Open Access
Add to BibSonomy
Abstract
We investigate the effects of a floating ring electrode (FRE) on the electro-optical performance of a hole-patterned liquid crystal (LC) lens and successfully reveal the design criterion of the FRE-embedded LC lens. Results show that when the FRE is close to the hole-patterned electrode, the addressing voltage of the LC lens decreases due to the strengthened electric potential in the center of aperture hole (AH). The tunable focal length range of the LC lens is also broadened. On the contrary, when the FRE is close to the LC layer, the wavefront aberration of the LC lens is suppressed because the embedded FRE increases the gradients of the fringing electric field and the associated phase profile near the AH periphery. The suppressed wavefront aberration exhibits a low root-mean-square error and an excellent modulation transfer function curve. However, the FRE close to the LC layer inevitably increases the addressing voltage of the LC lens because the FRE near the AH periphery gathers the fringing electric field around the AH periphery.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The focal length tunability and autofocusing capability of conventionally focus variable lenses were achieved by mechanically moving the lens module in a compact camera module (CCM). Nonetheless, the control of lens motion was complicated and the moving parts occupied a large space. To rectify these issues, a number of studies had been conducted on liquid crystal (LC) lens with electrically tunable focal length, aiming for the application of autofocusing CCM [1–7]. The first concept of LC lens was introduced by Sato in 1979 [8]. Since then, electrically tunable LC lenses had been intensively studied because of their autofocus property and compact and lightweight structure [9]. When LC lens was turned on, the LC reorientation induced a quadratic refractive index profile, which converged (or diverged) the incident light [10–12]. Extensive researches had been executed on numerous types of LC lenses, such as diffractive LC lenses [13,14], dual-frequency lenses [15], polymer network LC lenses [16], smectic [17], cholesteric LC lenses [18], and forked vortex LC lenses [19,20]. In addition, a number of LC lenses with different electrode designs, such as spherical, hexagonal patterned, hole-patterned, cylindrical, curved, and multiple-ring electrodes, were presented [21–27]. LC lenses with different aperture sizes had been investigated for the last two decades. With the development of mobile camera, large-aperture liquid crystal (LALC) lens with a millimeter-diameter size had been paid considerable attention. Various approaches were proposed to construct LALC lenses [28–37]. Among them, the hole-patterned LALC lens had the advantages of easy fabrication, simple operating system, high optical quality, and extensive tunable focusing, and it can be used in several areas, such as imaging systems, 3D displays, and biological applications [38–43]. However, for the hole-patterned LALC lens, a thick dielectric substrate had to be inserted between the LC layer and hole-patterned electrode to extend the fringing electric field into the aperture hole (AH) region. The embedded dielectric substrate also created a vertical electric field in the hole center, which resulted in the decreased phase difference from the center to the periphery of the LC lens and hence decreased the tunable focal length of the LC lens. The operation voltage of the LC lens was also increased. The modal LC lens had been demonstrated to decrease the operation voltage of the LC lens; a floating disk electrode (FDE) was further placed on the top of the hole-patterned electrode to correct the wavefront profile of the modal LC lens [44–46]. However, selecting the adopted high-impedance conductive electrode and operation frequency was a complicated task. Recently, we had proposed a novel concept to fabricate an LC lens that inserted a floating ring electrode (FRE) on the interface between the dielectric slab and LCs [36] [47]. In comparison with the modal LC lens with an FDE, our FRE LC lens had the advantage of using a conventional indium tin oxide (ITO) substrate and a driving scheme with fixed frequency.
Previously we had demonstrated the electro-optical properties of hole-patterned LC lenses upon variation of the inner diameter of the inserted FRE [36,47]. In those works, the FRE always positioned at the interface between the dielectric substrate and LC layer. It was concluded that LC lens with an inserted FRE in 2 mm-diameter provided excellent focusing quality, low wavefront error, and fast driving response. In the current study, the electro-optical properties of the hole-patterned LC lens cells with FRE embedded at different positions of the dielectric substrate were investigated. Optical interference fringes of the LC lenses were obtained to determine the minimum addressing voltage above which the LC lens can be considered an optical lens. Optical interference fringes were also used to estimate the wavefront errors of the LC lenses. The frequency-dependent capacitances of the lens cells were measured to discuss the difference in the addressing voltage of LC lens. Calculation in diffraction-limited value was used to analyze the focusing qualities of the LC lenses. Their image resolutions were examined by the obtained modulation transfer functions (MTFs). Finally, the design criterion for the FRE-embedded LC lens is revealed.
