Optical textures: characterizing spatiotemporal chaos

1.1 MB·

by  Marcel G. Clerc

Optical textures: characterizing
spatiotemporal chaos
MarcelG.Clerc,1GregorioGonza´lez-Corte´s,1VincentOdent,1,2and
MarioWilson1,3,∗
1DepartamentodeF´ısica,FacultaddeCienciasF´ısicasyMatema´ticas,UniversidaddeChile,
BlancoEncalada2008,Santiago,Chile
2Presentaddress:Universite´Lille1,LaboratoiredePhysiquedesLasers,Atomeset
Mole´cules,CNRSUMR8523,59655Villeneuved’AscqCedex,France
3CONACYT–CICESE,CarreteraEnsenada-Tijuana3918,ZonaPlayitas,C.P.22860,
6 Ensenada,Me´xico
1 ∗[email protected]
0
2

n Abstract: Macroscopicsystemssubjectedtoinjectionanddissipationof
u energycanexhibitcomplexspatiotemporalbehaviorsasresultofdissipative
J self-organization. Here, we report a one and two dimensional pattern

2 forming set up, which exhibits a transition from stationary patterns to

spatiotemporal chaotic textures, based on a nematic liquid crystal layer
]
S with spatially modulated input beam and optical feedback. Using an
P adequate projection of spatiotemporal diagrams, we determine the largest
. Lyapunov exponent. Jointly, this exponent and Fourier transform allow
n
i us to distinguish between spatiotemporal chaos and amplitude turbulence
l
n concepts,whichareusuallymerged.
[
© 2016 OpticalSocietyofAmerica

