Abstract

We study modulation instability (MI) of random-phase waves in nonlinear photonic lattices. We find that an incoherent superposition of extended nonlinear eigenstates of the system, that is, an incoherent extended stationary beam, may become unstable due to nonlinearity. The instability process depends on the nonlinearity, on the structure of the diffraction curves of the lattice, as well as on the properties of the beam, whose spectrum can be comprised of Bloch modes from different bands, and from different regions of diffraction (normal/anomalous). This interplay among diffraction, incoherence, and nonlinearity leads to a variety of phenomena, including the possibility of tailoring the diffraction curve of the lattice, or the coherence properties of the beam, to enhance or suppress the instability. We present several examples of such phenomena, including a case where increasing the lattice depth flattens the diffraction curve thereby enhancing the instability, ”locking” the most unstable mode to the edge of the 1st Brillouin zone for large nonlinearity, and incoherent MI in self-defocusing media.

© 2007 Optical Society of America

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2006

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
[CrossRef]

O. Morsch and M. Oberthaler, "Dynamics of Bose-Einstein condensates in optical lattices," Rev. Mod. Phys. 78, 179-215 (2006).
[CrossRef]

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

D. Tr¨ager, R. Fischer, D.N. Neshev, A.A. Sukhorukov, C. Denz, W. Krolikowski, and Y.S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14, 1913-1923 (2006).
[CrossRef] [PubMed]

2005

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 (2005).
[CrossRef] [PubMed]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, "Incoherent modulation instability in instantaneous nonlinear Kerr media," Opt. Lett. 30, 2143-2145 (2005).
[CrossRef] [PubMed]

R. Iwanow, G.I. Stegeman, R. Schiek, Y. Min, and W. Sohler, "Discrete modulational instability in periodically poled lithium niobate waveguide arrays," Opt. Express 13, 7794-7799 (2005).
[CrossRef] [PubMed]

M. Stepi’c, C. Wirth, C.E. R¨uter, and D. Kip, "Observation of modulational instability in discrete media with self-defocusing nonlinearity," Opt. Lett. 31247-249 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
[CrossRef] [PubMed]

2004

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

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

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

2003

D.N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide arrays," Nature (London) 424817-823 (2003).
[CrossRef]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

A.A. Sukhorukov and Y.S. Kivshar, "Multigap discrete vector solitons," Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

2002

K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Z. Chen, S.M. Sears, H. Martin, D. N. Christodoulides andM. Segev, "Clustering of solitons in weakly correlated wavefronts," P. Natl. Acad. Sci. USA 99, 5223-5227 (2002).
[CrossRef]

H. Buljan, A. ˇSiber, M. Soljaˇci’c, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601 (2002).
[CrossRef]

2001

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

2000

M. Soljaˇci’c, M. Segev, T.H. 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]

D. Kip, M. Soljaˇci’c, 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]

1998

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

V.V. Shkunov and D. Anderson, "Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media," Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

1997

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]

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997).
[CrossRef]

1992

Y.S. Kivshar and M. Peyrard, "Modulation instabilities in discrete lattices," Phys. Rev. A 46, 3198-3205 (1992).
[CrossRef] [PubMed]

1988

Aitchison, J.S.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Alberici, E.

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

Anderson, D.

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

V.V. Shkunov and D. Anderson, "Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media," Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

Assanto, G.

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

Bartal, G.

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
[CrossRef]

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

Boyd, A.R.

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Bräuer, A.

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
[CrossRef] [PubMed]

Buljan, H.

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
[CrossRef]

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[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 (2005).
[CrossRef] [PubMed]

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

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

H. Buljan, A. ˇSiber, M. Soljaˇci’c, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601 (2002).
[CrossRef]

Carmon, T.

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

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

Centurion, M.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Chen, Z.

Z. Chen, S.M. Sears, H. Martin, D. N. Christodoulides andM. Segev, "Clustering of solitons in weakly correlated wavefronts," P. Natl. Acad. Sci. USA 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]

Christodoulides, D.N.