2. Experimental preparations
The top substrate of the lens cells consisted of three pieces of 0.55 mm-thick glasses; a hole-patterned ITO electrode was deposited on the top surface of the middle glass. The dielectric constant of the used glass was 6.05 at 1 kHz, obtained using an LCR meter (Hioki 3532-50, Japan). A 50 μm-wide ring electrode with an inner ring diameter of 4 mm was placed at different positions in the top substrate of lens cells. In the present study, the ring width of the FRE was fixed to 50 μm, mainly due to the lithography limitation in our laboratory. The ring width of FRE may perturb the LC reorientation and associated refractive index profile around the FRE. If the FRE width is large, the LCs beneath the FRE will be significantly affected and the quadratic refractive index profile of LC lens will be distorted. Using thick glass dielectric layer effectively extends the fringing electric field into the AH center, decreasing the addressing voltage but also the lens power of LC lens. The bottom substrate of the lens cells consisted of a 0.55 mm glass deposited with planar ITO electrode on the top surface. The FRE was inserted at four different positions, namely, the FRE-top (Fig. 1(a)), FRE-inplane (Fig. 1(b)), FRE-middle (Fig. 1(c)), and FRE-bottom (Fig. 1(d)). The cell gap of the LC lens was fixed by 50 μm-thick Mylar spacers. It was noticed that the thinner cell gap resulted in the larger operation voltage for the hole-patterned LC lens, because the fringing electric field generated from the hole-patterned electrode was difficult to spread into the AH center. The inner surfaces of the top and bottom substrates of the lens were coated with homogeneous polyimide AL-1426CA (Daily Polymer, Taiwan) and underwent antiparallel rubbing. Nematic LC E7 (Daily Polymer, Taiwan) was initially heated up to isotropic phase and then injected into empty lens cells by capillary action. After injection, the LCs were cooled down to nematic phase. The LC E7 possessed a rotational viscosity of 232.6 mPa-s, dielectric anisotropy of 14.5, average elastic constant of ∼13 pN, and birefringence of 0.22 at room temperature. The 1 kHz alternating current (AC) square wave voltage (Vrms) was supplied to the LC lens cell across the hole-patterned electrode and planar ITO electrode. The ring electrode was always kept at floating potential (defined as FRE), which indicated that the ring electrode was not directly connected to the voltage source. The frequency-dependent capacitances of the LC lens cells were measured using the LCR meter with an applied AC field of 0.1 V/µm in a frequency ranging from 42 Hz to 5 MHz, by connecting the LCR meter across the top hole-patterned electrode and bottom planar ITO electrode of the LC lens. The conventional hole-patterned LC lens without embedded FRE was also fabricated and defined as FRE-free LC lens to compare the electro-optical properties. The electro-optical properties of FRE-free LC lens were almost similar to those of FRE-top LC lens.
Fig. 1. Schematics of (a) FRE-top, (b) FRE-inplane, (c) FRE-middle, and (c) FRE-bottom LC lenses. The red symbols indicate the FRE positions.
