  Phase contrast imaging seeks to reconstruct the complex refractive index of
an unknown sample from scattering intensities, measured for example under
illumination with coherent X-rays. By incorporating refraction, this method
yields improved contrast compared to purely absorption-based radiography but
involves a phase retrieval problem which, in general, allows for ambiguous
reconstructions. In this paper, we show uniqueness of propagation-based phase
contrast imaging for compactly supported objects in the near field regime,
based on a description by the projection- and paraxial approximations. In this
setting, propagation is governed by the Fresnel propagator and the unscattered
part of the illumination function provides a known reference wave at the
detector which facilitates phase reconstruction. The uniqueness theorem is
derived using the theory of entire functions. Unlike previous results based on
exact solution formulae, it is valid for arbitrary complex objects and requires
intensity measurements only at a single detector distance and illumination wave
length. We also deduce a uniqueness criterion for phase contrast tomography,
which may be applied to resolve the three-dimensional structure of micro- and
nano-scale samples. Moreover, our results may have some significance to
electronic imaging methods due to the equivalence of paraxial wave propagation
and Schr\"odinger's equation.
