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Journal of Electron Microscopy

Asymmetry of Electron Interference Fringes Produced by an Electron Biprism

Akinori Ohshita, Yasuhiro NUMATA, Yukihiro Mametani, Hiroshi Tomita

Published Apr 1, 1988• 4 citations
Publisher

Abstract

The intensity distributions of electron interference fringes produced by an electron biprism are usually asymmetric with respect to the central fringe. In order to investigate the effect of the noncircular cross-section of the central electrode of the electron biprism on this asymmetry, the intensity distributions of interference fringes were numerically calculated under the condition of the nonuniform radius of curvature of the central electrode. The calculated results were compared with the experimental ones. It was found that the asymmetry of interference fringes can be caused by the nonuniformity of the radius of curvature of the central electrode. Furthermore it was found that the average visibility of the central and adjacent fringes gives a good approximation to the degree of coherence of electron beam also for the asymmetric interference fringes.

DOI
10.1093/oxfordjournals.jmicro.a050662

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The intensity distributions of electron interference fringes produced by an electron biprism are usually asymmetric with respect to the central fringe. In order to investigate the effect of the noncircular cross-section of the central electrode of the electron biprism on this asymmetry, the intensity distributions of interference fringes were numerically calculated under the condition of the nonuniform radius of curvature of the central electrode. The calculated results were compared with the experimental ones. It was found that the asymmetry of interference fringes can be caused by the nonuniformity of the radius of curvature of the central electrode. Furthermore it was found that the average visibility of the central and adjacent fringes gives a good approximation to the degree of coherence of electron beam also for the asymmetric interference fringes.
We examine how partial coherence and wavefront curvature of beams affect interference fringes behind diffraction gratings. We simulate (1) the Talbot effect, (2) far-field diffraction, (3) Mach-Zehnader interferometers, (4) Talbot-Lau interferometers, and (5) Lau interferometers using a numerically efficient expression. We show how interference fringes in each case depend on the beam's initial width, its coherence width (transverse coherence length), and its wavefront curvature in directions both parallel and perpendicular to the grating bars. The separation between gratings and the rotational alignment of the gratings about the optical axis are also considered. The formula used for our simulations is derived using the mutual intensity function of a Gaussian Schell-model beam.
The paper describes a model of electron interference obtained using an electrostatic bi-prism. The model is an approximation to the actual interference process, in which two diffracted waves, each representing the diffraction of electrons emerging from a single source by an opaque half-plane, are superposed in such a way that the half-planes are of opposite orientation and overlap by a distance equal to the width of the bi-prism filament. The intensity distribution in the interference pattern calculated according to the formula derived from this model is in an exact agreement with the intensity distribution found experimentally. This makes it possible to present a consistent quantitative explanation of all the special properties of the interference pattern, including intensity modulation of the fringes. The model may also serve for detailed interpretation of some interference phenomena in light optics, such as those observed with the help of Fresnel mirrors or a Fresnel bi-prism.
The wave function describing the electron interference produced by an electrostatic biprism derived from the perturbation theory involves an integral which is not easy to evaluate numerically. Using the method of stationary phase it is shown in the paper that the expression of this integral in terms of the Fresnel integrals considerably facilitates the calculations. Physically this expression corresponds to the superposition of two waves emerging from two virtual images of the source diffracted by two opaque half-planes of opposite orientation. In such a way the model widely used for discussions of the interference of this type is justified.
Electron biprism technique, if extended to any charged particles, will make comparison possible among two-beam interference fringes of electrons and other particles under the same(including wave length)geometrical condition. Combination of the technique with sharply focusing mass spectroscopy will realize such experiments. Due to the tradition since Young-Fresnel, it is believed that two-beam visibilities of particles of various masses under the same geometrical setup are constant and independent of each total sum of macroscopically separated path-lengths, which is not to be confused with ordinary path-difference.(See Fig. 2.)But the assumption of such constant visibility has never been verified experimentally and seems to become gradually unnatural as the masses increase. Experimental reexamination of the assumption is proposed with method of observing an expected mass effect on interdependence between visibility and pathlengths. The smaller visibilities are expected for the heavier masses and the longer path-lengths, contrary to conventional constant visibility; e.g., ≅0.4 for protons of wave length 0.02Å and beam width0.2μ in a realizable experimental setup. Technical problems for the experiment and effects to the observation theory are discussed.
This article describes elucidations of conditions for electron interference observation, device designs, and numerical simulations for the solid-state biprism. The solid-state biprism is solid-state version of the already realized vacuum biprism, and is also proposed as a test ground for electron interference observation and a new conceptual device. Towards the realization of this device, the designed semiconductor device structures show that interference pattern with a pitch of 86 nm and a visibility of 0.6 can be realized with an electron energy of 20 meV, a transverse energy spread of 0.3 meV, and an operation temperature of 4 K.
Electron double-slit experiments, the essence of quantum mechanical “wave/particle duality,” have been extensively and continuously studied because their unrealistic interpretations have attracted many researchers. Here, we discuss recent sophisticated V-shaped double-slit experiments to elucidate electron-path information, i.e., which slit electrons pass through, which path they pursue, and where they arrive and interfere on the image plane. In a transmission electron microscope, an optically zero-propagation distance condition was realized, where the double-slit position was imaged just on the detector plane. Interference fringes composed of dot images were controlled by using two electron biprisms. Using a V-shaped double slit, we confirmed that it is possible to observe interference fringes only when the path information of individual electrons is not available. Furthermore, we studied how the interference fringes would shift their detected positions when electron wave phases were modulated before illumination to the double slit. Phase shifts were generated by tilting electron beams. We confirmed that each single electron as a wave changed its phase even when it traveled alone inside the microscope.