Abstract

Following our earlier work [F. Flossmann et al., Phys. Rev. Lett. 95 253901 (2005)], we describe the fine polarization structure of a beam containing optical vortices propagating through a birefringent crystal, both experimentally and theoretically.We emphasize here the zero surfaces of the Stokes parameters in three-dimensional space, two transverse dimensions and the third corresponding to optical path length in the crystal. We find that the complicated network of polarization singularities reported earlier –lines of circular polarization (C lines) and surfaces of linear polarization (L surfaces) – can be understood naturally in terms of the zeros of the Stokes parameters.

© 2006 Optical Society of America

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  1. J. F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations, (IoP Publishing, 1999).
  2. M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).
  3. J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).
  4. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
    [CrossRef]
  5. M. V. Berry, "Singularities in waves and rays," in Les Houches, Session XXV - Physics of Defects, R. Balian, M. Kléman, and J.-P. Poirier, eds., (North-Holland, 1981).
  6. K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 143039-3044 (2006).
    [CrossRef]
  7. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).
  8. L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
    [CrossRef]
  9. F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
    [CrossRef]
  10. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
    [CrossRef]
  11. J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).
  12. A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express 144208-4220 (2006).
    [CrossRef]
  13. M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).
  14. M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.
  15. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
    [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. 95253901 (2005).
    [CrossRef]
  17. B. Riemann, in Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber (Teubner, Leipzig, 1892) p. 301.
  18. R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).
  19. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy (Academic Press, San Diego, 1990).
  20. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27545-547 (2002).
  21. D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).
  22. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, "Generation of single-charge optical vortices with an uniaxial crystal" Opt. Express 143724-3729 (2006).
  23. M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
    [CrossRef]
  24. F. Flossmann, Singularit¨aten von Phase und Polarisation des Lichts (PhD Thesis, University of Regensburg, 2006).

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. 95253901 (2005).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

2004 (1)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

2003 (3)

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef]

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

2002 (2)

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27545-547 (2002).

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).

2000 (1)

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).

1992 (1)

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

1990 (1)

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).

1983 (1)

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).

Allen, L.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

Angelsky, O. V.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Biener, G.

Dennis, M. R.

K. O’Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 143039-3044 (2006).
[CrossRef]

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. 95253901 (2005).
[CrossRef]

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).

Desyatnikov, A. S.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef]

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

Fadeyeva, T.

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

Flossmann, F.

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. 95253901 (2005).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

Freund, I.

Hajnal, J. V.

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).

Hasman, E.

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Kivshar, Y. S.

Kleiner, V.

Krolikowski, W.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

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. 95253901 (2005).
[CrossRef]

Mokhun, A. I.

Mokhun, I. I.

Neshev, D. N.

Niv, A.

Nye, J.F.

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).

O’Holleran, K.

Padgett, M. J.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

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. 95253901 (2005).
[CrossRef]

Shvedov, V.

Soskin, M. S.

Spreeuw, R. J. C.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

Vasnetsov, M.

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).

Volyar, A.

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

Woerdman, J. P.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "Fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6S217-S228 (2004).
[CrossRef]

New J. Phys. (1)

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5134 (2003).
[CrossRef]

Opt. Commun. (2)

F. Flossmann, U. T. Schwarz, and M. Maier "Propagation dynamics of optical vortices in Laguerre-Gaussian beams," Opt. Commun. 250218-230 (2005).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213201-221 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A 458185-8189 (1992).
[CrossRef]

Phys. Rev. Lett. (2)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91233901 (2003).
[CrossRef]

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. 95253901 (2005).
[CrossRef]

Proc. R. Soc. Lond. A (4)

J. V. Hajnal, "Observation on singularities in the electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. Lond. A 430413-421 (1990).

M. V. Berry and M. R. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 4591261-1292 (2003).

J.F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. Lond. A 389279-290 (1983).

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A 4562059-2079 (2000).

Prog. Opt. (2)

M. S. Soskin and M. Vasnetsov, "Singular optics," Prog. Opt. 42219-276 (2001).

M. V. Berry and M. R. Jeffrey, "Conical diffraction: Hamilton’s diabolic point at the heart of crystal optics," Prog. Opt.in press.

Other (7)

B. Riemann, in Bernhard Riemann’s gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber (Teubner, Leipzig, 1892) p. 301.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).

D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized light in optics and spectroscopy (Academic Press, San Diego, 1990).

J. F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations, (IoP Publishing, 1999).

M. V. Berry, "Singularities in waves and rays," in Les Houches, Session XXV - Physics of Defects, R. Balian, M. Kléman, and J.-P. Poirier, eds., (North-Holland, 1981).

D. J. Struik, Lectures on Classical Differential Geometry (Dover, 1988).

F. Flossmann, Singularit¨aten von Phase und Polarisation des Lichts (PhD Thesis, University of Regensburg, 2006).

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