Simulations of spin/polarization-resolved laser–plasma interactions in the nonlinear QED regime

Wan Feng; Lv Chong; Xue Kun; Dou Zhen-Ke; Zhao Qian; Ababekri Mamutjan; Wei Wen-Qing; Li Zhong-Peng; Zhao Yong-Tao; Li Jian-Xing

doi:10.1063/5.0163929

Volume 8 Issue 6

Nov. 2023

Turn off MathJax

Article Contents

Article Navigation > Matter and Radiation at Extremes > 2023 > 8(6): 064002

Wan Feng, Lv Chong, Xue Kun, Dou Zhen-Ke, Zhao Qian, Ababekri Mamutjan, Wei Wen-Qing, Li Zhong-Peng, Zhao Yong-Tao, Li Jian-Xing. Simulations of spin/polarization-resolved laser–plasma interactions in the nonlinear QED regime[J]. Matter and Radiation at Extremes, 2023, 8(6): 064002. doi: 10.1063/5.0163929

Citation:

Wan Feng, Lv Chong, Xue Kun, Dou Zhen-Ke, Zhao Qian, Ababekri Mamutjan, Wei Wen-Qing, Li Zhong-Peng, Zhao Yong-Tao, Li Jian-Xing. Simulations of spin/polarization-resolved laser–plasma interactions in the nonlinear QED regime[J]. Matter and Radiation at Extremes, 2023, 8(6): 064002. doi: 10.1063/5.0163929

Citation:

Wan Feng, Lv Chong, Xue Kun, Dou Zhen-Ke, Zhao Qian, Ababekri Mamutjan, Wei Wen-Qing, Li Zhong-Peng, Zhao Yong-Tao, Li Jian-Xing. Simulations of spin/polarization-resolved laser–plasma interactions in the nonlinear QED regime[J]. Matter and Radiation at Extremes, 2023, 8(6): 064002. doi: 10.1063/5.0163929

PDF( 9280 KB)

Simulations of spin/polarization-resolved laser–plasma interactions in the nonlinear QED regime

doi: 10.1063/5.0163929

1.
Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an 710049, China
2.
Department of Nuclear Physics, China Institute of Atomic Energy, P.O. Box 275(7), Beijing 102413, China

More Information

Corresponding author: a)Author to whom correspondence should be addressed: jianxing@xjtu.edu.cn
Received Date: 2023-06-20
Accepted Date: 2023-08-08

Available Online: 2023-11-01

Publish Date: 2023-11-01

Abstract

Abstract

Strong-field quantum electrodynamics (SF-QED) plays a crucial role in ultraintense laser–matter interactions and demands sophisticated techniques to understand the related physics with new degrees of freedom, including spin angular momentum. To investigate the impact of SF-QED processes, we have introduced spin/polarization-resolved nonlinear Compton scattering, nonlinear Breit–Wheeler, and vacuum birefringence processes into our particle-in-cell (PIC) code. In this article, we provide details of the implementation of these SF-QED modules and share known results that demonstrate exact agreement with existing single-particle codes. By coupling normal PIC simulations with spin/polarization-resolved SF-QED processes, we create a new theoretical platform to study strong-field physics in currently running or planned petawatt or multi-petawatt laser facilities.

Conflict of Interest
The authors have no conflicts to disclose.
Feng Wan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Chong Lv: Formal analysis (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). Kun Xue: Software (equal); Writing – review & editing (equal). Zhen-Ke Dou: Software (equal); Writing – review & editing (equal). Qian Zhao: Writing – review & editing (equal). Mamutjan Ababekri: Conceptualization (equal); Writing – review & editing (equal). Wen-Qing Wei: Writing – review & editing (equal). Zhong-Peng Li: Writing – review & editing (equal). Yong-Tao Zhao: Conceptualization (equal); Writing – review & editing (equal). Jian-Xing Li: Conceptualization (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).
Author Contributions
The data supporting this study's findings are available from the corresponding author upon reasonable request.

