Abstract

The differential Mueller matrix is an important concept for analyzing the polarization properties of an optically homogeneous anisotropic sample, both nondepolarizing and depolarizing. In this work, we present a new method of interpreting Mueller matrix of anisotropic medium based on the relationships that exist between the components of a differential Mueller matrix and the polar components of the corresponding macroscopic Mueller matrix, and the necessary conditions are determined that guarantee the physical realizability of the generating matrices. Finally, a group of the experimental data of a sample from the literature with some known polarization properties was used to demonstrate the analysis. The work is helpful for obtaining new insights or new interpretations of the measured Mueller matrix of the medium.

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

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  1. 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(1), 190–202 (2006).
    [Crossref] [PubMed]
  2. S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
    [Crossref] [PubMed]
  3. S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
    [Crossref] [PubMed]
  4. W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
    [Crossref] [PubMed]
  5. B. Jirgensons, Optical Rotatory Dispersion of Proteins and Other Macromolecules, (Springer-Verlag,1960).
  6. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
    [Crossref] [PubMed]
  7. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. 40(22), 3810–3821 (2001).
    [Crossref] [PubMed]
  8. N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
    [Crossref] [PubMed]
  9. S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
    [Crossref] [PubMed]
  10. A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
    [Crossref] [PubMed]
  11. C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
    [Crossref] [PubMed]
  12. J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography - a review [Invited],” Biomed. Opt. Express 8(3), 1838–1873 (2017).
    [Crossref] [PubMed]
  13. R. A. Chipman, Handbook of Optics, (2nded., M. Bass, Ed.), Chap. 22.
  14. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, (Wiley, 1998).
  15. 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]
  16. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26(5), 1109–1118 (2009).
    [Crossref] [PubMed]
  17. 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]
  18. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36(12), 2330–2332 (2011).
    [Crossref] [PubMed]
  19. N. Ortega-Quijano and J. L. Arce-Diego, “Mueller matrix differential decomposition,” Opt. Lett. 36(10), 1942–1944 (2011).
    [Crossref] [PubMed]
  20. 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 (Stuttg.) 76, 67–71 (1987).
  21. 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]
  22. T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
    [Crossref] [PubMed]
  23. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4matrix calculus,” J. Opt. Soc. Am. 68(12), 1756–1767 (1978).
    [Crossref]
  24. N. Ortega-Quijano and J. L. Arce-Diego, “Depolarizing differential Mueller matrices,” Opt. Lett. 36(13), 2429–2431 (2011).
    [Crossref] [PubMed]
  25. V. Devlaminck, “Physical model of differential Mueller matrix for depolarizing uniform media,” J. Opt. Soc. Am. A 30(11), 2196–2204 (2013).
    [Crossref] [PubMed]
  26. T. A. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37(5), 921–923 (2012).
    [Crossref] [PubMed]
  27. M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39(7), 1779–1782 (2014).
    [Crossref] [PubMed]
  28. S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
    [Crossref] [PubMed]
  29. J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
    [Crossref] [PubMed]
  30. N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
    [Crossref] [PubMed]
  31. N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
    [Crossref]
  32. X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
    [Crossref] [PubMed]
  33. N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
    [Crossref] [PubMed]
  34. N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
    [Crossref] [PubMed]
  35. R. Ossikovski and V. Devlaminck, “General criterion for the physical realizability of the differential Mueller matrix,” Opt. Lett. 39(5), 1216–1219 (2014).
    [Crossref] [PubMed]
  36. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttg.) 75, 26 (1986).

2017 (2)

J. F. de Boer, C. K. Hitzenberger, and Y. Yasuno, “Polarization sensitive optical coherence tomography - a review [Invited],” Biomed. Opt. Express 8(3), 1838–1873 (2017).
[Crossref] [PubMed]

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

2014 (2)

2013 (1)

2012 (3)

T. A. Germer, “Realizable differential matrices for depolarizing media,” Opt. Lett. 37(5), 921–923 (2012).
[Crossref] [PubMed]

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

2011 (4)

2010 (1)

2009 (3)

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

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

2008 (1)

N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

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(1), 190–202 (2006).
[Crossref] [PubMed]

2005 (1)

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

2004 (2)

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

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]

2003 (1)

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

2002 (1)

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

2001 (2)

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid monte carlo diffusion model,” Appl. Opt. 40(22), 3810–3821 (2001).
[Crossref] [PubMed]

2000 (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

1999 (1)

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

1996 (1)

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 (Stuttg.) 76, 67–71 (1987).

