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Abstract
Although metalens has attracted many research interests for its advantages of light weight, ultrathin size, and high design freedom in realizing achromatic and aberration-free optical devices, it still lacks adjustability in zoomable optical systems. Moiré metalens, which consists of two cascaded metasurface layers, can realize large focus tuning range by the mutual rotation of the two layers, and becomes a possible solution to realize real application of reconfigurable metalenses. However, due to the spacing between the two metasurface layers, it suffers from aberration caused by diffraction, leading to a dramatically decreased efficiency with the spacing. In this paper, we propose a reinforced design method for moiré metalenses with large spacing based on diffraction optics. Simulation results demonstrate that at the wavelength of 810 nm, when the spacing of the two metasurfaces is 10λ, the focusing efficiency of the reinforced moiré metalens is 3.4 times larger than the traditional moiré metalens. Furthermore, in order to consider the situation that the spacing between the two metasurfaces cannot be controlled precisely, we also propose a reinforced design method for multiplex spacings, which can make the device maintain a high focusing efficiency (3 times larger than the traditional moiré metalens) for the spacing in a range of 6λ∼10λ. The new design method is anticipated to be applied in realizing tunable metalenses in integrated continuously zoomable optical systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurface is an artificially manufactured ultra-thin two-dimensional structure composed of nano-antenna arrays, which is used to modulate the wavefront of an electromagnetic wave [1–3]. It can achieve locally and spatially abrupt phase change within a distance much smaller than the incident wavelength, and has the characteristics of high design freedom and miniaturization [4], which has been widely used in wave plates [5,6], vortex beam generators [7,8], optical holographic imaging [9,10] and aberration control [11,12]. Metalens is an important branch of metasurfaces. As one of the most basic optical elements, the lens plays an irreplaceable role in a variety of imaging systems such as microscopes and telescopes. Traditional optical lenses relying on geometric curvature to tailor the wavefront have the limitations of bulky size, heavy weights, and large axial movement distance for zooming. A metalens is able to focus light on a diffraction-limited spot using metasurface structures [13–15], which is more conducive to promoting the integration of compact optical platforms, especially zoom systems that require multiple optical components. It has been demonstrated that metalenses can be designed with corrections of chromatic and monochromatic aberrations [13,16]. However, non-adjustment becomes the primary obstacle faced by metalens applications. Many approaches have been proposed to solve this problem, such as applying mechanical deformation or displacement of the metasurfaces [17,18], or non-mechanical tuning mechanisms, such as electro-refractive [19], thermo-optical [20] and all-optical [21] effects. However, these elements also suffer from the limited tuning range and low optical efficiency that cannot meet the design requirements. In 2008, Bernet proposed the concept of diffractive moiré elements [22,23], which can be extended to the moiré metalens (MML) composed of two parallel metasurfaces based on the phase modulation [15,24–28]. By designing the arrangement of the antenna array and controlling the relative rotation angle between the two metasurfaces, the focal length can be theoretically tuned from infinity to zero. Compared with traditional zoomable optical components, a MML is able to achieve the on-demand tuning range without an axial movement of the lens, thus the optical system can be controlled in a small size, which is an important feature for both imaging and non-imaging systems [29]. In 2019, Guo et al. adapted the moiré structure to design metalens that can continuously zoom in the microwave band [15]. They realized the diffraction-limited focusing and conducted the experiment. Similarly, in 2020, Wei et al. experimentally achieved tuning range from ±3 mm to ±54 mm through designing a moiré metalens at 1550 nm wavelength [27]. The focusing efficiency of the doublet has a minimum of 34% and a maximum of 83% for positive lengths. In the same year, Iwami et al. proposed a tunable moiré metalens at 900 nm that can realize the focal length changing from ±1.73 mm−5 mm with corresponding rotation angle between ±90 degrees [28].
However, due to the existence of the diffraction effect, the spacing between the two metasurface layers must be maintained in a very small range, otherwise the device will have the inevitable phase distortion and efficiency wane [22,24]. For example, in the study of Wei et al. [27], it was demonstrated that the focusing efficiency decreases and focusing behavior becomes weaker when the doublet’s spacing increases.