FullText(HTML)

References(86)

References

[1]	A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).10.1103/revmodphys.84.1177
[2]	A. R. Bell and J. G. Kirk, “Possibility of prolific pair production with high-power lasers,” Phys. Rev. Lett. 101, 200403 (2008).10.1103/physrevlett.101.200403
[3]	A. J. Gonsalves, K. Nakamura, J. Daniels, C. Benedetti, C. Pieronek, T. C. H. de Raadt, S. Steinke, J. H. Bin, S. S. Bulanov, J. van Tilborg, C. G. R. Geddes, C. B. Schroeder, C. Tóth, E. Esarey, K. Swanson, L. Fan-Chiang, G. Bagdasarov, N. Bobrova, V. Gasilov, G. Korn, P. Sasorov, and W. P. Leemans, “Petawatt laser guiding and electron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide,” Phys. Rev. Lett. 122, 084801 (2019).10.1103/physrevlett.122.084801
[4]	E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81, 1229–1285 (2009).10.1103/revmodphys.81.1229
[5]	V. I. Ritus, “Quantum effects of the interaction of elementary particles with an intense electromagnetic field,” J. Russ. Laser Res. 6, 497–617 (1985).10.1007/bf01120220
[6]	V. N. Baier, V. M. Katkov, and V. M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals (World Scientific, 1998).
[7]	J. D. Jackson, Classical Electrodynamics (John Wiley and Sons, 2021).
[8]	I. M. Ternov and A. A. Sokolov, Radiation from Relativistic Electrons, American Institute of Physics Translation Series (American Institute of Physics, 1986).
[9]	D. Del Sorbo, D. Seipt, T. G. Blackburn, A. G. R. Thomas, C. D. Murphy, J. G. Kirk, and C. P. Ridgers, “Spin polarization of electrons by ultraintense lasers,” Phys. Rev. A 96, 043407 (2017).10.1103/physreva.96.043407
[10]	Y.-F. Li, R. Shaisultanov, K. Z. Hatsagortsyan, F. Wan, C. H. Keitel, and J.-X. Li, “Ultrarelativistic electron-beam polarization in single-shot interaction with an ultraintense laser pulse,” Phys. Rev. Lett. 122, 154801 (2019).10.1103/physrevlett.122.154801
[11]	B. King and S. Tang, “Nonlinear Compton scattering of polarized photons in plane-wave backgrounds,” Phys. Rev. A 102, 022809 (2020).10.1103/physreva.102.022809
[12]	Y.-F. Li, R. Shaisultanov, Y.-Y. Chen, F. Wan, K. Z. Hatsagortsyan, C. H. Keitel, and J.-X. Li, “Polarized ultrashort brilliant multi-GeV γ-rays via single-shot laser–electron interaction,” Phys. Rev. Lett. 124, 014801 (2020).10.1103/physrevlett.124.014801
[13]	Z. Gong, K. Z. Hatsagortsyan, and C. H. Keitel, “Retrieving transient magnetic fields of ultrarelativistic laser plasma via ejected electron polarization,” Phys. Rev. Lett. 127, 165002 (2021).10.1103/physrevlett.127.165002
[14]	F. Wan, Y. Wang, R.-T. Guo, Y.-Y. Chen, R. Shaisultanov, Z.-F. Xu, K. Z. Hatsagortsyan, C. H. Keitel, and J.-X. Li, “High-energy γ-photon polarization in nonlinear Breit–Wheeler pair production and γ polarimetry,” Phys. Rev. Res. 2, 032049 (2020).10.1103/physrevresearch.2.032049
[15]	K. Xue, Z.-K. Dou, F. Wan, T.-P. Yu, W.-M. Wang, J.-R. Ren, Q. Zhao, Y.-T. Zhao, Z.-F. Xu, and J.-X. Li, “Generation of highly-polarized high-energy brilliant γ-rays via laser–plasma interaction,” Matter Radiat. Extremes 5, 054402 (2020).10.1063/5.0007734
