1. Introduction

The generation of spatial solitary waves (in the following simply solitons) stemming from the balance of diffractive beam spreading and self-focusing was one of the first topics studied in nonlinear optics [1]. In the simplest medium case, i.e. a Kerr medium with a local response, the so-called ”Townes solitons” are intrinsically unstable in two transverse dimensions and subject to beam filamentation and collapse [2, 3]. Several approaches have been investigated to avoid collapse, such as 1D geometries [4], higher order [5] and saturating nonlinearities [6], parametric interactions [7], nonlocality [8, 9]. Due to the inherent nonlinear nature of self-localized beams, a natural question is whether solitons can be bistable versus beam excitation. In his pioneering theoretical work, Kaplan stated that two stable solitons can coexist for the same input power when the optical nonlinear response exhibits a threshold in the relationship between refractive index and beam intensity [10]. Since then, bistability of bright solitons has been theoretically investigated in cubic-quintic systems [11] and in colloidal media [12, 13].

Thanks to the high nonlinear response typical of nematic liquid crystals (NLCs) [14], self-trapping of light beams in this class of materials has been extensively investigated in the last years. Stable (2+1)D spatial solitons, named nematicons, have been demonstrated in several configurations, including voltage biased and unbiased cells (see Refs. [15, 16] for an exhaustive review) and exploiting the reorientational molecular response [14]. The latter is subject to a threshold when the electric field of the incident beam and the NLC molecules are orthogonal, the so called Optical Fréedericskz Transition (OFT) [17]. In this Paper we demonstrate that in NLCs subject to OFT, i.e. exhibiting a second-order transition when excited by plane wave-like beams [18], a finite size beam (appreciably diffracting as it propagates in the medium) can undergo a first order transition when self-trapping is accounted for, eventually leading to the formation of reorientational solitons. Self-focusing was shown to yield hysteretic effects with an external mirror in previous demonstrations with NLCs [19]. Hereby we demonstrate that, beyond a threshold input power, the free energy of the system has two local minima versus maximum reorientation. Thus, the equilibrium point becomes dependent on the previous evolution history, leading to a hysteresis loop versus excitation. Finally, while Braun et al. used a similar configuration in NLC-filled capillaries for light self-trapping [20], our system benefits from an external voltage bias in order to minimize the optical power for OFT, and our planar geometry permits to avoid the instabilities deriving from spontaneous symmetry breaking.

2. Self-trapping of light in the presence of OFT

The basic principle behind the reorientational nonlinearity of (undoped) NLCs is simple: the (electric component of the) impinging optical field induces dipoles in the elongated organic NLC molecules along a preferential direction, as dictated by their anisotropy; these dipoles in turn tend to align to the field vector or polarization direction. The molecules tend to locally align with the same orientation due to strong intermolecular links; the macroscopic (average) alignment direction pointwise is called molecular director and usually indicated by n̂ [14]. Hence, light in ordered NLCs propagates in an uniaxial crystal with the optic axis corresponding to n̂ and refractive indices equal to n_⊥ and n_‖ for polarizations normal and parallel to n̂, respectively. The net effect of an (intense) electric field distribution, such as a light beam, on the NLC molecules is to reorient the director n̂ according to its intensity profile, thereby changing the local extraordinary refractive index n_e [15]. Let us consider a planar cell of thickness L_x = 100μm along x filled with the standard nematic liquid crystal E7, infinitely wide along y and extending from z = 0 to infinity (Figs. 1(a)–1(b)). We take the director n̂ to be parallel to z everywhere in the absence of excitations. The director distribution is given pointwise by the angle θ formed with ẑ, with θ_m the maximum θ in each section normal to ẑ. A linearly polarized Gaussian (TEM₀₀) beam of wavelength of λ = 1064nm propagates paraxially along z and impinges on the sample; thus, its electric field E can be written as E = A exp[ik₀n_e(θ_m)z] x̂ (k₀ = 2π/λ). To help the optical reorientation, a bias voltage V can be applied to the cell across x, providing a quasi-static electric field E_LF ≈ V/L_x. In the hypotheses above, the director can rotate in the plane xz and θ unambiguously determines the director as well as the extraordinary refractive index distributions. If spatial walk-off is ignored, the beam evolves according to

