solitons in nonlinear photonic lattices · first use: mitchell, segev, christodoulides 1997...
TRANSCRIPT
Moti Segev
Physics Department, Technion – Israel Institute of Technology
Nikolaos K. Efremidis Jared HudockDemetrios N. Christodoulides
School of Optics/CREOL, University of Central Florida
Jason W. Fleischer (now @ Princeton)Oren Cohen (now @ Univ. of Colorado)Hrvoje Buljan (now @ Univ. of Zagreb)Tal Carmon (now @ Caltech)Guy BartalBarak Freedman Ofer ManelaTal Schwartz
Physics Department, Technion
Solitons in Nonlinear Photonic Lattices
Outline
Nonlinear lattices in science: general problem of wave propagation, role of optics
Lattice solitons
Optical induction of nonlinear photonic lattices
1D spatial “gap” lattice solitons
2D lattice solitons
Vortex-ring lattice solitons ; higher-band vortices
Multi-band vector lattice solitons
Random-phase lattice solitons
Brillouin zone spectroscopy of linear and nonlinear photonic lattices
Photonic Quasi-Crystals
Conclusions
Coupled anharmonic oscillators
E. Fermi, J. Pasta, and S. UlamLos Alamos Report LA-1940 (1955).
Nonlinear Lattices in Science
Atomic chain
Charge density waves in transition metalsB.I. Swanson et al., PRL 82, 3288 (2000).
Spin waves in antiferromagnetsU.T. Schwartz et al., PRL 83, 223 (1999).
A. Xie et al., PRL 84, 5436 (2000).
Biology: phonon energy in α-helices
A.S. Davydov, J. Theor. Biol. 38, 559 (1973).
Myoglobin
• Potential applications (e.g. photonics)
D.N. Christodoulides et al., PRL 87, 233901 (2001).
• Easily control input
• Directly image output
Advantages:
D.N. Christodoulides et al., Opt. Lett. 13, 794 (1988).1D:
N.K. Efremidis et al., PRE 66, 046602 (2002).2D:
Optics
H.S. Eisenberg et al., PRL 81, 3383 (1998).
J.W. Fleischer et al., Nature 422, 147 (2003).
A. Xie et al., PRL 84, 5436 (2000).
Atomic chain
CDW waves in transition metalsB.I. Swanson et al., PRL 82, 3288 (2000).
Spin waves in antiferromagnetsU.T. Schwartz et al., PRL 83, 223 (1999).
Coupled anharmonic oscillators
Biology: phonon energy in α-helices
A.S. Davydov, J. Theor. Biol. 38, 559 (1973).
Nonlinear Lattices in Science
Myoglobin
BeamEnvelope Diffraction Kerr NL Periodic index
( ) 021 2
22 =∆++∇+
∂∂
⊥⊥ ψψψψψ rnnzi Array
Nonlinear Schrödinger equation
Optics
E. Fermi, J. Pasta, and S. UlamLos Alamos Report LA-1940 (1955).
A. Xie et al., PRL 84, 5436 (2000).
Atomic chain
CDW waves in transition metalsB.I. Swanson et al., PRL 82, 3288 (2000).
Spin waves in antiferromagnetsU.T. Schwartz et al., PRL 83, 223 (1999).
Coupled anharmonic oscillators
Biology: phonon energy in α-helices
A.S. Davydov, J. Theor. Biol. 38, 559 (1973).