[16]	S. Tang, B. King, and H. Hu, “Highly polarised gamma photons from electron-laser collisions,” Phys. Lett. B 809, 135701 (2020).10.1016/j.physletb.2020.135701
[17]	H.-H. Song, W.-M. Wang, and Y.-T. Li, “Dense polarized positrons from laser-irradiated foil targets in the QED regime,” Phys. Rev. Lett. 129, 035001 (2022).10.1103/physrevlett.129.035001
[18]	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).10.1088/0741-3335/57/11/113001
[19]	C. Birdsall and A. Langdon, Plasma Physics via Computer Simulation (CRC Press, 2018).
[20]	A. Gonoskov, S. Bastrakov, E. Efimenko, A. Ilderton, M. Marklund, I. Meyerov, A. Muraviev, A. Sergeev, I. Surmin, and E. Wallin, “Extended particle-in-cell schemes for physics in ultrastrong laser fields: Review and developments,” Phys. Rev. E 92, 023305 (2015).10.1103/physreve.92.023305
[21]	R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, and J. C. Adam, “OSIRIS: A three-dimensional, fully relativistic particle in cell code for modeling plasma based accelerators,” in Computational Science—ICCS 2002, edited by P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra (Springer, Berlin, Heidelberg, 2002), pp. 342–351.
[22]	H. Burau, R. Widera, W. Hönig, G. Juckeland, A. Debus, T. Kluge, U. Schramm, T. E. Cowan, R. Sauerbrey, and M. Bussmann, “PIConGPU: A fully relativistic particle-in-cell code for a GPU cluster,” IEEE Trans. Plasma Sci. 38, 2831–2839 (2010).10.1109/tps.2010.2064310
[23]	J. Derouillat, A. Beck, F. Pérez, T. Vinci, M. Chiaramello, A. Grassi, M. Flé, G. Bouchard, I. Plotnikov, N. Aunai, J. Dargent, C. Riconda, and M. Grech, “Smilei: A collaborative, open-source, multi-purpose particle-in-cell code for plasma simulation,” Comput. Phys. Commun. 222, 351–373 (2018).10.1016/j.cpc.2017.09.024
[24]	D. Wu, W. Yu, S. Fritzsche, and X. T. He, “Particle-in-cell simulation method for macroscopic degenerate plasmas,” Phys. Rev. E 102, 033312 (2020).10.1103/physreve.102.033312
[25]	Y.-F. Li, Y.-Y. Chen, K. Z. Hatsagortsyan, and C. H. Keitel, “Helicity transfer in strong laser fields via the electron anomalous magnetic moment,” Phys. Rev. Lett. 128, 174801 (2022).10.1103/physrevlett.128.174801
[26]	C. Danson, D. Hillier, N. Hopps, and D. Neely, “Petawatt class lasers worldwide,” High Power Laser Sci. Eng. 3, e3 (2015).10.1017/hpl.2014.52
[27]
[28]	J. Zou, C. Le Blanc, D. Papadopoulos, G. Chériaux, P. Georges, G. Mennerat, F. Druon, L. Lecherbourg, A. Pellegrina, P. Ramirez et al., “Design and current progress of the Apollon 10 PW project,” High Power Laser Sci. Eng. 3, e2 (2015).10.1017/hpl.2014.41
[29]
[30]	Z. Gan, L. Yu, C. Wang, Y. Liu, Y. Xu, W. Li, S. Li, L. Yu, X. Wang, X. Liu, J. Chen, Y. Peng, L. Xu, B. Yao, X. Zhang, L. Chen, Y. Tang, X. Wang, D. Yin, X. Liang, Y. Leng, R. Li, and Z. Xu, “The Shanghai superintense ultrafast laser facility (SULF) project,” in Progress in Ultrafast Intense Laser Science XVI, edited by K. Yamanouchi, K. Midorikawa, and L. Roso (Springer International Publishing, Cham, 2021), pp. 199–217.
[31]	O. Buneman, “Time-reversible difference procedures,” J. Comput. Phys. 1, 517–535 (1967).10.1016/0021-9991(67)90056-3
[32]
