An important effect of dynamical localization of light waves in liquid crystal electro-hydrodynamic instabilities is reported by investigating coherent backscattering effects. Recurrent multiple scattering in dynamic and chaotic complex fluids lead to a cone of enhanced backscattered light. The cone width and the related mean free path dependence on the dynamic scattering regimes emphasize the diverse light localization scales related to the internal structures present in the sample. The systems investigated up to now were mainly nano-powdered solutions or biological tissues, without any external control on the disorder. Here, an anisotropic complex fluid is “driven” throughout chaotic regimes by an external electric field, giving rise to dynamics that evolve through several spatio-temporal patterns.
©2009 Optical Society of America
Coherent backscattering (CBS), as precursor of the Anderson strong localization , has amplified the interest for photon weak localization phenomena in random media . CBS is a photon self-interference effect leading to an enhancement in the intensity cone profile, of width Δθ~λ/ℓ (where ℓ is the scattering mean free path inside the medium and λ is the incident wavelenght) for the backscattering direction. The analysis of the CBS cone gives information on the properties of random media . Among other materials, nematic liquid crystals (NLCs), with their anisotropic properties, represent an excellent candidate to investigate weak localization of light [4, 5]. NLCs are uniaxial fluids formed by rod-like molecules aligned on an average direction (described by a unit vector n(r) known as molecular director). Under the effect of a low frequency sinusoidal electric field, a planar aligned sample of NLC (with the director parallel to the glass substrates) having a negative dielectric anisotropy (Δε<0) can be driven through several regimes of increasing stochasticity by tuning the amplitude of the external field, and establishing a sequence of electro-hydrodynamic (EHD) instabilities. While increasing the electric field, the first encountered regime consists in a series of stationary convective rolls (Williams domains), whose periodicity is of the order of the sample thickness . This instability arises as result of two competing forces: a restoring dielectric torque owing to the negative dielectric anisotropy, and a force exerted on the bulk fluid due to the charge separation produced by the positive anisotropy of the conductivity (Δσ>0). These two forces lead to the formation of a recurrent pattern of convective roll structures, associated to the periodic distortion of the director field, n(r). When increasing the amplitude of the external field, bifurcations to more complicated spatio-temporal patterns are to be found (distortion of Williams domains and weak turbulence). At higher electric field a transient bimodality [7, 8], from a first turbulent regime (called DSM1) to the fully developed turbulent regime (DSM2), is finally observed [9, 10].
Anderson localization is expected to occur at very strong scattering, that is for ℓ≪λ. For light it is very difficult to find systems which satisfy this stringent condition. In multiple scattering systems it is possible to get close to the condition ℓ~λ, where light localization effects due to recurrent scattering have been already observed . In these systems, for each photon path γ there is a non zero probability that a -γ path will also exist. The interference between the two propagating waves results proportional to Φ~exp[i(k i+k f)·(x 0-x N)], where k i and k f represent the incident and the outgoing wave vectors respectively, while x 0 and x N represent the coordinates of the first and last scatterer . In the CBS condition (q=k i+k f=0), the amplitude of the two waves will be the same (Aγ=A-γ) and will interfere constructively, thus leading to an enhancement of the backscattering cone intensity by a factor |Aγ+A-γ|2=4|Aγ|. The mean free path ℓ is related to the scattering cross-section S of the scatterers by the following relation ℓ~1/mS (m is the concentration of the scatterers).
In a NLC sample thermal fluctuations of the molecular director n(r)=n 0+δ n(r) leads to fluctuations of the dielectric tensor εαβ=ε ⊥ δαβ+(ε ‖-ε ⊥)nαnβ, this effect being the main responsible of the recurrent multiple scattering and the localization of light observed in such systems. In the case of thermal fluctuations  (and in absence of external stimuli), the ratio of the scattering cross–sections due to the variation of refraction index caused by thermal fluctuation in NLC sample SNLC, and that owing to arbitrary isotropic scatterers Siso is of the order of SNLC/Siso≃106 . This indicates that, even in the absence of external stimuli, NLCs provide an important scattering environment, which already proved a striking optical feedback (localization) and is responsible for the random laser action observed in the case of several confinement geometries . Thus, liquid crystalline materials, being interesting reconfigurable media able to reveal dynamical localization of light waves are used as model systems for investigating multiple scattering induced by chaotic dynamics.
