Magnetic routing of light-induced waveguides

Among photofunctional materials that can be employed to control the propagation of light by modifying their properties, soft dielectrics such as nematic liquid crystals (NLCs) stand out for their large all-optical response. Through reorientation, the molecular distribution of NLCs can be modified by the electric field of light, permitting functional operations and supporting self-localized light beams or spatial optical solitons. To date, the generation and routing of such solitons have been limited by the boundary conditions employed to tailor the properties of NLCs in planar cells or capillaries. Here we report on spatial solitons in bulk NLCs with no lateral anchoring, where the application of an external magnetic field effectively controls the direction of propagation and the angular steering of the self-trapped wavepackets. Our results entail a completely new approach to the routing of self-localized beams and light-induced waveguides in three dimensions, without the usual limitations imposed by transverse boundary conditions.

T he main interest in practical applications of optical spatial solitons is caused by their ability to serve as light-induced channels for optical information and, therefore, as basic elements for all-optical signal processing [1][2][3] . The latter requires soliton waveguides to be reconfigurable in space by modifying the soliton trajectories through external stimuli. Developing efficient strategies to achieve such control is one of the challenges in nonlinear optics 4 . To date, the control of soliton trajectories has been achieved only over small steering angles in bulk solid-state media, such as photorefractive crystals 5,6 , or in specific planar geometries in soft matter, such as liquid crystals [7][8][9][10][11][12][13] . Photorefractive crystals allow for varying soliton path, but are rather expensive, require usage of high voltages and have relatively long relaxation times 5,6,14 . Conversely, liquid crystals are affordable and efficient because of their unique sensitivity to light-matter interactions [15][16][17] . The key to such sensitivity is the organization of nematic liquid crystal (NLC) anisotropic molecules, providing a reorientational response to the electric field of incident light 16 . The ensuing nonlinearity and the formation of spatial solitons normally require appropriate molecular alignment at the boundaries, with a well-defined orientation of the optical axis 8,15 . Hence, the steering of spatial solitons in NLCs, so-called nematicons 7 , has been limited by cell geometries, such as hollow channels 18 or planar cells with an NLC layer, typically 0.1 mm or less in thickness 8,19 . The most notable experimental results in nematicon steering have been reported in planar configurations 9,[20][21][22] , whereby molecular orientation is defined by anchoring on the cell glass/NLC interfaces 19 . In planar cells the NLC alignment can be modified by applying a low-frequency voltage across the layer by means of thin film or solid electrodes 8,9,23 , by changing the beam power 13,24 , by using additional beams 25,26 or by photoalignment layers 10,27,28 . However, fully three-dimensional (3D) soliton dynamics is hampered by the dimensional restrictions of these planar structures 29 , as beam trajectories can only vary within the NLC region about the mid-plane of the cell, with angular steering not exceeding fractions of degree orthogonally to the propagation plane parallel to the interfaces 8,9 . In this paper we demonstrate three-dimensional steering of self-trapped beam paths by employing bulk NLC samples without lateral boundary conditions and a magnetic field to modify their molecular orientation. In achieving this goal, we manage to exploit magnetic molecular reorientation 16 while preserving the nonlinear optical response of the material, introducing magneto-optic control of spatial solitons, realizing 3D soliton routing without limitations of voltage-controlled planar schemes, and hence paving the way to a whole new family of signal routers and processors.

Results
Operation with external magnetic field. The experimental geometry is sketched in Fig. 1. Unlike in earlier reports 7,9 , in the adopted configuration (see Methods) the sample was not subjected to boundary conditions on the sides (orthogonally to x and y), but exclusively at input and output facets orthogonal to the direction of propagation z of the incident light beam.
