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Abstract
Microlens has significant applications in integrated micro-optical systems. Recently, multifocal microlens arrays are expected to extend the depth of field for imaging systems and realize a highly efficient laser beam homogenizer. This work presents what we believe to be a novel approach for developing a tunable multifocal liquid crystal microlens array (TMLCMA), which can be operated in convex and concave modes through voltage control schemes. The TMLCMA is manufactured using nematic liquid crystals (LCs) with negative dielectric anisotropy, in conjunction with a triple-electrode structure consisting of top large-hole, middle small-hole array, and bottom planar electrodes. When a voltage is applied, the axially symmetric fringing electric field induced by the large-hole electrode causes the focal length of the microlens to gradually and radially change from the TMLCMA border toward the center. The gradient in the change of focal length is electrically tunable. The calculated spatial potential distributions qualitatively explain the multifocal characteristic and dual lens modes of the TMLCMA. The LC molecules in each microlens are reoriented in an axially symmetrical form, resulting in a polarization-insensitive TMLCMA. The imaging functions of the TMLCMA operated with dual lens modes are shown through practical demonstrations. The simple fabrication and versatile function make the developed TMLCMA highly promising for various optical system applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The development of optical systems has broadened the market for the use of microlens arrays. Microlens array has numerous applications, such as imaging systems [1], zoom system [2], beam steering [3], 2D/3D switchable display [4], astronomy [5], and aberration correction [6]. A microlens array with a tunable focus over a wide range is of great interest for increasing the efficiency of light detection, recording, imaging, and coupling. Liquid crystal (LC) materials possess large electrical and optical anisotropies, which are promising for fabricating micro-optical components for converging or diverging incident rays. In the past couple of years, the LC microlens array (LCMA) with a tunable focal length has gained popularity as an alternative approach, replacing the conventional focus-fixed microlens array. The non-mechanical beam steerer LCMA has played a crucial role in various applications, such as light detection [7] and telescope [8]. In LCMAs, a spatial gradient distribution of refractive index can be induced and continuously changed by applying voltage, eliminating the need for mechanical movement, which results in a compact structure and rapid response time. Several methods have been demonstrated to fabricate LCMAs, such as surface relief [9], hole-patterned electrode [10], spherical electrode [11], ink-jet printing [12], polymer dispersions [13,14], droplet evaporation [15], and nanoparticle-induced alignment [16]. However, these LCMAs provide a single focal length at a specific voltage, limiting their applications for optical systems. The vergence–accommodation conflict is a major problem in augmented reality (AR) and virtual reality (VR) displays [17,18]. This conflict occurs when the human brain receives mismatching hints between vergence and accommodation of the eye. The tunable multifocal LC optical device can be used to adjust the image plane, leading to the focus of user coordinates with the changing vergence, which mitigates the vergence–accommodation conflict [19]. In the homogenization of highly coherent laser beam [20,21], a microlens array with a single focal length produces a periodic lattice phenomenon, resulting in the degradation of the homogenization quality. The multifocal microlens array can be used to improve the beam homogenization effect. In the field of autostereoscopic vision [22], the depth of field can be enhanced by using a multifocal microlens array. An individually controllable optical tweezer is achieved by combining a microlens array and spatial light modulation [23]. The tunable multifocal LCMA can simplify the components and create a compact system of individually controllable optical tweezers without the need for spatial light modulation. To date, the tunable multifocal LCMAs based on array of interleaved microlenses with multiple aperture sizes have been reported [22,24]. The number of the focal lengths is mainly determined with the number of aperture sizes, thereby limiting the multifocal tunability of the LCMAs.
In contrast with previous multifocal LCMAs that utilized a multi-aperture-patterned electrode [22,24], where the microlenses were only functioned in a convex mode and created limited number of focal lengths, we propose a novel tunable multifocal LCMA (TMLCMA) based on the triple-electrode structure and the vertically-aligned LCs with negative dielectric anisotropy (defined as negative LCs hereafter). In the TMLCMA, the microlenses can be operated in concave and convex modes through voltage control schemes. The focal length of the microlens gradually and radially changes from the TMLCMA border toward the center due to the axially symmetric fringing electric field induced by the large-hole electrode. The gradient multifocal lengths can be tunable with the frequency and amplitude of applied voltage. The spatial potential profiles of the microlenses were calculated to understand the gradient multifocal property and lens modes. The interference fringes and the voltage-dependent focal lengths were measured to demonstrate the tunable gradient multifocal lengths. The optical performance through the TMLCMA were examined and demonstrated.
