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Liquid crystals in the nematic phase exhibit substantial reorientation when the molecules are driven by electric fields of any frequencies. Exploiting such a response at optical frequencies, self-focusing supports transverse localization of light and the propagation of self-confined beams and waveguides, namely “nematicons”. Nematicons can guide other light signals and interact with inhomogeneities and other beams. Moreover, they can be effectively deviated by using the electro-optic response of the medium, leading to several strategies for voltage-controlled reconfiguration of light-induced guided-wave circuits and signal readdressing. Hereby, we outline the main features of nematicons and review the outstanding progress achieved in the last twelve years on beam self-trapping and electro-optic readdressing.

In the sixties, the invention of the laser made available optical fields of very high, formerly inaccessible, intensities, allowing the birth of nonlinear optics [

Spatial optical solitons (hereafter, simply solitons) have been largely investigated since the inception of nonlinear optics [

In this paper, we present an overview of spatial solitons in NLC, in short, nematicons, specifically addressing their routing by external electric fields,

Let us start by recalling the basics of linear light propagation in standard NLC. Most NLC are highly birefringent and belong to the class of positive uniaxials [_{o}_{⊥}, whereas extraordinary waves depend on a refractive index _{e}

We now turn to the nonlinear optical regime, where a specific model is required to describe the (dominant) light-matter interaction. Hereby, we address molecular reorientation in uniaxial NLC. As briefly discussed above, NLC reorientation results from the non-resonant molecular interaction with electromagnetic fields, at either low or optical frequencies, whereby the induced dipoles coplanar with the electric field tend to align their long axis (director) to the field vector (in the case of positive anisotropy _{||}_{⊥}), as sketched in

where _{||}_{⊥} is the dielectric anisotropy. In

Reorientation in NLC is strongly polarization-dependent. When the electric field of the light beam is extraordinarily polarized with field and optic axis non-orthogonal to one another, reorientation does not experience the Freedericksz threshold [^{2} is, the higher the extraordinary refractive index becomes. This nonlinear mechanism is the dominant one in undoped NLC subject to extraordinary-wave illumination, in the absence of the Freedericksz threshold and below the transition temperature [

Let us focus on a bell-shaped light beam propagating in positive uniaxial NLC with the electric field in the extraordinary polarization. At low power,

With reference to _{0} with respect to the

In the paraxial regime and for

where _{0} is the vacuum wavenumber, _{y}

with _{0}_{a}_{2}) and the nematicon trajectory (determined by _{0}) can be managed in NLC by controlling _{0}. _{2} on the rest angle _{0}, respectively: both walk-off and nonlinearity are maximized for values of _{0} slightly above _{a}_{2} and _{a}

_{0} ≈ 3 μm. The director at rest is homogeneously aligned at _{0} = 45º. A CCD (Charge Coupled Device) camera allows collecting of the scattered light from above the cell and observing the beam propagation in the plane,

Experimental observations with various NLC mixtures, carried out to underline the role of material parameters, confirmed the theoretical predictions: _{||}_{⊥} = 0.05_{⊥} ≈ 1.5; consistently, the corresponding measured walk-off angles were _{2} is, the lower the excitation needed for nematicon formation.

As mentioned in the Introduction, the control of nematicon trajectory is essential for their use as readdressable waveguides in reconfigurable interconnects and switching circuits for optical signal processing. Nematicons can be steered in direction by two main approaches: modifying the walk-off or introducing index gradients in the medium, respectively. The walk-off _{0}, the direction of propagation (Poynting vector) would change according to _{0} depending on the spatial coordinates), thus creating index gradients able to deflect the self-guided wave packet according to the refractive index landscape in which it propagates.

To give a mathematical basis to these physical statements, let us refer to

where the subscript, _{xy}

Nematicons are beams that propagate according to input wavevector and polarization, as well as to birefringent walk-off and medium inhomogeneities (if present). Therefore, with the exception of specific launch conditions in uniform samples, in general, their energy flow in the NLC volume has a three-dimensional character [

Let us consider the planar cell described in _{0} = 45º and

The achievable nematicon deflection in a planar cell can be increased in geometries that maximize the refractive index gradients: in this case, the steering is mainly due to wavevector variations and depends on both the interval spanned by _{e}_{1} and _{2}, in regions 1 (on the left in _{1} and _{2} in sign and/or magnitude.

