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Phase contrast imaging seeks to reconstruct the complex refractive index ofan unknown sample from scattering intensities, measured for example underillumination with coherent X-rays. By incorporating refraction, this methodyields improved contrast compared to purely absorption-based radiography butinvolves a phase retrieval problem which, in general, allows for ambiguousreconstructions. In this paper, we show uniqueness of propagation-based phasecontrast imaging for compactly supported objects in the near field regime,based on a description by the projection- and paraxial approximations. In thissetting, propagation is governed by the Fresnel propagator and the unscatteredpart of the illumination function provides a known reference wave at thedetector which facilitates phase reconstruction. The uniqueness theorem isderived using the theory of entire functions. Unlike previous results based onexact solution formulae, it is valid for arbitrary complex objects and requiresintensity measurements only at a single detector distance and illumination wavelength. We also deduce a uniqueness criterion for phase contrast tomography,which may be applied to resolve the three-dimensional structure of micro- andnano-scale samples. Moreover, our results may have some significance toelectronic imaging methods due to the equivalence of paraxial wave propagationand Schr\"odinger's equation.

A uniqueness result for propagation-based phase contrast imaging from a single measurement