## 1 Introduction  
Emission elements and units consisting of a source of charged particles (cathode lens) and focusing electrodes are an important element of most vacuum electronic devices [1; 2; 3; 4; 5; 6; 7]. At the same time, the quality parameters and technical characteristics of electron-optical devices and technological installations like, for example, electron microscopes, mass-spectrometric devices, and microwave electronic devices, as well as electron-lithographic and ion-lithographic installations of nano- and micro-electronic technologies, largely depend on the focusing properties and characteristics of the emission units [8; 9; 10; 11; 12]. According to the structure of the focusing fields, the emission elements (cathode lenses) can be classified as immersion electron lenses, but due to the fact that these lenses, along with the focusing function, also perform the functions of extracting charged particles, they are usually distinguished separately and called immersion objectives. A number of works note that the methods of studying single and conventional immersion electron lenses are unsuitable for developing the theory of immersion objectives due to the fact that they do not take into account the specificity of the initial conditions during the formation of charged particle beams in the region of charged particle extraction. The creation of an effective theory of immersion objectives has long been hampered by mathematical difficulties associated with the potential on the cathode surface becoming zero, as well as with large slopes of trajectories in the vicinity of a surface with zero potential [13; 14; 15; 16; 17]. When upgrading vacuum devices and units, one of the most important tasks is to ensure minimum values of aberrations in focusing elements and units. The solution to this problem is achieved by various methods [18; 19; 20; 21; 22; 23], including the selection of the types of symmetry of the focusing fields and by using additional corrective electron lenses. For example, multipole electron lenses allow obtaining minimum values of aberrations in at least one of the directions of focusing of the charged particles. Moreover, additional corrective electron lenses are used to improve the focusing parameters. The correction of aberrations of vacuum electronic devices can also be accomplished by using lenses with a curvilinear axis or electron lenses with asymmetrical focusing fields. For example, in [22; 23] it is noted that the aberration values can be significantly reduced in asymmetrical electron lenses. At present, in connection with the use of an axisymmetric cathode lens as the first electron lens in charged particle sources, it is often necessary to additionally use single or immersion lenses with good corrective properties to ensure minimal values of emission unit aberrations. This significantly complicates the schemes of charged particle sources. When designing emission units, cathode lenses with different types of focusing field symmetry are practically not used due to the insufficiency of effective theoretical methods for their study. As noted above, this is due to the fact that classical methods for studying electron lenses are unsuitable for developing the theory of cathode lenses due to the fact that they do not take into account the specificity of their initial conditions. For instance, in single and immersion lenses, it is rightly assumed that the trajectories of charged particles everywhere have a small inclination toward the main optical axis. For a cathode lens, this assumption is not valid, since charged particles leave the cathode at an angle from 0 to 90 degrees and all particles emitted by the cathode subsequently participate in the formation of a crossover or an image of the cathode surface. In addition, in single and immersion lenses, the condition of a small scatter of energies of charged particles relative to the values of the axial potential is satisfied everywhere, and in a cathode lens in the cathode region the said condition is not satisfied.