Phase retrieval is concerned with the problem of finding the phases of a complex function from its absolute value. The square of the absolute value of a function may be considered as an energy density. Thus phase retrieval can be considered also as the problem of obtaining a field function from an energy density. Usually, in the one-dimensional case, one uses the method of Hilbert transforms on the logarithm of the Fourier transform. However, use of the Hilbert transform requires knowledge of the position of the zeros of the Fourier transform. From an operational point of view it is difficult to obtain the zeros in the complex plane. Moreover, the phase may be unstable with respect to errors in the position of the zeros. The present talk will concentrate on the problem of obtaining phases from energy densities only. A theorem will be given which shows that knowledge of the density in physical space (i.e., x-space) and a certain type of energy in frequency space is sufficient to determine the phase uniquely within a constant. One can also introduce energies in spaces other than in frequency space. An example is the energy associated with the Mellin transform. It is easy to give a meaning to energy density in frequency space as the energy associated with the frequency after the wave goes through a grating. The grating Fourier transforms the wave form. Other spaces, as the Mellin transform space, would need to have a physical interpretation analogous to that for the Fourier transform. Such an interpretation is given for the Mellin transform. Finally we consider generalization to electromagnetic fields in two and three dimensions. Here the phase retrieval problem is replaced by the problem of finding autocorrelation functions from suitable energy densities. A uniqueness theorem is given.
Harry E. Moses, Harry E. Moses,
"Conditions For The Uniqueness Of Phases And Correlation Functions In Wave Propagation", Proc. SPIE 0413, Inverse Optics I, (22 September 1983); doi: 10.1117/12.935855; https://doi.org/10.1117/12.935855