Abstract

We analyze the effects of turbulence on the detection probability spectrum and the mode weight of the orbital angular momentum (OAM) for Whittaker-Gaussian (WG) laser beams in weak non-Kolmogorov turbulence channels. Our numerical results show that WG beam is a better light source for mitigating the effects of turbulence with several adjustable parameters. The real parameters of WG beams γ and W0, which have significant effects on the mode weight, have no influence on the detection probability spectrum. Larger signal OAM quantum number, shorter wavelength, smaller beamwidth and coherence length will lead to the lower detection probability of the signal OAM mode.

© 2014 Optical Society of America

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References

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  • Publication
  1. S. Shwartz, M. Golub, and S. Ruschin, “Diffractive optical elements for mode-division multiplexing of temporal signals with the aid of Laguerre-Gaussian modes,” Appl. Opt. 52(12), 2659–2669 (2013).
    [Crossref] [PubMed]
  2. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
    [Crossref] [PubMed]
  3. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  4. X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
    [Crossref]
  5. 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]
  6. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
    [Crossref] [PubMed]
  7. Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
    [Crossref]
  8. J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
    [Crossref]
  9. 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(17), 3735–3737 (2012).
    [Crossref] [PubMed]
  10. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
    [Crossref] [PubMed]
  11. D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).
  12. M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008).
    [Crossref] [PubMed]
  13. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2th ed. (SPIE, 2005).
  14. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [Crossref] [PubMed]
  15. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
    [Crossref] [PubMed]
  16. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
    [Crossref]
  17. H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
    [Crossref]
  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).

2014 (2)

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

2013 (3)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

S. Shwartz, M. Golub, and S. Ruschin, “Diffractive optical elements for mode-division multiplexing of temporal signals with the aid of Laguerre-Gaussian modes,” Appl. Opt. 52(12), 2659–2669 (2013).
[Crossref] [PubMed]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

2012 (2)

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

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(17), 3735–3737 (2012).
[Crossref] [PubMed]

2011 (1)

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

2009 (2)

G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
[Crossref] [PubMed]

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).

2008 (3)

2005 (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

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

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Alfano, R. R.

Anguita, J. A.

Bandres, M. A.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008).
[Crossref] [PubMed]

Boyd, R. W.

Carrasco, S.

Chen, H.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

Gao, J.

Golub, M.

Gutiérrez-Vega, J. C.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).

M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008).
[Crossref] [PubMed]

He, Y.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Jia, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Jiang, Y.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Lavery, M. P. J.

Liao, J.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Liu, X.

Lopez-Mago, D.

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).

Malik, M.

Mirhosseini, M.

Neifeld, M. A.

O’Sullivan, M. N.

Ou, J.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Padgett, M.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Robertson, D. J.

Rodenburg, B.

Ruschin, S.

Sheng, X.

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Shwartz, S.

Sztul, H. I.

Tang, H.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Torner, L.

Torres, J. P.

Tyler, G. A.

Vasic, B. V.

Wang, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Wang, S.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Wang, X.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Wang, Y.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Wang, Z.

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Xu, J.

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Zhang, J.

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Zhang, Y.

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

Zhao, F.

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

Zhu, Y.

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Appl. Opt. (2)

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
[Crossref]

Opt. Commun. (3)

Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011).
[Crossref]

J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012).
[Crossref]

Phys. Rev. Lett. (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

SPIE Optical Engineering Applications. (1)

D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009).

Other (2)

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2th ed. (SPIE, 2005).

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

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Figures (7)
Fig. 1
Fig. 1 Detection probability of the signal OAM mode for WG beams (a) and LG beams (b) against z for l0

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Fig. 2
Fig. 2 Crosstalk probability of the signal OAM mode for WG beams for z against l0, with ∆l = 1

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Fig. 3
Fig. 3 Detection probability spectrum of OAM modes for WG beams against l for l0.

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Fig. 4
Fig. 4 Detection probability of the signal OAM mode for WG beams against C n 2 for w0.

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Fig. 5
Fig. 5 Detection probability of the signal OAM mode for WG beams against α for λ.

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Fig. 6
Fig. 6 Detection probability and Mode weight of the signal OAM mode for WG beams against γ for C n 2 .

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Fig. 7
Fig. 7 Detection probability and Mode weight of the signal OAM mode for WG beams against W0 for z.

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Equations (13)

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W G l 0 ,iγ ( r,φ,z )= C l 0 ,iγ ( 1−z/ q 0 * ) iγ/2−1/2 ( 1+z/ q 0 ) iγ/2+1/2 exp( ik r 2 2( z+ q 0 ) ) × ( P r 2 ) l 0 /2 F 1 1 ( β, l 0 +1,P r 2 )exp( i l 0 φ )
σ l 0 ,iγ 0 = π l 0 ! w 0 2 2 2 l 0 +1 ζ 2 l 0 | ( 1−i ζ 2 /4 ) 1+ l 0 −iγ | ×F( 1+ l 0 −iγ 2 , 1+ l 0 +iγ 2 ; l 0 +1; ζ 4 16+ ζ 4 )
P= 2 w 0 2 { ( z 2 +1 )[ ( 1/ R 2 )+4 ( k w 0 2 ) −2 ]+2z/R }
W G ′ l,iγ ( r,φ,z )=W G l 0 ,iγ ( r,φ,z )exp[ ψ 1 ( r,φ,z ) ]
W G ′ l,iγ ( r,φ,z )= 1 2π ∑ l=−∞ ∞ β l ( r,z )exp( −ilφ )
β l ( r,z )= 1 2π ∫ 0 2π W G ′ l,iγ ( r,φ,z ) exp[ ilφ ]dφ
| β l (r,z) | 2 = 1 2π ∫ 0 2π ∫ 0 2π W G ′ l,iγ ( r,φ,z )W G ′ l,iγ ∗ ( r,φ,z )  exp[ −il( φ− φ ′ ) ]d φ ′ dφ
〈 | β l (r,z) | 2 〉= 1 2π ∫ 0 2π ∫ 0 2π W G l 0 ,iγ ( r,φ,z )W G l 0 ,iγ ∗ ( r,φ,z ) exp[ −il( φ− φ ′ ) ]d φ ′  dφ  ×exp[ −( 2 r 2 −2 r 2 cos( φ ′ −φ ) )/ ρ 0 2 ]
ρ 0 = { 2Γ( 3−α 2 ) [ 8 α−2 Γ( 2 α−2 ) ] ( α−2 )/2 π 1/2 k 2 Γ( 2−α 2 ) ∫ 0 z C n 2 ( ξ, θ ) ( 1−ξ/z ) α−2 dξ } 1/(α−2)  , 3<α<4
∫ 0 2π exp[−in φ 1 +ηcos( φ 1 − φ 2 )] d φ 1 =2πexp(−in φ 2 ) I n (η),
〈 | β l (r,z) | 2 〉= ( P r 2 ) l 0 σ l 0 ,iγ 0 exp[ −γ( argδ+arg δ ′ ) ] ( 1− z 2 ) 2 + ( 2z/k w 0 2 ) 2 exp{ − 2 r 2 w 0 2 η[ ( z+1/Rη ) 2 + ( 2/kη w 0 2 ) 2 ] }     × | F 1 1 ( β, l 0 +1,P r 2 ) | 2 I l− l 0 ( 2 r 2 ρ 0 2 )
ℑ m = ∫ 0 ∞ 〈 | β m (r,z) | 2 〉rdr
℘ l ( z )= ℑ l ( z )/ ∑ m=−∞ ∞ ℑ m ( z )

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