Abstract

We present the first experimental observation of spontaneous pattern formation in a nonlinear optical cavity in which the circulating light is both spatially and temporally incoherent.

© 2004 Optical Society of America

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References

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  1. A.M. Turring, �??The Chemical Basis of Morphogenesis,�?? Phil. Trans. RS B (London), 237, 37 (1952).
    [CrossRef]
  2. M. A. Kessler, B. T. Werner, �??Self-organization of sorted patterned ground,�?? Science, 299, 380 (2003).
    [CrossRef] [PubMed]
  3. J. V. Hardenberg, E. Meron, M. Shachak. Y. Zarmi, �??Diversity of Vegetation Patterns and Desertification,�?? Phys. Rev. Lett. 87, 198101 (2001).
    [CrossRef]
  4. O. Zik, D. Levine, S.G. Lipson, S. Shtrikman, and J. Stavans, �??Rotationally Induced Segregation of Granular Materials�??, Phys. Rev. Lett. 73, 644 (1994).
    [CrossRef] [PubMed]
  5. F.T. Arecchi, S. Boccaletti and P.L. Ramazza, �??Pattern formation and competition in nonlinear optics,�?? Phys. Rep. 318, 1 (1999).
    [CrossRef]
  6. L.A. Lugiato and R. Lefever, �??Spatial dissipative structures in passive optical systems,�?? Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef] [PubMed]
  7. V.B. Taranenko, I. Ganne, R.J. Kuszelewicz, C.O. Weiss, �??Patterns and localized structures in bistable semiconductor resonators,�?? Phys. Rev. A 61, 063818 (2000).
    [CrossRef]
  8. S. Barland, J. R. Tredicce, M. Brambilla,L. A. Lugiato,S. Balle,M. Giudicl, T. Maggipinto, L. Spinelli,G. Tissoni, T. Knodl, M. Miller, R. Jager, �??Cavity solitons as pixels in semiconductor microcavities,�?? Nature (London) 419, 699 (2002).
    [CrossRef]
  9. W.J. Firth and G.K. Harkness, Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, Berlin, 2001), Chap. 13, pp. 343�??358
  10. C.O. Weiss et al., Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, Berlin, 2001), Chap. 15, pp. 395�??416
  11. T. Carmon, M. Soljacic, and M. Segev, �??Pattern formation in a cavity longer than the coherence length of the light in it,�?? Phys. Rev. Lett., 89, 183902 (2002).
    [CrossRef] [PubMed]
  12. T. Schwartz, J.W. Fleischer, O. Cohen, H. Buljan, T. Carmon, and M. Segev, �??Pattern formation in a ring cavity with temporally incoherent feedback,�?? J. Opt. Soc. Am. B, to appear.
  13. H. Buljan, M. Soljacic, T. Carmon, and M. Segev, Cavity pattern formation with incoherent light, Phys. Rev. E, 68, 016616, (2003).
    [CrossRef]
  14. M. Segev, B. Crosignani, P. DiPorto, G. C. Valley and A. Yariv, �??Steady state spatial screening-solitons in photorefractive media with external applied field,�?? Phys. Rev. Lett. 73, 3211 (1994).
    [CrossRef] [PubMed]
  15. D. N. Christodoulides, and M.I. Carvalho, �??�??Bright, Dark and Gray Spatial Soliton States in Photorefractive Media�??, J. Opt. Soc. Am. B, 12, 1628 (1995).
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  16. M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani and P. DiPorto, �??Observation of twodimensional steady-state photorefractive screening- solitons,�?? Electron. Lett. 31, 826 (1995).
    [CrossRef]
  17. D. Kip, M. Soljacic, M. Segev, S. M. Sears, and D. N. Christodoulides, �??(1+1) Dimensional modulation instability of spatially-incoherent light,�?? J. Opt. Soc. Am. B, 19, 502 (2002).
    [CrossRef]
  18. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, �??Modulation instability of incoherent beams in non-instantaneous nonlinear media,�?? Phys. Rev. Lett. 84, 467 (2000)
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  19. M. Segev, M.F. Shih, and G.C. Valley, �??Photorefractive screening solitons of high and low intensity,�?? J. Opt. Soc. Am. B, 13, 706 (1996).
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  20. D. Kip, M. Soljacic, M. Segev, E. Eugenieva and D. N. Christodoulides, �??Modulation instability and pattern formation in spatially incoherent light beams,�?? Science 290, 495 (2000).
    [CrossRef] [PubMed]
  21. S.M. Sears, M. Soljacic, D.N. Christodoulides, and M. Segev, �??Pattern formation via symmetry breaking in nonlinear weakly correlated systems,�?? Phys. Rev. E 65, 036620 (2002).
    [CrossRef]
  22. Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, �??Clustering of solitons in weakly correlated systems,�?? Proceedings of the US National Academy of Science (PNAS), 99, 5223 (2002).
    [CrossRef]
  23. D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, �??Features of modulational instability of partially coherent light: Importance of the incoherence spectrum,�?? Phys. Rev. E 69, 025601 (2004).
    [CrossRef]

Electron. Lett

M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani and P. DiPorto, �??Observation of twodimensional steady-state photorefractive screening- solitons,�?? Electron. Lett. 31, 826 (1995).
[CrossRef]

J. Opt. Soc. AM. B

J. Opt. Soc. Am. B.

T. Schwartz, J.W. Fleischer, O. Cohen, H. Buljan, T. Carmon, and M. Segev, �??Pattern formation in a ring cavity with temporally incoherent feedback,�?? J. Opt. Soc. Am. B, to appear.

