Abstract

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized are derived. We find that both the completely unpolarized stochastic electromagnetic Gaussian Schell-model beam and the completely polarized stochastic electromagnetic Gaussian Schell-model beam will keep their spectral degree of polarization or become partially polarized under different constraint conditions during their propagation in free space or through turbulent atmosphere. We give necessary theoretical explanation to the physical phenomena. They are considered as coherence-induced polarization changes and spectral density-induced polarization changes.

© 2008 Optical Society of America

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References

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  1. D. F. V. James, "Change of polarization of light beams on propagation in free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994).
    [CrossRef]
  2. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
    [CrossRef]
  3. O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
    [CrossRef]
  4. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]
  5. 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]
  6. 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]
  7. X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008).
    [CrossRef]
  8. X. Du and D. Zhao, "Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system," J. Opt. Soc. Am. A 25, 773-779 (2008).
    [CrossRef]
  9. E. Wolf, "Polarization invariance in beam propagation," Opt. Lett. 32, 3400-3401 (2007).
    [CrossRef] [PubMed]
  10. E. Wolf, "Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?" Opt. Lett. 33, 642-644 (2008).
    [CrossRef] [PubMed]
  11. D. Zhao and E. Wolf, "Light beams whose degree of polarization does not change on propagation," Opt. Commun. 281, 3067-3070 (2008).
    [CrossRef]
  12. M. Salem and E. Wolf, "Coherence-induced polarization changes in light beams," Opt. Lett. 33, 1180-1182 (2008).
    [CrossRef] [PubMed]
  13. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  14. E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995).
  16. H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
    [CrossRef]

2008 (5)

2007 (2)

2005 (3)

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

2004 (1)

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 (2)

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

1994 (1)

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Du, X.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

James, D. F. V.

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]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (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]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Salem, M.

M. Salem and E. Wolf, "Coherence-induced polarization changes in light beams," Opt. Lett. 33, 1180-1182 (2008).
[CrossRef] [PubMed]

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]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Wolf, E.

D. Zhao and E. Wolf, "Light beams whose degree of polarization does not change on propagation," Opt. Commun. 281, 3067-3070 (2008).
[CrossRef]

M. Salem and E. Wolf, "Coherence-induced polarization changes in light beams," Opt. Lett. 33, 1180-1182 (2008).
[CrossRef] [PubMed]

E. Wolf, "Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?" Opt. Lett. 33, 642-644 (2008).
[CrossRef] [PubMed]

E. Wolf, "Polarization invariance in beam propagation," Opt. Lett. 32, 3400-3401 (2007).
[CrossRef] [PubMed]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (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]

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

E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
[CrossRef] [PubMed]

Zhao, D.

J. Mod. Opt. (1)

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. A (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (5)

H. Roychowdhury and O. Korotkova, "Realizability conditions for electromagnetic Gaussian Schell-model sources," Opt. Commun. 249, 379-385 (2005).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (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]

X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008).
[CrossRef]

D. Zhao and E. Wolf, "Light beams whose degree of polarization does not change on propagation," Opt. Commun. 281, 3067-3070 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

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

Other (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

Illustrating the notation.

Fig. 2.
Fig. 2.

Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely unpolarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =Ay =1, Bxy =0, σx σy ,=1cm, δxx =2mm, and δyy is shown in the figure.

Fig.3. .
Fig.3. .

s Fig. 2, but passing through the turbulent atmosphere with C 2 n =10-12 m-2/3.

Fig. 4.
Fig. 4.

Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely polarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, Ax =2, Ay =1, Bxy =exp(/3), σx=1cm, δxx =δyy =δxy =2mm, and σy is shown in the figure.

Fig. 5.
Fig. 5.

As Fig. 4, but passing through turbulent atmosphere with C 2 n =10-12 m-2/3.

Equations (35)