 

Fig. 1 (a) Side and (b) top view of NLC planar cell; the blue arrows indicate the director distribution at rest. (c) Free energy F versus θ_m when K = 12×10⁻¹²N and the temperature is 18°C. (d) Soliton width versus input power P corresponding to (c). (e) Sketch of the hysteresis loop in the plane (P, θ_m). (f) Power threshold for a fixed Gaussian beam of waist 3.5μm versus applied bias V (blue line with crosses, left axis); the dotted lines with squares graph the loop width versus V between w_in = 5, 11, 35μm (from bottom to top, right axis) and a soliton with an average width of 3.5μm.

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Equation (1) is the nonlinear Schrödinger equation (NLSE) modified to account for the medium anisotropy and the strong perturbation regime; $D_x=n_e^2({\theta }_m)/(n_{\perp }^2{\text{sin}}^2{\theta }_m+n_{\parallel }^2\hspace{0.17em}{\text{cos}}^2{\theta }_m)$ is the diffraction coefficient along x, whereas $\mathrm{\Delta }{n}_e^2=n_e^2(\theta )-n_e^2({\theta }_m)$ is the nonlinear index well associated with the induced director rotation. Eq. (2) is cast in the single elastic constant K approximation [14, 15] and describes the director distribution affected by the external stimuli A and E_LF, weighted by the corresponding anisotropies ${\epsilon }_a=n_{\Vert }^2-n_{\perp }^2$ and Δε_LF, respectively. Thus, the nonlinear response is highly nonlocal owing to the Poisson-like form of the equation, and saturable owing to the sine term of the torque acting on the molecules [15]. Eqs. (1) and (2) model the propagation of self-confined waves in highly nonlocal NLC, namely nematicons [15].

In standard configurations, the nonlinear response governed by Eq. (2) is thresholdless because the angle θ is finite in the absence of external excitations [15], and Eq. (2) transforms into a Poisson equation for weak nonlinear perturbations. Conversely, when the (electric field) polarization of the input beam and the director are initially perpendicular to one another, θ_m is non-zero only above the OFT: in this case Eq. (2) cannot be linearized near threshold, with the breakdown of the analytical methods usually employed in nematicon-related models [15, 16]. In order to understand the transition in the presence of self-trapping (given by Eq. (1)), we need to compute the free energy of the system versus the maximum perturbation θ_m. The simplest approach is to find the equilibrium states studying the soliton case, i.e., with the ansatz A(x, y, z) = ϕ(x, y) exp[ik₀(n_S − n_e(θ_m))z], assuming θ invariant across z as well. For the sake of simplicity, we calculate the free energy in the unbiased case V = 0: the application of a bias below the electric Fréedericksz threshold reduces the threshold optical power, without affecting the qualitative trend of the free energy [18].

