Abstract

The source and receiver plane characteristics of flat topped (FT) beam propagating in turbulent atmosphere are investigated. To this end, source size, beam power and M 2 factor of source plane FT beam are derived. For a turbulent propagation medium, via Huygens Fresnel diffraction integral, the receiver plane intensity is found. Power captured within an area on the receiver plane is calculated. Kurtosis parameter and beam size variation along the propagation axis are formulated. Graphical outputs are provided displaying the variations of the derived source and receiver plane parameters against the order of flatness and propagation length. Analogous to free space behavior, when propagating in turbulence, the FT beam first will form a circular ring in the center. As the propagation length increases, the circumference of this ring will become narrower, giving rise to a downward peak emerging from the center of the beam, eventually turning the intensity profile into a pure Gaussian shape.

© 2006 Optical Society of America

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References

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    [CrossRef]
  2. D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F. M. Dickey and D. L. Shealy, eds., Proc. SPIE 5876, 1-11 (2005).
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    [CrossRef]
  4. Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  9. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2006

2005

2004

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004).
[CrossRef]

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, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (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]

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]

2003

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

2002

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

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

2000

B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A. 17, 2001-2004 (2000).
[CrossRef]

B. Lü and H. Ma, "Coherent and incoherent off-axial Hermite-Gaussian beam combinations," Appl. Opt. 39, 1279-1289 (2000).
[CrossRef]

1996

1994

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1984

J. E. Harvey and J. L. Forgham, "The spot of Arago: New relevance for an old phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

1980

Ambrosini, D.

Bagini, V.

Baykal, Y.

Borghi, R.

Cai, Y.

Carter, W.H.

Eyyuboglu, H. T.

Forgham, J. L.

J. E. Harvey and J. L. Forgham, "The spot of Arago: New relevance for an old phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

Ge, D.

Gori, F.

Harvey, J. E.

J. E. Harvey and J. L. Forgham, "The spot of Arago: New relevance for an old phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

He, S.

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]

Ji, X.

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

Jing, F.

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004).
[CrossRef]

Li, Y.

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

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

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, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (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]

Liu, H.

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004).
[CrossRef]

Lou, S.

B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A. 17, 2001-2004 (2000).
[CrossRef]

Lü, B.

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

B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A. 17, 2001-2004 (2000).
[CrossRef]

B. Lü and H. Ma, "Coherent and incoherent off-axial Hermite-Gaussian beam combinations," Appl. Opt. 39, 1279-1289 (2000).
[CrossRef]

Ma, H.

Mao, H.

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004).
[CrossRef]

Santarsiero, M.

Spagnolo, G. S.

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]

Zhao, D.

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (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]

Am. J. Phys.

J. E. Harvey and J. L. Forgham, "The spot of Arago: New relevance for an old phenomenon," Am. J. Phys. 52, 243-247 (1984).
[CrossRef]

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

H. Mao, D. Zhao, F. Jing, and H. Liu, "Propagation characteristics of the kurtosis parameters of flat-topped beams passing through fractional Fourier transformation systems with a spherically aberrated lens," J. Opt. A: Pure Appl. Opt. 6,640-650 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A: Pure Appl. Opt. 6, 390-395 (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]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

B. Lü and S. Lou, "General propagation equation of flattened Gaussian beams," J. Opt. Soc. Am. A. 17, 2001-2004 (2000).
[CrossRef]

Opt. Commun.

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[CrossRef]

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[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]

Opt. Express

Opt. Lett.

Optik

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

Other

F. M. Dickey and S. C. Holswade, Laser beam shaping: theory and techniques (Marcel Dekker, New York, 2000).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F. M. Dickey and D. L. Shealy, eds., Proc. SPIE 5876, 1-11 (2005).

J. Zhang and Y. Li, "Atmospherically turbulent effects on partially coherent flat-topped Gaussian beam," in Optical Technologies for Atmospheric, and Environmental Studies, D. Lu and G. G. Matvienko, eds., Proc. SPIE 5832, 48-55 (2005).
[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

FT source beam profiles at different flatness order.

Fig. 2.
Fig. 2.

