Flat-field superoscillation metalens

Dingpeng Liao; Fengliang Dong; Kun Zhang; Yi Zhou; Gaofeng Liang; Zhihai Zhang; Zhongquan Wen; Zhengguo Shang; Gang Chen; Luru Dai; Weiguo Chu

doi:10.1364/PRJ.462848

Abstract

Superoscillation metalenses have demonstrated promising prospects in breaking the theoretical diffraction limitations on the resolution of optical devices and systems. However, most reported superoscillation metalenses have a very small field of view of several tenths of a degree, which greatly limits their applications in imaging and microscopy. Therefore, it is of critical importance to achieve absolute high resolution by increasing the numerical apertures (NAs) of optical devices and systems. Unfortunately, similar to the case in traditional optics, it is challenging to realize a large field of view at high NA, especially in the superoscillation regime. To date, no attempt has been made to achieve flat-field focusing in the superoscillation regime, to our knowledge. Here, we demonstrate a high-NA superoscillation metalens with an entrance aperture stop, which is optimized for superoscillation performance with a comparatively large field of view. The proposed flat-field superoscillation metalens has an effective NA as large as 0.89 and achieves superoscillation focusing within a field of view of 9°. Such a superoscillation metalens may offer a promising way toward superoscillation imaging and fast-scanning label-free far-field superoscillation microscopy.

1. INTRODUCTION

Past decades have witnessed fast development and wide applications of optical super-resolution in bioscience. Fluorescent super-resolution techniques [1 –3] and structured illumination microscopy [4 –6] have achieved spatial resolution ranging from 100 nm down to several nanometers. However, there is an increasing demand in developing super-resolution without fluorescent labeling, which was believed forbidden by traditional optical theories. The resolution of conventional optical devices and systems is restricted to the Abbe diffraction limit of $0.5 λ /NA$ [7], where $λ$ and NA are the working wavelength and numerical aperture, respectively. Since the concept of superoscillation was proposed [8,9], there has been growing interest in developing optical super-resolution devices for far-field operation with propagation waves [10,11]. Theoretically, an arbitrary small point-spread function can be constructed by wavefront manipulation through a properly designed superoscillation lens [12]. According to the size of the central lobe, a point-spread function can be categorized into three different types, i.e., superoscillation, super-resolution, and diffraction limited. The point-spread function is superoscillation when the spot size is smaller than $0.38 λ /NA$ ; it is super-resolution when the spot size is greater than $0.38 λ /NA$ but smaller than $0.61 λ /NA$ ; and it is diffraction limited when the spot size is greater than $0.61 λ /NA$ [13]. Label-free optical super-resolution microscopes have been demonstrated in the confocal configuration with a superoscillation lens [14] or super-resolution lens [15] as an illumination lens. Various types of superoscillatory lenses have been investigated, including linear-focusing lenses [16,17], point-focusing lenses [18 –20], needle-shaped-focusing lenses [21 –23], and vector wave superoscillatory lenses [24 –29], for normally incident light, all of whose fields of view are limited to an order of one-tenth degree. This greatly restricts the performance of superoscillation lenses and the imaging acquisition rate of super-resolution confocal microscopes utilizing superoscillation lens illumination. Therefore, developing wide-angle superoscillation lenses or flat-field superoscillation lenses is of great importance for such applications.

