A lens is a fundamental element but plays a key role in optical system. Traditional lenses control the light wave-front based on the phase accumulation in propagation paths. Therefore, they contain the thickness differences of the geometric shapes. However, limited by the refractive index of the natural materials, the thickness of the traditional lenses is often much larger than the working wavelength, which impedes the miniaturization and integration of optical systems. Depending on abrupt phase discontinuity in subwavelength thickness, metasurfaces are ideal materials for integrated and miniaturized optical systems^1-4. The wave-front manipulations of the metasurfaces are relying on the change of the geometrical parameters, such as length^{5, 6}, width⁷, shape^{8, 9} and orientation angle¹⁰ of the local structures. A series of novel lenses with varied properties have been demonstrated using metasurfaces^11-13, such as achromatic lenses^{14, 15}, polarization sensitive/insensitive lenses^{16, 17}, switchable functionality lenses¹⁸, sub-diffraction lens^19-21, etc. However, a large proportion of contemporary metalenses are composed of discrete micro/nano structures, which may affect their final performances. On the one hand, the discretization accuracy will affect the focusing/imaging quality of the lens. Although small enough pixels can theoretically alleviate this kind of problem, metalens composed of small unit cells are hard to process. Significantly, the near-field coupling becomes more obvious with the decrease of the distance between adjacent elements. On the other hand, due to the resonance property of the commonly used discrete structure, the energy efficiency of the constructed metalens always only keeps a high value near a preset wavelength, which greatly limits the operating bandwidth of the lens.

Fortunately, the quasi-continuous metasurfaces (such as metallic catenary/trapezoid and dielectric nanoarc/streamlined shapes) proposed in recent years can relieve the above problems and realize broadband high-efficiency metadevices. High-performance metadevices, such as one-dimensional lens²², optical angular momentum generator²³, anomalous reflector^{24, 25}, chiral splitter and hologram²⁶ have been achieved via quasi-continuous metasurfaces. However, their abilities of engineering phase and amplitude need to be further upgraded. The main reason is the fact that the phase related to the catenary structure changes linearly along a certain direction in space, so if the target phase does not satisfy the linear distribution relation, the approximate processing must be carried out, which will cause some deviations from the target wave-front. Similarly, the phase manipulation ability for the trapezoid and nanoarc will also be limited by their fixed geometry. The metalens made up of streamlined structures²⁷ can achieve a high diffraction efficiency (approaching 100%), but it works in the infrared band. Consequently, it is of great significance to simultaneously realize the broadband and high-efficiency advantages and arbitrary phase manipulation using quasi-continuous metasurface in the visible light or other wavelength range.

In this paper, a dielectric metasurface composed of quasi-continuous nanostrips is introduced to realize the broadband high-efficiency lenses at the wavelength ranging from 450 nm to 1000 nm, which includes the visible and near-infrared region. Actually, the phase shift of the quasi-continuous metasurface derives from the Pancharatnam-Berry (PB) phase (also known as geometric phase) principle, which is related to the different orientation angles of the nanostrips at different spatial positions. Thanks to the broadband property of the PB phase, the phase shift of the proposed quasi-continuous metasurface is intrinsically insensitive to the illumination wavelength. In addition, the power efficiencies of the quasi-continuous metalenses are nearly achromatic. In theory, the phase of the lens changes continuously with the spatial position, except for a few phase jump points. Based on the PB phase principle²⁸ (Φ=2σφ, where σ=±1 denotes the left-hand circular polarization (LCP)/right-hand circular polarization (RCP) light incidence, φ is the orientation angle of the nanoapertures), the nanostrip with its orientation angle spatially continuous changing can achieve the continuous phase required by a focusing lens, which is different from the discontinuous phase of the discrete structures.

