Abstract

We introduce a consistent matrix formalism for the characterization of partial polarization and coherence of random, nonstationary electromagnetic beams in time and frequency domains. We derive the temporal and spectral degrees of polarization and the Stokes parameters in terms of the time-domain and frequency-domain polarization matrices. The connections between temporal polarization and spectral coherence on the one hand and spectral polarization and temporal coherence on the other hand are discussed. Additionally, we establish equivalence theorems for fields with different temporal coherence properties to have the same spectral polarization states and for fields with different spectral coherence properties to possess identical temporal polarization. The theory is illustrated by analyzing specific examples of time-domain and frequency-domain electromagnetic Gaussian Schell-model pulsed beams.

© 2012 Optical Society of America

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    [CrossRef]
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  21. T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
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  22. G. S. Agarwal, “Utility of 3×3 polarization matrices for partially polarized transverse electromagnetic fields,” J. Mod. Opt. 52, 651–654 (2005).
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  23. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248, 333–337 (2005).
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  24. A. Luis, “Degree of polarization for three-dimensional fields as a distance between correlation matrices,” Opt. Commun. 253, 10–14 (2005).
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    [CrossRef]
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    [CrossRef]
  28. J. J. Gil and I. San José, “3D polarimetric purity,” Opt. Commun. 283, 4430–4434 (2010).
    [CrossRef]
  29. C. J. R. Sheppard, “Partial polarization in three dimensions,” J. Opt. Soc. Am. A 28, 2655–2659 (2011).
    [CrossRef]
  30. In [20–29], several distinct approaches to introduce the three-dimensional (3D) degree of polarization have been presented. We note that, likewise, a number of mathematically and physically different definitions have been proposed in the literature for the degree of coherence of vectorial (electromagnetic, 2D or 3D) fields. Since this paper deals only with polarization, the degree of electromagnetic coherence plays no role in the analysis and is therefore not addressed. For a recent review on electromagnetic coherence see, e.g., [4].
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  32. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
  33. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [CrossRef]
  34. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [CrossRef]
  35. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
    [CrossRef]
  36. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
    [CrossRef]
  37. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
    [CrossRef]
  38. W. Huang, S. A. Ponomarenko, M. Cada, and G. P. Agrawal, “Polarization changes of partially coherent pulses propagating in optical fibers,” J. Opt. Soc. Am. A 24, 3063–3068 (2007).
    [CrossRef]
  39. C. Ding, L. Pan, and B. Lü, “Changes in the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. B 26, 1728–1735 (2009).
    [CrossRef]
  40. C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
    [CrossRef]
  41. M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18, 22503–22514(2010).
    [CrossRef]
  42. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
    [CrossRef]
  43. A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
    [CrossRef]
  44. 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]
  45. G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88, 097903 (2002).
    [CrossRef]
  46. Ch. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  47. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  48. E. Collett and E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
    [CrossRef]
  49. H. C. Kandpal and E. Wolf, “Partially coherent sources which generate far fields with the same spatial coherence properties,” Opt. Commun. 110, 255–258 (1994).
    [CrossRef]
  50. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  51. K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
  52. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [CrossRef]
  53. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
    [CrossRef]
  54. Ł. Michalik and A. W. Domański, “Effective state of polarization of a photon-beam,” Photon. Lett. Poland 3, 41–43 (2011).
    [CrossRef]
  55. M. Lahiri and E. Wolf, “Quantum analysis of polarization properties of optical beams,” Phys. Rev. A 82, 043805 (2010).
    [CrossRef]
  56. E. Brainis, “Quantum imaging with N-photon states in position space,” Opt. Express 19, 24228–24240 (2011).
    [CrossRef]
  57. A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
    [CrossRef]

2011

Ph. Refrégiér, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE 8171, 817102 (2011).
[CrossRef]

Ł. Michalik and A. W. Domański, “Effective state of polarization of a photon-beam,” Photon. Lett. Poland 3, 41–43 (2011).
[CrossRef]

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

E. Brainis, “Quantum imaging with N-photon states in position space,” Opt. Express 19, 24228–24240 (2011).
[CrossRef]

C. J. R. Sheppard, “Partial polarization in three dimensions,” J. Opt. Soc. Am. A 28, 2655–2659 (2011).
[CrossRef]

2010

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18, 22503–22514(2010).
[CrossRef]

M. Lahiri and E. Wolf, “Quantum analysis of polarization properties of optical beams,” Phys. Rev. A 82, 043805 (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]

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

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

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

2009

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

J. Tervo and J. Turunen, “Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1001 (2009).
[CrossRef]

E. Wolf, “Reply to Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1002 (2009).
[CrossRef]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Changes in the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. B 26, 1728–1735 (2009).
[CrossRef]

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[CrossRef]

M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
[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]

2008

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[CrossRef]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[CrossRef]

2007

2005

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

G. S. Agarwal, “Utility of 3×3 polarization matrices for partially polarized transverse electromagnetic fields,” J. Mod. Opt. 52, 651–654 (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]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

2004

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

2002

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88, 097903 (2002).
[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]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

2000

1994

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

H. C. Kandpal and E. Wolf, “Partially coherent sources which generate far fields with the same spatial coherence properties,” Opt. Commun. 110, 255–258 (1994).
[CrossRef]

1984

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).

