Coherent diffraction imaging (CDI), a lensless imaging technique, is mainly developed to reconstruct the complex amplitude of objects with simple optical paths. Combining CDI and Ptychography, Rodenburg proposed a Ptychographic Iterative Engine (PIE) in 2004, which scans the sample through a localized probe light to a raster of positions and records all diffraction intensities formed in a far field. The complex amplitude of the object under observation can be reconstructed accurately and promptly with proper overlapping between two neighboring illuminated regions. As a powerful phase retrieval method, the PIE algorithm has been widely applied in bioimaging, wavefront diagnosis, and optical elements measurement. Presently, the PIE has been successfully realized with a high energy electron beam, Xray, visible light, and terahertz wave.

Applying the multislice theory of electron microscopy, threedimensional (3D) imaging can also be realized with the PIE by regarding a 3D object as a series of 2D infinitely thin layers. In comparison with traditional 3D imaging methods such as optical coherent and magnetic resonant tomography, which generates intensity images, the 3D PIE (3PIE) can provide a high quality 3D phase image for a transparent volume object rapidly. A singleshot 3PIE was also realized by recording a subdiffraction patterns array with one detector exposure, making 3D phase imaging for dynamic imaging possible.

The coherence of synchrotron radiations is not as ideal as that of common laser beams, and a clear reconstruction was not available for the common GS algorithm, Fienup’s algorithms, standard PIE algorithm, and coherent modulation imaging (CMI) algorithm in most of cases. By treating the recorded incoherent intensity I(x, y) as the summation of several coherent diffraction patterns of different wavelengths as I0(x,y)=∑n I ( λ n, x, y ), the light field of each wavelength can be reconstructed separately using the synchronized constrain ∑n I '( λ n, x, y ) / I0 ( x, y ) · I '( λ n, x, y ). This is the principle of multimode PIE (mmPIE). It has been proved experimentally that this multimode PIE can achieve satisfying reconstruction with temporally or spatially incoherent illumination. Several studies on the influence of partial coherence on the reconstruction of CDI have been conducted, and various physical and numerical approaches have been proposed to improve the quality of reconstructed images with partially incoherent illumination.

The original PIE algorithm requires a known illumination, and this makes the use of PIE much difficult because the distribution of illumination cannot be accurately measured in majority of cases. In parallel PIE and nonlinear optimization algorithms, both profiles of the target object and probe light can be reconstructed in the absence of prior accurate information on illumination, and a similar strategy based on an extended PIE (ePIE) was proposed by Andy Madden to remarkably improve the convergence speed and robustness to noise. Because of the ability to retrieve illumination and objects simultaneously, ePIE has become the mainstream CDI algorithm.

CDI methods including PIE, CMI, and ePIE are all iterative phase retrieval algorithms, and no formula exists to compute the object under inspection directly from recorded intensity in previous decades. The underlying mathematical mechanisms and the existence of a unique solution PIE are problematic and controversial. This makes the error analysis on their reconstructed images impossible and hinders the application of CDI including PIEs optical measurement and metrology fields, where mathematical uniqueness and error analysis are crucial.

To investigate the underlying physics and mathematics of the CDI technique, a set of linear mathematical modes was set up and an efficient computing method to obtain analytical solutions was proposed. The diffraction intensities were written as a series of linear equations where the sample or illumination spectra were unknown, and the spatial components of the sample and illumination can be analytically determined by solving this linear equation set. The underlying mathematical mode and the existence of the unique solution for mmPIE, 3PIE, and ePIE algorithms were illustrated for the first time in this study. Furthermore, the influences of experimental factors such as detecting noise and positioning error were considered and the robustness to the noise of the proposed method was also testified. The studies have laid a physical and mathematical foundation for CDI as a measurement instrument and provided a guiding error analysis theory.

CDI technique as an important phase retrieval tool that can achieve high resolution imaging without imaging elements. The mathematical analytic solutions theory makes the diffraction imaging technique more universal and provides theoretical basis in the field of measurement and metrology. In addition, the CDI technique can be applied for beam measurement, wavefront diagnosis of highpower laser devices, thermal distortion measurement, stress measurement, and damage measurement of large aperture optical components, promoting the rapid development of various cuttingedge studies.