Comparing Fourier optics and contrast transfer function modeling of image formation in low energy electron microscopy.

A theoretical understanding of image formation in cathode lens microscopy can facilitate image interpretation. We compare Fourier Optics (FO) and Contrast Transfer Function (CTF) approaches that were recently adapted from other realms of microscopy to model image formation in low energy electron microscopy (LEEM). Although these two approaches incorporate imaging errors from several sources similarly, they differ in the way that the image intensity is calculated. The simplification that is used in the CTF calculation advantageously leads to its computational efficiency. However, we find that lens aberrations, and spatial and temporal coherence may affect the validity of the CTF approach to model LEEM image formation under certain conditions. In particular, these effects depend strongly on the nature of the object being imaged and also become more pronounced with increasing defocus. While the use of the CTF approach appears to be justified for objects that are routinely imaged with LEEM, comparison of theory to experimental observations of a focal image series for rippled, suspended graphene reveals one example where FO works, but CTF does not. This work alerts us to potential pitfalls and guides the effective use of FO and CTF approaches. It also lays the foundation for quantitative image evaluation using these methods.