Abstract

It is now well-established that a variety of singularities can be characterized and observed in optical wavefields. It is also known that these phase singularities, polarization singularities and coherence singularities are physically related, but the exact nature of their relationship is still somewhat unclear. We show how a Young-type three-pinhole interference experiment can be used to create a continuous cycle of transformations between classes of singularities, often accompanied by topological reactions in which different singularities are created and annihilated. This arrangement serves to clarify the relationships between the different singularity types, and provides a simple tool for further exploration.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. M.R. Dennis, K. O’Holleran, and M.J. Padgett, “Singular optics: optical vortices and polarization singularities,” in: Progress in Optics, edited by E. Wolf, ed. (Elsevier, 2001), 53, 293–363.
    [Crossref]
  3. J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1970).
    [Crossref]
  4. J.F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).
  5. M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457, 141–155 (2001).
    [Crossref]
  6. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
    [Crossref]
  7. I. Freund, “Polarization singularities in optical lattices,” Opt. Lett. 29, 875–877 (2004).
    [Crossref] [PubMed]
  8. R.W. Schoonover and T.D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733–5745 (2006).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  10. G. Gbur and T.D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [Crossref]
  11. D.G. Fischer and T.D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
    [Crossref]
  12. S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37, 4179–4181 (2012).
    [Crossref] [PubMed]
  13. S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Topological reactions of correlation functions in partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 30, 582–588 (2013).
    [Crossref]
  14. G. Gbur, T.D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wavefields,” Jnl. of Optics A 6, S239–S242 (2004).
    [Crossref]
  15. G. Gbur and T.D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
    [Crossref]
  16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [Crossref] [PubMed]
  17. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
    [Crossref] [PubMed]
  18. I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  26. G.C.G. Berkhout and M.W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
    [Crossref] [PubMed]
  27. G.C.G. Berkhout and M.W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” Jnl. of Optics 11, 094021 (2009).
  28. G.C.G. Berkhout and M.W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18, 13836–13841 (2010).
    [Crossref] [PubMed]
  29. R.W. Schoonover and T.D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009).
    [Crossref]
  30. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
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    [Crossref] [PubMed]
  32. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento Ser. X 13, 1165–1181 (1959).
    [Crossref]
  33. O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
    [Crossref]
  34. H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
    [Crossref]
  35. J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999). See Sec. 7.2.
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2014 (1)

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2010 (1)

2009 (2)

R.W. Schoonover and T.D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009).
[Crossref]

G.C.G. Berkhout and M.W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” Jnl. of Optics 11, 094021 (2009).

2008 (3)

O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
[Crossref]

G.C.G. Berkhout and M.W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
[Crossref] [PubMed]

T.D. Visser and R.W. Schoonover, “A cascade of singular field patterns in Young’s interference experiment,” Opt. Commun. 281, 1–6 (2008).
[Crossref]

2006 (3)

2005 (2)

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

2004 (4)

2003 (3)

H.F. Schouten, G. Gbur, T.D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

G. Gbur and T.D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

2002 (1)

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

2001 (2)

M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457, 141–155 (2001).
[Crossref]

I. Freund, “Poincaré vortices,” Opt. Lett. 26, 1996–1998 (2001).
[Crossref]

1970 (1)

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1970).
[Crossref]

1961 (1)

C. Jönsson, “Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten,” Zeitschrift für Physik 161, 454–474 (1961).An English translation was published by D. Brandt and S. Hirschi, in the American Journal of Physics 42, 4–11 (1974).
[Crossref]

1959 (1)

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

1938 (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Alkemade, P.F.A.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Beijersbergen, M.W.

G.C.G. Berkhout and M.W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18, 13836–13841 (2010).
[Crossref] [PubMed]

G.C.G. Berkhout and M.W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” Jnl. of Optics 11, 094021 (2009).

G.C.G. Berkhout and M.W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
[Crossref] [PubMed]

Berkhout, G.C.G.

G.C.G. Berkhout and M.W. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18, 13836–13841 (2010).
[Crossref] [PubMed]

G.C.G. Berkhout and M.W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” Jnl. of Optics 11, 094021 (2009).

G.C.G. Berkhout and M.W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
[Crossref] [PubMed]

Berry, M.V.

M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457, 141–155 (2001).
[Crossref]

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1970).
[Crossref]

Blok, H.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 1999).
[Crossref]

Dennis, M.R.

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
[Crossref] [PubMed]

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457, 141–155 (2001).
[Crossref]

M.R. Dennis, K. O’Holleran, and M.J. Padgett, “Singular optics: optical vortices and polarization singularities,” in: Progress in Optics, edited by E. Wolf, ed. (Elsevier, 2001), 53, 293–363.
[Crossref]

Dubois, G.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Eliel, E.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Fischer, D.G.

