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High-order harmonic generation with orbital angular momentum beams: Beyond-dipole corrections

Esra Ilke Albar1,*, Valeriia P. Kosheleva1, Heiko Appel1, Angel Rubio1,2, and Franco P. Bonafé1,†

  • *Contact author: esra-ilke.albar@mpsd.mpg.de
  • †Contact author: franco.bonafe@mpsd.mpg.de
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Phys. Rev. A 112, 063109 – Published 10 December, 2025

DOI: https://doi.org/10.1103/c1yg-2svv

Abstract

We study high harmonic generation with vortex beams beyond the dipole approximation. To do so, we employ the full minimal coupling approach to account for multipolar coupling without truncation and describe the full spatio-temporal properties of the electromagnetic field. This allows us to investigate the beyond-dipole deviations in electron trajectories and the harmonic spectra, where the influence of the orbital angular momentum contains both magnetic and quadrupolar effects. We show that when the system is driven by plane-wave light, nonlinear dipole oscillations occur along both the polarization and propagation directions, consistent with previous findings. In contrast, vortex beams also generate nonlinear dipole oscillations in the direction perpendicular to both the polarization and propagation of the incoming field. We identify the effects of the resulting symmetry breaking due to beyond-dipole corrections, which are especially visible in even harmonics.

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      References (83)

