1. INTRODUCTION

The dynamic tuning of an optical polarization vector to an arbitrary orientation has widespread and increasing application in modern optical systems for biological imaging, quantum processing, polarimetry, Berry-phase optics, and soft robotics. [1–12]. For instance, in bioimaging systems, the capability of dynamically changing the probe, excitation, or detection polarization of light aids in offering further insights into the structures and kinetics of biospecimens (such as tissues, proteins, and biomolecular complexes) [1–6]. However, the strong dispersion underlying almost all the mechanisms of such devices as half-wave plate, magneto-optic (Faraday) [13], and chiro-optic rotators [14,15] imposes many constraints on the system design and limits their usefulness to monochromatic applications. Polarization rotation generally originates from the linear or circular phase retardation $\delta$ in an anisotropic medium and so depends strongly on the operating wavelength $\lambda$ as $\delta = {2}\pi L\Delta n/\lambda$, where $L$ is the interaction length, and $\Delta n$ is the linear or circular birefringence. Many attempts have been made to realize dispersionless polarization rotators (PRs) [15–21]. Dispersion compensation, which involves a set of (dispersive) wave plates that cancel each other’s chromatic dispersion, is the most commonly used approach to expanding the bandwidth of half-wave rotation to several tens or hundreds of nanometers [16–19]. The angle of polarization rotation can in principle be tuned mechanically by reorienting the optic axis of the system. Nonetheless, the limited bandwidth, bulkiness, and complex design/alignment of optics still hinder their practical applications. To date, truly achromatic rotation of polarization in a single-element device has only been achieved using the so-called twisted-nematic liquid crystal (LC) cell, without the need for dispersion compensation [20–23]. The polarization at all wavelengths that fulfill the Mauguin condition ($\lambda \;{\ll}\;L\Delta n$) follows the twist of the LC optic axis and is thus rotated. However, such a PR allows only stepwise (on–off) switching between zero rotation and a structure-determined rotation (typically up to 90°), with no intermediate state. The realization of an achromatic PR with continuous angular tuning remains a great challenge and requires a completely different mechanism from all the aforementioned PRs. Here, we show by experiments and numerical modeling that a hybrid splay–twist (HST) LC with sub-100-µm thickness can function as an achromatic electrotunable PR across the entire visible spectrum. When an electric field is applied, the optic axes in the LC cell are spatially twisted and splayed along the thickness dimension. This allows an incident linearly polarized beam to experience an adiabatic rotation of the polarization vector as it propagates. The adiabatic polarization rotation is dispersionless and dynamically tunable from 0° to 90° through simple electrical control over both the azimuthal and polar orientations of the local optic axes across the cell. As will be discussed below, the variable optic-axis distribution makes HST-LC advantageous over and fundamentally different from its conventional (splay-free) twisted-nematic counterpart [20–23] and other state-of-the-art techniques (e.g.,  dispersion-compensated wave-plate sets [15–21] and chiro-optic metasurfaces [24]) for polarization manipulations. This paper elucidates the mechanism for the adiabatic optical rotation and associated continuous electro-optic tuning in an HST-LC, and a tandem-cell scheme to expand the tuning range up to 180° is also demonstrated experimentally.

2. WORKING PRINCIPLE AND DEMONSTRATIONS

An HST-LC cell comprises a large-area sub-100-µm-thick layer of dielectrically negative LC sandwiched between two indium–tin–oxide-coated glass windows. Prior to the assembly of the sandwich cell, one of the windows is coated with a planar-alignment (PA) polyimide, and the other is coated with a vertical-alignment (VA) polyimide to determine the polar anchoring condition of LC at the boundaries. The two alignment coatings are then rubbed in different directions to determine the azimuthal anchoring condition. The aperture size of the LC PR can be designed to meet different application needs (${{\rm mm}^2}$, ${{\rm cm}^2}$, or larger). In the absence of an applied field, the LC director (and the optic axis) varies continuously from an in-plane to out-of-plane orientation along the thickness dimension and exhibits a negligible azimuthal twist—the splay(-only) state depicted in Fig. 1(a). The LC cell can thus be seen as an optically uniaxial crystal but with its local birefringence gradually vanishing from one end to the other. An incident optical beam that is linearly polarized parallel (or perpendicular) to the LC director at the entrance surface will not experience any orientation or ellipticity change of the polarization during propagation. When an ac field ($\sim {\rm kHz}$) is applied, the electric torque pushes the LC directors away from the field axis on account of the negative dielectric anisotropy ($\Delta \varepsilon$), and the directors are reoriented with a twist to adapt to the azimuthal condition at the boundaries—the hybrid state depicted in Fig. 1(b). The degree of director reorientation increases with the applied field strength (${E_{{\rm ac}}})$. Ultimately (at ${E_{{\rm ac}}} = {E_{{\rm max}}}$), almost all the LC directors in the bulk are oriented parallel to the cell plane and the azimuthal twist reaches 90°—the twist(-only) state depicted in Fig. 1(c). In a twisted HST-LC (${E_{{\rm ac}}}\; \gt \;{0}\;{\rm V/}\unicode{x00B5}{\rm m}$), the input linearly polarized beam experiences an adiabatic rotation of the polarization vector following the twist of the optic axis [Figs. 1(a)–1(c)]. The angle between the input and output polarization vectors (${{\boldsymbol P}_{{\rm in}}}$ and ${{\boldsymbol P}_{{\rm out}}}$) is henceforth denoted as ${\Phi _{\textit{PR}}} = \sphericalangle{\boldsymbol P_{\rm in}}{\boldsymbol P_{\rm out}}$ [Fig. 1(d)]. In practice, due to the strong surface anchoring, the LC near the VA boundary can hardly be reoriented by the applied field. The maximum angle of the field-induced twist of the LC (and optic axis) is, therefore, always smaller than the angle formed by the two azimuthal alignment directions (${\Phi _{{\rm align}}}$), and so does the maximum angle of polarization rotation (${\Phi _{\textit{PR}}}$ at ${E_{{\rm max}}}$). For instance, the maximum ${\Phi _{\textit{PR}}}$ is only ${\sim}{87}^\circ$ if ${\Phi _{{\rm align}}} = {90}^\circ$. We found that single HST cells capable of producing a maximum ${\Phi _{\textit{PR}}}$ of 90° can be fabricated by doping the LC with a trace of chiral agent to determine the direction of the field-induced twist and setting ${\Phi _{{\rm align}}}$ to 93° (determined through trial and error). As will be demonstrated shortly, the ${\Phi _{\textit{PR}}}$ can be continuously and reversibly tuned between 0° and 90° by varying the applied field strength. According the Mauguin theorem (originally derived for splay-free twisted nematic LC [25]), the polarization rotation should also be independent of the operating wavelength ($\lambda$) when the condition $L\Delta n\;{\gg}\;\lambda$ is satisfied.

 

Fig. 1. Continuous electro-optical tuning of polarization rotation in HST-LC cell: proof-of-concept experiment and simulation. (a)–(c) Schematics of HST-LC in the (a) splay-only (${E_{{\rm ac}}} = {0}\;{\rm V/}\unicode{x00B5}{\rm m}$), (b) hybrid (${0}\;{\rm V/}\unicode{x00B5}{\rm m}\; \lt \;{E_{{\rm ac}}}\; \lt \;{E_{{\rm max}}}$), and (c) twist-only (${E_{{\rm ac}}} = {E_{{\rm max}}}$) states. ${E_{{\rm ac}}}$, applied ac field; ${E_{{\rm max}}}$, the field strength to achieve 90° rotation; PA/VA, planar-/vertical-alignment surface; ${P_{\rm in/out}}$, input/output optical polarization; ${\Phi _{\textit{PR}}}$, angle of polarization rotation; $k$, wave vector. (d) Polar plots of measured output polarizations at three applied field strengths, obtained for 20-µm-thick HST sample. Input polarization is at 0°. ${\varphi _A}$, analyzer angle with respect to input polarization orientation; $T$, polarized transmittance; light source, He−Ne laser at 633 nm. (e) Rotation angle and DOLP as functions of applied field strength (${E_{{\rm ac}}}$), experimental (Exp.) and simulation (Sim.) results for 20- and 70-µm-thick samples; inset, polar plots of measured output polarizations from 20-µm-thick sample at various field strengths, showing continuous ${\Phi _{\textit{PR}}}$ tuning between 0° and 90°; gray dashed line, DOLP = 100%. See Methods for more details.

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We first present exemplary experimental results obtained with a horizontally polarized laser probe at $\lambda = {633}\;{\rm nm}$ and a polarization analyzer to highlight the electro-optical tunability of HST-LC-based PR [material: R811-doped HNG726200-100; see Appendix A (Methods) for details]. Two HST-LC thicknesses are chosen for demonstration: $L = {20}$ and 70 µm, which approach the Mauguin condition ($L\;{\gg}\;\lambda /\Delta n\; \approx \;{1.57 - 2.75}\;\unicode{x00B5}{\rm m}$ for $\lambda$ from 400 to 700 nm and $\Delta n\; \approx \;{0.255}$) while maintaining the ease of LC alignment and cell fabrication. Each polar plot in Fig. 1(d) shows how the polarized transmittance ($T$) of an HST-LC at a particular field strength varies with the angle (${\varphi _A}$) between the transmission axis of the analyzer and the input polarization vector. A two-lobed pattern is observed, clearly indicative of linear polarization (according to Malus’s Law), and the polarization rotation angle ${\Phi _{\textit{PR}}}$ is equal to the ${\varphi _A}$ at which the $T$ reaches the maximum. With increasing strength of the applied field ${E_{{\rm ac}}}$, the pattern is rotated counterclockwise from 0° to 90° (viewing toward the light source). Taking the 20-µm sample as an example, the ${\Phi _{\textit{PR}}}$ is increased sharply with ${E_{{\rm ac}}}$ in the low field regime to ${\sim}{70}^\circ$ at 0.6 V/µm and then gradually reaches a full rotation of 90° at 3.0 V/µm [lower left panel in Fig. 1(e)]. The rotation direction is determined by the handedness of the LC twist: counterclockwise/clockwise rotation in a right-/left-handed HST-LC [inset in Fig. 1(e)]. With thickness increased to 70 µm, the ${\Phi _{\textit{PR}}}$ grows with ${E_{{\rm ac}}}$ in a similar fashion, but the required ${E_{{\rm ac}}}$ to reach a particular ${\Phi _{\textit{PR}}}$ becomes smaller due to the relatively weak influence of surface anchoring on the bulk LC molecules [lower right panel in Fig. 1(e)]. From the polar plots [e.g., inset in Fig. 1(e)], the degree of linear polarization (DOLP) can also be retrieved: ${\rm DOLP} = ({T_{{\rm max}}} - {T_{{\min}}})/({T_{{\max}}} + {T_{{\min}}})$, where ${T_{{\rm max}}}$ and ${T_{{\min}}}$ are the maximum and minimum polarized transmittances, respectively. Upper panels in Fig. 1(e) depict that the DOLP of light output from the 20 and 70-µm samples remain greater than 98% throughout their respective ranges of operation, indicating high polarization fidelity of such a PR.

 

Fig. 2. Mechanism underlying the continuous electro-optical tuning of polarization rotation in HST-LC cell. (a) Simulated distribution of optic axes at different applied field strengths. LC thickness $L = {20}\;\unicode{x00B5}{\rm m}$; (b) definition of azimuthal angle $\varphi$ and polar angle $\theta$. The $\varphi$ of a director is 0° if oriented parallel to ${P_{{\rm in}}}$. (c) Simulated polar angle; (d) effective birefringence $\Delta {n_{{\rm eff}}}$; and (e) azimuthal angle as functions of applied field strength and position in cell; (f) polar plots of simulated output polarizations at different field strengths; probe wavelength, 633 nm. See Appendix A (Methods) for simulation setting.

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To gain further insight into the complex optic-axis distribution of an HST-LC in the applied field, the field-induced LC deformation and corresponding optical responses are simulated using TechWiz LCD 3D v16 (software based on the finite-element method) and Jones calculus coded in MATLAB, respectively [Figs. 1(e), 2, and Fig. S1 of Supplement 1; see Appendix A (Methods) for numerical parameters]. The simulation results are generally in good agreement with the experiments [Fig. 1(e)], and the slight differences are attributed to spatial nonuniformity of the LC alignment in experiments and a lack of information about the surface anchoring properties of the polyimide coatings and the elastic constants of the LC for simulations. Figure 2(a) displays the director profile of an HST-LC at various ${E_{{\rm ac}}}$, capturing how the LC evolves from the splay through hybrid to twist alignment with increasing field. A more detailed analysis [Figs. 2(c)–2(f)] is conducted by decomposing the LC orientation into the polar and azimuthal components defined in Fig. 2(b). The polar angle $\theta$ controls the local effective birefringence,

In Fig. 3, we experimentally demonstrate the achromaticity of the tunable polarization rotation by the 20 and 70-µm HST-LC samples. The sample is sandwiched between crossed polarizers with the transmission axis of the front polarizer parallel to the azimuthal orientation of the LC at the entrance surface and examined by a broadband white light source with $\lambda$ ranging from 400 to 700 nm. In the crossed polarizer setup, the rotation angle at each wavelength can be retrieved from the polarized transmittance using Malus’s law: ${\Phi _{\textit{PR}}} = {{\rm \sin}^{- 1}}[{(T/{T_{{\rm max}}})^{1/2}}]$. The nearly flat ${\Phi _{\textit{PR}}}$ spectra throughout the electrical tuning range (${0 - 8}\;{\rm V}\;\unicode{x00B5} {{\rm m}^{- 1}}$) [Fig. 3(a)] and the colorless (white) transmission of the HST cell under a copolarized microscope [Fig. 3(e)] both indicate that the polarization rotation is almost independent of the operating wavelength. The slight fluctuations of ${\Phi _{\textit{PR}}}$ in Fig. 3(a) are attributable to the fact that the Mauguin condition ($L\;{\gg}\;\lambda /\Delta n$) [25] is not fully satisfied in the 20-µm cell [$L\Delta n/\lambda \;\sim\;{7.3}$ at $\lambda = {700}\;{\rm nm}$]. In this case, the wavelength-dependent phase retardation and the effects of material dispersion (see Fig. S2, Supplement 1) on the ${n_{{\rm eff}}}(\lambda)$ [26,27] and associated depth of effective polarization rotation come into play. The fluctuations are particularly noticeable in the high-field regime because the broadband probe beam experiences a relatively large birefringence [$\Delta n(\lambda) = {n_{{\rm eff}}}(\lambda) - {n_o}(\lambda)\; \to \;{n_e}(\lambda) - {n_o}(\lambda)$ at high fields] as it propagates and greater spectral variations in the effective polarization rotation depth than in the low-field regime. As expected from the Mauguin theorem, with an increase in the LC layer thickness to 70 µm [Fig. 3(b)], the spectral fluctuations are reduced; here, $L$ is about 25.5 times larger than $\lambda /\Delta n$ at $\lambda = {700}\;{\rm nm}$. The polarization fidelities of both samples are also examined at each wavelength and field strength. Figures 3(c) and 3(d) reveal that, at all operating wavelengths, the DOLPs are well preserved upon polarization rotation (mean: 98.9%, standard deviation: 0.1%). These findings are confirmed qualitatively in the simulation results with the material dispersion of refractive indices considered (see Fig. S1, Supplement 1) [26].

 

Fig. 3. Spectrally resolved polarization rotation angles measured with (a) 20- and (b) 70-µm HST-LCs at various electric field strengths, and the DOLP obtained for (c) 20- and (d) 70-µm samples; (e) optical micrographs of 20-µm HST-LC between parallel polarizers, captured with increasing applied field; P/A, transmission axis of polarizer/analyzer; ${{n}_{{\rm ent}}}$, LC orientation at entrance surface. See Appendix A (Methods) for more details.

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We then examine the switching dynamics of the 20-µm HST-LC PR by monitoring the variations in polarized transmittance with a linearly polarized 633-nm laser [Fig. 4]. The transmission axis of the analyzer is set perpendicular to the polarization vector of the laser source. When a 1-kHz field of 3.0 V/µm is applied, the HST-LC is switched from the splay through hybrid to twist state, causing an increase of ${\Phi _{\textit{PR}}}$ from 0° to 90° and thus a rise in polarized transmittance; the corresponding time constant is found to be ${\sim}{1.5}\;{\rm s}$ [solid curve in Fig. 4(a)]. Before it reaches the steady state, the transient fluctuations of the polarized transmittance imply the occurrence of field-induced director backflows [28]. The field is then removed, allowing the HST-LC to relax back to the splay state in ${\sim}{2.5}\;{\rm s}$ [solid curve in Fig. 4(b)]. To speed up the relaxation, one can replace the dielectrically negative nematic host with a dual-frequency LC [29]. Here, a 20-µm HST sample consisting of R811-doped HEF967100-100 is used for demonstration, which exhibits a crossover drive frequency (${f_c}$) at ${\sim}{3.5}\;{\rm kHz}$; the dielectric anisotropy $\Delta \varepsilon$ is negative for drive frequencies $f\; \gt \;{f_c}$ and positive for $f\; \lt \;{f_c}$. We first apply a 20-kHz field of 3.0 V/µm to switch the dual-frequency HST-LC to the twist state. The switching dynamic is similar to the case of dielectrically negative HST-LC but with a relatively short time constant of ${\sim}{0.5}\;{\rm s}$ (possibly due to lower viscosity and/or higher elastic constant) [dashed curve in Fig. 4(a)]. To return to the splay state, we switch the applied field from 3.0 V/µm at 20 kHz to 0.05 V/µm at 2 kHz instead of directly removing the field. This exerts a restoring torque on the reoriented LC and shortens the ${90}^\circ \to {0}^\circ$ switching process to ${\sim}{0.2}\;{\rm s}$ [dashed curve in Fig. 4(b)].

 

Fig. 4. Dynamics of normalized polarized transmittance showing the electro-optic responses of HST-LC PRs consisting of dielectrically negative LC and dual-frequency LC. (a) ${0}^\circ \to \;{90}^\circ$ (splay to twist) and (b) ${90}^\circ \to \;{0}^\circ$ (twist to splay) switching.

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To enhance the applicability of HST-LC-based PRs, Fig. 5 experimentally demonstrates a tandem-cell scheme that expands the range of dynamic ${\Phi _{\textit{PR}}}$ tuning to $\ge \;{180}^\circ$. This allows the access to all polarization orientations under the “headless vector” assumption made in most applications (i.e., a polarization state is considered unchanged upon $\pi$ rotation). Here, the PR consists of two 20 µm-thick HST cells [Fig. 5(a)]. The dynamic tuning of the output polarization from 0° to 90° is achieved by keeping the front HST-LC in the splay-only state [i.e., $E_{{\rm ac}}^{(f)} = {0}\;{\rm V/}\unicode{x00B5}{\rm m}$] and increasing the applied field across the rear HST-LC $E_{{\rm ac}}^{(r)}$ from 0 to 3 V/µm [Fig. 5(b), upper row; Fig. 5(c), left half]. To tune the ${\Phi _{\textit{PR}}}$ between 90° and 180°, $E_{{\rm ac}}^{(f)}$ is set to ${E_{{\rm max}}}$ to rotate the input polarization by 90° before the light enters the rear HST-LC, and $E_{{\rm ac}}^{(r)}$ is modulated to determine the orientation of the output polarization [Fig. 5(b), lower row; Fig. 5(c), right half]. Throughout the range of operation, the DOLP remains above 95% with slight fluctuations attributable to nonuniform LC reorientation, which can be improved by increasing the spatial uniformity of the bulk LC alignment [Fig. 5(c)]. Besides the tandem-cell scheme, a preliminary study on single-cell HST designs that (ultimately) allow a full $\pi$ or ${2}\pi$ tuning of the output polarization is underway. With ${\Phi _{{\rm align}}}$ much larger than 90°, HST-LC can naturally adopt a splay–twist alignment that results in a nonzero ${\Phi _{\textit{PR}}}$ at ${E_{{\rm ac}}} = {0}\;{\rm V/}\unicode{x00B5}{\rm m}$, and thus a mechanism for tuning between the zero-field ${\Phi _{\textit{PR}}}$ and 0° is needed. Dual-frequency LC [29] is found to be a promising candidate for such HST PR because it enables the tuning between ${\Phi _{\textit{PR}}}({0})$ and 0° when operating at $f\; \lt \;{f_c}$ (in which regime $\Delta \varepsilon \; \gt \;{0}$) and the tuning from ${\Phi _{\textit{PR}}}({0})$ to a larger angle with $f\; \gt \;{f_c}$ (in which regime $\Delta \varepsilon \; \lt \;{0}$), but detailed studies are required to determine the feasibility of achromatic full $\pi$ (or ${2}\pi$) tuning in such a geometry.

 

Fig. 5. (a) Two-stage 180° tuning of optical rotation angle by tandem HST-LC. Superscripts (${f}$) and (${r}$) denote front and rear HST cells. (b) Polar plots of output polarizations at various electric field strengths; input polarization is at 0°. (c) Rotation angle and DOLP as functions of applied field strength. Dashed line in upper panel, DOLP = 100%.

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3. CONCLUSION

In summary, electrotunable achromatic PRs have been developed based on a dielectrically negative LC in the HST configuration. The proposed polarization-rotation mechanism works for all wavelengths that satisfy $\lambda \;{\ll}\;L\Delta n$ without sacrificing the DOLP and transmitted power. In an HST cell, the orientational distribution of local optic axes exhibits a coupled splay–twist deformation and can be dynamically controlled by an applied electric field, thereby enabling adiabatic polarization rotation that is dispersionless and continuously tunable up to 90° (or 180° in the tandem-cell geometry). The capability of large achromatic polarization rotation in a compact single-element device greatly eases the integration into various optical systems and further system miniaturization. The dynamic electro-optical tunability also enables more versatile optical designs such as spatial and temporal multiplexings. These key features together make the proposed PR a promising alternative to the technologies currently implemented for polarization manipulations in a wide variety of optical and photonic systems. The HST-LC PR offers further design possibilities that are not feasible with other technologies. For instance, a pixelated HST-LC can function as a spatial light modulator (SLM) that achromatically converts an optical beam from simple (uniform) to complex vector fields. [Note that conventional SLMs enable only the conversions among the linear, elliptical, and circular polarizations that share a fixed principal vector orientation, usually at 45° with respect to the alignment axis.]