We stress that, due to the adiabaticity of the graded director distribution, in this geometry, light propagates in the Mauguin limit with no power coupling between ordinary and extraordinary components [_{2}_{1}_{r}_{r}

The 3D motion of nematicons discussed in Section 3.1 is quite sensitive to the launch conditions, such as spurious wavevector tilt and misalignment with respect to the cell mid-plane. More reliable control of the nematicon trajectory can be obtained in geometries ensuring director rotation within the propagation plane. To this extent, the electrodes need to be designed so that the applied electric field reorients _{0} = _{0}(

_{0} = _{0} (_{0} = _{0}(

Analyzing the beam evolution in the NLC, one can notice that also the magnitude of the nonlinearity changes, with stronger self-focusing for a given excitation. In fact, by varying _{0}, the effective nonlinear coefficient obeys

A natural extension of what is discussed in Section 3.2 is the use of interdigitated electrodes to define in-plane dielectric interfaces, analogously to Section 3.1.2. ; an in-plane interface is expected to maximize the angular steering via wavevector changes through an index gradient. Let us examine the geometry in _{1}_{2} and _{1}_{2}, respectively, with the output nematicon position and angle depending on both the bias difference Δ_{1}_{2} and the absolute value _{1}_{2}

In the experiments, we launched a TEM_{00} beam of waist _{0} ≈ 3 μm from region 1 (_{1}. _{1} = 0, the refractive index in region 2 is higher than in region 1: the nematicon is refracted in region 2 going through the interface. Increasing _{2}, the angle of refraction increases (_{2} = 0, the angle of incidence changes with _{1}, due to walk-off; in this case, the Poynting vector after the transition can point towards negative _{1} = 0.9 V) displays an outgoing nematicon that propagates nearly parallel to the interface [_{1}

The overall deflection _{t}_{r}

In the previous sections, we reviewed configurations that permit us to control the nematicon direction by varying the amplitude of the applied bias _{c}_{0}Δ^{2}. In the nematic mixtures, named Double Frequency Liquid Crystals (DFLC), the crossover frequency lies in the lower range, 1–10 kHz [

To exploit this effect we designed a planar cell containing the DFLC MLC-2048, as drawn in _{c}

A change in the frequency of the applied electric field could affect both transverse confinement and trajectory of the nematicons, as both the nonlinear coefficient _{2} and the walk-off depend on the angle between the director and the beam wavevector (see _{MAX}_{MAX}

For _{c}_{c}_{c}_{2}, according to

We have shown how all-optical reorientation in nematic liquid crystals sustains self-focusing and the formation of stable optical spatial solitons, stressing that material and geometric properties affect the nematicon propagation, including trajectory and width. Based on the electro-optic response of NLC, we have reported and discussed various strategies to achieve and maximize the voltage-controlled addressing of nematicons,

While spatial optical solitons in nematic liquid crystals form an ideal platform for the development of a new generations of photonic guided-wave networks for all-optical signal processing, switching and routing, the wealth of the reported approaches for the controlled steering of nematicons lets us envision further developments in neighboring areas of optics. Among them, we like to mention their use as intense optical probes for the physical characterization (thermal, dielectric and elastic properties) of new mixtures of liquid crystals, including the complex interaction between NLC (host) and dopant (guest) molecules [

(_{e}

(_{0} defines the birefringent walk-off _{0} = _{0}) between the Poynting vector _{0} for various degrees of birefringence and _{⊥} = 1.5 (the arrows indicate increasing _{||}

(

Beam propagation in three (undoped) NLC mixtures. As birefringence goes up, walk-off increases, whereas self-confinement is appreciable at lower powers. The rightmost graphs plot linear (dashed line) and nonlinear (solid line, corresponding to the highest excitation) output beam profiles across

(

(

(

(_{2} (red solid line) and the nematicon breathing period Ω (blue stars)

(_{1} = 3 V and several _{2}. The trajectory of a soliton undergoing (_{2}_{1}, with _{1} = 10º ) and (_{1}_{2}, with _{2} = 10º ); in panels (_{2} are indicated next to each line. The interface is located in

(_{2} when _{1} = 0; Corresponding graphs of (_{1}, when _{2} = 0. Corresponding graphs of (

(_{1}_{2}, with _{1} (_{2}) vanishing when _{1}_{2} (_{1}_{2}); (

(_{c}

(_{0} is the initial waist)

We are grateful to M. Peccianti and R. Barboza for their contributions in early and later stages of this work, respectively. We also thank M. Kaczmarek and O. Buchnev for sample preparation.

The authors declare no conflict of interest.