Abstract

The average intensity of the Bessel Gaussian beams propagating through the non-Kolmogorov turbulence based on Rytov theory is derived without the quantic approximation in this paper. Therefore, this result is comparatively more accurate than that calculated by the extended Huygens–Fresnel principle, especially when the inner scale of the turbulence is small or the beams width is large. There is an interesting finding which does not exist in Gaussian beams propagation. It is the intensity variation with the inner scale that displays different behaviors when the beams width is different. Moreover, there will be some beams with specific source width, whose average intensities on the axis do not affected by the turbulence after the inner scale increasing to a certain value as their turbulence perturbation is zero. And the beams here become to the flat top beams. In summary, this paper provides an accurate method for the investigation of the Bessel Gaussian beams propagation through the non-Kolmogorov turbulence and improves the theoretical basis for the applications.

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

Full Article  |  PDF Article
OSA Recommended Articles
Propagation of multiple Bessel Gaussian beams through weak turbulence

Wanjun Wang, Zhensen Wu, Qingchao Shang, and Bai Lu
Opt. Express 27(9) 12780-12793 (2019)

Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence

Huafeng Xu, Zhifeng Cui, and Jun Qu
Opt. Express 19(22) 21163-21173 (2011)

Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence

Xiaoyang Wang, Mingwu Yao, Zhiliang Qiu, Xiang Yi, and Zengji Liu
Opt. Express 23(10) 12508-12523 (2015)

References

  • View by:
  • |
  • |
  • |

  1. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
    [Crossref]
  2. H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
    [Crossref]
  3. 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]
  4. R. G. Frehlich and M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30(36), 5325–5352 (1991).
    [Crossref] [PubMed]
  5. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
    [Crossref] [PubMed]
  6. J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12(7), 1559–1570 (1995).
    [Crossref]
  7. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
    [Crossref]
  8. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref] [PubMed]
  9. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
    [Crossref] [PubMed]
  10. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [Crossref]
  11. R. Rao, “Scintillation index of optical wave propagating in turbulent atmosphere,” Chin. Phys. B 18(2), 581–587 (2009).
    [Crossref]
  12. N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
    [Crossref] [PubMed]
  13. X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
    [Crossref]
  14. I. P. Lukin, “Coherence of Bessel-Gaussian Beams Propagating in a Turbulent Atmosphere,” Atmos. Oceanic Opt. 31(1), 49–59 (2018).
    [Crossref]
  15. H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
    [Crossref]
  16. H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
    [Crossref]
  17. B. Chen and J. Pu, “Propagation of Gauss–Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
    [Crossref]
  18. A. Carbajal-Dominguez, J. Bernal, A. Martin-Ruiz, and G. M. Niconoff, “Generation of J(0) Bessel beams with controlled spatial coherence features,” Opt. Express 18(8), 8400–8405 (2010).
    [Crossref] [PubMed]
  19. R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
    [Crossref]
  20. M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
    [Crossref] [PubMed]
  21. A. Dudley, Y. Li, T. Mhlanga, M. Escuti, and A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38(17), 3429–3432 (2013).
    [Crossref] [PubMed]
  22. P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015).
    [Crossref] [PubMed]
  23. S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7(1), 43233 (2017).
    [Crossref] [PubMed]
  24. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003).
    [Crossref] [PubMed]
  25. Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
    [Crossref] [PubMed]
  26. 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]
  27. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  28. R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
    [Crossref] [PubMed]
  29. Y. Zhang, L. Shan, Y. Li, and Y. Lin, “Effects of moderate to strong turbulence on the Hankel-Bessel-Gaussian pulse beam with orbital angular momentum in the marine-atmosphere,” Opt. Express 25(26), 33469–33479 (2017).
    [Crossref]
  30. H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
    [Crossref]
  31. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
    [Crossref]
  32. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Elsevier/Academic, 2007).
  33. L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
    [Crossref]
  34. 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]
  35. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]

2018 (1)

I. P. Lukin, “Coherence of Bessel-Gaussian Beams Propagating in a Turbulent Atmosphere,” Atmos. Oceanic Opt. 31(1), 49–59 (2018).
[Crossref]

2017 (2)

2016 (3)

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]

Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
[Crossref] [PubMed]

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

2015 (1)

2013 (1)

2012 (2)

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
[Crossref] [PubMed]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

2011 (1)

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
[Crossref]

2010 (1)

2009 (3)

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

R. Rao, “Scintillation index of optical wave propagating in turbulent atmosphere,” Chin. Phys. B 18(2), 581–587 (2009).
[Crossref]

B. Chen and J. Pu, “Propagation of Gauss–Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

2008 (5)

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

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]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (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]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

2007 (4)

R. J. Noriega-Manez and J. C. Gutiérrez-Vega, “Rytov theory for Helmholtz-Gauss beams in turbulent atmosphere,” Opt. Express 15(25), 16328–16341 (2007).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
[Crossref]

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

2003 (1)

2002 (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

1995 (1)

1994 (1)

1992 (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

1991 (1)

1988 (1)

1987 (2)

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

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

Agnew, M.

