- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Analytical calculation of the spectrum of nonlinear Compton scattering beyond local approximations

M. P. Malakhov1,2,3,*, Th. Benahmed4,†, E. G. Gelfer4, A. M. Fedotov1,3, O. Klimo4,5, S. Weber4, and S. G. Rykovanov2,3

  • *Contact author: mak.malahov2000@gmail.com
  • †Present address: Deutsches Elektronen–Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany.
PDF

Phys. Rev. D 114, 056015 – Published 11 September, 2026

DOI: https://doi.org/10.1103/rnzy-mrz6

Abstract

We derive compact analytical formulas for the spectrum of nonlinear Compton scattering in a finite plane-wave pulse with a smooth temporal envelope. The strong-field QED probability is reduced to finite-pulse phase integrals, which are evaluated asymptotically for multicycle pulses with a broad class of smooth envelopes. We use the uniform approximation to remove the caustic divergences that appear at the nonlinear edges of broadened harmonics. Away from the caustics, it reduces to the standard saddle-point result. The behavior near the linear edge is further improved by an envelope-corrected saddle-point approximation. The approach retains the harmonic substructure in the spectral-angular region carrying the dominant part of the emitted radiation. The locally monochromatic approximation is recovered by averaging the finite-pulse interference. Within their asymptotic domain of applicability, the resulting formulas agree with direct numerical calculations and can be used to evaluate spectra from an electron beam.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (99)