Recent development in metasurfaces [30,31] has provided alternatives to realize diffractive lenses in more flexible ways through manipulation of amplitude [32], phase [33,34], polarization [35], and dispersion [36] at a sub-wavelength scale. Different approaches have been reported on realizing wide-angle metalenses. A 40°-wide-angle metalens with NA of 0.4 [37] has been demonstrated by optimizing the lens phase profile and an aperture stop; a doublet with NA of 0.44 has achieved 50°-wide-angle performance [38]; a doublet with NA of 0.49 has also been demonstrated for a wide angle of 60° [39]; using a quadratic metalens, wide-angle operation as large as 80° and a high NA of 0.89 have been achieved [40]. However, the resolution of all reported wide-angle metalenses is restricted by the fundamental Abbe diffraction limit. Recently, a few trials have been conducted to achieve flat-field operation in the super-resolution regime. A super-resolution singlet metalens [41] has been proved effective for obliquely incident light. The lens has a radius of $240 λ$ , focal length of $60 λ$ , and high NA of 0.97, and its spot size remains approximately $0.45 λ$ on the focal plane within an incident angle range of $\pm 2 °$ , showing little fluctuation. A 50°-wide-angle super-resolution doublet [42] has been reported with an NA of 0.45. Simulation results have shown that the spot size varies between $0.75 λ$ and $0.78 λ$ in $x$ direction and between $0.772 λ$ and $0.931 λ$ in $y$ direction (on the incident plane of $y o z$ ), clearly broadened in $y$ direction. The small NA results in a spot size greater than half-wavelength. Actually, even in conventional optics, it is also challenging to increase the field of view at large NA. For a commercialized objective lens with an NA of 0.95, its field of view is approximately 10°.

To date, no attempt has been made to achieve flat-field focusing in the superoscillation regime with a comparatively large field of view. In the present work, utilizing geometric phase metasurfaces, a high-NA dielectric singlet metalens with an entrance aperture stop has been demonstrated for superoscillation focusing within a field of view of 9° at a wavelength of $λ = 632.8 nm$ . Such a high-NA flat-field superoscillation metalens can be applied in superoscillation imaging and fast image acquisition for far-field label-free superoscillation confocal microscopy.

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2. PRINCIPLE AND DESIGN

Figure 1(a) schematically illustrates the focusing by the flat-field superoscillation metalens, and Fig. 1(b) shows the a-Si cubic geometric-phase meta-atom [43,44] for wavefront manipulation of the metalens. The metalens is composed of a metasurface behaving as a superoscillation focusing lens and an optical aperture stop aiming to further reduce the off-axis aberration, which are integrated on two sides of a ${SiO}_{2}$ glass substrate. Considering the latter’s fabrication capability, for a given pitch of $P = 270 nm$ and height of $H = 436 nm$ , the transverse size of the a-Si meta-atom has been optimized for reasonably high-efficiency phase modulation of cross-polarization under illumination of circularly polarized light at a wavelength of $λ = 632.8 nm$ by using the finite-difference time-domain (FDTD) method. The complex refractive index used for optimization is $3.02 + i 0.005$ for a-Si and 1.457 for ${SiO}_{2}$ substrate (v015.04-1107, Plan Optik AG). The refractive index of a-Si is experimentally measured by a spectroscopic ellipsometer (SE-850 DUV, SENTECH Instruments GmbH). The optimized sizes are $L_{x} = 186 nm$ and $L_{y} = 88 nm$ , resulting in an amplitude transmittance of 80%. By rotating the a-Si block in the $x y$ plane by an angle of $γ$ , the transmitted cross-polarized circularly polarized light gains a phase shift of $ϕ = 2 γ$ .

Flat-field superoscillation metalens. (a) Schematic illustration of focusing of off-axis light by a flat-field superoscillation metalens. (b) Schematic illustration of the a-Si dielectric meta-atom for geometric-phase manipulation in the metalens.

Figure 1.Flat-field superoscillation metalens. (a) Schematic illustration of focusing of off-axis light by a flat-field superoscillation metalens. (b) Schematic illustration of the a-Si dielectric meta-atom for geometric-phase manipulation in the metalens.

