Achieving an efficient capsule implosion in National Ignition Facility indirect-drive target experiments requires symmetric hohlraum x-ray drive for the duration of the laser pulse. This is commonly achieved using two-sided two-cone laser irradiation of cylindrical hohlraums that, in principle, can zero the time average of all spherical harmonic asymmetry modes <6 as well as the time dependence of the usually dominant mode 2. In practice, experimental evidence indicates that maintaining symmetric drive becomes limited late in the pulse due to the inward expansion of the hohlraum wall and outward expansion of the capsule ablator plasmas impairing the propagation of the inner-cone laser beams. This effect is enhanced in hohlraums employing low gas-fill, now used almost exclusively as these provide the highest performing implosions and reduce Stimulated Brillouin and Raman backscatter losses, since the gas plasma provides less back pressure to limit blow-in of the hohlraum wall and capsule ablator plasmas. In order to understand this dynamic behavior, we combined multi-keV X-ray imaging of the wall and imploded fuel plasmas as we changed a single parameter at a time: hohlraum gas-fill, laser outer cone picket energy, radius of high density carbon capsules used, and laser beam polar and azimuthal pointing geometry. We developed a physics-based multi-parameter experimental scaling to explain the results that extend prior scalings and compare those to radiation hydrodynamic simulations to develop a more complete picture of how hohlraum, capsule, and laser parameters affect pole vs equator drive symmetry.

## I. INTRODUCTION

The control of low mode drive asymmetry is crucial for achieving high performance implosions in inertial confinement fusion. The key of the implosion is conversion of kinetic energy coupled to the imploding shell into internal energy of the final stagnated cold fuel and core (or hot spot) plasma. Departure from spherical symmetry (i.e., uneven distribution of mass, kinetic energy, and/or momentum over the shell) can cause significant distortion from the ideal spherical shape and interfere with efficient conversion from kinetic energy into internal energy at the deceleration phase.^{1,2}

Current indirect-drive implosion experiments at the National Ignition Facility (NIF) use a spherical capsule, which contains DT fuel suspended in a cylindrically shaped, typically low Z gas-filled high Z hohlraum.^{3} Laser beams burn through a thin plastic membrane covering the two laser entrance holes (LEHs), propagate through an under-dense low Z plasma in the hohlraum, and heat up the hohlraum wall surface. The lasers create a radiation density of soft x rays inside the hohlraum typically equivalent to that from a blackbody with temperature up to 300 eV. A fraction of that x-ray power is absorbed on the outer surface of the ablator, accelerating the shell inwards with the pressure produced by mass ablation (up to 300 Mbar).

On the deceleration phase, the kinetic energy of the shell (20–30 kJ given to the ablator and the DT fuel) is converted to internal energy of the DT fuel through thermodynamic PdV work. If the deceleration is completed while maintaining spherical symmetry, stagnation of the shell happens simultaneously around the shell and the conversion to the internal energy is maximized (internal energy of the DT fuel ∼10–15 kJ). When the shell has low-mode asymmetries, such as the lowest P_{2} Legendre asymmetry intrinsic to cylindrical hohlraums, one part of the shell (such as the poles centered at θ = 0° and 180°, see Fig. 1) will decelerate earlier, while other parts of the shell (such as the equator regions at θ = 90°) are still moving even at the moment of maximum compression on poles. This asynchronous deceleration impairs efficient conversion of the shell's kinetic energy into the internal energy of the fuel and hot spot at stagnation (commonly referred to as bangtime). For example, the neutron yield of implosions relative to a 1D spherically symmetric implosion is calculated^{1,2} to degrade as [1 − (P_{2}/P_{0})^{2}] at current performance levels, where the stagnated core hot spot shape contour is defined as P_{0} + P_{2}(θ). Therefore, while a P_{2}/P_{0} = 30% x-ray core shape is calculated to reduce the yield by only 10%, it also greatly increases the probability of increasingly larger yield degradations when added in other plausible source of implosion imbalances and imperfections. Depending on the time history of the P_{2} drive asymmetry, the asynchrony between pole and equator trajectories can be further magnified and also lead to non-radial kinetic energy losses. These manifest in P_{2} areal density variations at stagnation in the cold fuel and shell,^{1,2,4,5} as recently observed by neutron downscattered imaging^{6} and Compton-mediated x-ray radiography of the stagnated fuel.^{7}

Most hohlraums currently used on NIF indirect-drive experiments are hollow cylinders with two apertures (laser entrance holes: LEHs) on the top and bottom. A NIF hohlraum is irradiated by 192 beams in four different cone angles *β*. The 23.5° and 30° cone beams (arranged in eight groups of four beams from the top and the bottom) are called “inner cone” beams and these are pointed to irradiate the equatorial part of the hohlraum as shown in Fig. 1. The 44.5° and 50° cone beams (arranged in 16 groups of four beams from the top and the bottom) irradiate two ring-like regions ≈ midway between the equatorial and LEH planes on the top and bottom side of the cylinder. Most of the experiments are designed to set the cone fraction, CF (defined as peak power of the inner cones beams/total peak power) in the vicinity of 33% because there are half as many inner cone beams as outer cone beams on the NIF.

Ideally, the outer surface of the shell should be irradiated uniformly by soft x-rays throughout the duration of the drive pulse t_{L} and this is most easily predictable if the laser beams deposit their energy at the hohlraum wall location to which the beams have been pointed. However, transport of the laser power in the hohlraum is impaired^{8,9} and affected by evolution in space and time of the low and high-Z plasma constituents and their density and temperature. Their evolution is magnified in the low-density (≤0.6 mg/cc He) hohlraum gas-fills, chosen currently because of their higher efficiency of coupling of laser to x-rays due to reduced power loss from heating the gas and laser plasma instability (LPI)-induced backscatter.^{10,11} The transport of the outer cone beams is relatively insensitive to dynamic changes of plasma in the hohlraum because of their shorter path length in the hohlraum (R_{hohl}/sin*β*_{o}) = 4.1 mm, where R_{hohl} is the hohlraum radius and *β*_{o} is the angle of the outer beam relative to the hohlraum axis. In contrast, the delivery of the inner cone beam power is more sensitive to the material distribution in the hohlraum due to their longer path length (5.8–7.5 mm) in the hohlraum. In addition to burning through LEH windows (which are needed to hold the initial pressure of the tamping gas), inner beams must penetrate through the compressed tamping gas (helium) and the high-Z plasma ablated off of the outer cone spots of the hohlraum wall (i.e., high-Z bubble).^{12} Furthermore, mid-Z plasma ablated off the capsule and stagnating against the hohlraum blow-off plasma can also absorb the laser power before it arrives at the hohlraum midplane wall, especially in hohlraums with low pressure tamping gas. Simulations predict that absorption of the inner beam laser power on its outboard and inboard edges as it passes through the high-Z bubble and the compressed mid-Z capsule ablator plasma, respectively, can become significant in the final part of the drive pulse and reduce the effective inner cone fraction reaching the hohlraum wall. This is shown schematically in Fig. 1, where, for clarity, we only show one set of inner and outer cones.

From Fig. 1, it should be clear how the dynamic blow-off plasma ingress from either side of the inner beams could dramatically change the hohlraum illumination from predictable wall-based sources to distributed volumetric sources that can also conduct heat to and, hence, induce reradiation from the neighboring higher density walls. The issue has become how well can we model in time this dynamic system so one can optimize the search for more efficient ignition design hohlraums while still maintaining sufficiently symmetric drive illumination. This has proven to be a difficult task. For example, the fluid approximation in large radiation-hydrodynamic simulation codes such as Hydra^{13} does not allow for interpenetration of species with long mean-free-paths. This led to overestimating^{14} inner beam propagation impairment at very low gas fills (0.03 mg/cc He). The minimum density at which a fluid treatment is believed to be valid is between 0.15 and 0.3 mg/cc fill. To provide a simplified description for guiding the hohlraum simulation design space principally between 0.3 and 0.6 mg/cc He gas fill where fluid treatment should be justifiable and laser plasma instability losses low, an empirical model was developed^{8} to predict the final core shape P_{2} asymmetry due to inner cone beam absorption in the outer Au bubble and capsule blow-off. The main parameters considered were the capsule initial outer radius R_{cap}′ and the initial outer pulse fluence E (during the initial ∼1 ns of the laser pulse called “the picket”), which is assumed to set the ballistic Au bubble inward distance traveled as $xbub\u221d\u2009t\u2032E$ (where t′ is the laser pulse length since the turn-on of the outer beams). Both are normalized to the initial hohlraum radius R_{hohl} since to a good approximation, beam pointing locations and hohlraum size scale together. A final parameter is included, the initial hohlraum gas-fill ρ_{0}, which serves to tamp the Au bubble growth, but which also has shown an unexpected behavior on P_{2} core asymmetry^{15} not captured by the simple model so far. Specifically, as shown in Eq. (1), the model assumes more bubble growth x_{bub} ∼ 1/√*ρ*_{0} and, hence, more negative core P_{2} (more oblate) at lower initial hohlraum gas-fill. This was shown to be true between 0.3 < *ρ*_{0} < 0.6 mg/cc, but the core P_{2} shape proved to be insensitive between 0.15 < *ρ*_{0} < 0.3 mg/cc. The full equation for the core P_{2} asymmetry for a given incident peak power inner cone fraction CF that has been consistent at the higher gas-fills is given by

where *A* and *B* are constants, and 270 *μ*m reflects the average empirically determined change in P_{2} with incident cone fraction for a nominal size $Rcap\u2032$ and ratio $Rcap\u2032Rhohl$. We define the current figure-of-merit (FoM) as $[B\u2032E\rho 0\u2009t\u2032RhohlRcap\u2032Rhohl]$. We note Eq. (1) has an arbitrary x-intercept when the FoM = 1.2 (conveniently close to a value of 1), which we will approximately keep as we update the model.

To further understand the approximate meaning of this FoM, we rewrite Eq. (1) more generally as follows:

To closely match the right hand expression of Eq. (1), we set A = 90, simplifying Eq. (2) to P_{2} = 90(3CF − FoM), = 90(1 − FoM) at the natural NIF CF = 1/3. Such a FoM is directly proportional to changes in the throughput of the inner beam peak power reaching the hohlraum equator, where a FoM value of one means baseline (no change in CF), 0.5 means the equivalent of 50% more CF, and 1.5 means the equivalent of 50% less CF.

Equation (1) illustrates a key fortunate trade-off. A larger picket fluence E for the same spot size means more picket power, hence a stronger, faster first shock driven in the capsule. This will increase the entropy of the imploding system, reducing compressibility and final core density and fusion rate for a given final core temperature. This will also reduce the required total laser pulse length ∼ t′ since shock break-out on the inside of a shell of a given thickness will occur earlier. Since shock speed scales as √E, Eq. (1) suggests the FoM and, hence, core P_{2} shape will be independent of shock speed and, hence, entropy of design. However, since there is uncertainty in the data inferred power laws for both E and t′, if we can improve the FoM power law accuracy through further experiments and modeling, it can not only help optimize empirical implosion design choices but also further motivate model changes in simulations. Similarly, the FoM dependence on the ratio of the capsule to hohlraum radius sets constraints on maximum coupling efficiency.

