Abstract

The classification of polarization properties of polarization elements is studied to derive data-reduction equations for extracting the diattenuation, retardance, and other polarization properties from their Jones matrices. Polarization elements, and Jones matrices as well, are divided into two classes: homogeneous, with orthogonal eigenpolarizations, and inhomogeneous, with nonorthogonal eigenpolarizations. The basic polarization properties, diattenuation and retardance, of homogeneous polarization elements are straightforward and well known; these elements are characterized by their eigenvalues and eigenpolarizations. Polarization properties of inhomogeneous polarization elements are not so evident. By applying polar decomposition, the definitions of diattenuation and retardance are generalized to inhomogeneous polarization elements, providing an understanding of their polarization characteristics. Furthermore, an inhomogeneity parameter is introduced to describe the degree of inhomogeneity in a polarization element. These results are then adapted to degenerate polarization elements, which have only one linearly independent eigenpolarization.

© 1994 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  3. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the new calculus,”J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  4. See, for example, R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  5. R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,”J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  6. C. Whitney, “Pauli-algebraic operators in polarization optics,”J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]
  7. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  8. J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,” Optik 76, 67–71 (1987).
  9. L. J. November, “Recovery of the matrix operators in the similarity and congruency transformations: applications in polarimetry,” J. Opt. Soc. Am. A 10, 719–739 (1993).
    [CrossRef]
  10. S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 86–109 (1955).
  11. S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. Sect. A 46, 280–302 (1957).
  12. See, for example, P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).
  13. R. C. Jones, “A new calculus for the treatment of optical systems, IV,”J. Opt. Soc. Am. 32, 486–493 (1942).
    [CrossRef]
  14. G. R. Bird, W. A. Shurcliff, “Pile-of-plates polarizers for the infrared: improvement in analysis and design,”J. Opt. Soc. Am. 49, 235–237 (1959).
    [CrossRef]
  15. These notations are borrowed from linear algebra; see, for example, Ref. 12, pp. 175 and 192.
  16. E. Collett, Polarized Light (Marcel Dekker, New York, 1992).

1993 (1)

1989 (1)

See, for example, R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

1987 (2)

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

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

1971 (1)

1959 (1)

1957 (1)

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. Sect. A 46, 280–302 (1957).

1955 (1)

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 86–109 (1955).

1942 (1)

1941 (2)

Azzam, R. M. A.

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

Barakat, R.

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

Bashara, N. M.

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

Bernabeu, E.

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

Bird, G. R.

Chipman, R. A.

See, for example, R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Collett, E.

E. Collett, Polarized Light (Marcel Dekker, New York, 1992).

Gil, J. J.

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

Jones, R. C.

Lancaster, P.

See, for example, P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

November, L. J.

Pancharatnam, S.

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. Sect. A 46, 280–302 (1957).

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 86–109 (1955).

Shurcliff, W. A.

Tismenetsky, M.

See, for example, P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

Whitney, C.

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

See, for example, R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Optik (1)

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

Proc. Indian Acad. Sci. Sect. A (2)

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 86–109 (1955).

S. Pancharatnam, “Light propagation in absorbing crystals possessing optical activity,” Proc. Indian Acad. Sci. Sect. A 46, 280–302 (1957).

Other (5)

See, for example, P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

These notations are borrowed from linear algebra; see, for example, Ref. 12, pp. 175 and 192.

E. Collett, Polarized Light (Marcel Dekker, New York, 1992).

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

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

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Tables (1)

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Table 1 Properties of Example Inhomogeneous Matrices

Equations (81)

