Abstract

The propagation of laser beams having orbital angular momenta (OAM) in the turbulent atmosphere is studied numerically. The variance of random wandering of these beams is investigated with the use of the Monte Carlo technique. It is found that, among various types of vortex laser beams, such as the Laguerre–Gaussian (LG) beam, modified Bessel–Gaussian beam, and hypergeometric Gaussian beam, having identical initial effective radii and OAM, the LG beam occupying the largest effective volume in space is the most stable one.

© 2014 Optical Society of America

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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, 8185–8189 (1992).
    [CrossRef]
  2. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
    [CrossRef]
  3. J. P. Torres and L. Torner, eds., Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
  4. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
    [CrossRef]
  5. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 220–276.
  6. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
    [CrossRef]
  8. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–229 (2008).
    [CrossRef]
  9. T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
    [CrossRef]
  10. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37, 3735–3737 (2012).
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    [CrossRef]
  12. M. Charnotskii, “Beam scintillations for ground-to-space propagation. Part I: path integrals and analytic techniques,” J. Opt. Soc. Am. A 27, 2169–2179 (2010).
    [CrossRef]
  13. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef]
  14. 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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [CrossRef]
  15. Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38, 4062–4065 (2013).
    [CrossRef]
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  17. H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
    [CrossRef]
  18. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
    [CrossRef]
  19. Y. Gu, “Statistics of optical vortex wander on propagation through atmospheric turbulence,” J. Opt. Soc. Am. A 30, 708–715 (2013).
    [CrossRef]
  20. V. P. Aksenov and C. E. Pogutsa, “Increase in laser beam resistance to random inhomogeneities of atmospheric permittivity with an optical vortex included in the beam structure,” Appl. Opt. 51, 7262–7269 (2012).
    [CrossRef]
  21. V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
    [CrossRef]
  22. H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel–Gaussian beams in turbulence,” Opt. Laser Technol. 40, 343–351 (2008).
    [CrossRef]
  23. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32, 742–744 (2007).
    [CrossRef]
  24. H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
    [CrossRef]
  25. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).
  26. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
    [CrossRef]
  27. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Vol. 3 of Integrals and Series (Gordon & Breach Science, 1990).
  28. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [CrossRef]
  29. P. A. Konyaev and V. P. Lukin, “Thermal distortions of focused laser beams in the atmosphere,” Appl. Opt. 24, 415–421 (1985).
    [CrossRef]
  30. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4 of Wave Propagation through Random Media (Springer, 1988).
  31. V. I. Talanov, “Focusing light in cubic media,” JETP Lett. 11, 199–201 (1970).
  32. G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and its Applications (Cambridge University, 1996).
  33. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 2nd ed. (Dover, 2000).

2013

Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. J. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38, 4062–4065 (2013).
[CrossRef]

Y. Gu, “Statistics of optical vortex wander on propagation through atmospheric turbulence,” J. Opt. Soc. Am. A 30, 708–715 (2013).
[CrossRef]

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
[CrossRef]

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
[CrossRef]

2012

2010

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

M. Charnotskii, “Beam scintillations for ground-to-space propagation. Part I: path integrals and analytic techniques,” J. Opt. Soc. Am. A 27, 2169–2179 (2010).
[CrossRef]

2009

T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
[CrossRef]

2008

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–229 (2008).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[CrossRef]

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel–Gaussian beams in turbulence,” Opt. Laser Technol. 40, 343–351 (2008).
[CrossRef]

2007

2005

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef]

2004

2002

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

1999

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

1992

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, 8185–8189 (1992).
[CrossRef]

1989

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1985

1983

1976

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1970

V. I. Talanov, “Focusing light in cubic media,” JETP Lett. 11, 199–201 (1970).

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Ahmed, N.

