Chapter 11: Power-Dependent Nematicon Self-Routing

Alessandro Alberucci, Armando Piccardi, AND Gaetano Assanto

Nonlinear Optics and OptoElectronics Lab, University ROMA TRE, Rome, Italy

11.1 Introduction

Most studies on reorientational nematicons were carried out in the limit of small nonlinear effects, an assumption often adopted in nonlinear optics [1]. Perturbational models in nonlinear optics are based on a nonlinear polarization much smaller than the linear counterpart , allowing the use of well-known techniques, for example, coupled-mode theory [2]. In the reorientational nonlinearity of nematic liquid crystals (NLC), perturbative approximations require light-induced variations in the director field to be negligible when compared with the molecular distribution in the absence of illumination. The first theoretical models on nematicons made extensive use of this hypothesis [3–5], making a drastic simplification of the problem. On the other hand, such models are inherently unable to describe a few phenomena expected on physical grounds, such as saturation of the nonlinearity and dependence of nematicon trajectory on power, the latter independent of excitation according to first-order perturbation theory. In particular, nematicon paths change with input power when the optical reorientation becomes comparable with the unperturbed director orientation angle, hence leading to an appreciable variation in soliton walk-off [6].

Nonlinear changes in walk-off are not the only potential cause of light self-steering in NLC. Other mechanisms are based on variations of the beam wave vector. Specifically, when suitable dopants are added to the NLC, the combination of a noncentrosymmetric response and beam nonparaxiality via a nonlinearity enhancement can give rise to a transverse force acting on the beam path [7]. Solitons in NLC can also self-deflect through interactions with the cell boundaries owing to the highly nonlocal response, breaking the translational symmetry, and inducing an index gradient responsible for bouncing off the interface(s) [8, 9].

Finally, the nonlinear Goos–Hänchen–like shift on total internal reflection (TIR) at graded interfaces makes nematicon trajectory dependent on power [10], and so does their interaction with finite-size linear inhomogeneities in director distribution [11], with a symmetry-breaking analogous to that mentioned earlier.

11.2 Nematicons: Governing Equations

Let us consider an NLC sample with molecular director lying in the plane yz owing to anchoring at the interfaces and to the excitation scheme; θ is the orientation angle of with . In this geometry, ordinary and extraordinary (e-) waves are decoupled provided that the spatial spectrum of the beam is not too extended in reciprocal k-space. With reference to the extraordinary wave component and indicating with a superscript “(b)” quantities computed in correspondence to the intensity peak, we name H_x the magnetic field along x and set , where A is the slowly varying envelope, k₀ is the vacuum wave number, is the carrier e-refractive index, and z is the direction of propagation. In the paraxial approximation and for a dielectric tensor that varies adiabatically across the transverse profile,¹ light propagation is governed by the Anisotropic NonLinear Schrödinger Equation (ANLSE) [12]

11.1

where we introduced the dielectric tensor dependent quantities (the square of the extraordinary refractive index), (the diffraction coefficient along y, in general different from one) and δ = arctan(ϵ_yz/ϵ_zz) (the walk-off angle); moreover, in Equation 11.1, we defined the perturbation of the extraordinary refractive index , containing both linear (elastic effects from boundaries or torque due to quasi-static electric or magnetic fields) and nonlinear (light-induced director reorientation or thermal effects) contributions, the latter responsible for self-focusing (defocusing).

In order to express the electric field corresponding to solutions of Equation 11.1, it is convenient to define the local reference system xts, obtained from xyz through a rotation by an angle δ^(b) around the x-axis; thus, and , being and the directions parallel to the electric field and the Poynting vector, respectively. We stress that the reference system xts can change in propagation: our treatment holds valid only if these variations are adiabatic. Using Maxwell's equations, the nonnegligible electric field components are

11.2

11.3

hence, for narrow beams, a longitudinal electric field component is present, with an odd (even) profile if H is even (odd), in agreement with the theory of nonparaxial optics developed by Lax, predicting the appearance of a longitudinal field as a first-order correction and longitudinal second-order derivatives at higher orders (neglected in Equation 11.1) [13–15]. It is worth mentioning that, in the limit , we get , with Z₀ the impedance of vacuum.

To complete our model of nonlinear optical propagation in NLC, an equation governing the material response under the action of light is still missing. For a reorientational nonlinearity, the equation stemming from the equilibrium between optical and elastic torques can be found by minimizing the NLC free energy [16, 17], yielding the Euler–Lagrange equation²

11.4

where is the interaction strength, with K the scalar Frank's elastic constant [16] and ϵ_a the optical anisotropy. We stress that the reorientational nonlinearity can only be focusing, both in positive and negative uniaxial NLC, because the optical torque acts in order to minimize the dielectric energy stored in the medium [16]. It is also apparent from Equation 11.4 that, for a given profile of the electromagnetic field, the reorientation depends on the normalized parameter γP, P being the beam power; furthermore, for NLC with equal birefringence ϵ_a, reorientational effects scale with the ratio P/K.