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

D.N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide arrays," Nature (London) 424817-823 (2003).
[CrossRef]

D.N. Christodoulides, E.D. Eugenieva, T.H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
[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).
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D.N. Christodoulides, and R.I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

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G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

Conti, C.

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[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]

Coskun, T.H.

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

Darmanyan, S.

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997).
[CrossRef]

Efremidis, N.K.

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

Eisenberg, H.S.

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Eugenieva, E.D.

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

Fleischer, J.W.

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
[CrossRef]

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

Frantzekakis, D.J.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Freedman, B.

Helczynski-Wolf, L.

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

Hulet, R.G.

K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Iwanow, R.

Joseph, R.I.

Kaiser, F.

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
[CrossRef] [PubMed]

Kevrekidis, P.G.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Kip, D.

D. Kip, M. Soljaˇci’c, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams", Science 290, 495-498 (2000).
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Kivshar, Y.S.

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
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A.A. Sukhorukov and Y.S. Kivshar, "Multigap discrete vector solitons," Phys. Rev. Lett. 91, 113902 (2003).
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Y.S. Kivshar and M. Peyrard, "Modulation instabilities in discrete lattices," Phys. Rev. A 46, 3198-3205 (1992).
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Klinger, J.

Kowarschik, R.

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
[CrossRef] [PubMed]

Lederer, F.

D.N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide arrays," Nature (London) 424817-823 (2003).
[CrossRef]

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997).
[CrossRef]

Lisak, M.

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

Manela, O.

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

Martin, H.

Z. Chen, S.M. Sears, H. Martin, D. N. Christodoulides andM. Segev, "Clustering of solitons in weakly correlated wavefronts," P. Natl. Acad. Sci. USA 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]

Meier, J.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

Michaelis, D.

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
[CrossRef] [PubMed]

Millot, G.

Min, Y.

Mitchell, M.

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

Morandotti, R.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
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O. Morsch and M. Oberthaler, "Dynamics of Bose-Einstein condensates in optical lattices," Rev. Mod. Phys. 78, 179-215 (2006).
[CrossRef]

Motzek, K.

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
[CrossRef] [PubMed]

Musslimani, Z.H.

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

Oberthaler, M.

O. Morsch and M. Oberthaler, "Dynamics of Bose-Einstein condensates in optical lattices," Rev. Mod. Phys. 78, 179-215 (2006).
[CrossRef]

Partridge, G.B.

K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Peccianti, M.

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

Peyrard, M.

Y.S. Kivshar and M. Peyrard, "Modulation instabilities in discrete lattices," Phys. Rev. A 46, 3198-3205 (1992).
[CrossRef] [PubMed]

Pezer, R.

Picozzi, A.

Pitois, S.

Porter, M.A.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Psaltis, D.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Pu, Y.

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
[CrossRef]

Relke, I.

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997).
[CrossRef]

Salamo, G.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

Sauter, A.

Schiek, R.

Schwartz, T.

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

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

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

Sears, S.M.

Z. Chen, S.M. Sears, H. Martin, D. N. Christodoulides andM. Segev, "Clustering of solitons in weakly correlated wavefronts," P. Natl. Acad. Sci. USA 99, 5223-5227 (2002).
[CrossRef]

Segev, M.

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
[CrossRef]

G. Bartal, R. Pezer, H. Buljan, O. Cohen, O. Manela, J.W. Fleischer, M. Segev, "Observation of random-phasegap solitons in photonic lattices", Opt. Lett. 31, 483-485 (2006).
[CrossRef] [PubMed]

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

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 (2005).
[CrossRef] [PubMed]

J.W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N.K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

R. Pezer, H. Buljan, J.W. Fleischer, G. Bartal, O. Cohen, M. Segev "Gap random-phase lattice solitons", Opt. Express 13, 5013-5023 (2005).
[CrossRef] [PubMed]

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

H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

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

Semenov, V.E.

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

V.V. Shkunov and D. Anderson, "Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media," Phys. Rev. Lett. 81, 2683-2686 (1998).
[CrossRef]

Silberberg, Y.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
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D.N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide arrays," Nature (London) 424817-823 (2003).
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H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Sohler, W.

