Abstract

The Mueller–Stokes formalism that governs conventional polarization optics is formulated for plane waves, and thus the only qualification one could require of a 4×4 real matrix M in order that it qualify to be the Mueller matrix of some physical system would be that M map Ω(pol), the positive solid light cone of Stokes vectors, into itself. In view of growing current interest in the characterization of partially coherent partially polarized electromagnetic beams, there is a need to extend this formalism to such beams wherein the polarization and spatial dependence are generically inseparably intertwined. This inseparability brings in additional constraints that a pre-Mueller matrix M mapping Ω(pol) into itself needs to meet in order to be an acceptable physical Mueller matrix. These additional constraints are motivated and fully characterized.

© 2010 Optical Society of America

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 6.
  2. C. Brosseau, Fundamentals of Polarized Light: A Statistical Approach (Wiley, 1998).
  3. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
    [CrossRef]
  4. D. F. V. James, “Change of polarization of light-beams on propagation through free space,” J. Opt. Soc. Am. A 11, 1641-1643 (1994).
    [CrossRef]
  5. F. Gori, “Matrix treatment of partially polarized, partially coherent beams,” Opt. Lett. 23, 241-243 (1998).
    [CrossRef]
  6. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).
  7. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  8. 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]
  9. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]
  10. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137-1142 (2003).
    [CrossRef] [PubMed]
  11. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space frequency domain,” Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  12. M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065-2069 (2007).
    [CrossRef]
  13. P. Refrégiér and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 1051-1060 (2005).
    [CrossRef]
  14. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
    [CrossRef] [PubMed]
  15. 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]
  16. A. Aiello and J. P. Woerdman, “Role of spatial coherence in polarization tomography,” Opt. Lett. 30, 1599-1601 (2005).
    [CrossRef] [PubMed]
  17. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
    [CrossRef] [PubMed]
  18. A. Luis, “Degree of coherence for vectorial electromagnetic fields as a distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063-1068 (2007).
    [CrossRef]
  19. R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471-1473 (2007).
    [CrossRef] [PubMed]
  20. R. Martínez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
    [CrossRef] [PubMed]
  21. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
    [CrossRef]
  22. N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” (Ed. note: commentary on ) Phys. Rev. 48, 696-702 (1935).
    [CrossRef]
  23. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
    [CrossRef]
  24. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
    [CrossRef]
  25. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).
  26. K. D. Abhyankar and A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
    [CrossRef]
  27. E. S. Fry and E. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811-2814 (1981).
    [CrossRef] [PubMed]
  28. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
    [CrossRef]
  29. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
    [CrossRef]
  30. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
    [CrossRef]
  31. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).
  32. S. R. Cloude, “Group-theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).
  33. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]
  34. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
    [CrossRef]
  35. A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps,” Phys. Rev. A 76, 032323 (2007).
    [CrossRef]
  36. Sudha, A. V. G. Rao, A. R. U. Devi, and A. K. Rajagopal, “Positive-operator-valued measure view of the ensemble approach to polarization optics,” J. Opt. Soc. Am. A 25, 874-880 (2008).
    [CrossRef]
  37. M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
    [CrossRef]
  38. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 809-812 (1979).
    [CrossRef] [PubMed]
  39. C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
    [CrossRef]
  40. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  41. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689-691 (2007).
    [CrossRef] [PubMed]
  42. V. Devlaminck and P. Terrier, “Definition of a parameter form for non-singular Mueller matrices,” J. Opt. Soc. Am. A 25, 2636-2643 (2008).
    [CrossRef]
  43. F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
    [CrossRef]
  44. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).
  45. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).
  46. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  47. N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416-426 (1985).
    [CrossRef]
  48. J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
    [CrossRef]

2009 (1)

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

2008 (2)

2007 (8)

2006 (1)

2005 (3)

2004 (1)

2003 (3)

2001 (1)

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]

1998 (4)

F. Gori, “Matrix treatment of 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-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).

1996 (1)

1994 (3)

1993 (2)

V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

1992 (1)

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

1987 (3)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
[CrossRef]

J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
[CrossRef]

1986 (1)

S. R. Cloude, “Group-theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).

1985 (2)

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

1981 (2)

E. S. Fry and E. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811-2814 (1981).
[CrossRef] [PubMed]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

1979 (1)

1969 (1)

K. D. Abhyankar and A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” (Ed. note: commentary on ) Phys. Rev. 48, 696-702 (1935).
[CrossRef]

1852 (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Abhyankar, K. D.

K. D. Abhyankar and A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

Aiello, A.

