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Workshop on Network Synchronization: from dynamical systems to neuroscienceLorentz Center, Leiden, 19-30 May 2008
Excitability mediated by dissipative solitons
Pere Colet
Adrian Jacobo, Damià Gomila, Manuel Matías
Claudio J. Tessone, Alessandro Sciré, Raúl Toral
http://ifisc.uib.es
• Introduction
• Dissipative solitons in a Kerr cavity
• Soliton instabilities
• Soliton excitability
• Effect of a localized pump
• Interaction of oscillating & excitable solitons
• Collective firing induced by noise or diversity.
Outline
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Dissipative solitons
Localized excitations in a vertically vibrated granular layer. P.B. Umbanhowar, F. Melo & H.L. Swinney Nature 382, 793 (1996).
Soliton in a Vertical Cavity Surface Emitting LaserS. Barland et al., Nature, 419, 699 (2002).
Dissipative solitons are localized spatial structures that appear in certain dissipative media:
Chemical reactions: J.E. Pearson, Science 261, 189 (1993); K.J. Lee & H.L. Swinney, Science 261, 192 (93).Gas discharges: I. Müller, E. Ammelt & H.G. Purwins, Phys. Rev. Lett. 73, 640, (1994).Fluids: O. Thual & S. Fauve, J. Phys. 49, 1829 (1988).
N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005);“Dissipative Solitons: From Optics to Biology and Medicine”, (Springer 2008)
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Pattern formation in nonlinear optical cavities
1. Driving
2. Dissipation
3. Nonlinearity
4. Spatial coupling
Spontaneous
pattern formation
Pumpfield
Non
line
ar m
ediu
m
Sodium vapor cell with single mirror feedback
Liquid crystal light valve
T. Ackemann and W. Lange, Appl. Phys. B 72, 21 (2001)
P.L. Ramazza et al., J. Nonlin. Opt. Phys. Mat. 8, 235 (1999)P.L. Ramazza, S. Ducci, S. Boccaletti & F.T. Arecchi, J. Opt. B 2, 399 (2000)
F.T. Arecchi, S. Boccaletti & P.L. Ramazza, Phys. Rep. 318, 1 (1999).L.A. Lugiato, M. Brambilla & A. Gatti, Adv. Atom. Mol. Opt. Phys. 40, 229 (1999)N.N. Rosanov, “Spatial Hysteresis and Optical Patterns”, Springer 2002.
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Dissipative solitons versus propagation solitons
“Dissipative solitons”
Dissipative.
Unique once the parameters of the system are fixed.
Potentially useful for optical storage & information processing.
Propagation solitons
Conservative
Continuous family of solutions depending on energy.
Useful for optical communication systems
N.N. Rosanov in Progress in Optics, 35 (1996).
M. Segev (ed.) Special Issue on Solitons, Opt. Photonics News 13(27), 2002
L.A. Lugiato (ed), Feature section on Cavity Solitons, IEEE J. Quantum Electron. 39(2) (2003);
N. Akhmediev & A. Ankiewicz (eds), “Dissipative solitons”, Lecture Notes in Physics 661 (Springer, Berlin, 2005).
Ackemann-Lange
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Scenarios for dissipative solitons
Stable droplets: Localized structures stabilised by nonlinear domain wall dynamics due curvature. Exist in 2d systems.
D. Gomila et al, PRL 87, 194101 (2001)
Homogeneous
Solution
Control Parameter
Am
plit
ude
BistabilityBistability Homogeneous
Solution
Homogeneous
Solution
Localized structures as single spot of a cellular pattern. Exist in 1d & 2d systems.
W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996)Homogeneous
Solution
Hexagonal Pattern
Subcritical Cellular PatternSubcritical Cellular Pattern
Control Parameter
Am
plit
ude
Localized structures stabilised by interaction of oscillatory tails. Exist in 1d & 2d systems.
P. Coullet, et al PRL 58, 431(1987)G.-L.Oppo et al. J. Opt. B 1, 133 (1999)
G.-L.Oppo et al. J. Mod Opt. 47, 2005 (2000)P. Coullet, Int. J. Bif. Chaos 12, 2445 (2002)
Excitability mediated by localized structures
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Excitability. General ideas
Excitability: has origin in Biology (action potential of nerve cells; also heart),also found in reaction-diffusion systems.
Simplest minimum ingredients in phase space for excitability: • Stable fixed point• Threshold• Reinjection mechanism in phase space (that leads to refractory period).
Different responses to sub/supra-threshold perturbations.
Three simplest excitability routes (2-D phase space), occur close to bifurcations leading to oscillatory behavior:
a) saddle-node in invariant circle (Andronov-Leontovich) (Adler equation)
b) saddle-loop (homoclinic) bifurcation
c) fast-slow systems with S nullcline (slow manifold): canard (Fitzhugh-Nagumo)
Excitable media: spatially extended systems in which the local dynamics is excitable.
J.D. Murray, Mathematical Biology, Springer 2002, 3rd ed.
