THE ELECTRIC field of an optical beam can change the

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY Time Dependence of Soliton Formation in Planar Cells of Nematic Liquid Crystals Jeroen Beeckman, Kristiaan Neyts, Xavier Hutsebaut, Cyril Cambournac, and Marc Haelterman Abstract Spatial optical solitons can be observed in bulk nematic-liquid-crystal cells with an entrance window and a bias voltage applied over the cell to enhance the optical nonlinear effect of laser-induced molecular reorientation. The soliton-induced waveguide is observed in transmission by use of a second light source and crossed polarizers. This setup allows us to investigate the time dependence of the molecular reorientation that sustains the soliton-like beam propagation. Results from a numerical simulation are in good agreement with the experimental results, which confirms the validity of our model. Index Terms Nematic liquid crystals, optical nonlinearity, spatial optical solitons, time dependence. I. INTRODUCTION THE ELECTRIC field of an optical beam can change the orientation of the molecules in a nematic liquid crystal. In return, the material properties seen by the optical beam will change. In particular, the refractive index becomes intensity dependent. This is an optical nonlinear effect denoted as optical field-induced director reorientation [1], [2],[3]. Although an optical beam focused in a liquid-crystal layer will diffract, with increasing optical power self-focusing (or defocusing) of light occurs due to this nonlinear effect, depending on the geometry. When self-focusing balances diffraction, the beam may propagate without change of its intensity profile, thus witnessing the formation of a spatial optical solitary wavepacket [4] [6]. Recently, such a soliton-like optical beam propagation was reported in nematic liquid crystals, with a few milliwatts of light power and propagation lengths up to one centimeter [7], [8], and it is believed that the laser-induced molecular reorientation is the responsible effect. The required optical power can be as low as a few milliwatts owing to the enhancement of the nonlinear effect by applying a voltage over the cell [7], [9]. However, a major issue for possible applications concerns the dynamics of the soliton formation. For applications in computing, optical interconnects can be useful to reduce the network latency due to a higher bandwidth of optical connections [10]. Manuscript received December 16, 2004; revised February 9, The work of J. Beeckman was supported in part by the Fonds voor Wetenschappelijk Onderzoek Vlaanderen (FWO Vlaanderen). The work of X. Hutsebaut was supported in part by the Fonds Pour la Formation à la Recherche dans l Industrie et l Agriculture (FRIA). J. Beeckman and K. Neyts are with the Liquid Crystals and Photonics Group, Department of Electronics and Information Systems, Ghent University, B-9000 Ghent, Belgium ( jeroen.beeckman@elis.ugent.be; Kristiaan.Neyts@elis.ugent.be). X. Hutsebaut, C. Cambournac, and M. Haelterman are with the Applied Sciences Faculty, Free University of Brussels, B-1050 Brussels, Belgium ( xhutseba@ulb.ac.be; cyril.cambournac@ulb.ac.be; marc.haelterman@ ulb.ac.be). Digital Object Identifier /JQE Indeed, adding reconfigurability to the optical interconnects can open new possibilities, e.g., by enhancing the speed of multiprocessor systems, but an adequate switching speed (of the order of a microsecond) is required to outdo the electrical interconnects [11]. Also, for building an optical network the speed and the ease of switching light between fibers is of importance. Displays made with nematic liquid crystals typically have a refresh rate of a few tens of milliseconds [12] and we can expect that the soliton formation time will be of the same order of magnitude. Up to now, research on spatial optical solitons in nematic liquid crystals was focused on the steady state, that is, experiments were analyzed after a time long enough to avoid transient effects [7] [9], [13] [15]. In this paper, we report on a numerical and experimental study of the dynamics of the soliton formation in biased nematic-liquid-crystal planar cells. To our best knowledge it is the first time that simulations and experiments specifically focus on the time dependence of soliton formation. The paper is organized as follows. Sections II and III, respectively, describe the experimental setup and observed results. In Section IV we describe the theoretical model on which the numerical simulation is based. Section V is devoted to the comparison between experiment and theory. Finally, our conclusions are drawn in Section VI. II. EXPERIMENTAL SETUP A He Ne laser beam (633 nm) is injected into a liquid-crystal cell by use