aBeijing Advanced Innovation Center for Materials Genome Engineering, Institute for Advanced Materials and Technology, University of Science and Technology Beijing, Beijing 100083, China
bSchool of Materials and Engineering, University of Science and Technology Beijing, Beijing 100083, China
cState Key Laboratory of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
Metalenses with achromatic performance offer a new opportunity for high-quality imaging with an ultra-compact configuration; however, they suffer from complex fabrication processes and low focusing efficiency. In this study, we propose an efficient design method for achromatic microlenses on a wavelength scale using materials with low dispersion, an adequately designed convex surface, and a thickness profile distribution. By taking into account the absolute chromatic aberration, relative focal length shift (FLS), and numerical aperture (NA), microlens with a certain focal length can be realized through our realized map of geometric features. Accordingly, the designed achromatic microlenses with low-dispersion fused silica were fabricated using a focused ion beam, and precise surface profiles were obtained. The fabricated microlenses exhibited a high average focusing efficiency of 65% at visible wavelengths of 410-680 nm and excellent achromatic capability via white light imaging. Moreover, the design exhibited the advantages of being polarization-insensitive and near-diffraction-limited. These results demonstrate the effectiveness of our proposed achromatic microlens design approach, which expands the prospects of miniaturized optics such as virtual and augmented reality, ultracompact microscopes, and biological endoscopy.
In modern optical systems, miniaturized optical devices with chromatic aberration correction play a crucial role in determining the performance of high-quality precision instruments, such as lithography, full-color imaging, and medical imaging systems [1], [2], [3]. To achieve achromatic focusing, conventional refractive optical components with macroscopic dimensions are combined with multiple lenses to form a bulky lens group that is difficult to miniaturize [4].
Recently, the emergence of optical metasurfaces offers new opportunities to address these issues [5], [6], [7], [8], [9], [10], [11], [12], [13]. Metasurfaces with wavelength-scale thicknesses can arbitrarily modulate the phase, amplitude, and polarization of light by adjusting the geometric parameters of their nanostructures [14]. In particular, by using artificially designed nanostructures to achieve a curved phase front for focusing with a certain group delay to eliminate the dispersion, a traditional, heavy, bulky achromatic lens can be compressed into a thin layer called an achromatic metalens, the thickness of which is on the wavelength scale [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. According to design theory, an achromatic metalens has exactly the same focal length in the entire working band, showing perfect achromatic imaging. However, in practice, the fabricated metalens still shows a varying range of focal lengths at different wavelengths because of either the optical response deviation or fabrication errors of the actual nanostructure [29], [30], [31], [32], [33], [34]. However, in the design process, the determination of a nanostructure with the desired optical response usually consumes tremendous computational resources and time, and the fabrication of a metalens with dense nanostructures is usually expensive [35], [36], [37], [38], [39], [40]. Moreover, the final imaging performance is limited by the weak focal efficiency due to strong scattering from the nanostructures [41], [42], [43], [44].
As alternative technology solutions for miniaturized optical devices, conventional lenses with convex structures are compressed down to the micron scale (termed microlenses), and these exhibit the advantages of a simple structure, compactness, and high integration. Additionally, their large arrays are also considered [45], [46], [47], [48], [49], [50]. By studying the relationship between the dispersion of the materials and the achromatic performance, we found that using materials with low dispersion, a properly designed convex surface, and the overall size, the focal length shift (FLS) at different wavelengths of a microlens can be greatly reduced to the level of achromatic imaging. Based on these findings, we developed a map of microlenses of different sizes that showed low chromatic aberration, meeting the needs of achromatic imaging. According to this map, several microlenses with different focal lengths and sizes made of fused silica, which has extremely low dispersion, were fabricated via efficient manufacturing with a focused ion beam (FIB) that was demonstrated to be achromatic, polarization-insensitive, and highly efficient at visible wavelengths of 410-680 nm.