3. Results and discussion
Figure 2 depicts the optical interference patterns of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses at various voltages. The LC lens cell was placed between a pair of crossed polarizers with a transmission axis of 45° with respect to the rubbing direction of the lens cell. A He–Ne laser with a wavelength of 632.8 nm was incident normally through the polarizers and lens cell. The interference fringes were captured using a high-resolution charge-coupled device (CCD) camera located behind the analyzer. The neighboring bright (dark) fringes indicated a 2π phase difference. The focal length (f) of LC lens correlated with the number (N) of interference fringes with an appropriately spatial distribution with reference to the following formula:
where r is the AH radius and λ denotes the wavelength of incident light. A large N indicates a large phase difference from the AH center to the AH periphery and hence a small f. The reciprocal of f is defined as lens power. Figures 2(a)–2(p) depict that the number of interference fringes initially increases and then decreases with the increased voltage. The focal length of the LC lens decreases as the voltage increases initially due to the increased LC tilt angle in the AH periphery measured from the substrate surface, resulting in the large phase difference from the AH center to the AH periphery. As the voltage increases further, the LC tilt angle in the AH periphery saturates and LCs in the AH center start to tilt high, which decreases the phase difference from the AH center to the AH periphery and hence increases the focal length again [48]. At low voltages, as shown in Figs. 2(a), 2(e), 2(i), and 2(m), the interference fringes gather near the AH periphery, indicating the flat phase in the AH center; hence, the resulted phase profile of the lens cell cannot be considered as an optical lens. As the voltage increases further, the interference fringes start to cover the entire AH, and then the LC lens can be used as the optical lens. Considering as an optical lens, the minimum voltage at which the interference fringes start to fulfill the entire AH is defined as the addressing voltage Vad of LC lens. Vad values of FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses are 38, 30, 40, and 45 V, respectively (Figs. 2(b), 2(f), 2(j), and 2(n), respectively). The FRE-inplane LC lens provides a relatively low Vad, which can be further verified by numerical calculation. Figs. 3(a)–3(d) show the calculated potential distributions in the LC layers of FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses with the commercial software LCDmaster 3D (Shintech, Inc., Japan). In the calculation, for simplifying the calculation complexity, the reduced structures were used to explain the underlying physics. The structure dimensions and their parameters were set as follows: the width of LC lens cell was 1000 μm; the diameter of hole-patterned electrode was 540 μm; the diameter and width of FRE were 520 and 50 μm, respectively; the LC layer and glass dielectric layer were 10 and 20 μm thick, respectively; the supplied voltage across the hole-patterned and bottom planar electrodes was 60 V. Consequently, as the FRE position approaches the hole-patterned electrode, the potential near the AH center (black dash-lines) strengthens, assists in the reorientation of LCs in the AH center and hence lowers Vad of LC lens. The strengthened potential indicates the spreading of fringing electric field into the AH center, increases the dielectric polarization in the AH and the associated the capacitance of lens cell [49], as shown in Fig. 3(e). The measured capacitances of the LC lenses demonstrate the similar tendency that the capacitance increases as the FRE position approaches the hole-patterned electrode, as shown in Fig. 3(f). Figures 2(e), 2(i), and 2(m) indicate that the interference fringes are difficult to spread toward the AH center if the FRE position is close to the LC layer, because the fringing electric field in the LC layer is gathered around the large inner diameter of FRE which is close to the AH periphery [36]. In this paper, Vad of FRE-bottom LC lens is higher than that of FRE-free LC lens. However, our previous work presents Vad of FRE-bottom LC lens is lower than that of FRE-free LC lens [33]. This is because in that paper the inner diameter of the inserted FRE is relatively small (2 mm), and then the fringing electric field in the LC layer is gathered around the FRE near the AH center, assisting in rotating LCs in the AH center and thus decreasing Vad of the LC lens. In the current study, as voltage increases, the maximum fringe numbers in the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses occur at 70, 50, 60, and 70 V, respectively (Figs. 2(c), 2(g), 2(k), and 2(o), respectively). At low voltages (Figs. 2(a), 2(e), 2(i), and 2(m)), the dark patterns near the AH center are attributed to the nonuniform cell gaps due to the handmade process. As shown in Figs. 2(q)–2(t), the change in the interference fringes of the FRE-free LC lens is almost similar to that of the FRE-top LC lens.Fig. 2. Interference fringes of the FRE-top LC lens at (a) 20, (b) 38, (c) 70, and (d) 100 V; interference fringes of the FRE-inplane LC lens at (e) 20, (f) 30, (g) 50, and (h) 100 V; interference fringes of the FRE-middle LC lens at (i) 20, (j) 40, (k) 60, and (l) 100 V; interference fringes of the FRE-bottom LC lens at (m) 20, (n) 45, (o) 70, and (p) 100 V; interference fringes of the FRE-free LC lens at (q) 20, (r) 38, (s) 70, and (t) 100 V.
