We present quantitative motivations and assessments of various proposed and ongoing directions to further improving yields and target gain of igniting indirect-drive implosions at the National Ignition Facility (NIF). These include increasing compression and confinement time, improving hohlraum and ablator efficiency, and further increasing peak power and laser energy. 1D hydroscaled simulations, augmented by analytic 1D theory, have been used to project yield improvements for each of these implosion optimization tracks, normalized to the best current performing 4 MJ shot. At current NIF capabilities of 2.2 MJ, 450 TW, we project several paths could reach 15 MJ yield levels. We also expect several key implosion physics questions will be addressed in attempting to reach this yield level. These include demonstrating to what extent lower adiabat designs leading to higher compression will increase gain and efficiency, and whether we can reduce residual kinetic energy and ablator-fuel mix that is probably limiting the current burn-up fraction. For an envisaged NIF upgrade to E_{L} = 3 MJ at fixed 450 TW peak power, scaling capsule size and fuel thicknesses faster than pure hydroscaling should allow for yields that could reach up to 60–80 MJ, depending on the efficiency gains realized in increasing deuterium-tritium fuel thickness, reducing hohlraum losses, and switching to lower Z ablators. The laser-plasma instability and beam transmission scaling in these larger hohlraums is shown to be favorable if the spot size is increased with hohlraum scale.

## I. INTRODUCTION

In inertial confinement fusion experiments at the National Ignition Facility (NIF), a spherical shell of deuterium-tritium (DT) fuel is imploded to reach the conditions needed for fusion, self-heating, and eventual ignition. In the principal approach to indirect-drive, x rays created by a laser-heated hohlraum ablatively drive a spherical shell surrounding a layer of frozen DT fuel (capsule). Since ignition (thermal instability according to Lawson criterion^{1}) and target gain > 1 (fusion energy > laser energy into target) were achieved recently,^{2} interest has now evolved beyond the pre-ignition research directions.^{3} The major current goals are further improving neutron yields and performance robustness using current NIF capabilities, and extrapolating to what might be achievable if a proposed NIF Extended Yield Capability (EYC) (an upgrade adding 40% more laser energy, from 2.2 to 3 MJ) is realized.

Increasing neutron yields will open up new high energy density (HED) science regimes and applications.^{4,5} For example, the large neutron fluxes for yields above 10 MJ will allow access to astrophysically relevant excited state and nonlinear reaction rate nuclear physics in the presence of dense plasma screening.^{6,7} More robustly burning designs may permit not just samples at the hohlraum wall,^{8} but doping low Z capsule ablators with otherwise ignition quenching radiochemical tracers.^{9} Probing the limits on achievable driver efficiency and target gain (ratio of fusion to laser energy) will aid in setting requirements for a future high yield facility, and is synergistic with the renewed interest in inertial fusion energy.^{10} Accompanying the high neutron flux is intense alpha particle heating of the fuel, driving a blast wave that leads to strong ablator reradiation.^{11} This has the potential to create hohlraum temperatures greater than by just using the 500 TW NIF laser, for studying extreme HED regimes in material opacity and equation of state.^{5,12} In all these ventures, increasing neutron yields can improve data signal-to-noise ratio and counting statistics, as well as allow for larger or more samples that can be simultaneously exposed to a given neutron or x-ray flux.

In this paper, we use 1D hydroscaled simulations augmented by analytic 1D theory to project yield improvements for each implosion optimization track, normalized to the best current performing 4 MJ shot. These tracks include increasing compression and confinement time, improving hohlraum and capsule ablator efficiency and increasing laser energy and peak power. Key tradeoffs emerge from simulations and 1D analytic theory that help explain the experimental strategies. At current NIF^{13,14} capabilities of 2 MJ, 450 TW, we have identified several paths to reaching 10–15 MJ yield levels, and presumably even higher by combining successful design optimizations. We discuss the challenges in each approach, and the key questions that need to be addressed. These first include principally laser pulse shape strategies, testing to what extent lower adiabat designs leading to higher compression will increase gain and efficiency, and whether we can reduce residual stagnated fuel non-uniformities and ablator-fuel mix that are probably limiting the current burn-up fraction. A second set of approaches seek to improve hohlraum and capsule efficiency, for which the challenges include maintaining sufficient drive symmetry control, mitigation of laser backscatter losses and the effects of capsule engineering features such as the support film.

In the last sections, we present an example scaling strategy for utilizing a proposed NIF upgrade to E_{L} = 3 MJ at fixed 450 TW peak power. The hohlraum and capsule radii are scaled as E_{L}^{1/2} while the DT fuel thickness is scaled even faster, linearly with E_{L}, compared to the usual hydroscaling as E_{L}^{1/3}. We show how this self-consistently leads to designs with ≈5%–10% lower hohlraum radiation temperatures T_{r}, and consequently capsules imploding with ≈5%–10% lower implosion velocities (v_{imp}) while still meeting the conditions for ignition. The larger capsule and fuel thicknesses should allow for yields that could reach up to 60–80 MJ, depending on the efficiency gains realized in reducing hohlraum losses and switching to lower average Z ablators (e.g., glow discharge polymer plastic, CH, vs current, high density carbon, HDC). Simple 1D Stimulated Brillouin Scattering (SBS) gain scaling in these larger hohlraums is shown to be favorable (i.e., less SBS losses) if the laser spot size is increased with hohlraum scale. The scaling for drive symmetry control is also favorable, with hotter coronal plasmas predicted due to reduced conduction losses in larger hohlraums allowing for more inner cone transmission. The challenges here will be controlling the radiation drive symmetry in smaller, more efficient hohlraums and for lower density more efficient ablators such as CH that require longer laser pulse drives. We emphasize throughout the importance attached to testing laser-plasma instability (LPI), hohlraum energetics and capsule ablator scalings and sensitivities using both integrated and focused experiments.

In Sec. II, we describe and validate the use of 1D theory and simulations to extrapolate yields as function of drive laser energy and capsule size. In Secs. III–V, we assess through simple analytic models, normalized to the existing highest performance implosions, the expected performance improvements through increased compression and confinement, hohlraum and ablator efficiency and laser energy, respectively. We summarize in Sec. VI, followed by three appendices with more details on derivations.

## II. 1D THEORY AND HYDROSCALING

_{imp}(typically greater than 330

*μ*m/ns at NIF-scale) is needed to bring the stagnated DT hotspot ion temperature T

_{i}above 4–5 keV. Second, the scale or inner radius R of the capsule ablator has to be sufficient to provide a stagnated DT hotspot areal density ρr

_{hs}above ≈0.3 g/cm

^{2}, comparable to the alpha range. The combination of those two

^{15}conditions will allow for a sufficient burn rate such that alpha heating will overcome radiative loss. The third condition is based on the requirement of having sufficient fuel and remaining ablator areal density ρr at stagnation, above ≈1.2 g/cm

^{2}based on existing data and 1D simulations,

^{16}so that the burn rate can exponentiate and propagate into the cold fuel before target disassembly. These conditions are illustrated by the following 1D model

^{16,17}of an indirectly driven implosion, validated by data and simulations:

_{no-α}is defined as the DT neutron yield in the absence of alpha heating. M

_{imp}is the mass inside the ablation front at peak velocity and ρr is the areal density in the absence of alpha heating of the combined DT and remaining ablator at minimum volume. The model is cast in terms of Y

_{no-α}, to avoid the complication of the power law dependencies increasing with increasing alpha-heating.

^{18,19}The middle expression assumes self-similarity in imploding shell aspect ratio such that M scales as R

^{3}. The final expression assumes ρr/R ∼ v

_{imp}/α

_{if}, where α

_{if}is the in-flight adiabat, defined as ratio of the mass-averaged DT fuel pressure to Fermi pressure

^{20}at peak velocity. It is based on an average sensitivity from 1D simulations and analytic derivations,

^{19}for which the power law dependencies on v

_{imp}and α

_{if}vary by ≈30%.

We do not need to dwell more on the exact Y_{no-α} dependencies since we are ultimately interested in scaling and extrapolation from current igniting capsules. However, it should be noted that the scaling with R and v_{imp} according to Eq. (1) has been previously shown to describe^{21} well the NIF layered implosion scaling performance envelope in the low alpha heating regime. However, for CH ablators, while the expected inverse scaling of ρr and α_{if} has been observed,^{22} the expected yield scaling with α_{if} (as it drops below ≈ 2) has not been demonstrated,^{17} even for the best performing implosions. This has been ascribed to large hydroinstability growth of capsule surface perturbations. Thus, the near-term path forward favors α_{if} > 2.5 implosions. By contrast, the assumed sensitivity of ρr to α_{if} for higher adiabat, high density carbon (HDC) ablator implosions has not been observed,^{22} even when well shock-timed. This has been attributed to potential ablator-fuel mix for crystalline HDC^{23} that is a current active area of study^{24} and mitigation.^{25–27}

To include alpha heating^{28–30} leading to ignition, propagating burn and eventually yield or fuel burn-up saturation, we use a family of igniting hydroscaled^{31–36} 1D clean capsule-only HYDRA^{37} simulations. They are based on a generic v_{imp} ≈ 400 *μ*m/ns HDC ablator design with M_{fuel} = 181 *μ*g, initial ablator thickness ΔR = 77 *μ*m and inner ablator radius R = 1.05 mm. Hydroscaling keeps v_{imp}, α_{if} and initial aspect ratio ΔR/R of the capsule fixed, such that the final energy density or pressure of the stagnated hotspot and shell is also held fixed, in the absence of alpha heating. In the case of an integrated (hohlraum and capsule) simulation, the x-ray drive power profiles are a fixed function of the ratio t/R, with the drive pulse length τ scaling as R, and with small corrections to account for material properties such as opacity that do not scale. This also ensures that the coast time, defined as time between laser turn off (the start of T_{r} dropping on Fig. 1) and minimum core volume or bangtime, that together with peak T_{r}, determines the important^{38–40} parameter of ablation pressure just before the rebound shock passes through the shell, also scales with R. In addition, for a given v_{imp}, the capsule trajectory can be assumed to remain self-similar throughout since τ ∼ R/v_{imp} ∼ R.

In practice, we use capsule-only simulations that allow for a simplified, validated version of hydroscaling. First, the implosion at a given α_{if} is simulated at the baseline R = 0.95 mm scale using the x-ray drive shown in Fig. 1, parametrized as an equivalent blackbody flux of temperature T_{r}. To ensure perfect hydroscaling, as soon as the implosion reaches peak velocity, all of the physical dimensions are scaled up by the same factor, and then the simulation is allowed to proceed until full burn completed.