of a 40 microscope objective (cf. Fig. 1). The beam waist of the optical field in the focal plane is estimated from the observed angular divergence of the Gaussian beam [16], which yields a value of about 3 m. The light is linearly polarized along the direction. The liquid-crystal cell is made up as described in [9]. It consists of two glass plates covered with a thin indium tin oxide (ITO) electrode and an alignment layer. Plates are glued together with Mylar spacer balls in between. The obtained 75- m-thick gap between the plates is filled with the liquid crystal E7 from Merck [17], and an entrance window is glued perpendicularly to the glass plates. The orientation of the molecules is controlled by the rubbing of the alignment layers. Hence, in absence of any electric field, the molecules are oriented along the direction, but they have a small pretilt of about 2. The input window is also treated with an alignment layer to ensure the homogeneity of the medium at the entrance and to maintain the polarization state of the incoming light. The second part of the setup consists of the visualization apparatus. By collecting the light laterally scattered by the liquid-crystal molecules, the beam propagation in the cell can be observed with the combination of a lens and a CCD camera. For low optical powers, the beam diffracts and the light is spreading out /$ IEEE

2 736 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005 Fig. 1. Experimental setup and indication of axes. (Drawing is not to scale.) Fig. 2. (a) (b) Steady-state light propagation in the cell for a 1-V voltage. (a) Diffraction regime, i.e., for low optical power. (b) Soliton regime at relatively high optical power (here 2.6 mw). The rectangles show the location where the light intensity is measured. (c) Temporal evolution of the beam intensity for a 2.6-mW optical beam. Light is switched on after the V -induced molecular orientation (orientation at rest) reached steady state. [Fig. 2(a)]. With increasing optical power the nonlinear effect becomes significant and, for a certain optical power, the beam propagates with an almost constant width, as can be seen in Fig. 2(b). For even higher optical powers, the beam starts to undulate and the undulations become stronger with increasing optical power (see [9] for details). Besides the light propagation, it is possible to observe the laser-induced molecular reorientation. This is performed by using polarized light from a xenon lamp. Polarized light traveling through the liquid-crystal layer undergoes a polarization change depending on the molecular orientation. In this way, the self-induced waveguide can be observed. For such purpose, two crossed polarizers and a bandpass filter (660 nm) are used, the directions of polarizer and analyzer being 45 with respect to the axis. III. EXPERIMENTAL RESULTS For our experimental beam size and a bias voltage of 1 V, soliton-like propagation can be observed for an optical power of 2.6 mw. When the optical power is switched on, the beam evolves from a diffracting regime when the molecules are not oriented yet to a soliton-like propagation regime. The evolution of the intensity profile of light scattered from the beam after a propagation distance of 1.6 mm is shown in Fig. 2(c). At 0 ms, the width of the beam is large and the intensity is low. After 1 s, the width of the beam has become smaller, the peak intensity higher and the intensity profile reaches a steady state. Fig. 3 shows the polarized transmission image of the cell for different powers of the laser beam. The latter is entering from the left side, where the entrance of the cell is visible. The black region is caused by the glue on top of the interface between the two glass plates. Due to the anisotropy, there is a phase retardation between ordinary and extraordinary polarizations of about after propagation through the 75 m-thick cell. 1 The laser-induced molecular reorientation reduces the retardation and consequently changes the transmission through the crossed polarizers in the region where the soliton-like beam propagates. In Fig. 3(a) the effect of the reorientation is small, but it increases with increasing optical power [Fig. 3(b) (c)]. With increasing molecular reorientation the image becomes black, white, and black again as can be seen in Fig. 3(c). This is because the retardation is larger than. This method of visualizing the self-induced waveguide is clearly not sensitive enough for the small reorientation occurring at 2.6 mw. For this purpose, an interferometric approach is more sensitive and thus more appropriate 1 This is because of the large thickness of the cell (75 m). Nematic-liquidcrystal displays typically have a thickness which is times smaller.

