The physics of gain relevant to inertial fusion energy target designs Free

In recent years, inertial confinement fusion (ICF) experiments have seen an upward trend in fusion yield. This culminated in a significant result last year where, for the first time, a laser-driven ICF target exceeded ignition conditions and achieved target gain.^1–3 In addition, laser-direct-drive implosions on OMEGA⁴ have reached the point at which the energy of fusion reaction products exceeded that of the hotspot internal energy.⁵ These achievements reflect the significant advancement in knowledge since the initial results from the Nation Ignition Campaign in 2012.⁶ The path of the most recent experiments on both the National Ignition Facility (NIF)⁷ and the omega Laser Facility, however, have pursued target regimes that will not scale to the high gains required for inertial fusion energy (IFE).

Performance of current ICF experiments is significantly degraded by poor laser coupling^8–11 and, as a result, increased sensitivity to hydrodynamic instabilities seeded by capsule imperfections,^12,13 engineering features^14,15 (fill tubes, stalks, etc.), and non-uniformities in the drive illumination.^16,17 This has led to the design of implosions that maximize shell implosion velocity and kinetic energy in an attempt to efficiently couple energy to the hotspot and reach ignition conditions. To accomplish this, the highest-performing implosions on the NIF and OMEGA use target designs with increased fuel entropy or adiabat¹⁸ in order to minimize the impact of the hydrodynamic instabilities. Fuel adiabat α is defined as the ratio of the target shell pressure to the Fermi-degenerate pressure of the fuel at solid density. The combination of high adiabat and high implosion velocity leads to a lower assembled fuel mass and lower fuel burnup fraction (because of reduced fuel areal density), limiting the target gain. To reach the performance required for IFE-relevant designs at megajoule-scale laser energies, it is necessary to develop a path toward the higher-mass, lower-adiabat regime where fuel gains are significantly higher.

A simple argument for the minimum gain required for IFE considers the power balance of a reactor. If the driver energy is given by the product of the power in $P in$ and driver “wall-plug” efficiency $η D$⁠, the fusion power output is then $G η D P in$⁠, where G is the target gain. Assuming the conversion of fusion power to electric power is 40%,¹⁹ the total electric power of a power plant is $P out ≈ 0.4 G η D P in$⁠. Then, if 1/4 of the output is circled back into powering the laser and other plant components, $P in = 0.25 P out = 0.1 G η D P in$⁠; this leads to $G η D ≈ 10$⁠. Finally, taking $η D ∼ 10 %$⁠, we arrive at the requirement that target gain be G > 100.

To date, the highest-yield targets have been achieved on the NIF using the indirect-drive approach to ICF. This is where laser light is first converted into a bath of x rays contained within a high-Z casing (or Hohlraum). In indirect drive, only ∼10% to 15% of the driver energy is incident on the capsule. Conversely, laser direct drive has almost all the energy focused on target since the focal spot size is similar to that of the initial capsule size. However, the presence of the laser-plasma instabilities (LPI), such as stimulated Brillouin scattering and stimulated Raman scattering, lead to coupling losses, significantly reducing drive (ablation) pressures and limiting fuel mass and areal density. Understanding and eliminating these losses is, therefore, a priority in the development of next-generation laser-driver technology. New broadband lasers have great promise in their ability to mitigate the deleterious effects of LPI.^20–23 It is expected that the use of bandwidth in combination with beam zooming will raise the absorption efficiency to upward of 90%. Laser focal spot zooming is the act of decreasing the spot size of the laser during the implosion. This helps prevent any loss of laser light that is not absorbed as the target decreases in size. A number of concepts have been proposed to achieve focal spot zooming. For excimer lasers, it has been shown that zooming can be achieved via induced spatial incoherence as demonstrated on Nike.^24,25 For the solid-state laser technology considered in this paper, zooming would have to be employed by the use of multiple beam lines. If the use of bandwidth is shown to increase the convergent ablation pressure to above 200 Mbar (and possibly above 300 Mbar with zooming) without excessive fuel preheat caused by hot-electrons generated by LPI, direct drive will be a very compelling candidate for an ICF driver in an IFE facility.

