Noise and transverse flow effects on spatio-temporal instabilities in a liquid crystal optical system Gonzague Agez

Directors:Pierre GlorieuxEric LouvergneauxChristophe Szwaj

Spontaneous pattern formationHomogeneous systemSpatial organization (ordered or disordered, dynamical or stationary)

Examples of natural patternscloudszebra skingator cuirassspiral galaxysand ripplesleopard skin

Outlinepattern formation mechanism I. Pattern formation in the liquid crystal deviceII. The noise effects below and near the thresholdIII. The effects of a transverse flowmodel The noise in the system noisy precursorsphase localization at the onset of the 1D instability Speckle analysis to determine dynamical constants of the system Introduction to the convective and absolute instabilitiesExperimental evidence of noise sustained structures in opticsThe properties of 1D and 2D pattern in presence of a transverse flowNoise sustained superlattice and quasicrystalsIV. Conclusion

The set up : 2D configurationLaserMirrorKerr Mediumn=f(I)

LaserMirrorKerr Mediumn=f(I)xyInput beamyxCameraxyOutput beamSpot lineThe set up : 1D configuration

The mechanism of selective amplification: the Talbot effect-A.M-P.M+A.M+PM+P.MzHomogeneous front planez=0z=lT/4z=2lT/4z=3lT/4z=lTAuto-reproduction Talbot lengthlT=22/0The distance of feedback sets the periodicityMirrorSpatial selective amplificationKerr mediumP.M :Phase modulationA.M :Amplitude modulation

The modelKerr mediumn=f(I)LaserMirror (R)dLwithMedium properties:: diffusion length: relaxation time: Kerr coefficientDevice properties:: feedback distance: sample width: wave number of light: reflectivity of the mirrorW.J.Firth, J. Mod. Opt. 37, 151 (1990)the modeld

Linear stability analysisthe modelSolution for the refractive indexMarginal curve of stabiltyI0I0Positive nonlinearityPositive nonlinearity

Theoretical bifurcation diagram-0.100.10.20.30.40.50.60.90.9511.051.11.151.2UModulation amplitudeI/IthHomogeneousthe modelDAlessandro et al., Phys. Rev. A, 46(1). (1992)Neubecker et al, Phys. Rev. E, 65, 066206 (2002)

The liquid crystal layerAdditive Gaussian white noiseFluctuation of the director axis around his mean value Homeotropic nematic cell= Kerr mediumStochastic termthe modelThermal fluctuations

II.The noise effects below and near the threshold

with noisewithout noiseAbove threshold (=1.05c)Below threshold (=0.95c)Near field intensity(at the ouput of the LC)Far field intensity(optical FT)Numerical simulationsExperiment80 W/cm85 W/cm00.511.5-0.5-1-1.5k (m-1)1-10x/w0.50.51-100.50.5x/w00.511.5-0.5-1-1.5k (m-1)kxxxxyykxkxkykyNoise effects on pattern formationNoise needs to be taken into account for achieving a realistic description

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

Properties of the precursors(u.a.)Analytical results

(u.a.)Analytical resultsProperties of the precursors

(u.a.)Analytical resultsProperties of the precursorsThe noisy precursors anticipate the wave number that appear at threshold

Experimental results608010012014016018020055657585I0 (W/cm)maximum intensity of the first peak (u.a.)Evolution of the fondamental Fourier componentEvolution of the time-average experimental optical FT intensity with input intensityExperiments

2D configuration: concentrical rings six spots The crossing of the thresholdNeeds a criterion to localize the threshold in the 1D configurationBelow thresholdAbove thresholdExperiments

Spatial phase localizationtxTemporal evolution of a 1D patternbelowaboveTime averageCrossing the threshold = phase localizationExperimentsinstantaneousSpatial phase

The crossing of the thresholdthreshold without noise0204060100951301641990,50,70,91,100,10,20,30,40,580Indicator for the level of noise

Standard deviation of the spatial phase (degrees)Input intensity (W/cm2)Input intensity (u.a.)ExperimentNumerical simulationThe localization of the spatial phase can be used to determine a threshold in presence of noiseExperimentsInflexion pointG. Agez et al. , Phys. Rev. A, 66, 063805 (2002)

Speckle analysis to determine dynamical constantsApplicationrelaxation timediffusion lengthStandard diffusive equation:In our case:

Speckle analysis to determine dynamical constantsxykxkyNear fieldFar fieldAnalytical expression:ainErLCoutEr (s-1)(u.a)Time-averaged far field intensityAnalytical expression:Square modulus of the double Fourier transform of the output intensity :Diffusion length:Relaxation time:experimentfitexperimentfitApplicationG. Agez et al. , Opt.Comm., (2005)

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

The system with nonlocal interactionLiquid crystalLasermirrorTheorySee Ramazza et al. Vorontsov et al. Ackemann et al.