Fig. 3. Calculated potential distribution in the LC layers of (a) FRE-top, (b) FRE-inplane, (c) FRE-middle, and (d) FRE-bottom LC lenses. The color indicates the potential intensity. (e) Calculated capacitances of the lens cells. (f) Measured frequency-dependent capacitances of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses.
Figure 4 shows the measured focal lengths of the LC lenses as a function of supplied voltages. A He–Ne laser with a wavelength of 632.8 nm was incident normally into the LC lens cell. The LC lens was placed behind a polarizer with a transmission axis parallel to the rubbing direction of the lens cell. The focal length was defined as the distance between the LC lens and focused point. The minimum supplied voltage for measuring the focal lengths was Vad of LC lens. In Fig. 4, the measured focal lengths initially decrease and then increase with the increasing voltage. The tunable focal lengths of FRE-top LC lens ranged from 55 to 75 cm at voltages of 38–140 V, those of FRE-inplane LC lens ranged from 56 to 82 cm at voltages of 30–50 V, those of FRE-middle LC lens ranged from 61 to 78 cm at voltages of 40–140 V, and those of FRE-bottom LC lens ranged from 58 to 76 cm at voltages of 45–140 V. In a summary, the FRE-inplane LC lens exhibited a relatively wide focus tunable range with low operation voltages and a narrow operation voltage range, whereas the FRE-top LC lens has the largest lens power. Notably, the operation voltage can be further reduced via optimization in the structure parameters of LC lens.
Fig. 4. Voltage-dependent focal lengths of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses.
Figure 5 presents the calculated focusing spot sizes of the LC lens operated at maximum lens power (MaxP). The full width at half maximum (FWHM) of the focusing spot could be used to analyze the focusing quality of the LC lens. In optics, the best focusing spot was described by an Airy disk that a perfect lens with a circular aperture could generate, limited by light diffraction [50]. The FWHM value of the diffraction-limited spot (Airy disk) can be calculated by using the following well-known formulas [51,52]:
where NA is the numerical aperture, r denotes the AH radius of the LC lens, f represents the focal length, λ indicates the wavelength of incident light, and dFWHM refers to the FWHM of the Airy disk. From Fig. 4, the measured focal lengths of the LC lenses addressed at MaxP are substituted into Eqs. (2) and (3) to calculate NA and dFWHM. If the obtained FWHM of the focusing spot is less than 1.38X dFWHM, then the LC lens is considered to possess a good focusing quality. Figure 5 shows that the measured FWHMs of the focusing spot of the FRE-top LC lens exceeds 1.38X dFWHM, which indicates a poor focusing quality. By contrast, those of the FRE-inplane, FRE-middle, and FRE-bottom LC lenses are smaller than 1.38X dFWHM, thereby showing excellent focusing qualities.Fig. 5. Measured focusing spot sizes and calculated diffraction-limited values of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses addressed at MaxP.
Figure 6(a) plots the phase retardation profiles of the LC lenses operated at MaxP by using the obtained optical interference fringes (Figs. 2(c), 2(g), 2(k), and 2(o)). In Fig. 6(a), symbols and solid lines indicate the measured results and quadratic fitting curves, respectively. Among these LC lenses, the FRE-top LC lens provides the largest phase difference between the AH center and periphery. The wavefront error of experimental data was then calculated from the ideal quadratic curve to examine the extent of the wavefront aberration of the LC lenses [15,53]. The wavefront error was defined as the root mean square (RMS) of the difference between the experimental data and fitted quadratic curve. The calculated RMS error was characterized with the unit of λ. A low RMS error implied superior lens quality. The RMS error of 0.07 λ was defined as the common standard for the conventional solid lens [15,53]. Table 1 shows the RMS errors of the LC lenses operated at MaxP. The FRE-bottom LC lens has a relatively low RMS error of 0.070 λ at MaxP due to the small deviation between the measured phase retardation and quadratic-fitting curve near the AH periphery (red dashed circle in Fig. 6(a)). In this study, the inner diameter of the FRE is large, such that the FRE is close to the AH periphery. As the voltage is supplied to the LC lens, the electric field is gathered near the AH periphery due to the embedded FRE, thereby increasing the gradients of the electric field and phase profile near the AH periphery and considerably decreasing the phase deviations in the AH periphery and the RMS error. Figure 6(b) shows that the RMS errors of the LC lenses increase with the supplied voltages. FRE-inplane LC lens has a low wavefront error of less than 0.1 λ in a narrow operation voltage range (30–50 V) with a wide focus tunable range. In the operation range, FRE-bottom structure also significantly suppresses the wavefront aberration of LC lens but with a narrow tunable focal range and a wide operation voltage range. Notably, the measured focusing spot size is also related to the wavefront error of lens. Large RMS error contributes wide focusing spot. As shown in Fig. 5, the measured focusing spot size of FRE-top LC lens exceeds the 1.38x dFWHM, owing to the large RMS error shown in Table 1. With the polarization of incident light parallel to the rubbing direction of LC lens, the focusing efficiency of LC lens is ∼72% [54], owing to the interfacial reflections.
