Fast deterministic single-exposure coherent diffractive imaging at sub-Ångström resolution

In coherent diffractive imaging (CDI), conventional image-forming optics are replaced by the solution of an inverse scattering problem which obtains the exit surface wave from the diffraction pattern. To date CDI implementations typically use nonlinear iterative solutions, often using more than one diffraction pattern, and have been mainly implemented in the optical and x-ray regimes. Here we present single-exposure CDI reconstructions of a region of a cerium dioxide nanocrystal, illuminated by a coherent electron probe, in which the positions of both cerium and the lighter oxygen atoms are recovered simultaneously. We employ a fast deterministic algorithm based on the iterative solution of a set of linear equations with input data obtained from the inverse Fourier transform of the diffraction pattern. The solution of the linear equations proceeds using a conjugate gradient least-squares method based on fast Fourier transforms. This allows regularization of the inversion problem by iteration number, making it robust to experimental noise. This approach is successful where standard nonlinear phase retrieval techniques for the same data set are not.