Abstract

The set of functions that appear in the correlation matrix of an electromagnetic source must satisfy the constraint of nonnegative definiteness. Here we derive a necessary and sufficient condition for nonnegativeness for the class of electromagnetic Schell-model sources. This result also suggests a possible synthesis procedure for this type of source. As an illustration, two specific examples of electromagnetic Schell-model sources are discussed.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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  10. J. Deschamps, D. Courjon, and J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256-261 (1983).
    [CrossRef]
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  17. M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 96, 183901-183904 (2006).
  18. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291-1293 (2000).
    [CrossRef]
  19. 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]
  20. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
    [CrossRef]
  21. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
    [CrossRef]
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    [CrossRef]
  25. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarisation matrix,” Pure Appl. Opt. 7, 941-951 (1998).
    [CrossRef]
  26. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1985).
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  30. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
    [CrossRef]
  31. L. Pan and B. Lü, “Polarization changes in vector Gaussian-Schell-model beams propagating through a thin lens,” Optik (Stuttgart) 113, 459-463 (2002).
  32. L. Wang and B. Lü, “Focal shift of partially polarized Gaussian Schell-model beams,” Optik (Stuttgart) 114, 169-174 (2003).
    [CrossRef]
  33. 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]
  34. 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]
  35. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
    [CrossRef]
  36. T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256, 197-209 (2005).
    [CrossRef]
  37. 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]
  38. F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
    [CrossRef] [PubMed]
  39. A. Luis, “Ray picture of polarization and coherence in a Young interferometer,” J. Opt. Soc. Am. A 23, 2855-2860 (2006).
    [CrossRef]
  40. R. Martinez-Herrero and A. F. Moreu, “On the polarization of non-paraxial transverse fields,” Opt. Commun. 267, 20-23 (2006).
    [CrossRef]
  41. 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]
  42. J. X. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
    [CrossRef]
  43. B. J. Davis, “Simulation of vector fields with arbitrary second-order correlations,” Opt. Express 15, 2837-2846 (2007).
    [CrossRef] [PubMed]
  44. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  45. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
    [CrossRef]

2007 (3)

2006 (6)

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
[CrossRef] [PubMed]

A. Luis, “Ray picture of polarization and coherence in a Young interferometer,” J. Opt. Soc. Am. A 23, 2855-2860 (2006).
[CrossRef]

R. Martinez-Herrero and A. F. Moreu, “On the polarization of non-paraxial transverse fields,” Opt. Commun. 267, 20-23 (2006).
[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]

R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901-183904 (2006).
[CrossRef] [PubMed]

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 96, 183901-183904 (2006).

2005 (5)

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

T. Shirai, “Polarization properties of a class of electromagnetic Gaussian Schell-model beams which have the same far-zone intensity distribution as a fully coherent laser beam,” Opt. Commun. 256, 197-209 (2005).
[CrossRef]

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 (2)

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, 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]

2003 (1)

L. Wang and B. Lü, “Focal shift of partially polarized Gaussian Schell-model beams,” Optik (Stuttgart) 114, 169-174 (2003).
[CrossRef]

2002 (2)

L. Pan and B. Lü, “Polarization changes in vector Gaussian-Schell-model beams propagating through a thin lens,” Optik (Stuttgart) 113, 459-463 (2002).

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

2001 (2)

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]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (2)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
[CrossRef]

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

1994 (2)

1993 (1)

1991 (1)

1989 (1)

G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251-259 (1989).
[CrossRef]

1986 (1)

1983 (1)

1982 (1)

R. Grella, “Synthesis of generalized Collett-Wolf sources,” J. Org. Chem. 13, 127-131 (1982).

1979 (1)

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
[CrossRef]

1967 (1)

A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187-188 (1967).
[CrossRef]

1959 (1)

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233-243 (1959).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
[CrossRef] [PubMed]

R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901-183904 (2006).
[CrossRef] [PubMed]

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 96, 183901-183904 (2006).

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

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]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291-1293 (2000).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Bulabois, J.

Cole, T. W.

T. W. Cole, “Quasi-optical techniques of radio astronomy,” in Progress in Optics, Vol. XV (Elsevier, 1977), pp. 187-244.
[CrossRef]

Courjon, D.

Davis, B. J.

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258-1262 (1986).
[CrossRef]

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Deschamps, J.