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J E q = ξ q E q ,             J E r = ξ r E r ,
E q E r = 0.
D ( J ) = ξ q 2 - ξ r 2 ξ q 2 + ξ r 2 ,             0 D 1 ,
R ( J ) = δ q - δ r ,             0 R π .
σ 1 = [ 1 0 0 - 1 ] ,             σ 2 = [ 0 1 1 0 ] ,             σ 3 = [ 0 - i i 0 ] .
J = k = 0 3 c k σ k .
J = ρ 0 exp ( i ϕ 0 ) ( σ 0 + c 1 σ 1 + c 2 σ 2 + c 3 σ 3 ) ,
D = 2 ( c 1 2 + c 2 2 + c 3 2 ) 1 / 2 1 + c 1 2 + c 2 2 + c 3 2 .
D c 1 ( c 1 2 + c 2 2 + c 3 2 ) 1 / 2 ,             D c 2 ( c 1 2 + c 2 2 + c 3 2 ) 1 / 2 ,             D c 3 ( c 1 2 + c 2 2 + c 3 2 ) 1 / 2
J = ρ 0 exp ( i ϕ 0 ) exp ( i k = 1 3 d k σ k ) ,
R = 2 ( d 1 2 + d 2 2 + d 3 2 ) 1 / 2 .
A = UH = H U .
H 2 = A A ,             H 2 = A A .
U = A H - 1 = H - 1 A .
J = J R J D = J D J R ,
J = VDW ,
J D = WDW ,             J D = VDV ,             J R = V W .
T ( E ) = JE 2 E 2 = E J JE E E .
T max = max [ T ( E ) ] = T ( E max ) , T min = min [ T ( E ) ] = T ( E min ) ,
T max , min = 1 2 ( tr ( J J ) ± { [ tr ( J J ) ] 2 - 4 det J 2 } 1 / 2 ) ,
E max E min = 0.
( J E max ) ( J E min ) = 0.
J D = T max E ^ max E ^ max + T min E ^ min E ^ min ,
J D = T max JE ^ max JE ^ max + T min JE ^ min JE ^ min ,
J R = JE ^ max E ^ max + JE ^ min E ^ min ,
J = T max JE ^ max E ^ max + T min JE ^ min E ^ min .
T ( E ^ ) = T max E ^ max E ^ 2 + T min E ^ min E ^ 2 .
T ( E ^ ) = E ^ max E ^ 2 = cos 2 θ ,
E ^ u = 1 2 ( E ^ max E ^ min ) ,
J E ^ u = 1 2 ( T max JE ^ max T min JE ^ min ) .
T ( E ^ u ) = 1 2 ( T max + T min ) ,
DOP ( J E ^ u ) = T max - T min T max + T min .
E ^ p = α E ^ ( 1 - α ) E ^ u ,
T ( E ^ p ) = α ( T max E ^ max E ^ 2 + T min E ^ min E ^ 2 ) + ( 1 - α ) T max + T min 2 .
D ( J ) = D ( J D ) = D ( J D ) ,
R ( J ) = R ( J R ) .
D ( J ) = T max - T min T max + T min ,
R ( J ) = 2 cos - 1 | 1 2 tr J R | = 2 cos - 1 ( 1 2 E ^ max JE ^ max + E ^ min JE ^ min ) .
R ( J ) = 2 cos - 1 E ^ max JE ^ max .
D = { 1 - 4 det J 2 [ tr ( J J ) ] 2 } 1 / 2 .
R = 2 cos - 1 | tr J + det J det J tr J | 2 [ tr ( J J ) + 2 det J ] 1 / 2 .
R = 2 cos - 1 tr J [ tr ( J J ) ] 1 / 2 .
η = E ^ q E ^ r ,             0 η 1.
η 2 = tr ( J J ) - 1 2 tr J 2 - 1 2 ( tr J ) 2 - 4 det J tr ( J J ) - 1 2 tr J 2 + 1 2 ( tr J ) 2 - 4 det J .
η = cos Θ q r 2 .
