [see Fig. 3(b)].

The small temporal variation of lowest-order ablation pressure asymmetry seen in Fig. 6(a) significantly differs from the cylindrical hohlraum database where large swings (in x-ray self-emission core symmetry) at peak power to pole-high (oblate or “pancaked”) asymmetry are routinely simulated and measured. Such large swings on the order of 10% over 1–2 ns give rise to nonradial flows and nonsymmetric implosions in general. The physical explanation behind this late-time symmetry swing is that the inner beams are absorbed well away from the hohlraum waist due to the formation and motion of a wall “bubble” originating from and associated with the outer-cone laser deposition. This effect is what principally inspired the I-Raum concept:⁷ force a delay of this bubble motion by recessing in radius the region of the hohlraum where the outer beams strike the wall. Simulations and data are consistent in delaying the loss of inner-cone propagation toward the waist by almost one nanosecond under nearly the same drive conditions. Use of a larger capsule only magnifies the loss of symmetry at late times as the extra ablated mass only increases inverse bremsstrahlung absorption of the inner-cone laser energy. From Figs. 6(a) and 7(b), it is evident that the (simulated) Frustraum does not suffer from a degradation of late-time symmetry control to the degree seen in cylinders —even with the far larger capsule.

The Frustraum clearly benefits from the large volume above the capsule near the waist to promote lower electron density and reduced inverse bremsstrahlung absorption of the inner cones, but is this the sole mechanism responsible for drastically improved symmetry control seen in the simulations? Figures 8(a) and 8(b) offers an additional mechanism potentially at play in the Frustraum. Figure 8(a) shows that considerable specular reflection of the outer cones at first bounce from the wall into the channel above the capsule leads to efficient laser absorption. Much of this specular reflection is due to the low flux limit (0.03–0.04) assumed in the model and corroborated by data as described in Sec. III. This heated region largely coincides with passage of the inner beams as seen in Fig. 8(b) and helps assist inner beam propagation toward the waist as desired. In cylinders, this boost from the outer cones is far smaller as we now argue. Recall that the turning point for a ray incident on a linearly ramped stratified plasma density profile at an angle θ occurs at $sin2(θ)nc$⁠, where $nc$ is the critical electron density.²³ Thus, the local inverse bremsstrahlung rate of absorption for an isothermal expansion of the wall scales as $sin4(θ)$ at constant temperature. Comparing this scaling for a cylinder vs a Frustraum at 22.3° wall angle for the 44.5° outer beams gives a factor of 4.6 $×$ less inverse bremsstrahlung absorption at the turning point in a Frustraum; for the 50° outer beams, the factor is 6.4 $×$ less. The difference is even larger if we take into account the colder temperature at the turning points of the more penetrating, higher angle beams in a cylinder and the resulting increase (decrease) in absorption (reflection). The conclusion is that if the low flux-limit model is largely correct, then the low absorption rate at the first turning point in the outer cone rays may indeed boost inner beam propagation to the waist to benefit symmetry tuning. The reduced absorption at a turning point and the glancing incidence angle of the outers both act in concert to lessen wall bubble growth compared with cylindrical hohlraums.⁴ On the flip side, the inner cone ray propagation has reduced specular reflection from the wall in a Frustraum compared with a cylinder at early time because the beams are pointed to cross the midplane and intercept the wall at an angle similar to the outers in a cylindrical hohlraum [see Figs. 5(a) and 5(b)].

Another challenge for a hohlraum design in general is ensuring that hohlraum-plasma-mediated backscatter of laser light remains at tolerably low levels. Beyond incurring a loss of drive from any reflected laser light an even larger threat is damage to the optics elements and turning mirrors on the NIF due to stimulated Brillouin scatter (SBS). Safeguarding against this source of optics damage has necessitated a “walk up” in energy, particularly for nonstandard hohlraum shapes such as the rugby.^24–26 Here, we provide peak linear gain estimates of stimulated Raman scatter (SRS) and SBS for each of the four cones. The design goal is to limit these values to under ∼10 and provide some confidence that backscatter reflectivity remains low for the Frustraum. Historically, calculated gains under ∼10 generally yield low reflectivity in cylinders—but not always. The expectation is that this criterion can be met for the Frustraum due to the design feature of reducing the outer-cone path lengths near the LEH by virtue of the tapered geometry. Figure 9 shows the simulated peak values of SRS and SBS gain from the laser-interaction Plasma postprocessor NEWLIP²⁷ of a radiation-hydrodynamic simulation of a Frustraum ignition design. The SBS gain is highest for the cone 44° beams but still within the design goal of ∼10. The SRS gain is highest for the cone 23° beams but also low. These results are encouraging but await experimental testing.

Another LPI phenomenon that can impact Frustraum performance is cross-beam energy transfer (CBET) of laser energy from one set of cones to another in the vicinity of the LEH. This effect has been exploited in the past to recover time-integrated symmetry in cylindrical hohlraums at high (density) gas fill by using large wavelength separations between the inner and outer cones.²⁸ This strategy for symmetry control may not be required in Frustraums, but beam energy transfer may still occur even at low gas-fill density (∼0.3 mg/cc) and under neutral conditions, i.e., when all the cones nominally have the same wavelength. This energy exchange could shift the implosion symmetry in a Frustraum to potentially require a small wavelength shift Δλ between the inner and outer cones on the order of 1 Å and restore favorable symmetry. Until recently, postprocessed radiation-hydrodynamic simulations for gauging CBET have neglected ray refraction, which may introduce significant errors in applying such an (“offline”) tool to a Frustraum. The use of an in-line CBET package in 2-D radiation-hydrodynamic simulations with 3-D ray tracking is now available for testing,²⁹ as well as the offline CBET code Vampire³⁰ that includes refraction. However, CBET modeling to date needs an ion-acoustic wave amplitude saturation clamp on $δne/ne$ to match implosion shape data. Also, laser speckle effects are ignored in backscatter gains and CBET modeling. For example, nearly 4% of a NIF laser beam can have intensity ∼7 times the average intensity, leading to beam self-focusing and rippling of the laser reflection surface.