2. Materials and methods
Three indium-tin-oxide (ITO) glass substrates were used for the TMLCMA cell. The glass substrate was 0.55 mm-thick, and the ITO layer deposited on the glass substrate was 100 nm-thick. The ITO layers on the top of the bottom substrate and on the bottom of the middle substrate were spin-coated with the vertical polyimide (AL-8088C-0000-21NI, Daily Polymer, Taiwan), as depicted in Fig. 1(a). The bottom and middle substrates were separated by 50 µm-thick Mylar spacers. The ITO layer of the top substrate was placed face down. Meanwhile, the ITO layers of the top and middle substrates were etched as large-hole electrode and small-hole array electrode by photolithography, respectively. Figure 1(b) shows the optical microscope (OM) image of the small-hole array of etched substrate. The diameters of the large and small holes were 6 mm and 300 µm, respectively. The pitch of the small hole array was 450 µm. A semiconductor zinc-tin-oxide (ZTO) thin film was spin-coated on the large-hole electrode of the top substrate as a modal layer to assist in the distribution of the fringing electric field throughout the 6 mm-diameter hole area [25]. The sheet resistivity of the ZTO thin film was ∼ 36 MΩ/sq. Nematic LC HNG736300-000 (Fusol Material Co., Ltd., Tainan, Taiwan) used in this experiment had a birefringence (Δn) of 0.089 (n0 = 1.479 and ne = 1.568) at the wavelength λ of 589 nm and a dielectric anisotropy (Δε) of −3.8 at the frequency of 1 kHz at room temperature. The LC mixture was initially heated up to the isotropic phase and injected into the empty TMLCMA cell through capillary action. After injection, the LC mixture was cooled down to the nematic phase. The actual photo of the assembled TMLCMA cell is shown in Fig. 1(c). The assembled TMLCMA was manipulated with various voltage control schemes. For the floating scheme, a square-wave AC voltage was supplied to the TMLCMA across the top large-hole and bottom planar ITO electrodes, and the middle small-hole array electrode was kept at a floating potential, as depicted in Fig. 1(d). If the middle small-hole array electrode was kept at equipotential with the bottom planar ITO electrode, then the voltage control was defined as the equipotential scheme, as depicted in Fig. 1(e). In the cases of floating and equipotential schemes, the microlenses were operated in concave and convex modes, respectively; the focal length of the microlens radially decreased and increased from the TMLCMA border toward the center, respectively. The detailed mechanism will be discussed in the following section.
Fig. 1. (a) Structure diagram of TMLCMA. (b) OM image of the small-hole array electrode. (c) Actual photo of the TMLCMA cell. Voltage controls of the (d) floating and (e) equipotential schemes.