Ahmed, N.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Almaiman, A.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
[Crossref]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

Anguita, J. A.

Ashrafi, S.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Baykal, Y.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

Bernal, J.

Birch, P.

Boyd, R. W.

Cai, Y.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Carbajal-Dominguez, A.

Chatwin, C.

Chen, B.

B. Chen and J. Pu, “Propagation of Gauss–Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

Chen, M.

Cheng, M.

Conan, J. M.

Deacon, K. S.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Dogariu, A.

Dudley, A.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
[Crossref] [PubMed]

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

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
[Crossref] [PubMed]

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

Escuti, M.

Eyyuboglu, H. T.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Forbes, A.

Frehlich, R. G.

Gori, F.

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

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]

Guattari, G.

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

Guo, L.

Gutiérrez-Vega, J. C.

Hardalaç, F.

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

Hardy, N. D.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
[Crossref]

Huang, H.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Huang, Q.

Ituen, I.

Ji, X.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

Kahn, J. M.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Kavaya, M. J.

Korotkova, O.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[Crossref] [PubMed]

Lavery, M. P. J.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Leach, J.

Li, J.

Li, L.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Li, S.

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7(1), 43233 (2017).
[Crossref] [PubMed]

Li, Y.

Liao, P.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Lin, Y.

Lü, B.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

Lukin, I. P.

I. P. Lukin, “Coherence of Bessel-Gaussian Beams Propagating in a Turbulent Atmosphere,” Atmos. Oceanic Opt. 31(1), 49–59 (2018).
[Crossref]

Madec, P. Y.

Martin-Ruiz, A.

McLaren, M.

Meyers, R. E.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Mhlanga, T.

Miceli, J.

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

Miceli, J. J.

Miller, W. B.

Molisch, A. F.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Neifeld, M. A.

Niconoff, G. M.

Nistazakis, H. E.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Noriega-Manez, R. J.

Padgett, M. J.

Padovani, C.

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

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Pu, J.

B. Chen and J. Pu, “Propagation of Gauss–Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

Rao, R.

R. Rao, “Scintillation index of optical wave propagating in turbulent atmosphere,” Chin. Phys. B 18(2), 581–587 (2009).
[Crossref]

Ren, Y.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Ricklin, J. C.

Rousset, G.

Roux, F. S.

Sermutlu, E.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Shan, L.

Shapiro, J. H.

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
[Crossref]

Shih, Y.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Shirai, T.

Tombras, G. S.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Tsiftsis, T. A.

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

Tunick, A. D.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Tur, M.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

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.

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7(1), 43233 (2017).
[Crossref] [PubMed]

Wang, Z.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Willner, A. E.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Willner, A. J.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Wolf, E.

Xie, G.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Yan, Y.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

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]

Young, R.

Zhang, Y.

Zhao, Z.

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

Zhu, X.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

Zhu, Y.

Appl. Opt. (2)

Appl. Phys. B (2)

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
[Crossref]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel–Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Appl. Phys. Lett. (1)

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Atmos. Oceanic Opt. (1)

I. P. Lukin, “Coherence of Bessel-Gaussian Beams Propagating in a Turbulent Atmosphere,” Atmos. Oceanic Opt. 31(1), 49–59 (2018).
[Crossref]

Chin. Phys. B (2)

R. Rao, “Scintillation index of optical wave propagating in turbulent atmosphere,” Chin. Phys. B 18(2), 581–587 (2009).
[Crossref]

B. Chen and J. Pu, “Propagation of Gauss–Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

IEEE Trans. Commun. (1)

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002).
[Crossref]

IET Commun. (1)

H. E. Nistazakis, T. A. Tsiftsis, and G. S. Tombras, “Performance analysis of free-space optical communication systems over atmospheric turbulence channels,” IET Commun. 3(8), 1402–1409 (2009).
[Crossref]

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

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

Opt. Express (5)

Opt. Laser Technol. (2)

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (1)

N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84(6), 063824 (2011).
[Crossref]

Phys. Rev. Lett. (1)

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

Proc. SPIE (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Sci. Rep. (2)

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7(1), 43233 (2017).
[Crossref] [PubMed]

N. Ahmed, Z. Zhao, L. Li, H. Huang, M. P. J. Lavery, P. Liao, Y. Yan, Z. Wang, G. Xie, Y. Ren, A. Almaiman, A. J. Willner, S. Ashrafi, A. F. Molisch, M. Tur, and A. E. Willner, “Mode-division-multiplexing of multiple Bessel-Gaussian beams carrying orbital-angular-momentum for obstruction-tolerant free-space optical and millimeter-wave communication links,” Sci. Rep. 6(1), 22082 (2016).
[Crossref] [PubMed]

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]

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Elsevier/Academic, 2007).

