Abstract

The goal of this paper is to propose a mathematical framework to define and analyze a general parametric form of an arbitrary nonsingular Mueller matrix. Starting from previous results about nondepolarizing matrices, we generalize the method to any nonsingular Mueller matrix. We address this problem in a six-dimensional space in order to introduce a transformation group with the same number of degrees of freedom and explain why subsets of O(5,1), the orthogonal group associated with six-dimensional Minkowski space, is a physically admissible solution to this question. Generators of this group are used to define possible expressions of an arbitrary nonsingular Mueller matrix. Ultimately, the problem of decomposition of these matrices is addressed, and we point out that the “reverse” and “forward” decomposition concepts recently introduced may be inferred from the formalism we propose.

© 2008 Optical Society of America

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References

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  1. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
    [CrossRef]
  2. Z. F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461-484 (1992).
    [CrossRef]
  3. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]
  4. J. W. Hovenier, “Structure of a general pure Mueller matrix,” Appl. Opt. 33, 8318-8324 (1994).
    [CrossRef] [PubMed]
  5. A. V. Gopala, K. S. Mallesh, and J. 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]
  6. M. S. Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
    [CrossRef]
  7. A. B. Kostinski, C. R. Given, and J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646-1651 (1993).
    [CrossRef] [PubMed]
  8. C. R. Givens and B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471-481 (1993).
    [CrossRef]
  9. R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).
  10. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
    [CrossRef]
  11. D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
    [CrossRef]
  12. C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248-2251 (1993).
    [CrossRef]
  13. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26-36 (1986).
  14. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
    [CrossRef]
  15. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328-334 (2000).
    [CrossRef]
  16. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67-71 (1987).
  17. 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]
  18. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition,” J. Opt. Soc. Am. A 25, 473-482 (2008).
    [CrossRef]
  19. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller polarimetric imaging system with liquid crystals,” Appl. Opt. 43, 2824-2832 (2004).
    [CrossRef] [PubMed]
  20. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688-1703 (2006).
    [CrossRef] [PubMed]
  21. P. Pellat Finet, “Geometrical approach to polarization optics: quaternionic representation of the polarized light,” Optik (Stuttgart) 87, 68-76 (1991).
  22. S. Baskal and Y. S. Kim, “De Sitter group as a symmetry for optical decoherence,” J. Phys. A 39, 7775-7788 (2006).
    [CrossRef]
  23. H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37-41 (1973).
    [CrossRef]
  24. Sudha and A. V. Gopala Rao, “Polarization elements, a group theoretical study,” J. Opt. Soc. Am. A 18, 3130-3134 (2001).
    [CrossRef]
  25. D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
    [CrossRef]
  26. R. Ossikovski, A. De Martino, and S. Guyot, “Forward and reverse product decompositions of depolarizing Mueller matrices,” Opt. Lett. 32, 689-691 (2007).
    [CrossRef] [PubMed]
  27. S. Sternberg, Group Theory and Physics (Cambridge U. Press, 1994).
  28. J. Bognar, Indefinite Inner Product Spaces (Springer, 1974).
  29. I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products (Biikhauser OT 8, 1983).
  30. D. H. Sattiger and O. L. Weaver, Lie Group and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1991).
  31. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).
  32. K. D. Abhyankar and A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
    [CrossRef]
  33. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
    [CrossRef]
  34. W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).
  35. A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras (Academic, 1973).
  36. 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] [PubMed]

2008 (1)

2007 (1)

2006 (2)

J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688-1703 (2006).
[CrossRef] [PubMed]

S. Baskal and Y. S. Kim, “De Sitter group as a symmetry for optical decoherence,” J. Phys. A 39, 7775-7788 (2006).
[CrossRef]

2004 (2)

2001 (1)

2000 (1)

1998 (1)

A. V. Gopala, K. S. Mallesh, and J. 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]

1997 (1)

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
[CrossRef]

1996 (1)

1994 (3)

1993 (4)

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

C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248-2251 (1993).
[CrossRef]

A. B. Kostinski, C. R. Given, and J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646-1651 (1993).
[CrossRef] [PubMed]

1992 (2)

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

M. S. Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

1991 (1)

P. Pellat Finet, “Geometrical approach to polarization optics: quaternionic representation of the polarized light,” Optik (Stuttgart) 87, 68-76 (1991).

