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Focusing of linearly polarized optical vortex and a Hall effect
V.V. Kotlyar 1,2, A.A. Kovalev 1,2, A.G. Nalimov 1,2

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

 PDF, 1110 kB

DOI: 10.18287/2412-6179-CO-1358

Pages: 26-34.

Full text of article: Russian language.

Abstract:
Polarization of a higher-order cylindrical vector beam (CVB) is known to be locally linear. The higher the beam order, the larger number of full circles the local linear polarization vector makes around the optical axis. It is also known that the CVB with radially symmetric amplitude has zero spin angular momentum (SAM) and zero orbital angular momentum (OAM) both in the initial plane and in the focal plane (because in both Cartesian components of the vector field, the angular derivative of phase is zero). We show here that near the focal plane of the CVB (i.e. before and beyond the focus), an even number of local subwavelength areas with rotating polarization vectors are generated. In addition, in the neighboring areas, the polarization vectors are rotating in the opposite directions. Thus, the longitudinal components of the SAM vector in such neighboring areas are of different sign. After passing through the focal plane, the rotation direction of the polarization vector at each point of the beam cross-section changes to the opposite one. Such a spatial separation of the left and right rotation of the polarization vectors is a manifestation of the optical spin Hall effect.

Keywords:
topological charge, optical vortex, Hall effect.

Citation:
Kotlyar VV, Kovalev AA, Nalimov AG. Focusing of linearly polarized optical vortex and a Hall effect. Computer Optics 2024; 48(1): 26-34. DOI: 10.18287/2412-6179-CO-1358.

Acknowledgements:
This work was partly funded by the Russian Science Foundation (Project No. 23-12-00236, theoretical background) and within the government project of the Federal Scientific Research Center "Crystallography and Photonics" of the Russian Academy of Sciences (numerical simulation).

References:
  1. Poynting JH. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarized light. Proc R Soc Lond A 1909; 82(557): 560-567. DOI: 10.1098/rspa.1909.0060.
  2. Beth RA. Mechanical detection and measurement of the angular momentum of light. Phys Rev 1936; 50(2): 115-125. DOI: 10.1103/PhysRev.50.115.
  3. Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys Rev A 1992; 45(11): 8185-8189. DOI: 10.1103/PhysRevA.45.8185.
  4. 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.
  5. Guo J-X, Wang W-Y, Cheng T-Y, Lü J-Q. 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. Youngworth KS, Brown TG. Focusing of high numerical aperture cylindrical-vector beams. Opt Express 2000; 7(2): 77-87. DOI: 10.1364/oe.7.000077.
  7. Han L, Liu S, Li P, Zhang Y, Cheng H, Zhao J. Catalystlike effect of orbital angular momentum on the conversion of transverse to three-dimensional spin states within tightly focused radially polarized beams. Phys Rev A 2018; 97(5): 053802. DOI: 10.1103/PhysRevA.97.053802.
  8. 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: 39419-39427.
  9. Zhang X, Shen B, Zhu Z, Rui G, He J, Cui Y, Gu B. Understanding of transverse spin angular momentum in tightly focused linearly polarized vortex beams. Opt Express 2022; 30(4): 5121-5130. DOI: 10.1364/OE.449583.
  10. Li M, Cai Y, Yan S, Liang Y, Zhang P, Yao B. Orbit-induced localized spin angular momentum in strong focusing of optical vectorial vortex beams. Phys Rev A 2018; 97(5): 053842. DOI: 10.1103/PhysRevA.97.053842.
  11. Meng P, Man Z, Konijnenberg AP, Urbach HP. Angular momentum properties of hybrid cylindrical vector vortex beams in tightly focused optical systems. Opt Express 2019; 27(24): 35336-35348. DOI: 10.1364/OE.27.035336.
  12. Li M, Yan S, Yao B, Liang Y, Zhang P. Spinning and orbiting motion of particles in vortex beams with circular or radial polarizations. Opt Express 2016; 24(18): 20604-20612. DOI: 10.1364/OE.24.020604.
  13. Chen R, Chew K, Dai C, Zhou G. Optical spin-to-orbital angular momentum conversion in the near field of a highly nonparaxial optical field with hybrid states of polarization. Phys Rev A 2017; 96(5): 053862. DOI: 10.1103/PhysRevA.96.053862.
  14. Hu K, Chen Z, Pu J. Tight focusing properties of hybridly polarized vector beams. J Opt Soc Am A 2012; 29(6): 1099-1104. DOI: 10.1364/JOSAA.29.001099.
  15. Huang S, Zhang G, Wang Q, Wang M, Tu C, Li Y, Wang H-T. Spin-to-orbital angular momentum conversion via light intensity gradient. Optica 2021; 8(9): 1231-1236. DOI: 10.1364/OPTICA.435475.
  16. Yu P, Liu Y, Wang Z, Li Y, Gong L. Interplay between spin and orbital angular momenta in tightly focused higher-order Poincarésphere beams. Ann Phys 2020; 532(8): 2000110. DOI: 10.1002/andp.202000110.
  17. Kotlyar VV, Stafeev SS, Kovalev AA, Zaitsev VD. Spin hall effect before and after the focus of a high-order cylindrical vector beam. Appl Sci 2022; 12(23): 12218. DOI: 10.3390/app122312218.
  18. Kotlyar VV, Stafeev SS, Kozlova ES, Butt MA. High-order orbital and spin hall effects at the tight focus of laser beams. Photonics 2022; 9(12); 970. DOI: 10.3390/photonics9120970.
  19. Kotlyar VV, Stafeev SS, Zaitsev VD, Telegin AM. Poincaré beams at the tight focus: Inseparability, radial spin hall effect, and reverse energy flow. Photonics 2022; 9(12): 969. DOI: 10.3390/photonics9120969.
  20. 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.
  21. Richards B, Wolf E. Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proc R Soc A 1959; 253: 358-379. DOI: 10.1098/rspa.1959.0200.
  22. Kotlyar VV, Nalimov AG, Stafeev SS. Exploiting the circular polarization of light to obtain a spiral energy flow at the subwavelength focus. J Opt Soc Am B 2019; 36(10): 2850-2855. DOI: 10.1364/JOSAB.36.002850.
  23. Kotlyar VV, Stafeev SS, Kovalev AA. Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area. Opt Express 2019; 27(12): 16689-16702. DOI: 10.1364/OE.27.016689.
  24. Barnett SM, Allen L. Orbital angular momentum and nonparaxial light beams. Opt Commun 1994; 110: 670-678. DOI: 10.1016/0030-4018(94)90269-0.
  25. Humblet J. Sur le moment d’impulsion d’une onde électromagnétique. Physica 1943; 10: 585-603. DOI: 10.1016/s0031-8914(43)90626-3.
  26. Kotlyar VV, Kovalev AA, Porfirev AP. Radial dependence of the angular momentum density of a paraxial optical vortex. Phys Rev A 2018; 97(5): 053833. DOI: 10.1103/PhysRevA.97.053833.

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