An unconventional optical sparse aperture metalens

Optical lenses as fundamental components are widely used in various optical systems, such as microscopy, display, and optical lithography.¹ The conventional lens has usually been made in the bulk form by shaping glasses or other transparent materials that hinder their miniaturization of optical components and limit their applications in compact optical systems. Recently, metalens has been used in the spotlight to develop next-generation optical elements due to its ultra-thin, lightweight, and tailorable characteristics.^2–5 Compared with conventional lenses, the phase shift of metalens is modulated by engineering the physical shape of artificial subwavelength nanostructures instead of relying upon gradual phase accumulation,⁶ which can achieve the thicknesses at the wavelength scale or below. However, billions of subwavelength nanostructures of the metalens pose a serious challenge for fabrication, especially for the large-scale ones.^7–10 

Most metalens devices are patterned using either electron beam lithography (EBL)^4,11 or focused ion beam (FIB)⁷ according to the high accuracy requirement of subwavelength nanostructures. However, EBL and FIB processing methods are high cost, and their time-consuming is not suitable for large-scale metalens. Therefore, several alternative techniques have been developed. For example, photolithography^9,12 and nanoimprint^3,13 technologies have been used to pattern large-scale nanostructures, but the exposure field size of the lithography hinders the metalens from further enlarging its size. Optical sparse aperture (OSA) systems have been widely used in both large-aperture telescopes^14,15 and space optical remote sensing systems¹⁶ due to the fact that they have distinct advantages of reduction in size and weight of filled aperture lens by combining several smaller sub-aperture lenses together.^17,18 This OSA method is very appealing in technology areas where a filled aperture is quite large or hard to fabricate. More importantly, due to the high precision of semiconductor processing technology, the sub-aperture of the metalens can be processed to any desired shape without considering the limitations of conventional lens processing technology, which provides a new dimension for the design of sparse aperture. In recent years, a previous study by our group has introduced the OSA system into metalens imaging system to enlarge the effective aperture by traditional Ring6 system,⁸ and it has shown that the effective aperture of OSA metalens has higher spatial resolution and better-restored image quality than the individual metalens. However, this work introduced only the OSA metalens concept with the traditional circular sub-aperture OSA system in simulation and without experiment validation.

In this paper, we designed and fabricated an unconventional optical sparse aperture (UOSA) (the OSA whose sub-apertures are not conventional circular) metalens that consists of four identical and concentric annular sectors sub-aperture metalens. Compared with the circular sub-aperture conventional OSA systems, our method extends a new degree of freedom for the design of the OSA metalens with the large nonzero domain of modulation transfer function (MTF) and enlarges the effective aperture of the metalens with low cost and high efficiency by reducing processing time. The UOSA metalens with a numerical aperture of 0.423 with an effective aperture of 0.762 mm was fabricated, and each sub-aperture metalens was composed of Au nano-cuboid array with a specific rotation angle on a SiO₂ substrate. The simulation and experiment both show that the UOSA metalens has the limited diffraction imaging resolution as full aperture conventional lenses.

To demonstrate the advantages of UOSA, the MTF of both OSA and UOSA metalens are theoretically studied. The MTF has commonly been used to evaluate the imaging quality of the imaging system, which is the Fourier transform of the impulse response function^17,19,20 (see  Appendix A). It specifies the weighting factor applied by the system to the various frequency components relative to the zero-frequency component. The MTF has an important geometrical interpretation, which represents the ratio of the area of overlap of two displaced pupil functions and the area of overlap by the total area of the pupil.²¹ Figure 1(a) shows the MTF nonzero domain boundaries of the OSA and the UOSA when the two kinds of apertures have the same aperture fill factor and it can be seen that the UOSA with annular sector subapertures has a larger nonzero domain (cut-off frequency) of MTF. This is because the UOSA has a larger displacing range when the overlap of two displaced pupil functions does not disappear, and simultaneously, it can keep the mid-frequency not being lost by adjusting the opening angle of annular sector sub-apertures in the UOSA metalens, as shown in Figs. 1(b) and 1(c). In the following parts, the imaging quality of UOSA metalens will be studied in detail.

The UOSA we designed is Quadrupole4 (Q4), as shown in Figs. 2(a) and 2(b). The sub-apertures of UOSA are four identical, evenly distributed, and concentric annular sectors. The outer radius (R₂), inner radius (R₁), and opening angle (β) of the annular sectors are 0.381, 0.05, and 30 mm respectively. This parameter ensures that the image can be restored while effectively reducing processing costs. The circumscribed circle of the Q4 pattern, as shown in Fig. 2(b), is the effective aperture [twice of outer radius (R₂) is 0.762 mm] that can represent the size of the aperture of UOSA. For comparison, the equal processing area circular aperture metalens [as shown in Fig. 2(b), the diameter is 0.436 mm] and the full aperture metalens (the diameter is the same as the effective aperture) are also simulated. For this imaging system, the background transmission is too weak, and it can be processed as low frequency noise by the Wiener filter algorithm described later.