1987 (2)

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

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433-437 (1987).
[CrossRef]

1986 (1)

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

1982 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

1973 (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37-41 (1973).
[CrossRef]

1969 (1)

K. D. Abhyankar and A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

Abhyankar, K. D.

K. D. Abhyankar and A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

Anderson, D. G. M.

Barakat, R.

D. G. M. Anderson and R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305-2319 (1994).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Baskal, S.

S. Baskal and Y. S. Kim, “De Sitter group as a symmetry for optical decoherence,” J. Phys. A 39, 7775-7788 (2006).
[CrossRef]

Bernabeu, E.

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

R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).

Bognar, J.

J. Bognar, Indefinite Inner Product Spaces (Springer, 1974).

Brosseau, C.

Chipman, R. A.

Cloude, S. R.

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

De Martino, A.

Drévillon, B.

Espinosa-Luna, R.

R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).

Fymat, A. L.

K. D. Abhyankar and A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

Gil, J. J.

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

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

Given, C. R.

Givens, C. R.

C. Brosseau, C. R. Givens, and A. B. Kotinski, “Generalized trace condition on the Mueller-Jones polarization matrix,” J. Opt. Soc. Am. A 10, 2248-2251 (1993).
[CrossRef]

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

Gohberg, I.

I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products (Biikhauser OT 8, 1983).

Gopala, A. V.

A. V. Gopala, K. S. Mallesh, and J. 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]

Gopala Rao, A. V.

Goudail, F.

Guyot, S.

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
[CrossRef]

Hinojosa-Ruiz, S.

R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).

Hovenier, J. W.

Kim, K.

Kim, Y. S.

S. Baskal and Y. S. Kim, “De Sitter group as a symmetry for optical decoherence,” J. Phys. A 39, 7775-7788 (2006).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
[CrossRef]

Kostinski, A. B.

Kostinski, B.

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

Kotinski, A. B.

Kumar, M. S.

M. S. Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

Kwiatkowski, J. M.

Lancaster, P.

I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products (Biikhauser OT 8, 1983).

Laude-Boulesteix, B.

Lu, S. Y.

Mallesh, K. S.

A. V. Gopala, K. S. Mallesh, and J. 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]

Mandel, L.

Morio, J.

Noz, M. E.

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
[CrossRef]

Ossikovski, R.

Pellat Finet, P.

P. Pellat Finet, “Geometrical approach to polarization optics: quaternionic representation of the polarized light,” Optik (Stuttgart) 87, 68-76 (1991).

Rodman, L.

I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products (Biikhauser OT 8, 1983).

Rodriguez-Carrera, D.

R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).

Sagle, A. A.

A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras (Academic, 1973).

Sattiger, D. H.

D. H. Sattiger and O. L. Weaver, Lie Group and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1991).

Schwartz, L.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

Simon, R.

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

M. S. Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

Sridhar, R.

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

Sternberg, S.

S. Sternberg, Group Theory and Physics (Cambridge U. Press, 1994).

Sudha,

Sudha, J.

A. V. Gopala, K. S. Mallesh, and J. 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]

Takenaka, H.

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37-41 (1973).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

van der Mee, V. M.

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

Walde, R. E.

A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras (Academic, 1973).

Weaver, O. L.

D. H. Sattiger and O. L. Weaver, Lie Group and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1991).

Wolf, E.

Wolfe, J. E.

Xing, Z. F.

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

Appl. Opt. (4)

J. Math. Phys. (2)

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

K. D. Abhyankar and A. L. Fymat, “Relations between the elements of the phase matrix for scattering,” J. Math. Phys. 10, 1935-1938 (1969).
[CrossRef]

J. Mod. Opt. (4)

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

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

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

A. V. Gopala, K. S. Mallesh, and J. 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]

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

J. Phys. A (1)

S. Baskal and Y. S. Kim, “De Sitter group as a symmetry for optical decoherence,” J. Phys. A 39, 7775-7788 (2006).
[CrossRef]

Nouv. Rev. Opt. (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouv. Rev. Opt. 4, 37-41 (1973).
[CrossRef]

Opt. Commun. (3)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293-297 (1982).
[CrossRef]

M. S. Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464-470 (1992).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (4)

P. Pellat Finet, “Geometrical approach to polarization optics: quaternionic representation of the polarized light,” Optik (Stuttgart) 87, 68-76 (1991).

