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CONVECTIVE INSTABILITY
and
NOISE-SUSTAINED STRUCTURES IN OPTICS
E. Louvergneaux
Laboratoire de Physique des Lasers Atomes MoléculesUniversity of Lille (France)
Colonia 2005 2
Pattern formation and non locality
Pattern formation ≡ spontaneous symmetry breaking (translation) of the ground state
System
System with transversenon local interaction
Pattern formation ≡ spontaneous symmetry breaking (translation) of the ground state + breaking of the reflexion symmetry
propagating pattern (drifting pattern)
Colonia 2005 3
Pattern formation and non locality
Analysis of pattern formation ≡ determination of wavenumber and growth rate at threshold
System
System with transversenon 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
Colonia 2005 4
Pattern formation and non locality
System
System with transversenon local interaction
Convective patterns / Absolute patternsNoise sustained patternsNew types of patternsNew features …
Well known absolute patterns
Colonia 2005 5
Pattern formation and non locality
System
System with transversenon local interaction
Convective patterns / Absolute patternsNoise sustained patternsNew types of patternsNew features …
Well known absolute patterns
Colonia 2005 6
Pattern formation and non locality in optics
System with transversenon local interaction
Translation Rotation Magnification
See works on LCLV, OPO : M.A. Vorontsov et alP.L. Ramazza et alM. San Miguel et al M. Taki et al
drrr
Colonia 2005 7
Pattern formation and non locality in optics
Translation
Drifting patterns
Translational shift in a feedback loopWalk off (anisotropic crystal)
System with transversenon local interaction
drrr
Colonia 2005 8
The optical system
Transverse coordinate x
Tim
e t
Liquidcrystal
Laser
mirrorLiquidcrystal
Laser
mirror
With transverse shift
Transverse coordinate x
Tim
e t
Transverse coordinate x
Numerics
Colonia 2005 9
Competition between the spreading and the drift
(diffraction vs advection)
Transverse coordinate x
Tim
e 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 wavenumberto account for spatial and temporal growth rates
Liquidcrystal
Laser
mirror
With transverse shift
Colonia 2005 10
The nature of the instabilities
Convective instability Absolute instability
→ Impulse response of the system
Initial perturbationTi
me
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
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
Colonia 2005 13
With noise
Transverse coordinate x Transverse coordinate x
Noise sustained pattern Absolute patternNoise sustained pattern
The noise sustained patterns
Without noise
Transverse coordinate x
Tim
e t
Transverse coordinate x
Absolute patternNo pattern
noise sustained patterns ≡
convective patterns sustained by noise
Numerics
