Abstract

The algebraic methods for serial and parallel decompositions of Mueller matrices are combined in order to obtain a general framework for a suitable analysis of polarimetric measurements based on equivalent systems constituted by simple components. A general procedure for the parallel decomposition of a Mueller matrix into a convex sum of pure elements is presented and applied to the two canonical forms of depolarizing Mueller matrices [Ossikovski, J. Opt. Soc. Am. A 27, 123 (2010).], leading to the serial–parallel decomposition of any Mueller matrix. The resultant model is consistent with the mathematical structure and the reciprocity properties of Mueller matrices.

© 2012 Optical Society of America

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  1. J. J. Gil, “Polarimetric characterization of light and media,” EPJ Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  2. J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
    [CrossRef]
  3. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  4. Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
    [CrossRef]
  5. C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
    [CrossRef]
  6. 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]
  7. J. J. Gil and E. Bernabéu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189(1986).
    [CrossRef]
  8. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]
  9. J. J. Gil, “Determination of polarization parameters in matricial representation. Theoretical contribution and development of an automatic measurement device,” Ph.D. thesis (Facultad de Ciencias, Univ. Zaragoza), Spain, 1983. Available from http://www.pepegil.es/PhD-Thesis-JJ-Gil-English.pdf .
  10. J. J. Gil and E. Bernabéu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from its Mueller matrix,” Optik 76, 67–71 (1987).
  11. S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  12. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  13. A. B. Kostinski and R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952(1992).
    [CrossRef]
  14. T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
    [CrossRef]
  15. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25, 473–482 (2008).
    [CrossRef]
  16. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29, 2234–2236 (2004).
    [CrossRef]
  17. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
    [CrossRef]
  18. M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” JEOS RP 2, 07018 (2007).
    [CrossRef]
  19. R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
    [CrossRef]
  20. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod Opt. 41, 1903–1915 (1994).
    [CrossRef]
  21. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
    [CrossRef]
  22. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).
    [CrossRef]
  23. A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
    [CrossRef]
  24. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
    [CrossRef]
  25. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–36 (1986).
  26. S. R. Cloude, “Conditions for the physical realizability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).
    [CrossRef]
  27. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .
  28. I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
    [CrossRef]
  29. J. J. Gil, “Parallel decompositions of Mueller matrices and polarimetric subtraction,” EPJ Web Conf. 5, 04002 (2010).
    [CrossRef]
  30. S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34, 498–518 (1996).
    [CrossRef]
  31. S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).
  32. W. A. Holm and R. M. Barnes, “On radar polarization mixed target state decomposition techniques,” in IEEE 1988 National Radar Conference (IEEE, 1988), pp. 249–254.
  33. J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” Monografías del Seminario Matemático García de Galdeano 27, 23–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .
  34. 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]
  35. V. M. Red’kov, A. A. Bogush, and N. G. Tokarevskaya, “On Parametrization of the linear GL(4,C) and unitary SU(4) groups in terms of Dirac matrices,” SIGMA 4, 021 (2008).
  36. L. Lavoura, “Parametrization of the four-generation quark mixing matrix by the moduli of its matrix elements,” Phys. Rev. D 40, 2440–2448 (1989).
    [CrossRef]
  37. S. R. Cloude, “The physical interpretation of eigenvalue problems in optical scattering polarimetry,” Proc. SPIE 3121, 88–99 (1997).
    [CrossRef]
  38. S. R. Cloude, “Special unitary groups in polarimetry theory,” Proc. SPIE 2265, 292–303 (1994).
    [CrossRef]
  39. G. Arfken, Mathematical Methods for Physicists (Academic, 1970), Chap. 4.
  40. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).
  41. S. R. Cloude, “Uniqueness of target decomposition theorems in radar polarimetry,” in Direct and Inverse Methods in Radar Polarimetry, Part I, W. M. Boerner, L. A. Cram, W. A. Holm, D. E. Stein, W. Wiesbeck, W. Keydel, D. Giuli, D. T. Gjessing, F. A. Molinet, and H. Brand, eds. (Kluwer, 1992).
  42. J. L. Alvarez-Perez, “Coherence, polarization, and statistical independence in Cloude–Pottier’s radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 49, 426–441 (2011).
    [CrossRef]

2011 (3)

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. L. Alvarez-Perez, “Coherence, polarization, and statistical independence in Cloude–Pottier’s radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 49, 426–441 (2011).
[CrossRef]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
[CrossRef]

2010 (2)

R. Ossikovski, “Canonical forms of depolarizing Mueller matrices,” J. Opt. Soc. Am. A 27, 123–130 (2010).
[CrossRef]

J. J. Gil, “Parallel decompositions of Mueller matrices and polarimetric subtraction,” EPJ Web Conf. 5, 04002 (2010).
[CrossRef]

2009 (2)

2008 (2)

R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25, 473–482 (2008).
[CrossRef]

V. M. Red’kov, A. A. Bogush, and N. G. Tokarevskaya, “On Parametrization of the linear GL(4,C) and unitary SU(4) groups in terms of Dirac matrices,” SIGMA 4, 021 (2008).

