1 July 2002

Physics Letters A 299 (2002) 212–216

www.elsevier.com/locate/pla

New laser induced spatio–temporal transition in nematics

E. Brasseleta,b,∗,1, B. Doyona, T.V. Galstiana,1, L.J. Dubéa

a Département de Physique, de Génie Physique et d’Optique, Université Laval, Cité Universitaire, Québec, QC, G1K 7P4 Canadab Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 94235 Cachan, France

Received 8 February 2002; accepted 22 April 2002

Communicated by A.P. Fordy

Abstract

We report the observation of a new light-induced spatio–temporal transition in homeotropically aligned nematic liquidcrystals under the action of circularly polarized light. We believe that the competition of symmetric and asymmetric excitationmodes is at the origin of this abrupt transition. A theoretical model explaining the main features of the observation is proposed. 2002 Elsevier Science B.V. All rights reserved.

PACS: 64.60.Ht; 64.70.Md; 61.30.Gd

The understanding of the role of dynamic fluctua-tions in the origin of phase transitions in nature is ofparamount importance [1]. Light-induced phase tran-sitions, which, in some cases, may bring dramaticchanges in the organization of matter, without using“traditional” (e.g., heat) control parameters, have at-tracted considerable interest over the years [2–10].A rich variety of such “isothermal” phase transitionswere demonstrated in liquid crystals, where the di-electric torque of light results in threshold-like tran-sitions with the collective molecular orientation act-ing as the order parameter [2,8,10]. In the special caseof normally incident circularly polarized light, it wasshown that a well-established sequence of transitionscan be observed in a homeotropically aligned nematic

* Corresponding author.E-mail address: ebrassel@lpqm.ens-cachan.fr (E. Brasselet).

1 E.B. and T.V.G. are members of the Centre d’Optique, Pho-tonique, et Laser, Université Laval, Québec, Canada.

liquid crystal (i.e., molecules oriented normal to thecell substrates) [11–14]. Two consecutive transitionswere experimentally isolated; the first one, the Fréed-ericksz transition, is reproducible with high accuracyand is well understood theoretically [12,13]. In con-trast, the second one is more delicate and still lacksa complete theoretical description [14]. However, wehave recently suggested a mechanism based on dy-namic fluctuations of the molecular orientation that ex-plains qualitatively the origin of the transition [15].2

2 In [15], the qualitative model introduced the ratio of theelectric coherence length,ξe and the characteristic length,�, ofthe nonuniform angular momentum deposition as physical relevantparameters. The numerical values given for�/ξe , in Section 3.5must be corrected. We should emphasize that the pertinent ratio forthe discussion is�eff/ξe where the effective length�eff takes intoaccount the relative “weight” of the zone with positive and negativeangular momentum (AM) deposition. Specifically, we define�eff =�× (AM of a negative zone)/(maximum AM of a zone). Thus, in thecase of polarizationσ = −1 and a ratioR = 0.5 of weak to strong

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E. Brasselet et al. / Physics Letters A 299 (2002) 212–216 213

Fig. 1. Experimental setup.(ex , ey , ez): Cartesian coordinate system; Verdi: Verdi laser (from Coherent) operating at 532 nm;λ/2: half waveplate; GP: Glan prism; BS: beamsplitter; M: mirror;λ/4 : quarter wave plate; L: lens; NLC: homeotropically aligned nematic liquid crystalfilm; WP: Wollaston prism;F–F andf–f : system of lenses;Dx : photodiode detectors. The solid line represents the transverse extension of beamin the absence of reorientation and the dashed line represents the beam in the presence of self-focusing.

In this Letter, we present results of further studiesthat have allowed us to identify a new intermediatephase transition located between the previously knowntwo thresholds and missed in earlier investigations.This new transition is well-defined in our experimentas a bifurcation point, where the modal compositionof the molecular organization is drastically changed.We will now describe the experiment and proposea qualitative theoretical model that accounts for themain physical features of this bifurcation.

We perform our experiment following the set-up sketched in Fig. 1. The sample is a film ofNLC E7 sandwiched between two glass substrates,located atz = 0 andz = L = 100 µm, respectively.The substrates were chemically treated to providealignment perpendicular (homeotropic) to the surfacesof the plates. The pumping laser beam is obtained froma Verdi laser (operating at 532 nm) and is focused atnormal incidence on the sample to a beam diameterof 60 µm. The polarization state of the excitationlight is made circular using a quarter wave (λ/4)plate placed before the lens L. The incident intensityis adjustable using a half wave plate (λ/2), whoseorientation is monitored by a stepping micromotor,located in front of a Glan polarizer (GP). Two pairsof lensesF–F and f –f with large diameter and aWollaston prism (WP) are used to measure the total

field intensity, we obtain�eff/ξe � 1 at the Fréedericksz transitionthreshold (I = 1), �eff/ξe = 0.76 at I = 1.5 corresponding to pointA of Fig. 9 and�eff/ξe < 0.2 at I = 1.5 corresponding to pointB of the same figure. Forσ = +1 and R = 0.5, one obtains�eff/ξe = 1.56 at I = 1. The conclusions of Section 3.5 of [15] arehowever unaffected: these values are in accord with our physicalinterpretation of a first-order transition mechanism.