In this article, we use a modified algorithm to propose a reinforced design method for a MML, which can still maintain a good focusing effect under a large spacing. Simulation results demonstrate that at the designed wavelength of 810 nm and the designed focal length of 6λ, when the spacing between the two metasurfaces is 10λ, it can still have a focusing efficiency of 16.77%, which is 3.4 times of the traditional MML. Besides, we have further proposed a reinforced moiré metalens for multiplex spacings, which does not need to strictly control the spacing between the two metasurface layers, and can perform well within a certain distance.
2. Focusing performance of a MML dependent on the spacing
A typical MML is composed of two cascaded metasurfaces based on the phase modulation, as illustrated in Fig. 1(a). The phase distributions of the two metasurfaces–metasurface 1 (MS1) and metasurface 2 (MS2), are denoted by φ1 and φ2, respectively. In order to realize adjustable focal length, φ1 and φ2 are required to be: φ1 = Fl(r)α and φ2 = -Fl(r)α, in which r and α represent the coordinates of a unit cell on the metasurface in the polar coordinate system, respectively. Fl(r) = −2π/λ*√(r2+f2)-f denotes the phase function of a lens with the focal length f. When MS2 is rotated with respect to MS1 by an angle of β around center, the phase of the two metasurface layers can be expressed as [22]:
At this time, the phase of the combined metalens increases linearly with the mutual rotation angle, the joint phase distribution is:
Fig. 1. (a) Schematic diagram of a moiré metalens, which is composed of two parallel metasurfaces with opposite phase distributions φ1 and φ2. The superposition of φ1 and φ2 constitute a lens phase, whose focus can be tuned by the mutual rotation of the two metasurfaces. (b) Phase distributions of $\varphi_{1}^{\prime \prime}$ obtained by the propagation of the phase φ1 on MS1 through a spacing d and the superposition of $\varphi_{1}^{\prime \prime}$ and φ2, when the spacing is 0, 0.1λ, 5λ, 10λ, respectively.
Therefore, the phase of the whole metalens is dependent on the mutual rotation. In the condition of paraxial approximation, the corresponding parabolic phase profile can be written in Fl(r) = ar2, where a is a free selectable constant. Thus, the focal length can be shown as
Equation (3) reveals that if the constant a and the wavelength λ are determined, the focal length is inversely proportional to the rotation angle, and it can be adjusted by tuning the rotation angle from infinite distance to nearly zero.
However, the above conclusions are obtained under ideal conditions, i.e., there is no diffraction between the two metasurface layers. When MS1 and MS2 are separated by a spacing d, the total phase modulation of the whole structure will be the superposition of $\varphi_{1}^{\prime \prime}$ and φ2, in which $\varphi_{1}^{\prime \prime}$ is the diffraction propagation of MS1 at the input surface of MS2, as shown in Fig. 1(a). Apparently, the spacing will cause φ1′′ deviates from φ1 due to the diffraction. In order to investigate the influence of the spacing between the two metasurfaces on the focusing, we perform theoretical calculations (the detail of the sampling method is introduced in section S1 of the Supplement 1). Without loss of generality, the working wavelength is selected as 810 nm. A MML is designed by the traditional method in Eq. (1), with two planes having the same spatial distribution parameters: the spatial period is 400 nm × 400 nm and the number of the sampling points is 30 in both x and y directions. The focal length is set to 6λ (4.86 µm), and the mutual rotation angle β is selected as 1 rad. Then the diffraction between MS1 and MS2 can be calculated by the Huygens-Fresnel principle [30],
where G = exp(j*2πr/λ)/r is the diffraction transform function, U1 = exp(jφ1) is the complex amplitude at the exit surface of MS1. r = √(x2 - x1)2 + (y2 - y1)2 + d2 represents the spacing between two arbitrary unit-cells on MS1 and MS2. The phase $\varphi_{1}^{\prime \prime}$ can be calculated by arg(U2). Then we can get the superposition of φ1′′ and φ2. Figure 1(b) shows phase $\varphi_{1}^{\prime \prime}$ and φ1''+φ2 with the increasing spacing from 0 to 10λ. A concentric circle-shaped phase distribution can be clearly observed when the two metasurfaces are sticked together, indicating a well-focused spot can be found. However, when the spacing between the two layers becomes larger, φ1′′ deviates greatly from φ1, resulting in the total phase φ1′′ + φ2 kept away from a lens phase.Using Eq. (4) and calculated $\varphi_{1}^{\prime \prime}$ +φ2, we can also calculate the electric field distributions of the x-z plane in the far-field behind MS2, which is dependent on d. The results are shown in Figs. 2(a)–2(c). It is found that when the spacing is increased from 0.1λ to 10λ, the focal length is decreased from 6.40λ to 3.92λ, which deviates greatly from the preset 6λ. More importantly, the focusing effect becomes worse as the spacing increases. From the calculated electric field intensities along the x-axis on the focal plane shown on the right column in Figs. 2(a)–2(c), in which the intensity is normalized to the maximum intensity of the focus in the ideal condition with no spacing (section S2 of the Supplement 1), we can find that the peak electric field intensity decreases from 0.5884 to 0.0274, and more side lobes appears with the spacing. In addition, the full width at half maximum (FWHM) is increased from 437 nm to 902 nm, and the corresponding focusing efficiency, defined as the ratio of focused light power to the total incident light power [24] (section S3 of the Supplement 1), drops from 40.62% to 15.20%.