1983

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

1979

1978

1963

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

1959

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

1956

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Agarwal, G. S.

G. S. Agarwal, “Utility of 3×3 polarization matrices for partially polarized transverse electromagnetic fields,” J. Mod. Opt. 52, 651–654 (2005).
[CrossRef]

Agrawal, G. P.

Andersen, U. L.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Barakat, R.

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

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).

Björk, G.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Brainis, E.

Brosseau, Ch.

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

Cada, M.

Cai, Y.

Collett, E.

Dennis, M. R.

Ding, C.

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Changes in the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. B 26, 1728–1735 (2009).
[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]

Domanski, A. W.

Ł. Michalik and A. W. Domański, “Effective state of polarization of a photon-beam,” Photon. Lett. Poland 3, 41–43 (2011).
[CrossRef]

Dong, R.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[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.

Ph. Refrégiér, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE 8171, 817102 (2011).
[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]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[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]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[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]

T. Setälä, F. Nunziata, and A. T. Friberg, “Partial polarization of optical beams: temporal and spectral descriptions,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 207–216.

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]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gori, F.

Heersink, J.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Huang, W.

James, D. F. V.

Kaivola, M.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[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]

Kandpal, H. C.

H. C. Kandpal and E. Wolf, “Partially coherent sources which generate far fields with the same spatial coherence properties,” Opt. Commun. 110, 255–258 (1994).
[CrossRef]

Klimov, A. B.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Korotkova, O.

M. Yao, Y. Cai, O. Korotkova, Q. Lin, and Z. Wang, “Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings,” Opt. Express 18, 22503–22514(2010).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Lahiri, M.

M. Lahiri and E. Wolf, “Quantum analysis of polarization properties of optical beams,” Phys. Rev. A 82, 043805 (2010).
[CrossRef]

M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
[CrossRef]

Lajunen, H.

Lassen, M.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Leuchs, G.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Lin, Q.

Lindfors, K.

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]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[CrossRef]

Lü, B.

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Changes in the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. B 26, 1728–1735 (2009).
[CrossRef]

Luis, A.

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

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

Madsen, L. S.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marquardt, Ch.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

Michalik, L.

Ł. Michalik and A. W. Domański, “Effective state of polarization of a photon-beam,” Photon. Lett. Poland 3, 41–43 (2011).
[CrossRef]

Nunziata, F.

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[CrossRef]

T. Setälä, F. Nunziata, and A. T. Friberg, “Partial polarization of optical beams: temporal and spectral descriptions,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 207–216.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Pan, L.

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

C. Ding, L. Pan, and B. Lü, “Changes in the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. B 26, 1728–1735 (2009).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

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]

Ponomarenko, S. A.

Priimägi, A.

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

Qian, X.-F.

Refrégiér, Ph.

Ph. Refrégiér, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE 8171, 817102 (2011).
[CrossRef]

Roy, R.

G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88, 097903 (2002).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Saastamoinen, K.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

San José, I.

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

Sánchez-Soto, L. L.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Setälä, T.

Ph. Refrégiér, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE 8171, 817102 (2011).
[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]

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[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]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[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]

T. Setälä, F. Nunziata, and A. T. Friberg, “Partial polarization of optical beams: temporal and spectral descriptions,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 207–216.

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]

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[CrossRef]

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

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

Söderholm, J.

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[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), reprinted in W. Swinell, ed., Polarized Light (Dowden, Hutchinson & Ross, 1975), pp. 124–141.

Tervo, J.

Turunen, J.

J. Tervo and J. Turunen, “Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1001 (2009).
[CrossRef]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
[CrossRef]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Vahimaa, P.

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

VanWiggeren, G. D.

G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88, 097903 (2002).
[CrossRef]

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]

Wang, Z.

Wolf, E.

