Abstract

Recently, an order-independent Mueller matrix decomposition was proposed in an effort to elucidate the nine depolarization degrees of freedom [Handbook of Optics, Vol. 1 of Mueller Matrices (2009)]. This paper addresses the critical computational issues involved in applying this Mueller matrix roots decomposition, along with a review of the principal matrix root and common methods for its calculation. The calculation of the pth matrix root is optimized around p = 105 for a 53 digit binary double precision calculation. A matrix roots algorithm is provided which incorporates these computational results. It is applied to a statistically significant number of randomly generated physical Mueller matrices in order to gain insight on the typical ranges of the depolarizing Matrix roots parameters. Computational techniques are proposed which allow singular Mueller matrices and Mueller matrices with a half-wave of retardance to be evaluated with the matrix roots decomposition.

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References

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    [CrossRef]
  8. C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. 432, 1905–1922 (2010).
    [CrossRef]
  9. R. D. Skeel and J. B. Keiper, Elementary Numerical Computing with Mathematica, Chapter 2 (McGraw Hill, 1993).
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. H. D. Noble, “Mueller matrix roots,” Doctoral dissertation (2011).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
    [CrossRef]
  17. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  18. K. M. Twietmeyer, R. A. Chipman, A. E. Elsner, Y. Zhao, and D. VanNasdale, “Mueller matrix retinal imager with optimized polarization conditions,” Opt. Express 16, 21339–21354 (2008).
    [CrossRef] [PubMed]

2010

C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. 432, 1905–1922 (2010).
[CrossRef]

2009

2008

2005

D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. 39, 349–378 (2005).
[CrossRef]

R. A. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. 44, 2490–2495 (2005).
[CrossRef] [PubMed]

2002

1997

N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor. 15, 227–242 (1997).
[CrossRef]

1996

1994

1989

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).

1987

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

1986

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

1948

Barakat, R.

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

Bini, D. A.

D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. 39, 349–378 (2005).
[CrossRef]

Chipman, R. A.

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).

Elsner, A. E.

Gil, J. J.

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

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

Guo, C.-H.

C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. 432, 1905–1922 (2010).
[CrossRef]

Higham, N. J.

D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. 39, 349–378 (2005).
[CrossRef]

N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor. 15, 227–242 (1997).
[CrossRef]

N. J. Higham, Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, 2008).
[CrossRef]

Jones, R. C.

Keiper, J. B.

R. D. Skeel and J. B. Keiper, Elementary Numerical Computing with Mathematica, Chapter 2 (McGraw Hill, 1993).

Lu, S.-Y.

Meini, B.

D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. 39, 349–378 (2005).
[CrossRef]

Noble, H. D.

H. D. Noble, “Mueller matrix roots,” Doctoral dissertation (2011).

Ossikovski, R.

Skeel, R. D.

R. D. Skeel and J. B. Keiper, Elementary Numerical Computing with Mathematica, Chapter 2 (McGraw Hill, 1993).

Smith, M. H.

Twietmeyer, K. M.

VanNasdale, D.

Zhao, Y.

Appl. Opt.

J. Mod. Opt.

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.

J. Opt. Soc. Am. A

Linear Algebra Appl.

C.-H. Guo, “On newton’s method and halley’s method for the principal pth root of a matrix,” Linear Algebra Appl. 432, 1905–1922 (2010).
[CrossRef]

Num. Algor.

D. A. Bini, N. J. Higham, and B. Meini, “Algorithms for the matrix pth root,” Num. Algor. 39, 349–378 (2005).
[CrossRef]

N. J. Higham, “Stable iterations for the matrix square root,” Num. Algor. 15, 227–242 (1997).
[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. Express

Proc. SPIE

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).

Other

R. A. Chipman, Handbook of Optics, Vol. 1 of Mueller Matrices (McGraw Hill, 2009), 3rd ed.

R. D. Skeel and J. B. Keiper, Elementary Numerical Computing with Mathematica, Chapter 2 (McGraw Hill, 1993).

N. J. Higham, Functions of Matrices: Theory and Computation (Society for Industrial and Applied Mathematics, 2008).
[CrossRef]

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

H. D. Noble, “Mueller matrix roots,” Doctoral dissertation (2011).

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Figures (9)

Fig. 1
Fig. 1

Taking the root of a uniform Mueller matrix is analogous to slicing it into very thin identical pieces.

Fig. 2
Fig. 2

The error of the root calculation vs. root order for a simple retarder converges to a minimum relative error just beyond the 105th root.