A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps,” Phys. Rev. A 76, 032323 (2007).
[CrossRef]

A. Aiello and J. P. Woerdman, “Role of spatial coherence in polarization tomography,” Opt. Lett. 30, 1599-1601 (2005).
[CrossRef] [PubMed]

Anderson, D. G. M.

Barakat, R.

D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
[CrossRef]

Bohr, N.

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” (Ed. note: commentary on ) Phys. Rev. 48, 696-702 (1935).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[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, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).

Boulvert, F.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Brosseau, C.

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

Cariou, J.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Group-theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).

De Martino, A.

Dennis, M. R.

Devi, A. R. U.

Devlaminck, V.

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Elachi, C.

J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
[CrossRef]

Friberg, A. T.

Fry, E. S.

Fymat, A. L.

K. D. Abhyankar and A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
[CrossRef]

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
[CrossRef]

Givens, C. R.

C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gopala Rao, A. V.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).

Gori, F.

Goudail, F.

P. Refrégiér and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 1051-1060 (2005).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

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

Guyot, S.

Howell, B. J.

James, D. F. V.

Kattawar, E. W.

Kim, K.

Korotkova, O.

Kostinski, B.

C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

Kumar, M. Sanjay

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

Le Brun, G.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Le Jeune, B.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Lu, S. Y.

Luis, A.

Mallesh, K. S.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

Mandel, L.

Martin, L.

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Martínez-Herrero, R.

Mejías, P. M.

Mondello, A.

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]

Mukunda, N.

Ossikovski, R.

Papas, C. H.

J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[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]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Puentes, G.

A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps,” Phys. Rev. A 76, 032323 (2007).
[CrossRef]

Rajagopal, A. K.

Rao, A. V. G.

Refrégiér, P.

P. Refrégiér and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 1051-1060 (2005).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young's fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588-590 (2007).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[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, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” J. Opt. A, Pure Appl. Opt. 7, 941-951 (1998).

Setälä, T.

Simon, R.

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[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]

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
[CrossRef]

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416-426 (1985).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

Sridhar, R.

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Sudarshan, E. C. G.

Sudha,

Sudha, A. V. G. Rao, A. R. U. Devi, and A. K. Rajagopal, “Positive-operator-valued measure view of the ensemble approach to polarization optics,” J. Opt. Soc. Am. A 25, 874-880 (2008).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).

Terrier, P.

Tervo, J.

van der Mee, V. M.

V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

van Zyl, J. J.

J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
[CrossRef]

Vicalvi, S.

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

Woerdman, J. P.

A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps,” Phys. Rev. A 76, 032323 (2007).
[CrossRef]

A. Aiello and J. P. Woerdman, “Role of spatial coherence in polarization tomography,” Opt. Lett. 30, 1599-1601 (2005).
[CrossRef] [PubMed]

Wolf, E.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198-200 (2005).
[CrossRef] [PubMed]

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

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 6.

Appl. Opt. (2)

Eur. Phys. J.: Appl. Phys. (1)

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J.: Appl. Phys. 40, 1-47 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarization of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818-825 (1987).
[CrossRef]

J. Math. Phys. (2)

K. D. Abhyankar and A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

J. Mod. Opt. (4)

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
[CrossRef]

C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955-987 (1998).

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics: II. Necessary and sufficient condition for Jones-derived Mueller matrix,” J. Mod. Opt. 45, 989-999 (1998).

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

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

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]

J. Opt. Soc. Am. A (10)

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially coherent partially polarized beams,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

D. F. V. James, “Change of polarization of light-beams on propagation through free space,” J. Opt. Soc. Am. A 11, 1641-1643 (1994).
[CrossRef]

M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065-2069 (2007).
[CrossRef]

A. Luis, “Degree of coherence for vectorial electromagnetic fields as a distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063-1068 (2007).
[CrossRef]

D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
[CrossRef]

Sudha, A. V. G. Rao, A. R. U. Devi, and A. K. Rajagopal, “Positive-operator-valued measure view of the ensemble approach to polarization optics,” J. Opt. Soc. Am. A 25, 874-880 (2008).
[CrossRef]

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
[CrossRef]

V. Devlaminck and P. Terrier, “Definition of a parameter form for non-singular Mueller matrices,” J. Opt. Soc. Am. A 25, 2636-2643 (2008).
[CrossRef]

S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996).
[CrossRef]

N. Mukunda, R. Simon, and E. C. G. Sudarshan, “Fourier optics for the Maxwell field: formalism and applications,” J. Opt. Soc. Am. A 2, 416-426 (1985).
[CrossRef]

Opt. Acta (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259-261 (1985).
[CrossRef]

Opt. Commun. (4)

M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

F. Boulvert, G. Le Brun, B. Le Jeune, J. Cariou, and L. Martin, “Decomposition algorithm for an experimental Mueller matrix,” Opt. Commun. 282, 692-704 (2009).
[CrossRef]

Opt. Express (2)

P. Refrégiér and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 1051-1060 (2005).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence of electromagnetic fields,” Opt. Express 11, 1137-1142 (2003).
[CrossRef] [PubMed]

Opt. Lett. (9)

Optik (Stuttgart) (2)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).