Flossmann, F.

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
[Crossref] [PubMed]

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Freund, I.

Friberg, A.T.

Gbur, G.

G. Gbur and T.D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

G. Gbur, T.D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wavefields,” Jnl. of Optics A 6, S239–S242 (2004).
[Crossref]

H.F. Schouten, G. Gbur, T.D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

G. Gbur and T.D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

Gbur, G.J.

G.J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, 2011). See Sec. 10.3.

Hooft, G.W.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Jackson, J.D.

J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999). See Sec. 7.2.

Jönsson, C.

C. Jönsson, “Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten,” Zeitschrift für Physik 161, 454–474 (1961).An English translation was published by D. Brandt and S. Hirschi, in the American Journal of Physics 42, 4–11 (1974).
[Crossref]

Korotkova, O.

O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
[Crossref]

Kuzmin, N.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Lenstra, D.

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

Maier, M.

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
[Crossref] [PubMed]

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Nye, J.F.

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London 336, 165–190 (1970).
[Crossref]

J.F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).

O’Holleran, K.

M.R. Dennis, K. O’Holleran, and M.J. Padgett, “Singular optics: optical vortices and polarization singularities,” in: Progress in Optics, edited by E. Wolf, ed. (Elsevier, 2001), 53, 293–363.
[Crossref]

Padgett, M.J.

M.R. Dennis, K. O’Holleran, and M.J. Padgett, “Singular optics: optical vortices and polarization singularities,” in: Progress in Optics, edited by E. Wolf, ed. (Elsevier, 2001), 53, 293–363.
[Crossref]

Raghunathan, S.B.

Samuel, J.

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

Sawant, R.

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

Schoonover, R.W.

R.W. Schoonover and T.D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009).
[Crossref]

T.D. Visser and R.W. Schoonover, “A cascade of singular field patterns in Young’s interference experiment,” Opt. Commun. 281, 1–6 (2008).
[Crossref]

R.W. Schoonover and T.D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733–5745 (2006).
[Crossref] [PubMed]

Schouten, H.F.

S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Topological reactions of correlation functions in partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 30, 582–588 (2013).
[Crossref]

S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37, 4179–4181 (2012).
[Crossref] [PubMed]

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

H.F. Schouten, G. Gbur, T.D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

Schwarz, U.T.

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).
[Crossref] [PubMed]

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Setala, T.

Sinha, A.

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

Sinha, S.

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

Sinha, U.

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

Soskin, M.S.

M.S. Soskin and M.V. Vasnetsov, “Singular Optics,” in: Progress in Optics, E. Wolf, ed. (Elsevier, 2001), 42, 219–276.
[Crossref]

Tervo, J.

Vasnetsov, M.V.

M.S. Soskin and M.V. Vasnetsov, “Singular Optics,” in: Progress in Optics, E. Wolf, ed. (Elsevier, 2001), 42, 219–276.
[Crossref]

Visser, T.D.

S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Topological reactions of correlation functions in partially coherent electromagnetic beams,” J. Opt. Soc. Am. A 30, 582–588 (2013).
[Crossref]

S.B. Raghunathan, H.F. Schouten, and T.D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37, 4179–4181 (2012).
[Crossref] [PubMed]

R.W. Schoonover and T.D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009).
[Crossref]

O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
[Crossref]

T.D. Visser and R.W. Schoonover, “A cascade of singular field patterns in Young’s interference experiment,” Opt. Commun. 281, 1–6 (2008).
[Crossref]

G. Gbur and T.D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

R.W. Schoonover and T.D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733–5745 (2006).
[Crossref] [PubMed]

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

G. Gbur, T.D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wavefields,” Jnl. of Optics A 6, S239–S242 (2004).
[Crossref]

D.G. Fischer and T.D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

H.F. Schouten, G. Gbur, T.D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

G. Gbur and T.D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

Wolf, E.

O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
[Crossref]

G. Gbur, T.D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wavefields,” Jnl. of Optics A 6, S239–S242 (2004).
[Crossref]

H.F. Schouten, G. Gbur, T.D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

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

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

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 1999).
[Crossref]

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” in Progress in Optics, E. Wolf (ed.), 50, (Elsevier, 2007), 251–273.
[Crossref]

Zernike, F.

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Il Nuovo Cimento Ser. X (1)

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

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

Jnl. of Optics (1)

G.C.G. Berkhout and M.W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” Jnl. of Optics 11, 094021 (2009).

