Abstract

Conventional generalized ellipsometry instrumentation is capable of measuring 12 out of the 16 elements of the Mueller matrix of the sample. The missing column (or row) of the experimental partial Mueller matrix can be analytically determined under additional assumptions. We identify the conditions necessary for completing the partial Mueller matrix to a full one. More specifically, such a completion is always possible if the sample is nondepolarizing; the fulfillment of additional conditions, such as the Mueller matrix exhibiting symmetries or being of a special two-component structure, are necessary if the sample is depolarizing. We report both algebraic and numerical procedures for completing the partial 12-element Mueller matrix in all tractable cases and validate them on experimental examples.

© 2019 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).
  2. R. A. Chipman, “Polarimetry,” in Handbook of Opics II, M. Bass, ed. (McGraw Hill, 1995), Sections 22.21, 22.22, and 22.26.
  3. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).
  4. H. G. Tomkins and J. N. Hilfiker, Spectroscopic Ellipsometry: Practical Application to Thin Film Characterization (Momentum, 2016).
  5. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [Crossref]
  6. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).
  7. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular value decomposition,” J. Opt. Soc. Am. A 25, 473–482 (2008).
    [Crossref]
  8. 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]
  9. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [Crossref]
  10. J. J. Gil and I. S. José, “Polarimetric subtraction of Mueller matrices,” J. Opt. Soc. Am. A 30, 1078–1088 (2013).
    [Crossref]
  11. J. J. Gil, I. S. José, and R. Ossikovski, “Serial-parallel decompositions of Mueller matrices,” J. Opt. Soc. Am. A 30, 32–50 (2013).
    [Crossref]
  12. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011).
    [Crossref]
  13. D. Goldstein, Polarized Light (Marcel Dekker, 2003).
  14. M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films 313-314, 323–332 (1998).
    [Crossref]
  15. T. Sato, T. Araki, Y. Sasaki, T. Tsuru, T. Tadokoro, and S. Kawakami, “Compact ellipsometer employing a static polarimeter module with arrayed polarizer and wave-plate elements,” Appl. Opt. 46, 4963–4967 (2007).
    [Crossref]
  16. G. E. Jellison and F. A. Modine, “Polarization modulation ellipsometry,” in Handbook of Ellipsometry, H. G. Tomkins and E. A. Irene, eds. (W. Andrew, 2005), Chap. 7.2.3.
  17. R. Ossikovski and O. Arteaga, “Complete Mueller matrix from a partial polarimetry experiment: the nine-element case,” J. Opt. Soc. Am. A36, 403–415 (2019).
    [Crossref]
  18. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
    [Crossref]
  19. E. A. Oberemok and S. N. Savenkov, “Determination of the polarization characteristics of objects by the method of three probing polarizations,” J. Appl. Spectrosc. 69, 72–77 (2002).
    [Crossref]
  20. E. A. Oberemok and S. N. Savenkov, “Structure of deterministic Mueller matrices and their reconstruction in the method of three input polarizations,” J. Appl. Spectrosc. 70, 224–229 (2003).
    [Crossref]
  21. E. A. Oberemok and S. N. Savenkov, “Recovery of the complete Mueller matrix of an arbitrary object in the method of three input polarizations,” J. Appl. Spectrosc. 71, 128–132 (2004).
    [Crossref]
  22. S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
    [Crossref]
  23. S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).
    [Crossref]
  24. J. J. Gil and R. Ossikovski, Polarized Light. The Mueller Matrix Approach (CRC Press, 2016).
  25. R. Ossikovski and J. J. Gil, “Basic properties and classification of Mueller matrices derived from their statistical definition,” J. Opt. Soc. Am. A 34, 1727–1737 (2017).
    [Crossref]
  26. O. Arteaga and A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010); (erratum) 3525.
    [Crossref]
  27. M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. 282, 735–741 (2009).
    [Crossref]
  28. M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
    [Crossref]
  29. R. Ossikovski, E. Garcia-Caurel, M. Foldyna, and J. J. Gil, “Application of the arbitrary decomposition to finite spot size Mueller matrix measurements,” Appl. Opt. 53, 6030–6036 (2014).
    [Crossref]
  30. E. Kuntman and O. Arteaga, “Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions,” Appl. Opt. 55, 2543–2550 (2016).
    [Crossref]
  31. E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
    [Crossref]
  32. M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
    [Crossref]
  33. R. Ossikovski, M. Stchakovsky, M. Kildemo, and M. Mooney, “Incoherent reflection model for spectroscopic ellipsometry of a thick semi-transparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
    [Crossref]
  34. S. Nichols, O. Arteaga, A. Martin, and B. Kahr, “Measurement of transmission and reflection from a thick anisotropic crystal modeled by a sum of incoherent partial waves,” J. Opt. Soc. Am. A 32, 2049–2057 (2015).
    [Crossref]
  35. O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
    [Crossref]
  36. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  37. O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51, 6805–6817 (2012).
    [Crossref]

