Abstract

We study the effects of non-diffraction on the pulse spreading and average intensity of the orbital angular momentum (OAM) mode of a Hankel-Bessel beam propagating through a strong anisotropic marine atmosphere. The average intensity of the OAM mode of the Hankel-Bessel pulse beam is modeled based on the modified Rytov approximation. We find that the pulse width and the average intensity of the received beam are affected by the refractive index structure constant of turbulence as well as the wavelength. The pulse widths of the received OAM signal modes have imperceptible changes for different OAM quantum numbers, and outer and inner scales of turbulence. Our results show that we can choose with longer wavelength light to reduce the impacts of turbulence on the pulse width and average intensity of received OAM signal modes.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  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. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
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  20. Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014).
    [Crossref] [PubMed]
  21. J. Gao, Y. Zhang, W. Dan, and Z. Hu, “Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams,” Opt. Express 23(13), 17024–17034 (2015).
    [Crossref] [PubMed]
  22. M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
  23. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
    [Crossref] [PubMed]
  24. Q. Hao, Y. Cheng, J. Cao, F. Zhang, X. Zhang, and H. Yu, “Analytical and numerical approaches to study echo laser pulse profile affected by target and atmospheric turbulence,” Opt. Express 24(22), 25026–25042 (2016).
    [Crossref] [PubMed]
  25. V. A. Banakh and L. O. Gerasimova, “Strong scintillations of pulsed Laguerrian beams in a turbulent atmosphere,” Opt. Express 24(17), 19264–19277 (2016).
    [Crossref] [PubMed]
  26. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A 29(5), 741–747 (2012).
    [Crossref] [PubMed]
  27. L. Torner, J. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005).
    [Crossref] [PubMed]
  28. L. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
  29. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th Ed. (Academic, 2000).
  30. 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]

2016 (9)

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[Crossref]

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

R. Neo, M. Goodwin, J. Zheng, J. Lawrence, S. Leon-Saval, J. Bland-Hawthorn, and G. Molina-Terriza, “Measurement and limitations of optical orbital angular momentum through corrected atmospheric turbulence,” Opt. Express 24(3), 2919–2930 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref] [PubMed]

Q. Hao, Y. Cheng, J. Cao, F. Zhang, X. Zhang, and H. Yu, “Analytical and numerical approaches to study echo laser pulse profile affected by target and atmospheric turbulence,” Opt. Express 24(22), 25026–25042 (2016).
[Crossref] [PubMed]

V. A. Banakh and L. O. Gerasimova, “Strong scintillations of pulsed Laguerrian beams in a turbulent atmosphere,” Opt. Express 24(17), 19264–19277 (2016).
[Crossref] [PubMed]

Y. Zhu, Y. Zhang, and Z. Hu, “Spiral spectrum of Airy beams propagation through moderate-to-strong turbulence of maritime atmosphere,” Opt. Express 24(10), 10847–10857 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (1)

2013 (1)

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88(2), 022326 (2013).
[Crossref]

2012 (3)

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(7), 488–496 (2012).
[Crossref]

C. Chen, H. Yang, Y. Lou, S. Tong, and R. Liu, “Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence,” Opt. Express 20(7), 7749–7757 (2012).
[Crossref] [PubMed]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A 29(5), 741–747 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (2)

2008 (2)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

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]

2005 (2)

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

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

1999 (1)

D. E. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulse,” Waves Random Media 9(3), 307–326 (1999).
[Crossref]

1998 (1)

1979 (1)

C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

Ahmed, N.

Alonso, J. R. G.

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88(2), 022326 (2013).
[Crossref]

Andrews, L. C.

Anguita, J. A.

Ashrafi, N.

Ashrafi, S.

Avramov-Zamurovic, S.

Banakh, V. A.

Bland-Hawthorn, J.

Bock, R.

Boyd, R. W.

Brun, T. A.

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88(2), 022326 (2013).
[Crossref]

Cao, J.

Cao, Y.

Carrasco, S.

Castillo-Vázquez, M.

Chen, C.

Cheng, M.

Cheng, Y.

Cvijetic, M.

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

Dan, W.

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(7), 488–496 (2012).
[Crossref]

Fan, C.

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[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(7), 488–496 (2012).
[Crossref]

Gao, J.

Garrido-Balsells, J. M.

Gerasimova, L. O.

Goodwin, M.

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Guo, L.

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).

Hao, Q.

Hu, Z.

Huang, H.

Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. J. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. 40(10), 2249–2252 (2015).
[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(7), 488–496 (2012).
[Crossref]

Huang, Q.

