Abstract

Various different parameters have been introduced to describe the degree of polarization of a partially polarized electromagnetic field in three dimensions. Of these, parameters based on the eigenvalues of the coherency matrix are invariant under a unitary transformation. Here, explicit expressions are presented for the eigenvalues, thus providing a geometrical interpretation of the behavior. These expressions are applied to the Huynen decomposition and allow interrelations between different parameters to be developed.

© 2011 Optical Society of America

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  1. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
    [CrossRef]
  2. M. R. Dennis, “Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization,” J. Opt. A 6, S26–S31 (2004).
    [CrossRef]
  3. K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
    [CrossRef]
  4. A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253, 10–14 (2005).
    [CrossRef]
  5. J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
    [CrossRef]
  6. P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. Am. A 23, 671–678 (2006).
    [CrossRef]
  7. M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065–2069 (2007).
    [CrossRef]
  8. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  9. J. R. Huynen, Phenomological Theory of Radar Targets(Drukkerij Bronder-Offset Rotterdam N. V., 1970).
  10. J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astron. Soc. 34, 403–419 (1973).
    [CrossRef]
  11. R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150(1977).
    [CrossRef]
  12. J. C. Samson, “Comments on polarization and coherence,” J. Geophys. 48, 195–198 (1980).
  13. J. C. Samson and J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
    [CrossRef]
  14. R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” J. Mod. Opt. 30, 1171–1182 (1983).
  15. W. A. Holm and R. M. Barnes, “On radiation polarization mixed target state decomposition techniques,” in Proceedings 1988, IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
    [CrossRef]
  16. M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
    [CrossRef]
  17. G. Birkhoff and S. MacLane, A Survey of Modern Algebra(Macmillan, 1996), p. 118.
  18. H. Baker, “Solution of the cubic equation with three real roots,” (2009), http://home.pipeline.com/~hbaker1/cubic3realroots.htm.
  19. F. Hioe, “Isotropy and polarization in a statistically stationary electromagnetic field in three-dimensions,” J. Mod. Opt. 53, 1715–1725 (2006)
    [CrossRef]
  20. J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010)
    [CrossRef]

2010 (1)

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010)
[CrossRef]

2009 (1)

H. Baker, “Solution of the cubic equation with three real roots,” (2009), http://home.pipeline.com/~hbaker1/cubic3realroots.htm.

2007 (2)

M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065–2069 (2007).
[CrossRef]

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

2006 (2)

P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. Am. A 23, 671–678 (2006).
[CrossRef]

F. Hioe, “Isotropy and polarization in a statistically stationary electromagnetic field in three-dimensions,” J. Mod. Opt. 53, 1715–1725 (2006)
[CrossRef]

2005 (3)

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253, 10–14 (2005).
[CrossRef]

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

2004 (1)

M. R. Dennis, “Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization,” J. Opt. A 6, S26–S31 (2004).
[CrossRef]

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

1996 (1)

G. Birkhoff and S. MacLane, A Survey of Modern Algebra(Macmillan, 1996), p. 118.

1988 (1)

W. A. Holm and R. M. Barnes, “On radiation polarization mixed target state decomposition techniques,” in Proceedings 1988, IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
[CrossRef]

1983 (1)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” J. Mod. Opt. 30, 1171–1182 (1983).

1980 (2)

J. C. Samson, “Comments on polarization and coherence,” J. Geophys. 48, 195–198 (1980).

J. C. Samson and J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[CrossRef]

1977 (1)

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150(1977).
[CrossRef]

1973 (1)

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astron. Soc. 34, 403–419 (1973).
[CrossRef]

1970 (1)

J. R. Huynen, Phenomological Theory of Radar Targets(Drukkerij Bronder-Offset Rotterdam N. V., 1970).

1962 (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Baker, H.

H. Baker, “Solution of the cubic equation with three real roots,” (2009), http://home.pipeline.com/~hbaker1/cubic3realroots.htm.

Barakat, R.

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” J. Mod. Opt. 30, 1171–1182 (1983).

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150(1977).
[CrossRef]

Barnes, R. M.

W. A. Holm and R. M. Barnes, “On radiation polarization mixed target state decomposition techniques,” in Proceedings 1988, IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
[CrossRef]

Birkhoff, G.

G. Birkhoff and S. MacLane, A Survey of Modern Algebra(Macmillan, 1996), p. 118.

Dennis, M. R.

M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065–2069 (2007).
[CrossRef]

M. R. Dennis, “Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization,” J. Opt. A 6, S26–S31 (2004).
[CrossRef]

Dogaruji, A.

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Ellis, J.

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Friberg, A. T.

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

Gell-Mann, M.

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Gil, J.

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

Gil, J. J.

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010)
[CrossRef]

Goudail, F.

Hioe, F.

