We review theoretical and experimental results on spatial solitons in nematic liquid crystalline cells, including two-dimensional solitons in bulk and discrete solitons in one-dimensional waveguide arrays. In bulk we describe the propagation of continuous solitons in the presence of adjustable walk-off, their interaction with light-induced defects and refraction-reflection at a voltage-tunable interface. In optical lattices we address the transition from discrete diffraction to localization, as well as all-optical beam steering.
©2007 Optical Society of America
Spatial optical solitons have become a standard topic in textbooks on nonlinear optics, inasmuch as light-induced waveguides seem to entail large potentials for applications in novel generations of re-addressable/reconfigurable networks.[1–4] In this context, nematic liquid crystals have offered an ideal workbench for the study of light localization, because they conjugate a giant molecular nonlinearity with a large electro-optic response, a mature technology and extended spectral transparency, allowing for the demonstration and the understanding of fundamental effects at relatively low powers. Being highly non-local anisotropic dielectrics, i.e. uniaxial media with a response which extends well beyond the electromagnetic excitation, nematic liquid crystals (NLC) not only support stable spatial solitons in two transverse dimensions (2D+1),[5–7] but they also allow to take full advantage of their inherent birefringence and walk-off to control the direction of energy flux, i. e. their Poynting vector, by acting on an external polarization (voltage) to reorient the constituent molecules. The latter electro-optic response is the key for definition and tuning of periodic waveguide arrays by the use of fingered electrodes, adjusting the degree of optical confinement as well as mutual (evanescent) coupling by way of an external voltage.
In this Paper, after a brief summary of material properties and basic equations describing molecular (re)orientation in §2, in §3 we summarize light propagation in a positive uniaxial medium and then, in §4, generation and properties of spatial solitons in the presence of voltage-adjustable walk-off as well as their interaction with light-induced lenses by an external beam or with a voltage-defined interface. Turning to discrete arrays, in § 5 we recall the basics of discrete light propagation in 1D lattices and, finally, in §6 we describe discrete diffraction, solitons and nonlinear beam steering in NLC.
2. Basic Physics
Nematic liquid crystals are aggregates of dielectrically anisotropic molecules in a fluid state; they tend to align to one another by elastic links, exhibiting a degree of orientational order. In such state the medium behaves as a birefringent uniaxial with optic axis along the mean molecular alignment, usually described by the director field n̂ (x,y,z). 
In general, the bulk alignment is perfectly compatible with neither the constraints imposed by the anchoring at the sample boundaries nor the presence of external fields (low-frequency or optical), the latter able to exert a torque through interaction with the induced-dipoles. In practical cases, the consequent alignment deformation occurs on scales larger than the molecular size and the Frank-Oseen elastic continuum theory applies.
Naming ∆ε LF = εLF ∥ -εLF ⊥ and ∆ε opt = n 2 ∥ -n 2 ⊥ the dielectric anisotropy at low (LF) and optical frequencies, respectively, and n ∥and n ⊥ the refractive indices for field components parallel and orthogonal to n̂, respectively, according to the Frank-Oseen model the Gibbs free energy density in a liquid crystal cell is:
The first three terms on the RHS are the energies associated with splay, twist and bend distortions of the molecules, with K 11, K 22, K 33 the elastic coefficients and
the field-matter interaction. For the latter we consider the action of a quasi-static electric field of RMS value ELF and an optical field of (complex) envelope A; g is a director-independent term. For positive uniaxials ∆εLF>0 and ∆εopt>0; energy minimization implies that the electric torque reorients the molecular director towards the field polarization while being counteracted by a restoring force (owing to the distortion energy). Such mechanism is responsible for an increase in the extraordinary-wave refractive index, i. e. a nonlinear optical response which is highly nonlocal, in the sense that the director distortion spreads from the excitation owing to the intermolecular links.
