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Bayesian optimization of electron energy from laser wakefield accelerators

P. Valenta1,*, T. Zh. Esirkepov2, J. D. Ludwig3, S. C. Wilks3, and S. V. Bulanov1

  • *Contact author: petr.valenta@eli-beams.eu
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Phys. Rev. Accel. Beams 28, 094601 – Published 12 September, 2025

DOI: https://doi.org/10.1103/knh7-hbr3

Abstract

We use Bayesian optimization in combination with three-dimensional particle-in-cell simulations to determine the optimal laser and plasma parameters that, for a given laser pulse energy, maximize the cut-off energy of an electron beam produced by laser wakefield accelerators. We assume a Gaussian laser driver with matched spot size and amplitude and investigate both self-guiding in a uniform-density plasma and guiding in a preformed plasma channel with matched radius. To interpret the simulation results quantitatively, we derive novel analytical expressions for the maximum electron energy and the corresponding acceleration distance, accounting for the effects of laser diffraction and energy depletion. Based on the results obtained, we discuss the potential scalability of the optimal input (plasma density, pulse duration, amplitude, spot size, and channel radius) and output (electron energy, electron charge, acceleration length, and acceleration efficiency) parameters to laser systems of arbitrary energy.

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References (75)

  1. T. Tajima and J. M. Dawson, Laser electron accelerator, Phys. Rev. Lett. 43, 267 (1979).
  2. E. Esarey, C. B. Schroeder, and W. P. Leemans, Physics of laser-driven plasma-based electron accelerators, Rev. Mod. Phys. 81, 1229 (2009).
  3. A. J. Gonsalves et al., Petawatt laser guiding and electron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide, Phys. Rev. Lett. 122, 084801 (2019).
  4. F. Grüner, S. Becker, U. Schramm, T. Eichner, M. Fuchs, R. Weingartner, D. Habs, J. Meyer-ter-Vehn, M. Geissler, M. Ferrario, L. Serafini, B. van der Geer, H. Backe, W. Lauth, and S. Reiche, Design considerations for table-top, laser-based VUV and X-ray free electron lasers, Appl. Phys. B 86, 431 (2007).
  5. S. Corde, K. Ta Phuoc, G. Lambert, R. Fitour, V. Malka, A. Rousse, A. Beck, and E. Lefebvre, Femtosecond x rays from laser-plasma accelerators, Rev. Mod. Phys. 85, 1 (2013).
  6. S. V. Bulanov, T. Z. Esirkepov, M. Kando, A. S. Pirozhkov, and N. N. Rosanov, Relativistic mirrors in plasmas. Novel results and perspectives, Phys. Usp. 56, 429 (2013).
  7. F. Albert, A. G. R. Thomas, S. P. D. Mangles, S. Banerjee, S. Corde, A. Flacco, M. Litos, D. Neely, J. Vieira, Z. Najmudin, R. Bingham, C. Joshi, and T. Katsouleas, Laser wakefield accelerator based light sources: Potential applications and requirements, Plasma Phys. Controlled Fusion 56, 084015 (2014).
  8. T. Kurz et al., Demonstration of a compact plasma accelerator powered by laser-accelerated electron beams, Nat. Commun. 12, 2895 (2021).
  9. A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, Charged particle motion and radiation in strong electromagnetic fields, Rev. Mod. Phys. 94, 045001 (2022).
  10. T.-P. Yu, K. Liu, J. Zhao, X.-L. Zhu, Y. Lu, Y. Cao, H. Zhang, F.-Q. Shao, and Z.-M. Sheng, Bright x/gamma-ray emission and lepton pair production by strong laser fields: A review, Rev. Mod. Plasma Phys. 8, 24 (2024).
  11. D. Terzani et al., Measurement of directional muon beams generated at the Berkeley Lab Laser Accelerator, arXiv:2411.02321.
  12. L. Calvin, E. Gerstmayr, C. Arran, L. Tudor, T. Foster, B. Bergmann, D. Doria, B. Kettle, H. Maguire, V. Malka, P. Manek, S. P. D. Mangles, P. McKenna, R. E. Mihai, C. Ridgers, J. Sarma, P. Smolyanskiy, R. Wilson, R. M. Deas, and G. Sarri, Experimental evidence of muon production from a laser-wakefield accelerator, arXiv:2503.20904.
