Abstract

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989).
  2. E. L. O’Neill, Introduction to Statistical Optics. (Addison-Wesley, Reading, Mass., 1963).
  3. D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, Boston, 1990).
  4. E. Collett, Polarized Light. Fundamentals and Applications (Dekker, New York, 1993).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957 [reprint, Dover, New York, 1981]).
  6. W. A. Shurcliff, Polarized Light. Production and Use (Harvard U. Press, Cambridge, Mass., 1962).
  7. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).
  8. R. Clark Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  9. R. Clark Jones, “A new calculus for the treatment of optical systems. V. A more general formulation, and description of another calculus,” J. Opt. Soc. Am. 37, 107–112 (1947).
    [CrossRef]
  10. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  11. A. B. Kostinski, B. D. James, W.-M. Boerner, “Optimal reception of partially polarized waves,” J. Opt. Soc. Am. A 5, 58–64 (1988).
    [CrossRef]
  12. R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
    [CrossRef]
  13. Z.-F. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
    [CrossRef]
  14. K. D. Abhyankar, A. L. Fymat, “Relationships between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935–1938 (1969).
    [CrossRef]
  15. E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef] [PubMed]
  16. 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).
  17. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  18. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
    [CrossRef]
  19. J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  20. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
  21. H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.
  22. J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. 26, 488–499 (1969).
    [CrossRef]
  23. J. W. Hovenier, “Principles of symmetry for polarization studies of planets,” Astron. Astrophys. 7, 86–90 (1970).

1992 (1)

Z.-F. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

1990 (1)

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

1988 (1)

1987 (1)

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

1986 (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).

1985 (1)

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

1983 (1)

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

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 elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef] [PubMed]

1970 (1)

J. W. Hovenier, “Principles of symmetry for polarization studies of planets,” Astron. Astrophys. 7, 86–90 (1970).

1969 (2)

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

J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. 26, 488–499 (1969).
[CrossRef]

1947 (1)

1941 (1)

Abhyankar, K. D.

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

Barakat, R.

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

Boerner, W.-M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989).

Clark Jones, R.

Collett, E.

E. Collett, Polarized Light. Fundamentals and Applications (Dekker, New York, 1993).

Fry, E. S.

Fymat, A. L.

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

Hovenier, J. W.

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. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

J. W. Hovenier, “Principles of symmetry for polarization studies of planets,” Astron. Astrophys. 7, 86–90 (1970).

J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. 26, 488–499 (1969).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

James, B. D.

Kattawar, G. W.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, Boston, 1990).

Kostinski, A. B.

Lewis, J. W.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, Boston, 1990).

O’Neill, E. L.

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

Randall, C. E.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, Boston, 1990).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light. Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

Simon, R.

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]

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).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957 [reprint, Dover, New York, 1981]).

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

van der Mee, C. V. M.

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. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989).

Xing, Z.-F.

Z.-F. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (3)

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. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

J. W. Hovenier, “Principles of symmetry for polarization studies of planets,” Astron. Astrophys. 7, 86–90 (1970).

J. Atmos. Sci. (1)

J. W. Hovenier, “Symmetry relations for scattering of polarized light in a slab of randomly oriented particles,” J. Atmos. Sci. 26, 488–499 (1969).
[CrossRef]

J. Math. Phys. (1)

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

J. Mod. Opt. (2)

Z.-F. Xing, “On the deterministic and nondeterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

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

Opt. Commun. (3)

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

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

Other (8)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989).

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

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, Boston, 1990).

E. Collett, Polarized Light. Fundamentals and Applications (Dekker, New York, 1993).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957 [reprint, Dover, New York, 1981]).

W. A. Shurcliff, Polarized Light. Production and Use (Harvard U. Press, Cambridge, Mass., 1962).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

H. C. van de Hulst, Multiple Light Scattering, Tables, Formulas, and Applications (Academic, New York, 1980), Vols. 1 and 2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

The 16 dots in each pictogram represent the elements of a pure Mueller matrix. A solid line or curve connecting two elements represents a positive product, and a dashed curve or line represents a negative product. In each pictogram, the sum of all positive and negative products vanishes. (a) Twelve pictograms that represent equations that carry corresponding products of any two chosen rows and columns. (b) Eighteen pictograms that demonstrate that the sum or difference of any chosen pair of complementary subdeterminants vanishes.

Equations (65)