Defining the coordinate system xyz with n̂(x,y,z) ≡ [sinξ,cosξcosρ,cosξsinρ]T, being ξthe elevation of the director out of plane yz and ρ its azimuth, we model the director distortion via a standard variational approach in ξ and ρ, writing the Fréchet derivatives of the total energy Fd:[10–11]
A complete electromagnetic description typically implies coupling system (3) with i) the Poisson equation ∇∙(ε͇E LF) = 0 to take into account the LF-field distribution, being εij = ε⊥δij+εa ni nj the dielectric tensor elements with i,j ∊ (x,y,z), and with ii) an evolution equation for light propagation in this non-homogenous anisotropic medium. Although such description can be quite complex, it often reduces to a subset of geometry-specific equations. It is worth nothing that, for homogenously aligned NLC, in the limit of null reorientation for electric fields orthogonal to n̂, system (3) yields an eigenvalue problem with non-trivial solutions only above a threshold [10–11].
3. Optical waves in nematic liquid crystals
Most NLC are positive uniaxials (n ∥ > n ⊥) with large optical birefringence (n ∥ - n ⊥≥ 0.2).  In bulk, the plane eigenwave with wavevector k∥z, i. e. E=A exp(ikz), can have extraordinary (e) and ordinary (o) components with linear polarizations lying in the plane defined by n̂ and ẑ or normal to it, respectively. Since o-waves are orthogonal to n̂ , at low powers (below the optical Fréedericks transition) they do not induce molecular reorientation.
The allowed wavevectors can be represented by fixing an origin O on xyz and plotting k = k 0 n[k̂)k̂, with n(k̂) the refractive index for waves propagating along k̂. Taking θ0 as the angle between n̂ (director) and k̂, for o-waves n(k̂) = no = n ⊥ whereas for e-waves n(k̂) = ne = [cos(θ 0)/n ⊥]2 +[sin(θ 0)/n ∥]2)-1/2. Hence, vector k describes a two sheet-surface [Fig. 1(a)] known as the Inverse Surface of Wave Normals (ISWN).  In several cases the propagation geometry forces k to belong to a specific plane, hence the ISWN reduces to a pair of ellipses [Fig. 1(b)].
For a given wavenormal, the Poynting vector S is orthogonal to the ISWN. In particular, for an e-wave Se is at the walk-off angle δ = arctan[∆εopt sin 2θ 0 /(∆εopt + 2n 2 ⊥ + ∆εopt cos 2θ 0)] with respect to k e.
4. Spatial solitons
Bright spatial optical solitons result from the interplay between diffraction and a (self-focusing) nonlinearity. Under a simple Kerr dependence with index increase linear with optical intensity, (2D+1) solitons are unstable, resulting in filamentation and catastrophic beam collapse. Nonlocality is one of the stabilizing mechanisms able to prevent such collapse, as clearly ascertained by the observation of three-dimensional spatial solitons supported by molecular reorientation in NLC, namely Nematicons.  Weakly nonlocal solitons have been reported in photorefractives; [14–15] more recently, highly nonlocal thermo-optic solitons and their long range interactions have been observed in lead glass. 
4.1 Bulk Nematicons
In order to model nematicon propagation, let us consider an optical field propagating in a planar NLC cell of thickness h, homogenously polarized with LF-voltage V across the thickness such that ξ=ξ(V) and ρ=ρ 0, as in Fig. 2. The boundaries, with properly treated coatings, provide the planar anchoring of the director at angle ρ in the plane yz.
Considering the e-wave A = A t̂, in the perturbation regime we can write the angle θ between n̂ and ẑ as the sum of bias- (θ 0) and light-dependent (ψ) contributions, i.e. θ≈ θ 0(V) + ψ, with cos(θ 0) = cosξ(V)sinρ o[see Fig. 2(c)]. Considering a rotated frame rts [Fig. 2(c)] with ŝ parallel to the Poynting vector S e and r̂ = t̂Ś, using a multiscale approach we can describe the beam propagation by: [6–8]
In the single constant approximation Κ= Κ11≈Κ22≈Κ33,  system (3) can be simplified to:
with Γ = 2ε 0∆ε;LF(V/h)2[sin(2θ 0)/2θ 0-cos(2θ 0)]/sin2(ρ). The system (4-5) rules the field-matter interaction in NLC and supports the propagation of stable self-localized solutions, i. e. nematicons, for θ 0≠π/2.  A class of solution of (4-5) can be found analytically for the highly nonlocal regime, i. e. beam waists much smaller than the transverse size of the perturbation ψ. When such hypothesis applies (as it does in several experimental cases) the optical field experiences a parabolic refractive profile, ψ ≅ ψ0 + ψ 2 (t 2 + r 2). Looking for solitary solutions in the form, with Z0 the vacuum impedance, the integration of (4-5) yields a wide class of self-localized solutions, the simplest with Gaussian intensity profiles
with , c the speed of light in vacuum, P an arbitrary soliton power and (Dt/Dr)1/4 the ellipticity of the transverse profile. The (6) represent a family of unconditionally stable solitons  which propagate with Poynting vector along ŝ and wave vector gradually tilted towards ŝ as power increases.
