Abstract

We study the long-range propagation of incoherent light following the modulation instability (MI) process in non-instantaneous nonlinear Kerr-type media. We find that the system eventually reaches a steady-state characterized by a lower degree of coherence than in the initial state, with small fluctuations around a pronounced mean value. We find that the average values of the spatial correlation distance at steady-state and the fluctuations around it, which are obtained either through ensemble averaging, or by spatial averaging, or via temporal averaging, are all identical. This feature may be viewed as indication of ergodic behavior, which occurs in the long-time evolution following incoherent MI. Finally, we find that the steady-state properties of the system depend on the initial coherence but not on the nonlinearity strength, although the system evolves faster to steady-state as the strength of the nonlinearity is increased.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  16. 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, 502-512 (2002).
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    [CrossRef]
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    [CrossRef] [PubMed]
  25. S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
    [CrossRef]
  26. E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995)
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    [CrossRef]
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    [CrossRef]
  32. V. G. Makhan�??kov and O. K. Pashaev, "Nonlinear Schrödinger equation with noncompact isogroup," Teor. Mat. Fiz. 53, 55 (1982) [Theor. Math. Phys. 53, 979-987 (1982)].
  33. M. Soljacic, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squie, "Collisions of Two Solitons in an Arbitrary Number of Coupled Nonlinear Schrödinger Equations," Phys. Rev. Lett. 90, 254102-254105 (2003).
    [CrossRef] [PubMed]
  34. H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
    [CrossRef] [PubMed]

2007 (1)

2006 (1)

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[CrossRef] [PubMed]

2005 (1)

H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
[CrossRef] [PubMed]

2004 (2)

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, 025601-025604 (2004).
[CrossRef]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

2003 (1)

M. Soljacic, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squie, "Collisions of Two Solitons in an Arbitrary Number of Coupled Nonlinear Schrödinger Equations," Phys. Rev. Lett. 90, 254102-254105 (2003).
[CrossRef] [PubMed]

2002 (5)

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (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, 502-512 (2002).
[CrossRef]

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (2002).
[CrossRef]

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[CrossRef]

2001 (2)

2000 (1)

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

1999 (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

1997 (2)

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880-883 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

1995 (2)

1994 (1)

L. A. Lugiato, Chaos Solitons Fractals 4, 1245-1251 (1994).

1993 (1)

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

1992 (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

1986 (1)

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

1982 (1)

V. G. Makhan�??kov and O. K. Pashaev, "Nonlinear Schrödinger equation with noncompact isogroup," Teor. Mat. Fiz. 53, 55 (1982) [Theor. Math. Phys. 53, 979-987 (1982)].

1966 (1)

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in nonlinear liquids," Zh. Eksperim. i Teor. Fiz.-Pis'ma Redakt. 3, 471 (1966); [translation: JETP Letters 3, 307-312 (1966)].

1957 (2)

E. T. Jaynes, "Information Theory and Statistical Mechanics," Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. T. Jaynes, "Information Theory and Statistical Mechanics. II," Phys. Rev. 108, 171-190 (1957).
[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, 025601-025604 (2004).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in nonlinear liquids," Zh. Eksperim. i Teor. Fiz.-Pis'ma Redakt. 3, 471 (1966); [translation: JETP Letters 3, 307-312 (1966)].

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Buljan, H.

H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
[CrossRef] [PubMed]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[CrossRef]

Carmon, T.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

Chavez-Cerda, S.

Chen, Z.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (2002).
[CrossRef]

J. Klinger, H. Martin, and Z. Chen, "Experiments on induced modulational instability of an incoherent optical beam," Opt. Lett. 26, 271-273 (2001).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Christodoulides, D. N.

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (2002).
[CrossRef]

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (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, 502-512 (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, 495-498 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

Coskun, T. H.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

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, 495-498 (2000).
[CrossRef] [PubMed]

Hall, B.

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

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, 025601-025604 (2004).
[CrossRef]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Iturbe-Castillo, M. D.

Jankovic, L.

Jauslin, H. R.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[CrossRef] [PubMed]

Jaynes, E. T.

E. T. Jaynes, "Information Theory and Statistical Mechanics," Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. T. Jaynes, "Information Theory and Statistical Mechanics. II," Phys. Rev. 108, 171-190 (1957).
[CrossRef]

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, 502-512 (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, 495-498 (2000).
[CrossRef] [PubMed]

Klinger, J.

Kuznetsov, E. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995)
[CrossRef]

Lagrange, S.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[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, 025601-025604 (2004).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, Chaos Solitons Fractals 4, 1245-1251 (1994).

Makhan???kov, V. G.

V. G. Makhan�??kov and O. K. Pashaev, "Nonlinear Schrödinger equation with noncompact isogroup," Teor. Mat. Fiz. 53, 55 (1982) [Theor. Math. Phys. 53, 979-987 (1982)].

