Skip to main content Accessibility help

The filamented electron bunch of the bubble regime

  • Lars Reichwein (a1), Johannes Thomas (a1) and Alexander Pukhov (a1)


We present a theory for describing the inner structure of the electron bunch in the bubble regime starting from a random distribution of electrons inside the bubble and subsequently minimizing the system's energy. Consequently, we find a filament-like structure in the direction of propagation that is surrounded by various shells consisting of further electrons. If we specify a two-dimensional (2D) initial structure, we observe a hexagonal structure for a high number of particles, corresponding to the close packing of spheres in two dimensions. The 2D structures are in agreement with the equilibrium slice model.

  • 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.

      The filamented electron bunch of the bubble regime
      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.

      The filamented electron bunch of the bubble regime
      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.

      The filamented electron bunch of the bubble regime
      Available formats


This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.

Corresponding author

Author for correspondence: L. Reichwein, Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany. E-mail:


Hide All
Apostol, M and Ganciu, M (2011) Polaritonic pulse and coherent x- and gamma rays from Compton (Thomson) backscattering. Journal of Applied Physics 109, 013307.
Barzilai, J and Borwein, JM (1988) Two-point step size gradient methods. IMA Journal of Numerical Analysis 8, 141148.
Baxevanis, P., Hogan, M. J., Huang, Z., Litos, M., O'Shea, B., Raubenheimer, T. O., Frisch, J. C., White, G., Xu, X. L. and Mori, W. (2017) Operation and applications of a plasma wakefield accelerator based on the density down-ramp injection technique. AIP Conference Proceedings 1812, 100013. doi:
Chen, M, Esarey, E, Geddes, CGR, Cormier-Michel, E, Schroeder, CB, Bulanov, SS, Benedetti, C, Yu, LL, Rykovanov, S, Bruhwiler, DL and Leemans, WP (2014) Electron injection and emittance control by transverse colliding pulses in a laser-plasma accelerator. Physical Review Special Topics – Accelerators and Beams 17, 051303.
Crandall, R and Williams, R (1971) Crystallization of electrons on the surface of liquid helium. Physics Letters A 34, 404405.
Dubin, DHE and O'Neil, TM (1999) Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states). Reviews of Modern Physics 71, 87172.
Faure, J, Rechatin, C, Norlin, A, Lifschitz, A, Glinec, Y and Malka, V (2006) Controlled injection and acceleration of electrons in plasma wakefields by colliding laser pulses. Nature 444, 737739.
Gonsalves, AJ, Nakamura, K, Lin, C, Panasenko, D, Shiraishi, S, Sokollik, T, Benedett, C, Schroeder, CB, Geddes, CGR, van Tilborg, J, Osterhoff, J, Esarey, E, Toth, C and Leemans, WP (2011) Tunable laser plasma accelerator based on longitudinal density tailoring. Nature Physics 7, 862866.
Gonsalves, A, Pollock, B and Lu, W (2017) Summary report of working group 1: Laser-plasma wakefield acceleration. AIP Conference Proceedings 1812, 030001.
Gordienko, S and Pukhov, A (2005) Scalings for ultrarelativistic laser plasma and quasimonoenergetic electrons. Physics of Plasmas 12, 043109.
Hamacher, K and Wenzel, W (1999) Scaling behavior of stochastic minimization algorithms in a perfect funnel landscape. Physical Review E 59, 938941.
Hidding, B, Pretzler, G, Rosenzweig, JB, Königstein, T, Schiller, D and Bruhwiler, DL (2012 a) Ultracold electron bunch generation via plasma photocathode emission and acceleration in a beam-driven plasma blowout. Physical Review Letters 108, 035001.
Hidding, B, Rosenzweig, JB, Xi, Y, OShea, B, Andonian, G, Schiller, D, Barber, S, Williams, O, Pretzler, G, Königstein, T, Kleeschulte, F, Hogan, MJ, Litos, M, Corde, S, White, WW, Muggli, P, Bruhwiler, DL and Lotov, K (2012 b) Beyond injection: Trojan horse underdense photocathode plasma wakefield acceleration. AIP Conference Proceedings 1507, 570.
Huang, S, Zhou, SY, Li, F, Wan, Y, Wu, YP, Hua, JF, Pai, CH, Lu, W, Wang, Z, Deng, HX, Liu, B, Wang, D, Zhao, ZT, An, WM, Xu, XL, Joshi, C and Mori, WB (2017) Study of high transformer ratio plasma wakefield acceleration for accelerator parameters of SXFEL facility at Sinap using 3D PIC simulations. Proceedings of the IPAC2017, pp. 1734–1736, Copenhagen, Denmark.
Ikegami, M, Okamoto, H and Yuri, Y (2006) Crystalline beams in dispersion-free storage rings. Physical Review Special Topics – Accelerators and Beams 9, 124201.
Jackson, JD, Witte, C and Diestelhorst, M (2013) Klassische Elektrodynamik. Berlin, Boston: Gruyter, Walter de GmbH.
James, D (1998) Quantum dynamics of cold trapped ions with application to quantum computation. Applied Physics B: Lasers and Optics 66, 181190.
Kalmykov, S, Yi, SA, Khudik, V and Shvets, G (2009) Electron self-injection and trapping into an evolving plasma bubble. Physical Review Letters 103, 135004.
Kostyukov, IY and Pukhov, A (2015) Plasma-based methods for electron acceleration: current status and prospects. Physics-Uspekhi 58, 1.
