Analysis of the Problems of the Research and Modernization of Emission Units of Analytical Devices of Vacuum Electronics

Abstract

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.

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. A number of authors [24,25,26,27,28] devoted their works to the development of the theory of cathode lenses, taking into account the noted features in their initial conditions. However, it was not possible to fully achieve the set goal in these works. In some of the mentioned works, the obtained formulas lead to a violation of the convergence of the results when the values of the initial velocity tend to zero; in other works, the cathode of the lens is assumed to be flat, which is a significant limitation, since, in practice, cathodes with a curved surface are often used. In the works [16,17,18], the authors of the current article proposed methods for solving existing difficulties in the study of cathode lenses with different types of focusing field symmetry.

This paper, using the method developed by the authors, examines theoretical problems in studying the processes of focusing charged particles in emission elements and units, and based on numerical calculations of specific electron-optical elements, methods are proposed for solving problems of upgrading emission units of vacuum electronics.

2. Materials and Methods

As is known, electron-optical systems with a straight optical axis are most often used in electronic instrument making. At the same time, in practice, there are many cases when it becomes necessary to use focusing systems with a curvilinear optical axis [29,30,31,32].

In this paper, we will first consider the problem of constructing a method for studying cathode lenses with a curvilinear optical axis. From the general theoretical materials obtained in this case, results can be derived that can also be used to study emission systems with a straight optical axis.

The motion of a charged particle with charge e and mass m in an electromagnetic field with electric field strength $\overrightarrow{E}$ and magnetic field strength $\overrightarrow{H}$, as is known, satisfies the equation

$${\displaystyle \frac{d\left(m\overrightarrow{V}\right)}{dt}}=e\overrightarrow{E}+{\displaystyle \frac{e}{c}}\left[\overrightarrow{V},\overrightarrow{H}\right]$$

(1)

where $\overrightarrow{V}$—the speed of the particle, $t$—time.

Note that in the relativistic case, according to the special theory of relativity, the value of the mass m of a particle moving at high speed and the value of the rest mass of the same charged particle $m_0$ are related by the equation

$$m={\displaystyle \frac{m_0}{\sqrt{1-\frac{V^2}{c^2}}}},$$

(2)

where c is the speed of light and V is the absolute value of the speed of a charged particle.

The magnitudes of the electric field strength and the magnetic field strength are generally determined as functions of spatial coordinates and time

$$\stackrel{⃑}{E}=\stackrel{⃑}{E}\left(\overrightarrow{R},t\right), \overrightarrow{H}=\overrightarrow{H}\left(\stackrel{⃑}{R},t\right).$$

(3)

Here, the spatial coordinates are specified using the radius vector $\overrightarrow{R}$.

Vectors $\overrightarrow{E}$ and $\overrightarrow{H}$, according to field theory, satisfy the equations

$$div\overrightarrow{H}=0, rot\overleftarrow{E}=-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \overrightarrow{H}}{\partial t}},$$

(4)

$$div\overrightarrow{E}=4\pi \rho , rot\overrightarrow{H}=-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \overrightarrow{E}}{\partial t}}+{\displaystyle \frac{4\pi }{c}}\overrightarrow{j}.$$

(5)

In the last equations, the symbol $\rho$ denotes the space charge distribution function, $\overrightarrow{j}$—beam current density. Note that Equation (4) constitutes the first pair of Maxwell’s equations, and Equation (5) constitutes the second pair of Maxwell’s equations.

Electromagnetic fields, however, are also characterized by scalar electrostatic potential $\varphi$ and vector magnetic potential $\overrightarrow{A}$, which are related to $\overrightarrow{E}$ and $\overrightarrow{H}$ as follows

$$\overrightarrow{E}=-{\displaystyle \frac{1}{c}}{\displaystyle \frac{\partial \overrightarrow{A}}{\partial t}}-grad\varphi , \overrightarrow{H}=rot\overrightarrow{A}.$$

(6)

The magnetic field strength in the absence of a vortex component can also be determined through the value of the scalar magnetic potential $\omega$

$$\overrightarrow{H}=-grad\omega .$$

(7)

Equation (1), taking into account (6), takes the form

$${\displaystyle \frac{d\left(m\overrightarrow{V}\right)}{dt}}=-e\left({\displaystyle \frac{\partial \overrightarrow{A}}{\partial t}}+grad\varphi \right)-{\displaystyle \frac{e}{c}}\left[\overrightarrow{V},\overrightarrow{H}\right].$$

(8)

Equation (8) is written in vector form and represents, as is known, the general equation of motion of charged particles in electromagnetic fields.

