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Plasma scale length and quantum electrodynamics effects on particle acceleration at extreme laser plasmas

Published online by Cambridge University Press:  15 November 2021

Ozgur Culfa Open the ORCID record for Ozgur Culfa [Opens in a new window]  and
Sinan Sagir Open the ORCID record for Sinan Sagir [Opens in a new window]
Ozgur Culfa*
Affiliation:
Department of Physics, Karamanoglu Mehmetbey University, Karaman 70200, Turkey
Sinan Sagir
Affiliation:
Vocational School of Health Services, Karamanoglu Mehmetbey University, Karaman, 70200, Turkey
*
†Email address for correspondence: ozgurculfa@kmu.edu.tr
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Abstract

In this work, simulations of multipetawatt lasers at irradiances ${\sim }10^{23} \ \mathrm {W}\ \mathrm {cm}^{-2}$, striking solid targets and implementing two-dimensional particle-in-cell code was used to study particle acceleration. Preformed plasma at the front surface of a solid target increases both the efficiency of particle acceleration and the reached maximum energy by the accelerated charged particles via nonlinear plasma processes. Here, we have investigated the preformed plasma scale length effects on particle acceleration in the presence and absence of nonlinear quantum electrodynamic (QED) effects, including quantum radiation reaction and multiphoton Breit–Wheeler pair production, which become important at irradiances ${\sim } 10^{23}\ \mathrm {W}\ \mathrm {cm}^{-2}$. Our results show that QED effects help particles gain higher energies with the presence of preformed plasma. In the results for all cases, preplasma leads to more efficient laser absorption and produces more energetic charged particles, as expected. In the case where QED is included, however, physical mechanisms changed and generated secondary particles ($\gamma$-rays and positrons) reversing this trend. That is, the hot electrons cool down due to QED effects, while ions gain more energy due to different acceleration methods. It is found that more energetic $\gamma$-rays and positrons are created with increasing scale length due to high laser conversion efficiency (${\sim }$24 % for $\gamma$-rays and $\sim$4 % for positrons at $L = 7\ \mathrm {\mu }\textrm {m}$ scale length), which affects the ion and electron acceleration mechanisms. It is also observed that the QED effect reduces the collimation of angular distribution of accelerated ions because the dominant ion acceleration mechanism is changing when QED is involved in the process.

Keywords

PIC simulationsparticle accelerationlaser-plasma interactions
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 6 , December 2021 , 905870602
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Albright, B., Yin, L., Bowers, K.J., Hegelich, B.M., Flippo, K.A., Kwan, T.J.T. & Fernández, J.C. 2007 Relativistic buneman instability in the laser breakout afterburner. Phys. Plasmas 14, 094502.10.1063/1.2768933CrossRefGoogle Scholar
Arber, T., Bennett, K., Brady, C., Lawrence-Douglas, A., Ramsay, M.G., Sircombe, N.J., Gillies, P., Evans, R.G., Schmitz, H., Bell, A.R. & Ridgers, C.P. 2015 Contemporary particle-in-cell approach to laser-plasma modelling. Plasma Phys. Control. Fusion 57, 113001.10.1088/0741-3335/57/11/113001CrossRefGoogle Scholar
