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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  1. A.M. Turring, “The Chemical Basis of Morphogenesis,” Phil. Trans. RS B (London),  237, 37 (1952).
    [CrossRef]
  2. M. A. Kessler and B. T. Werner, “Self-organization of sorted patterned ground,” Science,  299, 380 (2003).
    [CrossRef] [PubMed]
  3. J. V. Hardenberg, E. Meron, M. Shachak, and 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, and 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, and 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).
    [CrossRef]
  16. M. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, “Observation of two-dimensional 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);
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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]

2004 (1)

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]

2003 (2)

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

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

2002 (5)

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

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]

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]

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]

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]

2001 (1)

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

2000 (3)

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

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]

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]

1999 (1)

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

1996 (1)

1995 (2)

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

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).
[CrossRef]

1994 (2)

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]

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]

1987 (1)

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

1952 (1)

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

Anderson, D.

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]

Arecchi, F.T.

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

Balle, S.

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

Barland, S.

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

Boccaletti, S.

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

Brambilla, M.

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

Buljan, H.

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

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.

Carmon, T.

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

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]

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.

Carvalho, M.I.

Chen, Z.

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]

Christodoulides, D. N.

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]

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]

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]

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]

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).
[CrossRef]

Christodoulides, D.N.

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]

Cohen, O.

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.

Coskun, T.

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]

Crosignani, B.

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

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]

DiPorto, P.

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

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]

Eugenieva, E.

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]

Firth, W.J.

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;

Fleischer, J.W.

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.

Ganne, I.

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

Giudicl, M.

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

Hardenberg, J. V.

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

Harkness, G.K.

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;

Helczynski-Wolf, L.

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]

Jager, R.

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

Kessler, M. A.

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

Kip, D.

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]

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]

Knodl, T.

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

Kuszelewicz, R.J.

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

Lefever, R.

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

Levine, D.

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]

Lipson, S.G.

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]

Lisak, M.

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]

Lugiato, L. A.

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

Lugiato, L.A.

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

Maggipinto, T.

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

Martin, H.

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]

Meron, E.

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

Miller, M.

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

Ramazza, P.L.

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

Salamo, G.

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

Schwartz, T.

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.

Sears, S. M.

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]

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]

Sears, S.M.

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]

Segev, M.

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

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]

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]

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]

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]

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]

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]

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).
[CrossRef]

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

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. 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.

Semenov, V.

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]

Shachak, M.

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

Shih, M.

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

Shih, M.F.

Shtrikman, S.

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]

Soljacic, M.

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

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]

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]

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]

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]

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]

Spinelli, L.

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

Stavans, J.

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]

Taranenko, V.B.

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

Tissoni, G.

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

Tredicce, J. R.

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

Turring, A.M.

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

Valley, G. C.

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

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]

Valley, G.C.

Vishwanath, A.

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]

Weiss, C.O.

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

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

Werner, B. T.

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

Yariv, A.

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]

Zarmi, Y.

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

Zik, O.

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]

Electron. Lett. (1)

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

J. Opt. Soc. Am. B (3)

Nature (London) (1)

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

Phil. Trans. RS B (London) (1)

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

Phys. Rep. (1)

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 (1)

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

Phys. Rev. E (3)

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]

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

Phys. Rev. Lett. (6)

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, and 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]

Proceedings of the US National Academy of Science (PNAS) (1)

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]

Science (2)

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]

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

Other (3)

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.

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.

Supplementary Material (1)

» Media 1: MOV (2499 KB)     

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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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