Direct observation of curvature of the wave surface in transmission electron microscope using transport intensity equation

Highlights: • The first direct observation of curvature of field in transmission electron microscope. • Artifact-free solution of transport of intensity equation using a selected area aperture. • Acquisition of through-focus images by adjusting an intermediate lens focus. In optics it is well known that the image surface is curved, even when an illumination is an ideal plane wave. However, in transmission electron microscopy (TEM) the curvature of field, or wave surface, has not been discussed seriously. We have observed the curvature of field in TEM using the transport of intensity equation (TIE). The TIE describes the relation between the phase and intensity distributions of the wave that propagates in vacuum. The common way to obtain the phase distribution using the TIE requires solving the Poisson equation, which needs the information on the boundary (boundary condition). In this report, we observe the images that pass through a selected area (SA) aperture on the intermediate image plane in order to resolve the boundary value problem. Then, we have developed a new iterative scheme to solve the TIE based on the discrete cosine transform (DCT) valid for the Neumann boundary condition. Using the iterative DCT solver we have observed the phase modulation that shows curvature of field within the SA aperture. To the authors' knowledge this is the first direct observation of the additional phase term corresponding to the curvature of field on the image plane in transmission electron microscope. We have also explained why electron holography does not detect this spherical wave on the image plane.