Abstract

We obtain theoretical relationships to define topological charge (TC) of vortex laser beams devoid of radial symmetry, namely asymmetric Laguerre-Gaussian (LG), asymmetric Bessel-Gaussian (BG), and asymmetric Kummer beams, as well as Hermite-Gaussian (HG) vortex beams. Although they are obtained as superposition of respective conventional LG, BG, and HG beams, these beams have the same TC equal to that of a single mode, n. At the same time, the normalized orbital angular momentum (OAM) that the beams carry is different, differently responding to the variation of the beam’s asymmetry degree. However, whatever the asymmetry degree, TC of the beams remains unchanged and equals n. Although separate HG beam does not have OAM and TC, superposition of only two HG modes with adjacent numbers (n, n + 1) and a π/2-phase shift produces a modal beam whose TC is -(2n + 1). Theoretical findings are validated via numerical simulation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, Vortex laser beams (CRC Press: Boca Raton, 2019).
  2. S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
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  4. S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
    [Crossref]
  5. A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
    [Crossref]
  6. K. Dai, W. Li, K. S. Morgan, Y. Li, J. K. Miller, R. J. Watkins, and E. G. Johnson, “Second-harmonic generation of asymmetric Bessel-Gaussian beams carrying orbital angular momentum,” Opt. Express 28(2), 2536–2546 (2020).
    [Crossref]
  7. N. Dimitrov, M. Zhekova, G. G. Paulus, and A. Dreischuh, “Inverted field interferometer for measuring the topological charges of optical vortices carried by short pulses,” Opt. Commun. 456, 124530 (2020).
    [Crossref]
  8. R. J. Watkins, K. Dai, G. White, W. Li, J. K. Miller, K. S. Morgan, and E. G. Johnson, “Experimental probing of turbulence using a continuous spectrum of asymmetric OAM beams,” Opt. Express 28(2), 924–935 (2020).
    [Crossref]
  9. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
    [Crossref]
  10. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  11. V. V. Kotlyar, A. A. Kovalev, and A. P. Pofirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
    [Crossref]
  12. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
    [Crossref]
  13. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016).
    [Crossref]
  14. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27(8), 11236–11251 (2019).
    [Crossref]
  15. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
    [Crossref]
  16. A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
    [Crossref]
  17. A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
    [Crossref]
  18. A. E. Siegman, Lasers (University Science, 1986).
  19. F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [Crossref]
  20. M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [Crossref]
  21. A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre-Gaussian beams,” Phys. Rev. A 93(6), 063858 (2016).
    [Crossref]
  22. V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, and V. A. Soifer, “Asymmetric Bessel-Gauss beams,” J. Opt. Soc. Am. A 31(9), 1977–1983 (2014).
    [Crossref]
  23. V. V. Kotlyar, A. A. Kovalev, and E. G. Abramochkin, “Kummer laser beams with a transverse complex shift,” J. Opt. 22(1), 015606 (2020).
    [Crossref]
  24. V. V. Kotlyar and A. A. Kovalev, “Hermite-Gaussian modal laser beams with orbital angular momentum,” J. Opt. Soc. Am. A 31(2), 274–282 (2014).
    [Crossref]
  25. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
    [Crossref]
  26. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
    [Crossref]
  27. V. G. Volostnikov and E. G. Abramochkin, The modern optics of the Gaussian beams [In Russian] (Fizmatlit Publisher: Moscow, 2010).
  28. V. V. Kotlyar, A. A. Kovalev, and A. V. Volyar, “Topological charge of a linear combination of optical vortices: topological competition,” Opt. Express 28(6), 8266–8281 (2020).
    [Crossref]
  29. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic: New York, 1965).

2020 (8)

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
[Crossref]

K. Dai, W. Li, K. S. Morgan, Y. Li, J. K. Miller, R. J. Watkins, and E. G. Johnson, “Second-harmonic generation of asymmetric Bessel-Gaussian beams carrying orbital angular momentum,” Opt. Express 28(2), 2536–2546 (2020).
[Crossref]

N. Dimitrov, M. Zhekova, G. G. Paulus, and A. Dreischuh, “Inverted field interferometer for measuring the topological charges of optical vortices carried by short pulses,” Opt. Commun. 456, 124530 (2020).
[Crossref]

R. J. Watkins, K. Dai, G. White, W. Li, J. K. Miller, K. S. Morgan, and E. G. Johnson, “Experimental probing of turbulence using a continuous spectrum of asymmetric OAM beams,” Opt. Express 28(2), 924–935 (2020).
[Crossref]

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and E. G. Abramochkin, “Kummer laser beams with a transverse complex shift,” J. Opt. 22(1), 015606 (2020).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. V. Volyar, “Topological charge of a linear combination of optical vortices: topological competition,” Opt. Express 28(6), 8266–8281 (2020).
[Crossref]

2019 (3)

2018 (1)

A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
[Crossref]

2017 (1)

2016 (2)

2015 (1)

2014 (2)

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref]

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

2003 (1)

1998 (1)

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

1992 (1)

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1987 (1)

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Abramochkin, E. G.

