Meta Shack-Hartmann wavefront sensor with large sampling density and large angular field of view: phase imaging of complex objects

The paper addresses the challenge of accurate, high-resolution optical phase measurement, which is central to applications such as optical metrology, adaptive optics, biomedical imaging, and LiDAR. Conventional interferometry-based phase imaging achieves high accuracy and large space-bandwidth product but requires bulky, vibration-sensitive setups with a reference arm. Computational phase retrieval techniques (e.g., transport-of-intensity, ptychography, iterative algorithms) alleviate hardware complexity but often impose constraints such as weak-scattering samples and multiple measurements, limiting their suitability for high-speed, real-time applications. Wavefront sensing offers an indirect route to phase recovery by measuring local propagation directions (via focal spot displacements) and integrating phase gradients to obtain a 2D phase map, and is compatible with incoherent sources. However, typical wavefront sensors, especially conventional Shack-Hartmann wavefront sensors (SHWFS), are limited by MEMS-fabricated lenslets (~100 µm, low NA), restricting sampling density (~100 per mm²) and maximum measurable angle (~1°), thus confining them to slowly varying wavefronts. The study’s purpose is to overcome these limits using a metasurface-enhanced SHWFS (meta SHWFS) by optimizing lenslet diameter and focal length to simultaneously maximize acceptance angle, resolvable gradient levels, and sampling density, enabling single-shot phase imaging of complex phase objects, including biological tissues.

The authors review two major classes of phase imaging: interferometry-based methods that require a reference arm and are sensitive to environmental fluctuations, and computational phase retrieval methods (transport-of-intensity, ptychography, iterative approaches) that often require multiple measurements and impose sample constraints. They discuss the principles and advantages of wavefront sensing and its compatibility with incoherent light, while noting its traditionally lower spatial resolution compared to phase imaging. Conventional SHWFS using MEMS lenslet arrays are constrained by minimum feature size and maximum curvature, leading to ~100 µm lenslets with low NA, limiting sampling density to ~100 per mm² and maximum angle to ~1°, adequate only for low-order wavefronts (Zernike). Prior metasurface lens array efforts have focused on beam diagnosis, light-field imaging, and multiphoton quantum sources, with limited exploration for single-shot phase imaging via wavefront sensing. This work builds on metasurface capabilities to transcend traditional SHWFS limitations.

Design and principle: The meta SHWFS comprises a 100 × 100 array of metalenses. An image sensor is placed at the focal plane. Local wavefront angles θ produce focal spot displacements Δ according to Δ = f tan θ. The local phase gradient dφ relates to θ via dφ = k sin θ, with k = 2π/λ. A full 2D phase map is reconstructed by numerically integrating the measured gradient field. Performance metrics are: maximum acceptance angle θ_max ≈ tan⁻¹(D/2f), number of resolvable angles N_θ ≈ 2Δ_max/Δ_res where Δ_max = D/2 − S (S accounts for finite spot size), and sampling density N_s ∝ 1/D². Lenslet parameter selection: The metalens diameter D targets ~10 µm for high sampling density, set to 12.95 µm (least common multiple of metasurface lattice U = 0.35 µm and effective camera pixel P = 0.4625 µm). Effective pixel size P = 0.4625 µm is achieved via 4× imaging with a camera of 1.85 µm physical pixel (FLIR Blackfly S BFS-U3-120S4M-CS). D cannot be arbitrarily small due to diffraction degrading spot contrast as the diffraction-limited spot approaches lens size. Focal length f influences θ_max and N_θ. A practical choice balances spot size relative to pixel size to enable accurate subpixel localization. Δ_max is defined with a guard space S = 2×FWHM ≈ 2×[λ/(2NA)] to ensure >90% focal energy remains within the lenslet area. The localization error Δ_res is evaluated via simulations using radial symmetry-based subpixel localization (approaching the Cramér–Rao bound), across SNR conditions; SNR = 10 dB is used for optimization. The ratio Δ_max/Δ_res is maximized at f = 30 µm, chosen as the operating focal length (NA ≈ 0.21). With D = 12.95 µm, f = 30 µm, and Δ_res ≈ 0.13 µm (SNR = 10 dB), the wavefront measurement accuracy is ~0.1λ and N_θ ~ 3600, with θ_max ≈ ±8°. Metasurface design and fabrication: Metalenses are built from silicon nitride (SiN) rectangular cuboids on a square lattice of ~350 nm period. Meta-atom widths span 60–275 nm, achieving 0–2π phase coverage at λ = 532 nm with height 630 nm. Lens phase profiles sample a hyperbolic phase for a converging spherical wave. Each 12.95 µm lens comprises 37 × 37 meta-atoms. Arrays are 100 × 100 lenslets. Fabrication follows standard dielectric metasurface processes (details referenced in Supplementary Note 5). SEM confirms structural fidelity. Calibration and measurement: The array is imaged so each lens maps to 28 × 28 sensor pixels. Alignment uses focal spots at array corners. Incident plane-wave angles are controlled with two galvanometer mirrors. At normal incidence, focal spot FWHM is 1.80 µm (consistent with NA_metalens ≈ 0.21 and imaging NA ≈ 0.16). Focusing efficiency is measured as 47.7%, defined as power within a radius of 3×FWHM at the focal plane. For θ in −8° to +8°, focal spots translate linearly with Δ = f tan θ with small standard deviation; beyond ~8°, crosstalk with neighboring lenslets degrades localization stability. Off-axis aberrations mildly deform spots for |θ| > 5°, but symmetry and contrast remain sufficient for subpixel tracking via radial symmetry. Incoherent source handling: A green LED (size 0.3 mm; spectrum 520–535 nm) is used to demonstrate 3D position tracking. Chromatic dispersion of the metalens yields f = f_c λ_c/λ with f_c = 30 µm at λ_c = 532 nm; the very short focal length keeps focal shift over 432–632 nm to a few micrometers, below the depth of field DOF = λ/[2(1 − cos θ)] ≈ 11.8 µm, enabling operation across the visible band. Spatial coherence requirements are evaluated via the Van Cittert–Zernike theorem. The coherence area across the array must exceed the sensor size, yielding a minimum source-to-lens distance L ≳ 2.3 mm for the 1.295 mm array and 0.3 mm source size. Phase retrieval is performed by integrating the localized gradient field over the lenslet grid to reconstruct phase maps of complex samples.

- Implemented a metasurface-enhanced Shack-Hartmann wavefront sensor (meta SHWFS) with 100 × 100 metalenses, lens diameter D = 12.95 µm, focal length f = 30 µm (NA ≈ 0.21). - Achieved sampling density of 5963 per mm² (about 100× higher than conventional SHWFS) and maximum acceptance angle θ_max ≈ ±8° (about 10× larger than conventional systems). - Number of resolvable wavefront angle levels N_θ ~ 3600 at SNR = 10 dB; wavefront measurement accuracy ~0.1λ with localization accuracy Δ_res ≈ 0.13 µm. - Measured focal spot FWHM ≈ 1.80 µm; focusing efficiency ≈ 47.7% (power within 3×FWHM radius). - Focal spots translate uniformly with Δ = f tan θ across −8° to +8°, validating large-angle operation; beyond ~8°, crosstalk increases localization fluctuations. - Demonstrated viability for incoherent sources (LED, 520–535 nm) and across a broad visible spectrum (≈432–632 nm) due to short focal length and DOF (~11.8 µm) mitigating chromatic focal shifts. - Enabled single-shot phase imaging of complex objects, including histopathologic tissue-like patterns synthesized on an SLM, and wide-angle position detection for incoherent sources.

By reducing the lenslet diameter to 12.95 µm and optimizing the focal length to 30 µm, the meta SHWFS overcomes the long-standing trade-offs in conventional SHWFS between sampling density, acceptance angle, and resolvable gradient levels. The metasurface platform enables high-NA microlenses with subwavelength control of phase, delivering a sampling density two orders of magnitude higher and a tenfold larger angular range than traditional MEMS lenslet arrays. Accurate subpixel spot localization and calibration confirm linear spot displacement with angle up to ±8°, providing sufficient dynamic range and resolution (N_θ ~ 3600) for reconstructing complex phase maps. Compatibility with incoherent illumination is maintained, as demonstrated with an LED source, due to both the wavefront-sensing principle and the short focal length limiting chromatic defocus within the depth of field. These advances directly address the need for compact, robust, single-shot phase imaging capable of handling complex wavefronts and pave the way for broader deployment in adaptive optics, biomedical imaging, and beam diagnostics.

The study presents a metasurface-enhanced Shack-Hartmann wavefront sensor that achieves a sampling density of 5963 per mm² and a maximum acceptance angle of ±8°, surpassing conventional SHWFS by approximately 100× in spatial sampling and 10× in angular range. The device, comprising a 100 × 100 array of SiN metalenses (D = 12.95 µm, f = 30 µm, NA ≈ 0.21), delivers an angular resolution corresponding to N_θ ~ 3600 and a wavefront accuracy of ~0.1λ, enabling single-shot phase imaging of complex patterns, including biological tissue-like structures. Demonstrations include large-angle calibration, robust spot localization, and 3D position tracking of an incoherent LED source with broad spectral compatibility. Future directions include integrating achromatic metalens designs to further improve broadband performance, tailoring focal length/NA for application-specific dynamic range versus resolution trade-offs, and leveraging additional metasurface functionalities to extend capabilities in wavefront sensing and computational phase imaging.

- Minimum lenslet diameter D is constrained by diffraction and aperture-induced spot degradation; further reduction compromises spot contrast and increases crosstalk. - Off-axis aberrations cause slight spot deformation for |θ| > 5°, and crosstalk increases beyond ~8°, limiting the usable angular range. - Focusing efficiency (~47.7%) could limit SNR in low-light scenarios. - Chromatic dispersion remains (f ∝ 1/λ); although mitigated by short focal length and DOF, truly broadband accuracy may require achromatic metasurface designs. - For incoherent sources, sufficient source-to-sensor distance is required (per Van Cittert–Zernike) to ensure spatial coherence across the array (e.g., L ≳ 2.3 mm for a 0.3 mm LED and 1.295 mm array).