Ishimaru, A.

Ji, X.

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[Crossref]

Jurado-Navas, A.

Kelly, D. E.

D. E. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulse,” Waves Random Media 9(3), 307–326 (1999).
[Crossref]

Korotkova, O.

Kotlyar, V. V.

Kovalev, A. A.

Lavery, M. P. J.

Lawrence, J.

Leon-Saval, S.

Li, J.

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).

Li, L.

Li, M.

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

Li, Y.

Liu, C. H.

C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

Liu, R.

Lou, Y.

Lu, L.

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[Crossref]

Malek-Madani, R.

Molina-Terriza, G.

Neifeld, M. A.

Nelson, C.

Neo, R.

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]

Puerta-Notario, A.

Ren, B.

Ren, Y.

Shapiro, J. H.

Soifer, V. A.

Tong, S.

Torner, L.

Torres, J.

Tur, M.

Tyler, G. A.

Vasic, B. V.

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

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. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, Z.

Willner, A. E.

Willner, A. J.

Xie, G.

Yan, Y.

Yang, H.

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(7), 488–496 (2012).
[Crossref]

Yeh, K. C.

C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

C. Y. Young, L. C. Andrews, and A. Ishimaru, “Time-of-arrival fluctuations of a space–time Gaussian pulse in weak optical turbulence: an analytic solution,” Appl. Opt. 37(33), 7655–7660 (1998).
[Crossref] [PubMed]

Yu, H.

Yu, Z.

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[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. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zhang, F.

Zhang, P.

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[Crossref]

Zhang, X.

Zhang, Y.

Zhao, F.

Zhao, Z.

Zheng, J.

Zhu, Y.

Appl. Opt. (2)

IEEE Photonics J. (1)

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).

J. Opt. Soc. Am. A (2)

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(7), 488–496 (2012).
[Crossref]

Opt. Commun. (2)

M. Li, Z. Yu, and M. Cvijetic, “Influence of atmospheric turbulence on OAM-based FSO system with use of realistic link model,” Opt. Commun. 364, 50–54 (2016).
[Crossref]

Z. Wang, L. Lu, P. Zhang, C. Fan, and X. Ji, “Broadening of ultra-short pulses propagating through weak-to-strong oceanic turbulence,” Opt. Commun. 367, 95–101 (2016).
[Crossref]

Opt. Express (10)

C. Chen, H. Yang, Y. Lou, S. Tong, and R. Liu, “Temporal broadening of optical pulses propagating through non-Kolmogorov turbulence,” Opt. Express 20(7), 7749–7757 (2012).
[Crossref] [PubMed]

C. Chen, H. Yang, S. Tong, B. Ren, and Y. Li, “Characterization of temporal pulse broadening for horizontal propagation in strong anisotropic atmospheric turbulence,” Opt. Express 23(4), 4814–4828 (2015).
[Crossref] [PubMed]

R. Neo, M. Goodwin, J. Zheng, J. Lawrence, S. Leon-Saval, J. Bland-Hawthorn, and G. Molina-Terriza, “Measurement and limitations of optical orbital angular momentum through corrected atmospheric turbulence,” Opt. Express 24(3), 2919–2930 (2016).
[Crossref] [PubMed]

O. Korotkova, S. Avramov-Zamurovic, R. Malek-Madani, and C. Nelson, “Probability density function of the intensity of a laser beam propagating in the maritime environment,” Opt. Express 19(21), 20322–20331 (2011).
[Crossref] [PubMed]

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

Q. Hao, Y. Cheng, J. Cao, F. Zhang, X. Zhang, and H. Yu, “Analytical and numerical approaches to study echo laser pulse profile affected by target and atmospheric turbulence,” Opt. Express 24(22), 25026–25042 (2016).
[Crossref] [PubMed]

V. A. Banakh and L. O. Gerasimova, “Strong scintillations of pulsed Laguerrian beams in a turbulent atmosphere,” Opt. Express 24(17), 19264–19277 (2016).
[Crossref] [PubMed]

Y. Zhu, Y. Zhang, and Z. Hu, “Spiral spectrum of Airy beams propagation through moderate-to-strong turbulence of maritime atmosphere,” Opt. Express 24(10), 10847–10857 (2016).
[Crossref] [PubMed]

Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014).
[Crossref] [PubMed]

J. Gao, Y. Zhang, W. Dan, and Z. Hu, “Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams,” Opt. Express 23(13), 17024–17034 (2015).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (1)

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88(2), 022326 (2013).
[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]

Radio Sci. (1)

