The Question of Phase Retrieval in Optics

It is generally asserted that valuable phase information is irretrievably lost in the squaring operation required to obtain the intensity from the amplitude distribution in the image of a star formed by a lens. In this paper the phase reconstruction is studied under the constraint that the aperture of the lens be finite. It is shown that in some cases the phase reconstruction is unique; and that in other cases there is finite or denumerable infinite number of discrete solutions. Conversely, it is proved that every non-negative function that is integrable and band-limited can be realized as the intensity distribution in the image of a star formed by a lens.