  13. F. Zhang et al., Proof-of-principle demonstration of muon production with an ultrashort high-intensity laser, Nat. Phys. 21, 1050 (2025).
  14. J. D. Ludwig, S. C. Wilks, A. J. Kemp, G. J. Williams, N. Lemos, E. Rockafellow, B. Miao, J. E. Shrock, H. M. Milchberg, J.-L. Vay, A. Huebl, R. Lehe, A. Cimmino, R. Versaci, S. V. Bulanov, P. Valenta, V. Tang, and B. A. Reagan, Laser based 100 GeV electron acceleration scheme for muon production, Sci. Rep. 15, 25902 (2025).
  15. V. G. Nedorezov, S. G. Rykovanov, and A. B. Savel’ev, Nuclear photonics: Results and prospects, Phys. Usp. 64, 1214 (2021).
  16. D. Kolenatý, P. Hadjisolomou, R. Versaci, T. M. Jeong, P. Valenta, V. Olšovcová, and S. V. Bulanov, Electron-positron pairs and radioactive nuclei production by irradiation of high-Z target with γ-photon flash generated by an ultra-intense laser in the λ3 regime, Phys. Rev. Res. 4, 023124 (2022).
  17. W. Lu, C. Huang, M. Zhou, W. B. Mori, and T. Katsouleas, Nonlinear theory for relativistic plasma wakefields in the blowout regime, Phys. Rev. Lett. 96, 165002 (2006).
  18. M. H. Cho, V. B. Pathak, H. T. Kim, and C. H. Nam, Controlled electron injection facilitated by nanoparticles for laser wakefield acceleration, Sci. Rep. 8, 16924 (2018).
  19. B. Beaurepaire, A. Vernier, M. Bocoum, F. Böhle, A. Jullien, J.-P. Rousseau, T. Lefrou, D. Douillet, G. Iaquaniello, R. Lopez-Martens, A. Lifschitz, and J. Faure, Effect of the laser wave front in a laser-plasma accelerator, Phys. Rev. X 5, 031012 (2015).
  20. D. Oumbarek Espinos, A. Rondepierre, A. Zhidkov, N. Pathak, Z. Jin, K. Huang, N. Nakanii, I. Daito, M. Kando, and T. Hosokai, Notable improvements on LWFA through precise laser wavefront tuning, Sci. Rep. 13, 18466 (2023).
  21. S. Y. Kalmykov, A. Beck, X. Davoine, E. Lefebvre, and B. A. Shadwick, Laser plasma acceleration with a negatively chirped pulse: All-optical control over dark current in the blowout regime, New. J. Phys. 14, 033025 (2012).
  22. H. T. Kim, V. B. Pathak, K. Hong Pae, A. Lifschitz, F. Sylla, J. H. Shin, C. Hojbota, S. K. Lee, J. H. Sung, H. W. Lee, E. Guillaume, C. Thaury, K. Nakajima, J. Vieira, L. O. Silva, V. Malka, and C. H. Nam, Stable multi-GeV electron accelerator driven by waveform-controlled PW laser pulses, Sci. Rep. 7, 10203 (2017).
  23. P. Sprangle, B. Hafizi, J. R. Peñano, R. F. Hubbard, A. Ting, C. I. Moore, D. F. Gordon, A. Zigler, D. Kaganovich, and T. M. Antonsen, Wakefield generation and GeV acceleration in tapered plasma channels, Phys. Rev. E 63, 056405 (2001).
  24. E. Guillaume, A. Döpp, C. Thaury, K. Ta Phuoc, A. Lifschitz, G. Grittani, J.-P. Goddet, A. Tafzi, S. W. Chou, L. Veisz, and V. Malka, Electron rephasing in a laser-wakefield accelerator, Phys. Rev. Lett. 115, 155002 (2015).
  25. S. E. Perevalov, K. F. Burdonov, A. V. Kotov, D. S. Romanovskiy, A. A. Soloviev, M. V. Starodubtsev, A. A. Golovanov, V. N. Ginzburg, A. A. Kochetkov, A. P. Korobeinikova, A. A. Kuz’min, I. A. Shaikin, A. A. Shaykin, I. V. Yakovlev, E. A. Khazanov, and I. Y. Kostyukov, Experimental study of strongly mismatched regime of laser-driven wakefield acceleration, Plasma Phys. Controlled Fusion 62, 094004 (2020).
