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Orbital angular momentum of superpositions of optical vortices after passing through a sector diaphragm
A.A. Kovalev 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, 1004 kB

DOI: 10.18287/2412-6179-CO-1072

Pages: 196-203.

Full text of article: Russian language.

In optical communications, it is desirable to know some quantities describing a light field, that are conserved on propagation or resistant to some distortions. Typically, optical vortex beams are characterized by their orbital angular momentum (OAM) and/or topological charge (TC). Here, we study what happens with the OAM of a superposition of two or several optical vortices (with different TCs) when it is distorted by a hard-edge sector aperture. We discover several cases when such perturbation does not violate the OAM of the whole superposition. The first case is when the incident beam consists of two vortices of the same power. The second case is when the aperture half-angle equals an integer number of π divided by the difference between the topological charges. For more than two incident beams, this angle equals an integer number of π divided by the greatest common divisor of all possible differences between the topological charges. For two incident vortex beams with real-valued radial envelopes of the complex amplitudes, the OAM is also conserved when there is a ±(pi)/2 phase delay between the beams. When two beams with the same power pass through a binary radial grating, their total OAM is also conserved.

sector aperture, orbital angular momentum, optical vortex, superposition.

Kovalev AA. Orbital angular momentum of superpositions of optical vortices after passing through a sector diaphragm. Computer Optics 2022; 46(2): 196-203. DOI: 10.18287/2412-6179-CO-1072.

This work was supported by the Russian Science Foundation under Project No. 18-19-00595 (Sections "Normalized orbital angular momentum of a superposition of two optical vortices after passing through a sector diaphragm" and "Superposition of two vortex beams with equal power"), the RF Ministry of Science and Higher Education under a government project of the FSRC "Crystallography and Photonics" RAS (Section "Superposition of two vortex beams with different power"), and the grant for Samara University within the federal academic leadership program "Priority 2030" (Section "Normalized orbital angular momentum of a superposition of two optical vortices after passing through a binary radial grating").


  1. Bouchal Z, Wagner J, Chlup M. Self-reconstruction of a distorted nondiffracting beam. Opt Commun 1998; 151(4-6): 207-211.
  2. Pinnell J, Rodríguez-Fajardo V, Forbes A, Chabou S, Mihoubi K, Bencheikh A. Revealing the modal content of obstructed beams. Phys Rev A 2020; 102(3): 033524.
  3. Arrizon V, Mellado-Villaseñor G, Aguirre-Olivas D, Moya-Cessa H. Mathematical and diffractive modeling of self-healing. Opt Express 2018; 26: 12219-12229.
  4. Zambale N, Doblado G, Hermosa N. OAM beams from incomplete computer generated holograms projected onto a DMD. J Opt Soc Am B 2017; 34: 1905-1911.
  5. Zhang Y, Chen MLN, Jiang L. Extraction of the characteristics of vortex beams with a partial receiving aperture at arbitrary locations. J Opt 2021; 23(8): 085601.
  6. Zheng S, Hui X, Zhu J, Chi H, Jin X, Yu S, Zhang X. Orbital angular momentum mode-demultiplexing scheme with partial angular receiving aperture. Opt Express 2015; 23: 12251-12257.
  7. Volyar AV, Bretsko MV, Akimova YaE, Egorov YuA. Orbital angular momentum and informational entropy in perturbed vortex beams. Opt Lett 2019; 44: 5687-5690.
  8. Volyar A, Akimova Y. Structural stability of spiral vortex beams to sector perturbations. Appl Opt 2021; 60: 8865-8874.
  9. Socratovich B. Fresnel diffraction from sector apertures. Opt Express 2021; 29: 30419-30425.
  10. Franke-Arnold S, Barnett SM, Yao E, Leach J, Courtial J, Padgett M. Uncertainty principle for angular position and angular momentum. New J Phys 2004; 6: 103.
  11. Yao E, Franke-Arnold S, Courtial J, Barnett S, Padgett M. Fourier relationship between angular position and optical orbital angular momentum. Opt Express 2006; 14(20): 9071-9076.
  12. Berry MV, Jeffrey MR, Mansuripur M. Orbital and spin angular momentum in conical diffraction. J Opt A: Pure Appl Opt 2005; 7: 685-690.
  13. Kotlyar VV, Kovalev AA. Optical vortex beams with a symmetric and almost symmetric OAM spectrum. J Opt Soc Am A 2021; 38(9): 1276-1283. DOI: 10.1364/JOSAA.432623.
  14. Siegman AE. Lasers. Mill Valley, CA: University Science Books; 1986.
  15. Gori F, Guattari G, Padovani C. Bessel-Gauss beams. Opt Commun 1987; 64(6): 491-495.
  16. Kotlyar VV, Skidanov RV, Khonina SN, Soifer VA. Hypergeometric modes. Opt Lett 2007; 32(7): 742-744. DOI: 10.1364/OL.32.000742.
  17. Karimi E, Zito G, Piccirillo B, Marrucci L, Santamato E. Hypergeometric-Gaussian modes. Opt Lett 2007; 32: 3053-3055.
  18. Abramowitz M, Stegun IA. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. New York: Dover Publications Inc; 1970.
  19. Hebri D, Rasouli S, Yeganeh M. Intensity-based measuring of the topological charge alteration by the diffraction of vortex beams from amplitude sinusoidal radial gratings. J Opt Soc Am B 2018; 35(4): 724-730.
  20. Wang W, Liu D, Gu M, Han P, Xiao M. Generation of a sub-diffracted Bessel beam via diffraction interference in a combined amplitude structure. Opt Express 2021; 29: 597-603.
  21. Kotlyar VV, Kovalev AA, Volyar AV. Topological charge of optical vortices and their superpositions. Computer Optics 2020; 44(2): 145-154. DOI: 10.18287/2412-6179-CO-685.
  22. Kotlyar VV, Kovalev AA, Volyar AV. Topological charge of a linear combination of optical vortices: topological competition. Opt Express 2020; 28(6): 8266-8281. DOI: 10.1364/OE.386401.
  23. Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A: Pure Appl Opt 2004; 6(2): 259-268.

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