Abstract

There has been much discussion in the literature about rival measures of classical polarization in three dimensions. We gather and compare the various proposed measures of polarization, creating a geometric representation of the polarization state space in the process. We use majorization, previously used in quantum information, as a criterion to establish a partial ordering on the polarization state space. Using this criterion and other considerations, the most useful polarization measure in three dimensions is found to be one dependent on the Bloch vector decomposition of the polarization matrix.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. H. Poincaré, “Leçons sur la théorie mathématique de la lumière,” G. Carré. Paris 4, 408 (1889).
  3. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley-Interscience, 1998).
  4. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge, 2007).
  5. A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32, 1015–1016 (2007).
    [CrossRef]
  6. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).
  7. T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
    [CrossRef]
  8. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [CrossRef]
  9. X. F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [CrossRef]
  10. R. Barakat and C. Brosseau, “von Neumann entropy of n interacting pencils of radiation,” J. Opt. Soc. Am. A 10, 529–532 (1993).
    [CrossRef]
  11. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  12. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
    [CrossRef]
  13. M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065–2069 (2007).
    [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. O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
    [CrossRef]
  16. M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436–439 (1999).
    [CrossRef]
  17. H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions—beyond mean values,” Phys. Rev. A 63, 022301 (2001).
    [CrossRef]
  18. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [CrossRef]
  19. A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71, 063815 (2005).
    [CrossRef]
  20. T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
    [CrossRef]
  21. T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
    [CrossRef]
  22. C. J. R. Sheppard, “Geometric representation for partial polarization in three dimensions,” Opt. Lett. 37, 2772–2774 (2012).
    [CrossRef]
  23. J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
    [CrossRef]
  24. X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).
  25. M. Gell-Mann and Y. Ne’eman, The Eightfold Way (Benjamin, 1964).
  26. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [CrossRef]
  27. G. Kimura, “The Bloch vector for N-level systems,” Phys. Lett. A 314, 339–349 (2003).
    [CrossRef]
  28. E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. III. Teil,” Mathematische Annalen 65, 370–399 (1908).
    [CrossRef]
  29. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
    [CrossRef]
  30. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics (Springer, 1997).
  31. A. W. Marshall, I. Olkin, and B. C. Arnold, “Schur-convex functions,” in Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (Springer, 2011), pp. 79–154.

2012 (3)

O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
[CrossRef]

C. J. R. Sheppard, “Geometric representation for partial polarization in three dimensions,” Opt. Lett. 37, 2772–2774 (2012).
[CrossRef]

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

2011 (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

X. F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[CrossRef]

2010 (2)

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

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[CrossRef]

2009 (1)

2007 (3)

2005 (2)

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71, 063815 (2005).
[CrossRef]

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

2003 (1)

G. Kimura, “The Bloch vector for N-level systems,” Phys. Lett. A 314, 339–349 (2003).
[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]

2001 (1)

H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions—beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[CrossRef]

2000 (1)

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

1999 (1)

M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436–439 (1999).
[CrossRef]

1993 (1)

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).

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

1946 (1)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

1908 (1)

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. III. Teil,” Mathematische Annalen 65, 370–399 (1908).
[CrossRef]

1889 (1)

H. Poincaré, “Leçons sur la théorie mathématique de la lumière,” G. Carré. Paris 4, 408 (1889).

1852 (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).

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Al-Qasimi, A.

Arnold, B. C.

A. W. Marshall, I. Olkin, and B. C. Arnold, “Schur-convex functions,” in Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (Springer, 2011), pp. 79–154.

Barakat, R.

R. Barakat and C. Brosseau, “von Neumann entropy of n interacting pencils of radiation,” J. Opt. Soc. Am. A 10, 529–532 (1993).
[CrossRef]

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

Bergman, J.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Bhatia, R.

R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics (Springer, 1997).

Bloch, F.

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

Brosseau, C.

