1. Introduction

Nematic Liquid Crystals (NLC) are an interesting molecular system for optical and optoelectronic applications, as well as an ideal workbench for investigating a vast number of nonlinear optical effects.[1–4] Field-induced reorientation in NLC, in fact, results in both a large electro-optic response and a remarkably high nonlinear response, orders of magnitude in excess of reference materials such as CS2.[2] In undoped NLC such nonlinearity is polarization-dependent, self-focusing, non local and non-resonant, as recently addressed in the investigation of low-power (2+1)D spatial solitons -also known as nematicons-[5–9] and spatial modulational instability. [10]

The observation of spatial solitons often requires light propagation and an all-optical response in the presence of significant anisotropy, as it is well established for quadratic and photorefractive solitons. [11–13] While this is not an issue in NLC when invoking a thermo-optic response or a phase transition, [14–16] in the case of induced reorientation the propagation of “extraordinary” light-waves and their inherent walk-off has to be taken into careful account.

In nematic liquid crystals, the formation of spatial solitons requires light-matter interaction with an extraordinary-polarized beam of wavevector k at a non-negligible angle with respect to the optic axis. With reference to the sample geometry used to demonstrate nematicons, [6, 17] the optical excitation encompasses extraordinary components with wave vector lying in a principal plane, the latter defined by the direction of propagation Z and the NLC molecular director n (see Fig. 1). In addition, the low-frequency electric bias employed to avoid the Freedericks threshold gives rise to an index modulation across the NLC thickness X, i. e. an X-dependent distribution of the “director field” and, thereby, of birefringence and walk-off. In actual experimental arrangements the finite thickness of the sample as well as the inherent walk-off has a relevant impact on the propagation of nematicons. As the Poynting vector associated to an extraordinary wave departs from k, in fact, the nematicon is forced to interact with the bias-induced graded index profile and distort its trajectory. In this Paper we investigate the transverse dynamics of nematicons as they propagate in a standard NLC cell (as in Ref. 17). Such complex dynamics, depending on the degree of self-confinement, the transverse non locality of anchored NLC, the applied voltage, the walk-off distribution as well as on the beam interaction with graded-index boundaries, can be properly addressed only by means of a non-paraxial (vectorial) model. This is a formidable task and goes beyond the scope of this work. Here, using a scalar model to support the experimental evidence, for the first time we show that, unless walk-off is compensated at the input by a proper tilt, [17] nematicons oscillate in the principal plane where they are excited, their periodicity depending on the voltage-controlled birefringence.

2. Sample and model

With reference to the sample geometry sketched in Fig. 1, let us consider an X-polarized beam propagating with wavevector k=k Z and injected into a glass cell filled with nematic liquid crystals (Fig. 1). The cell consists of two glass slides which confine the NLC by capillarity, and a third slide sealing the input and providing a polarization-maintaining interface. [5] Kapton spacers define the NLC thickness L across X. Indium-tin-oxide electrodes enable to apply a low-frequency bias in order to orient the optic (director) axis in the plane X-Z, whereas rubbed polyimide films ensure the planar anchoring of the molecules with a pretilt of 2° in X=±L/2.

The bias induced orientation is governed by the Euler-Lagrange equation: [1–2]

being Θ=Θ(X) the angle between the molecular director n and the propagation versor Z=k/k, K₁ and K₃ the Frank elastic constants for splay and bend, respectively, ε _a =ε _‖-ε _⊥ the dielectric anisotropy (ε _⊥ and ε _‖ the low-frequency dielectric costants for fast and slow crystal axes, respectively) and V=V(X) the applied voltage. From ∇·D=0 along X (D is the electric displacement), we obtain the equation ruling the potential distribution:

 

Fig. 1. Sketch of the NLC cell and experimental geometry.

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From the integration of Eqs. (1) and (2) with boundary conditions V(L/2)=V₀, V(-L/2)=0, Θ(-L/2)=Θ(L/2)=2π/180, we derive the NLC orientation across its thickness L. In such a uniaxial crystal, the walk-off angle can be expressed in terms of Θ and the optical birefringence Δn as:

Here n _‖ and n _⊥ refer to light polarizations parallel or normal to the molecular axis (i.e., director), respectively. For the nematic E7 at room temperature and in the near infrared, the pertinent material parameters are K₁ =1.2 10^-11 N, K₃ =1.95 10^-11 N, ε _⊥=5.1, ε _‖=19.6, n _⊥=1.5038 and n _‖=1.6954 at λ=1.064µm. [18] For a thickness L=75µm and a bias V₀≈1.5V we obtain the maximum walk-off δ≈7° at Θ≈48° in the middle of the cell, i.e., X=0. Fig. 2 displays the calculated walk-off versus applied voltage V ₀.

 

Fig. 2. Calculated maximum (on-axis) walk-off versus cell bias

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Nonlinear light propagation in the sample is numerically investigated by means of a suitable beam propagator. To this extent, on the basis of the Frank free-energy formulation and by minimizing the total energy spent by the NLC to hold a specific director distribution, [1–2, 5] we can derive the evolution equation for an optical envelope E linearly polarized along X, and also account for walk-off δ in the plane X-Z by an additional phase term: [13]

with θ=Θ+θ’ the overall reorientation owing to both the low-frequency (V) and the optical field (E) contributions Θ and θ’, respectively, k ₀ the vacuum undulance and n²(θ)=cos² θ/${\mathrm{n}}_{\perp }^2$ +sin² θ/${\mathrm{n}}_{\Vert }^2$ the refractive index for an extraordinary wave. Note that, in the presence of both an external (low-frequency) field and a polarized light beam of envelope E, equation (1) is written in two transverse dimensions as:

with K the elastic constant in the single-constant approximation.[2] Externally-applied and propagating electric fields affect the orientation of the induced NLC dipoles, thereby distorting the refractive distribution in the transverse plane X-Y. The bias provides a graded-index well which is invariant across Y, whereas the injected beam gives rise to a bell shaped increase sustaining light confinement and a spatial soliton.

