1. Introduction

In the nonlinear regime of optical materials, the index of refraction may change due to the presence of an optical field. When the refractive index increases with increasing optical field, the beam undergoes a self-induced lensing effect and self-focusing occurs. Spatial optical solitons (SOS) appear when the self-focusing effect balances the diffraction of the beam [1]. The study of SOS gained interest because of the possible applications in all-optical signal handling and optical reconfigurable interconnects, as the waveguide induced by a soliton beam can be used to guide a second information-carrying beam [2].

SOS can be observed in liquid crystals (LCs) with milliwatts of light power due to their large nonlinearity. The nonlinearity can be 6 to 8 orders of magnitude bigger than in other materials such as glass [3], but occurs on a much larger time-scale as the effect is related to a reorientation of correlated molecules. Due to the torque induced by the optical field, the average orientation of the molecules changes and yields a nonlinear effect called optical field-induced director reorientation. This effect is similar to the switching of LC molecules in display applications, where a DC field is used to reorient the molecules.

Stability of SOS over an adequate propagation distance is required for applications. This requirement can be satisfied as propagation distances in the centimeter range have been reported [4]. On the other hand, it is desirable to minimize the optical power for soliton propagation. The required optical power can be reduced significantly by applying an external quasi static electric field to the LC cell in order to overcome the Fréedericksz threshold [5]. In this article we numerically and experimentally investigate the influence of different parameters on the required optical power for soliton propagation.

In Section 2 we present the experimental setup and the results are shown in Section 3. Section 4 describes the simulation model and section 5 contains the results from simulations. We finally discuss and conclude in Sections 6 and 7, respectively.

2. Setup

A collimated laser beam from a laser diode is launched into a liquid-crystal cell by a 20× objective lens (Fig. 1). The infrared laser beam of 980-nm wavelength is focused into a spot with a waist of approximately 2 µm, estimated from the observed divergence of the beam after the focal point. The light is linearly polarized along the x-direction.

 

Fig. 1. Experimental setup and indication of axes.

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The liquid-crystal cell consists of two glass plates glued together, with a gap between the plates which is controlled by mixing spacer balls in the glue. The glass plates’ inner surfaces are covered with an ITO layer, which is used as electrode. An alignment layer is deposited on top of this to control the alignment of the liquid-crystal molecules. The molecules are nearly parallel to the surface and parallel to the propagation direction of the optical beam (z-direction) but they have a small tilt of 2° in the xz-plane. A perpendicular glass plate is glued on one side to form the entrance window for the optical beam. This input window is also treated with an alignment layer to minimize the non-uniformity of the liquid crystal at the entrance and to maintain the polarization state of the incoming light. The cell is filled with the nematic liquid crystal E7 which has a birefringence of about 0.2 [6].

The light propagation in the cell is detected by a CCD camera in combination with a 5× objective lens and an optional attenuator. The attenuators have a transmission of 10% (D1), 1% (D2), and 0.1% (D3). The light can be detected because of scattering in the liquid crystal. The different parts of the setup are aligned by micro translation stages.

3. Experimental results

For low optical powers, the beam broadens due to diffraction [Fig. 2(a)]. At high enough powers, the self-focusing starts to become significant [Fig. 2(b)] and at a certain value, the observed width of the beam stays almost constant over the propagation distance and a stable soliton is formed [Fig. 2(c)]. As a matter of fact, the propagation distance increases significantly. For higher optical powers, the beam starts to fluctuate up and down and shows undulation [Fig. 2(d)].

In soliton regime the propagation distance of the beam is essentially limited by absorption and scattering of the light in the liquid crystal, but other mechanisms, such as the loss of polarization of the beam, are also possible.

 

Fig. 2. Light propagation in a 75 µm-thick cell for a voltage of 1 V and different optical powers: (a) 0.8 mW and D1; (b) 1.5 mW and D2; (c) 2.3 mW and D2; (d) 6 mW and D3.

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The behavior described here is similar for different bias voltages applied to the cell. For each voltage, some value of the optical power can be found for which the beam propagates without broadening over a relatively large propagation distance, as shown in Fig. 2(c). The definition of the minimal power is somewhat empirical, but the experimental values obtained this way are quite reproducible with an error on the order of less than 10%. Indeed, for a power under this value the beam intensity fluctuates in time, but becomes stable when slightly increasing the optical power. The obtained values for different cell thickness d are reported in Fig. 3.

For all three cells, the minima of the curves are situated between 1 and 1.5 V, as was already predicted in [7]. When the voltage is below 1 V the required optical power is higher and the propagation distance, i.e. the stability of the soliton beam, is smaller. It is possible that thermal effects are responsible for this behavior because of the higher required power to obtain a soliton.