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1 Average intensity on the axis calculated by different methods varying with inner scale in the Karman turbulence. (a) w = 0.02m, L = 500m, (b) w = 0.01m, L = 500m, (c) w = 0.02m, L = 1000m, (d) w = 0.02m, L = 1000m.
Fig. 2
Fig. 2 Relative error between the two methods. (a) variation with inner scale L = 500m, (b) variation with inner scale L = 1000m, (c) variation with outer scale L = 500m, (d) variation in non-Kolmogorov turbulence.
Fig. 3
Fig. 3 Average intensity of 0 order Bessel Gaussian beams calculated by different method at different propagating distance in the Karman turbulence. (a)L = 500m, (b) L = 1000m.
Fig. 4
Fig. 4 Average intensity of the first order Bessel Gaussian beams calculated by different method at different propagating distance in the Karman turbulence. (a)L = 500m, (b) L = 1000m.
Fig. 5
Fig. 5 Average intensity of Bessel Gaussian beams in the different atmospheric power spectrum turbulence. (a) 0 order Bessel Gaussian beams 2000m away, (b) different order Bessel Gaussian beams 1000m away.
Fig. 6
Fig. 6 Average intensity of high order Bessel Gaussian beams in slant path through the Karman turbulence.
Fig. 7
Fig. 7 Average intensity on the axis of 0 order Bessel Gaussian beams in the Karman turbulence. (a) large source width when L = 1000m. (b) small source width when L = 1000m. (c) large source width when L = 500m. (d) small source width when L = 500m.
Fig. 8
Fig. 8 Normalized average intensity of no turbulence affected Bessel Gaussian beams

Tables (1)