C. H. Liu and K. C. Yeh, “Pulse spreading and wandering in random media,” Radio Sci. 14(5), 925–931 (1979).
[Crossref]

Waves Random Complex Media (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Waves Random Media (1)

D. E. Kelly and L. C. Andrews, “Temporal broadening and scintillations of ultrashort optical pulse,” Waves Random Media 9(3), 307–326 (1999).
[Crossref]

Other (3)

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

L. C. Andrews, R. L. Philips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).

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

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Figures (5)

Fig. 1
Fig. 1 Pulse width of HBP as a function of wavelength for different quantum numbers of OAM signal.
Fig. 2
Fig. 2 Pulse width of HBP as a function of wavelength for different quantum numbers of OAM.
Fig. 3
Fig. 3 Pulse width of HBP as a function of turbulent inner-scale for different quantum number differences of OAM.
Fig. 4
Fig. 4 Pulse width of HBP as a function of turbulent outer-scale for different quantum number differences of OAM.
Fig. 5
Fig. 5 Pulse width of HBP as a function of the refractive index structure constant of turbulence from 10 15 to 10 12 for different quantum number differences of OAM.

Equations (27)

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u m ( r,φ,z;t )= u m 0 ( r,φ,z )f( t )exp[ ψ x ( r,φ,z,ω )+ ψ y ( r,φ,z,ω ) ],
u m 0 ( r,φ,z;t )= i 3 m 0 +1 m 0 ! A 0 π 2kz J m 0 /2 ( k r 2 /4z )exp[ i( kz π m 0 4 π 4 )+i m 0 φ t 2 τ 0 2 ],
u m ( r,φ,z;t )= m β m ( r,z;t )exp( imφ ) ,
β m ( r,z;t )= 1 2π 0 2π u m ( r,φ,z;t ) exp( imφ )dφ.
| β m (r,z;t) | 2 = ( 1 2π ) 2 0 2π 0 2π u m 0 ( r,φ,z ) u m 0 ( r, φ ,z ) × exp[ ψ x ( r,φ,z )+ ψ x ( r, φ ,z )+ ψ y ( r,φ,z )+ ψ y ( r, φ ,z ) ] ×exp[ im( φ φ ) ] exp[ 2 t 2 / τ 0 2 ] dφd φ .
exp[ ψ x ( r,φ,z )+ ψ x ( r, φ ,z )+ ψ y ( r,φ,z )+ ψ y ( r, φ ,z ) ] =exp[ 1 2 D( r,φ, φ ;z ) ],
D( r,φ, φ ;z )=4 π 2 k 2 z r 2 [ 1cos( φ φ ) ] 0 κ 3 ϕ n,eff ( κ )dκ ,
ϕ n,eff ( κ )=0.033 C n 2 κ 11/3 [ f( κ l 0 )g( κ L 0 ) G x ( κ )+ G y ( κ ) ],
f( κ l 0 )=exp( κ 2 κ H 2 )[ 10.061 κ κ H +2.836 ( κ κ H ) 7/6 ],
g( κ L 0 )= κ 11/3 ( κ 2 + κ 0 2 ) 11/6 , κ 0 = 8π L 0 .
G x ( κ )=exp( κ 2 / κ x 2 ),
G y ( κ )= κ 11/3 ( κ 2 + κ y 2 ) 11/6 ,
η x 2.61 Q 0 / [ 2.61+ Q 0 ( 1+0.65 d 2 +0.45 σ 1 2 Q H 1/6 ) ] ,
