Abstract

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system is derived. Some analyses are illustrated by numerical examples relating to changes in the spectral degree of polarization and in the spectral degree of coherence of an electromagnetic Gaussian–Schell-model beam propagating through such an optical system. We find that the degree of polarization in the neighboring areas of the focal plane is oscillating, and the effect of misalignment on coherence is not so evident as that on polarization.

© 2008 Optical Society of America

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References

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    [CrossRef]
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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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    [CrossRef]
  7. 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]
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    [CrossRef]
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    [CrossRef]
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  16. X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. (to be published).
  17. X. Du and D. Zhao, “Propagation of elliptical Gaussian beams in apertured and misaligned optical systems,” J. Opt. Soc. Am. A 23, 1946-1950 (2006).
    [CrossRef]

2007

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

2006

2005

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]

O. Korotkova and E. Wolf, “Effects of linear non-image-forming devices on spectra and on coherence and polarization properties of stochastic electromagnetic beams: part I: general theory,” J. Mod. Opt. 52, 2659-2671 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Effects of linear non-image-forming devices on spectra and on coherence and polarization properties of stochastic electromagnetic beams: part II: examples,” J. Mod. Opt. 52, 2673-2685 (2005).
[CrossRef]

2004

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

2001

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

1994

1990

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Jena) 85, 67-72 (1990).

1970

J. Mod. Opt.

O. Korotkova and E. Wolf, “Effects of linear non-image-forming devices on spectra and on coherence and polarization properties of stochastic electromagnetic beams: part I: general theory,” J. Mod. Opt. 52, 2659-2671 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Effects of linear non-image-forming devices on spectra and on coherence and polarization properties of stochastic electromagnetic beams: part II: examples,” J. Mod. Opt. 52, 2673-2685 (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]

J. Opt. A, Pure Appl. Opt.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270, 474-478 (2007).
[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. (to be published).

Opt. Lett.

Optik (Jena)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law,” Optik (Jena) 85, 67-72 (1990).

Phys. Lett. A

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

Other

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

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Figures (7)

Fig. 1
Fig. 1

Optical transforming system with a misaligned thin lens.

Fig. 2
Fig. 2

Matrix for the misaligned optical transforming system.

Fig. 3
Fig. 3

Changes in the spectral degree of polarization P along the z axis of a random electromagnetic beam passing through a misaligned optical system. The source is assumed to be the Gaussian–Schell-model source with λ = 632.8 nm , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π 3 ) , σ x = 10 mm , σ y = 20 mm , δ x x = 0.6 mm , δ y y = 0.6 mm , and δ x y = 1.2 mm . The different misalignment parameters are (a) ε x = 0 , (b) ε x = 0.2 mm , (c) ε x = 0.5 mm , (d) ε x = 1 mm .

Fig. 4
Fig. 4

As Fig. 3, but δ x x = 1 mm .

Fig. 5
Fig. 5

Changes in the spectral degree of coherence μ along the z axis of a random electromagnetic beam passing through a misaligned optical system with pairs of field points ρ 12 T = ( 0.3 mm , 0 , 0.3 mm , 0 ) . The source is assumed to be the Gaussian–Schell-model source with λ = 632.8 nm , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π 3 ) , σ x = 10 mm , σ y = 20 mm , δ x x = 0.6 mm , δ y y = 0.6 mm , and δ x y = 1.2 mm . The different misalignment parameters are (a) ε x = 0 , (b) ε x = 0.5 mm .

Fig. 6
Fig. 6

Changes in the spectral degree of coherence μ along the z axis of a random electromagnetic beam passing through a misaligned optical system with the same misalignment parameter ε x = 0.5 mm . The source is assumed to be the Gaussian–Schell-model source with λ = 632.8 nm , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π 3 ) , σ x = 10 mm , σ y = 20 mm , δ x x = 0.6 mm , δ y y = 0.6 mm , and δ x y = 1.2 mm . The different pairs of field points (a) ρ 12 T = ( 0.1 mm , 0 , 0.1 mm , 0 ) , (b) ρ 12 T = ( 0.2 mm , 0 , 0.2 mm , 0 ) , (c) ρ 12 T = ( 0.3 mm , 0 , 0.3 mm , 0 ) , (d) ρ 12 T = ( 0.5 mm , 0 , 0.5 mm , 0 ) .

Fig. 7
Fig. 7

As Fig. 6, but δ x x = 1 mm .

