The article is devoted to the theoretical problems of studying cathode lenses with two planes of symmetry. It is noted that the classical methods of studying conventional electron lenses are unsuitable for a detailed study of the focusing properties of cathode lenses because these methods do not take into account the specificity of the initial conditions of cathode lenses. For a cathode lens, it is inaccurate to assume that the trajectories of charged particles throughout have a small inclination to the main optical axis. In addition, in single and immersion lenses, the condition of the smallness of the spread of charged particle energies with respect to the values of the axial potential is satisfied throughout, while in the cathode lens, this condition is not satisfied in the cathode region. The method proposed in the article is based on the study of the parameters of the trajectory of an arbitrary particle in a beam of charged particles relative to the parameters of the trajectory of an axial particle with zero initial energy, the trajectory of which is chosen as a scalar motion. The results obtained in the article will expand the scope of the application of cathode lenses with two planes of symmetry.

Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences.

In various electron-optical and ion-beam devices and vacuum electronics installations, emission elements and units are used to extract and form beams of charged particles. Their focusing properties significantly affect the quality parameters and technical characteristics of the analytical devices and technological installations in which they are used. Due to the specificity of the initial conditions during the formation of the beams of charged particles in the area of charged particle extraction, the methods of studying single and conventional immersion electron lenses are not suitable for the high quality theoretical and numerical studies of the properties of immersion objectives. In this paper, theoretical and numerical studies were conducted on the problems in analyzing the focusing parameters of charged particles in emission elements and units when solving problems of designing and upgrading devices and installations of vacuum electronics. As a result of the studies, a theoretical method for studying cathode lenses with a curved optical axis was developed and options for solving problems in correcting aberrations of emission elements and units were proposed.