Abstract

We experimentally assess the validity of the symmetric decomposition of Mueller matrices [R. Ossikovski, J. Opt. Soc. Am A 26, 1109-1118 (2009)] into a sequence of five factors consisting of a diagonal depolarizer between two retarder and diattenuator pairs. The raw data were Mueller images of combinations of polarization components which were individually measured and then assembled in different combinations. The possibility to recover all the elements is discussed, including the experimentally relevant cases of “degenerate” depolarizers, with two equal eigenvalues, which were not explicitly considered in the general theory.

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References

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  1. J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).
  2. S. R. Cloude, “Physical Realisability of Matrix Operators in Polarimetry,” Proc. SPIE 1166, 177–185 (1989).
  3. S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13(5), 1106–1113 (1996).
    [CrossRef]
  4. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004).
    [CrossRef] [PubMed]
  5. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
    [CrossRef] [PubMed]
  6. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009).
    [CrossRef]
  7. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994).
    [CrossRef]
  8. 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).
  9. Z. Xing, “On the deterministic and non-deterministic Mueller matrices,” J. Mod. Opt. 39(3), 461–484 (1992).
    [CrossRef]
  10. 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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
    [CrossRef]
  11. B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
    [CrossRef]
  12. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999).
    [CrossRef]
  13. C. Brosseau, Polarized Light: A Statistical Optics Approach, (Wiley, 1998), Chap 4.1 and 4.4.
  14. R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008).
    [CrossRef]

2009 (1)

2008 (1)

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008).
[CrossRef]

2007 (2)

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
[CrossRef] [PubMed]

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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

2004 (2)

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[CrossRef]

J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29(19), 2234–2236 (2004).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

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).

1996 (1)

1994 (1)

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994).
[CrossRef]

1992 (1)

Z. Xing, “On the deterministic and non-deterministic Mueller matrices,” J. Mod. Opt. 39(3), 461–484 (1992).
[CrossRef]

1989 (1)

S. R. Cloude, “Physical Realisability of Matrix Operators in Polarimetry,” Proc. SPIE 1166, 177–185 (1989).

1986 (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 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. Europ. Opt. Soc. Rap. Public 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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Physical Realisability of Matrix Operators in Polarimetry,” Proc. SPIE 1166, 177–185 (1989).

Compain, E.

De Martino, A.

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
[CrossRef] [PubMed]

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[CrossRef]

Drevillon, B.

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[CrossRef]

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999).
[CrossRef]

Gil, J. J.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).

Gopala Rao, A. V.

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).

Goudail, F.

Guyot, S.

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
[CrossRef] [PubMed]

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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

Laude-Boulesteix, B.

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[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 I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).

Morio, J.

Ossikovski, R.

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009).
[CrossRef]

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32(6), 689–691 (2007).
[CrossRef] [PubMed]

Poirier, S.

Schwartz, L.

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[CrossRef]

Simon, R.

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994).
[CrossRef]

Sridhar, R.

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994).
[CrossRef]

Sudha,

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).

Xing, Z.

Z. Xing, “On the deterministic and non-deterministic Mueller matrices,” J. Mod. Opt. 39(3), 461–484 (1992).
[CrossRef]

Appl. Opt. (2)

B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2224–2832 (2004).
[CrossRef]

E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public (1)

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. Europ. Opt. Soc. Rap. Public 2, 07018 (2007).
[CrossRef]

J. Mod. Opt. (3)

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41(10), 1903–1915 (1994).
[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).

Z. Xing, “On the deterministic and non-deterministic Mueller matrices,” J. Mod. Opt. 39(3), 461–484 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (Lond.) (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical-system,” Opt. Acta (Lond.) 33, 185–189 (1986).

Opt. Commun. (1)

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281(9), 2406–2410 (2008).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

S. R. Cloude, “Physical Realisability of Matrix Operators in Polarimetry,” Proc. SPIE 1166, 177–185 (1989).

Other (1)

C. Brosseau, Polarized Light: A Statistical Optics Approach, (Wiley, 1998), Chap 4.1 and 4.4.

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Figures (3)

Fig. 1
Fig. 1

Scheme of the Mueller polarimeter.

Fig. 2
Fig. 2

Diattenuators used in transmission. Left: 1 mm thick glass plate coated with a thin film of amorphous silicon (D1). Right: pair of uncoated thick glass plates (D2).

Fig. 3
Fig. 3

Relative positions of the different available components. D1,D2: diattenuators described in Fig. 2 . R1, R2: mica retarders. Δp: plastic sheet depolarizer. Δs: suspension of latex microspheres (second depolarizer).

Tables (8)

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Table 1 Experimental matrices M mi) of the degenerate depolarizers measured alone.

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Table 2 Characteristics of individual polarization elements from direct measurements.

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Table 3 Rms distances between the directly measured parameters of the individual components listed in Table 2 and the results of the symmetric decomposition of M 0 m .

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Table 4 Error estimators for the polar (LC) and reverse (Rev) decompositions with the nondegenerate depolarizer

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Table 5 Experimental configurations with a single retarder and a degenerate depolarizer.

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Table 6 Experimental matrices measured in the configurations listed in Table 5.

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Table 7 Error estimators for symmetric decomposition of the data listed in Table 6.

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Table 8 Error estimators for the Lu-Chipman and reverse decompositions of the data listed in Table 6.

Equations (17)

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M = M ( Δ ) M ( R ) M ( D )
M ( D ) = [ 1 D T D m D ]
M ( R ) = [ 1 0 T 0 m R ]
R = cos 1 [ 1 2 Tr M ( R ) 1 ]
M ( Δ ) = [ 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ] w i t h | a | , | b | , | c | 1
M = M ( D 2 ) M ( R 2 ) M ( Δ ) M ( R 1 ) T M ( D 1 )
M = M 2 M ( Δ ) M 1
M Δ s = [ 1 0 0 0 0 a 0 0 0 0 a 0 0 0 0 b ]
M δ = [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ]
M Δ p = 1 Δ δ δ Δ δ / 2 δ + Δ δ / 2 M δ d δ = [ ¨ 1 0 0 0 0 1 0 0 0 0 a 0 0 0 0 a ] M δ
δ D i = 1 3 k = 1 3 D i , k m D i , k d e c 2
M m ( Δ t ) = [ 1 0.010 0.013 0.0057 0.009 0.579 0.017 0.023 0.018 0.012 0.472 0.004 0.001 0.002 0.000 0.379 ]
M 0 m = [ 1 0.193 0.02 0.021 0.023 0.312 0.023 0.026 0.018 0.008 0.410 0.183 0.042 0.122 0.204 0.422 ]
M ' = M ( D 2 ) 1 M M ( D 1 ) 1
M ' = U Σ V
R α = [ 1 0 0 0 0 cos α sin α 0 0 sin α cos α 0 0 0 0 1 ]
M ' = U Σ V = U R α Σ R α V = ( U R α ) Σ ( V R α )

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