Equations (23)

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W ( r 1 , r 2 , ω ) [ W i j ( r 1 , r 2 , ω ) ] = [ E i * ( r 1 , ω ) E j ( r 2 , ω ) ]
( i = x , y , j = x , y ) ,
E i ( ρ , z , ω ) = i λ [ Det ( B ) ] 1 2 E i ( 0 ) ( ρ , ω ) × exp [ i k 2 ( ρ T B 1 A ρ 2 ρ T B 1 ρ + ρ T DB 1 ρ ) ] × exp [ i k 2 ( ρ T B 1 e f + ρ T B 1 g h ) ] d ρ ,
A = [ a 0 0 a ] , B = [ b 0 0 b ] , C = [ c 0 0 c ] , D = [ d 0 0 d ] ,
e = 2 ( α T ε x + β T ε x ) , f = 2 ( α T ε y + β T ε y ) ,
g = 2 ( b γ T d α T ) ε x + 2 ( b δ T d β T ) ε x ,
h = 2 ( b γ T d α T ) ε y + 2 ( b δ T d β T ) ε y ,
α T = 1 a , β T = l b , γ T = c , δ T = ± 1 d .
W i j ( ρ 12 , z , ω ) = 1 λ 2 [ Det ( B ¯ ) ] 1 2 W i j ( 0 ) ( ρ 12 , ω ) × exp [ i k 2 ( ρ 12 T B ¯ 1 A ¯ ρ 12 2 ρ 12 T B ¯ 1 ρ 12 + ρ 12 T D ¯ B ¯ 1 ρ 12 ) ] × exp [ i k 2 ( ρ 12 T B ¯ 1 e ¯ f + ρ 12 T B ¯ 1 g ¯ h ) ] d 4 ρ 12 ,
A ¯ = [ A 0 0 A ] , B ¯ = [ B 0 0 B ] , C ¯ = [ C 0 0 C ] , D ¯ = [ D 0 0 D ] .
e ¯ f = [ e f e f ] , g ¯ h = [ g h g h ] .
W i j ( 0 ) ( ρ 1 , ρ 2 , ω ) = S i ( 0 ) ( ρ 1 , ω ) S j ( 0 ) ( ρ 2 , ω ) η 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 2 ρ 2 2 4 σ j 2 ) exp ( ρ 2 ρ 1 2 2 δ i j 2 ) .
W i j ( 0 ) ( ρ 12 , ω ) = A i A j B i j exp ( i k 2 ρ 12 T M i j 1 ρ 12 ) ,
M i j 1 = [ i 2 k σ i 2 i k δ i j 2 0 i k δ i j 2 0 0 i 2 k σ i 2 i k δ i j 2 0 i k δ i j 2 i k δ i j 2 0 i 2 k σ j 2 i k δ i j 2 0 0 i k δ i j 2 0 i 2 k σ j 2 i k δ i j 2 ] .
W i j ( ρ 12 , z , ω ) = A i A j B i j [ Det ( A ¯ + B ¯ M i j 1 ) ] 1 2 exp { i k 2 ρ 12 T [ D ¯ B ¯ 1 B ¯ 1 T ( B ¯ 1 A ¯ + M i j 1 ) 1 B ¯ 1 ] ρ 12 } exp { i k 2 ρ 12 T [ B ¯ 1 g ¯ h + B ¯ 1 T ( B ¯ 1 A ¯ + M i j 1 ) 1 B ¯ 1 e ¯ f ] } exp [ i k 8 e ¯ f T B ¯ 1 T ( B ¯ 1 A ¯ + M i j 1 ) 1 B ¯ 1 e ¯ f ] .
W i j ( ρ 12 , z , ω ) = A i A j B i j [ Det ( A ¯ + B ¯ M i j 1 ) ] 1 2 × exp { i k 2 ρ 12 T [ D ¯ B ¯ 1 B ¯ 1 T ( B ¯ 1 A ¯ + M i j 1 ) 1 B ¯ 1 ] ρ 12 } .
P ( ρ 12 , z , ω ) = 1 4 Det W ( ρ 12 , z , ω ) [ Tr W ( ρ 12 , z , ω ) ] 2 ,
μ ( ρ 12 , z , ω ) = Tr W ( ρ 12 , z , ω ) Tr W ( ρ 11 , z , ω ) Tr W ( ρ 22 , z , ω ) .
A = [ 1 z f 1 0 0 1 z f 1 ] , B = [ z 0 0 z ] ,
C = [ 1 f 1 0 0 1 f 1 ] , D = [ 1 0 0 1 ] .
α T = z f 1 , β T = 0 , γ T = 1 f 1 , δ T = 0 ,
e = 2 ε x z f 1 , f = 0 , g = 0 , h = 0 .

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