Anderson localization, wave diffusion and the effect of nonlinearity

in disordered lattices

Yoav Lahini1, Assaf Avidan1, Francesca Pozzi2 , Marc Sorel2, Roberto Morandotti4and 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 Recherche Scientifique, Varennes, Québec, Canada

Outline

• Introduction – Anderson localization• Motivation for this work• Tight-binding model with disorder• Experimental realization – disordered waveguide lattices• Linear measurements

– Eigenmodes– Effect of disorder in diffusion

• Nonlinear measurements

Introduction

• Periodic systems in QMExtended eigenmodes (Bloch modes) Bands of Eigenvalues (Energy)

Diffusion of electrons, conductivity.

• P.W. Anderson (1958): Disorder can suppress diffusion due to interference effects.Eigenmodes become localized in spaceIn 1,2 dimensions – for any disorder (infinite systems) In 3 dimensions – a metal insulator phase transition

• Similar description for classical waves.

Anderson, P. W. ‘Absence of Diffusion in Certain Random Lattices’ Phys. Rev. 109, 1492 (1958)

Anderson, P. W. ‘The question of classical localization: a theory of white paint?’ Phil. Mag. B, 52, 505-509 (1985).

Motivation for this work

1) The basic underling phenomena – localized modes and the decay of diffusion, are usually impossible to observe.

Localization was always deduced indirectly by measurements of macroscopic quantities of bulk samples such as conductance, transmission and reflection. (condensed-matter physics, optics)

2) Unresolved issues: the effect of nonlinearity on Anderson localization.

Nonlinearity introduces coupling between the propagating waves.

Nonlinearly accumulated phases of the propagating waves can significantly change their interference properties

Pertsch, T. et. al. Nonlinearity and Disorder in Fiber Arrays Phys. Rev. Lett. 93, 053901 (2004)

Emergence of localized Eigenmodes

• The discrete Schrödinger equation (Tight Binding model)

Un – amplitude at site nβn – on site eigenvalue (energy)C – tunneling rate between sites

• Solving a set of N equations to get eigenvalues and eigenmodes

• Perfectly periodic lattice:

• Disordered lattice : (‘Diagonal’ disorder)

[ ] 0111, =+++∂∂

−+± nnnnnnn UUCU

tUi β

ββ =n

nn Δ+= ββFlat box distribution

βn

n n+1

Un

Eigenmodes of a periodic lattice N=99

Eigenmodes of a disordered lattice 1=ΔC

10 20 30 40 50 60 70 80 90

−2

−1

0

1

2

Eig

enva

lue

/ C

Mode Number

(c)

20 40 60 80

0.4

0

−0.4

n (Site number)

Un

(b)

20 40 60 800

0.2

0.4

n (Site number)

Un

(a)

20 40 60 80

−0.1

0

0.1

n (Site number)

Un

(c)

The eigenvalue band

The waveguide array

Slab waveguide

2D corex

y

z

• Light is confined to travel in two dimentions

• x direction – ridges on top define an effective index of refraction in the slab - One dimensional lattice of weakly coupled waveguides.

• z direction – homogenous axis. Analogous to time evolution.

• Experiments - define initial condition across x , at z=0

4 μm 8 μm

Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behavior in linear and nonlinear waveguide lattices, Nature 424, 817 (2003)

The waveguide array

Slab waveguide

2D core

Linear regime: Discrete diffraction

[ ] 011 =+++∂∂

−+ nnnn UUCU

zUi β

Un – amplitude at site nβ – on site propagation constant (eigenvalue)C – coupling coefficient (tunneling)

Linear Model – the discrete Schrödinger equation

x

y

z

4 μm 8 μm

Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behavior in linear and nonlinear waveguide lattices, Nature 424, 817 (2003)

Effect of nonlinearity in a perfectly periodiclattice

[ ] 022011 =++++

∂∂

−+ nnnnnn UUnkUUCU

zUi β

The discrete nonlinear Schrödinger equation (DNLSE)

Un – amplitude in site nβ – on site propagation constantC – coupling coefficientn2 - Kerr coefficient

D. N. Christodoulides and R. I. Joseph (1988)H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison (1998)

• Randomly varying the width of each waveguide• Using numeric calculations we can determine:

βn waveguide n width / index stepCn,n+1 distance between waveguides n, n+1.