Fig. 2. Focusing performance of a moiré metalens dependent on the spacing between the two metasurfaces. (a)-(c) Theoretically calculated electric field distributions of the focus in the x-z plane and FWHM in the focal plane when the spacing is 0.1λ, 5λ, and 10λ, respectively. (d)-(f) Simulated electric field distributions of the focus in the x-z plane and FWHM on the focal plane when the spacing is 0.1λ, 5λ, and 10λ, respectively. The insets in the right figures demonstrate the actual phase modulations of the two metasurfaces.
Besides the theoretical calculations, we also use the finite-difference time domain (FDTD) method to conduct a simulation study (S1 of the Supplement 1). Both MS1 and MS2 of the MML designed by Eq. (1) are composed of amorphous silicon nanorods with the height of 600 nm and period of 400 nm, arranged on a silicon dioxide substrate. The radius of the nanorods varies from 20 nm to 115 nm, and the optical response of the unit-cells are adopted from one of our previous work [25]. The arrangement of the nanorods in the simulations is shown in Table 1. There are 17 circles from the center to the outermost in the antenna array. The distance between two circles is 400 nm, and the radius of the outermost circle is 6.4 µm. Thus, the diameter of the whole MML is 12.8 µm. The number of the nanorods of each circle is proportional to the cyclomatic number in order to guarantee the interval of two adjacent nanorods is close to 400 nm, which ranges from 1 to 96. The two metasurfaces are placed face to face as that in [25]. A monitor is placed at the exit surface of MS2, and the far fields are also calculated based on the electromagnetic fields recorded by this monitor.
Table 1. Arrangement of the Amorphous Silicon Nanorods of the Metasurfaces
Figures 2(d)–2(f) indicate the simulation results on the x-z plane of the focus when the spacing changes from d = 0.1λ, 5λ and 10λ, respectively. Similar to the theoretical calculation results, we can observe that the focal length decreased with the increasing spacing, from 6.20λ to 3.64λ. The FWHM is increased from 611 nm to 792 nm. The peak electric field intensity of the focus drops from 0.3794 to 0.0344, and the focusing efficiency is reduced from 17.05% to 4.87%. According to these results, we can conclude that with the increase of the spacing, the diffraction effect causes phase distortion and lower focusing efficiency, which will lead to the functionality decrease of the device.
3. Reinforced design method for moiré metalens
We then proposed a new design algorithm to rectify the phase distortion caused by the diffraction of wave that is transmitted from MS1 to MS2. Figure 3 illustrates the principle of the process. In the first step, we calculate the ideal phase profiles of the metasurfaces, φ1 and φ2, derived by Eq. (1). From the above study, it can be inferred that if we want to maintain a good focusing function of the device, a phase $\varphi_{1}^{\prime}$ of MS1 should be calculated to guarantee that when the light wave propagates from MS1 to the input plane of MS2, its complex amplitude U1 is exp(jφ1). Based on this, we regard φ1 as the target phase at the position of MS2. Therefore, the second step is to calculate the propagation process in the back direction from MS2 to MS1, and finally get $U_{1}^{\prime}$, which is the complex amplitude of MS1:
Fig. 3. Scheme of the reinforced design method for moiré metalens with large spacing. φ1 and φ2 represent the phase distributions of metasurfaces in an ideal moiré lens, respectively. φ1’ is the phase extracted from the complex amplitude that exp(jφ1) inversely propagates for a distance of the spacing d. Then the total phase modulation of the two metasurfaces MS1 and MS2 is $\varphi_{1}^{\prime \prime}+\varphi_{2}$, in which φ1′′ is the phase extracted from the complex amplitude that exp(jφ1’) forward propagates for a distance of d.