2007 (3)

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” JEOS RP 2, 07018 (2007).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media,” EPJ Appl. Phys. 40, 1–47 (2007).
[CrossRef]

2004 (2)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 29, 2234–2236 (2004).
[CrossRef]

2003 (2)

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” Monografías del Seminario Matemático García de Galdeano 27, 23–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
[CrossRef]

2000 (1)

1998 (2)

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
[CrossRef]

1997 (1)

S. R. Cloude, “The physical interpretation of eigenvalue problems in optical scattering polarimetry,” Proc. SPIE 3121, 88–99 (1997).
[CrossRef]

1996 (2)

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]

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34, 498–518 (1996).
[CrossRef]

1994 (3)

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod Opt. 41, 1903–1915 (1994).
[CrossRef]

S. R. Cloude, “Special unitary groups in polarimetry theory,” Proc. SPIE 2265, 292–303 (1994).
[CrossRef]

S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

1993 (2)

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

1992 (2)

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

A. B. Kostinski and R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952(1992).
[CrossRef]

1989 (2)

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

L. Lavoura, “Parametrization of the four-generation quark mixing matrix by the moduli of its matrix elements,” Phys. Rev. D 40, 2440–2448 (1989).
[CrossRef]

1987 (2)

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

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

1986 (2)

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

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

1971 (1)

Alvarez-Perez, J. L.

J. L. Alvarez-Perez, “Coherence, polarization, and statistical independence in Cloude–Pottier’s radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 49, 426–441 (2011).
[CrossRef]

Anastasiadou, M.

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” JEOS RP 2, 07018 (2007).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1970), Chap. 4.

Barakat, R.

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

Barnes, R. M.

W. A. Holm and R. M. Barnes, “On radar polarization mixed target state decomposition techniques,” in IEEE 1988 National Radar Conference (IEEE, 1988), pp. 249–254.

Ben Hatit, S.

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” JEOS RP 2, 07018 (2007).
[CrossRef]

Bernabéu, E.

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

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

Bogush, A. A.

V. M. Red’kov, A. A. Bogush, and N. G. Tokarevskaya, “On Parametrization of the linear GL(4,C) and unitary SU(4) groups in terms of Dirac matrices,” SIGMA 4, 021 (2008).

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “The physical interpretation of eigenvalue problems in optical scattering polarimetry,” Proc. SPIE 3121, 88–99 (1997).
[CrossRef]

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34, 498–518 (1996).
[CrossRef]

S. R. Cloude, “Special unitary groups in polarimetry theory,” Proc. SPIE 2265, 292–303 (1994).
[CrossRef]

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

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

S. R. Cloude, Polarisation: Applications in Remote Sensing (Oxford University, 2009).

S. R. Cloude, “Uniqueness of target decomposition theorems in radar polarimetry,” in Direct and Inverse Methods in Radar Polarimetry, Part I, W. M. Boerner, L. A. Cram, W. A. Holm, D. E. Stein, W. Wiesbeck, W. Keydel, D. Giuli, D. T. Gjessing, F. A. Molinet, and H. Brand, eds. (Kluwer, 1992).

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” Monografías del Seminario Matemático García de Galdeano 27, 23–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

De Martino, A.

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

Foldyna, M.

Garcia-Caurel, E.

Gil, J. J.

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

J. J. Gil, “Components of purity of a Mueller matrix,” J. Opt. Soc. Am. A 28, 1578–1585 (2011).
[CrossRef]

J. J. Gil, “Parallel decompositions of Mueller matrices and polarimetric subtraction,” EPJ Web Conf. 5, 04002 (2010).
[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]

J. J. Gil, “Polarimetric characterization of light and media,” EPJ Appl. Phys. 40, 1–47 (2007).
[CrossRef]

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” Monografías del Seminario Matemático García de Galdeano 27, 23–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
[CrossRef]

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

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

J. J. Gil, “Determination of polarization parameters in matricial representation. Theoretical contribution and development of an automatic measurement device,” Ph.D. thesis (Facultad de Ciencias, Univ. Zaragoza), Spain, 1983. Available from http://www.pepegil.es/PhD-Thesis-JJ-Gil-English.pdf .

Givens, C. R.

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Givens, R. C.

A. B. Kostinski and R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952(1992).
[CrossRef]

Gopala Rao, A. V.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
[CrossRef]

Goudail, F.

Guyot, S.

R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689–691 (2007).
[CrossRef]

M. Anastasiadou, S. Ben Hatit, R. Ossikovski, S. Guyot, and A. De Martino, “Experimental validation of the reverse polar decomposition of depolarizing Mueller matrices,” JEOS RP 2, 07018 (2007).
[CrossRef]

Holm, W. A.

W. A. Holm and R. M. Barnes, “On radar polarization mixed target state decomposition techniques,” in IEEE 1988 National Radar Conference (IEEE, 1988), pp. 249–254.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Kostinski, A. B.

