Abstract

A point of circular polarization embedded in a paraxial field of elliptical polarization is a polarization singularity called a C point. At such a point the major axis a and minor axis b of the ellipse become degenerate. Away from the C point this degeneracy is lifted such that surfaces a and b form nonanalytic cones that are joined at their apex (the C point) to produce a double cone called a diabolo. Typically, during propagation diabolo pairs are created or annihilated. We present rules based on geometry and topology that govern these events, provide initial experimental confirmation, and enumerate the allowed configurations in which diabolos can be created or annihilated.

© 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 and J. V. Hajnal, Proc. R. Soc. London, Ser. A 409, 21 (1987).
  3. M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 457, 141 (2001).
  4. M. R. Dennis, Opt. Commun. 213, 201 (2002).
  5. I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
  6. M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 459, 1261 (2003).
  7. I. Freund, Opt. Lett. 29, 875 (2004).
    [PubMed]
  8. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).
  9. R. I. Egorov, V. G. Denisenko, and M. S. Soskin, JETP Lett. 81, 375 (2005).
  10. D. W. Diehl, R. W. Schooner, and T. D. Visser, Opt. Express 14, 3030 (2006).
    [PubMed]
  11. M. V. Berry, Proc. R. Soc. London, Ser. A 461, 2071 (2005).
  12. R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
    [PubMed]
  13. M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).
  14. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
  15. F. Faure and B. Zhillinski, Phys. Rev. Lett. 85, 960 (2000).
    [PubMed]
  16. L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).
  17. A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
  18. M. V. Berry, M. J. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).
  19. I. Freund, Phys. Rev. E 52, 2348 (1995).
  20. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wessley, 1994), Chap. 6, pp. 174-180.

2006 (3)

2005 (2)

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

M. V. Berry, Proc. R. Soc. London, Ser. A 461, 2071 (2005).

2004 (2)

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).

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

2003 (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 459, 1261 (2003).

2002 (2)

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

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

2001 (1)

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

2000 (1)

F. Faure and B. Zhillinski, Phys. Rev. Lett. 85, 960 (2000).
[PubMed]

1995 (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).

1993 (1)

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

1987 (1)

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

1984 (1)

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).

1978 (1)

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

1977 (1)

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

Berry, M. V.

M. V. Berry, M. J. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).

M. V. Berry, Proc. R. Soc. London, Ser. A 461, 2071 (2005).

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 459, 1261 (2003).

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

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).

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

Canto, L. F.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Chu, S. Y.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Cooley, C. R.

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

Denisenko, V. G.

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

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 459, 1261 (2003).

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

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

Diehl, D. W.

Egorov, R. I.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[PubMed]

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

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).

Faure, F.

F. Faure and B. Zhillinski, Phys. Rev. Lett. 85, 960 (2000).
[PubMed]

Freund, I.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[PubMed]

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

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

I. Freund, Phys. Rev. E 52, 2348 (1995).

Hajnal, J. V.

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

Hannay, J. H.

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

Jeffrey, M. J.

M. V. Berry, M. J. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).

Lunney, J. G.

M. V. Berry, M. J. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).

Mokhun, A. I.

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

Nye, J. F.

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

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

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

Rasmussen, J. O.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Ring, P.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Schooner, R. W.

Soskin, M. S.

R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
[PubMed]

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

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

Stoyer, M. A.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wessley, 1994), Chap. 6, pp. 174-180.

Sun, Y.

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Thorndike, A. S.

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

Visser, T. D.

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).

Wilkinson, M.

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).

Zhillinski, B.

F. Faure and B. Zhillinski, Phys. Rev. Lett. 85, 960 (2000).
[PubMed]

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

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, J. Opt. A, Pure Appl. Opt. 6, S217 (2004).

J. Phys. A (2)

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

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

JETP Lett. (1)

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

Opt. Commun. (2)

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

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. C (1)

L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).

Phys. Rev. E (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).

Phys. Rev. Lett. (1)

F. Faure and B. Zhillinski, Phys. Rev. Lett. 85, 960 (2000).
[PubMed]

Proc. R. Soc. London, Ser. A (6)

M. V. Berry, M. J. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).

M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 459, 1261 (2003).

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

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

M. V. Berry, Proc. R. Soc. London, Ser. A 461, 2071 (2005).

M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).