Baeva, T., Gordienko, S., Robinson, A. & Norreys, P. 2011 Efficient laser-driven proton acceleration from a cryogenic solid hydrogen target. Phys. Plasmas 18, 056702.10.1063/1.3566068CrossRefGoogle Scholar
Bashinov, A. & Kim, A. 2013 On the electrodynamic model of ultra-relativistic laser-plasma interactions caused by radiation reaction effects. Phys. Plasmas 20, 113111.10.1063/1.4835215CrossRefGoogle Scholar
Bell, A. & Kirk, J. 2008 Possibility of prolific pair production with high-power lasers. Phys. Rev. Lett. 101, 200403.10.1103/PhysRevLett.101.200403CrossRefGoogle ScholarPubMed
Blaga, C., Xu, J., DiChiara, A., Sistrunk, E., Zhang, K., Agostini, P., Miller, T.A., DiMauro, L.F. & Lin, C.D. 2011 Imaging ultrafast molecular dynamics with laser-induced electron diffraction. Nature 483, 194–197.10.1038/nature10820CrossRefGoogle Scholar
Brady, C., Ridgers, C., Arber, T., Bell, A.R. & Kirk, J.G. 2012 Laser absorption in relativistically underdense plasmas by synchrotron radiation. Phys. Rev. Lett. 109, 245006.10.1103/PhysRevLett.109.245006CrossRefGoogle ScholarPubMed
Breit, G. & Wheeler, J.A. 1934 Collision of two light quanta. Phys. Rev. 46, 1087.10.1103/PhysRev.46.1087CrossRefGoogle Scholar
Brunel, F. 1987 Not-so-resonant, resonant absorption. Phys. Rev. Lett. 59, 52–55.10.1103/PhysRevLett.59.52CrossRefGoogle ScholarPubMed
Bulanov, S., Esirkepov, T., Hayashi, Y., Kando, M., Kiriyama, H., Koga, J.K., Kondo, K., Kotaki, H., Pirozhkov, A.S., Bulanov, S.S., et al. 2011 On the design of experiments for the study of extreme field limits in the interaction of laser with ultrarelativistic electron beam. Nucl. Instrum. Meth. Phys. Res. 660, 31–42.10.1016/j.nima.2011.09.029CrossRefGoogle Scholar
Bulanov, S.V., Wilkens, J.J., Esirkepov, T., Korn, G., Kraft, G., Kraft, S.D., Molls, M. & Khoroshkov, V.S. 2014 Laser ion acceleration for hadron therapy. Phys. Uspekhi 57, 1149–1179.10.3367/UFNe.0184.201412a.1265CrossRefGoogle Scholar
Culfa, O., Tallents, G.J., Korkmaz, M., Rossall, A.K., Wagenaars, E., Ridgers, C.P., Murphy, C.D., Booth, N., Carroll, D.C., Wilson, L.A., et al. 2017 Plasma scale length effects on protons generated in ultra intense laser plasmas. Laser Particle Beams 35, 58–63.10.1017/S0263034616000811CrossRefGoogle Scholar
Culfa, O., Tallents, G.J., Rossall, A.K., Wagenaars, E., Ridgers, C.P., Murphy, C.D., Dance, R.J., Gray, R.J., McKenna, P., Brown, C.D.R., et al. 2016 Plasma scale-length effects on electron energy spectra in high-irradiance laser plasmas. Phys. Rev. E 93, 043201.10.1103/PhysRevE.93.043201CrossRefGoogle ScholarPubMed
Culfa, O., Tallents, G.J., Wagenaars, E., Ridgers, C.P., Dance, R.J., Rossall, A.K., Gray, R.J., McKenna, P., Brown, C.D.R., James, S.F., et al. 2014 Hot electron production in laser solid interactions with a controlled pre-pulse. Phys. Plasmas 21, 043106.10.1063/1.4870633CrossRefGoogle Scholar
Daido, H., Nishiuchi, M. & Pirozhkov, A. 2012 Review of laser-driven ion sources and their applications. Rep. Prog. Phys. 75, 056401.10.1088/0034-4885/75/5/056401CrossRefGoogle ScholarPubMed
Danson, C.N., Haefner, C., Bromage, J., Butcher, T., Chanteloup, J-C. F., Chowdhury, E.A., Galvanauskas, A., Gizzi, L.A., Hein, J., Hillier, D.I., et al. 2019 Petawatt and exawatt class lasers worldwide. High Power Laser Sci. Engng 7, e54.10.1017/hpl.2019.36CrossRefGoogle Scholar
Dirac, P. 1938 Classical theory of radiating electrons. Proc. R. Soc. Lond. A 167, 148.Google Scholar
