Transcript What are solitons

What Are Solitons, Why Are They Interesting
And How Do They Occur in Optics?
George Stegeman
KFUPM Chair Professor
Professor Emeritus
College of Optics and Photonics, Un.
Central Florida, USA
High Power
Low Power
courtesy of Moti Segev
Material Requirement: The phase velocity of a beam (finite width in space or time)
must depend on the field amplitude of the wave!
All Wave Phenomena: A Beam Spreads in Time and Space on Propagation
Space: Broadening by Diffraction
Time: Broadening by Group Velocity Dispersion
Spatial/Temporal Soliton
Broadening +
Narrowing Via a Nonlinear Effect
= Soliton (Self-Trapped beam)
1.
An optical soliton is a shape invariant self-trapped beam of light
or a self-induced waveguide
2.
Solitons occur frequently in nature in all nonlinear wave phenomena
3.
Contribution of Optics: Controlled Experiments
Solitons Summary
exhibit both wave-like and particle-like properties
• solitons are common in nature and science
Self-consistency Condition
•any nonlinear mechanism leading to beam
narrowing will give bright solitons, beams
whose shape repeats after1 soliton period!
•solitons are the modes of nonlinear
(high intensity) optics
I(x)  Δn(x) = n2I(x)
Δn(x)
I(x)
• robustness (stay localized through
small perturbations)
• unique collision and interaction properties
• Kerr media
• no energy loss to radiation fields
• number of solitons conserved
• Saturating nonlinearities
• small energy loss to radiation fields
• depending on geometry, number of solitons
can be either conserved or not conserved.
x
x
Δn(x) traps beam
z
1D Bright Spatial Soliton
Diffraction in 1D only!
x Optical Kerr Effect → Self-Focusing: n(I)=n0+n2I, n2>0
c
c
Vp(I>0)
phase
velocity:
Vp (I)  
Vp(I0)
n n0  n2 I
Soliton Properties
1. No change in shape on
propagation
2.
Vp(soliton) < Vp(I0)
3. Flat (plane wave) phase front
I(x)
4. Nonlinear phase shift  z (not
obvious)
Self-focusing
Diffraction in space
n2>0
Vp(I0)>Vp(I>0)
Soliton!
Phase
front
First “Published” Scientific Record of Solitons
John Scott Russell in 1834 was riding a horse along a narrow and shallow
canal in Scotland when he observed a “rounded smooth well-defined heap
of water” propagating “without change of form or diminuation of speed”
Soliton
Russell, J. S., 1838, Report of committee
on waves. Report of the 7-th Meeting of
British Association for the Advancement of
Science, London, John Murray, 417-496.
Soliton on an Aqueduct
Union Canal, Edinburgh, 12 July 1995.
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
Temporal
Spatio-Temporal
Spatial
Homogeneous Media
Discrete Media
1D, 2D
Propagating Solitons
Cavity Solitons
Kerr n=n2I
Kerr-like
Photorefractive
Media
Local Non-local
Quadratic
Liquid Crystals
Gain Media
Optical Solitons
Temporal Solitons in Fibers
Spatial Solitons 1D
Supported by Kerr
nonlinearity nNL = n2I
heff
n2
n1
Spatial Solitons 2D
nonlinearity
NOT Kerr
n2>n1
Field distribution
along x-axis fixed
by waveguide mode
Discrete Spatial Solitons 1D
Two color solitons
Quadratic nonlinearity
Nonlinear Wave Equation
2 
2

 L  NL
1


2
 E  2 2 E  0 2 {P  P }
c t
t
 (1) 
0   E


E (r )  A( x, y ) exp[ i{kz  t}]

 ( 3)
 Kerr  EEE
depends on nonlinear
mechanism
2


 NL
n
2
2
2
0
 E  2  E   0 P
c

 NL
 
spatial
2
2
  2ik E   E   0 P
z diffraction nonlinearity
Slowly varying phase
and amplitude
approximation (SVEA,1st
order perturbation theory) temporal
 2ik
Plane Wave Solution?
Shape
invariance

| E | 0
z
 Unstable mode
 Filamentation
Zero diffraction
and/or dispersion
+
2 E  0 or k2  0
 NL
 
2 
2
E  kk2 2 E   0 P
z
T
Group velocity dispersion

| E | 0
z
Nonlinear Mode
Spatial soliton
1D Kerr Solitons: nNL = n2I= n2,E|E|2
Kerr Effect : P NL  2 0n0n2, E | E |2
“Nonlinear Schrödinger Equation” “NLSE”

2

2
2
NL
Space  2ik E  2 E  2k n n0 E Time  2ik
E  2 E  2k 2n NLn0 E
z
z
x
T
dispersion
nonlinearity
diffraction nonlinearity
Invariant shape
on propagation
Bright Soliton, n2>0
E ( x, T ) 
Nonlinear
phase shift
n0
1
( x, T )
z
sech{
} exp[ i
]
2
2
n2, E n0 kvac ( w0 , T0 )
( w0 , T0 )
2n0 kvac ( w0 , T0 )
Psol 
heff c 0
2
w0 k vac
( )n2 ||, E (;  )

heff c 0
dPsol
 2 2
0
dw0
w0 k vac ( )n2 ||, E (;  )
2(w0,T0)
dP
Remarkable stability comes from sol  0,
dw0
x, T i.e. if
Psol  w0  and vice - versa!
All other nonlinearities do NOT lead to analytical solutions and must be found numerically!
Stability of Kerr Self-Trapped Beams in 2D?
1 D Waveguide Case
h
w0
w02 n0
Diffraction length LD 
vac
Nonlinear length (/2) LNL 
dP
0
dw0
LD 2n2n0
 2
w0 P  constant
LNL vach
w0
2 D Bulk Medium Case
LD 2n2 n0