3. Results and discussions
The spatial potential distributions of the microlenses in the TMLCMA were calculated by using the finite element method software COMSOL to qualitatively explain the lens functions. In this calculation, a simplified cell structure was used to mimic the proposed TMLCMA, and its cell parameters were as follows. The thickness and dielectric constant of the top and middle glass substrates in the TMLCMA were 0.55 mm and 6.7, respectively. The diameter of the large-hole electrode was 6 mm. Fifteen 300 µm-diameter small holes were laid out along the x axis, as shown in Fig. 2(a). The pitch of the small hole array was set to 350 µm. A 30 nm-thick layer with a conductivity of 0.05 S/m was set on the surface of the large-hole electrode as a modal layer. The LC thickness was set to 60 µm. A 5 kHz (2 kHz) sinusoidal wave AC signal with an amplitude of 20 V (60 V) was set to address the TMLCMA with the floating (equipotential) scheme. When the TMLCMA was operated with the floating scheme, the potential increased with the radius in each microlens, as shown in Figs. 2(b)–2(d). In this case, the LCs at the microlens border were tilted parallel to the substrate surface, but those at the microlens center were vertically aligned. The effective refractive index for light that was transmitted through the microlens exhibited a gradient profile in an axially symmetrical radial form. The LCs at the microlens border had a larger refractive index than those at the microlens center. Consequently, the incident light diverged, and the microlenses operated in concave mode. By contrast, when the equipotential scheme was employed, the potential decreased with the radius in each microlens, indicating that the LCs at the microlens border had a smaller refractive index than those at the microlens center, as shown in Figs. 2(e)–2(g). The incident light converged, and the microlenses operated in convex mode. The potential gradient of the microlens increased (decreased) from the TMLCMA border toward the center with the floating (equipotential) scheme, indicating that the refractive index profile of the microlens and the associated focal length radially changed from the TMLCMA border toward the center, that is, gradient multifocal lengths. Figures 2(h)–2(i) demonstrate the tilt angle distributions of the LC directors when the TMLCMA were operated with the floating scheme at 20 V and 5 kHz and the equipotential scheme at 60 V and 2 kHz, respectively. In Fig. 2(h), the tilt angles of the LCs at the microlens border were smaller than those of LCs at the microlens center when the TMLCMA was operated with the floating scheme, indicating that the microlenses were operated in concave mode. By contrast, in Fig. 2(i), the tilt angles of the LCs at the microlens border were larger than those of LCs at the microlens center when the TMLCMA was operated with the equipotential scheme, indicating that the microlenses were operated in convex mode.
Fig. 2. (a) Microlenses located at the TMLCMA center, middle, and border are indicated as microlenses C, M, and B, respectively. Calculated potential distributions in microlenses (b) B, (c) M, and (d) C when the TMLCMA is operated with the floating scheme; calculated potential distributions in microlenses (e) B, (f) M, and (g) C when the TMLCMA is operated with the equipotential scheme. Tilt angles of the LC directors when the TMLCMA is operated in the (h) floating and (i) equipotential schemes. The scale bar indicates the magnitude of the tilt angle.
Figure 3 shows the measured interference fringes of the TMLCMA by using the typical setup [26]: a He–Ne laser with a wavelength of 632.8 nm was used as the incident light. The TMLCMA cell was placed between a pair of crossed polarizers. The transmitted image with interference fringes was recorded with a charge-coupled device (CCD) camera. The adjacent bright or dark fringes resembled a phase change of 2π. Three microlenses from the TMLCMA border toward the center were defined as microlenses B, M, and C, as indicated in Fig. 2(a). The 500 and 12 Hz square wave AC signals were used to address the TMLCMA with the floating and equipotential schemes, respectively. The signal frequencies used in the experiment were different from those (5 and 2 kHz) used in the calculation due to the simplified cell structure in the calculation. As shown in Figs. 3(a) and 3(b), in the case of the floating scheme, the fringe number (N) initially increases and thereafter decreases with the applied voltage. Microlenses B, M, and C have the same N below 15 V. N can be correlated with the focal length (f) of microlens according to Eq. (1): [27]
where r is the radius of microlens. Therefore, the microlenses exhibit single focal lengths at voltages lower than 15 V. As the voltage increases, the N of the microlens increases with the voltage, and also increases from the TMLCMA border toward the center (microlens B toward microlens C). The maximum N in microlens C appears around 30 V (Fig. 3(a)), where the LCs at the microlens border and center were aligned parallel and perpendicular to the substrate surface, respectively. When the applied voltage further increases (Fig. 3(b)), N starts to decrease because the LCs in the microlens center gradually tilted parallel to the substrate surface. Some fringes are observed between the adjacent microlenses due to the fringing electric field generated at the microlens border. As shown in Figs. 3(c) and 3(d), when the TMLCMA is operated with the equipotential scheme, the fringe starts to occur in microlens C at 55 V (Fig. 3(c)). N gradually increases with the applied voltage because the LCs at the microlens center are gradually tilted parallel to the substrate surface, and those at the microlens border remain vertical (Fig. 3(d)). In contrast to the result with the floating scheme, the N of the microlens decreases from the TMLCMA border toward the center with the equipotential scheme. The dark state appeared between the adjacent microlenses because the LCs in this area are vertically aligned. When various voltage controls are employed, the changes in N from the TMLCMA border toward the center were opposite, which is in agreement with those in the calculated potential gradient from the TMLCMA border toward the center (Fig. 2). The crossed dark pattern in each microlens indicates that the LC molecules are reoriented in an axially symmetrical form. The high operation voltages could be further improved by optimizing the modal layer and using the LC material with high dielectric anisotropy.Fig. 3. Interference fringe images of the TMLCMA operated with the floating scheme at (a) 30 V, 500 Hz and (b) 43 V, 500 Hz; interference fringe images of the TMLCMA operated with the equipotential scheme at (c) 55 V, 12 Hz and (d) 95 V, 12 Hz.
