What Are Solitons, Why Are They Interesting And How Do They Occur in Optics?

George StegemanKFUPM Chair Professor

Professor EmeritusCollege of Optics and Photonics, Un.

Central Florida, USA

Material Requirement: The phase velocity of a beam (finite width in space or time) must depend on the field amplitude of the wave!

High Power

Low Power

courtesy of Moti Segev

Space: Broadening by Diffraction Time: Broadening by Group Velocity Dispersion

All Wave Phenomena: A Beam Spreads in Time and Space on Propagation

Broadening +Narrowing Via a Nonlinear Effect

= Soliton (Self-Trapped beam)

Spatial/Temporal Soliton

1. An optical soliton is a shape invariant self-trapped beam of lightor a self-induced waveguide

2. Solitons occur frequently in nature in all nonlinear wave phenomena

3. Contribution of Optics: Controlled Experiments

Solitons Summary

• solitons are common in nature and science

•any nonlinear mechanism leading to beamnarrowing will give bright solitons, beamswhose shape repeats after1 soliton period!

•solitons are the modes of nonlinear(high intensity) optics

• robustness (stay localized through small perturbations)

• unique collision and interaction properties• Kerr media

• no energy loss to radiation fields• number of solitons conserved

exhibit both wave-like and particle-like properties

I(x)

x

Δn(x)

x

I(x) Δn(x) = n2I(x)

Δn(x) traps beam

• Saturating nonlinearities• small energy loss to radiation fields• depending on geometry, number of solitons can be either conserved or not conserved.

Self-consistency Condition

1D Bright Spatial Soliton

Optical Kerr Effect → Self-Focusing: n(I)=n0+n2I, n2>0

Soliton Properties

1. No change in shape on propagation

2. Vp(soliton) < Vp(I0)

3. Flat (plane wave) phase front

4. Nonlinear phase shift z (not obvious)

Soliton!Diffraction in space Phasefront

Diffraction in 1D only!

Self-focusing

x

z

n2>0

I(x)

Vp(I>0)Vp(I0)

Vp(I0)>Vp(I>0)

I)I(V

20p nn

c

n

c

phase velocity:

Soliton

John Scott Russell in 1834 was riding a horse along a narrow and shallowcanal in Scotland when he observed a “rounded smooth well-defined heapof water” propagating “without change of form or diminuation of speed”

First “Published” Scientific Record of Solitons

Russell, J. S., 1838, Report of committee on waves. Report of the 7-th Meeting ofBritish Association for the Advancement of Science, London, John Murray, 417-496.

Union Canal, Edinburgh, 12 July 1995.

Soliton on an Aqueduct

Solitons in Oceans: The “Rogue” Wave

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace”, Physics Letters, A 373 (2009) 675–678.

Soliton Sightings by Weather Satellites and/or Weather Planes

Optical Solitons

SpatialTemporal Spatio-Temporal

Homogeneous Media Discrete Media

1D, 2D

Propagating SolitonsCavity Solitons

MediaLocal Non-local

Photorefractive

Kerr n=n2I

Kerr-like

Quadratic Gain Media

Liquid Crystals

Optical Solitons

Temporal Solitons in Fibers

Spatial Solitons 1D

Discrete Spatial Solitons 1DSpatial Solitons 2D

nonlinearityNOT Kerr

Two color solitonsQuadratic nonlinearity

Supported by Kerr

nonlinearity nNL = n2I

heff

Field distributionalong x-axis fixedby waveguide mode

n2

n1 n2>n1

Field distributionalong x-axis fixedby waveguide mode

n2

n1 n2>n1

Nonlinear Wave Equation

}{1

2

2

02

2

22 NLL PP

tE

tcE

E

)1(0

Slowly varying phaseand amplitude approximation (SVEA,1st

order perturbation theory)