R. Espinosa-Luna, D. Rodrıguez-Carrera, E. Bernabeu, and S. Hinojosa-Ruız, “Transformation matrices for the Mueller-Jones formalism,” Optik (Stuttgart) (to be published).

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

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

Phys. Rev. E (1)

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as Minkowskian four-vectors,” Phys. Rev. E 56, 6065-6076 (1997).
[CrossRef]

Other (7)

S. Sternberg, Group Theory and Physics (Cambridge U. Press, 1994).

J. Bognar, Indefinite Inner Product Spaces (Springer, 1974).

I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products (Biikhauser OT 8, 1983).

D. H. Sattiger and O. L. Weaver, Lie Group and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1991).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

W. A. Shurcliff, Polarized Light (Harvard U. Press, 1962).

A. A. Sagle and R. E. Walde, Introduction to Lie Groups and Lie Algebras (Academic, 1973).

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Equations (58)

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[ x , x ] = g μ ν x μ x ν ,
g μ ν Γ λ μ Γ σ ν = g λ σ or Γ λ μ Γ σ ν g λ σ = g μ ν .
R = ( 1 [ 0 ] [ 0 ] T [ r ] ) ,
R = ( cosh ( u ) 0 0 sinh ( u ) 0 1 0 0 0 0 1 0 sinh ( u ) 0 0 cosh ( u ) ) .
Γ = R ( φ , θ , ψ ) L 3 ( u ) R 1 ( α , β , 0 ) ,
M = k [ R ( φ , θ , ψ ) L 3 ( 0 ) R 1 ( α , β , 0 ) ] [ R ( α , β , 0 ) L 3 ( u ) R 1 ( α , β , 0 ) ] ,
( Γ 0 0 ) 2 i = 1 3 ( Γ i 0 ) 2 + g 44 ( Γ 4 0 ) 2 + g 55 ( Γ 5 0 ) 2 = 1 .
P = ( Γ 1 0 ) 2 + ( Γ 2 0 ) 2 + ( Γ 3 0 ) 2 Γ 0 0 .
P 2 = 1 1 g 44 ( Γ 4 0 ) 2 g 55 ( Γ 5 0 ) 2 ( Γ 0 0 ) 2 .
J 03 = ( 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )
L 03 ( u ) = e i u J 03 = ( cosh ( u ) 0 0 sinh ( u ) 0 0 0 1 0 0 0 0 0 0 1 0 0 0 sinh ( u ) 0 0 cosh ( u ) 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ) ,
J 12 = ( 0 0 0 0 0 0 0 0 i 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )
R 12 ( ρ ) = e i ρ J 12 = ( 1 0 0 0 0 0 0 cos ( ρ ) sin ( ρ ) 0 0 0 0 sin ( ρ ) cos ( ρ ) 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ) ,
J 13 = ( 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )
R 13 ( ρ ) = e i ρ J 13 = ( 1 0 0 0 0 0 0 cos ( ρ ) 0 sin ( ρ ) 0 0 0 0 1 0 0 0 0 sin ( ρ ) 0 cos ( ρ ) 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ) ,
J 23 = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )
R 23 ( ρ ) = e i ρ J 23 = ( 1 0 0 0 0 0 0 1 0 0 0 0 0 0 cos ( ρ ) sin ( ρ ) 0 0 0 0 sin ( ρ ) cos ( ρ ) 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ) .