In light of the demonstration of laboratory ignition at moderate laser drive energies and new laser technology on the horizon, the time is apt to revisit the ICF ignition design parameter space relevant to IFE. This manuscript establishes scaling laws that relate target gain to ICF design parameters. This is achieved by using a combination of the conclusions of simple physics arguments with the results of radiation-hydrodynamic simulations. The outcome is a set of simple scaling laws that can be used to describe the parameter space of IFE target design.

Extensive previous work has been done on deriving scaling laws for ICF-relevant metrics by Betti and Zhou.^26–28 These scaling laws were derived using a flux-limited thermal conduction model with the value of the flux limiter chosen based on experiments conducted on the OMEGA laser. Significant progress in understanding laser coupling in laser-direct-drive ICF plasmas has been made over the last several years, so many of these scaling laws need to be revisited. This study re-derives these scaling laws considering the enhancement in ablation pressure due to broadband laser technology and beam zooming. This study also presents a key addition to the work, where the design space is constrained by considering the minimum implosion velocity required for ignition. In doing this, it is possible to express the target gain as a function of driver energy, incident laser intensity, drive pressure, and fuel adiabat.

This section summarizes the target design process that was used to generate a database of 1D implosion simulations. This study was carried out with the radiation-hydrodynamics code LILAC³⁴ that has been benchmarked against direct-drive experiments. In each case, the design of each implosion began by setting the thicknesses of the different capsule layers; the laser pulse profile was then shaped to set the fuel adiabat and ignite the target.

The targets consisted of central DT gas, surrounded by a frozen DT layer encased in a DT-wetted CH foam; a 100 mg/cm³ foam was used, so the wetted foam density was 0.33 g/cm³. The central region contained DT vapor at the triple point, which corresponds to a density of 0.67 mg/cc. The inner radius of the targets varied in 100 μm steps from 750 to 1350 μm, and fuel masses were set to the values of 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.5, 3.5, 5, and 7 mg. The thickness of the DT-wetted CH foam ablator was optimized in each implosion, such that the wetted-foam layer is entirely ablated during the early part of the drive pulse, taking advantage of the high ablation and hydrodynamic efficiency of the DT fuel layer. Crucially, the pure DT never drops to densities below critical in the plasma corona; therefore, laser absorption takes place only in the CH DT region, which has higher inverse bremsstrahlung absorption due to the higher Z. The initial beam spot shape was a super-Gaussian order 5, with the 1/e fraction of the peak intensity at 90% of the initial outer radius of the target. Higher ablation pressures would be achieved with smaller spot shapes; however, this choice of large beam radius is purposefully cautious, with the intention to minimize low-mode asymmetries that arise from the beam-port geometry. Note that the effect of multi-dimensional perturbations is not modeled in this study. In some implosions, one- and two-stage focal-spot zooming was employed to enhance absorption and drive pressure without sacrificing implosion symmetry. For the first stage, the beam radius was reduced to $0.55 ×$ the initial value at the time of shock breakout from the inner surface. By timing this at shock breakout, when the conduction zone and plasma are established, this aids in smoothing out nonuniformities that arise from the finite beam size. When a second zooming stage was used, the spot size was reduced to $0.275 ×$ the initial beam radius 1 to 2 ns after the first zooming stage. It should be noted that the aggressive zooming used in this study was chosen to ensure minimal refractive losses for all implosions. It is possible for strong zooming to introduce significant low-mode perturbations as the spot size decreases. A fully considered optimization of zooming requires multi-dimensional simulations; this work is currently being pursued by the authors.

The laser wavelength used in the simulations was 351 nm (frequency tripled ND:glass). The pulse shape in this study used a 300 ps FWHM picket followed by a foot, then a rise to the peak power. The pulse was shaped systematically, iterating over a series of simulations to find the maximum gain. First the picket power was tuned to set the mass-averaged adiabat of the fuel at a desired value that varied between 1 and 7. The power of the foot was then chosen such that the shock launched from the foot breaks out soon after the breakout of the shock launched from the picket. The rise of the main pulse then had a kidder-like³⁵ shape, with the timings chosen so the main shock is also synchronous with the picket and the peak in laser power occurs after the shock breakout. Finally, the power of the main pulse was increased until the shell velocity became sufficient to achieve ignition. In each case, the duration the peak was timed so that the pulse ended at the time the shell reaches maximum implosion velocity. The laser energy in these pulses ranged from 1 to 8 MJ. The implosion velocities varied between 223 and 473 km s⁻¹.

