Demonstrating ignition hydrodynamic equivalence in direct-drive cryogenic implosions on OMEGA

Achieving ignition in a direct-drive cryogenic implosion at the National Ignition Facility (NIF) requires reaching central stagnation pressures in excess of 100 Gbar, which is a factor of 3 to 4 less than what is required for indirect-drive designs. The OMEGA Laser System is used to study the physics of cryogenic implosions that are hydrodynamically equivalent to the spherical ignition designs of the NIF. Current cryogenic implosions on OMEGA have reached 56 Gbar, and implosions with shell convergence CR< 17 and fuel adiabat α > 3.5 proceed close to 1-D predictions. Demonstrating hydrodynamic equivalence on OMEGA will require reducing coupling losses caused by cross-beam energy transfer (CBET), minimizing long- wavelength nonuniformity seeded by power imbalance and target offset, and removing target debris occumulated during cryogenic target production.


INTRODUCTION
The main approach to ignition by means of laser-driven inertial confinement fusion [1] (ICF) currently pursued by the National Ignition Facility [2] (NIF) is x-ray (or indirect) drive. In the other ICF laser approach, direct drive, the target is driven by laser irradiation directly coupled to the plasma blowing off the imploding capsule. The main advantage of the indirect-drive approach is reduced sensitivity of drive uniformity to short-scale beam nonuniformities. The main advantage of direct drive is higher coupling efficiency (by a factor of 3 to 5) of the laser energy into kinetic energy of the shell (hydrodynamic efficiency) compared to that of x-ray drive. The OMEGA Laser System [3] and the KrF laser NIKE at the Naval Research laboratory [4] have been the principal facilities for direct-drive experiments in the United States. Early challenges in achieving beam uniformity required for ignition have been resolved over the last several decades by introducing several beam smoothing techniques. These include distributed  Figure 1.
Threshold hot-spot pressure p thr as a function of hotspot internal energy. Typical hot-spot energies in indirect-and direct-drive implosions for a NIF-scale laser energy are shown with blue and red boxes, respectively.
phase plates [5] (DPP's), polarization smoothing with birefringent wedges [6], and smoothing by spectral dispersion [7] (SSD). In addition, implementing adiabat-shaping techniques [8,9] significantly reduced the impact of Rayleigh-Taylor (RT) instability growth [1] during shell acceleration. Also, imprint reduction was demonstrated by using mid-Z-doped ablators [10] and high-Z target overcoats [11]. These developments have made the direct-drive approach very attractive. Such progress and the challenges in achieving ignition on the NIF using x-ray drive suggest considering direct drive as a viable alternative for developing a burning plasma platform in a laboratory.
Compared to the x-ray drive, direct-drive targets couple a larger fraction of laser energy into shell kinetic energy and internal energy of the neutron-producing central region of the target (hot spot) at peak fuel compression. This relaxes the requirement on shell convergence and hot-spot pressure in an igniting target. This can be shown with the help of a commonly used ignition condition (which can also be derived from generalized Lawson criterion [12]) according to which plasma self-heating is initiated by pdV work of the shell and alpha heating if the product of areal density and ion temperature inside the shot spot satisfies [1,13] (ρR) hs × T 0.3 g/cm 2 × 5 keV, where ρ, R hs , and T are the hot-spot density, radius, and temperature. Substituting expressions for the pressure p hs = (1 + Z)ρT /m i (Z is the average ion charge and m i is the average ion mass) and internal energy E hs = 3/2 p hs V hs (V hs is the neutron-averaged hot-spot volume) into Eq. (1) gives a minimum pressure requirement (threshold) for ignition, p hs > p thr = 250 Gbar E hs 10 kJ whereP is the ignition pressure parameter. The dependence of the threshold pressure p thr on the hot-spot internal energy is shown in Fig. 1. Spherically symmetric direct-drive cryogenic designs on OMEGA couple up to 0.44 kJ (out of 26-kJ incident laser energy) into the hot-spot internal energy. When hydrodynamically scaled to the NIF-size laser energy (1.5 MJ to 1.8 MJ), these designs are predicted to couple 5× to 10× more energy into the hot spot [25 kJ to 40 kJ, depending on laser coupling efficiency (see red-shaded region in Fig. 1)] compared to that of indirect drive [4 kJ to 5 kJ (see blue-shaded region in Fig. 1)], resulting in 2.5× to 3× lower hot-spot pressures required for ignition. The hot-spot size also gets larger with E hs , leading to smaller shell convergence (Cr∼ 22 compared to 35 to 40 in present x-ray-drive ignition designs) and less-demanding long-wavelength drive uniformity requirements.