The system composed of Equations 11.1–11.4 rules the propagation of reorientational solitons in a rather general way: we can distinguish three different regimes of nematicon evolution, depending on the injected beam power. To this extent, we assume a uniform director distribution θ = θ₀ in the absence of light (i.e., when A = 0) and write θ = θ₀ + ψ when A ≠ 0, ψ representing the all-optical perturbation of the director. In the linear regime, that is, at very low powers, ψ is negligible in each term of Equation 11.1, including self-focusing; thus, the beam spreads according to the diffraction coefficient D_y, with energy flux at an angle with respect to (Fig. 11.1a). As power increases, the all-optical response changes, with beam self-trapping but no power-dependent trajectory: this is the nonlinear perturbative regime distinguished by ψ θ₀ (optical reorientation negligible with respect to the unperturbed director distribution), corresponding to first-order nonlinear perturbation (Fig. 11.1b). As power increases further, ψ becomes comparable with θ₀ and the all-optical term affects the trajectory of the self-confined beam: this is the nonperturbative nonlinear regime, where both beam waist and trajectory depend on light power (Fig. 11.1c). We emphasize that the ranges identifying the three regimes depend on θ₀, as discussed in the following sections.

11.2.1 Perturbative Regime

In this section, we address some general properties of nematicons in the perturbative nonlinear regime, identifying two scalar parameters able to measure nonlinearity and nonlinear walk-off as functions of the initial orientation θ₀.

In the perturbative limit, the nonlinear response is not high enough to sustain narrow nematicons where E_s becomes appreciable; as ψ θ₀, we can substitute θ → θ₀ in the leftover sinusoidal term of Equation 11.4, obtaining:

11.5

where we set . Equation 11.5 is a linear Poisson equation, linking the nonlinear perturbation ψ to the beam intensity (proportional to ), which can be solved by using the Green function formalism; incidentally, such feature demonstrates the equivalence between thermal and reorientational nonlinearities in NLC in this regime [8]. After setting , we get

11.6

with G the Green function, which takes different shapes according to the examined geometry. We stress that G rules the nonlocality: for an infinitely extended medium, the nonlinear perturbation ψ is infinitely extended because ψ resembles the electrostatic potential of a pointlike charge in empty space [18, 19]. In actual experiments, the sample size is obviously finite, implying a different G with extension defined by the closest boundaries [8, 9].

For small nonlinear perturbations we can linearize , obtaining (the perturbation is proportional to power, as expected), with and having defined an effective (nonlocal) Kerr coefficient [12]

11.7

Recalling the definition of γ and considering that tanδ₀∝ϵ_a for realistic values of the anisotropy, we find that the maximum of n₂ versus θ₀ is proportional to for any given K. Thus, the nonlinear behavior of an NLC mixture depends quadratically on ϵ_a, because of the combined effects of a stronger optical torque and a deeper index well for a fixed perturbation ψ, whereas it is inversely proportional to the elastic constant because larger K imply stronger molecular interactions counteracting the light-induced rotation. Figure 11.2a shows n₂ versus θ₀ for a fixed γ: all curves, each of them corresponding to a different birefringence ϵ_a, show a relative maximum occurring very close to θ₀ = 45^° for small anisotropies, moving toward higher θ₀ for larger ϵ_a. As expected from Equation 11.7, the walk-off tanδ₀ displays a similar behavior (Fig. 11.2b). Having addressed the nematicon waist with the parameter n₂, we define a scalar figure c_δ to describe the nematicon trajectory in the perturbative limit; it is c_δ = dδ^(b)/dP calculated in θ = θ₀. From Equation 11.6 it is straightforward to derive

11.8

where

11.9

Figure 11.2c shows c_δ and c_θ for the commercial NLC E7: the nonlinear variations in walk-off depend on the trade-off between the sensitivity of δ versus θ (parameter c_θ), with absolute maxima around θ₀ = 0^° or θ₀ = 90^°, and the growth rate of θ versus power P, maximum for θ ≈ 45^° and nearly zero around θ₀ = 0^° or θ₀ = 90^°. As a consequence, c_δ is positive for θ₀ < 48^°, with a relative maximum for θ₀ ≈ 26^°; it is negative for θ₀ > 48^° and undergoes a relative minimum for θ₀ ≈ 71^°.

11.2.2 Highly Nonlinear Regime

Let us consider Equation 11.4 assuming E_s = 0 for simplicity; hence, . For the sake of generality, we will take θ₀ to be a space-dependent function, as experimentally obtainable, for example, by defining nonuniform anchoring at the boundary (Section 5.3.2) or using patterned electrodes for the application of a low-frequency electric field (Section 5.4) [20]. When ψ becomes comparable with θ₀, it is no longer possible to neglect the all-optical perturbation in the sine term, that is, the integral solution Equation 11.6 can be recast as a nonlinear integral equation

11.10

Setting ϵU = |A|² (ϵ is a smallness parameter) and , a standard perturbation approach yields [8]

11.11

where we neglected all derivatives of δ with respect to θ. Equations 11.11 are an infinite set of Poisson-like equations in agreement with the Born approximation, with a forcing term depending on the perturbation U (equal to |A|² in the limit ϵ → 1) and on the lower-order solutions. At order ϵ⁰, we get back the equation governing the director distribution in the absence of light. At order ϵ¹, we retrieve Equation 11.5. Recalling that , in the highly nonlinear regime, the nonlinear effect ψ is no longer directly proportional to power P owing to saturation in reorientation (i.e., a complicate power series with alternating signs).

