Abstract

Conditions ensuring that the polarization properties at the output plane of a Young interferometer fed by an electromagnetic partially coherent beam are the same as those at the pinholes are derived. Such a behavior is interpreted in terms of the vector modes of the electromagnetic source corresponding to the field emerging from the Young pinholes.

© 2007 Optical Society of America

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References

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  1. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
    [CrossRef]
  2. H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and polarization," Opt. Commun. 252, 268-274 (2005).
    [CrossRef]
  3. F. Gori, M. Santarsiero, and R. Borghi, "Vector mode analysis of a Young interferometer," Opt. Lett. 31, 858-860 (2006).
    [CrossRef]
  4. Y. Li, H. Lee, and E. Wolf, "Spectra, coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams," Opt. Commun. 265, 63-72 (2006).
    [CrossRef]
  5. T. Setälä, J. Tervo, and A. T. Friberg, "Stokes parameters and polarization contrasts in Young's interference experiment," Opt. Lett. 31, 2208-2210 (2006).
    [CrossRef] [PubMed]
  6. A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855-2860 (2006).
    [CrossRef]
  7. M. Santarsiero, F. Gori, R. Borghi and G. Guattari, "Vector-mode analysis of symmetric two-point sources," J. Opt. A, Pure Appl. Opt. 9, 593-602 (2007).
    [CrossRef]
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  9. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, "Effects of coherence on the degree of polarization in Young interference pattern," Opt. Lett. 31, 688-690 (2006).
    [CrossRef] [PubMed]
  10. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, "Coherent-mode decomposition of partially polarized, partially coherent sources," J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]
  11. J. Tervo, T. Setälä, and A. T. Friberg, "Theory of partially coherent electromagnetic fields in the space-frequency domain," J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  12. F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 41-43 (1998).
    [CrossRef]
  13. E. Wolf, "Unified theory of coherence and polarization of statistical electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  14. E. Wolf, "New theory of partial coherence in the spacefrequency domain. I. Spectra and cross-spectra of steady-state sources," J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  15. R. Horn and C. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991), Chap. 4.
    [CrossRef]

2007

M. Santarsiero, F. Gori, R. Borghi and G. Guattari, "Vector-mode analysis of symmetric two-point sources," J. Opt. A, Pure Appl. Opt. 9, 593-602 (2007).
[CrossRef]

2006

2005

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and polarization," Opt. Commun. 252, 268-274 (2005).
[CrossRef]

2004

2003

1998

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

F. Gori, "Matrix treatment for partially polarized, partially coherent beams," Opt. Lett. 23, 41-43 (1998).
[CrossRef]

1982

J. Opt. A, Pure Appl. Opt.

M. Santarsiero, F. Gori, R. Borghi and G. Guattari, "Vector-mode analysis of symmetric two-point sources," J. Opt. A, Pure Appl. Opt. 9, 593-602 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Roychowdhury and E. Wolf, "Young's interference experiment with light of any state of coherence and polarization," Opt. Commun. 252, 268-274 (2005).
[CrossRef]

Y. Li, H. Lee, and E. Wolf, "Spectra, coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams," Opt. Commun. 265, 63-72 (2006).
[CrossRef]

Opt. Lett.

Phys. Lett. A

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

Pure Appl. Opt.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, "Beam coherence polarization matrix," Pure Appl. Opt. 7, 941-951 (1998).
[CrossRef]

Other

R. Horn and C. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991), Chap. 4.
[CrossRef]

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

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Figures (5)

Fig. 1
Fig. 1

Typical intensity distributions produced across the output plane by the four modes. The phase ψ S has been set to zero and P = π K denotes the fringe period.

Fig. 2
Fig. 2

Polarization of the modes at the two Young pinholes for the source of Section 5.

Fig. 3
Fig. 3

Polarization direction of Ψ 1 for the source of Section 5.

Fig. 4
Fig. 4

Four eigenvalues as functions of the scalar degree of coherence between the fields at the Young pinholes for the source of Section 5.

Fig. 5
Fig. 5

Polarization of the modes at the two Young pinholes for the case of the source of Section 5 with μ 0 .

Equations (49)

