Main

Optical spatial solitons have been investigated in a variety of nonlinear materials, both in one- and two-dimensional geometries1,2,3. In a few cases, the problem of filaments reaching an interface has been tackled theoretically with reference to cubic6,7,8,9,10 and quadratic responses11, and experimental demonstrations have been reported in quadratically nonlinear media at grazing incidence12,13, in cubically nonlinear semiconductor waveguides with a prism electrode14 and in prism-shaped photovoltaic crystals15. The manner in which molecules reorient and arrange themselves in a nematic liquid crystal (NLC) gives rise to both a strong electro-optic response and a giant non-resonant optical nonlinearity16,17. The latter of these properties, in conjunction with the non-local nature of molecular ordering in NLCs, enables them to support stable, low-power two-dimensional solitons18,19,20,21, whereas the former allows distinct regions of a liquid crystalline structure to be independently tuned by the application of different voltages. This entails the definition of voltage-adjustable transitions between birefringent uniaxial media differing in refractive index, orientation and nonlinear response, which in turn offers a great deal of versatility in the study of light propagation and localization phenomena at and across interfaces.

NLCs consist of elongated rod-like molecules with refractive indices n|| and along and orthogonal to their main axis (or director n), respectively. Several liquid crystals in the nematic phase16 are positive uniaxial materials () and can exhibit a large birefringence and a giant molecular nonlinearity. An incident extraordinarily polarized optical beam, with electric field non-orthogonal to n, can interact with the induced dipoles and force the molecular reorientation. Such reorientational response enabled us to observe anisotropic spatial solitons in planar cells, the so-called nematicons18, as well as their interaction, periodic breathing, transverse dynamics and walk-off20,21,22. As reorientation can also be driven by a low-frequency electric field, for example, through a voltage applied across the thickness of the NLC cell16,17, linear, nonlinear and nonlocal properties of the medium can be externally adjusted23. As a result, the system is very adaptive, tunable and amenable to the exploration of various geometries, for example, defining specific regions and interfaces via tailored electrodes.

We examine a cell consisting of two parallel glass slides that confine a thick layer of NLCs. As shown in Fig. 1a, we take z as the direction normal to the cell input facet and x as the direction normal to the electrodes (that is, across the NLC thickness). Anchoring films at the cell boundaries and a voltage applied via transparent electrodes determine the director alignment in the bulk of the medium. As the top electrode is split in two, a different potential can be applied to each part with respect to the bottom ground plane (see Fig. 1a). These two regions of the liquid crystal cell are labelled as 1 and 2 and are ideally separated by the plane x p, with p being a straight line bisecting the ‘gap’ (of finite width G) at an angle Θ with respect to z in the plane y z. We consider the structure invariant along p and define a rotated framework x t p (see Fig. 1b) with . The director n is at an angle ξ above the plane t p and its azimuthal angle is ρ with respect to t, that is, n(x,t,p)≡n(sinξ,cosξcosρ,cosξsinρ). The reorientation induced by an externally applied low-frequency electric field E=(Ex,Et,0) is governed by the Poisson equation

with the elements of the dielectric tensor depending on the director projection onto the reference axes, that is, with i,j∈(x,t,p). Using a standard variational approach applied to the interaction energy between field and director (see Supplementary Information, Section S1), we derive:

K is a scalar elastic coefficient for splay, bend and twist of the NLC molecules16. Equation (1) can be integrated with boundary conditions ρ(x=0)=ρ(x=L)=ρ(V =0)=ρ0 and ξ(x=0)=ξ(x=L)=ξ(V =0)=ξ0 stemming from the anchoring at the interfaces, with L being the cell thickness.

Figure 1: The sample and its electro-optic response.
figure 1

a, Schematic diagram of the liquid crystalline cell with two regions. On the top side, two electrodes are separated by a straight gap of width G at an angle Θ with z. b, Unequal voltages V1 and V2 applied to regions 1 and 2, respectively, induce different reorientations ξ and ρ of the director n. As a result, these regions exhibit different extraordinary refractive indices ne1 and ne2, respectively. p and t are parallel and orthogonal to the gap line separating the electrodes, respectively. c–f, Calculated electrostatic and reorientational effects from equation (1) for V1=1.5 V and V2=0.7 V: low-frequency electric-field distribution in the plane x t (red is more intense) (c); reorientation ξ (d) and ρ (in degrees) versus x (e) and in the cell mid-plane (x=50 μm) (f), respectively, for ξ0=0.017π and ρ0=0.5π at V1=V2=0 V. Material parameters refer to liquid crystal E7 (see the Methods section).

