Abstract

We introduce soliton percolation phenomena in the nonlinear transport of light packets in suitable optical lattices with random properties. Specifically, we address lattices with a gradient of the refractive index in the transverse plane, featuring stochastic phase or amplitude fluctuations, and we discover the existence of a disorder-induced transition between soliton-insulator and soliton-conductor regimes. The soliton current is found to reach its maximal value at intermediate disorder levels and to drastically decrease in both, almost regular and strongly disordered lattices.

© 2007 Optical Society of America

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  16. V. A. Hopkins, J. Keat, G. D. Meegan, T. M. Zhang, and J. D. Maynard, "Observation of the predicted behavior of nonlinear pulse propagation in disordered media," Phys. Rev. Lett. 76, 1102 (1996).
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  17. V. V. Konotop, D. Cai, M. Salerno, A. R. Bishop, and N. Grønbech-Jensen, "Interaction of a soliton with point impurities in an inhomogeneous discrete nonlinear Schrödinger system," Phys. Rev. E 53, 6476 (1996).
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  21. P. L. Christiansen, Y. B. Gaididei, M. Johansson, K. O. Rasmussen, D. Usero, and L. Vazquez, "Stabilization of nonlinear excitations by disorder," Phys. Rev. B 56, 14407 (1997).
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  22. Y. B. Gaididei, D. Hendriksen, P. L. Christiansen, and K. O. Rasmussen, "Stationary states of the two-dimensio-nal nonlinear Schrodinger model with disorder," Phys. Rev. B 58, 3075 (1998).
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    [CrossRef] [PubMed]
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  43. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Oscillations of two-dimensional solitons in harmonic and Bessel optical lattices," Phys. Rev. E 71, 036621 (2005).
    [CrossRef]
  44. A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, "Transition radiation by matter-wave solitons in optical lattices," Phys. Rev. Lett. 91, 260402 (2003).
    [CrossRef]
  45. Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, "Spatial soliton switching in quasi-continuous optical arrays," Opt. Lett. 29, 766 (2004).
    [CrossRef] [PubMed]
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2007 (1)

T. Schwartz, G. Bartal, S. Fishman and M. Segev, "Transport and Anderson localization in disordered two-dimensional photonic lattices," Nature (London) 446, 52 (2007).
[CrossRef]

2006 (7)

A. A. Sukhorukov, "Enhanced soliton transport in quasiperiodic lattices with introduced aperiodicity," Phys. Rev. Lett. 96, 113902 (2006).
[CrossRef] [PubMed]

Y. J. Yun, I. C. Baek, and M. Y. Choi, "Phase transition and critical dynamics in site-diluted Josephson-junction arrays," Phys. Rev. Lett. 97, 215701 (2006).
[CrossRef] [PubMed]

G. Allison, E. A. Galaktionov, A. K. Savchenko, S. S. Safonov, M. M. Fogler, M. Y. Simmons, and D. A. Ritchie, "Thermodynamic density of states of two-dimensional GaAs systems near the apparent metal-insulator transition," Phys. Rev. Lett. 96, 216407 (2006).
[CrossRef] [PubMed]

V. I. Kozub, A. A. Zyuzin, Y. M. Galperin, and V. Vinokur, "Charge transfer between a superconductor and a hopping insulator," Phys. Rev. Lett. 96, 107004 (2006).
[CrossRef] [PubMed]

D. Clement, A. F. Varon, J. A. Retter, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, "Experimental study of the transport of coherent interacting matter-waves in a 1D random potential induced by laser speckle," New J. Phys. 8, 165 (2006).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef] [PubMed]

2005 (7)

S. Das Sarma, M. P. Lilly, E. H. Hwang, L. N. Pfeiffer, K. W. West, and J. L. Reno, "Two-dimensional metal-insulator transition as a percolation transition in a high-mobility electron system," Phys. Rev. Lett. 94, 136401 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Oscillations of two-dimensional solitons in harmonic and Bessel optical lattices," Phys. Rev. E 71, 036621 (2005).
[CrossRef]

R. C. Kuhn, C. Miniatura, D. Delande, O. Sigwarth, and C. A. Muller, "Localization of matter waves in two-dimensional disordered optical potentials," Phys. Rev. Lett. 95, 250403 (2005).
[CrossRef] [PubMed]

T. Schulte, S. Drenkelforth, J. Kruse, W. Ertmer, J. Arlt, K. Sacha, J. Zakrzewski, and M. Lewenstein, "Routes towards Anderson-like localization of Bose-Einstein condensates in disordered optical lattices," Phys. Rev. Lett. 95, 170411 (2005).
[CrossRef] [PubMed]

