Abstract

The coherence theory of random, vector-valued optical fields has been of great research interest in recent years. In this work we formulate the foundations of electromagnetic coherence theory both in the space–time and space–frequency domains, with particular emphasis on various types of optical interferometry. Analyzing statistically stationary, two-component (paraxial) electric fields in the classical and quantum-optical contexts we show fundamental connections between the conventional (polarization) Stokes parameters and the associated two-point (coherence) Stokes parameters. Measurement of the coherence and polarization properties of random vector beams by nanoparticle scattering and two-photon absorption is also addressed.

© 2016 Optical Society of America

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References

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    [Crossref]

2016 (3)

2015 (4)

2014 (5)

T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

O. Gamel and D. F. V. James, “Majorization and measures of classical polarization in three dimensions,” J. Opt. Soc. Am. A 31, 1620–1626 (2014).
[Crossref]

C. J. R. Sheppard, “Jones and Stokes parameters for polarization in three dimensions,” Phys. Rev. A 90, 023809 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

2013 (2)

2012 (4)

2011 (2)

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of the Stokes parameters,” Appl. Phys. B 105, 305–308 (2011).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref]

2010 (2)

A. Al-Qasimi, M. Lahiri, D. Kuebel, D. F. V. James, and E. Wolf, “The influence of the degree of cross-polarization on the Hanbury Brown–Twiss effect,” Opt. Express 18, 17124–17129 (2010).
[Crossref]

M. Lahiri and E. Wolf, “Relationship between complete coherence in the space-time and in the space-frequency domains,” Phys. Rev. Lett. 105, 063901 (2010).
[Crossref]

2009 (7)

2008 (4)

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]

F. Gori, “Partially correlated sources with complete polarization,” Opt. Lett. 33, 2818–2820 (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]

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A 10, 055001 (2008).
[Crossref]

2007 (7)

2006 (3)

2005 (5)

J. Ellis, D. Aristide, 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]

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref]

T. Jouttenus, T. Setälä, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[Crossref]

2004 (8)

2003 (4)

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[Crossref]

J. Grondalski and D. F. V. James, “Is there a fundamental limitation on the measurement of spatial coherence for highly incoherent fields?” Opt. Lett. 28, 1630–1632 (2003).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (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]

1996 (1)

B.-G. Englert, “Fringe visibility and which-way information,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

1995 (1)

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[Crossref]

1981 (1)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

1963 (3)

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

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[Crossref]

B. Karczewski, “Coherence theory of the electromagnetic field,” Nuovo Cimento 30, 906–915 (1963).
[Crossref]

1956 (3)

R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

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

1955 (1)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London Ser. A 230, 246–265 (1955).
[Crossref]

Al-Qasimi, A.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).

Aristide, D.

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

Auñón, J. M.

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[Crossref]

Blomstedt, K.

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: a second look,” J. Opt. Soc. Am. A 32, 718–732 (2015).
[Crossref]

T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
[Crossref]

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

K. Blomstedt, “Electromagnetic coherence theory, universality results, and effective degree of coherence,” Ph.D. dissertation (Aalto University, 2013).

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference” (submitted).

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5, 267–270 (2009).
[Crossref]

Borghi, R.

Brosseau, C.

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

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

Dennis, M. R.

Dogariu, A.

Ellis, J.

J. Ellis, D. Aristide, 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, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref]

Englert, B.-G.

B.-G. Englert, “Fringe visibility and which-way information,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

Fabre, C.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5, 267–270 (2009).
[Crossref]

Friberg, A. T.

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, J. Lehtolahti, A. T. Friberg, and T. Setälä, “Detection of partial polarization of light beams with dipolar nanocubes,” Opt. Express 24, 1472–1479 (2016).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Connection of electromagnetic degrees of coherence in space-time and space-frequency domains,” Opt. Lett. 41, 1821–1824 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
[Crossref]

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: a second look,” J. Opt. Soc. Am. A 32, 718–732 (2015).
[Crossref]

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
[Crossref]

J. Tervo, T. Setälä, J. Turunen, and A. T. Friberg, “Van Cittert-Zernike theorem with Stokes parameters,” Opt. Lett. 38, 2301–2303 (2013).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
[Crossref]

P. Refrégiér, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett. 37, 3750–3752 (2012).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref]

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of the Stokes parameters,” Appl. Phys. B 105, 305–308 (2011).
[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. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (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]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (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. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
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T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization constrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
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T. Jouttenus, T. Setälä, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–340 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
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T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
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A. T. Friberg, J. Tervo, and T. Setälä, “Young’s interference experiment reloaded,” in 6th International Workshop on Information Optics (WIO), J. A. Benediktsson, B. Javidi, and K. Gudmundsson, eds. (2007), pp. 131–137.