The polarizers used in the fringe measurement setup were removed to determine the focal length of the microlens. The CCD camera was initially focused on the microlens surface, and the TMLCMA cell was moved forward or backward the laser source until a sharp focal point was found. The distance that the TMLCMA cell traveled was equal to the focal length of the microlens [28–30]. The measured voltage-dependent focal lengths of microlenses B, M, and C are plotted in Fig. 4. When no voltage is applied, the microlenses do not have lens function because all the LC molecules are vertically aligned where their focal lengths are infinite. In the case of the equipotential scheme, the microlenses operate in convex mode and have positive focal lengths that decrease with the applied voltage because the LCs at the microlens center gradually tilt parallel to the substrate surface, as shown in Fig. 4(a). The focal length of the microlens C is infinite at voltages below 55 V because of the weak electric field in the TMLCMA center. As the voltage increases, the LCs at the center of the microlens B are tilted parallel to substrate surface prior to those at the center of the microlens C, due to the larger electric field at the TMLCMA border, while the LCs at the border of the microlenses are always aligned vertically. Once the voltage is sufficiently high, the focal length of microlens B will approach saturation earlier than that of microlens C. Therefore, the difference in focal lengths of microlenses B and C decreases with the applied voltage. By contrast, the manipulation of the floating scheme causes the microlenses to operate in concave mode and have negative focal lengths. The focal lengths initially decrease and thereafter increase with the applied voltage, as shown in Fig. 4(b). The TMLCMA starts to exhibit the multifocal characteristic above 15 V. Once the voltage exceeded 30 V, the focal lengths start to increase because the LCs at the microlens border and center were gradually tilted parallel to the substrate surface. Below 30 V, the microlens B experiences a less decrease in the absolute value of focal length compared to the microlens C with the applied voltage. This difference can be attributed to the lower potential gradient within microlens B (Figs. 2(b)–2(d)). When the voltage surpasses 30 V, the LCs within microlens B tilt in parallel to the substrate surface before those in microlens C, owing to the presence of a larger electric field at the TMLCMA border. As a result, the focal length of microlens B will reach infinite prior to that of microlens C as the voltage continues to rise, so that the difference in focal lengths of microlenses B and C increases with the applied voltage. The difference in the focal lengths of microlenses B, M and C can be comprehended by the potential calculations in Figs. 2(b)–2(g). The potential gradient of microlens increases (decreases) from the TMLCMA border toward the center with the floating (equipotential) scheme. This result indicates that the gradient in refractive index profile of microlens increases (decreases) so that the focal length radially decreases (increases) from the TMLCMA border toward the center with the floating (equipotential) scheme. Furthermore, the minimum focal length can be estimated by the radius (r) of the microlens, the LC cell gap (d), and the refractive index difference ($\delta n$) between the border and the center of microlens, as follows:
In an ideal condition that maximizing the LC lens efficiency, $\delta n$ is equal to Δn. In our design, each microlens had r = 150 µm and d = 50 µm, and the LC had a birefringence Δn of 0.089. Accordingly, the microlens will have an estimated minimum focal length f = 2.5 mm. The absolute values of the measured minimum focal lengths were 3 and 4 mm with the equipotential and floating schemes, respectively, which were close to the estimated one (2.5 mm) (Fig. 4). These results indicated that the proposed TMLCMA had a high utilization efficiency of LC birefringence.
