Abstract

The analytical expression for the spectral degree of coherence of partially coherent flat-topped beams propagating through the turbulent atmosphere is derived, and the spatial correlation properties are studied in detail. In particular, we find that the oscillatory behavior and phase singularities of the spectral degree of coherence may appear when partially coherent flat-topped beams propagate through the turbulent atmosphere; this behavior is very different from the behavior of Gaussian Schell-model beams. But the oscillatory behavior becomes weaker with increasing turbulence and even disappears when the turbulence is strong enough. The width of the spectral degree of coherence is always smaller than that of the spectral density in the far field when the turbulence is strong enough, whereas the width of the spectral degree of coherence in free space can be either larger or smaller than that of the spectral density in the far field.

© 2007 Optical Society of America

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References

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    [CrossRef]
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
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    [CrossRef]
  4. G. Gbur and E. Wolf, 'Spreading of partially coherent beams in random media,' J. Opt. Soc. Am. A 19, 1592-1598 (2002)
    [CrossRef]
  5. 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]
  6. A. Dogariu and S. Amarande, 'Propagation of partially coherent beam: turbulence-induced degradation,' Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  7. X. Ji and B. Lü, 'Turbulence-induced quality degradation of partially coherent beams,' Opt. Commun. 251, 231-236 (2005).
    [CrossRef]
  8. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, 'Turbulence induced beam spreading of higher order mode optical waves,' Opt. Eng. (Bellingham) 41, 1097-1103 (2002).
    [CrossRef]
  9. H. T. Eyyuboglu 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]
  10. X. Ji, E. Zhang, and B. Lü, 'Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,' J. Mod. Opt. 53, 2111-2127 (2006).
    [CrossRef]
  11. Y. Cai and S. He, 'Propagation of various dark hollow beams in a turbulent atmosphere,' Opt. Express 14, 1353-1367 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. M. Zahid and M. S. Zubairy, 'Directionality of partially coherent Bessel-Gaussian,' Opt. Commun. 70, 361-364 (1989).
    [CrossRef]
  14. Y. Li, 'New expressions for flat-topped light beam,' Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
  15. S. C. H. Wang and M. A. Plonus, 'Optical beam propagation for a partially coherent source in the turbulent atmosphere,' J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [CrossRef]
  16. A. Ishimaru, 'Phase fluctuations in a turbulent medium,' Appl. Opt. 16, 3190-3192 (1997).
    [CrossRef]
  17. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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    [CrossRef] [PubMed]
  19. G. Gbur and T. D. Visser, 'Coherence vortices in partially coherent beams,' Opt. Commun. 222, 117-125 (2003).
    [CrossRef]
  20. G. Gbur, T. D. Visser, and E. Wolf, 'Hidden singularities in partially coherent wavefields,' J. Opt. A, Pure Appl. Opt. 6, 5239-5242 (2004).
    [CrossRef]

2006 (2)

X. Ji, E. Zhang, and B. Lü, 'Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,' J. Mod. Opt. 53, 2111-2127 (2006).
[CrossRef]

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

2005 (2)

X. Ji and B. Lü, 'Turbulence-induced quality degradation of partially coherent beams,' Opt. Commun. 251, 231-236 (2005).
[CrossRef]

H. T. Eyyuboglu 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]

2004 (1)

G. Gbur, T. D. Visser, and E. Wolf, 'Hidden singularities in partially coherent wavefields,' J. Opt. A, Pure Appl. Opt. 6, 5239-5242 (2004).
[CrossRef]

2003 (4)

2002 (3)

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

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, 'Turbulence induced beam spreading of higher order mode optical waves,' Opt. Eng. (Bellingham) 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]

1997 (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 beam through turbulent media,' J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1989 (1)

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

1979 (1)

Appl. Opt. (1)

J. Mod. Opt. (3)

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

X. Ji, E. Zhang, and B. Lü, 'Propagation of multi-Gaussian beams in incoherent combination through turbulent atmosphere and their beam quality,' J. Mod. Opt. 53, 2111-2127 (2006).
[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)

G. Gbur, T. D. Visser, and E. Wolf, 'Hidden singularities in partially coherent wavefields,' J. Opt. A, Pure Appl. Opt. 6, 5239-5242 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

G. Gbur and T. D. Visser, 'Coherence vortices in partially coherent beams,' Opt. Commun. 222, 117-125 (2003).
[CrossRef]

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

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

X. Ji and B. Lü, 'Turbulence-induced quality degradation of partially coherent beams,' Opt. Commun. 251, 231-236 (2005).
[CrossRef]

Opt. Eng. (Bellingham) (1)

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

Opt. Express (1)

Opt. Lett. (2)

Other (3)

R. L. Fante, 'Wave propagation in random media: a systems approach,' in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), Chap. VI.
[CrossRef]

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (7)

Fig. 1
Fig. 1

Spectral degree of coherence μ ( x , x , z , ω ) of a partially coherent flat-topped beam with M = 12 and α = 1 at the plane z = 10 km for different values of C n 2 .

