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Abstract
We propose and demonstrate a high-performance refractive Fresnel liquid crystal (LC) lens with a simple electrode design. The interconnected circular electrodes enable the creation of a parabolic voltage distribution within each Fresnel zone using only two driving voltages. By controlling these voltages within the linear response region of LC material, the desired parabolic phase profile can be achieved. We provide a detailed discussion on the electrode structure design methodology and operating principles of the lens. In our experiments, we constructed a four-zone Fresnel LC lens with a total aperture of 8 mm. The results show that the optical power of the lens can be continuously adjusted from −1.30 D to +1.33 D. Throughout the process of electrically tuning the optical power, the phase distribution within each Fresnel zone maintains a parabolic profile. These results demonstrate the high-performance of the proposed Fresnel LC lens.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Liquid crystal (LC) lenses, originally proposed by Sato in 1979 [1], are optical devices that utilize the properties of LC materials to control their optical power. These lenses have found applications in various fields, including imaging systems [2,3], zooming systems [4,5], optical tweezer [6], augmented reality (AR)/virtual reality (VR) [7–9], 2D/3D displays [10,11] and depth measurement [12,13]. In AR/VR applications, two primary challenges exist: the vergence-accommodation conflict (VAC) and vision correction. However, these issues are difficult to solve using traditional solid-state lenses. LC lenses, with their ability to adjust the optical power, provide a potential solution for VAC problem [7]. Additionally, the tunable optical power of LC lenses can be utilized for personalized vision correction in individuals with different visual needs. As a result, LC lenses have emerged as a promising solution to address challenges in AR/VR, generating growing interest in their application in recent years. However, there are several requirements for LC lenses in AR/VR applications, including a larger aperture, greater optical power, and fast response. Currently, researchers are actively striving to develop LC lenses that can meet these requirements.
Traditional LC lenses, such as those based on hole-patterned electrodes [14–16] and modal lenses utilizing high-resistivity films [17–19], face challenges in achieving a large aperture. In these years, some LC lenses based on indium tin oxide (ITO) electrodes have been proposed [20–26] to overcome these limitations and enable larger apertures. However, this introduces another challenge where larger aperture LC lenses experience reduced optical power. To maintain favorable optical power, increasing the thickness of the LC layer is one solution, but this results in increased response time and optical scattering. Another method is to use high birefringence LC materials [27]. The ultrahigh birefringence of LC material has been demonstrated to be 0.7∼0.8 [28], but the associated issues include high viscosity and poor optical quality [27]. Additionally, stacking multiple LC cells [29,30] can increase the optical power, with the optical power being proportional to the number of cells. However, an increase in the number of LC cell will lead to a decrease in transmittance. In fact, as the aperture size of an LC lens increases while the LC layer thickness remains constant, the optical power decreases according to a quadratic relationship. Although these methods mentioned above can partially increase the optical power, they cannot fully compensate for the decrease in optical power caused by increasing the aperture size. As a result, they cannot fully meet the requirements of AR/VR and wearable eyeglasses.
Fresnel LC lenses have emerged as a promising strategy for achieving large-aperture LC lenses while maintaining favorable optical power and fast response. This is achieved by segmenting the aperture into concentric zones, where the optical phase is reset across the zones to reduce the total LC thickness. Numerous studies have explored the use of Fresnel LC lenses, encompassing diffractive Fresnel lenses [31–41] and refractive Fresnel lenses [42–52]. While diffractive Fresnel LC lenses offer larger optical power, fast response, and large aperture, they often encounter challenges such as strong chromatic aberrations, wavelength dependency, and the lack of continuous tunability of optical power, making their application in near-eye display devices quite challenging. On the other hand, refractive Fresnel LC lenses are designed with a segmented phase profile, where each segment is large enough to minimize diffraction effects [46]. Moreover, the optical power is continuously tunable. Among the designs for refractive Fresnel LC lenses, the discrete ring electrode-based segmented phase profile developed by the group from Kent State University is noteworthy [43,46,50,52–54]. These designs utilize multiple driving voltages to approximate a parabolic phase profile in each Fresnel zone, allowing the lens to have a large aperture and offering continuous optical power adjustment. Nonetheless, a current challenge associated with these designs is the complex and costly manufacturing process required due to the need for fabricating multiple layers of electrodes.
