Abstract

The vortex beam carrying single orbital angular momentum (OAM) propagating through a medium with a certain transmission function is investigated. We show that the OAM mode weights in the output OAM spectrum involve two factors: the radial distribution of output beam power and the proposed restriction-characterized function. Based on the restriction-characterized function, we show that the OAM mode weights can only vary in a limited range. We analyze the relationship between the radial distribution of the output beam power and the OAM mode weights in the output OAM spectrum. Finally, our theoretical analysis is illustrated numerically with the cases of eccentric circular aperture and atmospheric turbulence in a weak fluctuation regime. These results provide new insights into the characterization of the OAM spectrum and may find applications for fields involving OAM, such as an OAM-based optical communication link and object detection.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  4. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref]
  5. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
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    [Crossref] [PubMed]
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  8. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. S. Sasaki and I. McNulty, “Proposal for Generating Brilliant X-Ray Beams Carrying Orbital Angular Momentum,” Phys. Rev. Lett. 100(12), 124801 (2008).
    [Crossref] [PubMed]
  13. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  18. G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
    [Crossref]
  19. Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
    [Crossref] [PubMed]
  20. C. W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
    [Crossref] [PubMed]
  21. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [Crossref] [PubMed]
  22. Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015).
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  23. T. Zhang, Y. D. Liu, J. Wang, P. Liu, and Y. Yang, “Self-recovery effect of orbital angular momentum mode of circular beam in weak non-Kolmogorov turbulence,” Opt. Express 24(18), 20507–20514 (2016).
    [Crossref] [PubMed]
  24. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
    [Crossref]
  25. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

2017 (2)

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

C. W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (3)

2014 (1)

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

2013 (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

2012 (1)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

2011 (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

2010 (1)

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464, 737–739 (2010).
[Crossref] [PubMed]

2008 (2)

2007 (1)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

2005 (3)

2004 (1)

2003 (1)

D. G. Grier, “A Revolution in Optical Manipulation,” Nature 424, 21–27 (2003).
[Crossref]

1992 (1)

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

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Allen, L.

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

Alvarez-Herrero, A.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Andrews, D. L.

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Anguita, J. A.

Ariste, A. L.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Ashrafi, N.

Ashrafi, S.

Babiker, M.

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref]

Beijersbergen, M. W.

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

Belenguer, T.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Bernet, S.

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Carrasco, S.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

Courtial, J.

Dolinar, S.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fickler, R.

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

Franke-Arnold, S.

Furhapter, S.

Gao, W.

Gelly, B.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Gibson, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Grier, D. G.

D. G. Grier, “A Revolution in Optical Manipulation,” Nature 424, 21–27 (2003).
[Crossref]

Hu, Z.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Jesacher, A.

Krenn, M.

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

Lavery, M. P. J.

Lavery, M. P.J.

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

LeMen, C.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Li, F.

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Li, H.

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Liu, P.

Liu, R.

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Liu, Y. D.

Malik, M.

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

McNulty, I.

S. Sasaki and I. McNulty, “Proposal for Generating Brilliant X-Ray Beams Carrying Orbital Angular Momentum,” Phys. Rev. Lett. 100(12), 124801 (2008).
[Crossref] [PubMed]

Molina-Terriza, G.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Mu, C.

Neifeld, M. A.

Padgett, M J

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Padgett, M. J.

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref]

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref] [PubMed]

Phillips, D. B.

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Qiu, C. W.

C. W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
[Crossref] [PubMed]

Ramachandran, S.

Ramos, A. A.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Rebane, L.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Ritsch-Marte, M.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

Sainz, R. M.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Sasaki, S.

S. Sasaki and I. McNulty, “Proposal for Generating Brilliant X-Ray Beams Carrying Orbital Angular Momentum,” Phys. Rev. Lett. 100(12), 124801 (2008).
[Crossref] [PubMed]

Schlederer, F.

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Spreeuw, R. J. C.

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

Thirunavukkarasu, G.

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

Tonomura, A.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464, 737–739 (2010).
[Crossref] [PubMed]

Torner, L.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

Torres, J. P.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
[Crossref] [PubMed]

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Uchida, M.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464, 737–739 (2010).
[Crossref] [PubMed]

Uribe-Patarroyo, N.

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

Vasic, B. V.

Vasnetsov, M.

Wang, J.

Williams, M. D.

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Woerdman, J. P.

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

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, Y.

C. W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
[Crossref] [PubMed]

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

T. Zhang, Y. D. Liu, J. Wang, P. Liu, and Y. Yang, “Self-recovery effect of orbital angular momentum mode of circular beam in weak non-Kolmogorov turbulence,” Opt. Express 24(18), 20507–20514 (2016).
[Crossref] [PubMed]

Yao, A. M.

Yuan, J.

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

Yue, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zeilinger, A.

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

Zhang, L.

Zhang, T.

Zhang, Y.

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Zhu, Y.

Zhu, Z.

Adv. Opt. Photon. (2)

Appl. Opt. (1)

Astronom. Astrophys. (1)

N. Uribe-Patarroyo, A. Alvarez-Herrero, A. L. Ariste, A. A. Ramos, T. Belenguer, R. M. Sainz, C. LeMen, and B. Gelly, “Detecting photons with orbital angular momentum in extended astronomical objects: application to solar observations,” Astronom. Astrophys. 56, A526 (2011).

J. Europ. Opt. Soc. Rap. Publ. (1)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Europ. Opt. Soc. Rap. Publ. 2, 07014 (2007).
[Crossref]

J. Opt. (1)

R. Liu, D. B. Phillips, F. Li, M. D. Williams, D. L. Andrews, and M J Padgett, “Discrete emitters as a source of orbital angular momentum,” J. Opt. 17, 045608 (2015).
[Crossref]

Nat. Commun. (1)

Y. Yan, G. Xie, M. P.J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Nature (2)

D. G. Grier, “A Revolution in Optical Manipulation,” Nature 424, 21–27 (2003).
[Crossref]

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464, 737–739 (2010).
[Crossref] [PubMed]

New J. Phys. (1)

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18043019 (2016).
[Crossref]

Opt. Express (5)

Optica (1)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (3)

C. Paterson, “Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref] [PubMed]

S. Sasaki and I. McNulty, “Proposal for Generating Brilliant X-Ray Beams Carrying Orbital Angular Momentum,” Phys. Rev. Lett. 100(12), 124801 (2008).
[Crossref] [PubMed]

Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119, 094802 (2017).
[Crossref] [PubMed]

Science (2)

C. W. Qiu and Y. Yang, “Vortex generation reaches a new plateau,” Science 357(6352), 645 (2017).
[Crossref] [PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

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Figures (3)
Fig. 1
Fig. 1 Upper and lower bounds of output OAM mode weights for Δl = 0, 1 (defined by Eq. (10)) in the case of eccentric circular hole against the eccentric coefficient r0/R. There are indications of relative position between the optical axis (•) and the hole edge (○) for r0/R = 0, 1, 2, respectively.

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Fig. 2
Fig. 2 Restriction of the eccentric circular hole with R =0.05 m and {r0 = 0.05m, φ0 = π/4} upon output OAM distribution of LG beams. There are considerations on the LG mode with l0=0, radial index p =0 and waist radius 0.02m at z =0: (a) output intensity distribution, where dash black line indicates the hole edge, (b) curves of Ψ(r, z) (solid pink line) and ΩΔl(r) with Δl=0 (long dash blue line), Δl=1 (dot dash green line) and Δl=2 (short dash red line), (c) distribution of output OAM mode weights for Δl = 0 to 2, along with their upper bounds (solid red line) and lower bounds (dash green line). Moreover, there are considerations of LG modes respectively based on (d)–(f) l0=5 and p =0, (g)–(i) l0=15 and p =0, (j)–(l) l0=10 and p =1.

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Fig. 3
Fig. 3 (a)–(c) Restriction of the atmospheric turbulence in weak fluctuation regime upon output OAM distribution of an input LG beam with l0=0, p =0, ρ0=0.3m, waist radius 0.05 m and z =0: (a) output intensity distribution; (b) curves of Ψ (r, z) (solid pink line) and ΩΔl(r) with Δl=0 (long dash blue line), Δl=1 (dot dash green line) and Δl=2 (short dash red line); (c) distribution of output OAM mode weights for Δl = 0 to 2, along with their upper bounds (solid red line) and lower bounds (dash green line). Moreover, there are cases of LG modes based on (d)–(f) l0=30, (g)–(i) l0=150, respectively.

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Tables (1)

Tables Icon

Table 1 The upper bounds of output OAM mode weights for |Δl| =0 to 5 (defined by Eq. (10)) in the case of the atmospheric turbulence in weak fluctuation regime.

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Equations (12)

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Φ ( r , φ , z ) = Π ( r , φ ) R l 0 ( r , z ) exp ( i l 0 ϕ ) 2 π ,
Φ ( r , φ , z ) = 1 2 π ∑ l = − ∞ ∞ C l ( r , z ) exp ( i l φ ) .
C l ( r , z ) = 1 2 π ∫ 0 2 π Π ( r , φ ) R l 0 ( r , z ) exp ( − i Δ l φ ) d φ ,
A Δ l ( r ) = 1 4 π 2 ∫ 0 2 π ∫ 0 2 π 〈 Π ( r , φ 1 ) [ Π ( r , φ 2 ) ] * 〉 exp [ − i Δ l ( φ 1 − φ 2 ) ] d φ 1 d φ 2 ,
p l ( z ) = ∫ 0 ∞ 〈 | C l ( r , z ) | 2 〉 r d r ∑ Δ l = − ∞ ∞ ∫ 0 ∞ 〈 | C l ( r , z ) | 2 〉 r d r .
p l ( z ) = ∫ 0 ∞ | R l 0 ( r , z ) | 2 A Δ l ( r ) r d r ∫ 0 ∞ | R l 0 ( r , z ) | 2 ∑ Δ l = − ∞ ∞ A Δ l ( r ) r d r .
Ω Δ l ( r ) = A Δ l ( r ) ∑ Δ l = − ∞ ∞ A Δ l ( r ) ,
p l ( z ) = ∫ S Ψ ( r , z ) Ω Δ l ( r ) d r ,
Ψ ( r , z ) = | R l 0 ( r , z ) | 2 ∑ Δ l = − ∞ ∞ A Δ l ( r ) r ∫ S | R l 0 ( r , z ) | 2 ∑ Δ l = − ∞ ∞ A Δ l ( r ) r d r .
inf { Ω Δ l ( r ) | r ∈ S } ≤ p l ( z ) ≤ sup { Ω Δ l ( r ) | r ∈ S } .
Ω Δ l ( r ) = { Δ θ 2 Δ θ 2 + 4 ∑ n = 1 ∞ 1 − cos ( n Δ θ ) n 2 ( Δ l = 0 ) 2 [ 1 − cos ( Δ l Δ θ ) ] Δ l 2 Δ θ 2 + 4 Δ l 2 ∑ n = 1 ∞ 1 − cos ( n Δ θ ) n 2 ( Δ l ≠ 0 ) .
Ω Δ l ( r ) = exp ( − 2 r 2 ρ 0 2 ) I Δ l ( 2 r 2 ρ 0 2 ) .

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