  Promoted by the advent of coherent synchrotron light sources, phase contrast
tomography allows to resolve three-dimensional variations of an unknown
sample's complex refractive index from scattering intensities recorded at
different incident angles of an X-ray beam. By diffractive free-space
propagation of the transmitted wave field, this method is sensitive not only to
absorption but also to refractive phase shifts induced by the specimen,
permitting three-dimensional nanoscale imaging of quasi-transparent samples
such as biological cells. However, the reconstruction of the specimen structure
from the observed data constitutes an algorithmically challenging nonlinear
ill-posed inverse problem, mainly due to the characteristic loss of phase
information in the detection of the wave field.
  In this work, regularized Newton methods are developed for the solution of
this tomographic phase retrieval problem, based on a detailed analysis of its
mathematical structure. We consider both the near-field- or Fresnel regime
characterized by a moderate propagation length between sample and detector and
the far-field limit of large detector distances, where propagation is governed
by the Fourier transform. In the former setting, excellent numerical
reconstructions are obtained via the chosen Newton-type approach, supplemented
by novel theoretical results stating that measurements from a single detector
distance are sufficient to uniquely recover both refraction and absorption of a
sample. The proposed algorithm simultaneously performs tomographic- and phase
reconstruction, which is found to stabilize the latter by exploiting
correlations between the diffraction patterns recorded under different incident
angles.
