Dream fusion in octahedral spherical hohlraum

Energy from controlled nuclear fusion has been a quest of scientists worldwide for more than a half a century,^1–3 and achieving a predictable and reproducible fusion gain is the first step toward an inertial fusion energy power plant. In indirect-drive inertial fusion, a spherical encapsulated deuterium–tritium (DT) fuel pellet of millimeter-size is irradiated and imploded with a very high velocity by x-rays converted from lasers inside a centimeter-size hohlraum, thereby compressing it to extreme density, temperature, and pressure to release significant fusion energy. The world’s largest laser facility, the National Ignition Facility (NIF),^4–6 the size of three football fields, aims to achieve fusion ignition via an indirect-drive scheme. Recent striking progress on the NIF, with 1.37 MJ fusion yield^7–10 for ∼70% of the input laser energy, has come close to demonstrating the feasibility of the indirect drive for clean and sustainable energy production. However, it should be recognized that this remarkable achievement of the NIF has been the culmination of many years of experiments and simulations, including symmetry tuning campaigns,^10–15 determinations of the appropriate level of irradiation of the capsule from implosion performance,^6,16 investigations of the effects on the incidence angle of the laser pulse and the use of time-dependent multipliers,^10,17,18 and iterations to find the right design parameters to produce the required spherical radiation drive inside cylindrical hohlraums.^10,15,19 Note that the symmetry tuning techniques used on the NIF are aimed mainly at suppressing the Legendre polynomials P₂, which represent the inherent asymmetry of all kinds of hohlraums with two laser entrance holes (LEHs). It is clear that the NIF will meet even greater challenges on the road to achieving a predictable and reproducible fusion at high gain. Nevertheless, its historic achievement and lessons provide clues to enable further progress to be made toward a dream fusion at an upgraded facility with simple and robust target designs.

To achieve a dream fusion expected from an ideal design of a high-convergence spherical implosion, with predictable and reproducible ignition and gain, there are three prerequisites: (1) a credible one-dimensional (1D) theoretical design of a spherical implosion at a convergence ratio higher than 30, (2) sufficient laser energy to irradiate the hohlraum and generate an implosion producing the required radiation energy, and (3) a highly symmetric spherical radiation drive with the implosion required temporal and spectral characteristics. With regard to the first of these, a near-1D indirectly driven implosion has been successfully achieved at a convergence ratio 30 on the NIF, thus demonstrating the credibility of 1D implosion design at such a high convergence ratio.²⁰ With regard to the second prerequisite, a systematic analysis of the hohlraum energetics experiments has shown that the NIF energy is sufficiently high to generate the radiation energy required for ignition.²¹ However, it is unfortunately not easy to fulfill the third prerequisite and create spherically symmetric radiation on the NIF.

As is well known, the major obstacles preventing ignition at the NIF include asymmetry of target irradiation, laser plasma instabilities (LPIs), and hydrodynamic instabilities.^22–24 However, viewed from another angle, leaving engineering issues aside, the relevant problems are completely physical issues concerning the radiation drive. To achieve the dream fusion expected from an ideal design of high-convergence spherical implosion, an ideal and clear radiation drive is necessary. In other words, it is hard to achieve a dream fusion with an nonideal and unclear drive. An ideal drive must meet two requirements: (1) it must remain highly symmetric at all times and for all spectra; (2) it must have all the required characteristics with regard to time and spectrum. Thus, it is necessary to understand the real drives that are used in experiments and know how far away they are from what is to be expected to be needed for successful ignition. Two questions then arise: First, how can the desired highly symmetric radiation drive be generated at all times and for all spectra? Second, what are the three-dimensional (3D), temporal, and spectral characteristics of the radiation drives in real experiments? The solutions are strongly connected with the approach of hohlraum configuration and laser arrangement.

