Abstract

We propose a new algorithm, the pseudopolar decomposition, to decompose a Jones or a Mueller–Jones matrix into a sequence of matrix factors: JJRJDJ1CJ2C or MMRMDM1CM2C. The matrices JR (MR) and JD (MD) parameterize, respectively, the retardation and dichroic properties of J (M) in a good approximation, while JiC (MiC) are correction factors that arise from the noncommutativity of the polarization properties. The exponential versions of the general Jones matrix are used to demonstrate the pseudopolar decomposition and to calculate each one of the matrix factors. The decomposition preserves all the polarization properties of the system on the factorized JR (MR) and JD (MD) matrix terms. The algorithm that calculates the pseudopolar decomposition for experimentally determined Mueller matrices is presented.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P. K. Gupta, “Polar decomposition of 3×3 Mueller matrix: a tool for quantitative tissue polarimetry,” Opt. Express 14, 9324-9337 (2006).
    [CrossRef] [PubMed]
  8. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Gupta, and J. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190-202 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
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    [CrossRef]
  16. Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
    [CrossRef] [PubMed]
  17. O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
    [CrossRef]
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    [CrossRef]
  19. J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. (Washington, D.C.) 87, 1359-1399 (1987).
    [CrossRef]
  20. H. P. Jensen, J. A. Schellman, and T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192-200 (1978).
    [CrossRef]
  21. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962-982 (1967).
    [CrossRef]
  22. D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
    [CrossRef]
  23. M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
    [CrossRef]
  24. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689-691 (2007).
    [CrossRef] [PubMed]
  25. R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
  28. N. Go, “Optical activity of anisotropic solutions. I,” J. Chem. Phys. 43, 1275-1280 (1965).
    [CrossRef]

2008

R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25, 473-482 (2008).
[CrossRef]

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (2008).
[CrossRef]

N. Ghosh, M. F. G. Wood, and A. I. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[CrossRef] [PubMed]

2007

2006

2005

2004

2002

1996

1994

1987

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67-71 (1987).

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. (Washington, D.C.) 87, 1359-1399 (1987).
[CrossRef]

1978

1977

M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
[CrossRef]

1971

1967

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962-982 (1967).
[CrossRef]

1965

N. Go, “Optical activity of anisotropic solutions. I,” J. Chem. Phys. 43, 1275-1280 (1965).
[CrossRef]

1948

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (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 Publ. 2, 07018 (2007).
[CrossRef]

Arteaga, O.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
[CrossRef]

Barakat, R.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67-71 (1987).

Buddhiwant, P.

Canillas, A.

O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
[CrossRef]

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Cariou, J.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Chen, Z.

Chipman, R. A.

Chung, J.

Collet, C.

Crusats, J.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

De Martino, A.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (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] [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. Eur. Opt. Soc. Rapid Publ. 2, 07018 (2007).
[CrossRef]

B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824-2832 (2004).
[CrossRef] [PubMed]

Drévillon, B.

El-Hachemi, Z.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
[CrossRef]

Eliès, P.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Escudero, C.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Garcia-Caurel, E.

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (2008).
[CrossRef]

Ghosh, N.

Gil, J. J.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67-71 (1987).

Go, N.

N. Go, “Optical activity of anisotropic solutions. I,” J. Chem. Phys. 43, 1275-1280 (1965).
[CrossRef]

Griffiths, C. O.

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 Publ. 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] [PubMed]

Hammer-Wilson, M. J.

Harada, T.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Hatit, S. Ben

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (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 Publ. 2, 07018 (2007).
[CrossRef]

Holcomb, D. E.

Jellison, G. E.

Jensen, H. P.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. (Washington, D.C.) 87, 1359-1399 (1987).
[CrossRef]

H. P. Jensen, J. A. Schellman, and T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32, 192-200 (1978).
[CrossRef]

Jones, C. R.

Jung, W.

Kuroda, R.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Laude-Boulesteix, B.

Le Jeune, B.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Le Roy-Bréhonnet, F.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Lotrian, J.

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Lu, S.-Y.

Manhas, S.

Marienko, V. V.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E 74, 056607 (2006).
[CrossRef]

Muttiah, R. S.

Oberemok, E. A.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E 74, 056607 (2006).
[CrossRef]

Ossikovski, R.

