Abstract

On the basis of the generalized diffraction integral formula for an ABCD optical system in the spatial domain, a propagation law for the generalized Stokes parameters of a stochastic electromagnetic beam passing through an ABCD optical system is obtained. We describe the Stokes parameters of the source as linear combinations of the elements of the cross-spectral density matrix, and study the changes in the spectral degree of polarization and in the state of the polarization ellipse of a stochastic electromagnetic Gaussian Schell-model beam propagating through a gradient-index fiber with the help of generalized Stokes parameters and the cross-spectral density matrix. The medium has significant effect on the change of the spectral degree of polarization. However, when the correlation coefficients of the source satisfy the relation δxx=δyy=δxy=δyx, the medium does not influence the spectral degree of polarization.

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. 281, 2711-2715 (2008).
    [CrossRef]
  9. 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]
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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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  21. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067-3070 (2008).

2008 (4)

O. Korotkova, “Conservation laws for stochastic electromagnetic free fields,” J. Opt. A, Pure Appl. Opt. 10, 02503 (2008).
[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).

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]

2007 (2)

2006 (4)

2005 (3)

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

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]

2004 (1)

2003 (1)

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

2000 (1)

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

1994 (1)

1993 (1)

1991 (1)

1977 (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

1970 (1)

Agrawal, G. P.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Collins, S. A.

Du, X.

Friberg, A. T.

Gamliel, A.

James, D. F. V.

Korotkova, O.

Mendlovic, D.

Ozaktas, H. M.

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, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940-948 (2006).
[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]

Salem, M.

Setälä, T.

Tervo, J.

Wang, S.

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

Wolf, E.

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

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

M. Salem, O. Korotkova, and E. Wolf, “Can two planar sources with the same sets of Stokes parameters generate beams with different degrees of polarization?” Opt. Lett. 31, 3025-3027 (2006).
[CrossRef] [PubMed]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940-948 (2006).
[CrossRef]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[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, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

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

Zhao, D.

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

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]

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

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

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, Pure Appl. Opt. (1)

O. Korotkova, “Conservation laws for stochastic electromagnetic free fields,” J. Opt. A, Pure Appl. Opt. 10, 02503 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

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

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

Opt. Express (1)

Opt. Lett. (6)

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 (2)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

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Figures (5)

Fig. 1
Fig. 1

Notation illustrated.

Fig. 2
Fig. 2

Changes in (a) the Stokes parameters and (b) the normalized Stokes parameters of a stochastic electromagnetic beam on propagation through the lenslike medium. The source is assumed to be a Gaussian Schell-model source with λ = 632.8 nm , A x = 1.5 , A y = 1 , B x y = 0.3 exp ( i π 6 ) , B y x = 0.3 exp ( i π 6 ) , σ x = σ y = 15 μ m , δ x x = 0.15 μ m , δ y y = 0.225 μ m , and δ x y = δ y x = 0.25 μ m . The parameters for the lenslike medium are chosen from [18] with β ( ω 0 ) = 4.8 × 10 4 k 0 .

Fig. 3
Fig. 3

Changes in the polarization properties associated with a stochastic electromagnetic Gaussian Schell-model beam on propagation through the lenslike medium with the same values of the parameters used in Fig. 2. (a) Spectral degree of polarization, (b) orientation angle θ, (c) ellipticity angle ε.

Fig. 4
Fig. 4

Changes in the polarization ellipse associated with a stochastic electromagnetic Gaussian Schell-model beam on propagation through the lenslike medium with the same parameter values as used in Fig. 2.

Fig. 5
Fig. 5

Changes in the degree of polarization associated with a stochastic electromagnetic Gaussian Schell-model beam on propagation through the lenslike medium with parameters (a) δ x x = 0.15 μ m , δ y y = 0.225 μ m , δ x y = δ y x = 0.25 μ m ; (b) δ x x = 0.20 μ m , δ y y = 0.225 μ m , δ x y = δ y x = 0.25 μ m ; (c) δ x x = δ y y = 0.225 μ m , δ x y = δ y x = 0.25 μ m ; (d) δ x x = δ y y = δ x y = δ y x = 0.25 μ m . Other parameters are the same as in Fig. 2.

Equations (26)