NLPEc

nE

0

222

202 }]{exp[),()( tkziyxArE

depends on nonlinearmechanism

EEEKerr )3(

NLPEEz

ikspatial

0222

diffraction nonlinearity

NLPET

kkEz

iktemporal

02

2

2

22

Group velocity dispersion

0||

Ez

Nonlinear ModeSpatial soliton

0||

Ez

Shapeinvariance

+ 0or 0 22 kE

Zero diffractionand/or dispersion

Plane Wave Solution? Unstable mode Filamentation

1D Kerr Solitons: nNL = n2I= n2,E|E|2

Bright Soliton, n2>0

2(w0,T0)

All other nonlinearities do NOT lead to analytical solutions and must be found numerically!

x, T

2,200 ||2 :EffectKerr EnnP E

NL

]),(2

exp[}),(

),({sech

),(

1),(E

20

20000000,2

0

Twkn

zi

Tw

Tx

Twknn

nTx

vacvacE

Invariant shapeon propagation

Nonlinearphase shift

“Nonlinear Schrödinger Equation” “NLSE”

EnnkEx

Ez

ik 0NL2

2

2

22Space

diffraction nonlinearity

EnnkET

Ez

ik 0NL2

2

2

22Time

dispersion nonlinearity

versa!- viceand if i.e.

,0 from comesstability Remarkable

0);()();()(

0

0

E||,22vac

20

0

0E||,22vac0

0

wP

dw

dP

nkw

ch

dw

dP

nkw

chP

sol

sol

effsoleffsol

Stability of Kerr Self-Trapped Beams in 2D?

vacD

nwL

0

20Diffraction length

Pn

hwL vac

NL2

0

2

Nonlinear length (/2)

h

w0

1 D Waveguide Case

Pwh

nn

L

L

vacNL

D02

022

constant ! i.e. ,0

0robustStable

dw

dP

2 D Bulk Medium Case w0

vacD

nwL

0

20

Pn

wL vac

NL2

20

2

constant 2

202 P

nn

L

L

vacNL

D

!0

0Unstable

dw

dP

No Kerr solitons in 2D! BUT,2D solitons stable in other forms of nonlinearity

Fluctuation in power leads to either diffraction or narrowing dominating

Higher Order Solitons

- Previously discussed solitons were N=1 solitons where DNL2 LLN

)]4cos(3)2cosh(4)4[cosh(

)]cosh(3)3[cosh(4),( 2

)2/4

ii ee

uN0D

T

T

L

z

- Higher Order solitons obtained from Inverse Scattering or Darboux transforms

0.20.4

1.0

0.8

0.60/ zz

0-10 10

0/TT

4

2

0

Inte

nsity

N=32/ :) allfor (same periodSoliton D0 LzN

Need to refine “consistency condition”.Soliton shape must reproduce itself every soliton period!

Zoology of Spatial Soliton Systems

Soliton Type # Soliton Parameters Critical Trade-Off

1D Kerr 1* Diffraction vs self-focusing

1D & 2D Saturating Kerr 1* Diffraction vs self-focusing

1D & 2D Quadratic 2† Diffraction vs self-focusing

1D & 2D Photorefractive 1* Diffraction vs self-focusing

1D & 2D Liquid Crystals 1* Diffraction vs self-focusing

1D & 2D Dissipative 0 Diffraction vs self-focusing

+ Gain (e.g. SOA) vs loss

† Two of peak intensity, width and wavevector mismatch* Peak intensity or width

1D & 2D Discrete

Arrays of coupled waveguides

0, 1, 2 Discrete diffraction vs

self-focusing (or defocusing)

White Light (Incoherent) Photorefractive Solitons

12 m Self-Trapped OutputBeam with Voltage Applied

82 m DiffractedOutput Beam

14 m Input Beam

M. Mitchell and M. Segev, Nature, 387, 880 (1997)

But aren’t solitons supposed to be coherent beams?Most are, BUT that is NOT a necessary condition!Why? Because the nonlinear index change required depends on intensity I

i.e. n |E|2 not E2! No coherence required!

