1. Introduction

Recently, a great deal of attention has been devoted to light propagation in optical lattices, owing to their relevance in both basic and applied optics. [1–9] Such periodic structures, in fact, offer fascinating research opportunities, including the investigation of dynamics characteristic of multi-level quantum systems, a subject of great physical importance as witnessed by the contemporary literature. [5–8, 11, 12] Among the various phenomena reported to date, including but not limited to linear and nonlinear Bloch oscillations, [5] excitation of higher Floquet-Bloch (FB) modes [10, 11] and accelerated lattice dynamics, [6–8, 12] Landau-Zener (LZ) tunneling is one of the frontier topics in optical lattices. [7,8, 12] This phenomenon was discussed in a famous 1932 paper, where C. Zener investigated the dynamics of a two-level quantum system with an interaction region (the transition region). [13] When an external force is applied to the system, a non-adiabatic crossing of energy levels can be established and the (two) corresponding eigenfunctions are able to connect, in spite of their distinct properties and features. [10] In optical lattices, Landau-Zener tunneling can occur between diverse FB bands, provided that a non-adiabatic acceleration is available (e.g. a refractive index gradient in the transition region). Light waves initially coupled to a specific band can therefore be transmitted to another band, [7, 8] hence modify such properties as the position of energy maxima and/or the direction of propagation. Nematic liquid crystals (NLC) are excellent materials for linear, nonlinear and applied optics. [14, 15] Being characterized by a mature chemistry and technology, they encompass a large birefringence (≥ 0.2) and a giant non resonant nonlinearity, orders of magnitude higher than in glass or standard semiconductors. Nematic liquid crystals consist of elongated rod-like molecules, aligned towards a mean direction in space described by the director field. [14] In the presence of an applied electric field (either static or high-frequency or optical), NLC molecules can alter their mean angular orientation through the (induced) dipole-field reaction and tune their refractive index. This field-driven reorientational process is at the basis of the electro-optic and all-optical response of NLC and entails fully tunable architectures. [14,15] Linear and nonlinear phenomena in periodic geometries, such as discrete diffraction, self-localized waves [16–18] and interplay between transverse motion and localization [19] have been previously reported in voltage-controlled NLC arrays of identical channel waveguides.

 

Fig. 1. Sketch of optically induced Landau-Zener tunneling: an intense pump beam, launched straight into the lattice (left), impresses a non-adiabatic acceleration thru an index decrease in the transition regions (right). A probe, initially coupled to an upper FB band, is accelerated in one of the transition regions and transfers energy to a lower band through LZ tunneling.

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In this paper we investigate Landau-Zener light tunneling by impressing an all-optical acceler-ation onto a one dimensional NLC lattice (Fig. 1). A wide, intense gaussian beam (the pump) is injected along the waveguides (Fig. 1), producing a nonlinear refractive index decrease and defining two transition regions (Fig. 1 right). Such all-optical acceleration can act on a second beam (the probe) (Fig. 1 left) which, initially consisting of light coupled to a specific FB spectral band, crosses FB levels and exchanges energy with a lower FB band. In order to transfer light from an upper to a lower band, a negative index shift (Fig. 1 right) hence a self-defocusing nonlinearity need be exploited. Interestingly enough, after the transfer to a lower band, probe light is able to propagate through the lattice at a transverse velocity higher than the maximum defined by its initial FB band, undergoing angular steering and spatial switching (see Sec. 3).

 

Fig. 2. Sketch of the NLC periodic lattice: (a) front view; (b) side view.

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2. Sample and NLC model

Figure 2 is a sketch of the NLC cell. A planar symmetric dielectric waveguide consists of a thin film (thickness d) of the nematic liquid crystal PCB (n _⊥ = 1.516, n _∥ = 1.681 at λ = 1.064μm) sandwiched between two BK7 (n=1.507) glass plates; the top plate is coated with an array of parallel strips of Indium-Tin-Oxide (ITO) electrodes, the bottom one is uniformly covered by a grounded ITO film. Such electrodes, via the application of an external low-frequency (1kHz) voltage, can define a periodic set of identical channel waveguides evanescently coupled to one another, hence make a one dimensional dielectric lattice of constant Λ. When an electric field distribution is applied thru a bias V, in fact, molecular reorientation takes place in the principal plane (x,z), thereby increasing the refractive index $n_e^2$ = $n_{\perp }^2$ + $n_a^2$sin² θ ₀ (with $n_a^2$ = $n_{\parallel }^2$-$n_{\perp }^2$ the NLC optical birefringence) experienced by an x–polarized guided mode and originating a periodic index modulation across the sample. [17] A complete nonlinear model of the liquid crystals consists of three coupled equations, as detailed below. The linear propagation of a weak (signal) beam, the probe, is ruled by:

being A_probe its electric field envelope. The steady-state distribution of the molecular director θ ₀(x,y) is described by: [14,15]

K being the NLC mean elastic constant, Δε_RF = ε₀ (ε₂₂₂₅ - ε_⊥ ) the low-frequency birefringence and E_x the x-component of the external field. A self-defocusing nonlinearity is provided by the thermo-optic response of the liquid crystal. The temperature distribution T = T ₀ + ΔT, with T ₀ the temperature of the bulk and ΔT its pump-induced variation, is described by:

where κ is the thermal conductivity of the NLC and αI_pump the heat generated per unit time and space by the pump-intensity I_pump - By working near the NLC transition temperature [14] and acting on the bias V in order to operate at a large director angle, [15] we could maximize the thermal response with respect to reorientation, making the former the dominant all-optical effect and the latter entirely negligible in our experiments (and model). The index change can be expressed as $n_e^2$(r;V;T) = n ₀(r;V;T ₀)+ Δn_e (r;V; ΔT), with $\Delta n_e=\left[\frac{\partial n_{\perp }^2}{\partial T}+\frac{\partial n_a^2}{\partial T}{\sin }^2\left({\theta }_0\right)\right]\mathrm{\Delta T}.$

3. Theory

To reduce the number of degrees of freedom of Eqs. (1)–(3), we start by writing the dielectric constant as ε(x,y) = ε₀(x)+ Δε(x,y), with a periodic modulation Δε(x,y) = Δε(x,y + Δ), and perform a factorization A_probe (x,y,z) = Δ(y,z)B(x)exp(iβ_x z) with B(x)exp(iβ _x z) the solution of (1) for ε = ε₀, [4] thus reducing Eq. (1) to the one-dimensional Schrödinger equation:

To solve Eq. (2) we set θ ₀(x,y) = θ_r + γ(x,y), i.e. a mean value θ_r (≈ π/4) and a small periodic modulation γ(x,y) = γ(x,y + Λ). By applying the method of strained parameters, assuming an applied field of the form $E_x=\sqrt{1+\sigma F\left(Y\right)}$ with σ ≪ 1 and F(Y) = F(Y + Λ), [20] we obtain:

being ${\gamma }_1={\sum }_m\frac{{\Lambda }^2{\xi }_m}{4{\pi }^2{m}^2}\exp \left(i2\pi \frac{m}{\Lambda }y\right).$ and $\Delta {\epsilon }_{\mathit{RF}}E\cos {\theta }_{r}F\left(y\right)=K{\sum }_m{\xi }_m\exp \left(i2\pi \frac{m}{\Lambda }y\right).$ To reduce Eq. (3), we observe that the medium non locality (represented by the Laplacian operator) has a different impact on x than on y, due to the strong asymmetry of the cell, much wider than thick. An input beam with an x-waist comparable with the cell thickness d does not undergo a strong non locality owing to the boundaries at a fixed temperature. Conversely, no boundaries are present along y and the temperature distribution is free to widen. Therefore, we can conveniently take a local response along x (∂²/∂x ² ≈ 0), factorize I(x,y) = I(x)I(y) and expand I(y) in the eigenfunctions ϕ_v = exp(ivy) of the homogeneous kernel ∂²ϕ_v/∂y ² = -v²ϕ_v :

where I(y) = ∫Ĩϕ_vdv. In the case of a small transition region (≈ Λ), we can further expand the thermal shift ΔT at first-order. By substituting Eq. (5)–(6) into (4), after some lengthy but otherwise straightforward calculations we obtain the following dimension-less Schrödinger-like equation:

with $V_p=u_{-1}\left(Y\right)\mathit{\delta Y}+u_{-1}\left(Y-\frac{V_0}{\delta }\right)\left(V_0-\mathit{Y\delta }\right)$, having set $y=\frac{1}{\omega }\sqrt{\frac{2\int {\mid B\mid }^2\mathrm{dx}}{{\mu }_{0}W}}Y$ and $z=\frac{2{\beta }_x\int {\mid B\mid }^2\mathrm{dx}}{{\omega }^2{\mu }_{0}W}Z$ with W = $n_a^2$cosθ_r ∫γ(x)∥B∣²dx, $\delta =\frac{i\Delta {\epsilon }_T{\alpha }^2\sqrt{2\int {\mid B\mid }^2\mathrm{dx}}}{\omega {\kappa }^2{\mu }_0^{\frac{1}{2}}{W}^{\frac{3}{2}}}\int \int I\left(x\right){\mid B\mid }^2\tilde{I}\left(v\right)v^{-1}\mathrm{dxdv},$, $V_0=\frac{\alpha }{\kappa }\Delta {\epsilon }_T\int \int I\left(x\right){\mid B\mid }^2\tilde{I}\left(v\right)v^{-2}\mathrm{dxdv},$, $\Delta {\epsilon }_T={\epsilon }_0\frac{\partial n_e^2}{\partial T},$, $A=\frac{\int {\mid B\mid }^2\mathrm{dx}}{{\omega }^2{\mu }_{0}W}\exp \left[\mathrm{iZ}{V}_p\left(Y\right)+\mathrm{iZ}\frac{\int \left(n_{\perp }^2-{\epsilon }_0(x\right){\mid B\mid }^2\mathrm{dx}}{W}\right]\psi$ and having indicated with u _-1(Y) the Heavyside function. We can then Fourier expand the periodic term V(Y) in a series V = Σ_nv _n cos(2nY) and the field ψ on a plane-wave basis ψ = a ₁(Z) exp(iKY) + a ₂ (Z) exp(-iKY). After substituting and projecting on Eq. (7), we finally obtain the original Zener model: [13]

 

Fig. 3. (a) Band-gap diagram corresponding to Eq. (7) with V_p = 0,V ₀ = 1 and δ = 0.5; (b) propagation of FB modes in band 1 at the maximum transverse velocity; (c) Zener tunneling between band 1 and band 2: in the transition region (dotted line) light is accelerated and transfers energy to band 2.

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where $a_1=b_1\exp \left[i(\frac{2v_0-1}{2}Z-\frac{{\delta }^2}{6}{Z}^3+\int \Theta \left(z\right)\mathrm{dZ}\right]$, $a_2=b_2\exp \left[i(\frac{2v_0-1}{2}Z-\frac{{\delta }^2}{6}{Z}^3-\int \Theta \left(Z\right)\mathrm{dZ}\right]$ and Θ(Z) = - δZ. Eqs (8–9) predict tunneling between bands at the characteristic exponential rate exp(-π${\mathrm{v}}_1^2$/4δ). To verify that tunneling has occurred in an optical lattice, one should monitor the location of the energy maxima of the excited FB modes. In each band, in fact, FB modes exhibit maxima in different spatial positions, hence a close inspection of the insensity distribution can reveal the tunneling process [7] Such approach, however, cannot be effectively pursued when the period Λ and the wavelength λ are comparable. Since our lattice is characterized by a small periodicity, we alternatively exploit the dispersion of the eigenvalue spectrum and monitor the transverse velocity of the signal before and after tunneling. Each Floquet-Bloch band, in fact, is characterized by a specific maximum in propagation angle, given by the normal in the band-gap spectrum (see Fig. 3(a), black arrows). As the band-number (-order) increases, such maximum increases as well and light transmitted to higher-order bands (lower β in the diagram) can travel with larger angles (with respect to z) than initially imposed by excitation. To elucidate this concept, we numerically integrated Eq. (7) for V(Y) = sin²(Y), V ₀ = 1 and δ = 0.5. A linear superposition of FB modes belonging to band 1, launched with the maximum transverse velocity (Fig. 3(b)), LZ-tunnels to band 2 as it travels through the accelerated region (Fig. 3(c), dotted line). Clearly, the angle of propagation increases beyond the maximum dictated by band 1 (Fig. 3(b)–(c)), unambiguously witnessing an LZ tunneling.

