Abstract

A measure of the degree of polarization for the three-dimensional polarization matrix (coherence matrix) of an electromagnetic field is proposed, based on Rayleigh scattering. The degree of polarization at a point is defined as an average, over all scattering directions, of an imagined dipole scattering of the three-dimensional state of polarization. This gives a well-defined purity measure, which, unlike other proposed measures of the three-dimensional degree of polarization, is not a unitary invariant of the matrix. This is demonstrated and discussed for several examples, including a partially polarized transverse beam.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  2. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  3. M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
    [Crossref]
  4. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, "Degree of polarization of statistically stationary electromagnetic fields," Opt. Commun. 248, 333-337 (2005).
    [Crossref]
  5. A. Luis, "Degree of polarization for three-dimensional fields as a distance between correlation matrices," Opt. Commun. 253, 10-14 (2005).
    [Crossref]
  6. 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]
  7. J. Ellis and A. Dogariu, "On the degree of polarization of random electromagnetic fields," Opt. Commun. 253, 257-265 (2005).
    [Crossref]
  8. E. Wolf, "Coherence properties of partially polarized electromagnetic radiation," Nuovo Cimento 13, 1165-1181 (1959).
    [Crossref]
  9. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).
  10. J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).
  11. B. C. van Fraassen, Quantum Mechanics: An Empiricist View (Oxford U. Press, 1991).
  12. R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, 1982).
  13. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1998).
  14. S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).
  15. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, 1955).
  16. J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
    [Crossref]
  17. M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
    [Crossref]
  18. R. Y. Chiao and Y.-S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
    [Crossref] [PubMed]
  19. M. V. Berry, "Interpreting the anholonomy of coiled light," Nature 326, 277-278 (1987).
    [Crossref]
  20. J. H. Hannay, "Polarization of sky light from a canopy atmosphere," New J. Phys. 6, 197 (2004).
    [Crossref]
  21. M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
    [Crossref]
  22. M. R. Dennis, "Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere," J. Phys. A 37, 9487-9500 (2004).
    [Crossref]
  23. A. Gleason, "Measures on the closed subspaces of a Hilbert space," J. Math. Mech. 6, 885-893 (1957).
  24. M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space," J. Opt. Soc. Am. A 11, 2233-2243 (2004).
    [Crossref]

2005 (4)

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

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

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]

J. Ellis and A. Dogariu, "On the degree of polarization of random electromagnetic fields," Opt. Commun. 253, 257-265 (2005).
[Crossref]

2004 (5)

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[Crossref]

J. H. Hannay, "Polarization of sky light from a canopy atmosphere," New J. Phys. 6, 197 (2004).
[Crossref]

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
[Crossref]

M. R. Dennis, "Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere," J. Phys. A 37, 9487-9500 (2004).
[Crossref]

M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space," J. Opt. Soc. Am. A 11, 2233-2243 (2004).
[Crossref]

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

2001 (1)

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[Crossref]

1998 (2)

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

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1998).

1995 (1)

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

1994 (1)

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

1991 (1)

B. C. van Fraassen, Quantum Mechanics: An Empiricist View (Oxford U. Press, 1991).

1987 (2)

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[Crossref]

M. V. Berry, "Interpreting the anholonomy of coiled light," Nature 326, 277-278 (1987).
[Crossref]

1986 (1)

R. Y. Chiao and Y.-S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[Crossref] [PubMed]

1982 (1)

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, 1982).

1959 (1)

E. Wolf, "Coherence properties of partially polarized electromagnetic radiation," Nuovo Cimento 13, 1165-1181 (1959).
[Crossref]

1957 (1)

A. Gleason, "Measures on the closed subspaces of a Hilbert space," J. Math. Mech. 6, 885-893 (1957).

1955 (1)

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, 1955).

1950 (1)

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Alonso, M. A.

M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space," J. Opt. Soc. Am. A 11, 2233-2243 (2004).
[Crossref]

Berry, M. V.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
[Crossref]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[Crossref]

M. V. Berry, "Interpreting the anholonomy of coiled light," Nature 326, 277-278 (1987).
[Crossref]

Brosseau, C.

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

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

Chiao, R. Y.

R. Y. Chiao and Y.-S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[Crossref] [PubMed]

Dennis, M. R.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
[Crossref]

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[Crossref]

M. R. Dennis, "Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere," J. Phys. A 37, 9487-9500 (2004).
[Crossref]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[Crossref]

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]

J. Ellis and A. Dogariu, "On the degree of polarization of random electromagnetic fields," Opt. Commun. 253, 257-265 (2005).
[Crossref]

Ellis, J.

J. Ellis and A. Dogariu, "On the degree of polarization of random electromagnetic fields," Opt. Commun. 253, 257-265 (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]

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 in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

Gleason, A.

A. Gleason, "Measures on the closed subspaces of a Hilbert space," J. Math. Mech. 6, 885-893 (1957).

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[Crossref]

Hannay, J. H.

J. H. Hannay, "Polarization of sky light from a canopy atmosphere," New J. Phys. 6, 197 (2004).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1998).

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 in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

Lee, R. L.

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
[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]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, 1982).

Nye, J. F.

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[Crossref]

Peres, A.

