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Excitability mediated by dissipative solitons

Pere Colet

Adrian Jacobo, Damià Gomila, Manuel Matías

Claudio J. Tessone, Alessandro Sciré, Raúl Toral

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• 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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Azimuth instabilities

m=6

m=5 Unstable Eigenmode

t

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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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AND

1 0 1 1

0 1

Pulse on

Pulse off

Pulse on

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NOT

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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.