As we discuss below, the orientation of the NLC molecular director n (that is, the optical axis of the corresponding uniaxial medium) with respect to z can be adjusted in the bulk by applying an external magnetic field 16 . Throughout this work we assume that the magnetic field is sufficiently strong to completely reorient the molecular axes along the vector B in the absence of optical excitations. We consider a uniform field strength and, as a result, a homogeneous distribution of n in the sample. Although some line disclinations of the director may occur when rotating the field B with respect to z in the proximity of input and output interfaces, these are not expected to affect the bulk optical properties of NLCs. In our experiments we used cylindrical permanent magnets (20 mm diameter) placed at a distance of 10 mm from the entrance point of the beam. Considering that the magnet size is relatively big in comparison with the separation between the magnet and the beam path, the magnetic field could be assumed uniform at the propagation distance of the beam in the bulk NLC (limited to d ¼ 1 mm in our sample). The magnetic field strength measured at this distance was B ¼ 0.2 T.
Soliton formation. Let us consider a monochromatic Gaussian beam of wavelength l and a given direction of propagation with wavevector k along the z axis. At optical frequencies, NLCs behave as a typical birefringent uniaxial medium with positive dielectric anisotropy De ¼ n 2 8 À n 2 > , and refractive indices n > and n 8 (n > on 8 ) for electric fields of light perpendicular or parallel to the molecular director n, respectively. With proper anchoring at input and output interfaces to ensure a homeotropic orientation, the initial orientation of the NLC director n 0 (x,y,z), in the absence of a magnetic field and light, is set at the background value y b (x,y,z) ¼ 0 with respect to the z axis. Denoting by y m the polar angle between z and the applied magnetic field vector and assuming complete magnetic reorientation, the resulting optical axis n takes a new background orientation at y m . Hence, the light beam splits into two independent linearly polarized components, namely an ordinary (o) wave beam with electric field E o orthogonal to the plane kn and an extraordinary (e) wave with the electric field E e coplanar with k and n.
While NLCs are insensitive to the ordinary o-polarized beam below the so-called Freedericks transition 16 , the electric field E of the extraordinary e-polarized beam exerts an optical torque on the highly polarizable NLC molecules, which tends to align the director along the electric field vector. The torque tilts the director by a small (in the weak nonlinear regime) angle c in the region around the beam axis (at peak intensity), and the local director orientation becomes y ¼ y m þ c, with y m the uniform distribution of the director in the absence of optical stimuli. The extraordinary index n e ðyÞ ¼ ðcos 2 y=n 2 ? þ sin 2 y=n 2 jj Þ À 1=2 increases in the region surrounding and containing the beam, inducing a graded-index waveguide and allowing for self-confinement 7,18,19 . As the trapped beam propagates with wavevector k e ¼ n e ðyÞk 0ẑ , the Poynting vector tilts at the walk-off angle d ¼ arctan De sin 2y= De þ 2n 2 ? þ De cos 2y Â Ã À Á with respect to k e due to birefringence.
Soliton routing. The propagation direction of the e-polarized soliton-forming beam in our geometry is determined by the spatial orientation of the stationary magnetic field B. An arbitrary fixed direction of the magnetic vector defines both the director n and the walk-off orientation, with respect to chosen coordinates. While the z axis is always orthogonal to the entrance interface of the NLC and k jjẑ, the axes x and y with corresponding unit vectorsx andŷ can be chosen arbitrarily in the plane z ¼ const.
Let us assume that the B vector has fixed polar orientation with an angle y m and is initially coplanar with the xz plane, so B ¼ B sin y mx þ B cos y mẑ , as shown in Fig. 1c. After rotating by an azimuthal angle j around the z axis the magnetic field becomes coplanar with the new plane x 1 z: B ¼ B sin y mx1 þ B cos y mẑ in a new coordinate basisx 1 ¼x cos j þŷ sin j;ŷ 1 ¼ŷ cos j Àx sin j. As the molecular director n together with the electric field of extraordinary beam E e follow the plane zB, the trajectory of spatial soliton lies in the planes yz and y 1 z before and after the rotation, respectively. In our experimental conditions the effect of optical torque on the polar director orientation y is significantly small (co oy m ) in comparison with the magneto-induced torque as we are in the weak nonlinear regime 30 . Hence, the resulting walk-off angle d is practically determined by y m : In this equation models well our experimental observations of magnetic-field control of spatial solitons in bulk liquid crystals with no lateral boundary conditions. The model neglects the interplay between all-optical and magnetic reorientations, while describing the walk-off of a bell-shaped beam on axis, that is, in the framework of the highly nonlocal approximation 7,30-32 . In other words, we take a linearized superposition of all-optical and magnetic-driven orientations, analogous to the nonlinear optical response in the presence of an external voltage bias 7,19 .