V. V. Kotlyar, A. A. Kovalev, and E. G. Abramochkin, “Kummer laser beams with a transverse complex shift,” J. Opt. 22(1), 015606 (2020).
[Crossref]

V. G. Volostnikov and E. G. Abramochkin, The modern optics of the Gaussian beams [In Russian] (Fizmatlit Publisher: Moscow, 2010).

Akimova, Y. E.

A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
[Crossref]

A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
[Crossref]

Alagashev, G.

A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
[Crossref]

Allen, L.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Alperin, S. N.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Bekshaev, A. Y.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Brezko, M. V.

A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
[Crossref]

A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
[Crossref]

Chavez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref]

Dai, K.

Dimitrov, N.

N. Dimitrov, M. Zhekova, G. G. Paulus, and A. Dreischuh, “Inverted field interferometer for measuring the topological charges of optical vortices carried by short pulses,” Opt. Commun. 456, 124530 (2020).
[Crossref]

Dreischuh, A.

N. Dimitrov, M. Zhekova, G. G. Paulus, and A. Dreischuh, “Inverted field interferometer for measuring the topological charges of optical vortices carried by short pulses,” Opt. Commun. 456, 124530 (2020).
[Crossref]

Egorov, Y. A.

A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
[Crossref]

A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
[Crossref]

Falkovich, G.

A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
[Crossref]

Fickler, R.

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref]

Gopinath, J. T.

Gori, F.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic: New York, 1965).

Guattary, G.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref]

Hiekkamaki, M.

Jing, J.

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

Johnson, E. G.

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and E. G. Abramochkin, “Kummer laser beams with a transverse complex shift,” J. Opt. 22(1), 015606 (2020).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. V. Volyar, “Topological charge of a linear combination of optical vortices: topological competition,” Opt. Express 28(6), 8266–8281 (2020).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27(8), 11236–11251 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Pofirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref]

A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre-Gaussian beams,” Phys. Rev. A 93(6), 063858 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, and V. A. Soifer, “Asymmetric Bessel-Gauss beams,” J. Opt. Soc. Am. A 31(9), 1977–1983 (2014).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Hermite-Gaussian modal laser beams with orbital angular momentum,” J. Opt. Soc. Am. A 31(2), 274–282 (2014).
[Crossref]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, Vortex laser beams (CRC Press: Boca Raton, 2019).

Kovalev, A. A.

Li, S.

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Li, W.

Li, X.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Li, Y.

Liu, H.

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

Liu, M.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Miller, J. K.

Milyukov, V. V.

A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
[Crossref]

Morgan, K. S.

Niederriter, R. D.

Padovani, C.

F. Gori, G. Guattary, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Pan, X.

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

Paulus, G. G.

N. Dimitrov, M. Zhekova, G. G. Paulus, and A. Dreischuh, “Inverted field interferometer for measuring the topological charges of optical vortices carried by short pulses,” Opt. Commun. 456, 124530 (2020).
[Crossref]

Pofirev, A. P.

Porfirev, A. P.

Prabhakar, S.

Pryamikov, A.

A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
[Crossref]

Ren, Y.

S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic: New York, 1965).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Siemens, M. E.

Skidanov, R. V.

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using lights orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref]

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, R. V. Skidanov, and V. A. Soifer, “Asymmetric Bessel-Gauss beams,” J. Opt. Soc. Am. A 31(9), 1977–1983 (2014).
[Crossref]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

Soskin, M. S.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20(8), 1635–1643 (2003).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Sun, Q.

S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
[Crossref]

Turitsyn, S.

A. Pryamikov, G. Alagashev, G. Falkovich, and S. Turitsyn, “Light transport and vortex-supported wave-guiding in micro-strustured optical fibers,” Sci. Rep. 10(1), 2507 (2020).
[Crossref]

Vasnetsov, M. V.

Vastnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Volostnikov, V. G.

V. G. Volostnikov and E. G. Abramochkin, The modern optics of the Gaussian beams [In Russian] (Fizmatlit Publisher: Moscow, 2010).

