Abstract

The degree of polarimetric purity of a Mueller matrix, also called “depolarization index” [Opt. Acta 33, 185 (1986)] is expressed as a quadratic average of two contributions of different nature. The contribution due to the polarizance and diattenuation properties is given by a unique parameter called “degree of polarizance,” and the complementary contribution due to nonpolarizing properties is given by a parameter called “degree of spherical purity.” These two intrinsic quantities are useful in order to analyze the sources of the polarimetric purity of a material sample whose Mueller matrix has been measured and provide criteria for the classification of Mueller matrices.

© 2011 Optical Society of America

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  2. J. J. Gil and E. Bernabéu, “A depolarization criterion for Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  3. J. J. Gil and E. Bernabéu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [CrossRef]
  4. S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
    [CrossRef]
  5. 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]
  6. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
    [CrossRef]
  7. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. R. Espinosa-Luna, E. Bernabéu, and G. Atondo-Rubio, “Q(M) and the depolarization index scalar metrics,” Appl. Opt. 47, 1575–1580 (2008).
    [CrossRef] [PubMed]
  13. J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
    [CrossRef]
  14. R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
    [CrossRef]
  15. R. Ossikovski, “Alternative depolarization criteria for Mueller matrices,” J. Opt. Soc. Am. A 27, 808–814 (2010).
    [CrossRef]
  16. I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  17. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
    [CrossRef]
  26. Z-F Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
    [CrossRef]
  27. C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
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  31. S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).
  32. 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).
    [CrossRef]
  33. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “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).
    [CrossRef]
  34. G. R. Bird and W. A. Shurcliff, “Pile-of-plates polarizers for the infrared: improvement in analysis and design,” J. Opt. Soc. Am. 49, 235–237 (1959).
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    [CrossRef]
  37. B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
    [CrossRef] [PubMed]
  38. C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.
  39. S.-Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
    [CrossRef]
  40. Sudha and A. V. Gopala Rao, “Polarization elements: a group-theoretical study,” J. Opt. Soc. Am. A 18, 3130–3134(2001).
    [CrossRef]
  41. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
    [CrossRef]
  42. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088(1993).
    [CrossRef]

2011 (1)

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

2010 (3)

R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
[CrossRef]

R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
[CrossRef]

R. Ossikovski, “Alternative depolarization criteria for Mueller matrices,” J. Opt. Soc. Am. A 27, 808–814 (2010).
[CrossRef]

2009 (2)

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
[CrossRef]

J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
[CrossRef]

2008 (1)

2007 (3)

R. Espinosa-Luna and E. Bernabéu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256–258 (2007).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

S. N. Savenkov, O. I. Sydoruk, and R. S. Muttiah, “Eigenanalysis of dichroic, birefringent, and degenerate polarization elements: a Jones-calculus study,” Appl. Opt. 46, 6700–6709 (2007).
[CrossRef] [PubMed]

2006 (2)

T. Tudor, “Non-hermitian polarizers: A biorthogonal analysis,” J. Opt. Soc. Am. A 23, 1513–1522 (2006).
[CrossRef]

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.

2005 (3)

2004 (2)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.

B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
[CrossRef] [PubMed]

2003 (2)

2001 (1)

2000 (1)

1998 (3)

S.-Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[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).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “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).
[CrossRef]

1996 (1)

1995 (1)

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

1994 (2)

S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

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

1993 (2)

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

1992 (1)

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

1989 (1)

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).

1987 (2)

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

1986 (2)

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

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

1985 (1)

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

1959 (1)

1941 (1)

Aiello, A.

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94, 090406 (2005).
[CrossRef] [PubMed]

Atondo-Rubio, G.

R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
[CrossRef]

R. Espinosa-Luna, E. Bernabéu, and G. Atondo-Rubio, “Q(M) and the depolarization index scalar metrics,” Appl. Opt. 47, 1575–1580 (2008).
[CrossRef] [PubMed]

Bernabéu, E.

R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
[CrossRef]

R. Espinosa-Luna, E. Bernabéu, and G. Atondo-Rubio, “Q(M) and the depolarization index scalar metrics,” Appl. Opt. 47, 1575–1580 (2008).
[CrossRef] [PubMed]

R. Espinosa-Luna and E. Bernabéu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256–258 (2007).
[CrossRef]

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

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

Bird, G. R.

Chipman, R.

Chipman, R. A.

Cloude, S. R.

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

S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).

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

Correas, J. M.

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.

DeBoo, B.

Espinosa-Luna, R.

R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
[CrossRef]

R. Espinosa-Luna, E. Bernabéu, and G. Atondo-Rubio, “Q(M) and the depolarization index scalar metrics,” Appl. Opt. 47, 1575–1580 (2008).
[CrossRef] [PubMed]

R. Espinosa-Luna and E. Bernabéu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256–258 (2007).
[CrossRef]

Ferreira, C.

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.

Gdoutos, E. E.

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

Gil, J. J.

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
[CrossRef]

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

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

J. J. Gil, “Determinación de parámetros de polarización en representación matricial. Contribución teórica y realización de un dispositivo automático,” Ph.D. thesis, Facultad de Ciencias, Universidad de Zaragoza, 1983. Available from http://www.pepegil.es/PhD-Thesis-JJ-Gil.pdf.

Givens, C. R.

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

Gopala Rao, A. V.

Sudha and A. V. Gopala Rao, “Polarization elements: a group-theoretical study,” J. Opt. Soc. Am. A 18, 3130–3134(2001).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “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).
[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).
[CrossRef]

Hallikainen, M. T.

J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
[CrossRef]

Hinojosa-Ruiz, S.

R. Espinosa-Luna, G. Atondo-Rubio, E. Bernabéu, and S. Hinojosa-Ruiz, “Dealing depolarization of light in Mueller matrices with scalar metrics,” Optik 121, 1058–1068 (2010).
[CrossRef]

Jones, R. C.

Kim, K.

Koeniguer, E. C.

J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
[CrossRef]

Kostinski, A. B.

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

Lu, S.-Y.

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 conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
[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).
[CrossRef]

Mandel, L.

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monog. Sem. Mat. G. de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf.

Muttiah, R. S.

Ossikovski, R.

Pottier, E.

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

Praks, J.

J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
[CrossRef]

San Jose, I.

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

San José, I.

C. Ferreira, I. San José, J. J. Gil, and J. M. Correas, “Geometric modeling of polarimetric transformations,” Monog. Sem. Mat. G. de Galdeano 33, 115–119 (2006). Available from www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf.

Sasian, J.

Savenkov, S. N.

Shurcliff, W. A.

Simon, R.

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

Sridhar, R.

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

Sudha,

Sudha and A. V. Gopala Rao, “Polarization elements: a group-theoretical study,” J. Opt. Soc. Am. A 18, 3130–3134(2001).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “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).
[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).
[CrossRef]

Sydoruk, O. I.

Theocaris, P. S.

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

Tudor, T.

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]

Woerdman, J. P.

A. Aiello and J. P. Woerdman, “Physical bounds to the entropy-depolarization relation in random light scattering,” Phys. Rev. Lett. 94, 090406 (2005).
[CrossRef] [PubMed]

Wolf, E.

Xing, Z-F

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

Appl. Opt. (3)

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

J. J. Gil, “Polarimetric characterization of light and media. Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. Praks, E. C. Koeniguer, and M. T. Hallikainen, “Alternatives to target entropy and alpha angle in SAR polarimetry,” IEEE Trans. Geosci. Remote Sens. 47, 2262–2274 (2009).
[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. (5)

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).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “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).
[CrossRef]

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

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

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Figures (1)

Fig. 1
Fig. 1

Feasible region for P S , P P .

Equations (46)

