Abstract

Two single number metrics for depolarization of samples are contrasted: (1) the average degree of polarization of the exiting light averaged over the Poincaré sphere and (2) the depolarization index of Gill and Berbenau [ Opt. Acta 32, 259– 261 ( 1985); Opt. Acta 33, 185– 189 ( 1986)]. The depolarization index is a geometric measure that varies from 0 for the ideal depolarizer to 1 for nondepolarizing Mueller matrices. The average degree of polarization also varies from 0 to 1 and characterizes the typical level of depolarization. Although the depolarization index is very often close to the average degree of polarization, these two metrics can differ by more than 0.5 for certain Mueller matrices.

© 2005 Optical Society of America

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References

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  1. J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  2. J. J. Gil, E. Bernabeu, “Depolarization and polarization indices of an optical system,” Opt. Acta 33, 185–189 (1986).
    [CrossRef]
  3. R. A. Chipman, “Structure of the Mueller calculus,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 1–9 (2000).
    [CrossRef]
  4. D. A. Haner, R. T. Menzies, “Integrating sphere depolarization at 10 µm for a non-Lambertian wall surface: application to lidar calibration,” Appl. Opt. 32, 6804–6807 (1993).
    [CrossRef] [PubMed]
  5. S. C. McClain, C. L. Bartlett, J. L. Pezzaniti, R. A. Chipman, “Depolarization measurements of an integrating sphere,” Appl. Opt. 34, 152–154 (1995).
    [CrossRef] [PubMed]
  6. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995).
  7. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, New York, 2003), Chap. 9.
    [CrossRef]
  8. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  9. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  10. S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 177–185 (1989).
    [CrossRef]
  11. A. B. Kostinski, C. R. Clark, J. M. Kwiatkowski, “Some necessary conditions on Mueller matrices,” in Polarization Analysis and Measurement, D. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 213–220 (1992).
    [CrossRef]
  12. B. DeBoo, J. Sasian, R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941–4958 (2004), http://www.opticsexpress.org .
    [CrossRef] [PubMed]

2004 (1)

1995 (1)

1993 (1)

1987 (1)

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

1986 (1)

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

1985 (1)

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

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Barakat, R.

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

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Bartlett, C. L.

Bernabeu, E.

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

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

Chipman, R.

Chipman, R. A.

S. C. McClain, C. L. Bartlett, J. L. Pezzaniti, R. A. Chipman, “Depolarization measurements of an integrating sphere,” Appl. Opt. 34, 152–154 (1995).
[CrossRef] [PubMed]

R. A. Chipman, “Structure of the Mueller calculus,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 1–9 (2000).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995).

Clark, C. R.

A. B. Kostinski, C. R. Clark, J. M. Kwiatkowski, “Some necessary conditions on Mueller matrices,” in Polarization Analysis and Measurement, D. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 213–220 (1992).
[CrossRef]

Cloude, S. R.

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 177–185 (1989).
[CrossRef]

DeBoo, B.

Gil, J. J.

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

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

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, New York, 2003), Chap. 9.
[CrossRef]

Haner, D. A.

Kostinski, A. B.

A. B. Kostinski, C. R. Clark, J. M. Kwiatkowski, “Some necessary conditions on Mueller matrices,” in Polarization Analysis and Measurement, D. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 213–220 (1992).
[CrossRef]

Kwiatkowski, J. M.

A. B. Kostinski, C. R. Clark, J. M. Kwiatkowski, “Some necessary conditions on Mueller matrices,” in Polarization Analysis and Measurement, D. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 213–220 (1992).
[CrossRef]

McClain, S. C.

Menzies, R. T.

Pezzaniti, J. L.

Sasian, J.

Appl. Opt. (2)

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]

Opt. Acta (2)

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

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

Opt. Commun. (1)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Opt. Express (1)

Other (5)

S. R. Cloude, “Conditions for the physical realisability of matrix operators in polarimetry,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 177–185 (1989).
[CrossRef]

A. B. Kostinski, C. R. Clark, J. M. Kwiatkowski, “Some necessary conditions on Mueller matrices,” in Polarization Analysis and Measurement, D. Goldstein, R. A. Chipman, eds., Proc. SPIE1746, 213–220 (1992).
[CrossRef]

R. A. Chipman, “Structure of the Mueller calculus,” in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE4133, 1–9 (2000).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (Optical Society of America, Washington, D.C., 1995).