2. Results and discussion
2.1. Principle of achromatic microlens design
The principle of an achromatic microlens is illustrated in Fig. 1(a). Generally, the focusing behavior results in a convergent wavefront, whose phase profile ($\varphi \left( r,\lambda \right)$) is expressed as follows [51]:
where λ, f, and r represent the wavelength, focal length, and radial coordinate, respectively; and C(λ) is a phase independent of the radial position and can be chosen arbitrarily as a function of the incident wavelength. Here, we propose a generalized choice of $C\left( \lambda \right)=\frac{2\text{ }\!\!\pi\!\!\text{ }\alpha (\lambda )}{\lambda }$, where α(λ) is a positive value that depends on the wavelength. To meet the requirements of Eq. (1), specially designed nanostructures were used to obtain a certain phase modulation at each radius and thus compose a metalens. If we use a conventional convex lens with its material’s index n(λ) and a geometric surface profile determined by d and r, where d is the thickness of the lens at the radial position r, the phase modulation (${{\varphi }_{\text{p}}}$) can be expressed by
where d0 is the central thickness of the microlens, d is the thickness at a certain radius, and d1 is the distance from the surface of the microlens to plane 1, as shown in Fig. 1(a). Thus, ${{\varphi }_{d}}=\frac{2\text{ }\!\!\pi\!\!\text{ }}{\lambda }n(\lambda )d$ is the transmission phase phase of the microlens. ${{\varphi }_{{{d}_{1}}}}=\frac{2\text{ }\!\!\pi\!\!\text{ }}{\lambda }{{d}_{1}}$ is the accumulated transmission phase in the air between the lens surface and plane 1. Therefore, when the phase modulation in Eq. (2) is equal to the desired wavefront in Eq. (1), a convex lens with focal length f should have a shape determined by
For simplicity, we assume that α(λ) is equal to n(λ)·d0. As material’s index (n) varies with the incident wavelength owing to the dispersion of the material, the convex lens may have different shapes at different wavelengths to ensure the same f. Here, we need to choose an appropriate index to design the shape profile such that the overall deviation of f is sufficiently small and can be tolerated as approximately achromatic. From Eq. (3), a convex lens of a certain shape (central thickness d0 and outer radius r0) may have a dispersive focal length.
$|\mathrm{d} f|=\left|\frac{-r_{0}^{2}}{2(n-1)^{2} d_{0}}-\frac{d_{0}}{2}\right| \mathrm{d} n $
Eq. (4b) implies that the absolute chromatic aberration is directly caused by the material’s dispersion dn and is related to the refractive index of the material as well as the sizes of the r0 and d0 values of the lens. Theoretically, materials with a large refractive index and weak dispersion should be chosen, which is always contradictory in nature. The benefit of a high index is exhibited by a thinner material, as indicated in the second term in Eq. (4b), because we need to keep (n−1)d0 constant in Eq. (4a) to ensure that f remains the same as the designed value. Therefore, weak dispersion is the first consideration in material selection. Because transparent materials are always in a Cauchy-type dispersion in the visible range and the value of the index increases monotonically but slightly when the wavelength is red, once the material is determined, we should use the maximum refractive index at the shortest wavelength of the working bandwidth to design the lens in Eq. (3).
However, we can attempt to depress the chromatic aberration through the size r0. To further verify the effectiveness of the achromatic performance, we added the changing ratio of the focal length, based on Eqs. (4a) and (4b):
where Δn is the level of the index dispersion; η = Δn/(n−1) is a key factor, which is again contradictory in nature: between the high index and the low dispersion. At visible wavelengths of 400-700 nm, high dispersion materials have high η values. For example, TiO2 (n = 2.338, Δn = 0.225) has η = 0.168, and GaN (n = 2.522, Δn = 0.253) has η = 0.166. However, low-dispersion materials usually have much smaller η values. For example, the η values of SiO2 (n = 1.431, Δn = 0.017) and CaF2 (n = 1.442, Δn = 0.010) are 0.039 and 0.022, respectively. Therefore, regarding the FLS, the optimal choice of lens material also favors low dispersion, which is consistent with the previous rule.