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E ( r , t ) = ( E x ( r , t ) E y ( r , t ) ) ,
J ̂ ( r 1 , r 2 ) = E ( r 1 , t ) E ( r 2 , t ) = [ J x x ( r 1 , r 2 ) J x y ( r 1 , r 2 ) J y x ( r 1 , r 2 ) J y y ( r 1 , r 2 ) ] ,
I ( r ) = Tr { J ̂ ( r , r ) }
P ( r ) = 1 4 Det { J ̂ ( r , r ) } [ Tr { J ̂ ( r , r ) } ] 2 ,
J ̂ ( r 1 , r 2 ) Φ n ( r 2 ) d r 2 = Λ n Φ n ( r 1 ) ,
J ̂ ( r 1 , r 2 ) = n Λ n Φ n ( r 1 ) Φ n ( r 2 ) .
Ε ( t ) = ( E 1 ( t ) E 2 ( t ) ) = ( E x ( 1 ) ( t ) E y ( 1 ) ( t ) E x ( 2 ) ( t ) E y ( 2 ) ( t ) ) ,
ρ ̂ = Ε ( t ) Ε ( t ) = [ J ̂ 11 J ̂ 12 J ̂ 12 J ̂ 22 ] ,
ρ ̂ Ψ n = λ n Ψ n .
Ψ n = ( Φ n , 1 Φ n , 2 ) = ( φ n , x ( 1 ) φ n , y ( 1 ) φ n , x ( 2 ) φ n , y ( 2 ) ) ,
ρ ̂ = n = 1 4 λ n Ψ n Ψ n .
E out ( ξ ) = E 1 e i K ξ + E 2 e i K ξ ,
J ̂ out ( ξ 1 , ξ 2 ) = E out ( ξ 1 ) E out ( ξ 2 ) ,
P ̂ = [ a c c * b ] ,
J ̂ out ( ξ , ξ ) = E 1 E 1 + E 2 E 2 + E 1 E 2 e 2 i K ξ + E 2 E 1 e 2 i K ξ = J ̂ 11 + J ̂ 22 + J ̂ 12 e 2 i K ξ + J ̂ 12 e 2 i K ξ ,
J ̂ out ( ξ , ξ ) = ( α + β + 2 Re { γ e 2 i K ξ } ) P ̂ ,
I out ( ξ ) = ( α + β + 2 Re { γ e 2 i K ξ } ) ( a + b ) ,
ρ ̂ = ρ ̂ S P ̂ ,
ρ ̂ S = [ α γ γ * β ] ,
J ̂ ( r 1 , r 2 ) = J S ( r 1 , r 2 ) P ̂ ,
ρ ̂ = ρ ̂ S P ̂ = [ α γ γ * β ] [ a c c * b ] ,
λ 1 S = 1 2 [ ( α + β ) + ( α β ) 2 + 4 γ 2 ] ,
λ 2 S = 1 2 [ ( α + β ) ( α β ) 2 + 4 γ 2 ] ,
Ψ 1 S = N S ( η S e i ψ S 1 ) , Ψ 2 S = N S ( e i ψ S η S ) ;
η S = ( α β ) + ( α β ) 2 + 4 γ 2 2 γ ,
N S = 1 1 + η S 2 .
I 1 , 2 S ( ξ ) = λ 1 , 2 S N S 2 [ 1 + η S 2 ± 2 η S cos ( 2 K ξ + ψ S ) ] ,
λ 1 = λ 1 S λ 1 P , λ 2 = λ 1 S λ 2 P ,
λ 3 = λ 2 S λ 1 P , λ 4 = λ 2 S λ 2 P ;
Ψ 1 = Ψ 1 S Ψ 1 P , Ψ 2 = Ψ 1 S Ψ 2 P ,
Ψ 3 = Ψ 2 S Ψ 1 P , Ψ 4 = Ψ 2 S Ψ 2 P .
P out ( ξ ) = [ I 1 ( ξ ) + I 3 ( ξ ) ] [ I 2 ( ξ ) + I 4 ( ξ ) ] I 1 ( ξ ) + I 2 ( ξ ) + I 3 ( ξ ) + I 4 ( ξ ) = λ 1 P λ 2 P λ 1 P + λ 2 P ,
ρ ̂ = α [ 1 μ μ 1 ] [ a c c b ] ,
I ph = α ( a + b ) ,
P ph = ( a b ) 2 + 4 c 2 ( a + b ) .
λ 1 = α ( 1 + μ ) [ ( a + b ) + ( a b ) 2 + 4 c 2 ] ,
λ 2 = α ( 1 + μ ) [ ( a + b ) ( a b ) 2 + 4 c 2 ] ,
λ 3 = α ( 1 μ ) [ ( a + b ) + ( a b ) 2 + 4 c 2 ] ,
λ 4 = α ( 1 μ ) [ ( a + b ) ( a b ) 2 + 4 c 2 ] ,
Ψ 1 = N 2 ( η P 1 η P 1 ) , Ψ 2 = N 2 ( 1 η P 1 η P ) ,
Ψ 3 = N 2 ( η P 1 η P 1 ) , Ψ 4 = N 2 ( 1 η P 1 η P ) ;
η P = ( a b ) + ( a b ) 2 + 4 c 2 2 c ,
N = 1 1 + η P 2 .
ϑ = tan 1 ( 1 η P ) = tan 1 [ ( a b ) 2 c + 1 + ( a b 2 c ) 2 ] 1 ,
P out ( ξ ) = I 1 ( ξ ) I 2 ( ξ ) I 1 ( ξ ) + I 2 ( ξ ) = λ 1 λ 2 λ 1 + λ 2 = ( a b ) 2 + 4 c 2 ( a + b ) ,
Ψ 1 = Ψ 1 + Ψ 3 2 = N ( η P 1 0 0 ) ,
Ψ 2 = Ψ 2 + Ψ 4 2 = N ( 1 η P 0 0 ) ,
Ψ 3 = Ψ 1 Ψ 3 2 = N ( 0 0 η P 1 ) ,
Ψ 4 = Ψ 2 Ψ 4 2 = N ( 0 0 1 η P ) .

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