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, New York, 2003), Chap. 9.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Difference between the depolarization index and the average degree of polarization for an aperture divided between an ideal polarizer (aperture fraction a) and an ideal depolarizer.

Equations (19)

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S Exiting = M · S Incident = [ M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 22 M 23 M 30 M 31 M 32 M 33 ] · [ S 0 S 1 S 2 S 3 ] = [ S 0 S 1 S 2 S 3 ] .
I = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] , ID = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .
DoP [ S ] = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 S 0 .
4 M 00 2 = i = 0 3 j = 0 3 M i , j 2 .
AverageDoP [ M ] = 0 π π / 2 π / 2 DoP [ M · S ( θ , ϕ ) ] cos ( ϕ ) d θ d ϕ 4 π .
S ( θ , ϕ ) = [ 1 cos ( θ ) cos ( ϕ ) sin ( 2 θ ) cos ( ϕ ) sin ( ϕ ) ] .
WeightedDoP [ M ] = 0 π π / 2 π / 2 ( M [ 0 ] · S [ 0 , ϕ ] ) DoP [ M · S [ θ , ϕ ] cos [ ϕ ] d θ d φ 0 π π / 2 π / 2 ( M [ 0 ] · S [ θ , ϕ ] cos [ ϕ ] d θ d φ .
Di [ M ] = ( i , j = 0 3 M i , j 2 M 0 , 0 2 ) 1 / 2 3 M 0 , 0 .
UD [ a ] = [ 1 0 0 0 0 a 0 0 0 0 a 0 0 0 0 a ] , NDD [ a , b , c ] = [ 1 0 0 0 0 a 0 0 0 0 b 0 0 0 0 c ] .
DI [ NDD [ a , b , c ] ] = ( a 2 + b 2 + c 2 ) 1 / 2 3 .
DQWHLR [ δ , a ] = [ 1 0 0 0 0 a 0 0 0 0 a cos [ δ ] a sin [ δ ] 0 0 a sin [ δ ] a cos [ δ ] ] .
DI [ DQWHLR [ δ , a ] ] = AverageDoP [ DQWHLR [ δ , a ] ] = a .
DHLP [ a ] = [ 1 a 0 0 a a 0 0 0 0 0 0 0 0 0 0 ] .
D [ M ] = ( M 0 , 1 2 + M 0 , 2 2 + M 0 , 3 2 ) 1 / 2 M 0 , 0 , P [ M ] = ( M 1 , 0 2 + M 2 , 0 2 + M 3 , 0 2 ) 1 / 2 M 0 , 0 ,
DD 1 [ d , b ] = 1 1 + d [ 1 b d 0 0 d b 0 0 0 0 b ( 1 d 2 ) 1 / 2 0 0 0 0 b ( 1 d 2 ) 1 / 2 ] ,
DD 2 [ d , b ] = 1 1 + d [ 1 d 0 0 b d b 0 0 0 0 b ( 1 d 2 ) 1 / 2 0 0 0 0 b ( 1 d 2 ) 1 / 2 ] ,
DD 3 [ d , a ] = 1 1 + d [ 1 d ( 1 a ) 0 0 d ( 1 a ) ( 1 a ) 0 0 0 0 ( 1 a ) ( 1 d 2 ) 1 / 2 0 0 0 0 ( 1 a ) ( 1 d 2 ) 1 / 2 ] .
DI [ DD 1 [ 1 , 0 ] ] = 1 / 3 0.577 , AverageDOP [ DD 1 [ 1 , 0 ] ] = 1 ,
DI [ DD 3 [ 1 , 0 ] ] = 1 / 3 0.577 , AverageDOP [ DD 3 [ 1 , 0 ] ] = 0 ,

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