Figure 2(c) gives the 3D normalized modulation transfer function (MTF) of UOSA metalens. It is shown that there are several second peaks around the main peak that effectively extend the nonzero area of the MTF of UOSA metalens. For comparison, the 1D normalized MTFs of UOSA, equal processing area circular aperture, and full aperture metalens is simulated in Fig. 2(d). The MTF of UOSA has a rapid decline in the middle frequency, but no zero value appears in the curve before the maximum cut-off frequency. It means that the information is not lost and can be restored by image restoration algorithms, such as the Wiener filtering algorithm, which is an efficient restoration algorithm for the OSA system.^8,22,23 More importantly, the MTF of UOSA has the same maximum cut-off frequency as full aperture metalens, which is larger than the equal processing area of circular aperture metalens. In addition, it is worth noting that the MTF is just a normalized value relative to the zero-frequency component of optical systems itself, so the UOSA’s MTF can be higher than the full aperture lens at the high frequency, as shown in Fig. 2(d).

For a visual illustration of the USOA’s imaging performance, the simulations of standard 1951 USAF resolution target with the above three apertures with the same focus distance of 0.9 mm and working wavelength of 632.8 nm were performed and shown in Fig. 3. Among them, Fig. 3(a) shows the original image, and Figs. 3(b)–3(d) show the imaging results of the metalens based on the equal processing area circular aperture, the UOSA, and the full aperture metalens, respectively. Figures 3(e)–3(g) are restored results of them by the Wiener filtering algorithm (see  Appendix B). It is straightforward to show that the imaging result of the USOA becomes most blurred because its mid-frequency of MTF is the lowest among them. However, the information in the blurred image can be restored by the Wiener filtering algorithm because there is no zero value in the MTF of the USOA metalens, as shown in Fig. 2(d). Thus, the imaging resolution of the USOA is close to the full aperture and much better than the equal processing area circular aperture metalens, which can prove that the USOA can enlarge the effective aperture of the lens and improve the imaging resolution based on equal processing area.

To demonstrate the imaging performance of USOA metalens, it was achieved by Au nano-cuboid array with a specific rotation angle on a SiO₂ substrate based on the P–B phase theory.^24–27 To function like a spherical lens, the phase profile of the USOA metalens needs to follow^3,28

where f = 0.9 mm is the focal length of the metalens and λ = 632.8 nm is the design wavelength. In Fig. 4(a), the parameters W, L, H, and center-to-center spacing P of the cuboids in the USOA metalens are 80, 230, 80, and 320 nm, respectively. In the case of the circularly polarized light incident, these rotations (the rotation angle is θ) yield a phase shift, satisfying φ(r) = 2θ.^26,27 Simulations using a commercial finite difference time domain (FDTD) solver (Lumerical Inc., Vancouver) in Fig. 4(b) not only show that the structure with different rotation angle have equal conversion efficiencies but also the phase shifts were consistent with the theoretical hypothesis. The UOSA metalens was fabricated by standard electron beam lithography (EBL) and a lift-off process. First, the PMMA-A4 photoresist was deposited via spin coating on the SiO₂ substrate, and then EBL exposure was used to create the lens pattern in the photoresist. Third, develop the sample in a solution of isopropyl alcohol (IPA) and 4-Methyl-2-pentanone (MP). Fourth, the gold layer was deposited using an E-beam evaporator and the photoresist was then removed by the lift-off process in acetone. Figure 4(c) is image of the UOSA metalens obtained by optical microscopy. Figures 4(d) and 4(e) show scanning electron microscope (SEM) images.

To verify the imaging performance of the USOA metalens, the optical system is shown in Fig. 5(a). The light source was a supercontinuum laser (Fianium SC400–4) set at 632.8 nm with a bandwidth of 1 nm. Since the metalens we designed can only work on circularly polarized light, a polarizer (ThorLabs WP12L-VIS) and a λ/4 wave plate (ThorLabs AQWP05M-600) are used to allow circularly polarized light to pass through only. The next element is a ground glass lens with a high rotating speed that can convert incident light into an incoherent light. Then, a Fourier lens (ThorLabs AC508-080-A-ML) is used to focus the beam on the standard 1951 USAF resolution target (Edmund). The passed beam with resolution target information is collimated through a collimator with a length of 80 mm and a diameter of 20 mm (SHJingMi) and then imaged by the metalens. Finally, the imaging result is magnified by a 50× magnification and displayed on the CCD (Allied vision GT3400c). The other polarizer and a λ/4 wave plate before CCD are used to filter ambient light.