OMEGA cryogenic implosions
To separate one-dimensional (1-D) factors that limit the target performance (drive efficiency, adiabat, etc.) from three-dimensional (3-D) effects, dedicated experiments are performed on OMEGA with the purpose of improving physics understanding and accuracy of 1-D code predictions. To identify critical implosion parameters, the 1-D scaling laws for peak pressure, hot-spot energy, and the ignition pressure parameter are written in terms of implosion velocity v imp (defined as the peak mass-averaged shell velocity), the drive (ablation) pressure p a , and the in-flight shell adiabat α (ratio of the shell pressure to Fermi pressure at shell density) [14]: The implosion velocity and shell kinetic energy are inferred in an experiment by measuring ablation-front trajectory and mass ablation rate using self-emission imaging [15]. The ablation pressure is inferred from simulations that match the measured ablation-front trajectory, mass ablation rate, bang time [16], and scattered-light power and spectrum [17]. Finally, the shockinduced adiabat is inferred by measuring shock velocities early in the pulse using VISAR [18]. An additional fuel-adiabat increase caused by hot-electron preheat is estimated by measuring the hard x-ray signal [19] and areal density [20,21] in mid-to high-adiabat implosions (the areal density in 1-D, for a given laser energy, depends mainly on shell adiabat [22], ρR ∼ α −0.5 ). A detailed comparison of 1-D simulation results using the hydrocode LILAC [23] with the data [14] shows good agreement between the two for a variety of target designs and drive conditions. One-dimensional simulations include nonlocal thermal transport model [24], a ray-based cross beam energy transfer (CBET) model [25], and first-principle EOS (FPEOS) models [26] for both the DT ice and CD ablator. An analysis of direct-drive implosions on OMEGA has shown that coupling losses related to CBET [25] significantly limit the ablation pressure (as much as 40% on OMEGA and up to 60% on the NIF-scale targets), implosion velocity, and shell kinetic energy. Considering such losses, demonstrating the hydrodynamic equivalence of implosions on OMEGA to ignition designs on the NIF requires the shell IFAR to exceed the current stability threshold level (∼ 22) [14]. One of the CBET mitigation strategies [27] involves using laser illumination with a laser beam diameter that is smaller than the initial shell diameter. This strategy, as demonstrated both theoretically and experimentally, recovers some coupling losses and increases the ablation pressure. Since the effect of CBET is small early in the implosion when the density scalelength and laser intensity are small, beam zooming schemes [28] consider the beam focal spot at an early time to be at the initial target radius (to maximize the illumination uniformity), reducing it to 0.6× to 0.7× of that size at the beginning of the main drive.
