Correction of the Mathematical Method for Studying a Cathode Lens with Two Planes of Symmetry

Abstract

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.

1. Introduction

Cathode lenses are the main element of most analytical electron and ion ray instruments and technological devices. For example, they are used to build electron-optical converters and image intensifiers, electron microscopes, mass spectrometers, accelerators, and other devices in which they serve to form electronic images and focus charged particle beams [1,2,3].

Classical methods for studying electron lenses [4,5,6] are unsuitable for developing the theory of cathode lenses because they do not take into account the specific nature of their initial conditions. For example, in single and immersion lenses, it is accurately assumed that the trajectories of charged particles throughout have a small inclination to the main optical axis. For a cathode lens, this assumption is not valid since the charged particles leave the cathode at an angle of 0 to 90 degrees, and all the particles emitted by the cathode further participate in the formation of the crossover or the image of the cathode surface. 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. A more detailed explanation of these features of cathode lenses is given below. A number of works are devoted to the development of the theory of cathode lenses, taking into account the noted features in their initial conditions, for example, [7,8,9,10]. In these works, there is an important idea of taking into consideration the fact that when the time of flight of charged particles is equal, in addition to transverse spatial aberrations, there are also longitudinal aberrations. However, as indicated in [11,12], mathematical transformations in earlier works do not always fully reflect physical processes. For example, in the works [7,8,9], in the equations of the trajectories of charged particles, the decomposition in a series of small parameters for the analysis of the aberration characteristics of cathode lenses is carried out by analogy with the theory of electronic mirrors, which was developed a little earlier. Due to the fact that small parameters are associated with initial conditions, it was necessary to clarify the differences between the conditions of motion of charged particles in the area of their extraction in cathode lenses and in the area of reflection of charged particles in electronic mirrors. In the works [11,12], the features and differences of cathode lenses from other types of electronic lenses are taken into account in sufficient detail.

Currently, charged particle beams are widely used in many high-tech analytical devices and scientific and technological systems [13,14,15,16]; therefore, solving theoretical problems that have not yet been considered for the study of properties and design of various modified cathode lens circuits with improved focusing characteristics is an important task.

In [12], the author of this work proposed a method for correcting the theory of studying the properties of cathode lenses. As known, in article [12], a cathode lens with rotational symmetry was considered. In practice, it is necessary to use and design lens systems with other symmetries of focusing fields as well. The aim of this work is to develop a corrected theory of an electrostatic cathode lens with two planes of symmetry that adequately reflects the physical features of charged particle focusing in these lenses.

2. Materials and Methods

2.1. Trajectory Equations

To achieve the above goal, this paper uses a research method that has the following mathematical features. 1. The initial conditions for solving the system of equations for the trajectories of charged particles are given in terms of the values of the initial energies of the particles, which in practice are always much less than the energy of the focusing fields. This eliminates the disadvantages of using initial conditions associated with the angle of inclination between the initial directions of the particle trajectories and the main optical axis. The use of corrected initial conditions ensures the correctness of all expansions performed in power series and the correctness of all the results obtained on their basis. 2. The concept of the main trajectory is introduced, which is the trajectory of a particle with zero initial energy moving along the main optical axis. The coordinates of the trajectories of all other particles are determined relative to the coordinates of the main particle. 3. A complete set of variational parameters was analyzed, and all equations were obtained to determine the values of paraxial and aberration characteristics.

In the Cartesian coordinate system x, y, z, the z-axis of which coincides with the main optical axis of the corpuscular-optical system, the motion of a charged object with charge e and mass m is described by the following equations:

$$m\ddot{x}=-e\frac{\partial \phi }{\partial x},$$

(1)

$$m\ddot{y}=-e\frac{\partial \phi }{\partial y},$$

(2)

$${\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2=-\frac{2e}{m}(\phi +\epsilon ),$$

(3)

where the dots denote differentiation with respect to time, $\phi =\phi (x,y,z)$ is the distribution function of the electrostatic potential, and $-e\epsilon$ is the value of the initial energy of the charged particle.

To solve Equations (1)–(3), initial conditions must be specified. In this case, the type of initial conditions depends on the type of corpuscular-beam (electron-optical) element under study.

The initial conditions for a cathode lens have the form

$$x(t){|}_{t=0}=x_k,$$

(4)

$$y(t){|}_{t=0}=y_k,$$

(5)

$$z(t){|}_{t=0}=z_k,$$

(6)

$$\dot{x}(t){|}_{t=0}=\sqrt{-\frac{2e}{m}{\epsilon }_x}=\sqrt{-\frac{2e}{m}\epsilon }\sin \alpha \cos \beta ,$$

(7)

$$\dot{y}(t){|}_{t=0}=\sqrt{-\frac{2e}{m}{\epsilon }_y}=\sqrt{-\frac{2e}{m}\epsilon }\sin \alpha \cos \beta ,$$

(8)

$$\dot{z}(t){|}_{t=0}=\sqrt{-\frac{2e}{m}{\epsilon }_z}=\sqrt{-\frac{2e}{m}\epsilon }\cos \alpha ,$$

(9)

where α is the angle between the direction of emission of the particle emitted by the cathode and the main optical axis and β is the angle between the projection of the initial velocity vector onto the $xy$ plane and the x-axis; the index “k” denotes the value of the quantities at t = 0, i.e., on the cathode.

Under the given initial conditions, the time of flight and other parameters of the motion of charged particles are counted from the moment of their departure from the cathode surface, on which the condition is

$${\phi }_k=\phi (x_k,y_k,z_k)=0.$$

(10)

In electron mirrors, in the region reflecting charged particles, there is also a surface that satisfies the condition (10). Therefore, the initial conditions specified above for the cathode lens can, in principle, be used to derive the trajectory equations and study the properties of doubly symmetric electron (or ion) mirrors on their basis. However, it should be remembered here that, unlike a cathode lens, the full trajectory of charged particles in a mirror consists of its direct (incident) and backward (reflected) branches, and the turning point of a particle with nonzero initial energy, as a rule, is not located on the surface, whose potential satisfies the condition (10).

Considering the above, to study the properties of an electron mirror, the initial conditions for the system of Equations (1)–(3) can be set as follows:

$$x(z){|}_{z=z_H}=x_H,y(z){|}_{z=z_H}=y_H,t(z){|}_{z=z_H}=t_H.$$

(11)

$${\frac{dx}{dz}|}_{z=z_H}=x_H^{\prime },{\frac{dy}{dz}|}_{z=z_H}=y_H^{\prime },{\frac{dz}{dt}|}_{t=t_0}=0.$$

(12)

In the previous expressions and below, the index “H” denotes the values of the quantities in the object plane, $t_0$ is the moment in time when the particle passes the turning point, and $x_H,y_H,x_H^{\prime },y_H^{\prime }$ are the quantities of the first order of smallness.