Other (2)

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wessley, 1994), Chap. 6, pp. 174-180.

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

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Figures (4)

Fig. 1
Fig. 1

Diabolos. (a) The upper (lower) surface is major axis a (minor axis b); these surfaces touch at the C points, here diabolic points. The a cone (b cone) of the left diabolo tilts below (above) the horizontal x y plane that contains the surrounding ellipses, and the diabolo is hyperbolic, whereas the right diabolo is elliptic. (b), (c) Contours and topological indices. Horizontal level planes intersect the a cone to form contours that are the classic conic sections: (b) hyperbolas, (c) ellipses. b cone contours (not shown) are similar. The index I D of a diabolic point (filled circle) is the net signed rotation angle Θ of the tangent to the contours along a closed path that encircles the point [dotted ellipse in (b), any ellipse in (c)]. (b) Hyperbolic diabolo. The two special contours (thick lines) passing through the diabolic point are h lines. For hyperbolic diabolos Θ = 0 , and therefore also I D = 0 . (c) Elliptic diabolo. Here Θ = + 2 π , and I D = + 1 .

Fig. 2
Fig. 2

Loop rules. M, maximum; m, minimum. (a)–(d) Loop rules for saddles.[19] The four special contours that pass through the saddle point (open squares) are bifurcation lines (thick solid curves). Generically, these lines close to form bifurcation loops (dotted curves) that contain either a maximum or a minimum. (a), (b) Figure-eight saddles, S; both loops can contain either (a) maxima, or (b) minima. (c), (d) Re-entrant saddles, S . If (c) the inner loop contains a maximum, the outer must contain a minimum, and (d) vice versa. (e)–(h) Loop rules for diabolos. Shown is major axis a. (e), (f) Elliptic diabolo, E. (e) Cross section of cone. a increases in all directions away from the cone apex (diabolic point). (f) Contours. The diabolic point is a minimum. (g), (h) Hyperbolic diabolo. (g) Cross section of cone. a decreases (increases) to the right (left). (h) h lines. Generically, h lines, Fig. 1b, close to form an h-line loop (dotted lines). If closure is to the right (left), where a decreases (increases), this loop encloses a minimum (maximum), and for a given surface, a or b, we label the diabolic point and its loop H ( H ) . For minor axis b the cones in (e) and (g) point up, and in (f) and (h) minima become maxima and maxima become minima. The loops of a and b for a given hyperbolic diabolo need not be of the same type, so there are four geometrically distinct hyperbolics.

Fig. 3
Fig. 3

Experimentally measured diabolo configurations in a random speckle pattern. Shown are contour maps of major axis a, increasing gray to white. Elliptic (hyperbolic) diabolic points are shown by filled circles (diamonds), saddle points by open squares. Bifurcation loops of saddles and h-line loops of hyperbolic diabolos are shown by thick closed curves, ordinary contours by thin curves. C points and their diabolos are located at intersections of the zero lines of Stokes parameters S 1 and S 2 (Ref. [5]); these zero lines are shown by intermediate-thickness solid and dotted curves, respectively. Diabolo configurations, Table 1: (a) No. 1, [E]–S–[E]. (b) No. 9, H–[E]. (c) No. 10, H–[H(m)]. (d) No. 3, [E]–S–[H(m)].

Fig. 4
Fig. 4

Diabolo collisions and annihilations. All symbols are as in Fig. 3. (a)–(d) Calculated diabolos in a Gaussian laser beam collide and annihilate during propagation. Shown is major axis a. (a) Elliptic diabolos inside the loops of a figure-eight saddle, [E]–S–[E], collide with and annihilate the saddle and each other, leaving (b) a minimum, m; Table 1, No. 1. (c) Nested diabolos, H [ H ( M ) ] , collide and annihilate, leaving (d) a maximum, M, Table 1, No. 11. (e)–(g) Experimentally measured diabolos in a random field under calculated controlled perturbation.[9] Shown is minor axis b. (e) An elliptic diabolo inside the loop of a hyperbolic, H–[E], transforms into (f) another, inner, hyperbolic, generating the configuration H–[H(M)]. The inner hyperbolic then collides with and annihilates the outer hyperbolic, leaving (g) a maximum, M; Table 1, No. 10, with m replaced with M for axis b.

Tables (1)

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Table 1 Diabolo Annihilations (Major Axis a a )

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