(45-2) 02 * << * >> * Russian * English * Content * All Issues

Focusing fractional-order cylindrical vector beams
S.S. Stafeev 1,2, V.D. Zaitsev 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, 1043 kB

DOI: 10.18287/2412-6179-CO-805

Pages: 172-178.

Full text of article: Russian language.

Abstract:
By numerically simulating the sharp focusing of fractional-order vector beams (0≤m≤1, with azimuthal polarization at m=1 and linear polarization at m=0), it is shown that the shape of the intensity distribution in the focal spot changes from elliptical (m=0) to round (m=0.5) and ends up being annular (m=1). Meanwhile, the distribution pattern of the longitudinal component of the Poynting vector (energy flux) in the focal spot changes in a different way: from circular (m=0) to elliptical (m=0.5) and ends up being annular (m=1). The size of the focal spot at full width at half maximum of intensity for a first-order azimuthally polarized optical vortex (m=1) and numerical aperture NA=0.95 is found to be 0.46 of the incident wavelength, whereas the diameter of the on-axis energy flux for linearly polarized light (m=0) is 0.45 of the wavelength. Therefore, the answers to the questions: when the focal spot is round and when elliptical, or when the focal spot is minimal -- when focusing an azimuthally polarized vortex beam or a linearly polarized non-vortex beam, depend on whether we are considering the intensity at the focus or the energy flow.
     In another run of numerical simulation, we investigate the effect of the deviation of the beam order from m=2 (when an energy backflow is observed at the focal spot center). The reverse energy flow is shown to occur at the focal spot center until the beam order gets equal to m=1.55.

Keywords:
cylindrical vector beam, sharp focusing, Richard-Wolf formulas, energy backflow.

Citation:
Stafeev SS, Zaitsev VD. Focusing fractional-order cylindrical vector beams. Computer Optics 2021; 45(2): 172-178. DOI: 10.18287/2412-6179-CO-805.

