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Localized modes and solitonsin nonlinear discrete systems
Yuri KivsharNonlinear Physics Centre
Research School of Physics and EngineeringAustralian National University
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Outline
Solitons: historical remarksRecent advances: fields and conceptsOptical solitons in periodic structuresMulti-colour optical solitonsPlasmon solitonsControl of matter-wave solitons Self-trapped localized states
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What is “soliton” ?
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John Scott Russell (1808-1882)
Very bright engineer: invented an improved steam-driven road carriage in 1833. ``Union Canal Society'' of Edinburgh asked him to set up a navigation system with steam boats
During his investigations, 6 miles from the centre of Edinburgh,he observed a soliton for the first time in August 1834
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“Solitons”
>1400 citations
(1925-2006)0
241
xxxxt UUUU
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Solitons
Nonlinear localized waves existing when nonlinearity is balanced by dispersion
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sine-Gordon equation
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The latest observation:a huge “soliton mode”
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Recent advances
New media and materials: nonlinear optics: nonlocal media, discrete and subwavelength structures, slow light BEC: nonlinearity management nanostructures: graphene, carbon nanotubes
New types of localized modes: gap solitons, discrete breathers, compactons, self-trapped modes, azimuthons, etc
Importance of nonintergrable models
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Self-focusing and spatial optical solitons
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Photonic crystals and lattices
fabricated
Optically induced
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How does periodicity affect solitons ?
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Spatial dispersion and solitonsBulk media
Waveguide array
TIR GAPSPATIAL SOLITON
LATTICE SOLITON
Theory: Christodoulides & Joseph (1988), Kivshar (1993)Experiments: Eisenberg (1998), Fleischer (2003), Neshev (2003), Martin (2004)
TIR GAP
BR GAP
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Effective discrete systems
Self-focusingnonlinearity
Defocusingnonlinearity
DISCRETE SOLITONS
GAP SOLITONS
z
Kx
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Gap Solitons - defocusing case
low power 10nW high power 100WLiNbO3 waveguide array
TIR GAP
BR GAP
Opt. Exp. 14, 254 (2006)
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Polychromatic solitons
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Diffraction of polychromatic light
waveguides5 0 0 6 0 0 7 0 0 8 0 0
1 0 1
1 0 2
1 0 3
1 0 4
s u p e rc o n tin u u m in c a n d e s c e n t la m p
I(),
arb.
uni
ts
, n m
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Theory: coupled NLS equations
Optically-controlled separation and mixing of colors
Power
Micro-scale prism Filtering of redWhite-light input and output
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Experiment: polychromatic gap soliton
10W 6mW 11mW
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Plasmon solitonsand oscillons
Nonlinear Kerr-type dielectric
2|E| linear
Nonlinear modes in planar metal-dielectric waveguidesV.M. Agranovich et al., JETP (1980), G.I. Stegeman et al., J. Appl. Phys. (1985), A.R. Davoyan et al. Opt Express (2008)
Plasmon-solitonTemporal plasmon solitons - A.D. Boardman et al. Phys. Rev. B (1986)
Spatial plasmon soliton in a slot waveguideE. Feigenbaum and M. Orenstein, Opt. Lett. (2007)
Nonlinear plasmonics
Heat
Collisions
Interband
Ponderomotive force
Drude model works in case of Au for IR
Negligible in short pulse operation
Spectral band limited
Metal nonlinearities
- 1 8 2 2= 3 1 0 [ m / V ]
Orenstein, 2010
V. Drachev, A. Buin, H. Nakotte, and V. Shalaev, Nano Lett. 4, 1535 (2004)
10 nm radius Ag spheres possess high and purely real cubic susceptibility
Currently, there is no reliable theoretical models describing nonlinear opticalresponse of metal nanoparticles, however experimental data shows that it depends on many factors, including duration and frequency of the external excitation as well as particle characteristics themselves (metal type and size)
Arrays of nonlinear metal particles
An array of metal particles in external electromagnetic field
Modulational instability and oscillons
“oscillons” –nonlinear localized modes in externally driven systemsH. Swinney et al, Nature 382, 793 (1996) H. Arbell et al, Phys. Rev. Lett. 85, 756 (2000)
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Matter waves and BEC
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Solitons in Bose-Einstein condensates
Bright solitons – attractive interaction, negative scattering lengthAchieved through self focusing, modulational instability, collapse
L. Khaykovich et al., Science 296, 1290 (2002); K. E. Strecker et al., Nature 417, 150 (2002); S. Cornish et al., PRL 96, 170401 (2006)
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1D driven model for a BEC
All symmetries are broken, no damping
3D to 1D reduction due to trapping geometry
Normalization using typical scales of the system
T. Salger et al., PRL 99, 190405 (2007)
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• Being initially at rest, the soliton starts moving provided N larger than a certain critical value
• Cumulative velocity depends on the soliton mass (particle number); this effect can be explained by the effective particle approximation
The first example of the mass-dependent soliton ratchet
Collisions of driven solitons
Initially different values of N Initially equal values of N
www.rsphysse.anu.edu.au/nonlinearThe driving initiates and controls the dynamics
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Nonlinear self-trapped states
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Self-trapping in BEC
Th. Anker et al, PRL 94, 020403 (2005)
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Novel ‘broad’ gap states
two types of modes
truncated nonlinear Bloch modes
Darmanyan et al, 1999
Discrete NLSE
T.J. Alexander et al, Phys. Rev. Lett. 96, 140401 (2006)
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Nonadiabatic generation• Nonadiabatic loading into a 1D optical lattice produces broad states
t=0
t=25 ms
V0 4ER ; N ~ 103
V0
Experimental observation in optics
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Conclusions
• Discrete systems offer a simple ground for the study of many fundamental effects in physics of nonlinear waves
• Many novel types of localized waves discovered: compactons, azimuthons, etc
• Generalized concepts: nonlinearity management, ratchets, self-localized states
• Optical systems allows to observe and study many different types of nonlinear waves and solitons
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