T = e ( i 2 ) α m n J m n ,
[ J 12 , J 13 ] = i J 23 , [ J 12 , J 23 ] = i J 13 , [ J 13 , J 23 ] = i J 12 ,
[ J 04 , J 05 ] = i J 45 , [ J 05 , J 45 ] = i J 04 , [ J 04 , J 45 ] = i J 05 ,
[ J h 4 , J k m ] = i ε h k m J m 4 , h , k , m { 1 , 2 , 3 } ,
[ J h 5 , J k m ] = i ε h k m J m 5 , h , k , m { 1 , 2 , 3 } ,
[ J 0 h , J k m ] = i ε h k m J 0 m , h , k , m { 1 , 2 , 3 } ,
R J j 4 R 1 = J k 4 R j k ,
R J j 5 R 1 = J k 5 R j k ,
R J 0 j R 1 = J 0 k R j k ,
J n 4 = R J 34 R 1 = J k 4 R ( α , β , 0 ) 3 k = J k 4 n k ,
J n 5 = R J 35 R 1 = J k 5 R ( α , β , 0 ) 3 k = J k 5 n k ,
J 0 n = R J 03 R 1 = J 0 k R ( α , β , 0 ) 3 k = J 0 k n k ,
J n = R J 12 R 1 = J k R ( α , β , 0 ) 3 k = J k n k ,
T R = [ R ( φ , θ , 0 ) R 12 ( ψ ) R 1 ( φ , θ , 0 ) ] ,
T D = [ R ( α , β , 0 ) L 03 ( u ) R 1 ( α , β , 0 ) ] ,
T Δ 1 2 3 4 = [ R ( γ , δ , 0 ) R 34 ( μ ) R 1 ( γ , δ , 0 ) ] ,
T Δ 1 2 3 5 = [ R ( ε , η , 0 ) R 35 ( ν ) R 1 ( ε , η , 0 ) ] ,
T Δ 045 = [ L 0 4 ( v ) R 45 ( τ ) L 0 5 ( w ) ] ,
T R T D = ( M R M D 0 0 I 2 ) T D T R = ( M D M R 0 0 I 2 ) ,
T Δ = ( M Δ X Y Z ) .
M Δ 1 2 3 4 = ( 1 [ 0 ] [ 0 ] T m Δ 1 2 3 4 ) , M Δ 1 2 3 5 = ( 1 [ 0 ] [ 0 ] T m Δ 1 2 3 5 ) ,
M Δ 0 4 5 = ( a [ 0 ] [ 0 ] T I 3 ) ,
M Δ 12345 = ( 1 [ 0 ] [ 0 ] T m Δ 12345 ) ,
T Δ 12345 = [ R 12 ( ε ) R 13 ( η ) ] [ R 23 ( φ ) R 45 ( ψ ) ] R 34 ( μ ) R 25 ( ν ) [ R 23 ( φ ) R 45 ( ψ ) ] 1 [ R 12 ( ε ) R 13 ( η ) ] 1 .
a = cosh ( v ) cosh ( w ) .
T Δ 12345 T Δ 045 = ( M Δ 12345 G H J ) ( M Δ 045 K N Q ) = ( M Δ 12345 M Δ 045 + G N X Y Z ) ,
T Δ 045 T Δ 12345 = ( M Δ 045 K N Q ) ( M Δ 12345 G H J ) = ( M Δ 045 M Δ 12345 + K H X Y Z ) ,
M Δ f = M Δ 12345 M Δ 045 + G N ,
M Δ r = M Δ 045 M Δ 12345 + K H ,
G N = ( 0 [ 0 ] [ P Δ ] T [ 0 ] 3 ) , K H = ( 0 D Δ [ 0 ] T [ 0 ] 3 ) ,
M Δ f = ( a [ 0 ] [ P Δ ] T m Δ 12345 ) or M Δ r = ( a D Δ [ 0 ] T m Δ 12345 ) ,
1 cosh ( v ) cosh ( w ) , cos ( μ ) cosh ( v ) cosh ( w ) , cos ( ν ) cosh ( v ) cosh ( w ) .
M D M R M Δ r = ( a D Δ + D m R m Δ a D T D T D Δ + m D m R m Δ ) ,
M Δ f M R M D = ( a a D P Δ T + m Δ m R D T P Δ T D + m Δ m R m D ) ,
J = i ( i = 1 3 α i J i 4 + k = 1 3 β k J k 5 ) .
J = ( [ 0 ] 4 A T A [ 0 ] 2 ) with A = ( 0 a 12 a 13 a 14 0 a 22 a 23 a 24 ) , a k l R ,
J 2 = ( A T A [ 0 ] 2 , 4 T [ 0 ] 2 , 4 A A T ) ,
J 2 k = ( ( 1 ) k ( A T A ) k [ 0 ] 2 , 4 T [ 0 ] 2 , 4 ( 1 ) k ( A A T ) k ) ,
J 2 k + 1 = ( [ 0 ] 4 ( 1 ) k + 1 ( A T A ) k A T ( 1 ) k ( A A T ) k A [ 0 ] 2 ) .
e J = m = 0 + J m m ! .
( 1 [ 0 ] [ 0 ] T m ) ,

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