It should be noted that the classical methods for studying electron lenses were developed to analyze the focusing characteristics of single and immersion lenses, for which the initial conditions have the form

$$x(z){|}_{z=z_H}=x_H,y(z){|}_{z=z_H}=y_H,$$

$${\frac{dx}{dz}|}_{z=z_H}=x_H^{\prime },{\frac{dy}{dz}|}_{z=z_H}=y_H^{\prime }.$$

In this case, the general solutions of the paraxial equations of motion of charged particles are in the form

$$x(z)=x_H{p}_x(z)+x_H^{\prime }{g}_x(z),y(z)=y_H{p}_y(z)+y_H^{\prime }{g}_y(z).$$

Here, $p_x(z)$, $g_x(z)$, $p_y(z)$, and $g_y(z)$ are particular solutions of paraxial equations for $x_H=y_H=1$ and $x_H^{\prime }=y_H^{\prime }=0$. In ordinary single and immersion electronic lenses, $x_H^{\prime }$ and $y_H^{\prime }$ have small values. Therefore, in the theory of these lenses, in all mathematical operations, these initial conditions are used as small variation parameters. In cathode lenses and electron mirrors, the field potentials in the cathode and reflection regions are equal or almost equal to zero, and the potential energy of the focusing field in these regions is less or not greater than the values of the initial energies of the particles in the beam. In this case, the angles between the directions of the trajectories of charged particles and the main optical axis can have large and maximum values. Consequently, the parameters $x_H^{\prime }$ and $y_H^{\prime }$ for cathode lenses and electron mirrors cannot be considered small variation parameters. To overcome this problem, in the initial conditions of the cathode lens (7)–(9), instead of $x_H^{\prime }$ and $y_H^{\prime }$, the values of the initial energies of charged particles are used, which are much less than the energy of the focusing field of the lens as a whole. Later in this paper, it will be shown that the correction of the initial conditions and the introduction of the concept of the main trajectory makes it possible to construct a complete and correct theory of cathode lenses with two planes of symmetry.

The distribution of the electrostatic potential in doubly symmetrical corpuscular-optical systems satisfies the condition

$${\frac{\partial \phi (x,y,z)}{\partial x}|}_{x=y=0}={\frac{\partial \phi (x,y,z)}{\partial y}|}_{x=y=0}=0$$

(13)

near the main optical axis and can be represented as the following row:

$$\begin{gathered} \phi (x,y,z)=\Phi (z)-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y^2+\left(\frac{{\Phi }^{IV}}{64}-\frac{f_{KB}^{″}}{12}+f_{OK}\right)x^4+ \\ +\left(\frac{{\Phi }^{IV}}{64}+\frac{f_{KB}^{″}}{12}+f_{OK}\right)y^4+\left(\frac{{\Phi }^{IV}}{32}-6f_{OK}\right)x^2{y}^2+\dots . \end{gathered}$$

(14)

Here, the primes denote differentiation with respect to z, $f_{KB}=f_{KB}(z)$ and $f_{OK}=f_{OK}(z)$ are the functions characterizing the quadrupole and octupole components of the field, respectively, and $\Phi (z)=\phi (0,0,z)$ is the potential distribution along the main optical axis.

Substituting the value of the function $\phi (x,y,z)$ from (14) into Equations (1)–(3), we obtain

$$\ddot{x}=-\frac{e}{m}\left[-\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)x+\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x^3+\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x{y}^2\right],$$

(15)

$$\ddot{y}=-\frac{e}{m}\left[-\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)y+\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+4f_{OK}\right)y^3+\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x^{2}y\right],$$

(16)

$${\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2=-\frac{2e}{m}\left[\Phi -\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y^2+\epsilon \right].$$

(17)

In Equations (15)–(17), small terms up to the third order of smallness inclusive are retained on the right-hand sides.

2.2. Method for Analyzing Motions Relative to the Trajectory of the Main Particle

Let us consider the motion of an arbitrary particle in the beam with respect to the motion of the particle chosen as the main (reference). We call the motion of the main particle the reference motion. As the main one, we choose a particle that moves along the main optical axis and has zero initial energy (i.e., $\epsilon =0$). Then, the reference motion is described by the system of equations as

$$x_{on}(z_{on})=y_{on}(z_{on})=0 \mathrm{and}$$

(18)

$${\dot{z}}_{on}{}^2=-\frac{2e}{m}\Phi (z_{on}),$$

(19)

where the index “on” indicates that the quantities belong to the reference movement.

Here we note that the value $z_k$ for an arbitrary particle entering the initial condition (6) can coincide with the corresponding value for the main particle only in the case of a flat cathode. In general,

$$z(x_k,y_k)=z_k(0,0)+\frac{\partial z_k}{\partial x_k}{x}_k+\frac{\partial z_k}{\partial y_k}{y}_k+\frac{1}{2}\frac{{\partial }^2{z}_k}{\partial x_k{}^2}{x}_k{}^2+\frac{1}{2}\frac{{\partial }^2{z}_k}{\partial y_k{}^2}{y}_k{}^2+\dots ,$$

(20)

where we will accept

$$z_k(0,0)=z(x_k,y_k){|}_{x_k=y_k=0}=0.$$

(21)

Condition (21) means that the origin of the Cartesian coordinate system in the study of the properties of the cathode lens is located at the point of intersection of the main optical axis with the central point of the cathode.

Obviously, for a doubly symmetric lens, the condition is

$$\frac{\partial z_k}{\partial x_k}=\frac{\partial z_k}{\partial y_k}=0;$$

(22)

therefore, from (22), we have

$$z_k(x_k,y_k)=\frac{1}{2}\left(\frac{{\partial }^2{z}_k}{\partial x_k{}^2}{x}_k{}^2+\frac{{\partial }^2{z}_k}{\partial y_k{}^2}{y}_k{}^2\right).$$

(23)

Functions $\frac{{\partial }^2{z}_k}{\partial x_k{}^2}$ and $\frac{{\partial }^2{z}_k}{\partial y_k{}^2}$ in (23) describe the curvature of the cathode.

From Equation (19), we get

$${\dot{z}}_{on}=\sigma \sqrt{-\frac{2e}{m}\Phi (z_{on})}.$$

(24)

In the last equation, $\sigma$ is a sign coefficient and determines the direction of motion of charged particles. For a cathode lens, $\sigma =1$. In the electrode mirrors, the trajectories for the forward branch are $\sigma =+1$ and, for its reverse branch, —$\sigma =-1$.

The motion of an arbitrary particle in the systems under study relative to the coordinate of the main particle (reference motion) can be described by the following equations:

$$x=x\left(z_{on},x_k,y_k,{\epsilon }_x{}^{\frac{1}{2}},{\epsilon }_y{}^{\frac{1}{2}},{\epsilon }_z{}^{\frac{1}{2}}\right),$$

(25)

$$y=y\left(z_{on},x_k,y_k,{\epsilon }_x{}^{\frac{1}{2}},{\epsilon }_y{}^{\frac{1}{2}},{\epsilon }_z{}^{\frac{1}{2}}\right),$$

(26)

$$z=z\left(z_{on},x_k,y_k,{\epsilon }_x{}^{\frac{1}{2}},{\epsilon }_y{}^{\frac{1}{2}},{\epsilon }_z{}^{\frac{1}{2}}\right).$$

(27)

In expressions (25)–(27), there is no value of the parameter $z_k$. As can be seen from (20)–(23), $z_k$ is a quantity of the second order of smallness and determined by formula (23) through the values $x_k$ and $y_k$. Therefore, in the future, to take into account the value of $z_k$, the values of the radius of curvature of the cathode are used.