Jnl. of Optics A (1)

G. Gbur, T.D. Visser, and E. Wolf, “Hidden singularities in partially coherent and polychromatic wavefields,” Jnl. of Optics A 6, S239–S242 (2004).
[Crossref]

Opt. Commun. (5)

G. Gbur and T.D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

G. Gbur and T.D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

O. Korotkova, T.D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515–520 (2008).
[Crossref]

T.D. Visser and R.W. Schoonover, “A cascade of singular field patterns in Young’s interference experiment,” Opt. Commun. 281, 1–6 (2008).
[Crossref]

Opt. Express (3)

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Phys. Rev. A (1)

R.W. Schoonover and T.D. Visser, “Creating polarization singularities with an N-pinhole interferometer,” Phys. Rev. A 79, 043809 (2009).
[Crossref]

Phys. Rev. E (1)

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E,  67, 036608 (2003).
[Crossref]

Phys. Rev. Lett. (4)

H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

R. Sawant, J. Samuel, A. Sinha, S. Sinha, and U. Sinha, “Nonclassical paths in quantum interference experiments,” Phys. Rev. Lett. 113, 120406 (2014).
[Crossref] [PubMed]

G.C.G. Berkhout and M.W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
[Crossref] [PubMed]

F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref] [PubMed]

Physica (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

Proc. R. Soc. Lond. A (1)

M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457, 141–155 (2001).
[Crossref]

Proc. R. Soc. London (1)

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[Crossref]

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C. Jönsson, “Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten,” Zeitschrift für Physik 161, 454–474 (1961).An English translation was published by D. Brandt and S. Hirschi, in the American Journal of Physics 42, 4–11 (1974).
[Crossref]

Other (8)

J.F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).

M.S. Soskin and M.V. Vasnetsov, “Singular Optics,” in: Progress in Optics, E. Wolf, ed. (Elsevier, 2001), 42, 219–276.
[Crossref]

M.R. Dennis, K. O’Holleran, and M.J. Padgett, “Singular optics: optical vortices and polarization singularities,” in: Progress in Optics, edited by E. Wolf, ed. (Elsevier, 2001), 53, 293–363.
[Crossref]

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” in Progress in Optics, E. Wolf (ed.), 50, (Elsevier, 2007), 251–273.
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 1999).
[Crossref]

J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999). See Sec. 7.2.

G.J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge University Press, 2011). See Sec. 10.3.

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

Supplementary Material (11)

NameDescription
» Visualization 1: MP4 (333 KB)      Visualization 1
» Visualization 2: MP4 (224 KB)      Visualization 2
» Visualization 3: MP4 (312 KB)      Visualization 3
» Visualization 4: MP4 (186 KB)      Visualization 4
» Visualization 5: MP4 (276 KB)      Visualization 5
» Visualization 6: MP4 (95 KB)      Visualization 6
» Visualization 7: MP4 (313 KB)      Visualization 7
» Visualization 8: MP4 (160 KB)      Visualization 8
» Visualization 9: MP4 (300 KB)      Visualization 9
» Visualization 10: MP4 (144 KB)      Visualization 10
» Visualization 11: MP4 (304 KB)      Visualization 11

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Figures (10)