  1. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  2. 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 (2012).
  3. A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, Charged particle motion and radiation in strong electromagnetic fields, Rev. Mod. Phys. 94, 045001 (2022).
  4. W. P. Leemans, R. W. Schoenlein, P. Volfbeyn, A. H. Chin, T. E. Glover, P. Balling, M. Zolotorev, K. J. Kim, S. Chattopadhyay, and C. V. Shank, X-ray based subpicosecond electron bunch characterization using 90° Thomson scattering, Phys. Rev. Lett. 77, 4182 (1996).
  5. F. Albert, S. G. Anderson, D. J. Gibson, C. A. Hagmann, M. S. Johnson, M. Messerly, V. Semenov, M. Y. Shverdin, B. Rusnak, A. M. Tremaine, F. V. Hartemann, C. W. Siders, D. P. McNabb, and C. P. J. Barty, Characterization and applications of a tunable, laser-based, MeV-class Compton-scattering γ-ray source, Phys. Rev. ST Accel. Beams 13, 070704 (2010).
  6. S. G. Rykovanov, C. G. R. Geddes, J. L. Vay, C. B. Schroeder, E. Esarey, and W. P. Leemans, Quasi-monoenergetic femtosecond photon sources from Thomson Scattering using laser plasma accelerators and plasma channels, J. Phys. B 47, 234013 (2014).
  7. K. Khrennikov, J. Wenz, A. Buck, J. Xu, M. Heigoldt, L. Veisz, and S. Karsch, Tunable all-optical quasimonochromatic Thomson x-ray source in the nonlinear regime, Phys. Rev. Lett. 114, 195003 (2015).
  8. W. S. Graves, J. Bessuille, P. Brown, S. Carbajo, V. Dolgashev, K.-H. Hong, E. Ihloff, B. Khaykovich, H. Lin, K. Murari, E. A. Nanni, G. Resta, S. Tantawi, L. E. Zapata, F. X. Kärtner, and D. E. Moncton, Compact x-ray source based on burst-mode inverse Compton scattering at 100 kHz, Phys. Rev. ST Accel. Beams 17, 120701 (2014).
  9. K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J.-P. Goddet, R. Shah, S. Sebban, and A. Rousse, All-optical Compton gamma-ray source, Nat. Photonics 6, 308 (2012).
  10. W. Yan, C. Fruhling, G. Golovin, D. Haden, J. Luo, P. Zhang, B. Zhao, J. Zhang, C. Liu, M. Chen, S. Chen, S. Banerjee, and D. Umstadter, High-order multiphoton Thomson scattering, Nat. Photonics 11, 514 (2017).
  11. H. Tashima and T. Yamaya, Compton imaging for medical applications, Radiol. Phys. Technol. 15, 187 (2022).
  12. R. Tommasini, S. P. Hatchett, D. S. Hey, C. Iglesias, N. Izumi, J. A. Koch, O. L. Landen, A. J. MacKinnon, C. Sorce, J. A. Delettrez, V. Y. Glebov, T. C. Sangster, and C. Stoeckl, Development of Compton radiography of inertial confinement fusion implosions, Phys. Plasmas 18, 056309 (2011).
  13. S. Kulpe, M. Dierolf, B. Günther, J. Brantl, M. Busse, K. Achterhold, F. Pfeiffer, and D. Pfeiffer, Spectroscopic imaging at compact inverse Compton X-ray sources, Phys. Med. 79, 137 (2020).
  14. V. G. Nedorezov, S. G. Rykovanov, and A. B. Savel’ev, Nuclear photonics: Results and prospects, Phys. Usp. 64, 1214 (2021).
  15. E. S. Sarachik and G. T. Schappert, Classical theory of the scattering of intense laser radiation by free electrons, Phys. Rev. D 1, 2738 (1970).
  16. E. Esarey, S. K. Ride, and P. Sprangle, Nonlinear Thomson scattering of intense laser pulses from beams and plasmas, Phys. Rev. E 48, 3003 (1993).
  17. S. V. Popruzhenko and A. M. Fedotov, Dynamics and radiation of charged particles in ultra-intense laser fields, Phys. Usp. 66, 460 (2023).
  18. C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, and A. W. Weidemann, Observation of nonlinear effects in Compton scattering, Phys. Rev. Lett. 76, 3116 (1996).
  19. T. G. Blackburn, Radiation reaction in electron–beam interactions with high-intensity lasers, Rev. Mod. Plasma Phys. 4, 5 (2020).
  20. S. Bulanov, T. Z. Esirkepov, J. Koga, and T. Tajima, Interaction of electromagnetic waves with plasma in the radiation-dominated regime, Plasma Phys. Rep. 30, 196 (2004).
  21. E. Gelfer, N. Elkina, and A. Fedotov, Unexpected impact of radiation friction: Enhancing production of longitudinal plasma waves, Sci. Rep. 8, 6478 (2018).
  22. E. Gelfer, A. Fedotov, and S. Weber, Theory and simulations of radiation friction induced enhancement of laser-driven longitudinal fields, Plasma Phys. Controlled Fusion 60, 064005 (2018).
  23. E. G. Gelfer, A. M. Fedotov, and S. Weber, Radiation induced acceleration of ions in a laser irradiated transparent foil, New J. Phys. 23, 095002 (2021).