For higher voltages (above 1.5 V), the required optical power increases again and this is due to the saturation of the nonlinear effect. Indeed, for high voltages the liquid-crystal molecules are tilted almost 90° with respect to the z-axis. For voltages higher than 2 V no adequate value for the optical power could be found.

 

Fig. 3. Relationship between required optical power for soliton propagation and voltage for different cell thickness d. (Experiment.)

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For the 18-µm cell, periodic intensity fluctuations along the propagation distance can be observed (Fig. 4) at low power (i.e., without nonlinear effect) as well as at high power. As shown in Fig. 4(b), however, the width of the beam in the y-direction remains constant along the propagation distance, but the intensity fluctuates. The explanation for this phenomenon will be given further in this article.

 

Fig. 4. Light propagation in a 18 µm-thick cell for a voltage of 1.6 V: (a) for 1.5 mW and D1 filter; (b) for 4.5 mW and D3 filter. The scattering on the left side of the pictures comes from the entrance window.

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4. Model

To calculate the orientation of the molecules, the total free energy of the liquid crystal has to be minimized. The free energy consists in the distortion free energy, containing the contributions from splay (K ₁), twist (K ₂), bent (K ₃), and in the electric free energy. The latter contains the contribution from the quasi static electric field and the optical electric field which is responsible for the nonlinear effect. The orientation of the molecules is calculated for a two-dimensional situation in the xy-plane, using for simplicity a one-constant approximation with K=K ₁=K ₂=K ₃ [8]. It is assumed that the light remains linearly polarized when propagating through the liquid crystal and that the molecules rotate in the xz-plane only, as in Ref. [5]. In this approximation the director of the molecules n̄ can be fully described by the tilt angle θ, as shown in Fig. 1. The differential equation describing the system is given by [7]:

In this equation opt Δε^opt =$n_e^2$-$n_o^2$ with n_e and n_o being respectively the extraordinary and ordinary indices of refraction. Together with this equation the distribution of the DC electric field has to be calculated, as the DC permittivity changes when molecules reorient. Starting from ∇.(ε̿.∇V)=0, with V the potential distribution, the equation to be solved is:

The two differential equations are solved by using a relaxation method [9]. For the initial distribution of the optical field, a circular-symmetric Gaussian beam is considered:

with a value of 3 µm for r ₀. The propagation of the optical field is calculated by a scalar beam-propagation method (BPM) [10] with the following propagation equation:

At each step of the BPM algorithm, the director orientation is recalculated for the new optical field distribution. Assuming that the variations along the z-direction are much smaller than the variations along x and y, the two-dimensional calculation is a good approximation. Otherwise, a full three-dimensional calculation of the director would be necessary, taking into account also the distortion energy arising from variations in the z-direction.

All simulations have been performed with the parameters of the liquid crystal E7 [6] and a value of 12 pN for the K constant. Hard boundary values (θ=2°) are used for the orientation of molecules at the glass-liquid crystal interfaces. Periodic boundary conditions are used for the y-direction.

5. Simulation results

In Fig. 5 simulation results are shown for a 53 µm-thick cell and a bias voltage of 1 V. In Fig. 5(a) it can be seen that the optical field induces a higher refractive index in the middle of the layer, as the molecules are optically tilted. From this figure and Fig. 8, however, it can be seen that the nonlinear effect of optical field-induced reorientation is a nonlocal effect [11]. The beam has a width of a few microns whereas the director reorientation spreads out over a region of several tens of microns, as otherwise confirmed experimentally in [12].

Figure 5(b) shows the evolution of the maximal amplitude of the optical field, i.e., the amplitude in the middle of the layer (x=d/2,y=0). For low optical powers, the intensity decreases as the beam is broadening due to diffraction as can be seen in Fig. 5(c) and 5(d). For higher powers, the intensity drops more slowly and for even higher powers the intensity initially increases, due to the narrowing of the beam.

From Fig. 5(b), (c) and (d) it can be seen that no strict soliton propagation is possible for a Gaussian input beam. Indeed, for a soliton, the beam width and peak intensity should remain constant along with the propagation through the liquid crystal. Nevertheless, the graphs show that for a certain value of power the intensity and width are approximately constant for a propagation distance of four Rayleigh lengths, with a Rayleigh length for the simulated beam of 29 µm [13]. However, numerical simulations reveal that such a quasi invariant behavior remains over larger propagation distances, as observed experimentally as long as power-loss stays insignificant.

 

Fig. 5. (a) Tilt distribution in the presence of the optical field for a 53 µm-thick cell and 1-V voltage. (b) Evolution of the optical-field peak amplitude for different input powers. (c, d) Corresponding evolution of beam width in the y- and x-direction, respectively. [The arrows indicate an increasing initial optical field A (i.e. an increasing optical power).]

