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Toroidal polarization vortices in tightly focused beams with singularity
S.S. Stafeev 1,2, V.V. Kotlyar 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, 960 kB

DOI: 10.18287/2412-6179-CO-734

Pages: 685-690.

Full text of article: Russian language.

Abstract:
In this paper, we numerically investigated tight focusing of cylindrical vector beams of the sec-ond order using Richards-Wolf formulae. It was shown that intensity rings where the Poynting vector was equal to zero appeared not only in the focal plane but also in nearby planes. For example, a lens with numerical aperture NA=0.95 was shown to generate periodical toroidal vortices with a 0.8-mkm period along the z-axis at a distance of about 0.45 mkm from the axis. The vortices were generated pairwise, with the closest-to-focus vortex having clockwise helicity and the subsequent being anti-clockwise. The vortices were also characterized by saddle points. When focusing an optical beam passed through a narrow annular aperture, no toroidal vortices were observed.

Keywords:
tight focusing, Richards-Wolf formulae, energy backflow, toroidal vortices.

Citation:
Stafeev SS, Kotlyar VV. Toroidal polarization vortices in tightly focused beam with singularity. Computer Optics 2020; 44(5): 685-690. DOI: 10.18287/2412-6179-CO-734.

Acknowledgements:
:The work was partially funded by the Russian Science Foundation under project # 18-19-00595 (Section "Richards-Wolf formulas for narrow angular aperture"), by the Russian Foundation for Basic Research under project ## 18-07-01122, 18-07-01380, 18-29-20003 (Section "Numerical simulation for polarization vortex" and by the Russian Federation Ministry of Science and Higher Education within a state contract with the "Crystallography and Photonics" Research Center of the RAS (Section "Introduction").

References:
  1. Dorn R, Quabis S, Leuchs G. Sharper focus for a radially polarized light beam. Phys Rev Lett. 2003; 91(23): 233901.
  2. Chong CT, Sheppard C, Wang H, Shi L, Lukyanchuk B. Creation of a needle of longitudinally polarized light in vacuum using binary optics. Nat Photonics 2008; 2(8): 501-505.
  3. Yu Y, Huang H, Zhou M, Zhan Q. Engineering of multi-segmented light tunnel and flattop focus with designed axial lengths and gaps. Opt Commun 2018; 407: 398-401.
  4. Zheng C, Su S, Zang H, Ji Z, Tian Y, Chen S, Mu K, Wei L, Fan Q, Wang C, Zhu X, Xie C, Cao L, Liang E. Characterization of the focusing performance of axial line-focused spiral zone plates. Appl Opt 2018; 57(14): 3802-3807.
  5. Lin J, Chen R, Jin P, Cada M, Ma Y. Generation of longitudinally polarized optical chain by 4 π focusing system. Opt Commun 2015; 340: 69-73.
  6. Yu Y, Zhan Q. Generation of uniform three-dimensional optical chain with controllable characteristics. J Opt 2015; 17(10): 105606.
  7. Wang X, Zhu B, Dong Y, Wang S, Zhu Z, Bo F, et al. Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays. Opt Express 2017; 25(22): 26844.
  8. Chen H, Tripathi S, Toussaint KC. Demonstration of flat-top focusing under radial polarization illumination. Opt Lett 2014; 39(4): 834-837.
  9. Gao X-Z, Pan Y, Zhang G-L, Zhao M-D, Ren Z-C, Tu C-G, Li Y-N, Wang H-T. Redistributing the energy flow of tightly focused ellipticity-variant vector optical fields. Photonics Res 2017; 5(6): 640-648.
  10. Man Z, Bai Z, Zhang S, Li X, Li J, Ge X, Zhang Y, Fu S. Redistributing the energy flow of a tightly focused radially polarized optical field by designing phase masks. Opt Express 2018; 26(18): 23935.
  11. Man Z, Li X, Zhang S, Bai Z, Lyu Y, Li J, Ge X, Sun Y, Fu S. Manipulation of the transverse energy flow of azimuthally polarized beam in tight focusing system. Opt Commun 2019; 431: 174-180.
  12. Jiao X, Liu S, Wang Q, Gan X, Li P, Zhao J. Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles. Opt Lett 2012; 37(6): 1041-1043.
  13. Pan Y, Gao X-Z, Zhang G-L, Li Y, Tu C, Wang H-T. Spin angular momentum density and transverse energy flow of tightly focused kaleidoscope-structured vector optical fields. APL Photonics 2019; 4(9): 096102.
  14. Wu G, Wang F, Cai Y. Generation and self-healing of a radially polarized Bessel-Gauss beam. Phys Rev A 2014; 89: 043807.
  15. Stafeev SS, Kotlyar VV, Nalimov AG, Kozlova ES. The non-vortex inverse propagation of energy in a tightly focused high-order cylindrical vector beam. IEEE Photonics J 2019; 11(4): 4500810. DOI: 10.1109/JPHOT.2019.2921669.
  16. 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.
  17. Rondón-Ojeda I, Soto-Eguibar F. Properties of the Poynting vector for invariant beams: Negative propagation in Weber beams. Wave Motion 2018; 78: 176-184.
  18. 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.
  19. Berry M. Wave dislocation reactions in nonparaxial Gaussian beams. J Mod Opt 1998; 45(9): 1845-1858
  20. Volyar AV, Shvedov VG, Fadeeva TA The structure of a nonparaxial Gaussian beam near the focus: II. Optical vortices. Optics and Spectroscopy 2001; 90(1): 93-100.
  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(1274): 358-379.
  22. 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.
  23. Stafeev SS, Kotlyar VV. Elongation of the area of energy backflow through the use of ring apertures. Opt Commun 2019; 450: 67-71. DOI: 10.1016/j.optcom.2019.05.057.

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