Resolution in Emission Microscopy
Authors: Rempfer G. F. 
Published: June 2020

A commonly used estimate for the resolution limit imposed by spherical aberration in electron microscopes is the radius of the circle of least confusion rℓc. The radius of least confusion is calculated for a point object and a single energy, and is referred to object space. There are a number of reasons for questioning the usefulness of the radius of least confusion as a measure of resolution. If the intensity were uniformly distributed over the circle of confusion at different depths in the image it would be natural to assume that best resolution occurs in the plane in which the circle of confusion is smallest. However, the intensity is not uniform. Furthermore the effect of a distribution of electron energies and a non-zero object size (required for a non-zero current) should be included in calculating resolution, especially in emission microscopy, where the chromatic aberration can be very large, and low emission current density can limit the smallness of details which can be viewed or recorded. In an earlier work Storbeck, and recently my colleagues and I using a different approach, have taken these effects into account in resolution studies based on the intensity distribution in the image.

In emission microscopy the aberrations introduced by the accelerating field as well as those due to the objective lens must be considered. In our calculations the spherical aberration coefficients due to the field and the lens are referred to virtual specimen space (the image space of the accelerated electrons) at unit magnification, where they are combined, as are the chromatic aberration coefficients. The object for the microscope is a small disc centered on the axis. The emission current density is uniform, with a cosine angular distribution, and an emission energy distribution chosen to fit the particular application. The intensity distribution in the image plane is calculated first for monoenergetic beams, as a function of the axial position of the plane. The distribution curves in Fig. 1 exhibit the effects of spherical aberration and object size as the defocus changes. The shapes of the curves are due to the behavior of the image disc as a function of the emission angle αe. Between the plane of least confusion and the paraxial plane, as αe increases from 0° the image disc at first moves away from the axis in the azimuth of emission (retrograde direction). After reaching a maximum displacement, which depends on the distance from the paraxial plane, the image disc moves back to the axis and into the opposite azimuth as αe continues to increase. The intensity on the axis is highest when the retrograde displacement is equal to the image disc radius, Fig. 1c. This intensity distribution turns out to be more favorable for resolution than does the distribution in the plane of least confusion, Fig. If, even though the beam spreads over a larger area. The smaller the object radius and spherical aberration coefficient are, the closer the high-intensity plane is to the paraxial plane. For a monoenergetic beam the high-intensity plane for the smallest object which can provide the required current in the image is the optimum image plane for geometrical resolution. For a beam with a range of energies the total intensity distribution is obtained from the weighted sum of single-energy distributions calculated for a series of values in the energy range and for a given position of the image plane, Fig. 2. A good approximation to the plane providing best resolution for the beam as a whole is the high-intensity plane for the average energy.