Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices Yoav Lahini 1 , Assaf Avidan 1 , Francesca Pozzi 2 , Marc Sorel 2 , Roberto Morandotti 3 Demetrios N. Christodoulides 4 and Yaron Silberberg 1 1 Department of Physics of Complex Systems, the Weizmann Institute of Science, Rehovot, Israel 2 Department of Electrical and Electronic Engineering, University of Glasgow, Glasgow, Scotland 3 Institute National de la Recherché Scientifique, Varennes, Québec, Canada 4 CREOL/College of Optics, University of Central Florida, Orlando, Florida, USA www.weizmann.ac.il/~feyaron
Transcript
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
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
