Abstract

Polarized light imaging is a potential tool to obtain an adequate description of the properties of depolarizing media such as biological tissues. In many biomedical applications, for instance, dermatology, ophthalmology, or urology, imaging polarimetry provides a noninvasive diagnosis of a wide range of disease states, and, likewise, it could be applied to the study of internal tissues though the use of endoscopes that use optical fibers. We introduce an algebraic method, based on the Mueller-coherence matrix, for a clearer analysis of the polarization characteristics of depolarizing media via the entropy factor. First-order errors introduced by the measurement system are corrected. Entropy defines three kinds of media according to their depolarizing behavior, and several examples corresponding to each region are shown. The calculation of this factor provides clearer information than that provided by the traditional Mueller matrix in the analysis of biological tissue properties by polarization measurement techniques.

© 2005 Optical Society of America

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References

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    [CrossRef]
  2. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis Vol. TT38 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 2000).
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    [CrossRef] [PubMed]
  4. D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  5. W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  6. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31, 488–503 (1941).
    [CrossRef]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  8. A. Gerard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  9. J. Lee, J. Koh, R. W. Collins, “Multichannel Mueller matrix ellipsometer for real-time spectroscopy of anisotropic surfaces and films,” Opt. Lett. 25, 1573–1575 (2000).
    [CrossRef]
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    [CrossRef]
  15. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  16. S. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
    [CrossRef]
  17. D. Pereda Cubián, J. L. Arce Diego, “Variation of the Pauli matrices coefficients in a PCA system under non-desired effects for ellipsometric applications,” in Proceedings of IEEE Workshop on Fiber and Optical Passive Components (IEEE, Piscataway, N. J., 2002), pp. 182–188.
  18. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A Pure Appl. Opt. 2, 216–222 (2000).
    [CrossRef]
  19. A. H. Hielscher, A. A. Eick, J. R. Mourant, D. S. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1, 441–453 (1997).
    [CrossRef] [PubMed]
  20. J. M. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vision Res. 40, 3791–3799 (2000).
    [CrossRef] [PubMed]
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  22. F. Le Roy-Brehonnet, B. La Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
    [CrossRef]

2002 (1)

S. Jiao, L. V. Wang, “Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography,” J. Biomed. Opt. 7, 350–358 (2002).
[CrossRef] [PubMed]

2000 (5)

1999 (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

1998 (1)

G. A. Wagnieres, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef] [PubMed]

1997 (2)

F. Le Roy-Brehonnet, B. La Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

A. H. Hielscher, A. A. Eick, J. R. Mourant, D. S. Shen, J. P. Freyer, I. J. Bigio, “Diffuse backscattering Mueller matrices of highly scattering media,” Opt. Express 1, 441–453 (1997).
[CrossRef] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (1)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1986 (1)

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

1941 (1)

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31, 488–503 (1941).
[CrossRef]

Arce Diego, J. L.

D. Pereda Cubián, C. Vlcek, J. L. Arce Diego, Z. Zaoralek, “Variation of the Pauli matrices coefficients in Nd-doped fibers subjected to a magnetic field,” in Proceedings of IEEE Lasers and Electro-Optics Society, 2001, (IEEE, Piscataway, N. J., 2001), pp. 823–824.
[CrossRef]

D. Pereda Cubián, J. L. Arce Diego, “Variation of the Pauli matrices coefficients in a PCA system under non-desired effects for ellipsometric applications,” in Proceedings of IEEE Workshop on Fiber and Optical Passive Components (IEEE, Piscataway, N. J., 2002), pp. 182–188.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bigio, I. J.

Bohm, D.

D. Bohm, Quantum Theory (Dover, New York, 1989).

Bueno, J. M.

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A Pure Appl. Opt. 2, 216–222 (2000).
[CrossRef]

J. M. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vision Res. 40, 3791–3799 (2000).
[CrossRef] [PubMed]

Burch, J. M.

A. Gerard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cho, G. C.

Cloude, S. R.

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

Collins, R. W.

Eick, A. A.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Freyer, J. P.

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gerard, A.

A. Gerard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gutsche, A.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

Han, P. Y.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hielscher, A. H.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Jacques, S. L.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

Jiao, S.

S. Jiao, L. V. Wang, “Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography,” J. Biomed. Opt. 7, 350–358 (2002).
[CrossRef] [PubMed]

Jones, R. C.