      1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
      2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
      3. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, Opt. Express 12, 3548 (2004).
      4. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
      5. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 94, 231124 (2009).
      6. E. Prinz, M. Hartelt, G. Spektor, M. Orenstein, and M. Aeschlimann, ACS Photon. 10, 340 (2023).
      7. R. M. Kerber, J. M. Fitzgerald, X. Xiao, S. S. Oh, S. A. Maier, V. Giannini, and D. E. Reiter, New J. Phys. 20, 095005 (2018).
      8. E. I. Albar, F. P. Bonafé, V. P. Kosheleva, S. T. Ohlmann, H. Appel, and A. Rubio, Sci. Rep. 13, 14748 (2023).
      9. R. Chen, H. Zhou, M. Moretti, X. Wang, and J. Li, IEEE Communic. Surveys Tutor. 22, 840 (2020).
      10. X. Fang, H. Ren, K. Li, H. Luan, Y. Hua, Q. Zhang, X. Chen, and M. Gu, Adv. Opt. Photon. 13, 772 (2021).
      11. D. A. Grier, Nature (London) 424, 810 (2003).
      12. C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, Nat. Commun. 4, 2891 (2013).
      13. O. S. Magaña-Loaiza and R. W. Boyd, Rep. Prog. Phys. 82, 124401 (2019).
      14. D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, Phys. Rev. Appl. 11, 064058 (2019).
      15. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
      16. C. Maurer, A. Jesacher, S. Bernet, and M. Ritsch-Marte, Laser Photon. Rev. 5, 81 (2010).
      17. M. Drechsler, S. Wolf, C. T. Schmiegelow, and F. Schmidt-Kaler, Phys. Rev. Lett. 127, 143602 (2021).
      18. M. Solyanik-Gorgone, A. Afanasev, C. E. Carlson, C. T. Schmiegelow, and F. Schmidt-Kaler, J. Opt. Soc. Am. B 36, 565 (2019).
      19. C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G. Poschinger, and F. Schmidt-Kaler, Nat. Commun. 7, 12998 (2016).
      20. J. I. Fuks, G. F. Q. Rosen, H. Appel, and P. I. Tamborenea, Phys. Rev. B 107, L081111 (2023).
      21. A. Afanasev, Carl E. Carlson, and A. Mukherjee, Phys. Rev. A 88, 033841 (2013).
      22. A. Afanasev, C. E. Carlson, C. T. Schmiegelow, J. Schulz, F. Schmidt-Kaler, and M. Solyanik, New J. Phys. 20, 023032 (2018).
      23. G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, Phys. Rev. Lett. 119, 253203 (2017).
      24. J. L. Bégin, A. Jain, A. Parks, F. Hufnagel, P. Corkum, E. Karimi, T. Brabec, and R. Bhardwaj, Nat. Photon. 17, 82 (2023).
      25. A. A. Peshkov, Y. M. Bidasyuk, R. Lange, N. Huntemann, E. Peik, and A. Surzhykov, Phys. Rev. A 107, 023106 (2023).
      26. R. Lange, N. Huntemann, A. A. Peshkov, A. Surzhykov, and E. Peik, Phys. Rev. Lett. 129, 253901 (2022).
      27. A. A. Peshkov, E. Jordan, M. Kromrey, K. K. Mehta, T. E. Mehlstäubler, and A. Surzhykov, Ann. Phys. 535, 2300204 (2023),.
      28. V. P. Kosheleva, V. A. Zaytsev, R. A. Müller, A. Surzhykov, and S. Fritzsche, Phys. Rev. A 102, 063115 (2020).
      29. A. Picón, A. Benseny, J. Mompart, J. R. D. V. Aldana, L. Plaja, G. F. Calvo, and L. Roso, New J. Phys. 12, 083053 (2010).
      30. S. Giri, M. Ivanov, and G. Dixit, Phys. Rev. A 101, 033412 (2020).
      31. Y. Chen, Y. Zhou, M. Li, K. Liu, Marcelo F. Ciappina, and P. Lu, Phys. Rev. A 107, 033112 (2023).
      32. M. D. Kiselev, E. V. Gryzlova, and A. N. Grum-Grzhimailo, Phys. Rev. A 108, 023117 (2023).
      33. M. D. Kiselev, E. V. Gryzlova, and A. N. Grum-Grzhimailo, Phys. Rev. A 109, 023108 (2024).
      34. T. Kaneyasu, Y. Hikosaka, M. Fujimoto, T. Konomi, M. Katoh, H. Iwayama, and E. Shigemasa, Phys. Rev. A 95, 023413 (2017).
      35. G. D. Ninno, J. Wätzel, P. R. Ribič, E. Allaria, M. Coreno, M. B. Danailov, C. David, A. Demidovich, M. D. Fraia, L. Giannessi, K. Hansen, Š. Krušič, M. Manfredda, M. Meyer, A. Mihelič, N. Mirian, O. Plekan, B. Ressel, B. Rösner, A. Simoncig , Nat. Photon. 14, 554 (2020).
      36. A. A. Peshkov, A. V. Volotka, A. Surzhykov, and S. Fritzsche, Phys. Rev. A 97, 023802 (2018).
      37. A. A. Peshkov, S. Fritzsche, and A. Surzhykov, Phys. Scr. 94, 105402 (2019).
      38. V. G. Serbo, A. Surzhykov, and A. Volotka, Ann. Phys. 534, 2100199 (2022),.
      39. K. A. Forbes and A. Salam, Phys. Rev. A 100, 053413 (2019).
      40. C. Hernández-García, J. Vieira, J. T. Mendonça, L. Rego, J. San Román, L. Plaja, P. R. Ribic, D. Gauthier, and A. Picón, Photonics 4, 28 (2017).
      41. Y. Toda, S. Honda, and R. Morita, Opt. Express 18, 17796 (2010).
      42. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, Nat. Phys. 8, 743 (2012).
      43. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, Phys. Rev. Lett. 113, 153901 (2014).
      44. D. Gauthier, P. R. Ribic, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. F. Dimauro, G. Dovillaire, F. Frassetto, R. Géneaux, P. Miotti, L. Poletto, B. Ressel, C. Spezzani, M. Stupar, T. Ruchon, and G. D. Ninno, Nat. Commun. 8, 14971 (2017).
      45. A. K. Pandey, A. de las Heras, T. Larrieu, J. San Román, J. Serrano, L. Plaja, E. Baynard, M. Pittman, G. Dovillaire, S. Kazamias, C. Hernández-García, and O. Guilbaud, ACS Photon. 9, 944 (2022).