Duclous, R., Kirk, J. & Bell, A. 2011 Monte Carlo calculations of pair production in high-intensity laser–plasma interactions. Plasma. Phys. Control. Fusion 53, 015009.10.1088/0741-3335/53/1/015009CrossRefGoogle Scholar
Esarey, E., Schroeder, C. & Leemans, W. 2009 Physics of laser-driven plasma-based electron accelerators. Rev. Mod. Phys. 81, 1229–1285.10.1103/RevModPhys.81.1229CrossRefGoogle Scholar
Esirkepov, T., Borghesi, M., Bulanov, S., Mourou, G. & Tajima, T. 2004 Highly efficient relativistic-ion generation in the laser-piston regime. Phys. Rev. Lett. 92, 175003.10.1103/PhysRevLett.92.175003CrossRefGoogle ScholarPubMed
Esirkepov, T., Koga, J., Sunahara, A., Morita, T., Nishikino, M., Kageyama, K., Nagatomo, H., Nishihara, K., Sagisaka, A., Kotaki, H., et al. 2014 Prepulse and amplified spontaneous emission effects on theinteraction of a petawatt class laser with thin solid targets. Nucl. Instrum. Meth. Phys. Res. 745, 150–163.10.1016/j.nima.2014.01.056CrossRefGoogle Scholar
Fedotov, A., Narozhny, N.B., Mourou, G. & Korn, G. 2010 Limitations on the attainable intensity of high power lasers. Phys. Rev. Lett. 105, 080402.10.1103/PhysRevLett.105.080402CrossRefGoogle ScholarPubMed
Gibbon, P. 2000 Short Pulse Laser Interactions with Matter. Imperial College Press.Google Scholar
Hadjisolomou, P., Tsygvintsev, I., Sasorov, P., Gasilov, V., Korn, G. & Bulanov, S. 2020 Preplasma effects on laser ion generation from thin foil targets. Phys. Plasmas 27, 013107.10.1063/1.5124457CrossRefGoogle Scholar
Jiao, J., Zhang, B., Yu, J. , Zhang, Z., Yan, Y., He, S., Deng, Z., Teng, J., Hong, W. & Gu, Y. 2017 Generating high-yield positrons and relativistic collisionless shocks by 10 PW laser. Laser Particle Beams 35, 234–240.10.1017/S0263034617000106CrossRefGoogle Scholar
Jung, D., Yin, L., Gautier, D., Wu, H.-C., Letzring, S., Dromey, B., Shah, R., Palaniyappan, S., Shimada, T., Johnson, R.P., et al. 2013 Laser-driven 1 GeV carbon ions from preheated diamond targets in the break-out afterburner regime. Phys. Plasmas 20, 083103.10.1063/1.4817287CrossRefGoogle Scholar
Kirk, J., Bell, A. & Arka, I. 2009 Pair production in counter-propagating laser beams. Plasma Phys. Control. Fusion 51, 085008.10.1088/0741-3335/51/8/085008CrossRefGoogle Scholar
Kneip, S., Nagel, S., Martins, S., Mangles, S.P.D., Bellei, C., Chekhlov, O., Clarke, R.J., Delerue, N., Divall, E.J., Doucas, G., et al. 2009 Near-GeV acceleration of electrons by a nonlinear plasma wave driven by a self-guided laser pulse. Phys. Rev. Lett. 103 (3), 035002.10.1103/PhysRevLett.103.035002CrossRefGoogle ScholarPubMed
Kostyukov, I. & Nerush, E. 2016 Production and dynamics of positrons in ultrahigh intensity laser-foil interactions. Phys. Plasmas 23, 093119.10.1063/1.4962567CrossRefGoogle Scholar
Kruer, W. 1988 The Physics of Laser Plasma Interactions. Addison-Wesley.Google Scholar
Leemans, W., Nagler, B., Gonsalves, A., Tóth, Cs., Nakamura, K., Geddes, C.G.R., Esarey, E., Schroeder, C.B. & Hooker, S.M. 2006 GeV electron beams from a centimetre-scale accelerator. Nat. Phys. 2, 696–699.10.1038/nphys418CrossRefGoogle Scholar
Lefebvre, E. & Bonnaud, G. 1995 Transparency/opacity of a solid target illuminated by an ultrahigh-intensity laser pulse. Phys. Rev. Lett. 74, 2002.10.1103/PhysRevLett.74.2002CrossRefGoogle ScholarPubMed