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31, 488–503 (1941).
[CrossRef]

Koh, J.

La Jeune, B.

F. Le Roy-Brehonnet, B. La Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Le Roy-Brehonnet, F.

F. Le Roy-Brehonnet, B. La Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Lee, J.

Li, S.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Mourant, J. R.

Pereda Cubián, D.

D. Pereda Cubián, J. L. Arce Diego, “Variation of the Pauli matrices coefficients in a PCA system under non-desired effects for ellipsometric applications,” in Proceedings of IEEE Workshop on Fiber and Optical Passive Components (IEEE, Piscataway, N. J., 2002), pp. 182–188.

D. Pereda Cubián, C. Vlcek, J. L. Arce Diego, Z. Zaoralek, “Variation of the Pauli matrices coefficients in Nd-doped fibers subjected to a magnetic field,” in Proceedings of IEEE Lasers and Electro-Optics Society, 2001, (IEEE, Piscataway, N. J., 2001), pp. 823–824.
[CrossRef]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Schwartz, J.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

Shen, D. S.

Shuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Star, W. M.

G. A. Wagnieres, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef] [PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tittel, F. K.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

Tuchin, V.

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis Vol. TT38 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 2000).

Vlcek, C.

D. Pereda Cubián, C. Vlcek, J. L. Arce Diego, Z. Zaoralek, “Variation of the Pauli matrices coefficients in Nd-doped fibers subjected to a magnetic field,” in Proceedings of IEEE Lasers and Electro-Optics Society, 2001, (IEEE, Piscataway, N. J., 2001), pp. 823–824.
[CrossRef]

Vo-Dich, T.

T. Vo-Dich, Handbook on Biomedical Photonics (CRC Press, Boca Raton, United States, 2003).
[CrossRef]

Wagnieres, G. A.

G. A. Wagnieres, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef] [PubMed]

Wang, L.

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

Wang, L. V.

S. Jiao, L. V. Wang, “Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography,” J. Biomed. Opt. 7, 350–358 (2002).
[CrossRef] [PubMed]

Welch, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilson, B. C.

G. A. Wagnieres, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef] [PubMed]

Zaoralek, Z.

D. Pereda Cubián, C. Vlcek, J. L. Arce Diego, Z. Zaoralek, “Variation of the Pauli matrices coefficients in Nd-doped fibers subjected to a magnetic field,” in Proceedings of IEEE Lasers and Electro-Optics Society, 2001, (IEEE, Piscataway, N. J., 2001), pp. 823–824.
[CrossRef]

Zhang, X. C.

IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

IEEE Quantum Electron. (1)

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review on optical properties of biomedical tissues,” IEEE Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

J. Biomed. Opt. (1)

S. Jiao, L. V. Wang, “Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography,” J. Biomed. Opt. 7, 350–358 (2002).
[CrossRef] [PubMed]

J. Opt. A Pure Appl. Opt. (1)

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A Pure Appl. Opt. 2, 216–222 (2000).
[CrossRef]

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

S. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
[CrossRef]

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. A 31, 488–503 (1941).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (1)

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

Photochem. Photobiol. (1)

G. A. Wagnieres, W. M. Star, B. C. Wilson, “In vivo fluorescence spectroscopy and imaging for oncological applications,” Photochem. Photobiol. 68, 603–632 (1998).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

F. Le Roy-Brehonnet, B. La Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109–151 (1997).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Shuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Vision Res. (1)

J. M. Bueno, “Measurement of parameters of polarization in the living human eye using imaging polarimetry,” Vision Res. 40, 3791–3799 (2000).
[CrossRef] [PubMed]

Other (8)

S. L. Jacques, A. Gutsche, J. Schwartz, L. Wang, F. K. Tittel, “Video reflectometry to extract optical properties of tissue in in vivo,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE, Bellingham, Wash., 1992) pp. 211–226.

D. Pereda Cubián, J. L. Arce Diego, “Variation of the Pauli matrices coefficients in a PCA system under non-desired effects for ellipsometric applications,” in Proceedings of IEEE Workshop on Fiber and Optical Passive Components (IEEE, Piscataway, N. J., 2002), pp. 182–188.