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Figure 1c shows the calculated molecular reorientation for a cell with L=100 μm and pretilt ξ0=0.017π, with the liquid crystal director initially parallel to p (that is, ρ0=0.5π) and the electrode gap G=100 μm. For low-frequency (fbias=1 kHz) voltages V1=1.5 V and V2=0.7 V in phase at the electrodes, a maximum reorientation ξ(x=L/2)=0.3π is obtained in region 1, whereas the field in region 2 is below the reorientational (Freedericks) threshold17. This results in an index change for extraordinarily polarized beams propagating across p.

Owing to the significant NLC optical birefringence, a plane wave incident at the interface between regions 1 and 2 and extraordinarily (e) polarized (its electric field in the principal plane nk defined by the optical axis and the unit wavevector k/|k|) is expected to undergo refraction when V2>V1 (that is, refractive index ne1 smaller than refractive index ne2 for the same optical unit wavevector k/|k| in regions 1 and 2, respectively) and total internal reflection (TIR) when V1>V2 (ne1>ne2) for incidence angles larger than a critical value. The behaviour of a light filament encountering such an interface is truly intriguing as, to maintain its self-confinement by nonlinear reorientation, it must remain e-polarized on either refraction or reflection21.

We excited a nematicon by launching a 4.5 mW gaussian beam (λ=1,064 nm) into the NLC cell (ρ0=0.5π and ξ0=0.017π) with input wavevector k at an angle ΦI=0.44π (79∘) with t. Figure 2 shows images of the filament propagation in the plane y z for V2>V1 (Fig. 2a) and V1>V2 (Fig. 2b), respectively. Keeping V1=0.7 V and increasing V2 from 0.7 to 1.5 V, strong filament refraction occurred. The amount of refraction could be adjusted by varying ΔV =V1−V2 from 0 to −0.8 V across the gap G. Conversely, for V2=0.7 V and ΔV ≥0.5 V, we observed TIR of the self-confined beam (Fig. 2b). The latter is the first demonstration of TIR of a soliton waveguide at a nonlinear/nonlinear interface in cubically nonlinear media. Noticeably, the system could be continuously tuned from straight-on propagation to refraction or to reflection by simply varying ΔV. One striking feature of the nonlinear propagation of such a self-trapped filament through an interface is that its electric field remains e-polarized after refraction. In fact, the nematicon rotates its (extraordinary) electric-field vector to keep it in the principal plane n k, as the optical axis n adiabatically turns in the gap separating region 1 from 2 (see Supplementary Information, Section S2). Furthermore, even after TIR (V1>V2) the propagation of an ordinarily polarized field component is forbidden owing to momentum (wavevector) conservation along p (see Supplementary Information, Section S2). Hence, through polarization healing, all the power (ideally) remains bound with the e-wave soliton (Fig. 2b).

Figure 2: Intensity evolution of a 4.5 mW (input power) nematicon launched at ΦI=0.44π with respect to t in a sample with ρ0=0.5π and ξ0=0.017π.
figure 2

a, For ΔV =V1−V2≤0 V, the filament is refracted at the interface between regions 1 and 2. b, By making region 1 optically denser than region 2 (ΔV ≥0 V), the nematicon can undergo TIR for ΔV ≥0.5 V, as demonstrated in the bottom panels. For ΔV =0.2 V (b, upper right panel) the beam loses self-confinement and diffracts because nonlinearity in region 2 is minimized as the wavevector k approaches the director alignment n (that is, the e-wave turns into an ordinary ray).

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For the case of refraction (ΔV <0), acquired and recorded trajectories of light filaments are shown in Fig. 3a. The non-diffracting beam is angularly steered by as much as 18∘ by increasing the V2 voltage, which corresponds to a lateral shift Δy≈−210 μm after propagating for 1.4 mm. The latter is equivalent to more than 20 Rayleigh lengths of the linear beam.