Levy, M., Blackburn, T., Ratan, N., Sadler, J., Ridgers, C.P., Kasim, M., Ceurvorst, L., Holloway, J., Baring, M.G., Bell, A.R., et al. 2019 QED-driven laser absorption. arXiv:1609.00389v2 [physics.plasm-ph].Google Scholar
Lezhnin, K., Sasorov, P., Korn, G. & Bulanov, S. 2018 High power gamma flare generation in multi-petawatt laser interaction with tailored targets. Phys. Plasmas 25, 123105.10.1063/1.5062849CrossRefGoogle Scholar
Macchi, A., Borghesi, M. & Passoni, M. 2013 Ion acceleration by superintense laser-plasma interaction. Rev. Mod. Phys. 85, 751–793.10.1103/RevModPhys.85.751CrossRefGoogle Scholar
Max, C., Arons, J. & Langdon, A. 1974 Self-modulation and self-focusing of electromagnetic waves in plasmas. Phys. Rev. Lett. 33, 209–212.10.1103/PhysRevLett.33.209CrossRefGoogle Scholar
McKenna, P., Carroll, D., Lundh, O., Nürnberg, F., Markey, K., Bandyopadhyay, S., Batani, D., Evans, R.G., Jafer, R., Kar, S., et al. 2008 Effects of front surface plasma expansion on proton acceleration in ultraintense laser irradiation of foil targets. Laser Particle Beams 26 (4), 591–596.10.1017/S0263034608000657CrossRefGoogle Scholar
Mourou, G., Tajima, T. & Bulanov, S. 2006 Optics in the relativistic regime. Rev. Mod. Phys. 78, 309.10.1103/RevModPhys.78.309CrossRefGoogle Scholar
Najmudin, Z., Krushelnick, K., Tatarakis, M., Clark, E.L., Danson, C.N., Malka, V., Neely, D., Santala, M.I.K. & Dangor, A.E. 2003 The effect of high intensity laser propagation instabilities on channel formation in underdense plasmas. Phys. Plasmas 10, 438.10.1063/1.1534585CrossRefGoogle Scholar
Nakamura, T., Koga, J., Esirkepov, T.Z. , Kando, M., Korn, G. & Bulanov, S.V. 2012 High-power ${\gamma }$-ray flash generation in ultraintense laser-plasma interactions. Phys. Rev. Lett. 108, 195001.10.1103/PhysRevLett.108.195001CrossRefGoogle ScholarPubMed
Nerush, E.N., Kostyukov, I.Y., Fedotov, A.M., Narozhny, N.B., Elkina, N.V. & Ruhl, H. 2011 Laser field absorption in self-generated electron-positron pair plasma. Phys. Rev. Lett. 106, 035001.10.1103/PhysRevLett.106.035001CrossRefGoogle ScholarPubMed
Palaniyappan, S., Hegelich, B.M., Wu, H., Jung, D., Gautier, D., Yin, L., Albright, B., Johnson, R., Shimada, T., Letzring, S., et al. 2012 Dynamics of relativistic transparency and optical shuttering in expanding overdense plasmas. Nat. Phys. 8, 763–769.10.1038/nphys2390CrossRefGoogle Scholar
Peebles, J., Wei, M., Arefiev, A., McGuffey, C., Stephens, R.B., Theobald, W., Haberberger, D., Jarrott, L.C., Link, A., Chen, H., et al. 2017 Investigation of laser pulse length and pre-plasma scale length impact on hot electron generation on omega-ep. New J. Phys. 19, 023008.10.1088/1367-2630/aa5a21CrossRefGoogle Scholar
Piazza, A.D., Muller, C., Hatsagortsyan, K. & Keitel, C.H. 2012 Extremely high-intensity laser interactions with fundamental quantum systems. Rev. Mod. Phys. 84, 1177.10.1103/RevModPhys.84.1177CrossRefGoogle Scholar
Polz, J., Robinson, A., Kalinin, A., Becker, G.A., Costa Fraga, R.A., Hellwing, M., Hornung, M., Keppler, S., Kessler, A., Klöpfel, D., et al. 2019 Efficient laser-driven proton acceleration from a cryogenic solid hydrogen target. Sci. Rep. 9, 16534.10.1038/s41598-019-52919-7CrossRefGoogle ScholarPubMed