From this, we could get $\varphi_{1}^{\prime \prime}=\arg \left(U_{1}^{\prime \prime}\right)$, and its superposition with φ2 constituted the phase of the final reinforced MML.
In order to verify the effectiveness of the design method, we also conduct both theoretical and simulation studies. The corresponding parameters are selected as same as those in section 2. The theoretical calculation results of the electric field distributions of the focus are shown in Fig. 4(a)-(c). It can be seen that the focus is legible when the spacing is changed from 0.1λ to 10λ, and the focal length remains around 6.4λ. The FWHM is increased from 430 nm to 641 nm, which is smaller than that in Figs. 2(a)–2(c). The side lobes are not obvious even d is 10λ. The peak intensity of the focus changes from 0.2909 to 0.1919. The peak intensity of the focus for d = 10λ is dramatically higher than that of the general MML in Fig. 2. The focusing efficiency is changed from 27.36% to 34.70%. At the spacing of d = 10λ, the peak intensity is almost 7 times, and the focusing efficiency is 2 times of the general MML.
Fig. 4. Focusing performance of the reinforced moiré metalens dependent on the spacing between the two metasurfaces. (a)-(c) Theoretically calculated electric field distributions of the focus in the x-z plane and FWHM in the focal plane when the spacing is 0.1λ, 5λ, and 10λ, respectively. (d)-(f) Simulated calculated electric field distributions of the focus in the x-z plane and FWHM on the focal plane when the spacing is 0.1λ, 5λ, and 10λ, respectively. The insets in the right figures demonstrate the actual phase modulations of the two metasurfaces.
Figures 4(d)–4(f) show the corresponding simulation results. It is found that similar to the theoretical results, with the increased spacing, the focal length remains within a certain range changing from 6.32λ to 6.17λ. The FWHM increases from 575 nm to 790 nm. The peak electric field intensity is in the range from 0.2275 to 0.2430 and the focusing efficiency is increased from 10.30% to 16.77%. At the spacing of 10λ, the peak intensity is 7 times, and the focusing efficiency is almost 3.4 times of the general MML.
Figure 5 shows the comparison of the focusing performance of the general MML and the reinforced MML at different spacings. The designed focal length is set to 6λ. We selected 11 points of d from 0.1λ to 10λ for theoretical calculation, and 6 points for simulation. In Fig. 5(a), we can find that as the spacing increases, the focal length of the general MML gradually decreases (yellow line). When the spacing is 10λ, the focus is at 3.89λ, which is far smaller than the 6λ. The focal length of the reinforced MML remains basically unchanged (blue line), around 6.42λ. The theoretical calculation is basically consistent with the simulation results. In Fig. 5(b), it is found that for both theoretical and simulation results, except for the distance less than 1λ, the focusing efficiency of the general MML is smaller than that of the reinforced MML. Therefore, compared with the general MML, the greater the distance is, the more obvious the increase in efficiency of the reinforced MML is. A recent published paper shows that it is not adequate to evaluate the focusing performance of a metalens only using the focusing efficiency [31], thus similar to this work, we also calculated the modulation transfer function (MTF) curves of the focus (section S4 of the Supplement 1) when the spacing is d = 0.1λ and 10λ. The results are shown in Figs. 5(c) and 5(d), respectively. It can be found from both theoretical and simulation calculations, although the MTF curves corresponding to d = 10λ is lower than that those corresponding to d = 0.1λ, the MTF curves of the reinforced MMLs are higher than their corresponding MTF curves of the general MMLs, which demonstrates a better focusing performance is obtained by the reinforced MML design method. Finally, we borrow the concept of peak signal to noise ratios (PSNR) in the reproduction of a holographic image [32] to study the similarity of $\varphi_{1}^{\prime \prime}$ to φ1. The calculation method of PSNR is to calculate the mean square error (MSE) of φ1′′ and φ1, which can be expressed by the following formulas:
Fig. 5. Focal performance comparison between the general moiré metalens and the reinforced moiré metalens, when the spacing between the two elements varies from 0.1λ to 10λ. (a)-(b) Comparisons of the focal length, and the focusing efficiency between the general MML and the reinforced MML, respectively. (c)-(d) MTF curves obtained by theoretical and simulation results of the focus when the spacing between the two metasurfaces is d = 0.1λ and 10λ, respectively. (e) PSNR compared by the ideal phase φ1 and actual phase φ1′′ of the two MMLs.