F. Hioe, “Isotropy and polarization in a statistically stationary electromagnetic field in three-dimensions,” J. Mod. Opt. 53, 1715–1725 (2006)
[CrossRef]

Holm, W. A.

W. A. Holm and R. M. Barnes, “On radiation polarization mixed target state decomposition techniques,” in Proceedings 1988, IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
[CrossRef]

Huynen, J. R.

J. R. Huynen, Phenomological Theory of Radar Targets(Drukkerij Bronder-Offset Rotterdam N. V., 1970).

Kaivola, M.

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

Lindfors, K.

Luis, A.

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253, 10–14 (2005).
[CrossRef]

MacLane, S.

G. Birkhoff and S. MacLane, A Survey of Modern Algebra(Macmillan, 1996), p. 118.

Olson, J. V.

J. C. Samson and J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[CrossRef]

Ponomarenko, S.

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

Réfrégier, P.

Samson, J. C.

J. C. Samson, “Comments on polarization and coherence,” J. Geophys. 48, 195–198 (1980).

J. C. Samson and J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[CrossRef]

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astron. Soc. 34, 403–419 (1973).
[CrossRef]

San José, I.

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010)
[CrossRef]

Setälä, T.

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

Wolf, E.

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

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

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

Geophys. J. R. Astron. Soc. (2)

J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astron. Soc. 34, 403–419 (1973).
[CrossRef]

J. C. Samson and J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[CrossRef]

J. Geophys. (1)

J. C. Samson, “Comments on polarization and coherence,” J. Geophys. 48, 195–198 (1980).

J. Mod. Opt. (2)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” J. Mod. Opt. 30, 1171–1182 (1983).

F. Hioe, “Isotropy and polarization in a statistically stationary electromagnetic field in three-dimensions,” J. Mod. Opt. 53, 1715–1725 (2006)
[CrossRef]

J. Opt. A (1)

M. R. Dennis, “Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization,” J. Opt. A 6, S26–S31 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253, 10–14 (2005).
[CrossRef]

J. Ellis, A. Dogaruji, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
[CrossRef]

R. Barakat, “Degree of polarization and the principal idempotents of the coherency matrix,” Opt. Commun. 23, 147–150(1977).
[CrossRef]

J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010)
[CrossRef]

Phys. Rev. (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615(2002).
[CrossRef]

Other (4)

G. Birkhoff and S. MacLane, A Survey of Modern Algebra(Macmillan, 1996), p. 118.

H. Baker, “Solution of the cubic equation with three real roots,” (2009), http://home.pipeline.com/~hbaker1/cubic3realroots.htm.

W. A. Holm and R. M. Barnes, “On radiation polarization mixed target state decomposition techniques,” in Proceedings 1988, IEEE National Radar Conference (IEEE, 1988), pp. 249–254.
[CrossRef]

J. R. Huynen, Phenomological Theory of Radar Targets(Drukkerij Bronder-Offset Rotterdam N. V., 1970).

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Figures (3)

Fig. 1
Fig. 1

(a) Geometry to indicate the three real eigenvalues of the coherency matrix, (b) for the case of a single nonzero eigenvalue (fully polarized), and (c) for the case of two equal nonzero eigenvalues (fully 2D unpolarized).

Fig. 2
Fig. 2

(a) Behavior of α with Σ, Δ. (b) Behavior of P, P B with Σ, Δ. The values of the eigenvalues are also shown. (c) Behavior of P P P , P N U with Σ, Δ.

Fig. 3
Fig. 3

(a) Behavior of P P P , P N U with P, α. (b) Behavior of P B with P, α.

Tables (1)

Tables Icon

Table 1 Some Special Polarization Cases

Equations (23)

