Abstract

A depolarization scalar metric for Mueller matrices, named Q(M), is derived from the degree of polarization. Q(M) has been recently reported, and it has been deduced from the nine bilinear constraints between the sixteen elements of the Mueller–Jones matrix. We discuss the relations between Q(M) and the depolarization index.

© 2008 Optical Society of America

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References

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  1. J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259-261 (1985).
  2. J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical system,” J. Mod. Opt. 33, 185-189 (1986).
  3. 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]
  4. S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26-36 (1986).
  5. B. DeBoo, J. Sasian, and R. Chipman, “Degree of polarization surfaces and maps for analysis of depolarization,” Opt. Express 12, 4941-4958 (2004).
    [CrossRef] [PubMed]
  6. R. Chipman, “Depolarization index and the average degree of polarization,” Appl. Opt. 44, 2490-2495 (2005).
    [CrossRef] [PubMed]
  7. B. DeBoo, J. M. Sasian, and R. Chipman, “Depolarization of diffusely reflecting man-made objects,” Appl. Opt. 44, 5434-5445 (2005).
    [CrossRef] [PubMed]
  8. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688-1703(2006).
    [CrossRef] [PubMed]
  9. S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11-14 (1998).
    [CrossRef]
  10. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  11. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989).
  12. J. J. Gil, “polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1-47 (2007).
    [CrossRef]
  13. R. Espinosa-Luna and E. Bernabeu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256-258 (2007).
    [CrossRef]
  14. R. A. Chipman, Polarimetry, in Handbook of Optics (McGraw Hill, 1995), Vol. 2.
  15. S. Manhas, M. K. Swami, P. Buddhiwant, N. Ghosh, P. K. Guota, and K. Singh, “Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry,” Opt. Express 14, 190-202 (2006).
    [CrossRef] [PubMed]
  16. A. Aiello, G. Puentes, D. Voight, and J. P. Woerdman, “Maximum likelihood estimation of Mueller matrices,” Opt. Lett. 31, 817-819 (2006).
    [CrossRef] [PubMed]
  17. S. N. Savenkov, R. S. Muttiah, and Yu A. Oberemok, “Transmitted and reflected scattering matrices from an English oak leaf,” Appl. Opt. 42, 4955-4962 (2003).
    [CrossRef] [PubMed]
  18. S. N. Savenkov and K. E. Yushtin, “Pecularities of depolarization of linearly polarized radiation by a layer of the anisotropic inhomogeneous medium,” Ukr. J. Phys. 50, 235-239 (2005).
  19. S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
    [CrossRef]

2007 (3)

J. J. Gil, “polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1-47 (2007).
[CrossRef]

R. Espinosa-Luna and E. Bernabeu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256-258 (2007).
[CrossRef]

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

2006 (3)

2005 (3)

2004 (1)

2003 (1)

1998 (1)

S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11-14 (1998).
[CrossRef]

1994 (1)

1986 (2)

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26-36 (1986).

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical system,” J. Mod. Opt. 33, 185-189 (1986).

1985 (1)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259-261 (1985).

Aiello, A.

Anderson, D. G. M.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989).

Barakat, R.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989).

Bernabeu, E.

R. Espinosa-Luna and E. Bernabeu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256-258 (2007).
[CrossRef]

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical system,” J. Mod. Opt. 33, 185-189 (1986).

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259-261 (1985).

Brosseau, C.

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

Buddhiwant, P.

Chipman, R.

Chipman, R. A.

E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688-1703(2006).
[CrossRef] [PubMed]

S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11-14 (1998).
[CrossRef]

R. A. Chipman, Polarimetry, in Handbook of Optics (McGraw Hill, 1995), Vol. 2.

Cloude, S. R.

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26-36 (1986).

DeBoo, B.

Espinosa-Luna, R.

R. Espinosa-Luna and E. Bernabeu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256-258 (2007).
[CrossRef]

Ghosh, N.

Gil, J. J.

J. J. Gil, “polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1-47 (2007).
[CrossRef]

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical system,” J. Mod. Opt. 33, 185-189 (1986).

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259-261 (1985).

Guota, P. K.

Lu, S. Y.

S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11-14 (1998).
[CrossRef]

Manhas, S.

Muttiah, R. S.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

S. N. Savenkov, R. S. Muttiah, and Yu A. Oberemok, “Transmitted and reflected scattering matrices from an English oak leaf,” Appl. Opt. 42, 4955-4962 (2003).
[CrossRef] [PubMed]

Oberemok, Yu A.