[33]	H. Qin, S. Zhang, J. Xiao, J. Liu, Y. Sun, and W. M. Tang, “Why is Boris algorithm so good?,” Phys. Plasmas 20, 084503 (2013).10.1063/1.4818428
[34]	Y. Kane, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).10.1109/tap.1966.1138693
[35]	T. Esirkepov, “Exact charge conservation scheme for particle-in-cell simulation with an arbitrary form-factor,” Comput. Phys. Commun. 135, 144–153 (2001).10.1016/s0010-4655(00)00228-9
[36]	P. A. M. Dirac, “Classical theory of radiating electrons,” Proc. R. Soc. London, Ser. A 167, 148–169 (1938).10.1098/rspa.1938.0124
[37]	L. D. Landau and E. Lifshitz, The Classical Theory of Fields (Elsevier, Singapore, 1999).
[38]	R. Ekman, T. Heinzl, and A. Ilderton, “Reduction of order, resummation, and radiation reaction,” Phys. Rev. D 104, 036002 (2021).10.1103/physrevd.104.036002
[39]	R. Ekman, “Reduction of order and transseries structure of radiation reaction,” Phys. Rev. D 105, 056016 (2022).10.1103/physrevd.105.056016
[40]	A. Ilderton and G. Torgrimsson, “Radiation reaction in strong field QED,” Phys. Lett. B 725, 481–486 (2013).10.1016/j.physletb.2013.07.045
[41]
[42]	N. Neitz and A. Di Piazza, “Electron-beam dynamics in a strong laser field including quantum radiation reaction,” Phys. Rev. A 90, 022102 (2014).10.1103/physreva.90.022102
[43]	M. Tamburini, F. Pegoraro, A. Di Piazza, C. H. Keitel, and A. Macchi, “Radiation reaction effects on radiation pressure acceleration,” New J. Phys. 12, 123005 (2010).10.1088/1367-2630/12/12/123005
[44]	S. V. Bulanov, T. Z. Esirkepov, M. Kando, J. K. Koga, T. Nakamura, S. S. Bulanov, A. G. Zhidkov, Y. Kato, and G. Korn, “On extreme field limits in high power laser matter interactions: Radiation dominant regimes in high intensity electromagnetic wave interaction with electrons,” in High-Power, High-Energy, and High-Intensity Laser Technology; and Research Using Extreme Light: Entering New Frontiers with Petawatt-Class Lasers, edited by J. Hein, G. Korn, and L. O. Silva (SPIE, 2013).
[45]	A. I. Nikishov and V. I. Ritus, “Quantum processes in the field of a plane electromagnetic wave and in a constant field,” Sov. Phys. JETP 19, 529–541 (1964).
[46]	I. V. Sokolov, N. M. Naumova, J. A. Nees, G. A. Mourou, and V. P. Yanovsky, “Dynamics of emitting electrons in strong laser fields,” Phys. Plasmas 16, 093115 (2009).10.1063/1.3236748
[47]	A. Di Piazza, K. Z. Hatsagortsyan, and C. H. Keitel, “Quantum radiation reaction effects in multiphoton Compton scattering,” Phys. Rev. Lett. 105, 220403 (2010).10.1103/physrevlett.105.220403
[48]	A. G. R. Thomas, C. P. Ridgers, S. S. Bulanov, B. J. Griffin, and S. P. D. Mangles, “Strong radiation-damping effects in a gamma-ray source generated by the interaction of a high-intensity laser with a wakefield-accelerated electron beam,” Phys. Rev. X 2, 041004 (2012).10.1103/physrevx.2.041004
[49]	I. V. Sokolov, J. A. Nees, V. P. Yanovsky, N. M. Naumova, and G. A. Mourou, “Emission and its back-reaction accompanying electron motion in relativistically strong and QED-strong pulsed laser fields,” Phys. Rev. E 81, 036412 (2010).10.1103/physreve.81.036412
