Abstract

We measure the recently introduced electromagnetic temporal degree of coherence of a stationary, partially polarized, classical optical beam. Instead of recording the visibility of intensity fringes, the spectrum, or the polarization characteristics, we introduce a novel technique based on two-photon absorption. Using a Michelson interferometer equipped with polarizers and a specific GaAs photocount tube, we obtain the two fundamental quantities pertaining to the fluctuations of light: the degree of coherence and the degree of polarization. We also show that the electromagnetic intensity-correlation measurements with two-photon absorption require that the polarization dynamics, i.e., the time evolution of the instantaneous polarization state, is properly taken into account. We apply the technique to unpolarized and polarized sources of amplified spontaneous emission (Gaussian statistics) and to a superposition of two independent, narrow-band laser beams of different mid frequencies (non-Gaussian statistics). For these two sources femtosecond-range coherence times are found that are in good agreement with the traditional spectral measurements. Although previously employed for laser pulses, two-photon absorption provides a new physical principle to study electromagnetic coherence phenomena in classical and quantum continuous-wave light at extremely short time scales.

© 2015 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  2. J. Peřina, Coherence of Light (Reidel, 1985).
  3. E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
    [Crossref]
  4. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [Crossref]
  5. B. L. Morgan and L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
    [Crossref]
  6. H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
    [Crossref]
  7. S. Hooker and C. Webb, Laser Physics Sec. 17.4 (Oxford University, 2010).
  8. A. M. Weiner, Ultrafast Optics (Wiley, 2009).
    [Crossref]
  9. B. R. Mollow, “Two-photon absorption and field correlation functions,” Phys. Rev. 175, 1555–1563 (1968).
    [Crossref]
  10. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).
  11. J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27, 2076–2078 (2002).
    [Crossref]
  12. A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
    [Crossref]
  13. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nature Phys. 5, 267–270 (2009).
    [Crossref]
  14. F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
    [Crossref] [PubMed]
  15. A. Nevet, A. Hayat, and M. Orenstein, “Ultrafast pulse compression by semiconductor two-photon gain,” Opt. Lett. 35, 3877–3879 (2010).
    [Crossref] [PubMed]
  16. 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]
  17. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [Crossref] [PubMed]
  18. 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]
  19. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
    [Crossref]
  20. 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]
  21. 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]
  22. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [Crossref]
  23. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [Crossref]
  24. R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
    [Crossref]
  25. L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [Crossref]
  26. J. W. Goodman, Statistical Optics (Wiley, 2000).
  27. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown–Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
    [Crossref] [PubMed]
  28. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).
  29. T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  30. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Correlation matrix of a completely polarized, statistically stationary electromagnetic field,” Opt. Lett. 29, 1536–1538 (2004).
    [Crossref] [PubMed]

2014 (1)

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]

2011 (3)

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (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] [PubMed]

2010 (1)

2009 (2)

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]

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

2006 (2)

2004 (3)

2003 (1)

2002 (2)

J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. 27, 2076–2078 (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]

1977 (1)

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[Crossref]

1969 (1)

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

1968 (1)

B. R. Mollow, “Two-photon absorption and field correlation functions,” Phys. Rev. 175, 1555–1563 (1968).
[Crossref]

1966 (1)

B. L. Morgan and L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[Crossref]

1965 (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

1963 (1)

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

1956 (1)

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

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

Alley, C. O.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

Boitier, F.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

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

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Chang, R. F.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Dagenais, M.

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[Crossref]

Delaye, P.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

Detenbeck, R. W.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

Dogariu, A.

Dubreuil, N.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

Ellis, J.

Fabre, C.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

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

Friberg, A. T.

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

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

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space–frequency domain,” Opt. Lett. 29, 328–330 (2004).
[Crossref]

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

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

Ginzburg, P.

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
[Crossref]

Glauber, R. J.

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

Godard, A.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

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

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hanbury Brown, R.

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

Hassinen, T.

Hayat, A.

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
[Crossref]

A. Nevet, A. Hayat, and M. Orenstein, “Ultrafast pulse compression by semiconductor two-photon gain,” Opt. Lett. 35, 3877–3879 (2010).
[Crossref] [PubMed]

Hooker, S.

