## 3 Results
\]  
Here, the angular coordinates \(\psi\) and \(\theta\) for axisymmetric cathode lens are equal to zero, the indices \(k\) denote the belonging of the quantities to the cathode before the designations of the aberration coefficients \(\rm B_{ij}\), the values of the coordinates of the point of departure of charged particles and the values of their initial energies are given, and the indices at \(\rm B_{ij}\) denote the types of aberrations. When comparing the aberration characteristics of axially symmetrical lenses with the aberration characteristics of box (double-symmetrical) lenses, the above formula must be transformed by replacing the radial coordinate with the coordinates of the Cartesian coordinate system, which is used to study doubly symmetrical electron lenses. This makes it possible to compare the aberration characteristics of the lenses under consideration in a general form. In this case, the third-order aberrations of smallness \(D_{\bar{y}\bar{y}}\) doubly symmetrical cathode lenses have the form  
\[ \begin{array}{l}D_{x3}=x_{\rm N}^{3}D_{x31}+x_{\rm k}^{2}\sqrt{\varepsilon_{ x}}D_{x32}+x_{\rm k}\varepsilon_{x}D_{x33}+\sqrt[3]{\varepsilon_{x}}D_{x34}+\\ +x_{\rm k}y_{\rm k}^{2}D_{x35}+x_{\rm k}y_{\rm k}\sqrt{\varepsilon_{y}}D_{x36}+ x_{\rm k}\varepsilon_{y}D_{x37}+\sqrt{\varepsilon_{x}}y_{\rm k}^{2}D_{x38}+\\ +\sqrt{\varepsilon_{x}}y_{\rm k}\sqrt{\varepsilon_{y}}D_{x39}+\sqrt{\varepsilon _{x}}\varepsilon_{y}D_{x310}+x_{\rm k}\varepsilon_{x}D_{x311}+\sqrt{\varepsilon _{x}}\varepsilon_{x}D_{x312}\end{array}  \tag{29}\]  
\[ \begin{array}{l}D_{y3}=y_{\rm N}^{3}D_{y31}+y_{\rm k}^{2}\sqrt{\varepsilon_{ y}}D_{y32}+y_{\rm k}\varepsilon_{y}D_{y33}+\sqrt[3]{\varepsilon_{y}}D_{y34}+\\ +y_{\rm k}x_{\rm k}^{2}D_{y35}+y_{\rm k}x_{\rm k}\sqrt{\varepsilon_{x}}D_{y36} +y_{\rm k}\varepsilon_{x}D_{y37}+\sqrt{\varepsilon_{y}}x_{\rm k}^{2}D_{y38}+\\ +\sqrt{\varepsilon_{y}}x_{\rm k}\sqrt{\varepsilon_{x}}D_{y39}+\sqrt{\varepsilon _{y}}\varepsilon_{x}D_{y310}+y_{\rm k}\varepsilon_{x}D_{y311}+\sqrt{\varepsilon _{y}}\varepsilon_{x}D_{y312}\end{array}  \tag{30}\]  
Here, as above, the indices at the aberration coefficients indicate the type of aberration. The considered axisymmetric lens consists of a flat cathode and two electrodes with the same values of radius R, and the three-electrode box lens consists of a flat cathode and two pairs of orthogonally located flat electrodes along the main optical axis of the lens. The diagram and shapes of the electrodes of the three-electrode box cathode lens are shown in Figure 1.  
The dimensions of the focusing electrodes of a box lens in the vertical direction are designated \(\rm l_{y}\), and in the horizontal direction \(\rm l_{x}\). The results of the numerical studies are presented in the form of a table of graphs, which provide the values of aberrations for similar sizes of the studied three-electrode cathode lenses with different types of focusing field symmetry. Graphs of the aberration coefficients of three-electrode axisymmetrical and box-shaped cathode lenses are given in the Appendix A. Note that the values of the aberration coefficients were calculated under the condition that the studied lenses form a crossover at a point with coordinates \(\rm z=1\), \(\rm x=y=0\). In the graphs provided, the red line corresponds to the values \(\rm R=l_{x}=l_{y}=0.05\), the green line to the values \(\rm R=l_{x}=l_{y}=0.1\), and the blue line to the values \(\rm R=l_{x}=l_{y}=0.15\). As noted above, the analysis of the obtained results shows that a box lens with a square cross-section, compared to an axisymmetric lens, can provide a reduction in the absolute value of aberrations, i.e., it provides better focusing of charged particles.