Fig. 4. Focal lengths as a function of applied voltages when the TMLCMA was operated with (a) equipotential and (b) floating schemes.
The focal length of microlens is primarily determined with the size of aperture hole. Consequently, the number of focal lengths in LCMA depends on the number of the aperture hole sizes, i.e., LCMA comprising of two distinct sizes of aperture holes produces two focal lengths. In previous design [22], the multifocal LCMA was fabricated with limited number of aperture hole sizes, thereby limiting the number of focal lengths of the LCMA. Besides, the fabricated microlenses can be only operated in convex mode. By contrast, thanks to the fringing electric field induced by the large-hole electrode, our design enables the focal lengths of the microlenses to gradually and radially change from the TMLCMA border toward the center. The gradient multifocal lengths can be tunable with the frequency and amplitude of applied voltage. Moreover, the microlenses can be operated in concave and convex modes with voltage control schemes. In the application of laser beam homogenization, the multiple focal lengths of microlens array is beneficial to eliminate the lattice phenomenon [20]. Accordingly, the tunable gradient multifocal feature makes our TMLCMA a more suitable candidate for use as a laser homogenizer compared to the previous design. The axially-symmetric LC reorientation also promotes the TMLCMA device to other interesting applications, such as vortex beam generator [31], spatial polarization converter array [32], and inspections in optic axis and phase retardation of the phase compensation film [33].
The setup of focal length measurement was arranged to test the converging and diverging abilities of the microlenses. Figures 5(a)–5(b) show the images of the spot patterns produced through the TMLCMA. The CCD camera is focused on the focal plane of microlens B in the TMLCMA operated with the equipotential scheme (Fig. 5(a)). The tight focus spot is observed in microlens B and gradually defocused from the TMLCMA border toward the center due to the gradient multifocal convex function. When the TMLCMA is operated with the floating scheme, the focal plane of the CCD camera is set near the sample surface (Fig. 5(b)). The light rings are gradually diverged from the TMLCMA border toward the center due to the gradient multifocal concave function. Furthermore, a polarizer was installed in the front of the TMLCMA operated with the equipotential scheme to control the linear polarization of incident light. The transmission axis of the polarizer was initially parallel to the Y-axis (0°) and then rotated clockwise. The peak intensity of the focus spot through microlens B was quantified with incident polarizations. Figure 5(c) shows that the peak intensity remains almost constant, indicating that the focused intensity through the proposed TMLCMA is polarization-insensitive. When the floating scheme is employed, the LCs are also reoriented in an axially symmetrical form. Accordingly, the polarization-insensitive behavior of the TMLCMA is also expected to exist. Figure 5(d) shows the focus spots and the corresponding intensity distributions of microlenses B and C when the TMLCMA is operated at (95 V, 12 Hz) with the equipotential scheme. The full width at half maximum (FWHM) of the focus spot of a lens can be correlated with its focal length [34]. The FWHM (40 µm) of microlens C is larger than that (30 µm) of microlens B due to the longer focal length of microlens C under the same applied voltage. The optical transfer function (OTF) of microlens B was calculated by Fourier transform of the measured point spread functions to quantitatively characterize the image resolution. The normalized magnitude of the OTF was referred to as the modulation transfer function (MTF) [35,36]. If the MTF value was beyond 0.5 until the spatial frequency of 50 cycles/mm, then the microlens would have a high-performance imaging quality [36]. However, the MTF value of microlens B in the presented TMLCMA has been lower than 0.1 at the spatial frequency of 50 cycles/mm (Fig. 5(e)). The imperfect imaging quality is caused by the scattering by a thick LC layer and microlens aberrations. In addition, the microlenses in the TMLCMA converge or diverge only a portion of unpolarized incident light due to the axially-symmetric radial reorientation and the intrinsic birefringence of LCs, which also lowers the focusing efficiency and imaging quality. To enhance the light utilization efficiency, a possible approach is to create the radially–azimuthally twisted LC reorientation in each microlens [37]. Furthermore, the switching time of the TMLCMA was measured ∼10 s. Unlike the LC rotation in the homogeneous LC cell with planar electrodes, that in the microlens is mainly due to the reorientation process under the fringing electric field provided by the hole-patterned electrode [38,39]. For example, with equipotential scheme, when the TMLCMA is suddenly supplied with voltage, the electric field at the microlens border almost remains zero, and the LC rotation at the microlens border follows after the LC rotation at the microlens center. Once the supplied voltage suddenly declines, the LCs at the microlens border quickly rewind due to the low tilt angle, and the LC rotation at the microlens center follows after the LC rotation at the microlens border by the reorientation process. The reorientation process causes a slow switching time. Meanwhile, the glass dielectric layer placed between the large-hole and small-hole array electrodes in the TMLCMA decreases the effective voltage dropped in the LC layer, which also increases the switching time. Some approaches can improve the slow switching response, such as the optimization of the modal layer to omit the glass dielectric layer, the use of the low viscosity of the negative LCs [40] or dual-frequency LCs [41], and the doping of an organic material [39,42].
