Abstract

The intensity of a random optical field consists of bright speckle spots (maxima) separated from dark areas (minima and optical vortices) by saddle points. We show that hidden in this complicated landscape are umbilic points—singular points at which the eigenvalues Λ± of the Hessian matrix that measure the curvature of the landscape become degenerate. Although not observed previously in random optical fields, umbilic points are the most numerous of all special points, outnumbering maxima, minima, saddle points, and vortices. We show experimentally that the directions of principal curvature, the eigenvectors Ψ±, rotate about intensity umbilic points with positive or negative half-integer winding number, in accord with theory, and that Λ+ and Λ generate a double cone known as a diabolo. At optical vortices the curvature of the amplitude is singular, and we show from both theory and experiment that for this landscape Ψ± rotate about vortex centers with a positive integer winding number. Diabolos can be classified as elliptic or hyperbolic, and we present initial results for the measured fractions of these two different types of umbilic diabolos.

© 2007 Optical Society of America

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  1. J. W. Goodman, Statistical Optics (Wiley, 1985).
  2. M. V. Berry, Adv. Phys. 25, 1 (1976). Umbilic points on the geometrical optics wavefront that generate far-field caustics are discussed there.
    [Crossref]
  3. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
    [Crossref]
  4. A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
    [Crossref]
  5. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999). Umbilic points on the geometrical optics wavefront that generate near field caustics are discussed there.
  6. M. V. Berry and M. Wilkinson, Proc. R. Soc. London, Ser. A 392, 15 (1984).
    [Crossref]
  7. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, Proc. R. Soc. London, Ser. A 462, 1629 (2006).
    [Crossref]
  8. M. V. Berry and M. R. Jeffrey, J. Opt. A, Pure Appl. Opt. 8, 363 (2006).
    [Crossref]
  9. R. I. Egorov, M. S. Soskin, and I. Freund, Opt. Lett. 31, 2048 (2006).
    [Crossref] [PubMed]
  10. I. Freund, M. S. Soskin, R. I. Egorov, and V. Denisenko, Opt. Lett. 31, 2381 (2006).
    [Crossref] [PubMed]
  11. I. Freund, "Optical diabolos: configurations, nucleations, transformations, and reactions," Opt. Commun. (to be published).
  12. A. L. Sobolewski and W. Domcke, Europhys. News 37, 20 (2006).
    [Crossref]
  13. I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 036227 (1996).
    [Crossref]
  14. L. F. Canto, P. Ring, Y. Sun, J. O. Rasmussen, S. Y. Chu, and M. A. Stoyer, Phys. Rev. C 47, 2836 (1993).
    [Crossref]
  15. H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (1983).
    [Crossref]
  16. I. Freund, Opt. Lett. 26, 1996 (2001).
    [Crossref]
  17. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [Crossref] [PubMed]
  18. I. Freund, Waves Random Media 8, 119 (1998).
  19. I. Freund, Phys. Rev. E 52, 2348 (1995).
    [Crossref]
  20. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [Crossref]

2006 (5)

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

M. V. Berry and M. R. Jeffrey, J. Opt. A, Pure Appl. Opt. 8, 363 (2006).
[Crossref]

A. L. Sobolewski and W. Domcke, Europhys. News 37, 20 (2006).
[Crossref]

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

I. Freund, M. S. Soskin, R. I. Egorov, and V. Denisenko, Opt. Lett. 31, 2381 (2006).
[Crossref] [PubMed]

2002 (1)

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

2001 (1)

1998 (1)

I. Freund, Waves Random Media 8, 119 (1998).

1996 (1)

I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 036227 (1996).
[Crossref]

1995 (1)

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

1994 (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[Crossref] [PubMed]

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).
[Crossref]

1984 (1)

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

1983 (1)

H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (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]

1976 (1)

M. V. Berry, Adv. Phys. 25, 1 (1976). Umbilic points on the geometrical optics wavefront that generate far-field caustics are discussed there.
[Crossref]

Berry, M. V.

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

M. V. Berry and M. R. Jeffrey, J. Opt. A, Pure Appl. Opt. 8, 363 (2006).
[Crossref]

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

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

M. V. Berry, Adv. Phys. 25, 1 (1976). Umbilic points on the geometrical optics wavefront that generate far-field caustics are discussed there.
[Crossref]

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).
[Crossref]

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).
[Crossref]

Cooley, C. R.

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

Denisenko, V.

Dennis, M. R.

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

Domcke, W.

A. L. Sobolewski and W. Domcke, Europhys. News 37, 20 (2006).
[Crossref]

Egorov, R. I.

Freund, I.

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

I. Freund, M. S. Soskin, R. I. Egorov, and V. Denisenko, Opt. Lett. 31, 2381 (2006).
[Crossref] [PubMed]

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

I. Freund, Waves Random Media 8, 119 (1998).

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

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[Crossref] [PubMed]

I. Freund, "Optical diabolos: configurations, nucleations, transformations, and reactions," Opt. Commun. (to be published).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Hannay, J. H.

