Fig. 1. Working principle and schematic system design of (a) Phase-shifting interferometry-based transmission-mode iQPM; (b) Off-axis interferometry-based transmission-mode iQPM; (c) Linnik interferometry-based epi-mode iQPM

Fig. 2. (a) Time series of quantitative phase images and the temporal phase noise of the specified point; (b) The quantitative phase image and the spatial phase noise along the red line

Fig. 3. The noise power spectra density in phase measurement

Fig. 4. Sketch of the raw interferogram and the intensity profile distribution relative to the full electron well depth of the camera

Fig. 5. Noise reduction strategies with common-path interference. (a)-(b) The averaged frequency spectra (in logarithmic scale) of the phase maps measured with common-path and non-common-path QPM systems, respectively^[23] (figures adapted from reference [23]). Selected bandwidth: 15-30 Hz and 185-200 Hz. BS-beam splitter, DG-diffraction grating, and PH-pinhole; (c) The schematic design of interferometric imaging part of MISS microscopy

Fig. 6. The schematic of spatiotemporal filtering method. (a) Spatiotemporal spectrum along three different planes (k_x-k_y，k_x-f，k_y-f) in the k_x-k_y-f 3D frequency domain, where k_x and k_y stand for spatial frequencies in the x- and y-directions, respectively, and f axis stands for the temporal frequency in the t direction, color map is in the log scale; (b) Bandpass filtering over the selected spatiotemporal bands, where two temporal bandwidth regions of 15-55 Hz and 180-220 Hz and three spatial bandwidth regions with radii of 2π, π and π/2 are selected, respectively^[23]

Fig. 7. The schematic of frame summing method (a) Flow chart of the frame summing algorithm; (b) Histogram of the intensity distribution of a raw interferogram; (c) and (d) show the relations between the temporal phase sensitivity and the effective well capacity and frame numbers, respectively^[23] (figures adapted from reference [23])

Fig. 8. (a) Principle of dynamic range expansion in ADRIFT-QPI; (b) PD images measured by DH (left) and ADRIFT-DH (right) in the MIR OFF state; (c) Images of the photothermal OPD changes due to absorption of the MIR pump light measured by DH (left) and ADRIFT-DH (right) ^[25]

Fig. 9. Speckle noises whose illuminations are (a) spatiotemporally coherent, (b) temporally coherent and spatially low coherent, (c) spatially coherent and temporally low coherent, and (d) spatiotemporally low coherent; (e) σ_Φ against l_s with l_t = 63.3 μm and (f) is σ_Φ against l_t with l_s = 11.8 μm^[27] (figures adapted from reference [27])

Fig. 10. (a) Schematic of the SLIM add-on module integrated with a commercial microscope. The system employs white light illumination (central wavelength 552.3 nm) with ultra-short coherence length. A SLM is placed at the pupil plane to generate phase shifts in π/2 increments, and phase reconstruction can be realized by recording images under different phase shifts; (b) The OPD maps of the sample-free region obtained by (b) SLIM and (c) DPM reconstruction. Benefitting from high stability of common-channel interferometry, and the reduction of speckle noise with white light illumination, the phase image retrieved by SLIM has the superior spatial uniformity and measurement precision, where colorbar in nanometers; (d) OPD noise levels measured spatially and temporally, which are 0.3 nm and 0.03 nm, respectively. The solid lines indicate Gaussian fitting^[28]; (e) In P-DHM, the traditional coherent illumination module is replaced by a SWLL combined with an AOTF; (f) The average OPD curves for the control group (36 holograms under 540 nm illumination) and the P-DHM method (36 holograms under 500-850 nm multi-wavelength illumination) are shown in black, while the OPD curves of single-frame images used for frame averaging are shown in colors corresponding to the respective wavelengths; (g) The relationship between and the averaged frame number N in the control (green) and P-DHM (orange) methods^[31]. The black line is the theoretical curve, where A is value of the first data point

Fig. 11. (a) Schematic design of high-speed angle-scanning illumination based on dual DMD devices in HISTR-SAPM; (b)-(c) Phase maps of a COS-7 and a HeLa cell under normal illumination; (d)-(e) Phase maps reconstructed with HISTR-SAPM for the cells in (a), (d)^[32] (figures adapted from reference [32])

Fig. 12. (a) Dynamics monitoring of human red blood cells by high-sensitivity iQPM^[23] （Scale bar: 10 µm）; (b) Representative phase images for monocytes (red), granulocytes (green), B lymphocytes (blue), and T lymphocytes (orange); (c) t-SNE visualization of the feature extracted by the monocyte–granulocyte–lymphocyte classifier and the B-T lymphocyte classifier^[35]

Fig. 13. (a) The inspection of the cellular deformation across a neuron^[37] (Scale bar: 20 μm); (b) Comparison between the single-shot image acquired with 540 nm illumination, control image, and P-DHM image^[31] (Scale bar: 10 μm)

Fig. 14. (a)-(b) The thickness mapping results of a monolayer MoS₂ sample by AFM and TM-QPP, respectively (scale bars: 2 μm); (c) The histogram of the geometric thicknesses map shown in (b); (d) and (e) The thickness mapping results of a multilayer WSe₂ sample by AFM and TM-QPP, respectively (scale bars: 10 μm); (f) The geometric thickness values for ii-iv regions can be calculated, and the number of layers for each region can be determined by comparing the thickness values with the reference values (dashed lines). The error bars indicate the standard deviation values of the geometric thickness of the corresponding regions^[38] (figures adapted from reference [38])

Fig. 15. (a) 22 nm node wafer defect inspection^[39]; (b) 9 nm node wafer defect inspection^[40]

Fig. 16. The roadmap of phase sensitivity improvement iniQPM^{[21−24, 37, 41-42]}