Abstract

The model of partially coherent annular beams with linear non-uniformity field profile in the x direction is set up. The analytic expressions for the average intensity and the centre of gravity of partially coherent annular beams with decentered field propagating through atmospheric turbulence along a slant path are derived. The propagation equation governing the position of the intensity maximum is also given. It is found that the beam non-uniformity is amended gradually as the propagation distance and the strength of turbulence increase. The centre of beam gravity is independent of both the propagation distance and the turbulence. However, the position of the intensity maximum changes versus the propagation distance and the turbulence, and is farthest away from the propagation z-axis at a certain propagation distance. When the propagation distance is large enough, the position of the intensity maximum reaches an asymptotic value which increases with decreasing the zenith angle and is largest for the free space case. When the propagation distance is large enough, the position of the intensity maximum is not on the propagation z-axis, and is nearer to the propagation z-axis than the centre of beam gravity. On the other hand, changes in the intensity maximum in the far field are also examined in this paper.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed., (Cambridge U Press, 1997).
  2. E. Valdez and J. Agraz, Laser Communication Links for use Between Satellites (San Diego State U Press, 1999).
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  6. A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).
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    [CrossRef]
  8. X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281(18), 4765–4770 (2008).
    [CrossRef]
  9. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
    [CrossRef] [PubMed]
  10. H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
    [CrossRef]
  20. 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(3), 035706 (2011).
    [CrossRef]
  21. X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25(1), 74–79 (2008).
    [CrossRef] [PubMed]
  22. H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  25. M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10(5), 595–597 (1980).
    [CrossRef]

2011 (2)

X. Chu, “Evolution of an airy beam in turbulence,” Opt. Lett. 36(14), 2701–2703 (2011).
[CrossRef] [PubMed]

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(3), 035706 (2011).
[CrossRef]

2010 (2)

H. Mao and D. Zhao, “Second-order intensity-moment characteristics for broadband partially coherent flat-topped beams in atmospheric turbulence,” Opt. Express 18(2), 1741–1755 (2010).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

2009 (2)

2008 (3)

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281(18), 4765–4770 (2008).
[CrossRef]

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25(1), 74–79 (2008).
[CrossRef] [PubMed]

2006 (2)

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
[CrossRef]

2004 (1)

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

2002 (1)

Y. Li, “New expressions for flat-topped light beam,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027– S1049 (1992).
[CrossRef]

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gaussian,” Opt. Commun. 70(5), 361–364 (1989).
[CrossRef]

1980 (1)

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10(5), 595–597 (1980).
[CrossRef]

1979 (1)

1978 (1)

1973 (1)

A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).

Altay, S.

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
[CrossRef]

Andrews, L. C.

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

Baykal, Y.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10(5), 595–597 (1980).
[CrossRef]

Cai, Y.

Chen, X.

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281(18), 4765–4770 (2008).
[CrossRef]

Chu, X.

Dang, A.

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(3), 035706 (2011).
[CrossRef]

Eyyuboglu, H. T.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
[CrossRef]

Guo, H.

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(3), 035706 (2011).
[CrossRef]

He, S.

Ji, X.

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

X. Ji and X. Li, “Directionality of Gaussian array beams propagating in atmospheric turbulence,” J. Opt. Soc. Am. A 26(2), 236–243 (2009).
[CrossRef] [PubMed]

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281(18), 4765–4770 (2008).
[CrossRef]

Kon, A. I.

A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).

Korotkova, O.

Leader, J. C.

Li, X.

Li, Y.

Y. Li, “New expressions for flat-topped light beam,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

Liu, Z.

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(3), 035706 (2011).
[CrossRef]

Mao, H.

Mironov, V. L.

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10(5), 595–597 (1980).
[CrossRef]

A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).

Plonus, M. A.

Qu, J.

Tseitlin, V. E.

A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).

Vetelino, F. E. S.

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

Wang, S. C. H.

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027– S1049 (1992).
[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(3), 035706 (2011).
[CrossRef]

Wu, Y.

Yu, S.

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(3), 035706 (2011).
[CrossRef]

Yuan, Y.

Zahid, M.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gaussian,” Opt. Commun. 70(5), 361–364 (1989).
[CrossRef]

Zhao, D.