For refractive Fresnel LC lenses, generating a parabolic phase distribution within each zone is crucial. Additionally, it is challenging to harmonize the phase distributions across all zones to create a common focal point. Recently, we have introduced electrode structures with interconnected circular electrodes for the design of LC lenses [55–57]. These electrode structures have enabled the creation of a parabolic voltage distribution. In this work, through further exploration of these electrode configurations, we discovered that by segmenting the electrodes into multiple independently controllable zones following the Fresnel pattern and connecting one end of the electrodes within each Fresnel zone to a common voltage, while connecting the other end to another common voltage, a natural emergence of a parabolic voltage distribution occurs within all Fresnel zones. Moreover, the voltages within all Fresnel zones harmonize seamlessly, resulting in a unified parabolic voltage distribution when unwrapped. This suggests that all Fresnel zones share the same focal point, effectively overcoming challenges faced by current refractive Fresnel LC lenses. This type of LC lens requires only two driving voltages, and continuous focal length adjustment can be achieved by simply controlling the difference between these two driving voltages. The electrode structure can be developed with a single photolithography step, making the fabrication process simple and cost-effective. The approach we propose has the potential to achieve large aperture, both in terms of the fabrication process and the underlying principles involved. Considering the pupil size of the human eye varies from 2 to 8 mm, depending upon a person’s age and illumination conditions [58], we first attempted an 8 mm Fresnel LC lens to validate our design approach. Comprehensive tests were conducted on the lens, covering aspects such as phase information, focused spot characteristics, imaging tests, transmittance, response speed, and more. The results confirmed the high performance of the lens and aligned well with theoretical analyses.
2. Principle and electrode design
The schematic diagram of the proposed Fresnel LC lens is shown in Fig. 1(a). A planar LC cell is formed by two transparent substrates with an ITO layer on their inner surfaces. One substrate has a uniform plane electrode (not patterned) that is grounded, while the other substrate features a patterned electrode as depicted in Fig. 1(b). The dashed circles represent the boundaries of the Fresnel zones. This patterned electrode consists of numerous circular electrodes and two bus lines. Bus line #1 is used to apply voltage V1 to the inner boundaries of each Fresnel zone, while bus line #2 is used to apply voltage V2 to the outer boundaries of each Fresnel zone. To enhance clarity, Fig. 1(c) shows the dashed circles separately and marks the area and boundaries of each Fresnel zone. The patterned electrode can be decomposed into multiple electrodes as shown in Fig. 1(d), with each corresponding to a different Fresnel zone. In Fig. 1(d), rinner and router respectively represent the inner and outer boundary radius of each Fresnel zone. It can be observed that each Fresnel zone contains many circular electrodes, and they are interconnected to form a continuous electrode. One end of this continuous electrode is connected to bus line #1, while the other end is connected to bus line #2.
Fig. 1. Schematic diagram of the four-zone Fresnel lens. (a) The lens structure, featuring a patterned electrode on one substrate and a uniform plane electrode on the other substrate, (b) the top view of the electrode pattern is shown, with dashed circles representing the boundaries of the Fresnel zones, (c) the Fresnel zone area and Fresnel zone boundaries, (d) the electrode structure within each Fresnel zone.
In a Fresnel lens, the successive zones are numbered in increasing order radially outward, as shown in Fig. 1(c). Due to the small electrode gap, for two adjacent Fresnel zones, it can be assumed that the outer radius of the inner Fresnel zone is equal to the inner radius of the outer Fresnel zone. For the i-th Fresnel zone, the inner radius and the outer radius are
This equation demonstrates a parabolic voltage distribution within each Fresnel zone. Moreover, the voltage range of all Fresnel zones is between V1 and V2. Although the voltage distribution follows a parabolic pattern regardless of the values of V1 and V2, only when the voltages are within the linear response region of the LC material can the parabolic voltage distribution result in a parabolic phase distribution [59]. The LC material used in this work is HTG116900-100 from HCCH Co., Ltd. The linear response region has been measured in Ref. [60], which ranges from 1.6 Vrms to 2.5 Vrms. Therefore, the driving voltage should satisfy 1.6 Vrms ≤ V1 ≤ 2.5 Vrms and 1.6 Vrms ≤ V2 ≤ 2.5 Vrms. Under these conditions, the positive or negative state of the lens depends on V1 and V2. If V1 < V2, the lens behaves as a positive lens, whereas if V1 > V2, it functions as a negative lens. The voltage difference applied across each Fresnel zone is V2 − V1. Therefore, the optical power of the lens is proportional to V2 − V1.
