Abstract

We present an algorithm that decomposes a Mueller matrix into a sequence of three matrix factors: a diattenuator, followed by a retarder, then followed by a depolarizer. Those factors are unique except for singular Mueller matrices. Based on this decomposition, the diattenuation and the retardance of a Mueller matrix can be defined and computed. Thus this algorithm is useful for performing data reduction upon experimentally determined Mueller matrices.

© 1996 Optical Society of America

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References

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  1. S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).
  2. 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 (Stuttgart) 76, 67–71 (1987).
  3. Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
    [CrossRef]
  4. R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
    [CrossRef]
  5. D. G. Anderson, 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]
  6. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  7. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  8. P. Lancaster, M. T. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985).
  9. R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
    [CrossRef]
  10. H. de Lang, “Eigenstates of polarization in lasers,” Philips Res. Rep. 19, 429–440 (1964).
  11. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  12. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
  13. R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
    [CrossRef]
  14. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
    [CrossRef]

1994 (3)

1993 (1)

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

1992 (1)

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

1990 (1)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

1989 (1)

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

1987 (2)

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 (Stuttgart) 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 (1)

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

1964 (1)

H. de Lang, “Eigenstates of polarization in lasers,” Philips Res. Rep. 19, 429–440 (1964).

Anderson, D. G.

Barakat, R.

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

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 (Stuttgart) 76, 67–71 (1987).

Chipman, R. A.

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

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

Cloude, S. R.

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

de Lang, H.

H. de Lang, “Eigenstates of polarization in lasers,” Philips Res. Rep. 19, 429–440 (1964).

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 (Stuttgart) 76, 67–71 (1987).

Horn, R. A.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

Johnson, C. R.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

Lancaster, P.

P. Lancaster, M. T. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985).

Lu, S.-Y.

Shurcliff, W. A.

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

Simon, R.

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

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

Sridhar, R.

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

Tismenetsky, M. T.

P. Lancaster, M. T. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985).

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]

Xing, Z.-F.

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[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. (3)

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

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

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

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

Opt. Commun. (1)

R. Simon, “Nondepolarizing systems and degree of polarization,” Opt. Commun. 77, 349–354 (1990).
[CrossRef]

Opt. Eng. (1)

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

Optik (Stuttgart) (2)

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

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 (Stuttgart) 76, 67–71 (1987).

Philips Res. Rep. (1)

H. de Lang, “Eigenstates of polarization in lasers,” Philips Res. Rep. 19, 429–440 (1964).

Other (3)

P. Lancaster, M. T. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, New York, 1985).

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

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

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Tables (1)

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Table 1 Classification of Analyzers and Polarizers

Equations (74)