1986 (1)

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

1978 (1)

Alexandrakis, G.

Arce-Diego, J. L.

Azzam, R. M. A.

Backman, V.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Badizadegan, K.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

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 (Stuttg.) 76, 67–71 (1987).

Bouma, B. E.

Bouma, G. J.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Buddhiwant, P.

Busch, D. R.

Butler, J.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Cerussi, A. E.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Chipman, R. A.

Cloude, S. R.

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

Darling, C. L.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Dasari, R. R.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

de Boer, J. F.

De Martino, A.

Devlaminck, V.

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Elson, D. S.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

Faris, G. W.

Feld, M. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Fried, D.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Georgakoudi, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Germer, T. A.

Ghosh, N.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
[Crossref] [PubMed]

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(1), 190–202 (2006).
[Crossref] [PubMed]

Gil, J. J.

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 (Stuttg.) 76, 67–71 (1987).

Goudail, F.

Groner, W.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Guo, X.

Gupta, P. K.

Gurjar, R. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Guyot, S.

Hanna, C. F.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Harris, A. G.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Hitzenberger, C. K.

Hsiang, D.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Huynh, G. D.

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

Ince, C.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Itzkan, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Jacques, S. L.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Jakubowski, D.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Kantor, S.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Khalil, O. S.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Kumar, S.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

Lee, K.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Li, R.-K.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Li, S. H.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Lo, Y.-L.

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

Lu, S. Y.

Manhas, S.

Messmer, K.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Morio, J.

Nadeau, R. G.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Ortega-Quijano, N.

Ossikovski, R.

Patterson, M. S.

Perelman, L. T.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Pham, T. T.

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

Purwar, H.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

Qi, J.

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

Ramella-Roman, J. C.

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

Roman, J. R.

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Shah, N.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Singh, J.

Swami, M. K.

Tromberg, B. J.

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

Villiger, M.

Vitkin, I. A.

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
[Crossref] [PubMed]

Weisel, R. D.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Wilson, B. C.

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

Winkelman, J. W.

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

Wood, M. F. G.

X. Guo, M. F. G. Wood, N. Ghosh, and I. A. Vitkin, “Depolarization of light in turbid media: a scattering event resolved Monte Carlo study,” Appl. Opt. 49(2), 153–162 (2010).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
[Crossref] [PubMed]

Yasuno, Y.

Yeh, S. J.

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

Zhu, T. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Appl. Opt. (2)

Biomed. Opt. Express (1)

J. Appl. Phys. (1)

N. Ghosh, M. F. G. Wood, and I. A. Vitkin, “Polarimetry in turbid, birefringent, optically active media: A Monte Carlo study of Mueller matrix decomposition in the backscattering geometry,” J. Appl. Phys. 105(10), 102023 (2009).
[Crossref]

J. Biomed. Opt. (8)

N. Ghosh and I. A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16(11), 110801 (2011).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, and I. A. 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(4), 044036 (2008).
[Crossref] [PubMed]

C. L. Darling, G. D. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310 nm,” J. Biomed. Opt. 11(3), 034023 (2006).
[Crossref] [PubMed]

T. T. Pham and Y.-L. Lo, “Extraction of effective parameters of turbid media utilizing the Mueller matrix approach: study of glucose sensing,” J. Biomed. Opt. 17(9), 97002 (2012).
[Crossref] [PubMed]

S. Kumar, H. Purwar, R. Ossikovski, I. A. Vitkin, and N. Ghosh, “Comparative study of differential matrix and extended polar decomposition formalisms for polarimetric characterization of complex tissue-like turbid media,” J. Biomed. Opt. 17(10), 105006 (2012).
[Crossref] [PubMed]

