Excitability, dynamical instabilities and interaction of localized - - PowerPoint PPT Presentation

http://ifisc.uib-csic.es - Mallorca - Spain LENCOS ‘09 Sevilla, July 14 - 17, 2009

Excitability, dynamical instabilities and interaction of localized structures in a nonlinear optical cavity

Damià Gomila, Adrián Jacobo, Manuel Matías, Pere Colet

http://ifisc.uib-csic.es

• Introduction
• Dynamical instabilities of Kerr cavity solitons
• Excitability mediated by localized structures
• Interaction of oscillating localized structures
• Interaction of excitable localized states: logical gates
• Summary

Outline

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Pattern formation in nonlinear optical cavities

• 1. Driving
• 2. Dissipation
• 3. Nonlinearity
• 4. Spatial coupling

Spontaneous pattern formation Pump field

Nonlinear medium

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)

• U. Bortolozzo, R. Rojas, and S.

Residori, Phys. Rev. E 72, 045201(R) (2005)

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Localized Structures (or dissipative solitons)

Ackemann-Lange

Soliton in a Vertical Cavity Surface Emitting Laser

• S. Barland et al., Nature, 419, 699 (2002).

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Self-focusing Kerr cavity

L.A. Lugiato & R. Lefever, PRL 58, 2209 (1987)

: Detuning : Pump Homogeneous Steady State Solution Becomes unstable at leading to a subcritical hexagonal pattern

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Self-focusing Kerr cavity solitons

No solitons Hopf Azimuth inst. m=5 m=6 Saddle- Node W.J. Firth, G.K. Harkness, A. Lord, J. McSloy,

• D. Gomila & P. Colet, JOSA B 19, 747 (2002)

Is

Hopf Saddle- node Homogeneous solution

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Saddle-loop bifurcation

θ=1.30478592 θ=1.304788 θ=1.3047 θ=1.3

middle-branch cavity soliton

• scillating cavity soliton

Is =0.9

θ

L C

max(|E|)

Hopf

Saddle-loop

homogeneous solution

SN Homogeneous solution

• D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005)

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Saddle-loop bifurcation. Scaling law ( )

θ θ λ − − ≈

c

T ln 1

1

Close to bifurcation point: T: period of oscillation λ1 unstable eigenvalue of saddle (middle-branch soliton)

S.H. Strogatz, Nonlinear dynamics and chaos 2004

1/λ1

numerical simulations middle-branch soliton spectrum λ1

λ1=0.177 Saddle-node index: ν=-λs/λu < 1 (unstable limit cycle, but we observe a stable one)

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Phase space close to saddle-loop bifurcation

Only two localized modes.

ψu ψs middle-branch soliton spectrum

Close to saddle: dynamics takes place in the plane (ψu, ψs)

• D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75, 026217 (2007).

Saddle-node index: ν=-λs/λu=2.177/0.177>1 (stable limit cycle)

δA=(E-Esaddle)/Es

Beyond Saddle Loop Oscillatory regime Projection onto ψs Projection onto ψs Projection onto ψu Projection onto ψu

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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 spatial Coupling.

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

• scillating solitons

d → 0 for θ → ∞ and Is → 0 NLSE

saddle-node Hopf θ=1.5 θ=1.6 θ=1.7

The Hopf frequency when it meets the saddle- node is zero. Takens-Bogdanov point. Unfolding of TB yields a Saddle-Loop Saddle-loop bifurcation is not generic. Why it is present here? The NLSE has, at least, four zero eigenvalues:

• D. Skyrabin, JOSA B 19, 529 (2002)

θ max(|E|)

L C Hopf SN

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Interaction of two oscillating solitons

θ =1.27, Is=0.9, homogeneous pump

In-phase oscillation. T=8.59 Out-phase oscillation. T=10.45 Single structure period T=8.66

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Collective modes of oscillation

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Bifurcation diagram of two coupled oscillatory LS

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Dynamical regimes

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Coupled Landau-Stuart Equations

) )( ( ] ) ( [ ) )( ( ] ) ( [

2 1 2 2 2 2 1 2 2 1 1 1

A A i A i i A A A A i A i i A A κ δ β α γ ω μ κ δ β α γ ω μ − + + + − + = − + + + − + = & &

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Logical Operations

By coupling excitable Cavity Solitons it is possible to realize logical

• perations

Logical Operations Using Cavity Solitons:

Input 1 Input 2 Output OR 1 1 1 1 1 AND 1 1 1 1 1 NOT 1 1

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Logical Operations

• AND. 1 0 0:

1 0

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Logical Operations

• AND. 1 1 1:

1 1 1

Logic gates and hardware NAND and NOR universal logic gates are the two pillars of logic, in that all other types of Boolean logic gates (i.e., AND, OR, NOT, XOR, XNOR) can be created from a suitable network

• f just NAND or just NOR gate(s). They can be

built from relays or transistors, or ANY OTHER TECHNOLOGY that can create an inverter and a two-input AND or OR gate.

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Summary

• Dissipative solitons in a nonlinear Kerr cavity: subcritical cellular patterns
• Excitable regime associated with the existence of cavity solitons.
• Extended systems, in order to exhibit excitability, do not require local excitable

behavior.

• 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 θ → ∞ & Is

→ 0)

• We showed some evidence that this scenario comes from the NLSE
• Interacting oscillatory solitons lock to equilibrium distances given

by tail interaction:

• The interaction leads to two collective modes of oscillation.
• Depending on the locking distance solitons different

dynamical regimes are obtained.

• For the closest distance the dynamics might not be

reproduced by two simple coupled oscillator.

• Interaction of excitable solitons can be used to create logical

gates.