Fig. 6. (a) Phase retardations of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses addressed at MaxP. The symbols and solid lines represent the measured data and quadratic-fitting curve, respectively. (b) RMS errors of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses at various supplied voltages.
Table 1. Estimated RMS error of LC lens at MaxP
An edge pattern shown in the inset of Fig. 7 was used as the object and placed in front of the LC lens to discuss the image resolution. The distance between the object and LC lens cell was 10 cm. A CCD camera with a lens module was placed behind the LC lens cell to capture the object image. The distance between the CCD camera and LC lens cell was 1 cm. The CCD camera was attached with a polarizer having a transmission axis parallel to the rubbing direction of the LC lens cell. The image was blurred before the voltage was applied to the LC lens. When the LC lens was operated at MaxP, the obtained clear image was used to calculate the MTF by using the commercially available software Quick MTF. MTF was defined in the following way:
where Imax and Imin are the maximum and minimum intensities in the image, respectively. Generally, MTF falls from 100% to 0% as the spatial frequency increases. As shown in Fig. 7, the FRE-bottom LC lens performs the best MTF curve, which descends slowly with the increasing spatial frequency. At spatial frequencies below 0.3 cycles/pixel, the MTF curve of FRE-free LC lens is similar to that of FRE-top LC lens. However, at spatial frequencies above 0.3 cycles/pixel, the MTF of FRE-free LC lens descends rapidly. Therefore, FRE-top and FRE bottom LC lenses exhibit excellent image resolutions. By contrast, the FRE-middle LC lens shows the worst image resolution due to the rapidly descending MTF below 0.4 cycles/pixel. Results from the measured focal lengths and estimated MTFs imply that the FRE-bottom LC lens realizes a relatively excellent imaging resolution but a moderate focus tunable range. On the contrary, the FRE-inplane LC lens realizes the widest focus tunable range within low operation voltages but a moderate image resolution. The image performance can be affected by others factors, such as scattering of the LC cell and color dispersion of the LC material. Table 2 summarizes the electro-optical features of FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses.Fig. 7. MTFs of FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses. The inset indicates the edge pattern used as object.
Table 2. Summary of the FRE-top, FRE-inplane, FRE-middle, and FRE-bottom LC lenses
4. Conclusions
We investigated the influences of FRE at different positions of the hole-patterned LC lens and successfully revealed the design criterion of the FRE-embedded LC lens. The FRE must be placed in the same plane with the hole-patterned electrode to achieve the lowest operation voltage and widest tunable focal length capability. By contrast, the FRE must be placed near the LC layer to obtain the best image performance. The electro-optical performances of the LC lenses could be further improved by optimizing the material and geometry parameters of the used FRE, glass substrate, and LC layer.