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S 0 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E x ( r 2 , ω ) + E y * ( r 1 , ω ) E y ( r 2 , ω ) ,
S 1 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E x ( r 2 , ω ) E y * ( r 1 , ω ) E y ( r 2 , ω ) ,
S 2 ( r 1 , r 2 , ω ) = E x * ( r 1 , ω ) E y ( r 2 , ω ) + E y * ( r 1 , ω ) E x ( r 2 , ω ) ,
S 3 ( r 1 , r 2 , ω ) = i [ E y * ( r 1 , ω ) E x ( r 2 , ω ) E x * ( r 1 , ω ) E y ( r 2 , ω ) ] ,
E i ( ρ , z , ω ) = i λ B E i ( ρ , 0 , ω ) exp [ i k 2 B ( A ρ 2 2 ρ ρ + D ρ 2 ) ] d 2 ρ ( i = x , y ) ,
E i * ( ρ 1 , ρ 2 , z , ω ) E j ( ρ 1 , ρ 2 , z , ω ) = 1 ( λ B ) 2 E i * ( ρ 1 , ρ 2 , 0 , ω ) E j ( ρ 1 , ρ 2 , 0 , ω ) exp { i k 2 B [ A ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + D ( ρ 1 2 ρ 2 2 ) ] } d 2 ρ 1 d 2 ρ 2 .
S α ( ρ 1 , ρ 2 , z , ω ) = 1 ( λ B ) 2 S α ( ρ 1 , ρ 2 , 0 , ω ) exp { i k 2 B [ A ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + D ( ρ 1 2 ρ 2 2 ) ] } d 2 ρ 1 d 2 ρ 2 ( α = 0 , 1 , 2 , 3 ) .
P ( ρ , z , ω ) = S 1 2 ( ρ , z , ω ) + S 2 2 ( ρ , z , ω ) + S 3 2 ( ρ , z , ω ) S 0 ( ρ , z , ω ) ,
θ = 1 2 arctan ( S 2 ( ρ , z , ω ) S 1 ( ρ , z , ω ) ) ,
ε = 1 2 arcsin ( S 3 ( ρ , z , ω ) S 1 2 ( ρ , z , ω ) + S 2 2 ( ρ , z , ω ) + S 3 2 ( ρ , z , ω ) ) .
S 0 ( 0 ) ( ρ 1 , ρ 2 , ω ) = W x x ( 0 ) ( ρ 1 , ρ 2 , ω ) + W y y ( 0 ) ( ρ 1 , ρ 2 , ω ) ,
S 1 ( 0 ) ( ρ 1 , ρ 2 , ω ) = W x x ( 0 ) ( ρ 1 , ρ 2 , ω ) W y y ( 0 ) ( ρ 1 , ρ 2 , ω ) ,
S 2 ( 0 ) ( ρ 1 , ρ 2 , ω ) = W x y ( 0 ) ( ρ 1 , ρ 2 , ω ) + W y x ( 0 ) ( ρ 1 , ρ 2 , ω ) ,
S 3 ( 0 ) ( ρ 1 , ρ 2 , ω ) = i [ W y x ( 0 ) ( ρ 1 , ρ 2 , ω ) W x y ( 0 ) ( ρ 1 , ρ 2 , ω ) ] ,
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S i ( 0 ) ( ρ 2 , ω ) μ i j ( 0 ) ( ρ 2 ρ 1 , ω ) .
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = A i A j B i j exp ( ρ 1 2 4 σ i ) exp ( ρ 2 2 4 σ j ) exp ( ρ 2 ρ 1 2 2 δ i j 2 ) ,
W i j ( ρ 1 , ρ 2 , z , ω ) = A i A j B i j Δ i j 2 exp ( ρ 1 2 + ρ 2 2 4 Δ i j 2 σ 2 ) exp ( ρ 2 ρ 1 2 2 Δ i j 2 δ i j 2 ) exp ( i k ( ρ 2 2 ρ 1 2 ) 2 Φ i j ( z ) ) ,
Δ i j 2 = A 2 + B 2 4 σ 4 k 2 + B 2 σ 2 δ i j 2 k 2 ,
Φ i j ( z ) = B D ( 1 + A D Δ i j 2 A ) .
S 0 ( ρ , z , ω ) = A x 2 Δ x x 2 exp ( ρ 2 2 Δ x x 2 σ 2 ) + A y 2 Δ y y 2 exp ( ρ 2 2 Δ y y 2 σ 2 ) ,
S 1 ( ρ , z , ω ) = A x 2 Δ x x 2 exp ( ρ 2 2 Δ x x 2 σ 2 ) A y 2 Δ y y 2 exp ( ρ 2 2 Δ y y 2 σ 2 ) ,
S 2 ( ρ , z , ω ) = A x A y B x y Δ x y 2 exp ( ρ 2 2 Δ x y 2 σ 2 ) + A x A y B y x Δ y x 2 exp ( ρ 2 2 Δ y x 2 σ 2 ) ,
S 3 ( ρ , z , ω ) = i [ A x A y B y x Δ y x 2 exp ( ρ 2 2 Δ y x 2 σ 2 ) A x A y B x y Δ x y 2 exp ( ρ 2 2 Δ x y 2 σ 2 ) ] ,
Δ i j 2 = A 2 + B 2 4 σ 4 k 2 + B 2 σ 2 δ i j 2 k 2 .
n ( r , ω ) = n 0 ( ω ) [ 1 β ( ω ) r 2 ] .
( A B C D ) = ( cos ( l 2 β ( ω ) ) 1 n 0 ( ω ) 2 β ( ω ) sin ( l 2 β ( ω ) ) n 0 ( ω ) 2 β ( ω ) sin ( l 2 β ( ω ) ) cos ( l 2 β ( ω ) ) ) .

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