Abstract

Umbilic points—singular points of curvature characterized by a fractional topological charge q=±12—are the most numerous of all special points in the landscape of random optical fields (speckle patterns), outnumbering maxima, minima, saddle points, and optical vortices. To the best of our knowledge, we present the first experimental evidence that positive and negative umbilic points screen one another. Theory predicts that in the absence of screening the charge variance in a bounded region is proportional to the area of the region, whereas in the presence of screening the variance is drastically reduced and is proportional to the perimeter. Our data confirm this latter prediction and provide the first estimates of the screening lengths for umbilic points of the intensity and of the amplitude (field modulus).

© 2007 Optical Society of America

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References

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  1. B. I. Halperin, Physics of Defects, R.Balian, M.Kleman, and J.-P.Poirier, eds. (North-Holland, 1981), p. 814.
  2. B. W. Roberts, E. Bodenschatz, and J. P. Sethna, Physica D 99, 252 (1996).
    [CrossRef]
  3. I. Freund and M. Wilkinson, J. Opt. Soc. Am. A 15, 2892 (1998).
    [CrossRef]
  4. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
    [CrossRef]
  5. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 3048 (2000).
  6. G. Foltin, J. Phys. A 36, 1729 (2003).
    [CrossRef]
  7. G. Foltin, J. Phys. A 36, 4561 (2003).
    [CrossRef]
  8. M. R. Dennis, J. Phys. A 36, 6628 (2003).
  9. M. Wilkinson, J. Phys. A 37, 6763 (2004).
    [CrossRef]
  10. G. Foltin, S. Gnutzmann, and U. Smilansky, J. Phys. A 37, 11363 (2004).
    [CrossRef]
  11. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994).
  12. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 2165 (1974).
  13. N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii and V. V. Shkukov, JETP Lett. 33, 195 (1981).
  14. V. Yu Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).
  15. M. S. Soskin and M. V. Vasnetsov, Progress in Optics, E.Wolf, ed. (Elsevier, 2001), p. 219.
    [CrossRef]
  16. M. V. Berry and J. H. Hannay, J. Phys. A 10, 1809 (1977).
    [CrossRef]
  17. A. S. Thorndike, C. R. Cooley, and J. F. Nye, J. Phys. A 11, 1455 (1978).
    [CrossRef]
  18. M. V. Berry and M. Wilkinson, Proc. R. Soc. London Ser. A 392, 15 (1984).
    [CrossRef]
  19. L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comput. Graph. 3, 1 (1997).
    [CrossRef]
  20. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).
  21. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  22. M. V. Berry, J. Phys. A. 40, F185 (2007).
    [CrossRef]
  23. M. S. Soskin, R. I. Egorov, and I. Freund, Opt. Lett. 32, 891 (2007).
    [CrossRef] [PubMed]
  24. Because the absolute intensity of the optical field is generally of no interest, one can scale (attenuate) I so that ∣∇I∣⪡1, in which case Gaussian curvature K=(IxxIyy−Ixy2)/(1+∣∇I∣2)2=λ+λ−, and mean curvature C=(λ++λ−)/2, where λ+/- are the eigenvalues of M.
  25. M. Soskin, V. Denisenko, and R. Egorov, J. Opt. A 6, S281 (2004).
    [CrossRef]
  26. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
  27. Although branch cuts are convenient, visually prominent markers, any contour value can be used. The reason is that a suitable redefinition of the zero of phase can convert any contour into a branch cut.
  28. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef] [PubMed]

2007 (2)

2004 (3)

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

M. Wilkinson, J. Phys. A 37, 6763 (2004).
[CrossRef]

G. Foltin, S. Gnutzmann, and U. Smilansky, J. Phys. A 37, 11363 (2004).
[CrossRef]

2003 (3)

G. Foltin, J. Phys. A 36, 1729 (2003).
[CrossRef]

G. Foltin, J. Phys. A 36, 4561 (2003).
[CrossRef]

M. R. Dennis, J. Phys. A 36, 6628 (2003).

2002 (1)

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

2001 (1)

2000 (2)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 3048 (2000).

1998 (1)

1997 (1)

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comput. Graph. 3, 1 (1997).
[CrossRef]

1996 (1)

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, Physica D 99, 252 (1996).
[CrossRef]

1994 (1)

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

1990 (1)

V. Yu Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

1984 (1)

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

1981 (1)

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii and V. V. Shkukov, JETP Lett. 33, 195 (1981).

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]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 2165 (1974).