T. Vo-Dich, Handbook on Biomedical Photonics (CRC Press, Boca Raton, United States, 2003).
[CrossRef]

V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis Vol. TT38 of Tutorial Texts in Optical Engineering (SPIE, Bellingham, Wash., 2000).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

A. Gerard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

D. Bohm, Quantum Theory (Dover, New York, 1989).

D. Pereda Cubián, C. Vlcek, J. L. Arce Diego, Z. Zaoralek, “Variation of the Pauli matrices coefficients in Nd-doped fibers subjected to a magnetic field,” in Proceedings of IEEE Lasers and Electro-Optics Society, 2001, (IEEE, Piscataway, N. J., 2001), pp. 823–824.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Measurement system scheme. P, polarizer; QWP, quarter-wave plate; BS, beam splitter.

Fig. 2
Fig. 2

Entropy factor (continuous curve) and the horizontal-to-vertical cross-talk parameter (dashed curve) of the horizontal meridian with the pixels (illumination angle) along the middle of the Mueller image of a human eye.

Equations (18)

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

C 2 × 2 = 1 2 i = 0 3 S i σ i C 4 × 4 = 1 4 i , j = 0 3 m i j n i j .
c 11 = m 11 + m 12 + m 22 , c 31 = m 31 + m 32 + i ( m 41 + m 42 ) , c 12 = m 13 + m 23 + i ( m 14 + m 24 ) , c 32 = m 33 - m 44 + i ( m 34 + m 43 ) , c 13 = m 31 + m 32 - i ( m 41 + m 42 ) , c 33 = m 11 + m 12 - m 21 - m 22 , c 14 = m 33 + m 44 + i ( m 34 - m 43 ) , c 34 = m 13 - m 23 + i ( m 14 - m 24 ) , c 21 = m 13 + m 23 - i ( m 14 + m 24 ) , c 41 = m 33 + m 44 - i ( m 34 - m 43 ) , c 22 = m 11 - m 12 - m 21 - m 22 , c 42 = m 31 - m 32 + i ( m 41 - m 42 ) , c 23 = m 33 - m 44 - i ( m 34 + m 43 ) , c 43 = m 13 - m 23 - i ( m 14 - m 24 ) , c 24 = m 31 - m 32 - i ( m 41 - m 42 ) , c 44 = m 11 - m 12 - m 21 + m 22 .
C 4 × 4 = λ 1 C 4 × 4 1 + λ 2 C 4 × 4 2 + λ 3 C 4 × 4 3 + λ 4 C 4 × 4 4 .
H = - x i log 4 x i x i = λ i / j λ i ,             0 < x i < 1.
m 1 , 1 meas = 1.0227 m 1 , 1 corr - 0.1787 m 2 , 1 corr , m 2 , 1 meas = 0.9974 m 2 , 1 corr , m 3 , 1 meas = 1.0711 m 3 , 1 corr , m 4 , 1 meas = - 0.0964 m 1 , 1 corr - 0.0098 m 2 , 1 corr , - 0.0146 m 3 , 1 corr + 0.9691 m 4 , 1 corr ,
M P meas = [ 1 1.01 0.02 - 0.01 1.03 1.04 - 0.01 0.01 0.03 0.04 0.02 0.01 - 0.02 - 0.02 0.02 - 0.02 ] .
M P corr = [ 1 1 0.02 0.09 0.92 0.92 0.03 0.09 0.07 0.06 0.02 0.02 0.18 0.20 0.02 0 ] .
