Abstract

It was recently shown that so-called coherence vortices, singularities of the two-point correlation function, generally occur in partially coherent electromagnetic beams. We study the three-dimensional structure of these singularities and show that in successive cross sections of a beam a rich variety of topological reactions takes place. These reactions involve, apart from vortices, the creation or annihilation of dipoles, saddles, maxima and minima of the phase of the correlation function. Since these reactions happen generically, i.e., under quite general conditions, these observations have implications for interference experiments with partially coherent, electromagnetic beams.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
    [CrossRef]
  2. I. Freund, “Critical foliations,” Opt. Lett. 26, 545–547 (2001).
    [CrossRef]
  3. I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000).
    [CrossRef]
  4. I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71–90 (2001).
    [CrossRef]
  5. G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
    [CrossRef]
  6. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
    [CrossRef]
  7. A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. Chen, and Y. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14, 8317–8327 (2006).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
  9. G. P. Karman, A. van Duijl, and J. P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22, 1503–1505 (1997).
    [CrossRef]
  10. A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  11. H. F. Schouten, G. Gbur, T. D. Visser, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371–380 (2003).
    [CrossRef]
  12. H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B 6, S404–S409 (2004).
    [CrossRef]
  13. D. W. Diehl and T. D. Visser, “Phase singularities of the longitudinal field components in high-aperture systems,” J. Opt. Soc. Am. A 21, 2103–2108 (2004).
    [CrossRef]
  14. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).
  15. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  17. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  18. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2010), Vol. 55, pp. 285–341.
  19. 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]
  20. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
    [CrossRef]
  21. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  22. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef]
  23. G. A. Swartzlander and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
    [CrossRef]
  24. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
    [CrossRef]
  25. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
    [CrossRef]
  26. G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
    [CrossRef]
  27. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26, 741–744 (2009).
    [CrossRef]
  28. 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]
  29. T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
    [CrossRef]
  30. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010).
    [CrossRef]
  31. M. L. Marasinghe, M. Premaratne, D. M. Paganin, and M. A. Alonso, “Coherence vortices in Mie scattered nonparaxial partially coherent beams,” Opt. Express 20, 2858–2875 (2012).
    [CrossRef]
  32. W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
    [CrossRef]
  33. Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
    [CrossRef]
  34. M. L. Marasinghe, D. M. Paganin, and M. Premaratne, “Coherence-vortex lattice formed via Mie scattering of partially coherent light by several dielectric nanospheres,” Opt. Lett. 36, 936–938 (2011).
    [CrossRef]
  35. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, “Correlation singularities in partially coherent electromagnetic beams,” Opt. Lett. 37, 4179–4181 (2012).
    [CrossRef]
  36. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994), pp. 174–180.
  37. T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
    [CrossRef]
  38. R. Hanbury Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
    [CrossRef]
  39. 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]
  40. T. Hassinen, J. Tervo, T. Setälä, and A. T. Friberg, “Hanbury Brown-Twiss effect with electromagnetic waves,” Opt. Express 19, 15188–15195 (2011).
    [CrossRef]
  41. G. Gbur and G. A. Swartzlander, “Complete transverse representation of a correlation singularity of a partially coherent field,” J. Opt. Soc. Am. B 25, 1422–1429 (2008).
    [CrossRef]
  42. G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
    [CrossRef]
  43. G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
    [CrossRef]
  44. 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]
  45. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
    [CrossRef]
  46. C. Hsiung, A First Course in Differential Geometry (International, 1997), p. 266.

2012 (2)

2011 (2)

2010 (2)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010).
[CrossRef]

2009 (3)

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26, 741–744 (2009).
[CrossRef]

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[CrossRef]

2008 (3)

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]

G. Gbur and G. A. Swartzlander, “Complete transverse representation of a correlation singularity of a partially coherent field,” J. Opt. Soc. Am. B 25, 1422–1429 (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]

2007 (1)

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

2006 (4)

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef]

A. Bezryadina, D. N. Neshev, A. S. Desyatnikov, J. Young, Z. Chen, and Y. S. Kivshar, “Observation of topological transformations of optical vortices in two-dimensional photonic lattices,” Opt. Express 14, 8317–8327 (2006).
[CrossRef]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

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

2004 (8)

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

G. A. Swartzlander and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
[CrossRef]

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B 6, S404–S409 (2004).
[CrossRef]

D. W. Diehl and T. D. Visser, “Phase singularities of the longitudinal field components in high-aperture systems,” J. Opt. Soc. Am. A 21, 2103–2108 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (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]

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[CrossRef]

2003 (4)

2001 (3)

I. Freund, “Critical foliations,” Opt. Lett. 26, 545–547 (2001).
[CrossRef]

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71–90 (2001).
[CrossRef]

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

2000 (1)

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000).
[CrossRef]

1998 (1)

1997 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

1967 (1)

1956 (1)

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

Alonso, M. A.

