Abstract

We assess the degree of coherence of vectorial electromagnetic fields in the space–frequency domain as the distance between the cross-spectral density matrix and the identity matrix representing completely incoherent light. This definition is compared with previous approaches. It is shown that this distance provides an upper bound for the degree of coherence and visibility for any pair of scalar waves obtained by linear combinations of the original fields. This same approach emerges when applying a previous definition of global coherence to a Young interferometer.

© 2007 Optical Society of America

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  1. B. Karczewski, "Degree of coherence of the electromagnetic field," Phys. Lett. 5, 191-192 (1963).
    [CrossRef]
  2. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic fields," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  3. S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
    [CrossRef]
  4. J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
    [CrossRef]
  5. T. Setälä, J. Tervo, and A. T. Friberg, "Complete electromagnetic coherence in the space-frequency domain," Opt. Lett. 29, 328-330 (2004).
    [CrossRef]
  6. E. Wolf, "Comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1712 (2004).
    [CrossRef]
  7. T. Setälä, J. Tervo, and A. T. Friberg, "Reply to comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1713-1714 (2004).
    [CrossRef]
  8. F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica (Utrecht) 5, 785-795 (1938).
    [CrossRef]
  9. F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. London 61, 158-164 (1948).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, 1999).
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  12. A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  13. A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
    [CrossRef]
  14. A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
    [CrossRef]
  15. A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
    [CrossRef]
  16. A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
    [CrossRef]
  17. A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
    [CrossRef]
  18. A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
    [CrossRef]
  19. A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
    [CrossRef]
  20. A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855-2860 (2006).
    [CrossRef]
  21. F. Gori, M. Santarsiero, and R. Borghi, "Vector mode analysis of a Young interferometer," Opt. Lett. 31, 858-860 (2006).
    [CrossRef] [PubMed]
  22. 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]
  23. R. Barakat, "Degree of polarization and the principal idempotents of the coherency matrix," Opt. Commun. 23, 147-150 (1977).
    [CrossRef]
  24. J. C. Samson and J. V. Olson, "Generalized Stokes vectors and generalized power spectra for second-order stationary vector-processes," SIAM J. Appl. Math. 40, 137-149 (1981).
    [CrossRef]
  25. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
    [CrossRef] [PubMed]
  26. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization for optical near fields," Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  27. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.
  28. T. Saastamoinen and J. Tervo, "Geometric approach to the degree of polarization for arbitrary fields," J. Mod. Opt. 51, 2039-2045 (2004).
    [CrossRef]
  29. J. Ellis and A. Dogariu, "Complex degree of mutual polarization," Opt. Lett. 29, 536-538 (2004).
    [CrossRef] [PubMed]
  30. Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  31. T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
    [CrossRef]
  32. M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
    [CrossRef]
  33. S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
    [CrossRef]
  34. R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J. Opt. Soc. Am. 53, 317-323 (1963).
    [CrossRef]
  35. P. Réfrégier and F. Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  36. M. J. Bastiaans, "New class of uncertainty relations for partially coherent light," J. Opt. Soc. Am. A 1, 711-715 (1984).
    [CrossRef]
  37. H. Lajunen, J. Tervo, and P. Vahimaa, "Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses," J. Opt. Soc. Am. A 21, 2117-2123 (2004).
    [CrossRef]
  38. H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
    [CrossRef]
  39. P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
    [CrossRef]
  40. M. A. Alonso, "Radiometry and wide-angle wave fields III: partial coherence," J. Opt. Soc. Am. A 18, 2502-2511 (2001).
    [CrossRef]

2006 (2)

2005 (5)

H. Lajunen, P. Vahimaa, and J. Tervo, "Theory of spatially and spectrally partially coherent pulses," J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

P. Réfrégier and F. Goudail, "Invariant degrees of coherence of partially polarized light," Opt. Express 13, 6051-6060 (2005).
[CrossRef] [PubMed]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

2004 (11)

A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
[CrossRef]

T. Saastamoinen and J. Tervo, "Geometric approach to the degree of polarization for arbitrary fields," J. Mod. Opt. 51, 2039-2045 (2004).
[CrossRef]

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[CrossRef]

S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
[CrossRef]

P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
[CrossRef]