N. Shah, A. E. Cerussi, D. Jakubowski, D. Hsiang, J. Butler, and B. J. Tromberg, “Spatial variations in optical and physiological properties of healthy breast tissue,” J. Biomed. Opt. 9(3), 534–540 (2004).
[Crossref] [PubMed]

S. J. Yeh, O. S. Khalil, C. F. Hanna, and S. Kantor, “Near-infrared thermo-optical response of the localized reflectance of intact diabetic and nondiabetic human skin,” J. Biomed. Opt. 8(3), 534–544 (2003).
[Crossref] [PubMed]

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7(3), 329–340 (2002).
[Crossref] [PubMed]

J. Biophotonics (2)

J. Qi and D. S. Elson, “Mueller polarimetric imaging for surgical and diagnostic applications: a review,” J. Biophotonics 10(8), 950–982 (2017).
[Crossref] [PubMed]

N. Ghosh, M. F. G. Wood, S. H. Li, R. D. Weisel, B. C. Wilson, R.-K. Li, and I. A. Vitkin, “Mueller matrix decomposition for polarized light assessment of biological tissues,” J. Biophotonics 2(3), 145–156 (2009).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Lasers Surg. Med. (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Nat. Med. (2)

W. Groner, J. W. Winkelman, A. G. Harris, C. Ince, G. J. Bouma, K. Messmer, and R. G. Nadeau, “Orthogonal polarization spectral imaging: a new method for study of the microcirculation,” Nat. Med. 5(10), 1209–1212 (1999).
[Crossref] [PubMed]

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med. 7(11), 1245–1248 (2001).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (8)

Optik (Stuttg.) (2)

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

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 (Stuttg.) 76, 67–71 (1987).

Phys. Med. Biol. (1)

A. Dimofte, J. C. Finlay, and T. C. Zhu, “A method for determination of the absorption and scattering properties interstitially in turbid media,” Phys. Med. Biol. 50(10), 2291–2311 (2005).
[Crossref] [PubMed]

Other (3)

B. Jirgensons, Optical Rotatory Dispersion of Proteins and Other Macromolecules, (Springer-Verlag,1960).

R. A. Chipman, Handbook of Optics, (2nded., M. Bass, Ed.), Chap. 22.

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

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Figures (5)

Fig. 1
Fig. 1 Relationships between M R and m d e t .
Fig. 2
Fig. 2 Relationships between M D and m d e t .
Fig. 3
Fig. 3 Relationships between M Δ and m d e p .
Fig. 4
Fig. 4 The flow chart showing the outcome parameters and coefficients of the polar decomposition method. The coefficients in the red dotted blocks represent three different polarization properties.
Fig. 5
Fig. 5 The flow chart showing the outcome parameters and coefficients of our method based on the relationships of the components between the polar decomposition and the differential Mueller matrix method. The coefficients in black blocks can be derived or obtained from the yellow blocks using parameters of the differential Mueller matrix.

Tables (3)

Tables Icon

Table 1 Relationships between M R and m d e t

Tables Icon

Table 2 Relationships between M D and m d e t

Tables Icon

Table 3 Relationships between M Δ and m d e p

Equations (64)