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gaussian,” Opt. Commun. 70(5), 361–364 (1989).
[CrossRef]

Appl. Phys. B (1)

H. T. Eyyuboğlu, Y. Baykal, and X. Ji, “Radius of curvature variations for annular, dark hollow and flat topped beams in turbulence,” Appl. Phys. B 99, 801–807 (2010).

Izv. Vyssh. Uchebn. Zaved. Physica (1)

A. I. Kon, V. L. Mironov, and V. E. Tseitlin, “Turbulent broadening of the focal light spot formed by an annular aperture,” Izv. Vyssh. Uchebn. Zaved. Physica 11, 149–151 (1973).

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(3), 035706 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (4)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gaussian,” Opt. Commun. 70(5), 361–364 (1989).
[CrossRef]

H. T. Eyyuboğlu, S. Altay, and Y. Baykal, “Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence,” Opt. Commun. 264(1), 25–34 (2006).
[CrossRef]

Y. Li, “New expressions for flat-topped light beam,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281(18), 4765–4770 (2008).
[CrossRef]

Opt. Express (3)

Opt. Laser Technol. (1)

H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40(1), 156–166 (2008).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), S1027– S1049 (1992).
[CrossRef]

Proc. SPIE (1)

F. E. S. Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004).
[CrossRef]

Sov. J. Quantum Electron. (1)

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10(5), 595–597 (1980).
[CrossRef]

Other (6)

I. T. U.-R. Document, 3J/31-E, “On propagation data and prediction methods required for the design of space-to-earth and earth-to-space optical communication systems,” in Radio-Communication Study Group meeting, Budapest (2001), p. 7.

M. Born and E. Wolf, Principles of Optics, 6th ed., (Cambridge U Press, 1997).

E. Valdez and J. Agraz, Laser Communication Links for use Between Satellites (San Diego State U Press, 1999).

Free space optics, www. Free space optics. org.

A. E. Siegman, Laser (University Science Press, California, 1986).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, Bellingham, 2005)

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Figures (11)

Fig. 1
Fig. 1

3D intensity distributions I( x , y ,0) and counter lines at the initial plane z = 0 for different values of the decentered beam parameter s and the beam obscure ratio ε. M = N = 10, w0 = 0.05m.

Fig. 2
Fig. 2

The centre of beam gravity versus (a) the decentered beam parameter s, and (b) the obscure ratio ε. M = N = 10, λ = 1060nm.

Fig. 3
Fig. 3

(a) Two-dimensional (2D) average intensity distributions <I(x,0,L)> with and without the quadratic approximation, (b) Relative error of the quadratic approximation. σ0 = 0.03m, ζ = π/4, ε = 0.4, s = 0.06m.

Fig. 4
Fig. 4

Counter lines of the normalized average intensity distribution <I(x,y,L)>/Imax at different propagation distance in free space. σ0 = 0.02m, ε = 0.5, s = 0.06m

Fig. 5
Fig. 5

Counter lines of the normalized average intensity distribution distributions <I(x,y,L)>/Imax at different propagation distance in turbulence. σ0 = 0.02m, ζ = 0.45π, ε = 0.5, s = 0.06m.

Fig. 6
Fig. 6

2D normalized average intensity <I(x,0,L)>/Imax at different propagation distance in turbulence. σ0 = 0.03m, ζ = π/4, ε = 0.4, s = 0.06m.

Fig. 7
Fig. 7

Position xmax of the intensity distributions maximum versus the propagation distance L. σ0 = 0.03m, ε = 0.4, s = 0.06m.

Fig. 8
Fig. 8

2D average intensity distributions <I(x,0,L)> at different propagation distance in three propagation ways. σ0 = 0.03m, ζ = π/6, ε = 0.6, s = 0.08m

Fig. 9
Fig. 9

2D average relative intensity distributions <I(x,0,L)>/Ifmax in the far field for (a) the decentered beam parameter s = , and (b) s = 0.06m. σ0 = 1m, ζ = π/6, ε = 0.8, L = 20km.

Fig. 10
Fig. 10

2D average relative intensity distributions <I(x,0,L)>/Ifmax propagating in the far field for (a) the beam obscure ratio ε = 0.1, and (b) ε = 0.9. σ0 = 1m, ζ = π/6, s = 0.2m, L = 20km.