The last factor to be considered is the numerical aperture (NA), which is expressed as
Based on the above discussion, the lens is designed to solve Eq. (3) with certain limitations set by Eqs. (4)-(6). The focal length, FLS, dispersed focal length |df|, and NA values of lenses of different sizes (r0 and d0) were plotted using silica, and the results are shown in Figs. 1(b)-(e).
From the application view of compact scale, the lateral scale is ideally larger than 10 μm (r0 > 5 μm), the thickness is no larger than 1.5 μm, and the ratio of the radius to the thickness is greater than or equal to 5. Then, one can outline an initial region in Fig. 1(b) that covers a wide range of focal lengths from 19 μm to the macro scale. From the viewpoint of achromatic focusing, the FLS is less than 10% over the entire map of Fig. 1(c) because of the weak dispersion of silica. Fig. 1(d) shows that the absolute deviation of the focal length (|df|) increases sharply with f, which should be considered for lenses with long focal lengths. Here, we set the limit of |df| as 13.5 μm, according to the Rayleigh criterion $\frac{{{\lambda }_{\text{min}}}}{\text{NA}_{\text{max}}^{2}}$ [52], as indicated by the dashed line in Fig. 1(d). From the viewpoint of the NA value, lenses with NA > 0.08 are preferred to ensure a better imaging performance, as shown in Fig. 1(e). As shown in the inset of Fig. 1(b), a final region with achromatic performance is obtained from this map, according to the thickness (d0 ≤ 1.5 μm), the geometry feature ($\frac{{{r}_{0}}}{{{d}_{0}}}$≥ 5), and NA ≥ 0.08, where f ranges from 29 to 201 μm, as indicated by the white and red numbers, respectively.
2.2. Fabrication and characterization of achromatic microlenses
Practically, we consider a silica lens with f = 100 μm, according to the region in the inset in Fig. 1(b). The optical constants and dispersion of fused silica are n = 1.431 and Δn = 0.017, as measured by an ellipsometer (VASE Ellipsometer, J.A.Woollam, USA). Moreover, Fig. 2(a) shows that the average transmittance exceeds 94%. The size of the lens is chosen with d0 = 1.27 μm and r0 = 10.65 μm, and Fig. 2(b) shows the microlens profile calculated using Eq. (3). The lens is directly fabricated through grayscale milling using a FIB on a substrate of 100 μm. The surface profiles were characterized using scanning electron microscopy (SEM; Carl Zeiss AG, USA) and atomic force microscopy (AFM; OXFORD, USA), as shown in Figs. 2(c) and (d), respectively. Fig. 2(b) shows that the actual surface shape was close to the theoretical shape. It is worth noting that there are four slightly protruding lines on the microlens surface, and these are attributed to the scanning manner of the FIB partition processing. According to Fig. S1 in Appendix A, these lines have a depth of approximately 15 nm and a width of approximately 400 nm. The depths of these lines were significantly smaller than the working wavelength, which did not cause additional resonance inside the microlens. According to Eq. (2), the wavefront profile distortion of these traces on the propagation phase is approximately 1%; therefore, they have little effect on the focusing and imaging performance. Furthermore, we performed a full-wave simulation of the lens slice profile along the direction of the white dashed line, comparing the cases with and without the protruding lines. The results in Figs. S1(b)-(f) demonstrate that the lines have almost no effect on the lens focusing performance.
To experimentally characterize the optical performance of the microlens, we customized the optical setup, as shown in Fig. 3(a). The focal length of D20 is steady over a broad waveband of 410-680 nm, and the average focal length is measured as 98 μm, as depicted in Fig. 3(b). These results demonstrate excellent achromatic performance. The experimental results also agree well with the numerical simulations, as shown in Fig. S2 in Appendix A, which predicted high-level achromatic focusing.