For constant electromagnetic fields, the vectors $\overrightarrow{E}$ and $\overrightarrow{H}$ do not depend on time and are functions of spatial coordinates only. Then, from (8), taking into account (7), for charged particles in constant electric and magnetic fields without taking into account the relativistic effect, that is, under the condition $m=m_0$, we get

$$m{\displaystyle \frac{d\overrightarrow{V}}{dt}}=-egrad\varphi -{\displaystyle \frac{e}{c}}\left[\overrightarrow{V},grad\omega \right].$$

(9)

Next, we examine the properties of the emission and reflective corpuscular–optical systems based on the analysis of Equation (9).

Let us introduce a system of generalized curvilinear orthogonal coordinates $q_1,$ $q_2$, $q_3$. This coordinate system is characterized by unit vectors (orths) $i_1,$ $i_2$, $i_3$ and Lame coefficients $h_1$, $h_2$, $h_3$.

In the introduced coordinate system from Equation (9), we obtain the following system of equations

$$\begin{gathered} m{\displaystyle \frac{d}{dt}}\left(h_1{\dot{q}}_1\right)+m{h}_2{\dot{q}}_2\left({\displaystyle \frac{{\dot{q}}_1}{h_2}}{\displaystyle \frac{\partial h_1}{\partial q_2}}-{\displaystyle \frac{{\dot{q}}_2}{h_1}}{\displaystyle \frac{\partial h_2}{\partial q_1}}\right)+m{h}_3{\dot{q}}_3\left({\displaystyle \frac{{\dot{q}}_1}{h_3}}{\displaystyle \frac{\partial h_1}{\partial q_3}}-{\displaystyle \frac{{\dot{q}}_3}{h_1}}{\displaystyle \frac{\partial h_3}{\partial q_1}}\right)= \\ =e{E}_1+{\displaystyle \frac{e}{c}}\left(h_2{\dot{q}}_2{H}_3-h_3{\dot{q}}_3{H}_2\right), \end{gathered}$$

(10)

$$\begin{gathered} m{\displaystyle \frac{d}{dt}}\left(h_2{\dot{q}}_2\right)+m{h}_1{\dot{q}}_1\left({\displaystyle \frac{{\dot{q}}_2}{h_1}}{\displaystyle \frac{\partial h_2}{\partial q_1}}-{\displaystyle \frac{{\dot{q}}_1}{h_2}}{\displaystyle \frac{\partial h_1}{\partial q_2}}\right)+m{h}_3{\dot{q}}_3\left({\displaystyle \frac{{\dot{q}}_2}{h_3}}{\displaystyle \frac{\partial h_2}{\partial q_3}}-{\displaystyle \frac{{\dot{q}}_3}{h_2}}{\displaystyle \frac{\partial h_3}{\partial q_2}}\right)= \\ =e{E}_2+{\displaystyle \frac{e}{c}}\left(h_3{\dot{q}}_3{H}_1-h_1{\dot{q}}_1{H}_3\right), \end{gathered}$$

(11)

$$h_1^2{\dot{q}}_1^2+h_2^2{\dot{q}}_2^2+h_3^2{\dot{q}}_3^2=-{\displaystyle \frac{2e}{m}}\left[\varphi \left(q_1,q_2,q_3\right)+\epsilon \right],$$

(12)

where $e\epsilon$—initial energy of the particle, the dot above the symbol denotes differentiation with respect to time.

Equations (10)–(12) represent the general equations of motion, and Equation (12) is the equation of the conservation of energy.

We study the motion of an arbitrary particle relative to the motion of the main particle, which has zero initial energy and moves along the coordinate (line)$h_3{q}_3$, that is (in the general case), along the curvilinear–optical axis. In an equal period of time, the main particle flies a distance equal to $h_3{q}_{30}$; the projection of the trajectory of an arbitrary particle will be equal to $h_3{q}_{3\Sigma }=h_3{q}_3$. The difference between these values ${h_{3}q}_{\delta }$ is the longitudinal aberration displacement of an arbitrary particle along the main trajectory

$$h_3{q}_3=h_3{q}_{30}+h_3{q}_{\delta }$$

(13)

From (12) for the main trajectory we have

$$h_3^2{\dot{q}}_{30}^2=-{\displaystyle \frac{2e}{m}}\varphi \left(\mathrm{0,0},q_{30}\right)$$

(14)

where

$$h_3{\dot{q}}_{30}=\sigma \sqrt{-{\displaystyle \frac{2e}{m}}\varphi \left(\mathrm{0,0},q_{30}\right)}=\sigma \sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}.$$

(15)

Here $\sigma =\pm 1$—a sign coefficient indicating the direction of motion of a charged particle; ${\varphi }_0$—the distribution function of the electrostatic potential along the line (coordinate) of the main particle $h_3{q}_{30}$.