Let us now turn back to the case of a homogeneous θ₀ and discuss these results with reference to nematicons. In Section 11.2, we defined the transition from perturbative to highly nonlinear regimes referring to the soliton trajectory, that is, comparing nonlinear changes in walk-off with its linear value δ₀. From Equation 11.8 and Figure 11.2c, it is clear that the power P_tr corresponding to such transition depends on θ₀ and is inversely proportional to c_δ; hence, P_tr is maximum for θ₀ close to 45^°. Noteworthy, the higher-order terms in Equation 11.11 also affect the nematicon waist, inducing changes in the effective n₂. The net result consists in a saturation of the nonlinearity, implying a nonmonotonic trend of the nematicon existence curve in the plane waist-power [21], at variance with the perturbative theories predicting a soliton waist always decreasing with power [3, 5, 22].

11.2.3 Simplified (1+1)D Model in a Planar Cell

Hereafter, we focus on homogeneous planar cells, with the shortest dimension (thickness L) along x and parallel interfaces in x = 0 and x = L. The Green function defined in Equation 11.6 reads

11.12

where we set and L_n = L/(nπ); K₀ is the zeroth-order modified Bessel function of the second kind. In the framework of perturbative theories, for the sake of computational savings, in Equation 11.4 the second derivative along the Poynting vector can be neglected (this is rigorous for a soliton in bulk dielectric). This leads to several considerations: first, an a priori knowledge of the propagation direction is needed (an unknown variable in the highly nonlinear regime); second, the assumption of a zero longitudinal second derivative is equivalent to consider the medium local in this direction, an unphysical hypothesis because the Green function is radially symmetric in the plane yz (Eq. 11.12), with appreciable differences between the two models when the breathing period becomes very small; third, losses due to Rayleigh scattering are unavoidable in actual NLC samples, yielding a drop in beam power versus propagation and a weakening of the nonlinear effect, the latter resulting in a curved trajectory (via changes in δ^(b)) as well as longitudinal variations of the refractive index . Therefore, the longitudinal derivative needs to be included in the reorientational equation in order to account for all the properties of the original system (Equations 11.1 and 11.4), that is, nonlocality, self-steering and self-confinement, and their dependence on θ₀ [12].

Numerical simulations of the full 3D model composed by Equations 11.1 and 11.4 can be carried out with standard solvers relying on iterative algorithms; the magnetic field is computed by solving Equation 11.1 (to this extent, the splitting operator can be used by choosing the Crank–Nicolson algorithm for computing diffraction); the electric field components E_t and E_s are calculated from Equations 11.2 and 11.3; finally, the substitution of E_t and E_s into Equation 11.4 allows computing the director distribution (for instance, with an over relaxed Gauss–Seidel scheme), providing the dielectric parameters for the solution of Equation 11.1. In this procedure, the relaxation code in a 3D geometry is the bottleneck on computational time.

A simplified (1 + 1)D model can be obtained by writing the all-optical perturbation in a Fourier series along x

11.13

Substituting Equation 11.13 into Equation 11.4, neglecting E_s for simplicity (the generalization including the longitudinal field is straightforward) and considering a highly nonlocal response, we obtain

11.14

for every positive integer n, having exploited the orthogonality of the sine functions and having defined . Equation 11.14 is an infinite set of Yukawa equations, with the screening length in the nth equation given by L/(πn) [12].

By setting A(x, y, z) = X(x, z)u(y, z) (the maximum of X is one), we get , where . If we consider symmetric beams with respect to the cell midplane x = L/2 (, with |a| < L/2), light will propagate in this plane as an equal repulsion from the two parallel boundaries (placed in x = 0 and x = L) balances out (Section 11.6); moreover, will be nonzero only if n is odd; hence, we can set n = 2m + 1 (m = 0, 1, 2, … ). As shown in Figure 11.3a, for a bell-shaped beam is an alternating series and ψ_2m+1 from Equation 11.14 has the same trend. Equation 11.13 provides ; therefore, all terms on the right-hand side of the series have the same sign.

Let us now analyze the propagation Equation 11.1. In the highly nonlocal limit, integrating along x yields [12]

11.15

The next step is the introduction of an equivalent two-dimensional all-optical perturbation ψ_2D(y, z). Toward this goal, we set ψ_2D = ψ₁; thus, ψ_2D is underestimated with respect to the actual ψ and we have to compensate it by an effective two-dimensional power P_2D smaller than the actual power P. Defining the fit parameter σ via P = σP_2D, it is σ > 1; assuming equal amplitudes of all , we get a rough estimate for σ:

11.16

Figure 11.3b plots σ versus waist for a Gaussian beam. As expected, σ increases for narrower beams due to the spreading of I_n toward larger integers, as demonstrated in Figure 11.3a.