S. Hooker and C. Webb, Laser Physics Sec. 17.4 (Oxford University, 2010).

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, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Kimble, H. J.

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[Crossref]

Korenman, V.

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

Leppänen, L.-P.

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]

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

Mandel, L.

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[Crossref]

B. L. Morgan and L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[Crossref]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

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

Mollow, B. R.

B. R. Mollow, “Two-photon absorption and field correlation functions,” Phys. Rev. 175, 1555–1563 (1968).
[Crossref]

Morgan, B. L.

B. L. Morgan and L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[Crossref]

Murphy, T. E.

Nevet, A.

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
[Crossref]

A. Nevet, A. Hayat, and M. Orenstein, “Ultrafast pulse compression by semiconductor two-photon gain,” Opt. Lett. 35, 3877–3879 (2010).
[Crossref] [PubMed]

Orenstein, M.

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
[Crossref]

A. Nevet, A. Hayat, and M. Orenstein, “Ultrafast pulse compression by semiconductor two-photon gain,” Opt. Lett. 35, 3877–3879 (2010).
[Crossref] [PubMed]

Perina, J.

J. Peřina, Coherence of Light (Reidel, 1985).

Ponomarenko, S.

Rosencher, E.

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

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

Roth, J. M.

Saastamoinen, K.

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]

Setälä, T.

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

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, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts 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, 328–330 (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]

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

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

Shevchenko, A.

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, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Tervo, J.

Twiss, R. Q.

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

Webb, C.

S. Hooker and C. Webb, Laser Physics Sec. 17.4 (Oxford University, 2010).

Weiner, A. M.

A. M. Weiner, Ultrafast Optics (Wiley, 2009).
[Crossref]

Wolf, E.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Correlation matrix of a completely polarized, statistically stationary electromagnetic field,” Opt. Lett. 29, 1536–1538 (2004).
[Crossref] [PubMed]

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

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

Xu, C.

Nat. Commun. (1)

F. Boitier, A. Godard, N. Dubreuil, P. Delaye, C. Fabre, and E. Rosencher, “Photon extrabunching in ultrabright twin beams measured by two-photon counting in a semiconductor,” Nat. Commun. 2, 425 (2011).
[Crossref] [PubMed]

Nature (1)

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

Nature Phys. (1)

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

New J. Phys. (2)

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]

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]

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884–888 (1954).
[Crossref]

Opt. Commun. (1)

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]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. (3)

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

R. F. Chang, V. Korenman, C. O. Alley, and R. W. Detenbeck, “Correlations in light from a laser at threshold,” Phys. Rev. 178, 612–621 (1969).
[Crossref]

B. R. Mollow, “Two-photon absorption and field correlation functions,” Phys. Rev. 175, 1555–1563 (1968).
[Crossref]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (2)

B. L. Morgan and L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[Crossref]

H. J. Kimble, M. Dagenais, and L. Mandel, “Photon antibunching in resonance fluorescence,” Phys. Rev. Lett. 39, 691–695 (1977).
[Crossref]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

Semicond. Sci. Technol. (1)

A. Hayat, A. Nevet, P. Ginzburg, and M. Orenstein, “Applications of two-photon processes in semiconductor photonic devices: invited review,” Semicond. Sci. Technol. 26, 083001 (2011).
[Crossref]

Other (7)

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

J. W. Goodman, Statistical Optics (Wiley, 2000).

S. Hooker and C. Webb, Laser Physics Sec. 17.4 (Oxford University, 2010).

A. M. Weiner, Ultrafast Optics (Wiley, 2009).
[Crossref]

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University, 2000).