Compared with a previously reported super-resolution singlet metalens [41], the present superoscillation lens is designed with the same basic parameters, i.e., lens radius and focal length. For a given thickness of the ${SiO}_{2}$ substrate $d = 175 μm$ , radius of the focusing metalens $R_{F} = 240 λ$ , and focal length of $f = 60 λ$ , the phase profile $φ (x, y)$ and aperture stop radius $R_{A}$ of the flat-field high-NA superoscillation metalens are optimized by using the particle-swarm method (PSM) [45], aiming to minimize the off-axis aberration of superoscillation focusing within the target incident angle range of $\pm 4.5 °$ . In many practical applications, such as telescopes and microscopes, only transversely polarized fields participate in the final imaging. Therefore, in the present design, only the transverse polarization components are considered. The focused optical fields are obtained by using a two-dimensional angular spectrum method [46]. According to the diffraction patterns on the focal plane, the focusing performance parameters are obtained, including focal spot peak intensity, focal spot size, focal spot displacement, and sidelobe ratio (the intensity ratio between the maximum sidelobe and central lobe).

Figure 2.Results of the optimized flat-field superoscillation metalens. (a) Phase profile of the lens $φ (x, y)$ ; (b) peak intensity (red solid), full width at half-maximum ${FWHM}_{x}$ (blue solid) and ${FWHM}_{y}$ (blue open) in $x$ and $y$ directions, and sidelobe ratio in $x$ (green solid) and $y$ (green open) directions at different incident angles.

Figure 2(b) presents the focusing performance parameters, i.e., intensity, focal spot size, and sidelobe ratio, at different incident angles. The theoretical results of previously reported wide-angle super-oscillatory doublets [42] show a fast increase in spot size in the incident plane as the incident angle increases. However, in the present work, it is clearly seen that the intensity (red) varies between 190 and 260, and the spot size (blue) has a very small fluctuation between $0.391 λ$ and $0.42 λ$ , which is much smaller than the Abbe diffraction limit of $0.562 λ$ ( $0.5 λ /NA$ ) and is also smaller than the superoscillation criterion of $0.427 λ$ ( $0.38 λ /NA$ ) [13]. Therefore, the lens is superoscillatory. In optimization of the flat-field superoscillation metalens, we optimized only the spot size and sidelobe of the focal spot within the angular field of view, and did not put any restriction on the intensity fluctuation at different incident angles. Therefore, there exists a clear variation in intensity when the incident angle changes. Such fluctuation can be reduced by setting a restriction on intensity, but it will take more time for optimization of the lens. It is also noted that the spot size has a very small difference in $x$ and $y$ directions. Although the effective NA is comparatively smaller, the spot size is even smaller than the previously reported size of $0.46 λ$ [41], which demonstrates the advantage of using the optical aperture stop in the flat-field superoscillation metalens, realizing superoscillation focusing and greatly increasing the field of view as well. The sidelobe ratio (the ratio of maximum sidelobe intensity to the peak intensity, green) remains below 30% in the entire target incident angle range. According to our simulation, the spot size is $0.436 λ$ as the incident angle increases to 5°, but with a sharp increase in the sidelobe ratio up to 40%. Compared with the previously reported super-resolution metalens singlet for focusing of obliquely incident light, due to the optimized optical aperture stop, the field of view of the present lens has been enhanced by a factor of two. More importantly, the present lens already works in the superoscillation regime, and a field of view of 9° is comparable to commercialized diffraction-limited high-NA flat-field objective lenses.

3. LENS FABRICATION

A 175-μm-thick ${SiO}_{2}$ plate is first cleaned by traditional piranha solution (a 3:1 mixture of sulfuric acid and 30% hydrogen peroxide) and acetone. A 445-nm-thick a-Si layer is deposited on the front side of the ${SiO}_{2}$ substrate by using inductively coupled plasma enhanced chemical vapor deposition (Sentech SI 500D, SENTECH Instruments GmbH). Then a 50-nm-thick aluminum film hard mask is grown on both sides of the device by using magnetron sputtering. A double-side mask aligner is used to make the gold alignment marks on both sides of the device. The pattern of the optical aperture stop is transferred to the hard mask using e-beam lithography (Vistec EBPG 5000plus ES, Vistec Electron Beam GmbH), and inductively coupled plasma etching (Sentech PTSA SI 500, SENTECH Instruments GmbH) is applied to form the aperture stop on the front side of the device. The metalens is fabricated with the same method. Figure 3 presents a diagram of the focusing metalens and an SEM picture of the fabricated flat-field superoscillation metalens.