  26. K. Põder, J. C. Wood, N. C. Lopes, J. M. Cole, S. Alatabi, M. P. Backhouse, P. S. Foster, A. J. Hughes, C. Kamperidis, O. Kononenko, S. P. D. Mangles, C. A. J. Palmer, D. Rusby, A. Sahai, G. Sarri, D. R. Symes, J. R. Warwick, and Z. Najmudin, Multi-GeV electron acceleration in wakefields strongly driven by oversized laser spots, Phys. Rev. Lett. 132, 195001 (2024).
  27. R. J. Shalloo et al., Automation and control of laser wakefield accelerators using Bayesian optimization, Nat. Commun. 11, 6355 (2020).
  28. S. Jalas, M. Kirchen, P. Messner, P. Winkler, L. Hübner, J. Dirkwinkel, M. Schnepp, R. Lehe, and A. R. Maier, Bayesian optimization of a laser-plasma accelerator, Phys. Rev. Lett. 126, 104801 (2021).
  29. M. Kirchen, S. Jalas, P. Messner, P. Winkler, T. Eichner, L. Hübner, T. Hülsenbusch, L. Jeppe, T. Parikh, M. Schnepp, and A. R. Maier, Optimal beam loading in a laser-plasma accelerator, Phys. Rev. Lett. 126, 174801 (2021).
  30. S. Jalas, M. Kirchen, C. Braun, T. Eichner, J. B. Gonzalez, L. Hübner, T. Hülsenbusch, P. Messner, G. Palmer, M. Schnepp, C. Werle, P. Winkler, W. P. Leemans, and A. R. Maier, Tuning curves for a laser-plasma accelerator, Phys. Rev. Accel. Beams 26, 071302 (2023).
  31. A. Ferran Pousa, S. Jalas, M. Kirchen, A. Martinez de la Ossa, M. Thévenet, S. Hudson, J. Larson, A. Huebl, J.-L. Vay, and R. Lehe, Bayesian optimization of laser-plasma accelerators assisted by reduced physical models, Phys. Rev. Accel. Beams 26, 084601 (2023).
  32. F. Irshad, S. Karsch, and A. Döpp, Multi-objective and multi-fidelity Bayesian optimization of laser-plasma acceleration, Phys. Rev. Res. 5, 013063 (2023).
  33. F. Irshad, C. Eberle, F. M. Foerster, K. v. Grafenstein, F. Haberstroh, E. Travac, N. Weisse, S. Karsch, and A. Döpp, Pareto optimization and tuning of a laser wakefield accelerator, Phys. Rev. Lett. 133, 085001 (2024).
  34. R. Roussel et al., Bayesian optimization algorithms for accelerator physics, Phys. Rev. Accel. Beams 27, 084801 (2024).
  35. R. Roussel, J. P. Gonzalez-Aguilera, E. Wisniewski, A. Ody, W. Liu, J. Power, Y.-K. Kim, and A. Edelen, Efficient six-dimensional phase space reconstructions from experimental measurements using generative machine learning, Phys. Rev. Accel. Beams 27, 094601 (2024).
  36. G.-Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, Self-focusing of short intense pulses in plasmas, Phys. Fluids 30, 526 (1987).
  37. N. M. Naumova, S. V. Bulanov, K. Nishihara, T. Zh. Esirkepov, and F. Pegoraro, Polarization effects and anisotropy in three-dimensional relativistic self-focusing, Phys. Rev. E 65, 045402 (2002).
  38. P. Valenta, G. M. Grittani, C. M. Lazzarini, O. Klimo, and S. V. Bulanov, On the electromagnetic-electron rings originating from the interaction of high-power short-pulse laser and underdense plasma, Phys. Plasmas 28, 122104 (2021).
  39. S. V. Bulanov, M. Lontano, T. Zh. Esirkepov, F. Pegoraro, and A. M. Pukhov, Electron vortices produced by ultraintense laser pulses, Phys. Rev. Lett. 76, 3562 (1996).
  40. S. V. Bulanov, T. Zh. Esirkepov, N. M. Naumova, F. Pegoraro, and V. A. Vshivkov, Solitonlike electromagnetic waves behind a superintense laser pulse in a plasma, Phys. Rev. Lett. 82, 3440 (1999).