  24. A. Di Piazza, K. Hatsagortsyan, and C. H. Keitel, Quantum radiation reaction effects in multiphoton Compton scattering, Phys. Rev. Lett. 105, 220403 (2010).
  25. M. Vranic, T. Grismayer, R. A. Fonseca, and L. O. Silva, Quantum radiation reaction in head-on laser-electron beam interaction, New J. Phys. 18, 073035 (2016).
  26. J. M. Cole et al., Experimental evidence of radiation reaction in the collision of a high-intensity laser pulse with a laser-wakefield accelerated electron beam, Phys. Rev. X 8, 011020 (2018).
  27. K. Poder et al., Experimental signatures of the quantum nature of radiation reaction in the field of an ultraintense laser, Phys. Rev. X 8, 031004 (2018).
  28. E. E. Los et al., Observation of quantum effects on radiation reaction in strong fields, Nat. Commun. 17, 1157 (2026).
  29. D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D. A. Reis, and W. Ragg, Positron production in multiphoton light-by-light scattering, Phys. Rev. Lett. 79, 1626 (1997).
  30. C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D. A. Reis, W. Ragg, C. Bula, K. T. McDonald, E. J. Prebys, D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, and A. W. Weidemann, Studies of nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses, Phys. Rev. D 60, 092004 (1999).
  31. A. Mironov, E. Gelfer, and A. Fedotov, Onset of electron-seeded cascades in generic electromagnetic fields, Phys. Rev. A 104, 012221 (2021).
  32. A. Bell and J. G. Kirk, Possibility of prolific pair production with high-power lasers, Phys. Rev. Lett. 101, 200403 (2008).
  33. A. Fedotov, N. Narozhny, G. Mourou, and G. Korn, Limitations on the attainable intensity of high power lasers, Phys. Rev. Lett. 105, 080402 (2010).
  34. E. Gelfer, A. Mironov, A. Fedotov, V. Bashmakov, E. Nerush, I. Y. Kostyukov, and N. Narozhny, Optimized multibeam configuration for observation of QED cascades, Phys. Rev. A 92, 022113 (2015).
  35. A. Mercuri-Baron, A. Mironov, C. Riconda, A. Grassi, and M. Grech, Growth rate of self-sustained QED cascades induced by intense lasers, Phys. Rev. X 15, 011062 (2025).
  36. H. Hu, C. Müller, and C. H. Keitel, Complete QED theory of multiphoton trident pair production in strong laser fields, Phys. Rev. Lett. 105, 080401 (2010).
  37. F. Mackenroth and A. Di Piazza, Nonlinear trident pair production in an arbitrary plane wave: A focus on the properties of the transition amplitude, Phys. Rev. D 98, 116002 (2018).
  38. S. Tang and B. King, Pulse envelope effects in nonlinear Breit-Wheeler pair creation, Phys. Rev. D 104, 096019 (2021).
  39. A. I. Nikishov and V. I. Ritus, Quantum processes in the field of a plane electromagnetic wave and in a constant field. I, Sov. Phys. JETP 19, 529 (1964).
  40. V. I. Ritus, Quantum effects of the interaction of elementary particles with an intense electromagnetic field, J. Sov. Laser Res. 6, 497 (1985).
  41. K. Krajewska and J. Kamiński, Frequency scaling law for nonlinear Compton and Thomson scattering: Relevance of spin and polarization effects, Phys. Rev. A 90, 052117 (2014).
  42. W. H. Furry, On bound states and scattering in positron theory, Phys. Rev. 81, 115 (1951).
  43. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, Course of Theoretical Physics, Vol. 4 (Butterworth-Heinemann, London, 1982).
  44. M. Mirzaie, C. I. Hojbota, D. Y. Kim, V. B. Pathak, T. G. Pak, C. M. Kim, H. W. Lee, J. W. Yoon, S. K. Lee, Y. J. Rhee, M. Vranic, Ó. Amaro, K. Y. Kim, J. H. Sung, and C. H. Nam, All-optical nonlinear Compton scattering performed with a multi-petawatt laser, Nat. Photonics 18, 1212 (2024).
  45. O. Har-Shemesh and A. Di Piazza, Peak intensity measurement of relativistic lasers via nonlinear Thomson scattering, Opt. Lett. 37, 1352 (2012).
  46. T. Blackburn, E. Gerstmayr, S. Mangles, and M. Marklund, Model-independent inference of laser intensity, Phys. Rev. Accel. Beams 23, 064001 (2020).
  47. C. Z. He, A. Longman, J. A. Pérez-Hernández, M. de Marco, C. Salgado, G. Zeraouli, G. Gatti, L. Roso, R. Fedosejevs, and W. T. Hill, Towards an in situ, full-power gauge of the focal-volume intensity of petawatt-class lasers, Opt. Express 27, 30020 (2019).
  48. J. F. Dawson and Z. Fried, Plane-wave packets and their limitation in nonlinear Compton scattering, Phys. Rev. D 1, 3363 (1970).