Overlaid plots of four FT source beams belonging to Fig. 1 cut along the slanted axis.

Fig. 3.
Fig. 3.

Source beam size variation versus flatness order.

Fig. 4.
Fig. 4.

Source beam power variation versus flatness order.

Fig. 5.
Fig. 5.

Variation of Mx2 versus flatness order.

Fig. 6.
Fig. 6.

Propagation view of the FT beam for selected source and propagation parameters.

Fig. 7.
Fig. 7.

Overlaid plots of the FT beam belonging to Fig. 6 cut along the slanted axis.

Fig. 8.
Fig. 8.

Intensity distribution of an elliptical FT beam before and after propagation.

Fig. 9.
Fig. 9.

Receiver beam size variation versus flatness order.

Fig. 10.
Fig. 10.

Power in bucket variation versus flatness order.

Fig. 11.
Fig. 11.

Kurtosis parameter variation versus propagation length.

Equations (20)

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

u s ( s ) = u s ( s x , s y ) = 1 [ 1 exp ( s x 2 α sx 2 s y 2 α sy 2 ) ] N ,
α sxN = [ 2 s x 2 I s ( s ) d s x d s y I s ( s ) d s x d s y ] 1 2 ,
α syN = [ 2 s y 2 I s ( s ) d s x d s y I s ( s ) d s x d s y ] 1 2 ,
I s ( s ) = u s ( s ) u s * ( s ) .
α sxN = α sx [ 2 n = 1 N n 1 = 1 N ( 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) 2 n = 1 N n 1 = 1 N ( 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) ] 2 ,
P sN = 2 π α sx α sy n = 1 N n 1 = 1 N ( 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) .
M x 2 = 4 π ( σ sx 2 σ fx 2 ) 0.5 ,
M y 2 = 4 π ( σ sy 2 σ fy 2 ) 0.5 ,
σ sx 2 = s x 2 I s ( s x ) d s x I s ( s x ) d s x ,
σ fx 2 = f x 2 I f ( f x ) d f x I f ( f x ) d f x .
M x 2 = 2 [ n = 1 N n 1 = 1 N ( 1 ) n + n 1 ( N n ) ( N n 1 ) ( ( n n 1 ) ( 1 + n 1 ) 3 ) n = 1 N n 1 = 1 N ( 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) ] 1 2 .
< I r ( p ) > = < I r ( p x , p y ) > = k 2 ( 2 π L ) 2 d 2 s 1 d 2 s 2 u s ( s 1 ) u s ( s 2 ) exp { jk ( p s 1 ) 2 ( p s 2 ) 2 2 L }
× < exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] > ,
< exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] > = exp [ 0.5 D ψ ( s 1 s 2 ) ] = exp [ ρ 0 2 ( s 1 s 2 ) 2 ] ,
< I r ( p ) > = 1 n = 0 N ( 1 ) n ( N n ) β α sx α sy ( ζ nx ζ ny ) 1 2 exp ( β n p x 2 2 ζ nx β n p y 2 2 ζ ny ) n = 0 N ( 1 ) n ( N n ) β α sx α sy ( ζ nx * ζ ny * ) 1 2 exp ( β n p x 2 2 ζ nx * β n p y 2 2 ζ ny * )
+ n = 0 N n 1 = 0 n n 2 = 0 n 1 ( 1 ) n 1 ( N n ) ( n n 1 ) ( n 1 n 2 ) β α sx 2 α sy 2 ( γ n n 1 n 2 x γ n n 1 n 2 y ) 1 2 exp ( β n n 1 α sx 2 p x 2 2 γ n n 1 n 2 y + β n n 1 α sy 2 p y 2 2 γ n n 1 n 2 y ) ,
α pxN = [ 2 p x 2 I r ( p ) d p x d p y P sN ] 1 2 .
K x = p x 4 I r ( p ) d p x d p y P sN [ p x 2 I r ( p ) d p x d p y P sN ] 2 ,
P αN = 2 π 0 α r r I r ( p ) dr P sN .
Δ α pxN = ( α pxN α sxN ) α sxN .

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