The specific goals here were to gain further understanding by making additional measurements beyond the core emission shape and probe parameter space that had proven puzzling (<0.3 mg/cc hohlraum gas fill) over a wide picket fluence and capsule size range to increase experimental sensitivity. To facilitate data interpretation and, hence, understanding, a single parameter was changed per experiment on gas-filled high density carbon (HDC) ablator implosions. The variables were hohlraum initial gas-fill, laser outer cone picket energy, capsule size, and beam azimuthal geometry. We combined the data from three measurements discussed in an earlier paper^{15} (Fig. 2) and have added a fourth. First, the time-integrated P_{2} asymmetry of the x-ray drive on the capsule was evaluated by analyzing the shape of the x-ray self-emission image of the compressed core from an equator view.^{16} Second, the expansion of the high-Z bubble from the outer cone beam spots was observed through the LEH by a time gated multiframe camera looking down the hohlraum axis. Third, the energy deposition of the inner cone beam power on the equatorial region was observed by imaging x rays produced through an 8 *μ*m thin gold patch. Finally, the stagnation region between capsule ablation and Au wall blow-off was observed by x-ray imaging through the LEH at an angle to add depth perspective. Section II presents the experimental platform and parameter variations. Section III presents the results and physics-motivated changes to the P_{2} model. Section IV uses the model to make further predictions and is successfully applied to other data. Section V compares prior and new FoM and motivates renewed effort on advanced drive and hohlraum designs. Section VI summarizes and briefly discusses the need for further measurements.

## II. EXPERIMENTAL PLATFORM AND PARAMETER VARIATION

The targets and drive pulse are based on a design that experimentally demonstrated a symmetric implosion across the entire duration of the laser pulse but used a gold vs depleted uranium (DU) hohlraum walls^{17} for manufacturing convenience. Laser and capsule parameters are varied around a baseline 1 MJ, 400 TW peak power pulse illuminating a 10.13 mm length × 5.75 mm diameter 0.3 mg/cm^{3 4}He fill Au hohlraum with 3.37 mm diameter LEHs driving a 64-*μ*m-thick W-doped HDC capsule of 844 *μ*m inner radius R_{cap}. Figure 3 shows the laser pulse shape and capsule pie diagram for more details. Each higher step in power represents launch of the next shock, with final shell acceleration occurring during the peak power phase between 4.5 and 6.1 ns. The first peak of the outer cone laser power (which is called the picket) starts at 0.3 ns. This delay is to allow the time needed for inner cone beams to burnthrough the LEH windows and gas region and get to the hohlraum wall further away and to allow the window density to fall below quarter-critical density (n_{c}/4 = 2.3 × 10^{21}/cc) before the more intense outer beams turn on to avoid producing hot electrons from the 2ω_{p} instability.^{18}

We discuss results from seven shots (representing two shots added to the five prior shots that were presented in the context of technique development in Izumi 2018^{15}) for which only one parameter is varied relative to another shot in the set to help isolate the effects of each parameter change. The three parameter variations include the capsule size (1.2× vs 1×), the hohlraum gas-fill (*ρ*_{0} = 0.15 vs 0.3 mg/cc), and the outer fluence (picket power variation for E = 1.5× vs 1× and outer beam pointing variations reducing average intensity ≈25%).

The first shot N171010 is the baseline target design without a core image. N170427 was an identical baseline shot, which has a core image but no thin wall data. On the second shot N171212, we applied a 50% increase in the power of the picket portion of the drive pulse (first 1.3 ns to both inner and outer cone beams). On the third shot N180102, we used an 18% larger inside capsule radius but only 6% thicker ablator relative to the baseline, as shown in Fig. 3(b). The ablator thickness increase was kept minimal to reduce the time offset between the baseline and larger capsule trajectory. Such trajectory offsets will change the weighting of any time-dependent P_{2} asymmetry seen by the capsules (further discussed in Sec. III B 4) and, hence, bias comparisons of changes in hohlraum conditions as a consequence of capsule size changes. On the fourth shot N180103, the hohlraum fill gas density was lowered 2× to 0.15 mg/cc.

The next parameter variation involved the outer beam pointing. In conventional laser pointing on NIF hohlraum experiments, four outer-cone laser beams of the same group (called a “quad”) are aligned so that their centroids pass through the same point on the hohlraum axis. Due to the small angular offset of those four beams (vertical separation: 4.7° and horizontal separation: 4.2°), those beams land on four individual points on the hohlraum wall. In the case of 44.5° (50°) quads, the axial separation is 480 *μ*m (400 *μ*m), and the azimuthal separation is 285 *μ*m (260 *μ*m). On the fifth shot N180117, the overlapped peak laser intensity of the outer cone beams on the hohlraum wall was reduced by repointing the pairs of beams within a quad (this is described with the term “quad splitting”) in the azimuthal directions by ±400 *μ*m. Then to reduce the overlap of the 44.5° and the 50° quad beams, the center axes of 44.5° and 50° quads are moved closer to (further from) the equatorial plane by 200 *μ*m (this is described with the term “cone splitting”). Figure 4 shows the estimated laser intensity on the hohlraum wall at t = 0.5 ns before and after quad-and cone-splitting. The shrinking hohlraum diameter resulting from the wall trajectory causes the laser intensity at the wall to change with time, but we note that the wall trajectory is ballistic and mostly determined by the early time wall intensity. As a result of this repointing, we estimate that the average intensity of the outer cone beams during the picket (at a time before the hohlraum wall has moved inward much) is reduced by 27%. However, this is at the expense of an increased initial axial width Δz as defined in Fig. 1 and as shown in Fig. 4(b). Hence, we might expect that one would not get the full effect of decreased fluence by simply increasing the axial extent of the beams, as we will show later.

On the sixth shot N180705, we used both an 18% larger capsule radius, a corresponding 5% thicker ablator, and the lower hohlraum gas-fill. One top outer quad misfired 1 ns early on this shot only. On the seventh shot N180917, we used both an 18% larger capsule radius, a corresponding 5% thicker ablator, the lower hohlraum gas-fill, and quad splitting as on N180117. One top outer quad was missing on this shot, but its absence of direct shine or inner beam clipping by the local bubble is not visible from the compressed core azimuthal line-of-sight, and it represents a mode 1 not mode 2 distortion. Hence, we do not expect it to affect the P_{2} inferred from the imaging line-of-sight and do not attempt to correct for it in Secs. III–VI.

## III. DATA TRENDS AND P_{2} MODEL UPDATES

In this section, we present the data from the seven shots, one measurement type at a time. At each step, we extract data trends that allow us to update the P_{2} model accordingly based on simple physical arguments. The final version of the model divides the P_{2} ordinate by a *R _{cap}*

^{1.5}/

*R*

_{hohl}^{0.5}scaling and changes the abscissa FoM for HDC between 0.5 and 0.3 mg/cc hohlraum gas-fill to the following $:$

where *E* is the spatially averaged outer picket fluence over the first 1 ns, *R _{cap}* is the inside radius of the capsule, which approximates to its outer radius after the shock compression phase,

*R*(

_{hohl}*R*) are the initial hohlraum radii at the outer beam (equatorial) location, ρ

_{hohle}_{0}is the initial

^{4}He hohlraum gas-fill density, and

*τ*is now the smaller of the stagnation or bangtime t

_{BT}− 1 ns and midpoint time of peak power phase of inner beams, both defined relative to launch time of outer beam picket.

### A. Core emission shape sensitivities

X-ray self-emission from the compressed core (radius P_{0} = 60–70 *μ*m) is imaged onto a gated x-ray camera and imaging plates using an array of 10 *μ*m diameter pinholes located 10 cm from the object. After transmission through a 2.5-*μ*m-thick gold coating on a 0.6 × 0.3 mm 120-*μ*m-thick diamond window at the hohlraum equator, a polycarbonate blast shield in front of the pinhole array, and a Kapton filter in front of the detector, the nominal photon energy of the recorded image is about 15 keV. This image is used for evaluating the shape of the imploded core in terms of Legendre moments using the usual 17% contour^{16} for consistency. Extracted Legendre shape moments of time-integrated and time-resolved images at peak emission agree to a few *μ*m. We expect variations in the hohlraum conditions near the window that only subtends 0.2% of the full solid angle to be insignificant as we varied each laser and capsule parameter. Moreover, any differential asymmetry due to the window would manifest as a varying m1 drive asymmetry,^{19} not a varying core P_{2} since by definition the core limb observed is at right angles to the window line-of-sight.

In Fig. 5(a), we first plot the measured peak emission P_{2} component in *μ*m vs the previously described empirical FoM metric normalized as in Callahan 2018 as shown in Eq. (1). In general, the y-axis value shown in the plot includes a −2.7 *μ*m P_{2} correction (more oblate) for every 0.01 deviation in inner peak power cone fraction (CF) from the natural 0.333 [see Eq. (1)]. All seven shots in this study delivered an average peak CF = 0.333 ± 0.003, so no correction was added. The range of core P_{2} = ±20 *μ*m represents a range P_{2}/P_{0}= ±30%, avoiding the P_{2}/P_{0} < −30% region for which we expect a toroidal shaped core that leads to greater uncertainty in extracting a value for P_{2}. P_{2} becomes more negative with increasing FoM (x-axis), consistent with less equatorial drive as Au bubbles and capsules get larger and further impair inner beam propagation. The 0.3 mg/cc black and 0.15 mg/cc points each follow trend lines with a slope of ≈−100 *μ*m/unity change in FoM, consistent with prior and other published implosions^{8} principally operating at 0.3 mg/cc fill, and suggesting the scaling with *E* and *R _{cap}* will not change much. The increase in core P

_{2}with quad-splitting is ≈2× different between the two gas-fills, but this comparison is complicated by the fact that the capsule scale also increased in the 0.15 mg/cc case. More significantly, as has been noted previously, the 1/√

*ρ*scaling for gas tamping overestimates the inner beam clipping effect for the lower density 0.15 mg/cc case shown as red points displaced ≈1.4× to the right.

