Abstract

Two types of superposition, i.e., the superposition of the intensity and the superposition of the cross-spectral density function of off-axis partially coherent beams propagating through atmospheric turbulence, are analytically and numerically studied. The Gaussian–Schell model beam is taken as a typical example of the partially coherent beam, and analytical propagation equations for the resulting beam in turbulence are derived. The mean-squared beam width, the power in the bucket, and the β parameter are taken as the characteristic parameters of beam quality to compare the results of the two types of superimposed partially coherent beams in turbulence. It is shown that for the two types of superposition the smaller the coherence parameter α is, the less the resulting beam is sensitive to the effects of turbulence. The resulting beam of off-axis partially coherent beams for the superposition of the intensity is less affected by the turbulence than that for the superposition of the cross-spectral density function. The results are physically interpreted.

© 2008 Optical Society of America

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References

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  1. J. D. Strohschein, J. J. Herb, and E. C. Clarence, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
    [CrossRef]
  2. M. Brunel, A. L. Floch, F. Bretenaker, J. Marty, and E. Molva, “Coherent addition of adjacent lasers by forked eigenstate operation,” Appl. Opt. 37, 2402-2406 (1998).
    [CrossRef]
  3. R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
    [CrossRef]
  4. J. R. Andrews, “Interferometric power amplifiers,” Opt. Lett. 14, 33-35 (1989).
    [CrossRef] [PubMed]
  5. B. Lü and H. Ma, “Coherent and incoherent off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
    [CrossRef]
  6. B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” Appl. Opt. 39, 1279-1289 (2000).
    [CrossRef]
  7. A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Photonics Ser. 17, 184-199 (1998).
  8. B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
    [CrossRef]
  9. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985), Chap. 6.
    [CrossRef]
  10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  11. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  12. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  13. A. Dogariu and S. Amarande, “Propagation of partially coherent beam: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  14. X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231-236 (2005).
    [CrossRef]
  15. 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]
  16. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  17. A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).
  18. 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]
  19. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175-178 (1978).
    [CrossRef]
  20. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713-720 (1988).
    [CrossRef]

2006

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]

2005

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

2003

A. Dogariu and S. Amarande, “Propagation of partially coherent beam: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003).
[CrossRef] [PubMed]

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

2002

2000

1999

B. Lü and H. Ma, “Coherent and incoherent off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

1998

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Photonics Ser. 17, 184-199 (1998).

J. D. Strohschein, J. J. Herb, and E. C. Clarence, “Beam propagation constants for a radial laser array,” Appl. Opt. 37, 1045-1048 (1998).
[CrossRef]

M. Brunel, A. L. Floch, F. Bretenaker, J. Marty, and E. Molva, “Coherent addition of adjacent lasers by forked eigenstate operation,” Appl. Opt. 37, 2402-2406 (1998).
[CrossRef]

1990

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

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

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1989

1988

1979

1978

Amarande, S.

Andrews, J. R.

Andrews, L. C.

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

Bernard, J. M.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Bretenaker, F.

Brunel, M.

Chodzko, R. A.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Clarence, E. C.

Dogariu, A.

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985), Chap. 6.
[CrossRef]

Floch, A. L.

Friberg, A. T.

Garay, A.

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

Gbur, G.

Herb, J. J.

Ji, X.

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]

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

Leader, C.

Li, B.

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

Lü, B.

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]

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

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

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

B. Lü and H. Ma, “Coherent and incoherent off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Ma, H.

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

B. Lü and H. Ma, “Coherent and incoherent off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Marty, J.

Mirels, H.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Molva, E.

Phillips, R. L.

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

Plonus, M. A.

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Photonics Ser. 17, 184-199 (1998).

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Strohschein, J. D.

Turunen, J.

Wang, S. C. H.

Wolf, E.

Wu, J.

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

Zhang, E.

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]

Appl. Opt.

J. Mod. Opt.

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. Opt. A, Pure Appl. Opt.

