Neural phase microscopy with metasurface optics for real-time and nanoscale quantitative phase imaging

Domain: physics
Published: 2026-01-13
Editorial outcome: major_revision

Abstract
Quantitative phase imaging (QPI) enables non-invasive analysis of transparent specimens across biomedicine, materials science, and neuroscience. However, conventional hardware relies on complex architectures with multi-shot acquisition that preclude compact, real-time operation, while phase-retrieval software often yields degraded image quality, is environmentally sensitive, and runs slowly. Here, we introduce a compact, fast, high-resolution QPI platform that addresses these challenges by integrating nanophotonic metasurfaces with artificial intelligence (AI). Our metasurface optics simplifies the optical architecture by replacing bulky optics and modulators, enabling single-shot acquisition and a drastic reduction in form factor. We develop physics-informed AI models that correct optical aberrations, compensate for imperfections in nanofabrication and alignment, and restore nanoscale quantitative phase information in real-time. The system achieves nanoscale resolution better than 840 nm at 74 Hz within a single, thin optical layer. Our unique combination of nanophotonic hardware and AI algorithms advances QPI technology towards portable, precise, real-time phase imaging.

Introduction
The invention of the bright-field microscope in the 17th century revolutionized science by unlocking the exploration of the microscopic world through optical intensity1,2. Fritz Zernike’s development of phase contrast microscopy in the 1950s3, along with subsequent advancements4,5, enabled the observation of transparent and colorless specimens through the contrast or gradients of optical phase, revealing structures previously invisible to bright-field microscopes. While these methods provide valuable visualization, they remain at a qualitative level and lack precise, quantitative information about the optical properties and morphology of microscopic samples. Advanced applications require more accurate data, inspiring the development of quantitative phase imaging (QPI), which enables the quantitative visualization of optical phase. By offering non-invasive, label-free imaging, QPI has become indispensable across fields such as biomedical engineering, materials science, and neuroscience engineering, providing precise structural and functional information of microscopic samples critical to modern research6–8.

Emerging applications across various fields – from semiconductor inspection, in-vivo neuroimaging, and live cell bioimaging to nanometrology, mobile diagnostics, healthcare devices, and clinical microsurgery – demand compact, real-time QPI systems that deliver high-resolution, quantitative phase information without compromising system portability or acquisition speed. While there are various types of conventional QPI systems with different optical configurations and components, most remain far from miniaturization due to their reliance on interferometry or holography involving multi-element assemblies, relay optics, and long optical paths, resulting in substantial system bulk and complexity. Moreover, most of these systems require multi-shot measurements to capture the full light field information, utilizing additional opto-mechanical or electro-optical components such as piezo stages or spatial light modulators, which further increase system bulk, alignment sensitivity, and capture latency (Fig. 1a)9–11. Metasurface optics12–15 can enable system miniaturization with planar optics utilizing nanoscale light modulation capabilities in many applications, including augmented reality displays16,17, micro displays18, cameras19,20, image sensors21, endoscopy22,23, and phase contrast microscopy24–27. While promising, simplifying and miniaturizing QPI systems remains challenging due to the imperative need in QPI for sufficient degrees of freedom to complex light field reconstruction. Moreover, existing QPI methods are limited to low resolutions of only tens of microns, making them unsuitable for emerging nanotechnology applications, and they are susceptible to environmental disturbances and have long processing times7,28,29. While recent efforts have explored metasurface-assisted QPI using phase-only beam-splitting metasurface designs in combination with conventional or naive neural network-based reconstruction methods30,31, these systems still rely on bulky 4f architectures, and their algorithms have limited capability in phase retrieval, which significantly constrains their resolution, robustness, and potential for miniaturization. These challenges limit the development of real-time, portable, and precise QPI solutions, highlighting the demand for compact, high-performance optical systems coupled with precise and rapid reconstruction algorithms to unlock future applications in a variety of domains.

Here, we develop a QPI method that combines nanophotonic metasurface optics with artificial intelligence (AI)-driven algorithms to implement a compact QPI system with nanoscale resolution and real-time processing (Fig. 1b). Our system features a complex-amplitude metasurface32, which provides simultaneous nanoscale control over both the amplitude and phase of light, enabling single-shot measurements and nanoscale precision within a single-chip form factor. The precise complex-amplitude modulation at subwavelength scales provided by our metasurfaces simplifies microscope system configurations by eliminating the imperative need for complex interferometers or additional active components found in conventional systems, while maintaining competitive optical performance with optical aberration corrections. Computational algorithms have been developed for phase microscopy33–36, but previous approaches primarily focused on conventional bulk microscopes without accurate consideration of physics, limiting their scalability to miniaturized systems with complex physics. Furthermore, they are often limited to low resolutions of only tens of microns, making them unsuitable for emerging nanoscale applications, and they are typically susceptible to environmental disturbances and have long processing times, hindering practical applications. To address these challenges, we develop mathematical models that describe an aberration-corrected forward image formation and inverse phase retrieval within our miniaturized system by combining physical modeling and learnable AI components. The physics-informed training37,38 and experimentally captured datasets allow our models to account for system physics, hardware imperfections, and all optical phenomena in phase retrieval. Therefore, our AI-driven modeling approach, along with experimental datasets, provides high-quality QPI with fast processing times, enabling nanoscale and real-time QPI solutions within miniaturized systems. Our approach, combining nanophotonics and AI, represents a significant advance in optical engineering and machine learning for various application fields from life sciences to materials science.

Results
Miniaturized microscopy with complex-amplitude metasurfaces
Conventional QPI systems traditionally rely on capturing multiple intensity images or interference patterns to reconstruct a quantitative phase image, necessitating complex, bulky optical setups such as interferometers10,39, 4f optical systems with spatial filtering40, and multi-illumination sources41–43, or additional opto-mechanical actuators or electro-optic modulators39,44, all of which pose fundamental bottlenecks for miniaturization. This is because QPI inherently requires capturing multiple degrees of freedom to reconstruct the full complex-amplitude field, requiring multi-shot or multiplexed measurements. To address these challenges, we propose a metasurface optical system that can collect sufficient light field information through a single-layer device with a single-shot measurement, fundamentally resolving the bottlenecks in miniaturized QPI systems.

We develop a multi-functional metasurface that integrates multiple optical functions within a single device (Fig. 2a). Based on the generalized law of refraction45, an off-axis metasurface lens (or metalens) can be designed to focus light at specific locations when the incident wavefront arrives at an oblique incident angle of θ. In addition, arbitrary optical aberrations can be introduced to the wavefront around the focal point by adding phase delays through the metasurface. By combining two different off-axis metalenses with distinct focal points and optical aberrations, our multi-functional metasurface can direct light from different incident angles to separate focal points, with each focus engineered to have specific optical aberrations or Zernike coefficients. This multiplexing of two off-axis metalenses is implemented by designing the complex-amplitude profile of the metasurface as the sum of the phase profiles of each metalens, each corresponding to a distinct focal position and aberration coefficient:

U(x,y) = \sum_{j} a(x_j,y_j) \cdot e^{ikt_j \cdot r} \cdot e^{i \sum_{m} a_{j,m} Z_m}

where U(x,y) represents the total complex-amplitude profile resulting from the superposition of multiple contributions indexed by j. Here, a(x_j,y_j) denotes the complex-amplitude profile of the metalens with a focal length at (x_j,y_j). The first exponential term introduces phase gradients for oblique incidence, with k as the transverse wave vector and r=(x,y) as the spatial coordinate vector. The second exponential term accounts for the additional wavefront aberrations, where a_{j,m} are coefficients modulating the m-th Zernike polynomial Z_m, typically used to model optical aberrations, including defocus. Figure 2a shows the amplitude and phase profile of the designed complex-amplitude profile U(x,y). This design enables the encoding of multiple optical functions into a single-layer metasurface, shrinking conventional bulk optical systems into a single planar device and eliminating the need for bulky interferometric components or additional modulators. Furthermore, the elimination of physical movement or wavefront modulation to measure multiple different intensity images enables single-shot measurements in our system.

To realize the multi-functional metasurface concept, we employ complex-amplitude metasurfaces composed of X-shaped silicon nanostructures, designed to simultaneously modulate the local amplitude and phase of visible light32. These X-shaped nanostructures allow control over the full range of both amplitude (from 0 to 1) and phase (from −π to π) by adjusting their shapes and orientations (see Methods for details). The subwavelength nanostructures, with a period of 330 nm, are designed to operate at a wavelength of 532 nm, enabling precise control over complex-amplitude fields and the modulation of high-spatial frequency components. We fabricate the metasurfaces using electron beam lithography (see Methods for details), and Fig. 2b, c show the fabricated metasurface sample and its corresponding scanning electron microscope (SEM) image.

Our design employs complex-amplitude metasurfaces rather than phase-only metasurfaces to implement multi-functional optical elements. While phase-only metasurfaces have been explored for multi-functionality24,46,47, their limited modulation capability restricts accurate reconstruction of complex-valued fields, resulting in fewer degrees of freedom for implementing multiple optical functions. In contrast, our complex-amplitude metasurface effectively suppresses phase conjugate replications and diffraction artifacts, while accurately implementing dual focal points with distinct Zernike coefficients to enhance coherent imaging quality. For baseline comparison, we also design and fabricate a phase-only metasurface, which shares the same phase profile as the complex-amplitude metasurface but maintains a unity amplitude profile across the aperture. Figure 2d, e show experimentally captured point spread functions of complex-amplitude and phase-only metasurfaces. The results demonstrate that our complex-amplitude metasurface significantly reduces diffraction artifacts caused by coherent interference between adjacent focal points. To further support this claim, we include additional experiments in Supplementary Fig. 3. These results show that diffraction artifacts from the phase-only metasurface degrade phase retrieval quality, whereas the cleaner intensity profiles enabled by the complex-amplitude metasurface lead to improved reconstruction accuracy and robustness across all reconstruction methods, which is especially beneficial for noise-sensitive applications like ours.

Quantitative phase retrieval with artificial intelligence
The proposed metasurface system can simultaneously capture multiple intensity images with distinct optical aberrations, so that a quantitative phase image can be reconstructed via a phase retrieval algorithm. Conventional phase retrieval algorithms, such as the transport-of-intensity equation (TIE)28, can reconstruct the phase profile ϕ(x,y,z) of a specimen from multiple intensity images captured at different depth planes or with different optical aberrations48. The conventional TIE algorithm is expressed as

k \frac{\partial I}{\partial z} = −\nabla_{xy} \cdot (I\nabla_{xy} \phi)

where I(x,y,z) represents the intensity and k is the wavenumber of the incident light. Despite their widespread use, conventional TIE methods have significant drawbacks, including resolution limitations, susceptibility to noise and artifacts, and difficulties in handling the physical complexities of more compact systems49,50. This makes them less suitable for our metasurface-based system, where coherent wavefront propagation is sensitive to factors including optical aberrations, nanophotonic effects, and nanofabrication imperfections. To overcome these challenges, we develop an AI-driven algorithm that combines physically accurate modeling and AI. The core idea of our approach is to design both a physically accurate microscopy image formation model (forward model) and a phase retrieval algorithm (inverse model), and also to calibrate learnable model components through physics-informed learning and experimentally captured datasets. Due to the absence of publicly available training datasets for QPI and its vulnerability to mismatches between theory and physical systems, relying solely on a single inverse model is insufficient for our applications compared to other applications in holography17,51,52 and computational photography53,54. The forward model enables the simulation of realistic intensity patterns, allowing the inverse model to learn accurate phase retrieval despite optical aberrations and nanoscale defects, providing better robustness and quality. This strategy ensures a deeper understanding of the system’s optical behavior.

The forward model simulates the coherent light propagation through the metasurface microscope system to produce intensity images at specific depth planes as input to the inverse retrieval model. Based on the angular spectrum method55, this model calculates the complex-valued point spread functions (PSFs) of the metasurface system at a particular depth z, which is expressed as

U_z = F^{-1} \{ F \{ U(x,y) \cdot u_{inc} \} \cdot H_z \}

where U(x,y) is the complex-amplitude profile of the metasurface, u_{inc} denotes the incident field, F is the Fourier transform, and H_z is the transfer function for wave propagation at a distance z. The resulting intensity image at depth z can be given by

I_z = | F^{-1} \{ F \{ a_{target} e^{i\phi_{target}} \} \cdot F \{ u_z \} \} |^2 + \eta

where a_{target} and ϕ_{target} denote the target amplitude and phase, and η ∼ N(0,σ^2) is the Gaussian read noise of an image sensor. These intensity images generated by the forward model are crucial for training the inverse model, allowing it to learn accurate mappings from intensity to phase by accounting for the physical system’s behavior.

While conventional TIE-based methods can serve as phase retrieval techniques, they are inadequate for our metasurface QPI system due to their reliance on paraxial approximations, poor handling of high-frequency information, and susceptibility to artifacts from noise and boundary conditions. To overcome these limitations, we introduce an AI-driven inverse model in the form of a convolutional neural network (CNN). Our phase retrieval network, denoted as CNN_{phase}, replaces the conventional TIE by directly converting intensity images into a phase map, bypassing the drawbacks of the traditional approaches. The inverse model can be expressed as

ϕ_{recon} = CNN_{phase}(I_0, I_z)

where I_0 and I_z are intensity images at different depth planes. The network is designed to reconstruct the phase ϕ_{recon}, also compensating for optical aberrations and noise that traditional solvers struggle with (see Supplements for details on training and network architecture). In addition, we augment this inverse model with a secondary correction network, CNN_{corr}, that is trained on experimental datasets to handle discrepancies between our forward model simulations and real-world experimental conditions, including further aberrations, nanofabrication mismatch, and nanoscale system misalignment. By leveraging both forward and inverse models, the system can learn a more accurate representation of the optical system behavior, enhancing the overall phase retrieval process.

Accurate and robust phase retrieval requires a reliable training process. We utilize a physics-informed loss function to guide the learning of our neural networks, ensuring that the reconstructed phase is consistent with the underlying physical principles. Rather than directly comparing phase images, which could lead to ambiguities due to phase offsets, our loss function considers free-space wavefront propagation by evaluating the discrepancies between diffraction patterns at multiple depth planes generated by the reconstructed and target phase maps (ϕ_{target} and ϕ_{recon}):

L(ϕ_{target}, ϕ_{recon}) = \sum_{i} || f_{free}(a_0 e^{i\phi_{target,z_i}}) |^2 - | f_{free}(a_0 e^{i\phi_{recon,z_i}}) |^2 ||^2

where f_{free} represents free-space propagation, z_i is a particular depth for each depth plane i, and a_0 is the amplitude profile at the target plane. By comparing multiple diffraction patterns rather than pixel-wise phase values, this physics-informed loss ensures physical accuracy while improving the network performance and mitigating artifacts like boundary inconsistencies and high-frequency blurring. The training process follows two steps: pre-training with synthetic datasets generated from the forward model, and refinement with experimentally captured datasets. In the pre-training stage, we generate synthetic intensity images using the forward microscopy model, simulating a variety of phase and amplitude distributions under realistic optical conditions, including Gaussian blur and practical constraints on amplitude-phase correlation (see “Methods” for details). This pre-training allows the network to generalize across a wide range of phase objects and handle various aberrations. Once pre-trained, the inverse model is refined with experimental datasets captured from our metasurface system, enabling it to adapt to practical conditions. During the capture refinement, the parameters of the initially trained CNN_{phase} are frozen, and only the parameters of CNN_{corr} are further trained to learn experimental environments. By freezing model parameters after initial pre-training on a large synthetic dataset, we reduce the amount of experimentally captured data required to sufficiently train our inverse model, which is beneficial in this application because both public availability and manual data collection are highly constrained. This approach, incorporating physics-informed learning and training on both synthetic and experimental datasets, enables our model to better understand underlying physical phenomena efficiently, resulting in more accurate, robust learning and superior visual quality compared to conventional methods.

Experimental results
We build an optical microscope setup with our metasurface to compare the performance of our phase retrieval models with existing approaches. The setup includes a coherent laser source with a wavelength of 532 nm, illumination optics, a fabricated metasurface, and a camera. Our metasurface can create two separate intensity images at different depths within a single image plane, allowing the camera to simultaneously capture two images with a single shot. To benchmark our method, we qualitatively and quantitatively assess the retrieved phase image quality and compare several different models: a Fast Fourier Transform (FFT)-based direct TIE solver (TIE-Direct) offering fast processing at the cost of limited image quality, an iterative TIE solver (TIE-Iterative) with improved image quality but slow processing time and high sensitivity to optical aberrations, a non-physical inverse model with both physical and learned components but without physics-informed learning (see Supplements for details), and our physics-informed model with physics-informed learning. Note that the non-physical inverse model reflects the type of approach previously explored in deep learning-based QPI studies, which relied on naive forward models without physics-informed supervision or model calibration31,34. While the conventional TIE solvers are fundamentally limited in resolution and sensitive to experimental conditions, inverse models are expected to improve image quality with fast processing time. In particular, physics-informed learning procedures trained on experimental datasets are expected to enhance robustness to optical aberrations, system alignment, and nanofabrication defects, ultimately enabling better imaging performance.

Our initial experiments focus on assessing the performance of CNN_{phase}, initially trained with synthetic datasets generated from the forward microscopy model, using conventional microscopy setups. Figure 4 shows phase retrieval results from the intensity images with various methods. The existing TIE methods provide low image quality and suffer from background noise and artifacts, especially in the low-intensity region, where the solutions tend to diverge (see Supplements for detailed analysis of TIE solvers). In contrast, the inverse models show significant improvements by learning to compensate for aberrations and recover high-frequency details. This demonstrates that our physical modeling approach is effective in overcoming the resolution and robustness limitations of existing TIE methods and improving retrieval quality. In particular, the physics-based supervision enables very high-quality phase reconstruction through additionally supervising physical wave propagation.

Our compact metasurface microscope is composed of the fabricated metasurface with illumination optics and camera. Compared to conventional bulk microscopes, the compact metasurface microscope introduces more challenging conditions, with significantly increased optical aberrations, nanofabrication imperfections, and additional physical phenomena in nanophotonic devices. The correction CNN (CNN_{corr}) in the inverse model is designed to compensate for experiment mismatches and additional artifacts not accounted for in the forward model by refinement training with the experimental dataset captured through our compact metasurface microscope. Figure 5 shows the phase retrieval results, comparing the different inverse models. In all cases, our physics-informed inverse model outperforms baselines by a large margin. Our experimental results on the 1951 USAF resolution target further demonstrate the capability of our microscopy system to resolve element 2 in group 9, corresponding to a nanoscale spatial resolution of 870 nm or better. It is noteworthy that the image quality of the conventional TIE methods and the non-physical model is much worse than that using the conventional microscope in Fig. 4, due to more severe aberrations and additional artifacts caused by simplified design, imperfect nanofabrication, and nanophotonic phenomena of our metasurfaces. This highlights the strength of our approach in achieving robust, accurate phase retrieval even in the presence of complex system imperfections, which conventional methods fail to address, denoting our physics-informed supervision, along with experimental datasets, is inevitable for miniaturization and simplification of microscope systems.

We also demonstrate quantitative validation of our method compared to the others using custom target samples. As a ground truth phase object, we fabricate a custom-designed metasurface target with step-like phase regions defined by 3.10 μm features, equivalent to group 7, element 3 of a standard USAF 1951 target. This target manipulates local phase delay by engineering the effective refractive index of its nanostructures, making it physically equivalent to traditional stepped-height references (see Supplementary Note 3 for details). Figure 5c, d demonstrate that our model can recover quantitative phase information more accurately than other methods (see Supplements for a detailed quantitative comparison of reconstructed phase values). For real-time applications, we also measure the runtime of each method to see how fast our model is compared to others. Our model, with very high performance and robustness, offers fast processing times of 13.5 ms per frame or, similarly, 74 Hz, making it suitable for real-time applications. Notably, our method, based on single-shot measurement, has no additional latency other than computing runtime since the system measures multiple intensity images simultaneously. However, other methods based on multi-shot measurements incur additional time delay between measurements on top of the computational runtime, highlighting the rapid speed of our approach.

Methods
Metasurface design with X-shaped meta-atoms
To realize the proposed concept of the multi-functional metalens with two focal points with distinct Zernike coefficients, we design the complex amplitude profiles by summing two different complex amplitude profiles of off-axis metalenses with different focal positions and coefficients as expressed in Eq. (1). Figure 2 depicts the amplitude and phase profiles of the designed complex metasurfaces, which have a focal length of 1 mm for both two focal points. In this design, a Zernike defocus coefficient corresponding to the depth of 3 μm is added to one of the two focal points, and the phase gradients with the transverse wave vector of k_t = k_0 sin(θ) are designed such that each focal point has the separation angle (θ) of ±4°, respectively.

To implement the complex-amplitude metasurface, we utilize an X-shaped metasurface to manipulate complex-amplitude profiles on a subwavelength scale32. As an extension of the Pancharatnam–Berry phase principle, the X-shaped meta-atom allows for a full range of both amplitude and phase to be mapped and adjusted by tuning the orientation angles of its two arms. The X-shaped meta-atom can be modeled as two independent electric dipoles, and its behavior can be analyzed by the superposition of the geometric phase. The complex-amplitude modulation of X-shaped meta-atoms can be expressed as

E_{cross} / 2 cos(θ_1 - θ_2) exp(j(θ_1 + θ_2))

where θ_1 and θ_2 are the orientation angles of two arms consisting of the X-shaped structure and σ represents the handedness of the circular polarization. The adjustment of the two orientation angles of rods enables simultaneous modulation of amplitude and phase. The phase component of the cross-polarized component is proportional to the sum of their orientation angles, while the amplitude component is proportional to the cosine of the differences between their orientation angles. In this work, the X-shaped meta-atom is composed of hydrogenated amorphous-silicon (a-Si:H) with a period of 330 nm, which is smaller than the operating wavelength of 532 nm for satisfying Nyquist sampling theory (see Supplements for details).

Metasurface sample fabrication
We use electron beam lithography to fabricate our metasurface, as depicted in Supplementary Fig. 9. First, amorphous silicon (a-Si) with a thickness of 150 nm is deposited onto a glass wafer using plasma-enhanced chemical vapor deposition (PECVD) (HiDep-SC, BMR Technology). Then, positive e-beam resists PMMA 495 A4 and PMMA 950 A2 are spin-coated on the a-Si layer. Conductive polymer (E-spacer 300Z) is also coated to prevent charge accumulation. Our metasurface is patterned using an electron beam lithography (JBX-6300FS, JEOL), followed by developing of sample with a MIBK/IPA 1:3 developer. A chromium (Cr) layer with a thickness of 20 nm is deposited as a hard mask using electron beam evaporation (KVE-3004, Korea Vacuum Tech). The resists are removed using acetone, followed by etching the a-Si layer using an inductively coupled plasma (ICP) etcher (Multiplex ICP, STS). Finally, the Cr hard mask is removed using a Cr etchant.

Experimental measurements
Our experimental setups are designed to evaluate the performance of our metalens and quantitative phase imaging (QPI). Two experiments are conducted: measuring the point spread function (PSF) and demonstrating QPI. The experimental setup depicted in Supplementary Fig. 10 is utilized for the PSF measurements. The monochrome laser with a wavelength of 532 nm consists of supercontiuum laser source (SuperK, NKT Photonics) with a monochromator are utilized as a light source. This light passes through a cascaded linear polarizer and quarter waveplate for the generation of circular polarization as an input for our metalens. An objective lens (MPlanFLN, 50×, NA=0.8, Olympus) captures two PSFs of our metalens and is then focused by a tube lens (f=50 mm, Nikon). Next, a cascaded quarter waveplate and linear polarizer is inserted for capturing only cross-polarized light. The PSFs are finally captured with charge-coupled device (CCD) camera (Blackfly S BFS-U3-120S4C, Color, pixel pitch of 1.85 μm, FLIR).

The QPI demonstration is based on a setup depicted in Supplementary Fig. 11. The same light source and polarization components are utilized. An objective lens (MPlanFLN, 5×, NA=0.15, Olympus) is used as a condenser lens for focusing light to the samples. Then, our metalens acts as an objective lens and separate two-depth images. The same tube lens (f=50 mm, Nikon) and polarization components are inserted as in the PSF measurement. Two images are finally captured with a CCD camera (Blackfly S BFS-U3-17S7C-C, Color, pixel pitch of 9 μm, FLIR).

For the conventional experimental demonstration, a standard microscopy system is utilized. The laser-line filtered LED is utilized as a light source. The sample image is magnified with an objective lens (MPlanFLN, 50×, NA=0.8, Olympus) and a tube lens of 180 mm focal length. The image is captured with a CCD camera (Blackfly S BFS-U3-17S7C-C, Color, pixel pitch of 9 μm, FLIR).

Synthetic dataset for pre-training
A set of random phase data is generated by using Equations (7), (8), and (9) as simulated targets. Equation (7) represents a two-dimensional Gaussian function utilized for blurring. In this equation, m and n denote the coordinates of a pixel in 2D space, while R denotes a single random variable that governs the radius of the Gaussian functions. In addition, A represents the normalization constant used to ensure that the summation of Gaussian functions equals to 1.

g(m,n) = A exp(-(m^2+n^2)/R^2)

Y(m,n) = g(m,n) * X(m,n)

φ(m,n) = ϕ_0 tanh(2βY(m,n) - β) / tanh(2β)

X(m,n) in Equation (8) represents a random matrix where each component is independent and follows a uniform distribution in the range of [0,1]. This equation applies a Gaussian blur to the original random matrix. The * operation here denotes a convolution operator. By convolving the original data with a Gaussian function, we effectively blur or smooth the data into Y(m,n). Equation (9) finally generates the random phase data while considering physical constraints. The blurred data Y(m,n) undergoes the hyperbolic tangent function, where the parameter β controls the sharpness of the generated phase. Higher values of β result in sharper contrast. The examples of the random phase data are plotted in Supplementary Fig. 7.

Neural network architecture and training details
We use the U-net architecture for the learnable components in our models, noted for its proficiency in complex image reconstruction tasks, which is particularly advantageous for our phase-retrieval application. The details of the CNNs are explained in the Supplement. For the loss function, we employ an L1 loss function of intensity generated from the recovered phase using a free-space propagation model. The Adam optimizer is employed with a learning rate of 1e-4 for training the phase retrieval network. During training, a batch size of 4 is used, with each epoch consisting of 100 synthetic phase datasets. It is noteworthy that each epoch uses a distinct dataset to promote variation in training. The dimensions of the input data for intensity are 480 × 480. In addition, Gaussian noise with a standard deviation between 0 and 0.01 is included in the training data. The training is conducted on a system running Ubuntu, equipped with Intel Xeon(R) CPU E5-2698 and NVIDIA Tesla V100.

For pre-training, we use the synthetic dataset to produce network training data using the forward image formation model (Fig. 3a). Two point spread functions (PSFs) of our complex-amplitude metasurface are first simulated with the angular spectrum method (ASM). Before the convolution with the PSFs, the normalized amplitudes are combined with the randomly generated phase maps, creating the complex amplitude fields. The complex amplitudes are padded with 3 times of their original size using the replicate padding method. Subsequently, the padded complex amplitudes are convolved with two PSFs utilizing fast Fourier transform (FFT), and the central region is cropped with its original size. We apply an absolute-value-square operator for both cropped images to calculate the simulated intensity images, which are next normalized to have a maximum intensity of 1, maintaining the scale between two images. To account for the Gaussian read noise of an image sensor, η ∼ N(0,σ^2) is added to the intensity images, where σ is 0.01. These processed images are utilized as input data for training the phase retrieval network (CNN_{phase}). For the physics-informed supervision, We also pad the recovered output phase of the neural network image and propagate the padded data using the ASM before cropping it. In addition, we evaluate the loss on the intensity obtained from the absolute square of the propagated wavefronts. After the pre-training, the refinement procedure (Fig. 3b) trains the correction network (CNN_{corr}) with experimentally captured datasets through our inverse model with physics-informed learning, while the pre-trained parameters of the phase retrieval network (CNN_{phase}) are fixed. The experimental dataset contains intensity images at different depth planes captured by the metasurface microscope system, consisting of two intensity images obtained at the off-axis focal points of the proposed complex-amplitude metasurface and additional intensity images obtained via on-axis mechanical translation of the metasurface microscope system for further training data. Both the on-axis and off-axis intensity images are used in the capture refinement procedure. On-axis images have higher phase retrieval quality than off-axis images due to the smaller offset between intensity pairs, which increases the upper bound on the retrieval results of the inverse model during refinement (see Supplements for details).

Unlike other AI applications17,35,51–54, our application for nanoscale QPI makes it much more challenging to adopt AI in terms of dataset generation, given the difficulty in accurately and massively capturing quantitative phase information of microscopic samples at the nanoscale resolution. Our subsequent learning method with both synthetic and experimental datasets can alleviate the burden on the size of the experimentally captured dataset, enabling sufficient model training with a minimized experimental dataset size.

Sample preparation
For the QPI demonstrations shown in Figs. 4 and 5, we utilized a USAF 1951 resolution chart (Edmund Optics), and onion samples purchased from a grocery store. The phase samples used as ground truth in Fig. 5 are prepared through only a part of the E-beam lithography process. For the PMMA metasurface with rectangular holes, PMMA 495 A4 is applied by spin-coating onto a silicon nitride (SiN) on glass substrate. The patterning is accomplished via E-beam lithography, and then the MIBK/IPA 1:3 developer is utilized for development. Supplementary Fig. 15d displays a scanning electron microscopy (SEM) image of the fabricated ground-truth sample. The pixel pitch of the PMMA metasurface is 350 nm. The refractive index of PMMA 495 A4 at 532 nm wavelength, as stated in the datasheet, is roughly 1.49. The height of the PMMA layer is measured to be approximately 350 nm through SEM. Supplementary Fig. 15a displays the amplitude and phase modulation in relation to width, simulated using the rigorous coupled-wave analysis68.

Image Template Matching for Pre-processing
For the results captured through our metasurface in Figs. 4 and 5, we use image template matching to crop identical areas from two depth images in a single image plane, as described in

R(m,n) = \sum_{i,j} X(i,j) Y(i+m, j+n)

We fix one image’s (X) crop window position and roughly estimate the initial position of the other image’s (Y) crop window. By shifting Y by (m,n) pixels, we calculate the cross-correlation (R) of the two images. The location where R reaches its maximum is identified as the best match, and the cropping window of Y is aligned accordingly. When using a single image captured by the metalens through conventional z-stage translation, the identical crop windows are applied to both depth images. Additional details can be found in the Supplements.

Conventional phase retrieval algorithms
The phase retrieval algorithms utilized in Figs. 4 and 5 include the FFT-based direct TIE and iterative TIE. For the direct TIE, we utilize the method developed by Paganin and Nugent69. The solution of direct TIE can be expressed as

ϕ(x) = -k \nabla^{-2} \nabla \cdot ( \frac{1}{I(x)} \nabla \nabla^{-2} \frac{\partial I}{\partial z} )

where an inverse Laplacian ∇^{-2} can be translated into the Fourier domain by the equation ∇^{-2} = 1 / (4π^2 |u|^2). This enables utilizing the FFT computationally for rapid phase reconstruction. In this method, the FFT necessitates a periodic boundary condition. Due to 1/I(x), the solution would diverge when I has many near zero components.

For the iterative TIE, we use the iterative method developed by Zhang et al.70. This iteration-based algorithm can be expressed as

ϕ(x) = -k \nabla^{-2} ( \frac{1}{I_{max}} \frac{\partial I}{\partial z} )

which assumes uniform intensities. The retrieved phase from this equation is utilized to determine the discrepancy of the intensity derivative through TIE or angular spectrum method. Based on this, the correction term for the phase is derived using the preceding equation and iteratively converged to the optimized phase profile. This iterative method typically outperforms the direct TIE method due to its applicability to inhomogeneous boundary conditions and near-zero components.

Quantitative analysis
The quantitative analysis presented in Fig. 5 utilizes root mean square error (RMSE) and structural similarity index measure (SSIM) to compare the recovered phase image with the ground-truth phase template. Unlike intensity images, phase images can differ by an unknown constant offset while still conveying the same structural information. Accordingly, the same RMSE and SSIM formulas used for the intensity images are applied, but evaluated after adding a global phase offset to the recovered phase image so that RMSE is minimized and SSIM is maximized. Mathematically, the RMSE and SSIM can be expressed as

RMSE = \sqrt{ \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} (ϕ(i,j) - (ϕ_{gt}(i,j) + ϕ_c))^2 }

SSIM(x,y) = (2μ_x μ_y + c_1)(2σ_{xy} + c_2) / ((μ_x^2 + μ_y^2 + c_1)(σ_x^2 + σ_y^2 + c_2))

where the additive phase constant ϕ_c are determined to minimize RMSE and maximize SSIM. In addition, x and y denote the corresponding coordinates in the recovered phase image and the ground-truth image, respectively, within a window of size 11 × 11. Here, μ_x and μ_y are the average values of window x and y, respectively. Also, σ_x^2 and σ_y^2 are the variance of the windows x and y, respectively, and σ_{xy} is the covariance of the windows x and y, respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper reports on how to receive single shot quantitative phase measurements with much improved spatial resolution and decent temporal frame rate (74 Hz). The text is nicely written and supported with persuasive images. The level of spatial resolution around 900 nm is an amazing technical improvement for the phase restoration technique. There are no direct comments on the text itself.\nWhat the paper is lacking is the conceptual novelty. The approach given in the paper is based on two pillars: the single shot phase retrieving scheme with a single silicon metasurface and the AI algorithm assisting phase image rectification. However, both two parts have been reported in literature. The single-shot phase microscopy with an all-dielectric metasurface, exactly as depicted in Fig.1(b), was proposed in Engay, E.; Huo, D.; Malureanu, R.; Bunea, A.-I.; Lavrinenko, A. Polarization-Dependent All-Dielectric Metasurface for Single-Shot Quantitative Phase Imaging”, Nano Lett. 2021, 21, 3820– 3826. This principle has been updated in the very recent paper by Junxiao Zhou, Ang Li, Ming Lei, Jie Hu, Guanghao Chen, Zachary Burns, Fanglin Tian, Xinyu Chen, Yu-Hwa Lo, Din Ping Tsai, Zhaowei Liu, “Advanced Quantitative Phase Microscopy Achieved with Spatial Multiplexing and a Metasurface”, Nano Lett. 2025, 25, 2034–2040. The polarization sensitive metasurface is accompanied with the deep learning phase retrieval algorithm used for low-error fast and direct phase retrieving.\nThus, the reviewed paper reports on a very nice quantitative improvement of the phase imaging scheme which has not brought any new conceptual advances. I suggest resubmitting the paper in a more specialized optical journal.", "reviewer_2": "This manuscript introduces a compact and AI-driven quantitative phase imaging (QPI) system that leverages nanophotonic metasurfaces for real-time, high-resolution phase retrieval. The combination of single-layer complex-amplitude metasurfaces and physics-informed AI models offers a compelling advancement over conventional QPI approaches, which often suffer from bulkiness, slow acquisition speeds, and phase reconstruction artifacts.\nThe topic of research is timely, and highly innovative and provides a promising direction for miniaturized, high-speed QPI applications in biomedical imaging, nanomaterials research, and diagnostics. The use of a neural network-based phase retrieval pipeline, combined with physics-informed supervision, represents a noteworthy contribution to computational microscopy.\nThe authors present an impressive study that successfully integrates nanophotonic metasurfaces with AI-driven phase retrieval for real-time QPI applications. The proposed physics-informed neural network is an important step forward, offering an innovative alternative to conventional interferometric and TIE-based methods.\nWhile the study is well-structured and well-supported with theoretical analysis, simulations, and experimental validation, there are some areas that require further clarification and refinement to fully align with Nature Communications' scope and standards. In principle, I support the publication of the manuscript, provided that the following comments are addressed.\nPoints for Revision\n1. Clarification of Comparison with Conventional QPI (Figure 1a). In Figure 1a, the depiction of interferometric QPI systems may be misleading since not all interferometric methods require an actuator for phase imaging. The authors should explicitly define which types of conventional QPI methods they are comparing against.\n2. The manuscript introduces the terms \"complex-amplitude metasurface\" and \"phase-only metasurface\" without a clear, concise definition. While the explanation is provided later, an earlier clarification would improve readability. A brief definition or description in the Introduction would be beneficial.\n3. The architecture of the non-physical model (one of the inverse solvers) is not well described. The authors should explicitly state how this model is structured and how it differs from other models used in the study.\n4. The authors claim that the iterative TIE method performs better than the direct TIE, yet Figures 4 and 5 appear to suggest otherwise. The authors should clarify why this discrepancy arises and whether the observed results align with their expectations.", "reviewer_3": "This paper presents an approach to quantitative phase imaging (QPI) combining nanophotonic metasurfaces with artificial intelligence. The authors designed a complex-amplitude metasurface that captures two intensity images at different focal planes, using neural networks to reconstruct phase information similar to the transport of intensity equation (TIE) method. The integration of metasurface optics with AI-based reconstruction represents an interesting direction, though the implementation may fall short in several aspects listed below.\n1. The authors provide inadequate justification for implementing their device as a metasurface rather than using simpler alternatives. Holographic optical elements (HOEs) could achieve similar complex-amplitude modulation functionality without the severe fabrication constraints. While the authors emphasize nanoscale precision of their metasurface design, they don't demonstrate how this translates to performance advantages that couldn't be achieved with holographic plates. Holographic elements can modulate both amplitude and phase simultaneously through various established techniques, while offering significant advantages: they can be fabricated at much larger aperture sizes (compared to the severely limiting 1 mm aperture), produced with simpler equipment than electron-beam lithography, manufactured at lower cost, and potentially achieve higher optical efficiency.\n2. The use of intensity resolution targets to demonstrate phase imaging capabilities in Fig. 4, Fig. 5a-b, and supplementary figures may be inappropriate. For a paper claiming quantitative phase imaging, proper phase targets (such as the standard quantitative phase target from Benchmark Technologies) are essential. The authors may need to demonstrate different height measurements of known phase objects compared to ground truth values (for example, this can be done using the standard quantitative phase target with different heights from Benchmark Technologies).\n3. The approach of using two defocused intensity images for phase retrieval has inherent limitations that render the ‘quantitative’ claim questionable. For example, the reported method cannot recover linear phase ramps (like those from optical prisms) with arbitrary spatial frequencies, as the constant intensity patterns of the phase ramp contain no information about such phase profiles. If the system cannot handle such basic phase distributions, the claim of ‘quantitative phase imaging’ may not be a proper one. The approach may only work for a narrow subset of weakly scattering samples without phase wraps, which may restrict its practical utility.\n4. The quality of the recovered phase in Fig. 5 raises some questions, particularly regarding variations in the background and certain features. For biological cell samples, conventional understanding suggests that phase within cells should typically be higher than at the edges, which appears inconsistent with some of the presented results. These errors may indicate problems with the phase retrieval process.\n5. The paper only discusses bulky optical implementations. However, there are many single-shot phase retrieval methods that have been implemented in lensfree configurations, even on ordinary smartphones without using any lenses. These lensfree microscopy approaches typically offer fields of view orders of magnitude larger than the reported method (which is severely limited by the 1 mm aperture) and can be implemented at much lower cost with similar or better performance. The imaging throughput is also substantially higher than the present work.\n6. In discussing Fig. 2d, the authors make claims that their approach \"significantly reduces diffraction noise and improves the robustness of phase retrieval.\" This figure shows only intensity measurements without any phase retrieval. The nature of \"diffraction noise\" is undefined.\n7. While metasurfaces typically offer advantages for use with incoherent light, this work may be restricted to coherent illumination and specifically to weak phase objects without phase wraps or background variations. This may represent a drawback compared to other metasurface approaches that work with incoherent illumination. The requirement for coherent illumination combined with the weak phase assumption constrains the system's practical utility. These limitations would make the approach challenging to implement in reflection mode or with general objects that don't satisfy the weak phase approximation. Such restrictions could narrow the range of real-world applications where this technology would be effective, particularly in comparison to alternative approaches that can operate under more flexible illumination conditions."}, "author_response": "Point-by-Point Response\nWe thank reviewers for their insightful comments and appreciate their constructive feedback.\nWe are encouraged by reviewers commenting on our submission being “an impressive study\nthat successfully integrate nanophotonic metasurfaces with AI-driven phase retrieval for\nreal-time QPI applications” (R2) that achieves “an amazing technical improvement for the phase\nrestoration technique” (R1) and “an innovative alternative to conventional interferometric and\nTIE-based methods” (R2) with “the text is nicely written and supported with persuasive images”\n(R1) and “the study is well-structured and well supported with theoretical analysis, simulations\nand experimental validation” (R2).\nWe have substantially improved the manuscript and supplement to address all comments and\nconcerns. In the following, we provide a point-by-point response, addressing these comments.\nTo recap, our work introduces a compact, high-performance quantitative phase imaging (QPI)\nplatform that integrates nanophotonic metasurface optics with physics-informed AI models. By\ndesigning and fabricating a complex-amplitude metasurface lens tailored for simultaneous\nmulti-depth imaging, we significantly suppress diffraction artifacts and enhance the quality of\ncaptured intensity data, while enabling single-shot measurements and a compact form factor.\nThese improvements allow robust phase retrieval even in the presence of system imperfections\nand strong optical aberrations. Our AI model is trained on both simulated and experimentally\ncaptured data using physics-informed supervision that incorporates optical principles and\nsystem-specific imperfections, enabling the model to generalize across diverse imaging\nconditions. Together, these advances establish a scalable, miniaturized QPI system that offers\nperformance comparable to that of conventional bulky microscopes, without requiring\ninterferometry or precise mechanical or optoelectronic scanning. This combination of\nnanophotonics and AI not only opens new possibilities for compact biomedical imaging but also\nestablishes a fundamentally new paradigm for metasurface system applications.\nReferee #1 (Remarks to the Author):\n● The paper reports on how to receive single shot quantitative phase measurements with\nmuch improved spatial resolution and decent temporal frame rate (74 Hz). The text is\nnicely written and supported with persuasive images. The level of spatial resolution\naround 900 nm is an amazing technical improvement for the phase restoration\ntechnique. There are no direct comments on the text itself.\nWhat the paper is lacking is the conceptual novelty. The approach given in the\npaper is based on two pillars: the single shot phase retrieving scheme with a single\nsilicon metasurface and the AI algorithm assisting phase image rectification. However,\nboth two parts have been reported in literature. The single-shot phase microscopy with\nan all-dielectric metasurface, exactly as depicted in Fig.1(b), was proposed in Engay, E.;\nHuo, D.; Malureanu, R.; Bunea, A.-I.; Lavrinenko, A. Polarization-Dependent\nAll-Dielectric Metasurface for Single-Shot Quantitative Phase Imaging”, Nano Lett. 2021,\n21, 3820– 3826. This principle has been updated in the very recent paper by Junxiao\nZhou, Ang Li, Ming Lei, Jie Hu, Guanghao Chen, Zachary Burns, Fanglin Tian, Xinyu\nChen, Yu-Hwa Lo, Din Ping Tsai, Zhaowei Liu, “Advanced Quantitative Phase\nMicroscopy Achieved with Spatial Multiplexing and a Metasurface”, Nano Lett. 2025, 25,\n2034–2040. The polarization sensitive metasurface is accompanied with the deep\nlearning phase retrieval algorithm used for low-error fast and direct phase retrieving.\nThus, the reviewed paper reports on a very nice quantitative improvement of the phase\nimaging scheme which has not brought any new conceptual advances.\nWe thank the reviewer for their thoughtful feedback and for highlighting the “amazing” technical\nimprovements of our work, particularly the ~870 nm spatial resolution and real-time acquisition\nat 74 Hz. While these performance gains are substantial, we respectfully emphasize that our\ncontributions go beyond quantitative improvements. The core novelty of our work lies in the\nco-design of a nanophotonic imaging system and a physics-informed computational framework,\nwhich together achieve a level of compactness, performance, and robustness not demonstrated\nin prior metasurface-based QPI studies.\nAs the reviewer noted, the cited works by Engay et al. (Nano Lett. 2021) and Zhou et al. (Nano\nLett. 2025) also explore metasurface-assisted, single-shot QPI using either conventional TIE\nsolvers or naive neural network models. However, while the high-level goals may appear similar,\nthe core technologies, system architectures, and computational methods are fundamentally\ndifferent from those in our work, which is why none of the previous approaches have\ndemonstrated the level of performance achieved by our system.\nIn terms of hardware design, both prior works use phase-only metasurfaces placed inside\ntraditional bulky 4f relay optics. These metasurfaces function solely as polarization-dependent\nbeam splitters, without performing any imaging or aberration correction. The rest of the system\nrelies on conventional 4f optics with uncompromisable spaces between the components,\nresulting in a large and alignment-sensitive multi-layer architecture. In contrast, our metasurface\nis based on complex-amplitude modulation, allowing it to simultaneously perform imaging, beam\nsplitting, and spatial wavefront modulation within a single flat optical element. This integration\nenables us to eliminate all relay optics and construct a fully compact, single-layer QPI system\nwith high numerical aperture and high resolution, which is fundamentally not achievable using\nthe beam-splitting metasurface approaches embedded in 4f systems.\nIn terms of computational design, the previous works rely on either conventional TIE solvers or\nnaive neural network models trained on idealized datasets, which often result in limited spatial\nresolution, poor image quality, and high sensitivity to optical aberrations. These models struggle\nto handle real-world imperfections such as optical aberrations, fabrication defects, and system\nmisalignment, which are factors that are particularly pronounced in compact or chip-scale\nsystems. In fact, these approaches are comparable to the TIE baselines or the non-physical\nmodel baseline evaluated in our study, both of which we show to perform poorly in terms of\nreconstruction accuracy and robustness. By contrast, our method employs a physics-informed\nmodel that jointly learns from both a forward image formation model and inverse phase retrieval\nalong with experimentally captured datasets. This enables the model to understand the true\nphysical behavior of wave propagation, metasurface PSFs, fabrication-induced variations, and\nother system-specific aberrations or noises. Our approach allows for robust quantitative phase\nrecovery even in the presence of strong optical imperfections, without relying on paraxial\nassumptions or ignoring design-to-fabrication mismatch.\nTaken together, these hardware and algorithmic innovations result in a QPI platform that\nsupports single-shot acquisition, achieves nanoscale resolution (~870 nm), delivers real-time\nperformance at 74 Hz, maintains robustness to fabrication and alignment errors, and is compact\nand scalable in form factor. This combination of capabilities has not been realized in either of\nthe cited studies.\nFor clarity, we have also updated the manuscript to appropriately cite these previous works\nwhile clearly outlining the conceptual and technical advances introduced in our study.\nReferee #2 (Remarks to the Author):\n● 1. Clarification of Comparison with Conventional QPI (Figure 1a). In Figure 1a, the\ndepiction of interferometric QPI systems may be misleading since not all interferometric\nmethods require an actuator for phase imaging. The authors should explicitly define\nwhich types of conventional QPI methods they are comparing against.\nWe thank the reviewer for this valuable clarification and agree that not all interferometric QPI\nsystems require actuators. In Figure 1a, our original intent was to illustrate a representative\nclass of conventional QPI systems that rely on multi-shot acquisition through interferometers\nwith active modulation. These include systems that use mechanical scanning with a\npiezoelectric stage, spatial light modulators, or tunable optics to acquire multiple phase-shifted\ninterferograms or defocused intensity images. Representative examples include phase-shifting\ndigital holography, spatial light interference microscopy (SLIM), and transport-of-intensity-based\ntechniques that rely on axial scanning or structured illumination.\nWe note that Figure 1a is not meant to imply that all QPI techniques require interferometers or\nmechanical actuation. Rather, it is intended to highlight the level of optical complexity and large\nsystem footprint typically associated with many conventional QPI implementations. While there\nare various types of conventional QPI systems with different optical configurations and\ncomponents, most remain far from miniaturization due to their reliance on interferometry or\nholography involving multi-element assemblies, long optical paths, and dynamic optical\ncomponents.\nTo clarify this point in the revised manuscript, we now explicitly describe in the main text that\nFigure 1a depicts multi-shot interferometric QPI systems. We also cite representative examples\nfor this category, including Cuche et al. (Opt. Lett., 1999; DOI:10.1364/OL.24.000291), Marquet\net al. (Opt. Lett., 2005; DOI:10.1364/OL.30.000468), and Huang et al. (Light Sci. Appl., 2024;\nDOI:10.1038/s41377-024-01453-x), which reflects the type of architecture our schematic is\nintended to represent. We have revised the figure caption and the corresponding sentence in\nthe introduction to clarify this point and avoid overgeneralization.\n● 2. The manuscript introduces the terms \"complex-amplitude metasurface\" and\n\"phase-only metasurface\" without a clear, concise definition. While the explanation is\nprovided later, an earlier clarification would improve readability. A brief definition or\ndescription in the Introduction would be beneficial.\nWe appreciate the reviewer’s suggestion and agree that introducing key terminology earlier\nimproves clarity and accessibility for a broad readership. In the revised manuscript, we now\nprovide concise definitions of both “complex-amplitude metasurface” and “phase-only\nmetasurface” in the Introduction (page 3, paragraph 1) and the Results section (page 5,\nparagraph 1). Specifically, we clarify that the phase-only metasurface has a unity amplitude\nprofile and modulates only the phase of the transmitted light, using the same phase profile as\nthe complex-amplitude metasurface. In contrast, the complex-amplitude metasurface modulates\nboth amplitude and phase simultaneously, enabling more advanced wavefront control and\nsuppression of optical artifacts. This distinction is critical to understanding the advantages of our\noptical design, and we thank the reviewer for the opportunity to improve clarity.\n● 3. The architecture of the non-physical model (one of the inverse solvers) is not well\ndescribed. The authors should explicitly state how this model is structured and how it\ndiffers from other models used in the study.\nWe appreciate the reviewer’s helpful suggestion. To address this, we have added a dedicated\nsubsection in the revised Supplementary Information (Supplementary Note 2 and\nSupplementary Figure 6) that details the architecture and training procedure of the non-physical\nmodel. This model is implemented using a standard U-net architecture similar to CNN ,\nbut\nphase\nwithout any integration of physics-guided supervision, model calibration, or correction networks.\nIt is trained in a purely supervised manner using paired input–output data of intensity images\nand ground truth phase, without accounting for physical wave propagation, optical aberrations,\nor system-specific physical constraints.\nThis type of non-physical model reflects methodologies adopted in previous deep\nlearning-based QPI approaches (Light: Science & Applications 2018; DOI:\n10.1038/lsa.2017.141; Nano Letters 2025; DOI: 10.1021/acs.nanolett.4c06039), but typically\nyields limited performance in terms of spatial resolution, image quality, and robustness.\nAs\nshown in our results, this baseline model performs significantly worse than our physics-informed\napproach, which combines forward and inverse modeling with physics-guided supervision and\nmodel calibration using experimentally captured datasets. This integration substantially\nimproves image quality, resolution, and robustness.\nWe have also revised the main text (page 8, paragraph 1) to explicitly reference this new\nsupplementary section and clarify the key architectural and conceptual differences between the\nnon-physical baseline and our proposed physics-informed model.\nSupplementary Figure 6. Architecture of the non-physical inverse model for phase\nretrieval. The non-physical model is implemented using a standard U-net architecture that\nreceives two input intensity images captured at different depth planes and directly outputs the\nreconstructed phase. Unlike the physics-informed model, this approach does not incorporate\nwave propagation modeling, system calibration, or correction networks. Instead, it is trained in a\npurely supervised manner using a pixel-wise L1 loss between the predicted and ground truth\nphase images. This simplified data-driven architecture reflects methodologies commonly used in\nprevious deep learning-based QPI approaches but suffers from limited performance and\nrobustness under real experimental conditions.\n● 4. The authors claim that the iterative TIE method performs better than the direct TIE, yet\nFigures 4 and 5 appear to suggest otherwise. The authors should clarify why this\ndiscrepancy arises and whether the observed results align with their expectations.\nWe thank the reviewer for this insightful comment. We agree that Figures 4 and 5 may initially\nappear to contradict the general expectation that iterative TIE solvers outperform direct TIE\nmethods. However, the observed behavior aligns with our expectations when considering the\nspecific noise and aberration characteristics of our miniaturized experimental setup. To clarify\nthis point, we conducted new simulations and added a new supplementary note and figure to\ncompare the two solvers under various experimental conditions.\nIn theory, iterative TIE solvers can typically yield higher-quality phase reconstructions than direct\nmethods when operating under ideal or well-conditioned scenarios, especially when system\naberrations and noise levels are negligible. This expected behavior is confirmed in our new\nsimulations, as shown in the top row of Supplementary Figure 5a, where the iterative solver\noutperforms the direct method under ideal conditions.\nHowever, iterative solvers are also more sensitive to system imperfections such as optical\naberrations, fabrication errors, and measurement noise. These issues are particularly prominent\nin our metasurface-based QPI system due to its compact architecture with more aberrations and\nsubwavelength-scale design features. In the bottom row of Supplementary Figure 5a, we show\nthat the performance of the iterative solver drops significantly and becomes worse than that of\nthe direct solver under aberrated conditions. Moreover, this trend is further exacerbated when\nGaussian noise from the sensor is introduced as shown in Supplementary Figure 4.\nThis degradation in performance is primarily due to the iterative solver’s reliance on repeated\nPoisson-based updates, which become unstable in the presence of wavefront distortions or\nnoise, amplifying system-level artifacts and leading to poor convergence. We further quantify\nthese effects in Supplementary Figure 5b, which shows the histogram of peak signal-to-noise\nratio (PSNR) under aberrated and ideal conditions. When aberration is present, even at low\nnoise levels, the PSNR of the iterative solver decreases sharply, while the direct solver remains\nrelatively stable. Note that our physics-informed model always outperforms the TIE solvers,\nregardless of the conditions.\nThis analysis explains why Figures 4 and 5 show the iterative solver performing worse than the\ndirect solver in our system, and it is consistent with the behavior we observe across both\nsimulation and experiment. To support this clarification, we have added a new subsection in the\nSupplementary Information (Supplementary Note 1 and Supplementary Figures 4 and 5) that\npresents these comparisons in detail. We have also revised the main text (page 8, paragraph 2)\nto explicitly refer to this supplementary analysis and provide further explanation of the solvers’\nrelative performance under different imaging conditions. We sincerely thank the reviewer for\nraising this important point and allowing us to clarify the results with further validation.\nSupplementary Figure 5 Comparison of direct and iterative TIE solvers under varying\naberration conditions. (a) Phase retrieval results using direct and iterative TIE solvers, with\nand without metasurface-induced aberrations, under noise-free conditions. The\nphysics-informed model is also included for comparison. The black scale bar is 30 µm. (b)\nHistogram of the peak signal-to-noise ratio (PSNR) for reconstructed phase images using\nTIE-iterative method, evaluated under metasurface aberration and low Gaussian noise\n(standard deviation = 0.001).\nReferee #3 (Remarks to the Author):\n● 1. The authors provide inadequate justification for implementing their device as a\nmetasurface rather than using simpler alternatives. Holographic optical elements (HOEs)\ncould achieve similar complex-amplitude modulation functionality without the severe\nfabrication constraints. While the authors emphasize nanoscale precision of their\nmetasurface design, they don't demonstrate how this translates to performance\nadvantages that couldn't be achieved with holographic plates. Holographic elements can\nmodulate both amplitude and phase simultaneously through various established\ntechniques, while offering significant advantages: they can be fabricated at much larger\naperture sizes (compared to the severely limiting 1 mm aperture), produced with simpler\nequipment than electron-beam lithography, manufactured at lower cost, and potentially\nachieve higher optical efficiency.\nWe appreciate the reviewer’s interest in comparing our metasurface-based optical system with\nholographic optical elements (HOEs). However, we respectfully disagree with the\ncharacterization that HOEs offer superior fabrication scalability or optical efficiency relative to\nmetasurfaces. We provide the following clarifications:\nFirst, regarding fabrication feasibility: while our prototype was fabricated using electron-beam\nlithography (EBL) due to resource constraints at academic research institutions, the\nmetasurface was specifically designed to be compatible with industrial mass-production\nworkflows. Our design uses CMOS-compatible materials (e.g., amorphous silicon) and\nmaintains a minimum feature size above 30 nm—parameters that are well within the limits of\nscalable techniques such as nanoimprint lithography (NIL), deep-UV photolithography, and\nroll-to-roll printing. These techniques have been successfully demonstrated in prior works for\nmass-producing metasurfaces at wafer-scale or on flexible substrates (Nature Materials 2023;\nDOI: 10.1038/s41563-023-01485-5, Nature Reviews Electrical Engineering 2025; DOI:\n10.1038/s44287-024-00136-4). Recent commercial advances further confirm the viability of\nmetasurface mass production using existing semiconductor infrastructure. Therefore, we believe\nthe claim that metasurfaces are fundamentally limited by fabrication scalability does not\naccurately reflect the current state of the field. By contrast, HOEs require precision\ninterferometric recording setups that involve complex optical alignment and beam stability, and\ntheir replication processes often rely on distinct manufacturing chains that are not\nCMOS-compatible. While companies such as DigiLens have made admirable progress in\nscaling HOE fabrication, this typically involves specialized processes based on volume\nholograms, photopolymer materials, and custom replication systems. These methods are not\nuniversally accessible and may introduce additional environmental and cost considerations\ncompared to semiconductor-based manufacturing compatible with metasurfaces. Thus, it is not\nevident that HOEs offer a more scalable or sustainable fabrication path than metasurfaces; if\nanything, metasurfaces benefit from greater compatibility with global CMOS infrastructure and\nmass production chains.\nSecond, regarding scalability of aperture size: the 1 mm aperture of our prototype was chosen\nsolely based on practical constraints (e.g., fabrication time, cost, and limited access in academic\ncleanroom based on shared equipment usage), not due to any intrinsic design limitation. Our\nmetasurface design, governed by the wavefront engineering formulation in Equation 1, is fully\nscalable. Other studies in this field have also shown great scalability of metasurface designs,\nincluding scalable design to centimeter-scale apertures (e.g., Nature Communications 2018;\nDOI:10.1038/s41467-018-07011-5, Nano Letters 2019; DOI:10.1021/acs.nanolett.9b03333,\nNature Materials 2023; DOI:10.1038/s41563-023-01485-5, ACS Nano 2024;\nDOI:10.1021/acsnano.3c09462). This confirms that our approach can be extended to larger\napertures without degradation in function, provided that fabrication capacity is available.\nThird, in terms of programmability and wavefront complexity, our metasurface achieves spatially\nmultiplexed, complex-amplitude modulation, enabling simultaneous encoding of distinct focal\nlengths, wavefront curvatures, and aberration profiles. This level of programmable control is\ndifficult—if not infeasible—to achieve with standard HOE techniques. HOEs rely on replicating\nphysical wavefronts from existing optics via interference-based recording; they are inherently\nlimited in their ability to generate new or non-intuitive wavefronts. While some SLM-based\nholographic printers have been proposed to fabricate programmable HOEs, these methods\nsuffer from poor spatial resolution (e.g., hogel sizes on the order of hundreds of microns),\nphase-only modulation, and very slow printing speed due to the movement and stabilization\nprocess of mechanical stages (ACM TOG 2020; DOI: 10.1145/3414685.3417762). Hence, the\nfidelity and control of these printed HOEs remain insufficient for encoding densely multiplexed\nfunctions such as those implemented in our metasurface.\nFourth, in terms of optical efficiency, we respectfully disagree with the reviewer’s assertion that\nHOEs “potentially achieve higher optical efficiency.” While it is true that volume HOEs can\nexhibit high diffraction efficiency under specific conditions, they are highly angularly selective\ndue to Bragg diffraction. Efficient operation is typically confined to narrow angular windows\ndetermined by the reference beam during recording. In other words, only a single wave vector\ncomponent satisfying the Bragg’s equation can be diffracted without multiplexing or broadening.\nBroadening this angular range often sacrifices diffraction efficiency or introduces optical\naberrations. This trade-off is particularly limiting in wide-field or non-collimated applications,\nwhere angular tolerance is essential. In contrast, our metasurface design achieves a high\ndiffraction efficiency across a broad angular spectrum and can be further optimized using\nalternative dielectric materials such as crystalline Si, SiN, TiO₂, or synthetic structures (Science\n2016; DOI: 10.1126/science.aaf6644, Nature Materials 2023; DOI:\n10.1038/s41563-023-01485-5). In this work, we selected amorphous Si to ensure compatibility\nwith scalable, cost-efficient manufacturing rather than to maximize theoretical efficiency. Future\nversions could readily achieve higher performance using low-loss materials without changing\nthe design principles.\nIn summary, while we respect the value of HOE technologies and their promising applications,\nwe do not believe they are capable of realizing the programmable, compact, and scalable\nwavefront engineering demonstrated by our metasurface QPI system. Our work not only\npresents a high-performance imaging system but also provides a path forward for scalable,\nmass-producible, and functionally dense optical platforms. We appreciate the reviewer’s\ncomments, which helped us expand our discussion on practical design considerations.\nHowever, we believe it is not appropriate to make generalized comparisons by aggregating\nindividual advantages from various HOE implementations and using that idealized combination\nas a benchmark against a single metasurface device. Each optical technology involves\ntrade-offs, and meaningful comparisons must account for system-level constraints and design\nobjectives. To clarify these points for readers, we have added a dedicated discussion section to\nthe revised manuscript and Supplementary Note 5, where we compare HOEs and metasurfaces\nin greater detail.\n● 2. The use of intensity resolution targets to demonstrate phase imaging capabilities in\nFig. 4, Fig. 5a-b, and supplementary figures may be inappropriate. For a paper claiming\nquantitative phase imaging, proper phase targets (such as the standard quantitative\nphase target from Benchmark Technologies) are essential. The authors may need to\ndemonstrate different height measurements of known phase objects compared to\nground truth values (for example, this can be done using the standard quantitative phase\ntarget with different heights from Benchmark Technologies).\nWe respectfully note that the reviewer’s concern appears to come from a misunderstanding of\nour manuscript, as quantitative validation using a proper phase target is already presented in\nFigure 5c–e. Moreover, we argue that the use of intensity-based resolution targets, such as the\nUSAF-1951 target in Figure 4 and Figure 5a–b, also provides meaningful validation of spatial\nresolution and reconstruction fidelity.\nFirst, regarding the use of intensity resolution targets in phase imaging, intensity-based patterns\n(e.g., USAF-1951) are widely adopted in the QPI literature. They are commonly used in\nconjunction with phase retrieval algorithms to benchmark the system’s spatial resolution and to\nvalidate binarized or step-like phase profile reconstructions. This approach is well established in\nprior studies (Nat. Photonics 2013; DOI: 10.1038/nphoton.2013.187, Optica 2018; DOI:\n10.1364/OPTICA.5.000976, Adv. Photonics 2022; DOI: 10.1117/1.AP.4.5.056002), where the\nretrieved phase maps faithfully reproduce the phase profiles embedded in the intensity-defined\ntarget designs. Following this established approach, we employed the USAF resolution target\nalong with biological samples in Figures 4 and 5a-b of our manuscript to demonstrate that our\nphase retrieval pipeline can accurately reconstruct binarized phase distributions.\nSecond, regarding the use of proper quantitative phase targets, we would like to clarify that we\nhave already performed experimental validation using a dedicated phase-only target, as shown\nin Figures 5c–e of the main manuscript. Rather than using a commercial target such as those\nfrom Benchmark Technologies or other vendors, we fabricated a custom-designed dielectric\nmetasurface phase target ourselves, which also provides known and spatially varying phase\nlevels. This type of target has also been employed in recent studies to benchmark QPI\nperformance (e.g., Nat. Commun. 2022; DOI: 10.1038/s41467-022-34197-6). Our custom target\nwas fabricated using electron-beam lithography on a PMMA positive photoresist substrate,\nforming subwavelength nanohole arrays with varying widths to induce multilevel phase delays\n(Supplementary Figure 15). This design enables phase modulation performance equivalent to\nconventional height-based phase targets using diffractive optical elements or outperforming\nthem with a higher spatial resolution at the nanoscale.\nTo clarify this point, we have revised the Supplementary Information to include additional details\non the target’s design, fabrication, and optical characterization (see Supplementary Notes 3 and\nSupplementary Figures 15 and 16). Furthermore, we have added a new quantitative evaluation\ncomparing the reconstructed phase values to the ground truth design profile (Supplementary\nFigure 24). This expanded analysis demonstrates strong agreement, reinforcing the reliability\nand quantitative accuracy of our QPI system.\nSupplementary Figure 15 Details of custom-designed metasurface sample used as a\nphase target. (a) Simulated transmission (blue) and phase (red) responses of the metasurface\nas a function of the square-hole width. Inset: unit-cell geometry (material: PMMA on SiN/glass;\nthicknesses d = 600 nm, d = 350 nm). (b) Ground-truth phase map of the designed\n1 2\nmetasurface pattern. (c) Scanning electron microscopy (SEM) image of the fabricated\nmetasurface target, comprising three subarrays with different phase values. Colored boxes\ncorrespond to the regions shown in (d). Scale bar: 6 μm. (d) Magnified SEM images of the blue,\ngreen, and red regions in (c), showing the subwavelength structures. Scale bar: 250 nm.\nSupplementary Figure 16 Scanning electron microscopy image tilted at 30 degrees for\nthickness measurement of the custom metasurface phase target. The white arrow marks\nthe vertical distance from the substrate surface to the top surface of the nano-hole. The actual\nheight is obtained by dividing the measured length by sin(30°) The scale bar is 200 nm.\nSupplementary Figure 24 Extended quantitative-phase comparison using the\nmetasurface phase target. Scatter plot comparing retrieved phase values against the ground\ntruth design across the reconstruction methods. Black circles: design (ground truth); blue\nsquares: TIE-Direct; red triangles: TIE-Iterative; cyan stars: non-physical model; green\ndiamonds: our physics-informed model.\n● 3. The approach of using two defocused intensity images for phase retrieval has\ninherent limitations that render the ‘quantitative’ claim questionable. For example, the\nreported method cannot recover linear phase ramps (like those from optical prisms) with\narbitrary spatial frequencies, as the constant intensity patterns of the phase ramp contain\nno information about such phase profiles. If the system cannot handle such basic phase\ndistributions, the claim of ‘quantitative phase imaging’ may not be a proper one. The\napproach may only work for a narrow subset of weakly scattering samples without phase\nwraps, which may restrict its practical utility.\nThe ability to detect and reconstruct linear phase gradients, such as those introduced by a\nprism, is indeed a critical requirement for any quantitative phase imaging technique. However,\nthe example presented in the reviewer’s comment is based on an incorrect physical\ninterpretation. Intensity-based phase retrieval methods, including the TIE methods and our\napproach, are fully capable of detecting and reconstructing linear phase gradients.\nThe reviewer’s claim that a linear phase ramp with arbitrary spatial frequency contains no\nretrievable information for phase reconstruction overlooks a key point: a linear phase ramp\ninduces a lateral displacement of the beam profile as a function of axial position (z). The\ndirection and magnitude of this displacement depend on the spatial frequency components or\ngradient of the phase ramp, resulting in measurable intensity shifts between defocused images.\nThese variations are directly observable through capturing defocused intensity images, provided\nthat the shifted beam remains within the aperture and field of view of the imaging system.\nTherefore, phase gradients can be reliably inferred from intensity data using the TIE framework,\nand the scenario described does not represent a fundamental limitation of our method and does\nnot restrict its practical utility. The only case in which this may fail is when the beam profile\nextends beyond the aperture of the system, resulting in uniform intensity across the entire\ncaptured image. However, this does not reflect a theoretical limitation in the method’s ability to\nretrieve phase information—it simply defines the finite field of view of the system. As most\nimaging systems inherently have limited field of view, this constraint is not specific to our\napproach but applies broadly to practical optical implementations.\nTo clarify this point, we provide a theoretical explanation grounded in the transport of intensity\nequation. According to the TIE, the axial derivative of intensity is related to the divergence of the\ntransverse energy flux, involving both the phase gradient and curvature (see more details in this\nreference paper; Opt. Lasers Eng. 2020; DOI: 10.1016/j.optlaseng.2020.106233), and can be\nexpressed as\n𝑑𝐼 2\n− 𝑘 = ∇ · (𝐼∇ϕ)= ∇𝐼 · ∇ϕ + 𝐼∇ ϕ.\n𝑑𝑧\nThe first term on the right-hand side, ∇𝐼 · ∇ϕ, represents the contribution from the phase\n2\ngradient, and can be referred to as the \"prism term\". The second term, 𝐼∇ ϕ, corresponds to the\nphase curvature or so called “lens term”. As we can see from this equation, there is a phase\ngradient component at the first term, which relates to the defocused intensities or the axial\ndifferentiation of the intensity from the term on the left-hand side.\nTo analyze the problem that the reviewer discussed, let us consider a simple linear phase ramp:\n2\nϕ(𝑥) = 𝑎𝑥, where a is a constant coefficient. In this case, ∇ ϕ = 0, and the second term\nvanishes. Now, if the beam profile is spatially confined and remains entirely within the system’s\nfield of view and aperture, the gradient of the intensity ∇𝐼 will be nonzero at the beam edges,\nand thus the first term on the right-hand side in the equation becomes measurable. These edge\nvariations are sufficient to encode the phase ramp information. Conversely, if the beam is too\nwide or the shift caused by the phase gradient moves part of the intensity outside the detectable\nregion, no intensity variation is observed (i.e., ∇𝐼 = 0), and recovery becomes impossible. This\ndoesn’t mean a failure of the TIE method or of our approach, but rather a practical limitation\nimposed by the finite FoV of the optical system.\nThis analysis confirms that the ability of TIE-based methods to reconstruct linear phase ramps is\nentirely valid as long as the intensity variations caused by the phase gradient are captured\nwithin the system’s field of view. In our metasurface-based implementation, this condition is\nsatisfied for the majority of practical imaging scenarios. Moreover, the aperture and field of view\ncan be even further scaled up in future system designs by increasing the aperture size or larger\nFOV lens design with a smaller NA, further extending the accessible spatial frequency range. To\nhelp clarify this point, we have added a detailed discussion in the revised Supplementary Note\n5, where we explain the origin of this perceived limitation and relate it to field of view constraints\nrather than a failure of the quantitative phase imaging framework. We have also revised the\nmanuscript (page 9, paragraph 1) to clearly describe the experiments using phase targets and\nmention an extended quantitative analysis in the revised Supplementary Information.\n● 4. The quality of the recovered phase in Fig. 5 raises some questions, particularly\nregarding variations in the background and certain features. For biological cell samples,\nconventional understanding suggests that phase within cells should typically be higher\nthan at the edges, which appears inconsistent with some of the presented results. These\nerrors may indicate problems with the phase retrieval process.\nThe observation that phase values within biological cells are generally expected to be higher\nthan those at the edges is valid and well-established. However, we believe the concerns raised\nhere arise from a misinterpretation of the context in which the results are presented and\ncompared in our manuscript.\nIn Figure 5, we present phase reconstructions obtained using our compact metasurface-based\nmicroscope with multiple reconstruction methods. Among these, only our physics-informed\nmodel produces high quality phase maps that are consistent with biological expectations,\nshowing elevated phase within intracellular regions due to increased optical path lengths. In\ncontrast, the baseline methods (TIE-direct, TIE-iterative, and the non-physical model) yield\ndistorted or inconsistent phase distributions, likely due to their limited ability to account for the\nimperfections in our compact optical system or a mismatch between the physics and models. It\nseems the specific case the reviewer referred to may correspond to the non-physical model\nresults in Figure 5b or TIE-based results in Figure 5d, which serve as baseline comparisons\nrather than outputs from our proposed method.\nThis distinction becomes even clearer when compared with the results in Figure 4c, where the\nsame algorithms are applied to data from a conventional microscope with fewer aberrations and\nlower physics-model mismatch. In that setting, all methods perform comparably well, reinforcing\nthe idea that traditional algorithms are effective under ideal optical conditions. However, our\nmetasurface-based microscope, designed for maximum miniaturization and integration of\nmultiple optical functions, introduces greater sensitivity to optical aberrations, misalignment, and\nnanofabrication imperfections. These factors degrade the image quality when not properly\nmodeled or corrected, as seen in the outputs of conventional algorithms. Our physics-informed\nmodel addresses this by learning from both simulation and real experimental data. By\nincorporating an experimentally calibrated forward model into the inverse retrieval process, our\napproach captures system-specific aberrations and physical behaviors, resulting in consistent\nand reliable phase reconstructions. This robustness is especially critical for miniaturized\nsystems, where conventional methods tend to fail.\nTo further aid interpretation and emphasize the morphological validity of our results, we have\nadded new 3D visualizations of the retrieved phase maps in Supplementary Figure 21–23 and\nrevised the manuscript accordingly. These renderings clearly highlight the biological structure,\nincluding elevated intracellular phase regions, and further illustrate the limitations of baseline\nmethods. We hope these visualizations will help clarify any remaining concerns and reinforce\nthe robustness of our proposed framework.\nSupplementary Figure 21 Three-dimensional visualization of experimentally retrieved\nphase maps using different reconstruction methods. The phase retrieval results correspond\nto those shown in Figure 4c of the main manuscript, based on intensity images captured using a\nconventional microscope. Each row represents a different sample, while the columns compare\nreconstruction methods: TIE-Direct, TIE-Iterative, and our physics-informed model. The number\nat the top left of each image indicates the peak phase value (in radians) for the corresponding\nreconstruction.\nSupplementary Figure 22 Three-dimensional visualization of experimentally retrieved\nphase maps using different reconstruction methods. The phase retrieval results correspond\nto those shown in Figure 5b of the main manuscript, based on intensity images simultaneously\ncaptured at both focal planes of the compact metasurface microscope system. Each row\nrepresents a different sample, while the columns compare reconstruction methods: TIE-Direct,\nTIE-Iterative, and our physics-informed model. The number at the top left of each image\nindicates the peak phase value (in radians) for the corresponding reconstruction.\nSupplementary Figure 23 Three-dimensional visualization of experimentally retrieved\nphase maps using different reconstruction methods. The phase retrieval results correspond\nto those shown in Supplementary Figure 17, based on intensity images captured via axial\ntranslation at one of the two focal points in the compact metasurface microscope system. Each\nrow represents a different sample, while the columns compare reconstruction methods:\nTIE-Direct, TIE-Iterative, and our physics-informed model. The number at the top left of each\nimage indicates the peak phase value (in radians) for the corresponding reconstruction.\n● 5. The paper only discusses bulky optical implementations. However, there are many\nsingle-shot phase retrieval methods that have been implemented in lensfree\nconfigurations, even on ordinary smartphones without using any lenses. These lensfree\nmicroscopy approaches typically offer fields of view orders of magnitude larger than the\nreported method (which is severely limited by the 1 mm aperture) and can be\nimplemented at much lower cost with similar or better performance. The imaging\nthroughput is also substantially higher than the present work.\nWe respectfully disagree with the reviewer’s assertion that the manuscript \"only discusses bulky\noptical implementations.\" As shown in Figure 1, our work demonstrates a significant\nadvancement in miniaturized QPI by replacing conventional multi-element optical assemblies\nwith a single flat metasurface chip. This metasurface combines the functions of multiple discrete\noptical components such as a lens, beam splitter, and waveplates into a monolithic\nnanophotonic element. This integration significantly reduces the system's form factor and\ncomplexity. In contrast to traditional QPI systems that rely on large objective lenses, beam\nsplitters, interferometric systems, and spatial light modulators or actuators, our system achieves\nquantitative phase imaging using a streamlined design. Importantly, this miniaturization is\naccomplished without compromising optical performance, spatial resolution, robustness, or\nimaging throughput, as demonstrated in Figures 4 and 5. While further discussion of other\ncompact approaches is valuable, we believe our work already represents a substantial step\nforward in realizing miniaturized QPI systems suitable for portable and field-deployable\napplications.\nWe also wish to clarify that smartphone-based QPI technologies and lensfree QPI architectures\nrepresent fundamentally different approaches and should not be conflated. To the best of our\nknowledge, most prior works pursue either smartphone integration or lensless architecture, but\nrarely both simultaneously, as each approach involves distinct system-level trade-offs and\nhardware assumptions. We believe the reviewer’s comment merges these categories in a way\nthat may misrepresent the scope and limitations of existing systems. It is also important to\nemphasize that lensfree and smartphone-based QPI systems, although often perceived as\ncompact, typically involve nontrivial spatial or hardware requirements.\nSmartphone-based QPI systems frequently incorporate bulky optical setups with motorized\nstages, structured illumination, or coded aperture masks, and often require multi-frame\nacquisition or scanning mechanisms. These added components increase system complexity\nand decrease imaging throughput. For example, recent smartphone-compatible systems report\nacquisition times of up to a few seconds per frame (e.g., 10 seconds in ACS Photonics 2021;\nDOI: 10.1021/acsphotonics.1c00350, and up to 7 seconds in Lab Chip 2016; DOI:\n10.1039/c6lc01321j), depending on hardware configurations. In contrast, our system achieves\nreal-time 60 Hz throughput without mechanical scanning or active illumination control, enabling\ndynamic phase imaging under compact and robust hardware constraints.\nSimilarly, lensfree QPI systems could remove most of the optical elements including a lens in\nthe system, but are not inherently compact in practice. For example, source-to-sample distances\nin such systems are often large (e.g., ~150 mm) to ensure planar illumination or diffraction\npropagation conditions for spatial coherence, which counteracts the benefit of removing lenses\nin terms of system form factor. These systems also face fundamental trade-offs in resolution\n(limited by sensor pixel size or coherence of source), noise sensitivity, and algorithmic\nrobustness. For example, recent lensfree QPI implementations illustrate these limitations.\nLee et\nal. (Opt. Express 2021; DOI: 10.1364/oe.433719) employed sub-pixel shifting and multi-depth\nscanning with laser illumination, requiring a long source-to-sample distance, precision actuator\nstages, and multi-frame acquisition, which limits portability and speed. Chen et al. (Opt. Lett.\n2022; DOI: 10.1364/ol.474992) achieved single-shot imaging using LED illumination and\niterative reconstruction, but relied on a ~150 mm free-space propagation distance and suffered\nfrom limited imaging capabilities that were sensitive to coherence and worked only for sparse\nsamples but not for dense samples. Sun et al. (Light Sci. Appl. 2022; DOI:\n10.1038/s41377-022-00898-2) introduced a fiber-based QPI system using far-field speckle\nreconstruction, but resolution remained limited and background noise was substantial. These\nexamples indicate that, despite simplifying some optical components, lensless imaging systems\noften compromise on resolution, imaging speed, and robustness, which are critical factors for\nbiomedical applications.\nAdditionally, our system is not fundamentally limited by the 1 mm aperture used in the prototype.\nAs detailed in our response to Reviewer 3 – Question 1, the design principles described in\nEquation 1 are fully scalable and compatible with large-area design and fabrication. We selected\na 1 mm aperture for prototyping convenience and fabrication efficiency within the resource\nconstraints of an academic lab. However, the same multiplexed imaging principles can be\nconsistently applied at larger scales, with demonstrated scalability of metasurface designs up to\na few millimeter- or even tens of centimeter-scale apertures in previous works (e.g., Nature\nCommunications 2018; DOI:10.1038/s41467-018-07011-5, Nano Letters 2019;\nDOI:10.1021/acs.nanolett.9b03333, Nature Materials 2023; DOI:10.1038/s41563-023-01485-5,\nACS Nano 2024; DOI:10.1021/acsnano.3c09462). These precedents confirm that our approach\ncan be extended to larger apertures without compromising performance.\nRegarding fabrication and cost, although our sample was fabricated using electron-beam\nlithography at academic facilities, our design is CMOS-compatible and amenable to scalable\nmanufacturing processes in industry such as nanoimprint lithography and deep-UV\nphotolithography. These techniques have already been demonstrated at the wafer level (Nat.\nMater. 2023; DOI: 10.1038/s41563-023-01485-5) and are being adopted in commercial\nconsumer electronics, including the Apple iPad and iPhone. Therefore, contrary to the\nassumption that metasurface systems are inherently expensive or unscalable, our platform\naligns with the trajectory of low-cost, high-volume nanophotonic production.\nIn summary, our proposed system represents a miniaturized, scalable, and high-performance\nQPI platform that is fundamentally distinct from both bulky conventional optics and the practical\nlimitations observed in lensless or smartphone-based systems. We have clarified this point in\nthe revised manuscript and included additional discussion in Supplementary Note 5 for the\nbenefit of readers.\n● 6. In discussing Fig. 2d, the authors make claims that their approach \"significantly\nreduces diffraction noise and improves the robustness of phase retrieval.\" This figure\nshows only intensity measurements without any phase retrieval. The nature of\n\"diffraction noise\" is undefined.\nWe appreciate the reviewer’s careful reading and agree that the term “diffraction noise” used in\nthe original manuscript could be more precisely defined. In the revised version, we have\nreplaced it with “diffraction artifact,” which more accurately describes the structured intensity\nfluctuations near the focal plane caused by coherent interference between the adjacent focal\nspots produced by our metasurface design. These artifacts are distinct from random sensor\nnoise (e.g., Gaussian noise) and can significantly impact image quality and the robustness of\nphase retrieval algorithms. Please note that one of the key benefits of our complex-amplitude\nmetasurface, compared to the phase-only counterpart, is its ability to reduce such diffraction\nartifacts, resulting in higher-quality intensity images and consequently improved phase retrieval\nperformance.\nTo further illustrate the effect of diffraction artifacts on phase reconstruction, we have also added\nnew simulation results in Supplementary Figure 3. This figure compares intensity images and\nphase retrieval performance for two types of metasurface designs: a phase-only metasurface\nand a complex-amplitude metasurface, both corresponding to the designs introduced in Figure\n2d of the main manuscript. The phase-only metasurface exhibits noticeable interference\nartifacts in both PSF profiles (Figure 2d) and raw intensity images (I and I in Supplementary\n0 z\nFigure 3), which degrade phase reconstruction across all methods, including baseline\nalgorithms and our proposed model. In contrast, the complex-amplitude metasurface\nsuppresses these artifacts, producing cleaner intensity distributions and significantly improved\nphase reconstructions.\nThese results underscore the importance of minimizing diffraction artifacts at the image\nacquisition stage to enhance the fidelity and robustness of phase retrieval. Since our\nreconstruction method depends directly on the captured intensity images, improvements in raw\nimage quality directly translate to better phase recovery. We have revised the main text (page 5,\nparagraph 1) and included the expanded analysis in Supplementary Note 1.\nSupplementary Figure 3 Impact of diffraction artifacts on phase retrieval using\nphase-only and complex-amplitude metasurfaces from Figure 2d. (a) Captured intensity\nimages from the phase-only metasurface and corresponding phase retrieval results using\ndifferent reconstruction methods. The phase-only metasurface introduces significant diffraction\nartifacts around the PSFs and intensity profiles, caused by coherent interference between\nadjacent focal points. These artifacts lead to substantial distortions in the retrieved phase maps\nacross TIE-Direct, TIE-Iterative, and even our physics-informed model. (b) Captured intensity\nimages from the complex-amplitude metasurface and corresponding phase retrieval results.\nThe\ncomplex metasurface effectively suppresses these diffraction artifacts, yielding cleaner PSFs\nand improved intensity image quality. This results in more accurate and robust phase\nreconstruction across all methods. The black scale bar is 30 μm.\n● 7. While metasurfaces typically offer advantages for use with incoherent light, this work\nmay be restricted to coherent illumination and specifically to weak phase objects without\nphase wraps or background variations. This may represent a drawback compared to\nother metasurface approaches that work with incoherent illumination. The requirement\nfor coherent illumination combined with the weak phase assumption constrains the\nsystem's practical utility. These limitations would make the approach challenging to\nimplement in reflection mode or with general objects that don't satisfy the weak phase\napproximation. Such restrictions could narrow the range of real-world applications where\nthis technology would be effective, particularly in comparison to alternative approaches\nthat can operate under more flexible illumination conditions.\nWe respectfully disagree with the reviewer’s assertion that our work is limited due to its use of\ncoherent illumination or a reliance on the weak-phase approximation. In addition, we also\ndisagree with the implication that other metasurface-based QPI approaches operate under\nincoherent illumination. This comment appears to reflect a misunderstanding of both the\nfundamental requirements of quantitative phase imaging (QPI) and the capabilities of our\nproposed method.\nFirst, the suggestion that metasurface-based systems should preferentially operate under\nincoherent illumination overlooks a central point: quantitative phase imaging fundamentally\nrequires coherence. The vast majority of QPI techniques—regardless of whether they use\nconventional optics or metasurfaces—depend on either coherent or partially coherent\nillumination to extract phase information. This includes established methods such as TIE\nmicroscopy (Phys. Rev. Lett. 1998; DOI: 10.1103/PhysRevLett.80.2586, Opt. Lasers Eng. 2020;\nDOI: 10.1016/j.optlaseng.2020.106233), digital holography (Opt. Lett. 1999; DOI:\n10.1364/OL.24.000291, Opt. Lett. 2005; DOI: 10.1364/OL.30.000468, Light Sci. Appl. 2024;\nDOI: 10.1038/s41377-024-01453-x), Fourier ptychographic microscopy (Nat. Photonics 2013;\nDOI: 10.1038/nphoton.2013.187), and many more technologies (Nat. Photon 2018;\nDOI:10.1038/s41566-018-0253-x). Even recent metasurface-integrated QPI systems are all\ndemonstrated under coherent sources (Nat. Photonics 2020; DOI: 10.1038/s41566-019-0536-x,\nNano Lett. 2025; DOI: 10.1021/acs.nanolett.4c06039, ACS Appl. Mater. Interfaces 2025; DOI:\n10.1021/acsami.5c02901). These examples confirm that the requirement for coherence is not a\nlimitation of our specific approach, but a universal physical prerequisite for QPI across all optical\nplatforms. By contrast, imaging under fully incoherent illumination cannot support quantitative\nphase retrieval, as it lacks the phase correlations needed to infer optical path length differences.\nWhile incoherent illumination can be advantageous in general-purpose imaging (e.g., for\neliminating speckle in fluorescent or bright-field modalities), it is inherently incompatible with the\ngoals of QPI. Thus, it is incorrect to frame incoherent illumination as an advantage in this\ncontext.\nSecond, we clarify that our approach is not restricted to weak-phase objects. This assumption is\noften necessary in traditional analytical TIE methods, but our framework does not rely on the\nweak-phase or paraxial approximations. Instead, we employ a physics-informed deep learning\nmodel trained on both simulated and experimentally captured datasets. This model directly\nlearns the nonlinear forward and inverse mappings between intensity images and phase\nprofiles, effectively overcoming the limitations of conventional TIE by the paraxial\napproximations. As a result, our method enables robust phase retrieval even in the presence of\nlarge phase variations, moderate scattering, and severe optical aberrations in compact,\nintegrated systems. As demonstrated in Figure 5 of the manuscript, our method successfully\nreconstructs complex biological samples that contain significant phase variations across both\nintracellular and extracellular regions. These samples clearly exceed the weak-phase regime,\nyet our approach recovers high-quality phase maps with accurate morphological features and\nstrong biological interpretability. Furthermore, we have added 3D visualizations (Supplementary\nFigures 21–23) that emphasize these spatial variations, reinforcing that our system is capable of\nhandling a wide range of phase distributions.\nThird, while interferometric QPI methods can be difficult to scale to compact or reflection-mode\ngeometries due to coherence requirements and alignment sensitivity, our metasurface-based\nTIE framework is inherently more flexible. TIE is known to be highly tolerant of partial\ncoherence, and can be implemented using spatially incoherent but temporally coherent sources\nsuch as LED arrays. This makes it one of the most adaptable phase imaging techniques\navailable for compact hardware integration. For example, as described in Opt. Lasers Eng. 2020\n(DOI: 10.1016/j.optlaseng.2020.106233), TIE can operate effectively under low-coherence\nillumination conditions that are incompatible with full-field holography or interferometry.\nIn summary, the reviewer’s concern about the coherence requirement and weak-phase\nlimitations does not reflect the physical realities of QPI or the actual capabilities of our system.\nOur approach uses coherence in a way that is consistent with all mainstream QPI methods,\nwhile offering increased tolerance to system imperfections and flexibility in implementation.\nImportantly, our work fully leverages the unique benefits of metasurfaces—including\nmulti-functionality, complex-amplitude modulation, and compact integration—to enable an\noptical platform that is both scalable and high-performing. The framework is not restricted to\nweak-phase objects and remains applicable to real-world imaging scenarios involving strong\nphase gradients and moderate scattering, making it broadly useful across biological and\nbiomedical applications.\n---\nAgain, we appreciate the constructive feedback of all reviewers and have revised our\nmanuscript and supplement to clarify all of these comments.\nThank you very much for your consideration,\nThe authors"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I read the verbose authors' responses with interest. I find that the concerns raised by the reviewers are well-debated, and the revisions in the manuscript text and SI address most of them, profoundly deepening the description. In particular, I revised my statement that the paper lacks conceptual novelty. Therefore, the paper in the revised version can be accepted for publication.", "reviewer_2": "I appreciate the authors' thorough responses to the comments and the substantial revisions made to the manuscript and supplementary materials. The authors have successfully addressed all of my previous concerns in a clear and rigorous manner. The revised manuscript is significantly improved in clarity, technical accuracy, and presentation quality.\nI recommend acceptance of the manuscript in its current form.", "reviewer_3": "This reviewer appreciates the authors' efforts in revising the manuscript, including the addition of new supplementary notes, expanded experimental validation, and clarification of technical details. The revised manuscript shows improvement in presentation and addresses several reviewers' concerns. However, this reviewer still has reservations about several fundamental aspects of this work.\nMost concerning is the fundamental limitation revealed in the authors' response regarding linear phase ramps. Since linear phase ramps constitute the Fourier basis for arbitrary optical fields, the inability to accurately measure any of these components would imply that the system cannot accurately reconstruct complex phase patterns with different spatial frequencies. The authors claim their TIE-based approach can detect such gradients through the edge intensity variation term ∇I·∇ϕ, but this argument is not correct. For a uniform beam with a linear phase ramp, the intensity gradient ∇I equals zero throughput the bulk of the field, meaning that most of the field provides no phase information. This differs fundamentally from interferometric quantitative phase methods that directly measure phase across the entire field.\nThe mathematical implications of this edge-only measurement approach are significant. Recovering a global phase profile from boundary information alone constitutes an ill-posed problem, where multiple different phase distributions could produce identical edge intensity patterns. This makes the reconstruction highly sensitive to noise and boundary conditions, with small errors at edges propagating throughout the entire reconstructed phase map. Since the system cannot reliably measure any of these fundamental Fourier basis functions across the field, it cannot accurately measure complex phase patterns with different spatial frequencies. The authors' attempt to minimize this issue by labeling it a \"field of view limitation\" rather than acknowledging it as a fundamental measurement limitation may not fully capture the nature of this constraint. Unlike their system, interferometric and holographic QPI methods can accurately reconstruct linear gradients throughout their field of view, not just at boundaries.\nWhile the authors provide extensive arguments defending their metasurface choice over holographic optical elements, their comparison would benefit from a more balanced perspective. They emphasize HOE limitations such as Bragg selectivity and fabrication constraints while understating comparable metasurface challenges, including the need to maintain sub 100 nm feature uniformity across large areas and the associated fabrication costs. The claim that HOEs cannot achieve complex amplitude modulation could be better substantiated, as modern computer-generated HOEs have demonstrated sophisticated modulation capabilities.\nThe validation of quantitative phase accuracy remains insufficient. The authors' custom phase target consists exclusively of high-frequency diffractive structures implemented as varying nanohole arrays, rather than traditional stepped-height references. This approach only demonstrates accuracy for phase patterns with high spatial frequencies. Notably absent is any validation using uniform height steps in standard phase target that are characteristic of biological samples. Without testing against standard commercial phase targets or fabricated height steps of known dimensions, the claim of quantitative accuracy for general phase imaging applications remains unsubstantiated."}, "author_response": "Point-by-Point Response\nWe thank the editor and all reviewers for their insightful and constructive feedback on our\nmanuscript. We are encouraged by the positive comments from R1 and R2, who found that our\nprevious revision significantly improved the manuscript’s clarity, technical accuracy, and\npresentation quality.\nTo address the remaining concerns raised by R3, we have carefully revised the manuscript and\nsupplement. In particular, we have expanded our discussion to provide a more balanced\ncomparison between our chosen technology and alternative approaches, and we offer detailed\nclarifications on the fundamental principles of our validation methods. Below, we provide a\npoint-by-point response to each of R3’s comments.\nReferee #3 (Remarks to the Author):\n● 1. Most concerning is the fundamental limitation revealed in the authors' response\nregarding linear phase ramps. Since linear phase ramps constitute the Fourier basis for\narbitrary optical fields, the inability to accurately measure any of these components\nwould imply that the system cannot accurately reconstruct complex phase patterns with\ndifferent spatial frequencies. The authors claim their TIE-based approach can detect\nsuch gradients through the edge intensity variation term ∇I·∇ϕ, but this argument is not\ncorrect. For a uniform beam with a linear phase ramp, the intensity gradient ∇I equals\nzero throughput the bulk of the field, meaning that most of the field provides no phase\ninformation. This differs fundamentally from interferometric quantitative phase methods\nthat directly measure phase across the entire field. The mathematical implications of this\nedge-only measurement approach are significant. Recovering a global phase profile\nfrom boundary information alone constitutes an ill-posed problem, where multiple\ndifferent phase distributions could produce identical edge intensity patterns. This makes\nthe reconstruction highly sensitive to noise and boundary conditions, with small errors at\nedges propagating throughout the entire reconstructed phase map. Since the system\ncannot reliably measure any of these fundamental Fourier basis functions across the\nfield, it cannot accurately measure complex phase patterns with different spatial\nfrequencies. The authors' attempt to minimize this issue by labeling it a \"field of view\nlimitation\" rather than acknowledging it as a fundamental measurement limitation may\nnot fully capture the nature of this constraint. Unlike their system, interferometric and\nholographic QPI methods can accurately reconstruct linear gradients throughout their\nfield of view, not just at boundaries.\nWe thank the reviewer for this insightful comment and for allowing us to elaborate on the\ndifferences between TIE-based methods and interferometry; we agree that this is an important\ndiscussion. We would like to note that the outlined characteristics are not specific to our method,\nbut to all TIE methods. Due to the relative benefits and limitations of TIE and interferometry,\nboth approaches are popular choices in different applications.\nThe reviewer is correct that for an idealized, infinite, and uniformly illuminated beam, a linear\nphase ramp produces no intensity gradient (∇I=0) within the captured field of any classic TIE\nmethod, making phase recovery with this class of methods challenging in this specific condition.\nHowever, most practical imaging systems in the real world operate with a finite beam size.\nIn\nthis practical context, when the aperture size and the field of view is sufficiently large to capture\nthe light field information, a linear phase ramp should have a finite size of boundary and induce\nclear detectable information such as a depth-dependent lateral shift of the beam intensity\nprofile. This shift creates measurable intensity variations across the beam displacement or at\nthe beam's edges, which are captured in the defocused image pair. As discussed in the\nprevious revision, this information is encoded in the TIE’s \"prism term\" (∇I⋅∇ϕ), allowing the\nphase gradient to be recovered through the intensity gradient. Hence, we maintain our position\nthat the TIE method is theoretically capable of resolving all spatial frequency components of\narbitrary fields or Fourier basis functions within the field, under the assumption that the field is\nwithin the imaged field of view. We clarify this assumption in the discussion section of the\nrevised manuscript.\nWe agree with the reviewer that for conventional TIE solvers, the concentration of information at\nthe boundaries of the intensity gradient can make the reconstruction sensitive to noise and\nboundary conditions. This is a well-known challenge for analytical TIE methods and explains\ntheir performance degradation in the presence of aberrations or noise. However, this sensitivity\nreflects a limitation of the classic algorithm, not a fundamental inability of the underlying physics\nto measure a phase ramp. The problem only becomes ill-posed if the beam shift is so large that\nthe intensity variations occur entirely outside the detector's FOV. This is a practical FOV\nconstraint, not a theoretical one, and it applies to all imaging modalities; for instance, an\ninterferometer also cannot measure the tilt of a prism that deflects the beam entirely out of its\ndetection path with a very steep angle.\nCrucially, our work was also designed to overcome these limitations through a co-design of\nhardware and software. As motivated in our introduction section, the conventional TIEs are very\nsensitive to the noise and artifacts, and we have stated that this is also one of the main\nchallenges that we would like to tackle through this study. On the optics side, our\ncomplex-amplitude metasurface is engineered to suppress diffraction artifacts and other stray\nlight, providing the phase retrieval algorithm with cleaner, higher-fidelity intensity data to begin\nwith. This minimizes the noise that classic TIE solvers are so sensitive to. On the algorithm side,\nour physics-informed inverse model is not a classic TIE solver. It is a physical model composed\nof deep neural networks that learn a direct, nonlinear mapping from the two intensity images to\nthe final phase map in the real physical system. By being trained on a physics-based forward\nmodel and refined with real experimental data, our model learns all the physical and practical\nparameters. This learned inverse mapping is far more robust to noise, aberrations, and\nboundary conditions than any analytical solvers, as demonstrated by our experimental results\nwhere our method succeeds even when conventional TIE solvers fail.\nWe thank the reviewer for prompting a more comprehensive comparison with interferometric\nand holographic QPI methods. These approaches indeed measure the complex field directly\nand thus avoid the boundary sensitivity inherent to TIE. However, they introduce a different set\nof trade-offs, primarily related to hardware complexity. Interferometric and Holographic methods\ndirectly measure the complex field, which allows them to mitigate the boundary sensitivity\ninherent to TIE methods. On the other hand, they typically require bulky optical setups (e.g.,\nMach-Zehnder or Michelson interferometers), active components for phase shifting (e.g.,\nmechanical actuators or spatial light modulators), multi-frame acquisition, and high\ntemporal/spatial coherence of input sources. In contrast, TIE-based or intensity-based\napproaches offer a path toward simpler, more compact hardware that can eventually enable\ncompact and single-shot acquisition, trading hardware complexity for a greater reliance on\nsophisticated computational reconstruction algorithms.\nOur work sits firmly in the second category, leveraging nanophotonics and AI to maximize the\nadvantages of the TIE approach (system complexity and compactness) while mitigating its\ntraditional disadvantages (noise and boundary sensitivity). As the reviewer notes, each\nmethodology has its place, and the optimal choice depends on the application's specific\npriorities. The framework we developed—using a physics-informed model refined with\nexperimental data—is broadly applicable and could even be adapted to improve robustness in\nother QPI approaches.\nIn this revision, we have highlighted the limitations of the TIE method in the discussion section\nof the revised manuscript, provided a detailed discussion of these limitations in Supplementary\nNote 5, and presented a comprehensive comparison to other QPI methods in Supplementary\nTable 1. This will provide readers with a more objective understanding of the limitations of the\nvarious QPI methods and provide a balanced discussion, which will be of help for further\ninvestigation in this field.\n● 2. While the authors provide extensive arguments defending their metasurface choice\nover holographic optical elements, their comparison would benefit from a more balanced\nperspective. They emphasize HOE limitations such as Bragg selectivity and fabrication\nconstraints while understating comparable metasurface challenges, including the need\nto maintain sub 100 nm feature uniformity across large areas and the associated\nfabrication costs. The claim that HOEs cannot achieve complex amplitude modulation\ncould be better substantiated, as modern computer-generated HOEs have demonstrated\nsophisticated modulation capabilities.\nWe thank the reviewer for this valuable feedback and for encouraging a more balanced and\ndetailed comparison between metasurfaces and HOEs in general. We agree that a balanced\ndiscussion of the relative benefits and limitations of these technologies is a valuable addition to\nour paper.\nFirst, to provide a balanced view, we have revised Supplementary Note 5 to explicitly and more\nthoroughly acknowledge the challenges associated with metasurface technology. We now detail\nthat while our work demonstrates their feasibility and potential, scaling them for industrial mass\nproduction is not trivial. This includes the engineering hurdles of maintaining nanoscale feature\nresolution, ensuring large-area uniformity across centimeter-scale wafers, and addressing the\nyield and cost implications at a production scale. In previous revisions, we contextualized this by\ndiscussing recent metasurface fabrication technologies, noting that while prototyping often relies\non expensive, low-throughput electron-beam lithography, reliable large-area and\nmass-production pathways are emerging in the field through deep ultraviolet lithography and\nnanoimprinting technologies. These efforts are reflected in academic research [ref. 16, 56–58]\nand in industrial commercialization from representative companies such as Metalenz\n(https://metalenz.com/), NIL Technology (https://www.nilt.com/), and Samsung [ref. 18].\nHowever, we agree that significant gaps remain between these promising demonstrations and\nrobust, cost-effective commercialization, particularly concerning production yield and scalability,\nwhich should be addressed in the future.\nSecond, our revised discussion now better recognizes the impressive advances in modern\ncomputer-generated HOEs. We have added citations to introduce HOE’s state-of-the-art\ncapabilities in large-area replication, sophisticated wavefront design, and approximate\ncomplex-amplitude modulation. For example, recent efforts in print-based HOE research have\ndemonstrated programmable wavefront modulation through holographic printers [ref. 14 in\nsupplement]. That said, although modern computer-generated HOEs have made these\nimpressive advances, the literature also shows they still face inherent limitations. These include\ntrade-offs between spatial resolution (i.e., hogel size), programmability of each pixel, angular\nselectivity (the Bragg condition), diffraction efficiency, and the fidelity of arbitrary\ncomplex-amplitude modulation. Our system design demands simultaneous, programmable,\nsubwavelength-resolution control over both amplitude and phase within a single, monolithic\nelement for an integrated optical system with multiple functions. This is a set of requirements\nthat presents significant challenges for current HOE technologies but which our metasurface\ndesign is engineered to address. Therefore, for our specific and demanding goal of a chip-scale,\nsingle-element objective that simultaneously performs lensing, beam splitting, and\nprogrammable point-spread-function shaping with a short focal length and low coherent\ncrosstalk, we concluded that a complex-amplitude metasurface was the most suitable\ntechnological choice.\nWe note that using metasurfaces in our system was a deliberate engineering decision based on\na careful evaluation of the trade-offs discussed above. This does not imply that one platform is\nuniversally superior to the other. Depending on the application, there will undoubtedly be system\ndesigns where HOEs are advantageous (e.g., for optical elements with engineered angular\nselectivity, thanks to HOEs’ volumetric structures), in which case they would be a better choice\nthan metasurfaces. To ensure this context is clear to all readers, we have implemented these\nchanges by adding balanced language and new references to Supplementary Note 5.\n● 3. The validation of quantitative phase accuracy remains insufficient. The authors'\ncustom phase target consists exclusively of high-frequency diffractive structures\nimplemented as varying nanohole arrays, rather than traditional stepped-height\nreferences. This approach only demonstrates accuracy for phase patterns with high\nspatial frequencies. Notably absent is any validation using uniform height steps in\nstandard phase target that are characteristic of biological samples. Without testing\nagainst standard commercial phase targets or fabricated height steps of known\ndimensions, the claim of quantitative accuracy for general phase imaging applications\nremains unsubstantiated.\nWe thank the reviewer for their careful consideration of our validation methods. We believe the\nconcern regarding our phase target comes from a misunderstanding of our target’s design and\nits physical equivalence to traditional phase targets, which we would like to clarify in detail. Our\nvalidation is indeed quantitative, robust, and aligned with established approaches in the field.\nFirst, we would like to address the reviewer’s claim that our target consists \"exclusively of\nhigh-frequency diffractive structures.\" Although the metasurface is patterned with\nsubwavelength nanoholes (350 nm pitch), the effective optical features that they realize are\nmicron-scale, uniform phase regions that include low spatial frequencies. As shown in the\ndesign schematics and SEM images in Supplementary Figure 15, the target consists of three\ndistinct and multi-micron regions. Each region is composed of periodic, identical nanostructures\nto provide a uniform, step-like phase delay within that region. The spatial frequency is\nequivalent to group 7, element 3 of the 1951 USAF resolution test target (corresponding to\n161.3 cycles/mm or 3.10 µm feature size), which is highly relevant for typical biological\nstructures. The claim that our target lacks low-frequency content is therefore incorrect as it was\nexplicitly designed to exercise features at biologically meaningful scales.\nSecond, our metasurface phase target is physically equivalent, from the light’s perspective, to a\ntraditional stepped-height diffractive optical element (DOE) of commercial targets. Both realize\nphase modulation by controlling optical path length of transmission. The imparted phase delay,\nΔϕ, is expressed by the equation Δϕ= 2π/λ ⋅(n − n )⋅h, where λ is the wavelength, h is the\neff 0\nphysical height of the structure, n is the refractive index of the surrounding medium, and n is\n0 eff\nthe effective refractive index in the phase modulation region. Conventional DOE targets create\nphase delays by varying the physical height h at a fixed refractive index. In contrast, our target\nand most metasurfaces achieve the exact same phase delay effect by varying the effective\nrefractive index n by adjusting subwavelength nanostructure geometry while keeping h\neff\nconstant. In both cases, the transmitted wave experiences the same phase shift, and the phase\nsteps produced by both methods are indistinguishable for the purposes of QPI. The principle of\nusing subwavelength structures to engineer an effective refractive index and thereby modulating\nlocal phase delay is a standard, well-validated approach in nanophotonics and has been widely\nemployed in the field [ref. 13 and 14]. In other words, our target implements the same “uniform\nand known height steps” in phase domain, just using effective index engineering with\nmetasurfaces and nanofabrication rather than step-height engineering with DOEs and\nmicrofabrication.\nFinally, custom-fabricated targets are a common and valid approach for QPI validation as long\nas the target phase is known and verified [ref. 26]. Our in-house fabrication process allows us to\ndesign and fabricate high-precision targets matched to our system's resolution, avoiding long\nlead times and limited customization of commercial parts. Supplementary Figures 15 and 16\nconfirm fabrication quality and experimentally validate the “known” dimensions and phase level\nwith SEM measurements. Figure 5 and Supplementary Figure 24 then show excellent\nagreement between reconstructed phase and the known, designed phase profiles. We believe\nthis validation is physically sound and consistent with reliable approaches in the field. In this\nrevision, we have also expanded the manuscript text to provide additional implementation\ndetails.\n---\nAgain, we appreciate the constructive feedback of all reviewers and have revised our\nmanuscript and supplement to clarify all of these comments.\nThank you very much for your consideration,\nThe authors"}, {"round": 3, "reviewer_comments": {"reviewer_3": "The authors have provided detailed responses addressing each concern and have committed to expanding the supplementary material with balanced comparisons and additional technical details. As a result, I recommend its publication in NC.\n(Remarks on code availability)"}, "author_response": ""}]

Decision letter