Analysis of spatial point patterns in electron-counting images.

In this study, the spatial counting statistics of free electron beams, which were released via field emission from cold metal and propagated through a vacuum region, were investigated to examine the normal functioning of the counting equipment for electron correlation spectroscopy. The beam electrons were recorded separately according to the locations of individual events as they reached the direct detection transmission Complementary Metal Oxide Semiconductor (CMOS) sensor. We examined the spatial point patterns arising from the locations of the individual events of each primary electron being detected in the case of electrons in a state in which the wave function is constant on the sensor. The quadrat method, which compares the observed frequencies of the number of electron counts in the subsets of the study region with the predicted frequencies from a Poisson distribution, indicates a clustering-type departure from complete spatial randomness. To explore some of the basic principles governing the location of coherent electrons being counted, Ripley's K-function and the corresponding L-function of a stationary spatial point process were used to test the complete spatial randomness from the data. The maximum peak in the average of the L-functions was sensitive only to the mean counts per frame. Thus, clustering of spatial point patterns may result from abnormalities in the direct detection camera. When the interaction of the beam electrons with the sensor is included in the simulation, there is a reasonable match between the average of the L-functions and the experimental curves with the theoretically simulated curves.

Mechanism for correlation in a coherent electron beam.

The correlation of electron counts from two detectors illuminated by a coherent electron beam is analyzed by associating the path of the electron beam through the lenses with the direct Coulomb interaction between two individual electrons. This is shown to lead to a full statistical description of the electron counts. The dominant contribution to the correlation is found to be due to the trajectory displacement caused by the repulsive Coulomb interaction between the first anode and the cathode tip, and the correlation of electron counts is found to depend on the amount of defocusing on the shift of the virtual source for two electrons within the correlation time. The Coulomb scatterings, which altered the direction of two neighbor electrons, occur during the acceleration, leading to a significant decrease in the electron density. The Coulomb potential with no screening will then cause large-angle scattering of nearest-neighbor electrons within the correlation time. These results are consistent with those obtained by a previous experiment.

Properties of electrostatic correcting systems with annular apertures.

Image formation in electron microscopes with circular hole and annular apertures is studied theoretically. The apertures-the circular hole aperture being negative with respect to the annular aperture-produce an additional electrostatic field that exerts a force on the electrons directed toward the optical axis. The resulting deflection angle decreases with increasing distance from the optical axis. This electrostatic field results in a correcting effect of the unavoidable spherical aberration of round electron lenses; the deflection toward the optical axis increases stronger than linearly with increasing distance from the optical axis. Analytical formulae are given for the correcting effect of circular hole and annular apertures. The expressions are based on the Davisson-Calbick formula, which is used to calculate focal length of a simple electrostatic lens.

Electron holography on Fraunhofer diffraction.

Electron holography in Fraunhofer region was realized by using an asymmetric double slit. A Fraunhofer diffraction wave from a wider slit worked as an objective wave interfered with a plane wave from a narrower slit as a reference wave under the pre-Fraunhofer condition and recorded as a hologram. Here, the pre-Fraunhofer condition means that the following conditions are simultaneously satisfied: single-slit observations are performed under the Fraunhofer condition and the double-slit observations are performed under the Fresnel condition. Amplitude and phase distributions of the Fraunhofer diffraction wave were reconstructed from the hologram by the Fourier transform reconstruction method. The reconstructed amplitude and phase images corresponded to Fraunhofer diffraction patterns; in particular, the phase steps of π at each band pattern in the phase image were confirmed. We hope that the developed Fraunhofer electron holography can be extended to a direct phase detection method in the reciprocal space.

Phase reconstruction in annular bright-field scanning transmission electron microscopy.

A novel technique for reconstructing the phase shifts of electron waves was applied to Cs-corrected scanning transmission electron microscopy (STEM). To realize this method, a new STEM system equipped with an annular aperture, annularly arrayed detectors and an arrayed image processor has been developed and evaluated in experiments. We show a reconstructed phase image of graphite particles and demonstrate that this new method works effectively for high-resolution phase imaging.

Line profile reconstruction from simultaneously recorded secondary and backscattered electron signals.

We propose a reconstruction method of surface morphology using a combination of secondary and back-scattered electron signals from the scanning electron microscope (SEM). Compared with multiple-detector methods, the proposed system requires only conventional secondary and backscattered electron detectors for a line profile reconstruction in one direction. This method is an application of genetic algorithms to the measurement of surface morphology in SEM. We use the chi-square distribution of the reconstruction error as the objective function within a scheme to minimize the number of vertices in the reconstructed surface profile. (The reconstruction error is the relative difference between the calculated and experimental data.) To evaluate the efficacy of our method, a surface profile is successfully reconstructed from a pair of line scans across the center of a latex particle.

Two-electron interference in a coherent beam.

The interference between quantum amplitude for two electrons, emitted from two source points, to be detected at two detection points, is a direct result of quantum exchange statistics. Such interference is observed in the coincidence probability, compared with that of statistically independent electrons, by computing the time correlation function from the arrival times of the electrons. When the two detectors are separated by a distance less than the coherence length, the coincidence probability is suppressed for electrons (antibunching) due to the Pauli principle, even though they do not interact with each other. However, electrons are charged particles. The Coulomb potential, which governs the scattering of one charged particle by another, is so long ranged. It is obvious that we must consider the Pauli principle and the Coulomb interactions simultaneously. This paper deals with basic experimental and theoretical investigations of the antibunching behavior of electrons in a free beam by considering the Pauli principle and the direct Coulomb interaction between two individual electrons. The experimentally found dependences are described in a model which considers the Coulomb scattering and theoretical values of correlation signals evaluated by analytical calculations agree with those determined by experiment. A study of the time correlation function from the arrival times of the electrons will lead to an understanding of the physical processes that take place in electron guns.

Evolutionary computation applied to the reconstruction of 3-D surface topography in the SEM.

A genetic algorithm has been applied to the line profile reconstruction from the signals of the standard secondary electron (SE) and/or backscattered electron detectors in a scanning electron microscope. This method solves the topographical surface reconstruction problem as one of combinatorial optimization. To extend this optimization approach for three-dimensional (3-D) surface topography, this paper considers the use of a string coding where a 3-D surface topography is represented by a set of coordinates of vertices. We introduce the Delaunay triangulation, which attains the minimum roughness for any set of height data to capture the fundamental features of the surface being probed by an electron beam. With this coding, the strings are processed with a class of hybrid optimization algorithms that combine genetic algorithms and simulated annealing algorithms. Experimental results on SE images are presented.