Abstract

The degree of polarization is employed as a criterion to find the nine independent relations among the elements of the Mueller–Jones matrix. This procedure is applied by considering a previously determined, physically realizable Mueller matrix. On the other hand, the nine bilinear constrains are obtained by directly measuring the degree of polarization from an outgoing beam of light from an optical system by considering nine incident states of light taken from the Poincaré sphere. For practical purposes, all the incident polarization states must be scanned from the Poincaré sphere in order to satisfy the overpolarization and the overgain conditions, respectively, for the physical realizability of the Mueller matrix.

© 2007 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1989).
  2. E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, 1993).
  3. D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
    [CrossRef]
  4. See, for example, the following sites: http://www.gaertnerscientific.com/ellipsometers/lse.htm;
    http://www.lot-oriel.com/site/site_down/el_m2000_uken.pdf.
  5. A. C. Holland and C. Cagne, "The scattering of polarized light by polydisperse systems of irregular particles," Appl. Opt. 9, 1113-1121 (1970).
    [CrossRef] [PubMed]
  6. P. S. Hauge, "Mueller matrix ellipsometry with imperfect compensators," J. Opt. Soc. Am. 68, 1519-1528 (1978).
    [CrossRef]
  7. G. Atondo-Rubio, R. Espinosa-Luna, and A. Mendoza-Suárez, "Mueller matrix determination for one-dimensional rough surfaces with a reduced number of measurements," Opt. Commun. 244, 7-13 (2005).
    [CrossRef]
  8. R. Espinosa-Luna, A. Mendoza-Suárez, G. Atondo-Rubio, S. Hinojosa, J. O. Rivera-Vázquez, and J. T. Guillén-Bonilla, "Mueller matrix determination for one-dimensional rough surfaces: four reduced measurement equivalent sets," Opt. Commun. 259, 60-63 (2006).
    [CrossRef]
  9. W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices and polarized scattered light," Am. J. Phys. 53, 468-478 (1985).
    [CrossRef]
  10. R. Espinosa-Luna, "Scattering by rough surfaces in a conical configuration: experimental Mueller matrix," Opt. Lett. 27, 1510-1512 (2002).
    [CrossRef]
  11. R. Espinosa-Luna, G. Atondo-Rubio, and A. Mendoza-Suárez, "Complete determination of the conical Mueller matrix for one-dimensional rough metallic surfaces," Opt. Commun. 257, 62-71 (2006).
    [CrossRef]
  12. O. G. Rodríguez-Herrera and N. C. Bruce, "Mueller matrix for an ellipsoidal mirror," Opt. Eng. 45, 053602 (2006).
    [CrossRef]
  13. K. A. O'Donnell and M. E. Knotts, "Polarization dependence of scattering from one-dimensional rough surfaces," J. Opt. Soc. Am. A 8, 1126-1131 (1991).
    [CrossRef]
  14. N. C. Bruce, A. J. Sant, and J. Dainty, "The Mueller matrix for rough surface scattering using the Kirchhoff approximation," Opt. Commun. 88, 471-484 (1992).
    [CrossRef]
  15. C. Brosseau, Fundamentals of Polarized Light: Statistical Optics Approach (Wiley, 1998).
  16. R. C. Jones, "A new calculus for the treatment of optical systems. V. A more general formulation, and description of another calculus," J. Opt. Soc. Am. 37, 107-110 (1947).
    [CrossRef]
  17. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  18. R. Barakat, "Bilinear constraints between elements of the 4 × 4 Mueller-Jones transfer matrix of polarization optics," Opt. Commun. 38, 159-161 (1981).
    [CrossRef]
  19. E. S. Fry and G. Kattawar, "Relationships between elements of the Stokes matrix," Appl. Opt. 20, 2811-2814 (1981).
    [CrossRef] [PubMed]
  20. J. W. Hovenier, H. C. Van der Hulst, and C. V. M. Van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).
  21. J. W. Hovenier, "Structure of a general pure Mueller matrix," Appl. Opt. 33, 8318-8324 (1994).
    [CrossRef] [PubMed]
  22. C. Brosseau, "Mueller matrix analysis of light depolarization by a linear optical medium," Opt. Commun. 131, 229-235 (1996).
    [CrossRef]
  23. See page 175, Eqs. -(88) and (9.90), respectively, of Ref. .
  24. A. B. Kostinski, C. R. Givens, and J. M. Kwiatkowski, "Constraints on Mueller matrices of polarization optics," Appl. Opt. 32, 1646-1651 (1993).
    [CrossRef] [PubMed]
  25. J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
    [CrossRef]
  26. J. J. Gil and E. Bernabeu, "Depolarization and polarization indexes of an optical system," Opt. Acta 33, 185-189 (1986).
    [CrossRef]
  27. J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the polar decomposition of its Mueller matrix," Optik 76, 67-71 (1987).