J P = [ 1 0.016 - 0.046 i 0.036 + 0.102 i 0.005 - 0.001 i ] .
M C meas = [ 0.92 0.52 - 0.14 - 0.03 0.51 0.55 - 0.05 0.01 0.05 - 0.02 - 0.11 - 0.08 - 0.03 - 0.05 - 0.06 - 0.08 ] .
M C corr = [ 0.91 0.48 - 0.22 0.13 0.51 0.52 - 0.11 0.16 0.05 - 0.01 - 0.108 - 0.07 0.06 0.02 - 0.08 - 0.08 ] .
J C = [ 1 - 0.163 - 0.133 i 0.016 + 0.053 i - 0.088 - 0.14 i ] .
M E meas = [ 1 - 0.14 - 0.36 - 0.05 - 0.02 0.51 - 0.07 - 0.04 - 0.3 - 0.35 0.5 - 0.06 0.16 0.35 0.19 0.34 ] .
M E corr = [ 0.97 - 0.02 - 0.30 0.24 - 0.17 0.45 - 0.31 0.24 - 0.4 - 0.04 0.49 0.19 - 0.18 0.04 - 0.06 0.39 ] .
HVC = par - perp par + perp .
M = U · ( J J * ) · U - 1 ,
U = [ 1 0 0 1 1 0 0 - 1 0 1 1 0 0 i - i 0 ] .
m 1 , 1 = 1 8 ( 1 + ϕ 2 ) 2 ( 1 + γ ) 2 + 1 2 cos 2 ( 2 θ ) ( 1 - γ ) 2 ϕ 2 + 1 8 ( 1 + ϕ 2 ) [ sin 2 ( 2 θ ) + δ 2 ] - 1 8 ( 1 + ϕ 2 ) ( 1 - γ ) × δ ϕ cos ( 2 θ ) + 1 8 cos 2 ( 2 θ ) ( 1 - ϕ 2 ) 2 ( 1 - γ ) 2 , m 1 , 2 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) ( 1 - γ 2 ) , m 1 , 3 = 1 4 sin ( 2 θ ) ( 1 + ϕ 2 ) 2 ( 1 + γ ) , m 1 , 4 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) ( 1 - γ ) δ - 1 2 cos 2 ( 2 θ ) ϕ ( 1 - γ ) 2 × ( 1 - ϕ 2 ) , m 2 , 1 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) ( 1 - γ 2 ) , m 2 , 2 = 1 8 ( 1 + ϕ 2 ) 2 ( 1 + γ ) 2 - 1 2 cos 2 ( 2 θ ) ( 1 - γ ) 2 ϕ 2 - 1 8 ( 1 + ϕ 2 ) [ sin 2 ( 2 θ ) + δ 2 ] + 1 8 ( 1 + ϕ 2 ) × ( 1 - γ ) δ ϕ cos ( 2 θ ) + 1 8 cos 2 ( 2 θ ) ( 1 - ϕ 2 ) 2 ( 1 - γ ) 2 , m 2 , 3 = 1 4 cos ( 2 θ ) sin ( 2 θ ) ( 1 = ϕ 4 ) ( 1 - γ ) , m 2 , 4 = 1 4 ( 1 + ϕ 2 ) 2 ( 1 + γ ) δ - 1 2 cos ( 2 θ ) ϕ × ( 1 + ϕ 2 ) ( 1 - γ 2 ) , m 3 , 1 = 1 4 sin ( 2 θ ) ( 1 + ϕ 2 ) 2 ( 1 + γ ) , m 3 , 2 = 1 4 cos ( 2 θ ) sin ( 2 θ ) ( 1 - ϕ 4 ) ( 1 - γ ) , m 3 , 3 = 1 8 ( 1 + ϕ 2 ) 2 ( 1 + γ ) 2 - 1 2 cos 2 ( 2 θ ) ( 1 - γ ) 2 ϕ 2 + 1 8 ( 1 + ϕ 2 ) [ sin 2 ( 2 θ ) + δ 2 ] + 1 8 ( 1 + ϕ 2 ) × ( 1 - γ ) δ ϕ cos ( 2 θ ) - 1 8 cos 2 ( 2 θ ) × ( 1 - ϕ 2 ) 2 ( 1 - γ ) 2 , m 3 , 4 = 0 , m 4 , 1 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) ( 1 - γ ) δ - 1 2 cos 2 × ( 2 θ ) ϕ ( 1 - γ ) 2 ( 1 - ϕ 2 ) , m 4 , 2 = 1 4 ( 1 + ϕ 2 ) 2 ( 1 + γ ) δ - 1 2 cos ( 2 θ ) ϕ × ( 1 + ϕ 2 ) ( 1 - γ 2 ) , m 4 , 3 = 0 , m 4 , 4 = 1 8 ( 1 + ϕ 2 ) 2 ( 1 + γ ) 2 - 1 2 cos 2 ( 2 θ ) ( 1 - γ ) 2 ϕ 2 - 1 8 ( 1 + ϕ 2 ) [ sin 2 ( 2 θ ) + δ 2 ] + 1 8 ( 1 + ϕ 2 ) × ( 1 - γ ) δ ϕ cos ( 2 θ ) - 1 8 cos 2 ( 2 θ ) ( 1 - ϕ 2 ) 2 ( 1 - γ ) 2 .
