Abstract

The classical singularities of elliptically polarized light are points of circular (linear) polarization, characterized by a half-integer (integer) topological index. On average, in any plane of a random ellipse field there is of the order of one each of these classical singularities per coherence area. It is shown that every ellipse in such a field is a multiple singularity characterized by nine different topological indices: Three indices characterize rotations of the principal axis system of the surrounding ellipses, and six indices characterize a one- or two-turn spiral precession of these axes. The nine indices can divide the field into 32, 768 different volumes with different structures separated by singular surfaces (grain boundaries) on which an index becomes undefined. This unprecedented proliferation of singularities and structures can occur in other three-dimensional systems in which individual elements are described by unique principal axis systems, for example, liquid crystals, and should be sought in such systems.

© 2005 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).
  2. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999).
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    [CrossRef]
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  5. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
    [CrossRef]
  6. M. V. Berry, J. Opt. A 6, 675 (2004).
    [CrossRef]
  7. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
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  8. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  11. M. S. Soskin, V. Denisenko, and I. Freund, Opt. Lett. 28, 1475 (2003).
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  12. M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
    [CrossRef]
  13. R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
    [CrossRef]

2004

I. Freund, J. Opt. A 6, S229 (2004).
[CrossRef]

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

I. Freund, Opt. Lett. 29, 875 (2004).
[CrossRef] [PubMed]

M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
[CrossRef]

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

2003

2001

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

1987

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Boaz, M. F.

M. F. Boaz, Mathematical Methods in the Physical Sciences (Wiley, New York, 1966), pp. 256–257.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).

Dandliker, R.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

Denisenko, V.

M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, Opt. Lett. 28, 1475 (2003).
[CrossRef] [PubMed]

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Egorov, R.

M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
[CrossRef]

Freund, I.

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

Konukhov, A. I.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Marki, I.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

Melnikov, L. A.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Nesci, A.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

Nye, J. F.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

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

Salt, M.

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

Soskin, M. S.

M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, Opt. Lett. 28, 1475 (2003).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).

J. Opt. A

I. Freund, J. Opt. A 6, S229 (2004).
[CrossRef]

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

R. Dandliker, I. Marki, M. Salt, and A. Nesci, J. Opt. A 6, S189 (2004).
[CrossRef]

J. Opt. A.

M. S. Soskin, V. Denisenko, and R. Egorov, J. Opt. A. 6, S281 (2004).
[CrossRef]

J. Opt. B

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Opt. Lett.

Proc. R. Soc. London Ser. A

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 457, 141 (2001).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).

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

M. F. Boaz, Mathematical Methods in the Physical Sciences (Wiley, New York, 1966), pp. 256–257.

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Figures (6)

Fig. 1
Fig. 1

αα and ββ singularties. (a) [(b)] 3D view of the major (minor) axis α β of a representative ellipse (central figure) and the major (minor) axes α β of the surrounding ellipses. a b A singularity appears in the projection of α β onto a plane perpendicular to the axis of the central ellipse. By following the rotation of the lines about the center, the winding number of the αα ββ singularity in a b can be seen to be Iαα=-1Iββ=+1. The corresponding γγ singularity is not shown but is qualitatively similar to a,a and has index Iγγ=-1.

Fig. 2
Fig. 2

Stokes phase Φ12 coded -π to +π black to white for the αα,ββ, and γγ singularities of the representative ellipse shown in Fig. 1. (a) αα singularity corresponding to Figs. 1(a) and 1(a′). (b) ββ singularity corresponding to Figs. 1(b) and 1(b′). (c) γγ singularity (not shown in Fig. 1). In all cases the winding number Σ of the Stokes vortex is twice that of the corresponding ellipse singularity, as expected. Here, Σαα=2Iαα=-2, Σββ=2Iββ=+2, and Σγγ=2Iγγ=-2.

Fig. 3
Fig. 3

Distribution of topological indices (a) Iαα, (b) Iββ, and (c) Iγγ, in the plane Z=0 of the random ellipse field described in the text. Shown is a square containing 2.5 coherence areas. Regions of positive (negative) index are colored white (gray). Points at which C lines (L lines) pierce the plane are shown by black circles with white rims (white circles with black rims). By examining the index in corresponding grid squares, one can see that all eight possible sign combinations for the three indices Iαα, Iββ, and Iγγ are present.

Fig. 4
Fig. 4

One-turn αγ spiral surrounding the central ellipse in Fig. 1. (a) 3D view. a The Möbius strip in (a) viewed from above, or cut in two along its midline, yields two interlocking rings. ταγ=-1 for this right-handed spiral.

Fig. 5
Fig. 5

The αγ spiral shown in Fig. 4 is wrapped around the circle that contains the centers of the ellipses that surround the central figure. Here the cumulative winding number ταγΘ is shown as a function of position on the circle parameterized by angle Θ=0-2π. The inset shows the spiral when the circle is opened to a straight line.

Fig. 6
Fig. 6

Cumulative winding number ταβΘ of a two-turn αβ spiral. As in Fig. 3, the spiral is wrapped around a circle; the inset shows the spiral when the circle is opened to a straight line. The winding number of this spiral is ταβ=-2.

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