Tables Icon

Table 1 Parameters of different atmospheric power spectrum

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

U(r, φ r )= J n (βr)exp(kα r 2 )exp(in φ r )
U(r,L)= exp(ikL) 1+2iαL exp(in φ r )exp[ i β 2 L+2α k 2 r 2 2k( 1+2iαL ) ] J n ( βr 1+2iαL )
I(r, φ r )=U(r,L) U * (r,L)exp[2 E 1 (0,0)+ E 2 (r,r)]
E 1 (r,r)= Φ 2 (r,L)
E 2 ( r 1 , r 2 )= Φ 1 ( r 1 ,L) Φ 1 * ( r 2 ,L)
Φ 1 (r,L)= k 2 2π 0 L dz d 2 s exp[ ik(Lz)+ ik | sr | 2 2(Lz) ] U(s,z) U(r,L) n 1 (s,z) Lz
Φ 2 (r,L)= k 2 2π 0 L dz d 2 s exp[ ik(Lz)+ ik | sr | 2 2(Lz) ] U(s,z) U(r,L) Φ 1 (s,z) n 1 (s,z) Lz
0 x exp( γ x 2 ) J n (αx) J n ( βx )dx= 1 2γ exp( α 2 + β 2 4γ ) I n ( αβ 2γ )
I n (z)= i n J n ( iz )
0 2π exp(βcosx)exp(inx)dx= 2π I n (β)
Φ 1 (r,L)=ik J n ( βr 1+2iαL ) 1 0 L dz dv(K,z) exp[ iγKr i κ 2 γ 2k (Lz) ] × J n [ β(Lz) k( 1+2iαL ) | K kr Lz | ]exp[ in( φ Kr φ r ) ]
dv(K,z)d v * (K',z') = F n ( K,| z z ' | )δ(K K ' ) d 2 κ d 2 κ '
E 2 (r,r)=2π k 2 | J n ( βr 1+2iαL ) | 2 0 L dη d 2 κ Φ n (κ) ×exp[ i(γγ*)Kr i κ 2 2k (γγ*)(Lη) ] × J n [ β(Lη) k( 1+2iαL ) | K kr Lη | ] J n * [ β(Lη) k( 1+2iαL ) | K kr Lη | ]
Φ 2 (r,L)= i k 3 2π J n ( βr 1+2iαL ) 1 0 L dz 0 z d z ' dv(K,z)dv( K ' , z ' ) γ( Lz ) ×exp[ iγk r 2 2( Lz ) i γ ' κ ' 2 2k ( z z ' ) ]exp[ i β 2 (Lz) 2k( 1+2iαL )( 1+2iαz ) ] × d 2 s exp[ ik s 2 2γ( Lz ) ]exp[ is( K+ γ ' K ' kr Lz ) ] × J n [ β(zz') k( 1+2iαz ) | K ' ks zz' | ]exp[ in( φ s φ r )in( φ K's φ s ) ]
Φ 2 (r,L)= k 2 J n ( βr 1+2iαL ) 1 0 L dz 0 z d z ' dv(K,z)dv( K ' , z ' ) ×exp[ in( φ KK'r φ r ) ] J n [ β(Lz) k( 1+2iαL ) | K+ γ ' K ' kr Lz | ] ×exp[ iγ( K+ γ ' K ' )r i ( K+ γ ' K ' ) 2 γ 2k (Lz) i γ ' κ ' 2 2k ( z z ' ) ]
dv(K,z)d v * ( K ' , z ' ) = dv(K,z)d v * ( K ' , z ' ) = F n ( K,| z z ' | )δ(K+ K ' ) d 2 κ d 2 κ '
E 1 (r,r)= Φ 2 (r,L) = E 1 (0,0)=π k 2 0 L dη d 2 κ Φ n (K)
| K kr Lη | kr Lη
E 2 ' (r,r)=4 π 2 k 2 0 L dη dκκ Φ n (K) J 0 [ (γγ*)κr ]exp[ i κ 2 2k (γγ*)(Lη) ]
T=2 E 1 (0,0)+ E 2 (r,r) =4 π 2 k 2 0 L dη 0 dκ κ Φ n (κ){ exp[ i κ 2 2k (γγ*)(Lη) ] × J 0 [ β(Lη)κ k( 1+2iαL ) ] J 0 * [ β(Lη)κ k( 1+2iαL ) ]1 }
T=4 π 2 k 2 0 L dη 0 dκκ Φ n (κ) κ 2 (Lη) 2 2k( 1+4 α 2 L 2 ) [ β 2 ( 4 α 2 L 2 1 ) k( 1+4 α 2 L 2 ) 4α ]
16k α 3 L 2 4 α 2 β 2 L 2 +4kα+ β 2 =0
I(r, φ r )= b 2 π(kαib+1/ ρ 0 2 ) exp[ a b 2 +4b r 2 4(kαib+1/ ρ 0 2 ) ] × 0 0 2π d s 2 d φ s 2 J n ( a b s 2 )exp(in φ s 2 ) s 2 × [ibrexp(i φ r )+ s 2 exp(i φ s 2 )/ ρ 0 2 ] n [ b 2 r 2 + s 2 2 / ρ 0 4 2ibr s 2 cos( φ r φ s 2 )/ ρ 0 2 ] n/2 × J n { a b [ b 2 r 2 + s 2 2 / ρ 0 4 2ibr s 2 cos( φ r φ s 2 )/ ρ 0 2 ] 1/2 kαib+1/ ρ 0 2 } ×exp[ k α * s 2 2 (ibkα+kα/ ρ 0 2 + b 2 ) s 2 2 2br s 2 (ikα+b)cos( φ r φ s 2 ) kαib+1/ ρ 0 2 ]
ρ 0 2 = π 2 k 2 z/3 0 dκ κ 3 Φ n (κ)
I HGB (r,L)= k 2 w 2 4 L 2 0 QdQ J 0 ( krQ L )exp( k Q 2 4ΛL )exp[ 1 2 D sp (Q) ]
Λ= 2L k w 2 [ 1+ ( 2L ) 2 / ( k w 2 ) 2 ]
D sp (Q)=1.303 C n 2 k 2 L κ x 5/3 { Γ(5/6) [ 1 2 F 2 (5/6,1/2;1,3/2; κ x 2 Q 2 /4) ] + a 1 Γ(1/3)[ 1 2 F 2 (1/3,1/2;1,3/2; κ x 2 Q 2 /4) ] + a 2 Γ(1/4)[ 1 2 F 2 (1/4,1/2;1,3/2; κ x 2 Q 2 /4) ] 3 5 κ x 5/3 κ 0 1/3 Q 2 }
Φ(κ,z)=0.033 C n 2 (z)[ 1+ a 1 ( κ κ x )+ a 2 ( κ κ x ) 7/6 ] exp( κ 2 / κ x 2 ) ( κ 2 + κ 0 2 ) 11/6
Φ(κ,z)= 1 4 π 2 Γ(α1)cos( απ 2 ) C ˜ n 2 exp( κ 2 / κ x 2 ) ( κ 2 + κ 0 2 ) α/2 ,3<α<4
κ x = [ 1 6π Γ(α1)Γ( 5α 2 )cos( απ 2 ) ] 1/(α5) / l 0
C n 2 (h)=8.148× 10 56 v 2 h 10 e h/1000 +2.7× 10 16 e h/1500 +1.7* 10 14 e h/100

Metrics