Considering that $x_k,y_k,{\epsilon }_x{}^{\frac{1}{2}},{\epsilon }_y{}^{\frac{1}{2}},{\epsilon }_z{}^{\frac{1}{2}}$ are small parameters, it follows from the last equations that

$$\begin{gathered} x(z_{on})=x_{on}(z_{on})+{\displaystyle \sum _{i=1}^5\frac{\partial x_{oni}}{\partial {\delta }_i}{\delta }_i+\frac{1}{2}}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5\frac{{\partial }^2{x}_{onij}}{\partial {\delta }_i\partial {\delta }_j}{\delta }_i{\delta }_j+}} \\ +\frac{1}{6}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{k=1}^5\frac{{\partial }^2{x}_{onijk}}{\partial {\delta }_i\partial {\delta }_j\partial {\delta }_k}}{\delta }_i{\delta }_j{\delta }_k+\dots }}, \end{gathered}$$

(28)

$$\begin{gathered} y(z_{on})=y_{on}(z_{on})+{\displaystyle \sum _{i=1}^5\frac{\partial y_{oni}}{\partial {\delta }_i}{\delta }_i+\frac{1}{2}}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5\frac{{\partial }^2{y}_{onij}}{\partial {\delta }_i\partial {\delta }_j}{\delta }_i{\delta }_j+}} \\ +\frac{1}{6}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{k=1}^5\frac{{\partial }^2{y}_{onijk}}{\partial {\delta }_i\partial {\delta }_j\partial {\delta }_k}}{\delta }_i{\delta }_j{\delta }_k+\dots }}, \end{gathered}$$

(29)

$$\begin{gathered} z(z_{on})=z_{on}+{\displaystyle \sum _{i=1}^5\frac{\partial z_{oni}}{\partial {\delta }_i}{\delta }_i+\frac{1}{2}}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5\frac{{\partial }^2{z}_{onij}}{\partial {\delta }_i\partial {\delta }_j}{\delta }_i{\delta }_j+}} \\ +\frac{1}{6}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{k=1}^5\frac{{\partial }^2{z}_{onijk}}{\partial {\delta }_i\partial {\delta }_j\partial {\delta }_k}}{\delta }_i{\delta }_j{\delta }_k+\dots }}. \end{gathered}$$

(30)

Here, ${\delta }_1=x_k,{\delta }_2=y_k,{\delta }_3={\epsilon }_x{}^{\frac{1}{2}},{\delta }_4={\epsilon }_y{}^{\frac{1}{2}},{\delta }_5={\epsilon }_z{}^{\frac{1}{2}}$. Moreover, taking into account (18), $x_{on}(z_{on})=x(z_{on},0,0,0,0,0)=y_{on}\left(z_{on}\right)=y_{on}(z_{on},0,0,0,0,0)=0$.

Denoting

$$x_i=\frac{\partial x_{oni}}{\partial {\delta }_i},x_{ij}=\frac{{\partial }^2{x}_{onij}}{\partial {\delta }_i\partial {\delta }_j},x_{ijk}=\frac{{\partial }^3{x}_{onijk}}{\partial {\delta }_i\partial {\delta }_j\partial {\delta }_k},$$

(31)

$$y_i=\frac{\partial y_{oni}}{\partial {\delta }_i},y_{ij}=\frac{{\partial }^2{y}_{onij}}{\partial {\delta }_i\partial {\delta }_j},y_{ijk}=\frac{{\partial }^3{y}_{onijk}}{\partial {\delta }_i\partial {\delta }_j\partial {\delta }_k},$$

(32)

$$z_i=\frac{\partial z_{oni}}{\partial {\delta }_i},x_{ij}=\frac{{\partial }^2{z}_{onij}}{\partial {\delta }_i\partial {\delta }_j},$$

(33)

from (28)–(30) we get

$$x(z_{on})={\displaystyle \sum _{i=1}^5{x}_i{\delta }_i+{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{x}_{ij}\delta {}_i\delta {}_j+}}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{k=1}^5{x}_{ijk}\delta {}_i\delta {}_j{\delta }_k}}}},$$

$$y(z_{on})={\displaystyle \sum _{i=1}^5{y}_i{\delta }_i+{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{y}_{ij}\delta {}_i\delta {}_j+}}{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{\displaystyle \sum _{k=1}^5{y}_{ijk}\delta {}_i\delta {}_j{\delta }_k}}}},$$

$$\eta (z_{on})=z(z_{on})-z_{on}={\displaystyle \sum _{i=1}^5{z}_i{\delta }_i+{\displaystyle \sum _{i=1}^5{\displaystyle \sum _{j=1}^5{z}_{ij}\delta {}_i\delta {}_j}}}.$$

The last expressions, in view of the arbitrariness of the order of differentiation and the resulting equalities of the form $x_{abc}=x_{acb}=x_{bca}=x_{bac}=x_{cab}=x_{cba}$, take the following form:

$$\begin{gathered} x(z_{on})={\displaystyle \sum _{i=1}^5{x}_i{\delta }_i+x_{11}{x}_k{}^2}+x_{22}{y}_k{}^2+x_{33}{\epsilon }_x+x_{44}{\epsilon }_y+x_{55}{\epsilon }_z+2(x_{12}{x}_k{y}_k+x_{13}{x}_k\sqrt{{\epsilon }_x}+ \\ +x_{14}{x}_k\sqrt{{\epsilon }_y}+x_{15}{x}_k\sqrt{{\epsilon }_z}+x_{23}{y}_k\sqrt{{\epsilon }_x}+x_{24}{y}_k\sqrt{{\epsilon }_y}+x_{25}{y}_k\sqrt{{\epsilon }_z}+x_{34}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+ \\ +x_{35}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+x_{45}\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z})+x_{111}{x}_k{}^2+x_{222}{y}_k{}^2+x_{333}{\epsilon }_k{}^{\frac{3}{2}}+x_{444}{\epsilon }_y{}^{\frac{3}{2}}+x_{555}{\epsilon }_z{}^{\frac{3}{2}}+ \\ +3(x_{112}{x}_k{}^2{y}_k+x_{113}{x}_k{}^2\sqrt{{\epsilon }_x}+x_{114}{x}_k{}^2\sqrt{{\epsilon }_y}+x_{115}{x}_k{}^2\sqrt{{\epsilon }_z}+x_{221}{y}_k{}^2{x}_k+x_{223}{y}_k{}^2\sqrt{{\epsilon }_x}+ \\ +x_{224}{y}_k{}^2\sqrt{{\epsilon }_y}+x_{225}{y}_k{}^2\sqrt{{\epsilon }_z}+x_{331}{\epsilon }_x{x}_k+x_{332}{\epsilon }_x{y}_k+x_{334}{\epsilon }_x\sqrt{{\epsilon }_y}+x_{335}{\epsilon }_x\sqrt{{\epsilon }_z}+x_{441}{\epsilon }_y{x}_k+ \\ +x_{442}{\epsilon }_y{y}_k+x_{443}{\epsilon }_y\sqrt{{\epsilon }_x}+x_{445}{\epsilon }_y\sqrt{{\epsilon }_z}+x_{551}{\epsilon }_z{x}_k+x_{552}{\epsilon }_z{y}_k+x_{553}{\epsilon }_z\sqrt{{\epsilon }_x}+x_{554}{\epsilon }_z\sqrt{{\epsilon }_y})+ \\ +6(x_{123}{x}_k{y}_k\sqrt{{\epsilon }_x}+x_{124}{x}_k{y}_k\sqrt{{\epsilon }_y}+x_{125}{x}_k{y}_k\sqrt{{\epsilon }_z}+x_{134}{x}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+ \\ +x_{135}{x}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+x_{145}{x}_k\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}+x_{234}{y}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+x_{235}{y}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+ \\ +x_{245}{y}_k\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}+x_{345}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}) \end{gathered}$$

