Abstract

It is shown that any depolarizing Mueller matrix can be reduced, through a product decomposition, to one of a total of two canonical depolarizer forms, a diagonal and a non-diagonal one. As a consequence, depolarizing Mueller matrices can be divided into Stokes diagonalizable and Stokes non-diagonalizable ones. Properties characteristic of the two canonical depolarizers are identified and discussed. Both canonical depolarizer forms are illustrated in experimental examples taken from the literature.

© 2009 Optical Society of America

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References

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  1. C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).
  2. J. J. Gil and 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. 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]
  4. R. Sridhar and R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903-1915 (1994).
    [CrossRef]
  5. A. V. Gopala Rao, K. S. Malesh, 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).
  6. C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]
  7. Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.
  8. Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
    [CrossRef]
  9. 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]
  10. A. V. Gopala Rao, K. S. Malesh, 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).
  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. M. Renardy, “Singular value decomposition in Minkowski space,” Linear Algebr. Appl. 236, 53-58 (1996).
    [CrossRef]
  13. S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766-773 (1994).
    [CrossRef]
  14. Sudha and A. V. Gopala Rao, “Polarization elements: a group theoretical study,” J. Opt. Soc. Am. A 18, 3130-3134 (2001).
    [CrossRef]
  15. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109-1118 (2009).
    [CrossRef]
  16. A. S. Marathay, “Operator formalism in the theory of partial polarization,” J. Opt. Soc. Am. 55, 969-980 (1965).
  17. R. Ossikovski, C. Fallet, A. Pierangelo, and A. De Martino, “Experimental implementation and properties of Stokes nondiagonalizable depolarizing Mueller matrices,” Opt. Lett. 34, 974-976 (2009).
    [CrossRef] [PubMed]
  18. J. J. Gil and E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185-189 (1986).
    [CrossRef]
  19. Z. Sekera, “Scattering matrices and reciprocity relationships for various representations of the state of polarization,” J. Opt. Soc. Am. 56, 1732-1740 (1966).
    [CrossRef]
  20. R. Ossikovski, “Interpretation of nondepolarizing Mueller matrices based on singular value decomposition,” J. Opt. Soc. Am. A 25, 473-482 (2008).
    [CrossRef]
  21. F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
    [CrossRef]
  22. R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406-2410 (2008).
    [CrossRef]
  23. R. Ossikovski, M. Foldyna, C. Fallet, and A. De Martino, “Experimental evidence for naturally occurring nondiagonal depolarizers,” Opt. Lett. 34, 2426-2428 (2009).
    [CrossRef] [PubMed]

2009

2008

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406-2410 (2008).
[CrossRef]

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

2007

2001

1998

A. V. Gopala Rao, K. S. Malesh, 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).

A. V. Gopala Rao, K. S. Malesh, 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).

1997

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

1996

M. Renardy, “Singular value decomposition in Minkowski space,” Linear Algebr. Appl. 236, 53-58 (1996).
[CrossRef]

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]

1994

1993

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

1987

J. J. Gil and 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).

1986

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

1966

1965

Anastasiadou, M.

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406-2410 (2008).
[CrossRef]

Anderson, D. G. M.

Barakat, R.

Bernabeu, E.

J. J. Gil and 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).

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

Bolshakov, Y.

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).

Chipman, R. A.

De Martino, A.

Fallet, C.

Foldyna, M.

Gil, J. J.

J. J. Gil and 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).

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

Gopala Rao, A. V.

Sudha and A. V. Gopala Rao, “Polarization elements: a group theoretical study,” J. Opt. Soc. Am. A 18, 3130-3134 (2001).
[CrossRef]

A. V. Gopala Rao, K. S. Malesh, 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).

A. V. Gopala Rao, K. S. Malesh, 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).

Guyot, S.

Le Jeune, B.

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Lu, S.-Y.

Malesh, K. S.

A. V. Gopala Rao, K. S. Malesh, 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).

A. V. Gopala Rao, K. S. Malesh, 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).

Marathay, A. S.

Ossikovski, R.

Pierangelo, A.

Ran, A. C. M.

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

Reichstein, B.

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

Renardy, M.

M. Renardy, “Singular value decomposition in Minkowski space,” Linear Algebr. Appl. 236, 53-58 (1996).
[CrossRef]

Rodman, L.

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

Roy-Bréhonnet, F. Le

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Sekera, Z.

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,

Sudha and A. V. Gopala Rao, “Polarization elements: a group theoretical study,” J. Opt. Soc. Am. A 18, 3130-3134 (2001).
[CrossRef]

A. V. Gopala Rao, K. S. Malesh, 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).

A. V. Gopala Rao, K. S. Malesh, 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).

van der Mee, C. V. M.

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

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

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

Integral Equation Oper. Theory

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” Integral Equation Oper. Theory 27, 497-501 (1997).
[CrossRef]

J. Math. Phys.

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.

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

A. V. Gopala Rao, K. S. Malesh, 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).

A. V. Gopala Rao, K. S. Malesh, 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).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Linear Algebr. Appl.

M. Renardy, “Singular value decomposition in Minkowski space,” Linear Algebr. Appl. 236, 53-58 (1996).
[CrossRef]

Opt. Acta

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

Opt. Commun.

R. Ossikovski, M. Anastasiadou, and A. De Martino, “Product decompositions of depolarizing Mueller matrices with negative determinants,” Opt. Commun. 281, 2406-2410 (2008).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

J. J. Gil and 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).

Prog. Quantum Electron.

F. Le Roy-Bréhonnet and B. Le Jeune, “Utilization of Mueller matrix formalism to obtain optical targets depolarization and polarization properties,” Prog. Quantum Electron. 21, 109-151 (1997).
[CrossRef]

Other

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).