η y =3 ( σ 1 σ 2 ) 12/5 ( 1+0.69 σ 2 12/5 ),
σ 2 2 =3.86 σ 1 2 { 5.581 Q H 5/6 + [ sin( 11 6 tan 1 Q H ) 0.051sin( 4 3 tan 1 Q H ) ( 1+ Q H 2 ) 1/4 +sin( 11 6 tan 1 Q H ) 0.051sin( 4 3 tan 1 Q H ) ( 1+ Q H 2 ) 1/4 + 3.052sin( 5 4 tan 1 Q H ) ( 1+ Q H 2 ) 7/ 24 ] ( 1+ 1 Q H 2 ) 11/ 12 }.
ϕ n,eff ( κ )=0.033 C n 2 [ 1 ( κ 2 + κ 0 2 ) 11/6 exp( κ 2 κ xH 2 )+ 1 ( κ 2 + κ y 2 ) 11/6 + 2.836 κ 7/6 κ H 7/6 ( κ 2 + κ 0 2 ) 11/6 exp( κ 2 κ xH 2 ) 0.061κ κ H ( κ 2 + κ 0 2 ) 11/6 exp( κ 2 κ xH 2 ) ],
D( r,φ, φ ;z )=0.132 π 2 k 2 C n 2 r 2 z[ 1cos( φ φ ) ] 0 [ exp( κ 2 κ xH 2 ) κ 3 ( κ 2 + κ 0 2 ) 11/6 0.061 κ H exp( κ 2 κ xH 2 ) κ 4 ( κ 2 + κ 0 2 ) 11/6 + κ 3 ( κ 2 + κ y 2 ) 11/6 + 2.836 κ H 7/6 exp( κ 2 κ xH 2 ) κ 25/6 ( κ 2 + κ 0 2 ) 11/6 ]dκ.
0 κ 2μ exp( κ 2 / κ H 2 ) ( κ 2 + κ 0 2 ) 11/6 dκ= 1 2 κ 0 2μ8/3 Γ( μ+ 1 2 )U( μ+ 1 2 ;μ 1 3 ; κ 0 2 κ H 2 ),μ> 1 2 ,
0 x t μ1 ( 1+βt ) ν dt = x μ μ F 2 1 ( ν,μ;1+μ;βx ),μ>0,
1 2 D( r,φ, φ ;z )=2 r 2 π 2 k 2 C n 2 z[ 1cos( φ φ ) ] × [ 0.0165 κ 0 1/3 U( 2; 7 6 ; κ 0 2 κ xH 2 ) 0.013 κ 0 4/3 κ H U( 5 2 ; 5 3 ; κ 0 2 κ xH 2 ) + 0.0661 κ 0 3/2 κ H 7/6 U( 31 12 ; 7 4 ; κ 0 2 κ xH 2 )+ 0.0083 κ H 4 κ y 11/3 F 2 1 ( 11 6 ,2;3; κ H 2 κ y 2 ) ],
ρ 0 2 = π 2 k 2 z C n 2 [ 0.0165 κ 0 1/3 U( 2; 7 6 ; κ 0 2 κ xH 2 ) 0.013 κ 0 4/3 κ H U( 5 2 ; 5 3 ; κ 0 2 κ xH 2 ) + 0.0661 κ 0 3/2 κ H 7/6 U( 31 12 ; 7 4 ; κ 0 2 κ xH 2 ) + 0.0083 κ H 4 κ y 11/3 F 2 1 ( 11 6 ,2;3; κ H 2 κ y 2 ) ],
P m = | β m (r,z;t) | 2 = π 2kz ( m 0 ! A 0 ) 2 exp[ 2 t 2 τ 0 2 ] 0 D/2 r 2 m 0 +1 | J m 0 /2 [ k r 2 4z ] | 2 exp[ 2 r 2 ρ 0 2 ] I m m 0 ( 2 r 2 ρ 0 2 ) dr,
exp[ 2 t 2 τ 0 2 ] = τ 0 τ 1 exp[ 2 ( t-z/c ) 2 τ 1 2 ],
δ τ 2 = 2 π 2 z c 2 0 κ ϕ n,eff ( κ ) dκ.
δ τ 2 = 0.651 C n 2 z c 2 0 [ exp( κ 2 κ xH 2 ) κ ( κ 2 + κ 0 2 ) 11/6 + κ ( κ 2 + κ y 2 ) 11/6 0.061 κ H exp( κ 2 κ xH 2 ) κ 2 ( κ 2 + κ 0 2 ) 11/6 + 2.836 κ H 7/6 exp( κ 2 κ xH 2 ) κ 13/6 ( κ 2 + κ 0 2 ) 11/6 ]dκ.
δ τ 2 = 0.651 C n 2 z c 2 { 0.5 κ 0 5 3 U( 1; 2 3 ; κ 0 2 κ xH 2 ) 0.027 κ H κ 0 2 3 U( 3 2 ; 2 3 ; κ 0 2 κ xH 2 ) + 1.2645 κ H 7/6 κ 0 1 2 U( 19 12 ; 3 4 ; κ 0 2 κ xH 2 ) + κ H 2 2 κ y 11/3 F 2 1 ( 11 6 ,1;2; κ H 2 κ y 2 ) }.
P m ( t,z )= π τ 0 ( m 0 ! A 0 ) 2 4kz τ 1 m = p m 0 D 2 /4 x m 0 | J m 0 /2 [ kx 4z ] | 2 I m m 0 ( 2x ρ 0 2 )exp[ 2x ρ 0 2 2 ( tz/c ) 2 τ 1 2 ]dx.

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