Improved Hilbert phase contrast for transmission electron microscopy

Hilbert phase contrast has been recognized as a means of recording high resolution images with high contrast using a transmission electron microscope. This imaging mode could be used to image typical phase objects such as unstained biological molecules or cryo sections of biological tissue. According to the original proposal by (Danev et al., 2002) the Hilbert phase plate applies a phase shift of π to approximately half the focal plane (for example the right half excluding the central beam) and an image is recorded at Gaussian focus. After correction for the inbuilt asymmetry of differential phase contrast this image will have an almost perfect contrast transfer function (close to 1) from the lowest spatial frequency up to a maximum resolution determined by the wave length and spherical aberration of the microscope. In this paper I present theory and simulations showing that this maximum spatial frequency can be increased considerably almost without loss of contrast by using a Hilbert phase plate of half the thickness, leading to a phase shift of π/2, and recording images at Scherzer defocus. The maximum resolution can be improved even more by imaging at extended Scherzer defocus, though at the cost of contrast loss at lower spatial frequencies.