S. R. Cloude, “Group-theory and polarization algebra,” Optik (Stuttgart) 75, 26-36 (1986).

Phys. Lett. A (1)

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

Phys. Rev. (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” (Ed. note: commentary on ) Phys. Rev. 48, 696-702 (1935).
[CrossRef]

Phys. Rev. A (1)

A. Aiello, G. Puentes, and J. P. Woerdman, “Linear optics and quantum maps,” Phys. Rev. A 76, 032323 (2007).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399-416 (1852).

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 6.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (1)

Fig. 1
Fig. 1

The tetrahedron that corresponds to Mueller matrices.

Equations (78)

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E [ E 1 E 2 ] C 2 .
Φ E E = [ E 1 E 1 * E 1 E 2 * E 2 E 1 * E 2 E 2 * ] ,
tr Φ > 0 ,
det Φ 0 .
J = [ J 11 J 12 J 21 J 22 ] : E E = [ E 1 E 2 ] = J E Φ E E Φ = E E = J Φ J .
Φ = 1 2 a = 0 3 S a τ a S a = tr ( τ a Φ ) ; tr τ a τ b = 2 δ a b .
S 0 > 0 ,
S 0 2 S 1 2 S 2 2 S 3 2 0 .
J : S S = M ( J ) S .
Φ ̃ = [ Φ ̃ 0 Φ ̃ 1 Φ ̃ 2 Φ ̃ 3 ] [ E 1 E 1 * E 1 E 2 * E 2 E 1 * E 2 E 2 * ] = [ Φ 11 Φ 12 Φ 21 Φ 22 ] .
J J * [ J 11 J * J 12 J * J 21 J * J 22 J * ] = [ J 11 J 11 * J 11 J 12 * J 12 J 11 * J 12 J 12 * J 11 J 21 * J 11 J 22 * J 12 J 21 * J 12 J 22 * J 21 J 11 * J 21 J 12 * J 22 J 11 * J 22 J 12 * J 21 J 21 * J 21 J 22 * J 22 J 21 * J 22 J 22 * ] .
[ S 0 S 1 S 2 S 3 ] = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] [ Φ ̃ 0 Φ ̃ 1 Φ ̃ 2 Φ ̃ 3 ] ,
M ( J ) = A ( J J * ) A 1 ,
B ( M ) : Φ Φ , Φ ̃ = B ( M ) Φ ̃ ,
i.e. , Φ i j = k l = 1 2 B i j , k l ( M ) Φ k l ;
B ( M ) A 1 M A , M = A B ( M ) A 1 .
H i k , j l ( M ) = B i j , k l ( M ) .
H ( M ) : Φ Φ , Φ i j = k l H i k , j l ( M ) Φ k l .
U a b = 1 2 τ a τ b * = U a b = U a b 1 ,
tr ( U a b U c d ) = δ a c δ b d , a , b , c , d { 1 , 2 , 3 , 4 } .
H ( M ) = 1 2 a , b = 0 3 K a b τ a τ b * .
Φ i j = a b K a b k l ( τ a ) i j ( τ b ) k l * Φ k l .
H ( M ) = 1 2 a , b = 0 3 M a b τ a τ b * .
( M ( H ) ) a b = 1 2 tr ( H τ a τ b * ) , a , b = 0 , 1 , 2 , 3 .