This work considers a four-zone Fresnel LC lens with a diameter of 4 mm for the first zone (${r_{outer\_1}}$= 2 mm) and a total aperture size of 8 mm. The radii of the outer boundaries of the four Fresnel zones are 2 mm, $2\sqrt 2 $mm, $2\sqrt 3 $mm, and $4$ mm, respectively. For illustration purposes, we consider a positive lens with the highest optical power (V1 = 1.6 Vrms, V2 = 2.5 Vrms) and a negative lens with the highest optical power (V1 = 2.5 Vrms, V2 = 1.6 Vrms) to calculate the voltage distribution using Eq. (3). The results are presented in Fig. 2, where Fig. 2(a) and Fig. 2(b) show the two-dimensional voltage distributions, while Fig. 2(c) and Fig. 2(d) illustrate the voltage distributions at the cross-section where y = 0. It can be seen that the voltage distributions exhibit a parabolic shape, and the voltage range of all Fresnel zones is between V1 and V2. From Fig. 2(c) and Fig. 2(d), the change rate of voltage along the radial direction gradually increases. In fact, the derivative of the voltage is
which is a continuous linear function. Equation (4) indicates that the voltage (or phase) distributions within all Fresnel zones harmonize perfectly with each other, forming a seamless parabolic voltage (or phase) distribution when unwrapped. This suggests that all Fresnel zones share the same focal point. The optical power of the lens is proportional to the derivative of V(r), that is, the optical power of the Fresnel lens is proportional to ${V_2} - {V_1}$ and inverse proportion to the square of ${r_{outer\_1}}$.Fig. 2. Voltage distribution of the four-zone Fresnel lens. (a) Voltage distribution of positive lens, V1 = 1.6 Vrms, V2 = 2.5 Vrms, (b) voltage distribution of negative lens, V1 = 2.5 Vrms, V2 = 1.6 Vrms, (c) voltage of positive lens at the y = 0 cross-section, (d) voltage of negative lens at the y = 0 cross-section.
3. Experiments and results
Due to the simple electrode structure design, the electrode fabrication process only requires one step of photolithography. We first attempted an 8 mm Fresnel LC lens to validate our design approach in the experiment. The electrode on one substrate is a uniform plane electrode (without a pattern) that is grounded, while the other substrate features a patterned electrode shown in Fig. 1(b). The ITO electrode has a thickness of 100 nm and a sheet resistivity of 20 Ω/sq. The prepared glass substrate with a patterned electrode is shown in Fig. 3(a), and the diameter of the electrode pattern is 8 mm. Figures 3(b) and 3(c) respectively show magnified views of the central part of the first zone and the boundary between the first and second Fresnel zones. In these magnified views, the bright areas correspond to the ITO electrodes, while the darker regions indicate the etched ITO areas. As can be seen, bus line #1 connects to the inner boundary of Fresnel zones, while bus line #2 connects to the outer boundary of the Fresnel zones. In Fig. 3, both the width (w) and inter-line gap (d) of the circular ITO electrode are set at 5 µm. The width and gap of the two bus lines are also 5 µm. Subsequently, a Fresnel LC lens with an LC layer thickness of 30 µm was fabricated using the prepared substrates and the aforementioned LC material.
Fig. 3. (a) The substrate with the electrode pattern, (b) magnified view of the central part of the first zone, (c) magnified view of the electrode at the boundary between the first and second zones.
To generate a parabolic phase distribution corresponding to the parabolic voltage distribution, the two driving voltages were set within the linear response range of the LC material (1.6 ∼ 2.5 Vrms). For the positive lens, the voltage V1 is kept constant at 1.6 Vrms, while the focus is tuned by increasing the voltage V2 from 1.6 Vrms to 2.5 Vrms. Conversely, for the negative lens, the voltage V2 is fixed at 1.6 Vrms, and the focus is tuned by increasing the voltage V1 from 1.6 Vrms to 2.5 Vrms. The interference fringes of the lens are measured using polarization interferometry with a laser wavelength of 532 nm. (Continuously recorded interference fringes see Visualization 1 and Visualization 2). Here, we will analyze three different states for both the positive and negative lenses separately. When V2 is set to 2.1 Vrms, 2.3 Vrms, and 2.5 Vrms, the interference fringes of the positive lens are depicted in Fig. 4(a-c). Similarly, when V1 is set to 2.1 Vrms, 2.3 Vrms, and 2.5 Vrms, the interference fringes of the negative lens are illustrated in Fig. 4(d-f). It can be observed that as the absolute difference between V2 and V1 increases, the number of interference fringes gradually increases for both positive and negative lenses, indicating a change in optical power. Moreover, based on the distribution of fringe density, it can be initially inferred that the lens exhibits good performance.
