Abstract

The effect of astigmatism on states of polarization of aberrant stochastic electromagnetic beams in turbulent atmosphere is investigated. Using the Gaussian-Schell model source with astigmatism, the analytical formula for the degree of polarization, the orientation angle, and the degree of polarization ellipse are derived. Analytical results show that different strengths of astigmatism have different effects on states of polarization on propagation. It is also shown that when the astigmatic coefficient of sources is large enough, states of polarization are hardly affected by atmospheric turbulence and the free-space diffraction phenomenon. The sufficient conditions for propagating with invariant polarization are derived and discussed.

© 2009 Optical Society of America

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  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]
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    [CrossRef]
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    [CrossRef]
  4. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257-4262 (2009).
    [CrossRef] [PubMed]
  5. Y. Zhu and D. Zhao, “Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams,” Phys. Lett. A 373, 1595-1598 (2009).
    [CrossRef]
  6. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483-3485 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1130 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2009 (6)

X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257-4262 (2009).
[CrossRef] [PubMed]

Y. Zhu and D. Zhao, “Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams,” Phys. Lett. A 373, 1595-1598 (2009).
[CrossRef]

D. Zhao and Y. Zhu, “Generalized formulas for stochastic electromagnetic beams on inverse propagation through nonsymmetrical optical systems,” Opt. Lett. 34, 884-886 (2009).
[CrossRef] [PubMed]

Y. Zhu and D. Zhao, “Propagation of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere,” Appl. Phys. B: Photophys. Laser Chem. 96, 155-160 (2009).
[CrossRef]

L. Pan, M. Sun, C. Ding, Z. Zhao, and B. Lu, “Effect of astigmatism on spectral, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system,” Opt. Express 17, 7310-7321 (2009).
[CrossRef] [PubMed]

W. Chen, W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE 7200, 1-10 (2009).

2008 (6)

2007 (4)

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483-3485 (2007).
[CrossRef] [PubMed]

W. Gao, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270, 474-478 (2007).
[CrossRef]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1130 (2007).
[CrossRef]

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

2006 (1)

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749-754 (2006).
[CrossRef]

2005 (3)

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)

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

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57-60 (2003).
[CrossRef]

2000 (1)

1997 (1)

Agrawal, P.

Alda, J.

Alonso, A.

A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393-2396 (2008).
[CrossRef]

Alonso, J.

Baykal, Y.

Bernabeu, E.

Cai, Y.

Chen, W.

W. Chen, W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE 7200, 1-10 (2009).

Chen, Z.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1130 (2007).
[CrossRef]

Ding, C.

Du, X.

Enderlein, J.

Eyyuboglu, T.

Gao, W.

W. Gao, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270, 474-478 (2007).
[CrossRef]

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749-754 (2006).
[CrossRef]

Gradsteyn, S.

S. Gradsteyn and M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Gregor, I.

Hanson, G.

Haus, W.

W. Chen, W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE 7200, 1-10 (2009).

Jakobsen, L.

Kaivola, M.

Korotkova, O.

Y. Cai, O. Korotkova, T. Eyyuboglu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834-15846 (2008).
[CrossRef] [PubMed]

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483-3485 (2007).
[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]

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]

Lin, Q.

Lindfors, K.

Lu, B.

Pan, L.

Pu, J.

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1130 (2007).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57-60 (2003).
[CrossRef]

Ryzhik, M.

S. Gradsteyn and M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Salem, M.

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]

Setala, T.

Sun, M.

Takeda, M.

Wang, F.

Wang, W.

Wolf, E.

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

A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393-2396 (2008).
[CrossRef]

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

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483-3485 (2007).
[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]

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]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57-60 (2003).
[CrossRef]

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

P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019-2023 (2000).
[CrossRef]

Zhan, Q.

W. Chen, W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE 7200, 1-10 (2009).

Zhao, D.

X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257-4262 (2009).
[CrossRef] [PubMed]

Y. Zhu and D. Zhao, “Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams,” Phys. Lett. A 373, 1595-1598 (2009).
[CrossRef]

Y. Zhu and D. Zhao, “Propagation of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere,” Appl. Phys. B: Photophys. Laser Chem. 96, 155-160 (2009).
[CrossRef]

D. Zhao and Y. Zhu, “Generalized formulas for stochastic electromagnetic beams on inverse propagation through nonsymmetrical optical systems,” Opt. Lett. 34, 884-886 (2009).
[CrossRef] [PubMed]

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

D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32, 3483-3485 (2007).
[CrossRef] [PubMed]

Zhao, Z.