For many decades, the mainstream approach has been based on hohlraums with cylindrical symmetry,² such as the NIF, and the Laser Mega-Joule (LMJ)²⁵ and the SG series²⁶ laser facilities, all of which need supplementary technology to tune and create a spherical radiation drive inside hohlraums with two LEHs. However, it is hard to tune to obtain highly symmetric radiation at all times and for all spectra, let alone with the characteristics required by ignition capsules. Furthermore, the tuning itself is complicated and even unpredictable, which makes the radiation source more uncontrollable and unpredictable, further aggravating the problem.

As shown in Fig. 1, we proposed an octahedral spherical hohlraum^27–30 (hereinafter referred to simply as an octahedral hohlraum) in 2013 and have studied it for nearly a decade both theoretically and experimentally on the SG laser facilities. From our studies, we have found that the octahedral hohlraum is an attractive concept for the next generation of laser systems, with the merits that it provides an ideal answer to the two questions raised above and a practical way toward dream fusion. The octahedral hohlraum campaign started in 2014, including demonstrations of improved laser propagation inside spherical hohlraums by using cylindrical LEHs,^30–33 hohlraum energetics,^34–36 comparisons of LPI between spherical and cylindrical hohlraums and a demonstration of low LPI with an octahedral hohlraum design,^37,38 and a proof-of-concept experiment.³⁹ The campaign has successfully demonstrated the key designs and proof of concept of the octahedral hohlraum and has attracted broad interest from the fusion community.^40–55 

The laser arrangement is key to retaining the high symmetry of an octahedral hohlraum. The ideal laser arrangement design of an octahedral hohlraum²⁸ is presented in Fig. 1, which results from a comprehensive consideration of radiation symmetry, energy coupling efficiency, LPI, and hydrodynamic instabilities. Without any supplementary technology for symmetry tuning, it can naturally convert 3D lasers into quasi-1D spherical radiation. From our 3D view factor code VF3D, at a hohlraum-to-capsule radius ratio larger than 3.7, we find that the asymmetry of the octahedral hohlraum can be lower than 1%^28,56 without any symmetry tuning technology, easily meeting the ignition requirement.^2,57 This ideal laser arrangement provides the following enormous advantages.

First, the ideal laser arrangement perfectly retains the high symmetry of a 6-LEH spherical hohlraum and can naturally and robustly create highly symmetric radiation inside the hohlraum at all times and for all spectra. Relative to the six LEHs, the laser spots look like numerous stars dressing the hohlraum around the sky.^29,56 Hence, the radiation asymmetry is determined mainly by the two geometrical ratios of the target, i.e., the hohlraum-to-capsule radius ratio and the LEH-to-capsule radius ratio, while the contribution from the laser can be neglected.^28,56 As a result, the symmetry is robust and insensitive to laser power imbalance, laser pointing accuracy, and accuracy of assembly.^28,56,58 Under this ideal laser arrangement, the radiation distribution does not contain asymmetries corresponding to the spherical harmonic modes l = 2 and all odd l, and the asymmetry corresponding to l = 6 is at a very low level. Choosing the golden hohlraum-to-capsule radius ratio²⁸ of around 5, the mode l = 4 is completely suppressed, leaving l = 8 as the dominant mode.

Second, all laser quads inside the octahedral hohlraum are the same, with no laser crossing inside the hohlraum, no overlapping of laser spots on the hohlraum wall, and no interplay between laser propagation and ablated material from the capsule. This is very different from the case of the cylindrical hohlraums of the NIF.^59–61 As a result, there is no need to tune the symmetry via nonlinear cross-beam energy transfer, no need to adjust different temporal pulse shapes among beams with different injection angles, and no need to suppress the serious laser plasma instabilities of laser beams at small injection angles, resulting in more efficient energy coupling and significant suppression of potentially dangerous nonlinear processes. Furthermore, the relatively simple and clean environment makes it possible to find a physical solution that enables determination of the actual radiation drive on the capsule inside an octahedral hohlraum.