( 1 - η 2 ) T 2 - [ ξ q 2 + ξ r 2 - η 2 ( ξ q * ξ r + ξ q ξ r * ) ] T + ( 1 - η 2 ) ξ q 2 ξ r 2 = 0 ,
D = { 1 - 4 ( 1 - η 2 ) 2 ξ q 2 ξ r 2 [ ξ q 2 + ξ r 2 - η 2 ( ξ q * ξ r + ξ q ξ r * ) ] 2 } 1 / 2 .
R = 2 cos - 1 { [ ( 1 - η 2 ) ( ξ q + ξ r ) 2 ( ξ q + ξ r ) 2 - η 2 ( 2 ξ q ξ r + ξ q * ξ r + ξ q ξ r * ) ] | cos δ q - δ r 2 | } ,
R = 2 cos - 1 ( 1 - η 2 ) 1 / 2 = 2 sin - 1 η .
η = J F ^ q - ξ q F ^ q ,
η = j 12 + j 21 .
UJU = [ ξ q η 0 ξ q ] ,
D = η 2 ξ q 2 + η 2 ( 4 ξ q 2 + η 2 ) 1 / 2 ,
R = 2 cos - 1 ξ q ( ξ q 2 + η 2 / 4 ) 1 / 2 .
J 1 ( θ 2 ) = [ cos θ sin θ sin θ - cos θ ] [ 1 0 0 0 ] = [ cos θ 0 sin θ 0 ] ,
J 2 ( θ ) = [ cos 2 θ sin θ cos θ sin θ cos θ cos 2 θ ] [ 1 0 0 0 ] = cos θ [ cos θ 0 sin θ 0 ] ,
J 3 ( θ ) = 1 2 [ 1 + i cos 2 θ i sin 2 θ i sin 2 θ 1 - i cos 2 θ ] [ 1 0 0 0 ] = 1 2 [ 1 + i cos 2 θ 0 i sin 2 θ 0 ] ,
J 4 ( θ 2 ) = [ 1 0 0 0 ] [ cos θ - sin θ - sin θ - cos θ ] = [ cos θ - sin θ 0 0 ] ,
J 5 ( θ ) = [ 1 0 0 0 ] [ cos 2 θ - sin θ cos θ - sin θ cos θ cos 2 θ ] = cos θ [ cos θ - sin θ 0 0 ] ,
J 6 ( θ ) = [ 1 0 0 0 ] 1 2 [ 1 + i cos 2 θ - i sin 2 θ - i sin 2 θ 1 - i cos 2 θ ] = 1 2 [ 1 + i cos 2 θ - i sin 2 θ 0 0 ] .
J 1 = U 1 J 2 U 2 .
J 1 = U J 2 U .
( J - ξ q I ) 2 = 0 ,
J F ^ q = ξ q F ^ q + η E ^ q .
[ E ^ q , F ^ q ] J [ E ^ q , F ^ q ] = [ E ^ q , F ^ q ] [ ξ q E ^ q , ξ q F ^ q + η ' E ^ q ] = [ ξ q η 0 ξ q ] .
J 1 E ^ 1 q = ξ q E ^ 1 q ,             J 1 E ^ 1 r = ξ r E ^ 1 r ,
J 2 E ^ 2 q = ξ q E ^ 2 q ,             J 2 E ^ 2 r = ξ r E ^ 2 r ,
E ^ 1 q E ^ 1 r = E ^ 2 q E ^ 2 r ,
E ^ 1 q E ^ 1 r = E ^ 2 q E ^ 2 r .
U = [ E ^ 1 q , E ^ 1 r ] [ E ^ 2 q , E ^ 2 r ] - 1 .
U E ^ 2 q = E ^ 1 q ,             U E ^ 2 r = E ^ 1 r ,
U E ^ 1 q = E ^ 2 q ,             U E ^ 1 r = E ^ 2 r ,
J 1 = U J 2 U ,             J 2 = U J 1 U .
J = c 0 σ 0 + c T σ ,
tr J = 2 c 0 ,
det J = c 0 2 - ( c 1 2 + c 2 2 + c 3 2 ) = c 0 2 - c T c ,
tr ( J J ) = 2 ( c 0 2 + c 1 2 + c 2 2 + c 3 2 ) = 2 ( c 0 2 + c c ) .
D = [ 1 - ( c 0 2 - c T c c 0 2 + c c ) 2 ] 1 / 2 ,
R = 2 cos - 1 | c 0 + c 0 2 - c T c c 0 2 - c T c c 0 * | ( 2 c 0 2 + 2 c c + 2 c 0 2 - c T c ) 1 / 2 ,
R = 2 cos - 1 2 c 0 ( 2 c 0 2 + 2 c c ) 1 / 2 ,
η 2 = c c - c T c c c + c T c .

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