      46. C. Hernández-García, A. Picón, J. San Román, and L. Plaja, Phys. Rev. Lett. 111, 083602 (2013).
      47. R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, Nat. Commun. 7, 12583 (2016).
      48. F. Kong, C. Zhang, F. Bouchard, Z. Li, G. G. Brown, D. H. Ko, T. J. Hammond, L. Arissian, R. W. Boyd, E. Karimi, and P. B. Corkum, Nat. Commun. 8, 14970 (2017).
      49. F. Sanson, A. K. Pandey, F. Harms, G. Dovillaire, E. Baynard, J. Demailly, O. Guilbaud, B. Lucas, O. Neveu, M. Pittman, D. Ros, M. Richardson, E. Johnson, W. Li, P. Balcou, and S. Kazamias, Opt. Lett. 43, 2780 (2018).
      50. L. Rego, N. J. Brooks, Q. L. D. Nguyen, J. S. Román, I. Binnie, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, Sci. Adv. 8, eabj7380 (2022).
      51. L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. S. Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García, Science 364, eaaw9486 (2019).
      52. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
      53. A. Ludwig, J. Maurer, B. W. Mayer, C. R. Phillips, L. Gallmann, and U. Keller, Phys. Rev. Lett. 113, 243001 (2014).
      54. J. Maurer and U. Keller, J. Phys. B: At. Mol. Opt. Phys. 54, 094001 (2021).
      55. M. W. Walser, C. H. Keitel, A. Scrinzi, and T. Brabec, Phys. Rev. Lett. 85, 5082 (2000).
      56. M. Førre, J. P. Hansen, L. Kocbach, S. Selstø, and L. B. Madsen, Phys. Rev. Lett. 97, 043601 (2006).
      57. L. B. Madsen, Phys. Rev. A 106, 043118 (2022).
      58. M. C. Suster, J. Derlikiewicz, K. Krajewska, F. C. Vélez, and J. Z. Kamiński, Phys. Rev. A 107, 053112 (2023).
      59. S. V. B. Jensen, M. M. Lund, and L. B. Madsen, Phys. Rev. A 101, 043408 (2020).
      60. S. V. B. Jensen and L. B. Madsen, Phys. Rev. A 105, L021101 (2022).
      61. L. Rego, J. S. Román, A. Picón, L. Plaja, and C. Hernández-García, Phys. Rev. Lett. 117, 163202 (2016).
      62. N. Tancogne-Dejean, M. J. T. Oliveira, X. Andrade, H. Appel, C. H. Borca, G. Le Breton, F. Buchholz, A. Castro, S. Corni, A. A. Correa, U. De Giovannini, A. Delgado, F. G. Eich, J. Flick, G. Gil, A. Gomez, N. Helbig, H. Hübener, R. Jestädt, J. Jornet-Somoza et al., J. Chem. Phys. 152, 124119 (2020).
      63. A. Castro, A. Rubio, and E. K. U. Gross, Eur. Phys. J. B 88, 191 (2015).
      64. N. Tancogne-Dejean and A. Rubio, Sci. Adv. 4, eaao5207 (2018).
      65. O. Neufeld, Z. Nourbakhsh, N. Tancogne-Dejean, and A. Rubio, J. Chem. Theory Comput. 18, 4117 (2022).
      66. U. De Giovannini, A. H. Larsen, and A. Rubio, Eur. Phys. J. B 88, 56 (2015).
      67. A. Gómez Pueyo, M. A. L. Marques, A. Rubio, and A. Castro, J. Chem. Theory Comput. 14, 3040 (2018).
      68. J. Wätzel and J. Berakdar, Phys. Rev. A 101, 043409 (2020).
      69. F. P. Bonafé, E. I. Albar, S. T. Ohlmann, V. P. Kosheleva, C. M. Bustamante, F. Troisi, A. Rubio, and H. Appel, Phys. Rev. B 111, 085114 (2025).
      70. R. P. Schmidt, S. Ramakrishna, A. A. Peshkov, N. Huntemann, E. Peik, S. Fritzsche, and A. Surzhykov, Phys. Rev. A 109, 033103 (2024).
      71. S. A.-L. Schulz, A. A. Peshkov, R. A. Müller, R. Lange, N. Huntemann, C. Tamm, E. Peik, and A. Surzhykov, Phys. Rev. A 102, 012812 (2020).
      72. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific Press, Singapore, 1988).
      73. G. F. Quinteiro, C. T. Schmiegelow, D. E. Reiter, and T. Kuhn, Phys. Rev. A 99, 023845 (2019).
      74. J. D. Jackson, Electrodynamics, Classical, In digital Encyclopedia of Applied Physics (Wiley-VCH Verlag GmbH & Co. KGaA, New York, 2003).
      75. A. Gorlach, O. Neufeld, N. Rivera, O. Cohen, and I. Kaminer, Nat. Commun. 11, 4598 (2020).
      76. See Supplemental Material at http://link.aps.org/supplemental/10.1103/c1yg-2svv for the pulse description in position space, the harmonic spectra in terms of magnetic-dipole emission and directional components, the dipole-moment dynamics, and the field-gradient and angular-distribution dependence on the OAM number m.
      77. H. R. Reiss, Phys. Rev. Lett. 101, 043002 (2008).
      78. E. I. Albar, Animated trajectories for HHG with OAM light beyond dipole, Zenodo (2025), https://zenodo.org/record/14930614.
      79. E. S. Sarachik and G. T. Schappert, Phys. Rev. D 1, 2738 (1970).
      80. H. R. Reiss, J. Phys. B: At. Mol. Opt. Phys. 47, 204006 (2014).
      81. J. Wätzel and J. Berakdar, Opt. Express 28, 19469 (2020).
      82. A. de las Heras, A. K. Pandey, J. S. Román, J. Serrano, E. Baynard, G. Dovillaire, M. Pittman, C. G. Durfee, L. Plaja, S. Kazamias, O. Guilbaud, and C. Hernández-García, Optica 9, 71 (2022).
      83. E. I. Albar, Raw Data and Input Files for “High Harmonic Generation with Orbital Angular Momentum Beams: Beyond-dipole Corrections” (1.0) [Data set], Zenodo (2025), https://doi.org/10.5281/zenodo.15224430.

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      Phys. Rev. A 112, 063109– Published 10 December, 2025
      • Received 17 April 2025
      • Revised 2 September 2025
      • Accepted 13 November 2025
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