Funding
Ministry of Science and Technology, Taiwan (MOST) (104-2112-M-018-003-MY3, 107-2112-M-018-003-MY3, 107-2811-M-018-003).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. H. Ren, D. W. Fox, B. Wu, and S.-T. Wu, “Liquid crystal lens with large focal length tunability and low operating voltage,” Opt. Express 15(18), 11328–11335 (2007). [CrossRef]
2. C.-W. Chiu, Y.-C. Lin, P. C. P. Chao, and A. Y. G. Fuh, “Achieving high focusing power for a large-aperture liquid crystal lens with novel hole-and-ring electrodes,” Opt. Express 16(23), 19277–19284 (2008). [CrossRef]
3. H. Ren, Y.-H. Fan, and S.-T. Wu, “Tunable Fresnel lens using nanoscale polymer-dispersed liquid crystals,” Appl. Phys. Lett. 83(8), 1515–1517 (2003). [CrossRef]
4. W. Bin, Y. Mao, and S. Sato, “Liquid crystal lens with focal length variable from negative to positive values,” IEEE Photonics Technol. Lett. 18(1), 79–81 (2006). [CrossRef]
5. H. Ren and S.-T. Wu, Introduction to Adaptive Lenses (John Wiley & Sons, 2012), Vol. 75.
6. Y.-H. Lin, Y.-J. Wang, and V. Reshetnyak, “Liquid crystal lenses with tunable focal length,” Liq. Cryst. Rev. 5(2), 111–143 (2017). [CrossRef]
7. J. F. Algorri, D. C. Zografopoulos, V. Urruchi, and J. M. Sánchez-Pena, “Recent Advances in Adaptive Liquid Crystal Lenses,” Crystals 9(5), 272 (2019). [CrossRef]
8. S. Susumu, “Liquid-Crystal Lens-Cells with Variable Focal Length,” Jpn. J. Appl. Phys. 18(9), 1679–1684 (1979). [CrossRef]
9. S. Sato, “Applications of liquid crystals to variable-focusing lenses,” Opt. Rev. 6(6), 471–485 (1999). [CrossRef]
10. J. Prost, The Physics of Liquid Crystals (Oxford university press, 1995), Vol. 83.
11. D.-K. Yang, Fundamentals of Liquid Crystal Devices (John Wiley & Sons, 2014).
12. H.-C. Lin, M.-S. Chen, and Y.-H. Lin, “A review of electrically tunable focusing liquid crystal lenses,” IEEE Trans. Electr. Electron. Mater. 12(6), 234–240 (2011). [CrossRef]
13. J. Tan, Y. Song, J.-L. Zhu, S.-B. Ni, Y.-J. Wang, X.-Y. Sun, J.-G. Lu, B.-R. Yang, and H.-P. D. Shieh, “Blue phase LC/polymer Fresnel lens fabricated by holographics,” J. Disp. Technol. 10(2), 157–161 (2014). [CrossRef]
14. S.-H. Lin, B.-Y. Huang, C.-Y. Li, K.-Y. Yu, J.-L. Chen, and C.-T. Kuo, “Electrically and optically tunable Fresnel lens in a liquid crystal cell with a rewritable photoconductive layer,” Opt. Mater. Express 6(7), 2229–2235 (2016). [CrossRef]
15. Y.-Y. Kao and P. C.-P. Chao, “A new dual-frequency liquid crystal lens with ring-and-pie electrodes and a driving scheme to prevent disclination lines and improve recovery time,” Sensors 11(5), 5402–5415 (2011). [CrossRef]
16. H. Ren and S.-T. Wu, “Tunable electronic lens using a gradient polymer network liquid crystal,” Appl. Phys. Lett. 82(1), 22–24 (2003). [CrossRef]
17. F. Serra, M. A. Gharbi, Y. Luo, I. B. Liu, N. D. Bade, R. D. Kamien, S. Yang, and K. J. Stebe, “Curvature-Driven, One-Step Assembly of Reconfigurable Smectic Liquid Crystal “Compound Eye” Lenses,” Adv. Opt. Mater. 3(9), 1287–1292 (2015). [CrossRef]
18. P. Popov, L. W. Honaker, M. Mirheydari, E. K. Mann, and A. Jákli, “Chiral nematic liquid crystal microlenses,” Sci. Rep. 7(1), 1603 (2017). [CrossRef]