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference” (submitted).

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.

Gamel, O.

Gbur, G.

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T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of the Stokes parameters,” Appl. Phys. B 105, 305–308 (2011).
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T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
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T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
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T. Jouttenus, T. Setälä, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
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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).
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T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Polarization time and length for random optical beams,” Phys. Rev. A 78, 033817 (2008).
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T. Jouttenus, T. Setälä, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
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T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
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A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference” (submitted).

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L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

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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.

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Réfrégier, P.

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F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5, 267–270 (2009).
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Roueff, A.

Roussey, M.

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Saastamoinen, T.

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[Crossref]

Santarsiero, M.

Setälä, T.

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, J. Lehtolahti, A. T. Friberg, and T. Setälä, “Detection of partial polarization of light beams with dipolar nanocubes,” Opt. Express 24, 1472–1479 (2016).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Connection of electromagnetic degrees of coherence in space-time and space-frequency domains,” Opt. Lett. 41, 1821–1824 (2016).
[Crossref]

A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
[Crossref]

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: a second look,” J. Opt. Soc. Am. A 32, 718–732 (2015).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
[Crossref]

J. Tervo, T. Setälä, J. Turunen, and A. T. Friberg, “Van Cittert-Zernike theorem with Stokes parameters,” Opt. Lett. 38, 2301–2303 (2013).
[Crossref]

P. Refrégiér, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett. 37, 3750–3752 (2012).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (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]

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. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization constrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Jouttenus, T. Setälä, M. Kaivola, and A. T. Friberg, “Connection between electric and magnetic coherence in free electromagnetic fields,” Phys. Rev. E 72, 046611 (2005).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–340 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[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.

A. T. Friberg, J. Tervo, and T. Setälä, “Young’s interference experiment reloaded,” in 6th International Workshop on Information Optics (WIO), J. A. Benediktsson, B. Javidi, and K. Gudmundsson, eds. (2007), pp. 131–137.

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference” (submitted).

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C. J. R. Sheppard, “Geometric representation for partial polarization in three dimensions,” Opt. Lett. 37, 2772–2774 (2012).
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A. Shevchenko, M. Roussey, A. T. Friberg, and T. Setälä, “Ultrashort coherence times in partially polarized stationary optical beams measured by two-photon absorption,” Opt. Express 23, 31274–31285 (2015).
[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]

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

Shirai, T.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A 10, 055001 (2008).
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T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
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Tervo, J.

J. Tervo, T. Setälä, J. Turunen, and A. T. Friberg, “Van Cittert-Zernike theorem with Stokes parameters,” Opt. Lett. 38, 2301–2303 (2013).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
[Crossref]

T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
[Crossref]

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of the Stokes parameters,” Appl. Phys. B 105, 305–308 (2011).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

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

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization constrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–340 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[Crossref]

T. Saastamoinen and J. Tervo, “Geometric approach to the degree of polarization for arbitrary fields,” J. Mod. Opt. 51, 2039–2045 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref]

A. T. Friberg, J. Tervo, and T. Setälä, “Young’s interference experiment reloaded,” in 6th International Workshop on Information Optics (WIO), J. A. Benediktsson, B. Javidi, and K. Gudmundsson, eds. (2007), pp. 131–137.

Turunen, J.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
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R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
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Vahimaa, P.

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[Crossref]

Visser, T.

G. Gbur and T. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

Voipio, T.

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
[Crossref]

T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
[Crossref]

T. Voipio, “Partial polarization and coherence in stationary and nonstationary electromagnetic fields,” Ph.D. dissertation (University of Eastern Finland, 2015).

Volkov, S. N.

S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A 10, 055001 (2008).
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M. Lahiri and E. Wolf, “Statistical similarity and cross-spectral purity of stationary stochastic fields,” Opt. Lett. 37, 963–967 (2012).
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M. Lahiri and E. Wolf, “Relationship between complete coherence in the space-time and in the space-frequency domains,” Phys. Rev. Lett. 105, 063901 (2010).
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S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A 10, 055001 (2008).
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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. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[Crossref]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref]

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

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[Crossref]

M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414–2417 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett. 20, 623–625 (1995).
[Crossref]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London Ser. A 230, 246–265 (1955).
[Crossref]

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

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

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), Vol. 50, pp. 251–274.