Malendevich, R.

Marklund, M.

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Martin, H.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (2002).
[CrossRef]

J. Klinger, H. Martin, and Z. Chen, "Experiments on induced modulational instability of an incoherent optical beam," Opt. Lett. 26, 271-273 (2001).
[CrossRef]

Mikhailov, A. V.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995)
[CrossRef]

Mitchell, M.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880-883 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Newell, A. C.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

Pashaev, O. K.

V. G. Makhan�??kov and O. K. Pashaev, "Nonlinear Schrödinger equation with noncompact isogroup," Teor. Mat. Fiz. 53, 55 (1982) [Theor. Math. Phys. 53, 979-987 (1982)].

Picozzi, A.

A. Picozzi, "Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics," Opt. Express 15, 9063-9083 (2007).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[CrossRef] [PubMed]

Pitois, S.

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[CrossRef] [PubMed]

Pushkarev, A.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Sanchez-Mondragon, J. J.

Schwartz, T.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

Sears, S. M.

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (2002).
[CrossRef]

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (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, 502-512 (2002).
[CrossRef]

Segev, M.

H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
[CrossRef] [PubMed]

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[CrossRef]

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (2002).
[CrossRef]

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (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, 502-512 (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, 495-498 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880-883 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

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, 025601-025604 (2004).
[CrossRef]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Shimokhin, I. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995)
[CrossRef]

Šiber, A.

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[CrossRef]

Solja??i??, M.

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[CrossRef]

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (2002).
[CrossRef]

Soljacic, M.

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, 502-512 (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, 495-498 (2000).
[CrossRef] [PubMed]

Stegeman, G.

Stepanov, S. I.

Stewart Aitchison, J.

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in nonlinear liquids," Zh. Eksperim. i Teor. Fiz.-Pis'ma Redakt. 3, 471 (1966); [translation: JETP Letters 3, 307-312 (1966)].

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

Torres-Cisnero, G. E.

Torres-Cisneros, M.

Vardi, A.

H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
[CrossRef] [PubMed]

Vysloukh, V. A.

Zakharo, V. E.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

Chaos Solitons Fractals (1)

L. A. Lugiato, Chaos Solitons Fractals 4, 1245-1251 (1994).

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

Nature (London) (1)

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature (London) 387, 880-883 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rep. (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Phys. Rev. (2)

E. T. Jaynes, "Information Theory and Statistical Mechanics," Phys. Rev. 106, 620-630 (1957).
[CrossRef]

E. T. Jaynes, "Information Theory and Statistical Mechanics. II," Phys. Rev. 108, 171-190 (1957).
[CrossRef]

Phys. Rev. E (4)

S. M. Sears, M. Solja�?i�?, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620-9 (2002).
[CrossRef]

H. Buljan, A. Šiber, M. Solja�?i�?, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601-4 (2002).
[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, 025601-025604 (2004).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, "Statistical effects in the multistream model for quantum plasmas," Phys. Rev. E 65, 046417-5 (2002).
[CrossRef]

Phys. Rev. Lett. (7)

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent white light," Phys. Rev. Lett. 93, 223901-5 (2004).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, "Velocity locking of incoherent nonlinear wave packets," Phys. Rev. Lett. 97, 033902-5 (2006).
[CrossRef] [PubMed]

M. Soljacic, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squie, "Collisions of Two Solitons in an Arbitrary Number of Coupled Nonlinear Schrödinger Equations," Phys. Rev. Lett. 90, 254102-254105 (2003).
[CrossRef] [PubMed]

H. Buljan, M. Segev, and A. Vardi, "Incoherent Matter-Wave Solitons and pairing instability in an attractively interacting Bose-Einstein Condensate," Phys. Rev. Lett. 95, 180401-4 (2005).
[CrossRef] [PubMed]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[CrossRef] [PubMed]

Physica D (1)

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995)
[CrossRef]

Physica D (Amsterdam) (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharo., "Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear," Physica D (Amsterdam) 57, 96-160 (1992).
[CrossRef]

Pis'ma Redakt. (1)

V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in nonlinear liquids," Zh. Eksperim. i Teor. Fiz.-Pis'ma Redakt. 3, 471 (1966); [translation: JETP Letters 3, 307-312 (1966)].

PNAS (1)

Z. Chen, S. M. Sears, H. Martin, D. N. Christodoulides, and M. Segev, "Clustering of solitons in weakly correlated wavefronts," PNAS 99, 5223-5227 (2002).
[CrossRef]

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M. C. Cross and P. C. Hohenberg, "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Science (1)

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

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V. G. Makhan�??kov and O. K. Pashaev, "Nonlinear Schrödinger equation with noncompact isogroup," Teor. Mat. Fiz. 53, 55 (1982) [Theor. Math. Phys. 53, 979-987 (1982)].

