A broadband achromatic metalens for focusing and imaging in the visible

A key goal of metalens research is to achieve wavefront shaping of light using optical elements with thicknesses on the order of the wavelength. Such miniaturization is expected to lead to compact, nanoscale optical devices with applications in cameras, lighting, displays and wearable optics. However, retaining functionality while reducing device size has proven particularly challenging. For example, so far there has been no demonstration of broadband achromatic metalenses covering the entire visible spectrum. Here, we show that by judicious design of nanofins on a surface, it is possible to simultaneously control the phase, group delay and group delay dispersion of light, thereby achieving a transmissive achromatic metalens with large bandwidth. We demonstrate diffraction-limited achromatic focusing and achromatic imaging from 470 to 670 nm. Our metalens comprises only a single layer of nanostructures whose thickness is on the order of the wavelength, and does not involve spatial multiplexing or cascading. While this initial design (numerical aperture of 0.2) has an efficiency of about 20% at 500 nm, we discuss ways in which our approach may be further optimized to meet the demand of future applications. Controlling the geometry of each dielectric element of a nanostructured surface enables frequency-dependent group delay and group delay dispersion engineering, and the fabrication of an achromatic metalens for imaging in the visible in transmission.

or cascading. While this initial design (numerical aperture of 0.2) has an efficiency of about 20% at 500 nm, we discuss ways in which our approach may be further optimized to meet the demand of future applications.
Conventional refractive optical components are generally bulky, costly, and timeconsuming to manufacture with high precision 1 . These are significant limitations, particularly for applications such as portable and wearable devices. In recent years, metasurfaces have emerged as a versatile platform for wavefront shaping. Since the phase is accurately controlled by subwavelength-spaced structures with thicknesses at the wavelength scale or below, many compact optical devices based on metasurfaces have been demonstrated. These include flat lenses 2-4 , polarimeters [5][6][7] , axicons 8,9 , polarization elements 10-12 and holograms [13][14][15] . However, these devices are highly chromatic despite being comprised of weakly dispersive materials. This can be attributed to two separate factors: dispersion arising from a periodic lattice (see Fig. S1 for detailed discussion), as well as light confinement in either a resonant or guided manner. Previous works have addressed this challenge by using multiple coupled resonances to tailor phase profiles at several discrete frequencies 16,17 , stacking/stitching several layers of meta-surfaces 18 , increasing the phase modulation to be more than 2p radians (so-called "multi-order diffractive lenses") 19 or engineering the dispersion [20][21][22][23] . Recently, achromatic focusing at green wavelengths (with a 60 nm bandwidth) in a reflective metalens was achieved 24 . Other groups have experimentally demonstrated achromatic metalenses in the near-infrared with bandwidths of a few tens of terahertz (THz) [25][26][27] . However, none of these works demonstrated achromatic imaging.
Here, we demonstrate the ability to engineer the frequency dependent phase profile φ( ,ω), and thereby achieve arbitrary control of metalens dispersion over a large continuous bandwidth in the visible. This is made possible by separately engineering the group delay and group delay dispersion of each constituent nanostructure, independent of its phase, at a given frequency. This is distinct from other approaches, particularly that of Wang et. al. 27 . They utilized plasmonic resonances without considering group delay dispersion, and therefore lacked a systematic, general way to implement metalenses of different dispersion. As a proof of concept, we demonstrate metalenses with tailored dispersion, including achromatic metalenses with diffractionlimited focusing covering nearly the entire visible (from 470 nm to 670 nm). The achromatic metalens is also capable of performing white light imaging. Finally, we design and model a metasurface which, when patterned over a commercial spherical lens, renders it achromatic and diffraction-limited across the visible spectrum.

