On the uniqueness of the phase retrieval problem from far field amplitude-only data

The phase retrieval problem has gained considerable attention since the wave phase is lost or impractical to measure in several applications in Electromagnetic Engineering. Quite a few approaches have been proposed recently for the extraction of phase from amplitude-only data; however few of them investigate the uniqueness of the problem prior to the solution. In this communication, the problem of the uniqueness of the phase retrieval problem from far field amplitude-only data is examined in detail. It is shown that uniqueness is guaranteed if two experiments can be performed that differ by a quadratic term in the phase. Conversely, experiments creating only linear phase differences in the data do not guarantee uniqueness. This result is demonstrated by considering the problems of a scattering target illuminated in two experiments differing by a quadratic phase shift, corresponding to two different values of defocusing of the incident wave.