Fig. 1. Principle of KKSAI. (a) Schematic of experimental setup, where pupil modulation is achieved by an SLM-based module. (b) Simplified 4f system corresponding to (a). (c) Simulated complex-valued sample. (d) Amplitude pupil modulation indicated by the green circle, whose center is (ui,vi). (e) Measured images corresponding to (d). (f) KKSAI reconstruction algorithm flowchart. It finally recovers the pupil-limited sample spectrum. MMF, multimode fiber; CL, collimating lens; RL, relay lens; P, polarizer; BS, beam splitter; SLM, spatial light modulator; TL, tube lens.

Fig. 2. Analogy between KKSAI measurement and off-axis hologram. (a) Measurement I1 between its corresponding complex-valued spectrum subregion and the amplitude of its FT. (b) Shifted spectrum subregion still brings in the same measurement due to the frequency shifting property of FT and phase loss of the square-law detector. (c) The shifted spectrum subregion can be hypothetically decomposed into a Dirac delta function and the shifted scattered complex-valued function.

Fig. 3. Titchmarsh theorem applied to a band-limited signal. (a) Amplitude and (b) phase of s˜(l→) with bandwidth of ρNAape. (c) Logarithm of its 2D Fourier amplitude spectrum. (d) Logarithm of its 1D Fourier amplitude spectrum along the l∥ axis and its shifted copies by (e) |ρ→r|<ρNAape and (f) |ρ→r|=ρNAape.

Fig. 4. Scanning scheme examples to cover the entire pupil. (a) Four circular apertures; (b) two rectangular apertures. The circled numbers are used to label the measurement sequence.

Fig. 5. Reconstruction of phase-only samples by two existing methods and our proposed method. (a) Weak phase sample; (b) strong phase sample.

Fig. 6. Reconstruction of complex-valued sample by two existing methods and our proposed method. (a) Phase; (b) amplitude.

Fig. 7. Effect of distance between aperture edge and pupil center on the final reconstruction accuracy.

Fig. 8. KKSAI based on the scanning scheme with only two measurements. (a) Measurements; (b) reconstruction.

Fig. 9. Experimental results for a microlens array. (a) Reconstructions of a single microlens by PM-DPC, PM-FPM, and KKSAI using four measurements. (b) A close-up view of the SEM image of the microlens array (adapted from Thorlabs website). (c) Radial average profile of three phase recoveries compared with the ground truth (GT).

Fig. 10. Experimental results for a thyroid carcinoma pap smear slide. (a) Two out of four measurements acquired by KKSAI and their Fourier amplitude spectrum. (b) Amplitude reconstruction by PM-FPM and KKSAI compared with ground truth. (c) Phase reconstruction by PM-DPC, PM-FPM, and KKSAI compared with ground truth.

Fig. 11. Chromatic aberration correction by digital refocusing ability of KKSAI. (a) Reconstructed amplitudes by KKSAI of three channels. (b) Reconstructed phases by KKSAI of three channels. (c) Digitally refocused amplitudes with the corresponding refocusing distance labeled at the bottom. (d) Color composite of three channels before and after digital refocusing with the enlargements showing improved image quality. R, red (638 nm); G, green (532 nm); B, blue (405 nm).

Fig. 12. Complex wave-field reconstruction by KKSAI based on only two measurements. (a) Scanning scheme, raw measurements, and their spectrum amplitude; (b) and (c) reconstructed amplitude and phase, respectively, by KKSAI from two measurements, four measurements, and PM-FPM with 47 measurements. Here PM-FPM reconstruction is taken as the reference to calculate the FSIM metric for KKSAI reconstruction.