In this geometry θ0 changes towards π/2 as the bias V increases, modifying the walk-off δ . Moreover, as the director reorients, both the e-plane n̂k̂ and the reference rts also rotate with V. This results in a bias-controlled Poynting versor Ś, with the consequent steering of self-trapped nematicons.
We carried out experiments in a cell filled with E7 (Κ 11=10-11N, Κ 22=1.2 + 10-11 N, Κ 33=1.95 + 10-11 N, n ∥≈1.7, n ⊥≈1.5, ∆ε LF=14.5, ε⊥ LF=5.1) and consisting of two glass slides spaced by 75 μm, holding the NLC by capillarity. Conductive indium-tin-oxide (ITO) electrodes on top and bottom slides enabled the application of a 1kHz bias, while rubbed polymer coatings induced the planar alignment (ξ(V=0V)=0) of the director with ρ 0=45°. A nematicon was excited with wave vector k parallel to z and Rayleigh length of ≈30 μm by a Gaussian beam from a Nd:YAG laser (λ=1064nm), using a 20x microscope objective. Figure 3 shows the soliton evolution as acquired by means of the out-of-plane scattering for various V. The apparent walk-off α, i.e. the walk-off from the observation plane, decreases with bias from 7 to 0°, demonstrating the intrinsic robustness of nematicons against changes in optical properties as well as propagation direction. 
It is important to pin-point that the excitation of an e-wave nematicon requires a beam polarization in the plane n̂k̂. However, when fixing the orientation of the director at the input interface, the injected e-field polarization remains in the e-plane as it rotates within the NLC bulk, provided the transition is adiabatic. In such conditions the injected e-beam self-heals and remains confined for arbitrary biases. 
4.2 Interaction of nematicons with light-induced defects
Nonlocal spatial solitons can non-destructively interact with localized perturbations of linear and/or nonlinear properties of the medium, keeping their self-localization. Therefore it is possible to manipulate a nematicon (and the signal it carries) by creating one or more defects close to its propagation path. [18–19] We implemented a simple all-optical readdressing scheme by exploiting lens-like perturbations induced by an external beam on the NLC . In Fig. 4 an x-polarized nematicon at λ=1064nm propagates along z in a planar aligned NLC cell with ρ0=0. While both the voltage bias and the soliton beam increase the extraordinary index, a z-polarized control beam transmitted through the cell thickness counteracts the latter and induces a defocusing lens-like defect, splitting the nematicon into an effective Y junction [Fig. 4(a)] or deflecting it [Fig. 4(b)] depending on the relative transverse position of soliton and lens axes. In the second case, for a given beam size, the steering angle α monotonically depends on the external control power PC, with a maximum of α=2.5° for PC=35 mW [Fig. 4(c)].
4.3 Nematicons at an interface
Self-localized beams can also interact with interfaces between different media. Since NLC molecular orientation can be controlled by a low-frequency electric field, linear, nonlinear and non-local properties can be adjusted. In particular, our planar geometry enables the definition of distinct regions across an interface defined by properly tailored electrodes.
We adopted a structure with the top electrode split into two (Fig. 5), forming a rectilinear gap of width G=100μm and two NLC regions, namely region 1 and 2. In the rotated framework xtp with p along the graded interface and t̂ = p̂×x̂, (re-)defining ρ in p̂t̂ such that n̂(x,t,p) = [sinξcosξcosρ,cosξsinρ]T, we can consider the geometry invariant with p. By applying unequal biases V1≠V2 to the two regions, system (3) predicts a smooth reorientation (ξ,ρ) change in the bulk NLC, i.e. an extraordinary index variation from region 1 to region 2. Owing to the e-index dependence on bias, an e-polarized wave propagating in region 1 and impinging the interface is expected to either undergo refraction or total internal reflection (TIR) above a critical angle for V 1 > V 2. In either cases non-locality enables nematicons to self-heal in polarization  and survive the interaction with the interface.