U. Gavish and Y. Castin, "Matter-wave localization in disordered cold atom lattices," Phys. Rev. Lett. 95, 020401 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov and V. A. Vysloukh, "Anderson localization of solitons in optical lattices with random frequency modulation," Phys. Rev. E 72, 026606 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton control in chirped photonic lattices," J. Opt. Soc. Am. B 22, 1356 (2005).
[CrossRef]

2004 (5)

Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, "Spatial soliton switching in quasi-continuous optical arrays," Opt. Lett. 29, 766 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton trains in photonic lattices," Opt. Express 12, 2831 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced disloca-tions in partially coherent photonic lattices" Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

A. P. Vinogradov and A. M. Merzlikin, "Band theory of light localization in one-dimensional disordered systems," Phys. Rev. E 70, 026610 (2004).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, "Nonlinea-rity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

2003 (5)

P G. Kevrekidis, Y. S. Kivshar, and A. S. Kovalev, "Instabilities and bifurcations of nonlinear impurity modes," Phys. Rev. E 67, 046604 (2003).
[CrossRef]

A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, "Transition radiation by matter-wave solitons in optical lattices," Phys. Rev. Lett. 91, 260402 (2003).
[CrossRef]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147 (2003).
[CrossRef] [PubMed]

2001 (2)

E. Yablonovitch, "Photonic crystals: Semiconductors of light," Scientific American 285, 47 (2001).
[CrossRef]

J. Garnier, "Propagation of solitons in a randomly perturbed Ablowitz-Ladik chain," Phys. Rev. E 62, 026608 (2001).
[CrossRef]

1999 (2)

Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, "Different regimes of light localization in a disordered photonic crystal," Phys. Rev. B 60, 1555 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

1998 (1)

Y. B. Gaididei, D. Hendriksen, P. L. Christiansen, and K. O. Rasmussen, "Stationary states of the two-dimensio-nal nonlinear Schrodinger model with disorder," Phys. Rev. B 58, 3075 (1998).
[CrossRef]

1997 (2)

P. L. Christiansen, Y. B. Gaididei, M. Johansson, K. O. Rasmussen, D. Usero, and L. Vazquez, "Stabilization of nonlinear excitations by disorder," Phys. Rev. B 56, 14407 (1997).
[CrossRef]

J. D. Joannopoulos, "Photonic-bandgap microcavities in optical waveguides," Nature 386, 143 (1997).
[CrossRef]

1996 (2)

V. A. Hopkins, J. Keat, G. D. Meegan, T. M. Zhang, and J. D. Maynard, "Observation of the predicted behavior of nonlinear pulse propagation in disordered media," Phys. Rev. Lett. 76, 1102 (1996).
[CrossRef] [PubMed]

V. V. Konotop, D. Cai, M. Salerno, A. R. Bishop, and N. Grønbech-Jensen, "Interaction of a soliton with point impurities in an inhomogeneous discrete nonlinear Schrödinger system," Phys. Rev. E 53, 6476 (1996).
[CrossRef]

1995 (1)

R. Knapp, "Transmission of solitons through random-media," Physica D 85, 496 (1995).
[CrossRef]

1993 (2)

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, "Dynamics and interaction of solitons on an integrable inhomogeneous lattice," Phys. Rev. E 48, 563 (1993).
[CrossRef]

Y. S. Kivshar and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077 (1993).
[CrossRef]

1991 (1)

R. Scharf and A. R. Bishop, "Properties of the nonlinear Schrödinger equation on a lattice," Phys. Rev. A 43, 6535 (1991).
[CrossRef] [PubMed]

1990 (1)

Y. S. Kivshar, S. A. Gredeskul, A. Sanchez, and L. Vazquez, "Localization decay induced by strong nonlinearity in disordered systems," Phys. Rev. Lett. 64, 1693 (1990).
[CrossRef] [PubMed]

1987 (2)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

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

Nature (2)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147 (2003).
[CrossRef] [PubMed]

J. D. Joannopoulos, "Photonic-bandgap microcavities in optical waveguides," Nature 386, 143 (1997).
[CrossRef]

Nature (London) (1)

T. Schwartz, G. Bartal, S. Fishman and M. Segev, "Transport and Anderson localization in disordered two-dimensional photonic lattices," Nature (London) 446, 52 (2007).
[CrossRef]

New J. Phys. (1)

D. Clement, A. F. Varon, J. A. Retter, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, "Experimental study of the transport of coherent interacting matter-waves in a 1D random potential induced by laser speckle," New J. Phys. 8, 165 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

R. Scharf and A. R. Bishop, "Properties of the nonlinear Schrödinger equation on a lattice," Phys. Rev. A 43, 6535 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (3)

Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, "Different regimes of light localization in a disordered photonic crystal," Phys. Rev. B 60, 1555 (1999).
[CrossRef]

P. L. Christiansen, Y. B. Gaididei, M. Johansson, K. O. Rasmussen, D. Usero, and L. Vazquez, "Stabilization of nonlinear excitations by disorder," Phys. Rev. B 56, 14407 (1997).
[CrossRef]