R. Barakat and C. Brosseau, “von Neumann entropy of n interacting pencils of radiation,” J. Opt. Soc. Am. A 10, 529–532 (1993).
[CrossRef]

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

Carozzi, T.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Chuang, I.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

Dennis, M. R.

Dogariu, A.

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

Eberly, J. H.

Ellis, J.

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

Friberg, A. T.

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[CrossRef]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[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]

Gamel, O.

O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
[CrossRef]

Gell-Mann, M.

M. Gell-Mann and Y. Ne’eman, The Eightfold Way (Benjamin, 1964).

Gil, J. J.

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

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

Gu, M.

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

James, D.

James, D. F. V.

O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
[CrossRef]

Kaivola, M.

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

Karlsson, R.

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[CrossRef]

Kimura, G.

G. Kimura, “The Bloch vector for N-level systems,” Phys. Lett. A 314, 339–349 (2003).
[CrossRef]

Korotkova, O.

Lan, T.

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

Li, X.

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

Lindfors, K.

Lo, H.-K.

H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions—beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[CrossRef]

Luis, A.

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71, 063815 (2005).
[CrossRef]

Marshall, A. W.

A. W. Marshall, I. Olkin, and B. C. Arnold, “Schur-convex functions,” in Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (Springer, 2011), pp. 79–154.

Milione, G.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Ne’eman, Y.

M. Gell-Mann and Y. Ne’eman, The Eightfold Way (Benjamin, 1964).

Nielsen, M.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

Nielsen, M. A.

M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436–439 (1999).
[CrossRef]

Nolan, D. A.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Olkin, I.

A. W. Marshall, I. Olkin, and B. C. Arnold, “Schur-convex functions,” in Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (Springer, 2011), pp. 79–154.

Poincaré, H.

H. Poincaré, “Leçons sur la théorie mathématique de la lumière,” G. Carré. Paris 4, 408 (1889).

Ponomarenko, S.

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

Popescu, S.

H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions—beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[CrossRef]

Qian, X. F.

San José, I.

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

Schmidt, E.

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. III. Teil,” Mathematische Annalen 65, 370–399 (1908).
[CrossRef]

Setälä, T.

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[CrossRef]

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34, 3394–3396 (2009).
[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]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Sheppard, C. J. R.

Shevchenko, A.

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[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]

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).

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Tien, C. H.

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

Voipio, T.

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[CrossRef]

Wolf, E.

A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32, 1015–1016 (2007).
[CrossRef]

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge, 2007).

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

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]

G. Carré. Paris (1)

H. Poincaré, “Leçons sur la théorie mathématique de la lumière,” G. Carré. Paris 4, 408 (1889).

J. Mod. Opt. (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).

J. Opt. Soc. Am. A (2)

Mathematische Annalen (1)

E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. III. Teil,” Mathematische Annalen 65, 370–399 (1908).
[CrossRef]

Nat. Commun. (1)

X. Li, T. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3, 998 (2012).

Opt. Commun. (2)

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

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

Opt. Lett. (4)

Phys. Lett. A (1)

G. Kimura, “The Bloch vector for N-level systems,” Phys. Lett. A 314, 339–349 (2003).
[CrossRef]

Phys. Rev. (1)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

Phys. Rev. A (4)

A. Luis, “Polarization distribution and degree of polarization for three-dimensional quantum light fields,” Phys. Rev. A 71, 063815 (2005).
[CrossRef]

T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A 82, 063807 (2010).
[CrossRef]

O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
[CrossRef]

H.-K. Lo and S. Popescu, “Concentrating entanglement by local actions—beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[CrossRef]

Phys. Rev. E (2)

T. Carozzi, R. Karlsson, and J. Bergman, “Parameters characterizing electromagnetic wave polarization,” Phys. Rev. E 61, 2024–2028 (2000).
[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]

Phys. Rev. Lett. (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436–439 (1999).
[CrossRef]

Trans. Cambridge Philos. Soc. (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).

