Phase retrieval from image intensities: Why does exit wave restoration using IWFR work so well?

The iterative wave function reconstruction (IWFR) is one type of in-line holography and retrieves a complex wave function from a set of through-focus images. We have verified that the IWFR provides an extremely good estimate of an atomic-resolution exit wave function (EWF) simply from the Fourier transforms of observed intensities. Thus, the first guess of the EWF using only five images gives all the features of the final result, and the convergence of the IWFR is very quick. The IWFR accepts a wide variety of defocus step, and the total defocus span or the defocus step is not essential. The absolute defocus can be estimated by propagating the EWF to the plane where the propagated wave function gives the minimum amplitude variation. Even when there is some error in the spherical aberration coefficient, the EWF suffers from only the aberrations due to the estimation error. The residual error may be adjusted on the reconstructed complex wave function. With the development of a stable microscope it becomes more realistic to routinely record multiple images with good quality, which allows advanced image processing such as the IWFR. Such an exit wave reconstruction is also desirable to investigate a phase object using a Cs-corrected microscope, since the intentionally introduced aberrations to amplify the phase object contrast are desirable to be eliminated by post-processing.