• Keeping Cn,n+1 = Constant• Exact disorder realization is known – linear experiments can be

compared to theory

Experimental setup

Introducing disorder

Experimental results - linear regime

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

(a) Periodic array - diffusion(b) Disordered array - No overlap with a localized state - diffusion(c) Disordered array - Overlap with a localized state – localization

(a)

(b)

(c)

Exciting Pure Anderson localized modes

• We measure intensities• First modes are mostly flat phased• Occupy separated regions in space• Ideally – can be excited with a wide, flat phased initial

condition.

• Using a wide input beam for low mode content.

Exciting Pure Anderson localized modes

Exciting Pure Anderson localized modes

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90 a

Output Position

Inpu

t Pos

ition

c

d

b

f

e

10 20 30 40 50 60 70 80 90

g

Output Position

10 20 30 40 50 60 70 80 90

10

20

30

40

50

60

70

80

90 h

Inpu

t Pos

ition

Output Position

k

j

l

i

10 20 30 40 50 60 70 80 90Output Position

m

• Using a wide input beam for low mode content.

Flat phased initial conditions Staggered initial conditions

The effect of nonlinearity on pure Anderson localized modes

85 90 95 1000

1c

Output Position

Nor

mal

ized

Inte

nsity

0

1b

a

Out

put P

ower

(m

W)

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

10 20 30 40 50 60 70 80 90

−2

−1

0

1

2

Eig

enva

lue

/ C

Mode Number

(c)

20 40 60 80

0.4

0

−0.4

n (Site number)

Un

(b)

20 40 60 800

0.2

0.4

n (Site number)

Un

(a)

20 40 60 80

−0.1

0

0.1

n (Site number)

Un

(c)

The effect of nonlinearity on pure Anderson localized modes

Wave diffusion – single site excitation

• In one and Two dimensions - Strong localization for any disorder strength.BUT- this is true for infinite samples, long time limit.what happens on short time scales - ?

• How to do it?

- Excite a single site as an initial conditionAll modes (flat and staggered) with overlap ≠ 0 are excited

diffusion

- Average over disorder realizations.OR

- Average over lattice sites of the same realization.

Wave diffusion – single site excitation

0

1

Distance from input site

σ = 16.4d

Increasing disorder

−20 −10 0 10 200

1σ = 16.4h

0

0.7

σ = 22.6a

−20 −10 0 10 200

0.7σ = 21.5

e

−20 −10 0 10 200

0.5

σ = 19.3f

0

0.5

σ = 22b

−20 −10 0 10 200

0.5

σ = 19.1

Lin

ear

g

Ave

rage

d In

tens

ity (

arb

units

)

No

nlin

ear

0

0.5

σ = 21.2c

The effect of nonlinearity on wave diffusion

0

1

Distance from input site

σ = 16.4d

Increasing disorder

−20 −10 0 10 200

1σ = 16.4h

0

0.7

σ = 22.6a

−20 −10 0 10 200

0.7σ = 21.5

e

−20 −10 0 10 200

0.5

σ = 19.3f

0

0.5

σ = 22b

−20 −10 0 10 200

0.5

σ = 19.1

Lin

ear

g

Ave

rage

d In

tens

ity (

arb

units

)

No

nlin

ear

0

0.5

σ = 21.2c

Wave diffusion – single site excitation

Time (arb. units)

Site

Num

ber

100 200 300 400 500 600 700 800 900 1000

-30

-20

-10

0

10

20

30

Conclutions

• Direct observation of Anderson localized modes

• Characterization of the different regimes of diffusion - from ballistic transport to localization

• Effect of nonlinearity on eigenmodes - depends on eigenvalue

• Effect of nonlinearity on wave diffusion – always increased localization

• Instability of a nonlinear beam in disordered lattices

Thank you