2006 (3)

R. Espinosa-Luna, A. Mendoza-Suárez, G. Atondo-Rubio, S. Hinojosa, J. O. Rivera-Vázquez, and J. T. Guillén-Bonilla, "Mueller matrix determination for one-dimensional rough surfaces: four reduced measurement equivalent sets," Opt. Commun. 259, 60-63 (2006).
[CrossRef]

R. Espinosa-Luna, G. Atondo-Rubio, and A. Mendoza-Suárez, "Complete determination of the conical Mueller matrix for one-dimensional rough metallic surfaces," Opt. Commun. 257, 62-71 (2006).
[CrossRef]

O. G. Rodríguez-Herrera and N. C. Bruce, "Mueller matrix for an ellipsoidal mirror," Opt. Eng. 45, 053602 (2006).
[CrossRef]

2005 (1)

G. Atondo-Rubio, R. Espinosa-Luna, and A. Mendoza-Suárez, "Mueller matrix determination for one-dimensional rough surfaces with a reduced number of measurements," Opt. Commun. 244, 7-13 (2005).
[CrossRef]

2002 (1)

1996 (1)

C. Brosseau, "Mueller matrix analysis of light depolarization by a linear optical medium," Opt. Commun. 131, 229-235 (1996).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

N. C. Bruce, A. J. Sant, and J. Dainty, "The Mueller matrix for rough surface scattering using the Kirchhoff approximation," Opt. Commun. 88, 471-484 (1992).
[CrossRef]

1991 (1)

1987 (1)

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the polar decomposition of its Mueller matrix," Optik 76, 67-71 (1987).

1986 (2)

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

J. W. Hovenier, H. C. Van der Hulst, and C. V. M. Van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

1985 (2)

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

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices and polarized scattered light," Am. J. Phys. 53, 468-478 (1985).
[CrossRef]

1981 (2)

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

E. S. Fry and G. Kattawar, "Relationships between elements of the Stokes matrix," Appl. Opt. 20, 2811-2814 (1981).
[CrossRef] [PubMed]

1978 (1)

1970 (1)

1947 (1)

Am. J. Phys. (1)

W. S. Bickel and W. M. Bailey, "Stokes vectors, Mueller matrices and polarized scattered light," Am. J. Phys. 53, 468-478 (1985).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

J. W. Hovenier, H. C. Van der Hulst, and C. V. M. Van der Mee, "Conditions for the elements of the scattering matrix," Astron. Astrophys. 157, 301-310 (1986).

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

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

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

Opt. Commun. (6)

C. Brosseau, "Mueller matrix analysis of light depolarization by a linear optical medium," Opt. Commun. 131, 229-235 (1996).
[CrossRef]

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

N. C. Bruce, A. J. Sant, and J. Dainty, "The Mueller matrix for rough surface scattering using the Kirchhoff approximation," Opt. Commun. 88, 471-484 (1992).
[CrossRef]

R. Espinosa-Luna, G. Atondo-Rubio, and A. Mendoza-Suárez, "Complete determination of the conical Mueller matrix for one-dimensional rough metallic surfaces," Opt. Commun. 257, 62-71 (2006).
[CrossRef]

G. Atondo-Rubio, R. Espinosa-Luna, and A. Mendoza-Suárez, "Mueller matrix determination for one-dimensional rough surfaces with a reduced number of measurements," Opt. Commun. 244, 7-13 (2005).
[CrossRef]

R. Espinosa-Luna, A. Mendoza-Suárez, G. Atondo-Rubio, S. Hinojosa, J. O. Rivera-Vázquez, and J. T. Guillén-Bonilla, "Mueller matrix determination for one-dimensional rough surfaces: four reduced measurement equivalent sets," Opt. Commun. 259, 60-63 (2006).
[CrossRef]

Opt. Eng. (1)

O. G. Rodríguez-Herrera and N. C. Bruce, "Mueller matrix for an ellipsoidal mirror," Opt. Eng. 45, 053602 (2006).
[CrossRef]

Opt. Lett. (1)

Optik (1)

J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of a nondepolarizing optical system from the polar decomposition of its Mueller matrix," Optik 76, 67-71 (1987).