Focal tuning with the mutual rotation angle between the two metasurfaces is an important feature of a MML. We then conduct both theoretical and simulation calculations to compare the focal tuning ranges of the general MML and the reinforced MML (Electric field distributions are shown in section S5 of the Supplement 1). Seven rotation angles are selected in the theoretical calculations: π/180, π/32, π/16, π/8, 1, π/2 and 5*π/8; while 3 angles are selected in the simulation: π/180, 1 and π/2. The corresponding focal lengths at spacings of d = 0.1λ, 5λ, and 10λ are shown in Figs. 6(a) and 6(b). It is found that when the rotation angle β is increased, the focal length is decreased. For the theoretical calculation, when β is changed from π/180 to 5*π/8, the focus is changed 69λ for both the general MML and the reinforced MML at d = 0.1λ; while when the spacing is 10λ, the focal tuning is 73λ for the reinforced MML, and 18λ for the general MML. For the simulation results, when β is changed from π/180 to π/2, the focus is changed 90λ for the general MML, and 100λ for the reinforced MML at d = 0.1λ; while when the spacing is 10λ, the focal tuning is 18λ for the reinforced MML, and 133λ for the general MML. Figure 6(c) shows the focal tuning range for the general MML and the reinforced MML at various spacings. It is found that the reinforced MML design method is able to increase the focal tuning of a MML, especially for large spacings.
Fig. 6. Comparison of the focal tuning range of the general MML and the reinforced MML. (a) Theoretical calculation of focal length as a function of the mutual rotation angle β between the two metasurfaces when the spacing d is 0.1λ, 5λ, and 10λ. (b) Simulation calculation of focal length as a function of the mutual rotation angle β between the two metasurfaces when the spacing d is 0.1λ, 5λ, and 10λ. (c) Focal tuning ranges Δf of the general moiré metalenses and the reinforced moiré metalenses calculated by theory and simulation under spacings of 0.1λ, 5λ, and 10λ.
4. Reinforced moiré metalens for multiplex spacings
Although the design method proposed in Section 3 can improve the focusing performance of a MML, it still needs precise control of the spacing between the two layers. However, in actual operations, it is difficult to accurately place the two metasurfaces with the designed distance. Therefore, it is desired that a reinforced moiré metalens can improve the focusing performance not only at a specific spacing, but also in a spacing range, which means when the spacing d changes in an interval, the phase distribution φ1 in Eq. (1) can be all realized at the input surface of MS2 after the diffraction between the two metasurfaces. In 2015, Zhao et al. proposed a hologram design method to realize various intensity distributions at different distances [33]. Inspired by this work, in this section, we propose a reinforced algorithm for multiplex spacings that can maintain the focusing performance of the device within a certain spacing range, based on the concept of realizing the same phase distributions (φ1) at different distances (d).
The schematic of the design method is depicted in Fig. 7. Assuming that the spacing d between MS1 and MS2 is in a range from a to b, then we can divide the interval [a, b] into n - 1 segments with n junctions: d(1), d(2), d(3), …, d(n), in which d(1) = a and d(n) = b. After calculating the ideal phase distribution φ1 by Eq. (1), which is also the desired phase distribution at the input surface of MS2, we can make the complex amplitude U1 with the phase φ1 and the amplitude of 1 (U1 = exp(jφ1)) propagate backwards to the n junctions, and obtain a set of complex amplitudes U1'(i), in which I = 1, …, n. If we add all of these complex amplitudes up, we can get the superposition U1’ of the group of U1'(i), as shown in Eq. (8).
Fig. 7. Scheme of design method of the reinforced moiré metalens with multiplex spacings. φ1 = arg(U1) and φ2 represent the phase distributions of metasurfaces in an ideal moiré lens, respectively. φ1'(i) is the phase extracted from the complex amplitude that U1 inversely propagates for the ith distance of the spacing d(i). The phase of MS1 φ1’ is the superposition from φ1'(1) to φ1'(n). Then the total phase modulation of the two metasurfaces MS1 and MS2 are φ1''+φ2, in which φ1′′ is the phase extracted from the complex amplitude that exp(jφ1’) forward propagates for a distance of d.
