Abstract

The general model of beam wander for a coherent Gaussian beam propagating through atmospheric turbulence is extended to the case of a partially polarized electromagnetic Gaussian–Schell model (EGSM) beam. The expression of the beam wander is obtained by characterizing the EGSM beam with the effective beam parameters. The effects of initial spatially coherent lengths, degree of polarization, and phase curvature are examined in detail. A condition is derived under which beams with different spatial coherence and degrees of polarization will generate the same beam wander.

© 2012 Optical Society of America

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References

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  1. J. C. Ricklin and F. M. Davidson, “Atmospheric optical communication with a Gaussian Schell beam,” J. Opt. Soc. Am. A 20, 856–866 (2003).
    [CrossRef]
  2. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
    [CrossRef]
  3. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
    [CrossRef]
  4. C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
    [CrossRef]
  5. Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37, 1553–1555 (2012).
    [CrossRef]
  6. F. Dios, J. A. Rubio, A. Rodriguez, and A. Comeron, “Scintillation and beam-wander analysis in an optical ground station-satellite uplink,” Appl. Opt. 43, 3866–3873 (2004).
    [CrossRef]
  7. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
    [CrossRef]
  8. D. H. Tofsted, “Outer-scale effects in beam-wander and angle-of-arrival variances,” Appl. Opt. 31, 5865–5870(1992).
    [CrossRef]
  9. H. T. Eyyuboglu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
    [CrossRef]
  10. C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
    [CrossRef]
  11. G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
    [CrossRef]
  12. X. Xiao and D. G. Voelz, “Beam wander analysis for focused partially coherent beams propagating in turbulence,” Opt. Eng. 51, 026001 (2012).
    [CrossRef]
  13. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  14. G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
    [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]
  16. H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
    [CrossRef]
  17. G. Wu, Q. Lou, J. Zhou, H. Guo, H. Zhao, and Z. Yuan, “Beam conditions for radiation generated by an electromagnetic J0-correlated Schell-model source,” Opt. Lett. 33, 2677–2679(2008).
    [CrossRef]
  18. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
    [CrossRef]
  19. G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
    [CrossRef]
  20. J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
    [CrossRef]
  21. D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express 18, 20746–20758 (2010).
    [CrossRef]

2012 (2)

X. Xiao and D. G. Voelz, “Beam wander analysis for focused partially coherent beams propagating in turbulence,” Opt. Eng. 51, 026001 (2012).
[CrossRef]

Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37, 1553–1555 (2012).
[CrossRef]

2011 (3)

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

2010 (1)

2009 (1)

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

2008 (3)

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

H. T. Eyyuboglu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

G. Wu, Q. Lou, J. Zhou, H. Guo, H. Zhao, and Z. Yuan, “Beam conditions for radiation generated by an electromagnetic J0-correlated Schell-model source,” Opt. Lett. 33, 2677–2679(2008).
[CrossRef]

2007 (2)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
[CrossRef]

2006 (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

2005 (1)

H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
[CrossRef]

2004 (1)

2003 (2)

2002 (2)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[CrossRef]

1992 (1)

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[CrossRef]

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

Arpali, C.

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

Arpali, S. A.

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

Baykal, Y.

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

Berman, G. P.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
[CrossRef]

Borah, D. K.

Cai, Y.

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

Chumak, A. A.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
[CrossRef]

Cil, C. Z.

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

H. T. Eyyuboglu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

Comeron, A.

Dang, A.

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

Davidson, F. M.

Dios, F.

Dogariu, A.

Eyyuboglu, H. T.

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

H. T. Eyyuboglu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

Gbur, G.

Gorshkov, V. N.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
[CrossRef]

Gu, Y.

Guo, H.

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

G. Wu, Q. Lou, J. Zhou, H. Guo, H. Zhao, and Z. Yuan, “Beam conditions for radiation generated by an electromagnetic J0-correlated Schell-model source,” Opt. Lett. 33, 2677–2679(2008).
[CrossRef]

Korotkova, O.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[CrossRef]

Lou, Q.

Luo, B.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[CrossRef]

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

Ponomarenko, S. A.

H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
[CrossRef]

Pu, J.

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

Ricklin, J. C.

Rodriguez, A.

Roychowadhury, H.

H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
[CrossRef]

Rubio, J. A.

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Shirai, T.

Tofsted, D. H.

Voelz, D. G.

X. Xiao and D. G. Voelz, “Beam wander analysis for focused partially coherent beams propagating in turbulence,” Opt. Eng. 51, 026001 (2012).
[CrossRef]

D. K. Borah and D. G. Voelz, “Spatially partially coherent beam parameter optimization for free space optical communications,” Opt. Express 18, 20746–20758 (2010).
[CrossRef]

Wolf, E.

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003).
[CrossRef]

Wu, G.

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

G. Wu, Q. Lou, J. Zhou, H. Guo, H. Zhao, and Z. Yuan, “Beam conditions for radiation generated by an electromagnetic J0-correlated Schell-model source,” Opt. Lett. 33, 2677–2679(2008).
[CrossRef]

Xiao, X.

