Abstract

A complete and minimum set of necessary and sufficient conditions for a real 4×4 matrix to be a physical Mueller matrix is obtained. An additional condition is presented to complete the set of known conditions, namely, the four conditions obtained from the nonnegativity of the eigenvalues of the Hermitian matrix H associated with a Mueller matrix M and the transmittance condition. Using the properties of H, a demonstration is also presented of Tr(MTM)=4m002 as being a necessary and sufficient condition for a physical Mueller matrix to be a pure Mueller matrix.

© 2000 Optical Society of America

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References

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  1. S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).
  2. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  3. C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
    [CrossRef]
  4. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
    [CrossRef]
  5. A. V. Gopala, K. S. Mallesh, “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).
  6. A. V. Gopala, K. S. Mallesh, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
  7. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  8. J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  9. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
    [CrossRef]
  10. C. Brosseau, C. R. Givens, A. B. Kostinski, “Generalized trace condition on the Mueller–Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248–2251 (1993).
    [CrossRef]
  11. R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
    [CrossRef]
  12. B. Chakraborty, “Depolarizing effect of propagation of a polarized polychromatic beam through an optically active medium: a generalized study,” J. Opt. Soc. Am. A 3, 1422–1427 (1986).
    [CrossRef]
  13. N. G. Parke, “Matrix optics,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1948).
  14. E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  16. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  17. R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
    [CrossRef]
  18. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
    [CrossRef]
  19. U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1953).
    [CrossRef]
  20. R. C. Jones, “A new calculus for the treatment of optical systems. IV,” J. Opt. Soc. Am. 32, 486–493 (1942).
    [CrossRef]
  21. C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
    [CrossRef]
  22. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  23. A. B. Kostinski, R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952 (1992).
    [CrossRef]
  24. J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from its Mueller matrix,” Optik 76, 67–71 (1987).
  25. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  26. D. G. M. Anderson, 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]
  27. S. R. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
    [CrossRef]
  28. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 404.
  29. Z. Sekera, “Scattering matrices and reciprocity relationships for various representations of the state of polarization,” J. Opt. Soc. Am. 56, 1732–1740 (1966).
    [CrossRef]
  30. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4.
  31. K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
    [CrossRef]
  32. E. S. Fry, G. W. Kattawar, “Relationships between the elements of the Stokes matrix,” Appl. Opt. 20, 3428–3435 (1981).
    [CrossRef]
  33. J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).
  34. J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
    [CrossRef] [PubMed]
  35. H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
    [CrossRef]
  36. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  37. A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
    [CrossRef] [PubMed]
  38. C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
    [CrossRef]
  39. M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 11, 2305–2319 (1994).
  40. R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
    [CrossRef]
  41. J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [CrossRef]

1998 (2)

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

1996 (1)

C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

1995 (1)

S. R. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

1994 (5)

1993 (4)

C. Brosseau, C. R. Givens, A. B. Kostinski, “Generalized trace condition on the Mueller–Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248–2251 (1993).
[CrossRef]

A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
[CrossRef] [PubMed]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

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

1992 (1)

A. B. Kostinski, R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952 (1992).
[CrossRef]

1991 (1)

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

1990 (1)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

1987 (4)

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

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

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

1986 (4)

B. Chakraborty, “Depolarizing effect of propagation of a polarized polychromatic beam through an optically active medium: a generalized study,” J. Opt. Soc. Am. A 3, 1422–1427 (1986).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

1985 (2)

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

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

1982 (1)

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

1981 (2)

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

E. S. Fry, G. W. Kattawar, “Relationships between the elements of the Stokes matrix,” Appl. Opt. 20, 3428–3435 (1981).
[CrossRef]

1973 (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[CrossRef]

1969 (1)

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

1966 (1)

1963 (1)

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1953 (1)

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1953).
[CrossRef]

1942 (1)

Abhyankar, K. D.

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

Anderson, D. G. M.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barakat, R.

D. G. M. Anderson, 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]

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[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. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53, 317–323 (1963).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bernabeu, E.

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

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

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

Brosseau, C.

C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

C. Brosseau, C. R. Givens, A. B. Kostinski, “Generalized trace condition on the Mueller–Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248–2251 (1993).
[CrossRef]

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[CrossRef]

Chakraborty, B.

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

Fano, U.

