Abstract

Waves are superoscillatory where their local phase gradient exceeds the maximum wavenumber in their Fourier spectrum. We consider the superoscillatory area fraction of random optical speckle patterns. This follows from the joint probability density function of intensity and phase gradient for isotropic Gaussian random wave superpositions. Strikingly, this fraction is 13 when all the waves in the two-dimensional superposition have the same wavenumber. The fraction is 15 for a disk spectrum. Although these superoscillations are weak compared with optical fields with designed superoscillations, they are more stable on paraxial propagation.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. V. Berry, in Quantum Coherence and Reality, J.S.Anandan and J.L.Safko, eds. (World Scientific, 1994), pp. 55-65.
  2. A. Kempf, J. Math. Phys. 41, 2360 (2000).
    [CrossRef]
  3. M. V. Berry, and S. Popescu, J. Phys. A 39, 6965 (2006).
    [CrossRef]
  4. F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
    [CrossRef]
  5. L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
    [CrossRef]
  6. Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005), Chap. 16.
    [CrossRef]
  7. N. I. Zheludev, Nature Mater. 7, 420 (2008).
    [CrossRef]
  8. J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts and Co., 2007).
  9. K. J. Ebeling, in Physical Acoustics: Principles and Methods, W.P.Mason and R.N.Thurston, eds. (Academic, 1977), Vol. 17, p. 233.
  10. H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge U. Press, 1999).
  11. A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
    [CrossRef]
  12. A. C. Hamilton and J. Courtial, arXiv:0809.4370 [physics.optics] (submitted to New J. Phys).
  13. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  14. M. V. Berry, J. Phys. A 11, 27 (1978).
    [CrossRef]
  15. N. B. Baranova, B. Y. Zel'dovitch, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).
  16. M. V. Berry and M. R. Dennis, Proc. R. Soc. London, Ser. A 456, 2059 (2000).
    [CrossRef]
  17. E. Ochoa and J. W. Goodman, J. Opt. Soc. Am. 73, 943 (1983).
    [CrossRef]
  18. J. W. Goodman, Statistical Optics (Wiley, 1985).
  19. P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (Cambridge U. Press, 1993).
    [CrossRef] [PubMed]
  20. M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys A (to be published).

2008

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

2007

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

2006

M. V. Berry, and S. Popescu, J. Phys. A 39, 6965 (2006).
[CrossRef]

2002

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

2000

A. Kempf, J. Math. Phys. 41, 2360 (2000).
[CrossRef]

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

1983

1981

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

1978

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

1974

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Aharonov, Y.

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005), Chap. 16.
[CrossRef]

Baranova, N. B.

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

Berggren, K.-F.

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

Berry, M. V.

M. V. Berry, and S. Popescu, J. Phys. A 39, 6965 (2006).
[CrossRef]

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

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

M. V. Berry, in Quantum Coherence and Reality, J.S.Anandan and J.L.Safko, eds. (World Scientific, 1994), pp. 55-65.

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys A (to be published).

Boissel, Y.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

Courtial, J.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

A. C. Hamilton and J. Courtial, arXiv:0809.4370 [physics.optics] (submitted to New J. Phys).

Dennis, M. R.

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

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys A (to be published).

Ebeling, K. J.

K. J. Ebeling, in Physical Acoustics: Principles and Methods, W.P.Mason and R.N.Thurston, eds. (Academic, 1977), Vol. 17, p. 233.

Garcia De Abajo, F. J.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

Goodman, J. W.

E. Ochoa and J. W. Goodman, J. Opt. Soc. Am. 73, 943 (1983).
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts and Co., 2007).

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

Hamilton, A. C.

A. C. Hamilton and J. Courtial, arXiv:0809.4370 [physics.optics] (submitted to New J. Phys).

Holland, P. R.

P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (Cambridge U. Press, 1993).
[CrossRef] [PubMed]

Huang, F. M.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

Ishio, H.

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

Kempf, A.