In this paper we present the first experimental observations regarding the photon dynamical localization in several NLCs turbulent regimes. The investigated multiple scattering samples, presents in literature, were mainly characterized by the absence of external control on the disorder. Here, an anisotropic complex fluid is “driven” throughout chaotic regimes by an external electric field giving rise to several dynamical scattering regimes related to dynamical light localization effects. Even though the system is characterized by a spatial stochastic behaviour, constructive interference survives exactly in the backwards direction, forming a well defined CBS cone. The central motivation of this study was born from the importance of exploring the process of light waves dynamical localization in turbulent systems. This phenomenon could reveal significant conceived and still un-conceived features both scientifically and technologically in the field of fluid dynamics and applied optics and photonics. Indeed, the liquid crystals offer unparalleled opportunities to investigate chaotic regimes, since under the action of low frequency electrical stimuli provide a controllable cascade of EHD instabilities until the occurrence of a fully developed turbulence. Previous light depolarization studies [14, 15] emphasize that fully polarized light waves travelling through NLCs in chaotic regimes can overcome a complete depolarization. This depolarization effect is referable to a remarkable process of multiple scattering which induces substantial randomization of the phases in the wave field.
The experimental setup used for the investigations of CBS is represented by an Off-Centered Rotation (OCR) system . The light source was a He-Ne laser (λ=632.8nm) impinging onto the NLC sample by means of a preserving polarization beam splitter. The sample consists of a conventional nematic liquid crystal [N-(4-Methoxybenzyliden)-4-butylanilin] (MBBA) cell placed behind a quarter wave plate and a bi-convex lens (L1) used to focus circularly polarized light within the sample and also collect the backscattered light waves. We selected circular light polarization for maximizing the efficiency of the scattering events in the sample when the electric field is switched on and the molecular alignment is broken. An optical fibre is placed in the focus of the positive lens (L2) and connected to a photo multiplier tube (PMT) for collecting the signal. A linear polarizer is employed to select one polarization state for the scattered light while avoiding multiple reflections due to the optical elements. The system is placed on a rotating frame; the center of rotation O’ represents the center of rotation O of the sample surface mirrored with respect to the plane of the beam splitter 1(a). Figure 1(b) shows the setup after rotation. The incoming beam is directed to O, while after rotation it still arrives to the rotation center of the sample surface O’. With respect to the frame, the incoming direction has changed but the direction of detection is still the same. By rotating the sample around O, the sample surface is kept at a constant angle with respect to the incoming beam 1(b). The experiments were performed by driving the LC samples through the cascade of EHD instability regimes that were previously observed by means of an optical microscope to clearly individuate the electric field parameters (frequency and amplitude) and the relative dynamic regimes. The analyzed planarly aligned cell (about 52µm thickness), is placed between two semitransparent electrodes. A sinusoidal electric field E=(0,0,Ez), at frequency f=70Hz, is applied on the sample. The sample is illuminated by a white light beam linearly polarized along the anchoring direction, and it is observed by an optical microscope connected to an image acquisition system. The instantaneous transmitted light intensity reflects the spatial variation of the refractive index of the sample. In Fig.2 (a) the convective roll structures characteristic of the Williams domains are shown, while Fig.2 (b) reports the pattern which originates when the rolls become unstable by increasing the of the electric field amplitude (weak turbulent state). Finally, Fig.2 (c) shows the DSM1→DSM2 transition, with the DSM2 regime that starts to nucleate and develop inside the DSM1. In DSMs cases the structures that originates in the sample can be observed only if the focal plane of the microscope coincides with the sample . Furthermore, for well observing the small scales originates in the turbulent regimes we used an high numerical aperture microscope objective so that the focal depth of