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W ( r 1 , r 2 , ω ) [ W ij ( r 1 , r 2 , ω ) ] = [ E i * ( r 1 , ω ) E j ( r 2 , ω ) ] , ( i = x , y ; j = x , y ) ,
W ( ρ 1 , ρ 2 , z , ω ) = W ( 0 ) ( ρ 1 , ρ 2 , ω ) K ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) d 2 ρ 1 d 2 ρ 2 ,
K ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) = G * ( ρ 1 ρ 1 , z , ω ) G ( ρ 2 ρ 2 , z , ω ) ,
K rm ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) = G m * ( ρ 1 ρ 1 , z , ω ) G m ( ρ 2 ρ 2 , z , ω ) rm ,
P ( ρ , ρ , z , ω ) = 1 4 Det W ( ρ , ρ , z , ω ) [ T r W ( ρ , ρ , z , ω ) ] 2 ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = a ( ρ 1 , ρ 2 , ω ) [ 1 0 0 1 ] .
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ e x * ( ρ 1 , ω ) e x ( ρ 2 , ω ) e x * ( ρ 1 , ω ) e y ( ρ 2 , ω ) e y * ( ρ 1 , ω ) e x ( ρ 2 , ω ) e y * ( ρ 1 , ω ) e y ( ρ 2 , ω ) ] ,
W ij ( 0 ) ( ρ 1 , ρ 2 , ω ) = A i A j B ij exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ ij 2 ) ,
W ij ( 0 ) ( ρ , ρ , ω ) = A i A j B ij exp ( ρ 2 4 σ i 2 ρ 2 4 σ j 2 ) .
W ( 0 ) ( ρ , ρ , ω ) = A 2 exp ( ρ 2 2 σ 2 ) [ 1 0 0 1 ] .
max { δ xx , δ yy } δ xy min { δ xx B xy , δ yy B xy } ,
W ( 0 ) ( ρ , ρ , ω ) = [ A x A x B xx exp ( ρ 2 4 σ x 2 ρ 2 4 σ x 2 ) A x A y B xy exp ( ρ 2 4 σ x 2 ρ 2 4 σ y 2 ) A y A x B yx exp ( ρ 2 4 σ y 2 ρ 2 4 σ x 2 ) A y A y B yy exp ( ρ 2 4 σ y 2 ρ 2 4 σ y 2 ) ] .
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( I ¯ + BP ¯ + B ¯ M ij 1 ) ] 1 2 exp { ik 2 ρ 12 T [ ( B ¯ 1 + P ¯ ) ,
( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 + P ¯ + M ij 1 ) 1 ( B ¯ 1 1 2 P ¯ ) ] ρ 12 }
M ij 1 = [ i 2 k σ i 2 i k δ ij 2 0 i k δ ij 2 0 0 i 2 k σ i 2 i k δ ij 2 0 i k δ ij 2 i k δ ij 2 0 i 2 k σ j 2 i k δ ij 2 0 0 i k δ ij 2 0 i 2 k σ j 2 i k δ ij 2 ] .
B ¯ = [ z I 0 0 z I ] , P ¯ = 2 ik ρ 0 2 [ I I I I ] ,
W ij ( ρ 12 , z , ω ) = A i A j B ij [ Det ( I ¯ + B ¯ M ij 1 ) ] 1 2
× exp { ik 2 ρ 12 T [ B ¯ 1 B ¯ 1 T ( B ¯ 1 + M ij 1 ) 1 B ¯ 1 ] ρ 12 } ,
F ij ( ρ 12 , z , ω ) = [ Det ( I ¯ + BP ¯ + B ¯ M ij 1 ) ] 1 2 exp { ik 2 ρ 12 T [ ( B ¯ 1 + P ¯ ) ,
( B ¯ 1 1 2 P ¯ ) T ( B ¯ 1 + P ¯ + M ij 1 ) 1 ( B ¯ 1 1 2 P ¯ ) ] ρ 12 }
W xx ( ρ 12 , z , ω ) = W yy ( ρ 12 , z , ω ) = A 2 F ( ρ 12 , z , ω ) ,
W xy ( ρ 12 , z , ω ) = W yx ( ρ 12 , z , ω ) = 0 .
P ( ρ , ρ , z , ω ) = 1 4 Det W ( ρ , ρ , z , ω ) [ Tr W ( ρ , ρ , z , ω ) ] 2
= W xx ( ρ 12 , z , ω ) W yy ( ρ 12 , z , ω ) W xx ( ρ 12 , z , ω ) + W yy ( ρ 12 , z , ω ) ,
= 0
S i ( 0 ) ( ρ ' , ω ) = A i 2 exp ( ρ ' 2 2 σ i 2 ) , ( i = x , y ) .
W xx ( ρ 12 , z , ω ) = A x A x B xx F ( ρ 12 , z , ω ) ,
W xy ( ρ 12 , z , ω ) = A x A y B xy F ( ρ 12 , z , ω ) ,
W yx ( ρ 12 , z , ω ) = A y A x B yx F ( ρ 12 , z , ω ) ,
W yy ( ρ 12 , z , ω ) = A y A y B yy F ( ρ 12 , z , ω ) .
Det W ( ρ , ρ , z , ω ) = W xx ( ρ 12 , z , ω ) W yy ( ρ 12 , z , ω ) W xy ( ρ 12 , z , ω ) W yx ( ρ 12 , z , ω )
= A x 2 A y 2 ( B xx B yy B xy B yx ) F 2 ( ρ 12 , z , ω )
= 0
P ( ρ , ρ , z , ω ) = 1 0 [ Tr W ( ρ , ρ , z , ω ) ] 2 .
= 1

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