Aksenov, V. P.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
[CrossRef]

V. P. Aksenov and C. E. Pogutsa, “Increase in laser beam resistance to random inhomogeneities of atmospheric permittivity with an optical vortex included in the beam structure,” Appl. Opt. 51, 7262–7269 (2012).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[CrossRef]

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, 8185–8189 (1992).
[CrossRef]

Andrews, L. C.

R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Baker, G. A.

G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and its Applications (Cambridge University, 1996).

Barnett, S.

Baykal, Y.

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[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, 8185–8189 (1992).
[CrossRef]

Boyd, R. W.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Vol. 3 of Integrals and Series (Gordon & Breach Science, 1990).

Cai, Y.

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

Chandrasekaran, N.

Charnotskii, M.

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
[CrossRef]

Çil, C. Z.

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Courtial, J.

Dolinar, S.

Erkmen, B. I.

Eyyuboglu, H. T.

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel–Gaussian beams in turbulence,” Opt. Laser Technol. 40, 343–351 (2008).
[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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Franke-Arnold, S.

Freund, I.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

Gbur, G.

Gibson, G.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Graves-Morris, P.

G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed., Vol. 59 of Encyclopedia of Mathematics and its Applications (Cambridge University, 1996).

Gu, Y.

Hardalaç, F.

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel–Gaussian beams in turbulence,” Opt. Laser Technol. 40, 343–351 (2008).
[CrossRef]

Huang, H.

Ji, X.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

Khonina, S. N.

Kolosov, V. V.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
[CrossRef]

Konyaev, P. A.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 2nd ed. (Dover, 2000).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 2nd ed. (Dover, 2000).

Korotkova, O.

H. T. Eyyuboğlu, Y. Baykal, C. Z. Çil, O. Korotkova, and Y. Cai, “Beam wander characteristics of flat-topped, dark hollow, cos and cosh-Gaussian, J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere: a review,” Proc. SPIE 7588, 75880N (2010).
[CrossRef]

Kotlyar, V. V.

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4 of Wave Propagation through Random Media (Springer, 1988).

Lavery, M. P. J.

Lukin, V. P.

Malik, M.

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Vol. 3 of Integrals and Series (Gordon & Breach Science, 1990).

Mirhosseini, M.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Neifeld, M.

O’Sullivan, M. N.

Padgett, M.

Padgett, M. J.

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]

Phillips, R. L.

R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Pogutsa, C. E.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
[CrossRef]

V. P. Aksenov and C. E. Pogutsa, “Increase in laser beam resistance to random inhomogeneities of atmospheric permittivity with an optical vortex included in the beam structure,” Appl. Opt. 51, 7262–7269 (2012).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Special Functions, Vol. 3 of Integrals and Series (Gordon & Breach Science, 1990).

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
[CrossRef]

Ren, Y.

Robertson, D. J.

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Rodenburg, B.

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4 of Wave Propagation through Random Media (Springer, 1988).

Shapiro, J. H.

Skidanov, R. V.

Soifer, V. A.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 220–276.

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, 8185–8189 (1992).
[CrossRef]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1965).

Steinhoff, N. K.

Talanov, V. I.

V. I. Talanov, “Focusing light in cubic media,” JETP Lett. 11, 199–201 (1970).

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 4 of Wave Propagation through Random Media (Springer, 1988).

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Tur, M.

Tyson, R. K.

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 220–276.

Wang, J.

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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
[CrossRef]

Willne, A. E.

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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Willner, A. E.

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, 8185–8189 (1992).
[CrossRef]

Xie, G.

Yan, Y.

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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Appl. Phys. B

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Scintillations of Laguerre Gaussian beams,” Appl. Phys. B 98, 857–863 (2010).
[CrossRef]

J. Opt.

V. P. Aksenov, V. V. Kolosov, and C. E. Pogutsa, “The influence of the vortex phase on the random wandering of a Laguerre–Gaussian beam propagating in a turbulent atmosphere: a numerical experiment,” J. Opt. 15, 044007 (2013).
[CrossRef]

J. Opt. Soc. Am. A

JETP Lett.