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, “Synthetic-aperture optics,” in Progress in Optics, Vol. VIII (Elsevier, 1970), pp. 1-50.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531-3533 (2007).
[CrossRef] [PubMed]

R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901-183904 (2006).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
[CrossRef] [PubMed]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

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]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291-1293 (2000).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
[CrossRef]

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

P. De Santis, F. Gori, G. Guattari, and C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258-1262 (1986).
[CrossRef]

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Grella, R.

R. Grella, “Synthesis of generalized Collett-Wolf sources,” J. Org. Chem. 13, 127-131 (1982).

Guattari, G.

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

P. De Santis, F. Gori, G. Guattari, and C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258-1262 (1986).
[CrossRef]

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1985).

Indebetouw, G.

G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251-259 (1989).
[CrossRef]

James, D. F. V.

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, 1985).

Korotkova, O.

J. X. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (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]

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]

Lee, H.

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]

Li, Y.

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]

Lü, B.

L. Wang and B. Lü, “Focal shift of partially polarized Gaussian Schell-model beams,” Optik (Stuttgart) 114, 169-174 (2003).
[CrossRef]

L. Pan and B. Lü, “Polarization changes in vector Gaussian-Schell-model beams propagating through a thin lens,” Optik (Stuttgart) 113, 459-463 (2002).

Luis, A.

Mandel, L.

L. Mandel, “Fluctuations of photon beams: the distribution of the photo-electrons,” Proc. Phys. Soc. London 74, 233-243 (1959).
[CrossRef]

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

Martinez-Herrero, R.

R. Martinez-Herrero and A. F. Moreu, “On the polarization of non-paraxial transverse fields,” Opt. Commun. 267, 20-23 (2006).
[CrossRef]

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

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]

Moreu, A. F.

R. Martinez-Herrero and A. F. Moreu, “On the polarization of non-paraxial transverse fields,” Opt. Commun. 267, 20-23 (2006).
[CrossRef]

Mukunda, N.

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “Synthesis of partially coherent fields,” J. Opt. Soc. Am. A 3, 1258-1262 (1986).
[CrossRef]

P. de Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979).
[CrossRef]

Pan, L.

L. Pan and B. Lü, “Polarization changes in vector Gaussian-Schell-model beams propagating through a thin lens,” Optik (Stuttgart) 113, 459-463 (2002).

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

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]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291-1293 (2000).
[CrossRef]

Pu, J. X.

J. X. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

Riesz, F.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Dover, 1990).

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

Roychowdhury, H.

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]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (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]

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]

Santarsiero, M.

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531-3533 (2007).
[CrossRef] [PubMed]

R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901-183904 (2006).
[CrossRef] [PubMed]

M. Santarsiero and R. Borghi, “Measuring spatial coherence by using a reversed-wavefront Young interferometer,” Opt. Lett. 96, 183901-183904 (2006).

F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
[CrossRef] [PubMed]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
[CrossRef]

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]

G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291-1293 (2000).
[CrossRef]

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

Schell, A.

A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187-188 (1967).
[CrossRef]

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L. Wang and B. Lü, “Focal shift of partially polarized Gaussian Schell-model beams,” Optik (Stuttgart) 114, 169-174 (2003).
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J. X. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
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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).
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F. Gori, M. Santarsiero, R. Borghi, and E. Wolf, “Effects of coherence on the degree of polarization in a Young interference pattern,” Opt. Lett. 31, 688-690 (2006).
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[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
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[CrossRef]

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[CrossRef]

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A. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187-188 (1967).
[CrossRef]

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[CrossRef]

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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]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

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[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).
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G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. 195, 339-350 (2001).
[CrossRef]

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[CrossRef]

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]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208, 9-16 (2002).
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Opt. Express (1)

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L. Wang and B. Lü, “Focal shift of partially polarized Gaussian Schell-model beams,” Optik (Stuttgart) 114, 169-174 (2003).
[CrossRef]

Phys. Rev. E (1)

J. X. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. 96, 183901-183904 (2006).
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[CrossRef]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

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[CrossRef]

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Equations (38)

