Abstract

The effect of a linear random medium on the state of polarization of the transmitted light is investigated, and the connection between the Stokes vector formalism and the coherence or polarization matrix formalism is discussed. It is shown that an ensemble of Jones matrices corresponds to the Mueller matrix in general.

© 1987 Optical Society of America

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References

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  1. G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).
  2. N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 119–260 (1930), especially Sec. 9.
    [CrossRef]
  3. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [CrossRef]
  4. H. Mueller, “The foundation of optics,”J. Opt. Soc. Am. 38, 661 (A) (1948); see also N. G. Parke, “Optical algebra,”J. Math. Phys. (MIT) 28, 131–139 (1949).
  5. R. C. Jones, “A new calculus for the treatment of optical systems,”J. Opt. Soc. Am. 31, 488–493 (1941); J. Opt. Soc. Am. 32, 486–493 (1942); J. Opt. Soc. Am. 37, 107–110 (1947).
    [CrossRef]
  6. Although this matrix was originally called the coherency matrix (cf. Ref. 2), it actually describes the state of polarization of the wave; therefore we call it the polarization matrix from here on.
  7. R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,”J. Opt. Soc. Am. 53, 317–323 (1963).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sect. 2.12.
  9. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 809–812 (1979).
    [CrossRef] [PubMed]
  10. K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,”J. Math. Phys. 10, 1935–1938 (1969).
    [CrossRef]
  11. E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef] [PubMed]
  12. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
    [CrossRef]
  13. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  14. J. J. Gil, E. Bernabeu, “A depolarization criterion in Mueller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  15. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
    [CrossRef]
  16. G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
    [CrossRef]
  17. E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

1985

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

1982

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1981

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

E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef] [PubMed]

1979

1969

K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,”J. Math. Phys. 10, 1935–1938 (1969).
[CrossRef]

1963

1960

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

1959

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

1954

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

1948

H. Mueller, “The foundation of optics,”J. Opt. Soc. Am. 38, 661 (A) (1948); see also N. G. Parke, “Optical algebra,”J. Math. Phys. (MIT) 28, 131–139 (1949).

1941

1930

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 119–260 (1930), especially Sec. 9.
[CrossRef]

1852

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Abhyankar, K. D.

K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,”J. Math. Phys. 10, 1935–1938 (1969).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sect. 2.12.

Barakat, R.

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

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,”J. Opt. Soc. Am. 53, 317–323 (1963).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sect. 2.12.

Bernabeu, E.

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

Fry, E. S.

Fymat, A. L.

K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,”J. Math. Phys. 10, 1935–1938 (1969).
[CrossRef]

Gil, J. J.

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

Howell, B. J.

Jones, R. C.

Kattawar, G. W.

Mueller, H.

H. Mueller, “The foundation of optics,”J. Opt. Soc. Am. 38, 661 (A) (1948); see also N. G. Parke, “Optical algebra,”J. Math. Phys. (MIT) 28, 131–139 (1949).

O’Neil, E. L.

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Parrent, G. B.

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

Roman, P.

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

Simon, R.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Stokes, G. G.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Wiener, N.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 119–260 (1930), especially Sec. 9.
[CrossRef]

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

Acta Math.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 119–260 (1930), especially Sec. 9.
[CrossRef]

Appl. Opt.

J. Math. Phys.

K. D. Abhyankar, A. L. Fymat, “Relations between the elements of the phase matrix for scattering,”J. Math. Phys. 10, 1935–1938 (1969).
[CrossRef]

J. Opt. Soc. Am.

Nuovo Cimento

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
[CrossRef]

G. B. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 15, 370–388 (1960).
[CrossRef]

Opt. Acta

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

Opt. Commun.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[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]

Trans. Cambridge Philos. Soc.

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc. 9, 399–416 (1852).

Other

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Although this matrix was originally called the coherency matrix (cf. Ref. 2), it actually describes the state of polarization of the wave; therefore we call it the polarization matrix from here on.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sect. 2.12.