[50]	F. Niel, C. Riconda, F. Amiranoff, R. Duclous, and M. Grech, “From quantum to classical modeling of radiation reaction: A focus on stochasticity effects,” Phys. Rev. E 97, 043209 (2018).10.1103/physreve.97.043209
[51]	H. K. Avetissian, Relativistic Nonlinear Electrodynamics: The QED Vacuum and Matter in Super-Strong Radiation Fields (Springer, 2015), Vol. 88.
[52]	A. D. Piazza, “Exact solution of the Landau–Lifshitz equation in a plane wave,” Lett. Math. Phys. 83, 305–313 (2008).10.1007/s11005-008-0228-9
[53]	S. X. Hu and C. H. Keitel, “Spin signatures in intense laser–ion interaction,” Phys. Rev. Lett. 83, 4709–4712 (1999).10.1103/physrevlett.83.4709
[54]	M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, “Spin and radiation in intense laser fields,” Phys. Rev. A 65, 043410 (2002).10.1103/physreva.65.043410
[55]	G. R. Mocken and C. H. Keitel, “FFT-split-operator code for solving the Dirac equation in 2 + 1 dimensions,” Comput. Phys. Commun. 178, 868–882 (2008).10.1016/j.cpc.2008.01.042
[56]	H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, “Relativistic spin operators in various electromagnetic environments,” Phys. Rev. A 89, 052101 (2014).10.1103/physreva.89.052101
[57]	M. Wen, C. H. Keitel, and H. Bauke, “Spin-one-half particles in strong electromagnetic fields: Spin effects and radiation reaction,” Phys. Rev. A 95, 042102 (2017).10.1103/physreva.95.042102
[58]	V. N. Baĭer, “Radiative polarization of electron in storage rings,” Sov. Phys.-Usp. 14, 695–714 (1972).10.1070/pu1972v014n06abeh004751
[59]	R.-T. Guo, Y. Wang, R. Shaisultanov, F. Wan, Z.-F. Xu, Y.-Y. Chen, K. Z. Hatsagortsyan, and J.-X. Li, “Stochasticity in radiative polarization of ultrarelativistic electrons in an ultrastrong laser pulse,” Phys. Rev. Res. 2, 033483 (2020).10.1103/physrevresearch.2.033483
[60]	J. G. Kirk, A. R. Bell, and I. Arka, “Pair production in counter-propagating laser beams,” Plasma Phys. Controlled Fusion 51, 085008 (2009).10.1088/0741-3335/51/8/085008
[61]	C. Ridgers, J. Kirk, R. Duclous, T. Blackburn, C. Brady, K. Bennett, T. Arber, and A. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser–matter interactions,” J. Comput. Phys. 260, 273–285 (2014).10.1016/j.jcp.2013.12.007
[62]	W.-Y. Liu, K. Xue, F. Wan, M. Chen, J.-X. Li, F. Liu, S.-M. Weng, Z.-M. Sheng, and J. Zhang, “Trapping and acceleration of spin-polarized positrons from γ photon splitting in wakefields,” Phys. Rev. Res. 4, l022028 (2022).10.1103/physrevresearch.4.l022028
[63]	D. Green and C. Harvey, “SIMLA: Simulating particle dynamics in intense laser and other electromagnetic fields via classical and quantum electrodynamics,” Comput. Phys. Commun. 192, 313–321 (2015).10.1016/j.cpc.2015.02.030
[64]	F. Wan, R. Shaisultanov, Y.-F. Li, K. Z. Hatsagortsyan, C. H. Keitel, and J.-X. Li, “Ultrarelativistic polarized positron jets via collision of electron and ultraintense laser beams,” Phys. Lett. B 800, 135120 (2020).10.1016/j.physletb.2019.135120
[65]	V. I. Ritus and A. I. Nikishov, “Quantum Electrodynamics of Phenomena in a Strong Field,” Trudy Fiz. Inst. Akad. Nauk SSSR 111, (1979).
[66]