Volyar, A. V.

V. V. Kotlyar, A. A. Kovalev, and A. V. Volyar, “Topological charge of a linear combination of optical vortices: topological competition,” Opt. Express 28(6), 8266–8281 (2020).
[Crossref]

A. V. Volyar, M. V. Brezko, Y. E. Akimova, Y. A. Egorov, and V. V. Milyukov, “Sector perturbation of a vortex beam: Shannon entropy, orbital angular momentum and topological charge,” CO 43(5), 723–734 (2019).
[Crossref]

A. V. Volyar, M. V. Brezko, Y. E. Akimova, and Y. A. Egorov, “Beyond the intensity or intensity moments and measuring the spectrum of optical vortices in complex beams,” CO 42(5), 736–743 (2018).
[Crossref]

Wang, G.

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[Crossref]

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S. Li, X. Li, L. Zhang, G. Wang, L. Zhang, M. Liu, C. Zeng, L. Wang, Q. Sun, W. Zhao, and W. Zhang, “Efficient optical angular momentum manipulation for compact multiplexing and demultiplexing using a dielectric metasurface,” Adv. Opt. Mater. 8(8), 1901666 (2020).
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[Crossref]

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[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Patterns of intensity (a, c, e) and phase (b, d, f) for an aBG beam in three different planes. Simulation was conducted at a wavelength of λ = 532 nm, beam radius w0= 0.5 mm, an OV of the eighth order n = 8, scaling factor α = 1/w0, asymmetry parameter c = w0/40, propagation distance z = 0 (source plane) (a,b), z = z0 (Rayleigh range) (c,d), and z = 10z0 (far field) (e,f), for the calculation domain –R ≤ x, y ≤ R, where R = 10 mm (z = 0), R = 10 mm (z = z0), and R = 20 mm (z = 10z0), and the number of pixels 2048 × 2048. TC was calculated along a circle of x2 + y2 = R12, where R1 = 0.8R. The resulting values of TC are 11.9926 (a,b), 8.8837 (c,d), and 7.9393 (e,f).
Fig. 2.
Fig. 2. Patterns of intensity (a,c,e) and phase (b,d,f) for an aLG beam in three different planes. The simulation was conducted at a wavelength of λ = 532 nm, waist radius w0 = 0.5 mm, OV order n = 8, mode radial index m = 3, shift vector (x0, y0) = (0, iw0/4), propagation distance z = 0 (source plane) (a,b), z = z0 (Rayleigh range) (c,d), and z = 10z0 (far field) (e,f), computation domain –R ≤ x, y ≤ R, where R = 5 mm (z = 0), R = 5 mm (z = z0), R = 30 mm (z = 10z0), and the number of pixels 2048 × 2048. TC was calculated along a circle of radius x2 + y2 = R12, where R1 = 0.8R. The resulting values of TC are 7.9974 (a,b), 7.9925 (c,d), and 7.9226 (e,f).
Fig. 3.
Fig. 3. Patterns of intensity (a,c,e) and phase (b,d,f) for an asymmetric Kummer beam in three different planes. The simulation was conducted at a wavelength of λ = 532 nm, waist radius of the Gaussian beam, w0 = 0.5 mm, OV order n = 1, parameters m and γ taken to be m = 3 and γ = 0, shift parameter a = 0.2, propagation distance z = 0 (source plane) (a,b), z = z0 (Rayleigh length) (c,d), and z = 10z0 (far field) (e,f), computation domain –R ≤ x, y ≤ R, where R = 5 mm (z = 0), R = 5 mm (z = z0), R = 10 mm (z = 10z0), in each plot, the number of pixels is 2048 × 2048. The inset in Fig. 3(b) depicts a magnified central fragment. TC was calculated along a circle of radius x2 + y2 = R12, where R1 = 0.8R. The resulting values of TC are 0.9981 (a,b), 0.9992 (c,d), and 0.9999 (e,f).
Fig. 4.
Fig. 4. Patterns of intensity (a) and phase (b) for the sum of two HG modes in the source plane. The simulation was conducted at a wavelength of λ = 532 nm, waist radius of the Gaussian beam w0 = 0.5 mm, OV order n = 5, computation domain –R ≤ x, y ≤ R, where R = 5 mm, the number of pixels 2048 × 2048. TC was calculated along a circle of radius x2 + y2 = R12, where R1 = 0.8R. The resulting TC is -10.9550.
Fig. 5.
Fig. 5. Patterns of intensity (a) and phase (b) for a vortex HG mode in the source plane. The simulation was conducted at a wavelength of λ = 532 nm, waist radius of the Gaussian beam w0 = 0.5 mm, OV order n = 10, asymmetry parameter a = 0.3, computation domain –R ≤ x, y ≤ R, where R = 5 mm, and the number of pixels is 2048 × 2048. TC was numerically calculated along a circle x2 + y2 = R12, where R1 = 0.8R. The resulting TC value is -9.9993.