m 1 , 1 = 1 8 ( 1 + ϕ 2 ) 2 [ 1 + exp ( - 2 α ) + 2 exp ( - α ) cos ( δ ) ] + 1 8 cos 2 ( 2 θ ) ϕ 2 [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] + 1 8 cos 2 ( 2 θ ) ( 1 - ϕ 2 ) 2 + sin 2 ( 2 θ ) ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] . m 1 , 2 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) [ 1 - exp ( - 2 α ) ] + 1 4 sin ( 2 θ ) cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) [ 1 - exp ( - 2 α ) - 2 exp ( α ) cos ( δ ) ] , m 1 , 3 = 1 4 sin ( 2 θ ) ( 1 + ϕ 2 ) 2 [ 1 - exp ( - 2 α ) ] - 1 4 cos 2 ( 2 θ ) ϕ ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] , m 1 , 4 = 1 2 cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) exp ( - α ) cos ( δ ) , m 2 , 1 = 1 4 cos ( 2 θ ) ( 1 - ϕ 4 ) [ 1 - exp ( - 2 α ) ] - 1 4 sin ( 2 θ ) cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) - 2 exp ( - α ) cos ( δ ) ] , m 2 , 2 = 1 8 ( 1 + ϕ 2 ) 2 [ 1 + exp ( - 2 α ) + 2 exp ( - α ) cos ( δ ) ] - 1 8 cos 2 ( 2 θ ) ϕ 2 [ 1 + exp ( - 2 α ) - 2 exp ( - α ) cos ( δ ) ] , m 2 , 3 = 1 4 sin ( 2 θ ) cos ( 2 θ ) ( 1 - ϕ 4 ) [ 1 + exp ( - 2 α ) - 2 exp ( - α ) cos ( δ ) ] - 1 4 cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) [ 1 - exp ( - 2 α ) ] , m 2 , 4 = - 1 2 sin ( 2 θ ) exp ( - α ) ( 1 + ϕ 2 ) 2 sin ( δ ) , m 3 , 1 = 1 4 sin ( 2 θ ) ( 1 + ϕ 2 ) 2 [ 1 - exp ( - 2 α ) ] + 1 4 cos 2 ( 2 θ ) ϕ ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] , m 3 , 2 = 1 4 sin ( 2 θ ) cos ( 2 θ ) ( 1 - ϕ 4 ) [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] + 1 4 cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) × [ 1 - exp ( - 2 α ) ] , m 3 , 3 = 1 8 ( 1 + ϕ 2 ) 2 [ 1 + exp ( - 2 α ) + 2 exp ( - 2 α ) cos ( δ ) ] - 1 8 cos 2 ( 2 θ ) ϕ 2 [ 1 + exp ( - 2 α ) - 2 exp ( - 2 α ) cos ( δ ) ] - 1 8 cos 2 ( 2 θ ) ( 1 + ϕ 2 ) 2 - sin 2 ( 2 θ ) ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) - 2 exp ( - α ) cos ( δ ) ] , m 3 , 4 = 1 2 cos ( 2 θ ) exp ( - α ) sin ( δ ) ( 1 + ϕ 4 ) , m 4 , 1 = 1 2 cos ( 2 θ ) ϕ ( 1 + ϕ 2 ) 2 exp ( - α ) cos ( δ ) , m 4 , 2 = 1 2 sin ( 2 θ ) exp ( - α ) ( 1 + ϕ 2 ) 2 sin ( δ ) , m 4 , 3 = - 1 2 cos ( 2 θ ) exp ( - α ) sin ( δ ) ( 1 + ϕ 4 ) , m 4 , 4 = 1 8 ( 1 + ϕ 2 ) 2 [ 1 + exp ( - 2 α ) + 2 exp ( - α ) cos ( δ ) ] + 1 8 cos 2 ( 2 θ ) ϕ 2 [ 1 + exp ( - 2 α ) - 2 exp ( - α ) cos ( δ ) ] - 1 8 cos 2 ( 2 θ ) ( 1 + ϕ 2 ) 2 + sin 2 ( 2 θ ) ( 1 + ϕ 2 ) [ 1 + exp ( - 2 α ) ] - 2 exp ( - α ) cos ( δ ) ] .

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