Fig. 1
Fig. 1 Illustrating the continuous cycle in which the coherence parameter μ and the polarization angle θ are smoothly varied, and eventually brought back to their initial values. Stage I is at the top of the figure.
Fig. 2
Fig. 2 Three identical pinholes in an opaque screen occupying the plane z = 0. The pinholes are located symmetrically with respect to the origin O of a right-handed Cartesian coordinate system. An interference pattern is formed on a second, parallel screen a distance Δz away (a). The three pinholes with their relative phase, coherence parameter, and orientation of the electric field (b).
Fig. 3
Fig. 3 The spectral density S(x,y) = |E(x,y)|2 in arbitrary units (a), and the color-coded phase of Ex (x,y) on the observation screen (b). In this example the polarization angle θ = 0, μ = 1, λ = 0.5 × 10 6 m, ρ = 0.5 mm, and Δz = 1 m.
Fig. 4
Fig. 4 The local orientation of the major axis of the polarization ellipse after the angle of polarization at pinhole 1 has been increased from zero to θ = 0.03. The phase singularity of Ex at (0,0) in Fig. 3 has decayed into two polarization singularities: a lemon (top) and a star (bottom) (a). Selected contours of the Stokes parameters for the same region as in the left-hand panel: s 1 = 0 (blue), s 2 = 0 (green), s 3 = 0.998 (red), s 3 = 0.998 (black) and s 3 = 0 (orange) (b). All other parameters are the same as in Fig. 3.
Fig. 5
Fig. 5 Color-coded plot of the phase of the spectral degree of coherence η(x 1,y 1,x 2,y 2) on the observation screen. The reference point is taken as (x 1,y 1) = (0.45,0) mm. The polarization angle θ = 0 (a), 0.65 (b), 1.12 (c), and π/2 (d). All other parameters are the same as in Fig. 3.
Fig. 6
Fig. 6 Color-coded plot of the phase of the spectral degree of coherence η(x 1,y 1,x 2,y 2) on the observation screen. The reference point is taken as (x 1,y 1) = (0.45,0) mm. The polarization angle θ = π/2. The coherence parameter μ = 1 (a), 0.57 (b), 0.40 (c), and 0 (d). All other parameters are the same as in Fig. 3.
Fig. 7
Fig. 7 Contours of the Stokes parameters for the case μ = 0.51 and θ = π/2, just before the annihilation of pairs of C points with the same handedness (a). Shown are contours of s 3 = 0.9999 (red), s 3 = 0.9999 (black), s 1 = 0 (blue), s 2 = 0 (green), and s 3 = 0 (orange). In (b) the situation for μ = 0 and θ = π/2 is plotted. All other parameters are the same as in Fig. 3.
Fig. 8
Fig. 8 Showing the major axis of the polarization ellipse field for two values of the coherence parameter μ with the polarization angle θ = π/3. In panel (a) μ = 0.4 and no C points are exist. In panel (b) μ = 0.7 and several C points have been created. All other parameters are the same as in Fig. 3.
Fig. 9
Fig. 9 Contour lines of the Stokes parameters corresponding to Fig. 8(b), i.e., θ = π/3 and μ = 0.7. The different contours represent s 1 = 0 (blue), s 2 = 0 (green), s 3 = 0.995 (red), s 3 =0.995 (black), and s 3 = 0 (orange). All other parameters are the same as in Fig. 3.
Fig. 10
Fig. 10 Contour lines of the Stokes parameters when μ = 0.7 and θ = 0.90 (a) and θ = 0.75. The different contours represent s 1 = 0 (blue), s 2 = 0 (green), s 3 = 0.995 (red), s 3 = −0.995 (black), and s 3 = 0 (orange). All other parameters are the same as in Fig. 3.

Equations (53)

