Abstract

We theoretically show that optical vortices conserve the integer topological charge (TC) when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with same-sign different TC, the resulting TC of the beam is shown to equal the sum of all constituent TCs. If the beam is composed of an on-axis superposition of Laguerre-Gauss modes (n, 0), the resulting TC equals that of the mode with the highest TC. If the highest positive and negative TCs of the constituent modes are equal in magnitude, the “winning” TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. In the case of equal weight coefficients of both optical vortices, TC of the entire beam equals the greatest TC by absolute value. We have given this effect the name “topological competition of optical vortices”.

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

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References

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    [Crossref]
  2. A. Volyar, M. Bretsko, Y. Akimova, and , and Y. Egorov, “Vortex avalanche in the perturbed singular beams,” J. Opt. Soc. Am. A 36(6), 1064–1071 (2019).
    [Crossref]
  3. Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
    [Crossref]
  4. H. Zhang, X. Li, H. Ma, M. Tang, H. Li, J. Tang, and Y. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express 27(16), 22930–22938 (2019).
    [Crossref]
  5. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
    [Crossref]
  6. 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]
  7. V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. S. Kozlova, “Orbital angular momentum of a laser beam behind an off-axis spiral phase plate,” Opt. Lett. 44(15), 3673–3676 (2019).
    [Crossref]
  8. S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
    [Crossref]
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    [Crossref]
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    [Crossref]
  12. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
    [Crossref]
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    [Crossref]
  15. A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Asymmetric Laguerre- Gaussian beams,” Phys. Rev. A 93(6), 063858 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019).
    [Crossref]
  21. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
    [Crossref]
  22. M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
    [Crossref]
  23. D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
    [Crossref]
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    [Crossref]
  25. V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
    [Crossref]
  26. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
    [Crossref]
  27. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
    [Crossref]
  28. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  29. J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
    [Crossref]
  30. C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
    [Crossref]
  31. 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]
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    [Crossref]
  33. J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
    [Crossref]
  34. G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
    [Crossref]

2020 (1)

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

2019 (10)

J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Orbital angular momentum and informational entropy in perturbed vortex beams,” Opt. Lett. 44(23), 5687–5690 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and , and Y. Egorov, “Vortex avalanche in the perturbed singular beams,” J. Opt. Soc. Am. A 36(6), 1064–1071 (2019).
[Crossref]

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

H. Zhang, X. Li, H. Ma, M. Tang, H. Li, J. Tang, and Y. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express 27(16), 22930–22938 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
[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, A. P. Porfirev, and E. S. Kozlova, “Orbital angular momentum of a laser beam behind an off-axis spiral phase plate,” Opt. Lett. 44(15), 3673–3676 (2019).
[Crossref]

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

2018 (1)

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

2017 (4)

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Asymmetric Gaussian optical vortex,” Opt. Lett. 42(1), 139–142 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
[Crossref]

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[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]

2016 (1)

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

2014 (1)

2012 (1)

2009 (1)

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

2008 (2)

2007 (2)

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32(7), 742–744 (2007).
[Crossref]

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[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]

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 carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[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 (2)

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

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

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
[Crossref]

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[Crossref]

Abramochkin, E. G.

V. V. Kotlyar, A. A. Kovalev, A. P. Porfirev, and E. G. Abramochkin, “Fractional orbital angular momentum of a Gaussian beam with an embedded off-axis optical vortex,” Comput. Opt. 41(1), 22–29 (2017).
[Crossref]

Akimova, Y.

Alexeyev, C. N.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[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]

Bandres, M. A.

Barnett, S. M.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

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]

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]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
[Crossref]

Bretsko, M.

Brundavanam, M. M.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

Cai, Y.

Ceballos-Herrera, D. E.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Cheng, W.

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

Durnin, J.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Egorov, Y.

Egorov, Y. A.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

Ferrando, A.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Fonseca, E. J. S.

Franke-Arnold, S.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Gao, J.

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

Garcia-March, M. A.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Gbur, G.

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 carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Gotte, J. B.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[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]

Gutierrez-Vega, J. C.

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 carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Herbi, D.

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

Hickmann, J. M.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40(1), 73–87 (1993).
[Crossref]

Jacob, P.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[Crossref]

Jesus-Silva, A. J.

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

Kozlova, E. S.

Li, H.

Li, X.

Liang, G.

G. Liang and W. Cheng, “Splitting and rotating of optical vortices due to non-circular symmetry in amplitude and phase distributions of the host beams,” Phys. Lett. A 384(2), 126046 (2020).
[Crossref]

Ma, H.

Maji, S.

S. Maji, P. Jacob, and M. M. Brundavanam, “Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,” Phys. Rev. Appl. 12(5), 054053 (2019).
[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 carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Micely, J. J.

J. Durnin, J. J. Micely, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).
[Crossref]

Padovani, C.

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

Pofirev, A. P.

Porfirev, A. P.

Rasouli, S.

D. Herbi and S. Rasouli, “Combinaed half-integer Bessel-like beams: A set of solutions of the wave equation,” Phys. Rev. A 98(4), 043826 (2018).
[Crossref]

Ryzhik, I. M.

