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Gonzague Agez

Noise and transverse flow effects on spatio-temporal instabilities in a liquid crystal optical system. Gonzague Agez. Directors: Pierre Glorieux Eric Louvergneaux Christophe Szwaj. Spontaneous pattern formation. Spatial organization (ordered or disordered, dynamical or stationary).

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Gonzague Agez

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  1. Noise and transverse flow effects on spatio-temporal instabilities in a liquid crystal optical system Gonzague Agez Directors: Pierre Glorieux Eric Louvergneaux Christophe Szwaj

  2. Spontaneous pattern formation Spatial organization (ordered or disordered, dynamical or stationary) Homogeneous system

  3. Examples of natural patterns spiral galaxy sand ripples leopard skin gator cuirass clouds zebra skin

  4. Outline I. Pattern formation in the liquid crystal device pattern formation mechanism model The noise in the system II. The noise effects below and near the threshold noisy precursors phase localization at the onset of the 1D instability Speckle analysis to determine dynamical constants of the system III. The effects of a transverse flow Introduction to the convective and absolute instabilities Experimental evidence of noise sustained structures in optics The properties of 1D and 2D pattern in presence of a transverse flow Noise sustained superlattice and quasicrystals IV. Conclusion

  5. The set up : 2D configuration y x Camera y y x x Input beam Output beam Hexagonal lattice Laser Mirror Kerr Medium n=f(I)

  6. The set up : 1D configuration y x Laser Camera Mirror Kerr Medium n=f(I) y y x x Output beam Input beam Spot line

  7. The mechanism of selective amplification: the Talbot effect P.M :Phase modulation Kerr medium A.M :Amplitude modulation Homogeneous front plane +P.M -A.M -P.M +A.M +PM  Spatial selective amplification The distance of feedback sets the periodicity z z=0 z=lT/4 z=2lT/4 z=3lT/4 z=lT Mirror Auto-reproduction Talbot length lT=22/λ0

  8. The model the model Kerr medium n=f(I) Image of the mirror Mirror (R) Laser 4f changes the sign of the Kerr nonlinearity L d<0 d Medium properties: : diffusion length with : relaxation time : Kerr coefficient Device properties: : feedback distance : sample width : reflectivity of the mirror W.J.Firth, J. Mod. Opt. 37, 151 (1990) : wave number of light

  9. Linear stability analysis the model Solution for the refractive index Marginal curve of stabilty Positive nonlinearity I0 I0 Positive nonlinearity

  10. Theoretical bifurcation diagram Rolls Positive hexagons Square R S + H - H - Negative hexagons the model 0.6 0.5 0.4 0.3 Modulation amplitude 0.2 0.1 Homogeneous U 0 -0.1 0.9 0.95 1 1.05 1.1 1.15 1.2 I/I th D’Alessandro et al., Phys. Rev. A, 46(1). (1992) Neubecker et al, Phys. Rev. E, 65, 066206 (2002)

  11. The liquid crystal layer the model Thermal fluctuations Fluctuation of the director axis around his mean value Homeotropic nematic cell = Kerr medium Stochastic term Additive Gaussian white noise

  12. II. The noise effects below and near the threshold

  13. Noise effects on pattern formation Numerical simulations Experiment 85 W/cm² Above threshold (µ=1.05µc) Below threshold (µ=0.95µc) 80 W/cm² x/w x/w x x x -1 0.5 0 0.5 1 -1 0.5 0 0.5 1 Near field intensity (at the ouput of the LC) y y Far field intensity (optical FT) ky ky -1.5 -0.5 0.5 1.5 -1 0 1 -1.5 -0.5 0.5 1.5 -1 0 1 kx kx kx k (µm-1) k (µm-1) without noise with noise Noise needs to be taken into account for achieving a realistic description

  14. Analytical expression for the noisy precursors Analytical results Linearized expression of the evolution equation of the index fluctuations Δn in presence of noise: Experimentally observable quantity: the far field intensityIFF(k,t) Only the auto-correlation function of Δn can be analytically written : Analytical expression for the time-averaged far field intensity:

  15. Properties of the precursors Marginal stability curves kc:: critical wave number Analytical results (u.a.)

  16. Properties of the precursors Marginal stability curves kc:: critical wave number Analytical results (u.a.)

  17. Properties of the precursors Analytical results (u.a.) The noisy precursors anticipate the wave number that appear at threshold

  18. Experimental results -kc +kc Experiments Evolution of the time-average experimental optical FT intensity with input intensity Evolution of the fondamental Fourier component 200 180 160 140 120 100 80 maximum intensity of the first peak (u.a.) 60 55 65 75 85 I0 (W/cm²)

  19. The crossing of the threshold 1D configuration: no qualitative difference Intensity profile Intensity profile kc kc Experiments 2D configuration: concentrical rings six spots Above threshold Below threshold Needs a criterion to localize the threshold in the 1D configuration

  20. Spatial phase localization t x Spatial phase dispersion Experiments Temporal evolution of a 1D pattern Time average t x below above instantaneousSpatial phase Crossing the threshold = phase localization

  21. The crossing of the threshold Experiments Experiment Numerical simulation threshold without noise Standard deviation of the spatial phase (degrees) 100 0,5 Inflexion point 0,4 80 0,3 60 0,2 40 0,1 20 Indicator for the level of noise 0 0 0,5 0,7 0,9 1,1 95 130 164 199 Input intensity (u.a.) Input intensity (W/cm2) The localization of the spatial phase can be used to determine a threshold in presence of noise G. Agez et al. , Phys. Rev. A, 66, 063805 (2002)

  22. Speckle analysis to determine dynamical constants Quantitative comparisons between theory and experiments Experimental measurement of et Problem : No direct measurement of the intrinsic relaxation time (only response time measurements) New method Application Standard diffusive equation: In our case: relaxation time diffusion length

  23. Speckle analysis to determine dynamical constants (u.a) k (µm-1) Application Time-averaged far field intensity LC r r a E E out i n Analytical expression: y ky experiment x kx fit Near field Far field Diffusion length: Square modulus of the double Fourier transform of the output intensity : Analytical expression: (u.a) Relaxation time: experiment fit Ω (s-1) G. Agez et al. , Opt.Comm., (2005)

  24. III. Effects of a transverse flow (non local interaction)

  25. The system with nonlocal interaction Liquidcrystal mirror Laser Transverse flow Time t Transverse coordinate x Transverse coordinate x Theory Liquidcrystal mirror Laser See Ramazza et al. Vorontsov et al. Ackemann et al.

  26. The system with nonlocal interaction Liquidcrystal mirror Laser Time t Time t Transverse coordinate x Transverse coordinate x Transverse coordinate x Theory Liquidcrystal mirror Laser

  27. Absolute and convective regimes Time t Transverse coordinate x Transverse coordinate x Theory With transverse flow Liquidcrystal Competition between spatial amplification and drift mirror Laser Analysis of temporal evolution of an initial local perturbation (the pattern grows but is advected away by the drift) Convective instability Absolute instability (the pattern grows fighting the drift upstream)

  28. The impulse response of the system Theory Convective instability Absolute instability time Local perturbation Local perturbation x x Convective threshold Absolute threshold (x/t) (x/t) t t t t t (x/t) L L Evolution of the wave packet R (x/t) C (x/t) (x/t) R R x x x x x 0 0 0 0 0 A mode ka with zero group velocity and zero growth rate Only one critical mode kc with zero growth rate

  29. Determination of convective and absolute thresholds Convective threshold Absolute threshold (x/t) (x/t) t t t t t (x/t) L L R Evolution of the wave packet (x/t) C (x/t) (x/t) R R x x x x x 0 0 0 0 0 Theory l l l l l (x/t) growth rate λ x/t x/t x/t x/t x/t (x/t) (x/t) (x/t) (x/t) C 0 A R L R L (x/t) R C Dispersion relation :W(k) withW=Wr+iWiandk=kr+iki Spatial mode

  30. Conditions of threshold Theory

  31. The dispersion relation Ω(k) Theory Liquidcrystal mirror Laser h Evolution equation of the refractive index Δn in presence of noise and with a tilted mirror: Dn variation of therefractive index ld  diffusion length t  decay time n diffusion length R  mirror reflectivity c  nonlinear susceptibility (Kerr type) x(x,y,t) gaussian white noise e noise amplitude I=|F|2with F=F0e-(x/w)2gaussian pumping field s = d/k0with k0 laser wave number

