1. Introduction

Most of the experiments performed in the field of nonlinear optics of liquid crystals (LC) concern films in which the light-matter interaction occurs in a small volume. On this field and on the so called Optical Freedericksz Effect, there are many reviews [1–5] and some chapters of general textbooks on liquid crystals [6–8] that the reader can refer to. Recently, a tubular geometry has been considered, that allows the interaction to take place in a larger volume and some phenomena either usual in nonlinear optics, such as focusing and filamentation, or less usual, as undulation, have been observed [9] and then explained [9–13]. Although these studies have been undertaken from an academic point of view, it is worth noticing that such a cylindrical geometry fits any fiber optics based device. This is why we have considered an identical geometry, at a commercial fiber scale. We report on the possibility to induce some stable waveguiding structures in dye doped nematic liquid crystals. Under large pump power, one expects a large optically-induced reorientation up to saturation: the pump beam experiences an index profile, whose gradient is large enough to keep the energy trapped. In such a regime, the nematic is no longer strongly optically non-linear and the system becomes almost insensitive to power changes.

This paper is organized in three parts: in the first, we explain how we performed the experiments, the second one is devoted to the experimental results, which are then discussed in the third part.

2. Setup

The set up is shown in Fig. 1. The inner face of a capillary is treated to get the alignment we want, and then filled with the material we choose and illuminated with a source. A fiber which was tapered under specific conditions and then cleaved provides a beam whose width is smaller than the taper diameter (~10μm). Depending on the taper geometry, on the different indices of refraction of the fiber, and of the used liquid crystal, the outcoming beam can be very well collimated [14]. We have also used a simply cleaved fiber: in that case, the beam size is smaller (4 μm) but more divergent. Such a fiber, tapered or not, is fed with the green line of an Ar⁺ CW laser and is referred to as the source in the following. In addition, the polarization of this beam can be adjusted either with a normal polarizer or with a three loops system. The available power at the output of the taper is in the range of 0 to 100 mW, and more in the case of the non tapered fiber, which is more than enough to induce nonlinear effects and even to get the nematic to melt into the isotropic phase. By means of different micropositioning stages, this source can be oriented to emit in a direction parallel to the capillary axis (oz, Fig. 1), translated accurately in the section of the capillary (ox, oy, Fig. 1), and along with the capillary axis, allowing the source to enter the liquid crystal more or less deeply (oz, Fig.1). This latter adjustment is important in the sense that it allows to ensure a monomode behavior for the emerging beam out of the taper (it also depends on the geometry of the taper). All the reported experiments have been performed with 250 μm internal diameter capillaries and a beam width of about 10 μm (tapered fiber) or 4μm (non tapered fiber). This setup is installed on a polarizing microscope in such a way that the capillary axis lies parallel to the microscope stage. A second microscope is setup in the perpendicular direction (ox, Fig.1b) to spot the source beam properly and to observe the system from another point of view. The beams are traced out by analyzing the light scattered by the liquid crystal in the oy direction (Fig.1b). This is what we call “leakography”[15]. Let us finally add some dye specificity. First the absorption induces a z dependence of the different parameters of the system, that is essentially the local available power decrease along with the capillary axis. Second, the used argon green line is partly scattered by the liquid crystal and partly absorbed by the dye molecules. The excited molecules release some energy through fluorescence, in our case, in the red region. So, by properly filtering, one can observe either the scattered green light, the red fluorescence of the dye or both.

 

Fig. 1. Setup; 1.a: side view ; 1.b: section view.

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The LC that has been used is the well known 5CB from Merck. The ordinary index of the 5CB is 1.53, which means, in terms of numerical aperture, a slightly divergent beam travelling in the liquid crystal. These LC’s have been doped with small amounts of an anthraquinone dye (AQ1) in different concentrations, up to 0.7% w/w. The real part of the doped nematics refractive indices were measured using TIR [16] and are not different from those of the pure hosts. It is now well known that it is possible to observe the Optical Fréedericksz Transition with a mW laser [17] in these doped materials. The way to align the doped LC’s is described in the next section devoted to the experimental results.

3. Experimental results.

The inner face of the capillary is treated using the usual technique of the polyvinyl alcohol to get a planar alignment of the director on the capillary wall, parallel to its axis. The flow induced by blowing away the solvent plays the role of the conventional rubbing. With such treatment, the material is therefore seen by the beam as homeotropically aligned wherever the beam is located in the capillary. The quality of the alignment is checked by visual inspection through the polarizing microscope.