The overall free energy of the system consisting of NLCs and the electromagnetic field is given by the combination of elastic energy ℱ_K, given by $\frac{K}{2}(\nabla \theta \cdot \nabla \theta )$ in the limit of a single elastic constant, and Lagrangian of the light field L_opt, given by $\frac{1}{2}{\epsilon }_0{n}_{\perp }^2{\mathit{E}}^2+\frac{1}{2}{\epsilon }_0{\epsilon }_a{\left(\widehat{n}\cdot \mathit{E}\right)}^2-\frac{1}{2{\mu }_0}{\mathit{B}}^2$. Note that the light-matter interaction, responsible for torque and reorientation, is contained in the term L_opt. Due to the highly nonlocal character of the NLC reorientational response, ϕ can be taken Gaussian, i.e., $\varphi =\sqrt{4Z_{0}P/\left[\pi n_e({\theta }_m)w^2\right]}\text{exp}[-(x^2+y^2)/w^2]$ [9], with P and w the soliton power and waist, respectively; hence, the optical reorientation is $\theta =0.5\gamma Z_{0}P\text{sin}(2{\theta }_m){\mathrm{\Sigma }}_{l=1}^{\infty }{V}_l(y)\text{sin}\left[\frac{\pi l(x-L_x/2)}{L_x}\right]/n_e({\theta }_m)$, with γ = ε₀ε_a/(4K) and V_l(y) = sinc (πl/2) [F(y) + F(−y)] with $F(y)\equiv \text{erfc}[\sqrt{2y}/w+\pi lw/(2\sqrt{2}{L}_x)]e^{\pi ly/L_x}$ [21]. The overall free energy $F={\int }_{-\infty }^{\infty }{\int }_{{-L}_x/2}^{L_x/2}({ℱ}_{\text{K}}+L_{\text{opt}})\text{d}x\text{d}y$ is [21]

Figure 1(c) shows that, below a threshold power P_th, F monotonically increases with θ_m, thus reorientation does not occur and only the linear regime is stable (in the plot P_th ≈ 18mW). When the power overcomes P_th, a local minimum with θ_m ≠ 0 appears in F, yielding the formation of a shape-preserving nematicon with width dictated by the power P. Besides the minimum, a local maximum corresponds to an unstable nematicon. Fig. 1(d) compares the nematicon width calculated from Eq. (3) and from numerical simulation of Eqs. (1) and (2). Since for a fixed P every point in Fig. 1(c) corresponds to a different beam width [21], it is straightforward to describe the insurgence of hysteresis. We take a power larger than the threshold P_th and fix the input beam width to w_in. If the beam is wider than the unstable soliton (i.e., the input state is on the left of the local maximum), the beam evolves towards the linear regime θ_m = 0. Conversely, if the input beam is narrower than the unstable soliton (i.e., the initial state is placed on the right of the local maximum), the beam reaches the minimum and turns into a stable soliton. Thus, Eq. (3) predicts the occurrence of a hysteretic loop versus power: for a given input waist w_in a power $P_{\text{th}}^{\text{inc}}$ exists such that w_in corresponds to the local maximum. As power increases from zero to $P_{\text{th}}^{\text{inc}}(w_{\text{in}})$, reorientation and self-focusing do not take place, and light propagates in the linear regime. When $P=P_{\text{th}}^{\text{inc}}(w_{\text{in}})$, OFT occurs and the beam starts to self-focus, eventually forming a nematicon with width depending on P according to the existence curve in Fig. 1(d). Now, when decreasing the input power, the beam width differs from w_in. Since the OFT depends on beam width [22], the director remains reoriented at powers lower than $P_{\text{th}}^{\text{inc}}(w_{\text{in}})$, as long as P > P_th. The feedback required for memory effects is the nonlocal NLC response, which stores in the director distribution the information on previous optical excitations. Thus, a hysteretic loop for $P_{\text{th}}<P<P_{\text{th}}^{\text{inc}}(w_{\text{in}})$ is predicted (Fig. 1(e)): the two coexisting stable states of the system correspond to a diffracting beam (lower branch, increasing power) and a nematicon (upper branch, decreasing power). The application of a bias lowers P_th (Fig. 1(f)). Finally, since the larger the input width the larger $P_{\text{th}}^{\text{inc}}(w_{\text{in}})$ is, the loop width widens as w_in increases (Fig. 1(f)).