Figure 3.(a) Diagram of the superoscillation focusing metalens arrangement in square lattice array, where inset gives the arrangement of the meta-atoms located at the device center. (b) SEM picture of the central part of the fabricated flat-field superoscillation metalens.

4. RESULTS AND DISCUSSION

We characterized the fabricated flat-field superoscillation metalens by illuminating it with a collimated $λ = 632.8 nm$ laser beam at different incident angles and measuring its optical intensity distribution on the designed focal plane at $z = 60 λ$ . A linearly polarized 632.8 nm He–Ne laser (HNL210L, Thorlabs, Inc.) is used as the coherent light source; a linear polarizer is used as an intensity filter to control the illumination power on the metalens; the beam size is enlarged by a beam expander (BE02-05-A, Thorlabs, Inc.); the expanded beam is sent to mirror #2 mounted on a rotation stage with a rotation angle resolution of 5 arcmin (PR01/M, Thorlabs, Inc.); then the beam is reflected and impinged on the back side of the metalens. The diffraction pattern of the metalens is then captured by a high-NA-optical-microscope-based testing system, composed of an infinite objective lens (CF Plan $100 \times / 0.95$ , Nikon), a one-dimensional nanopositioner (EO-S1047, Edmund Optics), a tube lens (ITL200, Thorlabs, Inc.), and a high-resolution digital camera (acA3800-14 μm, Basler, Inc.). The incident angle can be precisely controlled by fine rotating the rotation stage. The images of the diffraction pattern on the $x y$ plane were recorded through the digital camera at different $z$ by scanning the one-dimensional nanopositioner.

The focal spots of the flat-field superoscillation metalens were measured with linearly polarized incident light and are shown along with simulation results in Figs. 4(a)–4(h) for different incident angles of 0°, 0.5°, 1.0°, 1.5°, 2.0°, 2.5°, 3.0°, 3.5°, 4.0°, and 4.5°. The incident plane is the $x z$ plane. Figures 4(a) and 4(e) present the optical intensity distribution, and Figs. 4(b) and 4(f) give the intensity curves in $x$ and $y$ directions on the designed focal plane, while Figs. 4(c), 4(d), 4(g), and 4(h) display the experimental results, which show good agreement with theoretical predictions, especially when the incident angle is smaller than 3.5°. In the sidelobe, there is a clear increase at incident angles greater than 3.5°. The focusing performance parameters are presented in Fig. 4(i). It shows that for all tested incident angles, the spot sizes (full width at half maximum) in both $x$ and $y$ directions vary between $0.37 λ$ and $0.507 λ$ , which are smaller than the Abbe diffraction limit ( $0.562 λ$ , $0.5 λ /NA$ ), as indicated by the black dashed line. However, compared with results obtained by simulations, there is a small increase in focal spot size. The sidelobe ratio is found to be smaller than 36% at a small incident angle, while it reaches 76% at the incident angle of 4.5°, which is also larger than numerical predictions. Figure 4(j) plots the focal spot displacement against the incident angle on the focal plane, where both numerical and experimental results are presented for comparison. Both results show good linear displacement with the incident angle. Although there is a small difference, the experimental results are consistent with the theoretical prediction. A displacement of approximately $9 λ$ is achieved. Compared with a previously reported super-resolution singlet metalens, the field of view and the displacement have been enhanced by a factor of two. The discrepancy between the experimental and theoretical results may partially result from the assumption of uniform distribution of the optical fields within each meta-atom on the metalens surface during the diffraction pattern calculation, while the misalignment between the aperture stop and focusing lens can also contribute to this deviation.