National Academy of Science (PNAS)

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides and M. Segev, �??Clustering of solitons in weakly correlated systems,�?? Proceedings of the US National Academy of Science (PNAS), 99, 5223 (2002).
[CrossRef]

Nature

S. Barland, J. R. Tredicce, M. Brambilla,L. A. Lugiato,S. Balle,M. Giudicl, T. Maggipinto, L. Spinelli,G. Tissoni, T. Knodl, M. Miller, R. Jager, �??Cavity solitons as pixels in semiconductor microcavities,�?? Nature (London) 419, 699 (2002).
[CrossRef]

Phil. Trans. RS B

A.M. Turring, �??The Chemical Basis of Morphogenesis,�?? Phil. Trans. RS B (London), 237, 37 (1952).
[CrossRef]

Phys. Rep.

F.T. Arecchi, S. Boccaletti and P.L. Ramazza, �??Pattern formation and competition in nonlinear optics,�?? Phys. Rep. 318, 1 (1999).
[CrossRef]

Phys. Rev. A

V.B. Taranenko, I. Ganne, R.J. Kuszelewicz, C.O. Weiss, �??Patterns and localized structures in bistable semiconductor resonators,�?? Phys. Rev. A 61, 063818 (2000).
[CrossRef]

Phys. Rev. E

H. Buljan, M. Soljacic, T. Carmon, and M. Segev, Cavity pattern formation with incoherent light, Phys. Rev. E, 68, 016616, (2003).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, �??Features of modulational instability of partially coherent light: Importance of the incoherence spectrum,�?? Phys. Rev. E 69, 025601 (2004).
[CrossRef]

S.M. Sears, M. Soljacic, D.N. Christodoulides, and M. Segev, �??Pattern formation via symmetry breaking in nonlinear weakly correlated systems,�?? Phys. Rev. E 65, 036620 (2002).
[CrossRef]

Phys. Rev. Lett.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley and A. Yariv, �??Steady state spatial screening-solitons in photorefractive media with external applied field,�?? Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

T. Carmon, M. Soljacic, and M. Segev, �??Pattern formation in a cavity longer than the coherence length of the light in it,�?? Phys. Rev. Lett., 89, 183902 (2002).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, �??Modulation instability of incoherent beams in non-instantaneous nonlinear media,�?? Phys. Rev. Lett. 84, 467 (2000)
[CrossRef] [PubMed]

L.A. Lugiato and R. Lefever, �??Spatial dissipative structures in passive optical systems,�?? Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

J. V. Hardenberg, E. Meron, M. Shachak. Y. Zarmi, �??Diversity of Vegetation Patterns and Desertification,�?? Phys. Rev. Lett. 87, 198101 (2001).
[CrossRef]

O. Zik, D. Levine, S.G. Lipson, S. Shtrikman, and J. Stavans, �??Rotationally Induced Segregation of Granular Materials�??, Phys. Rev. Lett. 73, 644 (1994).
[CrossRef] [PubMed]

Science

M. A. Kessler, B. T. Werner, �??Self-organization of sorted patterned ground,�?? Science, 299, 380 (2003).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva and D. N. Christodoulides, �??Modulation instability and pattern formation in spatially incoherent light beams,�?? Science 290, 495 (2000).
[CrossRef] [PubMed]

Other

W.J. Firth and G.K. Harkness, Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, Berlin, 2001), Chap. 13, pp. 343�??358

C.O. Weiss et al., Spatial Solitons, edited by S. Trillo and W. Torruellas (Springer-Verlag, Berlin, 2001), Chap. 15, pp. 395�??416

Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1.

The experimental setup.

Fig. 2.
Fig. 2.

(Movie 2.5 MB) Top row: Intensity patterns at the output of the crystal, with and without feedback; the strength of the nonlinearity is increased from left to right. Plot: Modulation depth of the emerging pattern as a function of nonlinearity for different values of feedback.

Fig. 3.
Fig. 3.

Upper picture: Characteristic (time-averaged) intensity pattern. Lower picture: The highly speckled structure of the instantaneous intensity of the incoherent beam obtained by stopping the rotating diffuser, and capturing the speckled pattern. The average size of the speckles, which corresponds to the spatial correlation distance, is several times smaller than the pattern periodicity

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