X. Xiao and D. G. Voelz, “Beam wander analysis for focused partially coherent beams propagating in turbulence,” Opt. Eng. 51, 026001 (2012).
[CrossRef]

Yu, S.

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

Yuan, Z.

Zhao, H.

Zhao, T.

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

Zhou, J.

Appl. Opt. (2)

Appl. Phys. B (3)

C. Arpali, S. A. Arpali, Y. Baykal, and H. T. Eyyuboglu, “Intensity fluctuations of partially coherent laser beam arrays in weak atmospheric turbulence,” Appl. Phys. B 103, 237–244 (2011).
[CrossRef]

H. T. Eyyuboglu and C. Z. Cil, “Beam wander of dark hollow, flat-topped and annular beams,” Appl. Phys. B 93, 595–604 (2008).
[CrossRef]

C. Z. Cil, H. T. Eyyuboglu, Y. Baykal, and Y. Cai, “Beam wander characteristics of cos and cosh-Gaussian beams,” Appl. Phys. B 95, 763–771 (2009).
[CrossRef]

J. Mod. Opt. (1)

H. Roychowadhury, S. A. Ponomarenko, and E. Wolf, “Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere,” J. Mod. Opt. 52, 1611–1618 (2005).
[CrossRef]

J. Opt. (1)

G. Wu, B. Luo, S. Yu, A. Dang, and H. Guo, “The propagation of electromagnetic Gaussian-Schell model beams through atmospheric turbulence in a slanted path,” J. Opt. 13, 035706 (2011).
[CrossRef]

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

Opt. Commun. (2)

G. Wu, B. Luo, S. Yu, A. Dang, T. Zhao, and H. Guo, “Effects of coherence and polarization on the beam spreading and direction through atmospheric turbulence,” Opt. Commun. 284, 4275–4278 (2011).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
[CrossRef]

Opt. Eng. (2)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

X. Xiao and D. G. Voelz, “Beam wander analysis for focused partially coherent beams propagating in turbulence,” Opt. Eng. 51, 026001 (2012).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E (2)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606 (2007).
[CrossRef]

Proc. SPIE (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmosphere: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1.
Fig. 1.

Dimensionless quantity Bw of collimated beams versus the structure parameter Cn2 for different coherent conditions. The parameters are L=10km, w0=0.025m, P=0, and λ=1060nm.

Fig. 2.
Fig. 2.

Dimensionless quantity Bw as a function of (a) Cn2 for different degrees of polarization P and (b) the degree of polarization P for different coherent lengths σyy. In both plots, the rest of the parameters are the same as those in Fig. 1.

Fig. 3.
Fig. 3.

Dimensionless quantity Bw as a function of Cn2. The coherent conditions and polarization states satisfy Eq. (19). The other parameters are w0=0.03m, λ=1060nm, L=10km, and F0=.

Fig. 4.
Fig. 4.

Dimensionless quantity Bw as a function of propagation distance L for collimated and focused beams. The parameters are σxx=σyy, Cn2=1014m2/3, and the others are the same as those in Fig. 1.

Equations (19)

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

rc2=4πk2WFS2(L)0L0κΦn(κ)HLS(κ,z){1exp[ΛLκ2(1z/L)2k]}dκdz,
Φn(κ)=0.033Cn2κ11/3,
HLS(κ,z)=exp[κ2WLT2(z)],
Λ=2LkWFS2(L).
WLT(z)=[2r2I(r,z)d2rI(r,z)d2r]1/2,
W(0)(r1,r2,0)=[Wxx(0)(r1,r2)00Wyy(0)(r1,r2)],
Wii(0)(r1,r2)=Ui(r1,0)Ui*(r2,0)exp[(r1r2)22σii2],(i=x,y),
Ui(r,0)=Aiexp[(1w02+jk2F0)r2],
I(r,z)=Ax2w02wxx2(z)exp[2r2wxx2(z)]+Ay2w02wyy2(z)exp[2r2wyy2(z)],
wii(z)=w0[(1zF0)2+(1+w02σii2+2w02ρsp(z)2)(2zkw02)2]1/2,i=x,y.
ρsp(z)=(0.55Cn2k2z)3/5.
WLT(z)=1+P2wxx2(z)+1P2wyy2(z),
P=|Ax2Ay2|Ax2+Ay2.
1exp[ΛLκ2(1z/L)2k]ΛLκ2(1z/L)2k.
rc2=7.25L2Cn20L(1zL)2WLT(z)1/3dz,
WLT(z)=w0[(1zF0)2+(1+w02δp2)Λ02(zL)2+2w02Λ02ρsp2(L)(zL)16/5]1/2,
ΔWcohw02[1+(1+w02δp2)Λ02(zL)2],
ΔWatm2w04Λ02ρsp2(L)(zL)16/5,
δp=1+Pσxx2+1Pσyy2=constant.

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