Abstract

Analytical formulas for the elements of the 2×2 cross-spectral density matrix of a kind of stochastic electromagnetic array beam propagating through the turbulent atmosphere are derived with the help of vector integration. Two types of superposition (i.e. the correlated superposition and the uncorrelated superposition) are considered. The changes in the spectral density and in the spectral degree of polarization of such an array beam generated by isotropic or anisotropic electromagnetic Gaussian Schell-model sources on propagation are determined by the use of the analytical formulas. It is shown by numerical calculations that for the array beam composed by isotropic Gaussian-Schell model sources, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic sources, the spectral degree of polarization in the far field approaches a fixed value that is different from the source.

© 2008 Optical Society of America

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  1. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972).
    [CrossRef] [PubMed]
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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. G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
    [CrossRef]
  6. 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]
  7. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  8. T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003).
    [CrossRef] [PubMed]
  9. O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
    [CrossRef]
  10. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
    [CrossRef]
  11. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  12. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
    [CrossRef]
  13. X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007).
    [CrossRef] [PubMed]
  14. X. Ji, E. Zhang, and B. Lü, "Superimposed partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. B 25, 825-833 (2008).
    [CrossRef]
  15. H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
    [CrossRef]
  16. Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
    [CrossRef]
  17. B. Li and B. Lü, "Characterization of off-axis superposition of partially coherent beams," J. Opt. A 5, 303-307 (2003).
    [CrossRef]
  18. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

2008 (2)

X. Ji, E. Zhang, and B. Lü, "Superimposed partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. B 25, 825-833 (2008).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

2007 (4)

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
[CrossRef]

2005 (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

2004 (2)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

2003 (4)

2002 (2)

1979 (1)

1972 (1)

Baykal, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Cai, Y.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Davidson, F. M.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003).
[CrossRef] [PubMed]

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]

Du, X.

Eyyuboglu, H. T.

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

Gbur, G.

Ji, X.

Korotkova, O.

X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007).
[CrossRef] [PubMed]

G. Gbur and O. Korotkova, "Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Li, B.

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

Lin, Q.

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Lü, B.

X. Ji, E. Zhang, and B. Lü, "Superimposed partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. B 25, 825-833 (2008).
[CrossRef]

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

Plonus, M. A.

Ricklin, J. C.

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Shirai, T.

Wang, S. C. H.

Wolf, E.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

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]

T. Shirai, A. Dogariu, and E. Wolf, "Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence," Opt. Lett. 28, 610-612 (2003).
[CrossRef] [PubMed]

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

Yura, H. T.

Zhang, E.

Zhao, D.

Appl. Opt. (1)

Appl. Phys. B (2)

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Degree of polarization for partially coherent general beams in turbulent atmosphere," Appl. Phys. B 89, 91-97 (2007).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and Y. Cai, "Scintillations of laser array beams," Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

J. Opt. A (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

Y. Cai, Q. Lin, Y. Baykal, and H. T. Eyyuboglu, "Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere," Opt. Commun. 278, 157-167 (2007).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Other (1)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

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Figures (3)

Fig. 1.
Fig. 1.

2D model of the M×N stochastic electromagnetic Gaussian Schell-model sources array.

Fig. 2.
Fig. 2.

Normalized spectral density for the array beam composed by uncorrelated superposition (solid curve) and correlated superposition (dashed curve) propagating in (a) free space and (b) turbulent atmosphere. The source parameters for each beamlet are: λ=632.8 nm, Ax =2, Ay =1, Bxy =0.2exp(/3), σx =1 cm, σy =1 cm, σxx =σyy =3 mm, σxy =6 mm. The other parameters are: x 0=y 0=σy , M=N=3, C 2 n =10-15m-2/3.

Fig. 3.
Fig. 3.

Changes in the spectral degree of polarization P along the z-axis of electromagnetic Gaussian Schell-model array beam propagating through the turbulent atmosphere. Solid curves: the uncorrelated superposition. Dashed curves: the correlated superposition. The source parameters are the same as in Fig. 2, but (a) σx =σy =1 cm, σxx =σyy =3 mm, σxy =6 mm; (b)σx =2cm, σy =1 cm, σxx =σyy =σxy =6 mm; (c) σx =2 cm, σy =1 cm, σxx =σyy =3 mm, σxy =6 mm;. The other parameters are: x 0=y 0=σy , M=N=3, C 2 n =10-15m-2/3.

Equations (25)