The two situations were experimentally demonstrated using a 4.5mW nematicon in a 100μm thick cell, incidence at an angle of 79° with respect to t, being ρ(V=0)=ρ0=π/2 and ξ (V=0)=ξ0=3°. Figure 6(a) shows the results for increasing V 2 with respect to V 1 (the latter kept close to the reorientation threshold), i. e. ∆V= V 1-V 2 ≤ 0: the soliton undergoes refraction as it crosses the interface, with angular steering as large as 18° for ∆V = - 0.8 V. In Fig. 6(b), TIR takes place for ∆V= V 1-V 2 ≥ 0.2 V and the nematicon emerges in region 1 with a deviation of 22° for ∆V = 0.8 V. As underlined above, in both cases the soliton maintains its e-polarization (hence confinement) through an adiabatic rotation; moreover, in TIR the propagation of o-waves is forbidden by (transverse) wavevector conservation along p. 
5. Optics in discrete systems
Ever since their introduction in late 80’s , discrete systems have stirred interest in optics due to the inherent physics and the possibility of applications. [9, 22–35] Discrete systems belong to the class of photonic crystals (PC), i.e. systems subject to a periodic index modulation along one, or more, spatial directions. At variance with PC, however, the index periodicity in discrete systems is comparable to (or even greater than) the wavelength of light and they effectively behave as arrays of identical waveguides. Optical discrete systems are also referred to as photonic lattices. Many of the interesting properties of such structures originate from discretized light-matter interactions occurring within a finite number of waveguides or quantum-units of the lattice. In analogy with molecular chains, photonic lattices allow to manipulate the propagation direction (i.e., group velocity) of light as well as its diffraction (i.e., effective mass). [21–29] Nonlinearity leads to even more fascinating dynamics, from discrete nonlinear waves to chaos. [21–25, 27, 31–34]
The success of discrete systems has been granted by the availability of a broad variety of materials and nonlinear mechanisms, including semiconductors, [26–27] glasses,  polymers,  photorefractives,  ferroelectrics  and liquid crystals [9, 33, 35] with either cubic or quadratic or molecular responses, respectively.
6. Discrete nonlinear optics in NLC
6.1 Optical lattices in nematic liquid crystals
To realize a periodic index modulation in NLC we exploited their electro-optic response by embedding a liquid crystal layer in an electromagnetic lattice [Fig. 7(a)]. The latter was obtained in a planar thin cell (see § 4) with a periodic set of fingered electrodes parallel to the propagation axis z. Via the application of a bias V, reorientation takes place in the principal plane xz thereby defining an array of identical channel waveguides, i.e. a one-dimensional photonic lattice supporting TM-polarized guided modes. For planar anchoring along z, the molecular director in plane x̂ẑ is conveniently described by the angle θ = ξ(ELF, A). To model such a medium, after calculating the steady state director distribution from system (3), i. e., a director field n̂ =[sinθ, 0,cosθ]T and neglecting bending effects (i. e., in the single constant approximation), a variational derivative applied to the free-energy density yields: 
being ELFx and Ax the x-components of the applied electric field ELF and the optical envelope A, respectively. The static field distribution ELF can be calculated from Maxwell divergence equation in terms of the potential V:
with ELF=-∇V. Finally, the refractive index experienced by a TM-polarized wave is . Equations (7–8), together with Maxwell equations, describe nonlinear light evolution in the NLC array.
Figure 7 shows sketch and photograph of a typical NLC array. We employed standard 5CB (Κ=3.2∙10-12N, n∥=1.5158, n⊥=1.6814 at λ,=1.064 μm), the physical properties of which are well known [10–11]. Top and bottom BK7 glass plates (ng= 1.50664 at λ=1.064 μm) provided planar anchoring along z, while ITO stripes of width ≈ 100nm on the top slide permitted the application of a periodic bias V(y) = V(y±∧) across the thickness h of the NLC layer.