Figure 3: Filament trajectories in the plane y z versus voltage difference ΔV between regions 1 and 2.
figure 3

a, Refraction of a nematicon. b, TIR of a nematicon. The dashed line represents the interface between regions 1 and 2. The initial trajectories near z=0 are affected by the director distortion at the cell input (see the Methods section).

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Figure 3b presents the filament trajectories for the case of TIR, which occurs for ΔV >0. For ΔV =0.8 V, mirror-like reflection is observed and the filament is deviated by as much as 22∘, corresponding to a measured lateral shift Δy≈200 μm after a propagation distance of 1.4 mm. For V1>1.5 V, the filaments were affected by significant transverse (x) dynamics, preventing them from propagating in the cell mid-plane through the gap (see Supplementary Information, Section S3). Whereas the measurements can only reveal the angular steering of the Poynting vector S (the energy flux) across the interface 1–2, the walk-off angle δ between S and the wavevector k depends on the angle θ between k and n (with S lying in the plane n k between n and k if )21. Therefore, with reference to the interface along p, the incidence angle ΦIS (with respect to t) of S differs from the incidence angle ΦI of the wavevector k by the apparent walk-off δy z (ref. 21), that is, the walk-off angle as visible in the observation plane y z (see Fig. 4). As an out-of-plane component of S corresponds to a filament up- or down-shift along x, such additional transverse dynamics affects the soliton x position across the interface and limits the bias mismatch |ΔV | to be used to observe TIR.

Figure 4: Asymmetric TIR in the presence of optical anisotropy.
figure 4

a, Photograph of a nematicon undergoing TIR for V1=1.5 V and V2=0 V. b, The Poynting vectors Sin and Sout of incident and reflected filaments differ in angle from their corresponding wavevectors kI and kR by the apparent walk-offs δy z_I and δy z_R, respectively.

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Refraction and reflection of a self-confined nematicon filament at a graded-index anisotropic interface between voltage-tunable nonlinear media encompass some remarkable physics. In the presence of optical anisotropy, the direction of transmitted and reflected rays depends on the orientation of the optical axis (see Supplementary Information, Section S2). In the geometry illustrated above (ρ0=0.5π), the angles (ΦIS, ΦTS, ΦRS) formed by the Poynting vectors of incident, transmitted and reflected filaments, respectively, are always larger than those (ΦI, ΦT, ΦR) of the corresponding wavevectors; as the director is aligned with p, for the reflected filament ΦR=ΦI and, therefore, ΦRS=ΦIS. It is worth noting, however, that for ρ0≠0.5π the walk-off of a nematicon can cause non-specular (ΦR≠ΦI) TIR, that is, the beam reflected at the interface (and eventually exiting a cell with parallel input and output facets) would not be the mirror-like reflection of the incident one. To this extent, let us consider the propagation of a filament in a cell with, for example, ρ0=0.28π and potentials V1=1.5 V and V2=0 V, respectively, as shown in Fig. 4. An e-wave filament is totally reflected with incidence and emergence angles ΦIS=79.0∘ and ΦRS=79.5∘, respectively. In such a case, ΦI=ΦIS+δy z_I, whereas ΦR=ΦRS−δy z_R, with δy z_I and δy z_R being the apparent walk-off angles associated with the incident and reflected beams, respectively. For nematicons propagating (at moderate ΔV) along the middle of the cell (the ‘mid-plane’ x=L/2), we estimated δy z_I and δy z_R and, correspondingly, ΦI=81.1∘ and ΦR=76.6∘, a net asymmetric reflection. However, as in general δy z_j(x,y,z)≥δy z_j(L/2,y,z) along the whole trajectory due to filament transverse dynamics across the thickness of the cell (out of the mid-plane), the actual difference ΦI−ΦR is generally larger than in our evaluation (see Supplementary Information, Section S2). Note that a small angular contribution to the angle of the reflected filament is due to the nonlinear response associated with filament propagation. The latter contribution, however, is much smaller than the anisotropic effect and below our experimental accuracy.