Puentes, G.

Sasian, J.

Sasian, J. M.

Savenkov, S. N.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, “Pecularities of depolarization of linearly polarized radiation by a layer of the anisotropic inhomogeneous medium,” Ukr. J. Phys. 50, 235-239 (2005).

S. N. Savenkov, R. S. Muttiah, and Yu A. Oberemok, “Transmitted and reflected scattering matrices from an English oak leaf,” Appl. Opt. 42, 4955-4962 (2003).
[CrossRef] [PubMed]

Singh, K.

Swami, M. K.

Voight, D.

Volchkov, S. A.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

Woerdman, J. P.

Wolfe, E.

Yushtin, K. E.

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

S. N. Savenkov and K. E. Yushtin, “Pecularities of depolarization of linearly polarized radiation by a layer of the anisotropic inhomogeneous medium,” Ukr. J. Phys. 50, 235-239 (2005).

Appl. Opt. (4)

Eur. Phys. J. Appl. Phys. (1)

J. J. Gil, “polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1-47 (2007).
[CrossRef]

J. Mod. Opt. (2)

J. J. Gil and E. Bernabeu, “A depolarization criterion in Mueller matrices,” J. Mod. Opt. 32, 259-261 (1985).

J. J. Gil and E. Bernabeu, “Depolarization and polarization indexes of an optical system,” J. Mod. Opt. 33, 185-189 (1986).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

S. N. Savenkov, R. S. Muttiah, K. E. Yushtin, and S. A. Volchkov, “Mueller-matrix model of an inhomogeneous, linear, birefringent medium: single scattering case,” J. Quant. Spectrosc. Radiat. Transfer 106, 475-486 (2007).
[CrossRef]

Opt. Commun. (2)

R. Espinosa-Luna and E. Bernabeu, “On the Q(M) depolarization metric,” Opt. Commun. 277, 256-258 (2007).
[CrossRef]

S. Y. Lu and R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11-14 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Optik (1)

S. R. Cloude, “Group theory and polarization algebra,” Optik 75, 26-36 (1986).

Ukr. J. Phys. (1)

S. N. Savenkov and K. E. Yushtin, “Pecularities of depolarization of linearly polarized radiation by a layer of the anisotropic inhomogeneous medium,” Ukr. J. Phys. 50, 235-239 (2005).

Other (3)

R. A. Chipman, Polarimetry, in Handbook of Optics (McGraw Hill, 1995), Vol. 2.

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

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989).

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Figures (2)

Fig. 1
Fig. 1

(a) Output degree of polarization of the optical system described by Eq. (24) as a function of the incident state of polarization parametrized by the ellipsometric angles χ and ψ. (b) Plot of the gain of the optical system described by Eq. (24) as a function of the incident state of polarization parametrized by the ellipsometric angles χ and ψ.

Fig. 2
Fig. 2

(a) Output degree of polarization of the optical system described by Eq. (26) as a function of the incident state of polarization parametrized by the ellipsometric angles χ and ψ. (b) Plot of the gain of the optical system described by Eq. (26) as a function of the incident state of polarization parametrized by the ellipsometric angles χ and ψ.

Equations (28)

