Abstract

Taking the partially coherent Hermite–sinh-Gaussian (H-ShG) beam as a more general type of partially coherent beams, a comparative study of the beam-width spreading of partially coherent H-ShG beams in atmospheric turbulence is performed by using the relative width, normalized beam width, and turbulence length. It is shown that the relative width versus the beam parameters, such as the spatial correlation length σ0, beam orders m, n, Sh-part parameter Ω0, and waist width w0, provides a simple and intuitive insight into the beam-width spreading of partially coherent H-ShG beams in turbulence, and the results are consistent with those using the turbulence length. The validity of our results is interpreted physically.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  3. A. Consortini and L. Ronchi, “Gaussian beams in turbulence media,” Appl. Opt. 9, 125-128 (1970).
    [CrossRef] [PubMed]
  4. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
    [CrossRef]
  5. H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  6. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
    [CrossRef] [PubMed]
  7. H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
    [CrossRef]
  8. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37-47 (2005).
    [CrossRef]
  9. Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568-570 (2006).
    [CrossRef] [PubMed]
  10. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
  11. J. Wu, “Propagation of a Gaussian-Schell through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  12. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  13. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  14. M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261-265 (2003).
    [CrossRef]
  15. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610-612 (2003).
    [CrossRef] [PubMed]
  16. 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, 1094-1102 (2003).
    [CrossRef]
  17. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  18. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic GSM beam in turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]
  19. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007).
    [CrossRef]
  20. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
    [CrossRef]
  21. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
    [CrossRef]
  22. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007).
    [CrossRef]
  23. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21-28 (2008).
    [CrossRef]
  24. H. T. Eyyuboğlu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10, 015005 (2008).
    [CrossRef]
  25. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954-961 (1998).
    [CrossRef]
  26. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
    [CrossRef]
  27. A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express 16, 8366-8381 (2008).
    [CrossRef] [PubMed]
  28. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).
  29. I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

2008 (3)

2007 (4)

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007).
[CrossRef]

2006 (3)

2005 (3)

2004 (1)

2003 (4)

2002 (2)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

1998 (1)

1991 (1)

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990 (1)

J. Wu, “Propagation of a Gaussian-Schell through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

1970 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Baykal, Y.

H. T. Eyyuboğlu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10, 015005 (2008).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Boardman, A. D.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

Cai, Y.

H. T. Eyyuboğlu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10, 015005 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568-570 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic GSM beam in turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Casperson, L. W.

Chen, X.

Consortini, A.

Dogariu, A.

Eyyuboglu, H. T.

H. T. Eyyuboğlu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10, 015005 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37-47 (2005).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Gbur, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Gradysteyn, I. S.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

He, S.

Ji, X.

Lin, Q.

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Lü, B.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Ronchi, L.

Ryzhik, I. M.

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Salem, M.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261-265 (2003).
[CrossRef]

Shirai, T.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tovar, A. A.

Wolf, E.

Wu, J.

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, “Propagation of a Gaussian-Schell through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

Yang, A.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Zahid, M.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Zhang, E.

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B: Lasers Opt. (1)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic GSM beam in turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

J. Mod. Opt. (2)

J. Wu, “Propagation of a Gaussian-Schell through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

H. T. Eyyuboğlu, Y. Cai, and Y. Baykal, “Spectral shifts of general beams in turbulent media,” J. Opt. A, Pure Appl. Opt. 10, 015005 (2008).
[CrossRef]

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

Opt. Commun. (5)

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37-47 (2005).
[CrossRef]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007).
[CrossRef]

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261-265 (2003).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989).
[CrossRef]

Opt. Eng. (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Other (4)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

I. S. Gradysteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

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

Beam widths w ( z ) turb , w ( z ) free and relative width w ( z ) turb w ( z ) free of partially coherent H-ShG beams versus the spatial correlation length σ 0 : (a) z = 2 km , (b) z = 7 km , (c) z = 30 km .