Powell, H., King, M., Gray, R., MacLellan, D.A., Gonzalez-Izquierdo, B., Stockhausen, L.C., Hicks, G., Dover, N.P., Rusby, D.R., Carroll, D.C., et al. 2015 Proton acceleration enhanced by a plasma jet in expanding foils undergoing relativistic transparency. New J. Phys. 17, 103033.10.1088/1367-2630/17/10/103033CrossRefGoogle Scholar
Qiao, B., Kar, S., Geissler, M., Gibbon, P., Zeph, M. & Borghesi, M. 2012 Dominance of radiation pressure in ion acceleration with linearly polarized pulses at intensities of $10^{21}\ \mathrm {W}\ \mathrm {cm}^{-2}$. Phys. Rev. Lett. 108, 115002.10.1103/PhysRevLett.108.115002CrossRefGoogle Scholar
Ridgers, C., Brady, C., Duclous, R., Kirk, J.G., Bennett, K., Arber, T.D., Robinson, A.P.L. & Bell, A.R. 2012 Dense electron-positron plasmas and ultraintense ${\gamma }$ rays from laser-irradiated solids. Phys. Rev. Lett. 108, 165006.10.1103/PhysRevLett.108.165006CrossRefGoogle ScholarPubMed
Ridgers, C., Kirk, J., Duclous, R., Blackburn, T.G., Brady, C.S., Blackburn, K., Arber, T.D. & Bell, A.R. 2014 Modelling gamma-ray photon emission and pair production in high-intensity laser–matter interactions. J. Comput. Phys. 260, 273–285.10.1016/j.jcp.2013.12.007CrossRefGoogle Scholar
Robinson, A., Zepf, M., Kar, S., Evans, R.G. & Bellei, C. 2008 Radiation pressure acceleration of thin foils with circularly polarized laser pulses. New J. Phys. 10, 01302.10.1088/1367-2630/10/1/013021CrossRefGoogle Scholar
Sadighi-Bonabi, R., Yazdani, E., Cang, Y. & Hora, H. 2010 Dielectric magnifying of plasma blocks by nonlinear force acceleration with delayed electron heating. Phys. Plasmas 17, 113108.10.1063/1.3497009CrossRefGoogle Scholar
Sahai, A., Tsung, F., Tableman, A., Mori, W.B. & Katsouleas, T.C. 2013 Relativistically induced transparency acceleration of light ions by an ultrashort laser pulse interacting with a heavy-ion-plasma density gradient. Phys. Rev. E 88, 043105.10.1103/PhysRevE.88.043105CrossRefGoogle ScholarPubMed
Santala, M., Zeph, M., Watts, I., Beg, F.N., Clark, E., Tatarakis, M., Krushelnick, K., Dangor, A.E., McCanny, T., Spencer, I., et al. 2000 Effect of the plasma density scale length on the direction of fast electrons in relativistic laser-solid interactions. Phys. Rev. Lett 84, 1459–1462.10.1103/PhysRevLett.84.1459CrossRefGoogle ScholarPubMed
Savin, A., Ross, A., Aboushelbaya, R., Mayr, M.W., Spiers, B., Wang, R.H.-W. & Norreys, P.A. 2019 Energy absorption in the laser-qed regime. Sci. Rep. 9, 8956.10.1038/s41598-019-45536-xCrossRefGoogle ScholarPubMed
Savin, A., Ross, A., Serzans, M., Trines, R.M.G.M., Ceurvorst, L., Ratan, N., Spiers, B., Bingham, R., Robinson, A.P.L. & Norreys, P.A. 2017 Attosecond-scale absorption at extreme intensities. Phys. Plasmas 24, 113103.10.1063/1.4989798CrossRefGoogle Scholar
Schwinger, J. 1951 On gauge invariance and vacuum polarization. Phys. Rev. 82, 664.10.1103/PhysRev.82.664CrossRefGoogle Scholar
Shen, B., Li, Y., Yu, M.Y. & Cary, J. 2007 Bubble regime for ion acceleration in a laser-driven plasma. Phys. Rev. E 76, 055402.10.1103/PhysRevE.76.055402CrossRefGoogle Scholar
Silva, L., Marti, M., Davies, J., Fonseca, R.A., Ren, C., Tsung, F.S. & Mori, W.B. 2004 Proton shock acceleration in laser-plasma interactions. Phys. Rev. Lett. 92, 015002.10.1103/PhysRevLett.92.015002CrossRefGoogle ScholarPubMed
Sokolov, I., Naumova, N.M. & Nees, J. 2011 Numerical modeling of radiation-dominated and quantum-electrodynamically strong regimes of laser-plasma interaction. Phys. Plasmas 18, 093109.10.1063/1.3638138CrossRefGoogle Scholar