In summary, nonlinear beam propagation in NLC can be effectively modeled in (1 + 1)D by the system of Equation 11.15 and the reorientational equation

11.17

We point out that Equation 11.17 retains the degree of nonlocality of the full 3D structure. Conversely, removing the term ∂²ψ/∂²x in Equation 11.4 or 11.5 implies the disappearance of the screening term in Equation 11.17: in such case, the nonlocality range would be the cell size along y [9, 18], that is, a significant misestimate of the nonlinear index well, both in shape and size [8] (Fig. 11.3c).

11.3 Single-Hump Nematicon Profiles

In this section, we discuss light self-confinement in the absence of losses, with specific reference to single-hump nematicons; we solve the nonlinear eigenvalue problem for various powers and initial orientation θ₀, comparing the results of the full 3D model with the simplified 2D model.

11.3.1 (2 + 1)D Complete Model

We consider nematicons with phase fronts normal to , that is, solitons that can be excited by beams impinging normally on the input interface of the NLC sample. Shape-preserving nonlinear wave packets will take the form A = A_s(x, y − tanδ^(b)z)exp(ik₀n_NLz), and the corresponding optical perturbation is ψ = ψ_s(x, y − tanδ^(b)z), with n_NLk₀ the variation induced by self-phase modulation on the propagation constant and tanδ^(b) the nonlinear change in walk-off. The nonlinear eigenvalue problem has the form

11.18

11.19

To illustrate the main properties of nonlocal spatial solitons in NLC, the system of Equations 11.18 and 11.19 can be solved in the highly nonlocal limit, that is, when the nonlocal index well can be satisfactorily approximated by a parabola. In fact, setting and neglecting tan²δ^(b) with respect to 1, Equation 11.19 provides , with A₀ = A_s(x = L/2, y = 0). The beam profile is , where (P_s is the nematicon power), and the soliton existence curve in the plane power-waist takes the expression

11.20

which can be recast as .

Figure 11.4 shows solutions of system (11.18 and 11.19) for θ₀ = 45^° versus nematicon power. At very low power, the soliton waist is comparable with the sample thickness L; hence, the nonlocality is of the order of unity and the refractive index profile is asymmetric due to the distinct boundary conditions along x (finite side) and y (infinite side). As power increases, the nematicon gets narrower because of a higher nonlinear index well; when the waist becomes much smaller than L, the highly nonlocal regime can be invoked, as confirmed by the conservation of the ψ_s profile versus power. For still higher powers, the soliton waist increases because of the saturation in the reorientational nonlinearity as stated by Equation 11.20, the latter also responsible for changes in soliton walk-off. Finally, consistently with Section 11.2.2, the soliton behavior with power depends on the initial angle θ₀, which determines how soon saturation occurs and the angular span of soliton deflection.

11.3.2 (1 + 1)D Simplified Model

Hereby, we carry out a procedure similar to the one of the former Section 11.3.1 to compute the soliton profile versus power using the simplified (1 + 1)D model described in Section 11.2.3. From Equations 11.15 and 11.17, the nematicon in the midplane x = L/2 is governed by the eigenvalue problem

11.21

11.22

Figure 11.5a shows the computed nonlinear index well at three excitations: the nonlinear reorientation ψ_s conserves the shape of the complete 3D model (Fig. 11.4) thanks to the presence of the screening term , that is, the nonlocality is conserved even if the dimensionality has been reduced. The corresponding soliton profiles |u_s|² are graphed in Figure 11.5b: for powers larger than 0.25 mW, the shape is very similar to Gaussian, in agreement with the highly nonlocal approximation. We checked the shape invariance of the calculated profiles by means of a bidimensional anisotropic BPM code (see Section 11.4.1 for details); the results for beam profile and director perturbation are plotted in Figure 11.5c and 11.5d, respectively. Obviously, beam self-steering due to nonlinear changes in walk-off still occurs, but its dynamics are shifted to lower powers, consistently with Section 11.2.3 (Fig. 11.5e). Furthermore, the soliton existence curve in the plane waist-power retains its nonmonotonic behavior because of saturation in the nonlinearity, although the beam dynamics versus initial condition remains the same (Fig. 11.5f). Figure 11.5g plots the minimum soliton waist for a given θ₀, demonstrating that the best confinement takes place for small θ₀ as it stems from the highly perturbative regime discussed in Section 11.2.2.

11.4 Actual Experiments: Role of Losses

In Section 11.3, we computed nematicon shape and trajectory versus power and initial director angle θ₀, assuming no losses in the sample. In typical experiments, the input can be controlled in profile, power, and polarization, whereas the beam evolution in yz is monitored by collecting the out-scattered light. The latter is a substantial cause of attenuation even in undoped NLC; hence, it affects actual solitons and their properties, even though many of the properties discussed in Section 11.3 hold valid (e.g., dependence of self-focusing and self-steering on θ₀, nonlocality dependence on the geometry, etc.). In this section, we account for scattering losses in Equations 11.15 and 11.17, comparing the results with experimental measurements.