Fig. 8. Focusing performance of the reinforced moiré metalens with multiplex spacings dependent on the spacing between the two metasurfaces. (a)-(c) Simulated electric field distributions of the focus in the x-z plane for a general MML and FWHM in the focal plane when the spacing is 6λ, 8λ, and 10λ, respectively. (d)-(f) Simulated electric field distributions of the focus in the x-z plane for a reinforced MML and FWHM on the focal plane when the spacing is 6λ, 8λ, and 10λ, respectively. The insets in the right figures demonstrate the actual phase modulations of the two metasurfaces. (g) MTF curves obtained by simulation results of the focus in (a)-(f). (h)-(i) Comparisons of the focal length, the FWHM of the focus, and the focusing efficiency between the general MML and the reinforced MML with multiplex spacings, respectively.
In order to study the effectiveness of the proposed design method, we choose the spacing interval of d ∈ [6λ, 10λ], and selected 5 spacings junctions of d(1) = 6λ, d(2) = 7λ, d(3) = 8λ, d(4) = 9λ, and d(5) = 10λ to superimpose the complex amplitudes of the inverse propagation. The designed focal length is 6λ, and the mutual rotation angle β is selected as 1 rad, which are the same as those selected in Sections 2 and 3. Consequently, the phase distribution of MS1 can be obtained by the multiplex spacing algorithm above. Based on the phase distributions of MS1 and MS2, we then conduct simulations to calculate the field distributions of the focuses corresponding to the spacing between the two metasurfaces is d = 6λ, 6.5λ, 7λ, 7.5λ, 8λ, 8.5λ, 9λ, 9.5λ, and 10λ, which are all in the spacing interval. Figures 8(a)–8(c) shows the changes in the focus and FWHM of a general MML at three spacings of 6λ, 8λ, and 10λ, respectively. It can be seen that as the distance increases, the focal length becomes shorter, from 4.62λ to 3.64λ. The normalized peak value of the electric field intensity decreases from 0.0482 to 0.0344. The FWHM becomes wider, from the minimum value of 784 nm to the maximum value of 1195 nm. Figures 8(d)–8(f) depicts field distributions of the focus and FWHM of a reinforced MML for multiplex spacings at the spacings of 6λ, 8λ, and 10λ, respectively. We can observe that the focal length slightly decreases with the increase of spacing, but it is closer to the designed focal length of 6λ. The peak electric field intensity is reduced from 0.2247 to 0.1659, and the value of FWHM does not change much, from 662 nm to 729 nm. From the MTF curves in Fig. 8(g), we can also find that the MTF curves obtained by the reinforced MMLs for multiplex spacings are higher than those of the general MMLs. We also compare the focal lengths, FWHM and focusing efficiencies of the general MML and the reinforced MML for multiplex spacings at all 9 spacings in Figs. 8(h)–8(j). It can be seen that within the spacing range, the focal length of the reinforced MML for multiplex spacings fluctuates around 6λ, which is close to the ideal value. The FWHM is smaller and more stable than that of the general MML. The focusing efficiency is around 15%, which is almost 3 times that of the general MML. Therefore, this multiplex distance correction algorithm can effectively correct the phase distortion caused by the spacing between the two metasurfaces within a certain range.
5. Conclusion
In summary, a reinforced design method for moiré metalenses with adjustable focal length that can maintain good performance at large spacing is designed. We calculated the phase distribution actually attached to the incident metasurface plane using both theoretical and simulation methods. At the operation wavelength of 810 nm, the metalens with a spacing of 10λ have a stable focal length of 6.17λ, and a much higher focusing efficiency of 16.77% than that of the general moiré metalens. In addition, we also design the enhanced moiré metalens for multiplex spacings with the focus at around 6λ and the focusing efficiency of 15%, which reduces the difficulty of accurately controlling the spacing between the two elements in actual experiments. It can be expected that the proposed method provides new opportunities to promote the application of ultrathin varifocal devices in compact optical platforms.
Funding
National Natural Science Foundation of China (61875010).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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.
Supplemental document
See Supplement 1 for supporting content.
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