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M = M R M D .
M = M L R M C R M D ,
d M d z = d M L R d z M C R M D + M L R d M C R d z M D + M L R M C R d M D d z .
m M = ( m L R M L R ) M C R M D + M L R ( m C R M C R ) M D + M L R M C R ( m D M D ) ,
M | z 0 = 0 = M L R | z 0 = 0 = M C R | z 0 = 0 = M D | z 0 = 0 = I .
m | z 0 = 0 = m L R + m C R + m D .
m i n i t i a l = [ α β γ δ β α μ ν γ μ α η δ ν η α ] ,
m = [ 0 β γ δ β 0 μ ν γ μ 0 η δ ν η 0 ] .
M = W d i a g ( exp ( σ 0 z ) , exp ( σ 1 z ) , exp ( σ 2 z ) , exp ( σ 3 z ) ) W - 1 ,
m L R = [ 0 0 0 0 0 0 0 ν 0 0 0 η 0 ν η 0 ] .
M L R ( η , ν ) = [ 1 0 0 0 0 η 2 + ν 2 cos ( i η 2 ν 2 ) η 2 + ν 2 η ν ( cos ( i η 2 ν 2 ) 1 ) η 2 + ν 2 ν i sin ( i η 2 ν 2 ) η 2 ν 2 0 η ν ( cos ( i η 2 ν 2 ) 1 ) η 2 + ν 2 ν 2 + η 2 cos ( i η 2 ν 2 ) η 2 + ν 2 η i sin ( i η 2 ν 2 ) η 2 ν 2 0 ν i sin ( i η 2 ν 2 ) η 2 ν 2 η i sin ( i η 2 ν 2 ) η 2 ν 2 cos ( i η 2 v 2 ) ] .
M L R ( φ , θ ) = [ 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos φ cos 2 θ sin 2 θ ( 1 cos φ ) sin 2 θ sin φ 0 cos 2 θ sin 2 θ ( 1 cos φ ) sin 2 2 θ + cos 2 2 θ cos φ cos 2 θ sin φ 0 sin 2 θ sin φ cos 2 θ sin φ cos φ ] ,
φ = i η 2 v 2 , cos 2 θ = η i η 2 ν 2 , sin 2 θ = ν i η 2 ν 2 .
m C R = [ 0 0 0 0 0 0 μ 0 0 μ 0 0 0 0 0 0 ] .
M C R ( μ ) = [ 1 0 0 0 0 cos μ sin μ 0 0 sin μ cos μ 0 0 0 0 1 ] .
M C R ( δ c ) = [ 1 0 0 0 0 cos δ c sin δ c 0 0 sin δ c cos δ c 0 0 0 0 1 ] .
δ c = μ ,
R = cos 1 ( t r ( M R ) 2 1 ) = cos - 1 ( ( cos μ + 1 ) ( 1 + cos ( i η 2 ν 2 ) ) 2 1 ) ,
m D = [ 0 β γ δ β 0 0 0 γ 0 0 0 δ 0 0 0 ] .
M D = T u [ 1 D T D m 3 ] ,
m 3 = a 1 I + b 1 ( D ^ D ^ T ) = ( 1 | D | 2 ) 1 / 2 I + [ 1 ( 1 | D | 2 ) 1 / 2 ] D ^ D ^ T ,
M D = [ 1 B 12 B 13 B 14 B 12 c + b B 12 2 a b B 12 B 13 a b B 12 B 14 a B 13 b B 12 B 13 a c + b B 13 2 a b B 13 B 14 a B 14 b B 12 B 14 a b B 13 B 14 a c + b B 14 2 a ] ,
D = [ D H D 45 D C ] = [ B 12 B 13 B 14 ] ,
D L = D H 2 + D 45 2 = ( β 2 γ 2 ) tan ( i β 2 + δ 2 + γ 2 ) β 2 + δ 2 + γ 2 = B 12 2 + B 13 2 .
D C = B 14 = δ i tan ( i β 2 + δ 2 + γ 2 ) β 2 + δ 2 + γ 2 .
D = D H 2 + D 45 2 + D C 2 = D L 2 + D C 2 = | D | = ± i tan ( i β 2 + δ 2 + γ 2 ) .
M = M Δ M R M D ,
M Δ = [ 1 0 0 0 p 1 e 1 0 0 p 2 0 e 2 0 p 3 0 0 e 3 ] ,
M = M Δ M L R M C R M D = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η M L R M C R M D ,
M Δ μ ν η = M Δ μ M Δ ν η = M Δ ν η M Δ μ ,
M = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η M L R M C R M D = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) M L R M C R M D .
d M d z = d M Δ α 1 α 2 α 3 , β γ δ d z M Δ - β - γ - δ M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ d M Δ - β - γ - δ d z M Δ - μ ν η M L R M C R M D M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ d ( M Δ μ ) d z M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ d ( M Δ ν η ) d z M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η d M L R d z M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R d M C R d z M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R M C R d M D d z .
m M = ( m Δ α 1 α 2 α 3 , β γ δ M Δ α 1 α 2 α 3 , β γ δ ) M Δ - β - γ - δ M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ ( m Δ - β - γ - δ M Δ - β - γ - δ ) M Δ - μ ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( m Δ - μ M Δ - μ ) M Δ - ν η M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ( m Δ - ν η M Δ - ν η ) M L R M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η ( m L R M L R ) M C R M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R ( m C R M C R ) M D + M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ - μ ν η M L R M C R ( m D M D ) .
M | z 0 = 0 = M Δ α 1 α 2 α 3 , β γ δ | z 0 = 0 = M Δ β γ δ | z 0 = 0 = M Δ - μ | z 0 = 0 = M Δ - ν η | z 0 = 0 = M L R | z 0 = 0 = M C R | z 0 = 0 = M D | z 0 = 0 = I .
m | z 0 = 0 = ( m Δ α 1 α 2 α 3 , β γ δ + m Δ β γ δ + m Δ - μ + m Δ - ν η ) + ( m L R + m C R + m D ) = m d e p + m d e t ,
m d e p = [ 0 β ' γ ' δ ' β ' α 1 μ ' ν ' γ ' μ ' α 2 η ' δ ' ν ' η ' α 3 ] = m Δ α 1 α 2 α 3 , β γ δ + m Δ β γ δ + m Δ - μ + m Δ - ν η ,
m Δ α 1 α 2 α 3 , β γ δ = [ 0 0 0 0 β ' α 1 0 0 γ ' 0 α 2 0 δ ' 0 0 α 3 ] , m Δ - μ = [ 0 0 0 0 0 0 μ ' 0 0 μ ' 0 0 0 0 0 0 ] , m Δ - ν η = [ 0 0 0 0 0 0 0 ν ' 0 0 0 η ' 0 ν ' η ' 0 ] .
M Δ - α 1 α 2 α 3 , β γ δ = [ 1 0 0 0 β ' ( exp ( α 1 ) 1 ) α 1 exp ( α 1 ) 0 0 γ ' ( exp ( α 2 ) 1 ) α 2 0 exp ( α 2 ) 0 δ ' ( exp ( α 3 ) 1 ) α 3 0 0 exp ( α 3 ) ] , M Δ - μ = [ 1 0 0 0 0 cos ( i μ ' ) i sin ( i μ ' ) 0 0 i sin ( i μ ' ) cos ( i μ ' ) 0 0 0 0 1 ] .
M Δ - ν η = [ 1 0 0 0 0 η ' 2 + ν ' 2 cos ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' ν ' ( cos ( i η ' 2 + ν ' 2 ) 1 ) η ' 2 + ν ' 2 ν ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 0 η ' ν ' ( cos ( i η ' 2 + ν ' 2 ) 1 ) η ' 2 + ν ' 2 ν ' 2 + η ' 2 cos ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 0 ν ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 η ' i sin ( i η ' 2 + ν ' 2 ) η ' 2 + ν ' 2 cos ( i η ' 2 + ν ' 2 ) ] .