19. W. Duan, P. Chen, S.-j. Ge, B.-y. Wei, W. Hu, and Y.-q. Lu, “Helicity-dependent forked vortex lens based on photo-patterned liquid crystals,” Opt. Express 25(13), 14059–14064 (2017). [CrossRef]
20. W. Duan, P. Chen, S.-J. Ge, X. Liang, and W. Hu, “A Fast-Response and Helicity-Dependent Lens Enabled by Micro-Patterned Dual-Frequency Liquid Crystals,” Crystals 9(2), 111 (2019). [CrossRef]
21. B. Wang, M. Ye, M. Honma, T. Nose, and S. Sato, “Liquid crystal lens with spherical electrode,” Jpn. J. Appl. Phys. 41(Part 2), L1232–L1233 (2002). [CrossRef]
22. M. Ye, S. Hayasaka, and S. Sato, “Liquid crystal lens array with hexagonal-hole-patterned electrodes,” Jpn. J. Appl. Phys. 43(9A), 6108–6111 (2004). [CrossRef]
23. T. Nose, S. Masuda, and S. Sato, “A liquid crystal microlens with hole-patterned electrodes on both substrates,” Jpn. J. Appl. Phys. 31(Part 1), 1643–1646 (1992). [CrossRef]
24. Y.-H. Lin, H. Ren, K.-H. Fan-Chiang, W.-K. Choi, S. Gauza, X. Zhu, and S.-T. Wu, “Tunable-focus cylindrical liquid crystal lenses,” Jpn. J. Appl. Phys. 44(1A), 243–244 (2005). [CrossRef]
25. G. Li, P. Valley, M. Giridhar, D. L. Mathine, G. Meredith, J. N. Haddock, B. Kippelen, and N. Peyghambarian, “Large-aperture switchable thin diffractive lens with interleaved electrode patterns,” Appl. Phys. Lett. 89(14), 141120 (2006). [CrossRef]
26. Y. Lou, L. Chen, C. Wang, and S. Shen, “Tunable-focus liquid crystal Fresnel zone lens based on harmonic diffraction,” Appl. Phys. Lett. 101(22), 221121 (2012). [CrossRef]
27. H. Ren and S.-T. Wu, “Adaptive liquid crystal lens with large focal length tunability,” Opt. Express 14(23), 11292–11298 (2006). [CrossRef]
28. B. Wang, M. Ye, and S. Sato, “Lens of electrically controllable focal length made by a glass lens and liquid-crystal layers,” Appl. Opt. 43(17), 3420–3425 (2004). [CrossRef]
29. O. Sova, V. Reshetnyak, T. Galstian, and K. Asatryan, “Electrically variable liquid crystal lens based on the dielectric dividing principle,” J. Opt. Soc. Am. A 32(5), 803–808 (2015). [CrossRef]
30. K. Asatryan, V. Presnyakov, A. Tork, A. Zohrabyan, A. Bagramyan, and T. Galstian, “Optical lens with electrically variable focus using an optically hidden dielectric structure,” Opt. Express 18(13), 13981–13992 (2010). [CrossRef]
31. H.-C. Lin and Y.-H. Lin, “An electrically tunable-focusing liquid crystal lens with a low voltage and simple electrodes,” Opt. Express 20(3), 2045–2052 (2012). [CrossRef]
32. L. Li, D. Bryant, T. Van Heugten, and P. J. Bos, “Physical limitations and fundamental factors affecting performance of liquid crystal tunable lenses with concentric electrode rings,” Appl. Opt. 52(9), 1978–1986 (2013). [CrossRef]
33. X.-Q. Wang, A. K. Srivastava, V. G. Chigrinov, and H.-S. Kwok, “Switchable Fresnel lens based on micropatterned alignment,” Opt. Lett. 38(11), 1775–1777 (2013). [CrossRef]
34. M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, and S. Sato, “Focus tuning by liquid crystal lens in imaging system,” Appl. Opt. 51(31), 7630–7635 (2012). [CrossRef]
35. C. J. Hsu and C. R. Sheu, “Using photopolymerization to achieve tunable liquid crystal lenses with coaxial bifocals,” Opt. Express 20(4), 4738–4746 (2012). [CrossRef]
36. C.-J. Hsu, J.-J. Jhang, and C.-Y. Huang, “Large aperture liquid crystal lens with an imbedded floating ring electrode,” Opt. Express 24(15), 16722–16731 (2016). [CrossRef]