Moreover, the other two microlenses with different sizes and NA values (e.g., D30 with diameter D = 29.4 μm, d0 = 1.54 μm, and NA = 0.097; D10 with D = 10.5 μm, d0 = 1.05 μm, and NA = 0.149) were fabricated and characterized for comparison. The obtained surface topography and achromatic-focusing performance results were similar to those for D20, as shown in Figs. S3-S5 in Appendix A, which proves the generality of the design principles. Additionally, a large microlens (D = 100 μm, NA = 0.264) was simulated and analyzed using the COMSOL Multiphysics software package (COMSOL Inc., Sweden), as shown in Fig. S6 in Appendix A. This demonstrates that our efficient achromatic design approach is also suitable for large aperture microlenses.
The measured intensity profiles on the focal plane of the microlens D20 at different incident wavelengths are shown in Fig. 3(c), and they reveal the symmetric distributions of focal spots and imply high-performance focusing quality. In addition, we further analyzed the focusing performance of the three sizes of microlenses, as shown in Fig. 4. The extracted full-width half-maximum (FWHM) of the three microlenses all show near-diffraction-limited (∼$\frac{\lambda }{2\text{NA}}$) focuses that are linearly proportional to the wavelength, as expected (dashed lines in Fig. 4(a)). For example, the FWHM at 410 nm is 1.99 μm, which is close to the diffraction limit of 1.88 μm. Fig. S7 in Appendix A shows that the modulation transfer function (MTF) at 410 nm was 262 line pairs (lp)∙mm−1 at 10% contrast. Further, the Strehl ratio of the microlens was greater than 0.8, which meets the condition of the diffraction-limit focal spot.
Based on the focal lengths shown in Fig. 4(b), the measured FLS values of the D30, D20, and D10 were 6%, 8%, and 11%, respectively. According to the normalized intensity profiles, the |df| values of the D30, D20, and D10 were 9, 10, and 4 μm, respectively, which are close to their theoretical values of 7, 4, and 1 μm, respectively.
Fig. 4(c) shows the focusing efficiency of the three achromatic microlenses, which is defined as the ratio of the focused spot power (integration of light intensity passing through an aperture with a radius three times the FWHM) positioned on the focal plane to the power of the incident light. The incident power was measured by focusing the camera on the transparent region of the quartz substrate and setting the area of integration to be equal to the diameter of the microlens [27], [33], [53]. This shows that the focusing efficiency ranges from 52% to 78% in the entire visible light range for D20, while the average efficiency is higher than 65% for the three microlenses with different diameters (D30, 66%; D20, 65%; D10, 66%). These values are significantly higher than those of recently reported achromatic metalenses [54], [55], [56]. Fig. 5 compares the performance of the achromatic bandwidth and focusing efficiency of ultrathin achromatic lenses over the visible waveband. Our microlenses have both a high focusing efficiency and a broadband operating wavelength of 410-680 nm, meaning that they perform with excellence in the trade-off between efficiency and bandwidth.
2.3. Achromatic imaging ability
To further characterize the achromatic imaging ability, another optical setup was constructed, as shown in Fig. S8 in Appendix A, in which a halogen lamp (GCI-060101, Daheng Optics, China) was used as the white light source. The United States Air Force (USAF) resolution target (R3L3S1N, Thorlabs, USA) was used as the object, as shown in Fig. 4(d). Depending on the sizes of the microlenses, each group of elements in the field of view (consisting of a single Arabic number or three parallel lines) was imaged separately. As shown in the image formed by the D20 in Fig. 4(e), the chromatic aberration was well-corrected, even under white-light illumination, and high-resolution polarization-independent imaging was achieved because the structure of the microlenses was quadruple-symmetric. The smallest feature size that can be clearly identified on the image is 3.34 μm, which is close to the aforementioned diffraction limit values. Moreover, the images exhibit a slight green color because of the high efficiency of the blue wavelengths in the illumination source, which can be modified by balancing the ratio of the three primary colors of the incident light. High-performance achromatic lenses are highly desirable for the color imaging of microscopic biological samples. Figs. 4(g)-(j) show a set of microscopic images of stained onion cells, flag feathers, berry fuzziness, and paramecium samples. Figs. 4(k)-(n) show the imaging results obtained by the D20 achromatic microlens under white light. In the case of onion cells, a relatively realistic morphology and cell wall structure can be clearly observed, indicating their excellent ability to correct chromatic and monochromatic aberrations.