R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25, 473-482 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (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 Publ. 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] [PubMed]

Ribó, J. M.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Rosa, M.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Rouleau, C. M.

Savenkov, S. N.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E 74, 056607 (2006).
[CrossRef]

S. N. Savenkov, O. I. Sydoruk, and R. S. Muttiah, “Conditions for polarization elements to be dichroic and birefringent,” J. Opt. Soc. Am. A 22, 1447-1452 (2005).
[CrossRef]

Schellman, J.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. (Washington, D.C.) 87, 1359-1399 (1987).
[CrossRef]

Schellman, J. A.

Scholz, D.

D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
[CrossRef]

Schwartz, L.

Singh, J.

Suzuki, M.

M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
[CrossRef]

Swami, M. K.

Sydoruk, O.

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E 74, 056607 (2006).
[CrossRef]

Sydoruk, O. I.

Takakura, Y.

Troxell, T.

Uppal, A.

Vitkin, A. I.

N. Ghosh, M. F. G. Wood, and A. I. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[CrossRef] [PubMed]

Weyrauch, M.

D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
[CrossRef]

Whitney, C.

Wilcox, R. M.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962-982 (1967).
[CrossRef]

Wilder-Smith, P.

Wood, M. F. G.

N. Ghosh, M. F. G. Wood, and A. I. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[CrossRef] [PubMed]

Zallat, J.

Appl. Opt.

Appl. Spectrosc.

Chem. Rev. (Washington, D.C.)

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. (Washington, D.C.) 87, 1359-1399 (1987).
[CrossRef]

Chem.-Eur. J.

Z. El-Hachemi, O. Arteaga, A. Canillas, J. Crusats, C. Escudero, R. Kuroda, T. Harada, M. Rosa, and J. M. Ribó, “On the mechano-chiral effect of vortical flows on the dichroic spectra of 5-phenyl-10,15,20-tris(4-sulfonatophenyl)porphyrin j-aggregates,” Chem.-Eur. J. 14, 6438-6443 (2008).
[CrossRef] [PubMed]

Commun. Math. Phys.

M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
[CrossRef]

J. Biomed. Opt.

N. Ghosh, M. F. G. Wood, and A. I. Vitkin, “Mueller matrix decomposition for extraction of individual polarization parameters from complex turbid media exhibiting multiple scattering, optical activity, and linear birefringence,” J. Biomed. Opt. 13, 044036 (2008).
[CrossRef] [PubMed]

J. Chem. Phys.

N. Go, “Optical activity of anisotropic solutions. I,” J. Chem. Phys. 43, 1275-1280 (1965).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ.

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 Publ. 2, 07018 (2007).
[CrossRef]

J. Math. Phys.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8, 962-982 (1967).
[CrossRef]

D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

F. Le Roy-Bréhonnet, B. Le Jeune, P. Eliès, J. Cariou, and J. Lotrian, “Optical media and target characterization by Mueller matrix decomposition,” J. Phys. D 29, 34-38 (1996).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Jena)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from the polar decomposition of its Mueller matrix,” Optik (Jena) 76, 67-71 (1987).

Phys. Rev. E

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, “Generalized matrix equivalence theorem for polarization theory,” Phys. Rev. E 74, 056607 (2006).
[CrossRef]

Phys. Status Solidi A

O. Arteaga, Z. El-Hachemi, and A. Canillas, “Application of transmission ellipsometry to the determination of CD spectra of porphyrin J-aggregates,” Phys. Status Solidi A 205, 797-801 (2008).
[CrossRef]

R. Ossikovski, M. Anastasiadou, S. Ben Hatit, E. Garcia-Caurel, and A. De Martino, “Depolarizing Mueller matrices: how to decompose them?” Phys. Status Solidi A 205, 720-727 (2008).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Evolution of the error function ϵ ( i ) with the increase of the number of filtering steps applied to the experimental Mueller matrices.

