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Focusing a second-order cylindrical vector beam with a gradient index Mikaelian lens

S.S. Stafeev 1,2, E.S. Kozlova 1,2, A.G. Nalimov 1,2

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

 PDF, 603 kB

DOI: 10.18287/2412-6179-CO-633

Pages: 29-33.

Full text of article: Russian language.

In this paper, we numerically simulate the focusing of a second-order cylindrical vector beam with a gradient index Mikaelian lens. It is shown that the lens forms a region of the reverse energy flow near its output surface. If the lens has an on-axis micropit, the region of the direct energy flow can be confined within the lens material, whereas that of the reverse energy flow is put out in free space.

Poynting vector, energy backflow, gradient index lens, cylindrical vector beam, scattering force.

Stafeev SS, Kozlova ES, Nalimov AG. Focusing a second-order cylindrical vector beam with a gradient index Mikaelian lens. Computer Optics 2020; 44(1): 29-33. DOI: 10.18287/2412-6179-CO-633.

The work was partly funded by the Russian Science Foundation under grant # 18-07-01122 ("Gradient index lens"), the Russian Foundation for Basic Research under grant # 18-19-00595 ("Gradient index lens with metallic layer"), and grant # 18-07-01380 (" Gradient index lens with a micropit"), and the RF Ministry of Science and Higher Education within a state contract with the "Crystallography and Photonics" Research Center of the RAS under agreement ("Introduction").


  1. Grosjean T, Gauthier I. Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths. Opt Commun 2013; 294: 333-337.
  2. Wu Z, Zhang K, Zhang S, Jin Q, Wen Z, Wang L, Dai L, Zhang Z, Chen H, Liang G, Liu Y, Chen G. Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams. Opt Express 2018; 26(13):16585.
  3. 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.
  4. 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.
  5. Zheng C, Su S, Zang H, et al. Characterization of the focusing performance of axial line-focused spiral zone plates. Appl Opt 2018; 57(14): 3802-3807.
  6. 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.
  7. Yu Y, Zhan Q. Generation of uniform three-dimensional optical chain with controllable characteristics. J Opt 2015; 17(10): 105606.
  8. 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.
  9. 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.
  10. 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. 10.1103/PhysRevA.99.033840.
  11. 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.
  12. Novotny L, Hecht B. Principles of nano-optics. Cambridge: Cambridge University Press; 2006.
  13. Sukhov S, Dogariu A. On the concept of “tractor beams.” Opt Lett 2010; 35(22): 3847–3849.
  14. Mikaelian AL. Application of stratified medium for waves focusing. Doklady Akademii Nauk SSSR 1951; 81: 569-571.
  15. Rivas-Moscoso JM, Nieto D, Gómez-Reino C, Fernández-Pousa CR. Focusing of light by zone plates in Selfoc gradient-index lenses. Opt Lett 2003; 28(22): 2180-2182.
  16. Hewak DW, Lit JWY. Solution deposited optical waveguide lens. Appl Opt 1989; 28(19): 4190-4198.
  17. Zentgraf T, Liu Y, Mikkelsen MH, Valentine J, Zhang X. Plasmonic Luneburg and Eaton lenses. Nat Nanotechnol 2011; 6(3): 151-155.
  18. Born M, Wolf E. Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. 6th (corrected) ed. Elsevier; 2013.
  19. Fathollahi Khalkhali T, Alipour-Beyraghi M, Lalenejad M, Bananej A. Polarization-independent and super broadband flat lens composed of graded index annular photonic crystals. Opt Commun 2019; 435: 202-211.
  20. Gaufillet F, Akmansoy É. Design of flat graded index lenses using dielectric graded photonic crystals. Opt Mater 2015; 47: 555-560.
  21. Gilarlue MM, Badri SH, Rasooli Saghai H, Nourinia J, Ghobadi C. Photonic crystal waveguide intersection design based on Maxwell’s fish-eye lens. Photon Nanostr 2018; 31: 154-159.
  22. Xia F, Li S, Zhang K, Jiao L, Kong W, Dong L, Yun M. Negative Luneburg lens based on the graded annular photonic crystals. Physica B 2018; 545: 233-236.
  23. Lin SCS, Huang TJ, Sun JH, Wu TT. Gradient-index phononic crystals. Phys Rev B 2009; 79(9): 094302.
  24. Zhu Y, Yuan W, Sun H, Yu Y. Broadband ultra-deep sub-diffraction-limit optical focusing by metallic graded-index (MGRIN) lenses. Nanomaterials 2017; 7(8): 221.
  25. Gilarlue MM, Nourinia J, Ghobadi C, Badri SH, Rasooli Saghai H. Multilayered Maxwell’s fisheye lens as waveguide crossing. Opt Commun 2019; 435: 385-393.
  26. Badri SH, Gilarlue MM. Maxwell’s fisheye lens as efficient power coupler between dissimilar photonic crystal waveguides. Optik 2019; 185: 566-570.
  27. Behera S, Kim K. Design and studies on gradient index metasurfaces for broadband polarization-independent, subwavelength, and dichroic focusing. Appl Opt 2019; 58(18): 5128-5135.
  28. Kotlyar VV, Stafeev SS, Nalimov AG. Tight focusing of laser light by microoptics. Samara: Novaya Technika Publisher; 2018.
  29. Zhang XA, Bagal A, Dandley EC, Zhao J, Oldham CJ, Wu BI, Parsons GN, Chang CH. Ordered 3D thin-shell nanolattice materials with near-unity refractive indices. Adv Funct Mater 2015; 25(42): 6644-6649.
  30. Kwon DH, Werner DH. Low-index metamaterial designs in the visible spectrum. Opt Express 2007; 15(15): 9267-9272.
  31. Kotlyar VV, Nalimov AG, Stafeev SS, O’Faolain L. Single metalens for generating polarization and phase singularities leading to a reverse flow of energy. J Opt 2019; 21(5): 055004.
  32. Novitsky AV, Novitsky DV. Negative propagation of vector Bessel beams. J Opt Soc Am A 2007; 24(9): 2844-2849.


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