In Fig. 5(b), we have updated the model in two physically motivated ways and obtain a better fit to the density dependence. It is useful to view the x-axis of Fig. 5 as the source of P_{2} asymmetry from the hohlraum wall and the y-axis as the response of the capsule in terms of its stagnated core P_{2}. The larger capsules do not just have the potential to affect inner beam impairment and, hence, hohlraum P_{2} asymmetry through the x-axis FoM. For a given change in hohlraum drive asymmetry (ΔP_{2}/P_{0})_{hohl} represented by a change in the x-axis Δx, the change in core P_{2} in *μ*m is given by simple viewfactor^{20} and kinematic estimates as ΔP_{2} ∼ R×S_{2}(ΔP_{2}/P_{0})_{hohl}. In this expression, R is the distance traveled by the shell during the main acceleration phase and, hence, scales with R_{cap} and S_{2} is the mode 2 transport smoothing factor^{21} between a P_{2} asymmetry at the hohlraum wall and P_{2} drive at the capsule surface, which scales as (R_{cap}/R_{hohl})^{0.5}, typically 0.4. Figure 5(a) implicitly assumes nominal R_{cap} and R_{cap}/R_{hohl} by using an average R and S_{2} sensitivity represented by the 270 *μ*m/CF factor. Thus, we are justified in Fig. 5(b) y-axis formally changed from P_{2} − 270ΔCF to (P_{2} − 270M^{1.5}ΔCF)/M^{1.5}, where M is defined as the capsule radius size multiplier and ΔCF = CF − 0.333. This downgrades the M = 1.2× larger capsule P_{2} asymmetry by ≈24% to put it on equal footing with the baseline capsule size, correcting for the fact that larger capsules will sense more drive asymmetry for a longer time or distance traveled and, hence, we expect them to have a steeper slope. As can be seen, that also reduces the ΔP_{2} between cone- and quad-split (N180917) and nominal pointing (N180705) data for the larger capsule pair at 0.15 mg/cc from 28.3 *μ*m to a normalized 22.3 *μ*m, closer to that of the ΔP_{2} = 13.6 *μ*m for the usual capsule size at 0.3 mg/cc (N180117 vs N170427). Finally, a better fit with a correlation coefficient R^{2} of 0.96 across all seven shots can be gained by assuming the FoM x-axis scales as (R_{cap}/R_{hohl})^{1.1 ± 0.1} and ρ_{0}^{0.13 ± 0.03} to be compared to the original R_{cap}/R_{hohl} and 1/√*ρ*_{0} scaling in Fig. 5(a). We note this is already more consistent with the core P_{2} shapes being insensitive to 0.3 vs 0.45 mg/cc gas-fill for another class of design.^{22} The line of best-fit shown in Fig. 5(b) has a slope of −115 *μ*m/unity, larger than found previously, −85 *μ*m/unity x-axis at 0.3 mg/cc per Eq. (1), not too surprising since we have redefined the FoM. We have intentionally weighted N180705 less in the fit due to its early top inner quad potentially affecting all later beam transport through the top LEH. Finally, to get a better fit between the cone- and quad-split and nominal pointing cases, we had to reduce the 27% fluence drop to 20% ± 3%. This could be attributed to having an axially longer bubble when clipping begins, though 2D simulations do not predict such an axial widening. Clearly, it would be interesting in the future to leave Δz fixed (no cone splitting) and just apply the azimuthal splitting within the four beams of each quad to check if reducing the eightfold intensity pattern, shown in Fig. 4(c) as peaked at the inner beam locations, reduces the inner beam impairment and increases core P_{2}.

In Fig. 6, we compare the measured vs simulated core P_{2} asymmetry using 2D integrated simulations with a heat conduction flux = 0.15 of maximum possible free streaming electron flux ∼ n_{e}T_{e}v_{e}, where n_{e}, T_{e}, and v_{e} are the free electron density, thermal temperature, and velocity. We have chosen to compare the data with this flux limiter as it gives the closest match to the quad and cone-split shot N180117 that is most 2D-like in illumination. For the 0.3 mg/cc gas-fill, the simulations predict on average 7 *μ*m more negative P_{2m}, 2× outside error bars. For the 0.15 mg/cc fill, the discrepancy between simulations and data has grown to 25 *μ*m more negative P_{2}, corresponding to a peak ΔCF = −0.1 relative to data. We note that changing the flux limiter from 0.15 to 0.03 does not alter the simulation vs data discrepancy in core P_{2} dependence on gas-fill. Indeed, inference of T_{e} from dopant dot spectroscopy experiments^{23} suggests f = 0.03 is more applicable at the Au bubble.

This weak empirical scaling with hohlraum gas-fill that also does not match core P_{2} simulation sensitivity (comparing, for example, N170427 to N180103) requires further study to understand from a physics perspective. To aid in this, we now look at the bubble x-ray imaging measurement from the pole to help construct a self-consistent model that explains both the unchanged picket power sensitivity and the new trend with gas-fill between 0.15 and 0.3 mg/cc.

### B. Outer Au bubble sensitivities and model

#### 1. Bubble growth sensitivity

We used an x-ray framing camera (gate-width of 100 ps) located 570 mm from the object to directly observe the hohlraum wall inward radial expansion through the top LEH. This camera is equipped with a pinhole array (hole diameter of 100 *μ*m) located 190 mm from the object that casts 16 images on the detector with 2× magnification filtered by 650-*μ*m-thick Kapton and 12.5-*μ*m-thick Fe. The exposure of the image is split between x rays from 5 to 7.1 keV (Au M-band recombination continuum below the Fe K-edge) and x rays well above the K-edge at 10–12 keV (Au L-shell lines^{24}). Figure 7(a) shows typical images obtained. The field of view is limited by the LEH window (diameter = 3.37 mm), indicating that the high-Z plasma ablated from the inner surface of the hohlraum wall has moved inward more than 1.2 mm before becoming visible. The tips of the eightfold pattern coincide with the 50° beams that have 24% higher intensity than the 45° beams due to a 7% smaller spot size and a smaller incidence angle. This is consistent with what would be expected from bubble expansion increasing with picket intensity.^{8,9,15} The images obtained represent a partial overlap of the emission from the outer cone bubbles from the top and bottom side separated by ≈6 mm, clocked 11.24° in azimuth, and imaged at slightly different magnifications (6 mm/190 mm = 3%); this explains the slight asymmetry to the bubbles. This will also fill in the eightfold modulation and make the far bubble images appear to protrude further inward than the near bubbles due to the lower magnification. The finite size of the pinhole array (subtending up to ±2.1°) causes up to 200 *μ*m of parallax over the 6 mm of the array. As a result, the emission profiles perpendicular to the parallax direction are used to infer the bubble tip location. Figure 7(b) shows such radial profiles of the images. Based on simulations that show that the emission profile exhibits a maximum fractional jump within 10–30 *μ*m of the Au-gas boundary throughout the bubble trajectory, we deemed it sufficient to determine the location of this boundary from the maximum in dln(Intensity)/dr in the lineouts.

Figure 8(a) compares the displacement of the Au-low-Z gas boundary from the initial hohlraum wall defined as x_{bub} for the seven shots. The linear fits if extrapolated to zero distance traveled have an average time intercept of 0.5 ± 0.1 ns. This is consistent with the Au bubble expanding at a near constant ballistic velocity launched by outer beams turning on at t + 0.3 ns as shown in Fig. 3, as previously noted.^{9} Several trends are visible in the results of Fig. 8(a). First, the bubbles for larger capsules (comparing large to small dots) expand the same for both gas-fills (differential of 10 ± 30 *μ*m). We attribute this to the peaks of the bubbles not being affected by the capsule ablator due to geometric isolation. Second, the bubbles in lower gas fill shots (shown in red) have expanded more than the higher gas-fill shots in black, by 250 ± 30 *μ*m, =14% ± 2% at average x_{bub} = 1.8 mm. This is consistent with less gas tamping of the Au wall blow-off, but at a significantly lower sensitivity than the prior 1/√*ρ*_{0} model. Specifically, the 14% difference in bubble travel for 2× in initial gas-fill in Fig. 8(a) suggests x_{bub} ∼1/*ρ*_{0}^{0.2 ±} ^{0.03}. Hence, we now have one indicator of why Fig. 5(b) required much less sensitivity to gas-fill for a better fit. Third, the 50% increase in the power of the picket portion of the drive pulse (black triangles) leads to the Au-gas boundary trajectory advanced on average 160 *μ*m further from the baseline. However, this is only a +9% ± 1.5% increase compared to the assumed √E = +23% scaling in Fig. 5(a), suggesting at first glance an E^{0.2 ±} ^{0.03} FoM scaling instead. On the cone- and quad-split shots where the overlapped laser fluence of the outer cone beams on the hohlraum wall was reduced 27%, the trajectory of the boundary is lagging 120 ± 30 *μ*m behind the baseline for a change of −6% ± 1.5%, also leading to a scaling E^{0.2 ±} ^{0.05}. It has been noted that this weaker scaling of bubble speed v_{bub} with E is more consistent with analytical isothermal shock release models.^{9} It is also consistent with v_{bub} scaling by mass conservation as dm/dt/*ρ*_{0}, assuming 1D bubble expansion valid over the 1 ns picket duration and invoking the planar laser driven mass ablation rate^{19} dm/dt ∼ [(1 − η_{x})E/1 ns]^{1/3}. Here, η_{x} is the fraction of energy converted to the >1 keV x rays that do not contribute to ablation at n_{c}, which can reach >50% in high Z Au irradiated^{25} at typical outer cone picket intensities of 10^{14} W/cm^{2}. Indeed, one would expect η_{x} to increase with plasma T_{e} and, hence, I and E, such that postulating (1 − η_{x}) ∼ E^{−0.4} would match the observed v_{bub} ∼ E^{0.2}.

Simulations using the 2D version of the radiation-hydrodynamics code Hydra^{13} of the trajectory of the Au bubble tip during the peak power phase are shown in Fig. 8(b) for comparison with Fig. 8(a). The relative ordering of bubble motion with picket power, capsule size, cone and quad-splitting, and gas-fill is reproduced. The simulated gas-fill dependence is a bit larger, ≈1/ρ_{0}^{0.25}, reflected by the simulated trajectories for the 0.3 mg/cc fill in black being on average 5% lower than the data. At this point, it is worth discussing the appropriate fluence to use in the bubble growth scaling in the presence of azimuthal fluence variations due to the standard pointing. Standard pointing has the inner beams closely clocked azimuthally in phase with the 50° outer beams per Fig. 4(c). These 50° beams provide the eightfold leading “spikes” seen in Fig. 7(a) due to the 50° beam smaller spot size and, hence, ≈24% higher fluence. However, by the time the bubble starts clipping the inner beams, the bubble has converged R_{hohl}/(R_{hohl} − R_{o}) ≈ 2×, ending up at 3× convergence per Fig. 8(a). At those reduced radii, the eight inner quads on each side overlap around the azimuth as shown in Fig. 9(a), so the phase of beam clocking is not important anymore and using an average intensity seems appropriate. We now consider why the √E scaling assumption for bubble speed and, hence, position x_{bub} works so well in the Fig. 5(b) abscissa metric given the x_{bub} scales as E^{0.2} per Fig. 8. We argue that the solution lies in the fact that the inner beam clipping only begins when the bubble has reached the outer edge R_{o} of the inner beam location, at about 1.3 mm from the hohlraum wall as shown in Fig. 1. The inners subtend a spot size of ΔR_{inner} = 1.1 mm, so the inners extend from 1.3 to 2.4 mm from the hohlraum wall at the outer bubble z location (z_{bub} = ±3 mm). The y-axis of Fig. 8 has been limited to mainly show the region intercepted by the inner beams and to make the point that during the full epoch of peak power, a fraction of the inner beam spatial profile will always intercept the Au bubble in all cases. Hence, a more natural metric for the relative clipping fraction f of the inner beam would be to normalize x_{bub} by ΔR_{inner} = R_{o} − R_{i} instead of R_{hohl}. More correctly, treating the inner beams as an annulus of outer radius R_{o} and inner radius R_{i} as schematically shown in Fig. 1 and which approximates real beam profiles shown in Fig. 9(a), we have

Specifically, ignoring for the moment the *ρ*_{0} and R_{cap} dependencies, a more cumbersome but geometrically motivated sensitivity for FoM would be

valid when some level of clipping occurs, that is, when x_{bub} > R_{hohl} − R_{o}; otherwise, P_{2} remains = A'.