Yields are then calculated as functions of scale R and α_{if} (by varying the preheat level) for a fixed NIF relevant v_{imp} ≈ 400 *μ*m/ns. The preheat is added to the DT fuel 200 ps before peak v_{imp} is reached, over a 100 ps duration, before the rescaling described above is applied. Preheat has been added in steps of 1.5 × 10^{5} J/g when the DT fuel is at T ≈ 10 eV, thus raising the entropy Δs in steps of ≈0.015 GJ/g/keV. A published fit^{20} between DT entropy and adiabat is then used to convert entropy s to α_{if}. One such series of simulations at α_{if} = 3.1 with Δs = 0.015 GJ/g/keV is shown in Fig. 2 as a curve of yield vs initial inner capsule ablator radius R. It is chosen because it approximately bisects the two highest performing and igniting^{2,41,42} HDC shots N221204 and N230729 at R = 1.05 mm.

Effectively, we have normalized a 1D hydroscaling curve to the best performing implosions. Since, as we will show later, the curves are self-similar on the ignition cliff with changes in in-flight adiabat equivalent to just a multiplicative factor in scale, the exact choice of α_{if} is not critical. The overlaid gray contours of burn-up fraction scale as mass of DT fuel M_{fuel} ∼ R^{3}, where M_{fuel} ≈ 220 *μ*g for the two data points. Also shown as alternate x axes are the hydroscaled values of laser energy E_{L} ∼ R^{2}τ ∼ R^{3} and ρr ∼ R, normalized to the experimental conditions, E_{L} ≈ 2 MJ and ρr ≈ 1.25 g/cm^{2} at R = 1.05 mm. The justification for the ρr estimate in the absence of alpha heating is included in Appendix A. The dashed curve shows the yield reaching the anticipated scaling Y_{max} ∼ M_{fuel}ϕ ≈ M_{fuel}ρr ∼ R^{4} for ρr ≪ 6 g/cm^{2}, for the expected^{32} burn-up fraction ϕ = ρr/(6 + ρr) ≈ 21% at ρr = 1.6 g/cm^{2}.

The R^{3} laser energy scaling is for a fixed hohlraum-to-capsule radius ratio R_{h}/R and hohlraum-to-capsule coupling efficiency. It also assumes, conservatively for the moment, that increases in hohlraum wall albedo with time^{43} due to the diffusive nature of the Marshak wave (and hence decreasing fractional wall losses with scale) is compensated by an accompanying increased loss from laser backscatter due to correspondingly larger coronal density scale-lengths increasing with hohlraum scale. Similarly, the hohlraum cooling rate for realistic laser drives that turn off before peak v_{imp} is assumed to scale^{44} with peak drive duration (i.e., with stored energy/area in hohlraum walls) and, hence, with R_{h,}.

^{17}increases the ratio of capsule to hohlraum radius for a given hohlraum geometry. For this scheme, the required E

_{L}scales desirably more slowly, as ∼τ ∼ R in the approximation x-ray losses to the capsule are small compared to losses to the hohlraum wall and out of the LEHs. Hence, for fixed hohlraum scale, the capsule absorbed energy can increase almost as fast as R

^{3}. However, this is at the expense of quickly reduced drive symmetry from more capsule ablator plasma filling, impairing inner laser beam propagation, and more imprinting of mid mode drive asymmetry from laser spots being closer to capsule.

^{45}

While the current approach uses a strictly 1D scaling, most 2D and 3D degradations can be easily incorporated into the 1D formalism.^{46} Low mode drive asymmetries (principally mode 1 and 2) hinder the full conversion of peak radial kinetic energy to stagnated internal energy. This leads to residual kinetic energy (RKE)^{47–49} by minimum radius, and a reduced^{50} ρr and v_{imp}. The effect can be included as either a reduction in efficiency^{51} η in Eq. (2), or directly as a reduction in yield and other plasma parameters (pressure, internal energy, etc.) via an analytic piston model.^{52} Ablator mixing into the colder fuel^{53} due to hydrodynamic instability or molecular diffusion increases the fuel adiabat^{54} and, hence, can also reduce ρr. Mitigating such mix^{25,27} can be viewed as simply decreasing α_{if} and thus increasing ρr uniformly (hence a 1D effect), as will be discussed in Sec. III. By contrast, if penetration and/or mixing of the ablator into the hotspot^{55–59} is significant enough to impede ignition by radiative cooling, it is assumed that such consequential hotspot mix should be mitigated first before pursuing other 1D improvements. This is because we are only considering the >1 MJ yield regime. The justification for approximating 2D and 3D perturbations as just a shift in the hydroscale curves has been borne out by comparing 1D, 2D and 3D simulation yields as vary capsule absorbed energy (effectively, as a function of R^{3}).^{60}

This extrapolation methodology using 1D scaling allows us to easily renormalize our projections based on any future improved yields. The first set of approaches for improving yield include increasing ρr or reducing RKE, α_{if,} initial DT gas-fill or ablator-fuel mix, as discussed in Sec. III.

## III. IMPROVED CONFINEMENT/COMPRESSION PATHS

There are multiple routes to improving confinement and compression. We discuss the expected performance gains using the hydroscaled formalism discussed above for four distinct paths, with the understanding that some of them have, in general, multiple possible implementations. These are implosions with thicker ablators, reduced in-flight adiabat, reduced initial gas-fill, and reduced ablator-fuel mix.

### A. Thicker ablator

One path to higher ρR and hence higher yield according to Eq. (1) is to increase the initial ablator thickness ΔR. The principal trade-off^{41,61} for a given inner capsule radius R and peak T_{r} is ΔR vs peak fuel velocity v_{imp}. NIF 3-shock HDC designs have progressed from delivering 1.3 to 4 MJ yield, thereby fully igniting, by increasing ΔR by 6 *μ*m (≈8%) and driving with 6% more laser energy (2.05 vs 1.91 MJ). The extra laser energy was used to extend the peak power of the pulse to accomodate thicker capsules imploding more slowly. The inferred ρr in the absence of alpha heating was ≈10% larger, so consistent with the predicted ≈3× yield increase as shown in Fig. 3, by jumping from a ρr = 1.15 g/cm^{2} to a 1.25 g/cm^{2} hydroscaling curve on the ignition cliff.

However, even at fixed peak power P_{L}, further thickening ablators might improve yield as per a thick shell model^{19} validated by numerical simulations, the inverse scaling of v_{imp} with ΔR is weak, ∼1/ΔR^{0.4}. This is due to two beneficial effects. First, a thicker capsule absorbs more energy since its ablation front remains at a larger radius for longer times. Second, as further discussed in a later section, the less efficient buried doped layer used for x-ray preheat shielding is ablated later in the implosion trajectory. This occurs when the capsule surface area is smaller and, hence, represents a smaller contribution to the ablative drive when more inefficient. The same thick shell^{19} model validated by simulations showed that ρr ∼ ΔR^{1.15}v_{imp}^{0.76}, hence increasing as ΔR^{0.85} by substituting for v_{imp} from above. Upon ignition, higher ρr is desirable to reduce the rate of fuel expansion to keep the burnrate higher for longer, thereby further increasing yield. However, we must also respect the constraint that theory validated by simulations^{51} predict ignition in 1D requires preserving a minimum ρrv_{imp}^{2}. Substituting for v_{imp} from above again, the ignition threshold ρrv_{imp}^{2} ∼ ΔR^{0.85}/ΔR^{0.8}, almost independent of ΔR. So, a practical question is: given a capsule size and T_{r}, what is the maximum increase in ΔR that still allows ignition as v_{imp} decreases? To answer this, experiments at the NIF are in progress^{63} using thicker capsules and a further 7% more energy from NIF, up to 2.2 MJ, 440 TW. This plan is overlaid pictorially on Fig. 2, where unfilled symbols are projections of expected performance. In addition, in-flight x-ray radiography^{64} is assessing mitigation techniques for RKE from time-dependent drive asymmetry, to further reduce the v_{imp} required for ignition.

The discussion so far assumed the drive pulse is extended to maintain similar coast time as ΔR increased, by increasing E_{L} at fixed P_{L}. Another related question is: How much further could ΔR be increased at fixed E_{L} and still achieve ignition, which also necessitates decreasing P_{L}? This is closely related to non-hydroscaled strategies discussed in later sections, so the analysis is deferred to Appendix C.

### B. Lower adiabat

Figure 4 shows that constant yield contours at fixed v_{imp} trade off R with α_{if}^{1/2}. We note that Eq. (1) differs by showing R ∼ α_{if}^{1/3}, which reinforces the point that yield scalings without alpha heating should only be used as a guide for dependencies. When including alpha heating, a better metric is the yield amplification (Y_{amp}) vs the ignition metric ITFX in the absence of alpha heating. ITFX is a figure of merit based on fitting simulation sensitivities that gives the distance relative to the ignition cliff of any design based on the product Yield x ρr_{fuel}^{2.1}. Y_{amp} curves vs ITFX exhibit steeper slopes^{16} as α_{if} decreases, which could explain the discrepancy between scalings of R vs α_{if} with and without alpha heating.

Using the same 2.05 MJ NIF energy and R = 1.05 mm scale as the igniting shots, the open square on Fig. 4 shows the predicted yield increase, doubling to 8 MJ for 10% lower adiabat.

One demonstrated path^{65} to lowering the mass-averaged fuel adiabat is by optimizing shock merge depths to be near the DT fuel/gas interface,^{66} and reducing the rise-rate to peak laser power. This is currently being pursued with the 3-Shock HDC design, first at 1.9 MJ laser energy, and then potentially at higher energy. Reductions of 10% in adiabat seem feasible. The main issue is whether increased sensitivity to 2D asymmetries and hydroinstability growth from higher convergence could thwart ignition. Indeed, experiments using a low adiabat 4-shock CH design (α_{if} < 1.8) that exhibit higher convergence have not shown increased yields,^{17,67} partially attributable to hydroinstability growth of features such as the capsule support tent at its liftoff location^{68} and/or ablator imperfections limiting the maximum v_{imp} that is tolerable. For HDC, compression had not gone up^{22} until recently at a given v_{imp}, an example of which is discussed in a later section.