2017 (2)

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

E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
[Crossref]

2016 (1)

2015 (1)

2014 (2)

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

R. Ossikovski, E. Garcia-Caurel, M. Foldyna, and J. J. Gil, “Application of the arbitrary decomposition to finite spot size Mueller matrix measurements,” Appl. Opt. 53, 6030–6036 (2014).
[Crossref]

2013 (2)

2012 (1)

2011 (2)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

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

2010 (1)

2009 (3)

2008 (1)

2007 (1)

2004 (1)

E. A. Oberemok and S. N. Savenkov, “Recovery of the complete Mueller matrix of an arbitrary object in the method of three input polarizations,” J. Appl. Spectrosc. 71, 128–132 (2004).
[Crossref]

2003 (1)

E. A. Oberemok and S. N. Savenkov, “Structure of deterministic Mueller matrices and their reconstruction in the method of three input polarizations,” J. Appl. Spectrosc. 70, 224–229 (2003).
[Crossref]

2002 (2)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Determination of the polarization characteristics of objects by the method of three probing polarizations,” J. Appl. Spectrosc. 69, 72–77 (2002).
[Crossref]

2000 (1)

1998 (2)

M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
[Crossref]

M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films 313-314, 323–332 (1998).
[Crossref]

1996 (1)

1989 (1)

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

1987 (1)

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

1980 (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[Crossref]

Araki, T.

Arteaga, O.

E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
[Crossref]

E. Kuntman and O. Arteaga, “Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions,” Appl. Opt. 55, 2543–2550 (2016).
[Crossref]

S. Nichols, O. Arteaga, A. Martin, and B. Kahr, “Measurement of transmission and reflection from a thick anisotropic crystal modeled by a sum of incoherent partial waves,” J. Opt. Soc. Am. A 32, 2049–2057 (2015).
[Crossref]

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51, 6805–6817 (2012).
[Crossref]

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010); (erratum) 3525.
[Crossref]

R. Ossikovski and O. Arteaga, “Complete Mueller matrix from a partial polarimetry experiment: the nine-element case,” J. Opt. Soc. Am. A36, 403–415 (2019).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

Bernabeu, E.

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

Canillas, A.

E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
[Crossref]

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

O. Arteaga and A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010); (erratum) 3525.
[Crossref]

Chipman, R. A.

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]

R. A. Chipman, “Polarimetry,” in Handbook of Opics II, M. Bass, ed. (McGraw Hill, 1995), Sections 22.21, 22.22, and 22.26.

Cloude, S. R.

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

De Martino, A.

M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. 282, 735–741 (2009).
[Crossref]

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[Crossref]

Foldyna, M.

Freudenthal, J.

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration,” Appl. Opt. 51, 6805–6817 (2012).
[Crossref]

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

Garcia-Caurel, E.

Gil, J. J.

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, 2003).

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[Crossref]

Hilfiker, J. N.

H. G. Tomkins and J. N. Hilfiker, Spectroscopic Ellipsometry: Practical Application to Thin Film Characterization (Momentum, 2016).

Jellison, G. E.

G. E. Jellison and F. A. Modine, “Polarization modulation ellipsometry,” in Handbook of Ellipsometry, H. G. Tomkins and E. A. Irene, eds. (W. Andrew, 2005), Chap. 7.2.3.

José, I. S.

Kahr, B.

Kawakami, S.

Kildemo, M.

R. Ossikovski, M. Stchakovsky, M. Kildemo, and M. Mooney, “Incoherent reflection model for spectroscopic ellipsometry of a thick semi-transparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[Crossref]

M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
[Crossref]

Klimov, A.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

Kuntman, E.

E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
[Crossref]

E. Kuntman and O. Arteaga, “Decomposition of a depolarizing Mueller matrix into its nondepolarizing components by using symmetry conditions,” Appl. Opt. 55, 2543–2550 (2016).
[Crossref]

Licitra, C.

M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. 282, 735–741 (2009).
[Crossref]

Lu, S.-Y.

Martin, A.