Discussion
We used circularly polarized beams (l ¼ 800 nm) yielding o-and e-components of equal powers in the NLC mixture 6CHBT (n > ¼ 1.5144 and n 8 ¼ 1.6714 at 20°C), independently of y.
While the electric field E e of the e-polarized beam tends to align NLC molecules along its own direction, the magnetic field determines the background alignment; the stronger the B, the higher the beam intensity/power required to produce an index change Dn 2 e for self-trapping. Hence, equation (1) is valid as long as the external magnetic field effects are comparable with those due to the electric field of light. In this perturbative regime, we observed diffractionless propagation of B6 mm diameter solitons over millimetre distances, at powers from a few to tens of mW depending on the magnetic field direction. Figure 2 shows the experimental results on soliton generation and routing. Beam trajectories (Fig. 2a-d) and output intensity profiles (Fig. 2f) are reported for a circularly polarized fundamental Gaussian input. When the magnetic field B is applied parallel to the input beam wavevector (y m ¼ 0) or is absent, the low-power beam (P ¼ 7 mW) rapidly diffracts (Fig. 2a). When the magnetic field is tilted with respect to the z axis (y m a0), the optical wavepacket splits into ordinary and extraordinary-wave components. The former still propagates along the original direction (z axis) experiencing diffraction, while the latter propagates at the walk-off angle d and forms a spatial soliton via reorientation-induced self-focusing. The evolution of the self-localized beam in our geometry depends on the direction of the magnetic field. As the magnetic field rotates by an angle j around the z axis, the beam trajectory follows this rotation and remains coplanar with them in the plane zB (see Fig. 1c). Figure 2b-d,f shows soliton trajectories for y m ¼ 45°with j ¼ 90°, 0°, 180°, 270°in the planes yz (side view) and xy (output), respectively. Here the circularly polarized input beam had a total power of 7 mW, the e-wave soliton B3.5 mW with its polarization in the plane zB, as only extraordinary waves could contribute to all-optical reorientation. Hence, we verified that the orientation of the magnetic field determined the principal plane for the e-wave component of the beam undergoing reorientational self-confinement.  Figure 3 illustrates the role of y m on the wavepacket walk-off and the soliton threshold power. We measured the walk-off by varying y m from À 90°to þ 90°in the plane yz and obtained results (Fig. 3b) in perfect agreement with equation (1).
In order to measure the soliton threshold power P th , the input beam was set to be an extraordinary-wave with electric field coplanar to the zB plane. The results are plotted in Fig. 3(c). The lowest experimental threshold was 1.9 ± 0.2 mW for a magnetic field oriented around y m ¼ 45°with respect to z. The experimental data follow closely the theoretical dependence of the soliton threshold on the orientation y m . The self-trapping threshold power, in fact, is approximately proportional to the inverse of the nonlinear index change 1 . The latter, in turn, depends on sin2y m (ref. 7), leading to P th p1/sin 2y m as indicated by the solid line in Fig. 3(c).
We also observed unstable self-confinement even for y m approaching 0°and 90°, but at much higher powers (440 mW). In both situations the beam experiences fluctuation of its trajectory near the z direction as well as temporal transitions between self-focusing and diffraction regimes due to the Freedericks transition threshold 16,33,34 or competition between reorientational and thermal responses through residual absorption 35,36 . While the self-confined beam for y m approaching 90°was observed in a linear state of polarization along the magnetic field, the self-confinement for y m ¼ 0°had a random polarization accompanying symmetry-breaking instabilities 34 . A detailed analysis of these unstable cases will be carried out in future work.
We also evaluated the time required by self-confined beams to relax to their original trajectories once the magnetic field was switched off, as illustrated in Fig. 4 for a 7 mW incident beam power. The plot in Fig. 4b shows an exponential decay d ¼ d 0 exp( À t/t) of the walk-off angle from its maximum value d 0 ¼ 5.8°at y m ¼ 50°.