H ( M ) = 1 2 [ M 00 + M 11 + M 01 + M 10 M 02 + M 12 + i ( M 03 + M 13 ) M 20 + M 21 i ( M 30 + M 31 ) M 22 + M 33 + i ( M 23 M 32 ) M 02 + M 12 i ( M 03 + M 13 ) M 00 M 11 M 01 + M 10 M 22 M 33 i ( M 23 + M 32 ) M 20 M 21 i ( M 30 M 31 ) M 20 + M 21 + i ( M 30 + M 31 ) M 22 M 33 + i ( M 23 + M 32 ) M 00 M 11 + M 01 M 10 M 02 M 12 + i ( M 03 M 13 ) M 22 + M 33 i ( M 23 M 32 ) M 20 M 21 + i ( M 30 M 31 ) M 02 M 12 i ( M 03 M 13 ) M 00 + M 11 M 01 M 10 ] .
M ( H ) = 1 2 [ H 00 + H 11 + H 22 + H 33 H 00 H 11 + H 22 H 33 H 01 + H 10 + H 23 + H 32 i ( H 01 H 10 ) i ( H 23 H 32 ) H 00 + H 11 H 22 H 33 H 00 H 11 H 22 + H 33 H 01 + H 10 H 23 H 32 i ( H 01 H 10 ) + i ( H 23 H 32 ) H 02 + H 20 + H 13 + H 31 H 02 + H 20 H 13 H 31 H 03 + H 30 + H 12 + H 21 i ( H 03 H 30 ) + i ( H 12 H 21 ) i ( H 02 H 20 ) + i ( H 13 H 31 ) i ( H 02 H 20 ) i ( H 13 H 31 ) i ( H 03 H 30 ) + i ( H 12 H 21 ) H 03 + H 30 H 12 H 21 ] .
Φ Φ = k = 1 n p k J ( k ) Φ J ( k ) ,
p k > 0 , k = 1 n p k = 1 .
M = k = 1 n p k M ( J k ) , H ( M ) = k = 1 n p k J ̃ ( k ) J ̃ ( k ) .
Ω ( pol ) = { S R 4 | S 0 > 0 , S T G S 0 } ,
G = diag ( 1 , 1 , 1 , 1 ) ,
S T G S = S 0 2 S 1 2 S 2 2 S 3 2 .
Type I : M = L l M ( 1 ) L r , L l , L r S O ( 3 , 1 ) ,
M ( 1 ) = diag ( d 0 , d 1 , d 2 , d 3 ) , d 0 d 1 d 2 | d 3 | ;
Type II : M = L l M ( 2 ) L r , L l , L r S O ( 3 , 1 ) ,
M ( 2 ) = [ d 0 d 0 d 1 0 0 0 d 1 0 0 0 0 d 2 0 0 0 0 d 3 ] , d 0 > d 1 > 0 , d 0 d 1 d 2 | d 3 | ;
Polarizer : M = L l M ( pol ) L r , L l , L r S O ( 3 , 1 ) ,
M ( pol ) = [ d 0 d 0 0 0 d 0 d 0 0 0 0 0 0 0 0 0 0 0 ] , d 0 > 0 ;
Pin Map : M = L l M ( pin ) L r , L l , L r S O ( 3 , 1 ) ,
M ( pin ) = [ d 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 ] , d 0 > 0 .
M ( depol ) = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ 1 0 0 0 ] , [ 0 1 0 0 ] .
E ( ρ ) = E 1 ( ρ ) x ̂ + E 2 ( ρ ) y ̂ E ( ρ ) = [ E 1 ( ρ ) E 2 ( ρ ) ] ,
E 1 ( ρ ) , E 2 ( ρ ) L 2 ( R 2 ) .
Φ ( ρ ; ρ ) = [ E 1 ( ρ ) E 1 ( ρ ) * E 1 ( ρ ) E 2 ( ρ ) * E 2 ( ρ ) E 1 ( ρ ) * E 2 ( ρ ) E 2 ( ρ ) * ] ,
Φ j k ( ρ ; ρ ) = Φ k j ( ρ ; ρ ) * , j , k = 1 , 2 ;
d 2 ρ d 2 ρ E ( ρ ) Φ ( ρ ; ρ ) E ( ρ ) 0 ,
i.e. , j k d 2 ρ d 2 ρ E j ( ρ ) * Φ j k ( ρ ; ρ ) E k ( ρ ) 0 ,
E ( ρ ) C 2 L 2 ( R 2 ) .
