A closer look at the effect of lens aberrations and object size on the intensity distribution and resolution in electron optics

Abstract

A geometrical optics approach is used to calculate the intensity distribution in the image of a small object centered on the axis. The calculations take into account the size of the object as well as the spherical and chromatic aberrations of the lens. The intensity distribution curves, calculated as a function of depth in the image, reveal the existence of a compact high intensity image peak in the caustic region between the paraxial image plane and the plane of least confusion. The intensity distribution curves show that the geometrical resolution is better in the plane with the high intensity peak than in the plane of least confusion. As the object radius approaches zero the plane in which the high intensity peaks occurs moves toward the paraxial plane, and the geometrical resolution approaches zero. For an object of finite brightness the intensity in the image peak also approaches zero. A geometrical error rg is introduced, which depends on the size of the smallest object providing the required intensity or current in the image peak. The geometrical error and the diffraction error are combined quadratically to obtain a resultant error. An adaptation of the geometrical approach, combined with wave optics, is explored in the case of a point source. Discussions of the intensity-distribution approach as it affects the resolution calculations in the transmission electron microscope and the emission electron microscope are included. In the case of a probe-type instrument a perhaps unexpected finding is that a larger source size along with a smaller angular aperture is preferable in some cases to a smaller source size and larger angular aperture.