Abstract

Coherence and polarization properties of the field generated by a partially polarized incoherent source are examined in terms of the 2×2 cross-spectral density matrix, with a view to deriving an electromagnetic counterpart of the van Cittert–Zernike theorem. Emphasis is placed on the polarization properties of the field. As some consequences of the extended theorem, we found that the spectral degree of polarization of the resultant field is generally different from that of the source and that both are the same in a special case. A different somewhat unconventional view of the extended theorem is also discussed in brief.

© 2009 Optical Society of America

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  1. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
    [CrossRef]
  2. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, Opt. Lett. 25, 1291 (2000).
    [CrossRef]
  3. E. Wolf, Phys. Lett. A 312, 263 (2003).
    [CrossRef]
  4. J. Tervo, T. Setälä, and A. T. Friberg, Opt. Express 11, 1137 (2003).
    [CrossRef] [PubMed]
  5. P. Réfrégier and F. Goudail, Opt. Express 13, 6051 (2005).
    [CrossRef] [PubMed]
  6. M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
    [CrossRef]
  7. S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
    [CrossRef]
  8. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, Opt. Express 17, 1746 (2009).
    [CrossRef] [PubMed]
  9. O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
    [CrossRef]
  10. T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
    [CrossRef]
  11. T. Shirai, Opt. Commun. 256, 197 (2005).
    [CrossRef]
  12. E. Wolf, Opt. Lett. 32, 3400 (2007).
    [CrossRef] [PubMed]
  13. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  14. A. Luis, J. Opt. Soc. Am. A 24, 1063 (2007).
    [CrossRef]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Sec. 6.6.
  16. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
    [CrossRef]

2009 (2)

2008 (1)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

2007 (2)

2006 (1)

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

2005 (2)

2004 (2)

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
[CrossRef]

2003 (2)

2000 (1)

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Alonso, M. A.

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, Opt. Lett. 25, 1291 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Friberg, A. T.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, Opt. Lett. 25, 1291 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Goudail, F.

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

James, D. F. V.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

Korotkova, O.

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Luis, A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Sec. 6.6.

Martínez-Niconoff, G.

Martínez-Vara, P.

Olvera-Santamaría, M. A.

Ostrovsky, A. S.

Piquero, G.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

Réfrégier, P.

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, Opt. Lett. 25, 1291 (2000).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

T. Shirai, Opt. Commun. 256, 197 (2005).
[CrossRef]

T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
[CrossRef]

Tervo, J.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

Wolf, E.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

E. Wolf, Opt. Lett. 32, 3400 (2007).
[CrossRef] [PubMed]

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

T. Shirai and E. Wolf, J. Opt. Soc. Am. A 21, 1907 (2004).
[CrossRef]

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Sec. 6.6.

J. Mod. Opt. (1)

M. A. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 (2006).
[CrossRef]

J. Opt. A (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, J. Opt. A 11, 085706 (2009).
[CrossRef]

J. Opt. A. (1)

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, J. Opt. A. 10, 055001 (2008).
[CrossRef]

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

Opt. Commun. (2)

T. Shirai, Opt. Commun. 256, 197 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, Opt. Commun. 233, 225 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Lett. A (1)

E. Wolf, Phys. Lett. A 312, 263 (2003).
[CrossRef]

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, Pure Appl. Opt. 7, 941 (1998).
[CrossRef]

Other (2)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Sec. 6.6.

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Equations (12)

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W ( ρ 1 , ρ 2 , ω ) = z = 0 W ( 0 ) ( ρ 1 , ρ 2 , ω ) G ( ρ 1 , ρ 1 , ω ) G ( ρ 2 , ρ 2 , ω ) d 2 ρ 1 d 2 ρ 2 ,
G ( ρ , ρ , ω ) = i k 2 π z exp [ i k 2 z ( ρ ρ ) 2 ] ,
W i j ( ρ 1 , ρ 2 , ω ) = E i ( ρ 1 , ω ) E j ( ρ 2 , ω ) , ( i = x , y ; j = x , y ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( 0 ) ( ρ 1 , ω ) δ ( ρ 2 ρ 1 ) ,
S ( 0 ) ( ρ , ω ) = [ S i j ( 0 ) ( ρ , ω ) ] ,     ( i = x , y ; j = x , y )
W ( ρ 1 , ρ 2 , ω ) = ( k 2 π z ) 2 exp [ i k 2 z ( ρ 2 2 ρ 1 2 ) ] D S ( 0 ) ( ρ , ω ) exp [ i k z ( ρ 2 ρ 1 ) ρ ] d 2 ρ .
η ( ρ 1 , ρ 2 , ω ) tr   W ( ρ 1 , ρ 2 , ω ) tr   W ( ρ 1 , ρ 1 , ω ) tr   W ( ρ 2 , ρ 2 , ω ) = exp [ i k 2 z ( ρ 2 2 ρ 1 2 ) ] D S ( 0 ) ( ρ , ω ) exp [ i k z ( ρ 2 ρ 1 ) ρ ] d 2 ρ D S ( 0 ) ( ρ , ω ) d 2 ρ ,
P ( ρ , ω ) = 1 4   det   W ( ρ , ρ , ω ) [ tr   W ( ρ , ρ , ω ) ] 2 ,
P ( 0 ) ( ρ , ω ) = [ S x x ( 0 ) ( ρ , ω ) S y y ( 0 ) ( ρ , ω ) ] 2 + 4 | S x y ( 0 ) ( ρ , ω ) | 2 S x x ( 0 ) ( ρ , ω ) + S y y ( 0 ) ( ρ , ω ) .
P ( ρ , ω ) = { D [ S x x ( 0 ) ( ρ , ω ) S y y ( 0 ) ( ρ , ω ) ] d 2 ρ } 2 + 4 | D S x y ( 0 ) ( ρ , ω ) d 2 ρ | 2 D [ S x x ( 0 ) ( ρ , ω ) + S y y ( 0 ) ( ρ , ω ) ] d 2 ρ .
P ( 0 ) ( ρ , ω ) = [ S x x ( 0 ) ( ω ) S y y ( 0 ) ( ω ) ] 2 + 4 | S x y ( 0 ) ( ω ) | 2 S x x ( 0 ) ( ω ) + S y y ( 0 ) ( ω ) .
E x ( ρ , ω ) = z = 0 E x ( 0 ) ( ρ , ω ) G x ( ρ , ρ , ω ) d 2 ρ ,

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