T. Setälä, J. Tervo, and A. T. Friberg, "Complete electromagnetic coherence in the space-frequency domain," Opt. Lett. 29, 328-330 (2004).
[CrossRef]

J. Ellis and A. Dogariu, "Complex degree of mutual polarization," Opt. Lett. 29, 536-538 (2004).
[CrossRef] [PubMed]

E. Wolf, "Comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1712 (2004).
[CrossRef]

T. Setälä, J. Tervo, and A. T. Friberg, "Reply to comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1713-1714 (2004).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, "Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses," J. Opt. Soc. Am. A 21, 2117-2123 (2004).
[CrossRef]

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]

2003 (6)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

J. Tervo, T. Setälä, and A. T. Friberg, "Degree of coherence for electromagnetic fields," Opt. Express 11, 1137-1143 (2003).
[CrossRef]

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

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

2002 (3)

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[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]

2001 (1)

2000 (1)

T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
[CrossRef]

1999 (1)

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

1998 (1)

Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

1995 (1)

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

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

1984 (1)

1981 (1)

J. C. Samson and J. V. Olson, "Generalized Stokes vectors and generalized power spectra for second-order stationary vector-processes," SIAM J. Appl. Math. 40, 137-149 (1981).
[CrossRef]

1977 (1)

R. Barakat, "Degree of polarization and the principal idempotents of the coherency matrix," Opt. Commun. 23, 147-150 (1977).
[CrossRef]

1963 (2)

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

R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J. Opt. Soc. Am. 53, 317-323 (1963).
[CrossRef]

1948 (1)

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. London 61, 158-164 (1948).
[CrossRef]

1938 (1)

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica (Utrecht) 5, 785-795 (1938).
[CrossRef]

Alonso, M. A.

Barakat, R.

R. Barakat, "Degree of polarization and the principal idempotents of the coherency matrix," Opt. Commun. 23, 147-150 (1977).
[CrossRef]

R. Barakat, "Theory of the coherency matrix for light of arbitrary spectral bandwidth," J. Opt. Soc. Am. 53, 317-323 (1963).
[CrossRef]

Bastiaans, M. J.

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
[CrossRef]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Borghi, R.

Born, M.

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

Brosseau, Ch.

Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
[CrossRef]

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

Dennis, M. R.

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[CrossRef]

Dogariu, A.

Ellis, J.

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

Friberg, A. T.

Gil, J. J.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

Gori, F.

Goudail, F.

Kaivola, M.

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]

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
[CrossRef]

Lajunen, H.

Leavy, J. V.

S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
[CrossRef]

Luis, A.

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

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
[CrossRef]

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[CrossRef]

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Mandel, L.

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

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

Olson, J. V.

J. C. Samson and J. V. Olson, "Generalized Stokes vectors and generalized power spectra for second-order stationary vector-processes," SIAM J. Appl. Math. 40, 137-149 (1981).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Réfrégier, P.

Saastamoinen, T.

T. Saastamoinen and J. Tervo, "Geometric approach to the degree of polarization for arbitrary fields," J. Mod. Opt. 51, 2039-2045 (2004).
[CrossRef]

Samson, J. C.

J. C. Samson and J. V. Olson, "Generalized Stokes vectors and generalized power spectra for second-order stationary vector-processes," SIAM J. Appl. Math. 40, 137-149 (1981).
[CrossRef]

Santarsiero, M.

Schirmer, S. G.

S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
[CrossRef]

Setälä, T.

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]

Tervo, J.

Vahimaa, P.

Wolf, E.

E. Wolf, "Comment on 'Complete electromagnetic coherence in the space-frequency domain'," Opt. Lett. 29, 1712 (2004).
[CrossRef]

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

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

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

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

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. London 61, 158-164 (1948).
[CrossRef]

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica (Utrecht) 5, 785-795 (1938).
[CrossRef]

Zhang, T.