The nature of CBET also depends sensitively on the flow velocities and material composition of the plasmas near the LEHs. In standard hohlraum modeling, the LEH zoning consists only of the 0.5 μm thick polyimide window overlaying the high-Ζ lip of the LEH [see Fig. 10(a)]. In practice, the LEH assembly is far more complicated due to the addition of a 0.1 μm thick “storm window” vacuum seal meant to prevent chamber gas condensation on the primary window and an associated reduction in hohlraum drive. Another feature of LEHs that has heretofore been neglected in integrated hohlraum modeling is the presence of a (∼100 μm) thick aluminum washer that mechanically supports each (primary) LEH window. For hohlraums with comparatively large LEHs the aluminum washer can have an inner radius close to the radius of the LEH lip.⁸ Figure 10(a) shows an example of a recent cylindrical hohlraum design where the LEH washer is only modestly larger than the LEH lip diameter (e.g., shot N180307). By 7 ns when ∼80% of the laser energy has been delivered, the materials distribution in the LEH region [Fig. 10(c)] is noticeably different than the control simulation without the LEH storm window assembly [Fig. 10(b)]. Thus, there is the potential for significant amounts of Al plasma migrating to the throat of the LEHs at late times and displacing the Au from the LEH lip region. Another possible complication is late-time refraction of the outer cone by the ablated storm-window CH retainer ring. A similar pair of simulations was performed for the Frustraum ignition design as well. Figure 11(b) shows the distribution of these LEH window features in a radiation-hydrodynamic simulation for a Frustraum after 0.5 ns compared with the standard simulation setup of the LEH shown in Fig. 11(a). By 7 ns the distribution of materials near the LEH as shown in Figs. 11(c) and 11(d) is only modestly different with the ablated Al remaining behind the LEH lip. From the standpoint of predicting CBET and SRS and SBS backscatter levels such small differences could still matter. Figure 12 shows the peak values of SRS and SBS gain. Compared with the values shown in Fig. 9 with standard zoning, a notable difference in peak levels is found. The higher inner-cone SBS gains found with the detailed LEH modeling are identified with the two expanding and colliding windows at early time well before peak laser power is reached. This more complicated picture of window evolution may also affect evaluation of two plasmon decay instability thresholds and preheat generation during the laser picket compared with the single-window simulation setup.³¹ To accommodate the predicted large values of capsule absorbed energy in a Frustraum the LEHs are designed to be significantly smaller compared with standard cylindrical hohlraum experiments at low gas fill.⁶ This design feature renders the Frustraum less sensitive to details of modeling the LEH assembly for a standard Al washer size—but not to the point that deleterious effects on drive and symmetry can always be ruled out. On physical grounds the presence of large amounts of Al and CH just outside the LEHs can lead to drive degradation owing to the relatively large laser ablation efficiency and low x-ray albedo of these materials compared with Au or DU. In addition, the ablated Al window washer and CH storm-window retainer ring can refract the outer edges of the outer cones (44° and 50°) and shift the average beam deposition closer to the LEHs.

Although the Frustraum design with small LEHs is meant to minimize drive and symmetry effects of the storm window components, their presence can affect the flow behavior just outside the LEH and potentially alter the conditions for CBET to occur. Figures 13(a)–13(d) compares the cone-by-cone effect of CBET with standard zoning that neglects the storm window and with detailed storm-window zoning. For the most part, the practice of standard zoning leads to exaggerated CBET compared with storm-window zoning, with or without inclusion of the storm-window materials. The effects of CBET are still potentially significant and warrant some symmetry tuning adjustment, whether via use of a nonzero Δλ or a dynamic cone fraction adjustment to compensate for the predicted CBET. More work is needed to benchmark the CBET modeling, particularly the choice of saturation parameter.

For the remainder of this paper, detailed storm-window zoning of the Frustraum will not be considered further; however, more work to accommodate the above-mentioned effects is warranted.

The potential for the Frustraum to deliver 500 kJ or more of absorbed capsule energy without putting the NIF optics at undue risk of damage invites a comparison with the standard cylindrical hohlraum geometry. Specifically, what differences with the comparable cylinder can be identified that are consistent with a nearly $3×$ or more increase in $Ecap$? To answer this question a pair of simulations comparing a Frustraum and cylindrical hohlraum under the same conditions in all respects except for a shape change under the constraint of equal surface area was conducted. To preserve the same LEH size (and laser clearances), the cylinder hohlraum radius at z = 0 is adjusted to a diameter of 6.7 mm [compared with 9.0 mm for the Frustraum, see Fig. 11(a)]. Figures 14(a) and 14(b) shows a side-by-side comparison of the electron number density in units of the critical electron density $ncr$ at 8 ns. Overall the wall bubble is sharper and more pronounced in the cylinder with a nearly 2 $×$ higher average electron density. Figure 15 shows a lineout of the electron density along the vertical dotted line drawn in Figs. 14(a) and 14(b). The bubble in the Frustraum has a lower electron density due to stretching along the z axis by the glancing incidence outer beams on the (slanted) Frustraum wall. By 10 ns when the laser is turned off, the Frustraum electron density map [Fig. 14(d)] shows lower density above the capsule compared with the cylinder version [Fig. 14(c)] by at least a factor of 2. Figures 16(a)–16(d) show the electron temperature maps at the same two times (8 and 10 ns). Two notable features are that the Frustraum wall bubble is nearly one keV hotter than in the cylinder case and a higher temperature ridge to the waist (z = 0) is maintained in a Frustraum. Both of these temperature properties in a Frustraum facilitate inner-cone propagation past the wall bubble. Together with the overall lower electron density predicted for the Frustraum, these properties help explain why symmetry control in a Frustraum may potentially be easier than in a comparable cylinder.

