Lossless imaging based on a donut-like optical sparse aperture metalens

With the continued expansion of optical devices and photonic integrated systems into new areas of application, such as wearable and portable optics, the limitations of conventional bulky, massive, and curved lenses have become increasingly apparent.^1–6 Recent progress in metalenses has provided new opportunities to overcome these limitations, owing to their ultrathin, lightweight, and tailorable characteristics.^7–12 By controlling the geometry and size of these two-dimensional planar nanostructures in a particular arrangement, electromagnetic waves can effectively be focused and manipulated from terahertz to visible light frequencies.^13–19 However, creating large-aperture metalenses still poses a significant challenge because of the time-consuming, expensive, and high-accuracy processing technology required to produce billions of nanoscale building blocks.

Conventional optical sparse aperture (COSA) lenses have been proven to be effective in reducing the fabrication area of full-aperture (FA) lenses by combining several small subaperture lenses.²⁰ The COSA metalens has also been proposed as a method of circumventing the aperture size limitations of optical metalenses.^21,22 Furthermore, based on the planar form factor of metalenses, unconventional optical sparse aperture (UOSA) metalenses with noncircular subapertures have also been developed to improve imaging performance by introducing more adjustable parameters for the subaperture.²³ However, due to the sparsity and discreteness of aperture arrays of COSA and UOSA metalenses, the outer edge of their effective pupil function is no longer a connected domain, which leads to a non-uniform imaging resolution at different azimuth angles and prevents them from fully achieving the corresponding FA metalens resolution. In addition, the mid-frequency modulation transfer function (MTF) decreases or even drops to zero, possibly leading to the loss of information contained in the image.^24–26 Hence, it is highly desirable to develop an appropriate method or architecture that can achieve the same or better imaging performance than an FA metalens while minimizing the fabrication area.

In this work, a shear design method is developed to design and fabricate a frequency lossless donut-like optical sparse aperture (DOSA) metalens. The fabricated DOSA metalens is composed of a GaN nanobrick array on an $Al 2 O 3$ substrate with only a quarter of the processing area, which can focus unpolarized visible light onto the same spot at a wavelength of 632.8 nm. Both simulation and experimental results demonstrate that the DOSA metalens can avoid the non-uniform imaging resolution of the COSA metalens at different azimuth angles and provide FA metalens resolution. Furthermore, due to the improvement in the MTF value in the high-frequency region, practical images from the DOSA metalens show a stronger noise-resistance capability.

To demonstrate the advantages of the DOSA metalens in optical imaging, five types of metalenses are designed for performance comparisons, as shown in Figs. 2(a)–2(e), respectively. The imaging principle is detailed in the supplementary material, S1. The simulated 2D MTFs of the FA metalens, COSA metalenses with fill factors of 40% and 25%, and DOSA metalenses with fill factors of 40% and 25% can be found in Figs. 2(f)–2(j). The yellow dashed line represents the cutoff frequencies (CFs) of the MTFs of these metalenses. The red auxiliary dashed lines are used to compare the MaxCFs of the MTFs. The MaxCFs of the COSA metalenses are much smaller than that of the FA metalens, while the MaxCFs of the DOSA metalenses with both the 40% and 25% fill factors can reach the same level as the FA metalens. The underlying physical mechanism can be explained using the geometric interpretation of the MTF, which is the ratio of the overlap area of two displaced pupil functions to the total pupil area.^24,28,29 In Fig. 2(k), the pupil functions of DOSA (red annular) has a larger range of shift in all directions when the two pupil functions overlap, so the DOSA metalens have larger CFs than that of the COSA metalenses (blue circle). While in Fig. 2(l), since the pupil function of DOSA and FA have the same shift range in all directions, they have the same CFs. Therefore, the properties of the OSA system can be intuitively judged by shifting the pupil function. For a more detailed comparison, the simulated 1D MTFs of these metalenses are presented in Fig. 2(m). The MTF of the COSA metalens with a 25% fill factor (orange line) first decreases sharply and then drops to zero at mid-frequency, implying that the mid-frequency information of the image is completely lost. Thus, the image can barely be recovered, and the metalens loses its imaging ability. The MTF of the COSA metalens with a 40% fill factor (blue line) first decreases sharply at the mid-frequency, then increases again, and finally reaches zero at a higher frequency. This narrow range implies that much of the high-frequency image information is lost, resulting in low resolution. However, the MTFs of the DOSA metalenses with fill factors of 40% (green line) and 25% (red line) first decrease sharply at mid-frequency, then maintain a non-zero state before reaching zero. The CF of the DOSA metalenses is much larger than that of the COSA metalenses and the same as that of the FA metalens (black line). Furthermore, the CF of the COSA metalens is inconsistent in different directions [Fig. 2(n), blue dash-dotted line], implying that the imaging capabilities of the metalens in different directions are not uniform. The MaxCF of the COSA metalens is 144 lp/mm at 30°, 90°, 150°, 210°, 270°, and 330°, and its minimum cutoff frequency is 114 lp/mm at 0°, 60°, 120°, 180°, 240°, and 300°. Thus, these drawbacks may greatly hinder the practical application of COSA metalenses. However, these disadvantages can be effectively avoided by using a DOSA metalens design with perfect circular symmetry. The CFs of the DOSA metalenses with 40% and 25% fill factors [Fig. 2(n), red dash-dotted line and green dashed line, respectively] are consistent for all directions and the same as those of the FA metalens [Fig. 2(n), black dotted line]. Unlike for the COSA metalenses, the MaxCFs of both DOSA metalenses can reach 157 lp/mm. This implies that the DOSA metalens can achieve the same imaging resolution as the FA metalens in all directions after image restoration.^25,30,31