B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A, Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

B. Lü and H. Ma, “Coherent and incoherent off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Opt. Lett.

OSA Trends Opt. Photonics Ser.

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Trends Opt. Photonics Ser. 17, 184-199 (1998).

Proc. SPIE

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers optical resonators,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

A. Garay, “Continuous wave deuterium fluoride laser beam diagnostic system,” Proc. SPIE 888, 17-22 (1998).

Other

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E.Wolf, ed. (Elsevier, 1985), Chap. 6.
[CrossRef]

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

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Figures (7)

Fig. 1
Fig. 1

Relative intensity distribution I ( x , z ) I 0 max of the resulting beam (a) for the superposition of the intensity and (b) for the superposition of the cross-spectral density function. The calculation parameters are z = 8 km , and α = 0.3 ; C n 2 = 0 (solid curve) and C n 2 = 10 13 m 2 3 (dashed curve).

Fig. 2
Fig. 2

Relative intensity distribution I ( x , z ) I 0 max of the resulting beam for the superposition of the cross-spectral density function. The calculation parameters are z = 8 km , (a) α = 0.05 , (b) α = 3 , and (c) α ; C n 2 = 0 (solid curve) and C n 2 = 10 13 m 2 3 (dashed curve).

Fig. 3
Fig. 3

Relative width w ( z ) w ( 0 ) of the resulting beam versus the propagation distance z (a) for the superposition of the intensity and (b) for the superposition of the cross-spectral density function; C n 2 = 0 (solid curve) and C n 2 = 10 13 m 2 3 (dashed curve).

Fig. 4
Fig. 4

PIB curves of the resulting beam at the plane z = 8 km (a) for the superposition of the intensity and (b) for the superposition of the cross-spectral density function; C n 2 = 0 (solid curve) and C n 2 = 10 13 m 2 3 (dashed curve).

Fig. 5
Fig. 5

PIB curves of the resulting beam at the plane z = 8 km for the two types of superposition (a) α = 0.8 and (b) α = 4 ; C n 2 = 0 (solid curve) and C n 2 = 10 13 m 2 3 (dashed curve).

Fig. 6
Fig. 6

β parameter of the resulting beam versus the coherence parameter α for the two types of superposition. The calculation parameters are z = 8 km and C n 2 = 10 13 m 2 3 .

Fig. 7
Fig. 7

Mean-squared beam width w ( α ) of the resulting beam in free space versus the coherence parameter α for the two types of superposition. The calculation parameters are z = 8 km and C n 2 = 0 .

Equations (42)