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1953).
[CrossRef]

Fry, E. S.

Fymat, A. L.

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

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4.

Gil, J. J.

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

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

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

Givens, C. R.

Givens, R. C.

A. B. Kostinski, R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952 (1992).
[CrossRef]

Gopala, A. V.

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

A. V. Gopala, K. S. Mallesh, “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).

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 404.

Hovenier, J. W.

J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318–8324 (1994).
[CrossRef] [PubMed]

J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Johnson, C. R.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 404.

Jones, R. C.

Kattawar, G. W.

Kim, K.

Kostinski, A. B.

A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
[CrossRef] [PubMed]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

C. Brosseau, C. R. Givens, A. B. Kostinski, “Generalized trace condition on the Mueller–Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248–2251 (1993).
[CrossRef]

A. B. Kostinski, R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952 (1992).
[CrossRef]

Kumar, M. S.

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 11, 2305–2319 (1994).

Kwiatkowski, J. M.

Lu, S.-Y.

Mallesh, K. S.

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

Mandel, L.

O’Neill, E. L.

E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Parke, N. G.

N. G. Parke, “Matrix optics,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1948).

Pottier, E.

S. R. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Sekera, Z.

Simon, R.

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

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 11, 2305–2319 (1994).

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

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

Sridhar, R.

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

Takenaka, H.

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4.

van de Hulst, H. C.

J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

van der Mee, C. V. M.

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

J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

Wolf, E.

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

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (1)

J. W. Hovenier, H. C. van de Hulst, C. V. M. Van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

J. Math Phys. (1)

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

J. Math. Phys. (1)

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

J. Mod. Opt. (7)

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

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

A. B. Kostinski, R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952 (1992).
[CrossRef]

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Nouv. Rev. Opt. (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37–41 (1973).
[CrossRef]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

Opt. Acta (3)

R. Barakat, “The statistical properties of partially polarized light,” Opt. Acta 32, 295–312 (1985).
[CrossRef]

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

J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
[CrossRef]

Opt. Commun. (6)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

C. Brosseau, R. Barakat, “Jones and Mueller polarization matrices for random media,” Opt. Commun. 84, 127–132 (1991).
[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]

C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 11, 2305–2319 (1994).

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

Opt. Eng. (1)

S. R. Cloude, E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Optik (2)

S. R. Cloude, “Group theory and polarisation algebra,” Optik 75, 26–36 (1986).

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

Phys. Rev. (1)

U. Fano, “A Stokes-parameter technique for the treatment of polarization in quantum mechanics,” Phys. Rev. 93, 121–123 (1953).
[CrossRef]

Other (5)

N. G. Parke, “Matrix optics,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1948).

E. L. O’Neill, Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985), p. 404.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4.

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Equations (73)

