(49-4) 01 * << * >> * Russian * English * Content * All Issues

Direction of energy flows at the focus of cylindrical vortex vector beams
V.V. Kotlyar 1,2, A.A. Kovalev 1,2, A.G. Nalimov 1,2, A.M. Telegin 2

1 Image Processing Systems Institute, NRC “Kurchatov Institute”,
Molodogvardeyskaya Str. 151, Samara, 443001, Russia;
2 Samara National Research University,
Moskovskoye Shosse 34, Samara, 443086, Russia

 PDF, 3123 kB

DOI: 10.18287/2412-6179-CO-1607

Pages: 531-540.

Full text of article: Russian language.

Abstract:
In this work, we show analytically using the Richards-Wolf formalism and through the numerical simulation that at the sharp focus of a circularly polarized optical vortex, three energy flows occur: the direct longitudinal, reverse longitudinal, and azimuthal transverse ones. Moreover, the rotation of energy at different distances from the optical axis is different-handed. Therefore, only a part of the initial energy of the beam intersects the focal plane along the optical axis per unit time. The same portion (all other things being equal) intersects the focal plane along the positive direction of the optical axis if an optical vortex with cylindrical polarization is focused. The difference is that in the presence of an optical vortex, the transverse energy flow at the focus rotates around the optical axis, and if an optical vortex is absent (a beam with only cylindrical polarization), then the transverse flow is, on average, zero in the focal plane. But in some areas in the focal plane the flow is directed toward the optical axis, and in some areas – away from the optical axis.

Keywords:
circular polarization, longitudinal energy flow, transverse energy flow, optical vortex, a variety of the Hall effect.

Citation:
Kotlyar VV, Kovalev AA, Nalimov AG, Telegin AM. Direction of energy flows at the focus of cylindrical vortex vector beams. Computer Optics 2025; 49(4): 531-540. DOI: 10.18287/2412-6179-CO-1607.

Acknowledgements:
The work was partly funded by the Russian Science Foundation under grant #23-12-00236 (Theoretical analysis) and within a government project of NRC “Kurchatov Institute” (Numerical simulation).

