Ph. D. Thesis Spatial Structures and Information Processing in Nonlinear Optical Cavities Adrian Jacobo

Dynamics of LS in Kerr Cavities 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) Bistability Stable droplets: Localized structures stabilised by nonlinear domain wall dynamics due curvature. Exist in 2d systems. D. Gomila et al, PRL 87, 194101 (2001) Subcritical Cellular Pattern 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) Excitability mediated by localized structures

Kerr Effect: Dynamics of LS in Kerr Cavities Is a change in the refractive index of a material in response to an electric field. Kerr Effect: : Detuning : Pump

Homogeneous Steady State Solution Dynamics of LS in Kerr Cavities Stability of the solutions: Homogeneous Steady State Solution Becomes unstable at leading to a subcritical hexagonal pattern L.A. Lugiato & R. Lefever, PRL 58, 2209 (1988).

Oscillating LS – Cross Section Dynamics of LS in Kerr Cavities Stability of the solutions: Oscillating LS – Cross Section Hopf Saddle-node Homogeneous solution D. Gomilla, P. Colet, M. Matías. Phys. Rev. Lett. 94 063905 (05) W.J. Firth, A. Lord & A.J. Scroggie, Phys. Scripta, T67, 12 (96) W.J. Firth & A. Lord, J. Mod. Opt. 43, 1071 (96)

Dynamics of LS in Kerr Cavities Saddle – Loop Bifurcation: Saddle-loop LC Hopf max(|E|) SN Homogeneous solution q

Minimum distance of oscillating LS to middle-branch LS Dynamics of LS in Kerr Cavities Saddle – Loop Bifurcation: oscillating LS q=1.3047 Minimum distance of oscillating LS to middle-branch LS q=1.30478592 middle-branch LS q=1.304788 homogeneous solution Is =0.9 D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, 063905 (2005).

Scaling Law of the SL Bifurcation: Dynamics of LS in Kerr Cavities Scaling Law of the SL Bifurcation: middle-branch LS spectrum lu Close to bifurcation point: T: period of oscillation lu unstable eigenvalue of saddle (middle-branch LS) S.H. Strogatz, Nonlinear dynamics and chaos 2004 1/u numerical simulations

Excitability: Dynamics of LS in Kerr Cavities Occurs beyond the saddle-loop bifurcation Small perturbations of homogeneous solution decay. Localized perturbations above middle branch LS send the system to a long excursion through phase-space. The system is not locally excitable. Excitability emerges from spatial coupling

Dynamics of LS in Kerr Cavities Takens-Bogdanov Point: Unfolding of the Takens-Bogdanov Point (normal form) TB Distance between saddle-node and Hopf Hopf saddle-loop saddle-node No LS LS oscillating LS Excitability The Hopf frequency when it meets the saddle-node is zero: Takens-Bogdanov point. Unfolding of TB yields a Saddle-Loop d → 0 for  → ∞ and Is → 0 NLSE

Oscillations Excitability Dynamics of LS in Kerr Cavities Effect of a Localized Pump: Excitability arising from a saddle-loop bifurcation have a large threshold. To reduce the threshold we consider for the pump: Oscillations Excitability Is max(|E|2) 1 Pattern Hom. pump SNIC Saddle Node Hopf Ish=0.7, q=1.34 Loc. pump

Dynamics of LS in Kerr Cavities SNIC Bifurcation: unstable upper branch LS Is=0.927 Projection onto yu Projection onto ys Is=0.907 Is=0.8871 Is=0.8 middle-branch cavity LS fundamental solution Ish =0.3 q=1.45

Scaling Law of the SNIC Bifurcation: Dynamics of LS in Kerr Cavities Scaling Law of the SNIC Bifurcation: Close to bifurcation point: T: period of oscillation S.H. Strogatz, Nonlinear dynamics and chaos 2004 numerical simulations

Dynamics of LS in Kerr Cavities Phase diagrams: H V III II c II III SL SN IV IV I I 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). Lower excitability threshold. I No soliton II Stationary soliton III Oscillating soliton, fundamental solution stable IV Excitablility V Oscillating soliton, no fundamental solution

Dynamics of LS in Kerr Cavities Exitability:

Dynamics of LS in Kerr Cavities Cusp Point: I No soliton II Stationary soliton

Dynamics of LS in Kerr Cavities SNSL Point:

Logical Operations Using LS: Dynamics of LS in Kerr Cavities By coupling excitable Localized Structures it is possible to realize logical operations Logical Operations Using LS: Input 1 Input 2 Output OR 1 AND NOT

Dynamics of LS in Kerr Cavities AND. 1 0 0:

Dynamics of LS in Kerr Cavities AND. 1 1 1:

Dynamics of LS in Kerr Cavities OR. 1 0 1:

Dynamics of LS in Kerr Cavities OR. 1 1 1:

Dynamics of LS in Kerr Cavities NOT:

Conclusions SHG can be used to perform all-optical image processing: Frequency transfer Contour recognition and contrast enhancement Denoising Image processing operations are also possible with spherical mirrors. This ideas can be applied to change-point detection in data series and to message decryption. LS in Kerr cavities display rich dynamics such as oscillations and excitability. The properties of the excitability of LS can be controlled by a localized pump. Logic gates can be built using the dynamical properties of LS. Oscillatory LS are a model of nonlocal oscillator and display nontrivial behavior when coupled.

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