11.4.1 BPM (1 + 1)D Simulations

Figure 11.6 displays the numerical simulations of a Gaussian beam input with waist w₀ = 5 μm propagating in NLC with attenuation α = 5 cm⁻¹, the latter directly related to experimental measurements: for a given θ₀, the beam spreads at low power (diffraction dominates over self-focusing); for increasing power, self-effects become strong enough to counteract diffraction, going from self-focusing (i.e., the waist increases more gradually than in the linear regime) to self-confinement (i.e., the waist diminishes along z). Such dynamics take place at lower powers for initial orientation θ₀ close to 45^°, as predicted by Figure 11.2a. At variance with the exact soliton solutions studied in Section 11.3, the beam trajectory is no longer a straight line but bends because of power attenuation along z: in fact, the perturbation ψ decays exponentially along z, thereby reducing the nonlinear walk-off. Hence, for z 1/α, the beam propagates at δ₀ with respect to z, that is, the evolution goes from highly nonlinear to perturbative nonlinear regimes along z. Figure 11.7 graphs ψ corresponding to the beam in Figure 11.6. Such curves underline the differences arising in the highly nonlinear regime between the case θ₀ = π/4 − Δθ and θ₀ = π/4 + Δθ (0 < Δθ < π/4): in the perturbative regime, self-focusing would be of equal strength in the two cases (Fig. 11.2a), whereas at high powers, the nonlinear effects are much stronger for small θ₀ than for large θ₀ because of saturation when all-optical reorientation becomes comparable with the initial angle θ₀. This is visible in Figure 11.8, plotting waist versus z for several θ₀. As regards the trajectory, differences versus θ₀ are apparent even in the nonlinear perturbative regime, as the nonlinear walk-off variations are positive (negative) when θ₀ is smaller (larger) than 45^° (Fig. 11.9).

11.4.2 Experiments

We verified our theoretical and numerical results by injecting a 5 − μm waist TEM₀₀ beam at wavelength λ = 1064 nm in an NLC cell and observing its evolution by acquiring the image of out-scattered light (see Fig. 11.10). Two comb electrodes realized in Indium Tin Oxide are deposited on each glass–NLC interface, in order to set and tune by voltage V the angle θ₀ in the midplane x = L/2; each finger of the combs is much smaller than L, ensuring a constant director angle inside the cell (Chapter 5). To quantify the dependence between θ₀ and V, we measured the walk-off in the linear regime versus bias; the results are shown in Figure 11.11 (points), whereas the solid lines stem from the reorientational equation, neglecting electric field components along x and assuming an exponential decay in the applied field toward the center x = L/2 [12]. Figure 11.10 plots the acquired evolution profiles in the plane yz. As expected, at low power, the beam diffracts according to D_y, forming an angle δ₀ with respect to the z-axis. As the power increases, the beam undergoes self-focusing and eventually self-confinement (top row in Fig. 11.8); self-focusing depends on θ₀ and is maximum for θ₀ ≈ 45^°. Solitons have bent trajectories because of nonlinear changes in walk-off, with negative and positive curvatures for θ₀ < 45^° and θ₀ > 45^°, respectively, in agreement with the theoretical predictions (Fig. 11.9).

11.5 Nematicon Self-Steering in Dye-Doped NLC

In this section, we investigate nematicons and their self-steering in dye-doped NLC. Dyes consist of molecules absorbing in a specific spectral range, altering the optical nonlinearity via the so-called Janossy effect [23, 24]. Such effect is modeled by inserting a material-dependent enhancement factor η in the expression of the optical torque; in the reorientation equation, for η > 1, the all-optical response is amplified; thus, the power can be scaled down by a factor 1/η in order to achieve the same nonlinear effects³. In this case, we consider an E7 mixture doped with 1%

in weight of 1-amino-anthraquinone (1-AAQ) [23, 25]. The input interface (placed in z = 0) is such to induce planar director orientation: at rest, the director angle θ versus z goes from θ_in = 90^° to θ₀ = 45^°, the latter determined by anchoring at the two interfaces parallel to the plane yz; we also recall that the extension of the transition region along z is equal to the cell thickness L (in our case L ≈ 100 μm). Thus, we can write , being the director distribution in the absence of illumination.