It is the aim of this study to relate design parameters (energy, adiabat, ablation pressure, and incident intensity) to target performance (gain and stability characteristics). This is achieved by relating various implosion metrics to one another through simple power-law scalings. In each case, basic physics arguments are used to argue the dependence of variables upon one another, resulting in a scaling law. Where necessary, fits are then employed to find exact dependencies of the variables from a database of 1D ICF implosion simulations. In each fit, least squared regression is used to find either the equation coefficients or exponents or both. In this study, scaling laws are found for unablated target mass, ablation pressure, assembled fuel burnup fraction, areal density, and implosion velocity, all of which relate to target gain. Additionally, a scaling law for IFAR is found since it (along with fuel adiabat) has been shown to be relevant to hydrodynamic stability.³⁶ 

In Sec. II, a simple scaling law was used to relate kinetic energy to work done by the ablative drive. In that derivation, the effects of mass ablation and convergence were omitted for simplicity. These effects can be accounted for with the addition of some corrective factors.

For each implosion, it is necessary find a value for the ablation pressure that is characteristic of the implosion. However, ablation pressure is a time-dependent variable that changes with laser power and capsule convergence. In this study, ablation pressure is defined as the pressure at the ablation front at the time where the front has converged to $1 / 3.5 ×$ the initial outer surface of the capsule. This definition of drive pressure uses the same convention as outlined in Ref. 41 and roughly corresponds to the point in the implosion where the effect of convergence causes the pressure inside the shell to exceed the ablation pressure. The ablation front itself is defined at the minimum of the electron temperature in the shell (⁠ $d T e / d r = 0$⁠). Details of this definition can be seen in Fig. 2.

The errors on the coefficients were ±4 Mbar, ±3 Mbar, and ±3 Mbar, respectively, and the errors on the exponents were all ±0.02. The variable $p a , 1 − zoom$ is taken from simulations, where the beam radius is decreased by a factor of $0.55 ×$ at the time of shock breakout. Then, $p a , 2 − zoom$ is taken from simulations using a second zooming stage, reducing to $0.275 ×$ the initial beam radius 1–2 ns after the first zoom stage. The ablation pressure with time profile in Fig. 2 shows a one-stage zoom implosion and highlights the jump in ablation pressure that is observed when the beam size is reduced. The scaling of ablation pressure with intensity from the simulation database and corresponding fits can be seen in Fig. 3. These relations are used later to explore IFE designs for different scalings of intensity and ablation pressure.

Burnup fraction is well understood to be directly dependent on ρR by $f burn = ρ R / ( ρ R + H B )$⁠,²⁹ where the variable $H B$ is dependent on the hotspot conditions and can vary between 6 and 9  $g / cm 2$⁠. For the results in this study, $H B = 6.67$ was used across all simulations. It should be noted that more accurate models for burnup fraction have been proposed,⁴² but the variance of the data in this study is large enough that such a model does not improve the quality of the fit to the data.

For any given target design, the choice of implosion velocity is a result of the balancing of ignition conditions in the hotspot, target gain, and shell stability properties. As will be shown below, satisfying the ignition conditions at peak compression requires that the implosion velocity $v imp$ exceed a threshold value $v min$⁠. Since the shell mass is inversely proportional to the implosion velocity, operating at the limit of this threshold maximizes fuel mass and, therefore, target gain. The downside of such a target, however, is that there is no margin to overcome losses to 3D effects (perturbation growth, which reduces hotspot pressure and energy). On the other hand, designs that operate with implosion velocity well above the minimum threshold are also less robust to nonuniformity growth because they typically have higher IFAR's, which are susceptible to short-wavelength perturbations during shell acceleration. With this in mind, it is proposed that operating with $v imp$ in close proximity to $v min$ is a good compromise between stability and target gain. The target design process outlined in Sec. III is intended to find targets that ignite at the minimum implosion velocity.