For fully scalable designs, R_{o}, R_{i}, and ΔR_{inner} would scale with R_{hohl}. However, we leave Eq. (4) as the general metric for bubble clipping as NIF hohlraum experiments do not scale individual spot sizes as hohlraum scale changes (thus changing ΔR_{inner}/R_{hohl}), and there are small relative z pointing changes between inners and outers and new hohlraum geometries^{26} being pursued (thus changing R_{o}/R_{hohl}).

It is instructive to ask how the FoM represented by Eq. (4) could be approximated by a mathematically more convenient power law of x_{bub} and, hence, of E and t. In this simplified totally opaque closing iris model, f remains zero until x_{bub} > R_{hohl} − R_{o} and then rises to a maximum full beam clipping of one when x_{bub} has reached R_{hohl} − R_{i} as shown by the black dashed lines in Fig. 9(b) for R_{hohl} − R_{o} = 1.3 mm and R_{hohl} − R_{i} = 2.4 mm. This represents a universal curve for a given hohlraum and beam geometry; for example, faster bubbles just reach further along the curve by the end of the drive pulse. The solid black line is a more realistic moving average over Δx_{bub} = 0.7 mm to account for the finite incident angle β of the inner beams over the bubble axial extent and for the range of x_{bub} sampled throughout peak power as shown in Fig. 8. The red curve is a power law fit [x_{bub}/(R_{hohl} − R_{i})]^{2.5}, which fits to 15% over the most relevant range 0.3 < f < 1. It is true that a power law fit (x_{bub}/C)^{n} around n ≈ 3.5 would fit better, but it should be recognized that the opaque iris model is itself an approximation. As discussed in Sec. III B 2, we go beyond the opaque iris approximation to include partial bubble transmission decreasing in time due to convergence increasing bubble density. This will alter the convex black curve in Fig. 9(b) by multiplying by a concave function, closer to the power law fit. The power law n = 2.5 has been chosen to satisfy FoM ∼ x_{bub}^{2.5} ∼ E^{0.5} to bridge the picket fluence sensitivities extracted from Figs. 8 and 5. In full, substituting for x_{bub} ∼ E^{0.2}τ, we have

where the second expression assumes R_{i} scales with R_{hohl}. It is important to recognize that τ is the time between the start of the outer picket and midway in peak power of drive, 5.3 − 0.3 = 5 ns in the cases here. To summarize in physical terms, although the bubble velocity is a 2.5× weaker function of laser outer picket fluence than previously assumed, this is compensated by 2.5× increased sensitivity to beam clipping when it occurs because it is localized over an inner beam size ΔR_{inner} ≈ R_{hohl}/2.5 at a distance R_{hohl} − R_{o} ≈ R_{hohl}/2.5 as shown visually in Fig. 9(a).

We can also connect to the prior model by expanding the difference in squares of Eq. (3) to linearize f as ∼[x_{bub} − (R_{hohl} − R_{o})]/(R_{o} − R_{i}). It is then easy to show by expanding around the f = 1 point that the best fit power law exponent n between the relevant range 0.5 < f < 1 will tend to n = (R_{hohl} − R_{o})/(R_{o} − R_{i}) + 1 ≈ 2.2. For example, in the limit inner beams are grazing hohlraum wall such that R_{o} = R_{hohl}, then we recover the original linear in time FoM. By contrast, the larger radii cylindrical and “Iraum” hohlraums using recessed walls just at the outer spot location^{26} that use the same NIF beam smoothing phase plates and, hence, maintain similar values of ΔR_{inner} should experience a greater power law sensitivity to fractional changes in picket energy and pulse duration once bubble clipping starts. Clearly, shots varying the pulse length or hohlraum radius with all else kept fixed would be valuable checks on this FoM. Section III B 2 will provide a more realistic 2D and physics-based model for finite bubble absorption and reemission to better explain the core P_{2} sensitivity to gas-fill and bubble growth deduced so far from data.

#### 2. Bubble absorption model

As part of understanding the sensitivity to bubble motion, we first investigate the reasons for the apparent insensitivity of the FoM to gas-fill [∼*ρ*_{0}^{0.1} for best fit Fig. 5(b)] despite the fact that the bubble motion per Fig. 8 is more at lower gas-fill (∼1/*ρ*_{0}^{0.2}). We would also like to reconcile this with data that show the FoM reverses in density sensitivity^{8} if the initial gas-fill density exceeds 0.45 mg/cc. To account for this, we need to consider the interplay between density-dependent bubble size and inverse Bremsstrahlung absorption in the Au bubble. The bubble tip inward radial motion is parameterized as x_{bub} = 271(E/*ρ*_{0})^{0.2}(t − 0.3) *μ*m/ns per fitting of Fig. 8 data, where E is in units of the baseline outer picket fluence, ρ_{0} is in mg/cc, and we ignore for the moment any dependence on the initial capsule size. The Au electron density is estimated by assuming the expanding Au bubble and He plasma electrons have equilibrated in pressure and temperature, so Au n_{e} ∼ (Z_{He}/A_{He})ρ. The He density ρ is initially set at Sρ_{0} where the factor of S accounts for shock compression of the He by the laser ablated Au front during the outer picket energy delivery. The He density and, hence, Au bubble density are then assumed to increase further through convergence (bubble tip inner toroidal circumference ∼ R_{bub} ∼ R_{hohl} − x_{bub}) by applying the density multiplier R_{hohl}/R_{bub} as a function of time. This 1D convergence treatment ignores the 3D effect of He plasma escaping toward the poles of the hohlraum. We justify neglecting this on the grounds that simulations show axial back pressure applied by volumetric expansion of the ablating capsule and the LEH window material bulging into the hohlraum. For calculating the Au inverse Bremsstrahlung absorption ∼ Z_{Au}n_{e}^{2}/T_{e}^{1.5}, the Au temperature is assumed to be 5 keV based on mid-Z dopant K-shell dot spectroscopy measurements^{27} and Z_{Au} is assumed to be 53 based on existing simulations and data.^{24} The fact that the same dopant He-like and H-like line ratios are observed^{28} at both gas-fill densities of 0.15 and 0.3 mg/cc and by inference T_{e} is consistent with Inverse Bremsstrahlung absorption largely balanced by optically thin coronal radiative losses in Au both scaling as Zn_{e}^{2}. We define the ratio of inner beam power absorbed by the bubble to the total laser power as CF_{bub}. The fraction of inner cone energy absorbed in the Au bubble during peak power is then given by

where f is the fraction of the inner beam clipped by the Au bubble from Eq. (3) and G incorporates the constants in the inverse Bremsstrahlung formula.^{29} The average axial (or poloidal) chord length of the clipped bubble traversed by the inner beams is taken to be Δz(x_{bub}/1 mm)^{0.5}, to reflect the approximately flattened toroidal bubble geometry, growing from an initial outer spot size Δz = 1 mm. Therefore, the average chord length = 1.1 mm at the start of bubble clipping, growing to 1.5 mm if bubbles were fully clipped (x_{bub} = 2.4 mm), a good approximation to simulations.

The time dependence of *ρ* is given by

To estimate S, we use the Rankine–Hugoniot equations^{30} across a pressure jump P_{1}/P_{0} for an ideal gas of γ = 5/3 that lead to S = (4P_{1}/P_{0} + 1)/(P_{1}/P_{0} + 4), where P_{1}/P_{0} = [S/(S − 1)](v_{bub}/c_{s})^{2} and c_{s} = √(ZT_{e}/m_{i}) is the isothermal sound speed of the unshocked but laser heated He. Assuming a picket T_{e} ≈ 400 eV and, hence, c_{s} ≈ 140 *μ*m/ns at baseline conditions^{9} and using the measured v_{bub} ≈ 345 *μ*m/ns leads to P_{1}/P_{0} ≈ 10 and S ≈ 3, consistent with simulations. Note that we do not expect S to vary with E and only as *ρ*_{0}^{−0.1}. This fortunate result comes about because S at values around three is a weak function of v_{bub}/c_{s}, itself ∼ (E^{0.2}/*ρ*_{0}^{0.2})/(E*ρ*_{0})^{0.2} ∼1/*ρ*_{0}^{0.4} by equating the He internal energy *ρ*_{0}T_{e} to the inverse Bremsstrahlung heating ∼Eρ_{0}^{2}/T_{e}^{1.5} and substituting for c_{s} ∼ √T_{e}. Therefore, if we reduce the hohlraum fill He density ρ_{0} by a factor of 2, the model predicts the density of the Au bubble tip also goes down by ≈ a factor of 2 and the energy deposition of inner cone beam power in the bubble per unit volume should go down by factor of ≈ 4 for a given R_{bub} and fixed T_{e}. The bubble temperature when observed is of course much larger as has been set by fixed peak powers >10× that of the picket power, limited by high Z radiative losses balancing Inverse Bremsstrahlung absorption.

Figure 10(a) shows that the time-dependent model CF_{bub}/CF per Eq. (6) for the baseline case at 0.3 mg/cc matches well the radiation hydrodynamics simulation, including the acceleration in CF_{bub}/CF attributable to the combination of increasing bubble density, chord length, and clipping fraction. Figure 10(b) compares the calculated transmitted CF − CF_{bub} per Eq. (6) vs time for baseline cases at *ρ*_{0} = 0.15 and 0.3 mg/cc. The initial CF is set at the peak power level of 0.33 to isolate the changing CF_{bub}. The change in transmitted CF for the two initial densities closely tracks, matching the weak *ρ*_{0}^{0.1} dependence of the data. The bubble transparency is finite, calculated at 29% (54%) at 5.5 ns for the 0.3 (0.15) mg/cc fill, so we expect the thin wall equatorial images discussed later to show only gradual dimming as opposed to a full eclipsing effect, as will be shown later. Figure 10(b) includes, for completeness, the assumed linear dependence of x_{bub} on t starting at the 0.3 ns outer beam delay, scaling as (E/*ρ*_{0})^{0.2}(t − 0.3). Figure 10(c) shows the calculated time-dependent increase in n_{e}/n_{c} per Eq. (6) assuming a fixed Z = 53, so strictly valid only at peak power (after 4.5 ns) but that is when it matters, when the bubble begins to clip inner beams as shown in Figs. 10(a) and 10(b) CF curves. Figure 11(d) shows that the model increase in n_{e}/n_{c} vs time reproduces the hydrocode simulated trend.

We note the accelerating drop in CF − CF_{bub} in Fig. 10(b) as the Au bubble proceeds radially inward and as the density builds up per Fig. 10(c). This is consistent with the measured spatially averaged hard x-ray signal at the equator regions where inner beams should be depositing their energy dropping faster than linearly in time.^{15} Such a behavior is also seen in simulations as a late positive swing in drive P_{2} asymmetry if not corrected by also varying the laser CF vs time.^{14} Indeed, the laser CF extractable from Fig. 3(a) that was kept fixed for all the current experiments and that is overplotted in blue in Fig. 10(a) is slightly increasing in time to partially counteract the decreasing effective CF from increasing bubble clipping.