37. H. Dou, F. Chu, Y.-Q. Guo, L.-L. Tian, Q.-H. Wang, and Y.-B. Sun, “Large aperture liquid crystal lens array using a composited alignment layer,” Opt. Express 26(7), 9254–9262 (2018). [CrossRef]
38. M.-S. Chen, P.-J. Chen, M. Chen, and Y.-H. Lin, “An electrically tunable imaging system with separable focus and zoom functions using composite liquid crystal lenses,” Opt. Express 22(10), 11427–11435 (2014). [CrossRef]
39. Y.-H. Lin and H.-S. Chen, “Electrically tunable-focusing and polarizer-free liquid crystal lenses for ophthalmic applications,” Opt. Express 21(8), 9428–9436 (2013). [CrossRef]
40. Y. Lei, Q. Tong, X. Zhang, H. Sang, A. Ji, and C. Xie, “An electrically tunable plenoptic camera using a liquid crystal microlens array,” Rev. Sci. Instrum. 86(5), 053101 (2015). [CrossRef]
41. J. F. Algorri, V. Urruchi, N. Bennis, P. Morawiak, J. M. Sánchez-Pena, and J. M. Otón, “Integral Imaging Capture System With Tunable Field of View Based on Liquid Crystal Microlenses,” IEEE Photonics Technol. Lett. 28(17), 1854–1857 (2016). [CrossRef]
42. Y.-J. Wang, X. Shen, Y.-H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015). [CrossRef]
43. A. Hassanfiroozi, Y.-P. Huang, B. Javidi, and H.-P. D. Shieh, “Dual layer electrode liquid crystal lens for 2D/3D tunable endoscopy imaging system,” Opt. Express 24(8), 8527–8538 (2016). [CrossRef]
44. T. Galstian, K. Asatryan, V. Presniakov, A. Zohrabyan, A. Tork, A. Bagramyan, S. Careau, M. Thiboutot, and M. Cotovanu, “High optical quality electrically variable liquid crystal lens using an additional floating electrode,” Opt. Lett. 41(14), 3265–3268 (2016). [CrossRef]
45. T. Galstian, O. Sova, K. Asatryan, V. Presniakov, A. Zohrabyan, and M. Evensen, “Optical camera with liquid crystal autofocus lens,” Opt. Express 25(24), 29945–29964 (2017). [CrossRef]
46. O. Sova and T. Galstian, “Liquid crystal lens with optimized wavefront across the entire clear aperture,” Opt. Commun. 433, 290–296 (2019). [CrossRef]
47. C.-J. Hsu, J.-J. Jhang, J.-C. Jhang, and C.-Y. Huang, “Influence of floating-ring-electrode on large-aperture liquid crystal lens,” Liq. Cryst. 45(1), 40–48 (2018). [CrossRef]
48. M. Ye and S. Sato, “Optical Properties of Liquid Crystal Lens of Any Size,” Jpn. J. Appl. Phys. 41(Part 2), L571–L573 (2002). [CrossRef]
49. T. B. Jones and N. G. Nenadic, Electromechanics and MEMS (Cambridge University Press, 2013).
50. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2008).
51. M. Honma, T. Nose, and S. Sato, “Optimization of Device Parameters for Minimizing Spherical Aberration and Astigmatism in Liquid Crystal Microlenses,” Opt. Rev. 6(2), 139–143 (1999). [CrossRef]
52. S. Masuda, S. Takahashi, T. Nose, S. Sato, and H. Ito, “Liquid-crystal microlens with a beam-steering function,” Appl. Opt. 36(20), 4772–4778 (1997). [CrossRef]
53. Y.-Y. Kao, P. C. P. Chao, and C.-W. Hsueh, “A new low-voltage-driven GRIN liquid crystal lens with multiple ring electrodes in unequal widths,” Opt. Express 18(18), 18506–18518 (2010). [CrossRef]
54. C. J. Hsu, K. Agrahari, P. Selvaraj, W. F. Chiang, C. Y. Huang, R. Manohar, and C. Y. Huang, “Application of ultra-thin indium–tin–oxide film in liquid crystal lens,” Opt. Laser Technol. 119, 105603 (2019). [CrossRef]