Kostyukov, I, Pukhov, A and Kiselev, S (2004) Phenomenological theory of laser-plasma interaction in bubble regime. Physics of Plasmas 11, 52565264.
Li, F, Hua, JF, Xu, XL, Zhang, CJ, Yan, LX, Du, YC, Huang, WH, Chen, HB, Tang, CX, Lu, W, Joshi, C, Mori, WB and Gu, YQ (2013) Generating high-brightness electron beams via ionization injection by transverse colliding lasers in a plasma-wakefield accelerator. Physical Review Letters 111, 015003.
Malka, V (2012) Laser plasma accelerators. Physics of Plasmas 19, 055501.
Meissner, G, Namaizawa, H and Voss, M (1976) Stability and image-potential-induced screening of electron vibrational excitations in a 3-layer-structure. Physical Review B 13, 13701376.
Metropolis, N, Rosenbluth, AW, Rosenbluth, MN, Teller, AH and Teller, E (1953) Equation of state calculations by fast computing machines. The Journal of Chemical Physics 21, 10871092.
Morfill, GE and Ivlev, AV (2009) Complex plasmas: an interdisciplinary research field. Reviews of Modern Physics 81, 13531404.
Muggli, P (2016) Beam-driven, plasma-based particle accelerators. CERN Yellow Reports, Vol. 1 (2016): Prague, Czech Republic: Proceedings of the 2014 CAS–CERN Accelerator School: Plasma Wake Acceleration.
Pak, A, Marsh, KA, Martins, SF, Lu, W, Mori, WB and Joshi, C (2010) Injection and trapping of tunnel-ionized electrons into laser-produced wakes. Physical Review Letters 104, 025003.
Petrillo, V, Bacci, A, Zinati, RBA, Chaikovska, I, Curatolo, C, Ferrario, M, Maroli, C, Ronsivalle, C, Rossi, AR, Serafini, L, Tomassini, P, Vaccarezza, C and Variola, A (2012) Photon flux and spectrum of compton sources. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 693, 109116.
Pukhov, A and Meyer-ter Vehn, J (2002) Laser wake field acceleration: the highly non-linear broken-wave regime. Applied Physics B: Lasers and Optics 74, 355361.
Pyka, K, Keller, J, Partner, HL, Nigmatullin, R, Burgermeister, T, Meier, DM, Kuhlmann, K, Retzker, A, Plenio, MB, del Campo, A, Zurek, WH and Mehlstaeubler, TE (2013) Topological defect formation and spontaneous symmetry breaking in ion coulomb crystals. Nature Communications 4, 2291.
Radzvilavičius, A and Anisimovas, E (2011) Topological defect motifs in two-dimensional coulomb clusters. Journal of Physics: Condensed Matter 23, 385301.
Reichwein, L, Thomas, J and Pukhov, A (2018) Two-dimensional structures of electron bunches in relativistic plasma cavities. Physical Review E 98, 013201.
Sävert, A, Mangles, SPD, Schnell, M, Siminos, E, Cole, JM, Leier, M, Reuter, M, Schwab, MB, Möller, M, Poder, K, Jäckel, O, Paulus, GG, Spielmann, C, Skupin, S, Najmudin, Z and Kaluza, MC (2015) Direct observation of the injection dynamics of a laser wakefield accelerator using few-femtosecond shadowgraphy. Physical Review Letters 115, 055002.
Schiffer, J (1996) Crystalline beams. Proceedings of the Particle Accelerator Conference. IEEE, Dallas, TX, USA.
Schnell, M, Sävert, A, Landgraf, B, Reuter, M, Nicolai, M, Jäckel, O, Peth, C, Thiele, T, Jansen, O, Pukhov, A, Willi, O, Kaluza, MC and Spielmann, C (2012) Deducing the electron-beam diameter in a laser-plasma accelerator using x-ray betatron radiation. Physical Review Letters 108, 075001.
Swanson, KK, Tsai, H -E, Barber, SK, Lehe, R, Mao, H -S, Steinke, S, van Tilborg, J, Nakamura, K, Geddes, CGR, Schroeder, CB, Esarey, E and Leemans, WP (2017) Control of tunable, monoenergetic laser-plasma-accelerated electron beams using a shock-induced density downramp injector. Physical Review Accelerators and Beams 20, 051301.
Thomas, J, Günther, MM and Pukhov, A (2017) Beam load structures in a basic relativistic interaction model. Physics of Plasmas 24, 013101.
Tochitsky, S, Fiuza, F and Joshi, C (2016) Prospects and directions of CO2 laser-driven accelerators. AIP Conference Proceedings 1777, 020005.
Tooley, MP, Ersfeld, B, Yoffe, SR, Noble, A, Brunetti, E, Sheng, ZM, Islam, MR and Jaroszynski, DA (2017) Towards attosecond high-energy electron bunches: Controlling self-injection in laser-wakefield accelerators through plasma-density modulation. Physical Review Letters 119, 044801.
Wang, J, Feng, J, Zhu, C, Li, Y, He, Y, Li, D, Tan, J, Ma, J and Chen, L (2018) Small energy spread electron beams from laser wakefield acceleration by self-evolved ionization injection. Plasma Physics and Controlled Fusion 60, 034004.
Wigner, E (1934) On the interaction of electrons in metals. Physical Review 46, 1002.
Xu, J, Buck, A, Chou, S-W, Schmid, K, Shen, B, Tajima, T, Kaluza, MC and Veisz, L (2017 a) Dynamics of electron injection in a laser-wakefield accelerator. Physics of Plasmas 24, 083106.
Xu, XL, Li, F, An, W, Dalichaouch, TN, Yu, P, Lu, W, Joshi, C and Mori, WB (2017 b) High quality electron bunch generation using a longitudinal density-tailored plasma-based accelerator in the three-dimensional blowout regime. Physical Review Accelerators and Beams 20, 111303.


The filamented electron bunch of the bubble regime

  • Lars Reichwein (a1), Johannes Thomas (a1) and Alexander Pukhov (a1)


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.