V. I. Talanov, “Focusing light in cubic media,” JETP Lett. 11, 199–201 (1970).

Laser Photonics Rev.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008).
[CrossRef]

Nat. Photonics

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. Willne, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Opt. Commun.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

T. Wang, J. Pu, and Z. Chen, “Beam-spreading and topological charge of vortex beams propagating in a turbulent atmosphere,” Opt. Commun. 282, 1255–1259 (2009).
[CrossRef]

H. T. Eyyuboğlu, “Scintillation analysis of hypergeometric Gaussian beam via phase screen method,” Opt. Commun. 309, 103–107 (2013).
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Figures (9)

Fig. 1.
Fig. 1.

Intensity and phase distributions of LG, BG, and HG beams with index l=8. (a) Intensity at the end of the turbulent layer, (b) phase at the end of the turbulent layer, (c) intensity in the far diffraction zone after the turbulent layer, and (d) phase in the far zone after the turbulent layer.

Fig. 2.
Fig. 2.

Variance of beam wandering normalized to the squared effective radius of the source at the end of the turbulent layer as a function of the Fried radius at the layer thickness z=8zd and the outer scale M0=20a. (curve 1) LG beam, (curve 2) BG beam, and (curve 3) HG beam.

Fig. 3.
Fig. 3.

Angular wandering of vortex beams in the far diffraction zone behind the turbulent layer as a function of the Fried radius at the layer thickness z=8zd and the outer scale M0=20a. (curve 1) LG beam, (curve 2) BG beam, and (curve 3) HG beam. σθc2=σc2/Z2, θd=1/ka is the angle of diffraction divergence.

Fig. 4.
Fig. 4.

Wandering of vortex beams at the boundary of the turbulent layer as a function of the diffraction conditions. Fried radius r0n=2.5. Solid curves are for the calculated effective volume occupied by the corresponding beams in the layer. (curves 1 and 1′) LG beam, (curves 2 and 2′) BG beam, and (curves 3 and 3′) HG beam.

Fig. 5.
Fig. 5.

Angular wandering of vortex beams in the far diffraction zone behind the turbulent layer as a function of the layer thickness. Fried radius r0n=2.5. Solid curves are for the calculated effective volume occupied in the layer by the corresponding beams. (curves 1 and 1′) LG beam, (curves 2 and 2′) BG beam, and (curves 3 and 3′) HG beam.

Fig. 6.
Fig. 6.

Variance of wandering of LG beam at the end of the turbulent layer as a function of diffraction conditions at different values of the outer scale of turbulence. Fried radius r0n=2.5. (curve 1) M0=20a, and (curve 2) M0=28a.

Fig. 7.
Fig. 7.

Variance of centroid displacements normalized to the squared beam radius a for the vortex BG and HG beams at the end of the turbulent layer as a function of the diffraction conditions for different values of the outer scale of turbulence. Fried radius r0n=2.5. (curve 1) BG beam, M0=20a, (curve 2) HG beam, M0=20a, (curve 3) BG beam, M0=28a, and (curve 4) HG beam, M0=28a.

Fig. 8.
Fig. 8.

Angular wandering of vortex LG beams behind the turbulent layer as a function of the layer thickness at different values of the outer scale of turbulence. Fried radius r0n=2.5. (curve 1) M0=20a, and (curve 2) M0=28a.

Fig. 9.
Fig. 9.

Angular wandering of vortex BG and HG beams behind the turbulent layer as a function of diffraction conditions at different values of the outer scale of turbulence. Fried radius r0n=2.5. (curve 1) BG beam, M0=20a, (curve 2) HG beam, M0=20a, (curve 3) BG beam, M0=28a, and (curve 4) HG beam, M0=28a.