### C. Lower initial capsule gas fill

There is another path to improving convergence. The igniting designs have the first 2 shocks merge ≈10 *μ*m before the DT fuel/gas interface, increasing the adiabat and hence temperature of that inner most DT fuel. This enhances the mass ablated^{69} into the hotspot during deceleration by thermal conduction after PdV heating. More hotspot mass M_{hs} at a given stagnated pressure translates to increased hotspot radius R_{hs} and hence a reduction in ρr_{fuel}. By contrast, reducing the initial cryogenic temperature of the layered capsule further below the DT triple point^{20} should lead to less M_{hs} and, hence, smaller R_{hs} and higher ρr_{fuel} at minimum volume. This should provide more inertia to fuel expansion and a more sustained high burnrate after ignition. Prior attempts on non-igniting designs have not improved yields. A repeat test, i.e., reducing the initial DT hotspot mass by up to 30% (known as “full quench”) is planned on an existing igniting design. Given that M_{hs} at minimum volume is composed of about 80% ablated inner fuel, one would expect “full quench” to reduce M_{hs} and R_{hs} by only about 6% and 2% for fixed ρ_{hs}, respectively. However, simulations predict that ρ_{hs} also increases some at full quench, so R_{hs} should become even smaller. The increase in ρr ∼ M_{fuel}/R_{hs}^{2} is ≈10%, and thus, we expect the yield to again increase by ≈2× according to Fig. 3. If both lower adiabat and gas fill techniques were successful at doubling yields, we would probably not get a full 4x improvement in yield, as we move beyond the ignition cliff and approach the maximum burn-up fraction.

### D. Lower ablator-fuel mix

It is instructive at this point to compare the measured burn-up fraction ϕ vs a 1D analytic expression that is a function of the burning DT thermal ion temperature T_{i}, the fuel density ρ_{fuel} and the stagnated spatial scale r. For small ϕ, it is given^{1} by the product of the burn rate ∼ρ_{fuel}T_{i}^{n} and the confinement time ∼r/v, the latter set by adiabatic cooling and density dropping as the fuel expands at speed v ∼ $ T$. Since simulations for implosions with strong alpha heating predict the fuel and stagnated ablator are of similar density by peak burn,^{16} we can replace ρ_{fuel} with simply ρ. We assume the burn is truncated after enough spherical adiabatic expansion. Specifically, we mark the end of burn when the pressure ∼1/r^{5} is reduced by 2x, so when r increases by 15%. Using a fit to reactivity^{70} valid between 2 < T_{i} < 30 keV, we arrive at ϕ = 1.0 × 10^{−5} $\rho r T i 7 / T i 0.2 \u2212 0.5$ with T_{i} in keV and ρr in g/cm^{2} (see Appendix A for details). The hotspot temperature T_{i} in the formula above uses the average value inferred from Doppler broadened^{71} DD neutron spectra measured from 2 to 3 different lines-of-sight (LoS). The DD spectrum is chosen over the DT spectrum because it does not show anomalous shifts^{72} and it is less susceptible (by a factor of 1/5^{th}) to extra broadening by non-thermal bulk kinetic or turbulent flows.^{73} The minimum DD T_{i} value would more accurately represent the thermal component. In practice, the standard deviation among all detectors is usually within the error bar (5%) of the average value. An understanding of space and time-resolved T_{i} would be useful for a better comparison to expected burn-up fractions. For example, accurate measurements of the width and shift of the backscattered DT neutron edge^{74} at ≈3.5 MeV could isolate the burn wave temperature in the dense fuel, that is expected to be lower than the average T_{i}. Though difficult, one could also develop time-resolved T_{i} measurements.^{75}

The data vs model burn-up fractions are plotted on Fig. 5 for HDC-based indirect-drive layered implosions at NIF with ϕ > 0.0001. The model includes a small correction for depletion^{76} of fusion products by plotting ϕ/(1 + ϕ). The ρr's with and without alpha heating are evaluated according to Appendix A based on the measured ratio (DSR) of 10–12 MeV scattered to 13–15 MeV unscattered DT neutrons, corrected for the remaining ablator contribution and, in the no alpha case, for the effects of expansion during burn. As expected,^{29} the measured burn-up fractions for lower yields fall well below the analytic model predictions that assume all the fuel is burning (defined as burning at 1% or more of the peak burn rate). At the highest yields and burn-up fractions, the data are approaching the model using the smaller ρr values that account for alpha heating induced expansion, that also fits the simulations better (see Fig. 11). However, even the highest data lie 2× below the model. More detailed burn-up fraction models are worth developing adding in non-ideal effects. For example, one potential limiter to ϕ is non-uniformities in ρr, that is thin spots or aneurysms^{77} expanding faster, for which a simple correction is to use the smaller angular weighted harmonic mean^{78} ρr, outside the scope of this paper. At the same time, the maximum burn-up fraction achievable with implosions at the current scale is clearly an important question to explore experimentally. This issue of thin spots degrading confinement time is being actively pursued by measuring and attempting to reduce time-dependent drive asymmetries, and thus ρr non-uniformities for the current igniting designs.

ϕ may also be eventually limited by ablator mix into the outer fuel regions. Simulations show that ablator can mix into the outer regions of DT fuel in-flight due to RM and RT growth of roughness,^{60} microstructure,^{23} and/or voids^{54} at or near doped/undoped and undoped/fuel interfaces. X ray preheated ablator mixed into colder DT fuel can heat and thus decompress the fuel, reducing ρr at bangtime. Additionally, ablator mix reduces the fuel burn rate and burn-up fraction by increasing the radiative and conduction losses, and by diluting the local fuel density if atomically mixed. There are ongoing efforts using monochromatic radiography^{24} for measuring the mixing extent by peak implosion velocity. In addition, we are planning to improve gamma imaging capabilities^{79} such that comparisons^{80,81} of DT neutron images with neutron mediated C gamma images can provide more accurate information on the mixed zone of ablator and fuel at peak burn. The ratio of reaction-in flight DT neutrons above 17 MeV (from knock-on D and T) to downscattered neutrons is also a promising technique for inferring the level and morphology of ablator-fuel mix.^{82}

Meanwhile, a new design, denoted SQ-n, seeks to reduce any ablator-fuel mix and improve ρr in HDC based implosions.^{83} Richtmyer–Meshkov (RM) growth of any perturbations at the ablator fuel interface after first shock passage is replaced by stable Rayleigh Taylor (RT) oscillations,^{25} by using a ramped drive promoting continuous acceleration instead of the usual 2 successive shocks. In addition, the inner undoped layer of a typical HDC target (Fig. 3) that leads to a classically unstable inner undoped/doped interface layer, has been eliminated, thus trading reduced instability growth for possibly larger radiation losses due to mixing of high-Z dopant with the fuel. A velocity interferometer system for any reflector (VISAR)^{84} measuring the speed of the first shock as traverses a surrogate D_{2} filled capsule,^{85,86} and ablator-fuel interface tracking by high magnification streaked x-ray radiography,^{87} both confirm the transition from uniform to accelerating shock and particle speeds. Such layered SQ-n implosions at 80% of full-scale have shown^{27} an ∼30% increase in ρr at a similar in-flight adiabat, v_{imp} and coast time as similar 80% ignition-scale 3-shock HDC implosions. Combined with an observed reduction in W dopant emission at stagnation time, this suggests substantial ablator-fuel mix has been mitigated. Optimization of the ramped drive and drive symmetry for 1.9 MJ full-scale tests (R = 1.05 mm) for comparison to the 3-shock N210808 design has now begun.

Figure 6 shows an expected ≈6x increase in yield if a 20%–25% increase in ρr persists, all else equal. If successful, SQ-n designs could further explore the benefits of increased ablator thickness, laser energy, and reduced initial hotspot density as described in Secs. III A–III C.

## IV. IMPROVED EFFICIENCY PATHS

Improved efficiency designs also have several distinct paths. We discuss the projected performance and efficiency gains for three ongoing approaches: reducing losses in the hohlraum, increasing the drive T_{r} and hence capsule ablation pressure, and optimizing the ablator efficiency.

### A. Smaller LEH and/or hohlraum wall area

^{43}as shown in Eq. (4). The hohlraum flux ∼T

_{r}

^{4}reached for a given peak laser power P

_{L}is approximately inversely proportional to the combination of peak power areas of hohlraum wall (A

_{h}), LEH (A

_{LEH}), and capsule ablation surface (A

_{c}) as

_{x}is the conversion efficiency from laser to x-ray power, and α

_{h}and α

_{c}are the hohlraum wall and capsule albedos

^{88}that are weak but increasing functions of T

_{r}and time, as discussed in later sections. Typically, for NIF-scale cylindrical hohlraum designs at peak power, α

_{h}≈ 0.85, α

_{c}≈ 0.3 and A

_{LEH}/A

_{h}and A

_{c}/A

_{h}are ≈5.5% and 2.5%, respectively. Thus, the relative sink contributions on the right side of Eq. (4) are in the approximate ratio 0.65:0.25:0.1.

The first shot, N210808, that met the Lawson criterion of reaching the ignition cliff by doubling the core temperature and raising the yield 6× to >1 MJ level, had reduced the A_{LEH} by 27%. Thus, the losses according to Eq. (4) were reduced by ≈8%. This allowed the P_{L} to be reduced by ≈8% according to Eq. (4) to maintain similar x-ray flux T_{r}^{4}. Hence, the peak power portion could be lengthened by ≈8% (≈300 ps), within the constraint of fixed 1.9 MJ laser energy at the time. The outcome using similar size capsules as prior shots (but with significantly reduced surface and volume imperfections) was a reduction in coast time, and an increase in the key metrics of peak implosion velocity [see Eq. (1)] and ablation pressure near peak velocity.^{40} Further gains using such high quality capsules are being pursued by reducing A_{LEH} a further 25%, and reducing the cylindrical hohlraum diameter and hence A_{h} by 3% to increase hohlraum efficiency η_{h} by ≈+9%. If successful at drive symmetry control, this could be used to drive even thicker capsules to higher ρr and hence predicted yield as in Sec. III A. Such a smaller LEH could also be fielded without reducing the hohlraum radius to ease drive symmetry control, bringing η_{h} only down to +6%. Alternatively, one could trade the efficiency gains of a smaller LEH to increase the hohlraum and capsule diameter (and hence ρr all else equal) by 5%, and increase E_{L} by an available +7%, thus moving up the hydroscale curve to R = 1.1 mm as shown on Fig. 7.

A second path is modifying the hohlraum geometry. As an example being currently evaluated, we shall use the dual frustum-shaped hohlraum^{44,89} (denoted “frustraum”) as shown in Fig. 7. This new geometry reduces A_{h} compared to the cylindrical hohlraums used for igniting shots (by 20% for the current design), while increasing the initial minimum clearance between capsule surface and hohlraum wall. The LEH area is kept fixed but subtends a larger solid angle since the frustraum, approximating a sphere, is shorter. The overall expected efficiency improvement according to Eq. (3) is η_{h} ≈ +10%. This includes a negative correction to account for less energy stored in the lower A_{h} wall, that requires a shorter coast time (longer laser drive) to compensate for faster hohlraum cooling.^{44,90} That 10% calculated and demonstrated^{91} efficiency gain and increased clearance is being used to drive a 5% larger capsule (R = 1.1 mm) at similar v_{imp} and α_{if}. This requires only 6% more x-ray fluence for sustaining a 5% longer drive and 1% higher capsule losses according to Eq. (4). Thus, in principle, less laser energy will be required. The projected 1D yield increase over N230729 is 2× as shown in Fig. 7, as one moves up the hydroscaled curve at fixed v_{imp} and α_{if} for 5% higher R and ρr.