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C = 1 3 [ Λ 0 + Λ 3 + 1 3 Λ 8 Λ 1 i Λ 2 Λ 4 i Λ 5 Λ 1 + i Λ 2 Λ 0 Λ 3 + 1 3 Λ 8 Λ 6 i Λ 7 Λ 4 + i Λ 5 Λ 6 + i Λ 7 Λ 0 2 3 Λ 8 ]
C = Λ 0 [ 1 0 0 0 1 0 0 0 1 ] + Λ 1 [ 0 1 0 1 0 0 0 0 0 ] + Λ 2 [ 0 i 0 i 0 0 0 0 0 ] + Λ 3 [ 1 0 0 0 1 0 0 0 0 ] + Λ 4 [ 0 0 1 0 0 0 1 0 0 ] + Λ 5 [ 0 0 i 0 0 0 i 0 0 ] + Λ 6 [ 0 0 0 0 0 1 0 1 0 ] + Λ 7 [ 0 0 0 0 0 i 0 i 0 ] + Λ 8 [ 1 / 3 0 0 0 1 / 3 0 0 0 2 / 3 ] .
x 3 tr C x 2 + ( tr C ) 2 tr C 2 2 x det C = 0 ,
tr C = Λ 0 , tr C 2 = 1 3 j = 0 8 Λ j = Σ 3 , det C = 1 243 ( 18 ( Λ 1 Λ 4 Λ 6 Λ 1 Λ 5 Λ 7 + Λ 2 Λ 5 Λ 6 Λ 2 Λ 4 Λ 7 ) 9 Λ 0 ( Σ Λ 0 2 ) + 9 Λ 3 [ ( Λ 4 2 + Λ 5 2 ) ( Λ 6 2 + Λ 7 2 ) ] + 3 Λ 8 { 6 Σ 9 [ ( Λ 4 2 + Λ 5 2 ) + ( Λ 6 2 + Λ 7 2 ) ] 8 Λ 8 2 } ) = Δ .
λ 1 + λ 2 + λ 3 = tr C = Λ 0 , λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 = ( tr C ) 2 tr C 2 2 = Λ 0 2 Σ / 3 2 , λ 1 λ 2 λ 3 = det C = Δ , λ 1 2 + λ 2 2 + λ 3 2 = tr C 2 = Σ / 3 .
P 2 = 1 2 ( 3 tr C 2 ( tr C ) 2 1 ) = 1 2 ( Σ Λ 0 2 1 ) = λ 1 2 + λ 2 2 + λ 3 2 λ 1 λ 2 λ 2 λ 3 λ 3 λ 1 ( λ 1 + λ 2 + λ 3 ) 2 .
P B 2 = 1 27 Δ Λ 0 3 = 1 27 ( λ 1 λ 2 λ 3 ) ( λ 1 + λ 2 + λ 3 ) 3 ,
U C U = D = [ λ 1 0 0 0 λ 2 0 0 0 λ 3 ] = λ 1 [ 1 0 0 0 0 0 0 0 0 ] + λ 2 [ 0 0 0 0 1 0 0 0 0 ] + λ 3 [ 0 0 0 0 0 0 0 0 1 ] ,
U = [ e 1 * e 2 * e 3 * ] .
e i e j = δ i j .
C = U D U = λ 1 e 1 * e 1 T + λ 2 e 2 * e 2 T + λ 3 e 3 * e 3 T ,
e 1 e 1 + e 2 e 2 + e 3 e 3 = I ,
C = ( λ 1 λ 2 ) e 1 * e 1 T + 2 ( λ 2 λ 3 ) ( e 1 * e 1 T + e 2 * e 2 T ) + 3 λ 3 I ,
C = ( λ 1 λ 2 ) e 1 * e 1 T + 2 ( λ 2 λ 3 ) ( I e 3 * e 3 T ) + 3 λ 3 I = C P + C U P + C U ,
P P P = λ 1 λ 2 λ 1 + λ 2 + λ 3 ,
P N U = 1 3 λ 3 λ 1 + λ 2 + λ 3 = λ 1 + λ 2 2 λ 3 λ 1 + λ 2 + λ 3
λ j = Λ 0 3 { 1 + 2 P cos [ α ( j 1 ) 2 π 3 ] } , j = 1 , 2 , 3 ,
α = 1 3 arccos ( D 4 P 3 Λ 0 3 ) , D = 54 Δ + Λ 0 ( 3 Σ 5 Λ 0 2 ) = 4 ( Λ 1 Λ 4 Λ 6 Λ 1 Λ 5 Λ 7 + Λ 2 Λ 5 Λ 6 Λ 2 Λ 4 Λ 7 ) + Λ 0 ( Σ 3 Λ 0 2 ) + 2 Λ 3 [ ( Λ 4 2 + Λ 5 2 ) ( Λ 6 2 + Λ 7 2 ) ] + 2 3 9 Λ 8 { 6 Σ 9 [ ( Λ 4 2 + Λ 5 2 ) + ( Λ 6 2 + Λ 7 2 ) ] 8 Λ 8 2 } .
2 ( Σ Λ 0 2 ) 3 / 2 54 Δ + Λ 0 ( 3 Σ 5 Λ 0 2 ) 2 ( Σ Λ 0 2 ) 3 / 2 ,
P P P = λ 1 λ 2 Λ 0 = P ( cos α 1 3 sin α ) , P N U = 1 3 λ 3 Λ 0 = P ( cos α + 3 sin α ) .
4 P 2 = 3 P P P 2 + P N U 2 , cos α = 3 P P P + P N U 2 3 P P P 2 + P N U 2 .
0 α π 3 , P 1 2 0 α π 3 arccos ( 1 2 P ) , 1 2 P 1 .
P B = 1 27 Δ Λ 0 3 = P 3 2 P cos 3 α ,

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