The relationship between the Ewald sphere and exit wave explored using focal series electron micrographs

CryoEM reconstructions must be performed along the Ewald spheres to achieve resolutions beyond the projection approximation limit. The linear theory of image formation models the scattering from the specimen and focusing by the objective lens as two conjugate Ewald spheres that correspond to the two halves of the contrast transfer function (CTF). A focal series of micrographs mimics the focal gradient inherent in the Ewald wavefront, and its 3D Fourier transform provides a visual representation of the conjugate Ewald spheres. To better understand the nature of the Ewald spheres, focal series of graphene oxide as a thin specimen and platinum-iridium as a strongly scattering specimen were acquired. An algorithm was developed to fit and correct the CTF of the whole focal series. Correcting for the one half of the CTF flattens one of the Ewald spheres in the 3D transform, while doubling the curvature of the other. This yields the best representation of the Ewald sphere and thus the `exit wave', and can be performed on every single micrograph. The 2D reconstruction as the sum of the partial CTF corrected transforms recovers missing information in the individual micrographs and increases the signal-to-noise ratio. Iterative refinement of the reconstructed image against the amplitudes of the original focal series recovers the second Ewald sphere only at frequencies where the two spheres overlap, but introduces artifacts. The conclusion is that the correct way to account for the Ewald spheres in cryoEM is to apply the two halves of the CTF to a particle image and incorporate the two results into their corresponding spheres within the cryoEM reconstruction.