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

Hutchinson, H. J.

H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (1983).
[Crossref]

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, J. Opt. A, Pure Appl. Opt. 8, 363 (2006).
[Crossref]

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

Lunney, J. G.

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

Nye, J. F.

H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (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 (IOP, 1999). Umbilic points on the geometrical optics wavefront that generate near field caustics are discussed there.

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).
[Crossref]

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).
[Crossref]

Rotter, I.

I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 036227 (1996).
[Crossref]

Sadreev, A. F.

I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 036227 (1996).
[Crossref]

Salmon, P. S.

H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (1983).
[Crossref]

Shvartsman, N.

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[Crossref] [PubMed]

Sobolewski, A. L.

A. L. Sobolewski and W. Domcke, Europhys. News 37, 20 (2006).
[Crossref]

Soskin, M. S.

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).
[Crossref]

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).
[Crossref]

Thorndike, A. S.

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

Wilkinson, M.

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

Adv. Phys. (1)

M. V. Berry, Adv. Phys. 25, 1 (1976). Umbilic points on the geometrical optics wavefront that generate far-field caustics are discussed there.
[Crossref]

Europhys. News (1)

A. L. Sobolewski and W. Domcke, Europhys. News 37, 20 (2006).
[Crossref]

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

M. V. Berry and M. R. Jeffrey, J. Opt. A, Pure Appl. Opt. 8, 363 (2006).
[Crossref]

J. Phys. A (2)

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

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

J. Struct. Mech. (1)

H. J. Hutchinson, J. F. Nye, and P. S. Salmon, J. Struct. Mech. 11, 371 (1983).
[Crossref]

Opt. Commun. (2)

I. Freund, "Optical diabolos: configurations, nucleations, transformations, and reactions," Opt. Commun. (to be published).

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

Opt. Lett. (3)

Phys. Rev. A (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[Crossref] [PubMed]

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).
[Crossref]

Phys. Rev. E (2)

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

I. Rotter and A. F. Sadreev, Phys. Rev. E 71, 036227 (1996).
[Crossref]

Proc. R. Soc. London, Ser. A (2)

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

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

Waves Random Media (1)

I. Freund, Waves Random Media 8, 119 (1998).

Other (2)

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999). Umbilic points on the geometrical optics wavefront that generate near field caustics are discussed there.

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Figures (3)

Fig. 1
Fig. 1

Experimental intensity umbilic diabolos in a random speckle pattern. Shown are silhouettes of (a) an elliptic and (b) a hyperbolic diabolo. The upper (lower) cone corresponds to major curvature Λ + (minor curvature Λ ). The corresponding gray-scale-coded contour maps show ( a ) the elliptic contours of the lower cone in (a) and ( b ) the hyperbolic contours of the lower cone in (b). Contour maps of the upper cones are similar and are not shown. The apex of an elliptic (hyperbolic) cone is shown by a circle (triangle), and contours are shown by the thin curves. The special V shaped contour that passes through the apex of the hyperbolic cone in ( b ) , shown by the thick curve, is an h-line.

Fig. 2
Fig. 2

Gray-scale-coded contour maps of experimental intensity umbilic points in a segment of a random field. (a) Φ 12 . Zero lines Z 1 ( Z 2 ) are shown by thick black (thick dotted) curves, positive (negative) Stokes vortices by white (black) filled circles, and saddle points by small white squares. As can be seen, adjacent vortices on Z 1 and Z 2 have opposite signs, in accord with the vortex sign rules.[18, 19] The vortices also obey the extended sign rule that requires vortices terminating adjacent bifurcation lines (the four special lines emanating from a saddle point) to have opposite signs. (b) Λ corresponding to (a). Elliptic (hyperbolic) cones are shown by circles (triangles), saddle points by small squares, their bifurcation lines by black curves of intermediate thickness, and h-loops by the thickest black curves. One complete H loop and one complete H loop are visible in the figure center. As can be seen, in accord with the loop rules[10, 11] the H-loop contains an (up-pointing) E cone, whereas the H loop contains three down-pointing minima located within loops (bifurcation loops) formed by the bifurcation lines of saddle points.

Fig. 3
Fig. 3

Experimental amplitude ( A ) umbilics. (a) Vortex amplitude cone. (b) Gray-scale-coded contour map of Stokes phase Φ 12 ( A ) . Two zero lines Z 2 ( A ) , thick dotted curves, that parallel the x y -axes, and two Z 1 ( A ) , thick black curves, oriented at ± 45 ° , pass through a vortex, shown by a double white circle that indicates its Stokes charge of + 2 . Also shown are three ordinary hyperbolic umbilics, one with Stokes charge + 1 (white circle), and two with Stokes charge 1 (black circles). As expected, all Stokes amplitude vorti ces obey the sign rules.[17, 18, 19]

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