Sorel, M.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

Stegeman, G.I.

R. Iwanow, G.I. Stegeman, R. Schiek, Y. Min, and W. Sohler, "Discrete modulational instability in periodically poled lithium niobate waveguide arrays," Opt. Express 13, 7794-7799 (2005).
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J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
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Strecker, K.E.

K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
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Streppel, U.

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
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Sukhorukov, A.A.

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
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A.A. Sukhorukov and Y.S. Kivshar, "Multigap discrete vector solitons," Phys. Rev. Lett. 91, 113902 (2003).
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K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

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 (2005).
[CrossRef] [PubMed]

Yang, H.

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
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Laser Phys. Lett.

M. Peccianti, C. Conti, E. Alberici, and G. Assanto, "Spatially incoherent modulational instability in a nonlocal medium," Laser Phys. Lett. 2, 25-29 (2005).
[CrossRef]

Nature

K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Nature (London)

O. Cohen, G. Bartal, H. Buljan, J.W. Fleischer, T. Carmon, M. Segev, and D.N. Christodoulides, "Observation of random-phase lattice solitons", Nature (London) 433, 500-503 (2005).
[CrossRef]

D.N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide arrays," Nature (London) 424817-823 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Optics Express

K. Motzek, A.A. Sukhorukov, F. Kaiser, and Y.S. Kivshar, "Incoherent multi-gap optical solitons in nonlinear photonic lattices", Optics Express 132916-2923 (2005).
[CrossRef] [PubMed]

P. Natl. Acad. Sci. USA

Z. Chen, S.M. Sears, H. Martin, D. N. Christodoulides andM. Segev, "Clustering of solitons in weakly correlated wavefronts," P. Natl. Acad. Sci. USA 99, 5223-5227 (2002).
[CrossRef]

Phys. Rev E

R. Pezer, H. Buljan, G. Bartal,M. Segev, and J.W. Fleischer, "Incoherent white-light solitons in nonlinear periodic lattices," Phys. Rev E 73, 056608 (2006).
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Phys. Rev. A

Y.S. Kivshar and M. Peyrard, "Modulation instabilities in discrete lattices," Phys. Rev. A 46, 3198-3205 (1992).
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Phys. Rev. E

S. Darmanyan, I. Relke, and F. Lederer, "Instability of continuous waves and rotating solitons in waveguide arrays," Phys. Rev. E 55, 7662-7668 (1997).
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H. Buljan, A. ˇSiber, M. Soljaˇci’c, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601 (2002).
[CrossRef]

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

D.N. Christodoulides, E.D. Eugenieva, T.H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601 (2001).
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Phys. Rev. Lett.

O. Cohen, T. Schwartz, J.W. Fleischer, M. Segev, and D.N. Christodoulides, "Multiband vector lattice solitons" Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

A.A. Sukhorukov and Y.S. Kivshar, "Multigap discrete vector solitons," Phys. Rev. Lett. 91, 113902 (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 (2005).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, H. Buljan, J.W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices", Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

U. Streppel, D. Michaelis, R. Kowarschik, and A. Bräuer, "Modulational instability in systems with integrating nonlinearity," Phys. Rev. Lett. 95, 073901 (2005).
[CrossRef] [PubMed]

H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

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

M. Soljaˇci’c, M. Segev, T.H. 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]

J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, "Experimental observation of discrete modulational instability," Phys. Rev. Lett. 92163902 (2004).
[CrossRef] [PubMed]

M. Centurion, M.A. Porter, Y. Pu, P.G. Kevrekidis, D.J. Frantzekakis, and D. Psaltis, "Modulational instability in a layered Kerr medium: Theory and experiment," Phys. Rev. Lett. 97234101 (2006).
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H. Buljan, O. Cohen, J.W. Fleischer, T. Schwartz, M. Segev, Z.H. Musslimani, N.K. Efremidis, and D.N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices", Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

V.V. Shkunov and D. Anderson, "Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media," Phys. Rev. Lett. 81, 2683-2686 (1998).
[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).
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Rev. Mod. Phys.