#### 3. Drive and core shape P_{2} sensitivity to cone fraction

We now relate the change in CF_{bub}/CF during the capsule acceleration phase at peak power to the expected change in drive P_{2} vs time and final implosion core shape P_{2}. We will assume implicitly that the inner beam energy clipped by the outer Au bubble converts as efficiently to useful x-ray drive as the portion that reaches the equator. Hence, the bubble absorbed fraction CF_{bub}/CF is also by definition an x-ray re-emitted fraction and the total x-ray peak power can be assumed to stay constant over all shots. This is justified by, for example, noting that as we changed the pointing geometry of the outer cone beams and, hence, Au bubble size and clipped fraction, the implosion bangtime did not change by more than 50 ps, translating to <3% change in total x-ray drive P_{0}. We begin by modifying the static two-cone hohlraum viewfactor model^{3} by recognizing that the inner cone x-ray sources will be, in general, subdivided into three locations: the targeted equatorial wall component CF − CF_{bub} at θ = 75° with respect to the hohlraum axis, the multi-keV (M-band) coronal bubble plasma component we define as mCF_{bub} at an average θ ≈ 20°, and the remaining thermal component (1 − m)CF_{bub} from the bubble conducting energy to the nearby wall (yellow arrow in Fig. 11) at the outer spot location θ = 40°. These source angles are overlaid on a plasma and ray density simulation map in Fig. 11. We note that the inner rays are predicted to be strongly absorbed soon after entering the Au bubble as seen by the darkening ray color scale, justifying the small polar angle (20°) assumed for the coronal radiation.

The factor m is formally the ratio of the coronal M-band bubble emission to total emission resulting just from inners absorbed in the bubble. A further justifiable assumption is that the harder coronal x-ray emission provides a similar level of useful ablation pressure as the softer thermal wall components. This leads to the following modified hohlraum analytical viewfactor:^{31}

where P_{2o,i}(40°, 75°, 20°) ≈ (0.4, −0.4, 0.8) are the components of P_{2} at the peak power of inner and outer x-ray source locations, S_{2} is again the mode two smoothing factor taken to be a fixed value of 0.4, and F is the ratio of recirculating to laser produced x-ray flux, ≈3 for these albedo ≈ 0.8 hohlraums. The prefactor five represents the normalization of P_{n} = 2n + 1. For more clarity, we have for the moment left off the fixed negative P_{2} contribution of the cold LEHs since we are just interested in derivatives with respect to CF and CF_{bub}/CF. Though the peak CF was intentionally kept fixed during this set of experiments, it is instructive to first examine the dependence on CF.

Differentiating Eq. (8) with respect to CF for any given value of CF_{bub}/CF and m and plugging in above numbers, the sensitivity of drive P_{2} to CF is given by

Figure 12 plots Eq. (9) P_{2} sensitivity normalized to the case of no inner beam clipping (ΔP_{2}/P_{0}/ΔCF = −0.4 at CF_{bub}/CF = 0) as a function of fraction CF_{bub}/CF of inner beam energy deposited in the bubble for various assumed values of coronal emission fraction m.

Figure 12(a) and Eq. (9) provide several key insights. First, the dependence of the P_{2} drive and, hence, imploded core asymmetry on the incident CF is not strictly a constant such as 270 *μ*m/CF as assumed in Eq. (1). In general, the sensitivity will be less than the static viewfactor model case and will depend on the implosion design, as suggested by Fig. 5 in Ref. 8. For example, a design with strong bubble clipping CF_{bub}/CF that still zeroes out P_{2} by increasing the incident or effective CF [by, for example, using cross beam energy transfer (CBET)] will have less sensitivity to CF changes than a design with no bubble clipping. Second, the assumed fraction m of coronal Au M-band drive vs thermal drive due to bubble absorption has a modest effect on the CF sensitivity for realistic values of CF_{bub}/CF < 0.5. This is fortunate as no quantitative data yet exist on the value of m, though there is new quantitative evidence^{32} from measuring the peak coronal emission P_{2} drive asymmetry that it is significantly non-zero. For the current design studied here, the baseline (time averaged over peak power) CF_{bub}/CF is ≈ 1/3 per Fig. 11(b) and as denoted by the vertical dashed line in Fig. 12, reducing the sensitivity to between 50% and 67% of the non-clipped case. The simple physical picture is as follows: if one increases the CF in the absence of bubble clipping, that is a transfer of source emission from *θ*_{o} = 40° to *θ*_{i} = 75°, from the pole-centric to equator-centric. Applying the same CF increase in the presence of bubble absorption of inners transfers the source emission from *θ*_{o} = 40° to a mixture of *θ*_{i} = 75°, 20°, and 40° and, hence, from the pole-centric to a mixture of equator- and pole-centric, so less of a change in P_{2}. By a similar argument, an increase in the fraction m means more transfer of inner energy to *θ*_{i} = 20° vs 40° and, hence, even more pole-centric. Indeed, for sufficiently large values of CF_{bub}/CF and non-zero m, Fig. 12 shows that the sensitivity of the CF should even change sign.

A key approximation so far in deriving Eq. (9) is that the ratio CF_{bub}/CF is independent of CF. This is equivalent to postulating that the clipped inner peak power does not appreciably affect the plasma conditions in the bubble. Simulations fully turning off the inner beams at the start of peak power predict inners provide a time-averaged peak power increase in ΔT_{e}/T_{e} of +30%, which translates to a decrease in 25% in the inverse Bremsstrahlung absorption ∼Z_{Au}^{3}n_{Au}^{2}/T_{e}^{1.5} ∼ 1/T_{e}^{0.9} after substituting for the calculated^{33} Au coronal plasma ionization sensitivity Z_{Au} ∼ T_{e}^{0.2}. While the level of ΔT_{e} remains to be verified experimentally since it may depend on an uncertain heat flux limiter discussed later, we should assume for the moment that the sensitivity slopes in Fig. 11(a) are likely overestimated by ≈30%. This has been corrected in Fig. 12(b), such that the normalized Δ(P_{2}/P_{0})/ΔCF sensitivity at CF_{bub}/CF = 1/3 assuming, for example, m = 0.5 increased from ≈60% to 70%.

The effective x-ray CF_{x} is ≈0.5 as the 2× fewer inner beams are compensated by their x-ray sources being closer to the capsule in a cylindrical hohlraum. Rewriting Eq. (9) including the −25% correction for self-heating,

where the last term assumes m = 0.5, varying only ±10% between 0 < m < 1. The change in core asymmetry ΔP_{2} then follows by simple kinematics as

where g is the relevant fraction of distance traveled by the shell relative to initial capsule radius after the start of peak power phase of interest. gR_{cap} ≈ R_{cap} − u(t_{BT} − t_{0}) ≈ 700 *μ*m, where R_{cap} is defined as the relevant inside radius post-shock phase ≈844 *μ*m, u ≈ 50 *μ*m/ns is the shock imparted speed at the start of acceleration t_{0,} and t_{BT} is the stagnation time (bangtime) 3.3 ns later. Hence, substituting Eq. (10) into Eq. (11) yields core ΔP_{2} (*μ*m) = 95ΔCF_{x}/CF. If we equate ΔCF_{x}/CF_{x} with ΔCF/CF, this model sensitivity is consistent with data^{8} showing an average core ΔP_{2} sensitivity of 90ΔCF/CF *μ*m for a variety of implosions in low gas-fill hohlraums.

#### 4. Drive and core P_{2} sensitivity to bubble model

Having validated by data the P_{2} model sensitivity to CF, we now calculate the derivative of Eq. (8) viewfactor with respect to what we have varied here, CF_{bub.} For fixed CF and for the plausible assumption that m is at best weakly dependent on CF_{bub,}

A glance at the ratio of Eq. (12) to Eq. (10) shows that the absolute P_{2} sensitivity to CF_{bub} is stronger than to CF for any finite value of CF_{bub} for a given m and also more sensitive to the assumed value of m. The simple physical picture is as follows. An increase in CF to zeroth order redistributes peak inner cone laser energy at both the hohlraum equator and bubble near poles, so partially canceling drive ΔP_{2}. By contrast, an increase in CF_{bub} intercepts more inner beam energy at the bubble that would have reached the equator. Furthermore, if m is non-zero, some of that bubble energy is radiated as coronal emission more directly on the capsule poles,^{32} further increasing ΔP_{2} per Eq. (12). Indeed, for the baseline case CF_{bub}/CF= 1/3 and assuming m = 0.5, the P_{2} sensitivity to CF_{bub} is as expected of opposite sign, but also significantly stronger, −1.8× that of the sensitivity to CF. Hence, using ΔCF_{bub}/ΔCF= −1.8 for a given ΔP_{2}/P_{0}, we arrive at

Figure 13(a) plots the core P_{2}/M^{1.5} inferred from substituting for CF_{bub}/CF from Eq. (5) into Eq. (13) vs the new FoM = (E^{0.2}τ/R_{hohl})^{2.5}(ρ/0.3)^{0.13} for the baseline capsule size. The P_{2} axis has been shifted by +64 *μ*m universally for an easier visual comparison to the core P_{2} data in Fig. 5(b). This positive shift can be mainly attributed to the absence of drive from the LEH regions, given by

where Ω_{LEH} and Ω_{w} are the solid angles ≈0.05 and 0.95 subtended by the LEHs and hohlraum wall at the capsule center and S_{2LEH} = 0.37. Inserting the Eq. (14) result into Eq. (11) leads to core ΔP_{2} = +50 *μ*m due to the LEHs alone, justifying much of the applied P_{2} shift.

We note the good correlation between core P_{2} and the FoM, and the full range of core ΔP_{2} = 36 *μ*m in Fig. 13(a) is in good agreement with the ΔP_{2} range in data that it represents in Fig. 5. This good fit is a consequence of choosing a high enough value of m. In effect, matching prior data^{8} on ΔP_{2}/ΔCF and current data on ΔP_{2}/ΔCF_{bub} has given us an estimate of the value of m. Moreover, the relative ordering of the 0.3 mg/cc points and the one 0.15 mg/cc fill point at standard and 1.2× capsule size [Fig. 13(b)] matches that of the data. Hence, we have found a physically motivated model for inner beam impairment by the outer Au bubble that provides a self-consistent quantitatively accurate explanation for the bubble growth and core P_{2} as functions of picket energy and gas-fill for a given capsule size. At lower density, the greater bubble travel and, hence, larger inner beam clipping fraction f are almost exactly canceled by the lower absorption in the bubble.