Nonlinear Lattices in Science
Myoglobin
BeamEnvelope Diffraction Kerr NL Periodic index
( ) 021 2
22 =∆++∇+
∂∂
⊥⊥ ψψψψψ rnnzi Array
Optics
Nonlinear Schrödinger equation
Bose-Einstein Condensates
~~~~~~
~~~~~~~~~
A. Trombettoni and A. Smerzi, PRL 86 2353 (2001).
B. Eiermann et al., PRL 92, 230401 (2004).
Mean-field interactions Periodic potential
Gross-Pitaevskii (NLS) equation
E. Fermi, J. Pasta, and S. UlamLos Alamos Report LA-1940 (1955).
ħ ( )ψψψψψrVU
mti ++∇−=
∂∂ 2
02
2
2ħ
Linear Waves in Periodic Media
Homogeneous: V = 0 Lattice: V(x+d) = V(d)
Translational symmetry
Fourier basis
Periodicity
Floquet-Bloch basis
β1.0
0.5
0
-0.5
-1.01.0 2.00.5 1.5
kxπ/d
( ) ( ) ziexzx βψ Φ=,( ) 02
2
=+∂∂
+∂∂
xVxz
iψψ
( ) ( )xudxuxx kk =+
( ){ }xikk
x
xexu{ }xikxe
1.0
0.5
0
-0.5
-1.01.0 2.00.5 1.5
kxπ/d
β
2xz kk −≅≡ ∆β[ /2k ]
(Bloch) momentum
θ
z
xx k
kzx
k ≅∆∆
=→ θQuadraticwavefront
x 0
1V
Linear Waves in Periodic Media
The modes (Floquet-Bloch waves) are extended waves
Transmission spectrum is divided into bands
Mode characterization: Band index and Bloch wavenumber kx {-π/d < kx < π/d}
Brillouin zone
π/d
β
-π/dkx
1a b
cd
Consider array where each WG supports a single guided mode
FB modes of single-mode WG array
kx = 0 kx = π/d
a b
cd
1st
band
2nd
band(For individual WG, highermodes would radiate away)
Linear Transport in Lattices
Eisenberg et al., PRL 85, 1863 (2000).
+ Anomalous
Homogeneous
Divergence of rays gives diffraction:
Normal of wavefront defines direction of transport (ray):
x
z
kk
zx
∂∂
=∆∆
=θ
2
2
x
z
kkD
∂∂
≡2
2
x∂∂ ψ ↔
β ≡ kz
kx
π/d
n=3
Normal
Periodic
-π/d
n=1
n=2
Diffraction
Linear Transport in Latticesβ ≡ kz
kx
π/d
n=3
Homogeneous Periodic
-π/d
n=1
n=2
Divergence of rays gives diffraction:
Normal of wavefront defines direction of transport (ray):
x
z
kk
zx
∂∂
=∆∆
=θ
2
2
x
z
kkD
∂∂
≡2
2
x∂∂ ψ ↔
2
2
21 1
kmeff ∂
∂≡− ε
Effective mass in lattice Dispersion of temporal pulse
2
21ωωωω ∂
∂=
⎠⎞
⎜⎝⎛
∂∂
∂∂
=⎠
⎞⎜⎜⎝
⎛
∂∂
≡ kkv
Dgr
⎜ ⎜
General definition of transport:
vgr
grating
Transport in Latticesβ ≡ kz
kx
π/d
n=3
Homogeneous Periodic
-π/d
n=1
n=2
Divergence of rays gives diffraction:
Normal of wavefront defines direction of transport (ray):
x
z
kk
zx
∂∂
=∆∆
=θ
2
2
x
z
kkD
∂∂
≡2
2
x∂∂ ψ ↔
Soliton formationwith focusing NL
Normal: self-focusing nonlinearity
Anomalous: self-defocusing nonlinearity
D.N. Christodoulides et al., Opt. Lett. 13, 794 (1988).H.S. Eisenberg et al., PRL 81, 3383 (1998).
Y.S. Kivshar, Opt. Lett. 18, 1147 (1993).J.W. Fleischer et al., PRL 90, 023902 (2003).
Linear transport of a wave-packet in a latticeHomogeneous system
Tra
nsve
rse
disp
lace
men
t
Develops two “lobes” Gaussian profile stays Gaussian
Lattice transport
Treats only bound states– decay between sites
No radiation modes
Tight-binding approximation
Continuous models are more general…
Coupling between nearest-neighbors
Analogous to mass-spring system
n-1 n n+1d
A.L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
1 1( ) 0nn n
dEi C E Edz − ++ + =
Discrete Model:
Non-linear Transport in Lattices
In-phase kx = 0 Staggered (GAP) kx= π/d
kk
kk
z
xxθ ≈=
Edge of Brillouin zone
Propagation distance
“Discrete” soliton
21 1( ) 0n
n n n ndEi C E E E Edz
γ− ++ + + =
when nonlinearitybalances diffraction
Christodoulides et al. (1988).Eisenberg et al., (1998).