intensity of the horizontal and vertical componentof the electric field at the output of the sample,namelyI‖(t) = |Ex(z = L, t)|2 andI⊥(t) = |Ey(z =L, t)|2. In the presence of self-focusing, the outputdivergence of the beam increases (see dashed linesin Fig. 1) and from simple geometrical optic raytracing, one can verify that the detectorsD‖ andD⊥ measureI‖(t) and I⊥(t), respectively. Since atz = L, the light-induced divergence of the output beamis not appreciable, one expects that our apparatusis insensitive to self-focusing effect. As expected,I‖(t) + I⊥(t) is a constant at fixed incident intensityI in the experimental range. The data were takenby slowly varying the input intensity. WhenI < IF ,whereIF is the Fréedericksz transition threshold,I‖(t)andI⊥(t) are constant, equal and proportional toI .

At I = IF , an almost uniform sinusoidal oscilla-tion is observed in the temporal variations ofI‖(t) andI⊥(t). This is the signature of the light-induced col-lective molecular precession [12]. When the intensityis progressively increased, the nature of the oscillationis qualitatively the same until, atI = IB , the dynamicsof the system changes abruptly. Evidence of this be-havior is obtained through the Fourier transform of theexperimental signalI‖(t) (Fig. 2). At I < IB , curve Aof this figure shows a main peak accompanied by twosatellites indicative of a small modulation of the ampli-tude of the fundamental oscillation (nutation, [8]). ForI > IB (curve B in Fig. 2), the spectrum ofI‖ has nowtwo dominant frequencies. A quantitative measure ofthe appearance of a second frequency is taken as theratio between the amplitude of the secondary peak (onthe right side of the fundamental frequency) and theamplitude of the fundamental peak. This ratio is plot-ted in Fig. 3 as function of the intensity normalized

214 E. Brasselet et al. / Physics Letters A 299 (2002) 212–216

Fig. 2. Fourier transform of the experimental signalI‖(t). The solid

line refers toI < IB (point A in Fig. 3) and the dashed line refers toI > IB (point B in Fig. 3). The thickness of the cell isL = 100 µm.

Fig. 3. Experimental ratio of the secondary peak divided by thefundamental peak of the spectrum ofI‖ versus the normalized

intensity I . The solid lines are linear fits for the two regionsI < IBandI > IB and are meant to guide the eye. The thickness of the cellis L = 100 µm.

to the first threshold intensity,I = I/IF . This graphemphasizes the sudden character of the transition be-tween two distinct nonlinear dynamical regimes. Forfurther increase of the intensity, the system switchesat I = I∗ > IB to a large reoriented state accompaniedwith a large number of self-diffraction rings (> 20)[14,15]. This transition is what has been referred topreviously as the “second” transition.

To the best of our knowledge, this light-inducedspatio–temporal transition has never been predicted inthe literature neither theoretically nor on the basis of aphysically intuitive argument, although the circularlypolarized Fréedericksz transition was discovered twodecades ago [2]. We propose hereafter a theoreticalmodel that describes such a transition and that sug-gests its physical interpretation.

The interaction of light impinging at normal in-cidence on a homeotropic cell of nematic liquidcrystal (NLC) is considered. In a Cartesian coor-dinate system(ex, ey, ez), where thez-axis is cho-sen as the direction of propagation of light, the di-rector (average orientation of the NLC molecules)can be defined by two angles,α1 and α2, as n =(sinα1,cosα1 sinα2,cosα1 cosα2). We restrict theproblem to the infinite plane wave model, all functionsdepending only on the spatial coordinatez and on thetime t . Additionally, we assume slowly varying spa-tial deformations compared to the wavelengthλ of thelight, |∂αi/∂z| � 1/λ, so that reflected waves can beneglected. The equations of motion for the director aregiven by the balance of the various torques involved,namely the elastic (Γel), electromagnetic (Γem) andviscous torque (Γvisc). The strong anchoring condition(i.e.,αi(z = 0, t)= αi(z = L, t) = 0) allows to expandthe anglesαi as Fourier series,

(1)αi(z, t)=∑

j

αi,j (t)sin(jπz/L),

whereL is the cell thickness. Aminimal model issuggested by the observation that the Fourier spectrumof Fig. 2 changes from symmetric to asymmetric asI

grows beyondIB . It is therefore necessary to retain atleast the first two modes of each angle, such that onemay only consider expressions of the form

α(2)i (z, t) ≡ αi,1(t)sin(πz/L)+ αi,2(t)sin(2πz/L).

(2)

Indeed, at the transition thresholdI = IF , the presenceof the asymmetric mode,αi,2, is already necessarysince there, the symmetry of the system is broken (i.e.,existence of reorientation means thatαi �= 0) implyingthat the polarization of light is becoming increasinglynon-uniform alongz, thereby exciting asymmetricallythe liquid crystal.