From a Fréedericksz transition point of view, this configuration is a threshold one and it is very similar to an homeotropic thin film illuminated under normal incidence, apart from the totally different boundary conditions, that make this geometry attractive. Actually, as opposed to the usual film geometry, the presence of the capillary wall stabilizes the radial structures and plays a major role in the nonlinearity of the nematic. Again, it should be taken into account in our geometry the presence of dye which makes the problem z dependent.

Although this geometry is the same as the one studied by the Libchaber’s group [9–13], some differences should be noticed. Both the radial size of the capillary and the beam size are smaller, though the ratio is quite the same. However we found different results. Thermal effects have to be considered for two reasons: the capillary is not cooled down and the absorption of the dye is responsible for a local heating. In addition, the range of energy explored is probably different in terms of ratio input energy/Fréedericksz threshold energy. This latter threshold has not been measured for our specific mixture but can be estimated from previous studies. It has been shown that the threshold for the mixture (I_{fr, mixture}) can be correlated with that one of the pure host (I_{fr, pure}) this following way: I_{fr, mixture} = I_{fr, pure}/(1+ζ), where ζ is proportional to the dye absorption [18]

The dye doped mixture we used exhibits a positive optical non linearity and can be assimilated to a Kerr material only in a small range of optical intensities. It becomes nonlinear above a threshold power which we call the Fréedericksz threshold power P_Fr. As the input power equals the critical power or self trapping power P_st, the focusing property of the nonlinear material just overcomes the natural diffraction of the beam. For input powers larger than this P_st, the beam is actually self-focused. For still higher powers, the beam exhibits a multifocus configuration [9] and can be confined in some tubes: it is the so called filamentation; also some undulations of these filaments have been reported [9]. What we have observed is quite close to this picture, however there are some differences, depending on the dye concentration.

For the pure nematic, and a normal source (i.e. non tapered), we have been able to reproduce all the reported features [9], although in a differently sized confinement. It is worth noticing that, in this case, the beam looks spatially unstable.

For a low dye concentration (0.2 % w/w), while increasing the input power, we have observed the already reported sequence of events: focusing, shortening of the focal distance, appearance of a multifocus regime (in this specific case up to three foci have been counted over a distance of around 750 μm). For higher powers, a new event occurs: the appearance of a “self waveguiding” structure. The beam is propagating straightforwardly in a very narrow tube, over a distance that depends directly on the input power. As opposed to the spatially unstable propagation in the pure material, this tube is extremely stable and there are practically no light escaping out from this tube: the nematic scattering is lower than usual and the tube looks more reddish, due to the fluorescence of the dye. This feature is more pronounced for higher dye concentrations.

For a quite high dye concentration (0.7%, w/w), while increasing the input power from 0 to 6mW, the observed sequence is a little bit different: focusing, shortening of the focal distance, the appearance of a “self waveguiding” structure and finally the appearance of the isotropic phase. While decreasing the input power, the sequence is reversed with an extra observation: some undulation of the beam occurs in the nematic region, as the isotropic phase is present. The multifocus pattern has not been visible for such a high concentration and in the “self waveguiding” structure, the tube is almost totally red, with some dark strips inside it, along with the tube. The Fig. 2 video shows the onset of focusing and the self waveguiding structure (the input power increasing is stopped just before reaching the isotropic phase).

We now focus on this specific event, which has not been reported in the previous studies. Due to a practically inevitable overexposure of the camera, what is actually viewed is sketched on Fig. 3 and the corresponding photograph extracted from the movie on the Fig. 4 (up). It consists in some green “brushes” (a, b Fig. 3) and a narrow, linear, well marked, mostly red and stripped tube (c, OO’, Fig. 3). The length of this tube is monotonically dependent on the pump power: as the power is increased or decreased, the length of the tube (l, OO’, Fig. 3) increases or decreases. This tube appears almost totally red: the fluorescence is prominent whereas the scattering is very weak. This red light is strongly linearly polarized. Some darker strips can be seen in this tube revealing an internal structure. Viewed from the second microscope, the tube looks the same as viewed from the upper microscope, apart a central narrow dark line. In all our experiments, this tube looks perfectly straight, we did not observe any undulations. This structure is stable for tens of minutes (for the higher dye concentration), then, due to the integral heating (not the local one), the index gradient is flatter and the structure destabilizes; i.e. the beam appears to be divergent as expected for the propagation in a medium with small index gradient. The waveguiding effect becomes observable again after the whole sample has been cooled down.

 

Fig. 2 Video recording showing the behavior of the source beam as the input power is increased then decreased. [Media 1]

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Fig. 3. Schematic representation of the observed “self waveguiding” structure, as shown on the upper photograph on the right.

