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1. Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic LatticesYoav Lahini1, Assaf Avidan1, Francesca Pozzi2 , Marc Sorel2, Roberto Morandotti3 Demetrios N. Christodoulides4 and Yaron Silberberg11Department of Physics of Complex Systems, the Weizmann Institute of Science, Rehovot, Israel 2Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow, Scotland3Institute National de la Recherché Scientifique, Varennes, Québec, Canada4CREOL/College of Optics, University of Central Florida, Orlando, Florida, USA www.weizmann.ac.il/~feyaron

2. The 1d waveguide lattice • The Tight Binding Model (Discrete Schrödinger Equation) • The discrete nonlinear Schrödinger equation (DNLSE)

3. Ballistic expansion in 1d periodic lattice

4. Nonlinear localization in a periodic lattice Solitons of the discrete nonlinear Schrödinger equation (DNLSE) Christodoulides and Joseph (1988) Eisenberg, Silberberg, Morandotti, Boyd, Aitchison, PRL (1998)

5.  (1/m) Band 1 Band 2 K (/period) Band 3 Band 1 Band 2 Band 4 Band 3 Band 4 Band 5 Beyond tight binding - Floquet-Bloch modes Low power High power

6. The disordered waveguide lattice βn – determined by waveguide’s width - diagonal disorder Cn,n±1 – separation between waveguides – off-diagonal disorder γ–nonlinear (Kerr) coefficient Samples can be prepared to match exactly a prescribed set of parameters

7. In this work • Realization of the Anderson model in 1D • An experimental study of the effect of nonlinearity on Anderson localization: • Nonlinearity introduces interactions between propagating waves. This can significantly change interference properties (-> localization). Pikovsky and Shepelyansky: Destruction of Anderson localization by weak nonlinearity arXiv:0708.3315 (2007) Kopidakis et. al. : Absence of Wavepacket Diffusion in Disordered Nonlinear Systems arXiv:0710.2621 (2007) Experiments: Light propagation in nonlinear disordered lattices: Eisenberg, Ph.D. thesis, Weizmann Institute of Science, (2002). (1D) Pertsch et. al. Phys. Rev. Lett. 93 053901 ,(2004). (2D) Schwartz et. al. Nature 446 53, (2007). (2D)

8. The Original Anderson Model in 1D • The discrete Schrödinger equation (Tight Binding model) • The Anderson model: • A measure of disorder is given by Flat distribution, width Δ P.W. Anderson, Phys. Rev. 109 1492 (1958)

9. Eigenmodes of a periodic lattice N=99

10. Eigenvalues and eigenmodes for N=99, Δ/C=0 Eigenvalues and eigenmodes for N=99, Δ/C=1 Eigenvalues and eigenmodes for N=99, Δ/C=3

11. Eigenmodes of a disordered lattice

12. Eigenmodes of a disordered lattice N=99, Δ/C=1 :Intensity distributions

13. Injecting a narrow beam (~3 sites) at different locations across the lattice Experimental setup (a) (b) (c) • Periodic array – expansion • Disordered array - expansion • Disordered array - localization

14. Exciting Pure localized eigenmodes • Using a wide input beam (~8 sites) for low mode content. Flat-phased localized eigenmodes Staggered localized eigenmodes Experiment Tight-binding theory

15. The effect of nonlinearity on localized eigenmodes – weak disorder Flat phased modes Staggered modes • Two families ofeigenmodes, with opposite response to nonlinearity • Delocalization through resonance with the ‘extended’ modes G. Kopidakis and S. Aubry, Phys. Rev. Lett. 843236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

16. The effect of nonlinearity on localized eigenmodes – weak disorder G. Kopidakis and S. Aubry, Phys. Rev. Lett. 843236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

17. The effect of nonlinearity on localized eigenmodes – strong disorder • Delocalization through resonance with nearby localized modes G. Kopidakis and S. Aubry, Phys. Rev. Lett. 843236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

18. The effect of nonlinearity on localized eigenmodes – strong disorder G. Kopidakis and S. Aubry, Phys. Rev. Lett. 843236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

19. Wavepacket expansion in disordered lattices The effect of nonlinearity on wavepacket expansion • Single-site excitation • Short time behavior – from ballistic expansion to localization

20. Wavepacket expansion in a 1D disordered lattice

21. Wavepacket expansion in 1D disordered lattices:experiments • Wavepacket expansion on short time scales • Exciting a single site as an initial condition • Averaging

22. Wavepacket expansion in 1D disordered lattices:nonlinear experiments • Wavepacket expansion on short time scales • Exciting a single site as an initial condition • Averaging • The effect of weak nonlinearity: accelerated transition into localization

23. Wavepacket expansion in a nonlinear disordered lattice Single site excitation, positive/negative nonlinearity D.L. Shepelyansky, Phys. Rev. Lett, 70 1787 (1993), Pikovsky and Shepelyansky, arXiv:0708.3315 (2007) Kopidakis et. al., arXiv:0710.2621 (2007) Two site in-phase excitation, positive nonlinearity Or Two site out-of-phase excitation, negative nonlinearity Two site out-of-phase excitation, positive nonlinearity Or Two site in-phase excitation, negative nonlinearity

24. Summary • Realization of the 1D Anderson model with nonlinearity. • Full control over all disorder parameters. • Selective excitation of localized eigenmodes. • The effect of nonlinearity on eigenmodes in the weak and strong disorder regimes. • Wavepacket expansion in 1D disordered lattices: the buildup of localization • co-existence of a ballistic and localized component • no diffusive dynamics in 1D • Effect of (weak) nonlinearity on wavepacket expansion in disordered lattices: an accelerated buildup of localization