Equations (13) and (15) allow us to eliminate time from Equations (10)–(12). We transform the time differentiation operators as follows

$$\begin{gathered} {\displaystyle \frac{d}{dt}}={\displaystyle \frac{d{q}_{30}}{dt}}{\displaystyle \frac{d}{d{q}_{30}}}=h_3{\dot{q}}_{30}{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}=\sigma \sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}, \\ {\displaystyle \frac{d^2}{d{t}^2}}=h_3{\dot{q}}_{30}{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\left(\sigma \sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\right)= \\ =-{\displaystyle \frac{e}{m}}\left[2{\varphi }_0{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\left({\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\right)+{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d{\varphi }_0}{d{q}_{30}}}{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\right]. \end{gathered}$$

(16)

It is easy to see that

$$h_i{\dot{q}}_i=h_i{\displaystyle \frac{d{q}_i}{dt}}=\sigma \sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}{\displaystyle \frac{h_i}{h_3}}{\displaystyle \frac{d{q}_i}{d{q}_{30}}}$$

(17)

where $i$ to indicate directions $i_1,i_2,i_3$ takes, respectively, the values 1,2,3.

Taking into account the last relations and using the notations

$$\begin{gathered} {q_i}^{\prime }={\displaystyle \frac{h_i{\dot{q}}_i}{h_3{\dot{q}}_{30}}}={\displaystyle \frac{h_{i}d{q}_i}{h_{3}d{q}_{30}}}, \\ {q_i}^{″}={\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d}{d{q}_{30}}}\left({\displaystyle \frac{h_i}{h_3}}{\displaystyle \frac{d{q}_i}{d{q}_{30}}}\right), {{\varphi }_0}^{\prime }={\displaystyle \frac{1}{h_3}}{\displaystyle \frac{d{\varphi }_0}{d{q}_{30}}}, \end{gathered}$$

(18)

we obtain the following relations

$${\displaystyle \frac{d}{dt}}\left(h_i{\dot{q}}_i\right)=-{\displaystyle \frac{e}{m}}\left(2{\varphi }_0{q_i}^{″}+{{\varphi }_0}^{\prime }{q_i}^{\prime }\right),$$

(19)

$$h_1{\dot{q}}_1^2+h_2{\dot{q}}_2^2+h_3{\dot{q}}_3^2=-{\displaystyle \frac{2e}{m}}{\varphi }_0\left[{q^{\prime }}_1^2+{q^{\prime }}_2^2+{\left(1+q_{\delta }^{\prime }\right)}^2\right].$$

(20)

In the last expressions and further, unless otherwise specified, primes denote differentiation with respect to the variable $q_{30}$.

Distributions of electrostatic potential, as is known, can be represented in the form of the following power series

$$\begin{gathered} \varphi =\varphi (q_1,q_2,q_{30}+q_{\delta })={\varphi }_{000}+{\varphi }_{100}{h}_1{q}_1+{\varphi }_{010}{h}_2{q}_2+{\varphi }_{001}{h}_3{q}_{\delta }+ \\ +{\varphi }_{200}{h}_1^2{q}_1^2+{\varphi }_{020}{h}_2^2{q}_2^2+{\varphi }_{002}{h}_3^2{q}_{\delta }^2+{\varphi }_{110}{h}_1{q}_1{h}_2{q}_2+ \\ +{\varphi }_{101}{h}_1{q}_1{h}_3{q}_{\delta }+{\varphi }_{011}{h}_2{q}_2{h}_3{q}_{\delta }+ \\ +{\varphi }_{300}{h}_1^3{q}_1^3+{\varphi }_{030}{h}_2^3{q}_2^3+{\varphi }_{003}{h}_3^3{q}_{\delta }^3+\dots \end{gathered}$$

(21)

where ${\varphi }_{000}={\varphi }_0=\varphi (\mathrm{0,0},q_{30})$, ${\varphi }_{ijk}={\varphi }_{ijk}(\mathrm{0,0},q_{30})$.

