Abstract

The turbulent effects of strong irradiance fluctuations on the probability densities and the normalized powers of the orbital angular momentum (OAM) modes are modeled for fractional Bessel Gauss beams in paraxial turbulence channel. We find that the probability density of signal OAM modes is a function of position deviation from the beam center, and the farther away from the beam center the detection position is, the smaller the probability density is. For fractional OAM quantum numbers, the average probability densities of signal/crosstalk modes oscillate along the beam radius except the half-integer. When the beam waist of source decreases or the irradiance fluctuation increases, the average probability density of the signal OAM mode drops. The peak of the average probability density of crosstalk modes shifts to outward of the beam center as beam waist gets larger. In the nearby region of beam center, the larger the quantum number deviation of OAM, the smaller the beam waist and the turbulence fluctuations are, the lower average probability densities of crosstalk OAM modes are. Especially, the increase of turbulence fluctuations can make the crosstalk stronger and more concentrated. Lower irradiance fluctuation can give rise to higher the normalized powers of the signal OAM modes, which is opposite to the crosstalk normalized powers.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence

Chunyi Chen, Huamin Yang, Shoufeng Tong, and Yan Lou
Opt. Express 24(7) 6959-6975 (2016)

Orbital-angular-momentum crosstalk and temporal fading in a terrestrial laser link using single-mode fiber coupling

Gustavo Funes, Matías Vial, and Jaime A. Anguita
Opt. Express 23(18) 23133-23142 (2015)

References

  • View by:
  • |
  • |
  • |

  1. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
    [Crossref]
  2. W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
    [Crossref]
  3. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
    [Crossref]
  4. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007).
    [Crossref] [PubMed]
  5. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
    [Crossref] [PubMed]
  6. F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α,” Opt. Lett. 36(5), 606–608 (2011).
    [Crossref] [PubMed]
  7. J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
    [Crossref]
  8. S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004).
    [Crossref] [PubMed]
  9. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
    [Crossref]
  10. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18(16), 16480–16485 (2010).
    [Crossref] [PubMed]
  11. S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
    [Crossref] [PubMed]
  12. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
    [Crossref]
  13. S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
    [Crossref]
  14. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
    [Crossref]
  15. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  16. F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
    [Crossref]
  17. F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
    [Crossref]
  18. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2th ed. (SPIE, 2005).
  19. C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16(15), 11411–11422 (2008).
    [Crossref] [PubMed]
  20. Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  21. C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
    [Crossref]
  22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2007).
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  24. 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]

2013 (3)

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

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

2012 (1)

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

2011 (1)

2010 (2)

2008 (4)

2007 (2)

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express 15(10), 6300–6313 (2007).
[Crossref] [PubMed]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2004 (5)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

S. H. Tao and X. Yuan, “Self-reconstruction property of fractional Bessel beams,” J. Opt. Soc. Am. A 21(7), 1192–1197 (2004).
[Crossref] [PubMed]

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

2003 (1)

2000 (1)

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Anguita, J. A.

Barnett, S. M.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Burnham, D.

Cornwall, J. M.

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

Donegan, J. F.

Eastham, P. R.

Fickler, R.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Flossmann, F.

Franke-Arnold, S.

Götte, J. B.

Gutiérrez-Vega, J. C.

Jiang, Y.

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

Krenn, M.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Lee, W. M.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
[Crossref] [PubMed]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
[Crossref]

C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16(15), 11411–11422 (2008).
[Crossref] [PubMed]

Lunney, J. G.

Masino, A. J.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

McGloin, D.

Mitri, F. G.

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

F. G. Mitri, “Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α,” Opt. Lett. 36(5), 606–608 (2011).
[Crossref] [PubMed]

Neifeld, M. A.

O’Dwyer, D. P.

O’Holleran, K.

Ou, J.

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

Padgett, M. J.

Phelan, C. F.

Plick, W. N.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Preece, D.

Rakovich, Y. P.

Ramelow, S.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Rudd, D.

Senthilkumaran, P.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Singh, R. K.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Subich, C. J.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Tang, H.

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

Tao, S. H.

Thomas, F. E.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Vasic, B. V.

Vyas, S.

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Wang, S.

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

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Yong, C. Y.

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Yuan, X.

Yuan, X. C.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Dynamic optical manipulation with a higher-order fractional bessel beam generated from a spatial light modulator,” Opt. Lett. 28(20), 1867–1869 (2003).
[Crossref] [PubMed]

Zambrini, R.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

Zeilinger, A.

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

Zhang, J.

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

Appl. Opt. (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Eur. Phys. J. D (1)

F. G. Mitri, “High-order Bessel non-vortex beam of fractional type α: II. Vector wave analysis for standing and quasi-standing laser wave tweezers,” Eur. Phys. J. D 67(7), 135 (2013).
[Crossref]

J. Mod. Opt. (1)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum for mulation of fractional orbital angular momentum,” J. Mod. Opt. 54(12), 1723–1738 (2007).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A: Pure Appl. Opt. 10(1), 015009 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6(1), 71 (2004).
[Crossref]

Opt. Commun. (2)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1–3), 129–135 (2004).
[Crossref]

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

Opt. Express (4)

Opt. Laser Technol. (1)

S. Vyas, R. K. Singh, and P. Senthilkumaran, “Fractional vortex lens,” Opt. Laser Technol. 42(6), 878–882 (2010).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (2)

W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87(3), 033806 (2013).
[Crossref]

F. G. Mitri, “High-order Bessel nonvortex beam of fractional type α,” Phys. Rev. A 85(2), 025801 (2012).
[Crossref]

Phys. Rev. D Part. Fields (1)

J. M. Cornwall, “Center vortices, nexuses, and fractional topological charge,” Phys. Rev. D Part. Fields 61(8), 085012 (2000).
[Crossref]

Waves Random Media (1)

C. Y. Yong, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14(1), 75–96 (2004).
[Crossref]

Other (3)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2007).

