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Although originally developed for coherent paraxial scalar electromagnetic radiation in the visible-light regime, phase retrieval using the transport-of-intensity equation has been successfully applied to a range of paraxial radiation and matter-wave fields. Such applications include using electron wave fields to quantitatively image magnetic skyrmions and spin ices, propagation-based phase-contrast imaging using cold neutrons and hard x-rays, and visible-light refractive imaging of the projected column density of cold-atom clouds. Teague's method for phase retrieval using the transport-of-intensity equation, which renders the phase of a paraxial complex wave indirectly measurable via the existence of a conserved current, has been applied to a broad variety of situations which include all of the experiments described above. However, these applications have been undertaken without a thorough analysis of the underlying validity of the method. Here we derive sufficient conditions for the phase-retrieval solution provided by Teague's method to coincide with the true phase of the paraxial radiation or matter-wave field. We also present a sufficient condition guaranteeing that the discrepancy between the true phase function and that reconstructed using Teague's solution is small. These conditions demonstrate that, in most practical cases, for phase-amplitude retrieval using the transport-of-intensity equation, the Teague solution is very close to the exact solution. However, we also describe a counter example in the context of phase-amplitude retrieval using hard x-rays, in which the relative root-mean-square difference between the exact solution and that obtained using Teague's method is 9%. These findings clarify the foundations of one of the most widely applied methods for propagation-based phase retrieval of both paraxial matter and radiation wave fields and define a region for its applicability. |
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