Other (6)

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000).

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge, 2007).

M. Gell-Mann and Y. Ne’eman, The Eightfold Way (Benjamin, 1964).

R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics (Springer, 1997).

A. W. Marshall, I. Olkin, and B. C. Arnold, “Schur-convex functions,” in Inequalities: Theory of Majorization and Its Applications, Springer Series in Statistics (Springer, 2011), pp. 79–154.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

Shaded region represents the space of allowable maximum and minimum eigenvalues, λ1 and λ3, of the three dimensional polarization matrix. The middle eigenvalue is then given by λ2=1λ1λ3. The shaded region satisfies the inequalities in Eq. (20) and non-negativity requirements. The labeled points A–J represent specific polarization states that are listed in Table 1 ahead. State A is the fully unpolarized state (1/3,1/3,1/3) and J the fully polarized state (1,0,0). To illustrate the majorization relations, we separate the area into quadrants centered around the state G given by (5/8,1/4,1/8).

Fig. 2.
Fig. 2.

Graphs of the state space with contours of constant degree of polarization for each of the measures under consideration. Each graph is essentially Fig. 1 with the contours of constant polarization superimposed. The measure used is in the top right of each diagram. The points A–J defined in Table 1 and plotted in previous figures are shown in the graphs. The measures Ps, Pbl, and Psc have the same contours since they are functions of one another.

Tables (1)

Tables Icon

Table 1. Five Independent Measures of Polarization Defined in Eqs. (4) and (7)–(10) Evaluated for 10 Polarization States of Interest

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

Φij(3)=EiEj*,i,j=1,2.3.
P(2)=14det(Φ)Tr[Φ]2,
λ1λ2λ30λ1+λ2+λ3=1.
Ps=λ12+λ22+λ32.
Pbl=12(3Ps1).
K=1κ14+κ24+κ34.
Psc=1Ps.
Pv=λ1log2λ1+λ2log2λ2+λ3log2λ3.
(λ1λ2)[100000000]rank1+(λ2λ3)[100010000]rank2+λ3[100010001]rank3.
P1=λ1λ2.
P2=λ1λ3.
Pb=127λ1λ2λ3.
maxψψ|Φ|ψ=λmax,
minψψ|Φ|ψ=λmin,
λ1μ1,λ3μ3.
i=1kyii=1kxi,k=1,,N,
λ1μ1,
λ1+λ2μ1+μ2,
λ1+λ2+λ3=μ1+μ2+μ3.
1λ113.
2λ1+λ3λ1+λ2+λ3=1λ1+2λ3,
12(1λ1)λ312λ1.
λ⃗Aλ⃗B,λ⃗Cλ⃗D,λ⃗E,λ⃗F,λ⃗Gλ⃗Hλ⃗J,λ⃗E,λ⃗Gλ⃗Iλ⃗J.
λ⃗=(λ1,λ2,λ3),μ⃗=(λ1+a,λ2a+b,λ3b),
Ps(μ⃗)Ps(λ⃗)=[(λ1+a)2+(λ2a+b)2+(λ3b)2](λ12+λ22+λ32)=2[aλ1+(ba)λ2bλ3]+a2+b2+(ba)2=2[a(2λ1+λ31)+b(1λ12λ3)]+a2+b2+(ba)20,
P2(μ⃗)P2(λ⃗)=[(λ1+a)(λ3b)](λ1λ3)=a+b0.
fv(λ1,λ3)λ1log2λ1+(1λ1λ3)log2(1λ1λ3)+λ3log2λ3.
fvλ1=log2λ1log2(1λ1λ3),fvλ3=log2λ3log2(1λ1λ3).
Ps=i=1Nλi2Tr[Φ2],
Pbl=NPs1N1,
Psc=1Ps,
Pv=i=1Nλilog2λi,
Pb=1NNλ1λ2λN.
Pk=λ1λk+1,k=1,,N1.

Metrics