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

J. Peřina, Coherence of Light (Reidel, 1985).

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Figures (5)

Fig. 1
Fig. 1 Michelson interferometer used in the experiments. Light is delivered to the setup in an optical fiber. BC is beam collimator, BS non-polarizing beam splitter, P polarizers, and M mirrors. The output beam is focused with a lens (L) onto a photomultiplier tube (PMT) leading to electric current J. The mirror in arm b is translated with a piezoelectric transducer altering τ. The inset shows the spectral densities (SD) of beams obtained from amplified spontaneous emission (ASE) of an Er-doped fiber amplifier (blue curve) and two independent lasers (red curve).
Fig. 2
Fig. 2 a) Two-photon absorption counts S(τ) measured for ASE source: x polarized field in both arms (blue XX curve), y polarized field in both arms (red YY curve), and x and y polarized fields in arms a and b, respectively (black XY curve). The inset shows the signals for small τ demonstrating their oscillatory behavior. b) ASE intensity correlation functions (ICFs) 〈Ix(t)Ix(t+τ)〉 (blue line), 〈Iy(t)Iy(t+τ)〉 (red line) and 〈Ix(t)Iy(t+τ)〉 (black line).
Fig. 3
Fig. 3 Measured electromagnetic degree of coherence γ(τ) of unpolarized (blue line) and polarized (red line) ASE light. The related stars show γ(τ) as obtained by Fourier transforming the ASE spectrum of Fig. 1 and assuming that the field is unpolarized and polarized, respectively. The coherence time in both cases is τc 200 fs.
Fig. 4
Fig. 4 a) Two-photon absorption counts S(τ) measured for the two-laser beam with the following polarizer orientations in the interferometer: x orientations in both arms (blue XX curve), y orientations in both arms (red YY curve), and x and y orientations in arms a and b, respectively (black XY curve). The inset shows the signals for small τ. b) Intensity correlation functions 〈Ix(t)Ix(t+τ)〉 (blue line), 〈Iy(t)Iy(t+τ)〉 (red line), and 〈Ix(t)Iy(t+τ)〉 (black line) obtained from the data shown in (a).
Fig. 5
Fig. 5 Electromagnetic degree of coherence γ(τ) for the nearly unpolarized two-laser field (blue line) and its x-polarized component (red line). The stars show the values of γ(τ) obtained from the spectrum. The oscillation period is T = 270 fs and the coherence time is τc = T/2 = 135 fs.

Equations (29)