Apart from high-resolution imaging, all the DOSA metalenses show MTFs much higher than that of the FA metalens in the high-frequency region [Fig. 2(m)]. This can also be determined using the geometric definition of the MTF. When calculating the cross correlation of the aperture functions, the overlap area of the DOSA metalenses [see the left side of Fig. 2(l), yellow area] decreases faster than that of the FA metalens in the low-frequency region and remains basically stable in the mid-frequency region. Thus, the MTF of the DOSA metalenses drops sharply and then tends to be invariant in the mid-frequency region, as shown in Fig. 2(m). While the overlap areas for both cases in the high-frequency region are equal [see the right side of Fig. 2(l), yellow area], because the total DOSA aperture area is smaller, the DOSA MTF is larger.²² Considering the enhancement of the MTF in the high-frequency region, the DOSA metalenses may provide better noise resistance and imaging resolution enhancement effects than the FA metalens.

To visually demonstrate the imaging performance of the DOSA metalens, simulations were performed of the point spread function (PSF) and standard 1951 USAF resolution target with the above five metalenses, as shown in Fig. 3. When the PSF is known, the image can be largely restored with a Wiener Filtering Algorithm (WFA, see details in the supplementary material, S2) as shown in Figs. 3(a)–3(e). Direct imaging results of the FA metalens, COSA metalenses with 40% and 25% fill factors, and DOSA metalens with 40% and 25% fill factors are shown in Figs. 3(f)–3(j). The corresponding recovery results from the WFA are shown in Figs. 3(k)–3(o). For clarity, magnified views of the red dashed frames in Figs. 3(k)–3(o) are shown in Figs. 3(p)–3(t). To compare the imaging capabilities of the metalenses in different orientations, the restored results of the resolution target rotated by 30° are shown in Figs. 3(u)–3(y). For the FA metalens, the narrowest element that could be resolved in all directions was the sixth element of group 3 [Figs. 3(p) and 3(u)], i.e., the maximum resolution of these metalenses. For the COSA metalens with 40% fill factor, in the recovery image [Fig. 3(q)], the horizontal lines in the fifth element of group 3 could be resolved, while the vertical lines could not be resolved well. After rotating the resolution target by 30° [Fig. 3(v)], the resolutions of the horizontal and vertical lines imaged by the COSA metalens were reversed. Thus, the narrowest element that could be resolved by the COSA metalens was the fourth element, whereas the widest element that could not be resolved was the fifth element of group 3. For the COSA metalens with a fill factor of only 25%, the image could not be recovered using the WFA because a large amount of image information was lost, which indicates that this COSA metalens lost its normal imaging capabilities [Figs. 3(s) and 3(x)]. Direct image observations of all-dielectric DOSA metalenses with both the 40% and 25% fill factors are more difficult because of their lower MTFs in the mid-frequency region. As there are no zero values in the MTFs of the all-dielectric DOSA metalenses, and their CFs are consistent in all directions, the details of the blurred image could effectively be restored by the WFA. The narrowest element that could be resolved in all directions was the sixth element of group 3 [Figs. 3(r), 3(w), 3(t), and 3(y)]. Therefore, all-dielectric DOSA metalenses show high imaging resolution equal to that of the FA metalens, which is much better than that of the COSA metalens (with a 40% fill factor). More importantly, it is worth mentioning that when the fill factor of the processing area is extremely low (25%), the DOSA metalens still maintains its good image recovery ability, while the COSA metalens fails. These features further demonstrate the potential superiority of the DOSA metalens.

Four types of metalenses (FA, COSA with a 40% fill factor, and DOSA with 40% and 25% fill factors) have been fabricated based on the standard lithographic processing technique (supplementary material S3). The photographs of these metalenses captured by scanning electron microscopy (SEM) are shown in Figs. 4(a)–4(d). All of the fabricated metalenses have the same circumcircle radius of 0.350 mm. The inner radii of the fabricated DOSA metalenses with 40% and 25% fill factors are 0.271 and 0.303 mm, respectively. The vertical GaN nanobricks with a height of 600 nm were arranged in a square lattice with a nearest-neighbor separation of 320 nm, with some discrepancies compared to the designed sizes [Figs. 4(e) and 4(f)].

