Abstract

Anisotropy coefficients α, β, and γ that describe the type and the relative amount of the three kinds of anisotropy generally present in a Mueller matrix are introduced. Their derivation, algebraic properties, and physical interpretation are discussed. In particular, they are shown to permit a geometrical representation for the anisotropy and polarizing characteristics of a Mueller matrix. Illustrative experimental examples are provided.

© 2011 Optical Society of America

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References

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  1. O. Arteaga and A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media,” Opt. Lett. 35, 559–561 (2010).
    [CrossRef] [PubMed]
  2. O. Arteaga and A. Canillas, “Analytic inversion of the Mueller-Jones polarization matrices for homogeneous media: erratum,” Opt. Lett. 35, 3525 (2010).
    [CrossRef]
  3. J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix,” Optik 76, 67–71(1987).
  4. C. R. Jones, “New calculus for the treatment of optical systems. VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685(1948).
    [CrossRef]
  5. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767(1978).
    [CrossRef]
  6. J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
    [CrossRef]
  7. A. S. Marathay, “Operator formalism in the theory of partial polarization,” J. Opt. Soc. Am. 55, 969–977 (1965).
    [CrossRef]
  8. O. Arteaga, “On the existence of Jones birefringence and Jones dichroism,” Opt. Lett. 35, 1359–1360 (2010).
    [CrossRef] [PubMed]
  9. H. Tompkins and E. A. Irene, Handbook of Ellipsometry (Materials Science and Process Technology) (William Andrew, 2006).
  10. O. Arteaga and A. Canillas, “Pseudopolar decomposition of the Jones and Mueller-Jones exponential polarization matrices,” J. Opt. Soc. Am. A 26, 783–793 (2009), Appendix A.
    [CrossRef]
  11. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).
    [CrossRef]
  12. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” J. Mod. Opt. 33, 185–189 (1986).
    [CrossRef]
  13. R. Ossikovski, “Alternative depolarization criteria for Mueller matrices,” J. Opt. Soc. Am. A 27, 808–814 (2010).
    [CrossRef]
  14. O. Arteaga, A. Canillas, and G. E. Jellison, “Determination of the components of the gyration tensor of quartz by oblique incidence transmission two-modulator generalized ellipsometry,” Appl. Opt. 48, 5307–5317 (2009).
    [CrossRef] [PubMed]

2010 (4)

2009 (2)

1987 (2)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix,” Optik 76, 67–71(1987).

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[CrossRef]

1986 (1)

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

1985 (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).
[CrossRef]

1978 (1)

1965 (1)

1948 (1)

Arteaga, O.

Azzam, R. M.

Bernabeu, E.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix,” Optik 76, 67–71(1987).

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

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).
[CrossRef]

Canillas, A.

Gil, J. J.

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix,” Optik 76, 67–71(1987).

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

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).
[CrossRef]

Irene, E. A.

H. Tompkins and E. A. Irene, Handbook of Ellipsometry (Materials Science and Process Technology) (William Andrew, 2006).

Jellison, G. E.

Jensen, H. P.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[CrossRef]

Jones, C. R.

Marathay, A. S.

Ossikovski, R.

Schellman, J.

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[CrossRef]

Tompkins, H.

H. Tompkins and E. A. Irene, Handbook of Ellipsometry (Materials Science and Process Technology) (William Andrew, 2006).

Appl. Opt. (1)

Chem. Rev. (1)

J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87, 1359–1399 (1987).
[CrossRef]

J. Mod. Opt. (2)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259–261 (1985).
[CrossRef]

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

J. Opt. Soc. Am. (3)

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

Opt. Lett. (3)

Optik (1)

J. J. Gil and E. Bernabeu, “Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix,” Optik 76, 67–71(1987).

Other (1)

H. Tompkins and E. A. Irene, Handbook of Ellipsometry (Materials Science and Process Technology) (William Andrew, 2006).

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Figures (6)

Fig. 1
Fig. 1

Poincaré sphere showing the states of polarization along the coordinate axes. H stands for linear horizontal polarization, V for linear vertical polarization, 45 ° for 45 ° linear polarization with respect to the horizontal, 135 ° for 135 ° linear polarization with respect to the horizontal, RH for right-handed circular polarization, and LH for left-handed circular polarization.

Fig. 2
Fig. 2

Rotations in the space of anisotropies. L stands for horizontal linear anisotropy, L for 45 ° linear anisotropy, and C for circular anisotropy. Combinations of elements invariant under the rotation of a given type of anisotropy belong to this type of anisotropy.

Fig. 3
Fig. 3

Ternary geometric representation of α 2 , β 2 , and γ 2 . Each vertex of the triangle corresponds to an elementary matrix M L , M L , or M C .

