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

2. 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, (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

3. 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

4. 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).

5. 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 (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)

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

7. 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=
middle-branch LS q=
homogeneous solution
Is =0.9
D. Gomila, M. Matias and P. Colet, Phys. Rev. Lett. 94, (2005).

8. 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

9. 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

10. 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 → NLSE

11. 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

12. 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

13. 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

14. 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

15. Dynamics of LS in Kerr Cavities
Exitability:

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

17. Dynamics of LS in Kerr Cavities
SNSL Point:

18. 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

19. Dynamics of LS in Kerr Cavities
AND :

20. Dynamics of LS in Kerr Cavities
AND :

21. Dynamics of LS in Kerr Cavities
OR :

22. Dynamics of LS in Kerr Cavities
OR :

23. Dynamics of LS in Kerr Cavities
NOT:

24. 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.

25. Thank you for your attention