IEEE Trans. Vis. Comput. Graph. (1)

L. Hesselink, Y. Levy, and Y. Lavin, IEEE Trans. Vis. Comput. Graph. 3, 1 (1997).
[CrossRef]

J. Opt. A (1)

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

J. Opt. Soc. Am. A (1)

J. Phys. A (7)

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]

G. Foltin, J. Phys. A 36, 1729 (2003).
[CrossRef]

G. Foltin, J. Phys. A 36, 4561 (2003).
[CrossRef]

M. R. Dennis, J. Phys. A 36, 6628 (2003).

M. Wilkinson, J. Phys. A 37, 6763 (2004).
[CrossRef]

G. Foltin, S. Gnutzmann, and U. Smilansky, J. Phys. A 37, 11363 (2004).
[CrossRef]

J. Phys. A. (1)

M. V. Berry, J. Phys. A. 40, F185 (2007).
[CrossRef]

JETP Lett. (2)

N. B. Baranova, B. Ya Zel'dovich, A. V. Mamaev, N. Pilipetskii and V. V. Shkukov, JETP Lett. 33, 195 (1981).

V. Yu Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Opt. Commun. (1)

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

Opt. Lett. (2)

Phys. Rev. A (1)

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

Physica D (1)

B. W. Roberts, E. Bodenschatz, and J. P. Sethna, Physica D 99, 252 (1996).
[CrossRef]

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

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 2165 (1974).

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

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 3048 (2000).

Other (6)

M. S. Soskin and M. V. Vasnetsov, Progress in Optics, E.Wolf, ed. (Elsevier, 2001), p. 219.
[CrossRef]

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994).

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

B. I. Halperin, Physics of Defects, R.Balian, M.Kleman, and J.-P.Poirier, eds. (North-Holland, 1981), p. 814.

Because the absolute intensity of the optical field is generally of no interest, one can scale (attenuate) I so that ∣∇I∣⪡1, in which case Gaussian curvature K=(IxxIyy−Ixy2)/(1+∣∇I∣2)2=λ+λ−, and mean curvature C=(λ++λ−)/2, where λ+/- are the eigenvalues of M.

Although branch cuts are convenient, visually prominent markers, any contour value can be used. The reason is that a suitable redefinition of the zero of phase can convert any contour into a branch cut.

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Figures (4)

Fig. 1
Fig. 1

(a) Measured Stokes phase Φ 12 of the amplitude of a random field coded black to white π to + π . Zero lines Z 1 of S 1 ( Z 2 of S 2 ) are shown by thick white (black) curves, and positive + 1 (negative 1 ) umbilics, located at the intersections of Z 1 and Z 2 , are shown by black (white) circles. A u 2 S point (Fig. 2 and its introductory text), which is located in the upper half of the figure to the left of center, is shown by two nested white circles that indicate its double + 2 charge. Passing through this point are two Z 1 (two Z 2 ) oriented at + 45 ° and 45 ° (0° and 90°) relative to the horizontal x axis, in accord with theory [23]. Here there are a total of 34 singularities: 1 doubly charged u 2 S point and 33 ordinary umbilics; 16 umbilics are positive, 17 are negative. The net charge of all singularities is Q = + 1 . (b) Cumulative charge Q ( P ) as a function of position P measured in pixels (px) along the outer boundary of (a). The net charge is + 1 , in accord with the direct count in (a).

Fig. 2
Fig. 2

Splitting of a u 2 S point by an incoherent background B. (a) B = 0 . (b) B I max = 0.03 , where I max is the maximum intensity. The u 2 S point splits into two ordinary + 1 umbilics, conserving charge. (c) B I max = 0.06 . The two umbilics separate, revealing an intervening saddle point (black diamond), whose presence is required by conservation of the Poincaré index I P [11].

Fig. 3
Fig. 3

Experimentally measured data (points) for the cumulative sum j Q j 2 for j = 1 to j = n versus j for umbilic points in n = 20 independent realizations of a random field. The net charge Q j was measured using a boundary walk, Fig. 1, for a circular area with radius R measured in pixels (px). The solid [dashed] curves are calculated using the theoretical result for screening, Eq. (2), [for no screening, Eq. (1).] (a),(b) Amplitude (Ampl) umbilics. (c),(d) Intensity (Int) umbilics. (a),(c) R = 240 px . (b),(d) R = 120 px . The screening lengths obtained from these data, Λ S ( Ampl ) = 15 px , Λ S ( Int ) = 23 px , are significantly less than the transverse coherence length L coh = 35 px . These lengths are accurate to about 10%.

Fig. 4
Fig. 4

Branch cuts (here 2 π phase jumps) [27], the sign rule, and screening. Shown within the square is the wavefield in Fig. 1a with the u 2 S point split into two umbilics by the background B I max = 0.04 . Branch cuts are shown by thick black curves, positive (negative) umbilics by white (black) circles. In a random phase field branch cuts, like all other contours, are either terminated by singularities or close on themselves. These latter make no contribution to the charge. The net charge within the square is the sum of the signed branch cuts that exit the square, where the sign of a branch cut is the sign of the singularity that terminates the end inside the square. This sum is + 3 2 = + 1 , in accordance with Fig. 1. The sign rule [28] guarantees that opposite ends of a branch cut are terminated by singularities of the opposite sign. This one-to-one local pairing of positive and negative charges is the structure underlying the screening. The pairing severely damps out charge fluctuations, and yields a net charge variance that depends not on the total number of charges, but on the number of charges close to the boundary that have branch cuts that cross the boundary.

Equations (3)

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Q 2 = ρ A = N ,
Q 2 = 1 4 ρ Λ S P N 1 2 ,
M ( x , y ) = ( M x x ( x , y ) M x y ( x , y ) M x y ( x , y ) M y y ( x , y ) )

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