Y. Bolshakov, C. V. M. van der Mee, A. C. M. Ran, B. Reichstein, and L. Rodman, “Polar decompositions in finite dimensional indefinite scalar product spaces: special cases and applications,” in Operator Theory: Advances and Applications, I.Gohberg, ed. (Birkhäuser Verlag, 1996), Vol. 87, pp. 61-94.

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Equations (40)

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G H T G = H
U T G U = G .
A = U H ,
M nd T G M nd = det ( M nd ) G .
H = U 1 H U
H 1 = [ ρ 0 0 0 0 0 ρ 1 0 0 0 0 ρ 2 0 0 0 0 ε ρ 3 ]
H 2 = [ ρ 0 + α α 0 0 α ρ 0 α 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] ,
N = M + M = G M T G M
M = L 1 K L 2
K 2 = [ n 0 ( ρ 0 n 0 ) n 0 0 0 0 ρ 0 n 0 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] ,
M D ( a ) = 1 1 a 2 [ 1 a 0 0 a 1 0 0 0 0 1 a 2 0 0 0 0 1 a 2 ] ,
K 2 = M D ( a 1 ) K 2 = [ ( n 0 + ρ 0 ) 2 ρ 0 ( ρ 0 n 0 ) 2 ρ 0 0 0 ( n 0 ρ 0 ) 2 ρ 0 ( 3 ρ 0 n 0 ) 2 ρ 0 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] .
M Δ nd = M D ( a 2 ) K 2 M D 1 ( a 2 ) = [ 2 ρ 0 ρ 0 0 0 ρ 0 0 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] .
N = M M + = M G M T G .
M = s 1 s 2 T ,
M D = T u [ 1 D T D m D ] ,
m D = 1 D 2 I 3 + ( 1 1 D 2 ) D ̂ D ̂ T ,
M D 1 s 1 = k T u ( 1 D 2 ) [ 1 D T D m D ] [ 1 D ] = [ 1 0 ] = s 0 ,
M DA = M D s 0 ( s 0 T s 2 ) s 2 T = M D ( s 0 s 0 T ) ( s 2 s 2 T ) = M D M ID M IP ,
M DP = s 1 ( s 1 T s 0 ) s 0 T M D = ( s 1 s 1 T ) ( s 0 s 0 T ) M D = M IP M ID M D ,
K = [ α 0 0 0 α 0 0 0 0 0 0 0 0 0 0 0 ] = α s H s 0 T ,
K = α s H s 0 T = α s H ( s 3 T s 0 ) s 0 T = α ( s H s 3 T ) ( s 0 s 0 T ) = α M IIP M ID ,
M Δ d = [ ρ 0 0 0 0 0 ρ 1 0 0 0 0 ρ 2 0 0 0 0 ± ρ 3 ] = ρ 0 [ 1 0 0 0 0 d 1 0 0 0 0 d 2 0 0 0 0 d 3 ] ,
M Δ nd = [ 2 ρ 0 ρ 0 0 0 ρ 0 0 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] = 2 ρ 0 [ 1 1 2 0 0 1 2 0 0 0 0 0 c 0 0 0 0 c ] ,
M Δ nd = ρ 0 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] + [ ρ 0 0 0 0 0 ρ 0 0 0 0 0 ρ 2 0 0 0 0 ρ 2 ] ,
P Δ = tr ( M T M ) M 11 2 3 M 11 2 ,
P Δ ( M Δ nd ) P Δ ( K 2 ) .
M r = Q M T Q ,
M Δ = [ 1 0 P Δ m Δ ] , M Δ r = [ 1 D Δ 0 m Δ r ] ,
M ( 1 ) = [ α 2 + β α 2 0 0 α 2 α 2 + β 0 0 0 0 β 0 0 0 0 β ] ,
M Δ nd ( 1 ) = M D ( a 1 ) M ( 1 ) M D 1 ( a 1 ) = [ 2 β β 0 0 β 0 0 0 0 0 β 0 0 0 0 β ] = β [ 2 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 ] .
M ( 2 ) = [ 1 0 0 α 0 β γ 0 0 0 0 γ β 0 α 0 0 α β γ ] ,
M R 1 = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] ,
M 1 ( 2 ) = M R 1 M ( 2 ) M R 1 1 = [ 1 α 0 0 α α β γ 0 0 0 0 γ β 0 0 0 0 β γ ] .
M 2 ( 2 ) = M R 2 M 1 ( 2 ) = [ 1 α 0 0 α β + γ α 0 0 0 0 β γ 0 0 0 0 β γ ] .
M Δ nd ( 2 ) = M ( a 2 ) M 2 ( 2 ) M 1 ( a 2 ) = [ 2 ( β + γ ) ( β + γ ) 0 0 β + γ 0 0 0 0 0 β γ 0 0 0 0 β γ ] ,
P Δ ( K 2 ) = ( ρ 0 n 0 ) 2 + ρ 0 2 + 2 ρ 2 n 0 3 n 0 2 = 1 3 [ ( ρ 0 n 0 1 ) 2 + ( ρ 0 n 0 ) 2 + 2 ρ 2 n 2 ]
P Δ ( M Δ nd ) = 2 ρ 0 + 2 ρ 2 12 ρ 0 = 1 + r 6 ,
P Δ min ( K 2 ) = 1 3 [ ( ρ 0 + ρ 2 2 ρ 0 ) 2 + ( ρ 0 ρ 2 2 ρ 0 ) 2 + ρ 2 ( ρ 0 ρ 2 ) ρ 0 2 ] = 1 + 2 r r 2 6 .
P Δ min 2 ( K 2 ) P Δ 2 ( M Δ nd ) = r ( 1 r ) 6 0 ,

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