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Bezryadina, A.

Blok, H.

Bogatyryova, G. V.

Boivin, A.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

Chen, Z.

Desyatnikov, A. S.

Diehl, D. W.

Dow, J.

Duan, Z.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Fel’de, C. V.

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[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]

Freund, I.

I. Freund, “Critical foliations,” Opt. Lett. 26, 545–547 (2001).
[CrossRef]

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71–90 (2001).
[CrossRef]

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000).
[CrossRef]

Friberg, A. T.

Gbur, G.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

G. Gbur and G. A. Swartzlander, “Complete transverse representation of a correlation singularity of a partially coherent field,” J. Opt. Soc. Am. B 25, 1422–1429 (2008).
[CrossRef]

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

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371–380 (2003).
[CrossRef]

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

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

Gu, Y.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

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]

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Hassinen, T.

Hernandez-Aranda, R. I.

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

Hsiung, C.

C. Hsiung, A First Course in Differential Geometry (International, 1997), p. 266.

James, D. F. V.

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]

Karman, G. P.

Kessler, D. A.

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71–90 (2001).
[CrossRef]

Kivshar, Y. S.

Lenstra, D.

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B 6, S404–S409 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371–380 (2003).
[CrossRef]

Maleev, I. D.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Mandel, L.

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

Marasinghe, M. L.

Marathay, A. S.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Miyamoto, Y.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Neshev, D. N.

Nye, J. F.

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

Paganin, D. M.

Palacios, D. M.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

Polyanskii, P. V.

Ponomarenko, S. A.

Premaratne, M.

Raghunathan, S. B.

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Schmit, J.

G. A. Swartzlander and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef]

Schoonover, R. W.

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]

Schouten, H. F.

Setälä, T.

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).
[CrossRef]

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994), pp. 174–180.

Swartzlander, G. A.

G. Gbur and G. A. Swartzlander, “Complete transverse representation of a correlation singularity of a partially coherent field,” J. Opt. Soc. Am. B 25, 1422–1429 (2008).
[CrossRef]

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

G. A. Swartzlander and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. A 21, 1895–1900 (2004).
[CrossRef]

Takeda, M.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef]

Tervo, J.

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

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]

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[CrossRef]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26, 741–744 (2009).
[CrossRef]

van Duijl, A.

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

Visser, T. D.

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

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[CrossRef]

T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26, 741–744 (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]

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

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[CrossRef]

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B 6, S404–S409 (2004).
[CrossRef]

D. W. Diehl and T. D. Visser, “Phase singularities of the longitudinal field components in high-aperture systems,” J. Opt. Soc. Am. A 21, 2103–2108 (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, 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]

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

H. F. Schouten, G. Gbur, T. D. Visser, D. Lenstra, and H. Blok, “Creation and annihilation of phase singularities near a sub-wavelength slit,” Opt. Express 11, 371–380 (2003).
[CrossRef]

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

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).
[CrossRef]

Wang, W.

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

Woerdman, J. P.

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[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]

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. 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]

G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[CrossRef]

A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

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

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

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Young, J.

J. Opt. A (2)

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]

G. Gbur, T. D. Visser, and E. Wolf, “Hidden singularities in partially coherent wavefields,” J. Opt. A 6, S239–S242 (2004).
[CrossRef]

J. Opt. B (1)

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B 6, S404–S409 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

Nature (1)

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

Opt. Commun. (7)

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[CrossRef]

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

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000).
[CrossRef]

I. Freund and D. A. Kessler, “Critical point trajectory bundles in singular wave fields,” Opt. Commun. 187, 71–90 (2001).
[CrossRef]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[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 (5)

Opt. Lett. (6)

Phys. Rev. A (1)

T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79, 033805 (2009).
[CrossRef]

Phys. Rev. Lett. (7)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96, 073902 (2006).
[CrossRef]

G. A. Swartzlander and R. I. Hernandez-Aranda, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef]

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

W. Wang and M. Takeda, “Coherence current, coherence vortex, and the conservation law of coherence,” Phys. Rev. Lett. 96, 223904 (2006).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef]

G. A. Swartzlander and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef]

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[CrossRef]

Proc. R. Soc. A (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP Publishing, 1999).

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42, pp. 83–110.

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

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

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

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994), pp. 174–180.

C. Hsiung, A First Course in Differential Geometry (International, 1997), p. 266.

Supplementary Material (1)

» Media 1: AVI (1137 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1.
Fig. 1.

Illustrating the notation. A partially coherent, electromagnetic GSM beam propagates in the z-direction. The source plane is taken to be at z=0. The vector ρ=(x,y) indicates a transverse position.

Fig. 2.
Fig. 2.

