Fig. 1. FPM-QPI has evolved into three branches stimulated by advances in the forward imaging model and the phase retrieval algorithm. Schematic of the three main branches: phase retrieval algorithm, intensity diffraction tomography, and deep learning are indicated. The progress of the main events is listed in the timeline. The citations are showcased as follows: Phase retrieval algorithm: sequential Gerchberg–Saxton^[39,40], adaptive system correction^[49], embedded pupil function recovery (EPRY)^[50], multiplexed coded illumination^[51], Wirtinger flow^[63], differential phase contrast (DPC) initialization^[69], adaptive step size^[52], sparse regularization^[67,68], deep learning^[86,91], field-dependent pupil recovery^[54], momentum acceleration^[53], feature domain^[63], and spatially-coded^[73]. Intensity diffraction tomography: multi-slice beam propagation (MSBP)^[81], Fourier ptychographic diffraction tomography (FPDT)^[75], high-resolution MSBP^[82], multi-layer Born^[83], high-resolution FPDT^[76], accelerated FPDT^[77], transport-of-intensity FPDT^[80], polarization-sensitive MSBP^[84], and isotropic-resolution FPDT^[79]. Artificial intelligence: PtychNet^[85], conditional generative adversarial network (cGAN)^[86], automatic differentiation^[97], classification^[102], visual staining^[103], sparse FPM-PINN^[93], Fourier ptychography multiparameter neural network (FPMN)^[98], FPM using untrained deep neural network priors (FPMUP)^[91], adaptive coded illumination^[99], and visual restaining^[104].

Fig. 2. Forward imaging model of FPM-QPI for thin samples. (a) Schematic diagram of system configuration. (b) Color high-resolution FPM intensity image of a thin blood smear. (c) The complex transmittance function consists of amplitude and phase at 630 nm wavelength. (d) Low-pass filtered spectrum at the pupil plane. (e) Captured intensity image on the image sensor.

Fig. 3. Iterative phase retrieval process of FPM-QPI.

Fig. 4. Forward imaging model of 3D FPM-QPI. (a) Forward imaging model of Fourier ptychographic diffraction tomography, where the scattering potential is used to describe the 3D sample, which is a function of refraction index (RI). (b) Forward imaging model of multi-slice beam propagation, where RI is used to describe the sample that is discretized as multiple infinitely thin slices with equivalent intervals.

Fig. 5. Modeling strategies for three categories of AI-based FPM-QPI methods. (a) End-to-end deep learning. (b) Physics-based neural network. (c) Physics-based automatic differentiation.

Fig. 6. Label-free cell monitoring and analysis with FPM-QPI. (a) Recovered phase images of unlabeled HeLa cells via the high-speed in vitro FPM implementation. The time-lapse phase images can be used to track mitosis of unstained cells and perform automated segmentation and dry mass quantification. (b) Long-term and high-speed live-cell monitoring across 51 h with adaptive optical FPM implementation, in which the spatially and temporally varying aberrations can be effectively corrected. The line plot indicates the focus drift during the process. (c) High-resolution Fourier ptychographic diffraction tomography using both brightfield and darkfield illuminations, in which the 3D refractive index (RI) of a large HeLa cell population is recovered from intensity measurements. 390 nm lateral resolution and 899 nm axial resolution were achieved across an FOV of 1.77mm2. (a) adapted with permission from Ref. [69], © Optical Society of America. (b) adapted from Ref. [55], CC-BY 4.0. (c) adapted from Ref. [76], CC-BY 4.0.

Fig. 7. Digital pathology and clinical diagnosis with FPM-QPI. (a) Large-field complex amplitude images recovered from a stained pathology slide. The recovered phase image shows the phase delays induced by the sample, and the reduced scattering map quantifies how much light has been scattered by the sample, revealing the microscale heterogeneity^[106]. (b) Virtual IHC staining images by utilizing the FPM reconstruction as the input of a network, in which the network outputs show high color fidelity when compared with the measurements of a 0.75 NA objective^[103]. (c) High-resolution phase image of an unstained renal tissue slide, where the morphologies of glomerulus and vasal sections can be observed clearly, showing the potential of FPM-QPI in characterizing the morphology of unstained pathology slides^[107]. (d) Hemozoin detection in a malaria-infected red blood cell (iRBC) using a conventional microscope (40×/0.65 NA) and polarization-sensitive FPM. In the conventional microscope image, hemozoin crystals in the schizont stage iRBC were visualized in dark brown through Wright–Giemsa staining. In PS-FPM, the raw image (4×/0.1 NA) could not resolve hemozoin crystals. After reconstruction, the birefringence signatures (0.51 NA) could be detected, where θ and δ denote the optical-axis retardation and the phase retardation, respectively. By overlaying birefringence information on the recovered phase image, structural information of normal red blood cells and the location of hemozoin crystals could be obtained simultaneously. (a) adapted with permission from Ref. [106], Elsevier: Computerized Medical Imaging and Graphics © 2015. (b) adapted with permission from Ref. [103], © Optical Society of America. (c) adapted from Ref. [107], CC-BY 4.0. (d) adapted from permission from Ref. [112], Copyright 2021, American Chemical Society.

Fig. 8. Topography measurement with FPM-QPI. (a) Topography reconstruction of a high-resolution phase-type sample with reflective Fourier ptychographic topography (FPT). The height profiles are compared with the measurement of an optical profilometer, demonstrating the effectiveness of FPT^[117]. (b) Statistics of 731 flaws identified on the reconstructed phase map of a glass surface. The color scheme represents the number of flaws in squares of size 500μm×500μm, and the ellipses denote the average orientation and eccentricity of these flaws^[126]. (c) FPM reconstruction of a 1.5 nm defect on an EUV photolithography mask, illustrating the exquisite sensitivity to small phase defects^[123]. (d) 3D topography map of a 1.804 µm step structure using dual-wavelength FPM^[120]. (e) Recovered phase image of an application-specific integrated circuit via X-ray FPM. The resolution is estimated to be 47 nm via the 1-bit criterion of Fourier ring correlation (FRC), that is, the intersection between the red curve and the dashed line. It is better than the Rayleigh resolution limit (RRL), marked by the dotted line^[125]. (f) Comparison of the measured and designed unwrapped phase images of a 0.59 NA focusing metalens and a focusing vortex metalens, where the phase error map reveals the fabrication errors^[127]. (a) adapted with permission from Ref. [117], © Optical Society of America. (b) adapted with permission from Ref. [126], © AIP Publishing. (c) adapted with permission from Ref. [123], © 2018 SPIE. (d) adapted with permission from Ref. [120], © Optical Society of America. (e) adapted from Ref. [125], CC-BY 4.0. (f) adapted with permission from Ref. [127], © Optical Society of America.