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Fig. 4. Photographs extracted from the movie: Up (= Fig. 2): the “self waveguiding structure”; Down: the isotropic zone has been underlined together with the fiber; (part of the movie not presented here).

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At the extrema of the tube two different sets of “brushes” are visible: first, a bundle emerging out from the end of the tube, linearly polarized along with the capillary axis (b, Fig. 3) and second, two beams emerging one from the starting point of the tube and the second from the end (a, from O and O’, Fig. 3), both with the same angle of emergence with respect to the capillary axis. These latter beams (a) are almost completely depolarized. While observing the capillary using the second microscope (viewing the oyz plane, Fig. 3), it turns out that the “a” brush system is developed on a cone with an irregular distribution of intensity, whereas the “b” system keeps propagating almost in the plane oxz.

As the input power is increased, up to around 5mW, the isotropic phase appears in such a way that it can not be missed (Fig. 4, down). Within the isotropic bulb, the beam is slightly divergent (normal numerical aperture of the taper) and then entering, the nematic phase, it is focused.

4. Discussion

This discussion does not focus on the already known events such as focusing, multifocal pattern and undulations but on the “self waveguiding” structure. However, it should be worthy to correlate the distance of the focusing point from the output of the taper (f, Fig. 3) with the pump power, in order to compare the herewith considered doped mixtures with the pure host behavior. In the latter case, linear or exponential dependences have been found, for thin (100 μm) or thick (1000 μm) films respectively [9]. The fourth event, namely the structure of the beam as an isotropic bulb being present right at the end of the fiber (Fig. 4, down), can be readily explained through simple geometrical optics. However, it contains interesting informations on the director distribution. The beam emerging out of the fiber enters into an isotropic material whose index is larger than the taper one. As a result, it diverges slightly. As the beam enters the nematic phase, it is focused due to the curved interface; however the angle of refraction through this interface depends on the director alignment. The director distribution can be estimated from measurements on such a photograph (Fig. 4, down), but such a study is beyond the scope of this paper.

More intriguing is what we called up to now the “self waveguiding” structure. First, looking at the fourth event, one can be sure that the tube is not a confined region where the material went to the isotropic phase. However, there is obviously some heating due to the dye absorption: the central part of the tube is hotter than the external part. This affects the extraordinary index for a tilted director through the principal values n_e(T) and n_o(T). The thermal gradient changes the index gradient induced by the optical reorientation. This modification is not qualitatively essential in the propagation processes. However, as it will be shown below, it can explain the observed strips in the tube. These remarks are valid as long as the heating remains local: as the temperature of the whole sample is increased, the birefringence decreases in the whole sample and the index gradient as well: the material looses its focusing property. This is confirmed by our observations reported above: after tens of minutes, the effects is no longer visible. After a cooling down to the room temperature, the guiding structure can be retrieved.

To meet the main experimental features of this tubular regime, namely a well organized, spatially stable and practically no scattering tube, which occurs above a specific threshold, we propose a simple picture based more on waveguiding than on nonlinear processes. This tube is originated right after the focus (O, Fig. 3) as the input power is larger than a specific power, which we call hereafter P_sw, (sw for self waveguiding; P_sw > P_st > P_Fr). In the region between the taper and this point O, there are large orientational gradients in both radial and axial directions. The resulting effect is like a strongly focusing lens, thus strongly aberrating and therefrom the brushes (a, Fig. 3). Moreover, the light escaping from the central beam passes through regions with a very complex optical axis pattern, leading to a phase shift of the polarization components varying from point to point. This could explain the observed depolarization of the light in these brushes. At the point O, the energy is highly concentrated: one can therefore expect that the director is totally or almost totally reoriented in this region (Fig. 5, up). Let us assume, for the sake of simplicity, that in this region, the director is uniformly reoriented perpendicular to the capillary axis. This creates an optical channel out of which the light cannot escape, due to the total internal reflection. So, if the light intensity is high enough to keep the nematic largely reoriented, this large reorientation keeps the light confined. Now, considering the dye absorption, even if confined in such a self waveguiding structure, the intensity decreases as light propagates along with the capillary axis. The optical torque therefore decreases and the director reorientation becomes smaller: first the radial size over which the director is totally reoriented becomes smaller (z and z’, Fig. 5b), and second, the maximum angle of reorientation is lower than 90° (z”, Fig 3b). The gradients become too small and the light is no longer trapped. This is the end of the tube where the power is equal to P_sw. Beyond this point O’, there exists again a perturbed zone with large gradients, very similar to the region around the point O, giving rise to another set of brushes (a, Fig. 3) with the same characteristics as the previous one. At this point O’, also the main beam is divided in a further set of brushes (b, Fig. 3). The latter set of brushes is much less divergent than the previous and maintains the beam polarization; thus it corresponds to the output of the modes propagating in the tube. In a first naive picture, we have considered the tube as a self built up waveguide with a constant core index surrounded with a graded index cladding as shown in Fig. 5. The tube behaves as a graded index waveguide and only few modes are propagating in it.

