Phase retrieval in TEM using Fresnel images

As an alternative to sideband holography in an electron microscope, methods for phase recovery by in-line holography utilising Fresnel images of aperiodic objects were tested with computed simulations and experimental data. Phases were recovered by minimising an error functional defined as a measure of the differences between experimental and calculated image intensities. The probability of convergence to local minima of the error function was reduced by increasing the ratio of known to unknown parameters, partly by use of several Fresnel images at different defoci, and also by an incremental relaxation of the phase bandwidth. Iterative methods did not converge reliably to a global minimum, but a conjugate gradient algorithm usually recovered the phases exactly, even for object arrays which included large phase variations. In practice, it was essential to use analytic expressions for the error gradients with respect to the phases, defoci and beam direction. Phase shifts of several radians were measured near the edge of a contaminated aperture. The factors that limit the accuracy and reliability of phase recovery from Fresnel images are discussed.