Sokolov, I., Nees, J.A., Yanovsky, V., Naumova, N.M. & Mourou, G.A. 2010 Emission and its back-reaction accompanying electron motion in relativistically strong and qed-strong pulsed laser fields. Phys. Rev. E 81, 036412.10.1103/PhysRevE.81.036412CrossRefGoogle ScholarPubMed
Sorbo, D.D., Blackman, D., Capdessus, R., Small, K., Slade-Lowther, C., Luo, W., Duff, M.J., Robinson, A.P.L., McKenna, P., Sheng, Z.-M., et al. 2018 Efficient ion acceleration and dense electron–positron plasma creation in ultra-high intensity laser-solid interactions. New J. Phys. 20, 033014.10.1088/1367-2630/aaae61CrossRefGoogle Scholar
Steinke, S., van Tilborg, J., Benedetti, C., Geddes, C.G.R., Schroeder, C.B., Daniels, J., Swanson, K.K., Gonsalves, A.J., Nakamura, K., Matlis, N.H., et al. 2016 Multi stage coupling of independent laser-plasma accelerators. Nature 530, 190–193.10.1038/nature16525CrossRefGoogle Scholar
Tajima, T. & Dawson, J. 1979 Laser electron accelerator. Phys. Rev. Lett. 43 (4), 267–270.10.1103/PhysRevLett.43.267CrossRefGoogle Scholar
Tamburini, M., Pegoraro, F., Piazza, A.D., Keitel, C.H. & Macchi, A. 2010 Radiation reaction effects on radiation pressure acceleration. New J. Phys. 12, 123005.10.1088/1367-2630/12/12/123005CrossRefGoogle Scholar
Vshivkov, V., Naumova, N., Pegoraro, F. & Bulanov, S. 1998 Nonlinear electrodynamics of the interaction of ultra-intense laser pulses with a thin foil. Phys. Plasmas 5, 2727.10.1063/1.872961CrossRefGoogle Scholar
Wang, W.-M., Gibbon, P., Sheng, M., Li, Y.-T. & Zhang, J. 2017 Laser opacity in underdense preplasma of solid targets due to quantum electrodynamics effects. Phys. Rev. E 96, 013201.10.1103/PhysRevE.96.013201CrossRefGoogle ScholarPubMed
Wilks, S., Kruer, W., Tabak, M. & Langdon, A.B. 1992 Absorption of ultra-intense laser pulses. Phys. Rev. Lett. 69, 1383.10.1103/PhysRevLett.69.1383CrossRefGoogle ScholarPubMed
Wilks, S., Langdon, A., Cowan, T., Roth, M., Singh, M., Hatchett, S., Key, M.H., Pennington, D., MacKinnon, A. & Snavely, R.A. 2001 Ion acceleration by superintense laser-plasma interaction. Phys. Plasmas 8, 542.10.1063/1.1333697CrossRefGoogle Scholar
Xu, Y., Wang, J., Hora, H., Qi, X., Xing, Y., Yang, L. & Zhu, W. 2018 Plasma block acceleration based upon the interaction between double targets and an ultra-intense linearly polarized laser pulse. Phys. Plasmas 25, 043102.10.1063/1.5024032CrossRefGoogle Scholar
Yazdani, E., Sadighi-Bonabi, R., Afarideh, H., Yazdanpanah, J. & Hora, H. 2014 Enhanced laser ion acceleration with a multi-layer foam target assembly. Laser Particle Beams 32, 509–515.10.1017/S0263034614000342CrossRefGoogle Scholar
Yin, L., Albright, B., Hegelich, B., Bowers, K.J., Flippo, K.A., Kwan, T.J.T. & Fernández, J.C. 2007 Monoenergetic and Gev ion acceleration from the laser breakout afterburner using ultrathin targets. Phys. Plasmas 14, 056706.10.1063/1.2436857CrossRefGoogle Scholar
Zhang, P., Ridgers, C. & Thomas, A. 2015 The effect of nonlinear quantum electrodynamics on relativistic transparency and laser absorption in ultra-relativistic plasmas. New J. Phys. 17, 043051.10.1088/1367-2630/17/4/043051CrossRefGoogle Scholar
Zhang, X., Shen, B., Ji, L., Wang, F., Wen, M., Wang, W., Xu, J. & Yu, Y. 2010 Ultrahigh energy proton generation in sequential radiation pressure and bubble regime. Phys. Plasmas 17, 123102.Google Scholar