A natural question is whether the cylinder version of the Frustraum can be tuned to give comparable symmetry. Figures 17(a)–17(c) shows the results of lowering (raising) the inner (outer) cone laser picket and trough power for the cylinder while also moving the outer (44° and 50°) cones inward by 400 μm. After 5 ns the symmetry history is unchanged, suggesting that the inner cone propagation in the cylinder remains stymied by the sharp wall bubble and high-density HDC blow-off near the waist. The net result of the cylinder symmetry is a nearly +4% offset of time-integrated ablation pressure symmetry compared with the Frustraum. Although this example hardly represents an exhaustive search for a tuned cylinder analog of a Frustraum, it provides further anecdotal support for why nominal-sized cylinders are intrinsically difficult to accommodate oversized capsules. Larger cylinders would naturally provide improved symmetry control but at the prohibitive cost of reduced drive.

A final comparison is the flow profile in a Frustraum compared with a cylinder. Figures 18(a) and 18(b) shows the flow velocity field near the LEH for the Frustraum and comparable cylinder at 6 ns. A notable feature is that the Frustraum flow is toward the symmetry axis, whereas the cylinder flow is inward. Thus, the Frustraum geometry promotes somewhat more nozzlelike flow through the LEH compared with a cylinder to possibly give a lower average channel density. This difference in flow pattern may also impact the onset and strength of CBET phenomena in Frustraums, as remarked earlier in Sec. II C.

As favorable as the simulations of an ignition-scale Frustraum appear in this work, skepticism is well justified. Consider the experience with modeling cylindrical hohlraum experiments on the NIF over the past ten years and the standard practice of invoking ad hoc time-dependent power and cone-fraction multipliers to approximate the wealth of data.¹⁰ The Frustraum geometry represents a considerable departure from the cylinder hohlraum, and one is justified to question the basis of the supporting simulations methods. Fortunately, a recent dataset with oversized capsules in a rugby-shaped hohlraum arguably provides support for the methods used to design the Frustraum ignition platform. This section describes the methods used to model these experiments while highlighting important differences and comparisons with specific Frustraum modeling.

Figure 19 shows the two NIF targets that were recently fielded and subsequently used to benchmark Frustraum modeling. The hohlraum shapes are two different rugbies with the same oversized aluminum capsule. This campaign was primarily intended to test a single-shell surrogate of the outer shell of an ignition double-shell design.²⁴ Consequently, the laser pulse shape for shot N171030 was a reverse ramp of nearly 1 MJ total energy that coupled ∼300 kJ to the nominal aluminum shell of 1.51 mm radius and 0.15 mm thickness. The subsequent lower energy shot (N180624) of ∼0.8 MJ was required to alleviate backscatter concerns with the requested rugby shape change, i.e., ∼20% smaller radius. Both targets used 0.3 mg/cc hohlraum fill of ⁴He as in the current Frustraum target designs (Sec. II). The aluminum capsules were filled with ∼7 mg/cc of DT gas to provide a measure of integrated drive on the capsule through the bang time [Gamma Reaction History (GRH)]³² and neutron yield. Gated x-ray backlighter imaging (2DConA³³) of the imploding aluminum shell assesses the symmetry of the implosion. The hohlaum drive was measured with Dante, an absolutely calibrated array of x-ray diodes with a field of view through one LEH at 37° from the symmetry axis. The laser backscatter was monitored with the standard (cone-specific) diagnostics, FABS and NBI, as well as the drive diagnostic (DrD) installed on at least one beam per quad.³⁴ 

The goal of modeling this pair of rugby hohlraums with low gas-fill density is to find a simulation methodology that is consistent with the data without the need to invoke drive and cone-fraction multipliers. A further requirement is that the outer-cone incidence angles be close to those for the Frustraum ignition designs. Figure 20 compares the outer cone incidence angles for N171030 and the Frustraum ignition design. For the 44° cones, the average incidence angles are quite close while for the 50° cones the difference is still less than 3°. Thus, the rugby geometry provides a reasonably close surrogate with respect to outer-cone glint in the absence of Frustraum data to date.

The glint model that is used in the radiation-hydrodynamic simulations consists of user-specified specular and diffuse fractional components at a laser turning point. If there is some diffuse reflection (“laser turning point scatter multiplier” or xltpsc ≥0), the distribution of scattered light is controlled by a cosⁿ(θ) dependence with n an additional free parameter. The absorption of light past the turning point is determined by inverse bremsstrahlung with the option of adding resonance absorption at near normal incidence. The degree of specular reflection at a laser turning point depends on the plasma density gradient, which in turn depends on the electron flux limit. The electron flux limit is a parameter used to approximate the effects of nonlocal thermal transport without unduly sacrificing computational speed. Its use remains the subject of ongoing debate, ranging from what global value to choose (0.02–0.15) to how nonuniform its value should be inside a hohlraum. On physical grounds, one expects variations due to the presence of MegaGauss-scale Biermann-type magnetic fields near the laser spots and the ensuing inhibited electron transport. Any enhanced laser absorption near the LEHs from the overlapping beam intensities can also lead to higher temperatures and an effectively lower flux limit. Measurements of plasma conditions at various points in a hohlraum, e.g., above the capsule and within the LEH, using microdot spectroscopy are consistent with such a picture of locally varying flux limit³⁵ that can mask myriad physical phenomenona. For this work, we will attempt to establish a range of average value of flux limit that is consistent with the pair of rugby hohlraum data. These data will comprise Dante measurements of x-ray drive, implosion symmetry, implosion bang time, and neutron yield.

Figures 21(a) and 21(b) shows the measured and simulated Dante histories for N171030 under several modeling assumptions. Figure 21(a) shows the sensitivity to pure specular reflection (xltpsc = 0) and mild diffuse scattering [xltpsc = 0.1 (n = 0)] for FL = 0.04. These data are most consistent with FL = 0.04 and xltpsc = 0.1. The principal differences between the models and the data occur between 2 and 4 ns when backscatter is occurring. For this shot moderate levels of SBS backscatter (∼4%) were inferred from the DrD,³⁴ but with a fairly large 2–3 $×$ uncertainty in the total reflectivity; the simulations used ∼9% total reflectivity. However, Fig. 21(b) shows that pure specular reflection and FL = 0.05 are also consistent with the data. To break this apparent degeneracy between FL and xltpsc, the second rugby shot N180624 was analyzed. This shot also benefitted from 2 to 3 $×$ less backscatter—presumably due to the shorter outer-cone path lengths in the smaller radius hohlraum. Figure 21(c) shows that FL = 0.04 and no diffuse scatter tracks the observed Dante history very well. Overall, the Dante data for these two shots are consistent with FL = 0.04–0.05 and xltpsc = 0.0–0.1.

The 2DConA gated imaging data also constrain the modeling parameters.³³ The imploding Al shell was backlit with Zr at 16.1 keV and imaged at ∼13 ns. The image analysis identified the best-fit contour of maximum steepest rise of transmission in the ablated aluminum. Ideally, a transmission minimum would be a truest measure of the shell distortion, but the high opacity Al shell markedly lowers the transmission and broadens its minimum—even for the comparatively high energy of Zr backlighting, cf. standard Zn backlighting at 9.1 keV. Simulation studies however show that deviations from round of the transmissivity inflection point contour correlates well with the transmission minimum contour that tracks the inner radius of the Al shell. A Legendre decomposition of the transmissivity inflection point contour was then performed and compared with simulations. Figures 22(a) and 22(b) show the measured average radius $a0$ of this contour alongside the postprocessed modeling results. The data are consistent with FL = 0.04–0.05 and pure specular reflection to modest diffuse scattering. Figures 22(c) and 22(d) show the lowest order distortion a₂ for the same two shots alongside the modeling curves. The data are consistent with FL = 0.035–0.04 and minimal diffuse scattering. It should be noted that Figs. 22(c) and 22(d) strongly rule out the high values (0.15) of FL needed for the high-flux model.¹² Also shown in Figs. 22(e) and 22(f) are the fourth Legendre distortion coefficients, showing overall consistency of the modeling with the data for FL = 0.04–0.05 and low diffuse scattering.

The next set of data to help constrain the modeling is the measured nuclear bang time from the gamma reaction history (GRH) diagnostic.³² Figures 23(a) and 23(b) displays the sensitivity of the modeling with flux limit and scattering fraction compared with the data for the two rugby shots. Both shots are consistent with a flux limit in the range of 0.03–0.04 and a (isotropic) diffuse scattering strength xltpsc = 0.0–0.1.

The final set of data for comparison with modeling is the neutron yield. The caveat with using neutron yield constrain modeling is the inherent challenge with representing instability growth and mix from first principles and avoiding ad hoc assumptions. Fortunately, the aluminum shell used in the two rugby hohlraum shots of interest led to a (radial) fuel convergence of less than ∼7, which is expected to lead to minimal yield degradation from mixing of aluminum shell material and DT fuel. All mix models to date use one or more free parameters that are constrained by applying the model to characteristically large sets of data. Here, we posit an alternative mix model that is parameter free provided the instability growth is dominated by Rayleigh-Taylor.

The proposed mix model is based on the “fall line,” or tangent line trajectory from the unstable interface at peak velocity.^36,37 The physical reasoning is that all infalling material should remain behind the fall line based on causality if the mixing process is strictly hydrodynamic in origin. The main challenge in applying a fall-line mix model is determining the effective penetration fraction, i.e., the fraction of the distance from the interface to the fall line where full atomic mixing is presumed to occur. This is generally accomplished by considering a body of data and choosing a constant penetration fraction η that best agrees with a prescribed set of observables. To sidestep this practice an alternative to the usual fall-line model is introduced. Consider a parabolic trajectory $R(t)$ of the unstable interface close to the time of stagnation $R′=0$⁠,

where g is the (constant) acceleration, $υimp$ is the peak implosion speed occurring at time $t0$ and radius $R0$⁠. An assumption of a parabolic trajectory should well hold when the time of stagnation $tf$ is close to deceleration onset, justifying a second order Taylor approximation to the trajectory near $tf.$ The fall-line trajectory follows $Rflt=υimp·t−t0+R0$⁠, giving for the distance from the trajectory to the fall line

where $g=υimp/tf−t0$ is used. The penetration fraction enters as a multiplier $0≤η≤1$ on Eq. (4). If we invoke well-developed Rayleigh-Taylor mixing that is well known to generate the mixing scale lengths³⁸ 

we can then analytically determine the penetration fraction. Here, j = b or s denotes the bubble (b) or spike (s) side of the interface, $αj$ are coefficients constrained by experiments and simulations, and $At$ is the Atwood number of the (clean or unmixed) interface. Equating Eq. (4) with an applied penetration fraction η and Eq. (5) gives an analytical form for the penetration fraction on either side of the interface

Equation (6) is then used in a 1-D radiation-hydrodynamic simulation with time-dependent Atwood number to gauge the impact of interfacial mix on the neutron yield from the Al encapsulated DT fuel. The only arguable assumption in this parameter-free analysis is that the mix be Rayleigh-Taylor dominated. Figures 24(a) and 24(b) shows the sensitivity of yield flux limit and diffuse scattering compared with the data. For both rugby hohlraum shots the data are consistent with a range of flux limit from 0.035 to 0.045 with the lower value correlating with weak diffuse scattering (xltpsc = 0.1). The mix modeling obtains a yield reduction of nearly 20% for this pair of low convergence implosions. Weaker mix will only lead to slightly lower inferred values of flux limit.