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

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]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

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 in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

van Fraassen, B. C.

B. C. van Fraassen, Quantum Mechanics: An Empiricist View (Oxford U. Press, 1991).

von Neumann, J.

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, 1955).

Wolf, E.

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, "Coherence properties of partially polarized electromagnetic radiation," Nuovo Cimento 13, 1165-1181 (1959).
[Crossref]

Wu, Y.-S.

R. Y. Chiao and Y.-S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[Crossref] [PubMed]

J. Math. Mech. (1)

A. Gleason, "Measures on the closed subspaces of a Hilbert space," J. Math. Mech. 6, 885-893 (1957).

J. Opt. A, Pure Appl. Opt. (1)

M. R. Dennis, "Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization," J. Opt. A, Pure Appl. Opt. 6, S26-S31 (2004).
[Crossref]

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

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]

M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free space," J. Opt. Soc. Am. A 11, 2233-2243 (2004).
[Crossref]

J. Phys. A (1)

M. R. Dennis, "Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere," J. Phys. A 37, 9487-9500 (2004).
[Crossref]

Nature (1)

M. V. Berry, "Interpreting the anholonomy of coiled light," Nature 326, 277-278 (1987).
[Crossref]

New J. Phys. (2)

J. H. Hannay, "Polarization of sky light from a canopy atmosphere," New J. Phys. 6, 197 (2004).
[Crossref]

M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., "Polarization singularities in the clear sky," New J. Phys. 6, 162 (2004).
[Crossref]

Nuovo Cimento (1)

E. Wolf, "Coherence properties of partially polarized electromagnetic radiation," Nuovo Cimento 13, 1165-1181 (1959).
[Crossref]

Opt. Commun. (3)

J. Ellis and A. Dogariu, "On the degree of polarization of random electromagnetic fields," Opt. Commun. 253, 257-265 (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]

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

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, "Degree of polarization in near fields of thermal sources: effects of surface waves," Phys. Rev. E 66, 016615 (2002).
[Crossref]

Phys. Rev. Lett. (1)

R. Y. Chiao and Y.-S. Wu, "Manifestations of Berry's topological phase for the photon," Phys. Rev. Lett. 57, 933-936 (1986).
[Crossref] [PubMed]

Proc. R. Soc. London, Ser. A (2)

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[Crossref]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[Crossref]

Other (8)

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

A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, 1995).

J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison-Wesley, 1994).

B. C. van Fraassen, Quantum Mechanics: An Empiricist View (Oxford U. Press, 1991).

R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer, 1982).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1998).

S. Chandrasekhar, Radiative Transfer (Clarendon, 1950).

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, 1955).

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 (3)

Fig. 1
Fig. 1

Geometric representation of polarization matrix ρ 3 as ellipsoid and vector. The ellipses and vectors for ρ 2 ( θ , ϕ ) in three orthogonal projection directions are also shown.

Fig. 2
Fig. 2

Contours of constant P Ray (thick curves) and P quad (thin green curves) in the S 1 S 3 plane, for 3 × 3 paraxial partial polarization. The contour lines of P quad depend only on the radius S 1 2 + S 3 2 , whereas the P Ray contours have weak angular dependence.

Fig. 3
Fig. 3

Illustrating the various 3-D polarization degree measures P 3 for paraxial partial polarization, parametrized by the paraxial measure P 2 . (a) P 3 plot; (b) plot of P 3 ( P 2 + 1 ) 2 (i.e., linear part subtracted). P 3 = P Ray with S 3 ( S 1 ) = 0 (thick solid (dashed) curve); P lin (thick gray solid line); P quad (thin solid curve); P vN (thin dashed curve).

Equations (12)

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

ρ 2 = E * E E 2 , ρ 2 , i j = E i * E j E 2 ,
P 2 2 = 2 tr ρ 2 2 ( tr ρ 2 ) 2 ( tr ρ 2 ) 2 .
ρ 2 ( θ , ϕ ) = 3 σ 8 π p ( θ , ϕ ) ρ 3 p ( θ , ϕ ) ,
I ( θ , ϕ ) = tr ρ 2 ( θ , ϕ )
P 2 2 ( θ , ϕ ) = 2 tr ρ 2 2 ( θ , ϕ ) [ tr ρ 2 ( θ , ϕ ) ] 2 [ tr ρ 2 ( θ , ϕ ) ] 2 ,
P Ray = 0 2 π d ϕ 0 π d θ sin θ P 2 ( θ , ϕ ) I ( θ , ϕ ) 0 2 π d ϕ 0 π d θ sin θ I ( θ , ϕ ) ,
ρ 3 = [ M 1 i N 3 i N 2 i N 3 M 2 i N 1 i N 2 i N 1 M 3 ] ,
P lin = λ 1 λ 3 ,
P quad 2 = [ 3 tr ρ 3 2 ( tr ρ 3 ) 2 ] 2 .
P vN 2 = 1 + 1 log 3 j = 1 3 λ j log λ j .
ρ 3 , trans = 1 2 [ 1 + S 1 i S 3 0 i S 3 1 S 1 0 0 0 0 ] .
P Ray , ex 1 = 0.5932 a , P Ray , ex 2 = 0.5 a ,

Metrics