Abstract

The issue of physical realizability constraints on depolarizing scattering or imaging systems is addressed. In particular, the overpolarization problem, i.e., the problem of ensuring that the output degree of polarization is always smaller than (or equal to) unity, is discussed in detail. A set of necessary conditions for the elements of a Mueller matrix is derived. These conditions can be used to test the accuracy of polarimetric measurements and computations. Several recent experimental examples from polarization optics and radar are discussed.

© 1993 Optical Society of America

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References

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  1. C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudopolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
    [Crossref]
  3. J. Cariou, B. Le Jeune, J. Lotran, Y. Guern, “Polarization effects of sea water and underwater targets,” Appl. Opt. 29, 1689–1695 (1990).
    [Crossref] [PubMed]
  4. R. F. Coleman, Kuo-Nan Liou, “Light scattering by hexagonal ice crystals,” J. Atmos. Sci. 38, 1260–1271 (1981).
    [Crossref]
  5. W. P. Van de Merwe, D. R. Huffman, B. V. Bronk, “Reproducability and sensitivity of polarized light scattering for identifying bacterial suspensions,” Appl. Opt. 28, 5052–5057 (1989).
    [Crossref] [PubMed]
  6. P. M. A. Sloot, A. G. Hoekstra, H. van der Liet, C. G. Figdor, “Scattering matrix elements of biological particles measured in a flow through system: theory and practice,” Appl. Opt. 28, 1752–1762 (1989).
    [Crossref] [PubMed]
  7. 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).
  8. K. Tragl, “Polarimetric radar backscattering from reciprocal random targets,” IEEE Trans. Geosci. Remote Sensing 28, 856 (1990).
    [Crossref]
  9. F. T. Ulaby, C. Elachi, eds., Radar Polarimetry for Geoscience Applications (Artech, Norwood, Mass., 1990).
  10. A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
    [Crossref]
  11. A. B. Kostinski, W. -M. Boerner, “Polarimetric contrast optimization,” IEEE Trans. Antennas Propag. AP-35, 988–990 (1987).
    [Crossref]
  12. B. J. Howell, “Measurements of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 809 (1979).
    [Crossref] [PubMed]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  14. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  15. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159 (1981).
    [Crossref]
  16. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [Crossref]
  17. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  18. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [Crossref]
  19. 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]
  20. E. Fry, G. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [Crossref] [PubMed]
  21. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [Crossref]
  22. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
    [Crossref]
  23. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
    [Crossref] [PubMed]
  24. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [Crossref]
  25. K. J. Voss, F. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 12, 1323–1332 (1984).
  26. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  27. S. Chandrasekar, Radiative Transfer (Dover, New York, 1960).
  28. J. van Zyl, C. Papas, C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987).
    [Crossref]

1992 (1)

1990 (3)

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudopolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[Crossref]

J. Cariou, B. Le Jeune, J. Lotran, Y. Guern, “Polarization effects of sea water and underwater targets,” Appl. Opt. 29, 1689–1695 (1990).
[Crossref] [PubMed]

K. Tragl, “Polarimetric radar backscattering from reciprocal random targets,” IEEE Trans. Geosci. Remote Sensing 28, 856 (1990).
[Crossref]

1989 (2)

1988 (1)

A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
[Crossref]

1987 (5)

A. B. Kostinski, W. -M. Boerner, “Polarimetric contrast optimization,” IEEE Trans. Antennas Propag. AP-35, 988–990 (1987).
[Crossref]

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]

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]

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

1986 (3)

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

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

1985 (1)

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

1984 (1)

K. J. Voss, F. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 12, 1323–1332 (1984).

1982 (1)

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

1981 (3)

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

E. Fry, G. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[Crossref] [PubMed]

R. F. Coleman, Kuo-Nan Liou, “Light scattering by hexagonal ice crystals,” J. Atmos. Sci. 38, 1260–1271 (1981).
[Crossref]

1979 (1)

Azzam, R. M. A.

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

Barakat, R.

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

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

Bashara, N. M.

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

Bernabeu, E.

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

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

Boerner, W. -M.