S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
[CrossRef]

J. Mod. Opt. (1)

T. Saastamoinen and J. Tervo, "Geometric approach to the degree of polarization for arbitrary fields," J. Mod. Opt. 51, 2039-2045 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

P. Vahimaa and J. Tervo, "Unified measures for optical fields: degree of polarization and effective degree of coherence," J. Opt. A, Pure Appl. Opt. 6, S41-S44 (2004).
[CrossRef]

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Phys. A (2)

S. G. Schirmer, T. Zhang, and J. V. Leavy, "Orbits of quantum states and geometry of Bloch vectors for N-level systems," J. Phys. A 37, 1389-1402 (2004).
[CrossRef]

A. Luis, "Visibility for anharmonic fringes," J. Phys. A 35, 8805-8815 (2002).
[CrossRef]

Opt. Commun. (5)

A. Luis, "Polarization correlations in quantum optics," Opt. Commun. 216, 165-172 (2003).
[CrossRef]

A. Luis, "Properties of spatial-angular Stokes parameters," Opt. Commun. 251, 243-253 (2005).
[CrossRef]

A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
[CrossRef]

S. A. Ponomarenko and E. Wolf, "The spectral degree of coherence of fully spatially coherent electromagnetic beams," Opt. Commun. 227, 73-74 (2003).
[CrossRef]

R. Barakat, "Degree of polarization and the principal idempotents of the coherency matrix," Opt. Commun. 23, 147-150 (1977).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Lett. (1)

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

Phys. Lett. A (2)

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

A. Luis, "Visibility for multi-particle interference," Phys. Lett. A 314, 197-202 (2003).
[CrossRef]

Phys. Rev. A (3)

A. Luis, "Classical and quantum polarization correlations," Phys. Rev. A 69, 023803 (2004).
[CrossRef]

A. Luis, "Polarization distribution and degree of polarization for three-dimensional quantum light fields," Phys. Rev. A 71, 063815 (2005).
[CrossRef]

A. Luis, "Degree of polarization in quantum optics," Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Phys. Rev. E (2)

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

T. Carozzi, R. Karlsson, and J. Bergman, "Parameters characterizing electromagnetic wave polarization," Phys. Rev. E 61, 2024-2028 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, "Experimental realization of any discrete unitary operator," Phys. Rev. Lett. 73, 58-61 (1994).
[CrossRef] [PubMed]

Physica (Utrecht) (1)

F. Zernike, "The concept of degree of coherence and its application to optical problems," Physica (Utrecht) 5, 785-795 (1938).
[CrossRef]

Proc. Phys. Soc. London (1)

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. London 61, 158-164 (1948).
[CrossRef]

SIAM J. Appl. Math. (1)

J. C. Samson and J. V. Olson, "Generalized Stokes vectors and generalized power spectra for second-order stationary vector-processes," SIAM J. Appl. Math. 40, 137-149 (1981).
[CrossRef]

Other (4)

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

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

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, "Generalized polarization algebra," in Proceedings of the 8th Conference Zaragoza-Pau of Applied Mathematics and Statistics (2003), pp. 161-167. Available on line at http://www.unizar.es/galdeano/actaslowbarpau/PDFVIII/pp161-167.pdf.

Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

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Equations (38)