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J ( ρ 1 , ρ 2 ) = s ( ρ 1 ) s ( ρ 2 ) j ( ρ 1 ρ 2 ) ,
s ( ρ ) = J ( ρ , ρ ) ,
J α β ( ρ 1 , ρ 2 ) = E α * ( ρ 1 , t ) E β ( ρ 2 , t ) ; ( α , β = x , y ) ,
J α β ( ρ 1 , ρ 2 ) = s α ( ρ 1 ) s β ( ρ 2 ) j α β ( ρ 1 ρ 2 ) , ( α , β = x , y ) ,
s α ( ρ ) = J α α ( ρ , ρ ) ; ( α = x , y ) .
Q = α = x , y β = x , y j α β ( ρ 1 , ρ 2 ) f α * ( ρ 1 ) f β ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
j α β ( ρ 1 ρ 2 ) = j ̃ α β ( η ) exp [ 2 π i η ( ρ 1 ρ 2 ) ] d 2 η ,
Q = α = x , y β = x , y f ̃ α * ( η ) f ̃ β ( η ) j ̃ α β ( η ) d 2 η .
j ̃ y x ( η ) = j ̃ x y * ( η ) ,
Q = [ f ̃ x ( η ) 2 j ̃ x x ( η ) + f ̃ y ( η ) 2 j ̃ y y ( η ) + 2 Re { f ̃ x * ( η ) f ̃ y ( η ) j ̃ x y ( η ) } ] d 2 η ,
j ̃ x y ( η ) j ̃ x x ( η ) j ̃ y y ( η ) ,
B α β = j α β ( 0 ) ,
B x y B x x B y y = 1 ,
B x y = j x y ( 0 ) = j ̃ x y ( η ) d 2 η ,
B x y = j ̃ x y ( η ) d 2 η j ̃ x y ( η ) d 2 η j ̃ x x ( η ) j ̃ y y ( η ) d 2 η .
j ̃ x x ( η ) j ̃ y y ( η ) j ̃ x x ( η ) + j ̃ y y ( η ) 2 ,
B x y j ̃ x x ( η ) j ̃ y y ( η ) d 2 η j ̃ x x ( η ) + j ̃ y y ( η ) 2 d 2 η = B x x + B y y 2 = 1 .
j ̃ x y ( η ) 2 d 2 η j ̃ x x ( η ) j ̃ y y ( η ) d 2 η .
j x y ( ρ ) 2 d 2 ρ j x x * ( ρ ) j y y ( ρ ) d 2 ρ .
j x y ( ρ ) 2 d 2 ρ j x x ( ρ ) 2 d 2 ρ j y y ( ρ ) 2 d 2 ρ .
J α β ( ρ 1 , ρ 2 ) = s α ( ρ 1 ) s β ( ρ 2 ) exp { i [ ϕ β ( ρ 2 ) ϕ α ( ρ 1 ) ] } j α β ( ρ 1 ρ 2 ) , ( α , β = x , y ) ,
ϕ α ( ρ ) = π λ z ρ 2 ,
s α ( ρ ) = A α exp ( ρ 2 4 σ α 2 ) ,
j α β ( ρ ) = B α β exp ( ρ 2 2 δ α β 2 ) ,
B x y δ x y 2 exp ( 2 π 2 δ x y 2 η 2 ) δ x x δ y y exp [ π 2 ( δ x x 2 + δ y y 2 ) η 2 ]
δ x x 2 + δ y y 2 2 δ x y δ x x δ y y B x y .
max { δ x x , δ y y } δ x y min { δ x x , δ y y } B x y .
δ x x 2 + δ y y 2 2 max { δ x x , δ y y } , δ x x δ y y min { δ x x , δ y y } ,
δ x x 2 + δ y y 2 2 δ x x δ y y B x y ,
B x y 2 δ x x δ y y + δ y y δ x x .
j α β ( ρ ) = B α β besinc ( ρ δ α β ) ,
besinc ( t ) = 2 J 1 ( π t ) π t ,
besinc ( ρ δ α β ) FT 4 δ α β 2 π circ ( 2 δ α β η ) ,
δ x x δ y y circ ( 2 δ M η ) B x y δ x y 2 circ ( 2 δ x y η ) ,
max { δ x x , δ y y } δ x y δ x x δ y y B x y .
B x y min { δ x x δ y y , δ y y δ x x } .
δ x x δ x y δ x x B x y ,
[ circ ( 2 δ x x η ) circ ( 2 δ x y η ) circ ( 2 δ x y η ) circ ( 2 δ x x η ) ] ,

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