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Topological charge of axial superposition of Gaussian optical vortices
V.V. Kotlyar1,2, A.A. Kovalev1,2, A.G. Nalimov1,2

1Image Processing Systems Institute, NRC "Kurchatov Institute", 443001, Samara, Russia, Molodogvardeyskaya 151;
2Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34

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DOI: 10.18287/COJ1659

Article ID: 1659

Language: English

Abstract:
The topological charge of a finite superposition of optical vortices with a Gaussian envelope is considered. It is shown theoretically and numerically that in the initial plane the topological charge of such a superposition is equal to the number of zeros of a complex polynomial of degree n, where n is the maximum topological charge of optical vortices in the superposition located in a unit radius disk including its boundary. When propagating in free space, the topological charge of such a superposition is always equal to n. If the modulus of the coefficient of the superposition term with the topological charge equal to k is greater than the sum of the moduli of all other coefficients of the superposition, then k zeros lie in the unit radius disk and the topological charge of the entire superposition in the initial plane is equal to k (k ≤ n). If all coefficients of the superposition are equal in modulus, then in the initial plane the topological charge is equal to half (n/2), but during propagation the topological charge is again equal to n. In this case, additional zeros of the superposition of optical vortices are formed almost immediately at a distance much smaller than the wavelength from the initial plane and at a distance from the optical axis greater than the radius of the limiting aperture of the initial field.

Keywords:
optical vortex, topological charge, superposition of optical vortices, Bessel beams, Laguerre-Gaussian beams.

Acknowledgements:
This work was supported by the Russian Science Foundation under grant No. 23-12-00236 (theory and simulations) and by the State assignment of the NRC "Kurchatov Institute" (Introduction and Conclusion).

Citation:
Kotlyar VV, Kovalev AA, Nalimov AG. Topological charge of axial superposition of Gaussian optical vortices. Computer Optics. 2026. 50(1): 1659. DOI: 10.18287/COJ1659.

References:

  1. Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys Rev A 1992; 45(11): 8185-8189. DOI: 10.1103/PhysRevA.45.8185.
  2. Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A 2004; 6(2): 259-268. DOI: 10.1088/1464-4258/6/2/018.
  3. Shabat BV. Vvedenie v kompleksnyi analiz. Nauka, Moscow; 1969. 576 p. (In Russian).
  4. Levin BYa. Raspredelenie kornei tselykh funktsii. Gos. izd. tekh.-teor. lit., Moscow; 1956. 632 p. (In Russian).
  5. Zauderer E. Complex argument Hermite-Gaussian and Laguerre-Gaussian beams. J Opt Soc Am A 1986; 3: 465-469.
  6. Gori F, Guattari G, Padovani C. Bessel-Gauss beams. Opt Commun 1987; 64(6): 491-495. DOI: 10.1016/0030-4018(87)90276-8.
  7. Supp S, Jahns J. Coaxial superposition of Bessel beams by discretized spiral axicons. J Eur Opt Soc-Rapid Publ 2018; 14: 51938687. DOI: 10.1186/s41476-018-0086-8.
  8. Ando T, Matsumoto N, Ohtake Y, Takiguchi Y, Inoue T. Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes. J Opt Soc Am A 2010; 27: 2602-2612.
  9. Volyar AV, Abramochkin EG, Razueva EV, Akimova YE, Bretsko MV. Structural stability of spiral beams and fine structure of an energy flow. Computer Optics 2021; 45(4): 482-489. DOI: 10.18287/2412-6179-CO-885.
  10. Volyar A, Abramochkin E, Bretsko M, Akimova Ya, Egorov Yu. Fine structure of perturbed Laguerre-Gaussian beams: Hermite-Gaussian mode spectra and topological charge. Appl Opt 2020; 59(25): 7680-7687. DOI: 10.1364/AO.396557.
  11. Hirst HP, Macey WT. Bounding the Roots of Polynomials. Coll Math J 1997; 28(4): 292-295. DOI: 10.1080/07468342.1997.11973878.
  12. Rouché É. Mémoire sur la série de Lagrange. J Éc Polytech 1862; 22: 193-224.
  13. Kotlyar V, Kovalev A, Kozlova E, Savelyeva A, Stafeev S. Geometric progression of optical vortices. Photonics 2022; 9: 407. DOI: 10.3390/photonics9060407.
  14. Indebetouw G. Optical vortices and their propagation. J Mod Opt 1993; 40: 73-87. DOI: 10.1080/09500349314550101.
  15. Abramochkin EG, Volostnikov VG. Spiral-type beams. Opt Commun 1993; 102: 336-342. DOI: 10.1016/0030-4018(93)90406-U.
  16. Kotlyar VV, Kovalev AA, Volyar AV. Topological charge of a linear combination of optical vortices: topological competition. Opt Express 2020; 28: 8266-8281. DOI: 10.1364/OE.384662.

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