(34)

$$\begin{gathered} y(z_{on})={\displaystyle \sum _{i=1}^5{y}_i{\delta }_i+y_{11}{x}_k^2+y_{22}{y}_k^2+y_{33}{\epsilon }_x+y_{44}{\epsilon }_y+y_{55}{\epsilon }_z+2(y_{12}{x}_k{y}_k+} \\ +y_{13}{x}_k\sqrt{e_x}+y_{14}{x}_k\sqrt{e_y}+y_{15}{x}_k\sqrt{e_z}+y_{23}{y}_k\sqrt{e_x}+y_{24}{y}_k\sqrt{e_y}+y_{25}{y}_k\sqrt{e_z}+ \\ +y_{34}\sqrt{e_x}\sqrt{e_y}+y_{35}\sqrt{e_x}\sqrt{e_z}+y_{45}\sqrt{e_y}\sqrt{e_z})+y_{111}{x}_k^3+y_{222}{y}_k^3+y_{333}{\epsilon }_k^{\frac{3}{2}}+y_{444}{\epsilon }_y^{\frac{3}{2}}+ \\ +y_{555}{\epsilon }_z^{\frac{3}{2}}+3(y_{112}{x}_k^2{y}_k+y_{113}{x}_k^2\sqrt{e_x}+y_{114}{x}_k^2\sqrt{e_y}+y_{114}{x}_k^2\sqrt{e_z}+y_{221}{y}_k^2{x}_k+ \\ +y_{223}{y}_k^2\sqrt{{\epsilon }_x}+y_{224}{y}_k^2\sqrt{{\epsilon }_y}+y_{225}{y}_k^2\sqrt{{\epsilon }_z}+y_{331}{\epsilon }_x{x}_k+y_{332}{\epsilon }_x{y}_k+y_{334}{\epsilon }_x\sqrt{{\epsilon }_y} \\ +y_{335}{\epsilon }_x\sqrt{{\epsilon }_z}+y_{441}{\epsilon }_y{x}_k+y_{442}{\epsilon }_y{y}_k+y_{443}{\epsilon }_y\sqrt{{\epsilon }_x}+y_{445}{\epsilon }_y\sqrt{{\epsilon }_z}+y_{551}{\epsilon }_z{x}_k+ \\ +y_{552}{\epsilon }_z{x}_k+y_{553}{\epsilon }_z\sqrt{{\epsilon }_x}+y_{554}{\epsilon }_z\sqrt{{\epsilon }_y})+6(y_{123}{x}_k{y}_k\sqrt{{\epsilon }_x}+y_{124}{x}_k{y}_k\sqrt{{\epsilon }_y}+ \\ +y_{125}{x}_k{y}_k\sqrt{{\epsilon }_z}+y_{134}{x}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+y_{135}{x}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+y_{145}{x}_k\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}+y_{234}{y}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+ \\ +y_{235}{y}_k\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+y_{245}{y}_k\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}+y_{345}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}), \end{gathered}$$

(35)

$$\begin{gathered} \eta (z_{on})={\displaystyle \sum _{i=1}^5{z}_i{\delta }_i+z_{11}{x}_k^2+z_{22}{y}_k^2+z_{33}{\epsilon }_x+z_{44}{\epsilon }_y}+z_{55}{\epsilon }_z+2(z_{12}{x}_k{y}_k+ \\ +z_{13}{x}_k\sqrt{{\epsilon }_x}+z_{14}{x}_k\sqrt{{\epsilon }_y}+z_{15}{x}_k\sqrt{{\epsilon }_z}+z_{23}{y}_k\sqrt{{\epsilon }_x}+z_{24}{y}_k\sqrt{{\epsilon }_y}+z_{25}{y}_k\sqrt{{\epsilon }_z}+ \\ +z_{34}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_y}+z_{35}\sqrt{{\epsilon }_x}\sqrt{{\epsilon }_z}+z_{45}\sqrt{{\epsilon }_y}\sqrt{{\epsilon }_z}). \end{gathered}$$

(36)

Taking into account (36), the functions of z can be expanded in the form

$$\Phi (z)=\Phi (z_{on})+{\Phi }^{\prime }(z_{on})\eta (z_{on})+\frac{{\Phi }^{″}(z_{on})}{2}{\eta }^2(z_{on})+\dots .$$

(37)

Using (24) and (37), from the Equations (15)–(17), we get

$$\begin{gathered} L_x\left[x\right]=-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }\eta x-{\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)}^{″}{\eta }^{2}x+\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x^3+ \\ +\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x{y}^2, \end{gathered}$$

(38)

$$\begin{gathered} L_y\left[y\right]=-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }\eta y-{\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)}^{″}{\eta }^{2}y+\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+4f_{OK}\right)y^3+ \\ +\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x^{2}y, \end{gathered}$$

(39)

$$L_{\eta }\left[\eta \right]=-\Phi {x^{\prime }}^2-\Phi {y^{\prime }}^2-\Phi {{\eta }^{\prime }}^2+\frac{{\Phi }^{″}}{2}{\eta }^2-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y^2+\epsilon ,$$

(40)

where $L_x,L_y,L_{\eta }$—operators, have the form

$$L_x\left[x\right]=2\Phi \frac{d^{2}x}{d{z}_{on}^2}+{\Phi }^{\prime }\frac{dx}{d{z}_{on}}+\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)x,$$

(41)

$$L_y\left[y\right]=2\Phi \frac{d^{2}y}{d{z}_{on}^2}+{\Phi }^{\prime }\frac{dy}{d{z}_{on}}+\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)y,$$

(42)

$$L_{\eta }\left[\eta \right]=2\Phi \frac{\partial \eta }{\partial z_{on}}-{\Phi }^{\prime }\eta .$$

(43)

In Equations (38)–(40) and further, until it is specifically stated, the arguments of all functions are $z_{on}$, and primes denote differentiation with respect to the parameter $z_{on}$.

Equations (38)–(40) are the equations of motion of charged particles in parametric form. For a comprehensive theoretical study of the properties of a doubly symmetric corpuscular-optical system, it is necessary to solve this system of equations.