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L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Temporal degree of coherence and Stokes-parameter modulations in Michelson’s interferometer,” Appl. Phys. B 122, 32 (2016).
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Eur. Phys. J. Appl. Phys. (1)

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T. Voipio, K. Blomstedt, T. Setälä, and A. T. Friberg, “Coherent modes of random electric and magnetic fields in a spherical volume,” J. Opt. 16, 125705 (2014).
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J. Opt. A (3)

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
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K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
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S. N. Volkov, D. F. V. James, T. Shirai, and E. Wolf, “Intensity fluctuations and the degree of cross-polarization in stochastic electromagnetic beams,” J. Opt. A 10, 055001 (2008).
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J. Opt. Soc. Am. A (10)

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
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K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: a second look,” J. Opt. Soc. Am. A 32, 718–732 (2015).
[Crossref]

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
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A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
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J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

M. Mujat, A. Dogariu, and E. Wolf, “A law of interference of electromagnetic beams of any state of coherence and polarization and the Fresnel-Arago interference laws,” J. Opt. Soc. Am. A 21, 2414–2417 (2004).
[Crossref]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382–2385 (2004).
[Crossref]

P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. Am. A 23, 671–678 (2006).
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O. Gamel and D. F. V. James, “Majorization and measures of classical polarization in three dimensions,” J. Opt. Soc. Am. A 31, 1620–1626 (2014).
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M. R. Dennis, “A three-dimensional degree of polarization based on Rayleigh scattering,” J. Opt. Soc. Am. A 24, 2065–2069 (2007).
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Nat. Phys. (1)

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R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
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New J. Phys. (2)

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A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. 11, 073004 (2009).
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E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 36, 247–249 (1981).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[Crossref]

H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[Crossref]

J. Ellis, D. Aristide, 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]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272, 289–292 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett. (24)

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref]

R. Martínez-Herrero and P. M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
[Crossref]

M. Lahiri and E. Wolf, “Statistical similarity and cross-spectral purity of stationary stochastic fields,” Opt. Lett. 37, 963–967 (2012).
[Crossref]

T. Hassinen, J. Tervo, and A. T. Friberg, “Cross-spectral purity of electromagnetic fields,” Opt. Lett. 34, 3866–3868 (2009).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–340 (2004).
[Crossref]

A. T. Friberg and E. Wolf, “Relationships between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett. 20, 623–625 (1995).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Connection of electromagnetic degrees of coherence in space-time and space-frequency domains,” Opt. Lett. 41, 1821–1824 (2016).
[Crossref]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
[Crossref]

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

J. Tervo, T. Setälä, J. Turunen, and A. T. Friberg, “Van Cittert-Zernike theorem with Stokes parameters,” Opt. Lett. 38, 2301–2303 (2013).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
[Crossref]

C. J. R. Sheppard, “Geometric representation for partial polarization in three dimensions,” Opt. Lett. 37, 2772–2774 (2012).
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J. M. Auñón and M. Nieto-Vesperinas, “On two definitions of the three-dimensional degree of polarization in the near field of statistically homogeneous partially coherent sources,” Opt. Lett. 38, 58–60 (2013).
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P. Refrégiér, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett. 37, 3750–3752 (2012).
[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]

F. Gori, “Partially correlated sources with complete polarization,” Opt. Lett. 33, 2818–2820 (2008).
[Crossref]

F. Gori, J. Tervo, and J. Turunen, “Correlation matrices of completely unpolarized beams,” Opt. Lett. 34, 1447–1449 (2009).
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T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization constrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
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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).
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M. Lahiri, “Polarization properties of stochastic light beams in the space-time and space-frequency domains,” Opt. Lett. 34, 2936–2938 (2009).
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L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
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J. Grondalski and D. F. V. James, “Is there a fundamental limitation on the measurement of spatial coherence for highly incoherent fields?” Opt. Lett. 28, 1630–1632 (2003).
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Phys. Lett. (1)

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Phys. Lett. A (1)

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M. Lahiri and E. Wolf, “Relationship between complete coherence in the space-time and in the space-frequency domains,” Phys. Rev. Lett. 105, 063901 (2010).
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Proc. R. Soc. London Ser. A (1)

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[Crossref]

Other (19)

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

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

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

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

K. Blomstedt, “Electromagnetic coherence theory, universality results, and effective degree of coherence,” Ph.D. dissertation (Aalto University, 2013).

T. Hassinen, “Studies on coherence and purity of electromagnetic fields,” Ph.D. dissertation (University of Eastern Finland, 2013).