Fig. 5. Images of the spot patterns through the TMLCMA operated with the (a) equipotential (70 V, 12 Hz) and (b) floating (30 V, 500 Hz) schemes. The red dash circle indicates microlens B. (c) Peak intensities of the focus spot through microlens B as a function of incident polarizations with the equipotential scheme. (d) Focus spots and the corresponding intensity distributions of microlenses B and C when the TMLCMA is operated at (95 V, 12 Hz) with the equipotential scheme. (e) MTF of microlens B.
Figure 6(a) shows the OM setup to demonstrate the imaging functions of the TMLCMA operated in dual lens modes. A transparent letter “7” with a height of 0.27 mm was used as a target and placed at the OM stage. The TMLCMA cell was placed at 10 and 15 mm above the target for the equipotential and floating schemes, respectively. A white light source passing through a red dichroic filter was incident from the bottom of the OM. The distance between the 5X objective and TMLCMA cell was set to 15 and 5 mm for the equipotential and floating schemes, respectively. A digital camera was installed atop the OM to record the images. The inverted real and upright virtual images were captured through the TMLCMA operated with the equipotential and floating schemes, respectively, due to the convex and concave modes, as shown in Figs. 6(b)–6(e). In the equipotential scheme, the camera is focused at the imaging plane of microlens B. Accordingly, the inverted real images captured near the TMLCMA border are clearer than those captured near the TMLCMA center. By contrast, if the TMLCMA is operated with the floating scheme, then the camera is focused at the imaging plane of microlens C. The relatively clear upright virtual images are observed in the TMLCMA center. The image gradually becomes blurry from the TMLCMA center out to the border. The results confirm the gradient multifocal convex and concave functions of the TMLCMA operated with the equipotential and floating schemes.
Fig. 6. (a) Target image and OM diagram. Imaging performance captured near the TMLCMA (b) border and (c) center through the TMLCMA operated with the equipotential scheme (70 V, 12 Hz); imaging performance captured near the TMLCMA (d) border and (e) center through the TMLCMA operated with the floating scheme (30 V, 500 Hz).
4. Conclusion
In summary, a versitile TMLCMA is successfully fabricated by using the triple-electrode structure and vertically-aligned negative LCs. In the equipotential and floating schemes, the calculated spatial potential distributions disclose that the microlenses operate in convex and concave modes, respectively; the potential gradient dropped in the microlens decreases and increases from the TMLCMA border toward the center, respectively; the experimental results reveal that the fringe number of microlens radially decreases and increases from the TMLCMA border toward the center, respectively; the difference in the focal lengths of microlenses B and C decreases and increases with the applied voltage, respectively. Moreover, the images of the spot patterns produced through TMLCMA confirm the gradient multifocal and polarization-insensitive characteristics. The imaging performance indicates the gradient multifocal convex and concave functions of TMLCMA on the basis of various voltage controls. The effects of signal frequency and waveform on the electro-optical characteristics of the proposed TMLCMA is currently under investigation.
Funding
National Science and Technology Council (112-2112-M-018-008, 112-2811-M-018-002, 111-2112-M-018-009, 111-2811-M-018-006).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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