Kivshar, (1993). Fleischer et al., (2003)
Lattice (“discrete”) solitons:
on-axis
Tra
nsve
rse
disp
lace
men
t
Propagation distance
Lattice (“discrete”) solitons
In-phase kx = 0 Gap (Staggered) kx= π/d
21 1( ) 0n
n n n ndEi C E E E Edz
γ− ++ + + =
D.N. Christodoulides et al. (1988).H.S. Eisenberg et al., (1998).
Y. Kivshar, (1993). J.W. Fleischer et al., (2003)
β
π/d-π/dkx
Lattice soliton is bound state of its own self-induced defect
β β ββββ
ββ ∆−
β β ββββ
ββ ∆+
Propagation distance
“Discrete” soliton
Tra
nsve
rse
disp
lace
men
t
Propagation distance
β
π/d-π/dkx In-phase kx = 0
D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988).
H.S. Eisenberg et al.PRL 81, 3383 (1998).
Y.S. Kivshar, Opt. Lett. 18, 1147 (1993). J.W. Fleischer et al.PRL 90, 23902 (2003)
Gap (Staggered) kx= π/d
Twisted (dipole) 0 < kx<< π/dDarmanyan et al.,Sov. Phys. JETP 86, 682 (1998)
Neshev et al., Opt. Lett. 9, 710 (2003)
2D: Yang et al., Opt. Lett. 29, 1662 (2004)
Vector at kx=0
Meier et al., PRL 91, 143907 (2003).
2D: Chen et al., Opt. Lett. 29, 1656 (2004)
Darmanyan et al.,Phys. Rev. E, 57, 3520 (1998)
Theory Experiment
Zoology of Lattice Solitons
2D: J.W. Fleischer et al., Nature 422, 147 (2003)More recent:
- Lattice solitonsin quadratic media
Stegeman’s group 2004
- Lattice solitonsin liquid crystals
Assanto’s group 2004
- Modulation instabilityStegeman’s group 2004
Finding Lattice Solitons: the self-consistency method
The soliton is a guided mode of its own induced waveguide
Idea: Askar’yan, 1962
Formulation: Snyder et al. 1991
First use: Mitchell, Segev, Christodoulides 1997
HomogeneousThe soliton is a localized mode of the
full potential (lattice + induced defect)
Lattice
Localized modeFull potential
(lattice + induced defect)
Intensity I
∆n
( ) IxVx
Φ=Φ⎥⎦
⎤⎢⎣
⎡++
∂∂ β2
22I Φ=
O. Cohen et al. PRL 91, 113901 (2003).
Intensity
∆nGuided mode
∆n = ∆n(I)
Self-Consistency: 2nd-band lattice solitonThe soliton is a localized mode of the full potential
(lattice + induced defect)
Single radiation modefrom second-band
O. Cohen et al. PRL 91, 113901 (2003).
Optical Solitons in Nonlinear Waveguide Arrays
D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988).
H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, PRL 81, 3383 (1998).
••
Motivation
• Fixed waveguide arrays
• 1D topologies
• Self-focusing nonlinearityonly certain classes of solitons
Previous experimental configurations: Challenges:
• Reconfigurable lattices
• 2D latticescollisions, angular momentum
• All classes of solitons allowed
T. Pertsch et al., OSA Trends inOptics and Photonics 80 (2002).
Transition to 2D – How?