3. Observation of bistability

The results shown in Sec. 2 assume (medium and beam) invariance along z, which is unrealistic in actual situations due to scattering losses, breathing and imperfections. Nonetheless, they provide a qualitative description of beam dynamics in actual experiments. First, the threshold power $P_{\text{th}}^{\text{inc}}$, monotonically increasing with w_in when z-invariance is invoked, shows a local minimum due to the interplay between the diffractive spreading -decreasing for larger w_in- and the peak intensity -decreasing with w_in-. Numerical simulations show that the power at OFT versus w_in initially diminishes, then has a local minimum when w_in is between 6 and 8μm and eventually grows indefinitely for further increases in w_in. Hence, in actual experiments, the hysteresis is maximized when w_in is as small as possible. Second, when self-focusing occurs, the input beam width does not change, as assumed in Sec. 2. Actually, the beam starts breathing (i.e. its size oscillates periodically) around the equilibrium state shown in Fig. 1(d) [16].

Experimentally, we launched a beam of waist ≈ 2μm at the input. For V = 0V nematicons beyond OFT were subject to temporal instabilities [20]. To circumvent this limitation, we biased the cell with a voltage at 1kHz: such additional electric torque helps molecular reorientation, lowering P_th and avoiding instabilities. At the same time, the loop width reduces as V increases (Fig. 1(f)). We found a satisfactory tradeoff for V = 0.92V and used it in all measurements.

The measured beam evolution in the plane yz is shown in Fig. 2. As predicted, the beam underwent a bistable behavior: when power was ramped up from zero, self-localization did not occur below 16.5mW, the latter corresponding to $P_{\text{th}}^{\text{inc}}$ in experiments. Starting from the self-confined state and decreasing power, the system conserved memory of the former state through optical reorientation, and self-trapping was maintained down to 14.5mW, corresponding to P_th.

 

Fig. 2 Observation of the hysteresis loop by imaging beam propagation in the plane yz. (a) The input power is 2mW, corresponding to standard diffraction; the power is increased (b and c) until OFT occurs above 16.5mW; then (d) a stable nematicon is formed. Starting from a stable nematicon, the power is ramped down (b’ and c’): the beam remains self-trapped for P > 14.5mW. Further decreases in power lead to linear diffraction (a) as in first half of the cycle. The beam width w normalized to the apparent input width w₀ [22] is plotted versus z for (e) P = 15.5mW and (f) P = 16.5mW; blue and red lines correspond to ramp-up (b–c) and ramp-down (b’–c’) halves of the cycle. Dashed and dotted-dashed lines are the beam widths for P = 2mW and 20mW, respectively. The observed states were stable over time intervals of the order of 30 minutes.

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The OFT is expected to depend on temperature [14]: Fig. 3 shows hysteresis loops versus temperature. The power threshold decreases as temperature increases, whereas the loop shrinks, nearly disappearing at room temperature. Such trend is confirmed by calculations accounting for the three different elastic constants and the temperature dependence of both refractive indices and elastic constants [22, 23]. Although the loop width is maximum at the lowest temperatures, side effects take place, including longer relaxation times reaching several minutes.

 

Fig. 3 Hysteresis of normalized beam width versus power in z = 930μm, at temperatures of (a) 16°, (b) 18° and (c) 23°C. Panel (b) corresponds to the experiment in Fig. 2. (d) Both the experimental (red stars) and the theoretical (black squares) data show that the loop shrinks at higher temperatures. The theoretical results are found with the z-invariant model [22] and effective widths 3.5 and 11μm for self-trapped and diffracting states, respectively.

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4. Conclusions

Using a Lagrangian approach for the theory and planar cells for the experiments with near-infrared light, we investigated the role of the optical Fréedericksz transition in the nonlinear propagation of finite-size optical beams. We predicted and observed a bistable behavior accompanied by hysteresis as power was ramped up and down, with stable states corresponding to diffracting and self-trapped beams. The role of power, bias voltage and temperature was addressed in detail and found consistent with simplified models. Our findings underline the role of nematic liquid crystals in the study of highly perturbed regimes in nonlinear optics and further illustrate the wealth of phenomena in nonlocal reorientational soft matter.