It is of great significance to expand the field-of-view (FOV) in the entire imaging field. The conventional way to expand FOV is utilizing multiple refractive lenses, which makes the system bulky and heavy. Metalens^[1-8], as a focusing phase designed metasurface^[9-16], features ultra-thin and ultra-light advantages, and holds great potential for compact and miniaturized imaging systems. However, due to the limited FOV of each metalens, the realization of wide-angle imaging has been hindered. Recently remarkable efforts have been made to achieve wide FOV imaging^[17-32], such as topology-optimized multilayered metalenses^[18], quadratic metalenses^[19], metalens doublets^[20,21], and hybrid metalenses^[22], while these designs are complex and usually need to sacrifice in certain aspects of imaging performance (e.g.,  degraded focusing quality, decreased entrance pupil). Recently researchers^[25] developed a metalens array to compose an array-based camera and obtained a 120°-wide viewing angle. This work optimizes the phase by introducing a tilted phase design, but only solves the problem of focusing phase matching. However, for a metalens with only a kind of nanopillar structure, e.g.,  normal structures perpendicular to the substrate, in addition to the aberration caused by phase at large incident angles, there inevitably exist efficiency decreases as incident angle increases. Therefore, to meet the requirement of high-performance imaging at large angle incidence, new designs are highly demanded to improve the lens efficiency at oblique incidences.

In this work, we propose a tilted structure of unit cell for the metalens design, which can efficiently improve the focusing efficiency at oblique light incidences. The tilted structure, which breaks the symmetry of the $z$-axis, is equivalent to a tilted dipole^[33-36]; thus, its strongest radiation lobe is tilted. Therefore, the focusing efficiency along the optical axis could be improved. In this work, proper focusing phase with respect to the incident light is introduced to each lens to correct aberrations. Then we systematically compare the focusing performances between the tilted design and the normal one with different incident angles, which is characterized by the transmission, diffraction efficiency, total efficiency, and corresponding modulation transfer function (MTF). Furthermore, we develop a hybrid metalens array consisting of tilted lenses and normal ones, which has higher imaging performance than the normal metalens array. Our results would pave the way to realize high-performance optical imaging devices in different scenarios.

For a normally incident light, the ideal hyperbolic phase profile of a focusing metalens is expressed as $$\varphi (x,y)=k(f-\sqrt{x^2+y^2+f^2}),$$(1)where $x$ and $y$ are the coordinates along the metalens plane, $k=2\pi /\lambda$ is the wavenumber at the design wavelength of $\lambda$, and $f$ is the focal length. Such a phase distribution can focus light under paraxial illumination; however, it introduces severe aberrations in cases of oblique incidence. Therefore, we need to redesign the phase to eliminate the aberrations. The scheme is depicted in Fig. 1(a). The green and red lines indicate ray-I and ray-II with incident angle $\theta$ starting from the same wavefront of the incident plane wave and ending at the same focal spot on the focal plane. Ray-II directly passes through the lens center, and the position of the focal point on the focal plane is at $P=f\tan \theta$, in which the lens can project an undistorted image on the image plane^[37]. Ray-I reaches the lens at the position $x$. The phase delay between ray-I and ray-II at the interface is $$\mathrm{\Delta }\varphi (x,\theta )=kx\sin \theta ,$$(2)where $\theta$ is the incident angle in the air. Since the optical paths between the isophase surfaces (yellow lines) are equal, we can get the following relationship: $$\mathrm{\Delta }\varphi (x,\theta )+{\varphi }_m(x,y,\theta )+k[\sqrt{{(f\tan \theta -x)}^2+y^2+f^2}-f]=\text{const}.$$(3)

By setting the constant to 0 and substituting Eq. (2) into Eq. (3), we can obtain the phase required by the metalens, $${\varphi }_m(x,y,\theta )=k[f-\sqrt{{(x-f\tan \theta )}^2+y^2+f^2}-x\sin \theta ].$$(4)

The metalens can be designed in principles of dynamic phase^[38-40] and geometric phase^[41-43]. Here, we begin from the dynamic phase. The working wavelength is defined at 850 nm. The unit cell of the metalens consists of Si rectangle structures with length $w_x$, width $w_y$, height $H$, and tilted angle $\alpha$ on the ${\mathrm{SiO}}_2$ substrate since Si has high refractive index and relatively low loss. Figure 1(b) shows a normal structure with $\alpha =0°$, and the tilted structure is shown in Fig. 1(c). A commercial solution of Lumerical FDTD is employed in the optical simulations. The simulated phase and transmission maps are shown in Figs. 1(d)–1(k). The light is $x$-polarized, and the incident angle $\theta$ varies with respect to the tilted angle of $\alpha$ (e.g.,  $\theta =0°$ when $\alpha =0°$, and $\theta =10°$ when $\alpha =10°$). The length $w_x$ and width $w_y$ of the structure are swept from 50 nm to 360 nm with a step of 5 nm, height $H=1\mathrm{\mu m}$, and period $P=400\mathrm{nm}$. For the four different slanted structures, the phase can cover $2\pi$, and the transmissions of most structures are close to unity. The green triangle marks indicate the selected 8 different structures, and the corresponding phase and transmission efficiency are shown in Figs. 1(l)–1(o). The average transmission efficiency of the 4 groups is 0.97, 0.97, 0.95, and 0.96, respectively.