• Direct manufacturing, etching
- difficult- possible on microwave scale, but linear- unknown process to date
• Naturally-occurring 2D structure
- atomic scale x-ray (again, linear)- unlikely- none found so far
• Strong nonlinear response – solitons at low intensity (mW)• Strength and sign of nonlinearity – adjustable• Dynamically adjustable – lattice spacing and potential well depth
All-optical
- induction technique (à la holography)
material {
Optically-induced Waveguide Arrays
c-axisWant strong optical anisotropy
e.g. photorefractive (SBN-75) crystal
• Array propagates linearly• Probe feels periodic potential, NL self-focusing
Polarize probe || to c-axis
Igrating:Iprobe
~ 5-10:1
Requirements
Create WGs by interfering pairs of plane waves– polarize ⊥ to c-axis V
Apply Voltage || to c-axisV > 0 ⇒ focus, V < 0 ⇒ defocus
• “Sharpens” array • Nonlinearity: ∆n ~ 10-3
NL requires very low power, but has slow response time
Optical Induction of 2D Array of 2D Waveguides
Crystal output
Representative square array WG diam = 7µm D=11µm
Igrating:Iprobe = 5:1
Interfere 2 pairs of plane waves to create dynamic 2D array
Strong anisotropy of crystal allows distinction betweeninduced lattice (waveguide array) and signal beam
Spatial scale of individual waveguides is small, but must consider diffraction of broad (in-phase) plane waves.
15mW in each plane wave
On-axis Propagation in 2D NL Photonic Lattice
8x reduction of Iprobe
8x red. of Iprobe at same voltage Interferogram
Lattice Diffraction (200V) 2D Lattice Soliton (800V) Soliton simulation
(800V)
Interferencewith plane waves
J.W. Fleischer et al.,Nature 422, 147 (2003)
2D Propagation at Edge of Brillouin Zone
8x red. of Iprobe at same voltage Interferogram
2D Lattice Soliton (-800V) Soliton simulation
Interferencewith plane waves
8x reduction of Iprobe
(-800V)
Lattice Diffraction (-200V)
J.W. Fleischer et al.,Nature 422, 147 (2003)
Spatial Gap Solitonsβ
π/d-π/dkx
β β ββββ
ββ ∆−
J.W. Fleischer et al., PRL 90, 023902 (2003).
J.W. Fleischer et al.,Nature 422, 147 (2003)
From 1st band with defocusing nonlinearity
1D:
2D:
β
π/d-π/dkx
β β ββββ
ββ ∆+
D. Mandelik et al., PRL 90, 053902 (2003).
From 2nd band with focusing nonlinearity
Vortex-Ring Lattice Solitons
J.W. Fleischer et al., PRL 92, 123904 (2004).-----------------------------* see also Neshev et al.PRL 92, 123903 (2004).
Advantage of 2D: solitons with angular momentumButlattice breaks continuous rotational symmetry of homogeneous medium
in general, angular momentum in lattice not conserved
Off-site On-site
Diffraction
Intensity
Phase
0, 2π
3π/2π
π/2
3π/2
π/2
π
0, 2π
Prediction of vortex lattice solitons:• Malomed & Keverkidis, PRE 2001• Yang & Musslimani, Opt. Lett. 2003
Experiments:
2nd Band vortex lattice soliton2D Lattice Transmission spectrum
X
MX
Phase structure – array of counter-rotating vortices
In the 2nd band the edge of BZ is at the X points
Solitonintensity
Manela, Cohen, Bartal, Fleischer, and Segev.Opt. Lett. 17, 2049 (2004)
Diffractionintensity
Evolution from a ring to a 2nd Band vortex lattice solitonExcitation (input)
OutputDiffraction
Solitonintensity
Powerspectrum
Soliton phase(interferogram)
Input phase(interferogram)
Inputintensity
Powerspectrum
1st band vortex vs. 2nd band vortex
1st band
Excitation(input) diffraction k-space
2nd band
Soliton Phase information(interferogram)
Bartal, Manela, Cohen, Fleischer, and Segev, submitted to PRL, Feb. 2005
Multiband vector lattice solitons
Localized modesFull potential
(lattice + induced defect)
Intensity
O. Cohen et al. PRL 91, 113901 (2003). [similar work: Sukhorukov et al. , PRL 91, 113902 (2003)]
( ) iiiIxVx
Φ=Φ⎥⎦
⎤⎢⎣
⎡++
∂∂ β2
222
ii
icI Φ= ∑ { }xkni ,=
Have vector soliton with components from different bands, each with same curvature
Fundamental + 2nd-band mode
self-trap as vector soliton
π/d
β
-π/d
kx
∆n
Self-consistencymethod
Multi-band vector lattice solitons
β 1
π/d-π/d kx
1st
2nd
-40
0
40
x
0 150z
-40
0
40x
Lineardiffraction
Vectorsoliton
Individualwith NL
BPMResults
Self-focusing nonlinearity
Noisy intensities at input
Intensities at output (on unperturbed sols.)