Other (6)

The only exception we know of is M. Rigol, et al., Phys. Rev. Lett. 98, 050405-5 (2007) where a quantum integrable system relaxes to a steady state, carrying memory of initial conditions.
[CrossRef] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, and M. Segev, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601-035603 (R) (2001).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, 1995).

E. Fermi, J. Pasta, and S. Ulam, "Studies of non linear problems," Los Alamos Rpt. LA-1940 (1955).

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in Noninstantaneous Nonlinear Media," Phys. Rev. Lett. 84, 467-470 �??2000).
[CrossRef] [PubMed]

R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1994).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Input field distribution E(x,z=0) and the total intensity at the input (z=0). The insert shows the linear phase upon all three plane waves, indicating that the modes propagate in different directions with respect to the z-axis (direction of the central plane wave). The fluctuations in all three field amplitudes reflect one particular realization of the input noise. (b) Spatial power spectrum of the input field (z=0). The power within the three plane waves conforms to a Gaussian distribution in kx -space. The insert zooms to show the presence of small initial noise in all spatial wave vectors.

Fig. 2.
Fig. 2.

(a) Emergence and dynamics of the incoherent MI intensity pattern, as the system evolves towards steady-state. (b) Evolution of the transverse correlation distance upon the partially-spatially-incoherent beam, as the system evolves towards the steady state. (c) Evolution of the transverse intensity-variance, M(z), which is a measure for the modulation depth (visibility) of the evolving MI pattern shown in Fig. 2(a).

Fig. 3.
Fig. 3.

(a) Ensemble-averaged value of 〈lc (x, z)〉 e=800 taken over 800 realizations of the initial noise The initial transient behavior for z<40, and the final steady-state are clearly observed. (b),(c) Dynamics of lc (x, z) along to arbitrary chosen x values, showing that the behavior in equilibrium is x-independent. (d) Dynamics of Lc (z) shows the same behavior as in (b) and (c). See text for details.

Fig. 4.
Fig. 4.

Emergence and dynamics of the incoherent MI pattern as the system evolves towards equilibrium. The intensity pattern is shown in the background, while superimposed on it are the average correlation distance Lc (z) (solid line; left axis), and the transversevariance of the intensity, M(z) (dashed line; right axis).

Fig. 5.
Fig. 5.

(a) Distribution of the lc values at the point (x 1,z 1)shown in Fig. 3a, from an ensemble of 800 samples. (b) Distribution of the lc (x, z) values chosen from a cut taken at a fixed x=x 1, and for various z. The values are chosen from a single-shot evolution, after the system has reached steady-state (i.e., only values for z>40 taken).

Fig. 6.
Fig. 6.

(a) Evolution of the power spectrum along Z. (b) Input power spectrum. (c) Power spectrum at z=150. (d) Power spectrum at various propagation distances. At z=5, the side-lobes are significant, indicating MI formation. For z>5, the power spectrum broadens gradually, until it reaches a constant width and fluctuates around that value, indicating the existence of a steady-state.

Fig. 7.
Fig. 7.

(a) Evolution of the correlation distance, Lc (z), for five values of the initial intensity; the plots are similar for z≥10. (b)<Lc (z)> z and the STD σz . Both are independent of the position of the center of the averaging window. (c) Zooming into the initial stages of evolution; the distance to equilibrium, increases as the total intensity decreases.

Fig.8.
Fig.8.

Relative loss of coherence ΔLc /Lc (z=0) (black curve), and the ratio σz /〈Lc (z)〉 z (red curve) at steadystate, both as functions of the initial coherence.

Equations (7)

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i Ψ n ( x , z ) z + 2 Ψ n ( x , z ) x 2 + I ( x , z ) Ψ n ( x , z ) = 0 n = 1 , 2 . . . . N ,
B ( x 1 , x 2 , z ) = E * ( x 2 , z , t ) E ( x 1 , z , t ) = n d n Ψ n ( x 1 , z ) Ψ n * ( x 2 , z )
B ( x 1 , x 2 , z ) z = i ( 2 B ( x 1 , x 2 , z ) x 1 2 2 B ( x 1 , x 2 , z ) x 2 2 ) + i ( B ( x 1 , x 1 , z ) B ( x 2 , x 2 , z ) ) B ( x 1 , x 2 , z )
l c ( x , z ) = μ ( x , x ' , z ) 2 x = 1 2 d d d μ ( x , x , z ) 2 d x
M ( z ) = ( I ( x , z ) I ¯ ( z ) ) 2 x = 1 2 d d d ( I ( x , z ) I ¯ ( z ) ) 2 d x
H = 1 2 d d d dx [ i ψ i ( x , z ) x 2 I ( x , z ) 2 ] T ( z ) + U ( z )
M ( z ) = 1 2 d d d dx [ I ( x , z ) 2 I ¯ ( z ) 2 ] = U ( z ) I 0 2 = H I 0 2 + T ( z )

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