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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…
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Outline of Talk
• Anderson Localization – Short Introduction
• Transverse Localization of Light
• Experiments
• Nonlinearity with Disorder
• Conclusion and Future Directions
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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
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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
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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
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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
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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
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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
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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
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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:
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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
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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
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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:
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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
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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]
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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
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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!
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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?)
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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