The imaging performance of the USOA metalens in the experiment is shown in Fig. 5. Figure 5(c) shows the point spread function (PSF) of the USOA metalens obtained in the experiment, which is consistent with the theoretical result in Fig. 5(b). Figures 5(d)–5(f) are three imaging results of the standard 1951 USAF resolution target with the above optical system, directly. Figures 5(g)–5(i) are restored results of them by the Wiener filtering algorithm with the experimental PSF [shown in Fig. 5(c)]. The elements of the “1” group and “2” group are shown in Figs. 5(i) and 5(g), respectively. The narrowest element that can be resolved and the widest element that cannot be resolved are the fourth element (0.99 µm) and the fifth element (0.89 µm) of the “2” group, respectively. According to the Rayleigh criterion, the theoretical highest resolution of the USOA metalens (calculated by the effective aperture) is about 0.91 µm. Thus, the experimental results prove that the resolution of the USOA metalens has almost reached its diffraction limit, which exceeds the highest resolution (1.59 µm) of equal processing area circular aperture metalens. The UOSA metalens can enlarge the effective aperture of the metalens with only 1/3 processing area that effectively shortens the processing time and reduces the cost. In addition, the UOSA metalens also breaks the limitation that the sub-aperture is circular, which extends a new degree of freedom for the design of the optical sparse aperture metalens.

In conclusion, we designed and fabricated an UOSA metalens that consists of four identical and concentric annular sectors sub-aperture metalens with a numerical aperture of 0.423 and an effective aperture of 0.762 mm. Compared with the circular sub-aperture conventional OSA systems, the UOSA metalens has a larger nonzero domain of MTF and can enlarge the effective aperture of the metalens. The experimental results show that the USOA metalens almost has a limited diffraction resolution (0.91 µm), which is consistent with the simulation results. The UOSA method not only can enlarge the effective aperture of the metalens with low cost and less processing time but also breaks the limitation that the sub-aperture is circular, which extends a new degree of freedom for the design of the optical sparse aperture metalens.

This work was supported by the Integrated Circuit Major Science and Technology Project of Shanghai (Grant No. 20501110600), the International Science and Technology Cooperation Program of Shanghai (Grant No. 20500711300), and the National Natural Science Foundation of China (Grant No. 61805264).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

According to imaging theory, the optical system can be regarded as a linear invariant system. The imaging system that uses incoherent illumination has been seen to obey the intensity convolution integral,

where f(x, y) is the original image and h(x, y) is the PSF of the real optical system that represents the diffuse light spot distribution imaged by the point light source in the optical system. n(x, y) is the additive noise, ⊗ is the convolution operation, and g(x, y) is the degraded image. Application of the convolution theorem to Eq. (A1) then yields the frequency–domain relation

where $Gfx,fy$⁠, $Ffx,fy$⁠, and $Nfx,fy$ are the Fourier transform of g(x, y), g(x, y), and n(x, y), respectively. By international agreement, the function H is known as the optical transfer function (OTF) of the system. Its modulus $|Hfx,fy|$ is known as the modulation transfer function (MTF). Note that $Hfx,fy$ simply specifies the complex weighting factor applied by the system to the frequency component at $fx,fy$⁠, relative to the weighting factor applied to the zero-frequency component. It can be described as

where $F$ is Fourier transform.

The Wiener filter was first proposed by Wiener, and it is widely used in image restoration. It has a good effect on the restoration of degraded images and has an excellent antinoise ability, but the time consumption of the Wiener filter is little because of its low calculation property. The Wiener filter is a kind of least square method filter whose basic idea is to minimize the mean square error (MSE) between the restored image f′(x, y) and the original image f(x, y),

The Wiener filter is an image restoration algorithm based on the frequency domain. The spectra of the restored image can be calculated by

where $Gfx,fy$ is the Fourier transform of the degraded image g(x, y), $Hfx,fy$ is the OTF calculated by Eq. (A3), $H*fx,fy$ is the complex conjugate of $Hfx,fy$⁠, and C value is the ratio of the power spectrum of the noise and the original image. In this paper, C is set to zero in the ideal image restoration simulation, while it is an optimized value based on the restoration results in the experiment.