Φ ( ρ ; ρ ) = 1 2 a = 0 3 S a ( ρ ; ρ ) τ a S a ( ρ ; ρ ) = tr ( Φ ( ρ ; ρ ) τ a ) .
S a ( ρ ; ρ ) = S a ( ρ ; ρ ) * , a = 0 , 1 , 2 , 3 ,
a = 0 3 g a d 2 ρ d 2 ρ S a ( ρ ; ρ ) S ̂ a ( ρ ; ρ ) 0 ,
M ( 1 ) = diag ( d 0 , d 1 , d 2 , d 3 ) : [ S 0 ( ρ ; ρ ) S 1 ( ρ ; ρ ) S 2 ( ρ ; ρ ) S 3 ( ρ ; ρ ) ] [ S 0 ( ρ ; ρ ) S 1 ( ρ ; ρ ) S 2 ( ρ ; ρ ) S 3 ( ρ ; ρ ) ] = [ d 0 S 0 ( ρ ; ρ ) d 1 S 1 ( ρ ; ρ ) d 2 S 2 ( ρ ; ρ ) d 3 S 3 ( ρ ; ρ ) ] .
Φ 11 ( ρ ; ρ ) = [ ( d 0 + d 1 ) Φ 11 ( ρ ; ρ ) + ( d 0 d 1 ) Φ 22 ( ρ ; ρ ) ] 2 ,
Φ 22 ( ρ ; ρ ) = [ ( d 0 + d 1 ) Φ 22 ( ρ ; ρ ) + ( d 0 d 1 ) Φ 11 ( ρ ; ρ ) ] 2 ,
Φ 12 ( ρ ; ρ ) = [ ( d 2 + d 3 ) Φ 12 ( ρ ; ρ ) + ( d 2 d 3 ) Φ 21 ( ρ ; ρ ) ] 2 ,
Φ 21 ( ρ ; ρ ) = [ ( d 2 + d 3 ) Φ 21 ( ρ ; ρ ) + ( d 2 d 3 ) Φ 12 ( ρ ; ρ ) ] 2 .
Φ ( 0 ) ( ρ ; ρ ) = E ( ρ ) E ( ρ ) ,
E ( ρ ) = [ ψ 1 ( ρ ) ψ 2 ( ρ ) ] , ψ j ( ρ ) ψ k ( ρ ) * d 2 ρ = δ j k .
d 2 ρ d 2 ρ Φ i j ( 0 ) ( ρ ; ρ ) Φ k l ( 0 ) ( ρ ; ρ ) * = δ i k δ j l .
E ( ± ) ( ρ ) = [ ψ 1 ( ρ ) ± ψ 2 ( ρ ) ] , F ( ± ) ( ρ ) = [ ψ 2 ( ρ ) ± ψ 1 ( ρ ) ] .
d 1 d 2 d 3 d 0 ,
d 1 + d 2 + d 3 d 0 ;
d 1 + d 2 d 3 d 0 ,
d 1 d 2 + d 3 d 0 .
H ( M ) = 1 2 [ d 0 + d 1 0 0 d 2 + d 3 0 d 0 d 1 d 2 d 3 0 0 d 2 d 3 d 0 d 1 0 d 2 + d 3 0 0 d 0 + d 1 ] .
M ( 2 ) : S ( ρ ; ρ ) S ( ρ ; ρ ) = [ S 0 ( ρ ; ρ ) S 1 ( ρ ; ρ ) S 2 ( ρ ; ρ ) S 3 ( ρ ; ρ ) ] = [ d 0 S 0 ( ρ ; ρ ) + ( d 0 d 1 ) S 1 ( ρ ; ρ ) d 1 S 1 ( ρ ; ρ ) d 2 S 2 ( ρ ; ρ ) d 3 S 3 ( ρ ; ρ ) ] .
Φ 11 ( ρ ; ρ ) = d 0 Φ 11 ( ρ ; ρ ) ,
Φ 22 ( ρ ; ρ ) = d 1 Φ 22 ( ρ ; ρ ) + ( d 0 d 1 ) Φ 11 ( ρ ; ρ ) ,
Φ 12 ( ρ ; ρ ) = [ ( d 2 + d 3 ) Φ 12 ( ρ ; ρ ) + ( d 2 d 3 ) Φ 21 ( ρ ; ρ ) ] 2 ,
Φ 21 ( ρ ; ρ ) = [ ( d 2 + d 3 ) Φ 21 ( ρ ; ρ ) + ( d 2 d 3 ) Φ 12 ( ρ ; ρ ) ] 2 .
E ( θ ) ( ρ ) = [ cos θ ψ 1 ( ρ ) sin θ ψ 2 ( ρ ) ] , F ( θ ) ( ρ ) = [ cos θ ψ 2 ( ρ ) sin θ ψ 1 ( ρ ) ] .
d 3 = d 2 , ( d 2 ) 2 d 0 d 1 .
H M ( 2 ) = [ d 0 0 0 1 2 ( d 2 + d 3 ) 0 0 1 2 ( d 2 d 3 ) 0 0 1 2 ( d 2 d 3 ) d 0 d 1 0 1 2 ( d 2 + d 3 ) 0 0 d 1 ] .
M = [ 1 0 0 0 0 d 1 0 0 0 0 d 3 0 0 0 0 d 3 ] .
M : Ω ( pol ) Ω ( pol ) 1 d k 1 , k = 1 , 2 , 3 .
E 1 = a E 1 + b E 2 ,
E 2 = c E 1 + d E 2 ,

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