Fig. 2
Fig. 2

Beam widths w ( z ) turb , w ( z ) free and relative width w ( z ) turb w ( z ) free of partially coherent H-ShG beams versus the beam orders m, n: (a) z = 2 km , (b) z = 7 km , (c) z = 30 km .

Fig. 3
Fig. 3

Beam widths w ( z ) turb , w ( z ) free and relative width w ( z ) turb w ( z ) free of partially coherent H-ShG beams versus the Sh-part parameter Ω 0 : (a) z = 2 km , (b) z = 7 km , (c) z = 30 km .

Fig. 4
Fig. 4

Beam widths w ( z ) turb , w ( z ) free and relative width w ( z ) turb w ( z ) free of partially coherent H-ShG beams versus the waist width w 0 : (a) z = 2 km , (b) z = 7 km , (c) z = 30 km .

Fig. 5
Fig. 5

Normalized beam widths w ( z ) turb w ( 0 ) , w ( z ) free w ( 0 ) of partially coherent beams versus the propagation distance z for selected values of σ 0 .

Fig. 6
Fig. 6

Normalized beam widths w ( z ) turb w ( 0 ) , w ( z ) free w ( 0 ) of partially coherent beams versus the propagation distance z for selected values of m, n.

Fig. 7
Fig. 7

Normalized beam widths w ( z ) turb w ( 0 ) , w ( z ) free w ( 0 ) of partially coherent beams versus the propagation distance z for selected values of Ω 0 .

Fig. 8
Fig. 8

Normalized beam widths w ( z ) turb w ( 0 ) , w ( z ) free w ( 0 ) of partially coherent beams versus the propagation distance z for selected values of w 0 .

Tables (4)

Tables Icon

Table 1 Turbulence Lengths z T of Partially Coherent H-ShG Beams for Selected Values of σ 0 and η %

Tables Icon

Table 2 Turbulence Lengths z T of Partially Coherent H-ShG Beams for Selected Values of m, n, and η %

Tables Icon

Table 3 Turbulence Lengths z T of Partially Coherent H-ShG Beams for Selected Values of Ω 0 and η %

Tables Icon

Table 4 Turbulence Lengths z T of Partially Coherent H-ShG Beams for Selected Values of w 0 and η %

Equations (47)