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1(t)=E1(t)+iE˜1(t),
2(t)=E2(t)+iE˜2(t),
(t)=T(t),
Φ=+=11*12*21*22*,
Φ=12k=03skσk=12s0+s1s2-is3s2+is3s0-s1,
σ0=1001,σ1=100-1,
σ2=0110,σ3=0-ii0.
Φ=+=T(T)+=T+T+=T+T+=TΦT+.
ϕ0ϕ11,ϕ1ϕ12,ϕ2ϕ21,ϕ3ϕ22.
s=Aϕ,
A=1001100-101100i-i0.
ϕ=(TT*)ϕ,
V(TT*).
N=AVA-1=A(TT*)A-1.
12{(|t11|2+|t12|2+|t21|2+|t22|2)+[(|t11|2-|t12|2+|t21|2-|t22|2)2+4|t11t12*+t21t22*|2]1/2}1.
gf=n00+(n012+n022+n032)1/21.
gf=n00+(n102+N202+n302)1/21,
(n012+n022+n032)=(n102+n202+n302).
pi=I(i)/I,i pi=1.
i(t)=T(i)[pi(t)],
ϕiii*=[T(i)pi][T(i)pi]*=[T(i)T(i)*]pi(*)=pi[T(i)T(i)*](*)=piV(i)ϕ.
ϕiϕi=i{pi[T(i)T(i)*](*)}=ipiV(i)ϕ=ipiV(i)ϕ,
LipiV(i)=ipi[T(i)T(i)*],
MALA-1=AipiV(i)A-1=Aipi[T(i)T(i)*]A-1=ipi{A(T(i)T(i)*)A-1}=ipiN(i).
xeipix(i),
mkl=nkle,k, l=0, 1, 2, 3.
L=T(i)T(i)*e=t0t0*et0t1*et1t0*et1t1*et0t2*et0t3*et1T2*et1t3*et2t0*et2t1*et3t0*et3t1*et2t2*et2t3*et3t2*et3t3*e,
hkl12tktl*e,k, l=0, 1, 2, 3.
H=14k,l=03mklEkl,
Ekl=σkσl(k, l=0, 1, 2, 3),
mkl=Tr(EklH).
hklμklσkσl,
σk2=hkk=tktk*e=|tk|2e.
ρkl|μkl|,βklarg(μkl).
h000;
1ρ01;
1+2ρ01ρ12ρ02 cos(β01+β12-β02)ρ012+ρ122+ρ022;
det(H)=1+2ρ12ρ23ρ13 cos(β12+β23-β13)+2ρ02ρ23ρ03 cos(β02+β23-β03)+2ρ01ρ13ρ03 cos(β01+β13-β03)+2ρ01ρ12ρ02 cos(β01+β12-β02)-2ρ01ρ02ρ13ρ23 cos(β01-β02+β13-β23)-2ρ01ρ03ρ12ρ23 cos(β01-β03+β12+β23)-2ρ02ρ03ρ12ρ13 cos(β02-β03-β12+β13)+ρ012ρ232+ρ022ρ132+ρ032ρ122-ρ012-ρ022-ρ032-ρ122-ρ132-ρ2320.
M=Ne=A[T(i)T(i)*]A-1e=AT(i)T(i)*eA-1,
gfm00+(m012+m022+m032)1/2.
mf(m01, m02, m03)T,nf(i)(n01(i), n02(i), n03(i))T,
gf=m00+|mf|=ipin00(i)+l=13i[pin0l(i)]21/2
l=13i[pin0l(i)]21/2=ipi2[n01(i)]2+ipi2[n02(i)]2+ipi2[n03(i)]2+jkpjpk[n01(j)n01(k)+n02(j)n02(k)+n03(j)n03(k)]1/2=ipi2|nf(i)|2+jkpj[nf(j)]Tpk[nf(k)]1/2=ipinf(i)ipi|nf(i)|.
gfipin00(i)+ipi|nf(i)|=ipi[n00(i)+|nf(i)|].
n00(i)+|nf(i)|1,
gf1.
grm00+(m102+m202+m302)1/2,
mr(m10, m20, m30)T,nr(i)(n10(i), n20(i), n30(i))T.
gr=ipi00(i)+l=13ipi[nl0(i)]21/2ipi[n00(i)+|nr(i)|],
ipi[n00(i)+|nr(i)|]=ipi[n00(i)+|nf(i)|]1,
gr1.
gf=m00+(m012+m022+m032)1/21
grm00+(m102+m202+m302)1/21,
gf=Tr (H)+[(h00-h11-h22+h33)2+4(h01+h23)(h01*+h23*)]1/21,
gr=Tr (H)+[(h00+h11-h22-h33)2+4(h02+h13)(h02*+h13*)]1/21.
MLILP1000100000000000.
H=WΛW+=λ0WD(1, 0, 0, 0)W++λ1WD(0, 1, 0, 0)W++λ2WD(0, 0, 1, 0)W++λ3WD(0, 0, 0, 1)W,
B=k=03λkB(k)=k=03λkTr(H)[Tr(H)B(k)]=k=03pkN(k),
pkλkTr(H),k=03pk=1,
N(k)[Tr(H)B(k)](k=0, 1, 2, 3),
b00+|bf|1,
b00=Tr(H),b00(k)=1,
n00(k)=Tr(H)(k=0, 1, 2, 3).
bf=k=03pknf(k)
|nf(k)|1-b00(k=0, 1, 2, 3),
|nf(k)|1-n00(k)(k=0, 1, 2, 3),
Tr(MTM)4m002
Tr(MTM)=4m002
Tr(MTM)=4 Tr(H2)=4i=03λi2,
m002=(Tr H)2=i=03λi2.
i=03λi2i=03λi2,
14Tr(MTM)m002.
P(M)Tr(MTM)-m0023m0021/2=134 Tr(H2)(Tr H)2-11/2.

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