Fig. 4
Fig. 4

The signed α, β, and γ coefficients define a sphere. Any nondepolarizing Mueller matrix is represented by a point of the surface of this sphere.

Fig. 5
Fig. 5

Spectroscopic α, β, and γ of a thin ( 1 mm ) α-quartz plate measured at various different small angles of incidence. The optical axis of the crystal was perpendicular to the surface of the plate, so at normal incidence, there is only the circular anisotropy produced by the optical activity of quartz.

Fig. 6
Fig. 6

α, β, and γ were calculated after polarimetric measurements of a stretched Melinex polyester film in the infrared. Mueller matrix measurements were performed at different angles of azimuthal rotation of the film.

Equations (20)

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M L = [ 1 m 01 0 0 m 10 1 0 0 0 0 m 22 m 23 0 0 m 32 m 33 ] ,
M L = [ 1 0 m 02 0 0 m 11 0 m 13 m 20 0 1 0 0 m 31 0 m 33 ] ,
M C = [ 1 0 0 m 03 0 m 11 m 12 0 0 m 21 m 22 0 m 30 0 0 1 ] ,
M L ( | S 1 | S 1 0 0 ) = ( | S 1 | S 1 0 0 ) ,
M L ( | S 2 | 0 S 2 0 ) = ( | S 2 | 0 S 2 0 ) ,
M L ( | S 3 | 0 0 S 3 ) = ( | S 3 | 0 0 S 3 ) .
1 1 A 2 [ 1 A 0 0 A 1 0 0 0 0 C B 0 0 B C ] [ m 00 m 01 m 02 m 03 m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ] [ 1 A 0 0 A 1 0 0 0 0 C B 0 0 B C ] = 1 1 A 2 [ m 00 + A ( m 01 m 10 A m 11 ) m 01 + A ( m 00 m 11 A m 10 ) m 10 + A ( m 11 m 00 A m 01 ) m 11 + A ( m 10 m 01 A m 00 ) C m 20 B m 30 A ( B m 31 C m 21 ) C m 21 B m 31 A ( B m 30 C m 20 ) B m 20 + C m 30 + A ( B m 21 + C m 31 ) B m 21 + C m 31 + A ( B m 20 + C m 30 ) C ( m 02 A m 12 ) B ( m 03 A m 13 ) B ( m 02 A m 12 ) + C ( m 03 A m 13 ) C ( m 12 A m 02 ) B ( m 13 A m 03 ) B ( m 12 A m 02 ) + C ( m 13 A m 03 ) B ( B m 33 C m 23 ) C ( B m 32 C m 22 ) C ( B m 22 + C m 23 ) B ( B m 32 + C m 33 ) C ( B m 22 + C m 32 ) B ( B m 23 + C m 33 ) B ( B m 22 + C m 32 ) + C ( B m 23 + C m 33 ) ] ,
( m 01 + m 10 ) transformed = 1 1 A 2 [ m 01 + m 10 A 2 ( m 01 + m 10 ) ] = m 01 + m 10 ,
( m 23 m 32 ) transformed = 1 1 A 2 [ ( m 23 m 32 ) ( B 2 + C 2 ) ] = m 23 m 32 .
α = 1 Σ [ ( m 01 + m 10 ) 2 + ( m 23 m 32 ) 2 ] ,
β = 1 Σ [ ( m 02 + m 20 ) 2 + ( m 13 m 31 ) 2 ] ,
γ = 1 Σ [ ( m 03 + m 30 ) 2 + ( m 12 m 21 ) 2 ] ,
Σ = 3 m 00 2 ( m 11 2 + m 22 2 + m 33 2 ) + 2 Δ ,
Δ = m 01 m 10 + m 02 m 20 + m 03 m 30 ( m 23 m 32 + m 13 m 31 + m 12 m 21 ) .
α 2 + β 2 + γ 2 = m 01 2 + m 10 2 + m 23 2 + m 32 2 + m 02 2 + m 20 2 + m 13 2 + m 31 2 + m 03 2 + m 30 2 + m 12 2 + m 21 2 + 2 Δ 3 m 00 2 ( m 11 2 + m 22 2 + m 33 2 ) + 2 Δ = tr ( M M T ) ( m 00 2 + m 11 2 + m 22 2 + m 33 2 ) + 2 Δ 4 m 00 2 ( m 00 2 + m 11 2 + m 22 2 + m 33 2 ) + 2 Δ .
α 2 + β 2 + γ 2 = 1 .
α 2 + β 2 + γ 2 < 1 .
sign ( α ) = sign ( m 01 + m 10 ± ( m 32 m 23 ) ) ,
sign ( β ) = sign ( m 02 + m 20 ± ( m 13 m 31 ) ) ,
sign ( γ ) = sign ( m 03 + m 30 ± ( m 12 m 21 ) ) ,

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