In summary, the totality of rugby hohlraum data with low gas-fill density are consistent with a flux limit in the range of 0.035 to 0.045 and weak diffuse scattering. Analyzing the rugby hohlraum data for gauging the fidelity of Frustraum modeling is justified on the basis of similar hohlraum gas-fill density and outer-cone beam angle incidence. The main caveat is that the rugby pulse shape is reverse ramped and leads to a high-adiabat implosion, in contrast to the lower adiabat, multishock Frustraum ignition designs (see Secs. II, IV, and VI).

The capsule design presented in Sec. II A represents a compromise between shell stability and simplified Frustraum performance. The shell designs are thinner than an Euler-based scale up from the standard 1.0 mm-scale capsule designs and allow an increased risk of shell breakup from hydrodynamic instability. On the other hand the associated higher implosion velocities promote increased 1-D performance margin (or ITF, more precisely) and a reduced risk of hohlraum filling when symmetry control and laser backscatter become potentially problematic. The other design trade-off is how much preheat to allow in the capsule designs: A high adiabat design helps control hydrodynamic instability growth but at the expense of 1-D margin. However, the Frustraum designs presented in Sec. II allow for a significantly larger (∼1.4 $×$⁠) capsule and an accompanying big increase (∼3 $×$⁠) in performance margin. The strategy is to leverage this available boost in margin to provide a significant increase in 2- and 3-D margin for added stability control (from a high fuel adiabat) while maintaining sufficient 1-D margin and potentially benign hohlraum behavior.

Figures 25(a) and 25(b) shows the pie diagram of the nominal and large scale capsule designs^5,16,39 used to compare stability properties over a range of fuel adiabats α. The large-scale capsule shown here differs from Fig. 3(a) by the use of modestly thicker fuel and ablator layers for greater shell stability without sacrificing 1-D margin $M1D=ITF−1∝α−1.91υimp3.58Pa1.16 Ecap−1:$⁵ The lower peak implosion speed $υimp$ is more than balanced by high capsule energy absorbed and a lower fuel adiabat α from improved pulse shaping. The drive history for the Frustraum is meant to minimize the “coast time” of the implosion, defined as the difference in laser turn off time and instant of peak implosion speed, cf. Figs. 3(b) and 4. Consequently, the ablation pressure scaling for the above 1-D margin formula should be evaluated near peak drive conditions.

Table II compares several performance metrics of the nominal design and two large capsule designs with intermediate and high fuel adiabat.^16,39 For all three cases as shown the requested laser energy and power are about 10% below the rated maximum values for the NIF, leaving some headroom for overcoming modest drive deficiencies should they arise. The favorable Atwood numbers on the fuel-ablator interface for the larger scale capsules in a DU Frustraum result from the several percent lower 1.8–3 keV x-ray preheat fractional levels (“Au M-band fraction”). Most of the M-band preheat is generated by the outer beams incident on the hohlraum wall. A lower (glancing) incidence angle in the Frustraum reduces the laser intensity on wall and helps reduce the preheat levels, affording more control of fuel preheat with the nominal amount of ablator dopant (0.3 at. % W) and a favorable A_t. The other takeway from Table I is that the generalized Lawson criterion GLC $≈2.1×ITF0.6$ significantly improves with size,^5,40 even providing appreciable margin at an ultrahigh adiabat of 5.75.

Figures 26(a)–26(c) suggest significant robustness of the large capsule designs to various degradations of 1-D margin: Deficit in peak velocity of the shell (unablated mass, including DT fuel), specific preheat of DT fuel and mixing of doped ablator material with hot-spot fuel.^5,39 Overall, the large capsule can withstand nearly twice a given amount of 1-D degradation and still maintain half yield.

Although Table I indicates favorable or neutral Atwood number on the DT-ablator interface, 2-D multimode simulations show a high degree of ablation-front growth and the potential for significant feed-thru to this interface. Figures 27(a)–27(c) show simulated ablation-front growth-factor spectra for the nominal and large capsules at intermediate and large fuel adiabat α.^5,16,40 A near doubling of the adiabat still shows significantly larger growth than for the nominal capsule. The reason for this increased growth is lower ablation-front stabilization from the lower hohlraum temperature in the Frustraum, a higher in-light aspect ratio, and reduced x-ray preheat for shorter (and destabilizing) density-gradient scale lengths at the ablation front.⁵ This increased growth is largely offset by the low Atwood numbers in the capsule designs at the fuel-ablator interface. This is confirmed by separate growth factor simulations at the fuel-ablator interface and by 2-D multimode simulations^5,39 as shown in Figs. 28(a) and 28(b). The multimode simulations include mode numbers 12–1000 with snapshots of the material distribution and density shown at the instant of peak shell velocity.⁵ The surface spectra used in the multimode simulations include a (measured) DT ice RMS roughness of 0.96 μm, an inner ablator RMS roughness of 9.4 nm and an outer surface roughness of 10 nm. The clean fuel fraction reaches 96% in the large capsule design, compared with 79.6% for the nominally sized capsule. The clean fuel fraction can be further improved but at the considerable cost of reduced 1-D performance margin. Moreover, this trade-off between stability and 1-D margin is comparatively tight at the nominal scale.

