Transport and Anderson Localization in Disordered 2D Photonic Lattices

Tal Schwartz, Guy Bartal, Shmuel Fishman and Mordechai Segev

Physics Department, Technion, Haifa 32000, Israel

Soon to be publishedin Nature

Magazine…

Outline of Talk

• Anderson Localization – Short Introduction

• Transverse Localization of Light

• Experiments

• Nonlinearity with Disorder

• Conclusion and Future Directions

Disorder and Anderson LocalizationPeriodic Potential:

Bloch waves(extended states) ( )xV

Ψ

A wave propagates freely through the medium Ballistic Transport/Diffraction

Disordered Potential:

Localized StatesTypical scale ξ

ξ

The wave can remain confined in some region of the potential

Philip W. Anderson, 1958 (Nobel Prize 1977)

- Localization Length

Why Localization?

∑≠

+=nm

mnmnnn uVuui εThe Anderson Model: A lattice with random on-site energies

The mixing between different sites:

Nearby sites - Large overlap, different energies

Far away sites – Similar energy, exponentially small overlap

mn

mn

mn

nm VVεεϕϕ

εε −=

−

Bound state of a single site

Energy of site n

Matrix elements, interaction among different sites

Why Localization? (2)Scattering by impurities generates random walk

Classical Diffusion

Constructive interference between recurrent scatterings

But: The wave is coherently scattered by defects

Higher return probability Quantum corrections to diffusion, Localization

Localization above critical disorder,

Phase Transition (Conductor-Insulator)

Almost all modes are localized, always

Dimensionality:

1-D l~ξ2-D ( )kll exp~ξ

3-D

mean free path

ξ1

Disorder Strength

Typically Large

Localization effects are hard to observe

Localization of Light – Brief Overview

Akkermans et al., J. Phys. Lett. 1985 (Coherent Back-Scattering)

John, PRL 1984Anderson, Phil. Mag. 1985

Pertsch, Lederer et al., PRL 2004Eisenberg, Silberberg, PhD Thesis, 2002

Propagation dynamics in disordered waveguide arrays. Fixed randomization of sites location due to fabrication process

THEORY

EXPERIMENT

EXPERIMENTS

Anderson Localization in disordered lattices never observed!

Albada & Lagendijk, PRL 1985Wolf & Maret, PRL 1985Etemad et al., PRL 1986

Wiersma et al., Nature 1997Chabanov et al., Nature 2000

Strong localization effects in transmission through 3-D

highly-scattering random medium

Experimental Observations of Coherent Back-Scattering

Well, why not?Thus far, Anderson localization in atomic lattices

was never observed

1. Phonons – the potential varies in time, electrons lose their phase coherence due to inelastic scattering

2. Many-body interactions – Nonlinearity effectively modifies the potential

The main problems (in solid state physics)

However: Localization is a WAVE phenomenon (not quantum)General for all wave systems! (e.g. Optics)

Anderson localization requires a disordered potential which is time invariant

Transverse Localization of LightSuggested by De Raedt, Lagengijk & de Vries (PRL 1989)

y

x

z

Localization = disorder eliminates diffraction

( ) ( ) ( )xezyxAzyxE wtkxi ˆ,,,, −=

cwnk 0= A- Slowly varying envelope of CW beam

( )A

nyxnA

kzA

ki

0

22

,2

1 Δ−∇−=

∂∂

⊥( ) ( )Ψ+Ψ∇−=∂

Ψ∂ rVmt

tri 22

2,

Optics Quantum Mechanicstz ↔1↔km↔12

2

2

2

yx ∂∂

+∂∂ Random refractive

index changez-independent

Time-invariant potential( )rVnn −↔Δ 0

Consider a beam, propagating along z in a bulk medium with transverse disorder,

uniform along propagation

y

x

z

Index modulation

410~ −

Relevant wavenumber -widthbeam

k 1∝⊥

⎟⎠⎞

⎜⎝⎛×= ⊥kd 2

exp2πξ

ℓ - mean free path

λπ 02 nkk =<<⊥

Transverse localization can be observed even with a very weak modulation of refractive index

Determined by fluctuations of Δn

Localization length:

Objective: We search for Anderson Localization in disordered 2D photonic lattices

(Periodic potential with super-imposed disorder)

But: It is a statistical problem (at finite distances): We must take ensemble averages (many realization of disorder)

DiffractionDiffraction

Transport Dynamics Along PropagationStudied numerically with typical experimental parameters

100

50

20

1 5 10

Disorder Level

Perfect Lattice Ballistic Transport

Diffusive Transport

21

~ zeffω

zeff ~ω

Average width ωeff

[µm]

