(47-4) 02 * << * >> * Russian * English * Content * All Issues

Focusing a vortex beam with circular polarization: angular momentum
V.V. Kotlyar 1,2, A.A. Kovalev 1,2, A.M. Telegin 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, 744 kB

DOI: 10.18287/2412-6179-CO-1289

Pages: 524-532.

Full text of article: Russian language.

Abstract:
Based on the Richards-Wolf formalism, we obtain two different exact expressions for the angular momentum (AM) density in the focus of a vortex beam with the topological charge n and with right circular polarization. One expression for the AM density is derived as the cross product of the position vector and the Poynting vector and has a nonzero value at the focus for an arbitrary integer number n. The other expression for the AM density is deduced as a sum of the orbital angular momentum (OAM) and the spin angular momentum (SAM). We reveal that at the focus of the light field under analysis, the latter turns zero at n = –1. While both these expressions are not equal to each other at each point of space, 3D integrals thereof are equal. Thus, exact expressions are obtained for densities of AM, SAM and OAM at the focus of a vortex beam with right-hand circular polarization and the identity for the densities AM = SAM + OAM is shown to be violated. Besides, it is shown that the expressions for the strength vectors of the electric and magnetic fields near the sharp focus, obtained by adopting the Richards-Wolf formalism, are exact solutions of the Maxwell's equations. Thus, Richards–Wolf theory exactly describes the behavior of light near the sharp focus in free space.

Keywords:
vortex beam; focusing; Richards-Wolf formalism; angular momentum; orbital angular momentum; spin angular momentum; Maxwell's equations.

Citation:
Kotlyar VV, Kovalev AA, Telegin AM. Focusing a vortex beam with circular polarization: angular momentum. Computer Optics 2023; 47(4): 524-532. DOI: 10.18287/2412-6179-CO-1289.

Acknowledgements:
This work was financially supported by the Russian Science Foundation under project No.22-12-00137 (theory). The work was also performed within the State assignment of Federal Scientific Research Center "Crystallography and Photonics" of Russian Academy of Sciences (simulation).

References:
  1. Beth RA. Direct detection of the angular momentum of light. Phys Rev 1935; 48: 471. DOI: 10.1103/PhysRev.48.471.
  2. Beth RA. Mechanical detection and measurement of the angular momentum of light. Phys Rev 1936; 50: 115-127. DOI: 10.1103/PhysRev.50.115.
  3. Barnett SM. On the six components of optical angular momentum. J Opt 2011; 13: 064010. DOI: 10.1088/2040-8978/13/6/064010.
  4. Mansuripur M. Spin and orbital angular momenta of electromagnetic waves in free space. Phys Rev A 2011; 84: 033838. DOI: 10.1103/physreva.84.033838.
  5. 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: 5121-5130. DOI: 10.1364/OE.449583.
  6. Bliokh KY, Bekshaev AY, Nori F. Dual electromagnetism: helicity, spin, momentum, and angular momentum. New J Phys 2013; 15: 033026. DOI: 10.1088/1367-2630/15/3/033026.
  7. Guo J, Wang W, Cheng T, Lu J. Interaction of spin-orbit angular momentum in the tight focusing of structured light. Frontiers in Physics 2022; 10: 1079265. DOI: 10.3389/fphy.2022.1079265.
  8. Bliokh KY, Alonso MA, Ostrovskaya EA, Aiello A. Angular momenta and spin-orbit interaction of nonparaxial light in free space. Phys Rev A 2010; 82: 063825. DOI: 10.1103/physreva.82.063825.
  9. Bekshaev AY, Bliokh KY, Soskin MS. Internal flows and energy circulation in light beams. J Opt 2011; 13: 053001. DOI: 10.1088/2040-8978/13/5/053001.
  10. Bliokh KY, Bekshaev AY, Nori F. Extraordinary momentum and spin in evanescent waves. Nat Commun 2014; 5: 3300. DOI: 10.1038/ncomms4300.
  11. 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.
  12. 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.
  13. Ma C, Song T, Chen R, Li H, Li X. Spin Hall effect of fractional order radially polarized beam in its tight focusing. Opt Commun 2022; 520: 128548. DOI: 10.1016/j.optcom.2022.128548.
  14. Ma C, Song T, Chen R, Hu H, Li H, Li X. Vortex-dependent spin angular momentum in tight focusing of power-exponent azimuthal-variant beams. Appl Phys B 2022; 128: 182. DOI: 10.1007/s00340-022-07902-y.
  15. 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. DOI: 10.1364/OE.443271.
  16. Graydon O. Photonic wheel. Nature Photon 2013; 7: 672. DOI: 10.1038/nphoton.2013.229.
  17. 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.
  18. Li H, Wang C, Tang M, Li X. Controlled negative energy flow in the focus of a radial polarized optical beam. Opt Express 2020; 28: 18607-18615. DOI: 10.1364/OE.391398.
  19. 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. DOI: 10.1098/rspa.1959.0200.
  20. Bekshaev AY, Soskin MS. Transverse energy flows in vectorial fields of paraxial beams with singularities. Opt Commun 2007; 271: 332-348. DOI: 10.1016/j.optcom.2006.10.057.
  21. Berry MV. Optical currents. J Opt A: Pure Appl Opt 2009; 11: 094001. DOI: 10.1088/1464-4258/11/9/094001.
  22. McDonald KT. Orbital and spin angular momentum of electromagnetic fields. March 2009, update October 2021. Source: <http://kirkmcd.princeton.edu/examples/spin.pdf>.
  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. Humblet J. Sur le moment d'impulsion d'une onde électromagnétique. Physica 1943; 10: 585-603. DOI: 10.1016/s0031-8914(43)90626-3.
  25. Feynman RP, Leighton RB, Sands M. The Feynman lectures on physics. Vol II. Boston: Addison-Wesley; 1964. ISBN 0-8053-9049-9.
  26. Zhao Y, Edgar JS, Jeffries GDM, McGloin D, Chiu DT. Spin-to-orbital angular momentum conversion in a strongly focused optical beam. Phys Rev Lett 2007; 99: 073901. DOI: 10.1103/physrevlett.99.073901.
  27. Kotlyar VV, Nalimov AG, Kovalev AA, Porfirev AP, Stafeev SS. Spin-orbit and orbit-spin conversion in the sharp focus of laser light: Theory and experiment. Phys Rev A 2020; 102(3): 033502. DOI: 10.1103/physreva.102.033502.
  28. Schaefer B, Collett E, Smyth R, Barrett D, Fraher B. Measuring the Stokes polarization parameters. Am J Phys 2007; 75: 163-168. DOI: 10.1119/1.2386162.
  29. Alperin SN, Niederriter RD, Gopinath JT, Siemens ME. Quantitative measurement of the orbital angular momentum of light with a single, stationary lens. Opt Lett 2016; 41: 5019-5022. DOI: 10.1364/OL.41.005019.
  30. Volyar A, Bretsko M, Akimova Y, Egorov Y. Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens. Appl Opt 2019; 58: 5748-5755. DOI: 10.1364/AO.58.005748.
  31. Kotlyar VV, Kovalev AA, Porfirev AP. Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments. Opt Express 2019; 27(8): 11236-11251. DOI: 10.1364/OE.27.011236.
  32. Volyar AV, Bretsko MV, Akimova YE, Egorov YA. Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams. Computer Optics 2018; 42(5): 736-743. DOI: 10.18287/2412-6179-2017-42-5-736-743.

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