Third, the ideal laser arrangement of an octahedral hohlraum can be applied to diverse fusion schemes,^28,29,52,56 such as indirect drive,² direct drive,⁶² hybrid-drive,⁶³ and laser-driven spherically convergent plasma fusion.^64,65 Note that the octahedral laser arrangement keeps all laser quads the same, and hence it can also retain symmetry for all nonlinear phenomena associated with LPI inside a spherical hohlraum for indirect drive.²⁸ For direct drive, a high degree of uniformity in laser energy deposition on the capsule targets is also required,^{51,52,66–68} and this can be achieved by using this ideal laser arrangement via laser shifting within the adjustments permitted by laser engineering. Indeed, the octahedral hohlraum represents an enormous advance for all indirect-drive or direct-drive based approaches, which need a spherical drive for spherical implosions. Thus, the ideal laser arrangement opens the way to a drive laser system applicable to multiple schemes.

Only by determining the real drive felt by the capsule in an experiment can we know how far away it is from the design and then accurately adjust it to meet requirements. Even for postshot simulations of hydrodynamic instability,⁶⁹ which is one of the major obstacles preventing ignition at the NIF, the real radiation drive is needed as an input. However, since the beginning of indirect-drive experiments, it has been a puzzle to accurately determine the actual characteristics of the radiation on the capsule.^70,71 First, it is hard to directly measure the radiation on such a small capsule of 2–3 mm inside a hohlraum 1 cm in size. Second, it is usual to measure the radiation flux streaming out of an LEH and then take the result at a single angle as the radiation drive.^2,57,70 However, owing to the limited field of view, the presence of cold plasma outside the hohlraum, and shrinking of the LEH, it is hard to find an angle at which the observed radiation at all times and for the full spectrum is the same as that seen by a capsule located at the center of the hohlraum here, let alone that the radiation distribution on capsule is actually 3D. Third, the ViewFactor method was proposed to characterize the capsule x-ray drive,⁷² but it needs an open geometry, which has a deleterious effect on the hohlraum radiation environment. Fourth, the witness method also changes the radiation drive on the capsule by placing a witness plate inside the hohlraum.^{2,36,57,70,73} Thus, up to now, the real radiation drives of the NIF hohlraums have not been entirely clear. In fact, owing to the very complicated environment inside a cylindrical hohlraum, it is hard to know the 3D, temporal, and spectral characteristics of the real radiation drive.

However, benefiting from the simple and clean hohlraum environment inside an octahedral hohlraum, there is a physical solution to the real drive problem. Inside an octahedral hohlraum, the radiation on the capsule is contributed mainly by the characteristic regions, including the hot laser spots, the cool re-emitting wall, and the cooler closing LEHs, as shown in Fig. 2. Hence, we can determine the radiation drive in the following way. First, we measure the temporal and spectral emissions of all kinds of characteristic regions using diagnostics outside the LEHs. Second, by putting the measured data into an extended view factor model,^28,74 we can reconstruct the whole radiation field inside the hohlraum and calculate the temporal and spectral radiation drive at any point on the capsule surface directly. The laser spot movement can be taken into consideration in the extended view factor model. This method had been successfully applied in analyzing an implosion experiment in an octahedral hohlraum.³⁹ 

We proposed to use the prepulse of an ignition pulse to determine the LEH size of an ignition-scale hohlraum via LEH closure behavior and carried out experiments in 2017 and 2020 at SGIII.⁷⁶ Convincing evidence from multiple diagnostics again showed that R_L ≈ 1 mm for R_Q = 0.6 mm, ɛ ∼ 0.08 mm, and the prepulse used in the experiment. Thus, the total area of the six LEHs with R_L = 1 mm is similar to that of the two LEHs of the cylindrical hohlraums used in the NIF experiments,^77–79 which indicates that octahedral hohlraums have similar radiation energy losses to cylindrical hohlraums. It might be asked why the NIF cylindrical hohlraums need much bigger LEHs. One of the reasons is that these hohlraums need much fatter inner beams^80,81 to suppress their serious LPI. In addition, the LEH size of the NIF cylindrical hohlraums is usually taken as 50%–60% of the hohlraum radius owing to symmetry considerations.⁸² 

