On the solution of the phase retrieval problem

It is shown that the intensity in the image plane of a microscope determines uniquely the phase of the corresponding image wavefunction up to an over-all phase. This result is obtained using the a priori information that both the image wavefunction and the unperturbed wavefunction in the Fraunhofer plane are band-limited and that we have some a priori knowledge about the intensity at the rim of the diaphragm in the Fraunhofer plane. If we have no useful a priori information about the wavefunction in the Fraunhofer plane, unique phase reconstruction is possible from two exposures, corresponding to two different values of the defocusing.