Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices

Yoav Lahini1, Assaf Avidan1, Francesca Pozzi2 , Marc Sorel2, Roberto Morandotti3Demetrios 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

• The Tight Binding Model (Discrete Schrödinger Equation)

[ ]111, −+± ++=∂∂

− nnnnnnn TE

ti ψψψψ

Slab waveguide

2D corex

y

z

4 µm 8 µm

• The discrete nonlinear Schrödinger equation (DNLSE)

[ ] nnnnnnnnn UUUUCU

zUi 2

111, γβ +++=∂∂

−+±

Ballistic expansion in 1d periodic lattice

Slab waveguide

2D corex

y

z

4 µm 8 µm

Propagation distance (AU)

Pos

ition

(site

num

ber)

100 200 300 400 500 600 700 800 900

80

60

40

20

0

-20

-40

-60

-80

Nonlinear localization in a periodic lattice

[ ] nnnnnn UUUUCU

zUi 2

11 γβ +++=∂∂

−+

Solitons of the discrete nonlinear Schrödinger equation (DNLSE)

Christodoulides and Joseph (1988)Eisenberg, Silberberg, Morandotti, Boyd, Aitchison, PRL (1998)

Beyond tight binding - Floquet-Bloch modesβ

(1/m

)

K (π/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

Slab waveguide

2D corex

y

z

4 µm 8 µm

[ ] nnnnnnnnn UUUUCU

zUi 2

111, γβ +++=∂∂

−+±

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

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 =+++∂∂

−+ nnnnn UUCU

zUi β

• The Anderson model:

• A measure of disorder is given by

.1, ConstC nn

nn

=∆+=

±

ββ Flat distribution, width ∆

c/∆

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

Eigenmodes of a periodic lattice N=99

Eigenvalues and eigenmodes for N=99, ∆/C=0Eigenvalues and eigenmodes for N=99, ∆/C=1Eigenvalues and eigenmodes for N=99, ∆/C=3

Eigenmodes of a disordered lattice 1=∆C

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

Experimental setup

• Injecting a narrow beam (~3 sites) at different locations across the lattice

(a)

(b)

(c)

(a) Periodic array – expansion(b) Disordered array - expansion(c) Disordered array - localization

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

Exciting Pure localized eigenmodes

Flat-phased localized eigenmodes Staggered localized eigenmodes

10 20 30 40 50 60 70 80 90Output Position

No

rma

lize

d in

ten

sity

Norm

aliz

ed in

tensi

ty

10 20 30 40 50 60 70 80 90Output Position

ExperimentTight-binding theory

The effect of nonlinearity on localized eigenmodes –weak disorder

85 90 95 1000

1c

Output Position

Nor

mal

ized

Inte

nsity

0

1b

aO

utpu

t Pow

er (

mW

)

0.1

0.2

0.3

0.4

d

0.2

0.4

0.6

0.8

0

1

60 70 80 900

1f

0

1e

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)

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

Position (AU)

Pow

er (

mW

)

100 200 300 400 500 600

3

2

1

Position (AU)

Pow

er (

mW

)

100 200 300 400 500 600

4

3

2

1

• 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

Wavepacket expansion in disordered lattices

The effect of nonlinearity on wavepacket expansion

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

Wavepacket expansion in a 1D disordered lattice

Propagation distance (AU)

Pos

ition

(site

num

ber)

100 200 300 400 500 600 700 800 900

80

60

40

20

0

-20

-40

-60

-80

Propagation distance (AU)

Pos

ition

(site

num

ber)

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

0.5f

0

0.5b

-10 0 10 0

0.5

Line

ar

g

Aver

aged

Inte

nsity

(arb

uni

ts)

Non

linea

r

0

0.5cσ=8.1 I

0=0.25 σ=7.9 I

0=0.32 σ=7.4 I

0=0.4 σ=5.9 I

0=1

σ=7.9 I0=0.24 σ=6.9 I

0=0.41 σ=6.6 I

0=0.48 σ=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

0.5f

0

0.5b

-10 0 10 0

0.5

Line

ar

g

Aver

aged

Inte

nsity

(arb

uni

ts)

Non

linea

r

0

0.5cσ=8.1 I

0=0.25 σ=7.9 I

0=0.32 σ=7.4 I

0=0.4 σ=5.9 I

0=1

σ=7.9 I0=0.24 σ=6.9 I

0=0.41 σ=6.6 I

0=0.48 σ=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 nonlinearityOr

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