Reverse flux of energy of a nonparaxial optical vortex in the near field
Kotlyar V.V., Kovalev A.A.


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


In this work, using an expansion in terms of plane waves, we obtain all six components of the electric and magnetic field vectors for an elliptically polarized optical vortex with an arbitrary integer topological charge. We also obtain expressions for the distributions of the field intensity and the Poynting vector components in the initial plane of this optical vortex. For the particular case of a narrow angular spectrum of plane waves (Bessel beam) and for circular polarization, it is shown that in the presence of inhomogeneous evanescent waves in the initial light field, a reverse flux of light energy can occur near the optical axis. It is also shown that in the initial plane the effect of an “angular tractor” occurs, when at different distances from the optical axis the transverse energy flux rotates clockwise or counterclockwise. Using the obtained general expressions, the light flux for other known exact solutions of the Maxwell equations can be analyzed.

optical vortex, angular spectrum of plane waves, evanescent waves, Poynting vector, reverse power flow, optical "tractor".

Kotlyar VV, Kovalev AA. Reverse flux of energy of a nonparaxial optical vortex in the near field. Computer Optics 2019; 43(1): 54-62. DOI: 10.18287/2412-6179-2019-43-1-54-62.


  1. Diekmann R, Wolfson DL, Spahn C, Heilemann M, Schuttpelz M, Huser T. Nanoscopy of bacterial cells immobilized by holographic optical tweezers. Nat Commun 2016; 7: 13711. DOI: 10.1038/ncomms13711.
  2. Mitri FC, Li RX, Guo LX, Ding CY. Optical tractor Bessel polarized beams. J Quant Spectr Rad Trans 2017; 187: 97-115. DOI: 10.1016/j.jqsrt.2016.09.023.
  3. Wilk SR. The pull of the tractor beam. Optics and Photonics News 2009; 20: 12-15.
  4. Pfeiffer C, Grbic A. Generating stable tractor beams with dielectric metasurfaces. Phys Rev B 2015; 91(11): 115408. DOI: 10.1103/PhysRevB.91.115408.
  5. Mitri FC. Superposition of nonparaxial vectorial complex-source spherically focused beams: Axial Pointing singularity and reverse propagation. Phys Rev A 2016; 94(2): 023801. DOI: 10.1103/PhysRevA.94.023801.
  6. Salem MA, Bağci H. Energy flow characteristics of vector X-waves. Opt Express 2011; 19(9): 8526-8532. DOI: 10.1364/OE.19.008526.
  7. Yuan GH, Zheludev N. Gigantic wavevectors and energy backflow in the focus of a superoscillatory lens. 2017 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC). DOI: 10.1109/CLEOE-EQEC.2017.8087772.
  8. Kotlyar VV, Kovalev AA Angular momentum density of a circularly polarized paraxial optical vortex. Computer Optics 2018; 42(1): 5-12. DOI: 10.18287/2412-6179-2018-42-1-5-12.
  9. Wang R, Li T, Shao X, Li X, Huang X, Shao J, Chen Y, Gong H. Subwavelength gold grating as polarizers integrated with InP-based InGaAs sensors. ACS Appl Mater Intefaces 2015; 7(26): 14471-14476. DOI: 10.1021/acsami.5b03679.
  10. Kotlyar VV, Kovalev AA, Nalimov AG, Stafeev SS. High resolution through gradient-index microoptics. Adv Opt Technol 2012; 2012: 647165. DOI: 10.1155/2012/647165.
  11. Kotlyar VV, Stafeev SS, Liu Y, O’Faolain L, Kovalev AA. Analysis of the shape of a subwavelength focal spot for the linear polarized light. Appl Opt 2013; 52(3): 330-339. DOI: 10.1364/AO.52.000330.
  12. Čižm< r T, Šiler M, Zem< nek P. An optical nanotrap array movable over a millimetre range. Appl Phys B 2006; 84(1-2): 197-203. DOI: 10.1007/s00340-006-2221-2.
  13. Schouten HF, Visser TD, Lenstra D. Optical vortices near sub-wavelength structures. J Opt B: Quant Semiclass Opt 2004; 6(5): S404-S409. DOI: 10.1088/1464-4266/6/5/031.
  14. Braunbek W, Laukien G. Einzelheiten zur Halbebenen-Beugung. Optik 1952; 9: 174-179.
  15. Novitsky AV, Novitsky DV. Negative propagation of vector Bessel beams. J Opt Soc Am A 2012; 24(9): 2844-2849. DOI: 10.1364/JOSAA.24.002844.
  16. 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.
  17. Vaveliuk P, Martinez-Matos O. Negative propagation effect in nonparaxial Airy beams. Opt Express 2012; 20(24): 26913-26921. DOI: 10.1364/OE.20.026913.
  18. Richards B, Wolf E. Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic systems. Proc R Soc A 1959; 253(1274): 358-379. DOI: 10.1098/rspa.1959.0200.
  19. Ignatovsky VS. Diffraction by a lens having arbitrary opening. Transactions of the Optical Institute of Petrograd 1919; 1: 1-36.
  20. Karman GP, Beijersbergen MW, van Duijl A, Woerdman JP. Opt Lett 1997; 22(9): 1503-1505. DOI: 10.1364/OL.22.001503.
  21. Berry MV. Wave dislocation reactions in non-paraxial Gaussian beams. J Mod Opt 1008; 45(9): 1845-1858. DOI: 10.1080/09500349808231706.
  22. Volyar AV. Nonparaxial Gausian beams. 1. Vector fields. Techn Phys Lett 2000; 26(7): 573-575. DOI: 10.1134/1.1262917.
  23. Volyar AV, Shvedov VG, Fadeeva TA. The structure of a nonparaxial Gaussian beam near the focus. II. Optical vortices. Opt Spectr 2001; 90(1): 93-100. DOI: 10.1134/1.1343551.
  24. Kotlyar VV, Nalimov AG. A vector optical vortex generated and focused using a metalens. Computer Optics 2017; 41(5): 645-654. DOI: 10.18287/2412-6179-2017-41-5-645-654.
  25. Monteiro PB, Neto PAM, Nussenzveig HM. Angular momentum of focused beams: Beyond the paraxial approximation. Phys Rev A 2009 79(3), 033830. DOI: 10.1103/PhysRevA.79.033830.
  26. Dogariu A, Sukhov S, Saenz JJ. Optically induced 'negative forces'. Nat Photon 2012; 7: 24-27. DOI: 10.1038/nphoton.2012.315.
  27. Shvedov V, Davoyan AR, Hnatovsky C, Engheta N, Krolikowski W. A long-range polarization-controlled optical tractor beam. Nat Photon 2014; 8: 846-850. DOI: 10.1038/nphoton.2014.242.
  28. Miller W. Symmetry and separation of variables. London: Addison-Wesley Publishing Company; 1977. ISBN: 978-0-521-30224-1.
  29. Kotlyar VV, Kovalev AA. Circularly polarized Hankel vortices. Opt Express 2017; 25(7): 7778-7790. DOI: 10.1364/OE.25.007778.
  30. Merlin R. Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing. Science 2007; 317(5840): 927-929. DOI: 10.1126/science.1143884.
  31. Grbic A, Jiang L, Merlin R. Near-field plates: subdiffraction focusing with patterned surfaces. Science 2008; 320(5875): 511-513. DOI: 10.1126/science.1154753.

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