As shown in Fig. 5(a), the optical imaging performances of the fabricated all-dielectric metalenses were characterized using a custom-built setup (supplementary material S4). The imaging performances of the FA, COSA, and DOSA metalenses in the experiment can be examined in Figs. 5(b)–5(u). As shown in Figs. 5(b)–5(e), the PSFs of the metalenses obtained in the experiment were consistent with the theoretical result. The standard 1951 USAF resolution target was imaged to compare the imaging capabilities of the prepared FA, COSA, and DOSA metalenses. The actual resolution of the metalenses could be identified based on the positions of the clear elements in the resolution target. The direct imaging performances for the standard 1951 USAF resolution target of the FA, COSA, and DOSA metalenses are shown in Figs. 5(f)–5(i). Figures 5(j)–5(m) shows the corresponding restored results by WFA with the experimental PSFs [shown in Figs. 5(b)–5(e)].

For the FA metalens, the narrowest element that could be resolved in all directions [Fig. 5(n) and 5(r)] was the fifth element of group 3 (136 lp/mm), whereas the widest element that could not be well resolved was the sixth element of group 3, which is inconsistent with the simulated results [Figs. 2(p) and 2(u)], possibly due to experimental noise and algorithm limitations. For the COSA metalens with a 40% fill factor, as seen in Fig. 5(o), the horizontal lines in the fifth element of group 3 (136 lp/mm) could be resolved, while the vertical lines could not be well resolved. After rotating the resolution target by 30° [Fig. 5(s)], the resolutions of the horizontal and vertical lines imaged by the COSA metalens were reversed. Thus, the narrowest element that could be well resolved by the COSA metalens (40% fill factor) was the fourth element of group 3 (121 lp/mm), and non-uniform imaging resolution at different azimuth angles occurred. However, for the fabricated DOSA metalenses with 40% and 25% fill factors, the narrowest element that could be resolved in all directions [Figs. 5(p), 5(t), 5(q), and 5(u)] was the sixth element of group 3 (153 lp/mm). It should be noted that some stray light appears in Fig. 5(m) and 5(q) due to the lower MTF in the mid-frequency region of the DOSA metalens with a 25% fill factor. This disturbance noise can be eliminated by optimizing the optical system and algorithms. In addition, we also calculated the intensity distribution and visibility of tri-stripe on the 1951 USAF target after the restoration (see the supplementary material, S5). The results indicate that DOSA metalens with low processing areas has better imaging performance than FA and COSA metalens. Furthermore, compared with COSA metalens, the DOSA metalens has consistent imaging performance in all directions.

In our experiment, the maximum optical resolution of both the DOSA metalenses with fill factors 40% and 25% was 153 lp/m, which not only exceeds the actual resolution of the FA metalens (136 lp/mm) but also almost reaches the theoretical highest resolution (158 lp/mm) of the FA metalens. This is because the annular aperture can obtain a higher MTF than the FA metalens in the high-frequency region through adjustment of its inner radius size to control the proportion of the high-frequency MTF. Therefore, compared with the FA metalens, the DOSA metalenses have a better noise-resistance ability, which makes them more suitable for practical applications. More importantly, due to the circular symmetry of the DOSA metalens, it avoids the non-uniform imaging resolution of the COSA metalens at different azimuth angles.

To further validate the practical application performance of the DOSA metalens, both the FA metalens and DOSA metalens were used to image the liquid crystal display (LCD) screen of a mobile phone. A clear image of the LC array could be obtained directly by the FA metalens [Fig. 6(a)], and the quality of the restored image was much better [Fig. 6(b)]. On the contrary, the direct imaging effect of the DOSA lens was really poor [Fig. 6(d)]. However, after image restoration processing, a very clear image could be obtained [Fig. 6(e)], and the imaging effect in all directions was consistent with the FA metalens. Furthermore, it can be seen from Figs. 6(c) and 6(f) that the contour of the LC block in the restored image of the DOSA metalens was clearer than that of the FA metalens, which confirms the anti-noise characteristics of the DOSA metalens in practical applications. It is worth mentioning that the DOSA metalens can not only image a single LC block clearly but can also image defects [such as holes, shown in the white dotted line box in Figs. 6(c) and 6(f)] clearly. This indicates that the DOSA metalens can be used to check the quality of devices such as mobile phone screens instead of the traditional bulky optical imaging lens.

In summary, we have proposed and fabricated a donut-like DOSA metalens with a lossless imaging property based on a shear design method. The fabricated DOSA metalens with an effective aperture of 0.7 mm achieves a diffraction-limited resolution of 153 lp/m at all azimuth angles with only one-fourth of the processing area of the FA metalens, and circumvents the non-uniform imaging resolution of the COSA metalens at different azimuth angles to achieve full FA resolution. Furthermore, the DOSA metalens has a stronger noise-resistance ability for practical images due to the improvement of the MTF value in the high-frequency region. Hence, our work paves the way for practical applications of optical sparse aperture metalenses in advanced optical devices and photonic integrated systems.