C. R. Givens and A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

A. B. Kostinski and R. C. Givens, “On the gain of a passive linear depolarizing system,” J. Mod. Opt. 39, 1947–1952(1992).
[CrossRef]

Lavoura, L.

L. Lavoura, “Parametrization of the four-generation quark mixing matrix by the moduli of its matrix elements,” Phys. Rev. D 40, 2440–2448 (1989).
[CrossRef]

Lu, S.-Y.

Mallesh, K. S.

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).
[CrossRef]

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004). Available from http://www.unizar.es/galdeano/actas_pau/PDFVIII/pp161-167.pdf .

J. M. Correas, P. A. Melero, and J. J. Gil, “Decomposition of Mueller matrices into pure optical media,” Monografías del Seminario Matemático García de Galdeano 27, 23–240 (2003). Available from http://www.unizar.es/galdeano/actas_pau/PDF/233.pdf .

Morio, J.

Ossikovski, R.

Pottier, E.

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34, 498–518 (1996).
[CrossRef]

Red’kov, V. M.

V. M. Red’kov, A. A. Bogush, and N. G. Tokarevskaya, “On Parametrization of the linear GL(4,C) and unitary SU(4) groups in terms of Dirac matrices,” SIGMA 4, 021 (2008).

San Jose, I.

I. San Jose and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for n×n covariance matrices,” Opt. Commun. 284, 38–47 (2011).
[CrossRef]

Simon, R.

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod Opt. 41, 1903–1915 (1994).
[CrossRef]

Sridhar, R.

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod Opt. 41, 1903–1915 (1994).
[CrossRef]

Sudha,

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix,” J. Mod. Opt. 45, 955–987 (1998).
[CrossRef]

A. V. Gopala Rao, K. S. Mallesh, and Sudha, “On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones derived Mueller matrices,” J. Mod. Opt. 45, 989–999 (1998).
[CrossRef]

Tokarevskaya, N. G.

V. M. Red’kov, A. A. Bogush, and N. G. Tokarevskaya, “On Parametrization of the linear GL(4,C) and unitary SU(4) groups in terms of Dirac matrices,” SIGMA 4, 021 (2008).

Tudor, T.

T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
[CrossRef]

van der Mee, C. V. M.

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

Whitney, C.

Xing, Z.-F.

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

EPJ Appl. Phys. (1)

J. J. Gil, “Polarimetric characterization of light and media,” EPJ Appl. Phys. 40, 1–47 (2007).
[CrossRef]

EPJ Web Conf. (1)

J. J. Gil, “Parallel decompositions of Mueller matrices and polarimetric subtraction,” EPJ Web Conf. 5, 04002 (2010).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34, 498–518 (1996).
[CrossRef]

J. L. Alvarez-Perez, “Coherence, polarization, and statistical independence in Cloude–Pottier’s radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 49, 426–441 (2011).
[CrossRef]

J. Math. Phys. (1)

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

J. Mod Opt. (1)

R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod Opt. 41, 1903–1915 (1994).
[CrossRef]

J. Mod. Opt. (6)

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Equations (94)