Propagation Distance [mm]Diffusive Regime

Breakdown of Diffusion

Localization

Beam width after a given, finite propagation distance

A. Diffusive broadening up to a width ~ ξB. Absence of diffusion, Localization

Disorder Level

100

150

50

0 0 10 20 30

Ballistic Transport

Diffusive Regime Localization

Level of disorder

At a finite distance:

Disorder Level

100

150

50

0 0 10 20 30

Transport Dynamics Along PropagationStudied numerically with typical experimental parameters

When looking at a finite distance (as in our experiments):

2~ln xI − xI −~ln

Ensemble-Averaged Intensity

Average width ωeff

[µm]

But how can we tell Localization from simply reduced diffusion

???

We have to reveal the transport properties of the lattice!

(without looking inside the lattice)

Exponential decay of intensity profile marks the

Anderson Localization

Making A Disordered LatticeOptical Induction Technique

(idea: Christodoulides 2002, 1st demo: Fleischer 2003):

Interfere 3 beams on a photo-sensitive anisotropic material (SBN:60)

Hexagonal Lattice

Add another beam, passed through a diffuser (=random speckles)

Controlled Level of Disorder

mμ10≈• Material translates intensity pattern into an index modulation

Making A Disordered Lattice - ProblemWe use a diffuser for generating disorder by

a speckled intensity patternNarrow speckles diffract

Non-stationary propagation of writing beams

The lattice changes along propagation direction

(= time-dependent potential)

Localization effects are destroyed!

=Decoherenceby phonons

Solution:

( )222yxz kkkk +−=

All Fourier components travel

with the same velocity(random superposition of

Bessel beams)

Z-Independent Disordered Lattice

Lattice Beams Far-Field

kx

ky

Construct a ring-shaped angular

spectrum, with random phase and amplitude

along the ring.Then:

Experimental SetupDIFFUSER

Rotating the diffuser – different realization, same statistical properties

4-f SYSTEMCONICAL LENS

CRYSTAL INPUT FACE

Photorefractive SBN

CRYSTAL

WRITING BEAMSWEAK

FOCUSED PROBE ~

10μm

IMAGING SYSTEM

DIFFUSER

HIGH VOLTAGE

Disorder Beam

The process is repeated for statistical data

Measure of Confinement:(calculated for intensity distribution at lattice output, 10mm propagation)

( ) [ ]areadxdyI

dxdyIP /12

2

∫∫=

Averaged Effective Width:

inverse participation

ratio

Experimental Results

Averaged over 100 realizations of disorder

21−= Peffω

100

180

160

140

120

Relative disorder level [%] 0 10 20 30 40 50

2

Relative disorder level [%] 0 10 20 30 40 50

12

10

8

6

4

Statistical STD – Relative fluctuations ~1 (Mirlin, Phys. Rep. 2000)

]10[ 25 −− mP μAveraged IPR: Average effective width ωeff [μm]

Numerics:

100μm

Clear lattice 2.5% Disorder 15% 45%

Averaged output intensity cross-section (100 samples)at the lattice output (after 10mm propagation)

x

( )xI

( )xIln

Gaussian Profile Diffusion

Exponential Decay Localization

30%

5%

Experimental Observation of Anderson Localization

Disorder and NonlinearityNonlinearities (Coulomb interactions, nonlinear optical response)

may play a fundamental role in disordered systems

Thus far, the combined effect (NL + disorder) is yet unclearBy increasing the probe intensity – modifies the local index change of the lattice – Nonlinear propagation

Linear 1=α

3=α

2=α

][ mx μ0-50 50

100

180

160

140

120

Ave

rage

eff

ectiv

e w

idth

[µm

]

Relative disorder level [%]0 10 20 30 40 50

LinearSelf-Focusing with increasing

Nonlinearity (focusing) and fixed disorder

(15%)

( )xIln

α = NL strength, relative to lattice

Self-focusing nonlinearity promotes Anderson localization!

Conclusions• Experimental and numerical study of light

propagation in disordered 2D photonic lattices

• Observation of diffusive transport and its arrest by disorder

• First observation of Anderson Localization in any disordered lattice

• Experimental study of nonlinearity combined with disorder: Self-focusing promotes localization

• Same equations – Same Solutions:The ideas and methods can be applied to BEC(Anderson Transition in 3D?)

Where do we go next?

• How does the lattice band-structure influences localization?

• Nonlinear effects – focusing/defocusing at different dispersion regimes – special transitions as the band-structure deforms

• Solitons in disordered medium – formation and transport properties