O. Morsch and M. Oberthaler, "Dynamics of Bose-Einstein condensates in optical lattices," Rev. Mod. Phys. 78, 179-215 (2006).
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Science

D. Kip, M. Soljaˇci’c, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams", Science 290, 495-498 (2000).
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Stud. Appl. Math.

H. Buljan, G. Bartal, O. Cohen, T. Schwartz, O. Manela, T. Carmon, M. Segev, J.W. Fleischer, D.N. Christodoulides, "Partially coherent waves in nonlinear periodic lattices", Stud. Appl. Math. 115, 173-208 (2005).
[CrossRef]

Other

G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001)

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Figures (4)

Fig. 1.
Fig. 1.

Features of incoherent lattice MI. (a) The c κ-spectrum of a stationary beam. (b) The intensity of the beam at z = 0 (red dashed line) and z = 283 mm (black solid line). (c) The linear diffraction curves βκ L vs. κ for p 0 = 0 (no lattice, dashed line) and p 0 = 3.4 ϗ 10-4 (solid line). (d) The d κ-spectrum of the noise intensity at z = 0 (red dotted line) and z = 236 mm (black solid line). (e) The exponential growth of the maximally destabilizing perturbations d κ; κ+2πn/D ,n = -1,0,1. (f) The growth rates g(α) reduced to the 1st BZ, as calculated from the numerical evolution (red dashed line) and Eq. (9) (black solid line).

Fig. 2.
Fig. 2.

(a) Gain curve for various values of the nonlinearity γI 0: 3.48 × 10-5 (black solid line), 3.74 × 10-5 (red dashed line), 3.87 × 10-5 (blue dotted line), and 4.2 × 10-5 (green dashed line). The most unstable mode locks at the edge of the first BZ, for sufficiently high values of the nonlinearity. (b) The power spectrum of the extended stationary state (see text for details).

Fig. 3.
Fig. 3.

Gain curves for two types of c κ-spectrum and various degrees of the coherence; a more incoherent beam has a broader c κ-spectrum. (a) The gain curves correspond to different widths of the bell-shaped and symmetric c κ-spectra in (b). (c) The gain curves correspond to multi-band beams with different widths of the two-humped c κ-spectra shown in (d).

Fig. 4.
Fig. 4.

Incoherent MI in self-defocusing nonlinear medium. (a) The c κ spectrum, and (b) the intensity structure of an unstable beam at z = 0 (red dashed line) and z = 263 mm (black solid line).

Equations (9)

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i B z + 1 2 k [ 2 x 1 2 2 x 2 2 ] B + k n 0 [ V ( x 1 , z ) V ( x 2 , z ) ] B = 0 .
B x 1 x 2 z = m d m ψ m ( x 1 , z ) ψ m * ( x 2 , z ) .
i ψ m z + 1 2 k 2 ψ m x 2 + V x z n 0 ψ m x z = 0 .
B 0 x 1 x 2 = κ c κ ϕ κ N ( x 1 ) ϕ κ N * ( x 2 ) ,
N ̂ ( x ) ϕ κ N ( x ) = β κ N ϕ κ N ,
N ̂ ( x ) = 1 2 k 2 x 2 + k n 0 [ p ( x ) + δn ( κ c κ ϕ κ N ( x ) 2 ) ] ;
i B 1 x 1 x 2 z z + [ N ̂ ( x 1 ) N ̂ ( x 2 ) ] B 1 ( x 1 , x 2 , z ) + n 0 [ B 1 ( x 1 , x 2 , z ) B 1 ( x 1 , x 1 , z ) ] B 0 x 1 x 2 = 0 ;
[ N ̂ ( x 1 ) N ̂ ( x 2 ) ] B α + n 0 [ B α x 1 x 1 B α x 2 x 2 ] B 0 x 1 x 2 = ig B α x 1 x 2 .
I α ( x ' ) = α 1 α 2 n 0 ϕ α 2 N * ( x ' ) ϕ α 1 N ( x ' ) ig + β ( α 1 ) β ( α 2 ) I α ( x ) ϕ α 2 N ( x ) ϕ α 1 N * ( x ) dx [ c α 2 c α 1 ] .

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