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

U ( ρ , z = 0 ) = H m ( 2 w 0 ρ x ) H n ( 2 w 0 ρ y ) exp ( ρ x 2 + ρ y 2 w 0 2 ) sinh ( Ω 0 ρ x + Ω 0 ρ y ) ,
W ( 0 ) ( ρ 1 , ρ 2 , z = 0 ) = H m ( 2 w 0 ρ 1 x ) H n ( 2 w 0 ρ 1 y ) exp ( ρ 1 x 2 + ρ 1 y 2 w 0 2 ) sinh ( Ω 0 ρ 1 x + Ω 0 ρ 1 y ) × H m ( 2 w 0 ρ 2 x ) H n ( 2 w 0 ρ 2 y ) exp ( ρ 2 x 2 + ρ 2 y 2 w 0 2 ) sinh ( Ω 0 ρ 2 x + Ω 0 ρ 2 y ) × exp [ ( ρ 1 x ρ 2 x ) 2 2 σ 0 2 ] exp [ ( ρ 1 y ρ 2 y ) 2 2 σ 0 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 W ( 0 ) ( ρ 1 , ρ 2 , z = 0 ) exp { ( i k 2 z ) [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } C ψ ( ρ 1 , ρ 2 , ρ 1 , ρ 2 , z ) ,
C ψ ( ρ 1 , ρ 2 , ρ 1 , ρ 2 , z ) = exp { 4 π 2 k 2 z 0 1 0 d κ d ξ κ Φ n ( κ ) [ 1 J 0 ( κ ( 1 ξ ) ( ρ 2 ρ 1 ) + ξ ( ρ 2 ρ 1 ) ) ] } ,
u = ρ 1 + ρ 2 2 , v = ρ 1 ρ 2 ,
I ( ρ , z ) = W ( ρ , ρ , z ) = 1 4 ( k 2 π z ) 2 d 2 u d 2 v H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ] × H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] exp ( 2 u 2 w 0 2 ) exp ( v 2 ϵ 2 ) exp ( i k z u v ) × exp ( i k z ρ v ) { exp [ 2 Ω 0 ( u x + u y ) ] exp [ Ω 0 ( v x + v y ) ] exp [ Ω 0 ( v x + v y ) ] + exp [ 2 Ω 0 ( u x + u y ) ] } ,
1 ϵ 2 = 1 2 w 0 2 + 1 2 σ 0 2 + T ,
T = 1 3 k 2 π 2 z 0 κ 3 Φ n ( κ ) d κ .
Φ n ( κ ) = 0.033 C n 2 κ 11 3 exp ( κ 2 κ m 2 ) ,
T = 0.5466 k 2 C n 2 l 0 1 3 z .
w ( z ) = ρ 2 I ( ρ , z ) d 2 ρ I ( ρ , z ) d 2 ρ .
w ( z ) turb = ( A + B z 2 + 2.18641 C n 2 l 0 1 3 z 3 ) 1 2 ,
A = R 1 R 0 ,
B = R 2 R 0 k 2 ,
R 0 = exp ( Ω 0 2 w 0 2 ) L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) 1 ,
R 1 = exp ( Ω 0 2 w 0 2 ) { w 0 2 2 ( L n 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 1 1 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 0 ( Ω 0 2 w 0 2 ) L n 1 1 ( Ω 0 2 w 0 2 ) ) + Ω 0 2 w 0 4 [ 1 2 L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 1 1 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 2 2 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 0 ( Ω 0 2 w 0 2 ) L n 1 1 ( Ω 0 2 w 0 2 ) + L m 0 ( Ω 0 2 w 0 2 ) L n 2 2 ( Ω 0 2 w 0 2 ) ] } w 0 2 2 ( m + n + 1 ) ,
R 2 = exp ( Ω 0 2 w 0 2 ) { 2 σ 0 2 L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + 2 w 0 2 [ L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 1 1 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 0 ( Ω 0 2 w 0 2 ) L n 1 1 ( Ω 0 2 w 0 2 ) ] } 2 σ 0 2 + 2 Ω 0 2 2 w 0 2 ( m + n + 1 ) ,
w ( z ) free = ( A + B z 2 ) 1 2 .
w ( 0 ) = A .
w ( z ) turb = ( A 1 + B 1 z 2 + 2.18641 C n 2 l 0 1 3 z 3 ) 1 2 ,
A 1 = exp ( Ω 0 2 w 0 2 ) ( w 0 2 2 + Ω 0 2 w 0 4 2 ) w 0 2 2 exp ( Ω 0 2 w 0 2 ) 1 ,
B 1 = ( 2 σ 0 2 + 2 w 0 2 ) k 2 + 2 Ω 0 2 k 2 exp ( Ω 0 2 w 0 2 ) 1 .