Fig. 3
Fig. 3

The norm of the depolarizing retarder’s diagonal depolarization parameters (D13, D14, and D15) converges to a steady value of approximately 3.03 beyond the 104th root.

Fig. 4
Fig. 4

The norm of the depolarizing retarder’s matrix roots retardance parameters (D4, D5, and D6) converges to a steady value of approximately 1.07 beyond the 104th root.

Fig. 5
Fig. 5

Relative error for Mathematica’s matrix root calculation.

Fig. 6
Fig. 6

Matrix Roots algorithm flow chart.

Fig. 7
Fig. 7

A histogram of the matrix root amplitude depolarization values for 76,336 randomly generated physical Mueller matrices.

Fig. 8
Fig. 8

A histogram of the matrix root phase depolarization values for 76,336 randomly generated physical Mueller matrices.

Fig. 9
Fig. 9

A histogram of the matrix root diagonal depolarization values for 76,336 randomly generated physical Mueller matrices.

Tables (1)

Tables Icon

Table 1 The sixteen polarization properties of the Mueller matrix given by the Mueller matrix roots decomposition.

Equations (34)

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N = M p ,
lim p M p = I .
N = M p .
N = d 0 ( 1 d 1 + d 7 d 2 + d 8 d 3 + d 9 d 1 d 7 1 f 13 d 6 + d 12 d 5 + d 11 d 2 d 8 d 6 + d 12 1 f 14 d 4 + d 10 d 3 d 9 d 5 + d 11 d 4 + d 10 1 f 15 ) ,
d 13 = f 14 f 13 2
d 14 = 2 3 ( 2 f 15 + f 14 + f 13 )
d 15 = 1 6 ( f 15 + f 14 + f 13 ) .
D n = p d n , n = 0 , 1 , 2 , , 15 .
A = Z diag ( λ i ) Z 1 ,
A 1 / p = Z diag ( λ i ) 1 / p Z 1 .
LP ( θ ) = 1 2 ( 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ 1 2 sin 4 θ 0 sin 2 θ 1 2 sin 4 θ sin 2 2 θ 0 0 0 0 0 ) .
LP ( θ ) LP ( θ ) = LP ( θ ) ,
LP ( θ ) 1 / p = LP ( θ ) .
θ = 1 2 arctan d H d 45 , and
η = arcsin d R d H 2 + d 45 2 + d R 2 .
δ = δ 1 + δ 2 .
LR ( δ , θ ) 1 / p = LR ( δ / p , θ ) .
LR ( π , 0 ° ) = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ,
θ = 1 2 arctan δ H δ 45 , and
η = arcsin δ R δ H 2 + δ 45 2 + δ R 2 .
ER ( η , θ , δ ) = LR ( η , θ + π / 4 ) LR ( δ , θ ) LR ( η , θ π / 4 ) ,
LR ( δ , θ ) = ( 1 0 0 0 0 cos 2 2 θ + cos δ sin 2 2 θ sin 2 δ 2 sin 4 θ sin δ sin 2 θ 0 sin 2 δ 2 sin 4 θ cos δ cos 2 2 θ + sin 2 2 θ cos 2 θ sin δ 0 sin δ sin 2 θ cos 2 θ sin δ cos δ ) .
ER ( η , θ , δ = π ɛ ) = LR ( η , θ + π / 4 ) LR ( π ɛ , θ ) LR ( η , θ π / 4 ) .
ER ( η , θ , δ = π ) 1 p = LR ( η , θ + π / 4 ) LR ( ( π ɛ ) , θ ) 1 p LR ( η , θ π / 4 ) .
HWR ( η , θ ) 1 p = LR ( ɛ , θ + π / 4 ) LR ( π p , θ ) LR ( η , θ π / 4 ) ,
( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ) = ( 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ) ,
M = M Δ M R M D .
M = M Δ M R M D .
M = M Δ M R M D .
M = M Δ M R M D .
Δ x = ( δ H D 4 ) 2 + ( δ 45 D 5 ) 2 + ( δ R D 6 ) 2 δ .
PD ( a , b , c ) = ( 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ) .
ED ( 0.210 , 0.033 , 1.003 ) PD ( 0.1 , 0.2 , 0.3 ) = ( 0.494 0.010 0.003 0.419 0.104 0.002 0.0 0.029 0.016 0.0 0.0 0.005 0.495 0.010 0.003 0.142 ) .
Δ root = M R ( M R p ) p 2 M R .

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