The optical design of our metalens was made using commercial optical design software (Zemax OpticStudio, Zemax LLC, Kirkland, WA, USA), combined with full-wave finite-difference-time-domain (FDTD) simulations to determine the amplitude and phase response of the nanoantennas. The design concept is based on an aperture stop located at the front focal plane of the lens, which results in a telecentric design (the chief ray exits parallel to optical axis), with good off-axis aberration correction [13], [17]. The layout of the optical system is shown in Figure 1B. The front aperture is 1.35 mm in diameter, and the focal length of the metalens is 3.36 mm (F-number 2.5). The design supports field angles of up to 40°, with near-diffraction-limited performance, using a quadratic diffractive phase function, as described by Eq. (1):

$$\varphi \mathrm{(}r\mathrm{)}=a{r}^{2}\text{,\hspace{0.17em}\hspace{0.17em}}a=-1098.2{\text{\hspace{0.17em}mm}}^{-2}.$$(1)

The main performance criterion for imaging lenses is the modulation transfer function (MTF), which describes the resolution of the lens. The MTF gives the modulation (i.e. contrast) attenuation factor for each spatial frequency, as a result of the lens blur spot. The function describing the blur spot is called the point spread function (PSF) and is a two-dimensional function of the transverse horizontal (*x*) and vertical (*y*) image plane coordinates. Integration of the PSF in the horizontal and vertical directions yields the vertical and horizontal one-dimensional line spread functions (LSFs), respectively. The MTF is the Fourier transform of the LSFs; thus, we have two MTFs: horizontal and vertical [18].

The MTF of the nominal metalens design, calculated in Zemax for monochromatic illumination at 850 nm, is shown in Figure 1C. For the on-axis image point, the nominal horizontal and vertical MTFs are identical because of symmetry; however, for the off-axis points, there are two MTF graphs. The direction of the off-axis excursion is called the tangential direction. In our case, this is defined as the *y*-axis direction, as denoted in Figure 1B. The perpendicular direction is called the sagittal direction, which is the *x*-axis direction in our case.

If higher-order diffractive coefficients were used (beyond quadratic), the spherical aberration could be perfectly corrected, so that the on-axis monochromatic MTF (blue line in Figure 1C) would coincide with the diffraction limit. However, when considering the entire FOV, the overall improvement is negligible, which is why we elected to remain with the parabolic phase. Furthermore, previous wide-FOV metalenses used a two-element design [13], [14], with the front element functioning as a Schmidt spherical aberration corrector [19]. This was necessary as they were operating at approximately *F*/1. However, as our metalens operates at a more moderate aperture of F/2.5, which is in line with common cell-phone camera lenses [15], the additional metasurface is not required, making the implementation of the imaging system much simpler.

In addition to MTF, it is also critical for an imaging lens to provide illumination that is both strong enough and reasonably uniform over the FOV. As our lens is designed to operate with broad-spectrum illumination, we must allow a sufficiently wide spectral range of light to pass through to obtain a good illumination level. Unfortunately, the wider the spectral range, the larger the chromatic blurring will be, resulting in a degradation of the monochromatic MTF shown in Figure 1C to the polychromatic MTFs that will be shown later (Figure 4A–C). This trade-off between resolution and illumination signal was explored from a theoretical point of view in our previous paper [16]. The results shown in this paper confirm our previous expectation that at outdoor illumination levels, one can obtain good signal and resolution by using moderate spectral bands of up to 40 nm.

An additional performance parameter, not accounted for by MTF, is the geometrical distortion, which is a distortion of the shape of the imaged objects, without affecting the image resolution. The relative distortion is defined as the ratio between the shift in position of an image point relative to the absolute ideal position. Our telecentric type of design exhibits negative (“barrel”) distortion, reaching 23% at 40° FOV, but only 3.4% at 15° FOV, as shown in Figure S1A. As a rule of thumb, a distortion of up to 10% is not disturbing to a standard viewer. Distortion can also be corrected using established image processing techniques [20].

To reduce the fabrication effort, the lens aperture diameter was limited to 2 mm. This resulted in blocking of some rays (which optical designers call “vignetting”) at off-axis incidence angles >8°, causing a gradual drop-off in illumination as shown in Figure S1B. However, at incidence angles up to 15°, we still have >65% relative illumination (this is without considering the Huygens antenna response to different incidence angles, which will be discussed in Section 3.2).

To implement the phase shifts required for the diffractive phase function, while maintaining high transmission, we used Huygens nanoantennas. The nanoantenna simulation was performed using a commercial three-dimensional FDTD code (Lumerical Inc., Vancouver, BC, Canada). The antennas are made of amorphous silicon on a glass substrate and are covered with a thin layer of polymethylmethacrylate (PMMA) (~300 nm thick). The lattice period was chosen to be 500 nm. This period was chosen considering sub-wavelength and phase sampling requirements[21] (see details in section 2 of the Supplementary Material), in addition to antenna coupling, which occurs at smaller periods, and interferes with achieving the Kerker condition [22].

To find the optimal antenna dimensions, we performed a numerical scan over the antenna radius and height, while monitoring the transmission and phase of a periodic antenna array. It turned out that for a hexagonal lattice with a period of 500 nm, the optimal antenna height (for which the electric and magnetic dipole resonances overlap at a wavelength of 850 nm) is 140 nm. The nominal transmission and phase response for a periodic nanoantenna array is shown in Figure 2. Field plots showing the electric and magnetic dipole resonances are presented in section 3 of the Supplementary Material.

Figure 2: Response of Huygens a-Si antenna array on glass.

(A) Nano-antenna unit cell design. *P*=500 nm, *H*=140 nm, and *D* is in the range of 200–328 nm. (B) Transmission as a function of wavelength and antenna radius. White vertical line is the section along which graphs C and D are drawn. (C) Transmission at 850 nm as a function of antenna radius. (D) Phase at 850 nm as a function of antenna radius. Red markers are at the location of the eight antenna radii used in our metalens.

To implement the desired quadratic phase function, eight discrete antenna radii were chosen, spanning a range of 100–164 nm, such that the phase shifts are equally spaced over the 2*π* range. It can be seen from Figure 2C that transmission >60% is maintained for the full range of radii. The parabolic phase function extracted from the above-mentioned Zemax design was used to determine which of the eight antenna radii should be placed at each transverse location across the lens aperture. The metalens graphic layout together with optical and scanning electron microscopy (SEM) images of the manufactured metalens are shown in Figure 3.

Figure 3: Metalens design pattern for e-beam lithography.

Each color represents a different antenna radius. The bulls-eye pattern in the center of the metalens are the Fresnel zones. Other bulls-eye patterns are aliasing artifacts. Insets: (A) Zoom-in on antenna pattern. (B) SEM image of metalens section. (C) Optical microscope image of metalens section.

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