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s = Ms ,
P Δ = [ i , j = 0 3 m i j 2 m 00 2 3 m 00 2 ] 1 / 2 = [ 4 i = 0 3 λ i 2 ( i = 0 3 λ i ) 2 3 ( i = 0 3 λ i ) 2 ] 1 / 2 ,
H i = 0 3 ( λ ^ i log 4 λ ^ i ) , λ ^ i λ i k = 0 3 λ k .
Q 3 P Δ 2 D 2 1 + D 2 .
M = M D 2 M R 2 M Δ M R 1 M D 1 ,
L 1 ( M ) ( ρ 1 + ρ 2 + ρ 3 3 ρ 0 ) 1 / 2 ,
L 2 ( M ) [ 4 i = 0 3 ρ i 2 ( i = 0 3 ρ i ) 2 3 ( i = 0 3 ρ i ) 2 ] 1 / 2 ,
M = m 00 ( 1 D T P m ) ; D 1 m 00 ( m 01 , m 02 , m 03 , ) T , P 1 m 00 ( m 10 , m 20 , m 30 , ) T , m 1 m 00 ( m 11 m 12 m 13 m 21 m 22 m 23 m 31 m 32 m 33 ) ,
M J r = diag ( 1 , 1 , 1 , 1 ) M J T diag ( 1 , 1 , 1 , 1 ) .
P ( M ) = D ( M T ) = D ( M r ) = 1 m 00 ( m 10 , m 20 , m 30 ) T ,
D ( M ) = P ( M T ) = P ( M r ) = 1 m 00 ( m 01 , m 02 , m 03 ) T .
P Δ ( M ) = P Δ ( M r ) = P Δ ( M T ) = ( M 2 2 M 0 2 3 M 0 2 ) 1 / 2 ,
P Δ 2 = i = 1 3 m i 0 2 3 m 00 2 + j = 1 3 m 0 j 2 3 m 00 2 + k , l = 1 3 m k l 2 3 m 00 2 .
P | P | = [ i = 1 3 m i 0 2 m 00 2 ] 1 / 2 , D | D | = [ j = 1 3 m 0 j 2 m 00 2 ] 1 / 2 .
P Δ 2 = 1 3 P 2 + 1 3 D 2 + P S 2 ,
P S [ k , l = 1 3 m k l 2 3 m 00 2 ] 1 / 2 = 1 3 m 2 .
P P [ 1 2 ( P 2 + D 2 ) ] 1 / 2 ,
P | P ( M ) | = 1 m 00 [ i = 1 3 m i 0 2 ] 1 / 2 , D | P ( M T ) | = 1 m 00 [ j = 1 3 m 0 j 2 ] 1 / 2 .
M = m 00 [ 1 0 T 0 m O ] ; 0 T ( 0 , 0 , 0 ) .
s 0 ( s ^ 1 , s ^ 2 , s ^ 3 ) ; s ^ i s i ( s 1 2 + s 2 2 + s 3 2 ) 1 / 2 , i = 1 , 2 , 3 ,
( s 0 , s 1 , s 2 , s 3 ) T M ( 1 , s 1 , s 2 , s 3 ) T , ( s 1 2 + s 2 2 + s 3 2 = 1 ) ,
M = m 00 [ 1 0 T 0 m R ] ; 0 T ( 0 , 0 , 0 ) ,
P Δ 2 = 2 3 P P 2 + P S 2 .
P P 2 1 2 ( 1 + 3 P S 2 ) .
0 tr ( GM T GM ) = m 00 2 [ ( 1 + 3 P S 2 ) 2 P P 2 ] .
1 = 2 3 P P 2 P Δ 2 + P S 2 P Δ 2 ,
2 P P max 2 3 P S 2 = 1.
M = m 00 ( 1 D T P 0 ) ,
M = m 00 ( 1 0 T P 0 ) .
M = m 00 ( 1 D T 0 0 ) .
M PO = m 00 [ 1 P T P m P ] diag ( 1 , 0 , 0 , 0 ) = m 00 [ 1 0 T P diag ( 0 , 0 , 0 ) ] , P = | P | = 1.
M OP = diag ( 1 , 0 , 0 , 0 ) m 00 [ 1 P T P m D ] = m 00 [ 1 P T 0 diag ( 0 , 0 , 0 ) ] , P = | P | = 1.
M DOP = m 00 [ 1 P T P m P ] diag ( 1 , 0 , 0 , 0 ) [ 1 D T D m D ] = m 00 [ 1 D T P P × D T ] , D = P = 1.
M = M J 1 M Δ M J 2 ,
M Δ d = diag ( d 0 , d 1 , d 2 , ε d 3 ) , d i 0 , ε det ( M ) | det ( M ) | ,
M Δ n d = [ 2 a 0 a 0 0 0 a 0 0 0 0 0 0 a 2 0 0 0 0 a 2 ] , a i 0.
d 0 1 , 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 d 1 + d 2 d 3 ,
0 a 2 a 0 1 / 3.
P P ( M Δ d ) = 0 , P S ( M Δ d ) = P Δ ( M Δ d ) = L 1 ( M ) .
P P ( M Δ n d ) = P ( M Δ n d ) = D ( M Δ n d ) = 1 2 , P S ( M Δ n d ) = 1 6 a 2 a 0 .
P Δ ( M Δ n d ) = [ 2 3 P P 2 ( M Δ n d ) + P S 2 ( M Δ n d ) ] 1 / 2 = [ 1 6 ( 1 + a 2 2 a 0 2 ) ] 1 / 2 .
c ( M Δ n d ) [ P S 2 ( M Δ n d ) 2 3 P P 2 ( M Δ n d ) ] 1 / 2 = a 2 a 0 1.
M = M RO M K M RI ,
P P ( M ) = P P ( M K ) , P S ( M ) = P S ( M K ) , P Δ ( M ) = P Δ ( M K ) .
M = M R 2 M D 2 M Δ M D 1 M R 1 ,
P P ( M ) = P P ( M C ) , P S ( M ) = P S ( M C ) , P Δ ( M ) = P Δ ( M C ) ; M C M D 2 M Δ M D 1 .

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