At this point, we should note that our P_{2} model ignores the relative angular spot motion^{34,35} Δθ_{o} of the outer beams potentially increasing with decreasing fill density (decreasing Au wall tamping). This would increase polar over equatorial drive, thus masking as a smaller CF and yielding a more negative core ΔP_{2}. We evaluate this contribution by once again differentiating the Eq. (8) cylindrical viewfactor model with respect to θ_{o} and combining with Eq. (11) yielding

where the last term uses the substitution ΔP_{2}(θ_{o})= −1.5cos θ_{o}sin θ_{o}Δθ_{o} = −0.74Δθ_{o} at θ_{o} = 40°. Since simulations and data to be presented in Sec. III C show only 450 *μ*m of inward motion of the n_{c}/4 surface that fully absorbs the laser, assuming the same 1/ρ_{0}^{0.2} expansion sensitivity as for the bubble tip yields 70 *μ*m of differential expansion between 0.3 and 0.15 mg/cc fill cases. Hence, Δθ_{o} will be of the order of −70/R_{nc/4} = −0.03, and by Eq. (15), ΔP_{2} ≈ −4 *μ*m. Refitting Fig. 5(b) after removing such a systematic −ΔP_{2} that shifts the red 0.15 mg/cc points by 4 *μ*m leads to a revised FoM ∼ ρ_{0}^{0.1}, within the fitting error bar.

Clearly, a comparison of Figs. 5 and 13 P_{2} shows that this model is not the full story since it cannot capture the negative P_{2} offset between the baseline shots and the 1.2× capsule shots for either gas-fill. Before invoking or searching for an Au bubble density or other inner beam impairment sensitivity to the initial size of the capsule, let us estimate the time-dependent capsule sensitivity to the hohlraum conditions. An M = 1.2× larger radius capsule will stagnate later for a given drive; hence, it will be affected more by any late time pole hot drive. Specifically, let us assume the reasonable approximation of uniform in time inward acceleration for indirect-drive implosions where loss of mass is compensated by convergence keeping the ablator areal density m_{0} about constant. The weighting of a peak drive asymmetry P_{n}(t) then just scales^{36} as (t_{BT} − t)/(t_{BT} − t_{0}). This reflects the fact that the velocity differential Δv ∼ ΔgΔt ∼ P_{n}Δt/m_{0} due to a transitory drive asymmetry P_{n} applied at time t over duration Δt will lead to a shape differential by bangtime ∼ Δv(t_{BT} − t). The experiments showed that (t_{BT} − t_{0}) ∼ M since the measured t_{BT} = 7.8 and 8.5 ns for M = 1 and M = 1.2 and t_{0} ≈ 4.5 ns (ignoring the slight delay in t_{0} (<20 ps) as the initial capsule thickness increased by only 5%). At this point, it is worthwhile considering what is the maximum time t in the drive that significantly matters to imparting an asymmetry since we just stated that the weighting scales as t_{BT} − t. Simulations predict that the final asymmetry of NIF-scale implosions is insensitive^{37} to drive conditions when reaching a radius just before peak velocity, 1 ns before t_{BT}, when the capsule radius and surface area are about 1/3rd and 1/9th of the initial conditions. Hence, we redefine the t in the FoM as the smaller of t_{BT} − 1 and duration t_{L} of the inners, both referenced relative to the start of the outer picket. For the present cases, t_{BT} − t_{L} ≥ 1.8 ns; hence, the full laser portion of the drive should matter to setting the final core P_{2} asymmetry.

The time-weighted core P_{2} asymmetry as a function of capsule size multiplier is then given by

where the integrals go from t_{0} = 4.5 ns to the end of laser pulse at t_{L} = 6.0 ns to represent the majority of the drive and t_{BT} is defined here as that of the baseline capsule.

It is instructive to consider the extreme limit of Eq. (16), that is, suppose the P_{2} drive asymmetry suddenly changed at the end of the laser pulse over a duration Δt, thus dominating the core P_{2} asymmetry. The fractional difference in weighted average drive P_{2} for the M = 1.2 vs M = 1 capsules would be ≈[(1 − 1.5/4)/(1 − 1.5/3.3) − 1](Δt/1.7) = 0.15Δt/1.7, so ≈ 9% for Δt even as long as 1 ns, too small to explain 100% core P_{2} differences between capsule scales as shown in Fig. 5. This confirms that the larger capsule must be affecting the hohlraum environment, as discussed in Sec. III C looking at the equatorial hohlraum region where inner beams would deposit their full energy if unimpeded.

### C. Hohlraum equator emission sensitivities

#### 1. Spatially integrated trends

Evolution of the blow-off plasmas near the hohlraum equator regions can also alter the transport of the laser power, especially late in the pulse. By using a “thin-wall” patch,^{38,39} which is still thicker than the sub-10-*μ*m Marshak radiation depth, it is possible to image higher energy x-rays transmitted through the thin-wall and visualize the region heated by laser energy deposition without affecting the thermal soft x-ray transport and environment. We replaced a section of the wall near the equator (thickness: 30 *μ*m) with a curved epoxy plate having an 8-*μ*m-thick Au layer deposited on the inside surface [Fig. 14(a)]. The x-ray signal through the 8-*μ*m-thick wall is dominated by photons just below the binding energy of the Au L shell (between 8.5 and 12 keV). The thin wall increases the transmitted signal by two orders of magnitude compared to the standard 30-*μ*m-thick walls. The estimated nonuniformity of the gold layer thickness is 200 nm peak-to-valley, corresponding to <5% transmission variability. We boost the signal further by using 100-*μ*m-diameter pinholes giving 230 *μ*m resolution imaging at 0.75× magnification. Figure 14(a) shows that the field of view through the thin patch is partially obscured by a vertical pillar (width: 1.0 mm) and horizontal flange (height 0.2 mm), which are required to establish the mechanical strength of the hohlraum, as seen in an example of the data in Fig. 14(b).

Figure 15 shows the time progression of the x-ray images obtained through the thin wall patch. Each frame originates from a different pinhole of the array. The difference of signal throughput of each pinhole was corrected for the actual diameter of each hole measured by a contact radiography system at General Atomics. The corrections were no more than ±50% with a residual error of 5%. The 2–3× variation of the MCP gain across the images due to attenuation of pulsed bias voltage along the strip line of the MCP-based framing camera is also compensated for with 10% accuracy.^{40} The same framing camera was used on all shots spaced over 1 year giving an absolute comparison accuracy of ±23%. Simultaneously, time integrated images were recorded using an Image Plate detector of ±10% relative shot-to-shot absolute accuracy. Figure 16(a) shows a plot of the average image plate signal level vs the new x-axis FoM for hohlraum filling. It shows the expected decrease in hard x-ray emission as inner beam impairment is increased (larger x value). A similar trend is seen plotting the framing camera signal strength in Fig. 16(b) from the earliest frames at 5.35 ns. A notable outlier as in Fig. 5(b) in the time-integrated data is N180705, perhaps due to the 1 ns early outer quad on this shot affecting equatorial conditions more before the gated images are taken mid-way through peak drive. For the 0.3 mg/cc fill, the larger capsule reduces the equatorial hard x-ray signal by 2–3×, which gives proof that the hohlraum environment is affected by the capsule size. Assuming for the moment that the inner beam energy clipping at the outer Au bubble is returned to thermal x rays (consistent with bangtime not changing for a given capsule size), the ±20 *μ*m range in core P_{2} can be related to ±0.07 change in the delivered inner CF. For an initial inner CF of 0.33, that is ±20% in inner beam intensity at the equator. As expected, these thin wall images transmitting >7 keV x rays are amplifying sensitivity to intensity, showing ±60% variations, consistent with an I^{3} dependence observed previously.^{39} The gated data at the earliest time recorded (5.3 ns) show on average a 30% greater sensitivity in hard x-ray emission vs gas-fill than predicted by the FoM.

#### 2. Spatially resolved information

We now look at the spatially resolved information for more insights. The image in Fig. 15 labeled as 5.35 ns shows the x-ray emission from the inner cone spots as a single ring because spots of the inner cone beams from the top side and the bottom side overlap on the equatorial region of the hohlraum at this time. The observed intensity of the signal gradually dims after 5.6 ns. Also, after 5.7 ns, the x-ray emission in the images can be seen to separate into a top side and bottom side ring.

In Fig. 17, we compare a time sequence of normalized vertical line-outs of the x-ray emission to various simulations varying the heat conduction^{23} flux limiter. Typically, the Au wall, the He gas-fill, and the ablator material can be given different flux limiting values. “Global,” as used in the legend of Fig. 17, refers to all hohlraum materials; “regional” refers to the Au wall and He gas-fill only. The case of 0.03 regional means the Au wall and gas-fill use the 0.03 flux limiter and the ablator uses 0.15. The flux limiter of the Au bubble affects the transmission of the inner beams through the bubble on their outboard side (denoted R_{o} in Fig. 1) that would reach the wall before reaching the equator. This is also the reason for the overall drop in x-ray emission with time seen in Fig. 15. Simulations show that the lower the flux limiter, the hotter the Au bubble and the higher the overall inner beam transmission and x-ray emission in this region “above” the equator. Rather than trying to compare absolute values of simulated vs measured hard x-ray levels, which has large uncertainties as it depends on complicated high Z NLTE (non-local thermodynamic equilibrium) emissivities, we concentrate on comparing the hard x-ray profiles. In particular, the widening gap in emission at the equator shown in Fig. 15 is the geometrical consequence of the inboard edge (denoted R_{i}) of the inner beams being intercepted before reaching the equator region (z = 0). We postulate this is due to laser absorption in the region where the capsule ablator plasma and outer Au bubble begin to stagnate on the He gas in between. Continual mass flow from the Au (wall) and C (ablator) causes the density near the boundary (both the gold and the carbon sides) to build up in time and eventually become dense enough to terminate the propagation of the inboard edge of the inner cone beams before they reach the equatorial plane of the hohlraum. This explains why the gap between the inner and outer laser cones grows wider toward the end of the drive pulse.

As expected, the simulations in Fig. 17 show that the gap size at the equator is mainly sensitive to the flux limiter assumed in the ablator plasma and not very sensitive to the flux limiter in the Au and He gas-fill. The simulations with a 0.15 flux limiter in all materials best reproduce the widening of the emission gap as a function of time; however, the case with a 0.15 flux limiter in the ablator and 0.03 flux limiter in the Au and He gas is not that different. The data clearly show that assuming a 0.03 flux limiter in the ablator plasma overestimates the inner cone power delivery near the equator by making the ablator plasma too hot and transmissive. This is consistent with modeling of the T_{e} inferred by dopant spectroscopy at the equator^{23} showing much better agreement with the 0.15 flux limiter that predicts low T_{e} and more absorption per the inverse Bremsstrahlung 1/T_{e}^{1.5} dependence than the more restrictive heat flux limiter =0.03 that predicts high T_{e}.

To examine the equatorial region further, Fig. 18 shows simulated vs measured hard x-ray images of the baseline target midway during peak power. We see two steep edges to the emission, at R = 2.4 and 2.1 mm. Hydrodynamic simulations show that the ablated wall plasma quarter-critical density point, the highest density contour the inner cone beams can reach consistent with the turning point density = n_{c}sin^{2}θ, has reached R_{nc/4} = 2.4 mm by this time. Figure 19 shows that the simulated and measured radial emission line-outs also agree on the main steep emissivity contour located at R = 2.1 mm. This contour location is not a function of flux limiter, and is attributed to the under-dense gold plasma at the equator stagnating against the carbon plasma ablated off the HDC shell, as shown schematically in Figs. 1 and 20. Laser power deposited in the low-Z materials (He and C) contributes insignificantly to the x-ray image observed through the thin-wall patch (limited to hν = 7 to 11 keV) due to the lack of emission lines and free–free emissivity scaling as Z for a given electron density. Therefore, the x-ray signal through the thin wall should be dominated by emission from the stagnated and n_{c}/4 regions of the gold plasma, with the n_{c}/4 region at a larger radius undergoing more reabsorption through the thicker wall chord.

