On the uniqueness of planar near-field phaseless antenna measurements based on two amplitude-only measurements

The promising application of phaseless measurement techniques has gained considerable attention among researchers. These techniques offer the hardware convenience along with the opportunity of operation at higher frequencies for the near-field antenna measurements. There are a number of interesting solutions for the problem of phase extraction from the amplitude-only data in the literature, but a very important question to the best knowledge of the authors is left unanswered. Provided the fact that one is dealing with the modulus of the near-field can one be assured of the existence of one and only one corresponding phase distribution. The near-field phase distribution can be formulated based on the near-field amplitude distribution through certain non-linear equations. Based on the type of measurement system (planar, cylindrical and spherical) this nonlinearity may have different order and form. Nonetheless they all raise a simple question and that is whether there is a unique solution, because in general non-linear equations are prone to having more than one solution. Hopefully in the case of antenna measurements there are some limiting factors including the analyticity, band-limitedness or, some extra information such as the geometry of the radiator, amplitude measurements over another surface, etc. These facts increase the possibility of having a unique solution. This paper is focused on a proof of the uniqueness for the case where two sets of near-field magnitude are available. At first an overview of the source of ambiguity is presented for one-dimensional signals (one set of intensities). Then the extension of the theory for the two-dimensional signals (one set of intensities) is presented. Subsequently, the problem of two-plane intensities (two set of intensities) is discussed for the one-dimensional signals. The complexity of the kernel linking the two planes is discussed in microwave and millimeter region appropriate for antenna measurements. Finally the uniqueness is proved when one has two sets of one-dimensional intensity function in the near-field.