Abstract

By using the symmetric serial decomposition of a normalized Mueller matrix M [J. Opt. Soc. Am. A 26, 1109 (2009)] as a starting point and by considering the reciprocity property of Mueller matrices, the geometrical features of the Poincaré sphere mapping by M are analyzed in order to obtain a new parameterization of M in which the 15 representative parameters have straightforward geometrical interpretations. This approach provides a new geometry-based framework, whereby any normalized Mueller matrix M is completely described by a set of three associated ellipsoids whose geometrical and topological properties are characteristic of M. The mapping analysis considers the cases of type-I and type-II, as well as singular and nonsingular Mueller matrices. The novel parameterization is applied to several illustrative examples of experimental Mueller matrices taken from the literature.

© 2013 Optical Society of America

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    [CrossRef]
  39. R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426–2428 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  42. 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).
  43. J. J. Gil, “Transmittance constraints in serial decompositions of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 30, 701–707 (2013).
    [CrossRef]
  44. I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  45. J. J. Gil and I. San José, “Polarimetric subtraction of Mueller matrices,” J. Opt. Soc. Am. A 30, 1078–1088 (2013).
    [CrossRef]
  46. S. Manhas, M. K. Swami, P. Buddhiwant, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
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2013 (3)

2011 (3)

2010 (1)

2009 (2)

2008 (3)

R. Bhandari, “Transpose symmetry of the Jones matrix and topological phases,” Opt. Lett. 33, 854–856 (2008).
[CrossRef]

R. Bhandari, “Transpose symmetry of the Jones matrix and topological phases: erratum,” Opt. Lett. 33, 2985 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406–2410 (2008).
[CrossRef]

2007 (3)

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Pub. 2, 07018 (2007).
[CrossRef]

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

2006 (2)

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), www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf .

S. Manhas, M. K. Swami, P. Buddhiwant, P. K. Gupta, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190–202 (2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (1)

T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
[CrossRef]

2001 (1)

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

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. G. 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. G. 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).

1997 (2)

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

1996 (1)

1994 (2)

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

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

1988 (1)

1987 (2)

A. Schönhofer and H.-G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[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).

1986 (3)

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

M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[CrossRef]

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

1966 (1)

1941 (1)

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406–2410 (2008).
[CrossRef]

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Pub. 2, 07018 (2007).
[CrossRef]

Ben Hatit, S.

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Pub. 2, 07018 (2007).
[CrossRef]

Bernabéu, E.

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]

Bhandari, R.

Boerner, W. M.

Brosseau, C.

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

Buddhiwant, P.

Cariou, J.

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

Chipman, R. A.

Cloude, S. R.

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), www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf .

Cubián, D. P.

De Martino, A.

R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426–2428 (2009).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406–2410 (2008).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Pub. 2, 07018 (2007).
[CrossRef]

DeBoo, B.

Diego, J. L. A.

Fallet, C.

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), www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf .

Foldyna, M.

Gdoutos, E. E.

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

Gerligand, P. Y.

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

Gil, J. J.

J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A 30, 32–50 (2013).
[CrossRef]

J. J. Gil, “Transmittance constraints in serial decompositions of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 30, 701–707 (2013).
[CrossRef]

J. J. Gil and I. San José, “Polarimetric subtraction of Mueller matrices,” J. Opt. Soc. Am. A 30, 1078–1088 (2013).
[CrossRef]

I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. 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), www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.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, “Mueller matrices,” in Light Scattering from Microstructures, Vol. 534 of Lecture Notes in Physics (Springer, 2000), Chap 4.

J. J. Gil, “Determination of polarization parameters in matricial representation: Theoretical contribution and development of an automatic measurement device,” Ph.D. thesis (Facultad de Ciencias, University Zaragoza, 1983), zaguan.unizar.es/record/10680/files/TESIS-2013-057.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]

Goldhar, J.

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

Goudail, F.

Gupta, P. K.

Guyot, S.

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” J. Eur. Opt. Soc. Rapid Pub. 2, 07018 (2007).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

James, B.

Jones, R. C.

Kostinski, A.

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]

Kuball, H.-G.

A. Schönhofer and H.-G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

Le Jeune, B.

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

Le Roy-Bréhonnet, F.

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Lotrian, J.

F. Le Roy-Brehonnet, B. Le Jeune, P. Y. Gerligand, J. Cariou, and J. Lotrian, “Analysis of depolarizing optical targets by Mueller matrix formalism,” Pure Appl. Opt. 6, 385–404 (1997).
[CrossRef]

Lu, S.-Y.

Mallesh, K. S.

A. V. G. 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. G. 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).

Manea, V.

Manhas, S.

Morio, J.

Ossikovski, R.

Rao, A. V. G.

A. V. G. 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).

A. V. G. 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).

Rentmeesters, R.

Richardson, C. J. K.

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

San José, I.