  49. F. V. Hartemann and A. K. Kerman, Classical theory of nonlinear Compton scattering, Phys. Rev. Lett. 76, 624 (1996).
  50. G. A. Krafft, Spectral distributions of Thomson-scattered photons from high-intensity pulsed lasers, Phys. Rev. Lett. 92, 204802 (2004).
  51. C. A. Brau, Oscillations in the spectrum of nonlinear Thomson-backscattered radiation, Phys. Rev. ST Accel. Beams 7, 020701 (2004).
  52. M. Boca and V. Florescu, Nonlinear Compton scattering with a laser pulse, Phys. Rev. A 80, 053403 (2009).
  53. F. Mackenroth and A. Di Piazza, Nonlinear Compton scattering in ultrashort laser pulses, Phys. Rev. A 83, 032106 (2011).
  54. D. Seipt and B. Kämpfer, Nonlinear Compton scattering of ultrashort intense laser pulses, Phys. Rev. A 83, 022101 (2011).
  55. K. Krajewska and J. Z. Kamiński, Compton process in intense short laser pulses, Phys. Rev. A 85, 062102 (2012).
  56. D. Seipt and B. Kämpfer, Nonlinear Compton scattering of ultrahigh-intensity laser pulses, Laser Phys. 23, 075301 (2013).
  57. B. King, Interference effects in nonlinear Compton scattering due to pulse envelope, Phys. Rev. D 103, 036018 (2021).
  58. T. N. Wistisen, Interference effect in nonlinear Compton scattering, Phys. Rev. D 90, 125008 (2014).
  59. A. Ilderton, B. King, and S. Tang, Toward the observation of interference effects in nonlinear Compton scattering, Phys. Lett. B 804, 135410 (2020).
  60. K. Krajewska, M. Twardy, and J. Kamiński, Global phase and frequency comb structures in nonlinear Compton and Thomson scattering, Phys. Rev. A 89, 052123 (2014).
  61. Z.-d. Zhao and S. Tang, Polarization-dependent interference in nonlinear Compton scattering, Phys. Rev. D 112, 056033 (2025).
  62. I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, Spectral bandwidth reduction of Thomson scattered light by pulse chirping, Phys. Rev. ST Accel. Beams 16, 030705 (2013).
  63. D. Seipt, V. Y. Kharin, and S. G. Rykovanov, Optimizing laser pulses for narrow-band inverse Compton sources in the high-intensity regime, Phys. Rev. Lett. 122, 204802 (2019).
  64. A. Timoshenko, M. Malakhov, A. Fedotov, and S. Rykovanov, Pulse-stacking technique for nonlinear Compton scattering: Compensation of nonlinear broadening and generation of multicolor gamma sources, Phys. Rev. A 112, L061501 (2025).
  65. M. A. Valialshchikov, V. Y. Kharin, and S. G. Rykovanov, Narrow bandwidth gamma comb from nonlinear Compton scattering using the polarization gating technique, Phys. Rev. Lett. 126, 194801 (2021).
  66. D. Seipt, V. Kharin, S. Rykovanov, A. Surzhykov, and S. Fritzsche, Analytical results for nonlinear Compton scattering in short intense laser pulses, J. Plasma Phys. 82, 655820203 (2016).
  67. M. P. Malakhov and A. M. Fedotov, Calculation of spectral and angular characteristics of nonlinear Compton scattering of a laser pulse, Bull. Lebedev Phys. Inst. 52, S291 (2025).
  68. A. G. R. Thomas, Algorithm for calculating spectral intensity due to charged particles in arbitrary motion, Phys. Rev. ST Accel. Beams 13, 020702 (2010).
  69. V. Y. Kharin, D. Seipt, and S. G. Rykovanov, Temporal laser-pulse-shape effects in nonlinear Thomson scattering, Phys. Rev. A 93, 063801 (2016).
  70. A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Implementing nonlinear Compton scattering beyond the local-constant-field approximation, Phys. Rev. A 98, 012134 (2018).
  71. A. Ilderton, B. King, and D. Seipt, Extended locally constant field approximation for nonlinear Compton scattering, Phys. Rev. A 99, 042121 (2019).
  72. E. G. Gelfer, A. M. Fedotov, A. A. Mironov, and S. Weber, Nonlinear Compton scattering in time-dependent electric fields beyond the locally constant crossed field approximation, Phys. Rev. D 106, 056013 (2022).
  73. 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).
  74. A. Di Piazza, M. Tamburini, S. Meuren, and C. H. Keitel, Improved local-constant-field approximation for strong-field QED codes, Phys. Rev. A 99, 022125 (2019).
  75. V. Dinu, C. Harvey, A. Ilderton, M. Marklund, and G. Torgrimsson, Quantum radiation reaction: From interference to incoherence, Phys. Rev. Lett. 116, 044801 (2016).
  76. C. N. Harvey, A. Ilderton, and B. King, Testing numerical implementations of strong-field electrodynamics, Phys. Rev. A 91, 013822 (2015).