3 BEECKMAN et al.: TIME DEPENDENCE OF SOLITON FORMATION IN PLANAR CELLS OF NEMATIC LIQUID CRYSTALS 737 TABLE I PROPERTIES OF E7 (T =20 C, =644nm) per unit volume consists of three terms, which can be written as follows [19]: Fig. 3. Steady-state transmission of xenon light through the cell when the laser beam with increasing power from (a) to (c) is launched into the cell. The gray arrow indicates the estimated entrance of the cell. (1a) (1b) (1c) Fig. 4. Evolution in time of transmitted xenon light corresponding to Fig. 3(b) and (c), after laser beam is switched on. [18], but to demonstrate the time effects, the described method is useful as the molecular reorientation can be immediately estimated from the transmission profiles. The evolution in time of the transmission close to the entrance of the 75- m-thick cell is illustrated in Fig. 4 for two different optical powers. Due to the glue at the entrance it is not possible to define the distance exactly, but it is estimated to be a few hundred micrometers. The two pictures show that the time to completely reorient the liquid crystal is on the order of 50 s. This is in contrast to Fig. 2(c) where the formation of the soliton-like beam takes on the order of one second. A plausible explanation will be given further in this article. IV. NUMERICAL MODEL In order to simulate the time dependence of soliton formation in a nematic-liquid-crystal layer, a two-dimensional calculation of the molecular director is used. In absence of any electric field, the molecules in the cell have an angle of 2 with respect to the axis due to the pretilt of the rubbing layers. Hard boundary conditions are assumed, which means that the molecules at the border have a fixed orientation of 2. A bias voltage applied over the cell gives rise to a static electric field along the axis. In addition, an optical electric field is also present and is oriented mainly along the axis. This field arises from an optical beam which is propagating in the direction and which has a polarization along. It is assumed that molecules tilt in the plane only, which means that the orientation can be described by one angle (see Fig. 1). With these constraints, the free energy The first term is the distortion free energy, with,, and the three elastic constants. The second and third terms are the free energies of the static and optical electric fields, respectively. Using the Euler Lagrange formula, we can derive the following equation which describes the time evolution of molecular reorientation [20] In this formula, is the rotational viscosity of the liquid crystal. When calculating the successive time steps, Maxwell s equation has to be fulfilled for each time step, with. To calculate the transmission of polarized light through the cell, the Jones calculation was numerically implemented [17]. For the calculation of the evolution of the optical-wave envelope, i.e., the soliton-like propagation, a scalar model can be used, governed by the following propagation equation [9] (3) where and are, respectively, the wave number and its vacuum counterpart, and the pretilt angle. For a full description of the light propagation, the anisotropy of the liquid crystal should be included, but this is out of the scope of the present article. The effects of such a vectorial description were indeed studied elsewhere [21]. V. SIMULATION RESULTS Numerical calculations are performed with the parameters of the liquid-crystal mixture E7 from Merck, listed in Table I [17], [22]. (2)

4 738 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005 Fig. 5. (a) Evolution of in the middle of a 75-m-thick liquid-crystal layer when switching on voltage at t =0s. (b) Corresponding rise time, i.e., the time for the midtilt to reach 99% of its steady-state value. Fig. 6. Transmission through the thick cell when switching off the bias voltage at t =0s. A. Switching Voltage In order to enhance the optical nonlinear effect, a voltage is applied over the cell to tilt the molecules a little. After all, the torque on the molecules is maximal when the angle between the director and the light polarization is 45 and it is null for 0 and 90 [2]. Fig. 5(a) shows the temporal evolution of director orientation in the middle of a 75 m-thick cell when a voltage around the threshold is switched on. Note that no laser-induced molecular reorientation is here considered, so that is invariant with respect to. Fig. 5(b) shows the evolution of the rise time arbitrarily defined as the time for the midtilt to reach 99% of its steady-state value as a function of the applied voltage. The switching time is long on the order of 100 s for 1V for two reasons. First, the cell is very thick, compared to displays, and second, the voltage is below threshold [23]. This results in a static electric field of V m, which gives rise to a very small driving force to reorient the molecules, and consequently to a slow process. The transmission through the cell when switching off the 1 V voltage still without laser-induced molecular reorientation was measured with the experimental setup and compared to the numerical simulations. In Fig. 6 it is shown that the agreement between the two is good. To achieve this agreement, the simulation was performed for a slightly smaller cell thickness (73.9 m). There is, however, no objection to our simulation Fig. 7. Evolution of the midtilt when switching on a 4.41-mW laser beam at t =0s. The voltage over the cell is 1 V. method. Indeed, the technology of making the cell introduces an uncertainty on the thickness on the order of a few percents. B. Soliton Formation in a 75 m-thick Cell Fig. 7 shows the evolution of the midtilt i.e., the tilt distribution in the middle of the cell, that is, in the plane after switching the laser beam on. The increase of the tilt in the middle gives rise to a nonhomogeneous increase of the refractive index seen by the same laser beam. This self-induced mechanism is responsible for the collimation of the beam, through the self-focusing effect. In other words, the beam is trapped in its own induced graded-index waveguide. The initial reorientation is relatively fast and happens on a time scale of a few seconds. Indeed, the reorientation in the middle reaches more than 50% of the final reorientation after 2.5 s. More important, it can be seen that the angular profile in the vicinity of the beam center stays almost unchanged as soon as 2.5 s. As the angular profile, hence the index modulation, determines the self-trapping of the beam, this time scale is consistent with the experimental time evolution of the beam, as discussed in Section III. After this, the reorientation of the molecules further away from the center occurs until an overall steady state is reached where the width of the induced tilt profile is much larger than that after a few seconds and a fortiori than the soliton-like beam itself. Let us recall that the initial optical