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

Sahu, S.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Siegman, A. E.

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

Skidanov, R. V.

Soifer, V. A.

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vastnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carring 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]

Tang, J.

Tang, M.

Tuson, R. K.

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 carring optical vortex,” Phys. Rev. A 56(5), 4064–4075 (1997).
[Crossref]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993).
[Crossref]

Volyar, A.

Volyar, A. V.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96(6), 063807 (2017).
[Crossref]

Wang, L.

Wen, J.

Woerdman, J.

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]

Yang, X.

J. Wen, L. Wang, X. Yang, J. Zhang, and S. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

Zacares, M.

M. A. Garcia-March, A. Ferrando, M. Zacares, S. Sahu, and D. E. Ceballos-Herrera, “Symmetry, winding number, and topological charge of vortex solitons in discrete-symmetry media,” Phys. Rev. A 79(5), 053820 (2009).
[Crossref]

Zambrini, R.

J. B. Gotte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Zhang, H.

Zhang, J.

Zhang, Y.

Y. Zhang, X. Yang, and J. Gao, “Orbital angular momentum transformation of optical vortex with aluminium metasurfaces,” Sci. Rep. 9(1), 9133 (2019).
[Crossref]

Zhu, S.

Appl. Opt. (2)

Comput. Opt. (2)

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Sectorial perturbation of vortex beams: Shannon entropy, orbital angular momentum and topological charge,” Comput. Opt. 43(5), 723–734 (2019).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Distributions of intensity (a,c,e,g) and phase (b,d,f,h) of a Gaussian optical vortex bounded by a sector-shape diaphragm in the initial plane z = 0 (a-d) and after propagation in free space (e-h) for two different angles of the sector aperture α = π/6 (a,b,e,f) and α = π/4 (c,d,g,h). Red rings (f,h) show the circle over which the TC was calculated. Yellow text (e,g) shows the TC.
Fig. 2.
Fig. 2. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an off-axis optical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space (c, d, g, h, k, l) for different lateral displacements of the vortex from the optical axis. Calculation parameters: waist radius w = 1 mm, TC is n = 7, displacement r0 = w0/4 (a-d), r0 = w0/2 (e-h), r0 = 2w0 (i-l); φ0 = 0 in all figures, the propagation distance in space is z = z0/2 (z0 is the Rayleigh distance). The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by formula (1).
Fig. 3.
Fig. 3. The intensity (a, c) and phase (b, d) of the axial superposition of two Gaussian OVs with TC 12 and 7, but with the same weight amplitudes (in (22)) in the initial plane (a, c) and at the Rayleigh distance (c, d). The red (dashed) rings on the phase distributions denote the radius of the ring by which the topological charge was calculated by the formula (1).
Fig. 4.
Fig. 4. Intensity distributions measured at a distance z = 200 mm (at a double focal length from a cylindrical lens) from a spiral phase plate with fractional order µ: (a) 2.3, (b) 2.5, (c) 2.7, (g) 2.9. The sizes of pictures are 4000 by 4000 microns.
Fig. 5.
Fig. 5. Distributions of intensity (a, c, e, g, i, k) and phase (b, d, f, h, j, l) of a Gaussian beam with an elliptical vortex in the initial plane (a, b, e, f, i, j) and after propagation in space z = z0/2 (c, d, g, h, k, l) for different ellipticities. The red (dashed) rings on the phase distributions denote the radius of the ring by which the TC was calculated by the formula (1).
Fig. 6.
Fig. 6. TC of a Gaussian beam with the fractional-order vortex computed by Eq. (47) (a), by Eq. (46) (b), and by Eqs. (45) and (1) (c).

Equations (52)