J. J. Gil and I. San José, “Polarimetric subtraction of Mueller matrices,” J. Opt. Soc. Am. A 30, 1078–1088 (2013).
[CrossRef]

J. J. Gil, I. San José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A 30, 32–50 (2013).
[CrossRef]

I. San José and J. J. Gil, “Invariant indices of polarimetric purity: generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[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), www.unizar.es/galdeano/actas_pau/PDFIX/FerSanGilCor05.pdf .

Sasian, J.

Saylors, M.

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

Schönhofer, A.

A. Schönhofer and H.-G. Kuball, “Symmetry properties of the Mueller matrix,” Chem. Phys. 115, 159–167 (1987).
[CrossRef]

Sekera, Z.

Simon, R.

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

Singh, K.

Sridhar, R.

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

Sudha,

A. V. G. 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. G. 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).

Swami, M. K.

Sylla, P. M.

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

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]

van Leeuwen, M.

P. M. Sylla, C. J. K. Richardson, M. van Leeuwen, M. Saylors, and J. Goldhar, “DOP ellipsoids for systems with frequency-dependent principal states,” IEEE Photon. Technol. Lett. 13, 1310–1312 (2001).
[CrossRef]

Williams, M. W.

Xing, Z.-F.

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

Appl. Opt. (2)

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Figures (26)

Fig. 1.
Fig. 1.

Polarization states represented in the Poincaré sphere.

Fig. 2.
Fig. 2.

Ellipsoid EΔP defined through the Poincaré sphere mapping by M.

Fig. 3.
Fig. 3.

Ellipsoid EΔD defined through the Poincaré sphere mapping by MT.

Fig. 4.
Fig. 4.

Ellipsoid EΔd.

Fig. 5.
Fig. 5.

Ellipsoid ERΔd.

Fig. 6.
Fig. 6.

Ellipsoid EDΔd.

Fig. 7.
Fig. 7.

Poincaré sphere mapping by A1 (left), MΔdIIA1 (middle), and MΔnd=A2MΔdIIA1 (right).

Fig. 8.
Fig. 8.

Ellipsoid EΔnd (left) is rotated by MR, resulting in the ellipsoid ERΔnd (right).

Fig. 9.
Fig. 9.

Ellipsoid EΔnd (left) is displaced and distorted by MD, resulting in the ellipsoid EDΔnd (right). Both EΔnd and EDΔnd have the genuine property of having a single point that touches the unit sphere.

Fig. 10.
Fig. 10.

Examples of mapping by MΔd (ellipsoid EΔd, left), MR2MΔd (middle), and MD2MR2MΔd (ellipsoid EI2, right), corresponding to a given type-I Mueller matrix M.

Fig. 11.
Fig. 11.

Examples of mapping by MΔd (ellipsoid EΔd, left), MR1TMΔd (middle), and MD1MR1TMΔd (ellipsoid EI1, right), corresponding to a given type-I Mueller matrix M.

Fig. 12.
Fig. 12.

Examples of ellipsoids EΔd (left), EI2 (middle), and EI1 (right) (degenerated into respective ellipses) for the case d3=0, d10, d20.

Fig. 13.
Fig. 13.

Examples of ellipsoids EΔd (left), EI2 (middle), and EI1 (right) (degenerated into respective line segments) for the case d2=d3=0, d10.

Fig. 14.
Fig. 14.

Examples of ellipsoids EΔd (left), EI2 (middle), and EI1 (right) (degenerated into respective single points) for the case d1=d2=d3=0.

Fig. 15.
Fig. 15.

Examples of ellipsoids EΔd (left), EI2 (middle), and EI1 (right) (degenerated into respective single points) of a given type-I Mueller matrix M with D1<1, D2=1.

Fig. 16.
Fig. 16.

Examples of ellipsoids EΔd (left), EI2 (middle), and EI1 (right) (degenerated into respective single points) of a given type-I Mueller matrix M with D1=1, D2<1.

Fig. 17.
Fig. 17.

Examples of ellipsoids EΔnd (left), EII2 (middle), and EII1 (right) of a given type-II Mueller matrix M.

Fig. 18.
Fig. 18.

Examples of ellipsoids EΔnd (left), EII2 (middle), and EII1 (right) (EII2 and EII1 degenerated into respective single points) for the case D1<1, D2=1).

Fig. 19.
Fig. 19.

Examples of ellipsoids EΔnd (left), EII2 (middle), and EII1 (right) (EII2 and EII1 degenerated into respective single points) for the case D1=1, D2<1).

Fig. 20.
Fig. 20.

Ellipsoid EΔd for a type-I canonical Mueller matrix MΔd with rankH(MΔd)=2. EΔd only touches the unit sphere at two antipodal points.

Fig. 21.
Fig. 21.

Ellipsoid EΔnd for a type-II canonical Mueller matrix MΔnd with rankH(MΔnd)=2. EΔnd touches the unit sphere at a single point.

Fig. 22.
Fig. 22.