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Equations (44)

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E = E 1 1 + E 2 2 ,
κ · j = 0             ( i = 1 , 2 )
i * · j = δ i j             ( i , j = 1 , 2 ) .
J i j = E i E j * ,
P = [ 1 - 4 det J / ( Tr J ) 2 ] 1 / 2 .
S 0 = E 1 E 1 * + E 2 E 2 * , S 1 = E 1 E 1 * - E 2 E 2 * , S 2 = E 1 E 2 * + E 2 E 1 * , S 3 = i [ E 2 E 1 * - E 1 E 2 * ]
P = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 / S 0 .
σ ( 0 ) = [ 1 0 0 1 ] , σ ( 1 ) = [ - 1 0 0 + 1 ] , σ ( 2 ) = [ 0 1 1 0 ] , σ ( 3 ) = [ 0 i - i 0 ] ,
J i j = 1 2 S μ σ i j ( μ )             ( μ = 0 , 1 , 2 , 3 ) .
Tr [ J σ ( ν ) ] = 1 2 S μ Tr [ σ ( μ ) σ ( ν ) ] .
S μ = Tr [ J σ ( μ ) ] = J i j σ j i ( μ )             ( μ = 0 , 1 , 2 , 3 ) ,
E = E 1 1 + E 2 2 ,
E i = T i j E j             ( i , j = 1 , 2 ) ,
J i j = E i E j * = T i m E m E n * T j n * = T i m J m n T n j
J = T J T ,
S μ = M μ ν S ν             ( μ , ν = 0 , 1 , 2 , 3 ) .
S μ = Tr [ J σ ( μ ) ] = Tr [ T J T σ ( μ ) ]
S μ = T i m J m n T j n * σ j i ( μ ) .
S μ = 1 2 Tr [ T σ ( ν ) T σ ( μ ) ] S ν .
M μ ν = 1 2 Tr [ T σ ( ν ) T σ ( μ ) ] = 1 2 Tr [ σ ( μ ) T σ ( ν ) T ] = 1 2 T n p T q m σ m n ( μ ) σ p q ( ν ) ,
J m n = I 1 δ m 1 δ n 1 .
J i j = T i 1 T j 1 * I 1
det J = I 1 ( T 11 T 11 * T 21 T 21 * - T 21 T 11 * T 11 T 21 * ) = 0.
M = [ K a 1 b 1 K 0 a 2 b 2 0 0 a 3 b 3 0 0 a 4 b 4 0 ]
Tr ( M T M ) = M μ ν M μ ν = 1 4 Tr [ T σ ( ν ) T σ ( μ ) ] Tr [ T σ ( ν ) T σ ( μ ) ] ,
Tr [ A ] Tr [ B ] = Tr [ A B ]
( A B ) ( C D ) = A C B D ,
Tr ( M T M ) = 1 4 Tr [ T σ ( ν ) T σ ( μ ) T σ ( ν ) T σ ( μ ) ] = 1 4 Tr [ ( T σ ( ν ) T σ ( ν ) ) ( T σ ( μ ) T σ ( μ ) ) ] = 1 4 Tr [ ( T T ) ( σ ( ν ) σ ( ν ) ) ( T T ) ( σ ( μ ) σ ( μ ) ) ] .
1 2 σ ( ν ) σ ( ν ) = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] ,
Tr ( M T M ) = Tr [ ( T T ) ( T T ) ]
Tr ( M T M ) = Tr ( T T T T ) = [ Tr ( T T ) ] 2 .
M 00 = ½ Tr ( T T )
Tr ( M M ) = 4 M 00 2 .
E i ( e ) = T i j ( e ) E j .
J i j = e p e E i ( e ) E j ( e ) * = e p e T i m ( e ) J m n T n j ( e )
J = e ( p e T ( e ) J T ( e ) ) = T ( e ) J T ( e ) e ,
det J = I 1 [ T 11 ( e ) T 11 ( e ) * e T 21 ( e ) T 21 ( e ) * e - T 21 ( e ) T 11 ( e ) * e T 11 ( e ) T 21 ( e ) * e ] ,
M μ ν = 1 2 Tr [ e p e σ ( μ ) T ( e ) σ ( ν ) T ( e ) ]
M μ ν = 1 2 T n p ( e ) T q m ( e ) e σ m n ( μ ) σ p q ( ν ) .
T n p q m = T n p ( e ) T q m ( e ) e ,
M μ ν = 1 2 T n p q m σ q m ( μ ) σ p q ( ν ) .
M μ ν σ i j ( μ ) * σ k l ( ν ) * = 1 2 T n p q m σ m n ( μ ) σ i j ( μ ) * σ p q ( ν ) σ k l ( ν ) * ,
σ m n ( μ ) σ i j ( μ ) * = 2 δ m i δ n j .
1 2 M μ ν σ i j ( μ ) * σ k l ( ν ) * = τ j k l i ,

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