m Δ β γ δ = [ 0 β ' γ ' δ ' 0 0 0 0 0 0 0 0 0 0 0 0 ] ,
d S d z = m S ,
d S 0 / d z = β ' S 1 γ ' S 2 δ ' S 3 d S 1 / d z = 0 d S 2 / d z = 0 d S 3 / d z = 0.
S 0 = S 0 0 ( β ' S 1 + γ ' S 2 + δ ' S 3 ) z ,
M Δ = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ M Δ μ ν η = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) = [ 1 0 0 0 β ' ( exp α 1 1 ) α 1 exp α 1 0 0 γ ' ( exp α 2 1 ) α 2 0 exp α 2 0 δ ' ( exp α 3 1 ) α 3 0 0 exp α 3 ] [ 1 0 0 0 0 cos ( i μ ' ) i sin ( i μ ' ) 0 0 i sin ( i μ ' ) cos ( i μ ' ) 0 0 0 0 1 ] [ 1 0 0 0 0 η ' 2 + ν ' 2 cos ( i z ) z η ' ν ' ( cos ( i z ) 1 ) η ' 2 + ν ' 2 ν ' i sin ( i z ) z 0 η ' ν ' ( cos ( i z ) 1 ) z ν ' 2 + η ' 2 cos ( i z ) z η ' ' i sin ( i z ) z 0 ν ' i sin ( i z ) z η ' ' i sin ( i η ' 2 + ν ' 2 ) z cos ( i z ) ] .
M Δ 1 = [ 1 0 p 1 m Δ 1 ] , m Δ 1 T = m Δ 1 ,
Δ 1 = 1 - | t r ( m Δ 1 ) | 3 = 1 - | t r ( M Δ 1 ) 1 | 3 = 1 - ( exp ( α 1 ) + exp ( α 2 ) ) cos ( i μ ' ) + exp ( α 3 ) cos ( i η ' 2 + ν ' 2 ) ) 3 , 0 Δ 1 1.
M ( z + Δ z ) = ( I + m d e t Δ z + m d e p Δ z ) M ( z ) ,
C ( I ) = [ 1 0 T 0 O 3 ] .
M ( z + Δ z ) = ( I + m L R Δ z ) M ( z ) .
C ( m L R ) = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .
C ( m C R ) = [ 0 0 0 μ i 2 0 0 0 0 0 0 0 0 μ i 2 0 0 0 ] , C ( m D ) = 1 2 [ 0 β γ δ β 0 0 0 γ 0 0 0 δ 0 0 0 ] .
C ( m Δ - μ ) = [ 0 0 0 0 0 0 μ ' 2 0 0 μ ' 2 0 0 0 0 0 0 ] , C ( m Δ - ν η ) = [ 0 0 0 0 0 0 0 ν ' 2 0 0 0 η ' 2 0 ν ' 2 η ' 2 0 ] .
C ( m Δ β γ δ ) = 1 4 [ 0 β ' γ ' δ ' β ' 0 i δ ' i γ ' γ ' i δ ' 0 i β ' δ ' i γ ' i β ' 0 ] = 1 4 ( [ 0 β ' γ ' δ ' β ' 0 0 0 γ ' 0 0 0 δ ' 0 0 0 ] + [ 0 0 0 0 0 0 i δ ' 0 0 i δ ' 0 0 0 0 0 ' 0 ] + [ 0 0 0 0 0 0 0 i γ ' 0 0 0 i β ' 0 i γ ' i β ' 0 ] ) .
C ( m Δ α 1 α 2 α 3 , β γ δ ) = 1 4 [ α 1 + α 2 + α 3 β ' γ ' δ ' β ' α 1 - α 2 - α 3 - i δ ' i γ ' γ ' i δ ' - α 1 + α 2 - α 3 - i β ' δ ' i γ ' i β ' - α 1 - α 2 + α 3 ] = 1 4 ( [ 0 β ' γ ' δ ' β ' 0 - i δ ' i γ ' γ ' i δ ' 0 - i β ' δ ' i γ ' i β ' 0 ] + [ α 1 + α 2 + α 3 0 0 0 0 α 1 - α 2 - α 3 0 0 0 0 - α 1 + α 2 - α 3 0 0 0 0 - α 1 - α 2 + α 3 ] ) .
M = [ 1 0.026 0.044 0.039 0.029 0.962 0.144 0.047 0.002 0.126 0.975 0.026 0.039 0.019 0.115 0.936 ] .