w ( z ) turb = ( A + B 2 z 2 + 2.18641 C n 2 l 0 1 3 z 3 ) 1 2 ,
B 2 = R 2 R 0 k 2 ,
R 2 = exp ( Ω 0 2 w 0 2 ) { ( 2 w 0 2 ) [ L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 1 1 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) + L m 0 ( Ω 0 2 w 0 2 ) L n 1 1 ( Ω 0 2 w 0 2 ) ] } + 2 Ω 0 2 ( 2 w 0 2 ) ( m + n + 1 ) .
w ( z ) turb w ( z ) free = 1 + 2.18641 C n 2 l 0 1 3 A + B z 2 z 3 .
w ( z ) turb w ( 0 ) = 1 + B A z 2 + 2.18641 C n 2 l 0 1 3 A z 3
w ( z ) free w ( 0 ) = 1 + B A z 2 ,
w 2 ( z T ) turb w 2 ( z T ) free w 2 ( z T ) turb = η % .
w ( z ) = I ρ 2 d 2 ρ I d 2 ρ = F F 0 ,
F 0 = W ( 0 ) ( ρ , ρ , z = 0 ) d 2 ρ = 1 2 2 m + n 1 m ! n ! w 0 2 π [ exp ( Ω 0 2 w 0 2 ) L m 0 ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) 1 ] ,
F = F 1 + F 2 + F 3 + F 4 ,
F l = ρ 2 I l ( ρ , z ) d 2 ρ , ( l = 1 , 2 , 3 , 4 ) ,
I 1 ( ρ , z ) = 1 4 ( k 2 π z ) 2 d 2 u d 2 v × H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ] H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] × exp ( 2 u 2 w 0 2 ) exp ( v 2 ϵ 2 ) exp ( i k z u v ) exp ( i k z ρ v ) exp [ 2 Ω 0 ( u x + u y ) ] ,
I 2 ( ρ , z ) = 1 4 ( k 2 π z ) 2 d 2 u d 2 v × H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ] H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] × exp ( 2 u 2 w 0 2 ) exp ( v 2 ϵ 2 ) exp ( i k z u v ) exp ( i k z ρ v ) exp [ Ω 0 ( v x + v y ) ] ,
x 2 exp ( i 2 π x s ) d x = 1 ( 2 π ) 2 δ ( s ) ,
F 1 = F 11 + F 12 ,
F 11 = 1 4 ( z k ) 2 d 2 u d 2 v H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ] H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] exp ( 2 u 2 w 0 2 ) exp ( v 2 ϵ 2 ) exp ( i k z u v ) exp [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ( v y ) ,
F 12 = 1 4 ( z k ) 2 d 2 u d 2 v H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ] H m [ 2 w 0 ( u x v x 2 ) ] H n [ 2 w 0 ( u y v y 2 ) ] exp ( 2 u 2 w 0 2 ) exp ( v 2 ϵ 2 ) exp ( i k z u v ) exp [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ( v y ) .
exp [ ( x y ) 2 ] H m ( x ) H n ( x ) d x = 2 n π m ! y n m L m n m ( 2 y 2 ) ,
exp ( x 2 ) H m ( x + y ) H n ( x + z ) d x
= 2 n π m ! y n m z n m L m n m ( 2 y z ) ,
f ( x ) δ ( x ) d x = f ( 0 ) ,
F 11 = 1 4 ( z k ) 2 w 0 2 2 n + m 1 π n ! m ! exp ( Ω 0 2 w 0 2 ) L n 0 ( Ω 0 2 w 0 2 ) × [ ( k 2 w 0 2 4 z 2 ϵ 2 ) L m 0 ( Ω 0 2 w 0 2 ) k 2 w 0 4 Ω 0 2 z 2 ( 1 4 L m 0 ( Ω 0 2 w 0 2 ) + L m 1 1 ( Ω 0 2 w 0 2 ) + L m 2 2 ( Ω 0 2 w 0 2 ) ) ( 2 w 0 2 + k 2 w 0 2 2 z 2 ) L m 1 1 ( Ω 0 2 w 0 2 ) ] .
F 2 = F 21 + F 22 ,
F 21 = 1 4 ( z k ) 2 w 0 2 2 n + m 1 π n ! m ! [ k 2 w 0 2 4 z 2 ϵ 2 + Ω 0 2 m ( 2 w 0 2 + k 2 w 0 2 2 z 2 ) ] .
w ( z ) = ( A + B z 2 + 2.18641 C n 2 l 0 1 3 z 3 ) 1 2 .

Metrics