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TC=limr12π02πdφφargE(r,φ)=12πlimrIm02πdφE(r,φ)/φE(r,φ).
E(r,φ)=A(r)exp(inφ)f(φ),
f(φ)={1,α<φ<α,δ1,otherwise.
TC=limrIm2π02πdφinE(r,φ)+A(r)einφf(φ)φE(r,φ)=Im2πlimr02πdφ(in+f(φ)φ1f(φ))=n.
f(r,φ)={1,(r,φ)Ω,δ1,(r,φ)Ω,
TC=Im2πlimr02πdφ(in+f(r,φ)φ1f(r,φ))=n.
Jz=Im002πE¯(r,φ,z)(E(r,φ,z)dφ)rdrdφ=Im0ααA(r)einφ(inA¯(r)einφ)rdrdφ=2αn0|A(r)|2rdr,JzW=n,W=2α0|A(r)|2rdr.
E(x,y,0)=exp[x2+y2w2+inarg(x+iy)]rect{arg[(xx0)+i(yy0)]2α},
En(r,φ)=(reiφr0eiφ0w)nA(r).
TC=limrIm2π02πdφinreiφreiφr0eiφ0=12πImlimr02πdφinreiφreiφr0eiφ0=n.
E(r,φ)=circ(rR)exp[inarctan(rsinφrcosφx0)],
TC=n2π02πr2rx0cosφR2+x022rx0cosφdφ={n,x0<R,n/2,x0=R.
JzW=n(1x02R2).
{x=acosφp,y=asinφp,
E(r,φ,z)=1σ(2w0)mexp(r2σw02)(rmeimφσmam),
TC=12πlimrIm{02πimσmrmeimφσmrmeimφamdφ}=m.
Em(r,φ,z=0)=A(r)p=1m(reiφrpeiφp)mp.
TC=12πlimrIm{02πireiφp=1mmpreiφrpeiφpdφ}=p=1mmp.
EN,M(r,φ,z=0)=exp(r2w2)n=MNCn(rw)|n|einφ.
TC=12πlimrIm{02πin=MNnCn(rw)|n|einφn=MNCn(rw)|n|einφdφ}.
TC=12πIm{02πiN(CNeiNφCNeiNφ)(CNeiNφ+CNeiNφ)dφ}.
E(r,φ)=(aeinφ+beimφ)er2/r2w2w2,
TC=12πlimrIm{02πE(r,φ)/E(r,φ)φφE(r,φ)dφ}=12πRe{02πnaeinφ+mbeimφaeinφ+beimφdφ}.
TC=12π02π(n+m2+nm2|a|2|b|2|a|2+|b|2+2|a||b|cost)dt.
TC=n+m2+12πnm2|a|2|b|2|a|2+|b|202πdt1+2|a||b||a|2+|b|2cost.
0πcos(nx)dx1+acosx=π1a2(1a21a)n[a2<1,n0].
TC=n+m2+nm2|a|2|b|2||a|2|b|2|.
OAM=na2+mb2a2+b2.
E(r,φ)=|a|(einφ+iarga+eimφ+iargb)er2/r2w2w2=2|a|cos(nφmφ+argaargb2)exp(r2w2+inφ+mφ+arga+argb2).
TC=limr12π02πφ(nφ+mφ+arga+argb2)dφ=n+m2.
E1(r,φ)=(eiφ+ei7φ)er2/r2w2w2,E2(r,φ)=(ei2φ+ei6φ)er2/r2w2w2,E3(r,φ)=(ei3φ+ei5φ)er2/r2w2w2,E4(r,φ)=ei4φer2/r2w2w2.
E(r,φ)=er2/r2w2w2+inφ,
Ez(ρ,θ)=(i)n+1π2z0Bq1ξexp(ikDρ22B+inθξ)[In12(ξ)In+12(ξ)],
ξ=(z0/z0BB)2(ρ/ρww)2/(z0/z0BB)2(ρ/ρww)2(2q1)(2q1),q1=1i(A/ABB)z0.
Ez(ρ,θ)=iπ2z0Bq1exp(ikDρ22B)ξexp(ξ)×{a(i)neinθ[In12(ξ)In+12(ξ)]+b(i)meimθ[Im12(ξ)Im+12(ξ)]}.
In12(ξ)In+12(ξ)eξ2πξ{[1(n1)218ξ][1(n+1)218ξ]}=neξ2ξ2πξ.
Ez(ρ,θ)=iπ2z0Bq1exp(ikDρ22B)ξexp(ξ)×[a(i)nexp(inθ)neξ2ξ2πξ+b(i)mexp(imθ)meξ2ξ2πξ]=iz04Bq1ξexp(ikDρ22B)[an(i)nexp(inθ)+bm(i)mexp(imθ)].
Eμ(r,φ,z)=exp(iμφ)Ψ(r,z)=eiπμsinπμπΨ(r,z)n=einφμn.
Jz=Im002πE¯(r,φ,z)(E(r,φ,z)φ)rdrdφ
Jz=Wsin2(πμ)π2n=n(μn)2,
W=002πE(r,φ,z)E¯(r,φ,z)rdrdφ.
n=1n2(n2±a2)2=π4a[±{cothπacotπa}a{cosech2πacosec2πa}],
JzW=μsin2πμ2π.
Eμ(r,φ,z=0)=exp(iμφ(rw)2)=eiπμsinπμπn=einφr2/w2μn.
E2(ρ,θ)=12π(iz0q1z)exp(ikρ22z+iπμ)sin(πμ)xexp(x)××m=(i)|m|exp(imθ)μm[I|m|12(x)I|m|+12(x)]
TC=Re2π{02π[n=(i)|n|n|n|einφμn][n=(i)|n||n|einφμn]1dφ}.
TC=Re2π{02π[n=neinφμn][n=einφμn]1dφ}.
TC=n=nrect(μn),rect(x)={1,|x|1/2,0,|x|>1/2
E(r,φ)=A(r)exp(inφ),
Ee(x,y)=A(x2+y2)(x+iαy)n==A(x2+y2)(x2+α2y2)n/2exp(inarctan(αyx)).
TC=12π02πdφφargEe(r,φ)=12π02πdφφ(narctan(αtanφ))==(nα2π)02πdφcos2φ+α2sin2φ=n.
JzW=nPn1(y)Pn(y)<n,

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