To determine the values of the x and y coordinates in explicit dependence on the coordinate of the main optical axis, we use Equation (36), which, for simplicity, we write in the form

$$z=z_{on}+\eta (z_{on}).$$

(44)

We solve (44) by the method of successive approximations. Then, keeping the terms not higher than the third order of smallness, we obtain

$$z_{on}=z-\eta (z-\eta (z-\eta (z)))=z-\eta (z)+{\eta }^{\prime }(z)\eta (z).$$

(45)

Taking into account (45), particular linearly independent solutions of the paraxial equations $x_1(z_{on}),y_2(z_{on}),x_3(z_{on})$, and $y_4(z_{on}).$ can be expanded in the form

$$x_1(z_{on})=x_1(z)-x_1^{\prime }(z)\eta (z)+\frac{1}{2}{x}_1^{″}(z){\eta }^2(z)+x_1^{\prime }(z){\eta }^{\prime }(z)\eta (z),$$

(46)

$$y_2(z_{on})=y_2(z)-y_2^{\prime }(z)\eta (z)+\frac{1}{2}{y}_2^{″}(z){\eta }^2(z)+y_2^{\prime }(z){\eta }^{\prime }(z)\eta (z),$$

(47)

$$\begin{gathered} x_3(z_{on})={\overline{x}}_3(z_{on})\sqrt{\Phi (z_{on})}=\left[{\overline{x}}_3(z_{on})-{\overline{x}}_3{}^{\prime }(z)\eta (z)+\frac{1}{2}{\overline{x}}_3{}^{″}(z){\eta }^2(z)+{x^{\prime }}_3(z){\eta }^{\prime }(z)\eta (z)\right]· \\ ·\sqrt{\Phi (z)-{\Phi }^{\prime }(z)\eta (z)+\frac{1}{2}{\Phi }^{″}(z){\eta }^2(z)+{\Phi }^{\prime }(z){\eta }^{\prime }(z)\eta (z)}, \end{gathered}$$

(48)

$$\begin{gathered} y_4(z_{on})={\overline{y}}_4(z_{on})\sqrt{\Phi (z_{on})}=\left[{\overline{y}}_4(z_{on})-{\overline{y}}_4{}^{\prime }(z)\eta (z)+\frac{1}{2}{\overline{y}}_4{}^{″}(z){\eta }^2(z)+{y^{\prime }}_4(z){\eta }^{\prime }(z)\eta (z)\right]· \\ ·\sqrt{\Phi (z)-{\Phi }^{\prime }(z)\eta (z)+\frac{1}{2}{\Phi }^{″}(z){\eta }^2(z)+{\Phi }^{\prime }(z){\eta }^{\prime }(z)\eta (z)}, \end{gathered}$$

(49)

where ${\overline{x}}_3$ and ${\overline{y}}_4$ are auxiliary functions having finite values under condition (10).

It is easy to see that, under the condition $\Phi (z)>>{\epsilon }_z^{\frac{1}{2}}$ for a cathode lens, Equations (48) and (49) take the form

$$x_3(z_{on})=x_3(z)-x_3^{\prime }(z)\eta (z)+\frac{1}{2}{x}_3^{″}(z){\eta }^2(z)+x_3^{\prime }(z){\eta }^{\prime }(z)\eta (z),$$

(50)

$$y_4(z_{on})=y_4(z)-y_4^{\prime }(z)\eta (z)+\frac{1}{2}{y}_4^{″}(z){\eta }^2(z)+y_4^{\prime }(z){\eta }^{\prime }(z)\eta (z).$$

(51)

The above conditions, as known, are not satisfied in the near cathode region for emission systems and in the reflection region for electron mirrors. Therefore, Equations (50) and (51) are valid at a distance from the surface for which condition (10) is valid.

When deriving Equations (48) and (49), no restrictions were introduced. Consequently, these equations are valid everywhere, including the near cathode region. However, due to the fact that the most interesting parts of the trajectories from a practical point of view are located in regions far enough from the surface with a zero potential value, for further studies, we use expansions of the forms (50) and (51).

3. Results

• Electron-optical properties.

To derive the equations of trajectories in parametric form, it is necessary to solve the system of parametric equations of motion of charged particles (38)–(40). First, let’s substitute the values $x(z_{on}),y(z_{on})$, and $\eta (z_{on})$ from expressions (34)–(36) into the right parts of these equations. After that, taking into account the linearity of the operators (41)–(43), the arbitrariness of the order of differentiation in (31)–(33), and keeping the terms not higher than the third order of smallness for the transverse coordinates and the second order of smallness for the longitudinal coordinate, from (38)–(40) we obtain the following series of independent equations:

$$L_x\left[x_i\right]=0,$$

(52)

$$L_y\left[y_i\right]=0,$$

(53)

$$L_{\eta }\left[z_i\right]=0,$$

(54)

$$L_x\left[x_{ij}\right]={\vartheta }_{ij}^{(2)},$$

(55)

$$L_y\left[y_{ij}\right]={\theta }_{ij}^{(2)},$$

(56)

$$L_{\eta }\left[z_{ij}\right]={\zeta }_{ij,}$$

(57)

$$L_x\left[x_{ijk}\right]={\vartheta }_{ijk}^{(3)},\mathrm{and}$$

(58)

$$L_y\left[y_{ijk}\right]={\theta }_{ijk}^{(3)}$$

(59)

Equations (52)–(54) describe the motions of charged particles in the paraxial approximation. Solutions to these equations are substituted into the right parts of Equations (55)–(59). In addition, solutions of Equations (55)–(57) are taken into account in the right parts of Equations (58) and (59). The values of ${\vartheta }_{ij}^{(2)}$, ${\theta }_{ij}^{(2)}$, ${\zeta }_{ij,}$${\vartheta }_{ijk}^{(3)},$ and ${\theta }_{ijk}^{(3)}$ are given below.

Let us define the initial conditions for Equations (52)–(54). To do this, taking into account (31)–(33), we differentiate the initial conditions (4)–(9):

$${x_2(z_{on})|}_{z_{on}=z_k}={x_4\left(z_{on}\right)|}_{z_{on}=z_k}={x_5\left(z_{on}\right)|}_{z_{on}=z_k}=0,$$

(60)

$${x_1(z_{on})|}_{z_{on}=z_k}=1,$$

(61)

$${y_2(z_{on})|}_{z_{on}=z_k}=1,$$

(62)

$${y_1(z_{on})|}_{z_{on}=z_k}={y_3\left(z_{on}\right)|}_{z_{on}=z_k}={y_5\left(z_{on}\right)|}_{z_{on}=z_k}=0,$$

(63)

$${z_1(z_{on})|}_{z_{on}=z_k}={z_2\left(z_{on}\right)|}_{z_{on}=z_k}={z_3\left(z_{on}\right)|}_{z_{on}=z_k}={z_4\left(z_{on}\right)|}_{z_{on}=z_k}=0.$$

(64)

When determining the initial conditions for functions $x_3,y_4,z_5$, it must be taken into account that the product of zero and infinity is an uncertainty that should be disclosed. For example, for $x_3$, taking into account (7) and (2), we obtain

$${{\dot{x}}_3(z_{on})|}_{z_{on}=z_k}={{\dot{z}}_{on}{x}_3^1\left(z_{on}\right)|}_{z_{on}=z_k}={\sigma \sqrt{-\frac{2e}{m}\Phi (z_{on})}{x}_3^1\left(z_{on}\right)|}_{z_{on}=z_k}=\sqrt{-\frac{2e}{m},}$$

whence

$${\sigma \sqrt{\Phi (z_{on})}{x}_3^1(z_{on})|}_{z_{on}=z_k}=1.$$

(65)

Likewise,

$${\sigma \sqrt{\Phi (z_{on})}{y}_4^1(z_{on})|}_{z_{on}=z_k}=1,\mathrm{and}$$

(66)

$${\sigma \sqrt{\Phi (z_{on})}{z}_5^1(z_{on})|}_{z_{on}=z_k}=1.$$

(67)