Zhu, Y.

Y. Zhu and D. Zhao, “Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams,” Phys. Lett. A 373, 1595-1598 (2009).
[CrossRef]

D. Zhao and Y. Zhu, “Generalized formulas for stochastic electromagnetic beams on inverse propagation through nonsymmetrical optical systems,” Opt. Lett. 34, 884-886 (2009).
[CrossRef] [PubMed]

Y. Zhu and D. Zhao, “Propagation of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere,” Appl. Phys. B: Photophys. Laser Chem. 96, 155-160 (2009).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (1)

Y. Zhu and D. Zhao, “Propagation of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere,” Appl. Phys. B: Photophys. Laser Chem. 96, 155-160 (2009).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A, Pure Appl. Opt. 9, 1123-1130 (2007).
[CrossRef]

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

Opt. Commun. (7)

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

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]

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]

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749-754 (2006).
[CrossRef]

W. Gao, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270, 474-478 (2007).
[CrossRef]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57-60 (2003).
[CrossRef]

A. Alonso and E. Wolf, “The cross-spectral density matrix of a planar, electromagnetic stochastic source as a correlation matrix,” Opt. Commun. 281, 2393-2396 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Phys. Lett. A (2)

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

Y. Zhu and D. Zhao, “Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams,” Phys. Lett. A 373, 1595-1598 (2009).
[CrossRef]

Proc. SPIE (1)

W. Chen, W. Haus, and Q. Zhan, “Propagation of scalar and vector vortex beams through turbulent atmosphere,” Proc. SPIE 7200, 1-10 (2009).

Other (1)

S. Gradsteyn and M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

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Figures (5)

Fig. 1
Fig. 1

Illustrating the notation of propagation of a beam in turbulent atmosphere.

Fig. 2
Fig. 2

Effect of different strengths of astigmatism on degree of polarization (on-axis) in free space (curve A) and atmosphere (curves B, C). The source parameters are chosen as λ = 632.8 nm , σ = 0.02 m , δ xx = 2.2 mm , δ xy = 3 mm , A x = 2 , A y = 1 , B = 0.5 , β = π 3 .

Fig. 3
Fig. 3

Degree of polarization (on-axis) for different values of astigmatic coefficient C 6 in (a) free space; (b) (c) turbulent atmosphere. Other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Degree of ellipticity (on-axis) for different values of astigmatic coefficient C 6 in (a) free space, (b) turbulent atmosphere. Other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Orientation angle θ (on-axis) for different values of astigmatic coefficient C 6 in (a) free space, (b) turbulent atmosphere. Other parameters are the same as in Fig. 2.

Equations (31)