Modine, F. A.

G. E. Jellison and F. A. Modine, “Polarization modulation ellipsometry,” in Handbook of Ellipsometry, H. G. Tomkins and E. A. Irene, eds. (W. Andrew, 2005), Chap. 7.2.3.

Mooney, M.

Muttiah, R.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

Nichols, S.

S. Nichols, O. Arteaga, A. Martin, and B. Kahr, “Measurement of transmission and reflection from a thick anisotropic crystal modeled by a sum of incoherent partial waves,” J. Opt. Soc. Am. A 32, 2049–2057 (2015).
[Crossref]

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

Oberemok, E.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

Oberemok, E. A.

E. A. Oberemok and S. N. Savenkov, “Recovery of the complete Mueller matrix of an arbitrary object in the method of three input polarizations,” J. Appl. Spectrosc. 71, 128–132 (2004).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Structure of deterministic Mueller matrices and their reconstruction in the method of three input polarizations,” J. Appl. Spectrosc. 70, 224–229 (2003).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Determination of the polarization characteristics of objects by the method of three probing polarizations,” J. Appl. Spectrosc. 69, 72–77 (2002).
[Crossref]

Ossikovski, R.

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

R. Ossikovski, E. Garcia-Caurel, M. Foldyna, and J. J. Gil, “Application of the arbitrary decomposition to finite spot size Mueller matrix measurements,” Appl. Opt. 53, 6030–6036 (2014).
[Crossref]

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

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

M. Foldyna, E. Garcia-Caurel, R. Ossikovski, A. De Martino, and J. J. Gil, “Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices,” Opt. Express 17, 12794–12806 (2009).
[Crossref]

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

M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. 282, 735–741 (2009).
[Crossref]

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. Stchakovsky, M. Kildemo, and M. Mooney, “Incoherent reflection model for spectroscopic ellipsometry of a thick semi-transparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[Crossref]

M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
[Crossref]

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

R. Ossikovski and O. Arteaga, “Complete Mueller matrix from a partial polarimetry experiment: the nine-element case,” J. Opt. Soc. Am. A36, 403–415 (2019).
[Crossref]

Sasaki, Y.

Sato, T.

Savenkov, S.

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

Savenkov, S. N.

E. A. Oberemok and S. N. Savenkov, “Recovery of the complete Mueller matrix of an arbitrary object in the method of three input polarizations,” J. Appl. Spectrosc. 71, 128–132 (2004).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Structure of deterministic Mueller matrices and their reconstruction in the method of three input polarizations,” J. Appl. Spectrosc. 70, 224–229 (2003).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Determination of the polarization characteristics of objects by the method of three probing polarizations,” J. Appl. Spectrosc. 69, 72–77 (2002).
[Crossref]

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[Crossref]

Schubert, M.

M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films 313-314, 323–332 (1998).
[Crossref]

Stchakovsky, M.

R. Ossikovski, M. Stchakovsky, M. Kildemo, and M. Mooney, “Incoherent reflection model for spectroscopic ellipsometry of a thick semi-transparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000).
[Crossref]

M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
[Crossref]

Tadokoro, T.

Tomkins, H. G.

H. G. Tomkins and J. N. Hilfiker, Spectroscopic Ellipsometry: Practical Application to Thin Film Characterization (Momentum, 2016).

Tsuru, T.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Wang, B.

Appl. Opt. (5)

Appl. Surf. Sci. (1)

E. Kuntman, A. Canillas, and O. Arteaga, “Retrieval of nondepolarizing components of depolarizing Mueller matrices by using symmetry conditions and least squares minimization,” Appl. Surf. Sci. 421, 697–701 (2017).
[Crossref]

J. Appl. Spectrosc. (3)

E. A. Oberemok and S. N. Savenkov, “Determination of the polarization characteristics of objects by the method of three probing polarizations,” J. Appl. Spectrosc. 69, 72–77 (2002).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Structure of deterministic Mueller matrices and their reconstruction in the method of three input polarizations,” J. Appl. Spectrosc. 70, 224–229 (2003).
[Crossref]

E. A. Oberemok and S. N. Savenkov, “Recovery of the complete Mueller matrix of an arbitrary object in the method of three input polarizations,” J. Appl. Spectrosc. 71, 128–132 (2004).
[Crossref]

J. Opt. Soc. Am. A (7)

J. Quant. Spectrosc. Radiat. Transfer (1)

S. Savenkov, R. Muttiah, E. Oberemok, and A. Klimov, “Incomplete Mueller polarimetry: measurement of the block-diagonal scattering matrix,” J. Quant. Spectrosc. Radiat. Transfer 112, 1796–1802 (2011).
[Crossref]