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

[ E r , 2 E l , 2 ] = [ J 1 J 4 J 3 J 2 ] [ E r , 1 E l , 1 ]
I = [ I Q U V ] = [ E l E l * + E r E r * E l E l * - E r E r * E l E r * + E r E l * i ( E l E r * - E r E l * ) ] ,
I 2 = M I 1 ,
S k j = ½ ( J k J j * + J j J k * ) ,
D k j = ½ i ( J k J j * - J j J k * ) ,
M 11 = ½ ( S 11 + S 22 + S 33 + S 44 ) ,
M 12 = ½ ( - S 11 + S 22 - S 33 + S 44 ) ,
M 13 = S 32 + S 41 ,
M 14 = D 32 - D 41 ,
M 21 = ½ ( - S 11 + S 22 + S 33 - S 44 ) ,
M 22 = ½ ( S 11 + S 22 - S 33 - S 44 ) ,
M 23 = S 32 - S 41 ,
M 24 = D 32 + D 41 ,
M 31 = S 42 + S 31 ,
M 32 = S 42 - S 31 ,
M 33 = S 21 + S 43 ,
M 34 = - D 21 - D 43 ,
M 41 = - D 42 + D 31 ,
M 42 = - D 42 - D 31 ,
M 43 = D 21 - D 43 ,
M 44 = S 21 - S 43 .
M = T [ J 1 J 1 * J 1 J 4 * J 4 J 1 * J 4 J 4 * J 1 J 3 * J 1 J 2 * J 4 J 3 * J 4 J 2 * J 3 J 1 * J 3 J 4 * J 2 J 1 * J 2 J 4 * J 3 J 3 * J 3 J 2 * J 2 J 3 * J 2 J 2 * ] T - 1 ,
T = [ 1 0 0 1 1 0 0 - 1 0 1 1 0 0 i - i 0 ]
T - 1 = 1 2 [ 1 1 0 0 0 0 1 - i 0 0 1 i 1 - 1 0 0 ] .
J J * .
d = det J ,
M 11 2 - M 21 2 - M 31 2 - M 41 2 = d 2 .
Tr M = Tr J 2 ,
det M = d 4 ,
J = [ J 1 J 4 J 3 J 2 ] ~ M .
α J ~ α 2 M ,
[ J 1 J 3 J 4 J 2 ] ~ Δ 4 M ˜ Δ 4 ,
[ J 1 * J 3 * J 4 * J 2 * ] ~ M ˜ .
[ J 1 - J 4 - J 3 J 2 ] ~ Δ 3 , 4 M Δ 3 , 4 ,
[ J 2 J 4 J 3 J 1 ] ~ Δ 2 M ˜ Δ 2 ,
[ J 1 - J 3 - J 4 J 2 ] ~ Δ 3 M ˜ Δ 3 ,
[ J 1 * J 4 * J 3 * J 2 * ] ~ Δ 4 M Δ 4 ,
J - 1 = 1 det J [ J 2 - J 4 - J 3 J 1 ] ~ 1 d 2 G M ˜ G ,
M - 1 = 1 d 2 G M ˜ G .
M 11 2 - M 21 2 - M 31 2 - M 41 2 = - M 12 2 + M 22 2 + M 32 2 + M 42 2 = - M 13 2 + M 23 2 + M 33 2 + M 43 2 = - M 14 2 + M 24 2 + M 34 2 + M 44 2 = M 11 2 - M 12 2 - M 13 2 - M 14 2 = - M 21 2 + M 22 2 + M 23 2 + M 24 2 = - M 31 2 + M 32 2 + M 33 2 + M 34 2 = - M 41 2 + M 42 2 + M 43 2 + M 44 2 .
M s = [ M 11 2 - M 12 2 - M 13 2 - M 14 2 - M 21 2 M 22 2 M 23 2 M 24 2 - M 31 2 M 32 2 M 33 2 M 34 2 - M 41 2 M 42 2 M 43 2 M 44 2 ]
M 11 M 12 - M 21 M 22 - M 31 M 32 - M 41 M 42 = 0 ,
M 11 M 22 - M 12 M 21 - M 33 M 44 + M 34 M 43 = 0.
M 11 0 ,
M 11 + M 22 - M 12 - M 21 0.
( M 11 + M 22 ) 2 - ( M 12 + M 21 ) 2 = ( M 33 + M 44 ) 2 + ( M 34 - M 43 ) 2 ,
( M 11 - M 12 ) 2 - ( M 21 - M 22 ) 2 = ( M 31 - M 32 ) 2 + ( M 41 - M 42 ) 2 ,
( M 11 - M 21 ) 2 - ( M 12 - M 22 ) 2 = ( M 13 - M 23 ) 2 + ( M 14 - M 24 ) 2 ,
( M 11 + M 22 - M 12 - M 21 ) ( M 13 + M 23 ) = ( M 31 - M 32 ) ( M 33 + M 44 ) - ( M 41 - M 42 ) ( M 34 - M 43 ) ,
( M 11 + M 22 - M 12 - M 21 ) ( M 34 + M 43 ) = ( M 31 - M 32 ) ( M 14 - M 24 ) + ( M 41 - M 42 ) ( M 13 - M 23 ) ,
( M 11 + M 22 - M 12 - M 21 ) ( M 33 - M 44 ) = ( M 31 - M 32 ) ( M 13 - M 23 ) - ( M 41 - M 42 ) ( M 14 - M 24 ) ,
( M 11 + M 22 - M 12 - M 21 ) ( M 14 + M 24 ) = ( M 31 - M 32 ) ( M 34 - M 43 ) + ( M 41 - M 42 ) ( M 33 + M 44 ) ,
( M 11 + M 22 - M 12 - M 21 ) ( M 31 + M 32 ) = ( M 33 + M 44 ) ( M 13 - M 23 ) + ( M 34 - M 43 ) ( M 14 - M 24 ) ,
( M 11 + M 22 - M 12 - M 21 ) ( M 41 + M 42 ) = ( M 33 + M 44 ) ( M 14 - M 24 ) - ( M 34 - M 43 ) ( M 13 - M 23 ) .
M 11 + M 22 + M 12 + M 21 0 ,
M 11 - M 22 - M 12 + M 21 0 ,
M 11 - M 22 + M 12 - M 21 0.
i = 1 4 j = 1 4 M i j 2 = 4 M 11 2
M ˜ GM = - ½ [ Tr ( M ˜ GM ) ] G ,
G = diag ( 1 , - 1 , - 1 , - 1 ) .
- ½ Tr ( M ˜ GM ) = d 2 .
MG M ˜ = - ½ [ Tr ( MG M ˜ ) ] G ,
- ½ Tr ( MG M ˜ ) = d 2 ,
[ 2 - 1 0 0 - 2 1 0 0 0 0 0 0 0 0 0 0 ]
[ 1 0 0 0 0 cos δ q sin δ 0 0 sin δ cos δ 0 0 0 0 1 ] ,

Metrics