Phase retrieval of radiated fields

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Abstract
The problem of determining radiated electromagnetic fields from phaseless distributions on one or more surfaces surrounding the source is considered. We first examine the theoretical aspects and basic points of an appropriate formulation and show the advantage of tackling the problem as the inversion of the quadratic operator, which, by acting on the real and imaginary parts of the field, provides square amplitude distributions. Next, useful properties and representations of both fields and square amplitude distributions are introduced, thus making it possible to come to a convenient finite-dimensional model of the problem, to recognize its ill-posed nature and, finally, to define an appropriate generalized solution. Novel uniqueness conditions for the solution of the problem and questions regarding the attainment of the generalized solution are discussed. The geometrical properties of the functional set corresponding to the range of the quadratic operator relating the unknowns to the data are examined. The question of avoiding local minima problems in the search for the generalized solution is carefully discussed and the crucial role of the ratio between the dimension of the data representation space and that of the unknowns is emphasized.