Fig. 11
Fig. 11

2D average relative intensity distributions <I(x,0,L)>/Ifmax propagating in the far field for (a) the spatial correlation length σ0 = , and (b) σ0 = 0.02m. ε = 0.4, ζ = π/6, s = 0.1m, L = 20km.

Equations (28)

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

U 0 ( ρ ,0 ) | FT = m=1 M α m exp[ ( m p m x 2 w 0 2 ) ] n=1 N α n exp[ ( n p n y 2 w 0 2 ) ],
α m = ( 1 ) m+1 M! m!( Mm )! , α m = ( 1 ) m+1 M! m!( Mm )! ,
U 0 ( ρ ,0 )=( 1 x /s ){ m=1 M α m exp[ ( m p m x 2 w 0 2 ) ] n=1 N α n exp[ ( n p n y 2 w 0 2 ) ] m=1 M α m exp[ ( m p m x 2 w 0 2 ) ] n=1 N α n exp[ ( n p n y 2 w 0 2 ) ] },
W 0 ( ρ 1 , ρ 2 ,0 )= U 0 * ( ρ 1 ,0 ) U 0 ( ρ 2 ,0 ) μ (0) ( ρ 1 , ρ 2 ,0 ) = W 011 ( ρ 1 , ρ 2 ,0 )+ W 022 ( ρ 1 , ρ 2 ,0 ) W 012 ( ρ 1 , ρ 2 ,0 ) W 021 ( ρ 1 , ρ 2 ,0 ),
W 011 ( ρ 1 , ρ 2 ,0 )= m=1 M m =1 M α m α m exp[ ( m p m x 1 2 w 0 2 + m P m x 2 2 w 0 2 ) ] exp[ ( x 1 x 2 ) 2 2 σ 0 2 ] × n=1 N n =1 N α n α n exp[ ( n p n y 1 2 w 0 2 +n p n y 2 2 w 0 2 ) ] exp[ ( y 1 y 2 ) 2 2 σ 0 2 ] ×( 1 x 1 /s )( 1 x 2 /s ),
W 012 ( ρ 1 , ρ 2 ,0 )= m=1 M m =1 M α m α m exp[ ( m p m x 1 2 w 0 2 + m P m x 2 2 w ' 0 2 ) ] exp[ ( x 1 x 2 ) 2 2 σ 0 2 ] × n=1 N n =1 N α n α n exp[ ( n p n y 1 2 w 0 2 +n p n y 2 2 w 0 2 ) ] exp[ ( y 1 y 2 ) 2 2 σ 0 2 ] ×( 1 x 1 /s )( 1 x 2 /s ),
I( ρ,L ) = ( k 2πL ) 2 d 2 ρ 1 1 d 2 ρ 2 W 0 ( ρ 1 , ρ 2 ,0 ) ×exp{ ik 2L [ ( ρ 1 2 ρ 2 2 )2( ρ ρ 1 ρ ρ 2 ) ] } exp[ ψ*( ρ, ρ 1 )+ψ( ρ, ρ 2 ) ] m ,
exp[ ψ*( ρ, ρ 1 )+ψ( ρ, ρ 2 ) ] m exp{ [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ]/ ρ 0 2 },
ρ 0 = [ 1.46 k 2 sec( ζ ) 0 H C n 2 ( h ) ( 1 h H ) 5/3 dh ] 3/5 ,
C n 2 ( h )=8.148× 10 56 V 2 h 2 exp( h/1000 ) +2.7× 10 16 exp( h/1500 )+ C 0 exp( h/100 ),
I( ρ,L ) = ( k 2πL ) 2 d 2 ρ d 2 ρ d W 0 ( ρ , ρ d ,0 ) ×exp( ik L ρ ρ d )exp( ik L ρ ρ d )exp( ρ d 2 / ρ 0 2 ),
I( ρ,L ) = I 11 ( ρ,L ) + I 22 ( ρ,L ) I 12 ( ρ,L ) I 21 ( ρ,L ) ,
I 11 ( ρ,L ) = ( k 2L ) 2 m=1 M m =1 M n=1 M n =1 M P w 0 2 Q 1 Q 3 A 11 B 11 { 1 s 2 [ w 0 2 2 Q 1 + 1 16 A 11 2 ( 2 A 11 k 2 x 2 L 2 ) ( w 0 4 Q 1 2 C 11 2 1 ) ] 1 s ( i w 0 2 kx C 11 2 Q 1 A 11 L )+1 }exp( k 2 x 2 4 A 11 L 2 )exp( k 2 y 2 4 B 11 L 2 ),