Fig. 4. The interference fringes of (a-c) positive and (d-f) negative lens states. (a) V1 = 1.6 Vrms, V2 = 2.1 Vrms, (d) V1 = 1.6 Vrms, V2 = 2.3 Vrms, (c) V1 = 1.6 Vrms, V2 = 2.5 Vrms, (d) V1 = 2.1 Vrms, V2 = 1.6 Vrms, (e) V1 = 2.3 Vrms, V2 = 1.6 Vrms, (f) V1 = 2.5 Vrms, V2 = 1.6 Vrms. (Continuously recorded fringes see Visualization 1 and Visualization 2)
Note that due to the relatively large range of interference pattern collection, the boundaries of the Fresnel zones are not particularly easy to observe in these interference patterns. In reality, they are unavoidable. In Fig. 5(a), the phase reset region at the zone boundary marked by the rectangular box in Fig. 4(c) is obtained using an objective lens with a higher magnification. Multiple fringes are present because the phase is reset in this region. Due to the steep phase variation in this region, light incidents on this area will be scattered. The width of the phase reset region is ∼25 µm, and it is a constant for a particular LC layer thickness [46]. Analytically, the diffraction efficiency for different zones can be expressed as
where Γd represents the width of phase reset region and Γi represents the width of i-th Fresnel zone [61]. The diffraction efficiency for the LC lens is shown in Fig. 5(b), with all four Fresnel zones having a diffraction efficiency greater than 90%. As the number of Fresnel zones increases, the diffraction efficiency gradually decreases due to the decreasing width of the Fresnel zones. Thus, if one aims to increase the number of Fresnel zones to achieve a larger aperture for the Fresnel lens, the diffraction efficiency of these Fresnel zones will continue to decrease.Fig. 5. (a) The phase reset region at the zone boundary marked by the rectangular box in Fig. 4(c). (b) the analytical diffraction efficiently of different zones.
In order to obtain more phase data, we used the free software DFTFringe to measure the phase distribution along the horizontal axis in the fringe patterns shown in Fig. 4, and the results are shown in Fig. 6. Note that the width of the phase reset region, relative to the width of the four Fresnel zones of the lens, is small, and thus, the phase information in these regions has not been extracted and shown. Figure 6(a) illustrates the phase within each Fresnel zone. The phase distribution is similar to the voltage distribution shown in Fig. 2, indicating a parabolic shape for the phase distribution. The positions of phase resets essentially align with the voltage reset positions shown in Fig. 2, but there is still a slight discrepancy. The main cause of this discrepancy is the influence of the phase reset region. For the positive lens, the maximum phase within the four Fresnel zones is mostly identical, and it increases with the increment of V2. This indicates an adjustable optical power as V2 changes. Similar patterns can be observed for a negative lens.
Fig. 6. Phase profile of the four-zone Fresnel lens (phase within the phase reset region not shown). (a) The phase profiles in four Fresnel zones, (b) the unwrapped phase profiles across the entire lens aperture.
To obtain a more intuitive understanding of the phase distribution across the entire lens, the phase distribution in the Fig. 6(a) was unwrapped. In the phase unwrapping process, the phase distribution within the fourth Fresnel zone remains unchanged. The phase distribution within the third zone is added to the maximum phase within the fourth zone for unwrapping. Similarly, the phase within the second zone is added to the sum of the maximum phases within the third and fourth zones. Finally, the phase within the first zone is added to the maximum phase within the second, third, and fourth zones respectively. The unwrapped phase profiles are presented in Fig. 6(b), with symbols representing the unwrapped phase and curves depicting the parabolic fittings derived from the unwrapped phase. To ensure that the symbols are not overly crowded, the spacing between them has been increased compared to Fig. 6(a). Under different driving voltages, the phase distributions within all Fresnel zones of the positive and negative lenses can harmonize with each other, resulting in an unwrapped phase distribution that exhibits a parabolic shape. This indicates that the lens can consistently maintain a parabolic phase profile and minimize aberrations throughout the continuous focus tuning.