2
OCIS codes: (190.0190) Nonlinear optics; (190.3100) Instabilities and chaos; (230.3720)
v
Liquid-crystaldevices.
4
4
8
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1. Introduction
Opticalsystemsmaintainedfarfromequilibrium,throughtheinjectionanddissipationofen-
ergy,canpresentspatiotemporalstructures,patterns[1–9].Thesestructuresappearasawayto
optimizeenergytransportandmomenta[10].Patternsaretheresultoftheinterplaybetweenthe
lineargainandthenonlinearsaturationmechanisms.Inmanyphysicalsystems,thesestructures
arestationaryandemergeasaspatialinstabilityofauniformstatewhenacontrolparameteris
changedandsurpassesacriticalvalue,whichusuallycorrespondstoimbalancesofforces.As
theparametersofthesystemarechanged,stationarypatternscanbecomeunstableandbifur-
cate to more complex patterns, even into aperiodic dynamics states [11–14]. This behavior is
characterizedbycomplexspatiotemporaldynamicsexhibitedbythepatternandacontinuous
coupling between spatial modes in time. Complex spatiotemporal dynamics of patterns have
been observed, for example, in fluids [15–19], chemical reaction-diffusion systems [20], car-
x
LCLV feedback loop
V
0
SLM z
L y PC L F
LCLV
FP L B d FB
BS
M
1-D 2-D
BS Far
Field
y
L
L
1 mm
Near
x Near Field M Field
Fig. 1. Schematic representation of the experimental setup. LCLV stands for the liquid
crystallightvalve,Lrepresentstheachromaticlenseswithafocaldistance f =25cm,M
arethemirrors,FBisanopticalfiberbundle,BSstandsforthebeamsplitters,PCrepresents
apolarizingcube,SLMisaspatiallightmodulatorandFPrepresentstheFourierplane.F
andBstandfortheforward(incoming)andbackward(reflected)beamrespectively.d is
theequivalentopticallength.Inthebottomleftpartoftheimage,twoexamplesofobtained
patternsandtextures.
diac fibrillation [21], electroconvection [22], fluidized granular matter [23], nonlinear optical
cavities [1–4] and in a liquid crystal light valve [6,24,25]. In most of these studies, complex
behaviors are characterized by spatial and temporal Fourier transforms, wave vector distribu-
tion, filtering spatiotemporal diagrams, power spectrum of spatial mode, length distributions,
Poincare´ maps and number of defects as a function of the parameters. However, in these ex-
perimentalstudies,spatiotemporalcomplexityhasnotbeencharacterizedusingrigoroustools
of dynamical systems theory as Lyapunov exponents [26,27]. These exponents characterize
the exponential sensitivity of the dynamical behaviors under study and in turn gives a char-
acteristic time scale on which one has the ability to predict the time evolution of the system.
When the largest Lyapunov exponent (LLE) is positive (negative) the system under study is
chaotic(stationary).Recently,wehavecomputedtheexperimentalLLEandcharacterizedthe
spatiotemporalchaosintwospatialdimensionsinaliquidcrystallightvalve(LCLV)withopti-
calfeedback[28].ThedynamicsexhibitedbytheLCLVwithopticalfeedbackischaracterized
bythechangesthatmolecularorientationinducesinthephaseofthereflectedlight,which,in
itsturn—opticalfeedback—producesavoltagethatreorientstheliquidcrystalmolecules.
Oneofthemostimportantconceptsincomplexspatiotemporaldynamicsisturbulence:the
powerspectrumasaresultofthetransportofsomephysicalquantityofdifferentscalesshows
apowerlawdecayasitsmainsignature[36].Theaimofthisarticleistoinvestigateoneand
twodimensionalpatternformingsysteminaLCLV,showninFig.1,whichexhibitsatransi-
tionfromstationarypatternstospatiotemporalchaotictexturesandtoquasiperiodicity.Using
adequatespatiotemporaldiagrams,weobtaintheLLE.Jointly,thisexponentandFouriertrans-
formallowustodistinguishbetweenspatiotemporalchaosandamplitudeturbulenceconcepts,
whichareusuallymerged.
2. Experimentaldetails
Theliquidcrystallightvalvewithanopticalfeedbackisaflexibleopticalexperimentalsetup
thatexhibitspatternformation[6][seeFig.1].TheLCLVisilluminatedbyanexpandedand
collimatedHe-Nelaserbeam,λ =633nm,with3cmtransversediameterandpowerI =6.5
in
mW/cm2, linearly polarized along the vertical axis. Once shone into the LCLV, the beam is
reflected by the dielectric mirror deposed on the rear part of the cell and, thus, sent to the
polarizingcube.Duetothephase-changethelightsuffersinthereflection,thepolarizingcube
will send the reflected light into the feedback loop. To close the feedback loop, a mirror and
an optical fiber bundle are used, these elements assure the light to reach the photoconductor
placed in the back part of the LCLV. In the feedback loop, a 4-f array is placed in order to
obtain a self-imaging configuration and access to the Fourier plane, this array is constructed
with2identicallenseswithfocallength f =25cmplacedinsuchawaythatbothsidesofthe
LCLVareconjugatedplanes.WefiltertheFourierplaneinordertoforcethesystemtoexhibit