Y. B. Gaididei, D. Hendriksen, P. L. Christiansen, and K. O. Rasmussen, "Stationary states of the two-dimensio-nal nonlinear Schrodinger model with disorder," Phys. Rev. B 58, 3075 (1998).
[CrossRef]

Phys. Rev. E (8)

J. Garnier, "Propagation of solitons in a randomly perturbed Ablowitz-Ladik chain," Phys. Rev. E 62, 026608 (2001).
[CrossRef]

V. V. Konotop, D. Cai, M. Salerno, A. R. Bishop, and N. Grønbech-Jensen, "Interaction of a soliton with point impurities in an inhomogeneous discrete nonlinear Schrödinger system," Phys. Rev. E 53, 6476 (1996).
[CrossRef]

P G. Kevrekidis, Y. S. Kivshar, and A. S. Kovalev, "Instabilities and bifurcations of nonlinear impurity modes," Phys. Rev. E 67, 046604 (2003).
[CrossRef]

A. P. Vinogradov and A. M. Merzlikin, "Band theory of light localization in one-dimensional disordered systems," Phys. Rev. E 70, 026610 (2004).
[CrossRef]

Y. V. Kartashov and V. A. Vysloukh, "Anderson localization of solitons in optical lattices with random frequency modulation," Phys. Rev. E 72, 026606 (2005).
[CrossRef]

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, "Dynamics and interaction of solitons on an integrable inhomogeneous lattice," Phys. Rev. E 48, 563 (1993).
[CrossRef]

Y. S. Kivshar and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Oscillations of two-dimensional solitons in harmonic and Bessel optical lattices," Phys. Rev. E 71, 036621 (2005).
[CrossRef]

Phys. Rev. Lett. (19)

A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, "Transition radiation by matter-wave solitons in optical lattices," Phys. Rev. Lett. 91, 260402 (2003).
[CrossRef]

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef] [PubMed]

S. Das Sarma, M. P. Lilly, E. H. Hwang, L. N. Pfeiffer, K. W. West, and J. L. Reno, "Two-dimensional metal-insulator transition as a percolation transition in a high-mobility electron system," Phys. Rev. Lett. 94, 136401 (2005).
[CrossRef] [PubMed]

Y. J. Yun, I. C. Baek, and M. Y. Choi, "Phase transition and critical dynamics in site-diluted Josephson-junction arrays," Phys. Rev. Lett. 97, 215701 (2006).
[CrossRef] [PubMed]

G. Allison, E. A. Galaktionov, A. K. Savchenko, S. S. Safonov, M. M. Fogler, M. Y. Simmons, and D. A. Ritchie, "Thermodynamic density of states of two-dimensional GaAs systems near the apparent metal-insulator transition," Phys. Rev. Lett. 96, 216407 (2006).
[CrossRef] [PubMed]

V. I. Kozub, A. A. Zyuzin, Y. M. Galperin, and V. Vinokur, "Charge transfer between a superconductor and a hopping insulator," Phys. Rev. Lett. 96, 107004 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83, 4752 (1999).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Soliton intensity distributions corresponding to different realizations of lattices with phase fluctuations at σ p=0.03 (a), σ p=0.2 (b) and amplitude fluctuations at σ a=0.05 (c). In all cases p=0.4, α=0.0025, Ω=6, and L cor=1.8. Distributions corresponding to different lattice realizations are superimposed.

Fig. 2.
Fig. 2.

Soliton current versus dispersion of phase fluctuations at α=0.0025 and various lattice depths (a) and versus slope of linear potential at σ p=0.03 and p=0.4 (b). Soliton current versus dispersion of amplitude fluctuations at α=0.0025 and various lattice depths (c) and versus slope of linear potential at σ a=0.007 and p=0.4(d). In all cases Ω=6, L cor=1.8, and soliton current is calculated for 103 lattice realizations.

Fig. 3.
Fig. 3.

Statistics of the distance ξ c for lattices with phase fluctuations at σ p=0.1 (a) and amplitude fluctuations at σ a=0.03 (b). In both cases α=0.0025, Ω=6, p=0.4, and L cor=1.8

Fig. 4.
Fig. 4.

(a) Lattice with random phase fluctuations at Ω=6, σ p=0.1, L cor=1.8. (b) Soliton propagation trajectories on (ηcc) plane corresponding to different realizations of lattice with phase fluctuations at σ p=0.2, p=0.4, α=0.0015, Ω=6, and L cor=1.8.

Equations (3)

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i q ξ = 1 2 Δ q q q 2 R q α η q .
η c = U 1 η | q | 2 d η    d ζ ,            ζ c = U 1 ζ | q | 2 d η    d ζ ,
U = | q | 2 d η    d ζ

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