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M=m00(1DTPm);D1m00(m01,m02,m03)T,P1m00(m10,m20,m30)T,m1m00(m11m12m13m21m22m23m31m32m33).
PΔi,j=03mij2m0023m002.
12{tr(TT)+[(tr(TT))24det(TT)]1/2}1,
T=TDTR=TRTD,
M(T)=MDMR=MRMD(MD=MR1MDMR),
M=MR2(MDMR)MR1=MR2(MRMD)MR1,
Mm00(1DTPm)=m00MΔPMRMD;MΔP(10TPΔPmΔP),MR(10T0mR),MD(1DTDmD),
M=MJ2MΔMJ1,
MΔd=diag(d0,d1,d2,εd3),di0,εdet(M)|det(M)|,
MΔnd=[2a0a000a000000a20000a2],0a2a0.
d01,
a01/3.
d0d1d2d3,d0d1+d2+d3,d0d1d2+d3,d0d1+d2d3,
0a2a0.
pi=IiI,ipi=1.
si=piMJis.
s=isi=i(piMJis)=Ms,M(ipiMJi);pi0,ipi=1.
H(M)=14k,l=03mkl(σkσl),
σ0=(1001),σ1=(1001),σ2=(0110),σ3=(0ii0).
mkl=tr[(σkσl)H].
H=Udiag(λ0,λ1,λ2,λ3)U,
H=λ0trH[Udiag(trH,0,0,0)U]+λ1trH[Udiag(0,trH,0,0)U]+λ2trH[Udiag(0,0,trH,0)U]+λ3trH[Udiag(0,0,0,trH)U],
H=i=03λitrHHi,Hi(trH)(uiui).
M=i=03λim00MJi.
H=i=03piHi;i=03pi=1,Hi=Hi=[(trH)(wiwi)],|wi|=1,rank(Hi)=1,trHi=trH,
pi=tr{diag(λ0,λ1,λ2,λ3)[wiwi]}/trH,pi>0,
M=i=03piMJi,pi>0,i=03pi=1,(MJi)00=m00,
ρ=(λ0λ1)Udiag(1,0,0)U+(λ1λ2)Udiag(1,1,0)U+λ2Udiag(1,1,1)U,
H=λ0λ1trHH0+2λ1λ2trHH1+3λ2λ3trHH2+4λ3trHH3H0trH[Udiag(1,0,0,0)U],H112trH[Udiag(1,1,0,0)U],H213trH[Udiag(1,1,1,0)U],H314trH[Udiag(1,1,1,1)U],
M=P1MJ0(H0)+(P2P1)M1(H1)+(P3P2)M2(H2)+(1P3)M3(H3),
M=MJ2MΔdMJ1,
H(MΔd)=14[d0+d100d2+d30d0d1d2d300d2d3d0d10d2+d300d0+d1],
H(MΔd)=WΛW;Λdiag(λ0,λ1,λ2,λ3),λ0λ1λ2λ30,W12[1100001100111100],λ0=14(d0+d1+d2+d3),λ1=14(d0+d1d2d3),λ2=14(d0d1+d2d3),λ3=14(d0d1d2+d3),
M=MJ2(i=03pΔd,iMΔd,i)MJ1=i=03pΔd,i(MJ2MΔd,iMJ1);i=03pΔd,i=1,(MJ2MΔd,iMJ1)00=m00,
M=i=03pi[MJ2(d0MRi)MJ1];i=03pi=1;d0(MJ2MRiMJ1)00=m00.
MΔd=i=03pi(d0MRi);pi=λid0,MR0diag(1,1,1,1),MR1=diag(1,1,1,1),MR2=diag(1,1,1,1),MR3=diag(1,1,1,1).
MO12(1000000000000000)=14{12(1100110000000000)+12(1100110000000000)+12(1100110000000000)+12(1100110000000000)}.
v0(cosα,sinαeiμ,0,0)T,v1(sinαeiμ,cosα,0,0)T,
MΔd2diag(d0,d0,d2,d2)=12(1+cosεcos2α){d01+cosεcos2α[1+cosεcos2αsinεsin2αcosμ00sinεsin2αcosμ1+cosεcos2α0000cosε+cos2αsinεsin2αsinμ00sinεsin2αsinμcosε+cos2α]}+12(1cosεcos2α){d01cosεcos2α[1cosεcos2αsinεsin2αcosμ00sinεsin2αcosμ1cosεcos2α0000cosεcos2αsinεsin2αsinμ00sinεsin2αsinμcosεcos2α]}.
d0+|d2cos2α|+d02d22|sin2αcosμ|1.
MΔd2diag(d0,d0,d2,d2)=12{d0[1000010000cosεsinε00sinεcosε]}+12{d0[1000010000cosεsinε00sinεcosε]}.
MΔd2diag(d0,d0,d2,d2)=12{d0[1sinε00sinε10000cosε0000cosε]}+12{d0[1sinε00sinε10000cosε0000cosε]}.
MΔd2diag(d0,d0,d2,d2)=d0+d22d0{d0[1000010000100001]}+d0d22d0{d0[1000010000100001]}.