Two surfaces for which the phase of η(ρ1,ρ2,z) equals π. In this case δyy=0.18mm. The other parameters are λ=632.8nm, δxx=0.2mm, σ=1mm, Ax=1, and Ay=3. The reference point ρ1=(2.5,0)mm (Media 1).

Fig. 3.
Fig. 3.

Two surfaces for which the phase of η(ρ1,ρ2,z) equals π. In this case δyy=0.14mm.

Fig. 4.
Fig. 4.

Closed string of coherence singularities (green curve).

Fig. 5.
Fig. 5.

Two surfaces for which the phase of η(ρ1,ρ2,z) equals π. In this case δyy=0.12mm. A closed string of coherence vortices (green curve) has come into existence. The right-hand phase sheet terminates on the string, creating a hole and a protrusion of the sheet.

Fig. 6.
Fig. 6.

Single surface for which the phase of η(ρ1,ρ2,z) equals π. In this case δyy=0.11mm. The string of coherence vortices (green curve) has expanded, causing the protrusion of Fig. 5 to grow. The two formerly disjointed phase sheets are now connected.

Fig. 7.
Fig. 7.

Two surfaces for which the phase of η(ρ1,ρ2,z) equals π. In this case δyy=0.06mm. The string of coherence vortices (green curve) has moved sideways and now only intersects the left-hand phase sheets. The two phase sheets are again disconnected.

Fig. 8.
Fig. 8.

Phase contours of η(ρ1,ρ2,z) in the plane z=0.1m. A minimum, a maximum, and two saddle points (intersections of the red curves) can be seen. In this and in the following examples we have taken ρ1=(2.5,0)mm, Ax=1, Ay=3, σ=1mm, δxx=0.2mm, δyy=0.12mm, and the wavelength λ=633nm.

Fig. 9.
Fig. 9.

Phase of η(ρ1,ρ2,z) at the minimum and at the phase saddle (visible on the left-hand side in Fig. 8), in various cross sections of the beam.

Fig. 10.
Fig. 10.

Position ρ2x of the minimum (blue curve) and that of the saddle (red curve) in various cross sections of the beam. Near z=2.06m the minimum and the saddle point annihilate each other.

Fig. 11.
Fig. 11.

Phase contours of η(ρ1,ρ2,z) in the plane z=1.12m. The right-hand side phase saddle of Fig. 8 has decayed into a minimum and two saddles (intersections of the two red curves).

Fig. 12.
Fig. 12.

Phase contours of η(ρ1,ρ2,z) in the plane z=1.1808m, containing a dipole and two saddle points (intersections of the red curves).

Fig. 13.
Fig. 13.

Phase contours of η(ρ1,ρ2,z) in the plane z=1.4m, containing two vortices (“coherence singularities”) and two saddle points (intersections of the red curves).

Fig. 14.
Fig. 14.

Phase contours of η(ρ1,ρ2,z) in the plane z=2.4m, containing a maximum, a minimum, and two saddle points (intersections of the green curves).

Fig. 15.
Fig. 15.

Phase contours of η(ρ1,ρ2,z) in the plane z=3.0m, containing a maximum and a saddle (intersection of the red curve).

Fig. 16.
Fig. 16.

Phase contours of η(ρ1,ρ2,z) in the plane z=3.33m, right after the final topological reaction. There are no more singularities or stationary points.

Tables (1)

Tables Icon

Table 1. Topological Charge and Index of Singular and Stationary Points

Equations (12)

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

W(r1,r2,ω)=(Wxx(r1,r2,ω)Wxy(r1,r2,ω)Wyx(r1,r2,ω)Wyy(r1,r2,ω)),
Wij(r1,r2,ω)=Ei*(r1,ω)Ej(r2,ω),(i,j=x,y).
η(r1,r2,ω)=TrW(r1,r2,ω)[TrW(r1,r1,ω)TrW(r2,r2,ω)]1/2,
|Wxx(ρ1,ρ2,z)|=|Wyy(ρ1,ρ2,z)|,
arg[Wxx(ρ1,ρ2,z)]arg[Wyy(ρ1,ρ2,z)]=π(mod2π),
Wij(ρ1,ρ2,z=0)=Si(ρ1)Sj(ρ2)μij(ρ2ρ1),(i,j=x,y)
Si(ρ)=Ai2exp(ρ2/2σi2),
μij(ρ2ρ1)=Bijexp[(ρ2ρ1)2/2δij2].
Wij(ρ1,ρ2,z)=AiAjBijΔij2(z)exp[(ρ1+ρ2)28σ2Δij2(z)]×exp[(ρ2ρ1)22Ωij2Δij2(z)]exp[ik(ρ22ρ12)2Rij(z)],
Δij2(z)=1+(z/kσΩij)2,
1Ωij2=14σ2+1δij2,
Rij(z)=[1+(kσΩij/z)2]z

Metrics