Other (7)

See page 175, Eqs. -(88) and (9.90), respectively, of Ref. .

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

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

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

E. Collett, Polarized Light Fundamentals and Applications (Marcel Dekker, 1993).

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

See, for example, the following sites: http://www.gaertnerscientific.com/ellipsometers/lse.htm;
http://www.lot-oriel.com/site/site_down/el_m2000_uken.pdf.

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Figures (3)

Fig. 1
Fig. 1

(a) Output degree of polarization of the optical system described by Eq. (40) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i . (b) Plot of the gain of the optical system described by Eq. (40) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i .

Fig. 2
Fig. 2

(a) Output degree of polarization of the optical system described by Eq. (41) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i . (b) Plot of the gain of the optical system described by Eq. (41) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i .

Fig. 3
Fig. 3

(a) Output degree of polarization of the optical system described by Eq. (42) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i . (b) Plot of the gain of the optical system described by Eq. (42) as a function of the incident state of polarization parametrized by the ellipsometric angles χ i and ψ i .

Equations (64)

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m 01 2 m 11 2 m 21 2 m 31 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 02 2 m 12 2 m 22 2 m 32 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 03 2 m 13 2 m 23 2 m 33 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 00 m 01 m 10 m 11 m 20 m 21 m 30 m 31 = 0 ,
m 00 m 02 m 10 m 12 m 20 m 22 m 30 m 32 = 0 ,
m 00 m 03 m 10 m 13 m 20 m 23 m 30 m 33 = 0 ,
m 01 m 02 m 11 m 12 m 21 m 22 m 31 m 32 = 0 ,
m 01 m 03 m 11 m 13 m 21 m 23 m 31 m 33 = 0 ,
m 02 m 03 m 12 m 13 m 22 m 23 m 32 m 33 = 0 .
0 DoP = ( s 1 ) 2 + ( s 2 ) 2 + ( s 3 ) 2 s 0 1 , S = ( s 0 s 1 s 2 s 3 ) ,
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 ) = ( m 00 s 0 i + m 01 s 1 i + m 02 s 2 i + m 03 s 3 i m 10 s 0 i + m 11 s 1 i + m 12 s 2 i + m 13 s 3 i m 20 s 0 i + m 21 s 1 i + m 22 s 2 i + m 23 s 3 i m 30 s 0 i + m 31 s 1 i + m 32 s 2 i + m 33 s 3 i ) ,
s 0 = E p E p * + E s E s * , s 1 = E p E p * E s E s * ,
s 2 = E p E s * + E s E p * , s 3 = i ( E p E s * E s E p * ) ,
E o = J E i ( E p o E s o ) = [ j 11 j 12 j 21 j 22 ] ( E p i E s i ) ,
M J = A ( J J * ) A 1 ; A = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] ;
A 1 = 1 2 ( A * ) T ,
M T G M = | det ( J ) | 2 G = ( det ( M ) ) 1 / 2 G ,
G = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 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 ( M , S ) = { ( M n S i ) T ( M n S i ) } 1 / 2 { ( M d S i ) T ( M d S i ) } 1 / 2 1 ,
( S i ) T [ M d T M d ] ( S i ) ( S i ) T [ M n T M n ] ( S i ) ,