M. Lahiri and E. Wolf, “Quantum analysis of polarization properties of optical beams,” Phys. Rev. A 82, 043805 (2010).
[CrossRef]

E. Wolf, “Reply to Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1002 (2009).
[CrossRef]

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008).
[CrossRef]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[CrossRef]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (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]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

H. C. Kandpal and E. Wolf, “Partially coherent sources which generate far fields with the same spatial coherence properties,” Opt. Commun. 110, 255–258 (1994).
[CrossRef]

E. Collett and E. Wolf, “New equivalence theorems for planar sources that generate the same distributions of radiant intensity,” J. Opt. Soc. Am. 69, 942–950 (1979).
[CrossRef]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyrowski, F.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Yao, M.

Zhao, D.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[CrossRef]

Zhao, Z.

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

Eur. Phys. J. Appl. Phys.

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

J. Mod. Opt.

G. S. Agarwal, “Utility of 3×3 polarization matrices for partially polarized transverse electromagnetic fields,” J. Mod. Opt. 52, 651–654 (2005).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Photonics

K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).

New J. Phys.

A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
[CrossRef]

Nuovo Cimento

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

Opt. Acta

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

Opt. Commun.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[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]

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

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

H. C. Kandpal and E. Wolf, “Partially coherent sources which generate far fields with the same spatial coherence properties,” Opt. Commun. 110, 255–258 (1994).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

C. Ding, Z. Zhao, L. Pan, and B. Lü, “Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” Opt. Commun. 283, 4470–4477 (2010).
[CrossRef]

Opt. Express

Opt. Lett

T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, and A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett 29, 2587–2589 (2004).
[CrossRef]

Opt. Lett.

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

T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space–time and space–frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
[CrossRef]

M. Lahiri, “Polarization properties of stochastic light beams in the space–time and space–frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
[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]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400–3401 (2007).
[CrossRef]

E. Wolf, “Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?” Opt. Lett. 33, 642–644 (2008).
[CrossRef]

J. Tervo and J. Turunen, “Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1001 (2009).
[CrossRef]

E. Wolf, “Reply to Comment on ‘Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?’” Opt. Lett. 34, 1002 (2009).
[CrossRef]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
[CrossRef]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

Photon. Lett. Poland

Ł. Michalik and A. W. Domański, “Effective state of polarization of a photon-beam,” Photon. Lett. Poland 3, 41–43 (2011).
[CrossRef]

Phys. Rev.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Phys. Rev. A

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

M. Lahiri and E. Wolf, “Quantum analysis of polarization properties of optical beams,” Phys. Rev. A 82, 043805 (2010).
[CrossRef]

K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A 80, 053804 (2009).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
[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]

Phys. Rev. E

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.

G. D. VanWiggeren and R. Roy, “Communication with dynamically fluctuating states of light polarization,” Phys. Rev. Lett. 88, 097903 (2002).
[CrossRef]

A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
[CrossRef]

Proc. Indian Acad. Sci. A

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Proc. R. Soc. A

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).

Proc. SPIE

Ph. Refrégiér, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE 8171, 817102 (2011).
[CrossRef]

Other

In [20–29], several distinct approaches to introduce the three-dimensional (3D) degree of polarization have been presented. We note that, likewise, a number of mathematically and physically different definitions have been proposed in the literature for the degree of coherence of vectorial (electromagnetic, 2D or 3D) fields. Since this paper deals only with polarization, the degree of electromagnetic coherence plays no role in the analysis and is therefore not addressed. For a recent review on electromagnetic coherence see, e.g., [4].

G. G. Stokes, “On the composition and resolution of streams of polarized light from different sources,” Trans. Cambridge Philos. Soc.9, 399–416 (1852), reprinted in W. Swinell, ed., Polarized Light (Dowden, Hutchinson & Ross, 1975), pp. 124–141.

T. Setälä, F. Nunziata, and A. T. Friberg, “Partial polarization of optical beams: temporal and spectral descriptions,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 207–216.

J. W. Goodman, Statistical Optics (Wiley, 1985).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).

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

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Figures (4)

Fig. 1.
Fig. 1.

Geometric illustration of a partially polarized state (A) and the related fully polarized state (B) using the Poincaré sphere. Points on the surface represent fully polarized states, the origin O corresponds to an unpolarized state, and other points represent partially polarized states. The ‘north’ and ‘south’ poles refer to right-hand and left-hand circularly polarized states, respectively, and the points on the intersection of the surface of the sphere with the equatorial plane (indicated with the dashed circle) represent linearly polarized states. The polarization state of the fully polarized part of the field indicated by point A is the same as the polarization state shown by point B. As shown in the figure, the Poincaré vector of the polarized part of a partially polarized state can be found by continuing the Poincaré vector of the partially polarized state (solid red line segment) to the surface of the sphere (dashed red line segment).

