## Introduction

Metasurfaces comprising sub-wavelength spaced nanostructures at an interface provide the means to accurately control the properties of light, including phase, amplitude, and polarization1,2,3,4. This allows for the possibility of highly compact and efficient devices5,6,7,8,9,10,11,12,13. Amongst these devices, metalenses have attracted intense interest due to their applicability to both consumer (phone cameras, virtual/augmented reality headsets) and industry products (microscopy, lithography, sensors, and displays)14,15,16,17,18,19,20,21,22,23. Recent works have focused on developing the broadband achromatic focusing capabilities of metalenses in the visible spectrum24,25. However, these metalenses suffer from polarization sensitivity, i.e., they can only focus light of a given circular polarization. This challenge can be overcome by using symmetric cylindrical or square-shaped nano-pillars in both the visible26 and the near-infrared regions27,28,29. However, by doing so, we lose a degree of freedom in the design space due to the symmetry of these constituent structures.

Here, counterintuitively, we show that it is indeed possible to simultaneously achieve an achromatic metalens capable of focusing any incident polarization in the visible using anisotropic TiO2 nanofins. This is a different solution compared to recent publications associated with spatial multiplexing and symmetry30,31,32. These anisotropic nanofins allow us to accurately and simultaneously implement the phase and its higher-order derivatives (i.e., group delay and group delay dispersion) with respect to frequency. We designed and fabricated a metalens with a numerical aperture (NA) of 0.2. The metalens exhibits a measured focal length shift of only 9% λ = 460–700 nm and has diffraction-limited focal spots across the entire range. The focusing efficiency of the metalens varies by only ~ 4% under various incident polarizations. To showcase the generality of our principle, we also demonstrate a polarization-insensitive metasurface with diffraction efficiency of about 92% at wavelength λ = 530 nm.

## Results

### Principle of polarization-insensitive and achromatic focusing

To achromatically focus a broadband incident beam in a diffraction limited spot, a metalens must impart a spatially and frequency-dependent phase profile given by

$$\varphi {\mathrm{(}}r,\omega {\mathrm{) = - }}\frac{\omega }{c}(\sqrt {r^2 + F^2} - F),$$
(1)

where r, ω, and F are the lens radial coordinate, angular frequency, and a constant focal length, respectively. The Taylor expansion of Eq. 1:

$$\varphi {\mathrm{(}}r,\omega {\mathrm{) = }}\varphi {\mathrm{(}}r,\omega _{\rm{d}}{\mathrm{) + }}\left. {\frac{{\partial \varphi }}{{\partial \omega }}} \right|_{\omega = \omega _d}(\omega - \omega _{\rm{d}}){\mathrm{ + }}\left. {\frac{{\partial ^2\varphi }}{{2\partial \omega ^2}}} \right|_{\omega = \omega _{\rm{d}}}(\omega - \omega _{\rm{d}})^2 + ...$$
(2)

identifies the required phase φ (r,ωd), group delay $$\left. {\frac{{\partial \varphi }}{{\partial \omega }}} \right|_{\omega = \omega _{\rm{d}}}$$, and group delay dispersion $$\left. {\frac{{\partial ^2\varphi }}{{\partial \omega ^2}}} \right|_{\omega = \omega _{\rm{d}}}$$ that needs to be fulfilled at every lens coordinate r. An intuitive way to understand each term in Eq. 2 is to treat the incident light as wavepackets. The required phase profile sends incident wavepackets towards the focal point, while the first and the higher order derivative terms ensure that the incident wavepackets arrive at the focal point simultaneously and identically in the time domain, respectively24. The challenge here lies in the fact that the chosen nanostructures must satisfy each derivative term in Eq. 2 at every lens coordinate. Previous designs made use of the geometric (or Pancharatnam-Berry) phase principle to decouple the phase, φ (r,ωd), from the dispersion (group delay and group delay dispersion)24,25,33. However, this approach also comes with an unwanted polarization-sensitivity, i.e. these achromatic metalenses can only focus incident light with a particular circular polarization.

Our design principle still involves Pancharatnam–Berry phase; however, we circumvent the aforementioned drawback by limiting the rotation angle of each anisotropic element to either 0 or 90 degrees. Each element is comprised of multiple nanofins to provide additional degrees of freedom to engineer the dispersion (Fig. 1a, inset). The layout of a quarter of our achromatic and polarization-insensitive metalens is depicted in Fig. 1a and a scanning electron microscope image from a region of our fabricated metalens is shown in Fig. 1b. To tune the phase and dispersion, each nanofin’s length and width is varied and the gap (g) between nanofins is set to be either 60 nm or 90 nm. By using anisotropic elements instead of standard symmetric circular or square pillars26,28, we have more geometric parameters to alter for better dispersion control. More importantly, the anisotropic elements offer the freedom to impart an additional π phase shift without changing their dispersion characteristics. This is essential in order to fulfill both the required phase and dispersion given by Eq. 2, and can be understood from the Pancharatnam–Berry phase34,35. When light passes through a nanofin, the transmitted electric field can be described by the Jones vector:36

$$\left[ {\begin{array}{*{20}{c}} {{\tilde{\mathrm E}}_x} \\ {\tilde E_y} \end{array}} \right] = \frac{{\tilde t_l + \tilde t_s}}{2}\left[ {\begin{array}{*{20}{c}} 1 \\ { \pm i} \end{array}} \right] + \frac{{\tilde t_l - \tilde t_s}}{2}\exp ^{ \pm i2\alpha }\left[ {\begin{array}{*{20}{c}} 1 \\ { \mp i} \end{array}} \right],$$
(3)

where $$\tilde t_l$$ and $$\tilde t_s$$ represent complex transmission coefficients when the normalized electric field of the incident light is polarized along the long and short axis of the nanofin, respectively. The α term is defined as the counterclockwise rotation angle of the nanofin with respect to the x-axis. The first term of Eq. 3 causes unwanted scattering and can be minimized if the nanofin is designed as a miniature half-waveplate. In this case, the amplitude of the second term $$abs(\frac{{\tilde t_l - \tilde t_s}}{2})$$ increases, corresponding to maximal polarization conversion efficiency. The exp±i2α in the second term is accompanied by a polarization converted term and illustrates the origin of Pancharatnam-Berry phase. Under left-handed circularly polarized incidence, a rotation of α imparts a frequency-independent phase of 2α to the right-handed circularly polarized output light ($$\left[ {\begin{array}{*{20}{c}} 1 \\ { - i} \end{array}} \right]$$) without affecting the dispersion, which is determined by $$\frac{{\tilde t_l - \tilde t_s}}{2}$$. This usually results in polarization-sensitivity because the values of expi2α and expi2α, obtained under left and right circular polarized (LCP and RCP) incident light, respectively, are not identical. However, if one arranges the nanofin with α=0° or α=90°, their values become equal. Therefore, both RCP and LCP incident light will experience the same phase profile upon interacting with a metalens consisting of either mutually parallel or perpendicular nanofins. Since any incident polarization can be decomposed into a combination of LCP and RCP, this property implies that the metalens is polarization insensitive, capable of focusing any incident polarization. Figure 1c confirms the results predicted by Eq. 3. A metalens element provides the same phase for both RCP (line) and LCP (circles) incidence, and, for a given circular polarization, a 90-degree rotation imparts a π phase shift without affecting group delay (slope) and group delay dispersion (curvature).

### Design of an achromatic and polarization-insensitive metalens

The design of our polarization-insensitive and achromatic metalens starts from a parameter sweep of the element shown in the inset of Fig. 1a to build a library. We used a finite-difference time-domain (FDTD) solver to obtain each element’s phase at λ= 530 nm, as well as its group delay and group delay dispersion. More simulation details can be found in our previous publication24. Figure 2a plots the three quantities of interest: phase, group delay, and group delay dispersion, at the design wavelength of 530 nm for each element. There are thousands of geometrical combinations, resulting in a dense scatter plot from which we identify the optimal elements to fine tune the dispersion. Note that due to the principle outlined in Fig. 1c, an element rotated by 90 degrees (i.e. purple points) will experience a π phase shift for all frequencies with no change in the values of its dispersion. As a result, the design library can be further extended, allowing for better implementation of the required phase and dispersion (black symbols), which were calculated based on Eq. 1 for an achromatic metalens with a diameter of 26.4 μm and an NA of 0.2. To realize the metalens, the elements selected must be those closest to the required (black) points in the 3-dimensional space of phase, group delay, and group delay dispersion displayed in Fig. 2a. Because only the relative values of these parameters are important, the library can be shifted in this 3-dimensional space to better fit the required values. A particle swarm optimization method was used to find the optimal shifts for phase, group delay, and group delay dispersion, which minimizes the distance between each required point and the values provided by the elements in our library. The final results can be better visualized in Fig. 2b–d. The phase, group delay, and group delay dispersion of the selected metalens elements are shown in blue, together with the corresponding required values (black curves). We only consider terms up to the group delay dispersion because the values of any higher orders for our selected elements are very small.

### Focal spot and focusing efficiency characterizations

We subsequently fabricated the achromatic and polarization insensitive metalens using electron beam lithography, followed by atomic layer deposition of TiO2 and resist removal37, and compared its performance to a chromatic metalens of the same diameter and NA. The chromatic metalens was designed using rotated  nanofins with the same length and width to impart the Pancharatnam-Berry phase. The chromatic metalens represents the case without dispersion engineering and has a focal length shift similar to a Fresnel lens. We also show in Supplementary Movie 1 simulation results for a complete metalens with a smaller lens diameter and a higher NA of 0.6, confirming its achromatic and polarization-insensitive focusing behavior (Supplementary Figure 1). The focal length shifts of the fabricated achromatic and chromatic metalenses were determined by measuring their point spread functions at each wavelength along the propagation direction (z-axis) with 1 μm resolution (Fig. 3a). The left panel in Fig. 3a demonstrates a small focal length variation of about 6 μm for the achromatic metalens compared to that of 30 μm in the chromatic metalens (right panel). The normalized intensity profiles along the white dashed lines can be seen in Fig. 3b and Supplementary Figure 2 for the achromatic and chromatic metalenses, respectively. The achromatic metalens is diffraction-limited and its focal spot sizes and Strehl ratios as a function of wavelength are given in Supplementary Figure 3. Figure 3c shows achromatic imaging of a USAF resolution target from blue to red wavelengths in the visible. The results of imaging colored objects are given in Supplementary Figure 4. The achromatic metalens was also characterized by measuring the focusing efficiency of the focal spot under different polarizations of incident light. The focusing efficiency is defined as the focal spot power divided by transmitted power through an aperture with the same diameter as the metalens. The measured focusing efficiencies weakly change with polarization, as shown in Fig. 3d. The inset shows focal spot profiles for different incident polarizations. These results experimentally prove that the metalens can focus any incident polarization. Note that the polarization state of the focal spot becomes the same as that of the incident light polarized along the axes of nanofins, which can be understood from the polarization converted term in Eq. 3. We attribute the efficiency variation to the interference of the focal spot with background light, i.e., the scattered light from the polarization conserved term (first term on the right-hand-side) in Eq. 3.

## Discussion

The diameter of the achromatic metalens is still small because it is limited by the achievable group delay in nanostructure elements24. The group delay is given as the height of the nanostructure divided by the group velocity of light; this height is limited due to fabrication constraints. Currently, we can achieve a group delay range of about 5 femto-second in our 600-nm-tall TiO2 nanofins. There are some possible ways to circumvent this limitation, e.g. through hybrid diffractive-refractive lens design38,39, high-aspect ratio nanofabrication to increase structure height40,41 and using hyperbolic metamaterials to engineer group velocity over a large range42,43,44.

It is worth noting that the metalens focusing efficiency shown in Fig. 3d is lower than our previous chromatic metalenses19,45,46. This can be explained by the fact that some elements with low polarization conversion efficiency were selected to cover a large range of dispersion values for achromaticity (see Supplementary Figure 5 for a plot of efficiency and dispersion). However, we emphasize that our approach does not preclude the design of highly efficient metasurfaces. For example, we show in Fig. 4a the layout of a conventional chromatic metasurface beam deflector designed for wavelength λ = 530 nm with an output diffraction angle of θ = 15°. Figure 4b shows the normalized far-field power across the visible under x-polarized incidence as a function of wavelength. The metasurface has mainly a single diffraction order over a bandwidth of 50 nm centered at 530 nm and results in a high diffraction efficiency of about 92%. The diffraction efficiency is defined as the power of the first (+1) diffraction order divided by that of transmitted power. We numerically verified in Fig. 4c that such a high diffraction efficiency is maintained under various linearly and circularly polarized incident beams. It can be seen that at a given wavelength, the diffraction efficiency remains relatively constant across all polarizations, highlighting the polarization insensitivity of the metasurface. The absolute efficiency at λ = 530 nm, i.e. the power diffracted to 15 degrees divided by total incident power, is about 70% (see Supplementary Figure 6 for a plot of the absolute efficiency of the metasurface).

We have demonstrated with both simulations and experiments, a general principle for designing polarization-insensitive metasurfaces using anisotropic nanostructures as building blocks. These anisotropic structures allow for a more accurate implementation of phase, group delay, and group delay dispersion, while simultaneously making it possible to realize a polarization-insensitive, diffraction-limited and achromatic metalens from wavelength λ = 460–700 nm. Our design approach of polarization-insensitivity is also valid for other metasurface devices with applications in imaging and augmented reality.