  41. T. Zh. Esirkepov, K. Nishihara, S. V. Bulanov, and F. Pegoraro, Three-dimensional relativistic electromagnetic subcycle solitons, Phys. Rev. Lett. 89, 275002 (2002).
  42. W. Leemans, C. Siders, E. Esarey, N. Andreev, G. Shvets, and W. Mori, Plasma guiding and wakefield generation for second-generation experiments, IEEE Trans. Plasma Sci. 24, 331 (1996).
  43. N. A. Bobrova, A. A. Esaulov, J.-I. Sakai, P. V. Sasorov, D. J. Spence, A. Butler, S. M. Hooker, and S. V. Bulanov, Simulations of a hydrogen-filled capillary discharge waveguide, Phys. Rev. E 65, 016407 (2001).
  44. P. Sprangle, C.-M. Tang, and E. Esarey, Relativistic self-focusing of short-pulse radiation beams in plasmas, IEEE Trans. Plasma Sci. 15, 145 (1987).
  45. B. Hafizi, A. Ting, P. Sprangle, and R. F. Hubbard, Relativistic focusing and ponderomotive channeling of intense laser beams, Phys. Rev. E 62, 4120 (2000).
  46. D. Terzani, C. Benedetti, S. S. Bulanov, C. B. Schroeder, and E. Esarey, Compact, all-optical positron production and collection scheme, Phys. Rev. Accel. Beams 26, 113401 (2023).
  47. W. Lu, M. Tzoufras, C. Joshi, F. S. Tsung, W. B. Mori, J. Vieira, R. A. Fonseca, and L. O. Silva, Generating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regime, Phys. Rev. ST Accel. Beams 10, 061301 (2007).
  48. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, Cambridge, MA, 2005).
  49. P. Auer, N. Cesa-Bianchi, and P. Fischer, Finite-time analysis of the multiarmed bandit problem, Mach. Learn. 47, 235 (2002).
  50. N. Srinivas, A. Krause, S. M. Kakade, and M. W. Seeger, Information-theoretic regret bounds for Gaussian process optimization in the bandit setting, IEEE Trans. Inf. Theory 58, 3250 (2012).
  51. H. Ishibashi, M. Karasuyama, I. Takeuchi, and H. Hino, A stopping criterion for Bayesian optimization by the gap of expected minimum simple regrets, in Proceedings of the 26th International Conference on Artificial Intelligence and Statistics (PMLR, Brookline, MA, 2023), pp. 6463–6497.
  52. J. T. Wilson, Stopping Bayesian optimization with probabilistic regret bounds, in Advances in Neural Information Processing Systems, edited by A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang (Curran Associates, Inc., Red Hook, NY, 2024), Vol. 37, pp. 98264–98296.
  53. T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, Contemporary particle-in-cell approach to laser-plasma modelling, Plasma Phys. Controlled Fusion 57, 113001 (2015).
  54. S. Wilks, T. Katsouleas, J. M. Dawson, P. Chen, and J. J. Su, Beam loading in plasma waves, IEEE Trans. Plasma Sci. 15, 210 (1987).
  55. S. Bulanov, N. Naumova, F. Pegoraro, and J. Sakai, Particle injection into the wave acceleration phase due to nonlinear wake wave breaking, Phys. Rev. E 58, R5257 (1998).
  56. J. P. Boris, Relativistic plasma simulation-optimization of a hybrid code, in Proceedings of the Fourth Conference on Numerical Simulation of Plasmas, edited by J. P. Boris and R. A. Shanny (Naval Research Laboratory, Washington, DC, 1971), pp. 3–67.
  57. K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14, 302 (1966).
  58. J. Faure, Y. Glinec, J. J. Santos, F. Ewald, J.-P. Rousseau, S. Kiselev, A. Pukhov, T. Hosokai, and V. Malka, Observation of laser-pulse shortening in nonlinear plasma waves, Phys. Rev. Lett. 95, 205003 (2005).
  59. J. Schreiber, C. Bellei, S. P. D. Mangles, C. Kamperidis, S. Kneip, S. R. Nagel, C. A. J. Palmer, P. P. Rajeev, M. J. V. Streeter, and Z. Najmudin, Complete temporal characterization of asymmetric pulse compression in a laser wakefield, Phys. Rev. Lett. 105, 235003 (2010).