the objective was 1/100 the sample thickness. Actually the DSM1 regime is a metastable turbulent state which decays into a DSM2 regime through a transient bimodality [7, 8]. At a fixed external voltage V 0 the DSM1 regime starts and after a certain lag-time which depends on the applied voltage, the DSM2 nucleates in some regions o f the sample and slowly this new state invades all the sample. The measured voltage range for the EHD instabilities at a frequency of 70 Hz were: Williams domains (6V→9V, Fig.2(a)), weak turbulence (10V→44V, Fig.2(b)) and DSMs (45V→70V, Fig.2(c)). The same samples were then analysed by using the CBS setup previously calibrated and tested on nano-dispersed powdered solutions (TiO 2) and other NLCs cells in static conditions. For avoiding any artifacts generated by the optical elements interface reflections suitable states of polarization were carefully selected. Additionally, the reflections from the sample glass plates were ruled out by rotating the sample of a small angle with respect to the incident light beam, in order to mediate on each photon path inside the sample . Upon switching on the low frequency electric field, the multiply scattered light waves surprisingly produced an intense backscattered cone which changed width and amplitude as function of the applied voltage. At low voltage values Williams instability appears. The scattering regime is still polarization sensitive because of the creation of anisotropic scattering areas which are extended over the entire cell thickness. The backscattering cone, with a full width at half maximum (FWHM) of about 6mrad, measured during the Williams domains, shows an important increment with respect the static case (FWHM about 2mrad) . The coherent cone width increment is strictly related to the scattering mean free path, ℓ. Only in the hyper-diffusive regime (DSM2) ℓ results comparable with the transport mean free path ℓ*. Figure 3 shows the backscattering cone in the case of the DSM2 regime, where the FWHM results maximum. The excursion through the cascade of EHD instabilities, as evidenced by Fig.4 (circles), gives rise to a monotonic increase of the coherent cone width. The light waves localized within the distorted loops of the weakly turbulent complex fluid (in the range 10V → 44V) show a scattering mean free path of about 15µm. The three-fold lowering of the mean free path, with respect to the static case, emphasizes the enhancement of the multiple scattering in this dynamic regime, not yet chaotic. In fact, the scattered light waves still preserve some polarization residual, even though the intensity is importantly reduced with respect theWilliams regime. The polarization resudual bring the information that the system is still in the structured and not fully developed 2D turbulence . At higher voltages, the spatio-temporal dynamics evolves trough the Hopf bifurcation and the first dynamic scattering mode (DSM1) appears as a rapidly moving grained texture, having a typical size comparable with visible wavelength. Finally, DSM2 nuclei appear and grow until they occupy the entire bulk (Fig.2(c)).
CBS experiments performed within this fully chaotic regime evidenced striking constructive interference of partial waves traversing momentum-reversed scattering paths in the backscattering direction, accompanied by an almost complete reduction in the amount of light transported throughout the turbulent media. The cone width enlargement during the transition DSM1-DSM2, for a fixed voltage above threshold, emphasizes that a dynamical critical behavior regulates the recurrent multiple scattering process inducing a decrement of the scattering mean free path (about 8µm). In fact, the DSMs bifurcation is regulated by a dramatic director field distortion, manifested as a very strong non-linear flow, since the stresses have surpassed the threshold value of the viscoelastic limit. In conclusion, a remarkable effect of light waves CBS in turbulent anisotropic complex fluids has been reported. The cascade of EHD instabilities in NLCs have been utilized as reconfigurable systems to investigate weak localization phenomena in dynamical regimes characterized by critical behavior. The net reduction of light transport in the forward direction and the robust interference phenomena that survive multiple scattering as evidenced by CBS measurements provide the signature of weak light waves localization effect. This interesting feature opens up fascinating horizons concerning the opportunity to study random laser action in chaotic systems in presence of high efficiency gain media.
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