As an example, a normal metalens composed of the quasi-continuous nanostrips metasurface is designed and fabricated to demonstrate the broadband and high-efficiency advantages of the proposed metadevice. The design method of the quasi-continuous metalens is given and the experimental results of the illumination wavelength from 500 nm to 1000 nm are listed. Table 1 shows some high-efficiency metalenses operating in the visible and near-infrared bands. The power efficiency in Table 1 is defined as the ratio of the power of the focused light at the focal plane (or within the circle of radius 3×FWHM (full-width-at-half-maximum) spanning the center of the focal spot) to the incident power, which have been marked in the last column of Table 1. Actually, high-efficiency metalenses can be categorized into two groups with different working mode. The first group uses reflection metastructures^{29, 30}, which includes metal and dielectric materials. The other group is using transmission metastructures. Large amounts of related works have been carried out owing to the convenience of using transmission lenses. Crystalline Si (c-Si) is often used for metalenses operating in the infrared band³¹, while titanium dioxide (TiO₂)/gallium nitride (GaN)/silicon nitride (Si₃N₄) are commonly used in visible and near-infrared range for their low absorption losses in the relevant bands^32-38. All these representations of high-efficiency metalenses are progressive and inspiring. However, current metalenses are often composed of discrete structures, which limits their operational bandwidth and efficiency. Increasing the thickness of the structural layer can alleviate this problem, but the fabrication of metalenses using high-aspect-ratio structures is very challenging. So many high-aspect-ratio metalenses are only theoretically proved^{37, 39, 40}. With utilizing the quasi-continuous nanostrips, the average efficiency of the metalens in this work experimentally reaches 54.24% in the wavelength range 500-1000 nm, showing a significant improvement compared to the previously reported metalenses with the same thickness.

With similar method, a superoscillatory lens with a focal spot 0.7 times of the diffraction limit is designed and fabricated to further verify the feasibility of the proposed scheme. Compared with reported metallic-type quasi-continuous metasurfaces^{41, 42}, the dielectric structures using titanium TiO₂ in this work can effectively improve the energy efficiency of the quasi-continuous metadevices for visible light. Compared with our previously published quasi-continuous metahologram with simulated results⁴³, this work experimentally verifies the broadband high-efficiency characteristics of normal focusing metalens and superoscillatory lens based on quasi-continuous nanostrips. In addition, the optimized width and minimum distance for the nanostrips based on material parameters in the actual fabrication process enable a better performance of the designed quasi-continuous metadevices, especially for the power efficiency in the short wavelength range. The proposed approach is significant for the practical development of metasurface. In addition, some other metadevices, such as broadband high-efficiency reflector, beam splitter, vortex beam generator and virtual shaping, can be designed with similar scheme.

The schematic illustration of the proposed quasi-continuous metalens is shown in Fig. 1. The broadband high-efficiency metalens comprises abundant quasi-continuous nanostrips (as shown in the inset of a higher magnified image for part of the metalens), which is different from the commonly used discrete structures. The LCP light is normally incident on the sample, and the modulated RCP light is collected in the transmitted space. The phase shift of the transmitted light originates from the PB phase corresponding to the curved trajectory of the quasi-continuous nanostrips. Based on the relationship between phase distribution and focal length²⁰, the focal plane of the quasi-continuous metalens will gradually move away from the sample with the decrease of the illumination wavelength.

The phase distribution of a normal focusing metalens should be satisfied as follows:

Φ(x,y)=k0(−(f02+x2+y2)1/2+f0)+2mπ,（1）

where k₀=2π/λ₀ is the wave vector, f₀=1 mm is the focal length, (x, y) are coordinates on the lens plane and m is an integer to ensure the phase Φ(x, y) in the range of –π to π. With the phase relation shown in Eq. (1), a normal metalens composed of quasi-continuous nanostrips can be designed, as the portrait in Fig. 2(a), and its zoomed-in image is shown in Fig. 2(b). More details about the design process of the quasi-continuous metalens can be found in Section 1 of Supplementary information. The phase shift profile of the designed quasi-continuous metalens is illustrated in middle panel of Fig. 2(a), and the corresponding phase distribution along the center line (white dotted line) of the sample is shown in low panel of Fig. 2(a). TiO₂ is used as the structure material for the compatibility with the CMOS technique and the low absorption in the visible and near-infrared range. The quasi-continuous metalens was fabricated on 600-nm-thick TiO₂ film on SiO₂ substrate. The fabrication process of the metalens can be found in Section 2 of Supplementary information. The SEM image of a small area of the fabricated metalens is shown in Fig. 2(c). The SEM image indicates the high-quality fabrication of the quasi-continuous nanostrips, which is essential for the power efficiency in the experiment.

