METHOD OF REDUCTION MODEL FOR CALCULATION OF ELECTROSTATIC FIELDS OF ELECTRONIC OPTICS SYSTEMS
DOI:
https://doi.org/10.15588/1607-3274-2019-1-4Keywords:
model system, method of integral equations, abelian group of symmetry, circular matrix, Fourier transform, equipotential surface.Abstract
Context. The rapid development of nanotechnology puts forward new requirements for electronic optics systems. In modern electron-optical systems, a significant amount of complex electrodes with a geometric symmetry is observed. When calculating the electrostatic fields of the corresponding systems require high accuracy of calculations. This can be achieved by developing new and
improving existing algorithms for calculating potential fields.
Objective. The goal of the work is to develop the method for reducing the model for calculating the electrostatic fields of modern systems of electronic optics.
Method. In order to confirm the efficiency of the proposed method, the finding of the parameters of the electrostatic field of a particular model system is considered. It is shown that the configuration of surfaces of electrodes has an Abelian cyclic group of symmetry of the fourth order. Fourier transform matrix for this group is found. It was succeeded, using the method of reduction of
the model, to base the apparatus of group theory on the basis of the system of four integral equations to a sequence of four independent integral equations, where integration is carried out on ¼ of the aggregate boundary surface. The maximum (repeated) consideration of the existing symmetry of the boundary surface in the mathematical modeling of the electrostatic field allows, in turn,
to significantly reduce the order of the model – to go to the integration, for example, 1/16, 1/64 of the boundary surface.
Results. In work, without decreasing the universality, on the example of a particular model system, the calculation of parameters of the electrostatic field was carried out. For the visual representation of which surfaces of equal potential are used. The results of numerical simulation are given at a different variation of the known potential values on the boundary surfaces of the electrodes. The obtained results can be used for the design of modern electronic optics systems.
Conclusions. The method of reduction of the model for calculating the electrostatic fields of electron-optical systems is based on the boundary integral equations of the theory of potentials in combination with the apparatus of group theory, which, unlike existing methods, allows to simplify the procedure of numerical analysis of electrostatic field parameters maximally taking into account the available symmetry in geometry boundary surfaces, avoid numerical instability of calculations and get higher accuracy of calculations. The class of electronic optics systems that allow mathematical modeling based on the method of integral equations is expanded.
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