Abstract

The canonical point singularity of elliptically polarized light is an isolated point of circular polarization, a C point. As one recedes from such a point the surrounding polarization figures evolve into ellipses characterized by a major axis of length a, a minor axis of length b, and an azimuthal orientational angle α: at the C point itself, α is singular (undefined) and a and b are degenerate. The profound effects of the singularity in α on the orientation of the ellipses surrounding the C point have been extensively studied both theoretically and experimentally for over two decades. The equally profound effects of the degeneracy of a and b on the evolving shapes of the surrounding ellipses have only been described theoretically. As one recedes from a C point, a and b generate a surface that locally takes the form of a double cone (i.e., a diabolo). Contour lines of constant a and b are the classic conic sections, ellipses or hyperbolas depending on the shape of the diabolo and its orientation relative to the direction of propagation. We present measured contour maps, surfaces, cones, and diabolos of a and b for a random ellipse field (speckle pattern).

© 2006 Optical Society of America

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  1. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).
  2. J. F. Nye, Proc. R. Soc. London, Ser. A 389, 279 (1983).
    [CrossRef]
  3. J. F. Nye and J. V. Hajnal, Proc. R. Soc. London, Ser. A 409, 21 (1987).
    [CrossRef]
  4. M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 457, 141 (2001).
    [CrossRef]
  5. I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  6. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  7. V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
    [CrossRef]
  8. I. Freund, Opt. Lett. 29, 875 (2004).
    [CrossRef] [PubMed]
  9. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
    [CrossRef]
  10. A. Niv, G. Biener, V. Kleiner, and E. Hasman, Opt. Lett. 30, 2933 (2005).
    [CrossRef] [PubMed]
  11. I. Freund, Opt. Lett. 29, 875 (2004).
    [CrossRef] [PubMed]
  12. R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
    [CrossRef]
  13. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
    [CrossRef]
  14. M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984). Many important properties of the double cone generated by the splitting of degenerate eigenvalues are discussed theoretically, and the term "diabolo" to describe this unusual structure is introduced.
    [CrossRef]
  15. A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
    [CrossRef]
  16. M. Berry, "Hamilton's diabolic point," colloquium (Bar-Ilan University, 2005).
  17. M. Born and E. W. Wolf, Principles of Optics (Pergamon, 1959). a and b can also be calculated theoretically from the eigenvalues of the coherency matrix or of one of its variations, such as the {pq} matrices in Refs. .

2005 (2)

R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
[CrossRef]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, Opt. Lett. 30, 2933 (2005).
[CrossRef] [PubMed]

2004 (4)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

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

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

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

2002 (2)

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

2001 (1)

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

1987 (1)

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

1984 (1)

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984). Many important properties of the double cone generated by the splitting of degenerate eigenvalues are discussed theoretically, and the term "diabolo" to describe this unusual structure is introduced.
[CrossRef]

1983 (1)

J. F. Nye, Proc. R. Soc. London, Ser. A 389, 279 (1983).
[CrossRef]

1978 (1)

A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
[CrossRef]

1977 (1)

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

Berry, M.

M. Berry, "Hamilton's diabolic point," colloquium (Bar-Ilan University, 2005).

Berry, M. V.

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

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984). Many important properties of the double cone generated by the splitting of degenerate eigenvalues are discussed theoretically, and the term "diabolo" to describe this unusual structure is introduced.
[CrossRef]

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

Biener, G.

Born, M.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, 1959). a and b can also be calculated theoretically from the eigenvalues of the coherency matrix or of one of its variations, such as the {pq} matrices in Refs. .

Cooley, C. R.

A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
[CrossRef]

Denisenko, V. G.

R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

Dennis, M. R.

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

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

Egorov, R. I.

R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

Freund, I.

Hajnal, J. V.

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

Hannay, J. H.

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

Hasman, E.

Kleiner, V.

Mokhun, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Niv, A.

Nye, J. F.

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

J. F. Nye, Proc. R. Soc. London, Ser. A 389, 279 (1983).
[CrossRef]

A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
[CrossRef]

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

Soskin, M. S.

R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
[CrossRef]

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Thorndike, A. S.

A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
[CrossRef]

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

Wilkinson, M.

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984). Many important properties of the double cone generated by the splitting of degenerate eigenvalues are discussed theoretically, and the term "diabolo" to describe this unusual structure is introduced.
[CrossRef]

Wolf, E. W.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, 1959). a and b can also be calculated theoretically from the eigenvalues of the coherency matrix or of one of its variations, such as the {pq} matrices in Refs. .

J. Opt. A (1)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A 6, S217 (2004).
[CrossRef]

J. Phys. A (2)

A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
[CrossRef]

M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
[CrossRef]

JETP Lett. (2)

V. G. Denisenko, R. I. Egorov, and M. S. Soskin, JETP Lett. 80, 17 (2004).
[CrossRef]

R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
[CrossRef]

Opt. Commun. (2)

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

Opt. Lett. (3)

Proc. R. Soc. London, Ser. A (4)

J. F. Nye, Proc. R. Soc. London, Ser. A 389, 279 (1983).
[CrossRef]

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]

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984). Many important properties of the double cone generated by the splitting of degenerate eigenvalues are discussed theoretically, and the term "diabolo" to describe this unusual structure is introduced.
[CrossRef]

Other (3)

M. Berry, "Hamilton's diabolic point," colloquium (Bar-Ilan University, 2005).

M. Born and E. W. Wolf, Principles of Optics (Pergamon, 1959). a and b can also be calculated theoretically from the eigenvalues of the coherency matrix or of one of its variations, such as the {pq} matrices in Refs. .

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

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Figures (4)

Fig. 1
Fig. 1

(a) Calculated diabolo. The upper (lower) cone is major axis a (minor axis b). The cones, which are highly anisotropic, touch at their apex where the C point is located. (b) Level planes intersect the diabolo to form contours that are conic sections. (c) Elliptic contours, (d) hyperbolic contours. Cones a and b always produce the same type of contour, i.e., either elliptic or hyperbolic. The diabolo in (a) produces elliptic contours.

Fig. 2
Fig. 2

Experimental results for minor axis b in a random field. Equivalent results (not shown) are obtained for major axis a. (a) Gray-scale coded contour map. b increases from dark to light gray. Elliptic C E points are shown by white circles, hyperbolic C H points by black circles. (b) Map of discriminant D C . Positive (negative) regions are colored light (dark) gray. In (a) and (b) thick white (thick black) curves are Z 1 ( Z 2 ) . C points are located at the intersections of these curves.

Fig. 3
Fig. 3

(a) Close-up of a measured contour map of minor axis b in a random field; b increases from black to white. Shown are an elliptic C point (open circle) and a hyperbolic C point (filled circle). The contours in the immediate vicinity of each C point may be compared with the theoretical contours in Figs. 1c, 1d. The thick white curve is a line of linear polarization, an L line,[1] which separates regions of opposite handedness (right/left). This, and all other L lines, lies on zero lines of the discriminant D C . (b) Measured elliptic optical diabolo. The upper surface corresponds to major axis a, the lower to minor axis b. The large anisotropy of the cones [Fig. 1a] is readily apparent.

Fig. 4
Fig. 4

Experimental three-dimentional surfaces of two different areas of a random field; (a) major axis a, (b) minor axis b. Cones of elliptic (hyperbolic) C points are enclosed in circles (squares). On average, we find that in these maps hyperbolic points outnumber elliptic points by 2 : 1 . But hyperbolic points are often hidden, because they lie on slopes on the backsides of the peaks shown. Because the cones of the major axis in (a) are all deep pockets, this surface is plotted inside out and upside down.

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