A simple model of holography and some enhanced resolution methods in electron microscopy

Abstract

A simple pictorial model of electron interference effects based on an extended representation of the autocorrelation function is described and developed. Unlike Abbe's theory of transmission imaging, the model incorporates fully the effect of the loss of phase that occurs in the detector plane. The aperture transfer function and information limit (envelope function) are also incorporated with reference to the simplest scattering geometry of Young's slits. The model is then applied to holography, the diffraction phase problem, ptychography, Wigner distribution deconvolution, conventional bright-field imaging, single side-band imaging and tilt-series reconstruction. Some of these methods require an understanding of four-dimensional integral functions, but the model reduces the problem into a projection of a two-dimensional space. It is hoped that the model will help material scientists who are not specialists in imaging and diffraction theory to understand some recent developments in advanced super-resolution imaging methods.

Introduction

There are a number of advanced transmission electron imaging techniques which may obtain better resolution, or more easily interpretable information, by using unusual scattering geometries combined with inverse computation methods. As computing power becomes cheaper and detector technology is improved, these techniques have increasing potential to deliver real gains in microscope performance. Indeed, it may even be appropriate to change the whole rationale of electron microscopy and steer the instrumentation development in favour of exploiting indirect methods. However, such techniques, especially those that rely on exploiting illumination tilt series or the microdiffraction plane of the scanning transmission electron microscope (STEM), are hard to understand for non-specialists. Furthermore, much scepticism is met by any method which claims to surpass the resolution limit defined in terms of the maximum spatial frequency that can pass through a limited aperture. In this sense, Abbe's theory has become an impediment to our understanding of some indirect super-resolution methods.

In this paper I propose a different way of thinking about transmission imaging theory. The model is simply an extension of Abbe's theory, but it automatically builds in the phase problem and the fact that other variables, such as the angle of illumination (or, in the case of STEM, the extent of the microdiffraction plane) can provide information that greatly surpasses the conventional resolution limits. The model is primarily a way of picturing the consequences of complicated interference effects that are generally expressed as rather complicated integral equations. For the purposes of clarity, I keep to single set of co-ordinates and use a simple pictorial allegory. In practice, the co-ordinates of the data are sometimes in reciprocal space and at other times in real space, but I do not express these differences in the mathematics. Note also that I do not strictly differentiate between convolution and correlation: depending on definitions of physical co-ordinates, the aperture function should in places be reflected along the x-axis, but this is not crucial to the main picture. What matters here is to give the broad scope of how some rather seemingly unrelated techniques–holography, ptychography, single side-band imaging, bright-field imaging, the classic diffraction phase problem, and Wigner distribution deconvolution, tilt series reconstruction–can all be represented in a single diagram.

In the next section, the pictorial representation is introduced without any justification or background but is used to explain the simplest reconstruction method: holography. We write down a simple version of the mathematics in Section 3. After a brief explanation of how the model relates to Young's slits, holography and the classic phase problem in Section 4, we discuss how the limits of interference and the problems of increased resolution impact upon the model in Section 5. Section 6 then describes some of the conventional and super-resolution techniques in the context of the model. Conclusions are presented in Section 7.