v1(cosαcosβ,sinαcosβeiμ,sinβeiν,0)T,v2(cosαsinβsinγei(θ+ν+η)sinαcosγei(μ+φ),sinαsinβsinγei(θ+μνη)+cosαcosγeiφ,cosβsinγei(θ+η),0)T,v3(cosαsinβcosγei(ν+η)+sinαsinγei(θμφ),sinαsinβcosγei(μνη)cosαsinγei(θφ),cosβcosγeiη,0)T,
MΔd3=d0+d2q2d0[d0diag(1,1,1,1)]+d0d22d0[d0diag(1,1,1,1)]+q2d0[d0diag(1,1,1,1)].
v(cosαcosβcosγ,sinαcosβcosγeiμ,sinβcosγeiν,sinγeiη)T,
Eij=σiσj(i,j=0,1,2,3),exceptE00,
MΔd4=i=03pΔd4,iMΔd4,i(β=γ=ν=0),MΔd4,0(β=γ=ν=0)=d0c0[t+ucos2αrsin2αcosμ00rsin2αcosμt+ucos2α0000u+tcos2αrsin2αsinμ00rsin2αsinμu+tcos2α],MΔd4,1(β=γ=ν=0)=d0c1[tucos2αrsin2αcosμ00rsin2αcosμtucos2α0000utcos2αrsin2αsinμ00rsin2αsinμutcos2α],MΔd4,2(β=γ=ν=0)=d0diag(1,1,1,1),MΔd4,3(β=γ=ν=0)=d0diag(1,1,1,1),c0t+ucos2α,c1tucos2α,c2d0d1+d2d3=4λ2,c3d0d1d2+d3=4λ3,td0+d1=2(λ0+λ1),ud2+d3=2(λ0λ1),rt2u2=4λ0λ1,pΔd4,i=ci4d0(i=0,1,2,3).
M=P1(d0I)+(P2P1)M1+(P3P2)M2+(1P3)M3,Idiag(1,1,1,1),M1d0diag(1,1,0,0),M2d0diag(1,1/3,1/3,1/3),M3d0diag(1,0,0,0).
P1=12d2+d3d0,P2=122d1d2+d3d0,P3=d1+d2d3d0.
M1=12{d0[1sin2αcosμ00sin2αcosμ10000cos2αsin2αsinμ00sin2αsinμcos2α]}+12{d0[1sin2αcosμ00sin2αcosμ10000cos2αsin2αsinμ00+sin2αsinμcos2α]}
M=MJ2MΔndMJ1,
H(MΔnd)=12(a000a202a0000000a200a0),0a2a0,
H(MΔnd)=WΛW,Λdiag(λ0,λ1,λ2,λ3),λ0λ1λ2λ30,W[01/21/201000000101/21/20],λ0=a0,λ1=12(a0+a2),λ2=12(a0a2),λ3=0.
M=MJ2(i=02pΔnd,iMΔnd,i)MJ1=i=02pΔnd,i(MJ2MΔnd,iMJ1);i=02pΔnd,i=1,(MJ2MΔnd,iMJ1)00=m00,
MΔnd=i=02piMCi;p0=12,p1=14a0+a2a0,p2=14a0a2a0,MC02a0[1100110000000000],MC1=2a0[1000010000100001],MC2=2a0[1000010000100001].
v0(cosα,sinαeiμ,0,0)T,v1(sinαeiμ,cosα,0,0)T,
MΔnd2=12[2a0(22cos2αsin2αcosμsin2αsinμ2cos2α2cos2αsin2αcosμsin2αsinμsin2αcosμsin2αcosμ2sin2α0sin2αsinμsin2αsinμ02sin2α)]+12[2a0(22sin2αsin2αcosμsin2αsinμ2sin2α2cos2αsin2αcosμsin2αsinμsin2αcosμ+sin2αcosμ2cos2α0sin2αsinμ+sin2αsinμ02cos2α)].
a012[1+cosα(1+sin2α)1/2],a012[1+sinα(1+cos2α)1/2],
MΔnd2(α=π/4)=12[a0(212cosμ2sinμ102cosμ2sinμ2cosμ2cosμ102sinμ2sinμ01)]+12[a0(212cosμ2sinμ102cosμ2sinμ2cosμ+2cosμ102sinμ+2sinμ01)]
MΔnd2=12{2a0[1100110000000000]}+12{2a0[1000010000100001]},
MΔnd3=pΔnd3,0MΔnd3,0+(1pΔnd3,0)Mr;rankH(Mr)=2,(MΔnd3,0)00=(Mr)00=(MΔnd3)00=2a0,
MΔnd3=pΔnd3,0MΔnd3,0(α=π/2,β=0)+(1pΔnd3,0)Mr,MΔnd3,0(α=π/2,β=0)=2a0diag(1,1,1,1),Mr=2a0(3a0a2)[(3a0a2)2a0002a0(a0+a2)0000(a2a0)0000(a2a0)],pΔnd3,0=(a0+a2)/4a0.
MΔnd3=pΔnd3,0MΔnd3,0(β=π/2)+(1pΔnd3,0)Mr,MΔnd3,0(β=π/2)=2a0diag(1,1,1,1),Mr=2a03a0+a2[3a0+a22a0002a0a2a00000a0+a20000a0+a2],pΔnd3,0=a0a24a0,a0>a2.