Equations (37)

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E ( x , y , z ) = w ( 0 ) w ( z ) [ 2 w ( z ) ] | l | [ ( x x 0 ) + i θ ( l ) ( y y 0 ) ] | l | × × L p | l | [ 2 ρ 2 w 2 ( z ) ] exp [ ρ 2 w 2 ( z ) + i k ρ 2 2 R ( z ) i ( | l | + 2 p + 1 ) ζ ( z ) ] ,
ρ 2 = ( x x 0 ) 2 + ( y y 0 ) 2 , w ( z ) = w 1 + ( z z R ) 2 , R ( z ) = z [ 1 + ( z R z ) 2 ] , ζ ( z ) = arctan ( z z R ) ,
J z = Im R 2 E ( x E y y E x ) d x d y ,
W = R 2 E E d x d y .
J z W = l + 2 Im ( x 0 y 0 ) w 2 [ L p 1 ( Q 2 2 w 2 ) L p ( Q 2 2 w 2 ) + L p + l 1 ( Q 2 2 w 2 ) L p + l ( Q 2 2 w 2 ) 1 ] .
Q = 2 i ( Im x 0 ) 2 + ( Im y 0 ) 2 .
T C = lim r 1 2 π 0 2 π d φ φ arg E ( r , φ ) = 1 2 π lim r Im 0 2 π d φ E ( r , φ ) / φ E ( r , φ ) .
E ( r , φ , z ) φ = i l E ( r , φ , z ) [ 1 w 2 ( z ) + i k 2 R ( z ) ] ( 2 i a w r e i φ ) E ( r , φ , z ) 4 i a w r e i φ w 2 ( z ) 1 L p | l | ( ξ ) d d ξ L p | l | ( ξ ) E ( r , φ , z ) ,
ξ = 2 ρ 2 w 2 ( z )
T C = 1 2 π lim r Im 0 2 π d φ { i l [ 1 w 2 ( z ) + i k 2 R ( z ) ] ( 2 i a r w e i φ ) i 4 a w r e i φ w 2 ( z ) 1 L m | l | ( ξ ) L m | l | ( ξ ) ξ } = = l + lim r Re [ ( 2 w 2 ( z ) + i k R ( z ) ) r a w 0 2 π d φ e i φ ] lim r Re 0 2 π d φ 2 r | l | a w e i φ r ( r 2 a w e i φ ) = l .
E n ( r , φ , z = 0 ) = exp ( r 2 ω 0 2 + i n φ ) J n ( α r ) ,
E n ( r , φ , z ) = q 1 ( z ) exp ( i k z i α 2 z 2 k q ( z ) ) exp ( r 2 ω 0 2 q ( z ) + i n φ ) J n [ α r q ( z ) ] ,
E n ( r , φ , z ; c ) = 1 q ( z ) exp ( i k z i α 2 z 2 k q ( z ) r 2 q ( z ) ω 0 2 + i n φ ) × × [ α r α r 2 c q ( z ) exp ( i φ ) ] n / 2 J n { 1 q ( z ) α r [ α r 2 c q ( z ) exp ( i φ ) ] } .
k = 0 t k k ! J k + ν ( x ) = x ν / 2 ( x 2 t ) ν / 2 J ν ( x 2 2 t x ) .
J z W = n + [ p = 0 c 2 p p I n + p ( y ) ( p ! ) 2 ] [ p = 0 c 2 p I n + p ( y ) ( p ! ) 2 ] 1 ,
E n ( r , φ , z ; c ) φ = i n E n ( r , φ , z ; c ) + + i n c q ( z ) e i φ ( α r 2 c q ( z ) exp ( i φ ) ) E n ( r , φ , z ; c ) i c ( α r ) 1 / 2 e i φ J n 1 ( x ) ( α r 2 c q ( z ) exp ( i φ ) ) 1 / 2 J n ( x ) x E n ( r , φ , z ; c ) ,
T C = 1 2 π lim r Re 0 2 π d φ { n + n c q ( z ) e i φ [ α r 2 c q ( z ) e i φ ] c ( α r ) 1 / 2 e i φ [ α r 2 c q ( z ) e i φ ] 1 / 2 1 J n ( x ) J n ( x ) x } = = n + lim r Re [ c J n 1 ( x ) 2 π J n ( x ) x 0 2 π d φ e i φ ] = n .
