Abstract

The complete Mueller matrix for an English oak (Quercus robur) leaf for a fixed azimuth angle (90°) was determined immediately after plucking and a day following exposure to normal room temperature and pressures. The Mueller matrices were determined for transmitted light at observation angles ranging from 0° to 24° and for reflected backscattering angles from 153° to 170°. All the measurements were taken with a He-Ne laser light source at 0.63 µm. Since positive eigenvalues were obtained for the coherence matrix, the polarimetric measurements were physically realizable. The anisotropy parameters were determined from the Jones matrices by use of the decomposition theorem. From the M33 and M44 components of the Mueller matrices, it was found that nonspherical structures within the leaf were primarily responsible for observed transmitted light scatter, and spherical structures were mostly responsible for observed backscatter. Variations in backscatter Mueller matrix elements from a fresh leaf to a second day of observation were assumed because of changes to water vapor concentration in the leaf.

© 2003 Optical Society of America

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References

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  1. R. N. Clark, T. L. Roush, “Reflectance spectroscopy: quantitative analysis techniques for remote sensing applications,” J. Geophys. Res. 89, 6329–6340 (1984).
    [CrossRef]
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
    [CrossRef]
  4. R. S. Muttiah, “Light scattering from a leaf using the T-matrix method,” in From Spectroscopy to Remotely Sensed Spectra of Terrestrial Ecosystems, R. S. Muttiah, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2002), pp. 67–75.
  5. M. G. Holmes, D. R. Keiller, “Effects of pubescence and waxes on the reflectance of leaves in the ultraviolet and photosynthetic wavebands: a comparison of a range of species,” Plant Cell Environ. 25, 85–93 (2002).
    [CrossRef]
  6. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
    [CrossRef]
  7. S. R. Cloude, “Group theory and polarization algebra,” Optik (Stuttgart) 75, 26–36 (1986).
  8. M. I. Mishchenko, J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).
    [CrossRef] [PubMed]
  9. S. R. Cloude, E. Pottier, “Concept of polarization anisotropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
    [CrossRef]
  10. C. Brosseau, Fundamentals of Polarized Light, a Statistical Approach (Wiley, New York, 1998).
  11. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1987).
  12. V. V. Mar’enko, S. N. Savenkov, “Representation of arbitrary Mueller matrix on the basis of circular and linear anisotropy,” Opt. Spectrosc. 76, 94–96 (1994).
  13. G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computation (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  14. Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.
  15. Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.
  16. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 40–59.
  17. M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., (Academic, San Diego, Calif., 2000), pp. 525–554.
    [CrossRef]
  18. R. P. Feynman, QED: the Strange Theory of Light and Matter (Princeton University Press, Princeton, N.J., 1985), pp. 54–60.
  19. T. C. Vogelmann, “Plant tissue optics,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 44, 231–251 (1993).
    [CrossRef]
  20. O. Björkman, “Responses to different quantum flux densities,” in Encyclopedia of Plant Physiology, O. L. Lange, P. S. Nobel, H. Ziegler, eds. (Springer-Verlag, New York, 1981), Vol. 12, pp. 57–106.
  21. N. K. Boardman, “Comparative photosynthesis of sun and shade plants,” Annu. Rev. Plant Physiol. 28, 355–377 (1977).
    [CrossRef]
  22. R. W. Pearcy, “Sunflecks and photosynthesis in plant canopies,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 41, 421–453 (1990).
    [CrossRef]

2002

P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
[CrossRef]

M. G. Holmes, D. R. Keiller, “Effects of pubescence and waxes on the reflectance of leaves in the ultraviolet and photosynthetic wavebands: a comparison of a range of species,” Plant Cell Environ. 25, 85–93 (2002).
[CrossRef]

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

1999

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

1998

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

1995

M. I. Mishchenko, J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).
[CrossRef] [PubMed]

S. R. Cloude, E. Pottier, “Concept of polarization anisotropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

1994

V. V. Mar’enko, S. N. Savenkov, “Representation of arbitrary Mueller matrix on the basis of circular and linear anisotropy,” Opt. Spectrosc. 76, 94–96 (1994).