roll-patterns in a given direction. Thanks to this configuration the free propagation length in
thefeedbackloopcanbeeasilyadjusted.Fortheperformedexperimentsanopticalequivalent
lengthofd=−4cmwasused.Aspatiallightmodulator(SLM)wasplacedintheinputbeam
opticalpathwitha1:1imagingbetweentheSLMandthefrontalpartoftheLCLV.Withthe
aidofaspecializedsoftware,asquaremaskwasproducedandsenttotheSLM.TheSLMand
the polarizing cube combination allow to impose an arbitrary shape to the input beam. For a
uniform mask of 160 gray-value, the typical input intensity would be I =0.83mW/cm2. To
w
obtaintheshapeusedintheexperiments,oneandtwo-dimensionalmasks,I(x,y),werecreated
and,bymeansofthesemasks,oneandtwodimensionalpatternscanbeobtainedascanbeseen
in the bottom left part of Fig. 1. The system dynamics is controlled by adjusting the external
voltageV appliedtotheLCLV.
0
3. Fromstationarytodisordereddynamics
The presented dynamics in the LCLV have been explored in two different configurations, the
firstoneusinganintensitymaskofzero-levelintensityeverywhereexceptforacentralsquare
partwithlengtha =2.5mm(2-Dmask),andthesecondonewithazero-levelintensityexcept
0
onanarrowchannelof150µmwidthand2.5mmlength(1-Dmask).Theinjectedintensityis
spatiallymodulatedasI =I (x,y),whereI canbecontrolledbychangingthemaskcreatedin
in 0 0
theSLM,and{x,y}arethetransversecoordinatesofthesample.I ismeasuredwhenimposing
0
agivengray-valuetotheilluminatedarea,thatis,forthe2-Dmask
(cid:26)
I +b |x|≤a , and |y|≤a
I (x,y)= 0 0 0 0
0 b else
0
whenb isconstantthroughoutthesampleand|x|>0,|y|>0.Thesameappliesto1-Dmask
0
withtheonlydifferencethat|y|=150µm,whichissmallenough,comparedwiththepattern
wavelength, to neglect its size and consider it as a 1-D mask. In the presented configurations
I =0.9mW/cm2andb =0.1mW/cm2.ThealternatingvoltageV hasbeenvariedbetween3
0 0 0
and7V ,ataconstantfrequency f =5kHz,startingwiththeappearanceofstationaryroll-
rms 0
patterns.FordifferentV values,thedynamicalbehaviorobtainedinthesystemwasrecorded
0
with a CCD camera. Figure 2 shows the spatiotemporal evolution of the observed patterns in
oneandtwodimensions,respectively.Thisevolutionischaracterizedbyprojectedspatiotem-
poraldiagrams,whichareconstructed,inthe2-Dexperiments,bypickinganarbitraryline—
transversal to the rolls direction—in the illuminated zone and superposing it as time evolves;
in the 1-D experiments this construction is simpler, is enough to superpose the pattern as the
time evolves. The system exhibits stationary stripe patterns [cf. Fig. 2(a)]. These patterns are
a) b) c)
1-D 1-D 1-D
t
2-D 2-D 2-D
x
d)
y
x
t
Fig. 2. Spatial textures in LCLV with optical feedback at different voltageV . Top pan-
0
els correspond to spatiotemporal diagrams of observed dynamics working with one-
dimensional patterns. Middle panels panels stand for projected spatiotemporal diagrams
oftwo-dimensionaltextures.a)Periodicregime,b)chaoticbehavior,c)quasi-periodicdy-
namics,andd)a3-Dspatiotemporaldiagramofacomplextexturefoundintheintermittent
regimeatV =4.3Vrms.Allvaluesweretakenduring100sandarenormalized.
0
inducedbyaspatialfilteringintheFourierplane[cf.FPinFig.1].Actually,throughaslit,we
canfilterspatialmodes.
IncreasingV ,thedynamicsshownbythepatternbecomesabruptlycomplex[cf.Figs.2(b)
0
and 2(d)]. Clearly, in the projected spatiotemporal diagram, we detected an intermittent be-
havior. That is, the pattern exhibits aperiodic oscillations invaded by large fluctuations, gen-
erating several spatial and temporal dislocations. Likewise, the system exhibits a high spa-
tiotemporal complexity. This kind of disorder is usually associated to spatiotemporal chaotic
textures[28,32–34].Figure2(d)showsa3-Dspatiotemporaldiagram,fromthisdiagramitis
clearthatanarbitrarilychosenlinerepresentsthedynamics.
Further increasingV , the pattern begins to oscillate in a complex manner [see Fig. 2(c)].
0
We observe in the projected spatiotemporal diagram local waves, oscillations and spatiotem-
poral dislocations. Similar dynamics has been reported in one-dimensional inhomogeneous
systems[29–31].Inourexperimentstheseinhomogeneitiescanbecausedbytheinherentim-
perfections and inhomogeneities induced by the filter in the Fourier plane. Hence, this kind
of dynamical behavior could be expected. The complex dynamics exhibited by this pattern
is constantly repeated over time. Which leads us to infer that this kind of behavior could be
quasiperiodicity.
AmathematicaltoolforanalyzethespatialmodesinteractionistheFourierspectrum.Figure
3showstheFourierspectraofdifferentdynamicalregimes.Showingthatthedynamicschanges
between stripe patterns, quasi-periodicity and spatiotemporal chaotic textures. The stationary
104 V=3.25
f ’ V=3.80
V=4.35
f f ’’ f ’’’
]
s
A [102
100