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S o = M S i ( s 0 o s 1 o s 2 o s 3 o ) = [ 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 i s 1 i s 2 i s 3 i ) ,
S α = ( s 0 α s 1 α s 2 α s 3 α ) = ( E p α E p α * + E s α E s α * E p α E p α * E s α E s α * E p α E s α * + E s α E p α * ± i ( E p α E s α * E s α E p α * ) ) ,
S = s 0 ( 1 cos 2 χ cos 2 ψ cos 2 χ sin 2 ψ sin 2 χ ) .
0 DI ( M ) = { j , k = 0 3 m j k 2 m 00 2 } 1 / 2 / 3 m 00 1 .
0 DoP ( M , S ) = ( s 1 o ) 2 + ( s 2 o ) 2 + ( s 3 o ) 2 s 0 o = [ j = 1 3 ( m j 0 s 0 i + m j 1 s 1 i + m j 2 s 2 i + m j 3 s 3 i ) 2 ] 1 / 2 m 00 s 0 i + m 01 s 1 i + m 02 s 2 i + m 03 s 3 i 1 .
M M n + M d ,
M d = [ m 00 m 01 m 02 m 03 0 0 0 0 0 0 0 0 0 0 0 0 ] ;
M n = [ 0 0 0 0 m 10 m 11 m 12 m 13 m 20 m 21 m 22 m 23 m 30 m 31 m 32 m 33 ] .
DoP ( M , S ) = [ j = 1 3 ( m j 0 s 0 i + m j 1 s 1 i + m j 2 s 2 i + m j 3 s 3 i ) 2 ] 1 / 2 m 00 s 0 i + m 01 s 1 i + m 02 s 2 i + m 03 s 3 i = { ( M n S i ) T ( M n S i ) } 1 / 2 { ( M d S i ) T ( M d S i ) } 1 / 2 .
{ DoP } 2 = ( S i ) T ( M n ) T ( M n ) ( S i ) ( S i ) T ( M d ) T ( M d ) ( S i ) 1 .
Tr [ ( M n ) T M n ] = j = 1 , k = 0 3 m j k 2 = j , k = 0 3 m j k 2 k = 0 3 m 0 k 2 = { j , k = 0 3 m j k 2 m 00 2 } k = 1 3 m 0 k 2 .
Tr [ ( M n ) T M n ] = { 3 m 00 2 [ DI ( M ) ] 2 } k = 1 3 m 0 k 2 = m 00 2 { 3 [ DI ( M ) ] 2 [ D ( M ) ] 2 } .
Tr [ ( M d ) T M d ] = m 00 2 + k = 1 3 m 0 k 2 = m 00 2 { 1 + [ D ( M ) ] 2 } ,
0 D ( M ) = m 01 2 + m 02 2 + m 03 2 / m 00 1 .
Q ( M ) j = 1 , k = 0 3 m j k 2 k = 0 3 m 0 k 2 = 3 [ DI ( M ) ] 2 [ D ( M ) ] 2 1 + [ D ( M ) ] 2 .
Q ( M ) = { j , k = 1 3 m j k 2 } / m 00 2 + [ P ( M ) ] 2 1 + [ D ( M ) ] 2 ,
0 P ( M ) = m 10 2 + m 20 2 + m 30 2 / m 00 1 .
0 Q ( M ) = j = 1 , k = 0 3 m j k 2 k = 0 3 m 0 k 2 = 3 [ DI ( M ) ] 2 [ D ( M ) ] 2 1 + [ D ( M ) ] 2 = { j , k = 1 3 m j k 2 } / m 00 2 + [ P ( M ) ] 2 1 + [ D ( M ) ] 2 3 ,
M = 1 2 [ 1 + b 1 b 0 0 1 b 1 + b 0 0 0 0 2 b 0 0 0 0 2 b ] .
DI ( M ) = 1 + 2 b + b 2 1 + b = 1 , for     0 b 1 ,
( s 0 o s 1 o s 2 o s 3 o ) = 1 2 [ 1 + b 1 b 0 0 1 b 1 + b 0 0 0 0 2 b 0 0 0 0 2 b ] ( s 0 i s 1 i s 2 i s 3 i ) = 1 2 ( ( 1 + b ) s 0 i + ( 1 b ) s 1 i ( 1 b ) s 0 i + ( 1 + b ) s 1 i 2 b s 2 i 2 b s 3 i ) ,
DoP ( M , S ) = [ ( 1 b ) s 0 i / 2 + ( 1 + b ) s 1 i / 2 ] 2 + [ b s 2 i ] 2 + [ b s 3 i ] 2 ( 1 + b ) s 0 i / 2 + ( 1 b ) s 1 i / 2 .
DoP ( M ) = 1 for     0 b 1 ,
Q ( M ) = 1 + 4 b + b 2 1 + b 2 = { 3 for     b = 1 1 for     b = 0 1 < Q ( M ) < 3 for     0 < b < 1 ,
M = [ 1 0 0 0 0.001 0.258 0.01 0.009 0.028 0.01 0.241 0.015 0.064 0.009 0.015 0.541 ] .
DI ( M ) = 0.375 , DoP ( M , S r ) = 0.605 , Q ( M ) = 0.423 ,
M = [ 0.7599 0.0257 0.1206 0.0576 0.0372 0.5285 0.0001 0.0496 0.1208 0.0001 0.6184 0.1920 0.0554 0.0572 0.1794 0.4822 ] .
DI ( M ) = 0.7623 , DoP ( M , S + 45 ) = 0.8819 , Q ( M ) = 1.6579 ,

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