1993

T. C. Vogelmann, “Plant tissue optics,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 44, 231–251 (1993).
[CrossRef]

1990

R. W. Pearcy, “Sunflecks and photosynthesis in plant canopies,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 41, 421–453 (1990).
[CrossRef]

1986

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

1984

R. N. Clark, T. L. Roush, “Reflectance spectroscopy: quantitative analysis techniques for remote sensing applications,” J. Geophys. Res. 89, 6329–6340 (1984).
[CrossRef]

1977

N. K. Boardman, “Comparative photosynthesis of sun and shade plants,” Annu. Rev. Plant Physiol. 28, 355–377 (1977).
[CrossRef]

Astrashevski, Y. I.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

Azzam, R. M.

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

Bashara, N. M.

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

Björkman, O.

O. Björkman, “Responses to different quantum flux densities,” in Encyclopedia of Plant Physiology, O. L. Lange, P. S. Nobel, H. Ziegler, eds. (Springer-Verlag, New York, 1981), Vol. 12, pp. 57–106.

Boardman, N. K.

N. K. Boardman, “Comparative photosynthesis of sun and shade plants,” Annu. Rev. Plant Physiol. 28, 355–377 (1977).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light, a Statistical Approach (Wiley, New York, 1998).

Clark, R. N.

R. N. Clark, T. L. Roush, “Reflectance spectroscopy: quantitative analysis techniques for remote sensing applications,” J. Geophys. Res. 89, 6329–6340 (1984).
[CrossRef]

Cloude, S. R.

S. R. Cloude, E. Pottier, “Concept of polarization anisotropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

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

Feynman, R. P.

R. P. Feynman, QED: the Strange Theory of Light and Matter (Princeton University Press, Princeton, N.J., 1985), pp. 54–60.

Forsythe, G. E.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computation (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Holmes, M. G.

M. G. Holmes, D. R. Keiller, “Effects of pubescence and waxes on the reflectance of leaves in the ultraviolet and photosynthetic wavebands: a comparison of a range of species,” Plant Cell Environ. 25, 85–93 (2002).
[CrossRef]

Hovenier, J. W.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hull, P. G.

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., (Academic, San Diego, Calif., 2000), pp. 525–554.
[CrossRef]

Hunt, A. J.

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., (Academic, San Diego, Calif., 2000), pp. 525–554.
[CrossRef]

Jordan, D. L.

P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
[CrossRef]

Keiller, D. R.

M. G. Holmes, D. R. Keiller, “Effects of pubescence and waxes on the reflectance of leaves in the ultraviolet and photosynthetic wavebands: a comparison of a range of species,” Plant Cell Environ. 25, 85–93 (2002).
[CrossRef]

Malcolm, M. A.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computation (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Mar’enko, V. V.

V. V. Mar’enko, S. N. Savenkov, “Representation of arbitrary Mueller matrix on the basis of circular and linear anisotropy,” Opt. Spectrosc. 76, 94–96 (1994).

Mishchenko, M. I.

Moler, C. B.

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computation (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Muttiah, R. S.

R. S. Muttiah, “Light scattering from a leaf using the T-matrix method,” in From Spectroscopy to Remotely Sensed Spectra of Terrestrial Ecosystems, R. S. Muttiah, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2002), pp. 67–75.

Pearcy, R. W.

R. W. Pearcy, “Sunflecks and photosynthesis in plant canopies,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 41, 421–453 (1990).
[CrossRef]

Pottier, E.

S. R. Cloude, E. Pottier, “Concept of polarization anisotropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Quinby-Hunt, M. S.

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., (Academic, San Diego, Calif., 2000), pp. 525–554.
[CrossRef]

Raven, P. N.

P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
[CrossRef]

Roush, T. L.