For both endeavors, cylindrical and frustum-shaped hohlraums, the main challenge is preserving sufficient drive symmetry control through the pulse, in the face of more plasma filling due to the smaller hohlraum volumes, smaller LEHs and in the latter case, also larger capsules. Indeed, given the proportionately greater ablator plasma filling from having a larger ratio of the capsule-to-hohlraum volume, the optimum initial hohlraum gas-fill density is also being revisited. Other proposed hohlraum design modifications for reducing hohlraum wall area while maintaining drive symmetry include LEH shields^{92,93} and foam-filled hohlraums.^{93} The symmetry control challenges are further discussed in Sec. V D.

### B. Higher capsule ablation pressure

_{r}drive.

^{94,95}To evaluate the potential gains, we use the 1D version of the Generalized Lawson Criterion (GLC) metric, closely related to the Ignition Threshold Factor (ITF). GLC and ITF are single figures of merit that give the distance relative to the ignition cliff of any design based on the product of power laws of independent input values of an implosion. They are based on fitting simulation sensitivities, can be tracked back to analytic theory

^{96}and are given as follows:

^{16}

_{imp}, α

_{if}, and P

_{abl}. The latter is defined as the ablation pressure at δt = Δr

_{if}/c

_{s}≈ 0.5 ns before peak velocity, typically the sound crossing time in the capsule of in-flight thickness Δr

_{if}. This is typically near the maximum T

_{r}and P

_{abl}in high performing implosions that have 1 ns or less coast times. At peak power when ablating partially ionized and reemitting doped ablator, simulations show P

_{abl}scales as T

_{r}

^{3.5}(1 – α

_{c}) ∼ T

_{r}

^{3.3}. The strategy is then to significantly increase T

_{r}

^{4}at fixed P

_{L}by decreasing both A

_{h}and A

_{c}to preserve sufficient drive symmetry control, as shown pictorially in Fig. 8. To understand tradeoffs, it is useful to rewrite Eq. (5) as

_{if}term and the initial ablator density as assumed fixed for the moment. A further constraint between M

_{imp}/M

_{0}, v

_{imp}and P

_{abl}is provided by the indirect-drive Rocket equation,

_{abl}/(dm/dt) equates to the plasma exhaust velocity ∼ $ \Sigma ( Z + 1 T r / A$ for ablated plasma thermal temperatures assumed ∼T

_{r}. We have also introduced the mean atomic weight per particle (electrons and ions) term

*μ*= A/Σ(Z + 1) in preparation for comparing ablators in a later section. Σ(Z +1) is the sum of ions and free electron over each element of charge state Z comprising the ablator. The middle expression contains a convenient power law approximation

^{22}to the ln term, valid for the ignition-relevant indirect-drive range 5 < M

_{0}/M

_{imp}< 20. The last expression assumes fixed fractional mass remaining M

_{imp}/M

_{0}, to keep the threat of feedthrough of hydrodynamic instabilities or preheat low. Finally, we need to substitute for ΔR in Eq. (6). Since most of the capsule mass M

_{0}is ablated in indirect-drive, at a rate

^{97}$ m \u2009 \u0307$∼

*μ*T

_{r}

^{3}(1 − α

_{c}) over a duration ∼R/v

_{imp}, we can approximate for the ablated mass M

_{a}∼ M

_{0}as follows:

_{imp}using Eq. (7) and for (1 – α

_{c}) ∼ T

_{r}

^{−0.2}from above. Substituting for ΔR, v

_{imp}and P

_{abl}in Eq. (6) yields the trade-off R ∼ 1/T

_{r}

^{3}at a given value of

*μ*and ITF. Since the required peak power duration τ

_{peak}scales as R/v

_{imp}; hence, as ∼1/T

_{r}

^{3.5}for a given ITF, the efficiency advantage lies in the fact that a lower E

_{L}∼ τ

_{peak}can be used.

A current design uses a 25% lower A_{h}, and 18% higher A_{LEH} to ease symmetry control, for driving a 30% smaller A_{c}. According to Eq. (4), for fixed P_{L} ≈ 450 TW, this should lead to a 4% higher T_{r} and, hence, a 14% increase in energy efficiency. The optimum capsule initial aspect ratio ΔR/R is accordingly 9% larger according to Eq. (8). v_{imp} should be 2% higher according to Eq. (7), and ρr/R ∼ (ΔR/R)^{1/3}v_{imp}^{0.48}P_{abl}^{0.26} according to Eq. (12) should be 7.5% higher than hydroscaled. The higher mass ablation rate should also increase ablative stabilization of RT ablation front growth.

The dashed–dotted curve on Fig. 8 represents the approximate yield curve for a 4% higher T_{r} by just shifting the solid curve to 12% smaller R per the above derived scaling. This is a slight overestimate because ITF scales with yield amplification due to alpha heating (Y_{amp}), while what is plotted on Fig. 8 is the Y_{α}, a product of Y_{no-a} and Y_{amp}. Y_{no-a} according to Eq. (1) and, hence, Y_{α} will be 23% less, principally due to a 12% smaller capsule containing less fuel and stagnating to lower ρr. The predicted yield for the current R = 0.91 mm design is hence shown 23% below the dashed-dotted curve, but still predicted to ignite with target gain > 1 (yield of 2 MJ at E_{L} = 1.8 MJ). If we successfully ignite at a smaller scale using E_{L} below the current 2.2 MJ NIF capability, we unlock the potential to scale up to higher yields. The open diamond symbol shows a path to 15 MJ yield by using a higher efficiency frustraum with more laser energy (up to 2.2 MJ) to drive a 10% thicker version of the current R = 1.05 mm capsule to higher v_{imp} and higher ρr.

### C. Improved ablator efficiency

_{if}≈ 2.8, v

_{imp}≈ 390

*μ*m/ns and M

_{imp}∼0.5 mg. It is now worth revisiting if, by matching those 1D parameters, can we also ignite using other ablators that can provide higher efficiency by virtue of having a lower average Z, in particular Si-doped CH or Cu-doped Be.

^{98}CH and Be designs have only been tested at lower and higher adiabat and at smaller scale. The choice of ablator and preheat shield dopant will affect the implosion dynamics and efficiency principally through differences in

*μ*, ionization potentials IP and capsule albedo α

_{c}. The general form for P

_{abl}that encompasses all three of these ablator-specific parameters is given by

^{19}

_{e}= T

_{r}in the ablated material. ΣIP is the sums of ionization potential energies from neutral to fully ionized for each element in the ablator. There is a further important connection between α

_{c}and

*μ*. For example, for a given T

_{r}, pure CH has a higher exhaust velocity v

_{ex}∼ $ P abl / m \u0307$ ∼ 1/ $ \mu $, by virtue of the higher Z/A of H. This leads to less ablated mass ∼

*μ*and, hence, a smaller ablated optical depth and lower albedo α

_{c}than designs using a pure C ablator.

Table I shows the estimated metrics for CH, Be and B_{4}C-based designs relative to C, for an undoped ablator at T_{r} ≈ 300 eV, where it is assumed that all low Z ablators are fully ionized.^{99} For comparing the undoped albedo of ablated CH vs C, we assume α_{c} scales linearly with the mass of ablated C which itself scales as product of mass ablated ∼*μ* and mass fraction of C, hence ∼1/Σ(Z + 1). This linear approximation is justified in the low albedo limit (<0.3) of spherically expanding low Z ablators where the transport is not yet strongly diffusive. To compare absorption defined as (1 − α_{c}) between C and CH, we have approximated α_{c} as 2/Σ(Z + 1) such that α_{c} = 2/7 for C and 2/9 for CH, based on simulations. Hence a CH ablator has less absorption and less ionization losses per particle than C given the presence of H. A lower Z Be ablator^{100} has similar *μ* to C. To be consistent with what has been inferred in planar ablation rate experiments,^{101} we estimate for Be a 50% lower undoped albedo, that combined with an ≈(4/6)^{2} lower ΣIP with Z + 1 = 5 gives pure Be an ≈30% higher P_{abl} at a given T_{r} than pure C. As would be expected, the intermediate Z B_{4}C candidate ablator falls in between C and Be efficiency.

_{imp}v

_{imp}

^{2}using the middle expression in Eq. (7), for M

_{0}using Eq. (8) and included the IP loss term first introduced in Eq. (9). Thus, for fixed KE, v

_{imp}and scale R, the relative ablator efficiency η

_{c}can be defined as

*μ*, where the lower

*μ*of CH leads to reduced initial and ablated mass and hence lower capsule albedo. In addition, since all non-hydrogen containing ablators have similar

*μ*, they also have P

_{abl}/T

_{r}

^{3.5}scaling with η

_{c}according to Eq. (9). Table I shows that pure CH, Be and B

_{4}C should have 15, 31 and 16% higher η

_{c}. There is also a small correction for rocket efficiency (defined as implosion KE/blow-off KE) that could be applied to Eq. (11). The nominal igniting C-based implosions have M

_{imp}/M

_{0}≈ 0.1, on the low side of the peak in the rocket efficiency

^{32}at M

_{imp}/M

_{0}≈ 0.2. For the same M

_{imp,}and v

_{imp}as discussed above, equivalent CH designs will have M

_{imp}/M

_{0}= 0.1/

*μ*= 0.116, closer to the peak of the rocket efficiency adding ∼5% in relative efficiency for CH over C designs. Simulations with the full ignition relevant pulse shapes and buried dopant layers show that CH with 4% Si by atomic % exhibit a 16% higher η

_{c}than C with 0.4% W. This is just a little higher (somewhat fortuitously) than the +15% undoped estimate listed in Table I.

Since the largest capsule microphysics uncertainty is likely to be the higher Z dopant albedo contribution, side-by-side CH(Si) and C(W) streaked x-ray burnthrough measurements to infer mass ablation rate^{101} and P_{abl} near T_{r} = 300 eV would be valuable. In addition, one could compare, at fixed hohlraum and capsule scale, relative peak KE vs laser energy and peak power by in-flight radiography.^{102} A revisit of a CH ablator design will also entail checking that hydroinstability growth of discontinuities such as at the support tent liftoff location can be mitigated by switching to the current higher igniting adiabat design α_{if} (≈3), using a smaller tent contact area^{103,104} and larger capsules. The perturbation growth can be checked and compared by face-on gated x-ray radiography.^{105,106} The issue of drive symmetry control with longer pulses required of lower density ablators such as CH and Be is discussed in Sec. V C. Higher η_{c} can also be used to drive larger capsules and/or at lower T_{r} as discussed in Sec. V B.