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W m ( 0 ) ( x 1 , x 2 , z = 0 ) = exp [ ( x 1 m x d ) 2 + ( x 2 m x d ) 2 w 0 2 ] exp [ ( x 1 x 2 ) 2 2 σ 0 2 ] ,
m [ M 1 2 , M 1 2 ] ,
I ( x , z = 0 ) = m = M 1 2 M 1 2 exp [ 2 ( x m x d ) 2 w 0 2 ] .
W m ( x 1 , x 2 , z ) = k 2 π z d x 1 d x 2 W m ( 0 ) ( x 1 , x 2 , z = 0 ) 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 [ 0.5 D Ψ ( v , v ) ] ,
D Ψ ( v , v ) = 2.91 k 2 z 0 1 C n 2 ( η z ) η v + ( 1 η ) v 5 3 d η ,
exp [ ψ ( x 1 , x 1 ) + ψ * ( x 2 , x 2 ) ] m exp [ v 2 + v v + v 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 )
I m ( x , z ) = W m ( x , x , z ) = B A exp [ 2 B 2 A 2 ( x w 0 m x d w 0 ) 2 ] ,
A = 1 w 0 4 + 2 w 0 2 ρ 0 2 + 1 w 0 4 α 2 + B 2 ,
B = k 2 z ,
α = σ 0 w 0 ,
I ( x , z ) = m = M 1 2 M 1 2 I m ( x , z ) = B A m = M 1 2 M 1 2 exp [ 2 B 2 A 2 ( x w 0 m x d w 0 ) 2 ] .
w 2 ( z ) = 4 x 2 I ( x , z ) d x I ( x , z ) d x .
w 2 ( z ) = 4 M m = M 1 2 M 1 2 [ ( w 0 A 2 B ) 2 + ( m x d ) 2 ] .
PIB = h h I ( x , z ) d x I ( x , z ) d x ,
PIB = 1 2 M m = M 1 2 M 1 2 { erf [ 2 B A ( m x d w 0 + h w 0 ) ] erf [ 2 B A ( m x d w 0 h w 0 ) ] } ,
W ( 0 ) ( x 1 , x 2 , z = 0 ) = m = M 1 2 M 1 2 n = M 1 2 M 1 2 exp [ ( x 1 m x d ) 2 + ( x 2 n x d ) 2 w 0 2 ] exp { [ ( x 1 m x d ) ( x 2 n x d ) ] 2 2 σ 0 2 } ,
m , n [ M 1 2 , M 1 2 ] .
I ( x , z = 0 ) = m = M 1 2 M 1 2 n = M 1 2 M 1 2 exp [ ( x m x d ) 2 + ( x n x d ) 2 w 0 2 ] exp [ ( n x d m x d ) 2 2 σ 0 2 ] .
W ( x 1 , x 2 , z ) = k 2 π z d x 1 d x 2 W ( 0 ) ( x 1 , x 2 , z = 0 ) 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 .
I ( x , z ) = W ( x , x , z ) = B A m = M 1 2 M 1 2 n = M 1 2 M 1 2 exp ( P x 2 + Q x + R ) ,
P = B 2 [ ( C G A ) 2 + ( 1 C ) 2 ] ,
Q = i B x d ( C 2 D G A 2 H C 2 ) ,
R = x d [ m n w 0 2 α 2 1 w 0 2 ( 1 + 1 2 α 2 ) ( m 2 + n 2 ) + ( C D 2 A ) 2 + ( H 2 C ) 2 ] ,
C = 1 w 0 2 + 1 2 w 0 2 α 2 + 1 ρ 0 2 i B ,
D = 2 n w 0 2 + ( n m ) w 0 2 α 2 + H C 2 ( 1 ρ 0 2 + 1 2 w 0 2 α 2 ) ,
G = 1 1 C 2 ( 1 ρ 0 2 + 1 2 w 0 2 α 2 ) ,
H = 2 m w 0 2 + ( m n ) w 0 2 α 2 .
I ( x , z ) = B A m = M 1 2 M 1 2 n = M 1 2 M 1 2 exp ( P x 2 + Q x + R ) ,
A = 1 w 0 4 + 2 w 0 2 ρ 0 2 + B 2 ,
P = B 2 [ ( C G A ) 2 + ( 1 C ) 2 ] ,
Q = i B x d ( C 2 D G A 2 2 m w 0 2 C 2 ) ,
R = x d [ m 2 + n 2 w 0 2 + ( C D 2 A ) 2 + ( m w 0 2 C ) 2 ] ,
C = 1 w 0 2 + 1 ρ 0 2 i B ,
D = 2 n w 0 2 + 2 m w 0 2 ρ 0 2 C 2 ,
G = 1 1 ρ 0 2 C 2 .
w 2 ( z ) = 4 m = M 1 2 M 1 2 n = M 1 2 M 1 2 S ( 1 2 P + Q 2 4 P 2 ) m = M 1 2 M 1 2 n = M 1 2 M 1 2 S ,
S = exp ( R + Q 2 4 P ) .
PIB = m = M 1 2 M 1 2 n = M 1 2 M 1 2 S { erf [ P ( Q 2 P + h ) ] erf [ P ( Q 2 P h ) ] } 2 m = M 1 2 M 1 2 n = M 1 2 M 1 2 S .
β = A A 0 ,

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