A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
[Crossref]

A. B. Kostinski, W. -M. Boerner, “Polarimetric contrast optimization,” IEEE Trans. Antennas Propag. AP-35, 988–990 (1987).
[Crossref]

Bohren, C.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Bronk, B. V.

Cariou, J.

Chandrasekar, S.

S. Chandrasekar, Radiative Transfer (Dover, New York, 1960).

Chipman, R. A.

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudopolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[Crossref]

Cloude, S. R.

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

Coleman, R. F.

R. F. Coleman, Kuo-Nan Liou, “Light scattering by hexagonal ice crystals,” J. Atmos. Sci. 38, 1260–1271 (1981).
[Crossref]

Elachi, C.

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

Figdor, C. G.

Fry, E.

Fry, F. S.

K. J. Voss, F. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 12, 1323–1332 (1984).

Gil, J.

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

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

Guern, Y.

Hoekstra, A. G.

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).

Howell, B. J.

Huffman, D.

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Huffman, D. R.

James, B.

A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
[Crossref]

Kattawar, G.

Kim, K.

Kostinski, A.

A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
[Crossref]

Kostinski, A. B.

A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
[Crossref] [PubMed]

A. B. Kostinski, W. -M. Boerner, “Polarimetric contrast optimization,” IEEE Trans. Antennas Propag. AP-35, 988–990 (1987).
[Crossref]

Le Jeune, B.

Liou, Kuo-Nan

R. F. Coleman, Kuo-Nan Liou, “Light scattering by hexagonal ice crystals,” J. Atmos. Sci. 38, 1260–1271 (1981).
[Crossref]

Lotran, J.

Mandel, L.

McGuire, J. P.

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudopolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[Crossref]

Papas, C.

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

Simon, R.

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]

Sloot, P. M. A.

Tragl, K.

K. Tragl, “Polarimetric radar backscattering from reciprocal random targets,” IEEE Trans. Geosci. Remote Sensing 28, 856 (1990).
[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 (Dover, New York, 1981).

Van de Merwe, W. P.

van der Liet, H.

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).

van Zyl, J.

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

Voss, K. J.

K. J. Voss, F. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 12, 1323–1332 (1984).

Wolf, E.

Appl. Opt. (7)

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).

IEEE Trans. Antennas Propag. (2)

A. B. Kostinski, W. -M. Boerner, “Polarimetric contrast optimization,” IEEE Trans. Antennas Propag. AP-35, 988–990 (1987).
[Crossref]

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

IEEE Trans. Geosci. Remote Sensing (1)

K. Tragl, “Polarimetric radar backscattering from reciprocal random targets,” IEEE Trans. Geosci. Remote Sensing 28, 856 (1990).
[Crossref]

J. Atmos. Sci. (1)

R. F. Coleman, Kuo-Nan Liou, “Light scattering by hexagonal ice crystals,” J. Atmos. Sci. 38, 1260–1271 (1981).
[Crossref]

J. Mod. Opt. (2)

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. Opt. Soc. Am. A (1)

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

A. Kostinski, B. James, W. -M. Boerner, “Optical reception of partially polarized waves,” J. Opt. Soc. Am. A. 5, 58–64 (1988).
[Crossref]

Opt. Acta (2)

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

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

Opt. Commun. (2)

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

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

Opt. Eng. (1)

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudopolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[Crossref]

Optik (1)

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

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

S. Chandrasekar, Radiative Transfer (Dover, New York, 1960).

C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

F. T. Ulaby, C. Elachi, eds., Radar Polarimetry for Geoscience Applications (Artech, Norwood, Mass., 1990).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

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Equations (36)

