Fig. 1. Flow chart of G-S algorithm

Fig. 2. Schematic diagram of the optical path of PIE algorithm

Fig. 3. Optical path diagram of CMI algorithm

Fig. 4. Computation process of matrix vectorization in CDI algorithm

Fig. 5. Unique reconstruction of CDI under random illumination. (a) Phase distribution of the original sample; (b) spectrum (in logarithmic scale) of the original sample; (c) the distribution of the random illumination; (d) phase distribution of recovered sample; (e) spectrum of the recovered sample; (f) the difference between the original and recovered samples

Fig. 6. Simulation results of Ptychography algorithm. (a) A common setup for Ptychography; (b) phase of the original sample; (c) illumination used in experiment; (d) one of the produced diffraction patterns; (e) phase of recovered sample from Fig. (d) by solving the linear equations; (f) phase of recovered sample from nine diffraction patterns by the proposed method; (g) phase difference between original and recovered samples

Fig. 7. Simulation results of axial multiple-plane phase retrieval. (a) Experimental schematic of axial multiple-plane phase retrieval; (b) produced four diffraction patterns; (c) phase of recovered sample from first diffraction pattern; (d) phase of recovered sample from four diffraction patterns by solving the linear system; (e) phase difference between original sample and Fig. (d)

Fig. 8. Relationship between rank of transform matrix and convergence ability of iterative phase retrieval algorithm. (a)‒(c) Random illumination, pentagonal illumination, and square illumination, the ranks of corresponding transform matrix are rr,rp,rs,rr>rp>rs; (d)‒(f) phase of recovered sample by ER algorithm; (g) convergence curves

Fig. 9. Generation of diffractive intensity of different pixels on the diffraction patterns

Fig. 10. Convolution computation of objects (left) and illumination of different spatial incoherent modes (right) under the multi-wavelength illumination

Fig. 11. Color imaging optical path based on multi-modes PIE algorithm and color objects and illuminated light. (a) PIE schematic diagram of color imaging; (b) amplitude and phase of the RGB components of colorful objects; (c)‒(h) amplitudes and phases of illuminated light at wavelengths of 632.8, 532, 471 nm, respectively

Fig. 12. Diffraction patterns in colorful imaging optical path and reconstructed objects. (a)‒(c) Diffraction patterns of three different wavelengths; (d) recorded diffraction pattern by CCD; (e) RGB components of the objects reconstructed based on the multi-modal PIE algorithm; (f) reconstructed colorful objects

Fig. 13. Illuminations with three different spatial modes. (a)‒(b) Amplitude and phase of TEM11; (c)‒(d) amplitude and phase of TEM01; (e)‒(f) amplitude and phase of TEM10

Fig. 14. Object and diffraction patterns. (a)‒(b) Amplitude and phase of the object; (c)‒(f) sub-diffraction patterns of TEM11, TEM01 and TEM10 and their incoherent superposition diffraction intensities

Fig. 15. Reconstructed multi-modal illuminations. (a)‒(c) Reconstructed spectral distributions of TEM11, TEM01andTEM10; (d)‒(f) spatial amplitude distributions of reconstructed spectra; (g)‒(i) spatial phase distributions of reconstructed spectra; (j)‒(l) calculated intensity differences

Fig. 16. Optical path schematic diagram of 3PIE

Fig. 17. Amplitude and spectral function of two layers of objects. (a)‒(b) Amplitude functions of two layers of objects; (c)‒(d) spectral distributions of two layers of objects; (e) product of the spectra of the two layers of objects

Fig. 18. Formation of light fields U˜2fm,fn and U˜3fm,fn at different pixels

Fig. 19. Distributions of the object and illumination in the simulation. (a)‒(d) Amplitude and phase information of two layers samples; (e)‒(f) amplitude and phase information of the illumination; (g)‒(i) spectral functions of the two layers of samples and illumination, respectively

Fig. 20. Seven diffraction patterns applied in the calculation

Fig. 21. Reconstruction results of two layers of samples. (a)‒(b) Reconstructed spectra of two layers of samples; (c)‒(d) differences between the reconstructed and the original spectra; (e)‒(f) amplitude distributions of the reconstructed two layers of samples; (h)‒(i) phase distributions of the reconstructed two layers of samples; (g)(j) reconstruction errors of two layers of samples

Fig. 22. Reconstruction results of two layers of samples in the presence of Gaussian noise in the diffraction patterns. (a) Gaussian noise on the detector; (b)‒(h) seven new diffraction patterns; (i) (m) reconstructed spectra of the two layer of samples; (j)‒(k) reconstructed amplitude distributions of the two layers of samples; (n)‒(o) reconstructed phase distributions of the two layers of samples; (l)(p) reconstruction errors

Fig. 23. Reconstruction results of two layers of samples in the presence of Poisson noise in the diffraction patterns. (a) Poisson noise on the detector; (b)‒(h) seven new diffraction patterns; (i)(m) reconstructed spectra of the two layer of samples; (j)‒(k) reconstructed amplitude distributions of the two layers of samples; (n)‒(o) reconstructed phase distributions of the two layers of samples; (l)(p) reconstruction errors

Fig. 24. Generation of diffractive light field U(x,y) in ePIE

Fig. 25. Distributions of object and illumination used in ePIE

Fig. 26. Recorded diffractive light intensity and reconstructed results. (a)‒(g) Seven diffraction patterns in computation; (h)(l) spectra of illumination and object; (i)(j) amplitude and phase information of recovered illumination; (m)(n) amplitude and phase information of recovered object; (k)(o) reconstruction differences for illumination and object

Fig. 27. Effect of reconstruction from the diffractive light intensity images with the noisy signal. (a)(b) Reconstructed spectra of illumination and object; (c)(d) amplitude and phase information of the reconstructed illumination; (e)(f) amplitude and phase information of recovered object

Fig. 28. Hybrid diffraction patterns and reconstructed results using sever hybrid diffraction patterns. (a)‒(g) Seven newly generated diffraction patterns; (h)(l) spectra of reconstructed illumination and object; (i)(j) amplitude and phase information of recovered illumination; (m)(n) amplitude and phase information of recovered object; (k)(o) differences for the reconstructed illumination and object

Fig. 29. Direct reconstruction results in the presence of position errors. (a)(b) Reconstructed spectra of illumination and object;

Fig. 30. Spectra and spatial complex amplitude distributions of illumination and object reconstructed by proposed calculation method under the condition of position error. (a)(b) Reconstructed spectra; (c)(d) errors of the reconstructed spectra; (e)(f) amplitude and phase information of the recovered illumination; (g)(h) amplitude and phase information of the recovered object; (i)(j) errors for the reconstructed illumination light and object