T. Voipio, “Partial polarization and coherence in stationary and nonstationary electromagnetic fields,” Ph.D. dissertation (University of Eastern Finland, 2015).

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

G. Gbur and T. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.

P. Hariharan, “The geometric phase,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2005), Vol. 48, pp. 149–201.

A. T. Friberg, J. Tervo, and T. Setälä, “Young’s interference experiment reloaded,” in 6th International Workshop on Information Optics (WIO), J. A. Benediktsson, B. Javidi, and K. Gudmundsson, eds. (2007), pp. 131–137.

L.-P. Leppänen, “Polarization and electromagnetic coherence of light fields probed with nanoscatterers,” Ph.D. dissertation (University of Eastern Finland, 2016).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier, 2005).

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), Vol. 50, pp. 251–274.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), Vol. 49, pp. 315–380.

J. J. Gil Perez and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

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.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference” (submitted).

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Figures (7)

Fig. 1.
Fig. 1. Poincaré sphere. The vector s = s ( r , ω ) determines the degree ( P ) and the state of polarization of a partially polarized electric field, while s p = s p ( r , ω ) (of unit length) identifies the polarization state of the fully polarized part.
Fig. 2.
Fig. 2. Young’s interferometer. Two pinholes sample the incident electric field at points r 1 and r 2 in screen A and interference on screen B is analyzed.
Fig. 3.
Fig. 3. Generation of beams with desired degrees of polarization. One of the orthogonal components is delayed exceeding the coherence time of the laser and its polarization direction is controllably rotated prior to recombination.
Fig. 4.
Fig. 4. Degree of polarization P as a function of θ h specifying the wave-plate orientation in the delay line. Measurements by beam self-interference (blue dots), traditional measurements (triangles), and theoretical prediction (solid line).
Fig. 5.
Fig. 5. Michelson’s interferometer. Beam splitter and two mirrors sample the incident electric field at two time instants and interference on the detector is analyzed when one of the mirrors is translated.
Fig. 6.
Fig. 6. Hanbury Brown–Twiss interferometer. The detectors at r 1 and r 2 convert the incident radiation into photocurrents. Delaying one by τ , the currents are correlated and averaged.
Fig. 7.
Fig. 7. Degree of coherence γ ( τ ) for an ASE source: measured values (solid line) and values calculated from the spectrum (stars). The inset displays the ASE spectrum. The coherence time is τ c 200    fs .

Equations (45)