Equations (23)

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uLG(r,ϕ,0)=4aΦcp!(p+|l|)!(2ra)|l|×Lp|l|(2r2a2)exp{r2a2}exp{ilϕ},
uBG(r,ϕ,0)=4Φc1a[Il(β2a24)]1exp(β2a28)Il(βr)×exp(r2a2+ilϕ),
uHG(r,ϕ,0)=8Φc2q+|l|+12(2q+|l|)!r2q+|l|a2q+|l|+1exp(r2a2+ilϕ),
Re2=2d2rI(r;z)·r2/d2rI(r;z),
Re,LG=a(1+4z2/k2a4)(2p+|l|+1).
I(r;z)=|u(r;z)|2=(k2πz)2×Γ2(r1,r2;0)exp[ikz(r1r2)r+ik2z(r12r22)]dr1dr1,
Γ2(r1,r2;0)=A(r1)A(r2)exp(i[ϕ(r1)ϕ(r2)]).
Re2=z2k2P0dRxdRy{A(R)[2A(R+ρ/2)ρx2+2A(Rρ/2)ρx2+2A(R+ρ/2)ρy2+2A(Rρ/2)ρy2][ρx[ϕ(R+ρ/2)ϕ(Rρ/2)]+kzRx]2A2(R)[ρy[ϕ(R+ρ/2)ϕ(Rρ/2)]+kzRy]2A2(R)+i{[2ρx2[ϕ(R+ρ/2)ϕ(Rρ/2)]+2ρy2[ϕ(R+ρ/2)ϕ(Rρ/2)]]A2(R)+[ρx[ϕ(R+ρ/2)ϕ(Rρ/2)]+kzRx]×[A(R+ρ/2)ρx+A(Rρ/2)ρx]A(R)+[ρy[ϕ(R+ρ/2)ϕ(Rρ/2)]+kzRy]×[A(R+ρ/2)ρy+A(Rρ/2)ρy]A(R)}}ρ=0,
Re,BG2=14a2(44l+4μ)+a2μIl1(μ)Il(μ)+z2(4+2l+4μk2a2+4μIl1(μ)k2a2Il(μ)21leμμlk2a2Il(μ)Γ(l)×[22F2(l,1/2+l;2l,1+l;2μ)F22(l,1/2+l;1+2l,1+l,;2μ)]),
μ=14β2a2.
Re,HG2=a2(1+4z2(2q+l(1+l))k2a4(2q+l)(2q+l+1))(2q+l+1).
μ(1+Il+1(μ)Il(μ))2p=0.
S(ρ)=FT{ΦS(κ)exp[iϕ(κ)]},
Φs(κ)=2πk2ΔzΦn(κ,0).
Φn(κ,0)=0.033Cn2exp(κ2/κa2)(κ2+κ02)11/6×[1+1.802κκa0.254(κκa)7/6],
rc(j)(z)=1P0d2rI(j)(r,z)·r.
V=π·0z1Re2(z)dz,
I8(x)=k=0ckxkx810321920+x10371589120+x1229727129600+x143923981107200+x16753404372582400+x18195885136871424000+x2065817405988798464000+O(x21).
FL,M(z)=a0+a1z++aLzL1+b1z++bMzM,
[cLM+1cLM+2cLM+3cLcLM+2cLM+3cLM+4cL+1cLM+3cLM+4cLM+5cL+2cLcL+1cL+2cL+M1][bMbM1bM2b1]==[cL+1cL+2cL+3cL+M].
a0=c0,a1=c1+b1c0,a2=c2+b1c1+b2c0,aL=cL+k=1min(L,M)bkcLk.
I8(x)110321920x8+21809092931622635110400x10×(163427931557680x2+12495917065x413770190886118400x6+1213152342446080x846051716635795111936000x10)1.
I8(x)ex2πx[12558x1+62985128x248498451024x3+100391791532768x435137127025262144x5+15811707161254194304x61965169318612533554432x7+6092024887698752147483648x8+223374245882287517179869184x9+46908591635280375274877906944x10].

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