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W ijmn ( 0 ) ( ρ 1 ' , ρ 2 ' , ω ) = A i A j B ij exp [ ( ρ 1 ' ρ 0 mn ) 2 4 σ i 2 ( ρ 2 ' ρ 0 mn ) 2 4 σ j 2 ] exp ( ρ 2 ' ρ 1 ' 2 2 δ ij 2 ) ,
W ijmn ( 0 ) ( ρ ¯ 12 ' , ω ) = A i A j B ij exp [ i k 2 ( ρ ¯ 12 ' ρ ¯ 0 mn ) T M ij ' 1 ( ρ ¯ 12 ' ρ ¯ 0 mn ) ] ,
M ij ' 1 = [ ( i 2 k σ i 2 i k δ ij 2 ) I ( i k δ ij 2 ) I ( i k δ ij 2 ) I ( i 2 k σ j 2 i k δ ij 2 ) I ] ,
W ij ( 0 ) ( ρ ¯ 12 ' , ω ) = m = M 1 2 M 1 2 n = N 1 2 N 1 2 W ijmn ( 0 ) ( ρ ¯ 12 ' , ω )
= m = M 1 2 M 1 2 n = N 1 2 N 1 2 { A i A j B ij exp [ i k 2 ( ρ ¯ 12 ' ρ ¯ 0 mn ) T M ij ' 1 ( ρ ¯ 12 ' ρ ¯ 0 mn ) ] } .
W ij ( ρ 1 , ρ 2 , z , ω ) = k 2 4 π 2 z 2 W ij ( 0 ) ( ρ 1 ' , ρ 2 ' , ω ) exp [ i k 2 z ( ρ 1 ρ 1 ' ) 2 + i k 2 z ( ρ 2 ρ 2 ' ) 2 ]
× exp [ ψ * ( ρ 1 , ρ 1 ' , z , ω ) + ψ ( ρ 2 , ρ 2 ' , z , ω ) ] m d 2 ρ 1 ' d 2 ρ 2 ' .
exp [ ψ * ( ρ 1 , ρ 1 ' , z , ω ) + ψ ( ρ 2 , ρ 2 ' , z , ω ) ] m = exp [ ( 1 2 ) D ψ ( ρ d ' , ρ d ) ]
exp [ ( 1 ρ 0 2 ) ( ρ d ' 2 + ρ d ' · ρ d + ρ d 2 ) ] ,
W ij ( ρ ¯ 12 , z , ω ) = k 2 4 π 2 [ Det ( B ¯ ) ] 1 2 W ij ( 0 ) ( ρ ¯ 12 ' , ω ) exp [ ik 2 ( ρ ¯ 12 T B ¯ 1 ρ ¯ 12 ' 2 ρ ¯ 12 ' T B ¯ 1 ρ ¯ 12 + ρ ¯ 12 T B ¯ 1 ρ ¯ 12 ) ]
exp [ i k 2 ( ρ ¯ 12 ' T P ¯ ρ ¯ 12 ' + ρ ¯ 12 ' T P ¯ ρ ¯ 12 + ρ ¯ 12 T P ¯ ρ ¯ 12 ) ] d 4 ρ ¯ 12 '
B ¯ = [ z I 0 0 z I ] , P ¯ = 2 i k ρ 0 2 [ I I I I ] .
W ij ( ρ ¯ 12 , ω ) = m = M 1 2 M 1 2 n = N 1 2 N 1 2 W ijmn ( ρ ¯ 12 , ω )
= m = M 1 2 M 1 2 n = N 1 2 N 1 2 A i A j B ij [ Det ( I ¯ + B ¯ P ¯ + B ¯ M ij ' 1 ) ] 1 2 exp ( i k 2 ρ ¯ 12 T M 2 ij 1 ρ ¯ 12 )
× exp { i k ρ ¯ 0 mn T [ I ¯ + ( B ¯ 1 + P ¯ ) M ij ' ] 1 ( B ¯ 1 1 2 P ¯ ) ρ ¯ 12 } exp { i k 2 ρ ¯ 0 mn T [ M ij ' + ( B ¯ 1 + P ¯ ) 1 ] 1 ρ ¯ 0 mn }
M 2 ij 1 = ( B ¯ 1 + P ¯ ) ( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 + P ¯ + M ij ' 1 ) 1 ( B ¯ 1 1 2 P ¯ ) .
W ij ( 0 ) ( ρ 1 ' , ρ 2 ' , ω ) = ( m = M 1 2 M 1 2 n = N 1 2 N 1 2 E i ( ρ 1 ' ρ 0 mn ) ) * ( p = M 1 2 M 1 2 q = N 1 2 N 1 2 E j ( ρ 2 ' ρ 0 pq ) )
= m = M 1 2 M 1 2 n = N 1 2 N 1 2 p = M 1 2 M 1 2 q = N 1 2 N 1 2 E i * ( ρ 1 ' ρ 0 mn ) E j ( ρ 2 ' ρ 0 pq )
= m = M 1 2 M 1 2 n = N 1 2 N 1 2 p = M 1 2 M 1 2 q = N 1 2 N 1 2 { A i A j B ij exp [ ( ρ 1 ' ρ 0 mn ) 2 4 σ i 2 ( ρ 1 ' ρ 0 pq ) 2 4 σ j 2 ]
× exp ( ( ρ 2 ' ρ 0 mn ) ( ρ 1 ' ρ 0 pq ) 2 2 δ ij 2 ) }
W ij ( 0 ) ( ρ ¯ 12 ' , ω ) = m = M 1 2 M 1 2 n = N 1 2 N 1 2 p = M 1 2 M 1 2 q = N 1 2 N 1 2 A i A j B ij exp [ i k 2 ( ρ ¯ 12 ' ρ ¯ 0 mnpq ) T M ij ' 1 ( ρ ¯ 12 ' ρ ¯ 0 mnpq ) ] ,
W ij ( ρ ¯ 12 , ω ) = m = M 1 2 M 1 2 n = N 1 2 N 1 2 p = M 1 2 M 1 2 q = N 1 2 N 1 2 A i A j B ij [ Det ( I ¯ + B ¯ P ¯ + B ¯ M ij ' 1 ) ] 1 2 exp ( i k 2 ρ ¯ 12 T M 2 ij 1 ρ ¯ 12 )
× exp { i k ρ ¯ 0 mnpq T [ I ¯ + ( B ¯ 1 + P ¯ ) M ij ] 1 ( B ¯ 1 1 2 P ¯ ) ρ ¯ 12 } exp { i k 2 ρ ¯ 0 mnpq T [ M ij ' + ( B ¯ 1 + P ¯ ) 1 ] 1 ρ ¯ 0 mnpq }
S ( ρ , z , ω ) = Tr W ( ρ , ρ , z , ω ) ,
P ( ρ , z , ω ) = 1 4 Det W ( ρ , ρ , z , ω ) [ Tr W ( ρ , ρ , z , ω ) ] 2 .

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