References:
  1. Richards B, Wolf E. Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proc R Soc A Math Phys Eng Sci 1959; 253(1274): 358-379. DOI: 10.1098/rspa.1959.0200.
  2. Bliokh KY, Ostrovskaya EA, Alonso MA, Rodríguez-Herrera OG, Lara D, Dainty C. Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems. Opt Express 2011; 19(27): 26132-26149. DOI: 10.1364/OE.19.026132.
  3. Li H, Ma C, Wang J, Tang M, Li X. Spin-orbit Hall effect in the tight focusing of a radially polarized vortex beam. Opt Express 2021; 29(24): 39419-39427. DOI: 10.1364/OE.443271.
  4. Arzola AV, Chvátal L, Jákl P, et al. Spin to orbital light momentum conversion visualized by particle trajectory. Sci Rep 2019; 9: 4127. DOI: 10.1038/s41598-019-40475-z.
  5. Guo JX, Wang WY, Cheng TY, Lü JQ. Interaction of spin-orbit angular momentum in the tight focusing of structured light. Front Phys 2022; 10: 1079265. DOI: 10.3389/fphy.2022.1079265.
  6. Wu Y, Yu P, Liu Y, Wang Z, Li Y, Gong L. Time-varying optical spin-orbit interactions in tight focusing of self-torqued beams. J Lightw Technol 2023; 41(7): 2252-2258. DOI: 10.1109/JLT.2022.3210953.
  7. Jera ES, Ragb HK, Kyamo MJ, Darwish OM, Buaossa N. Spin-orbital angular momentum conversion under high NA focusing of vertically polarized vortex beam. NAECON 2021 – IEEE National Aerospace and Electronics Conference, Dayton, OH, USA 2021: 150-153. DOI: 10.1109/NAECON49338.2021.9696383.
  8. Graydon O. Photonic wheel. Nature Photon 2013; 7: 672. DOI: 10.1038/nphoton.2013.229.
  9. Aiello A, Banzer P, Neugebauer M, et al. From transverse angular momentum to photonic wheels. Nat Photonics 2015; 9: 789-795. DOI: 10.1038/nphoton.2015.203.
  10. Miao W, Pang X, Liu W. Photonic wheels and their topological reaction in a strongly focused amplitude tailored beam. IEEE Photon J 2020; 12(2): 6500709. DOI: 10.1109/JPHOT.2020.2981347.
  11. Berškys J, Orlov S. Interaction of photonic wheel with cluster of nanoparticles. 2021 Conference on Lasers and Electro-Optics Europe and European Quantum Electronics Conference, OSA Technical Digest 2021: eg_p_10.
  12. Galvez EJ, Dutta I, Beach K, et al. Multitwist Möbius strips and twisted ribbons in the polarization of paraxial light beams. Sci Rep 2017; 7: 13653. DOI: 10.1038/s41598-017-13199-1.
  13. Bauer T, et al. Observation of optical polarization Möbius strips. Science 2015; 347: 964-966. DOI: 10.1126/science.1260635.
  14. Angelsky OV, Mokhun II, Bekshaev AYa, Zenkova CYu, Zheng J. Polarization singularities: Topological and dynamical aspects. Front Phys 2023; 11. 1147788. DOI: 10.3389/fphy.2023.1147788.
  15. Wan C, Zhan Q. Generation of exotic optical polarization Möbius strips. Opt Express 2019; 27(8): 11516-11524. DOI: 10.1364/OE.27.011516.
  16. Bliokh KY, Alonso MA, Sugic D, Perrin M, Nori F, Brasselet E. Polarization singularities and Möbius strips in sound and water-surface waves. Phys Fluids 2021; 33(7): 077122. DOI: 10.1063/5.0056333.
  17. Freund I. Polarization Möbius strips on elliptical paths in three-dimensional optical fields. Opt Lett 2020; 45(12): 3333-3336. DOI: 10.1364/OL.392331.
  18. Shu W, Lin C, Wu J, Chen S, Ling X, Zhou X, Luo H, Wen S. Three-dimensional spin Hall effect of light in tight focusing. Phys Rev A 2020; 101(2): 023819. DOI: 10.1103/PhysRevA.101.023819.
  19. Zhang M, Ren H, Xu O, Jiang M, Lu Y, Hu Y, Fu S, Li Z, Chen Z, Guan BO, Cao Y, Li X. Nanointerferometric discrimination of the spin-orbit Hall effect. ACS Photon 2021; 8(4): 1169-1174. DOI: 10.1021/acsphotonics.1c00087.
  20. Wang W, Zhao R, Kang Q, Wang R, Liu X, Liu T, Fan SW, Guo Z. Photonic spin Hall effect driven broadband multi-focus dielectric metalens. Appl Opt 2023; 62(30): 8159-8167. DOI: 10.1364/AO.502888.
  21. Neugebauer M, et al. Experimental demonstration of the geometric spin Hall effect of light in highly focused vector beams. 2012 Conf on Lasers and Electro-Optics (CLEO), San Jose, CA, USA 2012: 1-2. DOI: 10.1364/QELS.2012.QW1E.4.