The presence of hydrodynamic jets from capsule-fielding engineering features such as capsule tent supports and fill tubes is known to degrade performance at the nominal scale.^40,41 The larger capsules should be more immune to these “machining” defects since the solid angle of such features is comparatively smaller and the shell is thicker for weaker feed-thru of such perturbations. Figure 29 compares the effect of a 50 nm-thick tent support feature on the large-sized capsules. The tent imprint is weak and leads to essentially no yield degradation.^5,39 Figures 30(a) and 30(b) show the distribution of materials and density at the time of peak implosion velocity for the large capsule and shot N170601 that used a 1 mm-scale capsule—both with a 5 μm-thick fill tube. The simulations are used to monitor the amount of fill-tube material that is injected into the hot spot at ignition. For the large-scale capsule design the amount of fill-tube mix is only 6 ng, compared with 42 ng for a nominal-sized capsule. This amount of mixing yields essentially no yield degradation for the large capsule, but less than 10% yield loss for the nominal capsule design.

The data available for ICF experiments using hohlraum shapes other than a cylinder at low gas-fill density are sparse. In the case of cylinders the preponderance of the data to date have led to the use of multipliers to account for errors in the modeling. Section III examined a pair of rugby shots and found overall good consistency with the modeling used for the Frustraum design. The largest difference between the rugby data and the Frustraum ignition designs presented here is the pulse shape used: a ∼5 ns long reverse-ramp for the rugby vs the ∼9 ns duration high contrast, ignition pulse shape for the Frustraum. Prudence and best practice dictate that a recovery strategy be in place in the event that future Frustraum data show a significant discrepancy with modeling. The largest perceived risk is that the glint modeling used in the Frustraum simulations is not as beneficial as indicated in Sec. II, leading to oblate implosions and a barrier to ignition. This section investigates recovery strategies and options for compensating for differences between the data and modeling as they arise.

Figures 31(a) and 31(b) show the effect of changing only the wall angle in a Frustraum on $a2/a0$ drive symmetry and drive temperure history. The simulations were applied to a scaled-down version of the Frustraum ignition designs described in Sec. II to accommodate the universal thermal mechanical package (UTMP) on the NIF that currently allows 7 mm diameter hohlraums at most. Figure 31(a) exhibits the ratio of running integral (or fluence) of $a2/a0$ vs time and an early-time rendering of the hohlraum geometry. Just a 2.4° difference in the tilt of the Frustraum wall leads to a ∼2% change in time-integrated asymmetry. Thus, if the implosion comes in more oblate than simulations predict, a larger tilt angle can help offset if not entirely neutralize the observed asymmetry. However, this correction comes at the cost of slightly more hohlraum wall area and reduced drive as shown in Fig. 31(b). Also shown is a snapshot of the implosion for the two Frustraum wall angles, demonstrating the favorably large leverage on symmetry if needed.

Another potential lever on controlling Frustraum symmetry is the use of a cone fraction picket during the final rise to maximum laser power. This method is meant to briefly turn up the inner cone power to prepare a less-absorptive channel of lower-density, higher temperature plasma through the intervening wall bubble if present. Figure 32(a) shows the instantaneous inner-cone power fraction vs time with a brief spike in inner cone power at ∼3.5 ns. Figure 32(b) shows the inner and outer cone powers with and without application of the cone-fraction picket. Finally, Fig. 32(c) depicts the resulting ablation pressure asymmetry history and the modest benefit on overall symmetry improvement with the cone-fraction picket.

A third option for helping control Frustraum symmetry is varying the gas-fill density. We have focussed so far on the value of 0.3 mg/cc ⁴He fill gas, but there is recent evidence that lowering the fill-gas density may help yield a prolate implosion if the Frustraum data were to show a tendency toward oblateness.¹⁷ 

A fourth technique for controlling drive symmetry is the use of a small wavelength separation between the outer and inner cones to effect energy transfer. This is an established procedure that has been used on cylinders to favorably impact time-integrated symmetry.²⁸ 

The biggest perceived threat to Frustraum performance is premature inner-beam stoppage near the wall bubble or ablator-wall capsule. Near-vacuum (0.03 mg/cc) hohlraum fills in cylinders have previously shown more prolateness than the fluid-based codes predict. This trend is believed due to interpenetrating ablator and wall plasmas, reducing the laser absorption in this region and facilitating inner beam propagation to the hohlraum midplane.¹⁵ Unfortunately, this finite ion mean-free-path effect is not currently captured in mainline fluid-based simulations unless ad hoc assumptions are made. The idea would be to leverage this portion of parameter space to facilitate symmetry control, but at the cost of losing some fidelity in predictive capability. This has been the recent case with use of 0.15 mg/cc ⁴He fills in cylinders where the implosion was measured to be more prolate than predicted.¹⁸ The idea is to exploit this trend in Frustraums if needed to augment the two techniques for symmetry control described above.

A recovery option for overcoming insufficient hohlraum symmetry control at late times is to shorten the laser pulse length and limit the degree of hohlraum filling. This strategy is often implemented by increasing the laser picket strength, shortening the duration of the laser trough, and maintaining a similar peak power level and overall laser energy. At the same time, the associated fuel adiabat increases, providing more margin to instability growth but at the cost of 1-D performance margin (see Sec. IV). At nominal scale the margin for such an increase in fuel adiabat puts ignition in jeopardy, but the larger scale proposed here at ∼500 kJ E_cap provides adequate margin to such a higher adiabat—and even the prospect for overcoming unknown but suspected sources of preheat-driven adiabat increases inferred in implosion experiments to date.⁴² Here, we provide some hohlraum simulation results for a design that uses a ∼2.5 ns shorter laser pulse length than used in Sec. II.