 

Fig.5. Schematic representation of the ≪ tube ≫. This is a picture valid only in the plane of the input beam polarization, namely oxz (Fig. 1). Up: radial director distribution. Down: associated extraordinary index profiles for different sections.

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Actually, we have calculated the index profile using a simple elastic model yielding to the director distribution whose the central part is totally reoriented (90°) whereas the director is strongly anchored to the capillary wall. In this naive model, we have inserted a linear temperature decrease from the central part down to the capillary wall, using measured values for the principal indices versus temperature [6]. The obtained index profiles are plotted in the Fig. 6. From such a simple calculation, it comes out that it is possible to have a flat profile and even a camel hump shaped profile as a temperature gradient is inserted. In the latter case, it is known that odd guided modes, yielding an output intensity profile with a central minimum, are favored with respect to the even ones, which inversely yield a central maximum. This can explain the observed stripped internal structure and the set of brushes (b). This hump profile can be understood if one considers that it is possible to have the following relation for the extraordinary index: n(θ - δθ, T - δT) > n(θ,T), valid for the 5CB nematic and provided that θ, δθ and δT are properly chosen. Obviously this model is extremely simple and concerns the index profile in the plane of the polarization of the input light. A more accurate model should take into account the non cylindrical symmetry of the system: the self-built-up waveguide has not the same index profile in the ox and oy directions: it is somehow a polarization maintaining waveguide. Furthermore, a model should account for the dye dichroism: the absorption is different as the dye molecules are not yet reoriented nor excited or, on the contrary, at the end of the process as the dye molecules are both excited and aligned along the optical field.

In accordance with the proposed mechanism, the term filamentation seems to us not appropriate for this tube. Actually, though this model is naive, it stresses on the limit of the nonlinearity of the nematic. As the field is large enough, the director in the central part is almost perpendicular to the capillary axis and cannot be changed anymore: the structure is less sensitive to the optical intensity. Schematically, the system viewed in the oxz plane (Fig. 5), can be reduced to two adjacent hybrid cells and the optical torque (on the axis x = 0, Fig. 5) plays the role of a surface torque. The structure of the film is ruled by the elastic theory together with specific boundary conditions, the anchoring energy on the central part being linked to the light intensity: a modelisation of such an assumption is underway. To illustrate this in a simple way, let us consider the saturation regime sketched in the first plot of Fig. 5 down. In the oxz plane, we have a homogeneous high index region between the two hybrid cells. This region is no longer a Kerr like medium, being now the core of a linear waveguide, for which an increase of the light intensity can only increase the propagation length, as shown in the movie (Fig. 2). Finally, it should be stressed that this behavior is due to the specific boundary conditions and thus to the chosen geometry.

 

Fig. 6 Calculated index profiles, including T(r), with T(r) = 33°C (no gradient) and a linear gradient with T(0) = 33°C; T(120) = 25°C; a maximum occurs around 12 μm

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5. Conclusions

In this paper, we report on some experimental results that we have obtained on nonlinear optics of dye doped liquid crystals confined in capillaries in a planar axial configuration. A narrow and collimated beam induces a reorientation of the positive Kerr like dye doped nematic LC. Apart from some features already observed in a similar geometry although differently sized (focusing, multifoci pattern, undulations), we observed for high pump powers the appearance of a self-waveguiding structure. We propose, on the basis of the main experimental features, a possible picture for this behavior. The high optical torque totally reorients the director in a small region whose index becomes the LC extraordinary index and is therefore larger than that in the remaining part of the capillary, creating a waveguide. The light is thus trapped with an intensity high enough to maintain this large reorientation and in turn this large reorientation keeps the light trapped. We stress the difference between the usual self-trapping threshold, above which undulations and eventually a multifoci pattern are observed, and the self-waveguiding threshold, above which a linear waveguide is created in the middle of the nonlinear medium. This model should be now confirmed through more detailed and numerical analysis. The most pertinent parameters of these guiding behavior are obviously the index gradients ruled mainly by the birefringence of the material. Such an analysis is underway, addressing a specific question: do our observation can be a spatial soliton? Finally, given the used geometry that fits the fiber size, such a self waveguide structure can be of interest in liquid crystal and fibers based devices such as connectors or routers.