To summarize, the relative efficiency of ablators in this model can be ascribed to differences in x-ray drive lost to ionization and x-ray reemission (i.e., albedo) that depend on the Z of the ablator, and for reemission, that also depends on the level of mass ablated that all else equal, scales as *μ* = A/Σ(Z + 1). Lower Z ablators are more efficient, but have only been tested in implosion configuration at either low^{107–109} or high adiabat^{35} and at less robust sub-mm scale. Hence, there is a strong case to revisit the use of lower Z ablators for larger capsule, intermediate adiabat igniting designs that can also mitigate ablation-front and ablator-fuel interface hydroinstability growth.

## V. IMPROVED LASER POWER AND ENERGY PATHS

_{L}and energy E

_{L}of the NIF laser can be increased by adding final amplifiers and further improving optics quality.

^{110}A prior study mapped out the hohlraum design space

^{45}for a NIF upgraded capability. Here, we compare the projected yield capabilities for two proposed performance tracks: a hydroscale track where energy and power increases ≈30% and 20% to 2.6 MJ and 540 TW, and an energy track where energy can be increased to 3 MJ with power clamped at 450 TW. Extrapolated designs must first meet the ignition threshold according to Eq. (5), for which GLC =1 corresponds to a yield amplification of ≈30, as on shot N210808. We normalize to the best-performing shots N221204 and N230729 with an estimated GLC = 1.1, v

_{imp}= 375

*μ*m/ns, T

_{r}= 310 eV, and assume conservatively that the in-flight adiabat of about 3 is held fixed. To evaluate the ρr that determines burn-up fraction, we use the following scaling fit to 1D simulations:

^{16}

### A. Hydroscale track

The hydroscale track by definition keeps peak T_{r}, α_{if}, v_{imp}, and M_{imp}/M_{0} fixed. In addition, the relative fraction of ablator mass remaining M_{r} and fuel mass M_{fuel} is kept fixed. It assumes that the required P_{L} and E_{L} scale as R^{2} and R^{3} since the hohlraum radius R_{h} is also assumed to scale as R. All ρr [according to Eq. (12)] and times also scale as R, while M_{0} and M_{imp} scale as R^{3}. Equation (5) shows that GLC ∼ M_{imp}^{0.54} then scales favorably as R^{1.6}. Hence, ignition achieved on N221204 automatically ensures ignition in 1D at larger scale, assuming no failure from non-hydroscaling physics. We can then assume robust ignition and simply read off the yields at (2.6/2)^{1/3} larger radii than current igniting capsules, as shown by the open symbols under the 2.6 MJ bracket on Fig. 9. Specifically, a 2.6 MJ cylindrical hohlraum and a dual frustum-shaped hohlraum at R_{h} and R = +10% relative to existing 2 MJ designs are predicted to reach fusion yields between 15 and 20 MJ, consistent with 1D simulations.

Full radiation-hydrodynamics simulations comparing hohlraum conditions at full vs 0.78 scale of an early ignition design had also been done for NIF, when full laser energy was not yet available.^{111} T_{r} was 7% higher at full-scale due to the rising hohlraum wall albedo in time, while the coronal plasma n_{e} and T_{e} were slightly lower and higher respectively. Both deviations from strict hydroscaling should make larger scale even more promising, as further discussed in the next sections.

### B. Non-hydroscale track

The 3 MJ option with a maximum 450 TW peak power, only slightly more than for the current 2 MJ designs (435 TW after removing few % backscatter losses), calls for deviating from hydroscaling. Most importantly, to maintain low coast times^{39,40} such that the laser turns off shortly before capsule deceleration for sustaining ablation pressure and maximizing compression,^{112} the capsule acceleration time τ_{acc} ∼ R/v_{imp} should scale with peak power duration τ_{peak} ∼ E_{L}/450 TW.

One could first postulate fixed v_{imp} such that R and hence R_{h} scale linearly with E_{L}. This, as we will now show, turns out to be both too aggressive and too inefficient a strategy, but nicely illustrates some of the other physics considerations. First, the peak hohlraum temperature T_{r} which scales approximately as (P_{L}/R_{h}^{2})^{1/4} would drop as 1/E_{L}^{1/2}, as would (M_{r} + M_{fuel})/M_{0} according to Eq. (7). Second, the areal mass density ablated M_{a}/R^{2} = M_{0}/R^{2} – (M_{r} + M_{fuel})/R^{2} scaling as T_{r}^{3}τ_{peak} would also drop as 1/E_{L}^{1/2}. Based on the two constraints, the current igniting designs at E_{L} = 2 MJ with 5% M_{r}/M_{0} and 5% M_{fuel}/M_{0} would transition at E_{L} = 3 MJ to, for instance, 4% M_{r}/M_{0} and 4% M_{fuel}/M_{0}, 20% less M_{0}/R^{2}, and 34% less M_{r}/R^{2} and M_{fuel}/R^{2}. This means that there would be less fuel and remaining ablator areal density to mitigate hydroinstability feedthrough and/or x-ray fuel preheat. In addition, since we seek to maximize the saturated yield ∼M_{fuel}ρr, the above scaling strategy provides no room to trade some inert M_{r} for fusionable M_{fuel}.

We, hence, adopt an intermediate strategy, kept simple to be representative but not unique, that scales R and R_{h} as E_{L}^{1/2} while scaling the initial fuel thickness ΔR_{fuel} ∼ M_{fuel}/R^{2} as E_{L}, compared to E_{L}^{1/3} for both when just hydroscaling. In addition, we choose an initial capsule mass M_{0} such that M_{r}/M_{0} remains ≈ constant at 5%–6% as current igniting designs, to keep the threat of feedthrough of hydrodynamic instabilities fixed.

In practice, we have improved the scaling to realistically increase yield prospects. First, we account for the laser energy of the earlier parts of the pulse increasing as ≈R_{h}^{2}ΔR_{fuel} to maintain fixed shock speeds and optimized shock timing, hence as E_{L}^{2}. This reduces the fraction of laser energy available for peak drive and hence τ_{peak}. This could be viewed as an unfavorable scaling. However, a shorter τ_{peak} translates to forcing v_{imp} and hence T_{r} to increase for a given R so that τ_{acc} will still scale with τ_{peak}, increasing the design's ignition metric according to Eq. (6). Second, we account for the increasing wall albedo in time. This reduces the contribution of wall losses in time relative to LEH losses such that T_{r} is now ∼(P_{L}τ_{peak}^{0.3}/R_{h}^{2})^{1/3.7}. The middle columns in Table II show the resultant scalings, with τ_{peak} ∼ E_{L}^{0.9}/P_{L}, and generalizing by including a hohlraum efficiency multiplier η_{h}. T_{r} and v_{imp} drop 5(7)% and 9(11)% respectively, with(without) η_{h} = +10%, as R increases by 28(22)% while still keeping at least 5% M_{r}/M_{0}. We note this near linear scaling of v_{imp} with T_{r} is consistent with a prior scaling^{32} keeping the inflight aspect ratio fixed. This is not surprising since that is equivalent to keeping M_{imp}/M_{0} fixed. These designs satisfy the GLC multiplier ≈ 1 according to Eq. (5), and increase the projected yield, above even the hydroscaling curve, to 30(40) MJ as shown on Fig. 9 under the E_{L} = 3 MJ bracketed region. The inherent assumption is that shot N230729 is already near the maximum yield for its value of ρr (discussed further in Appendix A). Hence, future designs at larger ρr as given in Table II should reach the saturated yield regime even more easily for GLC > 1 due to the expected steepening^{16,113} of the ignition cliff with increasing ρr. So, to summarize, satisfying the GLC ignition threshold set by Eq. (5) allows for larger M_{imp}, that scales approximately as R^{4} since ΔR_{fuel} ∼ R^{2}, driven to lower v_{imp}^{8} that scales approximately as 1/R^{4}, and at lower P_{abl}^{0.77} ∼ T_{r}^{1.7} ∼1/R. Comparing Figs. 9 and 2, the non-hydroscaled design drives the same size capsule R = 1.3 mm with E_{L} = 3 MJ instead of almost 4 MJ if just hydroscaled. This 25% savings in laser energy required is the difference between the capability of a NIF EYC upgrade vs building a new facility.

We have also included scalings in the three Table II right hand columns for an improved ablator efficiency η_{c} discussed in Sec. IV C. This allows R and R_{h} to further scale up as η_{c}^{1/2} and ΔR_{fuel} as η_{c}. The red symbols on Fig. 9 denote the projected yields for η_{c} = +16% as inferred from simulations. According to Eq. (11), to maintain the same peak kinetic energy to ensure ignition, a higher η_{c} allows for a lower peak x-ray flux T_{r}^{4} and hence lower P_{L}. Accordingly, the optimum P_{L} for 3 MJ CH designs drops to 420 TW, allowing for longer peak power durations driving ≈ 8% larger capsules than for C designs. The larger capsule increases projected yields 40–60 MJ. A further advantage of operating at both lower P_{L} and T_{r} is the expected reduction in hard (>1.8 keV) x-ray contribution. Hence, the required dopant level to maintain multi-keV x-ray shielding of the inner undoped regions is reduced, thus reducing the albedo losses of the capsule all else equal.

We also note the expected higher slope in Y_{max} ∼ M_{fuel}ρr ∼ R^{2}ΔR_{fuel}ρr ∼ R^{4.5}, vs the hydroscaled R^{4} presented on Fig. 2. This explains why the fixed power extrapolated yields progressively rise faster than the hydroscale curve. Nevertheless, the dominant sensitivity to maximizing yield is the capsule radius. Further quantitative explanation of the derivations and self-consistency of the scalings in Table II, and how they are used to construct the projected yields, are given in Appendix B.

There is another aspect to this non-hydroscaled strategy that could be key to further increasing ρr and GLC, and hence further improve design robustness and yields, not included in Table II scalings. Simulations^{19} increasing M_{fuel} by replacing ablator with fuel while keeping other parameters (R, M_{imp}, v_{imp,} T_{r}, and α_{if}) fixed show ρr, stagnation pressure and Y_{no-α} increasing as ΔR_{fuel}^{0.3}, ΔR_{fuel}^{0.2}, and ΔR_{fuel}^{0.5}, respectively (see Fig. 10). This is in contrast to Eq. (12), where ρr is only dependent on M_{imp}. Equation (12) was originally adapted from direct-drive^{114} theory and simulations. where the majority of the mass remaining is DT fuel, to fit indirect-drive design simulations^{16} for which significant changes in M_{fuel} are also accompanied by significant changes for example in the adiabat. The simulations also show ρr_{fuel} scaling close to linear with ΔR_{fuel} (see Fig. 9). The increases in stagnated ρr and pressure are consistent with a concomitant decrease in the final hotspot radius with increasing ΔR_{fuel}. These trends can be attributed to DT fuel reaching a higher peak density, allowing for both a more compact assembly, higher compressional pressure ρv_{imp}^{2}, and increase in areal density, compared to the x-ray preheated remaining ablator. Hence, the proposed strategy of increasing the M_{fuel}/M_{r} ratio as increase E_{L} should further increase ρr, pressure, and yield, beyond the hydroscaled M_{imp}^{1/3} term of Eq. (11) that weights fuel and remaining ablator equally. Specifically, the additional sensitivity translates to ρr ∼ (ΔR_{fuel}/R)^{0.32} ∼E_{L}^{0.16}. This represents a significant further 7% increase in ρr transitioning from E_{L}= 2 to 3 MJ, relative to the +10% increase for η_{h} = 1 shown in Table II.