We define a hohlraum filling time $τf$ to be when the wall bubble has reached an average hohlraum radius R. The speed of the bubble is approximated as the ion acoustic speed of the ablated Au front $Z+3Ti/TekBTe/M$⁠, where Z is average the charge state of a Au ion, $kB$ is Boltzmann's constant and M the ion mass. Here, we use the ion acoustic speed for the bubble evolution instead of the wall sound speed in order to include (plasma) ambipolar diffusive effects in the Au blowoff. Since the Au bubble is directly driven by the laser (in constrast to x-ray driven), the electron temperature follows the scaling $Te∼Iλ22/3∼EL/τLAλ22/3$⁠, with A the beam area on the hohlraum wall. Thus, $τf∼τL1/3/EL1/3Z+11/2$⁠. If we next define a normalized filling time $η≡τf/τL$⁠, we find the favorable scaling of hohlraum filling with laser pulse length: $η ∼ 1/τL2/3EL1/3Z+11/2$⁠. This scaling may be applied to the picket plus trough portion of the laser power history or to the entire pulse length. In the former case, compression of the trough and increased picket strength are often used to shorten the implosion time and raise the fuel adiabat. For the example of a shorter $τL$ design presented in Figs. 33(a) and 33(b), $EL$ is ∼67% greater and $τL$∼40% shorter duration trough, giving still a ∼10% increase in η compared with Fig. 3(b) after accounting for the higher $Z ∼ TR0.45$ (Ref. 43) found in the higher laser picket case. The total energy is a manageable 1.63 MJ with a peak picket power of 100 TW and maximum power of 460 TW. The inner cone fraction is at the ideal 1/3 fraction at peak power for efficient use of the NIF. The Frustraum geometry used here is identical to that shown in Figs. 5(a)–5(d), while the capsule is identical to that depicted in Fig. 25(b). Figures 34(a) and 34(b) show the two lowest-order ablation pressure Legendre coefficients vs time. Note the compressed scale and the minimal temporal variations. Figure 34(c) depicts the fairly symmetric fuel configuration at the time of ignition onset – when the central gas reaches 12 keV. The capsule yield in the 2-D integrated hohlraum simulation is ∼13 MJ, the fuel adiabat is 4.6, the mass remaining in 1-D simulations is 6%, and the peak radiation temperature is 288 eV.

We have focussed on E_cap designs of nearly 500 kJ until now, but it is worth asking what is the credible limit for a Frustraum driven on the NIF with ∼2 MJ and ∼500 TW available energy and power at 3ω. In addressing this question, we can gather some knowledge over how close to a performance cliff we are for 500 kJ-class E_cap Frustraum designs and how much further can we realistically advance this concept.

Figures 35(a) and 35(b) show a pie diagram of a 3.5 mm outer diameter capsule that absorbs over 650 kJ in 1-D simulations. The ablator and fuel thicknesses are close to those shown in Fig. 25(b) except for the extra 0.5 mm in diameter. The pulse shape is also identical to that shown in Fig. 3(b) to lower the risk of premature hohlraum filling and accompanying loss of symmetry control. The trade-off in this design is a larger shell aspect ratio for increased exposure to hydo-instability, but greater peak velocity for higher 1-D performance margin. The resulting fuel shape at ignition is shown in Fig. 35(b) with still a prolate shape overall but with higher mode perturbations, e.g., $ℓ=8,$ that are evident in the imploded configuration. Such higher mode structure can be expected due to the reduced case-to-capsule ratio and the resulting decrease in smoothing of lower-order radiation modes. Figures 36(a) and 36(b) show the 2nd and 4th Legendre coefficients for describing the ablation pressure nonuniformity. The lowest order (⁠$ℓ=2)$ asymmetry is mildly varying over the entire duration of the laser pulse as desired, while the $ℓ=4$ asymmetry shows more appreciable variations in time due to the reduced case-to-capsule ratio (CCR). The fuel adiabat is ∼3.0, including 2-D flux asymmetry and shock mistiming. The hot-spot convergence (defined as the initial capsule radius divided by the 1 keV fuel contour radius when the central ion temperature exceeds 12 keV) is less than 25, reflecting the lower fuel-density requirement for ignition [see Fig. 35(b)] at larger scale, i.e., ∼500 g/cc vs ∼900 g/cc at nominal scale. The shell used in the 2-D simulation has nearly the same thickness as at the smaller 500 kJ $Ecap$ scale, thereby providing more exposure to hydro-instability growth and potential shell breakup. This risk could be remedied by use of a thicker shell, albeit at the cost of possible loss of late-time hohlraum symmetry control. The peak implosion speed is a robust ∼440 um/ns, more than enough to accommodate a modestly thicker shell as needed.

If we increase the radius of the capsule by another 100 μm while still maintaining the same shell thickness, the integrated hohlraum simulations (with unchanged pulse shape, Frustraum geometry and laser pointing) still result in ignition but with degraded drive symmetry. Figures 37(a) and 37(b) show the $a2$ and $a4$ symmetry history, indicating that the challenge is controlling the large $a4$ symmetry swings near the time of rise to peak power (∼±10% over 2 ns). The $a2$ asymmetry history is little unchanged from the smaller scale version shown in Fig. 37(a). Figure 37(c) shows the resulting mildly oblate implosion at the onset of ignition. The 2-D yield is 27.6 MJ compared with an ideal 1-D yield of 45 MJ. The peak implosion velocity in 1-D is still high at 427 μm/ns despite the reduced peak channel radiation temperature of 280 eV. The rather high implosion velocity results from the longer duration of acceleration at this scale despite the lower peak drive temperature.

If we further increase the outer radius of the capsule by another 100 μm to 1950 μm while keeping the same shell thickness, the integrated hohlraum simulations (with identical pulse shape, Frustraum geometry and laser pointing) now show a strong degradation of drive symmetry. Now, the $a2$ and $a4$ asymmetry leads to a highly oblate implosion that barely fails to ignite. Figures 38(a) and 38(b) show that the $a2$ time-dependent asymmetry is modestly worse than in Fig. 37(a), but the most notable change is the large $a4$ swing in time. Such symmetry swings impart nonradial flows that lead to largely distorted fuel cores even when the time-integrated $a2$ and $a4$ drive asymmetries are nearly zero by the time the capsule implosion has become highly ballistic [see Fig. 38(c)]. The peak drive temperature in the simulations has now dropped to 275 eV as would be expected with such a large capsule in the same sized hohlraum, but the peak (1-D) implosion speed remains high at ∼430 μm/ns.