m = [ -0 .0011 0 .0237 0 .0487 -0 .0405 0 .0286 -0 .0292 -0 .1457 -0 .0467 0 .0008 0 .1292 -0 .0177 0 .0303 -0 .0407 0 .0126 0 .1223 -0 .0684 ] = m d e t + m d e p .
m d e t = [ 0 β γ δ β 0 μ ν γ μ 0 η δ ν η 0 ] = [ 0 0 .0262 0 .0247 -0 .0406 0 .0262 0 -0 .1374 -0 .0296 0 .0247 0 .1374 0 -0 .046 -0 .0406 0 .0296 0 .046 0 ] .
m d e p = [ 0 β ' γ ' δ ' β ' α 1 μ ' ν ' γ ' μ ' α 2 η ' δ ' ν ' η ' α 3 ] = [ 0 -0 .0025 0 .024 0 .0001 0 .0025 -0 .0281 -0 .0083 -0 .0170 -0 .024 -0 .0083 -0 .0166 0 .0763 -0 .0001 -0 .0170 0 .0763 -0 .0673 ] .
M Δ = [ 1 0 0 0 0.008 0.976 0.01 0.021 0.023 0.01 0.982 0.073 0.009 0.022 0.073 0.941 ] , M R = [ 1 0 0 0 0 0.99 0.138 0.027 0 0.136 0.99 0.048 0 0.033 0.044 0.998 ] , M D = [ 1 0.026 0.044 0.039 0.026 0.998 0.001 0.001 0.044 0.001 0.999 0.001 0.039 0.001 0.001 0.999 ] .
η = - 0. 048 ; ν = -0 .027 ; μ = - 0.1374 ; β = 0.026 ; γ = 0.044 ; δ = - 0.039 ,
α 3 = - 0.0608 ; α 2 = -0 .0182; α 1 = 0.0243 ; η = 0.0776 ; ν = 0.0234 ; μ = - 0.0102 ; β = 0.008 ; γ = - 0.023 ; δ = - 0.009.
M R ( η , ν , μ ) = [ 1 0 0 0 0 η 2 + ν 2 cos ( i x ) x cos μ + η ν ( cos ( i x ) 1 ) x sin μ η 2 + ν 2 cos ( i x ) x sin μ η ν ( cos ( i x ) 1 ) x cos μ ν i sin ( i x ) x 0 η ν ( cos ( i x ) 1 ) x cos μ + ν 2 + η 2 cos ( i x ) x sin μ η ν ( cos ( i x ) 1 ) x sin μ ν 2 + η 2 cos ( i x ) x cos μ η i sin ( i x ) x 0 ν i sin ( i x ) x cos μ η i sin ( i x ) x sin μ ν i sin ( i x ) x sin μ + η i sin ( i x ) x cos μ cos ( i x ) ] .
M D ( β , γ , δ ) = [ 1 β i tan ( i y ) y γ i tan ( i y ) y δ i tan ( i y ) y β i tan ( i y ) y δ 2 + γ 2 + β 2 cos ( i y ) y cos ( i y ) β γ ( cos ( i y ) 1 ) y cos ( i y ) β δ ( cos ( i y ) 1 ) y cos ( i y ) γ i tan ( i y ) y β γ ( cos ( i y ) 1 ) y cos ( i y ) δ 2 + β 2 + γ 2 cos ( i y ) y cos ( i y ) δ γ ( cos ( i y ) 1 ) y cos ( i y ) δ i tan ( i y ) y β δ ( cos ( i y ) 1 ) y cos ( i y ) δ γ ( cos ( i y ) 1 ) y cos ( i y ) β 2 + γ 2 + δ 2 cos ( i y ) y cos ( i y ) ] .
M Δ = M Δ α 1 α 2 α 3 , β γ δ M Δ - β - γ - δ ( M Δ μ M Δ ν η ) = [ 1 0 0 0 β ' ( e α 1 1 ) α 1 e α 1 ( z 3 η ' 2 + ν ' 2 z 1 z i z 4 η ' ν ' ( z 1 1 ) z ) e α 1 ( z 3 η ' ν ' ( z 1 ) 1 ) z i z 4 ν ' 2 + η ' 2 z 1 ) z ) e α 1 ( z 3 ν ' i z 2 z z 4 η ' z 2 ) z ) γ ' ( e α 2 1 ) α 2 e α 2 ( z 3 η ' ν ' ( z 1 1 ) z i z 4 η ' 2 + ν ' 2 z 1 z ) e α 2 ( z 3 ν ' 2 + η ' 2 z 1 z i z 4 η ' ν ' ( z 1 1 ) z ) e α 2 ( z 4 ν ' z 2 z z 3 η ' i z 2 z ) δ ' ( e α 3 1 ) α 3 e α 3 ν ' i z 2 z e α 3 η ' i z 2 z e α 3 z 1 ]

Metrics