1D simulations also suggest that the proposed increase in scaled fuel thickness ΔR_{fuel}/R ∼ (η_{h}η_{c}E_{L})^{0.5} up to +40% will not exceed the level of mass that can be stagnated, as set by the traversal time of the return shock^{115} by minimum volume or bangtime. One could test ΔR_{fuel} scaling with current NIF implosions using intentionally dudded “THD” implosions,^{116} to remove the ambiguity of ignition leading to higher burn-up fraction and yield just from having more fuel, and the inferred ρr also being affected^{16} by ignition.

### C. Laser plasma instability scaling

The major physics concern with efficiently scaling to higher laser energy is widely recognized to be keeping backscatter losses due to laser plasma instabilities (LPI) minimal. Since the main source of LPI at the low hohlraum gas-fill densities (0.3 mg/cc He) used for ignition designs has been Stimulated Brillouin Scatter (SBS),^{117} we first briefly review SBS scaling. The SBS convective gain for a given plasma species scales as the product of laser intensity I_{L}, velocity gradient scale length L_{v}, and ratio of electron density to temperature.^{32} We will assume the laser spot size is scaled with R_{LEH} to maintain proportional spatial clearance, combined with R_{LEH} itself scaling with R_{h}. It then follows that the peak intensity is fixed for hydroscaling while drops as 1/R_{h}^{2} hence as 1/E_{L} for fixed P_{L} scaling. If the velocity gradient scale-lengths are assumed to scale with R_{h}, then the product I_{L}L_{v} will increase as E_{L}^{1/3} for hydroscaling and drop as 1/E_{L}^{1/2} for fixed P_{L} scaling. The ratio of coronal plasma fill density to temperature in the laser channels should drop some for the hydroscaling case due to less thermal conduction losses as the laser spots get larger.^{32} So, we expect to zeroth order a < 10% increase in SBS gain even for the hydroscaling case where E_{L} and R_{h} increase by 30% and 10%, respectively. Cross Beam Energy Transfer (CBET), which, is also a 3-wave process mediated by ion waves, might be expected to have similar weak scaling with E_{L}, since the beam overlap lengths would also be directly proportional to spot size and hence R_{h} and L_{v}.

However, since plasma flows occur due to the presence of the LEHs, and the various plasma species (wall, fill, capsule ablator and LEH window coronal plasma) collide and possibly interpenetrate, the detailed gain regions for SBS and CBET are more complex and time-dependent. In addition, since controlling low mode drive symmetry may require more CBET, some of the cones will inherently accumulate higher intensity and be more susceptible to LPI losses. Experiments are being designed with relevant 10% larger hohlraums to first test SBS scaling on the inner cones that exhibit most of the current SBS. The effective peak power on the inner cones will be increased from the current 480 TW full NIF equivalent (FNE) limits by using CBET to reach and surpass the 540 TW peak power of a 2.6 MJ hydroscaled design. Such experiments must be conducted by progressively increasing effective peak power and energy to reduce the risk of laser optics damage from SBS back reflections.^{118} Since most of the SBS on the inner beams (those incident at 23.5° and 30° angle to hohlraum axis) occurs during the rise to and early peak power, the fact that current NIF can only provide just over 2 MJ currently does not impede studying that phase of the drive. Of course, by reversing the wavelength separation between cones, the potential appearance of significant SBS on outer cones at 540 TW FNE power and greater will also be checked. If the SBS remains low at higher intensities than by strictly hydroscaling laser spot sizes, smaller fractional LEH sizes could be planned in scaled-up designs, which could potentially increase x-ray drive power and yields. Simultaneously, SRS levels will also be monitored, especially from inner beams, as well as any associated hot electron production by monitoring hohlraum wall Bremsstrahlung using hard x-ray detectors.^{119,120}

For the constant P_{L} designs at even larger scale but lower drive T_{r} and I_{L}, the extrapolation uncertainties are likely larger. Specifically, the coronal plasma will be cooler for lower I_{L}, but also less dense because there will be less wall and capsule ablation due to the lower T_{r}. We can estimate the SBS gain scaling using a model^{32} that invokes pressure equilibrium between the laser channels heated by inverse Bremsstrahlung and cooled by conduction, and the regions outside the channels filled primarily by x-ray ablation of the hohlraum walls. The model (with the constraint v_{imp} ∼ T_{r}^{0.9} to keep fixed in-flight aspect ratio and τ_{peak} ∼ τ_{acc} that approximates current scaling) calculates that the SBS gain has a principal dependence ∼η_{h}^{0.4}I_{L}; hence, ∼η_{h}^{0.4}/R_{h}^{2} ∼ η_{h}^{0.4}/E_{L} for the fixed P_{L} case. The scaling with η_{h} results from a higher T_{r} leading to more wall ablation per unit area and, hence, higher fill density for a given size hohlraum. So even for a 10% increase in η_{h}, the lower I_{L} and T_{r} associated with driving a larger hohlraum with fixed P_{L} but more E_{L} should reduce SBS gain.

It should also be noted that NIF indirect-drive 3-shock HDC implosion designs at 0.3 mg/cc fill have already been scaled up by ≈15% (from^{121} R_{h} = 5.75 to^{122} 6.4–6.72 mm) and by 70% in E_{L} (from 1.2 to 2.05 MJ) without observing significant changes in SBS fraction. This was done keeping laser spot size fixed, such that P_{L} and I_{L} increased by up to 20%, representing a less conservative approach than we have just described for a smaller fractional scale-up in E_{L} to 3 MJ.

The ablator choice can also affect LPI losses. Assuming all else is equal, CH designs have lower hohlraum coronal plasma fill densities due to less ablated mass than HDC designs as explained in Sec. IV C, reducing the threat of SBS. Also the portion of inner beams passing directly through capsule ablated plasma will experience more stimulated ion wave damping and hence potentially less SBS gain in multispecies CH vs single species C.^{123} Be ablators,^{98} by contrast, have more mass ablation at a given T_{r} than C due to lower albedo as discussed earlier and should have less ion wave damping than CH, increasing the threat of SBS. This should be alleviated in Be designs by virtue of their lower Z being suited to driving even larger hohlraums at lower T_{r} and I_{L}.

### D. Drive symmetry control scaling

The second widely recognized issue with projecting to larger scale is maintaining sufficient drive symmetry control. We first consider the hydroscaling case. The main concern is the laser's Inverse Bremsstrahlung (IB) absorption length L_{IB} ∼ T_{e}^{3/2}/Zn_{e}^{2} in the coronal plasmas potentially decreasing relative to the spatial scale length L ∼ R_{h} ∼ E_{L}^{1/3}. According to the same pressure balance model used for SBS scaling, and for hydroscaling, L_{IB}/L ∼ E_{L}^{0.16}/T_{r}^{2.3}, and only weakly dependent on η_{h}. Hence, we expect slightly better transmission at 30% higher E_{L} for fixed T_{r}. Past experience on symmetry control scaling at NIF when increased R_{h} by 15% and E_{L} by up to 70% showed that more CBET had to be used to transfer energy to the inner cones at larger scale. However, that can be at least partially attributed to simultaneously increasing R/R_{h} by ≈ 10% and decreasing R_{LEH}/R_{h} by ≈20% to improve coupling efficiency at the expense of more pole-hot drive.

For the fixed P_{L} case, one has to consider the consequences of peak T_{r} decreasing as E_{L}^{−0.18} and ΔR_{fuel}/R_{h} increasing as E_{L}^{1/2}. For the former, per the IB scaling for fixed P_{L} and L ∼ E_{L}^{1/2}, we find L_{IB}/L ∼1/E_{L}^{0.16}T_{r}^{2.9} ∼ E_{L}^{0.36}, hence again suggesting better transmission for higher E_{L} at peak power conditions. However, this has to be tempered with the latter condition that the foot drive duration will increase by ≈15% to accommodate a proportionately longer first shock travel time, increasing hohlraum filling before reaching peak power. This will likely require revisiting the hohlraum fill gas density choice, and applying drive symmetry control techniques such as beam phasing^{124} to delay and counter effects of hohlraum high Z wall plasma ingress^{125–127} that lead to undesirable interception of inner beams and angular motion of laser spot deposition regions. Switching to CH designs presents both advantages and disadvanatges with respect to controlling drive symmetry. The CH ablated mass is reduced 20% relative to HDC designs at a given R and T_{r}, reducing plasma filling. However, CH designs require ≈60% longer foot drives τ_{foot} for a given adiabat due to having a 3.3× less dense, 2.6× thicker ablator increasing first shock propagation time. This is largely mitigated because to drive a similar strength first shock in lower density CH only requires approximately 3× lower foot power P_{foot} than for C. As a result, the level of coronal plasma filling by the end of the foot epoch ∼τ_{foot} $ P foot$ is approximately constant.

## VI. SUMMARY

We have presented quantitative assessments of various physics-motivated directions to increasing yields and target gain of igniting NIF indirect-drive implosions, beyond the 4 MJ, gain ≈ 1.7 best performance target of 2023. These include increasing compression and confinement time, improving hohlraum and capsule ablator efficiency and increasing laser energy and peak power. 1D hydroscaled simulations augmented by analytic 1D theory have been used to project yield improvements for each of these implosion optimization tracks, normalized to the best current performing 4 MJ shot. Several key tradeoffs emerged from simulations and 1D analytic theory, namely, capsule scale R ∼ $ \alpha if$ and ∼1/T_{r}^{3} for a given igniting yield. At the current NIF capabilities of 2 MJ and 450 TW, we expect 15 MJ yields are attainable by employing successful design optimizations. We also anticipate that addressing key implosion physics questions will be necessary to achieve this yield level. First, this includes determining the extent to which lower adiabat designs leading to higher compression can increase gain and efficiency. Second, we will investigate whether we can decrease excessive RKE and/or ablator-fuel mix if it ultimately restricts the burn-up fraction that is currently somewhat lower than 1D predictions.

