Skip to main content Accessibility help

Proton acceleration in a laser-induced relativistic electron vortex

  • L. Q. Yi (a1), I. Pusztai (a1), A. Pukhov (a2), B. F. Shen (a3) and T. Fülöp (a1)...


We show that when a solid plasma foil with a density gradient on the front surface is irradiated by an intense laser pulse at a grazing angle, ${\sim}80^{\circ }$ , a relativistic electron vortex is excited in the near-critical-density layer after the laser pulse depletion. The vortex structure and dynamics are studied using particle-in-cell simulations. Due to the asymmetry introduced by non-uniform background density, the vortex drifts at a constant velocity, typically $0.2{-}0.3$ times the speed of light. The strong magnetic field inside the vortex leads to significant charge separation; in the corresponding electric field initially stationary protons can be captured and accelerated to twice the velocity of the vortex (100–200 MeV). A representative scenario – with laser intensity of $10^{21}~\text{W}~\text{cm}^{-2}$ – is discussed: two-dimensional simulations suggest that a quasi-monoenergetic proton beam can be obtained with a mean energy 140 MeV and an energy spread of ${\sim}10\,\%$ . We derive an analytical estimate for the vortex velocity in terms of laser and plasma parameters, demonstrating that the maximum proton energy can be controlled by the incidence angle of the laser and the plasma density gradient.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Proton acceleration in a laser-induced relativistic electron vortex
      Available formats

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Proton acceleration in a laser-induced relativistic electron vortex
      Available formats

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Proton acceleration in a laser-induced relativistic electron vortex
      Available formats


This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence:


Hide All
Albertazzi, B., d’Humières, E., Lancia, L., Dervieux, V., Antici, P., Böcker, J., Bonlie, J., Breil, J., Cauble, B., Chen, S. N. et al. 2015 A compact broadband ion beam focusing device based on laser-driven megagauss thermoelectric magnetic fields. Rev. Sci. Instrum. 86 (4), 043502.
Angus, J. R., Richardson, A. S., Ottinger, P. F., Swanekamp, S. B. & Schumer, J. W. 2014 Nonquasineutral electron vortices in nonuniform plasmas. Phys. Plasmas 21 (11), 112306.
Arber, T. D., Bennett, K., Brady, C. S., Lawrence-Douglas, A., Ramsay, M. G., Sircombe, N. J., Gillies, P., Evans, R. G., Schmitz, H., Bell, A. R. et al. 2015 Contemporary particle-in-cell approach to laser–plasma modelling. Plasma Phys. Control. Fusion 57 (11), 126.
Bin, J. H., Ma, W. J., Wang, H. Y., Streeter, M. J. V., Kreuzer, C., Kiefer, D., Yeung, M., Cousens, S., Foster, P. S., Dromey, B. et al. 2015 Ion acceleration using relativistic pulse shaping in near-critical-density plasmas. Phys. Rev. Lett. 115, 064801.
Borghesi, M. & Macchi, A. 2016 Laser-Driven Ion Accelerators: State of the Art and Applications. pp. 221247. Springer.
Borisov, A. B., Shiryaev, O. B., McPherson, A., Boyer, K. & Rhodes, C. K. 1995 Stability analysis of relativistic and charge-displacement self-channelling of intense laser pulses in underdense plasmas. Plasma Phys. Control. Fusion 37 (5), 569597.
Bulanov, S. S., Brantov, A., Bychenkov, V. Y., Chvykov, V., Kalinchenko, G., Matsuoka, T., Rousseau, P., Reed, S., Yanovsky, V., Krushelnick, K. et al. 2008 Accelerating protons to therapeutic energies with ultraintense, ultraclean, and ultrashort laser pulses. Med. Phys. 35 (5), 17701776.
Bulanov, S. S., Bychenkov, V. Y., Chvykov, V., Kalinchenko, G., Litzenberg, D. W., Matsuoka, T., Thomas, A. G. R., Willingale, L., Yanovsky, V., Krushelnick, K. et al. 2010 Generation of gev protons from 1 PW laser interaction with near critical density targets. Phys. Plasmas 17 (4), 043105.
Bulanov, S. V. & Esirkepov, T. Z. 2007 Comment on ‘collimated multi-MeV ion beams from high-intensity laser interactions with underdense plasma’. Phys. Rev. Lett. 98, 049503.
Bulanov, S. V., Lontano, M., Esirkepov, T. Z., Pegoraro, F. & Pukhov, A. M. 1996 Electron vortices produced by ultraintense laser pulses. Phys. Rev. Lett. 76, 35623565.
Daido, H., Nishiuchi, M. & Pirozhkov, A. S. 2012 Review of laser-driven ion sources and their applications. Rep. Prog. Phys. 75 (5), 056401.
Fiuza, F., Fonseca, R. A., Tonge, J., Mori, W. B. & Silva, L. O. 2012 Weibel-instability-mediated collisionless shocks in the laboratory with ultraintense lasers. Phys. Rev. Lett. 108, 235004.
Fuchs, J., Antici, P., d’Humières, E., Lefebvre, E., Borghesi, M., Brambrink, E., Cecchetti, C., Kaluza, M., Malka, V., Manclossi, M. et al. 2006 Laser-driven proton scaling laws and new paths towards energy increase. Nat. Phys. 2, 4854.
Gauthier, M., Lévy, A., d’Humières, E., Glesser, M., Albertazzi, B., Beaucourt, C., Breil, J., Chen, S. N., Dervieux, V., Feugeas, J. L. et al. 2014 Investigation of longitudinal proton acceleration in exploded targets irradiated by intense short-pulse laser. Phys. Plasmas 21 (1), 013102.
Gordeev, A. V. & Levchenko, S. V. 1998 Is the Abrikosov model applicable for describing electronic vortices in plasmas? J. Expl. Theor. Phys. Lett. 67 (7), 482488.
Haberberger, D., Tochitsky, S., Fiuza, F., Gong, C., Fonseca, R. A., Silva, L. O., Mori, W. B. & Joshi, C. 2012 Collisionless shocks in laser-produced plasma generate monoenergetic high-energy proton beams. Nat. Phys. 8 (1), 9599.
Higginson, A., Gray, R. J., King, M., Dance, R. J., Williamson, S. D. R., Butler, N. M. H., Wilson, R., Capdessus, R., Armstrong, C., Green, J. S. et al. 2018 Near-100 MeV protons via a laser-driven transparency-enhanced hybrid acceleration scheme. Nat. Commun. 9, 724.
Hilz, P., Ostermayr, T. M., Huebl, A., Bagnoud, V., Borm, B., Bussmann, M., Gallei, M., Gebhard, J., Haffa, D., Hartmann, J. et al. 2018 Isolated proton bunch acceleration by a petawatt laser pulse. Nat. Commun. 9, 423.
Khudik, V., Yi, S. A., Siemon, C. & Shvets, G. 2014 The analytic model of a laser-accelerated plasma target and its stability. Phys. Plasmas 21 (1), 013110.
Lemos, N., Martins, J. L., Dias, J. M., Marsh, K. A., Pak, A. & Joshi, C. 2012 Forward directed ion acceleration in a lwfa with ionization-induced injection. J. Plasma Phys. 78 (4), 327331.
Liu, B., Meyer-ter Vehn, J., Bamberg, K.-U., Ma, W. J., Liu, J., He, X. T., Yan, X. Q. & Ruhl, H. 2016 Ion wave breaking acceleration. Phys. Rev. Accel. Beams 19, 073401.
Liu, B., Wang, H. Y., Liu, J., Fu, L. B., Xu, Y. J., Yan, X. Q. & He, X. T. 2013 Generating overcritical dense relativistic electron beams via self-matching resonance acceleration. Phys. Rev. Lett. 110, 045002.
Ma, W. J., Kim, I. J., Yu, J. Q., Choi, I. W., Singh, P. K., Lee, H. W., Sung, J. H., Lee, S. K., Lin, C., Liao, Q. et al. 2019 Laser acceleration of highly energetic carbon ions using a double-layer target composed of slightly underdense plasma and ultrathin foil. Phys. Rev. Lett. 122, 014803.
Macchi, A.2017 A review of laser–plasma ion acceleration. arXiv:1712.06443.
Macchi, A., Borghesi, M. & Passoni, M. 2013 Ion acceleration by superintense laser–plasma interaction. Rev. Mod. Phys. 85, 751.