 

Fig. 4. Signal beam propagating in the cell for V = 1.19V: (a) discrete diffraction for P_probe = 1mW and (b) self-defocusing for P_probe = 6mW.

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Fig. 5. (1.012 KB). Movie displaying the optically driven LZ-tunneling between FB bands 1 and 2. Light is initially coupled to band 1 with P_probe = 1mW,P_pump = 0mW and V = 1.19V at the maximum transverse velocity. As the pump (dotted line) is turned on (green line) and reaches P_pump = 25mW (red line), the probe is accelerated and LZ-tunnels to band 2.

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4. Experimental results

Samples with Λ = 4μm and d = 6μm were designed and realized with the nematic PCB (5CB). [21] We employed two incoherent laser beams of λ = 1.064μm and acquired images of the light scattered from the (y,z) plane with a microscope and a high resolution CCD camera. The pump was mechanically modulated and the CCD synchronized in order to acquire images of the (cw) probe only when I_pump = 0. The response of NLC is slow enough to permit the use of a standard chopper to implement this temporal separation. To characterize the nonlinear response, we performed a first series of experiments injecting just the probe in a single channel of the array. As its power was raised from P_probe = 1mW to 6mW, the refractive change reduced the transverse modulation causing a wider spreading of the beam in the plane (y,z) (Fig. 4(a)–(b)), demonstrating self-defocusing due to the dominant thermal response. Landau-Zener tunneling was then implemented by launching an intense pump in order to lower the NLC refractive index and induce transition regions around the accelerated portion of the array [21]. The pump was a gaussian beam of y-waist ω _y = 15μm, with a diffraction length of about 900μm. The latter condition prevents any overlap with the probe after tunneling. A clear demonstration of all-optical LZ tunneling is visible in the photo sequence displayed in Fig. 5, showing the linear propagation of a signal beam (y-waist ω_y = 1.5 Λ, power P_probe = 1mW) in the presence of the pump (dotted line) with ω_y = 3Λ, 0 ≤ P_pump ≤ 30mW). Light, initially coupled to band 1 at the maximum transverse velocity (for P_pump = 0), discretely diffracts owing to evanescent coupling. Once the pump is turned on (Fig. 5 dotted green line) no changes are appreciable until its power reaches P_pump = 25mW (Fig. 5 dotted red line). Beyond this value, the nonlinear acceleration causes the signal to LZ-tunnel to band 2 and propagate at a larger angle in the observation plane. Such visible increment over the maximum imposed by the initial band (Fig. 5) unambiguously demonstrates that the probe has changed state, tunneling to a higher-order band in the spectrum. The LC transition region is smoother than the employed first order potential step (Fig. 3(b)), hence it reduces reflections. As apparent in Fig. 5, the tunneling rate is quite high as the residual light in band 1 can be hardly distinguished from the noisy background.

5. Conclusions

In conclusion, for the first time we have investigated all-optical Landau-Zener tunneling in a one dimensional array of liquid crystalline waveguides. We derived the original Zener model stemming from the equations ruling NLC in the thermo-optical regime. The experimental results are in agreement with both the theoretical analysis and the numerical simulations, demonstrating a novel approach to FB interband tunneling and all-optical switching/steering.

We thank M. A. Karpierz (Warsaw University of Technology) for providing the samples.