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Γ ( r 1 , r 2 , τ ) = E * ( r 1 , t 1 ) E T ( r 2 , t 2 ) ,
W ( r 1 , r 2 , ω ) = 1 2 π Γ ( r 1 , r 2 , τ ) e i ω τ d τ .
γ ( r 1 , r 2 , τ ) = Γ ( r 1 , r 2 , τ ) F [ I ( r 1 ) I ( r 2 ) ] 1 / 2 ,
μ ( r 1 , r 2 , ω ) = W ( r 1 , r 2 , ω ) F [ S ( r 1 , ω ) S ( r 2 , ω ) ] 1 / 2 ,
J ( r ) = 0 Φ ( r , ω ) d ω .
P 2 ( r , ω ) = 1 4 det Φ ( r , ω ) tr 2 Φ ( r , ω ) = 2 [ tr Φ 2 ( r , ω ) tr 2 Φ ( r , ω ) 1 2 ] = 2 [ μ 2 ( r , r , ω ) 1 2 ] ,
S 0 ( r , ω ) = Φ x x ( r , ω ) + Φ y y ( r , ω ) ,
S 1 ( r , ω ) = Φ x x ( r , ω ) Φ y y ( r , ω ) ,
S 2 ( r , ω ) = Φ x y ( r , ω ) + Φ y x ( r , ω ) ,
S 3 ( r , ω ) = i [ Φ y x ( r , ω ) Φ x y ( r , ω ) ] ,
P 2 ( r , ω ) = n = 1 3 s n 2 ( r , ω ) ,
P 2 ( r ) = 1 4 det J ( r ) tr 2 J ( r ) = 2 [ tr J 2 ( r ) tr 2 J ( r ) 1 2 ] = 2 [ γ 2 ( r , r , 0 ) 1 2 ] ,
S 0 ( r ) = J x x ( r ) + J y y ( r ) ,
S 1 ( r ) = J x x ( r ) J y y ( r ) ,
S 2 ( r ) = J x y ( r ) + J y x ( r ) ,
S 3 ( r ) = i [ J y x ( r ) J x y ( r ) ] ,
P 2 ( r ) = n = 1 3 s n 2 ( r ) ,
W ( r 1 , r 2 , ω ) = 1 2 n = 0 3 S n ( r 1 , r 2 , ω ) σ n ,
S 0 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) + W y y ( r 1 , r 2 , ω ) ,
S 1 ( r 1 , r 2 , ω ) = W x x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) ,
S 2 ( r 1 , r 2 , ω ) = W x y ( r 1 , r 2 , ω ) + W y x ( r 1 , r 2 , ω ) ,
S 3 ( r 1 , r 2 , ω ) = i [ W y x ( r 1 , r 2 , ω ) W x y ( r 1 , r 2 , ω ) ] .
μ 2 ( r 1 , r 2 , ω ) = 1 2 n = 0 3 | s n ( r 1 , r 2 , ω ) | 2 ,
Γ ( r 1 , r 2 , τ ) = 1 2 n = 0 3 S n ( r 1 , r 2 , τ ) σ n ,
S n ( r 1 , r 2 , τ ) = 0 S n ( r 1 , r 2 , ω ) e i ω t d ω ,
S 0 ( r 1 , r 2 , τ ) = Γ x x ( r 1 , r 2 , τ ) + Γ y y ( r 1 , r 2 , τ ) ,
S 1 ( r 1 , r 2 , τ ) = Γ x x ( r 1 , r 2 , τ ) Γ y y ( r 1 , r 2 , τ ) ,
S 2 ( r 1 , r 2 , τ ) = Γ x y ( r 1 , r 2 , τ ) + Γ y x ( r 1 , r 2 , τ ) ,
S 3 ( r 1 , r 2 , τ ) = i [ Γ y x ( r 1 , r 2 , τ ) Γ x y ( r 1 , r 2 , τ ) ] ,
γ 2 ( r 1 , r 2 , τ ) = 1 2 n = 0 3 | s n ( r 1 , r 2 , τ ) | 2 ,
S n ( r , ω ) = S n ( r , ω ) + S n ( r , ω ) + 2 [ S 0 ( r , ω ) S 0 ( r , ω ) ] 1 / 2 × | s n ( r 1 , r 2 , ω ) | cos [ θ n ( r 1 , r 2 , ω ) k ( R 1 R 2 ) ] ,
V n ( r , ω ) = max [ S n ( r , ω ) ] min [ S n ( r , ω ) ] max [ S 0 ( r , ω ) ] + min [ S 0 ( r , ω ) ] ,
V n ( r , ω ) = C ( r , ω ) | s n ( r 1 , r 2 , ω ) | , n ( 0 , , 3 ) ,
C ( r , ω ) = 2 [ ξ ( r , ω ) ] 1 / 2 ξ ( r , ω ) + 1 ,
μ 2 ( r 1 , r 2 , ω ) = C 2 ( r , ω ) 1 2 n = 0 3 V n 2 ( r , ω ) .
P 2 ( r 0 , ω ) = n = 1 3 V n 2 ( r , ω ) .
S n ( r ) = S n ( r ) + S n ( r ) + 2 [ S 0 ( r ) S 0 ( r ) ] 1 / 2 × | s n ( r 1 , r 2 , τ ) | cos { arg [ s n ( r 1 , r 2 , τ ) ] } ,
V n ( r ) = max [ S n ( r ) ] min [ S n ( r ) ] max [ S 0 ( r ) ] + min [ S 0 ( r ) ] ,
P 2 ( r 0 ) = n = 1 3 V n 2 ( r ) ,
S n ( τ ) = 2 S n + 2 S 0 | s n ( τ ) | cos { arg [ s n ( τ ) ] + Δ φ } ,
V n ( τ ) = max [ S n ( τ ) ] min [ S n ( τ ) ] max [ S 0 ( τ ) ] + min [ S 0 ( τ ) ] = | s n ( τ ) | ,
γ 2 ( τ ) = 1 2 n = 0 3 V n 2 ( τ ) ,
Δ S ( r 1 , ω ) Δ S ( r 2 , ω ) S ( r 1 , ω ) S ( r 2 , ω ) = μ 2 ( r 1 , r 2 , ω ) ,
γ 2 ( τ ) = Δ I ( t ) Δ I ( t + τ ) I ( t ) 2 = I ( t ) I ( t + τ ) I ( t ) 2 1 .
G ( r 1 , r 2 , τ ) = tr [ ρ ^ E ^ ( ) ( r 1 , t ) E ^ ( + ) ( r 2 , t + τ ) ] ,

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