MΔnd3=pΔnd3,0MΔnd3,0(α=β=0)+(1pΔnd3,0)Mr,MΔnd3,0(α=β=0)=2a0[1100110000000000],Mr=2diag(a0,a0,a2,a2),pa=1/2.
MΔnd3=a0a24a0{2a0[1100110000000000]}+a2a0{[2a0a000a000000a00000a0]}+3(a0a2)4a0{23[3a0a000a0a00000000000]}.
D1/2diag(λ0,,λk1,0,,0).
V=(Vk000)
D=D1/2VV+D1/2,
H=UDU=UD1/2VV+D1/2U=UD1/2(i=0k1(vivi))D1/2U=i=0k1(UD1/2vi)(UD1/2vi),
H=i=0k1pi[trH(wiwi)],
wiUD1/2viD1/2vi2(i=0.k1),
piD1/2vi22trH(i=0,,k1),
i=0k1pi=1.
MΔd3=i=02pΔd3,iMΔd3,i.
(MΔd3,0)00=kΔd3,0[d0cos2β+d2cos2αcos2βq(cos2αcos2βsin2β)],(MΔd3,0)11=kΔd3,0[d0cos2β+d2cos2αcos2βq(cos2αcos2β+sin2β)],(MΔd3,0)22=kΔd3,0[d0cos2αcos2β+d2cos2βq(cos2αcos2βsin2β)],(MΔd3,0)33=kΔd3,0[d0cos2αcos2β+d2cos2βq(cos2αcos2β+sin2β)],(MΔd3,0)01=(MΔd3,0)10=kΔd3,0(d02d22)q(d0d2)sin2αcos2βcosμ,(MΔd3,0)02=(MΔd3,0)20=kΔd3,0q(d0+d2q)cosαsin2βcosν,(MΔd3,0)03=(MΔd3,0)30=kΔd3,0q(d0d2)sinαsin2βsin(μν),(MΔd3,0)12=(MΔd3,0)21=kΔd3,0q(d0d2)sinαsin2βcos(μν),(MΔd3,0)13=(MΔd3,0)31=kΔd3,0q(d0+d2q)cosαsin2βsinν,(MΔd3,0)23=(MΔd3,0)32=kΔd3,0(d02d22)q(d0d2)sin2αcos2βsinμ,
kΔd3,0=d0/[d0cos2β+d2cos2αcos2βq(cos2αcos2βsin2β)]
pΔd3,0=1/(2kΔd3,0).
x1sin2αcos2γ+cos2αsin2βsin2γ,x1cos2αcos2γ+sin2αsin2βsin2γ,x2sin2αsin2γ+cos2αsin2βcos2γ,x2cos2αsin2γ+sin2αsin2βcos2γ,y12sin2αsinβsin2γcos(μ+ν+ηφ+θ),l1sin2αcosμ(sin2βsin2γcos2γ),l1sin2αsinμ(sin2βsin2γcos2γ),l2sin2αcosμ(sin2βcos2γsin2γ),l2sin2αsinμ(sin2βcos2γsin2γ),mcos2αsinβsin2γcos(ν+ηφ+θ),mcos2αsinβsin2γsin(ν+ηφ+θ),nsin2αsinβsin2γcos(2μ+ν+ηφ+θ),nsin2αsinβsin2γsin(2μ+ν+ηφ+θ),r1cosαsin2βsin2γcosν,r1cosαsin2βsin2γsinν,r2cosαsin2βcos2γcosν,r2cosαsin2βcos2γsinν,s1sinαsin2βsinγ2sin(μν),s1sinαsin2βsinγ2cos(μν),s2sinαsin2βcosγ2sin(μν),s2sinαsin2βcosγ2cos(μν),tsinαcosβsin2γcos(μ+ηφ+θ),tsinαcosβsin2γsin(μ+ηφ+θ),ucosαcosβsin2γsin(ηφ+θ),ucosαcosβsin2γcos(ηφ+θ).
(MΔd3,1)00=kΔd3,1[(d0+d2q)(x1+y)+(d0d2)(x1y)+qcos2βsin2γ],(MΔd3,1)11=kΔd3,1[(d0+d2q)(x1+y)+(d0d2)(x1y)qcos2βsin2γ],(MΔd3,1)22=kΔd3,1[(d0+d2q)(x1+y)(d0d2)(x1y)+qcos2βsin2γ],(MΔd3,1)33=kΔd3,1[(d0+d2q)(x1+y)(d0d2)(x1y)qcos2βsin2γ],(MΔd3,1)01=(MΔd3,1)10=kΔd3,1(d02d22)q(d0d2)[l1m+n],(MΔd3,1)02=(MΔd3,1)20=kΔd3,1q(d0+d2q)[r1+t],(MΔd3,1)03=(MΔd3,1)30=kΔd3,1q(d0d2)[s1+u],(MΔd3,1)12=(MΔd3,1)21=kΔd3,1q(d0d2)[s1+u],(MΔd3,1)13=(MΔd3,1)31=kΔd3,1q(d0+d2q)[r1t],(MΔd3,1)23=(MΔd3,1)32=kΔd3,1(d02d22)q(d0d2)[l1mn],
pΔd3,1=12kΔd3,1=12d0[(d0+d2q)(x1+y)+(d0d2)(x1y)+qcos2βsin2γ].