Based on the selected structures, metalenses are designed with $\alpha =0°$, 10°, 20°, 30° for the focusing analyses with incident angle $\theta$ ranging from $-20°$ to 60°, respectively. All the structures of a metalens slant in the same direction and with a same angle of $\alpha$. The lens size and focal length are $R=7.4\mathrm{\mu m}$ and $f=35\mathrm{\mu m}$. The lens has a numerical aperture of 0.2. Figures 2(a)–2(d) show the corresponding light intensity distributions in the $x–z$ plane (top) and $x–y$ plane (bottom) for each metalens. It can be found that lenses with slanted structures show a brighter focus spot at oblique incidences. To better compare the metalens performance, the cross-sectional intensity distributions of focal spots in the $x–y$ plane are shown in Figs. 2(e)–2(h). When $\alpha =0°$, the focal spot energy shows maximal value at $\theta =0°$ and decreases significantly with increasing $\theta$, while as expected, when slanted angle $\alpha$ increases, the maximum peak shifts to the larger incident angle ($\theta >0°$) cases.

Besides the focal spot intensity, we also quantify the transmission ($T$), diffraction efficiency (${\eta }_{\mathrm{dif}}$), and total efficiency (${\eta }_{\mathrm{tot}}$) of each lens, as shown in Figs. 3(a)–3(c). The ${\eta }_{\mathrm{dif}}$ is defined as ${\eta }_{\mathrm{dif}}=I_{\text{spot}}/I_{\text{plane}}$, where $I_{\text{spot}}$ is the total light intensity in the area with diameter $3\lambda /2\mathrm{NA}$ on the focal plane and $I_{\text{plane}}$ is the total light intensity on the focal plane. The total efficiency ${\eta }_{\mathrm{tot}}$ is obtained by multiplying $T$ and ${\eta }_{\mathrm{dif}}$. For lenses with normal structures ($\alpha =0°$), $T$ is maximal at normal incidence ($T\sim 84.86\%$ at $\theta =0°$) and drops with the increasing incident angle ($T\sim 68.79\%$ at $\theta =30°$). Similarly, the diffraction efficiency decreases with increasing $\theta$ (${\eta }_{\mathrm{dif}}\sim 76.95\%$ at $\theta =0°$, and ${\eta }_{\mathrm{dif}}\sim 43.54\%$ at $\theta =30°$). As a result, the total efficiency significantly degrades at oblique incidence (${\eta }_{\mathrm{tot}}\sim 65.29\%$ at $\theta =0°$, and ${\eta }_{\mathrm{tot}}\sim 29.95\%$ at $\theta =30°$). However, we find that the lens performance at oblique incidence can be improved by utilizing the slanted structure. For example, ${\eta }_{\mathrm{tot}}$ of lens with $\alpha =10°$, $\alpha =20°$, and $\alpha =30°$ at $\theta =30°$ (51.31%, 56%, and 51.12%, respectively) are much higher than that of lenses with normal structures, which is attributed to the improved $T$ and ${\eta }_{\mathrm{dif}}$. The improvement of the efficiency can be explained as follows. The strongest radiation direction of the normal structure, which can be approximated as a dipole, is at $\theta =0°$, and the transmission reaches its maximum. However, for the oblique incidences, the emission direction is not along the main lobe of the dipole, which results in the decrease of the efficiency. By tilting the structure, the equivalent main lobe of the dipole is rotated and redirects the maximum focusing. Therefore, the transmission of the tilted design is improved over the normal one. In addition, compared with normal structures, the shadow effect^[44] of these slanted structures may be smaller under oblique incidence, which could also contribute to the improvement in $T$ and ${\eta }_{\mathrm{dif}}$. Figure 3(c) clearly shows the tendency of the efficiency maximum shifts to the larger incident angles. Our results indicate that the performance of the metalens can be improved at certain oblique incidences when the designed structure is tilted correspondingly with the incidence angle.

The MTF, characterizing the imaging performance of the lens, is also calculated, with the results shown in Figs. 3(d)–3(g), in which the diffraction limit (solid black line) is calculated by the focal spot of a plane wave passing through an ideal lens. For lenses with $\alpha =0°$ [Fig. 3(d)], the MTF curve at $\theta =0°$ is close to the diffraction limit. However, the focusing performance of the lens deteriorates as the angle increases; as a result, the MTFs at a large frequency drop away from the diffraction limit. The MTFs of the lens with $\alpha =10°$, $\alpha =20°$, and $\alpha =30°$ at a large spatial frequency are obviously higher than that of the lens with $\alpha =0°$, which indicates better focus performance of the lens with $\alpha =10°$, $\alpha =20°$, and $\alpha =30°$ at large incident angle.

In addition, we design and characterize the metalens based on geometric phase. The $w_x$ and $w_y$ of the tilted structure ($\alpha =20°$, $\theta =20°$) are 115 nm and 305 nm, the corresponding transmittance is 92.16%, and the polarization conversion rate (PCR) is nearly 1. The $R$ and $f$ of the metalenses based on geometric phase are the same as that of metalenses based on dynamic phase. The cross-sectional intensity distributions of the focal spot in the $x–y$ plane are shown in Figs. 4(a)–4(d). Compared to the lens with $\alpha =0°$ shown in Fig. 4(a), the spot intensity of the lenses with $\alpha =10°$, 20°, 30° [Figs. 4(b)–4(d)] improves at $\theta =10°$ to $\theta =60°$, which implies that lenses with tilted structures perform better at oblique incidence. Figures 4(e)–4(h) show $T$, ${\eta }_{\mathrm{dif}}$, ${\eta }_{\mathrm{tot}}$, and PCR of the lenses. The ${\eta }_{\mathrm{dif}}$ of all lenses with different $\alpha$ are similar under the same incident angle [Fig. 4(f)]. Notably, compared to the normal metalens, lenses with tilted structures show similar performance in PCR [Fig. 4(h)] but higher transmission [Fig. 4(e)] at oblique incidence, and thus exhibit higher total efficiency [Fig. 4(g)]. For example, ${\eta }_{\mathrm{tot}}$ of lens with $\alpha =30°$ is increased by 17.98% at $\theta =20°$ over the normal metalens. The improvement of total efficiency leads to better MTF at oblique incidences, as shown in Figs. 4(i)–4(l).

We first analyze the aberration of metalenses designed in different angles by Zemax software, as shown in Fig. 5(a). From bottom to the top the metalens is designed with respect to angles from 0° to 60°. From left to right are focal spots with incident angles from 0° to 60°. When the incident angles are close to the designed angles, the spots are small and have little aberrations, while the focal spots become much larger and show obvious coma aberrations when the deviation increases. We note that the spots smaller than 3 times of the Airy disk (black circle markers) have good focusing performance. We obtained the imaging range of each lens by analyzing the size of the focal spots, and it shows these lenses can cover an FOV of 60°, as shown in Fig. 5(b). In order to cover the angle of view of $\pm 60°$, we line up 13 metalenses with designed angles of 0°, $\pm 10°$, $\pm 20°$, $\pm 30°$, $\pm 40°$, $\pm 50°$, $\pm 60°$ to form an array. The average total efficiency of the hybrid metalens array composed of normal and tilted metalenses is 13.59% higher than that of the normal metalens array only composed of normal metalenses, as shown in Fig. 5(c). Finally, we compare the images of objects composed of long letters such as “NANJINGUNIVERSITY” with the normal metalens array and the hybrid metalens array, respectively. The obtained sub-images at each angle can only show part of the letters, and by stitching these images together we can finally get the complete image, similar to the reported work^[25]. Nevertheless, it can be seen that the hybrid metalens array presents a higher intensity image than the normal metalens array at large incidences ($\pm 20°$ to $\pm 60°$), as shown in Fig. 5(d), indicating the efficiency advantages of the hybrid lens array at large incident angles.

In summary, metalenses with tilted structures are systematically modeled and analyzed theoretically, which show better focusing performances under oblique incidences. Characterizations, including transmission, focusing efficiency, total efficiency, and MTF, of the lenses designed based on dynamic phase and geometric phase are investigated in detail. For dynamic phase, the total efficiency of the tilted design is improved by 26.05% over the normal lens at 30° incidence. As for geometric phase, the focus performance of the tilted design is also improved by 17.98% higher than the normal lens at 20° incidence. Compared with normal lenses, the significantly improved MTF curves demonstrate its superiority for oblique incidences. Moreover, the total efficiency of the tilted design based on dynamic phase and geometric phase slowly decreases about 9% and 7% as the angle of incidence varies from 0° to 30°, respectively. Finally, we propose a metalens array with 13 hybrid lenses and obtain a 120°-wide viewing angle with 13.59% higher average efficiency over the normal metalens array. We believe this design may provide some insights to high-efficiency meta devices related with wide angles in the future.