The optical performance of the fabricated sample was measured by a homebuilt microscope, as the schematic shown in Fig. 3(a). The illuminated light generated by a white light laser (NKT photonics EXR-15) normally impinged on the sample and the transmitted light was captured by a charge coupled device (CCD). The cascaded LP₁ (LP: linear polarizer) and QWP₁ (QWP: quarter waveplate) in front of the sample were used to convert the illuminated light to the required LCP light. In order to obtain the intensity of the transmitted RCP light, the cascade QWP₂ and LP₂ were placed between the sample and CCD. For the convenience of the measurement for the focal spot with CCD, an ×40 (numerical aperture = 0.65) objective was applied in the experiment. Through moving the sample along the propagating axis, the intensity distributions of the focal spot at different distances from the sample surface could be obtained. At the incident wavelength of 632.8 nm, the intensity distributions of the transmitted RCP light along the propagation direction (x-z plane and y-z plane) in the experiment are shown in Fig. 3(b). The measured cross-polarized intensity at the focal plane (z =1000 μm) of the calculated wavelength 632.8 nm is plotted in Fig. 3(c). The measured focusing full-width-at-half-maximum (FWHM) at the wavelength of 632.8 nm along x-axis (y-axis) is 2.11 μm (2.15 μm), approaching the Abbe diffraction limit of 0.5λ/NA=2.13 μm (deviations < 0.94%). By changing the emission wavelength of the used white light laser, the broadband performances of the fabricated metalens can be achieved, as the intensity profiles at different wavelengths shown in Fig. 3(d−i). Through measuring the cross-polarized intensity distributions along the propagation direction, the focal lengths for different wavelengths can be achieved, as labelled in the Fig. 3(d−i). The incident wavelength in the experiment changes from 500 nm to 1000 nm and the measured focal length decreases with the increase of the illuminated wavelength, which agrees with the theoretical prediction⁴⁴. The FWHMs of focal spots in experiment for the wavelengths of 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1000 nm along x-axis (y-axis) are 2.13 μm (2.14 μm), 2.10 μm (2.11 μm), 2.10 μm (2.23 μm), 2.38 μm (2.36 μm), 2.05 μm (2.16 μm), 2.22 μm (2.21 μm), respectively. All the FWHMs deviate < 9.68% from the theoretical values, which demonstrates the diffraction limited characteristic of the fabricated quasi-continuous metalens (more details about the FWHMs and Abbe diffraction limit for the quasi-continuous metalens can be found in Section 3 of Supplementary information). The quasi-continuous metalens here can achieve good focusing effects under single-wavelength laser illumination. If white light illumination is used, due to the difference of the focal length, FWHM of the focal spot and power efficiency for different wavelengths, a color spot with optical dispersion will be obtained.

In order to further verify the feasibility of the arbitrary phase manipulation with the proposed quasi-continuous nanostrips, a superoscillatory lens (SOL) for sub-diffraction limit focusing is designed and fabricated. In theory, the required phase profile of a SOL can be regarded as a combination of a lens phase profile (Φ_lens) and an additional phase modulation (Φ_SOL(r), r is the radial coordinate), which provides modulations of the high and low spatial frequencies, and the field distribution at the focal plane can be calculated as^{19, 20}:

I(ρ)∝(1λf)2|∫0Rexp[iΦSOL(r)]J0(2πrρλf)rdr|2,（2）

where ρ is radial coordinate on the focal plane, f=f₀ is the focal length and λ=λ₀ is the calculated wavelength, which is consistent with the up-mentioned normal metalens. R=150 μm is the aperture radius of the SOL and J₀ is zero-order Bessel function. In the design process, the full width of the central spot for the field distribution I(ρ) is set as 0.7 times the counterpart of Airy pattern. In addition, the M value, defined as the ratio of the maximum side-lobe intensity to the central intensity, is set as 0.15 in this work. The additional phase modulation Φ_SOL(r) can be optimized as the binary phase Φ_binary (0 or π) through a reverse design based on the linear programming method^{45, 46}, and the finally optimized π-phase-jump positions are 0.175R, 0.406R and 0.645R (the down panel of Fig. 4(b)). The design process of the quasi-continuous SOL is consistent with the quasi-continuous normal metalens design, except that the required phase in Eq. (1) is replaced by the following formula:

ΦSOL(x,y)=k0(−(f02+x2+y2)1/2+f0)+Φbinary+2mπ.（3）

The designed quasi-continuous SOL is shown in Fig. 4(a). A higher magnified image and the corresponding phase distribution for part of the SOL are displayed in the top-left inset and down-right inset of Fig. 4(a). The fabrication process of quasi-continuous SOL is the same with the above-mentioned metalens fabrication and the SEM image of the fabricated SOL is shown in Fig. 4(c). Using the homebuilt microscope (Fig. 3(a)), the optical properties of the SOL are tested. The measured cross-polarized intensity distributions along the propagation direction (Fig. 4(d)) show a needle-like focusing behavior, which is in accord with some related works²⁰. The focal intensity distributions at various incident wavelengths are shown in Fig. 4(e–k). The measured focusing FWHM at the calculated wavelength 632.8 nm is 1.72 μm (1.62 μm) along x-axis (y-axis), about 0.808 (0.761) times the Abbe diffraction limit. Theoretically, the focal FWHM is 0.775 times the Abbe diffraction limit. The experiment results show a good agreement with the theoretical results, except some experiment errors in the test process. In addition, the focal FWHMs in experiment at the wavelengths of 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1000 nm along x-axis (y-axis) are 1.55 μm (1.54 μm), 1.61 μm (1.69 μm), 1.69 μm (1.72 μm), 1.71 μm (1.72 μm), 1.63 μm (1.63 μm), 1.74 μm (1.67 μm), which are 0.731 (0.726) times, 0.759 (0.797) times, 0.793 (0.808) times, 0.792 (0.796) times, 0.758 (0.758) times, 0.798 (0.766) times of the related Abbe diffraction limit, respectively (more details about the FWHMs and Abbe diffraction limit for the quasi-continuous SOL can be found in Section 4 of Supplementary information). The experiment results clearly demonstrate the sub-diffraction focusing feature in a broadband wavelength range and prove the feasibility of the SOL based on the quasi-continuous nanostrips.

The wonderful feature of the proposed quasi-continuous nanostrips metadevices is the realization of high-efficiency in a broadband wavelength range. For comparison, the simulated and experimental power efficiency are shown in Fig. 5. The experimental power efficiency is defined as the ratio of the power of focused light (RCP light) at the focal plane to the incident light with opposite helicity (LCP light). It’s worth noting that the distance between adjacent nanostrips is different at different positions of the proposed quasi-continuous metasurface. For convenience, the quasi-continuous metasurface is simplified as a grating with period P in the simulation, where P corresponds to the distance between adjacent nanpstrips. The commercial software CST Microwave Studio is used to calculate the optical responses of the cross-polarized light for the simplified grating. The LCP light is normally illuminated on the structure from the SiO₂ substrate side. The boundary conditions along x-axis, y-axis and z-axis are set with unit-cell, unit-cell and open, respectively. Considering the distance features of the designed metalens (Fig. 2(a)) and SOL (Fig. 4(a)), the cross-polarized efficiency (simulated power efficiency) of the grating for P=160 to 300 nm are calculated and depicted in the blue region in Fig. 5. Obviously, the simulated power efficiency changes slowly with the change of grating period. More importantly, the simulated power efficiency remains above 37% at the wavelength ranging from 450 to 1000 nm, which is different from the discrete metasurface that often maintains high efficiency near a preset wavelength. It should be mentioned that the simulated power efficiency of the grating is the approximate results of the electromagnetic response of the quasi-continuous metasurface. However, this approximate method is effective and has been widely used in other works^{23, 41, 47, 48}. The red diamonds and black squares in Fig. 5 show the measured power efficiency for quasi-continuous metalens and quasi-continuous SOL, respectively. The highest experimental efficiency is 75.78% at 632.8 nm for metalens and the lowest experimental effiecieny is 29.23% at 1000 nm for SOL. The average efficiencies at the wavelength range 500−1000 nm of the demonstrated normal metalens and SOL are as high as 54.24% and 48.48%, respectively. Obviously, the experimental efficiencies are well matched with the simulated results and the average efficiencies keep at high value at the wavelength range 500−1000 nm, which verifies the excellent characteristics of maintaining high-efficiency in a broadband wavelength range. In fact, due to the difference for the distances between adjacent nanostrips, the parasitic propagation phase gradients may be leaded, which will reduce the performance of the quasi-continuous device here. Using the composite phase control method^{27, 49-52} to suppress the parasitic propagation phase, the power efficiency of the proposed quasi-continuous metalens/SOL can be further improved.

In summary, the quasi-continuous metasurface composed of all-dielectric nanostrips is proposed to simultaneously realize the broadband and high-efficiency advantages. For demonstrating the arbitrary phase manipulation of the quasi-continuous metasurface, two functional devices (normal metalens and SOL) are designed and fabricated. The fabricated quasi-continuous samples can work in a broadband wavelength range and the average efficiencies in the experiment are 54.24% and 48.48% at wavelength range 500−1000 nm. We believe that the proposed quasi-continuous metasurface with flexible and arbitrary phase manipulation and broadband high-efficiency advantage is significant for the development of metasurface. The proposed scheme here can be easily used to design other quasi-continuous phase meta-devices, such as metahologram, vortex beam generator, beam splitter and so on.