We then projected performance for a proposed NIF upgrade to either 2.6 MJ, 540 TW or E_{L} = 3 MJ at fixed 450 TW peak power. The 2.6 MJ track lends itself to hydroscaling by +10% in scale R, +20% in peak power and +30% in E_{L}, reaching projected yields of over 20 MJ. For the E_{L} = 3 MJ at fixed 450 TW peak power track, we have worked through an example strategy for which the hohlraum and capsule radii scale as E_{L}^{1/2} while the DT fuel thickness scales even faster, linearly with E_{L}, compared to the usual hydroscaling as E_{L}^{1/3}. This leads to ≈6% lower T_{r} driving capsules to ≈9% lower v_{imp} that still meet the conditions for ignition. The larger capsule radii and fuel thicknesses should allow for yields reaching up to 60 MJ, depending on the efficiency gain factors η_{h} and η_{c} realized in reducing hohlraum losses and switching to lower Z ablators (e.g., CH vs current C), respectively. In particular, we show that the optimum peak power at a given energy scales as ≈1/η_{c}^{0.8}. Further extrapolating to a Be ablator with an estimated 15% higher η_{c} than CH, M_{fuel} and ρr can be increased a further 32% and 7% corresponding to yields of ≈75–85 MJ.

The LPI scaling in all these larger hohlraums driven at 3 MJ looks favorable if the spot size is increased with hohlraum scale. Nonetheless, LPI tests are planned in the near term in larger hohlraums to probe the SBS and SRS thresholds as function of gas-fill, and of laser intensity by intentionally increasing the CBET to a given cone. The scaling for drive symmetry control is also favorable, with hotter coronal plasmas predicted due to reduced conduction losses in larger hohlraums allowing for more inner cone transmission.

However, controlling the drive symmetry in smaller and more efficient hohlraums, and for lower density and more efficient ablators such as CH that require longer pulse drives, present and will continue to present challenges. Addressing these issues will require focused experiments using existing and improved optical, x-ray, and nuclear techniques to better understand and characterize hohlraum and capsule dynamics. The likelihood of improving yields, from most credible to most risky, can be currently ordered as using more energy, improving compression, switching to a more efficient ablator, and finally improving hohlraum efficiency.

We have presented conservative scalings in the sense that they do not include reducing the in-flight adiabat lower than approximately 2.8. This is done on purpose to avoid the threat of instability growth of imperfections that can thwart ignition at higher levels of compression, as for the original low-foot NIF implosion campaign.^{67} We have restricted ourselves in this paper to moderate adiabat designs which are igniting. These extrapolate to providing target gains of ≈7 at current scale with the improved designs presented. Increasing gain can in principle be accomplished by either reducing adiabat or increasing scale to increase ρr. The main track presented here for reaching target gains of 25 is using larger capsules that contain more DT fuel. We also expect any non-ideal engineering features of fixed size will be less of a degradation factor as we increase scale.

It should also be understood that the 1D extrapolations presented here are only meant to serve as a guide to the potential performance improvements for the various design optimization directions. Each design direction is and will be using 2D and 3D radiation-hydrodynamic simulations that are a prerequisite for proposing and launching any new implosion design at NIF. This includes computer assisted searches for the optimum design space and tradeoffs, that also incorporate the knowledge gained from empirical trends. Moreover, focused and integrated implosion experiments, often iterative, will be needed for the foreseeable future to refine those simulation predictions and fully explore the yield potential of a given design.

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

## ACKNOWLEDGMENTS

This work was performed under the auspices of U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. 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 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 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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**O. L. Landen:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). **P. A. Amendt:** Conceptualization (equal); Investigation (equal); Resources (equal). **D. D.-M Ho:** Resources (equal); Validation (equal). **J. L. Milovich:** Conceptualization (equal); Formal analysis (equal); Validation (equal); Writing – review & editing (equal). **J. E. Ralph:** Conceptualization (equal); Formal analysis (equal); Resources (equal). **D. S. Clark:** Conceptualization (equal); Investigation (equal); Resources (equal). **K. D. Humbird:** Conceptualization (equal); Investigation (equal). **M. Hohenberger:** Investigation (equal); Resources (equal). **C. R. Weber:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Validation (equal). **R. Tommasini:** Formal analysis (equal); Resources (equal); Writing – review & editing (equal). **D. T. Casey:** Conceptualization (equal); Formal analysis (equal); Resources (equal); Writing – review & editing (equal). **R. C. Nora:** Data curation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **C. V. Young:** Conceptualization (equal); Investigation (equal); Validation (equal). **D. J. Schlossberg:** Data curation (equal); Resources (equal); Validation (equal). **S. A. MacLaren:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Validation (equal). **E. L. Dewald:** Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). **P. F. Schmit:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Validation (equal). **T. Chapman:** Conceptualization (equal); Formal analysis (equal); Methodology (equal). **D. E. Hinkel:** Investigation (equal); Supervision (equal). **J. D. Moody:** Conceptualization (equal); Validation (equal); Writing – review & editing (equal). **V. A. Smalyuk:** Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). **O. A. Hurricane:** Conceptualization (equal); Formal analysis (equal); Methodology (equal). **J. D. Lindl:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **R. P. J. Town:** Conceptualization (equal); Supervision (lead). **A. L. Kritcher:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Resources (equal). **S. W. Haan:** Resources (equal); Validation (equal). **M. D. Rosen:** Conceptualization (equal); Formal analysis (equal); Methodology (equal). **A. Pak:** Conceptualization (equal); Resources (equal); Writing – review & editing (equal). **L. Divol:** Formal analysis (equal); Methodology (equal). **K. L. Baker:** Data curation (equal); Methodology (equal); Resources (equal).

## DATA AVAILABILITY

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

### APPENDIX A: BURN-UP FRACTION MODEL

^{70}⟨σv

_{i}⟩ prefactor = 4 × 10

^{−21 }cm

^{3}/s at T

_{i}= 1 keV, for the D or T number density n

_{i}(cm

^{−3}) = 1.2 × 10

^{23}ρ (g/cm

^{3}), and for the confinement time τ

_{burn}at peak T

_{i}= 015r/v as discussed in main text. Time-resolved x-ray images of the exploding fuel

^{42}show an expansion rate v of about 230

*μ*m/ns at T

_{i}= 10 keV, so further substituting for v = 73 × 10

^{5}$ T i$ cm/s in Eq. (A1) leads to the final expression. For a typical measured r = 50

*μ*m, we note that τ

_{burn}is a plausible 33 ps.

Figure 11 plots the 1D Lasnex^{128} simulated burn-up fractions^{16} vs the model according to Eq. (A1) for a variety of CH and C-based designs with R = 0.9 to 1 mm and ranging from low (1.5) to medium adiabat (3). The peak drive T_{r} and, hence, v_{imp} is varied to provide a range of burn-up fractions. The solid (open) circles represent the model ϕ/(1+ϕ) assuming the ρr in the absence of (with) alpha heating, respectively. We see that both models track the simulated burn-up fractions for ϕ > 0.05, when all the DT fuel is participating in the burn.

The burn-up model at small T_{i} or ρr overestimates in Fig. 11 the simulated burn-up fraction because under those conditions, only a small fraction of the fuel is burning, and therefore, the burn averaged temperature is not representative of the average fuel temperature. The offset between the two burn-up fraction models is given by the ratio of ρr burn-on/ρr burn-off ≈ 0.6 over the range 70 < Y_{amp} < 500, relevant to the best NIF performing shots. The model using the ρr that includes the expansion due to alpha heating matches well 1D simulations.

^{16}cross-section for C vs DT, we arrive at the following formula for the ρr in the absence of alpha heating:

_{no-α}, that has been shown through 1D simulations to be a universal function of Y

_{amp}for a range of implosion adiabats.

^{16}The prefactor of 27.5 in Eq. (A1) includes an average 35% contribution to ρr from the stagnated ablator. This is based on simulations

^{41}for relevant M

_{r}≈ M

_{fuel}and measurements of the total C areal density on a subset of the implosions using an energy resolving gamma detector

^{129}measuring the 4.4 MeV gamma yield from the C(n, n′)γ reaction.

^{130}That prefactor multiplied by the measured DSR of 0.042 ± 0.0015 on the dudded THD version of N210808 (experiment N220220)

^{131}leads to the ρr of 1.15 g/cm

^{2}listed on Fig. 3.

_{amp}is derived from its one-to-one correspondence

^{16,132}with the ignition metric ITFX

_{α}that is a function of the observables of 13– 15 MeV neutron yield and DSR. This can be fit piecewise by the following, with M

_{fuel}in

*μ*g:

_{amp}> 6 threshold was first reached on shot N210808 that met the Lawson criterion and according to Eq. (A2), is now ≈100 on N230729. 1D simulations

^{16}predict a maximum Y

_{amp}of ∼150 at the estimated ρr = 1.25 g/cm

^{2}of N230729, so suggest we are close to reaching the Y

_{max}plateau at this value of ρr.

The uncertainty in the corrected ρr is estimated at 10%. This is a small contributor to the 20% uncertainty in the maximum burn-up fraction as shown on Fig. 5, dominated by the uncertainty in measured T_{i}. The solid and open circles represent the model ϕ assuming the no-α ρr and the burn-averaged ρr that does not use the Y_{amp} correction from Eq. (A3), respectively. At low burn-up fractions and hence low Y_{amp}, as for Fig. 11, there is no distinction in ρr's; hence, the open and closed circles overlap.

### APPENDIX B: HYDROSCALING TO E_{L} = 2.6 MJ AND NON-HYDROSCALING TO 3 MJ

We explain here in detail the generalized scalings to E = 2.6 and 3 MJ on Table II, that led to the projected yields and capsule scales on Fig. 9 for both C and CH designs with and without improved hohlraum efficiency. The algebraic expressions are given as power laws of E_{L}, η_{h}, η_{c} and *μ*. The values listed are the calculated multipliers relative to the existing igniting C designs with η_{h}, η_{c} and *μ* defined as 1, E_{L} = 2 MJ, P_{L} = 435 TW, R = 1.05 mm, R_{h} = 3.2 mm, T_{r} = 310 eV, v_{imp} = 390 *μ*m/ns, ΔR_{fuel} = 65 *μ*m, M_{imp}/M_{0} ∼ 10% equally divided between ablator M_{r} and fuel M_{fuel}, ρr ≈ 1.25 g/cm^{2}, and GLC ≈ 1.1. The hydroscaling track assuming a potential NIF upgrade to E_{L} = 2.6 MJ and P_{L} = 520 TW is shown in the first two columns. This design projects to R = 1.15 mm, R_{h} = 3.5 mm, yielding GLC ≈ 1.4. To maintain strict hydroscaling as discussed earlier, we have temporarily assumed fixed fractional losses, which can be viewed for example as fixing hohlraum wall albedo in T_{r} and τ_{peak} such that T_{r} ∼ (P_{L}/R_{h}^{2})^{1/4} is also fixed We note the ignition robustness GLC is well past what should be needed to ignite (1), but Y_{max} is only scaling as E_{L}^{4/3}, so only increasing by 1.4× for a 30% increase in E_{L}. Also shown on the first column is the strategy for using an improved efficiency hohlraum (multiplier η_{h} > 1) at full E_{L} while still maintaining hydroscaling. Specifically, the peak power and duration τ_{peak} are reduced and increased by η_{h}^{1/3} respectively, but the gain in Y_{max} is only improved by η_{h}^{4/3}.