m 00 2 m 10 2 + m 20 2 + m 30 2 ,
m 01 2 m 11 2 + m 21 2 + m 31 2 ,
m 02 2 m 12 2 + m 22 2 + m 32 2 ,
m 03 2 m 13 2 + m 23 2 + m 33 2 ,
m 00 m 01 m 10 m 11 + m 20 m 21 + m 30 m 31 ,
m 00 m 02 m 10 m 12 + m 20 m 22 + m 30 m 32 ,
m 00 m 03 m 10 m 13 + m 20 m 23 + m 30 m 33 ,
m 01 m 02 m 11 m 12 + m 21 m 22 + m 31 m 32 ,
m 01 m 03 m 11 m 13 + m 21 m 23 + m 31 m 33 ,
m 02 m 03 m 12 m 13 + m 22 m 23 + m 32 m 33 .
( S o ) p = [ 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 ] ( 1 1 0 0 ) = ( m 00 + m 01 m 10 + m 11 m 20 + m 21 m 30 + m 31 )
0 ( DoP ) p = ( m 10 + m 11 ) 2 + ( m 20 + m 21 ) 2 + ( m 30 + m 31 ) 2 m 00 + m 01 1.
( m 00 + m 01 ) 2 ( m 10 + m 11 ) 2 + ( m 20 + m 21 ) 2 + ( m 30 + m 31 ) 2 .
( m 00 m 01 ) 2 ( m 10 m 11 ) 2 + ( m 20 m 21 ) 2 + ( m 30 m 31 ) 2 ,
( m 00 + m 02 ) 2 ( m 10 + m 12 ) 2 + ( m 20 + m 22 ) 2 + ( m 30 + m 32 ) 2 ,
( m 00 m 02 ) 2 ( m 10 m 12 ) 2 + ( m 20 m 22 ) 2 + ( m 30 m 32 ) 2 ,
( m 00 + m 03 ) 2 ( m 10 + m 13 ) 2 + ( m 20 + m 23 ) 2 + ( m 30 + m 33 ) 2 ,
( m 00 m 03 ) 2 ( m 10 m 13 ) 2 + ( m 20 m 23 ) 2 + ( m 30 m 33 ) 2 .
m 01 2 m 11 2 m 21 2 m 31 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 02 2 m 12 2 m 22 2 m 32 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 03 2 m 13 2 m 23 2 m 33 2 + m 00 2 m 10 2 m 20 2 m 30 2 = 0 ,
m 00 m 01 m 10 m 11 m 20 m 21 m 30 m 31 = 0 ,
m 00 m 02 m 10 m 12 m 20 m 22 m 30 m 32 = 0 ,
m 00 m 03 m 10 m 13 m 20 m 23 m 30 m 33 = 0.
( m 22 + m 33 ) 2 + ( m 23 m 32 ) 2 = ( m 00 + m 11 ) 2 ( m 01 + m 10 ) 2 ,
( m 20 m 21 ) 2 + ( m 30 m 31 ) 2 = ( m 00 m 01 ) 2 ( m 11 m 10 ) 2 ,
( m 02 m 12 ) 2 + ( m 03 m 13 ) 2 = ( m 00 m 10 ) 2 ( m 11 m 01 ) 2 ,
( m 02 + m 12 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 20 m 21 ) ( m 23 m 32 ) ( m 30 m 31 ) ,
( m 22 m 33 ) ( m 00 + m 11 m 01 m 10 ) = ( m 02 m 12 ) × ( m 20 m 21 ) + ( m 03 m 13 ) ( m 30 m 31 ) ,
( m 20 + m 21 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 02 m 12 ) + ( m 23 m 32 ) ( m 03 m 13 ) ,
( m 23 + m 32 ) ( m 00 + m 11 m 01 m 10 ) = ( m 03 m 13 ) × ( m 20 m 21 ) + ( m 02 m 12 ) ( m 30 m 31 ) ,
( m 03 + m 13 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 30 m 31 ) + ( m 23 m 32 ) ( m 20 m 21 ) ,
( m 30 + m 31 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 03 m 13 ) ( m 23 m 32 ) ( m 02 m 12 ) .
( m 22 m 33 ) ( m 00 + m 11 m 01 m 10 ) = ( m 20 m 21 ) × ( m 02 m 12 ) ( m 30 m 31 ) ( m 03 m 13 ) ,
( m 03 + m 13 ) ( m 00 + m 11 m 01 m 10 ) = ( m 20 m 21 ) × ( m 23 m 32 ) + ( m 30 m 31 ) ( m 22 + m 33 ) ,
( m 20 + m 21 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 02 m 12 ) + ( m 23 m 32 ) ( m 03 m 13 ) ,
( m 30 + m 31 ) ( m 00 + m 11 m 01 m 10 ) = ( m 22 + m 33 ) × ( m 03 m 13 ) ( m 23 m 32 ) ( m 02 m 12 ) .
[ 4 0 3 1 0 0 3 3 2 0 2 3 2 0 2 3 0 0 ]
[ 1.000 0.019 0.021 0.130 0.024 0.731 0.726 0.005 0.008 0.673 0.688 0.351 0.009 0.259 0.247 0.965 ] .
[ 0.737 0.005 0.006 0.067 0.005 0.987 0.024 0.131 0.006 0.024 0.989 0.304 0.067 0.131 0.304 0.674 ] .

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