Fig. 2.
Fig. 2.

(a) Spectra and (b)–(d) spectral coherence functions of four different pulsed beams with the same temporal polarization matrices. The curves are for Ωc/Ωc values of 0.75 (blue dotted curve), 0.9 (green dash-dotted curve), 1.0 (red dashed curve), and 1.15 (turquoise solid curve). (e) Spectral profile of the integrated spectral coherence matrix elements W¯xx(Ω) (turquoise solid curve), W¯yy(Ω) (red dashed curve), and W¯xy(Ω) (dot-dashed green curve). (f) Temporal degree of polarization Pt(t) (solid curve) and the spectral degree of polarization shown for comparison (dashed line).

Fig. 3.
Fig. 3.

Degree of polarization for different values of the intercomponent delay τd/T0 and coherence time Tc/T0. In (a) τd/T0=0.25 and in (b) τd/T0=1. The thin blue (solid) and green (dashed) curves show the normalized amplitudes of the x and y components, respectively. The thick turquoise (solid, top), red (dotted), green (dashed), and blue (solid, bottom) curves indicate the degree of polarization Pt(t) calculated for coherence times Tc/T0 of 2, 1, 0.5, and 0.33, respectively. In (b) the curve for Tc/T0=0.5 overlaps with that of Tc/T0=0.33 and is not distinguished.

Fig. 4.
Fig. 4.

Variation of the spectral polarization state with the product ωτd. The polarization state changes from linear polarization (45° angle to the x axis) to elliptical polarization (major axis at 45°, right-handed rotation), right-hand circular polarization, right-hand elliptical polarization (major axis at 135°), orthogonal linear polarization at 135°, left-hand elliptical polarization (major axis at 135°), left-hand circular polarization, left-hand elliptical polarization (major axis at 45°), and finally back to linear polarization at 45° angle to the x axis.

Equations (70)

