Imaging with straight-edge phase plates in the TEM

The image of a simple phase object produced by a round lens with a Foucault or Hilbert phase plate can be determined with Abbe imaging theory and a 2D transform expressed in cylindrical coordinates. The contributions to the image amplitude from a uniform disc object and an azimuthally varying plate can then be distinguished and their phases relative to the incident wave can be compared. It appears that the usual choice of added phase for a Hilbert plate causes the image of a weak disc object to vanish as the plate edge approaches the axis, but a different choice of plate thickness can enable a weak phase object to provide a linear contribution to image intensity.