Fig. 2
Fig. 2

Spectral degree of coherence μ ( x , x , z , ω ) of a partially coherent flat-topped beam with M = 12 and α = 0.2 at the plane z = 10 km for different values of C n 2 .

Fig. 3
Fig. 3

Spectral degree of coherence μ ( x , x , z , ω ) of partially coherent flat-topped beams with M = 12 at the plane z = 10 km for different values of α. (a) C n 2 = 0 ; (b) C n 2 = 10 14 m 2 3 .

Fig. 4
Fig. 4

Spectral degree of coherence μ ( x , x , z , ω ) of a partially coherent flat-topped beam with α = 0.3 and M = 12 for different values of the propagation distance. (a) C n 2 = 0 ; (b) C n 2 = 10 14 m 2 3 .

Fig. 5
Fig. 5

Spectral degree of coherence μ ( x , x , z , ω ) of partially coherent flat-topped beams with α = 0.5 at the plane z = 10 km for different values of M. (a) C n 2 = 0 ; (b) C n 2 = 10 14 m 2 3 .

Fig. 6(a), 6(b), 6(c), and 6(d)
Fig. 6(a), 6(b), 6(c), and 6(d)

Spectral degree of coherence μ ( x , x , z , ω ) of partially coherent flat-topped beams with M = 12 at the plane z = 10 km. (a) C n 2 = 0 , α = 3 ; (b) C n 2 = 0 , α = 1.5 ; (c) C n 2 = 0 , α = 0.3 ; (d) C n 2 = 0 , α = 0.1 ; (e) C n 2 = 10 15 m 2 3 , α = 3 ; (f) C n 2 = 10 15 m 2 3 , α = 1 ; (g) C n 2 = 10 15 m 2 3 , α = 0.3 ; (h) C n 2 = 10 15 m 2 3 , α = 1.5 .

Fig. 6(e), 6(f), 6(g), and 6(h)
Fig. 6(e), 6(f), 6(g), and 6(h)

Continued.

Equations (38)

