Abstract

We present results of experiments of diffraction by an amplitude screen, made of randomly distributed circular holes. By careful selection of the experimental parameters we obtain an intensity pattern strongly connected to the Voronoi diagram (VD) generated by the centers of the apertures. With the help of simulations we give a description of the observed phenomenon and elucidate the optimal parameters for its observation. Finally, we also suggest how it can be used for a fast, all-optical generation of VDs.

© 2008 Optical Society of America

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  1. G. Voronoi and J. Reine Angew, "Nouvelles applications des parametres continus à la theorie des formes quadratiques," Math. 134, 198 (1908).
  2. A. Okabe, B. Boots, and K. Sugihara, Spatial Tesselations: Concept and Applications of Voronoi Diagrams (Wiley, Chichester, 1992).
  3. G. Schliecker, "Structure and dynamics of cellular systems," Adv. Phys. 51,1319-1378 (2002).
    [CrossRef]
  4. H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
    [CrossRef]
  5. J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).
  6. D. Tolmachiev and A. Adamatzky, "Chemical processor for computation of Voronoi diagram," Adv. Mater. Opt. Electron. 6, 191 (1996).
    [CrossRef]
  7. M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
    [CrossRef]
  8. A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
    [CrossRef]
  9. M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Speckle Phenomena in Optics (Greenwood Village, CO, 2007).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill), Chap. 4.
  12. B. Jähne, Digital Image Processing (Springer, 1995), Chap. 12.
  13. T. Bräunl, Parallel Image Processing (Springer, 2001), Chap. 5.

2002 (2)

G. Schliecker, "Structure and dynamics of cellular systems," Adv. Phys. 51,1319-1378 (2002).
[CrossRef]

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

2001 (1)

H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
[CrossRef]

2000 (1)

M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
[CrossRef] [PubMed]

1998 (1)

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

1996 (1)

D. Tolmachiev and A. Adamatzky, "Chemical processor for computation of Voronoi diagram," Adv. Mater. Opt. Electron. 6, 191 (1996).
[CrossRef]

1993 (1)

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

1908 (1)

G. Voronoi and J. Reine Angew, "Nouvelles applications des parametres continus à la theorie des formes quadratiques," Math. 134, 198 (1908).

Adamatzky, A.

D. Tolmachiev and A. Adamatzky, "Chemical processor for computation of Voronoi diagram," Adv. Mater. Opt. Electron. 6, 191 (1996).
[CrossRef]

Ammi, M.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Bideau, D.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Carpineti, M.

M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
[CrossRef] [PubMed]

Doi, M.

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

Gervois, A.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Giglio, M.

M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
[CrossRef] [PubMed]

Katsuki, F.

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

Koyama, T.

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

Lemaítre, J.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Liehr, A. W.

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

Moskalenko, A. S.

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

Oger, L.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Purwins, H.-G.

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

Reine Angew, J.

G. Voronoi and J. Reine Angew, "Nouvelles applications des parametres continus à la theorie des formes quadratiques," Math. 134, 198 (1908).

Rivier, N.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Schliecker, G.

G. Schliecker, "Structure and dynamics of cellular systems," Adv. Phys. 51,1319-1378 (2002).
[CrossRef]

Suzuki, Y.

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

Thorpe, S. M.

H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
[CrossRef]

Tolmachiev, D.

D. Tolmachiev and A. Adamatzky, "Chemical processor for computation of Voronoi diagram," Adv. Mater. Opt. Electron. 6, 191 (1996).
[CrossRef]

Troadec, J. P.

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Vailati, A.

M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
[CrossRef] [PubMed]

Voronoi, G.

G. Voronoi and J. Reine Angew, "Nouvelles applications des parametres continus à la theorie des formes quadratiques," Math. 134, 198 (1908).

Windle, A. H.

H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
[CrossRef]

Zanin, A. L.

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

Zhu, H. X.

H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
[CrossRef]

Adv. Mater. Opt. Electron. (1)

D. Tolmachiev and A. Adamatzky, "Chemical processor for computation of Voronoi diagram," Adv. Mater. Opt. Electron. 6, 191 (1996).
[CrossRef]

Adv. Phys. (1)

G. Schliecker, "Structure and dynamics of cellular systems," Adv. Phys. 51,1319-1378 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

A. L. Zanin, A. W. Liehr, A. S. Moskalenko, and H.-G. Purwins, "Voronoi diagrams in barrier gas discharge," Appl. Phys. Lett. 81, 3338 (2002).
[CrossRef]

Math. (1)

G. Voronoi and J. Reine Angew, "Nouvelles applications des parametres continus à la theorie des formes quadratiques," Math. 134, 198 (1908).

Philos. Mag. A (1)

H. X. Zhu, S. M. Thorpe, and A. H. Windle, "The geometrical properties of irregular two-dimensional Voronoi tessellations," Philos. Mag. A 81, 2765 (2001).
[CrossRef]

Philos. Mag. B (1)

J. Lemaítre, A. Gervois, J. P. Troadec, N. Rivier, M. Ammi, L. Oger, and D. Bideau, "Arrangement of cells in Voronoi tesselations of monosize packing of discs," Philos. Mag. B 81, 347 (1993).

Philos. Mag. Lett. (1)

M. Doi, Y. Suzuki, T. Koyama, and F. Katsuki, "Pattern evolution of crystalline Ge aggregates during annealing of an Al/Ge bilayer film deposited on a SiO2 substrate," Philos. Mag. Lett. 78, 241 (1998).
[CrossRef]

Phys. Rev Lett. (1)

M. Giglio, M. Carpineti, and A. Vailati, "Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r)," Phys. Rev Lett. 85, 1416 (2000).
[CrossRef] [PubMed]

Other (5)

J. W. Goodman, Speckle Phenomena in Optics (Greenwood Village, CO, 2007).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill), Chap. 4.

B. Jähne, Digital Image Processing (Springer, 1995), Chap. 12.

T. Bräunl, Parallel Image Processing (Springer, 2001), Chap. 5.

A. Okabe, B. Boots, and K. Sugihara, Spatial Tesselations: Concept and Applications of Voronoi Diagrams (Wiley, Chichester, 1992).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic representation of the optical setup. The light diverging from a single mode optical fiber (1) coupled with a super-luminous Light Emitting Diode (spectral center 675 nm, spectral bandwidth, FWHM, 13.1 nm) is collimated by an achromatic doublet (2) of focal length f=20 cm. This collimated beam is sent trough an opaque metallic screen with non overlapping circular pinholes of radius R=150 µm chemically etched in random positions. The minimum distance allowed between the centers of the holes is 400 µm. The surface fraction covered by the holes is around 20%. A second lens (4) conjugates a plane lying at distance z from the mask to the plane of a CCD camera sensor (5).

Fig. 2.
Fig. 2.

(a). Measured intensity distributions of the light scattered from an opaque screen with circular pinholes of radius R=150 µm on a plane z=1.7 cm far from the screen. (b). Same of (a) with a γ=0.25 correction. (c). Same of (b) with the calculated VD generated by the holes centers superimposed.

Fig. 3.
Fig. 3.

(a). Transmittance profile of an opaque screen with two circular apertures (d=2.5R). (b). Numerically evaluated intensity distribution of the light diffracted by the holes of panel (a) on the plane z=R 2/3λ. (c). Numerically evaluated intensity distribution on the same plane when additional holes are introduced on the screen depicted in panel (a). (d). Same of (c) when a Canny-Deriche edge detection filter is applied. (e). Skeleton of the filtered image shown in panel (d).

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