The conclusion from this scaling exercise is that a decreasing CCR exacerbates the $a6$ and $a8$ drive asymmetry modes, but more so for $a4$ and requires remedial measures. A first strategy would be to move the outer beams further out toward the LEH, but this likely would worsen the $a2$ asymmetry. A larger Frustraum would help lower the CCR, but the price of a lowered drive may be too severe. Overall, this exercise suggests that a maximum operating space for the Frustraum in a 2 MJ-class laser facility is around 700 kJ of absorbed energy and a coupling efficiency of nearly 40%.

A novel hohlraum geometry is proposed that minimizes the wall surface area for reduced Marshak wave losses while providing a larger volume over the capsule waist to accommodate significantly larger capsules than the standard 1 mm (radius) scale. The ability to potentially field larger capsules in ignition experiments that may absorb ∼0.5 MJ or more of soft x rays enables significantly higher performance margin to possible modeling errors in equations of state, opacity, thermal transport, x ray and hot electron preheat, and deviations from the hydrodynamic limit, i.e., kinetic effects. The candidate Frustraum geometry is simply constructed from two truncated conical halves (or frusta) joined at the waist. A major challenge with using such a large capsule in a standard cylinder hohlraum would be ensuring sufficient symmetry control at late times as the ablator plasma stagnates against the hohlraum wall and impedes inner-beam laser propagation to the waist. A Frustraum geometry delays the interaction of capsule blow off and wall material to provide a channel of relatively low-density plasma for more robust laser propagation. In addition, the outer laser cones strike the ends of the Frustraum at a more glancing angle compared with a cylinder and are more prone to specular reflection or glint. This first-bounce reflected laser energy may heat the channel ahead of the inner beams and reduce inverse bremsstrahlung absorption within the hohlraum wall bubble, according to radiation-hydrodynamic simulations. This potential scenario for boosted symmetry control from the outer cones reflecting off a glancing angle hohlraum wall depends on the choice of electron flux limit used in the radiation-hydrodynamic simulations. A lower value (0.03–0.05) gives rise to steeper temperature and density gradients to promote specular reflection. Recent data from the National Ignition Facility using oversized aluminum shells in rugby-shaped hohlraums¹⁴ come closest to approximating a Frustraum geometry and are consistent with such a range of flux limit in matching the simulated Dante radiation temperature history, the backlit trajectory of the Al shell, neutron yields and implosion times. Application of this simulation methodology to hot-spot ignition designs in a Frustraum show effective symmetry control and sufficient drive to enable high yield, moderate convergence implosions. The high-density carbon ablator capsule designs used for the Frustraum are thinner than an Euler-type scale-up of standard 1 mm-scale capsules, providing added margin to hohlraum filling with commensurately short 8–9 ns laser drives—but at a higher risk of shell breakup. However, multimode hydro-stability simulations show high margin to perturbation growth, including short-wavelength surface modes, as well as to the impact of capsule-fielding engineering features such as fill tubes and capsule-support tents. Other advantages of the Frustraum include relatively shorter outer-cone path lengths for mitigating laser glass damage risks from stimulated Brillouin scattering, and lower x-ray preheat generation from the glancing incidence outer cones near the hohlraum wall. A lower preheat fraction translates into requiring less mid- and high-Z dopant in the capsule ablator for higher 1-D performance margin. The envisioned main risk to the Frustraum design is loss of inner-cone propagation at late times from inaccurate modeling of the absolute strength and relative balance of specular and diffuse reflection of the outer cones at first laser bounce. However, simulations suggest that adjusting the obliquity of the Frustraum wall is a robust lever for symmetry tuning. Another strategy to this end is to shorten the laser pulse and reduce the degree of hohlraum filling. A design that uses a ∼6.8 ns laser pulse length is able to preserve good symmetry and high yield (∼13 MJ) along with providing a high fuel adiabat (∼4.6) for high margin to preheat.

The largest uncertainty in whether the Frustraum can deliver the needed drive symmetry is the degree of specular glint in the simulations. Two NIF shots with oversized Al shells in a rugby hohlraum geometry are consistent with low values of flux limit and the associated high fractions of specular glint. However, this pair of shots employed a reverse-ramp pulse shape—in contrast to low-adiabat, stepped laser pulses needed for achieving central hot-spot ignition. Nevertheless, there is recent experimental evidence on the NIF that the levels of glint inferred for an ignitionlike pulse shape may be substantially greater than what mainline simulations using a high flux limit (FL = 0.15) predict.⁴⁴ Specifically, the inferred density and temperature gradients were claimed to be stronger than what simulations give – which is consistent with a lower flux limit and its associated higher levels of glint.

Given these caveats in the integrity of the mainline simulations used to model Frustraums, the obvious step forward is to field a Frustraum on the NIF and baseline the modeling tools for this novel hohlraum geometry. The ICF program will shortly be conducting testing of this novel hohlraum geometry in order to address the accuracy of the modeling with regard to key physics issues and to test potential mitigation strategies. A parallel effort will also perform basic physics experiments at smaller scale laser facilities, e.g., Omega at the Univ. Rochester, to improve our understanding of the balance of specular vs diffuse reflection in subscale hohlraums as a function of laser intensity, angle of incidence and radiation temperature. We have outlined some robust recovery strategies in the event that the data differ significantly from the model predictions. If the data and simulations turn out to be in reasonable agreement, a path forward for accessing high capsule absorbed energies and substantially improved performance margins may be at hand.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 Lawrence Livermore National Security, LLC and supported by No. LDRD-17-ERD-119. This document was prepared as an account of 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 an legal liability of reponsibility for the accuracy, completemess, 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 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.