By finding a solution to the equations that determine the minimum implosion velocity required for ignition [Eqs. (27) and (30)], it is possible to constrain the model and express implosion velocity as a function of energy adiabat pressure and intensity. With that, the gain of a target can be calculated from the laser energy, ablation pressure incident intensity, and adiabat in the limit of the lowest implosion velocity required for ignition. Figure 9 shows the gain scaling from Eq. (36) with different levels of zooming [calculated by using the ablation pressure scaling Eqs. (14)–(16)]. It can be seen that a one-stage zoom at 1 MJ is enough to reach gain ∼100 at low adiabat $α ∼ 1$⁠. Increasing the energy to 2 MJ significantly expands the parameter space and enables the adiabat to be increased without sacrificing gain. Gain can also be increased by applying a second zooming stage and/or reducing the incident laser intensity. For instance, adding a second zoom stage and decreasing the intensity to $0.5 × 10 15$ W/cm² allows a gain of 100 to be achieved at 2 MJ with an adiabat above 3. However, one must be cautious in lowering the incident intensity because this may sacrifice target stability. Previous work has shown that the IFAR and adiabat of a target are related to the growth of high-mode instabilities.³⁶  Figure 10 compares how the IFAR predicted from Eq. (35) varies over the IFE target space for two different intensities, each using a two-stage zoom. Targets with lower incident intensity may achieve higher gains, but will make a sacrifice in the form of increasing IFAR. At present, it is not possible to estimate an upper limit on the IFAR since this will depend on the level of laser imprint present in next-generation laser technology. The addition of bandwidth is expected to reduce imprint below the levels seen in current experiments. The amount of imprint present in broadband laser sources will ultimately determine the IFAR limit for high-gain IFE targets. It should be noted that the current limit on IFAR in direct-drive implosions is dominated by imprint. Even in the event that next-generation laser technology significantly reduced imprint, targets would likely still be susceptible to other sources of high-mode perturbation such as imperfections from target manufacture. However, the low IFARs achieved when using MJ energies and higher ablation pressures will help mitigate the design sensitivity to those imperfections.

A reasonable estimate for the laser energy that may be available in a full-scale IFE reactor is 1–2 MJ. With that in mind, we present four IFE example targets that achieve a gain of 100 over a range of adiabats and drive pressures. The simulation characteristics are summarized in Table I. It should be noted that multi-dimensional effects in the targets summarized in Table I will reduce the total yields and as such should not be taken as suggestions for IFE targets. Instead, these results are intended to demonstrate the trade-offs that can be expected in 1D implosion performance when energy, adiabat, and ablation pressure are adjusted at the MJ scale.

The work presented here has explored the design space of future IFE facilities that will make use of next-generation laser technologies. The addition of bandwidth to the laser spectrum, along with focal-spot zooming, are technologies that will significantly boost the drive pressures in laser-direct-drive. This enhancement in ablation pressure will make it possible to reach a gain of 100 with targets that are substantially more hydrodynamically stable. A gain of 100 is the minimum required for the power flow of a plant with a 10% wall-plug efficiency. However, the economics of plant will also depend on the financial cost of the driver per megajoule and target gain would need to be adjusted accordingly.

In this work, it has been shown how simple scaling laws can be used to map out the implosion parameter space. For each scaling law, a simple physics model was used to justify the dependences of implosion parameters upon one another. The terms of those simply derived scaling laws were then fit to a simulation database of ICF implosions spanning a wide range of laser energies and target scales. A hotspot model was used to constrain the parameter space to implosions that operate at the minimum possible implosion velocities. Such implosions are the most robust and have the highest performance due to the resulting large mass assemblies. Additionally, the τ parameter was proposed as a correction to previous hotspot models that accounts for the amplified shell density due to convergent effects. With the scaling relations derived in this manuscript, it is possible to predict the 1D gain and IFAR for a given energy, adiabat, and ablation pressure scaling.

This study was funded by the ARPA-E BETHE (Grant No. DE-FOA-0002212). This work was based upon the work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, 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 paper.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express 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 U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The authors have no conflicts to disclose.

William Thomas Trickey: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (lead). Valeri N. Goncharov: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Riccardo Betti: Investigation (supporting); Writing – review & editing (supporting). E. Michael Campbell: Investigation (supporting); Writing – review & editing (supporting). Timothy J. B. Collins: Investigation (supporting); Writing – review & editing (equal). Russell K. Follett: Investigation (supporting); Writing – review & editing (equal).

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