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

Γij(r1,r2,t1,t2)=Ei*(r1,t1)Ej(r2,t2),
Γ(r1,r2,t1,t2)=Γ(r2,r1,t2,t1),
m,n=12a(rm,tm)Γ(rm,rn,tm,tn)a(rn,tn)0,
J(r,t)=Γ(r,r,t,t).
J(unpol)(r,t)=A(r,t)[1001],
J(pol)(r,t)=[B(r,t)D(r,t)D*(r,t)C(r,t)],
J(r,t)=J(unpol)(r,t)+J(pol)(r,t).
Pt(r,t)=trJ(pol)(r,t)trJ(r,t)=14detJ(r,t)tr2J(r,t)=2trJ2(r,t)tr2J(r,t)1.
St0(r,t)=Jxx(r,t)+Jxx(r,t)=Ix(r,t)+Iy(r,t),
St1(r,t)=Jxx(r,t)Jxx(r,t)=Ix(r,t)Iy(r,t),
St2(r,t)=Jxy(r,t)+Jyx(r,t)=Iα(r,t)Iβ(r,t),
St3(r,t)=i[Jyx(r,t)Jxy(r,t)]=Ir(r,t)Il(r,t).
i=13Sti2(r,t)=Pt2(r,t)St02(r,t),
σ0=[1001],σ1=[1001],σ2=[0110],σ3=[0ii0]
J(r,t)=12i=03Sti(r,t)σi.
sti(r,t)=Sti(r,t)/St0(r,t).
i=13sti2(r,t)=Pt2(r,t).
Ei(r,t)=0E˜i(r,ω)eiωtdω,
E˜i(r,ω)=12πEi(r,t)eiωtdt.
Wij(r1,r2,ω1,ω2)=E˜i*(r1,ω1)E˜j(r2,ω2),
W(r1,r2,ω1,ω2)=W(r2,r1,ω2,ω1),
m,n=12a(rm,ωm)W(rm,rn,ωm,ωn)a(rn,ωn)0,
W(r1,r2,ω1,ω2)=14π2Γ(r1,r2,t1,t2)ei(ω1t1+ω2t2)dt1dt2,
Γ(r1,r2,t1,t2)=00W(r1,r2,ω1,ω2)ei(ω1t1+ω2t2)dω1dω2,
0Si(r,ω)dω=12πIi(r,t)dt,
Φ(r,ω)=Φ(unpol)(r,ω)+Φ(pol)(r,ω).
Ps(r,ω)=trΦ(pol)(r,ω)trΦ(r,ω)=14detΦ(r,ω)tr2Φ(r,ω)=2trΦ2(r,ω)tr2Φ(r,ω)1.
Ss0(r,ω)=Φxx(r,ω)+Φyy(r,ω)=Sx(r,ω)+Sy(r,ω),
Ss1(r,ω)=Φxx(r,ω)Φyy(r,ω)=Sx(r,ω)Sy(r,ω),
Ss2(r,ω)=Φyx(r,ω)+Φxy(r,ω)=Sα(r,ω)Sβ(r,ω),
Ss3(r,ω)=i[Φyx(r,ω)Φxy(r,ω)]=Sr(r,ω)Sl(r,ω).
ssi(r,ω)=Ssi(r,ω)/Ss0(r,ω).
i=13ssi2(r,ω)=Ps2(r,ω),
J(r,t)=00W(r,r,ω1,ω2)ei(ω2ω1)tdω1dω2,
Φ(r,ω)=14π2Γ(r,r,t1,t2)eiω(t2t1)dt1dt2.
Φ(r,ω)=12πΓ¯(r,r,τ)eiωτdτ,
Γ¯(r,r,τ)=12πΓ(r,r,t¯τ/2,t¯+τ/2)dt¯
J(r,t)=W¯(r,r,Ω)eiΩtdΩ,
W¯(r,r,Ω)=14π2|Ω|/2W(r,r,ω¯Ω/2,ω¯+Ω/2)dω¯
max(Ωcxx,Ωcyy)Ωcxymin(Ωcxx|Bxy|,Ωcyy|Bxy|).
(rc24+1)1/2<c(1b21rxy2)1/2.
Jii(t)=2πAi2Ω02(1+4rc2)1/2exp(t2/T02),i=x,y,
Jxy(t)=2πAxAyBxyΩ02(1+4rc2rxy2)1/2exp(t2/T0xy2),
Ps2=14AxAy(Ax+Ay)2(1b2),
Pt2(t)=14AxAy(Ax+Ay)2(1|γxy(t)|2),
γxy(t)=Bxy(1+4rc21+4rc2rxy2)1/2exp[(T0xy2T02)t2].
I(t)=A02exp(t2/T02).
γ(t1,t2)=γ(t2t1)=exp[(t2t1)2/Tc2]exp[iω0(t2t1)],
Γ(t1,t2)=[E*(t1)E(t2)E*(t1)E(t2τd)E*(t1τd)E(t2)E*(t1τd)E(t2τd)]=[Γ(t1,t2)Γ(t1,t2τd)Γ(t1τd,t2)Γ(t1τd,t2τd)],
J(t)=[I(t)[I(t)I(tτd)]1/2γ(τd)[I(t)I(tτd)]1/2γ(τd)I(tτd)].
Pt(t)=[tanh2(τd22tτd2T02)+sech2(τd22tτd2T02)exp(2τd2/Tc2)]1/2.
St0(t)=A02exp(t2/T02){1+exp[(τd22tτd)/T02]},
St1(t)=A02exp(t2/T02){1exp[(τd22tτd)/T02]},
St2(t)=2A02exp{[t2+(tτd)2]/2T02}cos(ω0τd)exp(τd2/Tc2),
St3(t)=2A02exp{[t2+(tτd)2]/2T02}sin(ω0τd)exp(τd2/Tc2).
st1(t)=tanh(τd22tτd2T02),
st2(t)=cos(ω0τd)exp(τd2/Tc2)cosh[(τd22tτd)/2T02],
st3(t)=sin(ω0τd)exp(τd2/Tc2)cosh[(τd22tτd)/2T02].
st1(pol)(t)=sinh[(tτdτd2/2)/T02]{sinh2[(tτdτd2/2)/T02]+exp(2τd2/Tc2)}1/2,
st2(pol)(t)=cos(ω0τd)exp(τd2/Tc2){sinh2[(tτdτd2/2)/T02]+exp(2τd2/Tc2)}1/2,
st3(pol)(t)=sin(ω0τd)exp(τd2/Tc2){sinh2[(tτdτd2/2)/T02]+exp(2τd2/Tc2)}1/2.
W(ω1,ω2)=W(ω1,ω2)[1eiω2τdeiω1τdei(ω2ω1)τd].
W(ω1,ω2)=[S(ω1)S(ω2)]1/2exp[(ω2ω1)2Ωc2],
μ(ω1,ω2)=exp[(ω2ω1)2/Ωc2].
S(ω)=A02T04πδexp[14δ2(ωω0)2],
Φ(ω)=W(ω,ω)=S(ω)[1eiωτdeiωτd1],
ss1(ω)=0,
ss2(ω)=cos(ωτd),
ss3(ω)=sin(ωτd).
Γ¯(τ)=T0A022πeiω0τ[eδ2τ2eδ2(ττd)2eiω0τdeδ2(τ+τd)2eiω0τdeδ2τ2].

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