Abstract

We propose a novel, computationally efficient integral decomposition of depolarizing Mueller matrices allowing for the obtainment of a nondepolarizing estimate, as well as for the determination of the elementary polarization properties, in terms of mean values and variances-covariances of their fluctuations, of a weakly anisotropic depolarizing medium. We illustrate the decomposition on experimental examples and compare its performance to those of alternative decompositions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).
  2. M. Wurm, J. Endres, J. Probst, M. Schoengen, A. Diener, and B. Bodermann, “Metrology of nanoscale grating structures by UV scatterometry,” Opt. Express 25(3), 2460–2468 (2017).
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  3. M. Kupinski, M. Boffety, F. Goudail, R. Ossikovski, A. Pierangelo, J. Rehbinder, J. Vizet, and T. Novikova, “Polarimetric measurement utility for pre-cancer detection from uterine cervix specimens,” Biomed. Opt. Express 9(11), 5691–5702 (2018).
    [Crossref]
  4. R. Ossikovski and O. Arteaga, “Integral decomposition and polarization properties of depolarizing Mueller matrices“,” Opt. Lett. 40(6), 954–957 (2015).
    [Crossref]
  5. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4(3), 433–437 (1987).
    [Crossref]
  6. C. Brosseau, Fundamentals of Polarized Light. A Statistical Optics Approach (Wiley, 1998).
  7. V. Devlaminck, “Physical model of differential Mueller matrix for depolarizing uniform media,” J. Opt. Soc. Am. A 30(11), 2196–2204 (2013).
    [Crossref]
  8. R. Ossikovski and O. Arteaga, “Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media,” Opt. Lett. 39(15), 4470–4473 (2014).
    [Crossref]
  9. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  10. R. Ossikovski and J. J. Gil, “Basic properties and classification of Mueller matrices derived from their statistical definition,” J. Opt. Soc. Am. A 34(9), 1727–1737 (2017).
    [Crossref]
  11. E. Kuntman, M. Ali Kuntman, and O. Arteaga, “Vector and matrix states for Mueller matrices of nondepolarizing optical media,” J. Opt. Soc. Am. A 34(1), 80–86 (2017).
    [Crossref]
  12. O. Arteaga, E. Garcia-Caurel, and R. Ossikovski, “Anisotropy coefficients of a Mueller matrix,” J. Opt. Soc. Am. A 28(4), 548–553 (2011).
    [Crossref]
  13. H. P. Jensen, J. A. Schellman, and T. Troxell, “Modulation techniques in polarization spectroscopy,” Appl. Spectrosc. 32(2), 192–200 (1978).
    [Crossref]
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    [Crossref]
  15. R. Ossikovski, “Retrieval of a nondepolarizing estimate from an experimental Mueller matrix through virtual experiment,” Opt. Lett. 37(4), 578–580 (2012).
    [Crossref]
  16. R. Ossikovski and O. Arteaga, “Instrument-dependent method for obtaining a nondepolarizing estimate from an experimental Mueller matrix,” Opt. Eng. 58(08), 1–6 (2019).
    [Crossref]
  17. R. Ossikovski, “Differential matrix formalism for anisotropic depolarizing media,” Opt. Lett. 36(12), 2330–2332 (2011).
    [Crossref]
  18. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011).
    [Crossref]
  19. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36(13), 2429–2431 (2011).
    [Crossref]
  20. H. D. Noble and R. A. Chipman, “Mueller matrix roots algorithm and computational considerations,” Opt. Express 20(1), 17–31 (2012).
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  21. O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51(28), 6805–6817 (2012).
    [Crossref]
  22. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33(2), 185–189 (1986).
    [Crossref]
  23. S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of the polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
    [Crossref]

2019 (1)

R. Ossikovski and O. Arteaga, “Instrument-dependent method for obtaining a nondepolarizing estimate from an experimental Mueller matrix,” Opt. Eng. 58(08), 1–6 (2019).
[Crossref]

2018 (1)

2017 (4)

2015 (1)

2014 (1)

2013 (1)

2012 (3)

2011 (4)

1996 (1)

1987 (1)

1986 (2)

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

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33(2), 185–189 (1986).
[Crossref]

1978 (1)

Ali Kuntman, M.

Arce-Diego, J. L.

Arteaga, O.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33(2), 185–189 (1986).
[Crossref]

Bodermann, B.

Boffety, M.

Brosseau, C.

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

Chipman, R. A.

Cloude, S. R.

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

Devlaminck, V.

Diener, A.

Endres, J.

Freudenthal, J.

Garcia-Caurel, E.

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of the polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

O. Arteaga, E. Garcia-Caurel, and R. Ossikovski, “Anisotropy coefficients of a Mueller matrix,” J. Opt. Soc. Am. A 28(4), 548–553 (2011).
[Crossref]

Gil, J. J.

R. Ossikovski and J. J. Gil, “Basic properties and classification of Mueller matrices derived from their statistical definition,” J. Opt. Soc. Am. A 34(9), 1727–1737 (2017).
[Crossref]

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33(2), 185–189 (1986).
[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

Goudail, F.

Jensen, H. P.

Kahr, B.

Kim, K.

Kuntman, E.

Kupinski, M.

Lu, S.-Y.

Mandel, L.

Noble, H. D.

Novikova, T.

Ortega-Quijano, N.

Ossikovski, R.

