Sharp focusing of a light field with polarization and phase singularities of an arbitrary order
Kotlyar V.V., Stafeev S.S., Kovalev A.A.


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


Using the Richards-Wolf formalism, we obtain general expressions for all components of the electric and magnetic strength vectors near the sharp focus of an optical vortex with the topological charge m and nth-order azimuthal polarization. From these equations, simple consequences are derived for different values of m and n. If m=n>1, there is a non-zero intensity on the optical axis, like the one observed when focusing a vortex-free circularly polarized light field. If n=m+2, there is a reverse flux of light energy near the optical axis in the focal plane. The derived expressions can be used both for simulating the sharp focusing of optical fields with the double singularity (phase and polarization) and for a theoretical analysis of focal distributions of the intensity and the Poynting vector, allowing one to reveal the presence of rotational symmetry or the on-axis reverse energy flux, as well as the focal spot shape (a circle or a doughnut).

sharp focusing, Richards-Wolf formulae, optical vortex, topological charge, phase singularity, polarization singularity, Poynting vector, reverse flux of energy, focal spot symmetry

Kotlyar VV, Stafeev SS, Kovalev AA. Sharp focusing of a light field with polarization and phase singularities of an arbitrary order. Computer Optics 2019; 43(3): 337-346. DOI: 10.18287/2412-6179-2019-43-3-337-346.


  1. Richards B, Wolf E. Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 1959; 253(1274): 358-379. DOI: 10.1098/rspa.1959.0200.
  2. 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.
  3. Zhan Q, Leger JR. Focus shaping using cylindrical vector beams. Opt Express 2002; 10(7): 324-331. DOI: 10.1364/OE.10.000324.
  4. Zhan Q. Cylindrical vector beams: from mathematical concepts to applications. Adv Opt Photon 2009; 1(1): 1-57. DOI: 10.1364/AOP.1.000001.
  5. Chen B, Pu J. Tight focusing of elliptically polarized vortex beams. Appl Opt 2009; 48(7): 1288-1294. DOI: 10.1364/AO.48.001288.
  6. Rashid M, Marago OM, Jones PH. Focusing of high order cylindrical vector beams. J Opt A: Pure Appl Opt 2009; 11(6): 065204. DOI: 10.1088/1464-4258/11/6/065204.
  7. Milione G, Sztul HI, Nolan DA, Alfano RR. Higher-order Poincaré sphere, Stokes parameters, and angular momentum of light. Phys Rev Lett 2011; 107(5): 053601. DOI: 10.1103/PhysRevLett.107.053601.
  8. Holleczek A, Aiello A, Gabriel C, Marquardt C, Leuchs G. Classical and quantum properties of cylindrically polarized states of light. Opt Express 2011; 19(10): 9714-9736. DOI: 10.1364/OE.19.009714.
  9. Chen S, Zhou X, Liu Y, Ling X, Luo H, Wen S. Generation of arbitrary cylindrical vector beams on the higher order Poincaré sphere. Opt Lett 2014; 39(18): 5274-5276. DOI: 10.1364/OL.39.005274.
  10. Wang T, Kuang C, Hao X, Liu X. Focusing properties of cylindrical vector vortex beams with high numerical aperture objective. Optik 2013; 124(21): 4762-4765. DOI: 10.1016/j.ijleo.2013.01.070.
  11. Gong L, Ren Y, Liu W, Wang M, Zhong M, Wang Z. Generation of cylindrical polarized vector vortex beams with digital micromirror device. J Appl Phys 2014; 116: 183105. DOI: 10.1063/1.4901574.
  12. Zhang X, Chen R, Wang A. Focusing properties of cylindrical vector vortex beams. Opt Commun 2018; 414: 10-15. DOI: 10.1016/j.optcom.2017.12.076.
  13. Han Y, Chen L, Liu YG, Wang Z, Zhang H, Yang K, Chou KC. Orbital angular momentum transition of light using a cylindrical vector beam. Opt Lett 2018; 43(9): 2146-2149. DOI: 10.1364/OL.43.002146.
  14. Li Y, Zhu Z, Wang X, Gong L, Wang M, Nie S. Propagation evolution of an off-axis high-order cylindrical vector beam. J Opt Soc Am A 2014; 31(11): 2356-2361. DOI: 10.1364/JOSAA.31.002356.
  15. Matsusaka S, Kozawa Y, Sato S. Micro-hole drilling by tightly focused vector beams. Opt Lett 2018; 43(7): 1542-1545. DOI: 10.1364/OL.43.001542.
  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. Kotlyar VV, Nalimov AG, Kovalev AA. Helical reverse flux of light of a focused optical vortex. J Opt 2018; 20(9): 095603. DOI: 10.1088/2040-8986/aad606.
  18. Stafeev SS, Nalimov AG, Kotlyar VV. Energy backflow in a focal spot of the cylindrical vector beam. Computer Optics 2018; 42(5): 744-750. DOI: 10.18287/2412-6179-2018-42-5-744-750.
  19. Pal SK, Ruchi, Senthilkumaran P. C-point and V-point singularity lattice formation and index sign conversion methods. Opt Commun 2017; 393: 156-168. DOI: 10.1016/j.optcom.2017.02.048.
  20. Ruchi, Pal S, Senthilkumaran P. Generation of V-point polarization singularity lattices. Opt Express 2017; 25: 19326-19331. DOI: 10.1364/OE.25.019326.
  21. Stafeev SS, Kotlyar VV. Tight focusing of a quasi-cylindrical optical vortex. Opt Commun 2017; 403: 277-282. DOI: 10.1016/j.optcom.2017.07.054.

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