1. Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices Yoav Lahini 1, Assaf Avidan 1, Francesca Pozzi 2, Marc Sorel 2, Roberto Morandotti 3 Demetrios N. Christodoulides 4 and Yaron Silberberg 1 1 Department of Physics of Complex Systems, the Weizmann Institute of Science, Rehovot, Israel 2 Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow, Scotland 3 Institute National de la Recherché Scientifique, Varennes, Québec, Canada 4 CREOL/College of Optics, University of Central Florida, Orlando, Florida, USA www.weizmann.ac.il/~feyaron

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

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

6. The disordered waveguide lattice β n – determined by waveguide’s width - diagonal disorder C n,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 1.Realization of the Anderson model in 1D 2.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=1Eigenvalues 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. Experimental setup Injecting a narrow beam (~3 sites) at different locations across the lattice (a)Periodic array – expansion (b)Disordered array - expansion (c)Disordered array - localization (a) (b) (c)

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

15. The effect of nonlinearity on localized eigenmodes – weak disorder Flat phased modes Staggered modes Two families of eigenmodes, with opposite response to nonlinearity Delocalization through resonance with the ‘extended’ modes G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84 3236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

16. The effect of nonlinearity on localized eigenmodes – weak disorder

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. 84 3236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

18. The effect of nonlinearity on localized eigenmodes – strong disorder

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

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 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 D.L. Shepelyansky, Phys. Rev. Lett, 70 1787 (1993), Pikovsky and Shepelyansky, arXiv:0708.3315 (2007) Kopidakis et. al., arXiv:0710.2621 (2007)

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