I 12 ( ρ,L ) = ( k 2L ) 2 m=1 M m =1 M n=1 M n =1 M P ε 2 w 0 2 Q 5 Q 7 A 12 B 12 { 1 s 2 [ ε 2 w 0 2 2 Q 5 + 1 16 A 12 2 ( 2 A 12 k 2 x 2 L 2 ) ( ε 4 w 0 4 Q 5 2 C 12 2 1 ) ] 1 s [ i ε 2 w 0 2 kx C 12 2 Q 5 A 12 L ]+1 }exp( k 2 x 2 4 A 12 L 2 )exp( k 2 y 2 4 B 12 L 2 ),
A 11 = Q 1 4 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 w 0 2 4 Q 1 C 11 2 ,
B 11 = Q 3 4 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 w 0 2 4 Q 3 ( Q 4 w 0 2 i k L ) 2 ,
A 12 = Q 5 4 ε 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 ε 2 w 0 2 4 Q 5 C 12 2 ,
B 12 = Q 7 4 ε 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 ε 2 w 0 2 4 Q 7 ( Q 8 ε 2 w 0 2 i k L ) 2 ,
C 11 = Q 2 w 0 2 i k L , C 12 = Q 6 ε 2 w 0 2 i k L , P= α m α m α n α n ,
Q 1 =m p m + m p m , Q 2 =m p m m p m , Q 3 =n p n + n p n , Q 4 =n p n n p n ,
Q 5 = ε 2 m p m + m p m , Q 6 = ε 2 m p m m p m , Q 7 = ε 2 n p n + n p n , Q 8 = ε 2 n p n n p n .
x ¯ = xI(x,y,L)dxdy I(x,y,L)dxdy , y ¯ = yI(x,y,L)dxdy I(x,y,L)dxdy .
x ¯ ={ m=1 M m =1 M n=1 N n =1 N P s [ w 0 4 ( 1+ ε 4 )+ i w 0 2 Q 2 ( 1+ ε 2 )L /k Q 1 Q 1 Q 3 + w 0 4 + i w 0 2 Q 6 L /k Q 5 Q 5 Q 7 + w 0 4 + i w 0 2 Q 10 L /k Q 9 Q 9 Q 11 ] } /C ,
y ¯ =0.
C= m=1 M m =1 M n=1 N n =1 N P{ w 0 2 Q 1 Q 3 [ 1+ ε 2 + w 0 2 ( 1+ ε 4 ) 2 s 2 Q 1 ] w 0 2 [ 1+ w 0 2 / ( 2 s 2 Q 5 ) ] Q 5 Q 7 w 0 2 [ 1+ w 0 2 / ( 2 s 2 Q 9 ) ] Q 9 Q 11 },
d<I(x,0,L)> dx = d< I 11 (x,0,L)> dx + d< I 22 (x,0,L)> dx d< I 12 (x,0,L)> dx d< I 21 (x,0,L)> dx ,
d< I 11 (x,0,L)> dx = ( k 2L ) 3 m=1 M m =1 M n=1 N n =1 N { P w 0 2 Q 1 Q 3 A 11 B 11 [ D 11 k 3 x 3 16 A 11 3 L 3 s 2 + i C 11 w 0 2 k 2 x 2 4 A 11 2 L 2 s Q 1 kx 4 A 11 L ( 5 D 11 4 A 11 s + w 0 2 Q 1 s 2 +1 ) i w 0 2 C 11 Q 1 A 11 s ] }exp( k 2 x 2 4 A 11 L 2 ),
d< I 12 (x,0,L)> dx = ( k 2L ) 3 m=1 M m =1 M n=1 N n =1 N { P ε 2 w 0 2 Q 5 Q 7 A 12 B 12 [ D 12 k 3 x 3 16 A 12 3 L 3 s 2 + i C 12 ε 2 w 0 2 k 2 x 2 4 A 12 2 L 2 s Q 5 kx 4 A 12 L ( 5 D 12 4 A 12 s + ε 2 w 0 2 Q 5 s 2 +1 ) i ε 2 w 0 2 C 12 Q 5 A 12 s ] }exp( k 2 x 2 4 A 12 L 2 ),

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