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γ 2 ( τ ) = tr [ Γ ( τ ) Γ ( τ ) ] I ( t ) 2 ,
γ 2 ( τ ) = Δ I ( t ) Δ I ( t + τ ) I ( t ) 2 = I ( t ) I ( t + τ ) I ( t ) 2 1 ,
γ 2 ( τ ) = 2 I ( t ) I ( t + τ ) + g I ( t ) I ( t + τ ) min I ( t ) I ( t + τ ) min + I ( t ) I ( t + τ ) max 1 ,
S ( τ ) I 2 ( t ) = I a 2 ( t ) + I b 2 ( t ) + 2 I a ( t ) I b ( t + τ ) + 2 | A a T ( t ) A b * ( t + τ ) | 2 + Re [ F ( 1 ) ( τ ) e i ω 0 τ ] + Re [ F ( 2 ) ( τ ) e i 2 ω 0 τ ] ,
I a i ( t ) I b j ( t + τ ) = C [ S i j ( τ ) S a i S b j ] / η i j .
E ( t ) = A 1 ( t ) e i ω 1 t + A 2 ( t ) e i ω 2 t ,
Γ ( t ) = Γ 1 ( τ ) e i ω 1 τ + Γ 2 ( τ ) e i ω 2 τ ,
tr [ Γ ( τ ) Γ ( τ ) ] = tr [ Γ 1 ( τ ) Γ 1 ( τ ) ] + tr [ Γ 2 ( τ ) Γ 2 ( τ ) ] + tr [ Γ 1 ( τ ) Γ 2 ( τ ) ] e i Δ ω τ + tr [ Γ 2 ( τ ) Γ 1 ( τ ) ] e i Δ ω τ ,
tr [ Γ ( τ ) Γ ( τ ) ] = tr ( J 1 2 ) + tr ( J 2 2 ) + 2 tr ( J 1 J 2 ) cos ( Δ ω τ ) ,
γ 2 ( τ ) = I 1 ( t ) 2 + I 2 ( t ) 2 I ( t ) 2 + 2 tr ( J 1 J 2 ) cos ( Δ ω τ ) I ( t ) 2 ,
I ( t ) = E T ( t ) E * ( t ) = I 1 ( t ) + I 2 ( t ) + A 1 T ( t ) A 2 * ( t ) e i Δ ω t + A 2 T ( t ) A 1 * ( t ) e i Δ ω t ,
I ( t ) I ( t + τ ) = I 1 ( t ) I 1 ( t + τ ) + I 1 ( t ) I 2 ( t + τ ) + I 2 ( t ) I 1 ( t + τ ) + I 2 ( t ) I 2 ( t + τ ) + [ A 1 T ( t ) A 2 * ( t ) ] [ A 2 T ( t + τ ) A 1 * ( t + τ ) ] e i Δ ω τ + [ A 2 T ( t ) A 1 * ( t ) ] [ A 1 T ( t + τ ) A 2 * ( t + τ ) ] e i Δ ω τ ,
I ( t ) I ( t + τ ) = I 1 ( t ) 2 + I 2 ( t ) 2 + 2 I 1 ( t ) I 2 ( t ) + 2 | A 1 T ( t ) A 2 * ( t ) | 2 cos ( Δ ω τ ) .
I ( t ) I ( t + τ ) = I ( t ) 2 + 2 tr ( J 1 J 2 ) cos ( Δ ω τ ) ,
I ( t ) I ( t + τ ) min = I ( t ) 2 2 tr ( J 1 J 2 ) ,
I ( t ) I ( t + τ ) max = I ( t ) 2 + 2 tr ( J 1 J 2 ) ,
I ( t ) I ( t + τ ) min + I ( t ) I ( t + τ ) max = 2 I ( t ) 2 .
γ 2 ( τ ) = 2 [ I 1 ( t ) 2 + I 2 ( t ) 2 ] + 2 I ( t ) I ( t + τ ) I ( t ) I ( t + τ ) min + I ( t ) I ( t + τ ) max 1.
J 1 = ( tr J 1 ) e ^ * e ^ T ,
J 2 = ( tr J 2 ) e ^ * e ^ T ,
γ 2 ( τ ) = 2 I ( t ) I ( t + τ ) + 2 I ( t ) I ( t + τ ) min I ( t ) I ( t + τ ) min + I ( t ) I ( t + τ ) max 1 ,
I 1 ( t ) 2 + I 2 ( t ) 2 = 1 2 I 2 ( t ) = 1 2 I ( t ) I ( t + τ ) max = 1 2 I ( t ) I ( t + τ ) min ,
γ 2 ( τ ) = 2 I ( t ) I ( t + τ ) + I ( t ) I ( t + τ ) min I ( t ) I ( t + τ ) min + I ( t ) I ( t + τ ) max 1 ,
I ( t ) = I a ( t ) + I b ( t + τ ) + 2 Re [ E a T ( t ) E b * ( t + τ ) ] ,
S ( τ ) I 2 ( t ) = I a 2 ( t ) + I b 2 ( t ) + 2 I a ( t ) I b ( t + τ ) + 4 [ I a ( t ) + I b ( t + τ ) ] Re [ E a T ( t ) E b * ( t + τ ) ] + 4 { Re [ E a T ( t ) E b * ( t + τ ) ] } 2 ,
E a T ( t ) E b * ( t + τ ) = F ( t ; τ ) e i ω 0 τ ,
4 [ I a ( t ) + I b ( t + τ ) ] Re [ E a T ( t ) E b * ( t + τ ) ] = Re [ F ( 1 ) ( τ ) e i ω 0 τ ] ,
4 { Re [ E a T ( t ) E b * ( t + τ ) ] } 2 = 4 { Re [ F ( t ; τ ) ] cos ( ω 0 τ ) Im [ F ( t ; τ ) ] sin ( ω 0 τ ) } 2 ,
4 { Re [ E a T ( t ) E b * ( t + τ ) ] } 2 = 2 | A a T ( t ) A b * ( t + τ ) | 2 + Re [ F ( 2 ) ( τ ) e i 2 ω 0 τ ] ,

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