(MΔd3,2)00=kΔd3,2[(d0+d2q)(x2y)+(d0d2)(x2+y)+qcos2βcos2γ],(MΔd3,2)11=kΔd3,2[(d0+d2q)(x2y)+(d0d2)(x2+y)qcos2βcos2γ],(MΔd3,2)22=kΔd3,2[(d0+d2q)(x2y)(d0d2)(x2+y)+qcos2βcos2γ],(MΔd3,2)33=kΔd3,2[(d0+d2q)(x2y)(d0d2)(x2+y)qcos2βcos2γ],(MΔd3,2)01=(MΔd3,2)10=kΔd3,2(d02d22)q(d0d2)(l2+mn),(MΔd3,2)02=(MΔd3,2)20=kΔd3,2q(d0+d2q)(r2+t),(MΔd3,2)03=(MΔd3,2)30=kΔd3,2q(d0d2)(s2+u),(MΔd3,2)12=(MΔd3,2)21=kΔd3,2q(d0d2)(s2+u),(MΔd3,2)13=(MΔd3,2)31=kΔd3,2q(d0+d2q)(r2+t),(MΔd3,2)23=(MΔd3,2)32=kΔd3,2(d02d22)q(d0d2)(l2+m+n),
pΔd3,2=12kΔd3,2=12d0[(d0+d2q)(x2y)+(d0d2)(x2+y)+qcos2βcos2γ].
MΔd4=pΔd4,0MΔd4,0+(1pΔd4,0)Mr,rankH(Mr)=3,(MΔd4,0)00=(Mr)00=(MΔd4)00=d0.
(MΔd4,0)00=4kΔd4,0[λ0cos2αcos2βcos2γ+λ1sin2αcos2βcos2γ+λ2sin2βcos2γ+λ3sin2γ],(MΔd4,0)11=4kΔd4,0[λ0cos2αcos2βcos2γ+λ1sin2αcos2βcos2γλ2sin2βcos2γλ3sin2γ],(MΔd4,0)22=4kΔd4,0[λ0cos2αcos2βcos2γλ1sin2αcos2βcos2γ+λ2sin2βcos2γλ3sin2γ],(MΔd4,0)33=4kΔd4,0[λ0cos2αcos2βcos2γλ1sin2αcos2βcos2γλ2sin2βcos2γ+λ3sin2γ],(MΔd4,0)01=kΔd4,0[(d0+d1)2(d2+d3)2sin2αcos2βcos2γcosμ(d0d1)2(d2d3)2sinβsin2γcos(νη)],(MΔd4,0)10=kΔd4,0[(d0+d1)2(d2+d3)2sin2αcos2βcos2γcosμ+(d0d1)2(d2d3)2sinβsin2γcos(νη)],(MΔd4,0)02=kΔd4,0[(d0+d2)2(d1+d3)2cosαsin2βcos2γcosν+(d0d2)2(d1d3)2sinαcosβsin2γcos(μη)],(MΔd4,0)20=kΔd4,0[(d0+d2)2(d1+d3)2cosαsin2βcos2γcosν(d0d2)2(d1d3)2sinαcosβsin2γcos(μη)],(MΔd4,0)03=kΔd4,0[(d0d3)2(d1d2)2sinαsin2βcos2γsin(μν)(d0+d3)2(d1+d2)2cosαcosβsin2γsinη],(MΔd4,0)30=kΔd4,0[(d0d3)2(d1d2)2sinαsin2βcos2γsin(μν)(d0+d3)2(d1+d2)2cosαcosβsin2γsinη],(MΔd4,0)12=kΔd4,0[(d0d3)2(d1d2)2sinαsin2βcos2γcos(μν)+(d0+d3)2(d1+d2)2cosαcosβsin2γcosη],(MΔd4,0)21=kΔd4,0[(d0d3)2(d1d2)2sinαsin2βcos2γcos(μν)(d0+d3)2(d1+d2)2cosαcosβsin2γcosη],(MΔd4,0)13=kΔd4,0[(d0+d2)2(d1+d3)2cosαsin2βcos2γsinν+(d0d2)2(d1d3)2sinαcosβsin2γsin(μη)],(MΔd4,0)31=kΔd4,0[(d0+d2)2(d1+d3)2cosαsin2βcos2γsinν(d0d2)2(d1d3)2sinαcosβsin2γsin(μη)],(MΔd4,0)23=kΔd4,0[(d0+d1)2(d2+d3)2sin2αcos2βcos2γsinμ+(d0d1)2(d2d3)2sinβsin2γsin(νη)],(MΔd4,0)32=kΔd4,0[(d0+d1)2(d2+d3)2sin2αcos2βcos2γsinμ(d0d1)2(d2d3)2sinβsin2γsin(νη)],
kΔd4,0=d0/[4(λ0cos2αcos2βcos2γ+λ1sin2αcos2βcos2γ+λ2sin2βcos2γ+λ3sin2γ)]
pΔd4,0=1/(2kΔd4,0).
