Solitons in Nonlinear Photonic Lattices · First use: Mitchell, Segev, Christodoulides 1997...

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).






Myoglobin

BeamEnvelope Diffraction Kerr NL Periodic index

( ) 021 2

22 =∆++∇+

∂∂

⊥⊥ ψψψψψ rnnzi Array

Nonlinear Schrödinger equation

Optics


A. Xie et al., PRL 84, 5436 (2000).

Atomic chain

CDW waves in transition metalsB.I. Swanson et al., PRL 82, 3288 (2000).






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


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



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



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


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 ∆+


“Discrete” soliton

Tra

nsve

rse

disp

lace

men

t


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


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)


Spatial Gap Solitonsβ

π/d-π/dkx

β β ββββ

ββ ∆−

J.W. Fleischer et al., PRL 90, 023902 (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


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


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)


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…

Solitons in Nonlinear Photonic Lattices · First use: Mitchell, Segev, Christodoulides 1997...

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Transcript of Solitons in Nonlinear Photonic Lattices · First use: Mitchell, Segev, Christodoulides 1997...