While zooming implementation on OMGEA is still a few years away, a test of the CBET reduction strategy was performed using "static" DPP's that produce the focal spots smaller than the initial target size through the entire drive pulse. The focal-spot radius (defined as the radius of a 95% beam energy contour) in these experiments was fixed at R b = 410 µm. The ratio of R b to target size (R target ) was changed by varying the R target from 400 µm to 500 µm. SSD was off during the main pulse for larger targets with R target = 450 µm, 480 µm, and 500 µm (to increase on-target laser energy), while the targets with R target = 400 µm, 430 µm, and 450 µm used SSD for the entire pulse. Hence, the ratio R b /R target changed from 1.025 to 0.78. According to simulation results (which matched the observables), the smallest target (R target = 400 µm) had v imp = 3.5 to 3.6×10 7 cm/s and a hydroefficiency of f hydro = 3.5%, while the largest target had similar implosion velocity v imp = 3.6 to 3.7 × 10 7 cm/s, but more than twice larger hydroefficiency f hydro = 7.2%. Such an increase in hydroefficiency was partially caused by smaller refraction losses experienced by the larger target (smaller R b /R target and larger density scale length) and partially by reduced CBET losses. To quantify each effect, a simulation was performed with R target = 500 µm, where the R b was increased to match R target and the laser energy was increased from 27.5 kJ (used in an experiment) to 32 kJ to avoid coasting chase. In such a simulation, implosion velocity was dropped by 11% to v imp = 3.2 × 10 7 cm/s and the shell hydroefficiency was reduced by 26% to f hydro = 5.3%. Figure 2 shows the target performance (p hs ) for different target diameters. The hot-spot pressure is inferred [29] by using the measured neutron yield, burn duration ∆t burn [using both the neutron temporal diagnostics (NTD) [16] and framing-camera measurement of the x-rayburn duration], neutron-average ion temperature T i n , and hot-spot size R 17 (defined as the radius of 17% of peak emission contour for x rays in the 4-keV to 7-keV energy range) at bang time using high-resolution (∼ 6µm in space and ∼ 30 ps in time) time-resolved Kirkpatrick-Baez framing camera [30]. Assuming an isobaric hot spot and fitting the burn history with a Gaussian with FWHM= ∆t burn , the maximum burn rate N max is related to neutron yield Y as N max = 2Y ln 2/π/∆t burn , where N max = n T n D T 2 V hs dV σv /T 2 . Therefore, pressure at bang time can be determined using p hs 8Y ln 2/π/(f D f T ∆t burn V hs dV σv /T 2 ) 1/2 , where σv is the cross section for D-T reactions, and f D and f T are the fractions of D and T in the fuel, respectively. In evaluating the spatial integral in the latter equation, the following spatial profile for the ion temperature (obtained using simulation results) is assumed: where T c is the maximum hot-spot temperature, determined by matching ( V hs dV σv /T )/( V hs dV σv /T 2 ) with the measured T i n ], and, as follows from code predictions, R hs and R 17 are related using R hs = 1.06 R 17 .
To understanding trends shown in Fig. 2, the effect of shell nonuniformity must be considered. The evolution of long-wavelength nonuniformities seeded by target offset, beam geometry, beampower imbalance, and mispointing is studied using the 3-D hydrocode AST ER [31]. These simulations show that the bubbles (region of low-density material that protrudes from the central region into the higher-density shell), which develop because of the RT growth of longwavelength perturbations (l 5) during shell deceleration, increase the volume of central region V cntr , reducing the hot-spot pressure (p hs ∼ 1/V 5/3 cntr ) and neutron yield. As the shell continues to converge, the bubbles eventually break out of the shell, quenching hot-spot confinement and neutron yield (see Fig. 3). Earlier peak burn leads to sampling implosion conditions when shell convergence and central pressure have not yet reached the peak values. The 3-D effects also increase the central region volume, reducing the hot-spot pressure and preventing fuel material from reaching stagnation. This limits conversion efficiency of shell kinetic energy into the internal  Figure 3. Neutron-production rate calculated using the code AST ER without (labeled "1-D") and with (labeled "3-D") effects of long-wavelength nonuniformity growth. Also shown are simulated shell-density maps at times labeled by (1) and (2). energy of the hot spot. To account for the first effect (early pressure sampling), Fig. 4 plots the inferred hot-spot pressure normalized to the predicted pressure at the measured (earlier) bang time as a function of 1-D shell convergence calculated at the experimental bang time. Figure  4 shows that implosions with a fuel adiabat of α > 3.5 proceed close to 1-D predictions up to a shell convergence of C r ∼ 17. An additional shell convergence does not lead to an increased pdV work on the hot-spot because of the RT growth of low-l modes. Further reduction in the target performance at lower fuel adiabat is caused by compromised shell integrity due to short-wavelength nonuniformity growth during shell acceleration seeded by shell imperfections (mainly target debris).
In summary, the cryogenic campaign with reduced beam size relative to the target size (R b /R target < 1), performed on OMEGA to reduce CBET losses, demonstrated increased laser coupling and hydrodynamic efficiency. This coupling enhancement, however, did not improve the target performance. Numerical simulations indicate that long-wavelength nonuniformities caused by target offset and power imbalance lead to an increased target central volume and early burn truncation. This effect is exacerbated by a reduction in beam overlap when the target size is increased relative to beam size.

Acknowledgments
This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0001944, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.