We can further validate this cylinder-in cylinder model of the equatorial high Z plasmas by comparing forward fits of assumed cylindrically symmetric radial profiles, which include finite 230 *μ*m spatial blurring and radially dependent Au wall limb re-absorption, to Fig. 19 radial lineouts. Figure 21 shows we can reproduce the measured radial lineouts just by assuming the 2 zonal rings of emission at R_{stag} = 2.1 mm and R_{nc/4} = 2.4 mm. We now look for more evidence of this stagnation emission at R_{stag} = 2.1 mm.

### D. Stagnation region imaging

In addition to the three gated microchannel plate x-ray imagers, the NIF also has a gated 100 *μ*m resolution CMOS imager^{41,42} viewing through the bottom LEH at 19° angle to the hohlraum axis as shown schematically in Fig. 22. Also shown is an image obtained for the baseline target through a 250-*μ*m-thick Be filter dominated by photon energies over 5 keV and gated to integrate the signal from t = 4.5 to t = 5.5 ns.

The bright arc region observed on the upper side of the image corresponds to x-ray emission from the outer cone beam spots on the far side (shown by the dark blue mesh). Because the outer beam intensity is >3× that of inner beams and at more normal incidence, this is the brightest feature in the image. The radius of curvature of the wideband arc observed on the lower side of the image is consistent with x-ray emission from the R = 2.1 mm stagnation layer (shown by the yellow mesh on the left-hand side). By this time, density buildup at the stagnation layer is large enough to significantly absorb the inner cone beams before they reach the equatorial plane and so the x-ray emission is separated into two rings, as seen in Figs. 15 and 18. The stagnation ring of the far side is not seen because it is overlapped with much brighter emission from the outer cone spots. This gap is consistent with the thin wall hard x-ray imaging data. The existence of the dark region “gap” on the center of the image is consistent with the widening equatorial gap observed through the thin-wall. The band like emission from the stagnation layer is specific to the hohlraum with intermediate fill pressure. For the hohlraums that have higher gas pressure (>0.3 mg/cc), we see the individual spots of inner cone quads directly hitting the n_{c}/4 contour surface of the hohlraum wall.^{43}

In Fig. 22, the edge of the near-side outer cone bubble eightfold pattern is also seen on this line of sight. It is of similar intensity as the stagnation emission, with an average radius of ≈1 mm, corresponding to x_{bub} = 1.9 mm, consistent with Fig. 8(a). The carbon plasma ablated off the capsule collides against Au plasma both on the equatorial region (R_{stag} = 2.1 mm cylinder) and the inside facing surface of the outer cone bubbles near z = ±2.0 mm. The collision of ablator plasma against the outer cone bubble happens later because the distance from the initial ablator surface to the capsule side edge of the outer cone (∼2 mm) is a factor 2 larger than the distance from the initial ablator surface to the stagnation point on the equatorial region.

The question remains, why does the best fit in Fig. 5 to the capsule size normalized core P_{2} asymmetry scale as R_{cap}^{1.1}? To shed information on this, Fig. 23 plots the stagnation radius for all seven shots extracted from analyses such as shown in Fig. 21 vs capsule initial radius. The data only show R_{stag} sensitivity to R_{cap} and not to initial hohlraum fill or outer picket fluence, as expected. A local power law fit in Fig. 23 between the M = 1 and M = 1.185 data yields ΔR_{stag}/R_{stag} = (1/4)(ΔR_{cap}/R_{cap}). To relate to the fractional inner beam clipping that sets the P_{2} asymmetry, we start by using a similar rationale as for the outer bubble. Specifically, just as the derivative of Eq. (6) yields ΔCF_{bub}/CF ∼ Δf ∼ Δx_{bub}/(R_{o} − R_{i}), so ΔCF_{stag}/CF ∼ ΔR_{stag}/(R_{o} − R_{i}). Hence, since (R_{o} − R_{i})/R_{stag} ≈ ½, ΔCF_{stag}/CF ≈ 2ΔR_{stag}/R_{stag},= ΔR_{cap}/2R_{cap} from above, which suggests a weaker FoM contribution ∼ R_{cap}^{0.5}.

To finish this line of reasoning, we now estimate the change in core P_{2} as a function of ΔCF_{stag}. We will assume the simplest scenario of no x-ray reemission from C stagnation absorption of inner beams. We note at this point that this assumption will lead to a fractional change in average P_{0} drive ≈−ΔCF_{x} ≈ −1.5ΔCF_{stag}, ≈ −1.5CF(ΔR_{cap}/2R_{cap}) from above and, hence, −5% for ΔR_{cap}/R_{cap} = 0.2. This drive deficit translates to an ≈3% peak implosion velocity differential over a 3 ns acceleration time and, hence, in principle, a measurable +0.1 ns in bangtime. However, one would need to correct for other larger sources of the measured +0.7 ns t_{BT} differential: longer travel time, thicker capsule, and larger surface area and, hence, energy sink. Returning to P_{2} sensitivity to CF_{stag}, we account for this further reduction in transmitted inner beams to the equator by replacing the (CF − CF_{bub})P_{2i}(75°) term in Eq. (8) by (CF − CF_{bub} − CF_{stag})P_{2i}(75°) and taking the derivative with respect to CF_{stag,}

Hence, for the current case of ΔR_{cap}/R_{cap} = M − 1 = 0.185, we predict ΔP_{2} = −13 *μ*m, almost 2× smaller than the ΔP_{2} of −23 *μ*m shown in Fig. 5(b) between 0.3 mg/cc M = 1 and M = 1.2 data that have appropriately corrected for M-dependent sensitivity. This estimate is also as expected consistent with the 2× lower FoM exponent sensitivity mentioned above. The hypothesis for the underestimate in ΔP_{2}/ΔM is that apart from variations in R_{stag}, we are currently blind to the potential increases with M in density of pileup of absorbing capsule ablator plasma pictorially shown in Fig. 20 that could make CF_{stag} a stronger function of R_{stag} and, therefore, of R_{cap}. Hence, developing techniques to isolate the stagnation region profile and emissivity could become essential for understanding the changes in P_{0} and P_{2} drive as capsule size is increased in efforts^{44} to increase coupling efficiency.

## IV. MODEL PREDICTIONS AND LIMITATIONS

The model should be extendable to and, hence, testable at lower and higher gas-fills, constrained on the low end by the bubble not extending beyond the inboard edges of the inner beams or stagnating on axis. Figure 24(a) overplots the data with the model core P_{2} vs gas-fill for the baseline case, reemphasizing the good match in sensitivities shown in Fig. 13. The figure x-axis is the delivered picket fluence at the wall, so implicitly assuming the laser picket energy is increased at the higher gas fills to account for the initial energy lost to heating and ionizing the gas before reaching the hohlraum wall. The model predicts that the inner beam propagation, for the moment assuming all else being equal, is only moderately sensitive to all gas-fills, and most impaired at intermediate densities, ≈ 0.4 mg/cc. Above 0.4 mg/cc, a reversal in sensitivity is predicted (with FoM asymptoting to ∼1/*ρ*_{0}^{0.2}), consistent in sign with the prior model^{8} scaling as 1/*ρ*_{0}^{0.5} that was calibrated for gas-fills between 0.3 and 0.6 mg/cc. These power laws for intermediate gas fills will be brought into closer agreement by recalling that the He gas sound speed rising as *ρ*_{0}^{0.2} will lead to a progressively reduced initial He shock compression and, hence, a few *μ*m more positive core P_{2} than shown in Fig. 24(a) at 0.6 mg/cc fill. At even higher gas-fills, the bubble density becomes immaterial as will have reached the opaque iris limit. In addition, at high enough gas fills, by mass conservation, the initial Au bubble density 3*ρ*_{0}(Z_{He}/A_{He})/(Z_{Au}/A_{Au}) ≈ 6*ρ*_{0} (assuming 50% Au ionization for the picket sub-keV T_{e} plasma) cannot be supported when it exceeds dm/dt/v_{bub} even if there were no He tamping limiting v_{bub}. Quantifying at a 0.35 *μ*m laser wavelength, dm/dt = 1 × 10^{6}(I(1 − η_{x}))^{1/3} g/cm^{2}/s with intensity I in 10^{15} W/cm^{2} units^{31} and η_{x} the fraction of energy lost to penetrating radiation. For a typical outer picket intensity =0.1 × 10^{15} W/cm^{2} and assuming η_{x} = 75% and v_{bub} = 3 × 10^{7} cm/s at the higher gas-fills, dm/dt/v_{bub} = 3 × 10^{5}/3 × 10^{7} = 10 mg/cc. Hence, we can expect the ballistic v_{bub} to fall below the low gas-fill scaling above 1.6 mg/cc He gas-fill, when we expect even less clipping and even higher core P_{2} than shown in Fig. 24.

The other assumption of the bubble model is that continual ablation of Au can sustain the increasing Au density at and behind the bubble front even after the end of the picket phase. This appears satisfied for the dense HDC designs that universally have a short epoch between end of picket at 1 ns and second rise at 2.5 ns at almost the same intensity as the picket as shown in Fig. 3(a). However, shock timed CH ablator designs require a significantly longer delay between first and second launch as their ablators are ≈3.5× lower density ρ, since for a given ablator areal density ρδR to maintain the same ablated mass, the shock travel time for a given pressure is ∼ δR√*ρ* ∼ 1/√*ρ*. In addition, the optimized CH designs use a decaying first shock^{45,46} for greater hydrodynamic stability, which require up to 10× lower intensity between the picket and second shock launch. This leads to predicted deceleration of the bubble midway in the pulse due to insufficient Au blow-off, before a second weaker bubble acceleration phase on final rise to peak power results in the final bubble speeds^{9} of <200 *μ*m/ns vs v_{bub} > 300 *μ*m/ns for HDC, beyond the scope of the present model.

Figure 24(b) plots the calculated P_{2} vs picket fluence delivered at the hohlraum wall at 0.3 mg/cc fill for nominal capsule size. Sensitivity to picket fluence should remain similar at both lower and higher multipliers than tested. At very low picket fluences, the model should break down as the bubble blow-off trajectory by peak power will become increasingly dependent on higher power intermediate sections of the pulse as discussed above for CH designs, such as during the second shock launch at 2.5 ns shown in Fig. 3(a).

