## 4 Discussion  
It was noted above that, due to the specificity of the initial conditions during formation of charged particle beams in the region of charged particle extraction, the methods for studying single and conventional immersion electron lenses are not suitable for high-quality theoretical and numerical studies of the properties of immersion objectives. The specific features of  
Figure 1: Three-electrode box cathode lens.  
the initial conditions of cathode lenses are related to the fact that in the cathode region, at practically zero potential values, the trajectory slopes have a large slope. As a result, the use of the theory for studying conventional single or immersion lenses, in which the trajectory slopes have small values everywhere and the accelerating or focusing potentials of the fields have sufficiently high values, is practically unsuitable for a full-fledged study of cathode lenses. The noted difficulties and insufficiency of theoretical and numerical studies on the modernization of emission systems still do not allow many design and research teams to actively carry out work on improving the sources of charged particles and units, the quality of which determines the quality of the designed devices and installations as a whole.  
It should be noted that miniature electron and ion-optical systems that form beams of small-sized charged particles are also widely used at present. They are used in the construction of small-sized inexpensive analytical devices and can find wide application in technological systems of nano- and microelectronics. Therefore, the problem of ensuring minimal aberrations with a small number of electrodes and simpler schemes of emission units with different types of symmetry or the asymmetry of focusing fields continues to be an urgent problem.  
In this article, in order to solve the problems of expanding the volume of the theoretical base and numerical studies of specific types of cathode lenses, the authors have developed and tested a method for studying emission elements and units.  
The article develops a theoretical basis for an emission lens with a curvilinear axis. This makes it possible to use one element with a curvilinear axial trajectory instead of several electron-optical elements for deflecting (turning) charged particle flows to ensure their movement along curvilinear trajectories.  
It should be noted that before this article, the authors conducted theoretical and numerical studies of other types of lenses. For example, studies of the properties of transaxial lenses showed that their use instead of axisymmetric cathode lenses can lead to the construction of a source of charged particles with a higher value of the beam current.  
Based on the graphs provided in Appendix A, a comparative analysis of the aberration coefficients of the axially symmetrical and box-shaped cathode lenses with a straight optical axis in the charged particle beam crossover formation mode can be performed. The graphs show that the use of cathode lenses with a field symmetry different from rotational axial symmetry can lead to a noticeable improvement in the focusing quality, at least in one of the directions in space. It can also be concluded that for the modernization of emission units, it is possible and necessary to use focusing fields with the most suitable types of symmetries for the designed devices. In this case, the method for studying the properties of emission lenses and units proposed in this article can be used to design specific sources of charged particles.