The more promising but riskier non-hydroscaled tracks, which scale R and ΔR_{fuel} faster than E_{L}^{1/3} as explained in main text, are shown in the other columns of Table II. The strategy here is to find a scaling that maximizes R and ΔR_{fuel} and hence M_{fuel} and Y_{max}, while keeping the ignition metric GLC multiplier close to 1. A 3 MJ upgrade limited to 450 TW means that for projecting from existing 2 MJ, 435 TW igniting C designs, one has to make use of 1.5× more laser energy but only a few % more peak power, hence 1.4× longer peak power duration τ_{peak}. To optimally use such a longer peak drive requires proportionately longer capsule acceleration times hence larger capsules to keep v_{imp} ∼ R/τ_{peak} high enough to ensure ignition per the GLC metric. Thus, we arrived at R and R_{h} ∼ E_{L}^{1/2} that maintain fixed coupling efficiency between hohlraum and capsule while decreasing T_{r} and v_{imp} by 5%–10%. We note that this strategy also keeps the fractional hohlraum plasma inward expansion from x-ray ablation at peak power close to constant, scaling as T_{r}^{2}τ_{peak}/R_{h}. The other choice was the scaling of ΔR_{fuel}, which we set to rise even faster, ∼E_{L}, since M_{fuel}/M_{0} and M_{fuel}/M_{imp} that are currently at 5% and 50% provide margin for replacing ablator with extra DT fuel.

_{imp}, ρr, GLC and Y

_{max}at stagnation is as follows: We substitute for M

_{0}≈ M

_{a}according to Eq. (8) into Eq. (7) to give a second constraint on v

_{imp}based on the Rocket model, and equate the two,

*μ*that cancels out in the substitutions but will reappear in the context of ablation pressure. Substituting for the previously derived scaling in Table II of each parameter (R, T

_{r}, τ

_{peak}) with η

_{h}and E

_{L}, we arrive at a generalized scaling for M

_{imp}∼ R

^{2}ΔR

_{fuel}

^{n}to best satisfy Eq. (B1). For the hydroscaling case as expected, the exact fit exponent is n = 1, reflecting the fact that M

_{r}/R

^{2}and ΔR

_{fuel}∼ M

_{fuel}/R

^{2}and hence (M

_{r}+ M

_{fuel})/R

^{2}= M

_{imp}/R

^{2}are all increasing by the same factor by definition. For the non-hydroscaled cases, n will in general, vary as scale-up, but we find on average n = 0.67 and 0.5 for C and CH designs for best fit to the Eq. (B1) equality. Thus, to summarize, the M

_{imp}sensitivity to ΔR

_{fuel}is set for consistency between two constraints on v

_{imp}. For example, if the ablator areal mass remaining ∼M

_{r}/R

^{2}were fixed as change scale, then expanding around M

_{r}= M

_{fuel}leads to M

_{imp}∼ ΔR

_{fuel}

^{0.5}. The exponent n is probably the most uncertain parameter in this table. The effective exponent could be higher, per prior discussion that simulations with all else being equal suggest fuel mass has more weighting than remaining ablator mass in setting ρr, final pressure, and yield. Thus, we consider the current scaling of parameters depending on M

_{imp}(ρr, Y

_{max}and GLC) in Table II to be on the conservative side. This is the justification for letting the GLC multiplier fall slightly below 1 for η

_{h}= 1. The middle columns in Table II include the non-hydroscaling case for C designs with enhanced η

_{h}. In the spirit of maximizing Y

_{max}, we assume M

_{fuel}∼ R

^{2}ΔR

_{fuel}further increases by a factor η

_{h}

^{2}. In contrast to the hydroscaling case, given a higher η

_{h}, there is no impetus to lower P

_{L}and increase τ

_{peak}since we see that GLC is marginal for η

_{h}= 1. Instead, by keeping P

_{L}at its maximum value, τ

_{peak}independent of η

_{h}, and scaling R as η

_{h}

^{0.5}to take advantage of T

_{r}also increasing as η

_{h}

^{0.27}, we increase v

_{imp}(for either of its solutions). GLC then increases significantly due to its strong dependence on v

_{imp}and hence η

_{h}. This highlights why we remain interested in further improving η

_{h}, even just by 5%–10%, to increase the x-ray power ∼η

_{h}P

_{L}.

The three right hand columns show the non-hydroscaling sensitivities for a change in ablator characterized by multipliers in ablator efficiency η_{c} and mean atomic weight per plasma particle *μ* as defined and described in Sec. IV C. We begin by assuming that equivalent peak velocity conditions (same v_{imp}, M_{imp,} α_{if}) as igniting 2 MJ, 435 TW C designs can be reached at 2/η_{c} MJ and 435/η_{c} TW using the same R, R_{h}, and τ_{peak}. Integrated 2D Hydra^{37} simulations have demonstrated this equivalence for a doped CH design (*μ* ≈ 0.84) at E_{L} = 1.72 MJ and P_{L} = 376 TW, hence corresponding to η_{c} = 1.16. The strategy for projecting to 3 MJ is again different as the 450 TW limit is now well above the 375 TW starting design. If we want to maximize M_{fuel} by further scaling R as η_{c}, using the full 450 TW available leads to too high a R/τ_{peak} ratio and hence v_{imp}, and too little M_{r}/R^{2}. To maintain an adequate M_{r}/R^{2} requires an intermediate P_{L} scaling multiplier, ∼(η_{c}E_{L})^{0.2} such that P_{L} = 420 TW. This increases τ_{peak}, and reduces v_{imp} and T_{r} by ≈ the same 10 and 7% as for the C design. The exponent (currently 0.2) is in general, an increasing function of η_{c}. In absolute value, P_{L} ∼ (η_{c}E_{L})^{0.2}/η_{c} ∼ 1/η_{c}^{0.8}, highlighting the well-known design strategy that more efficient lower Z ablators are best suited to lower P_{L} and T_{r}. The faster optimum P_{L} scaling for η_{c} > 1 (E_{L}^{0.2} vs E_{L}^{0.1}) is a direct consequence of not being limited by the 450 TW peak power. The additional η_{c} dependence for optimum P_{L} as shown in Table II propagates throughout. By contrast, the *μ* dependence only appears for the parameters that depend on P_{abl}, ρr, GLC and Y_{max}. The final column combines both hohlraum and ablator efficiency, with the understanding that approximate since the non-hydroscaled C and CH designs have slightly different dependencies on E_{L}.

Finally, to plot the projected yields relative to the hydroscaling curve of Fig. 9 at their given value of R, we use the ratio between non-hydroscaled Y_{max} and hydroscaled Y_{max} ∼ R^{4}, given by the last row in Table II. This is only a strictly valid procedure for fully saturated yields, that fortunately encompass all the non-hydroscaled designs (Y > 30 MJ, R > 1.3 mm according to Fig. 2). Coincidentally that Y_{max} ratio is identical to the ρr multiplier tabulated in an earlier row because the current non-hydroscaling strategy assumes M_{fuel} ∼ R^{2}ΔR_{fuel} ∼ R^{4}.

The choice to plot yields vs capsule scale rather than laser or drive energy as is usually done in textbooks^{33} and early publications^{31} is worth a note. The advantage is seen most clearly in Fig. 9 where even non-hydroscaled predictions lie close to a single hydroscale curve at a given v_{imp} and α_{if}, as given by the simple scaling in the last row of Table II. A comparison of Figs. 2 and 9 shows, that for the same E_{L} = 3 MJ, the constant peak power scaling strategy provides higher yields by allowing an increase in R from 1.2 to 1.3 mm, hence in yield ∼R^{5} of ≈50%.

### APPENDIX C: IGNITION MARGIN VERSUS CAPSULE THICKNESS

We present an addendum to Sec. III A on calculating the margin for ignition as one increases the initial capsule thickness ΔR under the condition of fixed E_{L}, hohlraum and capsule radii. This is best addressed using ITF as the ignition metric, and the associated equations in Sec. IV B.

_{r}and v

_{imp}on ΔR. As ΔR is increased, the laser peak power P

_{L}at fixed E

_{L}also has to drop to accommodate a longer capsule acceleration phase of duration τ

_{peak}∼ R/v

_{imp}. According to Table II, for a given R

_{h}, E

_{L}∼ T

_{r}

^{3.7}τ

_{peak}

^{0.7}; hence, T

_{r}

^{3.7}∼ v

_{imp}

^{0.7}at fixed E

_{L}. Substituting for T

_{r}in Eq. (C1) and dropping the assumed fixed R term,

^{19}validated by numerically solving the Rocket equations, that gave v

_{imp}∼ 1/ΔR

^{0.4}for fixed P

_{L},. Similarly, for fixed ΔR, the same model finds v

_{imp}∼ T

_{r}

^{2.4}, consistent with integrated simulations giving v

_{imp}∼ P

_{L}

^{0.7}. Hence, equating the above constraint of v

_{imp}∼ T

_{r}

^{3.5}to match acceleration time, to the Rocket model constraints that v

_{imp}∼ T

_{r}

^{2.4}/ΔR

^{0.4}, we can eliminate T

_{r}and arrive at v

_{imp}∼1/ΔR

^{0.7}. Substituting for v

_{imp}in Eq. (C2),

So given that the estimated ITF margin above ignition for existing igniting shots is about 20% (=10% in GLC) as stated in the introduction of Sec. V, we would expect to fail to ignite for a 5% further increase in ΔR at fixed E_{L}. This is consistent with designs based on simulations only considering further ΔR increases of 5%–10% at a given E_{L}. This also explains why the non-hydroscaled paths in Sec. V B for which T_{r} and v_{imp} are also decreasing use larger capsules (ITF ∼ R^{2}) to regain ignition margin.

## REFERENCES

*Plasma Science: Enabling Technology, Sustainability, Security, and Exploration*

*Fundamental Research in High Energy Density Science*

*An Assessment of the Prospects for Inertial Fusion Energy*

*Frontiers in High Energy Density Physics: The X-Games of Contemporary Science*

*Plasma Physics and Controlled Nuclear Fusion Research 1990*

*The Physics of Inertial Fusion*