Figure 11.12a–c shows the expected behavior on the basis of the standard Janossy effect: linear diffraction at low power, perturbative and highly nonlinear effects at higher powers, with the corresponding changes in beam waist and trajectory in analogy to the case of undoped NLC. The main difference between the doped and undoped NLC should be the lower power needed to observe nonlinear effects and the propagation losses due to dye absorption. We launched along z in the sample described above an extraordinarily polarized fundamental Gaussian beam at λ = 442.5 nm with a waist of about 5 μm. We name ϕ the angle of the Poynting vector with . The linear walk-off is nearly 9^° because of the higher NLC birefringence in the blue spectrum. Losses are measured to be α_dye ≈ 50 cm⁻¹ because of the dye. Photos of the beam evolution are displayed in Figure 11.12d: clearly, the path is not as anticipated based on standard effects. Even though self-confinement takes place above a low input power of about P₀ = 80 μW (i.e., η ≈ 50, consistently with values reported in literature [23]), the output slope of the self-confined waves ϕ_output (i.e., ϕ measured for z much larger than the absorption length) is not δ₀, as it should in the linear limit. In particular, as power is increased, the beam gets closer to the z-axis; when P₀ approaches 240 μW, we observe a thresholdlike effect with the sudden appearance of an ordinary component diffracting along z; at the same time, the extraordinary component remains self-confined but takes negative ϕ_output, the latter reaching its maximum ϕ_output = − 30^° for P₀ = 400 μW (Fig. 11.13a and c). Figure 11.12e shows photos of the red light (acquired through a red filter at 632.8 nm) photoemitted by the dye and eventually guided by the blue soliton; the photoluminescence is stronger corresponding to the extraordinary blue component forming a nematicon.

The above-described phenomena are in contrast with the hypothesis that self-bending is only caused by nonlinear changes in walk-off. In fact, they can be explained only if an effective force directed along negative y acts on the beam wave vector, thereby inducing the observed deflections even for large z. In other words, the light–matter coupling has to produce a symmetry breaking between and − , consistently with the power dependence of the deflection angle.

Thermal effects cannot explain the observations, given the lack of self-focusing on the ordinary component. A source of asymmetry could be the inherent anisotropy in reorientational dynamics that, for angles θ ≠ 0 and in conjunction with the boundary at the input, could misalign the beam intensity peak and the maximum of the nonlinear index well . Numerical simulations, however, confirm that, even if such effects took place, they could account for deflections up to 1^°, in contrast to the experimental data. A source of asymmetry is the optical field itself: in fact, as described by Equation 11.3, in the nonparaxial regime the longitudinal field E_s becomes relevant and exhibits an odd parity profile if the transverse field E_t is even, as in our case. Noteworthy, solitons in doped NLC are narrow owing to the enhancement η. As the presence of E_s alone is not sufficient to explain the observations, the material itself has to be sensitive to the sign of E_s, that is, needs to be noncentrosymmetric. We model such noncentrosymmetry by writing

11.23

with I the intensity and I_sat its saturation value. The first term on the right-hand side of Equation 11.23 accounts for saturation in the Janossy response at high peak intensities. The second term models the noncentrosymmetric response and we write Δη = c₀Im(E_se^−iβz), where β is the propagation constant of the self-confined wave and c₀ is found by fitting the data. In the highly nonlocal limit, the soliton profile can be assumed Gaussian, that is, , where w_b and y_b are soliton waist and position, respectively, both of them varying with z. Under such approximations and defining the power in each section z = const as P = P₀exp( − α_dyez), the beam evolution can be calculated by solving the ODE system [7, 26]

11.24

11.25

where w₀(P) is the soliton waist in the highly nonlocal and lossless (α_dye = 0) case for a power P, whereas σ and χ are two constants depending on the dielectric properties of the medium [7].

The second term on the right-hand side of Equation 11.25 represents the transverse force responsible for the wave vector deviation [7]. Figure 11.13b shows the calculated trajectories: owing to losses, the wave packet self-steers in the proximity of the input interface, with paths in good agreement with the model and consistent with the change in c₀ graphed in Figure 11.13d. c₀ undergoes a Freedericksz-like threshold, suggesting that a torque tilts the molecular director out of the plane yz, thus explaining the appearance of the ordinary component when the sudden deviation occurs in the beam. Note that, according to the Mauguin limit, coupling between extraordinary and ordinary components can only occur for fast variations in the director distribution as compared with the wavelength: nonadiabatic variations are permitted just near the input interface, at distances smaller than the thickness L.

11.6 Boundary Effects

From Section 11.2.3–11.5, we limited ourselves to the study of nematicons propagating in the cell midplane x = L/2. Here, we discuss, both experimentally and theoretically, nematicons launched away from this plane, addressing the role of a finite cell (along x) and the consequent break in translational symmetry.

Let us consider a planar cell with director homogeneously distributed in the absence of external excitations, lying on the plane yz and forming an angle θ₀ with . Assuming that the nonlinear index well conserves parity with respect to y (i.e., for an infinite sample along y), for small wave vector deviations, the application of the Ehrenfest's theorem to Equation 11.1 provides [27, 28]

11.26

where we set , , and , having defined the effective potential and . In the following discussion, we neglect nonlinear changes in walk-off in order to focus on the role played by the boundaries. In the highly nonlocal limit, in the series on the RHS of Equation 11.26, only the term for m = 0 survives, that is, the nematicon trajectory obeys geometric optics [29] (Chapter 5). To decouple the beam motion along x and in the plane yz, we exploit the separation of variables by setting |A|²/∫|A|²dxdy = φ_x(x, z)φ_y(y, z); thus, it is straightforward to get a simplified two-dimensional model for nematicon trajectory in the plane xz

11.27

where , being the equivalent 1D potential. In deriving Equation 11.27, we neglected the dependence of φ_y on z and dropped the dependence on , the latter determined only by linear walk-off δ₀ due to the y-invariance of the nonlinear index well.

To complete the model, we also need an expression for the nonlinear force W₀. To this extent, we use the perturbative expansion Equation 11.11 in conjunction with the Green formalism in a planar cell of finite thickness along x, as stated in Equation 11.12 for unbound geometries in yz. After the transformation y → y − ztanδ₀, neglecting the partial derivatives along the propagation coordinate in the walk-off reference system of Equation 11.4, the Green function reads:

11.28

To obtain a closed form, for the nematicon we take a Gaussian ansatz, consistent with the high nonlocality [22]

11.29

Computing the convolution integral (Eq. 11.6) employing the Green function (Eq. 11.28) provides for θ₁:

11.30

with

11.31

11.32

Equation 11.30 with Equations 11.31 and 11.32 represent a closed-form solution for invariant nematicons along z propagating in planar cells, in the limit of high nonlocality and the perturbative regime, for beams launched in x = L/2 with appropriate waist (Eq. 11.20). In the more general case, these equations model a self-induced index well depending on beam position via the coefficients 11.31, that is, indicating that the nonlinear potential loses its symmetry with respect to when beams are not injected in the midplane x = L/2. As a consequence, the gradient of the nonlinear index center is nonzero on the beam axis, yielding a nonzero equivalent force W₀ ≠ 0 on the wave packet. In the perturbative regime, we can set ; then, from Equation 11.30, we can compute V_eq:

11.33

where we introduced , having defined the function . Finally, W₀ reads

11.34

We stress once again that expression (11.34) was derived in the perturbative regime only for the sake of simplicity. The same derivation can be carried out considering the square terms in P stemming from and θ₂ [8]; it can be found that the shape of the self-induced well (i.e., the effective force W₀) is nearly unchanged owing to the high nonlocality, with the peak of the optical perturbation ψ changing by less than 10% for θ₀ = 45^° and P = 2 mW.

Force − pW₀ versus beam position is plotted in Figure 11.14: the effective potential acting on the nematicon is anharmonic and even with respect to , with the force increasing as the beam gets closer to one of the interfaces. Noteworthy, the sign of the force (negative for , positive or zero otherwise) is such as to repel out the beam from the boundaries, thus favoring light trapping within the NLC layer. The solutions of Equation 11.27 together with Equation 11.34 predict periodic oscillations in the nematicon trajectory; some examples are graphed in Figure 11.14b for a zero initial velocity (i.e., input beam with a zero wave vector component along x) and various input positions . The trajectory tends to a sinusoidal form in the case of small displacements from the midplane. Conversely, the oscillation period Λ decreases as the nematicon is launched further and further away from the equilibrium position x = L/2; furthermore, Λ depends on excitation because of its nonlinear origin, decreasing approximately by P^−1/2 (Fig. 11.14c) [8]. We numerically simulated the PDE system of the reorientation equation and the NLSE in a 3D geometry, neglecting the second-order derivatives along the Poynting vector. We rotated the reference system xyz by δ₀ around the x-axis to obtain xts. Figure 11.14d and 11.14e shows that nematicons undergo oscillations in good agreement with the theoretical predictions based on Equations 11.27 and 11.34.

A series of experiments were carried out to verify these findings, using an L_z = 4 − mm − long NLC cell of thickness L = 100 μm and width > 1 cm along y, with the commercial mixture E7 (Fig. 11.15a and 11.15b). The glass interfaces were treated to ensure θ₀ = 30^°. The soliton evolution in yz as well as the output profiles in xy (z = L_z) were imaged via CCD cameras. A small offset with respect to x = L/2 and an angular tilt ξ (in our experiments, and ξ ≈ 0.6^°, respectively) were impressed on the input wave vector to maximize the nematicon x-displacement versus power; the beam at 1.064 μm was linearly polarized parallel to to maximize coupling to the extraordinary wave. Clearly, the soliton is expected to interact with the boundary-driven potential and oscillate for a fraction of the period Λ, shifting along x in z = L_z as power changes. The photographs of the output profiles in yz are superimposed in Figure 11.15c for several input powers (from 0.5 to 6 mW), demonstrating the predicted power-dependent repulsion from the boundary. Such nonlinear transverse dynamics along x are in excellent agreement with the integration of Equation 11.27 (using Equation 11.34 for the force) considering the actual sample parameters and a nonzero initial velocity , as displayed in Figure 11.15d [27].

11.7 Nematicon Self-Steering Through Interaction with Linear Inhomogeneities

Till now, we addressed power self-steering of nematicons when they propagate in a homogeneous molecular dielectric. In this section, we discuss how dielectric inhomogeneities affect the dependence of the beam trajectory on power. We address the nonlinear Goos–Hänchen shift on TIR at an interface, that is, a power-dependent lateral beam shift at a graded interface between two nonlinear NLC regions. We also illustrate power-dependent nematicon self-ejection from a finite index well. In both cases, perturbations in director distribution are impressed via the application of suitable biases.