E. Meron; Pattern formation in excitable media; Phys. Rep. 218, 1 (1992).
B. Lindner, J. García-Ojalvo, A. Neiman & L. Schimansky-Geier; Effects of noise in excitable systems; Phys. Rep. 392, 321 (2004).
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Excitability in optical systems
Some examples of excitability in optical systems (mostly active systems):
•Systems with thermal effects (slow variable) that interplay with a hysteresis cycle of a fast variable. Leads to (c), FHN-like excitability. Cavity with T-dependent absorption (Lu et al, PRA 58, 809 (1998)). Semiconductor optical amplifier (Barland et al, PRE 68, 036209 (2003)).
•Lasers with saturable absorber (Dubbeldam et al, PRE 60, 6580 (1999)); lasers with optical feedback (Giudicci et al, PRE 55, 6414 (1997); Yacomotti et al, PRL 83, 292 (1999)); lasers with injected signal (Coullet et al, PRE 58, 5347 (1998); Goulding et al, PRL 98, 153903 (2007)) . These lead to (a): saddle-node in an invariant circle.
•Lasers with intracavity saturable absorber (Plaza et al, Europhys. Lett. 38, 85 (1997)). Excitability mediated by a saddle-loop bifurcation.
•Semiconductor DFB laser (interaction of 2 modes) (Wuensche et al, PRL 88, 023901 (2002)). Homoclinic bifurcation slightly different than (b).
Possible applications: optical switch (responding to sufficiently high optical input signals); optical communications: pulse reshaping.
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Self-focusing Kerr cavity
3xoutput field
input field E0
z
y
EEiEEiEit
E 20
21 : detuning
Homogeneous solution2
0 )],(1[ ssss EIIiEE
E0: pump
Control parameters
Lugiato-Lefever model
L.A. Lugiato & R. Lefever, PRL 58, 2209 (1988).
It becomes unstable at Is=1 leading to a subcritical hexagonal pattern
field envelope
yxx
tzkietxEtzxE z
,
,,,
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Self-focusing Kerr cavity solitons
W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (1996); W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta, T67, 12 (96)
Cavity soliton
AAAAAAiIArrr
iAit
As
222*2
222
11
Can be seen as a solution connecting a cell of the pattern with the homogeneous solution
)1( AEE s
Radial equation:
Soliton profile can be found solving the l.h.s. equated to zero with 00
rr r
A
r
A
Numerical solutions with arbitrary precision:•Discretize r set of nonlinear ordinary eqs. Spatial derivatives computed in Fourier space•Solve using Newton-Raphson•Continuation methods can be used•Linear stability analysis can be performed
W.J. Firth & G.K. Harkness, Asian J. Phys 7, 665 (1998); G.-L. Oppo, A.J. Scroggie & W.J. Firth, PRE 63, 066209 (2001);J.M. McSloy, W.J. Firth, G.K. Harkness & G.-L. Oppo, PRE 66, 046606 (2002)
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Stability of Kerr cavity solitons
3.1
Hopf instability observed in W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta,T67,12 (96)
Stable
Soliton amplitude
Unstable
Hom. solution
Is
Saddle-Node
Hopf
No solitons
Hopf
Azimuth inst.
m=5m=6
Saddle-Node
W.J. Firth, G.K. Harkness, A. Lord, J. McSloy, D. Gomila & P. Colet, JOSA B 19, 747 (2002)
Is
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Cross-section
9.0,3.1 I
middle branch soliton
Oscillating soliton still useful for applications since its amplitude is bounded below by middle branch soliton.
Hopf instability
No solitons
Azimuth inst.
Hopf
Saddle-node
solitons
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Saddle-loop bifurcation
=1.30478592
=1.304788
=1.3047
=1.3
Is =0.9
middle-branch cavity soliton
oscillating cavity soliton
max
(|E
|)
LC
Hopf
Saddle-loop
homogeneous solution
SN
Homogeneous solution
Minimum distance of oscillating soliton to middle-branch soliton
D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
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Saddle-loop bifurcation. Scaling law
cT ln1
1
Close to bifurcation point:
T: period of oscillation1 unstable eigenvalue of saddle (middle-branch soliton)
S.H. Strogatz, Nonlinear dynamics and chaos 2004
1/1
numerical simulations
middle-branch soliton spectrum
1
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Phase space close to saddle-loop bifurcation
Only two localized modes.
u
s
middle-branch soliton spectrum
Close to saddle: dynamics takes place in the plane (u, s)
Saddle-node index: =-s/u=2.177/0.177>1 (stable limit cycle)
D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75, 026217 (2007).
A=(E-Esaddle)/Es
Beyond Saddle Loop
Oscillatory regime
Pro
ject
ion
on
to
sP
roje
ctio
n o
nto
s
Projection onto u
Projection onto u
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Small perturbations of homogeneous solution decay.
Localized perturbations above middle branch soliton send the system to a long excursion through phase-space.
Excitability
D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).