  77. T. Heinzl, B. King, and A. J. MacLeod, Locally monochromatic approximation to QED in intense laser fields, Phys. Rev. A 102, 063110 (2020).
  78. T. G. Blackburn, A. J. MacLeod, and B. King, From local to nonlocal: Higher fidelity simulations of photon emission in intense laser pulses, New J. Phys. 23, 085008 (2021).
  79. C. F. Nielsen, R. Holtzapple, and B. King, High-resolution modeling of nonlinear Compton scattering in focused laser pulses, Phys. Rev. D 106, 013010 (2022).
  80. T. G. Blackburn, B. King, and S. Tang, Simulations of laser-driven strong-field QED with Ptarmigan: Resolving wavelength-scale interference and γ-ray polarization, Phys. Plasmas 30, 093903 (2023).
  81. N. Larin and D. Seipt, Extended locally monochromatic approximations of strong-field QED processes, Phys. Rev. A 112, 032819 (2025).
  82. N. B. Narozhnyi and M. S. Fofanov, Photon emission by an electron in a collision with a short focused laser pulse, J. Exp. Theor. Phys. 83, 14 (1996).
  83. V. Y. Kharin, D. Seipt, and S. G. Rykovanov, Higher-dimensional caustics in nonlinear Compton scattering, Phys. Rev. Lett. 120, 044802 (2018).
  84. D. Seipt and B. Kämpfer, Asymmetries of azimuthal photon distributions in nonlinear Compton scattering in ultrashort intense laser pulses, Phys. Rev. A 88, 012127 (2013).
  85. D. M. Volkov, The solution for wave equations for a spin-charged particle moving in a classical field, Z. Phys. 94, 250 (1935).
  86. A. Ilderton, Trident pair production in strong laser pulses, Phys. Rev. Lett. 106, 020404 (2011).
  87. V. Dinu, T. Heinzl, and A. Ilderton, Infrared divergences in plane wave backgrounds, Phys. Rev. D 86, 085037 (2012).
  88. J. N. L. Connor, P. R. Curtis, and D. Farrelly, The uniform asymptotic swallowtail approximation: Practical methods for oscillating integrals with four coalescing saddle points, J. Phys. A 17, 283 (1984).
  89. C. Ferreira, J. L. López, and E. Pérez Sinusía, The asymptotic expansion of the swallowtail integral in the highly oscillatory region, Appl. Math. Comput. 339, 837 (2018).
  90. M. V. Fedoryuk, Saddle-Point Method (Nauka, Moscow, 1977) (in Russian); M. V. FedoryukAsymptotics: Integrals and Series (Nauka, Moscow, 1987) (in Russian).
  91. C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Math. Proc. Cambridge Philos. Soc. 53, 599 (1957).
  92. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, IEEE Press Series on Electromagnetic Wave Theory (Wiley-IEEE Press, Piscataway, NJ, USA, 1994), p. 924.
  93. D. B. Milošević, Application of the uniform approximation to integrals occurring in ionization by a strong elliptically polarized laser field, Phys. Rev. A 111, 053105 (2025).
  94. E. Gelfer, A. Fedotov, O. Klimo, and S. Weber, Coherent radiation of an electron bunch colliding with an intense laser pulse, Phys. Rev. Res. 6, L032013 (2024).
  95. E. G. Gelfer, A. M. Fedotov, M. P. Malakhov, O. Klimo, and S. Weber, Coherently enhanced radiation friction in laser-plasma collisions, Phys. Rev. Lett. 137, 095001 (2026).
  96. E. G. Gelfer, A. M. Fedotov, M. P. Malakhov, T. Benahmed, J. Custodio, O. Klimo, and S. Weber, Analytical theory of coherent radiation and radiation friction in laser-plasma collisions, Phys. Rev. E 114, 025209 (2026).
  97. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
  98. S. M. Wiggins, R. C. Issac, G. H. Welsh, E. Brunetti, R. P. Shanks, M. P. Anania, S. Cipiccia, G. G. Manahan, C. Aniculaesei, B. Ersfeld, M. R. Islam, R. T. L. Burgess, G. Vieux, W. A. Gillespie, A. M. MacLeod, S. B. van der Geer, M. J. de Loos, and D. A. Jaroszynski, High quality electron beams from a laser wakefield accelerator, Plasma Phys. Controlled Fusion 52, 124032 (2010).
  99. P. Winkler, M. Trunk, L. Hübner, A. Martinez de la Ossa, S. Jalas, M. Kirchen, I. Agapov, S. A. Antipov, R. Brinkmann, T. Eichner, A. Ferran Pousa, T. Hülsenbusch, G. Palmer, M. Schnepp, K. Schubert, M. Thévenet, P. A. Walker, C. Werle, W. P. Leemans, and A. R. Maier, Active energy compression of a laser-plasma electron beam, Nature (London) 640, 907 (2025).

Outline

Information

Phys. Rev. D 114, 056015– Published 11 September, 2026
  • Received 22 July 2026
  • Accepted 20 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation