Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Growth of concomitant laser-driven collisionless and resistive electron filamentation instabilities over large spatiotemporal scales


Collective processes in plasmas often induce microinstabilities that play an important role in many space or laboratory plasma environments. Particularly notable is the Weibel-type current filamentation instability, which is believed to drive the creation of collisionless shocks in weakly magnetized astrophysical plasmas. Here, this instability class is studied through interactions of ultraintense and short laser pulses with solid foils, leading to localized generation of megaelectronvolt electrons. Proton radiographic measurements of both low- and high-resistivity targets show two distinct, superimposed electromagnetic field patterns arising from the interpenetration of the megaelectronvolt electrons and the background plasma. Particle-in-cell simulations and theoretical estimates suggest that the collisionless Weibel instability building up in the dilute expanding plasmas formed at the target surfaces causes the observed azimuthally symmetric electromagnetic filaments. For a sufficiently high resistivity of the target foil, an additional resistive instability is triggered in the bulk target, giving rise to radially elongated filaments. The data reveal the growth of both filamentation instabilities over large temporal (tens of picoseconds) and spatial (hundreds of micrometres) scales.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Sketch of the experimental set-up.
Fig. 2: Proton radiographs showing filamentation instabilities.
Fig. 3: Numerical simulation and theory of the current filamentation instabilities.
Fig. 4: PIC-MHD simulations of the resistive filamentation and synthetic radiographs.

Similar content being viewed by others

Data availability

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

Code availability

The code used to generate Fig. 3 and Fig. 4. is CALDER. All simulated proton radiographs, shown in Fig. 2 and Fig. 4, are generated using the ILZ code. Both codes are detailed in the Methods section.


  1. Shkarofsky, I., Johnston, T. & Bachynski, M. The Particle Kinetics of Plasmas (Addison-Wesley, 1966).

  2. Belmont, G., Roland, G., Mottez, F., Pantellini, F. & Pelletier, G. Collisionless Plasmas in Astrophysics (Wiley, 2013).

  3. Davidson, R. C. in Handbook of Plasma Physics Vol. 1 (eds Rosenbluth, M. N. & Galeev, R. Z.) 519–585 (North-Holland, 1983).

  4. Weibel, E. S. Spontaneous growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys. Rev. Lett. 2, 83–84 (1959).

    ADS  Google Scholar 

  5. Fried, B. D. Mechanism for instability of transverse plasma waves. Phys. Fluids 2, 337–337 (1959).

    ADS  Google Scholar 

  6. Davidson, R. C., Hammer, D. A., Haber, I. & Wagner, C. E. Nonlinear development of electromagnetic instabilities in anisotropic plasmas. Phys. Fluids 15, 317–333 (1972).

    ADS  Google Scholar 

  7. Lee, R. & Lampe, M. Electromagnetic instabilities, filamentation, and focusing of relativistic electron beams. Phys. Rev. Lett. 31, 1390–1393 (1973).

    ADS  Google Scholar 

  8. Adam, J. C., Héron, A. & Laval, G. Dispersion and transport of energetic particles due to the interaction of intense laser pulses with overdense plasmas. Phys. Rev. Lett. 97, 205006 (2006).

    ADS  Google Scholar 

  9. Marcowith, A. et al. The microphysics of collisionless shock waves. Rep. Prog. Phys. 79, 046901 (2016).

    ADS  Google Scholar 

  10. Schlickeiser, R. & Shukla, P. K. Cosmological magnetic field generation by the Weibel instability. Astrophys. J. Lett. 599, L57–L60 (2003).

    ADS  Google Scholar 

  11. Allen, B. et al. Experimental study of current filamentation instability. Phys. Rev. Lett. 109, 185007 (2012).

    ADS  Google Scholar 

  12. Fox, W. et al. Filamentation instability of counterstreaming laser-driven plasmas. Phys. Rev. Lett. 111, 225002 (2013).

    ADS  Google Scholar 

  13. Huntington, C. M. et al. Observation of magnetic field generation via the Weibel instability in interpenetrating plasma flows. Nat. Phys. 11, 173–176 (2015).

    Google Scholar 

  14. Albertazzi, B. et al. A compact broadband ion beam focusing device based on laser-driven megagauss thermoelectric magnetic fields. Rev. Sci. Instrum. 86, 043502 (2015).

    ADS  Google Scholar 

  15. Schoeffler, K. M., Loureiro, N. F., Fonseca, R. A. & Silva, L. O. Magnetic-field generation and amplification in an expanding plasma. Phys. Rev. Lett. 112, 175001 (2014).

    ADS  Google Scholar 

  16. Mondal, S. et al. Direct observation of turbulent magnetic fields in hot, dense laser produced plasmas. Proc. Natl Acad. Sci. USA 109, 8011–8015 (2012).

    ADS  Google Scholar 

  17. Romagnani, L. et al. Dynamics of the electromagnetic fields induced by fast electron propagation in near-solid-density media. Phys. Rev. Lett. 122, 025001 (2019).

    ADS  Google Scholar 

  18. Gremillet, L., Bonnaud, G. & Amiranoff, F. Filamented transport of laser-generated relativistic electrons penetrating a solid target. Phys. Plasmas 9, 941–948 (2002).

    ADS  Google Scholar 

  19. Fiore, M., Fiuza, F., Marti, M., Fonseca, R. A. & Silva, L. O. Relativistic effects on the collisionless collisional transition of the filamentation instability in fast ignition. J. Plasma Phys. 76, 813–832 (2010).

    ADS  Google Scholar 

  20. Yang, X. H. et al. Effects of filamentation instability on the divergence of relativistic electrons driven by ultraintense laser pulses. Phys. Plasmas 23, 103110 (2016).

    ADS  Google Scholar 

  21. Fuchs, J. et al. Spatial uniformity of laser-accelerated ultrahigh-current MeV electron propagation in metals and insulators. Phys. Rev. Lett. 91, 255002 (2003).

    ADS  Google Scholar 

  22. MacLellan, D. A. et al. Annular fast electron transport in silicon arising from low-temperature resistivity. Phys. Rev. Lett. 111, 095001 (2013).

    ADS  Google Scholar 

  23. Storm, M. et al. High-current, relativistic electron-beam transport in metals and the role of magnetic collimation. Phys. Rev. Lett. 102, 235004 (2009).

    ADS  Google Scholar 

  24. Wei, M. S. et al. Observations of the filamentation of high-intensity laser-produced electron beams. Phys. Rev. E 70, 056412 (2004).

    ADS  Google Scholar 

  25. Quinn, K. et al. Weibel-induced filamentation during an ultrafast laser-driven plasma expansion. Phys. Rev. Lett. 108, 135001 (2012).

    ADS  Google Scholar 

  26. Metzkes, J. et al. Experimental observation of transverse modulations in laser-driven proton beams. New J. Phys. 16, 023008 (2014).

    ADS  Google Scholar 

  27. Göde, S. et al. Relativistic electron streaming instabilities modulate proton beams accelerated in laser–plasma interactions. Phys. Rev. Lett. 118, 194801 (2017).

    ADS  Google Scholar 

  28. Scott, G. G. et al. Diagnosis of Weibel instability evolution in the rear surface density scale lengths of laser solid interactions via proton acceleration. New J. Phys. 19, 043010 (2017).

    ADS  Google Scholar 

  29. Héron, A. & Adam, J. C. Physics of the interaction of ultra intense laser pulses with cold collisional plasma using large scale kinetic simulations. Phys. Plasmas 22, 072306 (2015).

    ADS  Google Scholar 

  30. Lefebvre, E. et al. Electron and photon production from relativistic laser–plasma interactions. Nucl. Fusion 43, 629–633 (2003).

    ADS  Google Scholar 

  31. Ren, C. et al. A global simulation for laser-driven MeV electrons in 50-μm fast ignition targets. Phys. Plasmas 13, 056308 (2006).

    ADS  Google Scholar 

  32. Dieckmann, M. E., Kourakis, I., Borghesi, M. & Rowlands, G. One-dimensional particle simulation of the filamentation instability: electrostatic field driven by the magnetic pressure gradient force. Phys. Plasmas 16, 074502 (2009).

    ADS  Google Scholar 

  33. Bret, A., Gremillet, L. & Dieckmann, M. E. Multidimensional electron beam-plasma instabilities in the relativistic regime. Phys. Plasmas 17, 120501 (2010).

    ADS  Google Scholar 

  34. Mora, P. Thin-foil expansion into a vacuum. Phys. Rev. E 72, 056401 (2005).

    ADS  Google Scholar 

  35. Lee, Y. T. & More, R. M. An electron conductivity model for dense plasmas. Phys. Fluids 27, 1273–1286 (1984).

    ADS  MATH  Google Scholar 

  36. McKenna, P. et al. Effect of lattice structure on energetic electron transport in solids irradiated by ultraintense laser pulses. Phys. Rev. Lett. 106, 185004 (2011).

    ADS  Google Scholar 

  37. Doumy, G. et al. Complete characterization of a plasma mirror for the production of high-contrast ultraintense laser pulses. Phys. Rev. E 69, 026402 (2004).

    ADS  Google Scholar 

  38. Sarri, G. et al. Dynamics of self-generated, large amplitude magnetic fields following high-intensity laser matter interaction. Phys. Rev. Lett. 109, 205002 (2012).

    ADS  Google Scholar 

  39. Chen, S. N. et al. Absolute dosimetric characterization of Gafchromic EBT3 and HDv2 films using commercial flat-bed scanners and evaluation of the scanner response function variability. Rev. Sci. Instrum. 87, 073301 (2016).

    ADS  Google Scholar 

  40. Sokolov, I. V. Alternating-order interpolation in a charge-conserving scheme for particle-in-cell simulations. Comput. Phys. Commun. 184, 320–328 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  41. Lehe, R., Lifschitz, A., Thaury, C., Malka, V. & Davoine, X. Numerical growth of emittance in simulations of laser-wakefield acceleration. Phys. Rev. Spec. Top. Accel. Beams 16, 021301 (2013).

    ADS  Google Scholar 

  42. Vay, J.-L., Geddes, C. G. R., Cormier-Michel, E. & Grote, D. P. Numerical methods for instability mitigation in the modeling of laser wakefield accelerators in a Lorentz-boosted frame. J. Comput. Phys. 230, 5908–5929 (2011).

    ADS  Google Scholar 

  43. Friedman, A. A 2nd-order implicit particle mover with adjustable damping. J. Comput. Phys. 90, 292–312 (1990).

    ADS  MATH  Google Scholar 

  44. Pérez, F., Gremillet, L., Decoster, A., Drouin, M. & Lefebvre, E. Improved modeling of relativistic collisions and collisional ionization in particle-in-cell codes. Phys. Plasmas 19, 083104 (2012).

    ADS  Google Scholar 

  45. Cohen, B., Kemp, A. & Divol, L. Simulation of laser plasma interactions and fast-electron transport in inhomogeneous plasma. J. Comput. Phys. 229, 4591–4612 (2010).

    ADS  MATH  Google Scholar 

  46. Stephens, R. B. et al. Kα fluorescence measurement of relativistic electron transport in the context of fast ignition. Phys. Rev. E 69, 066414 (2004).

    ADS  Google Scholar 

  47. Ping, Y. et al. Absorption of short laser pulses on solid targets in the ultrarelativistic regime. Phys. Rev. Lett. 100, 085004 (2008).

    ADS  Google Scholar 

  48. Decoster, A., Markowich, P., Perthame, B. & Raviart, P. Modeling of Collisions (Series in Applied Mathematics, Gauthier-Villars, 1998).

  49. Highland, V. L. Some practical remarks on multiple scattering. Nucl. Instrum. Methods 129, 497–499 (1975).

    ADS  Google Scholar 

  50. Groom, D. E. & Klein, S. R. Passage of particles through matter. Eur. Phys. J. C 15, 163–173 (2000).

    ADS  Google Scholar 

Download references


We thank F. Amiranoff, F. Fiuza, E. d’Humières, V. Gubchenko and V. T. Tikhonchuk for insightful discussions. We also acknowledge the support of the JLF-Titan technical teams. The simulations were performed using HPC resources at TGCC/CCRT. We acknowledge PRACE for awarding us access to TGCC/Curie (grant 2014112576). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement 654148 Laserlab-Europe and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement 787539. The research leading to these results is supported by Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project cofinanced by the Romanian Government and European Union through the European Regional Development Fund. This work was partly done within the LABEX Plas@Par project. It was supported by grants 11-IDEX-0004-02 and ANR-17-CE30-0026-Pinnacle from Agence Nationale de la Recherche. This work was also partly supported by the DFG GRK 1203 and SFB/TR18 programmes and by EPSRC grants EP/K022415/1 and EP/J002550/1. It was supported in part by the Ministry of Education and Science of the Russian Federation under contract 14.Z50.31.0007. The use of the Jupiter Laser Facility was supported by the US Department of Energy, Lawrence Livermore National Laboratory, under contract DE-AC52-07NA27344.

Author information

Authors and Affiliations



J.F. conceived the project. B.A., S.N.C., P.A., J.B., V.D., L.L., M.N., L.R., M. Swantusch, M.B., H.P. and J.F. performed the experiment, with support from R.S., O.W. and M. Starodubtsev. B.A., S.B. and J.F. analysed the data. C. Ruyer and L.G. developed the theoretical framework and performed the simulations, with discussions with M.G. and C. Riconda. S.B. computed the synthetic proton radiographs. J.F., C. Ruyer and L.G. wrote the paper. All authors commented on the paper in its various stages.

Corresponding authors

Correspondence to C. Ruyer, L. Gremillet or J. Fuchs.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Francesco Pegoraro, Luis Silva and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–10 and discussion.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ruyer, C., Bolaños, S., Albertazzi, B. et al. Growth of concomitant laser-driven collisionless and resistive electron filamentation instabilities over large spatiotemporal scales. Nat. Phys. 16, 983–988 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing