Electrically tunable liquid crystal lens with suppressed axial chromatic aberration

Most of the dielectric materials are dispersive. The refractive index of the material is dependent on the wavelength of incident light. This intrinsic material property results in the refractive and diffractive optical components having different responses to light at various wavelengths. The wavelength dispersion of a dielectric material causes chromatic aberrations in lenses. There are two types of chromatic aberrations appearing in the lens: axial chromatic aberration (ACA) and lateral chromatic aberration.¹ Axial chromatic aberration is the phenomenon of focal length variation in a lens with varying wavelengths of light at normal incidence. Lateral chromatic aberration is the phenomenon of forming focal points with different wavelengths at different points on one plane with the same oblique incidence. For the refractive type of lens, focal lengths are longer with longer wavelengths since most dielectric materials have a smaller refractive index at longer wavelengths. As for the diffractive type of lens, the relationship between focal lengths and wavelengths is inverse. Both types reduce contrast and deteriorate image quality. The current commercialized solution for chromatic aberration is post-image processing, achromatic doublet or triplet.^2–4 The achromatic doublet and triplet implement the idea of compensation between a concave lens with flint glass (high refractive index and high dispersion) and a convex lens with crown glass (relatively low refractive index and low dispersion). However, this solution results in a bulky optical system design. In order to shrink the size of optical components, several approaches were reported. Recently, Aieta et al.⁵ introduced using metasurfaces for chromatic aberration correction. Wang et al.⁶ proposed that scalar diffractive optics with the appropriate design can also have chromatic-aberration-corrected lens for broad band focusing. Both designs aimed for flat and compact optical systems but their lenses are not tunable.

Tunable lenses are intensively developed thanks to the necessity of building compact optical systems such as augmented reality and virtual reality wearable display devices. Liquid crystal is a great candidate for tunable lenses owing to its optical anisotropy and electrically tunable orientation. Numerous LC lens and microlens arrays are reported for 3D imaging and sensing system construction,⁷ electrically switchable 2D/3D modes,⁸ ophthalmic applications,⁹ picoprojectors,¹⁰ and optical zoom systems.¹¹ The focal length of the reported LC lenses and LC microlens arrays (LCMLA) are reported using a single wavelength. With the high wavelength dispersion of the LC material, the focal length of a nematic LC lens depends on the wavelength of light, which deteriorates the imaging quality.

In this work, we demonstrate that the twisted vertical alignment (TVA) configuration can be used in electrically tunable liquid crystal lens suppressing the axial chromatic aberration. The TVA mode was proposed by Rosenblatt et al. in 1993 for developing liquid crystal displays with a high contrast ratio, vanishing threshold, and achromatic dark and bright states.^12,13 The high contrast ratio comes from the great dark state at the field-off homeotropic alignment. The achromatic bright state comes from the achromatic polarization following the field-on twisted nematic structure. The imaging quality improvement is verified by comparing it with the traditional liquid crystal lens without the field-on twisted nematic structure.

In the control group, we utilized two-electrode structures to achieve both positive and negative switching characteristics of LC lens. The designed electrode pattern is illustrated in Fig. 1(a). The diameter D was 2.35 mm and the electrode gap S between the inner electrode and the outer electrode was 50 μm. The voltage applied onto the inner electrode is denoted as V_in, while that which was applied onto the outer electrode is named as V_out. The LC lens cell thickness was controlled by a 50 μm Mylar film with the accuracy of ±2 μm. The bottom substrate has polyimide [SE-1211, Nissan Chemical Industries, Ltd., denoted as VA-PI in Fig. 1(b)] that provides homeotropic alignment on top of the indium-tin-oxide (ITO) conductive electrode, while the top substrate has VA-PI coated on the non-ITO side and the patterned ITO electrode is facing the outside of the lens cell. A parabolic phase profile can be generated within the LC layer across the region D when there is a dielectric layer between the patterned electrodes and the LC layer and when V_in and V_out are properly chosen.¹⁴ In this study, the dielectric layer is the top glass substrate with a thickness of 0.4 mm. A cholesteric LC mixture filled in the LC cell composed of a nematic liquid crystal with negative dielectric anisotropy (HNG 715600–100, Δn = 0.153 at 589 nm, Δε = −12.2, T_NI = 87.9 °C, HCCH, China) with a concentration of 99.83 wt. % and a chiral dopant (CB-15, Merck) with a concentration of 0.17 wt. %. The helical pitch of the mixture was controlled at 100±4 μm to have a cell thickness-to-chiral pitch ratio (d/p) to be approximately 0.5. The LC used in this study is a highly dispersive material with the Abbe number being in the range of ∼5 to ∼6, which is calculated from the refractive index measurement with results fitting the Cauchy equation.¹⁵ To study the d/p effect on the image quality, the weight ratio of a chiral dopant with the fixed cell thickness is altered. The electric field inside the cell is nearly uniform when V_out is equal to V_in, and therefore, the LC cell is similar to the electrically controllable birefringence (ECB) cell without a lens effect. For the TVA mode, when the electric field is smaller than the threshold, the director orientation remains vertical. When the electric field exceeds the threshold, the angle between the liquid crystal director and the normal of the substrate increases as the field strength increases. The twist structure is contributed by the chiral dopant added to the LC mixture with a fixed d/p ratio. The electric field distribution with the selected V_in and V_out generated parabolic optical phase difference profile in the liquid crystal layer. With V_out>V_in, the tilt angle of the liquid crystal director from the normal exhibits a gradient increase from the center to the edge of the lens as depicted in Fig. 1(c). A gradient decrease in the tilt angle from the center to the edge of the lens appears when V_out is smaller than V_in as shown in Fig. 1(d). The optical path difference between the center and the edge of the lens is achieved via an increase of the effective refractive index of the liquid crystal when the local director tilts away from the normal direction. The maximum wavefront propagation delay appears when the liquid crystal director is tilted 90° away from the normal with the polarization direction of incident light parallel to the long axis of the liquid crystal molecule. With V_out > V_in, the light propagates faster in the center than at the edge, and an effect of a negative lens is created. A positive lens is obtained when V_in > V_out. The driving scheme of this liquid crystal lens is applying both V_in and V_out with an equally high voltage, which in this study is 40 V at 1 kHz of the square wave form, eliminating defects during switching. The twisted structure of cholesteric LC remains the same when the applied voltage is larger than 40 V. For positive tuning, the voltage at V_in is fixed at 40 V and the voltage at V_out is reduced to obtain the desired positive focal length. For negative tuning, the voltage at V_out is fixed at 40 V and the voltage at V_in is reduced to obtain the target negative focal length.

The voltage-dependent focal lengths of the LC lens with dual switching ability were measured by obtaining the lens profiles. A pair of convex lens with a focal length of 20 cm and a concave lens with a focal length of −1.8 cm were arranged in the Galilean beam expander configuration in front of the laser to expand the beam size. A LC lens was placed between a pair of parallel polarizers with the rubbing direction parallel to the transmission axis of the polarizers. The LC lens is polarization-dependent. A convex lens with a focal length of 7.6 cm is used to magnify the lens profile. The magnified lens profiles taken at 514 nm are projected onto the screen as shown in Fig. 2. The profiles in Figures 2(a)–2(c) were taken under positive lens operation, while those in Figs. 2(d)–2(f) were taken under negative lens operation. The neighboring rings of transmission maxima have an optical phase difference (OPD) of 1.05 λ (λ is the wavelength of light), which can be calculated with the optical axis of linear polarizers, the rubbing direction, and the total twist angle in the LC cell.^16,17 The OPD between the center of the lens and distance r with respect to the center of the lens can be measured through the phase profiles. The focal length (f) of the lens can be calculated by fitting the parabolic phase profile with the equation $OPD(r)≈− r 2 2 f$⁠.¹⁸ The lens profiles are taken at three different wavelengths: 633 nm, 514 nm, and 488 nm, to investigate the axial chromatic aberration. In this work, we focused on the wavelength dispersion study. The electrically tunable focal lengths of the LC lens are measured when the voltage combinations generate parabolic phase profiles. The lens quality improves the perspective of image contrast since the focal points formed by different wavelengths are closer. Other types of monochromatic error of lenses are not discussed in this work. The chromatic aberration comparison is between LC lens with a twist structure (proposed LC lens) and without a twist structure (conventional LC lens). We assume that the monochromatic errors are from the electric field distribution for phase profile generation. Since the electric field is controlled, the monochromatic error difference between proposed LC lens and conventional LC lens can be minimized.

Figure 3 shows the plot of the electrically tunable focal length as a function of applied voltages measured at different wavelengths for both positive and negative lenses with the upper half of the plot being the positive tuning of the LC lens (i.e., V_in > V_out. V_in is equal to 40 V and V_out is varying) and the lower half being the negative tuning (i.e., V_out > V_in. V_out is equal to 40 V and V_in is varying), as shown in Figs. 3(a) and 3(b). We compared two different cases: LC lens with d/p equal to 0.5 [Fig. 3(a)] and conventional liquid crystal lens [Fig. 3(b)] with d/p equal to 0 (i.e., pure nematic liquid crystal without twist). In Fig. 3(a), the suppressed wavelength dispersion of positive lens operation appears when V_out is from 0 V to 20 V and that of negative lens operation appears when V_in is from 10 V to 20 V. The focal length tunable range is from 18 cm to 40 cm (lens power 5.6 D ∼ 2.5 D) for the positive lens and from −22 cm to −32 cm (lens power −4.5 D ∼ −3.1 D) for the negative lens. In comparison, when the twist structure did not exist in the LC lens, the focal lengths show higher wavelength dispersion as shown in Fig. 3(b), especially in the negative lens operation part. We further define the axial chromatic aberration (ACA) to be $| f 633 n m − f 488 n m | | f 514 n m | ×100%$ which is modified from the definition in Ref. 19 due to the experimental limitation of light sources. The results are presented in Fig. 3(c). In Fig. 3(c), positive lens tuning is denoted as (P) with open circles for d/p = 0.5 and open triangles for d/p = 0, while negative lens tuning is denoted as (N) with solid circles for d/p = 0.5 and solid triangles for d/p = 0. The ACA for LC lens with the ∼180° twist structure at negative lens tuning is ∼8%, while for that with d/p = 0 is ∼30% in average. The maximum suppression of ACA (i.e., difference between ACA_d/p=0 and ACA_d/p=0.5) for negative tuning is 30%. The maximum suppression of axial chromatic aberration for LC lens at positive tuning with a twist structure is 8%.