Figure 4.Focusing performance of the flat-field superoscillation metalens. (a), (e) Simulation results of the optical intensity profile on the designed focal plane for different incident angles of 0°, 0.5°, 1.0°, 1.5°, 2.0°, 2.5°, 3.0°, 3.5°, 4.0°, and 4.5°; (b), (f) intensity distribution curves in $x$ direction (blue) and $y$ direction (red), both crossing the focal spot center; (c), (g) experimental results of the optical intensity profile on the designed focal plane for the 10 incident angles; (d), (h) intensity distribution curves in $x$ direction (blue) and $y$ direction (red), both crossing the focal spot center; (i) focusing performance parameters obtained from the experimental results, including peak intensity (red), spot size (FWHM) in $x$ (blue solid) and $y$ (blue open) directions and sidelobe ratio in $x$ (green solid) and $y$ (green open) directions; (j) spot displacement on the focal plane for the simulation result (red) and experimental result (blue).

Figure 5 presents the numerical and experimental results of intensity distribution and major focusing parameters, including peak intensity, spot size, and sidelobe ratio, on the $x z$ plane at different incident angles, where the vertical dashed lines denote the position of the focal plane, and the Abbe diffraction limit of $0.562 λ (0.5 λ /NA)$ and superoscillation criterion of $0.427 λ (0.38 λ /NA)$ are given by the dashed curve and the dashed-dotted curve, respectively. Figures 5(a) and 5(e) illustrate the transversely polarized optical field intensity on the propagation plane obtained by the angular spectrum method; Figs. 5(b) and 5(f) plot the focusing parameters along the optical axis; Figs. 5(c) and 5(g) depict the transversely polarized optical field intensity on the propagation plane obtained in experiments; and Figs. 5(d) and 5(h) present the experimental focusing parameters along the optical axis. It is found that, different from previously report flat-field super-resolution metalenses [40,41], the flat-field superoscillation lens generates a unique needle-shaped non-diffraction beam with a propagation distance of $20 λ$ at all tested incident angles. According to the simulation, in many parts of the needle, the transverse size is superoscillation. It is also interesting to find that all the needles are shifted in a way that they are parallel to each other at different incident angles. It seems that the lens can be treated as a superposition of series of local super-resolution axicons, or super-resolution Bessel beam generators. As pointed out above, the experimental results show a clear broadening in the transverse size of the needles, especially at larger incident angles.

$Optical propagation properties on the incident plane (xz plane) at different incident angles. Simulation results: (a), (e) intensity distribution on xz propagation plane; (b), (f) peak intensity (red), FWHM (blue), and sidelobe ratio (green) along the propagation direction. Experimental results: (c), (g) intensity distribution on the xz propagation plane; (d), (h) peak intensity (red), FWHM (blue), and sidelobe ratio (green) along the propagation direction. The Abbe diffraction limit and superoscillation criterion are displayed by the dashed curve and dashed-dotted curve, respectively. The focal plane is denoted by the vertical dashed line.$

Figure 5.Optical propagation properties on the incident plane ( $x z$ plane) at different incident angles. Simulation results: (a), (e) intensity distribution on $x z$ propagation plane; (b), (f) peak intensity (red), FWHM (blue), and sidelobe ratio (green) along the propagation direction. Experimental results: (c), (g) intensity distribution on the $x z$ propagation plane; (d), (h) peak intensity (red), FWHM (blue), and sidelobe ratio (green) along the propagation direction. The Abbe diffraction limit and superoscillation criterion are displayed by the dashed curve and dashed-dotted curve, respectively. The focal plane is denoted by the vertical dashed line.