5 BEECKMAN et al.: TIME DEPENDENCE OF SOLITON FORMATION IN PLANAR CELLS OF NEMATIC LIQUID CRYSTALS 739 Fig. 8. Simulated temporal evolution of the transmission through the cell after switching on laser beam of (a) 2.25 mw and (b) 4.41 mw. beam is a 3- m waist beam. This reorientation occurs on a much larger time scale and the steady state is reached after about 50 s. This means that initially the reorientation is fast and the nonlocality of the nonlinear effect is small. Then the nonlocality increases in time until a maximum is achieved. This effect is plausible as only a low energy is required to reorient the liquid crystal in the immediate vicinity of the optical beam. The complete reorientation, whose spatial extent is much wider, requires a much larger energy and, consequently, a longer time. With the values of the director orientation, the evolution of the transmission can be calculated to yield a result comparable to the experiment shown in Fig. 4. These numerical results are shown in Fig. 8. The numerical and experimental results correspond well, if we compare Fig. 4 with Fig. 8. Both figures show that the reorientation is completed after about 50 s. A first difference, however, is the larger width of the observed index profile. Probably this is due to the imaging method because in addition to a polarization effect also a diffractive effect may play a role. Conversely, the results of the interferometric method correspond better with the simulations [18]. A second difference is a mismatch in the power of the laser beam. Indeed, the experimental optical power is higher than the numerical one by more than a factor of two. Two reasons may be responsible for this mismatch. First, the optical power in the experiment is the optical power that is coming out of the laser. It is not known how much light is effectively entering the liquid-crystal cell. Reflections and scattering at the entrance wall of the cell indeed lower the power significantly. Second, the transmission is measured a few hundred micrometers from the entrance plane of the cell, because of the glue on top of the glass plates. Hence, it is not possible to visualize the transmitted light at the very entrance of the cell, as mentioned before. After this propagation distance, a part of the light may be lost due to scattering and absorption and to polarization scrambling of the beam in the liquid crystal. C. Soliton Formation in Thinner Cells The long time to reach the steady state in thick cells (about 50 s) originates mainly from the large scale of the nonlocal reorientation. Hence, it can be expected that the reorientation time be much smaller in thinner cells, owing to a less wide molecular reorientation (see [9], where the steady state reorientation in function of cell thickness is calculated). This behavior is illustrated in Fig. 9, where the evolution of the maximal tilt i.e., the tilt at, where the light intensity is maximal is shown. For the 75- m-thick cell the reorientation takes about 50 s, as illustrated before, but for the thinnest cell (18 m) the reorientation already reaches its steady-state value after 2 s. For even thinner cells, the time for overall molecular reorientation to Fig. 9. Simulation of the evolution of the midtilt at y =0mwhen a 2.25-mW laser beam is switched on at t =0s, and for different cell thicknesses. settle is even smaller. However, the initial dynamics is quite similar for thick and thin cells. Indeed, reorientation in the vicinity of the light beam completes in less than 2 s. VI. CONCLUSION In this article, we reported on a numerical and experimental study of the formation dynamics of soliton-like optical beams in biased nematic-liquid-crystal planar cells. The good agreement between experimental and numerical results implies that our model of laser-induced director reorientation is realistic, as was suggested in earlier publications on the same geometry [7] [9]. The results also reveal that soliton formation time in a bulk nematic liquid crystal is on the order of seconds. Due to the large nonlocal molecular reorientation in thick cells, however, the steady state is only reached after several tens of seconds. In thinner cells 18 m this nonlocal reorientation is much smaller in extent and hence the time to reach steady state is also smaller (on the order of seconds). For even thinner cells, switching times of less than a second are expected. For thin cells, however, when the width of the optical beam becomes comparable to the layer thickness, the solitons are no longer bulk, or two-dimensional solitons, but become one-dimensional in a waveguide geometry, for which a considerable amount of the light is not in the liquid crystal anymore but in the cladding layers. The description of these solitons is completely different [24]. Despite the fundamental importance of 2-D solitons, the results of the present study show that practical applications in a nematic-liquid-crystal configuration are encumbered by large switching times. Hence, it is more likely that solitons in a liquidcrystal waveguide geometry, that is, with a thin enough liquidcrystal layer, will find their way to applications, due to a smaller switching time. ACKNOWLEDGMENT This research is a result of a collaboration within the framework of the Interuniversity Attraction Poles program Photon Network of the Belgian Science Policy. The authors would also like to acknowledge the European Research Training Network SAMPA.