R. N. Clark, T. L. Roush, “Reflectance spectroscopy: quantitative analysis techniques for remote sensing applications,” J. Geophys. Res. 89, 6329–6340 (1984).
[CrossRef]

Savenkov, S. N.

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

V. V. Mar’enko, S. N. Savenkov, “Representation of arbitrary Mueller matrix on the basis of circular and linear anisotropy,” Opt. Spectrosc. 76, 94–96 (1994).

Shuplyak, V. I.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

Sikorski, A. B.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

Sikorski, V. V.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

Smith, C. E.

P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
[CrossRef]

Stelmakh, G. F.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 40–59.

Vogelmann, T. C.

T. C. Vogelmann, “Plant tissue optics,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 44, 231–251 (1993).
[CrossRef]

Annu. Rev. Plant Physiol.

N. K. Boardman, “Comparative photosynthesis of sun and shade plants,” Annu. Rev. Plant Physiol. 28, 355–377 (1977).
[CrossRef]

Annu. Rev. Plant Physiol. Plant Mol. Biol.

R. W. Pearcy, “Sunflecks and photosynthesis in plant canopies,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 41, 421–453 (1990).
[CrossRef]

T. C. Vogelmann, “Plant tissue optics,” Annu. Rev. Plant Physiol. Plant Mol. Biol. 44, 231–251 (1993).
[CrossRef]

J. Appl. Spectrosc.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, V. I. Shuplyak, “Spectrophotometric characteristics of radiation scattered from plant leaves under stressed conditions,” J. Appl. Spectrosc. 65, 103–113 (1998), in Russian.

Y. I. Astrashevski, A. B. Sikorski, V. V. Sikorski, G. F. Stelmakh, “Some peculiarities of light reflection and scattering by plant leaf,” J. Appl. Spectrosc. 66, 100–108 (1999), in Russian.

J. Geophys. Res.

R. N. Clark, T. L. Roush, “Reflectance spectroscopy: quantitative analysis techniques for remote sensing applications,” J. Geophys. Res. 89, 6329–6340 (1984).
[CrossRef]

Opt. Eng.

P. N. Raven, D. L. Jordan, C. E. Smith, “Polarized directional reflectance from laurel and mullein leaves,” Opt. Eng. 41, 1002–1012 (2002).
[CrossRef]

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

S. R. Cloude, E. Pottier, “Concept of polarization anisotropy in optical scattering,” Opt. Eng. 34, 1599–1610 (1995).
[CrossRef]

Opt. Lett.

Opt. Spectrosc.

V. V. Mar’enko, S. N. Savenkov, “Representation of arbitrary Mueller matrix on the basis of circular and linear anisotropy,” Opt. Spectrosc. 76, 94–96 (1994).

Optik (Stuttgart)

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

Plant Cell Environ.

M. G. Holmes, D. R. Keiller, “Effects of pubescence and waxes on the reflectance of leaves in the ultraviolet and photosynthetic wavebands: a comparison of a range of species,” Plant Cell Environ. 25, 85–93 (2002).
[CrossRef]

Other

R. S. Muttiah, “Light scattering from a leaf using the T-matrix method,” in From Spectroscopy to Remotely Sensed Spectra of Terrestrial Ecosystems, R. S. Muttiah, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2002), pp. 67–75.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

G. E. Forsythe, M. A. Malcolm, C. B. Moler, Computer Methods for Mathematical Computation (Prentice-Hall, Englewood Cliffs, N.J., 1977).

C. Brosseau, Fundamentals of Polarized Light, a Statistical Approach (Wiley, New York, 1998).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 40–59.

M. S. Quinby-Hunt, P. G. Hull, A. J. Hunt, “Polarized light scattering in the marine environment,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., (Academic, San Diego, Calif., 2000), pp. 525–554.
[CrossRef]

R. P. Feynman, QED: the Strange Theory of Light and Matter (Princeton University Press, Princeton, N.J., 1985), pp. 54–60.