The system with nonlocal interactionLiquid crystalLasermirrorTheory

Absolute and convective regimesLiquid crystalLasermirrorWith transverse flowConvective instability Absolute instability (the pattern grows fighting the drift upstream)(the pattern grows but is advected away by the drift)Competition between spatial amplification and driftAnalysis of temporal evolution of an initial local perturbationTheory

The impulse response of the systemtimeConvective instability Absolute instability Local perturbationLocal perturbationxtxtxtxtxt00000(x/t)CLREvolution of the wave packetConvective threshold(x/t)(x/t)R(x/t)R(x/t)L(x/t)Absolute thresholdxxOnly one critical mode kc with zero growth rateA mode ka with zero group velocity and zero growth rateTheory

Dispersion relation : W(k) with W=Wr+iWi and k=kr+iki

Spatial modelx/tlx/tlx/tlx/tlx/t(x/t)L(x/t)R(x/t)R(x/t)R(x/t)LCA0growth rate (x/t)CDetermination of convective and absolute thresholds Theory

Conditions of thresholdTheory

ConvectiveAbsolute Unstable wavenumber Wavenumber defined by Threshold defined by

The dispersion relation (k)Liquid crystalLasermirror

Dnvariation of the refractive index lddiffusion length tdecay time ndiffusion length R mirror reflectivityc nonlinear susceptibility (Kerr type)x(x,y,t) gaussian white noisee noise amplitudeI=|F|2 withF=F0 e-(x/w)2 gaussian pumping fields = d/k0 with k0 laser wave number

hTheoryEvolution equation of the refractive index n in presence of noise and with a tilted mirror:

The 1D configuration

Experimental evidence of convective structuresconvective regionAnalytical prediction for the liquid crystal device with tilted mirror Absolute thresholdConvective threshold

Transverse coordinate xTimeTransverse coordinate xNo patternAbsolute patternWithout noisenoise sustained structuresTime

Experimental evidence of noise sustained structuresTimexLouvergneaux et al, Phys. Rev. Lett. 92(4), 043901 (2004)Experiments

Conditions of thresholdAnalytical results++++---Convective thresholdMarginal stability curves c=f(k)

Conditions of thresholdAnalytical resultsConvective threshold

Properties of 1D noise sustained patterns Convective threshold of the 5 first tongues(p=1 to 5)Critical wavenumber of the 5 first tongues(p=1 to 5)tongue n:12345

Analytical resultsh

Properties of 1D noise sustained patterns group velocityVg(kc)phase velocityV(kc)Convective thresholds of the 5 first tongues(p=1 to 5)Critical wavenumbers of the 5 first tongues(p=1 to 5)tongue n:12345

Analytical results

Experimental stationary noise sustained patternExperimentsGenerator of stationary patterns with discrete wavelengths ajustable with the drift strength (h)

The 2D configuration

The different types of 2D convective structuresExperimentsConvective conditions with a non locality along the x-directionThe 1D type:The 2D type:Vertical rolls (as in the 1D case)Horizontal rollsn = 0Rectangular latticen > 0Near fieldFar fieldRamazza et al, Phys. Rev. A 54(4), 3472(1996)

2D type convective thresholdsAnalytical resultsh11.21.41.61.8202468101214p=1p=2p=3p=1p=2p=3n=4n=3n=2n=1n=0n=2n=0n=1n=0n=1n=2p: tongue indexn: from

1D type convective thresholdsAnalytical resultsConvective thresholdConvective threshold for the vertical rolls

Convective threshold of 1D type patterns (vertical rolls)Convective threshold of 2D type patterns(horizontal rolls and rectangular lattices)Experimental stationary noise sustained patternNear fieldFar fieldNear fieldFar fieldNear fieldFar field

Dynamical properties of the 2D structures

Noise sustained superlattice and quasicrystalsPatterns composed of at least 2 different wavelengths (i.e. composed of 2 previous modes- vertical, horizontal rolls and rectangular lattices)kxky

Experimental superlatticeExperimentst(s)x/w02000-0.10.10.2-0.2kx (m-1)kystationarityFarfieldkykx

Noise sustained superlatticeNumerical simulationsWith noiseWithout noise=1.05=1.05FarfieldNearfieldNo structures at long time

Noise sustained quasicrystalAnalytical resultsNumericalsimulationsNear fieldPatterncompositionFar field filterFar field

Examples of superlatticesNumerical simulation01-1x/w01-1-22h=12=-1701-1x/w01-1-22h=21s=-1701-1x/w01-1-22h=21.8s=+1701-1x/w01-1-22h=5.7s=-17k (en ld-1)k (en ld-1)k (en ld-1)k (en ld-1)s

ConclusionI. Noisy precursorsII. Transverse flow effectsEvidence of noise effect below the thresholdComplete analytical characterization and very good matching with experimentsSpatial phase localization during the onset of the 1D patternSpeckle analysis to determine experimentally dynamical constants of the systemDetermination of convective and absolute threshold Experimental evidence of noise sustained structures in opticsDynamical study of the 1D patterns (stationary, drifting, different wavenumber)Three different families of 2D pattern (horizontal and vertical rolls, rectangular lattice)No absolute threshold for horizontal rolls and rectangular lattice Resonance condition to build noise sustained superlattice and quasicrystals

PerspectivesNoise sustained pattern properties away from threshold

Pattern nonlinear interaction (between 2 structures either convective or absolute)

Experimental evidence of quasicrystals

Thank you for your attention !And others effects one funny example

The 2D patterns precursorsExperimental resultsVertical rollsHorizontal rollsRectangular latticesExperiments

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