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W ( r 1 , r 2 , z ; ω ) = W i j ( x 1 , y 1 , x 2 , y 2 , z ; ω ) = E i * ( x 1 , y 1 , z ; ω ) E j ( x 2 , y 2 , z ; ω ) , ( i = x , y ; j = x , y ) ,
W i j 0 ( r 1 , r 2 , 0 ; ω ) = S i ( 0 ) ( r 1 ; ω ) S j ( 0 ) ( r 2 ; ω ) μ i j ( 0 ) ( r 1 , r 2 ; ω ) ,
( i = x , y ; j = x , y ) ,
Φ ( x , y ) = C 6 ( x 2 y 2 ) ,
W i j ( r 1 , r 2 , z ; ω ) = d r 1 W i j ( 0 ) ( r 1 , r 2 ; ω ) × K ( r 1 , r 2 , r 1 , r 2 ; ω ) d r 2 ,
K ( r 1 , r 2 , r 1 , r 2 ; ω ) = ( k 2 π z ) 2 exp [ i k ( r 1 r 1 ) 2 ( r 2 r 2 ) 2 2 z ] × exp { i k [ Φ ( r 1 ) Φ ( r 2 ) ] } × exp [ ψ * ( r 1 , r 1 , z ; ω ) + ψ ( r 2 , r 2 , z ; ω ) ] .
exp [ ψ * ( r 1 , r 1 , z ; ω ) + ψ ( r 2 , r 2 , z ; ω ) ] exp [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ρ 0 2 ( z ) ] ,
ρ 0 ( z ) = ( 0.55 C n 2 k 2 z ) 3 5
S i ( 0 ) ( r , ω ) = A i 2 exp ( r 2 2 σ i 2 ) , i = ( x , y ) ,
μ i j ( 0 ) ( r 1 , r 2 ; ω ) = B i j exp [ ( r 2 r 1 ) 2 2 δ i j 2 ] , ( i = x , y ; j = x , y ) .
W i j ( x , y , z ; ω ) = A i A j B i j Δ 1 i j Δ 2 i j exp [ 1 2 σ 2 ( x 2 Δ 1 i j + y 2 Δ 2 i j ) ] ,
Δ 1 i j = ( 1 2 z C 6 ) 2 + α i j z 2 + T z 16 5 Δ 2 i j = ( 1 + 2 z C 6 ) 2 + i j z 2 + T z 16 5 ,
α i j = ( 1 k σ ) 2 ( 1 4 σ 2 + 1 δ i j 2 ) , T = 2 ( 0.55 C n 2 ) 6 5 k 2 5 σ 2 .
P ( r , ω ) = 1 4 Det W ( r , r , ω ) [ Tr ( W ( r , r , ω ) ) ] 2 ,
ϵ = A minor A major ,
A major 2 ( r , z , ω ) = ( ( W x x W y y ) 2 + 4 | W x y | 2 + ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ) 2 ,
A minor 2 ( r , z , ω ) = ( ( W x x W y y ) 2 + 4 | W x y | 2 ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ) 2 ,
θ ( r , z , ω ) = 1 2 arctan ( 2 Re [ W x y ] W x x W y y ) , π 2 θ π 2 .
W i j ( x , y , z ; ω ) = ( k 2 π z ) 2 A i A j B i j d x 1 d y 1 exp { i k 2 x ( x 2 x 1 ) ( x 2 2 x 1 2 ) + 2 y ( y 2 y 1 ) ( y 2 2 y 1 2 ) 2 z + i k ( x 1 2 y 1 2 x 2 2 + y 2 2 ) ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ρ 0 2 ( z ) x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ 2 ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 2 δ i j 2 } d x 2 d y 2 .
x 2 x 1 = u , x 2 + x 1 = v , y 2 y 1 = s , y 2 + y 1 = t .
W i j ( x , y , z ; ω ) = ( k 2 π z ) 2 A i A j B i j d u d v exp { u 2 + v 2 + s 2 + t 2 8 σ 2 u 2 + s 2 ρ 0 2 ( z ) i k 2 x u u v + 2 y s s t 2 z + i k C 6 ( s t u v ) } d s d t ,
exp ( p 2 x 2 ± q x ) d x = exp ( q 2 4 p 2 ) π p , [ Re p 2 > 0 ] .
P ( 0 , z ; ω ) = ( 9 25 + 4 25 Δ 1 x x Δ 2 x x Δ 1 y y Δ 2 y y ) 1 2 ,
Δ 1 x x Δ 2 x x Δ 1 y y Δ 2 y y = F x x F x y ,
F i j = T 2 z 6.4 + ( 8 C 6 2 + 2 α i j ) T z 5.2 + 2 T z 3.2 + ( 4 C 6 2 + α i j ) 2 z 4 + ( 2 α i j 8 C 6 2 ) z 2 + 1 .
( F x x F x y ) T < 0 .
2 ( F x x F x y ) T C 6 < 0 .
Δ 1 x x Δ 2 x x Δ 1 y y Δ 2 y y ( 4 C 6 2 + α x x ) 2 z 4 + ( 2 α x x 8 C 6 2 ) z 2 + 1 ( 4 C 6 2 + α x y ) 2 z 4 + ( 2 α x y 8 C 6 2 ) z 2 + 1 ,
P ( x 0 , y 0 , z 0 ; ω ) = { 9 25 + 4 25 Δ 1 x x Δ 2 x x Δ 1 x y Δ 2 x y exp [ 1 σ 2 S x 0 2 1 σ 2 S + y 0 2 ] } 1 2 ,
S = 1 Δ 1 x y 1 Δ 1 x x , S + = 1 Δ 2 x y 1 Δ 2 x x ,
S 1 ( 4 C 6 2 + α x y ) z 0 2 4 z 0 C 6 + 1 1 ( 4 C 6 2 + α x x ) z 0 2 4 z 0 C 6 + 1 .

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