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An orbital energy flow and a spin flow at the tight focus
S.S. Stafeev 1,2

IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151,
Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34

 PDF, 861 kB

DOI: 10.18287/2412-6179-CO-867

Pages: 520-524.

Full text of article: Russian language.

We have shown that a reverse energy flow (negative projection of the Poynting vector onto the optical axis) at the sharp focus of an optical vortex with topological charge 2 and left-hand circular polarization arises because the axial spin flow has a negative projection onto the optical axis and is greater in magnitude than positive projection onto the optical axis of the orbital energy flow (canonical energy flow). Also, using the Richards-Wolf formulas, it is shown that when focusing a left-handed circularly polarized light, in the region of the on-axis reverse energy flow, the light is right-handed circularly polarized.

orbital energy flow, spin flow, tight focusing, energy backflow, optical vortex.

Stafeev SS. An orbital energy flow and a spin flow at the tight focus. Computer Optics 2021; 45(4): 520-524. DOI: 10.18287/2412-6179-CO-867.

This work was supported by the Russian Science Foundation (grant 18-19-00595) in terms of modeling, the Russian Foundation for Basic Research (grant 18-29-20003) in the theoretical part, and the Ministry of Science and Higher Education of the Russian Federation in the framework of works on the State task of the Federal Research Center "Crystallography and Photonics" RAS.


  1. Ignatowsky VS. Diffraction by a lens having arbitrary opening. Trans Opt Inst Petrogr 1920; 1: 4.
  2. Richards B, Wolf E. Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proc Math Phys Eng Sci 1959; 253(1274): 358-379.
  3. Karman GP, Beijersbergen MW, van Duijl A, Woerdman JP. Creation and annihilation of phase singularities in a focal field. Opt Lett 1997; 22(19): 1503-1505.
  4. Berry MV. Wave dislocation reactions in non-paraxial gaussian beams. J Mod Opt 1998; 45(9): 1845-1858.
  5. Vasnetsov MV, Gorshkov VN, Marienko IG, Soskin MS. Wavefront motion in the vicinity of a phase dislocation: “optical vortex.” Opt Spectrosc 2000; 88(2): 260-265.
  6. Volyar AV. Nonparaxial gaussian beams: 1. Vector fields. Tech Phys Lett 2000; 26(7): 573-575.
  7. Volyar AV, Shvedov VG, Fadeeva TA. Structure of a nonparaxial gaussian beam near the focus: III. Stability, eigenmodes, and vortices. Opt Spectrosc 2001; 91(2): 235-245.
  8. Novitsky AV, Novitsky DV. Negative propagation of vector Bessel beams. J Opt Soc Am A 2007; 24(9): 2844-2849.
  9. Salem MA, Bağcı H. Energy flow characteristics of vector X-Waves. Opt Express 2011; 19(9): 8526-8532.
  10. Vaveliuk P, Martinez-Matos O. Negative propagation effect in nonparaxial Airy beams. Opt Express 2012; 20(24): 26913-26921.
  11. 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.
  12. Kotlyar VV, Kovalev AA, Nalimov AG. Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy. Opt Lett 2018; 43(12): 2921-2924. DOI: 10.1364/OL.43.002921.
  13. Kotlyar VV, Stafeev SS, Nalimov AG. Energy backflow in the focus of a light beam with phase or polarization singularity. Phys Rev A 2019; 99(3): 033840. DOI: 10.1103/PhysRevA.99.033840.
  14. Kotlyar VV, Stafeev SS, Nalimov AG, Kovalev AA, Porfirev AP. Mechanism of formation of an inverse energy flow in a sharp focus. Phys Rev A 2020; 101(3): 033811. DOI: 10.1103/PhysRevA.101.033811.
  15. Bekshaev AY, Soskin MS. Transverse energy flows in vectorial fields of paraxial beams with singularities. Opt Commun 2007; 271(2): 332-348.
  16. Bekshaev AY. Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows. J Opt 2013; 15(4): 044004.
  17. Bliokh KY, Alonso MA, Ostrovskaya EA, Aiello A. Angular momenta and spin-orbit interaction of nonparaxial light in free space. Phys Rev A 2010; 82(6): 063825.
  18. Berry MV. Optical currents. J Opt A-Pure Appl Opt 2009; 11(9): 094001.
  19. Bliokh KY, Bekshaev AY, Nori F. Extraordinary momentum and spin in evanescent waves. Nat Commun 2014; 5(1): 3300.
  20. Nalimov AG,  Kozlova ES. Inversion of the longitudinal component of spin angular momentum in the focus of a left-handed circularly polarized beam. Computer Optics 2020; 44(5): 699-706. DOI: 10.18287/2412-6179-CO-761.

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