A. Kempf, J. Math. Phys. 41, 2360 (2000).
[CrossRef]

Mamaev, A. V.

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

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Ochoa, E.

Pilipetskii, N.

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

Popescu, S.

M. V. Berry, and S. Popescu, J. Phys. A 39, 6965 (2006).
[CrossRef]

Rohrlich, D.

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005), Chap. 16.
[CrossRef]

Saddreev, A. F.

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

Saichev, A. I.

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

Shkukov, V. V.

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

Stöckmann, H. J.

H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge U. Press, 1999).

Thomson, L. C.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

Whyte, G.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

Yao, E.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

Zel'dovitch, B. Y.

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

Zheludev, N.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

Zheludev, N. I.

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

Appl. Phys. Lett.

F. M. Huang, N. Zheludev, Y. Chen, and F. J. Garcia De Abajo, Appl. Phys. Lett. 90, 091119 (2007).
[CrossRef]

J. Math. Phys.

A. Kempf, J. Math. Phys. 41, 2360 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. A

M. V. Berry, and S. Popescu, J. Phys. A 39, 6965 (2006).
[CrossRef]

A. I. Saichev, H. Ishio, A. F. Saddreev, and K.-F. Berggren, J. Phys. A 35, L87 (2002).
[CrossRef]

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

JETP Lett.

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

Nature Mater.

N. I. Zheludev, Nature Mater. 7, 420 (2008).
[CrossRef]

New J. Phys.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, New J. Phys. 10, 023015 (2008).
[CrossRef]

Proc. R. Soc. London Ser. A

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Proc. R. Soc. London, Ser. A

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

Other

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

P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics (Cambridge U. Press, 1993).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, “Natural superoscillations in monochromatic waves in D dimensions,” J. Phys A (to be published).

M. V. Berry, in Quantum Coherence and Reality, J.S.Anandan and J.L.Safko, eds. (World Scientific, 1994), pp. 55-65.

A. C. Hamilton and J. Courtial, arXiv:0809.4370 [physics.optics] (submitted to New J. Phys).

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley, 2005), Chap. 16.
[CrossRef]

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts and Co., 2007).

K. J. Ebeling, in Physical Acoustics: Principles and Methods, W.P.Mason and R.N.Thurston, eds. (Academic, 1977), Vol. 17, p. 233.

H. J. Stöckmann, Quantum Chaos: An Introduction (Cambridge U. Press, 1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Contour plot of the joint probability density function P ( I , χ ) of Eq. (4). It is clearly unbounded both in I and χ , although high values of these are anticorrelated.

Fig. 2
Fig. 2

Random superposition of 100 two-dimensional plane waves with the same wavenumber k. (a) Phase pattern (hues); (b) intensity pattern (grayscale). The white contour denotes the line χ = k . The suboscillatory region, occupying statistically 2 3 of the area, is shaded with a dark filter in (a). Several phase singularities can be seen in the superoscillatory region of (a). The intensity contour (dashed cyan curve) enclosing the lowest 1 3 of the intensity pattern is also shown in (b). This is close to the white superoscillation contour, but the two are clearly different. The area plotted is ( 4 π k ) 2 .

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

superoscillatory where χ 2 k max 2 > 0 ,
P ( I , J ) = 1 ( 2 π ) 3 d s d 2 t exp ( i s I + i t J ) × exp ( i s ψ 2 + ( 1 / 2 ) t ( ψ * ψ ψ ψ * ) ) ,
P ( I , J ) = J I I 0 2 k 2 exp ( 1 2 I 0 ( 2 I + J 2 I k 2 ) ) .
P ( I , χ ) = I χ I 0 2 k 2 exp ( I I 0 ( 1 + χ 2 2 k 2 ) ) .
P ( χ ) = 4 k 2 χ ( 2 k 2 + χ 2 ) 2 .
f = k max d χ P ( χ ) .
χ 2 k 2 = 2 ρ ρ .

Metrics