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

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 - Mallorca - Spain

Outline • 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 http://ifisc.uib-csic.es

Pattern formation in nonlinear optical cavities Sodium vapor cell with single mirror feedback T. Ackemann and W. Lange, Appl. Phys. B 72, 21 (2001) Nonlinear medium Pump field Liquid crystal light valve 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) 1. Driving Spontaneous pattern formation 2. Dissipation 3. Nonlinearity 4. Spatial coupling U. Bortolozzo, R. Rojas, and S. Residori, Phys. Rev. E 72 , 045201(R) (2005) http://ifisc.uib.es

Localized Structures (or dissipative solitons) Soliton in a Vertical Cavity Surface Emitting Laser S. Barland et al., Nature, 419 , 699 (2002). Ackemann-Lange http://ifisc.uib.es

Self-focusing Kerr cavity : Detuning : Pump Homogeneous Steady State Solution Becomes unstable at leading to a subcritical hexagonal pattern L.A. Lugiato & R. Lefever, PRL 58 , 2209 (1987) http://ifisc.uib.es

Self-focusing Kerr cavity solitons Azimuth inst. m=6 m=5 Hopf I s No solitons Saddle- Node Hopf Saddle- node Homogeneous solution W.J. Firth, G.K. Harkness, A. Lord, J. McSloy, D. Gomila & P. Colet, JOSA B 19 , 747 (2002) http://ifisc.uib.es

Saddle-loop bifurcation I s =0.9 oscillating cavity soliton θ =1.3 middle-branch cavity soliton θ =1.3047 L C max(| E |) Hopf θ =1.30478592 SN Saddle-loop θ Homogeneous solution θ =1.304788 homogeneous solution D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94 , 063905 (2005) http://ifisc.uib.es

Saddle-loop bifurcation. Scaling law middle-branch soliton spectrum 1 ( ) ≈ − θ − θ Close to bifurcation point: ln T λ c 1 T: period of oscillation λ 1 unstable eigenvalue of saddle (middle-branch soliton) λ 1 S.H. Strogatz, Nonlinear dynamics and chaos 2004 λ 1 =0.177 numerical simulations 1/ λ 1 Saddle-node index: ν =- λ s / λ u < 1 (unstable limit cycle, but we observe a stable one) http://ifisc.uib.es

Phase space close to saddle-loop bifurcation δ A =( E - E saddle )/ E s Oscillatory regime middle-branch Projection onto ψ s soliton spectrum ψ u Projection onto ψ u Beyond Saddle Loop ψ s Projection onto ψ s Only two localized modes . Close to saddle: dynamics takes place in the plane ( ψ u , ψ s ) Projection onto ψ u Saddle-node index: ν =- λ s / λ u =2.177/0.177>1 (stable limit cycle) D. Gomila, A. Jacobo, M. Matias and P. Colet, PRA 75 , 026217 (2007). http://ifisc.uib.es

Excitability Beyond saddle-loop bifurcation Small perturbations of homogeneous solution decay. Localized perturbations above middle branch soliton send the system to a long excursion through phase-space. The system is not locally excitable. Excitability emerges from spatial Coupling. D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94 , 063905 (2005). http://ifisc.uib.es

Takens-Bogdanov point Hopf saddle-node Saddle-loop bifurcation is not generic. Why it is present here? θ =1.7 solitons oscillating solitons θ =1.6 θ =1.5 L saddle-loop C saddle-node max(| E |) Hopf Hopf SN No solitons θ Distance between saddle-node and Hopf The Hopf frequency when it meets the saddle- Takens-Bogdanov point. node is zero. Unfolding of TB yields a Saddle-Loop TB d → 0 for θ → ∞ and I s → 0 NLSE The NLSE has, at least, four zero eigenvalues: D. Skyrabin, JOSA B 19 , 529 (2002) http://ifisc.uib.es

Interaction of two oscillating solitons θ =1.27, I s =0.9, homogeneous pump Single structure period T=8.66 Out-phase oscillation . T=10.45 In-phase oscillation . T=8.59 http://ifisc.uib.es

Collective modes of oscillation http://ifisc.uib.es

Bifurcation diagram of two coupled oscillatory LS http://ifisc.uib.es

Dynamical regimes http://ifisc.uib.es

Coupled Landau-Stuart Equations & = μ + ω − γ + α 2 + β + δ − κ [ ( ) ] ( )( ) A A i i A i A A 1 1 1 2 1 & = μ + ω − γ + α 2 + β + δ − κ A A [ i ( i ) A ] ( i )( A A ) 2 2 2 1 2 http://ifisc.uib.es

Logical Operations Logical Operations Using Cavity Solitons: By coupling excitable Cavity Solitons it is possible to realize logical operations Input 1 Input 2 Output OR 0 0 0 1 0 1 1 1 1 AND 0 0 0 1 1 0 1 1 1 NOT 0 1 1 0 http://ifisc.uib.es

Logical Operations AND. 1 0 0: 1 0 0 http://ifisc.uib.es

Logical Operations AND. 1 1 1: 1 1 Logic gates and hardware 1 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 of 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. http://ifisc.uib.es

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 → ∞ & I s • → 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 . http://ifisc.uib.es