We can further examine the radiation asymmetry inside such an octahedral hohlraum. The radiation flux on the capsule surface can be expanded as $∑l=0∞∑m=−llalmYlm(θ,ϕ)$⁠, where Y_lm(θ, ϕ) is the spherical harmonic of the polar mode l (viewed from the equator) and the azimuthal mode m (viewed from the pole) and a_lm is the spherical harmonic decomposition. We define C_l0 = a_l0/a₀₀ and C_lm = 2a_lm/a₀₀ for m > 0, and calculate C_lm with the 3D view factor code VF3D by taking the relative fluxes of the laser spot, hohlraum wall, and LEH as 2:1:0. Inside the above octahedral hohlraum with R_H,oct/R_C = 4 and R_L,oct/R_C = 0.84, VF3D gives C_2m = 0, C_lm = 0 for all odd l, C₄₀ = 0.36%, C₄₄ = 0.43%, C₈₀ = 4.8 × 10⁻⁴, C₈₄ = 3.6 × 10⁻⁴, and C₈₈ = 5.7 × 10⁻⁴, without any symmetry tuning technology.

As mentioned above, the laser absorption efficiency is determined mainly by LPI. For the NIF cylindrical hohlraums, LPI of the inner beams is serious and considered to a major obstacle to achieving ignition,^22,59 although LPI of the outer beams is low and acceptable. On average, according to Ref. 18, η_aL ∼ 84% ± 3% over a wide range of laser parameters in the NIF cylindrical hohlraums filled with He gas with a density of 0.96 mg/cm³. In contrast to the complicated environment of a cylindrical hohlraum, an octahedral hohlraum with the ideal laser arrangement has a relatively simple and clean environment, with no beam crossing, no laser spot overlapping, and no interplay between laser beam propagation and ablated material from the capsule. In addition, the octahedral hohlraum also has the following advantages. First, the lasers enter the hohlraum at θ_L = 55°, with a shorter distance of laser propagation inside the hohlraum than lasers at a small θ_L, which is similar to the outer beams of the NIF cylindrical hohlraums. Second, the plasma filling inside a sphere is the lowest among all kinds of hohlraum configurations of the same area, because of the greatest volume of the sphere compared with all other configurations. Third, cylindrical LEHs can be used with an octahedral hohlraum to alleviate the potential influence of wall plasmas on laser transportation. All these advantages imply a lower LPI and hence a higher η_aL in octahedral hohlraums than in cylindrical hohlraums.

To experimentally compare LPI between spherical and cylindrical hohlraums, we performed experiments on the SGIII laser facility in 2015.^37,38 The experimental results with the SGIII cylindrical hohlraums are similar to those from the NIF and indicated much higher LPI of the inner beams than the outer beams inside a gas-filled hohlraum. In particular, the experiments successfully demonstrated a low level of LPI in a spherical hohlraum for beams at θ_L = 55°, even with a capsule inside and filled with C₅H₁₂ gas at 0.9 mg/cm³. In 2017, we performed further LPI experiments in a spherical hohlraum at a high laser intensity at the ignition level. From our observations, we found that stimulated Raman scattering decreases with increasing laser intensity, while stimulated Brillouin scattering increases, and the LPI fraction at 55° in a spherical hohlraum is lower than 8% at up to 1.73 × 10¹⁵ W/cm². Usually, we take η_aL = 90% for an octahedral hohlraum in estimating its required laser energy and power.

