What Are Solitons, Why Are They Interesting

1
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
Material Requirement The phase velocity of a
beam (finite width in space or time)
must depend on the
field amplitude of the wave!
2
All Wave Phenomena A Beam Spreads in Time and
Space on Propagation
Space Broadening by Diffraction Time
Broadening by Group Velocity Dispersion

• An optical soliton is a shape invariant
  self-trapped beam of light
• or a self-induced waveguide
• Solitons occur frequently in nature in all
  nonlinear wave phenomena
• Contribution of Optics Controlled Experiments

3
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 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

Self-consistency Condition

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

4
1D Bright Spatial Soliton
Diffraction in 1D only!
Optical Kerr Effect ? Self-Focusing
n(I)n0n2I, n2gt0
Phase front
Diffraction in space
Soliton!
5
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.
6
Soliton on an Aqueduct
Union Canal, Edinburgh, 12 July 1995.
7
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) 675678.
8
Soliton Sightings by Weather Satellites and/or
Weather Planes
9
Optical Solitons
Spatial
10
Optical Solitons
Temporal Solitons in Fibers
Spatial Solitons 1D
Supported by Kerr nonlinearity ?nNL n2I
Discrete Spatial Solitons 1D
Two color solitons Quadratic nonlinearity
11
Nonlinear Wave Equation
Slowly varying phase and amplitude approximation
(SVEA,1st order perturbation theory)
? Unstable mode ? Filamentation
Plane Wave Solution?
12
1D Kerr Solitons ?nNL n2I n2,EE2
Nonlinear Schrödinger Equation NLSE
x, T
All other nonlinearities do NOT lead to
analytical solutions and must be found
numerically!
13
Stability of Kerr Self-Trapped Beams in 2D?
Fluctuation in power leads to either diffraction
or narrowing dominating
No Kerr solitons in 2D! BUT,2D solitons
stable in other forms of nonlinearity
14
Higher Order Solitons
- Higher Order solitons obtained from Inverse
Scattering or Darboux transforms
N3
Need to refine consistency condition. Soliton
shape must reproduce itself every soliton period!
15
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
16
White Light (Incoherent) Photorefractive Solitons
But arent 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 ?
E2 not E2! No coherence required!
M. Mitchell and M. Segev, Nature, 387, 880 (1997)
17
Optical Bullets Spatio-Temporal Solitons
Electromagnetic pulses that do not spread in time
and space
Require dispersion length (time) ? diffraction
length (space) ? nonlinear length
18
Quasi-1D Optical Bullets Frank Wises Group
x
z
y
19
Particle or Wave?
Kerr Nonlinearity Remains Highly Spatially
Localized Number of Particles Conserved on
Collision
BOTH!
Diffraction   Interference Refraction
20
Coherent Kerr Soliton Collisions Particles or
Waves?
Incoherent Soliton Interaction

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

21
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)

Soliton birth a third soliton appears!
22
Dissipative Solitons AlGaAs Semiconductor
Optical Amplifier
Diffraction vs self-focusing Gain (e.g. SOA) vs
loss
23
Waveguide Arrays Discrete Solitons
Discrete diffraction
24
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
Single channel soliton gt50 of power at output In
input channel
25
Interface Solitons Between Two Dissimilar Arrays

• 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.

26
2D Edge and Corner Discrete Solitons
K.G. Makris, J. Hudock, D.N. Christodoulides,
G.I. Stegeman M. Segev et. al, Opt. Lett. 31,
2774-6 (2006).
27
2D Edge and Corner Discrete Solitons Experiment
Theory
Experiment
Power
Experiment A. Szameit, et. al., Phys. Rev.
Lett., 98, 173903 (2007) Z. Chen, et. al., Phys.
Rev. Lett., 98, 123903 (2007)
28
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!!