From (65)–(67), it can be seen that the derivatives of functions $x_3,y_{4,}{z}_5$ with respect to the parameter $z_{on}$ at a singular point $(z_{on}=z_k)$ are equal to the same expression $\sigma {\left[\Phi \left(z_{on}\right)\right]}^{-\frac{1}{2}}$ and, in view of condition (10), tend to infinity in the vicinity of the point $z=z_k$. But from the same expressions (65)–(67), it is easy to see that this singularity in the vicinity of the point $z_{on}=z_k$ can be avoided by representing the desired functions in the form

$$x_3(z_{on})=C_x\sqrt{\Phi \left(z_{on}\right)}{\overline{x}}_3(z_{on}),$$

(68)

$$y_4(z_{on})=C_y\sqrt{\Phi \left(z_{on}\right)}{\overline{y}}_4(z_{on}),$$

(69)

$$z_5(z_{on})=C_z\sqrt{\Phi \left(z_{on}\right)}{\overline{z}}_5(z_{on}).$$

(70)

Here, $C_x,C_y,C_z$ are constant values to be determined, and ${\overline{x}}_3(z_{on})$, ${\overline{y}}_4(z_{on})$, ${\overline{z}}_5(z_{on})$ are auxiliary functions that do not have the features noted above.

Accept

$${{\overline{x}}_3(z_{on})|}_{z_{on}=z_k}={{\overline{y}}_4(z_{on})|}_{z_{on}=z_k}={{\overline{z}}_5(z_{on})|}_{z_{on}=z_k}=1.$$

(71)

Then, substituting (68)–(70) into (65)–(67), we have

$$C_x=C_y=C_z=\frac{2}{{\Phi }_k^{\prime }}\sigma .$$

(72)

Taking into account (68)–(70) from (44)–(46), we find equations that are satisfied by the functions ${\overline{x}}_3$, ${\overline{y}}_4$, and ${\overline{z}}_5$,

$$2\Phi {{\overline{x}}_3}^{″}+3{\Phi }^{\prime }{\overline{x}}_3{}^{\prime }+\left(\frac{3}{2}{\Phi }^{″}-2f_{KB}\right){\overline{x}}_3=0,$$

(73)

$$2\Phi {\overline{y}}_4{}^{″}+3{\Phi }^{\prime }{\overline{y}}_4{}^{\prime }+\left(\frac{3}{2}{\Phi }^{″}-2f_{KB}\right){\overline{y}}_4=0,$$

(74)

$${\overline{z}}_5=1.$$

(75)

From (75) and (70), it follows that

$$z_5(z_{on})=\frac{2}{{\Phi }_k^{\prime }}\sigma \sqrt{\Phi (z_{on})}.$$

(76)

Using (60)–(64) and (71), we determine the missing initial conditions for Equations (52), (53), (73), and (74):

$${x_1^{\prime }(z_{on})|}_{z_{on}=z_k}=-\frac{{\Phi }_k^{″}-4f_{KBk}}{2{\Phi }_k^{\prime }}=\frac{1}{R_x},$$

(77)

$${y_2^{\prime }(z_{on})|}_{z_{on}=z_k}=-\frac{{\Phi }_k^{″}+4f_{KBk}}{2{\Phi }_k^{\prime }}=\frac{1}{R_y},$$

(78)

$${{\overline{x}}_3{}^{\prime }(z_{on})|}_{z_{on}=z_k}=-\frac{3{\Phi }_k^{″}-4f_{KBk}}{6{\Phi }_k^{\prime }{}^2},\mathrm{and}$$

(79)

$${{\overline{y}}_4{}^{\prime }(z_{on})|}_{z_{on}=z_k}=-\frac{3{\Phi }_k^{″}+4f_{KBk}}{6{\Phi }_k^{\prime }{}^2},$$

(80)

where $R_x,R_y$ are the radii of curvature of the cathode surface, respectively, in the horizontal and vertical directions.

All other initial conditions for Equations (52)–(54) will be zero. Due to the linearity of the operators (41)–(43), all functions with zero initial conditions will be identically equal to zero.

In this way,

$$x_2=x_4=x_5=0,$$

(81)

$$y_1=y_3=y_5=0,\mathrm{and}$$

(82)

$$z_1=z_2=z_3=z_4=0.$$

(83)

Highlighting the first-order terms in the right-hand sides of Equations (34)–(36) and taking into account (76), (81)–(83), we obtain the parametric equations of trajectories in the paraxial approximation:

$$x^{(1)}(z_{on})=x_k{x}_1(z_{on})+{\epsilon }_x^{\frac{1}{2}}{x}_3(z_{on}),$$

(84)

$$y^{(1)}(z_{on})=y_k{y}_2(z_{on})+{\epsilon }_y^{\frac{1}{2}}{y}_4(z_{on}),$$

(85)

$${\eta }^{(1)}(z_{on})=\sigma {\epsilon }_z^{\frac{1}{2}}\frac{2}{{\Phi }_k^{\prime }}\sqrt{\Phi (z_{on})}.$$

(86)

Substituting (84)–(86) in Equations (55)–(57), which are solved under zero initial conditions, taking into account (38)–(40), we determine the values of the functions ${\vartheta }_{ij}^{(2)},{\theta }_{ij}^{(2)},{\zeta }_{ij}.$

We give only nonzero values of these functions:

$${\vartheta }_{15}^{(2)}=-\frac{1}{{\Phi }_k^{\prime }}({\Phi }^{‴}-4f_{KB}^{\prime })\sqrt{\Phi }{x}_1,$$

(87)

$${\vartheta }_{15}^{(2)}=-\frac{1}{{\Phi }_k^{\prime }}({\Phi }^{‴}-4f_{KB}^{\prime })\sqrt{\Phi }{x}_3,$$

(88)

$${\theta }_{25}^{(2)}=-\frac{1}{{\Phi }_k^{\prime }}({\Phi }^{‴}-4f_{KB})\sqrt{\Phi }{y}_2,$$

(89)

$${\theta }_{25}^{(2)}=-\frac{1}{{\Phi }_k^{\prime }}({\Phi }^{‴}-4f_{KB})\sqrt{\Phi }{y}_4,$$

(90)

$${\zeta }_{11}=-\Phi {x_1^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right){x_1}^2,$$

(91)

$${\zeta }_{13}=-\Phi x_1{}^{\prime }{x}_3{}^{\prime }-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x_1{x}_3,$$

(92)

$${\zeta }_{33}=-\Phi {x_3^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x_3^2+1,$$

(93)

$${\zeta }_{22}=-\Phi {y_2^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y_2^2,$$

(94)

$${\zeta }_{24}=-\Phi y_2^{\prime }{y}_4^1-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y_2{y}_4,$$

(95)

$${\zeta }_{44}=-\Phi {y_4^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y_4^2+1,$$

(96)

$${\zeta }_{55}=1-\frac{1}{{{\Phi }_k^{\prime }}^2}\left({{\Phi }^{\prime }}^2-2\Phi {\Phi }^{″}\right).$$

(97)

In view of the linearity of operators (41)–(43) and zero initial conditions for Equations (55)–(57) and taking into account (87)–(97), it follows that all functions of the second order of smallness in expansions (34)–(36), except for $x_{15},x_{35},y_{25},y_{45},z_{11},z_{13},z_{33},z_{22},z_{24},z_{44},z_{55}$, will be equal to zero.

