1. CONVECTIVE INSTABILITY and NOISE-SUSTAINED STRUCTURES IN OPTICS E. Louvergneaux Laboratoire de Physique des Lasers Atomes Molécules University of Lille (France)

2. Colonia 2005 2 Pattern formation and non locality Pattern formation ≡ spontaneous symmetry breaking (translation) of the ground state System System with transverse non local interaction Pattern formation ≡ spontaneous symmetry breaking (translation) of the ground state + breaking of the reflexion symmetry propagating pattern (drifting pattern)

3. Colonia 2005 3 Pattern formation and non locality Analysis of pattern formation ≡ determination of wavenumber and growth rate at threshold System System with transverse non local interaction Analysis of pattern formation ≡ determination of wavenumber and growth rate at threshold + analysis of pattern emergence and spreading How the emerging pattern will invade or not the whole space

4. Colonia 2005 4 Pattern formation and non locality System System with transverse non local interaction Convective patterns / Absolute patterns Noise sustained patterns New types of patterns New features … Well known absolute patterns

5. Colonia 2005 5 Pattern formation and non locality System System with transverse non local interaction Convective patterns / Absolute patterns Noise sustained patterns New types of patterns New features … Well known absolute patterns

6. Colonia 2005 6 Pattern formation and non locality in optics System with transverse non local interaction TranslationRotationMagnification See works on LCLV, OPO : M.A. Vorontsov et al P.L. Ramazza et al M. San Miguel et al M. Taki et al

7. Colonia 2005 7 Pattern formation and non locality in optics Translation Drifting patterns Translational shift in a feedback loop Walk off (anisotropic crystal) System with transverse non local interaction

8. Colonia 2005 8 The optical system Transverse coordinate x Time t Liquid crystal Laser mirror Liquid crystal Laser mirror With transverse shift Transverse coordinate x Time t Transverse coordinate x Numerics

9. Colonia 2005 9 Competition between the spreading and the drift (diffraction vs advection) Transverse coordinate x Time t The convective and absolute patterns Absolute pattern Convective pattern Transverse coordinate x (the pattern grows fighting the drift upstream) (the pattern grows but is advected away by the drift) Complex frequency and wavenumber to account for spatial and temporal growth rates Liquid crystal Laser mirror With transverse shift

10. Colonia 2005 10 The nature of the instabilities Convective instability Absolute instability → Impulse response of the system Initial perturbation Time

11. Colonia 2005 11 The convective and absolute instabilities Convective instability Absolute instability → Impulse response of the system No pattern at long time Patterned state Final state

12. Colonia 2005 12 The convective and absolute instabilities Convective instability Absolute instability → Impulse response of the system No pattern at long time Patterned state Final state If noise Noise sustained patterned state

13. Colonia 2005 13 With noise Transverse coordinate x Noise sustained patternAbsolute pattern Noise sustained pattern The noise sustained patterns Without noise Transverse coordinate x Time t Transverse coordinate x Absolute pattern No pattern noise sustained patterns ≡ convective patterns sustained by noise Numerics