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For thin cells the situation is even more pronounced. Figure 6 shows the evolution of the beam width for a thin cell of 18 µm, in the y- and the x-direction. The graph shows that for the simulated optical field, the width of the beam in the y-direction remains approximately constant whereas the width along the x-direction shows oscillation. Even at very low power (i.e., out of the nonlinear regime), the beam width shows oscillation in the x-direction. The explanation for this effect is that the orientation of the LC molecules arising from the DC voltage already induces a waveguide which confines the beam in the x-direction [7].

 

Fig. 6. Evolution of the width of a beam propagating in a 18 µm-thick cell, for a voltage of 1.6 V and an optical power of 3.66 mW.

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All this illustrates that it is not possible to unambiguously define a value for the parameter A for which soliton propagation occurs. Therefore, to define an optimal value for the optical power we take the lowest A for which the beam width in the y-direction remains within 20% of the initial beam width for a propagation distance of four Rayleigh lengths (about 120 µm). Due to the nature of the simulation method a maximal relative error of 5% for this critical optical power should be taken into account.

 

Fig. 7. Optimal optical power in function of voltage for different cell thickness with a maximal relative error of 5%. (Numerical simulations.)

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Figure 7 shows the numerically-obtained optimal optical power in function of the applied voltage. It can be seen that the optical power behaves in a similar way for different cell thickness. In the absence of an optical field the midtilt is just a function of voltage, because the thickness is canceled out in Eq. (1) [7]. From Fig. 8 we see how the optical field modifies the tilt in the case of a 1-V applied voltage. The midtilt is smaller for thinner cells, which is due to the bigger influence of the surface forces. However, the shape of the curve in the direct neighborhood of the beam is approximately the same for each thickness. This is important because the propagation of the light is mainly determined by the variation of the index of refraction in its direct neighborhood. This explains why the optimal optical power is here almost the same for different cell thickness. Only for voltages below 1 V a significant difference occurs.

 

Fig. 8. Tilt distribution in the middle of the layer along the y-axis for a voltage of 1 V and for the same parameter A for the different cell thicknesses. The shape of the optical-field distribution is also shown as indication.

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6. Discussion

When comparing Fig. 3 with Fig. 7, it can be seen that the agreement between the curves is good. This indicates that our model is a good approximation and that the nonlinear effect of optical field-induced director reorientation is indeed the responsible effect. It is confirmed that, for E7, whatever the LC-cell thickness, the lowest optical powers for soliton-like propagation occurs for applied DC voltages between 1 and 1.5 V. For such voltages and appropriate optical power, the beam width remains within 20% of its initial value for four Rayleigh lengths and even larger propagation distances, as observed experimentally. This is in contrast with the behavior at low voltages where it is not possible to keep the beam width within the 20% range for long propagation distances (larger than four Rayleigh lengths). Numerical simulations indeed reveal that the soliton-like propagation is here much more sensitive to power fluctuations and that a slight deviation to the optimal optical power can lead either to a diffracting or to a self-focusing propagation regime. Hence, soliton-like propagation can rapidly be lost because of small power changes which may come, e.g., from scattering losses inside the cell. We believe that the short propagation distances observed experimentally for low voltages are thus not only caused by a thermal effect, as suggested earlier, but are inherent to the underlying nonlinear effect of optically-induced director reorientation.

The longitudinal intensity modulation observed for thin cells (cf. Fig. 4) can also be explained by our model. It is due to an oscillation of the beam width in the x-direction when propagating through the cell, as discussed in Section 5. From the experiment, the distance between the maxima of the intensity modulation is approximately constant and has a value of 53 µm (for the upper part in Fig. 4). The simulated distance is 58 µm and these values are obtained by a linear fit for the positions of the maxima of intensity. The agreement is satisfactory, taking into account the approximations of our simulation method.

Finally, it can be seen from Fig. 8 that it is possible to tune the degree of nonlocality of the nonlinear effect by varying the cell thickness. Though beyond the scope of the present paper, this remark is worth mentioning, as soliton-beam parameters and dynamics are generally ruled by the degree of nonlocality of the self-focusing nonlinearity (see, e.g., Ref. [11] and references therein).

7. Conclusion

In this paper we reported on a numerical and experimental investigation of optimal conditions for soliton-beam propagation in planar cells of nematic liquid crystals. Both the simulations and the experiments have shown that a DC voltage over the LC cell can indeed lower the required optical powers for soliton propagation, which is an important issue for possible applications. For the described geometry, the minimum is situated between a voltage of 1 and 1.5 V. Moreover, the minimal required power is approximately independent of the layer thickness, but for thin cells large intensity fluctuations can be observed, which proves the existence of oscillations of the beam width along the thickness when propagating through the cell. In addition, the simulations have shown the possibility to tune the degree of nonlocality of the nonlinear effect (the optically-induced director reorientation) by varying the cell thickness.