The energy efficiency from laser to hot spot is given by the product η_aLη_LXη_HCη_CH. Among all these four efficiencies, η_aL, η_LX, and η_HC are relatively stable, while η_CH seems the most “elusive” and changeable, being quite sensitive to the symmetry of the radiation drive and to target fabrication. From the above discussions, the cylindrical and octahedral hohlraums have the same or similar η_LX (∼87%) and η_HC (10%–20%), while the octahedral hohlraum has a slight advantage in terms of η_aL because of its lower LPI. We can roughly take η_aL ∼ 85% for the cylindrical hohlraum and $∼90%$ for an octahedral hohlraum in an initial design.^83,84 However, the octahedral hohlraum has an absolute advantage in terms of η_CH owing to its very high symmetry. It is η_CH that strongly influences the neutron yield in experiments, or the ratio of the measured neutron yield to the 1D-calculated neutron yield, often called the “yield over clean” (YOC). Between a good and a bad symmetry, η_CH and YOC can jump by orders of magnitude. In other words, it is the high symmetry of the octahedral hohlraum that makes η_CH much more stable, and hence increases the predictability and reproducibility of fusion gain, with the uncertainties being left to engineering.

In fact, as shown in Fig. 4, among all spherical hohlraums with different number of LEHs, the octahedral hohlraum gives the best tradeoff between radiation symmetry and energy coupling efficiency. From the point designs,^75,87,88 the required laser energy for an octahedral hohlraum is within the capabilities of currently available laser facilities.

As mentioned above, the radiation asymmetry of an octahedral hohlraum is determined mainly by R_H/R_C and R_L/R_C. In fact, from Eq. (3), η_HC of an octahedral hohlraum is also determined by these two ratios at given albedos. Together with the requirement on R_L from Eq. (1), we can define a design island of octahedral hohlraum in the plane of R_H/R_C and R_L/R_C by considering the limitations on C_lm and η_HC of an ignition target. We assume the following limitations: (1) C₄₀ ≤ 0.8% at the initial time t = 0; (2) C₄₀ ≤ 0.8% at the time at which R_C*/R_C = 0.25; (3) η_HC ≥ 10%; and (4) R_L/R_C ≥ 0.8. We then obtain the design island shown in Fig. 5. Here, we denote by R_C* the radius of an imploding capsule of initial radius R_C. We consider the limitation on asymmetry given by R_C*/R_C = 0.25, because a capsule is usually compressed 3–4 times in radius at the end of an ignition laser pulse according to our simulations, and, furthermore, the smoothing factor^2,3,56 of C_4m changes little at R_H/R_C > 10. With a stringent limitation on asymmetry at both the initial time and the end of the laser pulse, it is possible to maintain high symmetry during the whole implosion process. The limitation on R_L is related to the details of ignition target design and laser facility, including the laser beam sizes, LPIs, LEH closure, and laser beam pointing error. Here, we take R_L/R_C ≥ 0.8 just as an example. We consider the limitation to C₄₀, whose absolute value is larger than C_4±4. Here, it is worth mentioning that the mode Y_4±4 has completely different polar and azimuthal angle ranges from the mode Y₄₀, and so C₄₀ and C_4±4 should not be added together. Finally, inside the design island, there is a region (blue in Fig. 5) where C₄₀ ≤ 0.1%, and inside this region, C_4m = 0 at the golden radius ratio of R_H/R_C ∼ 5.

As an example, we consider an octahedral hohlraum for the CH Rev5 ignition capsule of the NIF and estimate its asymmetry and required laser energy E_L. From Ref. 80, the CH Rev5 capsule has R_C = 1.108 mm and E_C = 165 kJ. For an octahedral hohlraum with R_H/R_C = 4 and R_L/R_C = 1, we have C₄₀ ∼ 4 × 10⁻³ at t = 0 and C₄₀ ∼ 5 × 10⁻³ at R_C*/R_C = 0.25 from VF3D, and η_HC ∼ 18% from Eq. (3). We then have E_L = E_C/η_aLη_LXη_HC ∼ 1.2 MJ by taking η_aL = 90% and η_LX = 87%. For the case of R_H/R_C = 5 and R_L/R_C = 0.9, we have C₄₀ ∼ 9 × 10⁻⁵ at t = 0, C₄₀ ∼ 3 × 10⁻³ at R_C*/R_C = 0.25, η_HC ∼ 13.3%, and E_L ∼ 1.6 MJ. Note that η_HC varies with α_W and α_C, while α_W and α_C are functions of the target material and radiation pulse. The albedos should be adjusted to match experiments to estimate E_L more accurately.

