Abstract

The relationship between the visibility of fringes and the degree of spatial coherence in electromagnetic two-pinhole interference is assessed. It is demonstrated that the customary definition of the degree of coherence of an electromagnetic field is flawed and a new quantity, free of the formal drawbacks, is introduced. The new definition, which is shown to be consistent with known results for Gaussian statistics, has some unusual properties characteristic only for electromagnetic fields. The degree of coherence is measurable by a sequence of interference experiments.

© 2003 Optical Society of America

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  1. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  2. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
    [CrossRef]
  3. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
    [CrossRef]
  4. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theoremfor partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
    [CrossRef]
  5. G. Gbur and D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).
  6. 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]
  7. R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
    [CrossRef]
  8. A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
    [CrossRef] [PubMed]
  9. J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
    [CrossRef] [PubMed]
  10. T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
    [CrossRef] [PubMed]
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  12. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [CrossRef]
  13. B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
    [CrossRef]
  14. B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
    [CrossRef]
  15. W. H. Carter and E. Wolf, “Coherence properties of lambertian and non-lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  16. W. H. Carter and E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
    [CrossRef] [PubMed]
  17. T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
    [CrossRef]
  18. G. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).
  19. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.
  20. C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical fields,” Phys. Rev. 134, A1143–A1149 (1964).
    [CrossRef]
  21. J. C. Samson and J. V. Olson, “Some comments on the description of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–130 (1980).
    [CrossRef]
  22. R. Barakat, “n-Fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
    [CrossRef]
  23. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]

2003 (2)

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]

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

2002 (3)

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

2000 (4)

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

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

G. Gbur and D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

1999 (2)

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
[CrossRef]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

1998 (1)

1987 (1)

W. H. Carter and E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

1983 (1)

R. Barakat, “n-Fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

1980 (1)

J. C. Samson and J. V. Olson, “Some comments on the description of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–130 (1980).
[CrossRef]

1975 (1)

1964 (1)

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical fields,” Phys. Rev. 134, A1143–A1149 (1964).
[CrossRef]

1963 (2)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[CrossRef]

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Agrawal, G. P.

Arfken, G.

G. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Barakat, R.

R. Barakat, “n-Fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

Blomstedt, K.

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

Borghi, R.

Carminati, R.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

Carter, W. H.

W. H. Carter and E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

W. H. Carter and E. Wolf, “Coherence properties of lambertian and non-lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

Chen, Y.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

Friberg, A. T.

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Gbur, G.

G. Gbur and D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).

Gori, F.

Greffet, J.-J.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

Guattari, G.

James, D. F. V.

G. Gbur and D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).

Joulain, K.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

Kaivola, M.

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

Mainguy, S.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

Mandel, L.

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

Mehta, C. L.

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical fields,” Phys. Rev. 134, A1143–A1149 (1964).
[CrossRef]

Mulet, J.-P.

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

Olson, J. V.

J. C. Samson and J. V. Olson, “Some comments on the description of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–130 (1980).
[CrossRef]

Piquero, G.

Samson, J. C.

J. C. Samson and J. V. Olson, “Some comments on the description of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–130 (1980).
[CrossRef]

Santarsiero, M.

Setälä, T.

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Shchegrov, A. V.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Simon, R.

Weber, H. J.

G. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Wolf, E.

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

W. H. Carter and E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

W. H. Carter and E. Wolf, “Coherence properties of lambertian and non-lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical fields,” Phys. Rev. 134, A1143–A1149 (1964).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

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

Geophys. J. R. Astron. Soc. (1)

J. C. Samson and J. V. Olson, “Some comments on the description of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–130 (1980).
[CrossRef]

J. Mod. Opt. (1)

G. Gbur and D. F. V. James, “Unpolarized sources that generate highly polarized fields outside the source,” J. Mod. Opt. 47, 1171–1177 (2000).

J. Opt. Soc. Am. (1)

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

Nature (1)

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[CrossRef] [PubMed]

Nuovo Cimento (2)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[CrossRef]

Opt. Acta (1)

R. Barakat, “n-Fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Phys. Rev. (1)

C. L. Mehta and E. Wolf, “Coherence properties of blackbody radiation. I. Correlation tensors of the classical fields,” Phys. Rev. 134, A1143–A1149 (1964).
[CrossRef]

Phys. Rev. A (1)

W. H. Carter and E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

Phys. Rev. E (2)

T. Setälä, K. Blomstedt, M. Kaivola, and A. T. Friberg, “Universality of electromagnetic-field correlations within homogeneous and isotropic sources,” Phys. Rev. E 67, 026613 (2003).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

Other (3)

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

G. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, New York, 2000), Sec. 13.2.

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Figures (1)

Fig. 1.
Fig. 1.

The geometry of Young’s interference experiment.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

I ( r , t ) = I 1 ( r , t ) + I 2 ( r , t ) + 2 I 1 ( r , t ) I 2 ( r , t ) { γ ( r 1 , r 2 , ( R 1 R 2 ) c ) } ,
γ ( r 1 , r 2 , τ ) = Γ ( r 1 , r 2 , τ ) I ( r 1 , t ) I ( r 2 , t ) ,
( r 1 , r 2 , τ ) = [ i j ( r 1 , r 2 , τ ) ] = [ E i * ( r 1 , t ) E j ( r 2 , t + τ ) ] ,
ζ ( r 1 , r 2 , τ ) = tr ( r 1 , r 2 , τ ) I ( r 1 , t ) I ( r 2 , t ) ,
' ( r 1 , r 2 , τ ) = ( r 1 ) ( r 1 , r 2 , τ ) 1 ( r 2 ) .
γ 2 ( r 1 , r 2 , τ ) = tr [ ( r 1 , r 2 , τ ) ( r 2 , r 1 , τ ) ] I ( r 1 , t ) I ( r 2 , t )
= i , j ij ( r 1 , r 2 , τ ) 2 i , j ii ( r 1 , r 1 , 0 ) jj ( r 2 , r 2 , 0 ) ,
P 3 2 ( r ) = 3 2 [ tr Φ 2 ( r ) tr 2 Φ ( r ) 1 3 ] = 3 2 [ γ 2 ( r ) 1 3 ] ,
P 2 2 ( r ) = 2 [ tr Φ 2 ( r ) tr 2 Φ ( r ) 1 2 ] = 2 [ γ 2 ( r ) 1 2 ] .
γ ( r 1 , r 2 , τ ) 2 = Δ I ( r 1 , t ) Δ I ( r 2 , t + τ ) I ( r 1 , t ) I ( r 2 , t ) .
Δ I ( r 1 , t ) Δ I ( r 2 , t + τ ) = i , j ij ( r 1 , r 2 , τ ) 2 .

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