The system is not locally excitable.Excitability emerges from spatialcoupling
Beyond saddle-loop bifurcation
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Takens-Bogdanov point
TB
Distance between saddle-node and Hopf
Hopf
saddle-loop
saddle-node
No solitons
solitons oscillating solitons
d → 0 for → ∞ and Is → 0 NLSE
saddle-node Hopf
=1.5
=1.6=1.7
The Hopf frequency when it meets the saddle-node is zero. Takens-Bogdanov point.
Unfolding of TB yields a Saddle-Loop
=1.5
Saddle-loop bifurcation is not generic. Why it is present here?
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Pump: Plane wave + Localized Gaussian Beam
20
2 /0
rrI HeEE ]1)[( 2
shsshs IIIIH
Is
max
(|E|2 )
1
Exc
itabi
lity
Pat
tern
Is
max
(|E|2 )
1
Exc
itabi
lity
Pat
tern
Osc
illat
ions
Hom. pump
SNIC
Saddle Node
Hopf
Ish=0.7, =1.34
Excitability arising from a saddle-loop bifurcation have a large threshold. To reduce the threshold we consider for the pump:
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Saddle-node in the circle (SNIC) bifurcation
From the new oscillatory regime to the excitable regime.
Is=0.927
Ish=0.3, =1.45
middle-branch cavity soliton
fundamental solution
Close to bifurcation point:
2/12 c
ss IIT
Projection onto u
Pro
ject
ion
on
to
s
unstable upper branch soliton
Is=0.907
Is=0.8871
Is=0.8
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Full scenario
Ish=0.3
I Only fundamental solutionII Stationary DS, fundamental solution stableIII Oscillating DS, fundamental solution stableIV Excitable DS, fundamental solution stableV Oscillating DS, no fundamental solution
Excitability can appear as a result of:•Saddle loop (oscillating and middle branch solitons collide)•Saddle node on the invariant circle (fundamental solution and middle branch soliton collide).
Controllable excitability threshold.
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Noise effects, coherence resonance
A. S. Pikovsky and J. Kurths, Phys. Rev. Lett. 78, 775 (1997).
( )VarR
Introducing white spatiotemporal noise excitable solitons show coherence resonance.
In excitable systems a moderate level of noise induces a more regular firing (coherence resonance)
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Interaction of two oscillating solitons
=1.27, Is=0.9, homogeneous pump
Oscillating solitons move until they reach equilibrium positions given by tails interaction.Three equilibrium distances are found:
Single structure period T=8.66
In-phase oscillation. T=8.93
Out-phase oscillation. T=8.94
Strong interaction. In & out-phase oscillation depending on initial condition.Tin=8.59 < Tout=10.45
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Interaction of excitable solitons
Pulse on
Pulse off
Firing
Firing induced by interaction
Pulse on
Firing
Firing induced by interaction
1 0
1
Firing bit 1
1 1
1
OR logical gate
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Collective firing induced by noise or diversity
Globally coupled active rotators
Diversity: natural frequencies noisej<1 excitable.
j>1 rotates.
j
titi jeN
et)()( 1
)( Kuramoto order parameterGlobal variables: )(t
Approximate equation )(sin)( tt
•Global phase dynamics similar to individual units but with scaled frequency.
•A degradation in entrainment lowers excitablity threshold allowing for
synchronous firing.
•The precise origin of the degradation of is irrelevant.
C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007).
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Numerical simulations
Diversity and noise play a similar role and induce coherent firing.
diversity noise
=0
=1.6
=3.0
D=0.4
D=1.0
D=5.0
No firing
Synchronized firing
Desynchronized firing
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Self-consistent approximation
)()( )()( titi etet Shinomoto-Kuramoto order parameter
No firing Collective firing Desynchronized firing
C.J. Tessone, A. Scirè, R. Toral and P. Colet, Phys. Rev. E 75, 016203 (2007).
Self-consistent approx.
N=50 x N=100
N=1000N=10000
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Summary
• Dissipative solitons in a nonlinear Kerr medium: subcritical cellular patterns
• Oscillating solitons: Still useful for applications envisioned for static solitons. New ones?
• Excitable regime associated with the existence of cavity solitons.
• Extended systems, in order to exhibit excitability, do not require local excitable behavior. Excitability in a whole new class of systems.
• For homogeneous pump excitability appears as a result of a saddle-loop bifurcation: oscillating and middle-branch soliton collide.
• Scenario organized by a Takens-Bogdanov codimension 2 point (at → ∞ & Is → 0)
• For pump composed of a Gaussian localized beam on top of homogeneous background excitability also mediated by a SNIC: fundamental solution and middle branch soliton collide.
• Lower (controllable) excitability threshold.• A suitable amount of white noise induces coherence resonance.
• Coupled oscillatory solitons lock to distances given by tail interaction. • Depending on the locking distance solitons oscillate in or out-of-phase.• For strong coupling in-phase and out-of phase oscillations coexists.
• Interaction of excitable solitons may be used for logical gates.• In coupled excitable systems disorder can induce collective firing.
• Any source of disorder plays a similar role.