Similarly, to determine the magnetic field, the function can be represented $\omega$

$$\begin{gathered} \omega =\omega \left(q_1,q_2,q_{30}+q_{\delta }\right)={\omega }_{000}+{\omega }_{100}{h}_1{q}_1+{\omega }_{010}{h}_2{q}_2+{\omega }_{001}{h}_3{q}_{\delta }+ \\ +{\omega }_{200}{h}_1^2{q}_1^2+{\omega }_{020}{h}_2^2{q}_2^2+{\omega }_{002}{h}_3^2{q}_{\delta }^2+{\omega }_{110}{h}_1{q}_1{h}_2{q}_2+ \\ +{\omega }_{101}{h}_1{q}_1{h}_3{q}_{\delta }+{\omega }_{011}{h}_2{q}_2{h}_3{q}_{\delta }+ \\ +{\omega }_{300}{h}_1^3{q}_1^3+{\omega }_{030}{h}_2^3{q}_2^3+{\omega }_{003}{h}_3^3{q}_{\delta }^3+\dots \end{gathered}$$

(22)

Here ${\omega }_{000}={\omega }_0=\omega (\mathrm{0,0},q_{30})$, ${\omega }_{ijk}={\omega }_{ijk}(\mathrm{0,0},q_{30})$.

Equations (10)–(12), taking into account (17)–(20), can be reduced to the form

$$\begin{gathered} 2{\varphi }_0{q_1}^{″}+{{\varphi }_0}^{\prime }{q_1}^{\prime }+2{\varphi }_0{q_2}^{\prime }\left({\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial h_1}{\partial q_2}}{\displaystyle \frac{{q_1}^{\prime }}{h_1}}-{\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial h_2}{\partial q_1}}{\displaystyle \frac{{q_2}^{\prime }}{h_2}}\right)+ \\ +2{\varphi }_0\left(1+{q_{\delta }}^{\prime }\right)\left({\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial h_1}{\partial q_3}}{\displaystyle \frac{{q_1}^{\prime }}{h_1}}-{\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial h_3}{\partial q_1}}{\displaystyle \frac{\left(1+{q_{\delta }}^{\prime }\right)}{h_3}}\right)= \\ ={\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial \varphi }{\partial q_1}}-{\displaystyle \frac{\sigma }{c}}\sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}\left({q_2}^{\prime }{H}_3-\left(1+{q_{\delta }}^{\prime }\right)H_2\right), \end{gathered}$$

(23)

$$\begin{gathered} 2{\varphi }_0{q_2}^{″}+{{\varphi }_0}^{\prime }{q_2}^{\prime }+2{\varphi }_0{q_1}^{\prime }\left({\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial h_2}{\partial q_1}}{\displaystyle \frac{{q_2}^{\prime }}{h_2}}-{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial h_1}{\partial q_2}}{\displaystyle \frac{{q_1}^{\prime }}{h_1}}\right)+ \\ +2{\varphi }_0\left(1+{q_{\delta }}^{\prime }\right)\left({\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial h_2}{\partial q_3}}{\displaystyle \frac{{q_2}^{\prime }}{h_2}}-{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial h_3}{\partial q_2}}{\displaystyle \frac{\left(1+{q_{\delta }}^{\prime }\right)}{h_3}}\right)= \\ ={\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial \varphi }{\partial q_2}}-{\displaystyle \frac{\sigma }{c}}\sqrt{-{\displaystyle \frac{2e}{m}}{\varphi }_0}\left(\left(1+{q_{\delta }}^{\prime }\right)H_1-{q_1}^{\prime }{H}_3\right), \end{gathered}$$

(24)

$${\varphi }_0\left[{q^{\prime }}_1^2+{q^{\prime }}_2^2+{\left(1+q_{\delta }^{\prime }\right)}^2\right]=\varphi \left(q_1,q_2,q_3\right)+\epsilon .$$

(25)

Equations (23)–(25) are the equations of motion of charged particles in parametric form, where the parameter in these equations is the coordinate of the main trajectory $h_3{q}_{30}$.