For the sample described above, the calculated refractive index profile in the absence of a light beam is shown in Fig. 3 for a bias V=1.48V and K=K₁ (in order to match calculated and measured bias threshold for reorientation). Correspondingly, a 1.064µm beam of waist w=3µm and polarization along X, when injected in X=0 with k-vector parallel to Z and power P=3mW generates a nematicon, as yielded by numerical simulations and shown in Fig. 4. If the beam phase front is flat and orthogonal to Z (Fig. 4(a)), the optical excitation self-traps though reorientation, but walk-off forces the Poynting vector off-axis. Thereby the localized beam interacts with the bias-controlled graded-index distribution, resulting into an oscillatory motion in the plane X-Z as it propagates forward. Clearly, the sinusoidal path described by the nematicon is characterized by a period which depends on voltage V and inherent NLC anisotropy. Conversely, if walk-off is compensated by a phase front tilt at the input (or a small offset in X), the nematicon can propagate on-axis, as shown in Fig. 4(b) and first reported in Ref. [17]. Note that the periodic variation in nematicon waist (but not in its transverse position) still visible in Fig. 4(b) has a different origin and underlying physics: such breathing is power-dependent and intrinsic to the (highly) non local response of the medium, as pointed out in Ref. [8] and experimentally investigated in Ref. [9].

 

Fig. 3. Bias induced index profile in a cell filled with E7 and an applied voltage V=1.48V, providing maximum walk-off.

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Fig. 4. Simulated propagation of a 3mW X-polarized gaussian beam launched in a biased cell (as in Fig. 3) with k-vector parallel to Z and a) no phase front tilt ; b) a 7° tilt in order to compensate walk-off on axis.

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Since in the NLC cell the index profile across X can be adjusted by acting on bias, the oscillation periodicity due to walk-off is expected to be modified accordingly.

3. Experimental results and discussion

For the experiments we used a linearly-polarized Nd:YAG laser beam (λ=1064nm) injected into the NLC cell with a 20x objective. The evolution of the optical envelope in the NLC was acquired by collecting the scattered light from the top of the cell with a microscope and a high-resolution CCD camera. In order to view the beam trajectory in the transverse plane X-Z, the microscope-axis was set at an angle ϕ≈45° with respect to X, i.e. looking at a σ-projection of the propagation plane X-Z onto the microscope image-plane.

 

Fig. 5. Nematicon transverse profile in the observation plane at ϕ=45° with respect to X. a) For V ₀=1.0V the small walk-off (about 2°) mediates an oscillation of modest amplitude; b) at V ₀=1.6V a larger walk-off (about 7°) corresponds to a shorter period with larger elongation across X. c) By launching the input beam with a phase front tilt in order to compensate the walk-off, the nematicon at V ₀=1.6V can be generated with no motion across X

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Figure 5 shows a few examples of nematicon propagation at an input power P=3.5mW. The transverse oscillation in the plane X-Z is visible at both small (V ₀=1.0V, δ≈2°, Fig. 5(a)) and large (V ₀=1.6V, δ≈7°, Fig. 5(b)) on-axis walk-off, being more pronounced and with shorter period in the second case. As predicted by the simple model outlined above, in both cases walk-off causes the extraordinarily-polarized beam to acquire a transverse velocity along X and interact with the graded index profile, resulting in an oscillatory behavior in the plane X-Z. By tilting the input wave front, however, the walk-off can be balanced out, and a nematicon propagate straight along Z (Fig. 5(c)).

 

Fig. 6. Soliton trajectories for P=3.2mW versus bias V ₀. The scale Δx quantifies the deviation from input position X=0.

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Figure 6 graphs the acquired nematicon trajectories for various biases. At low voltages (0.7<V ₀ <1.5V) the oscillation period decreases with bias, consistently with the increment in both walk-off and index perturbation. A higher bias (V ₀ >1.6V), however, weakens the oscillatory character, walk-off angles become smaller (see Fig. 2) and the refractive profile tends to flatten.

Finally, Fig. 7 compares the calculated periodicity after integration of Eqs. (4) and (5) (solid line) with the measured nematicon oscillations (symbols and dashed line) across X versus applied voltage. Despite the adopted scalar approximation (Eq. (4), see also Ref. [13])) and the experimental uncertainties, the agreement between data and model is excellent.

 

Fig. 7. Calculated (solid line) and measured (dashed line with dots) periodicity Λ of the nematicon transverse oscillation versus applied bias V₀.

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4. Conclusions

We have investigated the transverse dynamics of spatial solitons propagating in anisotropic nematic liquid crystals. Walk-off plays a relevant role in determining the trajectory of nematicons which, interacting with a graded index potential, oscillate with a voltage-tunable periodicity. Routing schemes based on nematicon position (across X) at the cell-output can be envisioned and combined with nematicon steering [19] and nematicon-nematicon collisions in the plane Y-Z. [20]