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W ( 0 ) ( x 1 , x 2 , z = 0 , ω ) = 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 ] ,
α t = ( 1 ) t + 1 M ! t ! ( M t ) ! , p t = t = 1 M α t t ( t = m , m ) .
W ( x 1 , x 2 , z , ω ) = k 2 π z d x 1 d x 2 W ( 0 ) ( x 1 , x 2 , z = 0 , ω ) exp { ( i k 2 z ) [ ( x 1 2 x 2 2 ) 2 ( x 1 x 1 x 2 x 2 ) + ( x 1 2 x 2 2 ) ] } exp [ ψ ( x 1 , x 1 ) + ψ * ( x 2 , x 2 ) ] m ,
exp [ ψ ( x 1 , x 1 ) + ψ * ( x 2 , x 2 ) ] m exp [ ( x 1 x 2 ) 2 + ( x 1 x 2 ) ( x 1 x 2 ) + ( x 1 x 2 ) 2 ρ 0 2 ] ,
ρ 0 = ( 0.545 C n 2 k 2 z ) 3 5 ,
exp ( β 2 x 2 + γ x ) d x = π β exp ( γ 2 4 β 2 ) ,
W ( x 1 , x 2 , z , ω ) = k 2 z m = 1 M m = 1 M α m α m β 1 β 2 exp { i k 2 z ( x 1 2 x 2 2 ) ( x 1 x 2 ) 2 ρ 0 2 + [ ( x 1 x 2 ) 2 ρ 0 2 β 1 + i k 2 z β 1 x 1 ] 2 } exp { [ ( 1 η ) ( x 1 x 2 ) 2 ρ 0 2 β 2 + i k ( x 2 η x 1 ) 2 z β 2 ] 2 } ,
β 1 2 = m p m w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 i k 2 z ,
β 2 2 = m p m w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 + i k 2 z η 2 β 1 2 ,
η = 1 2 β 1 2 ( 1 σ 0 2 + 2 ρ 0 2 ) .
S ( x , z , ω ) = W ( x , x , z , ω ) = k 2 z m = 1 M m = 1 M α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 2 w 0 2 ] .
μ ( x 1 , x 2 , z , ω ) = W ( x 1 , x 2 , z , ω ) [ S ( x 1 , z , ω ) S ( x 2 , z , ω ) ] 1 2 .
μ ( x 1 , x 2 , z , ω ) = m = 1 M m = 1 M α m α m β 1 β 2 exp { i k 2 z ( x 1 2 x 2 2 ) ( x 1 x 2 ) 2 ρ 0 2 + [ ( x 1 x 2 ) 2 ρ 0 2 β 1 + i k 2 z β 1 x 1 ] 2 + [ ( 1 η ) ( x 1 x 2 ) 2 ρ 0 2 β 2 + i k ( x 2 η x 1 ) 2 z β 2 ] 2 } { m = 1 M m = 1 M α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 1 2 w 0 2 ] m = 1 M m = 1 M α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 2 2 w 0 2 ] } 1 2 .
μ ( x , x , z , ω ) = m = 1 M m = 1 M α m α m β 1 β 2 exp { [ 4 w 0 2 ρ 0 2 + ( m p m + m p m ) β 1 2 β 2 2 ( k 2 4 z 2 1 ρ 0 4 ) + k w 0 2 z β 1 2 β 2 2 ( 2 k z ρ 0 2 + k 2 z σ 0 2 + i ( m p m m p m ) w 0 2 ρ 0 2 ) ] x 2 w 0 2 } m = 1 M m = 1 M α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 2 w 0 2 ] .
μ ( x , x , z , ω ) f r e e = m = 1 M m = 1 M α m α m β 1 f r e e β 2 f r e e exp { k 2 4 z 2 β 1 2 f r e e β 2 2 f r e e [ ( m p m + m p m ) + 2 w 0 2 σ 0 2 ] x 2 w 0 2 } m = 1 M m = 1 M α m α m β 1 f r e e β 2 f r e e exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 f r e e β 2 2 f r e e x 2 w 0 2 ] ,
β 1 2 f r e e = m p m w 0 2 + 1 2 σ 0 2 i k 2 z ,
β 2 2 f r e e = m p m w 0 2 + 1 2 σ 0 2 + i k 2 z 1 4 σ 0 4 β 1 2 f r e e .
μ ( x , x , z , ω ) G S M = exp { [ 4 w 0 2 ρ 0 2 + w 0 2 β 1 2 G S M β 2 2 G S M ( k 2 2 z 2 σ 0 2 + 2 k 2 z 2 ρ 0 2 2 w 0 2 ρ 0 4 ) ] x 2 w 0 2 } ,
β 1 2 G S M = 1 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 i k 2 z ,
β 2 2 G S M = 1 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 + i k 2 z η 2 β 1 2 G S M .
S ( x , z , ω ) = k 2 z 1 ( β 1 β 2 ) M = 1 exp [ k 2 2 z 2 ( β 1 2 β 2 2 ) M = 1 x 2 w 0 2 ] ,
( β 1 2 β 2 2 ) M = 1 = 1 w 0 4 + ( k 2 z ) 2 + 2 w 0 2 ( 1 2 σ 0 2 + 1 ρ 0 2 ) .
S ( x , z , ω ) = k 2 z m = 1 N 1 m = 1 N 1 α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 2 w 0 2 ]
S ( x , z , ω ) = m = 1 N m = 1 N f m m
f m m = k 2 z α m α m β 1 β 2 exp [ k 2 ( m p m + m p m ) 4 z 2 β 1 2 β 2 2 x 2 w 0 2 ] ,
β 1 2 β 2 2 = ( m p m ) ( m p m ) w 0 4 + m p m + m p m w 0 2 ( 1 2 σ 0 2 + 1 ρ 0 2 ) + ( k 2 z ) 2 + i ( m p m m p m w 0 2 ) ( k 2 z ) .
S ( x , z , ω ) = m = 1 N f m N + m = 1 N f N m + m = 1 N 1 m = 1 N 1 f m m .
W ( x 1 , x 2 , z , ω ) = k 2 π z d x 1 d x 2 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 ] exp { ( i k 2 z ) [ ( x 1 2 x 2 2 ) 2 ( x 1 x 1 x 2 x 2 ) + ( x 1 2 x 2 2 ) ] } exp [ ( x 1 x 2 ) 2 + ( x 1 x 2 ) ( x 1 x 2 ) + ( x 1 x 2 ) 2 ρ 0 2 ] .
h h f ( x ) d x = h h f ( x ) d x ,
W ( x , x , z , ω ) = k 2 π z d x 1 d x 2 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 2 + 2 x 1 x 2 + x 2 2 2 σ 0 2 ] exp { ( i k 2 z ) [ ( x 1 2 + 2 x 1 x ) ( x 2 2 + 2 x 2 x ) ] } exp [ x 1 2 + 2 x 1 x 2 + x 2 2 2 ( x 1 + x 2 ) x ρ 0 2 ] .
W ( x , x , z , ω ) = k 2 π z m = 1 M m = 1 M α m α m d x 1 d x 2 exp [ a ( x 1 ) + i b ( x 1 ) ] exp [ a ( x 2 ) i b ( x 2 ) ] exp [ q x 1 x 2 ]
a ( x 1 ) = m p m x 1 2 w 0 2 x 1 2 2 σ 0 2 x 1 2 2 x 1 x ρ 0 2 ,
b ( x 1 ) = ( k 2 z ) ( x 1 2 + 2 x 1 x ) ,
q = 1 σ 0 2 2 ρ 0 2 .
F = d x 1 d x 2 exp [ a ( x 1 ) + i b ( x 1 ) ] exp [ a ( x 2 ) i b ( x 2 ) ] = A 2 + B 2 ,
A = d x exp [ a ( x ) ] cos b ( x ) ,
B = d x exp [ b ( x ) ] sin b ( x ) .
exp [ q x 1 x 2 ] = 1 + q x 1 x 2 + ( q x 1 x 2 ) 2 2 ! + + ( q x 1 x 2 ) n n ! + ( < x 1 , x 2 < ) ,

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