O. Cohen et al. PRL 91, 113901 (2003).
Richer dynamics than traditional solid-state (with transitions between passive bands)
• Both bands contribute actively to NL• Bands interact with each other (through I)
Multi-band vector lattice solitons
β 1
π/d-π/d kx
1st
2nd
-40
0
40
x
0 150z
-40
0
40x
Lineardiffraction
Vectorsoliton
Individualwith NL
-50
50
x
0 z
50
-50
x
150
βkx
1
π/d-π/d
BPMResults
Lineardiffraction
Vectorsoliton
Individualwith NL
Self-focusing nonlinearity Self-defocusing nonlinearity
Spatially-incoherent lattice solitons
d
lc
• Interference phenomena drive dynamics
– greatly affected by coherence of waves
Propagating waves undergo multiple reflections
Correlation length lc vs. lattice spacing d
• Explore with statistical (spatially-incoherent) light
All previous research on lattice solitons performed with coherent waves
Phase is perfectly correlated (space & time) – staggered, vortex
Random nature of light vs. periodic constraint
(Bloch’s theorem)(Mutual correlation)
Have lattice soliton with randomly-varying phase front
( ) ( )τ
ψψ tzxtzx ,,*,, 21 )()( dxfxf +=
System: partially coherent light in nonlinear waveguide arrays
State of the system (intensity and statistics at a given z)
mτ
*, ,z,t)(x,z,t)EE(xz),xB(x 2121 =
Equation of motion for the mutual coherence function:
x
z
x1
x2
Spatially incoherent light beam
Periodically modulatednoninstantaneous nonlinear medium
0
21
2122110
210
22
2
21
2
=−
+−+⎥⎦
⎤⎢⎣
⎡∂∂
−∂∂
+∂∂
,z),x,z)]}B(x,xδn[B(x,z)],xn[B(x{nk
)}Bp(x){p(xnk
xB
xB
kzBi
δ
Diffraction Periodicity
Nonlinear term
lc
Note ls(x) = ls(x+mD)
Mutual coherence/correlation function
H. Buljan et al., Physical Review Letters 92, 223901 (2004).
( ) ( )( ) ( )δ
δψψµ
+
+=
xIxI
xx Xnn
Typical results: Random-phase lattice solitons
Transmission spectrum
π/d-π/d
β
kx
∆n
x
Many modes low lc Slowest-decaying mode
higher lc
Interpretation using modal theory
kxπ/d
Fourier
Floquet-Bloch
Power spectrum
I
I
lc
Note ls(x) = ls(x+mD)
Mutual coherence/correlation function
H. Buljan et al., Physical Review Letters 92, 223901 (2004).
( ) ( )( ) ( )δ
δψψµ
+
+=
xIxI
xx Xnn
Typical results: Random-phase lattice solitons
Transmission spectrum
π/d-π/d
β
kxkxπ/d
Fourier
Floquet-Bloch
Power spectrum
I
I
Intensity profiles, power spectra, andstatistics (coherence properties)
all conform to the lattice periodicity
Incoherent lattice solitons – Experimental setup
2. Spatially incoherent (probe) beam
4f system
1. Optical induction of lattice
Rotating diffuser
laser
Spatial filteraperture controlspower spectrum
a. Real space
b. Fourier space
3. Imaging into CCD camera
f
Optical Fourier transform
f
Fourier space and diffusion
Brillouin zones
1st zone
2nd 2nd
2nd2nd
Points from array
Array beams define Bragg anglesWith diffuser(incoherent)
Diffraction in homogeneous medium
Lλ .