0 265 530 795
K [µm−1]
Fig. 3. Fourier spectra for three different dynamical regimes. In gray (blue) the Fourier
spectrumofastationarypattern,inlight-gray(red)aquasi-periodicregime,itcanbeob-
servedtheemergenceofincommensurablefrequencies(f(cid:48), f(cid:48)(cid:48)and f(cid:48)(cid:48)(cid:48)).Inblack,abroad-
enedspectrumthatcorrespondstoachaotictexture,presentinganexponentialdecayofthe
modesmarkedbythedashedline.
patternischaracterizedbyadominantwavelength f.Thewidthofthispeakisduetotempera-
turefluctuationsanddynamicsofdefectssuchasdislocationsandboundarygrains.Thequasi
periodictextureischaracterizedbytheappearanceofincommensurablewavelengths,{f(cid:48),f(cid:48)(cid:48)(cid:48)},
withrespecttothemainwavelength f(cid:48)(cid:48) anditsharmonics.Thespatiotemporalchaotictexture
ischaracterizedby presentinganenlargedspectrum asaresultof theinteractionbetweenthe
main incommensurablemodes [35]. Note thatin this regime, themodes are coupledwith ex-
ponential decay [see the dashed line in Fig. 3]. Therefore, the system does not exhibit power
spectrumbehaviorwhichisthehallmarkofturbulencedynamics[36].
4. Quantifyingthedynamics
Acharacterizationofcomplexdynamicslikechaosandspatiotemporalchaoscanbedoneby
means of Lyapunov exponents. There are as many exponents as the dimension of the system
understudy.TheanalyticalstudyofLyapunovexponentsisaparamountendeavorandinprac-
ticeinaccessible,thenthepragmaticstrategyisanumericalderivationoftheexponents.From
experimentaldata,inthecaseoflow-dimensionaldynamicalsystems,bymeansofrecognition
of initial conditions one can determine the LLE [37]. This exponent accounts for the greatest
exponentialgrowthanditsdefinedby
1 (cid:20) ||u(x,t)−u(cid:48)(x,t)|| (cid:21)
λ = lim lim ln , (1)
0 t→∞∆0→0t ||u(x,to)−u(cid:48)(x,to)||
where u(x,t) and u(cid:48)(x,t) are given fields, ∆ ≡ ||u(x,t )−u(cid:48)(x,t )|| and ||f(x,t)||2 ≡
o 0 0
(cid:82)|f(x,t)|2dx is a norm. ∆(t)≡||u(x,t)−u(cid:48)(x,t)|| stands for the global evolution of the dif-
ferencebetweenthefields.
Whenλ ispositiveornegative,theperturbationofagiventrajectoryischaracterizedbyan
0
exponentialseparationorapproach,respectively.DynamicalbehaviorswithzeroLLEscorre-
spondtoequilibriumwithinvariantdirections,suchasperiodicorquasi-periodicsolutionsand
non-chaoticattractors[38].Hence,theLLEisanexceptionalorderparameterforcharacterizing
transitionsfromstationarytocomplexspatiotemporaldynamics.
Experimentally, to estimate the LLE, it is mandatory to have two close initial conditions
a) line A line B
1200
x [μm] A B
0
0 0 x [μm] 1200 1000 x [μm] 1200 200t[s]
0 A 0 B
1
t[s] t[s]
0
0
80 80
b) 0.6 line A
line B
I [a.u.]
0.2