Optical Bullets: Spatio-Temporal Solitons

Electromagnetic pulses that do not spread in time and space

Require: dispersion length (time) diffraction length (space) nonlinear length

2/ :periodSoliton

)()( :Soliton

]/[ :LengthNonlinear

2/)( :nDiffractio Spatial

||/)( :Dispersion Temporal

D0

DDNL

1effpeak2vacNL

20D

22

0D

Lz

rLTLL

APnkL

kwrL

kTTL

Characteristic Lengths

t

x

600

400

200

00 5 10 15 20 25

Propagation Distance

Puls

e D

urat

ion

(fs)

Dispersion

0 5 10 15 20 25Propagation Distance

200

0

300

100

Bea

m W

aist

(m

) Diffraction alongsoliton dimension

DiffractiveBroadening

Dispersive Broadening

Spatiotemporal Soliton”Light Bullet

Quasi-1D Optical Bullets: Frank Wise’s Group

x

yz

x

y

Particle or Wave?

Kerr Nonlinearity:

Remains Highly Spatially Localized

Number of Particles Conserved on Collision

Diffraction Interference

Refraction

BOTH!

Coherent Kerr Soliton Collisions: Particles or Waves?

Phase 1 Phase 2

-500 0 500

0

20

40

60

80

100

0

50

100

=0

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

=

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

40

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

40

=/2 =3/2

1. Number of solitons in = Number of solitons out particle-like behavior2. For 0, also wave-like behavior - energy exchange occurs via nonlinear mixing

Incoherent Soliton Interaction

Soliton Collisions Soliton “Birth”: Non-Kerr Media

•horizontal colliding angle 0.90

• in vertical plane not collided center to center

(vertical center to center separation 10m)

Soliton birth – a third soliton appears!

Observed at Output

Cu sheets

TE coolerInsulator

Al mount

Au wires

I Current source

Control gain versus loss by adjusting width of electrode strips

Diffraction vs self-focusing

+ Gain (e.g. SOA) vs loss

Dissipative Solitons: AlGaAs Semiconductor Optical Amplifier

Waveguide Arrays: Discrete Solitons

Discrete diffraction

Theoretical prediction: Nonlinear surfacewaves exist above a power threshold!

Discrete Spatial Surface Solitons

Input power is increased slowlyand output from array is recorded

Single channel soliton>50% of power at outputIn input channel

Without normalization

Observation plane

Input beamSingle channel excitation

1. Two discrete interface solitons with power thresholds propagate along 1D interfaces2. In 1D, two different surface soliton families exist with peaks on or near the boundary channels. One family experiences an attractive potential near the boundary, and the second a repulsive potential.3. Single channel excitation can lead to the excitation of single channel solitons peaked on channels different from the excitation channel.

Interface Solitons Between Two Dissimilar Arrays

2D Edge and Corner Discrete Solitons

Corner solitonEdge soliton

K.G. Makris, J. Hudock, D.N. Christodoulides, G.I. Stegeman M. Segev et. al, Opt.Lett. 31, 2774-6 (2006).

Experiment: A. Szameit, et. al., Phys. Rev. Lett., 98, 173903 (2007);Z. Chen, et. al., Phys. Rev. Lett., 98, 123903 (2007)

2D Edge and Corner Discrete Solitons: Experiment

TheoryExperiment

Power

Soliton Intensity Profile

Excitation channel Discrete Diffraction

Edge Soliton

Solitons Summary

• solitons are common in nature and science• any nonlinear mechanism leading to beam narrowing will give bright solitons, beams whose shape on propagation is either constant or repeats after 1 soliton period!•they arise due to a balance between diffraction (or dispersion) and nonlinearity in both homogeneous and discrete media. Dissipative solitons also require a balance between gain and loss.• solitons are the modes (not eigenmodes) of nonlinear (high intensity) optics• an important property is robustness (stay localized through small perturbations)• unique collision and interaction properties

• Kerr media• no energy loss to radiation fields• number of solitons conserved

exhibit both wave-like and particle-like properties

• Saturating nonlinearities• small energy loss to radiation fields• depending on geometry, number of solitons can be either conserved or not conserved.

• Solitons force you to give up certain ideas which govern linear optics!!