To evaluate the chromatic aberration and focusing performance of the lens, let parallel polarized (polarization direction is the same as the rubbing direction of the LC lens) laser beams of different wavelengths enter the positive lens (V1 = 1.6 Vrms, V2 = 2.5 Vrms) and collect the focused spots on their respective focal planes. A black mask is attached around the active area of the LC lens to eliminate stray light, and a complementary metal oxide semiconductor (CMOS) camera with a pixel size of 2.2 × 2.2 µm was used to capture the focused spots. Figure 7(a)-(c) depict the focused spots captured at the corresponding focal planes of the blue (457 nm), green (532 nm), and red (632.8 nm) lasers, demonstrating the excellent focusing capability of the Fresnel lens across different laser wavelengths. There is some noise present around the focused spot, primarily attributed to the diffraction effect caused by the black mask surrounding the aperture. Additionally, scattering from the phase reset region and phase variations near the bus lines, as well as diffraction induced by periodic electrode gaps, can also contribute to the noise.
Fig. 7. Focused spots and intensity distributions of laser beams with different wavelengths after passing through the positive lens (V1 = 1.6 Vrms and V2 = 2.5 Vrms) on their focal planes. (a) Blue laser, λ = 457 nm, f = 64.2 cm, (b) green laser, λ = 532 nm, f = 75.0 cm, (c) red laser, λ = 632.8 nm, f = 85.5 cm, (d, e, f) intensity distributions of focused spots for blue, green, and red lasers, respectively.
From the intensity distributions shown in Fig. 7(d)-(f), the diameter of the main lobe is measured in the x-direction (Dx) and y-direction (Dy). Comparatively, the Airy disk diameter was calculated by utilizing the formula 2.44 λ f / D, where λ represents the laser wavelength, f denotes the focal length, and D signifies the aperture size of the lens. The measured focal lengths, measured main lobe diameters, and theoretical Airy disk diameters for laser beams of different wavelengths are presented in Table 1. It can be seen that the difference between Dx and Dy values is minimal, indicating that the lens has similar focusing effects in both the horizontal and vertical directions. Additionally, the measured main lobe diameter of the focused spot for different wavelengths is comparable to the theoretical Airy disk diameter, indicating lens has good performance for monochromatic light of different wavelengths.
Table 1. The measured focal lengths, measured main lobe diameters, and theoretical Airy Disk diameter values for different wavelength lasers
The measured focal lengths for the three wavelengths of lasers are 64.2 cm, 75.0 cm, and 85.5 cm, respectively. The difference in focal lengths indicates a notable chromatic aberration of the lens, primarily caused by the severe dispersion of the LC material. To reduce chromatic aberration, an effective approach involves combining a diffractive LC lens with the proposed Fresnel LC lens. The diffractive LC lens exhibits negative chromatic aberration, while the refractive Fresnel LC lens demonstrates positive aberration. Through the integration of these two lenses, chromatic aberration can be notably minimized [62]. Additionally, opting for an LC material with lower dispersion offers a viable method to reduce chromatic aberration.
To quantitatively evaluate the resolving ability of the LC lens, the modulation transfer function (MTF) for the positive LC lens (V1 = 1.6 Vrms, V2 = 2.5 Vrms) is calculated from the intensity distribution in the focal plane [63]. For instance, the cross-sectional intensity distribution at y = 0 in Fig. 7(e) is depicted in Fig. 8(a). Additionally, the intensity distribution of an ideal lens with a focal length of 75 cm and an aperture of 8 mm is also presented for comparison. The main lobe of the measured intensity distribution is nearly identical to that of an ideal lens, but with a slightly larger diameter. The calculated MTF is shown in Fig. 8(b), with the MTF50 value of the lens measured at 8.4 cycle/mm, slightly smaller than the MTF50 value of 9.3 cycle/mm obtained for the ideal lens. Moreover, the cutoff frequency is measured to be approximately 19.5 cycle/mm, which closely aligns with the theoretical prediction of D / (f λ) = 20 cycle/mm [64]. This suggests that the resolving capability of the lens closely approximates that of an ideal lens. The MTF value of the Fresnel LC lens is lower than that of an ideal lens at all spatial frequencies, primarily due to diffraction caused by phase variations in electrode gaps [65] and light scattering resulting from steep phase variations in phase reset regions. To enhance the lens performance, it is advisable to minimize the electrode gap as much as possible. Moreover, the scattered light in the phase reset region can be effectively absorbed by incorporating a light-absorbing layer, such as a black mask [52].