The formation of a periodic lattice at a suitable voltage was ascertained by placing the sample between crossed polarizers under an optical microscope. Typical results are visible in Fig. 8. For V=0 V, for a sample with ∧= 8μm and d=6 μm, no modulation is appreciable; for V≥0.7 V reorientation induces an array of identical channel waveguides.
6.2 Discrete nematicons
Discrete nematicons are nondispersive waves originated when the nonlinear response of the NLC is able to balance discrete diffraction. Regardless the adopted mathematical approach [perturbative analysis, Lagrangian mechanics [22, 34] or numerical simulations from equations (7–8)], the formation of discrete solitons is well illustrated by coupled mode theory (CMT).  The optical lattice of identical (i. e., resonant) waveguides can be modeled as an N-core directional coupler (with N as large as needed), with light propagation described by the superposition of guided-modes of individual channels (for simplicity we assume each channel to be mono-mode). Light injected in the structure excites a set of waveguides and propagates by tunneling to neighboring sites, yielding a characteristic evolution pattern known as discrete diffraction (Fig. 9 at low power). As the input power increases, the self-focusing nonlinearity introduces a wavevector mismatch between launch and nearby waveguides, making light unable to resonantly tunnel across the array and trapping it within the initially excited channel(s). Eventually, a nonlinear wave or discrete soliton forms and propagates in the lattice (Fig. 9 at high power).
with ϕm being the modal amplitude evolving along a dimensionless coordinate ξ, in the m-th channel and κ a parameter describing the degree of nonlocality, the latter modeled by a peaked function with a Lorentzian spectrum. Equation (9) is non-integrable, i. e. only approximate solutions can be derived. Using a Lagrangian approach, the one-parameter family of solitary waves is:
with μ. being an arbitrary constant.
Figure 9(b) displays the generation of a discrete nematicon as computed by a nonlinear beam propagator in an NLC cell with ∧=h =6 μm and V=0.74 V.
Experiments were carried out in the near infrared (λ=1.064μm), in-coupling light with a microscope objective and collecting out-of-plane scattered photons with a high resolution CCD camera. Typical results for light evolution versus excitation are displayed in Fig. 10. In agreement with theory and simulations, as the power is raised the central waveguide is more and more mismatched from the remaining array until a discrete nematicon is formed.
6.3 Nonlinear beam steering
Quite interestingly, discrete solitons are forced to propagate straight, following the waveguide axis. As shown in Fig. 11, an input gaussian beam (wy =10μm), impinging with a tilt γ=λ/4 ∧= 1.90° along y on the waveguide array (V=8 μm, V=0.77 V), in the linear regime (P=PL=0.1 mW) forms a nearly non-diffracting beam “walking” across the array [Fig. 11(a)] as expected based on CMT and the periodic dispersion of a linear photonic lattice excited at the maximum transverse velocity.[29, 35] As the power increases, the input waveguide gets detuned until, at sufficiently high power (P=PH=2mW), traps light into a discrete soliton which can only propagate straight along the channel, as in Fig. 11(b).
We verified this nonlinear phenomenon, known as discrete beam steering,  in our samples with ∧=8μm and V=0.77V using a Gaussian input beam of waist wy =10μm at a tilt of 1.90° with respect to z. Figure 12 summarizes the results: light discretely diffracts for P=PL=1mW [Fig. 12(a)] and is localized as P=PH=7mW [Fig. 12(b)]. The intensity cross-sections in z=2mm clearly show nonlinear steering [Fig. 12(c)].
Nonlinearity, non locality, birefringence and electro-optics of nematic liquid crystals provide an excellent platform for spatial solitons, both in the continuous and discrete limits. Anisotropy entails the use of voltage to control walk-off as well as to define dielectric interfaces or periodic waveguide arrays, effectively coupling an external control to nonlinear optics and light localization in these structures. We anticipate a variety of specific light routing structures for the exploitation of these concepts in optical signal processing.
We are grateful to C. Umeton, M. Kaczmarek and M. Karpierz for providing the samples. Partial funding was provided by the Italian MIUR (PRIN 2005098337).
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