R. Ossikovski and O. Arteaga, “Instrument-dependent method for obtaining a nondepolarizing estimate from an experimental Mueller matrix,” Opt. Eng. 58(08), 1–6 (2019).
[Crossref]

M. Kupinski, M. Boffety, F. Goudail, R. Ossikovski, A. Pierangelo, J. Rehbinder, J. Vizet, and T. Novikova, “Polarimetric measurement utility for pre-cancer detection from uterine cervix specimens,” Biomed. Opt. Express 9(11), 5691–5702 (2018).
[Crossref]

R. Ossikovski and J. J. Gil, “Basic properties and classification of Mueller matrices derived from their statistical definition,” J. Opt. Soc. Am. A 34(9), 1727–1737 (2017).
[Crossref]

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of the polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

R. Ossikovski and O. Arteaga, “Integral decomposition and polarization properties of depolarizing Mueller matrices“,” Opt. Lett. 40(6), 954–957 (2015).
[Crossref]

R. Ossikovski and O. Arteaga, “Statistical meaning of the differential Mueller matrix of depolarizing homogeneous media,” Opt. Lett. 39(15), 4470–4473 (2014).
[Crossref]

R. Ossikovski, “Retrieval of a nondepolarizing estimate from an experimental Mueller matrix through virtual experiment,” Opt. Lett. 37(4), 578–580 (2012).
[Crossref]

O. Arteaga, E. Garcia-Caurel, and R. Ossikovski, “Anisotropy coefficients of a Mueller matrix,” J. Opt. Soc. Am. A 28(4), 548–553 (2011).
[Crossref]

R. Ossikovski, “Differential matrix formalism for anisotropic depolarizing media,” Opt. Lett. 36(12), 2330–2332 (2011).
[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

Pierangelo, A.

Probst, J.

Rehbinder, J.

Schellman, J. A.

Schoengen, M.

Troxell, T.

Vizet, J.

Wang, B.

Wolf, E.

Wurm, M.

Yoo, S. H.

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of the polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Spectrosc. (1)

Appl. Surf. Sci. (1)

S. H. Yoo, R. Ossikovski, and E. Garcia-Caurel, “Experimental study of thickness dependence of the polarization and depolarization properties of anisotropic turbid media using Mueller matrix polarimetry and differential decomposition,” Appl. Surf. Sci. 421, 870–877 (2017).
[Crossref]

Biomed. Opt. Express (1)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33(2), 185–189 (1986).
[Crossref]

Opt. Eng. (1)

R. Ossikovski and O. Arteaga, “Instrument-dependent method for obtaining a nondepolarizing estimate from an experimental Mueller matrix,” Opt. Eng. 58(08), 1–6 (2019).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Optik (1)

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

Other (2)

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

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

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Figures (2)

Fig. 1.
Fig. 1. A single cross (left) and two stacked crosses (right) made of scotch tape strips stuck on a glass slide.
Fig. 2.
Fig. 2. Depolarization index DI (a) and the three elementary birefringence (B) properties, CB (b), LB (c) and LB’ (d), obtained from three different decompositions as functions of the number of crosses.

Equations (21)

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

J=[c0+c1c2ic3c2+ic3c0c1]
C=cc+=cmcm++ΔcΔc+=Cm+ΔC
c=[c0c1c2c3]T=cm+Δc=[c0cm1cm2cm3]T+[0Δc1Δc2Δc3]T
ΔC=[00000|Δc1|2Δc1Δc2Δc1Δc30Δc2Δc1|Δc2|2Δc2Δc30Δc3Δc1Δc3Δc2|Δc3|2]
C=14i,jMijT(σiσj)T1i,j=0,1,2,3
T=[1001100101100ii0]
M=Mm+ΔM
cm=12t[M00+M11+M22+M33M01+M10+i(M23M32)M02+M20i(M13M31)M03+M30+i(M12M21)]=12t[tαAβAγA]
|Δc1|2=14(M00+M11M22M33)|cm1|2|Δc2|2=14(M00M11+M22M33)|cm2|2|Δc3|2=14(M00M11M22+M33)|cm3|2Δc1Δc2=14[M12+M21+i(M03M30)]cm1cm2Δc1Δc3=14[M13+M31i(M02M20)]cm1cm3Δc2Δc3=14[M23+M32+i(M01M10)]cm2cm3
J=a[cosT2iLTsinT2C+iLTsinT2CiLTsinT2cosT2+iLTsinT2]
Jwad=a[1i12L12Ci12L12Ci12L1+i12L]
pm=[LmLmCm]T=2it[αAβAγA]T
ΔPkΔPl=16tΔckΔclk,l=1,2,3
PIk=2PkTsinT2=PksincT2
M(c)=[10.0000.0060.0120.0020.9110.0500.0150.0040.0470.8900.1660.0180.0020.1820.877]
Mm(c)=[0.9280.0010.0020.0150.0010.9270.0480.0110.0000.0490.9110.1740.0150.0020.1740.912]
ΔM(c)=[0.0720.0010.0040.0030.0010.0160.0020.0040.0040.0020.0210.0080.0030.0040.0080.035]
M(1)=[10.0000.0120.0010.0000.9970.0060.0440.0000.0060.9850.0190.0000.0450.0220.979]
M(2)=[10.0000.0110.0010.0000.9450.0140.0910.0000.0110.9350.0450.0000.0900.0470.924]
M(3)=[10.0020.0100.0010.0020.7760.0190.1090.0010.0110.7710.0560.0000.1080.0520.741]
M(4)=[10.0020.0060.0000.0020.5310.0170.0880.0010.0100.5300.0470.0010.0930.0410.506]