Acknowledgements:
This work was supported by the Russian Science Foundation (Project No. 18-19-00595) in part of “Focusing of cylindrical vector beams of order varying from zero to unity”, Russian Foundation of Basic Research (Project No. 18-29-20003 in part of “Simulation”), and Ministry of Science and Higher Education within the State assignment FSRC "Crystallography and Photonics" RAS in part of “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. Khonina SN. Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions. Opt Eng 2013; 52(9): 091711. DOI: 10.1117/1.OE.52.9.091711.
  3. Grosjean T, Gauthier I. Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths. Opt Commun 2013; 294: 333-337.
  4. Guan J, Lin J, Chen C, Ma Y, Tan J, Jin P. Transversely polarized sub-diffraction optical needle with ultra-long depth of focus. Opt Commun 2017; 404: 118-123.
  5. 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.
  6. 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: 3802-3807.
  7. 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.
  8. Yu Y, Zhan Q. Generation of uniform three-dimensional optical chain with controllable characteristics. J Opt 2015; 17(10): 105606.
  9. Xiaoqiang Z, Ruishan C, Anting W. Focusing properties of cylindrical vector vortex beams. Opt Commun 2018; 414: 10-15.
  10. Khonina SN, Ustinov AV, Volotovsky SG. Shaping of spherical light intensity based on the interference of tightly focused beams with different polarizations. Opt Laser Technol 2014; 60: 99-106. DOI: 10.1016/j.optlastec.2014.01.012.
  11. Khonina SN, Savelyev DA. High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam. J Exp Theor Phys 2013; 117: 623-630. DOI: 10.1134/S1063776113120157.
  12. Rashid M, Maragò OM, Jones PH. Focusing of high order cylindrical vector beams. J Opt A–Pure Appl Opt 2009; 11: 065204.
  13. 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: 2356-2361.
  14. Qi J, Wang W, Pan B, Deng H, Yang J, Shi B, Shan H, Zhang L, Wang H. Multiple-slit diffraction of high-polarization-order cylindrical vector beams. Proc SPIE 2017; 10339; 1033927.
  15. Wang X-L, Ding J, Ni W-J, Guo C-S, Wang H-T. Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement. Opt Lett 2007; 32: 3549-3551.
  16. Chen H, Hao J, Zhang B-F, Xu J, Ding J, Wang H-T. Generation of vector beam with space-variant distribution of both polarization and phase. Opt Lett 2011; 36; 3179-3181.
  17. Liu Y, Ke Y, Zhou J, Liu Y, Luo H, Wen S, Fan D. Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements. Sci Rep 2017; 7: 44096.
  18. Khonina SN. Vortex beams with high-order cylindrical polarization: features of focal distributions. Appl Phys B 2019; 125: 100. DOI: 10.1007/s00340-019-7212-1.
  19. 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: 5274-5276.
  20. Khonina SN, Ustinov A V., Fomchenkov SA, Porfirev AP. Formation of hybrid higher-order cylindrical vector beams using binary multi-sector phase plates. Sci Rep 2018; 8: 14320. DOI: 10.1038/s41598-018-32469-0.
  21. D’Errico A, Maffei M, Piccirillo B, de Lisio C, Cardano F, Marrucci L. Topological features of vector vortex beams perturbed with uniformly polarized light. Sci Rep 2017; 7: 40195.
  22. 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: 640-648.
  23. Khonina SN, Ustinov AV, Porfirev AP. Vector Lissajous laser beams. Opt Lett 2020; 45(15): 4112-4115. DOI: 10.1364/OL.398209.
  24. Stafeev SS, Nalimov AG, Kotlyar VV. Energy backflow in the focal spot of a cylindrical vector beam. Computer Optics 2018; 42(5): 744-750. DOI: 10.18287/2412-6179-2018-42-5-744-750.
  25. 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 Photon J 2019; 11(4): 4500810. DOI: 10.1109/JPHOT.2019.2921669.
  26. 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.
  27. Degtyarev S, Savelyev D, Khonina S, Kazanskiy N. Metasurfaces with continuous ridges for inverse energy flux generation. Opt Express 2019; 27(11): 15129-15135. DOI: 10.1364/OE.27.015129.
  28. Khonina SN, Ustinov AV. Increased reverse energy flux area when focusing a linearly polarized annular beam with binary plates. Opt Lett 2019; 44(8): 2008-2011. DOI: 10.1364/OL.44.002008.
  29. Machavariani G, Lumer Y, Moshe I, Meir A, Jackel S. Efficient extracavity generation of radially and azimuthally polarized beams. Opt Lett 2007; 32: 1468-1470.
  30. Machavariani G, Lumer Y, Moshe I, Meir A, Jackel S. Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams. Opt Commun 2008; 281: 732-738.
  31. Alferov SV, Karpeev SV, Khonina SN, Moiseev OY. Experimental study of focusing of inhomogeneously polarized beams generated using sector polarizing plates. Computer Optics 2014; 38(1): 51-56. DOI: 10.18287/0134-2452-2014-38-1-51-56.
  32. Khonina SN, Karpeev SV, Porfirev AP. Sector sandwich structure: an easy-to-manufacture way towards complex vector beam generation. Opt Express 2020; 28(19): 27628-27643. DOI: 10.1364/OE.398435.
  33. Imai R, Kanda N, Higuchi T, Zheng Z, Konishi K, Kuwata-Gonokami M. Terahertz vector beam generation using segmented nonlinear optical crystals with threefold rotational symmetry. Opt Express 2012; 20: 21896-21904.
  34. Man Z, Min C, Zhang Y, Shen Z, Yuan X-C. Arbitrary vector beams with selective polarization states patterned by tailored polarizing films. Laser Phys 2013; 23: 105001.
  35. Nalimov AG, O’Faolain L, Stafeev SS, Shanina MI, Kotlyar VV. Reflected four-zones subwavelength microoptics element for polarization conversion from linear to radial. Computer Optics 2014; 38(2): 229-236. DOI: 10.18287/0134-2452-2014-38-2-229-236.
  36. Stafeev SS, Nalimov AG, Kotlyar MV, Gibson D, Song S, O’Faolain L, Kotlyar VV. Microlens-aided focusing of linearly and azimuthally polarized laser light. Opt Express 2016; 24(26): 29800-29813. DOI: 10.1364/OE.24.029800.
  37. Kotlyar VV, Stafeev SS, Kotlyar MV, Nalimov AG, O’Faolain L. Subwavelength micropolarizer in a gold film for visible light. Appl Opt 2016; 55(19): 5025-5032. DOI: 10.1364/AO.55.005025.
  38. 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.
  39. 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.

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