Ellipsoid EΔd for a type-I canonical Mueller matrix MΔd with rankH(MΔd)=3. EΔd is strictly inside the unit sphere.

Fig. 23.
Fig. 23.

Ellipsoid EΔnd for a type-II canonical Mueller matrix MΔnd with rankH(MΔnd)=3. EΔnd touches the unit sphere at a single point.

Fig. 24.
Fig. 24.

Characteristic ellipsoids EΔd (left), EI2 (middle), and EI1 (right) of an inorganic sample of rusty steel.

Fig. 25.
Fig. 25.

Characteristic ellipsoids EΔd (left), EI2 (middle), and EI1 (right) of an organic sample composed of a suspension of 2.0 μm polystyrene spheres in aqueous solution of glucose [46].

Fig. 26.
Fig. 26.

Characteristic ellipsoids EΔd (left), EI2 (middle), and EI1 (right) of a biological sample of cancerous tissue [47].

Tables (3)

Tables Icon

Table 1. Geometrical Parameterization of a Nonsingular Normalized Type-I Mueller Matrix M

Tables Icon

Table 2. Geometrical Parameterization of a Nonsingular Normalized Type-II Mueller Matrix M

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Table 3. Summary of the Topological Properties of the Characteristic Ellipsoids of a Mueller Matrix in Terms of the Rank of Its Associated Covariance Matrix H(M)

Equations (33)

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M=m00(1DTPm);D1m00(m01,m02,m03)T,P1m00(m10,m20,m30)T,m1m00(m11m12m13m21m22m23m31m32m33).
PΔ=(D2+P2+m22)/3,
s(s0,s1,s2,s3)T[IIPu];I=s0,P(s12+s22+s32)1/2s0,u1IP(s1s2s3)=1IP(cos2χcos2φcos2χsin2φsin2χ),
u1(Ms)0[(Ms)1,(Ms)2,(Ms)3]T(s1,s2,s3)T/s0;s(1,uT)T,|u|=1.
Mm00(1DTPm)=m00M^ΔPMRM^D;M^ΔP(10TPΔPmΔP),MR(10T0mR),M^D(1DTDmD),
mΔPT=mΔP;mR1=mRT;mD(1D2)1/2I+1D2[1(1D2)1/2]D×DT,Idiag(1,1,1,1),
M^ΔP(10TPΔI3)(10T0mΔ),
M^ΔPs=[1PΔ+mΔu],
MJr=diag(1,1,1,1)MJTdiag(1,1,1,1).
MTm00(1PTDmT)=m00M^ΔDMRM^P;M^ΔD(10TDΔDmΔD),MR(10T0mR),M^P(1PTPmP).
MΔd=diag(d0,d1,d2,d3)=d0diag(1,d^1,d^2,d^3),0|d^i|1,
MΔnd=[2a0a000a000000a20000a2],0a2a0.
M^Δnd=A2MΔdIIA1;A2[10001/210000100001],MΔdII[100001/40000k0000k],A1[11/200010000100001],ka22a0.
(u12/3)2(1/3)2+u22(a2/3a0)2+u32(a2/3a0)2=1;uisi/s0=(Ms)i/(Ms)0(i=1,2,3).
M=MJ2MΔMJ1,
MJ=MDMR=MRMD(MD=MR1MDMR),
M=MD2MR2MΔMR1MD1.
MTGMG(1,D1T)T=d02(1,D1T)T,MGMTG(1,D2T)T=d02(1,D2T)T,
MTGMG(1,D1T)T=d02(1,D1T)T,(1,D2T)T=MG(1,D1T)T/[MG(1,D1T)T]0,MGMTG(1,D2T)T=d02(1,D2T)T,(1,D1T)T=MTG(1,D2T)T/[MTG(1,D2T)T]0.
MT=MD1TMR1TMΔdMR2TMD2T=MD1MR1TMΔdMR2TMD2.
MΔd=MR21MD21MMD11MR11=MR2TMD21MMD11MR1T.
MD1=11D[1DTDmD],
MΔd=MΔdT=MR1MD11MMD21MR2,
MJ1=MD1MR1,MJ2MR2MD2.
M=MD2diag(1,0,0,0)MD1=m00[1D1TD2D2×D1T]=m00[1D2]×[1D1]T.
MT=MD1TMR1TMΔndMR2TMD2T=MD1MR1TMΔndMR2TMD2.
MΔnd=MR21MD21MMD11MR11.
MΔndT=MR1MD11MMD21MR2.
H(M)=14i,j=03mij(σiσj),
mij=tr[(σiσj)H].
M=[1.00000.01700.02030.00610.01990.63480.00920.00310.00760.01400.61120.00380.00570.00020.00580.4220],
M=[1.0000.1150.0660.0230.1110.7590.0610.0010.0180.1510.4350.1390.0460.0060.1280.334],
MCcorr=[0.910.480.220.130.510.520.110.160.050.010.100.070.060.020.080.08],

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