(MΔnd3,0)00=kΔnd3,0[2a0cos2αcos2β+(a0+a2)sin2αcos2β+(a0a2)sin2β],(MΔnd3,0)11=kΔnd3,0[2a0cos2αcos2β+(a0+a2)sin2αcos2β+(a0a2)sin2β],(MΔnd3,0)22=kΔnd3,0[(a0+a2)sin2αcos2β(a0a2)sin2β],(MΔnd3,0)33=kΔnd3,0[(a0+a2)sin2αcos2β(a0a2)sin2β],(MΔnd3,0)01=kΔnd3,0[2a0cos2αcos2β+(a02a22)sinαsin2βcos(μν)],(MΔnd3,0)10=kΔnd3,0[2a0cos2αcos2β+(a02a22)sinαsin2βcos(μν)],(MΔnd3,0)02=kΔnd3,0[a0(a0+a2)sin2αcos2βcosμ+a0(a0a2)cosαsin2βcosν],(MΔnd3,0)20=kΔnd3,0[a0(a0+a2)sin2αcos2βcosμa0(a0a2)cosαsin2βcosν],(MΔnd3,0)03=kΔnd3,0[a0(a0+a2)sin2αcos2βsinμ+a0(a0a2)cosαsin2βsinν],(MΔnd3,0)30=kΔnd3,0[a0(a0+a2)sin2αcos2βsinμa0(a0a2)cosαsin2βsinν],(MΔnd3,0)12=kΔnd3,0[a0(a0+a2)sin2αcos2βcosμ+a0(a0a2)cosαsin2βcosν],(MΔnd3,0)21=kΔnd3,0[a0(a0+a2)sin2αcos2βcosμ+a0(a0a2)cosαsin2βcosν],(MΔnd3,0)13=kΔnd3,0[a0(a0+a2)sin2αcos2βsinμ+a0(a0a2)cosαsin2βsinν],(MΔnd3,0)31=kΔnd3,0[a0(a0+a2)sin2αcos2βsinμ+a0(a0a2)cosαsin2βsinν],(MΔnd3,0)23=(MΔnd3,0)32=kΔnd3,0[(a02a22)sinαsin2βsin(μν)],
kΔnd3,0=2a0/[2a0cos2αcos2β+(a0+a2)sin2αcos2β+(a0a2)sin2β]
pΔnd3,0=1/(2kΔnd3,0).
(MΔnd3,1)00=kΔnd3,1[2a0(x1+y)+(a0+a2)(x1y)+(a0a2)cos2βsin2γ],(MΔnd3,1)11=kΔnd3,1[2a0(x1+y)+(a0+a2)(x1y)+(a0a2)cos2βsin2γ],(MΔnd3,1)22=(MΔnd3,1)33=kΔnd3,1[(a0+a2)(x1y)(a0a2)cos2βsin2γ],(MΔnd3,1)01=kΔnd3,1[2a0(x1+y)+(a02a22)(us1)],(MΔnd3,1)10=kΔnd3,1[2a0(x1+y)+(a02a22)(us)],(MΔnd3,1)02=kΔnd3,112[a0(a0+a2)(l1m+n)a0(a0a2)(r1+t)],(MΔnd3,1)20=kΔnd3,112[a0(a0+a2)(l1m+n)+a0(a0a2)(r1+t)],(MΔnd3,1)03=kΔnd3,112[a0(a0+a2)(l1mn)a0(a0a2)(r1t)],(MΔnd3,1)30=kΔnd3,112[a0(a0+a2)(l1mn)+a0(a0a2)(r1t)],(MΔnd3,1)12=kΔnd3,112[a0(a0+a2)(l1m+n)a0(a0a2)(r1+t)],(MΔnd3,1)21=kΔnd3,112[a0(a0+a2)(l1m+n)a0(a0a2)(r1+t)],(MΔnd3,1)13=kΔnd3,112[a0(a0+a2)(l1mn)a0(a0a2)(r1t)],(MΔnd3,1)31=kΔnd3,112[a0(a0+a2)(l1mn)a0(a0a2)(r1t)],(MΔnd3,1)23=(MΔnd3,1)32=kΔnd3,1(a02a22)(s1u),
pΔnd3,1=12kΔnd3,1=14a0[2a0(x1+y)+(a0+a2)(x1y)+(a0a2)cos2βsin2γ].
(MΔnd3,2)00=kΔnd3,2[2a0(x2y)+(a0+a2)(x2+y)+(a0a2)cos2βcos2γ],(MΔnd3,2)11=kΔnd3,2[2a0(x2y)+(a0+a2)(x2+y)+(a0a2)cos2βcos2γ],(MΔnd3,2)22=(MΔnd3,2)33=kΔnd3,2[(a0+a2)(x2+y)(a0a2)cos2βcos2γ],(MΔnd3,2)01=kΔnd3,2[2a0(x2y)(a02a22)(u+s2)],(MΔnd3,2)10=kΔnd3,2[2a0(x2y)(a02a22)(u+s2)],(MΔnd3,2)02=kΔnd3,212[a0(a0+a2)(l2+mn)a0(a0a2)(r2t)],(MΔnd3,2)20=kΔnd3,212[a0(a0+a2)(l2+mn)+a0(a0a2)(r2t)],(MΔnd3,2)03=kΔnd3,212[a0(a0+a2)(l2+m+n)a0(a0a2)(r2+t)],(MΔnd3,2)30=kΔnd3,212[a0(a0+a2)(l2+m+n)+a0(a0a2)(r2+t)],(MΔnd3,2)12=kΔnd3,212[a0(a0+a2)(l2+mn)a0(a0a2)(r2t)],(MΔnd3,2)21=kΔnd3,212[a0(a0+a2)(l2+mn)a0(a0a2)(r2t)],(MΔnd3,2)13=kΔnd3,212[a0(a0+a2)(l2+m+n)a0(a0a2)(r2+t)],(MΔnd3,2)31=kΔnd3,212[a0(a0+a2)(l2+m+n)a0(a0a2)(r2+t)],(MΔnd3,2)23=(MΔnd3,2)32=kΔnd3,2(a02a22)(s2+u),
pΔnd3,2=12kΔnd3,2=14a0[2a0(x2y)+(a0+a2)(x2+y)+(a0a2)cos2βcos2γ].

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