3. Discussion
We proposed general conditions for designing achromatic microlenses with low-dispersion materials and convex surfaces in the visible range. By considering the overall size of the lens, the absolute and relative chromatic aberrations, and the NA, we developed a map for the design of microlenses with focal lengths ranging from 29 to 201 μm while also showing good achromatic focusing and imaging performance in an experiment. Theoretically, a metalens shows perfect achromatic focusing; however, it still contains a certain inevitable FLS due to the limited phase modulation ability of an actual nanostructure deviating from the ideal phase and group delay as well as the errors from the complicated fabrication process. Compared to the metalens, our microlens demonstrates a certain FLS theoretically; however, it can be sufficiently compressed using a low-dispersion material, convex surface design, and lens size. In contrast, the microlens shows an even better focusing efficiency and NA and finally achieves competent achromatic performance. The thicknesses of the lenses can be controlled to be approximately 1 μm so that they are thin enough for applications with compact designs. Our design and fabrication processes have been greatly simplified, which allows the focal length, dispersion parameters, and efficiency of the lens to be tailored to achieve predictable results through material selection and structure customization. The adopted FIB process has clear advantages in terms of step simplification, precise surface profile construction, time, and cost. Finally, an average efficiency of achromatic focusing above 65% was achieved for various NA lenses in the visible range. However, there are some limitations to our design that need to be further addressed in future work. Due to the limitation of the refractive index of the material, the NA and size of the microlens are limited to a certain extent to maintain the thickness of the wavelength scale. This leads to a limited imaging range. In future work, we will investigate how to achieve an increased field of view using ultrathin achromatic microlenses.
In summary, we designed and prepared a broadband achromatic microlens with high efficiency in visible light. This paves the way for miniaturized optical devices and achromatic imaging with potential applications in highly integrated semiconductors, ultracompact microscopes, wearable devices, fiber-optic integration, and biological endoscopy.
4. Methods
4.1. Microlens fabrication
A ZEISS CrossBeam 340 (Germany) focused ion beam apparatus was used to define the surface profile of the microlens at a current of 100 pA. A 15 nm gold film was deposited onto the fused silica substrate via ion sputtering to make the material conductive to satisfy the FIB process requirements, and the gold film in the microlens area was removed during subsequent ion bombardment processing. The beam stream was then aligned, and the image dispersion was corrected to prevent the ion bombardment position from shifting with the partition scan. The scanning mode was fixed as bidirectional, and the ion beam dose was precisely controlled to prepare an accurate and consecutive microlens surface profile.
4.2. Optical characterization
In the experimental setup (Fig. 3(a)), a supercontinuum laser (SuperK FIU-15, NKT Photonics, Denmark) was used as the light source (410-680 nm), and each incident wavelength with a bandwidth of 10 nm was selected using an acousto-optical tunable filter. Then, an objective (40×, NA = 0.65, Olympus, Japan) paired with a tube lens (f = 200 mm, TTL200-A, Thorlabs) was used to receive the light passing through the microlens, and the objective was moved along the z-axis in a range of 0-200 μm with a step of 1 μm. Finally, the three-dimensional intensity distribution of the transmitted light was recorded using a charge-coupled device.
Acknowledgments
This work was supported by grants from the National Key Research and Development Program of China (2022YFB3806000), the National Natural Science Foundation of China (52325208 and 11974203), and the Beijing Municipal Science and Technology Project (Z191100004819002).
Compliance with ethics guidelines
Xueqian Wang, Chuanbao Liu, Feilou Wang, Weijia Luo, Chengdong Tao, Yuxuan Hou, Lijie Qiao, Ji Zhou, Jingbo Sun, and Yang Bai declare that they have no conflict of interest or financial conflicts to disclose.
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