P  constant
2
LNL
vac
LD 
dP
0
dw0
2n2 P
Stable, i.e. robust!
w02 n0
vac
vacw0 h
LNL 
vacw02
2n2 P
Unstable!
Fluctuation in power leads to either diffraction or narrowing dominating
No Kerr solitons in 2D!
BUT,2D solitons stable in other forms of nonlinearity
Higher Order Solitons
2
- Previously discussed solitons were N=1 solitons where N LNL  LD
- Higher Order solitons obtained from Inverse Scattering or Darboux transforms
N 2
4[cosh( 3 )  3e 4i cosh( )]ei / 2)
u ( , ) 
[cosh( 4 )  4 cosh( 2 )  3 cos( 4 )]

z
LD

T
T0
N=3
Soliton period (same for all N ) : z0  LD / 2
Intensity
4
2
z / z0
0
-10
0
T / T0
10
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
1D & 2D Discrete
Arrays of coupled waveguides
0, 1, 2
Discrete diffraction vs
self-focusing (or defocusing)
† Two of peak intensity, width and wavevector mismatch
* Peak intensity or width
White Light (Incoherent) Photorefractive Solitons
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!
14 m Input Beam
82 m Diffracted
Output Beam
M. Mitchell and M. Segev, Nature, 387, 880 (1997)
12 m Self-Trapped Output
Beam with Voltage Applied
Optical Bullets: Spatio-Temporal Solitons
x
t
Electromagnetic pulses that do not spread in time and space
Characteristic Lengths
Temporal Dispersion : LD (T )  T02 / | k 2 |

Spatial Diffractio n : LD (r )  kw02 / 2
Nonlinear Length : LNL  [k vac n2 Ppeak / Aeff ]1

Soliton : LNL  LD (T )  LD (r )
Soliton period : z0  LD / 2
Require: dispersion length (time)  diffraction length (space)  nonlinear length
Quasi-1D Optical Bullets: Frank Wise’s Group
Diffractive
Broadening
x
x
z
y
y
Dispersive Broadening
Spatiotemporal Soliton
”Light Bullet
300
Dispersion
400
200
0
0
10 15 20 25
5
Propagation Distance
Beam Waist (m)
Pulse Duration (fs)
600
Diffraction along
soliton dimension
200
100
0
0
10 15
20
5
Propagation Distance
25
Particle or Wave?
Kerr Nonlinearity:
Remains Highly Spatially Localized
Number of Particles Conserved on Collision
BOTH!
Diffraction
Interference
Refraction
Coherent Kerr Soliton Collisions: Particles or Waves?
Phase 2
Phase 1
=
=0
100
Incoherent Soliton Interaction
80
60
100
30
4020
50
0
-500
2010
0
0
500
=/2
0
-500
600
0
500
=3/2
6
500
5
400
4
40
40
30
300
3
30
20
200
2
10
100
20
10
0
-500
0
0
0
-500
500
1
0
0
500
1. Number of solitons in = Number of solitons out particle-like behavior
2. For  0,  also wave-like behavior - energy exchange occurs via nonlinear mixing
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 10m)
Observed at Output
Soliton birth – a third soliton appears!
Dissipative Solitons: AlGaAs Semiconductor Optical Amplifier
Diffraction vs self-focusing
+ Gain (e.g. SOA) vs loss
Control gain versus loss by
adjusting width of electrode strips
Au wires
Cu sheets
TE
cooler
Insulator
Al mount
I Current source
Waveguide Arrays: Discrete Solitons
Discrete diffraction
Discrete Spatial Surface Solitons
Theoretical prediction: Nonlinear surface
waves exist above a power threshold!
Input power is increased slowly
and output from array is recorded
Observation plane
Input beam
Single channel excitation
Without normalization
Single channel soliton
>50% of power at output
In input channel
Interface Solitons Between Two Dissimilar Arrays
1.
2.
3.
Two discrete interface solitons with power thresholds
propagate along 1D interfaces
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.
Single channel excitation can lead to the excitation of single
channel solitons peaked on channels different from the
excitation channel.
2D Edge and Corner Discrete Solitons
Corner soliton
Edge soliton
K.G. Makris, J. Hudock, D.N. Christodoulides, G.I. Stegeman M. Segev et. al, Opt.
Lett. 31, 2774-6 (2006).
2D Edge and Corner Discrete Solitons: Experiment
Experiment
Theory
Excitation channel
Discrete Diffraction
Edge Soliton
Soliton Intensity Profile
Power
Experiment: A. Szameit, et. al., Phys.
Rev. Lett., 98, 173903 (2007);
Z. Chen, et. al., Phys. Rev. Lett., 98, 123903 (2007)
Solitons Summary
exhibit both wave-like and particle-like properties
• 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
• 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!!