Abstract

The Mueller matrix roots decomposition recently proposed by Chipman in [1] and its three associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) are explored. Degree of polarization maps are used to differentiate among the three families and demonstrate the unity between phase and diagonal depolarization, while amplitude depolarization remains a distinct class. Three families of depolarization are generated via the averaging of different forms of two nondepolarizing Mueller matrices. The orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. The depolarization forms of Mueller matrices from two scattering measurements are analyzed with the matrix roots decomposition—a sample of ground glass and a graphite and wood pencil tip.

© 2012 Optical Society of America

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References

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  1. R. A. Chipman, Handbook of Optics, Vol. 1 of Mueller Matrices, 3rd ed. (McGraw Hill, 2009).
  2. 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]
  3. R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,” J. Opt. Soc. Am. A 26, 1109–1118 (2009).
    [CrossRef]
  4. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” Proc. SPIE 1166, 177–185 (1989).
  5. R. C. Jones, “A new calculus for the treatment of optical systems. vii. properties of the n-matrices,” J. Opt. Soc. Am. 38, 671–683 (1948).
    [CrossRef]
  6. 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]
  7. R. Ossikovski, “Differential matrix formalism for depolarizing anisotropic media,” Opt. Lett. 36, 2330–2332 (2011).
    [CrossRef]
  8. H. Noble and R. Chipman, “The Mueller matrix roots algorithm and computational considerations,” Opt. Express20, 17–31 (2012).
  9. B. Deboo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
    [CrossRef]
  10. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  11. J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
    [CrossRef]
  12. H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).
  13. B. J. DeBoo, J. M. Sasian, and R. A. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. 44, 5434–5445 (2005).
    [CrossRef]

2011

2009

2005

2004

2000

1996

1989

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

1985

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

1978

1948

Azzam, R. M. A.

Bernabeu, E.

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

Chipman, R.

B. Deboo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004).
[CrossRef]

H. Noble and R. Chipman, “The Mueller matrix roots algorithm and computational considerations,” Opt. Express20, 17–31 (2012).

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).

Deboo, B.

DeBoo, B. J.

Gil, J. J.

J. J. Gil, “Characteristic properties of Mueller matrices,” J. Opt. Soc. Am. A 17, 328–334 (2000).
[CrossRef]

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

Jones, R. C.

Lam, W. S.

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

Lu, S.-Y.

McClain, S.

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

Noble, H.

H. Noble and R. Chipman, “The Mueller matrix roots algorithm and computational considerations,” Opt. Express20, 17–31 (2012).

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

Ossikovski, R.

Sasian, J.

Sasian, J. M.

Smith, G. A.

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Express

Opt. Lett.

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, 3rd ed. (McGraw Hill, 2009).

H. Noble and R. Chipman, “The Mueller matrix roots algorithm and computational considerations,” Opt. Express20, 17–31 (2012).

H. Noble, G. A. Smith, W. S. Lam, S. McClain, and S. McClain, “Polarization imaging light scattering facility,” in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Vol. 6682 (SPIE, 2007).

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Figures (6)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

Degree of polarization maps for depolarizing Mueller matrices M7,15 through M15.

Fig. 3.
Fig. 3.

Measurement geometry for the University of Arizona scattering infrared polarimeter.

Fig. 4.
Fig. 4.

Mueller matrix for a graphite and wood pencil, measured at 505 nm with a Mueller matrix imaging polarimeter.

Fig. 5.
Fig. 5.

Nondepolarizing matrix roots parameters for a graphite and wood pencil, measured at 505 nm with a Mueller matrix imaging polarimeter.

Fig. 6.
Fig. 6.

Depolarizing matrix roots parameters for a graphite and wood pencil, measured at 505 nm with a Mueller matrix imaging polarimeter.

Tables (5)

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Table 1. Nondepolarizing Mueller Matrix Generators G1(d1) through G6(d6) and Their First-Order Taylor Series Approximations

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Table 2. Depolarizing Mueller Matrix Generators G7(d7) through G15(d15) and Their First-Order Taylor Series Approximations

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Table 3. Notation for the Basis Diattenuator and Retarder Mueller Matrices Oriented Along the Three Stokes Axes (Horizontal/Vertical, 45°/135° and Right/Left Circular), as well as an Attenuating Identity Matrixa

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Table 4. Depolarization Properties (Shown by Parameters D7 through D15) Produced by Averaging Two Nondepolarizing Mueller Matrices

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Table 5. Matrix Roots Parameters from a Ground Glass Sample for Specular {70°,70°} and Nonspecular Angle Pairs {70°,10°}

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

N=Mp,
limpMp=I.
Mp=ed0(i=115Gi(di)).
M=eD0(i=115Gi(Dip))p.
DoP(MS(θ,ϕ))=DoP((m0,0m0,1m0,2m0,3m1,0m1,1m1,2m1,3m2,0m2,1m2,2m2,3m3,0m3,1m3,2m3,3)(1cos2θcosϕsin2θcosϕsinϕ)).0θππ2ϕπ2
M15=(G15(D15p))p.
Mi,15=(Gi(Dip)G15(D15p))pi=7,8,14.
σ0=(1001)σ1=(1001)σ2=(0110)σ3=(0ii0)
σiσj=ϵi,j,kσkϵi,j,k={+1(i,j,k)=(1,2,3),(3,1,2),(2,3,1)1(i,j,k)=(1,3,2),(3,2,1),(2,1,3)0i=jj=kk=i.
RR/L(π)=R45°/135°(π)·RH/V(π).
MA=βM1+(1β)M2
RH(δ=π)=(1000010000100001).
Mn=(10.4370.0320.0040.4000.7810.0580.020.0190.0530.2160.1790.0010.0150.1800.138),
Ms=(10.2450.0100.0030.2140.9470.0260.0350.0100.0370.6100.3700.0000.0370.3390.598),

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