J n ( x >> 1 ) 2 π x cos ( x n π 2 π 4 ) , d J n d x ( x >> 1 ) 2 π x sin ( x n π 2 π 4 ) ,
E s ( r , φ , z ) = ( i ) n + 1 n ! Γ ( m + n + 2 + i γ 2 ) ( z 0 z q ( z ) ) q ( m + i γ ) / 2 ( z ) × × ( k w r 2 z q ( z ) ) n exp ( i n φ + i k s 2 2 z ) 1 F 1 ( m + n + 2 + i γ 2 , n + 1 , ξ ) ,
E s ( r , φ , z = 0 ) = r n e i n φ w n ( s w ) m n + i γ exp ( s 2 w 2 ) ,
E φ = i n E ( m n + i γ ) i r a w e i φ s 2 E + 2 i r a e i φ w E .
T C = 1 2 π lim r Im 0 2 π d φ ( i n i ( m n + i γ ) r a w e i φ r ( r 2 a w e i φ ) + 2 i r a e i φ w ) = = n + a r π w 0 2 π cos φ d φ = n .
E ( x , y , 0 ) = exp [ w 2 2 ( x 2 + y 2 ) ] × × [ H n ( w x ) H n + 1 ( w y ) + i γ H n + 1 ( w x ) H n ( w y ) ] ,
J z W = ( n + 1 ) .
i m H n ( ξ ) H m ( η ) = k = 0 [ ( n + m ) / 2 ] ( 2 ) k k ! P k ( n k , m k ) ( 0 ) × × [ ( ξ + i η ) n + m 2 k + ( 1 ) m ( ξ i η ) n + m 2 k ] L k n + m 2 k ( ξ 2 + η 2 ) ,
H n ( w x ) H n + 1 ( w y ) = i n 1 k = 0 n ( 2 ) k k ! P k ( n k , n + 1 k ) ( 0 ) w 2 n + 1 2 k × L k 2 n + 1 2 k ( w 2 x 2 + w 2 y 2 ) [ ( x + i y ) 2 n + 1 2 k + ( 1 ) n + 1 ( x i y ) 2 n + 1 2 k ] ,
H n + 1 ( w x ) H n ( w y ) = i n k = 0 n ( 2 ) k k ! P k ( n + 1 k , n k ) ( 0 ) w 2 n + 1 2 k × L k 2 n + 1 2 k ( w 2 x 2 + w 2 y 2 ) [ ( x + i y ) 2 n + 1 2 k + ( 1 ) n ( x i y ) 2 n + 1 2 k ] .
E ( r , φ , 0 ) = ( i ) n + 1 exp ( w 2 r 2 2 ) k = 0 n ( 2 ) k k ! ( w r ) 2 n + 1 2 k L k 2 n + 1 2 k ( w 2 r 2 ) × [ A k e i ( 2 n + 1 2 k ) φ ( 1 ) n A k + e i ( 2 n + 1 2 k ) φ ] .
P k ( α , β ) ( z ) = ( 1 ) k P k ( β , α ) ( z ) .
T C = ( 2 n + 1 ) .
k = 0 n n ! t k k ! ( n k ) ! H k ( x ) H n k ( y ) = ( 1 + t 2 ) n / 2 H n ( t x + y 1 + t 2 ) ,
U n ( x , y , z ) = i n exp ( x 2 + y 2 2 ) ( 1 a 2 ) n / 2 H n ( i a x + y 1 a 2 ) .
J z W = 2 a n 1 + a 2 .
cos α = a 1 + a 2 , sin α = 1 1 + a 2 .
T C = 1 2 π lim r Im 0 2 π d φ E ( r , φ ) / φ E ( r , φ ) = = n 2 π lim r Im 0 2 π d φ H n 1 ( i r cos φ cos α + r sin φ sin α ) ( i r sin φ cos α + r cos φ sin α ) H n ( i r cos φ cos α + r sin φ sin α ) .
T C = 1 2 π Im 0 2 π d φ n ( i sin φ cos α + cos φ sin α ) ( i cos φ cos α + sin φ sin α ) = = n 2 π 0 2 π d φ tan α cos 2 φ + tan 2 α sin 2 φ = n .
0 2 π d φ cos 2 φ + γ 2 sin 2 φ = 2 π γ .