Introduction

Publisher Summary

The paraxial form of the Schrodinger’s equation helps to understand the elementary image formation in terms of the wavefunction. This chapter explains the study of electron propagation through static electric or magnetic fields based on Schrodinger’s equation and discusses the elementary image formation in terms of the wave function. There is a discussion of the laws of diffraction and interference. Interference effects are divided into interferometry and holography. Image processing divides naturally into four large sections: acquisition, sampling, quantization, and coding; enhancement; restoration; and image analysis. Ideas of mathematical morphology, phase problems, and three-dimensional reconstruction are also discussed in the chapter.

54.1 Organization of the subject

The behaviour of beams of free electrons, released from a source and propagating through a vacuum region in some device, is of interest in many diverse fields of instrumentation and technology. The study of such beams forms the subject of electron optics, which divides naturally into geometrical optics, where effects due to wavelength are neglected, and wave optics, where these effects are considered. Volumes 1 and 2 were devoted to geometrical optics. This final volume is concerned with wave optics. A knowledge of this branch of the subject is essential in microscopy, to understand the propagation of electrons from the source to the specimen, through the latter and from it to the image plane of the instrument. It is also needed to explain all interference phenomena, notably holography, and in formal coherence theory.

The various branches of the subject have reached different degrees of sophistication. The laws that govern wave propagation are closely analogous to those already familiar in light microscopy, provided that electron spin is neglected, and can be regarded as well-established. Some of the applications are, conversely, in rapid evolution and new developments are to be anticipated. We have therefore concentrated on the principles, which should remain largely unaffected by the passage of time. We have not, of course, neglected their practical exploitation, in holography for example, or in image formation and processing, and the lifetime of the corresponding sections will no doubt prove limited.

We shall not repeat here the general remarks on the classification of electron optical studies to be found in the Preface to Volumes 1 and 2. Our theme in this third volume is the study of electron propagation through static electric or magnetic fields, including those inside specimens, based on Schrödinger’s equation. This is complemented by a Part on digital image processing which, if not in the main line of electron optics, is an inevitable preoccupation of anyone concerned with electron microscope imaging.

The wave theory of electron optics is founded on the Dirac equation but, in practice, it is almost always permissible to replace this by the relativistic form of Schrödinger’s equation, for spin is negligible except in a few very specialized situations. The book therefore opens (Part XI) with an account of the relevant material from quantum mechanics. A paraxial form of Schrödinger’s equation is derived in Chapter 58, which enables us to understand elementary image formation in terms of the wavefunction. In the last two chapters (59 and 60), the laws of diffraction and interference are studied.

In the remainder of the book, the laws of propagation established in Part XI are applied to a variety of different situations, directly in the case of Parts XII, XIII, XIV and XVI, indirectly in the case of Part XV on image processing. Interference effects are the subject of Part XII, which is divided into interferometry and holography. The distinction between the two is not sharp but, in interferometry, we are not concerned with fine diffraction effects in the specimen, whereas in holography it is precisely these effects that render the technique valuable. Holography is an example of a topic of which our account cannot be definitive, for the techniques are just beginning to emerge from the confines of a few research laboratories and, even there, have by no means attained their full potential. We therefore present only the fundamentals of the procedures that at present seem the most promising, but the situation may well change.

The next Part, which fills a substantial fraction of the book, is devoted to image formation, in the transmission electron microscope and, more briefly, in the scanning transmission electron microscope (STEM). Here, the relation between the intensity at the image and the wavefunction at the specimen is explored in great detail and the linear theory that is applicable to a certain class of specimen is presented at length. The effects of source-size and energy spread are examined, as are less conventional imaging modes, using tilted or hollow-cone illumination in particular. The chapter on the STEM concentrates on the differences between this instrument and the conventional microscope, notably, the possibility of controlling the detector response, either by configuring the detector surface or by recording the two-dimensional signal generated by each specimen-element as though it were an image and combining the intensity values of this image in any way that seems helpful. We also draw attention to the information that can be obtained about crystalline specimens when the area illuminated coherently by the probe is appreciably smaller than the unit cell.

Part XIV is a brief reminder of the ways in which the propagation of the electron wavefunction through the specimen is analysed. Superficial though this presentation inevitably is, for the subject is not central to the theme of the book, we felt that some account of this material was indispensable, for without it certain notions introduced elsewhere, the specimen transparency for example, would remain mysterious. The theory is presented separately for amorphous and crystalline specimens, for the collective effects in the latter require us to analyse them in terms of concepts totally inappropriate to amorphous materials. We do insist, however, that this Part can do no more than bridge the gap between our detailed presentation of imagery and other specialized texts on the microscopy of specimens of a particular kind.

In Part XV, we turn to digital image processing. We cover, even if unevenly, the whole field of image processing and make no apology for including this in a book on electron optics for much of the material presented has been a major preoccupation of microscopists over the years: the work on the phase problem and that on three-dimensional reconstruction are obvious examples and the current studies that aim to use all the information from every object-element in the STEM, so that the image becomes four-dimensional, provide an even more persuasive justification. At a humbler level, image enhancement has been practised in scanning electron microscopy since the earliest days of the instrument. Image processing divides naturally into four large sections: acquisition, sampling, quantization and coding; enhancement; restoration; and image analysis. We have adopted these divisions, adding to them a chapter on instrument control and on the measurement of microscope operating parameters, during image acquisition in particular. We have also included a short introduction to image algebra for, although this subject is too young to have had much impact on electron image processing as yet, we anticipate that some familiarity with it will be required to read the image processing literature of the future. In this Part, we describe many of the procedures that are used to improve images in some way or render them more informative; in particular, we devote considerable space to the ideas of mathematical morphology, which are already important in scanning electron microscopy, to the work on the phase problem and to three-dimensional reconstruction.

The book concludes with a Part of a rather theoretical nature devoted to coherence and in particular, to the relation between coherence and radiometry (Chapter 78). A short chapter is also devoted to instrumental aspects of coherence, notably the effect of partial coherence on image formation in terms of the transmission cross-coefficient. The discussion of the various brightness functions in Chapter 78 is inspired by the work of Wolf and his school, who were concerned with light sources. The translation to electron sources is, however, immediate, since the latter are quasi-monochromatic and only the spatial partial coherence raises problems: the contributions from different wavelengths can safely be addedx ‘incoherently’. Nevertheless, some questions still remain without a fully satisfactory answer; in particular, we have preferred to describe the work of Agarwal et al. (1987) out of context in Section 78.10.1, for although it provides a transparent gateway between light and electron optics, some further elucidation is required before we dare pass through.

Most aspects of wave electron optics have thus been covered here, some more thoroughly than others. The emphasis throughout is on physical principles and on their theoretical formulation while technical details of microscopes or ancillary equipment are kept to a minimum. Even more than...