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# Tripled yield in direct-drive laser fusion through statistical modelling

## Abstract

Focusing laser light onto a very small target can produce the conditions for laboratory-scale nuclear fusion of hydrogen isotopes. The lack of accurate predictive models, which are essential for the design of high-performance laser-fusion experiments, is a major obstacle to achieving thermonuclear ignition. Here we report a statistical approach that was used to design and quantitatively predict the results of implosions of solid deuterium–tritium targets carried out with the 30-kilojoule OMEGA laser system, leading to tripling of the fusion yield to its highest value so far for direct-drive laser fusion. When scaled to the laser energies of the National Ignition Facility (1.9 megajoules), these targets are predicted to produce a fusion energy output of about 500 kilojoules—several times larger than the fusion yields currently achieved at that facility. This approach could guide the exploration of the vast parameter space of thermonuclear ignition conditions and enhance our understanding of laser-fusion physics.

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## Data availability

Raw data were generated at the LLE’s OMEGA Laser Facility. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This material is based on work supported by the Department of Energy National Nuclear Security Administration under award numbers DE-NA0003856 and DENA0001944, the Department of Energy Office of Fusion Energy Sciences under award number DE-FC02-04ER54789, the University of Rochester and the New York State Energy Research and Development Authority. This report was prepared as an account of work sponsored by an agency of the US Government. Neither the US 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 favouring by the US Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the US Government or any agency thereof.

### Reviewer information

Nature thanks I. Kaganovich, R. Scott and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Affiliations

1. #### Laboratory for Laser Energetics, University of Rochester, Rochester, NY, USA

• V. Gopalaswamy
• , R. Betti
• , J. P. Knauer
• , N. Luciani
• , D. Patel
• , K. M. Woo
• , A. Bose
• , I. V. Igumenshchev
• , E. M. Campbell
• , K. S. Anderson
• , K. A. Bauer
• , M. J. Bonino
• , D. Cao
• , A. R. Christopherson
• , G. W. Collins
• , T. J. B. Collins
• , J. R. Davies
• , J. A. Delettrez
• , D. H. Edgell
• , R. Epstein
• , C. J. Forrest
• , D. H. Froula
• , V. Y. Glebov
• , V. N. Goncharov
• , D. R. Harding
• , S. X. Hu
• , D. W. Jacobs-Perkins
• , R. T. Janezic
• , J. H. Kelly
• , O. M. Mannion
• , A. Maximov
• , F. J. Marshall
• , D. T. Michel
• , S. Miller
• , S. F. B. Morse
• , J. Palastro
• , J. Peebles
• , S. P. Regan
• , S. Sampat
• , T. C. Sangster
• , A. B. Sefkow
• , W. Seka
• , R. C. Shah
• , A. Shvydky
• , C. Stoeckl
• , A. A. Solodov
• , W. Theobald
•  & J. D. Zuegel
2. #### Department of Mechanical Engineering, University of Rochester, Rochester, NY, USA

• V. Gopalaswamy
• , R. Betti
• , N. Luciani
• , D. Patel
• , A. R. Christopherson
• , A. Maximov
•  & S. Miller
3. #### Department of Physics and Astronomy, University of Rochester, Rochester, NY, USA

• R. Betti
• , K. M. Woo
•  & O. M. Mannion

• N. Luciani
5. #### Massachusetts Institute of Technology, Cambridge, MA, USA

• A. Bose
• , M. Gatu Johnson
• , R. D. Petrasso
• , C. K. Li
•  & J. A. Frenje

### Contributions

All authors contributed to the work presented in this paper. A.A.S., A.M. and J. Palastro were responsible for theoretical and experimental work on hot-electron pre-heating, and J. Peebles investigated high--mode instabilities due to laser imprinting, both of which were used in constraining the optimization process of this work.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to V. Gopalaswamy.

## Extended data figures and tables

1. ### Extended Data Fig. 1 Posterior predictive distributions for the parameters in equation (6).

The intercept parameter refers to the constant Cj in equation (5), ε refers to the noise parameter in equation (8) and the remaining parameters are the μi values for the corresponding simulated quantities, as in equation (5). The Markov chains for each parameter are converged over 5,000 steps. The tails of the distributions have the same sign as the mean and are not excessively wide when compared to the mean. All these indicate that the model is well specified.

2. ### Extended Data Fig. 2 Posterior predictive distributions for the parameters in equation (7).

The intercept parameter refers to the constant Cj in equation (5), ε refers to the noise parameter in equation (8) and the remaining parameters are the μi values for the corresponding simulated quantities, as in equation (5). As in Extended Data Fig. 1, these results indicate that the model is well specified; we note that this does not guarantee that the model will have any predictive power.

3. ### Extended Data Fig. 3 The effect of RT on the yield is less than 10% for the vast majority of implosions.

Vertical error bars are one standard deviation of $${R}_{T}^{-0.6}$$. Because the yield is proportional to $${R}_{T}^{-0.6}$$ in equation (7), and the vast majority of implosions carried out at OMEGA are repeatable, the empirical correction factor in equation (7) varies by less than 10% for most OMEGA implosions. This justifies the use of this parameter to account for the occasional shot with large random nonuniformities.

4. ### Extended Data Fig. 4 Areal densities for high-convergence implosions are well predicted.

The dashed line is the y = x line on which the data points would lie if the experimental areal density were perfectly modelled by the power-law dependence. As the convergence increases, the safety margin for a nominal implosion becomes thinner and makes designing these implosions more challenging. A predictive model for the areal density was built using the framework presented in this paper (training data are shown by blue circles). The areal density was increased from that of shot 87266 (green circle) over four shot days (orange circles) until the highest values were reached in July 2018. The yields for these implosions are marked.

5. ### Extended Data Fig. 5 Comparison of prediction accuracies of the statistical model and 1D simulations.

a, b, Predictions for the areal density (a) and neutron yield (b) including the second phase of the Optimization Campaign. The predictions from the statistical model (blue) remain accurate for implosions from both phases of the Optimization Campaign, whereas the corresponding simulated 1D quantities from LILAC (red) remain inaccurate and overpredict potential increases in performance. Horizontal error bars and centre values represent one standard deviation and mean, respectively, for 500 draws from the posterior distribution of the statistical model. Vertical error bars represent one standard deviation for the neutron yield and areal density detectors. The dashed line is the y = x line on which the data points would lie if the prediction perfectly matched the measurement.

6. ### Extended Data Fig. 6 Power-law dependence of the simulated yield on simulated parameters for initial conditions used in the Optimization Campaign.

The dashed line is the y = x line on which the data points would lie if the simulated yield were perfectly modelled by the power-law dependence. The simulated yield (Ysim) is well represented by power-law relations to the simulated implosion velocity ($${V}_{{\rm{i}}{\rm{m}}{\rm{p}}}^{{\rm{s}}{\rm{i}}{\rm{m}}}$$), mass ($${M}_{{\rm{s}}{\rm{t}}{\rm{a}}{\rm{g}}}^{{\rm{s}}{\rm{i}}{\rm{m}}}$$) and areal density (ρRsim). Although the exponents are not identical to those from analytical theory, physical intuition has provided a good basis for variable selection.

### DOI

https://doi.org/10.1038/s41586-019-0877-0