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

E l * ( r j , ω ) E m ( r k , ω ) = d τ E l * ( r j , t ) E m ( r k , t + τ ) exp ( i w τ ) ,
M ( r 1 , r 2 ) = [ Γ ( r 1 , r 1 ) Γ ( r 1 , r 2 ) Γ ( r 1 , r 2 ) Γ ( r 2 , r 2 ) ] ,
Γ ( r i , r j ) = [ E x * ( r i ) E x ( r j ) E x * ( r i ) E y ( r j ) E y * ( r i ) E x ( r j ) E y * ( r i ) E y ( r j ) ] .
D = 4 3 tr [ ( 1 4 I 1 tr M M ) 2 ] = 4 3 [ tr ( M 2 ) ( tr M ) 2 1 4 ] ,
tr ( M 2 ) = 1 2 { ( 1 + P 1 2 ) I 1 2 + ( 1 + P 2 2 ) I 2 2 + 4 I 1 I 2 μ ̃ 2 } ,
P 2 ( r j ) = 2 tr Γ 2 ( r j , r j ) [ tr Γ ( r j , r j ) ] 2 1 ,
μ ̃ 2 ( r 1 , r 2 ) = tr [ Γ ( r 1 , r 2 ) Γ ( r 1 , r 2 ) ] tr Γ ( r 1 , r 1 ) tr Γ ( r 2 , r 2 ) .
ν 0 ( r i , r j ) = Γ x , x ( r i , r j ) + Γ y , y ( r i , r j ) ,
ν 1 ( r i , r j ) = Γ x , y ( r i , r j ) + Γ y , x ( r i , r j ) ,
ν 2 ( r i , r j ) = i [ Γ x , y ( r i , r j ) Γ y , x ( r i , r j ) ] ,
ν 3 ( r i , r j ) = Γ x , x ( r i , r j ) Γ y , y ( r i , r j ) .
E 1 = E x ( r 1 ) , E 2 = E y ( r 1 ) , E 3 = E x ( r 2 ) , E 4 = E y ( r 2 ) .
M 2 = [ E 1 * E 1 E 1 * E 2 E 2 * E 1 E 2 * E 2 ] = q 0 I 2 + q σ ,
q 0 = 1 2 ( E 1 2 + E 2 2 ) , q 1 = 1 2 ( E 1 * E 2 + E 2 * E 1 ) ,
q 2 = i 2 ( E 1 * E 2 E 2 * E 1 ) , q 3 = 1 2 ( E 1 2 E 2 2 ) ,
D 2 = 2 [ tr ( M 2 2 ) ( tr M 2 ) 2 1 2 ] = q 2 q 0 2 ,
μ 2 ( E ̃ ) = E ̃ 1 * E ̃ 2 2 E ̃ 1 2 E ̃ 2 2 = q ̃ 1 2 + q ̃ 2 2 q ̃ 0 2 q ̃ 3 2 q ̃ 2 q ̃ 0 2 ,
v 2 ( E ̃ ) = 4 E ̃ 1 * E ̃ 2 2 ( E ̃ 1 2 + E ̃ 2 2 ) 2 = q ̃ 1 2 + q ̃ 2 2 q ̃ 0 2 q ̃ 2 q ̃ 0 2 ,
M ̃ = ( M N N K ) ,
M = ( E ̃ 1 * E ̃ 1 E ̃ 1 * E ̃ 2 E ̃ 2 * E ̃ 1 E ̃ 2 * E ̃ 2 ) ,
M ̃ = q 0 Λ 0 + 2 j = 1 15 q ̃ j Λ j ,
Λ 0 = 1 2 I , Λ j = 1 2 [ σ j 0 0 0 ] ,
μ 2 = q ̃ 1 2 + q ̃ 2 2 q ̃ 0 2 q ̃ 3 2 ,
μ 2 q ̃ 1 2 + q ̃ 2 2 + q ̃ 3 2 q ̃ 0 2 q ̃ 2 q ̃ 0 2 ,
D = 4 3 [ tr ( M ̃ 2 ) ( tr M ̃ ) 2 1 4 ] = 2 3 q ̃ 2 q 0 2 .
μ 2 3 q 0 2 2 q ̃ 0 2 D ,
v 2 = q ̃ 1 2 + q ̃ 2 2 q ̃ 0 2 3 q 0 2 2 q ̃ 0 2 D .
μ ̃ 2 ( r 1 , r 2 ) = [ tr Γ ( r 1 , r 2 ) ] 2 tr Γ ( r 1 , r 1 ) tr Γ ( r 2 , r 2 ) .
Γ 1 2 ( r 2 , r 2 ) Γ ( r 1 , r 2 ) Γ 1 2 ( r 1 , r 1 ) ,
E x ( r 1 , t ) = C e i ω t , E y ( r 2 , t ) = C e i ω t ,
M = C 2 [ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 ] ,
M = C 2 [ 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 ] ,
μ g 2 = d 2 r 1 d 2 r 2 tr Γ ( r 1 , r 1 ) tr Γ ( r 2 , r 2 ) μ ̃ 2 ( r 1 , r 2 ) [ d 2 r tr Γ ( r , r ) ] 2 .
E t ( r ) t ( r a ) E i ( a ) + t ( r + a ) E i ( a ) ,
Γ t ( r 1 , r 2 ) j , l = ± t ( r 1 a j ) t ( r 2 a l ) Γ i ( a j , a l ) ,
μ g 2 = tr ( M i 2 ) ( tr M i ) 2 ,
D = 4 3 ( μ g 2 1 4 ) .
μ g 2 = I + 2 ( 1 + P + 2 ) + I 2 ( 1 + P 2 ) + 4 I + I μ ̃ 2 2 ( I + I + ) 2 ,

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