cvsw l
λ λ⎛ ⎞⎜ ⎟⎝ ⎠
determined by
incoherent
input
reflection
No diffuser(coherent)
coherent
Homogeneous
Diffraction
Fourierspace
Real Spaceoutput
Inputbeam
Incoherent lattice solitons – Experimental results
Rotating diffuser
laser
Realspace
Random-phase lattice solitonDiffractionInput
1-peak 2-peak
d = 11.5 microns
correlation length lc ~ d Brillouin zones
1st zone
2nd 2nd
2nd2nd
Incoherent probe Soliton output
Brillouin zone excitement !!
Varying the spatial correlation distance
Low intensityDiffraction
Input Real space
d = 11.5 microns
Correlationlength lc ~ d
Input Fourier space
SolitonReal space
Soliton Fourier space
Correlationlength lc ~1.2 d
Latticeoff
Zero voltageDiffraction
Input Real space
Input Fourier space
OutputReal space
OutputFourier space
Gap Random-Phase Lattice Solitons (under self-defocusing nonlinearity)
Low intensityDiffraction
Input Real space
Input Fourier space
SolitonReal space
Soliton Fourier space
What happens if we apply a self-focusing nonlinearity
on an input with a square hole in the spectrum?
General technique: Bloch-wave spectroscopy
Spatially-incoherent probe beam
wide in both real spaceand Fourier space
Experimental lattice
Theoretical Brillouin zones
Image of output
4-fold symmetry 3-fold symmetry
Trigonal lattice
Made in Israel
1st zone2nd 2nd
2nd 2n
d
3rd
3rd
3rd
3rd
3rd
3rd
3rd 3rd
Photonic lattices with defectsOur experiments
Positive defect
Negativedefect
No defect
K-space
GuidedModes
Real spacePhotonic Crystal Fibers
P.S. Russell, Science 299, 258 (2003).
Far-fielddispersion Real space
Bloch-wave spectroscopy: Nonlinear effect
1st zone
2nd 2nd
2nd2nd
De-focusing nonlinearity
De-focusing
kxπ/d 2π/d
Focusing
kz• Exchanges energy between
linear modes
• Sensitive to band curvature
• Maps regions of differenttransport
Nonlinearity:
Self focusing nonlinearityNonlinearity off
35µm
30µm 35µm
Linear Diffraction
a b
Interference of 5 plane waves
a
b
85µm
Theory Experiment
Penrose Tiling
Photonic Quasi-Crystals
Under self-focusing
Towards solitonsin
Quasi-Crystals
The k-space picture of photonic Quasi-Crystals
Quasi Brillouin Zones
Theory Experiment
Same scattering at different spots Statistical similarity Crystal
Summary: recent group progress on waves in nonlinear photonic lattices
• First experimental demonstration, in any medium– 1D spatial “gap” lattice solitons [PRL 90, 023902 (2002).]– 2D in-phase and “gap” lattice solitons [Nature 422, 147 (2003).]– vortex-ring lattice solitons [PRL 92, 123904 (2004).]– incoherent (random-phase) lattice solitons [Nature, Feb. 2005]– 2D second-band (vortex) lattice solitons [submitted to PRL, Feb. 2005]
– random-phase gap lattice solitons [in preparation, Feb. 2005]
• Theory:– self-consistency method + multi-band lattice solitons [PRL 91, 113901 (2003)]– analysis of 2D lattice solitons [PRL 91, 213906 (2003)]– grating-mediated waveguiding + first experiment [PRL 93, 103902 (2004)]– incoherent (random-phase) lattice solitons [PRL 92, 223901 (2004)]– second-band vortex lattice solitons [Opt. Lett. 17, 2049 (2004)]
• Technique of Bloch-wave spectroscopy of photonic lattice [submitted to PRL, Dec. 2004]
• Brand new results: photonic quasi-crystals, Penrose Tiling, etc.
Summary
Induction techniqueallows 2D lattices
Optics allows directand k-space imaging
Optical physics + general physics using opticsTheory and experiment ongoing…