The use of Equations (13), (15), and (16) to eliminate time from the equations of motion allows us to avoid restrictions on the slopes of the trajectories in the region of the emission systems (cathode lenses) adjacent to the cathode and the reflecting region of the electron mirrors, i.e., it makes it possible to mathematically correctly describe and study the physical processes in the systems under study. Equations (23)–(25) are then solved taking into account (21) and (22) by the method of successive approximations.

3. Results

It was noted above that the problems of minimizing aberrations can be solved by using emission units with multipole focusing fields. In this connection, we note that the Equations of motion in parametric form for an electrostatic cathode lens with two planes of symmetry can be derived as a special case from Equations (23)–(25).

In the Cartesian coordinate system, we take $q_1=x,q_2=y,q_3=z,h_1=1,h_2=1,h_3=1$. The potential distribution from the form (21), taking into account the properties of the field symmetry under consideration, is reduced to the form

$$\begin{gathered} 2\Phi {\displaystyle \frac{d^{2}x}{d{z}_{o\Pi }^2}}+{\Phi }^{\prime }{\displaystyle \frac{dx}{d{z}_{on}}}+\left({\displaystyle \frac{{\Phi }^{″}}{2}}-2f_{KB}\right)x= \\ =-{\left({\displaystyle \frac{{\Phi }^{″}}{2}}-2f_{KB}\right)}^{\prime }\eta x-{\left({\displaystyle \frac{{\Phi }^{″}}{4}}-f_{KB}\right)}^{″}{\eta }^{2}x+\left({\displaystyle \frac{{\Phi }^{IV}}{16}}-{\displaystyle \frac{f_{KB}^{″}}{3}}+4f_{OK}\right)x^3+ \\ +\left({\displaystyle \frac{{\Phi }^{IV}}{16}}-12f_{OK}\right)x{y}^2, \end{gathered}$$

(26)

$$\begin{gathered} 2\Phi {\displaystyle \frac{d^{2}y}{d{z}_{o\Pi }^2}}+{\Phi }^{\prime }{\displaystyle \frac{dy}{d{z}_{on}}}+\left({\displaystyle \frac{{\Phi }^{″}}{2}}+2f_{KB}\right)y= \\ =-{\left({\displaystyle \frac{{\Phi }^{″}}{2}}+2f_{KB}\right)}^{\prime }\eta y-{\left({\displaystyle \frac{{\Phi }^{″}}{4}}+f_{KB}\right)}^{″}{\eta }^{2}y+\left({\displaystyle \frac{{\Phi }^{IV}}{16}}+{\displaystyle \frac{f_{KB}^{″}}{3}}+4f_{OK}\right)y^3+ \\ +\left({\displaystyle \frac{{\Phi }^{IV}}{16}}-12f_{OK}\right)x^{2}y \end{gathered}$$

(27)

$$\begin{gathered} 2\Phi {\displaystyle \frac{d\eta }{d{z}_{on}}}-{\Phi }^{\prime }\eta = \\ =-\Phi x^{\prime 2}-\Phi y^{\prime 2}-\Phi {\eta }^{\prime 2}+{\displaystyle \frac{{\Phi }^{″}}{2}}{\eta }^2-\left({\displaystyle \frac{{\Phi }^{″}}{4}}-f_{KB}\right)x^2-\left({\displaystyle \frac{{\Phi }^{″}}{4}}+f_{KB}\right)y^2+\epsilon . \end{gathered}$$

(28)

where $f_{KB}$—quadrupole component of focusing fields, $f_{OK}$—octupole component of focusing fields.

In Equations (26)–(28), the arguments of all the functions are $z_{on}$, and the primes denote differentiation with respect to the parameter $z_{on}$.

Let us recall that to replace the time value with a spatial variable (i.e., the coordinate of the main optical axis) in the initial equations of motion, Equations of the form (13), (15), and (16) were used. In this case, for electron lenses with a straight optical axis, as noted above, the Lame coefficients $h_1$, $h_2$, $h_3$ are taken to be equal to one.

It should be noted that in [16], a corrected method for studying cathode lenses with two planes of symmetry was presented. In this work, equations were also obtained that completely coincide with (26)–(28), which confirms the correctness of the above-proposed method for studying emission systems with a curvilinear axis.

Let us now present the results of numerical studies of focusing parameters in doubly symmetrical and axisymmetric cathode lenses, based on the formulas obtained in [16]. The calculations carried out and a comparative analysis of the values of the aberration coefficients of three-electrode axially symmetrical and box-shaped (double-symmetrical) immersion lenses confirms the assumption that multipole cathode lenses can be used to construct sources of charged particles with improved focusing in at least one of the directions of space.