O. Björkman, “Responses to different quantum flux densities,” in Encyclopedia of Plant Physiology, O. L. Lange, P. S. Nobel, H. Ziegler, eds. (Springer-Verlag, New York, 1981), Vol. 12, pp. 57–106.

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Figures (6)

Fig. 1
Fig. 1

Polarimeter setup for determination of the Mueller matrices for transmitted and reflected light scatter. For each observation angle, phase plate 6 was rotated to obtain the complete Stokes vector: 1, laser; 2, polarizer; 3, controlled quarter-wave phase plate; 4, beam extender; 5, sample (oak leaf); 6, rotable quarter-wave phase plate; 7, analyzer; 8, photoamplifier; 9, computer.

Fig. 2
Fig. 2

Mueller matrix element plots as a function of observation angle during (a) light transmission and (b) backscattered events.

Fig. 3
Fig. 3

Degree of polarization [from Eq. (2)] as a function of input radiation azimuth θ and ellipticity ε for (a) a fresh leaf and (b) a day old leaf for transmitted light at observation angle 0° for average measurement values.

Fig. 4
Fig. 4

Degree of polarization [from Eq. (2)] as a function of input radiation azimuth θ and ellipticity ε for (a) a fresh leaf and (b) a day old leaf for backscattered light at observation angle 0° for average measurement values.

Fig. 5
Fig. 5

Linear and circular depolarization for (a) transmitted and (b) backscattered light for average measurement values.

Fig. 6
Fig. 6

Anisotropy parameters as a function of observation angle for (a) transmitted and (b) backscattered light for average measurement values.

Tables (2)

Tables Icon

Table 1 Values of Entropy H and Eigenvalues λi of Coherence Matrix T for Transmitted Light for Fresh Leaf (Day 1) and for Leaf a Day after Picking

Tables Icon

Table 2 Values of Entropy H and Eigenvalues λi of Coherence Matrix T for Backscattering Light for Fresh Leaf (Day 1) and for Leaf a Day after Picking

Equations (17)

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

S1=Ex2+Ey2=I, S2=Ex2-Ey2=IP cos2θcos2ε, S3=2ExEy cosχ=IP sin2θcos2ε, S4=2ExEy sinχ=IP sin2ε,
P=S22+S32+S421/2S1.
Sout=MSinp, where M=M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44.
δL=S1-S2S1+S2,
δC=S1-S4S1+S4.
cos2α+sin2αexp-iΔcosαsinα1-exp-iΔcosαsinα1-exp-i·Δcos2αexp-iΔ+sin2α cosϕsinϕ-sinϕcosϕ cos2γ+p sin2γcosγsinγ1-pcosγsinγ1-pp cos2γ+sin2γ 1-iRiR1.
Lin.PhaseCirc.PhaseLin.Amp.Circ.Amp..
δM=Mexact-MexpMexact,
M=i,j |Mij|21/2.
MTsc=M11Tsc0000M22Tsc0000M33Tsc0000M44Tsc.
Mbscatt=M11bscattM12bscatt00M21bscattM22bscatt0000M33bscatt0000M44bscatt.
T11=14M11+M22+M33+M44,T14=14M14-iM23+iM32+M41,T22=14M11+M22-M33-M44,T23=14iM14+M23+M32-iM41,T33=14M11-M22+M33-M44,T32=14-iM14+M23+M32+iM41,T44=14M11-M22-M33+M44,T41=14M14+iM23-iM32+M41,T12=14M12+M21-iM34+iM43,T13=14M13+M31+iM24-iM42,T21=14M12+M21+iM34-iM43,T31=14M13+M31-iM24+iM42,T34=14iM12-iM21+M34+M43,T24=14-iM13+iM31+M24+M42,T43=14-iM12+iM21+M34+M43,T42=14iM13-iM31+M24+M42.
H=-r=1N Kr logNKr, Kr=λrj λj,
M=r=14 λrMJr.
MJ=AJ×J*A-1,
A=1001100-101100-ii0.
J=D1+D2D3-iD4D3+iD4D1-D2,

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