  60. J. L. Shaw, N. Lemos, L. D. Amorim, N. Vafaei-Najafabadi, K. A. Marsh, F. S. Tsung, W. B. Mori, and C. Joshi, Role of direct laser acceleration of electrons in a laser wakefield accelerator with ionization injection, Phys. Rev. Lett. 118, 064801 (2017).
  61. A. Pukhov, Z.-M. Sheng, and J. Meyer-ter-Vehn, Particle acceleration in relativistic laser channels, Phys. Plasmas 6, 2847 (1999).
  62. R. Babjak, L. Willingale, A. Arefiev, and M. Vranic, Direct laser acceleration in underdense plasmas with multi-PW lasers: A path to high-charge, GeV-class electron bunches, Phys. Rev. Lett. 132, 125001 (2024).
  63. P. Valenta, D. Maslarova, R. Babjak, B. Martinez, S. V. Bulanov, and M. Vranić, Direct laser acceleration: A model for the electron injection from the walls of a cylindrical guiding structure, Phys. Rev. E 109, 065204 (2024).
  64. J. Faure, D. Gustas, D. Guénot, A. Vernier, F. Böhle, M. Ouillé, S. Haessler, R. Lopez-Martens, and A. Lifschitz, A review of recent progress on laser-plasma acceleration at kHz repetition rate, Plasma Phys. Controlled Fusion 61, 014012 (2019).
  65. C. M. Lazzarini, G. M. Grittani, P. Valenta, I. Zymak, R. Antipenkov, U. Chaulagain, L. V. N. Goncalves, A. Grenfell, M. Lamač, S. Lorenz, M. Nevrkla, A. Špaček, V. Šobr, W. Szuba, P. Bakule, G. Korn, and S. V. Bulanov, Ultrarelativistic electron beams accelerated by terawatt scalable kHz laser, Phys. Plasmas 31, 030703 (2024).
  66. P. Valenta, T. Z. Esirkepov, J. K. Koga, A. Nečas, G. M. Grittani, C. M. Lazzarini, O. Klimo, G. Korn, and S. V. Bulanov, Polarity reversal of wakefields driven by ultrashort pulse laser, Phys. Rev. E 102, 53216 (2020).
  67. J. Huijts, L. Rovige, I. A. Andriyash, A. Vernier, M. Ouillé, J. Kaur, Z. Cheng, R. Lopez-Martens, and J. Faure, Waveform control of relativistic electron dynamics in laser-plasma acceleration, Phys. Rev. X 12, 011036 (2022).
  68. D. Farina and S. V. Bulanov, Relativistic electromagnetic solitons in the electron-ion plasma, Phys. Rev. Lett. 86, 5289 (2001).
  69. S.-Y. Zhou, W. Yu, X. Yuan, H. Xu, L. H. Cao, H. B. Cai, and C. T. Zhou, Study on the effects of ion motion on laser-induced plasma wakes, Phys. Plasmas 19, 093101 (2012).
  70. A. F. Lifschitz, X. Davoine, E. Lefebvre, J. Faure, C. Rechatin, and V. Malka, Particle-in-cell modelling of laser–plasma interaction using Fourier decomposition, J. Comput. Phys. 228, 1803 (2009).
  71. J.-L. Vay, Noninvariance of space- and time-scale ranges under a Lorentz transformation and the implications for the study of relativistic interactions, Phys. Rev. Lett. 98, 130405 (2007).
  72. L. Labun, M. Gracia-Linares, O. Z. Labun, and S. V. Milton, Performance envelope of laser wakefield accelerators, Nucl. Instrum. Methods Phys. Res., Sect. A 1076, 170468 (2025).
  73. S. V. Bulanov, T. Zh. Esirkepov, Y. Hayashi, H. Kiriyama, J. K. Koga, H. Kotaki, M. Mori, and M. Kando, On some theoretical problems of laser wake-field accelerators, J. Plasma Phys. 82, 905820308 (2016).
  74. F. Nogueira, Bayesian optimization: Open source constrained global optimization tool for python, https://github.com/bayesian-optimization/BayesianOptimization (2014).
  75. P. Valenta, Bayesian optimization of electron energy from laser wakefield accelerators [Data set], Zenodo, 10.5281/zenodo.16941558 (2025).

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Phys. Rev. Accel. Beams 28, 094601– Published 12 September, 2025
  • Received 10 January 2025
  • Accepted 21 August 2025
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