Figure 6 presents the Poynting vector on the incident plane ( $x z$ plane) between $z = 30 λ$ and $100 λ$ at different incident angles of 0°, 0.5°, 1.0°, 1.5°, 2.0°, 2.5°, 3.0°, 3.5°, 4.0°, and 4.5°, where the blue arrows give the direction of energy flow, and the strength of the energy flow is normalized. In all cases, there is a clear comparatively strong energy flow propagating along $z$ direction. The length of the strong energy flow changes approximately from $45 λ$ to $25 λ$ as the incident angle increases from 0° to 4.5°. It is also noted that there is a deviation from the straight line in energy flow at $z = 70 λ$ , especially when the incident angle is greater than 2.0°. However, the energy flow remains almost in a straight line in the nearby region at the designed focal point of $z = 60 λ$ . This again indicates the unique property of the flat-field superoscillation lens. In our design, we made requirements only on the focusing parameters on the target focal plane without any restriction on energy flow in the propagation direction. According to our experience, the better the off-axis aberration is corrected, the smoother the generated focused field along the propagation direction will be. Therefore, we believe that such deviation can be further reduced when adopting a doublet metalens configuration, which can provide more freedom in the design to correct the off-axis aberration.

Figure 6.Poynting vector on the incident plane ( $x z$ plane) at different incident angles, where the blue arrows give the direction of energy flow.

In the present design, the focusing efficiency is only approximately 1%. The low efficiency is due to three reasons. First, when the spot size decreases beyond the diffraction limit, the focusing efficiency usually drops sharply as the spot size decreases, especially when the spot size gets into the superoscillation regime. Second, there is only one functional phase surface in the present lens; the comparative large field of view is realized majorly by the aperture stop, which greatly limits the put-through of the entire incident light. Finally, the design of flat-field super-oscillatory lens is time communing, which makes it difficult to find the best optimal result in the design. The low efficiency may place a restriction on many practical applications. However, in certain cases, where the increase in resolution is priority, the low efficiency is not a big obstacle, such as in super-resolution microscopy demonstrated in label-free super-resolution microscopy based on super-resolution lenses [14,15].

5. CONCLUSION

In conclusion, we have demonstrated a dielectric metalens consisting of a-Si cubic geometric-phase meta-atoms and an optical aperture stop integrated on two sides of a 175-μm-thick ${SiO}_{2}$ substrate at wavelength $λ = 632.8 nm$ , with a high NA of 0.89, focal length of 38 μm ( $60 λ$ ), radius of 152 μm ( $240 λ$ ), and field of view of 9°. The metalens with an optical aperture stop can effectively reduce the off-axis aberration, which not only allows it to work in the superoscillation regime but also greatly increases the field of view. Due to the availability of substrate thickness, the distance between the optical aperture stop and the focusing lens is not optimized. Including this optimization in the design may further significantly improve the superoscillation focusing performance and field of view. In the present design, the designed spot size (smaller than $0.42 λ$ ) has already been in the superoscillation regime (smaller than $0.38 λ /NA = 0.427 λ$ ), which leads to comparatively low focusing efficiency and also comparatively low signal-to-noise-ratio in the experiments. The co-axis alignment of the optical stop and the focusing metalens is challenging in fabrication, which will significantly influence the actual performance of the high-NA flat-field superoscillation metalens, especially when all parameters, including focal length, radius of the aperture stop, radius of the focusing lens, and substrate thickness, are comparatively small. This problem may become ignorable for a lens with a large size. Our work presents a promising theoretical and experimental proof of achieving a high-NA flat-field superoscillation metalens with large field of view, which can be further improved by achieving a flat-field super-resolution doublet metalens, or even multiple metalenses. Such a superoscillation metalens might offer a promising way toward superoscillation imaging and fast-scanning label-free far-field superoscillation microscopy.

Acknowledgment

Acknowledgment. The authors extend sincere thanks to Prof. Ting Jiang and Prof. Yuanpeng Zou at the School of Foreign Languages and Cultures, Chongqing University, for their assistance in language polishing.

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