[67]	V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics: Volume 4 (Butterworth-Heinemann, 1982), Vol. 4.
[68]	T. N. Wistisen and U. I. Uggerhøj, “Vacuum birefringence by Compton backscattering through a strong field,” Phys. Rev. D 88, 053009 (2013).10.1103/physrevd.88.053009
[69]	S. Bragin, S. Meuren, C. H. Keitel, and A. Di Piazza, “High-energy vacuum birefringence and dichroism in an ultrastrong laser field,” Phys. Rev. Lett. 119, 250403 (2017).10.1103/physrevlett.119.250403
[70]	Y.-Y. Chen, P.-L. He, R. Shaisultanov, K. Z. Hatsagortsyan, and C. H. Keitel, “Polarized positron beams via intense two-color laser pulses,” Phys. Rev. Lett. 123, 174801 (2019).10.1103/physrevlett.123.174801
[71]	Y.-Y. Chen, K. Z. Hatsagortsyan, C. H. Keitel, and R. Shaisultanov, “Electron spin- and photon polarization-resolved probabilities of strong-field QED processes,” Phys. Rev. D 105, 116013 (2022).10.1103/physrevd.105.116013
[72]	K. Xue, R.-T. Guo, F. Wan, R. Shaisultanov, Y.-Y. Chen, Z.-F. Xu, X.-G. Ren, K. Z. Hatsagortsyan, C. H. Keitel, and J.-X. Li, “Generation of arbitrarily polarized GeV lepton beams via nonlinear Breit–Wheeler process,” Fundam. Res. 2, 539–545 (2022).10.1016/j.fmre.2021.11.022
[73]	F. Wan, C. Lv, M. Jia, H. Sang, and B. Xie, “Photon emission by bremsstrahlung and nonlinear Compton scattering in the interaction of ultraintense laser with plasmas,” Eur. Phys. J. D 71, 236 (2017).10.1140/epjd/e2017-70805-7
[74]	S. Agostinelli, J. Allison, K. Amako, J. Apostolakis et al., “Geant4—A simulation toolkit,” Nucl. Instrum. Methods Phys. Res., Sect. A 506, 250–303 (2003).10.1016/s0168-9002(03)01368-8
[75]	Y.-S. Tsai, “Pair production and bremsstrahlung of charged leptons,” Rev. Mod. Phys. 46, 815–851 (1974).10.1103/revmodphys.46.815
[76]
[77]	S. M. Seltzer and M. J. Berger, “Bremsstrahlung energy spectra from electrons with kinetic energy 1 keV–10 GeV incident on screened nuclei and orbital electrons of neutral atoms with Z = 1–100,” At. Data Nucl. Data Tables 35, 345–418 (1986).10.1016/0092-640x(86)90014-8
[78]	L. Kim, R. H. Pratt, S. M. Seltzer, and M. J. Berger, “Ratio of positron to electron bremsstrahlung energy loss: An approximate scaling law,” Phys. Rev. A 33, 3002–3009 (1986).10.1103/physreva.33.3002
[79]	G. M. Shore, “Superluminality and UV completion,” Nucl. Phys. B 778, 219–258 (2007).10.1016/j.nuclphysb.2007.03.034
[80]	F. Karbstein, “Photon polarization tensor in a homogeneous magnetic or electric field,” Phys. Rev. D 88, 085033 (2013).10.1103/physrevd.88.085033
[81]	V. Dinu, T. Heinzl, A. Ilderton, M. Marklund, and G. Torgrimsson, “Vacuum refractive indices and helicity flip in strong-field QED,” Phys. Rev. D 89, 125003 (2014).10.1103/physrevd.89.125003
[82]
[83]	C. Sanderson and R. Curtin, “Armadillo: A template-based C++ library for linear algebra,” J. Open Source Softw. 1, 26 (2016).10.21105/joss.00026
[84]
[85]
[86]	C. P. Ridgers, C. S. Brady, R. Duclous, J. G. Kirk, K. Bennett, T. D. Arber, A. P. L. Robinson, and A. R. Bell, “Dense electron–positron plasmas and ultraintense γ rays from laser-irradiated solids,” Phys. Rev. Lett. 108, 165006 (2012).10.1103/physrevlett.108.165006