TIR is normally defined by the linear index mismatch between two media. Although it is understood that TIR of finite-size beams at the boundary between two nonlinear materials of different linear indices depends on excitation7,8, in our case the effective nonlinearity is higher in region 2 (ref. 23), hence filament-driven reorientation tends to reduce the index mismatch. Assuming the medium stratified along t with extraordinary index constant in each layer t=tj, the filament trajectory bends24 with an emerging (reflected) beam being displaced more and more along p as the input power increases. This nonlinear Goos–Hänchen shift25, typically limited to small values even at large (grazing) incidence angles, can be addressed experimentally with (2+1)D spatial solitons in our non-local voltage-tunable NLC sample (see Supplementary Information, Section S4). Figure 5 shows filament trajectories and nonlinear Goos–Hänchen shift in the sample of Fig. 4 (ρ0=0.28π), for ΦIS=79.5∘ and increasing optical excitations.

Figure 5: Nonlinear Goos–Hänchen shift under TIR.
figure 5

a–d, Photographs of filament trajectories for various optical excitations (input powers), namely P=1.60 (a), 2.67 (b), 4.40 (c) and 9.33 mW (d). The displacement of the reflected filament along p increases with power. The horizontal dashed line corresponds to the gap line p. e, Filament position along t at p=1.6 mm for input powers ranging from 0.05 (linear limit) to 9.33 mW.

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Our study of light filaments and their self-induced waveguides in nonlocal and nonlinear media revealed some very unique properties. First, light filaments can propagate across an interface, not only surviving changes in linear refractive properties, but also maintaining their self-confined character as nonlinearity and anisotropy vary. Second, by demonstrating such phenomena in NLCs and across a voltage-defined boundary, we achieved an unprecedented soliton steering from −18 to 22∘ using moderate voltages and milliwatt optical powers. Furthermore, by making the material optical anisotropy work to our advantage, we verified that optical filaments not only maintain the extraordinary polarization required for self-guiding, but they also experience voltage-adjustable as well as nonlinear Goos–Hänchen shift and non-specular reflection. Such findings bridge some of the gaps between various soliton-related phenomena of nonlinear physics in anisotropic media and unveil rich scenarios for the implementation of soliton-based optical circuits and reconfigurable photonic interconnects.

Methods

Here we present some details on sample preparation and experiments. Supplementary Information, Section S1 contains details on the model and its derivation.

Samples

For our experimental study, we constructed cells consisting of two glass slides coated with transparent indium tin oxide (ITO) electrodes for the application of low-frequency voltage biases, with an NLC inbetween (see Fig. 1a). Separate electrodes defining regions 1 and 2 were created on one glass substrate through standard lithography and chemical etching. On the second substrate (parallel to the first one, that is, in the plane x y), see Fig. 1, the original ITO electrode was kept uniform to serve as the ground plane. The ITO interfaces were polymer-coated with polyimide PI-2555 and mechanically rubbed at predefined angles with respect to t to impose the director alignment and anchoring (the cases 0.5π and 0.28π are addressed in the main text). A further glass slide, parallel to plane x y (see Fig. 1), was added to seal the edge of the cell and serve as a launch window. In this way, lens-like effects and depolarization of the input beams through an NLC meniscus could be avoided. The window was covered with polyimide and rubbed at 0.25π with respect to x and y to provide a planar director alignment at the input interface. The thickness of the NLC layer (that is, across x) was L=100 μm, as defined by mylar spacers. The cell was filled by capillary action with an NLC E7, with refractive indices n||=1.6954 and at λ=1,064 nm and Frank elastic constant K≈10−11 N.

Experiments

We injected extraordinarily polarized (fundamental order) gaussian beams of wavelength λ=1,064 nm and waist w≈3.7 μm using a ×20 microscope objective. Photographic images of light evolution in the plane y z (or p t) were acquired by means of a high-resolution charge-coupled-device camera and a microscope, collecting the light scattered by the NLC out of the plane y z. The images were filtered, digitized and colour-coded before reproduction. The beam trajectories were determined as the mean position of the transverse intensity distribution. The cells were thermally stabilized at 26 ∘C. All reported powers were measured in front of the sample, following the objective lens.