We now come back to the model prediction on pulse length sensitivity. Increasing the pulse length will not be exactly the same as increasing picket fluence in the model since the latter leads to a faster Au bubble of velocity ∼E^{0.2} that will converge more during the peak power phase. Figure 25(a) plots the model predicted core ΔP_{2} vs pulse length variations around the *τ* = 5 ns baseline 0.3 mg/cc, nominal size capsule case. The P_{2} axis has again been shifted +64 *μ*m to place the baseline model P_{2} close to the data value. The pulse length sensitivity shows an S curve, over a much smaller fractional range than for picket energy in Fig. 24, a consequence of a higher power law sensitivity to τ than E. There is smaller sensitivity for short enough pulses when the bubble only clips the inner beams by the end of peak power. There is less negative slope at long enough pulse durations representing the inners fully clipped even by start of peak power. If we plot P_{2} as a power law function of *τ*, as shown in Fig. 25(b), a best fit constrained to match the FoM slope of −115 *μ*m/unity and x-intercept ≈1.2 yields *τ*^{2.4}, valid between 4.5 < *τ* < 6.3 ns when we expect some level of clipping throughout peak power. Thus, it is reassuring that the 2D physics-based model yields a similar pulse duration power law dependence for the FoM as our earlier opaque iris model (Fig. 9).

Figure 25(a) predicts that just a 10% increase in τ from the current shots will lead to a large drop in core ΔP_{2} of −23 *μ*m in the new model. This is well outside what is tolerable for high performance implosions (±5 *μ*m P_{2}) and fixing requires a CF modification of the incident laser and/or crossbeam energy transfer (CBET).^{47} However, as explained earlier, CF sensitivity is 1.8× less effective at correcting core P_{2} than the effects of bubble clipping. This highlights why low gas-fill designs are particularly difficult to implement when lengthening the pulse in effort to reduce shock strengths and entropy and thus increase capsule compressibility or when using thick capsule CH designs. It is also interesting to review the new FoM prediction of ΔP_{2} vs capsule size multiplier M for fixed hohlraum size, fill, and drive design. If the capsule thickness is also scaled by M for similar in-flight aspect ratio, then the drive duration τ should also scale with M such that Eq. (5) x-axis FoM ∼ M^{3.6}. In addition, the response of capsule P_{2} on the y-axis will vary as M^{1.5}. Hence, just a 10% increase in M to increase hohlraum coupling efficiency by ≈30% is predicted to decrease P_{2} from say, an optimum 0 to −[115(1.1^{3.6} − 1)]1.1^{1.5} ≈ −50 *μ*m. Conversely, this large sensitivity to capsule size can also serve as the impetus to revisit long duration, low adiabat four-shock CH pulse^{46} drives in a low laser plasma instability (LPI) loss, low gas-fill hohlraum of same size (5.75 mm diameter) by using subscale (M < 1) capsules.^{48}

Finally, we test the model predictions vs more recent data of core P_{2} asymmetry observed in 17% larger scale Au-lined DU cylindrical hohlraums (6.72 vs 5.75 mm diameter) employing quad-splitting.^{22,49} These implosions used larger scale DT layered HDC capsules with capsule size multiplier M varying between 1.18 and 1.30 in both 0.3 mg/cc and 0.45 mg/cc gas-fill hohlraums, also varying the laser entrance hole size (3.64 vs 4 mm diameter). The pulse duration between outer picket launch and midpoint time of inner peak power was τ = 6.3 ns. Small few % corrections for variations in peak CF, outer cone picket energy, and pulse length in one case have been applied to the observed P_{2} and x-axis FoM. Figure 26 shows that the FoM just assuming ρ_{0}^{0.13} scaling persists to 0.45 mg/cc gas-fill reproduces the core P_{2} trend, including the y-intercept and slope to 5%. Interestingly, assuming no ρ_{0} dependence per the bubble model Fig. 24 between 0.3 and 0.45 mg/cc gas fill is a worse fit, with the caveat that only two data points exist at 0.3 mg/cc. The dashed line fit also highlights the expected more negative core P_{2} per Eq. (14) when using a 1 − (3.64/4)^{2} = 17% smaller solid angle LEH. Further experiments raising the fill density beyond 0.45 mg/cc would be required to fully test the model prediction of non-monotonic core P_{2} dependence on gas-fill. The model could also be applied to alternate hohlraum surface wall materials^{50} of different A and Z and, hence, different bubble speeds and inverse Bremsstrahlung absorption ∼ Z and to designs using foam-liners as a means to reduce wall expansion.^{51}

## V. COMPARISON OF MODELS AND RAMIFICATIONS

At this point, it is useful to summarize comparisons to the prior model.^{8} We first list below in Table I the updated power laws with fit error bars and the physics they highlight.

Model term . | Dependence . | Physics . | |
---|---|---|---|

Y-axis | |||

R_{cap} | R_{cap}^{−1} | Capsule distance traveled | |

R_{cap}/R_{hohl} | (R_{cap}/R_{hohl}) ^{−0.5} | Hohlraum P_{2} asymmetry smoothing | |

X-axis | |||

ρ | ρ^{0.13 ± 0.03} | Competition between IB absorption ∼ρ^{2} and bubble speed ∼1/ρ^{0.2} affecting inner beams and core P_{2} | |

E^{0.2}τ/R_{hohl} | (E^{0.2 ± 0.04}τ/R_{hohl})^{2.5} | Time-delayed and spatially compressed bubble clipping of inner beams | |

R_{cap}/R_{hohle} | (R_{cap}/R_{hohle})^{1.1 ± 0.2} | Stagnation absorption further affecting inner beams and core P_{2} |

Model term . | Dependence . | Physics . | |
---|---|---|---|

Y-axis | |||

R_{cap} | R_{cap}^{−1} | Capsule distance traveled | |

R_{cap}/R_{hohl} | (R_{cap}/R_{hohl}) ^{−0.5} | Hohlraum P_{2} asymmetry smoothing | |

X-axis | |||

ρ | ρ^{0.13 ± 0.03} | Competition between IB absorption ∼ρ^{2} and bubble speed ∼1/ρ^{0.2} affecting inner beams and core P_{2} | |

E^{0.2}τ/R_{hohl} | (E^{0.2 ± 0.04}τ/R_{hohl})^{2.5} | Time-delayed and spatially compressed bubble clipping of inner beams | |

R_{cap}/R_{hohle} | (R_{cap}/R_{hohle})^{1.1 ± 0.2} | Stagnation absorption further affecting inner beams and core P_{2} |

The original model showed 8 *μ*m rms P_{2} core variations at a given FoM and quoted a picket fluence exponent error of ±33%. The larger variation was to be expected as comparing different implosion designs sometimes varying multiple FoM parameters at once including incident peak cone fraction and parameters such as outer beam average z pointing not in the models. For example, per Eq. (14), the typical ±10% changes in the LEH solid angle across designs can account for ±6 *μ*m changes in expected core P_{2} and all else held constant. The correction to the capsule response now applied to y-axis as R_{cap}/R_{hohl} varied by up to ±10% and can account for ±3 *μ*m in P_{2} at P_{2} = ±20 *μ*m. The apparent range of ±40 *μ*m core P_{2} was ≈1.5–2 × larger than the raw data range because it had been corrected for the incident cone fraction in the y-axis. Therefore, the core P_{2} ranges probed between the prior and updated model were similar. The gas-fill sensitivity comparison was between 0.3 and 0.6 mg/cc, so does not overlap this study, but the current model does predict the gas-fill sensitivity to reverse sign above 0.3 mg/cc as observed for HDC. The FoM capsule sensitivity to (R_{cap}/R_{hohle})^{1.1} is consistent with the prior model quoting a power law between 1 and 1.5. Relative to the prior model, the core P_{2} picket fluence sensitivity E^{0.5} that appeared to be independent of gas-fill at least up to 0.6 mg/cc has been retained, so validated that aspect of the prior model. More importantly, combined with the bubble data scaling as only E^{0.2} for HDC designs led to the prediction of a more constraining τ^{2.5} instead of linear in time scaling, backed up by the bubble model at least between 0.15 and 0.3 mg/cc. This is the most fundamental prediction, which puts a premium on developing innovative hohlraum designs to reduce outer beam fluence and/or bubble growth (by, for example, recessed^{26} and tilted^{52} hohlraum walls, and/or azimuthal quad time staggering^{53}) or to avoid inner beam propagation impairment (by, for example, avoiding using “inner” beams by a 3D LEH arrangement^{54}).

## VI. SUMMARY

An updated fit for the P_{2} imploded core asymmetry as a function of picket energy/fluence, gas-fill, and capsule size has been derived by combining x-ray core shape and hohlraum plasma imaging data with analytical and geometric arguments, backed up by comparison to simulations. The fit is explained by an analytical model for outer Au bubble absorption of inner beams, benchmarked by HDC designs imploded in cylindrical hohlraums with gas fills between 0.15 and 0.3 mg/cc of current interest for ICF as efficient, low backscatter hohlraums. The model also highlights how the core P_{2} asymmetry is typically ≈2× more sensitive to fractional changes in inner beam absorption in the Au bubble than to the incident laser cone fraction changes routinely applied for compensation and predicts greater sensitivity to drive pulse length than previously assumed. Furthermore, hard x-ray images from the equator region provide the first experimental proof of the existence of a predicted stagnation region between capsule and hohlraum blow-off that can further impair inner beam propagation that is sensitive to the initial capsule size and can qualitatively explain the measured core P_{2} asymmetry dependence on the initial capsule size. This equatorial data also discriminate between models of the flux limiter in the ablator plasma, suggesting that its value is high (0.15) compared to the 0.03 value for the outer Au bubble regions inferred by tracer spectroscopy in other experiments.^{28} We have not attempted to understand how the flux limiter in the high-Z Au could explain residual discrepancies between simulated and measured core P_{2} asymmetry as besides altering bubble transmission, it also affects the level of predicted^{28,55} reflected laser rays or glint^{56} during peak power not measurable in this set of experiments. Ongoing experiments are checking levels of peak power glint. The fact that the simulations predict an opposite core P_{2} trend vs initial gas-fill compared to data (Fig. 6), despite matching measured bubble growth (Fig. 8) and the bubble model absorption [Fig. 10(a)] and density [Fig. 10(d)], suggests simulations overestimate the effect of initial gas-fill on the growth of the stagnation region. We also note our model explains trends without invoking any sensitivity of CBET and, hence, ensuing core P_{2} asymmetry to the parameter changes we made. To test CBET sensitivity, we would need to repeat some of these shots choosing a different wavelength separation between inner and outer beams. One could then claim a CBET sensitivity if the differential core P_{2} asymmetry between the baseline design and one of the variants changed significantly with wavelength separation. Shots are also being devoted to measuring the absolute level of time-dependent peak power CBET using the +50% picket, 0.3 mg/cc design that could then be applied to the other designs.

## ACKNOWLEDGMENTS

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 and General Atomics under Contract No. DE-NA0001808. 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. We thank J. Lindl for useful comments and R. Tommasini for an independent check of the inverse Abel transform used to generate Figs. 21 and 23. We are deeply indebted to support from the target assembly, the laser operation, and the target diagnostics group. We are also grateful to S. Johnson and S. Vonhof for the design and precise assembly of the hohlraum with thin-wall patches. Special thanks to M. Farrell for the precise characterization of the pinhole throughput and H. Huang and M. Yamaguchi for the measurement of spectroscopic transmission of the x-ray filters.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.