From a physics-based viewpoint, an octahedral hohlraum can realize an ideal and clear drive without the use of symmetry tuning technology, with the remaining uncertainties of fusion gain moved to engineering. To achieve dream fusion, there are still many challenges in the path forward. The following tasks are worthy of efforts.

1. A fusion laser facility should be constructed with the ideal laser arrangement for an octahedral hohlraum.

2. A target chamber with octahedral configuration should be designed and constructed, the diagnostics for which should address the important physical quantities of all kinds of characteristic region inside the octahedral hohlraum. Here, the capsule is also regarded as a characteristic region.

3. Key diagnostics should be developed, with high spatial, temporal, and spectral resolutions.⁸⁹ These need to be such that their fields of view cover several LEHs⁹⁰ or such that they can be focused on the small size of a characteristic region.⁹¹ 

4. Novel capsule support methods that can retain the perfect symmetry of the octahedral hohlraum should be explored, for example, the use of superconducting magnetic levitation.⁹² 

5. The fabrication of beryllium or beryllium-based capsules needs to be improved to provide superior ablation properties^93–95 that can be fully exploited inside the ideal radiation environment of an octahedral hohlraum.⁸⁷ 

6. Optimum target designs with unconventional ablator capsules, such as aluminum capsules⁹⁶ or the recently proposed HDC-CH capsules,⁸⁸ with nonequilibrium between ions and electrons in the hot spot,⁹⁷ should be investigated.

7. Hohlraum walls with special structures, such as sandwich walls,^98,99 foam walls,^100–102 or honeycomb walls, should be investigated with the aim of increasing the laser absorption efficiency. Honeycomb walls differ from foam walls in their density requirement. There is no requirement on the density of a honeycomb wall, whereas foam walls must usually have a density less than 0.5 g/cm³.

8. An extended view factor model should be developed, with input consisting of the measured radiation from all characteristic regions and output consisting of the 3D spatial, temporal, and spectral characteristics of the radiation drive.

9. A 3D Monte Carlo radiation hydrodynamic code³¹ with more accurate physics models^103–110 should be developed for the octahedral hohlraum to improve target design and reveal details of novel physics.

10. Novel laser technologies with high laser absorption efficiency and low LPI, including but not limited to broadband^111,112 or sunlight-like lasers,¹¹³ should be developed for lasers at those wavelengths with a high damage threshold for optical components, such as lasers at 0.53 µm^101,102 or even longer wavelengths, to greatly increase the daily shot number operation at the full energy of a laser facility.

11. Finally, besides indirect drive, target designs for direct-drive, hybrid-drive, and laser-driven spherically convergent plasma fusion should be explored on the octahedral-configured fusion laser facility.

In summary, the octahedral hohlraum provides an ideal and practicle approach for the next generation of laser systems to achieve predictable and reproducible fusion gain via multiple schemes. This may open a new era in the development of controlled nuclear fusion and aid progress toward the realization of an airbus era of controlled nuclear fusion and laser inertial fusion energy power plants.

The author appreciate the entire octahedral spherical hohlraum team at the Institute of Applied Physics and Computational Mathematics in Beijing and the Research Center of Laser Fusion in Mianyang for their outstanding work in theory, simulation, experiment, diagnostics, and laser operations in studies of octahedral spherical hohlraums. The author also thanks Dr. Hui Cao for his help with Figs. 2 and 4. This work is supported by the National Natural Science Foundation of China (Grant No. 12035002).

The author has no conflicts to disclose.

Ke Lan: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

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