Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices Yoav Lahini...

Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices

Yoav Lahini1, Assaf Avidan1, Francesca Pozzi2 , Marc Sorel2, Roberto Morandotti3 Demetrios N. Christodoulides4 and Yaron Silberberg1

1Department of Physics of Complex Systems, the Weizmann Institute of Science, Rehovot, Israel 2Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow, Scotland

3Institute National de la Recherché Scientifique, Varennes, Québec, Canada4CREOL/College of Optics, University of Central Florida, Orlando, Florida, USA

www.weizmann.ac.il/~feyaron

The 1d waveguide lattice

nnnnnnnnn UUUUCUz

Ui

2

111,

• The discrete nonlinear Schrödinger equation (DNLSE)

Slab waveguide

2D corex

y

z

4 μm 8 μm

111,

nnnnnn

n TEt

i

• The Tight Binding Model (Discrete Schrödinger Equation)

Ballistic expansion in 1d periodic lattice

Slab waveguide

2D corex

y

z

4 μm 8 μm

Propagation distance (AU)

Pos

ition

(si

te n

umbe

r)

100 200 300 400 500 600 700 800 900

80

60

40

20

0

-20

-40

-60

-80

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)

nnnnnn UUUUCUz

Ui

2

11

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

The disordered waveguide lattice

nnnnnnnnn UUUUCUz

Ui

2

111,

βn – determined by waveguide’s width - diagonal disorderCn,n±1 – separation between waveguides – off-diagonal disorderγ – nonlinear (Kerr) coefficient

Samples can be prepared to match exactly a prescribed set of parameters

Slab waveguide

2D corex

y

z

4 μm 8 μm

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)

The Original Anderson Model in 1D

• The discrete Schrödinger equation (Tight Binding model)

011

nnnn

n UUCUz

Ui

• The Anderson model:

• A measure of disorder is given by

.1, ConstC nn

nn

Flat distribution, width Δ

P.W. Anderson, Phys. Rev. 109 1492 (1958)

c/

Eigenmodes of a periodic lattice N=99

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

Eigenmodes of a disordered lattice 1C

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

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(

• Using a wide input beam (~8 sites) for low mode content.

Exciting Pure localized eigenmodes

Flat-phased localized eigenmodes Staggered localized eigenmodes

ExperimentTight-binding theory

The effect of nonlinearity on localized eigenmodes

– weak disorder

Flat phased modesStaggered 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)

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84 3236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

The effect of nonlinearity on localized eigenmodes – weak disorder

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)

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84 3236 (2000) ; Physica D 139 247; 2000)) 130 155 (1999)

The effect of nonlinearity on localized eigenmodes – strong disorder

• Single-site excitation• Short time behavior – from ballistic expansion to localization

Wavepacket expansion in disordered lattices

The effect of nonlinearity on wavepacket expansion

Propagation distance (AU)

Pos

ition

(si

te n

umbe

r)

100 200 300 400 500 600 700 800 900

80

60

40

20

0

-20

-40

-60

-80

Wavepacket expansion in a 1D disordered lattice

Propagation distance (AU)

Pos

ition

(si

te n

umbe

r)

100 200 300 400 500 600 700 800 900

80

60

40

20

0

-20

-40

-60

-80

Wavepacket expansion in 1D disordered lattices:experiments

• Wavepacket expansion on short time scales • Exciting a single site as an initial condition• Averaging

0

1

Distance from input site (number of sites)

d

Increasing disorder

-10 0 100

1h

0

0.6 a

-10 0 10 0

0.6 e

-10 0 10 0

0.5

f

0

0.5

b

-10 0 10 0

0.5

Lin

ea

r

g

Ave

rag

ed

Inte

nsi

ty (

arb

un

its)

No

nlin

ea

r

0

0.5c=8.1 I

0=0.25 =7.9 I

0=0.32 =7.4 I

0=0.4 =5.9 I

0=1

=7.9 I0=0.24 =6.9 I

0=0.41 =6.6 I

0=0.48 =5.9 I

0=0.95

Wavepacket expansion in 1D disordered lattices:

nonlinear experiments

• Wavepacket expansion on short time scales • Exciting a single site as an initial condition• Averaging

0

1

Distance from input site (number of sites)

d

Increasing disorder

-10 0 100

1h

0

0.6 a

-10 0 10 0

0.6 e

-10 0 10 0

0.5

f

0

0.5

b

-10 0 10 0

0.5

Lin

ea

r

g

Ave

rag

ed

Inte

nsi

ty (

arb

un

its)

No

nlin

ea

r

0

0.5c=8.1 I

0=0.25 =7.9 I

0=0.32 =7.4 I

0=0.4 =5.9 I

0=1

=7.9 I0=0.24 =6.9 I

0=0.41 =6.6 I

0=0.48 =5.9 I

0=0.95

• The effect of weak nonlinearity: accelerated transition into localization

Wavepacket expansion in a nonlinear disordered lattice

Single site excitation, positive/negative nonlinearity

Two site in-phase excitation, positive nonlinearityOr

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)

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