Opt. Commun. (1)

M. Foldyna, A. De Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. 282, 735–741 (2009).
[Crossref]

Opt. Eng. (1)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Optik (1)

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

Proc. SPIE (1)

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

Surf. Sci. (1)

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[Crossref]

Thin Solid Films (3)

M. Schubert, “Generalized ellipsometry and complex optical systems,” Thin Solid Films 313-314, 323–332 (1998).
[Crossref]

M. Kildemo, R. Ossikovski, and M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near-IR ellipsometry,” Thin Solid Films 313-314, 108–113 (1998).
[Crossref]

O. Arteaga, J. Freudenthal, S. Nichols, A. Canillas, and B. Kahr, “Transmission ellipsometry of anisotropic substrates and thin films at oblique incidence. Handling multiple reflections,” Thin Solid Films 571, 701–705 (2014).
[Crossref]

Other (9)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

G. E. Jellison and F. A. Modine, “Polarization modulation ellipsometry,” in Handbook of Ellipsometry, H. G. Tomkins and E. A. Irene, eds. (W. Andrew, 2005), Chap. 7.2.3.

R. Ossikovski and O. Arteaga, “Complete Mueller matrix from a partial polarimetry experiment: the nine-element case,” J. Opt. Soc. Am. A36, 403–415 (2019).
[Crossref]

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

D. Goldstein, Polarized Light (Marcel Dekker, 2003).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1987).

R. A. Chipman, “Polarimetry,” in Handbook of Opics II, M. Bass, ed. (McGraw Hill, 1995), Sections 22.21, 22.22, and 22.26.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

H. G. Tomkins and J. N. Hilfiker, Spectroscopic Ellipsometry: Practical Application to Thin Film Characterization (Momentum, 2016).

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Figures (4)

Fig. 1.
Fig. 1. Complete Mueller matrix of the diffraction grating (solid line) and the recovery of its last column from the 12-element partial one by using the algebraic procedure from Appendix A (red crosses) and the numerical approach (green circles).
Fig. 2.
Fig. 2. Complete Mueller matrix of the mica slab (solid line) and the recovery of its M44 element from the 12-element partial one by using the algebraic procedure from Appendix B (red crosses) and the numerical approach (green circles). The elements M14, M24, and M34 have been recovered by exploiting the symmetries of the sample.
Fig. 3.
Fig. 3. Complete Mueller matrix of the grating–substrate mixture (solid line) and the recovery of its last column from the 12-element partial one by using the algebraic procedure from Appendix C (red crosses) and the numerical approach (green circles).
Fig. 4.
Fig. 4. Complete Mueller matrix of the grating–substrate mixture (solid line) and the recovery of its M44 element from the 12-element partial one by using the algebraic procedure from Appendix B (red crosses) and the numerical approach (green circles). The elements M14, M24, and M34 have been recovered by exploiting the symmetries of the sample.

Tables (1)

Tables Icon

Table 1. Instruments and 12-Element Partial Mueller Matrices They Measurea

Equations (57)