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Trace ( M T M ) = 4 m 00 2 .
ρ = ( s 1 2 + s 2 2 + s 3 2 ) 1 / 2 s 0 ,
t T = [ 1 u T ] , s T = [ a υ T ] ,
[ a u ] = [ 1 u ] + [ a 1 0 ] = [ 1 u ] + a 1 2 [ 1 w ] + a 1 2 [ 1 w ] ,
s = s 1 + s 2 + s 3 ,
M s = M s 1 + M s 2 + M s 3
l ( t 1 + t 2 ) = l ( t 1 ) + l ( t 2 ) + 2 t 1 T L t 2 .
M = [ m 00 m 1 T m 2 M 3 ] .
M t = [ m 00 + m 1 T u m 2 + M 3 u ]
| m 2 + M 3 u | m 00 + m 1 T u , for all u ' s such that | u | = 1 .
| m 2 | 2 + 2 u T M 3 T m 2 + u T M 3 T M 3 u m 00 2 + 2 m 00 m 1 T u + ( m 1 T u ) 2 ,
| m 2 | 2 2 u T M 3 T m 2 + u T M 3 T M 3 u m 00 2 2 m 00 m 1 T u + ( m 1 T u ) 2 .
u T M 3 T M 3 u + | m 2 | 2 m 00 2 + ( m 1 T u ) 2 , for all | u | = 1 .
k = 1 3 e k T ( M 3 T M 3 ) e k = Trace ( M 3 T M 3 ) .
Trace ( M 3 T M 3 ) + 3 | m 2 | 2 3 m 00 2 + | m 1 | 2 .
M = [ m 00 m 1 T m 2 M 3 ]
Trace ( M T M ) = m 00 2 + | m 1 | 2 + | m 2 | 2 + Trace ( M 3 T M 3 ) ,
Trace ( M T M ) 4 m 00 2 + 2 { | m 1 | 2 | m 2 | 2 } ,
M = [ m 00 m 1 T m 01 a 1 T m 02 a 2 T m 03 a 3 T ] ,
( s 0 , s 1 , s 2 , s 3 ) T = [ m 00 + m 1 T u m 01 + a 1 T u m 02 + a 2 T u m 03 + a 3 T u ] .
i = 1 3 ( m 0 i + a i T u ) 2 ( m 00 + m 1 T u ) 2 .
i = 1 3 ( m 0 i + a i T m ˆ 1 ) 2 ( m 00 | m 1 | ) 2 .
M = [ 0 . 7599 0 . 0623 0 . 0295 0 . 1185 0 . 0573 0 . 4687 0 . 1811 0 . 1863 0 . 0384 0 . 1714 0 . 5394 0 . 0282 0 . 1240 0 . 2168 0 . 0120 0 . 6608 ] .
Trace ( M T M ) = 1 . 709 , 4 m 00 2 + 2 [ | m 1 | 2 | m 2 | 2 ] = 4 m 00 2 + Δ = 2 . 307 .
k = 1 3 ( m 0 i a i T m ˆ 1 ) 2 = 0 . 4539 , ( m 00 | m 1 | ) 2 = 0 . 3879 .
M = [ 1 . 0000 0 . 0118 0 . 0279 0 . 0001 0 . 0045 0 . 9956 0 . 0013 0 . 0350 0 . 0012 0 . 0341 0 . 9838 0 . 0083 0 . 0092 0 . 0178 0 . 0002 0 . 9956 ] .
Trace ( M T M ) = 3 . 954 , 4 m 00 2 + Δ = 4 . 002 .
i = 1 3 ( m 0 i a i T m ˆ 1 ) 2 = 0 . 9481 , ( m 00 | m 1 | ) 2 = 0 . 9403 .
M = [ 1 . 000 0 . 0762 0 . 1399 0 . 0264 0 . 0762 0 . 7862 0 . 3832 0 . 0615 0 . 1399 0 . 3832 0 . 2302 0 . 0596 0 . 0264 0 . 0615 0 . 596 0 . 4619 ] .
Trace ( M T M ) = 2 . 245 , 4 m 00 2 + Δ = 4 . 000 .
i = 1 3 ( m 0 i a i T m ˆ 1 ) 2 = 0 . 4076 , ( m 00 | m 1 | ) 2 = 0 . 7031 .
0 | m 1 | m 00 .
0 | m 1 | 1 m 00 .
μ m 1 T u μ ,
2 α 1 α 1 + α 2 2 μ α 1 1 2 μ ,
2 α 2 α 2 + α 3 2 μ α 2 1 2 μ .

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