Principle of achromatic metalenses
As an initial example, consider the achromatic metalens shown in Fig. 1(a). The relative phase provided by the metalens elements with respect to the center follows 3 : where w, c, r and F are angular frequency, light speed, radial coordinate, and focal length, respectively. This spatial-and frequency-dependent phase profile ( , ) r j w implies that at a given r, the metalens provides different transverse wavevectors so that different wavelengths are deflected by the same angle. Equation   1 can be Taylor-expanded near a design frequency d w as  Fig. 1(a). The first term leads to a spherical wavefront (yellow line in Fig. 1(a)). The group delay term compensates for the difference in the wavepackets' arrival times at the focus, while the high-order derivative terms (group delay dispersion etc.) ensure that the outgoing wavepackets are identical. The net effect is the minimization of the spread in the arrival times of wavepackets at the focus to ensure they constructively interfere. The smaller the time spread, the larger the bandwidth achievable. Therefore, in order to realize diffraction-limited focusing for a broad bandwidth, both phase and group delay, as well as higher order terms, need to be considered.
To account for the dispersion of a metalens, the focal length F in Eq. 1 can be parametrized as: where k is a positive constant and n is a real number. The dispersion of a metalens can thus be designed to arbitrary specification by substituting different values for n (a real number). We refer to the metalenses with n = 0 and n = 1 as achromatic and diffractive

Independent control of phase and group delay
To increase the degrees of freedom in our design, we utilized coupled phase-shift elements: two nanofins in close proximity, acting as coupled waveguides. Their geometric parameters are defined in Fig. 2(a), and scanning electron microscopic images from a fabricated metalens are provided in Fig. 2(b) and Fig. S2. It has been previously shown that coupled waveguides can support tunable dispersion, e.g. near zero group delay dispersions for a wide bandwidth 28,29 . For simplicity and without loss of generality, we first consider the optical properties of a single TiO 2 nanofin, which can be fabricated using electron beam lithography followed by atomic layer deposition 30 . When a left-handed circularly polarized beam passes through the nanofin, the transmitted light can be described by the Jones vector 31 : where L t and S t represent complex transmission coefficients when the incident light is polarized along the long and short axis of the nanofin, and a is the rotation angle.
The second term in Eq. 4 is cross-polarized; we refer to its normalized amplitude squared as the polarization conversion efficiency hereafter. The phase shift is determined by the product ( -) exp( 2 ) where 2a is a frequency-independent 8 geometric phase equal to twice the rotation angle. This allows us to decouple the target phase profile from the required group delay and group delay dispersion (controlled by -L S t t ). Figure 2(c) shows phase spectra for a nanofin with different rotation angles.
The slope is approximately linear within a given bandwidth, and is independent of the rotation angle of the nanofin. This property allows us to design achromatic metalenses with a large bandwidth.
To gain physical insight into the dispersion design, each TiO 2 nanofins can be regarded as truncated waveguide. Neglecting end reflections, the phase of the transmitted light after passing through the structure at a given coordinate r is where n eff and h represent the effective index and the height of the nanofin, respectively. The derivative with respect to angular frequency yields the group delay: this is the ratio of the nanofin height to group velocity, which can be controlled by the nanofin dimensions and/or material used. Figure

Achromatic focusing and imaging
To demonstrate the versatility of our approach, we designed and fabricated an achromatic metalens (n = 0) as well as two other metalenses with n = 1 and 2. They all possess NA = 0.2 at wavelength l = 530 nm. For n = 1, i.e. a regular diffractive metalens, the phase profile was imparted by identical nanofins using the geometric phase 34 . The achromatic metalenses were designed by digitizing the required phase and group delay, µm, as shown in Fig. 3(b) to 3(d). The z-coordinate corresponding to the peak intensity value gives the focal length for a given wavelength.
We characterized the performance of these diffractive and achromatic metalenses in terms of their focal spot profiles (Fig. 3(e) and 3(f)). They were measured at the focal plane corresponding to an illumination wavelength of 470 nm (white dashed lines in Fig. 3(b) and 3(c)). The diffractive metalens shows significant defocusing when the wavelength is larger than 550 nm (Fig. 3(e)). In contrast, the focal spots of Control experiments using the diffractive metalens are in Fig. S6. The test target was fixed at the focal plane corresponding to illumination at l = 470 nm. In Fig. 4(b), a slight decrease of contrast in the images at red wavelengths is observed since the feature size of the target (~ 15 µm) is close to the diffraction limit of the achromatic metalens; there is also a decrease in efficiency of the metalens at red wavelengths.
Additionally, we demonstrate white light imaging using a broadband illumination source (white light laser) from 470-670 nm. The images of the USAF target and Siemens star are shown in Fig. 4(c) and 4(d). These images show that chromatic aberration is well-corrected even under white light illumination, and that the metalens is able to achieve high imaging quality over a few square millimeters, corresponding to a 30 degrees field of view. Note that the patterns at the center of the USAF target and the Siemens star have feature sizes smaller than the resolution of the achromatic metalens. We measured the focal spots of the metalens for different angles of incidence, and found that their Strehl ratios are larger than 0.8 till an incident angle of about 15 degrees (Fig. S7). The color of these white light images has a bluish tinge because of the wavelength-dependent efficiency of the metalens (see Methods for efficiency measurement). We measured about 20% efficiency around 500 nm, against a theoretically predicted value of 50% ( Fig. S8(a)). This deviation likely results from fabrication errors and the coupling between metalens elements. We simulate the phase of each element (coupled waveguides) using periodic boundary conditions. This approximation ignores near-field coupling to adjacent metalens elements, thereby 13 introducing a perturbation on the wavefront and an attendant reduction in efficiency.
This effect can be taken into account with comprehensive full lens simulation followed by optimization to increase efficiency 35,36 . Note that the metalens efficiency is lower compared to our prior work 37 , because in order to cover a larger range of group delay, some low polarization conversion efficiency elements must be chosen (see Fig. S3).
The metalens efficiency can be increased by introducing more complicated nanostructures to increase the freedom of design parameters, or by choosing highly efficient elements at the expense of reducing metalens diameter. We fabricated and measured two metalenses with smaller diameters with efficiencies of about 40% ( Fig.   S8(b)).
Our design principle can be applied to other regions of the electromagnetic spectrum 38,39 . In addition, as mentioned previously, realizing achromatic metalenses with larger diameters and higher numerical apertures requires a larger range of group delay supported by various combinations of nanofins with different dimensions. This can be realized, as shown in Eq. 5, by different dispersion engineering approaches [40][41][42][43]  To design the achromatic meta-lenses, we linearly fit the phase spectrum of each element in our library at the design wavelength (530 nm) within a 120 nm bandwidth to obtain the group delay. Any element that has an R-squared value and polarization conversion efficiency less than 0.99 and 5% was dropped. The linear fitting with high R-squared value ensures that all selected elements from the library fulfill the requirement for realizing achromatic meta-surfaces with larger than 120 nm bandwidth.
Our simulations and measurements of the point spread function have shown that the metalens remains achromatic with up to a bandwidth of 200 nm. Finally, for each coordinate of the achromatic metalens, the required group delay was achieved by positioning the corresponding suitably rotated element in order to satisfy the required phase profile at wavelength l = 530 nm. For the metalens with n = 2, we fit the phase spectrum with a quadratic polynomial to obtain group delay and group delay dispersion.
Two different offset values are chosen by the particle swarm algorithm to best satisfy the required relative group delay and group delay dispersion (see Fig. S10 for details).
Relation between required group delay and metalens radius and NA. The required range of group delay for an achromatic metalens is given by where c is light speed and R is the metalens radius, as obtained by substituting For all measurements, a pair of crossed circular polarizers were used to reduce background noise. A schematic of the experimental setup and details for Fig. 4 can be found in Fig. S11.
To measure metalens efficiency, we measured the power in the focal spot (using an iris aperture to block background light at the imaging plane) and divided it by the power of circularly polarized incident light passing through an aperture with the same diameter as the metalens.

Data availability
The libraries that support the design of metalenses are available from the corresponding author on reasonable request.

Additional Information
Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to W. T. C. and F. C.
Y. Z. and E. L. measured the metalenses. W. T. C., A. Y. Z., M. K. and F. C. wrote the manuscript. All authors discussed the results and commented on the manuscript.

Competing financial interests
The authors declare no competing financial interests