The measured relaxation time t ¼ 11 min needs be compared with the theoretical estimation of director relaxation time 37 t r ¼ gd 2 /Kp 2 of a homeotropically aligned NLC. For cell thickness d ¼ 1 mm on the basis of the known rotation viscosity coefficient gE16 mPa s and elastic constants KB10 pN of the liquid crystal 6CHBT 38,39 , t r E2.7 min. The discrepancy can be attributed to the fact that the theory is based on long-range mean field potential 40 in thin (o100 mm) films of NLC. In our thick NLC sample (1 mm) geometry, the short-range intermolecular correlations apparently play a significant role in the molecular reorientations and the field turn-off relaxation becomes a slower process. The relaxation time is mostly determined by the director behaviour in the cell's middle layers, since the restoration torque, which returns molecular orientations to their equilibrium positions, mainly comes from the homeotropic anchoring boundaries once the magnetic field is switched off.
Nevertheless, the beam path dynamics mediated by the magnetic field is rather fast (Fig. 2e). As the applied magnetic field rapidly changes its orientation, the molecular director in the LC middle layers experiences magnetic torque, which results in an accelerated relaxation time. When the magnetic field direction was changed (inE1 s) from y m ¼ À 45°to y m ¼ 45°, the soliton trajectory adjusted within 18 s, a time delay much shorter than obtained by photo-orientation 27 and comparable with voltage-driven switching in planar geometries [7][8][9] .
In summary, employing a bulk NLC geometry with neither lateral anchoring nor external voltage, we demonstrated that optical spatial solitons can be generated and routed by applying a moderate magnetic field. The magnetic field in such a bulk configuration provides fully controllable 3D molecular  orientation, entailing extra flexibility when redirecting self-confined light beams and re-addressing all-optical waveguides in quasi-static layouts. The substantial freedom from lateral bounds and corresponding anchoring conditions unveils rich scenarios for soliton-based optical circuits and reconfigurable photonic interconnects with the aid, for example, of electrically driven coils.

Methods
Sample preparation. The planar cell used in our experiments consists of two parallel optical grade glass slides placed normal to the z axis, measuring 10 Â 10 mm across x and y: such dimensions along both the transverse coordinates prevented boundary effects. The slide surfaces (at the glass-NLC interfaces) were chemically treated to induce homeotropic molecular alignment, thus ensuring a homogeneous distribution with n parallel to the z axis in the absence of external stimuli. The slide treatment also avoided undesired beam depolarization. We employed the NLC mixture 6CHBT. Its thickness d ¼ 1 mm along z was defined by rectangular spacers, so that we could observe the beam output position and its profile despite scattering. The two leak-proof spacers were attached using ultraviolet curable glue. To seal the cell, two additional 100-mm-thick glass slides were attached to the other two edges and served as side-windows. A small aperture was left at the top edge to permit cell filling by capillarity.
Experimental set-up. We injected either circularly or linearly extraordinary-wave polarized Gaussian beams (fundamental mode) of wavelength l ¼ 800 nm and waist wE3 mm using a 0.45 numerical aperture dry microscope objective (20 Â ). The orthogonality between the input wavevector and the NLC-glass input interface was ensured within B0.1°by measuring back-reflection. The evolution of the beam trajectory in 3D was imaged through the observation side-windows with two optical microscopes, collecting the light scattered by the NLC out of the planes xz and yz, respectively. The output profile and position of the beam were imaged by a third optical microscope collecting light at the exit facet (plane xy). A cylindrical permanent neodymium magnet (10 mm thick and 20 mm in diameter), mounted on a six-dimensional mechanical stage, was used to control the molecular orientation. The magnet axis was always pointed at the beam entrance in order to avoid potential artefacts due to field inhomogeneities and edge effects. The magnetic field strength was constant (B ¼ 0.2 T) in the volume of the NLC. The beam trajectories in the NLC bulk were acquired for several launch positions in order to average out contributions from noise and imperfections at the entrance facet.
Data availability. The authors declare that the data supporting the findings of this study are available within the paper. All relevant additional data are available from the authors.