Macchi, A., Nindrayog, A. S. & Pegoraro, F. 2012 Solitary versus shock wave acceleration in laser–plasma interactions. Phys. Rev. E 85, 046402.
Mora, P. 2003 Plasma expansion into a vacuum. Phys. Rev. Lett. 90, 185002.
Nakamura, T., Bulanov, S. V., Esirkepov, T. Z. & Kando, M. 2010 High-energy ions from near-critical density plasmas via magnetic vortex acceleration. Phys. Rev. Lett. 105, 135002.
Nakamura, T. & Mima, K. 2008 Magnetic-dipole vortex generation by propagation of ultraintense and ultrashort laser pulses in moderate-density plasmas. Phys. Rev. Lett. 100, 205006.
Naumova, N. M., Koga, J., Nakajima, K., Tajima, T., Esirkepov, T. Z., Bulanov, S. V. & Pegoraro, F. 2001 Polarization, hosing and long time evolution of relativistic laser pulses. Phys. Plasmas 8 (9), 41494155.
Nycander, J. & Pavlenko, V. P. 1991 Stationary propagating magnetic electron vortices. Phys. Fluids B: Plasma Phys. 3 (6), 13861391.
Pak, A., Kerr, S., Lemos, N., Link, A., Patel, P., Albert, F., Divol, L., Pollock, B. B., Haberberger, D., Froula, D. et al. 2018 Collisionless shock acceleration of narrow energy spread ion beams from mixed species plasmas using 1  $\unicode[STIX]{x03BC}$ m lasers. Phys. Rev. Accel. Beams 21, 103401.
Palmer, C. A. J., Schreiber, J., Nagel, S. R., Dover, N. P., Bellei, C., Beg, F. N., Bott, S., Clarke, R. J., Dangor, A. E., Hassan, S. M. et al. 2012 Rayleigh–Taylor instability of an ultrathin foil accelerated by the radiation pressure of an intense laser. Phys. Rev. Lett. 108, 225002.
Pegoraro, F. & Bulanov, S. V. 2007 Photon bubbles and ion acceleration in a plasma dominated by the radiation pressure of an electromagnetic pulse. Phys. Rev. Lett. 99, 065002.
Priest, E. R., Hornig, G. & Pontin, D. I. 2003 On the nature of three-dimensional magnetic reconnection. J. Geophys. Res. 108 (A7), doi:10.1029/2002JA009812.
Pukhov, A. 2001 Three-dimensional simulations of ion acceleration from a foil irradiated by a short-pulse laser. Phys. Rev. Lett. 86, 35623565.
Pukhov, A., Sheng, Z.-M. & Meyer-ter Vehn, J. 1999 Particle acceleration in relativistic laser channels. Phys. Plasmas 6 (7), 28472854.
Pukhov, A. & Meyer-ter Vehn, J. 1996 Relativistic magnetic self-channeling of light in near-critical plasma: three-dimensional particle-in-cell simulation. Phys. Rev. Lett. 76, 39753978.
Richardson, A. S., Angus, J. R., Swanekamp, S. B., Ottinger, P. F. & Schumer, J. W. 2013 Theory and simulations of electron vortices generated by magnetic pushing. Phys. Plasmas 20 (8), 082115.
Roth, M., Blazevic, A., Geissel, M., Schlegel, T., Cowan, T. E., Allen, M., Gauthier, J.-C., Audebert, P., Fuchs, J., Meyer-ter Vehn, J. et al. 2002 Energetic ions generated by laser pulses: a detailed study on target properties. Phys. Rev. ST Accel. Beams 5, 061301.
Rust, D. M. & Kumar, A. 1996 Evidence for helically kinked magnetic flux ropes in solar eruptions. Astrophys. J. 464 (2), L199L202.
Schreiber, J., Bolton, P. R. & Parodi, K. 2016 Invited review article: ‘Hands-on’ laser-driven ion acceleration: a primer for laser-driven source development and potential applications. Rev. Sci. Instrum. 87 (7), 071101.
Silva, L. O., Marti, M., Davies, J. R., Fonseca, R. A., Ren, C., Tsung, F. S. & Mori, W. B. 2004 Proton shock acceleration in laser–plasma interactions. Phys. Rev. Lett. 92, 015002.
Sylla, F., Flacco, A., Kahaly, S., Veltcheva, M., Lifschitz, A., Malka, V., d’Humières, E., Andriyash, I. & Tikhonchuk, V. 2013 Short intense laser pulse collapse in near-critical plasma. Phys. Rev. Lett. 110, 085001.
Thaury, C., Quéré, F., Geindre, J.-P., Levy, A., Ceccotti, T., Monot, P., Bougeard, M., Réau, F., d’Oliveira, P., Audebert, P. et al. 2007 Plasma mirrors for ultrahigh-intensity optics. Nat. Phys. 3, 424429.
Toncian, T., Borghesi, M., Fuchs, J., d’Humières, E., Antici, P., Audebert, P., Brambrink, E., Cecchetti, C. A., Pipahl, A., Romagnani, L. et al. 2006 Ultrafast laser-driven microlens to focus and energy-select mega-electron volt protons. Science 312 (5772), 410413.
Wiegelmann, T. & Büchner, J. 2001 Evolution of magnetic helicity in the course of kinetic magnetic reconnection. Nonlinear Process. Geophys. 8 (3), 127140.
Wilks, S. C., Langdon, A. B., Cowan, T. E., Roth, M., Singh, M., Hatchett, S., Key, M. H., Pennington, D., MacKinnon, A. & Snavely, R. A. 2001 Energetic proton generation in ultra-intense laser–solid interactions. Phys. Plasmas 8 (2), 542.
Yadav, S. K., Das, A. & Kaw, P. 2008 Propagation of electron magnetohydrodynamic structures in a two-dimensional inhomogeneous plasma. Phys. Plasmas 15 (6), 062308.
Yamada, M., Kulsrud, R. & Ji, H. 2010 Magnetic reconnection. Rev. Mod. Phys. 82, 603664.
Yi, L. Q., Shen, B. F., Pukhov, A. & Fülöp, T. 2018 Relativistic magnetic reconnection driven by a laser interacting with a micro-scale plasma slab. Nat. Commun. 9, 1601.
Zhai, S. H., Shen, B. F., Borghesi, M., Wang, W. P., Zhang, H., Kar, S., Ahmed, H., Li, J. F., Li, S. S., Zhang, H. et al. 2019 Proton array focused by a laser-irradiated mesh. Appl. Phys. Lett. 114 (1), 013509.
MathJax is a JavaScript display engine for mathematics. For more information see


Type Description Title
Supplementary materials

Yi et al. supplementary material
Yi et al. supplementary material 1

 Unknown (9.3 MB)
9.3 MB


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed