Fig. 1. Working principle of the PIC-based QPI technique. (a) Schematic of the chip-based illumination configuration for quantitative phase imaging. Straight purple arrow, incident beam; wavy light purple arrows, scattered field; top left image, FDTD simulation of the intensity image of the object illuminated with an oblique illumination compatible with the phase retrieval approach based on the Kramer–Kronig relations; top right image, retrieved phase image via numerical post-processing of the top left image. (b) Illustration of the KK-based QPI technique in k-space. Purple disk, aperture of the microscope objective with kmax the maximum modulus of the transmitted transverse component of the wave vectors; left, standard normal illumination, with incident transverse k-vector kinc=0; right, oblique illumination with kinc=kmax, i.e., compatible with the KK relations. (c) Log-scale Fourier domain of the retrieved field image after merging the frequency bands of the four directions of illumination di=1 to 4. (d) Amplitude and phase images obtained by inverse Fourier transform of (c). (e) Schematic of the photonic integrated circuit used in (a) including a cross section of the aluminum oxide waveguide (green). The diffraction gratings provide the oblique illuminations that are switched on (current Ion) and off (current Ioff) with an integrated 1×4 switch made of 1×2 or 2×2 multimode interferometer (MMI) and thermal phase shifters (yellow). Inset: optical image of the entire photonic chip bounded on a PCB board and electrically connected via gold wires.

Fig. 2. High-precision control methodology of the beam angle diffracted by the photonic chip. (a) Optical image of a typical photonic chip including six PICs. Zoomed image, atomic force microscopy (AFM) image of one of the shallow etched grating; etch depth, he=18.1±0.7nm; grating pitch, Λ=183  nm; fill factor, ff = 0.8. (b) Optical image of the back focal plane of the microscope objective when all gratings are excited. The clear aperture of the exit pupil is image with a light emitting diode operating at a central wavelength of 370 nm. (c) Optical image of the back focal plane of the microscope objective as in (b) but with only the d1 oblique illumination. In the zoom-in image δNA=n·cos(θmax)·δθ indicates the divergence of the beam in unit of NA where θmax is the maximum aperture angle. In (b) and (c), the lower right schematics show the operating state of the integrated optical switch. (d) Relationship between the NA mismatch ratio (NAex-NAco)/NAco and the retrieved phase error |1−hretrieved/hGT| simulated with the FDTD method (red). The experimental data points (black) are obtained with the PICs presented in (a). (e), (f) Topography of a test object imaged by AFM and by the proposed PIC-based QPI technique, respectively. At the bottom of (f), histogram of the height distribution extracted from the QPI image.

Fig. 3. High spatial resolution and low spatial phase noise. (a) PIC-based QPI images of strips patterns etched in the same glass substrate with spacings of 600 nm, 400 nm, and 300 nm. (b) Cross section of the topological profiles along the dashed black lines in (a). (c) PIC-based QPI image of the surface of a SiOx layer deposited on a glass substrate. The standard deviation of the phase noise σφexp=6.3  mrad is determined over the whole image. (d) Left: histogram plots of the fluctuations of the SiOx surface height measured with AFM and of the associated phase simulated with FDTD. Top right: topographic AFM image. Bottom right: KK-based phase image simulated with FDTD and produced by a surface whose roughness properties are identical to those of the measured AFM image. The root-mean-square (RMS) of the surface roughness σAFM=0.4  nm and the standard deviation of the phase fluctuations σφsim=3.0  mrad.

Fig. 4. Quantitative phase imaging of monolayer graphene. (a) Topographic image of a monolayer graphene patch measured with the PIC-based QPI (pixel size 100 nm). (b) Cross section along the dashed black line in (a). (c) Topographic image of the monolayer graphene patch measured by AFM (pixel size 78 nm). (d) Cross section along the dashed black line in (c). The cross section profiles are averaged over a vertical distance of 1.5 μm in both cases.

Fig. 5. Quantitative phase imaging of bacteria cells. (a) Phase image of Escherichia coli bacteria cells on a microscope cover glass. (b), (c) Zoom-in images of the area located in the dashed orange boxes showing the dividing cells and an individual cell, respectively. (d) 3D visualization of (a).

Fig. 6. Process flow of the algorithm to retrieve phase images by applying the Kramers–Kronig relations to intensity images.

Fig. 7. Robust phase retrieval based on KK relations. (a) Log-scale Fourier domain of the field image of a square pillar object with an illumination orientation along d1. (b) Phase image of a 50-nm-thick square pillar on cover glass, simulated using FDTD method. (c) Retrieved phase image of the object in (b) based on KK relations. (d) Cross section along the dashed lines in (b) and (c), respectively.

Fig. 8. Impact of the NA divergence on the retrieved topography. (a) Retrieved height images of a square pillar in the cases without beam divergence and with beam divergences δNA of 0.007 and 0.015, respectively. (b) Cross section along the solid lines in height images in (a). An offset of 5 nm along the y axis is set between the curves for better visualization. (c) Relationship between the height ratio of hretrieved/hGT and beam divergence, where hretrieved is the retrieved height and hGT is the ground truth simulated with FDTD method. The error bars correspond to the distribution of the ripple fluctuations observed in (b).

Fig. 9. Impact of depolarization on the retrieved topography. (a) Simulated intensity image of a square pillar obtained by using a linear polarizer along the y axis. The illumination beam is oriented along d1, while its polarization is oriented along the y axis. (b) Simulated intensity image obtained with a linear polarizer along the x axis. (c), (d) Retrieved height images using intensity images acquired with and without a linear polarizer, respectively. (e) Height profiles along the solid lines in (c) and (d), respectively. A 5 nm offset is set for better visualization.

Fig. 10. On-chip beam switching. (a) Relationship between the intensity at outputs and the length of 2×2 multi-mode interferometer when a light source excites the fundamental transverse electric mode in one of the inputs. (b) Simulated light field propagating in the optimized MMI with one of the inputs being excited. (c) Heat transfer map of the on-chip thermal phase shifter. (d) Optical microscopy image of the 1×2 switch. (e) Optical image of the light propagation in a 1×2 switch when the applied voltages is Vπ and V2π, respectively. The plot shows the intensity profiles along the red and blue solid lines in the image. (f) Relationship between the scattered intensity of the output₁ in (d) and the voltage applied on the phase shifter(c) Heat transfer map of the on-chip thermal phase shifter. (d) Relationship between the scattered intensity of the output1 and the voltage applied on the phase shifter. The black curve shows the scattered intensity of the output1 in Fig. 2(c) in the main text, and the red curve shows the applied voltage.

Fig. 11. Intensity profile of the illumination beam. Intensity image of a typical diffracted beam at the object plane. The intensity profiles along the x and y axes are summed on the top and left side of the image, respectively.

Fig. 12. Image processing steps. (a) Measured optical intensity images illuminated with oblique beams along the orientation d1–d4. (b) Log-scale fast Fourier transform of the intensity images in (a). (c) Log-scale amplitude image of merged light field in Fourier domain. (d) Retrieved amplitude image. (e) Retrieved height image.

Fig. 13. Schematic of the phase delay for oblique illumination. h, height of the object; n1, n2, refractive indices of the environment and object, respectively. The beam that is incident with an angle of θ1 is refracted with an angle θ2 inside the object. Red dashed line: optical path without object.

Fig. 14. Impact of the NA mismatch. (a) Relationship between the grating pitch and the numerical aperture NAex of the diffracted beam for filling factors of 0.6 and 0.8, respectively. (b) Histogram plot of the height profile of a square pillar measured by AFM in Fig. 2(e) in the main text. (c) Height images of the same object in (b), retrieved with PIC-based QPI for different NA mismatches. The NA mismatch is defined as (NAex-NAco)/NAco and given in percentage. (d) Histogram plots corresponding to the images in (c).

Fig. 15. Photograph of the PIC-based QPI setup. PIC, photonic integrated circuit; PCB, printed circuit board. The diameter of the 5-cent Euro is 21.25 mm.