  32. The 1D configuration

  33. Experimental evidence of convective structures Analytical prediction for the liquid crystal device with tilted mirror Absolute threshold Convective threshold convective region

  34. noise sustained structures Noise sustained pattern Absolute pattern Transverse coordinate x Transverse coordinate x With noise Local perturbation → finding a new criterion No pattern Absolute pattern Time Time Transverse coordinate x Transverse coordinate x Without noise

  35. Experimental evidence of noise sustained structures Edge detection Numerical simulations Experiments H: homogeneous state C: convective regime (noise sustained structures) A: absolute regime ) s t i n u H . b C C r a H ( A A 1.02 1.12 1.22 100 110 120 130 2 2 (W/cm) F I 0 0 Experiments Time x Louvergneaux et al, Phys. Rev. Lett.92(4), 043901 (2004)

  36. Conditions of threshold feedback distance h : nonlocality Analytical results Convective threshold Marginal stability curves µc=f(k) + - + - + - +

  37. Conditions of threshold Absolute threshold Large convective region Analytical results Convective threshold

  38. Properties of 1D noise sustained patterns 0 3 6 9 12 Theory Experimental data Analytical results tongue n°: 1 2 3 4 5 Convective threshold of the 5 first tongues (p=1 to 5) Critical wavenumber of the 5 first tongues (p=1 to 5) h

  39. Properties of 1D noise sustained patterns Stationary noise sustained patternsfor local minimum in the threshold curve Analytical results Convective thresholds of the 5 first tongues (p=1 to 5) tongue n°: 1 2 3 4 5 Critical wavenumbers of the 5 first tongues (p=1 to 5) phase velocity Vφ(kc) group velocity Vg(kc)

  40. Experimental stationary noise sustained pattern Experiments Generator of stationary patterns with discrete wavelengths ajustable with the drift strength (h)

  41. The 2D configuration

  42. The different types of 2D convective structures Experiments Convective conditions with a non locality along the x-direction Near field Far field The 1D type: Vertical rolls (as in the 1D case) Horizontal rolls n = 0 The 2D type: Rectangular lattice n > 0 Ramazza et al, Phys. Rev. A54(4), 3472(1996)

  43. 2D type convective thresholds p=2 p=1 p=3 A B C C,C’ E B,B’ D D E A B’ C’ 1.09 1.28 1.46 1.65 1.83 2.02 Analytical results p: tongue index n: from n=4 n=0 n=1 n=2 n=3 n=0 n=0 2 p=3 p=3 1.8 1.6 n=1 p=2 p=2 n=1 1.4 p=1 1.2 p=1 1 n=2 0 2 4 6 8 10 12 14 n=2 h

  44. 1D type convective thresholds Analytical results Convective threshold for the vertical rolls Convective threshold

  45. Experimental stationary noise sustained pattern Horizontal rolls Rectangular lattice Vertical rolls experiments numerical simulations Near field Near field Far field Far field Near field Far field Convective threshold of 1D type patterns (vertical rolls) Convective threshold of 2D type patterns (horizontal rolls and rectangular lattices)

  46. Dynamical properties of the 2D structures Horizontal rolls and rectangular lattice Vertical rolls Purely convective structures (no absolute threshold) Convective and absolute regime + + Stationary at the convective threshold (null phase velocity) Drifting or stationary

  47. Noise sustained superlattice and quasicrystals Resonance condition : Patterns composed of at least 2 different wavelengths(i.e. composed of 2 previous modes- vertical, horizontal rolls and rectangular lattices) ky kx

  48. Experimental superlattice Near field y -1 -0.5 0 0.5 1 -0.5 0 0.5 x/w Experiments ky kx (µm-1) -0.1 0 0.1 0.2 -0.2 kx Far field ky 200 stationarity t(s) 0 x/w

  49. Noise sustained superlattice Numerical simulations With noise Without noise Far field Near field No structures at long time µ=1.05 µ=1.05

  50. Noise sustained quasicrystal Analytical results Pattern composition Far field filter Numerical simulations Near field Far field

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