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2th ed. (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 (6)

Fig. 1
Fig. 1 The figure shows beam intensity simulations for fractional FBG beam along the direction z to describe how the beam spreads when the FBG beam transmits in Free Space Optics (FSO). For short distance transmission ( z <200m), the beam is approximate to non-diffracting; for long distance transmission ( z >200m), the diffractive spread of beams become obvious (for example, as γ = 5.1, when z increases from 300m to 1000m, the first ring radius of the fractional FBG beam spreads from 0.13m to 0.48m).
Fig. 2
Fig. 2 The average mode probability P l (r,z) of FBG beam along the direction of the beam radius r for different values of the quantum number deviation Δl .
Fig. 3
Fig. 3 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of γ when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities; (c) γ is half-integer, and Δl=0 signal probability densities; (d) γ is half-integer, and Δl=1 crosstalk probability densities.
Fig. 4
Fig. 4 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of beam waist w 0 when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities.
Fig. 5
Fig. 5 The average probability densities P l (r,z) of OAM modes for fractional FBG beam along the direction of the beam radius r with different values of Rytov variance σ R 2 when (a) Δl=0 , signal probability densities; (b) Δl=1 , crosstalk probability densities.
Fig. 6
Fig. 6 The normalized powers L z (l) of fractional OAM modes for fractional FBG beam along beam waist w 0 with different values of turbulence fluctuation σ R 2 when (a) Δl=0 , signal normalized powers L z ( l 0 ) ; (b) Δl=1 , crosstalk normalized powers L z (l l 0 ) .

Equations (18)

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

E( r,φ,z )= E free ( r,φ,z )exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z ) ],
E free ( r,φ,z )= 1 μ( z ) exp[ i k r 2 z 2kμ( z ) r 2 μ( z ) w 0 2 ] l 0 = C γ J | l 0 | [ k r r μ( z ) ]exp( i l 0 φ ) ,
E( r,φ,z )= l= β l ( r,z ) exp( ilφ ),
β l ( r,z )= 1 2π 0 2π E( r,φ,z ) exp( ilφ )dφ.
P l (r,z)= | β l (r,z) | 2 at = ( 1 2π ) 2 0 2π 0 2π E free ( r,φ,z ) E free ( r , φ ,z ) exp[ il( φ φ ) ] × exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z )+ ψ x ( r , φ ,z )+ ψ y ( r , φ ,z ) ] at dφd φ ,
exp[ ψ x ( r,φ,z )+ ψ y ( r,φ,z )+ ψ x ( r , φ ,z )+ ψ y ( r , φ ,z ) ] at =exp[ 1 2 D( r,φ, r , φ ;z ) ],
Φ n =0.033 C n 2 κ 11/3 [ f( κ l i )g( κ L o ) G x + G y ],
G x ( κ )=exp[ κ 2 / κ x 2 ], G y ( κ )= κ 11/3 ( κ 2 + κ x 2 ) 11/6 ,
D x ( r, r ,φ, φ ;z )=0.604 C n 2 k 2 z κ xm 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ xm 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 },
D y ( r, r ,φ, φ ;z )=0.604 C n 2 k 2 z κ m 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ m 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 } 0.604 C n 2 k 2 z κ ym 1/3 { [ r 2 + r 2 2r r cos( φ φ ) ] × [ 1+0.033 κ ym 2 [ r 2 + r 2 2r r cos( φ φ ) ] ] 1/6 },
D x ( r, r ,φ, φ ;z )0.604 C n 2 k 2 z κ xm 1/3 [ r 2 + r 2 2r r cos( φ φ ) ],
D y ( r, r ,φ, φ ;z )0.867 σ R 2 k z [ r 2 + r 2 2r r cos( φ φ ) ] 0.604 C n 2 k 2 z κ ym 1/3 [ r 2 + r 2 2r r cos( φ φ ) ].
exp[ 1 2 D( r,φ, r , φ ;z ) ]=exp{ [ 0.604 C n 2 k 2 z κ xm 1/3 +( 0.867 σ R 2 k z 0.604 C n 2 k 2 z κ ym 1/3 ) ] ×[ r 2 + r 2 2r r cos( φ φ ) ] } =exp[ r 2 + r 2 2r r cos( φ φ ) ρ 0 2 ],
ρ 0 = { 0.491 σ R 2 k z { ( 35.046 η x z k η x l i 2 +35.046z ) 1/6 +[ 1.766 ( 35.046 η y z k η y l i 2 +35.046z ) 1/6 ] } } 1/2 .
0 2π exp[in φ 1 +ηcos( φ 1 φ 2 )] d φ 1 =2πexp(in φ 2 ) I n (η),
P l (r,z)= w 0 2 w 2 ( z ) l 0 = C γ 2 | J | l 0 | ( k r r μ( z ) ) | 2 exp[ ( 1 w 2 ( z ) + 1 ρ 0 2 )2 r 2 ] I l l 0 ( 2 r 2 / ρ 0 2 ),
L z (l)= l= lB P l l= B P l .
B P l = w 0 2 w 2 ( z ) l 0 = C γ 2 0 | J | l 0 | ( k r r μ( z ) ) | 2 exp[ ( 1 w 2 ( z ) 1 ρ 0 2 )2 r 2 ] I l l 0 ( 2 r 2 / ρ 0 2 )rdr ,

Metrics