  22. Zhao K, Zhang Z, Zang H, Du J, Lu Y, Wang P. Generation of pure longitudinal magnetization focal spot with a triplex metalens. Opt Lett 2021; 46(8): 1896-1899. DOI: 10.1364/OL.422351.
  23. Ignatyeva DO, Davies CS, Sylgacheva DA, et al. Plasmonic layer-selective all-optical switching of magnetization with nanometer resolution. Nat Commun 2019; 10(1): 4786. DOI: 10.1038/s41467-019-12699-0.
  24. Hendriks F, Rojas-Lopez RR, Koopmans B, Guimarães MHD. Electric control of optically-induced magnetization dynamics in a van der Waals ferromagnetic semiconductor. Nat Commun 2024; 15(1): 1298. DOI: 10.1038/s41467-024-45623-2.
  25. Zhou J, Ma H, Zhang Y, Zhang S, Min C, Yuan X. Energy flow inversion in an intensity-invariant focusing field. Opt Lett 2022; 47(6): 1494-1497. DOI: 10.1364/OL.449056.
  26. Wu Y, Hu X, Li Y, Chen R. Energy backflow in tightly focused fractional order vector vortex beams with binary topological charges. Photonics 2023; 10(7): 820. DOI: 10.3390/photonics10070820.
  27. Li H, Wang C, Tang M, Li X. Controlled negative energy flow in the focus of a radial polarized optical beam. Opt Express 2020; 28(13): 18607-18615. DOI: 10.1364/OE.391398.
  28. Mitri FG. Reverse propagation and negative angular momentum density flux of an optical nondiffracting nonparaxial fractional Bessel vortex beam of progressive waves. J Opt Soc Am A 2016; 33(9): 1661-1667. DOI: 10.1364/JOSAA.33.001661.
  29. Shen Y, Zhang Q, Shi P, et al. Optical skyrmions and other topological quasiparticles of light. Nat Photonics 2024; 18: 15-25. DOI: 10.1038/s41566-023-01325-7.
  30. Zeng Y, Yu Y, Shen X, Chen J, Zhan Q. Tightly focused optical skyrmions and merons formed by electric-field vectors with prescribed characteristics. Nanophotonics 2024; 13(2): 251-261. DOI: 10.1515/nanoph-2023-0741.
  31. Barnett SM, Cisowski CM, McWilliam A, Speirits FC, Ye Z, Götte JB, Franke-Arnold S. Optical skyrmions. Proc SPIE 2023; 12647: 126470A. DOI: 10.1117/12.2676688.
  32. Moh KJ, Yuan XC, Bu J, Burge RE, Gao BZ. Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams. Appl Opt 2007; 46(30): 7544-7551. DOI: 10.1364/ao.46.007544.
  33. Zhan Q. Cylindrical vector beams: from mathematical concepts to applications. Adv Opt Photon 2019; 1(1): 1-57. DOI: 10.1364/AOP.1.000001.
  34. Beckley AM, Brown TG, Alonso MA. Full Poincaré beams. Opt Express 2010; 18(10): 10777-10785. DOI: 10.1364/OE.18.010777.
  35. Naidoo D, Roux F, Dudley A, et al. Controlled generation of higher-order Poincaré sphere beams from a laser. Nat Photonics 2016; 10(5): 327-332. DOI: 10.1038/nphoton.2016.37.
  36. Kotlyar VV, Kovalev AA, Telegin AM. Angular and orbital angular momenta in the tight focus of a circularly polarized optical vortex. Photonics 2023; 10(2): 160. DOI: 10.3390/photonics10020160.
  37. Kotlyar VV, Stafeev SS, Zaitsev VD, Telegin AM, Kozlova ES. Spin-orbital transformation in a tight focus of an optical vortex with circular polarization. Appl Sci 2023; 13(14): 8361. DOI: 10.3390/app13148361.
  38. Kotlyar VV, Stafeev SS, Telegin AM. Spin angular momentum at the tight focus of a cylindrical vector beam with an imbedded optical vortex. Optik 2023; 287: 171103. DOI: 10.1016/j.ijleo.2023.171103.
  39. Kovalev AA, Kotlyar VV. Spin Hall effect of double-index cylindrical vector beams in a tight focus. Micromachines 2023; 14(2): 494. DOI: 10.3390/mi14020494.
  40. Ghosh B, Daniel A, Gorzkowski B, Bekshaev AY, Lapkiewicz R, Bliokh KY. Canonical and Poynting currents in propagation and diffraction of structured light: tutorial. J Opt Soc Am B 2024; 41(6): 1276-1289. DOI: 10.1364/JOSAB.522393.

© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: journal@computeroptics.ru ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20