Tables (5)

Tables Icon

Table 1 Symbols Used and Definitions

Tables Icon

Table 2 Relation to the Notation of Lu–Chipman a

Tables Icon

Table 3 Factorized Jones Matrix, J J R J D J 1 C J 2 C

Tables Icon

Table 4 Decomposition of the Mueller Matrix of a Polaroid Film (Measured at 800 nm )

Tables Icon

Table 5 Decomposition of the Mueller Matrix of a Polyacrylamide Gel (Measured at 632.8 nm )

Equations (57)

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

A = U P , A = P U ,
J = J R J D = J D J R ,
J ( ω , z ) = exp [ z N ( ω ) ] ,
N ( ω ) = lim z 0 J ( ω , z ) I z .
N = 1 2 z ( LD i LB 2 ( i η + k ) LD i LB + CB i CD LD i LB CB + i CD LD + i LB 2 ( i η + k ) ) .
L LB i LD ,
L LB i LD ,
C CB i CD .
N = i 2 z ( χ + L L + i C L i C χ L ) .
J = exp [ i R ] ,
R = 1 2 ( χ + L L + i C L i C χ L ) .
R = 1 2 ( χ σ 0 + σ T ) ,
σ 0 = [ 1 0 0 1 ] , σ 1 = [ 1 0 0 1 ] ,
σ 2 = [ 0 1 1 0 ] , σ 3 = [ 0 i i 0 ] .
J = exp [ i R ] = e i χ 2 e i ( σ T ) 2 = e i χ 2 e i ( σ T R + σ ( i T D ) ) 2 .
e t ( X + Y ) = e t X e t Y e ( t 2 2 ) [ X , Y ] e ( t 3 6 ) ( 2 [ Y , [ X , Y ] ] + [ X , [ X , Y ] ] ) e t 4 ,
J J R J D J 1 C J 2 C ,
J R = e i η exp ( i T R 2 T R T R σ ) = e i η [ σ 0 cos T R 2 i T R T R σ sin T R 2 ] ,
J D = e k exp ( T D 2 T D T D σ ) = e k [ σ 0 cosh T D 2 1 T D T D σ sinh T D 2 ] ,
J 1 C = exp ( A 8 [ A A σ ] ) = σ 0 cosh A 8 + 1 A A σ sinh A 8 ,
J 2 C = exp ( i B 48 [ B B σ ] ) = σ 0 cos B 48 + i B B σ sin B 48 ,
J = e i χ 2 exp ( i T 2 T T σ ) = e i χ 2 [ σ 0 cos T 2 i T T σ sin T 2 ] ,
J J R J D J 1 C J 2 C J D J R J 1 C 1 J 2 C 1 .
T D ( J ) = T D ( J D ) , T R ( J ) = T R ( J R ) ,
CBLD LB CD = 0 ,
CDLB LDCB = 0 ,
LBLD LB LD = 0 .
M = A ( J J * ) A 1 ,
A = 1 2 ( 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ) .
M i j = 1 2 Tr ( σ i J σ j J ) ,
M R = ( 1 0 0 0 0 cos T R + LB 2 α LBLB α + CB β LBCB α + LB β 0 LBLB α CB β cos T R + LB 2 α LB CB α LB β 0 LBCB α LB β LB CB α + LB β cos T R + CB 2 α ) ,
M D = e 2 k ( cosh T D LD ν LD ν CD ν LD ν 1 + LD 2 μ LDLD μ LDCD μ LD ν LDLD μ 1 + LD 2 μ LD CD μ CD ν LDCD μ LD CD μ 1 + CD 2 μ ) ,
M 1 C = ( cosh 2 A 8 + sinh 2 A 8 A 1 γ A 2 γ A 3 γ A 1 γ 1 + 2 A 1 2 δ 2 A 1 A 2 δ 2 A 1 A 3 δ A 2 γ 2 A 1 A 2 δ 1 + 2 A 2 2 δ 2 A 2 A 3 δ A 3 γ 2 A 1 A 3 δ 2 A 2 A 3 δ 1 + 2 A 3 2 δ ) ,
γ = 1 A sinh A 4 , δ = 1 A 2 sinh 2 A 8 .
