Abstract

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
    [CrossRef]
  13. Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A Pure Appl. Opt. 6, 1061-1066 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
    [CrossRef]
  17. M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
    [CrossRef]
  18. A. Ishimaru, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radio Sci. 4, 295-305 (1969).
    [CrossRef]
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  20. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

2005 (4)

2004 (6)

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Y. Baykal, "Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere," J. Opt. Soc. Am. A 21, 1290-1299 (2004).
[CrossRef]

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

2003 (1)

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik (Jena) 114, 394-400 (2003).
[CrossRef]

2002 (2)

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

2001 (1)

1999 (1)

1969 (1)

A. Ishimaru, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radio Sci. 4, 295-305 (1969).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Young, Laser Beam Scintillation with Applications, Press Monograph Vol. PM99 (SPIE Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Press Monograph Vol. PM53 (SPIE Press, 1998).

Baykal, Y.

Baykal, Y. K.

Borghi, R.

Cai, Y.

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Eyyuboglu, H. T.

Ge, D.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Hu, L.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Ishimaru, A.

A. Ishimaru, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radio Sci. 4, 295-305 (1969).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jil, X. L.

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik (Jena) 114, 394-400 (2003).
[CrossRef]

Li, Y.

Lin, Q.

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Liu, T. N.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

Lu, B.

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik (Jena) 114, 394-400 (2003).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Press Monograph Vol. PM53 (SPIE Press, 1998).

L. C. Andrews, R. L. Phillips, and C. Y. Young, Laser Beam Scintillation with Applications, Press Monograph Vol. PM99 (SPIE Press, 2001).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Santarsiero, M.

Shen, M.

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Tang, H. Q.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

Tovar, A. A.

Wang, S.

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Wang, X. W.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

Young, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Young, Laser Beam Scintillation with Applications, Press Monograph Vol. PM99 (SPIE Press, 2001).
[CrossRef]

Zeng, G.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Zhou, N.

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

Zhu, K. C.

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

Appl. Opt. (2)

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

Y. Cai and Q. Lin, "A partially coherent elliptical flattened Gaussian beam and its propagation," J. Opt. A Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

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

N. Zhou, G. Zeng, and L. Hu, "Algorithms for flattened Gaussian beams passing through apertured and unapertured paraxial ABCD optical systems," Opt. Commun. 240, 299-306 (2004).
[CrossRef]

M. Shen and S. Wang, "Decentered elliptical flattened Gaussian beam," Opt. Commun. 240, 245-252 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (Jena) (2)

K. C. Zhu, H. Q. Tang, X. W. Wang, and T. N. Liu, "Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams," Optik (Jena) 113, 222-226 (2002).
[CrossRef]

X. L. Jil and B. Lu, "Propagation of a flattened Gaussian beam through multi-apertured optical ABCD systems," Optik (Jena) 114, 394-400 (2003).
[CrossRef]

Radio Sci. (1)

A. Ishimaru, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radio Sci. 4, 295-305 (1969).
[CrossRef]

Other (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

L. C. Andrews, R. L. Phillips, and C. Y. Young, Laser Beam Scintillation with Applications, Press Monograph Vol. PM99 (SPIE Press, 2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Press Monograph Vol. PM53 (SPIE Press, 1998).

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Figures (5)

Fig. 1
Fig. 1

Electric field view of flat-topped Gaussian beams at N = 1, 5, and 15.

Fig. 2
Fig. 2

(a) Variation of scintillation index against propagation length at α s = 1 cm. (b) Variation of scintillation index, against propagation length at α s = 1 cm (exploded view around turning point).

Fig. 3
Fig. 3

Variation of scintillation index against propagation length at α s = 1.8 cm.

Fig. 4
Fig. 4

Variation of scintillation index against source size.

Fig. 5
Fig. 5

Variation of scintillation index against wavelength of operation.

Equations (20)

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m 2 = 4 B χ ( p 1 , p 2 , z = L ) ,
B χ ( p 1 , p 2 , z = L ) = χ ( p 1 , L ) χ ( p 2 , L )
χ ( p , L ) = 0.5 [ ψ ( p , L ) + ψ * ( p , L ) ] ,
ψ ( p , L ) = k 2 2 π u FS ( p , L ) V d 3 r n 1 ( p , z ) u FS ( p , z ) × exp ( i k | r r | ) | r r | ,
n 1 ( p x , p y , z ) = exp ( i κ x p x + i κ y p y ) × d Z n ( κ x , κ y , z ) .
u FS ( p , z ) = k exp ( i k z ) 2 π i z d 2 s u inc ( s , z = 0 )
× exp [ i k ( p s ) 2 / ( 2 z ) ] ,
u inc ( s , z = 0 ) = n = 1 N ( 1 ) n 1 N ( N n ) exp ( n s 2 2 α s     2 ) .
u FS ( p , z ) = k exp ( i k z ) 2 π i z d 2 s n = 1 N ( 1 ) n 1 N ( N n ) × exp ( n s 2 2 α s     2 ) exp [ i k ( p s ) 2 / ( 2 z ) ] .
u FS ( p , z ) = e i k z n = 1 N ( 1 ) n 1 N ( N n ) 1 ( 1 + i α n z ) × exp [ k α n 2 ( 1 + i α n z ) p x     2 ] ×   exp [ k α n 2 ( 1 + i α n z ) p y     2 ] ,
m 2 = 0.264 π 2 C n     2   Re { 0 L d η 0 d κ κ 8 / 3 [ G 1 ( η , κ ) + G 2 ( η , κ ) ] } ,
G 1 ( η , κ ) = k 2 [ Y ( η , κ ) ] 2 D 2 ,
G 2 ( η , κ ) = k 2 | Y ( η , κ ) | 2 | D | 2 ,
Y ( η , κ ) = n = 1 N ( 1 ) n 1 N ( 1  +  i α n L ) ( N n ) × exp [ γ n ( η L ) 2 i k κ 2 ] ,
D = n = 1 N ( 1 ) n 1 N ( 1 + i α n L ) ( N n ) ,
γ n = ( 1 + i α n η ) ( 1 + i α n L ) .
m 2 = 4.884 C n 2 k 7 / 6 L [ Re ( 1 n =     1 N p =     1 N ( 1 ) n + p 2 [ 1 n p α 2 L 2 + i α L ( n + p ) ] ( N n ) ( N p )
× n = 1 N p = 1 N ( 1 ) n + p 2 [ 1 n p α 2 L 2 + i α L ( n + p ) ] ( N n ) ( N p ) L 5 / 6 [ 1 + n 2 p 2 α 4 L 4 + α 2 L 2 ( n 2 + p 2 ) ] 5 / 6
× 0 1 d t { α L ( n + p ) ( 1 + n p α 2 L 2 ) ( 1 t ) 2 + i [ 2 ( 1 + n 2 p 2 α 4 L 4 t ) ( 1 t ) + α 2 L 2 ( n 2 + p 2 ) ( 1 t 2 ) ] } 5 / 6 )
1 n =     1 N p =     1 N ( 1 ) n + p 2 [ 1 + n p α 2 L 2 + i α L ( n + p ) ] ( N n ) ( N p ) α 5 / 6 L 5 / 3 3 8 n = 1 N p = 1 N ( 1 ) n + p 2 ( n + p ) 5 / 6 [ 1 + n p α 2 L 2 + i α L ( n p ) ] 11 / 6 ( N n ) ( N p ) ] .

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