6 740 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 5, MAY 2005 REFERENCES [1] Y. R. Shen, The Principles of Nonlinear Optics. New York: Wiley, [2] I.-C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena. New York: Wiley, [3] R. W. Boyd, Nonlinear Optics, 2nd ed. San Diego: Academic, [4] G. I. Stegeman, D. N. Christodoulides, and M. Segev, Optical spatial solitons: historical perspectives, IEEE J. Sel. Topics Quantum Electron., vol. 6, pp , Nov. Dec [5] G. Assanto and M. Peccianti, Spatial solitons in nematic liquid crystals, IEEE J. Quantum Electron., vol. 39, no. 1, pp , Jan [6] G. Assanto, M. Peccianti, and C. Conti, Nematicons: optical spatial solitons in nematic liquid crystals, Opt. Photon. News, vol. 14, pp , Feb [7] M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I.-C. Khoo, Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells, Appl. Phys. Lett., vol. 77, pp. 7 9, Jul [8] X. Hutsebaut, C. Cambournac, M. 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Quantum Electron., vol. 30, pp , Oct Jeroen Beeckman was born in Zottegem, Belgium, in He received the B.S. degree in physics engineering from the Department of Applied Sciences, Ghent University, Ghent, Belgium, in 2002, where he is currently working toward the Ph.D. degree in the Department of Electronics and Information Systems. His research interests include lateral light propagation in liquid crystal cells and the development of reconfigurable optical interconnections with the use of liquid crystals. Kristiaan Neyts was born in Oostende, Belgium, in He received the B.S. degree in physics engineering in 1987 and the Ph.D. degree in engineering in 1992, both from the Department of Applied Sciences, Ghent University, Ghent. Belgium. He was with the Flemish Fund for Scientific Research from 1988 until 2000, when he became a full-time Professor in the Electronics and Information Systems Department (ELIS), Ghent University. He has been involved in the research of ac high-field electroluminescent displays, light emission from microcavity structures, and electrooptic behavior of display devices. He was on leave in at the Institute for Physical Electronics in Stuttgart, Stuttgart, Germany, and in at the Lawrence Berkeley National Laboratory, Berkeley, CA. Since 2000, he has been heading the Liquid Crystals and Photonics Group, ELIS, Ghent University. Xavier Hutsebaut was born in 1979 in Brussels, Belgium. He received the B.S. degree in physical sciences from the Free University of Brussels, Brussels, Belgium., where he is currently working toward the Ph.D. degree in the Optics and Acoustics Department. His main research interests are the spatial optical nonlinear phenomena, namely spatial solitons and modulation instability. Cyril Cambournac was born in 1975 in Saint-Mandé, France, and received the B.S. degree in physics and the Ph.D. degree in engineering science from the Franche-Comté University, Bescancon, France in 1998 and 2003, respectively. His Ph.D. work was supported by a doctoral fellowship from the French Ministry of Education, and his dissertation Spatial instabilities in Kerr media: Arrays of spatial solitons and symmetry breaking of multimode solitons in a planar waveguide was awarded the 2003 Saint-Gobain Young Researcher Prize from the French Physical Society. In October 2002, he started a Postdoctoral Fellowship in the Applied Sciences Faculty, Free University of Brussels, Brussels, Belgium, researching spatial phenomena in nonlinear optics, and particularly those related to soliton beams and instabilities. Marc Haelterman was born in Brussels, Belgium, in He received the B.S. and Ph.D. degrees in applied physical sciences and the Ph.D. degree from the Applied Sciences Faculty, Free University of Brussels (ULB), Brussels, in 1984 and 1989, respectively. He received the Pankowski Kipffer Prize of the Applied Sciences Faculty for his final year project dedicated to the theoretical and experimental study of light propagation in optical fibers. He started his career of researcher in the Optics and Acoustics Department of the Applied Sciences Faculty, ULB, under a doctoral fellowship of the Belgian National Fund for Scientific Research (FNRS). Since then, his work is devoted to theoretical and experimental studies of nonlinear light propagation in optical resonators and fibers. After receiving the Ph.D. degree, he started a series of three postdoctoral stays abroad to complete his education in the field of nonlinear optics. He went successively to the Laboratory of Electrooptics, Optics and Microwaves, Institut National Polytechnique de Grenoble, Grenoble, France, (with Prof. R. Reinisch), to the Fondazione Ugo Bordoni, Rome, Italy, (with Prof. B. Daino) and to the Optical Sciences Centre, Australian National University, Canberra, Australia, (with Prof. A. Snyder). When he returned to Brussels in 1994, he received a permanent position as a Chercheur Qualifié with the FNRS. Since 1996, he has been Professor at the Applied Sciences Faculty, ULB, where he teaches physics.