Finally, the original balance equation is reduced toa set of four coupled equations for the time derivative

E. Brasselet et al. / Physics Letters A 299 (2002) 212–216 215

Fig. 4. Theoretical Fourier spectrum ofI‖(t) in the neighborhood

of the bifurcation pointIB for L = 100 µm in units of inversecharacteristic timeτ = γL2/(K3π

2) whereγ is the orientationalviscosity andK3 is the elastic constant of bend deformations [16].In our case,τ ∼ 10 s.

amplitudes∂αi,j /∂t with i, j = (1,2) with a simul-taneous solution of the macroscopic Maxwell equa-tions for the electric field. Together, these sets of equa-tions describe the reorientation profile of the direc-tor [6]. Without going into the detailed derivation ofthese equations, which will be presented shortly in afull length publication, one can easily state the essen-tial difference between our model and previous ap-proaches [14,15]. In the latter, the system is restrictedto a limit cycle behaviour corresponding to a uniformprecession of the director around thez-axis and associ-ated with atime-independent three-dimensional mole-cular spatial organization. The proposed model, onthe other hand, allows for non-uniform precession andtime-dependent three-dimensional reorientation.

As a first test of the applicability of our model,we calculateI‖(t) versus normalized intensityI tocompare with the experimental data shown in Fig. 2.We find a single dominant frequency below a criticalvalue IB ∼ 1.75 and the abrupt appearance (within a∆I of order 10−2) of a second significant frequencyjust aboveIB as well as a slight shift of the mainpeak to lower frequency. These findings are displayedin Fig. 4 and it is clear that, even at this level ofapproximation, the model has captured the essentialelements found in the experiment.

Quantitatively, the bifurcation intensity valueIB ∼1.75 calculated from our model (Fig. 4) is not in

Fig. 5. Theoretical values of maxima of the amplitudesαi,1 andαi,2versus the normalized intensityI for L = 100 µm. The solid linerepresents symmetric modes (αi,1) and the dashed line representsthe asymmetric modes (αi,2).

complete agreement with the experimental oneIB ∼1.25 (Fig. 3). Nevertheless, this value remains in arealistic experimental range of light intensity. Caremust be exercised in comparing Figs. 3 and 4 sincethe relative change of the normalized intensities isapproximately 7 times larger for the experimental datathan for the theoretical curve. This affects in particularthe size of the frequency shift of the dominant peak.Fig. 4 has been constructed to display the abruptnessof the transition; we have however verified that alarger frequency shift is obtained with a larger∆I .We believe that part of the remaining discrepanciesmay be removed by retaining higher order modes ofreorientation. This extension will be carried out in asubsequent publication.

By looking at the maxima of the mode amplitudes,αmaxi,j ≡ maxt |αi,j (t)|, as function ofI , one can fur-

ther extract a physical scenario for the production ofthe transition. The results of Fig. 5 allow us to proposethe following. At the Fréedericksz threshold (I = 1),only the symmetric modes are excited, as expectedfrom the quasi in-plane configuration taken by theNLC molecules atI = 1 [15]. Then, as the light in-tensity is further increased above threshold, the mole-cular configuration becomes more and more twisted(three-dimensional), thereby associated with signifi-cant asymmetry [15,17]. This point is well illustratedin the figure where the contribution of asymmetric

216 E. Brasselet et al. / Physics Letters A 299 (2002) 212–216

modes to the total reorientation increases almost lin-early with the pump intensity up to a critical valueIB .At I = IB , αmax

i,j have a singular point that correspondsto the bifurcation observed experimentally. The advan-tage of this “two-mode” approximation is therefore toreveal the underlying physical process at the origin ofthe observed bifurcation, namely, the competition be-tween the symmetric and asymmetric modes of reori-entation. Such a competition was already pointed outin an other light-NLC interaction, where an ordinarywave at oblique incidence impinges on a homeotropi-cally aligned NLC film [18].

Furthermore, the minimal model also predicts thetransition to a large reorientation atI = I∗ althoughthe calculated value(I∗ ∼ 2.12) is somewhat largerthan the experimental one. Our previous commentswith regards to the discrepancy inIB apply herealso. For this transition as well, the model confirmsthe relevance of a mechanism based on dynamicalfluctuations of the molecular organization and therole of its three-dimensional spatial arrangement, aspointed out in [15].

In conclusion we have observed experimentally andinterpreted theoretically the existence of a new kindof bifurcation above the Fréedericksz transition. Thisbifurcation is shown to be related to the asymmetryof the light-induced reorientation. Our model exhibitsa rich dynamics and may be further exploited forthe analysis of a variety of light-induced nonlinearbehaviors, for instance, the detailed nature of thedirector dynamics at the bifurcation or the transitionto large reorientation regimes for different interactiongeometries.

Acknowledgements

E.B. wishes to thank G. Melkonyan for stimulatingdiscussions during this work.

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