Substituting (87)–(97) into Equations (55)–(57) and solving them by the method of variation of arbitrary constants, we find

$$x_{15}=\frac{1}{{\Phi }_k^{\prime }}\left(\sqrt{\Phi }{x}_1^{\prime }+\frac{x_3}{R_x}\right),$$

(98)

$$x_{35}=-\frac{1}{2}{x}_1+\frac{1}{{\Phi }_k^{\prime }}\sqrt{\Phi }{x}_3^{\prime },$$

(99)

$$y_{25}=\frac{1}{{\Phi }_k^{\prime }}\left(\sqrt{\Phi }{y}_2^{\prime }+\frac{y_4}{R_y}\right),$$

(100)

$$y_{45}=-\frac{1}{2}{y}_2+\frac{1}{{\Phi }_k^{\prime }}\sqrt{\Phi }{y}_4^{\prime },$$

(101)

$$z_{11}=-\frac{1}{2}{x}_1{x}_1^{\prime }+\sqrt{\Phi }{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\frac{x_1{x}_1^{″}}{\sqrt{\Phi }}d{z}_{on}},$$

(102)

$$z_{13}=\frac{\sqrt{\Phi }}{2}{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\left(x_1^{″}\overline{x_3}-x_1^{\prime }{\overline{x_3}}^{\prime }\right)d{z}_{on}},$$

(103)

$$z_{33}=\frac{\sqrt{\Phi }}{2}{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\frac{1}{\Phi \sqrt{\Phi }}\left[1-\Phi {x_3^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)x_3^2\right]d{z}_{on}},$$

(104)

$$z_{22}=-\frac{1}{2}{y}_2{y}_2^{\prime }+\sqrt{\Phi }{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\frac{y_2{y}_2^{\prime }}{\sqrt{\Phi }}d{z}_{on}},$$

(105)

$$z_{24}=\frac{1}{2}\sqrt{\Phi }{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\left(y_2^{″}\overline{y_4}-y_2^{\prime }{\overline{y_4}}^{\prime }\right)d{z}_{on}},$$

(106)

$$z_{44}=\frac{\sqrt{\Phi }}{2}{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\frac{1}{\Phi \sqrt{\Phi }}\left[1-\Phi {y_4^{\prime }}^2-\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)y_4^2\right]d{z}_{on}},$$

(107)

$$z_{55}=\frac{\sqrt{\Phi }}{2}{\displaystyle \underset{z_k}{\overset{z_{on}}{\int }}\frac{1}{\Phi \sqrt{\Phi }}\left[1-\frac{1}{{{\Phi }_k^{\prime }}^2}\left({{\Phi }^{\prime }}^2-2\Phi {\Phi }^{″}\right)\right]d{z}_{on}}.$$

(108)

Thus, the total aberrations of the second order have the form

$$x^{(2)}(z_{on})=2\left[x_k{\epsilon }_z^{\frac{1}{2}}{x}_{15}(z_{on})+{\epsilon }_x^{\frac{1}{2}}{\epsilon }_z^{\frac{1}{2}}{x}_{35}(z_{on})\right],$$

(109)

$$y^{(2)}(z_{on})=2\left[y_k{\epsilon }_z^{\frac{1}{2}}{y}_{25}(z_{on})+{\epsilon }_y^{\frac{1}{2}}{\epsilon }_z^{\frac{1}{2}}{y}_{45}(z_{on})\right],$$

(110)

$$\begin{gathered} {\eta }^{(2)}(z_{on})=x_k^2{z}_{11}(z_{on})+2x_k{\epsilon }_x^{\frac{1}{2}}{z}_{13}(z_{on})+{\epsilon }_x{z}_{33}(z_{on})+y_k^2{z}_{22}(z_{on})+ \\ +2y_k{\epsilon }_y^{\frac{1}{2}}{z}_{24}(z_{on})+{\epsilon }_y{z}_{44}(z_{on})+{\epsilon }_z{z}_{55}(z_{on}). \end{gathered}$$

(111)

Expressions (109)–(110) represent the values of the total transverse aberrations of the second order, respectively, in the horizontal and vertical directions. The total longitudinal aberration of the second order is presented in expression (111).

The values of the functions ${\vartheta }_{ijk}^{(3)}$ and ${\theta }_{ijk}^{(3)}$ in Equations (58) and (59), taking into account (84)–(86) and (98)–(108), take the form

$${\vartheta }_{111}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x_1^3-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_1{z}_{11},$$

(112)

$${\vartheta }_{113}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x_1^2{x}_3-\frac{1}{3}{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }(x_1{z}_{13}+x_3{z}_{11}),$$

(113)

$${\vartheta }_{133}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x_1{x}_3^2-\frac{1}{3}{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }(x_1{z}_{13}+x_3{z}_{13}),$$

(114)

$${\vartheta }_{333}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}-\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x_3^3-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_3{z}_{33},$$

(115)

$${\vartheta }_{122}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_1{y}_2^2-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_1{z}_{22}\right],$$

(116)

$${\vartheta }_{124}^{(3)}=\frac{1}{6}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_1{y}_2{y}_4-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_1{z}_{24}\right],$$

(117)

$${\vartheta }_{144}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_1{y}_4^2-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_1{z}_{44}\right],$$

(118)

$${\vartheta }_{322}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_3{y}_2^2-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_3{z}_{22}\right],$$

(119)

$${\vartheta }_{324}^{(3)}=\frac{1}{6}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_3{y}_2{y}_4-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_3{z}_{24}\right],$$

(120)

$${\vartheta }_{344}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)x_3{y}_4^2-{\left(\frac{{\Phi }^{″}}{2}-2f_{KB}\right)}^{\prime }{x}_3{z}_{44}\right],$$

(121)

$${\vartheta }_{155}^{(3)}=\frac{1}{3}\left[{\left(\frac{{\Phi }^{IV}}{2}-2f_{OK}\right)}^{\prime }{x}_1{z}_{55}-{\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)}^{″}{x}_1{z}_5^2\right],$$

(122)

$${\vartheta }_{355}^{(3)}=\frac{1}{3}\left[{\left(\frac{{\Phi }^{IV}}{2}-2f_{OK}\right)}^{\prime }{x}_3{z}_{55}-{\left(\frac{{\Phi }^{″}}{4}-f_{KB}\right)}^{″}{x}_3{z}_5^2\right],$$

(123)

$${\theta }_{222}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+4f_{OK}\right)x_2^3-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_2{z}_{22},$$

(124)

$${\theta }_{224}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+4f_{OK}\right)y_2^2{y}_4-\frac{1}{3}{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }(y_2{z}_{24}+y_4{z}_{22}),$$

(125)

$${\theta }_{244}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+f_{OK}\right)y_2{y}_4^2-\frac{1}{3}{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }(y_2{z}_{44}+y_4{z}_{24}),$$

(126)

$${\theta }_{444}^{(3)}=\left(\frac{{\Phi }^{IV}}{16}+\frac{f_{KB}^{″}}{3}+4f_{OK}\right)y_4^3+{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_4{z}_{44},$$

(127)

$${\theta }_{211}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_2{x}_1^2-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_2{z}_{11}\right],$$

(128)

$${\theta }_{213}^{(3)}=\frac{1}{6}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_2{x}_1{x}_3-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_2{z}_{13}\right],$$