0 x [μm] 1200
c) d) 55
diffs.
exp. fit
(I) [a.u.]Δ1500 (I) [a.u.]Δ30
7.5
50
V [Vrm5s.]5 3.5 10 t [s] 50 t 1[s0]0 200
Fig. 4. LLE estimation for the LCLV with optical feedback withV0=4.70Vrms. a) Pro-
jectedspatiotemporaldiagramoftheLCLV.Thebottompanelsaccountforspatiotemporal
diagramwithcloseinitialconditions(lines1and2),b)Intensityprofilesoflines1and2,c)
A3-Dgraphthatshowsevolutionofdifferences∆(t,V )fordifferentappliedvoltagesV ,
0 0
itcanbenotedthatnotalwaysthetrajectoriesareexponentiallyseparated,andd)Tempo-
ralevolutionofglobaldifference∆(t),dotsstandforexperimentaldataandthecontinuous
curveistheexponentialfitting∆(t)≈aebt+cedt witha=0.0045,b=8.813,c=16.57
andd=0.2773.
and observe if their evolution diverge at large times [28]. The implemented method needs,
as a first step, to find two close fields [see lines 1 and 2 in Figs. 4(a) and 4(b)] along the
projected spatiotemporal diagrams and compute their difference ∆ . The temporal evolution
0
ofthedifferenceshouldbe givenby∆(t)≈∆0eλ0t forlarget [cfFigs.4(c)and 4(d)].Dueto
the complexity of evolution of the difference between fields—clearly the number of positive
Lyapunovexponentsishuge—wewillconsideratleasttwounstablegrowthdirections,thatis
∆(t)≈aebt+cedt [cf.Fig.4].
A bifurcation diagram was constructed with the obtained LLEs as can be seen in Fig. 5.
The system starts with stationary stripe patterns at V =3.0V and the dynamics remains
0 rms
unchangeduntiltheappliedvoltagereachedV =3.5V .Atthisvoltage,theLLEgoestozero,
0 rms
meaning that the system exhibits a bifurcation. Experimentally, we observed that the steady
patternchangestoanaperiodicregime.Thechaoticbehaviorremainsuntilthemeanintensity
intheLCLVdestroysthechaoticattractorduetodestructiveinterferenceatV =3.9V [see
0 rms
Fig.5],causingacrisis.Oncethelightisrecovered,thesystementersinanintermittentregime
betweenchaosandquasi-periodicity.Afterthiswindowthesystembecomeschaoticuntilthe
attractorisannihilatedbydestructiveinterferenceatV =5.35V .Oncethelightisrecovered
0 rms
Intermittency
0.8 Chaos
Quasi-periodicity
u.]
a.
1/s]0.4 nsity [
λ [0 01.5n inte
a
0 0 me
−0.4
3 3.5 4 4.5 5 5.5 6 6.5 7
Voltage [Vrms]
Fig.5.BifurcationdiagramconstructedwiththeestimatedLLEasafunctionofapplied
voltageV of the LCLV with optical feedback. The diagram is clearly separated in four
0
dynamicalregimes,stationarypattersbeforeV0=3.5Vrms,chaosshadowedingray(green),
intermittencybetweenchaosandquasi-periodicityindarkgray(gray)areaandawindowof
quasi-periodicityshadowedattheendinlightgray.Thestarscorrespondtothecalculated
LLEwhilethecirclesshowthenormalizedmeanintensitypresentintheLCLV.
the system remains in a quasi-periodic regime until the next cycle of destructive interference
arrives. This dynamic regime is characterized by having an oscillatory pattern which LLE is
zero.
Givenatemporalsignal,theattractorofthesystemcanbebuiltbytakingthesignalatdiffer-
entperiodictimes(arbitraryperiodicseparationτ)andconstructingthevector{I(x,t),I(x,t+
τ,x),I(x,t+2τ),···} with x as a fixed position, phase space reconstruction [39]. The three
differentattractorsthatcanbereconstructedusingthisembeddingmethodareafixedpoint,a
torusandastrangeattractor.Forlowvoltageafixedpointcanbeseen(stationarypattern).In-
creasingthetensionV ,thephasespacereconstructionexhibitsatorus(quasi-periodicpattern)
0
andstrangeattractor(spatiotemporaltexture)[28].
5. Conclusions
Our study provides clear evidence that the LCLV with optical feedback is spatiotemporally
chaoticinacertainrangeofparameters.TheLLEsareexperimentallyaccessibleandallowus
tocharacterizethetransitionsfromstationarytocomplexspatiotemporaldynamics.Certainly
new concepts in the theory of dynamical systems must be developed to achieve a better ex-
perimentalcharacterizationofspatiotemporalcomplexbehaviors.Notwithstanding,jointlythe
LLEsandpowerspectrumallowusdistinguishingwell-establisheddynamicalbehaviorssuch
asamplitudeturbulenceandspatiotemporalchaos,whichareoftenmergedandconfused.
Acknowledgments
MGCandMWacknowledgethesupportofFONDECYTN◦1150507andN◦3140387respec-
tively.VOacknowledgesthesupportofthe‘Re´gionNord-Pas-de-Calais’.