Fig. 8. (a) Measured normalized intensity distribution in the focal plane of the Fresnel LC lens and the theoretical intensity distribution of an ideal lens, (b) measured MTF of the Fresnel LC lens with typical normalization approach, compared to an ideal lens.
The relationship between optical power and voltage is depicted in Fig. 9(a). The optical power can vary from −1.30 D to +1.33 D, exhibiting a linear relationship with V2 − V1. The Root-Mean-Square (RMS) error of the wavefront is extracted from interference fringes and presented in Fig. 9(b). The positive lens exhibits a maximum RMS value of approximately 0.20 λ, while the negative lens reaches a maximum RMS value of around 0.31 λ. The RMS error of the proposed LC lens is smaller than that of the refractive Fresnel LC lens proposed in Ref. [49,51]. For a variable-focus LC lens designed for the human eye, the maximum acceptable RMS wavefront error is typically λ/4 [66], as indicated by the dashed line in Fig. 9(b).The proposed LC lens almost meets this requirement. To quantify aberrations, Zernike coefficients are extracted using the software DFTFringe and shown in Fig. 10. It is evident that the primary aberration for both positive and negative lenses is x-axis astigmatism, with a magnitude close to 0.5 λ. However, apart from x-axis astigmatism, the majority of the Zernike coefficients remain below 0.2 λ.
Fig. 9. (a) Relationship between optical power and driving voltages, (b) relationship between RMS wavefront error and optical power.
In addition, the imaging performance of the positive LC lens (V1 = 1.6 Vrms, V2 = 2.5 Vrms) is evaluated by incorporating it into an imaging system. The system comprises of another glass lens with a fixed focal length of 16 mm. The LC lens is placed in front of the glass lens and turned off initially. An ISO 12233 chart is positioned about 32.5 cm away from the CMOS camera, and the captured image (out of focus) is displayed in Fig. 11(a). Subsequently, the LC lens is turned on, and the image is brought into focus by the positive lens, as shown in Fig. 11(b). It is evident that there is a significant enhancement in the quality of the recorded image.
The transmittance of the proposed LC lens within the visible light wavelength range was measured and presented in Fig. 12(a). The transmittance consistently exceeds 80%, with an average transmittance of 88.4%. The transmittance of an LC lens is influenced by factors such as the thickness of the ITO electrode, the LC material used, and the thickness of the glass substrate. By optimizing these parameters, it is possible to further improve the transmittance of the proposed LC lens.
Finally, the response time of the lens was measured by placing a photodiode at the focal plane of the positive lens. Initially, the lens was in the off state (V1 = V2 = 1.6 Vrms), and then V2 was adjusted to 2.5 Vrms. After the current signal output from the photodiode stabilized, V2 was readjusted back to 1.6 Vrms. Throughout this process, an oscilloscope (TBS1102B, Tektronix) was used to simultaneously record the voltage amplitude of V2 and the signal output from the photodiode. The results are presented in Fig. 12(b). The rise time and decay time were measured as 3.5 s and 0.5 s, respectively. The response time of an LC lens is directly proportional to the square of the cell thickness. Therefore, employing thinner cells can effectively reduce the response time. Additionally, the use of materials with lower viscosity or special lens structure design [67] can also contribute to reducing the response time.
4. Conclusion
In summary, our research introduces a novel refractive-type Fresnel LC lens. The proposed design features a simple interconnected circular electrode structure that can be fabricated using a single photolithography process. The driving mechanism of the lens is extremely simple, requiring only two driving voltages to achieve continuous adjustment of optical power and switching between positive and negative lenses. Throughout the entire process, the lens maintains a parabolic phase distribution. Experimental results demonstrate that the optical power of the lens can be continuously adjusted within the range of −1.30D to +1.33D, with the extracted phase closely matching the expected parabolic distribution. Additionally, the lens exhibits excellent focusing and imaging performance, which has been validated through experiments. The method proposed in this work will greatly reduce the cost of refractive Fresnel LC lenses and improve their performance. Therefore, it can be considered as a practical option for producing low-cost, high-performance refractive Fresnel LC lenses. Future work can focus on optimizing the lens structure and addressing issues such as chromatic aberration, Fresnel boundary scattering, diffraction effects between electrodes, capacitive effects, and other related challenges.
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 requests.
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