(129)

$${\theta }_{233}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_2{x}_3^2-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_2{z}_{33}\right],$$

(130)

$${\theta }_{411}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_4{x}_1^2-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_4{z}_{11}\right],$$

(131)

$${\theta }_{413}^{(3)}=\frac{1}{6}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_4{x}_1{x}_3-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_4{z}_{13}\right],$$

(132)

$${\theta }_{433}^{(3)}=\frac{1}{3}\left[\left(\frac{{\Phi }^{IV}}{16}-12f_{OK}\right)y_4{x}_3^2-{\left(\frac{{\Phi }^{″}}{2}+2f_{KB}\right)}^{\prime }{y}_4{z}_{33}\right],$$

(133)

$${\theta }_{255}^{(3)}=\frac{1}{3}\left[{\left(\frac{{\Phi }^{IV}}{2}+2f_{OK}\right)}^{\prime }{y}_4{z}_{55}-{\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)}^{″}{y}_2{z}_5^2\right],$$

(134)

$${\theta }_{455}^{(3)}=\frac{1}{3}\left[{\left(\frac{{\Phi }^{IV}}{2}+2f_{OK}\right)}^{\prime }{y}_4{z}_{55}-{\left(\frac{{\Phi }^{″}}{4}+f_{KB}\right)}^{″}{y}_4{z}_5^2\right].$$

(135)

All functions ${\vartheta }_{ijk}^{(3)}$ and ${\theta }_{ijk}^{(3)}$, except for those given in formulas (112)–(135), have zero values.

4. Discussion

In view of the fact that Equations (58) and (59) are solved under zero initial conditions, the nonzero values of the functions $x_{ijk}$ and $y_{ijk}$ will be determined by the method of variation of arbitrary constants. After a series of cumbersome transformations, the expressions characterizing the total aberrations of the third order in the horizontal and vertical directions of the lens under study are reduced to the form

$$\begin{gathered} x^{(3)}(z_{on})=x_k^3{x}_{111}(z_{on})+3\left[x_k^2{\epsilon }_k^{\frac{1}{2}}{x}_{113}(z_{on})+x_k{\epsilon }_x{x}_{133}(z_{on})\right]+{\epsilon }_x^{\frac{1}{2}}{x}_{333}(z_{on})+3x_k{y}_k^2{x}_{122}(z_{on})+ \\ +6x_k{y}_k{\epsilon }_y^{\frac{1}{2}}{x}_{124}(z_{on})+3\left[x_k{\epsilon }_y{x}_{144}(z_{on})+{\epsilon }_x^{\frac{1}{2}}{y}_k^2{x}_{322}(z_{on})\right]+6{\epsilon }_x^{\frac{1}{2}}{\epsilon }_y^{\frac{1}{2}}{y}_k{x}_{324}(z_{on})+ \\ +3\left[{\epsilon }_x^{\frac{1}{2}}{\epsilon }_y{x}_{344}(z_{on})+x_k{\epsilon }_z{x}_{155}(z_{on})+{\epsilon }_x^{\frac{1}{2}}{\epsilon }_z{x}_{355}(z_{on})\right], \end{gathered}$$

(136)

$$\begin{gathered} y^{(3)}(z_{on})=y_k^3{x}_{222}(z_{on})+3\left[y_k^2{\epsilon }_y^{\frac{1}{2}}{x}_{224}(z_{on})+y_k{\epsilon }_y{y}_{244}(z_{on})\right]+{\epsilon }_y^{\frac{3}{2}}{y}_{444}(z_{on})+ \\ +3y_k{x}_k^2{y}_{211}(z_{on})+6y_k{x}_k{\epsilon }_x^{\frac{1}{2}}{y}_{213}(z_{on})+3\left[y_k{\epsilon }_x{y}_{233}(z_{on})+{\epsilon }_y^{\frac{1}{2}}{x}_k^2{y}_{411}(z_{on})\right]+ \\ +6{\epsilon }_y^{\frac{1}{2}}{\epsilon }_x^{\frac{1}{2}}{x}_k{y}_{413}(z_{on})+3\left[{\epsilon }_y^{\frac{1}{2}}{\epsilon }_x{x}_{433}(z_{on})+y_k{\epsilon }_z{y}_{255}(z_{on})+{\epsilon }_y^{\frac{1}{2}}{\epsilon }_z{y}_{455}(z_{on})\right]. \end{gathered}$$

(137)

Here, to ensure a reasonable length of the article and due to the large number and volume of formulas, we do not give expressions for $x_{ijk}$ and $y_{ijk}$.

As shown above, relation (45) is used to pass to equations in explicit dependence on the coordinate of the main optical axis.

5. Conclusions

Summing up, we note that the following conclusions can be drawn from the results obtained above.

From (84) and (85) it follows that with the help of a doubly symmetrical cathode lens, the following can be obtained:

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a stigmatic electronic image of the object. Moreover, the lens makes it possible to convert image scales in mutually perpendicular directions.

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stroke tricks in vertical and horizontal directions.

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a single crossover of the flow of charged particles.

From (109) and (110), it can be seen that the total aberration of the second order consists of spherochromatic aberration and chromatic positional aberration.

Analysis of expressions (136) and (137) allows us to conclude that the use of cathode lenses with two planes of symmetry provides additional opportunities to eliminate or compensate for aberrations, at least in one of two mutually perpendicular directions.

The results obtained in the article will allow us to carry out effective numerical studies of the focusing properties of various schemes of cathode lenses with two planes of symmetry and expand their scope.

As noted above, in this work, a research method is applied that has the following mathematical features:

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The initial conditions for solving the system of equations for the trajectories of charged particles are given through the values of the initial energies of the particles, which in electronic lenses are much less than the energy of the focusing fields.

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The trajectory of a charged particle with zero initial energy moving along the main optical axis is chosen as the main trajectory, and the coordinates of the trajectories of an arbitrary charged particle are determined relative to the coordinates of the main particle.

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A complete analysis of all variational parameters was carried out, and all equations were obtained for determining the values of paraxial and aberration characteristics.

These features made it possible to provide a mathematical basis for constructing a theory that takes into account the specifics of modeling and studying the properties of cathode lenses. Using the theory developed in this article will lead to the following important results:

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Theoretical difficulties that currently lead to the practical application of only two-dimensional cathode lenses with planar or axial symmetry will be overcome.

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As for other types of electronic lenses, it will be possible to conduct numerical studies of the parameters of specific cathode lenses with two planes of symmetry and design emission systems based on them with significantly improved focusing characteristics.

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Since the process of formation of charged particle flows begins in the emission system, the design of emission systems with improved characteristics will lead to a noticeable improvement in the technical characteristics and capabilities of a large number of high-tech analytical instruments and technological installations in the fields of nanotechnology, medicine, instrumentation, and other industries.

Limitations of the Study

Like all scientific works, this article has certain limitations of its application related to various problems, the solutions for which will be presented in the following works of the author. These problems include:

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taking into account in the theory of emission lenses and nodes of focusing properties of more complex fields containing electrostatic and magnetic components with different and more complex types of symmetry;

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additional consideration in the theory of emission systems of the influence of fields of space charges in the formation of intense flows of charged particles; and

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construction of a detailed relativistic theory of emission systems.