M 2 C = ( ρ + ( ξ 2 ) ( B * B ) ε B 1 i + ζ B 1 r + ξ ( B 3 r B 2 i B 2 r B 3 i ) ε B 2 i + ζ B 2 r + ξ ( B 1 r B 3 i B 3 r B 1 i ) ε B 3 i + ζ B 3 r + ξ ( B 1 r B 2 i + B 2 r B 1 i ) ε B 1 i + ζ B 1 r ξ ( B 3 r B 2 i B 2 r B 3 i ) ρ + ξ ( ( B 1 i ) 2 + ( B 1 r ) 2 ) ( ξ 2 ) ( B * B ) ε B 3 r + ζ B 3 i + ξ ( B 1 r B 2 r + B 1 i B 2 i ) ε B 2 r ζ B 2 i ξ ( B 3 r B 1 r B 3 i B 1 i ) ε B 2 i + ζ B 2 r ξ ( B 1 r B 3 i B 3 r B 1 i ) ε B 3 r ζ B 3 i + ξ ( B 1 r B 2 r + B 1 i B 2 i ) ρ + ξ ( ( B 2 r ) 2 + ( B 2 i ) 2 ) ( ξ 2 ) ( B * B ) ε B 1 r + ζ B 1 i ξ ( B 3 r B 2 r B 3 i B 2 i ) ε B 3 i + ζ B 3 r ξ ( B 1 r B 2 i + B 2 r B 1 i ) ε B 2 r + ζ B 2 i ξ ( B 3 r B 1 r B 3 i B 1 i ) ε B 1 r ζ B 1 i ξ ( B 3 r B 2 r B 3 i B 2 i ) ρ + ξ ( ( B 3 i ) 2 + ( B 3 r ) 2 ) ( ξ 2 ) ( B * B ) ) ,
ε = ( B i sinh B i 24 + B r sin B r 24 ) B B * ,
ζ = ( B i sin B r 24 B r sinh B i 24 ) B B * ,
ξ = ( cosh B i 24 cos B r 24 ) B B * ,
ρ = ( cosh B i 24 + cos B r 24 ) 2 ,
M M R M D M 1 C M 2 C ,
X = ( i = 0 n 1 j = 0 n 1 x i j 2 ) 1 2 = [ Tr ( X * X ) ] 1 2 ,
M e = M Δ M ̃ R M ̃ D .
LD = m ̃ D 01 arctanh ( D ) D ,
LD = m ̃ D 02 arctanh ( D ) D ,
CD = m ̃ D 03 arctanh ( D ) D ,
LB = ( m ̃ R 32 m ̃ R 23 ) R 2 sin R ,
LB = ( m ̃ R 13 m ̃ R 31 ) R 2 sin R ,
CB = ( m ̃ R 12 m ̃ R 21 ) R 2 sin R ,
M e ( 1 ) = M e ( M 2 C ( 0 ) ) 1 ( M 1 C ( 0 ) ) 1 .
M e ( i ) = M e ( M 2 C ( i 1 ) ) 1 ( M 1 C ( i 1 ) ) 1 ,
ε ( i ) = M e M Δ ( i ) M ̃ R ( i ) M ̃ D ( i ) M 1 C ( i ) M 2 C ( i ) .
M = e 2 k ( X + ( W 2 ) ( T * T ) U LD V LB + W ( CBLD LB CD ) U LD V LB + W ( LBCD CBLD ) U CD + V CB + W ( LBLD LB LD ) U LD V LB W ( CBLD LB CD ) X + W ( LD 2 + LB 2 ) ( W 2 ) ( T * T ) U CB V CD + W ( LBLB + LDLD ) U LB V LD W ( CBLB + CDLD ) U LD V LB W ( LBCD CBLD ) U CB + V CD + W ( LBLB + LDLD ) X + W ( LB 2 + LD 2 ) ( W 2 ) ( T * T ) U LB + V LD W ( CBLB + CDLD ) U CD + V CB W ( LBLD LB LD ) U LB + V LD W ( CBLB + CDLD ) U LB V LD W ( CBLB + CDLD ) X + W ( CD 2 + CB 2 ) ( W 2 ) ( T * T ) ) ,
U = ( T i sinh T i + T r sin T r ) T T * ,
V = ( T i sin T r T r sinh T i ) T T * ,
W = ( cosh T i cos T r ) T T * ,
X = ( cosh T i + cos T r ) 2 .
M = e 2 k ( 1 + 1 2 ( LD 2 + LD 2 ) LD LD CD + 1 2 ( LBLD LB LD ) LD 1 + 1 2 ( LD 2 LB 2 ) CB + 1 2 ( LBLB + LDLD ) LB LD CB + 1 2 ( LBLB + LDLD ) 1 + 1 2 ( LD 2 LB 2 ) LB CD 1 2 ( LBLB LB LD ) LB LB 1 + 1 2 ( LB 2 + LB 2 ) ) .

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