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[M11M12M13M21M22M23M31M32M33M41M42M43]
[M11M12M13M14M21M22M23M24M31M32M33M34]
θm=0°(90°)[M11M12M13M31M32M33M41M42M43]&θm=±45°[M11M12M13M21M22M23M41M42M43][M11M12M13M21M22M23M31M32M33M41M42M43]
θm=0°(90°)[M11M13M14M21M23M24M31M33M34]&θm=±45°[M11M12M14M21M22M24M31M32M34][M11M12M13M14M21M22M23M24M31M32M33M34]
J=[J1J3J4J2],
Ei=|Ji|2,
Fij=Fji=ReJi*Jj,
Gij=Gji=ImJi*Jj
M=[12(E1+E2+E3+E4)12(E1E2E3+E4)F13+F42G13G4212(E1E2+E3E4)12(E1+E2E3E4)F13F42G13+G42F14+F32F14F32F12+F34G12+G34G14+G32G14G32G12+G34F12F34],
H=14i,jMij(σiσj)=12[E1F13iG13F14iG14F12iG12F13+iG13E3F34iG34F32iG32F14+iG14F34+iG34E4F42iG42F12+iG12F32+iG32F42+iG42E2],
[M11M12M13M14M12M22M23M24M13M23M33M34M14M24M34M44].
[M11M12M13M14M12M22M23M24M13M23M33M34M14M24M34M44].
Mbd=[M11M1200M12M110000M33M3400M34M33].
M14=G13G42=2Im(H12+H34),
M24=G13+G42=2Im(H12H34),
M34=G12+G34=2Im(H14H23),
M44=F12F34=2Re(H14H23).
H14+H23=12(F12iG12)+12(F34iG34)=12(F12+F34)i12(G12+iG34)=12(M33iM43).
H(34)(34)=|H11H12H21H22|=H11H22H12H21=H11H22|H12|2=H11H22(ReH12)2(ImH12)2=0
H(12)(12)=|H33H34H43H44|=H33H44H34H43=H33H44|H34|2=H33H44(ReH34)2(ImH34)2=0,
ReH12=12F13=14(M13+M23)
ReH34=12F42=14(M13M23)
H(23)(14)=|H12H13H42H43|=H12H43H13H42=H12H34*H13H24*=0
H13=12(F14iG14)=14[M31+M32i(M41+M42)]
H24=12(F32iG32)=14[M31M32i(M41M42)]
H(13)(12)=|H23H24H43H44|=H23H44H24H43=H23H44H24H34*=0
H(34)(24)=|H11H13H21H23|=H11H23H13H21=H11H23H13H12*=0,
H23=H24H34*H44=H13H12*H11=H24H34*+H13H12*H44+H11,
H44=12E2=14(M11+M22M12M21)
H11=12E1=14(M11+M22+M12+M21).
M44=F12F34=2(ReH14ReH23)
ReH14+ReH23=12(F12+F34)=12M33.
H(1)=|H22H23H24H32H33H34H42H43H44|=H22H33H44+H24*H34H23+H24H34*H23*H22|H34|2H33|H24|2H44|H23|2=0
H(4)=|H11H12H13H21H22H23H31H32H33|=H11H22H33+H13*H12H23+H13H12*H23*H33|H12|2H22|H13|2H11|H23|2=0,
H11H(1)H44H(4)=(H11H24*H34H44H13*H12)H23+(H11H24H34*H44H13H12*)H23*H11(H22|H34|2+H33|H24|2)+H44(H11|H23|2+H22|H13|2)=0.
ReH23=A+2Im(H11H24*H34H44H13*H12)ImH232Re(H11H24*H34H44H13*H12),
A=H11(H22|H34|2+H33|H24|2)H44(H11|H23|2+H22|H13|2).
H11=12E1=14(M11+M22+M12+M21),
H22=12E3=14(M11M22M12+M21),
H33=12E4=14(M11M22+M12M21),
H44=12E2=14(M11+M22M12M21),
H13=12(F14iG14)=14[M31+M32i(M41+M42)],
H24=12(F32iG32)=14[M31M32i(M41M42)],
H12=12(F13iG13)=14[M13+M23+i(M14+M24)],
H34=12(F42iG42)=14[M13M23+i(M14M24)],
ImH23=12G34=14(M34+M43).
H(1)+H(4)=H22H33(H11+H44)+(H13*H12+H24*H34)H23+(H13H12*+H24H34*)H23*H33(|H12|2+|H24|2)H22(|H13|2+|H34|2)(H11+H44)|H23|2=0,
H(2)+H(3)=H11H44(H22+H33)+(H13*H34*+H12*H24*)H14+(H13H34+H12H24)H14*H11(|H34|2+|H24|2)H44(|H12|2+|H13|2)(H22+H33)|H14|2=0,
ImH14=12G12=14(M34M43).
H=H1+Hbd=[H11H12H13H14H21H22H23H24H31H32H33H34H41H42H43H44]+[H1100H1400000000H4100H44],
H1(24)(24)=|H11H13H31H33|=H11H33H13H31=H11H33|H13|2=0
H1(13)(13)=|H22H24H42H44|=H22H44H24H42=H22H44|H24|2=0,
H1(34)(34)=|H11H12H21H22|=H11H22H12H21=H11H22|H12|2=H11H22(ReH12)2(ImH12)2=0
H1(12)(12)=|H33H34H43H44|=H33H44H34H43=H33H44|H34|2=H33H44(ReH34)2(ImH34)2=0,
H1(13)(12)=|H23H24H43H44|=H23H44H24H43=H23H44H24H34*=0
H1(34)(24)=|H11H13H21H23|=H11H23H13H21=H11H23H13H12*=0,
H23=H24H34*H44=H13H12*H11=H24H34*+H13H12*H44+H11.

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