I have followed some of the research on phase retrieval methods of Prof. Rodenburg who works at the University of Sheffield. As he states in his webpage, he is
… currently working on phase-retrieval methods, principally Ptychography at electron, visible light and X-ray wavelengths. I am currently seconded part-time to a spin-out company I helped found called Phase Focus Ltd.
Check for example his paper in Nature Communications where the authors show that it is posible to recover the phase of the electron wave in a SEM in transmission mode (see the picture above). The method uses diffractive imaging and an inverse computation using scattered intensity data to overcome the limitations of magnetic lenses. In the paper they explain that in diffraction mode in scanning mode it is possible to recover amplitude and phase of the electron wave. In this manner it is possible to compensate for the aberrations using iterative algorithms, and to obtain images at atomic resolution (sub-nm) even though the acceleration voltage is 30 kV.
MJ Humphry, B Kraus, AC Hurst, AM Maiden and JM Rodenburg “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging” Nature Communications, 3 2012. doi:10.1038/ncomms1733.
For more articles click here the departamental webpage of J Rodenburg at the University of Sheffield.
Check the concept behind this developments in the next video of Phase Focus