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D T q - T r T q + T r = ξ q 2 - ξ r 2 ξ q 2 + ξ r 2 ,             0 D 1.
D D D ^ = ( D d 1 D d 2 D d 3 ) ( D H D 45 D C ) .
D L D H 2 + D 45 2 .
D = D H 2 + D 45 2 + D C 2 = D L 2 + D C 2 = D .
J D exp ( α 2 D ^ · σ ) = σ 0 cosh ( α 2 ) + ( D ^ · σ ) sinh ( α 2 ) σ 0 + D · σ 1 + 1 - D 2 ,
σ 0 = [ 1 0 0 1 ] ,             σ 1 = [ 1 0 0 - 1 ] , σ 2 = [ 0 1 1 0 ] ,             σ 3 = [ 0 - i i 0 ] .
R δ q - δ r ,             0 R π ,
R R R ^ = ( R a 1 R a 2 R a 3 ) ( R H R 45 R C ) .
R L R H 2 + R 45 2 .
R = R H 2 + R 45 2 + R C 2 = R L 2 + R C 2 = R .
J exp ( i 2 R · σ ) = σ 0 cos ( R 2 ) + i ( R ^ · σ ) sin ( R 2 ) .
J = J R J D ,
D ( J ) D ( J D ) ,             R ( J ) R ( J R ) .
M R = [ 1 0 T 0 m R ] ,
( m R ) i j = δ i j cos R + a i a j ( 1 - cos R ) + k = 1 3 i j k a k sin R ,             i , j = 1 , 2 , 3 ,
M R ( 1 ± R ^ ) = ( 1 ± R ^ ) ,             m R R ^ = R ^ .
R = cos - 1 [ tr ( M R ) 2 - 1 ] , a i = 1 2 sin R j , k = 1 3 i j k ( m R ) j k .
M D = T u [ 1 D T D m D ] ,
m D = 1 - D 2 I + ( 1 - 1 - D 2 ) D ^ D ^ T ,
M D ( 1 ± D ^ ) = T u ( 1 ± D ) ( 1 ± D ^ ) ,
m D D = D ,             m D D = 1 - D 2 D ,
S = T u ( 1 + D · s ( D + s D ) + 1 - D 2 s D ) ,
cos Θ DS = D + cos Θ DS 1 + D cos Θ DS ,             Θ DS Θ DS ,
T ( S ^ ) = T u ( 1 + D s cos Θ DS ) ,
T max min = m 00 ± m 01 2 + m 02 2 + m 03 2 ,
S ^ max = ( 1 m 01 / m 01 2 + m 02 2 + m 03 2 m 02 / m 01 2 + m 02 2 + m 03 2 m 03 / m 01 2 + m 02 2 + m 03 2 ) , S ^ min = ( 1 - m 01 / m 01 2 + m 02 2 + m 03 2 - m 02 / m 01 2 + m 02 2 + m 03 2 - m 03 / m 01 2 + m 02 2 + m 03 2 ) .
D = T max - T min T max + T min = 1 m 00 m 01 2 + m 02 2 + m 03 2 .
D = ( D H D 45 D R ) = 1 m 00 ( m 01 m 02 m 03 ) .
S ^ max = ( 1 D ^ ) ,             S ^ min = ( 1 - D ^ ) .
T H - T V T H + T V = m 01 m 00 = D H , T 45 - T 135 T 45 + T 135 = m 02 m 00 = D 45 , T R - T L T R + T L = m 03 m 00 = D C .
P = 1 m 00 m 10 2 + m 20 2 + m 30 2 ,             0 P 1.
P ( P H P 45 P R ) = 1 m 00 ( m 10 m 20 m 30 ) ,             P = P .
M = M R M D ,
T u = m 00 ,             D = 1 m 00 ( m 01 m 02 m 03 ) .
M R = M M D - 1 .
M = m 00 [ 1 D T P m ] ,
m = m R m D = 1 - D 2 m R + ( 1 - 1 - D 2 ) P ^ D ^ T .
m R = 1 1 - D 2 [ m - ( 1 - 1 - D 2 ) P ^ D ^ T ] .
R = P × D P × D cos - 1 ( P · D ) .
M ND polarizer ND analyzer = m 00 ( 1 P ) ( 1 D ) T = m 00 [ 1 D T P P D T ] ,             D = P = 1.
P = m R D .
m 01 2 + m 02 2 + m 03 2 = m 10 2 + m 20 2 + m 30 2 .
[ 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ] ,             a , b , c 1.
Δ 1 - a + b + c 3 ,             0 Δ 1.
[ 1 0 T 0 m Δ ] ,             m Δ T = m Δ ,
[ 1 0 T P Δ m Δ ] = M Δ ,             m Δ T = m Δ ,
M M M D - 1 .
M Δ M R = [ 1 0 T P Δ m Δ ] [ 1 0 T 0 m R ] = [ 1 0 T P Δ m Δ m R ] = [ 1 0 T P Δ m ] = M .
M = M Δ M R M D ,
P Δ = P - m D 1 - D 2 ,
m = m Δ m R .
m Δ = ± [ m ( m ) T + ( λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 ) I ] - 1 × [ ( λ 1 + λ 2 + λ 3 ) m ( m ) T + λ 1 λ 2 λ 3 I ] .
M R = M Δ - 1 M .
Δ = 1 - tr ( m Δ ) 3 = 1 - tr ( M Δ ) - 1 3 ,             0 Δ 1.
M = M Δ 2 M D 2 M R 2 ,
M = M R 3 M D 3 M Δ 3 ,
M = M D 4 M R 4 M Δ 4 ,
M = M R 5 M Δ 5 M D 5 ,
M = M D 6 M Δ 6 M R 6 .
M Δ 2 = M Δ ,             M R 2 = M R ,             M D 2 = M R M D M R T .
M Δ 5 = M R T M Δ M R ,             M R 5 = M R ,             M D 5 = M D .
T ( S a ) = T max 0 ,             T ( S a ) = T min = 0.
m 00 [ 1 D T P P D T ] = { M analyzer , D = 1 M polarizer , P = 1 .
m 00 [ 1 D T P P D T ] = [ 1 0 T 0 P I ] × m 00 [ 1 D T P ^ P ^ D T ] = [ 1 0 T 0 P I ] × M R × m 00 [ 1 D T D D D T ] ,             D = 1 ,
R = P ^ × D P ^ × D cos - 1 ( P ^ · D ) .
m = ( v ^ 1 v ^ 2 v ^ 3 ) diag ( λ 1 , λ 2 , λ 3 ) ( u ^ 1 u ^ 2 u ^ 3 ) T = λ 1 v ^ 1 u ^ 1 T + λ 2 v ^ 2 u ^ 2 T + λ 3 v ^ 3 u ^ 3 T .
m Δ = ± ( λ 1 v ^ 1 v ^ 1 T + λ 2 v ^ 2 v ^ 2 T + λ 3 v ^ 3 v ^ 3 T ) ,
m R = ± ( v ^ 1 u ^ 1 T + v ^ 2 u ^ 2 T + v ^ 3 u ^ 3 T ) .
M Δ = [ 1 0 T P Δ 0 ] ,             M R = I .
m = λ 1 v ^ u ^ T = ( λ 1 v ^ v ^ T ) m R .
m Δ = λ 1 v ^ v ^ T = m ( m ) T tr [ m ( m ) T ] .
R = cos - 1 ( v ^ · u ^ ) = cos - 1 { tr ( m ) tr [ m ( m ) T ] } , R ^ = v ^ × u ^ v ^ × u ^ .
m Δ = ( λ 1 + λ 2 ) [ m ( m ) T + λ 1 λ 2 I ] - 1 m ( m ) T .
m R = v ^ 1 u ^ 1 T + v ^ 2 u ^ 2 T + v ^ 1 × v ^ 2 v ^ 1 × v ^ 2 ( u ^ 1 × u ^ 2 ) T u ^ 1 × u ^ 2 .

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