Note that in the rotational coordinate system r, z, the equations of the trajectories of charged particles up to the third order of smallness have the form

$$\begin{gathered} r{e}^{i\left(\psi +\theta \right)}=r_{k}u+b_{1}v+r_k\sqrt{{\epsilon }_z}{{\rm B}}_{21}+ \\ +b_1\sqrt{{\epsilon }_z}{{\rm B}}_{22}+{r_k}^3{{\rm B}}_{31}+{r_k}^2{\overline{b}}_1{{\rm B}}_{32}+{r_k}^2{b}_1{{\rm B}}_{33}+ \\ +r_k{b_1}^2{\mathsf{{\rm B}}}_{34}+r_k{b}_1{\overline{b}}_1{\mathsf{{\rm B}}}_{35}+{b_1}^2{\overline{b}}_1{\mathsf{{\rm B}}}_{36}+r_k{\epsilon }_z{\mathsf{{\rm B}}}_{37}+b_1{\epsilon }_z{\mathsf{{\rm B}}}_{38} \end{gathered}$$

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 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 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_ij doubly symmetrical cathode lenses have the form

$$\begin{array}{l}{D}_{x3}=x_k^3{D}_{x31}+x_k^2\sqrt{{\epsilon }_x}{D}_{x32}+x_k{\epsilon }_x{D}_{x33}+\sqrt[3]{{\epsilon }_x}{D}_{x34}+\\ +x_k{y}_k^2{D}_{x35}+x_k{y}_k\sqrt{{\epsilon }_y}{D}_{x36}+x_k{\epsilon }_y{D}_{x37}+\sqrt{{\epsilon }_x}{y}_k^2{D}_{x38}+\\ +\sqrt{{\epsilon }_x}{y}_k\sqrt{{\epsilon }_y}{D}_{x39}+\sqrt{{\epsilon }_x}{\epsilon }_y{D}_{x310}+x_k{\epsilon }_z{D}_{x311}+\sqrt{{\epsilon }_x}{\epsilon }_z{D}_{x312}\end{array}$$

(29)

$$\begin{array}{l}{D}_{y3}=y_k^3{D}_{y31}+y_k^2\sqrt{{\epsilon }_y}{D}_{y32}+y_k{\epsilon }_y{D}_{y33}+\sqrt[3]{{\epsilon }_y}{D}_{y34}+\\ +y_k{x}_k^2{D}_{y35}+y_k{x}_k\sqrt{{\epsilon }_x}{D}_{y36}+y_k{\epsilon }_x{D}_{y37}+\sqrt{{\epsilon }_y}{x}_k^2{D}_{y38}+\\ +\sqrt{{\epsilon }_y}{x}_k\sqrt{{\epsilon }_x}{D}_{y39}+\sqrt{{\epsilon }_y}{\epsilon }_x{D}_{y310}+y_k{\epsilon }_z{D}_{y311}+\sqrt{{\epsilon }_y}{\epsilon }_z{D}_{y312}\end{array}$$

(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 l_y, and in the horizontal direction 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 z = 1, x = y = 0.

In the graphs provided, the red line corresponds to the values R = l_x = l_y = 0.05, the green line to the values R = l_x = l_y = 0.1, and the blue line to the values 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.

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 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.

5. Conclusions

At present, various analytical instruments and technological installations of vacuum electronics are widely used in various innovative technological complexes. Emission elements and units are used to extract and form beams of charged particles in electron–ion-beam devices and vacuum electronics devices. Their focusing properties significantly affect the quality parameters and technical characteristics of the devices and installations in which they are used. For example, the values of the sensitivity and resolution of mass-spectrometric devices, electron microscopes, electron-lithographic and ion-lithographic installations of nano- and micro-electronic technologies largely depend on the focusing properties and characteristics of emission units and elements. Emission systems are based on cathode lenses; a number of authors also call these lenses immersion objectives.

In this paper, a theoretical basis for studying an emission element with a curvilinear optical axis is developed. In addition, numerical studies of the aberration characteristics of three-electrode cathode lenses with different types of focusing field symmetry are carried out. A comparative analysis of the obtained numerical results confirms the relevance and necessity of developing new types of cathode lenses with different types of field symmetry for the significant modernization of existing vacuum electronics instrument systems, which still use charged particle sources (cathode lenses) only with axial or planar symmetry.