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A high-resolution image may be derived from a large number of `dark-field' images of normal resolution, obtained by varying the angle of incidence of the electron beam in a standard electron microscope. The intensity distribution in each dark-field image is multiplied by an appropriate cosine function. Summing gives the function, for one dimension, C(x) = ∫ V(X) G(-X) V(X + x) G(x - X) dX where V(x) is the potential distribution in the specimen and G(x) defines the loss of resolution due to lens imperfections in normal micrographs. From this, V(x) may be derived. The practical applications of the method are limited by requirements of specimen and instrumental stability.

Single-crystal electron diffraction patterns with doubled spots obtained from zinc oxide smoke particles originate from pairs of thin sheets which are commonly seen in electron micrographs of this material joining up the members of a group of four spines. The plane of each sheet is parallel to a (1010)-type lattice plane. The sheets are continuous with one of the spines and spaced at 120° intervals around this spine. The other three spines, and additional spines which grow from corners of the sheets, are related to this one spine and the sheets by twinning on (1122) planes. Intensity measurements of single-crystal patterns allow the shape-transform and thickness of the sheets to be deduced. Sharp lines and extra spots indicate the occasional presence of lattice imperfections in the form of parallel faults. Electron micrographs show wide deviations from the idealized group of spines and sheets.

The theoretical and experimental investigation of Fourier images, described in Parts I and II, is extended to the case of finite sources. The Fourier images produced by a pair of point sources, a rectangular source and a Gaussian source are considered in detail. The dependence of focus on source dimensions for the first two of these cases is confirmed experimentally with light optics. From the results obtained for these idealized sources, the contrast and resolution of Fourier images from experimentally realizable sources is deduced. The possibility is considered of an improvement in resolution in the electron microscopy of thin crystals when the first stages of magnification are performed by forming magnified Fourier images. It is shown that the resolution that can be attained depends on the sharpness with which the boundaries of the source are defined. In particular, if the effective source is formed by using electron lenses to demagnify a larger source, the resolution may be improved by a factor of 2-1/2.

The theoretical and experimental investigation of Fourier images described in Part I is extended to the patterns on planes other than focal planes. The theory is developed from a new statement of the diffraction equations correct to the Fresnel approximations, and experimentally observed patterns are shown to be in good agreement with the results of the theory. It is concluded that, at least within the angular range of the experiments, some fifteen degrees, the Fresnel approximation gives an adequate description of the phenomenon.

The problem of designing a light or electron optical system specifically for the imaging of periodic objects, such as real crystals, is discussed. A qualitative appeal to communication theory suggests that it should be possible to devise systems of higher efficiency than the conventional microscope by using the a priori knowledge of periodicity. The possibility of deforming the incident wavefront in such a way that the periodic object acts as its own imaging system is considered and a formalism is set up. The particular case of the spherical wavefront is then examined in detail and it is predicted that a new type of image should be formed on certain planes. Since this image is in many ways analogous to the Fourier projection of crystallography, and since it can only be formed by periodic objects, it is named the Fourier image. Fourier images produced experimentally with light optics are presented and shown to be in agreement with theoretical predictions. Patterns on planes other than Fourier image planes are described briefly, but detailed treatment is deferred until Part II. The possibility of application to the imaging of crystal lattices by electron optics is discussed, but a quantitative treatment of the crucial problems of finite source size and coherence are deferred until Part III.