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s = 1 2 π C d ϕ ,
W ( r 1 , r 2 , ω ) = ( W x x ( r 1 , r 2 , ω ) W x y ( r 1 , r 2 , ω ) W y x ( r 1 , r 2 , ω ) W y y ( r 1 , r 2 , ω ) ) ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) , ( i , j = x , y ) .
S ( r , ω ) = | E ( r , ω ) | 2 = Tr W ( r , r , ω ) ,
η ( r 1 , r 2 , ω ) = Tr W ( r 1 , r 2 , ω ) [ Tr W ( r 1 , r 1 , ω ) Tr W ( r 2 , r 2 , ω ) ] 1 / 2 .
η ( r 1 , r 2 , ω ) = E X * ( r 1 , ω ) E x ( r 2 , ω ) + E y * ( r 1 , ω ) E y ( r 2 , ω ) | E ( r 1 , ω ) | | E ( r 2 , ω ) | .
E * ( r 1 , ω ) E ( r 2 , ω ) = 0.
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ,
μ ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) S ( r 1 , ω ) S ( r 2 , ω ) ,
S ( r , ω ) = W ( r , r , ω ) .
W ( r , r , ω ) = W ( p ) ( r , r , ω ) + W ( u ) ( r , r , ω ) .
W ( p ) ( r , r , ω ) = [ B ( r , r , ω ) D ( r , r , ω ) D * ( r , r , ω ) C ( r , r , ω ) ] ,
W ( u ) ( r , r , ω ) = [ A ( r , r , ω ) 0 0 A ( r , r , ω ) ] ,
B C D D * = 0.
A ( r , r , ω ) = 1 2 [ W x x + W y y ( W x x W y y ) 2 + 4 | W x y | 2 ¯ ] ,
B ( r , r , ω ) = 1 2 [ W x x W y y + ( W x x W y y ) 2 + 4 | W x y | 2 ¯ ] ,
C ( r , r , ω ) = 1 2 [ W y y W x x + ( W x x W y y ) 2 + 4 | W x y | 2 ¯ ] ,
D ( r , r , ω ) = W X Y .
S 0 ( r , ω ) = B + C = ( W x x W y y ) 2 + 4 | W x y | 2 ¯ ,
S 1 ( r , ω ) = B C = W x x W y y ,
S 2 ( r , ω ) = D + D * = W x y + W y x ,
S 3 ( r , ω ) = i ( D * D ) = i ( W y x W x y ) .
ψ = 1 2 arctan ( S 2 S 1 ) , ( 0 ψ < π ) .
s i ( r , ω ) S i ( r , ω ) / S 0 ( r , ω ) , ( i = 1 , 2 , 3 ) .
ρ 1 = ρ ( 0 , 1 ) ,
ρ 2 = ρ ( 3 / 2 , 0.5 ) ,
ρ 3 = ρ ( 3 / 2 , 0.5 ) ,
E ( r , ω ) = i = 1 3 K 1 ( r , ω ) E ( ρ i , ω ) ,
K i ( r , ω ) = 1 λ e i k R i R i d A ( i = 1 , 2 , 3 ) ,
E ( ρ 1 , ω ) = C ( cos θ x ^ + sin θ y ^ ) ,
E ( ρ 2 , ω ) = C e i 2 π / 3 x ^ ,
E ( ρ 3 , ω ) = C e i 4 π / 3 x ^ ,
E x ( 0 , 0 , z ) = 0.
W X X ( ρ 2 , ρ 3 ) = | C | 2 μ e i 2 π / 3 ,
E x ( ρ 1 ) = cos θ e i 4 π / 3 E x ( ρ 3 ) ,
W x x ( ρ 2 , ρ 1 ) = cos θ e i 4 π / 3 W x x ( ρ 2 , ρ 3 ) ,
= | C | 2 μ cos θ e i 2 π / 3 .
E y ( ρ 1 ) = ( sin θ / cos θ ) E x ( ρ 1 ) ,
W x y ( ρ 1 , ρ 1 ) = | C | 2 sin θ cos θ ,
W x y ( ρ 2 , ρ 1 ) = ( sin θ / cos θ ) W x x ( ρ 2 , ρ 1 ) ,
= | C | 2 μ sin θ e i 2 π / 3 .
E x * ( ρ 1 ) = cos θ e i 4 π / 3 E x * ( ρ 3 ) .
W x x ( ρ 1 , ρ 3 ) = cos θ e i 4 π / 3 W x x ( ρ 3 , ρ 3 ) ,
= | C | 2 cos θ e i 4 π / 3 .
W y x ( ρ 1 , ρ 3 ) = ( sin θ / cos θ ) W x x ( ρ 1 , ρ 3 ) ,
= | C | 2 sin θ e i 4 π / 3 .
S x ( ρ 1 ) = W x x ( ρ 1 , ρ 1 ) = | C | 2 cos 2 θ ,
S y ( ρ 1 ) = W y y ( ρ 1 , ρ 1 ) = | C | 2 sin 2 θ ,
S x ( ρ 2 ) = W x x ( ρ 2 , ρ 2 ) = | C | 2 ,
S x ( ρ 3 ) = W x x ( ρ 3 , ρ 3 ) = | C | 2 .
S ( x , y ) = | E x ( x , y ) | 2 + | E y ( x , y ) | 2 ,
= | C | 2 [ | K 1 ( x , y ) | 2 + | K 2 ( x , y ) | 2 + | K 3 ( x , y ) | 2 + 2 μ cos θ Re { K 2 * ( x , y ) K 1 ( x , y ) e i 2 π / 3 } + 2 cos θ Re { K 1 * ( x , y ) K 3 ( x , y ) e i 4 π / 3 } + 2 μ Re { K 2 * ( x , y ) K 3 ( x , y ) e i 2 π / 3 } ] .
Tr W ( x 1 , y 1 , x 2 , y 2 ) = | C | 2 { K 1 * ( x 1 , y 1 ) K 1 ( x 2 , y 2 ) + K 2 * ( x 1 , y 1 ) K 2 ( x 2 , y 2 ) + K 3 * ( x 1 , y 1 ) K 3 ( x 2 , y 2 ) + μ cos θ [ K 1 * ( x 1 , y 1 ) K 2 ( x 2 , y 2 ) e i 2 π / 3 + K 2 * ( x 1 , y 1 ) K 1 ( x 2 , y 2 ) e i 2 π / 3 + cos θ [ K 1 * ( x 1 , y 1 ) K 3 ( x 2 , y 2 ) e i 4 π / 3 + K 3 * ( x 1 , y 1 ) K 2 ( x 2 , y 2 ) e i 4 π / 3 + μ [ K 2 * ( x 1 , y 1 ) K 3 ( x 2 , y 2 ) e i 2 π / 3 + K 3 * ( x 1 , y 1 ) K 2 ( x 2 , y 2 ) e i 2 π / 3 ] } .

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