11.7.1 Interfaces: Goos–Hänchen Shift

Let us consider a nematicon propagating in a planar cell and encountering a dielectric interface in its path, as sketched in Figure 11.16a. We assume that the director orientation changes linearly from θ₁ for y < − d/2 to θ₂ for y > d/2. The transition width d has to be comparable with the shortest cell dimension L, in agreement with Equation 11.12. Figure 11.16b–d shows the results of BPM simulations, with nematicons surviving the interaction thanks to the adiabatic character of the dielectric perturbation as compared to the soliton waist. Self-confinement changes within the refractive barrier because the effective Kerr coefficient given by Equation 11.7 follows the variations in θ.

If the index contrast across the interface is high enough to provide TIR (for wave vectors along , the condition θ₁ > θ₂ is sufficient to ensure TIR), in the highly nonlinear regime, we expect the reflected beam path to depend on power because of nonlinear walk-off. In particular, for θ₂ < θ₁ < 45^°, the changes in walk-off are positive (if optical powers are low enough to ensure a director angle belonging to the domain where walk-off angle is monotonically growing with θ, see Fig. 11.2c); therefore, nematicons undergoing TIR move toward positive y, giving rise to a nonlinear Goos–Hänchen shift [10].

To verify the predictions, we performed experiments in a cell with two (top) electrodes separated by a 50 − μm gap and a common ground (bottom) electrode. Voltages V₁ and V₂ applied to these electrodes could lift the molecular director out of the plane yz by different amounts in the two NLC regions, defining a graded interface (Fig. 11.16e). The gap was realized in order to provide an incidence angle of 79.5^° with respect to the interface normal for beams launched with wave vector along z. Although this geometry is intrinsically 3D as walk-off acts also in xz, we can use our results to qualitatively explain the nematicon behavior: when the two biases are V₁ = 1.5V and V₂ = 0V, the director is lifted out of the plane yz only in the first region; hence, when the soliton propagates within the graded potential, the optical torque increases the walk-off and the self-confined beam penetrates deeper inside the second region, as shown in Figure 11.16f–h, with a larger lateral shift on TIR. We stress that, for a given position and transverse velocity, the force due to the index gradient depend slightly on power as we are in the highly nonlocal case (i.e., Equation 11.27 holds valid) [12]. Finally, the beam waist is much narrower than d; thus, the asymmetry in the nonlinear index well provoked by spatial variations of the effective Kerr effect can be neglected [30, 31].

11.7.2 Finite-Size Defects: Nematicon Self-Escape

After the interaction of a nematicon with an interface, we direct our attention to a localized defect in the dielectric tensor. We consider a perturbing region invariant along z and centered in y = 0, a nematicon with wave vector parallel to the symmetry axis and initial position in y = 0. If the coefficients c_δ (see Equation 11.8) corresponding to the unperturbed (i.e., in the absence of light) director distribution θ₀(y) are positive, we expect the force stemming from nonlinear walk-off to be able to “push” the nematicon out of the linear trapping potential, where no restoring forces are present; thereby, above a certain excitation, the nematicon should self-escape the trap [11, 32].

For the experiments carried out by Peccianti et al. [11], we employed the cell sketched in Figure 11.17a, similar to the one employed in Section 11.7.1 but with the two applied voltages V₁ and V₂ of opposite signs, in order to have the low-frequency electric field mainly in the plane yz and in the gap between the top electrodes [11]. This geometry allows us to consider molecular reorientation only in the plane yz. Figure 11.17b–d plots the experimental results: for excitations up to 4mW, the propagation in the perturbation regime allows the beam to remain trapped in the linear potential. As power is increased, the nematicon begins to bend toward positive y, until it eventually escapes the confining index well for powers larger than 13mW, as shown in Figure 11.17d.

11.8 Conclusions

In this chapter, we discussed nematicon properties in the highly nonlinear regime, the latter easily accessible in NLC because of their large reorientational response. When no losses are present, nematicon evolution strongly depends on initial conditions, as we discussed with reference to saturation effects and self-steering via nonlinear walk-off. We generalized our findings to include the role of scattering losses, unavoidable in the nematic phase, and compared the theoretical predictions with experimental results. We showed how nematicons can undergo nonlinear changes in trajectory because of interaction with the boundaries, nonparaxiality and asymmetry in light–matter interaction in guest–host dye-doped NLC, interaction with inhomogeneities in the dielectric tensor. These results point out several routes toward the realization of all-optical guided-wave signal processors, allowing the design of communication networks where light itself can define and modify the topology of the implemented circuits.

Acknowledgments

Special thanks to Marco Peccianti and Malgosia Kaczmarek for their important role in the work here discussed. A.A. thanks Regione Lazio for financial support.

Notes

¹ This implies that walk-off and diffraction can be assumed constant across the beam section; see Chapter 5 for details.

² For thermo-optic effects, we should resort to the heat transmission equation; see Chapter 9.

³ Given the resonant nature of the dye, the increase in losses has to be accounted for.

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