To demonstrate the imaging of LC lens in positive lens and negative lens operation modes, we use the USAF 1951 IX resolution chart as a target. A LC lens with d/p = 0.5 is attached on a webcam (Logitech HD Pro Webcam c920) with manual focusing function. On the left hand side in Figs. 4(a)–4(d) is the target that is closer to the LC lens, which we named as target 1, and on the right hand side of the photos is the farther target, which we named as target 2. The distance between LC lens and target 1 is defined as d₁, while the distance between lens and target 2 is defined as d₂. In the positive lens operation [Figs. 4(a) and 4(b)], d₁ is equal to 19 cm and d₂ is equal to 75 cm. The voltage condition in Fig. 4(a) is V_in = 40 V and V_out = 40 V without the lens effect. The webcam was set to focus on target 2 for positive lens demonstration in Fig. 4(a). When V_out was switched to 0 V, the LC lens is focused on target 1 as shown in Fig. 4(b). In the negative lens operation [Figs. 4(c) and 4(d)], d₁ is set as 19 cm and d₂ is set as 65 cm. Figure 4(c) is the lens-off-state for negative lens demonstration with V_in = V_out = 40 V. The webcam was set to focus on target 1 without the lens effect. When V_in was switched to 10 V, the image on target 2 came into focus. To investigate the axial chromatic aberration at different twist angles in the LC lens, we made the LC lenses with d/p = 0 (pure nematic), d/p = 0.25, and d/p = 0.5 with a fixed cell gap and acquired the positive lens and negative lens focused images using the same apparatus and lighting conditions. With the same cell gap control, the tunable focal length range is comparable. We analyzed the contrast ratio of the highlighted pattern in Fig. 4(b) for positive lens operation and in Fig. 4(d) for negative lens operation and use the same part of the resolution chart for analyzing the image quality of LC lenses with different d/p ratios. The contrast ratios of different lenses with different d/p ratios are presented in Fig. 4(e). With similar focal length tunability, the contrast ratio of d/p = 0.5 LC lens is 15% higher than the one without a chiral twist. With a d/p ratio close to 0.5, the liquid crystal structure has a 180° twist. The response time for focusing and blurring is 12 s in total. It can be further improved upon by the polymer-sustained alignment approach with the appropriate polymer concentration and curing condition.²⁰ 

To understand the mechanism of the suppressed axial chromatic aberration behavior, we made a single pixel TVA cell of the top-down square electrode with the same cell thickness and cholesteric pitch with vertical alignment as the liquid crystal lens (d/p = 0.5 and d/p = 0.25) and prepared a LC cell with a nematic liquid crystal without a twist (d/p = 0). We measured the static voltage response of the cells at the three wavelengths we used in the study as shown in Fig. S1 (supplementary material). The experimental results show that the wavelength dispersion of phase retardation-voltage response is smaller in the TVA cell with d/p = 0.5 than in the LC cell with a pure nematic LC. As a result, we can conclude that with the same voltage gradient within the liquid crystal lens cell, the TVA mode has less chromatic aberration than conventional LC lens filled with a pure nematic LC since the spatial distribution of the optical phase retardation within the cell is less sensitive to the wavelength of light.

In summary, we demonstrate a liquid crystal lens with reduced axial chromatic aberration. With the wavelength-insensitive optical path difference profile of a long pitch chiral nematic LC lens, the lens with the field-on twist structure has electrically tunable focal lengths, which is insensitive to the wavelength of light. The tunable lens power of positive LC lens is from 2.5 D to 5.6 D, and that of negative type is from −3.1 D to −4.5 D. The image quality of the proposed LC lens compared with conventional LC lens with the same LC material without a chiral dopant is proven to be improved by analyzing the contrast ratio of the focused image of the resolution chart. The contrast ratio increases up to 15% by introducing a twist structure to the LC lens. This work provides a solution for electrically tunable liquid crystal lens to achieve a better image quality with the high dispersion liquid crystal materials.

See supplementary material for the static voltage response of a single pixel cell for the suppressed wavelength dispersion study.

The authors would like to thank Mr. Vinay Joshi of Liquid Crystal Institute, Kent State University for useful discussion. This work was supported by the Ohio Third Frontier (OTR) Venture Startup Fund under Grant No. TECG 2015-0128.