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We propose and numerically investigate a versatile and easy-to-realize configuration for a guided-wave voltage-tunable distributed feedback grating based on reorientation in nematic liquid crystal and coplanar comb electrodes. The device has a wide tuning range exceeding 100 nm and covers C and L bands for wavelength division multiplexing.
©2010 Optical Society of America
1. Introduction
Bragg reflectors are key elements for a variety of applications such as spectral filtering [1,2], tunable lasers [3], polarization dispersion compensation and manipulation [4], multi/demultiplexing [5], spectrometry [6], and sensing [7]. Among the waveguides employed to date we mention those in polymers [2,3,8,9], silicon-on-insulator (SOI) [1,5,10,11], hollow capillaries [4,6,12], lithium niobate [13–15], silica [16–18], metal-insulator-metal [19,20] and liquid crystals [21–25].
Bragg tuning has been proposed/implemented thermo-optically [3,8,10,11], mechanically [2,12,16,18], acousto-optically [14,15], electro-optically [13,17,24,25] and opto-optically [26–29]. The largest thermo-optic tunings were obtained in SOI rib guides (18 nm) [11] and in polymeric gratings (between 20 and 30 nm) [3,8]. Wavelength shifts of about 45 nm were reported by tensile strain on a flexible polymeric waveguide [2] and of more than 90 nm in silica via mechanical beam-bending [18]. Tuning over 76 nm was achieved acoustically in LiNbO3 [15], and over 160 nm by piezoelectric actuators in hollow waveguides [12]. Tunable Bragg gratings were also realized with liquid crystals exploiting their large electro-optic response [24,25].
In this paper we propose and numerically investigate an electro-optically tunable integrated Bragg reflector in a liquid crystalline cell with coplanar comb electrodes. At variance with previously proposed geometries [30], this design allows a transverse electric (TE) mode to undergo distributed feedback with a Bragg wavelength adjustable in the near infrared over more than 100 nm with the application of modest voltages. The latter also ensures bi-dimensional signal confinement in the planar NLC waveguide.
2. Device geometry and physics
The device structure is a planar waveguide as sketched in Fig. 1 . It consists of a nematic liquid crystal (NLC) layer sandwiched between two borosilicate (BK7) glass plates (refractive index 1.50 at λ = 1550 nm). Liquid crystals are mesophase dielectrics with intermediate properties between solids and liquids [31], their nematic phase featuring orientational order and uniaxial symmetry, with the optic axis along the molecular director and a dielectric tensor with two eigenvalues ε|| = and ε ⊥ = along and normally to , respectively. The inner face of one of the plates is patterned with comb-shaped Indium Tin Oxide (ITO) transparent electrodes as in Fig. 1(b).
Fig. 1 (a) 3D sketch of the Bragg reflector and (inset) molecular director , (b) top view of the coplanar comb electrode pattern in ITO.
Both electrodes are periodic along z and symmetric with respect to y = 0. The electrode topology and the NLC parameters need be selected in order to achieve high coupling between an injected transverse electric (TE) optical beam and the voltage induced Bragg grating over a finite propagation distance. Moreover, the applied voltage is intended to increase the index in an NLC finite region along y and so ensure two-dimensional (2D) transverse confinement of the light injected in the thin film slab. We considered the commercial NLC mixture E7 (supplied by Merck). In order to obtain single mode operation at λ = 1550nm we designed an NLC thickness h = 1µm and electrode dimensions a = b = 500nm, c = 250nm, t = 250 nm and T = 500nm. An NLC layer with molecular director in the plane yz can support the propagation of transverse electric (TE) waves in the guide. We also assumed a pre-twist angle φ0 of about 4° with respect to z in order to eliminate the Fréedericks threshold [31], and ITO electrodes 100 nm thick, with complex refractive index 1.3 + i0.1 at λ = 1550nm.
The desired pre-twist condition can be obtained by using a thin (≈50nm) film of Nylon 6 for the alignment, rubbing it along z. In the regime of strong anchoring, the rubbed layer determines the boundary conditions for the molecular director in x = 0 and x = 1μm. The external voltage forces the rotation of the director in the bulk, yielding the formation of a periodic grating defined by the electrode topology. Nylon 6 has a refractive index of 1.52 (at λ = 1550nm); hence, it does not affect the propagation of light in the higher index NLC.
Figure 2 displays molecular reorientation without and with applied voltage. At V = 0V the director is aligned along z because of the anchoring. An applied voltage can induce reorientation. The resulting twist is larger in the regions where the inter-electrode separation is minimum (b) as compared to those where the separation is maximum (b + 2c).
Fig. 2 Sketch of liquid crystal reorientation above the electrode area (in brown) for (a) V = 0 and (b) V > 0V.
The low-frequency voltage applied between the electrodes perturbs the NLC molecular orientation; as anticipated, it creates an average increase of refractive index in a channel finite in y and parallel to z with an additional periodic modulation, as shown in Fig. 2, producing a 2D channel waveguide with a superimposed phase grating for TE propagation along z. The latter can operate as a distributed-feedback guided-wave reflector with voltage-controlled effective index and contrast of the periodic modulation. The electro-optic orientation of the NLC director corresponds to the minimum of the free energy, which includes elastic and electrostatic terms:
with the applied electric field, ε ⊥ = 7 the permittivity for a field perpendicular to the optic axis and the dielectric anisotropy. The elastic (Oseen-Frank) energy can be expressed as:where K 11 = 12pN, K 22 = 7.3pN, K 33 = 17pN are the splay, twist and bend elastic constants determining the restoring torques when an equilibrium configuration is perturbed. The minimization of F is achieved by solving the Euler–Lagrange equation for the free energy density. We couple the problem of a stationary F with the Poisson equation for the electric field distribution:Minimization of Eq. (2) and the solution of Eq. (3) yield the spatial distribution of the NLC director tilt (θ) and twist (φ) or, equivalently, its distribution (sinθ, cosθ cosφ, cosθ sinφ) in the reference system of Fig. 1. Due to the particular geometry of the cell and the nonlocal electro-optic response of NLC, the distribution of the refractive index for the extraordinarily (e-) polarized electric field of TE modes in the planar waveguide (confinement across x) can be approximated by solving:
with K = K 11≈K 22≈K 33 and Ey = Ey(x,y,z) the dominant component of the applied (low-frequency) electric field. Boundary conditions are dictated by surface anchoring (ϕ0) in x = 0 and x = h.Equation (4) is derived from Eq. (2) in the frequently used single constant approximation (K 11~K 22~K 33). The latter applies well to the present case as the NLC director undergoes a pure twist-deformation.
3. Design and analysis
Starting with the voltage-dependent molecular reorientation, we obtain the director distribution and finally the profile of the refractive index ne for extraordinarily (e-) polarized light, i.e. for electric field vectors in the plane xy. In the calculations we neglected the walk-off inherent to extraordinary-wave propagation in uniaxial dielectrics. Figures 3(a) thru 3(f) show the e-index profile for three values of the bias in two transverse sections along z, where the electrode separations are b + 2c (z = 0μm) and b (z = 0.25μm), respectively.
Fig. 3 Refractive index profile for e-polarization in (a,c,e) z = 0μm and (b,d,f) z = 0.25μm. The bias is 2.4V in (a,b), 5V in (c,d) and 13V in (e,f). The electrodes are in white. (g) Intensity profile of the TE00 eigenmode at 1550nm and V = 5V.
A mode solver or, equivalently, a beam propagator can then yield the transverse profile of the guided field distribution at a given wavelength and for each applied voltage. Due to the particular choice of parameters, only the fundamental order TE00 mode propagated in the structure. Figure 3(g) shows the TE00 transverse profile for a bias V = 5V, with an effective index NTE00 = 1.5474 at λ = 1550nm. The periodic separation between the electrodes yields a modulated strength of Ey and, correspondingly, the sought index grating along z. Figure 4(a) displays the index modulation in a 6-period region along z, evaluated in x = 0.2μm and y = 0 for various biases between 2.8 and 4.5V. As expected, the modulation is sinusoidal with a 0.5μm period, maximum when the electrode spacing is minimum (i.e. equals b). Figure 4(b) graphs the resulting index contrast versus voltage between 0 and 13V. For V = 5V the NLC director is entirely reoriented (//y) in the regions with minimum inter-electrode separation (b), but only partially reoriented where the electrodes are separated by b + 2c. Therefore, by increasing the voltage above 5V only the un-saturated NLC regions can still reorient, resulting in a progressive reduction of the index modulation for V > 5V. For V ≥ 12.4V the whole NLC has reoriented with director //y and a negligible index contrast.
Fig. 4 (a) Refractive index modulation along z for applied voltages between 2.8V (bottom line) and 4.5V (top line) in 0.1V steps, evaluated 200nm above the electrodes and in the symmetry axis between them (y = 0). A video (Media 1) shows a top-view of the index distribution versus applied voltage. (b) Corresponding longitudinal modulation versus applied voltage.
Based on the e-index distribution and using coupled mode theory, we calculated the resonant Bragg wavelength (Fig. 5 ) as well as the back-reflected power for TE light propagating over 3000 periods, i. e. a grating length of 1.5mm (Fig. 6 ).
Fig. 6 Spectral reflectivity for various voltages and propagation over 1.5mm (3000 periods). FWHM from left to right are 0.56nm (1.52076μm), 0.85nm (1.5299μm), 1.3nm (1.54219μm), 1.7nm (1.55394μm), 2.1nm (1.56731μm), 2.3nm (1.58108μm), 2.2nm (1.59565μm), 1.6nm (1.61032μm), 1.0nm (1.62052μm) and 0.60nm (1.6246μm).
The reflected wavelength at resonance red-shifts with applied biases of a few volts, providing an extended tunability of ≈104 nm with a reflectivity R ≥ 50% (98nm for R ≥ 90%) while maintaining good spectral selectivity. Graphs of Bragg reflectivity and spectral selectivity (FWHM) versus bias are visible in Figs. 7(a) , 7(b) for various reflector lengths, from 0.5 to 8.0mm. Furthermore, for a given propagation length, a modest change in bias can alter the reflectivity; for instance, for L = 1.0mm (2.0mm) R increases from 30 to 100% (70 to 100%) as the voltage changes from 2.8 to 4.5V.
Fig. 7 (a) Bragg reflectivity and (b) FWHM versus voltage for a few propagation lengths L in mm: 0.5 (blue), 1.5 (red), 3 (green), 5 (violet), 8 (black).
4. Conclusions
We have proposed, designed and numerically investigated a voltage-tunable distributed feedback grating in a liquid crystal waveguide with coplanar comb-shaped electrodes. The device provides transverse light confinement and periodic index modulation, yielding high reflectivity in a wide tuning range of ≈104nm for voltages between 2.5 and 10.2V. This novel geometry provides ease of fabrication and a simple electro-optic control, while ensuring Bragg reflection and spectral filtering over a wide range of wavelengths in the whole C + L band for optical communications.
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