Figure 1a displays a schematic image of the meta SHWFS, comprised of an array of metalenses. Similar to conventional SHWFS, an image sensor is placed at the focal plane of metalenses in the meta SHWFS. In this configuration, the focal spot displacement Δ can be related to the incident angle θ of the local wavefront on each lens, with $$\Delta =f\times \tan \theta$$ where f is the focal length of the metalenses. Subsequently, the wavefront angle θ is related to the local phase gradient ∂ϕ by ∂ϕ = k sin θ where k is the magnitude of wavevector (i.e., $$k=2\pi /\lambda$$, where λ is wavelength). Therefore, the local phase gradient can be derived from the focal spot displacement on each lens so that the 2D phase information of the incoming beam can be retrieved by numerically integrating the local gradient (details in Supplementary Note 1 in Supplementary Information).

Fig 1  Design principle of meta Shack-Hartmann wavefront sensor.

a Schematic of the meta SHWFS. b Effect of sampling density of SHWFS in phase reconstruction. c Illustration of a sampling unit of the meta SHWFS with related lenslet parameters. The maximum allowable displacement ∆_max is set with a minimum space S to take account of the finite width of the focal spot. d Maximum allowable displacement ∆_max, localization accuracy ∆_res, and their ratio ∆_max/∆_res for different focal lengths. The ∆_max/∆_res is maximized at the focal length of 30 μm

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In general, the performance of SHWFS is defined by the three key parameters—the maximum acceptance angle θ_max, the number of resolvable angles (i.e., the number of resolvable levels of phase gradient) N_θ, and the sampling density N_l. To avoid the cross-talk between neighboring lenses, the incident angle θ should be set within the angular range where the corresponding displacement Δ does not exceed the boundary of each lens (i.e., [−D/2, D/2] where D is the size of each meta lens). Therefore, θ_max is roughly set as $${\tan }^{-1}\left(D/2f\right)$$. On the other hand, N_θ is given as the number of resolvable spot positions within the region of each lens. Because the resolvable spot displacement is roughly given as the spot size, N_θ can be approximated to $$D^2/{(\lambda /2\mathrm{N}\mathrm{A})}^2$$ where NA is expressed as $$\sin [{\tan }^{-1}(D/2f)]$$. Finally, N₁ is simply proportional to 1/D² as each lens locally samples the phase gradient information at a point.

The metalens diameter is the key parameter that determines the sampling density of meta SHWFS, as illustrated in Fig. 1b. For phase imaging applications, we aimed to fabricate the diameter of metalens, D, in the range of 10 µm, which is ten times smaller than that of conventional SHWFS. In this study, we determined the metalens diameter D as 12.95 μm to be the least common multiple of the metasurface lattice size U = 0.35 μm and the effective pixel size of P = 0.4625 μm, aiming for high sampling density with easy alignment. The effective pixel size of 0.4625 μm, which is comparable to the pixel size of recently developed CMOS sensors, was accomplished through a 4× imaging system with a physical pixel size of 1.85 μm (FLIR Blackfly S BFS-U3-120S4M-CS). It should be noted that D cannot be set to an arbitrarily small value because the undesired diffraction effect significantly affects the focusing quality, such as spot-to-background contrast, when the diffraction-limited spot size gets closer to the size of the lens itself.

Along with the lenslet diameter D, the focal length of the metalens f serves as a crucial parameter that determines θ_max and N_θ. In our configuration with a small D, it is particularly challenging to achieve practically usable N_θ as N_θ has at least a power-of-two dependence on D. For phase imaging, N_θ needs to be sufficiently high to resolve various levels of phase slopes presented in complex structures like biological objects. Although, in the general relation, θ_max and N_θ monotonically increase with decreasing f, the ratio of the imaging pixel size P to the focal spot size should be taken into consideration in a practical scenario with a finite pixel size of an image sensor. If the spot size is smaller than a single pixel, it becomes impossible to accurately localize the spot for a subpixel displacement. Conversely, when the spot size is significantly larger than a single pixel, the presence of numerous noise sources, such as shot noise and dark noise, hinders the accurate tracking of the centroid position.

Therefore, we determined the optimum value of focal length considering the spot localization errors at various signal-to-noise levels. First, we defined the maximum allowable displacement of the focal spot Δ_max more accurately with a minimum space S to take account of the finite width of the focal spot (i.e., $${\Delta }_{max}=D/2-S$$). As shown in Fig. 1c, we set S to be twice the full width at half maximum (FWHM), ~λ/(2NA), to ensure that more than 90% of the focal intensity is distributed within the corresponding region of a lenslet. Then, we assessed the root mean square of localization error, Δ_res, for various focal lengths and signal-to-noise ratios (SNR) (details in Supplementary Note 2 in the Supplementary Information). For this purpose, we determined the position of the simulated focal spot through the calculation of radial symmetry with subpixel accuracy, closely approaching the theoretical lower limit of the error as defined by the Cramér-Rao bound (details in Supplementary Note 3 in the Supplementary Information)

The lenslet array in our meta SHWFS, composed of 100 × 100 metalenses, shares the same working principle with dielectric metasurfaces reported in previous studies

Fig 2  Large angle calibration of the meta SHWFS.

a Experimental setup for large angle calibration and the SEM image of the fabricated metalenses. b Focal intensity distribution created by the metalens array. The numbers outside the image denote the order of the metalenses. White dotted lines indicate the central lines of each metalens. The lower plot depicts the intensity profile along the yellow line. c Focal spot behaviors of a central metalens for different angles of incidence, following the equation ∆_i = f tanθ_i (i = x, y). d Focal spot displacements for various incident angles. The 100 × 100 focal spots uniformly move together as the incident angle changes. The large fluctuations in spot localization for the large angles beyond 8° is attributed to cross-talk between the focal spots of adjacent metalenses

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To characterize the meta SHWFS, we measured the behaviors of focal spots for varying incidence angles in the configuration where each metalens corresponds to the sensor area of 28 × 28 pixels, as shown in Fig. 2a. The meta SHWFS can be accurately aligned by observing the focal spots of the metalenses at four corners (details in the Supplementary Note 6 of the Supplementary Information). The incidence angle of a plane wave was controlled using two galvanometer mirrors (omitted in Fig. 2a). Figure 2b presents the intensity distribution of the focal spots at normal incidence. The FWHM of the focal spot profile is measured to be 1.80 µm, which is consistent with the theoretical expectation based on the NA of the metalenses and the imaging system, 0.21 and 0.16, respectively. Also, the diffraction pattern of the rectangular aperture appeared along horizontal and vertical directions. This diffraction effect becomes more significant for a smaller diameter, implying that there is a fundamental limit in reducing D. The focusing efficiency of our metalens was experimentally measured to be 47.7%. Here, the focusing efficiency is defined as the fraction of the light in the focal plane with a radius equal to three times the FWHM of the focal spot

To verify the capability of the proposed meta SHWFS for an incoherent light source, we demonstrated 3D position tracking of a light-emitting diode (LED) with a size of 0.3 mm and an emission spectrum of 520–535 nm. Since our metalens is designed for 532 nm, chromatic aberration can occur. This aberration arises from dispersion within the periodic lattice, akin to Fresnel lenses, resulting in different focal lengths given by $$f=f_{\mathrm{c}}{\lambda }_c/\lambda$$, where f_c is the focal length of 30 μm and λ_c is the center wavelength of 532 nm, respectively. Due to the extremely short focal length of our metalens, the shift in focal length is limited to a few micrometers, even across the wide wavelength range from 432 nm to 632 nm. Because this change in focal distance is smaller than the depth of field, $$\mathrm{D}\mathrm{O}\mathrm{F}=\lambda /\left[2(1-\cos \theta )\right]=11.8\mu \mathrm{m}$$, the impact of achromatic aberration remains marginal, enabling the use of the proposed meta SHWFS in the entire visible spectrum of 432–632 nm (details in Supplementary Note 7 in the Supplementary Information). To achieve better accuracy in a wider wavelength range, one can employ achromatic metalens

Figure 3a illustrates the schematic of the experimental setup. We note that similar configurations based on SHWFS have been preferred over other interferometric methods in the position tracking and alignment of optical components and beam quality assessment, especially when dealing with LED-based optical systems. Figure 3b, c shows the focal intensity distribution for the LED located at (S_X, S_Y, S_Z) = (0, 0, 15) mm and (8, 0, 70) mm, respectively. The radial shifts of the focal spots were more prominent when the LED got closer to the meta SHWFS, while the transverse shifts became more pronounced when the LED was translated in x- or y-directions. From the displacements of the 100 × 100 focal spots, we reconstructed the wavefront w (x, y) by integrating the gradient of wavefront, $$\sin ({\tan }^{-1}\frac{\Delta }{f})$$, as shown in Fig. 3d, e. As shown in Fig. 3e, the measurable total wavefront change over the aperture was larger than 150 µm which corresponds to ~300λ. The incoming wavefront from the LED located at (S_X, S_Y, S_Z) is expected to have a quadratic form as:

Fig 3  3D position tracking of incoherent light source.

a Experiment setup to demonstrate position tracking of a LED. b, c The behaviors of focal spots at (S_X, S_Y, S_Z) = (0, 0, 15) mm and (S_X, S_Y, S_Z) = (8, 0, 70) mm, respectively. d, e Wavefronts reconstructed by the focal spot displacements, presented in (b, c), respectively. f, g Measured LED positions versus actual positions. The LED positions were determined by quadratic fitting of the reconstructed wavefronts. h Distribution of measured LED positions in the x–y plane (left) and x–z plane (right), respectively. The dotted grid represents the actual position of the LED. Note that the sensing area of meta SHWFS is much larger than that of conventional SHWFS

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Therefore, the LED position (S_X, S_Y, S_Z) was determined through quadratic fitting of the measured wavefront. Figure 3f, g illustrate that the meta SHWFS accurately detects the LED position for various positions. Figure 3h shows the measured LED positions in the x–y plane (left) and x–z plane (right). The dotted grid lines represent the real position of the LED light source. Exploiting the large acceptance angle, the angular field of view of meta SHWFS is extended to the range that is 10× larger than that of conventional SHWFS. The observed lower accuracy for S_Z value above 85 mm is attributed to the small aperture size, 1.295 mm, of the entire meta SHWFS. As expected, at S_Z = 85 mm, the maximum incidence angle near the edge of the aperture, ~0.44° $$(={\tan }^{-1}(\frac{1.295}{2\times 85}))$$, is in the comparable range with the resolvable angle of our meta SHWFS, ~0.27° (= 8°/30). This axial range can be extended up to ~10 cm with large-area metasurfaces that can be fabricated by novel lithography techniques in cost-effective manner

Finally, we validated the phase imaging capability of the meta SHWFS. The phase imaging targets were synthesized using a spatial light modulator (SLM), as shown in Fig. 4a. The modulated phase pattern was directly projected on the input surface of the meta SHWFS through a 4f relay system with a spatial filter to get rid of undesired diffraction patterns from the SLM. The phase images were obtained from the measured phase gradient via a global least squares minimizer developed for reconstructing surface from gradients

Fig 4  Demonstration of phase imaging using meta SHWFS.

a Experiment setup for phase imaging via meta SHWFS. b Result of phase imaging. The first, second, and third rows represent the ground truth image (input image of SLM), the reconstructed phase, and the difference between them, respectively

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Meta SHWFS consisted of a silicon nitride rectangular cuboid arranged on a subwavelength square lattice with a periodicity of U = 350 nm. The width of each metaatom is precisely controlled within a range from 60 nm to 275 nm to achieve 2π phase coverage within a height of 630 nm at a wavelength of 532 nm. The phase delay imparted by the metaatoms were retrieved by using rigorous coupled-wave analysis (Supplementary Note 4 in the Supplementary Information). The meta SHWFS was fabricated on 630 nm thick silicon nitride on 500 μm thick fused silica. Silicon nitride film was deposited by plasma-enhanced chemical vapor deposition. The sample was spin-coated with a 300 nm thick positive electron beam resist (AR-P 6200) and the pattern was generated using electron beam lithography. After development, an electron-beam-evaporated aluminum oxide layer was used to reverse the generated pattern using a lift-off process and was used as a hard mask for dry etching of the underlying silicon nitride layer. The dry etching was performed using an inductively coupled plasma reactive ion etching process. The aluminum oxide layer was then dissolved in buffered oxide etchant. Figure S5 in the Supplementary Information shows the fabrication flow.

Figure 2a shows schematics of the optical setups. A single-mode laser (λ = 532 nm) was used to characterize the meta SHWFS. Two 1-axis Galvano mirrors were used to control the angle of incidence. Each mirror is located, following a 4f relay system. Meta SHWFS is positioned in a conjugate plane of the second Galvano mirror. The focal spots of meta SHWFS were measured using an imaging system with an objective lens (Olympus; UPlanXAPO 4×, NA = 0.16). The effective pixel size of 0.4625 μm, which is comparable to the pixel size of recently developed CMOS sensors, was accomplished through a 4× imaging system with a physical pixel size of 1.85 μm (FLIR Blackfly S BFS-U3-120S4M-CS).